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https://www.nebula-graph.io/posts/graph-embeddings
Contact Enterprise Sales What are graph embeddings ? In the modern world of big data, graphs are undoubtedly essential data representation and visualization tools. Imagine navigating a city without a map. When working with complicated networks, such as social relationships, molecular structures, or recommendation systems, data analysts frequently encounter similar difficulties. Here's where graph embeddings come into play. They allow researchers and data analysts to map nodes, edges, or complete graphs to continuous vector spaces for in-depth data analysis. What are graph embeddings and how do they work? In this guide, we examine the fundamentals of graph embeddings, including: What are graph embeddings How graph embeddings work Benefits of graph embeddings Trends in graph embeddings This guide will help you uncover the mysteries contained in graphs, whether you are a data analyst, researcher, or someone just interested in learning more about the potential of network analysis. Continue reading! Fundamentals of graphs A graph is a slightly abstract representation of objects that are related to each other in some way, and of these relationships. Typically, the objects in a graph database are drawn as dots called vertices or nodes. A line (or curve) connects any two vertices representing objects that are related or adjacent; such a line is called an edge. It is a simplified map where lines represent relationships and dots represent items. These dots, or vertices, hold information about the entities, while the lines, or edges, represent the connections between the entities. After learning about the vertices and edges that comprise a graph, let's investigate some of its unrealized possibilities: graph embeddings. What are graph embeddings? Graph embedding refers to the process of representing graph nodes as vectors which encode key information of the graph such as semantic and structural details, allowing machine learning algorithms and models to operate on them. In other words they are basically low-dimensional, compact graph representations that store relational and structural data in a vector space. Graph embeddings, as opposed to conventional graph representations, condense complicated graph structures into dense vectors while maintaining crucial graph features, potentially saving time and money in processing. Ever wondered how your social media knows to suggest perfect friends you never knew existed? Or how your phone predicts the traffic jam before you even hit the road? The answer lies in a hidden world called graphs, networks of connections, like threads linking people, places, and things. And to understand these webs, we need a translator: graph embeddings. Think of them as a magic trick that transforms intricate networks of vertices and edges into compact numerical representations. These "embeddings" capture the essence of each node (vertex) and its relationship to others, distilling the complex network into a format readily understood by machine learning algorithms. Benefits of graph embeddings Being able to represent data using graph embedding offers great benefits, including: Graph embeddings allow researchers and data scientists to explore hidden patterns within large networks of data. This greatly enhances the accuracy and efficiency of machine learning algorithms By identifying hidden patterns, researchers can make informed decisions and come up with better solutions for complex problems. Graph embedding distills complex graph-structured data and represents them as simple numerical figures, making computation operations on them very easy and fast. This benefit allows even the most complex algorithms to be scaled to fit all sorts of datasets. Techniques for generating graph embeddings Graph embedding algorithms and node embedding techniques are the two main kinds of techniques used to construct graph embeddings. 1. Node embedding techniques These techniques focus on representing individual nodes within the graph as unique vectors in a low-dimensional space. Imagine each node as a distinct character in a complex story, and these techniques aim to capture their essence and relationships through numerical encoding. i). DeepWalk Inspired by language modeling, DeepWalk treats random walks on the graph as sentences and learns node representations based on their "context" within these walks. Think of it as understanding a word better by its surrounding words in a sentence. ii). Node2Vec Building on DeepWalk, Node2Vec allows for flexible exploration of the graph by controlling the balance between breadth-first and depth-first searches. This "adjustable lens" allows for capturing both local and global structural information for each node. iii). GraphSAGE This technique focuses on aggregating information from a node's local neighborhood to create its embedding. Imagine summarizing a person based on their close friends and associates. GraphSAGE efficiently handles large graphs by sampling fixed-size neighborhoods for each node during training. 2. Graph embedding algorithms While node embedding techniques concentrate on specific nodes, graph algorithms try to capture the interactions and general structure of the entire network. Think of them as offering a thorough summary of the network that accounts for each node individually as well as its connections. i). Graph Convolutional Networks (GCNs) GCNs function directly on the graph structure, executing convolutions on adjacent nodes to represent their interconnection. They were inspired by convolutional neural networks for images. Consider applying a filter to an image that takes into account not only a pixel but also the pixels surrounding it. ii). Graph Attention Networks (GATs) Expanding upon GCNs, GATs incorporate an attention mechanism that enables the network to concentrate on the most pertinent neighbors for every node, perhaps resulting in more precise depictions. iii). Graph Neural Networks (GNNs) Refers to a variety of graph data processing and node representation learning frameworks. Their approach blends concepts from conventional neural networks with graph-specific processes to extract structure information as well as node attributes. Keep in mind that the subject of graph embedding is continually changing, with new methods and improvements appearing on a regular basis. Applications of graph embeddings Due to their ability to turn graph data into a computationally processable format, graph embedding is useful in graph pre-processing. Before we get to the use cases, let's look at the capabilities that provide the foundation that inform the use cases for graph embeddings: | | | --- | | Graph Analytics | Graph embeddings make it easy to gain insight into the structure, patterns and relationships in graphs | | Machine Learning and Deep Learning | Graph embeddings make it possible to represent graph data as continuous data, making it useful in natural language processing and training of various models such as recurrent neural networks | | Measuring the similarity between two vectors | Graph embedding makes it easy to understand how users interact with items | Based on the above capabilities, graph embeddings find wide-ranging applications in several disciplines due to their capacity to capture intricate interactions inside graphs and represent them in low-dimensional vector spaces. Here are some important use cases. 1. Social Network Analysis In social network analysis, graph embeddings facilitate community detection, user behavior prediction, and identification of influential nodes. Consider Facebook as an example, where graph embeddings help uncover communities of users, predict friendship connections, and identify influential users based on their interactions and network centrality. 2. Recommendation Systems Graph embeddings power recommendation systems by modeling user-item interactions and capturing recommendation graph structures. For example, systems like Netflix and YouTube use graph embeddings to recommend movies and videos based on users' past movie ratings among other metrics 3. Knowledge Graphs In knowledge graphs, graph embeddings enable query response, entity linking, and semantic similarity computation. The accessibility and interpretability of knowledge graphs are improved by integrating entities and relations. 4. Biological Networks and Bioinformatics Graph embeddings are used in the analysis of biological networks, including gene regulatory networks and protein-protein interactions. They can be used to detect gene-disease connections, accelerate drug discovery, and predict targets and protein functions by foreseeing target-drug interactions. 5. Fraud Detection and Anomaly Detection It is critical to safeguard users and financial systems against fraud. Graph embeddings play a critical role in fraud and anomaly detection systems by enabling the identification of anomalous patterns in networks such as social networks and financial transactions. Also Read: Fraud Detection With Graph Analytics Metrics for evaluating graph embeddings It's not enough to only create strong graph embeddings; we also need instruments to evaluate them. Metrics for Evaluating Graph Embeddings measure how well graph embedding methods capture and maintain the relational and structural information in graphs. Important measurements include: Node Classification Accuracy: Indicates how well nodes' properties and relationships can be captured by using learned embeddings to predict their labels. Link Prediction Accuracy measures how well a graph's learned embeddings may be used to predict future or missing edges, demonstrating how well graph topology is captured. Graph Reconstruction: Measures the degree to which graph attributes can be preserved by reconstructing the original graph structure using learned embeddings. Downstream Task Performance: Measuring downstream task performance with learned embeddings shows how useful downstream machine learning tasks are in practical applications. Embedding Quality: Assesses the degree of similarity preservation, dimensionality reduction, and computing efficiency that make up learned embeddings. All things considered, these measures offer thorough insights into the effectiveness and generalizability of graph embedding methods across a range of fields and applications. Challenges in generating effective graph embeddings and how to overcome them It can be difficult to generate efficient graph embeddings since analysts have to reduce the dimensionality of the graph while maintaining its structural information. As a result, creating graph embeddings often encounters these primary issues: Scalability issues Accurate and efficient embeddings of graphs can be challenging to produce due to their huge size and complexity. Scalability is an important concern, especially in the processing of real-world applications where the graphs might be of huge size and keep on changing. Heterogeneity issues You can encounter challenges in representing the structural information of a network in a low-dimensional space because nodes and edges in graphs have varying types and properties. Sparsity issues A large number of nodes and edges in some networks may lack connections, making them extremely sparse. Because of this, it could be challenging to represent the graph's structural information in a low-dimensional space. To overcome these challenges, researchers have developed several techniques and algorithms for generating effective graph embeddings including sampling strategies, skip connections, inductive learning, and adversarial training. Future trends in graph embeddings The future of graph embeddings is shaped by these emerging trends, mostly aimed at addressing evolving complexities. Dynamic graphs require adaptive embedding techniques. Interpretability is vital, with many techniques seen as opaque "black boxes." Demand will continue to grow for interpretable methods providing transparent insights. Efficiency will remain crucial amid growing graph complexity. Techniques must balance accuracy with computational efficiency. Scalable algorithms tailored to dynamic and multi-modal graphs will rise. They prioritize interpretability, offering clear explanations. In summary, the future of graph embeddings relies on adaptive, interpretable, and efficient techniques navigating dynamic, multi-modal graph data, fostering innovation across domains. Conclusion Graph embeddings have made it possible to untangle complex networks and reveal hidden connections. From social media analysis to drug discovery, their applications are vast. While challenges like scalability and interpretability persist, the future shines bright with dynamic and multi-modal techniques. NebulaGraph supports graph embeddings and is available in AWS and Azure. Get started with a Free Trial and witness the power of connections truly unveiled. Tag Features System-testing Dev-log Performance Use-cases Community Deployment Tools LLM Query-language Graph-computing Tech-talk Release-notes Architecture News Source-code Success-stories Knowledge-graph GQL Top Articles 1 Benchmarking the Mainstream Open Source Distributed Graph Databases at Meituan: NebulaGraph vs Dgraph vs JanusGraph 2 Graph Database vs Relational Database: What to Choose? 3 Difference Between Relational and Non Relational Database 4 Use cases of graph databases in real-time recommendation Contents Fundamentals of graphs What are graph embeddings? Benefits of graph embeddings Techniques for generating graph embeddings 1. Node embedding techniques i). DeepWalk ii). Node2Vec iii). GraphSAGE 2. Graph embedding algorithms i). Graph Convolutional Networks (GCNs) ii). Graph Attention Networks (GATs) iii). Graph Neural Networks (GNNs) Applications of graph embeddings 1. Social Network Analysis 2. Recommendation Systems 3. Knowledge Graphs 4. Biological Networks and Bioinformatics 5. Fraud Detection and Anomaly Detection Metrics for evaluating graph embeddings Challenges in generating effective graph embeddings and how to overcome them Scalability issues Heterogeneity issues Sparsity issues Future trends in graph embeddings Conclusion Recommended for you News NebulaGraph Cloud Achieves SOC 2 Type I Compliance, Strengthening Trust in Secure Graph Database-as-a-Service News NebulaGraph Ranks #2 Globally Among Graph Databases: A Milestone Achieved Together LLM Tech-talk Architecture When AI Meets Graph Databases: Innovating with Multimodal Data Fusion (Part I)
14001
https://byjus.com/chemistry/monopotassium-phosphate/
What is Monopotassium Phosphate? Monopotassium Phosphate is the monopotassium salt of phosphoric acid with the formula KH2PO4. It is freely soluble in water and insoluble in ethanol. Commercially available as pure KH2PO4, when reacted with MgO, produces high-quality ceramics. It is formed by the chloride or potassium carbonate reaction with phosphoric acid, and the phosphate is derived in a pure form as a crystalline material. Table of Contents Monopotassium phosphate is prepared industrially by adding 1 mol of a water solution of potassium hydroxide to 1 mol of phosphoric acid. Other names – Potassium dihydrogen phosphate, Potassium phosphate monobasic, Potassium phosphate monobasic | | | --- | | KH2PO4 | Monopotassium Phosphate | | Density | 2.34 g/cm³ | | Molecular Weight/ Molar Mass | 136.086 g/mol | | Boiling Point | 400 °C | | Melting Point | 252.6 °C | | Chemical Formula | KH2PO4 | Monopotassium Phosphate Structure – KH2PO4 Physical Properties of Monopotassium Phosphate – KH2PO4 | | | --- | | Odour | No odour | | Appearance | White powder, deliquescent | | Covalently-Bonded Unit | 2 | | Heavy Atom Count | 6 | | Complexity | 61.2 | | Solubility | Soluble in water | Chemical Properties of Monopotassium Phosphate – KH2PO4 KH2PO4 + H2O → H3PO4 + KOH KH2PO4 + 2NaOH → H2O + KOH + Na2HPO4 Uses of Monopotassium Phosphate – KH2PO4 Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin! Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz Congrats! Visit BYJU’S for all Chemistry related queries and study materials Your result is as below Request OTP on Voice Call Comments Leave a Comment Cancel reply Your Mobile number and Email id will not be published. Required fields are marked Request OTP on Voice Call Website Post My Comment Register with BYJU'S & Download Free PDFs Register with BYJU'S & Watch Live Videos
14002
https://www.ck12.org/flexi/physics/position-and-displacement/define-distance-and-displacement/
Flexi answers - Define distance and displacement. | CK-12 Foundation Subjects Explore Donate Sign InSign Up All Subjects Physics Position and Displacement Question Define distance and displacement. Distance is the total path length traveled by an object, while displacement is the shortest path between the initial and final positions of the object. Distance is the shortest path between two points, while displacement is the total path length traveled. Distance and displacement both refer to the total path length traveled by an object. Distance and displacement both refer to the shortest path between the initial and final positions of an object. Flexi Says: A. Distance is the total path length traveled by an object, while displacement is the shortest path between the initial and final positions of the object. In physics, distance and displacement are two distinct concepts. Distance refers to the total length of the path traveled by an object. It doesn't account for the direction of motion and is always a positive quantity. If, for example, a person goes on a 5km jog around a park, the total path length (distance) they covered would be 5km. Displacement, on the other hand, is the shortest path between the object's initial and final positions. Unlike distance, displacement can be positive, negative, or zero and does account for direction. For instance, if the person starts and ends their jog at the same position, despite having covered a distance of 5km, their displacement would be 0km as both their initial and final positions are the same. Analogy / Example Try Asking: Can one-dimensional motion have a zero distance but a nonzero displacement?An object's displacement could be greater than the distance the object travels. True/False.Calculate the distance and displacement for a car moving 30 meters east and then 20 meters west How can Flexi help? By messaging Flexi, you agree to our Terms and Privacy Policy
14003
https://math.stackexchange.com/questions/4240957/fixed-points-of-permutation-groups
Skip to main content Asked Modified 2 years, 9 months ago Viewed 1k times This question shows research effort; it is useful and clear 5 Save this question. Show activity on this post. I was studying permutation groups and I founs this question- Let Sn be the group of all permutations of the set X={1,2,…,n}. Given a permutation σ∈Sn, let f(σ) denote the number of fixed points of σ. a. Show that the average number of fixed points is 1, i.e., 1|Sn|∑σ∈Snf(σ)=1 b. Find the average value of f(σ)2. All that comes to my mind is to use Inclusion Exclusion Principle to calculate the number of combinations for a given value of f(σ). That is, explicitly calculate the number of permutations of X with exactly r fixed points, denoted by Sn(r). But, that is not a very easy task since we are doing it for a general n which means Sn(r) will be in the form of a summation, all of which needs to be summed again over all r. Also, this approach is not quite elegant. It becomes a real headache however in b since there you need to take a square as well. Also, we are never really using any property of permutation groups while solving this problem. Is there any other approach that can make life easier? While it is suggested in the comments and in an answer to use expectations of random variables, I don't think that is what the question asks of me considering the fact that the course in which I got the problem (it's a group theory course by the way) is far away from that. Is there any other ways to go about it? combinatorics group-theory permutations combinations fixed-points Share CC BY-SA 4.0 Follow this question to receive notifications edited Sep 12, 2021 at 18:59 Sayan Dutta asked Sep 3, 2021 at 14:55 Sayan DuttaSayan Dutta 10.6k33 gold badges1818 silver badges5353 bronze badges 8 3 Try Linearity of Expectation, and use indicator variables. – lulu Commented Sep 3, 2021 at 15:02 2 If you count |{(i,σ)∣σ(i)=i}| in two ways (first i then σ, and vice versa) you ought to get (a) easily. For (b) I'll leave you to decide what to count. – ancient mathematician Commented Sep 3, 2021 at 15:34 1 The point "a" is just Burnside's Lemma for the natural action of Sn on X, which is transitive. – user943729 Commented Sep 3, 2021 at 16:53 3 Point b is difficult to write down in general, but if G is 2-transitive then the answer is simple. The 'correct' proof of both of these is to use character theory. – David A. Craven Commented Sep 3, 2021 at 18:00 1 If you haven't heard of character theory (i.e., representation theory) then you shouldn't be using it. – David A. Craven Commented Sep 3, 2021 at 22:17 | Show 3 more comments 5 Answers 5 Reset to default This answer is useful 8 Save this answer. Show activity on this post. As in many cases where a random variable is equal to the number of occurrences of several events, a fruitful technique is to write that random variable as a sum of indicator random variables. Let Xi be a random variable which is 1 if σ(i)=i, and 0 otherwise. Letting X=X1+⋯+Xn, this means that f(σ)=X. It follows that E[f(σ)]=E[X]=E[X1]+⋯+E[Xn]=nE[X1], E[f(σ)2]=E[(X1+⋯+Xn)2]=∗nE[X21]+n(n−1)E[X1X2] In =∗, I expanded out (X1+⋯+Xn)2, distributed the E[], and then collected identically distributed terms. All that remains is to compute E[X1], E[X21], and E[X1X2]. These random variables only take the values 0 and 1, so computing their expectations involves computing a certain probability. I leave the rest to you. Share CC BY-SA 4.0 Follow this answer to receive notifications answered Sep 3, 2021 at 15:47 Mike EarnestMike Earnest 85.2k1212 gold badges8282 silver badges157157 bronze badges 6 While this is a nice solution (+1), I don't think, this is what was expected of me. I don't think the question wants me to use expectations. This course is far away from it. – Sayan Dutta Commented Sep 3, 2021 at 22:09 Well, what is your course about? You say "permutation groups," but I do not know what that usually entails. @SayanDutta – Mike Earnest Commented Sep 3, 2021 at 22:30 @SayanDutta In particular, have you learned about Burnside's Lemma? – Mike Earnest Commented Sep 3, 2021 at 22:41 it's a group theory course and being at the introductory phase, I haven't learnt Burnside Lemma as of now. But, we have it further down in our syllabus, so it seems I'll be learning it soon. You can add that version of the proof as well if you want to. – Sayan Dutta Commented Sep 4, 2021 at 5:44 1 @SayanDutta I can't say it better than this MSE answer. – Mike Earnest Commented Sep 4, 2021 at 5:57 | Show 1 more comment This answer is useful 5 Save this answer. Show activity on this post. Let G be a group, X a set, and: G×X(g,x)⟶⟼Xgx(1) a G-action on X. Then: Claim. For Xi=X (i=1,…,k), ΠX:=∏ki=1Xi, and x¯:=(x1,…,xk)∈ΠX, the map: G×ΠX(g,x¯)⟶⟼ ΠX g⋆x¯:=(gx1,…,gxk)(2) is a G-action on ΠX. Proof. We have to confirm that the map "⋆" fulfils the two properties for a group action. In fact: e⋆x¯=(ex1,…,exk)=(x1,…,xk)=x¯ ∀x¯∈ΠX; 2. g⋆(h⋆x¯)=g⋆(hx1,…,hxk)=(g(hx1),…,g(hxk))=((gh)x1,…,(gh)xk)=(gh)⋆x¯ ∀g,h∈G,∀x¯∈ΠX. □ The pointwise stabilizer for the action "⋆" reads: Stab⋆(x¯)={g∈G∣g⋆x¯=x¯}={g∈G∣(gx1=x1)∧⋯∧(gxk=xk)}=⋂i=1kStab(xi)(3) Furthermore: Fix⋆(g)={x¯∈ΠX∣g⋆x¯=x¯}={x¯∈ΠX∣(gx1=x1)∧⋯∧(gxk=xk)}(4) whence x¯∈Fix⋆(g)⟺xi∈Fix(g),i=1,…,k. So, every k-tuple (= ordered arrangement of k elements of a set, where repetition is allowed) of elements of Fix(g) gives rise to a x¯∈Fix⋆(g), and viceversa. Thus, for finite X: |Fix⋆(g)|=|Fix(g)|k(5) (see this Wiki page, section "Permutations with repetition"). For your case b, take G=Sn, X={1,…,n} and k=2. By (3), Stab⋆((i,j))=Stab(i)∩Stab(j), whence |Stab⋆((i,j))|=(n−1)! for j=i, and |Stab⋆((i,j))|=(n−2)! for j≠i. Therefore, there must be precisely two orbits for the action "⋆"†. Now, by applying Burnside's Lemma to the action "⋆": 1|Sn|∑σ∈Sn|Fix⋆(σ)|=2(6) and finally, by recalling (5): 1|Sn|∑σ∈Sn|Fix(σ)|2=2(7) which is your point b. (Your point a is the same result applied to the transitive action on one single copy of X.) †In fact, by the Orbit-Stabilizer Theorem, |Stab⋆((i,j))|=(n−1)! implies |O⋆((i,j))|=n, and |Stab⋆((i,j))|=(n−2)! implies |O⋆((i,j))|=n(n−1). But the set of orbits is a partition of the acted on set, whose size is n2, whence kn+ln(n−1)=n2 or, equivalently, k+l(n−1)=n. For n=2, this yields k+l=2; for n>2, l=n−kn−1 integer implies k=1, which in turn implies l=1, and then again k+l=2. So, the action "⋆" has two orbits for every n≥2. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Sep 8, 2021 at 7:08 answered Sep 6, 2021 at 19:38 user943729user943729 1 2 That's a nice solution (+1). Thanks. – Sayan Dutta Commented Sep 8, 2021 at 14:05 Add a comment | This answer is useful 2 Save this answer. Show activity on this post. Here's a linear algebra based proof. It uses Kronecker Products (denoted ⊗) and has a hint of representation theory but really doesn't require any knowledge other than basic group theory and somewhat sophisticated linear algebra. notation: Pg for g∈Sn denotes the standard (n×n) permutation matrix representation for some permutation g. Part 1: Idempotence Z:=(1|Sn|∑g∈SnPg⊗Pg) Then Z is idempotent. Proof: Z2 =(1|Sn|∑g∈SnPg⊗Pg)(1|Sn|∑g′∈SnPg′⊗Pg′) =1|Sn|2∑g∈Sn((Pg⊗Pg)∑g′∈SnPg′⊗Pg′) =1|Sn|2∑g∈Sn(∑g′∈Sn(PgPg′)⊗(PgPg′)) =1|Sn|2∑g∈Sn(∑g′′∈Sn(Pg′′)⊗(Pg′′)) =1|Sn|2∑g∈Sn(|Sn|⋅Z) =Z so Z has all eigenvalues 0 or 1 and trace(Z)=1|Sn|∑g∈Sntrace(Pg⊗Pg)=1|Sn|∑g∈Sntrace(Pg)2=1|Sn|∑g∈Snf(g)2 counts the number of eigenvalues equal to 1 for the matrix Z. We aim to show this =2. In particular since Z is a sum of real non-negative matrices so it is real non-negative and we may use Perron-Frobenius Theory. Part 2: Perron Roots / Orbits Via examining the (matrix form of) Sn's action on the standard basis vectors for the vector space Rn×n Consider the ordered Set M, which contains the standard basis vectors for vector space Rn×n, with the first n mi being the 'symmetric' ones (i.e. with trace 1). Now consider how Sn acts on M by conjugation. Again using our std permutation representation we have, for X∈M PgXP−1g=PgXPTg∈M. In particular the action by conjugation on M is a linear transformation, so we may write M=[M1M2] where M1 has the first n vectors placed next to each other and M2 has the others, observing the aforementioned ordering for g∈Sn TgM=MUg where Ug is the permutation matrix associated with Tg's action on M. Focusing on the action of conjugation by elementary type 2 matrices, we can show that all vectors in M1 are in the same conjugacy class (a single elementary type 2 action will do it) and all vectors in M2 are in the same conjugacy class (composing conjugation by 2 elementary type 2 matrices will do it). So Sn acts transitively on M1 and transitively on M2, i.e. 1|Sn|∑g∈SnTgM=M(1|Sn|∑g∈SnUg)=[Cn00Bn2−n×n2−n] i.e. a block diagonal matrix with 2 blocks on the diagonal, each being a positive matrix. Perron-Frobenius theory tells us that the RHS has exactly 2 orthonormal Perron vectors, one for each communicating class, and since the RHS is a convex combination of doubly stochastic matrices, we know the RHS is doubly stochastic. Hence Cn1n=1⋅1n and Bn2−n×n2−n1n2−n=1⋅1n2−n. Making use of the vec operator and Kronecker products, we can concretely close by exploiting the identity vec(ABC)=(CT⊗A)vec(B). I.e. by applying the vec operator to each X∈M and arranging them in the same ordering as before -- calling this new basis M′, we have TgM=MUg⟺(Pg⊗Pg)M′=M′Ug where Ug is the same permutation matrix as before. So, summing over the group and making use of the fact that M′ is an invertible matrix we have (M′)−1(1|Sn|∑g∈Sn(Pg⊗Pg))M′=(1|Sn|∑g∈SnUg)=[Cn00Bn2−n×n2−n] Thus the matrix Z is similar to (i.e. the RHS) has exactly 2 eigenvalues equal to 1, hence Z has two eigenvalues equal to 1 and all else are zero since it is idempotent. ⟹trace(Z)=2 which completes the proof of the second claim. Note: the proof of the 1st claim mimics this but is much easier. Just consider Z′:=(1|Sn|∑g∈SnPg) and (i) show it is idempotent (ii) show Z is a positive matrix, and being doubly stochastic that means exactly one Perron root =1. It isn't needed but if you like you can consider the way Sn acts on the set of n standard basis vectors and show one orbit. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Sep 8, 2021 at 5:55 answered Sep 8, 2021 at 1:53 user8675309user8675309 12.4k22 gold badges1212 silver badges2727 bronze badges 1 That's a nice solution (+1). Thanks. – Sayan Dutta Commented Sep 8, 2021 at 14:05 Add a comment | This answer is useful 1 Save this answer. Show activity on this post. Using combinatorial classes we have the following class P of permutations with fixed points marked: P=SET(U×CYC=1(Z)+CYC=2(Z)+CYC=3(Z)+⋯). This gives the EGF G(z,u)=exp(uz+z22+z33+z44+⋯)=exp(log11−z+(u−1)z)=11−zexp((u−1)z)=11−zexp(−z)exp(uz). It follows that for n≥1 the expectation for the number of fixed points is E[X]=[zn]∂∂uG(z,u)∣∣∣u=1=[zn]11−zexp(−z)exp(z)z=[zn]z1−z=1. We also get with n≥2 E[X(X−1)]=zn2G(z,u)∣∣∣u=1=[zn]11−zexp(−z)exp(z)z2=[zn]z21−z=1. We thus conclude that with n≥2 E[X2]=2. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Sep 7, 2021 at 16:02 answered Sep 7, 2021 at 14:13 Marko RiedelMarko Riedel 64.8k55 gold badges6767 silver badges147147 bronze badges 1 That's a nice solution (+1). Thanks. – Sayan Dutta Commented Sep 8, 2021 at 14:06 Add a comment | This answer is useful 1 Save this answer. Show activity on this post. Let then f(σ) be the number of fixed points of any permutation in Sn. For any i∈{1,2,…,n} write δi,σ=1 if σ(i)=i, 0 otherwise. Then ∑σf(σ)=∑σ∑iδi,σ=∑i∑σδi,σ=∑i(n−1)!=n(n−1)!=n! for the number of permutations with σ(i)=i is just the number of permutations of the n−1 remaining elements. Now ∑σ(f(σ))2=∑σ(∑iδi,σ)2=∑σ(∑iδi,σ)(∑jδj,σ). Note that δi,σδi,σ=δi,σ, so ∑σ(f(σ))2=∑σ∑iδi,σ+∑σ(∑i≠jδi,σδj,σ)=n!+∑i≠j(∑σδi,σδj,σ). As before, a permutation that fixes both i and j is just a permutation of the rest, counting (n−2)!. And the number of pairs (i,j) with i≠j is clearly n(n−1), so ∑σ(f(σ))2=n!+∑i≠j(n−2)!=n!+n(n−1)(n−2)!=2n!. Summarizing, the average of f(σ) is 1, and the average of (f(σ))2 is 2. Share CC BY-SA 4.0 Follow this answer to receive notifications edited Nov 1, 2022 at 9:27 answered Nov 1, 2022 at 9:20 Jose L. ArreguiJose L. Arregui 34322 silver badges77 bronze badges 1 Thanks for the answer (+1)! – Sayan Dutta Commented Nov 1, 2022 at 9:42 Add a comment | You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions combinatorics group-theory permutations combinations fixed-points See similar questions with these tags. Featured on Meta Will you help build our new visual identity? Upcoming initiatives on Stack Overflow and across the Stack Exchange network... Community help needed to clean up goo.gl links (by August 25) Linked 29 Show that : ∑σ∈Sn(number of fixed points of σ)2=2n! 5 Derivation of the Partial Derangement (Rencontres numbers) formula 3 Why is the formula 1|G|∑π∈Gϕ(π) more useful than 1|G|∑s∈Sη(s)? (Burnside's Lemma) 5 Group Action and the Bell Number 4 Average number of the maximum amount of fixed points of permutation in a partition of Sn 2 rank of permutation bundle Related 29 Show that : ∑σ∈Sn(number of fixed points of σ)2=2n! 4 Proving Average Fixed Points of a Permutation on n Letter is 1 1 Average of permutation group character over permutation subgroup 6 Minimal degree faithful permutation representations of some finite simple groups 4 Sum of sign of permutation number of fixed points over Sn 4 Do sets of commuting permutations with no fixed points generate Abelian groups with no fixed points? 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https://math.stackexchange.com/questions/970035/solve-a-system-of-equations-when-one-is-linear-and-the-other-is-quadratic
Solve a system of equations when one is linear and the other is quadratic - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Solve a system of equations when one is linear and the other is quadratic Ask Question Asked 10 years, 11 months ago Modified10 years, 11 months ago Viewed 80 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. x+y=3 m x+y=3 m x y=2 m 2 x y=2 m 2, m m is the parameter. I came to this 2 m 2−3 m x+x y=0 2 m 2−3 m x+x y=0. The solutions have to be:(m,2 m),(2 m,m)(m,2 m),(2 m,m). But I can't understand what is the role of this parameters, I don't know how to come to the solutions. Can someone help me I would appreciate that. systems-of-equations quadratic-forms Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Oct 17, 2014 at 16:43 Harry Peter 8,223 1 1 gold badge 25 25 silver badges 51 51 bronze badges asked Oct 12, 2014 at 12:06 dona12dona12 183 1 1 gold badge 3 3 silver badges 11 11 bronze badges 1 The second equation is not said to be quadratic Jasser –Jasser 2014-10-12 12:10:30 +00:00 Commented Oct 12, 2014 at 12:10 Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. Note that x,y x,y are the solutions of (t−x)(t−y)=0⇒t 2−(x+y)t+x y=0⇒t 2−3 m t+2 m 2=0.(t−x)(t−y)=0⇒t 2−(x+y)t+x y=0⇒t 2−3 m t+2 m 2=0. So, we have t=3 m±(−3 m)2−4⋅2 m 2−−−−−−−−−−−−−−√2=3 m±m 2=m,2 m.t=3 m±(−3 m)2−4⋅2 m 2 2=3 m±m 2=m,2 m. Hence, we have (x,y)=(m,2 m),(2 m,m).(x,y)=(m,2 m),(2 m,m). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Oct 12, 2014 at 12:10 mathlovemathlove 154k 10 10 gold badges 126 126 silver badges 313 313 bronze badges 3 But what did you replace with t?dona12 –dona12 2014-10-12 12:24:39 +00:00 Commented Oct 12, 2014 at 12:24 @dona12: I didn't. I just think about the equation about t t. You may want to see Vieta's formulas.mathlove –mathlove 2014-10-12 12:28:03 +00:00 Commented Oct 12, 2014 at 12:28 @dona12: Do you know the following? : If α,β α,β be the solutions of a x 2+b x+c=0 a x 2+b x+c=0, then α+β=−b/a,α+β=c/a α+β=−b/a,α+β=c/a. (This is one of the Vieta's formulas) We know x+y=3 m,x y=2 m 2 x+y=3 m,x y=2 m 2, then x,y x,y are the solutions of t 2−3 m t+2 m 2=0 t 2−3 m t+2 m 2=0. This is the same as (t−x)(t−y)=0(t−x)(t−y)=0.mathlove –mathlove 2014-10-12 12:34:53 +00:00 Commented Oct 12, 2014 at 12:34 Add a comment| This answer is useful 0 Save this answer. Show activity on this post. The easiest way is to think about this system of equations graphically. These are two intersecting curves (two points of intersection), one of which is linear, another is hyperbola. When you change your parameter you shift this curves (increasing parameter, shift linear curve up, hyperbola also up but faster). In terms of solving, the easiest way would be just to get y=3 m−x y=3 m−x from the first equation and plug it to the second one: x(3 m−x)=2 m 2 x(3 m−x)=2 m 2, then you get quadratic equation for x x: x 2−3 m x+2 m 2=0 x 2−3 m x+2 m 2=0 which gives exactly two solutions (D=9 m 2−8 m 2=m 2 D=9 m 2−8 m 2=m 2, then x 1=3 m+m 2=2 m x 1=3 m+m 2=2 m and x 2=3 m−m 2=m x 2=3 m−m 2=m. Correspondingly you get two y y s: y 1=3 m−x 1=3 m−2 m=m y 1=3 m−x 1=3 m−2 m=m and y 2=3 m−m=2 m y 2=3 m−m=2 m. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Oct 12, 2014 at 12:40 mathisfunmathisfun 319 1 1 silver badge 14 14 bronze badges Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions systems-of-equations quadratic-forms See similar questions with these tags. 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http://www.360doc.com/content/23/1229/10/46601607_1109175218.shtml
高中数学超几何分布知识点全解析 搜索 我的图书馆 查看信箱 系统消息 官方通知 设置 开始对话 有 11 人和你对话，查看 忽略 历史对话记录 通知设置 发文章 发文工具 撰写网文摘手文档视频思维导图随笔相册原创同步助手 其他工具 图片转文字文件清理AI助手 留言交流 搜索 分享 QQ空间QQ好友新浪微博微信 生成长图转Word打印朗读全屏修改转藏+1 × 微信扫一扫关注 查看更多精彩文章 × 微信扫一扫 将文章发送给好友 高中数学超几何分布知识点全解析 当以读书通世事 2023-12-29 发布于甘肃来源|170阅读|1 转藏 大 中 小 转藏全屏朗读打印转Word生成长图分享 QQ空间QQ好友新浪微博微信 展开全文 一、引言 超几何分布是概率论中的重要概念，它描述了在无放回抽样条件下成功次数的概率分布。在高中数学中，超几何分布是离散型随机变量分布的重要内容之一，对于理解概率论的基本原理和解决实际问题具有重要意义。本文将详细解析高中数学中超几何分布的相关知识点，帮助读者更好地理解和掌握这一概念。 二、超几何分布的定义与基本概念 无放回抽样：在总体中抽取样本时，如果每次抽取后不放回总体，则称为无放回抽样。 超几何分布的定义：设有N个元素，其中M个为成功元素。在无放回抽样的条件下，抽取n个元素，则成功元素恰好为k个的概率P(X=k)可以表示为P(X=k)=(C(M,k)C(N-M,n-k))/C(N,n)，其中k=0,1,2,...,min(n,M)。称这样的随机变量X服从参数为N、M和n的超几何分布。 三、超几何分布的性质与特点 概率分布特性：超几何分布的概率分布呈现一定的偏态，随着n的增加，偏态逐渐减小，当n较大时，超几何分布近似于二项分布。 期望与方差：超几何分布的期望E(X)=(n M)/N，方差D(X)=(n M(N-M)(N-n))/(N^2(N-1))。这些性质在实际问题中具有广泛的应用价值。 四、超几何分布在实际问题中的应用 质检问题：在产品质量检验中，如果总体中有M个不合格品，N个合格品，则在不放回抽样的条件下，抽取n个产品时恰好抽到k个不合格品的概率服从超几何分布。通过计算不同k值的概率，可以评估抽样结果的可靠性。 医学诊断试验：在医学诊断中，如果某项试验在患者群体中的阳性率为p，则在无放回抽样的条件下，从该患者群体中抽取n个样本时恰好有k个阳性样本的概率服从超几何分布。通过计算不同k值的概率，可以评估该诊断试验的准确性。 生态学问题：在生态学中，研究某种物种在特定区域内的分布情况时，可以通过超几何分布来描述该物种在不同区域内的分布情况。例如，在调查某区域内某种植物的分布情况时，可以设该区域内总共有N个植物个体，其中有M个属于目标植物种类。在无放回抽样的条件下，抽取n个植物样本时恰好有k个目标植物样本的概率服从超几何分布。通过计算不同k值的概率，可以分析目标植物在该区域内的分布情况。 五、总结与展望 通过本文的解析和应用举例，我们深入了解了高中数学中超几何分布的相关知识点。掌握超几何分布的定义、性质、特点以及在实际问题中的应用对于提高学生的数学素养和解决实际问题具有重要意义。在未来的学习和研究中，我们可以进一步探索复杂离散型随机变量的分布特性及其在实际问题中的应用。通过不断学习和实践我们可以更好地运用数学知识解决实际问题推动科学技术的进步和发展。 本站是提供个人知识管理的网络存储空间，所有内容均由用户发布，不代表本站观点。请注意甄别内容中的联系方式、诱导购买等信息，谨防诈骗。如发现有害或侵权内容，请点击 一键举报。 转藏分享 QQ空间QQ好友新浪微博微信 献花（0） +1 来自： 当以读书通世事>《073-数学（大中小学）》 举报/认领 上一篇： 高中数学样本相关系数知识点详解 下一篇： 直击中考压轴题-二次函数与直角三角形存在性问题 猜你喜欢 0 条评论 发表 请遵守用户评论公约 查看更多评论 类似文章更多 高中数学易混概念超级纠错之概率篇 高中数学易混概念超级纠错之概率篇。二项分布与超几何分布到底有着怎样的关系？超几何分布是一种不放回抽样中求概率情形，其抽样中每一... 二项分布和超几何分布，五分钟让你再也不迷糊！ 二项分布和超几何分布，五分钟让你再也不迷糊！2.每次抽取一件产品，抽完放回，抽n次(这里的n与N无关)，共抽到k次合格品的概率是？也可... 超几何分布与二项分布的区别联系 超几何分布与二项分布是学生容易混淆的两种概率分布类型。本文通过简明的论述这两者之间的区分与联系，并配有相关习题加以巩固。 概率分布列的第一座大山——超几何分布 概率分布列的第一座大山——超几何分布。接下来叶老师将通过：超几何分布的概念及个人深入理解、超几何分布的特点及适用范围、超几何分... 高中数学：随机变量、离散型随机变量的概念以及概率分布列的求解 高中数学：随机变量、离散型随机变量的概念以及概率分布列的求解。定义3：对于随机变量可能取的值，可以取某一区间内的一切值，这样的变量就叫做连续型随机变量。为随机变量X的概率分布，简称X的分布列... 第67课时 离散型随机变量及其分布列 1.随机变量：如果随机试验的结果可以用一个变量来表示，那么这样的变量叫做随机变量。2.离散型随机变量:对于随机变量可能取的值，可以按一定次序一一列出，这样的随机变。3.连续型随机变量:对于随机变... 二项分布与超几何分布的联系与区别 二项分布与超几何分布的联系与区别。经常有学生问二项分布与超几何分布到底怎么区分，是利用二项分布的公式去解决这道概率题目，还是利... 每天一点统计学——随机变量与概率分布 每天一点统计学——随机变量与概率分布。根据随机变量所代表数值的不同，随机变量分为两类：离散型随机变量和连续型随机变量。连续型随机变量是指在某一段区间上可以取无限多个数值的随机变量。根据随... 彩票概率计算方法示例 [彩票概率计算方法示例彩票概率计算方法示例。公式1] 中奖概率 P=Combin（N，T）＊Combin（M－N，S－T）／Combin（M，S）P(35,7,7,0)=Combin(7,0)Combin(35-7,7-0)/Combin(35,7)＝1/6.P(35,7,7,1)=Comb...]( 个图VIP年卡，限时优惠价168元>>x 当以读书通世事 关注对话 TA的最新馆藏 人生至理名言 高考语文10组高级思辨作文素材 2026高考作文思维训练：立意选择题25道，附深度解析 答案！提分更高效！ 高中导数的压轴题的解题逻辑思维 高考历史 | 中国近代史必背知识提纲大梳理！ 春晚史上最好的开场联唱，经典的旋律依然耳熟能详！ 喜欢该文的人也喜欢更多 生死与共阅33 一般不告诉别人的胡牌技巧，手气不好时可以这样阅134 中国特有的文化与文明-印纽【雕刻工艺】（13）阅95 遇见黄西，梦想小镇未来科学π阅66 这种世界最高、最长寿的树，它们进化到需要森林火灾才能更好的繁...阅213 热门阅读换一换 凤凰卫视资讯台直播「高清」阅612810 当前营商环境存在的困难问题及对策建议阅17734 120个常见文言实词阅457355 【初二物理】浮力计算典型题型20道（含详细解析））阅50015 数字神断基础 三角定律阅52554 复制 打印文章 发送到手机微信扫码，在手机上查看选中内容 全屏阅读 朗读全文 分享文章QQ空间QQ好友新浪微博微信 AI解释 复制 打印文章 adsbygoogle.js 发送到手机微信扫码，在手机上查看选中内容 全屏阅读 朗读全文 AI助手 阅读时有疑惑？点击向AI助手提问吧 联系客服 在线客服： 360doc小助手2 客服QQ： 1732698931 联系电话：4000-999-276 客服工作时间9:00-18:00，晚上非工作时间，请在QQ留言，第二天客服上班后会立即联系您。
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https://pmc.ncbi.nlm.nih.gov/articles/PMC5624146/
Maxillary first premolars with three root canals: two case reports - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice J Istanb Univ Fac Dent . 2017 Oct 2;51(3):50–54. doi: 10.17096/jiufd.03732 Search in PMC Search in PubMed View in NLM Catalog Add to search Maxillary first premolars with three root canals: two case reports Zeliha Ugur Zeliha Ugur 1 Department of Endodontics, Faculty of Dentistry, Abant İzzet Baysal University Turkey Find articles by Zeliha Ugur 1,, Kerem Engin Akpinar Kerem Engin Akpinar 2 Department of Endodontics, Faculty of Dentistry, Cumhuriyet University Turkey Find articles by Kerem Engin Akpinar 2, Demet Altunbas Demet Altunbas 2 Department of Endodontics, Faculty of Dentistry, Cumhuriyet University Turkey Find articles by Demet Altunbas 2 Author information Article notes Copyright and License information 1 Department of Endodontics, Faculty of Dentistry, Abant İzzet Baysal University Turkey 2 Department of Endodontics, Faculty of Dentistry, Cumhuriyet University Turkey To whom correspondence should be addressed: Dr. Zeliha Ugur Department of Endodontics Faculty of Dentistry Abant İzzet Baysal University 14000 Bolu Turkey Phone: +90 374 254 10 00 zlhugur@gmail.com Received 2017 Feb 13; Accepted 2017 Jun 11; Collection date 2017. Copyright © 2017 Journal of Istanbul University Faculty of Dentistry This article is licensed under Creative Commons License Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0) license ( Users must give appropriate credit, provide a link to the license, and indicate if changes were made. Users may do so in any reasonable manner, but not in any way that suggests the journal endorses its use. The material cannot be used for commercial purposes. If the user remixes, transforms, or builds upon the material, he/she may not distribute the modified material. No warranties are given. The license may not give the user all of the permissions necessary for his/her intended use. For example, other rights such as publicity, privacy, or moral rights may limit how the material can be used. PMC Copyright notice PMCID: PMC5624146 PMID: 29114431 Abstract It is very important that the dentists have sufficient information about possible variations in the expected root canal configurations in order to achieve success in endodontic treatment. In addition to having adequate knowledge on the variations of the root canal anatomy, periapical radiographs from different angles, careful examination of the pulp chamber floor, and use of dental operation microscope during the procedure are also important factors that contribute to the diagnosis of the additional roots and canals. The aims of this article are to present the diagnostic approach and root canal treatments of two maxillary first premolar teeth with three canals in two patients. Keywords: Root canals, variation, premolar tooth, endodontic treatment, anatomy Introduction The root canal system can present different anatomical variations. Those that go unnoticed can cause failures, particularly in the root canal treatment of teeth with pulp necrosis. Clinician should therefore possess necessary knowledge concerning the normal root canal anatomy, morphology and most importantly, be ready to deal with frequent variations. All of these factors are essential for predictable success in endodontic treatment (1). Various studies have investigated the differences in the external and internal anatomy of maxillary first premolars (2, 3, 4). In a Turkish sample, Kartal et al. (5) compared the frequencies of maxillary premolar teeth having one, two or three canals and roots. Authors reported that 8.66% of the premolars had one, 89.64% had two, and 1.66% had three root canals. In terms of root number, 37.31%, of the maxillary first premolars was found to have one, 61.32% to have two, and 1.33% to have three roots. Three-rooted maxillary premolars are occasionally referred to as small molars or as radiculous because of their anatomical similarity to maxillary molars (6). This variation causes additional challenge in access cavity design, localization, refining and forming of endodontic therapy procedures (7). Clinicians should consider extra roots and canals to prevent endodontic infections and related symptoms in patients (8). The aim of this case report is to present a rare anatomical variation of maxillary first premolars characterized by having two roots and canals buccally, and one root and canal palatally. Case reports Case 1 A 20 years old male was referred to Cumhuriyet University, Faculty of Dentistry, Department of Endodontics. Patient’s medical history was non-contributory. His main complaint was the spontaneous pain on the left side of his upper jaw. Clinical and radiographic examination revealed a deep disto-occlusal carious lesion in left maxillary first premolar. The tooth was not sensitive to percussion and exhibited normal mobility. Radiographic appearance of the periapical region was normal. Based on these findings, a diagnosis of symptomatic irreversible pulpitis was made (Figure 1). Figure 1. Open in a new tab Periapical radiograph, showing the complex root morphology of the maxillary first premolar suggesting the existence of three root canals. Endodontic access cavity was prepared after the injection of local anesthesia and rubber dam isolation. During the inspection of the pulp chamber floor, a second canal orifice was seen in the buccal section of the tooth. Positions of the three canal orifices were as follows: one mesiobuccal, one distobuccal and one palatinal. The working lengths of three root canals were determined by using Raypex 5 (VDW, Munich, Germany) apex locator, and checked with a radiograph (Figure 2). Canals were instrumented using Twisted File rotary system (TF; SybronEndo, Orange, CA, USA). Patency was achieved in all the canals and was maintained with a 10 K-file (DentsplyMaillefer, Ballaigues, Switzerland). 2.5% sodium hypochlorite (NaOCl) solution was used as the main irrigant during biomechanical preparation. Root canal filling was completed by using the cold lateral compaction method with AH Plus (Dentsply DeTrey, Konstanz, Germany) sealer and gutta-percha (Diadent Group International, Chungcheongbuk-do, Korea). The root canal treatment was completed in a single-visit appointment (Figure 3). The tooth was later restored with composite resin (3M, St. Paul, MN, USA). Figure 2. Open in a new tab Radiographical confirmation of three root canals and determination of the working lengths. Figure 3. Open in a new tab Periapical radiograph following the obturation of the three root canals. Case 2 A 25 years old male patient was referred to the same institution for the treatment of his maxillary right premolar tooth. Patient’s medical history was non-contributory. Intraoral examination showed a deep disto-occlusal carious lesion. No sensitivity to percussion was observed. No periapical lesion was found in the radiographic examination (Figure 4). Based on these findings, a diagnosis of symptomatic irreversible pulpitis was made. After the injection of local anesthesia and rubber dam isolation, endodontic access cavity was prepared. The inspection of the pulp chamber floor revealed the presence of three separate root canal orifices: one mesiobuccal, one distobuccal and one palatinal. Canals were instrumented with the TF after determining their working lengths with Raypex 5 apex locator (Figure 5). Biomechanical preparation was done using %2.5 NaOCl solution as the main irrigant. Root canal filling was completed by cold lateral compaction method using AH Plus (Dentsply DeTrey, Konstanz, Germany) sealer and gutta-percha (Diadent Group International, Chungcheongbuk-do, Korea). The root canal therapy was completed in a single-visit appointment (Figure 6). The tooth was then restored with composite resin (3M, St. Paul, MN, USA). Figure 4. Open in a new tab Periapical radiograph, showing the complex root morphology of the maxillary first premolar suggesting the existence of three root canals. Figure 5. Open in a new tab Radiographical confirmation of the three root canals and determination of the working lengths. Figure 6. Open in a new tab Periapical radiograph following the obturation of the three root canals. Discussion The anatomical variations of the teeth are very important in endodontics in terms of diagnosis, treatment planning and prognosis. In the present case reports, three roots and canals, which is an anatomical variation in of maxillary first premolar teeth, were shown in two different patients. Several studies regarding the presence of three roots and canals in maxillary first premolar teeth are available in the literature (9, 10, 11). In studies conducted on the Turkish population, Bulut et al . (12) reported that the incidence of three roots and canals in maxillary first premolar teeth was 1% while Ok et al. (13) reported the same incidence as 1.2%. Cleaning of the root canal system is important to achieve success in endodontic treatment. Magnifying glasses and the radiographs from different angles can be used during endodontic procedures to detect anatomical variations. However, they may not provide complete information about the canals as the resulting images are 2-dimensional. With the recent development of CBCT and its use in endodontic therapy, it is now possible to easily identify canals that can be missed even with periapical radiographs obtained from different angles (14). Furthermore, preparing a well-shaped endodontic access cavity and investigating its floor carefully are also effective in the detection of additional canal orifices. Balleri et al. (15) reported that T shaped endodontic cavity is ideal in terms of cleaning and gaining easy access to the pulp chamber and canals of the premolar teeth with three roots and canals. As suggested by Balleri et al. (15) a cut at the bucco-proximal angle, from the entrance of buccal canals to cavo-surface angle, was made in the present cases. So the outline of endodontic cavity was formed. This T-shaped access technique is useful to reach all the root canals properly (15). In straight-on radiographs of maxillary premolars, it was reported by Sieraski et al. (16) that whenever the mesio-distal width of the mid-root image was equal to the mesio-distal width of the crown or greater than it, the tooth probably has three roots. Conclusion Presence of a second buccal canal and root is a rare anatomical variation that can be seen in maxillary first premolars. Endodontists should always consider the possibility of unusual number of roots and canals to overcome infections and related symptoms. It is of utmost importance to diagnose and treat the teeth having such variations in order to maintain the balance of oral environment. Footnotes Source of funding: None declared. Conflict of interest: None declared. One of these cases was presented as a poster at the 13th International Scientific Congress of the Turkish Endodontic Society which was held at Kapadokya, Turkey, May 26-29, 2016. References 1.Cantatore G, Berutti E, Castellucci A. Missed anatomy: Frequency and clinical impact. Endod Topics. 2006;15(1):3–31. doi: 10.1111/j.1601-1546.2009.00240.x. [DOI] [Google Scholar] 2.Marca C, Dummer PM, Bryant S, Vier-Pelisser FV, So MV, Fontanella V, Dutra VD, de Figueiredo JA. Three-rooted premolar analyzed by high-resolution and cone beam ct. Clin Oral Investig. 2013;17(6):1535–1540. doi: 10.1007/s00784-012-0839-5. [DOI] [PubMed] [Google Scholar] 3.Pecora JD, Saquy PC, Sousa Neto MD, Woelfel JB. Root form and canal anatomy of maxillary first premolars. Braz Dent J. 1992;2(2):87–94. [PubMed] [Google Scholar] 4.Vier-Pelisser FV, Dummer PM, Bryant S, Marca C, So MV, Figueiredo JA. The anatomy of the root canal system of three-rooted maxillary premolars analysed using high-resolution computed tomography. Int Endod J. 2010;43(12):1122–1131. doi: 10.1111/j.1365-2591.2010.01787.x. [DOI] [PubMed] [Google Scholar] 5.Kartal N, Ozcelik B, Cimilli H. Root canal morphology of maxillary premolars. J Endod. 1998;24(6):417–419. doi: 10.1016/S0099-2399(98)80024-1. [DOI] [PubMed] [Google Scholar] 6.Javidi M, Zarei M, Vatanpour M. Endodontic treatment of a radiculous maxillary premolar: A case report. J Oral Sci. 2008;50(1):99–102. doi: 10.2334/josnusd.50.99. [DOI] [PubMed] [Google Scholar] 7.Orucoglu H, Kont Cobankara F. Maxillary first premolar with three roots: A case report. Hacettepe Diş Hek. Fak Derg. 2005;29(4):26–29. [Google Scholar] 8.Vertucci FJ. Root canal morphology and its relationship to endodontic procedures. Endod Topics. 2005;10(1):3–29. doi: 10.1111/j.1601-1546.2005.00129.x. [DOI] [Google Scholar] 9.Ahmad IA, Alenezi MA. Root and root canal morphology of maxillary first premolars: A literature review and clinical considerations. J Endod. 2016;42(6):861–872. doi: 10.1016/j.joen.2016.02.017. [DOI] [PubMed] [Google Scholar] 10.Ozcan E, Colak H, Hamidi MM. Root and canal morphology of maxillary first premolars in a turkish population. J Dent Sci. 2012;7(4):390–394. doi: 10.1016/j.jds.2012.09.003. [DOI] [Google Scholar] 11.Sert S, Bayirli GS. Evaluation of the root canal configurations of the mandibular and maxillary permanent teeth by gender in the turkish population. J Endod. 2004;30(6):391–398. doi: 10.1097/00004770-200406000-00004. [DOI] [PubMed] [Google Scholar] 12.Bulut DG, Kose E, Ozcan G, Sekerci AE, Canger EM, Sisman Y. Evaluation of root morphology and root canal configuration of premolars in the turkish individuals using cone beam computed tomography. Eur J Dent. 2015;9(4):551–557. doi: 10.4103/1305-7456.172624. [DOI] [PMC free article] [PubMed] [Google Scholar] 13.Ok E, Altunsoy M, Nur BG, Aglarci OS, Colak M, Gungor E. A cone-beam computed tomography study of root canal morphology of maxillary and mandibular premolars in a turkish population. Acta Odontol Scand. 2014;72(8):701–706. doi: 10.3109/00016357.2014.898091. [DOI] [PubMed] [Google Scholar] 14.Kakkar P, Singh A. Maxillary first molar with three mesiobuccal canals confirmed with spiral computer tomography. J Clin Exp Dent. 2012;4(4):e256–259. doi: 10.4317/jced.50850. [DOI] [PMC free article] [PubMed] [Google Scholar] 15.Balleri P, Gesi A, Ferrari M. Primer premolar superior com tres raices. Endod Pract. 1997;3:13–15. [Google Scholar] 16.Sieraski SM, Taylor GN, Kohn RA. Identification and endodontic management of three-canalled maxillary premolars. J Endod. 1989;15(1):29–32. doi: 10.1016/S0099-2399(89)80095-0. [DOI] [PubMed] [Google Scholar] Articles from Journal of Istanbul University Faculty of Dentistry are provided here courtesy of Istanbul University Faculty of Dentistry ACTIONS View on publisher site PDF (599.7 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Case reports Discussion Conclusion Footnotes References Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
14007
https://goopenva.org/courseware/lesson/1543/overview
Comparing and Ordering Fractions, Decimals, and Percents | #GoOpenVA Virginia Department of EducationAn official website of the Commonwealth of Virginia Here's how you knowAn official websiteHere's how you know Find a Commonwealth Resource Discover Resources Collections Providers Hubs Sign in to see your Hubs Login Featured HubsComputer Science Advancing Computer Science Education (ACSE) Internet Safety Resources Public Media Partners Health Education Resource Hub CodeVA See all Hubs Groups Sign in to see your Groups Login Featured GroupsVirginia Society for Technology in Education See all Groups Learn More About Help Center Add OER Open Author Create a standalone learning module, lesson, assignment, assessment or activity Create Resource Submit from Web Submit OER from the web for review by our librarians Add Link Learn more about creating OER Add OER Add Link Create Resource About creating OER Notifications Sign In/Register Search Advanced Search Sign In/Register Discover Resources Collections Providers Hubs Sign in to see your Hubs Login Featured HubsComputer Science Advancing Computer Science Education (ACSE) Internet Safety Resources Public Media Partners Health Education Resource Hub CodeVA See all Hubs Groups Sign in to see your Groups Login Featured GroupsVirginia Society for Technology in Education See all Groups Learn More About Help Center Add OER Open Author Create a standalone learning module, lesson, assignment, assessment or activity Create Resource Submit from Web Submit OER from the web for review by our librarians Add Link Learn more about creating OER Add OER Add Link Create Resource About creating OER Student View Preview Copy Save Please log in to save materials. Log in Report Details Standards Resource Library Author:Tiffany VietinghoffSubject:Mathematics Material Type:Activity/Lab, Assessment, Homework/Assignment Level:Middle School Tags: Compare Decimals Fractions Hanover-project Mini-grant Number Sense Order Percents Rational Numbers Log in to add tags to this item. License:Creative Commons AttributionLanguage:English Media Formats:Interactive Show MoreShow Less Education Standards DOCX Comparing and Ordering Fractions, Decimals, and Percents - Grade 7 Download Link to Online Desmos Lesson View Link to online Desmos Polygraph Partner Activity View Comparing and Ordering Fractions, Decimals, and Percents Overview Within this interactive lesson, students will navigate and explore comparing positive and negative fractions, deicmals, and percents through Desmos. Students will also gain practice from finding equivalencies, using picture representations and number lines, and through mathematical discourse. Students will be able to do the following: Use a number line to sort, compare, and order integers, fractions, decimals, and percents. Compare integers, fractions, decimals, and percents. Order integers, fractions, decimals, and percents. Mathematics Lesson Plan - Grade 7 Comparing and Ordering Fractions, Decimals, and Percents Strand:Number and Number Sense Topic:Comparing and Ordering Fractions, Decimals, and Percents Primary SOL:.7.1c The student will compare and order rational numbers. Vertically Aligned SOLs:5.2b The student will compare and order fractions, mixed numbers, and/or decimals in a given set, from least to greatest and greatest to least. 6.2bThe student will compare and order positive rational numbers. 6.3b The student will compare and order integers 8.1The student will compare and order real numbers. Materials Link to online Desmos Lesson Link to online Desmos Polygraph Partner Activity Vocabulary Rational number Whole Number Percent Fraction Decimal Integer Greater than Less than Student Learning Intentions: I can use a number line to sort, compare, and order integers, fractions, decimals, and percents. I can compare integers, fractions, decimals, and percents. I can order integers, fractions, decimals, and percents. Student/Teacher Actions: What should students be doing? What should teachers be doing? 1.Using the Desmos Link, the teacher will prompt students about the circle images. Students will collaborate and discuss how many ways they can write ½, 1, 0 and 2 (using integers, fractions, decimals and percentages). The teacher should guide the students accordingly. Students will also find equivalencies among these numbers before moving on in the lesson. This can be done as a class discussion or individually through Desmos, Teachers may consider the following question prompts: How many ways can you write this number (½, 1, 0, 2)? What would it look like as a decimal? A fraction? A percent? How do you know? What other ways can we draw that number? When may you see that number in real life? How can you compare this number to others (ie ½ to 1 or ½ to 0 or ¼ ) Students will then move to labeling a number line using integers and benchmark numbers (such as 0.5). The teacher should facilitate student discussion by asking students about what numbers are useful on a number line when comparing rational numbers. Teachers may consider the following question prompts: What are integers? What are whole numbers? How do we label a number line? Where are the smallest numbers located on a number line? The largest numbers? What are some numbers, “benchmarks”, that would help us navigate a number line? How can we write those numbers (decimals, fractions, percents)? Students will use number lines, either from Desmos or the blank number line below, compare fractions, decimals and percents before moving on to ordering them. After students complete slide 13, the teacher should lead a discussion to reflect on the sort. The teacher may use the following question prompts: How did you sort the numbers? What did you notice about the categories? What strategies did you use to sort the numbers? What was the easiest number to place and why? What was the most difficult number to place and why? After students complete slides 14 and 15, continue to monitor students progress and have class discussions using questions similar to the ones above to help students navigate through any obstacles. Place students in pairs. Students will then go to the Desmos Polygraph Activity Link. There, they will learn how to ask questions to guess their partner’s mystery number and then reflect. Teachers should monitor students progress (this can be done easily through the dashboard on Desmos) and provide feedback on the students’ questions. The teacher should review what questions and vocabulary the students are using. For example, the teacher can look to see if students are using benchmarks like 0, ½, and 1, describing numbers as decimals, fractions, percents, or a combination, and providing accurate feedback to each other. Once students have played the game one round, teachers should highlight, discuss, and encourage good descriptors, strategies, and questions that the students have used. Then, have the students play a few more rounds to practice using these questions and vocabulary. Assessment Questions/Journal/Reflection What do you know about integers? Decimals? Fractions? Percents? How many ways can you write or describe ½? 0? 1? 2? What do you know about a number line? How can number lines be useful when comparing or ordering integers, fractions, decimals, and percentages? What are some strategies for comparing integers, fractions, decimals, and percents? What are some strategies for ordering integers, fractions, decimals, and percents? What symbols are used for comparing numbers? Other Distribute the cards below and have students select 3-4 at random and place in either ascending or descending order. Extensions and Connections The cards below (Appendix A) can be used for hands-on activities such as war, a human number line, or a paper-version of the polygraph activity. The Desmos activity can be used as a powerpoint if computers or tablets are not available for every student. Strategies for Differentiation The teacher may use the Desmos lesson as a self-paced activity for students. Teachers can provide fraction bar visuals and equivalency charts to help students compare, such as number lines with tenths marked on them, fraction circles, or equivalent fraction charts. Appendix A -0.42 0.240.99 1.23 0.50-1- -0.002 -33%76%-0.74-0.5 4.5 0.12 60% -0.45 45%-12%6.5% Discover Resources Collections Providers Advanced Search Create Open Author Submit a Resource Connect Groups Hubs Learn More About Help Center Virginia Department of Education Contact Us Connect with #GoOpenVA Powered By Terms of Agreement The Virginia Department of Education (VDOE) is a collaborative partner in the #GoOpenVA initiative. The VDOE has not evaluated these resources for content nor accessibility but they have generally been reviewed by educational peers. Use your professional and subject expertise in evaluating and using these resources. For more details, see About #GoOpenVA Resources. GoOpenVA was last updated: 12/20/2019 × Sign in / Register Your email or username: Did you mean ? 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14008
https://bio.libretexts.org/Workbench/Bio_11A_-_Introduction_to_Biology_I/27%3A_Mendelian_Genetics/27.05%3A_A_Closer_Look_at_Probabilities
Skip to main content 27.5: A Closer Look at Probabilities Last updated : Nov 3, 2024 Save as PDF 27.4: Monohybrid Cross and the Punnett Square 27.6: Laws of Inheritance Page ID : 148204 ( \newcommand{\kernel}{\mathrm{null}\,}) Introduction The Punnett square is a valuable tool, but it's not ideal for every genetics problem. For instance, suppose you were asked to calculate the frequency of the recessive class not for an Aa x Aa cross, not for an AaBb x AaBb cross, but for an AaBbCcDdEe x AaBbCcDdEe cross. If you wanted to solve that question using a Punnett square, you could do it – but you'd need to complete a Punnett square with 1024 boxes. Probably not what you want to draw during an exam, or any other time, if you can help it! The five-gene problem above becomes less intimidating once you realize that a Punnett square is just a visual way of representing probability calculations. Although it’s a great tool when you’re working with one or two genes, it can become slow and cumbersome as the number goes up. At some point, it becomes quicker (and less error-prone) to simply do the probability calculations by themselves, without the visual representation of a clunky Punnett square. In all cases, the calculations and the square provide the same information, but by having both tools in your belt, you can be prepared to handle a wider range of problems in a more efficient way. In this article, we’ll review some probability basics, including how to calculate the probability of two independent events both occurring (event X and event Y) or the probability of either of two mutually exclusive events occurring (event X or event Y). We’ll then see how these calculations can be applied to genetics problems, and, in particular, how they can help you solve problems involving relatively large numbers of genes. Probability basics Probabilities are mathematical measures of likelihood. In other words, they’re a way of quantifying (giving a specific, numerical value to) how likely something is to happen. A probability of 1 for an event means that it is guaranteed to happen, while a probability of 0 for an event means that it is guaranteed not to happen. A simple example of probability is having a 1/2 chance of getting heads when you flip a coin. The product rule One probability rule that's very useful in genetics is the product rule, which states that the probability of two (or more) independent events occurring together can be calculated by multiplying the individual probabilities of the events. For example, if you roll a six-sided die once, you have a 1/6(27.5.1) chance of getting a six. If you roll two dice at once, your chance of getting two sixes is: (probability of a six on die 1) x (probability of a six on die 2) = (1/6)⋅(1/6)=1/36(27.5.2) . In general, you can think of the product rule as the “and” rule: if both event X and event Y must happen in order for a certain outcome to occur, and if X and Y are independent of each other (don’t affect each other’s likelihood), then you can use the product rule to calculate the probability of the outcome by multiplying the probabilities of X and Y. We can use the product rule to predict frequencies of fertilization events. For instance, consider a cross between two heterozygous (Aa) individuals. What are the odds of getting an aa individual in the next generation? The only way to get an aa individual is if the mother contributes an a gamete and the father contributes an a gamete. Each parent has a 1/2(27.5.3) chance of making an a gamete. Thus, the chance of an aa offspring is: (probability of mother contributing a) x (probability of father contributing a) = (1/2)⋅(1/2)=1/4(27.5.4) . This is the same result you’d get with a Punnett square, and actually the same logical process as well—something that took me years to realize! The only difference is that, in the Punnett square, we'd do the calculation visually: we'd represent the 1/2 probability of an a gamete from each parent as one out of two columns (for the father) and one out of two rows (for the mother). The 1-square intersect of the column and row (out of the 4 total squares of the table) represents the 1/4 chance of getting an a from both parents. The sum rule of probability In some genetics problems, you may need to calculate the probability that any one of several events will occur. In this case, you’ll need to apply another rule of probability, the sum rule. According to the sum rule, the probability that any of several mutually exclusive events will occur is equal to the sum of the events’ individual probabilities. For example, if you roll a six-sided die, you have a 1/6(27.5.5) chance of getting any given number, but you can only get one number per roll. You could never get both a one and a six at the same time; these outcomes are mutually exclusive. Thus, the chances of getting either a one or a six are: (probability of getting a 1) + (probability of getting a 6) = (1/6) +(1/6) = 1/3. You can think of the sum rule as the “or” rule: if an outcome requires that either event X or event Y occur, and if X and Y are mutually exclusive (if only one or the other can occur in a given case), then the probability of the outcome can be calculated by adding the probabilities of X and Y. As an example, let's use the sum rule to predict the fraction of offspring from an Aa x Aa cross that will have the dominant phenotype (AA or Aa genotype). In this cross, there are three events that can lead to a dominant phenotype: Two A gametes meet (giving AA genotype), or A gamete from Mom meets a gamete from Dad (giving Aa genotype), or a gamete from Mom meets A gamete from Dad (giving Aa genotype) In any one fertilization event, only one of these three possibilities can occur (they are mutually exclusive). Since this is an “or” situation where the events are mutually exclusive, we can apply the sum rule. Using the product rule as we did above, we can find that each individual event has a probability of 1/4. So, the probability of offspring with a dominant phenotype is: (probability of A from Mom and A from Dad) + (probability of A from Mom and a from Dad) + (probability of a from Mom and A from Dad) = (1/4) + (1/4) + (1/4)= 3/4. Once again, this is the same result we’d get with a Punnett square. One out of the four boxes of the Punnett square holds the dominant homozygote, AA. Two more boxes represent heterozygotes, one with a maternal A and a paternal a, the other with the opposite combination. Each box is 1 out of the 4 boxes in the whole Punnett square, and since the boxes don't overlap (they’re mutually exclusive), we can add them up 1/4 + 1/4 + 1/4 = 3/4 to get the probability of offspring with the dominant phenotype. The product rule and the sum rule | Product rule | Sum rule | --- | | For independent events X and Y, the probability ( P(27.5.6) ) of them both occurring (X and Y) is P(X)⋅P(Y)(27.5.7) . | For mutually exclusive events X and Y, the probability ( P(27.5.8) ) that one will occur (X or Y) is P(X)+P(Y)(27.5.9) . | Applying probability rules to dihybrid crosses Direct calculation of probabilities doesn’t have much advantage over Punnett squares for single-gene inheritance scenarios. (In fact, if you prefer to learn visually, you may find direct calculation trickier rather than easier.) Where probabilities shine, though, is when you’re looking at the behavior of two, or even more, genes. For instance, let’s imagine that we breed two dogs with the genotype BbCc, where dominant allele B specifies black coat color (versus b, yellow coat color) and dominant allele C specifies straight fur (versus c, curly fur). Assuming that the two genes assort independently and are not sex-linked, how can we predict the number of BbCc puppies among the offspring? One approach is to draw a 16-square Punnett square. For a cross involving two genes, a Punnett square is still a good strategy. Alternatively, we can use a shortcut technique involving four-square Punnett squares and a little application of the product rule. In this technique, we break the overall question down into two smaller questions, each relating to a different genetic event: What’s the probability of getting a Bb genotype? What’s the probability of getting an Cc genotype? In order for a puppy to have a BbCc genotype, both of these events must take place: the puppy must receive Bb alleles, and it must receive Cc alleles. The two events are independent because the genes assort independently (don't affect one another's inheritance). So, once we calculate the probability of each genetic event, we can multiply these probabilities using the product rule to get the probability of the genotype of interest (BbCc). To calculate the probability of getting a Bb genotype, we can draw a 4-square Punnett square using the parents' alleles for the coat color gene only, as shown above. Using the Punnett square, you can see that the probability of the Bb genotype is 1/2. (Alternatively, we could have calculated the probability of Bb using the product rule for gamete contributions from the two parents and the sum rule for the two gamete combinations that give Bb.) Using a similar Punnett square for the parents' fur texture alleles, the probability of getting an Cc genotype is also 1/2. To get the overall probability of the BbCc genotype, we can simply multiply the two probabilities, giving an overall probability of 1/4. You can also use this technique to predict phenotype frequencies. Attributions Probability in Genetics by KhanAcademy is license CCBY-NC-SA 4.0. 27.4: Monohybrid Cross and the Punnett Square 27.6: Laws of Inheritance
14009
https://eyewiki.org/Optic_Nerve_and_Retinal_Nerve_Fiber_Imaging
Please note: This website includes an accessibility system. Press Control-F11 to adjust the website to people with visual disabilities who are using a screen reader; Press Control-F10 to open an accessibility menu. Popup heading Create account Log in View form View source View history Optic Nerve and Retinal Nerve Fiber Imaging From EyeWiki Jump to:navigation, search All content on Eyewiki is protected by copyright law and the Terms of Service. This content may not be reproduced, copied, or put into any artificial intelligence program, including large language and generative AI models, without permission from the Academy. Article initiated by: Samuel P. Solish,MD All contributors: Brad H. Feldman, M.D., Samuel P. Solish,MD, Sarwat Salim MD, FACS, Aaron M. Miller, MD, MBA, FAAP, Ahmad A. Aref, MD, MBA, Daniel B. Moore, MD, Tala Al-Khaled, MD, Shivani Kamat, MD Assigned editor: Shivani Kamat, MD Review: Assigned status Up to Date by Shivani Kamat, MD on April 28, 2025. | | | add | | Contributing Editors: | add | Contents 1 Summary 2 Optic Nerve Anatomy 2.1 Optic Nerve in the Detection of Glaucoma 2.2 Optic Nerve Imaging Technologies 2.3 Photography of the Optic Nerve 2.3.1 Advantages of Optic Disc Photography 2.3.2 Disadvantages of Optic Disc Photography 2.4 Computed Analysis of the Optic Nerve Head and Retinal Nerve Fiber Layer 2.5 Confocal Scanning Laser Ophthalmoscopy 2.6 Scanning Laser Polarimetry 2.7 Optical Coherence Tomography 2.7.1 General Concepts of OCT Analysis for Glaucoma 3 Acknowledgement 4 Conclusion 5 Additional Resources 6 References Summary Since the mid-19th century, it has been recognized that changes in the optic nerve appearance correlate with vision and visual field loss in glaucoma. Although there have been variations in the definition of glaucoma over time, increased attention to the structure and appearance of the optic nerve has been a hallmark in understanding glaucoma. Direct observation with notation progressed to accurate photographic techniques. Recently, techniques utilizing sophisticated laser scanning with digital image processing have been used to assist clinical evaluation of the optic nerve and the retinal nerve fiber layer (RNFL). This article will discuss information on evaluation of the intra-bulbar structure of the optic nerve as it relates to glaucoma, with particular emphasis on three imaging devices, confocal scanning laser ophthalmoscopy, optical coherence tomography, and scanning laser polarimetry. Optic Nerve Anatomy The optic nerve (optic disc, optic disk, optic nerve head [ONH]) area is approximately 2.1-2.8 mm2 in whites who are not highly myopic depending on the measurement method utilized. The optic nerve size changes in early life and is likely stable after age 10. There are significant variations in optic nerve structural parameters in homogenous populations as well as with refractive error and race. Optic Nerve in the Detection of Glaucoma Changes in the optic nerve in glaucoma are classically considered to be cupping and pallor of the disc. The term pallor is likely to be misinterpreted to mean generalized pallor of the optic nerve. The pallor of glaucomatous optic nerves relates to the increased visibility of the lamina cribosa as compared to the neuroretinal rim (containing the ganglion cells). As the rim thins from ganglion cell loss, the cup becomes larger and often deeper exposing the lamina with increased pallor of the cup. Pallor of the neuroretinal rim indicates previous insufficiency of the vascular supply of the optic nerve which is often unrelated to glaucoma. The retinal ganglion cells coursing into the optic nerve are responsible for the appearance of the neuroretinal rim. The decrease in vision from glaucoma is related to the loss of retinal ganglion cells. Elevated intraocular pressure is the greatest risk factor for development of ganglion cell loss. However there appear to be a number of different pathophysiologic mechanisms by which ganglion cell loss may occur. The optic nerve appearance often provides evidence of the presence and progression of glaucoma. | Clinical Signs of Glaucomatous Optic Neuropathy | | Generalized | | Large optic cup Asymmetry of the cups (usually > 0.2) | | Focal | | Vertical elongation of the cup Regional thinning Cupping to the rim margin (notching) Splinter hemorrhage Nerve fiber layer loss | | Less specific | | Exposed lamina cribrosa Nasal displacement of vessels Baring of the circumlinear vessels Peripapillary atrophy | Optic Nerve Imaging Technologies This is the left optic disc of a patient with POAG. Note the increased vertical cup:disc ratio and thinning of the superior neuro-retinal rim. Although other technologies exist for the evaluation of the optic nerve and nerve fiber, the following are the major technologies with the greatest installed user base in 2015. Disc Photography Confocal Scanning Laser Ophthalmoscopy Scanning Laser Polarimetry Optical Coherence Tomography Photography of the Optic Nerve Fundus photography has been the primary method of documenting the optic nerve until the advent of computerized imaging techniques. Fundus photography was developed in the 1920’s. The standard optic nerve photographic technique of 35mm film and now digital photography has been used extensively since the 1960’s. Photographic documentation of the optic nerve serves multiple purposes. Single images can be used to closely evaluate the structural relationships within the nerve. Extended examination is difficult for patients and practitioners. Thus, photography allows the practitioner to evaluate fine details of the anatomy not easily seen on examination. It is not uncommon for practitioners to identify feature such as disc hemorrhages on subsequent evaluation of a photograph. In addition, photographs allow for analysis of changes in the anatomy of the optic nerve over time. Stereo disc photography and red-free nerve fiber layer photography are additional techniques to enhance the evaluation of the disc photograph. Stereo disc photography has greater utility in determining size and shape of the neuroretinal rim and the depth of the cup than two-dimensional photography. Stereo disc photography can be achieved utilizing several methods. A common technique is to take two photographs of the same nerve shifting the angle of the camera slightly between exposures. The resulting photographs can then be viewed with an inexpensive stereoscope viewer or similar device . Although recent advances in computerized analysis of the ONH have assisted the practitioner, these methods have not completely supplanted photographic analysis of the optic nerve. In particular, the cost of optic nerve photography is substantially lower than the newer techniques. Advantages of Optic Disc Photography Similar to clinical exam; comfort and easy interpretation for some clinicians Full color helps to distinguish between cupping and pallor Better detection of disc hemorrhages Aids in detection of peripapillary atrophy Stable technology Less expensive compared to other imaging devices Disadvantages of Optic Disc Photography Qualitative, not quantitative description Interobserver variability High quality of photographs required for accurate interpretation May be difficult to detect subtle changes with a photograph Requires special hand-held lenses for viewing and interpretation Computed Analysis of the Optic Nerve Head and Retinal Nerve Fiber Layer Digital evaluation of the optic nerve has progressed over the last thirty years. Various techniques for analyzing photographs have been utilized to provide objective data from photography. Techniques such as stereophotogrammetry, stereo chronometry, raster stereography were developed to analyze the ONH and/or disc photographs. Often expensive and complicated, these techiques failed to reach the level of common use. In the 1990’s, three new technologies were introduced for optic nerve imaging which garnered significant interest. These are Confocal Scanning Laser Ophthalmoscopy, Optical Coherence Tomography and Scanning Laser Polarimetry. | QUANTITATIVE IMAGING | | | --- | Principles | | Clinical Parameters Measured | | HRT | Confocal Scanning Laser Ophthalmoscopy | Optic Disc Tomography | | GDx | Scanning Laser Polarimetry/Birefringence | Retinal Nerve Fiber Layer Thickness | | OCT | Interferometry | Retinal Nerve Fiber Layer Thickness | Confocal Scanning Laser Ophthalmoscopy Confocal Laser Ophthalmoscopy (CSLO) is the concept behind the Heidelberg Retinal Tomograph. This technique for precise microscopic imaging was patented in 1955 by Marvin Minsky, now a Professor at M.I.T. Since the original patent, numerous instruments in many areas of science and engineering have utilized this technique to perform precision microscopy. Confocal microscopy offers several advantages over conventional optical microscopy, including controllable depth of field, the elimination of image degrading out-of-focus information, and the ability to collect serial optical sections from thick specimens. The key to the confocal approach is the use of spatial filtering to eliminate out-of-focus light or flare in specimens that are thicker than the plane of focus. Confocal scanning laser ophthalmoscopy generates up to 64 transaxial laser scans through the ONH and peripapillary retina to reconstruct a high-resolution 3-dimensional image. A 670-nm diode laser emits a beam that is focused in the x-axis and y-axis (horizontal and vertical dimensions) of the ONH, perpendicular to the z-axis (axis along the optic nerve). The reflected image from this plane is captured as a 2-dimensional scan. Successive equidistant images are obtained, up to 64 in total, depending on the cup depth. These sections are then combined to form a 3-dimensional construct of the ONH region. Surfaces of the optic cup, optic rim, and peripapillary retina are determined by a change in reflectance intensity along the z-axis at each point. This creates a topographic map for the calculation of cup-to-disc (C/D) ratio, rim area, and other optic disc parameters. The HRT II and III instruments have become the standard instruments for CSLO scanning of the ONH in glaucoma. Description of the instrument and the concepts underlying the analysis software is available. Scanning Laser Polarimetry Scanning laser polarimetry (SLP) (GDx Nerve Fiber Analyzer, Carl Zeiss Meditec) measures peripapillary RNFL thickness by sending a laser beam to the posterior retina and assessing the change in polarization (retardation) of the reflected beam. This retardation of the scanning beam results from the birefringent properties of the neurotubules contained within the ganglion cell axons. The laser scanning is also based on CSLO and has a wavelength of 780 nm. A high-resolution image of 256 by 256 pixels is created of the optic nerve and peripapillary retina. Each point is a measure of the retardation of the laser scan at its location. Three serial scans are obtained with each test. Although SLP measures RNFL thickness throughout the entire image, the RNFL thickness for the double hump is determined along a 3.2-mm-diameter 8-pixel-wide circle, centered on the disc (calculation circle). The double hump is a graphic plot of the RNFL thickness around the optic nerve that is observed in most normal individuals, with the superior and inferior poles having the greatest RNFL thickness compared with the nasal and temporal poles. Some of the parameters presented are based on the RNFL thickness measurements within the calculation circle alone, but the nerve fiber indicator (a summary value that is intended to represent the likelihood of glaucomatous RNFL loss) is based on the entire RNFL thickness map. In addition, comparison of serial scans with normative data to help determine progression is available, and this is based on the entire image, not just the calculation circle. Prior versions of the machine provided a fixed compensation for the corneal birefringence that contributes to the retardation of the laser signal (fixed corneal compensation [FCC]). However, the corneal effect may differ significantly among individuals, change over time, and be substantially altered after ocular surgery, particularly LASIK. The updated device, GDx with variable corneal compensation (VCC), incorporates individualized compensation for the corneal component. GDx Printout: Patient with Early POAG Optical Coherence Tomography Optical Coherence Tomography (OCT) is an imaging technique which utilizes the concepts of interferometry as described by Albert Abraham Michelson. The device he designed, later known as an interferometer, sent a single source of white light through a half-silvered mirror that was used to split it into two beams travelling at right angles to one another. After leaving the splitter, the beams travelled out to the ends of long arms where they were reflected back into the middle on small mirrors. They then recombined on the far side of the splitter in an eyepiece, producing a pattern of constructive and destructive interference based on the spent time to transit the arms.Most of Michelson’s work was concerning the measurement of the speed of light. The use of Optical Coherence in biological systems was described by Huang et al in 1991. Since that time, instrumentation has been developed by multiple companies to utilize this technique for measurement of ophthalmic structures. A major event in the evolution of OCT was the use of light wavelengths instead of time delay to determine the spatial location of reflected light. Through the use of Fourier transformation, this took the technology from the original method of time-domain OCT (TD-OCT) to the development of spectral-domain OCT (SD-OCT). The original OCT method, known as TD-OCT, encoded the location of each reflection in the time information relating the position of a moving reference mirror to the location of the reflection. SD-OCT, instead, acquires all information in a single axial scan through the tissue simultaneously by evaluating the frequency spectrum of the interference between the reflected light and a stationary reference mirror. This method enables much faster acquisition times, resulting in a large increase in the amount of data that can be obtained during a given scan duration using SD-OCT. A comprehensive review of SD OCT by Joel Schuman is available. OCT imaging of the RNFL, ONH, and macula has been employed as a tool to aid in the diagnosis of glaucoma. A series of devices have been manufactured to analyze each of these components. Studies have demonstrated the high analytical and diagnostic performance of OCT imaging of the RNFL, ONH, and macula for the recognition of glaucomatous and non-glaucomatous eyes. General Concepts of OCT Analysis for Glaucoma Circular scans are centered around the ONH in order to capture RNFL and ONH measurements. Based on the amount of light reflected between the outer edge of the RNFL and the internal limiting membrane (ILM), the thickness of the RNFL is captured on OCT. With regard to the Cirrus 5000 and Spectralis devices, the thickness of the neuroretinal rim is analyzed based on the minimum rim width (MRW), which is the distance from the ILM to the Bruch membrane opening (BMO). With regard to the 3D OCT-2000, the distance between the retinal pigment epithelium and the ILM is measured. Macular evaluation on OCT involves analysis of the ganglion cell - internal plexiform layer (GC-IPL) thickness, as well as analysis of the ganglion cell complex (GCC) that encompasses the RNFL, GC, and IPL. Analysis of the macula occurs within a rectangular area that is either centered on or near the fovea. Thickness measurements are often shown in a TSNIT (temporal, superior, nasal, inferior, temporal) orientation and are compared to age-matched controls. Age-matched control analysis generally utilizes the following criteria: the Green area is the 5th-95th percentile by age, the Yellow Area is 1st-5th percentile, and the Red Area is below the 1st percentile. | OCT Device | Analysis | | Cirrus HD-OCT 5000 (Carl Zeiss Meditec, Inc., Dublin, CA) | Parapapillary circle diameter: 3.46 mm Data points: 40,000 RNFL analysis: o Horizontal, vertical, and circular tomograms o RNFL thickness maps o RNFL thickness plots o TSNIT & clock-hour maps o RNFL deviation map (for areas outside the parapapillary circle) o Table of measurements: RNFL thickness and RNFL symmetry ONH analysis: o Table of measurements: optic nerve rim area, disc area, average and vertical C:D, and cup volume o Neuro-retinal rim thickness plots Dimensions of analysis area—centered on fovea: 2 mm x 2.4 mm Macula analysis: o GC-IPL thickness maps o Horizontal B-scans o Sectoral plot and GC-IPL thickness table o Deviation map | | Spectralis (Heidelberg Engineering, Inc., Franklin, MA) | Parapapillary circle diameter: 3.5 mm Data points: 760 RNFL analysis: o Infrared scout images o Tomograms o RNFL thickness plots o TSNIT plots o Asymmetry plot with quadrant & sectoral maps ONH analysis: o MRW analysis of 9 cross-sectional images o MRW TSNIT plots Dimensions of analysis area—centered on fovea: 10 x 10 mm Macula analysis: o Asymmetry maps and plots for retinal thickness | | Avanti Widefield OCT (Optovue, Inc., Fremont, CA) | Parapapillary circle diameter: 4.0 mm RNFL analysis: o Tomograms o RNFL thickness plot o Asymmetry Plot o Table of measurements: RNFL thickness ONH analysis: o Table of measurements: C:D (area, vertical, horizontal), disc area, rim area, and cup volume Dimensions of analysis area—1 mm temporal to fovea: 7 x 7 mm Macula analysis: o GCC thickness maps o Table of measurements: ganglion cell complex thickness | | Optical Coherence Tomography RS-3000 Advance 2(NIDEK CO., LTD., Gamagori, Japan) | Parapapillary circle diameter: 3.45 mm RNFL analysis: o Scout images o Circumferential tomograms o RNFL thickness maps o RNFL thickness plots o Whole, superior/inferior, TSNIT, and clock-hour maps o Asymmetry plot ONH analysis: o Table of measurements: C:D (horizontal, vertical), R:D (minimum, angle), disc area, and cup area Dimensions of analysis area—centered on fovea: 9 x 9 mm Macular analysis: o Thickness maps (RNFL, GCL, and IPL combined) o Deviation maps | | 3D OCT-2000 (TOPOCON CORPORATION, Tokyo, Japan) | Parapapillary circle diameter: 3.4 mm RNFL analysis: o Color and red-free fundus photos o RNFL circular tomograms o RNFL thickness maps o RNFL thickness plot o TSNIT, clock-hour, and 36-sector maps o Symmetry analysis o Table of measurements: RNFL thickness (total, superior, inferior) ONH analysis: o Horizontal tomograms o Table of measurements: rim area, disc area, C:D (linear and vertical), and cup volume Dimensions of analysis area—centered on fovea: 7 x 7 mm Macula analysis: o GCC thickness maps o Asymmetry map o Table of measurements: GCC thickness | | data compared with age-matched controlsC:D = cup to disc ratio; GCC = ganglion cell complex; GC-IPL = ganglion cell – inner plexiform layer; MRW = minimum rim width; OCT = optical coherence tomography; ONH = optic nerve head; R:D = rim to disc ratio; RNFL = retinal nerve fiber layer; TSNIT = temporal, superior, nasal, inferior, temporal | | This printout reveals blunting of "double hump" appearance of RNFL in the left eye and RNFL thinning superiorly in the right eye. Acknowledgement Photos Courtesy of Samuel Solish, MD, EyeCare Medical Group, Portland, Maine Conclusion Integration of imaging modalities into the clinical evaluation for glaucoma can guide diagnostic and treatment decisions. Evaluation by OCT has shown to be particularly effective. Additional Resources Medeiros, FA, Prata, JA, Tavares, IM. Optic Disc and Retinal Nerve Fiber Layer Analyzers in Glaucoma. American Academy of Ophthalmology. Current Insight 2006. Accessed March 25, 2019. Glaucoma Diagnosis, Structure and Function Reports and Consensus Statements of the 1st Global AIGS Consensus Meeting on Structure and Function in the Management of Glaucoma. Weinreb & Greve (ed) Kugler: The Hague, Netherlands 2004 References ↑ Jonas et al Survey of Ophthalmology 43; 4, 1999 ↑ ↑ ↑ ↑ Jump up to: 5.0 5.1 Lin, SC, Singh, K, Jampel, HD et al Ophthalmic Technology Assessment:Optic Nerve Head and Retinal Nerve Fiber Analysis. A Report by the American Academy of Ophthalmology, Ophthalmology 2007;114;1937-1949 ↑ Allingham, R. Rand , et al Shield’s Textbook of Glaucoma, 5th ed, Lippincott Williams & Wilkins, 2004 ↑ ↑ experiment He won the Nobel Prize in Physics in 1907. ↑ Huang D, Swanson EA, Lin CP, Schuman JS, Stinson WG, Chang W, et al., Science 1991; 254:1178-81 ↑ Joel S. Schuman, MD, Spectral Domain Optical Coherence Tomography For Glaucoma (An AOS Thesis) Trans Am Ophthalmol Soc 2008;106:426-458 ↑ Jump up to: 11.0 11.1 Aref A, Rosdahl J. Focal Points 2019 Module: Optical Coherence Tomography in Glaucoma Diagnosis. In: Eric P. Purdy M, ed. Vol XXXVII. San Francisco, CA: American Academy of Ophthalmology; 2019. Retrieved from " The Academy uses cookies to analyze performance and provide relevant personalized content to users of our website. Categories: Articles Glaucoma
14010
https://forums.swift.org/t/dropping-curly-braces-from-single-statement-guard-else-blocks/70164
Skip to main content The Swift Forums are governed by the Swift Code of Conduct Dropping Curly Braces From Single Statement guard … else Blocks EvolutionDiscussion You have selected 0 posts. select all cancel selecting Feb 2024 1 / 24 Feb 2024 Jul 25 dimi Dimitri Bouniol Feb 2024 The vast majority of the guard statements I write contain a single statement in its else block: return return value throw ErrorType() continue break loopLabel etc… This is fine and good, and I would never suggest we rename or shorten these phrases as their intent is particularly clear, but years later, I still end up not typing the curly braces on first pass, only remembering to add them after the fact. Since I don't think this particular iteration of a guard shorthand has been pitched yet, how do people feel about optionally dropping the curly braces when only a single statement lives in the else block? Some before and afters: Simple Unwrapping, Return nil ``` guard let unwrappedValue else { return nil } ``` ``` guard let unwrappedValue else return nil ``` Multiple Checks and Throw Error ``` guard currentLoad > watermark, let excessLoadManager else { throw LoadError.missingExcessManager } ``` ``` guard currentLoad > watermark, let excessLoadManager else throw LoadError.missingExcessManager ``` Loops ``` for job in sortedJobs { guard job.isValid else { break } job.start() } ``` ``` for job in sortedJobs { guard job.isValid else break job.start() } ``` Multi-statement else blocks would continue to require curly brackets as is the case today: ``` var firstInvalidJob: Job? for job in sortedJobs { guard job.isValid else { firstInvalidJob = job break } job.start() } ``` Although I understand why we don't allow single statements without braces for if-statements, loops, and functions, and would never argue to remove braces there, I think the unique spelling of guard ... else statements actually makes things more readable when the else phrase can flow directly into the single statement without interruption, much in the same way dropping parentheses from if conditionals helps readability there. I don't believe introducing this new sugar would break any existing code by introducing ambiguity, nor would using it cause any parsing issues, but I could very much be wrong here. All this said, this is my first time submitting a pitch to the language, so I would appreciate any help I could get if there is any shared interest in making this happen. [Pitch] Last expression as return value 2 read 6 min
14011
https://math.stackexchange.com/questions/2217576/do-all-odd-functions-have-symmetric-zeros
Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Do all odd functions have symmetric zeros? Ask Question Asked Modified 8 years, 5 months ago Viewed 3k times 12 $\begingroup$ Is there odd functions where a zero of positive x-axis doesn't have necessarily a symmetric? What I mean is, consider an odd function where its positive zeros are x=1, x=2, x=3. Since it is odd, it has a symmetry related to origin of referencial, so there are also the zeros x=-1, x=-2, x=-3. But is it possible to have an odd function with x=1,x=2,x=3, x=-1 and x=-2? Where x=3 doesn't have a symmetric. functions roots parity Share edited Apr 4, 2017 at 13:54 StackTD 28.4k3838 silver badges6767 bronze badges asked Apr 4, 2017 at 13:07 Vitor AguiarVitor Aguiar 43744 silver badges1111 bronze badges $\endgroup$ 0 Add a comment | 3 Answers 3 Reset to default 23 $\begingroup$ It's correct that for odd functions, $f(a) = 0 \iff f(-a) = 0$. We call $f$ odd if $f(-x) = -f(x)$, so if $f(a) = 0$, then $f(-a) = -f(a) = 0$. Share answered Apr 4, 2017 at 13:09 AJYAJY 9,11633 gold badges2424 silver badges4646 bronze badges $\endgroup$ Add a comment | 12 $\begingroup$ If an odd function exists in $x=a$ and in $x=-a$ and if it has a zero in one of them, then necessarily it has one in its symmetric counterpart since $f(-a)=-f(a)$ by definition of odd. The only way to make this go wrong, is if $f$ would be defined in only one of those points. This is something you may or may not (want to) allow for a function to be called odd. What I mean is, consider an odd function where its positive zeros are x=1, x=2, x=3. Since it is odd, it has a symmetry related to origin of referencial, so there are also the zeros x=-1, x=-2, x=-3. But is it possible to have an odd function with x=1,x=2,x=3, x=-1 and x=-2? Where x=3 doesn't have a symmetric. With the usual understanding (definition) of odd, you cannot have the situation you describe. We call a function odd if $f(-x)=-f(x)$ "for all $x$", but this is only meaningful if for every $x$ in the domain of $f$, you also have $-x$ in that domain. This means we're only considering functions defined on either $\mathbb{R}$ or with a domain that is symmetric with respect to the origin such as intervals of the form $[-a,a]$, with $a \in \mathbb{R}$. Take for example the odd function which has (at least) the zeroes you describe: $$f : \mathbb{R} \to \mathbb{R} : x \mapsto (x+3)(x+2)(x+1)x(x-1)(x-2)(x-3)$$ You could limit the domain to an asymmetric interval, such as $\left[ -2.5, 3.5 \right]$: $$g : \left[ -2.5, 3.5 \right] \to \mathbb{R} : x \mapsto (x+3)(x+2)(x+1)x(x-1)(x-2)(x-3)$$ Now this function $g$ has a zero in $x=3$ without a symmetric counterpart, since it is undefined in $x=-3$. Whether or not you would still call this function odd depends on the definition: do you require the domain to be of the specific form as described above or not? Share edited Apr 4, 2017 at 22:46 answered Apr 4, 2017 at 13:20 StackTDStackTD 28.4k3838 silver badges6767 bronze badges $\endgroup$ 3 1 $\begingroup$ +1 $f$ can be odd on a non-symmetric domain, in this case not every symmetric of zero is a zero $\endgroup$ Random Tourist – Random Tourist 2017-04-05 08:34:54 +00:00 Commented Apr 5, 2017 at 8:34 $\begingroup$ very good StackTD, thank you :) $\endgroup$ Vitor Aguiar – Vitor Aguiar 2017-04-07 21:48:59 +00:00 Commented Apr 7, 2017 at 21:48 $\begingroup$ @VitorAguiar Thanks and you're welcome. $\endgroup$ StackTD – StackTD 2017-04-10 11:13:47 +00:00 Commented Apr 10, 2017 at 11:13 Add a comment | 7 $\begingroup$ An odd function $f $ satisfies $f (-x)=-f (x) $, so if $3$ is a zero of $f$, $f (-3)=-f (3)=-0=0$. This works for any zero, so yes, the zeroes are symmetric. Share answered Apr 4, 2017 at 13:10 Mark S.Mark S. 26k22 gold badges5353 silver badges117117 bronze badges $\endgroup$ 3 $\begingroup$ But this is true also for even functions. Righ? $\endgroup$ Alexander Cska – Alexander Cska 2018-03-29 17:49:38 +00:00 Commented Mar 29, 2018 at 17:49 $\begingroup$ @Alexander The general equation for "even" would be different, but yes, even functions have symmetric zeros for a similar reason. $\endgroup$ Mark S. – Mark S. 2018-03-30 11:09:34 +00:00 Commented Mar 30, 2018 at 11:09 $\begingroup$ Thank you for your reply. $\endgroup$ Alexander Cska – Alexander Cska 2018-03-30 11:58:19 +00:00 Commented Mar 30, 2018 at 11:58 Add a comment | You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions functions roots parity See similar questions with these tags. 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14012
https://www.britannica.com/science/barium
SUBSCRIBE Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos barium chemical element Also known as: Ba Written by Timothy P. Hanusa Professor, Department of Chemistry, Vanderbilt University, Nashville, Tennessee. Timothy P. Hanusa Fact-checked by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Article History Key People: : Sir Humphry Davy Related Topics: : chemical element : alkaline-earth metal : psilomelane See all related content barium (Ba), chemical element, one of the alkaline-earth metals of Group 2 (IIa) of the periodic table. The element is used in metallurgy, and its compounds are used in pyrotechnics, petroleum production, and radiology. Element Properties | atomic number | 56 | | atomic weight | 137.327 | | melting point | 727 °C (1,341 °F) | | boiling point | 1,805 °C (3,281 °F) | | specific gravity | 3.51 (at 20 °C, or 68 °F) | | oxidation state | +2 | | electron configuration | [Xe]6s2 | Occurrence, properties, and uses Barium, which is slightly harder than lead, has a silvery white luster when freshly cut. It readily oxidizes when exposed to air and must be protected from oxygen during storage. In nature it is always found combined with other elements. The Swedish chemist Carl Wilhelm Scheele discovered (1774) a new base (baryta, or barium oxide, BaO) as a minor constituent in pyrolusite, and from that base he prepared some crystals of barium sulfate, which he sent to Johan Gottlieb Gahn, the discoverer of manganese. A month later Gahn found that the mineral barite is also composed of barium sulfate, BaSO4. A particular crystalline form of barite found near Bologna, Italy, in the early 17th century, after being heated strongly with charcoal, glowed for a time after exposure to bright light. The phosphorescence of “Bologna stones” was so unusual that it attracted the attention of many scientists of the day, including Galileo. Only after the electric battery became available could Sir Humphry Davy finally isolate (1808) the element itself by electrolysis. Barium minerals are dense (e.g., BaSO4, 4.5 grams per cubic centimetre; BaO, 5.7 grams per cubic centimeter), a property that was the source of many of their names and of the name of the element itself (from the Greek barys, “heavy”). Ironically, metallic barium is comparatively light, only 30 percent denser than aluminum. Its cosmic abundance is estimated as 3.7 atoms (on a scale where the abundance of silicon = 106 atoms). Barium constitutes about 0.03 percent of Earth’s crust, chiefly as the minerals barite (also called barytes or heavy spar) and witherite. Between six and eight million tons of barite are mined every year, more than half of it in China. Lesser amounts are mined in India, the United States, and Morocco. Commercial production of barium depends upon the electrolysis of fused barium chloride, but the most effective method is the reduction of the oxide by heating with aluminum or silicon in a high vacuum. A mixture of barium monoxide and peroxide can also be used in the reduction. Only a few tons of barium are produced each year. Britannica Quiz Facts You Should Know: The Periodic Table Quiz The metal is used as a getter in electron tubes to perfect the vacuum by combining with final traces of gases, as a deoxidizer in copper refining, and as a constituent in certain alloys. The alloy with nickel readily emits electrons when heated and is used for this reason in electron tubes and in spark plug electrodes. The detection of barium (atomic number 56) after uranium (atomic number 92) had been bombarded by neutrons was the clue that led to the recognition of nuclear fission in 1939. Naturally occurring barium is a mixture of six stable isotopes: barium-138 (71.7 percent), barium-137 (11.2 percent), barium-136 (7.8 percent), barium-135 (6.6 percent), barium-134 (2.4 percent), and barium-132 (0.10 percent). Barium-130 (0.11 percent) is also naturally occurring but undergoes decay by double electron capture with an extremely long half-life (more than 4 × 1021 years). More than 30 radioactive isotopes of barium are known, with mass numbers ranging from 114 to 153. The isotope with the longest half-life (barium-133, 10.5 years) is used as a gamma-ray reference source. Compounds In its compounds, barium has an oxidation state of +2. The Ba2+ ion may be precipitated from solution by the addition of carbonate (CO32−), sulfate (SO42−), chromate (CrO42−), or phosphate (PO43−) anions. All soluble barium compounds are toxic to mammals, probably by interfering with the functioning of potassium ion channels. Access for the whole family! Bundle Britannica Premium and Kids for the ultimate resource destination. Subscribe Barium sulfate (BaSO4) is a white, heavy insoluble powder that occurs in nature as the mineral barite. Almost 80 percent of world consumption of barium sulfate is in drilling muds for oil. It is also used as a pigment in paints, where it is known as blanc fixe (i.e., “permanent white”) or as lithopone when mixed with zinc sulfide. The sulfate is widely used as a filler in paper and rubber and finds an important application as an opaque medium in the X-ray examination of the gastrointestinal tract. Most barium compounds are produced from the sulfate via reduction to the sulfide, which is then used to prepare other barium derivatives. About 75 percent of all barium carbonate (BaCO3) goes into the manufacture of specialty glass, either to increase its refractive index or to provide radiation shielding in cathode-ray and television tubes. The carbonate also is used to make other barium chemicals, as a flux in ceramics, in the manufacture of ceramic permanent magnets for loudspeakers, and in the removal of sulfate from salt brines before they are fed into electrolytic cells (for the production of chlorine and alkali). On heating, the carbonate forms barium oxide, BaO, which is employed in the preparation of cuprate-based high-temperature superconductors such as YBa2Cu3O7−x. Another complex oxide, barium titanate (BaTiO3), is used in capacitors, as a piezoelectric material, and in nonlinear optical applications. Barium chloride (BaCl2·2H2O), consisting of colorless crystals that are soluble in water, is used in heat-treating baths and in laboratories as a chemical reagent to precipitate soluble sulfates. Although brittle, crystalline barium fluoride (BaF2) is transparent to a broad region of the electromagnetic spectrum and is used to make optical lenses and windows for infrared spectroscopy. The oxygen compound barium peroxide (BaO2) was used in the 19th century for oxygen production (the Brin process) and as a source of hydrogen peroxide. Volatile barium compounds impart a yellowish green color to a flame, the emitted light being of mostly two characteristic wavelengths. Barium nitrate, formed with the nitrogen-oxygen group NO3−, and chlorate, formed with the chlorine-oxygen group ClO3−, are used for this effect in green signal flares and fireworks. Timothy P. Hanusa periodic table Table of Contents Introduction & Top Questions History of the periodic law Classification of the elements The first periodic table Other versions of the periodic table Predictive value of the periodic law Discovery of new elements Significance of atomic numbers Elucidation of the periodic law The periodic table Periods Groups Classification of elements into groups Periodic trends in properties The basis of the periodic system Electronic structure Periodicity of properties of the elements Other chemical and physical classifications References & Edit History Quick Facts & Related Topics Images, Videos & Interactives For Students periodic table summary Quizzes 118 Names and Symbols of the Periodic Table Quiz Facts You Should Know: The Periodic Table Quiz Science Quiz Periodic Table of the Elements 36 Questions from Britannica’s Most Popular Science Quizzes Related Questions What is the periodic table? Where does the periodic table come from? Why does the periodic table split? How is the atomic number of an atom defined? periodic table chemistry Also known as: periodic table of the elements Written by Linus C. Pauling Research Professor, Linus Pauling Institute of Science and Medicine, Palo Alto, California, 1973–94. Professor of Chemistry, Stanford University, California, 1969–74. Nobel Prize for Chemistry, 1954; Nobel... Linus C. Pauling , J.J. Lagowski Professor of Chemistry, University of Texas at Austin. Author of The Chemical Bond. J.J. Lagowski •All Fact-checked by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Last Updated: • Article History In full: : periodic table of the elements Key People: : Dmitri Mendeleev : Lothar Meyer : Paul-Émile Lecoq de Boisbaudran Related Topics: : atom : When Was the Periodic Table Invented? : group : periodic law : period On the Web: : NIST - Henry Moseley and the Periodic Table of the Elements (Aug. 07, 2025) See all related content Top Questions What is the periodic table? The periodic table is a tabular array of the chemical elements organized by atomic number, from the element with the lowest atomic number, hydrogen, to the element with the highest atomic number, oganesson. The atomic number of an element is the number of protons in the nucleus of an atom of that element. Hydrogen has 1 proton, and oganesson has 118. What do periodic table groups have in common? The groups of the periodic table are displayed as vertical columns numbered from 1 to 18. The elements in a group have very similar chemical properties, which arise from the number of valence electrons present—that is, the number of electrons in the outermost shell of an atom. Where does the periodic table come from? The arrangement of the elements in the periodic table comes from the electronic configuration of the elements. Because of the Pauli exclusion principle, no more than two electrons can fill the same orbital. The first row of the periodic table consists of just two elements, hydrogen and helium. As atoms have more electrons, they have more orbits available to fill, and thus the rows contain more elements farther down in the table. Why does the periodic table split? The periodic table has two rows at the bottom that are usually split out from the main body of the table. These rows contain elements in the lanthanoid and actinoid series, usually from 57 to 71 (lanthanum to lutetium) and 89 to 103 (actinium to lawrencium), respectively. There is no scientific reason for this. It is merely done to make the table more compact. periodic table, in chemistry, the organized array of all the chemical elements in order of increasing atomic number—i.e., the total number of protons in the atomic nucleus. When the chemical elements are thus arranged, there is a recurring pattern called the “periodic law” in their properties, in which elements in the same column (group) have similar properties. The initial discovery, which was made by Dmitry I. Mendeleev in the mid-19th century, has been of inestimable value in the development of chemistry. It was not actually recognized until the second decade of the 20th century that the order of elements in the periodic system is that of their atomic numbers, the integers of which are equal to the positive electrical charges of the atomic nuclei expressed in electronic units. In subsequent years great progress was made in explaining the periodic law in terms of the electronic structure of atoms and molecules. This clarification has increased the value of the law, which is used as much today as it was at the beginning of the 20th century, when it expressed the only known relationship among the elements. History of the periodic law The early years of the 19th century witnessed a rapid development in analytical chemistry—the art of distinguishing different chemical substances—and the consequent building up of a vast body of knowledge of the chemical and physical properties of both elements and compounds. This rapid expansion of chemical knowledge soon necessitated classification, for on the classification of chemical knowledge are based not only the systematized literature of chemistry but also the laboratory arts by which chemistry is passed on as a living science from one generation of chemists to another. Relationships were discerned more readily among the compounds than among the elements; it thus occurred that the classification of elements lagged many years behind that of compounds. In fact, no general agreement had been reached among chemists as to the classification of elements for nearly half a century after the systems of classification of compounds had become established in general use. J.W. Döbereiner in 1817 showed that the combining weight, meaning atomic weight, of strontium lies midway between those of calcium and barium, and some years later he showed that other such “triads” exist (chlorine, bromine, and iodine [halogens] and lithium, sodium, and potassium [alkali metals]). J.-B.-A. Dumas, L. Gmelin, E. Lenssen, Max von Pettenkofer, and J.P. Cooke expanded Döbereiner’s suggestions between 1827 and 1858 by showing that similar relationships extended further than the triads of elements, fluorine being added to the halogens and magnesium to the alkaline-earth metals, while oxygen, sulfur, selenium, and tellurium were classed as one family and nitrogen, phosphorus, arsenic, antimony, and bismuth as another family of elements. Attempts were later made to show that the atomic weights of the elements could be expressed by an arithmetic function, and in 1862 A.-E.-B. de Chancourtois proposed a classification of the elements based on the new values of atomic weights given by Stanislao Cannizzaro’s system of 1858. De Chancourtois plotted the atomic weights on the surface of a cylinder with a circumference of 16 units, corresponding to the approximate atomic weight of oxygen. The resulting helical curve brought closely related elements onto corresponding points above or below one another on the cylinder, and he suggested in consequence that “the properties of the elements are the properties of numbers,” a remarkable prediction in the light of modern knowledge. Britannica Quiz Facts You Should Know: The Periodic Table Quiz Classification of the elements In 1864, J.A.R. Newlands proposed classifying the elements in the order of increasing atomic weights, the elements being assigned ordinal numbers from unity upward and divided into seven groups having properties closely related to the first seven of the elements then known: hydrogen, lithium, beryllium, boron, carbon, nitrogen, and oxygen. This relationship was termed the law of octaves, by analogy with the seven intervals of the musical scale. Then in 1869, as a result of an extensive correlation of the properties and the atomic weights of the elements, with special attention to valency (that is, the number of single bonds the element can form), Mendeleev proposed the periodic law, by which “the elements arranged according to the magnitude of atomic weights show a periodic change of properties.” Lothar Meyer had independently reached a similar conclusion, published after the appearance of Mendeleev’s paper. Access for the whole family! Bundle Britannica Premium and Kids for the ultimate resource destination. Subscribe Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Hanusa, Timothy P.. "barium". Encyclopedia Britannica, 5 Mar. 2025, Accessed 2 September 2025. Share Share to social media Facebook X External Websites CAMEO Chemicals - Barium Lenntech - Barium - Ba Chemicool - Barium Royal Society of Chemistry - Barium Nature - Nature Chemistry - Barium bright and heavy National Center for Biotechnology Information - Toxicological Profile for Barium and Barium Compounds New Jersey Department of health - Barium Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. barium - Student Encyclopedia (Ages 11 and up) Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Pauling, Linus C., Lagowski, J.J.. "periodic table". Encyclopedia Britannica, 7 Aug. 2025, Accessed 2 September 2025. Share Share to social media Facebook X URL External Websites UEN Digital Press with Pressbooks - Introductory Chemistry - The Periodic Table Ohio State University - Origins - Mendeleev's Periodic Table LiveScience - Periodic Table of the Elements NIST - Henry Moseley and the Periodic Table of the Elements Royal Society of Chemistry - Periodic Table Britannica Websites Articles from Britannica Encyclopedias for elementary and high school students. periodic table - Children's Encyclopedia (Ages 8-11) periodic table - Student Encyclopedia (Ages 11 and up)
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https://www.ilearnengineering.com/mechanical/what-is-the-work-energy-theorem
Download Prospectus Homepage Learning Resources Our Courses Qualifications Summary of our Qualifications Engineering Diplomas Higher International Certificate Higher International Diploma International Graduate Diploma Levels Credits Accreditation Learning Pathway Sectors Engineering (General) Mechanical Engineering Electrical Engineering Civil Engineering Computing Engineering Aerospace Engineering Industrial (Manufacturing) Engineering Reviews Learner Reviews Student Work Further Information Faculty FAQ About Us How You Learn Employers Fees Contact Us Courses Login Enrol Download our Engineering Courses Prospectus 23/24 Download Prospectus Courses Login Enrol Learning Resources Our Courses Qualifications Summary of our Qualifications Engineering Diploma Higher International Certificate Higher International Diploma International Graduate Diploma Levels Credits Accreditation Learning Pathway Sectors Engineering (General) Mechanical Engineering Electrical Engineering Civil Engineering Computer Engineering Aerospace Engineering Industrial (Manufacturing) Engineering Reviews Learner Reviews Student Work Further Information Faculty FAQ About Us How You Learn Employers Fees Contact Us +44 (0)333 772 9274 Homepage > What Is The Work-Energy Theorem? What Is The Work-Energy Theorem? The Work-Energy Theorem is a fundamental principle in classical mechanics that connects the concepts of work and kinetic energy. It provides a powerful tool for analysing the motion of objects without relying solely on Newton’s laws in their force-acceleration form. The principle of work and kinetic energy (also known as the work-energy theorem) states that the work done by the sum of all forces acting on a body equals the change in the kinetic energy of the particle. The work done, W, by the net force on a body equals the change in the particle’s kinetic energy KE: W = ∆KE = ½ mvf2 – ½ mvi2 where vi and vf are the speeds of the particle before (initial) and after (final) the application of force, and m is the mass of the body. Furthermore, the work done can also be stated as: W = Fd for a force, F moving a body through a distance d. Example 1 Consider a motor car travelling at 30mph which then crashes into a stationary object such as a tree! What force is exerted on the car during this collision assuming that the vehicle compresses by 300 mm upon impact. A typical mass of a car is around 1500kg. 1 mile = 1609 metres Velocity = 30mph = (30×1609) metres per hour = (30×1609)/3600 m/s = 13.4m/s Initial kinetic energy of the car travelling at 30mph: ½ mvf2 – ½ mvi2 KEi = ½ mvi2 = ½ (1500)(13.4)2 KEi = 134.8kJ Final kinetic energy of the car is zero because it will have zero velocity. KEf = 0 The change in kinetic energy can be found: ∆KE = 134.8 – 0 = 134.8kJ By the work-energy principle, the work done and the energy are related by: W = ∆KE Hence W = 134.8kJ However, recall the work done is also defined as: W = Fd where F is the force and d is the distance moved, which we have assumed to be the crumple zone of the car and is taken as 300 mm in this example. Therefore: Fd = 134800 F(0.3) = 134800 F = 134800 / 0.3 F = 450000 = 450kN The above definition can be extended to rigid bodies by considering the work of the torque and rotational kinetic energy: W = ∆KE linear + ∆KE rotational The rotational kinetic energy of a body in angular motion, is analogous to the linear equivalent: Now, we turn attention to the distance, x. From above, we know that: Linear KE = ½ mv2 and Rotational KE = ½ Iω2 where m is the mass (kg) and v is the velocity (m/s), I is the moment of inertia (kgm2) and ω is the angular velocity (rad/s). Energy has SI units of Joules (J). A body can of course also have gravitational potential energy. This is the work that has been done in raising the body to a height h above some given datum or ground level. i.e. PE = mgh We can now combine all of the above in to an equation for work done: W = ∆KE linear + ∆KE rotational + ∆PE The principle of conservation of energy is particularly useful when solving problems involving a combination of linear and angular motion. A typical example is a hoist, where a load is raised by a cable on a winding drum. The driving motor must supply a torque that is sufficient to overcome the inertia of both the winding drum and the load, the weight of the load and also frictional resistance. As x = 0 when t = 0, we can substitute this into the above equation to give: Such a mechanical system can be represented as a ‘system’ as follows: Where and diagrammatically as: An alternative way of expressing the work-energy principle is: Total work and energy input = Total work and energy output Work input + Initial PE + Initial KE = Final PE + Final KE + Friction work Wi + mgh1 + ½ mv12 + ½ Iω12 = mgh2 + ½ mv22 + ½ Iω22 + Wf This can be rearranged to give an equation for the input work: Wi = – mgh1 – ½ mv12 – ½ Iω12 + mgh2 + ½ mv22 + ½ Iω22 + Wf Wi = mg (h2-h1) + ½ m (v22 – v12) + ½I( ω22-ω12) + Wf If the frictional resistance force Ff is known, the work done in overcoming friction can be calculated using the formula below, where s is the linear distance travelled. Wf = Ffs If the friction torque Tf in the bearings of a system is known, the work done in overcoming friction can be calculated using the formula below where θ is the angle turned (radians): Wf = Tfθ Example 2 A hoist raises a load of 75 kg from rest to a speed of 2.5 m/s whilst travelling through a distance of 6 m. The load is attached to a light cable wrapped around a winding drum of diameter 500 mm. The mass of the drum is 60 kg and its radius of gyration is 180 mm. The bearing friction torque is 5 Nm. Determine (a) the angle turned by the drum (b) the final angular velocity of the drum (c) the work input from the driving motor (d) input torque to the winding drum (e) the maximum power developed by the motor. Solution: a) The load traverses 6 m in a linear direction. This is connected to the drum and so we must relate the linear distance of 6m to the associated angular distance on the drum. The drum has a diameter, d, of 500 mm = 0.5 m. The arc length, S, of a circle is defined as a section of the circle’s circumference and it is related to the angle and radius of a circle as follows: Hence: θ = s / r In our case, the arc length will be the length of the rope that wraps around the drum which we are told is 6 m. The radius, r, is half of the drum diameter. Therefore θ = 6 / 0.25 θ = 24 radians (b)The final angular velocity ω2 = v2 / r ω2 = 2.5 / 0.25 = 10 rad/s c) To find the work input we must first find the moment of inertia of the drum: I = mk2 I = (60)(0.18)2 I = 1.94 kgm2 and the work done in overcoming friction: Wf = Tf θ Wf = (5)(24) = 120J Finally, the input work can be calculated given that the initial linear and angular velocities are zero (starts from rest) and the initial height can be taken as zero also: Wi = mg (h2-h1) + ½ m (v22 – v12) + ½I( ω22-ω12) + Wf Wi = (75)(9.81)( 6-0) + ½ (75)(2.52 – 02) + ½ (1.94)(102 -02) + 120 Wi = 4414.5 + 234.375 + 97 + 120Wi = 4866.9J (d)The input torque required to operate the drum can now be easily found: W = Tθ Ti = Wi / θ Ti = 4866.8 / 24 Ti = 202.8 Nm (e)Finally, the power developed by the motor must be: P = Tω Pi = (202.8)(10) Pi = 2028W or 2.03kW Interested in our engineering courses? 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https://brilliant.org/wiki/applying-the-difference-of-two-squares-identity/
Difference Of Squares | Brilliant Math & Science Wiki HomeCourses Sign upLog in The best way to learn math and computer science. Log in with GoogleLog in with FacebookLog in with email Join using GoogleJoin using email Reset password New user? Sign up Existing user? Log in Difference Of Squares Sign up with FacebookorSign up manually Already have an account? Log in here. Sandeep Bhardwaj, 敬全 钟, Sam Reeve, and 10 others Ashley Toh Jongheun Lee Mahindra Jain Lucerne O' Brannan Tara Kappel Satyabrata Dash Ben Sidebotham Sandeep Bhardwaj Derek Guo Jimin Khim contributed The difference of two squaresidentity is a squared number subtracted from another squared number to get factorized in the form of a 2−b 2=(a+b)(a−b).a^2-b^2=(a+b)(a-b).a 2−b 2=(a+b)(a−b). We will also prove this identity by multiplying polynomials on the left side and getting equal to the right side. This identity is often used in algebra where it is useful in applications to integer factorization, the quadratic sieve, the algebraic factorization, etc. Contents Identity Example Problems Further Extension Problem Solving See Also Identity The difference of two squares identity is (a+b)(a−b)=a 2−b 2(a+b)(a-b)=a^2-b^2(a+b)(a−b)=a 2−b 2. We can prove this identity by multiplying the expressions on the left side and getting equal to the right side expression. Here is the proof of this identity. Let's begin with the left side of the expression. We have (a+b)(a−b)=a(a−b)+b(a−b)=a 2−a b+a b−b 2=a 2−b 2, \begin{aligned} (a+b)(a-b) &= a(a-b) + b(a-b) \ &= a^2 - ab + ab - b^2 \ & = a^2 - b^2, \end{aligned} (a+b)(a−b)​=a(a−b)+b(a−b)=a 2−ab+ab−b 2=a 2−b 2,​ which is equal to the right side of the identity. Hence proved. □_\square□​ Example Problems This section contains examples and problems to boost understanding in the usage of the difference of squares identity: a 2−b 2=(a+b)(a−b)a^2-b^2=(a+b)(a-b)a 2−b 2=(a+b)(a−b). Here are the examples to learn the usage of the identity. Rewrite 5 2−2 2 5^2-2^2 5 2−2 2 as a product. We have 5 2−2 2=(5−2)×(5+2)=3×7.□5^2-2^2 = (5-2) \times (5+2) = 3\times 7. \ _\square 5 2−2 2=(5−2)×(5+2)=3×7.□​ Calculate 299×301 299\times 301 299×301. You can brute force the answer to this problem by using a calculator, but we have a sweeter way. We can apply the difference of two squares identity. At first we may think about using the long multiplication method, but it wastes time and is, of course, boring. Notice that 299=300−1 299=300-1 299=300−1 and 301=300+1 301=300+1 301=300+1, so 299×301=(300−1)(300+1)=300 2−1 2=89999.□\begin{aligned}299\times 301&=(300-1)(300+1)\&=300^2-1^2\&=89999. \ _\square \end{aligned}299×301​=(300−1)(300+1)=30 0 2−1 2=89999.□​​ Show that any odd number can be written as the difference of two squares. Let the odd number be n=2 b+1 n = 2b + 1 n=2 b+1, where b b b is a non-negative integer. Then we have n=2 b+1=[(b+1)+b][(b+1)−b]=(b+1)2−b 2.□ n = 2b+1 = [ (b+1) + b ] [ (b+1) - b ] = (b+1)^2 - b^2. \ _\square n=2 b+1=[(b+1)+b][(b+1)−b]=(b+1)2−b 2.□​ What is 234567 2−234557×234577?234567^2-234557\times 234577\ ?23456 7 2−234557×234577? Using the same method as the example above, 234567 2−234557×234577=234567 2−(234567 2−10 2)=234567 2−234567 2+10 2=100.□\begin{aligned} 234567^2-234557\times 234577&=234567^2-\big(234567^2-10^2\big)\ &=234567^2-234567^2+10^2\ &=100.\ _\square \end{aligned}23456 7 2−234557×234577​=23456 7 2−(23456 7 2−1 0 2)=23456 7 2−23456 7 2+1 0 2=100.□​​ Solve the following problems: b−a b-a b−a a−b a-b a−b a+b a+b a+b a 2+b 2{ a }^{ 2 }{ +b }^{ 2 }a 2+b 2 Reveal the answer Which of the following equals a 2−b 2 a−b\dfrac { { a }^{ 2 }-{ b }^{ 2 } }{ a-b } a−b a 2−b 2​ for a≠b a\neq b a=b? The correct answer is: a+b a+b a+b 198 199 197 187 Reveal the answer What is 99 2−98 2?99^2 - 98^2 \, ?9 9 2−9 8 2? Note: Try it without using a calculator. The correct answer is: 197 Using the identity x 2−y 2=(x+y)(x−y),x^2 - y^2 = (x + y)(x - y),x 2−y 2=(x+y)(x−y), we let x=99 x = 99 x=99 and y=98 y = 98 y=98 to obtain the following: 99 2−98 2=(99+98)(99−98)=(197)(1)=197 99^2 - 98^2 = (99 + 98)(99 - 98) = (197)(1) = \boxed{197}9 9 2−9 8 2=(99+98)(99−98)=(197)(1)=197​ Reveal the answer 2014 2014×2014 2014−2014 2013×2014 2015=?\large \color{#3D99F6}{2014}\color{#3D99F6}{2014} \times \color{#3D99F6}{2014}\color{#3D99F6}{2014} - \color{#3D99F6}{2014}\color{#D61F06}{2013} \times \color{#3D99F6}{2014}\color{fuchsia}{2015} = ? 2014 2014×2014 2014−2014 2013×2014 2015=? Don't use a calculator! The correct answer is: 1 Further Extension Since the two factors are different by 2 b 2b 2 b, the factors will always have the same parity. That is, if a−b a-b a−b is even then a+b a+b a+b must also be even, so the product is divisible by four. Or neither are divisible by 2, so the product is odd. This implies that numbers which are multiple of 2 but not 4 cannot be expressed as the difference of 2 squares. The product of two differences of two squares is itself a difference of two squares in two different ways: (a 2−b 2)(c 2−d 2)=(a c)2−(a d)2−(b c)2+(b d)2=(a c)2−(a d)2−(b c)2+(b d)2+2 a b c d−2 a b c d=(a c)2+2 a b c d+(b d)2−[(a d)2+2 a b c d+(b c)2]=(a c+b d)2−(a d+b c)2=(a c)2−2 a b c d+(b d)2−[(a d)2−2 a b c d+(b d)2]=(a c−b d)2−(a d−b c)2.\begin{array} { l l l } \left(a^2-b^2\right)\left(c^2-d^2\right) &= (ac)^2-(ad)^2-(bc)^2+(bd)^2 \ &= (ac)^2-(ad)^2-(bc)^2+(bd)^2+2abcd-2abcd \ &= (ac)^2+2abcd+(bd)^2-\left[(ad)^2+2abcd+(bc)^2\right] &= (ac+bd)^2 - (ad+bc)^2 \ &= (ac)^2-2abcd+(bd)^2-\left[(ad)^2-2abcd+(bd)^2\right] &= (ac-bd)^2 - (ad-bc)^2. \ \end{array}(a 2−b 2)(c 2−d 2)​=(a c)2−(a d)2−(b c)2+(b d)2=(a c)2−(a d)2−(b c)2+(b d)2+2 ab c d−2 ab c d=(a c)2+2 ab c d+(b d)2−[(a d)2+2 ab c d+(b c)2]=(a c)2−2 ab c d+(b d)2−[(a d)2−2 ab c d+(b d)2]​=(a c+b d)2−(a d+b c)2=(a c−b d)2−(a d−b c)2.​ Problem Solving The examples and problems in this sections are a bit harder for enhancing the problem solving skills. Let's give them a try. Here are the examples to go through. Simplify (1−1 2 2)(1−1 3 2)(1−1 4 2)⋯(1−1 n 2).\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{4^2}\right)\cdots\left(1-\frac{1}{n^2}\right).(1−2 2 1​)(1−3 2 1​)(1−4 2 1​)⋯(1−n 2 1​). This is a very direct application of the identity mentioned in this text. We have (1−1 2 2)(1−1 3 2)⋯(1−1 n 2)=(1−1 2)(1+1 2)(1−1 3)(1+1 3)⋯(1−1 n)(1+1 n)=1 2⋅3 2⋅2 3⋅4 3⋯n−1 n⋅n+1 n.\begin{aligned} \left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\cdots\left(1-\frac{1}{n^2}\right) &=\left(1-\frac{1}{2}\right)\left(1+\frac{1}{2}\right)\left(1-\frac{1}{3}\right)\left(1+\frac{1}{3}\right)\cdots\left(1-\frac{1}{n}\right)\left(1+\frac{1}{n}\right)\ &=\frac{1}{2}\cdot\frac{3}{2}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdots\frac{n-1}{n}\cdot\frac{n+1}{n}. \end{aligned}(1−2 2 1​)(1−3 2 1​)⋯(1−n 2 1​)​=(1−2 1​)(1+2 1​)(1−3 1​)(1+3 1​)⋯(1−n 1​)(1+n 1​)=2 1​⋅2 3​⋅3 2​⋅3 4​⋯n n−1​⋅n n+1​.​ Notice that the product from the second term to the (n−1)th(n-1)^\text{th}(n−1)th term is equal to 1, and therefore the final product is n+1 2 n\frac{n+1}{2n}2 n n+1​. □_\square□​ Simplify (5+6+7)(5+6−7)(5−6+7)(−5+6+7).\left(\sqrt5+\sqrt6+\sqrt7\right)\left(\sqrt5+\sqrt6-\sqrt7\right)\left(\sqrt5-\sqrt6+\sqrt7\right)\left(-\sqrt5+\sqrt6+\sqrt7\right).(5​+6​+7​)(5​+6​−7​)(5​−6​+7​)(−5​+6​+7​). We may choose to expand it out, but that is time intensive and very error prone. Let's have the identity tackle this problem. We have (5+6+7)(5+6−7)=(5+6)2−(7)2=5+6+2 30−7=4+2 30.\begin{aligned}\big(\sqrt5+\sqrt6+\sqrt7\big)\big(\sqrt5+\sqrt6-\sqrt7\big)&=\big(\sqrt5+\sqrt6\big)^2-\big(\sqrt7\big)^2\&=5+6+2\sqrt{30}-7\&=4+2\sqrt{30}.\end{aligned}(5​+6​+7​)(5​+6​−7​)​=(5​+6​)2−(7​)2=5+6+2 30​−7=4+2 30​.​ Likewise, the product of the last two terms is (5−6+7)(−5+6+7)=(7+(5−6))(7−(5−6))=−4+2 30. \begin{aligned} \big(\sqrt5-\sqrt6+\sqrt7\big)\big(-\sqrt5+\sqrt6+\sqrt7\big) &= \left(\sqrt7+\big(\sqrt5-\sqrt6\big)\right)\left(\sqrt7-\big(\sqrt5-\sqrt6\big)\right) \ &=-4+2\sqrt{30}. \end{aligned} (5​−6​+7​)(−5​+6​+7​)​=(7​+(5​−6​))(7​−(5​−6​))=−4+2 30​.​ The final product is (4+2 30)(−4+2 30)=4(30)−16=104.\left(4+2\sqrt{30}\right)\left(-4+2\sqrt{30}\right)=4(30)-16=104.(4+2 30​)(−4+2 30​)=4(30)−16=104.□_\square□​ Try the following problems at your own. Reveal the answer 10 x=(10 624+25)2−(10 624−25)2\large 10^x=\Big(10^{624}+25\Big)^2-\Big(10^{624}-25\Big)^2 1 0 x=(1 0 624+25)2−(1 0 624−25)2 What is the value of x x x that satisfies the equation above? The correct answer is: 626 Reveal the answer Evaluate the following expression: 123456789 2−(123456788×123456790).123456789^{2} - (123456788 \times 123456790).12345678 9 2−(123456788×123456790). If you use a calculator whose precision is not strong enough to answer this question, then you will answer this problem incorrectly. The correct answer is: 1 25 125 625 5 Reveal the answer If x=4(5+1)(5 4+1)(5 8+1)(5 16+1),x=\frac{4}{\big(\sqrt{5}+1\big)\big(\sqrt{5}+1\big)\big(\sqrt{5}+1\big)\big(\sqrt{5}+1\big)},x=(5​+1)(4 5​+1)(8 5​+1)(16 5​+1)4​, then what is the value of (1+x)48?(1+x)^{48}?(1+x)48? The correct answer is: 125 Reveal the answer If you were only given that 438271606 2=192082000625819236,438271606^{2} = 192082000625819236,43827160 6 2=192082000625819236, can you find the sum of digits of the exact value of 561728395 2 561728395^{2}56172839 5 2? Notes: Yes, don't use Wolfram Alpha or a calculator to solve this (though you may use a calculator to calculate the digit sum). The correct answer is: 91 Reveal the answer 2 x−2 y=1 4 x−4 y=5 3 x−y=? \begin{aligned} \large \color{#3D99F6}2^{\color{#624F41}x} - \color{#3D99F6}2^{\color{#624F41}y} & = & \large 1 \ \ \large \color{#20A900}4^{\color{#624F41}x}- \color{#20A900}4^{\color{#624F41}y} & = & \large { \frac 5 3} \ \ \ \large {\color{#624F41}x} - {\color{#624F41}y} & = & \large \, ? \end{aligned} 2 x−2 y 4 x−4 y x−y​===​1 3 5​?​ Details and Assumptions: x x x and y y y are real numbers. The correct answer is: 2 512 128 256 32 14 20 64 54 Reveal the answer It is given that x,y∈Z+x,y\in\Bbb{Z^+}x,y∈Z+ and they satisfy the equation given below: ∏k=0 5(5 2 k+6 2 k)=6 x−5 y.\large\prod_{k=0}^5\left(5^{2^k}+6^{2^k}\right)=6^x-5^y.k=0∏5​(5 2 k+6 2 k)=6 x−5 y. What is the value of x+y?x+y?x+y? Note: This problem is not original. It is adapted from a question posted on Math SE. The correct answer is: 128 Reveal the answer (1+x)(1+x 2)(1+x 4)…(1+x 128)=∑r=0 n x r (1+x)\big(1+x^2\big)\big(1+x^4\big) \ldots \big(1+x^{128}\big) = \displaystyle \sum_{r=0}^n x^r (1+x)(1+x 2)(1+x 4)…(1+x 128)=r=0∑n​x r Given the above equation, what is n?n?n? The correct answer is: 255 See Also Factoring Polynomials Applying the Perfect Square Identity Applying the Perfect Cube Identity Simplifying Rational Expressions Cite as: Difference Of Squares. Brilliant.org. Retrieved 21:53, September 28, 2025, from Join Brilliant The best way to learn math and computer science.Sign up Sign up to read all wikis and quizzes in math, science, and engineering topics. Log in with GoogleLog in with FacebookLog in with email Join using GoogleJoin using email Reset password New user? Sign up Existing user? Log in
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https://en.khanacademy.org/math/up-class-11-12-bridge/xdf46f4ba3c9638cb:algebra/xdf46f4ba3c9638cb:algebraic-identities/v/simplifying-expressions-using-algebraic-identities
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14016
https://www.youtube.com/watch?v=-oVL0CDduAY
Respiratory | Mechanics of Breathing: Expiration | Part 3 Ninja Nerd 3850000 subscribers 20246 likes Description 556212 views Posted: 8 Jul 2017 Official Ninja Nerd Website: Ninja Nerds! In this lecture, Professor Zach Murphy will conclude our three-part series on the mechanics of breathing. We will discuss part 3, the expiration mechanism, along with the pressure changes that occur. We will also use an anatomy model to provide a better visual of the anatomy of this process. We hope you enjoy this lecture, and be sure to support us below! 🌐 Official Links Website: Podcast: Store: 📱 Social Media 💬 Join Our Community Discord: ninjanerd #Expiration #MechanicsofBreathing 540 comments Transcript: Intro I ninja nerds in this video we're going to go ahead and talk and part three of the mechanics of breathing alright so if you guys are here thanks again for sticking in there with us we just really want to make this stuff makes sense ice if you guys remember we left off talking about what happened during the inspiration process and if you guys remember really briefly to quickly recap that at the end at the a peak point of inspiration what was the pressure inside of the intrapleural cavity if you guys remember we called it the P I P if you guys remember we've denoted it as negative four millimeters of mercury during the rest right so it was negative four millimeters of mercury during rest then what happened when we inspired the actual thoracic cavity volume increased and the pressure decreased what did it decrease to if you guys remember that the intrapleural pressure is fix this here the intrapleural pressure all right it dropped down to negative six millimeters of mercury and this was during inspiration was that peak point of inspiration so it dropped down to you guys remember then we said not only did this change let's show this arrow here that this is to what is actually how it changed it went from at rest negative 4 to negative 6 during the inspiration process then what happened to the intrapulmonary pressure will thoracic cavity volume increase so the long started to expand right what happened the visceral pleura was pulled pulled closer towards the parietal pleura if you guys remember thoracic cavity volume increases the actual lung volume increases so the intrapulmonary pressure should decrease but originally at rest what do we say it was we said at rest the people which was denoted as was zero millimeters of mercury right in comparison to the atmospheric pressure right and then this was at rest but then at the peak point of inspiration what do we say we said it actually thoracic cavity volume what the thoracic cavity volume increase so the intrapulmonary pressure should decrease what do we say it was during that actual process we said it changed to about what negative one millimeter mercury so this was a during the inspiration process but that wasn't it remember we said that because the atmospheric pressure if you guys remember the atmospheric pressure was actually what 760 millimeters of mercury and if you guys remember how can we rewrite these numbers so you know we say zero millimeters of mercury doesn't mean that there's no difference between that in the atmospheric pressure so it's actually 760 millimeter of mercury here but over here it's one less than the atmospheric pressure so it's 7:59 millimeter mercury right so there's a pressure difference what do we say we said that the air would flow in actually right into the alveoli until the pressure in the alveoli equalled the pressure in the atmosphere so would rise right so the peak point of inspiration what would they convert to we said the peak point of inspiration it would switch and it would actually allow air to keep flowing into the alveoli until the intrapulmonary pressure was at 7:59 or negative one same thing went back to what it was out actually in the atmosphere so what should the people be at the end point of inspiration it should go back to zero millimeter of mercury which again if we rewrite it in terms of this it would actually be 760 millimeter mercury okay so that's what Quiet Inspiration happened there now one more thing before you move into expiration is if you guys understand something real quick this is what happens during quiet inspiration so there's all these events that we just talked about here this was occurring during quiet inspiration why am i mentioning that because you know there's what's called forced inspiration let me think about it but I'm just sitting here just normally breathing I'm not really Forced Inspiration putting a lot of effort into it right but if I really want to take some air in let's say I really want to take some air I'm getting ready to like I don't know lift something really heavy and I want to get some air going into me right I take as much air as possible I was a little bit more forced so a forced inspiration doesn't just involve my diaphragm on the external intercostals if you guys remember because they were the ones that were contracting and changing the thoracic cavity volume and therefore the pressures they know this other muscles involved so again in quiet an inspiration what muscles were involved in quiet inspiration the muscles that were involved in quiet inspiration was the external intercostals and what was the other one the diaphragm right the diaphragm was the big one that was actually one of the inspired 20 muscles during the actual quiet inspiration but whenever we have to force the inspire we have to pull in some accessory muscles so during this forced inspiration process let's write this one down during the forced inspiration process it requires some other muscles you know I'm the sternocleidomastoid it actually connects Other Muscles to the sternum and so actually what helps when it contract it actually could help to pull the sternum out a little bit and there's some other muscles too they're actually here in the lateral side of the neck they call them the scalene the anterior middle and posterior scalene they also are connected to the rib so they can help to pull the ribs a little bit and if you're very well-developed of individual you can have what's called the pectoralis minor they connect to the third fourth and fifth ribs so they help to be able to pull the ribs up just a little bit okay so they can play a role in a forced inspiratory process so again what were those muscles that I mentioned again one of them was the sternal glide oh massive we're just going to write SCM okay the other one was the scalenes and again there was an anterior a middle and a post here scalene and then the third one was going to be again the pectoralis minor okay why am i mentioning this because all it does is just going to add on to this so if you think about these guys if these guys are involved during forced inspiration what are they going to do let's otis i said that the sternocleidomastoid pulls the sternum out a little bit so helps to elevate that pull it up a little bit outward so that increases our thoracic cavity volume scaling is also pull up in the ribs petra also pulls up on the ribs a little bit if that's doing that what is it actually doing to the thoracic cavity volume all of these things are working to do what their primary function is to do all of these guys increase thoracic cavity volume even more than the diaphragm and the external causes going to do are doing so that's the case let's pretend for a second that those muscles contract what's going to happen to the thoracic cavity volume it's going to increase greater than normal so what would you expect so let's actually have another line here let's do this one in green for us actually no let's do this one in this maroonish color let's say that this is during forest inspiration so this is during forced inspiration so it goes from here rest we take in a certain amount of air but we go even more than that let's say that the interaural pressure decreases even a little bit more because the volume is increasing a little bit more so the pressure should decrease a little bit more so say it was negative sex let's say it goes down to negative seven now just suppose that it goes down to negative seven millimeter of mercury and during what is this during this is during forced inspiration alright sweet okay same thing what about for the people well we said it was a zero during rest okay well let's pretend for a second that we pull on the sternocleidomastoid scalings pectoralis minor and we've taken a little bit more air and what's going to happen to the thoracic cavity volume is going to increase so what should happen to the actual pressure inside of the actual alveoli it should decrease even more how much more should a decrease well decrease the negative wondering quiet inspiration let's add one more negative point so say we went down to negative two so that's the case it goes down to negative two millimeter of mercury and this is during forced inspiration but again you guys already know that whenever we reach that point of where there is a pressure difference between alveoli or the intrapulmonary pression the atmospheric pressure what's going to happen air has to rush in to equalize so think about this for a second this was a negative one so compare this 760 to 759 if this one drops down to negative two what is that in this terms of 750 and all that stuff this would be 758 millimeters of mercury who has this deeper pressure gradient this one into 759 or this one into 758 obviously 758 what does that mean it more air is going to flow down into the lungs until this pressure equals the atmospheric pressure until what and we can show this arrow going back to this point here so I can show it going right back to this point alright okay so that should make sense and that's why if there's a greater pressure difference what does that mean that means more air is going to flow in you know the relationship between that I don't leave you guys hanging there there's a relationship it's called a specifically flow of gases is equal to the change in pressure over the resistance so if you have a greater pressure difference what's going to happen to the flow so if this increases what's going to happen to this this will increase there's a greater pressure difference so what's going to happen more gas will flow in that's the whole point so whenever you're breathing in normally versus I'm taking more earing it's simple right okay that's force Muscles inspiration now here's the next question I want to ask you guys I want to pose a question to you guys I want you guys to think for a second all right so all these muscles are involved during the inspiration process a lot of muscles what muscles are involved in an exploration process if you guys said any muscle I want you to remember that's not correct there is no muscles that are involved during the expiratory process if it's completely passive why is it passive I'm glad you asked I'm going to tell you so you know our lungs are very elastic so during expiration now we're talking about quiet we'll talk about force in a second but we're talking right now about quiet expiration so during quiet expiration are there any muscles involved no muscles involved no muscles involved need you guys to remember that because it's the passive process so again what is it here it's a passive process so it doesn't require the activation of our muscles or skeletal muscles to contract what does it depend upon let's write this on a super bright color it completely depends upon the elasticity of the lungs what is elasticity elasticity is this desire of a structure in this case the Elasticity lungs to resist being stretched in other words it always wants to snap back and what's to reclone go back to the smallest size possible okay and you can determine that because you know elasticity is equal to the change in pressure over the change in volume so what happens if your volume decreases your elasticity increases so you want the volume to decrease that your elasticity can increase so that's the whole goal here that's what we want to happen so how does this happen let's explain that if you guys remember we're not going to take a significant amount of time here we're going to be pretty quick let me get this brown marker over here if we have over here if you guys remember we have a structure here inside of the medulla you know this is the midbrain this is the pons this is the medulla oblongata in the medulla you have this specialized structure it's a mixture of expiratory an inspiratory neurons this structure here is called the v RG the ventral respiratory group okay so now you know that there are from the v RG dis coming these downward presynaptic neurons they're sending these presynaptic neurons down to specific sections of the spinal cord specifically in the ventral gray horn in the ventral grey horn you have these somatic neurons especially especially around c3 c4 and c5 they actually send these axons out and they come onto the diaphragm as you guys remember right so this is the phrenic nerve and that was intervening the phrenic nerve i'm sorry that was intubating the diaphragm to cause it a contract and trigger the inspiration process then if you guys remember if it came down even more if I came down even more I'm not going to draw all of them I'm just gonna do a couple of them but you know that this is c3 to c5 this is t 1 to t 11 this is the intercostal nerves right and we'll draw the intercostal nerves in a different color let's do these in like a red color or something I don't know do the green so again these right here your intercostal nerves right and this would be T 1 2 T 11 and you get that these guys would come up here and they would enter v8 what the actual external intercostals right and cause those that contract whenever we start getting ready to expire the whole point of expiration Stretch Receptor is there's no muscles involved so we want to shut down these actual nerve impulses we don't want those nerve impulses to keep sending signals how do we do that there's special stretch receptors inside of the actual bronchi and around the actual lung area that pick up that increase in stretch because we inspire inspiration process and how our body deals with that let's say we have here let's draw a little stretch receptor out here look at him he's sitting here and he's having his feet in here and he feels some actual stretching so what he does is this stretch receptor this is our stretch receptor this little dude right here and what he's going to do he's going to pick up this stretch and he's going to send these signals into the medulla we're not going to go over the mechanism right now all we're going to say is that it inhibits the vrg the inspiratory centers of the medulla we'll talk specifically about the mechanisms in other videos but for right now there's what you guys to know that the stretch receptors is sending inhibitory signals into the actual module one inhibiting these inspire Tory centers and then if that's the case what's going to happen to these action potentials coming down they're going to drop so the action potentials going down this actual axons is going to decrease and if all of these action potentials decrease what happens to the action potential is going to the diaphragm it decreases what happens to the action potential is going to the external intercostals it decreases if all of this decreases what happens to these muscles they relax so if the external intercostal muscles what happens to this one it relaxes ah it relaxes right like I'm done contracting what happens to the diaphragm it relaxes he's like oh thank goodness I get relaxed now what happens when these muscles relax if you guys remember what Diaphragm do we say about the diaphragm let's let's show it actually oh just so a small diagram right here let's say I do a smaller a quick diagram crude diagram don't judge me guys it's going to be a really quick one and let's say right here I have the diaphragm right here right so here's your diaphragm if you guys remember whenever the diaphragm was contracting what was happening to the diaphragm is remember it was actually it was actually depressing downwards right so because it was depressing down which what was that doing sarasu cavity volume it was increasing it when this relaxes he goes back up so now if you can imagine what would happen then this would go back up and what would that do to the thoracic cavity volume it would decrease the thoracic cavity volume what's that relationship with Boyle's law okay why no Boyle's law Boyles Law that let's write that down Boyle's law says that pressure and volume are inversely proportional so there is an in inversely proportional relationship between pressure and volume if pressure increases what happens to the volume it decreases if the actual pressure decreases then the volume increases you get the point well now we said that we're bringing the diaphragm back up so what should happen to the volume it should decrease then what should happen to the pressure it should increase and that's what happened let's note that so again what happens here when the diaphragm and the excellent Urkel's contract the thoracic cavity volume decreases and then what does that do to the pressure and therefore the threat of the actual pressures would increase and we're going to see how that's actually happening okay before we do that let's say what the external air castles do remember what they were doing we showed Bucket Handle Movement you on the skeleton model right they were creating that that bucket handle movement they were pulling the ribs outwards and pulling the tsardom outwards and forward right upward and forward that was the increase in thoracic cavity boiling if they relaxed what's going to happen they're going to recoil they're going to come back down the Xoom is going to come back to this position what's gonna happen to the resting evety volume it's going to decrease and what's gonna happen to the pressures inside of thoracic cavity they're going to increase so let's see how those pressures are increasing on this diagram over here okay so now if we Pressure Changes take what we were during this actual inspiration process let's write that down over actually no we'll keep it and we'll show how it's happening over here right so during the inspiration process our intrapleural pressure was negative six millimeters of mercury right then what happens to this thoracic cavity volume it decreases so what should happen to this pressure it should increase to what point is that pressure increased well as the diaphragm is going back up and as the ribs are at the chest walls recoiling this intrapleural pressure should actually go up to about negative four millimeters of mercury of the original volume there the original pressure so what should that pressure now be where's my blue mark this is actually going to be enter plural pressure is equal to negative four millimeters of mercury and this is during the actual change during this change in the pressures right so now whenever this actual volume decreases the pressure should increase and it should go to negative four millimeters of mercury during the expiration process sweet nothing nothing too bad about that right okay so now why is this pressure decreasing it because the volume of our lives this pressure increasing because the volume decreased and the volume decreased because the actual chest wall was recoiling back as the chest wall is recoiling back also the diaphragms doning up and you know the lungs they're trying to recoil because the elasticity they're trying to recall so they're trying to pull the visceral pleura away from the actual parietal pleura right but remember these two layers are sticky to one another so what does that mean it's going to pull in the parietal pleura also so when it pulls on the parietal pleura it decreases the actual volume and increases the pressure okay but then inter pulmonary pressure okay let's take a look at that originally what was it well let's come back and look at it real quick before we come over here so we don't lose sight of this it was over here at approximately about 760 millimeters of mercury or zero right well now we're actually decreasing a thoracic cavity volume so the pressure should increase you know what the pressure actually increases - it increases to approximately about one millimeter of mercury above so it becomes positive one so now the p-pull the intrapulmonary pressure should go to positive one millimeter of mercury and again how can we write that with respect to the atmospheric pressure if we compare it with that what is the atmospheric pressure again the atmospheric pressure was 760 millimeter mercury this one is one above that so it should be 760 one millimeter mercury huh well now the pressure gradient difference is from here out there that's the diffusion principle things like to move from areas of high pressure to low pressure so where should the air go the air should go out out into the atmosphere where the pressure is lower and how long will it actually keep doing that it'll keep doing that until the pressure inside of the alveoli are inside of the lungs equalizes with the pressure in the atmosphere and then there's no net diffusion right so now let's say that that happens let's do this in pink let's show the change so during the exploration process so during this process when the thoracic cavity volume is decreasing right this is what you're going to get so this is during the actual beginning of exploration right so this is at the beginning point of exploration but when the air is actually leaving when the air is actually leaving because when thoracic cavity volume is decreasing that's when the pressure increases but then where does the air have to go as to go out into the atmosphere so then the pepole will change an air will start flowing out on pill the intra pulmonary pressure equalizes with the atmosphere so it should become zero millimeter of mercury or again how would you rewrite this this will be rewritten as 760 millimeter mercury and this is at the end of inspiration so I'm sorry exploration this is at the end of expiration and of expiration because the moment Winship when it becomes equal there's no going to be no net diffusion because again things like to go from high to low pressure when it's equal there's no net movement it's the equilibrium okay so that's the process there okay so what Quiet Expiration things were contributing to this if we remember let's come back review and then we're going to go into one last thing and then we're done here all right what do we say quiet expiration no muscles it's passive why because of the natural elasticity lungs it wants to recoil when it recoils it pulls the parietal pleura right and start pulls the visceral pleura away from the parietal pleura but if you remember those layers are sticky so it's going to try to pull the parietal pleura with it not only that but the chest wall is going to wreak whoa why because the external intercostals are relaxing and the diaphragm is relaxing so therefore the thoracic cavity volume will decrease and all those pressures will increase okay but you know our body has another way of dealing with when we need to get extra hours so if I'm just trying to breathe Norlin just trying to expire so let me Forced Expiration just stay here for a second go just a normal that's was just a normal inspiration exploration but now I want to breathe out let's say I'm doing ABS I'm doing some ABS I'm trying to get that you know I'm trying to get more fingers than ABS right and I'm sorry more abs and fingers can right and I'm trying to contract my abs if I do that I'm going to exert I'm going to try to breathe out as hard as I can so I have to exert more effort so it has exert more effort to exhale that extra air it's require some extra muscles so forced expiration does require muscles quiet expiration does not involve any muscles so let's write that down so again what did I say forced expiration involves muscles so what muscles would be involved in this primarily the abdominal wall muscles so the abdominal wall muscles so which ones you know like the it could be the external oblique the internal oblique and even the transverse abdominus into a little degree the rectus abdominis until a little degree of the rectus of them so you all can say the transverse transverse abdominis and a little bit of the rectus abdominis okay so now if these muscles are contracting how is this actually house it's actually happening there's one more muscle I Internal Intercostals forgot I'm sorry one more muscle it's called the internal intercostals we'll talk about these and we'll say how this plays a role in this relationship but what what's the overall effect of these guys okay let me explain the internal cost is first because it's the easier one and then I'll explain the abdominal wall muscles alright so I wanted to bring in an actual skeleton model so we could go again so we did with expert R we did with inspiration let's see how that works with expiration so if you guys remember here's our ribs right here right and if you guys remember there was the muscles that were the external hostels right and they were actually elevating the rib well we have other muscles which are in between the ribs which are called the internal intercostals the only difference is if you remember the external cause we're pulling the up the lower rib upwards the internal intercostals will pull the upper rib downwards when that happens it pulls the actual ribs downwards a little bit more and as it pulls the ribs downwards a little bit more and tries to push the sternum inwards as well as trying to push the ribs a little bit more inwards what happens to this thoracic cavity volume it it would decrease and just to give you a little bit more of an example we'll bring this here chest plate here and so you can see again if you look we're going to have here's your ribs and here's the internal and our costal muscles and again when they contract they pull the upper rib downwards depressing the actual rib cage and by doing that they decrease the thoracic cavity volume okay so I just want to give you guys that as an example let me get back get this out of the way and we'll get back into this video all right so now that we know that we know that the internal intercostals are actually helping to decrease the thoracic cavity volume that was their main goal so they're doing that so let's write this down Oh fix my spelling here internal and did it again and internal intercostals all right now these are just designed to be able to decrease thoracic cavity volume all right so we deal now these other ones so let's explain how these work so these are pretty cool when they contract the IntraAbdominal Pressure external oblique internal bleed transverse abdominus and the rectus abdominus when they contract they help they actually create a pressure they increase the pressure inside of the abdomen so it's called the intra-abdominal pressure right when that pressure increases it pushes upwards upwards and backwards on the diaphragm so imagine here's the diaphragm right here and the abdominal wall muscles are trying to push on the this diaphragm in its actual resting-state er this actual expiratory state right so I think it's actually relaxing I said the diaphragm is relaxing but then the abdominal wall muscles contract and they increase the pressure inside the abdomen and they start pushing and pushing and pushing and pushing imagine me yanking this diaphragm up I'm yanking the diaphragm up what am i doing to the thoracic cavity volume I'm decreasing it so I'm trying to push the diaphragm up to decrease thoracic cavity volume what is that going to do the pressure it's going to increase the pressure let me write this down here with these guys again what is the goal of all of these muscles here all of these ones is to increase the intra-abdominal pressure intra-abdominal pressure which will then push on the diaphragm and then when it pushes on the diaphragm it actually does what to the thoracic cavity volume it decreases the thoracic cavity volume alright so we deal now when that happens let's say that there the actual thoracic cavity volume generally whenever the diaphragm Thoracic Cavity Pressure is just normally relaxing and the external intercostals are normally relaxing this pressure was negative for right and then this pressure went to what positive one right now eventually went to zero well now we're going to decrease the volume even more so now this pressure should actually go up a little bit more it should actually become a little bit more positive let's say for example that it happens during the forced expiratory process so during the forced expiratory process what would we expect it to change a little bit from let's say that it changes to about negative three just suppose negative three millimeter mercury so not even a little bit more positive and again this is during the forced expiration and this is because the actual this volume here is decreasing because the chest wall was trying to push inwards and then also the diaphragm is trying to push upwards also so the volume is decreasing the pressure things get increased a little bit more with this one it was actually positive one right during that act for exploration process let's say that we actually decrease the thoracic cavity I'm even more his pressure should actually go up a little bit more maybe like +2 so let's say for example that this one actually does increase let's say it increases to about +2 from the actual just normal expiration project so we're just comparing them this is negative for negative 3 this one will be +1 this one should be +2 millimeter mercury and this is during what forced expiration okay why am I telling you this again what is the atmospheric pressure let's write this out here atmospheric pressure is 760 millimeter mercury compare the differences here 761 vs. what is this again if we rewrite this what would this actually be written as we could technically say this is actually 760 - because it's a 2 above the normal atmospheric pressure so 7 62 millimeter of mercury compare 760 - from 761 which one is actually going to flow out more well this is going to have more air to flow out so this is going to have more air so more flow from this lung and that's what explains during this actual forced exploration process why more air goes out so more air leaves because why the gradient here is to the gradient here is one more air will flow out until the pressure inside of the lungs equalizes with the pressure of the atmosphere which should be what it should go until it equalizes with the atmosphere so the intrapulmonary pressure at the end of the forced expiration should be zero millimeter mercury zero millimeter of mercury and again this should be at the end of forced expiration and again what is if we just reiterate again what is this 760 I mean what is this zero millimeter mercury it is just same thing as writing 760 millimeter mercury alright an engineers we cover a lot of information in this video I really appreciate you guys sticking in there with me I hope all of this stuff made sense I know it was a lot of stuff I Anna and I hope it made sense now I hope you guys if it did please hit the like button comment down the comment section and subscribe all right ninja nerds as always until next time
14017
https://www.sciencedirect.com/topics/biochemistry-genetics-and-molecular-biology/thiolase
Thiolase - an overview | ScienceDirect Topics Skip to Main content Journals & Books Thiolase In subject area:Biochemistry, Genetics and Molecular Biology Acetoacetyl thiolase (Acetyl-CoA:acetyl-CoA C-acetyl transferase) then catalyzes the condensation of two moles of acetyl coenzyme A to form acetoacetyl coenzyme A and one equivalent of free coenzyme A. Acetoacetyl coenzyme A is reduced by 3-hydroxybutyryl coenzyme A dehydrogenase with NADH to 3-hydroxybutyryl coenzyme A. The corresponding animal enzyme (l-3-hydroxyacyl-CoA:NAD oxidoreductase) produces the l(+) isomer of 3-hydroxybutyric acid. From:Bacterial Physiology and Metabolism, 1969 About this page Add to MendeleySet alert Discover other topics 1. On this page On this page Definition Chapters and Articles Related Terms Recommended Publications Featured Authors On this page Definition Chapters and Articles Related Terms Recommended Publications Featured Authors Chapters and Articles You might find these chapters and articles relevant to this topic. Review article Understanding the function of bacterial and eukaryotic thiolases II by integrating evolutionary and functional approaches 2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub 1 Thiolase II background Thiolase is a conserved enzyme present in the three domains of life (Bacteria, Archaea and Eukarya). This ubiquitous enzyme catalyzes the reversible thiolytic cleavage of 3-ketoacyl-CoA into acyl-CoA and acetyl-CoA, a two-step reaction involving a covalent intermediate formed with a catalytic cysteine. There are two major types of thiolases, (i) acyl-CoA:acetyl-CoA C-acyltransferase (EC 2.3.1.16), also named thiolase I or 3-oxoacyl-CoA thiolase, and (ii) acetyl-CoA:acetyl-CoA C-acetyltransferase (EC 2.3.1.9), also called thiolase II or acetoacetyl-CoA thiolase. While both classes of thiolase catalyze reversible reactions, thiolase I is associated with catabolic processes, whereas thiolase II usually shows anabolic functions under physiological conditions. Contrary to thiolase I, which shows broad chain-length specificity for its substrates (C4–C22) (Yang et al., 1990), thiolase II has high substrate specificity (Merilainen et al., 2008). Thus, thiolase II is specific for C4 chains and catalyzes the condensation of two molecules of acetyl-CoA to give acetoacetyl-CoA and CoA. Thiolase II is involved in the biosynthesis of various highly reduced compounds, depending on their genomic background. For instance, in bacteria, thiolase II catalyzes the first step in the production of the polyhydroxybutyrate (PHB) via the ABC pathway (Steinbuchel and Hein, 2001), whereas in eukaryotes it catalyzes the production of isoprenoids through the mevalonate (MVA) pathway (Kirby and Keasling, 2009) (Fig.1). In this article, we review and present new data suggesting that thiolase II is a conserved enzyme that catalyzes the rate-limiting step in the biosynthesis of PHB and isoprenoids during abiotic stress adaptation, and propose an integrative scenario to explain the conserved sequence and function of thiolase II, where this enzyme senses the tricarboxylic acid (TCA) cycle for the maintenance of the redox balance. Sign in to download hi-res image Fig.1. Thiolases II probably play a key role in the interpretation of metabolic state in response to abiotic stress in bacteria and plants. Proposed model for the rerouting of metabolites from the TCA cycle to antioxidants biosynthetic pathways in response to an abiotic stress in plant (left) and bacteria (right). Red lines show the inhibitory effect of abiotic stress over the TCA cycle, the concomitant availability of acetyl-CoA (AACT substrate) and the decreased of CoA levels (AACT inhibitor). The increase of AACT activity promoted the biosynthesis of isoprenoids and PHB that help to maintain the cell redox balance opposing the oxidative effect of the stress. Abbreviations: AACT, acetoacetyl-CoA thiolase; ABC, phbA, phbB, phbC; GA-3P, glyceraldehyde 3-phosphate; HMG-CoA, 3-hydroxy-3-methylglutaryl-CoA; HMGR, HMG-CoA reductase; HMGS, HMG-CoA synthase gene; H 2 O 2, hydrogen peroxide; MVA, mevalonate; NADP+, nicotinamide adenine dinucleotide phosphate; 1 O 2, singlet oxygen; O•−, superoxide; •OH, hydroxyl radical; PHB, polyhydroxybutyrate; TCA, tricarboxylic acid. Show more View article Read full article URL: Journal2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Review article Dynamic balancing of intestinal short-chain fatty acids: The crucial role of bacterial metabolism 2020, Trends in Food Science & TechnologyYouqiang Xu, ... Baoguo Sun 5.3 Thiolase (THL) THLs are a group of ubiquitous enzymes that play important roles in many core metabolic pathways, such as the well-known β-oxidation pathway for degrading fatty acids. The currently known enzymes of the THL superfamily from bacteria can be divided into two categories: biosynthetic THLs (also named acetyl-CoA C-acetyltransferase, acetoacetyl-CoA THL or THL II, EC 2.3.1.9) and degradative THLs (also called acetyl-CoA C-acyltransferase, 3-ketoacyl-CoA THL, THL I, or β-ketothiolase, EC 2.3.1.16) (Modis & Wierenga, 2000). The degradative THLs can also catalyse the reverse Claisen condensation reaction, promoting carbon–carbon bond formation (Fage, Meinke, & Keatinge-Clay, 2015). Some SCFA-producing gut species only harbour one copy of THL, for example Anaerostipes hadrus and Clostridium acetobutylicum, and some carry as many as three copies, like Blautia obeum (Fig. S3). It is interesting that no specific THL I or II is annotated in butyrate-producing Bacteroides spp., but after careful analysis, it is found that Bacteroides species each carry multiple copies of putative acetyltransferases, such as Bacteroides fragilis (putative acetyltransferases with GenBank accession no. BF3333, BF3500, BF3685, BF3855 and BF4204). Some of the putative acetyltransferases may play crucial roles in butyrate synthesis. Clostridium asparagiforme and Ruminococcus gnavus also carry multiple-copies of THLs. Up to now, knowledge is still insufficient about the key THLs in SCFA-producing gut bacteria. THL Is (AIN34153 [E. coli], PPA35241 [B. subtilis] and RAP25323 [B. pumilus]) and THL IIs (AMM90683 [B. pumilus], AMM94437 [B. simplex]) and AFI23434 [Betula platyphylla] are selected as references for evolutionary relationship analysis with the THLs from SCFA-producing gut bacteria. Results indicate that many of THLs are only distantly related to the selected references, and no distinct divergence is observed between type I and II THLs based on evolutionary relationship analysis (Fig. S3). Structural analysis shows that some enzymes of the THL superfamily are dimers, whereas tetramers are also found (Modis & Wierenga, 1999). The oligomer is formed by the interaction of the extended loop domain of the monomer (Kim et al., 2015). Each subunit has the N- and C-terminal domains, and shows the same βαβαβαββ topology (White, Zheng, Zhang, & Rock, 2005). The THL superfamily enzymes catalyse carbon–carbon-bond formation via a Claisen condensation (Haapalainen et al., 2006), and this is the key step in the biosynthesis of n-butyrate, n-valerate and n-caproate. THL I has broad chain-length specificity toward substrates, whereas THL II is specific for the thiolysis of acetoacetyl-CoA. The enzyme reaction occurs in two steps and follows ping-pong kinetics (Modis & Wierenga, 2000). Using the THL from E. coli (EcTHL) as an example, the reaction is initiated after the sulphur atom of Cys-88 loses a proton through nucleophilic attack by His-349 (Fig. 4 a I and II). Thereafter, nucleophilic attack of the substrate on the acetyl–enzyme complex is activated by Cys-379, that obtains a proton from the substrate (Fig. 4 a III and IV) (Ithayaraja, Janardan, Wierenga, Savithri, & Murthy, 2016). Sign in to download hi-res image Fig. 4. Catalysis of thiolase (THL) from Escherichia coli (a), and schematic diagram of redox-switch catalytic regulation of the THL from Clostridium acetobutylicum (b). A redox-switch modulation mechanism in the THL from Clostridium acetobutylicum (CaTHL) is found with high identity with THLs from other species of the same genus. A disulphide bond forms between two catalytic cysteine residues, providing redox regulation of the enzyme (Fig. 4 b). Such a disulphide bond formation has not been observed in other type II THLs (Kim et al., 2015). The catalytic residues of CaTHL are Cys-88, His-348 and Cys-378. In the oxidized state, formation of the disulphide bond between Cys-88 and Cys-378 leads to a conformational change in the loop region of the C-terminus and blocks access of the substrate to the catalytic domain. When in the reduced state (i.e., with no disulphide bond between Cys-88 and Cys-378), the catalytic reaction is initiated and carried out through the same process as in EcTHL. Based on systematic analysis of the four rate-limiting enzymes involved in the synthesis of propionate, n-butyrate, n-valerate and n-caproate in SCFA-producing gut bacteria, we realized the lack of knowledge so far on this topic. The identification and characterization of the main functional microorganisms, their specific metabolic pathways and rate-limiting enzymes for SCFAs synthesis will help us understand the formation of intestinal SCFAs and develop proper control strategies to rationally regulate the balance of intestinal SCFAs and safeguard gut health. Show more View article Read full article URL: Journal2020, Trends in Food Science & TechnologyYouqiang Xu, ... Baoguo Sun Review article β-Oxidation – strategies for the metabolism of a wide variety of acyl-CoA esters 2000, Biochimica et Biophysica Acta (BBA) - Molecular and Cell Biology of LipidsJ.Kalervo Hiltunen, Yong-Mei Qin The thiolase protein family can be divided into two categories, degradative thiolases and synthetic thiolases. Degradative thiolase can catalyze reaction with a fatty acyl-CoA molecules of variable length whereas synthetic thiolases can use as substrate C4-CoA molecules . The mammalian mitochondrial long-chain acyl-CoA thiolase is the small subunit of the trifunctional β-oxidation protein [35,40], whereas the short-chain thiolase (acetoacetyl-CoA thiolase) is soluble and tetrameric. View article Read full article URL: Journal2000, Biochimica et Biophysica Acta (BBA) - Molecular and Cell Biology of LipidsJ.Kalervo Hiltunen, Yong-Mei Qin Review article Understanding the function of bacterial and eukaryotic thiolases II by integrating evolutionary and functional approaches 2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Highlights • We have reported the evolutionary and functional equivalence of thiolases II. • We propose that thiolase II could sense changes in the acetyl-CoA/CoA ratio. • Thiolase II is probably an ancestral enzyme involved in abiotic stress adaptation. View article Read full article URL: Journal2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Review article Understanding the function of bacterial and eukaryotic thiolases II by integrating evolutionary and functional approaches 2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub 7 A hypothetical interpretation of the metabolic state by thiolase II The hypothesis that thiolase II is a conserved enzyme that catalyzes the rate-limiting step in the biosynthesis of PHB and isoprenoids via the ABC and MVA pathways during abiotic stress adaptation is supported by several evidences (Fig.1): (i) bacterial and plant thiolases II catalyze the first step and the highest endergonic reaction (Δ G=+25 kJ/mol) in these anabolic processes (Fig. S1) (Kadouri et al., 2005), (ii) plant and bacterial thiolases II are transcriptionally regulated in response to adverse environmental conditions (Fig.1) (Kadouri et al., 2005; Soto et al., 2011) and (iii) the activity of bacterial and plant thiolases II is post-transcriptionally regulated by CoA (Figs.1 and 3) (Senior and Dawes, 1973; Soto et al., 2011). This negative regulation of thiolase II activity by CoA could be the result of a negative feedback regulation and/or the metabolic state of bacterial and eukaryotic cells (Fig.1). However, these evidences are not enough to explain why thiolase II was evolutionarily selected as a key enzyme in the biosynthesis of antioxidant compounds during abiotic stress adaptation in so divergent organisms such as Pseudomonas and alfalfa. The hypothesis that thiolase II is a conserved enzyme that catalyzes the rate-limiting step in the biosynthesis of PHB and isoprenoids via the ABC and MVA pathways during abiotic stress adaptation is supported by several evidences (Fig.1): (i) bacterial and plant thiolases II catalyze the first step and the highest endergonic reaction (Δ G=+25 kJ/mol) in these anabolic processes (Fig. S1) (Kadouri et al., 2005), (ii) plant and bacterial thiolases II are transcriptionally regulated in response to adverse environmental conditions (Fig.1) (Kadouri et al., 2005; Soto et al., 2011) and (iii) the activity of bacterial and plant thiolases II is post-transcriptionally regulated by CoA (Figs.1 and 3) (Senior and Dawes, 1973; Soto et al., 2011). This negative regulation of thiolase II activity by CoA could be the result of a negative feedback regulation and/or the metabolic state of bacterial and eukaryotic cells (Fig.1). However, these evidences are not enough to explain why thiolase II was evolutionarily selected as a key enzyme in the biosynthesis of antioxidant compounds during abiotic stress adaptation in so divergent organisms such as Pseudomonas and alfalfa. To explain the conserved function of thiolase II, we propose to analyze an integrative scenario, where this enzyme senses the tricarboxylic acid (TCA) cycle for the maintenance of the redox balance during abiotic stress adaptation (Fig.1). The first argument supporting this hypothesis is related to the ancient origin of the TCA cycle. In fact, this metabolic pathway is a truly ancestral cycle present in the three domains of life. The second argument is related to the impact of the regulation of the TCA cycle on the metabolic status of the cell. Acetyl-CoA (the thiolase II substrate) is oxidized by the TCA cycle, whereas CoA (a thiolase II inhibitor) is released by this pathway during optimal growth conditions (Fig.1). Thus, the production of antioxidant compounds (PHB and isoprenoids) by thiolase II is repressed under favorable conditions, characterized by high respiratory rates (Fig.1) (Ayub et al., 2009; Soto et al., 2011). On the other hand, several evidences indicate that the TCA cycle is inhibited by the oxidative stress induced under abiotic stress exposure in highly divergent organisms such as yeast, animals, plants and bacteria (Fig.1) (Baxter et al., 2007; Godon et al., 1998; Grant, 2008; Liu et al., 2005; Pomposiello and Demple, 2002). Therefore, the levels of acetyl-CoA would be increased and the amount of free CoA would be decreased under abiotic stress conditions (Fig.1). Thus, bacteria and plants can redirect the flow of reducing power from the TCA cycle to the ABC and MVA pathways by thiolases II, and then, mitigate the oxidative stress induced by abiotic stress (Fig.1). Show more View article Read full article URL: Journal2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Review article Coenzyme A thioester-mediated carbon chain elongation as a paintbrush to draw colorful chemical compounds 2020, Biotechnology AdvancesShenghu Zhou, ... Yu Deng 3.1 Thiolases Three thiolases, 3-oxoacyl-CoA thiolase, β-ketoacyl-ACP synthase (KAS) and 3-hydroxyl-3-methylglutaryl-CoA synthase (HMGS), are the most commonly used enzymes in the Claisen-condensation based CoA-dependent carbon chain elongation process (Haapalainen et al., 2006; Harijan, 2017). Generally, 3-oxoacyl-CoA thiolase catalyzes CC bond formation between acetyl-CoA and acyl-CoA in a reverse β-oxidative process (Kallscheuer et al., 2017). While, KAS and HMGS are responsible for the carbon chain elongation process in the biosynthesis of fatty acids and isoprenoids, respectively (Clomburg et al., 2019; Heil et al., 2019). There are two oxyanion holes in all thiolases to stabilize the transition state of the substrates and facilitate the Claisen-condensation reaction (Fig. 3 A) (Heath and Rock, 2002). Extender units such as acetyl-CoA and malonyl-ACP are transformed into an enolate intermediate in oxyanion hole I. Then, the α-carbanion of acetyl-CoA enolate attacks the carbonyl carbon of acyl-Cys89 to form a CC bond (Haapalainen et al., 2006). Subsequently, a tetrahedral intermediate is formed and stabilized in oxyanion hole II (Fig. 3 A) (Meriläinen et al., 2009). In the Claisen-condensation reaction, formation of the α-carbanion is the most important step in the nucleophilic process. There are two pathways for α-carbon activation, decarboxylation and deprotonation. When acetyl-CoA is used as an extender unit, the α-carbon is deprotonated by a cysteine or glutamate residue in 3-oxoacyl-CoA thiolase or HMGS, respectively. Whereas, when malonyl-ACP is used as an extender unit, decarboxylation occurs to activate the α-carbon in KAS (Fig. 3 B) (Meriläinen et al., 2009; Meriläinen et al., 2008). Sign in to download hi-res image Fig. 3. The catalytic mechanisms of thiolase (A), PKSs (B), CCR (C) and ACC (D). The blue and red colored molecules represent the extender and primer units, respectively. Enzyme annotation: Polyketide synthases (PKSs), crotonyl-CoA carboxylase/reductase (CCR) and acetyl-CoA carboxylase (ACC). For the proposed catalytic mechanism of CCR please refer to the publication of Ray et al. (Ray and Moore, 2016) (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) Most of thiolases prefer to use acetyl-CoA or malonyl-ACP as extender units for straight-chain elongation. While, branched condensation enzymes, which use longer acyl-CoAs as extenders unit, have gained an increasing amount of interest in recent years (Blaisse et al., 2017; Kresovic et al., 2015). To extend the substrate scope of thiolase, Blaisse et al. identified five thiolases (Acat 1–5) from A. suum, among them Acat 2 and Acat 3 prefer to use propionyl-CoA as an extender unit for the production of α-methyl acids via a non-decarboxylative condensation reaction (Blaisse et al., 2017). Furthermore, longer acyl-CoAs (C12 to C14) can also be considered as extender units when OleA (Goblirsch et al., 2016), PpyS (Kresovic et al., 2015) and CorB (Zocher et al., 2015) are used to produce α-branched-β-ketoacids via the condensation reaction of two fatty acyl-CoAs. Based on crystal structure analysis, these enzymes were revealed to share a similar structure in the active pocket. Generally, a T-shaped configuration in the active site consists of three channels functions to accommodate phosphopantetheine and two alkyls groups (Goblirsch et al., 2016; Kresovic et al., 2015). Furthermore, two important residues, Cys and Glu, were identified to participate in the acyl transfer and α-carbon deprotonation steps of the Claisen-condensation process, respectively. Rational and non-rational protein engineering strategies are often used to improve the performance of thiolases. The Claisen-condensation reaction requires the consumption of ATP and thus, it usually acts as the rate limiting-step in biosynthetic pathways (Zhou et al., 2019). To further improve the activity, Wei et al. rationally engineered a homoserineacetyltransferase (MetXlm) based on evolutionary conservation sequence analysis and structure-guided engineering, and the activity of the best mutant (F147L/M182I/M240A) increased 12.5-fold (Wei et al., 2019). Furthermore, most thiolases display low substrate specificity, which significantly reduces the reaction efficiency. Bulky residues in the active pocket may have an influence on the accommodation of the substrate. Hence, to modify the substrate scope, Torres-Salas et al. rationally designed an Erg10 thiolase from S. cerevisiae and the results showed that F293 mutants were the most active and selective biocatalysts (Torres-Salas et al., 2018). Furthermore, based on the model- or crystal structure-guided rational design strategies, the selectivity of thiolases was significantly improved and results the titer of 3-hydroxy-hexanoic acid (Bonk et al., 2018) and α-methyl substituted substrates (Fage et al., 2015) increasement of 30-fold or so (Table 1). However, thiolases are naturally inhibited by CoA in vivo. To reduce feedback inhibition, Mann et al. engineered a ThlA thiolase from Clostridium acetobutylicum ATCC 824 using a directed evolution strategy (Table 1) (Mann and Luetke-Eversloh, 2013). Table 1. Strategies used to enhance the performance of thiolases and PKSs'. | Category/gene | Sources | Purpose | Strategy | Mutants | Results | References | --- --- --- | Thiolase/PhbA | Zoogloea ramigera | Increase the ratio of C6/C4 products | Structural guided engineering | M157A/G, M288S/A/G | The ratio of C6/C4 improved 30-fold | Bonk et al., 2018 | | Thiolase/ThlA | Clostridium acetobutylicum | Reduce CoA feedback inhibition | Directed evolution | R133G, H156N, G222V | Reduced feedback inhibition; ethanol (4.9 g/L); butanol (12.4 g/L) | Mann and Luetke-Eversloh, 2013 | | Thiolase/BktB | Ralstonia eutropha | Extend the substrate scope | Structural guided engineering | M290A | Significant activity toward α-methyl substituted substrates | Fage et al., 2015 | | Thiolase/BktB | Cupriavidus necator | Increase the ratio of C6/C4 products | Computational aided engineering | M158G, M158S | Ratio improved 10-fold | Bonk et al., 2018 | | PKSs I, II/eryA | Saccharopolyspora erythraea JC2 | Obtaining a chiral α-methyl group | Keto-reductase domain swapping | Amph KRs 1 and 11 introduced into A1-modular PKSs | Obtained a chiral α-methyl group | Annaval et al., 2015 | | PKSs I, II/pikAIII and IV | pET28-PikAIII, pET24-PikAIV | Generate functional chimeric PKSs | Homologous recombination in Saccharomyces cerevisiae | p3p3 and p3p4 hybrid PKSs | Active chimeric enzymes; 3-dehydro-10-deoxymethynolide (25±8 mg/L) | Chemler et al., 2015 | | PKSs I, II/rapAB | pPF137, pWV165 | Rapidly generate diverse and highly productive PKSs assembly lines | Homologous recombination in Streptomyces fradiae | Diverse combination of different regions of rapA and rapB | Generate new rapalogs | Wlodek et al., 2017 | | PKSs III/PhlD | Pseudomonas fluorescens Pf-5 | Increase stability | Directed evolution | M21T, N27D, A82T, A181S | Half-life improved 24-fold; phloroglucinol (3.65 g/L) | Rao et al., 2013 | | PKSs III/g2ps1 | Gerbera hybrida | Increase both the catalytic efficiency and in vivo stability | Structural guided engineering | C35S, C372S | Catalytic efficiency and stability improved 2- and 5-fold, respectively. | Vickery et al., 2018 | | PKSs III/chs | Glycyrrhiza uralensis | Improve the activity | Web-based service PROSITE to analyze the functional sites | S165M | Pinocembrin (67.81 mg/L) | Cao et al., 2016b | Show more View article Read full article URL: Journal2020, Biotechnology AdvancesShenghu Zhou, ... Yu Deng Review article Biotechnology for the Sustainability of Human Society 2009, Biotechnology AdvancesR. Gheshlaghi, ... C.P. Chou 3.5 EC 2.3.1.9: thiolase (acetyl-CoA acetyltransferase) Thiolase carries out the thermodynamically unfavorable condensation of two molecules of acetyl-CoA to one molecule of acetoacetyl-CoA, which is the precursor of the four-carbon products of C. acetobutylicum (Bennett and Rudolph, 1995). The change in the level of this enzyme in a batch of C. acetobutylicum has been studied (Hartmanis and Gatenbeck, 1984). They found high levels of thiolase activity throughout the fermentation, with the maximal activity occurring shortly after growth arrest. Thiolase competes with phosphate acetyltransferase (EC 2.3.1.8) during acidogenesis and with acetaldehyde dehydrogenase (EC 1.2.1.10) during solventogenesis for the available pool of acetyl-CoA. The latter could influence the ratio of butanol plus acetone to ethanol. Formation of acetic acid yields twice as much ATP per mole of acetyl-CoA compared with butyric acid formation. Thus, the yield of ATP is highly dependent on the regulation of thiolase. Thiolase from C. acetobutylicum ATCC 824 has been purified and characterized (Wiesenborn et al., 1988). This enzyme appears to be a tetramer with an estimated monomer size of 44 kDa. Thiolase activity from C. acetobutylicum has a very broad pH optimum ranging from pH 5.5 to 7.9. The internal pH of C. acetobutylicum remained within the range of 5.6 to 6.2 during the different stages of a typical batch fermentation (Bowles and Ellefson, 1985; Gottwald and Gottschalk, 1985). Thus, the change in internal pH does not appear to affect the regulation of thiolase. The Lineweaver–Burk plots for the thiolysis reaction suggested a ping-pong binding mechanism with the K m values of 4.7 and 32 μmol respectively for CoA and acetoacetyl-CoA (Wiesenborn et al., 1988). The rate of condensation reaction was studied by varying the acetyl-CoA concentration from 0.099 to 1 mM. In order to study the effect of CoA on the condensation reaction, three concentrations of CoA (9.7, 19.3, and 38.6 μM) were employed. It was observed that with increasing concentration of CoA the acetyl-CoA saturation curve became increasingly sigmoidal. Consequently, the Lineweaver–Burk plot deviated from a straight line. This phenomenon has been reported with other thiolases (Oeding and Schlegel, 1979; Berndt and Schlegel, 1975). The kinetic data revealed that this thiolase was sensitive even to very low ratio of CoA to acetyl-CoA and in the presence of free CoA the reaction rate was less. We may conclude that this ratio is an important factor in modulation of the net rate of the condensation reaction. The intracellular concentrations of CoA and acetyl-CoA in extracts of C. kluyveri were estimated to be 0.24 and 0.9 mM, respectively (Decker et al., 1976). This level of CoA concentration in C. acetobutylicum would result in a strong product inhibition. In the absence of CoA a Hill coefficient of 1 was observed, while this coefficient approached 1.45 in the presence of CoA. The above mentioned observations suggest a multi-substrate ping pong mechanism (Fromm, 1975). The following rate equation for two identical substrate molecules with product inhibition is proposed: (2)1 v 9=1 V max,9+K m,AcCoA V max,9[AcCoA][1+[CoA]K I,CoA]+K′9 V max,9[CoA][AcCoA]2 The kinetic constants of Eq. (2) were determined using the kinetic data reported by Wiesenborn et al. (1988) at 30°C and pH 7.4 (Fig. 2 of their work). In the absence of CoA, the linear behavior of the condensation reaction on the double-reciprocal plot proved a Michaelis–Menten mechanism with K m,AcCoA value of 0.275 mM. The maximum reaction rate, V max,9, was determined to be 0.0102 μmol/min. It can be seen from Eq. (2) that at fixed concentration of acetyl-CoA, when 1/v 9 is graphed as a function of CoA, the slope term is: (3)Slope=K m,AcCoA V max,9 K I,P 1[AcCoA]+K′9 V max,9 1[AcCoA]2 A secondary plot of Slope [AcCoA] versus 1/[AcCoA] showed a linear behavior. The K I,P and K′9 constants were determined to be 7.033 and 0.0086 mM, respectively. Consequently, the rate of condensation reaction can be written as: (4)ν 9=V max,9[AcCoA]2[AcCoA]2+0.275AcCoA+0.0086[CoA] Butyryl-CoA or ATP may act as feedback inhibitors if they accumulate to high internal concentrations. Relative to the control, the activity with butyryl-CoA was 58% at 1 mM and 85% at 0.2 mM, and activity with ATP was 59% at 10 mM and 90% at 2 mM (Wiesenborn et al., 1988). Show more View article Read full article URL: Journal2009, Biotechnology AdvancesR. Gheshlaghi, ... C.P. Chou Chapter Cholesterol synthesis 2023, Sweet Biochemistry (Second Edition)Asha Kumari First enzyme thiolase Thiolase carries Claisen condensation of two acetyl-CoA molecules in three steps. You will see this reaction in many biosynthetic pathways. Thiolase acetylates its special cysteine residue at an active site by acetyl-CoA. In the second step, the enzyme attaches second acetyl-CoA and binds, and enolisation takes place. Third step involves Claisen’s condensation between two acetyl-CoA still hanging at the cysteine residue. (Imagine how an enzyme will glue two substrates held in its amino acid residues.) Acetoacetyl-CoA is formed here. View chapterExplore book Read full chapter URL: Book2023, Sweet Biochemistry (Second Edition)Asha Kumari Review article Understanding the function of bacterial and eukaryotic thiolases II by integrating evolutionary and functional approaches 2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Abstract Acetoacetyl-CoA thiolase (EC 2.3.1.9), commonly named thiolase II, condenses two molecules of acetyl-CoA to give acetoacetyl-CoA and CoA. This enzyme acts in anabolic processes as the first step in the biosynthesis of isoprenoids and polyhydroxybutyrate in eukaryotes and bacteria, respectively. We have recently reported the evolutionary and functional equivalence of these enzymes, suggesting that thiolase II could be the rate limiting enzyme in these pathways and presented evidence indicating that this enzyme modulates the availability of reducing equivalents during abiotic stress adaptation in bacteria and plants. However, these results are not sufficient to clarify why thiolase II was evolutionary selected as a critical enzyme in the production of antioxidant compounds. Regarding this intriguing topic, we propose that thiolase II could sense changes in the acetyl-CoA/CoA ratio induced by the inhibition of the tricarboxylic acid cycle under abiotic stress. Thus, the high level of evolutionary and functional constraint of thiolase II may be due to the connection of this enzyme with an ancient and conserved metabolic route. View article Read full article URL: Journal2014, GeneAna Romina Fox, ... Nicolás Daniel Ayub Review article Peroxisomes: Morphology, Function, Biogenesis and Disorders 2006, Biochimica et Biophysica Acta (BBA) - Molecular Cell ResearchSébastien Léon, ... Suresh Subramani The assembly of thiolase in Yarrowia lipolytica employs a more specific mechanism. Here the PTS2 accessory protein, Pex20p, is required for the dimerization of the protein in the cytosol . In pex20 mutants, thiolase is found in the cytosol in monomeric form. In the wild-type strain, hetero-oligomers of Pex20p and thiolase form, and this is necessary for both thiolase dimerization and peroxisomal import. View article Read full article URL: Journal2006, Biochimica et Biophysica Acta (BBA) - Molecular Cell ResearchSébastien Léon, ... Suresh Subramani Related terms: Coenzyme A Dehydrogenase Acetyl-CoA Synthase Beta Oxidation Acyl-CoA Peroxisome Acetoacetyl-CoA Enzyme Oxidoreductase View all Topics Recommended publications Journal of Biological ChemistryJournal Metabolic EngineeringJournal Molecular Genetics and MetabolismJournal Biochimica et Biophysica Acta (BBA) - Molecular Cell ResearchJournal Browse books and journals Featured Authors Ferdinandusse, SachaAmsterdam Gastroenterology Endocrinology Metabolism, Amsterdam, Netherlands Citations7,684 h-index47 Publications48 Kmoch, StanislavWake Forest University School of Medicine, Winston Salem, United States Citations5,173 h-index40 Publications46 Kadoya, RyousukeTokyo University of Agriculture, Tokyo, Japan Citations560 h-index11 Publications1 About ScienceDirect Remote access Advertise Contact and support Terms and conditions Privacy policy Cookies are used by this site. Cookie Settings All content on this site: Copyright © 2025 or its licensors and contributors. 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SUBSCRIBE e-Anatomy The Anatomy of Imaging HOME e-Anatomy Anatomical structures ... Peritoneum Omental bursa; Lesser sac Human anatomy 2 ha2 Human body General Anatomy View the module Omental bursa Bursa omentalis Synonym: Epiploic bursa; Lesser sac Related terms: Omental bursa; Lesser sac Definition IMAIOS On the posterior abdominal wall the peritoneum of the general cavity is continuous with that of the Omental Bursa(bursa omentalis; lesser peritoneal sac) in front of the inferior vena cava. Starting from here, the bursa may be traced across the aorta and over the medial part of the front of the left kidney and diaphragm to the hilus of the spleen as the anterior layer of the phrenicolienal ligament. From the spleen it is reflected to the stomach as the posterior layer of the gastrosplenic ligament. It covers the postero-inferior surfaces of the stomach and commencement of the duodenum, and extends upward to the liver as the posterior layer of the lesser omentum; the right margin of this layer is continuous around the hepatic artery, bile duct, and portal vein, with the wall of the general cavity. Theepiploic foramen(foramen epiploicum; foramen of Winslow) is the passage of communication between the general cavity and the omental bursa. It is bounded in front by the free border of the lesser omentum, with the common bile duct, hepatic artery, and portal vein between its two layers;behind by the peritoneum covering the inferior vena cava;above by the peritoneum on the caudate process of the liver, and below by the peritoneum covering the commencement of the duodenum and the hepatic artery, the latter passing forward below the foramen before ascending between the two layers of the lesser omentum. The boundaries of the omental bursa will now be evident. It is bounded in front, from above downward, by the caudate lobe of the liver, the lesser omentum, the stomach, and the anterior two layers of the greater omentum.Behind, it is limited, from below upward, by the two posterior layers of the greater omentum, the transverse colon, and the ascending layer of the transverse mesocolon, the upper surface of the pancreas, the left suprarenal gland, and the upper end of the left kidney. To the right of the esophageal opening of the stomach it is formed by that part of the diaphragm which supports the caudate lobe of the liver.Laterally, the bursa extends from the epiploic foramen to the spleen, where it is limited by the phrenicolienal and gastrolienal ligaments. The omental bursa, therefore, consists of a series of pouches or recesses to which the following terms are applied: (1) thevestibule,a narrow channel continued from the epiploic foramen, over the head of the pancreas to thegastropancreatic fold;this fold extends from the omental tuberosity of the pancreas to the right side of the fundus of the stomach, and contains the left gastric artery and coronary vein; (2) thesuperior omental recess,between the caudate lobe of the liver and the diaphragm; (3) thesplenic recess (lienal recess),between the spleen and the stomach; (4) theinferior omental recess,which comprises the remainder of the bursa. In the fetus the bursa reaches as low as the free margin of the greater omentum, but in the adult its vertical extent is usually more limited owing to adhesions between the layers of the omentum. During a considerable part of fetal life the transverse colon is suspended from the posterior abdominal wall by a mesentery of its own, the two posterior layers of the greater omentum passing at this stage in front of the colon. This condition occasionally persists throughout life, but as a rule adhesion occurs between the mesentery of the transverse colon and the posterior layer of the greater omentum, with the result that the colon appears to receive its peritoneal covering by the splitting of the two posterior layers of the latter fold. In the adult the omental bursa intervenes between the stomach and the structures on which that viscus lies, and performs therefore the functions of a serous bursa for the stomach. References Gallery View the module Anatomical hierarchy Human anatomy 2 Human body> Visceral systems> Abdominopelvic cavity> Peritoneal cavity> Omental bursa Underlying structures: Omental foramen Vestibule of omental bursa Superior recess of omental bursa Inferior recess of omental bursa Splenic recess of omental bursa Human anatomy 1 Systemic anatomy> Abdominopelvic cavity> Peritoneum> Omental bursa; Lesser sac Underlying structures: Omental foramen; Epiploic foramen Vestibule Superior recess Inferior recess Splenic recess Gastropancreatic fold Hepatopancreatic fold Comparative anatomy in animals Omental bursa Translations MEDIASTINUM-HEART CT chestCT PREMIUM CT axial chestCT PREMIUM CTA coronary arteriesCT PREMIUM MediastinumIllustrations PREMIUM HeartIllustrations FREE CoronarographyAngiography PREMIUM CT body (lymph nodes)CT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Radiography chest abdomen pelvisRadiographs FREE ABDOMEN Female abdomen-pelvis CTCT PREMIUM CT axial male abdomen and pelvisCT PREMIUM CT body (lymph nodes)CT PREMIUM CT peritoneal cavityCT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Digestive systemIllustrations PREMIUM Radiography chest abdomen pelvisRadiographs FREE Magnetic Resonance CholangiopancreatographyMRI PREMIUM PELVIS MRI female pelvisMRI PREMIUM MRI male pelvisMRI PREMIUM Female pelvisIllustrations PREMIUM Male pelvisIllustrations PREMIUM CT axial male abdomen and pelvisCT PREMIUM CT body (lymph nodes)CT PREMIUM CT peritoneal cavityCT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Radiography chest abdomen pelvisRadiographs FREE SHOULDER MR arthrography shoulderMRI PREMIUM MRI shoulderMRI PREMIUM CT arthrography shoulderCT arthrogram PREMIUM MRI upper extremityMRI PREMIUM Radiography upper extremityRadiographs PREMIUM Upper extremityIllustrations PREMIUM MRI brachial plexusMRI PREMIUM ELBOW MRI elbowMRI PREMIUM CT arthrography elbowCT arthrogram PREMIUM MRI upper extremityMRI PREMIUM Radiography upper extremityRadiographs PREMIUM Upper extremityIllustrations PREMIUM WRIST-HAND MRI wristMRI PREMIUM MRI handMRI PREMIUM MRI thumbMRI PREMIUM MRI finger of handMRI PREMIUM MRI upper extremityMRI PREMIUM Radiography upper extremityRadiographs PREMIUM Upper extremityIllustrations PREMIUM HIP Hip MRIMRI PREMIUM MRI lower extremityMRI PREMIUM Radiography lower extremityRadiographs FREE Lower extremityIllustrations PREMIUM KNEE Knee MRIMRI PREMIUM CT arthrography kneeCT arthrogram PREMIUM MRI lower extremityMRI PREMIUM Radiography lower extremityRadiographs FREE Lower extremityIllustrations PREMIUM ANKLE-FOOT MRI ankle and hindfootMRI PREMIUM Forefoot MRIMRI PREMIUM MRI lower extremityMRI PREMIUM Radiography lower extremityRadiographs FREE Lower extremityIllustrations PREMIUM UPPER LIMB MRI upper extremityMRI PREMIUM MRI shoulderMRI PREMIUM MRI wristMRI PREMIUM MRI elbowMRI PREMIUM MRI handMRI PREMIUM Radiography upper extremityRadiographs PREMIUM Upper extremityIllustrations PREMIUM Arteriography upper extremityAngiography FREE Visible Human ProjectPhotography PREMIUM LOWER LIMB Lower extremityIllustrations PREMIUM Radiography lower extremityRadiographs FREE MRI lower extremityMRI PREMIUM Hip MRIMRI PREMIUM Knee MRIMRI PREMIUM CT arthrography kneeCT arthrogram PREMIUM MRI ankle and hindfootMRI PREMIUM Forefoot MRIMRI PREMIUM Lower limb CTACT PREMIUM Leg arteries and bonesCT FREE Arteriography lower extremityAngiography FREE Select a zone WHOLE BODY CT body (lymph nodes)CT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Visible Human ProjectPhotography PREMIUM SPINE MRI cervical spineMRI PREMIUM MRI lumbar spineMRI PREMIUM CT lumbar spineCT PREMIUM MRI brachial plexusMRI PREMIUM SpineIllustrations PREMIUM Spinal cordIllustrations PREMIUM Radiography spineRadiographs FREE CT head and neckCT PREMIUM MRI head and neckMRI PREMIUM HEAD AND NECK CT head and neckCT PREMIUM CT brainCT PREMIUM MRI brainMRI PREMIUM MRI axial brainMRI FREE CT faceCT PREMIUM MRI head and neckMRI PREMIUM MRA brainMRI PREMIUM BrainIllustrations PREMIUM CT temporal boneCT PREMIUM SkullIllustrations PREMIUM BRAIN CT brainCT PREMIUM MRI brainMRI PREMIUM MRI axial brainMRI FREE MRA brainMRI PREMIUM BrainIllustrations PREMIUM Cranial nervesIllustrations PREMIUM Arteriography brainAngiography PREMIUM MR cerebral venographyMRI PREMIUM MRI inner ear and IACMRI PREMIUM Cranial nerves MRIMRI PREMIUM EYE EyeIllustrations FREE Orbit MRIMRI PREMIUM MRI brainMRI PREMIUM CT head and neckCT PREMIUM CT faceCT PREMIUM Cranial nervesIllustrations PREMIUM EAR CT temporal boneCT PREMIUM MRI inner ear and IACMRI PREMIUM EarIllustrations PREMIUM Temporomandibular jointMRI FREE CT head and neckCT PREMIUM MOUTH AND NOSE CT head and neckCT PREMIUM CT faceCT PREMIUM Dental CBCTCT PREMIUM TeethIllustrations FREE Oral cavityIllustrations FREE Nasal cavityIllustrations FREE Nasal fibroscopyEndoscopy FREE NECK CT head and neckCT PREMIUM MRI head and neckMRI PREMIUM Carotid artery - 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Surgical approachPhotography FREE CT body (lymph nodes)CT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM CHEST CT chestCT PREMIUM CT body (lymph nodes)CT PREMIUM CT axial chestCT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Radiography chest abdomen pelvisRadiographs FREE LungsIllustrations PREMIUM Thoracic wall-BreastIllustrations PREMIUM BronchoscopyEndoscopy FREE MEDIASTINUM-HEART CT chestCT PREMIUM CT axial chestCT PREMIUM CTA coronary arteriesCT PREMIUM MediastinumIllustrations PREMIUM HeartIllustrations FREE CoronarographyAngiography PREMIUM CT body (lymph nodes)CT PREMIUM FDG-PET/CT whole bodyPET-CT PREMIUM Radiography chest abdomen pelvisRadiographs FREE ‹› Spotted a mistake? 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https://chem.libretexts.org/Courses/Thompson_Rivers_University/PASS_Chemistry_Book_CHEM_1500/02%3A_Quantum_Theory_and_Electronic_Structure/2.04%3A_Question_2.E.58_PASS_-_electron_configuration_orbital_diagrams_Aufbau's_Principle_Hund's_Rule
2.4: Question 2.E.58 PASS - electron configuration, orbital diagrams, Aufbau's Principle Hund's Rule - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 2: Quantum Theory and Electronic Structure PASS Chemistry Book CHEM 1500 { } { "2.01:Question_2.E.06_PASS-Energy_wavelength_frequency_and_colour_of_emitted_lithium_photons" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "2.02:_Question_2.E.26_PASS-Bohr_Model_quantized_energy_change" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "2.03:_Question_2.E.40_PASS-quantum_numbers_orbital_characteristics" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "2.04:_Question_2.E.58_PASS-_electron_configuration_orbital_diagrams_Aufbau\'s_Principle_Hund\'s_Rule" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } { "00:Front_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "01:_Introduction" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "02:_Quantum_Theory_and_Electronic_Structure" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "03:_Periodic_Relationships_Among_the_Elements" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "04:_Chemical_Bonding_I-Basic_Concepts" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "05:_Chemical_Bonding_II-Molecular_Geometry_and_Hybridization_of_Atomic_Orbitals" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "06:_Intermolecular_Forces_and_Liquids_and_Solids" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "07:_Organic_Chemistry_I-Bonding_and_Structure" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "08:_Organic_Chemistry_II-Stereochemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "09:_Organic_Chemistry_III-_Conformational_Analysis" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "zz:_Back_Matter" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Thu, 31 Aug 2023 18:14:12 GMT 2.4: Question 2.E.58 PASS - electron configuration, orbital diagrams, Aufbau's Principle Hund's Rule 452270 452270 Joshua Halpern { } Anonymous Anonymous User 2 false false [ "article:topic", "license:ccbyncsa", "licenseversion:40", "librenet:thompson-rivers-university" ] [ "article:topic", "license:ccbyncsa", "licenseversion:40", "librenet:thompson-rivers-university" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Campus Bookshelves 3. Thompson Rivers University 4. PASS Chemistry Book CHEM 1500 5. 2: Quantum Theory and Electronic Structure 6. 2.4: Question 2.E.58 PASS - electron configuration, orbital diagrams, Aufbau's Principle Hund's Rule Expand/collapse global location PASS Chemistry Book CHEM 1500 Front Matter 1: Introduction 2: Quantum Theory and Electronic Structure 3: Periodic Relationships Among the Elements 4: Chemical Bonding I - Basic Concepts 5: Chemical Bonding II - Molecular Geometry and Hybridization of Atomic Orbitals 6: Intermolecular Forces and Liquids and Solids 7: Organic Chemistry I - Bonding and Structure 8: Organic Chemistry II - Stereochemistry 9: Organic Chemistry III - Conformational Analysis Back Matter 2.4: Question 2.E.58 PASS - electron configuration, orbital diagrams, Aufbau's Principle Hund's Rule Last updated Aug 31, 2023 Save as PDF 2.3: Question 2.E.40 PASS - quantum numbers, orbital characteristics 3: Periodic Relationships Among the Elements Page ID 452270 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Exercise 2.4.2.⁢E⁢.58) Exercise 2.4.2.⁢E⁢.58 Consider magnesium (Mg), silicon (Si), and sulfur (S). Which of the following atoms has two unpaired electrons? a) Mg b) Si c) S d) Both Mg and S e) Both Si and S Answer e) Both Si and S See LibreText 2.6 Electronic Structure of Atoms Strategy Map | Step | Hint | --- | | 1. Find your element on the Periodic Table and note its atomic number | The atomic number (Z) describes the number of electrons the atom has Recall (LibreText section 3.1): The Periodic Table | | 2. Create an orbital box diagram for each elements electrons | Orbital diagrams are pictorial representations of the electron configuration, showing the individual orbitals as boxes and the pairing arrangement of electrons as arrows Begin to fill your box orbital diagram starting with the lowest energy '1s' orbital and stop when you “run out” of electrons Follow Hund's Rule and Aufbau Principle | | 3. Count the number of unpaired electrons | Any orbitals containing one single electron are described as unpaired electrons | Solution Magnesium (Mg): 12 electrons Mg has no unpaired electrons Silicon (Si): 14 electrons Silicon has two unpaired electrons Sulfur (S): 16 electrons Sulfur has 2 unpaired electrons Answer e) Both Si and S Guided Solution | Guided Solution | Hint | --- | | This problem relies on concepts from Electronic Structure of Atoms (Electron Configurations) | Refer to LibreText 2.6, Electronic Structure of Atoms (Electron Configurations) | | What does the question ask you to find? Question Consider magnesium (Mg), silicon (Si), and sulfur (S). Which of the following atoms has two unpaired electrons? Consider the circumstances in which electrons are paired or unpaired | Aufbau Principle: Fill electrons into the lowest available energy orbital Paired electrons: If the lowest energy orbital available already has one electron, you will add a second electron to it with an opposite spin represented by an arrow pointed in the opposite direction Hund's Rule: when degenerate (same energy) orbitals are available, maximize the number of unpaired electrons with the same spin Unpaired electrons: pay attention when filling the first, second, and third electrons into degenerate p-orbitals | | How many electrons does each element have? Mg: 12 electrons Si: 14 electrons S: 16 electrons | Find the atomic number (Z) on the Periodic Table to determine the number of electrons for each element | | Complete a Box Orbital diagram to determine the number of unpaired electrons for each element | Common mistake: having an even number of electrons does not always mean that they are all paired! The orbitals in the box diagram are arranged from left to right in order of increasing energy See LibreText Figure 2.6.2 - The Aufbau Principle | | Fill the appropriate number of electrons into each box diagram | Add electrons from left to right maximize the number of electrons in each lowest energy available orbital before moving on each orbital can hold a maximum of two electrons the two electrons must have opposite spin (pointed up and down) if degenerate empty orbitals are available, fill those first with unpaired electrons with the same spin before pairing electrons See LibreText Example 2.6.1: Quantum Numbers and Electron Configurations | | Complete Solution: Mg Magnesium, atomic number 12, has a total of twelve electrons. The diagram is filled starting with a pair of electrons in the 1s orbital box, followed by 2s, the 2p then 3s. All electrons are paired in magnesium. Si Silicon, atomic number 14, has a total of fourteen electrons. The diagram is filled starting with the 1s orbital box, followed by 2s, 2p, 3s then 3p. The 3p orbital has 2/6 electrons, meaning silicon has two unpaired electrons. S Sulfur, atomic number 16, has a total of sixteen electrons. The diagram is filled starting with the 1s orbital box, followed by 2s, 2p, 3s then 3p. The 3p orbital has 4/6 electrons, meaning sulfur has two unpaired electrons. | When filling degenerate (same enegy) orbitals, Hund's Rule must be obeyed to maintain the most stable (lowest energy) electron configuration. Hund's Rule is shown below: | | Check your work! When completing the box orbital diagram, count the number of arrows; be sure it matches the number of electrons determined from the atomic number! | | (question source adapted question 6.4.14 from 6.E: Electronic Structure and Periodic Properties (Exercises): shared under a CC BY 4.0 license, authored, remixed, and/or curated by OpenStax, original source question 59 Access for free at 2.4: Question 2.E.58 PASS - electron configuration, orbital diagrams, Aufbau's Principle Hund's Rule is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. Back to top 2.3: Question 2.E.40 PASS - quantum numbers, orbital characteristics 3: Periodic Relationships Among the Elements Was this article helpful? Yes No Recommended articles 2: Quantum Theory and Electronic Structure Article typeSection or PageLicenseCC BY-NC-SALicense Version4.0 Tags librenet:thompson-rivers-university © Copyright 2025 Chemistry LibreTexts Powered by CXone Expert ® ? 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https://math.stackexchange.com/questions/2187644/combining-functions-through-domain-and-range
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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Combining Functions through domain and range. Ask Question Asked 8 years, 6 months ago Modified1 year ago Viewed 1k times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. I have searched everywhere on ways to do these Combining Functions problem so here goes. Everywhere I look, this is only explained through actual functions. Ive yet to see one done with only the domain and range of a function. Let f,g,h be functions with domain and ranges below: f has domain [-1,1) and range [0,2) g has domain [0,2) and range [-1,1) h has domain [1,3) and range [1,2) For each of the following proposed new functions, specify its domain if it exists, otherwise state that the function does not exist. (f+g), (f+h), (g o h), (h o g) Any advice would be a godsend. I already have the answers, just looking for explanations for them. functions Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Mar 15, 2017 at 11:12 user421546user421546 21 3 3 bronze badges 3 1 Well, take f+g f+g, say. What might that mean? Well, for some x x we'd hope to write (f+g)(x)=f(x)+g(x)(f+g)(x)=f(x)+g(x) but for that to make sense we'd need to have both f(x)f(x) and g(x)g(x) defined. Is that possible? Yes! the intersections in the domains of f,g f,g is [0,1)[0,1). So f+g f+g has domain [0,1)[0,1). I don't believe it is possible to specify the range without further information. We know it is contained in [−1,3)[−1,3) but we don't know exactly what it is. the others are similar.lulu –lulu 2017-03-15 11:32:43 +00:00 Commented Mar 15, 2017 at 11:32 1 Should say: the definition of "range" isn't universally agreed on. I believe most people say it means the set of values taken by function. Others say it is simply a set that the function is said to map to, with no assumption that it hits every value. Do you know which definition you are using?lulu –lulu 2017-03-15 11:34:50 +00:00 Commented Mar 15, 2017 at 11:34 1 The Wiki article on Range gives a useful discussion of the ambiguity between the two possible definitions.lulu –lulu 2017-03-15 11:42:03 +00:00 Commented Mar 15, 2017 at 11:42 Add a comment| 1 Answer 1 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. (f + g) has domain [0,1) and range [-1,3). (f + h) has domain [-1,3) and range [1,4). (g o h) has domain [1,3) and range [-1,1). (h o g) isn't well defined.(It may or may not exist.) Note that domain of a function is the set of all values that the function can accept. Asking for the value of a function outside its domain is absurd. The range is the super-set of the outputs for the function. In the first two cases, for the sum of functions to be defined, I have taken the intersection of the domains of the two functions (for points outside this intersection, at-least one function is going to be not defined). I could have set the domain of the function to R R, but I chose the set such that it is impossible for the function to attain values outside this set. In the third and fourth problems, for (f o g) to be well defined on the domain of g, the range of g must be a subset of the domain of f. We can also restrict the domain to only include points such that g(x) ∈∈ domain(f). Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Mar 15, 2017 at 12:30 Udayan JoshiUdayan Joshi 93 4 4 bronze badges Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions functions See similar questions with these tags. 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14021
https://www.orpha.net/en/disease/detail/99927
Share Share Knowledge on rare diseases and orphan drugs COVID-19 & Rare diseases Rare Diseases Resources for Refugees/Displaced Persons Homepage > Rare diseases > Search Search for a rare disease Hydatidiform mole Suggest an update Your message has been sent Your message has not been sent. Please contact an administrator. Comment Form X Disease definition A rare, benign gestational trophoblastic disease that develops during pregnancy and is characterized by the abnormal fertilization, trophoblastic proliferation, and abnormal or absent embryo development. Hydatidiform moles can be either complete or partial. ORPHA:99927 Classification level: Disorder Synonym(s): Molar pregnancy Prevalence: Unknown Inheritance: Autosomal recessive, Not applicable Age of onset: Adolescent, Adult ICD-10: O01.0 , O01.1 , O01.9 ICD-11: JA02 OMIM: 231090 614293 UMLS: C0020217 MeSH: D006828 GARD: 10263 MedDRA: 10020481 Summary Epidemiology In Europe, the condition occurs in approximately 1/1,000 pregnancies. Clinical description Complete moles are asymptomatic in 40% of cases. Most often, a mole is detected upon suspicion of miscarriage in the first trimester, with bleeding and pelvic pain. The clinical signs in the second trimester (vomiting, metrorrhagia, abnormal increase in the size of the uterus, and more rarely anemia or preeclampsia) are observed less often, due to early detection by ultrasound examination. Hyperthyroidism is exceptional. The clinical signs of a partial mole (metrorrhagia, vomiting, etc.) are rare. A mole is usually detected histologically on analysis of aspiration samples from a suspected miscarriage. Etiology The moles are caused by abnormal fertilization with an excess of paternal chromosome material. Complete moles result from fertilization of an enucleated ovocyte by one or two haploid spermatozoa. The karyotype is 46,XX (75% of cases) or 46,XY (25%). The mole is characterized by trophoblastic hyperplasia associated with generalized degeneration of chorionic villi and absence of an amniotic cavity and embryonal tissue. Partial moles result from fertilization of a normal ovocyte by two spermatozoa or one abnormal spermatozoon. This type of mole is characterized by focal trophoblastic hyperplasia, localized degeneration of chorionic villi and identifiable embryonal tissue. The karyotype is triploid in 99% of cases. Diagnostic methods Ultrasound of a complete mole may show a classic ''snow storm'' appearance (solid, hyperechoic areas of varying forms interspersed with liquid areas of various sizes) occupying the entire uterine cavity. Earlier ultrasound before 9-10 weeks of pregnancy can show a limited vesicular appearance of the placenta. Ultrasound of a partial mole may sometimes show focal vesicular change. Embryonic structures without an increase in uterine size are commonly found. Diagnosis is based on histological examination of the product of fertilization. Expert pathology review is often useful. When hydatidiform mole is suspected, determination of total chorionic gonadotropin (hCG) must be performed. Differential diagnosis Moles may be, but should not be, confused with gestational trophoblastic neoplasms or with prolonged retention of a ''classic'' spontaneous miscarriage. Genetic counseling Aside from very rare cases of recurrent moles in the same patient or in the same family (1% of cases, in which a mutation in the NLRP7, KHDC3L, MEI1 or C11orf80 genes have sometimes been found), genetic counseling is not required. Management and treatment Treatment of moles consists of suction evacuation of the uterine contents. Ultrasound guidance may be useful. Evacuation must be scheduled rapidly due to the risk of complications which increases with gestational age. Due to possible progression to gestational trophoblastic neoplasia (GTN), plasma hCG should be monitored until the levels normalize. Diagnosis of GTN warrants disease staging and appropriate chemotherapy. Prognosis After removal, the prognosis is excellent. The main risk is retention by incomplete aspiration (up 25% of cases) which may justify ultrasound follow-up in the weeks following aspiration. Retention (ultrasound image of more than 17 mm in anteroposterior diameter) may require a repeated aspiration in certain clinical circumstances. In about 15 % of cases of complete moles and in 0.5 to 5% of partial moles, the condition leads to gestational trophoblastic neoplasia in the weeks and months following mole evacuation. Last update: June 2022 - Expert reviewer(s): Dr John COULTER A summary on this disease is available in Français, Español, Deutsch, Italiano, Português, Nederlands, Polski, Ελληνικά, Detailed information Guidelines Clinical practice guidelines Français (2010) - INCa Deutsch (2022) - AWMF English (2020) - Eur J Cancer : produced/endorsed by ERN(s) : produced/endorsed by FSMR(s) Additional information Further information on this disease Classification(s) (2) Gene(s) (4) Disability Clinical Signs and Symptoms Patient-centred resources for this disease Expert centre(s) (104) Networks of expert centre (6) Diagnostic tests (13) Patient organisation(s) (113) Federation/alliance(s) (45) Research activities on this disease Research project(s) (41) Clinical trial(s) (3) Biobank(s) (7) Registry(ies) (48) Network of experts (9) Newborn screening Newborn screening library The documents contained in this website are presented for information purposes only. The material is in no way intended to replace professional medical care by a qualified specialist and should not be used as a basis for diagnosis or treatment.
14022
https://www.getlabtest.com/news/post/marasmus-vs-kwashiorkor-differences
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Without completing the health questionnaire, the doctor won't be able to describe your test results Fill in the dietary questionnaire to receive a diet plan from a nutritionist Back to all articles Home News Diseases & Symptoms Understanding Marasmus vs Kwashiorkor: Key Differences in Childhood Malnutrition Understanding Marasmus vs Kwashiorkor: Key Differences in Childhood Malnutrition Illustration comparing marasmus and kwashiorkor symptoms in children with muscle wasting, edema, and skin changes. Explore marasmus vs kwashiorkor differences in symptoms and treatment strategies for childhood malnutrition. Severe malnutrition in children can manifest in two distinct forms: marasmus and kwashiorkor. These conditions represent different ends of the protein-energy malnutrition spectrum, each with unique characteristics and treatment approaches. Understanding these differences is crucial for healthcare providers and caregivers to ensure proper diagnosis and treatment. While both conditions stem from inadequate nutrition, their presentation, underlying causes, and management strategies differ significantly. This comprehensive guide explores the key distinctions between marasmus and kwashiorkor, their impacts on child health, and essential prevention strategies. Clinical Characteristics and Physical Presentation Marasmus and kwashiorkor present with distinctly different physical characteristics, making them relatively easy to distinguish clinically: Marasmus Features Children with marasmus typically show: Severe muscle wasting Minimal to no subcutaneous fat Visible rib cage and bones Loose, wrinkled skin Alert and irritable behavior Severe growth stunting Kwashiorkor Features Children with kwashiorkor typically present with: Bilateral pitting edema Distended abdomen Changes in hair color and texture Skin lesions and discoloration Apathetic behavior Moon-like facial features Nutritional Deficiency Patterns The underlying nutritional deficiencies in these conditions differ significantly, leading to their distinct presentations: Marasmus Nutritional Profile Marasmus results from overall caloric deficiency, including: Insufficient total energy intake Inadequate protein consumption Reduced carbohydrate intake Limited fat consumption Kwashiorkor Nutritional Profile Kwashiorkor primarily stems from protein deficiency, characterized by: Adequate or near-adequate caloric intake Severe protein deficiency Diet high in carbohydrates Insufficient essential amino acids Diagnosis and Treatment Approaches The treatment strategies for these conditions require different approaches, though both necessitate careful medical supervision: Treating Marasmus Treatment focuses on gradual refeeding with: Carefully calculated caloric increases Balanced macro and micronutrient supplementation Regular monitoring of weight gain Prevention of refeeding syndrome Treating Kwashiorkor Management requires special attention to: Protein repletion Management of edema Correction of electrolyte imbalances Treatment of concurrent infections Prevention Strategies Preventing these severe forms of malnutrition requires comprehensive approaches: Regular growth monitoring Adequate maternal nutrition during pregnancy Proper breastfeeding practices Appropriate complementary feeding Access to diverse, nutrient-rich foods Community education and support Frequently Asked Questions What are the main differences between marasmus and kwashiorkor in terms of symptoms and causes? Marasmus results from overall caloric deficiency, presenting with severe wasting and minimal body fat. Kwashiorkor primarily stems from protein deficiency while maintaining adequate calories, characterized by edema, distended abdomen, and skin changes. How do you diagnose and treat marasmus versus kwashiorkor? Diagnosis is primarily clinical, based on physical appearance and symptoms. Treatment for marasmus focuses on gradual caloric repletion, while kwashiorkor treatment emphasizes protein restoration and edema management. What are the key nutritional deficiencies that lead to marasmus and kwashiorkor? Marasmus develops from overall insufficient caloric intake affecting all macronutrients. Kwashiorkor results primarily from severe protein deficiency despite potentially adequate caloric intake from carbohydrates. What can parents do to prevent marasmus and kwashiorkor in their children? Parents should ensure proper breastfeeding, appropriate complementary feeding, regular growth monitoring, and access to diverse, nutrient-rich foods. Educational support and regular healthcare visits are also crucial. How do kwashiorkor and marasmus affect a child's growth and development long-term? Both conditions can lead to stunted growth, delayed development, cognitive impairment, and increased susceptibility to infections. Early intervention is crucial to minimize long-term impacts on physical and cognitive development. 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https://math.stackexchange.com/questions/4057243/help-understanding-two-theorems-on-primes-and-residue-systems
discrete mathematics - Help understanding two theorems on primes and residue systems - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. 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Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Help understanding two theorems on primes and residue systems Ask Question Asked 4 years, 6 months ago Modified4 years, 6 months ago Viewed 89 times This question shows research effort; it is useful and clear -1 Save this question. Show activity on this post. My textbook lists two theorems and I'm not sure how I'm supposed to interpret them. I don't need a proof; I'm only trying to figure out what information I'm being told by each theorem. Let p p be a prime and let a a be an integer not divisible by p p; that is, g c d(a,p)=1 g c d(a,p)=1. Then {a,2 a,3 a,...,p a}{a,2 a,3 a,...,p a} is a complete residue system modulo p p So for this first theorem, I believe the canonical complete residue system p p would be {0,1,2,...,p−1}{0,1,2,...,p−1}. So the set {a,2 a,3 a,...,p a}{a,2 a,3 a,...,p a} would be the integers that satifsy the congruences, a≡0(m o d p),2 a≡1(m o d p),3 a≡2(m o d p),...,p a≡(p−1)(m o d p)a≡0(m o d p),2 a≡1(m o d p),3 a≡2(m o d p),...,p a≡(p−1)(m o d p). Is my interpretation correct? Let p p be a prime and let a a be an integer not divisible by p p. Then, a⋅2 a⋅3 a⋅...⋅(p−1)a≡1⋅2⋅3⋅...⋅(p−1)(m o d n)a⋅2 a⋅3 a⋅...⋅(p−1)a≡1⋅2⋅3⋅...⋅(p−1)(m o d n) For this second theorem, is this theorem telling me that I can decompose congruence integers into products of primes? Thanks for any help. discrete-mathematics modular-arithmetic Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Follow Follow this question to receive notifications asked Mar 11, 2021 at 3:09 Bobby BBobby B 417 2 2 silver badges 16 16 bronze badges 2 1 "So the set {a,2a,3a,...,pa} would be the integers that satifsy the congruences, a≡0 (mod p),2a≡1 (mod p),3a≡2 (mod p),...,pa≡(p−1) (mod p)." They don't have to be (and actually can't) be in order. The theorem is saying a≡k 1(mod p)a≡k 1(mod p) for some k 1∈{0,1,2....,p−1}k 1∈{0,1,2....,p−1} (but not 0 0) and 2 a≡k 2(mod p)2 a≡k 2(mod p) for some k 2∈{0,1,2,...,p−1}k 2∈{0,1,2,...,p−1} but k 2≠k 1 k 2≠k 1 (and if a≢1 a≢1 then 2 a≢2 2 a≢2... but that's not stated in the theorem). For example if p=5 p=5 and a=7 a=7 then {7,14,21,28,35}≡{2,4,1,3,0}{7,14,21,28,35}≡{2,4,1,3,0} respectively.fleablood –fleablood 2021-03-13 16:36:44 +00:00 Commented Mar 13, 2021 at 16:36 1 Second Theorem is just saying that as {a,2 a,3 a,.....,(p−1)a}{a,2 a,3 a,.....,(p−1)a} is a residue system with every element bot the 0 0 one. (a p≡a×0≡0 a p≡a×0≡0 of course) then {a,2 a,3 a,....,(p−1)a}{a,2 a,3 a,....,(p−1)a} is equivalent to {1,2,3,....,p−1}{1,2,3,....,p−1} (but not in order!) then product of one set will be congruent to the product of the other set. This is a trivial result if think of it that way, but it means a p−1(p−1)!≡(p−1)!(mod p)a p−1(p−1)!≡(p−1)!(mod p) which implies (if we could do division, which we can't; or if we know (p−1)!≡1(mod p)(p−1)!≡1(mod p) which we don't know...yet) then a p−1≡1(mod p)a p−1≡1(mod p). Which is FLT.(almost)fleablood –fleablood 2021-03-13 16:48:08 +00:00 Commented Mar 13, 2021 at 16:48 Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 1 Save this answer. Show activity on this post. Okay, what does it mean. A complete residue system mod p mod p. What's that? If you compare any integer n(mod p)n(mod p) that are only p p options. Either n≡0(mod p)n≡0(mod p) or n≡1(mod p)n≡1(mod p) or n≡2(mod p)n≡2(mod p) or ...... or n≡p−1(mod p)n≡p−1(mod p). So the list {0,1,2,.....,p−1}{0,1,2,.....,p−1} is called a complete residue system because it contains a representation, a residue, for every case. But that there nothing special about choosing just those (the smallest non-negative) residues. We could have chosen any {r 1,r 2,.....,r p}{r 1,r 2,.....,r p} so that every integer is equivalent to one of them (and we can list them in any order). For example to find a complete residue system mod 6 mod 6 we can represent all the integers that have remainder of 0 0 by 78 78. (Because 6|78 6|78). And we can represent all the integers that have a remainder of 2 2 with −22−22 (because −22=−4∗6+2−22=−4∗6+2). And we can represent all the integers that have remainder of 5 5 with 647 647 (because 647=642+5=6∗107+5 647=642+5=6∗107+5) and so on. So {78,−22,647,3001,−2,10}{78,−22,647,3001,−2,10} is a complete residue system (mod 6)(mod 6) because every interger is congruent to exact one of those (mod 6)(mod 6). (test them: 0≡78(mod 6);1≡3001(mod 6);2≡−22(mod 6);3≡−2(mod 6);4≡10(mod 6);5≡647(mod 6);6≡78(mod 6);7≡3001(mod 6)0≡78(mod 6);1≡3001(mod 6);2≡−22(mod 6);3≡−2(mod 6);4≡10(mod 6);5≡647(mod 6);6≡78(mod 6);7≡3001(mod 6) and so on. [they just cycle and cycle through]. (Admittedly, it's a fairly irrelevant and awkward way to represent them... but it does represent all 6 6 possible congruences.) So the theorem is saying: If p p is a prime and a a is a number that .... isn't a multiple of p p... then the set {a,2 a,3 a,....,p a}{a,2 a,3 a,....,p a} is a complete residue system mod p mod p and that every integer is congruent to exactly one of those number. For example: mod 7 mod 7 and the number 3 3 we'd have {3,6,9,12,15,18,21}{3,6,9,12,15,18,21} is a complete residue system. And if we test it. 0≡21(mod 7);1≡15(mod 7);2≡9(mod 7);3≡3(mod 7);4≡18(mod 7);5≡12(mod 7);6≡6(mod 7);7≡21(mod 7);8≡15(mod 7)0≡21(mod 7);1≡15(mod 7);2≡9(mod 7);3≡3(mod 7);4≡18(mod 7);5≡12(mod 7);6≡6(mod 7);7≡21(mod 7);8≡15(mod 7) and so on.... We could to it with any number (as long as it's not a multiple of 7 7). For example...... 93 93.... {93,186,279,372,465,558,651}{93,186,279,372,465,558,651} should work. Does it? I'm sure it does because I have complete faith in the theorem but looking at that I'd have no idea if it weren't for the theorem but: 651=7∗93 651=7∗93 that was a gimme; 372=53∗7+1 372=53∗7+1 and 93=7∗13+2 93=7∗13+2 and 465=7∗67+3 465=7∗67+3 and 186=7∗26+4 186=7∗26+4 and 558=7∗79+5 558=7∗79+5 and 279=7∗39+6 279=7∗39+6 and so on. It does work. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Mar 12, 2021 at 19:23 answered Mar 12, 2021 at 19:12 fleabloodfleablood 132k 5 5 gold badges 52 52 silver badges 142 142 bronze badges 3 I know it's a heavy-handed approach, but writing out all the congruences modulo p p for your examples really helped me see what pattern was actually occurring.Bobby B –Bobby B 2021-03-13 01:11:03 +00:00 Commented Mar 13, 2021 at 1:11 @BobbyB You are supposed to have already had practice with complete residue systems before tackling these problems. My guess in comments on my answer appears appears to be correct - you are putting the cart before the horse. It is absolutely crucial to be sure you understand prior topics before moving forward to topics depending on such. Mathematical exposition exploits these dependencies to the hilt.Bill Dubuque –Bill Dubuque 2021-03-13 02:38:23 +00:00 Commented Mar 13, 2021 at 2:38 My issue was not understanding complete residue systems. I've asked questions already on the basics of complete residue systems and received some great answers. In fact, my understanding of the first theorem was nearly correct. Fleablood's answer simply filled in the pieces I was missing with his well detailed examples. The language of modular arithmetic can be very confusing for beginners as to what literal arithmetic is actually occurring. Formal definitions need to be tied together with simple and direct examples, not a flurry of abstract terms. It's hard to find those examples.Bobby B –Bobby B 2021-03-13 02:52:41 +00:00 Commented Mar 13, 2021 at 2:52 Add a comment| This answer is useful 0 Save this answer. Show activity on this post. No, you have misunderstood both. Let's consider a simple special case: a=2,p=5.a=2,p=5. i.e. we consider the doubling map n→2 n mod 5 n→2 n mod 5 whose action is as follows. mod 5:≡2{1,2,3,4}{2,4,1,3}mod 5:2{1,2,3,4}≡{2,4,1,3} i.e. 2⋅1 2⋅2 2⋅3 2⋅4≡2≡4≡1≡3 2⋅1≡2 2⋅2≡4 2⋅3≡1 2⋅4≡3 Multiplying them all yields product of all on LHS is congruent to product of all om RHS, i.e. ≡≡(2⋅1)(2⋅2)(2⋅3)(2⋅4)(2)⋅(4)⋅(1)⋅(3)4!≡−1(2⋅1)(2⋅2)(2⋅3)(2⋅4)≡(2)⋅(4)⋅(1)⋅(3)≡4!≡−1 by the Congruence Product Rule. That's what the theorem is telling you (and how it is proved). Remark The key idea is: gcd(a,p)=1⇒gcd(a,p)=1⇒ the map x→a x x→a x is invertible mod p mod p so it is a bijection, so it simply permutes the complete set of nonzero residues 1,2,…,p−1,1,2,…,p−1, which does not alter their product (p−1)!,(p−1)!, e.g. above the doubling map permuted 1,2,3,4 1,2,3,4 to 2,3,1,3 2,3,1,3. The multuiplication is a congruence generalization of the fact that equal finite sets of integers have equal products, by replacing "equal" by "congruent" (congruence can be viewed as a generalized equality - in fact it becomes an equality if we work in the quotient ring Z p Z p of cosets [k]=k+p Z)[k]=k+p Z) For another common example of this idea see the Wilson reflection formula where the permutation arises from an additive shift by a a (vs. multiplicative scaling by a)a). Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Follow Follow this answer to receive notifications edited Mar 12, 2021 at 17:03 answered Mar 11, 2021 at 8:14 Bill DubuqueBill Dubuque 284k 42 42 gold badges 339 339 silver badges 1k 1k bronze badges 4 This explanation is beyond my current level of number theory. Invertibility, bijection, and permutation has not been discussed yet in my textbook. I'm at the very basic introduction to Fermat's Little Theorem and it's proof whereby these theorems are simply meant to help with that introduction.Bobby B –Bobby B 2021-03-12 01:53:36 +00:00 Commented Mar 12, 2021 at 1:53 @BobbyB Then simply ignore that paragraph - it is not needed in order to understand the example. Ignore the downvote too - it has nothing to do with mathematiucs Bill Dubuque –Bill Dubuque 2021-03-12 02:37:55 +00:00 Commented Mar 12, 2021 at 2:37 No, the down vote was warranted because your answer is not useful for beginners in Number Theory. You did not actually explain any of your ideas, you only stated other terms with the incorrect assumption that one would immediately know what you're talking about.Bobby B –Bobby B 2021-03-12 15:57:19 +00:00 Commented Mar 12, 2021 at 15:57 @BobbyB Not true, the main part of the answer is about as simple as one can get. If you can't follow that then you need to go back and review your notes / textbook. It seems as if perhaps you are trying to learn to run before you have learned to walk (esp. considering the major misunderstandings in your question). If you explain precisely what you cannot follow in the first paragraph then I will be happy to elaborate. I have no idea what that might be. fyi: many downvotes here are, alas, politically based and have nothing to do with the answer.Bill Dubuque –Bill Dubuque 2021-03-12 16:41:38 +00:00 Commented Mar 12, 2021 at 16:41 Add a comment| You must log in to answer this question. 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14024
https://www.digipac.ca/chemical/mtom/contents/chapter3/cdhint1.htm
## The Chromate - Dichromate Equilibrium What happens if we add some hydrochloric acid? | | | For simplicity, the symbol H+ is being used for the acid proton, rather than the Brönsted-Lowry hydronium ion H3O+. All ions in these equations are aqueous, but the (aq) symbol has been omitted for brevity. | There are always a few H+ ions in water. Hydrochloric acid is a concentrated source of H+ ions. If there was no chemical reaction then adding HCl should increase the amount of H+. The original H+ in the water would increase to become H+ However, we wouldn't see any color change since H+ is colorless. However, there is going to be a chemical reaction, since CrO42- reacts with H+. From le Chtelier's principle, we know the reaction will try to remove some of the H+ we have added. Some of the added H+ reacts with the CrO42-. This removes some of both the H+ and CrO42-, and make more Cr2O72-. For the chromate solution (remember that the original solution is an equilibrium, so it contains mostly yellow chromate, but a bit of orange dichromate): | | | --- | | add some H+ (aq) and it becomes | a much more orange solution | | | | | --- | | Original condition | 2 CrO42- + 2 H+ Cr2O72- + H2O (l) | | Immediately after some H+ is added (since reactions take time, we can imagine what the system is like before it has time to react) | 2 CrO42- + 2 H+ Cr2O72- + H2O(l) | | New equilibrium is established, removing some of the added H+ | 2 CrO42- + 2 H+ Cr2O72- + H2O(l) | Notice, that the H+(aq) is less than it would be if there was no reaction, but greater than it was at the beginning. There has been a big increase in the amount of orange dichromate. | For the dichromate solution (remember that the original solution is also an equilibrium, so it contains mostly orange dichromate, but a bit of yellow chromate): | | | --- | | add some H+ (aq) and it becomes | a slightly more orange solution, and probably not noticeable | | | | | --- | | Original condition | 2 CrO42- + 2 H+ Cr2O72- + H2O (l) | | Immediately after some H+ is added (since reactions take time, we can imagine what the system is like before it has time to react) | 2 CrO42- + 2 H+ Cr2O72- + H2O (l) | | New equilibrium is established, removing some of the added H+ | 2 CrO42- + 2 H+ Cr2O72- + H2O(l) | Notice, that the H+ is less than it would be if there was no reaction, but greater than it was at the beginning. There has been an increase in the dichromate ion concentration, but since there was a lot to begin with, the change isn't very noticeable. |
14025
https://www.frontiersin.org/journals/pharmacology/articles/10.3389/fphar.2024.1452300/full
Your new experience awaits. Try the new design now and help us make it even better ORIGINAL RESEARCH article Front. Pharmacol., 11 September 2024 Sec. Inflammation Pharmacology Volume 15 - 2024 | Safety assessment of sulfasalazine: a pharmacovigilance study based on FAERS database Wangyu Ye†Yuan Ding†Meng LiZhihua TianShaoli WangZhen Liu Guang’anmen Hospital, China Academy of Chinese Medical Sciences, Beijing, China Background: Sulfasalazine is a widely used anti-inflammatory medication for treating autoimmune disorders such as ulcerative colitis (UC), Crohn’s disease, and rheumatoid arthritis. However, its safety profile has not been systematically evaluated in real-world settings. By analyzing the FDA Adverse Event Reporting System (FAERS) database, we identified risk signals associated with adverse reactions to sulfasalazine, offering valuable insights for clinical decision-making and risk management. Methods: Reports of adverse events (AEs) associated with sulfasalazine, covering the period from Q1 2004 to Q4 2023, were extracted from the FAERS database. Detailed case information was aggregated to assess demographic characteristics. The associations between sulfasalazine and adverse events were evaluated using the Proportional Reporting Ratio (PRR), Reporting Odds Ratio (ROR), Bayesian Confidence Propagation Neural Network (BCPNN), and Empirical Bayes Geometric Mean (EBGM). Results: We extracted 7,156 adverse event reports from the FAERS database where sulfasalazine was identified as the “Primary Suspect (PS)” drug. Using disproportionality analysis, we identified 101 preferred terms (PT) related to sulfasalazine across 24 organ systems. Notable adverse reactions consistent with the drug’s labeling were observed, including Stevens-Johnson syndrome, agranulocytosis, eosinophilic pneumonia, and crystalluria. Additionally, novel positive signals not previously documented in the drug label were identified, including acute febrile neutrophilic dermatosis, aseptic meningitis, glomerulonephritis, and hepatosplenic T-cell lymphoma. Conclusion: Most of the adverse reaction findings in this study are consistent with previous clinical research, and we have also identified new potential AEs associated with sulfasalazine. These findings provide valuable insights for the safety monitoring and clinical application of sulfasalazine. 1 Introduction Sulfasalazine is a widely used anti-inflammatory and immunomodulatory medication, composed of sulfapyridine and 5-aminosalicylic acid (5-ASA), linked by an azo bond that imparts distinctive pharmacological properties (Azadkhan et al., 1982). In the intestine, sulfasalazine is poorly absorbed but undergoes enzymatic cleavage by azoreductases from intestinal bacteria, producing the active constituents sulfapyridine and 5-ASA (Das and Dubin, 1976). Sulfapyridine primarily exerts systemic anti-inflammatory effects by suppressing the synthesis of inflammatory mediators, reducing leukocyte infiltration at inflammatory sites, and modulating cytokine secretion (Hoult, 1986). In contrast, 5-ASA offers localized protection to the intestinal mucosa, potentially through mechanisms such as scavenging oxidative free radicals, inhibiting neutrophil infiltration, and preserving mucosal barrier integrity. The combined action of these components endows sulfasalazine with dual systemic and localized therapeutic efficacy (Nissim-Eliraz et al., 2021; Pruzanski et al., 1997; Joshi et al., 2005). In clinical practice, sulfasalazine is widely used to manage rheumatic diseases such as rheumatoid arthritis, ankylosing spondylitis, and inflammatory bowel diseases (IBD) (Chen et al., 2014; Rains et al., 1995). Rigorous randomized controlled trials and case studies have validated sulfasalazine’s efficacy in significantly alleviating symptoms such as joint swelling, pain, and intestinal mucosal damage (Suarez-Almazor et al., 2000; Lim et al., 2016). Recent research has also explored its potential use in other autoimmune conditions, including psoriasis and alopecia areata. However, these applications require further validation through evidence-based studies (Aghaei, 2008; Menter et al., 2009). In summary, due to its unique chemical structure and diverse mechanisms of action, sulfasalazine has become a pivotal therapeutic option for managing chronic inflammation and autoimmune disorders. In recent years, despite the emergence of biologics and targeted therapies that offer more effective treatments for autoimmune diseases (Davis and Ballas, 2017), sulfasalazine remains a mainstay in clinical practice due to its cost-effectiveness and proven efficacy (Lichtenstein et al., 2018). However, the unavoidable adverse reactions associated with sulfasalazine should not be overlooked. The primary cause is its metabolite, sulfapyridine, which frequently leads to toxic reactions characteristic of sulfonamide drugs, such as headaches, nausea, vomiting, and various allergic responses, often manifesting within the first few months of treatment (Rains et al., 1995; Taffet and Das, 1983; Navarro and Hanauer, 2003). Less common adverse effects include hematological abnormalities, hepatic impairment, pulmonary complications, and hypersensitivity reactions (Ransford and Langman, 2002). Long-term use of sulfasalazine may also reduce sperm count and motility, potentially leading to infertility (Bermas, 2020). Even under stringent controls, clinical trials often fail to accurately predict the true risks encountered by patients in real-world clinical settings. Therefore, leveraging data mining from the FAERS is crucial, as it provides a wealth of real-world data on patient medication use and adverse reactions (Alomar et al., 2020). This study aims to analyze adverse reaction reports related to sulfasalazine in the FAERS database, using signal detection methodologies to identify potential adverse drug signals and provide essential insights for drug safety evaluations. 2 Methods 2.1 Data source The FAERS database is used by the United States FDA to monitor adverse events related to drugs and therapeutic products. This database includes adverse event information reported by patients, healthcare professionals, and manufacturers. The primary purpose of FAERS is to identify potential safety signals and assist the FDA in evaluating and monitoring drug safety. A retrospective analysis was conducted using the publicly accessible FAERS database ( to examine adverse reaction reports associated with sulfasalazine. Adverse reaction reports related to sulfasalazine from Q1 2004 to Q4 2023 were extracted from the FAERS database and imported into R Studio 4.3.3 for comprehensive extraction, organization, and analysis. 2.2 Data processing and standardization To ensure data accuracy and reliability, we extracted the most recent report for each case based on the case ID, retaining only the latest report and discarding earlier versions. We standardized drug names in the reports using the Medex_UIMA_1.3.8 system. We mapped both generic and brand names of drugs to a uniform standardized name to eliminate variability. We extracted reports related to sulfasalazine labeled as “Primary Suspect” to create our research dataset. During data processing, we classified adverse events (AEs) using preferred terms (PTs) according to the Medical Dictionary for Regulatory Activities (MedDRA, version 26.1) to ensure consistent terminology. We also categorized adverse events based on the System Organ Class (SOC). We collected information from sulfasalazine-related adverse event reports, including patient age, sex, reporter type, report time, and outcomes (e.g., death, hospitalization, life-threatening events), for descriptive statistical analysis. 2.3 Signal detection and analysis Disproportionality analysis is a data mining technique used to determine if there is an abnormal association between a specific drug and the adverse reactions reported. Disproportionality analysis is a commonly used method to evaluate the association between drugs and AEs. Its core principle involves comparing the observed frequencies of adverse events in exposed versus non-exposed groups using a contingency table, thereby quantifying the association between drugs and AEs (Supplementary Table S1). Our study utilized primary methods including the Proportional Reporting Ratio (PRR), Reporting Odds Ratio (ROR), Bayesian Confidence Propagation Neural Network (BCPNN), and Empirical Bayes Geometric Mean (EBGM) to assess the association between the exposed drug and AEs. A positive signal for adverse reactions is indicated if the lower limit of the 95% confidence interval (CI) for the ROR is greater than one and there are at least three reports of the AE. A significant association is indicated when PRR > 0, the chi-square test value exceeds 4, and there are at least three reports of the adverse event. Higher PRR and ROR values indicate a stronger association between the drug and the AE. BCPNN and EBGM are Bayesian statistical methods that use confidence intervals to evaluate the stability and significance of estimates. BCPNN uses the confidence interval of the Information Component (IC), where an IC025 > 0 indicates a statistically significant association. EBGM05, the lower limit of the 95% confidence interval for EBGM, indicates a significant statistical association between the drug and AEs if EBGM05 > 2. The calculation methods and criteria for positive signals in disproportionality analysis are detailed in Supplementary Table S2. The overall process of this study is illustrated in Figure 1. Figure 1 Figure 1. The flow diagram of selecting sulfasalazine-related AEs from FAERS database. 3 Results 3.1 Basic information on AEs of sulfasalazine From the first quarter of 2004 to the fourth quarter of 2023, we extracted 16, 800, 135 adverse event reports from the FAERS database. Among these reports, we identified 7,156 instances where sulfasalazine was the primary drug associated with adverse reaction events. In the sulfasalazine-related adverse event reports (Supplementary Table S3), the proportion of female patients (68.06%) was significantly higher than that of male patients (26.41%). The incidence of AEs was most common in the 45–65 age group (35.03%). Analyzing the sources of reports, the majority came from physicians (38.39%). Notably, sulfasalazine was most frequently used for the treatment of rheumatoid arthritis (33.49%). Regarding severe adverse outcomes, the clinical outcomes caused by sulfasalazine mainly included other serious - Important Medical Events (51.95%), hospitalization (32.13%), life-threatening (6.06%), disability (4.55%), death (4.00%), congenital anomaly (1.00%), and required intervention to prevent permanent impairment/damage (0.31%). 3.2 Detection of adverse signals for sulfasalazine Our study shows that sulfasalazine-related adverse reaction reports mainly involve 24 SOCs. The results (Supplementary Table S4) indicate that the three most frequently affected systems by sulfasalazine are general disorders and administration site conditions (n = 4,262, ROR 1.08, PRR 1.06, IC 0.09, EBGM 1.06), gastrointestinal disorders (n = 2,387, ROR 1.23, PRR 1.21, IC 0.27, EBGM 1.21), and skin and subcutaneous tissue disorders (n = 2,182, ROR 1.85, PRR 1.77, IC 0.82, EBGM 1.77). Although the EBGM values for the aforementioned three SOC categories are below 2, other signal detection algorithms, such as the ROR, PRR, and IC, offer varying degrees of risk indication. Each of these algorithms has its own strengths in signal detection, providing insights into potential risks from different perspectives. In this study, the results of some SOCs are consistent with the drug label. Notably, congenital, familial and genetic disorders, musculoskeletal and connective tissue disorders, and infections and infestations are not mentioned in the sulfasalazine drug label, indicating a need for further research and verification. In this study, we analyzed the AE signals of sulfasalazine across six SOCs and identified the characteristics of adverse events within each SOC (Figure 2). Using ROR and Chi-square analysis, we found that oculomucocutaneous syndrome exhibited the most significant signal among the skin and subcutaneous tissue disorders. For infectious diseases, necrotising fasciitis streptococcal and mononucleosis syndrome were identified as the primary adverse reactions. In the investigations category, human herpes virus six serology positive, abnormal lymphocyte morphology, and Epstein-Barr virus antibody positive demonstrated strong positive signal associations with sulfasalazine. Additionally, pulmonary eosinophilia and acute interstitial pneumonitis were the most prominent signal features within the respiratory system. In the blood and lymphatic system disorders, the main adverse reactions included anaemia folate deficiency, lymphocytosis, and agranulocytosis. In the renal and urinary system disorders, crystalluria and membranous glomerulonephritis were the most prominent signals. Figure 2 Figure 2. Distribution of AES among six SOCs. (A) Skin and subcutaneous tissue disorders; (B) Infections and infestations; (C) Investigations; (D) Respiratory, thoracic and mediastinal disorders; (E) Blood and lymphatic system disorders; (F) Renal and urinary system disorders. We used four different signal detection algorithms to evaluate the strength of the association signals between the drug and adverse reaction events, ultimately identifying 101 PTs (Supplementary Table S1). Subsequently, based on the results of the ROR algorithm, we ranked these PTs and selected the top 30 PTs with the strongest signal strength (Supplementary Table S5). Among them, the top 10 PTs with the strongest signal strength are anaemia folate deficiency, necrotising fasciitis streptococcal, oculomucocutaneous syndrome, mononucleosis syndrome, teratogenicity, human herpes virus six serology positive, eosinophilic myocarditis, pulmonary eosinophilia, pleuropericarditis, and lymphocyte morphology abnormal. 4 Discussion 4.1 General analysis of ADE reports This study analyzed post-marketing AEs associated with sulfasalazine by mining the FAERS database. As of the fourth quarter of 2023, there were 7,156 AE reports related to sulfasalazine. Clinically, sulfasalazine is primarily used to treat inflammatory diseases such as ankylosing spondylitis, Crohn’s disease, ulcerative colitis, and rheumatoid arthritis, aligning with the current indications of the drug (Plosker and Croom, 2005; Toussirot and Wendling, 2001; Zenlea and Peppercorn, 2014). The patients’ ages ranged from 18 to 65 years, with a majority being female. This higher prevalence of female patients may be attributed to the greater incidence of autoimmune diseases among women (Fish, 2008; Whitacre, 2001). Among the reporting countries, excluding those with unknown locations, the highest proportion of reports came from the United States. This was followed by Canada, Japan, the United Kingdom, and other developed countries. This distribution may be influenced by factors such as the development of the FAERS database, the level of national development, and the awareness of adverse drug reactions among the populations of these countries (Madhushika et al., 2022). 4.2 Skin and subcutaneous tissue disorders Sulfasalazine is an effective anti-inflammatory drug, but it can also cause adverse reactions in the skin and subcutaneous tissue. These adverse reactions can vary widely, ranging from mild rashes and itching to severe systemic exfoliative skin reactions (Atheymen et al., 2013; Iemoli et al., 2006). The results of this study indicate that drug reaction with eosinophilia and systemic symptoms, dermatitis exfoliative, hypersensitivity vasculitis, and Stevens-Johnson syndrome are high-risk signals for sulfasalazine, consistent with common skin adverse reactions recorded in the drug label. Previous studies have reported cases of different types of severe skin reactions, including Stevens-Johnson syndrome, exfoliative dermatitis, and Drug Reaction with Eosinophilia and Systemic Symptoms (Atheymen et al., 2013; Manvi et al., 2022). The potential mechanisms of these severe skin adverse reactions mainly include metabolite-mediated cytotoxic effects and immune-mediated hypersensitivity reactions (Atheymen et al., 2013). Although the incidence of these reactions is relatively low, their serious and potentially fatal consequences require significant attention in clinical use (Tremblay et al., 2011; Teo and Tan, 2006; Girelli et al., 2013). Additionally, acute febrile neutrophilic dermatosis, oculomucocutaneous syndrome, and granulomatous dermatitis are not listed on the sulfasalazine drug label. Sweet’s syndrome is a rare inflammatory skin disease characterized by fever, elevated white blood cell count, red plaques, and neutrophilic infiltration. Clinical case reports have indicated that some patients developed Sweet’s syndrome after taking sulfasalazine, with symptoms including fever, elevated white blood cell count, and red plaques (Romdhane et al., 2016). Although the exact mechanism is not yet fully understood, it may be related to allergic reactions and neutrophil dysfunction induced by sulfasalazine and its metabolites (Romdhane et al., 2016; Yamamoto, 2014; Ayyash and Sampath, 2021). 4.3 Blood system disorders Sulfasalazine can cause varying degrees of damage to the hematopoietic system, resulting in various hematologic abnormalities. The most common reported hematologic adverse reactions include granulocytopenia, folate deficiency anemia, hemolysis, and eosinophilia. Agranulocytosis is a rare but serious adverse reaction associated with sulfasalazine use, as indicated by current research (Jick et al., 1995). Sulfasalazine may inhibit granulocyte production or enhance their peripheral destruction via immune-mediated mechanisms (Dery and Schwinghammer, 1988). Furthermore, studies have identified that agranulocytosis associated with sulfasalazine is primarily linked to the MHC region on chromosome 6, which encodes HLA genes. Specifically, certain HLA alleles, such as HLA-B08:01 and HLA-A31:01, significantly increase the risk of this adverse reaction. This underscores the critical role of genetic factors in disease susceptibility and emphasizes the importance of personalized treatment strategies (Wadelius et al., 2018; Fathallah et al., 2015). Studies indicate that sulfasalazine can interfere with the absorption and metabolism of folate, and this effect is dose-dependent, with higher doses significantly increasing the risk of folate deficiency (Grindulis and McConkey, 1985). Hemolytic anemia is another potential hematologic toxicity associated with sulfasalazine use, particularly in patients with glucose-6-phosphate dehydrogenase (G6PD) deficiency. The deficiency of this enzyme makes red blood cells more susceptible to oxidative damage caused by sulfasalazine (Youngster et al., 2010; Sahm et al., 2021). 4.4 Infections and infestations Sulfasalazine can impair immune system function, reducing the body’s defense against various pathogens. This increases susceptibility to bacteria, viruses, and fungi (Chiu and Chen, 2020; Andrès and Maloisel, 2008; Nasiri-Jahrodi et al., 2023). Sulfasalazine can also impair neutrophil function, leading to decreased levels of these crucial anti-infective cells in peripheral blood. This further increases the risk of infection (Andrès and Maloisel, 2008; Farr et al., 1991). In this study, positive signals were observed for infections such as streptococcal necrotizing fasciitis, human herpesvirus six infection or reactivation, pneumococcal infection, cryptococcal pneumonia, neutropenic sepsis, and latent tuberculosis. Previous studies suggest that sulfasalazine may activate the HHV-6 virus, which can trigger and exacerbate hypersensitivity syndromes in susceptible patients. This may lead to recurrent conditions and multiple drug hypersensitivity (Tan and Chan, 2016; Komura et al., 2005). This highlights the importance of closely monitoring and managing the risk of HHV-6 reactivation during sulfasalazine treatment. We also identified a potential association between sulfasalazine use and aseptic meningitis, which aligns with existing clinical case reports. Aseptic meningitis is characterized by non-infectious inflammation of the meninges. Symptoms typically include severe headache, fever, neck stiffness, nausea, and vomiting, and may also involve altered consciousness or seizures. Although relatively rare in clinical practice, aseptic meningitis warrants attention in patients undergoing sulfasalazine treatment (Yelehe-Okouma et al., 2018; Tay et al., 2012; Salouage et al., 2013). 4.5 Respiratory, thoracic and mediastinal disorders Studies have demonstrated that sulfasalazine has pulmonary toxicity, directly damaging the respiratory system. Clinically, sulfasalazine-induced pulmonary toxicity manifests in various forms, commonly including interstitial lung disease, pneumonia, pulmonary fibrosis, and eosinophilic pneumonia. In this study, AEs associated with sulfasalazine included acute interstitial pneumonitis, eosinophilic pneumonia, alveolitis, pleural fibrosis, and pulmonary granuloma, consistent with previous clinical reports (Moss and Ind, 1991; Gabazza et al., 1992; Averbuch et al., 1985; Moseley et al., 1985). These conditions typically present as progressive dyspnea, cough, fever, and chest pain. In severe cases, they can lead to respiratory failure. Imaging studies, such as chest X-rays, often reveal diffuse alveolar interstitial infiltrates or patchy shadows, indicating abnormal lung patterns (Parry et al., 2002). Interstitial lung disease is a common and severe manifestation of sulfasalazine-related pulmonary toxicity. Pathological features include alveolar septal thickening, inflammatory cell infiltration, and fibrotic tissue formation. The development of interstitial lung disease may be related to the immunomodulatory effects of sulfasalazine, which can induce inflammatory cell infiltration and fibrotic reactions in the lungs, leading to tissue damage (Hamadeh et al., 1992). Sulfasalazine may also induce eosinophil accumulation in lung tissue, leading to eosinophilic pneumonia. In these cases, peripheral blood eosinophil counts are usually elevated, indicating an allergic or immune-mediated response (Nasim et al., 2021; Editorial, 1974). In clinical management, given sulfasalazine’s pulmonary toxicity, physicians should exercise heightened vigilance. If patients develop unexplained respiratory symptoms such as dyspnea, cough, and fever, the possibility of drug-induced toxicity should be considered. 4.6 Renal and urinary disorders Evidence suggests that sulfasalazine may cause severe renal damage. Sulfasalazine’s nephrotoxicity is primarily mediated through oxidative stress, as evidenced by increased serum creatinine and blood urea nitrogen levels, elevated reactive oxygen species, enhanced lipid peroxidation, and glutathione depletion. Sulfasalazine is metabolized into mesalazine and sulfapyridine, both of which may contribute to renal injury. Sulfapyridine, a sulfonamide, is particularly known for its potential adverse effects on the kidneys. Renal damage caused by sulfasalazine is often irreversible, with no specific treatment currently available for sulfasalazine-induced renal injury (Linares et al., 2011; Heidari et al., 2016a). This study reports adverse effects on the renal and urinary systems, primarily including crystalluria, membranous glomerulonephritis, ureteral stones, and glomerulonephritis. Current research primarily focuses on sulfasalazine’s impact on renal function and interstitial damage, with little direct evidence of glomerular injury, indicating a need for further verification. Additionally, studies have shown that sulfasalazine’s metabolite sulfapyridine is excreted in the urine. When the drug highly concentrates in the renal tubules and collecting ducts, it may form crystals. These crystals are more likely to form under urine supersaturation, particularly in conditions of reduced body fluids or low urine pH, leading to tubular obstruction and crystalluria, which may aggravate renal damage and potentially cause anuric renal failure (Perazella and Rosner, 2022; Durando et al., 2017). Therefore, for patients receiving sulfasalazine treatment, particularly those with a history of kidney stones, renal insufficiency, or dehydration, timely renal function monitoring and preventive measures are crucial (Durando et al., 2017). 4.7 Other controversial and new adverse reactions Assessing the teratogenic risks of sulfasalazine during pregnancy is complex and highly debated. Nørgård et al. conducted a study to evaluate the teratogenic risks of sulfasalazine during pregnancy. They used a case-control study design, analyzing data from the Hungarian Congenital Abnormality Registry, which included 22,865 newborns or fetuses with congenital anomalies (case group) and 38,151 without congenital anomalies (control group). The results showed that the incidence of congenital anomalies in pregnant women treated with sulfasalazine was not significantly higher than in those who were not treated. However, the limited data necessitates further studies to completely rule out its teratogenic effects (Nørgård et al., 2001). Notably, our study indicates a strong positive signal between sulfasalazine and teratogenicity. Experimental studies have shown that sulfasalazine may cause skeletal abnormalities and cleft palate in the fetuses of pregnant mice and rats (Kato, 1973). Previous clinical case reports documented two pregnant women with inflammatory bowel disease who were treated with sulfasalazine during pregnancy, and three newborns were observed with major congenital anomalies (Newman and Correy, 1983). A study aimed to analyze the medication patterns of women and men using antirheumatic drugs before and during pregnancy in Norway and investigate the association between these medications and the risk of congenital anomalies in infants. The study found that five children whose mothers used sulfasalazine within 3 months before or during pregnancy were born with congenital anomalies (Viktil et al., 2012). A meta-analysis on the use of 5-ASA drugs in pregnant women with IBD found an association with an increased risk of congenital anomalies, although this risk increase does not exceed 1.16 times. However, this risk increase is not statistically significant (Rahimi et al., 2008). Therefore, the potential teratogenic risks of sulfasalazine to the fetus require further investigation. It is well known that sulfonamides have hepatotoxic effects, and liver injury induced by sulfasalazine is a serious AE associated with its clinical use. Although the exact mechanism is not fully understood, defects in cellular defense functions and oxidative stress likely play significant roles. Excessive accumulation of reactive oxygen species can lead to organ dysfunction and parenchymal damage (Linares et al., 2011; Heidari et al., 2016b). Sulfasalazine-induced liver injury is mainly characterized by elevated transaminases, increased bilirubin levels, and hepatocellular necrosis. In severe cases, it may progress to granulomatous hepatitis or cholestatic liver cirrhosis (Heidari et al., 2016b; Linares et al., 2009). Granulomatous hepatitis is a specific type of liver damage typically associated with immune reactions. Sulfasalazine may induce the aggregation of inflammatory cells in the liver, leading to granuloma formation by affecting the immune system (Namias et al., 1981; Fich et al., 1984). Notably, in this study’s adverse event reports, hepatosplenic T-cell lymphoma showed a strong positive signal. Although IBD itself is not considered a risk factor for gastrointestinal lymphoma, long-term use of immunosuppressants and TNF inhibitors may increase the risk of lymphoma (Thai and Prindiville, 2010; Kotlyar et al., 2011). Suzuki et al. reported a case of a patient with UC who had been treated with 5-ASA for an extended period and eventually developed diffuse large B-cell lymphoma (DLBCL), leading to intestinal perforation. This case emphasizes that lymphoma can occur in IBD patients even without the use of traditional immunosuppressants. Although the case did not specifically mention HSTCL, we know that HSTCL is indeed associated with IBD, particularly in patients using certain treatments, suggesting that the underlying disease itself may to some extent increase the risk of lymphoma (Suzuki et al., 2021; Ashrafi et al., 2014). The results of this study indicate a positive signal association between sulfasalazine and HSTCL, challenging the previous view that HSTCL is primarily associated with the use of thiopurines and TNF inhibitors. This suggests that sulfasalazine, a common drug in the treatment of IBD, may have a potential link with the occurrence of HSTCL. However, there is currently no clear evidence or reports directly linking it to the occurrence of HSTCL, so more research is needed to establish a causal relationship. 5 Limitations FAERS is a pharmacovigilance system that relies primarily on voluntary reports, which presents inherent limitations in extracting accurate adverse drug reaction information. Firstly, FAERS data predominantly originate from voluntary reports, which may introduce biases in both the quantity and quality of the reports. This bias can result in an incomplete representation of the adverse reactions experienced by all patients. Secondly, voluntary reports often lack standardized criteria, leading to incomplete or inaccurate information. This lack of standardization affects the overall reliability of the data. Moreover, subjective judgment by reporters can lead to misreporting or underreporting, further affecting the outcomes of data analysis. Therefore, cross-validating adverse drug reactions with other pharmacovigilance databases, such as VigiBase and Canada Vigilance, is crucial. These databases offer alternative reporting sources and standards that can complement and validate findings from FAERS data, thereby enhancing the accuracy and reliability of drug safety assessments. Furthermore, conducting additional prospective studies and randomized controlled trials is essential for validating potential associations identified in FAERS and clarifying causal relationships. This approach ensures a comprehensive and scientific evaluation of drug safety. 6 Conclusion This study conducted a systematic analysis of AEs related to sulfasalazine using the FAERS database. The analysis revealed potential serious risks associated with the drug across multiple organ systems. Despite being a widely used anti-inflammatory medication, sulfasalazine is associated with a range of complex adverse reactions. Significant risk signals were identified particularly in the skin, hematologic, infectious, respiratory, and urinary systems. Sulfasalazine has been linked to severe adverse reactions in the skin and subcutaneous tissues, such as Stevens-Johnson syndrome and exfoliative dermatitis. Hematologic adverse reactions associated with sulfasalazine include agranulocytosis and folate deficiency anemia. Moreover, sulfasalazine may increase the risk of infections, as indicated by significant signals for severe conditions such as neutropenic sepsis and cryptococcal pneumonia. In the respiratory system, sulfasalazine may lead to pulmonary toxicities, such as interstitial lung disease and eosinophilic pneumonia. Similarly, sulfasalazine’s nephrotoxic effects, including crystalluria and glomerulonephritis, require close attention. The study also suggests a potential teratogenic risk associated with sulfasalazine use during pregnancy; however, this finding requires further validation. Furthermore, the potential association between sulfasalazine and hepatosplenic T-cell lymphoma requires further investigation. In conclusion, this study highlights the serious risks of adverse reactions associated with sulfasalazine across multiple systems and emphasizes the need for vigilant monitoring in clinical practice. Future research should focus on elucidating the mechanisms underlying these adverse reactions, validating the identified risks through large-scale prospective studies, and developing personalized treatment strategies for high-risk populations to improve drug safety management. Data availability statement The original contributions presented in the study are included in the article/Supplementary Material, further inquiries can be directed to the corresponding authors. Author contributions WY: Writing–original draft, Data curation. YD: Writing–original draft. ML: Writing–review and editing, Software. ZT: Writing–review and editing, Methodology. SW: Writing–review and editing, Project administration. ZL: Writing–review and editing, Supervision. Funding The author(s) declare that no financial support was received for the research, authorship, and/or publication of this article. Acknowledgments We acknowledge the use of generative AI technology (ChatGPT, GPT-3.5, OpenAI) for language refinement during the preparation of this manuscript. The AI tool was employed to enhance the clarity and coherence of the text, while the authors remain fully responsible for the scientific content and integrity of the work. Conflict of interest The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest. Publisher’s note All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher. Supplementary material The Supplementary Material for this article can be found online at: References Aghaei, S. (2008). An uncontrolled, open label study of sulfasalazine in severe alopecia areata. Indian J. Dermatol Venereol. Leprol. 74 (6), 611–613. doi:10.4103/0378-6323.45103 PubMed Abstract | CrossRef Full Text | Google Scholar Alomar, M., Tawfiq, A. M., Hassan, N., and Palaian, S. (2020). Post marketing surveillance of suspected adverse drug reactions through spontaneous reporting: current status, challenges and the future. Ther. Adv. Drug Saf. 11, 2042098620938595. doi:10.1177/2042098620938595 PubMed Abstract | CrossRef Full Text | Google Scholar Andrès, E., and Maloisel, F. (2008). Idiosyncratic drug-induced agranulocytosis or acute neutropenia. Curr. Opin. Hematol. 15 (1), 15–21. doi:10.1097/MOH.0b013e3282f15fb9 PubMed Abstract | CrossRef Full Text | Google Scholar Ashrafi, F., Kowsari, F., Darakhshandeh, A., and Adibi, P. (2014). Composite lymphoma in a patient with ulcerative colitis: a case report. Int. J. Hematol. Oncol. Stem Cell Res. 8 (4), 45–48. PubMed Abstract | Google Scholar Atheymen, R., Affes, H., Ksouda, K., Mnif, L., Sahnoun, Z., Tahri, N., et al. (2013). Adverse effects of sulfasalazine: discussion of mechanism and role of sulfonamide structure. Therapie 68 (6), 369–373. doi:10.2515/therapie/2013064 PubMed Abstract | CrossRef Full Text | Google Scholar Averbuch, M., Halpern, Z., Hallak, A., Topilsky, M., and Levo, Y. (1985). Sulfasalazine pneumonitis. Am. J. Gastroenterol. 80 (5), 343–345. PubMed Abstract | Google Scholar Ayyash, A. M., and Sampath, R. (2021). Acute febrile neutrophilic dermatosis. N. Engl. J. Med. 385 (4), e14. doi:10.1056/NEJMicm2033219 PubMed Abstract | CrossRef Full Text | Google Scholar Azadkhan, A. K., Truelove, S. C., and Aronson, J. K. (1982). The disposition and metabolism of sulphasalazine (salicylazosulphapyridine) in man. Br. J. Clin. Pharmacol. 13 (4), 523–528. doi:10.1111/j.1365-2125.1982.tb01415.x PubMed Abstract | CrossRef Full Text | Google Scholar Bermas, B. L. (2020). Paternal safety of anti-rheumatic medications. Best. Pract. Res. Clin. Obstet. Gynaecol. 64, 77–84. doi:10.1016/j.bpobgyn.2019.09.004 PubMed Abstract | CrossRef Full Text | Google Scholar Chen, J., Lin, S., and Liu, C. (2014). Sulfasalazine for ankylosing spondylitis. Cochrane Database Syst. Rev. 2014 (11), CD004800. doi:10.1002/14651858.CD004800.pub3 PubMed Abstract | CrossRef Full Text | Google Scholar Chiu, Y.-M., and Chen, D.-Y. (2020). Infection risk in patients undergoing treatment for inflammatory arthritis: non-biologics versus biologics. Expert Rev. Clin. Immunol. 16 (2), 207–228. doi:10.1080/1744666X.2019.1705785 PubMed Abstract | CrossRef Full Text | Google Scholar Das, K. M., and Dubin, R. (1976). Clinical pharmacokinetics of sulphasalazine. Clin. Pharmacokinet. 1 (6), 406–425. doi:10.2165/00003088-197601060-00002 PubMed Abstract | CrossRef Full Text | Google Scholar Davis, B. P., and Ballas, Z. K. (2017). Biologic response modifiers: indications, implications, and insights. J. Allergy Clin. Immunol. 139 (5), 1445–1456. doi:10.1016/j.jaci.2017.02.013 PubMed Abstract | CrossRef Full Text | Google Scholar Dery, C. L., and Schwinghammer, T. L. (1988). Agranulocytosis associated with sulfasalazine. Drug Intell. Clin. Pharm. 22 (2), 139–142. doi:10.1177/106002808802200208 PubMed Abstract | CrossRef Full Text | Google Scholar Durando, M., Tiu, H., and Kim, J. S. (2017). Sulfasalazine-induced crystalluria causing severe acute kidney injury. Am. J. Kidney Dis. 70 (6), 869–873. doi:10.1053/j.ajkd.2017.05.013 PubMed Abstract | CrossRef Full Text | Google Scholar Editorial (1974). Editorial: sulphasalazine-induced lung disease. Lancet. 2 (7879), 504–505. PubMed Abstract | Google Scholar Farr, M., Kitas, G. D., Tunn, E. J., and Bacon, P. A. (1991). Immunodeficiencies associated with sulphasalazine therapy in inflammatory arthritis. Br. J. Rheumatol. 30 (6), 413–417. doi:10.1093/rheumatology/30.6.413 PubMed Abstract | CrossRef Full Text | Google Scholar Fathallah, N., Slim, R., Rached, S., Hachfi, W., Letaief, A., and Ben Salem, C. (2015). Sulfasalazine-induced DRESS and severe agranulocytosis successfully treated by granulocyte colony-stimulating factor. Int. J. Clin. Pharm. 37 (4), 563–565. doi:10.1007/s11096-015-0107-2 PubMed Abstract | CrossRef Full Text | Google Scholar Fich, A., Schwartz, J., Braverman, D., Zifroni, A., and Rachmilewitz, D. (1984). Sulfasalazine hepatotoxicity. Am. J. Gastroenterol. 79 (5), 401–402. PubMed Abstract | Google Scholar Fish, E. N. (2008). The X-files in immunity: sex-based differences predispose immune responses. Nat. Rev. Immunol. 8 (9), 737–744. doi:10.1038/nri2394 PubMed Abstract | CrossRef Full Text | Google Scholar Gabazza, E. C., Taguchi, O., Yamakami, T., Machishi, M., Ibata, H., Suzuki, S., et al. (1992). Pulmonary infiltrates and skin pigmentation associated with sulfasalazine. Am. J. Gastroenterol. 87 (11), 1654–1657. PubMed Abstract | Google Scholar Girelli, F., Bernardi, S., Gardelli, L., Bassi, B., Parente, G., Dubini, A., et al. (2013). A new case of DRESS syndrome induced by sulfasalazine and triggered by amoxicillin. Case Rep. Rheumatol. 2013, 409152. doi:10.1155/2013/409152 PubMed Abstract | CrossRef Full Text | Google Scholar Grindulis, K. A., and McConkey, B. (1985). Does sulphasalazine cause folate deficiency in rheumatoid arthritis? Scand. J. Rheumatol. 14 (3), 265–270. doi:10.3109/03009748509100404 PubMed Abstract | CrossRef Full Text | Google Scholar Hamadeh, M. A., Atkinson, J., and Smith, L. J. (1992). Sulfasalazine-induced pulmonary disease. Chest 101 (4), 1033–1037. doi:10.1378/chest.101.4.1033 PubMed Abstract | CrossRef Full Text | Google Scholar Heidari, R., Rasti, M., Shirazi Yeganeh, B., Niknahad, H., Saeedi, A., and Najibi, A. (2016b). Sulfasalazine-induced renal and hepatic injury in rats and the protective role of taurine. Bioimpacts 6 (1), 3–8. doi:10.15171/bi.2016.01 PubMed Abstract | CrossRef Full Text | Google Scholar Heidari, R., Taheri, V., Rahimi, H. R., Shirazi Yeganeh, B., Niknahad, H., and Najibi, A. (2016a). Sulfasalazine-induced renal injury in rats and the protective role of thiol-reductants. Ren. Fail 38 (1), 137–141. doi:10.3109/0886022X.2015.1096731 PubMed Abstract | CrossRef Full Text | Google Scholar Hoult, J. R. (1986). Pharmacological and biochemical actions of sulphasalazine. Drugs 32 (Suppl. 1), 18–26. doi:10.2165/00003495-198600321-00005 PubMed Abstract | CrossRef Full Text | Google Scholar Iemoli, E., Piconi, S., Ardizzone, S., Bianchi Porro, G., and Raimond, F. (2006). Erythroderma and toxic epidermal necrolysis caused by to 5-aminosalacylic acid. Inflamm. Bowel Dis. 12 (10), 1007–1008. doi:10.1097/01.mib.0000231569.43065.a4 PubMed Abstract | CrossRef Full Text | Google Scholar Jick, H., Myers, M. W., and Dean, A. D. (1995). The risk of sulfasalazine- and mesalazine-associated blood disorders. Pharmacotherapy 15 (2), 176–181. doi:10.1002/j.1875-9114.1995.tb04352.x PubMed Abstract | CrossRef Full Text | Google Scholar Joshi, R., Kumar, S., Unnikrishnan, M., and Mukherjee, T. (2005). Free radical scavenging reactions of sulfasalazine, 5-aminosalicylic acid and sulfapyridine: mechanistic aspects and antioxidant activity. Free Radic. Res. 39 (11), 1163–1172. doi:10.1080/10715760500177880 PubMed Abstract | CrossRef Full Text | Google Scholar Kato, T. (1973). Production of congenital anomalies in fetuses of rats and mice with various sulfonamides. Cong Anom. 13, 7–23. Google Scholar Komura, K., Hasegawa, M., Hamaguchi, Y., Yukami, T., Nagai, M., Yachie, A., et al. (2005). Drug-induced hypersensitivity syndrome associated with human herpesvirus 6 and cytomegalovirus reactivation. J. Dermatol 32 (12), 976–981. doi:10.1111/j.1346-8138.2005.tb00885.x PubMed Abstract | CrossRef Full Text | Google Scholar Kotlyar, D. S., Osterman, M. T., Diamond, R. H., Porter, D., Blonski, W. C., Wasik, M., et al. (2011). A systematic review of factors that contribute to hepatosplenic T-cell lymphoma in patients with inflammatory bowel disease. Clin. Gastroenterol. Hepatol. 9 (1), 36–41. doi:10.1016/j.cgh.2010.09.016 PubMed Abstract | CrossRef Full Text | Google Scholar Lichtenstein, G. R., Loftus, E. V., Isaacs, K. L., Regueiro, M. D., Gerson, L. B., and Sands, B. E. (2018). ACG clinical guideline: management of Crohn's disease in adults. Am. J. Gastroenterol. 113 (4), 481–517. doi:10.1038/ajg.2018.27 PubMed Abstract | CrossRef Full Text | Google Scholar Lim, W.-C., Wang, Y., MacDonald, J. K., and Hanauer, S. (2016). Aminosalicylates for induction of remission or response in Crohn's disease. Cochrane Database Syst. Rev. 7 (7), CD008870. doi:10.1002/14651858.CD008870.pub2 PubMed Abstract | CrossRef Full Text | Google Scholar Linares, V., Alonso, V., Albina, M. L., Bellés, M., Sirvent, J. J., Domingo, J. L., et al. (2009). Lipid peroxidation and antioxidant status in kidney and liver of rats treated with sulfasalazine. Toxicology 256 (3), 152–156. doi:10.1016/j.tox.2008.11.010 PubMed Abstract | CrossRef Full Text | Google Scholar Linares, V., Alonso, V., and Domingo, J. L. (2011). Oxidative stress as a mechanism underlying sulfasalazine-induced toxicity. Expert Opin. Drug Saf. 10 (2), 253–263. doi:10.1517/14740338.2011.529898 PubMed Abstract | CrossRef Full Text | Google Scholar Madhushika, M. T., Weerarathna, T. P., Liyanage, PLGC, and Jayasinghe, S. S. (2022). Evolution of adverse drug reactions reporting systems: paper based to software based. Eur. J. Clin. Pharmacol. 78 (9), 1385–1390. doi:10.1007/s00228-022-03358-3 PubMed Abstract | CrossRef Full Text | Google Scholar Manvi, S., Mahajan, V. K., Mehta, K. S., Chauhan, P. S., Vashist, S., Singh, R., et al. (2022). The clinical characteristics, putative drugs, and optimal management of 62 patients with stevens-johnson syndrome and/or toxic epidermal necrolysis: a retrospective observational study. Indian Dermatol Online J. 13 (1), 23–31. doi:10.4103/idoj.idoj_530_21 PubMed Abstract | CrossRef Full Text | Google Scholar Menter, A., Korman, N. J., Elmets, C. A., Feldman, S. R., Gelfand, J. M., Gordon, K. B., et al. (2009). Guidelines of care for the management of psoriasis and psoriatic arthritis: section 4. Guidelines of care for the management and treatment of psoriasis with traditional systemic agents. J. Am. Acad. Dermatol 61 (3), 451–485. doi:10.1016/j.jaad.2009.03.027 PubMed Abstract | CrossRef Full Text | Google Scholar Moseley, R. H., Barwick, K. W., Dobuler, K., and DeLuca, V. A. (1985). Sulfasalazine-induced pulmonary disease. Dig. Dis. Sci. 30 (9), 901–904. doi:10.1007/BF01309523 PubMed Abstract | CrossRef Full Text | Google Scholar Moss, S. F., and Ind, P. W. (1991). Time course of recovery of lung function in sulphasalazine-induced alveolitis. Respir. Med. 85 (1), 73–75. doi:10.1016/s0954-6111(06)80215-5 PubMed Abstract | CrossRef Full Text | Google Scholar Namias, A., Bhalotra, R., and Donowitz, M. (1981). Reversible sulfasalazine-induced granulomatous hepatitis. J. Clin. Gastroenterol. 3 (2), 193–198. doi:10.1097/00004836-198106000-00017 PubMed Abstract | CrossRef Full Text | Google Scholar Nasim, F., Paul, J. A., Boland-Froemming, J., and Wylam, M. E. (2021). Sulfa-induced acute eosinophilic pneumonia. Respir. Med. Case Rep. 34, 101496. doi:10.1016/j.rmcr.2021.101496 PubMed Abstract | CrossRef Full Text | Google Scholar Nasiri-Jahrodi, A., Sheikholeslami, F.-M., and Barati, M. (2023). Cladosporium tenuissimum-induced sinusitis in a woman with immune-deficiency disorder. Braz J. Microbiol. 54 (2), 637–643. doi:10.1007/s42770-023-00978-4 PubMed Abstract | CrossRef Full Text | Google Scholar Navarro, F., and Hanauer, S. B. (2003). Treatment of inflammatory bowel disease: safety and tolerability issues. Am. J. Gastroenterol. 98 (12 Suppl. l), S18–S23. doi:10.1016/j.amjgastroenterol.2003.11.001 PubMed Abstract | CrossRef Full Text | Google Scholar Newman, N. M., and Correy, J. F. (1983). Possible teratogenicity of sulphasalazine. Med. J. Aust. 1 (11), 528–529. doi:10.5694/j.1326-5377.1983.tb136199.x PubMed Abstract | CrossRef Full Text | Google Scholar Nissim-Eliraz, E., Nir, E., Marsiano, N., Yagel, S., and Shpigel, N. Y. (2021). NF-kappa-B activation unveils the presence of inflammatory hotspots in human gut xenografts. PLoS One 16 (5), e0243010. doi:10.1371/journal.pone.0243010 PubMed Abstract | CrossRef Full Text | Google Scholar Nørgård, B., Czeizel, A. E., Rockenbauer, M., Olsen, J., and Sørensen, H. T. (2001). Population-based case control study of the safety of sulfasalazine use during pregnancy. Aliment. Pharmacol. Ther. 15 (4), 483–486. doi:10.1046/j.1365-2036.2001.00962.x PubMed Abstract | CrossRef Full Text | Google Scholar Parry, S. D., Barbatzas, C., Peel, E. T., and Barton, J. R. (2002). Sulphasalazine and lung toxicity. Eur. Respir. J. 19 (4), 756–764. doi:10.1183/09031936.02.00267402 PubMed Abstract | CrossRef Full Text | Google Scholar Perazella, M. A., and Rosner, M. H. (2022). Drug-induced acute kidney injury. Clin. J. Am. Soc. Nephrol. 17 (8), 1220–1233. doi:10.2215/CJN.11290821 PubMed Abstract | CrossRef Full Text | Google Scholar Plosker, G. L., and Croom, K. F. (2005). Sulfasalazine: a review of its use in the management of rheumatoid arthritis. Drugs 65 (13), 1825–1849. doi:10.2165/00003495-200565130-00008 PubMed Abstract | CrossRef Full Text | Google Scholar Pruzanski, W., Stefanski, E., Vadas, P., and Ramamurthy, N. S. (1997). Inhibition of extracellular release of proinflammatory secretory phospholipase A2 (sPLA2) by sulfasalazine: a novel mechanism of anti-inflammatory activity. Biochem. Pharmacol. 53 (12), 1901–1907. doi:10.1016/s0006-2952(97)00137-8 PubMed Abstract | CrossRef Full Text | Google Scholar Rahimi, R., Nikfar, S., Rezaie, A., and Abdollahi, M. (2008). Pregnancy outcome in women with inflammatory bowel disease following exposure to 5-aminosalicylic acid drugs: a meta-analysis. Reprod. Toxicol. 25 (2), 271–275. doi:10.1016/j.reprotox.2007.11.010 PubMed Abstract | CrossRef Full Text | Google Scholar Rains, C. P., Noble, S., and Faulds, D. (1995). Sulfasalazine. A review of its pharmacological properties and therapeutic efficacy in the treatment of rheumatoid arthritis. Drugs 50 (1), 137–156. doi:10.2165/00003495-199550010-00009 PubMed Abstract | CrossRef Full Text | Google Scholar Ransford, R. A. J., and Langman, M. J. S. (2002). Sulphasalazine and mesalazine: serious adverse reactions re-evaluated on the basis of suspected adverse reaction reports to the Committee on Safety of Medicines. Gut 51 (4), 536–539. doi:10.1136/gut.51.4.536 PubMed Abstract | CrossRef Full Text | Google Scholar Romdhane, H. B., Mokni, S., Fathallah, N., Slim, R., Ghariani, N., Sriha, B., et al. (2016). Sulfasalazine-induced Sweet's syndrome. Therapie 71 (3), 345–347. doi:10.1016/j.therap.2015.11.006 PubMed Abstract | CrossRef Full Text | Google Scholar Sahm, J., de Groot, K., and Schreiber, J. (2021). Sulfasalazine-induced mononucleosis-like-illness and haemolysis. Scand. J. Rheumatol. 50 (1), 83–84. doi:10.1080/03009742.2020.1747533 PubMed Abstract | CrossRef Full Text | Google Scholar Salouage, I., El Aïdli, S., Cherif, F., Kastalli, S., Zaiem, A., and Daghfous, R. (2013). Sulfasalazine-induced aseptic meningitis with positive rechallenge: a case report and review of the literature. Therapie 68 (6), 423–426. doi:10.2515/therapie/2013065 PubMed Abstract | CrossRef Full Text | Google Scholar Suarez-Almazor, M. E., Belseck, E., Shea, B., Wells, G., and Tugwell, P. (2000). Sulfasalazine for rheumatoid arthritis. Cochrane Database Syst. Rev. 1998 (2), CD000958. doi:10.1002/14651858.CD000958 PubMed Abstract | CrossRef Full Text | Google Scholar Suzuki, T., Iwamoto, K., Nozaki, R., Saiki, Y., Tanaka, M., Fukunaga, M., et al. (2021). Diffuse large B-cell lymphoma originating from the rectum and diagnosed after rectal perforation during the treatment of ulcerative colitis: a case report. BMC Surg. 21 (1), 50. doi:10.1186/s12893-021-01060-2 PubMed Abstract | CrossRef Full Text | Google Scholar Taffet, S. L., and Das, K. M. (1983). Sulfasalazine. Adverse effects and desensitization. Dig. Dis. Sci. 28 (9), 833–842. doi:10.1007/BF01296907 PubMed Abstract | CrossRef Full Text | Google Scholar Tan, S.-C., and Chan, G. Y. L. (2016). Relapsing drug-induced hypersensitivity syndrome. Curr. Opin. Allergy Clin. Immunol. 16 (4), 333–338. doi:10.1097/ACI.0000000000000288 PubMed Abstract | CrossRef Full Text | Google Scholar Tay, S. H., Lateef, A., and Cheung, P. P. (2012). Sulphasalazine-induced aseptic meningitis with facial and nuchal edema in a patient with spondyloarthritis. Int. J. Rheum. Dis. 15 (4), e71–e72. doi:10.1111/j.1756-185X.2012.01705.x PubMed Abstract | CrossRef Full Text | Google Scholar Teo, L., and Tan, E. (2006). Sulphasalazine-induced DRESS. Singap. Med. J. 47 (3), 237–239. PubMed Abstract | Google Scholar Thai, A., and Prindiville, T. (2010). Hepatosplenic T-cell lymphoma and inflammatory bowel disease. J. Crohns Colitis 4 (5), 511–522. doi:10.1016/j.crohns.2010.05.006 PubMed Abstract | CrossRef Full Text | Google Scholar Toussirot, E., and Wendling, D. (2001). Therapeutic advances in ankylosing spondylitis. Expert Opin. Investig. Drugs 10 (1), 21–29. doi:10.1517/13543784.10.1.21 PubMed Abstract | CrossRef Full Text | Google Scholar Tremblay, L., Pineton de Chambrun, G., De Vroey, B., Lavogiez, C., Delaporte, E., Colombel, J.-F., et al. (2011). Stevens-Johnson syndrome with sulfasalazine treatment: report of two cases. J. Crohns Colitis 5 (5), 457–460. doi:10.1016/j.crohns.2011.03.014 PubMed Abstract | CrossRef Full Text | Google Scholar Viktil, K. K., Engeland, A., and Furu, K. (2012). Outcomes after anti-rheumatic drug use before and during pregnancy: a cohort study among 150,000 pregnant women and expectant fathers. Scand. J. Rheumatol. 41 (3), 196–201. doi:10.3109/03009742.2011.626442 PubMed Abstract | CrossRef Full Text | Google Scholar Wadelius, M., Eriksson, N., Kreutz, R., Bondon-Guitton, E., Ibañez, L., Carvajal, A., et al. (2018). Sulfasalazine-induced agranulocytosis is associated with the human leukocyte antigen locus. Clin. Pharmacol. Ther. 103 (5), 843–853. doi:10.1002/cpt.805 PubMed Abstract | CrossRef Full Text | Google Scholar Whitacre, C. C. (2001). Sex differences in autoimmune disease. Nat. Immunol. 2 (9), 777–780. doi:10.1038/ni0901-777 PubMed Abstract | CrossRef Full Text | Google Scholar Yamamoto, T. (2014). Simultaneous occurrence of Sweet's syndrome and erythema nodosum possibly associated with sulfasalazine. Int. J. Dermatol 53 (4), e263–e265. doi:10.1111/ijd.12260 PubMed Abstract | CrossRef Full Text | Google Scholar Yelehe-Okouma, M., Czmil-Garon, J., Pape, E., Petitpain, N., and Gillet, P. (2018). Drug-induced aseptic meningitis: a mini-review. Fundam. Clin. Pharmacol. 32 (3), 252–260. doi:10.1111/fcp.12349 PubMed Abstract | CrossRef Full Text | Google Scholar Youngster, I., Arcavi, L., Schechmaster, R., Akayzen, Y., Popliski, H., Shimonov, J., et al. (2010). Medications and glucose-6-phosphate dehydrogenase deficiency: an evidence-based review. Drug Saf. 33 (9), 713–726. doi:10.2165/11536520-000000000-00000 PubMed Abstract | CrossRef Full Text | Google Scholar Zenlea, T., and Peppercorn, M. A. (2014). Immunosuppressive therapies for inflammatory bowel disease. World J. Gastroenterol. 20 (12), 3146–3152. doi:10.3748/wjg.v20.i12.3146 PubMed Abstract | CrossRef Full Text | Google Scholar Keywords: sulfasalazine, FDA adverse events reporting system, adverse drug reaction, adverse event, pharmacovigilance Citation: Ye W, Ding Y, Li M, Tian Z, Wang S and Liu Z (2024) Safety assessment of sulfasalazine: a pharmacovigilance study based on FAERS database. Front. Pharmacol. 15:1452300. doi: 10.3389/fphar.2024.1452300 Received: 20 June 2024; Accepted: 28 August 2024; Published: 12 September 2024. Edited by: Bernd Rosenkranz, Fundisa African Academy of Medicines Development, South Africa Reviewed by: Jefman Efendi Marzuki HY, University of Surabaya, Indonesia Romina Martinelli, DeltaPV, Spain Nadine De Godoy Torso, State University of Campinas, Brazil Copyright © 2024 Ye, Ding, Li, Tian, Wang and Liu. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Correspondence: Zhen Liu, doctorliuzhen@126.com; Shaoli Wang, drshaoliwang@163.com †These authors share first authorship Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.
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https://mathoverflow.net/questions/127560/references-on-techniques-for-solving-equations-with-discontinuous-functions-such
nt.number theory - References on techniques for solving equations with discontinuous functions such as floor and ceiling? - MathOverflow Join MathOverflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community MathOverflow helpchat MathOverflow Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more References on techniques for solving equations with discontinuous functions such as floor and ceiling? Ask Question Asked 12 years, 5 months ago Modified12 years, 5 months ago Viewed 2k times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. Here I describe the sort of reference I'm after with a motivating example. I am not seeking solutions to my equations on this forum; I'm quite happy to do that myself. Rather, I'm asking for some good references on solving equations of this sort. I had an equation 2⌊a⌋c−a=d−c 2⌊a⌋c−a=d−c with a,c,d∈Z a,c,d∈Z, and where ⌊n⌋k⌊n⌋k is my notation for k⌊n/k⌋k⌊n/k⌋ — essentially the floor function down to the nearest multiple of k k. I wished to solve for a a. Now, I didn't know how to tackle this algebraically, as the usual technique of bringing the a a's together does not seem to be available. However, I had some notion of what form a solution was likely to take. After some guesswork and experimentation with Maxima, I found the solution: a=2⌈d⌉c−(c+d)a=2⌈d⌉c−(c+d) This appears to be correct, but I have not yet found a way to prove this — but that's not my question. This approach is very unsatisfactory to me. I would much rather solve the problem algbraically. I'd like to know if there are any recommended references, either books or on-line, about techniques that can be used to solve equations involving the floor (⌊⋅⌋⌊⋅⌋), ceiling (⌈⋅⌉⌈⋅⌉), fraction-part, and similar functions, either in Z Z, Q Q or R R. Beyond my particular equation of interest, I'd be interested to learn how to tackle this sort of equation more generally. (In case you're interested why I was looking at this equation: I have recently encountered the remarkable Stern diatomic sequence. The equation in question is related to the successor function on ratios of consecutive terms; I wished to find the inverse function.) diophantine-equations nt.number-theory co.combinatorics discrete-mathematics recurrences Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Improve this question Follow Follow this question to receive notifications edited Apr 14, 2013 at 22:25 RhubbarbRhubbarb asked Apr 14, 2013 at 22:19 RhubbarbRhubbarb 534 7 7 silver badges 15 15 bronze badges 5 1 Look for literature on brackets in number theory (as opposed to brackets in other fields). Two texts that touch on the treatment are Concrete Mathematics by Graham, Knuth, and Patashnik, and (title something like) Number Theory, an elementary approach by Joe Roberts. The latter is the only calligraphed science text I know, and if you can find its bibliography and a decent citation index, you should likely succeed in finding newer texts that deal with floor and ceiling. Gerhard "Ask Me About Elementary Mathematics" Paseman, 2013.04.14 Gerhard Paseman –Gerhard Paseman 2013-04-14 22:55:58 +00:00 Commented Apr 14, 2013 at 22:55 @Gerhard Thanks. Please feel free to upgrade that to an answer.Rhubbarb –Rhubbarb 2013-04-15 08:43:23 +00:00 Commented Apr 15, 2013 at 8:43 2 Your special equation you can first solve mod c mod c, and then use a solution s∈Z s∈Z to write a=s+a′a=s+a′ where a′a′ is a multiple of c c. Now solve for a′a′. Someone –Someone 2013-04-15 12:14:02 +00:00 Commented Apr 15, 2013 at 12:14 @Someone thanks for your comment - a useful suggestion associated with my question Rhubbarb –Rhubbarb 2013-04-16 12:28:04 +00:00 Commented Apr 16, 2013 at 12:28 UPDATE: it turns out that the '2' in my equation is significant. I've put some notes on this on my blog rhubbarb.wordpress.com/2013/04/30/inverting-floorRhubbarb –Rhubbarb 2013-04-30 23:02:32 +00:00 Commented Apr 30, 2013 at 23:02 Add a comment| 3 Answers 3 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. Joe Roberts provided the words for the calligraphed book Elementary Number Theory: A Problem Oriented Approach, which was printed in the 1970's. This book has a chapter on brackets, which in some of the number theory literature is an older name for one or both of the floor and ceiling functions. While not providing as focused a treatment of brackets, Concrete Mathematics, which was authored by Graham, Knuth, and Patashnik, also gives some service to the handling of floor and ceiling. With the bibliographies of those two books and a decent citation index, you may find more recent treatments. There may be other search terms to use, but I would start with "brackets +number theory -Lie" or something like that in a web search. Gerhard "Is It Forty Years Already?" Paseman, 2013.04.15 Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Apr 15, 2013 at 20:03 Gerhard PasemanGerhard Paseman 1,081 1 1 gold badge 9 9 silver badges 8 8 bronze badges 1 @Gerhard and @Aaron - you might like to know that I have just bought myself a (used) copy of Joe Roberts' Elementary Number Theory, and am very please with it. (I do also have a copy of Concrete Mathematics, and agree that that too is an excellent source, which I have also gone back to.)Rhubbarb –Rhubbarb 2013-04-28 19:50:28 +00:00 Commented Apr 28, 2013 at 19:50 Add a comment| This answer is useful 3 Save this answer. Show activity on this post. Expressions formed by composing polynomials and the integer-part operator are refered to in numerous papers by the not very google-friendly name ``generalized polynomials''. The problem of determining whether a generalized polynomial equation has integer solutions includes Hilbert's Tenth problem, and is therefore effectively unsolvable. On the other hand there are some interesting results on the distribution of values of generalized polynomials, which you might find relevant: Bergelson and Leibman's paper ``Distribution of values of bounded generalized polynomials'' available here. Leibman's paper ``A canonical form and the distribution of values of generalized polynomials'' available here. There are related papers on Leibman's website and also by Haland and McCutcheon. Leibman's paper gives a cannonical form for generalized polynomials that helps to grasp what values the gp can assume mod 1. His paper is a follow-up to the Bergelson-Leibman paper, in which Bergelson shows very roughly speaking that every bounded generalized polynomial can be thought of as a matrix power map composed with a piecewise-polynomial function. Bergelson shows how tools from Ergodic Theory and Lie Theory can be brough to bear on the study of generalized polynomials. Incidentally, the problem of which equations g=0 g=0 are identities (i.e. hold for all integer values of the variables), where g is a gp, is also effective unsolvable by reduction to Hilbert's Tenth Problem: Let f f be any polynomial with integer coefficients. Then the equation ⌊2–√f(x¯)⌋+⌊−2–√f(x¯)⌋+1=0⌊2 f(x¯)⌋+⌊−2 f(x¯)⌋+1=0 is an identity if and only if f f has no integer zeros. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications answered Apr 16, 2013 at 12:51 Sidney RafferSidney Raffer 6,229 1 1 gold badge 29 29 silver badges 42 42 bronze badges Add a comment| This answer is useful 2 Save this answer. Show activity on this post. I agree that the Joe Roberts book is a gem. I've bought two copies over the years. I'd buy a third if I could. Here is a link to a scan I leave ethical issues to you, I got this from another MO answer . That said, the Wikipedia article may cover the same material (but check for yourself). The main reference for that article seems to be the Graham, Knuth, and Patashnik book so that would be more reliable. There are various easily verified equations and inequalities such as ⌈m n⌉=⌊m+n−1 n⌋=⌊m−1 n⌋+1⌈m n⌉=⌊m+n−1 n⌋=⌊m−1 n⌋+1 (which would give an alternate form to your answer which you might or might not prefer.) Then it is a matter of practice. Actually I see that that particular page say "then use a geometric arguement" in discussing quadratic reciprocity. The Joe Roberts book does give the arguement. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Improve this answer Follow Follow this answer to receive notifications edited Apr 13, 2017 at 12:57 CommunityBot 1 2 2 silver badges 3 3 bronze badges answered Apr 16, 2013 at 7:41 Aaron MeyerowitzAaron Meyerowitz 30.3k 2 2 gold badges 50 50 silver badges 105 105 bronze badges 1 1 @Aaron and @Gerhard - you might like to know that I have just bought myself a (used) copy of Joe Roberts' Elementary Number Theory, and am very please with it. I also made use of your scan link as having an electronic copy is also very useful.Rhubbarb –Rhubbarb 2013-04-28 22:23:44 +00:00 Commented Apr 28, 2013 at 22:23 Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions diophantine-equations nt.number-theory co.combinatorics discrete-mathematics recurrences See similar questions with these tags. Featured on Meta Spevacus has joined us as a Community Manager Introducing a new proactive anti-spam measure Linked 77Writing papers in pre-LaTeX era? Related 16Real algebraic sets bounded away from integer points 2Help with this system of Diophantine equations 0A square-squareroot integer race sequence involving primes Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why are you flagging this comment? It contains harassment, bigotry or abuse. This comment attacks a person or group. Learn more in our Code of Conduct. It's unfriendly or unkind. This comment is rude or condescending. Learn more in our Code of Conduct. Not needed. This comment is not relevant to the post. Enter at least 6 characters Something else. A problem not listed above. 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14027
https://brainly.com/question/21691260
[FREE] Express 490 as a product of its prime factors. - brainly.com 1 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +73,6k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +25,9k Ace exams faster, with practice that adapts to you Practice Worksheets +7,9k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Express 490 as a product of its prime factors. 2 See answers Explain with Learning Companion NEW Asked by nadianadianadia • 02/23/2021 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 4 people 4 5.0 2 Upload your school material for a more relevant answer 2×5×7² Explanation You will start by using the factor tree. 490 ^ 2 245 ``` ^ 5 49 ^ 7 7 ``` Answered by gladysobiriyeboah0 •1 answer•4 people helped Thanks 2 5.0 (2 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 4 people 4 5.0 2 Fundamentals of General Organic and Biological Chemistry - Mcmurry Et Al. Big Ideas in Cosmology - Kim Coble, Kevin McLin, Lynn Cominsky Quantitative Methods for Plant Breeding - Walter Suza Upload your school material for a more relevant answer To express 490 as a product of its prime factors, we factor it into prime numbers: 490=2×5×7 2. This means that 2, 5, and 7 are the prime factors of 490, with 7 being squared. The methods used include dividing by the smallest primes until reaching the prime factors. Explanation To express 490 as a product of its prime factors, we will follow a systematic approach using the factorization method: Start with the number: Begin with the number 490. Divide by the smallest prime: The smallest prime number is 2. Since 490 is even, we can divide it by 2: 490÷2=245 Continue factoring: Next, we factor 245. The smallest prime number that divides 245 is 5: 245÷5=49 Factoring further: Now, factor 49. The smallest prime number that divides 49 is 7: 49÷7=7 Final step: Since 7 is a prime number, we stop here. Now, we can write the prime factorization of 490 as: 490=2×5×7×7 This can also be expressed using exponents, as: 490=2×5×7 2 So, the prime factors of 490 are 2, 5, and 7, with 7 appearing twice. Examples & Evidence For example, if we took another number like 60, we could express it as 60=2 2×3×5 by similarly dividing it by its smallest prime factors until we reach prime numbers. The process of prime factorization is a fundamental method in mathematics to break down numbers into their basic building blocks, known as primes, which is a widely accepted principle in number theory. Thanks 2 5.0 (2 votes) Advertisement Community Answer This answer helped 1096922 people 1M 0.0 0 The prime factorization of 490 is 2 x 5 x 7 x 7 or 2 x 5 x 7².To express 490 as a product of its prime factors, divide it by the smallest prime numbers until only 1 is left. To express 490 as a product of its prime factors, you have to break it down into prime numbers that, when multiplied together, give the original number (490). Here is how you do it step-by-step: Start with the smallest prime number which divides 490, which is 2: 490 / 2 = 245. 245 is not an even number, so 2 will not divide into it. The next smallest prime is 3, but 245 is not a multiple of 3 either. Checking divisibility with prime number 5, you can see that 245 /5 = 49. So the next factor is 5. Now, we need to factor 49. It is not divisible by 2, 3, or 5, but 49 is 7 times 7. So we have 49 as 7 x 7. Putting it all together, we can express 490 as a product of its prime factors: 2 x 5 x 7 x 7 which can also be written in exponential form as 2 x 5 x 7². Answered by PragatiR •13.6K answers•1.1M people helped Thanks 0 0.0 (0 votes) Advertisement ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics The value of the expression x 2 x 2​+x(100−15 x) when x=5 is The multimedia department includes CDs, DVDs, audiobooks, and video games. The video games are one-fourth of the multimedia department while audiobooks are five-twelfths of the department. What fraction of the multimedia department is made up of video games and audiobooks? Provide your answer in the simplest form. A. Two-thirds B. Seven-twelfths C. Three-eighths D. Three-fourths Given f(z)=z 2−5 x and g(x)=2 x+1. Find (f+g)(−3). If x represents the amount Rita earns each week, which expression represents the amount she earns in a year? A. x+52 B. 52−x C. 52 x​ D. 52 x Find an ordered pair (x,y) that is a solution to the equation: x−5 y=5 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
14028
https://www.goodreads.com/book/show/585557.A_Classical_Introduction_to_Modern_Number_Theory
Jump to ratings and reviews Rate this book Graduate Texts in Mathematics #84 A Classical Introduction to Modern Number Theory Kenneth F. Ireland, Michael I. Rosen 60 ratings4 reviews Rate this book This well-developed, accessible text details the historical development of the subject throughout. It also provides wide-ranging coverage of significant results with comparatively elementary proofs, some of them new. This second edition contains two new chapters that provide a complete proof of the Mordel-Weil theorem for elliptic curves over the rational numbers and an overview of recent progress on the arithmetic of elliptic curves. GenresMathematicsNumber TheoryTextbooksScienceNonfictionAlgebraReference 408 pages, Hardcover First published January 1, 1982 12 people are currently reading 304 people want to read About the author Kenneth F. Ireland 2 books Ratings & Reviews What do you think? Rate this book Friends & Following Create a free account to discover what your friends think of this book! Community Reviews 4.38 60 ratings4 reviews 5 stars 33 (55%) 4 stars 20 (33%) 3 stars 5 (8%) 2 stars 1 (1%) 1 star 1 (1%) Displaying 1 - 4 of 4 reviews Lixin 1 review January 3, 2023 Book takes too long. First chapters are fairly accessible, for a graduate level text. Andrés Martínez de la Cárcel 35 reviews August 11, 2023 Un libro nada pedagógico. Dan 320 reviews81 followers March 31, 2009 I just read the chapter about the elliptic curve y^2 = x^3 + Dx, it was pretty good. maths Lulu 1,916 reviews Want to read January 29, 2019 Number Theory john-baez-physics-1 Displaying 1 - 4 of 4 reviews Join the discussion a quotea discussiona question Can't find what you're looking for? Get help and learn more about the design. Help center
14029
https://www.writersdigest.com/write-better-fiction/remove-that-from-your-writing-grammar-rules
Remove That From Your Writing (Grammar Rules) - Writer's Digest Cookie Policy This website uses cookies to ensure you get the best experience on our website. Learn more DeclineAllow cookies ScriptEventsCompetitionsArchiveSubscribeShop Online Writing Workshops Sept/Oct Cover Reveal Online Membership Write Better FictionWrite Better NonfictionWrite Better PoetryGet PublishedBe InspiredWD CompetitionsResources Remove That From Your Writing (Grammar Rules) There are many common ways to write with more concision. For instance, if you remove that from your writing, you’ll still retain the same meaning with fewer characters. Learn more, including examples, here. Robert Lee Brewer Published Jul 27, 2020 4:15 PM EDT Share this story I'm better about it now, but one problem I've struggled with for years is using the word "that" a little too much. Or maybe a lot too much. As such, that is a word that I'm constantly removing from sentences. (I mean, "As such, that is a word I'm constantly removing from sentences.") (Click here for more grammar rules for writers.) Often, the word "that" is a placeholder for nothing in particular. For me, it's almost like a railing that I constantly use in my language. But more times than not, I find that if I remove all instances of that from my writing, it still makes sense (and saves space). For instance, remove the first instance of "that" in the previous sentence. Still makes sense, right? Examples of Removing That From Your Writing Example #1: You know that some people are afraid of clowns. Better example:You know some people are afraid of clowns. Example #2: Let her know that you love her and that she means the world to you. Better example: Let her know you love her and she means the world to you. We could run through so many more examples, but that would get repetitive after a while. And yes, there are times when using "that" makes sense. Just like using the word "it." After all, the movie and song, That Thing You Do, wouldn't have the same ring if it were titled: Thing You Do. The main point of this post is to help others who suffer from the "that" affliction (as I do) by calling it out for what it is. So the next time you write something, search for the word "that" and remove it unless it's absolutely necessary for the sentence to make sense. It's small adjustments like these that can make all the difference. No matter what type of writing you do, mastering the fundamentals of grammar and mechanics is an important first step to having a successful writing career. Click to continue. Share this story Grammar Rules Robert Lee BrewerAuthor Robert Lee Brewer is Senior Editor of Writer's Digest, which includes managing the content on WritersDigest.com and programming virtual conferences. He's the author of Solving the World's Problems, The Complete Guide of Poetic Forms: 100+ Poetic Form Definitions and Examples for Poets, Poem-a-Day: 365 Poetry Writing Prompts for a Year of Poeming, and more. Also, he's the editor of Writer's Market, Poet's Market, and Guide to Literary Agents. Follow him on Twitter @robertleebrewer. Product Recommendations Here are some supplies and products we find essential. We may receive a commission from sales referred by our links; however, we have carefully selected these products for their usefulness and quality. Marvel Avengers T-Shirt Shop now at Amazon.comLamp Book Large Size Folding Mood Lig... Shop now at Amazon.comFIFINE Studio Condenser USB Microphon... Shop now at Amazon.com © 2025 Active Interest Media All rights reserved. About UsTerms of ServicePrivacy PolicyActive Interest MediaScriptMag.comDigital Editions2nd Draft CritiquesWebinarsTutorialsWD UniversityWriter's Digest ShopNovel Writing ConferenceWriter's Digest Annual ConferenceCompetitionsGive a Gift SubscriptionMedia KitBack IssuesCurrent IssueCustomer ServiceRenew a subscriptionSubscribeAdvertise With UsWrite For UsMeet UsSite MapAI PolicyCode of Conduct More AIM Sites [+] Antique TraderArts & Crafts HomesBank Note ReporterCabin LifeCuisine at HomeFine GardeningFine HomebuildingFine WoodworkingGreen Building AdvisorGarden GateKeep Craft AliveLog Home Living Military Trader/VehiclesNumismatic News/VehiclesNumismasterOld Cars WeeklyOld House JournalPeriod HomesPopular WoodworkingScriptShopNotesSports Collectors DigestThreadsTimber Home LivingTraditional Building WoodsmithWoodshop NewsWorld Coin NewsWriter's Digest ;
14030
https://www.pemphigus.org/p-p-clinical-information/
Pemphigus Pemphigus is defined as a group of life-threatening blistering disorders characterized by acantholysis resulting in the formation of intraepithelial blisters in mucous membranes and skin. Acantholysis is the loss of keratinocyte to keratinocyte adhesion, or the skin cells no longer being held together. Patients with pemphigus develop mucosal erosions and/or flaccid bullae (blisters), erosions, or pustules on skin (small bumps that fill with pus or fluid). The intraepidermal blistering observed in pemphigus occurs due to an immune response that results in the deposition of autoantibodies against epidermal cell surface antigens within the epithelium of mucous membranes or skin. The mechanism through which acantholysis occurs is not fully understood. The four major types of pemphigus include pemphigus vulgaris, pemphigus foliaceus, IgA pemphigus, and paraneoplastic pemphigus. The different forms of pemphigus are distinguished by their clinical features, associated autoantigens, and laboratory findings. Pemphigus vulgaris generally is more severe than pemphigus foliaceus. Pemphigus vulgaris usually presents with widespread mucocutaneous blisters and erosions. Cutaneous blistering in pemphigus foliaceus tends to occur in a seborrheic distribution. Blistering in pemphigus foliaceus is more superficial compared with pemphigus vulgaris. The diagnosis of pemphigus is based upon the recognition of consistent clinical, histological, and direct immunofluorescence findings, as well as the detection of circulating autoantibodies against cell surface antigens in serum. Laboratory studies are useful for distinguishing pemphigus from other blistering and erosive diseases. Types of pemphigus Key features:Mucosal or mucosal and cutaneous involvement, blisters on the upper skin layer, autoantibodies attack desmoglein 3 or both desmoglein 1 and desmoglein 3 Clinical variants: Pemphigus vegetans, pemphigus herpetiformis The pain associated with mucosal involvement of pemphigus vulgaris can be severe. Oral pain is often augmented by chewing and swallowing, which may result in poor alimentation, weight loss, and malnutrition. Pemphigus vulgaris is the most common form of pemphigus. However, in certain areas, particularly in locations where an endemic form of pemphigus foliaceus occurs, pemphigus foliaceus is more prevalent. Almost all patients with pemphigus vulgaris develop mucosal involvement. The oral cavity is the most common site of mucosal lesions and often represents the initial site of disease. Mucous membranes at other sites are also often affected, including the conjunctiva, nose, esophagus, vulva, vagina, cervix, and anus. In women with cervical involvement, the histological findings of pemphigus vulgaris may be mistaken for cervical dysplasia in Papanicolaou (Pap) smears. Since mucosal blisters erode quickly, erosions are often the only clinical findings. The buccal mucosa and palatine mucosa are the most common sites for lesion development in the oral cavity. Most patients also develop cutaneous involvement manifesting as flaccid blisters on normal-appearing or erythematous skin. The blisters rupture easily, resulting in painful erosions that bleed easily. Pruritus usually is absent. Although any cutaneous site may be affected, the palms and soles are usually spared. The Nikolsky sign (induction of blistering via mechanical pressure at the edge of a blister or on normal skin) often can be elicited. Key features:Cutaneous (skin) involvement only, subcorneal acantholytic blisters, autoantibodies against desmoglein 1 Clinical variants:Endemic pemphigus foliaceus (fogo selvagem), pemphigus erythematosus (Senear-Usher syndrome), pemphigus herpetiformis Pain or burning sensations frequently accompany the cutaneous lesions. Systemic symptoms are usually absent. The clinical manifestations of drug-induced pemphigus foliaceus are similar to idiopathic disease. Pemphigus foliaceus is a superficial variant of pemphigus that presents with cutaneous lesions. The mucous membranes are typically spared. Pemphigus foliaceus usually develops in a seborrheic distribution. The scalp, face, and trunk are common sites of involvement. The skin lesions usually consist of small, scattered superficial blisters that rapidly evolve into scaly, crusted erosions. The Nikolsky sign often is present. The skin lesions may remain localized or may coalesce to cover large areas of skin. Occasionally, pemphigus foliaceus progresses to involve the entire skin surface as an exfoliative erythroderma. Key features: Grouped vesicles or pustules and erythematous plaques with crusts, subcorneal or intraepidermal blisters, autoantibodies against desmocollin 1 Subtypes: Subcorneal pustular dermatosis-type IgA pemphigus (Sneddon-Wilkinson disease), intraepidermal neutrophilic IgA dermatosis The subcorneal pustular dermatosis type of IgA pemphigus is clinically similar to classic subcorneal pustular dermatosis (Sneddon-Wilkinson disease). Immunofluorescence studies are necessary to distinguish these diseases. Both the subcorneal pustular dermatosis and intraepidermal neutrophilic IgA dermatosis types of IgA pemphigus are characterized by the subacute development of vesicles that evolve into pustules. The vesicles and pustules are usually, but not always, accompanied by erythematous plaques. A herpetiform, annular, or circinate pattern may be present. The trunk and proximal extremities are common sites for involvement. The scalp, postauricular skin, and intertriginous areas are less common sites for lesion development. Pruritus may or may not be present. Mucous membranes are usually spared. Key features: Extensive, intractable stomatitis and variable cutaneous findings; associated neoplastic disease; suprabasal acantholytic blisters; autoantibodies against desmoplakins or other desmosomal antigens Life-threatening pulmonary involvement consistent with bronchiolitis obliterans also may be seen. Paraneoplastic pemphigus is the rarest form of the pemphigus types. Paraneoplastic pemphigus (also known as paraneoplastic autoimmune multiorgan syndrome) is an autoimmune multi-organ syndrome associated with neoplastic disease. Typically, patients suffer from severe and acute mucosal involvement with extensive, intractable stomatitis. The cutaneous manifestations are variable, and include blisters, erosions, and lichenoid lesions that may resemble other autoimmune blistering diseases, erythema multiforme, graft versus host disease, or lichen planus. Epidemiology Pemphigus vulgaris (the most common form of pemphigus) occurs worldwide and the frequency is influenced by geographic location and ethnicity. Incidence rates are between 0.1 and 2.7 per 100,000 people per year. The higher rates have been documented in certain populations. People of Jewish ancestry, particularly Ashkenazi Jews, and inhabitants of India, Southeast Europe, and the Middle East have the greatest risk for pemphigus vulgaris. In certain locations, such as North Africa, Turkey, and South America, the prevalence of pemphigus foliaceus exceeds pemphigus vulgaris. Pemphigus usually occurs in adults, with an average age of onset between 40 to 60 years of age for pemphigus vulgaris and nonendemic pemphigus foliaceus. Pemphigus is rare in children, with the exception of endemic pemphigus foliaceus, which affects children and young adults in endemic areas. Neonatal pemphigus is a rare transient form of pemphigus that occurs as a consequence of placental transmission of autoantibodies to the fetus from a mother with the disease. A few studies have found large imbalances in the sex distribution, such as a study that found a 4:1 ratio of females to males with pemphigus foliaceus in Tunisia and a study that found a 19:1 ratio of males to females in an endemic location in Columbia. Epidemiological information on IgA pemphigus is sparse. The disorder may occur at any age and may be slightly more common in females. Paraneoplastic pemphigus is extremely rare and is more common in middle-aged adults, but may occur in children. Pathogenesis The molecular mechanisms where the binding of autoantibodies to epithelial cells leads to acantholysis are still intensively debated. Several mechanisms for antibody-mediated acantholysis have been proposed, including the induction of signal transduction events that trigger cell separation and the inhibition of adhesive molecule function through steric hindrance. In particular, the theory of apoptolysis suggests that acantholysis results from autoantibody-mediated induction of cellular signals that trigger enzymatic cascades that lead to structural collapse of cells and cellular shrinkage. Autoantibodies against a variety of epithelial cell surface antigens have been identified in patients with pemphigus. Desmogleins are the antigens that have been most extensively studied in pemphigus vulgaris and pemphigus foliaceus. Desmogleins are components of desmosomes, integral structures for cell-to-cell adhesion. As with many other autoimmune diseases, the precipitating factors of pemphigus diseases are poorly understood. Both genetic and environmental factors may influence the development of pemphigus. Ultraviolet radiation has been proposed as an exacerbating factor for pemphigus foliaceus and pemphigus vulgaris, and pemphigus has been reported to develop following burns or cutaneous electrical injury. Viral infections, certain food compounds, ionizing radiation, and pesticides have been suggested as additional stimuli for this disease. Pemphigoid Pemphigoid is a group of subepidermal, blistering autoimmune diseases that primarily affect the skin, especially the lower abdomen, groin, and flexor surfaces of the extremities. Here, autoantibodies (anti-BPA-2 and anti-BPA-1) are directed against the basal layer of the epidermis and mucosa. The condition tends to persist for months or years with periods of exacerbation and remission. Localized variants of the condition have been reported, most often limited to the lower extremities and usually affecting women. There are two predominant types of pemphigoid: mucous membrane pemphigoid (MMP) also called cicatricial pemphigoid, and bullous pemphigoid (BP). Pathogenesis and management are quite different for these conditions. Scar formation in mucous membrane pemphigoid can lead to major disability. Types of pemphigoid Mucous membrane pemphigoid (MMP) is a chronic autoimmune disorder characterized by blistering lesions that primarily affect the various mucous membranes of the body, but also affects the skin (MMP is now the preferred term for lesions only involving the mucosa). It is also known as Cicatricial Pemphigoid (CP), as it is often scarring. The mucous membranes of the mouth and eyes are most often affected, but those of the nose, throat, genitalia, and anus may also be affected. The symptoms of MMP vary among affected individuals depending upon the specific site(s) involved and the progression of the disease. Disease onset is usually between 40 and 70 years and oral lesions are seen as the initial manifestation of the disease in about two thirds of the cases. Blistering lesions eventually heal, sometimes with scarring. Progressive scarring may potentially lead to serious complications affecting the eyes and throat. There is no racial or ethnic predilection although most studies have demonstrated a female to male ratio of approximately 2:1. The diagnosis of MMP is mainly based on history, clinical examination and biopsy of the lesions. Bullous Pemphigoid (BP) is a subepidermal blistering autoimmune disease that primarily affects the skin, especially the lower abdomen, groin, and flexor surfaces of the extremities. Mucous membrane involvement is seen in 10%-40% of patients. The disease tends to persist for months or years with periods of exacerbation and remission. The spectrum of presentations is extremely broad, but typically there is an itchy eruption with widespread blistering, and tense vesicles and bulla (blisters), with clear fluid (can be hemorrhagic) on apparently normal or slightly erythematous skin. Lesions normally appear on the torso and flexures, particularly on the inner thighs. Blisters can range in size from a few millimeters to several centimeters, and although pruritic, typically heal without scarring. Sometimes erosions and crusting is seen. Also itchy bumps (papules) and crusts (plaques) can be seen with an annular or figurate pattern. A characteristic feature is that multiple bullae usually arise from large (palm-sized or larger), irregular, urticarial plaques. Mucosal (oral, ocular, genital) involvement is also sometimes present, but ocular involvement, is rare. BP can be difficult to diagnose in its ‘non-blistering’ stage, when just itchy, red, elevated patches are visible. Erosions are much less common than in pemphigus, and the Nikolsky sign is negative. BP is characterized by spontaneous remissions followed by flares in disease activity that can persist for years. Even without therapy, BP is often self-limited, resolving after a period of many months to years, but may become very extensive. Localized variants of the disease have been reported, most often limited to the lower extremities and usually affecting women. One such variant, localized vulvar pemphigoid, reported in girls aged 6 months to 8 years, presents with recurrent vulvar vesicles and ulcerations that do not result in scarring. Bullous pemphigoid is distinguished from other blistering skin diseases, such as linear IgA dermatosis, epidermolysis bullosa acquisita, and MMP/cicatricial pemphigoid, by the following clinical items (it can also be distinguished by biopsy and certain immunological tests): Absence of atrophic scars; Absence of head and neck involvement; Relative absence of mucosal involvement. Pemphigoid gestationis (PG) is a rare autoimmune bullous dermatosis of pregnancy. The disease was originally named herpes gestationis on the basis of the morphological herpetiform feature of the blisters, but this term is a misnomer because PG is not related to or associated with any active or prior herpes virus infection. PG typically manifests during late pregnancy, with an abrupt onset of extremely pruritic urticarial papules and blisters on the abdomen and trunk, but lesions may appear any time during pregnancy, and dramatic flares can occur at or immediately after delivery. PG usually resolves spontaneously within weeks to months after delivery. Epidemiology Bullous pemphigoid: BP is the most frequent blistering disease of the skin (and mucosa) affecting typically the elderly (>65 years), but can occur at any age and in any race. Overall incidence: ± 7-10 new cases per million inhabitants per year. After the age of 70 incidence significantly increases. Relative risk for patients > 90 y have a 300-fold higher than those < 60. Women and men equally afflicted. Precipitating factors include trauma, burns, ionizing radiation, ultraviolet light, and certain drugs such as neuroleptics and diuretics, particularly lasix (furosemide), thiazides, and aldosterone antagonists. Correlations between BP flare activity and recurrence of underlying cancer suggest such an association in some patients. Even without therapy, BP can be self-limited, resolving after a period of many months to years, but is still a serious condition especially in the elderly. 1-y survival probability may be as high as 88.96% (standard error 5.21%), with a 95% confidence interval (75.6%, 94.2%) but other analyses have documented 1 year mortalities of as much as 25-30% in moderate to severe pemphigus even on therapy. Genetics:Genetic predisposition, but not hereditary Pemphigoid gestationis: Is a condition of pregnancy (childbearing age women). In the United States, PG has an estimated prevalence of 1 case in 50,000-60,000 pregnancies. No increase in fetal or maternal mortality has been demonstrated, although a greater prevalence of premature and small-for-gestational-age (SGA) babies is associated with PG. Patients with PG have a higher relative prevalence of other autoimmune diseases, including Hashimoto thyroiditis, Graves disease, and pernicious anemia. Histology The earliest lesion of BP is a blister arising in the lamina lucida, between the basal membrane of keratinocytes and the lamina densa. This is associated with loss of anchoring filaments and hemidesmosomes. Histologically, there is a superficial inflammatory cell infiltrate and a subepidermal blister without necrotic keratinocytes. The infiltrate consists of lymphocytes and histiocytes and is particularly rich in eosinophils. There is no scarring. Approximately 70% to 80% of patients with active BP have circulating antibodies to one or more basement membrane zone antigens. Autoantibodies to BP180 (and BP230). BP180 and BP230 are two components of hemidesmosomes, junctional adhesion complexes. T cell autoreactive response to BP180 and BP230 regulate autoantibody production. On direct immunofluorescence, the antibodies are deposited in a thin linear pattern; and on immune electron microscopy, they are present in the lamina lucida. (By contrast, the antibodies to basement membrane zone antigens that are present in cutaneous lupus erythematosus are deposited in a granular pattern). English▼ Arabic Bengali Chinese (Simplified) English French German Greek Hebrew Hindi Italian Japanese Persian Portuguese Punjabi Russian Spanish Thai Vietnamese
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Oral management strategies for radiotherapy of head and neck cancer - ScienceDirect Skip to main contentSkip to article Journals & Books ViewPDF Download full issue Search ScienceDirect Outline Summary Keywords 1. Introduction 2. Oral management strategies for head and neck cancer radiotherapy 3. After radiotherapy Conflict of interest Acknowledgment References Show full outline Cited by (62) Tables (1) Table 1 Japanese Dental Science Review Volume 56, Issue 1, November 2020, Pages 62-67 Review Article Oral management strategies for radiotherapy of head and neck cancer Author links open overlay panel Yumiko Kawashita a, Sakiko Soutome a, Masahiro Umeda b, Toshiyuki Saito c Show more Outline Add to Mendeley Share Cite rights and content Under a Creative Commons license Open access Summary Radiotherapy, often with concomitant chemotherapy, has a significant role in the management of head and neck cancer, however, radiotherapy induces adverse events include oral mucositis, hyposalivation, loss of taste, dental caries, osteoradionecrosis, and trismus, all of which have an impact on patients’ quality of life. Therefore, it is necessary to implement oral management strategies prior to the initiation of radiotherapy in patients with head and neck cancer. Since 2014, the National Comprehensive Cancer Network Clinical Practice Guidelines in Oncology (NCCN Guidelines) have enumerated the “Principles of Dental Evaluation and Management (DENT-A)” in the section on head and neck cancers, however, oral management was not explained in detail. Oral management has not been achieved a consensus protocol. The aim of this literature is to show that oral management strategy include removal infected teeth before the start of radiotherapy to prevent osteoradionecrosis, oral care for preventing severe oral mucositis to support patient complete radiotherapy during radiotherapy, and prevent of dental caries followed by osteoradionecrosis after radiotherapy. Previous article in issue Next article in issue Keywords Head and neck cancer Radiotherapy Oral management 1. Introduction Cancers of the head and neck cancer represent 5% of all cancers. In 2018, they accounted for an estimated 887,649 new cancer cases and 453,307 cancer-related deaths worldwide . Head and neck cancer is a broad term that encompasses epithelial malignancies arising from the paranasal sinuses, nasal cavity, oral cavity, pharynx, larynx, and salivary glands. Almost all of these epithelial malignancies are squamous cell carcinomas of the head and neck, for which the most important risk factors are tobacco and alcohol consumption . Although their incidence is rare, salivary gland tumors are head and neck cancers that include various histopathological subtypes. Radiotherapy (‘radiation therapy’ or ‘irradiation’) is defined as ‘the use of high-energy radiation from X-rays, gamma rays, neutrons, protons, and other sources to kill cancer cells and shrink tumors’ . Radiotherapy, often with concomitant chemotherapy, has a significant role in the ‘curative’ management of head and neck cancer. Primary chemo-radiation allows preservation of organ function, and is the treatment of choice for tumors arising in the oropharynx, nasopharynx, hypopharynx, and larynx [4,5]. In oral cavity cancers, the best cure rates are obtained using surgical techniques with adjuvant or post-operative radiotherapy (with or without chemotherapy) . Radiotherapy also plays an important role in the palliation of symptoms in patients with advanced/incurable head and neck cancer, offering shrinkage of tumors, prevention of ulceration, prevention of bleeding, and pain control [7,8]. Radiotherapy to the head and neck region may cause undesirable radiotherapy-induced changes in the surrounding tissues . Radiotherapy-induced adverse events include oral mucositis, hyposalivation, loss of taste, dental caries, osteoradionecrosis, and trismus, all of which have an impact on patients’ quality of life. Therefore, it is necessary to implement oral management strategies prior to the initiation of radiotherapy in patients with head and neck cancer. Since 2014, the National Comprehensive Cancer Network Clinical Practice Guidelines in Oncology (NCCN Guidelines) have enumerated the “Principles of Dental Evaluation and Management (DENT-A) ” in the section on head and cancers. Kawashita et al. have also demonstrated the benefits of the use of the prophylactic bundle in oral management prior to initiation of radiotherapy . In this review, we discuss the oral management strategies that are in use for head and neck cancer, both, prior to the initiation of radiotherapy and following treatment. 2. Oral management strategies for head and neck cancer radiotherapy 2.1. Prior to radiotherapy 2.1.1. Tooth extraction before the start of radiotherapy to prevent osteoradionecrosis Osteoradionecrosis (ORN) of the jaw is defined as a non-healing exposure of the bone with necrosis, which starts with a breach in the oral mucosa and persists for at least 3 months in a patient who has undergone previous radiotherapy. The necrosis, however, must be evidently different from a recurrent, vestigial, or metastatic tumor . Prior to the well-known alterations of the “three–H concept” (hypoxia, hypocellularity, hypovascularity), that become apparent in the vascular system , radiogenic effects initially appear in osteoclasts [16,17]; reports suggest that microorganisms do not play any causative role in ORN, but have a contaminant role instead. The bone shows significant fibrosis with a loss of remodeling elements. The aim of oral management before the start of radiotherapy is to prevent ORN. Although the incidence of ORN is low, it rarely cures spontaneously once it occurs; in patients with advanced lesions, surgical resection of the jaw becomes necessary . Therefore, dental evaluation of the source of infection and the need for dental extractions need to be determined . If necessary, extractions should be completed at least 2 weeks prior to the start of radiotherapy. As the source of infection in the jaw, pre-radiotherapy periapical foci were reported to be an independent risk factor for the development of ORN . Since most cases of ORN occur in the molar region of the lower jaw, the mandibular molar, with the periapical focus, may need to be extracted. Furthermore, tooth extraction after radiotherapy has also been found to significantly correlate with the development of ORN. Therefore, teeth that cannot be preserved for a long time should be extracted before the start of radiotherapy. Periodontal diseases are required extraction before radiotherapy to avoid future dental extraction and to reduce the development of ORN [20,21]. The German Society of Dental, and Oral and Craniomandibular Sciences show criteria for tooth removal before radiotherapy are periodontal probing depth equal or greater than 5 mm and furcation involvement. Another important risk factor for the development of osteonecrosis is the radiation dose to the bone, particularly to the less vascular mandible [22,23]. A radiation dose of 50 Gy or higher to the mandible significantly increased the risk of ORN [24,25]. 2.1.2. Preparation of spacers to prevent serious oral mucositis Radiotherapy for head and neck cancer is broadly classified into two types, namely, external irradiation and brachytherapy. External irradiation is most commonly employed. It generally involves the use of linear accelerators that direct X-ray and/or electron beams from outside the body into the tumor. Any existing dental metals produce an electronic backscatter, which may damage the surrounding soft tissue . Backscatter effects on the surface of dental materials cause an increase of up to 170% of the radiation dose, measured without materials. It has also been reported that the extent of the backscatter effect reaches maximal levels within a distance of 4 mm. Therefore, in some cases, a spacer retainer, also known as a spacer, is placed as appropriate. The thickness of the spacers are typically 3 mm, reaching up to 5 mm for cases with metal restorations . Recently, high-precision radiotherapy, such as intensity modulated radiotherapy (IMRT) is being widely used owing to the superior efficacy of IMRT in avoiding side effects compared with three-dimensional conformal radiotherapy (3D-CRT). IMRT has the advantage of allowing more precise dose delivery to the tumor site, while simultaneously reducing the exposure of normal tissues to radiation. Therefore, more accurate and reproducible patient fixation is considerably more important in IMRT than in 3D-CRT. In radiotherapy treatment plans, the target volumes to be irradiated are clearly defined as the gross tumor volume (GTV), clinical target volume (CTV), and planning target volume (PTV). The GTV is the tumor volume, which is generally delineated by diagnostic images, inspection, and/or palpation of both the primary and metastatic lesions. The CTV is the tissue volume that encompasses the GTV and any regions of subclinical disease, including lymph node areas designated for prophylactic irradiation. In radiotherapy for head and neck cancers, usually a 5- to 10-mm (or greater) margin is added around the GTVs. In view of internal organ motion and variations in daily setup (set-up margin), the PTV is expanded between 5 and 10 mm around the CTVs. In high-precision radiotherapy such as IMRT, it is important to decrease the PTV margin accounting for organ movement and set-up errors. Therefore, ready-made patient immobilization spacers are employed in head and neck IMRT to decrease rotation, flexion, and extension of the head. 2.2. During radiotherapy Acute adverse events associated with radiotherapy include oral mucositis, xerostomia, and loss of taste. Radiation-induced oral mucositis is considerably unpleasant and is more pronounced in patients undergoing chemoradiotherapy. In extreme cases, lesions are characterized by large and painful ulcers that have a significant impact on the patient’s quality of life, and may considerably restrict activities such as eating, speaking, and even swallowing saliva . Oral mucositis involves breaks in the tight junction between cells, which allows the development of bacterial infections [28,29]. The aim of oral management during radiotherapy is to prevent severe oral mucositis and related secondary infections; it also intends to control pain and support ingestion. Oral mucositis management should involve a defined preventive oral care regimen that includes aggressive implementation of oral hygiene procedures including brushing, flossing, and the use of bland rinses. Although these methods do not prevent mucositis or impact the severity of the lesions per se, they provide essential support to the oral cavity and may indirectly improve treatment compliance by decreasing the risks of infection . 2.2.1. Use of pilocarpine hydrochloride Salivary gland dysfunction is a predictable side effect of radiotherapy to the head and neck region . Salivary glands are sensitive to radiation, and even low doses result in a rapid decline in function [32,33]. This develops soon after the initiation of radiotherapy, progresses during treatment (and for some time after treatment), and it essentially permanent in cases where cumulative doses exceed 30 Gy . Patients develop xerostomia (subjective symptoms of dryness) and hyposalivation (objective reduction in salivary flow). Hyposalivation may further aggravate inflamed tissues, increase the risk of local infection, and make mastication difficult. Many patients complain about the thickening of salivary secretions owing to a decrease in the serous component of saliva. In healthy people, the average salivary flow rate is approximately 1.0 ml/min. This rate dramatically declines to well below 0.5 ml/min within 1–2 weeks of initiating radiotherapy . Pilocarpine is a parasympathomimetic agent that functions primarily as a muscarinic agonist, causing pharmacological stimulation of exocrine glands in humans; this results in salivation . As per the Cochrane Collaboration, pilocarpine hydrochloride is more effective than placebo and is at least as effective as artificial saliva ; the response rate ranges from 42% to 51% with a time to response of up to 12 weeks. The overall side effect rate has been found to be high, with side effects being the main reason for withdrawal in study patients (6%–15% of participants taking 5 mg thrice a day had to withdraw). The side effects usually result from generalized parasympathomimetic stimulation and include sweating, headaches, urinary frequency, and vasodilatation. Dose dependence has not been noted in response rates, but in the rates of side effects. Study patients with side effects have been administered a dose of 2.5 mg 4 times daily . Pilocarpine is contraindicated in patients with obstructive pulmonary disease, severe ischemic heart disease, stricture of the gastrointestinal tract or the bladder neck, Parkinson’s disease, and iritis. 2.2.2. Oral care Patients usually receive professional oral care by a dental hygienist at least once a week until completion of radiotherapy . Modalities for oral care typically include the removal of dental plaque using professional mechanical tooth-cleaning methods and the gentle removal of mucosal debris using a wet sponge to keep the oral cavity as clean as possible. For palliation of a dry mouth, it is advisable to sip water as needed to alleviate mouth dryness . Several supportive products including artificial saliva are also available. In addition, it is also advisable to rinse the mouth with a solution made from ½ a teaspoon of baking soda (and/or ¼ or ½ a teaspoon of table salt) in 1 cup of warm water several times a day to clean and lubricate the oral tissues and to buffer the oral environment. Chlorhexidine and povidone-iodine are 2 of the most commonly used antiseptic agents in this setting . Chlorhexidine and povidone-iodine have different mechanisms of action and different spectrums of efficacy. Chlorhexidine damages the outer layers of the microbial cell membrane, upsetting resting membrane potentials, whereas povidone-iodine uncouples iodine, which is absorbed by microbes, resulting in the inactivation of key cytoplasmic pathways . Povidone-iodine is useful for the prevention of oral infections. However, the MASCC/ISOO (Multinational Association of Supportive Care in Cancer in Cancer and International Society of Oral Oncology) Clinical Practice Guidelines for oral mucositis suggest that chlorhexidine mouthwashes should not be used to prevent oral mucositis in patients receiving radiation therapy for head and neck cancer . Oral viscous lidocaine is useful for the treatment of symptoms related to inflamed oral mucosa, including radiation- or chemotherapy-induced mucositis . Since the patient is instructed to spit out the solution after each use, the number of daily treatments with viscous lidocaine mouthwash are not restricted. However, particular attention must be paid to aspiration and to any biting of the buccal mucosa or tongue. 2.3. Pathogenesis of oral mucositis Radiotherapy for head and neck cancer almost always induces oral mucositis. The pathophysiologic progression that results in mucositis may be described in 5 phases: initiation, upregulation and message generation, signaling and amplification, ulceration, and healing [37,42]. 2.3.1. Initiation of tissue injury Radiation and/or chemotherapy induce cellular damage, which results in the death of the basal epithelial cells. The generation of reactive oxygen species (free radicals) by radiation or chemotherapy is also believed to exert a role in the initiation of mucosal injury. These small highly reactive molecules are byproducts of oxygen metabolism and may cause significant cellular damage. 2.3.2. Upregulation of inflammation via generation of messenger signals In addition to causing direct cell death, free radicals activate second messengers that transmit signals from receptors on the cellular surface to the inside of the cell. This leads to upregulation of pro-inflammatory cytokines, tissue injury and cell death. 2.3.3. Signaling and amplification Upregulation of proinflammatory cytokines such as tumor necrosis factor- alpha (TNF-α), produced mainly by macrophages, causes injury to mucosal cells, and also activates molecular pathways that amplify mucosal injury 2.3.4. Ulceration and inflammation Mucosal ulcerations are associated with a significant inflammatory cell infiltrate, partly related to the metabolic byproducts of the colonizing oral microflora. The secondary infection also further upregulates the production of pro-inflammatory cytokines. 2.3.5. Healing The healing phase is characterized by epithelial proliferation and cellular and tissue differentiation, restoring the integrity of the epithelium. 2.4. Clinical course of oral mucositis The lesions of radiation-induced oral mucositis are limited to tissues within the field of radiation, with the involvement of non-keratinized tissue being more common. The initial clinical signs of oral mucositis include mucosal erythema and superficial sloughing, which may occur with cumulative radiation doses of 20−30 Gy, at which, the intact mucosa begins to break down; this is followed by ulceration . The ulcerations are typically covered by a white fibrinous pseudomembrane. The clinical severity is directly proportional to the dose of radiation administered. Most patients who receive more than 50 Gy to the oral mucosa develop severe ulcerative oral mucositis . The lesions typically heal within approximately 2–4 weeks after the last fraction of radiotherapy. 2.5. Evaluation of oral mucositis A wide variety of scales have been used in clinical practice and research to record the extent and severity of oral mucositis (Table 1) [45,46]. The World Health Organization (WHO) Oral Toxicity Scale and the National Cancer Institute-Common Toxicity Criteria for Adverse Events (NCI-CTCAE) system are two most commonly used scales (Table 1). The WHO Oral Toxicity Scale is a simple and easy-to-use tool that is suitable for wide implementation in both, research and clinical practice settings. To assign a grade, this scale combines objective mucosal changes (such as erythema and ulceration) with functional outcomes (such as the ability to eat) . The NCI-CTCAE, a longstanding empirically developed system, has similar impact and applicability as the dedicated WHO Oral Toxicity Scale. In terms of the specific criteria for grading radiotherapy-related oral mucositis in head and neck cancer, differences have been observed in terms of subjective variables such as pain, dysphagia, and eating behavior in version 4.0; version 3.0 (clinical exam) mainly assessed oral mucositis based on objective signs including erythema, ulceration, and bleeding . The NCI-CTCAE v5.0 is currently available . Table 1. Definition of oral mucositis by grading scales. | Grading Scale | Grade 1 | Grade 2 | Grade 3 | Grade 4 | Grade 5 | --- --- --- | | World Health Organisation (WHO) | Soreness, erythema | Ulcers but able to eat solid foods | Oral ulcers and able to take liquids only | Oral alimentation impossible | – | | NCI-CTCAE v3.0 (clinical exam) | Erythema of the mucosa | Patcy ulcerations or pseudomembranes | Confluent ulcerations or pseudomembranes; bleeding with minor trauma | Tissue necrosis; significant spontaneous bleeding; life-threatening consequenced | Death | | NCI-CTCAE v3.0 (functional/symptomatic) | Minimal symptoms, normal diet; minimal respiratory symptoms but not interfering with function | Symptomatic but can eat and swallow modified diet; respiratory symptoms interfering with function but not interfering with ADL | Symptomatic and unable to adequately aliment or hydrate orally; respiratory symptoms interfering with ADL | Symptoms associated with life-threatening consequence | Death | | NCI-CTCAE v4.0 | Asymptomatic or mild symptoms; intervention not indicated | Moderate pain; not interfering with oral intake; modified diet indicate | Sever pain; interfering with oral intake | Life-threatening consequences; urgent intervention idicated | Death | | NCI-CTCAE v5.0 | Asymptomatic or mild symptoms; intervention not indicated | Moderate pain or ulcer that does not interfere with oral intake; modified diet indicate | Sever pain; interfering with oral intake | Life-threatening consequences; urgent intervention idicated | Death | 2.6. Prevention and treatment of oral mucositis A systematic review has showed that non-opioid interventions, including topical mouthwashes: doxepin, amitriptyline, diclofenac, and benzydamine, provided relief of pain due to radiotherapy-induced oral mucositis with or without chemotherapy for head and neck cancer . However, these topical mouthwashes are not readily available in Japan. Low-level laser therapy has been suggested for the prevention of oral mucositis in patients undergoing radiotherapy without concomitant chemotherapy. However, the impact of low-level laser therapy on tumor behavior and response to treatment remains unclear . Orally administered systemic zinc supplements may offer benefit in the prevention of oral mucositis in patients receiving radiation therapy or chemoradiation for oral cancer. Currently, there is no consensus-based protocol for the prophylaxis and treatment of chemoradiotherapy-induced oral mucositis in patients with head and neck cancer . Oral mucositis should be treated with anti-inflammatory drugs. Rugo et al. showed dexamethasone mouthwash reduced the incidence and severity of oral mucositis in patients receiving chemotherapy for breast cancer . Radiotherapy-induced oral mucositis is more severe than chemotherapy-induced oral mucositis. Therefore, radiotherapy-induced oral mucositis might be treated with steroid ointment which can adhere to oral membranes allowing for longer contact. Steroid ointment therapy for radiotherapy-induced oral mucositis has been used in Japan since the 1980s. Dexamethasone, triamcinolone acetonide, and beclomethasone dipropionate ointments were suggested to be useful for radiotherapy-induced oral mucositis. A multicenter phase II randomized controlled trial showed that a combination of adequate oral hydration, optimal oral cleanliness and hygiene, and topical dexamethasone therapy was effective for preventing severe oral mucositis caused by radiotherapy alone but not chemoradiotherapy . Steroid ointments of medium potency (dexamethasone) may not prevent severe oral mucositis caused by chemoradiotherapy for head and neck cancer. However, strong or very strong topical steroid therapy may prevent severe oral mucositis. 2.7. Oral fungal infections The risk for oral infections increases during and after therapy for oropharyngeal cancer because the oral microbial flora is altered by myelosuppression, and the oral cleansing property of saliva is diminished owing to the reduced salivary flow . Candida is a normal oral commensal in healthy individuals, and hence candidiasis is one of the most frequent oral infections during therapy for oropharyngeal cancer . Oral candidiasis usually presents as a removable white pseudomembrane or erythematous patch on the tongue, palate, and labial commissures. It causes alterations in taste, mucosal soreness, and an oral burning sensation . The diagnosis of oral candidiasis is largely based on clinical features. However, occasionally, confirmatory laboratory investigations are required. Topical antifungal therapy is very effective in controlling oral candidiasis 3. After radiotherapy The goals of post-treatment dental management include the prevention and treatment of dental caries, and the prevention of post-radiation osteonecrosis . Radiotherapy to the head and neck induces xerostomia and hyposalivation. This induces the development of severe dental caries and the introduction of infections in the jaw. The status of oral health after radiotherapy has been found to be a significant risk factor for the development of ORN. Radiation-related dental caries prevention programs are therefore crucial in the control of ORN [19,61,62]. Resistance to dental caries may be enhanced by the application of topical fluorides [63,64]; fluoride toothpaste has been demonstrated to provide significant benefit in preventing and remineralising root caries in patients undergoing radiation for head and neck cancer . The efficacy of fluoride in these patients may be limited by the lack of calcium and phosphate secondary to hyposalivation . Remineralization cannot occur if the saliva lacks sufficient levels of calcium and phosphate relative to tooth minerals. Exogenous calcium and phosphate may hypothetically improve dental outcomes by allowing remineralization of dental surfaces. It has been observed that after radiotherapy-induced hyposalivation, the colony counts of microorganisms in the oral microflora demonstrate a shift with an increase in cariogenic bacteria including Streptococcus mutans and Lactobacillus species [67,68]. Therefore, reductions in the rates of dental caries should involve the reduction of colony counts of cariogenic bacteria. Chlorhexidine suspensions have been shown to reduce colonization by cariogenic flora in patients undergoing radiotherapy to the head and neck. Unfortunately, the effects are not sustained and cariogenic bacterial counts have been noted to rapidly recover. This suggests the need for ongoing therapy to control the oral flora and reduce the risks of caries . The use of topical fluorides and chlorhexidine mouth rinses may have an impact on the prevention of dental caries . In conclusion, it is essential that the dental care provider motivates patients to adopt stringent plaque control. In addition, medications should be prescribed to stimulate salivary flow, and nutritional counseling should be offered to limit cariogenic diets. These measures are essential in the reduction of radiotherapy-induced dental caries, and will serve to improve the quality of life in head and neck cancer survivors . Conflict of interest The authors declare that they have no competing interests. Acknowledgment This work was supported by JSPS KAKENHI Grant Number JP18K10275. Recommended articles References F. Bray, J. Ferlay, I. Soerjomataram, R.L. Siegel, L.A. Torre, A. Jemal Global cancer statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries CA Cancer J Clin, 68 (2018), pp. 394-424 CrossrefGoogle Scholar A. Argiris, C. Eng Epidemiology, staging, and screening of head and neck cancer Cancer Treat Res, 114 (2003), pp. 15-60 View in ScopusGoogle Scholar K. Newbold, K. Harrington Overview of complications of radiotherapy (radiation therapy) A.N. Davies, J.B. Epstein (Eds.), Oral complications of cancer and its management, Oxford University Press (2010) Google Scholar R. Kazi, R. Venkitaraman, C. Johnson, V. Prasad, P. Clarke, P. Rhys-Evans, et al. Electroglottographic comparison of voice outcomes in patients with advanced laryngopharyngeal cancer treated by chemoradiotherapy or total laryngectomy Int J Radiat Oncol Biol Phys, 70 (2008), pp. 344-352 View PDFView articleView in ScopusGoogle Scholar J.L. Lefebvre Laryngeal preservation in head and neck cancer: multidisciplinary approach Lancet Oncol, 7 (2006), pp. 747-755 View PDFView articleView in ScopusGoogle Scholar C. Scully, J.V. Bagan Recent advances in oral oncology 2008; squamous cell carcinoma imaging, treatment, prognostication and treatment outcomes Oral Oncol, 45 (2009), pp. e25-30 View in ScopusGoogle Scholar R. Ordonez, A. Otero, I. Jerez, J.A. Medina, Y. Lupianez-Perez, J. Gomez-Millan Role of radiotherapy in the treatment of metastatic head and neck cancer Onco Targets Ther, 12 (2019), pp. 677-683 CrossrefView in ScopusGoogle Scholar S.V. Porceddu, B. Rosser, B.H. Burmeister, M. Jones, B. Hickey, K. Baumann, et al. Hypofractionated radiotherapy for the palliation of advanced head and neck cancer in patients unsuitable for curative treatment—"Hypo Trial" Radiother Oncol, 85 (2007), pp. 456-462 View PDFView articleView in ScopusGoogle Scholar A. Vissink, F.R. Burlage, F.K. Spijkervet, J. Jansma, R.P. Coppes Prevention and treatment of the consequences of head and neck radiotherapy Crit Rev Oral Biol Med, 14 (2003), pp. 213-225 CrossrefView in ScopusGoogle Scholar National Comprehensive Cancer Network. Pricniples of Dental Evaluation and Management (DENT-A). NCCN Guidelines Version 3. 2019 Head and Neck Cancers. Available from: [Accessed 13 January 2020]. Google Scholar Y. Kawashita, S. Hayashida, M. Funahara, M. Umeda, T. Saito Prophylactic bundle for radiation-induced oral mucositis in oral or oropharyneal cancer patients J Cancer Res Ther, 2 (2014), pp. 9-13 Google Scholar C. Madrid, M. Abarca, K. Bouferrache Osteoradionecrosis: an update Oral Oncol, 46 (2010), pp. 471-474 View PDFView articleView in ScopusGoogle Scholar N. Rice, I. Polyzois, K. Ekanayake, O. Omer, L.F. Stassen The management of osteoradionecrosis of the jaws—a review Surgeon, 13 (2015), pp. 101-109 View PDFView articleCrossrefView in ScopusGoogle Scholar S. Nabil, N. Samman Incidence and prevention of osteoradionecrosis after dental extraction in irradiated patients: a systematic review Int J Oral Maxillofac Surg, 40 (2011), pp. 229-243 View PDFView articleView in ScopusGoogle Scholar R.E. Marx Osteoradionecrosis: a new concept of its pathophysiology J Oral Maxillofac Surg, 41 (1983), pp. 283-288 View PDFView articleView in ScopusGoogle Scholar L.A. Assael New foundations in understanding osteonecrosis of the jaws J Oral Maxillofac Surg, 62 (2004), pp. 125-126 View PDFView articleView in ScopusGoogle Scholar B. Al-Nawas, H. Duschner, K.A. Grotz Early cellular alterations in bone after radiation therapy and its relation to osteoradionecrosis J Oral Maxillofac Surg, 62 (2004), p. 1045 View PDFView articleView in ScopusGoogle Scholar M. Akashi, S. Wanifuchi, E. Iwata, D. Takeda, J. Kusumoto, S. Furudoi, et al. Differences between osteoradionecrosis and medication-related osteonecrosis of the jaw Oral Maxillofac Surg, 22 (2018), pp. 59-63 View in ScopusGoogle Scholar Y. Kojima, S. Yanamoto, M. Umeda, Y. Kawashita, I. Saito, T. Hasegawa, et al. Relationship between dental status and development of osteoradionecrosis of the jaw: a multicenter retrospective study Oral Surg Oral Med Oral Pathol Oral Radiol, 124 (2017), pp. 139-145 View PDFView articleView in ScopusGoogle Scholar M.S. Irie, E.M. Mendes, J.S. Borges, L.G. Osuna, G.D. Rabelo, P.B. Soares Periodontal therapy for patients before and after radiotherapy: a review of the literature and topics of interest for clinicians Med Oral Patol Oral Cir Bucal, 23 (2018), pp. e524-e530 View in ScopusGoogle Scholar H.Y. Sroussi, J.B. Epstein, R.J. Bensadoun, D.P. Saunders, R.V. Lalla, C.A. Migliorati, et al. Common oral complications of head and neck cancer radiation therapy: mucositis, infections, saliva change, fibrosis, sensory dysfunctions, dental caries, periodontal disease, and osteoradionecrosis Cancer Med, 12 (2017), pp. 2918-2931 CrossrefView in ScopusGoogle Scholar R.B. Morrish Jr., E. Chan, S. Silverman Jr., J. Meyer, K.K. Fu, D. Greenspan Osteonecrosis in patients irradiated for head and neck carcinoma Cancer, 47 (1981), pp. 1980-1983 View in ScopusGoogle Scholar A.A. Owosho, C.J. Tsai, R.S. Lee, H. Freymiller, A. Kadempour, S. Varthis, et al. The prevalence and risk factors associated with osteoradionecrosis of the jaw in oral and oropharyngeal cancer patients treated with intensity-modulated radiation therapy (IMRT): the Memorial Sloan Kettering Cancer center experience Oral Oncol, 64 (2017), pp. 44-51 View PDFView articleView in ScopusGoogle Scholar I.J. Lee, W.S. Koom, C.G. Lee, Y.B. Kim, S.W. Yoo, K.C. Keum, et al. Risk factors and dose-effect relationship for mandibular osteoradionecrosis in oral and oropharyngeal cancer patients Int J Radiat Oncol Biol Phys, 75 (2009), pp. 1084-1091 View PDFView articleView in ScopusGoogle Scholar D.R. Gomez, C.L. Estilo, S.L. Wolden, M.J. Zelefsky, D.H. Kraus, R.J. Wong, et al. Correlation of osteoradionecrosis and dental events with dosimetric parameters in intensity-modulated radiation therapy for head-and-neck cancer Int J Radiat Oncol Biol Phys, 81 (2011), pp. e207-213 View in ScopusGoogle Scholar B. Reitemeier, G. Reitemeier, A. Schmidt, W. Schaal, P. Blochberger, D. Lehmann, et al. Evaluation of a device for attenuation of electron release from dental restorations in a therapeutic radiation field J Prosthet Dent, 87 (2002), pp. 323-327 View PDFView articleView in ScopusGoogle Scholar A.L. Watters, J.B. Epstein, M. Agulnik Oral complications of targeted cancer therapies: a narrative literature review Oral Oncol, 47 (2011), pp. 441-448 View PDFView articleView in ScopusGoogle Scholar N. Al-Dasooqi, S.T. Sonis, J.M. Bowen, E. Bateman, N. Blijlevens, R.J. Gibson, et al. Emerging evidence on the pathobiology of mucositis Support Care Cancer, 21 (2013), pp. 3233-3241 CrossrefView in ScopusGoogle Scholar Y. Kawashita, M. Funahara, M. Yoshimatsu, N. Nakao, S. Soutome, T. Saito, et al. A retrospective study of factors associated with the development of oral candidiasis in patients receiving radiotherapy for head and neck cancer: Is topical steroid therapy a risk factor for oral candidiasis? Medicine (Baltimore), 97 (2018), Article e13073 CrossrefView in ScopusGoogle Scholar T. Yokota, H. Tachibana, T. Konishi, T. Yurikusa, S. Hamauchi, K. Sakai, et al. Multicenter phase II study of an oral care program for patients with head and neck cancer receiving chemoradiotherapy Support Care Cancer, 24 (2016), pp. 3029-3036 View in ScopusGoogle Scholar H.J. Guchelaar, A. Vermes, J.H. Meerwaldt Radiation-induced xerostomia: pathophysiology, clinical course and supportive treatment Support Care Cancer, 5 (1997), pp. 281-288 View in ScopusGoogle Scholar S.F. Cassolato, R.S. Turnbull Xerostomia: clinical aspects and treatment Gerodontology, 20 (2003), pp. 64-77 CrossrefView in ScopusGoogle Scholar L. Franzen, U. Funegard, T. Ericson, R. Henriksson Parotid gland function during and following radiotherapy of malignancies in the head and neck. A consecutive study of salivary flow and patient discomfort Eur J Cancer, 28 (1992), pp. 457-462 View PDFView articleView in ScopusGoogle Scholar J.T. Johnson, G.A. Ferretti, W.J. Nethery, I.H. Valdez, P.C. Fox, D. Ng, et al. Oral pilocarpine for post-irradiation xerostomia in patients with head and neck cancer N Engl J Med, 329 (1993), pp. 390-395 View in ScopusGoogle Scholar A.N. Davies, J. Thompson Parasympathomimetic drugs for the treatment of salivary gland dysfunction due to radiotherapy Cochrane Database Syst Rev, 10 (2015), Article CD003782 View in ScopusGoogle Scholar H. Iwabuchi, E. Iwabuchi, K. Uchiyama, T. Fujibayashi Study on reducing adverse reactions to piloarpine hydrochloride -possibility of reducing sweating by adoption of multiple divided low-dose treatment J Jpn Soc Oral Mucous Membr, 16 (2010), pp. 17-23 [in Japanese] CrossrefGoogle Scholar R.V. Lalla, S.T. Sonis, D.E. Peterson Management of oral mucositis in patients who have cancer Dent Clin North Am, 52 (2008), pp. 61-77 viii View PDFView articleView in ScopusGoogle Scholar J.C. Dumville, E. McFarlane, P. Edwards, A. Lipp, A. Holmes, Z. Liu Preoperative skin antiseptics for preventing surgical wound infections after clean surgery Cochrane Database Syst Rev (2015), Article CD003949 View in ScopusGoogle Scholar C. Anderson, G. Uman, A. Pigazzi Oncologic outcomes of laparoscopic surgery for rectal cancer: a systematic review and meta-analysis of the literature Eur J Surg Oncol, 34 (2008), pp. 1135-1142 View PDFView articleView in ScopusGoogle Scholar R.V. Lalla, J. Bowen, A. Barasch, L. Elting, J. Epstein, D.M. Keefe, et al. MASCC/ISOO clinical practice guidelines for the management of mucositis secondary to cancer therapy Cancer, 120 (2014), pp. 1453-1461 CrossrefView in ScopusGoogle Scholar E.B. Rubenstein, D.E. Peterson, M. Schubert, D. Keefe, D. McGuire, J. Epstein, et al. Clinical practice guidelines for the prevention and treatment of cancer therapy-induced oral and gastrointestinal mucositis Cancer, 100 (2004), pp. 2026-2046 View in ScopusGoogle Scholar S.T. Sonis A biological approach to mucositis J Support Oncol, 2 (2004), pp. 21-32 discussion 35–26 View in ScopusGoogle Scholar S.M. Bentzen, M.I. Saunders, S. Dische, S.J. Bond Radiotherapy-related early morbidity in head and neck cancer: quantitative clinical radiobiology as deduced from the CHART trial Radiother Oncol, 60 (2001), pp. 123-135 View PDFView articleView in ScopusGoogle Scholar J.B. Epstein, M. Gorsky, A. Guglietta, N. Le, S.T. Sonis The correlation between epidermal growth factor levels in saliva and the severity of oral mucositis during oropharyngeal radiation therapy Cancer, 89 (2000), pp. 2258-2265 View in ScopusGoogle Scholar S.T. Sonis, L.S. Elting, D. Keefe, D.E. Peterson, M. Schubert, M. Hauer-Jensen, et al. Perspectives on cancer therapy-induced mucosal injury: pathogenesis, measurement, epidemiology, and consequences for patients Cancer, 100 (2004), pp. 1995-2025 View in ScopusGoogle Scholar H.M. Chaudhry, A.J. Bruce, R.C. Wolf, M.R. Litzow, W.J. Hogan, M.S. Patnaik, et al. The incidence and severity of oral mucositis among allogeneic hematopoietic stem cell transplantation patients: a systematic review Biol Blood Marrow Transplant, 22 (2016), pp. 605-616 View PDFView articleView in ScopusGoogle Scholar WHO Handbook for reporting results of cancer treatment World Health Organization, Geneva, Switzerland (1979) Google Scholar Y.J. Liu, G.P. Zhu, X.Y. Guan Comparison of the NCI-CTCAE version 4.0 and version 3.0 in assessing chemoradiation-induced oral mucositis for locally advanced nasopharyngeal carcinoma Oral Oncol, 48 (2012), pp. 554-559 View PDFView articleView in ScopusGoogle Scholar National Cancer Institute, National Institutes of Health, U.S. Department of Health and Human Services. Common Terminology Criteria for Adverse Events (CTCAE) Version 5.0 2017. Available from: [Accessed 19 August 2019]. Google Scholar J. Christoforou, J. Karasneh, M. Manfredi, B. Dave, J.S. Walker, P.D. Dios, et al. World workshop on oral medicine VII: non-opioid pain management of head and neck chemo/radiation-induced mucositis: a systematic review Oral Dis, 25 Suppl. 1 (2019), pp. 182-192 CrossrefView in ScopusGoogle Scholar J.A. Zecha, J.E. Raber-Durlacher, R.G. Nair, J.B. Epstein, S.T. Sonis, S. Elad, et al. Low level laser therapy/photobiomodulation in the management of side effects of chemoradiation therapy in head and neck cancer: part 1: mechanisms of action, dosimetric, and safety considerations Support Care Cancer, 24 (2016), pp. 2781-2792 CrossrefView in ScopusGoogle Scholar D. Moslemi, A.M. Nokhandani, M.T. Otaghsaraei, Y. Moghadamnia, S. Kazemi, A.A. Moghadamnia Management of chemo/radiation-induced oral mucositis in patients with head and neck cancer: a review of the current literature Radiother Oncol, 120 (2016), pp. 13-20 View PDFView articleView in ScopusGoogle Scholar J.E. Clarkson, H.V. Worthington, S. Furness, M. McCabe, T. Khalid, S. Meyer Interventions for treating oral mucositis for patients with cancer receiving treatment Cochrane Database Syst Rev (2010), Article CD001973 View in ScopusGoogle Scholar S.B. Jensen, V. Jarvis, Y. Zadik, A. Barasch, A. Ariyawardana, A. Hovan, et al. Systematic review of miscellaneous agents for the management of oral mucositis in cancer patients Support Care Cancer, 21 (2013), pp. 3223-3232 CrossrefView in ScopusGoogle Scholar A. Rodriguez-Caballero, D. Torres-Lagares, M. Robles-Garcia, J. Pachon-Ibanez, D. Gonzalez-Padilla, J.L. Gutierrez-Perez Cancer treatment-induced oral mucositis: a critical review Int J Oral Maxillofac Surg, 41 (2012), pp. 225-238 View PDFView articleView in ScopusGoogle Scholar H.S. Rugo, L. Seneviratne, J.T. Beck, J.A. Glaspy, J.A. Peguero, T.J. Pluard, et al. Prevention of everolimus-related stomatitis in women with hormone receptor-positive, HER2-negative metastatic breast cancer using dexamethasone mouthwash (SWISH): a single-arm, phase 2 trial Lancet Oncol, 18 (2017), pp. 654-662 View PDFView articleView in ScopusGoogle Scholar Y. Kawashita, Y. Koyama, H. Kurita, M. Otsuru, Y. Ota, M. Okura, et al. Effectiveness of a comprehensive oral management protocol for the prevention of severe oral mucositis in patients receiving radiotherapy with or without chemotherapy for oral cancer: a multicentre, phase II, randomized controlled trial Int J Oral Maxillofac Surg, 48 (2019), pp. 857-864 View PDFView articleView in ScopusGoogle Scholar L. Turner, M. Mupparapu, S.O. Akintoye Review of the complications associated with treatment of oropharyngeal cancer: a guide for the dental practitioner Quintessence Int, 44 (2013), pp. 267-279 View in ScopusGoogle Scholar H. Ahadian, S. Yassaei, F. Bouzarjomehri, M. Ghaffari Targhi, K. Kheirollahi Oral complications of the oromaxillofacial area radiotherapy Asian Pac J Cancer Prev, 18 (2017), pp. 721-725 View in ScopusGoogle Scholar J.B. Epstein, M. Gorsky, J. Caldwell Fluconazole mouthrinses for oral candidiasis in postirradiation, transplant, and other patients Oral Surg Oral Med Oral Pathol Oral Radiol Endod, 93 (2002), pp. 671-675 View PDFView articleView in ScopusGoogle Scholar K. Katsura, K. Sasai, K. Sato, M. Saito, H. Hoshina, T. Hayashi Relationship between oral health status and development of osteoradionecrosis of the mandible: a retrospective longitudinal study Oral Surg Oral Med Oral Pathol Oral Radiol Endod, 105 (2008), pp. 731-738 View PDFView articleView in ScopusGoogle Scholar W. Kobayashi, B.G. Teh, H. Kimura, S. Kakehata, H. Kawaguchi, Y. Takai Comparison of osteoradionecrosis of the jaw after superselective intra-arterial chemoradiotherapy versus conventional concurrent chemoradiotherapy of oral cancer J Oral Maxillofac Surg, 73 (2015), pp. 994-1002 View PDFView articleView in ScopusGoogle Scholar V.C. Marinho, J.P. Higgins, A. Sheiham, S. Logan Combinations of topical fluoride (toothpastes, mouthrinses, gels, varnishes) versus single topical fluoride for preventing dental caries in children and adolescents Cochrane Database Syst Rev (2004), Article CD002781 View in ScopusGoogle Scholar V.C. Marinho, J.P. Higgins, A. Sheiham, S. Logan One topical fluoride (toothpastes, or mouthrinses, or gels, or varnishes) versus another for preventing dental caries in children and adolescents Cochrane Database Syst Rev (2004), Article CD002780 View in ScopusGoogle Scholar A. Papas, D. Russell, M. Singh, R. Kent, C. Triol, A. Winston Caries clinical trial of a remineralising toothpaste in radiation patients Gerodontology, 25 (2008), pp. 76-88 CrossrefView in ScopusGoogle Scholar N.J. Cochrane, F. Cai, N.L. Huq, M.F. Burrow, E.C. Reynolds New approaches to enhanced remineralization of tooth enamel J Dent Res, 89 (2010), pp. 1187-1197 CrossrefView in ScopusGoogle Scholar J.B. Epstein, B.C. McBride, P. Stevenson-Moore, H. Merilees, J. Spinelli The efficacy of chlorhexidine gel in reduction of Streptococcus mutans and Lactobacillus species in patients treated with radiation therapy Oral Surg Oral Med Oral Pathol, 71 (1991), pp. 172-178 View PDFView articleView in ScopusGoogle Scholar J.B. Epstein, R. Loh, P. Stevenson-Moore, B.C. McBride, J. Spinelli Chlorhexidine rinse in prevention of dental caries in patients following radiation therapy Oral Surg Oral Med Oral Pathol, 68 (1989), pp. 401-405 View PDFView articleView in ScopusGoogle Scholar J. Deng, L. Jackson, J.B. Epstein, C.A. Migliorati, B.A. Murphy Dental demineralization and caries in patients with head and neck cancer Oral Oncol, 51 (2015), pp. 824-831 View PDFView articleView in ScopusGoogle Scholar Cited by (62) Association of neutrophil-to-lymphocyte ratio with severe radiation-induced mucositis in pharyngeal or laryngeal cancer patients: a retrospective study 2021, BMC Cancer Show abstract The neutrophil-to-lymphocyte ratio (NLR) is a marker of systemic inflammation that informs clinical decisions regarding recurrence and overall survival in most epithelial cancers. Radiotherapy for head and neck cancer leads to mucositis in almost all patients and severe radiation-mucositis affects their quality of life (QOL). However, little is known about the NLR for severe mucositis. Therefore, this study aimed to show the association between the NLR and severe radiation-induced mucositis in hypopharyngeal or laryngeal cancer patients. In this retrospective study, we determined the incidence of grade 3 mucositis in 99 patients who were receiving definitive radiotherapy or chemoradiotherapy (CRT) for hypopharyngeal or laryngeal cancer. We performed univariate and multivariate logistic regression analyses to investigate the characteristics of grade 3 mucositis. Kaplan–Meier curves and log-rank tests were used to evaluate the occurrence of grade 3 mucositis between two groups with high (NLR > 5) or low (NLR<5) systemic inflammation. The incidence of grade 3 mucositis was 39%. Univariate logistic regression analysis showed that the NLR (Odd ratio [OR] = 1.09; 95% confidence interval [CI] = 1.02–1.16; p = 0.016) and smoking (OR = 1.02; 95% CI = 1.00–1.03; p = 0.048) were significantly associated with grade 3 mucositis. Multivariate logistic regression analysis showed that the NLR was independently associated with grade 3 mucositis (OR = 1.09; 95% CI = 1.01–1.17; p = 0.021). Kaplan–Meier curves also showed that patients with higher NLR (NLR > 5) prior to radiotherapy developed grade 3 mucositis more frequently than those with lower NLR during radiotherapy (p = 0.045). This study suggests that a higher NLR is a risk factor and predictor of severe radiation-induced mucositis in hypopharyngeal or laryngeal cancer patients. ### Prognostic role of the prognostic nutritional index (PNI) in patients with head and neck neoplasms undergoing radiotherapy: A meta-analysis 2021, Plos One Show abstract This novel meta-analysis was conducted to systematically and comprehensively evaluate the prognostic role of the pretreatment PNI in patients with head and neck neoplasms (HNNs) undergoing radiotherapy. Three databases, PubMed, Embase, and Web of Science, were used to retrieve desired literature. Hazard ratios (HRs) with 95% confidence intervals (CIs) were extracted and pooled by fixed-effects or random-effects models to analyze the relationship between the PNI and survival outcomes: overall survival (OS), distant metastasis-free survival (DMFS), and progression-free survival (PFS). Ten eligible studies involving 3,458 HNN patients were included in our analysis. The robustness of the pooled results was ensured by heterogeneity tests (I 2 = 22.6%, 0.0%, and 0.0% for OS, DMFS, and PFS, respectively). The fixed-effects model revealed a lower pretreatment PNI was significantly related to a worse OS (HR = 1.974; 95% CI: 1.642–2.373; P<0.001), DMFS (HR = 1.959; 95% CI: 1.599–2.401; P<0.001), and PFS (HR = 1.498; 95% CI: 1.219–1.842; P<0.001). The trim-and-fill method (HR = 1.877; 95% CI: 1.361–2.589) was also used to prove that the existing publication bias did not deteriorate the reliability of the relationship. The pretreatment PNI is a promising indicator to evaluate and predict the long-term prognostic survival outcomes in HNN patients undergoing radiotherapy. ### The Trajectory of Oral Mucositis in Head and Neck Cancer Patients Undergoing Radiotherapy and its Influencing Factors 2025, Ear Nose and Throat Journal ### The relationship between UGT1A1 gene & various diseases and prevention strategies 2022, Drug Metabolism Reviews ### Prophylactic and Therapeutic Effects of Curcumin on Treatment-Induced Oral Mucositis in Patients with Head and Neck Cancer: A Meta-Analysis of Randomized Controlled Trials 2021, Nutrition and Cancer ### Natural Products for the Prevention and Treatment of Oral Mucositis—A Review 2022, International Journal of Molecular Sciences View all citing articles on Scopus Substances (3) Generated by ​, an expert-curated chemistry database. Recommended articles Masticatory muscle function affects the pathological conditions of dentofacial deformities Japanese Dental Science Review, Volume 56, Issue 1, 2020, pp. 56-61 Tomohiro Yamada, …, Yoshihide Mori View PDF ### Oral toxicity management in head and neck cancer patients treated with chemotherapy and radiation: Dental pathologies and osteoradionecrosis (Part 1) literature review and consensus statement Critical Reviews in Oncology/Hematology, Volume 97, 2016, pp. 131-142 Michela Buglione, …, Stefano M.Magrini ### Honey in the management of side effects of radiotherapy- or radio/chemotherapy-induced oral mucositis. A systematic review Complementary Therapies in Clinical Practice, Volume 34, 2019, pp. 145-152 Karsten Münstedt, …, Jutta Hübner ### Oral toxicity management in head and neck cancer patients treated with chemotherapy and radiation: Xerostomia and trismus (Part 2). Literature review and consensus statement Critical Reviews in Oncology/Hematology, Volume 102, 2016, pp. 47-54 Michela Buglione, …, Stefano M.Magrini ### Oral mucositis in head and neck cancer: Evidence-based management and review of clinical trial data Oral Oncology, Volume 95, 2019, pp. 29-34 Adriana Blakaj, …, Dukagjin M.Blakaj ### Effect of titanium dioxide nanoparticle reinforcement on flexural strength of denture base resin: A systematic review and meta-analysis Japanese Dental Science Review, Volume 56, Issue 1, 2020, pp. 68-76 Madhu Keshava Bangera, …, Ravishankar N View PDF Show 3 more articles Article Metrics Citations Citation Indexes 60 Policy Citations 1 Captures Mendeley Readers 246 Mentions News Mentions 2 Social Media Shares, Likes & Comments 41 View details About ScienceDirect Remote access Contact and support Terms and conditions Privacy policy Cookies are used by this site.Cookie Settings All content on this site: Copyright © 2025 Elsevier B.V., its licensors, and contributors. 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https://www.elsevier.com/resources/clinicalkey-ai/what-is-the-pathognomonic-sign-of-measles-78cb
What is the pathognomonic sign of Measles? | ClinicalKeyAI Skip to main content Unfortunately we don't fully support your browser. If you have the option to, please upgrade to a newer version or use Mozilla Firefox, Microsoft Edge, Google Chrome, or Safari 14 or newer. If you are unable to, and need support, please send us your feedback. We'd appreciate your feedback.Tell us what you think! Academic & GovernmentAcademic & Government HealthHealth IndustryIndustry Elsevier ConnectInsights AboutAbout Customer supportSupport Publish with us Open Search Location Selector Show Menu Home Resources ClinicalKey AI example questions What is the pathognomonic sign of Measles? AI-Generated response: What is the pathognomonic sign of Measles? Answer generated on June 4, 2024 The AI-generated responses are for illustration purposes only and should not be used for medical purposes. Trusted content. Powered by responsible AI.Try it for free The pathognomonic sign of measles is Koplik spots. Koplik spots are small, white lesions with a blue-gray center on an erythematous base, typically located adjacent to the molars on the buccal mucosa.[1-2] These spots are often visible before the eruption of the measles rash. Koplik spots are considered pathognomonic for measles, meaning their presence is highly specific to this disease and can be used to distinguish measles from other similar presentations.[1-2] They appear 1-4 days before the onset of the rash and may also be observed on other parts of the oropharyngeal mucosa. Despite their diagnostic importance, Koplik spots may be missed or misdiagnosed as thrush. Additional info References Follow up questions Slide Slide Additional info Koplik spots are crucial for the clinical diagnosis of measles, especially in the early stages before the characteristic rash appears. Their early identification can prompt timely isolation measures and appropriate management, reducing the risk of transmission. Measles is a highly contagious disease, and early diagnosis is essential for effective control and prevention. The presence of Koplik spots, combined with other clinical features such as fever, cough, and conjunctivitis, strengthens the clinical suspicion of measles, guiding further diagnostic and management steps. Show more about Additional info References Reference 1 1. Measles Gans, Hayley A. (2025). In Nelson Textbook of Pediatrics (pp. 1958). DOI: 10.1016/B978-0-323-88305-4.00293-5 Measles is a serious infection characterized by high fever, an enanthem, cough, coryza, conjunctivitis, and a prominent exanthem (Fig. 293.2). After an incubation period of 8-12 days, the prodromal phase begins with a mild fever followed by the onset of conjunctivitis with photophobia, coryza, a prominent cough, and increasing fever.Koplik spotsrepresent the enanthem and are the pathognomonic sign of measles, appearing 1-4 days before the onset of the rash (Fig. 293.3). They first appear as discrete red lesions with bluish-white spots in the center on the inner aspects of the cheeks at the level of the premolars. They may spread to involve the lips, hard palate, and gingiva. They also may occur in conjunctival folds and in the vaginal mucosa. Koplik spots have been reported in 50–70% of measles cases but probably occur in the great majority. Symptoms increase in intensity for 2-4 days until the first day of the rash. The rash begins on the forehead (around the hairline), behind the ears, and on the upper neck as a red maculopapular eruption. It then spreads downward to the torso and extremities, reaching the palms and soles in up to 50% of cases. The exanthem frequently becomes confluent on the face and upper trunk (Fig. 293.4). With the onset of the rash, symptoms begin to subside. The rash fades over about 7 days in the same progression as it evolved, often leaving a fine desquamation of skin in its wake. Of the major symptoms of measles, the cough lasts the longest, often up to 10 days. In more severe cases, generalized lymphadenopathy may be present, with cervical and occipital lymph nodes especially prominent. Measles should be suspected in the presence of fever and a maculopapular (nonvesicular) rash. Measles has a 7- to 21-day incubation period, with onset of fever and malaise (prodrome), as well as the more specific combination of a “croupy or brassy” cough, coryza, conjunctivitis, and photophobia beginning about 10 days after exposure (Fig. 338-1). Koplik spots (Fig. 338-2),which are raised bluish-white papules inside both cheeks, may appear at this time; though pathognomonic, they are often missed or misdiagnosed as thrush, and they fade as the rash emerges. The classic erythematous blanching maculopapular rash begins on the face, spreads down the body, and becomes confluent and darker in color over days (Fig. 338-3A).The rash is more subtle in dark-skinned patients (Fig. 338-3B), in whom the diagnosis may be delayed. In malnourished and immunosuppressed individuals, a prolonged desquamating dermatitis is commonly seen (Fig. 338-3C). Body temperature is high, 39° to 40.5° C, beginning with the prodrome and continuing at least 4 days into the rash. Patients are contagious to others from 4 days prior until 4 days after onset of the rash. As the rash darkens and fades, the skin will often flake and peel. In individuals with incomplete postimmunization protection, clinical symptoms may be milder, with less fever and catarrh, a later onset (12 to 16 days), and markedly reduced infectivity. Show excerpt Reference 2 2. Measles (Rubeola) Measles (Rubeola), Elsevier ClinicalKey Clinical Overview DiagnosisPatients often appear quite ill Conjunctival injection may be present Cervical lymphadenopathy may be present Koplik spots Pathognomonic for measles Often visible before rash eruption Lesions are small and white with a blue-gray center on an erythematous base Typically located adjacent to the molars on buccal mucosa; may be observed elsewhere on oropharyngeal mucosa Rash Often morbilliform early in its evolution May become confluent, especially on head and neck Palms and soles are often involved Becomes more brown than red over course of several days before fading Areas of desquamation may develop Less prevalent respiratory signs in young children Stridor associated with crouplike presentation Scattered, coarse wheezing; tachypnea; and retractions associated with bronchiolitislike presentation Show excerpt Follow up questions What are common differential diagnoses when Koplik spots are suspected? What are the typical complications associated with measles? How is measles diagnosed if Koplik spots are not visible? What are the recommended isolation measures upon suspecting measles? What treatments are available for managing measles symptoms? What preventive measures are effective against measles transmission? Useful links Submit your paper Shop Books & Journals Open access View all products Elsevier Connect About About Elsevier Careers Global Press Office Advertising, reprints & supplements Modern slavery act statement Support Customer support Resource center Global | English Copyright © 2025 Elsevier, its licensors, and contributors. All rights are reserved, including those for text and data mining, AI training, and similar technologies. Terms & Conditions Privacy policy Accessibility Cookie settings
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https://brainly.com/question/1899843
[FREE] What is the converse of the conditional statement? If x is even, then x + 1 is odd. - brainly.com 4 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +77,3k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +16,9k Ace exams faster, with practice that adapts to you Practice Worksheets +8,6k Guided help for every grade, topic or textbook Complete See more / Mathematics Expert-Verified Expert-Verified What is the converse of the conditional statement? If x is even, then x+1 is odd. 2 See answers Explain with Learning Companion NEW Asked by nakyahcarter29 • 10/03/2016 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 270410444 people 270M 4.6 10 Upload your school material for a more relevant answer Answer: If x is odd then x+1 is even. Explanation The converse statement of a conditional statement "If p then q" is given by "If q then p". The given conditional statement : If x is even, then x + 1 is odd. Then, the converse statement of the given conditional statement will be : If x is odd then x+1 is even. Answered by JeanaShupp •10.7K answers•270.4M people helped Thanks 10 4.6 (9 votes) Expert-Verified⬈(opens in a new tab) This answer helped 270410444 people 270M 4.6 10 Upload your school material for a more relevant answer The converse of the conditional statement 'If x is even, then x + 1 is odd' is 'If x is odd, then x + 1 is even.' This is achieved by reversing the hypothesis and conclusion of the original statement. Explanation To find the converse of a conditional statement, we switch the positions of the hypothesis and the conclusion. The given conditional statement is: If x is even, then x+1 is odd. In this case: The hypothesis (the part after 'If') is that x is even. The conclusion (the part after 'then') is that x+1 is odd. To form the converse, we reverse these parts: If x+1 is odd, then x is even. However, a more straightforward way to express the converse based on the values of x would be: If x is odd, then x+1 is even. This highlights the logical relationship that occurs when x takes on odd values. Thus, we can summarize that the converse of the given conditional statement is: If x is odd, then x+1 is even. Examples & Evidence For example, if we take x=3, which is odd, then x+1=4, which is indeed even. This confirms the converse as a true statement when tested with an odd number. The logical relationship between even and odd numbers is well-established: the sum of an odd number and one is always even. Thanks 10 4.6 (9 votes) Advertisement Community Answer This answer helped 201866519 people 201M 4.4 63 The converse of that statement is If x + 1 is odd, then x will be even The converse statement is just switching their sides. Answered by WorldlyGlass49 •9.9K answers•201.9M people helped Thanks 63 4.4 (27 votes) Advertisement ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics Given that the point (8,3) lies on the graph of g(x)=lo g 2​x, which point lies on the graph of f(x)=lo g 2​(x+3)+2 ? A. (5,1) B. (5,5) C. (11,1) D. (11,5) For what value of c​ is the function one-to-one?{(1,2),(2,3),(3,5),(4,7),(5,11),(6,c)}◯ 2◯ 5◯ 11◯ 13 Country A has a population of 2 x 10^9 people. Country B has a population of 5 x 10^7 people. Choose which country has the larger population. Then fill in the blank with a number written in standard notation. A. Country A has the larger population. The population of Country A is _ times as large as the population of Country B. B. Country B has the larger population. The population of Country B is _ times as large as the population of Country A. Two cars raced at a race track. The faster car traveled 20 mph faster than the slower car. In the time that the slower car traveled 165 miles, the faster car traveled 225 miles. If the speeds of the cars remained constant, how fast did the slower car travel during the race? | | Distance (mi) | Rate (mph) | Time (h) | :--- :--- | | Slower Car | 165 | r | r 165​ | | Faster Car | 225 | r+20 | r+20 225​ | A. 55 mph B. 60 mph C. 75 mph D. 130 mph 10 x 3+4 x 3−3 x 2+9 x 2 Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
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https://chem.libretexts.org/Courses/University_of_Missouri/MU%3A__1330H_(Keller)/25%3A_Chemistry_of_Life%3A_Organic_and_Biological_Chemistry/25.04%3A_Unsaturated_Hydrocarbons
25.4: Unsaturated Hydrocarbons - Chemistry LibreTexts Skip to main content Table of Contents menu search Search build_circle Toolbar fact_check Homework cancel Exit Reader Mode school Campus Bookshelves menu_book Bookshelves perm_media Learning Objects login Login how_to_reg Request Instructor Account hub Instructor Commons Search Search this book Submit Search x Text Color Reset Bright Blues Gray Inverted Text Size Reset +- Margin Size Reset +- Font Type Enable Dyslexic Font - [x] Downloads expand_more Download Page (PDF) Download Full Book (PDF) Resources expand_more Periodic Table Physics Constants Scientific Calculator Reference expand_more Reference & Cite Tools expand_more Help expand_more Get Help Feedback Readability x selected template will load here Error This action is not available. chrome_reader_mode Enter Reader Mode 25: Chemistry of Life: Organic and Biological Chemistry MU: 1330H (Keller) { } { "25.01:General_Characteristics_of_Organic_Molecules" : "property get Map 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MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "24:_Chemistry_of_Coordination_Chemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1", "25:_Chemistry_of_Life:_Organic_and_Biological_Chemistry" : "property get Map MindTouch.Deki.Logic.ExtensionProcessorQueryProvider+<>c__DisplayClass230_0.b__1" } Tue, 29 Aug 2017 15:03:00 GMT 25.4: Unsaturated Hydrocarbons 91379 91379 Delmar Larsen { } Anonymous Anonymous User 2 false false [ "article:topic", "showtoc:no", "license:ccbyncsa", "licenseversion:40" ] [ "article:topic", "showtoc:no", "license:ccbyncsa", "licenseversion:40" ] Search site Search Search Go back to previous article Sign in Username Password Sign in Sign in Sign in Forgot password Contents 1. Home 2. Campus Bookshelves 3. University of Missouri 4. MU: 1330H (Keller) 5. 25: Chemistry of Life: Organic and Biological Chemistry 6. 25.4: Unsaturated Hydrocarbons Expand/collapse global location Home Campus Bookshelves Alma College American River College Anoka-Ramsey Community College Arkansas Northeastern College Athabasca University Barry University Barstow Community College Bellarmine University Bellingham Technical College Bennington College Bethune-Cookman University Bloomsburg - Commonwealth University of Pennsylvania Monsignor Bonner & Archbishop Prendergast Catholic High School Brevard College BridgeValley Community and Technical College British Columbia Institute of Technology California Polytechnic State University, San Luis Obispo California State University, Chico Cameron University Cañada College Case Western Reserve University Centre College Chabot College Chandler Gilbert Community College Chippewa Valley Technical College City College of San Francisco Clackamas Community College CLC-Innovators Cleveland State University College of Marin College of the Canyons Colorado College Colorado State University Colorado State University Pueblo Community College of Allegheny County Cornell College CSU Chico CSU Fullerton CSU San Bernardino DePaul University De Anza College Diablo Valley College Douglas College Duke University Earlham College Eastern Mennonite University East Tennessee State University El Paso Community College Erie Community College Fordham University Franklin and Marshall College Fresno City College Fullerton College Furman University Galway-Mayo Institute of Technology Georgian College Georgia Southern University Gettysburg College Grand Rapids Community College Grand Valley State University Green River College Grinnell College Harper College Heartland Community College Honolulu Community College Hope College Howard University Indiana Tech Intercollegiate Courses Johns Hopkins University Kenyon College Knox College Kutztown University of Pennsylvania Lafayette College Lakehead University Lansing Community College Lebanon Valley College Lewiston High School Lock Haven University of Pennsylvania Los Angeles Trade Technical College Los Medanos College Louisville Collegiate School Lubbock Christian University Lumen Learning Madera Community College Manchester University Martin Luther College Maryville College Matanuska-Susitna College McHenry County College Mendocino College Meredith College Metropolitan State University of Denver Middle Georgia State University Millersville University Minnesota State Community and Technical College Modesto Junior College Montana State University Monterey Peninsula College Mountain View College Bookshelves Learning Objects 25.4: Unsaturated Hydrocarbons Last updated Aug 29, 2017 Save as PDF 25.3: Alkanes 25.5: Functional Groups Page ID 91379 ( \newcommand{\kernel}{\mathrm{null}\,}) Table of contents 1. Learning Objectives 2. Example 25.4.1 1. Solution Exercise 25.4.1 Example 25.4.2 Solution Exercise 25.4.2 Key Takeaway Learning Objectives To name alkenes given formulas and write formulas for alkenes given names. As noted before, alkenes are hydrocarbons with carbon-to-carbon double bonds (R 2 C=CR 2) and alkynes are hydrocarbons with carbon-to-carbon triple bonds (R–C≡C–R). Collectively, they are called unsaturated hydrocarbons because they have fewer hydrogen atoms than does an alkane with the same number of carbon atoms, as is indicated in the following general formulas: Some representative alkenes—their names, structures, and physical properties—are given in Table 25.4.1. Table 25.4.1: Physical Properties of Some Selected Alkenes| IUPAC Name | Molecular Formula | Condensed Structural Formula | Melting Point (°C) | Boiling Point (°C) | --- --- | ethene | C 2 H 4 | CH 2=CH 2 | –169 | –104 | | propene | C 3 H 6 | CH 2=CHCH 3 | –185 | –47 | | 1-butene | C 4 H 8 | CH 2=CHCH 2 CH 3 | –185 | –6 | | 1-pentene | C 5 H 10 | CH 2=CH(CH 2)2 CH 3 | –138 | 30 | | 1-hexene | C 6 H 12 | CH 2=CH(CH 2)3 CH 3 | –140 | 63 | | 1-heptene | C 7 H 14 | CH 2=CH(CH 2)4 CH 3 | –119 | 94 | | 1-octene | C 8 H 16 | CH 2=CH(CH 2)5 CH 3 | –102 | 121 | We used only condensed structural formulas in Table 25.4.1. Thus, CH 2=CH 2 stands for The double bond is shared by the two carbons and does not involve the hydrogen atoms, although the condensed formula does not make this point obvious. Note that the molecular formula for ethene is C 2 H 4, whereas that for ethane is C 2 H 6. The first two alkenes in Table 25.4.1, ethene and propene, are most often called by their common names—ethylene and propylene, respectively (Figure 25.4.1). Ethylene is a major commercial chemical. The US chemical industry produces about 25 billion kilograms of ethylene annually, more than any other synthetic organic chemical. More than half of this ethylene goes into the manufacture of polyethylene, one of the most familiar plastics. Propylene is also an important industrial chemical. It is converted to plastics, isopropyl alcohol, and a variety of other products. Figure 25.4.1: Ethene and Propene. The ball-and-spring models of ethene/ethylene (a) and propene/propylene (b) show their respective shapes, especially bond angles. Although there is only one alkene with the formula C 2 H 4 (ethene) and only one with the formula C 3 H 6 (propene), there are several alkenes with the formula C 4 H 8. Here are some basic rules for naming alkenes from the International Union of Pure and Applied Chemistry (IUPAC): The longest chain of carbon atoms containing the double bond is considered the parent chain. It is named using the same stem as the alkane having the same number of carbon atoms but ends in -ene to identify it as an alkene. Thus the compound CH 2=CHCH 3 is propene. If there are four or more carbon atoms in a chain, we must indicate the position of the double bond. The carbons atoms are numbered so that the first of the two that are doubly bonded is given the lower of the two possible numbers.The compound CH 3 CH=CHCH 2 CH 3, for example, has the double bond between the second and third carbon atoms. Its name is 2-pentene (not 3-pentene). Substituent groups are named as with alkanes, and their position is indicated by a number. Thus, the structure below is 5-methyl-2-hexene. Note that the numbering of the parent chain is always done in such a way as to give the double bond the lowest number, even if that causes a substituent to have a higher number. The double bond always has priority in numbering. Example 25.4.1 Name each compound. Solution The longest chain containing the double bond has five carbon atoms, so the compound is a pentene (rule 1). To give the first carbon atom of the double bond the lowest number (rule 2), we number from the left, so the compound is a 2-pentene. There is a methyl group on the fourth carbon atom (rule 3), so the compound’s name is 4-methyl-2-pentene. The longest chain containing the double bond has five carbon atoms, so the parent compound is a pentene (rule 1). To give the first carbon atom of the double bond the lowest number (rule 2), we number from the left, so the compound is a 2-pentene. There is a methyl group on the third carbon atom (rule 3), so the compound’s name is 3-methyl-2-pentene. Exercise 25.4.1 Name each compound. CH 3 CH 2 CH 2 CH 2 CH 2 CH=CHCH 3 Answer Just as there are cycloalkanes, there are cycloalkenes. These compounds are named like alkenes, but with the prefix cyclo- attached to the beginning of the parent alkene name. Example 25.4.2 Draw the structure for each compound. 3-methyl-2-pentene cyclohexene Solution a. First write the parent chain of five carbon atoms: C–C–C–C–C. Then add the double bond between the second and third carbon atoms: Now place the methyl group on the third carbon atom and add enough hydrogen atoms to give each carbon atom a total of four bonds. b First, consider what each of the three parts of the name means. Cyclo means a ring compound, hex means 6 carbon atoms, and -ene means a double bond. Exercise 25.4.2 Draw the structure for each compound. 2-ethyl-1-hexene cyclopentene Key Takeaway Alkenes are hydrocarbons with a carbon-to-carbon double bond. 25.4: Unsaturated Hydrocarbons is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by LibreTexts. 13.1: Alkenes- Structures and Names by Anonymous is licensed CC BY-NC-SA 3.0. Original source: Toggle block-level attributions Back to top 25.3: Alkanes 25.5: Functional Groups Was this article helpful? Yes No Recommended articles 25.1: General Characteristics of Organic MoleculesOrganic chemistry is the study of carbon compounds, nearly all of which also contain hydrogen atoms. 25.2: Introduction to HydrocarbonsHydrocarbons are organic compounds that contain only carbon and hydrogen. The four general classes of hydrocarbons are: alkanes, alkenes, alkynes and ... 25.3: AlkanesSimple alkanes exist as a homologous series, in which adjacent members differ by a CH2 unit. 25.5: Functional GroupsFunctional groups are atoms or small groups of atoms (two to four) that exhibit a characteristic reactivity. A particular functional group will almost... 25.6: Compounds with a Carbonyl GroupAldehydes and ketones are characterized by the presence of a carbonyl group (C=O), and their reactivity can generally be understood by recognizing tha... Article typeSection or PageLicenseCC BY-NC-SALicense Version4.0Show Page TOCno on page Tags This page has no tags. © Copyright 2025 Chemistry LibreTexts Powered by CXone Expert ® ? The LibreTexts libraries arePowered by NICE CXone Expertand are supported by the Department of Education Open Textbook Pilot Project, the UC Davis Office of the Provost, the UC Davis Library, the California State University Affordable Learning Solutions Program, and Merlot. We also acknowledge previous National Science Foundation support under grant numbers 1246120, 1525057, and 1413739. Privacy Policy. Terms & Conditions. Accessibility Statement.For more information contact us atinfo@libretexts.org. Support Center How can we help? Contact Support Search the Insight Knowledge Base Check System Status× contents readability resources tools ☰ 25.3: Alkanes 25.5: Functional Groups
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https://mathoverflow.net/questions/457566/limit-of-a-integral-whose-integrand-diverges-under-the-limit
measure theory - Limit of a integral whose integrand diverges under the limit - MathOverflow Join MathOverflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community MathOverflow helpchat MathOverflow Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Limit of a integral whose integrand diverges under the limit Ask Question Asked 1 year, 11 months ago Modified1 year, 11 months ago Viewed 245 times This question shows research effort; it is useful and clear 2 Save this question. Show activity on this post. \begingroup I am trying to simplify the following limit of integral where \mu is given: p(y) = \lim_{\sigma \to 0} \int_{\mathbb R} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^2} f(x) dx, however I am not sure if there is a way to simplify it, as the integrand does not converge under the limit \sigma \to 0 and the interchangeablity of limit and integral fails here: \lim_{\sigma \to 0} |x| \cdot \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^2} f(x) = \begin{cases} \inf, & x \ne \mu /y; \ 0, & x = \mu /y \end{cases}. However, I DO know the answer of the limit of the integral, as naturally it is the probablity density function of Y = T/X, where X has PDF f(x) and T \sim N(\mu, \sigma^2) is a normal distribution. The limit of integral at \sigma \to 0 then essentially means the PDF of \mu / X and can be easily calculated. So how do I calculate this limit of integral analytically, without the help of the intuition from probablity? measure-theory probability-distributions integration limits-and-convergence lebesgue-measure Share Cite Improve this question Follow Follow this question to receive notifications asked Nov 1, 2023 at 20:21 user482401user482401 43 4 4 bronze badges \endgroup Add a comment| 2 Answers 2 Sorted by: Reset to default This answer is useful 3 Save this answer. Show activity on this post. \begingroup \newcommand\si\sigma\newcommand\R{\mathbb R}We have to find \lim_{\si\downarrow0}p_\si(y) for all real y such that the limit exists, where p_\si(y):=\int_\R\frac{|x|\,dx}{\si\sqrt{2\pi}}e^{-(xy-\mu)^2/(2\si^2)}\,f(x). Assuming that f is bounded, and using the substitution x=(\mu+\si z)/y and dominated convergence, we get p_\si(y)=\frac1{y^2}\,\int_\R\frac{|\mu+\si z|\,dz}{\sqrt{2\pi}}e^{-z^2/2} f\Big(\frac{\mu+\si z}y\Big) \to p(y):=\frac{|\mu|}{y^2}\,f\Big(\frac\mu y\Big) \tag{1}\label{1} (as \si\downarrow0) for all real y\ne0 such that \mu/y is a point of continuity of f. If y=0, then \lim_{\si\downarrow0}p_\si(y)=\infty. If y\ne0 and \mu/y is not a point of continuity of f, then \lim_{\si\downarrow0}p_\si(y) does not exist in general. However, \eqref{1} holds for every real y\ne0 such \mu/y is a Lebesgue point of continuity of f. So, \eqref{1} holds for almost all real y. Share Cite Improve this answer Follow Follow this answer to receive notifications edited Nov 2, 2023 at 3:02 answered Nov 1, 2023 at 20:55 Iosif PinelisIosif Pinelis 141k 9 9 gold badges 120 120 silver badges 251 251 bronze badges \endgroup Add a comment| This answer is useful 0 Save this answer. Show activity on this post. \begingroup The Gaussian tends to a delta function in the \sigma\rightarrow 0 limit, \lim_{\sigma \to 0} \frac{1}{\sqrt{2\pi\sigma^2} } e^{-\frac{1}{2\sigma^2} (xy - \mu)^2} =\delta(xy-\mu)=|y|^{-1}\delta(x-\mu/y), so the integral evaluates to p(y) = |y|^{-1}\int_{-\infty}^\infty |x| f(x)\delta(x-\mu/y) \,dx=|y|^{-1}|\mu/y|f(\mu/y). Share Cite Improve this answer Follow Follow this answer to receive notifications edited Nov 1, 2023 at 21:08 answered Nov 1, 2023 at 20:45 Carlo BeenakkerCarlo Beenakker 199k 19 19 gold badges 479 479 silver badges 697 697 bronze badges \endgroup 3 \begingroup This answer incorrect for any real y<0, because the limit (if exists) must be \ge0. Also, as noted in my answer, if y\ne0 and \mu/y is not a point of continuity of f, then the limit does not exist in general. One also needs a condition such as the boundedness of f to make the limit transition. Also, you need to specify in what sense and why the delta function is the limit.\endgroup Iosif Pinelis –Iosif Pinelis 2023-11-01 21:04:10 +00:00 Commented Nov 1, 2023 at 21:04 \begingroup you're right, I missed an absolute value sign in the delta function, thank you for correcting me.\endgroup Carlo Beenakker –Carlo Beenakker 2023-11-01 21:07:35 +00:00 Commented Nov 1, 2023 at 21:07 \begingroup There are a number of other points as well, listed above, missing in your answer.\endgroup Iosif Pinelis –Iosif Pinelis 2023-11-01 21:08:45 +00:00 Commented Nov 1, 2023 at 21:08 Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions measure-theory probability-distributions integration limits-and-convergence lebesgue-measure See similar questions with these tags. 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What is the mass of sodium chloride? - Quora Something went wrong. Wait a moment and try again. Try again Skip to content Skip to search Sign In Chemistry Masses Molar Weight Sodium Chloride Inorganic Compounds Salt Chemical Elements Chemical Science Sodium 5 What is the mass of sodium chloride? 15 Answers Sort Recommended Stuart Herring B.S. in Information Technology, American InterContinental University Atlanta (Graduated 2004) · Author has 11.7K answers and 8.2M answer views ·1y What is the mass of sodium chloride in 200cm or 0.5m NaCl? Originally Answered: What is the mass of sodium chloride in 200cm or 0.5m NaCl? · What is the mass of sodium chloride in 200cm or 0.5m NaCl? Be more careful when copying your homework problems here. This should read: “What is the mass of sodium chloride in 200 cm³ of 0.5M NaCl?” (You left out the ‘3’ in ‘cm³’; you left out the required space before a unit symbol (it’s ‘200 cm³’, not ‘200cm³’); and you did not capitalize the molarity symbol (‘M’, not ‘m’). Accuracy of terms and symbols is important, especially in the sciences. If you omitted the ‘3’ exponent because you did not have superscripts available, use the ^ symbol as a substitute: ‘200 cm^3’.) Now, remember the defini Continue Reading What is the mass of sodium chloride in 200cm or 0.5m NaCl? Be more careful when copying your homework problems here. This should read: “What is the mass of sodium chloride in 200 cm³ of 0.5M NaCl?” (You left out the ‘3’ in ‘cm³’; you left out the required space before a unit symbol (it’s ‘200 cm³’, not ‘200cm³’); and you did not capitalize the molarity symbol (‘M’, not ‘m’). Accuracy of terms and symbols is important, especially in the sciences. If you omitted the ‘3’ exponent because you did not have superscripts available, use the ^ symbol as a substitute: ‘200 cm^3’.) Now, remember the definition of molarity: the number of moles of solute in one litre of solvent (such as water). Therefore 0.5M means 0.5 mol of solute per litre of solvent. Do you remember the relationship between cm³ and L? (1000 cm³ = 1000 mL = 1 L.) So, 200 cm³ is what fraction of 1000 cm³? Apply this to the number of moles that would be in one full litre here. Then use the molar mass of NaCl (look it up) to find the mass that is present in this solution. Note: all of these techniques were already discussed in your chemistry book before this assignment was given. Please go (re)read it. . . . Please do not post homework questions here without doing some work, and showing what you have already done. We can give guidance if you have difficulty, but we are not here to simply do all of your work for you. You are not a cheater. Upvote · Promoted by Coverage.com Johnny M Master's Degree from Harvard University (Graduated 2011) ·Updated Sep 9 Does switching car insurance really save you money, or is that just marketing hype? This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. Continue Reading This is one of those things that I didn’t expect to be worthwhile, but it was. You actually can save a solid chunk of money—if you use the right tool like this one. I ended up saving over $1,500/year, but I also insure four cars. I tested several comparison tools and while some of them ended up spamming me with junk, there were a couple like Coverage.com and these alternatives that I now recommend to my friend. Most insurance companies quietly raise your rate year after year. Nothing major, just enough that you don’t notice. They’re banking on you not shopping around—and to be honest, I didn’t. It always sounded like a hassle. Dozens of tabs, endless forms, phone calls I didn’t want to take. But recently I decided to check so I used this quote tool, which compares everything in one place. It took maybe 2 minutes, tops. I just answered a few questions and it pulled up offers from multiple big-name providers, side by side. Prices, coverage details, even customer reviews—all laid out in a way that made the choice pretty obvious. They claimed I could save over $1,000 per year. I ended up exceeding that number and I cut my monthly premium by over $100. That’s over $1200 a year. For the exact same coverage. No phone tag. No junk emails. Just a better deal in less time than it takes to make coffee. Here’s the link to two comparison sites - the one I used and an alternative that I also tested. If it’s been a while since you’ve checked your rate, do it. You might be surprised at how much you’re overpaying. Upvote · 999 485 999 103 99 17 Related questions More answers below What is the molecular mass of sodium chloride? How do I calculate the molecular mass of sodium chloride? What type of crystal is sodium chloride? How many grams of sodium are in a pound of sodium chloride (salt)? What is the formula mass for sodium chloride? Pooh 3y What is the formula mass for sodium chloride? Originally Answered: What is the formula mass for sodium chloride? · The formula mass of sodium chloride is 58.44g/mol Upvote · 9 3 Trevor Hodgson Knows English · Author has 11.8K answers and 12.3M answer views ·3y What is the mass of NaCl? Originally Answered: What is the mass of NaCl? · In order to answer your question as best I could - bearing in mind that abolutely no infiormation is given to me to assist me in this task - I decided to check my kitchen cupboard Happily I found a new , unopened packet of NaCl - marked “fine table salt” which my wife had purchased this morning I checked the mass of NaCl - The label read - very clearly : Mass of salt =100 g So I am able to answer your question with total confidence : The mass of NaCl is 100 g . Your response is private Was this worth your time? This helps us sort answers on the page. Absolutely not Definitely yes Upvote · 9 1 Rakesh Shukla 1y The atomic mass of sodium (Na) is about 22.99 grams per mole, the atomic mass of chlorine (Cl) is about 35.45 grams per mole. When added these atomic masses together, we get the molar mass of sodium chloride :So, the molar mass of sodium chloride is approximately 58.44 grams per mole. This means that one mole of sodium chloride weighs 58.44 grams.22.99 g/mol+35.45 g/mol=58.44 g/mol22.99g/mol+35.45g/mol=58.44g/mol Upvote · Related questions More answers below What is the percentage of sodium chloride (NaCl) by mass and volume? What are the primary compositions of sodium chloride? A solution is prepared by dissolving 15 grams of sodium chloride in 20 grams of water. What is the mass, by mass percentage, of salt in this solution? What mass of water must be added to 35.0 g of sodium chloride to make a 15.0 m/m% solution? What is sodium chloride, and what is it commonly used for? Guy Clentsmith Chemistry tutor... at Self-Employment (2018–present) · Author has 26.5K answers and 19.7M answer views ·5y What is the mass number of NaCl? Originally Answered: What is the mass number of NaCl? · Mass number is an atomic property … and refers to the mass of an isotope. Molecular mass or formula mass is a property compounds. And sodium chloride has a formula mass of 58.44∙g∙m o l−1 58.44•g•m o l−1…and this is the SUM of the average masses of sodium, and chloride isotopes… Upvote · Sponsored by Grammarly Stuck on the blinking cursor? Move your great ideas to polished drafts without the guesswork. Try Grammarly today! Download 99 35 Philip Howie materials scientist, academic, researcher · Author has 3K answers and 13M answer views ·8y What is the molar mass for NaCl? Originally Answered: What is the molar mass for NaCl? · You can find the molar masses of the elements on any good periodic table. The molar mass of sodium is 22.990 g mol−1−1. The molar mass of chlorine is 35.45 g mol−1−1. Adding them together, the molar mass of NaCl is 58.44 g mol−1−1. Thanks for the A2A. Upvote · 9 1 Vedu Dg Lecturer at The Team Academy (2023–present) ·1y What is the mass of sodium chloride in 200cm or 0.5m NaCl? Originally Answered: What is the mass of sodium chloride in 200cm or 0.5m NaCl? · (200 Cm3 of 0.5m NaCl …..ask like this) Answer will be 0.5m NaCl soln means 29.25 g NaCl in 1 lit, 0.5m NaCl soln of 200 ml/cm3 requires (200×29.25/1000) = 5.85g. Upvote · Promoted by Network Solutions® Zachary Tidwell Senior Communications Manager at Network Solutions (2015–present) ·Wed Who is the best domain registrar and why? Who would you recommend for a new website owner, and why? When you’re looking for the best domain provider, it’s important to take the full range of capabilities and options available through that provider into account. Why? Because an online journey, especially for a small business owner or entrepreneur, is rarely a straight line from A to B. I’ve worked in the web presence industry for a decade now, and I’ve seen business owners who called in to buy the same product, who in fact had very different needs. I’ve seen sales agents help customers who didn’t even have a name for their business yet, and I’ve seen successful businesses with numerous locatio Continue Reading When you’re looking for the best domain provider, it’s important to take the full range of capabilities and options available through that provider into account. Why? Because an online journey, especially for a small business owner or entrepreneur, is rarely a straight line from A to B. I’ve worked in the web presence industry for a decade now, and I’ve seen business owners who called in to buy the same product, who in fact had very different needs. I’ve seen sales agents help customers who didn’t even have a name for their business yet, and I’ve seen successful businesses with numerous locations finally get around to their online presence, long after achieving brick-and-mortar success. So, I’m going to focus on how to find the best domain provider for small businesses in particular, because that’s what my company, Network Solutions, specializes in, and that’s where my expertise lies. With that in mind, here are some considerations to keep in mind when deciding where to purchase a domain. Simplified Search Anyone who’s spent time searching for a domain name can tell you that finding a great option isn’t always easy. In many cases, that shiny, perfect domain name you envisioned may already be taken. This is where a feature like an AI domain name generator can be extremely helpful. I cover this in a lot more detail in this answer, but the main thing to know is that AI domain name generation helps you find helpful, pleasantly surprising alternatives to your preferred domain name, immensely simplifying the search process. Reputation and Credibility Whether you’re looking for a taco stand or a car dealership, you always want to make sure you choose an established company with a long-time presence and strong reputation. The same thing applies to the world of domains. In general, large, well-established companies will have the expertise and strong support to get you where you want to be online. Look for a company with a strong score (definitely 4.0+) on Trustpilot, and check to see how long that company has been in business. In my view, longevity and established trust in the domain industry should be always be two major points of consideration when choosing a provider. A Full Suite of Products Maybe today you just want a domain name. But you really should try to work with a company that can support your end-to-end online journey, whatever direction it goes. Look for additional features like an AI-powered website builder, reliable web hosting, robust cybersecurity products and online marketing tools, so you can take that next step when you’re ready. Another factor to consider here is professional services. DIY products are hugely popular, and for good reason, but it’s really nice to have the ability to turn to professional website designers and online marketing specialists if needed. The Journey So often, I see online journeys start with finding a great domain name, and progress from there to a truly expansive and successful digital endeavor. One of my favorite things is seeing small businesses make the most of their websites and eCommerce stores to get an edge on the competition and even compete with some of the bigger players out there, with the innovative tools available today really acting as an equalizer for so many businesses. As you search for an exceptional domain provider, I recommend you keep your options open and leave yourself and your business room to grow. Wherever your online journey takes you, I wish you the best of luck, and I hope finding your perfect domain name is just the start for your web presence. Upvote · Ronald L Klaus Taught chemical engineering and worked in the space program · Author has 365 answers and 667.2K answer views ·8y What is the molar mass for NaCl? Originally Answered: What is the molar mass for NaCl? · Keep a table of elemental molar masses handy. That will let you find the molar mass of any compound by adding the molar masses of the elements and multiplying each of these by the number of each atom in the molecule. Philip Howie’s post gives the answer to this specific question. Upvote · Guy Grotke B.S. in Biology&Chemistry, San Diego State University (Graduated 1979) · Author has 2K answers and 1.2M answer views ·8y If a pure sample of NaCl is 4.35 g, what is the mass of chloride? Originally Answered: If a pure sample of NaCl is 4.35 g, what is the mass of chloride? · I’m not doing your homework for you. Look up the atomic weights of sodium and chlorine, and divide 4.35 g by that ratio. Upvote · Sponsored by StealthGPT All your schoolwork, one app. Trusted by millions of students, writing countless essays and answering any question. Learn More 99 83 Sudhir Raj Studied at GOAL Institute ,patna ·3y What is the formula mass for sodium chloride? Originally Answered: What is the formula mass for sodium chloride? · Atomic weight of sodium is 23 and atomic weight of chlorine is 35.5 so formula mass will be 58.5 Upvote · Anida colonoscopy Lives in Toronto, ON ·7y What is the mass of 2.40 moles NaCl? Originally Answered: What is the mass of 2.40 moles NaCl? · Use m=nM where m is mass, n is number of moles, and M is molar mass. m=2.40 mol(58.4g/mol) m= 1.40x10^2g P.S. Because you must have 3 significant digits, the value must be written in scientific notation. Upvote · Shocking Apple am studying new things constantly ·3y What is the molar mass of NaCl? Originally Answered: What is the molar mass of NaCl? · Molar mass for NaCl = Gram atomic mass of Na + Gram atomic mass of Cl Molar mass of NaCl = 23 + 35.5 Molar mass of NaCl = 58.5 g Upvote · 9 1 Bhavik Makwana Research Scholar from Veer Narmad South Gujarat University ·3y What is the formula mass for sodium chloride? Originally Answered: What is the formula mass for sodium chloride? · NaCl Upvote · 9 3 Related questions What is the molecular mass of sodium chloride? How do I calculate the molecular mass of sodium chloride? What type of crystal is sodium chloride? How many grams of sodium are in a pound of sodium chloride (salt)? What is the formula mass for sodium chloride? What is the percentage of sodium chloride (NaCl) by mass and volume? What are the primary compositions of sodium chloride? A solution is prepared by dissolving 15 grams of sodium chloride in 20 grams of water. What is the mass, by mass percentage, of salt in this solution? What mass of water must be added to 35.0 g of sodium chloride to make a 15.0 m/m% solution? What is sodium chloride, and what is it commonly used for? What is the classification of sodium chloride? How was sodium chloride discovered? Is sodium chloride healthy? What are some examples of sodium chloride? Why is sodium chloride salt? Related questions What is the molecular mass of sodium chloride? How do I calculate the molecular mass of sodium chloride? What type of crystal is sodium chloride? How many grams of sodium are in a pound of sodium chloride (salt)? What is the formula mass for sodium chloride? What is the percentage of sodium chloride (NaCl) by mass and volume? Advertisement About · Careers · Privacy · Terms · Contact · Languages · Your Ad Choices · Press · © Quora, Inc. 2025
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Linear Equation Formulas How to Solve System of Linear Equations Solved Examples on Linear Equations Practice Problems on Linear Equations Frequently Asked Questions on Linear Equations What Is a Linear Equation? A linear equation or one-degree equation is the equation with degree 1. The degree of an equation is the highest power of variables present in the equation. Linear equation is an algebraic equation in which the highest degree of a variable is 1. How can we know if an equation is linear? No variable in a linear equation has an exponent or power greater than 1. They are called linear equations since the graph of a linear equation in one or two variables always forms a straight line. We can write linear equations in different ways, depending on the number of variables present. For example, a linear equation in one variable can be written in the form ax + b = 0, where a, b, and c are real numbers and x and y are variables with the highest power 1. Examples of a linear equation: $y = 2x$ $x + y = 2$ $y = 3x \;-\; 2$ Non-examples of a linear equation: $y^{2} = 3x$ $y^{3} = 2 + x$ Recommended Games Count Objects in Linear Arrangement Game Play Count Objects in Non-Linear Arrangement Game Play Count Objects in Non-Linear Arrangement within 10 Game Play Determine the Area of Rectilinear Shapes Game Play Find the Missing Number in Addition Equations Game Play Solve the Equations by Regrouping Ones and Tens Game Play Subtract 2-Digit Numbers to Complete the Equations Game Play Subtract 3-Digit Numbers to Complete the Equations Game Play Subtraction Equations Game Play Use Subtraction to Complete the Equations Game Play More Games Definition of a Linear Equation A linear equation is an algebraic equation in which each variable term is raised to the exponent or power of 1. A linear equation in one or two variables always represents a straight line when graphed. Example: $x + 2y = 4$ is a linear equation and the graph of this linear equation is a straight line. Recommended Worksheets More Worksheets Linear Equation in Standard Form Linear equations are a combination of constants and variables. Linear equation in one variable has the standard form $Ax + B = 0$, where A and B are constants (A≠0), x is the variable. A linear equation in n variables $x_{1},\;x_{2},\; …,x_{n}$is of the form $a_{1}\;x_{1} + a_{2}\;x_{2}$ … $+ a_{n}\;x_{n} = c$ where $x_{1},\; …, x_{n}$ are variables, coefficients $a_{0},\; …,\; a_{n}$ are constants (not all zero), c is a constant, a1 is the leading coefficient, x1 is the leading variable. Linear Equations in One Variable The standard form of linear equations in one variable is given by $Ax + B = 0$ where A and B are real numbers and $A \neq 0$ x is the variable. It has only one solution. Example of linear equation in one variable:$5x + 3 = 0$ Linear Equations in Two Variables The standard form of linear equations in two variables is given by $Ax + By + C = 0$ where A, B, and C are real numbers and $A neq 0,\; B \neq 0$ x and y are the variables. A linear equation in two variables has infinitely many solutions. Every solution of a linear equation can be represented by a unique point on the graph of the equation. Example of linear equation in two variables: $2x + 7y + 3 = 0$ Linear Equation Formulas As mentioned earlier, a linear equation in one variable or two variables represents a straight line. We can write the equation of a line in different forms. The formula for linear equations refers to the standard form, slope-point form, slope-intercept form, etc. General Form (Bold) A linear equation in two variables forms a straight line. Its general form is given by $Ax + By + C = 0$ A, B, C are constants. A and B cannot both be 0 simultaneously. Slope Intercept Form (Bold) Linear equations can be written in a simple slope-intercept form as $y = mx + b$, where x and y are the variables, m is the slope of the line, and b is the y-intercept. Example: Equation of a line with slope 5 and y-intercept 2 is $y = 5x + 2$. Slope Point Form (Bold) The slope of a line having slope m and passing through a point (x1,y1) is given by $y \;−\; y_{1} = m(x \;−\; x_{1})$ where $m = slope$ Example: Slope of a line $= 4$, Given point on the line $= (1,\; 2)$ Equation: $y \;−\;2 = 4(x \;−\;1)$ How to Graph Linear Equations Graphing linear equations is a simple process. The graph of a linear equation in one variable or two variables forms a straight line. A linear equation in one variable x forms a vertical line that is parallel to the y-axis. A linear equation in one variable y forms a horizontal line that is parallel to the x-axis. Let’s understand how to graph the linear equations with examples. Example 1: x – 8 = 0 is a linear equation in one variable. Isolate x by adding 8 to both sides of this equation. $x \;-\; 8 + 8 = 0 + 8$ $x = 8$ Example 2: Plot a graph for a linear equation in two variables: $4x \;-\; 2y = 8$ Let us plot the linear equation graph using the following steps. Step 1: Convert the equation in the form of $y = mx + b$. This will give: $y = 2x \;-\; 4$. Step 2: Put $x = 0$ in the equation to get the y-intercept. We get $y = 0 \;-\; 4 = \;-\;4$ Similarly, if we substitute $y = 0$, we get the x-intercept $x = 2$ (Note that you can replace the value of x for different numbers and get the resulting value of y to create the coordinates. Once you get two points, you can draw a line passing through these points.)List the coordinates of points that satisfy the given linear equation $y = 2x \;-\; 4$ as shown in the following table. Only two points are enough to graph the line. Here are the ordered pairs that satisfy the given linear equation. | | | | | | --- --- | x | 0 | 2 | 4 | $\;-\;2$ | | y | $\;-\;4$ | 0 | 4 | $\;-\;8$ | Plot these points on a graph and draw a line passing through these points. How to Solve Linear Equations in One Variable The value of the variable that satisfies the linear equation is called the solution of the linear equation. An equation is a mathematical statement in which the left hand side (L.H.S.) is equal to the right hand side (R.H.S.). Different properties of equality are used to solve algebraic equations. We isolate the variable or bring the variables to one side of the equation and the constant to the other side and then find the value of the unknown variable by simplifying. This is the method of solving linear equations in one variable. Example: Solve the linear equation $5x \;-\; 12 = 18$. Step 1: Add 12 on both sides. $5x \;-\; 12 + 12 = 18 + 12$ $5x = 30$ Step 2: Divide both sides by 5 to isolate x. $\frac{5x}{5} = \frac{30}{5}$ $x = 5$ Thus, we have $x = 5$. This is one of the methods of solving linear equations in one variable. How to Solve System of Linear Equations A system of linear equations consists of two or more linear equations made up of two or more variables such that all equations in the system are considered simultaneously. A system of linear equations in two variables can be solved using different methods like substitution, graphing method, matrix, cross multiplication, etc. Let’s understand the simplest method—the substitution method—with an example. Example: Solve the system of linear equations in two variables. $y = 3x\;-\;1$ $4x + y = \;-\; 8$ Solution: Consider the first equation: $y = 3x\;-\;1$. Substitute the value of y from this equation into the second equation$4x + y = \;-\;8$. $4x + 3x\;-\;1 = \;-\;8$ $7x = \;-\;7$ $x = \;-\; 1$ Put $x = \;-\;1$ in the first equation $y = 3x\;-\;1$. $y = 3(\;-\;1)\;-\;1=\;-\;5$ Thus, we get $x = \;-\;1$ and $y = \;-\;5$ Facts about Linear Equations Linear equation is defined as an equation of degree one. The graph obtained from a linear equation is always a straight line. The value of the variable that makes a linear equation true is called the solution or root of the linear equation. The solution of a linear equation is unaffected if a balanced operation of addition, subtraction, multiplication or division is performed on both sides of the equation. The graph of a linear equation in one or two variables always forms a straight line and it can be extended indefinitely in both directions. Conclusion In this article, we learned linear equations, their standard form, formulas associated with linear equations. We also learned how to graph linear equations. Let’s solve a few examples and practice MCQs for more clarity. Solved Examples on Linear Equations 1. Solve for x: $\frac{2x + 5}{3} = x \;-\; 5$. Solution: $\frac{2x + 5}{3} = x \;-\; 5$ Multiply both sides of equation by 3 $\frac{2x + 5}{3} \times 3 = (x \;-\; 5)\times 3$ $\Rightarrow 2x + 5 = 3x \;-\; 15$ Subtract 5 from both sides of equation $\Rightarrow 2x + 5\;-\;5 = 3x \;-\; 15\;-\;5$ $\Rightarrow 2x = 3x \;-\; 20$ $\Rightarrow 2x \;-\; 3x = \;-\; 20$ $\Rightarrow \;-\; x = \;-\; 20$ $\Rightarrow x = 20$ 2. The sum of two numbers is 55. If one number is 11 less than the other, find the numbers by framing a linear equation. Solution: Let the first number be x. Then the second number is $x \;-\; 11$. According to question, $x + x \;-\; 11 = 55$ $\Rightarrow 2x \;-\; 11 = 55$ Add 11 to both sides of the equation. $\Rightarrow 2x \;-\; 11 + 11 = 55 + 11$ $\Rightarrow 2x = 66$ Divide both sides by 2. $\Rightarrow x = 33$ Therefore, $x = 33$ is the first number. Second number $= 33 \;-\; 11 = 22$. 3. Solve the given linear equation: $4x + 92 = 72$. Solution: Given equation: $4x + 92 = 72$ Add $\;-\;92$ on both sides.$\Rightarrow 4x + 92 \;-\; 92 = 72 \;-\; 92$$\Rightarrow 4x = \;-\;20$ Divide both sides by 4. $\Rightarrow \frac{4x}{4} = \frac{\;-\;20}{4}$ $\Rightarrow x = \;-\;5$ 4. Five times of a number is equal to 45. Form a linear equation and find the unknown number. Solution: Let the unknown number be x. Five times of this number is equal to 45. $5x = 45$. So, the linear equation is $5x = 45$ Divide both sides by 5. $\Rightarrow \frac{5x}{5} = \frac{45}{5}$ $\Rightarrow x = 9$ Therefore, the unknown number is 9. 5. Solve the equation $8 \;–\; 2x = 12 \;–\; 4x$, and verify the result. Solution: Given equation: $8 \;–\; 2x = 12 \;–\; 4x$ Subtract 8 from on both sides of the equation. $8 \;–\; 2x \;–\; 8 = 12 \;–\; 4x \;–\; 8$ $\Rightarrow \;–\; 2x = 4 \;–\; 4x$ Transpose $-\;4x$ to left side $\Rightarrow \;–\; 2x + 4x = 4$ $\Rightarrow 2x = 4$ Divide both sides by 2 $\Rightarrow x = 2$ Verification: Put $x = 2$ in the equation $8 \;–\; 2x = 12 \;–\; 4x$ $\Rightarrow 8 \;-\; 2 x (2) = 12 \;-\; 4 x (2)$ $\Rightarrow 8 \;-\; 4 = 12 \;-\; 8$ $\Rightarrow 4 = 4$ Hence, LHS $=$ RHS. Practice Problems on Linear Equations Linear Equations - Definition, Graph, Examples, Facts, FAQs Attend this quiz & Test your knowledge. 1 Which of the following is NOT a linear equation? $x = 2$ $x \;-\; y = 5$ $x^{2} + y = 5$ $x + 3y = 5$ CorrectIncorrect Correct answer is: $x^{2} + y = 5$$x^{2} + y = 5$ is a quadratic equation, as the highest degree of variable x is 2$. 2 What is the slope of the line whose equation is $y \;-\; 2x = 12$. $\;-\;2$ 2 $\;-\;6$ 6 CorrectIncorrect Correct answer is: 2Given equation is $y\;-\;2x = 12$. $\Rightarrow y\;-\;2x = 12$ Compare this linear equation with $y = mx + b$. $\Rightarrow m = 2$ Therefore, the slope of the line is 2. 3 What is the value of x in the given linear equation $4x + 6 = 5x \;-\; 8$? CorrectIncorrect Correct answer is: 14Given equation: $4x + 6 = 5x \;–\; 8$ Subtract 6 from both sides of the equation. $4x + 6 \;–\; 6 = 5x \;–\; 8 \;–\; 6$ $\Rightarrow 4x = 5x \;–\; 14$ $\Rightarrow 14 = 5x \;-\; 4x$ $\Rightarrow 14 = x$ 4 Jenny's father’s age is 4 times Jenny's. If the sum of both ages is 40, what is Jenny's age? 8 years 10 years 40 years 25 years CorrectIncorrect Correct answer is: 8 yearsLet Jenny age is x years. Jenny father’s age $= 4x$ $4x + x = 40$ $\Rightarrow 5x = 40$ $\Rightarrow x = 8$ Hence, Jenny's age is 8 years. 5 Solve the linear equation in one variable: $7x + 6 = 13$ 7 6 CorrectIncorrect Correct answer is: 1$7x + 6 = 13$ $\Rightarrow 7x + 6 \;-\; 6 = 13 \;-\; 6$ $\Rightarrow 7x = 7$ $\Rightarrow \frac{7x}{7} = \frac{7}{7}$ $\Rightarrow x = 1$ Therefore, the solution of the equation $7x + 6 = 13$ is $x = 1$. Frequently Asked Questions on Linear Equations What is the graph of linear equations in one variable? The graph of a linear equation in one variable is a straight line, either vertical or horizontal. The graph of linear equations in two variables is a straight line passing through x-axis or y-axis. What is the use of linear equations in our daily life? With the help of a linear equation, we can find the value of any unknown quantity, like: age, wages of an employee, doses of medicines, etc. What is the meaning of the solution of a linear equation? The solution of a linear equation is the value of the variable that satisfies the given linear equation. How to know if an equation is linear? The degree of a linear equation is 1. There is no term involving the product of variables. What are the parts of linear equations? A linear equation consists of several parts such as variables, constants, coefficients, operators, equality sign (=), that play different roles in defining the equation and its properties. 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14038
https://www.grc.nasa.gov/www/k-12/BGP/Donna/airtemp_intro.htm
Beginner's Guide to Propulsion: Air Temperature and Kinetic Energy - Intro + Text Only Site + Non-Flash Version + Contact Glenn ### Beginner's Guide to Propulsion Air Temperature and Kinetic Energy Subject Area(s):Mathematics (Pre-Alegebra) Grade Level: 7 - 9 National Standards: > Mathematics > > Mathematics as Problem Solving - Use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content. > > > > Mathematics as Communication - Formulate mathematical definitions and express generalizations discovered through investigations. > > > > Algebra - Represent situations that involve variable quantities with expressions and equations. > > > > Functions - Represent and analyze relationships using tables, verbal rules, equations and graphs. > > > Technology > > > > Research Tools - Use content-specific tools, software and simulations (e.g., environmental probes, graphing calculators, exploratory environments, Web tools) to support learning and research. > > > > Problem-Solving and Decision-Making Tools - Routinely and efficiently use on-line information resources to meet needs for collaboration, research, publications, communications, and productivity. Objectives: > After reading an explanation from a NASA Web site called The Beginner's Guide to Propulsion, you will demonstrate an understanding of the text by completing a worksheet using the kinetic energy formula along with scientific notation. The Beginner's Guide to Propulsion is a Web site of information prepared at NASA Glenn Research Center to help you better understand aircraft engine propulsion. Click Beginner's Guide to Propulsionto access the list of slides. In the "Short Index" Under the heading Static Gases, click on the slide called Air Temperature. Read the explanation to see how air temperature and kinetic energy are related to aircraft propulsion. Using this information, complete the Activity and Worksheet to demonstrate your ability to use the kinetic energy formula. Assessment: > You, or you and your partner(s), will be evaluated on the accuracy and/or feasibility of your answers. Evaluation: > You will demonstrate the ability to use data on various gases and apply it to the kinetic energy formula using scientific notation. Submitted by: Donna Langenderfer, Lorain Southview High School, Lorain, Ohio Related Pages: Activity Worksheet Answers Propulsion Activity Index Propulsion Index + Inspector General Hotline + Equal Employment Opportunity Data Posted Pursuant to the No Fear Act + Budgets, Strategic Plans and Accountability Reports + Freedom of Information Act + The President's Management Agenda + NASA Privacy Statement, Disclaimer, and Accessibility CertificationEditor: Tom Benson NASA Official: Tom Benson Last Updated: Thu, May 13 02:38:38 PM EDT 2021 + Contact Glenn
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https://calculus.subwiki.org/wiki/Parametric_derivative
Parametric derivative - Calculus Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Report errors/view log Most viewed methods Integration by parts First derivative test Product rule Second derivative test Piecewise differentiation Most viewed general theorems Clairaut's theorem on equality of mixed partials Inverse function theorem Lagrange mean value theorem Most viewed core concepts Derivative Limit Popular functions sin^2 sec^2 sinc cos^2 tan^2 sin^3 cos^3 Calculus Search Search Log in [x] Personal tools Log in Want site search autocompletion? See here Encountering 429 Too Many Requests errors when browsing the site? See here [x] Contents move to sidebar hide Beginning 1 DefinitionToggle Definition subsection 1.1 Algebraic definition 2 Relation with ordinary derivativeToggle Relation with ordinary derivative subsection 2.1 Parametric expressions aren't necessarily functions 2.2"Nice" parametric expressions define functions locally at most points 2.3 If '"UNIQ--postMath-00000035-QINU"' is nonzero (and of constant sign) for '"UNIQ--postMath-00000036-QINU"' around '"UNIQ--postMath-00000037-QINU"', the parametric expression defines a function locally around '"UNIQ--postMath-00000038-QINU"') 2.4 The parametric derivative makes sense and is the correct expression when it is defined 2.5 When the parametric derivative is undefined, the ordinary derivative may or may not be defined 2.5.1 Cases of undefined parametric derivative 2.5.2 Cases of defined parametric derivative 2.5.3 Limiting behavior on undefined cases Toggle the table of contents [x] Toggle the table of contents Parametric derivative Page Discussion [x] English Read View source View history [x] Tools Tools move to sidebar hide Actions Read View source View history Refresh General What links here Related changes Special pages Printable version Permanent link Page information Browse properties From Calculus Definition Algebraic definition The parametric derivatived y/d x{\displaystyle dy/dx} for a parametric curve x=f(t),y=g(t){\displaystyle x=f(t),y=g(t)} at a point t=t 0{\displaystyle t=t_{0}} is given as follows, where f′(t 0){\displaystyle f'(t_{0})} and g′(t 0){\displaystyle g'(t_{0})} both exist and f′(t){\displaystyle f'(t)} is nonzero and of constant sign for an open interval around t 0{\displaystyle t_{0}}: d y d x|t=t 0=g′(t 0)f′(t 0){\displaystyle !{\frac {dy}{dx}}|{t=t{0}}={\frac {g'(t_{0})}{f'(t_{0})}}} As a general function of t{\displaystyle t}, the parametric derivative d y/d x{\displaystyle dy/dx} is defined as g′(t)/f′(t){\displaystyle g'(t)/f'(t)}. NOTE: When calculating the general expression for the parametric derivative, before canceling any factors between g′(t){\displaystyle g'(t)} and f′(t){\displaystyle f'(t)}, it is important to separate out the cases where that common value is zero. For any points where f′(t)=0{\displaystyle f'(t)=0}, the parametric derivative is not defined (the ordinary derivativemay still be defined, but we would need another method to calculate it). MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with a {}0{\displaystyle {}_{0}} subscript) in the domain of the relevant functions, and then discuss the version that deals with a point that is free to move in the domain, by dropping the subscript. Why do we do this? The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i.e., it does not move around while we are defining the construct or applying the fact. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering. This is the purpose of the second, generic point version. Relation with ordinary derivative Parametric expressions aren't necessarily functions The notation d y/d x{\displaystyle dy/dx} should be used only in the context that y{\displaystyle y} is a function of x{\displaystyle x}, i.e., a single value of x{\displaystyle x} gives rise to a single value of y{\displaystyle y}. Generally speaking, this is not guaranteed with parametric curves. For instance, for a parametric curve x=f(t),y=g(t){\displaystyle x=f(t),y=g(t)}, y{\displaystyle y} is expressible as a function of x{\displaystyle x} only if g(t){\displaystyle g(t)} is completely determined by f(t){\displaystyle f(t)}. This is the case, for instance, when f{\displaystyle f} is a one-one function. But it's not always the case. Consider a circle: x=cos⁡t,y=sin⁡t{\displaystyle x=\cos t,y=\sin t} In this case, a single value of x{\displaystyle x}, in most cases, corresponds to two values of y{\displaystyle y} that are negatives of each other. That's because if x 0=cos⁡t 0{\displaystyle x_{0}=\cos t_{0}}, we have also x=cos⁡(−t 0){\displaystyle x=\cos(-t_{0})}, so both sin⁡t 0{\displaystyle \sin t_{0}} and −sin⁡t 0{\displaystyle -\sin t_{0}} work. The only case where y{\displaystyle y} is unique in terms of x{\displaystyle x} is when x=1{\displaystyle x=1} and x=−1{\displaystyle x=-1}. Geometrically, vertical secant lines intersect the upper semicircle and lower semicircle, and these are the two y{\displaystyle y}-values for a given x{\displaystyle x}-value. "Nice" parametric expressions define functions locally at most points As described above, for a parametric curve, y{\displaystyle y} need not globally be a function of x{\displaystyle x}. However, even in the presented example of a circle, for most t 0{\displaystyle t_{0}}, if we restrict t{\displaystyle t} to a small enough open interval around t 0{\displaystyle t_{0}}, y{\displaystyle y} is a function of x{\displaystyle x} for the part of the curve where t{\displaystyle t} is restricted to that interval. So it's "locally" a function. Differentiation being a local operation, it still makes sense to take the derivative at the point. The parametric derivative should therefore be understood as the derivative for the function obtained by taking the local restriction. In the circle example, the only points at which y{\displaystyle y} is locally not a function of x{\displaystyle x} are the ones at the far left and far right: x=1{\displaystyle x=1} and x=−1{\displaystyle x=-1} (ironically, these are the only points with unique global y{\displaystyle y}-values). If f′(t){\displaystyle f'(t)} is nonzero (and of constant sign) for t{\displaystyle t} around t 0{\displaystyle t_{0}}, the parametric expression defines a function locally around t 0{\displaystyle t_{0}} If f′(t){\displaystyle f'(t)} is nonzero and of constant sign around t 0{\displaystyle t_{0}}, then (depending on the sign)f{\displaystyle f} is an increasing or decreasing function around t_0. In particular, within that open interval, f{\displaystyle f} is a one-one function, so no value of x{\displaystyle x} repeats. Thus locally y{\displaystyle y} is a function of x{\displaystyle x} on that interval. NOTE: To deduce increasing/decreasing behavior, it is not enough to assume that f′(t 0)≠0{\displaystyle f'(t_{0})\neq 0}. For a counterexample, see positive derivative at a point not implies increasing around the point. NOTE 2: The assumption of constant sign is not necessary; it can be deduced from the derivative being nonzero. That's because derivative of differentiable function satisfies intermediate value property. The parametric derivative makes sense and is the correct expression when it is defined We have now come full circle. In the case that f′(t)≠0{\displaystyle f'(t)\neq 0} around t 0{\displaystyle t_{0}}, y{\displaystyle y} is locally a function of x{\displaystyle x} around t 0{\displaystyle t_{0}}. In such cases, if g′(t 0){\displaystyle g'(t_{0})} also exists, the expression g′(t 0)f′(t 0){\displaystyle {\frac {g'(t_{0})}{f'(t_{0})}}} gives the parametric derivative. The proof that the parametric derivative expression is correct follows from the chain rule for differentiation. As established above, y{\displaystyle y} is locally a function of x{\displaystyle x} around t 0{\displaystyle t_{0}}. Let's call this local function h{\displaystyle h}. Locally around t 0{\displaystyle t_{0}}, we have: g=h∘f{\displaystyle g=h\circ f} By the chain rule for differentiation: g′(t)=h′(f(t))f′(t){\displaystyle g'(t)=h'(f(t))f'(t)} At t=t 0{\displaystyle t=t_{0}}, we get: g′(t 0)=h′(f(t 0))f′(t 0){\displaystyle g'(t_{0})=h'(f(t_{0}))f'(t_{0})} Rearranging, we get: h′(f(t 0))=g′(t 0)f′(t 0){\displaystyle h'(f(t_{0}))={\frac {g'(t_{0})}{f'(t_{0})}}} The left side is the expression we are trying to calculate, and the right side is the expression that we want to prove it to be. When the parametric derivative is undefined, the ordinary derivative may or may not be defined Consider the example of the astroid as a parametric curve: x=f(t)=cos 3⁡t,y=g(t)=sin 3⁡t{\displaystyle x=f(t)=\cos ^{3}t,y=g(t)=\sin ^{3}t} The derivatives are as follows: f′(t)=−3 cos 2⁡t sin⁡t,g′(t)=3 sin 2⁡t cos⁡t{\displaystyle f'(t)=-3\cos ^{2}t\sin t,g'(t)=3\sin ^{2}t\cos t} The parametric derivative is therefore: 3 sin 2⁡t cos⁡t−3 cos 2⁡t sin⁡t{\displaystyle {\frac {3\sin ^{2}t\cos t}{-3\cos ^{2}t\sin t}}} We can cancel the 3: sin 2⁡t cos⁡t−cos 2⁡t sin⁡t{\displaystyle {\frac {\sin ^{2}t\cos t}{-\cos ^{2}t\sin t}}} However, we should not cancel the sin⁡t{\displaystyle \sin t} and cos⁡t{\displaystyle \cos t} until we have isolated the cases where either of them is zero. Cases of undefined parametric derivative The cases where the derivative is undefined correspond precisely to the cases where cos⁡t=0{\displaystyle \cos t=0} or sin⁡t=0{\displaystyle \sin t=0}, which correspond precisely to the integer multiples of π/2{\displaystyle \pi /2}. At these points, the parametric derivative is not defined. The actual point coordinates are (1,0), (0,1), (-1,0), and (0,-1). Cases of defined parametric derivative When the parametric derivative is defined, we can cancel the common factors from the numerator and denominator. After simplifying, we get: sin 2⁡t cos⁡t−cos 2⁡t sin⁡t=sin⁡t−cos⁡t=−tan⁡t{\displaystyle {\frac {\sin ^{2}t\cos t}{-\cos ^{2}t\sin t}}={\frac {\sin t}{-\cos t}}=-\tan t} Limiting behavior on undefined cases When t=0{\displaystyle t=0} or t=π{\displaystyle t=\pi }, the limit of the parametric derivative expression (−tan⁡t{\displaystyle -\tan t}) equals zero. However, despite the limit existing, the parametric derivative is undefined at the point. In fact, the parametric curve is not locally a function at these points: geometrically, the x{\displaystyle x}-coordinate doubles back, forming a horizontal cusp. When t=π/2{\displaystyle t=\pi /2} or t=3 π/2{\displaystyle t=3\pi /2}, the parametric derivative does not have a defined limit, with the left-hand and right-hand limits being opposite signs of infinity. Geometrically, we have vertical cusps at these points. NOTE: In general, when the parametric derivative is not defined at a point, but there is a limit for it at that point, then the ordinary derivative, if defined, must equal that value. Retrieved from " This page was last edited on 27 September 2021, at 14:46. Content is available under Creative Commons Attribution Share Alike unless otherwise noted. Privacy policy About Calculus Disclaimers Mobile view Toggle limited content width
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https://www.geeksforgeeks.org/maths/bodmas-rule/
BODMAS Rule - Order of Operations in Maths Last Updated : 23 Jul, 2025 Suggest changes Like Article BODMAS rule is a set of guidelines used to determine the sequence in which mathematical operations must be performed when solving an expression. Following the correct order of operations is vital to getting accurate results. The term BODMAS is an acronym used to remember the order of operations to be followed when solving arithmetic expressions involving multiple operations. It stands for Brackets, Orders (i.e., powers and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Here’s the order of operations according to BODMAS: Brackets → Orders (Exponents and Roots) → Division → Multiplication → Addition → Subtraction. BODMAS Rule - Order of Operations To get accurate results always follow the order sequence to avoid confusion. Order and Operations in BODMAS rule is shown below as, | Rules of BODMAS in Order | Operations Rules | | --- | B - Brackets | Evaluate expressions within brackets first. | Example: 23 + (5 - 3) - 16/2 + 4×3 + 1 First solve (5 - 3) | | O - Orders | Evaluate expressions with exponents or roots. | Example: 23 + 2 - 16/2 + 4×3 + 1 Then solve (23) | | D - Division | Perform division from left to right. | Example: 8 + 2 - 16/2 + 4×3 + 1 Then solve (16/2) | | M - Multiplication | Perform multiplication from left to right. | Example: 8 + 2 - 8 + 4×3 + 1 Then solve (4×3) | | A - Addition | Perform addition from left to right. | Example: 8 + 2 - 8 + 12 + 1 Then solve 8 + 2 + 12 + 1 | | S - Subtraction | Perform subtraction from left to right. | Example: 23 - 8 At last, solve 23 - 8 = 15 | When to Use BODMAS Rule? BODMAS Rule is used when there are multiple arithmetic operations (Divide, Multiply, Addition, and Subtraction) in one equation only and the preference of solving then impact the result of the equation then we use the BODMAS rule to solve are equation correctly. Conditions to follow while solving using the BOADMAS rule are the following Bracket is to be simplified first. In bracket also first —(Bar) is simplified then ()(Parentheses) is simplified, then {}(Curly bracket) is simplified, and at last are simplified. Negative sign ahead of any bracket changes the internal sign of the bracket(positive to negative and vice-versa) when the bracket is opened. Example: - (b - c + d) = - b + c - d Any term outside the bracket is multiplied using the distributive property of multiplication. Example: a(b + c) = ab + ac Steps for Solving Problems using BODMAS Rule Step 1: Brackets: Evaluate expressions within brackets first. Step 2: Orders: Simplify expressions with exponents or roots. Step 3: Division: Perform division from left to right. Step 4: Multiplication: Perform multiplication from left to right. Step 5: Addition: Add numbers from left to right. Step 6: Subtraction: Subtract numbers from left to right. Simplification of Bracket BODMAS is used to simplify various arithmetic problems and simplifying the bracket is the first priority and the priority order of the bracket is (), {}, and []. That is we first solve for the bracket (), then {} and last we solve the bracket []. This is explained by the example added below as, Example: Simplify [2 + {3 × 4}]/(5-2) = [2 + {3 × 4}]/(5-3) = [2 + 12]/2 = 14/2 = 7 BODMAS, PEMDAS and BIDMAS PEMDAS and BIDMAS are variations of the BODMAS acronym, each emphasizing the same order of operations but using slightly different terminology. | BODMAS | BIDMAS | PEMDAS | --- | B - Brackets ( ( ), { }, [ ] ) | B - Brackets ( ), { }, [ ] | P - Parentheses ( ), { }, [ ] | | O - Order ( √ ) | I - Indices ( xn) | E - Exponents (xn) | | D - Division (÷) | D - Division (÷) | M - Multiplication (×) | | M - Multiplication (×) | M - Multiplication (×) | D - Division (÷) | | A - Addition (+) | A - Addition (+) | A - Addition (+) | | S - Subtraction (-) | S - Subtraction (-) | S - Subtraction (-) | Read More: Division Multiplication Addition Subtraction BODMAS Rule Solved Examples Example 1: Solve 2+7×8-5 Solution: Applying BODMAS 2 + (7 × 8) - 5 = 2 + 56 -5 = (2 + 56) - 5 = 58 - 5 = 53 Example 2: Find the value of the expression : (8 × 6 - 7) + 65 Solution: As brackets are provided here, solve them first (8 × 6 - 7) in this, multiplication operator has the highest priority therefore it will be = (48 - 7) = 41 So, the final result will be 41 + 65 = 106 Example 3: Find the value of 6× 6+ 6× 6+ 6× 6 Solution: Here, only have two operators that is addition and multiplication. Therefore, solve multiplication first 6× 6+ 6× 6+ 6× 6 = 36 + 36 + 36 =108 Example 4: Evaluate 8/4 × 6/3 × 7 + 8 - (70/5 - 6) Solution: Evaluate 8/ 4 × 6/3 × 7 + 8 - (70/5 - 6 ) We can rewrite expression as (8/4 × 6/3 × 7 + 8) - (70/5 - 6) Now we will solve respective brackets , = (2 × 2 × 7 + 8) - (14 - 6) = (4 × 7 + 8) - (8) = (28 + 8) - (8) = (36) - (8) = 28 R rida07 Improve Article Tags : Mathematics School Learning Class 7 Maths MAQ Maths-Formulas Maths-Calculators Maths-Class-8 Maths-Class-7 Similar Reads Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields 5 min read Basic Arithmetic What are Numbers? Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or 15+ min readArithmetic Operations Arithmetic Operations are the basic mathematical operations€”Addition, Subtraction, Multiplication, and Division€”used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in 9 min readFractions - Definition, Types and Examples Fractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E 7 min readWhat are Decimals? Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t 10 min readExponents Exponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t 9 min readPercentage In mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a 5 min read Algebra Variable in Maths A variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch 5 min readPolynomials| Degree | Types | Properties and Examples Polynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one 9 min readCoefficient A coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 €” it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is 8 min readAlgebraic Identities Algebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari 14 min read Properties of Algebraic Operations Algebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (ˆ’) 3 min read Geometry Lines and Angles Lines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang 9 min readGeometric Shapes in Maths Geometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two 2 min readArea and Perimeter of Shapes | Formula and Examples Area and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of 10 min readSurface Areas and Volumes Surface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the 10 min readPoints, Lines and Planes Points, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction 14 min readCoordinate Axes and Coordinate Planes in 3D space In a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su 6 min read Trigonometry & Vector Algebra Trigonometric Ratios There are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give 4 min readTrigonometric Equations | Definition, Examples & How to Solve Trigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might 9 min readTrigonometric Identities Trigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t 10 min readTrigonometric Functions Trigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in 6 min readInverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function 11 min readInverse Trigonometric Identities Inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv 9 min read Calculus Introduction to Differential Calculus Differential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia 6 min readLimits in Calculus In mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp 12 min readContinuity of Functions Continuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li 13 min readDifferentiation Differentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee 2 min readDifferentiability of a Function | Class 12 Maths Continuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f 11 min readIntegration Integration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y 3 min read Probability and Statistics Basic Concepts of Probability Probability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors 7 min readBayes' Theorem Bayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba 13 min readProbability Distribution - Function, Formula, Table A probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows 13 min readDescriptive Statistic Statistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat 5 min readWhat is Inferential Statistics? Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi 7 min readMeasures of Central Tendency in Statistics Central tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o 11 min readSet Theory Set theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple, 3 min read Practice NCERT Solutions for Class 8 to 12 The NCERT Solutions are designed to help the students build a strong foundation and gain a better understanding of each and every question they attempt. This article provides updated NCERT Solutions for Classes 8 to 12 in all subjects for the new academic session 2023-24. The solutions are carefully 7 min readRD Sharma Class 8 Solutions for Maths: Chapter Wise PDF RD Sharma Class 8 Math is one of the best Mathematics book. It has thousands of questions on each topics organized for students to practice. RD Sharma Class 8 Solutions covers different types of questions with varying difficulty levels. The solutions provided by GeeksforGeeks help to practice the qu 5 min readRD Sharma Class 9 Solutions RD Sharma Solutions for class 9 provides vast knowledge about the concepts through the chapter-wise solutions. These solutions help to solve problems of higher difficulty and to ensure students have a good practice of all types of questions that can be framed in the examination. Referring to the sol 10 min readRD Sharma Class 10 Solutions RD Sharma Class 10 Solutions offer excellent reference material for students, enabling them to develop a firm understanding of the concepts covered. in each chapter of the textbook. As Class 10 mathematics is categorized into various crucial topics such as Algebra, Geometry, and Trigonometry, which 9 min readRD Sharma Class 11 Solutions for Maths RD Sharma Solutions for Class 11 covers different types of questions with varying difficulty levels. Practising these questions with solutions may ensure that students can do a good practice of all types of questions that can be framed in the examination. This ensures that they excel in their final 13 min readRD Sharma Class 12 Solutions for Maths RD Sharma Solutions for class 12 provide solutions to a wide range of questions with a varying difficulty level. With the help of numerous sums and examples, it helps the student to understand and clear the chapter thoroughly. Solving the given questions inside each chapter of RD Sharma will allow t 13 min read We use cookies to ensure you have the best browsing experience on our website. By using our site, you acknowledge that you have read and understood our Cookie Policy & Privacy Policy Improvement Suggest Changes Help us improve. Share your suggestions to enhance the article. Contribute your expertise and make a difference in the GeeksforGeeks portal. Create Improvement Enhance the article with your expertise. 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https://medbox.org/index.php/dl/5e148832db60a2044c2d2852
WHO Guidelines for the management of severe erythema nodosum leprosum (ENL) reactions General principles: Severe ENL reaction is often recurrent and chronic and may vary in its presentation. The management of severe ENL is best undertaken by physician at a referral centre. The dose and duration of anti-reaction drugs used may be adjusted by the physician according to individual patient's needs. Definition: Severe ENL reactions include: Numerous ENL nodules with high fever ENL nodules and neuritis Ulcerating and pustular ENL Recurrent episodes of ENL Involvement of other organs (e.g. eyes, testes, lymph nodes and joints) Management with corticosteroids: If still on antileprosy treatment, continue the standard course with MDT. Use adequate doses of analgesics to control fever and pain. Use standard course of prednisolone in dosage per day not exceeding 1 mg per Kg body weight. Total duration 12 weeks. Management with clofazimine and corticosteroids: This is indicated - In cases with severe ENL who are not responding satisfactorily to treatment with corticosteroids or where the risk of toxicity with corticosteroids is high. If still on antileprosy treatment, continue the standard course with MDT. Use adequate doses of analgesics to control fever and pain Use standard course of prednisolone in dosage per day not exceeding 1 mg per Kg body weight Start clofazimine 100 mg three times a day for maximum of 12 weeks Complete the standard course of prednisolone. Continue clofazimine as below. Taper the dose of clofazimine to 100 mg twice a day for 12 weeks and then 100 mg once a day for 12-24 weeks. Management with only clofazimine: This is indicated - In cases with severe ENL where use of corticosteroids is contraindicated. If still on antileprosy treatment, continue the standard course with MDT. Use adequate doses of analgesics to control fever and pain. Start clofazimine 100 mg three times a day for maximum of 12 weeks Taper the dose of clofazimine to 100 mg twice a day for 12 weeks and then 100 mg once a day for 12-24 weeks. Note: If the MDT treatment is already completed the management of ENL should follow the guidelines. There is no need to restart MDT. The total duration of a standard course of corticosteroids (prednisolone) is 12 weeks. 3. The total duration of treatment with high dosage clofazimine should not exceed 12 months. It takes about 4-6 weeks for clofazimine to take full effect in controlling ENL. Other drug claimed to be useful in ENL is pentoxifylline, alone or in combination with clofazimine/prednisolone. For the reason of well-known teratogenic side effects WHO does not support use of thalidomide for the management of ENL in leprosy. References: WHO Expert Committee on Leprosy. TRS 874, 1998, WHO, Geneva Leprosy for medical practitioners and paramedical workers by Yawalkar SJ. Seventh Edition 2002, Basle Manson's textbook of tropical medicine. 21st Edition 2002. Guide to eliminate leprosy as a public health problem. First Edition 2000. WHO, Geneva. The final push strategy to eliminate leprosy as a public health problem: Questions & Answers. First Edition 2002. WHO, Geneva. Nery JA et al. The use of pentoxifylline in the treatment of type 2 reactional episodes in leprosy. Indian J. Leprosy, 2000, 72 (4): 457-467. Welsh O et al. A new therapeutic approach to type II leprosy reaction. International J. Dermatology, 1999,38: 931-933.
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https://brainly.com/question/30916106
[FREE] Determine the value of k for which the inequality x - 2 \leq k . - brainly.com 4 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +86,6k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +40,7k Ace exams faster, with practice that adapts to you Practice Worksheets +5,4k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Determine the value of k for which the inequality x−2≤k. 1 See answer Explain with Learning Companion NEW Asked by elhabibi • 02/26/2023 0:00 / -- Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 5376246 people 5M 0.0 0 Upload your school material for a more relevant answer In order for the inequality {x1 - 2k = -3} to have no solution, k must be equal to 1.5. What is value? Value is the regard that something is held to deserve; the importance, worth, or usefulness of something. Value is subjective and can differ from one individual to another. Values can be influenced by cultural, religious, and personal beliefs, as well as life experiences. Values can also refer to economic worth or the worth of something in terms of money. In economics, value is a measure of the benefit or utility of a good or service. This can be determined by solving the equation for k. To do this, first subtract -3 from both sides of the equation so that the left side is equal to 0. This leaves {x1 - 2k = 0}. Then divide both sides of the equation by -2 to solve for k. This leaves {k = -1/2 x1}. Since -1/2 x1 must equal 1.5 for the equation to have no solution, k must equal 1.5. To know more about value click- brainly.com/question/25184007 SPJ1 Answered by MonikaNandal •9.7K answers•5.4M people helped Thanks 0 0.0 (0 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 5376246 people 5M 0.0 0 Principles of Macroeconomics Fundamentals of Calculus - Joel Robbin, Sigurd Angenent Introductory Physics - Building Models to Describe Our World - Ryan D. Martin, Emma Neary, Joshua Rinaldo, Olivia Woodman Upload your school material for a more relevant answer To find the value of k for the inequality x−2≤k, we rearrange it to k≥x−2. This indicates that k must be greater than or equal to the value of x−2 for any given input of x. For specific values like x=2, k must be at least 0; for x=5, k must be at least 3. Explanation To determine the value of k for which the inequality x−2≤k holds, we need to isolate k. The inequality states that the expression on the left must be at most equal to k. We can rearrange the inequality as follows: Start with the inequality: x−2≤k To find the maximum value of k, we can express it as: k≥x−2 This means that for any value of x, k must be at least x−2. Therefore, we can conclude that the value of k needs to be no less than the expression x−2 calculated at any desired point (or range) of x. For example, if we specifically want to find k when x=2: k≥2−2 k≥0 Similarly, if we wanted to find k when x=5: k≥5−2 k≥3 Thus, k must be at least equal to the maximum value of x−2 based on the values that x can take. Examples & Evidence For example, if we calculate when x=3, then k must be at least 3−2=1. This shows that the inequality must hold true for different values of x while adjusting k accordingly. This conclusion is based on basic principles of inequalities in algebra, which state that rearranging terms can help isolate variables and determine their necessary constraints. Thanks 0 0.0 (0 votes) Advertisement elhabibi has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics Which expression has the same value as the expression shown below? −8 3​−8 7​ A. 8 3​+8 7​ B. −8 3​+8 7​ C. 8 3​+(−8 7​) D. −8 3​+(−8 7​) Hasmeet walks once round a circle with diameter 80 metres. There are 8 points equally spaced on the circumference of the circle. Find the distance Hasmeet walks between one point and the next point. f(x)=sec x Show that f′′(x)=2 sec 3 x−sec x Consider the following incomplete deposit slip: | Description | Amount ($) | | ---: | Cash, including coins | 150 | 75 | | Check 1 | 564 | 81 | | Check 2 | 2192 | 43 | | Check 3 | 4864 | 01 | | Subtotal | ???? | ?? | | Less cash received | ???? | ?? | | Total | 7050 | 50 | | | | | How much cash did the person who filled out this deposit slip receive? Which sequence is generated by the function f(n+1)=f(n)−2 for f(1)=10? A. −2,8,18,28,38,… B. 10,8,6,4,2,… C. 8,18,28,38,48,… D. −10,−12,−14,−16,−18,… Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com Dismiss Materials from your teacher, like lecture notes or study guides, help Brainly adjust this answer to fit your needs. Dismiss
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https://digitalcommons.wayne.edu/cgi/viewcontent.cgi?article=1331&context=jmasm
Journal of Modern Applied Statistical Methods Volume 10 | Issue 2 Article 14 11-1-2011 A Comparison of Factor Rotation Methods for Dichotomous Data W. Holmes Finch Ball State University , whfinch@bsu.edu Follow this and additional works at: Part of the Applied Statistics Commons, Social and Behavioral Sciences Commons, and the Statistical Theory Commons This Regular Article is brought to you for free and open access by the Open Access Journals at DigitalCommons@WayneState. It has been accepted for inclusion in Journal of Modern Applied Statistical Methods by an authorized editor of DigitalCommons@WayneState. Recommended Citation Finch, W. Holmes (2011) "A Comparison of Factor Rotation Methods for Dichotomous Data," Journal of Modern Applied Statistical Methods : Vol. 10 : Iss. 2 , Article 14. DOI: 10.22237/jmasm/1320120780 Available at: Journal of Modern Applied Statistical Methods Copyright © 2011 JMASM, Inc. November 2011, Vol. 10, No. 2, 549-570 1538 – 9472/11/$95.00 549 A Comparison of Factor Rotation Methods for Dichotomous Data W. Holmes Finch Ball State University, Muncie, IN Exploratory factor analysis (EFA) is frequently used in the social sciences and is a common component in many validity studies. A core aspect of EFA is the determination of which observed indicator variables are associated with which latent factors through the use of factor loadings. Loadings are initially extracted using an algorithm, such as maximum likelihood or weighted least squares, and then transformed - or rotated - to make them more interpretable. There are a number of rotational techniques available to the researcher making use of EFA. Prior work has discussed the advantages of a number of these criteria from a theoretical perspective, but few previous studies compare their performance across a broad range of conditions. This simulation study compared eight factor rotation criteria in terms of how well they were able to group dichotomous indicator variables correctly on the same factor, order the indicators by the magnitude of the factor loadings (identifying those indicators that were most strongly associated with the factors) and estimate the inter-factor correlations. Results reveal a mixed pattern of performance among the various rotations with the orthogonal Equamax consistently near the top in terms of correctly grouping and ordering indicator variables, and the orthogonal Facparsim performing well with more observed indicators. Advice regarding possible rotations to use for researchers conducting EFA with dichotomous indicators is provided. Key words: Factor rotation, dichotomous data, exploratory factor analysis, EFA. Introduction Exploratory Factor Analysis (EFA) of items on an instrument is a tool employed by psychometricians in the investigation of validity evidence for cognitive and affective measures (Zumbo, 2007; McDonald, 1999). In conjunction with subject matter expertise regarding the purpose of the instrument and its assumed structure, EFA can be used to identify the latent constructs underlying the observed items (McLeod, Swygert & Thissen, 2001). When items are found to group in conceptually meaningful ways based on content, instrument developers can conclude that the traits the scale W. Holmes Finch is a Professor of Psychology in the Department of Educational Psychology, and Educational Psychology Director of Research in the Office of Charter School. Email him at: whfinch@bsu.edu. is intended to measure are actually being represented. Conversely, when individual items are found to load on multiple factors - or to group in ways that do not conform to their content or intent - developers may target them for revision or removal from the instrument (Sass & Schmitt, 2010). Given its role in validity assessment, psychometricians must have a full understanding regarding the performance of EFA in the context of item level data under a variety of conditions. The objective of this simulation study was to investigate one important aspect of the EFA analysis process: factor rotation. A variety of factor rotation methods were compared with respect to how well they recovered the underlying latent structure for a set of dichotomous indicators like those that might comprise a psychological or educational scale. (Readers interested in learning more about the basic factor analysis model are encouraged to read one of several excellent references including: Gorsuch, 1983; Thompson, 2004; McLeod, Swygert & Thissen, 2001.) A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 550 Factor Analysis of Dichotomous Data The original EFA model was based on the presumption that observed indicators were continuous variables, calling into question its applicability for dichotomous data such as that from item responses (Gorsuch, 1983). Early analyses applying the standard linear EFA model to dichotomous item response data consistently identified a factor reflecting item difficulty, having nothing to do with substantive dimensions related to item content (Hattie, 1985; Guilford, 1941; Spearman, 1927). Furthermore, the use of linear factor analysis with dichotomous items was found to produce distorted factor loading estimates for very difficult and very easy items (Hattie, 1985). In response to these problems, McDonald introduced nonlinear factor analysis based on the normal ogive (McDonald, 1967; 1962). In the case of dichotomous variables such as item responses, this factor model takes the form 0 1 21 { 1| } ( ... )j j j jm P U N θ β β θ β θ β θ= = + + + + (1) where Uj is the response to item with a 1indicating correct, βj0 is the intercept for item j and βj1 is the factor loading for item j with latent trait m. Parameter estimation in this Normal Ogive Harmonic Analysis Robust Method (NOHARM) is conducted using unweighted least squares (ULS), allowing for analysis of large sets of items exhibiting high dimensionality (McDonald, 1981; 1967). This model was implemented in the NOHARM software package (Fraser & McDonald, 1988) and features both Varimax and Promax rotations. Bock and Aitkin (1981) developed an alternative model for the factor analysis of dichotomous item response data that takes the form: ( ) ( ) 22 11| 2 i jtij jzP x e dt θ θ π∞ −−= =  (2) w here ( ) ( )i j j i jz a b ι θθ= − , ja is the slope for item j, jb is the threshold for item j, and j ι θ is the latent trait for subject i on item j. In this conceptualization of the model, a j corresponds to item discrimination and b j corresponds to item difficulty, in the context of item response theory. This full information factor model underlies the TESTFACT software (Bock, et al., 2003) and is estimated using marginal maximum likelihood (MML), in contrast to the ULS used with NOHARM. Researchers comparing these approaches have found that ULS tends to provide more accurate parameter estimation for a smaller number of items, although MML is generally more accurate for more items (Gosz & Walker, 2002). As with NOHARM, TESTFACT allows for either VARIMAX or PROMAX rotations. Christofferson (1975) also introduced a factor model for item response data based on the normal ogive model, as was McDonald’s approach. The Christofferson model is expressed as ( ) 22 11 2itizP u e dt π∞ −= =  (3) where zi is the threshold for item i. This model was expanded upon by Muthén (1978) and has been shown to be equivalent to McDonald’s model (McDonald, 1997). Another approach to factor analysis for dichotomous data, such as item responses, is based on robust weighted least squares (RWLS). Weighted least squares (WLS) estimation has been shown to perform poorly for categorical variables in the context of factor analysis with small to moderate sample sizes (Flora & Curran, 2004). Muthén, du Toit and Spisic (1997) and Muthén (1993) extended the WLS approach in the form of RWLS, which does not require the inversion of the weight matrix used in the standard WLS approach, leading to very stable parameter estimation for samples as small as 100 with dichotomous indicator variables (Flora & Curran, 2004). The RWLS approach can also be used in the context of EFA with the MPLus software package (Muthén & Muthén, 2007) as was done herein. Factor Rotation The estimation of factor loadings in EFA typically occurs in two stages, the first of W. HOLMES FINCH 551 which - factor extraction - involves the initial estimation of loadings based on the covariance matrix for the indicator variables. The second step in an EFA - factor rotation - involves the transformation of the initial factor loadings in order to make them more interpretable in terms of (ideally) clearly associating an indicator variable with a single factor (Gorsuch, 1983). Although a large number of rotation algorithms have been described in the literature, these criteria all have the common goal of reducing a complexity function, f(Λ), so that the loadings approximate a simple structure and are thus more interpretable in practice. The notion of simple structure has been discussed extensively in the factor analysis literature, and though there is a common sense as to its meaning, there is no agreement regarding exact details. Thurstone (1947) first described simple structure as occurring when each row in the factor loading matrix has at least one zero, where rows represent indicator variables and columns represent factors. He also included 4 other rules that were initially intended to yield over-determination and stability of the factor loading matrix, but which were subsequently used by other researchers to define simple structure for methods of rotation (Browne, 2001). Subsequent to Thurstone’s work, others varying definitions of simple structure have been provided. For example, Jennrich (2007) defined perfect simple structure as occurring when each indicator has only one nonzero factor loading and compared it to Thurstone simple structure in which there are a fair number of zeros in the factor loading matrix, but not as many as in perfect simple structure. Conversely, Browne (2001) defined the complexity of a factor pattern as the number of nonzero elements in the rows of the loading matrix. These many varying definitions of simple structure have led to the development of a number of rotational criteria with the overarching goal of obtaining the most interpretable solution possible for a set of data (Asparouhov & Muthén, 2009). Factor rotations are broadly classified as either: (1) orthogonal, in which the factors are constrained to be uncorrelated, or (2) oblique, in which this constraint is relaxed. Within each of these classes, several options are available. Browne (2001) provides an excellent discussion of a number of these rotational criteria along with a history of their development and concluded that, when the factor pattern in the population conformed to what is termed above as pure simple structure, most of the rotation methods reviewed produced acceptable solutions. However, when there was greater complexity in the factor pattern, the rotations did not all perform equally well and - in some cases - the majority of them produced unacceptable results (Browne, 2001). For this reason, he argued for the need of educated human judgment in the selection of the best factor rotation solution for a given problem. In a similar vein, Yates (1987) stated that some rotations are designed to find a perfect simple structure solution in all cases, even when this may not be appropriate for the data at hand. Several excellent discussions of these rotation criteria are available in the literature, including two recently published manuscripts which provide detailed descriptions for interested readers (Sass & Schmitt, 2010; Asparouhov & Muthén, 2009). The rotations included in this study are summarized in Table 1. Many of these methods are readily available in common statistical software packages such as MPlus (Muthén & Muthén, 2007), which is featured in this study, as well as SAS and SPSS. Perhaps the most popular method in applied practice is the orthogonal Varimax rotation (Kaiser, 1958), which is a member of a larger group of criteria known collectively as the Orthomax family of rotations. The goal in Varimax rotation is to create simple structure by maximizing differences among loadings within factors across variables. Other notable Orthomax rotations include Quartimax, Equamax, Parsimax and Factor Parsimony. Promax is a two-stage oblique Procrustean rotation in which loadings are first obtained from the orthogonal Varimax rotation and then transformed based upon a target matrix of loadings raised to a particular power (typically the 4 th power), after which a transformation matrix is obtained using least squares (Hendrickson & White, 1964). Other Procrustean rotations include Promaj (Trendafilov, 1994) and Promin (Lorenzo-Seva, 1999). Another group of factor rotations is the Crawford-Ferguson (CF) family (Crawford & A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 552 Table 1: Summary of Studied Rotation Methods Rotation Criteria Definition Comments Varimax ( ) ( ) ( ) 22 22 2 21 1/ ppij ij ii f p p λλ= =    Λ = −       Spreads variance across factors Quartimin ( ) 2 21 1 pmmij il ijlj f λ λ = = ≠Λ =  Designed to minimize complexity of loadings across indicator variables. Quartimax ( ) 4 2 21 1 1 1 ppmmmij ij il ijijlj f λλ λ = = = = ≠Λ = +  Spreads variance across indicators Equamax 2 2 2 21 1 1 11 2 2 pppmmmij il ij lj ijljiilj m mp p λ λ λ λ = = ≠ = = ≠   − +     Combines Quartimax and Varimax criteria Parsimax ( ) 2 2 2 21 1 1 11 11 2 2 pppmmmij il ij il ijljiilj m mf p m p m λ λ λ λ = = ≠ = = ≠   − −Λ = − +    − + −     Equal weight is given to factor and indicator complexity. Geomin ( ) ( ) 1211 pmmij ij f λ= =   Λ = + ∈    ∏ Accommodates factor complexity while still providing interpretable solution. Promax Raise loadings from Varimax to some power (e.g., 4) and rotate the resulting matrix allowing for correlated factors. Based on Varimax rotation, but allows for correlated factors. Facparsim ( ) 2 21 1 ppmij il iilj f λ λ = = ≠Λ =  Minimizes loading complexity across factors. p= Number of indicators, m= Number of factors, λ=Extracted factor loading W. HOLMES FINCH 553 Ferguson, 1970). This criterion accounts for complexity across both the indicator variables and the factors. Members of the CF family differ in terms of a parameter, k, ranging between 0 and 1, where larger values of k place greater weight on minimizing factor complexity, whereas lower values emphasize the minimization of indicator variable complexity (Crawford & Ferguson, 1970). Other rotations that have been discussed widely in the literature are oblique Quartimin (Carroll, 1957), which seeks to minimize complexity only within the indicator variables, and Geomin (Yates, 1987) which also was designed to minimize variable complexity, but which allows for more such complexity than does Quartimin. There are a number of other rotation criteria extant in the literature. However, given that the current study is focused on comparing methods that are available to practitioners in commonly available software, they will not be discussed here. The interested reader is invited to read Mulaik (2010) and Browne (2001) for excellent descriptions of these alternative methods of rotation. Prior Research on Factor Rotations As noted, a large number of rotational criteria are available to a researcher interested in using EFA. Some of these, such as Varimax and Promax, are well known and frequently used, while others may be less well known but offer statistical advantages over the more commonly used approaches (Asparouhov & Muthén, 2009). Despite the abundance of available rotational methods, a great deal of empirical research has not been conducted regarding which might be best in a given research context (Sass & Schmitt, 2010). In addition, virtually none of the prior work examining the performance of these various rotation methods has been conducted with dichotomous indicator variables (the focus of this study). Therefore, earlier work using continuous indicators provides the only extant evidence regarding the comparative behavior of factor rotation methods, all of which can be applied to both EFA with continuous or dichotomous indicators. Thus, although they did not utilize dichotomous indicators, earlier studies provide researchers with some insights into what might be expected with regard to the performance of these rotation methods in general. Nevertheless, it is not clear to what extent earlier research with continuous indicators may be applicable. Therefore, this article builds upon this earlier research in an attempt to extend these results based on continuous variables to the case of dichotomous indicators. One recent Monte Carlo study (Sass & Schmitt, 2010) compared the ability of four rotational methods in terms of their abilities to reproduce the population factor loadings used to generate the data. This study involved 30 standard normally distributed observed indicators with 2 factors, and 4 different types of factor structure including perfect simple, approximate simple, complex and general (a single common factor) structures; note that the variables used in this study were continuous and not categorical. Sass and Schmidt focused on the performance of these rotation methods for normally distributed indicator variables; however, their study is relevant to this research with dichotomous indicators in that it is one of the few to systematically compare multiple rotational criteria. Furthermore, several of the rotations considered by Sass and Schmidt are also included in this study. Therefore, although their results with continuous, normally distributed variables may not be directly applicable to situations involving dichotomous indicators, their study does provide some potential insights into the performance of the rotational criteria that may in turn inform this research. Sass and Schmidt generated a sample of 300, with correlations between the factors (0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6 and 0.7) and used four oblique single stage rotational criteria, including Quartimin, CF-Equamax, CF-Facparsim and Geomin. They found that in the perfect simple structure condition all of the methods performed equally well, echoing Browne (2001). In the more complex cases, however, CF-Equamax and CF-Facparsim demonstrated somewhat less bias in factor loading estimates than did the other rotations. These authors concluded that researchers must be careful not to think of a particular rotational solution as inherently right or wrong, given that model fit does not change based on rotation. Echoing Browne (2001), Sass and Schmitt argued that the selection of the best A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 554 rotation must be made by the researcher using informed judgment, and cannot be done deterministically based solely on statistical results. A similar finding was reported by Asparouhov & Muthén (2009), who stated that based on their own simulated comparisons of the Geomin and Quartimin rotation criteria with loading bias as the primary outcome variable, the researcher in the end is responsible for determining what constitutes a simple and interpretable solution. Consistent with Sass and Schmitt (2010), they found that for simple factor patterns the rotation criteria performed similarly, but for more complex patterns the results across rotational methods (and even for the same method using different settings) might differ substantially. As noted, although the previous simulation research comparing factor rotation performance was focused on continuous indicator variables, it remains relevant for this study in that it provides the only published evidence regarding the behavior of these rotation criteria, all of which can be used with dichotomous indicators. The goal of this simulation study was to extend upon this earlier work by comparing the performance of several methods of factor rotation with dichotomous, rather than normally distributed continuous, indicator variables, and by including several more rotation criteria, including the very popular Varimax and Promax methods as well as others that have been shown to be effective previously. Furthermore, the current study extends upon these earlier efforts by including a broader range of conditions with respect to number of indicator variables, sample sizes and number of factors. Finally, the focus of this study in terms of outcomes is different than that of the previously mentioned research. Methodology A Monte Carlo simulation study was conducted to compare the performance of several methods of factor rotation in four areas: (1) proportion of correctly grouped indicator variables, (2) proportion of incorrectly grouped indicator variables, (3) proportion of indicator variables correctly ordered based on their population factor loading values, and (4) for oblique rotations, bias in the estimates of inter-factor correlations. Outcome 1 was the proportion of all item pairs that should have been grouped together that actually were, and outcome 2 was the proportion of all item pairs that should have been kept separate that actually were. Outcome 3 was the proportion of cases in which the item with the larger factor loading in the population also had the larger loading in the sample. Outcome 4 was the degree of accuracy of the inter-factor correlation estimate, which was calculated as ro −rp, where ro = sample estimate of inter-factor correlation between two factors and rp = population inter-factor correlation used in data simulation. In addition, the standardized bias of the correlation estimates was also calculated as the bias defined previously divided by the standard deviation of the correlation estimates. These outcomes were selected because they reflect issues that applied researchers might be interested in; that is, how accurately are the factors defined by appropriately grouped variables, how well ordered are the indicators in terms of the magnitude of their relationships to the factors and how well estimated are the correlations among the factors. Although all of these outcomes may be important in specific contexts, one could argue that the ability to accurately identify the factor structure by correctly grouping the items together may be the most crucial. Given that validity assessment is typically based on the extent to which the empirically identified factors reflect what would be expected for the constructs in question based on substantive content of the items, the accuracy of an EFA solution from a sample to reproduce the population factor structure would seem to be paramount. However, in certain circumstances each of these outcomes would be important to researchers using EFA. For each combination of the simulation conditions, 1,000 replications were generated using MPlus, version 5.1 (Muthén & Muthén, 2007) and all study conditions were completely crossed with one another. Dichotomous indicators were generated in MPlus using threshold values of 0.25 and were held constant across the observed variables. The relationship between the threshold ( τ) value and the probability (P i) of a respondent endorsing aW. HOLMES FINCH 555 dichotomous item is 1 .1 i P e−τ = + The threshold value of 0.25 corresponds to a probability of endorsing an item of 0.56 and was selected because it has been used in other simulation research involving factor analysis of dichotomous data (French & Finch, 2006; Meade & Lautenschlager, 2004). For each replication, exploratory factor analysis with Robust Weighted Least Squares (WLSMV) extraction was conducted using the MPlus software because it has been supported for use with categorical data in prior research (e.g., Muthén & Muthén, 2007; Flora & Curran, 2004). In conducting EFA with dichotomous data, MPlus first calculates the tetrachoric correlation matrix among the variables and then uses it to estimate the factor analysis parameters (factor loadings, inter-factor correlations). The commands to run the analysis requested the extraction of the correct number of factors (2 or 4) for a given replication but because the analysis was EFA, individual indicators were not linked to specific factors as they would have been in a confirmatory factor analysis. For example, when the data generated were from a 2 factor condition, the MPlus commands to run the EFA on the sample requested the extraction of 2 factors, but the individual indicators were not linked to a given factor. Data were generated for either 2 or 4 factors in the population, and for each factor there were either 6 or 12 observed indicator variables, leading to the following combinations: 2 factors with 6 indicators each, 2 factors with 12 indicators each, 4 factors with 6 indicators each and 4 factors with 12 indicators each. Four inter-factor correlation conditions were simulated: 0.1, 0.3, 0.5 and 0.7. All pairs of factors were correlated at the same level for a given combination of study conditions. For example, in the 4 factor, 6 indicator condition with r = 0.3, each pair of the 4 factors were generated with a correlation of 0.3. Four sample size conditions were simulated, 100, 200, 500 and 1,000. Prior research studying the minimum sample size necessary for EFA to provide reliable results with continuous indicators has found that when communalities are relatively high (e.g., 0.5), and most of the factors have a large number of indicators population factor are recovered well with samples as small as 100 subjects (MacCallum, et al., 1999). Conversely, MacCallum, et al. (1999), found that for low communalities and many factors, each of which has a small number indicators, samples of 500 or more are necessary. Preacher and MacCallum (2002) found that for sample sizes as low as 30, factor structure recovery was good (low root mean square error) provided that communalities were high (e.g., 0.8), the number of factors retained was 4 or fewer and the total number of indicators was 25 or more. Subsequently, other researchers investigating the impact of sample size on factor analysis have reported similar findings with regard to the need for larger samples with relatively poorly conditioned solutions (fewer indicators with low factor loadings, low communalities and many factors), and the positive performance with smaller samples (fewer than 50) when factors are well conditioned (de Winter, Dodou & Wieringa, 2009; Gagné & Hancock, 2006; Mundfrom, Shaw & Ke, 2005). Of particular interest given the inclusion of non-simple structure conditions in the current research are the results of de Winter, et al., who found that in the presence of non-simple structure, EFA performs worse with relatively smaller samples in terms of factor structure recovery, particularly when factors are correlated at 0.5 or greater. Given these earlier studies, sample sizes selected for the current research range from what might be considered somewhat small (100) to very large (1,000). Finally, the data were generated with 4 levels of factor structure complexity, reflecting different degrees to which individual indicators cross-loaded with a secondary factor. Table 2 provides an example of these patterns for each level of structural complexity in the 2 factor 6 indicator condition. For example, in complexity condition 1 each indicator has non-zero loadings for only one factor, whereas in the other 3conditions, each indicator has an additional non-zero loading on one other factor with complexity conditions differing based upon the magnitude of these non-zero loadings. In the 4 factor conditions, each indicator variable had only 2 non-zero loadings, one for its primary factor and A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 556 Table 2: Example Factor Loading Patterns Used In the Simulations Complexity Condition 1 Complexity Condition 2 Indicator Factor 1 Factor 2 Indicator Factor 1 Factor 2 Y1 0.8 0 Y1 0.8 0.1 Y2 0.8 0 Y2 0.8 0.1 Y3 0.6 0 Y3 0.6 0.1 Y4 0.6 0 Y4 0.6 0.1 Y5 0.4 0 Y5 0.4 0.1 Y6 0.4 0 Y6 0.4 0.1 Y7 0 0.8 Y7 0.1 0.8 Y8 0 0.8 Y8 0.1 0.8 Y9 0 0.6 Y9 0.1 0.6 Y10 0 0.6 Y10 0.1 0.6 Y11 0 0.4 Y11 0.1 0.4 Y12 0 0.4 Y12 0.1 0.4 Complexity Condition 3 Complexity Condition 4 Indicator Factor 1 Factor 2 Indicator Factor 1 Factor 2 Y1 0.8 0.2 Y1 0.8 0.3 Y2 0.8 0.2 Y2 0.8 0.3 Y3 0.6 0.2 Y3 0.6 0.3 Y4 0.6 0.2 Y4 0.6 0.3 Y5 0.4 0.2 Y5 0.4 0.3 Y6 0.4 0.2 Y6 0.4 0.3 Y7 0.2 0.8 Y7 0.3 0.8 Y8 0.2 0.8 Y8 0.3 0.8 Y9 0.2 0.6 Y9 0.3 0.6 Y10 0.2 0.6 Y10 0.3 0.6 Y11 0.2 0.4 Y11 0.3 0.4 Y12 0.2 0.4 Y12 0.3 0.4 W. HOLMES FINCH 557 the other for a single secondary factor. For example, in complexity condition 2 with 4factors and 12 indicators for each, indicator 1 had a loading of 0.8 for factor 1, a loading of 0.1 for factor 2 and loadings of 0 for factors 3 and 4. On the other hand, indicator 48 had a loading of 0.4 for factor 4, a loading of 0.1 for factor 3 and 0 loadings for factors 1 and 2. The decision to allow indicators in the 4 factor conditions to cross load with only one other factor was made to avoid confounding the number of cross loadings with the number of factors, making it impossible to directly compare results in the 2 and 4 factors cases. Similar factor loading patterns were used with the other factor and indicator combinations included in this study. Although a very large number of different such factor patterns could have been simulated using the number of factors and indicators included in this study, these patterns were selected because it was felt that they represented a range of non-simple structure conditions, were few enough so as to keep the study manageable and allowed for investigation of the impact of progressively greater factor complexity. The methods of factor rotation included the study were Quartimin (oblique), Varimax (orthogonal), Quartimax (orthogonal), Equamax (orthogonal), Parsimax (oblique), Geomin (oblique), Promax (oblique) and Facparsim (oblique). The selection of these particular rotations was made based upon a combination of prior research results, popularity in use and availability in statistical software. Again, though prior research comparing performance of rotational criteria used continuous indicators, these are the only available studies examining this issue; therefore, it was determined that these earlier studies did provide some insights into which rotations should be used. Sass and Schmitt (2010) used only oblique rotations, including Quartimin, oblique CF-Equamax, CF-Facparsim and Geomin, and found that Geomin and Quartimin performed slightly better in a pure simple structure condition (Complexity condition 1 in the current study), whereas oblique CF-Equamax and CF-Facparsim were somewhat better in the more complex cases. Asparouhov and Muthén (2009) compared Quartimin with Geomin using two values of the constant ε, 0.01 and 0.0001 and reported that Geomin with ε = 0.001 consistently produced the least bias in factor loading estimates. Based on these results, the current study included Geomin with ε = 0.001, Quartimin, and Facparsim. In addition, three orthogonal rotations (i.e., Varimax, Quartimax and Equamax) were included because heretofore their performance has not been investigated in such a study and they are very commonly used in practice. Similarly, Promax was included in the study because of its popularity and ubiquity in statistical software, and the fact that it was not included in the earlier work. For each included rotation criterion, except for Geomin as noted above, the default settings in MPlus were used in conducting the analyses in order to mimic what researchers are likely to do in practice. In addition to the Monte Carlo simulation, this study also included the use of EFA with item responses from a sample of 1,000 examinees who took the Law School Admissions Test (LSAT). These data, which have been discussed previously in the literature, have been shown to contain 4 separate factors corresponding to the 4 reading passages contained in the exam (Stout, et al., 1996). For these data, EFA using the RWLS method of extraction was followed with each of the rotations included in the simulation study. Note that analysis was conducted on the raw binary data. Results Because an initial examination of the simulation outcomes revealed that the results for factors 1, 2, 3 and 4 were similar in terms of the grouping of indicators and the ordering of indicators by factor loading magnitude, results are presented for the first factor only. Similarly, estimates of the inter-factor correlation between factors 1 and 2 were similar to those for the other factor pairs (where applicable), thus, only the results for this correlation will be presented. Factor Grouping A repeated measures Analysis of Variance (ANOVA) was used to identify which of the manipulated conditions and their interactions were significantly associated with the proportion of item pairs correctly grouped together, which served as the dependent A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 558 variable. These conditions included type of rotation, number of observed indicators per factor, number of factors, factor complexity, sample size and inter-factor correlation. Assumptions of equality of variance and normality of errors were assessed using Levene’s test and QQ plots, respectively, and were found to have been met. The results of the ANOVA indicated that the highest order significant ( α = 0.05) interaction was type of rotation by number of factors, number of indicators and factor complexity ( η2 = 0.112). In addition, the interaction among type of rotation, inter-factor correlation and factor complexity was also significantly related to the proportion of indicators correctly grouped ( η2 = 0.482), as was the main effect of sample size ( η2 = 0.801). All other significant main effects and interactions were subsumed in one of these three terms and will therefore not be discussed. Table 3 shows the proportion of observed indicator variables correctly and incorrectly grouped by the number of factors, number of indicators per factor, factor complexity and method of rotation. An examination of these results reveals that across methods of rotation the proportion of variables correctly grouped declined as the factor structure became more complex, but the proportion incorrectly grouped together increased. (Note that the numbers for complexity conditions presented in subsequent tables correspond to the numbers in Table 2). This decrease in indicator grouping accuracy with increased structural complexity was less marked for the Quartimin (QMIN) rotation across the number of factors and number of indicators, and the Facparsim (FAC) when there were 12 indicators per factor, regardless of the number of factors. Indeed, when there were 12 indicators per factor the decline in grouping accuracy for QMIN was very small, 0.04 for 2 factors and 0.02 for 4 factors. By contrast, QMIN also demonstrated a much higher rate of incorrectly grouping indicator variables together for more complex factor patterns, across numbers of factors and indicators. The other rotations generally demonstrated comparable levels of grouping accuracy across the conditions contained in Table 3. The only exceptions to this general result were for Varimax (VAR) and Parsimax (PAR) with 4 factors, both of which had somewhat larger declines in the proportion of correctly grouped indicators than the other approaches in the presence of 4 factors, and for Equamax (EQU), which consistently demonstrated among the lowest rates of incorrectly grouping indicators together, and comparable rates of correctly grouping indicators with one another. Table 4 presents the proportions of correctly and incorrectly grouped indicators by method of rotation, inter-factor correlation and factor complexity. As evident in Table 3, with increasing model complexity QMIN displayed a smaller decline in the proportion of correctly grouped indicators and a greater increase in the proportion of incorrectly indicators, than did the other rotation methods. Of particular interest is that two of the orthogonal rotations, VAR and EQU, did not show any greater diminution in the proportion of correctly grouped indicators than the oblique rotations as the inter-factor correlations increased, nor did they have greater increases in the proportion of incorrectly grouped items. By contrast, the orthogonal method QUA exhibited among the highest rates of incorrectly grouped indicators for the more complex factor patterns when the inter-factor correlation was 0.5 or 0.7. EQU and PAR consistently demonstrated among the lowest rates of incorrect indicator grouping, while being comparable to the other rotational methods (except QMIN) in terms of correctly grouped indicator variables. The impact of the factor pattern on correct indicator grouping was essentially the same regardless of the inter-factor correlation, with decreases in the proportion of correctly grouped item pairs and increases in the proportion of correctly grouped item pairs. For all methods of rotation, the proportion of correctly grouped indicator variables increased concomitantly with increases in sample size, whereas the proportion of incorrectly grouped indicators declined (see Table 5). Factor Loading Magnitudes As with the proportion of correctly grouped items, repeated measures ANOVA was used to determine which of the study conditions W. HOLMES FINCH 559 Table 3: Proportion of Variables Correctly | Incorrectly Grouped into Factors by Number of Factors (F), Number of Indicators per Factor (I) and Population Factor Complexity (C) F I C EQU GEO PAR PRO QUA QMIN VAR FAC 2 61 .94|.10 .94|.11 .94|.10 .91|.10 .94|.11 .93|.14 .94|.10 .88|.10 2 .91|.16 .90|.18 .91|.16 .87|.17 .90|.18 .91|.31 .91|.16 .79|.17 3 .86|.27 .85|.31 .86|.28 .82|.29 .85|.33 .88|.57 .85|.27 .69|.26 4 .78|.45 .77|.51 .77|.45 .74|.48 .78|.56 .87|.83 .75|.46 .66|.49 2 12 1 .97|.02 .97|.03 .97|.02 .95|.02 .97|.03 .96|.12 .97|.02 .99|.03 2 .95|.04 .95|.05 .95|.04 .93|.05 .95|.05 .96|.32 .95|.04 .98|.06 3 .89|.10 .88|.11 .89|.09 .86|.10 .89|.22 .92|.66 .88|.10 .98|.13 4 .80|.24 .80|.28 .80|.21 .77|.23 .83|.48 .92|.96 .78|.22 .95|.29 4 61 .92|.13 .91|.14 .91|.13 .90|.14 .91|.14 .91|.21 .90|.13 .82|.15 2 .90|.17 .89|.17 .89|.17 .87|.16 .89|.18 .90|.30 .88|.16 .73|.19 3 .86|.25 .86|.26 .85|.25 .83|.24 .86|.27 .90|.43 .83|.25 .63|.26 4 .82|.38 .82|.41 .79|.38 .82|.41 .85|.42 .90|.59 .73|.42 .51|.43 4 12 1 .96|.05 .95|.05 .95|.05 .95|.15 .95|.05 .95|.07 .95|.14 .99|.06 2 .94|.06 .94|.07 .94|.06 .94|.19 .94|.06 .95|.13 .94|.18 .96|.08 3 .89|.13 .92|.18 .88|.12 .92|.32 .90|.16 .94|.28 .93|.31 .95|.18 4 .82|.22 .88|.31 .79|.20 .88|.45 .85|.28 .93|.41 .83|.45 .93|.31 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 560 Table 4: Proportion of Variables Correctly | Incorrectly Grouped into Factors by Inter-Factor Correlations (r) and Population Factor Complexity (C) r C EQU GEO PAR PRO QUA QMIN VAR FAC 0.1 1 .97|.04 .97|.04 .97|.04 .95|.08 .97|.04 .97|.04 .97|.07 .98|.03 2 .96|.05 .95|.05 .95|.05 .94|.08 .95|.05 .95|.05 .95|.07 .95|.05 3 .93|.10 .92|.11 .92|.10 .91|.13 .92|.10 .92|.11 .92|.12 .92|.11 4 .85|.21 .84|.24 .84|.21 .83|.30 .84|.23 .91|.60 .85|.30 .80|.26 0.3 1 .96|.05 .96|.05 .96|.05 .95|.07 .96|.05 .96|.05 .96|.07 .96|.05 2 .94|.07 .94|.08 .94|.08 .92|.10 .94|.08 .94|.08 .94|.09 .92|.08 3 .91|.13 .90|.15 .90|.14 .88|.18 .89|.15 .93|.40 .89|.17 .86|.16 4 .83|.26 .84|.31 .84|.26 .82|.36 .82|.30 .91|.68 .86|.36 .84|.37 0.5 1 .95|.08 .94|.08 .94|.08 .93|.10 .94|.08 .94|.08 .94|.09 .95|.08 2 .94|.10 .93|11 .93|.10 .91|.13 .93|.11 .92|.22 .93|.12 .91|.10 3 .88|.19 .87|.23 .87|.20 .85|.27 .87|.22 .93|.66 .87|.27 .86|.24 4 .80|.34 .85|.41 .78|.33 .83|.42 .85|.55 .94|.73 .84|.41 .80|.48 0.7 1 .91|.14 .90|.16 .90|.15 .88|.17 .90|.16 .94|.38 .89|.16 .87|.16 2 .87|.21 .87|.24 .87|.21 .84|.27 .87|23 .94|.75 .86|.26 .85|.24 3 .82|.32 .85|.38 .81|.31 .82|.38 .85|.52 .92|.80 .83|.37 .79|.51 4 .76|.49 .75|.55 .74|.43 .74|.49 .77|.69 .88|.82 .74|.48 .73|.50 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. W. HOLMES FINCH 561 and their interactions were significantly related to the proportion of correctly ordered factor indicators based on their loading magnitudes in the sample. The highest order significant interaction was the rotation by inter-factor correlation by factor pattern ( η2 = 0.201). In addition, the 2-way interactions of rotation by number of indicators per factor ( η2 = 0.236) and rotation by number of factors ( η2 = 0.275) were also statistically significant, as was the main effect of sample size ( η2 = 0.858). For all of the rotations, results demonstrate (see Table 6) that the proportion of correctly ordered factor indicators by loading magnitude declines with increases in the inter-factor correlation and with increased factor complexity (reflected through higher numbers for the factor complexity condition). In addition, the deleterious impact of greater factor complexity was more pronounced for larger values of the inter-factor correlation. For example, in the simple structure condition (C = 1) with correlations of 0.1 and 0.3, the rotations performed similarly with respect to correct ordering of the factor indicators by loading magnitude, whereas for r = 0.5 FAC displayed a higher proportion of correctly ordered factor loadings, and for r = 0.7, FAC, QMIN and VAR all had somewhat higher proportions of correctly ordered loadings. On the other hand, for the greatest factor complexity (C = 4) VAR consistently had the highest proportion of correctly ordered loadings, with a variety of other rotations performing comparably for agiven inter-factor correlation. For example, QMIN performed similarly to VAR in the most complex case for inter-factor correlations of 0.1, 0.3 and 0.7, and FAC had similar values to VAR for proportion of correctly ordered loadings in the most complex case when r = 0.3. Results in Table 7 show that all of the rotations were more accurate in terms of correctly ordering indicators by the magnitude of factor loadings for 12 indicators, for 2 factors and for larger sample sizes. FAC was the rotation method whose performance was most strongly influenced by the number of indicators. For 6 indicators per factor, it performed the worst in terms of correctly ordering loadings, whereas for 12 indicators it performed the best. QMIN and VAR consistently produced among the most accurate ordering of loadings by magnitude across all of the conditions contained in Table 7. The performances of the other rotation methods were generally similar to one another, and somewhat worse than that of QMIN and VAR. Inter-Factor Correlation Bias A repeated measures ANOVA identified the 3-way interaction of rotation method by Table 5: Proportion of Variables Correctly | Incorrectly Grouped into Factors by Sample Size N EQU GEO PAR PRO QUA QMIN VAR FAC 100 .83|.31 .85|.33 .82|.30 .82|34 .84|.33 .88|.45 .84|.33 .84|.32 200 .87|.21 .88|.24 .86|.21 .86|.25 .88|.24 .92|.40 .88|.24 .86|.23 500 .92|.11 .92|.14 .91|.11 .90|.17 .93|.17 .95|.37 .92|.16 .89|.13 1000 .94|.07 .94|.09 .94|.07 .93|.13 .94|.14 .97|.37 .94|.13 .94|.12 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 562 Table 6: Proportion of Factor Loadings Correctly Ordered by Magnitude by Inter-Factor Correlations (r) and Population Factor Complexity (C) r C EQU GEO PAR PRO QUA QMIN VAR FAC 0.1 1 .94 .94 .94 .93 .94 .94 .94 .96 2 .93 .92 .93 .91 .93 .93 .93 .94 3 .90 .89 .89 .88 .91 .93 .90 .92 4 .83 .81 .81 .81 .81 .84 .84 .81 0.3 1 .93 .92 .93 .91 .92 .93 .93 .94 2 .91 .90 .90 .89 .90 .91 .91 .91 3 .87 .85 .85 .84 .85 .87 .87 .84 4 .78 .76 .75 .77 .76 .81 .81 .82 0.5 1 .90 .89 .89 .88 .89 .90 .90 .95 2 .89 .87 .88 .86 .87 .89 .90 .92 3 .81 .79 .79 .79 .78 .84 .83 .81 4 .73 .70 .68 .68 .73 .70 .77 .70 0.7 1 .83 .81 .81 .81 .80 .84 .85 .84 2 .79 .75 .75 .77 .75 .82 .81 .80 3 .72 .69 .67 .71 .70 .76 .76 .69 4 .65 .63 .57 .65 .64 .70 .70 .64 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. Table 7: Proportion of Factor Loadings Correctly Ordered by Magnitude by Number of Indicators per Factor (I), Number of Factors (F), and Sample Size I EQU GEO PAR PRO QUA QMIN VAR FAC 6 .75 .72 .72 .74 .72 .76 .77 .66 12 .93 .92 .91 .90 .92 .94 .93 .97 F EQU GEO PAR PRO QUA QMIN VAR FAC 2 .89 .86 .87 .86 .88 .91 .90 .86 4 .78 .77 .76 .79 .76 .79 .80 .78 N EQU GEO PAR PRO QUA QMIN VAR FAC 100 .69 .67 .66 .67 .67 .71 .70 .68 200 .80 .78 .77 .78 .78 .82 .82 .79 500 .91 .89 .89 .90 .89 .94 .93 .90 1000 .95 .94 .94 .94 .94 .96 .96 .92 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. W. HOLMES FINCH 563 inter-factor correlation by factor complexity (η2 = 0.049) as the highest order significant term. In addition, the main effects of number of factors ( η2 = 0.313), number of indicators per factor ( η2 = 0.041), and sample size ( η2 = 0.021) were also statistically significant. Table 8contains the mean raw bias and the standardized bias values across replications by the inter-factor correlation and the degree of model complexity. For r = 0.1, the sample correlation estimates displayed a positive bias across rotations, except for the simple structure condition (C = 1). In addition, as the degree of complexity increased, so did both raw and standardized bias, except for PRO. When r = 0.3, the negative bias in the simple structure condition was greater than for r = 0.1, and the positive bias for more complex models was lower, across rotation methods. For r = 0.5 and 0.7, bias was uniformly negative across levels of factor complexity, with greater negative bias associated with the largest population correlation. In addition, for r = 0.5 all rotation methods, except PAR, displayed greater negative bias for simple structure data (C = 1) or for the most complex structure (C = 4). In contrast, when r = 0.7, bias was generally higher for simple structure than for the next level of factor complexity (C = 2), after which bias increased concomitantly with increased model complexity. None of the rotation criteria consistently produced the least raw or standardized biased estimates. Table 9 shows that inter-factor correlation bias was more pronounced (and negative) when more indicators were present. In addition, the degree of bias for most of the rotation methods was slightly greater (and negative) for 4 factors as compared to 2, where the bias was positive. Finally, bias in the inter-factor correlation estimates declined with increased sample size, and across all conditions PAR produced somewhat more negatively biased estimates than the other criteria. Otherwise, differences in estimation accuracy across the conditions were relatively minor. Analysis of LSAT Data In order to demonstrate the relative performance of the rotation criteria on an actual, well studied data set, EFA was run on the LSAT data described in Stout, et al. (1996). Given that these authors, and others, reported the presence of 4 stable dimensions, 4 factors were extracted in this analysis, and each rotation was applied. Table 10 contains the factor loadings only for the primary factor for each item in order to save space. There were no cross-loaded items for any of the rotation criteria, defined as having multiple factors for which the loading values were great than 0.32 (Tabachnick & Fidell, 2007). A perusal of these results demonstrates that across items and factors, the loading values for the 8 different rotations were very similar to one another. There is no discernible pattern of difference in loadings by rotation, suggesting that a researcher using any of these criteria would reach the same substantive conclusions regarding both how items grouped together, and the strength of relationships between items and factors. Table 11 includes the correlation estimates for the 4 factor solution of the LSAT data for each of the oblique rotations studied here, and their standard errors with the exception of PROMAX, for which standard errors are not calculated in MPlus. These results demonstrate a greater degree of variation across rotation criteria than was evident for the factor loadings. For example, PROMAX had much larger inter-factor correlation estimates than the other methods for factor 1 with 3, 1 with 4 and 3 with 4. By contrast, PARSIMAX had much lower correlation estimates than the other methods for factors 1 with 3, 1 with 4, 2 with 4 and 3 with 4. GEOMIN, QUARTIMIN and FACPARSIM had very similar inter-factor correlation estimates to one another for this sample. Conclusion This study extends previous research comparing rotations in EFA, which focused on continuous factor indicator variables by comparing the performance of 8 factor rotation criteria with dichotomous indicator variables using the WLSMV initial extraction method in MPlus across a variety of conditions. Among the rotations included were some that had previously been found to be promising in terms of accuracy of factor loading estimates such as Geomin and Facparsim, and others that had not been studied before but which are very commonly used in practice, including Varimax and Promax. The outcomes of interest included A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 564 Table 8: Inter-Correlation Bias (Standardized Bias) by Inter-Factor Correlations (r) and Population Factor Complexity (C) r C GEO PAR PRO QMIN FAC 0.1 1 -0.04 (-0.43) -0.05 (-0.75) -0.02 (-0.23) -0.04 (-0.44) -0.03 (-0.33) 2 0.08 (0.45) 0.05 (0.34) 0.12 (0.69) 0.08 (0.49) 0.09 (0.50) 3 0.18 (0.65) 0.14 (0.55) 0.22 (0.84) 0.19 (0.73 0.19 (0.73) 4 0.24 (0.73) 0.21 (0.73) 0.17 (0.52) 0.25 (0.80) 0.26 (0.79) 0.3 1 -0.12 (-0.84) -0.16 (-0.93) -0.10 (-0.71) -0.11 (-0.79) -0.11 (-0.78) 2 -0.01 (-0.04) -0.07 (-0.38) 0.03 (0.13) -0.01 (-0.02) 0.01 (0.01) 3 0.07 (0.22) 0.01 (0.05) 0.08 (0.30) 0.08 (0.27) 0.09 (0.29) 4 0.09 (0.25) 0.07 (0.23) -0.02 (-0.07) 0.12 (0.35) 0.11 (0.32) 0.5 1 -0.21 (-0.95) -0.27 (-1.53) -0.18 (-0.94) -0.20 (-0.92) -0.21 (-0.92) 2 -0.09 (-0.34) -0.17 (-0.77) -0.08 (-0.31) -0.09 (-0.32) -0.09 (-0.33) 3 -0.08 (-0.22) -0.12 (-0.46) -0.14 (-0.43) -0.06 (-0.18) -0.08 (-0.19) 4 -0.13 (-0.35) -0.09 (-0.29) -0.20 (-0.58) -0.19 (-0.60) -0.20 (-0.59) 0.7 1 -0.31 (-1.00) -0.37 (-1.65) -0.31 (-1.20) -0.30 (-1.07) -0.32 (-1.06) 2 -0.26 (-0.78) -0.31 (-1.15) -0.32 (-1.00) -0.25 (-0.76) -0.27 (-0.79) 3 -0.30 (-0.80) -0.28 (-0.80) -0.35 (-1.06) -0.36 (-1.06) -0.36 (-1.08) 4 -0.38 (-0.99) -0.31 (-0.81) -0.33 (-1.05) -0.33 (-1.54) -0.36 (-1.44) GEO = Geomin, PAR = Parsimax, PRO = Promax, QMIN = Quartimin, FAC = Facparsim. W. HOLMES FINCH 565 the proportion of accurately grouped indicator variables, the proportion of indicators correctly ordered by the magnitude of their loading values and, for the oblique methods, the accuracy of inter-factor correlation estimates. It is hoped that this study builds upon earlier work by focusing on dichotomous indicators (i.e., items), by including outcomes that would be of interest to practitioners interested in using these methods to identify potential latent variables in existing measures and by expanding the range of conditions under which the rotations are examined, including the rotations themselves. Implications for Practice One implication of this study for researchers using EFA with categorical indicator variables is that when they know, or suspect, that the correlations among the factors will be upwards of 0.5, they should expect to have problems not only with appropriately grouping variables together, but also with accurately ordering variables in terms of the importance of their relationships with the factors. These problems are likely to be particularly acute if the factor pattern structure is very complex. It does seem however, that having a larger sample may ameliorate these problems to some extent, so that when it is likely the factors will be highly correlated and/or the factor pattern may be complex in nature, researchers should ideally try to obtain samples of 500 or more. These results are similar to those reported in de Winter, Dodou and Wieringa (2009) for continuous data. A second implication is that - for the oblique methods of rotation studied - there may be problems with accurately estimating inter-factor correlations across conditions like those simulated here. When these correlations were greater than 0.3, all of the criteria produced underestimates of r, whereas for lower correlations r was overestimated for more complex factor patterns and underestimated for the less complex patterns. These correlation estimation bias results are similar to those Table 9: Inter-Correlation Bias by Magnitude by Number of Indicators Per Factor (I), Number of Factors (F), and Sample Size I GEO PAR PRO QMIN FAC 6 0.03 -0.03 0.02 0.04 0.06 12 -0.13 -0.15 -0.15 -0.14 -0.13 F GEO PAR PRO QMIN FAC 2 0.17 0.10 0.16 0.11 0.12 4 -0.16 -0.17 -0.17 -0.14 -0.15 N GEO PAR PRO QMIN FAC 100 -0.11 -0.12 -0.08 -0.10 -0.11 200 -0.10 -0.12 -0.08 -0.09 -0.10 500 -0.06 -0.09 -0.08 -0.06 -0.08 1000 -0.03 -0.08 -0.08 -0.04 -0.06 GEO = Geomin, PAR = Parsimax, PRO = Promax, QMIN = Quartimin, FAC = Facparsim. A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 566 Table 10: Rotated Factor Loading Matrices for LSAT Data Item EQU GEO PAR PRO QUA QMIN VAR FAC Factor 1 1 0.35 0.33 0.33 0.33 0.34 0.33 0.35 0.32 2 0.40 0.40 0.40 0.41 0.40 0.40 0.40 0.39 3 0.43 .045 0.45 0.47 0.43 0.45 0.43 0.45 4 0.36 0.39 0.38 0.40 0.36 0.38 0.36 0.38 5 0.40 0.38 0.38 0.39 0.39 0.38 0.39 0.38 6 0.51 0.53 0.52 0.55 0.51 0.53 0.51 0.53 7 0.33 0.30 0.31 0.30 0.31 0.30 0.33 0.30 Factor 2 8 0.52 0.54 0.54 0.56 0.51 0.54 0.52 0.53 9 0.38 0.40 0.40 0.41 0.37 0.39 0.38 0.39 10 0.52 0.55 0.55 0.57 0.51 0.55 0.53 0.54 11 0.28 0.27 0.28 0.28 0.27 0.27 0.28 0.27 12 0.37 0.40 0.39 0.42 0.37 0.40 0.37 0.39 13 0.38 0.37 0.37 0.39 0.38 0.38 0.37 0.38 Factor 3 14 0.54 0.55 0.54 0.58 0.54 0.56 0.54 0.55 15 0.53 0.54 0.53 0.56 0.53 0.54 0.53 0.54 16 0.44 0.46 0.45 0.48 0.44 0.46 0.44 0.46 17 0.16 0.15 0.15 0.15 0.16 0.15 0.16 0.15 18 0.48 0.48 0.45 0.49 0.49 0.49 0.47 0.49 19 0.51 0.50 0.47 0.51 0.52 0.51 0.50 0.51 Factor 4 20 0.42 0.41 0.38 0.41 0.43 0.41 0.42 0.41 21 0.56 0.56 0.53 0.57 0.57 0.57 0.55 0.56 22 0.59 0.60 0.56 0.61 0.60 0.60 0.58 0.60 23 0.47 0.48 0.45 0.49 0.48 0.48 0.47 0.48 24 0.50 0.52 0.49 0.53 0.50 0.52 0.50 0.52 EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QUA = Quartimax, QMIN = Quartimin, VAR = Varimax, FAC = Facparsim. W. HOLMES FINCH 567 reported by Sass and Schmitt (2010) for the case of continuous indicators. A third implication for practitioners is that including more indicator variables (assuming that they are of good quality) will yield better solutions both in terms of correctly grouping the indicators and accurately ordering them in terms of their relationships to the factors. This result seems reasonable given that including more indicators for each factor provides a greater amount of information for the EFA extraction algorithm as well as for the rotations. The number of indicators was particularly important for the FAC technique, particularly in the case of a more complex factor pattern structures with more factors. Based on these results, researchers may consider using FAC when they have at least 12 indicators per factor, as it demonstrated better performance in terms of grouping the variables as well as ordering them, particularly in the 4 factor case. On the other hand, FAC would not appear to be optimal with fewer indicators per factor. A final implication of these results is that, in terms of both indicator grouping and ordering of importance in terms of factor relationships, researchers may generally find orthogonal and oblique rotations will produce similar results. Indeed, one of the consistently best performers in this study was the orthogonal rotation EQU. This result is not completely surprising, as EQU was designed to spread loading variation more equally across factors than several of the other rotations studied here (Saunders, 1962) by combining the VAR and QUA criteria. Thus, although VAR seeks to maximize the variation of loadings for factors, and QUA seeks to simplify loadings for the observed variables, EQU combines these two goals. This is not to suggest that researchers should only use EQU as the rotation of choice for all problems. When factors are thought to be correlated, the choice of an orthogonal rotation may not be appropriate, regardless of how well it performs. However, when the inter-factor correlation is low and the primary goal of a study is to identify which indicators are associated with which factors, EQU would be a reasonable choice. When a researcher is interested in estimating inter-factor correlations, or they believe that these correlations may be fairly large (greater than 0.5), several of the oblique rotations studied here would appear to be appropriate. In particular, PAR and FAC (for situations with a larger number of indicator Table 11: Inter-Factor Correlation (Standard Error) Estimates for LSAT Data by Oblique Rotations Factor Pair GEO PAR PRO QMIN FAC 1 with 2 0.35 (0.05) 0.30 (0.04) 0.32 (NA) 0.34 (0.06) 0.34 (0.06) 1 with 3 0.28 (0.05) 0.20 (0.04) 0.42 (NA) 0.28 (0.05) 0.29 (0.05) 1 with 4 0.26 (0.05) 0.18 (0.04) 0.35 (NA) 0.26 (0.05) 0.26 (0.06) 2 with 3 0.32 (0.05) 0.35 (0.04) 0.36 (NA) 0.33 (0.05) 0.31 (0.05) 2 with 4 0.42 (0.05) 0.23 (0.04) 0.38 (NA) 0.42 (0.05) 0.42 (0.05) 3 with 4 0.30 (0.04) 0.20 (0.03) 0.50 (NA) 0.32 (0.04) 0.33 (0.05) EQU = Equamax, GEO = Geomin, PAR = Parsimax, PRO = Promax, QMIN = Quartimin, FAC = Facparsim. A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 568 variables) demonstrated consistently strong performance in terms of correctly grouping and ordering indicator variables. On the other hand, QMIN may not be reliable for researchers interested in finding the correct groupings of factor indicators, as it (or the equivalent methods of oblique Quartimax and Oblimin) appears to reduce dimensionality in the sample too much by grouping most of the variables into a single factor. As a consequence, researchers using QMIN may come to the conclusion that, based on the sample there are a smaller number of factors present than is actually true for the population. Limitations As with any research effort, limitations to this study that must be considered when interpreting the results. First, for all of the rotations the MPlus system defaults were used. This was a decision made for two reasons: (1) It was desired to mimic what might be most commonly done in practices, and (2) In many cases there are a very large number of alternative settings that could have been used for some of the rotations. Therefore, in order to keep the study to a manageable size and the interpretation of the results fairly straightforward, it was felt that only a limited number of options could be used. Nonetheless, in practice researchers can choose from abroader range of settings when using many of these rotational criteria. A second limitation relates to the conditions simulated, including the factor patterns used and the number of indicators. In both cases, the selections made for this study were designed to mimic what would be seen in practice. However, clearly many other factor patterns and numbers of indicators could have been included, which may well have provided different results. Future studies should focus on both of these issues in order to expand upon what was learned here. Finally, these results were based on dichotomous indicator variables, which may not translate directly to ordinal data, such as that commonly found in many psychological scales. It should be noted that because rotations focus on loadings rather than the raw data, it is not clear how important this issue might be. Nonetheless, future research should verify to what extent the nature of the categorical data has an impact on the performance of rotational criteria. Summary In the final analysis, the admonition offered by Browne (2001) for researchers to use their expert judgment in conjunction with statistical results is definitely supported by these results. It is clearly not possible to state that any single rotational criterion will fit all EFA problems adequately, although in practice researchers often appear to use favorites regardless of the context. However, these results do suggest that certain features of the data will support the use of one or more such methods studied here. Clearly the ubiquitous VAR and PRO rotations must be used with caution when at all, as often they do not produce optimal results in terms of accurately reflecting the underlying factor structure. With the increased availability of other rotations in software packages such as MPlus, researchers are no longer limited to a small number of available options, and can thus experiment with a broader array of tools than could be done previously. References Asparouhov, T., & Muthén, B. (2009). Exploratory structural equation modeling. Structural Equation Modeling , 16 , 397-438. Bock, R. D. & Aitkin, M. (1981). Marginal maximum likelihood estimation of item parameters: Application of an EM algorithm. Psychometrika , 46 (4), 443-459. Bock, R. D., Gibbons, R., Schilling, S. G., Muraki, E., Wilson, D. T. & Wood, R. (2003). TESTFACT 4. Lincolnwood, IL: Scientific Software International. Browne, M. W. (2001). An overview of analytic rotations in exploratory factor analysis. Multivariate Behavioral Research , 36 (1), 111-150. Carroll, J. B. (1957). Biquartimin criterion for rotation to oblique simple structure in factor analysis. Science , 126 , 1114-1115. Christofferson, A. (1975). Factor analysis of dichotomized variables. Psychometrika , 40 , 5-32. W. HOLMES FINCH 569 Crawford, C. B., & Ferguson, G. A. (1970). A general rotation criterion and its use in orthogonal rotation. Psychometrika , 35 , 321-332. de Winter, J. C. F., Dodou, D., &Wieringa, P. A. (2009). Exploratory factor analysis with small sample sizes. Multivariate Behavioral Research , 44 , 147-181. Flora, D. B., & Curran, P. J. (2004). An empirical evaluation of alternative methods of estimation for confirmatory factor analysis with ordinal data. Psychological Methods , 9, 466-491. Fraser, C., & McDonald, R. P. (1988). NOHARM: Least squares item factor analysis. Multivariate Behavioral Research , 23 , 267-269. French, B. F. & Finch, W. H. (2006). Confirmatory factor analytic procedures for the determination of measurement invariance. Structural Equation Modeling , 13 (3), 378-402. Gagné, P., & Hancock, G. R. (2006). Measurement model quality, sample size, and solution propriety in confirmation factor models. Multivariate Behavioral Research , 41 , 65-83. Gorsuch, R. L. (1983). Factor analysis .Hillsdale, NJ: Lawrence Erlbaum Associates. Gosz, J. K., & Walker, C. M. (2002). An empirical comparison of multidimensional item response data using TESTFACT and NOHARM . Paper presented at the annual meeting of the National Council on Measurement in Education, New Orleans, LA. Guilford, J. P. (1941). The difficulty of a test and its factor composition. Psychometrika , 6, 67-78. Hattie, J. (1985). Methodology review: Assessing unidimensionality of tests and items. Applied Psychological Measurement , 9(2), 139-164. Jennrich, R. I. (2007). Rotation methods, algorithms, and standard errors. In R. Cudek & R. C. MacCallum (Eds.), Factor analysis at 100: Historical developments and future directions , 315-335. Mahwah, NJ: Lawrence Erlbaum Associates, Publishers. Kaiser, H. F. (1958). The varimax criterion for analytic rotation in factor analysis. Psychometrika , 23 (3), 187-200. Lorenzo-Seva, U. (1999). Promin: Amethod for oblique factor rotation. Multivariate Behavioral Research , 34 , 347-365. MacCallum, R. C., Widaman, K. F., Zhang, S., & Hong, S. (1999). Sample size in factor analysis. Psychological Methods , 4(1), 84-99. McDonald, R. P. (1962). A general approach to nonlinear factor analysis .Psychometrika , 27 , 297-415. McDonald, R. P. (1967). Nonlinear factor analysis . Psychometric Monographs , 15 .McDonald, R. P. (1981). The dimensinality of tests and items. British Journal of Mathematical and Statistical Psychology , 34 ,100-117. McDonald, R. P. (1997). Normal-ogive multidimensional model. In W. J. van der Linden & R. K. Hambleton (Eds.), Handbook of Modern Item Response Theory , 257-269. New York: Springer. McDonald, R. P. (1999). Test theory: A unified treatment . Mahwah, NJ: Lawrence Erlbaum Associates. McLeod, L. D., Swygert, K. A., &Thissen, D. (2001). Factor analysis for items scored in two categories. In D. Thissen & H. Wainer (Eds.), Test scoring , 189-216. Mahwah, NJ: Lawrence Erlbaum Associates. Meade, A. W., & Lautenschlager, G. J. (2004). A Monte-Carlo study of confirmatory factor analytic tests of measurement equivalence/invariance. Structural Equation Modeling, 11 , 60-72. Mulaik, S. A. (2010). Foundations of factor analysis . Boca Raton, FL: Chapman & Hall/CRC. Mundfrom, D. J., Shaw, D. G., & Ke, T. L. (2005). Minimum sample size recommendations for conducting factor analyses. International Journal of Testing , 5,159-168. Muthén, B. O. (1978). Contributions to factor analysis of dichotomous variables. Psychometrika , 43 , 531-560. Muthén, B. O. (1993). Goodness of fit with categorical and other non-normal variables. In K. A. Bollen, & J. S. Long (Eds.), Testing structural equation models , 205-243. Newbury Park, CA: Sage. A COMPARISON OF FACTOR ROTATION METHODS FOR DICHOTOMOUS DATA 570 Muthén, B. O., du Toit, S. H. C., & Spisic, D. (1997). Robust inference using weighted least squares and quadratic estimating equations in latent variable modeling with categorical and continuous outcomes .Unpublished manuscript. Muthén, L. K., & Muthén, B. O. (2007). MPlus, version 5: User’s guide . Los Angeles: Author. Preacher, K. J. & MacCallum, R. C. (2002). Exploratory factor analysis in behavior genetics research: Factor recovery with small sample sizes. Behavior Genetics , 32 (2), 153-161. Sass, D. A., & Schmitt, T. A. (2010). A comparative investigation of rotation criteria within exploratory factor analysis. Multivariate Behavioral Research , 45 , 73-103. Saunders, D. R. (1962). Trans-varimax: Some properties of the Ratiomax and Equamax criteria for blind orthogonal rotation . Paper presented at the Meeting of the American Psychological Association, St. Louis, MO, September. Spearman, C. (1929). The abilities of man. New York: Macmillan. Stout, W., Habing, B., Douglas, J., Kim, H. R., Roussos, L., & Zhang, J. Conditional covariance-based nonparametric multidimensionality assessment. Applied Psychological Measurement , 19 (4), 331-354. Tabachnick, B. G., & Fidell, L. S. (2007). Using Multivariate Statistics . Boston: Pearson. Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications . Washington, DC: American Psychological Association. Thurstone, L. L. (1947). Multiple factor analysis. Chicago: University of Chicago press. Trendafilov, N. T. (1994). A simple method for procrustean rotation in factor analysis using majorization theory. Multivariate Behavioral Research , 29 , 385-408. Yates, A. (1987). Multivariate exploratory data analysis: A perspective on exploratory factor analysis. Albany: State University of New York Press. Zumbo, B. D. (2007). Validity: Foundational issues and statistical methodology. In C. R. Rao (Series Ed.) & S. Sinharay (Volume Ed.), Handbook of statistics: Vol 25 Psychometrics , 45-80. Amsterdam: Elsevier.
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https://baike.baidu.com/item/%E4%B8%A4%E7%82%B9%E9%97%B4%E8%B7%9D%E7%A6%BB%E5%85%AC%E5%BC%8F/6773405
网页 新闻 贴吧 知道 网盘 图片 视频 地图 文库 资讯 采购 百科 帮助 近期有不法分子冒充百度百科官方人员，以删除词条为由威胁并敲诈相关企业。在此严正声明：百度百科是免费编辑平台，绝不存在收费代编服务，请勿上当受骗！详情>> 首页 秒懂百科 特色百科 知识专题 加入百科 : 进阶成长 任务广场 百科团队 : 校园团 分类达人团 热词团 繁星团 蝌蚪团 权威合作 : 合作模式 常见问题 联系方式 个人中心 两点间距离公式 播报 锁定 讨论6 上传视频 数学名词 高中数学基础知识导引:139 两点之间的距离公式 04:32 两点间的距离公式 05:30 272 两点间距离公式 07:57 知识点讲解：两点间的距离公式 01:14 两点间距离公式推导：从坐标到勾股定理 02:25 高中数学必备：两点间距离公式详解 01:26 数轴上两点距离公式 02:15 两点之间距离公式是什么 01:10 【中考数学】中考生应该知道的中点坐标公式+两点间距离公式详细讲解,包听包会! 12:06 河北单招网课---两点间距离公式2 12:38 解析几何基础：两点距离公式及应用 02:17 平面两点间距离公式 01:34 中职数学《基础模块》下册6.1两点间距离公式和线段的中点坐标公式 28:15 中职数学—6·1两点间距离公式和线段中点坐标公式 42:48 坐标系中两点距离公式 01:23 收藏 查看我的收藏 3180 有用+1 217 本词条由《中国科技信息》杂志社 参与编辑并审核，经科普中国·科学百科认证 。 两点间距离公式常用于函数图形内求两点之间距离、求点的 坐标 的基本公式，是距离公式之一。两点间距离公式叙述了点和点之间距离的关系。 中文名 : 两点间距离公式 外文名 : Distance between two points 学 科 : 数学 性 质 : 叙述了点和点之间距离的关系 应 用 : 用于函数图形内求两点之间距离 分 类 : 平面和空间两点间距离公式 目录 1 定义 2. 2 平面坐标形式 3. 3 公式 4. 4 推论 5 实例 2. 6 三维坐标形式 3. ▪ 公式 4. ▪ 推导过程 5. 7 极坐标形式 8 异面直线上两点间距离 定义 播报 两点间距离公式常用于函数图形内求两点之间距离、求点的坐标的基本公式，是距离公式之一。两点间距离公式叙述了点和点之间距离的关系。 平面坐标形式 播报 下面不加证明地给出几个公式。 公式 播报 设两个点A、B以及坐标分别为 、 ，则A和B两点之间的距离为： 推论 播报 直线上两点间的距离公式： 设直线 的方程为 ，点 ， 为该线上任意两点，则 这一公式即所谓 圆锥曲线 的弦长公式。若记 为直线AB的 倾斜角 ，则 同时，若已知直线公式和其中一个点，并且给定了距离，可以反求另一个点的坐标。 实例 播报 现在有一只工程队要铺设一条网络，连接A，B两城。他们首先要知道两城之间的距离，才能准备材料。他们用全球定位系统将两城的位置在 平面直角坐标系 中表示出来。我们就来试试看能不能帮他们求出A、B两城之间的距离。 在黑板上画出A，B两点，如图1： 图1 那么，我们怎么求出AB之间的距离呢？ 我们来试试看，能不能通过添加一些辅助线，来解答问题呢？ 首先我们作点A关于X轴的垂线，设垂足为A’，再作B关于Y轴的垂线，设垂足为B’；延长AA’和BB’使之交与C点。 如图2： 图2 显然角C等于90度，这样我们就构造出了一个三角形ABC，而我们要求的AB就在这 个直角三角形上。因此我们是不是可以考虑看看用勾股定理来求出AB呢？ 由勾股定理可以得知： 由A（-20,20）和B（20,-10），所以可知C（-20，-10）。现在我们可以将AB平移到Y轴上，设这两个对应的点为N1，N2，所以： 因此可知： AB 2 =|20-（-20）| 2 +|（-10）-20| 2 =2500 所以。 我们已经求出了A、B两城的距离。我们来思考一个问题：是不是任意两点，只要知道这两点的坐标，就可以求出这两点之间的距离呢？我们能不能找到一个公式来求两点之间的距离呢？ 不妨设A（x1,y1），B（x2,y2）。因此可以推出 三维坐标形式 播报 公式 设 ， ，则 推导过程 在三维坐标中，首先计算两点在平面坐标中（即 ， 轴上）的距离，再计算两点在 轴上的垂直距离 。再次用 勾股定理 ，即证。 极坐标形式 播报 公式 下面不加证明地给出该公式。设极坐标系中两点 ， ，则 异面直线上两点间距离 播报 异面直线上两点间距离公式是 。 词条图册 更多图册 概述图册 (1张) 词条图片 (4张) 1/1 参考资料 1 孙彩平 ．数学八（下） ．人民教育出版社 ．2008-6-1 2 "Calculate distance, bearing and more between Latitude/Longitude points". Retrieved 10 Aug 2013. 3 刘玉兰.异面直线上两点间距离公式的应用[J].数理化解题研究：高中版,2006(4):30-30 两点间距离公式的概述图（1张） 科普中国 致力于权威的科学传播 本词条认证专家为 肖志勇 副教授 审核 江南大学 权威合作编辑 : 《中国科技信息》杂志社 《中国科技信息》创刊于1989年10月，是... : 什么是权威编辑 词条统计 浏览次数：3737963次 编辑次数：52次历史版本 最近更新： 冰淇淋会流泪 （2025-03-06） 突出贡献榜 cjw30 1 定义 2 平面坐标形式 3 公式 4 推论 5 实例 6 三维坐标形式 公式 推导过程 7 极坐标形式 8 异面直线上两点间距离 相关搜索 两点间的距离公式 高中三角函数公式 距离公式 圆的面积怎么算 数学公式大全 高中数学公式汇总 数学 数学题 初一数学一元一次方程 高中必背88个数学公式 两点间距离公式 0 成长任务 编辑入门 编辑规则 本人编辑 内容质疑 在线客服 官方贴吧 意见反馈 举报不良信息 未通过词条申诉 投诉侵权信息 封禁查询与解封 ©2025 Baidu 使用百度前必读 | 百科协议 | 隐私政策 | 百度百科合作平台 | 京ICP证030173号
14046
https://www.semanticscholar.org/paper/7bcfc3f6e790e92962be7e1a252df851f7ee5c0f
Subungual Glomus Tumor: Clinical Manifestations and Outcome of Surgical Treatment | Semantic Scholar Skip to search formSkip to main contentSkip to account menu Search 229,291,877 papers from all fields of science Search Sign In Create Free Account DOI:10.1111/j.1346-8138.2004.tb00643.x Corpus ID: 20082187 Subungual Glomus Tumor: Clinical Manifestations and Outcome of Surgical Treatment @article{Moon2004SubungualGT, title={Subungual Glomus Tumor: Clinical Manifestations and Outcome of Surgical Treatment}, author={Sang Eun Moon and Jong Hyun Won and Oh Sang Kwon and Jeong-Aee Kim}, journal={The Journal of Dermatology}, year={2004}, volume={31}, url={ } S. Moon, J. Won, +1 authorJeong-Aee Kim Published in Journal of dermatology (Print…1 December 2004 Medicine The Journal of Dermatology TLDR This series suggests that a transungual approach with nail avulsion and an incision selected according to the tumor location can produce an excellent outcome with minimal postoperative complications.Expand View on Wiley ncbi.nlm.nih.gov Save to Library Save Create Alert Alert Cite Share 48 Citations Highly Influential Citations 3 Background Citations 15 Methods Citations 1 Results Citations 1 View All 48 Citations 25 References Related Papers 48 Citations Citation Type Has PDF Author More Filters More Filters Filters ### Subungual Glomus Tumors: Surgical Approach and Outcome Based on Tumor Location Soo Hyun LeeM. RohK. Chung Medicine Dermatologic Surgery 2013 TLDR Careful dissection and complete removal of the tumor offered cure without recurrence; anatomic location of the subungual glomus tumor at initial presentation may predict postoperative complications.Expand 26 1 Excerpt Save ### Treatment of Subungual Glomus Tumors Using the Nail Bed Margin Approach Pei-Ji WangYong ZhangJiajun Zhao Medicine Dermatologic Surgery 2013 TLDR The nail bed margin approach is a simple, feasible, effective new method with a low complication and recurrence rate and is expected to be an excellent alternative approach for the excision of subungual glomus tumors.Expand 16 Save ### Surgical Treatment of Subungual Glomus Tumor: A Unique and Simple Method Margaret SongH. KoK. KwonM. Kim Medicine Dermatologic Surgery 2009 TLDR In treatment of subungual glomus tumor, meticulous simple blunt dissection using a transungual approach led the tumor to “pop up” from the tumor bed and showed low recurrence and minimal complications.Expand 42 2 Excerpts Save ### Outcome of Microscopic Excision of a Subungual Glomus Tumor: A 12-Year Evaluation Hsiao-Peng HuangM. Tsai+5 authorsShyi-Gen Chen Medicine Dermatologic Surgery 2015 TLDR Microscopic surgical excision enables the surgeon to completely remove a glomus tumor while minimizing damage to the nail unit, thereby resulting in significantly decreased recurrence and nail deformity.Expand 15 PDF Save ### Subungueal glomus tumors treated by the mohs micrographic surgery : recurrence index and literature review N. D. Chiacchio Medicine 2010 TLDR To compare the rates of recurrence of conventional surgeries described in the literature with the ones found after the Mohs micrographic surgery (MMS), ten patients diagnosed with subungueal glomus tumor were compared.Expand 2 Save ### Transungual Excision of Glomus Tumors: A Treatment and Quality of Life Study. Edouard F H ReindersK.M.G. KlaassenM. Pasch Medicine Dermatologic Surgery 2019 TLDR This study showed that glomus tumors cause impairment on QoL, mostly due to severe pain, and surgical excision with the transungual approach is an effective treatment, without permanent damage to the nail unit that gives relief of pain and improves QOL.Expand 13 Highly Influenced 5 Excerpts Save ### Glomus tumor of hand: A rarity treated by transungual approach Anand DeorajK. BaithaP. PrakashS. Anand Medicine International Journal of Surgery Science 2020 TLDR The aim of this study is to review the time from onset of symptoms to the diagnosis, management of problem and recurrence retrospectively, at tertiary care center of eastern India between May 2017 to May 2019.Expand PDF 1 Excerpt Save ### Nail-preserving excision for subungual glomus tumour of the hand Hyun-Joo LeeP. KimH. KyungHyung-sub KimI. Jeon Medicine Journal of Plastic Surgery and Hand Surgery 2014 TLDR The nail preserving transungual approach provides several advantages, that is, better nail bed exposure, resulting in easier tumour excision, and less damage to the nail bed with less deformity of the nail.Expand 16 1 Excerpt Save ### Clinical Features of Multiple Glomus Tumors W. ParkerT. SchieferC. InwardsR. SpinnerP. Amadio Medicine Dermatologic Surgery 2008 TLDR Patients with multiple glomus tumors are frequently misdiagnosed, and proper recognition and diagnosis would lead to improved management, according to this retrospective review of patients seen at this institution over the past 25 years.Expand 47 Save ### Subungual glomus tumours: diagnosis and microsurgical excision through a lateral subperiosteal approach. K. MuramatsuK. IharaT. HashimotoY. TominagaT. Taguchi Medicine Journal of Plastic, Reconstructive & Aesthetic… 2014 18 Save ... 1 2 3 4 5 ... 25 References Citation Type Has PDF Author More Filters More Filters Filters ### Treatment of subungual glomus tumour. H. TakataY. IkutaO. IshidaK. Kimori Medicine Hand Surgery 2001 TLDR Patients with subungual glomus tumours of the hand operated on between 1964 and 1997 were reviewed, and it was found that to avoid nail deformity, it is better to apply a periungual approach for tumours developing in the peripheral region, and a transungUAL approach followed by meticulous repair of the nail bed for tumour developed in the central region.Expand 49 Save ### Glomus tumor: a clinicopathologic and electron microscopic study. M. TsuneyoshiM. Enjoji Medicine Cancer 1982 TLDR The clinicopathologic evidence presented supports the hypothesis that the glomus tumor is a tumor-like lesion of mesodermal disorder rather than a true neoplasm.Expand 215 Save ### Glomus Tumors of the Nail Unit: A Plastic Surgeon's Approach E. Wegener Medicine Dermatologic Surgery 2001 TLDR Glomus tumors may occur anywhere on the body, and the vast majority of these are located in the fingertips, but this lesion is frequently misdiagnosed as neuroma, and even as gout, arthritis, or causalgia.Expand 13 Save ### Glomus Tumours of the Hand J. GeertruydenP. Loréa+5 authorsJ. Moermans Medicine Journal of Hand Surgery-American Volume 1996 121 Save ### Prevention of postoperative nail deformity after subungual glomus resection. Hiroshi TadaTakakazu HiraymaYoshiharu Takemitsu Medicine Journal of Hand Surgery-American Volume 1994 40 Save ### Nail surgery: an assessment of indications and outcome. DA De BerkerMG DahlJS ComaishCM. Lawrence Medicine Acta Dermato-Venereologica 1996 TLDR Information from this study demonstrates the diagnostic and therapeutic value of nail surgery within dermatology.Expand 19 PDF Save ### Glomus tumor of the digits. A. RettigJ. W. Strickland Medicine Journal of Hand Surgery-American Volume 1977 113 Save ### The Use of Ultrasonography in Preoperative Localization of Digital Glomus Tumors S. ChenYaobin ChenM. ChengK. YeowHung-Chi ChenF. Wei Medicine Plastic and Reconstructive Surgery 2003 TLDR Low resolution ultrasonography showed a hypoechoic nodule with prominent vascularity between the nail body and the dorsal cortex of the distal phalanx in all subungual tumors, and complete resection was possible in all 35 glomus tumors, with assistance by accurate preoperative ultrasound localization.Expand 110 Save ### Glomus tumors of the hand: review of the literature and report on twenty-eight cases. R. E. CarrollA. T. Berman Medicine Journal of Bone and Joint Surgery. American… 1972 Twenty-eight cases of glomus tumor of the hand have been studied from various aspects including a review of the pertinent literature, clinical manifestations, treatment, and the incidence of this… Expand 314 Save ### Subungual Glomus Tumours: A Different Approach to Diagnosis and Treatment A. EkinM. ÖzkanT. Kabaklioğlu Medicine Journal of Hand Surgery-American Volume 1997 49 Save ... 1 2 3 ... Related Papers Showing 1 through 2 of 0 Related Papers Stay Connected With Semantic Scholar Sign Up What Is Semantic Scholar? Semantic Scholar is a free, AI-powered research tool for scientific literature, based at Ai2. 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14047
https://www.youtube.com/watch?v=wdEg4d2IQFg
Calculus: Natural Logs and Tangent Lines! Mr. Murray's Mathland 350 subscribers 3 likes Description 149 views Posted: 15 Oct 2023 This video contains 2 examples of natural log functions and tangent lines: one about writing the equation of a tangent line, and one about finding points where a function has horizontal tangent lines! Includes using the Chain Rule, Product Rule, and using properties of logarithms. Transcript: hey what's up everybody Mr Murray here Mr Murray's mathland and uh we're just doing a couple other uh natural log problems uh that uh apply to tangent lines so uh in video one we we showed how the to get the derivative of a natural log function and now we're just going to apply it to a couple tangent line you know classic tangent line types of problems uh just for a little extra practice here so number one we'll just Jump Right In Here write an equation for the line tangent to this function at xal 6 so uh you know if you're in my class or if you've been watching these videos and you know for all seven of you out there who do that um here is uh you know you I you hear me say this a lot if you want to write the equation of a tangent line you need two things right you need a point and the slope so the point they've given you the x coordinate for the point of tangency but we need to get the Y coord coordinates so we're going to plug 6 into the function and so that's going to be Ln of 6^ 2 which is 36 - 6 6 which is 36 + 1 and with a +4 after the log we simplify that and we get to Ln of 1 + 4 and hopefully everybody's thinking right along with me that hey Ln of one that's a value of zero right because that same e to what power gives you one and so that's the zero power so 0 + 4 = 4 of course and so our point of tangency is 6 comma 4 okay so we got that point hang on to that and now we need the slope and so if you've come this far in calculus you know the slope comes from the derivative so the derivative of this function will be y Prime equals and taking the derivative of a natural log log you do 1 over that quantity 1 / x^2 - 6X + 1 multiplied by that quantity's derivative so the derivative is that is 2x - 6 and you know just to uh Focus you know not ignore things there is this plus4 on the end but the derivative of that of course is zero being a constant so that's it for the derivative but you know watch out if there had been like an e to the x or some additional stuff there on the end so that is your derivative but remember that's just the slope function so if you want the slope at six you got to actually plug in six so when we plug in six to this 2 6 is 12 - 6 over and again in the bottom we get that same you know 36 - 36 + 1 and of course if you can do some of this in your head that's fine you're going to get 6 over one which is really just going to give you six so that is your slope and so your tangent line is y - 4 = the slope of 6 x - 6 and if you want to use a slope intercept form or some other version that's uh obviously fine but there's one version of the tangent line okay so pretty pretty straightforward just a little practice with the the Ln Rule and here's just one more here uh that has to do with tangent lines and this is like a classic a calculus problem that every time you learn a new derivative rule you kind you tend to see this kind of question uh for this function find where it has if any any horizontal tangent lines and you know that you're looking for essentially spots where the graph turns around relative Maxes relative mins or maybe any saddle points just places where the graph is flat where it's got a horizontal tangent line and so the strategy is always the same for these kinds of problems you're going to find find the derivative and set it equal to zero because you're basically going to ask when does this have a slope of zero okay so of course the the first half of the battle is getting the derivative and so you're doing the derivative of f ofx = x 4 Ln of x uh hopefully you're recognizing that this requires the product rule and this you know a fairly straightforward product rule to do but however you like to get that done you have the two functions to worry about you have X 4th and its derivative will be represented that'll be 4X cubed the other function is of course Ln of X and its derivative is a nice simple 1 /x so when you use your product rule you know these iners plus the product of these outers you're going to get 4X cubed Ln of X Plus now when you multiply this x 4 1/x actually works out kind of nicely you get X 4th overx which reduces to X cubed and that's actually uh if you're thinking ahead going to make our solving work out a little nicer and now we're going to set this derivative equal to zero and see if there's any X values where this is equal to zero and factoring is one of the classic things we use to solve these so let's look for that greatest common factor we can take out X cubed and then we will have 4 Ln of x + 1 and when we set each of those equal to zero you're going to get a solution of x equals 0 and here you're going to get a solution of well Ln of x = -4 When You Subtract 1 and divide by four and then to solve for x if you're a little rusty with your LNS uh this is where you can raise e to each side or you know it's called exponentiating equation uh because that's the inverse operation that will leave you with X on this side and you get e to the - 1/4 it's really just taking your thing from log form to exponential form if you like to think of it that way um and you know we tend to not like negative exponents so we might write that as 1 over e to the 1/4 or one over uh the fourth root of e if you like so we got two uh X values here that solutions to this equation uh but if you're thinking ahead that you looked at this direction we want any points that generally means we want the coordinates the ordered pairs so you're going to need the y-coordinates and I think I hope well maybe not I don't know but if you were to go and plug zero in you get 0 Ln of 0 and Ln of 0 is undefined so if you're you know it's been a while since you've done some things with logs like since last year in pre-cal uh you might take note of these kinds of things that the domain of a log function remember is that the inner uh expression the argument of the log must always be greater than zero can't be equal to zero even because then you're saying e to what power gives you zero and and nothing exists so so you know xal 0 is not in the domain so it's not going to be a valid solution here and uh we're only looking for values that are positive and 1 over e to 1/4th is a positive value so that's a solution uh we just now have to get the y-coordinate for it and like I said if you didn't think about the domain right off the bat hopefully you would notice when you went and plugged in zero but if you were a little careless you might plug in and think 0 0 was the the point there so just watch out for those kinds of things you know logs have domain issues and so now we're going to plug this 1 over e to 1/4 we're going to plug this back into the original and so that means the function is X 4th Ln of X so you get 1 over e to the 1/4 or if you want to think of that as 1 over the 4th root of e that's going to make this part look nicer for you times Ln of 1 E 1/4 and it's actually you know kind of a nice little simplifying uh algebra practice here that 1 over the 4th root of e rais to 4th is going to be 1 4th is 1 and the 4th root of e 4th is just e multiplied by Ln of 1 E 1/4 and this can also be simplified a little bit and might help you to visualize putting it right back to um e to the -14 if you helps you to see that because remember Ln and E are inverse operations so they cancel each other out Ln of e to the x is always that x value so you get 1/ e -4 and that of course will give you -1 over 4 e so your coordinates I would say in the nicest form possible are 1/ e to 1/4 or again 4th root of e comma -1 over 4 e and there is one point and those are uh some very small decimals and just as a little confirmation here you could always check the graph out on Desmos or on your you know TI calculator and there is one minimum notice and and notice that domain of course it doesn't exist to the left of zero or even at zero well this says it does but Desmos has issues uh but this little minimum here 779 negative 092 that's this point in decimal form and so there we go so hopefully you saw a couple problems brushed up on your calculus but also some pre-cal properties and uh all good if you have any questions just let me know and keep calm and do Cal kids see you later
14048
https://www.imdb.com/title/tt10995072/
"Shantaram" Dead Man Walking (TV Episode 2022) - IMDb Menu [x] Movies Release calendarTop 250 moviesMost popular moviesBrowse movies by genreTop box officeShowtimes & ticketsMovie newsIndia movie spotlight TV shows What's on TV & streamingTop 250 TV showsMost popular TV showsBrowse TV shows by genreTV news Watch What to watchLatest trailersIMDb OriginalsIMDb PicksIMDb SpotlightFamily entertainment guideIMDb Podcasts Awards & events OscarsEmmysToronto Int'l Film FestivalIMDb TIFF Portrait StudioHispanic Heritage MonthSTARmeter AwardsAwards CentralFestival CentralAll events Celebs Born todayMost popular celebsCelebrity news Community Help centerContributor zonePolls For industry professionals Language English (United States) [x] Language Fully supported English (United States) Partially supported Français (Canada) Français (France) Deutsch (Deutschland) हिंदी (भारत) Italiano (Italia) Português (Brasil) Español (España) Español (México) - [x] All All 8 suggestions available Watchlist Sign in Sign in New customer?Create account EN - [x] Fully supported English (United States) Partially supported Français (Canada) Français (France) Deutsch (Deutschland) हिंदी (भारत) Italiano (Italia) Português (Brasil) Español (España) Español (México) Use app Shantaram S1.E6 All episodesAll Cast & crew User reviews IMDbPro All topics Dead Man Walking Episode aired Nov 4, 2022 TV-MA 52m IMDb RATING 7.6/10 446 YOUR RATING Rate ActionAdventureCrimeDramaThriller Lin stops a reporter from running a story on his work but fails to extinguish her interest in discovering the identity of the man using a dead man's passport.Lin stops a reporter from running a story on his work but fails to extinguish her interest in discovering the identity of the man using a dead man's passport.Lin stops a reporter from running a story on his work but fails to extinguish her interest in discovering the identity of the man using a dead man's passport. Director Iain B. MacDonald Writers Marion Dayre Eric Warren Singer Steve Lightfoot Stars Charlie Hunnam Fayssal Bazzi Sujaya Dasgupta See production info at IMDbPro IMDb RATING 7.6/10 446 YOUR RATING Rate Director Iain B. MacDonald Writers Marion Dayre Eric Warren Singer Steve Lightfoot Stars Charlie Hunnam Fayssal Bazzi Sujaya Dasgupta STREAMING Set your preferred services Add to Watchlist Mark as watched 1 User review 3 Critic reviews See production info at IMDbPro See production info at IMDbPro ### Photos 3 Add photo ### Top cast 32 Edit Charlie Hunnam Lin Ford Fayssal Bazzi Abdullah Sujaya Dasgupta Kavita Antonia Desplat Karla Elham Ehsas Modena David Field Wally Nightingale Matthew Joseph Ravi Rachel Kamath Parvati Alyy Khan Qasim Ali Elektra Kilbey Lisa Shiv Palekar Vikram Luca Pasqualino Maurizio (as Luke Pasqualino) Vincent Perez Didier Shubham Saraf Prabhu Gabrielle Scharnitzky Madame Zhou Alexander Siddig Khader Khan Justin Amankwah Raheem Arka Das Nishant Patel Director Iain B. MacDonald Writers Marion Dayre Eric Warren Singer Steve Lightfoot All cast & crew Production, box office & more at IMDbPro ### User reviews 1 Review 7.6 446 12345678910 Featured reviews 10moviesfilmsreviewsinc ### Im just wowed and Haven't seen show this good in a long time. In the opening sequence of Shantaram episode 6, we see the backstory of why Lin ran away from the prison. The detective questioning him gave him one last chance to come clean about Officer Flores' murder but Lin declines the opportunity. Instead, the detective asks a prisoner to bring in beers for him and Lin, making him believe that Lin is a rat. If he would have stayed, Lin would have been dead before the day ended. Lin takes Prabhu to Reynaldo's to meet with Kavita. He apologizes for his behavior the last time and tries to come up with an explanation for why he does not want Kavita to cover him in the paper. Prabhu too plays his part and says it is because of Sagarwada's people that Lin does not want to glorify his work. "This city strikes at the poor, not the poverty", Kavita wisely says but keeps up the appearance of believing them. It is a ploy by Kavita to keep Lin in Mumbai until she investigates further. Lin gets his photos clicked and expresses his intention to leave Bombay. Prabhu tries to persuade Lin but he says he cannot risk it. Sebastian tells Lisa that Maurizio gets all his heroin from Zhou. Khaderbhai does not know about it yet and when he does, they all will be in danger. Abdullah tracks down Gaurav, Minster Pandey's driver, and forces him to reveal Pandey's schedule. He does so on Khader's instructions of finding dirt on Pandey to blackmail him. Prabhu is talked to by Nandita about bringing back Parvati to the house. She is spending the entire time in the clinic and they want her back. Despite the promise that they will give their blessings for marriage if Prabhu brings her back, he declines. Parvati admires his support and expresses her intention to marry him regardless. Episode 6 solidifies the notion that nice guys always finish last. Despite Lin's selfless, good work, he is being punished by his Karma for not bringing the murderer of Officer Flores, Charlie, to justice. Things keep going against him in Bombay. The new threat that looms over his head is a big one. He is marginally close to getting caught. Kavita is determined to get her big break and Nishant is helping him too. Lisa's handling by Modena was a setback indeed. For a while, he deceived himself into thinking that he was anything other than just a pimp. But he had to give in to Maurizio and allow the woman he loves to sleep with another man. Shocks like these keep Shantaram interesting, howsoever scattered the narrative might seem on the surface. ### Related interests Action Adventure Crime Drama Thriller ### Storyline Edit Did you know Edit GoofsAs Lin asks Karla to meet him the next day, he begs on his knees in front of a food cart. When he stands up they are no longer in front of the food cart. ### Top picks Sign in to rate and Watchlist for personalized recommendations Sign in Details Edit Release date November 4, 2022 (United States) Filming locations Bangkok, Thailand Production company Thai Occidental Productions See more company credits at IMDbPro ### Tech specs Edit Runtime 52m ### Contribute to this page Suggest an edit or add missing content Learn more about contributing Edit page More to explore Photos Greatest Character Actors of All Time See the gallery Photos Streaming Stars, Then and Now See the gallery ### User lists Related lists from IMDb users Create a list 2023-də baxdıqlarım created 2 years ago•181 titles Olmuş Hadisələr Əsasında Çəkilmiş Filmlər created 8 years ago•264 titles Watched created 10 years ago•2382 titles ♚ Johnny S┼ assis┼idos. created 7 years ago•3834 titles TV shows I've seen created 2 years ago•5382 titles Kitab əsasında çəkilmişlər created 9 years ago•68 titles Photos Hispanic and Latino Stars to Watch See the stars Recently viewed Clear all Dead Man Walking Get the IMDb App Sign in for more accessSign in for more access Follow IMDb on social Get the IMDb App For Android and iOS Help Site Index IMDbPro Box Office Mojo License IMDb Data Press Room Advertising Jobs Conditions of Use Privacy Policy Your Ads Privacy Choices © 1990-2025 by IMDb.com, Inc. 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https://mathspace.co/textbooks/syllabuses/Syllabus-1190/topics/Topic-22479/subtopics/Subtopic-285872/
Textbooks :: Mathspace Book a Demo Topics 6.C o n g r u e n c e&T r i a n g l e s 6.0 1 C o n g r u e n c e t r a n s f o r m a t i o n s 6.0 2 C o r r e s p o n d i n g p a r t s o f c o n g r u e n t t r i a n g l e s 6.0 3 S S S a n d S A S c o n g r u e n c e c r i t e r i a 6.0 4 A S A a n d A A S c o n g r u e n c e c r i t e r i a LessonPractice 6.0 5 R i g h t t r i a n g l e c o n g r u e n c e Log inSign upBook a Demo United States of AmericaVirginia Geometry 6.04 ASA and AAS congruence criteria LessonPractice Ideas ASA and AAS congruence criteria ASA and AAS congruence criteria Exploration We are given two triangles with 2 2 2 pairs of congruent angles, and the corresponding sides between those angles are congruent. Sketch two triangles to fit the description. Label the triangles G H I GHI G H I and L M N LMN L MN, so that ∠G≅∠L\angle G \cong \angle L∠G≅∠L,∠H≅∠M\angle H \cong \angle M∠H≅∠M, and G H‾≅L M‾\overline{GH} \cong \overline{LM}G H≅L M. Formulate a plan for proving that there is a sequence of rigid transformations that will map △G H I\triangle GHI△G H I to △L M N\triangle LMN△L MN and explain how you know one or more vertices will align at each step. If we are given two congruent corresponding angles and one congruent corresponding side, then we will be proving the triangles congruent by angle-side-angle or angle-angle-side congruency depending on the position of the given side. Included side The side between two angles of a polygon is the included side of those two angles. Angle-Side-Angle (ASA) congruency theorem If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Angle-Angle-Side (AAS) congruency theorem If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA AAS When proving triangles congruent, it can be difficult to distinguish between ASA and AAS congruence. That's usually due to a result of a corollary to the triangle sum theorem: Third angles theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Because of this theorem, any triangles that can be proven by ASA congruence can also be proven by AAS congruence and vice versa without any additional information. Examples Example 1 Use rigid transformations to prove the ASA congruency theorem: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Worked Solution Create a strategy We can set up a figure so we can do the proof in terms of a particular diagram. Since we can't use the ASA congruency theorem to prove itself, we must use another strategy. In this case, we need to identify a series of rigid transformations that will map △A B C\triangle ABC△A BC to △D E F\triangle DEF△D EF. Our rigid transformations are: Translation Rotation Reflection Apply the idea Given: ∠A B C≅∠D E F\angle{ABC}\cong \angle{DEF}∠A BC≅∠D EF B C‾≅E F‾\overline{BC}\cong \overline{EF}BC≅EF ∠A C B≅D F E‾\angle{ACB}\cong \overline{DFE}∠A CB≅D FE Prove: △A B C≅△D E F\triangle ABC \cong \triangle DEF△A BC≅△D EF We need to consider that the two triangles could be any orientation. There is a rigid transformation that will map B C‾\overline{BC}BC onto E F‾\overline{EF}EF because B C‾≅E F‾\overline{BC}\cong \overline{EF}BC≅EF. First, we can rotate and reflect the triangle to get B C‾∥E F‾\overline{BC}\parallel \overline{EF}BC∥EF so that and ∠B\angle B∠B and ∠E\angle E∠E are oriented in the same direction. Call this triangle △A′B′C′\triangle A'B'C'△A′B′C′. If A′A'A′ and D D D are on the same side, reflect over E F‾\overline{EF}EF until the figure is as shown. 2. Since these transformations are all rigid transformations, we have that: ∠A B C≅∠A′B′C′and∠A C B≅∠A′C′B′\angle{ABC}\cong \angle{A'B'C'} \text{ and }\angle{ACB}\cong \angle{A'C'B'}∠A BC≅∠A′B′C′and∠A CB≅∠A′C′B′ Using the given information and the transitive property of congruence, we have that:∠D E F≅∠A′B′C′and∠D F E≅∠A′C′B′\angle{DEF}\cong \angle{A'B'C'} \text{ and }\angle{DFE}\cong \angle{A'C'B'}∠D EF≅∠A′B′C′and∠D FE≅∠A′C′B′ Point A′A'A′ can be mapped on to point D D D by a reflection across E F‾\overline{EF}EF. We can justify that point A′A'A′ coincides with point D D D after the reflection as follows: If the reflection of △A′F E\triangle{A'FE}△A′FE is △A′′F E\triangle{A''FE}△A′′FE, points E E E and F F F will remain fixed during the reflection. E A′′→\overrightarrow{EA''}E A′′ will coincide with E D→\overrightarrow{ED}E D, and F A′′→\overrightarrow{FA''}F A′′ will coincide with F D→\overrightarrow{FD}F D because reflections preserve angles, and rays coming from the same point at the same angle will coincide. We have that E A′′→\overrightarrow{EA''}E A′′ and F A′′→\overrightarrow{FA''}F A′′ can only intersect at point A′′A''A′′, and E D→\overrightarrow{ED}E D and F D→\overrightarrow{FD}F D can only intersect at point D D D. Since the rays coincide, their intersection points, A′′A''A′′ and D D D, will also coincide. We have now shown that: A A A maps to D D D using rigid transformations B B B maps to E E E using rigid transformations C C C maps to F F F using rigid transformations So, we have that △A B C\triangle ABC△A BC can be mapped onto △D E F\triangle DEF△D EF using rigid motions, so: △A B C≅△D E F\triangle ABC \cong \triangle DEF△A BC≅△D EF Watch question walkthrough Example 2 Prove that the two triangles are congruent. Worked Solution Create a strategy Notice that one triangle is equilateral, while the other is equiangular. The corollary to the base angles theorem and its converse tells us that both triangles will be equilateral and equiangular. Apply the idea Both triangles are equiangular, which means that the two triangles share three common angle measures. Since they both also have a corresponding side of length 8 8 8, we can use the AAS test to justify that they are congruent. Reflect and check We could also show that the triangles are congruent by stating that both triangles are equilateral, which means that both must have three sides of length 8 8 8. We can then use the SSS test to justify that they are congruent. Watch question walkthrough Example 3 In the following diagram, A D‾\overline{AD}A D and B C‾\overline{BC}BC are both straight line segments. Prove that △A B X≅△D C X\triangle{ABX}\cong\triangle{DCX}△A BX≅△D CX. Worked Solution Create a strategy We want to find as much information as we can in order to satisfy one of the congruence tests. Since A D‾\overline{AD}A D and B C‾\overline{BC}BC are straight line segments, we can find vertically opposite angles, and since A B‾∥D C‾\overline{AB}\parallel\overline{DC}A B∥D C we can find alternate angles on parallel lines. Apply the idea To prove: △A B X≅△D C X\triangle{ABX}\cong\triangle{DCX}△A BX≅△D CX| | Statements | Reasons | --- | 1. | A D‾\overline{AD}A D and B C‾\overline{BC}BC are straight line segments | Given | | 2. | ∠A X B\angle{AXB}∠A XB and ∠D X C\angle{DXC}∠D XC are vertically opposite angles | Opposite angles between straight line segments | | 3. | ∠A X B≅∠D X C\angle{AXB}\cong\angle{DXC}∠A XB≅∠D XC | Vertically opposite angles are congruent | | 4. | ∠A B X\angle{ABX}∠A BX and ∠D C X\angle{DCX}∠D CX are alternate interior angles | B C‾\overline{BC}BC is a transversal of A B‾\overline{AB}A B and D C‾\overline{DC}D C | | 5. | A B‾∥D C‾\overline{AB}\parallel\overline{DC}A B∥D C | Given | | 6. | ∠A B X≅∠D C X\angle{ABX}\cong\angle{DCX}∠A BX≅∠D CX | Alternate interior angles are congruent | | 7. | A B‾≅D C‾\overline{AB}\cong\overline{DC}A B≅D C | Given | | 8. | △A B X≅△D C X\triangle{ABX}\cong\triangle{DCX}△A BX≅△D CX | AAS congruence | Reflect and check We could also use ASA to prove that △A B X≅△D C X\triangle{ABX} \cong \triangle{DCX}△A BX≅△D CX, ignoring vertical angles in a proof like the one that follows: To prove: △A B X≅△D C X\triangle{ABX}\cong\triangle{DCX}△A BX≅△D CX| | Statements | Reasons | --- | 1. | A D‾\overline{AD}A D and B C‾\overline{BC}BC are straight line segments | Given | | 2. | ∠A B X\angle{ABX}∠A BX and ∠D C X\angle{DCX}∠D CX are alternate interior angles | B C‾\overline{BC}BC is a transversal of A B‾\overline{AB}A B and D C‾\overline{DC}D C | | 3. | A B‾∥D C‾\overline{AB}\parallel\overline{DC}A B∥D C | Given | | 4. | ∠A B X≅∠D C X\angle{ABX}\cong\angle{DCX}∠A BX≅∠D CX | Alternate interior angles are congruent | | 5. | ∠B A X≅∠C D X\angle{BAX}\cong\angle{CDX}∠B A X≅∠C D X | Alternate interior angles are congruent | | 6. | A B‾≅D C‾\overline{AB}\cong\overline{DC}A B≅D C | Given | | 7. | △A B X≅△D C X\triangle{ABX}\cong\triangle{DCX}△A BX≅△D CX | ASA congruence | Watch question walkthrough Example 4 Find the value of x x x that makes the triangles congruent. Worked Solution Create a strategy The triangles are congruent by ASA, so we can identify corresponding parts and create an equation to solve for x x x. Based on the diagram, we see that∠E≅∠Y\angle E \cong \angle Y∠E≅∠Y and ∠D≅∠X\angle D \cong \angle X∠D≅∠X, so the included side for the first triangle is D E‾\overline{DE}D E and the included side for the second triangle is X Y‾\overline{XY}X Y. By the definition of congruence, if D E‾≅X Y‾\overline{DE}\cong \overline{XY}D E≅X Y, then the two segments will be equal in length. Therefore, we need to find x x x so that 15=4 x−1 15=4x-1 15=4 x−1. Apply the idea 15\displaystyle 15 15=\displaystyle ==4 x−1\displaystyle 4x-1 4 x−1 D E=X Y DE=XY D E=X Y 16\displaystyle 16 16=\displaystyle ==4 x\displaystyle 4x 4 x Add 1 1 1 to both sides 4\displaystyle 4 4=\displaystyle ==x\displaystyle x x Divide by 4 4 4 on both sides Watch question walkthrough Example 5 Construct a copy of the triangle shown using two angles and one side. Worked Solution Create a strategy We will use two angles and an included side. To create a copy of △A B C\triangle ABC△A BC, we will first create a copy of A C‾\overline{AC}A C. The endpoints of this new segment are the two vertices of the triangle. Then we will create copies of ∠A\angle A∠A on one of these vertices and a copy of ∠C\angle C∠C on the other vertex. The intersection of the sides of the angles will determine the third vertex. Apply the idea We will use GeoGebra to implement the steps. We start by constructing a point D D D that will become one of the vertices of the triangles. We can do this using the Point tool as shown. Next, we want to create a copy of A C‾\overline{AC}A C. To do this, we can use the Compass tool. Select vertices A A A and C C C first then select point D D D. This creates a circle centered at D D D with a radius of length A C AC A C. We can then use the Point tool to create a new point E E E on the arc. Connect D D D and E E E using the Segment tool. This is a copy of A C‾\overline{AC}A C. Because D E‾\overline{DE}D E is the radius of circle D D D it must be congruent to A C‾\overline{AC}A C. To create a copy of ∠A\angle A∠A and ∠C\angle C∠C, we need to create an arc centered at A A A that intersects both A B‾\overline{AB}A B and A C‾\overline{AC}A C. To do this, use the Circle with Center through tool. Select point A A A and any other point on A B‾\overline{AB}A B which we will call F F F. Find the point of intersection of the arc and A C‾\overline{AC}A C which we will call G G G and add it using the Point tool. Use the Compass tool to create a copy of circle A A A centered at point D D D. To do this, select A A A and F F F, and then select D D D. Locate the intersection of the arc of the smaller circle D D D and D E‾\overline{DE}D E using the Point tool. This point H H H is a copy of G G G. Create an arc using the Compass tool. Select F F F and G G G, then select H H H. Use the point tool to add the point I I I at the intersection of the arcs of circles D D D and H H H above D E‾\overline{DE}D E. We have created the point I I I to be the same distance from H H H as point F F F is from G G G. This means F G⌢≅I H⌢\overset{\large\frown}{FG} \cong \overset{\large\frown}{IH}FG⌢≅I H⌢. Use the Line tool. Select I I I and then select D D D.∠I D H\angle IDH∠I DH is a copy of ∠A\angle A∠A. We need to repeat the previous steps to copy ∠C\angle C∠C. Use the Circle with Center through tool. Select C C C and then another point J J J on B C‾\overline{BC}BC. Locate K K K, the intersection of the arc and A C‾\overline{AC}A C, using the Point tool. Use the Compass tool to create a copy of the arc on E E E. Select C C C and J J J, and then select E E E. Locate the intersection of the arc and D E‾\overline{DE}D E using the Point tool. This point L L L is a copy of K K K. Create an arc using the Compass tool. Select J J J and K K K, then select L L L. Use the Line tool. Select M M M, the intersection of the arc centered at E E E and the arc centered at L L L, and then select E E E.∠M E L\angle MEL∠ME L is a copy of ∠C\angle C∠C. Locate N N N, the intersection of D I↔\overleftrightarrow{DI}D I and E M↔\overleftrightarrow{EM}EM, using the Point tool. Finally, we can create △D N E\triangle DNE△D NE using the Polygon tool. This △D N E\triangle DNE△D NE is a copy of △A B C\triangle ABC△A BC. We know the triangles are congruent by ASA because we constructed copies of two angles and their included side. Watch question walkthrough Example 6 Justify that the following steps construct the perpendicular bisector of A B‾\overline{AB}A B. Using a compass, construct an arc centered at A A A with a radius with length that is more than half the length of A B‾\overline{AB}A B. Using the same compass setting, construct an arc centered at B B B. The two arcs must intersect at two points C C C and D D D. Draw C D‾\overline{CD}C D. This is a perpendicular bisector of A B‾\overline{AB}A B. Worked Solution Create a strategy We can use the fact that we used the same setting of the compass when drawing the arcs to justify that some segments are congruent. Using SSS, ASA and CPCTC, we will prove that C D‾\overline{CD}C D is a bisector of A B‾\overline{AB}A B and that the segments are perpendicular, and therefore that C D‾\overline{CD}C D is a perpendicular bisector if A B‾\overline{AB}A B. Apply the idea Start by marking point E E E, the intersection of C D‾\overline{CD}C D and A B‾\overline{AB}A B. C C C and D D D lie on the arc centered at A A A, and so A C‾≅A D‾\overline{AC} \cong \overline{AD}A C≅A D.C C C and D D D also lie on the arc centered at B B B, and so B C‾≅B D‾\overline{BC} \cong \overline{BD}BC≅B D. Since by construction the two arcs have the same radius, we know that A C‾≅A D‾≅B C‾≅B D‾\overline{AC} \cong \overline{AD} \cong \overline{BC} \cong \overline{BD}A C≅A D≅BC≅B D. This means △A C D≅△B C D\triangle ACD \cong \triangle BCD△A C D≅△BC D by SSS congruence. Using CPCTC, ∠A C D≅∠B C D\angle ACD \cong \angle BCD∠A C D≅∠BC D. This also means that ∠A C E≅∠B C E\angle ACE \cong \angle BCE∠A CE≅∠BCE. △A C B\triangle ACB△A CB is isosceles, so ∠C A E≅∠C B E\angle CAE \cong \angle CBE∠C A E≅∠CBE. By ASA congruence, we know that △A C E≅△B C E\triangle ACE \cong \triangle BCE△A CE≅△BCE. By CPCTC, we have A E‾≅B E‾\overline{AE} \cong \overline{BE}A E≅BE. This means C D‾\overline{CD}C D is a bisector of A B‾\overline{AB}A B. Using the same congruence statement and CPCTC, we know that ∠A E C≅∠B E C\angle AEC \cong \angle BEC∠A EC≅∠BEC. Since ∠A E C\angle AEC∠A EC and ∠B E C\angle BEC∠BEC are also linear pair, then m∠A E C=m∠B E C=90°m\angle AEC = m\angle BEC=90 \degree m∠A EC=m∠BEC=90°. This means that C E‾\overline{CE}CE, and by extension C D‾\overline{CD}C D, is perpendicular to A B‾\overline{AB}A B. This proves that C D‾\overline{CD}C D is a perpendicular bisector of A B‾\overline{AB}A B. Reflect and check It is important that the compass width is greater than half of A B AB A B to ensure that there will be two points of intersection. Setting the width to excatly half A B AB A B will only result in one intersection point and setting the width to less than half of A B AB A B will result in no intersection points. Watch question walkthrough Idea summary To show that two triangles are congruent, it is sufficient to demonstrate the following: Angle-side-angle, or ASA: If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. Angle-Angle-Side, or AAS: If two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. Outcomes G.TR.2 The student will, given information in the form of a figure or statement, prove and justify two triangles are congruent using direct and indirect proofs, and solve problems involving measured attributes of congruent triangles. G.TR.2a Use definitions, postulates, and theorems (including Side-Side-Side (SSS); Side-Angle-Side (SAS); Angle-Side-Angle (ASA); Angle-Angle-Side (AAS); and Hypotenuse-Leg (HL)) to prove and justify two triangles are congruent. G.TR.2b Use algebraic methods to prove that two triangles are congruent. G.TR.2d Given a triangle, use congruent segment, congruent angle, and/or perpendicular line constructions to create a congruent triangle (SSS, SAS, ASA, AAS, and HL). What is Mathspace About Mathspace
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https://www.math-only-math.com/worksheet-on-heights-and-distances.html
Subscribe to our ▶️ YouTube channel 🔴 for the latest videos, updates, and tips. Worksheet on Heights and Distances In worksheet on heights and distances we will practice different types of real life word problem trigonometrically using a right-angled triangle, angle of elevation and angle of depression. 1. A ladder rests against a vertical wall such that the top of the ladder reaches the top of the wall. The ladder is inclined at 60° with the ground, and the bottom of the ladder is 1.5 m away from the foot of the wall. Find (i) the length of the ladder, and (ii) the height of the wall. 2. An aeroplane takes off at an angle of 30° with the horizontal ground. Find the height of the aeroplane above the ground when it has travelled 184 m without changing direction. 3. The angle of elevation of the top of a vertical cliff from a point 15 m away from the foot of the cliff is 60°. Find the height of the cliff to the nearest metre. 4. The length of the shadow of a pillar is (\frac{1}{\sqrt{3}}) times the height of the pillar. Find the angle of elevation of the sun. 5. A ship is at a distance of 200 m from a tall tower. What is the angle of depression (to the nearest degree) of the ship found by a man after climbing 50 m up the tower? 6. The top of a tall vertical palm tree having been broken by the wind struck the ground at an angle of 60° at a distance of 9 m from the foot of the tree. Find the original height of the palm tree. 7. A 10-m-height pole is kept vertical by a steel wire. The wire is inclined at an angle of 40° with the horizontal ground. If the wire runs from the top of the pole to the point on the ground where its other end is fixed, find the length of the wire. 8. A tower is 64 m tall. A man standing erect at a distance of 36 m from the tower observes the angle of elevation of the top of the tower to be 60°. Find the height of the man. 9. From the top of a tall building of height 24 m, the angle of depression of the top of another building is 45° whose height is 10 m. Find the distance between the two buildings. 10. A tower stands by the side of a river at P. On the other side of the river, Q is a point on the bank such that PQ is the width of the river. R is the point on the bank of Q such that P, Q and R are in the same straight line. If QR = 5 metre and angles of elevation of top of the tower from Q area R are 60° and 45° respectively, find the width of the river and the height of the tower. 11. The angles of depression of two boats on a river from the top of a pole 30 metres high on the bank of the river are 60° and 75°. If the boats are in line with the pole, find the distance between the boats to the nearest metre. 12. A man standing on a cliff observes a ship at an angle of depression 30°, approaching the shore just beneath him. Three minutes later, the angle of depression of the ship is 60°. How soon will it reach the shore? 13. A man on the bank of a stream of observes a tree on the opposite bank exactly across the stream. He finds the angle of elevation of the top of the tree to be 45°. On receding perpendicularly a distance of 4 metre from the bank, he finds that the angle of elevation reduces by 15°. Is this information sufficient for the man to determine the height of the tree and the width of the stream? If so find them. 14. From the top of a light house the angles of depression of two ships on opposite sides of the light house were observed to be 60° and 45°. If the height of the light house is 100 m and the foot of the light house is in line with the ships, find the distance between the two ships. 15. From the top of a tower 40 m tall the angle of depression of the nearer of the two points P and Q on the ground on diametrically opposite sides of the tower is 45°. Find the angle of depression of the other point to the nearest degree if the distances of the two points from the base of the tower are in the ratio 1 : 2. 16. In the figure MN is a tower X and Y are two places on the ground on the either side of the tower such that XY subtends a right angle at M. If the distances of X and Y from the base N of the tower are 40 m and 90 m respectively. Find the height of the tower. 17. The angle of elevation of the top of an unfinished tower from a place at a distance of 50 m from the tower is 44° 40’. To what further height the unfinished tower should me raised so that the angle of elevation of the top of the tower from the same place would become 59° 30’? 18. A flagstaff, 5 m tall, stands on a vertical pole. The angles of elevation of the top and the bottom of the flagstaff from a point on the ground are found to be 60° and 30° respectively. Find the height of the pole. 19. A vertical pole fixed to the ground is divided into two parts by a mark on it. Each of the parts subtends an angle 30° at a place on the ground. (i) Find the ratio of the two parts. (ii) If the place on the ground is 15 m away from the base of the pole, find the lengths of the two parts of the pole. 20. A flagstaff is fixed on the top of mound and the angles of elevation of the top and the bottom of the flagstaff are 60° and 30° respectively at a point on the ground. Show that the length of the flagstaff is twice the height of the mound. 21. A man P walking towards a building AB finds that the building disappears from his view when the angle of elevation of the top C of a wall is x°, where tan x° = 1/3. The wall is 1.8 m high, and the distance between the wall and the building is 3.6 m. Find the height of the building. 20. A flagstaff is fixed on the top of mound and the angles of elevation of the top and the bottom of the flagstaff are 60° and 30° respectively at a point on the ground. Show that the length of the flagstaff is twice the height of the mound. 21. A man P walking towards a building AB finds that the building disappears from his view when the angle of elevation of the top C of a wall is x°, where tan x° = 1/3. The wall is 1.8 m high, and the distance between the wall and the building is 3.6 m. Find the height of the building. 22. A vertical tower subtends a right angle at the top of a vertical flag on the ground, the height of the flag being 10 m .If the distance between the tower and the flag be 20 m, find the height of the tower. 23. A vertical pole on one side of a street subtends a right angle at the top of a lamp post exactly on the opposite side of the street. If the angle of elevation of the top of the lamp post from the base of the pole is 58° 30’ and the width of the street is 30 m, find the heights of the pole and the lamp post. 24. From the top of a hill 200 m height, the angles of depression of the top and the bottom of a pillar are 45° and 59° 36’respectively. Find the height of the pillar and its distance from the hill. 25. A bird is perched on the top of a tree 20 m high and its angle of elevation from a point on the ground is 45°. The bird flies off horizontally straight away from the observer and in 1 sec the angle of elevation of the bird reduces to 35°. Find the speed of the bird. 26. The angles of depression and elevation of top of 12 m high wall from the top and the bottom of a tree are 60° and 30° respectively. Find (i) the height of the tree, and (ii) the distance of the tree from the wall. 27. Two pillars of equal height stand on either side of a road which is 40 m wide. From a point on the road between the pillars, the angles of elevation of tops of the pillars are 30° and 60°. Find (i) the position of the point of the point on the road, and (ii) the height of each pillar. 28. A ladder rests against a house on one side of a street. The angle of elevation of the top of the ladder is 60°. The ladder is turned over to rest against a house. On the other side of the street and the elevation now becomes 42° 50’. If the ladder is 40 m long, find the breadth of the street. 29. The angle of elevation of a cloud from a point h metre above a lake is 30° and the angle of depression of its reflection is 45°. If the height of the cloud be 200 metres, Find h. 30. A house, 15 metres high, stands on one side of a park and from a point on the roof of the house, the angle of depression of the foot of a chimney is 30° and the angle of elevation of the top of the chimney from the foot of the house is 60°. What is the height of the chimney? What is the distance between the house and the chimney? 27. Two pillars of equal height stand on either side of a road which is 40 m wide. From a point on the road between the pillars, the angles of elevation of tops of the pillars are 30° and 60°. Find (i) the position of the point of the point on the road, and (ii) the height of each pillar. 28. A ladder rests against a house on one side of a street. The angle of elevation of the top of the ladder is 60°. The ladder is turned over to rest against a house. On the other side of the street and the elevation now becomes 42° 50’. If the ladder is 40 m long, find the breadth of the street. 29. The angle of elevation of a cloud from a point h metre above a lake is 30° and the angle of depression of its reflection is 45°. If the height of the cloud be 200 metres, Find h. 30. A house, 15 metres high, stands on one side of a park and from a point on the roof of the house, the angle of depression of the foot of a chimney is 30° and the angle of elevation of the top of the chimney from the foot of the house is 60°. What is the height of the chimney? What is the distance between the house and the chimney? Answers on worksheet on heights and distances are given below to check the exact answers of the questions. Answers: 1. (i) 3 metres. (ii) 2.6 metres. 2. 92 metres 3. 26 metres 4. 60° 5. 14° 6. 33.6 metres. 7. 15.6 metres. 8. 1.65 metres. 9. 14 metres. 10. 6.83 metres, 11.83 metres respectively. 11. 9 metres. 12. 4½ minutes after the first observation. 13. Yes; Each = 5.46 metres. 14. 157.74 metres. 15. 27° 16. 60 metres. 17. 35.47 metres. 18. 2.5 metres. 19. (i) Lower part : Upper part = 1 : 2 (ii) Lower part = 8.66 metres, upper part = 17.32 metres. 21. 3 metres. 22. 50 metres. 23. 67.34 metres, 48.96 metres respectively. 24. 82.2 metres, 117.8 metres. 25. 8.56 m/sec. 26. (i) 48 metres. (ii) 20.78 metres. 27. (i) 10 metres and 30 metres from the pillars (two positions) (ii) 17.32 metres. 28. 49.33 metres. 29. 53.6 metres. 30. 45 metres, 15√3 metres 1. (i) 3 metres. (ii) 2.6 metres. 2. 92 metres 3. 26 metres 4. 60° 5. 14° 6. 33.6 metres. 7. 15.6 metres. 8. 1.65 metres. 9. 14 metres. 10. 6.83 metres, 11.83 metres respectively. 11. 9 metres. 12. 4½ minutes after the first observation. 13. Yes; Each = 5.46 metres. 14. 157.74 metres. 15. 27° 16. 60 metres. 17. 35.47 metres. 18. 2.5 metres. 19. (i) Lower part : Upper part = 1 : 2 (ii) Lower part = 8.66 metres, upper part = 17.32 metres. 21. 3 metres. 22. 50 metres. 23. 67.34 metres, 48.96 metres respectively. 24. 82.2 metres, 117.8 metres. 25. 8.56 m/sec. 26. (i) 48 metres. (ii) 20.78 metres. 27. (i) 10 metres and 30 metres from the pillars (two positions) (ii) 17.32 metres. 28. 49.33 metres. 29. 53.6 metres. 30. 45 metres, 15√3 metres 10th Grade Math FromWorksheet on Heights and Distancesto HOME PAGE 10th Grade Math FromWorksheet on Heights and Distancesto HOME PAGE Didn't find what you were looking for? Or want to know more information about Math Only Math. Use this Google Search to find what you need. New! Comments Share this page: What’s this? | | | --- | | E-mail Address | | | | | | First Name | | | | | | Then Don't worry — your e-mail address is totally secure. I promise to use it only to send you Math Only Math. | | © and ™ math-only-math.com. All Rights Reserved. 2010 - 2025.
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https://www.unh.edu/halelab/ANFS933/Readings/Topic9_Reading1.pdf
1 Topic 9: Factorial treatment structures Introduction A common objective in research is to investigate the effect of each of a number of variables, or factors, on some response variable. In earlier times, factors were studied one at a time, with separate experiments devoted to each one. But RA Fisher pointed out that important advantages are gained by combining the study of several factors in the same experiment. In a factorial experiment, the treatment structure consists of all possible combinations of all levels of all factors under investigation. Factorial experimentation is highly efficient because each experimental unit provides information about all the factors in the experiment. Factorial experiments also provide a systematic method of investigating the relationships among the effects of different factors (i.e. interactions). Terminology The different classes of treatments in an experiment are called factors (e.g. Fertilization, Medication, etc.). The different categories within each factor are called levels (e.g. 0, 20, and 40 lbs N/acre; 0, 1, and 2 doses of an experimental drug, etc.). We will denote different factors by upper case letters (A, B, C, etc.) and different levels by lower case letters with subscripts (a1, a2, etc.). The mean of experimental units receiving the treatment combination aibi will be denoted "Mean(aibi)". We will refer to a factorial experiment with two factors and two levels for each factor as a 2x2 factorial experiment. An experiment with 3 levels of Factor A, 4 levels of Factor B, and 2 levels of Factor C will be referred to as a 3x4x2 factorial experiment. Etc. Example of a 2x2 factorial An example of a CRD involving two factors: Nitrogen levels (N0 and N1) and phosphorous levels (P0 and P1), applied to a crop. The response variable is yield (lbs/acre). The data: Factor A = N level Level a1 = N0 a2 = N1 Mean (abi) a2-a1 B = P level b1 = P0 40.9 47.8 44.4 6.9 (se A,b1) b2 = P1 42.4 50.2 46.3 7.8 (se A,b2) Mean (aib) 41.6 49 45.3 7.4 (me A) b2-b1 1.5 (se B,a1) 2.4 (se B,a2) 1.9 (me B) The differences a2 - a1 (at each leavel of B) and b2 - b1 (at each level of A) are called the simple effects of a and b, respectively, denoted (se A) and (se B). The averages of the simple effects are the main effects of a and b, respectively, denoted (me A) and (me B). 2 One way of using these data is to consider the effect of N on yield at each P level separately. This information could be useful to a grower who is constrained to use one or the other P level. This is called analyzing the simple effects (se) of N. The simple effects of applying nitrogen are to increase yield by 6.9 lb/acre for P0 and 7.8 lb/acre for P1. It is possible that the effect of N on yield is the same whether or not P is applied. In this case, the two simple effects estimate the same quantity and differ only due to experimental error. One is then justified in averaging the two simple effects to obtain a mean yield response of 7.4 lb/acre. This is called the main effect (me) of N on yield. If the effect of P is independent of N level, then one could do the same thing for this factor and obtain a main effect of P on yield response of 1.9 lb/acre. Interaction If the simple effects of Factor A are the same across all levels of Factor B, the two factors are said to be independent. In such cases, it is appropriate to analyze the main effects of each factor. It may, however, be the case that the effects are not independent. For example, one might expect the application of P to permit a higher expression of the yield potential of the N application. In that case, the effect of N in the presence of P would be much larger than the effect of N in the absence of P. When the effect of one factor depends on the level of another factor, the two factors are said to exhibit an interaction. An interaction is a measure of the difference in the effect of one factor at the different levels of another factor. Interaction is a common and fundamental scientific idea. One of the primary objectives of factorial experiments, other than efficiency, is to study the interactions among factors. The sum of squares of an interaction measures the departure of the group means from the values expected on the basis of purely additive effects. In common biological terminology, a large positive deviation of this sort is called synergism. When drugs act synergistically, the result of the interaction of the two drugs may be above and beyond the simple addition of the separate effects of each drug. When the combination of levels of two factors inhibit each other’s effects, we call it interference. Both synergism and interference increase the interaction SS. 3 These differences between the simple effects of two factors, also known as first-order interactions or two-way interactions, can be visualized in the following interaction plots: In interaction plots, perfect additivity (i.e. no interaction) is indicated by perfectly parallel lines. Significant departures from parallel indicate significant interactions. a1 a2 se A,b1 b1 b2 Y se B,a1 a. High me B, no interaction a1 a2 b1 b2 Y b. Low me B, no interaction a1 a2 b1 b2 Y c. An interaction may be a difference in magnitude of the response a1 a2 b1 b2 Y d. It may also be a difference in direction of response e. Synergism a1 a2 b1 b2 Y f. Interference a1 a2 b1 b2 Y 4 Reasons for carrying out factorial experiments 1. To investigate interactions: If factors are not independent, single factor experiments provide a disorderly, incomplete, and often quite misleading picture of the system. More than this, most of the interesting questions today concern interactions. 2. To establish the dependence or independence of factors of interest: In the initial phases of an investigation, pilot or exploratory factorial experiments can establish which factors are independent and can therefore be justifiably analyzed in separate experiments. 3. To offer recommendations that must apply over a wide range of conditions: One can introduce "subsidiary factors" (e.g. soil type) into an experiment to ensure that any recommended results apply across a necessary range of circumstances. Some disadvantages of factorial experiments 1. The total possible number of treatment level combinations increases rapidly as the number of factors increases. For example, to investigate 7 factors (3 levels each) in a factorial experiment requires, at minimum, 2187 experimental units. 2. Higher order interactions (three-way, four-way, etc.) are very difficult to interpret. So a large number of factors complicates the interpretation of results. Differences between nested and factorial experiments (Biometry 322-323) Looking at data in a table, it is easy to get confused between nested and factorial experiments. Consider a factorial experiment in which leaf discs are grown in 10 different tissue culture media (all possible combinations of 5 different types of sugars and 2 different pH levels). In what way does this differ from a nested design in which each sugar solution is prepared twice, so there are two batches of sugar for each treatment? The following tables represent both designs, using asterisks to represent measurements of the response variable (leaf growth). 2x5 factorial experiment Nested experiment Sugar Type Sugar Type 1 2 3 4 5 1 2 3 4 5 pH1 Batch 1 pH2 Batch 2 The data tables look very similar, so what's the difference here? The factorial analysis implies that the two pH classes are common across the entire study (i.e. pH level 1 is a specific pH level that is the same across all sugar treatments). By analogy, if you were to analyze the nested experiment as a two-way factorial ANOVA, it would imply that Batches are common across the 5 entire study. But this is not so. Batch 1 for Treatment 1 has no closer relation to Batch 1 for Treatment 2 than it does to Batch 2 for Treatment 2. "Batch" is an ID, and Batches 1 and 2 are simply arbitrary designations for two randomly prepared sugar solutions for each treatment. Now, if all batches labeled 1 were prepared by the same technician on the same day, while all batches labeled 2 were made by someone else on another day, then “1” and “2” would represent meaningfully common classes across the study. In this case, the experiment could properly be analyzed using a two–way ANOVA with Technicians/Days as blocks (RCBD). While they both require two-way ANOVAs, RCBD's differ from two-way factorial treatment structures in their objective. In this example, we are not interested in the effect of the batches or in the interaction between batches and sugar types. Our main interest is to control for this additional source of variation so that we can better detect the differences among treatments; toward this end, we assume there to be no interactions. When presented with an experimental description and its accompanying dataset, the critical question to be asked to differentiate factors from experimental units or subsamples is this: Do the classes in question have a consistent meaning across the experiment, or are they simply ID's? Notice that ID (or dummy) classes can be swapped without affecting the analysis (switching the names of "Batch 1" and "Batch 2" within any given Sugar Type has no consequences) whereas factor classes cannot (switching "pH1" and "pH2" within any given Sugar Type will completely muddle the analysis). The two-way factorial analysis (Model I ANOVA) The linear model The linear model for a two-way factorial analysis is Yijk = µ + tAi + tBj + (tAtB)ij + eijk Here tAi represents the main effect of factor A (i = 1,...,a), tBj represents the main effect of factor B, (j = 1,...,b), (tAtB)ij represents the interaction of factor A level i with factor B level j, and eijk is the error associated with replication k of the factor combination ij (k = 1,..,r). In dot notation: main effect main effect interaction experimental factor A factor B effect (A:B) error The null hypotheses for a two-factor experiment are tAi = 0, tBj = 0, and (tAtB)ij = 0. The F statistics for each of these hypotheses may be interpreted independently due to the orthogonality of their respective sums of squares. TSS = SSA + SSB + SSAB + SSE ) ( ) ( ) ( ) ( . ... . . .. . ... . . ... .. ... ij ijk j i ij j i ijk Y Y Y Y Y Y Y Y Y Y Y Y -+ + --+ -+ -+ = 6 The ANOVA In the ANOVA for two-way factorial experiments, the Treatment SS is partitioned into three orthogonal components: a SS for each factor and an interaction SS. This partitioning is valid even when the overall F test among treatments is not significant. Indeed, there are situations where one factor, say B, has no effect on the response variable and hence contributes no more to the SST than one would expect by chance along. In such a circumstance, a significant response to A might well be lost in an overall test of significance. In a factorial experiment, the overall SST is rightly understood to be an intermediate computational quantity rather than an end product (i.e. a numerator for an F test). In a two factor experiment (A x B), there are a total of ab treatment combinations and therefore (ab – 1) treatment degrees of freedom. The main effect of factor A has (a – 1) df and the main effect of factor B has (b – 1) df. The interaction (AxB) has (a – 1)(b – 1) df. With r replications per treatment combination, there are a total of (rab) experimental units in the study and, therefore, (rab – 1) total degrees of freedom. General ANOVA table for a two-way CRD factorial experiment: Source df SS MS F Factor A a - 1 SSA MSA MSA/MSE Factor B b - 1 SSB MSB MSB/MSE AxB (a - 1)(b - 1) SSAB MSAB MSAB/MSE Error ab(r - 1) SSE MSE Total rab - 1 TSS The interaction SS is the variation due to the departures of group means from the values expected on the basis of additive combinations of the two factors' main effects. The significance of the interaction F test determines what kind of subsequent analysis is appropriate: No significant interaction: Subsequent analysis (mean comparisons, contrasts, etc.) are performed on the main effects (i.e. one may compare the means of one factor across all levels of the other factor). Significant interaction: Subsequent analysis (mean comparisons, contrasts, etc.) are performed on the simple effects (i.e. one must compare the means of one factor separately for each level of the other factor). 7 Relationship between factorial experiments and experimental design Experimental designs are characterized by the method of randomization: how were the treatments assigned to the experimental units? In contrast, factorial experiments are characterized by a certain treatment structure, with no requirements on how the treatments are randomly assigned to experimental units. A factorial treatment structure may occur within any experimental design. Example of a 4 x 2 factorial experiment within three different experimental designs: Since Factor A has 4 levels (1, 2, 3, 4) and Factor B has 2 levels (1, 2), there are eight different treatment combinations: (11, 12, 13, 14, 21, 22, 23, 24). CRD with 3 replications 24 23 13 23 24 14 13 23 11 24 12 14 22 13 12 21 21 11 22 12 11 22 21 14 RCBD with 3 blocks 13 12 21 23 11 24 14 22 12 11 24 23 13 22 21 14 24 14 22 21 11 13 23 12 8 x 8 Latin Square 24 11 22 12 13 14 23 21 21 23 13 14 22 12 11 24 12 14 24 11 23 21 22 13 13 22 21 24 11 23 14 12 23 12 11 13 21 22 24 14 14 24 23 22 12 13 21 11 11 21 12 23 14 24 13 22 22 13 14 21 24 11 12 23 8 Example of a 2 x 3 factorial experiment within an RCBD with no significant interactions (ST&D 391) Data: The number of quack-grass shoots per square foot after spraying with maleic hydrazide. Treatments are maleic hydrazide applications rates (R: 0, 4, and 8 lbs/acre) and delay in cultivation after spraying (D: 3 and 10 days). D R Block 1 Block 2 Block 3 Block 4 Means 3 0 15.7 14.6 16.5 14.7 15.38 4 9.8 14.6 11.9 12.4 12.18 8 7.9 10.3 9.7 9.6 9.38 10 0 18.0 17.4 15.1 14.4 16.23 4 13.6 10.6 11.8 13.3 12.33 8 8.8 8.2 11.3 11.2 9.88 Means 12.30 12.62 12.72 12.60 12.56 The R Code #The ANOVA quack_mod<-lm(Number ~ D + R + D:R + Block, quack_dat) ß WHAT'S MISSING?? anova(quack_mod) Note: If there were only 1 replication per D-R combination (i.e. only 1 block), you could not include the D:R interaction in the model. There would not be enough error df. The output Analysis of Variance Table Df Sum Sq Mean Sq F value Pr(>F) D 1 1.500 1.500 0.5713 0.4614 R 2 153.663 76.832 29.2630 6.643e-06 Block 3 0.582 0.194 0.0738 0.9731 D:R 2 0.490 0.245 0.0933 0.9114 NS Residuals 15 39.383 2.626 Note that the 15 error df = Block:D (3 df) + Block:R (6 df) + Block:D:R (6 df) 9 Essentially parallel lines in an interaction plot, as those observed in this case, indicate the absence of an interaction. The lines of this plot are essentially parallel because the difference between D levels is roughly the same at all R levels. This non-interaction can be seen from the perspective of either factor: Here, the lines are essentially parallel because the difference between R levels is approximately the same at all levels of D. Interactions of R and D D Number 10 12 14 16 3 10 R 0 4 8 Interactions of D and R R Number 10 12 14 16 0 4 8 D 3 10 10 If no interaction is present, you proceed by analyzing the main effects. If there is no significant interaction, you are justified in analyzing the effect of R without regard for the level D because the effect of R does not depend on the level of D, and vice versa. Detailed comparisons of the mean effects can be performed using contrasts or an appropriate multiple comparison test. Representative analyses: #Analyzing the main effects library(agricolae) HSD.test(quack_mod, "D") HSD.test(quack_mod, "R") Tukey means separations D, means alpha: 0.05 ; Df Error: 15 Critical Value of Studentized Range: 3.014325 Honestly Significant Difference: 1.409971 Means with the same letter are not significantly different. Groups, Treatments and means a 10 12.81 a 3 12.31 R, means alpha: 0.05 ; Df Error: 15 Critical Value of Studentized Range: 3.673378 Honestly Significant Difference: 2.104414 Means with the same letter are not significantly different. Groups, Treatments and means a 0 15.8 b 4 12.25 c 8 9.625 #Performing a trend analysis on the factor R # Contrast ‘Linear’ -1,0,1 # Contrast ‘Quadratic’ 1,-2,1 contrastmatrix<-cbind(c(-1,0,1),c(1,-2,1)) contrasts(quack_dat$R)<-contrastmatrix quack_Rcontrast_mod<-aov(Number ~ D + R + D:R + Block, quack_dat) summary(quack_Rcontrast_mod, split = list(R = list("Linear" = 1, "Quadratic" = 2))) 11 Contrasts (trend analysis of R) Df Sum Sq Mean Sq F value Pr(>F) D 1 1.50 1.50 0.571 0.461 R 2 153.66 76.83 29.263 6.64e-06 R: Linear 1 152.52 152.52 58.092 1.56e-06 R: Quadratic 1 1.14 1.14 0.435 0.520 Block 3 0.58 0.19 0.074 0.973 D:R 2 0.49 0.25 0.093 0.911 D:R: Linear 1 0.12 0.12 0.047 0.832 D:R: Quadratic 1 0.37 0.37 0.140 0.714 Residuals 15 39.38 2.63 9.7.4.2. Partitioning the Interaction Sum of Squares It is possible to find significant interaction components within an overall non-significant interaction! In Topic 4, we discussed how it is possible to find a significant 1 df contrast despite an overall non-significant treatment F test. The concept here is similar. When you divide the Interaction SS by the Interaction df to determine the Interaction MS, you are cutting that SS into equal parts. But it is possible that one component of the interaction (e.g. D:R Linear) is bigger than another (e.g. D:R quadratic), and that that part is significant. Look again at the contrast output above. The trend analysis using orthogonal contrasts partitioned not only the SS for the factor R but also the SS of the interaction D:R. In this way, R makes partitioning the Interaction SS very easy. This can be done another way as well, by "opening up" the factorial treatment structure, as described below: To manually partition the D:R interaction (2 df), you first need to create a variable, say "TRT," whose values are the full set of factorial combinations of D and R levels. The values of TRT for this example would be: D3 R0 = TRT 1 D10 R0 = TRT 4 D3 R4 = TRT 2 D10 R4 = TRT 5 D3 R8 = TRT 3 D10 R8 = TRT 6 Now we are back in familiar territory. We have "opened up" the factorial treatment structure, redefining it as a simple one-way classification. Now we can simply analyze TRT and use contrasts to partition the interaction, as you've seen before. 12 Modifying the original data table: D R TRT Block Number 3 0 1 1 15.7 3 0 1 2 14.6 3 0 1 3 16.5 ... ... ... ... ... 10 8 6 2 8.2 10 8 6 3 11.3 10 8 6 4 11.2 Representative code for this approach: #Performing a trend analysis on the factor TRT # TRT Levels: 1 2 3 4 5 6 # Contrast ‘R Linear’ -1 0 1 -1 0 1 # Contrast ‘R Quadratic’ 1 -2 1 1 -2 1 # Contrast ‘D’ 1 1 1 -1 -1 -1 # Contrast ‘R Lin D’ -1 0 1 1 0 -1 # Contrast ‘R Quad D’ 1 -2 1 -1 2 -1 contrastmatrix<-cbind(c(-1,0,1,-1,0,1),c(1,-2,1,1,-2,1), c(1,1,1,-1,-1,-1),c(-1,0,1,1,0,-1),c(1,-2,1,-1,2,-1)) contrasts(quack_dat$TRT)<-contrastmatrix quack_Rcontrast_mod<-aov(Number ~ TRT + Block, quack_dat) summary(quack_Rcontrast_mod, split = list(TRT = list("Lin R" = 1, "Quad R" = 2, "D" = 3, "Lin R D" = 4, "Quad R : D" = 5))) The output: Df Sum Sq Mean Sq F value Pr(>F) TRT 5 155.65 31.13 11.857 8.92e-05 TRT: Lin R 1 152.52 152.52 58.092 1.56e-06 TRT: Quad R 1 1.14 1.14 0.435 0.520 TRT: D 1 1.50 1.50 0.571 0.461 TRT: Lin R : D 1 0.12 0.12 0.047 0.832 TRT: Quad R : D 1 0.37 0.37 0.140 0.714 Block 3 0.58 0.19 0.074 0.973 Residuals 15 39.38 2.63 Here we have successfully partitioned the Treatment SS into its five single-df components, two of which are interaction components. Compare this output to that of the previous factorial analysis: 13 Df Sum Sq Mean Sq F value Pr(>F) D 1 1.50 1.50 0.571 0.461 R 2 153.66 76.83 29.263 6.64e-06 R: Linear 1 152.52 152.52 58.092 1.56e-06 R: Quadratic 1 1.14 1.14 0.435 0.520 Block 3 0.58 0.19 0.074 0.973 D:R 2 0.49 0.25 0.093 0.911 D:R: Linear 1 0.12 0.12 0.047 0.832 D:R: Quadratic 1 0.37 0.37 0.140 0.714 Residuals 15 39.38 2.63 By "opening up" the factorial treatment structure, we have successfully partitioned the SS of R into its two single-df components and the SS of the D:R interaction into its two single-df components. In this case, no significant interaction components were found "hiding" inside the overall non-significant interaction. Is it worth partitioning the Interaction SS? To answer this, divide it by 1 and test for significance. If that is not significant, it is not worth partitioning the Interaction SS because no significance is found even when all the variation is assigned to one component (1 df) of the interaction.
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https://www.youtube.com/watch?v=fCOHUBTxwfE
ACT Math Practice Test 2.48: Reciprocals Euler's Academy 9400 subscribers 1 likes Description 114 views Posted: 1 Nov 2017 Check out my new website: www.EulersAcademy.org This video focuses on the forty eighth problem from the second ACT practice test. This problem looks at the basic properties of reciprocals and asks us to determine what range the reciprocal of a number would fall into if the original number is greater than 1. You can find a copy of this practice test here: The math section starts at page 24 of the pdf. Here is a link for more practice tests: 2 comments Transcript: for xlviii problem on the second a CT practice test two numbers are reciprocals if their product is equal to one if x and y are reciprocals and X is bigger than 1 then y must be what so let's first look at the concept of a reciprocal and that might be easiest if we first just look at a few different examples so let's look at reciprocal examples so let's first just pick a number let's say we have the number 3 then the reciprocal of 3 what we're essentially going to do is first let's turn it into a fraction we can make it 3 over 1 and by finding its reciprocal all we're essentially going to do is just flip this fraction upside down so the reciprocal of 3 would be 1/3 and let's say we pick some other number like 5 halves and we want to find this reciprocal so again all you're really going to do is just flip this number upside down so the reciprocal of 5 halves would be 2/5 and as one final example let's say we have the fraction 3/7 then this reciprocal what we would do is again just flip it upside down so that we get 7/3 so finding the reciprocal of a number all you're essentially doing is just flipping that number upside down and if it's a whole number to start like 8 all you have to do is divide it by 1 to turn it into a fraction and then to find its reciprocal flip it upside down to make it 1/8 so with this in mind now let's go back to our problem we're told that X and y are reciprocals and the definition of a reciprocal is that their product is equal to 1 so if you were to go back to any of these four examples here and you multiply the number by its reciprocal what you'll see is you always give back the number one so for instance five halves times two fifths if we were to actually multiply these together you get ten in the numerator and ten in the denominator and ten divided by ten is one so this would actually happen with all of these examples or in general with any number and it's reciprocal so if x and y are reciprocals so we know that x times y is equal to one since they're reciprocals and X is bigger than one so we have to figure out what Y would be and if you just look at our examples where in this case this case in this case we started with a number that was bigger than one and then looking at its reciprocal we can see that it's always a number less than one but it's still a number greater than zero like when you take a number's reciprocal that started as a positive number you never get back a negative reciprocal because all you're doing is flipping the fraction so if it didn't have a negative sign to start then it wouldn't have one to finish and for the only number we started with that was less than one and between one and zero when we took its reciprocal we got a number greater than one so you can conclude just looking at these examples that our Y value is going to be between zero and one but another way to prove that is to just solve this equation for y so Y would be equal to one divided by x if we divide each side of this equation by X and since X is bigger than one we're going to have a fraction where we have a numerator of one in a denominator that's bigger than one so that means that this number here can never be greater than one since our denominator is always bigger than our numerator so we can conclude that choice letter J would be the final answer since why would always be between zero and one
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https://pubmed.ncbi.nlm.nih.gov/32940400/
Detection of multinucleated giant cells in differentiated keratinocytes with herpes simplex virus and varicella zoster virus infections by modified Tzanck smear method - PubMed Clipboard, Search History, and several other advanced features are temporarily unavailable. Skip to main page content An official website of the United States government Here's how you know The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site. The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely. 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Detection of multinucleated giant cells in differentiated keratinocytes with herpes simplex virus and varicella zoster virus infections by modified Tzanck smear method Takenobu Yamamoto12,Yumi Aoyama1 Affiliations Expand Affiliations 1 Department of Dermatology, Kawasaki Medical School, Kurashiki, Japan. 2 Department of Dermatology, Kawasaki Medical School General Medical Center, Okayama, Japan. PMID: 32940400 DOI: 10.1111/1346-8138.15619 Item in Clipboard Detection of multinucleated giant cells in differentiated keratinocytes with herpes simplex virus and varicella zoster virus infections by modified Tzanck smear method Takenobu Yamamoto et al. J Dermatol.2021 Jan. Show details Display options Display options Format J Dermatol Actions Search in PubMed Search in NLM Catalog Add to Search . 2021 Jan;48(1):21-27. doi: 10.1111/1346-8138.15619. Epub 2020 Sep 17. Authors Takenobu Yamamoto12,Yumi Aoyama1 Affiliations 1 Department of Dermatology, Kawasaki Medical School, Kurashiki, Japan. 2 Department of Dermatology, Kawasaki Medical School General Medical Center, Okayama, Japan. PMID: 32940400 DOI: 10.1111/1346-8138.15619 Item in Clipboard Cite Display options Display options Format Abstract Herpes simplex virus (HSV) and varicella zoster virus (VZV) infections induce the formation of intraepidermal vesicles containing acantholytic cells and multinucleated giant cells in the skin. The Tzanck smear is most commonly used to diagnose cutaneous herpetic infections, but it leads to many false-positive and -negative results. This study aimed at establishing a method detecting much larger multinucleated giant cells using the Tzanck smear because these cells characterize the viral cytopathic effect in skin infections. Morphological changes were analyzed among several layers of keratinocytes with HSV- or VZV-related cutaneous lesions, clinically and in vitro. We compared the sensitivity of the Tzanck smear to detect large acantholytic cells using both the removed roof tissue part (our approach) and the floor of the lesion (conventional approach) of a fresh vesicle. Large acantholytic cells were detected 2.0-times more frequently in the removed roof tissue part of the vesicle than in the floor of the lesion. Round cells were much larger in the removed roof tissue part of the vesicle corresponding to the granular or prickle layer of the epidermis than in its floor of the lesion corresponding to the basal or prickle layer with the Tzanck smear. Differentiated cultured keratinocytes formed multinucleated giant cells by cell-to-cell fusion with resolution of cell membrane with VZV infection. Differentiated keratinocytes promote multinucleated giant cell formation by cell-to-cell fusion with HSV-1 or VZV infection. To increase the sensitivity, the Tzanck smear should be prepared from the removed roof tissue part of a fresh vesicle to detect multinucleated giant cells in herpetic infections. Keywords: Tzanck smear; acantholytic cell; herpes simplex virus; multinucleated giant cell; varicella zoster virus. © 2020 Japanese Dermatological Association. PubMed Disclaimer Similar articles Comparison of Tzanck smear, viral culture, and DNA diagnostic methods in detection of herpes simplex and varicella-zoster infection.Nahass GT, Goldstein BA, Zhu WY, Serfling U, Penneys NS, Leonardi CL.Nahass GT, et al.JAMA. 1992 Nov 11;268(18):2541-4.JAMA. 1992.PMID: 1328700 Clinical Trial. Detection of herpes simplex and varicella-zoster infection from cutaneous lesions in different clinical stages with the polymerase chain reaction.Nahass GT, Mandel MJ, Cook S, Fan W, Leonardi CL.Nahass GT, et al.J Am Acad Dermatol. 1995 May;32(5 Pt 1):730-3. doi: 10.1016/0190-9622(95)91450-1.J Am Acad Dermatol. 1995.PMID: 7722016 Genital Herpes Zoster as Possible Indicator of HIV Infection.Ljubojević Hadžavdić S, Kovačević M, Skerlev M, Zekan Š.Ljubojević Hadžavdić S, et al.Acta Dermatovenerol Croat. 2018 Dec;26(4):337-338.Acta Dermatovenerol Croat. 2018.PMID: 30665486 Tests for detecting herpes simplex virus and varicella-zoster virus infections.Cohen PR.Cohen PR.Dermatol Clin. 1994 Jan;12(1):51-68.Dermatol Clin. 1994.PMID: 8143385 Review. Epidermal multinucleated giant cells are not always a histopathologic clue to a herpes virus infection: multinucleated epithelial giant cells in the epidermis of lesional skin biopsies from patients with acantholytic dermatoses can histologically mimic a herpes virus infection.Cohen PR, Paravar T, Lee RA.Cohen PR, et al.Dermatol Pract Concept. 2014 Oct 31;4(4):21-7. doi: 10.5826/dpc.0404a03. eCollection 2014 Oct.Dermatol Pract Concept. 2014.PMID: 25396080 Free PMC article.Review. See all similar articles Cited by Varicella challenges: A case of respiratory tract complications in an elderly patient.Rahmi AM, Prakasita KA, Damayanti D.Rahmi AM, et al.Narra J. 2024 Dec;4(3):e1150. doi: 10.52225/narra.v4i3.1150. Epub 2024 Nov 1.Narra J. 2024.PMID: 39816110 Free PMC article. Morphological and tissue-based molecular characterization of oral lesions in patients with COVID-19: A living systematic review.Silveira FM, Mello ALR, da Silva Fonseca L, Dos Santos Ferreira L, Kirschnick LB, Martins MD, Schuch LF, de Arruda JAA, Soares CD, de Oliveira Sales A, Bologna-Molina R, Santos-Silva AR, Vasconcelos ACU.Silveira FM, et al.Arch Oral Biol. 2022 Apr;136:105374. doi: 10.1016/j.archoralbio.2022.105374. Epub 2022 Feb 12.Arch Oral Biol. 2022.PMID: 35180550 Free PMC article. Visual clues - dermatological manifestations of sexually transmitted infections in men.Mass Lindenbaum M, Calderón D, Aslot V, Ljubetic B, Harlamova D, Bole R, Bajic P, Navarrete J.Mass Lindenbaum M, et al.Nat Rev Urol. 2025 Aug 13. doi: 10.1038/s41585-025-01071-1. Online ahead of print.Nat Rev Urol. 2025.PMID: 40804205 Review. Detection of Circulating VZV-Glycoprotein E-Specific Antibodies by Chemiluminescent Immunoassay (CLIA) for Varicella-Zoster Diagnosis.Kombe Kombe AJ, Xie J, Zahid A, Ma H, Xu G, Deng Y, Nsole Biteghe FA, Mohammed A, Dan Z, Yang Y, Feng C, Zeng W, Chang R, Zhu K, Zhang S, Jin T.Kombe Kombe AJ, et al.Pathogens. 2022 Jan 5;11(1):66. doi: 10.3390/pathogens11010066.Pathogens. 2022.PMID: 35056014 Free PMC article. Molecular Mechanisms of Cell-to-Cell Transmission in Human Herpesviruses.Yan L, Guo J, Zhong Y, Wei J, Wang Z.Yan L, et al.Viruses. 2025 May 22;17(6):742. doi: 10.3390/v17060742.Viruses. 2025.PMID: 40573333 Free PMC article.Review. See all "Cited by" articles References REFERENCES Durdu M, Baba M, Seçkin D. The value of Tzanck smear test in diagnosis of erosive, vesicular, bullous, and pustular skin lesions. J Am Acad Dermatol 2008; 59: 958-964. Eryılmaz A, Durdu M, Baba M, Yıldırım FE. Diagnostic reliability of the Tzanck smear in dermatologic diseases. Int J Dermatol 2014; 53: 178-186. Calonje E. Histopathology of the skin: General principles. In: Burns T, Breathnach S, Neil C, Griffiths C (eds). Rook's Textbook of Dermatology, Vol 1, 8th edn. UK: Wiley-Blackwell, 2010; 10.28-10.29. Stewart MI, Bernhard JD, Cropley TG, Fitzpatrick TB. Clinical-pathologic correlations of skin lesions: Approach to diagnosis. In: Freedberg IM, Eisen AZ, Wolff K, Austen KF, Goldsmith LA, Katz SI, eds. Fitzpatrick's Dermatology in General Medicine, 6th edn, Vol. 1. New York: MGCraw-Hill, 2003; 11-30. Ruocco V, Ruocco E. Tzanck smear, an old test for the new millennium: when and how. Int J Dermatol 1999; 38: 830-834. Show all 18 references MeSH terms Giant Cells Actions Search in PubMed Search in MeSH Add to Search Herpes Simplex / diagnosis Actions Search in PubMed Search in MeSH Add to Search Herpes Zoster / diagnosis Actions Search in PubMed Search in MeSH Add to Search Herpesvirus 3, Human Actions Search in PubMed Search in MeSH Add to Search Humans Actions Search in PubMed Search in MeSH Add to Search Keratinocytes Actions Search in PubMed Search in MeSH Add to Search Simplexvirus Actions Search in PubMed Search in MeSH Add to Search Related information MedGen Grants and funding 19K08761/Japan Society for the Promotion of Science R01-058/Kawasaki Medical School R02-052/Kawasaki Medical School LinkOut - more resources Medical MedlinePlus Health Information [x] Cite Copy Download .nbib.nbib Format: Send To Clipboard Email Save My Bibliography Collections Citation Manager [x] NCBI Literature Resources MeSHPMCBookshelfDisclaimer The PubMed wordmark and PubMed logo are registered trademarks of the U.S. Department of Health and Human Services (HHS). 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14054
https://www.britannica.com/science/bacillus-bacteria
SUBSCRIBE Ask the Chatbot Games & Quizzes History & Society Science & Tech Biographies Animals & Nature Geography & Travel Arts & Culture ProCon Money Videos bacillus bacteria Print Also known as: Bacillus, bacilli Written by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Article History Related Topics: : Bacillus thuringiensis : Bacillus anthracis : Bacillus fusiformis : Bacillus cereus : Bacillus alvie See all related content bacillus, (genus Bacillus), any of a genus of rod-shaped, gram-positive, aerobic or (under some conditions) anaerobic bacteria widely found in soil and water. The term bacillus has been applied in a general sense to all cylindrical or rodlike bacteria. The largest known Bacillus species, B. megaterium, is about 1.5 μm (micrometres; 1 μm = 10−6 m) across by 4 μm long. Bacillus frequently occur in chains. In 1877 German botanist Ferdinand Cohn provided an authoritative description of two different forms of hay bacillus (now known as Bacillus subtilis): one that could be killed upon exposure to heat and one that was resistant to heat. He called the heat-resistant forms “spores” (endospores) and discovered that these dormant forms could be converted to a vegetative, or actively growing, state. Today it is known that all Bacillus species can form dormant spores under adverse environmental conditions. These endospores may remain viable for long periods of time. Endospores are resistant to heat, chemicals, and sunlight and are widely distributed in nature, primarily in soil, from which they invade dust particles. Some types of Bacillus bacteria are harmful to humans, plants, or other organisms. For example, B. cereus sometimes causes spoilage in canned foods and food poisoning of short duration. B. subtilis is a common contaminant of laboratory cultures (it plagued Louis Pasteur in many of his experiments) and is often found on human skin. Most strains of Bacillus are not pathogenic for humans but may, as soil organisms, infect humans incidentally. A notable exception is B. anthracis, which causes anthrax in humans and domestic animals. B. thuringiensis produces a toxin (Bt toxin) that causes disease in insects. Medically useful antibiotics are produced by B. subtilis (bacitracin). In addition, strains of B. amyloliquefaciens bacteria, which occur in association with certain plants, are known to synthesize several different antibiotic substances, including bacillaene, macrolactin, and difficidin. These substances serve to protect the host plant from infection by fungi or other bacteria and have been studied for their usefulness as biological pest-control agents. A gene encoding an enzyme known as barnase in B. amyloliquefaciens is of interest in the development of genetically modified (GM) plants. Barnase combined with another protein synthesized by B. amyloliquefaciens known as barstar, forming the barnase-barstar gene system, was used to develop a line of non-self-fertilizing transgenic mustard (Brassica juncea) plants with enhanced outbreeding capability. The gene controlling production of the Bt toxin in B. thuringiensis has been used in the development of GM crops such as Bt cotton (see genetically modified organism). The Editors of Encyclopaedia Britannica This article was most recently revised and updated by Kara Rogers. antibiotic resistance Introduction Mechanisms of resistance Prevention and drug development References & Edit History Related Topics Images & Videos Quizzes A Visit with the Word Doctor: Medical Vocabulary Quiz antibiotic resistance medicine Written by Douglas Morier Graduate student, University of California, Los Angeles. Douglas Morier Fact-checked by The Editors of Encyclopaedia Britannica Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree.... The Editors of Encyclopaedia Britannica Last Updated: • Article History Key People: : Lancelot Thomas Hogben Related Topics: : bacteria : antibiotic : totally drug-resistant tuberculosis : extremely drug-resistant tuberculosis : vancomycin-resistant Staphylococcus aureus See all related content antibiotic resistance, loss of susceptibility of bacteria to the killing (bacteriocidal) or growth-inhibiting (bacteriostatic) properties of an antibiotic agent. When a resistant strain of bacteria is the dominant strain in an infection, the infection may be untreatable and life-threatening. Examples of bacteria that are resistant to antibiotics include methicillin-resistant Staphylococcus aureus (MRSA), penicillin-resistant Enterococcus, and multidrug-resistant Mycobacterium tuberculosis (MDR-TB), which is resistant to two tuberculosis drugs, isoniazid and rifampicin. MDR-TB is particularly dangerous because it can give rise to extensively drug-resistant M. tuberculosis (XDR-TB), which requires aggressive treatment using a combination of five different drugs. The potential for antibiotic resistance was recognized in the early 1940s, almost immediately after the first large-scale clinical applications of penicillin, the first antibiotic. Mass production of penicillin was part of the greater war effort of World War II, when the drug was used widely by military populations and by some small civilian populations. Along with penicillin’s effectiveness in the treatment of the wounded, the drug was lauded for lowering the rate of venereal disease among military personnel, since it was particularly potent against the bacterial organisms notorious for causing syphilis and gonorrhea. However, even before the war had ended, resistance to penicillin was already reported—first in 1940 by British biochemists Sir Ernst Boris Chain and Sir Edward Penley Abraham, who published a report about an enzyme capable of destroying penicillin, and again in 1944 by several scientists working independently, who reported a penicillin-inactivating enzyme that was secreted by certain bacteria. In the following decades, overuse and repeated exposure to antibiotic agents favoured the selection and replication of numerous strains of antibiotic-resistant bacteria. Mechanisms of resistance There are several genetic mechanisms by which resistance to antibiotics can develop in bacteria. These mechanisms give rise to resistance because they result in biochemical modifications that alter certain bacterial cell properties that normally render the cell sensitive to an antibiotic. Examples of biochemical modifications that lead to resistance include the production of enzymes that inactivate the drug; the alteration of the protein, enzyme, or receptor targeted by the drug; the activation of drug efflux pumps that deliberately remove the drug from the cell; and the alteration of cell-wall proteins that inhibit drug uptake. Britannica Quiz A Visit with the Word Doctor: Medical Vocabulary Quiz There are two important types of genetic mechanisms that can give rise to antibiotic resistance: mutation and acquisition of new genetic material. In the case of mutation, the rate at which resistance develops can be attributed to the rate at which bacteria mutate. A mutation is a permanent change in an organism’s genetic material. Mutations occur naturally when cells divide. Bacteria are especially prone to mutation because their genome consists of a single chromosome and because they have a high rate of replication. The more replications a cell undergoes, the higher the chance it has to mutate. The acquisition of new genetic material also is a naturally occurring process in bacteria. This process appears to be the most common mechanism by which resistance develops; it is facilitated by the fact that bacteria are prokaryotic organisms (which means that they do not have a nucleus protecting the genome) and by the presence of small pieces of DNA called plasmids that exist in a bacterial cell separate from the chromosome. Thus, the genetic material of bacteria is free-floating within the cell, making it open to gene transfer (the movement of a segment of genetic material from one bacterial cell to another), which often involves the transmission of plasmids. In nature, the primary mechanisms of bacterial gene transfer are transduction and conjugation. Transduction occurs when a bacterial virus, called a bacteriophage, detaches from one bacterial cell, carrying with it some of that bacterium’s genome, and then infects another cell. When the bacteriophage inserts its genetic content into the genome of the next bacterium, the previous bacterium’s DNA also is incorporated into the genome. Conjugation occurs when two bacteria come into physical contact with each other and a plasmid, sometimes carrying a piece of the chromosomal DNA, is transferred from the donor cell to the recipient cell. Plasmids often carry genes encoding enzymes capable of inactivating certain antibiotics. The original source of the genes for these enzymes is not known with certainty; however, mobile genetic elements, called transposons (“jumping” genes), may have played a role in their appearance and may facilitate their transfer to other bacterial species. Because many of the plasmids carrying antibiotic-resistant genes can be transferred between different species of bacteria, widespread resistance to a specific antibiotic can develop rapidly. The transmission of plasmids during conjugation has been associated with the generation of many different types of antibiotic-resistant bacteria. For example, conjugation involving a plasmid carrying the gene for resistance to methicillin (an antibiotic derived from penicillin) is suspected to have resulted in the generation of MRSA. Penicillin and methicillin work by weakening the wall of the bacterial cell; when the wall is compromised, the osmotic gradient between a bacterial cell’s cytoplasm and its environment forces the cell to lyse (break open). In MRSA the gene acquired through conjugation encodes a protein capable of inhibiting methicillin binding, preventing the drug from attaching to and disrupting its target protein in the bacterial cell wall. Another example is a plasmid carrying a gene that encodes the enzyme beta-lactamase. Beta-lactamase alters the structure of the penicillin molecule, rendering it inactive. Transduction and conjugation result in a process called recombination. The new bacterial genomes that are produced from genetic recombination are called recombinants. Antibiotics do not create recombinants—antibiotic-resistant recombinants exist naturally by way of normal gene transfer events. However, antibiotics, and particularly the improper use of these drugs, provide selective pressure to bacterial colonies, whereby the most sensitive organisms are killed quickly, and the most resistant organisms are able to survive and replicate. Access for the whole family! Bundle Britannica Premium and Kids for the ultimate resource destination. Subscribe Prevention and drug development The prospects of scientists developing new antibiotics as fast as bacteria develop resistance are poor. Therefore, other measures have been undertaken, including educating the public about the proper use of antibiotics and the importance of completing a full regimen as prescribed. Improvements in diagnostic equipment to facilitate the isolation and detection of resistant bacteria such as MRSA in hospital settings have enabled rapid identification of these organisms within hours rather than days or weeks. In addition, although efforts to fight bacteria by targeting them with bacteriophages were largely abandoned with the discovery of penicillin and broad-spectrum antibiotics in the 1940s, the growing presence of resistance has renewed interest in these methods. In addition, a significant amount of phage-therapy research was conducted throughout the 20th century in regions within the former Soviet Union. As a result, today in Georgia, which was once under Soviet rule, bandages saturated with bacteriophages against staphylococcus are commercially available as topical treatments for wounds and burns. In the 21st century, researchers worldwide were working to develop other topical and systemic phage therapies. A practical and extremely effective tool against the spread of antibiotic resistance is hand washing. The importance of hand washing was first realized in the 1840s by German-Hungarian physician Ignaz Philipp Semmelweis. Today, hand washing among medical personnel still is not as routine and thorough as it should be. In the early 2000s American critical-care physician Peter Pronovost developed a checklist for intensive care units that attending personnel could follow to ensure that every hand washing, antiseptic scrub, and surface disinfection required during medical procedures was performed, in order to prevent the spread of infection to hospitalized patients. Hospitals that have adopted these methods have lost fewer patients to complications caused by bacterial infections. Douglas Morier Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style The Editors of Encyclopaedia Britannica. "bacillus". Encyclopedia Britannica, 7 May. 2020, Accessed 22 August 2025. Share Share to social media Facebook X External Websites National Center for Biotechnology Information - Bacillus Feedback Thank you for your feedback Our editors will review what you’ve submitted and determine whether to revise the article. print Print Please select which sections you would like to print: verifiedCite While every effort has been made to follow citation style rules, there may be some discrepancies. Please refer to the appropriate style manual or other sources if you have any questions. Select Citation Style Morier, Douglas. "antibiotic resistance". Encyclopedia Britannica, 22 Jul. 2025, Accessed 22 August 2025. Share Share to social media Facebook X URL External Websites Cleveland Clinic - Antibiotic Resistance World Health Organization - Antibiotic Resistance Centers for Disease Control and Prevention - Antibiotic Resistance National Center for Biotechnology Information - PubMed Central - Antibiotic resistance: The challenges and some emerging strategies for tackling a global menace
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https://pmc.ncbi.nlm.nih.gov/articles/PMC3153167/
Diversity and Impact of Prokaryotic Toxins on Aquatic Environments: A Review - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice Toxins (Basel) . 2010 Oct 18;2(10):2359–2410. doi: 10.3390/toxins2102359 Search in PMC Search in PubMed View in NLM Catalog Add to search Diversity and Impact of Prokaryotic Toxins on Aquatic Environments: A Review Elisabete Valério Elisabete Valério 1 Centro de Recursos Microbiológicos (CREM), Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal; Email: evalerio@fct.unl.pt Find articles by Elisabete Valério 1, Sandra Chaves Sandra Chaves 2 Centro de Biodiversidade, Genómica Integrativa e Funcional (BioFIG), Faculdade de Ciências, Universidade de Lisboa, Edificio ICAT, Campus da FCUL, Campo Grande, 1740-016 Lisboa, Portugal; Email: sichaves@fc.ul.pt Find articles by Sandra Chaves 2, Rogério Tenreiro Rogério Tenreiro 2 Centro de Biodiversidade, Genómica Integrativa e Funcional (BioFIG), Faculdade de Ciências, Universidade de Lisboa, Edificio ICAT, Campus da FCUL, Campo Grande, 1740-016 Lisboa, Portugal; Email: sichaves@fc.ul.pt Find articles by Rogério Tenreiro 2, Author information Article notes Copyright and License information 1 Centro de Recursos Microbiológicos (CREM), Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa, Quinta da Torre, 2829-516 Caparica, Portugal; Email: evalerio@fct.unl.pt 2 Centro de Biodiversidade, Genómica Integrativa e Funcional (BioFIG), Faculdade de Ciências, Universidade de Lisboa, Edificio ICAT, Campus da FCUL, Campo Grande, 1740-016 Lisboa, Portugal; Email: sichaves@fc.ul.pt Author to whom correspondence should be addressed; Email: rptenreiro@fc.ul.pt; Tel.: +351-21-750-00-06; Fax: +351-21-750-0172. Received 2010 Aug 21; Revised 2010 Oct 1; Accepted 2010 Oct 13; Collection date 2010 Oct. © 2010 by the authors; licensee MDPI, Basel, Switzerland This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license ( PMC Copyright notice PMCID: PMC3153167 PMID: 22069558 Abstract Microorganisms are ubiquitous in all habitats and are recognized by their metabolic versatility and ability to produce many bioactive compounds, including toxins. Some of the most common toxins present in water are produced by several cyanobacterial species. As a result, their blooms create major threats to animal and human health, tourism, recreation and aquaculture. Quite a few cyanobacterial toxins have been described, including hepatotoxins, neurotoxins, cytotoxins and dermatotoxins. These toxins are secondary metabolites, presenting a vast diversity of structures and variants. Most of cyanobacterial secondary metabolites are peptides or have peptidic substructures and are assumed to be synthesized by non-ribosomal peptide synthesis (NRPS), involving peptide synthetases, or NRPS/PKS, involving peptide synthetases and polyketide synthases hybrid pathways. Besides cyanobacteria, other bacteria associated with aquatic environments are recognized as significant toxin producers, representing important issues in food safety, public health, and human and animal well being. Vibrio species are one of the most representative groups of aquatic toxin producers, commonly associated with seafood-born infections. Some enterotoxins and hemolysins have been identified as fundamental for V. cholerae and V. vulnificus pathogenesis, but there is evidence for the existence of other potential toxins. Campylobacter spp. and Escherichia coli are also water contaminants and are able to produce important toxins after infecting their hosts. Other bacteria associated with aquatic environments are emerging as toxin producers, namely Legionella pneumophila and Aeromonas hydrophila, described as responsible for the synthesis of several exotoxins, enterotoxins and cytotoxins. Furthermore, several Clostridium species can produce potent neurotoxins. Although not considered aquatic microorganisms, they are ubiquitous in the environment and can easily contaminate drinking and irrigation water. Clostridium members are also spore-forming bacteria and can persist in hostile environmental conditions for long periods of time, contributing to their hazard grade. Similarly, Pseudomonas species are widespread in the environment. Since P. aeruginosa is an emergent opportunistic pathogen, its toxins may represent new hazards for humans and animals. This review presents an overview of the diversity of toxins produced by prokaryotic microorganisms associated with aquatic habitats and their impact on environment, life and health of humans and other animals. Moreover, important issues like the availability of these toxins in the environment, contamination sources and pathways, genes involved in their biosynthesis and molecular mechanisms of some representative toxins are also discussed. Keywords: diversity of toxins, impact of toxins, prokaryotes, aquatic, molecular mechanisms 1. Introduction Toxins are any poisonous substance produced by a living organism that is capable of causing disease or death in other organisms. In several cases, the same organism can produce more than one toxin at the same time. These active products can be considered as part of survival strategies of the producers, as they constitute competitive advantages in the environment. It is not always straightforward to understand the benefit conferred by a certain toxin to a microorganism, but this can be mostly attributed to our limited knowledge about many ecologic, functional and evolutionary aspects of toxin-producing species. In fact, toxins can be considered evolutionary advantages, as they contribute to the survival and/or dominance of a particular organism in a particular environment. Microorganisms are recognized for their metabolic versatility and ability to produce diverse bioactive compounds such as hydrolytic enzymes, antibiotics, antitumorals and also toxins. Toxins can be produced by prokaryotes such as bacteria , in particular cyanobacteria , but also by eukaryotes such as dinoflagellates , diatoms , fungi (mycotoxins) [4,5], and animals (zootoxins or venoms) . With such a broad range of producers, it is expected that toxins present high diversity in chemical composition and mode of action. The ubiquity of microorganisms in the environment makes them important causes of water and foodborne intoxications, representing central issues in food safety, public health and human and animal well-being. Many of these microorganisms may be present in drinking water supplies or recreational waters. Moreover, the toxins have also great economical impact due to their deleterious effects. Due to their importance, this review will focus on toxins produced by prokaryotic microorganisms in aquatic environments. Toxins can have diverse natures, including small molecules, peptides, cyclic peptides, lipopeptides, alkaloids, carbamate alkaloids, organophosphates and proteins. Several hundreds of them are known and some have variants with different levels of toxicity. Toxins present different modes of action and cellular targets, resulting from the chemical nature of the toxin and from their interaction with the target cell. Classification of toxins is not a consensual subject: clinicians often arrange them by the organ they affect (hepatotoxin, neurotoxins, etc.), cell biologists prefer to group them according to their effect in the cell (mutagens or carcinogens) and biochemists refer to toxins by chemical origin (e.g., amino acids, peptides, lactones, etc). Other possible classification schemes can be defined based on the toxin mode of action, which may be considered a more wide-ranging criterion. Thus, a brief overview of their diversity will be presented based on the type of action that toxins play in the cell. Membrane permeabilization: These toxins start binding to the membrane in their monomeric form. Afterwards, self-oligomerization occurs resulting in the formation of pores that are permeable to ions and small metabolites. Ultimately, this leads to impaired membrane permeability, membrane disruption or osmotic lysis of the cell . Toxins affecting membrane traffic: Some toxins can interfere with several components of vesicle-associated membrane protein system, altering the traffic of molecules like neurotransmitters across the membrane (e.g., botulinum toxin) [7,8]. Toxins affecting signal transduction: There are toxins that target the intestinal epithelial cells where, after a complex series of events, they activate adenylate cyclase, interfering with signal transduction (e.g., cholera toxin) . Other natural toxins act by modifying key functions of the phosphorylation-based signaling machinery, thus affecting the signal transduction pathways (e.g., microcystins) . Toxins affecting protein synthesis: This group of toxins can present more than one mechanism to inhibit protein synthesis. Two examples are the cleavage of several nucleobases from the 28S rRNA (e.g., Shiga toxins) or the inactivation of elongation factor 2 (eEF-2) by transferring the adenosine diphosphate ribose moiety (ADP-ribose) of NAD to eEF-2 (e.g., Pseudomonas exotoxin A) . Cytoskeleton-affecting toxins: These toxins can induce structural changes in the cytoskeleton and consequently inhibit its functions. Cytoskeleton modifications include the disaggregation of actin microfilaments (e.g., Toxin B from Clostridium dificille) or the induction of the formation of giant multinucleated cells, leading to changes in actin and tubulin organization (e.g., cytotoxic necrotizing factor of Escherichia coli) . Voltage-gated ions channels blockers: These toxins have the ability to interact with the specific receptors associated with neurotransmitter receptors, or with voltage-sensitive ion channels, therefore inhibiting the nervous signaling (e.g., saxitoxin, kalkitoxin and jamaicamides) [3,14]. 2. Toxins Produced by Cyanobacteria Some of the most common toxins present in water are produced by cyanobacterial strains of several species. Cyanobacteria represent one of the major bacterial phyla, being an ancient group of prokaryotic microorganisms exhibiting the general characteristics of gram-negative bacteria whose fossil registers date to 3.5 billion years [15,16]. Cyanobacteria constitute an extraordinarily diverse group of prokaryotes. Due to their particular features, they have successfully colonized a wide range of habitats such as fresh, brackish and marine waters, nonacidic hot springs, hypersaline environments, Antarctic soils, rocks, ice and deserts [17,18,19,20]. Only pH seems to restrict the distribution of cyanobacteria, since they tend to prefer neutral or basic conditions and are less common at low pH . They are unique among the prokaryotes, as they have the ability of performing oxygenic photosynthesis, being presumably the first oxygen-evolving photosynthetic organisms during the Precambrian era. They are thought to be also responsible for the transition of the atmosphere of the Earth from its primordial anaerobic state to the current aerobic condition . 2.1. Blooms and toxicity Cyanobacterial cell numbers in water bodies vary seasonally as a consequence of changes in water temperature and irradiance, as well as meteorological conditions and nutrient supply. Interactions among phytoplankton organisms in freshwater ecosystems have been detected through changes in the relative abundance of microalgae populations within the phytoplankton communities. In temperate regions, seasonal successions of organisms belonging to different phytoplankton taxa are often observed. Whereas at the beginning of summer a great variety of microalgae and cyanobacteria usually co-exist in the same water body, towards the end of summer this diversity may drop drastically as the result of the massive development of cyanobacterial communities (blooms). One the most known phenomena are the dense blooms of Trichodesmium erythraeum that produce a red discoloration of the water and gave the Red Sea its name . Detrimental effects of such cyanobacterial blooms and toxin production are of major concern for water managers. They have become a worldwide increasing problem in aquatic habitats (lakes, rivers, estuaries, and oceans) and in man-made water storage reservoirs. These occurrences can be partially attributed to the gradual eutrophication of the waterways, exposure to constant sunshine, warmth and availability of nutrients like phosphates and nitrates . For example, a low ratio between nitrogen and phosphorous concentrations is one important factor that seems to favor the development of cyanobacterial blooms [18,22]. Since cyanobacteria possess maximum growth rates at temperatures higher than those of green algae and diatoms, the cyanobacterial blooms in temperate water bodies occur mostly during summer months [21,22]. However, there is an unpredictable nature in cyanobacterial blooms and the underlying factors that trigger these phenomena are still poorly understood. As a consequence, the erratic behavior of blooms, in respect to their occurrence, composition, intensity and persistency, demands careful attention in assessing risks for animal and human health. 2.2. Importance and impact of the cyanotoxins occurring in aquatic environments One of the habitats where microorganisms are highly abundant is water. Since many of these microorganisms can produce toxins, they may have a great impact in several living organisms. Toxins produced by cyanobacteria and other microorganisms in sea, rivers, lakes and reservoirs can create adverse effects worldwide. These impact on health and wellbeing, because they are able to induce illness or even death. However, the toxins have also great economic impact due to their deleterious effects. Contrarily to several other waterborne microbial and toxicant health hazards, which are undetectable to the human eye, cyanobacteria are often readily apparent to the human eye and sometimes olfaction. This is due to the water discoloration, formation of blooms and production of smelling compounds. Cyanobacterial toxic blooms create major threats to animal and human health, tourism, recreation and aquaculture (Figure 1). The occurrence of toxic mass populations appears to have a global distribution [2,23]. The first documented case of a lethal livestock intoxication occurred after consumption of water from a lake heavily populated with cyanobacteria. This was reported in a lake of the Murray River estuary (Australia) by Francis in the 1800s . Nowadays, incidents including both human and animal intoxications have been reported around the world. Lethal animal cases include death of sheep, cattle, horses, pigs, dogs, fish, rodents, amphibians, waterfowl, bats, flamingos, zebras and rhinoceroses [23,24,25,26,27,28,29,30,31,32]. Aquatic recreational activities involving direct contact with contaminated water such as swimming, sailboarding, canoeing and paddling may lead to ingestion, aspiration/inhalation or skin contact with toxic cyanobacterial cells and/or with their toxins. There have been reports about the effects on exposed humans, including respiratory irritation, eye inflammation and severe contact dermatitis. The severity of these effects depends on the toxin dose exposure [23,33,34]. Figure 1. Open in a new tab Routes for animal and human intoxication with cyanobacterial toxins. Although most reports of human intoxication caused by cyanobacteria are due to direct ingestion of contaminated water [35,36], chronic intoxication may occur by ingestion of food containing cyanobacterial toxins. For instance, it is known that several cyanobacterial toxins can be accumulated and transferred through the food chain. Microcystins can accumulate in mussels [37,38,39], fish and crustaceans , crayfish and even plants that are irrigated with contaminated water [39,42]. Paralytic shellfish poisons (PSP) toxins can accumulate in cladoceran Daphnia magna , clams, crabs and freshwater mussels [44,45]. Cylindrospermopsin accumulates in mussels . However, there is no evidence of human intoxication risk via bioaccumulation in cattle fed, in neither milk nor beef . So far, numerous bioactive metabolites produced by cyanobacteria have been described, including non-ribosomal peptides, lipopeptides, alkaloids and polyketides that present a vast diversity of structures and variants. Some of them are potent toxins [23,48,49,50,51]. The majority of these peptides are assumed to be synthesized by NRPS (non-ribosomal peptide synthesis, involving peptide synthetases) or NRPS/PKS hybrid pathways, involving peptide synthetases (PS) and polyketide synthases (PKS) . Non-ribosomal peptide synthetases (NRPSs) are multimodular enzymes, found in fungi, cyanobacteria and other bacteria, which biosynthesize peptides without the aid of ribosomes. This kind of biosynthesis allows reaching structures not possible to be obtained by ribosomal peptide synthesis. Most of the non-ribosomal peptides from microorganisms are classified as secondary metabolites, meaning that they do not have a role in primary metabolism, growth or reproduction, but have evolved to somehow benefit the organism that produces it. Since 2000, major efforts have been made to disclose the genetic basis of the biosynthesis of the compounds produced by cyanobacteria, some of them with unique structures. Here we overview the main gene clusters responsible for cyanotoxins biosynthesis; and discuss similarities and differences among them. Usually the toxins produced by cyanobacteria are classified according to the effect that they provoke in mammals and vertebrates, where hepatotoxins (liver damaging), cytotoxins (cell damaging), neurotoxins (nerve damaging) and toxins responsible for allergenic reactions (dermatotoxins) have been isolated and characterized from several cyanobacteria . A single species may contain toxic and non-toxic strains; therefore identification at the species level by microscopic morphology does not indicate the potential for toxin production of a given strain. Toxic variations, between and within species of cyanobacteria, are well known from laboratory studies based on isolated cultured strains [53,54,55,56]. So far, an organism able to produce all the variants of each type of toxins or all the types of cyanotoxins has not been described. However, there are some reports on Cylindrospermopsis raciborskii strains able to produce several toxins such as cylindrospermopsin, PSPs and unknown compounds [57,58], and also some cases of Microcystis strains able to produce microcystins and/or anatoxin-a . 2.3. Hepatotoxins 2.3.1. Microcystins One of the most abundant types of cyanotoxins worldwide are microcystins (MC) and are consequently the more intensely studied. An increased incidence of primary liver cancer in China has been associated with the chronic ingestion of sublethal doses of microcystins in raw drinking water [23,60]. The direct uptake of water contaminated with these cyanotoxins through renal dialysis also resulted in some human deaths [36,61]. Microcystins are cyclic peptides with a molecular mass ranging from 900–1100 Da. They share a common structure constituted by Adda-D-Glu-Mdha-D-Ala-L-X-D-MeAsp-L-Z, where X and Z are variable L-amino acids, Adda is the unusual C20 amino acid (3-amino-9-methoxy-2,6,8-trimethyl-10-phenyl-4,6-decadienoic acid), D-MeAsp is 3-methylaspartic acid, and Mdha is N-methyl-dehydroalanine . About 80 different variants of microcystins have been described [23,62], with different levels of toxicity. The most common microcystins are MC-LR, MC-RR and MC-YR having the L-amino acids leucine (L), arginine (R) or tyrosine (Y), respectively, in the X position. MC-LR is the most studied variant because of its ubiquity, abundance and toxicity. MC are known to be produced by several cyanobacterial genera including Microcystis, Planktothrix, Oscillatoria, Anabaena, Anabaenopsis, Nostoc, Hapalosiphon, Snowella and Woronichinia [18,23,27,34]. The main target of MC is the hepatocyte, the most common cell type in the liver. MC inhibit eukaryotic protein phosphatases and also activate the enzyme phosphorylase b, which results in an excessive phosphorylation of cytoskeletal filaments triggering apoptosis . Death of hepatocytes leads to the destruction of the finer blood vessels of the liver and to massive hepatic bleeding. Some in vivo and in vitro studies show that organs like kidney and colon can also be affect by the exposure of humans to these toxins . MC cannot diffuse through the plasma membrane because of their high molecular weight and structure. However, cell specificity and organotropism of MC-LR suggested that a selective pathway for MC uptake would probably exist. Several studies point to the intake of MC through the plasma membrane by a member of the organic anion transporting polypeptide superfamily (OATP) (Figure 2) . Concerning the molecular mechanism of MC toxicity, it is a multi-pathway process, in which the inhibition of serine/threonine protein phosphatases type 1 and type 2A (PP1/PP2A) leads to a cascade of events responsible for the MC cytotoxic and genotoxic effects in animal cells (Figure 2). The mechanisms of tumor promotion are unclear, but apparently they are related to protein phosphatase inhibition leading to hyperphosphorylation of many cellular proteins and deregulation of cell-cycle control. Cell-cycle progression is largely controlled by reversible phosphorylation of regulatory enzymes on their serine/threonine residues. Accordingly, it has been proposed that microcystin induces an increase of oxidative stress, leading to a raise of reactive oxygen species, which can cause DNA damage and is associated with microcystin-induced liver carcinogenesis. In fact, in vitro and in vivo studies have found oxidative DNA damage in the form of 8-oxo-7,8 dihydro 2’-deoxyguanosine associated with microcystin exposure . Through the inhibition of PP1 and PP2A, MC seems to control several cellular processes, e.g., activation of the calcium-calmodulin-dependent multifunctional protein kinase II (CaMKII) by inhibiting its dephosphorylation. The activation of CaMKII may further regulate downstream events such as ROS formation and phosphorylation of proteins including myosin light chain . MC-LR can also activate Nek2 kinase by binding to Nek2 kinase complex with PP1 holoenzyme . Nek2 kinase is a member of the NIMA-related serine/threonine kinase family that participates in the control of mitotic progression and chromosome segregation. This interaction may have implications in the cell viability, tissue injury and tumor development . Moreover, mitogen-activated protein kinases (MAPKs) are serine/threonine-specific protein kinases that regulate several cellular activities, such as proto-oncogenes expression, mitosis, differentiation, proliferation, and cell survival/apoptosis. PP2A mediates MAPKs expression. Therefore the presence of MC probably regulates MAPKs expression as well . MC genotoxicity is also associated with its ability to inhibit two DNA repair systems: nucleotide excision repair (NER) and DNA double strand break (DSB) repair by the nonhomologous end joining (NHEJ). Both systems are regulated by phosphorylation and the inhibition of PP1/PP2A significantly decreases their activity. Furthermore, the inhibition of the DSB-NHEJ pathway is a consequence of loss of activity of the DNA-dependent protein kinase (DNA-PK) resultant from its phosphorylation, after the inhibition of PP2A like enzymes . Additionally, an increase in serine phosphorylation of the nuclear phosphoprotein P53 was identified following both lethal and sublethal MC-LR exposure in mice . This protein is a substrate of PP2A and plays a role as a transcriptional trans-activator in DNA repair, apoptosis and tumor suppression pathways . P53 is also a regulator of the expression of the anti and proapoptotic genes including members of the Bcl-2 family such as Bax and Bid. Bax and Bid, play important roles in apoptosis, especially in mitochondria-dependent pathway. Studies indicate that MC-LR can induce mitochondria-dependent apoptosis via the regulation of Bcl-2 family members (Figure 2). The role of ROS and related mechanisms in MC-LR-induced liver injury in vivo are not completely understood. Two possible pathways are mentioned; one of them concerns the outer-membrane permeabilization of the mitochondria after a MC induced massive Ca 2+ influx, thereby triggering the process of apoptosis. Another plausible mechanism for ROS generation is the increase of NADPH oxidase activity (Figure 2). Besides what is here presented, there is still much to be done to completely unveil the molecular mechanisms of MC toxicity. Figure 2. Open in a new tab Schematic representation of the molecular mechanisms of microcystins (MC) toxicity. After intake through the plasma membrane by the organic anion transporting polypeptide system (OATP), MC binds specifically to the serine/threonine protein phosphatases (PP1/PP2A), inhibiting them and leading to a cascade of events responsible for the MC cytotoxic and genotoxic effects in animal cells (see text for details). Microcystins are produced non-ribosomally via a thio-template mechanism, by a multienzyme complex consisting of peptide synthetases (PS), polyketide synthases (PKS) and tailoring enzymes. The gene cluster for microcystin biosynthesis was the first to be completely sequenced from a cyanobacterium. It contains approximately 55 kb and is one of the largest bacterial gene clusters described so far. This cluster has been identified and sequenced in three phylogenetic distantly related strains, Microcystis aeruginosa PCC 7806 , Planktothrix agardhii CYA 126 and Anabaena sp. strain 90 . Its schematic representation is displayed in Figure 3. This gene cluster consists of nine (Planktothrix) or ten (Microcystis and Anabaena) open reading frames (ORFs). In each module there are specific domains for activation (aminoacyl adenylation domain) and thioesterification (peptide carrier domain) of the amino acid substrate and for elongation (condensation domain) of the growing peptide that is being assembled . The organization of the genes clearly differs among genera. In Microcystis and Anabaena, the genes are transcribed from a central bidirectional promoter region, whereas in Planktothrix all mcy genes except mcyT seem to be transcribed unidirectionally from a promoter located upstream of gene mcyD . However, the multienzyme components are highly similar in the different genera. Except for the tailoring enzymes mcyI, mcyF and mcyT, all other genes mcyABCDEGH are always present. However, only the mcyA-C arrangement appears to be fairly conserved among toxic strains of the different genera. The mcyH gene is an ABC-transporter-like gene and it is thought to be involved in the transport of microcystin . It is assumed that this transporter may be responsible for the localization of the toxin in thylakoids [76,77] or for its extrusion under certain growth conditions . 2.3.2. Nodularin Nodularin is a pentapeptide with a molecular mass of 824 Da. Comparison with microcystin shows the presence of N-methyl-dehydrobutyrine (Mdhb) instead of Mdha, and the lack of D-Ala and X residues. So far, this toxin has only been found in Nodularia spumigena . Like microcystins, nodularin is a potent tumor promoter that may also act as a carcinogen/tumor initiator and inhibits serine/threonine protein phosphatase-1 and 2A. However, it does not covalently bind to PP1 or PP2A . Due to its structural similarity with microcystins, nodularin is expected to present molecular mechanisms of toxicity similar to those of MC (Figure 2). A mcy homologous gene cluster (nda) described in Nodularia spumigena NSOR10 is considered responsible for the synthesis of the pentapeptide nodularin . The 48 kb region of the genome consists of nine ORFs (ndaA-I) as depicted in Figure 3. Functional assignment of the enzymes was based on bioinformatic analysis and homology to microcystin synthetase enzymes. The nda cluster also encodes several putative monofunctional enzymes that may have a role in the modification (NdaE and NdaG) and transport (NdaI) of nodularin. Studies that have been conducted on the detection and regulation of the genes that are involved in hepatotoxins production are beyond the present scope of this review. The interested reader should see recent reviews [74,81]. Figure 3. Open in a new tab Schematic representation of theorganization of cyanobacterial gene clusters responsible for the biosynthesis of hepatotoxins (microcystins, nodularin, cylindrospermopsin). Different ORFs are indicated as arrows and domains integrated within proteins as circles. 2.4. Cytotoxins: cylindrospermopsin Cylindrospermopsin (CYN) is also the object of several studies due its impact, namely in Australia. It is a potent alkaloid, with a molecular mass of 415 Da, consisting of a tricyclic guanidine moiety combined with hydroxymethyluracil. In contrast to MC, the structural variability is much lower. So far, only three variants of the cylindrospermopsin molecule have been described, including deoxy-cylindrospermopsin and 7-epi-cylindrospermopsin, with CYN being more toxic than deoxy-cylindrospermopsin. The presence of guanidino and sulfate groups makes CYN a zwitterionic molecule and hence more soluble in water. Moreover, being a small compound, it is likely to be taken by the cells through diffusion. CYN and its analogues are known to be produced by some cyanobacterial species, namely, Cylindrospermopsis raciborskii [82,83,84], Umezakia natans , Aphanizomenon ovalisporum , Raphidiopsis curvata , Anabaena bergii , and more recently Aphanizomenon flos-aquae and Lyngbya wollei . Terao et al. described the liver as the main target of this cyanotoxin but other histopathological studies showed that kidneys, thymus and heart are also affected [92,93]. The first clinical symptoms of CYN ingestion are kidney and liver failure . CYN toxicity results in four pathological changes in the liver: protein synthesis inhibition, membrane proliferation, fat droplet accumulation, and cell death. Despite extensive research, the specific molecular interactions that result in CYN-mediated toxicity are currently unknown. However, it is recognized that this toxin is genotoxic, hepatotoxic in vivo and is also a general cytotoxin that blocks protein synthesis. Its toxicity is due to the inhibition of glutathione (GSH) and protein synthesis as well as the inhibition of cytochrome P450 (CYP450) (Figure 4) . There is also evidence of its carcinogenic potential in mice . GSH seems to be required to inactivate cylindrospermopsin, but in the presence of CYN, GSH synthesis is inhibited in the hepatocytes . CYN can also covalently bind to DNA and there is evidence that CYN causes DNA breakage . Therefore, mutagenic activity of the toxin can also be expected. Nevertheless, the exact mode by which CYN causes DNA damage has yet to be determined. The disclosure of the genes responsible for the biosynthesis of cylindrospermopsin began with Schembri et al. . They showed a direct link between the presence of polyketide synthases (PKS) and peptide synthetases (PS) genes in C. raciborskii isolates and the ability of those isolates to produce cylindrospermopsin. Later on, Shalev-Alon et al. identified amidinotransferase genes (AoaA, AoaB, and AoaC) in an Aphanizomenon ovalisporum strain that could be implicated in the cylindrospermopsin synthesis. Recently, using adaptor-mediated gene walking technology, a polyketide biosynthetic pathway, thought to be responsible for the production of cylindrospermopsin, has been described in C. raciborskii [94,99]. This cluster spans 43 kb and most of the allocated genes are of PKS nature (Figure 3). However, the biochemical proof for the role of this gene cluster in cylindrospermopsin biosynthesis is still lacking, mostly due to the absence of suitable tools for genetic transformation of Cylindrospermopsis. Figure 4. Open in a new tab Schematic representation of the known molecular mechanisms involved in Cylindrospermopsin (CYN) toxicity. There is inhibition of glutathione (GSH) and protein synthesis, as well as cytochrome P450 (CYP450), and CYN interaction with DNA. 2.5. Neurotoxins 2.5.1. Anatoxin-a and homoanatoxin-a Anatoxin-a and homoanatoxin-a are unusual alkaloids, secondary amines, with low molecular masses (165 and 179 Da, respectively) exclusively produced by cyanobacteria. Anatoxin-a is synthesized by various members of the genera Anabaena , Cylindrospermum , Microcystis , Oscillatoria , Raphidiopsis , Planktothix and Aphanizomenon . Homoanatoxin-a is produced by species of the genera Oscillatoria , Anabaena , Raphidiopsis and Phormidium . Some strains are able to produce simultaneously anatoxin-a and homoanatoxin-a [101,106]. Anatoxin-a, also previously known as “Very Fast Death Factor”, acts as a post-synaptic neuromuscular blocking agent. Anatoxin-a and homoanatoxin-a are potent agonists of the muscular and neuronal nicotinic acetylcholine receptor. The toxin irreversible binding to the nicotinic acetylcholine receptor causes sodium channel opening and a constant inflow of sodium ions to cells (Figure 5). Overstimulation of the muscle cells occurs as a result of membrane depolarization and desensitization. When respiratory muscles are affected, the lack of oxygen in the brain may lead to convulsions and finally to death of animals by acute asphyxia . Figure 5. Open in a new tab Schematic representation of Anatoxin-a/Homoanatoxin-a and Anatoxin-a(s) molecular mechanisms of toxicity. During “normal events” acetylcholine is released from the neurons, binds to the acetylcholine-receptors on the postsynaptic muscle cell thereby inducing the influx of Na+ into the cell. Acetylcholine is degraded by the enzyme acetylcholinesterase in the synaptic cleft into acetate, which is eliminated, and cholin, which is taken up into the neuron by specific carriers. However, in the presence of anatoxin-a and homoanatoxin-a, these toxins bind irreversibly to the nicotinic acetylcholine receptor causing sodium channel opening and the constant inflow of sodium ions to cells. Anatoxin-a(s) causes an irreversible inhibition of the acetylcholinesterase preventing degradation of acetylcholine. The muscles become constantly stimulated. In 2009, the efforts of Cadel-Six et al. and Méjean et al. showed evidence linking the presence of a 29 kb DNA fragment containing polyketide synthases and the production of anatoxin-a and homoanatoxin-a (Figure 6) [107,108]. The sequence of the gene cluster, assumed to be involved in the production of these toxins, was also obtained by adaptor-mediated gene walking technology. The function of the several identified ORFs was deduced by comparison with other annotated genes and, to date, attempts to genetically confirm the role of this gene cluster in the biosynthesis of these toxins by specific gene disruption have been unsuccessful. Figure 6. Open in a new tab Schematic representation of theorganization of cyanobacterial gene clusters responsible for the biosynthesis of neurotoxins (anatoxin-a and homoanatoxin-a, saxitoxin, jamaicamide), curacin A and barbamide. Different ORFs are indicated as arrows and domains integrated within proteins as circles. 2.5.2. Anatoxin-a(s) Anatoxin-a(s) is a unique organophosphate with a molecular mass of 252 Da. It is synthesized by Anabaena flos-aquae and Anabaena lemmermanni . This toxin causes an irreversible inhibition of acetylcholinesterase, which consequently cannot perform the degradation of acetylcholine that is bound to the acetylcholine-receptor . As a result, muscles become constantly stimulated (Figure 5). Its functional consequences are comparable to the one of organophosphorous and carbamate insecticides like paraoxon, physostigmine, pyridostigmine and the chemical warfare agent sarin . 2.5.3. Saxitoxin Saxitoxins (STX), commonly known has paralytic shellfish poisons (PSPs), are tricyclic perhydropurine alkaloids that have a molecular mass of 299 Da. They can be non-sulfated (saxitoxins and neosaxitoxin), single sulfated (gonyautoxins) or doubly sulfated (C-toxins) and the possible substitutions at various positions of the molecule results in more than 30 structural variants [2,113,114]. The toxicity of the STX derivatives is different and depends on the type of variant produced; saxitoxin (STX), neosaxitoxin (NEO), and gonyautoxins (GTX1-4) are the most toxic molecules. Saxitoxins are produced by marine dinoflagellates and cyanobacteria. Members of the freshwater cyanobacteria genera Anabaena, Aphanizomenon, Cylindrospermopsis, Lyngbya and Planktothrix are able to produce these kinds of toxins . Saxitoxin binds to the sodium and calcium channels of the nerve axon membranes, preventing the passage of these ions through the cell membrane and therefore blocking the transfer of the nerve impulse [115,116] (Figure 7). It also extends the gating of potassium channels in heart cells . This action results in a disturbance in the propagation of action potential to muscle cells. Depending on the dose, saxitoxin poisoning may cause symptoms such as tingling and numbness around the lips or, in extreme situations, neuromuscular paralysis and death caused by respiratory failure. It has also been shown to exhibit a cardio depressant effect . Figure 7. Open in a new tab Schematic representation of saxitoxin toxicity mechanism. Saxitoxin binds to the sodium or calcium channels of the nerve axon membranes, preventing the passage of these ions through the cell membrane thus blocking the transfer of the nerve impulse. Saxitoxins are among the most toxic compounds known. Nowadays, saxitoxins are already included in the Schedule 1 of the Chemical Weapons Convention together with warfare agents such as mustard gas, sarin, ricin and others [118,119]. In 2008, Kellman et al. revealed the biosynthetic pathway of STX production using reverse genetics to identify the candidate STX biosynthetic gene cluster (sxt) in the cyanobacterium Cylindrospermopsis raciborskii T3 . Functional assignment of the enzymes was based on bioinformatic analysis combined with the liquid chromatography-tandem mass spectrometry analysis of the biosynthetic intermediates; however, the biochemical proof for the role of this gene cluster in saxitoxin biosynthesis is lacking, because C. raciborskii is not genetically transformable. In C. raciborskii, the sxt gene cluster spans approximately 35 kb and presents genes responsible for toxin biosynthesis, regulation and export (Figure 6). However, some of the identified genes have not yet been assigned to a function due to their low level of homology with proteins present in databases. Later on, in order to obtain the sequences of the gene cluster also responsible for SXT biosynthesis in Anabaena and Aphanizomenon, a gene walking technique (pan-handle PCR) was employed by Mihali et al. . It revealed a 29 kb gene cluster in Anabaena circinalis, and a slightly smaller cluster in Aphanizomenon of 27.5 kb (Figure 6). The bioinformatically-deduced functions reveal that the cluster presents some variations between the genera, namely in the genes assumed to be involved in toxin regulation, and there are also some differences regarding the genes supposedly involved in toxin transport. 2.6. Lipopeptides from marine cyanobacteria Marine cyanobacteria are amazing in the diversity of new biologically active natural products synthesized using mixed NRPS/PKS systems. Several lipopeptides have been purified from the marine cyanobacteria Lyngbya majuscula. For example, Kalkitoxin and jamaicamides A, B, and C are neurotoxins that block voltage-gated sodium channels, while antillatoxins A and B activate them . Metabolites with pharmacological importance, like barbamide (used in the biological control of snails), the anticancer compound curacin A and antifungal agents as hectochlorin, have also been identified. 2.6.1. Jamaicamides Jamaicamide A is a highly functionalized lipopeptide containing an alkynyl bromide, vinyl chloride, β-methoxy eneone system, and pyrrolinone ring. Jamaicamide B is a debromo analogue of jamaicamide A, while in jamaicamide C, which also lacks the bromine atom, a terminal olefin replaces the terminal alkyne of jamaicamide B. Jamaicamides A, B and C have all been isolated from Lyngbya majuscula. Jamaicamides have been demonstrated to present a sodium channel blocking activity (Figure 7) and fish toxicity. The biosynthetic pathway responsible for jamaicamides synthesis has been recently investigated . Based on feeding precursor experiments to jamaicamide-producing cultures, an effective cloning strategy for the biosynthetic gene cluster discovery was developed. The 58 kb gene cluster has 17 open reading frames (Figure 6), the majority of PKS nature, showing a notable co-linear arrangement with respect to its proposed utilization during biosynthesis . 2.6.2. Kalkitoxin Kalkitoxin is a thiazoline-containing lipid derivative also produced by the pantropical marine cyanobacterium Lyngbya majuscula. It has been indirectly shown that kalkitoxin blocks voltage-gated sodium channels (Figure 7) . Furthermore, this toxin has proven to be ichthyotoxic to the goldfish Carassius auratus and toxic to the crustacean brine shrimp Artemia salina . 2.6.3. Antillatoxins An extremely potent ichthyotoxic L. majuscula metabolite, antillatoxin A, was firstly reported in 1995 . Antillatoxin is a structurally remarkable lipopeptide, presenting a high degree of methylation. It is among the most ichthyotoxic metabolites isolated, and is only exceeded in potency by the brevetoxins . The studies performed so far to determine its mechanism of action showed that it activates the mammalian voltage-gated sodium channel at a pharmacological site that is distinct from any previously described . Antillatoxin B is a variant of antillatoxin A, which has reduced sodium channel-activation properties and exhibits less ichthyotoxic activity . 2.6.4. Curacin A Curacin A is a unique natural product presenting a structure with two lipid chains and sequential thiazoline and cyclopropyl rings. Curacin is promising as an antiproliferative agent due to its inhibitory action on tubulin polymerization. Since it has been shown to block cell cycle progression by interacting with the colchicines binding site on tubulin and inhibiting microtubule polymerization, this compound may have value in the treatment of neoplasic disorders . The genetic basis involved in curacin production has been described by Chang et al. . A combined approach employing isotope incorporation and molecular genetics was employed to reveal the biosynthetic pathway of curacin. The bioinformatic analysis showed that this gene cluster spans 63.7 kb, containing 14 ORFs (Figure 6), making this the largest gene cluster described in cyanobacteria. The genetic architecture of the cluster shows a co-linear arrangement with respect to its expected utilization during biosynthesis. This amazing cluster is almost only composed by PKS modules, with the exception of curF, which is a hybrid PKS/NRPS bimodule. 2.6.5. Barbamide Barbamide is a chlorinated lipopeptide that has been isolated due to its molluscicidal activity . The biosynthetic pathway responsible for this toxin production has started to be revealed based on incorporation studies using isotope-labeled precursors. In 2002, Chang et al. have described the complete sequence of this unusual gene cluster containing NRPS and PKS modules . Sequence comparison with databases showed the existence of 12 putative ORFs allocated in the 26 kb cluster (Figure 6). However, clear evidence of the involvement of each ORF of this gene cluster in barbamide biosynthesis will only be achieved by performing heterologous expression or gene disruption assays. In this review, the diversity of the NRPS or NRPS/PKS hybrid pathways, involving peptide synthetases and polyketide synthases used by cyanobacteria to produce toxins with unusual structures (Figure 3 and Figure 6), is shown. Most of the toxins (microcystins, nodularin, cylindrospermopsin, jamaicamide, curacin A and barbamide) have two main modular biosynthetic systems: (i) the non-ribosomal peptide synthetases (NRPSs), responsible for assembling amino acids leading to peptide formation, and (ii) the polyketide synthases (PKSs), used to link together acetate as the primary building block. Since NRPS and PKS enzymes are able to accept a wide range of different substrates, a huge number of possible different structures can be reached using these systems. In NRPS, a minimal elongation module, that is responsible for one elongation step, comprises three catalytic domains: an adenylation domain (A), responsible for substrate recognition and activation by adenylation; a thiolation domain (T), needed for the covalent incorporation as thioesters; and one condensation (C) domain, for condensation of the precursor (e.g., P. agardhii mcyB gene in Figure 3). Epimerization (E) or N-methylation (NMT) domains may also be present in the module, leading to further substrate modification (P. agardhii mcyA gene in Figure 3). The cyanobacterial PKS consists of multiple sets of domains and modules, which normally correspond to the number of acyl units in the product . One module comprises a set of domains that are responsible for the activation, modification and elongation of a single amino acid or carbon unit. A minimal multifunctional module is composed of a ketoacyl synthase (KS) domain, an acyltransferase (AT) domain and an acyl carrier protein (ACP) domain (e.g., the cyrF gene of C. raciborskii cylindrospermopsin cluster). Frequently, ketoreductase (KR), dehydratase (DH) and enoyl reductase (ER) domains are constituents of megasynthases, as the anaE gene of Oscillatoria anatoxin-a and homoanatoxin-a cluster. In the case of the neurotoxins clusters, namely saxitoxin, an extraordinary number of secondary tailoring manipulations, including oxidation, methylation and diverse forms of halogenations are also present. With the exception of the Lyngbya majuscula clusters, all the other described clusters include ORFs responsible for the toxin transport (red arrows in Figure 3 and Figure 6). This evidence indicates that the release of the cyanobacterial toxins to water does not necessary implies cell lysis, since some toxins may be actively exported from the cells. Another interesting feature of cylindrospermopsin, anatoxin-a/homoanatoxin-a and saxitoxin gene clusters is the presence of ORFs correspondent to transposases, which can lead to speculation that the presence of these transposases may have an important role in the horizontal transfer of these genes. 2.7. Lipopolysaccharides Endotoxic lipopolysaccharides (LPS) are part of the outer membrane of gram-negative bacteria, including cyanobacteria. LPS and its effects are well known from bacteria such as Escherichia coli, Salmonella spp., Vibrio cholera, Yersina pestis and Pseudomonas aeruginosa. LPS composition includes lipid A, core polysaccharides and an outer polysaccharide chain. Opposing to the other bacteria, LPS from cyanobacteria have a higher diversity of long chain unsaturated fatty acids, hydroxyl fatty acids and lack phosphate . It is accepted that LPS cause fever in mammals and are involved in septic shock syndrome . Besides their action on the immune system, LPS from bacteria and cyanobacterial origin also affect the detoxication system of different organisms . However, LPS from bacterial origins have shown to be more toxic than the cyanobacterial ones. 3. Toxins Produced by Other Bacteria in Aquatic Environments Although cyanobacteria are the most important group of aquatic toxigenic prokaryotes, there are other bacteria, present in aquatic environments, which can produce toxins with high relevance for human and animal health. Vibrio spp. are one of the most representative groups of aquatic toxin producers, commonly associated with seafood-born infections and intoxications. Aeromonas hydrophila is also associated with aquatic habitats and has been described as responsible for the production of toxins . Furthermore, recognized pathogens like Escherichia coli, Campylobacter spp. or Legionella pneumophila are also water contaminants and have been described as emergent toxin producers. An overview about several aspects of these water associated toxin producers will be presented, including the diversity of toxins, their impact on environment and human life, molecular mechanisms, cellular consequences, pathways and genes involved in their biosynthesis. The diversity of bacterial toxins is high and more potentially toxic molecules are emerging. In this review we will focus on some representative examples, chosen based on the specificity of their molecular mechanisms and their impact on human and animal life. 3.1. Vibrio spp. Vibrio species belong to Gamma-proteobacteria, are curved usually motile rods, mesophilic and presenting a chemoorganotrophic or facultative fermentative metabolism . They are highly abundant in aquatic environments, including estuaries, marine coastal waters and aquaculture facilities [136,137,138,139]. They also appear to be highly associated with marine organisms like fish [140,141], mollusks [142,143] and shrimps [144,145], which are important food products for human consumption. Vibrio organisms present another important feature: they can attach to the exoskeletons of crustaceans and other marine organisms of the zooplankton, producing biofilms . Their close relationship with zooplankton can be a survival strategy to resist to environmental stresses like starvation or antibiotic presence . This strong association to zooplankton is of utmost importance, as they can easily enter into the human and animal food webs. On the other hand, some Vibrio species are recognized as relevant pathogens for animals reared in aquaculture [147,148]. Fish and shellfish mortality caused by vibrios is very frequent in early larval stages [149,150]. Sometimes Vibrio spp. infections can lead to the death of entire populations, with high economical consequences. Three Vibrio species (Vibrio cholerae, Vibrio parahaemolyticus and Vibrio vulnificus) are considered serious human pathogens. Both V. cholerae and V. vulnificus produce toxins that are fundamental as virulence factors. Some of the most relevant toxins produced by these species will be addressed in more detail. 3.1.1. Vibrio cholerae This species is the causative agent of cholera, a severe disease that had a central role in the history of infectious diseases. Cholera outbreaks are reported since 1817. Presently, Cholera continues to be responsible for thousands of deaths, especially in developing countries, where poor water supply and poor sanitation are unsolved problems [151,152]. V. cholerae is found in coastal, estuarine and marine environments, often associated with aquatic fauna such as copepods and shellfish and is transmitted to humans by contaminated water and food [151,152]. Close relationships with zooplankton are also established and cholera outbreaks have been associated with planktonic blooms and sea surface temperatures. The wide ecological relationships of V. cholerae, the ability to form biofilms and to adapt to environmental changes have highlighted the pathogenic potential of this species. The main virulence factor associated to V. cholerae pathogenesis is the production of the potent cholera toxin (CT). Cholera is characterized by a voluminous watery diarrhea, leading to rapid dehydration. Patients can lose as much as 20 liters of fluid in 24 hours and more than 50% of them die if not treated. The clinical aspects of the disease are primarily induced by the activity of this toxin , but not all strains are able to produce it. In fact, more than 200 V. cholerae serotypes have been described, but only two (O1 and O139) can produce the CT-toxin [151,153]. The genes for toxin synthesis (ctx AB) are carried by the lysogenic bacteriophage CTXΦ and only strains with the integrated phage are able to produce CT-toxin . The expression of these genes is coordinated with the expression of other virulence factors like TCP (toxin-coregulated pilus, coded by tcp genes and required for intestinal colonization) and the accessory colonization factors (coded by the acj A–D genes). Virulence genes are located in a pathogenicity island of V. cholera genome . The primary direct activator of virulence genes transcription (including ctx ABC) is ToxT protein. It belongs to a large protein family (AraC/XylS) that shares a domain of 100 amino-acids, which corresponds to a helix-turn-helix DNA binding motif and has transcription activation functions . AraC domain is nearly invariant among all ToxT sequences, but a second domain (NTD) was identified in these proteins, being much less conserved. Its function is not clear, but it is thought that this domain is the binding site for a natural effector, which is proposed to be bile[156,157]. ToxT activates transcription by binding to a degenerate 13-bp DNA sequence (the toxboxes), located upstream of all genes activated by this protein. These toxboxes can occur in different configurations at different promoters and can be organized as direct or inverted repeats, never overlapping the -35 promoter element . The activity of ToxT in ctx AB promoter is mainly to counteract the H-NS histone-like protein that binds to the same region, but strongly represses the ctx AB gene expression. ToxT also interacts directly with the α subunit of RNA polymerase, activating transcription . The regulatory pathways of CT toxin are complex, involving upstream regulation of toxT expression (regulatory proteins as ToxR, TcpP and ToxS have already been identified and characterized), interaction with effectors and coordination with the expression of other virulence genes. The regulatory pathways of CT production and expression of other virulence factors were recently reviewed by Matson et al. . After entry in the intestinal lumen, V. cholera interacts with intestinal microenvironment, sensing a luminal factor that induces a low expression level of TCP and enables the adherence of the bacteria to the intestinal mucosa, probably to the glycocalyx. A second environmental signal, dependent on microbial adherence to the mucosa, is detected and enhances the level of TCP expression, promoting the intestinal colonization and finally inducing the production and secretion of CT toxin [160,161]. ToxR and TcpP, both inner membrane proteins, are critical for these events but the nature of the signals that activate them remain unclear. CT-toxin is a bipartite molecule belonging to the AB 5 family, which also includes Shiga and pertussis toxins. CT combines one A active subunit and five identical peptides (~11 kDa) that is assembled into a highly stable pentameric ring named B subunit (~55 kDa). The A subunit presents two domains: A 1 and A 2. These two peptides are linked by an exposed loop containing a protease-sensitive “nick” site and a single disulfide bond. The B subunit has a high affinity to the oligosaccharide domain of the G M1 ganglioside on the surface of the intestinal epithelial cells, allowing the binding of the toxin to the plasma membrane of host cells. B pentamer binds stoichiometrically to five G M1 gangliosides at cell surface . G M1 functions to concentrate CT in glycolipid-rich apical microdomains (“lipid rafts”) located in the cell surface . These lipid rafts are distinct cholesterol rich membrane structures that act as membrane organizing centers for signal transduction, protein and lipid sorting, endocytosis and transcytosis. The raft structure and role depends on the specific lipids that compose the microdomain and the specific binding of the toxin to GM 1 gangliosides depends on these lipids. This step is considered to be the critical step for the subsequent targeting of the toxin into all intracellular compartments, required to trigger the cellular response. A stable binding between B subunit and G M1-receptor complexes is essential for CT function. Considering the A subunit, the A 2 peptide (~5 kDa) attaches the A 1 peptide (~22 kDa) to the B subunit and presents a COOH-terminal KDEL motif that extends from B pentamer on the side that binds G M1 . This KDEL motif is known to be a sorting signal that allows endogenous proteins from endoplasmic reticulum (ER) to be retrieved efficiently from post-ER compartments. A 1 is the enzymatic active subunit of the toxin and must dissociate from B subunit and translocate across a cellular membrane to act on its cellular target. CT is not a pore forming toxin. Rather it uses the host cell membrane traffic machinery, entering into the intestinal cell through a complex mechanism. The proposed pathway (Figure 8) starts with the binding of CT holotoxin to G M1 in the apical membrane. The G M1-CT complex enters the cell by apical endocytosis and traffics retrograde through Golgi cisternae into the ER lumen. Then A 1 peptide unfolds, is translocated to the cytosol, and breaks away from the membrane after translocation. Then it can move by diffusion to its cellular target (adenylate cyclase complex), located on the cytoplasmatic surface of the basolateral membrane. Alternatively, the active subunit of the toxin can remain membrane associated. In this case, it moves back out of secretory pathway into the cytosol by vesicular transport back . The B subunit does not translocate across cell membranes, remaining membrane associated (probably bound to G M1 receptors) and moving back out of the secretory pathway by vesicular traffic to the cell surface. This mechanism allows B subunit to move from its original binding site (apical/mucosal) to the basolateral site (basolateral/serosal) through Golgi cisternae and ER. The cellular target of CT is the adenylate cyclase complex at the basolateral membrane. The active subunit A 1 acts as an enzyme that is able to specifically transfer ADP-ribose group to an arginine residue of the α subunit of the GTP-binding protein Gs. Once activated, this Gs α subunit dissociates from the Gs membrane-bound subunit and crosses the cell to attach to the catalytic unit of adenylate cyclase in the basolateral membrane. Once activated, this enzyme induces the formation of cAMP and the subsequent activation of cAMP-dependent protein kinase. Finally, the phosphorylation of membrane proteins involved in the transepithelial ion transfer induces changes in the ion transport. The final consequences of this process are the inhibition of Na+ and Cl− absorption in villous cells and the stimulation of secretion of Cl−, HCO 3− and water in epithelial cells resulting in massive electrolyte loss and dehydration [151,153,164]. Figure 8. Open in a new tab Proposed mechanism of cholera toxin (CT) traffic into intestinal epithelial cells. The CT holotoxin binds to GM 1 in the apical membrane. After endocytosis, the CT-GM 1 complex traffics retrograde through Golgi cisterna into endoplasmic reticulum (ER). Here, the A 1 subunit is unfolded and dissociated from the B pentamer. The unfolded A 1 peptide is probably translocated to the cytosol and may then gain access to its substrate, the heterotrimeric GTPase Gsα on the cytoplasmic surface of the basolateral membrane, by diffusion through the cytosol (if the A1 peptide breaks away from the membrane after translocation) or by membrane traffic back out of the secretory pathway (if the A1 peptide remains membrane associated). The B subunit is not unfolded in the ER, remaining membrane associated, probably bound to GM 1. It moves to the basolateral membrane by trafficking back out the secretory pathway by indirect transcytosis. Strains not carrying ctx genes can also be pathogenic, as they have several virulence factors and can produce other toxins like RTX cytotoxins , which will be discuss further in the context of toxins produced by V. vulnificus. Another V. cholera toxic metabolite is the cholix toxin . The existence of the cholix toxin was discovered by detailed analysis of non-01 and non-139 strain genomes that revealed the presence of a gene encoding a putative new exotoxin, similar to ExoA from Pseudomonas aeruginosa . Similarly to ExoA toxin from P. aeruginosa, cholix toxin is recognized by the lipoprotein receptor-related protein (LRP) of the host cells, enabling access to the cell cytoplasm. However, some recent studies performed in LPR receptor-deficient cells point to the existence of other paths for toxin entry, as these cells showed some sensitivity to the cholix molecule. Cholix toxin presents a KDEL sequence, responsible to direct it to ER, and is activated within the host cell by furin cleavage in an Arg-rich loop, together with the reduction of critical disulfide bridges . The toxin is an ADP-ribosylating protein that is specific to ribosomal eEF-2 elongation factor. It recognizes eEF-2 as the target protein substrate and has both glycohydrolase and ADP-ribosylating activities that are necessary to change diphthamide, the post-translationally modified histidine residue of this eukaryotic elongation factor. The modification mechanism involves the transfer of the ADP-ribose complex of NAD+ to the diphthamide imidazole in eEF-2 via a nucleophilic substitution were diphthamide imidazole is the nucleophile that replaces the nicotinamide base (leaving group) in NAD+. Exactly how ADP-ribosylation of the diphthamide inhibits eEF-2 function remains to be determined. Binding experiments of ADP-ribosylated eEF-2 to the ribosome show a reduction of affinity for the pre-translocational ribosome, but no changes are observed for the post-translocational ribosome. Competition and co-sedimentation experiments have indicated that the ADP-ribosylated eEF-2 is able to form a stable complex with the ribosome. However, other binding experiments have shown that ADP-ribosylated eEF-2 still has ribosome-dependent GTPase activity and can dissociate from the ribosome. These contradictory results confirm that further studies are necessary to obtain a better understanding of how these ribosylating toxins inactivates the elongation factor . Nevertheless, independently from the mechanism of eEF-2 inactivation, the final consequence for the host cell is the inhibition of protein synthesis and cell death. This toxic protein is coded by the chx A gene, corresponding to a 666 amino acid residue product (634 residues in the mature protein). The first 32 residues correspond to a leader sequence. It presents a sequence identity of 33.5% to ExoA (the catalytic domains alone have a sequence identity of 41.3%). In addition, chx A contains conserved residues previously shown to be crucial for catalytic activity . Full length structure of the protein demonstrated that cholix toxin is composed by three different domains (I–III) that are responsible by receptor binding, membrane translocation and enzyme catalysis, respectively . After diphtheria toxin and ExoA from P. aeruginosa, the V. cholerae cholix toxin is the third member of this group of protein synthesis inhibitors described so far. The chx A gene is widely distributed in V. cholerae populations, both in clinical and environmental strains. However, the specific targets of the toxin and the symbiotic interactions associated with its activity have yet to be determined. 3.1.2. Vibrio vulnificus This Vibrio species is part of the natural microbiota of coastal marine environments and is frequently present in water and seafood products like shrimp, fish, oysters and clams [167,168,169]. It is an opportunistic human pathogen, responsible for severe and fulminant systemic infections that are highly lethal . Consumption of seafood, especially raw oysters, is the main human contamination source. The characteristics of V. vulnificus infections include fever, chills, nausea, hypotensive septic shock and secondary lesion formation on the extremities of the body . Primary septicemia is the most lethal infection, with about 50% mortality rate. In addition, this bacterium can cause serious wound infections as a result of exposure to contaminated waters. Wounds are often acquired during recreational swimming, fishing or seafood handling . Two cytotoxins (an elastolytic protease, VvpE, and a hemolysin, VvhA) were first suggested to be responsible for the toxic effects of V. vulnificus [173,174]. However double mutants for the corresponding genes did not present significant differences in cytotoxicity from the wild type strain . Later it was shown that V. vulnificus strains are cytotoxic due the production of a repeats-in-toxin (RTX) exoprotein . This toxin is also present in several bacterial species including V. cholerae, and was first detected in this species, but is the main toxin in V. vulnificus, being essential for its virulence and infection. RTX toxins, including the RtxA1 from V. vulnificus, primarily promote pore formation in cellular membranes, but can also induce changes in cytoskeletal structure, bleb formation and aggregation of actin, resulting in cell rounding. In V. vulnificus infection, actin aggregation occurs without covalent cross-linking and is a depolymerization process that involves a change in the F/G actin dynamics for which Rho GTPases play important roles. These changes can lead to cellular necrosis and are probably related with the bacteria’s ability to destroy gastrointestinal epithelial/mucosal barrier and invade blood stream soon after infection . RTX operon contains its own processing and ABC transporter genes. After production RtxA1 is autoprocessed in at least two parts (approximately 350 and 130 kDa) and this activation mechanism is induced by the contact with host factors . Only the processed protein can act as cytotoxic factor. RTX proteins are usually large molecules, presenting GD-rich nonapeptide repeats (GGXGXDX[L/I/V/W/Y/F]X, where X is any amino acid). This structure is thought to be involved in the insertion of the toxin in the eukaryotic plasma membrane . However, the toxin of V. vulnificus and V. cholerae has some particular characteristics. The repeats are different, falling into three classes, presenting divergent sequences, but retaining a central conserved motif (G-7X-GXXN). The class A motive is also located in the N-terminus of the protein and not in the C-terminal part. Furthermore, the presence of the multifunctional autoprocessing domain in these toxins shows that they are part of a particular group of RTX toxins, called multifunctional autoprocessing repeats-in-toxins (MARTX). These toxins present several structural features that are different from other RTX toxins . The RtxA1 from V. vulnificus is estimated to be the largest RTX toxin known so far. The predicted amino acid sequence showed high homology with RtxA from V. cholerae. However, two domains show no identity with RtxA and it was hypothesized that these two regions may confer distinct activities to these toxins. RtxA1 toxin is likely to be much more cytotoxic to the host cells than RtxA from V. cholerae. These two domains may play an important role in pore formation in host cell membrane, which can be related with the cytotoxic mechanisms of RtxA1 . Identification of this toxin was possible by random chromosomal mutagenesis that allowed the detection of the corresponding gene (rtx A1). Two additional rtxA genes have also been identified, but their function is not clear, as they are not directly involved in cytotoxic activity. The first portion of the rtx operon (rtxBDE) includes genes for the type I secretion system that is responsible for the toxin secretion . The second part of the operon (rtxAC) includes the rtxC gene that codes for an activator of RtxA in V. cholerae, but probably has other function in V. vulnificus, as a mutation in this gene does not affect toxicity. Other potential regulator of rtxA (HlyU) was also identified in V. vulnificus. It is a transcriptional regulator that binds upstream of the rtx operon, initiating transcription . 3.2. Aeromonas hydrophila Aeromonas spp. are members of Aeromonadaceae that cause both intestinal and systemic infections in humans. Aeromonas hydrophila colonizes aquatic environments and is also isolated from food products . Although gastroenteritis occurs generally in young children, it has been frequently associated with the travel’s diarrhea . Furthermore, the cases of septicemia are often fatal. This species can express several virulence factors, including hemolysins, proteases, adhesins, lipases/phospholipases and toxins . This bacterium also has the ability to lyse erythrocytes, and has shown to be invasive and effective in triggering the proinflammatory response in macrophages and epithelial cell lines [179,182]. Act is an aerolisyn-related pore-forming toxin that is responsible for the hemolytic, cytotoxic and enterotoxic activities of A. hydrophila, being its main virulence factor. Hemolysis involves pore formation in the membrane of the target cell and water entry from the external media, resulting in swelling of the cells and subsequent lysis. The toxin interacts with the membranes of erythrocytes, inserts into the lipid bilayer as oligomers, and creates pores in the range of 1.14 to 2.8 nm. Cholesterol serves as the receptor for Act and the 3’-OH group of this membrane constituent is important for the interaction. Once Act has interacted with cholesterol on the cell membranes, the toxin is activated with subsequent oligomerization and pore formation . The toxin activity also includes tissue damage and high fluid secretion in intestinal epithelial cells, resulting from the induction of a proinflammatory response in the target cells (Figure 9). Act upregulates the production of proinflammatory cytokines such as tumor necrosis factor alpha (TNF-α), interleukin-1 beta (IL-1β) and IL-6 in macrophages. TNF-α and IL-1β stimulate the production of the inducible nitric oxide synthase (iNOS) that, through nitric oxide (NO) production, is an essential element of antimicrobial immunity and host-induced tissue damage. Simultaneously, Act has the ability to activate arachidonic acid (AA) metabolism in macrophages that leads to the production of eicosanoids (e.g., prostaglandin E2 [PGE 2]) coupled to cyclooxygenase-2 (COX-2) pathway. AA is a substrate for PGE2 production, but is present at limited concentrations in cells . Act increases the amount of AA from phospholipids by inducing group V secretory phospholipase A 2 (sPLA 2), which acts in the membrane of eukaryotic cells. Act increases cyclic AMP (cAMP) production in macrophages by indirect activation of adenylate cyclase by PGE 2. The A. hydrophyla toxin also induces the production of antiapoptotic protein Bcl-2 in macrophages, preventing the occurrence of massive apoptosis resulting from the induction of the inflammatory response, which would be undesirable for the bacteria. Act also promotes an increased translocation of the nuclear factor kB (NF-kB) and cAMP-responsive element binding protein (CREB) to the nucleus . Transcription factor NF-kB is important in a number of inflammation-related pathways. The enhancer/promoter regions of some immunoregulatory cytokine genes, including the TNF-α, IL-1β, and IL-6, present binding elements for NF-kB and CREB . These transcription factors have also important regulatory functions in the transcription of cox-2 and are implicated in the induction of Act cytotoxic activities. The production of proinflammatory cytokines and iNOS causes an extensive tissue injury in the intestinal loops. It also loosens the tight junctions around epithelial cells, allowing the influx of inflammatory cells into the intestinal lumen and increasing the uptake of Act into lamina propria, where inflammatory cells can be activated. Moreover, PGE2 along with cAMP leads to the stimulation of fluid secretory response and the subsequent fluid loss . The mature protein is 52 kDa and contains 493 amino acids. It is secreted as an inactive precursor and undergoes processing at both the N- and C-terminal ends to demonstrate biological activity. It has a leader sequence of 23 amino acids that allows the protein to transverse the inner membrane. This leader peptide is removed when the toxin enters the periplasmic space. After the secretion of Act into the medium, a polypeptide of approximately 4–5 kDa (about 45 amino acids) is cleaved from its C-terminus by a protease produced by A. hydrophila, resulting in the mature form of the toxin. This toxic protein is coded by A. hydrophila act gene, corresponding to a 1479 bp open reading frame. The first 69 bp codes for the N-terminal signal peptide. The typical regulatory -35, -10 and Shine-Delgarno sequences, as well as the promoter, were identified upstream of the coding region [179,184,185]. More recently, two other toxins were detected in A. hydrophila. Alt is a 44 kDa protein, with 368 amino acid residues, coded by the corresponding alt gene. The other is Ast, the product of a 1911 bp open reading frame that originates a protein of 71 kDa and 636 amino acids. Both represent new molecules with no significant homology to other bacterial toxins. Although there is evidence of their contribution to the elevated cAMP and prostaglandin E2 levels in infected cells, their specific roles must be clarified . Figure 9. Open in a new tab Act mediated pathways for activation of inflammatory response and apoptosis inhibition (see text for details). 3.3. Escherichia coli Escherichia coli is a genetically heterogeneous group of bacteria from the Enterobacteriaceae families, whose members are generally non-pathogenic and are part of the normal microbiota of the intestinal tract of humans and animals . However, some strains have acquired genes that enable them to cause diseases . These strains can be divided into two types (pathotypes), based on the mechanisms and virulence factors by which they cause disease . One of these E. coli pathotypes (STEC) corresponds to the enterohemorrhagic strains and is characterized by the production of at least one type of Shiga toxin, a family of structurally and functionally related exotoxins, which includes the toxin produced by Shigella dysenteriae . This E. coli toxin is also known as verotoxin due to its effect in Vero cells. Ruminants, in particular cattle, constitute a vast source of STEC and is frequent that human infections are originated in food and water contaminated with cattle manure, especially because they are carried by healthy animals . Infections in humans may result in water diarrhea, bloody diarrhea or in the hemolytic uremic syndrome (HUS), characterized by acute renal failure, hemolytic anemia and other severe symptoms. The kidney and the gastrointestinal tract are the most affected organs, but lungs, heart, central nervous system and pancreas can also be targeted. HUS develops in 5–10% of individuals infected with STEC O157:H7, the most frequent serotype causing this infection in humans . Shiga toxins are encoded in the genome of several lambda-like bacteriophages, which represent highly mobile genetic elements and play a central role in horizontal gene transfer . Phage stx genes are located in the late region, downstream of the late promoters, and are highly expressed during the lytic cycle . The folding and assemblage of the toxins are only possible in the particular conditions of bacterial periplasm and their release seems to occur by phage mediated bacterial lysis. Two types of Shiga toxin are recognized: Stx1 and Stx2, coded by the corresponding genes stx1 and stx2. Stx1 is highly conserved in structure and similar to Shiga toxin from S. dysenteriae. However, a variant Stx1c was already identified and has been mainly associated to ovine origins . In contrast, there are several antigenic variants of Stx2: Stx2c, Stx2d, Stx2d-activatable and Stx2e. Stx2d and Stx2e are believed to be associated to mild or asymptomatic diseases and Stx2c is believed to be less frequent in patients presenting severe symptoms . The Stx molecules (approximately 70 kDa) present an A 1 B 5 hexameric structure of toxins from AB 5 family, in which A subunit (32 kDa) is non-covalently linked to the pentamer of B subunits (7.7 kDa each). Subunit A (StxA) is enzymatically active and the B fragments (StxB) are responsible for host cell binding [186,188]. StxB binds to the neutral glycosphingolipid globotriaosylceramide (Gb3), which is present at the surface of susceptible cells, allowing the internalization of the toxin . Each StxB fragment binds to three trisaccharide sites. Sites 1 and 2 mediate high affinity receptor binding and are the most relevant for cell cytotoxicity. The third site mediates the recognition of additional low affinity Gb3 epitopes . Distinct variants of the Stx toxin present some conformational divergences in site 2 and show affinity for Gb3 receptors with different fatty acid chains . After binding to the specific receptor in the plasma membrane, Stx enters the cell by endocytosis. The toxin can induce endocytic invaginations in the plasma membrane without the help of cell machinery. Membrane blending results from the glycosphingolipid receptors aggregation, mediated by Stx . The membrane invaginations are then processed by the cell mechanisms, involving dynamin, actin and membrane cholesterol. After entry into the cell, Shiga toxin localizes in early endosomes, but it escapes the late endocytic pathway and is directly transferred to trans-Golgi network (TGN) and then to ER. Dynamin and retromer are molecules implicated in membrane blending and have shown to be important for the direct transfer of Stx from early endosome to TGN [195,196]. The transport mechanism from TGN to ER is unknown, but is independent from the coat protein complex I (COPI). Shiga toxin does not induce pore formation in the cell membrane. Like RtxA from V. cholerae, it relies on host cell machinery to translocate the A active subunit to the cytosol. During the early entry process, a protease sensitive loop in the C-terminal region of StxA (residues 242–261) is cleaved by a membrane-associated endoprotease (furin), originating two StxA fragments: the catalytic fragment A1 (amino acids 1–251) and the StxB associated fragment A2 (amino acids 252–293). A1 domain remains linked to StxA2-StxB complex by a disulfide bond, which is reduced in ER lumen, releasing the catalytic domain that is subsequently translocated to the cytosol [186,194] (Figure 10). Figure 10. Open in a new tab Trafficking mechanism of Shiga toxins. Toxin binding to the plasma membrane induces membrane-mediated clustering and the toxin-driven endocytosis. The toxin then undergoes retrograde sorting in early endosomes, in a dynamin-dependent process. Shiga toxins bypass the late endocytic pathway and are transferred directly from the early endosome to the trans-Golgi network (TGN) and from there to the endoplasmic reticulum (ER). Finally, Shiga toxins use the ER-associated degradation (ERAD) machinery to facilitate retrotranslocation into the host cell cytosol. There it can reach its cellular targets. Shiga toxin has a highly specific RNA-glycosidase activity that cleaves an adenine base from the 28S ribosomal RNA (rRNA) of the eukaryotic ribosome . The 3’ end of 28S rRNA functions in the aminoacyl t-RNA binding, peptidyltransferase activity and ribosomal translation. This modification of ribosomes inhibits the tRNA binding to the ribosome and the subsequent chain elongation. It also triggers a signaling response termed ribotoxic response that includes the activation of the JUN N-terminal kinase and p38 (a mitogen-activated protein kinase—MAPK), altering the signaling of the extracellular signal-regulating kinases ERK1 and ERK2 . The toxin also activates several cellular kinases, like tyrosine kinases that phosphorylate several proteins including dynamin, which favors toxin uptake, and p38, implicated in ribotoxic response . Shiga toxin damages the microcirculation and the intestinal mucosa, leading to bleeding into the bowel and bloody diarrhea. This provides essential nutrients to the bacteria, favoring its survival . Shiga also induces cytokine synthesis and release. Some monocytes and macrophages are resistant to the toxin and respond to toxin binding and internalization by producing and releasing pro-inflammatory cytokines, which in turn stimulate the Gb3 biosynthesis and expression in several endothelial cells, promoting the cytotoxic action of the toxin. After crossing the intestinal epithelium and entering in circulatory system, Stx stimulates the monocytes to secrete cytokines like GM-CFS and TNF . Interleukin 8 (IL-8) probably plays a central role in this process. All these reactions can contribute to the endothelial cell damage. Furthermore, Stx can signal apoptosis by a mechanism that requires retrograde transport through Golgi apparatus and ER and the activation of caspase 3. Caspases are cysteine-dependent, aspartate-specific proteases that are a central component of the programmed cell death pathway. Shiga toxin induces the ER stress response, a cellular mechanism that is usually activated under the accumulation of unfolded and misfolded proteins, leading to Ca 2+ release from ER stores [201,202]. Ca 2+ activates cysteine protease calpain that activates caspase 8 through cleavage. This last protease directly activates caspase 3, initiating the apoptotic process (Figure 11). Figure 11. Open in a new tab Apoptosis pathway triggered by induction of ER stress response and Ca 2+ mediated calpain activation. Bacterial ribosomes are also susceptible to Stx toxin , decreasing the proliferation of susceptible bacteria from the intestinal microbiota and allowing a more efficient propagation of the pathogen. 3.4. Legionella pneumophila Legionella pneumophila is the pathogenic organism responsible for the Legionnaires disease, a potential lethal pneumonia that results from the ability of this bacterium to survive and replicate in macrophages. No animal reservoir is known, but its natural hosts and environmental source are aquatic protozoa (e.g., amoebae), in which they replicate and seems to enhance its ability to infect mammalians cells . The human infection occurs mostly by inhalation of aerosols generated by domestic and environmental water sources . L. pneumophila cells enter the macrophages by vacuoles that are immediately surrounded by vesicles or mitochondria and move towards the endoplasmic reticulum, escaping from fusion with lysosomes and reaching a perfect niche for their multiplication . L. pneumophila also produces a toxin of the RTX family (of which the general structure is discussed above), a pore-forming protein that has an important role in adherence to host membranes and in the molecular traffic of the bacteria during the infection process . Its role in bacterial adherence and traffic may be due to the ability of these proteins to bind β 2 integrin receptors in the target cell membranes . After entering the cell, this toxin induces pore formation in the vacuoles, preventing proper docking of lysosomes and fusion. This process allows bacterium survival inside the host cell. In this species, RTX toxin (RtxA) is a large protein (about 7000 amino acids), with a variable number of repeated units and a modular structure. The toxin appears to be clearly divided in two regions: the N-terminal, involved in cell adhesion and the C-terminal, involved both in adhesion and pore-forming. The repeats at the N-terminal end are also highly variable among strains . Analysis of the RtxA genes in several strains showed the existence of long tandem repeats (460–549 bp), variable in number and sequence, and with a high level of rearrangements and diversity when compared with the flanking regions . The corresponding ORFs range between 4669 to 7910 bp. Different kinds of cell adhesion domains were identified in the N-terminal region of rtx genes of the diverse strains. Two of these motifs were always present: (i) the von Willebrand factor A (VWA) domain, involved in cell membrane adhesion processes, and (ii) other tandem repeat domain with homology with the hemolysin calcium-binding site, which is related with adhesion to other host surfaces and pore formation . Recently a glucosyltransferase from L. pneumophila (Lgt1) was identified. This Legionella enzyme (a protein with 60 kDa) modifies the eukaryotic eEF-1A. This elongation factor is a GTP-binding protein, possessing GTPase activity. Lgt1 alters the serine-53 of eEF-1A, located in the GTPase domain. The modification results in inhibition of protein synthesis both in vitro and in vivo and causes extreme changes in cellular morphology and, ultimately, death of intoxicated eukaryotic cells . This bacterium is able to multiply inside phagocytes. Toxin expression is only induced after a successful replication, when bacteria go out to the surrounding medium, being important to the transmission to a new host. L. pneumophila have several open reading frames encoding unknown proteins that can correspond to unknown toxins and justify further studies on this species. 3.5. Campylobacter spp. The genus Campylobacter includes curved, S-shaped and spiral rods, with a microaerophilic metabolism and presenting spherical or coccoid cells in old cultures or under oxygen exposure . Two species of this genus (Campylobacter jejuni and Campylobacter coli) are important causes of diarrheal diseases worldwide. Campylobacter spp. infections usually cause severe gastroenteritis, but can also be responsible for prolonged enteritis, bacteremia, septic arthritis and other infections . Campylobacter spp. are zoonotic and many animals are natural reservoirs for human disease, including poultry, cows, sheep and pigs. Campylobacters are frequently isolated from water, and water supplies have been a source of some outbreaks . The pathogenic mechanisms and virulence factors responsible for Campylobacter spp. gastroenteritis are not completely understood, but like other enteric pathogens, Campylobacter spp. colonize, invade and transmigrate across human intestinal cells. The interaction of the bacteria with the intestinal epithelium induces the production of several pro-inflammatory cytokines, including interleukin-8 (IL-8), a major cytokine secreted by the intestinal epithelial cells. IL-8 functions as a chemoattractant, recruiting neutrophils to the site of infection, contributing to the inflammatory response. One virulence factor already identified is a cytolethal distending toxin (CTD) that induces distention of the cytoplasm of infected cell, increase in their DNA content and accumulation of the inactive tyrosine phosphorylated form of Cde2, a key regulator of cell cycle progression . This type of toxin has already been found in other microorganisms like Shigella spp. or Helicobacter hepaticus [214,215]. This toxin is also implicated in the induction mechanism of IL-8 production . CTD is required for IL-8 production, in addition to toll-like receptors (TLRs), which play a central role in initiating the inflammatory response. Campylobacter jejuni cytotoxic strains secrete CTD by a flagellum mediated mechanism and activate NF-kB. The production of this nuclear transcription factor is stimulated both by CTD and TLRs, via MyD88 signaling TLRs adaptor, and triggers IL-8 . CDT is an AB 2 heterodimeric tripartite toxin, composed of three subunits: CdtA, CdtB and CdtC. Amino acid sequence of the CtdB subunit shows high similarity with members of DNase A nuclease family. Although limited to some residues, they correspond to a motif that is essential for nuclease activity. Thus, CtdB is the active subunit of CTD, acting by DNA damaging and stopping cell cycle in G 2/M phase . CtdA and CtdC fragments are involved in the delivery of the active subunit CtdB into the cell . The toxin is encoded by three genes, ctdA, ctdB and ctdC that are located in a chromosomal operon. 4. Water Contaminating Toxin Producing Bacteria Other toxigenic species like Clostridium spp. and Pseudomonas spp. are ubiquitous in the environment. Even though they are not aquatic microorganisms, they can easily contaminate drinking or irrigation waters and become hazards for humans and animals. Some representative examples of toxins produced by these bacteria will be briefly referred below. 4.1. Clostridium spp. Clostridium is a genus of pleomorphic, gram-positive, anaerobic rods that are widespread in several habitats, including soils, wetlands, lakes, coastal waters, intestinal track of fish and gills and viscera of crabs and other shellfish . They are highly resistant to adverse environmental conditions as they are able to produce endospores. These bacterial spores are cellular structures that can resist to extreme temperatures, desiccation, chemicals and radiation. When favorable growth conditions are reestablished, the spores can germinate, originating viable vegetative cells . Recreation and drinking water can became contaminated with Clostridium spp. spores from sources like soils and dust, and insects also contribute for their spread. Clostridium spp. cells are not pathogenic by themselves. However several species may produce exotoxins that can be extremely hazardous . 4.1.1. Clostridium botulinum Botulism, the disease cause by C. botulinum, is a severe neurological illness that causes paralysis. It is originated by a potent neurotoxin produced by this bacterium. Seven antigenically distinct botulism toxins (types A, B, C1, D, E, F and G) were identified and are considered among the most toxic substances known. Botulism toxins (BoNTs) are water soluble large molecules (150 kDa), produced as single peptides. The active toxin is composed by a heavy chain (H) and a light chain (L), linked by a disulfide bond. It is originated after proteolytic cleavage by endogenous bacterial proteases . BoNTs may form oligomers, which can be involved in channel formation and translocation of the protein into the host cytoplasm. The toxin can also form complexes with other proteins such as nonhemagglutinin (NTNH) and hemagglutinines [221,222]. These spontaneously formed complexes confer toxin resistance to proteases and their association is pH dependent. Complexes are maintained at pH lower than 7.2 and spontaneously dissociate at higher values. The toxin acts by blocking the neurotransmission at peripheral motor nerve terminals. They selectively hydrolyze proteins involved in the fusion of synaptic vesicles with the presynaptic plasma membrane, preventing acetylcholine release . This process occurs in four steps: binding, membrane translocation, internalization and intracellular action . The H chain is responsible for the selective binding to neurons, internalization of the total toxin and intraneuronal sorting. L chain blocks exocytosis after toxin release in the cytoplasm . The neurotoxin genes are located in a transcriptional unit, together with the genes encoding NTNH and hemagglutinins. This unit is referred as BoNT gene complex . The location of BoTN genes and associated nontoxic proteins depends on the type of toxin. BoNT A, B, E and F genes are located in the bacterial chromosome together with the genes of the corresponding associated proteins . BoNT gene complexes of serotypes C1 and D are coded by a bacteriophage . Genes of serotype G are located in a plasmid . In each gene complex, NTNH gene is located immediately upstream of BoNT genes in all toxin types . Genes for the other components are clustered upstream of NTNH gene for most of the toxins. Only C1 type presents a different arrangement, including three transcriptional units: (i) the first including NTNH and BoTN genes, (ii) the second composed by three genes coding hemagglutinins and (iii) the last one, with only one gene, probably having a regulatory function . 4.1.2. Clostridium perfringens The virulence of Clostridium perfringens largely results from its ability to produce toxins. At least 14 different toxins have been identified, but each strain only carries genes for a defined subset of the total repertoire. This characteristic provides the basis for a toxin typing system that groups C. perfringens isolates into five types (A to E), depending on its ability to produce four (alpha, beta, epsilon and iota) of the 14 toxins . The total set of toxins includes two that are active in the human intestinal tract: C. perfringens enterotoxin and beta-toxin, each one associated with a distinct disease. One is necrotic enteritis, caused by type C strains, with the beta-toxin considered as the main virulence factor. This toxin is related to alpha-toxin and leukocidin from Staphylococcus aureus. Beta-toxin is considered a single component toxin, presenting a molecular mass of about 35 kDa and 334 amino acids . The first 27 residues correspond to a signal peptide, directing the export of the toxin across the cell membrane. The toxin is encoded by a single open reading frame of approximately 1006 bp (cpb gene) with the Shine-Dalgarno sequence located at 7 bp upstream the ATG start codon . C. perfringens type A is responsible for most of the human diseases caused by this species. The symptoms of the infection (diarrhea and cramping) are due to the C. perfringens enterotoxin (CPE), a peptide presenting a unique amino acid sequence and mechanism of action. CPE is a thermo-labile protein with 319 amino acids, without N-terminal secretion sequence and presenting a molecular mass of 35 kDa . The production of this enterotoxin is associated with the sporulation process of C. perfringens and can represent more than 10% of the total protein in sporulating cells . The toxin is released when the mother cell lyses at the end of the sporulation. Thus, during this process, much CPE is accumulated in paracrystalline inclusion bodies in the cytoplasm of C. perfringens cells . The action of CPE is a multistep process. It starts with the binding of the toxin to its intestinal receptors, inducing the formation of a small complex that entraps CPE at the membrane surface. This complex interacts with other proteins, forming a second complex of intermediate size, which can interact with occludin (a tight junction protein) or other proteins of the eukaryotic cell. This causes the loss of membrane permeability characteristics, probably because the complex has pore-like properties or by inducing tight junction rearrangements . The N-terminal end of the toxin has the cytotoxic activity and the C-terminus is likely to be involved in binding to cell membrane. The CPE coding gene (cpe) can be located either in the bacterial chromosome or in a conjugative plasmid. The corresponding ORFs are identical. The expression of cpe is induced by regulators (e.g., alternative sigma factor) that are involved in the sporulation process, explaining its high production in sporulating cultures . One possible regulator is the Hpr global regulator, as Hpr-like binding sequences have been identified upstream and downstream of cpe ORF . Since Hpr is known to repress the expression of some genes during exponential growth of Bacillus subtilis, the role of Hpr in CPE expression could explain the lack of expression of the toxin in vegetative cells. 4.2. Pseudomonas aeruginosa Pseudomonas is a genus of gram-negative, rod-shaped and mobile bacteria that demonstrate a great metabolic diversity and ability to use a high number of organic substrates, including phenol derivatives and hydrocarbons . A significant number of species can also produce exopolysaccharides and create biofilms . These two characteristics highly contribute to their ability to colonize a wide range of habitats, being widespread in the environment, including plants, animals, soils and water. Pseudomonas aeruginosa is increasingly recognized as an emerging opportunistic pathogen of clinical relevance . As with other species, it is ubiquitous in the environment and can easily adapt to adverse conditions. The human infections range from acute infections like endocarditis, meningitis and septicemia to chronic lung infections in cystic fibrosis patients. Most infections occur in immunocompromised patients, like AIDS patients, burn victims or patients undergoing chemotherapy . P. aeruginosa presents several virulence factors, including the exotoxin A (ExoA/PE). It belongs to the same family of mono-ADP-ribosyltransferases of V. cholerae cholix toxin, which catalyses the ADP rybosilation of eukaryotic eEF-2 and consequently inhibits protein synthesis . PE is translated from an ORF with 2760 bp as a monocistronic message and is a protein of 613 amino acids, resulting from a 638 amino acid precursor that includes a hydrophobic leader peptide of 25 amino acids . As described before for cholix toxin, PE presents three distinct functional domains. The first, the N-terminal Ia domain (a.a. 1–252) is responsible for cell recognition . Afterwards, the toxin enters the cell and is internalized in early endosomes, where it is cleaved by the protease furin, originating two fragments: a N-terminal with 28 kDa and a C-terminal with 37 kDa . This last one is the active fragment and is released in a pH dependent process. Then it is transported to the endoplasmatic reticulum via late endosomes and Rab9-dependent route to the trans-Golgi network and finally traveling by the KDEL receptor mediator pathway [241,242]. When the enzymatic subunit of PE becomes present in the cytosol, it promotes the ADP ribosylation of the eEF-2. The ADP-ribosylation mechanism develops as previously described for cholix toxin, inactivating eEF-2 and inhibiting protein synthesis. Consequently, it leads to cell death. 5. Final Remarks Bacteria are ubiquitous in the environment and have the ability to adapt to very different habitats. Their survival is often dependent on the production of compounds that help them to attach to substrates, compete for nutrients and inhibit the growth of other microorganisms. Many virulence factors are products that give them advantages in their original environment, but in a host they function as pathogenic mechanisms of disease development. Some of these products represent powerful toxins that can lead to host disease and, frequently, to death. On the other hand, water is not only a crucial resource but is also a requisite to maintain life. The contamination of water systems by toxigenic microorganisms can have a catastrophic impact in health and well being of all living organisms. Toxins or toxin producers in water systems can arise either by contamination from other environmental sources (e.g., Clostridium spp. , P. aeruginosa or E. coli ) or because aquatic systems are their natural habitats (e.g., cyanobacteria, Vibrio spp. or Aeromonas hydrophila ). Microbial toxins are very diverse and there are many microorganisms recognized as producers. Although this review only focuses on the more relevant toxins and toxin producers, more toxic molecules are emerging in other organisms. An overview of the toxins produced by bacteria related to aquatic environments is summarized in Table 1. In fact, a better knowledge about all aspects of these microorganisms seems crucial. Identification of routes and sources of water contamination (Figure 12) will allow the design of preventive actions to avoid this contamination or to prevent conditions that favor the development of toxic microorganisms and toxin production. Table 1. Toxins produced by prokaryotes related to aquatic environments. | Mode of action | Toxin name | Produced by | References | :---: :---: | | Membrane permeabilizing toxins | Act | A. hydrophila | | | | α-Hemolysin | E. coli | | | | Bifermentolysin | C. bifermentans | | | Botulinolysin | C. botulinum | | Chauveolysin | C. chauvoei | | Histolyticolysin O | C. hystolyticum | | Novyilysin | C. novyi A | | Perfringolysin O | C. perfringens | | Septicolysin O | C. septicum | | Toxins affecting membrane traffic | Botulinum neurotoxin | C. botulinum | | | Toxins affecting signal transduction | Cholera toxin | V. cholerae | | | Heat-labile enterotoxin | E. coli | | | Toxins affecting protein synthesis | Cholix toxin | V. cholera | | | | Exotoxin A | P. aeruginosa | | | | Shiga toxin (verotoxin) | E. coli | | | | Lgt1 | L. pneumophila | | | | RtxA | V. vulnificus | | | | RtxA | L. pneumophila | | | Toxins inhibiting protein function | Cylindrospermopsin | Cyl. raciborskii | [82,83,84,85,86,87,88,89,90] | | Umezakia natans | | Aph. ovalisporum | | Raph. curvata | | A. bergii | | Aph. flos-aquae | | Lyngbya wollei | | | Microcystins | Microcystis | [27,51] | | Planktothrix | | Oscillatoria | | Nostoc | | Anabaena | | Anabaenopsis | | Hapalosiphon | | Snowella | | Woronichinia | | Arthrospira | | Phormidium | | Plectonema | | Pseudoanabaena | | Synechococcus | | Synechocystis | | | Nodularins | Nodularia spumigena | | | Cytoskeleton-affecting toxins | Anatoxin-a and homoanatoxin-a | Anabaena | [32,59,100,101,102,103,104,105,106] | | Oscillatoria | | Cylindrospermum | | Microcystis | | Aphanizomenon | | Planktothrix | | | C2 toxin | C. botulinum | | | | Cytotoxic necrotizing factors | E. coli | | | DNA damaging | Cytolethal distending toxin | Campylobacter spp. | | | Voltage-gated ions channel blockers | Saxitonin and gonyautoxins | A. circinalis | | | Aph. gracile | | C. raciborskii | | L. wollei | | Planktothrix | | | Jamaicamides | Lyngbya majuscula | | | Kalkitoxin | | Antillatoxin | | Unknown | Lypopolysaccharides (LPS) | E. coli | [39,133] | | Salmonella spp. | | V. cholera, | | P. aeruginosa | | Cyanobacteria | Open in a new tab Figure 12. Open in a new tab Natural reservoirs of the main bacterial toxin producers and routes for water, animal and human contamination. Clarification of the mechanisms of toxin action will certainly open new perspectives for efficient antidote as well as vaccine development [246,247]. Furthermore, the potential of these toxic products as anticancer, antifungal or antibiotic drugs are now recognized and can be an important source for biomedical and biotechnological applications in the near future [188,237,246]. References 1.Rappuoli R., Montecucco C., editors. Guidebook to Protein Toxins and Their Use in Cell Biology. Sambrook & Tooze Publications at Oxford University Press; New York, NY, USA: 1997. [Google Scholar] 2.Sivonen K., Jones G. Cyanobacterial toxins. In: Chorus I., Bartram J., editors. Toxic Cyanobacteria in Water: A Guide to Their Public Health Consequences, Monitoring and Management. Chapter 3 WHO, E & FN Spon; London, UK: 1999. pp. 41–111. [Google Scholar] 3.Wang D. Neurotoxins from marine dinoflagellates: A brief review. Mar. Drugs. 2008;6:349–371. doi: 10.3390/md20080016. [DOI] [PMC free article] [PubMed] [Google Scholar] 4.Herrero-Galán E., Álvarez-García E., Carreras-Sangrà N., Lacadena J., Alegre-Cebollada J., del Pozo A.M., Oñaderra M., Gavilanes J.G. Fungal ribotoxins: structure, function and evolution. In: Proft T., editor. Microbial Toxins: Current Research and Future Trends. Caister Academic Press; Wymondham, UK: 2009. [Google Scholar] 5.Berestetskiy A.O. A review of fungal phytotoxins: from basic studies to practical use. Appl. Biochem. Microbiol. 2008;44:453–465. [PubMed] [Google Scholar] 6.Zitzer A., Wassenaar T., Walev I., Bhakdi S. Potent membrane-permeabilizing and cytocidal action of Vibrio cholerae cytolysin on human intestinal cells. Infect. Immun. 1997;65:1293–1298. doi: 10.1128/iai.65.4.1293-1298.1997. [DOI] [PMC free article] [PubMed] [Google Scholar] 7.Martinez-Arca S., Alberts P., Zahraoui A., Louvard D., Galli T. Role of tetanus neurotoxin insensitive vesicle-associated membrane protein (TI-VAMP) in vesicular transport mediating neurite outgrowth. J. Cell Biol. 2000;149:889–900. doi: 10.1083/jcb.149.4.889. [DOI] [PMC free article] [PubMed] [Google Scholar] 8.Chen S., Barbieri J.T. Engineering botulinum neurotoxin to extend therapeutic intervention. Proc. Natl. Acad. Sci. USA. 2009;106:9180–9184. doi: 10.1073/pnas.0903111106. [DOI] [PMC free article] [PubMed] [Google Scholar] 9.Lencer W.I., Strohmeier G., Moe S., Carlson S.L., Constable C.T., Madara J.L. Signal transduction by cholera toxin: processing in vesicular compartments does not require acidification. Am. J. Physiol. Gastrointest. Liver Physiol. 1995;269:G548–G557. doi: 10.1152/ajpgi.1995.269.4.G548. [DOI] [PubMed] [Google Scholar] 10.Cherla R.P., Lee S.-Y., Mees P.L., Tesh V.L. Shiga toxin 1-induced cytokine production is mediated by MAP kinase pathways and translation initiation factor eIF4E in the macrophage-like THP-1 cell line. J. Leukoc. Biol. 2006;79:397–407. doi: 10.1189/jlb.0605313. [DOI] [PubMed] [Google Scholar] 11.Menestrina G., Pederzolli C., Forti S., Gambale F. Lipid interaction of Pseudomonas aeruginosa exotoxin A. Acid-triggered permeabilization and aggregation of lipid vesicles. Biophys. J. 1991;60:1388–1400. doi: 10.1016/S0006-3495(91)82176-2. [DOI] [PMC free article] [PubMed] [Google Scholar] 12.Pothoulakis C., LaMont J.T., Eglow R., Gao N., Rubins J.B., Theoharides T.C., Dickey B.F. Characterization of rabbit lleal receptors for Clostridium difficile toxin A. Evidence for a receptor-coupled G protein. J. Clin. Invest. 1991;88:119–125. doi: 10.1172/JCI115267. [DOI] [PMC free article] [PubMed] [Google Scholar] 13.Fiorentini C., Arancia G., Caprioli A., Falbo V., Ruggeri F.M., Donelli G. Cytoskeletal changes induced in HEp-2 cells by the cytotoxic necrotizing factor of Escherichia coli. Toxicon. 1988;26:1047–1056. doi: 10.1016/0041-0101(88)90203-6. [DOI] [PubMed] [Google Scholar] 14.Aráoz R., Molgó J., Tandeau de Marsac N. Neurotoxic cyanobacterial toxins. Toxicon. 2010;56:813–828. doi: 10.1016/j.toxicon.2009.07.036. [DOI] [PubMed] [Google Scholar] 15.Castenholz R.W. Species usage, concept, and evolution in the cyanobacteria (blue-green algae) J. Phycol. 1992;28:737–745. [Google Scholar] 16.Schopf J.W. The fossil record: tracing the roots of the cyanobacterial lineage. In: Whintton B.A., Potts M., editors. The Ecology of Cyanobacteria. Kluwer; Dordrecht, the Netherlands: 2000. pp. 13–35. [Google Scholar] 17.Castenholz R.W., Waterbury J.B. Group I. Cyanobacteria. In: Krieg N.R., Holt J.G., editors. Bergey’s Manual of Systematic Bacteriology. Vol. 3. Williams & Wilkins; Baltimore, MD, USA: 1989. pp. 1710–1728. [Google Scholar] 18.Falconer I.R. Cyanobacterial Toxins of Drinking Water Supplies—Cylindrospermopsins and Microcystins. CRC Press; Boca Raton, FL, USA: 2005. [Google Scholar] 19.Sze P. A Biology of the Algae. 2nd. WCB Publishers; Boston, MA, USA: 1986. Prokaryotic algae (Cyanophyta, Prochlorophyta) pp. 19–34. [Google Scholar] 20.Madigan M.T., Martinko J.M., Parker J. Brock Biology of Microorganisms. 9th. Prentice Hall; Upper Saddle River, NJ, USA: 2000. [Google Scholar] 21.Castenholz R.W. Bergey’s Manual of Systematic Bacteriology. Springer; New York, NY, USA: 2001. Phylum BX. Cyanobacteria; pp. 473–599. [Google Scholar] 22.Mur L.R., Skulberg O.M., Utkilen H. Cyanobacteria in the environment. In: Chorus I., Bartram J., editors. Toxic Cyanobacteria in Water: A guide to Their Public Health Consequences, Monitoring and Management. Chapter 2 WHO, E & FN Spon; London, UK: 1999. [Google Scholar] 23.Codd G.A., Morrison L.F., Metcalf J.S. Cyanobacterial toxins: risk management for health protection. Toxicol. Appl. Pharmacol. 2005;203:264–272. doi: 10.1016/j.taap.2004.02.016. [DOI] [PubMed] [Google Scholar] 24.Francis G. Poisonous Australian lake. Nature. 1878;18:11–12. [Google Scholar] 25.Mahmood N.A., Carmichael W.W., Pfahler D. Anticholinesterase poisonings in dogs from a cyanobacterial (blue-green algae) bloom dominated by Anabaena flos-aquae. Am. J. Vet. Res. 1988;49:500–503. [PubMed] [Google Scholar] 26.Codd G.A., Bell S.G., Brooks W.P. Cyanobacterial toxins in water. Water Sci. Technol. 1989;21:1–13. [Google Scholar] 27.Carmichael W.W. Cyanobacterial secondary metabolites—the cyanotoxins. J. Appl. Bacteriol. 1992;72:445–459. doi: 10.1111/j.1365-2672.1992.tb01858.x. [DOI] [PubMed] [Google Scholar] 28.Negri A.P., Jones G.J., Hindmarsh M. Sheep mortality associated with paralytic shellfish poisons from the cyanobacterium Anabaena circinalis. Toxicon. 1995;33:1321–1329. doi: 10.1016/0041-0101(95)00068-w. [DOI] [PubMed] [Google Scholar] 29.Henriksen P., Carmichael W.W., An J., Moestrup Ø. Detection of an anatoxin-a(s)-like anticholinesterase in natural blooms and cultures of cyanobacteria/blue-green algae from danish lakes and in the stomach contents of poisoned birds. Toxicon. 1997;35:901–913. doi: 10.1016/S0041-0101(96)00190-0. [DOI] [PubMed] [Google Scholar] 30.Matsunaga H., Harada K.-I., Senma M., Ito Y., Yasuda N., Ushida S., Kimura Y. Possible cause of unnatural mass death of wild birds in a pond in Nishinomiya, Japan: sudden appearance of toxic cyanobacteria. Nat. Toxins. 1999;7:81–84. doi: 10.1002/(sici)1522-7189(199903/04)7:2<81::aid-nt44>3.0.co;2-o. [DOI] [PubMed] [Google Scholar] 31.Krienitz L., Ballot A., Kotut K., Wiegand C., Putz S., Metcalf J.S., Codd G.A., Pflugmacher S. Contribution of hot spring cyanobacteria to the mysterious deaths of Lesser Flamingos at Lake Bogoria, Kenya. FEMS Microbiol. Ecol. 2003;43:141–148. doi: 10.1111/j.1574-6941.2003.tb01053.x. [DOI] [PubMed] [Google Scholar] 32.Wood S.A., Selwood A.I., Rueckert A., Holland P.T., Milne J.R., Smith K.F., Smits B., Watts L.F., Cary C.S. First report of homoanatoxin-a and associated dog neurotoxicosis in New Zealand. Toxicon. 2007;50:292–301. doi: 10.1016/j.toxicon.2007.03.025. [DOI] [PubMed] [Google Scholar] 33.Dennison W.C., O’Neil J.M., Duffy E.J., Oliver P.E., Shaw G.R. Blooms of the cyanobacterium Lyngbya majuscula in coastal waters of Queensland, Australia. Bull. Inst. Oceanogr. Monaco. 1999;19:501–506. [Google Scholar] 34.WHO, authors. Guidelines for Safe Recreational Water Environments. Coastal and Freshwaters. Vol. 1. World Health Organization; Geneva, Switzerland: 2003. pp. 136–158. [Google Scholar] 35.Teixeira M.G.L.C., Costa M.C.N., Carvalho V.L.P., Pereira M.S., Hage E. Gastroenteritis epidemic in the area of the Itaparica Dam, Bahia, Brazil. Bull. Pan Am. Health Organ. 1993;27:244–253. [PubMed] [Google Scholar] 36.Carmichael W.W., An J.S., Azevedo S.M.F.O., Lau S., Rinehart K.L., Jochisen E.M., Holmes C.E.M., Silva J.B. Analysis for microcystins involved in outbreak of liver failure and death of humans at a hemodialysis center in Caruaru, Pernambuco, Brazil; Proceedings of the IV Symposium of the Brazilian Society of Toxinology; São Paulo, Brazil. 6–11 October, 1996. [Google Scholar] 37.Vasconcelos V.M. Uptake and depuration of the heptapeptide toxin microcystin-LR in Mytilus galloprovincialis. Aquat. Toxicol. 1995;32:227–237. [Google Scholar] 38.Amorim A., Vasconcelos V. Dynamic of microcystins in the mussel Mytilus galloprovincialis. Toxicon. 1999;37:1041–1052. doi: 10.1016/s0041-0101(98)00231-1. [DOI] [PubMed] [Google Scholar] 39.Wiegand C., Pflugmacher S. Ecotoxicological effects of selected cyanobacterial secondary metabolites a short review. Toxicol. Appl. Pharmacol. 2005;203:201–218. doi: 10.1016/j.taap.2004.11.002. [DOI] [PubMed] [Google Scholar] 40.Magalhães V.F., Marinho M.M., Domingos P., Oliveira A.C., Costa S.M., Azevedo L.O., Azevedo S.M.F.O. Microcystins (cyanobacteria hepatotoxins) bioaccumulation in fish and crustaceans from Sepetiba Bay (Brasil, RJ) Toxicon. 2003;42:289–295. doi: 10.1016/s0041-0101(03)00144-2. [DOI] [PubMed] [Google Scholar] 41.Vasconcelos V., Oliveira S., Teles F.O. Impact of a toxic and a non-toxic strain of Microcystis aeruginosa on the crayfish Procambarus clarkii. Toxicon. 2001;39:1461–1470. doi: 10.1016/s0041-0101(01)00105-2. [DOI] [PubMed] [Google Scholar] 42.Saqrane S., El ghzali I., Ouahid Y., El Hassni M., El Hadrami I., Bouarab L., del Campo F.F., Oudra B., Vasconcelos V. Phytotoxic effects of cyanobacteria extract on the aquatic plant Lemna gibba: Microcystin accumulation, detoxication and oxidative stress induction. Aquat. Toxicol. 2007;83:284–294. doi: 10.1016/j.aquatox.2007.05.004. [DOI] [PubMed] [Google Scholar] 43.Nogueira I.C.G., Pereira P., Dias E., Pflugmacher S., Wiegand C., Franca S., Vasconcelos V.M. Accumulation of paralytic shellfish toxins (PST) from the cyanobacterium Aphanizomenon issatschenkoi by the cladoceran Daphnia magna. Toxicon. 2004;44:773–780. doi: 10.1016/j.toxicon.2004.08.006. [DOI] [PubMed] [Google Scholar] 44.Negri A.P., Jones G.J. Bioaccumulation of paralytic shellfish poisoning (PSP) toxins from the cyanobacterium Anabaena circinalis by the freshwater mussel Alathyria condola. Toxicon. 1995;33:667–678. doi: 10.1016/0041-0101(94)00180-g. [DOI] [PubMed] [Google Scholar] 45.Pereira P., Dias E., Franca S., Pereira E., Carolino M., Vasconcelos V. Accumulation and depuration of cyanobacterial paralytic shellfish toxins by the freshwater mussel Anodonta cygnea. Aquat. Toxicol. 2004;68:339–350. doi: 10.1016/j.aquatox.2004.04.001. [DOI] [PubMed] [Google Scholar] 46.Saker M.L., Metcalf J.S., Codd G.A., Vasconcelos V.M. Accumulation and depuration of the cyanobacterial toxin cylindrospermopsin in the freshwater mussel Anodonta cygnea. Toxicon. 2004;43:185–194. doi: 10.1016/j.toxicon.2003.11.022. [DOI] [PubMed] [Google Scholar] 47.Dittmann E., Wiegand C. Cyanobacterial toxins—occurrence, biosynthesis and impact on human affairs. Mol. Nutr. Food Res. 2006;50:7–17. doi: 10.1002/mnfr.200500162. [DOI] [PubMed] [Google Scholar] 48.Stewart I., Webb P., Schluter P., Shaw G. Recreational and occupational field exposure to freshwater cyanobacteria—a review of anecdotal and case reports, epidemiological studies and the challenges for epidemiologic assessment. Environ. Health. 2006;5:6. doi: 10.1186/1476-069X-5-6. [DOI] [PMC free article] [PubMed] [Google Scholar] 49.Okino T. Heterocycles from Cyanobacteria. Top. Heterocycl. Chem. 2006;5:1–19. [Google Scholar] 50.Berry J.P., Gantar M., Perez M.H., Berry G., Noriega F.G. Cyanobacterial toxins as allelochemicals with potential applications as algaecides, herbicides and insecticides. Mar. Drugs. 2008;6:117–146. doi: 10.3390/md20080007. [DOI] [PMC free article] [PubMed] [Google Scholar] 51.Sivonen K., Börner T. Bioactive compounds produced by cyanobacteria. In: Herrero A., Flores E., editors. The Cyanobacteria: Molecular Biology, Genomics and Evolution. Caister Academic Press; Norfolk, UV, USA: 2008. pp. 159–197. [Google Scholar] 52.Welker M., von Döhren H. Cyanobacterial peptides—Nature’s own combinatorial biosynthesis. FEMS Microbiol. Rev. 2006;30:530–563. doi: 10.1111/j.1574-6976.2006.00022.x. [DOI] [PubMed] [Google Scholar] 53.Rapala J., Sivonen K., Lyra C., Niemelä S.I. Variation of microcystins, cyanobacterial hepatotoxins, in Anabaena spp. as a function of growth stimuli. Appl. Environ. Microbiol. 1997;63:2206–2212. doi: 10.1128/aem.63.6.2206-2212.1997. [DOI] [PMC free article] [PubMed] [Google Scholar] 54.Welker M., Brunke M., Preussel K., Lippert I., von Döhren H. Diversity and distribution of Microcystis (Cyanobacteria) oligopeptide chemotypes from natural communities studied by single-colony mass spectrometry. Microbiology. 2004;150:1785–1796. doi: 10.1099/mic.0.26947-0. [DOI] [PubMed] [Google Scholar] 55.Kameyama K., Sugiura N., Inamori Y., Maekawa T. Characteristics of microcystin production cell cycle of Microcystis viridis. Environ. Toxicol. 2004;19:20–25. doi: 10.1002/tox.10147. [DOI] [PubMed] [Google Scholar] 56.Saker M.L., Fastner J., Dittmann E., Christiansen G., Vasconcelos V.M. Variation between strains of the cyanobacterium Microcystis aeruginosa isolated from a Portuguese river. J. Appl. Microbiol. 2005;99:749–757. doi: 10.1111/j.1365-2672.2005.02687.x. [DOI] [PubMed] [Google Scholar] 57.Fastner J., Heinze R., Humpage A.R., Mischke U., Eaglesham G.K., Chorus I. Cylindrospermopsin occurrence in two German lakes and preliminary assessment of toxicity and toxin production of Cylindrospermopsis raciborskii (cyanobacteria) isolates. Toxicon. 2003;42:313–321. doi: 10.1016/s0041-0101(03)00150-8. [DOI] [PubMed] [Google Scholar] 58.Saker M.L., Nogueira I.C.G., Vasconcelos V.M., Neilan B.A., Eaglesham G.H., Pereira P. First report and toxicological assessment of the cyanobacterium Cylindrospermopsis raciborskii from Portuguese freshwaters. Ecotoxicol. Environ. Saf. 2003;55:243–250. doi: 10.1016/s0147-6513(02)00043-x. [DOI] [PubMed] [Google Scholar] 59.Park H.-D., Watanabe M.F., Harada K.-I., Nagai H., Suzuki M., Watanabe M., Hayashi H. Hepatotoxin (microcystin) and neurotoxin (anatoxin-a) contained in natural blooms and strains of cyanobacteria from Japanese freshwaters. Nat. Toxins. 1993;1:353–360. doi: 10.1002/nt.2620010606. [DOI] [PubMed] [Google Scholar] 60.Yu S., Zhao N., Zi X. The relationship between cyanotoxin (microcystin, MC) in pond-ditch water and primary liver cancer in China. Zhonghua Zhong Liu Za Zhi. 2001;23:96–99. [PubMed] [Google Scholar] 61.Azevedo S.M.F.O., Rinehart K.L., Eaglesham G.K., Lau S., Carmichael W.W., Jochimsen E.M., Shaw G.R. Human intoxication by microcystins during renal dialysis treatment in Caruaru-Brazil. Toxicolog. 2002;181–182:441–446. doi: 10.1016/S0300-483X(02)00491-2. [DOI] [PubMed] [Google Scholar] 62.Hotto A.M., Satchwell M.F., Boyer G.L. Molecular characterization of potential microcystin-producing cyanobacteria in Lake Ontario embayments and nearshore waters. Appl. Environ. Microbiol. 2007;73:4570–4578. doi: 10.1128/AEM.00318-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 63.Batista T., de Sousa G., Suput J.S., Rahmani R., Suput D. Microcystin-LR causes the collapse of actin filaments in primary human hepatocytes. Aquat. Toxicol. 2003;65:85–91. doi: 10.1016/s0166-445x(03)00108-5. [DOI] [PubMed] [Google Scholar] 64.Campos A., Vasconcelos V. Molecular mechanisms of Microcystin toxicity in animal cells. Int. J. Mol. Sci. 2010;11:268–287. doi: 10.3390/ijms11010268. [DOI] [PMC free article] [PubMed] [Google Scholar] 65.Clark S.P., Ryan T.P., Searfoss G.H., Davis M.A., Hooser S.B. Chronic microcystin exposure induces hepatocyte proliferation with increased expression of mitotic and cyclin-associated genes in P53-deficient mice. Toxicol. Pathol. 2008;36:190–203. doi: 10.1177/0192623307311406. [DOI] [PubMed] [Google Scholar] 66.Krakstad C., Herfindal L., Gjertsen B.T., Boe R., Vintermyr O.K., Fladmark K.E., Doskeland S.O. CaM-kinaseII-dependent commitment to microcystin-induced apoptosis is coupled to cell budding, but not to shrinkage or chromatin hypercondensation. Cell Death Differ. 2005;13:1191–1202. doi: 10.1038/sj.cdd.4401798. [DOI] [PubMed] [Google Scholar] 67.Li M., Satinover D.L., Brautigan D.L. Phosphorylation and functions of inhibitor-2 family of proteins. Biochemistry. 2007;46:2380–2389. doi: 10.1021/bi602369m. [DOI] [PubMed] [Google Scholar] 68.Gehringer M.M. Microcystin-LR and okadaic acid-induced cellular effects: a dualistic response. FEBS Lett. 2004;557:1–8. doi: 10.1016/s0014-5793(03)01447-9. [DOI] [PubMed] [Google Scholar] 69.Guzman R.E., Solter P.F., Runnegar M.T. Inhibition of nuclear protein phosphatase activity in mouse hepatocytes by the cyanobacterial toxin microcystin-LR. Toxicon. 2003;41:773–781. doi: 10.1016/s0041-0101(03)00030-8. [DOI] [PubMed] [Google Scholar] 70.Weng D., Lu Y., Wei Y., Liu Y., Shen P. The role of ROS in microcystin-LR-induced hepatocyte apoptosis and liver injury in mice. Toxicology. 2007;232:15–23. doi: 10.1016/j.tox.2006.12.010. [DOI] [PubMed] [Google Scholar] 71.Tillett D., Dittmann E., Erhard M., von Döhren H., Börner T., Neilan B.A. Structural organization of microcystin biosynthesis in Microcystis aeruginosa PCC 7806: an integrated peptide-polyketide synthetase system. Chem. Biol. 2000;7:753–764. doi: 10.1016/s1074-5521(00)00021-1. [DOI] [PubMed] [Google Scholar] 72.Christiansen G., Fastner J., Erhard M., Börner T., Dittmann E. Microcystin biosynthesis in Planktothrix: genes, evolution, and manipulation. J. Bacteriol. 2003;185:564–572. doi: 10.1128/JB.185.2.564-572.2003. [DOI] [PMC free article] [PubMed] [Google Scholar] 73.Rouhiainen L., Vakkilainen T., Siemer B.L., Buikema W., Haselkorn R., Sivonen K. Genes coding for hepatotoxic heptapeptides (microcystins) in the cyanobacterium Anabaena strain 90. Appl. Environ. Microbiol. 2004;70:686–692. doi: 10.1128/AEM.70.2.686-692.2004. [DOI] [PMC free article] [PubMed] [Google Scholar] 74.Kurmayer R., Christiansen G. The genetic basis of toxin production in Cyanobacteria. Freshwater Rev. 2009;2:31–50. [Google Scholar] 75.Pearson L.A., Hisbergues M., Börner T., Dittmann E., Neilan B.A. Inactivation of an ABC transporter gene, mcyH, results in loss of microcystin production in the cyanobacterium Microcystis aeruginosa PCC 7806. Appl. Environ. Microbiol. 2004;70:6370–6378. doi: 10.1128/AEM.70.11.6370-6378.2004. [DOI] [PMC free article] [PubMed] [Google Scholar] 76.Shi L., Carmichael W.W., Miller I. Immuno-gold localization of hepatotoxins in cyanobacterial cells. Arch. Microbiol. 1995;163:7–15. doi: 10.1007/BF00262197. [DOI] [PubMed] [Google Scholar] 77.Young F.M., Thomson C., Metcalf J.S., Lucocq J.M., Codd G.A. Immunogold localisation of microcystins in cryosectioned cells of Microcystis. J. Struct. Biol. 2005;151:208–214. doi: 10.1016/j.jsb.2005.05.007. [DOI] [PubMed] [Google Scholar] 78.Kaebernick M., Neilan B., Borner T., Dittmann E. Light and the transcriptional response of the microcystin biosynthesis gene cluster. Appl. Environ. Microbiol. 2000;66:3387–3392. doi: 10.1128/aem.66.8.3387-3392.2000. [DOI] [PMC free article] [PubMed] [Google Scholar] 79.Bagu J.R., Sykes B.D., Craig M.M., Holmes C.F.B. A molecular basis for different interactions of marine toxins with protein phosphatase-1. J. Biol. Chem. 1997;272:5087–5097. doi: 10.1074/jbc.272.8.5087. [DOI] [PubMed] [Google Scholar] 80.Moffitt M.C., Neilan B.A. Characterization of the Nodularin synthetase gene cluster and proposed theory of the evolution of cyanobacterial hepatotoxins. Appl. Environ. Microbiol. 2004;70:6353–6362. doi: 10.1128/AEM.70.11.6353-6362.2004. [DOI] [PMC free article] [PubMed] [Google Scholar] 81.Pearson L.A., Moffitt M.C., Ginn H.P., Neilan B.A. The molecular genetics and regulation of cyanobacterial peptide hepatotoxin biosynthesis. Crit. Rev. Toxicol. 2008;38:847–856. doi: 10.1080/10408440802291513. [DOI] [PubMed] [Google Scholar] 82.Hawkins P.R., Runnegar M.T.C., Jackson A.R.B., Falconer I.R. Severe hepatotoxicity caused by the tropical cyanobacterium (blue-green Alga) Cylindrospermopsis raciborskii (Woloszynska) Seenaya and Subba Raju isolated from a domestic water supply reservoir. Appl. Environ. Microbiol. 1985;50:1292–1295. doi: 10.1128/aem.50.5.1292-1295.1985. [DOI] [PMC free article] [PubMed] [Google Scholar] 83.Ohtani I., Moore R.E., Runnegar M.T.C. Cylindrospermopsin: A potent hepatotoxin from the blue-green alga Cylindrospermopsis raciborskii. J. Am. Chem. Soc. 1992;114:7941–7942. [Google Scholar] 84.Li R., Carmichael W.W., Brittain S., Eaglesham G.K., Shaw G.R., Mahakhant A., Noparatnaraporn N., Yongmanitchai W., Kaya K., Watanabe M.M. Isolation and identification of the cyanotoxin cylindrospermopsin and deoxy-cylindrospermopsin from a Thailand strain of Cylindrospermopsis raciborskii (Cyanobacteria) Toxicon. 2001;39:973–980. doi: 10.1016/s0041-0101(00)00236-1. [DOI] [PubMed] [Google Scholar] 85.Harada K.-I., Ohtani I., Iwamoto K., Suzuki M., Watanabe M.F., Watanabe M., Terao K. Isolation of cylindrospermopsin from a cyanobacterium Umezakia natans and its screening method. Toxicon. 1994;32:73–84. doi: 10.1016/0041-0101(94)90023-x. [DOI] [PubMed] [Google Scholar] 86.Banker R., Carmeli S., Hadas O., Teltsch B., Porat R., Skenik A. Identification of cylindrospermopsin in the cyanobacterium Aphanizomenon ovalisporum (Cyanophyceae) isolated from lake Kinneret, Israel. J. Phycol. 1997;33:613–616. [Google Scholar] 87.Li R., Carmichael W.W., Brittain S., Eaglesham G.K., Shaw G.R., Yongding L., Watanabe M.M. First report of the cyanotoxins cylindrospermopsin and deoxycylindrospermopsin from Raphidiopsis curvata (cyanobacteria) J. Phycol. 2001;37:1121–1126. [Google Scholar] 88.Schembri M.A., Neilan B.A., Saint C.P. Identification of genes implicated in toxin production in the cyanobacterium Cylindrospermopsis raciborskii. Environ. Toxicol. 2001;16:413–421. doi: 10.1002/tox.1051. [DOI] [PubMed] [Google Scholar] 89.Preuβel K., Stüken A., Wiedner C., Chorus I., Fastner J. First report on cylindrospermopsin producing Aphanizomenon flos-aquae (Cyanobacteria) isolated from two German lakes. Toxicon. 2006;47:156–162. doi: 10.1016/j.toxicon.2005.10.013. [DOI] [PubMed] [Google Scholar] 90.Seifert M., McGregor G., Eaglesham G., Wickramasinghe W., Shaw G. First evidence for the production of cylindrospermopsin and deoxy-cylindrospermopsin by the freshwater benthic cyanobacterium, Lyngbya wollei (Farlow ex Gomont) Speziale and Dyck. Harmful Algae. 2007;6:73–80. [Google Scholar] 91.Terao K., Ohmori S., Igarashi K., Ohtani I., Watanabe M.F., Harada K.I., Ito E., Watanabe M. Electron microscopic studies on experimental poisoning in mice induced by cylindrospermopsin isolated from blue-green alga Umezakia natans. Toxicon. 1994;32:833–843. doi: 10.1016/0041-0101(94)90008-6. [DOI] [PubMed] [Google Scholar] 92.Harada K.-I., Kimura Y., Ogawa K., Suzuki M., Dahlem A.M., Beasley V.R., Carmichael W.W. A new procedure for the analysis and purification of naturally occurring anatoxin-a from the blue-green alga Anabaena flos-aquae. Toxicon. 1989;27:1289–1296. doi: 10.1016/0041-0101(89)90060-3. [DOI] [PubMed] [Google Scholar] 93.Falconer I.R., Hardy S.J., Humpage A.R., Froscio S.M., Tozer G.J., Hawkins P.R. Hepatic and renal toxicity of the blue-green alga (cyanobacterium) Cylindrospermopsis raciborskii in male Swiss albino mice. Environ. Toxicol. 1999;14:143–150. [Google Scholar] 94.Mihali T.K., Kellmann R., Muenchhoff J., Barrow K.D., Neilan B.A. Characterization of the gene cluster responsible for cylindrospermopsin biosynthesis. Appl. Environ. Microbiol. 2008;74:716–722. doi: 10.1128/AEM.01988-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 95.Falconer I.R., Humpage A.R. Preliminary evidence for in vivo tumour initiation by oral administration of extracts of the blue-green alga Cylindrospermopsis raciborskii containing the toxin cylindrospermopsin. Environ. Toxicol. 2001;16:192–195. doi: 10.1002/tox.1024. [DOI] [PubMed] [Google Scholar] 96.Runnegar M.T., Kong S.-M., Zhong Y.-Z., Lu S.C. Inhibition of reduced glutathione synthesis by cyanobacterial alkaloid cylindrospermopsin in cultured rat hepatocytes. Biochem. Pharmacol. 1995;49:219–225. doi: 10.1016/s0006-2952(94)00466-8. [DOI] [PubMed] [Google Scholar] 97.Shen X., Lam P.K.S., Shaw G.R., Wickramasinghe W. Genotoxicity investigation of a cyanobacterial toxin, cylindrospermopsin. Toxicon. 2002;40:1499–1501. doi: 10.1016/s0041-0101(02)00151-4. [DOI] [PubMed] [Google Scholar] 98.Shalev-Alon G., Sukenik A., Livnah O., Schwarz R., Kaplan A. A novel gene encoding amidinotransferase in the cylindrospermopsin producing cyanobacterium Aphanizomenon ovalisporum. FEMS Microbiol. Lett. 2002;209:87–91. doi: 10.1111/j.1574-6968.2002.tb11114.x. [DOI] [PubMed] [Google Scholar] 99.Kellmann R., Mills T., Neilan B. Functional modeling and phylogenetic distribution of putative cylindrospermopsin biosynthesis enzymes. J. Mol. Evol. 2006;62:267–280. doi: 10.1007/s00239-005-0030-6. [DOI] [PubMed] [Google Scholar] 100.Sivonen K., Himberg K., Luukkainen R., Niemelä S.I., Poon G.K., Codd G.A. Preliminary characterization of neurotoxic cyanobacteria blooms and strains from Finland. Toxic. Assess. 1989;4:339–352. [Google Scholar] 101.Namikoshi M., Murakami T., Watanabe M.F., Oda T., Yamada J., Tsujimura S., Nagai H., Oishi S. Simultaneous production of homoanatoxin-a, anatoxin-a, and a new non-toxic 4-hydroxyhomoanatoxin-a by the cyanobacterium Raphidiopsis mediterranea Skuja. Toxicon. 2003;42:533–538. doi: 10.1016/s0041-0101(03)00233-2. [DOI] [PubMed] [Google Scholar] 102.Viaggiu E., Melchiorre S., Volpi F., Corcia A.D., Mancini R., Garibaldi L., Crichigno G., Bruno M. Anatoxin-a toxin in the cyanobacterium Planktothrix rubescens from a fishing pond in northern Italy. Environ. Toxicol. 2004;19:191–197. doi: 10.1002/tox.20011. [DOI] [PubMed] [Google Scholar] 103.Selwood A.I., Holland P.T., Wood S.A., Smith K.F., McNabb P.S. Production of Anatoxin-a and a novel biosynthetic precursor by the cyanobacterium Aphanizomenon issatschenkoi. Environ. Sci. Technol. 2006;41:506–510. doi: 10.1021/es061983o. [DOI] [PubMed] [Google Scholar] 104.Skulberg O.M., Skulberg R., Carmichael W.W., Andersen R.A., Matsunaga S., Moore R.E. Investigations of a neurotoxic oscillatorialean strain (Cyanophyceae) and its toxin. Isolation and characterization of homoanatoxin-a. Environ. Toxicol. Chem. 1992;11:321–329. doi: 10.1002/etc.5620110306. [DOI] [Google Scholar] 105.Furey A., Crowley J., Shuilleabhain A.N., Skulberg O.M., James K.J. The first identification of the rare cyanobacterial toxin, homoanatoxin-a, in Ireland. Toxicon. 2003;41:297–303. doi: 10.1016/s0041-0101(02)00291-x. [DOI] [PubMed] [Google Scholar] 106.Aráoz R., Nghiem H.-O., Rippka R., Palibroda N., de Marsac N.T., Herdman M. Neurotoxins in axenic oscillatorian cyanobacteria: coexistence of anatoxin-a and homoanatoxin-a determined by ligand-binding assay and GC/MS. Microbiology. 2005;151:1263–1273. doi: 10.1099/mic.0.27660-0. [DOI] [PubMed] [Google Scholar] 107.Cadel-Six S., Iteman I., Peyraud-Thomas C., Mann S., Ploux O., Mejean A. Identification of a polyketide synthase coding sequence specific for anatoxin-a-producing Oscillatoria cyanobacteria. Appl. Environ. Microbiol. 2009;75:4909–4912. doi: 10.1128/AEM.02478-08. [DOI] [PMC free article] [PubMed] [Google Scholar] 108.Méjean A., Mann S., Maldiney T., Vassiliadis G., Lequin O., Ploux O. Evidence that biosynthesis of the neurotoxic alkaloids anatoxin-a and homoanatoxin-a in the cyanobacterium Oscillatoria PCC 6506 occurs on a modular polyketide synthase initiated by L-proline. J. Am. Chem. Soc. 2009;131:7512–7513. doi: 10.1021/ja9024353. [DOI] [PubMed] [Google Scholar] 109.Mahmood N.A., Carmichael W.W. The pharmacology of anatoxin-a(s), a neurotoxin produced by the freshwater cyanobacterium Anabaena flos-aquae NRC 525-17. Toxicon. 1986;24:425–434. doi: 10.1016/0041-0101(86)90074-7. [DOI] [PubMed] [Google Scholar] 110.Matsunaga S., Moore R.E., Niemczura W.P., Carmichael W.W. Anatoxin-a(s), a potent anticholinesterase from Anabaena flos-aquae. J. Am. Chem. Soc. 1989;111:8021–8023. [Google Scholar] 111.Cook W.O., Beasley V.R., Dahlem A.M., Dellinger J.A., Harlin K.S., Carmichael W.W. Comparison of effects of anatoxin-a(s) and paraoxon, physostigmine and pyridostigmine on mouse brain cholinesterase activity. Toxicon. 1988;26:750–753. doi: 10.1016/0041-0101(88)90282-6. [DOI] [PubMed] [Google Scholar] 112.Pita R., Anadón A., Martínez-Larrañaga M.R. Neurotoxins with anticholinesterase activity and their possible use as warfare agents. Med. Clin. (Barc.) 2003;121:511–517. doi: 10.1157/13053143. [DOI] [PubMed] [Google Scholar] 113.Kellmann R., Neilan B.A. Biochemical characterization of paralytic shellfish toxin biosynthesis in vitro. J. Phycol. 2007;43:497–508. [Google Scholar] 114.Mihali T.K., Kellmann R., Neilan B.A. Characterisation of the paralytic shellfish toxin biosynthesis gene clusters in Anabaena circinalis AWQC131C and Aphanizomenon sp. NH-5. BMC Biochem. 2009;10:8. doi: 10.1186/1471-2091-10-8. [DOI] [PMC free article] [PubMed] [Google Scholar] 115.Kao C.Y. Structure-activity relations of tetrodotoxin, saxitoxin, and analogues. Ann. N. Y. Acad. Sci. 1986;479:52–67. doi: 10.1111/j.1749-6632.1986.tb15561.x. [DOI] [PubMed] [Google Scholar] 116.Chang Z., Sitachitta N., Rossi J.V., Roberts M.A., Flatt P.M., Jia J., Sherman D.H., Gerwick W.H. Biosynthetic pathway and gene cluster analysis of curacin A, an antitubulin natural product from the tropical marine cyanobacterium Lyngbya majuscula. J. Nat. Prod. 2004;67:1356–1367. doi: 10.1021/np0499261. [DOI] [PubMed] [Google Scholar] 117.Tamplin M.L. A bacterial source of tetrodotoxins and saxitoxins. In: Hall S., Strichartz G., editors. Marine Toxins: Origin, Structure, and Molecular Pharmacology. Vol. 418. American Chemical Society; Washington, DC, USA: 1990. pp. 78–86. [Google Scholar] 118.Llewellyn L.E. Saxitoxin, a toxic marine natural product that targets a multitude of receptors. Nat. Prod. Rep. 2006;23:200–222. doi: 10.1039/b501296c. [DOI] [PubMed] [Google Scholar] 119.Toxins: Potential chemical weapons from living organisms. [(accessed on 24 June 2010)]. Available online: 120.Kellmann R., Mihali T.K., Jeon Y.J., Pickford R., Pomati F., Neilan B.A. Biosynthetic intermediate analysis and functional homology reveal a saxitoxin gene cluster in cyanobacteria. Appl. Environ. Microbiol. 2008;74:4044–4053. doi: 10.1128/AEM.00353-08. [DOI] [PMC free article] [PubMed] [Google Scholar] 121.Edwards D.J., Marquez B.L., Nogle L.M., McPhail K., Goeger D.E., Roberts M.A., Gerwick W.H. Structure and biosynthesis of the jamaicamides, new mixed polyketide-peptide neurotoxins from the marine cyanobacterium Lyngbya majuscula. Chem. Biol. 2004;11:817–833. doi: 10.1016/j.chembiol.2004.03.030. [DOI] [PubMed] [Google Scholar] 122.LePage K.T., Goeger D., Yokokawa F., Asano T., Shioiri T., Gerwick W.H., Murray T.F. The neurotoxic lipopeptide kalkitoxin interacts with voltage-sensitive sodium channels in cerebellar granule neurons. Toxicol. Lett. 2005;158:133–139. doi: 10.1016/j.toxlet.2005.03.007. [DOI] [PubMed] [Google Scholar] 123.Wu M., Okino T., Nogle L.M., Marquez B.L., Williamson R.T., Sitachitta N., Berman F.W., Murray T.F., McGough K., Jacobs R., Colsen K., Asano T., Yokokawa F., Shioiri T., Gerwick W.H. Structure, synthesis, and biological properties of Kalkitoxin, a novel neurotoxin from the marine cyanobacterium Lyngbya majuscula. J. Am. Chem. Soc. 2000;122:12041–12042. [Google Scholar] 124.Orjala J., Nagle D.G., Hsu V., Gerwick W.H. Antillatoxin: an exceptionally ichthyotoxic cyclic lipopeptide from the tropical cyanobacterium Lyngbya majuscula. J. Am. Chem. Soc. 1995;117:8281–8282. [Google Scholar] 125.Li W.I., Berman F.W., Okino T., Yokokawa F., Shioiri T., Gerwick W.H., Murray T.F. Antillatoxin is a marine cyanobacterial toxin that potently activates voltage-gated sodium channels. Proc. Natl. Acad. Sci. USA. 2001;98:7599–7604. doi: 10.1073/pnas.121085898. [DOI] [PMC free article] [PubMed] [Google Scholar] 126.Berman F.W., Gerwick W.H., Murray T.F. Antillatoxin and kalkitoxin, ichthyotoxins from the tropical cyanobacterium Lyngbya majuscula, induce distinct temporal patterns of NMDA receptor-mediated neurotoxicity. Toxicon. 1999;37:1645–1648. doi: 10.1016/s0041-0101(99)00108-7. [DOI] [PubMed] [Google Scholar] 127.Calabresi P., Chabner B. Chemotherapy of neoplastic diseases. In: Hardman J.G., Limbird L.E., Molinoff P.B., Ruddin R.W., Guilman A.G., editors. The Pharmacological Basis of Therapeutics. 9th. Pergamon Press; New York, NY, USA: 1996. pp. 1225–1287. [Google Scholar] 128.Chang Z., Sitachitta N., Rossi J.V., Roberts M.A., Flatt P.M., Jia J., Sherman D.H., Gerwick W.H. Biosynthetic pathway and gene cluster analysis of curacin A, an antitubulin natural product from the tropical marine cyanobacterium Lyngbya majuscula. J. Nat. Prod. 2004;67:1356–1367. doi: 10.1021/np0499261. [DOI] [PubMed] [Google Scholar] 129.Orjala J., Gerwick W.H. Barbamide, a chlorinated metabolite with molluscicidal activity from the Caribbean cyanobacterium Lyngbya majuscula. J. Nat. Prod. 1996;59:427–430. doi: 10.1021/np960085a. [DOI] [PubMed] [Google Scholar] 130.Chang Z., Flatt P., Gerwick W.H., Nguyen V.-A., Willis C.L., Sherman D.H. The barbamide biosynthetic gene cluster: A novel cyanobacterial system of mixed polyketide synthase (PKS)-non-ribosomal peptide synthetase (NRPS) origin involving an unusual trichloroleucyl starter unit. Gene. 2002;296:235–247. doi: 10.1016/s0378-1119(02)00860-0. [DOI] [PubMed] [Google Scholar] 131.Doekel S., Marahiel M.A. Biosynthesis of natural products on modular peptide synthetases. Metab. Eng. 2001;3:64–77. doi: 10.1006/mben.2000.0170. [DOI] [PubMed] [Google Scholar] 132.Jenke-Kodama H., Sandmann A., Muller R., Dittmann E. Evolutionary implications of bacterial polyketide synthases. Mol. Biol. Evol. 2005;22:2027–2039. doi: 10.1093/molbev/msi193. [DOI] [PubMed] [Google Scholar] 133.Choi S.H., Kim S.G. Lipopolysaccharide inhibition of rat hepatic microsomal epoxide hydrolase and glutathione S-transferase gene expression irrespective of nuclear factor-[κ]B activation. Biochem. Pharmacol. 1998;56:1427–1436. doi: 10.1016/s0006-2952(98)00204-4. [DOI] [PubMed] [Google Scholar] 134.Biscardi D., Castaldo A., Gualillo O., Fusco R. The occurrence of Aeromonas hydrophila strains in Italian mineral and thermal waters. Sci. Total Environ. 2002;292:255–263. doi: 10.1016/s0048-9697(01)01132-9. [DOI] [PubMed] [Google Scholar] 135.Holt J.G., Krieg N.R., Sneath P.H.A., Staley J.T., Williams S.T. Bergey’s Manual of Determinative Bacteriology. 9th. Williams & Wilkins; Baltimore, MD, USA: 1994. [Google Scholar] 136.Barbieri E., Falzano L., Fiorentini C., Pianetti A., Baffone W., Fabbri A., Matarrese P., Casiere A., Katouli M., Kuhn I., et al. Occurrence, diversity, and pathogenicity of halophilic Vibrio spp. and non-O1 Vibrio cholerae from estuarine waters along the Italian Adriatic coast. Appl. Environ. Microbiol. 1999;65:2748–2753. doi: 10.1128/aem.65.6.2748-2753.1999. [DOI] [PMC free article] [PubMed] [Google Scholar] 137.Heidelberg J.F., Heidelberg K.B., Colwell R.R. Seasonality of Chesapeake Bay bacterioplankton species. Appl. Environ. Microbiol. 2002;68:5488–5497. doi: 10.1128/AEM.68.11.5488-5497.2002. [DOI] [PMC free article] [PubMed] [Google Scholar] 138.Ortigosa M., Esteve C., Pujalte M.J. Vibrio species in seawater and mussels: abundance and numerical taxonomy. Syst. Appl. Microbiol. 1989;12:316–325. [Google Scholar] 139.Ortigosa M., Garay E., Pujalte M.J. Numerical taxonomy of Vibrionaceae isolated from oysters and seawater along an annual cycle. Syst. Appl. Microbiol. 1994;17:216–225. [Google Scholar] 140.Arias C.R., Garay E., Aznar R. Nested PCR method for rapid and sensitive detection of Vibrio vulnificus in fish, sediments, and water. Appl. Environ. Microbiol. 1995;61:3476–3478. doi: 10.1128/aem.61.9.3476-3478.1995. [DOI] [PMC free article] [PubMed] [Google Scholar] 141.Grisez L., Reyniers J., Verdonck L., Swings J., Ollevier F. Dominant intestinal microflora of sea bream and sea bass larvae from two hatcheries, during larval development. Aquaculture. 1997;155:387–399. doi: 10.1016/S0044-8486(97)00113-0. [DOI] [Google Scholar] 142.Bauer A., Østensvik Ø., Florvåg M., Ørmen Ø., Rørvik L.M. Occurrence of Vibrio parahaemolyticus, V. cholerae, and V. vulnificus in Norwegian blue mussels (Mytilus edulis) Appl. Environ. Microbiol. 2006;72:3058–3061. doi: 10.1128/AEM.72.4.3058-3061.2006. [DOI] [PMC free article] [PubMed] [Google Scholar] 143.Roque A., Lopez-Joven C., Lacuesta B., Elandaloussi L., Wagley S., Furones, Ruiz-Zarzuela I., de Blas I., Rangdale R., Gomez-Gil B. Detection and identification of tdh- and trh-positive Vibrio parahaemolyticus Strains from four species of cultured bivalve molluscs on the Spanish Mediterranean coast. Appl. Environ. Microbiol. 2009;75:7574–7577. doi: 10.1128/AEM.00772-09. [DOI] [PMC free article] [PubMed] [Google Scholar] 144.Gomez-Gil B., Roque A., Turnbull J.F., Tron-Mayen L. Species of Vibrio isolated from hepatopancreas, haemolymph and digestive tract of a population of healthy juvenile Penaeus vannamei. Aquaculture. 1998;163:1–9. [Google Scholar] 145.Vandenberghe J., Li Y., Verdonck L., Sorgeloos P., Xu H.S., Swings J. Vibrios associated with Penaeus chinensis (Crustacea: Decapoda) larvae in Chinese shrimp hatcheries. Aquaculture. 1998;169:121–132. [Google Scholar] 146.Wai S.N., Mizunoe Y., Yoshida S. How Vibrio cholerae survive during starvation. FEMS Microbiol. Lett. 1999;180:123–131. doi: 10.1111/j.1574-6968.1999.tb08786.x. [DOI] [PubMed] [Google Scholar] 147.Lightner D.V., Redman R.M. Shrimp diseases and currentdiagnostic methods. Aquaculture. 1998;164:201–220. [Google Scholar] 148.Bergh O., Nilsen F., Samuelsen O.B. Diseases, prophylaxis and treatment of the Atlantic halibut Hippoglossus hippoglossus: a review. Dis. Aquat. Org. 2001;48:57–74. doi: 10.3354/dao048057. [DOI] [PubMed] [Google Scholar] 149.Diggles B.K., Carson J., Hine P.M., Hickman R.W., Tait M.J. Vibrio species associated with mortalities in hatchery-reared turbot (Colistium nudipinnis) and brill (C. guntheri) in New Zealand. Aquaculture. 2000;183:1–12. doi: 10.1016/S0044-8486(99)00280-X. [DOI] [Google Scholar] 150.Hansen G.H., Olafsen J.A. Bacterial interactions in early life stages of marine cold water fish. Microb. Ecol. 1999;38:1–26. doi: 10.1007/s002489900158. [DOI] [PubMed] [Google Scholar] 151.Matson J.S., Withey J.H., DiRita V.J. Regulatory networks controlling Vibrio cholerae virulence gene expression. Infect. Immun. 2007;75:5542–5549. doi: 10.1128/IAI.01094-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 152.Thompson F.L., Tetsuya I., Swings J. Biodiversity of Vibrios. Microbiol. Mol. Biol. Rev. 2004;68:403–431. doi: 10.1128/MMBR.68.3.403-431.2004. [DOI] [PMC free article] [PubMed] [Google Scholar] 153.Jørgensen A.R., Purdy R.E., Fieldhouse J., Kimber M.S., Bartlett D.H., Merrill A.R. Cholix toxin, a novel ADP-ribosylating factor from Vibrio cholerae. J. Biol. Chem. 2008;283:10671–10678. doi: 10.1074/jbc.M710008200. [DOI] [PubMed] [Google Scholar] 154.Davis B.M., Moyer K.E., Boyd E.F., Waldor M.K. CTX prophages in classical biotype Vibrio cholerae: functional phage genes but dysfunctional phage genomes. J. Bacteriol. 2000;182:6992–6998. doi: 10.1128/jb.182.24.6992-6998.2000. [DOI] [PMC free article] [PubMed] [Google Scholar] 155.Higgins D., Nazareno E., DiRita V.J. The virulence gene activator ToxT from Vibrio cholerae is a member of the AraC family of transcriptional activators. J. Bacteriol. 1992;174:6974–6980. doi: 10.1128/jb.174.21.6974-6980.1992. [DOI] [PMC free article] [PubMed] [Google Scholar] 156.Schuhmacher D.A., Klose K.E. Environmental signals modulate ToxT-dependent virulence factor expression in Vibrio cholerae. J. Bacteriol. 1999;181:1508–1514. doi: 10.1128/jb.181.5.1508-1514.1999. [DOI] [PMC free article] [PubMed] [Google Scholar] 157.Prouty M.G., Osorio C.R., Klose K.E. Characterization of functional domains of the Vibrio cholerae virulence regulator ToxT. Mol. Microbiol. 2005;58:1143–1156. doi: 10.1111/j.1365-2958.2005.04897.x. [DOI] [PubMed] [Google Scholar] 158.Withey J.H., DiRita V.J. The toxbox: specific DNA sequence requirements for activation of Vibrio cholerae virulence genes by ToxT. Mol. Microbiol. 2006;59:1779–1789. doi: 10.1111/j.1365-2958.2006.05053.x. [DOI] [PubMed] [Google Scholar] 159.Yu R.R., DiRita V.J. Regulation of gene expression in Vibrio cholerae by ToxT involves both antirepression and RNA polymerase stimulation. Mol. Microbiol. 2002;43:119–134. doi: 10.1046/j.1365-2958.2002.02721.x. [DOI] [PubMed] [Google Scholar] 160.Lee S.H., Hava D.L., Waldor M.K., Camilli A. Regulation and temporal expression patterns of Vibrio cholerae virulence genes during infection. Cell. 1999;99:625–634. doi: 10.1016/s0092-8674(00)81551-2. [DOI] [PubMed] [Google Scholar] 161.Lencer W.I. Microbes and microbial toxins: paradigms for microbial-mucosal interactions V. Cholera: invasion of the intestinal epithelial barrier by a stably folded protein toxin. Am. J. Physiol. Gastrointest. Liver Physiol. 2001;280:781–786. doi: 10.1152/ajpgi.2001.280.5.G781. [DOI] [PubMed] [Google Scholar] 162.Lencer W.I., Hirst T.R., Holmes R.K. Membrane traffic and cellular uptake of cholera toxin. Biochim. Biophys. Acta. 1999;1450:177–190. doi: 10.1016/s0167-4889(99)00070-1. [DOI] [PubMed] [Google Scholar] 163.Sixma T.K., Pronk S.E., Kalk H.H., Wartna E.S., van Zanten B.A.M., Witholt B., Hol W.G.J. Crystal structure of a cholera toxin-related heat-labile enterotoxin from E. coli. Nature. 1991;351:371–377. doi: 10.1038/351371a0. [DOI] [PubMed] [Google Scholar] 164.Fasano A. Cellular microbiology: can we learn cell physiology from microorganisms? Am. J. Physiol. Cell Physiol. 1999;276:765–776. doi: 10.1152/ajpcell.1999.276.4.C765. [DOI] [PubMed] [Google Scholar] 165.Satchell K.J.F. MARTX, multifuntional autoprocessing repeats-in-toxin toxins. Infect. Immun. 2007;75:5079–5084. doi: 10.1128/IAI.00525-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 166.Jorgensen R., Merrill A.R., Andersen G.R. The life and death of translation elongation factor 2. Biochem. Soc. Trans. 2006;34:1–6. doi: 10.1042/BST20060001. [DOI] [PubMed] [Google Scholar] 167.DePaola A., Capers G.M., Alexander D. Densities of Vibrio vulnificus in the intestines of fish from the U.S. Gulf Coast. Appl. Environ. Microbiol. 1994;60:984–988. doi: 10.1128/aem.60.3.984-988.1994. [DOI] [PMC free article] [PubMed] [Google Scholar] 168.do Nascimento S.M., Fernandes Vieira R.H., Theophilo G.N., Rodrigues P.D., Vieira G.H. Vibrio vulnificus as a health hazard for shrimp consumers. Rev. Inst. Med. Trop. Sao Paulo. 2001;43:263–266. doi: 10.1590/s0036-46652001000500005. [DOI] [PubMed] [Google Scholar] 169.Baffone W., Tarsi R., Pane L., Campana R., Repetto B., Mariottini G.L., Pruzzo C. Detection of free-living and plankton-bound vibrios in coastal waters of the Adriatic Sea (Italy) and study of their pathogenicity associated properties. Environ. Microbiol. 2006;8:1299–1305. doi: 10.1111/j.1462-2920.2006.01011.x. [DOI] [PubMed] [Google Scholar] 170.Jones K.J., Oliver J.D. Vibrio vulnificus: disease and pathogenesis. Infect. Immun. 2009;77:1723–1733. doi: 10.1128/IAI.01046-08. [DOI] [PMC free article] [PubMed] [Google Scholar] 171.Chuang Y.C., Yuan C.Y., Liu C.Y., Lan C.K., Huang A.H. Vibrio vulnificus infection in Taiwan: report of 28 cases and review of clinical manifestations and treatment. Clin. Infect. Dis. 1992;15:271–276. doi: 10.1093/clinids/15.2.271. [DOI] [PubMed] [Google Scholar] 172.Oliver J.D. Wound infections caused by Vibrio vulnificus and other marine bacteria. Epidemiol. Infect. 2005;133:383–391. doi: 10.1017/s0950268805003894. [DOI] [PMC free article] [PubMed] [Google Scholar] 173.Gray L.D., Kreger A.S. Purification and characterization of an extracellular cytolysin produced by Vibrio vulnificus. Infect. Immun. 1985;48:62–72. doi: 10.1128/iai.48.1.62-72.1985. [DOI] [PMC free article] [PubMed] [Google Scholar] 174.Kothary M.H., Kreger A.S. Purification and characterization of an elastolytic protease of Vibrio vulnificus. J. Gen. Microbiol. 1987;133:1783–1791. doi: 10.1099/00221287-133-7-1783. [DOI] [PubMed] [Google Scholar] 175.Shao C.P., Hor L.I. Metalloprotease is not essential for Vibrio vulnificus virulence in mice. Infect. Immun. 2000;68:3569–3573. doi: 10.1128/iai.68.6.3569-3573.2000. [DOI] [PMC free article] [PubMed] [Google Scholar] 176.Kim Y.R., Lee S.E., Kook H., Yeom J.A., Na H.S., Kim S.Y., Chung S.S., Choy H.E., Rhee J.H. Vibrio vulnificus RTX toxin kills host cells only after contact of the bacteria with host cells. Cell. Microbiol. 2008;10:848–862. doi: 10.1111/j.1462-5822.2007.01088.x. [DOI] [PubMed] [Google Scholar] 177.Gulig P.A., Bourdage K.L., Starks A.M. Molecular pathogenesis of Vibrio vulnificus. J. Microbiol. 2005;43:118–131. [PubMed] [Google Scholar] 178.Lee B.C., Lee J.H., Kim M.W., Kim B.S., Oh M.H., Kim K.S., Kim T.S., Choi S.H. Vibrio vulnificus rtxE is important for virulence and its expression is induced by exposure to host cells. Infect. Immun. 2008;76:1509–1517. doi: 10.1128/IAI.01503-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 179.Chopra A.K. Hyper production, purification, and mechanism of action of the cytotoxic enterotoxin produced by Aeromonas hydrophila. Infect. Immun. 1997;65:4299–4308. doi: 10.1128/iai.65.10.4299-4308.1997. [DOI] [PMC free article] [PubMed] [Google Scholar] 180.Yamada S., Matsushita S., Dejsirilet S., Kudoh Y. Incidence and clinical symptoms of Aeromonas-associated traveller’s diarrhoea in Tokyo. Epidemiol. Infect. 1997;119:121–126. doi: 10.1017/s0950268897007942. [DOI] [PMC free article] [PubMed] [Google Scholar] 181.Sha J., Kozlova E.V., Chopra K. Role of various enterotoxins in Aeromonas hyrophila-induced gastroenteritis: generation of enterotoxin gene-deficient mutants and evaluation of their enterotoxin activity. Infect. Immun. 2002;70:1924–1935. doi: 10.1128/IAI.70.4.1924-1935.2002. [DOI] [PMC free article] [PubMed] [Google Scholar] 182.Chopra A.K., Xu X.J., Ribardo D., Gonzalez M., Kuhl K., Peterson J.W., Houston C.W. The cytotoxic enterotoxin of Aeromonas hydrophila induces proinflammatory cytokine production and activates arachidonic acid metabolism in macrophages. Infect. Immun. 2000;68:2808–2818. doi: 10.1128/iai.68.5.2808-2818.2000. [DOI] [PMC free article] [PubMed] [Google Scholar] 183.Shenkar R., Abraham E. Mechanisms of lung neutrophil activation after hemorrhage or endotoxemia: roles of reactive oxygen intermediates, NF-kB, and cyclic AMP response element binding protein. J. Immunol. 1999;163:954–962. [PubMed] [Google Scholar] 184.Howard S.P., Garland W.J., Green M.J., Buckley J.T. Nucleotide sequence of the gene for the hole-forming toxin aerolysin of Aeromonas hydrophila. J. Bacteriol. 1987;169:2869–2871. doi: 10.1128/jb.169.6.2869-2871.1987. [DOI] [PMC free article] [PubMed] [Google Scholar] 185.Ashok K.C., Houston C.W., Kurosky A. Genetic variation in related cytolytic toxins produced by different species of Aeromonas. FEMS Microbiol. Lett. 1991;78:231–238. doi: 10.1016/0378-1097(91)90163-5. [DOI] [PubMed] [Google Scholar] 186.Gyles C.L. Shiga toxin-producing Escherichia coli: an overview. J. Anim. Sci. 2007;85:E45–E62. doi: 10.2527/jas.2006-508. [DOI] [PubMed] [Google Scholar] 187.Kaper J.B., Nataro J.P., Mobley H.L. Pathogenic Escherichia coli. Nat. Rev. Microbiol. 2004;2:123–140. doi: 10.1038/nrmicro818. [DOI] [PubMed] [Google Scholar] 188.Johannes L., Romer W. Shiga toxins-from cell biology to biomedical applications. Nat. Rev. Microbiol. 2010;8:105–116. doi: 10.1038/nrmicro2279. [DOI] [PubMed] [Google Scholar] 189.Herold S., Karch H., Schmidt H. Shiga toxinencoding bacteriophages–genomes in motion. Int. J. Med. Microbiol. 2004;294:115–121. doi: 10.1016/j.ijmm.2004.06.023. [DOI] [PubMed] [Google Scholar] 190.Zhang W., Bielaszewska M., Kuczius T., Karch H. Identification, characterization, and distribution of a Shiga toxin 1 gene variant (stx(1c)) in Escherichia coli strains isolated from humans. J. Clin. Microbiol. 2002;40:1441–1446. doi: 10.1128/JCM.40.4.1441-1446.2002. [DOI] [PMC free article] [PubMed] [Google Scholar] 191.Waddell T., Cohen A., Lingwood C.A. Induction of verotoxin sensitivity in receptor-deficient cell lines using the receptor glycolipid globotriosylceramide. Proc. Natl. Acad. Sci. USA. 1990;87:7898–7901. doi: 10.1073/pnas.87.20.7898. [DOI] [PMC free article] [PubMed] [Google Scholar] 192.Soltyk A.M., Mackenzie C.R., Wolski V.M., Hirama T., Kitov P.I., Bundle D.R., Brunton J.L. A mutational analysis of the globotriaosylceramide-binding sites of verotoxin VT1. J. Biol. Chem. 2002;277:5351–5359. doi: 10.1074/jbc.M107472200. [DOI] [PubMed] [Google Scholar] 193.Kiarash A., Boyd B., Lingwood C.A. Glycosphingolipid receptor function is modified by fatty acid content. Verotoxin 1 and verotoxin 2c preferentially recognize different globotriaosyl ceramide fatty acid homologues. J. Biol. Chem. 1994;269:11138–11146. [PubMed] [Google Scholar] 194.Sandvig K., Spilsberg B., Lauvrak S.U., Torgersen M.L., Iversen T.G., van Deurs B. Pathways followed by protein toxins into cells. Int. J. Med. Microbiol. 2004;293:483–490. doi: 10.1078/1438-4221-00294. [DOI] [PubMed] [Google Scholar] 195.Lauvrak S.U., Torgersen M.L., Sandvig K. Efficient endosome-to-Golgi transport of Shiga toxin is dependent on dynamin and clathrin. J. Cell Sci. 2004;117:2321–2331. doi: 10.1242/jcs.01081. [DOI] [PubMed] [Google Scholar] 196.Popoff V., Mardones G.A., Tenza D., Rojas R., Lamaze C., Bonifacino J.S., Raposo G., Johannes L. The retromer complex and clathrin define an early endosomal retrograde exit site. J. Cell Sci. 2007;120:2022–2031. doi: 10.1242/jcs.003020. [DOI] [PubMed] [Google Scholar] 197.Endo Y. Site of action of a Vero toxin (VT2) from Escherichia coli O157:H7 and of Shiga toxin on eukaryotic ribosomes. RNA N-glycosidase activity of the toxins. Eur. J. Biochem. 1988;171:45–50. doi: 10.1111/j.1432-1033.1988.tb13756.x. [DOI] [PubMed] [Google Scholar] 198.Iordanov M.S., Pribnow D., Magun J.L., Dinh T.H., Pearson J.A., Chen S.L., Magun B.E. Ribotoxic stress response: activation of the stress-activated protein kinase JNK1 by inhibitors of the peptidyl transferase reaction and by sequence-specific RNA damage to the α-sarcin/ricin loop in the 28S rRNA. Mol. Cell Biol. 1997;17:3373–3381. doi: 10.1128/mcb.17.6.3373. [DOI] [PMC free article] [PubMed] [Google Scholar] 199.Foster G.H., Tesh V.L. Shiga toxin 1-induced activation of c-Jun NH2-terminal kinase and p38 in the human monocytic cell line THP-1: possible involvement in the production of TNF-α. J. Leukoc. Biol. 2002;71:107–114. [PubMed] [Google Scholar] 200.Yamasaki C., Natori Y., Zeng X.T., Ohmura M., Yamasaki S., Takeda Y., Natori Y. Induction of cytokines in a human colon epithelial cell line by Shiga toxin 1 (Stx1) and Stx2 but not by non-toxic mutant Stx1 which lacks N-glycosidase activity. FEBS Lett. 1999;442:231–234. doi: 10.1016/s0014-5793(98)01667-6. [DOI] [PubMed] [Google Scholar] 201.Smith W.E., Kane A.V., Campbell S.T., Acheson D.W., Cochran B.H., Thorpe C.M. Shiga toxin 1 triggers a ribotoxic stress response leading to p38 and JNK activation and induction of apoptosis in intestinal epithelial cells. Infect. Immun. 2003;71:1497–1504. doi: 10.1128/IAI.71.3.1497-1504.2003. [DOI] [PMC free article] [PubMed] [Google Scholar] 202.Lee S.Y., Lee M.S., Cherla R.P., Tesh V.L. Shiga toxin 1 induces apoptosis through the endoplasmic reticulum stress response in human monocytic cells. Cell. Microbiol. 2008;10:10770–10780. doi: 10.1111/j.1462-5822.2007.01083.x. [DOI] [PubMed] [Google Scholar] 203.Suh J.K., Hovde C.J., Robertus J.D. Shiga toxin attacks bacterial ribosomes as effectively as eucaryotic ribosomes. Biochemistry. 1998;37:9394–9398. doi: 10.1021/bi980424u. [DOI] [PubMed] [Google Scholar] 204.Brieland J., McClain M., Heath L., Chrisp C., Huffnagle G., LeGendre M., Hurley M., Fantone J., Engleberg C. Coinoculation with Hartmannella vermiformis enhances replicative Legionella pneumophila lung infection in a murine model of Legionnaires’ disease. Infect. Immun. 1996;64:2449–2456. doi: 10.1128/iai.64.7.2449-2456.1996. [DOI] [PMC free article] [PubMed] [Google Scholar] 205.Cirillo S.L.G., Littman L.Y.M., Samrakandi M.M., Cirillo J.D. Role of the Legionella pneumophila rtxA gene in amoebae. Microbiology. 2002;148:1667–1677. doi: 10.1099/00221287-148-6-1667. [DOI] [PubMed] [Google Scholar] 206.Horwitz M.A. Formation of a novel phagosome by the Legionnaires’ disease bacterium (Legionella pneumophila) in human monocytes. J. Exp. Med. 1983;158:1319–1331. doi: 10.1084/jem.158.4.1319. [DOI] [PMC free article] [PubMed] [Google Scholar] 207.Samrakandi M.M., Cirillo S.L., Ridenour D.A., Bermudez L.E., Cirillo J.D. Genetic and phenotypic differences between Legionella pneumophila strains. J. Clin. Microbiol. 2002;40:1352–1362. doi: 10.1128/JCM.40.4.1352-1362.2002. [DOI] [PMC free article] [PubMed] [Google Scholar] 208.Ambagala T.C., Ambagala A.P.N., Srikumaran S. The leukotoxin of Pasteurella haemolytica binds to b2 integrins on bovine leukocytes. FEMS Microbiol. Lett. 1999;179:161–167. doi: 10.1111/j.1574-6968.1999.tb08722.x. [DOI] [PubMed] [Google Scholar] 209.D’Auria G., Jimenez N., Peris-Bondia F., Pelaz C., Latorre A., Moya A. Virulence factor rtx in Legionella pneumophila, evidence suggesting it is a modular multifunctional protein. BMC Genomics. 2008;9:14. doi: 10.1186/1471-2164-9-14. [DOI] [PMC free article] [PubMed] [Google Scholar] 210.Belyi Y., Niggeweg R., Opitz B., Vogelsgesang M., Hippenstiel S., Wilm M., Aktories K. Legionella pneumophila glucosyltransferase inhibits host elongation factor 1A. Proc. Natl. Acad. Sci. USA. 2006;103:16953–16958. doi: 10.1073/pnas.0601562103. [DOI] [PMC free article] [PubMed] [Google Scholar] 211.Nachamkin I. Campylobacter jejuni. In: Doyle M.P., Beuchat L.R., Montville T.J., editors. Food Microbiology, Fundamentals and Frontiers. 2nd. ASM Press; Washington, DC, USA: 2001. [Google Scholar] 212.Skirrow M.B., Blaser M.J. Clinical aspects of Campylobacter infection. In: Nachamkin I., Blaser M.J., editors. Campylobacter. 2nd. ASM Press; Washington, DC, USA: 2000. pp. 69–88. [Google Scholar] 213.Whitehouse C.A., Balbo P.B., Pesci E.C., Cottle D.L., Mirabito P.M., Pickett C.L. Campylobacter jejuni cytolethal distending toxin causes a G2-phase cell cycle block. Infect. Immun. 1998;66:1934–1940. doi: 10.1128/iai.66.5.1934-1940.1998. [DOI] [PMC free article] [PubMed] [Google Scholar] 214.Okuda J., Kurazono H., Takeda Y. Distribution of the cytolethal distending toxin A gene (cdtA) among species of Shigella and Vibrio, and cloning and sequencing of the cdt gene from Shigella dysenteriae. Microb. Pathog. 1995;18:167–172. doi: 10.1016/s0882-4010(95)90022-5. [DOI] [PubMed] [Google Scholar] 215.Young V.B., Knox K.A., Schauer D.B. Cytolethal distending toxin sequence and activity in the enterohepatic pathogen Helicobacter hepaticus. Infect. Immun. 2000;68:184–191. doi: 10.1128/iai.68.1.184-191.2000. [DOI] [PMC free article] [PubMed] [Google Scholar] 216.Zheng J., Meng J., Zhao S., Singh R., Song W. Campylobacter-induced interleukin-8 secretion in polarized human intestinal epithelial cells requires Campylobacter-secreted cytolethal distending toxin- and Toll-like receptor-mediated activation of NF-kB. Infect. Immun. 2008;76:4498–4508. doi: 10.1128/IAI.01317-07. [DOI] [PMC free article] [PubMed] [Google Scholar] 217.Tejero L.M., Galan J.E. A bacterial toxin that controls cell cycle progression as a deoxyribonuclease I-like protein. Science. 2000;290:354–357. doi: 10.1126/science.290.5490.354. [DOI] [PubMed] [Google Scholar] 218.Tejero L.M., Galan J.E. CdtA, CdtB, and CdtC form a tripartite complex required for cytolethal distending toxin. Infect. Immun. 2001;69:4358–4365. doi: 10.1128/IAI.69.7.4358-4365.2001. [DOI] [PMC free article] [PubMed] [Google Scholar] 219.Long S.C., Tauscher T. Watershed issues associated with Clostridium botulinum: A literature review. J. Water Health. 2006;04.3:277–288. doi: 10.2166/wh.2006.516. [DOI] [PubMed] [Google Scholar] 220.DasGupta B.R., Sugiyama H. A common subunit structure in Clostridium botulinum type A, B and E toxins. Biochem. Biophys. Res. Commun. 1972;48:108–112. doi: 10.1016/0006-291x(72)90350-6. [DOI] [PubMed] [Google Scholar] 221.Tsuzuki K., Kimura K., Fujii N., Yokosawa N., Indoh T., Murakami T., Oguma K. Cloning and complete nucleotide sequence of the gene for the main component of hemagglutinin produced by Clostridium botulinum type C. Infect. Immun. 1990;58:3173–3177. doi: 10.1128/iai.58.10.3173-3177.1990. [DOI] [PMC free article] [PubMed] [Google Scholar] 222.Tsuzuki K., Kimura K., Fujii N., Yokosawa N., Oguma K. The complete nucleotide sequence of the gene coding for the nontoxic-nonhemagglutinin component of Clostridium botulinum type C progenitor toxin. Biochem. Biophys. Res. Commun. 1992;183:1273–1279. doi: 10.1016/s0006-291x(05)80328-6. [DOI] [PubMed] [Google Scholar] 223.Pellizzari R., Guidi-Rontani C., Vitale G., Mock M., Montecucco C. Anthrax lethal factor cleaves MKK3 in macrophages and inhibits the LPS/IFN gamma-induced release of NO and TNFalpha. FEBS Lett. 1999;462:199–204. doi: 10.1016/s0014-5793(99)01502-1. [DOI] [PubMed] [Google Scholar] 224.Montecucco C., Schiavo G., Rossetto O. The mechanism of action of tetanus and botulinum neurotoxins. Arch. Toxicol. Suppl. 1996;18:342–354. doi: 10.1007/978-3-642-61105-6_32. [DOI] [PubMed] [Google Scholar] 225.Montecucco C., Schiavo G. Mechanism of action of tetanus and botulinum neurotoxins. Mol. Microbiol. 1994;13:1–8. doi: 10.1111/j.1365-2958.1994.tb00396.x. [DOI] [PubMed] [Google Scholar] 226.East F.A.K., Collins M.D. Conserved structure of genes encoding components of botulinum neurotoxin complex M and the sequence of the gene coding for the nontoxic component in nonproteolytic Clostridium botulinum type. Curr. Microbiol. 1994;29:69–77. doi: 10.1007/BF01575751. [DOI] [PubMed] [Google Scholar] 227.Binz T., Kurazono H., Wille M., Frevert J., Wernars K., Niemann H. The complete sequence of botulinum neurotoxin type A and comparison with other clostridial neurotoxins. J. Biol. Chem. 1990;265:9153–9158. [PubMed] [Google Scholar] 228.Hauser D., Eklund M.W., Boquet P., Popoff M.R. Organization of the botulinum neurotoxin C1 gene and its associated non-toxic protein genes in Clostridium botulinum C 468. Mol. Gen. Genet. 1994;243:631–640. doi: 10.1007/BF00279572. [DOI] [PubMed] [Google Scholar] 229.Zhou Y., Sugiyama H., Nakano H., Johnson E.A. The genes for the Clostridium botulinum type G toxin complex are on a plasmid. Infect. Immun. 1995;63:2087–2091. doi: 10.1128/iai.63.5.2087-2091.1995. [DOI] [PMC free article] [PubMed] [Google Scholar] 230.Songer J.G., Meer R.M. Genotyping of Clostridium perfringens by polymerase chain reaction is a useful adjunct to diagnosis of clostridial enteric disease in animals. Anaerobe. 1996;2:197–203. [Google Scholar] 231.Hunter S.E., Brown J.E., Oyston P.C., Sakurai J., Titball R.W. Molecular genetic analysis of beta-toxin of Clostridium perfringens reveals sequence homology with alpha-toxin, gamma-toxin, and leukocidin of Staphylococcus aureus. Infect. Immun. 1993;61:3958–3965. doi: 10.1128/iai.61.9.3958-3965.1993. [DOI] [PMC free article] [PubMed] [Google Scholar] 232.Czeczulin J.R., Hanna P.C., McClane B.A. Cloning, nucleotidesequencing, and expression of the Clostridium perfringens enterotoxin gene in Escherichia coli. Infect. Immun. 1993;61:3429–3439. doi: 10.1128/iai.61.8.3429-3439.1993. [DOI] [PMC free article] [PubMed] [Google Scholar] 233.Brynestad S., Iwanejko L.A., Stewart G.S.A.B., Granum P.E. A complex array of Hpr consensus DNA recognition sequences proximal to the enterotoxin gene in Clostridium perfringens type A. Microbiology. 1994;140:97–104. doi: 10.1099/13500872-140-1-97. [DOI] [PubMed] [Google Scholar] 234.Anzai Y.K.H., Park J.Y., Wakabayashi H. Phylogenetic affiliation of the pseudomonads based on 16S rRNA sequence. Int. J. Syst. Evol. Microbiol. 2000;50:1563–1589. doi: 10.1099/00207713-50-4-1563. [DOI] [PubMed] [Google Scholar] 235.Hassett D., Cuppoletti J., Trapnell B., Lymar S., Rowe J., Yoon S., Hilliard G., Parvatiyar K., Kamani M., Wozniak D., Hwang S., McDermott T., Ochsner U. Anaerobic metabolism and quorum sensing by Pseudomonas aeruginosa biofilms in chronically infected cystic fibrosis airways: rethinking antibiotic treatment strategies and drug targets. Adv. Drug. Deliv. Rev. 2002;54:1425–1443. doi: 10.1016/s0169-409x(02)00152-7. [DOI] [PubMed] [Google Scholar] 236.Van Eldere J. Multicentre surveillance of Pseudomonas aeruginosa susceptibility patterns in nosocomial infections. J. Antimicrob. Chemother. 2003;51:347–352. doi: 10.1093/jac/dkg102. [DOI] [PubMed] [Google Scholar] 237.Wolf P., Elsasser-Beile Ú. Pseudomonas exotoxin A: From virulence actor toanti-cancera gent. Int. J. Med. Microbiol. 2009;299:161–176. doi: 10.1016/j.ijmm.2008.08.003. [DOI] [PubMed] [Google Scholar] 238.Domenighini M., Rappuoli R. Three conserved consensus sequences identify the NAD-binding site of ADP-ribosylating enzymes, expressed by eukaryotes, bacteria and T-even bacteriophages. Microbiology. 1996;21:667–674. doi: 10.1046/j.1365-2958.1996.321396.x. [DOI] [PubMed] [Google Scholar] 239.Siegall C.B., Chaudhary V.K., FitzGerald D.J., Pastan I. Functional analysis of domains II, Ib, and III of Pseudomonas exotoxin. J. Biol. Chem. 1989;264:14256–14261. [PubMed] [Google Scholar] 240.Ogata M., Fryling C.M., Pastan I., FitzGerald D.J. Cell-mediated cleavage of Pseudomonas exotoxin between Arg279 and Gly280 generates the enzymatically active fragment which translocates to the cytosol. J. Biol. Chem. 1992;267:25396–25401. [PubMed] [Google Scholar] 241.Lombardi D., Soldati T., Riederer M.A., Goda Y., Zerial M., Pfeffer S.R. Rab9 functions in transport between late endosomes and the trans Golgi network. EMBO J. 1993;12:677–682. doi: 10.1002/j.1460-2075.1993.tb05701.x. [DOI] [PMC free article] [PubMed] [Google Scholar] 242.Jackson M.E., Simpson J.C., Girod A., Pepperkok R., Roberts L.M., Lord J.M. The KDEL retrieval system is exploited by Pseudomonas exotoxin A, but not by Shiga-like toxin-1, during retrograde transport from the Golgi complex to the endoplasmic reticulum. J. Cell Sci. 1999;112:467–475. doi: 10.1242/jcs.112.4.467. [DOI] [PubMed] [Google Scholar] 243.Duris J.W., Haack S.K., Fogarty L.R. Gene and antigen markers of Shiga-toxin producing E. coli from Michigan and Indiana river water: Occurrence and relation to recreational water quality criteria. J. Environ. Qual. 2009;38:1878–1886. doi: 10.2134/jeq2008.0225. [DOI] [PubMed] [Google Scholar] 244.Belkin S., Colwell R.J. Oceans and Health: Pathogens in the Marine Environment. Springer Science+Business Media; New York, NY, USA: 2005. [Google Scholar] 245.Janda J.M., Abbott S.L. The genus Aeromonas: taxonomy, pathogenicity, and infection. Clin. Microbiol. Rev. 2010;23:35–73. doi: 10.1128/CMR.00039-09. [DOI] [PMC free article] [PubMed] [Google Scholar] 246.Barth H., Aktories K., Popoff M.R., Stiles B.G. Binary Bacterial toxins: biochemistry, biology, and applications of common Clostridium and Bacillus proteins. Microbiol. Mol. Biol. Rev. 2004;68:373–402. doi: 10.1128/MMBR.68.3.373-402.2004. [DOI] [PMC free article] [PubMed] [Google Scholar] 247.Glenn G.M., Francis D.H., Danielsen E.M. Toxin-mediated effects on the innate mucosal defenses: implications for enteric vaccines. Infect. Immun. 2009;77:5206–5215. doi: 10.1128/IAI.00712-09. [DOI] [PMC free article] [PubMed] [Google Scholar] Articles from Toxins are provided here courtesy of Multidisciplinary Digital Publishing Institute (MDPI) ACTIONS View on publisher site PDF (1.2 MB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract 1. Introduction 2. Toxins Produced by Cyanobacteria 3. 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Calculate normality of the following: a. 0.74g of Ca(OH)2 in 5mL of solution. More from this Exercise To calculate the normality of the solution, we will follow these steps: Step 1: Identify the given values - Weight of solute (Ca(OH)₂) = 0.74 g - Volume of solution = 5 mL Step 2: Convert volume from mL to liters Normality (N) is defined in terms of liters, so we need to convert the volume from mL to liters: Volume in liters=5 mL1000=0.005 L Step 3: Calculate the molecular weight of Ca(OH)₂ To find the molecular weight of calcium hydroxide (Ca(OH)₂), we add the atomic masses of its constituent elements: - Atomic mass of Calcium (Ca) = 40 g/mol - Atomic mass of Oxygen (O) = 16 g/mol (there are 2 O atoms) - Atomic mass of Hydrogen (H) = 1 g/mol (there are 2 H atoms) Calculating the molecular weight: Molecular weight of Ca(OH)₂=40+(2×16)+(2×1)=40+32+2=74 g/mol Step 4: Determine the valency factor The valency factor for Ca(OH)₂ is determined by the number of hydroxide ions (OH⁻) it can provide. Since there are 2 OH⁻ ions in Ca(OH)₂, the valency factor is: Valency factor=2 Step 5: Apply the normality formula The formula for normality (N) is given by: N=Weight of solute (g)×Valency factorMolecular weight (g/mol)×Volume (L) Substituting the values: N=0.74 g×274 g/mol×0.005 L Step 6: Calculate the normality Now, we can perform the calculations: N=1.480.37=4 N Final Answer The normality of the solution is 4 N. --- To calculate the normality of the solution, we will follow these steps: Step 1: Identify the given values - Weight of solute (Ca(OH)₂) = 0.74 g - Volume of solution = 5 mL Step 2: Convert volume from mL to liters Normality (N) is defined in terms of liters, so we need to convert the volume from mL to liters: Volume in liters=5 mL1000=0.005 L Step 3: Calculate the molecular weight of Ca(OH)₂ To find the molecular weight of calcium hydroxide (Ca(OH)₂), we add the atomic masses of its constituent elements: - Atomic mass of Calcium (Ca) = 40 g/mol - Atomic mass of Oxygen (O) = 16 g/mol (there are 2 O atoms) - Atomic mass of Hydrogen (H) = 1 g/mol (there are 2 H atoms) Calculating the molecular weight: Molecular weight of Ca(OH)₂=40+(2×16)+(2×1)=40+32+2=74 g/mol Step 4: Determine the valency factor The valency factor for Ca(OH)₂ is determined by the number of hydroxide ions (OH⁻) it can provide. Since there are 2 OH⁻ ions in Ca(OH)₂, the valency factor is: Valency factor=2 Step 5: Apply the normality formula The formula for normality (N) is given by: N=Weight of solute (g)×Valency factorMolecular weight (g/mol)×Volume (L) Substituting the values: N=0.74 g×274 g/mol×0.005 L Step 6: Calculate the normality Now, we can perform the calculations: N=1.480.37=4 N Final Answer The normality of the solution is 4 N. Topper's Solved these Questions Explore 6 Videos Explore 3 Videos Explore 2 Videos Explore 25 Videos Similar Questions Calculate the normality of a solution containing 15.8 g of KMnO4 in 50 mL acidic solution. Calculate the molarity of each of the following solutions : a.30g of Co(NO3)2.6H2O in 4.3L of solution b.30mL of 0.5MH2SO4 diluted to 500mL. Calculate the pH of the following solutions : 0.3 g of Ca(OH)2 dissolved in water to give 500 mL of solutions. Calculate the molarity of a solution containing 5g of NaOH in 450mL solution. Calculate the normality of a solution containing 13.4 g fo sodium oxalate in 100mL solution. Calculate the normality of the solution obtained by mixing 10 mL of N/5 HCl and 30 mL of N/10 HCl. Calculate the molarity of the following solution: (a) 4g of caustic soda is dissolved in 200mL of the solution. (b) 5.3 g of anhydrous sodium carbonate is dissolved in100 mL of solution. (c ) 0.365 g of pure HCl gas is dissolved in 50 mL of solution. Calculate the molarity of the following solution: (a) 2g of caustic soda is dissolved in 200mL of the solution. Calculate the PH of the resultant mixture : 10 mL of 0.2 M Ca(OH)2 + 25 mL of 0.1 M HCl (a) A solution is prepared by dissolving 3.65g of HCl in 500 mL of the solution. Calculate the normality of the solution (b) Calculate the volume of this solution required to prepare 250 mL of 0.05 N solution. CENGAGE CHEMISTRY ENGLISH-SOME BASIC CONCEPTS AND MOLE CONCEPT-Archives Subjective Calculate normality of the following: a. 0.74 g of Ca(OH)(2) in 5 mL... What is the molarity and molality of a 13% solution (by weight) of sul... The density of ammonia at 30^(@)C and 5 atm pressure is 4.215 g a metallic carbonate was heated in a hard glass tube and CO2 e... The density of a 3 M Na(2) S(2) O(3) (sodium thiosulphate) solution is... A sugar syrup of weight 214.2 g contains 34.2 g of sugar (C(12)H(2... Calculate the molality of 1 litre solution of 93% H2SO4 (weight/volume... Calculate the volume occupied by 5.0 g of acetylene gas at 50^(@)C and... On mixing 45.0 mL of 0.25 M lead nitrate solution with 25.0 mL of 0.10... When 0.575 xx 10^(-2) kg of Glaube's salt is dissolved in water, we ge... Calculate the molarity of water if its density is 1000 kg m^(-3) Calculate the amount of calcuium oxide required when it reacts with 85... 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14057
https://www.scribd.com/document/611793140/Chapter-5-Exercises
Chapter 5 - Exercises | PDF | Bonds (Finance) | Yield (Finance) Opens in a new window Opens an external website Opens an external website in a new window This website utilizes technologies such as cookies to enable essential site functionality, as well as for analytics, personalization, and targeted advertising. To learn more, view the following link: Privacy Policy Open navigation menu Close suggestions Search Search en Change Language Upload Sign in Sign in Download free for 30 days 0 ratings 0% found this document useful (0 votes) 160 views 2 pages Chapter 5 - Exercises This document contains 14 concept questions and exercises about bond valuation from Corporate Finance 11e by Ross, Westerfield, and Jaffe. The questions cover topics such as valuing bonds, c… Full description Uploaded by Vân Nhi Phạm AI-enhanced title and description Go to previous items Go to next items Download Save Save Chapter 5_Exercises For Later Share 0%0% found this document useful, undefined 0%, undefined Print Embed Ask AI Report Download Save Chapter 5_Exercises For Later You are on page 1/ 2 Search Fullscreen C o n c e p t Q u e s t i o n s a n d E x e r c i s e s C O R P O R A T E F I N A N C E 1 1 e b y R o s s, W e s t e r f i e l d, J a f f e CHAPTER5 BOND VALUATION Val uin g Bon ds What is the price of a 15-year, zero coupon bond paying $1,000 atmaturity, assuming semiannual compounding, if the YTM is: a. 6 percent? b. 8 percent? c. 10 percent? Valuing Bonds Microhard has issued a bond with the following characteriscs:Par: $1,000 Time to maturity: 20 years Coupon rate: 7 percent Semiannual payments Calculate the price of this bond if the YTM is: a. 7 percent b. 9 percent c. 5 percent Bond Yields Waers Umbrella Corp. issued 15-year bonds 2 years ago at a coupon rateof 5.9 percent.The bonds make semiannual payments. If these bonds currently sell for105 percent of par value, what is the YTM? Bond Yields A Japanese company has a bond outstanding that sells for 106 percentof its ¥100,000 par value. The bond has a coupon rate of 2.8 percent paid annually andmatures in 21 years. What is the yield to maturity of this bond? Zero Coupon Bonds You nd a zero coupon bond with a par value of $10,000 and17 years to maturity. If the yield to maturity on this bond is 4.9 percent, what is the dollarprice of the bond?Assume semiannual compounding periods. Valuing Bonds Yan Yan Corp. has a $2,000 par value bond outstanding with a couponrate of 4.9 percent paid semiannually and 13 years to maturity. The yield to maturity ohe bond is 3.8 percent. What is the dollar price of the bond? Valuing Bonds Union Local School District has bonds outstanding with a couponrate of 3.7 percent paid semiannually and 16 years to maturity. The yield to maturityon these bonds is 3.9 percent, and the bonds have a par value of $5,000. What is thedollar price of the bond? adDownload to read ad-free C o n c e p t Q u e s t i o n s a n d E x e r c i s e s C O R P O R A T E F I N A N C E 1 1 e b y R o s s, W e s t e r f i e l d, J a f f e Zero Coupon Bonds You buy a zero coupon bond at the beginning of the year that has a face value of$1,000, a YTM of 6.3 percent, and 25 years to maturity. If you hold the bond for the enre year, how much in interest income will you have to declare on your tax return? Assume semiannual compounding. Bond Yields Hacker Soware has 6.2 percent coupon bonds on the market with9 years to maturity. The bonds make semiannual payments and currently sell for104 percent of par. What is the current yield on the bonds? The YTM? The eecveannual yield? Bond Yields RAK Co. wants to issue new 20-year bonds for some much-needed expansion projects. The company currently has 6.4 percent coupon bonds on the market that sell for $1,063, make semiannual payments, and mature in 20 years. What coupon rate should the company set on its new bonds if it wants them to sell at par? Finding the Bond Maturity Erna Corp. has 9 percent coupon bonds making annual payments with a YTM of 7.81 percent. The current yield on these bonds is 8.42 percent.How many years do these bonds have le unl they mature? Findi ng the Maturity You’ve just found a 10 percent coupon bond on the market that sells for par value. What is the maturity on this bond? Zero Coupon Bonds Suppose your company needs to raise $50 million and you want to issue 30-year bonds for this purpose. Assume the required return on your bond issue will be 6 percent, and you’re evaluang two issue alternaves: A semiannual coupon bond with a 6 percent coupon rate and a zero coupon bond. Your company’s tax rate is 35 percent. a. How many of the coupon bonds would you need to issue to raise the $50 million? How many of the zeroes would you need to issue? b. In 30 years, what will your company’s repayment be if you issue the coupon bonds?What if you issue the zeroes? c. Based on your answers in (a) and (b), why would you ever want to issue the zeroes?To answer, calculate the rm’s ae rta x cas h oulow s for the rs t yea r und er the two diere nts cena rio s. Ass ume the IRS amorzaon rules apply for the zero coupon bonds. Valuing Bonds The Frush Corporaon has two dierent bonds currently outstanding.Bond M has a face value of $30,000 and matures in 20 years. The bond makes no paymentsfor the rst six years, then pays$800 every six months over the subsequent eight years, andnally pays $1,000 every six months over the last six years. Bond N also has a face value of$30,000 and a maturity of 20 years; it makes no coupon payments over the life of the bond.If the required return on both these bonds is 6.4 percent compounded semiannually, what isthe current price of Bond M? Of Bond N? 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14058
https://nrich.maths.org/problems/hidden-squares
Problem-Solving Schools can now access the Hub! Contact us if you haven't received login details Or search by topic Number and algebra Properties of numbers Place value and the number system Calculations and numerical methods Fractions, decimals, percentages, ratio and proportion Patterns, sequences and structure Coordinates, functions and graphs Algebraic expressions, equations and formulae Geometry and measure Measuring and calculating with units Angles, polygons, and geometrical proof 3D geometry, shape and space Transformations and constructions Pythagoras and trigonometry Vectors and matrices Probability and statistics Handling, processing and representing data Probability Working mathematically Thinking mathematically Mathematical mindsets Advanced mathematics Calculus Decision mathematics and combinatorics Advanced probability and statistics Mechanics For younger learners Early years foundation stage Hidden squares Can you find the squares hidden on these coordinate grids? Age 11 to 14 Challenge level Secondary GEOMETRY Angles, Polygons and Geometrical Proof Exploring and noticing Working systematically Conjecturing and generalising Visualising and representing Reasoning, convincing and proving Being curious Being resourceful Being resilient Being collaborative Problem Hidden Squares printable sheet You might like to play Square Itbefore working on this problem.28 points have been marked on the axes below. Image The points mark the vertices of eight hidden squares.Each of the four red points is a vertex shared by two squares.Each square shares exactly one vertex with another square.Can you find the eight hidden squares?Each square is a different size, and there are no points marked on the side of any square.Once you've found them all, take a look at the grid below: Image This time, there are $34$ points marking the vertices of ten hidden squares.There are $6$ red vertices, which are shared by two squares.All of the squares share at least one vertex with another square.Can you find the ten hidden squares?Once again, each square is a different size, and there are no points marked on the side of any square, although some points come very close!Why not have another go at Square It and see if you have improved your square-spotting skills? Or take a look at Square Coordinates for a similar challenge. Getting Started Start by choosing two points. Where would the other points be to make a square? Student Solutions Lots of school groups handed in excellent solutions to this problem.Well done particularly to Ramu and Atasha from Brookland Primary School for their answers: Square 1 - (1,0) , (1,3) , (4,0) , (4,3)Square 2 - (1,3) , (0,7) , (5,4) , (4,8)Square 3 - (20,1) , (16,1) , (16, 5) , (20,5)Square 4 - (0, 13) , (8,7) , (20,5) , (15,12)Square 5 - (12,7) , (12,14) , (19,7) , (19,14)Square 6 - (12,14) , (8,18) , (12,22) , (16,18)Sqaure 7 - (5,11) , (11,12) , (10,18) , (4,17)Sqaure 8 - (0,14) , (3,10) , (7,13) , (4,17) The Year 4s from Bradfield Dungworth sent in their picture: As Image As did the Year 6s from Wingrave C of E: Image For the second part, Harjun sent in this image to show us where the ten squares are: Image There is just one square which you've mis-drawn a line in I think, Harjun. Can you see which one it is? Akalya and Michelle from Devonshire Primary School and Esther sent in the coordinates of the 10 squares so we can check that way, too: 1.(2,-3) (-1,-3) (-1,-6) (2,-6)2.(0,0) (5,-3) (2,-8) (-3,-5)3.(0,0) (2,2) (0,4) (-2,2)4.(1,0) (4,1) (5,-2) (2,-3)5.(2,10) (8,10) (8,4) (2,4)6.(2,10) (7,8) (5,3) (0,5)7.(-1,-1) (-1,-3) (-3,-3) (-3,-1)8.(-4,2) (-9,2) (-9,-3) (-4,-3)9.(-7,5) (-10,2) (-7,-1) (-4,2)10.(-7,5) (-5,4) (-4,6) (-6,7) Thank you to all of you. Teachers' Resources Why do this problem? Students often get hung up on shapes being oriented in a particular way. This problem involves squares arranged in all sorts of orientations, so students will need a secure understanding of the properties of squares. There is also the opportunity to practise working with coordinates in all four quadrants. Possible approach This printable worksheet may be useful:Hidden SquaresIt might be useful to start by playing a few games of Square It so that students are challenged to think about squares which are not in the usual orientation.Hand out the worksheet and explain the problem. Encourage students to work in pairs to find some of the hidden squares, and while they work, circulate to listen to discussions and see what strategies are emerging.After students have had a chance to find a couple of squares, bring the class together to share strategies. You may wish to ask:"If these two points are corners of the same square, where would the other two corners be?"Or choose a student who you know has found a square that not many others have found yet:"Tell me the coordinates of two adjacent corners of your square. Now, can everyone else work out where the other two corners must be?"Next, give students some time to finish the first task and then apply their strategies to the second task using coordinates in all four quadrants.A nice way to finish would be by going back to Square It to see if they are more successful now that they have had more experience looking for tilted squares. Further lessons might look at the problems Square Coordinates, Opposite Vertices, or Tilted Squares. Possible support Suggest that students start by looking for the three squares (four in the second problem) whose sides are parallel to the coordinate axes. Possible extension Square Coordinates encourages exploration of the relationship between coordinates of the vertices of squares.Opposite Vertices challenges students to find squares given two opposite vertices.Tilted Squares encourages exploration of the area of squares and leads to Pythagoras' Theorem.
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https://web.williams.edu/Mathematics/sjmiller/public_html/math/papers/PrimeSqFreeWalkRevisedV22.pdf
WALKING TO INFINITY ALONG SOME NUMBER THEORY SEQUENCES Steven J. Miller Department of Mathematics and Statistics, Williams College, Williamstown, MA, USA sjm1@williams.edu Fei Peng Department of Mathematics, National University of Singapore, Singapore, Singapore pfpf@u.nus.edu Tudor Popescu Department of Mathematics, Brandeis University, Waltham, MA, USA tudorpopescu@brandeis.edu Joshua M. Siktar Department of Mathematics, Texas A&M University, TX, USA jmsiktar@tamu.edu Nawapan Wattanawanichkul Department of Mathematics, University of Illinois Urbana-Champaign, Urbana, IL, USA nawapan2@illinois.edu Received: , Revised: , Accepted: , Published: Abstract An interesting open conjecture asks whether it is possible to walk to infinity along primes, where at each step we add one more digit somewhere in the prime. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtrack-ing one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to the primes. We introduce stochastic models and analyze the walks’ expected lengths and the frequencies of the digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences, such as perfect squares, and primes in different bases. INTEGERS: 24 (2024) 2 1. Introduction 1.1. Background This paper is motivated by a simple question: is it possible to walk to infinity along the primes? By this we mean starting with a prime number, appending one digit to it to form a new prime, and repeating endlessly. Note that if each time we are appending to the left an unlimited number of digits, the answer would be positive. One can prove this using Dirichlet’s theorem for primes in arithmetic progression: given any prime p other than 2 and 5, choose some integer m so that 10m > p. As 10m and p are relatively prime, there are infinitely many primes congruent to p modulo 10m. Any such prime is obtainable by appending digits to the left of p. We can then repeat this process to walk to infinity. Another natural interpretation is to append one digit at a time, to the right. This greatly reduces the likelihood of an infinite walk. For example, one may start a walk as 3, 31, 317, and find that 317 cannot be extended further (by one digit to stay a prime). In fact, starting with any one-digit prime, the longest “prime walk” (via appending one digit a time to the right) always has length 8. For example, the optimal walk sequence starting with 3 is {3, 37, 373, 3733, 37337, 373379, 3733799, 37337999}. To see this, we introduce the notion of a right truncatable prime, which is a prime that remains prime after removing the rightmost digits successively. It is known that there are exactly 83 right truncatable primes, with the largest one being 73939133 . Every right truncatable prime with d digits corresponds to a prime walk of length d starting with a one-digit prime (and vice versa), so the longest such walk has length 8. Without the one-digit starting-point restriction, it is possible to have longer walks: {19, 197, 1979, 19793, 197933, 1979339, 19793393, 197933933, 1979339333} is a walk with step size 1 and length 9, while {409, 4099, 40993, 409933, 4099339, 40993391, 409933919, 4099339193, 40993391939, 409933919393, 4099339193933} is one of length 11. In particular, an exhaustive search shows that the above is the longest prime walk with a starting point less than one million, tied with {68041, 680417, 6804173, 68041739, 680417393, 6804173939, 68041739393, 680417393939, 6804173939393, 68041739393933, 680417393939333}. INTEGERS: 24 (2024) 3 On the other hand, some small primes do not have long walks, such as 11, whose longest walk is {11, 113}, and 53, which fails immediately as the numbers between 530 and 539 are all composite. This discussion suggests two central questions. • Is it possible to walk to infinity along the primes, where each prime in the sequence is the result of appending one digit to the right of the previous? From the last observation, we cannot do so by starting with a one-digit prime. Some remainder analysis shows that if there is an infinite prime walk (in base 10), it eventually only appends 3’s and 9’s. To clarify, in view of remainders modulo 2 and 5, we can never append an even number or a 5. Then consider remainders modulo 3. As 1 and 7 are both congruent to 1 modulo 3, appending them would increase the remainder by 1. Appending 3 or 9 would leave the remainder unchanged. As we need to avoid any multiple of 3 at all times, we can append 1 or 7 at most once (twice when starting at 3, but 3 is already not a promising starting point), and must only use 3’s and 9’s afterward. • What if, instead of appending just one digit, we append at most a bounded number of digits to the right? More generally, what if the number of digits we append in moving from pn to pn+1 is at most f(pn) for some function f tending to infinity? Unlike the case of appending to the left, we cannot immediately deduce the answer by appealing to Dirichlet’s theorem for primes in arithmetic progressions. 1.2. Stochastic Models Like most problems in number theory, the aforementioned questions are easy to state but resist progress. We thus consider instead related random problems to try and get a sense of what might be true. Such models have been used elsewhere with great success, from suggesting there are only finitely many Fermat primes to the veracity of the Twin Prime and Goldbach conjectures. For example, recall the nth Fermat number is Fn = 22n + 1. The prime number theorem says that the number of primes up to x is about x/ log x, and thus one often models a randomly chosen number of order x as being prime with probability 1/ log x. This is the famous Cram´ er model; while it is known to have some issues [12, pp. 507–514], it gives reasonable answers for many problems. If we let {Zn} be independent Bernoulli random variables where Zn = 1 with probability 1/ log Fn, then the expected number of Zn’s that are 1 (and thus the expected number of Fermat primes) is E " ∞ X n=0 Zn # = ∞ X n=0 1 log(22n + 1) ≈2.24507722, INTEGERS: 24 (2024) 4 which is reasonably close to the number of known Fermat primes, five, coming from n ∈{0, 1, 2, 3, 4}. As primes lack much inherent structure, we ask related questions of other se-quences, such as square-free numbers. From our heuristic model and numerical explorations, we do not believe one can walk to infinity through the primes by adding a bounded number of digits to the right; however, we believe it is possible for square-free numbers. For example, starting with 2, we can get a really long walk just by always appending the smallest digit that yields a square-free number. The following sequence only shows the first 17 numbers obtained using this greedy approach. We do not expect this sequence to terminate soon. {2, 21, 210, 2101, 21010, 210101, 2101010, 21010101, 210101010, 2101010101, 21010101010, 210101010101, 2101010101010, 2101010101010102, 210101010101021, 2101010101010210, 21010101010102101, . . .}. While the fraction of numbers at most x that are prime is approximately 1/ log x, which tends to zero, the fraction that are square-free tends to 1/ζ(2) = 6/π2, or about 60.79% (for more details, see Section 3.1). Thus, there are tremendously more square-free numbers available than primes. In particular, once our number is large, it is unlikely that any digit can be appended to create another prime. Thus, it should be impossible to walk to infinity among the primes by appending just one digit on the right. However, for square-free numbers, we expect to have several digits that we can append and stay square-free, leading to exponential growth in the number of paths. Computationally, a bottleneck of investigating prime or square-free walks is the hardness of factorization, which is necessary to determine whether the current num-ber is prime/square-free. To overcome this difficulty, we describe fast, stochastic models that approximate the actual walks. Explicitly, we consider the following random processes: given a sequence whose last term is x, we want to assign an appropriate probability of being able to append an additional digit to the right. We assume each term is independent of the previous, and the probability that a digit can be appended to x is p(x). Thus, the probability will decrease as x increases for primes but is essentially constant for square-free numbers. Furthermore, for prime walks, we present two different models: the first one randomly selects a digit among 1, 3, 7, and 9 and appends it to the number, while the second (refined) random model first checks what digits yield a prime number in the next step and then randomly selects one from the set. We assume all numbers with the same number of digits are equally likely to be in the sequence for simplicity. For the primes base 10, we cannot append a digit that is even or a 5, whereas, for square-free numbers, we cannot append a digit such that the sum of the digits is 9 or the last two digits are a multiple of 4. One could consider more involved models taking these into account. INTEGERS: 24 (2024) 5 We approximate that if a number has k digits, the number of primes of k digits in base b is bk log bk − bk−1 log bk−1 = bk−1 log b ·  b k − 1 k −1  = bk−1((k −1)b −k) k(k −1) log b ≈(b −1) · bk−1 k log b . Since there are (b −1)bk−1 numbers with exactly k digits in base b, we assume the probability that a k−digit number is prime is 1/(k log b), and assume that the events of two distinct numbers being prime are independent. Our main focus is the expected value and distribution of lengths of walks under these stochastic settings. Such probabilistic models have had remarkable success in modeling other problems, such as the 3x + 1 map and its generalizations . They also have several issues. In particular, we assume that the numbers formed by appending the digits under consideration are all independent in our desired sequence. However, this yields a simple model with easily computed results on how long we expect to be able to walk in the various sequences from different starting points. In the rest of the paper, the expected walk length naturally refers to the expected value of the length of the walk. Typically, there is either an explicit, finite collection (or sample) of walks to empirically determine the expected length from, or a corre-sponding model for a walk, under which the expected walk length can be computed theoretically from the model’s definition. 1.3. Results We compare the random model with observations of the actual sequences. We present the two random models for prime walks and show that the refined one is very close, in some sense, to the actual sequence. In particular, when considering prime walks with starting point less than a million, the difference between the experimental expected length for the careful greedy model and the real expected value for the primes is 0.14, less than 7 percent of the expected length of the real prime walks of 2.07. Furthermore, we note that the model becomes more precise as the starting point increases, and the prime numbers become more sparse. As the starting point in-creases, the number of primes from which we randomly choose to continue decreases. Then, we also look at the frequency of the digits added at each step and see that the refined model approximates the real world extremely well. Lastly, while we discuss infinite prime walks, we extend our predictions for the case where we are allowed to insert a digit anywhere, rather than only to the right. On the other hand, when investigating square-free walks, we present the ex-perimental expected length of our random models. Furthermore, we remark on the discrepancies in the frequencies of added digits, and give the number-theoretic reasons for these discrepancies. INTEGERS: 24 (2024) 6 Although we use stochastic models for prime and square-free walks, there are some sequences and restrictive scenarios for which we can prove several results regarding walks to infinity, for example, prime walks in base 2, 3, 4, 5, and 6, and on perfect squares. A related problem is the Gaussian Moat problem, which asks whether it is possi-ble to walk to infinity on Gaussian primes with steps of bounded length. Extensive research has been done on this. For example, Gethner, Wagon, and Wick in and Loh in proved numerous results related to the problem. Some of the authors of this article examined the behavior of prime walks in different number fields and proved that it is impossible to walk to infinity on primes in Z[ √ 2] if the walk remains within some bounded distance from the asymptotes y = ±x/ √ 2. The main results of the current study are summarized as follows. Prime walks • Expressions for the expected prime walk lengths under different models are given by Equations (2.4) and (2.5). • Comparisons of the two prime walk models and the actual primes can be found in Tables 2, 3, 4, and 5. • A proof that it is impossible to walk to infinity on primes in base 2 by ap-pending no more than 2 digits is given in Theorem 2.7, while Lemma A.1 shows that it is impossible to walk to infinity on primes in bases 3, 4, 5, or 6 by appending one digit to the right. Square-free walks • The expected lengths of square-free walks given by our models are presented in Tables 6 and 8, while Theorem 3.4 shows that there exists an infinite random square-free walk from most starting points. • Table 7 presents the frequencies of the digits added in square-free walks, and Remark 3.12 explains why some digits appear more often than others. • Theorems 3.8 and 3.9 give tight bounds on the theoretical expected lengths of square-free walks in bases 2 and 10, respectively. 2. Initial Models of Prime Walks 2.1. Models We now estimate the length of these random walks in base b, so there are b digits we can append. If our number has k digits, then from Section 1.2, the probability a digit yields a successful appending is approximately 1/(k log b), as we are assuming INTEGERS: 24 (2024) 7 all possible numbers are equally likely to be prime. For example, if b = 10, we are not removing even numbers or 5 or numbers that make the sum a multiple of 3. Thus, the probability that at least one of the b digits works is 1 minus the probability they all fail, or 1 −  1 − 1 k log b b . (2.1) The first stochastic model for prime walks can be described as follows. Algorithm 2.1 (Greedy Prime Walk in Base b). Each possible appended number is independently declared to be a random prime with probability as described by the reasoning used to deduce (2.1). Choose one of the admissible digits uniformly at random and check if the obtained number is prime; if it is not, stop and record the length; otherwise, continue the process. This algorithm can be imagined as a greedy prime walk, as we are not looking further down the line to see which of many possible random primes would be best to choose to get the longest walk possible. We call this the greedy model. Furthermore, note that we may easily improve the model in base 10 by appending from {1, 3, 7, 9}. We discuss this improvement later in this section, and compare it to this initial greedy model. In order to compute the theoretical expected length of such a walk, starting at a one-digit random prime in base b, we count the probabilities in two different ways; note that the expected length is just the infinite sum of the probabilities that we stop at the nth step times n. For brevity, let An denote the event that the walk has length at least n, and Bn denote the event that the walk has length exactly n. Since it is clear that the collection of events {Bi}∞ i=1 is pairwise independent and that An = ∪∞ i=nBi, it follows that ∞ X n=1 P[walk has length at least n] = ∞ X n=1 nP[walk has length exactly n]. (2.2) Note that the sum on the right hand side of (2.2) is the expected walk length in our greedy model, while the sum on the left equals ∞ X n=0 n−1 Y k=1 1 −  1 − 1 k log b b! , where each term in the sum represents the probability that there is a random prime with which we can extend the walk for the first n −1 steps, without considering the nth step. In particular, the expected length in base 10 when starting with a single digit is 4.690852, however we are also interested in other bases. With that in mind, we denote by Ys,b the random variable indicating the length of a prime walk INTEGERS: 24 (2024) 8 with a chosen prime starting point with at most s digits in base b. Furthermore, by multiplying by the approximate number of primes with exactly r digits and dividing by the approximate number of primes with at most s digits, we get that the theoretical expected length of a walk with a starting point at most s digits in base b is E[Ys,b] = 1 bs s log b s X r=1 (b −1)br−1 r log b ∞ X n=0 n−1 Y k=r 1 −  1 − 1 k log b b!!! , (2.3) which simplifies to E[Ys,b] = s(b −1) bs s X r=1 br−1 r ∞ X n=0 n−1 Y k=r 1 −  1 − 1 k log b b!!! . (2.4) Table 1 contains the expected lengths as we vary the starting point and base. For base 10, we also include the blind limited model in the table, which is described in more detail in Section 3.4. One can view this model as a greedy random prime walk because we always take another step if possible, with no regard to how many steps we may be able to take afterward; thus, all decisions are “local.” Note that the theoretical expected length of the walk in base 10 starting with a one-digit number, 4.22, is different than the one we computed earlier, 4.69. This is because we multiplied 4.69 by the approximation factor (b −1)/b, i.e., 0.9. More importantly, note that in base 10 we can only append {1, 3, 7, 9} and hope to stay prime since primes greater than 5 are odd and not divisible by 5. This suggests a simple improvement to the model base 10: we only allow the four digits 1, 3, 7, and 9 to be appended on the right. Henceforth, we will only use this improved version. To do this, we have to make a couple of changes in Equation (2.4): first, we replace the numerator of 1/(k log b) with 10/4, as we have the same number of primes despite having less freedom; furthermore, instead of raising 1 −10/(4k log b) to the bth power (in this case, 10), we raise it to the fourth power as only 4 options are left. We shall call this the greedy model. The expected walk length under this model is presented in Table 1 as 10’. Denote by g Ys,b the random variable denoting the length of walking according to this modified algorithm. By modifying our earlier analysis, we obtain the formula for the expected length of this model in base b to be E[g Ys,b] = s(b −1) bs s X r=1 br−1 r ∞ X n=0 n−1 Y k=r 1 −  1 − b φ(b)k log b φ(b)!!! . (2.5) where φ(n) denotes Euler’s totient function. Our second model is the careful greedy model. Algorithm 2.2 (Careful greedy algorithm). At each step, we check whether ap-pending any of 1, 3, 7, or 9 to the right yields a prime. If there are multiple digits INTEGERS: 24 (2024) 9 Maximum number of digits of starting point 1 2 3 4 5 6 7 Base 2 5.20 9.90 11.62 11.45 10.40 9.08 7.79 3 5.05 7.75 7.60 6.53 5.40 4.49 3.80 4 4.87 6.55 5.86 4.79 3.92 3.29 2.85 5 4.71 5.79 4.92 3.96 3.25 2.78 2.45 6 4.57 5.27 4.34 3.48 2.89 2.49 2.22 7 4.46 4.89 3.95 3.17 2.65 2.31 2.08 8 4.37 4.59 3.67 2.95 2.49 2.19 1.98 9 4.29 4.36 3.45 2.79 2.37 2.09 1.91 10 4.22 4.17 3.28 2.66 2.28 2.20 1.85 10’ 4.54 4.55 3.55 2.83 2.38 2.09 1.90 Table 1: Expected length of prime walks given by Equation (2.3), 10’ is the blind limited model that yield primes, the model selects one of them uniformly at random, and continues the process. This is a more refined version of Algorithm 2.1. Indeed, we first check whether any of the numbers obtained after appending an admissible digit is prime; if there is more than one such number, select one at random, and if there are none, stop the process. Obviously, this algorithm approximates the real-world data better than Algorithm 2.1, but this comes at a computational cost, as at each step we have to check whether four numbers are prime instead of just one. Lastly, in the primes model, we use backtracking to find the longest walk starting at a prime. While Table 1 presents the expected length of prime walks given by formulas (2.4) and (2.5), Tables 2, 3, 4, 5, and 9 show the data obtained by computer simulations on our previously described models. We note that this latter table records walks based on the exact number of starting digits. 2.2. Results and Comparison of Models According to the random probabilistic model of prime walks in Section 2.1, the expected length of a greedy prime walk, starting with a single digit prime in base 10, is 4.69. We compare this heuristic estimate with the primes. We present the results of our computer simulations for our blind unlimited and INTEGERS: 24 (2024) 10 careful limited models in Tables 2, 3, 4, and 5. The careful greedy model is rather close to the real data whereas the greedy one still predicts some behaviors of the walks. The data for the actual primes is computed by the program that exhaustively searches for the longest prime walk given a starting point. First, let us observe how the number of digits of the starting point affects the expected walk length of the models in Table 2: it shows that the expected length of the walks decreases as the starting point increases both theoretically and in our random model. Start has r digits 0 1 2 3 4 5 6 Blind unlimited model 1.00 1.86 1.60 1.41 1.31 1.25 1.21 Careful limited model 4.77 5.01 3.41 2.79 2.38 2.09 1.88 Primes 8.00 8.00 4.71 3.48 2.71 2.28 2.03 Table 2: Comparison of the expected walk lengths Furthermore, we analyze the frequency of digits added in the prime walks under base 10, both for the actual primes and in our models. We originally hypothesized that 3 and 9 appear more often than 1 and 7. This is because 1 and 7 cannot be appended when we start with a prime that is congruent to 2 modulo 3. We present the frequency of digits in Table 4 when the starting point is less than 1,000,000. As expected, in both our models and the real prime walks, the number of appended 3’s is very close to the number of appended 9’s while the number of appended 1’s is very close to the number of appended 7’s. One surprising observation is that there are significantly more 7’s in the random models than in the real prime walks. We observe how the starting point affects the frequency of the digits added in Tables 3, 4, and 5. As mentioned above, we observe that the number of appended 3’s and 9’s is larger than the number of appended 1’s and 7’s. This is due to the fact that by modulo 3 considerations, we can only append 3 or 9 to a number that is congruent to 2 modulo 3 to keep it a prime. In particular, when the starting number is congruent to 2 modulo 3, we must append 3’s and 9’s, and when it is congruent to 1 modulo 3, we can append 1 or 7 at most once, and every other digit appended must be 3 or 9.1 We present our model when starting with a number that is congruent to 2 modulo 3 in the following section. Furthermore, this bias will be seen in our models: indeed, if a number is composite after appending a digit, the digit will not be counted for. As the probability of a number being prime immediately after appending 3 or 9 is 1Since appending 2, 5 or 8 is forbidden for being divisible by 2 or 5, appending a digit either preserves the remainder modulo 3 (the case when appending 3 or 9), or increments it by 1 (the case when appending 1 or 7). In a prime walk, the remainder must never be zero, hence leaving at most one chance of appending 1 or 7 (combined) when the starting number is congruent to 1 modulo 3, and no chance at all when it is congruent to 2 modulo 3. INTEGERS: 24 (2024) 11 higher than that of being prime immediately after appending 1 or 7, the frequency of 3’s and 9’s will be higher than that of 1’s and 7’s, as can be seen in Tables 3, 4, and 5. Number appended 1’s 3’s 7’s 9’s Blind unlimited model 15.6% 33.0% 19.9% 31.3% Careful limited model 11.8% 36.7% 14.2% 37.1% Primes 12.1% 40.2% 11.1% 36.5% Table 3: Frequency of added digits in prime walks with starting point less than 100,000 Number appended 1’s 3’s 7’s 9’s Blind unlimited 15.4% 32.7% 18.5% 33.2% Careful limited model 12.5% 35.9% 14.7% 36.8% Primes 13.1% 38.8% 12.2% 35.6% Table 4: Frequency of added digits in prime walks with starting point less than 1,000,000 Number appended 1’s 3’s 7’s 9’s Blind unlimited model 16.3% 32.3% 18.5% 32.8% Careful limited model 12.7% 35.8% 14.8% 36.4% Primes 13.3% 38.6% 12.4% 35.5% Table 5: Frequency of added digits in prime walks with starting point greater than 100,000 but less than 1,000,000 We defer a more in-depth discussion of the case where our starting number is congruent to 2 modulo 3 to Appendix A.1. In the meantime, we use the stochastic prime walks presented thus far to motivate the results of Subsection 2.3, which will give conditions on which prime walks are impossible. 2.3. Main Results for Prime Walks As mentioned in the introduction, it is possible to walk to infinity on primes by appending an unbounded number of digits to the left at each step. We now show INTEGERS: 24 (2024) 12 that this statement’s counterpart is also true, namely that it is possible to walk to infinity on primes by appending an unbounded number of digits to the right. Theorem 2.3. Let p0 be a prime. Then there exists a sequence of infinitely many primes p0, p1, . . . such that for all i ≥1, pi is equal to 10ni · pi−1 + ki, for positive integers ni and ki with ki < 10ni. Proof. We can restate our goal as follows: given a fixed prime p, we must show that there exists an n such that there is a prime in the interval [p10n, (p + 1)10n). To do so, we note that for a given p and any fixed r ∈[0, 1], there exists an n such that p < 10 1−r r n −1. (2.6) Moreover, given such an n, it is then possible to find x > 0 such that p10n = x −xr. (2.7) Then, using (2.6) and (2.7), we have that x −xr < 10 n r −10n. The above inequality implies that xr < 10n, for when xr = 10n, then x −xr = 10 n r −10n, and moreover, x −xr is strictly increasing (once it is positive). Given that xr < 10n, then x −xr > x −10n. This means that p10n > x −10n, and so x < (p + 1)10n. All that remains is finding an r such that there is always a prime in the interval [x −xr, x]. Results of this nature are plentiful; most recently, Baker, Harman, and Pintz showed that a value of r = 0.525 suffices for x greater than some lower bound x0 . Since there exists a prime in the interval [x −x0.525, x] for x > x0, then by our previous definitions there must be a prime contained in the interval [p10n, (p + 1)10n). Note that in order to guarantee x > x0, it is necessary to choose an n such that n > log10((x0 −xr 0)/p) (and such that (2.6) holds as well). That there is a prime in [p10n, (p + 1)10n) implies that there exist n and k such that p10n + k is prime, with k < 10n. This gives the next prime in our sequence, which thus goes on infinitely. Now we define the extended Cunningham Chain, which ultimately serves as a tool for proving that one cannot walk to infinity on primes in base 2 when appending up to only 2 digits at a time to the right. Definition 2.4. An extended Cunningham chain is the infinite sequence e1, e2, . . ., generated by an initial prime p and the relation ek = 2ek−1 + 1 (for k ≥1 and INTEGERS: 24 (2024) 13 e0 = p). In other words, we have that e0 = p, e1 = 2p + 1, e2 = 4p + 3, . . . ei = 2ip + 2i −1, . . . We show that extended Cunningham chains, no matter their initial prime p, contain a sequence of consecutive composite ei’s of arbitrary length. To do so, we begin with the following lemma. Lemma 2.5. Let p be a prime and k be an integer. If k ≥⌈log2(p + 1)⌉+ 2, then 2k −(p + 1) is not a power of 2. Proof. Suppose that there exist k and n such that 2k −(p + 1) = 2n. Then it is the case that 2k −2n = p + 1. Moreover, we have that 2k −2n ≥2k −2k−1 = 2k−1. We can thus find a solution for n only if k < ⌈log2(p + 1)⌉+ 2, for if we take k ≥⌈log2(p + 1)⌉+ 2, then 2k−1 ≥2⌈log2(p+1)⌉+1 > p + 1. We thus have that p + 1 = 2k −2n ≥2k−1 > p + 1, which is a contradiction. The next result illustrates the power of Cunningham chains, which in turn are used to prove the main result of this section. Theorem 2.6. In any extended Cunningham chain {ek}∞ k=1, given any n ∈Z+, there exists i ∈Z+ such that {ei, ei+1, . . . , ei+n−1} are composite. Proof. Set k = ⌈log2(p + 1)⌉+ 2, and let us consider i = c · φ(2k −p −1) · φ(2k+1 −p −1) · · · φ(2k+n−1 −p −1), where c ∈Z+ is arbitrary. Moreover, for each of 2k+j −p −1 (with 0 ≤j ≤n −1), take an odd positive divisor dj | 2k+j −p −1 that is greater than 1. We can find such dj because we have chosen k via Lemma 2.5 such that none of 2k+j −p −1 are powers of 2. Because p + 1 ≡2k+j (mod 2k+j −p −1), it is also the case that p + 1 ≡2k+j (mod dj). Thus we have that ei−(k+j) = 2i−(k+j)(p + 1) −1 ≡2i−(k+j)2k+j −1 ≡2i −1 (mod dj). However, as 2 is coprime with dj, Euler’s theorem gives 2φ(dj) ≡1 (mod dj). Moreover, it is the case that φ(dj) | φ(2k+j −p−1), since dj | (2k+j −p−1). Hence INTEGERS: 24 (2024) 14 we have that ei−(k+j) ≡ 2i −1 ≡2c·φ(2k−p−1)·φ(2k+1−p−1)···φ(2k+n−1−p−1) −1 ≡ (2φ(dj))Kj −1 ≡0 (mod dj), such that Kj is an integer (Kj = c · [φ(2k −p −1) · · · φ(2k+n−1 −p −1)]/φ(dj)). We have thus shown that ei−k, ei−k−1, . . . , ei−(k+n−1) are composite. Notice that with c sufficiently large, i can be made greater than k + n −1, allowing us to find a subsequence of n composite elements for any n. Renaming the indices gives the desired result. Using this result we can prove that appending 2 digits at a time to the right is insufficient to walk to infinity on primes in base 2. Theorem 2.7. It is impossible to walk to infinity on primes in base 2 by appending no more than 2 digits at a time to the right. Proof. Since we are appending at most 2 digits in base 2, the allowed blocks are 02, 12, 002, 012, 102, and 112. Avoiding even numbers, we are left with 12, 012 and 112. Appending 012 to a prime p gives 4p+1. If p ≡2 (mod 3), then 4p+1 is divisible by 3 and is thus not prime. Moreover, given p ≡2 (mod 3), then 2p + 1 and 4p + 3 are equivalent to 2 modulo 3 as well. Thus if p ≡2 (mod 3), we can walk to infinity from that point onward only by appending 12 or 112. If p ≡1 (mod 3), then 4p+1 ≡2 (mod 3). This brings us to the above case, now applied to 4p+1. No matter the value of our initial prime p, we can therefore append 012 at most once in our walk to infinity. It is thus sufficient to consider the point at which we append only 12 or 112 to eternity. We can then apply Theorem 2.6 with n = 2. Namely, continuously appending 1 to a prime in base 2 creates a generalized Cunningham chain, which we know contains prime gaps of size 2; hence there will be some point in the prime sequence for which 2p+1 and 4p+3 are both composite, and we can walk no further. Applying the ideas of the above results allows us to make observations in bases 3, 4, 5, and 6; the analogous results and their proofs can be found in Section A. Section A.2 also discusses the inability to create walks on the (very sparse) Mersenne primes. In conclusion, the use of stochastic models suggests that there is no infinite prime walk given by adding one digit at a time to the right, but that it is likely to have one if we can add a digit anywhere. INTEGERS: 24 (2024) 15 3. Modeling Square-free Walks 3.1. Model We now turn our attention to square-free walks, whose density is positive, in contrast to sequences that have density zero like primes. Due to this difference, we expect that we can construct a square-free walk to infinity. Before discussing the details, we precisely define the square-free integers. Definition 3.1. A square-free integer is an integer that is not divisible by any perfect square other than 1. If Q(x) denotes the number of square-free positive integers less than or equal to x, it is well-known that Q(x) ≈x Y p prime  1 −1 p2  = x Y p prime 1 1 + 1 p2 + 1 p4 + · · · = x ζ(2) = 6x π2 . In this setting, each possible appended number is independently declared to be a square-free number with probability p = 6/π2. We now present our first model for estimating the length of square-free walks. Algorithm 3.2 (Blind Unlimited Square-Free Walk). Choose one digit uniformly at random from the set {0, 1, . . . , 9} and append it: if the obtained number is not square-free, stop and record the length; otherwise, continue the process. We present the experimental expected lengths under this model, starting with different number of digits, in Table 6. Akin to Table 9 we opt to provide data based on the exact number of digits in the starting number, rather than the maximal possible number. Start has exactly r digits 0 1 2 3 4 5 6 Blind unlimited square-free walk 1.68 2.79 2.76 2.72 2.71 2.71 2.71 Table 6: Experimental expected lengths of the square-free walks in base 10 To gain more understanding of the behavior of square-free walks, we first find the probability that the square-free walk is of length exactly k, where the number of starting digits is fixed. Lemma 3.3. Let Xm denote the length of our random square-free walk, starting with exactly m digits. Then the theoretical expected value of Xm is E[Xm] ≈ ( 1.55, m = 0, 2.55, m ≥1. INTEGERS: 24 (2024) 16 Proof. This proof is standard via the technique of differentiating identities, but we present it anyway for the sake of completeness. Note that, for all k ≥0 and positive m, Xm = k corresponds to k −1 successful appendings followed by an un-successful one. However, X0 = k corresponds to k successful ones and 1 unsuccessful one. Thus, we have that Pr[X0 = k] = pk(1 −p) = Pr[Xm = k + 1], and so X is a geometric random variable. Using the fact that ∞ X k=0 pk = 1 1 −p and differentiating term by term (which is permissible due to absolute convergence), we obtain ∞ X k=0 kpk−1 = 1 (1 −p)2 , which implies that ∞ X k=0 kpk(1 −p) = p 1 −p. Therefore, we find that E[X0] = ∞ X k=0 k Pr[X0 = k] = ∞ X k=0 kpk(1 −p) = p 1 −p = 6 π2 −6 ≈1.55, and E[Xm] = 1 + ∞ X k=0 k Pr[Xm −1 = k] = 1 + ∞ X k=0 kpk(1 −p) ≈2.55 (m ≥1). Let us now compute the probability that the longest walk starting with a given square-free number is at most k. Let Pi be the probability that the longest square-free walk has length at most i. In particular, P1 is the probability that the longest square-free walk has a length of exactly one, i.e., the walk is the starting point. In other words, appending any digit yields a non-square-free number, so P1 = (1 −p)10 = (π2 −6)10 π20 ≈8.58357 × 10−5. (3.1) Now, consider the probability that the longest square-free walk has length at most 2. There are 10 possible cases where exactly i digits work in the first appending, INTEGERS: 24 (2024) 17 i.e., 0 ≤i ≤9. Then, by using (3.1) and the Binomial Theorem we have that P2 = P1 + 10 1  (1 −p)9pP1 + 10 2  (1 −p)8(pP1)2 + · · · + 10 10  p10P 10 1 = 10 0  (1 −p)10 + 10 1  (1 −p)9pP1 + · · · + 10 10  p10P 10 1 = (1 −p + pP1)10 ≈8.59501 × 10−5. (3.2) To compute P3, let i denote the number of digits that we can append in the first step while remaining square-free. Then, there are 10i possible numbers after the second appendage. Like P2, we consider cases when there are exactly 0 ≤k ≤10i numbers that work. Note that, when i = 0 or j = 0, we have a walk of length 1, 2 respectively, so such cases are included in P2. Therefore, by (3.1) and (3.2), we have that P3 = P2 + 10 X i=1 10 i  pi(1 −p)10−i 10i X k=1 10i k  (1 −p)10i−k(pP1)k ! = P2 + 10 X i=1 pi(1 −p)10−i ((1 −p) + pP1)10i −(1 −p)10i = P2 + 10 X i=1 pi(1 −p)10−i P i 2 −P i 1  = P2 + 10 X i=1 pi(1 −p)10−iP i 2 − 10 X i=1 pi(1 −p)10−iP i 1 = P2 + (1 −p + pP2)10 −(1 −p)10) −((1 −p + pP1)10 −(1 −p)10 = (1 −p + pP2)10 ≈8.59502 × 10−5. The next step is to compute Pk for an arbitrary k ∈N+, which can be done by induction. Suppose that, for 2 ≤m ≤k, we have Pk = (1 −p + pPk−1)10. Similar to the idea used to compute P2 and P3, we have that Pk+1 = Pk + 10 X a1=1 10 a1  pa1(1 −p)10−a1 10a1 X a2=1 10a1 a2  pa2(1 −p)10a1−a2 · · · · · · 10ak−2 X ak−1=1 10ak−2 ak−1  pak−1(1 −p)10ak−2−ak−1 10ak−1 X ak=1 10ak−1 ak  pak(1 −p)10ak−1−akpak 1 !! . . . ! . We observe that the last sum in the above equation equals (1 −p + pP1)10ak−1 −(1 −p)10ak−1 = P ak−1 2 −P ak−1 1 . INTEGERS: 24 (2024) 18 Thus we obtain that Pk+1 = Pk + 10 X a1=1 10 a1  pa1(1 −p)10−a1 10a1 X a2=1 10a1 a2  pa2(1 −p)10a1−a2 · · · 10ak−2 X ak−1 10ak−2 ak−1  pak−1(1 −p)10ak−2−ak−1P ak−1 2 −P ak−1 1  ! . (3.3) By repeating the same procedure as in calculating P3, we are able to reduce the expression (3.3) to Pk+1 = Pk + 10 X a1=1 10 a1  pa1(1 −p)10−a1P a1 k −P a1 k−1  = Pk + (1 −p + pPk)10 −(1 −p + pPk−1)10 = (1 −p + pPk)10, (3.4) which holds true for any positive integer k ≥1. We now prove that Pk approaches some constant as k →∞. Using (3.4), we have that Pk = (1 −p + pPk−1)10 ≥0. Furthermore, if Pk−1 ≤1/2, then Pk ≤  1 −p + p 2 10 =  1 −3 π2 10 < 0.710 < 1 2. Then, by induction, when the base case is P1 ≈8.5835 × 10−5 (from (3.1)), we have that Pk ≤1/2 for any k ≥1. Lastly, note that P2 > P1, and using strong induction and (3.4), we get that Pk+1 = (1 −p + pPk)10 ≥(1 −p + pPk−1)10 = Pk. In other words, Pk  k≥1 is an increasing sequence. By the Monotone Convergence Theorem [1, Theorem 2.4.2], we get that there exists l ∈[0, 0.5] such that lim k→∞Pk = l. Sending k →∞in (3.4), we get that l = (1 −p + pl)10. Using Mathematica, we see that the only rational root in the range [0, 0.5] is l ≈8.59502 × 10−5. INTEGERS: 24 (2024) 19 The limit l stands for the probability that, starting at some fixed number x, there is a bounded limit N, which can be very large, such that no square-free walk can exceed length N. That is, if the limit of Pk is as small as 8.59502 × 10−5, it implies the following theorem. Theorem 3.4. Given we append one digit at a time, the theoretical probability that there is an infinite random square-free walk from any starting point is as least 1−l ≈0.99991. In other words, there is such a walk from almost any starting point. Remark 3.5. Although Theorem 3.4 suggests that stochastically, the probability of walking to infinity on square-free numbers is high, there exist square-free numbers that can’t be extended. For example, 231546210170694222 is a square-free number, such that if we append any digit to the right we get a non-square-free number. In particular, if we delete any number of digits to the right we get a square-free number as well, so this proves we can reach a stopping point when starting with 2 and append digits to the right randomly. Furthermore, our example implies that the walk is not constructive, i.e., if we start with a square-free walk and append a digit at random that yields a new square-free number, we may reach a point where we could not move forward. 3.2. Quantitative Results From Section 3.1, according to the blind unlimited model of square-free walks, the expected length of square-free walks is 6/(π2 −6) in any base. In reality, however, this is not always the case. Dropping the probabilistic assumption about the square-free numbers, we assume that a random square-free walk starts with the empty string, then randomly selected digits are appended to the right, and the process stops when the number obtained is not square-free. We let Eb denote the theoretical expected length of such a walk in base b, and SF the set of square-free numbers. We supplement this notation with another definition. Definition 3.6 (Right Truncatable Square Free). We denote by RTSFb the set of square-free numbers base b such that if we successively remove the rightmost digit, each resulting number is still square-free. Equivalently, let bk−1 ≤x < bk. Then, x ∈RTSFb if and only if for all ℓ∈{0, 1, . . . , k −1} we have ⌊x/bℓ⌋is square-free. We also define Lb,k := RTSFb ∩[bk−1, bk) . This quantity counts the number of right-truncatable square-free numbers with exactly k digits in base b. Lemma 3.7. We have Eb = ∞ X k=1 Lb,k bk −bk−1 . INTEGERS: 24 (2024) 20 Proof. The proof follows from the same reasoning used to prove (2.2). Theorem 3.8. We have E2 satisfies the following bounds: 2.31435013 < 636163720502 238 ≤E2 ≤636163930777 238 < 2.31435090. Proof. A straightforward calculation yields (L2,n)1≤n≤40 = (1, 2, 3, 5, 7, . . . , 168220). Let S1 := 40 X i=1 L2,i 2i = 318081860251 237 , S2 := ∞ X i=41 L2,i 2i , and note that S1 + S2 = E2. Moreover, let LO 2,k = RTSF2 ∩[2k−1, 2k) ∩(2Z + 1) be the number of odd right truncatable square-free binary numbers of length-k binary numbers, and similarly LE 2,k = RTSF2 ∩[2k−1, 2k) ∩2Z the even ones. By modulo 4 considerations, we have that LO 2,k+1 ≤LO 2,k + LE 2,k and therefore, we have LE 2,k+1 ≤LO 2,k. Putting the above observations together, we have S2 = LO 2,41 + LE 2,41 241 + ∞ X i=41 LO 2,i+1 + LE 2,i+1 2i+1 ≤LO 2,41 + LE 2,41 241 + ∞ X i=41 2LO 2,i + LE 2,i 2i+1 = LO 2,41 + LE 2,41 241 + S2 2 + LO 2,41 242 + ∞ X i=41 LO 2,i+1 2i+2 ≤3LO 2,41 + 2LE 2,41 242 + S2 2 + ∞ X i=41 LO 2,i + LE 2,i 2i+2 ≤5LO 2,40 + 3LE 2,40 242 + 3S2 4 . Thus, we have S2 ≤5LO 2,40 + 3LE 2,40 240 ≤5L2,40 240 = 210275 238 . As clearly S2 ≥0, it follows that 40 P i=1 L2,i 2i ≤E2 = 40 P i=1 L2,i 2i + S2. Substituting the numerical results from (3.2) yields the bound. INTEGERS: 24 (2024) 21 Although we do not use the base b = 2 model for square-free walks elsewhere in this paper, the proof is outlined here since it can be adapted to other bases. Theorem 3.9. 2.63297479 ≤E10 ≤2.720303756. Proof. The proof is similar to that of Theorem 3.8, using (L10,n)1≤n≤8 = (6, 39, 251, 1601, 10143, 64166, 405938, 2568499) and the inequalities LO 10,k+1 ≤5LO 10,k + 5LE 10,k, LE 10,k+1 ≤3LO 10,k + 2LE 10,k. 3.3. Additional Remarks on the Behavior of Square-free Walks We first introduce some notation. Given a number x and a digit i in base b, we denote xi = b · x + i; in other words, we append i to the right of x. The following are some remarks relating to the behavior of square-free walks. Remark 3.10. The fact that E10 > 6/(π2 −6) was expected, since we know that xi is more likely to be square-free if x is square-free. This is because if x is square-free, then x ̸≡0 (mod p2) for every prime p. In particular, this implies that [x0, x9] can be any segment of Z/p2Z except [0, 9], hence the chance that xi ̸≡0 (mod p2) for all p is slightly bigger. Notice that this behavior is consistent for any base b. Numerical evidence of these observations is provided in a Python program2. In particular, it yields that when x ∈{1, 2, . . . , 1,000,000} is square-free, the probabil-ity of xi also being square-free is around 0.631, and when x ∈{1, 2, . . . , 1,000,000} is not square-free, the probability of xi being square-free is around 0.571. As x increases, we expect the two probabilities to decrease, but they still have a small difference. Remark 3.11. We also explored how the starting point affects the length of the walk. Like in the prime walks, the experimental expected length of the square-free walk decreases as the starting point increases since small numbers have a bigger chance of being square-free. This is shown in Table 6. Note that the expected length of around 2.71 (when the starting point increases) is inside the interval given by Theorem 3.9. Remark 3.12. We also consider the frequency of the digits added in our square-free walk and how this changes when we vary the walk’s starting point. The results are shown in Table 7, and we also make the following related, qualitative observations. 2This script is available at INTEGERS: 24 (2024) 22 • Odd digits appear more often than even digits. This is because if x is square-free, then it cannot be a multiple of 4, hence even digits appear less. • The frequencies of 2 and 6 are less than 0, 4, and 8. This is because if x and xi are square-free and i is even, then if x is odd, by modulo 4 considerations i is 0, 4, or 8, and if x is even, then i is 2 or 6. However, x is almost twice more likely to be odd; hence the frequency of 0, 4, and 8 is bigger than that of 2 and 6. • We have that 5 appears less often than any other odd digit. Similar to the above, x5 is not square-free if x ends with 2 or 7. • We have that 9 appears more often than any other digit. This is because if x is square-free, then x ̸≡0 (mod 9), hence x9 ̸≡0 (mod 9). • As the starting point increases, the frequencies stabilize. Remark 3.13. By looking at the last digit, we can make informed decisions on what digit to append at each step to increase the chance the number is square-free using Remark 3.12. Number of digits of starting point 1 2 3 4 5 6 Digit added 0 10.1% 7.4% 7.6% 7.5% 7.5% 7.5% 1 14.0% 13.6% 13.2% 13.4% 13.4% 13.4% 2 8.4% 5.5% 5.3% 5.3% 5.3% 5.3% 3 13.5% 13.5% 13.4% 13.4% 13.4% 13.3% 4 5.1% 8.1% 8.0% 8.0% 8.0% 8.0% 5 12.1% 10.8% 10.9% 10.8% 10.8% 10.8% 6 8.3% 5.5% 5.4% 5.3% 5.3% 5.3% 7 13.4% 13.5% 13.2% 13.3% 13.3% 13.3% 8 4.9% 7.4% 8.0% 8.0% 8.0% 8.0% 9 9.7% 14.2% 14.5% 14.6% 14.6% 14.6% Table 7: Comparing the frequency of the digits of blind unlimited square-free walks in base 10 INTEGERS: 24 (2024) 23 3.4. Blind Limited Model Lastly, we present an alternative to the blind unlimited square-free walk. As stated in Remark 3.12, odd digits appear most frequently. This observation inspires a different model: if we start with an odd square-free number not divisible by 5, we can always append 0 to get a square-free number, since the initial number is not divisible by 2 or 5. Then, randomly append one of 1, 3, 7, and 9. If the number is square-free, repeat the process, otherwise stop and record the length. Using (3.1), we get that the probability that a random odd integer, non-divisible by 5, is square-free is p = Y p prime ̸=2,5  1 −1 p2  = 1 ζ(2) · 1 1 −1 4 · 1 1 −1 25 = 25 3π2 . Algorithm 3.14 (Blind Limited Square-Free Walk). Start with an odd square-free number not divisible by 5, then append 0 to it. Choose one digit uniformly at random from the set {1, 3, 7, 9} and append it to the right; if the resulting number is still square-free, append another 0 and repeat the process. Let X denote the length of the blind limited square-free walk, starting with the empty string. This is different from starting with one digit: with a 1-digit start, the starting point is 1, 3 or 7 with 1/3 probability each, whereas this time our first append is 1, 3, 7 or 9 with 1/4 probability each, so it has 1/4 chance not surviving the first step due to 9 not being square-free. In estimating the theoretical expected value of Z we assume that the result of every appending will be square-free with probability p, and all the events are independent. Note that Z is always even, since we append a 0 at every second digit. Therefore, the theoretical probability is P[X = 2k] = pk(1 −p) = 25k 3kπ2k · 3π2 −25 3π2 = 25k(3π2 −25) 3k+1π2k+2 . Analogously to (3.1), we have that E[X] = 2p 1 −p = 50 3π2 −25 ≈10.84, which is a lot larger than the expected walk length in the original model computed in (3.1). We present the experimental comparison in Table 8. The value 11.12 is close to the theoretical 10.84, which indicates that square-free numbers have good uniformity. Observe that the earlier comment suggests that the expected length with 1-digit starts should be 4/3 times that with the empty-string start, and this is confirmed by the experimental values. INTEGERS: 24 (2024) 24 Start: exactly r digits 0 1 2 3 4 5 6 Greedy sq-free walk 1.68 2.79 2.76 2.72 2.71 2.71 2.71 Alternative sq-free walk 11.12 14.82 13.24 13.37 13.47 13.49 13.50 Table 8: Comparing the expected walk lengths of greedy square-free models in base 10 4. Conclusion In the exploration to find a walk to infinity along some number theory sequences, given we append a bounded number of digits, we have established several results for different sequences. We chose to study prime and square-free walks in part be-cause the primes have zero density, whereas the square-free numbers have a positive density. Where we could not prove exact results, we used stochastic models that approximated the corresponding “true” values fairly well. Our stochastic models motivated a conjecture that there is no walk to infinity for primes, a sequence of zero density with no discernible pattern in its occurrence, while a walk to infinity exists for square-free numbers, whose density is a positive constant. We verified this conjecture for these and other sequences, namely perfect squares and primes in smaller bases; indeed, it is impossible to walk to infinity on primes in base 2 if appending up to 2 digits at a time, or in bases 3, 4, 5, or 6 if appending 1 digit at a time. Lastly, we found a way to append an even bounded number of digits indefinitely for perfect squares. Stochastic models give us a strong inclination to determine whether we can walk to infinity along certain number theoretic sequences. The results presented in this paper suggest simple speculation that small density leads to the absence of the walks to infinity. However, as we mainly observe sequences based on their density, it remains to be determined how much other factors, such as the sequence’s pattern or structure, may contribute as well. As one possible case study, one could consider walks on the Carmichael numbers, which were recently shown to have the property that the ratio of consecutive Carmichael numbers converge to 1; see . Another option is to give more flexibility in where digits are appended; this paper only discussed fixed-position models, where we either append digits only to the left, or only to the right. Allowing digits to be appended to either side, or even in the middle, generates a new set of conjectures to be studied. Acknowledgement. The authors would like to thank the anonymous reviewers of multiple drafts of this paper, for their numerous and extremely valuable com-ments and suggestions, especially those on the proof of Lemma A.1 parts 1 and 4. INTEGERS: 24 (2024) 25 Furthermore, we would like to thank the other Polymath REU Walking to Infinity group members in summer 2020 for their helpful discussions leading to this work. The group consisted of William Ball, Corey Beck, Aneri Brahmbhatt, Alec Critten, Michael Grantham, Matthew Hurley, Jay Kim, Junyi Huang, Bencheng Li, Tian Lingyu, Adam May, Saam Rasool, Daniel Sarnecki, Jia Shengyi, Ben Sherwin, Yit-ing Wang, Lara Wingard, Chen Xuqing, and Zheng Yuxi. This work was partially supported by NSF grant DMS1561945, Carnegie Mellon University, and Williams College. References S. Abbott, Understanding Analysis, Springer, New York, 2015. I.O. Angell and H.J. Godwin, On truncatable primes, Math. Comp. 31 (137) (1977), 265-267. R.C. Baker, G. Harman, and J. Pintz, The difference between consecutive primes, II, Proc. Lond. Math. Soc. 83 (3) (2001), 532-562. E. Gethner and H.M. Stark, Periodic Gaussian moats, Exp. Math. 6 (4) (1997), 289-292. E. Gethner, S. Wagon, and B. Wick, A stroll through the Gaussian primes, Amer. Math. Monthly 105 (4) (1998), 327-337. A. Kontorovich and J. Lagarias, Stochastic models for the 3x + 1 and 5x + 1 problems and related problems, in The Ultimate Challenge: The 3x + 1 Problem, Amer. Math. Soc., Provi-dence, RI, 2010. D. Larsen, Bertrand’s postulate for Carmichael numbers, Int. Math. Res. Not. IMRN 2023 (15) (2023), 13072-13098. B. Li, S.J. Miller, T. Popescu, D. Sarnecki, and N. Wattanawanichkul, Modeling random walks to infinity on primes in Z[ √ 2], J. Integer Seq. 25 (2022), 21 pp. P.R. Loh, Stepping to infinity along Gaussian primes, Amer. Math. Monthly 114 (2) (2007), 142-151. S.J. Miller, F. Peng, T. Popescu, and N. Wattanawanichkul, Walking to infinity on the Fibonacci sequence, in Proceedings of the 20th International Conference on Fibonacci Numbers and Their Applications, The Fibonacci Quarterly, 2022. S.J. Miller and R. Takloo-Bighash, An Invitation to Modern Number Theory, Princeton University Press, Princeton, NJ, 2006. H.L. Montgomery and K. Soundararajan, Beyond pair correlation, in Paul Erd˝ os and His Mathematics, I, J´ anos Bolyai Math. Soc., Budapest, 2002. Appendix A Impossibility of Walks We demonstrate that it is impossible to walk to infinity in bases 3, 4, 5, and 6 when adding only one digit at a time to the right. INTEGERS: 24 (2024) 26 Lemma A.1. The following statements hold. 1 It is impossible to walk to infinity on primes in base 3 by appending a single digit at a time to the right. 2 It is impossible to walk to infinity on primes in base 4 by appending a single digit at a time to the right. 3 It is impossible to walk to infinity on primes in base 5 by appending a single digit at a time to the right. 4 It is impossible to walk to infinity on primes in base 6 by appending a single digit at a time to the right. Proof. To prove 1, first note that we can only append a 2 in base 3, as appending a 0 would yield a number divisible by 3, while appending a 1 would yield an even number. Therefore, at each step we can only append a 2. Let p1, p2, . . . be the sequence formed by appending 2 at each step. We have that p1 = p1, p2 = 3p1 + 2, p3 = 9p1 + 8, . . . pi = 3i−1p1 + 3i−1 −1, . . . However, by Fermat’s little theorem, we have that pp1 ≡3p1−1p1 + 3p1−1 −1 ≡0 (mod p1). Hence pp1 is composite, and it is impossible to walk to infinity on primes in base 3 by appending just one digit at a time. To prove 2, we confine ourselves to considering only odd digits. Since 4 is con-gruent to 1 modulo 3, appending 1 to a prime p ≡2 (mod 3) gives 4p + 1 ≡0 (mod 3), a composite. One can thus append 1 at most a single time in walking to infinity, and so it suffices to consider the infinite subsequence over which only 3’s are appended. Denote the elements of this subsequence as p1, p2, . . .. Then, in a INTEGERS: 24 (2024) 27 similar fashion to the extended Cunningham chains, these elements take the form p1 = p1, p2 = 4p1 + 3, p3 = 16p1 + 15, . . . pi = 4i−1p1 + 4i−1 −1, . . . Again, by Fermat’s little theorem, pp1 is composite. Thus it is impossible to walk to infinity on primes in base 4 by appending just one digit at a time. To prove 3, note that parity mandates we append either 2 or 4 at each step. If we have a prime p ≡1 (mod 3), then 5p + 4 ≡0 (mod 3), and so we must append a 2. Moreover, if p ≡1 (mod 3) then 5p + 2 ≡1 (mod 3), so we must append another 2, and so on until infinity. If p1 ≡1 (mod 3), then we have that pi = 5i−1p1 + 5i−1 −1 2 . Then it is the case that 2pi ≡5i−1 −1 (mod p1), and so 2pp1 is divisible by p1 according to Fermat’s little theorem. Therefore pp1 is composite. On the other hand, if we have a p ≡2 (mod 3), then 5p + 2 ≡0 (mod 3), so we must append a 4. But 5p + 4 ≡2 (mod 3) when p ≡2 (mod 3), thus requiring that we append 4’s unto infinity. Given p1 ≡2 (mod 3), we find that pi = 5i−1p1 + 5i−1 −1. Fermat’s little theorem, therefore, allows us to conclude that pp1 is composite. The exception is when p1 = 5, in which case we write pi = 5i−1p1 + 5i−1 −1 = 5i−2(5p1 + 4) + 5i−2 −1 = 5i−2p2 + 5i−2 −1, and observe that pp2+1 is divisible by p2. Hence, regardless of our initial prime, there must be a composite element in the sequence produced by appending one digit to the right. Finally, we prove 4. By parity and modulo 3 considerations, we can only append 1 or 5 at each step. However, note that 1 can be appended at most 3 times, as 6p + 1 ≡p + 1 (mod 5). Assume that we have reached a point where we can only append 5 and let p1 be this prime. Let p1, p2, . . . be the sequence formed by INTEGERS: 24 (2024) 28 appending 5 at each step. We have that p1 = p1, p2 = 6p1 + 5, p3 = 36p1 + 35, . . . pi = 6i−1p1 + 6i−1 −1, . . . By Fermat’s little theorem, we have that 6p1−1 ≡1 (mod p1). Thus, we have that pp1 ≡6p1−1p1 + 6p1−1 −1 ≡0 (mod p1), and so pp1 is composite. Therefore, it is impossible to walk to infinity on primes in base 6 by appending just one digit at a time. A.1 Starting with 2 (mod 3) In this subsection, we compare our models with the primes when our starting num-ber is congruent to 2 modulo 3. The motivation is that we can only append 3 or 9 to such a prime while hoping to remain prime; any other digit would lead to a composite number divisible by 2, 3, or 5. Therefore, we refine our model to only append 3 or 9. In this case, the walks are shorter, but the model predictions are closer to the primes. Note that the longest prime walk with starting point that is congruent to 2 modulo 3 less than 1,000,000 has length 10, and is {809243, 8092439, 80924399, 809243993, 8092439939, 80924399393, 809243993933, 8092439939333, 80924399393333, 809243993933339}. Since there are now only two possible digits to append, instead of the four that appeared in Equations (2.4) and (2.5), the theoretical expected length of the walk is given by 9s 10s s X r=1 10r−1 r ∞ X n=0 n−1 Y k=r 1 −  1 − 10 2k log 10 2!!! . (A.1) We compare our model (A.1) to the primes in Table 9. The careful greedy model approximates the real world extraordinarily well, especially as the initial number increases. This is due to the sparsity of the primes, as usually at most one of {1, 3, 7, 9} can be appended as the number increases. INTEGERS: 24 (2024) 29 Start has exactly r digits 1 2 3 4 5 6 Greedy model 3.34 1.95 1.64 1.45 1.34 1.28 Careful greedy model 5.25 3.22 2.43 2.04 1.77 1.62 Primes 8.00 3.81 2.64 2.12 1.81 1.64 Table 9: Expected length of the prime walks with starting point that is congruent to 2 modulo 3 A.2 Walking on Mersenne Primes We now study walks on Mersenne Primes. Recall the following definition. Definition A.2. A Mersenne prime is a prime of the form Mn = 2n −1, n ∈N. A necessary (but not sufficient) condition for Mn to be prime is that n is prime. We now prove the following result on Mersenne Primes. Theorem A.3. The only nontrivial walk on Mersenne primes is 3 →31, i.e., if Mq = 10Mp + i, then p = 2, q = 5, i = 1. Proof. Let Mq = 10Mp + i be a Mersenne prime walk, where p and q are prime and i ∈{0, 1, . . . , 9}. We have the equality chain Mq = 10Mp + i 2q −1 = 10(2p −1) + i 9 −i = 5 · 2p+1 −2q. (A.2) Taking both sides of the last equation of (A.2) modulo 2, we see that i must be odd. We now consider the remaining possible values of i. • If i = 1, then 8 = 5 · 2p+1 −2q. If q ≤5, it is easy to check that the only pair that works is (p, q) = (2, 5). If q > 5, then p = 2 since the right hand side of (A.2) is divisible by 8 but not by 16. But then the right-hand side of (A.2) is negative, whereas the left-hand side is positive, which is a contradiction. • If i = 3 or i = 7 then the left-hand side of (A.2) is divisible by 2 but not by 4. Since 8 divides 5 · 2p+1 (as p ≥2), we must have that q = 1, which is false. • If i = 5, the left-hand side is divisible by 4 but not by 8. Since 8 divides 5·2p+1 (as p ≥2), we must have that q = 2. But then 8 = 5 · 2p+1, which obviously has no integer solutions. Therefore, the only nontrivial walk on Mersenne Primes is 3 →31. Remark A.4. Another interesting future direction would be to study walks on perfect numbers.
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https://physics.nist.gov/cuu/pdf/JPCRD2018CODATA.pdf
J. Phys. Chem. Ref. Data 50, 033105 (2021); 50, 033105 © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved. CODATA Recommended Values of the Fundamental Physical Constants: 2018 Cite as: J. Phys. Chem. Ref. Data 50, 033105 (2021); Submitted: 21 December 2020 • Accepted: 02 February 2021 • Published Online: 23 September 2021 Eite Tiesinga, Peter J. Mohr, David B. Newell, et al. COLLECTIONS This paper was selected as Featured This paper was selected as Scilight ARTICLES YOU MAY BE INTERESTED IN Latest values of fundamental physics constants Scilight 2021, 391101 (2021); Laser-Based Primary Thermometry: A Review Journal of Physical and Chemical Reference Data 50, 031501 (2021); https:// doi.org/10.1063/5.0055297 Equations of State for the Thermodynamic Properties of Three Hexane Isomers: 3-Methylpentane, 2,2-Dimethylbutane, and 2,3-Dimethylbutane Journal of Physical and Chemical Reference Data 50, 033103 (2021); https:// doi.org/10.1063/1.5093644 CODATA Recommended Values of the Fundamental Physical Constants: 2018 Cite as: J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 Submitted: 21 December 2020 • Accepted: 2 February 2021 • Published Online: 23 September 2021 Eite Tiesinga,a) Peter J. Mohr,b) David B. Newell,c) and Barry N. Taylord) AFFILIATIONS Joint Quantum Institute and Joint Center for Quantum Information and Computer Science, College Park, Maryland 20742, USA and National Institute of Standards and Technology, Gaithersburg, Maryland 20899, USA a)Author to whom correspondence should be addressed: eite.tiesinga@nist.gov b)mohr@nist.gov c)dnewell@nist.gov d)barry.taylor@nist.gov ABSTRACT We report the 2018 self-consistent values of constants and conversion factors of physics and chemistry recommended by the Committee on Data of the International Science Council. The recommended values can also be found at physics.nist.gov/constants. The values are based on a least-squares adjustment that takes into account all theoretical and experimental data available through 31 December 2018. A discussion of the major improvements as well as inconsistencies within the data is given. The former include a decrease in the uncertainty of the dimensionless fine-structure constant and a nearly two orders of magnitude improvement of particle masses expressed in units of kg due to the transition to the revised International System of Units (SI) with an exact value for the Planck constant. Further, because the elementary charge, Boltzmann constant, and Avogadro constant also have exact values in the revised SI, many other constants are either exact or have significantly reduced uncertainties. Inconsistencies remain for the gravitational constant and the muon magnetic-moment anomaly. The proton charge radius puzzle has been partially resolved by improved measurements of hydrogen energy levels. © 2021 by the U.S. Secretary of Commerce on behalf of the United States. All rights reserved. Key words: fundamental constants; precision measurements; QED; revised SI; conventional and SI electrical units; proton radius; fine-structure constant; Rydberg constant; electron and muon g-factors. CONTENTS I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3 II. Purpose of the Adjustment and Overview of Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 III. Least-Squares Adjustments . . . . . . . . . . . . . . . . 5 IV. Overview of Notable Changes . . . . . . . . . . . . . . 5 A. Electrical units . . . . . . . . . . . . . . . . . . . . . 5 B. Particle and relative atomic masses and the atomic mass constant . . . . . . . . . . . . . . . . . . . . . . 6 C. Proton charge radius and Rydberg constant or frequency . . . . . . . . . . . . . . . . . . . . . . . . . 6 D. Fine-structure constant and electron magnetic-moment anomaly . . . . . . . . . . . . . . . . . . . . 6 E. Muon magnetic-moment anomaly . . . . . . . . . 7 F. Newtonian constant of gravitation . . . . . . . . . 7 V. Outline of Paper . . . . . . . . . . . . . . . . . . . . . . . 7 VI. Relationships among the Rydberg Constant, Fine-Structure Constant, Electron Mass, and Atomic Mass Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 VII. Atomic Hydrogen and Deuterium Transition Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 A. Theory of hydrogen and deuterium energy levels 9 1. Dirac eigenvalue . . . . . . . . . . . . . . . . . . 9 2. Relativistic recoil . . . . . . . . . . . . . . . . . . 10 3. Self-energy . . . . . . . . . . . . . . . . . . . . . . 10 4. Vacuum polarization . . . . . . . . . . . . . . . 10 5. Two-photon corrections . . . . . . . . . . . . . 11 6. Three-photon corrections . . . . . . . . . . . . 12 7. Finite nuclear size and polarizability . . . . . . 13 8. Radiative-recoil corrections . . . . . . . . . . . 14 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-1 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr 9. Nucleus self-energy . . . . . . . . . . . . . . . . 14 B. Total theoretical energies and uncertainties . . . 14 C. Experimentally determined transition energies in hydrogen and deuterium . . . . . . . . . . . . . . . 15 1. Measurement of the hydrogen 2S−4P transition . . . . . . . . . . . . . . . . . . . . . . . 15 2. Measurement of the hydrogen two-photon 1S−3S transition . . . . . . . . . . . . . . . . . . 16 3. Measurement of the hydrogen 2S−2P Lamb shift . . . . . . . . . . . . . . . . . . . . . . . . . . 17 VIII. Electron Magnetic-Moment Anomaly . . . . . . . . . 18 IX. Relative Atomic Masses . . . . . . . . . . . . . . . . . . 20 X. Atom-Recoil Measurements . . . . . . . . . . . . . . . . 22 XI. Atomic g-Factors in Hydrogenic 12C and 28Si ions . 23 A. Theory of the bound-electron g-factor . . . . . . 23 B. Measurements of precession and cyclotron fre-quencies of 12C5+ and 28Si13+ . . . . . . . . . . . . 26 C. Observational equations for 12C5+ and 28Si13+ experiments . . . . . . . . . . . . . . . . . . . . . . . 26 XII. Muonic Hydrogen and Deuterium Lamb Shift . . . . 27 A. Muonic hydrogen Lamb shift . . . . . . . . . . . . 27 B. Muonic deuterium Lamb shift . . . . . . . . . . . . 28 C. Deuteron-proton charge radius difference . . . . 28 XIII. Electron-Proton and Electron-Deuteron Scattering . 28 A. Proton radius from e-p scattering . . . . . . . . . 29 B. Deuteron radius from e-d scattering . . . . . . . . 30 XIV. Magnetic-Moment Ratios of Light Atoms and Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 A. Definitions of bound-state and free g-factors . . 30 B. Theoretical ratios of g-factors in H, D, 3He, and muonium . . . . . . . . . . . . . . . . . . . . . . . . 30 C. Theoretical ratios of nuclear g-factors in HD and HT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 D. Ratio measurements in atoms and molecules . . 31 XV. Proton Magnetic Moment in Nuclear Magnetons . . 32 XVI. Muon Magnetic-Moment Anomaly . . . . . . . . . . . 32 A. Measurement of the muon anomaly . . . . . . . . 32 B. Theory of the muon anomaly . . . . . . . . . . . . 32 C. Analysis of experiment and theory for the muon anomaly . . . . . . . . . . . . . . . . . . . . . . . . . 36 XVII. Electron-to-Muon Mass Ratio and Muon-to-Proton Magnetic-Moment Ratio . . . . . . . . . . . . . . . . . . 36 A. Theory of the muonium ground-state hyperfine splitting . . . . . . . . . . . . . . . . . . . . . . . . . . 37 B. Measurements of muonium transition energies . 40 C. Analysis of the muonium hyperfine splitting and mass ratio mμ/me . . . . . . . . . . . . . . . . . . . . 40 XVIII. Lattice Spacings of Silicon Crystals . . . . . . . . . . . 41 XIX. Newtonian Constant of Gravitation . . . . . . . . . . . 41 A. Corrected value of the 2010 measurement at JILA 41 B. Measurements from the Huazhong University of Science and Technology . . . . . . . . . . . . . . . 42 XX. Electroweak Quantities . . . . . . . . . . . . . . . . . . . 43 XXI. The 2018 CODATA Recommended Values . . . . . . 44 A. Tables of values . . . . . . . . . . . . . . . . . . . . . 44 XXII. Summary and Conclusion . . . . . . . . . . . . . . . . . 44 A. Comparison of 2014 and 2018 CODATA recom-mended values . . . . . . . . . . . . . . . . . . . . . 44 B. Implications of the 2018 adjustment for metrology and physics . . . . . . . . . . . . . . . . . . . . . . . 53 1. Electrical metrology . . . . . . . . . . . . . . . . 53 2. Electron magnetic-moment anomaly, fine-structure constant, and QED theory . . . . . . 54 3. Proton radius and Rydberg constant . . . . . . 56 4. Muon mass and magnetic moment . . . . . . 56 5. Newtonian constant of gravitation . . . . . . . 56 6. Proton mass . . . . . . . . . . . . . . . . . . . . . 56 7. Physics in general . . . . . . . . . . . . . . . . . 57 List of Symbols and Abbreviations . . . . . . . . . . . 57 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . 58 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 List of Tables I. Exact quantities and their mathematical symbols relevant for the revised SI . . . . . . . . . . . . . . . . 4 II. Relevant values of the Bethe logarithms . . . . . . . 10 III. Values of the function π 3 GREC(x α) . . . . . . . 11 IV. Values of the function GSE(α) . . . . . . . . . . . . . 11 V. Values of the function G(1) VP(α) . . . . . . . . . . . . . 11 VI. Values of the function N(nℓ) . . . . . . . . . . . . . . 12 VII. Values of B60 and B71(nS1/2) . . . . . . . . . . . . . . 13 VIII. Input data for the additive energy corrections to account for missing contributions to the theoretical description of the electronic hydrogen and deuterium energy levels . . . . . . . . . . . . . . . . . . . . . . . . 15 IX. Correlation coefficients for data in Tables VIII and X 16 X. Measured transition energies for electronic hydrogen and deuterium considered as input data for the determination of the Rydberg constant R∞. . . . . 17 XI. Twenty-five additive adjusted constants for the H and D energy levels . . . . . . . . . . . . . . . . . . . . . . . 19 This review is being published simultaneously by Reviews of Modern Physics. This report was prepared by the authors under the auspices of the CODATA Task Group on Fundamental Constants. The members of the task group are: F. Bielsa, Bureau International des Poids et Mesures K. Fujii, National Metrology Institute of Japan, Japan S. G. Karshenboim, Pulkovo Observatory, Russian Federation and Max-Planck-Institut f¨ ur Quantenoptik, Germany H. Margolis, National Physical Laboratory, United Kingdom P. J. Mohr, National Institute of Standards and Technology, United States of America D. B. Newell, National Institute of Standards and Technology, United States of America F. Nez, Laboratoire Kastler-Brossel, France R. Pohl, Johannes Gutenberg-Universit¨ at Mainz, Germany K. Pachucki, University of Warsaw, Poland J. Qu, National Institute of Metrology of China, China A. Surzhykov, Physikalisch-Technische Bundesanstalt, Germany E. Tiesinga, National Institute of Standards and Technology, United States of America M. Wang, Institute of Modern Physics, Chinese Academy of Sciences, China B. M. Wood, National Research Council, Canada J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-2 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr XII. Coefficients for the QED contributions to the electron anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 XIII. Input data for relative atomic masses . . . . . . . . . 21 XIV. Correlation coefficients for data in Table XIII . . . . 21 XV. Ionization energies for 1H, 3H, 3He, 4He, 12C, and 28Si . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 XVI. Theoretical contributions and total value for the g-factor of hydrogenic 12C5+ . . . . . . . . . . . . . . 26 XVII. Theoretical contributions and total value for the g-factor of hydrogenic 28Si13+ . . . . . . . . . . . . . . 26 XVIII. Input data for the μH and μD Lamb shift and e-p and e-d scattering . . . . . . . . . . . . . . . . . . . 28 XIX. Fifty of the 75 adjusted constants in the 2018 CODATA least-squares minimization . . . . . . . . 29 XX. Theoretical values for various bound-particle to free-particle g-factor ratios . . . . . . . . . . . . . . . . . . 32 XXI. Input data to determine the fine-structure constant, muon mass, masses of nuclei with Z ≤2, and mag-netic-moment ratios among these nuclei as well as those of leptons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 XXII. Correlation coefficients among the input data in Table XXI . . . . . . . . . . . . . . . . . . . . . . . . . . 34 XXIII. Observational equations for input data on H/D spec-troscopy, muonic-H and -D Lamb shifts, and elec-tron-proton or deuteron scattering . . . . . . . . . . 35 XXIV. Mass-dependent QED contributions to the muon anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 XXV. Fractional contribution of mass-dependent QED contributions to the muon anomaly . . . . . . . . . . 36 XXVI. Observational equations for input data in Tables XXI and XXVII . . . . . . . . . . . . . . . . . . . . . . . . . 38 XXVII. Input data for the determination of the lattice spacings of an ideal natural Si crystal and x-ray units . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 XXVIII. Correlation coefficients for data in Table XXVII . 42 XXIX. Input data for the Newtonian constant of gravitation 43 XXX. An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chemistry based on the 2018 adjustment . . . . . . . 45 XXXI. The CODATA recommended values of the funda-mental constants of physics and chemistry based on the 2018 adjustment . . . . . . . . . . . . . . . . . . . 46 XXXII. The relative uncertainties and correlation coefficients of the values of a selected group of constants . . . . 52 XXXIII. Values of some x-ray-related quantities based on the 2018 CODATA adjustment of the constants . . . . 52 XXXIV. Non-SI units based on the 2018 CODATA adjust-ment of the constants, . . . . . . . . . . . . . . . . . . 53 XXXV. The values of some energy equivalents derived from the relations E mc2 hc/λ hn kT, part I . . . . . . 54 XXXVI. The values of some energy equivalents derived from the relations E mc2 Zc/λ hn kT, part II . . 55 List of Figures 1. Covariance error ellipses for the proton radius rp and the Rydberg constant R∞from the 2014 and the current 2018 CODATA adjustment . . . . . . . . . . . . . . . . . . . . . . 7 2. Results of measurements relevant for determining the recommended value of the fine-structure constant α 7 3. Relationships in the determinations of Eh, α, me, and mu as well as the theoretical and experimental means to de-termine their values . . . . . . . . . . . . . . . . . . . . . . . 9 4. Experimental hydrogen and deuterium transition energies used as input data in the 2018 least-squares adjustment 18 5. Fourteen fractional contributions to the theoretical anom-aly of the electron . . . . . . . . . . . . . . . . . . . . . . . . 21 6. Input data for the determination of the relative atomic mass of the proton . . . . . . . . . . . . . . . . . . . . . . . . 22 7. Comparison of recent determinations of the leading-order hadronic vacuum-polarization and light-by-light contri-butions to the muon anomaly . . . . . . . . . . . . . . . . . 36 8. Comparison of the experimental and theoretical value for the muon anomaly . . . . . . . . . . . . . . . . . . . . . . . . 37 9. The 16 input data determining the Newtonian constant of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 10. Comparison of a representative group of fundamental constants from the 2014 and 2018 CODATA adjustments 55 I. Introduction This report gives a detailed account of the 2018 least-squares adjustment of over 300 recommended values of basic fundamental constants in nature based on the latest relevant precision measure-ments and improvements of theoretical calculations. The work has been carried out under the auspices of the Task Group on Funda-mental Constants (TGFC) of the Committee on Data of the International Science Council (CODATA). The cutoff date for ac-cepted data was at the close of 31 December 2018, and the new set of values became available on World Metrology Day, 20 May 2019, at a website of the Fundamental Constants Data Center of the National Institute of Standards and Technology (NIST), Gaithersburg, Maryland, USA. The compilation of values of fundamental constants arguably started with Birge (1929) and afterwards occurred at irregular in-tervals until 1998. Since that year, updated and improved adjustments have been published every four years (Mohr and Taylor, 2000, 2005; Mohr, Taylor, and Newell, 2008a, 2008b, 2012a, 2012b; Mohr, Newell, and Taylor, 2016a, 2016b). In 2017, a special adjustment was done to provide values for the redefinition of the International System of Units (SI) (Mohr et al., 2018; Newell et al., 2018). Specifically, rec-ommended exact numerical values for the Planck constant h, ele-mentary charge e, Boltzmann constant k, and Avogadro constant NA were provided. See Mills et al. (2011) for a review of the proposals that led to the redefinitions. The revised SI units for time, length, mass, current, temperature, amount of substance, and luminous intensity based on these exact values together with the already exactly defined J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-3 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr frequency of the ground-state cesium hyperfine splitting and speed of light in vacuum c officially became effective on World Metrology Day. Table I lists the values of the defining constants including that of the luminous efficacy, a measure of light intensity as observed by the human eye. The revision has played an important role in the 2018 least-squares adjustment. The four newly fixed defining constants h, e, k, and NA within the revised SI replace four constants that previously helped define the SI. These were the mass of the international prototype of the kilogram (IPK) m(K), the permeability of vacuum (magnetic constant) μ0, the temperature at the triple point of water TTPW, and the molar mass of a carbon 12 atom at rest and in its ground state, M(12C). In the revised SI, these must now be determined experimentally and are no longer fundamental (Mohr et al., 2018; Newell et al., 2018). For example, the permeability of vacuum and the molar mass of carbon 12 are cal-culable from other (inexact) recommended values; specifically, these are the measurable fine-structure constant and the mass of a single carbon 12 atom, respectively. In this adjustment, we find μ0 4π 3 10−7[1 + 55(15) 3 10−11] N A−2 and M(12C) 0.012 3 [1 −35(30) 3 10−11]kg mol−1. The quantities TTPW and m(K) cannot be determined from other fundamental constants. Of course, the triple point of water can still be regarded “fundamental” in that this point has a well described definition that can be realized by any interested party. To date, however, no theoretical model can reach the accuracy of the best experimental determinations and, thus, TTPW is no longer relevant for the adjustment. The prototype of the kilogram is also no longer relevant for the adjustment, but for a different reason. In this case, the prototype is no longer fundamental. That is, it is no longer unique among massive objects. For further information see the Mise en pratique for the definition of the kelvin and kilogram in the online version of the SI brochure found at https:// www.bipm.org/en/publications/si-brochure. The cornerstone of this 2018 CODATA adjustment, as in previous adjustments, is the validity of physical theory as un-derstood today. Prominent in these theories are the concepts of energy and momentum. For example, the energy of a particle of mass m at rest is mc2 from special relativity. The energy of a single photon with angular frequency ω is Zω from quantum electro-dynamics (QED). Here, Z is the reduced Planck constant. From quantum mechanics we know about the particle-wave duality and that the momentum of a massive or massless object is p Zk, where the wave vector k has a length |k| 2π/λ and λ is the particle’s wavelength. Of course, energy and momentum con-servation then ensures, for example, that when an atom absorbs a photon (without ionizing) its momentum changes and its mass slightly increases. Finally, statistical mechanics and thermody-namics tell us that the mean kinetic energy of a three-dimensional classical gas of noninteracting atoms is 3kT/2 per atom at tem-perature T. It is worth noting that the possible time variation of the fine-structure constant α, proton-to-electron mass ratio, and other di-mensionless constants or ratios (Safronova et al., 2018) does not affect the 2018 adjustment. That is, our final uncertainty for these quantities is orders of magnitude larger than current upper bounds on their time variation. II. Purpose of the Adjustment and Overview of Constants Our periodic CODATA evaluations of the fundamental con-stants of physics and chemistry serve two purposes. First, they provide a self-consistent set of recommended values of the constants for all to use. Second, because they necessitate a summary and analysis of a wide range of experimental and theoretical data, they can identify possible inconsistencies among the data and suggest areas for future work. A constant is only fundamental as a matter of convention. For our adjustment, obvious constants are those that appear in basic physical and chemical theory, such as h, c, e, and k as well as the Newtonian constant of gravitation G and the dimensionless fine-structure constant α. Products and ratios of these constants, like the Josephson constant KJ 2e/h, the molar gas constant R NAk, and the Planck mass (Zc/G)1/2, are natural extensions. Over the years, many such products and ratios have been given dedicated names as these combinations appear as natural units for measurement observables. Masses and magnetic moments of the lightest charged leptons, i.e., the electron and muon, and of light nuclei also fall within the scope of our work as their precise evaluation often involves knowledge of the fine-structure and other constants. Our Task Group only publishes updated values for the neutron and nuclei with charge number Z 1 or 2. We provide masses in the SI unit kg and as relative atomic masses in the atomic mass unit 1 u mu (i.e., in units of one-twelfth of the mass of a neutral 12C atom). An extensive listing of relative atomic masses for stable and unstable atoms in the periodic table can be found in the Atomic-Mass-Data-Center publications (Huang et al., 2017; Wang et al., 2017). Particle properties relevant for high-energy physics, such as the masses of the W, Z, and Higgs particles, the Fermi coupling constant, decay modes of mesons, and many other quantities are collected by the Particle Data Group (Tanabashi et al., 2018). We also maintain values for the lattice constant of natural silicon single crystals and the shielded magnetic moments of the proton in liquid-water and the helion in 3He gas. For the TABLE I. Exact quantities and their mathematical symbols relevant for the revised SI Quantity Symbol Value Unit hyperf. transition freq. of 133Cs ΔnCs 9 192 631 770 Hz speed of light in vacuum c 299 792 458 m s−1 Planck constanta h 6.626 070 15 3 10−34 J Hz−1 Z 1.054 571 817 . . . 3 10−34 J s elementary charge e 1.602 176 634 3 10−19 C Boltzmann constant k 1.380 649 3 10−23 J K−1 Avogadro constant NA 6.022 140 76 3 1023 mol−1 luminous efficacy Kcd 683 lm W−1 aThe energy of a photon with frequency n expressed in unit Hz is E hn in unit J. Unitary time evolution of the state of this photon is given by exp(−iEt/Z)|φ〉, where |φ〉is the photon state at time t 0 and time is expressed in unit s. The ratio Et/Z is a phase. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-4 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr adjustment, the former are relevant for the calibration of x-rays. Before the redefinition of the SI, the precise values of the Si lattice constants in natural and enriched silicon crystals were used to help measure the Avogadro and Planck constants. The shielding factors are relevant because often only shielded magnetic-moment ratios are available. For conciseness, this review summarizes results from the four years before our 31 December 2018 closing date, as previous CODATA reports describe older data. Detailed discussions of the-oretical calculations and experiments are omitted and only note-worthy features are mentioned. Often a result is identified by an abbreviation for the institution at which it was obtained and the last two digits of the year in which the result was published in an archival journal. However, a result does not have to be published in such a journal to be considered as having met the 31 December 2018 closing date of the adjustment if it was available by this date in a detailed preprint. Any input datum with a 20 or earlier date after its institutional abbreviation has met this requirement. A comprehensive list of Symbols and Abbreviations is given near the end of this report. III. Least-Squares Adjustments The least-squares procedure for the determination of the values of fundamental constants is based on the assumption of a normal probability distribution for correlated input data and is described in detail in Appendix E of Mohr and Taylor (1999) and Mohr and Taylor (2000). Key points are as follows. Experiment as well as theory provide input data that are used to determine a set of independent quantities, the unknowns or variables of the ad-justment. They will be called adjusted constants. The expression that relates an input datum to the adjusted constants is its ob-servational equation, and the one-standard-deviation un-certainties of and covariances among the input data determine the weights of the data contributing to χ2 (chi squared), which is minimized in the least-squares adjustment. Observational equations are given by X ≐F(A1, A2, . . .), (1) where X and F(· · ·) are the input datum and its relationship to adjusted constants Aj with j 1, 2 . . ., respectively. The symbol ≐implies that the quantities on either side are equal in principle but need only agree to within the constraints of the adjustment. In its simplest form, the observational equation is X ≐A. We simplify to X ≐X when no confusion can arise. A good example of such a case is Newton’s gravitational constant G, where experimen-talists directly measure G. One-standard-deviation uncertainties will also be called stan-dard uncertainties. For quantity X they are presented as either an absolute standard uncertainty u(X) with the same unit as X or a dimensionless relative standard uncertainty ur(X) u(X)/|X|. Throughoutthisarticle,covariancesu(X,Y) betweenquantitiesX andY are specified in terms of correlation coefficients r(X,Y) u(X,Y)/ [u(X)u(Y)] with values between −1 and 1. Theoretical expressions, say for the g-factor of the electron, often have uncertainties due to inexact numerical calculations or uncalculated terms whose size cannot be ignored. They are dealt with by introducing an additive correction δth to the relevant theoretical expression and including δ as an input datum with magnitude zero and an uncertainty equal to that of the theoretical expression. An observational equation δ ≐δth is then added to χ2. Corrections δth are thus adjusted constants whose values and uncertainties are found in the least-squares procedure. Corre-lations, sometimes significant, among the δ due to common sources of uncertainty are taken into account in χ2 where appropriate. A measure of the consistency of our least-squares adjust-ment for the ith input datum Xi is its normalized residual ri (Xi −〈Xi〉)/u(Xi), where 〈Xi〉is its fitted, or adjusted, value. An absolute value greater than two is problematic and is reduced to less than two by the application of a multiplicative expansion factor to the initially assigned uncertainties of the input datum in question as well as related input data. For data pair Xi and Xj, expansion factors are applied in such a way that their correlation coefficient r(Xi, Xj) is unchanged. This procedure makes the effective data consistent. Several expansion factors have been used in this adjustment. After the application of all expansion factors, we charac-terize the quality of an adjustment with N input data and M adjusted constants by the probability p(χ2|n) of obtaining a value of χ2 by chance that large or larger, where n N −M and the Birge ratio RB   χ2/n . For the 2018 adjustment, the input data and adjusted constants separate into three independent data sets, corresponding to input data related to the determination of the gravitational constant, input data related to natural-silicon lattice spacings, and, finally, all remaining input data and adjusted constants. Each data set is treated separately. The gravitational constant is determined from N 16 measurements and an expansion factor of 3.9 is needed to decrease the residuals to below two. This modification leads to χ2 12.9, p(χ2|n) 0.61, and RB 0.93. For the natural-silicon lattice-spacing determination, there are N 21 input data and M 12 adjusted constants. No expansion factor is needed and χ2 7.3, p(χ2|n) 0.60, and RB 0.90. The third least-squares adjustment has N 105 and M 62 with χ2 31.5, p(χ2|n) 0.88, RB 0.87. Two expansion factors are included. A factor of 1.6 is applied to the 62 input data determining the Rydberg constant and proton and deuteron charge radii. A factor 1.7 is used for the two input data that determine the relative atomic mass of the proton. The input data for the 2018 CODATA adjustment can be found in Tables VIII, X, XVIII, XXI, XXVII, and XXIX. Links to tables with correlation coefficients are given in the captions of these tables. The adjusted constants are given in Tables XI and XIX. Observational equations are found in Tables XXIII and XXVI. IV. Overview of Notable Changes A. Electrical units The introduction of the revised SI has brought electrical metrology back into the SI. Between 1988 and 2018, on the rec-ommendation of the Consultative Committee for Electricity (CCE) and adopted by the International Committee for Weights and Measures (CIPM) (Quinn, 1989; Taylor and Witt, 1989), the J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-5 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr electrical units of current, voltage, resistance, etc. were the ampere-90, volt-90, ohm-90, etc. derived by fixing the Josephson and von Klitzing constants to the exact, conventional values KJ−90 483 597.9 GHz/V and RK−90 25 812.807 Ω, respectively, instead of using KJ 2e/h and RK h/e2 based on the most ac-curate values for hand e. Then, for example, a measurement of the resistance of a resistor would result in a number times RK−90, which is then expressed in the unit Ω90 (often the subscript 90 would be dropped) using the value of RK−90. Now these con-ventional 1990 electrical units are obsolete, because with exact values for h and e in SI units, the Josephson and von Klitzing constants are exact. This leads to fractional changes of two to twenty times 10−8 when reexpressing values of electrical quan-tities from conventional 1990 to the revised SI units. These changes, however, are generally much smaller than the relative uncertainties associated with most everyday measurements of electrical quantities and are only noticeable when comparing quantum electrical standards. B. Particle and relative atomic masses and the atomic mass constant Overnight, the revision of the SI has led to almost two orders of magnitude improvement in the uncertainties of the electron, neutron, and nuclear and atomic masses in the SI unit kg when compared to those found in the 2014 CODATA adjustment. The atomic mass constant, one-twelfth of the mass of the 12C atom in its ground state, has similarly become more accurate. These masses are now often known with relative uncertainties of a few times 10−10. By fixing h and e, the reduced uncertainty is achieved by combining the results of several distinct measurements with equally accurate theoretical calculations for these measurements. For example, in the revised SI the atomic mass constant is most accurately determined through mu 1 12 m(12C) 1 Ar(e) me 2hR∞ Ar(e)α2c, (2) where the adjusted constants are the Rydberg constant R∞, the fine-structure constant α, and the relative atomic mass of the electron Ar(e). Here, we use the Rydberg energy hcR∞ α2mec2/2, and me is the mass of the electron. The Rydberg constant is mainly constrained by measurements of the 1S-2S transition energy in hydrogen. (In practice, this transition energy is measured as a two-photon process.) The fine-structure constant is determined from a combination of calculations and measurements of the electron g-factor as well as atom-recoil measurements. Finally, the relative atomic mass of the electron (not me in kg) is found from spin-precession and cyclotron-frequency-ratio measurements on hydrogenic 12C5+. Of the three adjusted constants on the right-hand side of Eq. (2), the fine-structure constant α is by far the least well known with a still-impressive relative standard uncertainty of 1.5 3 10−10. The relative uncertainty of Ar(e) is 2.9 3 10−11, while that for R∞is 1.9 3 10−12. We find that the relative uncertainty for mu is slightly less than twice that of α once the small covariances among the three adjusted constants are taken into account. The mass for a neutral atom X is most accurately found from m(X) mX Ar(X)mu, (3) where we rely on the 2016 Atomic-Mass-Data-Center (AMDC16) values of relative atomic masses for neutral atoms throughout the periodic table (Huang et al., 2017; Wang et al., 2017). These relative atomic masses often have a smaller relative uncertainty than mu, even though the accuracy of mu has im-proved significantly. The masses of nuclei can be found by ac-counting for the electron masses and electron removal energies where available. In 2016, the Atomic Mass Data Center updated the relative atomic mass of hydrogen based on the then-available data. In 2017, Heiße et al. (2017) made an accurate measurement of the cyclotron frequency ratio of the proton and the 12C6+ nucleus. The implied relative atomic masses of the proton and hydrogen atom from these two sources are inconsistent and require an expansion factor in our least-squares adjustment. The uncertainties added by accounting for the electron mass and binding energy are negligible. C. Proton charge radius and Rydberg constant or frequency The disagreement between the (root-mean-square) charge radius of the proton rp obtained from Lamb-shift measurements in muonic hydrogen (a muon bound to a proton) and the value obtained from transition frequency measurements in hydrogen and electron-proton elastic scattering data, sometimes referred to as the “proton-radius puzzle,” has been partly resolved. Therefore, for this 2018 CODATA adjustment, the TGFC decided that the muonic hydrogen data, some of which were already available in 2010, as well as related muonic deuterium data, should no longer be excluded. The reduced disagreement in the determinations of the proton charge radius is mainly due to two new hydrogen spectroscopic measurements (Beyer et al., 2017; Bezginov et al., 2019), as they imply a smaller rp closer to that found from muonic hydrogen data. Figure 1 illustrates the improved agreement for rp as well as its strong cor-relation with the determination of the Rydberg constant R∞. We observe that our 2018 value for rp has a three-times improved un-certainty compared to that found in the 2014 CODATA evaluation. Moreover, the correlation coefficient between rp and R∞has sig-nificantly decreased. The covariance error ellipse is more circular in the 2018 adjustment. Similar observations hold for the determination of the deuteron charge radius rd. Our 2018 relative standard un-certainties for rp, rd, and R∞are 2.2 3 10−3, 3.5 3 10−4, and 1.9 3 10−12, respectively. The tension between the two approaches determining rp and rd has not been fully resolved. In fact, to obtain consistency among the many input data that contribute to the determination of R∞, rp, and rd, a multiplicative expansion factor of 1.6 is applied to their un-certainties. Further experiments are needed. D. Fine-structure constant and electron magnetic-moment anomaly The fine-structure constant, the dimensionless coupling constant in QED, is determined primarily by measuring either the electron magnetic-moment anomaly ae or the recoil momentum J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-6 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr of an atom from emitting or absorbing a resonant photon. To date, the two approaches lead to a roughly equal uncertainty for α. Figure 2 summarizes these data and the 2018 recommended value of α. The relative standard uncertainty of the 2018 recommended value of α is 1.5 3 10−10, a value that has improved steadily over the past hundred years, since its definition by Sommerfeld (1916). The uncertainty of the theoretical expression for the electron magnetic-moment anomaly ae, mainly a function of α, has now been reduced to the point where it contributes negligibly to the determination of the fine-structure constant α obtained by equating the experimental value of ae to the theoretical expression. For example, Laporta (2017) evaluated the four-virtual photon QED coefficient virtually exactly and hadronic corrections have been updated. The most recent experimental value for ae has a relative standard uncertainty of 2.4 3 10−10 (Hanneke, Fogwell, and Gabrielse, 2008). Its derived value for α is shown in Fig. 2 as Harvard-08. An important new atom-recoil input datum is that by Parker et al. (2018) measured at the University of California at Berkeley, USA. Using atom interferometry with laser-cooled 133Cs, the quotient h/m(133Cs) was measured with ur 4.0 3 10−10. It pro-vides a value of α with ur 2.0 3 10−10, which is the smallest un-certainty of all relevant measurements. It agrees with the less-accurate value of α from a 87Rb atom-interferometry measurement (Bouchendira et al., 2011) made at the Laboratoire Kastler-Brossel (LKB), France. Both data are shown in Fig. 2 and labeled by Berkeley-18 and LKB-11, respectively. We also observe that there exists tension between the ae and h/m(133Cs) measurements; their inferred values of α differ by five times the uncertainty of the 2018 recommended value of α. Nev-ertheless, no expansion factor for the uncertainties of these three input data is required. E. Muon magnetic-moment anomaly The theoretical expression for the muon magnetic-moment anomaly aμ is omitted from this CODATA adjustment as in the two previous adjustments. Although there has been progress in the theory in the past four years, there are still concerns about the hadronic and light-by-light vacuum-polarization contributions, and the 3σ to 4σ disagreement between theory and experiment remains. Currently, researchers at the Experiment E989 (Keshavarzi, 2019) of the Fermi National Accelerator Laboratory, USA and the muon g −2 J-PARC experiment (Abe et al., 2018) of the High Energy Accelerator Research Organization (KEK), Japan hope to resolve this discrepancy. F. Newtonian constant of gravitation Inconsistencies among measurements of the Newtonian con-stant of gravitation G have long been a problem. This is no different in the 2018 adjustment. Sixteen measurements lead to a relative un-certainty ur 2.2 3 10−5, a factor of two reduction compared to our previous adjustment. An expansion factor of 3.9, however, is needed to reduce the absolute value of all residuals below two. Two recent results, both with relative standard uncertainties of 1.2 3 10−5 (Li et al., 2018), have contributed to the improved recommended value. The two values differ by 2.7 times the root-mean square of their uncertainties. V. Outline of Paper The remainder of the paper describes the input data in the 2018 CODATA adjustment, analyzes these data where appropriate, and explains the observational equations. Recommended values of the FIG. 1. Covariance error ellipses for the proton radius rp and the Rydberg constant R∞from the 2014 (blue marker and curve) and the current 2018 (red marker and curve) CODATA adjustment. The black marker and ellipses correspond to a 2018 adjustment where the experimental data from muonic hydrogen and muonic deuterium have not been included. Solid and dashed curves correspond to the one- and two-standard-uncertainty ellipses,respectively. The x- and y-axis data are shifted and normalized by the 2018 recommended values and standard uncer-tainties of rp and R∞, respectively. FIG. 2. Results of measurements relevant for determining the 2018 CODATA recommended value of the fine-structure constant α. Error bars correspond to one-standard-deviation uncertainties. Labels “Harvard-08,” “LKB-11,” and “Berkeley-18” denote the laboratories and the last two digits of the year in which the result was reported. The individual values for (α−1 −137.03) 3 105 are 599.9150(33), 599.8998(85), and 599.9048(28) for Harvard-08, LKB-11, and Berkeley-18, re-spectively. See discussion of this figure for references. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-7 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr fundamental constants and conversion factors of energy equivalents are presented and discussed. We begin by describing the relationship among four important adjusted constants in the CODATA adjustments. Section VI shows how the determination of the Rydberg constant, the Hartree energy, the fine-structure constant, the electron mass, and the atomic mass constant are interconnected. The next five sections describe five types of experiments that determine the values of these five fundamental constants. Section VII explains the theory for and measurements of transition energies in hydrogen and deuterium relevant to the determination of the Rydberg constant or the Hartree energy. Section VIII summarizes the theory for the magnetic-moment anomaly or g-factor of the electron. In addition, the sole direct measurement of the anomaly is discussed. This measurement is one of two ways to determine the fine-structure constant. Section IX describes input data for the relative atomic masses of various nuclei and atoms, i.e., masses specified in units of the atomic mass constant or, equivalently, atomic mass units. Electron ionization and removal energies of H, 3H, 3He, 4He, 12C, and 28Si are also specified. Section X describes atom-recoil experiments, which determine the mass of neutral 87Rb and 133Cs atoms in SI unit kg. Section XI explains the theoretical calculations of the g-factor of the electron in hydrogenic 12C5+ and 28Si13+. In addition, the section describes measurements of the ratio of precession to cyclotron fre-quencies of these hydrogenic ions. Together, these theoretical g-factors and measurements, after accounting for electron removal energies, are the most accurate means to determine the electron mass in atomic mass units (or the atomic mass constant in units of the electron mass). The nexttwo sectionsdescribeinput datathatdeterminethe proton and deuteron charge radii. Section XII summarizes theory for and spectroscopic measurements of the Lamb shift for muonic hydrogen and deuterium. Proton and deuteron charge radii from electron-proton and electron-deuteron elastic scattering data are described in Sec. XIII. Sections XIV and XV describe the input data for magnetic-moment ratios of light nuclei. Both theoretical estimates and ex-perimental data for these ratios are given. The g-factor and mass of the muon are discussed in the next two sections. Section XVI describes both theoretical calculations and mea-surements of the magnetic-moment anomaly of the muon. Due to long-standing discrepancies between the theory and experiments, the Task Group has decided to only use the experimental data to determine the muon anomaly. Section XVII describes the input data for the determination of the mass of the muon relative to that of the electron. Data rely on measurements and theoretical calculations of the hyperfine splitting of ground-state muonium, an electron bound to an antimuon. These data also fix the muon-to-proton magnetic-moment ratio. Section XVIII summarizes the input data that determine the lattice spacing of natural silicon. Section XIX describes the input data for the determination of the Newtonian constant of gravitation. Section XX gives values for some electroweak quantities, i.e., the Fermi coupling constant and the weak mixing angle. Section XXI lists the 2018 CODATA Recommended Values. Tables of values and some calculational details are given. Section XXII gives a summary and conclusion based on a comparison of 2014 and 2018 CODATA recommended values. Changes in values are either due to the revision of the SI or due to newly available input data. We give implications of the 2018 adjustment for electrical metrology, the proton radius and Rydberg constant, the fine-structure constant, and Newton’s gravitational constant. We also make suggestions for future work. VI. Relationships among the Rydberg Constant, Fine-Structure Constant, Electron Mass, and Atomic Mass Constant Several sections in this article describe, in detail, how the Rydberg constant R∞, the Hartree energy Eh, fine-structure constant α, the atomic mass constant mu, and the electron mass me are de-termined. Their determinations are interrelated in CODATA ad-justments and involve five distinct measurements combined with state-of-the-art theoretical calculations within QED. A succinct, simplified flow diagram of the most important relationships and measurements is shown in Fig. 3. At the heart of the diagram are the relationships Eh ≡2R∞hc α2mec2, (4) where h and c are exact in the revised SI. The relationships, for example, imply that measuring two of Eh, α, or me in SI units de-termines the third. (Of course, the dimensionless fine-structure constant will have the same numerical value in any complete set of units.) Alternatively, measuring all three constants confirms the validity of the equation. Spectroscopy on the hydrogen atom, discussed in Sec. VII, and, in particular, the measurement of the 1S-to-2S transition energy or frequency determines the Rydberg constant or, equivalently, the Hartree energy in SI units. In fact, R∞or Eh has a unique place in the adjustment. Its relative uncertainty is orders of magnitude smaller than that of our other adjusted constants. The measurement of the ratio of spin-precession and cyclotron frequencies of a single, free electron in a magnetic flux density gives an accurate value for its g-factor. Combined with theoretical calculations of g as a function of α, this gives a competitive value for α. Details are given in Sec. VIII. Of the adjusted constants, the fine-structure constant has the second smallest relative uncertainty. Currently, the two types of mea-surementscombinedwithEq.(4)givethemostaccuratevalueforme inkg. Measurements of the ratio of precession and cyclotron fre-quencies of hydrogenic 12C5+ (and to a lesser extent 28Si13+) are used to determine the relative atomic mass of the electron, Ar(e) me/mu. Here, theoretical calculations of the g-factor of the bound electron (as a function of α) are also essential. Details can be found in Sec. XI.From the measurement of Ar(e) and the value for the electron mass, an accurate value for the atomic mass constant mu is derived. Finally, Fig. 3 shows how atom-recoil experiments that measure the mass of 87Rb and 133Cs in kg combined with mea-surements of their relative atomic masses as compiled by the Atomic Mass Data Center form a second pathway to determine me, but most importantly, a second competitive determination of the fine-structure constant. These experiments and data are discussed in Secs. IX and X, respectively. The directions of the arrows in Fig. 3 indicate the paths traversed to find the most accurate values for our four constants. The figure, however, does not show all relationships. For example, J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-8 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr atom-recoil experiments and the data from the Atomic Mass Data Center can be used to determine mu as well. Its value, however, would be less accurate. The transition energies among the eigenstates in hydrogen also depend on α and, thus, could con-strain its value. Still, the measurement of the g-factor of the free-electron and atom-recoil experiments are currently the best means to determine α. Moreover, hydrogen spectroscopy is also used to constrain the proton radius. VII. Atomic Hydrogen and Deuterium Transition Energies The comparison of theory and experiment for electronic transition energies in atomic hydrogen and deuterium is currently the most precise way to determine the Rydberg constant, or equivalently the Hartree energy, and to a lesser extent the charge radii of the proton and deuteron. Here, we summarize the theory of and the experimental input data on H and D energy levels in Secs. VII.A and VII.C, respectively. The charge radii of the proton and deuteron are also constrained by data and theory on muonic hydrogen and muonic deuterium as well as by those from electron scattering. These data are discussed in Secs. XII and XIII, respectively. The electronic eigenstates of H and Dare conveniently labeled by nℓj, where n 1, 2, . . . is the principal quantum number, ℓ 0, 1, . . . , n −1 is the quantum number for the electron orbital angular momentum L, and j ℓ± 1/2 is the quantum number of the total electronic angular momentum J. Following the usual conven-tion, we use S, P, D, . . . to denote ℓ 0, 1, 2, . . . states. Theoretical values for the energy levels of H and D are de-termined by the Dirac eigenstate energies, QED effects such as self-energy and vacuum-polarization corrections, as well as proton size and nuclear recoil effects. The expression for energy levels quickly becomes complex. The energies, however, do satisfy E −Eh 2n2 (1 + F) −R∞hc n2 (1 + F), (5) where Eh α2mec2 2R∞hc is the Hartree energy, R∞is the Ryd-berg constant, and α is the fine-structure constant. The dimensionless F, small compared to one, is determined by QED, recoil corrections, etc. Consequently, the measured H and D transition energies de-termine Eh and R∞as h and c are exact in the SI. The transition energy between states i and i′ with energies Ei and Ei′ is given by ΔEii′ Ei′ −Ei. (6) Alternatively, we write ΔEii′ ΔE(i −i′). A. Theory of hydrogen and deuterium energy levels Thissectiondescribesthetheoryofhydrogenanddeuteriumenergy levels. References to the original literature are generally omitted; these may be found in the recent review by Yerokhin, Pachucki, and Patk´ oˇ s (2019), on which we rely for recent developments, but also in earlier CODATA reports, Sapirstein and Yennie (1990) and Eides, Grotch, and Shelyuto (2001, 2007). Literature references to new developments are given where appropriate. Nine contributions to the energies with different physicaloriginshavebeenisolated.Eachisdiscussedinoneofthefollowing subsections. Moreover, each contribution has “correlated” and/or “un-correlated” uncertainties due to limitations in the calculations. An im-portantcorrelateduncertaintyiswhereacontributiontotheenergyhasthe form C/n3 with a coefficient C that is the same for states with the same ℓ and j. The uncertainty in C leads to correlations among energies of states with the same ℓand j. Such uncertainties are denoted as uncertainty type u0 in the text. Uncorrelated uncertainties, i.e., those independent of the quantum numbers, are denoted as type un. Other correlations are those between corrections for the same state in different isotopes, where the differenceinthecorrectionisonlyduetothedifferenceinthemassesofthe isotopes. Calculations of the uncertainties of the energy levels and the corresponding correlation coefficients are further described in Sec. VII.B. 1. Dirac eigenvalue The largest contribution to the energies is the Dirac eigenvalue for an electron bound to an infinitely heavy point nucleus or a sta-tionary point nucleus. It is ED f(n, κ)mec2, (7) where f(n, κ) 1 + (Zα)2 (n −δ)2 −1/2 , (8) with δ |κ| −   κ2 −(Zα)2  and κ is the angular momentum-parity quantum number (κ −1, 1, −2, 2, −3 for ℓj S1/2, P1/2, P3/2, D3/2, and D5/2 states, respectively). States with the same n and j |κ| −1/2 have degenerate eigenvalues. Finally, ℓ |κ + 1/2| −1/2 and we retain FIG. 3. Relationships in the determinations of Eh, α, me, and mu (red text and symbols) as well as the theoretical and experimental means (black text with orange measured quantity) to determine their values. Blue directed arrows give the most commonly traversed connections between the constants and measured quantities. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-9 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr the atomic number Z in the equations in order to classify the various contributions to the energies in this and other sections. For a nucleus with a finite mass mN, we have EM(H) Mc2 + [f(n, κ) −1]mrc2 −[f(n, κ) −1]2m2 rc2 2M + 1 −δℓ0 κ(2ℓ+ 1) (Zα)4m3 rc2 2n3m2 N + · · · (9) for hydrogen and EM(D) Mc2 + [f(n, κ) −1]mrc2 −[f(n, κ) −1]2m2 rc2 2M + 1 κ(2ℓ+ 1) (Zα)4m3 rc2 2n3m2 N + · · · (10) for deuterium, where δℓℓ′ is the Kronecker delta, M me + mN, and mr memN/(me + mN) is the reduced mass. Note that in this equation the energy of nS1/2 states differs from that of nP1/2 states. It is worth noting that in Eqs. (9) and (10) we follow a slightly different classification of terms when compared to that used by Yerokhin, Pachucki, and Patk´ oˇ s (2019). Specifically, contributions of order (me/mN)2(Zα)4mec2 in our equations are classified as rela-tivistic-recoil corrections that are second order in the mass ratio by Yerokhin, Pachucki, and Patk´ oˇ s (2019). The remaining difference between the CODATA expressions for the Dirac energy and those of Yerokhin, Pachucki, and Patk´ oˇ s (2019) is of order (me/mN)2(Zα)6mec2, negligible for our current purposes. 2. Relativistic recoil The leading relativistic-recoil correction, to lowest order in Zα and all orders in me/mN, is (Erickson, 1977; Sapirstein and Yennie, 1990) ES m3 r m2 emN (Zα)5 πn3 mec2 31 3δℓ0 ln(Zα)−2 −8 3 ln k0(n, ℓ) −1 9δℓ0 −7 3an − 2 m2 N −m2 e δℓ0m2 N ln me mr −m2 eln mN mr  , (11) where an −2ln(2/n) −2 + 1/n −2 n i1(1/i) for ℓ 0 and an 1/[ℓ(ℓ+ 1)(2ℓ+ 1)] otherwise. Values for the Bethe logarithms lnk0(n, ℓ) are given in Table II. Additional contributions to lowest order in the mass ratio and of higher order in Zα are ER me mN (Zα)6 n3 mec2[D60 + ZαGREC(Zα)], (12) where D60 4 ln 2 −7/2 for ℓ 0 and D60 2[3 −ℓ(ℓ+ 1)/n2]/ [(2ℓ−1)(2ℓ+ 1)(2ℓ+ 3)] otherwise. The function GREC(x) is GREC(x) D72ln2(x−2) + D71ln(x−2) + D70 + · · · , (13) where D72 −11/(60π)δℓ0. Other D7x coefficients are not known an-alytically. Instead, we use the numerically computed GREC(x) of Yerokhin and Shabaev (2015, 2016) for nS states with n 1, ... , 5 as well as for the 2P1/2 and 2P3/2 states. For x α, these values and uncertainties (both multipliedbyπ)arereproducedinTableIII.FornS stateswithn 6, 8,we extrapolate GREC(α) using g0 + g1/n, where coefficients g0 and g1 are found from fitting to the n 4 and 5 values of GREC(α). The values are 14.8(1) and 14.7(2) for n 6 and 8, with uncertainties based on com-parisontovaluesobtainedbyfittingg0 + g1/n + g2/n2 tothen 3, 4,and 5values.Fortheotherℓ> 0 states,weuseGREC(x) 0 and anuncertainty in the relativistic-recoil correction ES + ER equal to 0.01ER. The covariances for ES + ER between pairs of states with the same ℓ and j follow the dominant 1/n3 scalingof the uncertainty,i.e., areof type u0. 3. Self-energy The one-photon self-energy of an electron bound to a stationary point nucleus is E(2) SE α π (Zα)4 n3 F(Zα)mec2, (14) where the function F(x) is F(x) A41ln(x−2) + A40 + A50x + A62x2 ln2(x−2) + A61x2ln(x−2) + GSE(x)x2, (15) with A41 (4/3)δℓ0, A40 −(4/3)ln k0(n, ℓ) + 10/9 for ℓ 0 and A40 −(4/3)ln k0(n, ℓ) −1/[2κ(2ℓ+ 1)] otherwise. Next, A50 (139/32 −2 ln 2)πδℓ0, A62 −δℓ0, and A61 41 + 1 2 + · · · + 1 n + 28 3 ln 2 −4 ln n −601 180 −77 45n2 δℓ0 + n2 −1 n2  2 15 + 1 3δj 1 2 δℓ1 + 96n2 −32ℓ(ℓ+ 1) 3n2(2ℓ−1)(2ℓ)(2ℓ+ 1)(2ℓ+ 2)(2ℓ+ 3). Values for GSE(α) in Eq. (15) are listed in Table IV. The un-certainty of the self-energy contribution is due to the uncertainty of GSE(α) listed in the table and is taken to be type un. See Mohr, Taylor, and Newell (2012a) for details. Following convention, F(Zα) is multiplied by the factor (mr/me)3, except the magnetic-moment term −1/[2κ(2ℓ+ 1)] in A40, which is instead multiplied by the factor (mr/me)2, and the argument (Zα)−2 of the logarithms is replaced by (me/mr)(Zα)−2. 4. Vacuum polarization The stationary point nucleus second-order vacuum-polarization level shift is E(2) VP α π (Zα)4 n3 H(Zα)mec2, (16) where H(x) H(1)(x) + H(R)(x) with TABLE II. Relevant values of the Bethe logarithms ln k0(n, ℓ). Missing entries are for states for which no experimental measurements are included n S P D 1 2.984 128 556 2 2.811 769 893 −0.030 016 709 3 2.767 663 612 4 2.749 811 840 −0.041 954 895 −0.006 740 939 6 2.735 664 207 −0.008 147 204 8 2.730 267 261 −0.008 785 043 12 −0.009 342 954 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-10 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr H(1)(x) V40 + V50x + V61x2 ln(x−2) + G(1) VP(x)x2. Here, V40 −(4/15)δℓ0, V50 (5π/48)δℓ0, and V61 −(2/15)δℓ0. Values of G(1) VP(α) are given in Table V. Moreover, H(R)(x) G(R) VP (x)x2 with G(R) VP (x) 19 45 −π2 27 + 1 16 −31π2 2880πx + · · · (17) for ℓ 0. Higher-order and higher-ℓterms are negligible. We mul-tiply Eq. (16) by (mr/me)3 and include a factor of (me/mr) in the argument of the logarithm of the term proportional to V61. Vacuum polarization from μ+μ−pairs is E(2) μVP α π (Zα)4 n3 −4 15δℓ0 me mμ  2 mr me  3 mec2, (18) while the hadronic vacuum polarization is given by E(2) had VP 0.671(15)E(2) μVP. (19) Uncertainties are of type u0. The muonic and hadronic vacuum-polarization contributions are negligible for higher-ℓstates. 5. Two-photon corrections The two-photon correction is E(4) α π 2(Zα)4 n3 F(4)(Zα)mec2, (20) where F(4)(x) B40 + B50x + B63x2ln3(x−2) + B62x2ln2(x−2) + B61x2ln(x−2) + B60x2 + B72x3ln2(x−2) + B71x3ln(x−2) + · · · (21) with B40 3π2 2 ln 2 −10π2 27 −2179 648 −9 4 ζ(3)δℓ0 + π2ln 2 2 −π2 12 −197 144 −3ζ(3) 4  1 −δℓ0 κ(2ℓ+ 1), B50 −21.554 47(13)δℓ0, B63 −(8/27)δℓ0, B62 16 9 71 60 −ln 2 + ψ(n) + γ−lnn −1 n + 1 4n2 δℓ0 + 4 27 n2 −1 n2 δℓ1. Here, ζ(z), γ, and ψ(z) are the Riemann zeta function, Euler’s constant, and the psi function, respectively, and TABLE III. Values of the function π 3 GREC(x α) from Yerokhin and Shabaev (2015, 2016). Numbers in parentheses are the one-standard-deviation uncertainty in the last digit of the value. [The definitions of GREC(x) in this adjustment and that of Yerokhin and Shabaev (2015, 2016) differ by a factor π.] Missing entries are states for which data are not available from these references n S P1/2 P3/2 1 9.720(3) 2 14.899(3) 1.5097(2) −2.1333(2) 3 15.242(3) 4 15.115(3) 5 14.941(3) TABLE IV. Values of the function GSE(α) n S1/2 P1/2 P3/2 D3/2 D5/2 1 −30.290 240(20) 2 −31.185 150(90) −0.973 50(20) −0.486 50(20) 3 −31.047 70(90) 4 −30.9120(40) −1.1640(20) −0.6090(20) 0.031 63(22) 6 −30.711(47) 0.034 17(26) 8 −30.606(47) 0.007 940(90) 0.034 84(22) 12 0.009 130(90) 0.035 12(22) TABLE V. Values of the function G(1) VP (α) n S1/2 P1/2 P3/2 D3/2 D5/2 1 −0.618 724 2 −0.808 872 −0.064 006 −0.014 132 3 −0.814 530 4 −0.806 579 −0.080 007 −0.017 666 −0.000 000 6 −0.791 450 −0.000 000 8 −0.781 197 −0.000 000 −0.000 000 12 −0.000 000 −0.000 000 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-11 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr B61 413 581 64 800 + 4N(nS) 3 + 2027π2 864 −616 ln 2 135 −2π2 ln 2 3 + 40ln22 9 + ζ(3) + 304 135 −32 ln 2 9  3 4 + γ + ψ(n) −ln n −1 n + 1 4n2 −43 36 + 133π2 864 δℓ0 +4 3 N(nP) + n2 −1 n2  31 405 + 1 3δj 1 2 −8 27 ln 2 δℓ1, wheretherelevantvaluesanduncertaintiesforthefunctionN(nℓ) aregivenin Table VI. The last two terms contributing to B61 for S states are recently computedlight-by-lightcorrectionsobtainedbyCzarneckiandSzafron(2016). Before describing the next term in Eq. (21), i.e., B60, it is useful to observe that Karshenboim and Ivanov (2018b) have derived that B72 −139 48 + 4 ln 2 3 −5 72π δℓ0. In addition, they find the difference B71(nS) −B71(1S) π427 36 −16 3 ln 2 3 4 −1 n + 1 4n2 + ψ(n) + γ−ln n (22) for S states, but also that B71(nP) π 139 144 −4 ln 2 9 + 5 2161 −1 n2 for P states, and B71(nℓ) 0 for states with ℓ> 1. We determine the coefficients B60(1S) and B71(1S) by com-bining the analytical expression for B72 and the values and un-certainties for the remainder GQED2(x) B60 + B72x ln 2(x−2) + B71x ln(x−2) + · · · (23) for the 1S state extrapolated to x ≤2α by Yerokhin, Pachucki, and Patk´ oˇ s (2019) from numerical calculations of GQED2(x) as a function of x for x Zα with Z ≥15 given by Yerokhin, Indelicato, and Shabaev (2008) and Yerokhin (2009, 2018). Specifically, the remainder has three contributions. The largest by far has been evaluated at x 0 and α. The remaining two are available for x α and 2α. Fits to each of the three contributions give corresponding contributions to B60(1S) and B71(1S). We assign a type-u0 state-in-dependent standard uncertainty of 9.3 for B60(1S) and a 10% type-u0 uncertainty to B71(1S). The difference B60(nS) −B60(1S), given by Jentschura, Czarnecki, and Pachucki (2005), is then used to obtain B60(nS) for n > 1 and adds an additional small state-dependent un-certainty. Similarly, the expression for B71(nS) −B71(1S) in Eq. (22) is used to determine B71(nS). Values for B60 for nP and nD states with n 1, . . . , 6 are those published by Jentschura, Czarnecki, and Pachucki (2005) and Jentschura (2006), but using in place of the results in Eqs. (A3) and (A6) of the latter paper the corrected results given in Eqs. (24) and (25) by Yerokhin, Pachucki, and Patk´ oˇ s (2019). For n > 6, we use B60 g0 + g1/n with g0 and g1 determined from the values and uncertainties of B60 at n 5 and 6. Relevant values and uncertainties for B60(nℓ) and B71(1S) are listed in Table VII. For the B60 of S states, the first number in parentheses is the state-dependent uncertainty of type un, while the second number in parentheses is the state-independent uncertainty of type u0. Note that the extrapolation procedure for nS states is by no means unique. In fact, Yerokhin, Pachucki, and Patk´ oˇ s (2019) used a different approach that leads to consistent and equally accurate values for B60(nS). For B71(1S) and B60(nℓ) with ℓ> 0, the un-certainties are of type u0. As with the one-photon correction, the two-photon correction is multiplied by the reduced-mass factor (mr/me)3, except the mag-netic-moment term proportional to 1/[κ(2ℓ+ 1)] in B40 which is multiplied by the factor (mr/me)2, and the argument (Zα)−2 of the logarithms is replaced by (me/mr)(Zα)−2. 6. Three-photon corrections The three-photon contribution in powers of Zα is E(6) α π 3(Zα)4 n3 F(6)(Zα)mec2, (24) where F(6)(x) C40 + C50x + C63x2ln3(x) + C62x2ln2(x) + C61x2 ln x + C60x2 + · · · . (25) The leading term C40 is C40 −568a4 9 + 85ζ(5) 24 −121π2ζ(3) 72 −84 071ζ(3) 2304 −71ln42 27 −239π2ln22 135 + 4787π2ln 2 108 + 1591π4 3240 −252 251π2 9720 + 679 441 93 312 δℓ0 + −100a4 3 + 215ζ(5) 24 −83π2ζ(3) 72 −139ζ(3) 18 −25ln42 18 + 25π2ln22 18 + 298π2ln 2 9 + 239π4 2160 −17 101π2 810 −28 259 5184  1 −δℓ0 κ(2ℓ+ 1), where a4  ∞ n11/(2nn4) 0.517 479 061 . . .. Partial results for C50 have been calculated by Eides and Shelyuto (2004, 2007). We use C50 0 with uncertainty 30δℓ0 of type u0. Karshenboim and Ivanov (2018b) derived that C63 0 and TABLE VI. Values of N(nℓ) used in the 2018 adjustment and from Jentschura (2003) and Jentschura, Czarnecki, and Pachucki (2005) n N(nS) N(nP) 1 17.855 672 03(1) 2 12.032 141 58(1) 0.003 300 635(1) 3 10.449 809(1) 4 9.722 413(1) −0.000 394 332(1) 6 9.031 832(1) 8 8.697 639(1) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-12 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr C62 −2 3 −1523 648 −10π2 27 + 3 2π2ln 2 −9 4 ζ(3) −82 81δℓ0. They also presented an expression for the difference C61(nS) −C61(1S) as well as C61(nP) 2 9 n2 −1 n2 −1523 648 −10π2 27 + 3 2π2ln 2−9 4 ζ(3) −82 81 , and C61(nℓ) 0 for ℓ> 1. We do not use the expression for the difference. Instead, we assume that C61(nS) 0 with an uncertainty of 10 of type un. Finally, we set C60 0 with uncertainty 1 of type un for P and higher ℓstates. For S states we also use C60 0, but do not need to specify an uncertainty as the uncertainty of their three-photon correction is determined by the uncertainties of C50 and C61. The dominant effect of the finite mass of the nucleus is taken into account bymultiplying terms proportional to δℓ0 by thereduced-mass factor (mr/me)3 and the term proportional to 1/[κ(2ℓ+ 1)], the magnetic-moment term, by the factor (mr/me)2. The contribution from four photons is expected to be negligible at the level of uncertainty of current interest. 7. Finite nuclear size and polarizability Finite-nuclear-size and nuclear-polarizability corrections are ordered by powers in α, following Yerokhin, Pachucki, and Patk´ oˇ s (2019), rather than by finite size and polarizability. Thus, we write for the total correction Enucl  ∞ i4 E(i) nucl, (26) where index i indicates the order in α. The first and lowest-order contribution is E(4) nucl 2 3mec2(Zα)4 n3 mr me  3 rN ƛC  2 δℓ0 (27) and is solely due to the finite root-mean-square (rms) charge radius rN of nucleus N. Here, ƛC Z/mec is the reduced Compton wave-length of the electron. The α5 correction has both nuclear-size and polarizability contributions and has been computed by Tomalak (2019). For hy-drogen, the correction is parametrized as E(5) nucl(H) −1 3mec2(Zα)5 n3 mr me  3 rpF ƛC  3 δℓ0 (28) with effective Friar radius for the proton rpF 1.947(75) fm. (29) The functional form of Eq. (28) is inspired by the results of Friar (1979) and his definition of the third Zemach moment. For deuterium, the α5 correction is parametrized as (Yerokhin, Pachucki, and Patk´ oˇ s, 2019) E(5) nucl(D) −1 3mec2(Zα)5 n3 mr me  3 3 ⎡ ⎣Z rpF ƛC  3 + (A −Z) rnF ƛC  3 ⎤ ⎦δℓ0 + E(5) pol(D) (30) with atomic number A, effective Friar radius for the neutron rnF 1.43(16) fm, (31) and two-photon polarizability E(5) pol(D)/h −21.78(22)δℓ0 n3 kHz. (32) In principle, the effective Friar radius for the proton might be different in hydrogen and deuterium. Similarly, the Friar radius of the neutron extracted from electron-neutron scattering can be different from that in a deuteron. We assume that such changes in the Friar radii are smaller than the quoted uncertainties. Theα6 correctionhasfinite-nuclear-size,nuclear-polarizability,and radiative finite-nuclear-size contributions and can thus be written as E(6) nucl E(6) fns + E(6) pol + E(6) rad. The finite-nuclear-size and nuclear-polariz-abilitycontributionsaregivenbyPachucki,Patk´ oˇ s,and Yerokhin(2018). The finite-nuclear-size contribution is E(6) fns mec2(Zα)6 n3 mr me  3 rN ƛC  2  −2 3 9 4n2 −3 −1 n + 2γ−ln(n/2) + ψ(n) + ln mr me rN2 ƛC Zαδℓ0 + 1 6  1 −1 n2 δκ1, (33) and the polarization contribution for hydrogen is TABLE VII. Values of B60 and B71(nS1/2) used in the 2018 adjustment. The uncertainties of B60 are explained in the text n B60(nS1/2) B60(nP1/2) B60(nP3/2) B60(nD3/2) B60(nD5/2) B71(nS1/2) 1 −78.7(0.3)(9.3) −116(12) 2 −63.6(0.3)(9.3) −1.8(3) −1.8(3) −100(12) 3 −60.5(0.6)(9.3) −94(12) 4 −58.9(0.8)(9.3) −2.5(3) −2.5(3) 0.178(2) −91(12) 6 −56.9(0.8)(9.3) 0.207(4) −88(12) 8 −55.9(2.0)(9.3) 0.245(5) 0.221(5) −86(12) 12 0.259(7) 0.235(7) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-13 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr E(6) pol(H)/h 0.393 δℓ0 n3 kHz (34) with a 100% uncertainty and for deuterium E(6) pol(D)/h −0.541 δℓ0 n3 kHz (35) with a 75% uncertainty. The effective radius rN2 describes high-energy contributions and is given by rN2 1.068497rN. (36) The radiative finite-nuclear-size contribution of order α6 is (Eides, Grotch, and Shelyuto, 2001) E(6) rad 2 3 mec2α(Zα)5 n3 mr me  3 rN ƛC  2 (4 ln 2 −5)δℓ0. (37) Next-order radiative finite-nuclear-size corrections of order α7 also have logarithmic dependencies on α; see Yerokhin (2011). In fact, for nS states we have E(7) nucl 2 3 mec2 α(Zα)6 πn3 mr me  3 rN ƛC  2 3 −2 3 ln2{(Zα)−2} + ln2 mr me rN ƛC . (38) We assume a zero value with uncertainty 1 for the uncomputed co-efficient of ln(Zα)−2 inside the square brackets. For nPj states we have E(7) nucl 1 6 mec2 α(Zα)6 πn3 mr me  3 rN ƛC  2 1 −1 n2 3 8 9 ln{(Zα)−2} −8 9 ln 2 + 11 27 + δκ1 + 4n2 n2 −1 N(nP) (39) with a zero value for the uncomputed coefficient of Zα inside the square brackets with an uncertainty of 1. [This equation fixes a ty-pographical error in Eq. (64) of Yerokhin, Pachucki, and Patk´ oˇ s (2019). See also Eq. (31) of Jentschura (2003).] We assume a zero value for states with ℓ> 1. Uncertainties in this subsection are of type u0. Higher-order corrections are expected to be negligible. 8. Radiative-recoil corrections Corrections for radiative-recoil effects are ERR m3 r m2 emN α(Zα)5 π2n3 mec2δℓ0 6ζ(3) −2π2 ln 2 + 35π2 36 −448 27 + 2 3 π(Zα)ln2{(Zα)−2} + · · · . (40) We assume a zero value for the uncomputed coefficient of (Zα) ln(Zα)−2 inside the square brackets with an uncertainty of 10 of type u0 and 1 for type un. Corrections for higher-ℓstates are negligible. 9. Nucleus self-energy The nucleus self-energy correction is ESEN 4Z2α(Zα)4 3πn3 m3 r m2 N c2ln mN mr(Zα)2δℓ0 −ln k0(n, ℓ), (41) with an uncertainty of 0.5 for S states in the constant (α-independent) term in square brackets. This uncertainty is of type u0 and given by Eq. (41) with the factor in the square brackets replaced by 0.5. For higher-ℓstates, the correction is negligibly small compared to current experimental uncertainties. B. Total theoretical energies and uncertainties The theoretical energy of centroid En(L) of a relativistic level L nℓj is the sum of the contributions given in Secs. VII.A.1–VII.A.9. Here, atom X H or D. Uncertainties in the adjusted constants that enter the theoretical expressions are found by the least-squares ad-justment. Here, the most important adjusted constants are R∞ α2mec2/2hc, α, rp, and rd. The uncertainty in the theoretical energy is taken into account by introducing additive corrections to the energies. Specifically, we write EX(L) →EX(L) + δth(X, L) for relativistic levels L nℓj in atom X. Here, energy δth(X, L) is treated as an adjusted constant and we include δX(L) as an input datum with zero value and an uncertainty that is the square root of the sum of the squares of the uncertainties of the individual contributions. That is, u2[δX(L)]  i [u2 0i(X, L) + u2 ni(X, L)], (42) where energies u0i(X, L) and uni(X, L) are type-u0 and -un un-certainties of contribution i. The observational equation δX(L) ≐δth(X, L) is added to χ2. Covariances among the corrections δX(L) are accounted for in the adjustment. We assume that nonzero covariances for a given atom X only occur between states with the same ℓand j. We then have u[δX(n1ℓj), δX(n2ℓj)]  i u0i(X, n2ℓj)u0i(X, n1ℓj), when n1 ≠n2 and only uncertainties of type u0 are present. Co-variances between the corrections δ for hydrogen and deuterium in the same electronic state L are u[δH(L), δD(L)]  i{ic} [u0i(H, L)u0i(D, L) + uni(H, L)uni(D, L)] and for n1 ≠n2 u[δH(n1ℓj), δD(n2ℓj)]  i{ic} u0i(H, n1ℓj)u0i(D, n2ℓj), where the summation over i is only over the uncertainties common to hydrogen and deuterium. This excludes, for example, contributions that depend on the nuclear-charge radii. Values and standard uncertainties of δX(nℓj) are given in Table VIII and the non-negligible covariances of the corrections δ are given as correlation coefficients in Table IX. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-14 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr C. Experimentally determined transition energies in hydrogen and deuterium Table X gives the measured transition energies as well as measured weighted differences between transition energies in hydrogen and deuteriumusedasinputdatainthe2018adjustment.Allbutfourdataare the same as in the 2014 CODATA report. The new results in hydrogen are reviewed in the next three subsections. The transition energies were measured at the Max-Planck-Institut f¨ ur Quantenoptik (MPQ), Garching, Germany, the Laboratoire Kastler-Brossel (LKB), Paris, France, and York University (York), Toronto, Canada. These researchers considered the 2S−4P, 1S−3S, and 2S−2P1/2 transitions. Observational equations for the data are given in Table XXIII. Values for additive corrections δX(nℓj) and δhfs,H[nℓj(f)] to account for the uncertainties in the theoretical expressions are given in Table VIII. Some of the data are correlated and their correlation coefficients when greater than 0.0001 are given in Table IX. The H and D input data are displayed in Fig. 4. The first thing to note is that the data separate into 1S −2S transition energies measured to approximately h 3 10 Hz and those that have been measured to ∼h 3 10 kHz. The uncertainties of these input data are shown without the 1.6 expansion factor applied to these data in the least-squares ad-justment. Secondly, the figure shows the adjusted, or fitted, transition energies and their standard uncertainties for these input data after the application of the 1.6 expansion factor. The values and standard un-certainties of the fitted 1S −2S transition energies are in agreement with those of the experimental data. The standard uncertainties of the fitted values for most of the other data are an order of magnitude smaller than the uncertainties of the corresponding input data. The exceptions are three of the four newly added data. They are indicated as MPQ(2017), LKB(2018), and York(2019) in Fig. 4. In summary, the 1S −2S transition energies,thesethreeinputdata,andthemuonic-Handmuonic-DLamb-shiftmeasurementstobediscussed in Sec.XIIdeterminethe valuesof the Rydberg constant and charge radii. 1. Measurement of the hydrogen 2S−4P transition The hydrogen transition energy from the 2S1/2 hyperfine cen-troid to the 4P fine-structure centroid was measured by Beyer et al. (2017) at the MPQ. This new datum is item A9 in Table X. Here, the fine-structure centroid of a level nℓis EX(nℓ) 1 j(2j + 1) j (2j + 1)EX(nℓj), (43) where the sum over quantum number j runs from |ℓ−1/2| to ℓ+ 1/2 and EX(nℓj) is the hyperfine centroid of level nℓj. In the experiment, cold ground-state hydrogen atoms emerge from a copper nozzle held at a temperature of 5.8 K. These atoms are excited to the metastable 2S1/2(f 0) hyperfine level by a Doppler-free two-photon excitation using 243 nm light, chopped on and off at 160 Hz, enablingathoroughstudyofDopplershifts.Startingfromthismetastable state, transition energies for the hyperfine-resolved transitions 2S1/2(f 0) →4P1/2(f 1) and 2S1/2(f 0) →4P3/2(f 1) were measured to about 1 part in 10000 of the linewidth using a stable ret-roreflected 486 nm laser (Beyer et al., 2016) oriented perpendicular to the propagation direction of the atoms. Here, crucially dipole selection rules forbid excitations to P3/2(f 0) and P3/2(f 2) states. Atomic hydrogen in the 4P state mainly decays to the ground 1S state by emission of a Lyman-γ 97 nm photon. At MPQ, the emission rate of these photons as a function of the 486 nm laser frequency was detected. Lyman-γ radiation ejects electrons from graphite, which, in turn, can be efficiently counted with channel electron multipliers. Two such detectors were used to retain some directional information about the emitted Lyman-γ photons. Important for the experiments was an analysis of line-shape shifts and distortions of the two measured transitions due to the presence of neighboring resonances. Following Jentschura and Mohr (2002) but also Horbatsch and Hessels (2010, 2011), the MPQ researchers de-veloped a line-shape model that accounted for these so-called quantum interference effects as well as demonstrated its validity based on di-rectional information of the Lyman-γ photons as a function of the direction of the linear polarization of the 486 nm light. Quantum interference effects in precision spectroscopic mea-surements have a long history starting with Kramers and Heisenberg (1925) and Low (1952) in the context of QED. For a review of early observations of these effects, see Marrus and Mohr (1979). Jentschura and Mohr (2002) gave an early theoretical analysis of the effect and noted that these interferences are enhanced in differential or angular-dependent measurements. TABLE VIII. Summary of input data for the additive energy corrections to account for missing contributions to the theoretical description of the electronic hydrogen (H) and deuterium (D) energy levels. These correspond to 25 additive corrections δH,D(nℓj) for the centroids of levels nℓj. The label in the first column is used in Table IX to list correlation coefficients among these dataand in Table XXIII for observational equations. Relative uncertainties are with respect to the binding energy Value Rel. stand. Input datum (kHz) uncert. ur B1 δH(1S1/2)/h 0.0(1.6) 4.9 3 10−13 B2 δH(2S1/2)/h 0.00(20) 2.4 3 10−13 B3 δH(3S1/2)/h 0.000(59) 1.6 3 10−13 B4 δH(4S1/2)/h 0.000(25) 1.2 3 10−13 B5 δH(6S1/2)/h 0.000(12) 1.3 3 10−13 B6 δH(8S1/2)/h 0.0000(51) 9.9 3 10−14 B7 δH(2P1/2)/h 0.0000(39) 4.8 3 10−15 B8 δH(4P1/2)/h 0.0000(16) 7.6 3 10−15 B9 δH(2P3/2)/h 0.0000(39) 4.8 3 10−15 B10 δH(4P3/2)/h 0.0000(16) 7.6 3 10−15 B11 δH(8D3/2)/h 0.000 000(13) 2.6 3 10−16 B12 δH(12D3/2)/h 0.000 0000(40) 1.8 3 10−16 B13 δH(4D5/2)/h 0.000 00(17) 8.2 3 10−16 B14 δH(6D5/2)/h 0.000 000(58) 6.3 3 10−16 B15 δH(8D5/2)/h 0.000 000(22) 4.2 3 10−16 B16 δH(12D5/2)/h 0.000 0000(64) 2.8 3 10−16 B17 δD(1S1/2)/h 0.0(1.5) 4.5 3 10−13 B18 δD(2S1/2)/h 0.00(18) 2.2 3 10−13 B19 δD(4S1/2)/h 0.000(23) 1.1 3 10−13 B20 δD(8S1/2)/h 0.0000(49) 9.6 3 10−14 B21 δD(8D3/2)/h 0.000 0000(95) 1.8 3 10−16 B22 δD(12D3/2)/h 0.000 0000(28) 1.2 3 10−16 B23 δD(4D5/2)/h 0.000 00(15) 7.5 3 10−16 B24 δD(8D5/2)/h 0.000 000(19) 3.8 3 10−16 B25 δD(12D5/2)/h 0.000 0000(58) 2.5 3 10−16 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-15 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The line-shape model indicated that the two measured transition energies shifted up to h 3 40 kHz by quantum interference, which is much larger than the proton-radius discrepancy of h 3 9 kHz. More importantly, the two transitions shift in opposite directions. In fact, by constructing the hyperfine and fine-structure centroid energies from the measurementstheshiftscanceltoalargeextent.ThisledtothefinalMPQ result for the 2S1/2 −4P transition energy with u(ΔE/h) 2.3 kHz and a relative uncertainty of 3.7 3 10−12. In addition to the quantum in-terferencecorrections,Beyeretal.(2017) investigated 13othersystematic shifts and corrections. The first-order Doppler shift is negligible, but its h 3 2.1 kHz uncertainty is by far the largest contributor to the final uncertainty. 2. Measurement of the hydrogen two-photon 1S−3S transition The hydrogen 1S−3S transition energy was measured by Yost et al. (2016) at the MPQ and Fleurbaey et al. (2018) at the LKB. These new data are items A8 and A23 in Table X, respectively. The measurement un-certaintyoftheLKBgroupissignificantlysmallerthanthatobtainedatthe MPQ and, hence, we only describe details of the LKB experimental setup. The researchers at the LKB used two-photon spectroscopy. In this technique, the first-order Doppler shift is eliminated by having room-temperature atoms simultaneously absorb photons from counter-propagating laser beams. The measured transition energy has a five times smaller uncertainty than two older measurements of the same transition energy. The latter are listed as items A8 and A22 in Table X. Fleurbaey (2017) and Thomas et al. (2019) give more in-formation about the LKB measurement. A history of Doppler-free spectroscopy is given by Biraben (2019). The development of a continuous-wave laser source at 205 nm for the two-photon excitation by Galtier et al. (2015) contributed significantly to the fivefold uncertainty reduction by improving the signal-to-noise ratio compared to previous LKB experiments with a chopped laser source. The frequency of the 205 nm laser was de-termined with the help of a transfer laser, several Fabry-Perot cavities, and a femtosecond frequency comb whose repetition rate was ref-erenced to a Cs-fountain frequency standard. The laser frequency was scanned to excite the 1S1/2(f 1) −3S1/2(f 1) transition and the resonance was detected from the 656 nmradiationemitted bytheatomswhentheydecayfromthe3Stothe2P level. The well-known 1S and 3S hyperfine splittings were used to obtain TABLE IX. Correlation coefficients r(xi, xj) > 0.0001 among the input data for the hydrogen and deuterium energy levels given in Tables VIII and X. Coefficients r are strictly zero between input data An and Bm for positive integers n and m r(A1, A2) 0.1049 r(A1, A3) 0.2095 r(A1, A4) 0.0404 r(A2, A3) 0.0271 r(A2, A4) 0.0467 r(A3, A4) 0.0110 r(A6, A7) 0.7069 r(A10, A11) 0.3478 r(A10, A12) 0.4532 r(A10, A13) 0.1225 r(A10, A14) 0.1335 r(A10, A15) 0.1419 r(A10, A16) 0.0899 r(A10, A17) 0.1206 r(A10, A18) 0.0980 r(A10, A19) 0.1235 r(A10, A20) 0.0225 r(A10, A21) 0.0448 r(A11, A12) 0.4696 r(A11, A13) 0.1273 r(A11, A14) 0.1387 r(A11, A15) 0.1475 r(A11, A16) 0.0934 r(A11, A17) 0.1253 r(A11, A18) 0.1019 r(A11, A19) 0.1284 r(A11, A20) 0.0234 r(A11, A21) 0.0466 r(A12, A13) 0.1648 r(A12, A14) 0.1795 r(A12, A15) 0.1908 r(A12, A16) 0.1209 r(A12, A17) 0.1622 r(A12, A18) 0.1319 r(A12, A19) 0.1662 r(A12, A20) 0.0303 r(A12, A21) 0.0602 r(A13, A14) 0.5699 r(A13, A15) 0.6117 r(A13, A16) 0.1127 r(A13, A17) 0.1512 r(A13, A18) 0.1229 r(A13, A19) 0.1548 r(A13, A20) 0.0282 r(A13, A21) 0.0561 r(A14, A15) 0.6667 r(A14, A16) 0.1228 r(A14, A17) 0.1647 r(A14, A18) 0.1339 r(A14, A19) 0.1687 r(A14, A20) 0.0307 r(A14, A21) 0.0612 r(A15, A16) 0.1305 r(A15, A17) 0.1750 r(A15, A18) 0.1423 r(A15, A19) 0.1793 r(A15, A20) 0.0327 r(A15, A21) 0.0650 r(A16, A17) 0.4750 r(A16, A18) 0.0901 r(A16, A19) 0.1136 r(A16, A20) 0.0207 r(A16, A21) 0.0412 r(A17, A18) 0.1209 r(A17, A19) 0.1524 r(A17, A20) 0.0278 r(A17, A21) 0.0553 r(A18, A19) 0.5224 r(A18, A20) 0.0226 r(A18, A21) 0.0449 r(A19, A20) 0.0284 r(A19, A21) 0.0566 r(A20, A21) 0.1412 r(A24, A25) 0.0834 r(B1, B2) 0.9946 r(B1, B3) 0.9937 r(B1, B4) 0.9877 r(B1, B5) 0.6140 r(B1, B6) 0.6124 r(B1, B17) 0.9700 r(B1, B18) 0.9653 r(B1, B19) 0.9575 r(B1, B20) 0.5644 r(B2, B3) 0.9937 r(B2, B4) 0.9877 r(B2, B5) 0.6140 r(B2, B6) 0.6124 r(B2, B17) 0.9653 r(B2, B18) 0.9700 r(B2, B19) 0.9575 r(B2, B20) 0.5644 r(B3, B4) 0.9869 r(B3, B5) 0.6135 r(B3, B6) 0.6119 r(B3, B17) 0.9645 r(B3, B18) 0.9645 r(B3, B19) 0.9567 r(B3, B20) 0.5640 r(B4, B5) 0.6097 r(B4, B6) 0.6082 r(B4, B17) 0.9586 r(B4, B18) 0.9586 r(B4, B19) 0.9704 r(B4, B20) 0.5605 r(B5, B6) 0.3781 r(B5, B17) 0.5959 r(B5, B18) 0.5959 r(B5, B19) 0.5911 r(B5, B20) 0.3484 r(B6, B17) 0.5944 r(B6, B18) 0.5944 r(B6, B19) 0.5896 r(B6, B20) 0.9884 r(B7, B8) 0.0001 r(B9, B10) 0.0001 r(B11, B12) 0.6741 r(B11, B21) 0.9428 r(B11, B22) 0.4803 r(B12, B21) 0.4782 r(B12, B22) 0.9428 r(B13, B14) 0.2061 r(B13, B15) 0.2391 r(B13, B16) 0.2421 r(B13, B23) 0.9738 r(B13, B24) 0.1331 r(B13, B25) 0.1352 r(B14, B15) 0.2225 r(B14, B16) 0.2253 r(B14, B23) 0.1128 r(B14, B24) 0.1238 r(B14, B25) 0.1258 r(B15, B16) 0.2614 r(B15, B23) 0.1309 r(B15, B24) 0.9698 r(B15, B25) 0.1459 r(B16, B23) 0.1325 r(B16, B24) 0.1455 r(B16, B25) 0.9692 r(B17, B18) 0.9955 r(B17, B19) 0.9875 r(B17, B20) 0.5821 r(B18, B19) 0.9874 r(B18, B20) 0.5821 r(B19, B20) 0.5774 r(B21, B22) 0.3407 r(B23, B24) 0.0729 r(B23, B25) 0.0740 r(B24, B25) 0.0812 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-16 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr the final transition energy between the hyperfine centroids with u(ΔE/h) 2.6 kHz and ur 8.9 3 10−13. The distribution of velocities of the atoms in the room-tem-perature hydrogen beam led to a second-order Doppler shift of roughly −140 kHz, or 500 parts in 1013, and was the largest systematic effectintheexperiment.Toaccountforthisshift,thevelocitydistribution ofthehydrogenatomswasmappedoutbyapplyingasmallmagneticflux density B perpendicular to the hydrogen beam. In addition to Zeeman shifts,thefluxdensityleadstoStarkshiftsof3Shyperfinestatesbymixing with the nearby 3P1/2 level via the motional electric field perceived by the atoms. Both this motional Stark shift and the second-order Doppler shift have a quadratic dependence on velocity. Then the LKB researchers fit resonance spectra obtained at different B to a line-shape model averaged overamodifiedMaxwellianvelocitydistributionofaneffusivebeam.The fit gives the temperature of the H beam, distortion parameters from a Maxwellian distribution, and a line position with the second-order Doppler shift removed. Finally, the observed line position was corrected for light shifts due to the finite 205 nm laser intensity and pressure shifts due to elastic collisions with background hydrogen molecules. Light shifts increase the apparent transition energy by up to h 3 10 kHz depending on the laser intensity in the data runs, while pressure shifts decrease this energy by slightly less than h 3 1 kHz/(10−5 hPa). Pressures up to 20 3 10−5 hPa were used in the experiments. Quantum interference effects, mainly from the 3D state, are small for the 1S −3S transition and led to a correction of h 3 0.6(2) kHz. 3. Measurement of the hydrogen 2S−2P Lamb shift The hyperfine-resolved hydrogen 2S1/2(f 0) −2P1/2 3 (f 1) transition energy or Lamb shift was measured byBezginovetal. (2019) at TABLE X. Summary of measured transition energies ΔEX(i −i′) between states i and i′ for electronic hydrogen (X H) and electronic deuterium (X D) considered as input data for the determination of the Rydberg constant R∞. The label in the first column is used in Table IX to list correlation coefficients among these data and in Table XXIII for observational equations. Columns two and three give the reference and an abbreviation of the name of the laboratory in which the experiment has been performed. An extensive list of abbreviations is found at the end of this report Reported value Rel. stand. Reference Lab. Energy interval(s) ΔE/h (kHz) uncert. ur A1 Weitz et al. (1995) MPQ ΔEH(2S1/2 −4S1/2) −1 4 ΔEH(1S1/2 −2S1/2) 4 797 338(10) 2.1 3 10−6 A2 ΔEH(2S1/2 −4D5/2) −1 4 ΔEH(1S1/2 −2S1/2) 6 490 144(24) 3.7 3 10−6 A3 ΔED(2S1/2 −4S1/2) −1 4 ΔED(1S1/2 −2S1/2) 4 801 693(20) 4.2 3 10−6 A4 ΔED(2S1/2 −4D5/2) −1 4 ΔED(1S1/2 −2S1/2) 6 494 841(41) 6.3 3 10−6 A5 Parthey et al. (2010) MPQ ΔED(1S1/2 −2S1/2) −ΔEH(1S1/2 −2S1/2) 670 994 334.606(15) 2.2 3 10−11 A6 Parthey et al. (2011) MPQ ΔEH(1S1/2 −2S1/2) 2 466 061 413 187.035(10) 4.2 3 10−15 A7 Matveev et al. (2013) MPQ ΔEH(1S1/2 −2S1/2) 2 466 061 413 187.018(11) 4.4 3 10−15 A8 Yost et al. (2016) MPQ ΔEH(1S1/2 −3S1/2) 2 922 743 278 659(17) 5.8 3 10−12 A9 Beyer et al. (2017) MPQ ΔEH(2S1/2 −4P) 616 520 931 626.8(2.3) 3.7 3 10−12 A10 de Beauvoir et al. (1997) LKB/ ΔEH(2S1/2 −8S1/2) 770 649 350 012.0(8.6) 1.1 3 10−11 A11 SYRTE ΔEH(2S1/2 −8D3/2) 770 649 504 450.0(8.3) 1.1 3 10−11 A12 ΔEH(2S1/2 −8D5/2) 770 649 561 584.2(6.4) 8.3 3 10−12 A13 ΔED(2S1/2 −8S1/2) 770 859 041 245.7(6.9) 8.9 3 10−12 A14 ΔED(2S1/2 −8D3/2) 770 859 195 701.8(6.3) 8.2 3 10−12 A15 ΔED(2S1/2 −8D5/2) 770 859 252 849.5(5.9) 7.7 3 10−12 A16 Schwob et al. (1999) LKB/ ΔEH(2S1/2 −12D3/2) 799 191 710 472.7(9.4) 1.2 3 10−11 A17 SYRTE ΔEH(2S1/2 −12D5/2) 799 191 727 403.7(7.0) 8.7 3 10−12 A18 ΔED(2S1/2 −12D3/2) 799 409 168 038.0(8.6) 1.1 3 10−11 A19 ΔED(2S1/2 −12D5/2) 799 409 184 966.8(6.8) 8.5 3 10−12 A20 Bourzeix et al. (1996) LKB ΔEH(2S1/2 −6S1/2) −1 4 ΔEH(1S1/2 −3S1/2) 4 197 604(21) 4.9 3 10−6 A21 ΔEH(2S1/2 −6D5/2) −1 4 ΔEH(1S1/2 −3S1/2) 4 699 099(10) 2.2 3 10−6 A22 Arnoult et al. (2010) LKB ΔEH(1S1/2 −3S1/2) 2 922 743 278 678(13) 4.4 3 10−12 A23 Fleurbaey et al. (2018) LKB ΔEH(1S1/2 −3S1/2) 2 922 743 278 671.5(2.6) 8.9 3 10−13 A24 Berkeland, Hinds, and Boshier (1995) Yale ΔEH(2S1/2 −4P1/2) −1 4 ΔEH(1S1/2 −2S1/2) 4 664 269(15) 3.2 3 10−6 A25 ΔEH(2S1/2 −4P3/2) −1 4 ΔEH(1S1/2 −2S1/2) 6 035 373(10) 1.7 3 10−6 A26 Hagley and Pipkin (1994) Harvard ΔEH(2S1/2 −2P3/2) 9 911 200(12) 1.2 3 10−6 A27 Newton, Andrews, and Unsworth (1979) Sussex ΔEH(2P1/2 −2S1/2) 1 057 862(20) 1.9 3 10−5 A28 Lundeen and Pipkin (1981) Harvard ΔEH(2P1/2 −2S1/2) 1 057 845.0(9.0) 8.5 3 10−6 A29 Bezginov et al. (2019) York ΔEH(2P1/2 −2S1/2) 1 057 829.8(3.2) 3.0 3 10−6 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-17 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr York University to help resolve the proton-radius puzzle. This new datum is item A29 in Table X. The Dirac equation predicts that the 2S1/2 and 2P1/2 energy levels in hydrogen are degenerate, but because of vacuum fluctuations and vacuum polarization, the 2S1/2 level lies h 3 1058 MHz abovethe2P1/2 levelandh 3 9911 MHz belowthe2P3/2 level. In fact, historically the discovery of the Lamb shift led to the de-velopment of QED. Previous determinations of the Lamb shift are items A27 and A28 in Table X. A determination of the 2S1/2 −2P3/2 transition energy is given as item A26. The York researchers had to overcome the constraints that arise from the 1.6 ns natural lifetime of the 2P1/2 state and the minimal dimensions of the ≈1 GHz microwave cavities of several centimeters. They solved this by preparing fast mono-energetic beams of 2S1/2(f 0) hydrogen atoms with velocities up to 0.32 cm/ns or 1% of thespeedoflightinvacuum.Thisbeamwasobtainedbypassingprotons with a kinetic energy up to 55 keV through a H2 molecular gas and by rejecting H atoms in unwanted states, especially those in the three metastable 2S1/2(f 1) Zeeman states. The York researchers then used a modified version of the separated oscillatory field method to measure the Lamb shift, as described by Vutha and Hessels (2015). In this design, the frequencies of the microwave radiation applied to the two spatially separated field regions have a fixed small frequency difference and only the carrier frequency is scanned. Crucial for the effectiveness of the method is that the researchers could alternate between whether the atoms encounter the lower or higher frequency radiation first. This change occurred every few seconds. Also, part of the apparatus could be physically rotated by 180°, done about once per hour, so that the atoms encounter the separate oscillatory fields in reverse order. Data from these four cases were used to eliminate shifts due to imperfections in state preparation and microwave cavities. The frequency difference of the radiation in the two field regions leads to a time-dependent signal of 2S1/2 population that oscillates at the difference frequency with a phase offset that is proportional to the difference of the applied carrier frequency and the frequency equivalent of the Lamb shift. The sign of the slope depends on whether the atoms encounter the lower or higher frequency radiation first. The number of remaining H atoms in the 2S1/2 state at the end of the beam line was measured by applying an electric field in a detection zone and collecting the 121.6 nm Lyman-α photon emitted by the atoms. Data were obtained with 18 different combinations of beam velocity, strength of the 910 MHz microwave field, and distance between the separated field regions. No dependence on these pa-rameters was observed. The final h 3 3.2 kHz uncertainty for the 2S1/2(f 0) −2P1/2 (f 1) transition energy, which corresponds to ur 3.6 3 10−6, arises from an h 3 1.4 kHz statistical uncertainty and uncertainties from several systematic effects: h 3 2.3 kHz from the AC Stark shift, h 3 1.5 kHz from the measurement of phase, and h 3 1.0 kHz from the second-order Doppler shift. Quantum interference from hy-perfine states with n ≥3 had no discernible effect on the measurement. Marsman et al. (2018) reevaluated the experiment of Lundeen and Pipkin (1981, 1986), input datum A28 in Table X. They suggested that the transition energy should be reduced by h 3 6 kHz and the uncertainty increased from h 3 9 kHz to h 3 20 kHz. For the 2018 CODATA adjustment, the results of Lundeen and Pipkin (1981, 1986) have not been modified. VIII. Electron Magnetic-Moment Anomaly The interaction of the magnetic moment of a charged lepton ℓin a magnetic flux density (or magnetic field) B is described by the Hamiltonian H −μℓ· B, with μℓ gℓ e 2mℓ s, (44) where ℓ e ± , μ ± , or τ ± , gℓis the g-factor, with the convention that it has the same sign as the charge of the particle, e is the positive elementary charge, mℓis the lepton mass, and s is its spin. Since the FIG. 4. Experimental hydrogen and deuterium transition energies and differences of transition energies (yellow-filled red circles with red error bars) used as input data in the 2018 least-squares adjustment. For all data, the 2018 adjusted value of the transition energy has been subtracted. Data new to this adjustment have been indicated with the abbreviation of the name of the laboratory and year of publication in parentheses. An extensive list of abbreviations is found at the end of this report. Panel (a) shows data for the 1S−2S transition with one-standard-deviation un-certainties on the order of tens of h 3 Hz. Panel (b) shows the remaining input data with uncertainties on the scale of tens of h 3 kHz. Labels on the left-hand side of the figure group data belonging to the same class of transitions, i.e., nℓ−n′ℓ′ transitions. Input data without such label correspond to data that depend on (weighted) differences of four energy levels. Finally, the yellow-filled black circles with black error bars are the fitted values and their uncertainties. In the figure, the uncertainties of the input data have not been multiplied by 1.6, the expansion factor in this adjustment to make the H and D spectroscopic and muonic Lamb-shift data consistent. Fitted values are for the data when multiplied by this factor. Blue and black labels An on the right-hand side of the figure correspond to hydrogen and deuterium entries in Table X, respectively. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-18 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr spin has projection eigenvalues of sz ± Z/2, the magnitude of a magnetic moment is μℓ gℓ 2 eZ 2mℓ . (45) The lepton magnetic-moment anomaly aℓis defined by the relationship |gℓ| ≡2(1 + aℓ), (46) based on the Dirac g-value of −2 and +2 for the negatively and positively charged lepton ℓ, respectively. The Bohr magneton is defined as μB eZ 2me , (47) and the theoretical expression for the anomaly of the electron ae(th) is ae(th) ae(QED) + ae(weak) + ae(had), (48) where terms denoted by “QED,” “weak,” and “had” account for the purely quantum electrodynamic, predominantly electroweak, and predominantly hadronic (that is, strong interaction) contributions, respectively. The QED contribution may be written as ae(QED)  ∞ n1 C(2n) e α π n , (49) where the index n corresponds to contributions with n virtual photons and C(2n) e A(2n) 1 + A(2n) 2 (xeμ) + A(2n) 2 (xeτ) + · · · (50) with mass-independent coefficients A(2n) 1 and functions A(2n) 2 (x) evaluated at mass ratio x xeX ≡me/mX ≪1 for lepton X μ or τ. For n 1, we have A(2) 1 1/2, (51) and function A(2) 2 (x) 0, while for n > 1 coefficients A(2n) 1 include vacuum-polarization corrections with virtual electron/positron pairs. In fact, A(4) 1 −0.328 478 965 579 193 . . . , (52) A(6) 1 1.181 241 456 587 . . . , (53) A(8) 1 −1.912 245 764 . . . , (54) A(10) 1 6.675(192). (55) The functions A(2n) 2 (x) for n > 1 are vacuum-polarization corrections due to heavier leptons. For x →0, we have A(4) 2 (x) x2/45 + O(x4) and A(6) 2 (x) x2(b0 + b1lnx) + O(x4) with b0 0.593274 . . . and b1 23/135 (Laporta, 1993; Laporta and Remiddi, 1993). The O(x4) contributions are known and included in the calculations but not reproduced here. The functions A(8) 2 (x) and A(10) 2 (x) are also O(x2) for small x, but not reproduced here (Kurz et al., 2014a; Aoyama et al., 2015). Currently, terms with n > 5 and vacuum-polarization corrections that depend on two lepton mass ratios can be neglected. Table XII summarizes the relevant QED coefficients and summed C(2n) e with their one-standard-deviation uncertainties where appropriate as used in the 2018 CODATA adjustment. Additional references to the original literature can be found in descriptions of previous CODATA adjustments. It is worth noting that since 2014 the coefficient A(8) 1 has been evaluated by Laporta (2017), while the value for A(10) 2 has been updated by Aoyama, Kinoshita, and Nio (2018). Recently, the value for A(10) 2 has been refined by Aoyama, Kinoshita, and Nio (2019), although Volkov (2019) found a value for A(10) 2 , absent lepton loop contributions, that is significantly discrepant with that based on results in Aoyama, Kinoshita, and Nio (2018, 2019). Both Aoyama, Kinoshita, and Nio (2019) and Volkov (2019) were published after our closing date. The electroweak contribution is ae(weak) 0.030 53(23) 3 10−12 (56) and is calculated as discussed in the 1998 CODATA adjustment, but with the 2018 values of the Fermi coupling constant GF/(Zc)3 and the weak mixing angle θW (Tanabashi et al., 2018). Jegerlehner (2019) has provided updates to hadronic contri-butions to the electron anomaly. Currently, four such contributions have been considered. They are ae(had) aLO,VP e (had) + aNLO,VP e (had) + aNNLO,VP e (had) + aLL e (had) (57) corresponding to leading-order (LO), next-to-leading-order (NLO), and next-to-next-to-leading-order (NNLO) hadronic vacuum-polarization corrections and a hadronic light-by-light (LL) scattering term, re-spectively. Contributions are determined from analyzing experimental cross sections for electron-positron annihilation into hadrons and tau-lepton-decay data. The values in the 2018 adjustment are aLO,VP e (had) 1.849(11) 3 10−12, aNLO,VP e (had) −0.2213(12) 3 10−12, aNNLO,VP e (had) 0.028 00(20) 3 10−12, aLL e (had) 0.0370(50) 3 10−12 (58) leading to the total hadronic contribution ae(had) 1.693(12) 3 10−12. (59) A first-principle lattice quantum chromodynamics (QCD) evaluation of the leading-order hadronic correction aLO,VP e (had) to TABLE XI. Twenty-five of the 75 adjusted constants in the 2018 CODATA least-squares minimization. These variables account for missing contributions to the theoretical description of the electronic hydrogen (H) and deuterium (D) energy levels. Their input data are given in Table VIII Atom Level nℓj H δH 1S1/2, 2S1/2, 3S1/2, 4S1/2, 6S1/2, 8S1/2, 2P1/2, 2P3/2, 4P1/2, 4P3/2, 4D5/2, 6D5/2, 8D3/2, 8D5/2, 12D3/2, 12D5/2 D δD 1S1/2, 2S1/2, 4S1/2, 8S1/2, 4D5/2, 8D3/2, 8D5/2, 12D3/2, 12D5/2 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-19 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr the electron anomaly was published in 2018 (Borsanyi et al., 2018). The value is aLO,VP e (had) 1.893(26)(56) 3 10−12, (60) where the first and second numbers in parentheses correspond to the statistical and systematic uncertainty, respectively. The systematic uncertainty is dominated by finite-volume artifacts. The combined uncertainty is six times larger than that obtained by analyzing electron-positron scattering data. Figure 5 shows a graphical representation of 14 contributions to the electron anomaly. The QED corrections decrease roughly ex-ponentially in size with order n for both mass-independent and -dependent contributions. Contributions from virtual loops con-taining τ leptons are mostly negligible. The theoretical uncertainty of the electron anomaly (apart from uncertainty in the fine-structure constant) is dominated by two contributions: the mass-independent n 5 QED correction and the hadronic contribution. In fact, its value is u[ae(th)] 0.018 3 10−12 1.5 3 10−11ae, (61) and is shown in Fig. 5 as well. This theoretical uncertainty is significantly smaller than the uncertainty 2.4 3 10−10ae of the best by far experimental value for the electron anomaly from Hanneke, Fogwell, and Gabrielse (2008). Consequently, the relative uncertainty of the fine-structure constant based on only this experimental input datum would be the same as that for this experiment. Atom-recoil experiments, discussed in Sec. X, form a second competitive means to determine α. For the least-squares adjustment, we use the observational equations ae(exp) ≐ae(th) + δth(e) (62) and δe ≐δth(e) (63) with additive adjusted constant δth(e). Input datum ae(exp) is from Hanneke, Fogwell, and Gabrielse (2008), while input datum δe 0 with u[δe] 0.018 3 10−12 accounts for the uncertainty of the the-oretical expression. The input data are entries D1 and D2 in Table XXI. Relevant observational equations are found in Table XXVI. IX. Relative Atomic Masses In this section, we discuss the input data that determine the relative atomic masses of various nuclei and atoms relevant to the adjustment. Specifically, we focus on light nuclei, i.e., neutron n, proton p, deuteron d, triton t, helion h, and the alpha particle α. These are the nuclei of hydrogen 1H, deuterium 2H, tritium 3H, helium-3 3He, and helium-4 4He, respectively. This section also summarizes corresponding input data for the atoms 12C, 28Si, 87Rb, and 133Cs as they are relevant for the determination of the mass of the electron and the fine-structure constant discussed in Sec. VI. The input data for the mass of the muon are discussed in Sec. XVII. Table XIII gives the relative atomic masses of the neutron and six neutral atoms that are used as input data in the 2018 CODATA adjustment. The carbon-12 relative atomic mass is by definition simply the number 12. The remaining values have been taken from the 2016 Atomic Mass Evaluation (Huang et al., 2017; Wang et al., 2017). Task Group and Atomic-Mass-Data-Center (AMDC) member M. Wang supplied extra digits to reduce rounding errors. Correlation coefficients with r(Xi, Xj) > 0.0001 among these relative atomic masses are given in Table XIV. These input data are also given as items D5, D6, D11, and D18–D20 in Table XXI. The relative atomic masses of n, 87Rb, and 133Cs are adjusted constants and their observational equations are simply Ar(X) ≐Ar(X). On the other hand, we find it more convenient to use the relative atomic masses of the proton p, the alpha particle α, and the hydrogenic 28Si13+ as adjusted constants, rather than those of neutral 1H, 4He, and 28Si. Since the mass of an atom or atomic ion is the sum of the nuclear mass and the masses of its electrons minus the mass equivalent of the binding energy of the electrons, the observational equation for the relative atomic mass of a neutral atom X in terms of that of ion Xn+ in charge state n 1, 2, . . . is Ar(X) ≐Ar(Xn+) + nAr(e) −ΔEB(Xn+) muc2 , (64) where Ar(e) is the relative atomic mass of the electron and ΔEB(Xn+) > 0 is the binding or removal energy needed to remove n electrons from the neutral atom. This binding energy is the sum of the electron ionization energies EI(Xi+) of ion Xi+. That is, ΔEB(Xn+)  n−1 i0 EI(Xi+). (65) For a bare nucleus n Z, while for a neutral atom n 0 and ΔEB(X0+) 0. With our definition of observational equations, the quantities Ar(e) and ΔEB(Xn+) are adjusted constants. In addition to the input data in Table XIII, we also use mea-surements of four cyclotron frequency ratios as input data to further TABLEXII. Coefficientsfor theQED contributions tothe electronanomaly. ThecoefficientsA(2n) 1 and functions A(2n) 2 (x), evaluated at massratiosxeμ me/mμ and xeτ me/mτ for the muon and tau lepton, respectively; summed values C(2n) e , based on values for lepton mass ratios from the 2018 CODATA adjustment, are listed as accurately as needed for the tests described in this article. Missing values indicate that their contribution to the electron anomaly is negligible n A(2n) 1 A(2n) 2 (xeμ) A(2n) 2 (xeτ) C(2n) e 1 1/2 0 0 0.5 2 −0.328 478 965 579 193 . . . 5.197 386 74(23) 3 10−7 1.837 90(25) 3 10−9 −0.328 478 444 00 3 1.181 241 456 587 . . . −7.373 941 69(24) 3 10−6 −6.582 73(79) 3 10−8 1.181 234 017 4 −1.912 245 764 . . . 9.161 970 80(33) 3 10−4 7.428 93(88) 3 10−6 −1.911 322 138 91(88) 5 6.675(192) −0.003 82(39) 6.67(19) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-20 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr constrain the relative atomic mass of the proton and determine those of the remaining three light nuclei: the deuteron, triton, and helion. These measurements rely on the fact that ions Xn+ with charge ne in a homogeneous flux density or magnetic field of strength B undergo circular motion with a cyclotron frequency ωc(Xn+) neZB/m(Xn+) that can be accurately measured. With the right experimental design, ratios of cyclotron frequencies for ions Xn+ and Yp+ in the same magnetic-field environment then satisfy ωc(Xn+) ωc(Yp+) nAr(Yp+) pAr(Xn+) (66) independent of field strength. For ease of reference, the four cyclotron frequency ratios are summarized in Table XXI as items D14–D17. Observational equations are given in Table XXVI. The first of these measurements is relevant for the determination of the relative atomic mass of the proton. In 2017, the ratio of cy-clotron frequencies of the proton and the 12C6+ ion, ωc(12C6+)/ωc(p), was measured at a Max-Planck Institute in Heidelberg, Germany (MPIK) (Heiße et al., 2017). Their ratio has a relative uncertainty of 3.3 3 10−11, mostly limited by residual magnetic-field in-homogeneities in the multi-zone cryogenic Penning trap. Optimized for measuring the cyclotron frequencies of light ions, the trap has three separate but connected areas that are coaxial with an applied magnetic field. A single 12C6+ ion and a proton are then shuttled in and out of the central measurement trap. Heiße et al. (2017) recognized that their value of Ar(p) does not agree with that implied by Ar(1H) in Table XIII. As a check on their experiment, they carried out measurements on other ions but found results consistent with literature values. Figure 6 gives a graphical representation of the two discrepant input data as well as our fitted values for these data. Our predicted value for Ar(1H) is significantly smaller than that from the 2016 Atomic Mass Evaluation. For our 2018 CODATA adjustment, we have applied an expansion factor of 1.7 to the uncertainties of these two input data, also shown in the figure, in order to obtain a consistent least-squares adjustment. The 2014 cyclotron-frequency-ratio measurement for the deuteron d and 12C6+ essentially determines Ar(d). Reported by Zafonte and Van Dyck (2015) and identified with UWash-15, the result was already discussed in the 2014 CODATA adjustment. The measurement has a relative uncertainty of 2.0 3 10−11 and agrees with a preliminary value (Van Dyck et al., 2006) based on only 30% of the data. The 2016 AMDC evaluation of Ar(2H) is not included in our CODATA adjustment, as it was based on this preliminary determination. The final two cyclotron-frequency-ratio measurements de-termine the triton and helion relative atomic masses, Ar(t) and Ar(h), respectively. These masses are primarily determined by the ratios ωc(t)/ωc(3He+) and ωc(HD+)/ωc(3He+), both of which were mea-sured at Florida State University, Florida, USA. The ratios have been reported by Myers et al. (2015) and Hamzeloui et al. (2017), re-spectively. The former was already discussed in the 2014 CODATA adjustment. See also the recent review by Myers (2019). The quantity ωc(t)/ωc(3He+) is not directly measured by Myers et al. (2015), but determined from the quotient of ratios ωc(HD+)/ωc(3He+) and ωc(HD+)/ωc(t). While ur 4.8 3 10−11 for each of these directly measured ratios, ur 2.4 3 10−11 for their quotient because of a cancellation of several uncertainty components from systematic effects common to both. The 2016 AMDC evaluations of Ar(3H) and Ar(3He) are not included in this CODATA adjustment. They were primarily de-termined by ωc(HD+)/ωc(3He+) and ωc(HD+)/ωc(t) from Myers TABLE XIII. Relative atomic masses used as input data in the 2018 CODATA adjustment and taken from the 2016 Atomic-Mass-Data-Center (AMDC) mass evaluation (Huang et al., 2017; Wang et al., 2017). Correlations among these data are given in Table XIV Relative atomic Relative standard Atom massa Ar(X) uncertainty ur n 1.008 664 915 82(49) 4.9 3 10−10 1H 1.007 825 032 241(94) 9.3 3 10−11 4He 4.002 603 254 130(63) 1.6 3 10−11 12C 12 exact 28Si 27.976 926 534 99(52) 1.9 3 10−11 87Rb 86.909 180 5312(65) 7.4 3 10−11 133Cs 132.905 451 9610(86) 6.5 3 10−11 aThe relative atomic mass Ar(X) of particle X with mass m(X) is defined by Ar(X) m(X)/mu, where mu m(12C)/12 is the atomic mass constant. TABLE XIV. Correlation coefficients r(Xi, Xj) > 0.0001 among the input data for the relative atomic masses Ar(X) given in Table XIII based on covariances from the 2016 AMDC mass evaluation available in Supplementary files at web/masseval.html or at r(n, 1H) −0.1340 r(n, 28Si) −0.0198 r(n, 87Rb) −0.0070 r(n, 133Cs) −0.0070 r(1H, 28Si) 0.1934 r(1H, 87Rb) 0.0657 r(1H, 133Cs) 0.0602 r(28Si, 87Rb) 0.0495 r(28Si, 133Cs) 0.0402 r(87Rb, 133Cs) 0.1004 FIG. 5. Fourteen fractional contributions to the theoretical anomaly of the electron |δae|/ae(th). QED contributions are due to the mass-independent A(2n) 1 (yellow-filled black circles), to the muon-dependent A(2n) 2 (xeμ) (yellow-filled red circles), and to the tau-dependent A(2n) 2 (xeτ) (yellow-filled blue circles) corrections, respectively. Weak and hadronic corrections are also shown. The horizontal orange line shows the theoretical relative uncertainty of ae(th). The 2018 CODATA values for the fine-structure constant and lepton mass ratios are used here. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-21 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr et al. (2015). The former ratio is now superseded by the twice as accurate corresponding value from Hamzeloui et al. (2017). Binding energies are most accurately tabulated in terms of wave number equivalents ΔEB(Xn+)/hc but are needed as their relative atomic mass unit equivalents ΔEB(Xn+)/muc2. Given that the Ryd-berg energy hcR∞ α2mec2/2, the last term in Eq. (64) is then re-written as ΔEB(Xn+) muc2 α2Ar(e) 2R∞ ΔEB(Xn+) hc . (67) Bindingenergiesfor 1H+, 3He+, 4He2+, 12C5+, 12C6+,and 28Si13+ areused inthisCODATAadjustment.Theirvaluesaredeterminedorconstructed from ionization energies in Table XV taken from the 2018 NIST Atomic Spectra Database (ASD) at The rel-evantbindingenergiesarelistedinTableXXIasitemsD8,D12,andD21-24. Corresponding observational equations are given in Table XXVI. The uncertainties of the ionization data are sufficiently small that correlations among them or with any other data used in the 2018 adjustment are inconsequential. Nevertheless, the binding or removal energies of 12C5+ and 12C6+ are highly correlated with a correlation coefficient of 0.999 98, due to the uncertainties in the common ionization energies at lower stages of ionization. The observational equations for binding energies are simply ΔEB(Xn+)/hc ≐ΔEB(Xn+)/hc, (68) thereby allowing all binding-energy uncertainties and covariances to be properly taken into account. A word on the relative atomic mass of the molecular ion HD+ is in order. Its value helps determine the relative atomic mass of the 3He nucleus. We take Ar(HD+) Ar(p) + Ar(d) + Ar(e) −ΔEI(HD+) muc2 , (69) and have used the wave number equivalent of the ionization energy of the HD+ ion, ΔEI(HD+)/hc, as an adjusted constant whose value is constrained by the measurement or input datum ΔEI(HD+)/hc 13 122 468.415(6) m−1 (70) from Liu et al. (2010) and Sprecher et al. (2010). This input datum is item D25 in Table XXI. X. Atom-Recoil Measurements Atom-recoil measurements with rubidium and cesium atoms from the stimulated absorption and emission of photons are relevant for the CODATA adjustment as they determine the electron mass, the atomic mass constant, and the fine-structure constant (Peters et al., 1997; Young, Kasevich, and Chu, 1997; Mohr and Taylor, 2000). This can be understood as follows. First and foremost, recoil measure-ments determine the mass m(X) of a neutral atom X in kg using interferometers with atoms in superpositions of momentum states and taking advantage of the fact that photon energies can be precisely measured. Equally precise photon momenta p follow from their dispersion or energy-momentum relation E pc. In practice, Bloch oscillations are used to transfer a large number of photon momenta to the atoms in order to improve the sensitivity of the measurement (Clad´ e, 2015; Estey et al., 2015). Before the adoption of the revised SI on 20 May 2019, these experiments only measured the ratio h/m(X), since the Planck constant h was not an exactly defined constant. Second, atom-recoil measurements are a means to determine the atomic mass constant, mu m(12C)/12, and the mass of the electron, me, in kg. This follows, as many relative atomic masses Ar(X) m(X)/mu of atoms X are well known. For 87Rb and 133Cs, the relative atomic masses have a relative uncertainty smaller than 1 3 10−10 from the 2016 recommended values of the AMDC (see Table XIII). The relative atomic mass of the electron can be determined even more precisely with spin-precession and cyclotron-frequency-ratio measure-ments on hydrogenic 12C5+ and 28Si13+ as discussed in Sec. XI. We thus have mu m(X)/Ar(X) (71) and me Ar(e) Ar(X) m(X) (72) from a measurement of the mass of atom X. Finally, the fine-structure constant follows from the observation that the Rydberg constant R∞ α2mec/2h has a relative standard uncertainty of 1.9 3 10−12 based on spectroscopy of atomic hydrogen discussed in Sec. VII. The expression for R∞can be rewritten as α   2hcR∞ m(X)c2 Ar(X) Ar(e)  (73) and a value of α with a competitive uncertainty can be obtained from a measurement of m(X). Two m(X) measurements, represented by values for h/m(X), are input data in the current least-squares adjustment: A mass for 87Rb measured at the LKB, France by Bouchendira et al. (2011) and a mass for 133Cs measured at the University of California at Berkeley, USA by Parker et al. (2018). The rubidium mass was already available for previous adjustments, while this value for the cesium mass is a new input datum. The results are items D3 and D4 in Table XXI and satisfy the relevant observational equations in Table XXVI. FIG. 6. Input data for the determination of the relative atomic mass of the proton p. The input data (yellow-filled red circles with red error bars) and fitted values (yellow-filled black circles with black error bars) of the cyclotron frequency ratio of 12C 6+ and p and the relative atomic mass of the hydrogen atom Ar(1H) are shown. Error bars correspond to one-standard-deviation uncertainties. Datum X is shifted by the fitted value 〈X〉and normalized by the standard uncertainty of the input datum. Thus, fitted values shift to zero and input data become normalized residuals. Dashed orange lines are the standard uncertainties of the input data multiplied by the 1.7 expansion factor that ensures a consistent fit. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-22 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The values of α inferred from the two atom-recoil measurements are shown in Fig. 2, together with that inferred from an electron magnetic-moment anomaly ae measurement. Their comparison provides a useful test of the QED-based determination of ae and is discussed in Sec. IV.D. The new University-of-California-at-Berkeley value of m(133Cs) has ur 4.0 3 10−10 and currently provides a value of α with the smallest uncertainty. Thirteen systematic effects were investigated and included in the uncertainty budget. In parts in 1010, the net correction from systematic effects is −91.6(2.4). The two largest systematic corrections by far are −35.8(4) in parts in 1010 from acceleration gradients and −52.0(6) in parts in 1010 from wave front curvature and the Gouy phase of their Gaussian laser beams. The relative statistical uncertainty is 3.2 3 10−10. See also the review by Yu et al. (2019). Generalizations of the Gouy phase are of particular interest in atom-recoil experiments. In efforts to improve their rubidium ap-paratus, the researchers at LKB realized that small-scale intensity fluctuations in laser beams at the atomic positions lead to additional contributions to the Gouy phase (Bade et al., 2018; Clad´ e et al., 2019). In fact, in the new apparatus they expect to study this systematic effect in detail. Clad´ e et al. (2019) also concluded that the 2011 evaluation remains the most accurate determination of m(87Rb), unaffected by generalizations of the Gouy phase. Acknowledging the insights of Bade et al. (2018), Parker et al. (2018) at Berkeley realized that their relevant laser propagates a considerable distance before reaching the cesium atoms and small-scale intensity fluctuations smooth out, thereby significantly reducing the size of the effect. XI. Atomic g-Factors in Hydrogenic 12C and 28Si ions The most accurate value for the relative atomic mass of the electron is obtained from measurements of the ratio of spin-precession and cyclotron frequencies in hydrogenic carbon and silicon and theoretical expressions for the g-factors of their bound electron. See, for example, the recent analysis by Zatorski et al. (2017). These measurements also play an important role in de-termining the fine-structure constant using atom-recoil experi-ments discussed in Sec. X. For a hydrogenic ion X in its electronic ground state 1S1/2 and with a spinless nucleus, the Hamiltonian in an applied magnetic flux density B is H −ge(X) e 2me J · B, (74) where J is the electron angular momentum and ge(X) is the bound-state g-factor for the electron. The electron angular momentum projection is Jz ± Z/2 along the direction of B, so the energy splitting between the two levels is ΔE |ge(X)| eZ 2me B, (75) and the spin-flip precession frequency is ωs ΔE Z |g(X)| eB 2me . (76) In the same flux density, the ion’s cyclotron frequency is ωc qXB mX , (77) where qX (Z −1)e, Z, and mX are its net charge, atomic number, and mass, respectively. The frequency ratio ωs/ωc is then independent of B and satisfies ωs ωc |ge(X)| 2(Z −1) mX me |ge(X)| 2(Z −1) Ar(X) Ar(e) , (78) where Ar(X) is the relative atomic mass of the ion. We summarize the theoretical computations of the g-factor in Sec. XI.A and describe the experimental input data and observational equations in Secs. XI.B and XI.C. A. Theory of the bound-electron g-factor The bound-electron g-factor is given by ge(X) gD + Δgrad + Δgrec + Δgns + · · · , (79) where the individual terms on the right-hand side are the Dirac value, radiative corrections, recoil corrections, and nuclear-size corrections, and the dots represent possible additional corrections not already included. The Dirac value is (Breit, 1928) gD −2 3 1 + 2  1 −(Zα)2  −2 1 −1 3(Zα)2 −1 12(Zα)4 −1 24(Zα)6 + · · · , (80) where the only uncertainty is due to that in α. The radiative correction is given by the series Δgrad  ∞ n1 Δg(2n), (81) where Δg(2n) −2C(2n) e (Zα)α π n (82) TABLE XV. Ionization energies for 1H, 3H, 3He, 4He, 12C, and 28Si. The full description of unit m−1 is cycles or periods per meter. Covariances among the data in this table have not been included in the adjustment. See text for explanation. EI/hc (107 m−1) EI/hc (107 m−1) 1H 1.096 787 717 4307(10) 3H 1.097 185 4390(13) 3He+ 4.388 891 936(3) 4He 1.983 106 6637(20) 4He+ 4.389 088 785(2) 12C 0.908 203 480(90) 12C+ 1.966 634(1) 12C2+ 3.862 410(20) 12C3+ 5.201 753(15) 12C4+ 31.624 233(2) 12C5+ 39.520 616 7(5) 28Si 0.657 4776(25) 28Si+ 1.318 381(3) 28Si2+ 2.701 393(7) 28Si3+ 3.640 931(6) 28Si4+ 13.450 7(3) 28Si5+ 16.556 90(40) 28Si6+ 19.887(4) 28Si7+ 24.4864(42) 28Si8+ 28.333(5) 28Si9+ 32.374(3) 28Si10+ 38.414(2) 28Si11+ 42.216 3(6) 28Si12+ 196.610 389(16) 28Si13+ 215.606 31(2) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-23 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr with coefficients C(2n) e (x) that depend on x Zα. The first or one-photon coefficient in the series has self-energy (SE) and vacuum-polarization (VP) contributions, i.e., C(2) e (x) C(2) e,SE(x) + C(2) e,VP(x). The self-energy coefficient is (Faus-tov, 1970; Grotch, 1970; Close and Osborn, 1971; Pachucki, Jent-schura, and Yerokhin, 2004; Pachucki et al., 2005) C(2) e,SE(x) 1 21 + x2 6 + x4 32 9 ln(x−2) + 247 216 −8 9 ln k0 −8 3 ln k3 + x5RSE(x) ⎫ ⎬ ⎭, (83) where ln k0 2.984 128 556, (84) ln k3 3.272 806 545, (85) RSE(6α) 22.1660(10), (86) RSE(14α) 21.0005(1). (87) Values for the remainder function RSE(x) for carbon and silicon have been taken from Yerokhin and Harman (2017) and correspond to an almost tenfold improvement over the values used in the previous adjustment. It is worth noting that Pachucki and Puchalski (2017) have derived that RSE(0) π89 16 + 8 3 ln 2. (88) Finally, we have C(2) e,SE(6α) 0.500 183 607 131(80), C(2) e,SE(14α) 0.501 312 638 14(56). (89) The lowest-order vacuum-polarization coefficient C(2) e,VP(x) has a wave-function and a potential contribution, each of which can be separated into a lowest-order Uehling-potential contribution and a higher-order Wichmann-Kroll contribution. The wave-function correction is (Beier, 2000; Beier et al., 2000; Karshenboim, 2000; Karshenboim, Ivanov, and Shabaev, 2001a, 2001b) C(2) e,VPwf(6α) −0.000 001 840 343 1(43), C(2) e,VPwf(14α) −0.000 051 091 98(22). (90) For the potential correction, the Uehling contribution vanishes Beier et al. (2000), and for the Wichmann-Kroll part we take the value of Lee et al. (2005), which has a negligible uncertainty from omitted binding corrections for the present level of uncertainty. This leads to C(2) e,VPp(6α) 0.000 000 008 201(11), C(2) e,VPp(14α) 0.000 000 5467(11), (91) and for the total lowest-order vacuum-polarization coefficient C(2) e,VP(6α) − 0.000 001 832 142(12), C(2) e,VP(14α) − 0.000 050 5452(11). (92) Moreover, we have C(2) e (6α) C(2) e,SE(6α) + C(2) e,VP(6α) 0.500 181 774 989(81), C(2) e (14α) C(2) e,SE(14α) + C(2) e,VP(14α) 0.501 262 0929(12). (93) The two-photon n 2 correction factor for the ground S state is (Pachucki et al., 2005; Jentschura et al., 2006) C(4) e (x) 1 + x2 6 C(4) e + x414 9 ln(x−2) + 991 343 155 520 −2 9 ln k0 −4 3 ln k3 + 679π2 12 960 −1441π2 720 ln 2 + 1441 480 ζ(3) + 16 −19π2 216  + 1 2x5R(4)(x), (94) where C(4) e −0.328 478 444 00 . . .. The last term in square brackets for the contribution of order x4, absent in the previous adjustment, is the light-by-light scattering contribution (Czarnecki and Szafron, 2016). The term x5R(4)(x) in Eq. (94) is the contribution of order x5 and higher from diagrams with zero, one, or two vacuum-polarization loops. Yerokhin and Harman (2013) haveperformed nonperturbative calculations for many of the vacuum-polarization contributions to this function, denoted here by R(4) VP, with the results R(4) VP(6α) 14.28(39), R(4) VP(14α) 12.72(4) (95) for our two ions. These vacuum-polarization values are the sum of three contributions. The first, denoted with subscript SVPE, is from self-energy vertex diagrams with a free-electron vacuum-polarization loop included in the photon line and magnetic interactions on the bound-electron line. This calculation involves severe numerical cancellations when lower-order terms are subtracted for small Z. The results R(4) SVPE(6α) 0.00(15), R(4) SVPE(14α) −0.152(43) (96) were extrapolated from results for Z ≥20. The second contribution, denoted with subscript SEVP, is from screening-like diagrams with separate self-energy and vacuum-polarization loops. The vacuum-polarization loop includes the higher-order Wichmann-Kroll terms and magnetic interactions are only included in the bound-electron line. This set gives R(4) SEVP(6α) 7.97(36), R(4) SEVP(14α) 7.62(1). (97) The third contribution, denoted with subscript VPVP, comes from twice-iterated vacuum-polarization diagrams and from the K¨ all´ en-Sabry corrections with free-electron vacuum-polarization loops, all with magnetic interactions on the bound-electron line. This set gives R(4) VPVP(6α) 6.31, R(4) VPVP(14α) 5.25. (98) The results for this latter contribution are consistent with a pertur-bative result at x 0 given by (Jentschura, 2009) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-24 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr R(4) VPVP(0) 1 420 807 238 140 + 832 189 ln 2 −400 189 π π 7.4415 . . . . (99) Czarnecki et al. (2018) performed perturbative calculations at x 0 for a complementary set of diagrams contributing to R(4)(x). These calculations include self-energy diagrams without vacuum-polarization loops, with the combined result ΔR(4)(0) 4.7304(9). (100) This value has three contributions. One is from self-energy diagrams without vacuum-polarization loops given by R(4) SE (0) 0.587 35(9)π2. (101) The second set has light-by-light diagrams with nuclear interactions in a vacuum-polarization loop inserted into the photon line in a self-energy diagram, which gives R(4) LBL(0) −0.172 4526(1)π2. (102) The remaining contribution with external magnetic-field coupling to a virtual-electron loop is given by R(4) ML(0) −101 698 907 3 402 000 + 92 368 2025 ln 2 −7843 16 200 π π 0.064 387 . . . π2. (103) The results by Yerokhin and Harman (2013) and Czarnecki et al. (2018) can be combined to give R(4)(x) R(4) VP(x) + ΔR(4)(0), (104) which has uncertainty computed in quadrature from that of R(4) VP(x) and, following Czarnecki et al. (2018), u[ΔR(4)(0)] |x ln3(1/x2)| (105) taken to be on the order of the contribution of the next-order term. For x 6α and 14α, this uncertainty is approximately twice ΔR(4)(0). Finally, we have for the two-photon coefficients C(4) e (6α) −0.328 579 22(86), C(4) e (14α) −0.329 161(54). (106) For n > 2 contributions Δg(2n) to the radiative correction, it is sufficient to use the observations of Eides and Grotch (1997) and Czarnecki, Melnikov, and Yelkhovsky (2000), who showed that C(2n) e (Zα) 1 + (Zα)2 6 + · · · C(2n) e (107) for all n. The values for constants C(2n) e for n 1 through 5 are given in Table XII. This dependence for n 1 and 2 can be recognized in Eqs. (83) and (106), respectively. For n 3 we use C(6) e (Zα) 1.181 611 . . . for Z 6, 1.183 289 . . . for Z 14, (108) while for n 4 we have C(8) e (Zα) −1.911 933 . . . for Z 6, −1.914 647 . . . for Z 14, (109) and, finally, for n 5 C(10) e (Zα) 6.67(19) . . . for Z 6 6.68(19) . . . for Z 14. (110) Recoil of the nucleus gives a correction Δgrec proportional to the electron-nucleus mass ratio and can be written as Δgrec Δg(0) rec + Δg(2) rec + · · ·, where the two terms are zero and first order in α/π, respectively. The first term is (Eides and Grotch, 1997; Shabaev and Yerokhin, 2002) Δg(0) rec −(Zα)2 + (Zα)4 3[1 +  1 −(Zα)2  ]2 −(Zα)5P(Zα) me mN +(1 + Z)(Zα)2 me mN  2 , (111) where mN is the mass of the nucleus. Mass ratios, based on the current adjustment values of the constants, are me/m(12C6+) 0.000 045 727 5 . . . and me/m(28Si14+) 0.000 019 613 6 . . .. For carbon P(6α) 10.493 95(1), and for silicon we use the interpolated value P(14α) 7.162 23(1). For Δg(2) rec we have Δg(2) rec α π (Zα)2 3 me mN + · · · . (112) The uncertainty in Δg(2) rec is negligible compared to that of Δg(2) rad. Glazov and Shabaev (2002) have calculated the nuclear-size correction Δgns,LO within lowest-order perturbation theory based on a homogeneous-sphere nuclear-charge distribution and Dirac wave functions for the electron bound to a point charge. To good ap-proximation, the correction is (Karshenboim, 2000) −8 3(Zα)4 RN ƛC  2 , (113) where RN is the root-mean-square nuclear-charge radius and ƛC is the reduced Compton wavelength of the electron. In the CODATA ad-justment, we scale the values of Glazov and Shabaev (2002) with the squares of updated values for the nuclear radii RN 2.4702(22) fm and RN 3.1224(24) fm from the compilation of Angeli and Marinova (2013) for 12C and 28Si, respectively. Recently, higher-order contributions of the nuclear-size correction have been computed by Karshenboim and Ivanov (2018a). They are Δgns,NLO − 2 3 Zα RN ƛC CZF + α 4πΔgns,LO, (114) where CZF 3.3 is the ratio of the Zemach or Friar moment (Friar and Payne, 1997) to R3 N for a homogeneous-sphere nuclear-charge dis-tribution. We assume that Δgns,NLO has a 10% uncertainty. The sum of the scaled nuclear-size correction of Glazov and Shabaev (2002) and Eq. (114) yields Δgns −0.000 000 000 407(1) for 12C5+, Δgns −0.000 000 020 48(3) for 28Si13+ (115) for the total nuclear-size correction. Tables XVI and XVII list the contributions discussed above to ge(X) for X 12C5+ and 28Si13+, respectively. The final values are ge(12C 5+) −2.001 041 590 153(25), ge(28Si 13+) − 1.995 348 9571(17) (116) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-25 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr with uncertainties that are dominated by that of the two-photon radiative correction Δg(4). This uncertainty is dominated by terms proportional to (Zα)6 multiplying various powers of ln[(Zα)−2]. We shall assume that the uncertainties for this contribution are correlated with a correlation coefficient of r 0.80 (117) for our two hydrogenic ions. The derived value for the electron mass depends only weakly on this assumption; the value for the mass changes by only 2 in the last digit and the uncertainty varies by 1 in its last digit. B. Measurements of precession and cyclotron frequencies of 12C5+ and 28Si 13+ The experimentally determined quantities are ratios of the electron spin-precession (or spin-flip) frequency in hydrogenic carbon and silicon ions to the cyclotron frequency of the ions, both in the same magnetic flux density. The input data used in the 2018 adjustment for hydrogenic carbon and silicon are ωs(12C5+) ωc(12C5+) 4376.210 500 87(12) [2.8 3 10−11] (118) and ωs(28Si13+) ωc(28Si13+) 3912.866 064 84(19) [4.8 3 10−11] (119) with correlation coefficient rωs(12C5+) ωc(12C5+) , ωs(28Si13+) ωc(28Si13+)  0.347, (120) both obtained at MPIK using a multi-zone cylindrical Penning trap operating at B 3.8 T and in thermal contact with a liquid helium bath (Sturm et al., 2013, 2014; K¨ ohler et al., 2015; Sturm, 2015). The de-velopment of this trap and associated measurement techniques has occurred over a number of years, leading to the current uncertainties below 5 parts in 1011. A detailed discussion of the uncertainty budget and covariance and additional references can be found in the 2014 CODATA adjustment. We identify the results in Eqs. (118) and (119) by MPIK-15. C. Observational equations for 12C 5+ and 28Si 13+ experiments The observational equations that apply to the frequency-ratio experiments on hydrogenic carbon and silicon and theoretical computations of their g-factors follow from Eq. (78) when it is expressed in terms of the adjusted constants. That is, ωs(12C5+) ωc(12C5+) ≐−ge(12C5+) + δth(C) 10Ar(e) 3 12 −5Ar(e) + α2Ar(e) 2R∞ ΔEB(12C5+) hc  (121) for 12C5+ using Ar(12C) ≡12, Eq. (64), and Eq. (67). Similarly, ωs(28Si13+) ωc(28Si13+) ≐−ge(28Si13+) + δth(Si) 26Ar(e) Ar(28Si 13+) (122) for 28Si13+. In these two equations, α, R∞, the relative atomic masses Ar(e) and Ar(28Si13+), binding energy ΔEB(12C5+), and additive corrections δth(C) and δth(Si) to the theoretical g-factors of 12C5+ and 28Si13+ are adjusted constants. Of course, the observational equation Ar(28Si) ≐Ar(28Si13+) + 13Ar(e) −α2Ar(e) 2R∞ ΔEB(28Si13+) hc (123) relates the relative atomic mass of the silicon ion to that of the input datum of the neutral atom and ΔEB(28Si13+) is an adjusted constant. The theoretical expressions for g-factors ge(12C5+) and ge(28Si13+) are functions of adjusted constant α. The observational equations for the additive corrections δth(C) and δth(Si) for these g-factors are δX ≐δth(X) for X C and Si with input data δC 0.0(2.5) 3 10−11, δSi 0.0(1.7) 3 10−9, (124) and u(δC, δSi) 3.4 3 10−20 from Eqs. (116) and (117). TABLE XVI. Theoretical contributions and total value for the g-factor of hydrogenic 12C5+ based on the 2018 recommended values of the constants Contribution Value Source Dirac gD −1.998 721 354 3910(4) Eq. (80) Δg(2) SE −0.002 323 672 4382(5) Eq. (89) Δg(2) VP 0.000 000 008 511 Eq. (92) Δg(4) 0.000 003 545 708(25) Eq. (106) Δg(6) −0.000 000 029 618 Eq. (108) Δg(8) 0.000 000 000 111 Eq. (109) Δg(10) −0.000 000 000 001 Eq. (110) Δgrec −0.000 000 087 629 Eqs. (111) and (112) Δgns −0.000 000 000 407(1) Eq. (115) g(12C5+) −2.001 041 590 153(25) Eq. (116) TABLE XVII. Theoretical contributions and total value for the g-factor of hydrogenic 28Si 13+ based on the 2018 recommended values of the constants Contribution Value Source Dirac gD −1.993 023 571 552(2) Eq. (80) Δg(2) SE −0.002 328 917 509(3) Eq. (89) Δg(2) VP 0.000 000 234 81(1) Eq. (92) Δg(4) 0.000 003 5530(17) Eq. (106) Δg(6) −0.000 000 029 66 Eq. (108) Δg(8) 0.000 000 000 11 Eq. (109) Δg(10) −0.000 000 000 00 Eq. (110) Δgrec −0.000 000 205 88 Eqs. (111) and (112) Δgns −0.000 000 020 48(3) Eq. (115) g(28Si13+) −1.995 348 9571(17) Eq. (116) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-26 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The input data are summarized as entries D7 through D13 in Table XXI and observational equations can be found in Table XXVI. XII. Muonic Hydrogen and Deuterium Lamb Shift Muonic hydrogen and deuterium, μH and μD, respectively, are atoms consisting of a proton or a deuteron and a negatively charged muon. Since the mass of a muon is just over 200 times larger than that of the electron, the muonic Bohr radius is 200 times smaller than the electronic Bohr radius and the muon wave-function overlap with the proton or deuteron is more than a million times larger than in normal H or D. Consequently, the “muonic” Lamb shift, the energy difference ΔELS(X) E2P1/2(X) −E2S1/2(X) between the 2S1/2 and 2P1/2 levels, is much more sensitive to the proton and deuteron charge radii, rp and rd. The energy of the 2S1/2 level in H and D is higher than that of 2P1/2. Because of the much larger electron vacuum-polarization contribution, however, the energy of the 2S1/2 level in μH and μD lies below that of 2P1/2. In normal H and D, the Lamb shift is about h 3 1 GHz or 0.004 meV, while in μH and μD it is about h 3 50 THz or 200 meV. The first successful measurement of the Lamb shift of μH was carried out by the Charge Radius Experiment with Muonic Atoms (CREMA) collaboration at the Paul Scherrer Institute, Switzerland, in 2010 (Pohl et al., 2010). (Strictly speaking, the authors measured the transition energy between the 2S1/2 and 2P3/2 levels. The 2P1/2-2P3/2 fine-structure interval is sufficiently well known from theory that the uncertainty budget for the Lamb shift is not affected.) Based on the theory of ΔELS(μH) as it existed at the time, the CREMA collabo-ration derived that rp 0.841 84(67) fm. This value was inconsistent with the 2006 CODATA recommended value based on hydrogen spectroscopic and e-p elastic scattering data and gave rise to the “proton-radius puzzle.” For the CODATA 2010 adjustment, new elastic e-p scattering data from Bernauer et al. (2010) also became available. Their derived value for rp agreed with the CODATA 2006 recommended value. Because of the strong disagreement of rp derived from μH spec-troscopy and the value of rp derived from hydrogen spectroscopic and e-p scattering data, the Task Group decided not to include μH data in 2010. As a consequence, the disagreement between rp based on the μH Lamb shift and the CODATA 2010 recommended value increased to seven standard deviations. In 2013, the CREMA collaboration reported a second experi-mental value for ΔELS(μH) (Antognini et al., 2013; Antognini, Kottmann et al., 2013), as well as advances in the theory of μH, which together yielded a value for rp that was consistent with their 2010 estimate and had an even smaller uncertainty. Thus it did not alter the status of the proton-radius puzzle and the Task Group decided to omit μH data from the 2014 adjustment as well. In simplest terms, the puzzle was that there are two plausible values for rp: a “low” value of about 0.84 fm and a CODATA recommended “high” value of about 0.88 fm. Efforts to solve the proton-radius puzzle have continued. For example, a value for the deuteron radius rd, obtained from a mea-surement of ΔELS(μD), has been reported by the CREMA collabo-ration (Pohl et al., 2016). Their value for rd also confirmed the value for rp based on μH data when it was combined with a measurement of the difference of the Lyman-α transition energy of normal H and D by Parthey et al. (2010), item A5 in Table X, and the theory of H and D. The Task Group believes that the muonic data have been suf-ficiently verified and has decided to include the μH and μDLamb-shift data in the 2018 CODATA adjustment. Moreover, three measure-ments of transition energies in hydrogen have become available since the previous adjustment. Their contributions decrease the value of rp based solely on hydrogen spectroscopy. See also the discussions in Sec. IV.C and VII.C. Inconsistencies that exist among data that relate to the determination of rp and rd are dealt with by applying a mul-tiplicative expansion factor to the uncertainties of the relevant data. We review the μH and μD Lamb-shift data and relevant theory in the next two sections. Input data from the Lamb-shift measurements, theoretical additive constants, and theoretical parameters are sum-marized in Table XVIII. Observational equations are found in Table XXIII. A. Muonic hydrogen Lamb shift The CREMA collaboration measured the μH Lamb shift ΔELS(μH) 202.3706(23) meV with ur 1.1 3 10−5 (Antognini et al., 2013). The value was derived from the measured hyperfine-resolved 2S1/2(f 0) →2P3/2(f 1) transition energy, the pre-viously reported CREMA value of the 2S1/2(f 1) →2P3/2(f 2) transition energy (Pohl et al., 2010) updated as described by Antognini et al. (2013), and the sufficiently accurate theoretical es-timates of the 2P fine-structure and 2P3/2 hyperfine splittings by Antognini, Kottmann et al. (2013). The two experimental transition energies also led to the determination of the magnetic Zemach radius of the proton. Details regarding the CREMA experiment have been described in the 2014 CODATA publication. The measured value is datum C1 in Table XVIII. We use the theoretical expression for the muonic hydrogen Lamb shift from Peset and Pineda (2015) in order to derive a value for the proton charge radius rp. It is based on perturbation theory in a nonrelativistic effective field theory derived from higher-energy QED and QCD descriptions. For example, QED contributions up to α5mμc2 and α6ln(α−2)mμc2 have been included. Unlike for the the-oretical description of the H and D energy levels in Sec. VII.A, where we add many contributions to find level energies, we use D0H + D2Hr2 p (125) for the theoretical Lamb shift in the least-squares adjustment. The values and uncertainties for D0H and D2H are taken from Peset and Pineda (2015) and given as items C3 and C4 in Table XVIII. This simpler procedure is justified, as nearly 95% of the muonic Lamb shift is due to the electron vacuum-polarization correction of order α3mμc2 in D0H and the uncertainty of D0H is due to uncertainties in proton-structure corrections that are independent of rp. The corresponding relative standard uncertainty is orders of magnitude larger than those in α and mμ. Approximately 5% of the Lamb shift is due to the second term in Eq. (125). An early description of the theory for the muonic Lamb shift was published by Pachucki (1996). For the CODATA adjustment, the relevant observational equations are ΔELS(μH) ≐D0H + D2Hr2 p + δth(μH) (126) and δELS(μH) ≐δth(μH). (127) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-27 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Here, the proton charge radius rp and additive constant δth(μH) are adjusted constants and input datum δELS(μH) 0.0000(129) meV accounts for the uncertainty from uncomputed terms in D0H and the uncertainty of D2H, although the latter uncertainty is currently in-consequential. Substitution of input data C1, C3, C4, and C7 from Table XVIII into Eq. (126) yields rp 0.8413(15) fm. In 2013, Antognini et al. (2013) used theoretical estimates for D0H and D2H by Antognini, Kottmann et al. (2013) to publish a value for rp. ThevalueforD0H isconsistentwith thatofPesetand Pineda (2015).The uncertainty from Antognini, Kottmann et al. (2013), however, is five times smaller. The theory of Peset and Pineda (2015) is chosen over that of Antognini, Kottmann et al. (2013) as their estimate and uncertainty of hadronic corrections provide a more conservative value of rp. Similarly, Karshenboim et al. (2015) gave smaller uncertainties on the quantities D0H and D2H. Because the proton-radius puzzle is only partly resolved, a more conservative approach seems warranted. It, however, does point to the need for future research and possible future improvements in the accuracy of the proton charge radius. B. Muonic deuterium Lamb shift The CREMA collaboration measured the μD Lamb shift ΔELS(μD) 202.8785(34) meV with ur 1.7 3 10−5 (Pohl et al., 2016). In fact, the data were acquired during the same measurement period and using the same general method as for the muonic hydrogen data described in the previous section. The result is based on the measurement of the three hyperfine-resolved transition energies 2S1/2(f 3/2) →P3/2(f 5/2), 2S1/2(f 1/2) →2P3/2(f 3/2), and 2S1/2(f 1/2) →2P3/2(f 1/2). As with the μH data, Pohl et al. (2016)made use of the sufficiently well-known 2P fine-structure splitting and the 2P3/2 hyperfine splitting, both due to Krauth et al. (2016), to derive the Lamb shift. The 0.0034 meV total uncertainty is the root-sum-square of a 0.0031 meV statistical component and a 0.0014 meV component from systematic effects. The measured value is datum C2 in Table XVIII. The observational equations for ΔELS(μD) are based on the recent theoretical treatment of the n 2 energy levels of μD by Krauth et al. (2016) and Kalinowski (2019). That is, ΔELS(μD) ≐D0D + D2Dr2 d + δth(μD) (128) and δELS(μD) ≐δth(μD), (129) where the deuteron charge radius rd and additive constant δth(μD) are adjusted constants. Values and uncertainties for D0D and D2D are given as items C5 and C6 in Table XVIII. The coefficient D2D is due to Krauth et al. (2016). The coefficient D0D is the sum of two terms. The first is 228.776 66(96) meV also due to Krauth et al. (2016) and accounts for all contributions that do not explicitly depend on rd. The second is 1.748(21) meV from Kalinowski (2019), which we use for the nuclear-polarizability contribution instead of the corresponding value by Krauth et al. (2016). Input datum δELS(μD) 0.0000(210) meV in-corporates the uncertainty from uncomputed terms in the theoretical energy D0D and the uncertainty of the coefficient D2D, although the latter uncertainty has currently no influence on the adjustment. Substitution of input data C2, C5, C6, and C8 from Table XVIII into Eq. (128) yields rd 2.127 10(81) fm. C. Deuteron-proton charge radius difference The deuteron-proton radius difference r2 d −r2 p is constrained by the μH and μD Lamb-shift measurements, but also by the mea-surement of the isotope shift of the 1S-2S transition in H and D by Parthey et al. (2010), item A5 in Table X. From the 2018 CODATA adjustment, its recommended value is r2 d −r2 p 3.820 36(41) fm2, (130) mainly constrained by the H to D isotope shift measurement. XIII. Electron-Proton and Electron-Deuteron Scattering In electron-proton and electron-deuteron elastic scattering ex-periments, the differential scattering cross section for the electron is measured as a function of the incident energy of the electrons, Einc, and theelectron scatteringangleθ.Fromthese data,the electric formfactorof the proton GE(Q2) as a function of the negative of the squared four-momentum transfer Q2 can be extracted. Here, Q2 is uniquely specified TABLE XVIII. Input data for the experimental determinations of muonic hydrogen and muonic deuterium Lamb shifts ΔELS(μX), theoretical coefficients DiX for these Lamb shifts, additive energy corrections δELS(μX), as well as the proton (p) and deuteron (d) root-mean-square charge radii rN based on electron-proton and electron-deuteron scattering. The label in the first column is used in Table XXIII for observational equations. Only items C1, C2, and C7–C10 are input data in the adjustment. Columns two and three give the reference and an abbreviation of the name of the laboratory in which the experiment has been performed. An extensive list of abbreviations is found at the end of this report. The role of the expansion coefficients, items C3–C6, and the rationale for the values and uncertainties of the radii, C9 and C10, are discussed in the text. Relative standard uncertainties in square brackets are relative to the value of the theoretical quantity to which the additive correction corresponds. There are no correlations among these data Reference Lab. Input datum Value Rel. stand. unc. ur C1 Antognini et al. (2013) CREMA ΔELS(μH) 202.3706(23) meV 1.1 3 10−5 C2 Pohl et al. (2016) CREMA ΔELS(μD) 202.8785(34) meV 1.7 3 10−5 C3 Peset and Pineda (2015) UBarc D0H 206.0698(129) meV 6.2 3 10−5 C4 Peset and Pineda (2015) UBarc D2H −5.2270(7) meV fm−2 1.3 3 10−4 C5 Kalinowski (2019) WarsU D0D 230.5247(210) meV 9.1 3 10−5 C6 Krauth et al. (2016) MPQ D2D −6.110 25(28) meV fm−2 4.6 3 10−5 C7 theory δELS(μH) 0.0000(129) meV [6.4 3 10−5] C8 theory δELS(μD) 0.0000(210) meV [1.0 3 10−4] C9 rp 0.880(20) fm 2.3 3 10−2 C10 rd 2.111(19) fm 9.0 3 10−3 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-28 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr byEinc andθ,asintheseexperimentstheinitialmomentumoftheproton is negligibly small and the incident and final electron energies are much largerthantherestenergyoftheelectron[see,forexample,Bernaueretal. (2014)]. A typical upper bound for the incident electron energy is the rest energy of the proton or deuteron. A function is then fit to the data for the form factor GE(Q2) and the root-mean-square charge radius rN is calculated from the slope of GE(Q2) at Q2 0. Because cross-section measurements are not possible at Q2 0, the function chosen to extrapolate to this limit and the largest Q2 value in the data set are critical for the determination of the un-certainty budget for rN. In addition, various systematic effects must be accounted for in the procedure to extract the form factor from the cross section. We review rp and rd obtained from scattering data in the next two sections. Input data and observation equations are summarized in Tables XVIII and XXIII, respectively. A. Proton radius from e-p scattering Currently, the most extensive e-p scattering data are those obtained by the A1 Collaboration at Mainz University, Germany, with the Mainz linear accelerator (MAMI). Their data have been published by Bernauer et al. (2010, 2014). About 1400 cross sections were measured at six electron beam energies ranging from 180 MeV to 855 MeV with Q2 from 0.003(GeV/c)2 to1(GeV/c)2.The2010value rp 0.8791(79) fm from these authors was used in the CODATA 2010 adjustment, as was the value rp 0.895(18) fm due to Sick (2003, 2007, 2008). The only scattering value of the proton radius used as an input datum in the 2014 adjustment was rp 0.879(11) fm, a weighted mean of the values by Arrington and Sick (2015) and Bernauer and Distler (2015). The un-certainty was the simple average of the individual uncertainties because each value was based on essentially the same data. Before the closing date for new data for the 2018 adjustment, various authors reanalyzed the e-p scattering data with a variety of methods. Four such values are rp 0.840(16) fm given by Griffioen, Carlson, and Maddox (2016), obtained from the Mainz data with values of Q2 below 0.02(GeV/c)2; rp 0.844(7) fm obtained by Alarc´ on et al. (2019) using chiral effective field theory; rp 0.845(1) fm from Zhou et al. (2019) employing constrained Gaussian processes; and rp 0.855(11) fm due to Horbatsch, Hessels, and Pineda (2017) using chiral perturbation theory. Larger values, for example, rp 0.916(24) fm obtained by Lee, Arrington, and Hill (2015), were found by only analyzing the e-p scattering data of Bernauer et al. (2010). Most recently, Hayward and Griffioen (2020) found rp 0.841(4) fm from characterizing the effects of bias when omitting large-Q2 data. Based on these new analyses and the input data used for the 2010 and 2014 adjustments, the Task Group has decided to adopt as the only e-p scattering input datum rp 0.880(20) fm. This value and uncertainty are chosen so that all evaluations of rp lie within two standard deviations from this mean value. The value is essentially the same value as used in the 2014 adjustment but with an uncertainty that is approximately twice as large. For completeness, we note that results for rp from two new e-p scattering experiments have become available after the 31 December 2018 closing date of the 2018 adjustment. Xiong et al. (2019) report rp 0.831(24) fm determined by the PRad Collaboration at the Thomas Jefferson National Accelerator Laboratory, USA; and TABLE XIX. Fifty of the 75 adjusted constants in the 2018 CODATA least-squares minimization. Other variables in the adjustment are given in Table XI. Adjusted constant Symbol fine-structure constant α Rydberg constant R∞ proton rms charge radius rp deuteron rms charge radius rd Newtonian constant of gravitation G electron relative atomic mass Ar(e) proton relative atomic mass Ar(p) neutron relative atomic mass Ar(n) deuteron relative atomic mass Ar(d) triton relative atomic mass Ar(t) helion relative atomic mass Ar(h) alpha particle relative atomic mass Ar(α) 28Si13+ relative atomic mass Ar(28Si13+) 87Rb relative atomic mass Ar(87Rb) 133Cs relative atomic mass Ar(133Cs) 1H+ electron removal energy ΔEB(1H+) HD+ electron ionization energy ΔEI(HD+) 3He+ electron ionization energy ΔEI(3He+) 4He2+ electron removal energy ΔEB(4He2+) 12C5+ electron removal energy ΔEB(12C5+) 12C6+ electron removal energy ΔEB(12C6+) 28Si13+ electron removal energy ΔEB(28Si13+) additive correction to ae(th) δth(e) muon magnetic-moment anomaly aμ additive correction to gC(th) δth(C) additive correction to gSi(th) δth(Si) additive correction to ΔnMu(th) δth(Mu) electron-muon mass ratio me/mμ additive correction to μ-H Lamb shift δth(μH) additive correction to μ-D Lamb shift δth(μD) deuteron-electron magnetic-moment ratio μd/μe− electron-proton magnetic-moment ratio μe−/μp electron to shielded proton μe−/μ ′ p magnetic-moment ratio shielded helion to shielded proton μ ′ h/μ ′ p magnetic-moment ratio neutron to shielded proton μn/μ ′ p magnetic-moment ratio triton to proton magnetic-moment ratio μt/μp shielding difference of d and p in HD σdp shielding difference of t and p in HT σtp d220 of an ideal natural Si crystal d220 d220 of Si crystal ILL d220(ILL) d220 of Si crystal MO∗ d220(MO∗) d220 of Si crystal N d220(N) d220 of Si crystal NR3 d220(NR3) d220 of Si crystal NR4 d220(NR4) d220 of Si crystal WASO 04 d220(W04) d220 of Si crystal WASO 17 d220(W17) d220 of Si crystal WASO 4.2a d220(W4.2a) Copper Kα1 x unit xu(CuKα1) ˚ Angstrom star ˚ A∗ Molybdenum Kα1 x unit xu(MoKα1) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-29 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Mihoviloviˇ c et al. (2019) report rp 0.870(28) fm from a recent experiment performed at MAMI. B. Deuteron radius from e-d scattering Since 1998, the input datum for the charge radius of the deuteron obtained from elastic e-d scattering data in the CODATAadjustments is rd 2.130(10) fm as determined by Sick and Trautmann (1998) and Sick (2001). This value is based on some 340 cross-section data points for momentum transfers less than 2 GeV/c. Recently, Hayward and Griffioen (2020) determined with a novel algorithm the structure function A(Q2), a combination of electric, magnetic, and quadrupole form factors, from elastic e-d scattering data and extrapolated to Q2 0. The radius rd is then determined from the slope of A(Q2) at Q2 0. Only the data set of Simon, Schmitt, and Walther(1981),however,couldbeusefullyanalyzedwiththeiralgorithm. This yielded rd 2.092(19) fm. In view of this result and the many questions raised concerning the extraction of reliable values of rp and rd from scattering data, the value rd 2.111(19) fm is adopted as the e-d scattering input datum for the 2018 adjustment. It is the average of rd 2.092(19) fm and the long-used historical value rd 2.130(10) fm with an uncertainty of one-half their difference. Coincidentally, this uncertainty is the same as that of Hayward and Griffioen (2020). XIV. Magnetic-Moment Ratios of Light Atoms and Molecules The CODATA Task Group recommends values for the free-particle magnetic moments of leptons, the neutron, and light nuclei. The most precise means to determine the free magnetic moments of the electron, muon, and proton are discussed in Secs. VIII, XVI, and XV, respectively. In this section, we describe the determination of the neutron, deuteron, triton, and helion magnetic moments. The magnetic moment of the 4He nucleus or α particle is zero. Nuclear magnetic moments are determined from hydrogen and deuterium maser experiments and nuclear-magnetic-resonance (NMR) experimentsonatomsandmolecules.Bothtypesofexperimentsmeasure ratios of magnetic moments to remove the need to know the strength of the applied magnetic field. We rely on NMR measurements for ratios of nuclear magnetic moments in the HD and HT molecules as well as the ratio of the magnetic moment of the neutron and the helion in 3He with respect to that of the proton in H2O. For these molecules, the electronic ground state is an electron spin singlet. The magnetic moment of a nucleus or electron in an atom or molecule, however, differs from that of a free nucleus or electron and theoretical binding corrections are used to relate bound moments to free moments.Intheremainderofthissection,wegivetherelevanttheoretical binding corrections to magnetic-moment ratios and describe experi-mentalinputdata.Wealsodescribethebindingcorrectionsformagnetic-moment ratios of an antimuon and electron bound in muonium (Mu). These willbe relevant in the determination of the electron-to-muon mass ratio in Sec. XVII. A. Definitions of bound-state and free g-factors We recall that the Hamiltonian for a magnetic moment μ in a magnetic flux density B is H −μ · B. For lepton ℓ, the magnetic moment μℓ gℓ(e/2mℓ)s, where gℓ, mℓ, and s are its g-factor, mass, and spin, respectively. By convention, the magnetic moment of a neutron or nucleus with spin I is denoted by μ g e 2mp I, (131) where g is the g-factor of the neutron or nucleus. The charge and mass of the proton mp appear in the definition, regardless whether or not the particle in question is a proton. The magnitude of the magnetic moment of a charged lepton is μℓ 1 2gℓ eZ 2mℓ , (132) while that for the neutron or a nucleus is defined as μ gμNi, (133) where μN eZ/2mp is the nuclear magneton and integer or half-integer i is the maximum positive spin projection of I given by iZ. When electrons bind with nuclei to form ground-state atoms or molecules, the effective g-factors change. For atomic H and D in their electronic ground state, the Hamiltonian is H ΔωX Z s · I −ge(X) e 2me s · B −gN(X) e 2mp I · B, (134) where (X, N) (H, p) or (D, d) and the coefficients ge(X) and gN(X) are bound-state g-factors. For muonium, an atom where an electron is bound to an antimuon, the corresponding Hamiltonian is HMu ΔωMu Z se · sμ −ge(Mu) e 2me se · B −gμ(Mu) e 2mμ sμ · B. (135) B. Theoretical ratios of g-factors in H, D, 3He, and muonium Theoretical binding corrections to g-factors in the relevant atoms and muonium have already been discussed in previous CODATA reports. Relevant references can be found there as well. Here, we only give the final results. For atomic hydrogen, we have ge(H) ge 1 −1 3(Zα)2 −1 12(Zα)4 + 1 4(Zα)2α π + 1 2(Zα)2me mp + 1 2 A(4) 1 −1 4 Zα 2α π 2 −5 12(Zα)2α π me mp + · · · (136) and gp(H) gp 1 −1 3 α(Zα) −97 108 α(Zα)3 + 1 6 α(Zα)me mp 3 + 4ap 1 + ap + · · · , (137) where A(4) 1 is given in Eq. (52) and the proton magnetic-moment anomaly is ap μp/(eZ/2mp) −1 ≈1.793. For deuterium, we have J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-30 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr ge(D) ge 1 −1 3(Zα)2 −1 12(Zα)4 + 1 4(Zα)2α π + 1 2(Zα)2me md + 1 2 A(4) 1 −1 4 Zα 2α π 2 −5 12(Zα)2α π me md + · · · (138) and gd(D) gd 1 −1 3 α(Zα) −97 108 α(Zα)3 + 1 6 α(Zα)me md 3 + 4ad 1 + ad + · · · , (139) where the deuteron magnetic-moment anomaly is ad μd/(eZ/md) −1 ≈−0.143. For helium-3, we have μh(3He) μh 1 −59.967 43(10) 3 10−6 (140) for the magnitude of the magnetic moments (Rudzi´ nski, Puchalski, and Pachucki, 2009). This ratio, however, is not used as an input datum. It is not coupled to any other data, but allows the Task Group to provide a recommended value for the unshielded helion magnetic moment along with other related quantities. Finally, for muonium we have ge(Mu) ge 1 −1 3(Zα)2 −1 12(Zα)4 + 1 4(Zα)2α π + 1 2(Zα)2me mμ + 1 2 A(4) 1 −1 4 Zα 2α π 2 −5 12(Zα)2α π me mμ −1 2(1 + Z)(Zα)2 me mμ  2 + · · · (141) and gμ(Mu) gμ 1 −1 3 α(Zα) −97 108 α(Zα)3 + 1 2 α(Zα)me mμ + 1 12 α(Zα)α π me mμ −1 2(1 + Z)α(Zα) me mμ  2 + · · · . (142) Numerical values for the corrections in Eqs. (136) to (142) based on 2018 recommended values for α, mass ratios, etc. are listed in Table XX; uncertainties are negligible. See Ivanov, Karshenboim, and Lee (2009) for a negligible additional term. C. Theoretical ratios of nuclear g-factors in HD and HT Bound-state corrections to the magnitudes of nuclear magnetic moments in the diatomic molecules HD and HT are expressed as μN(X) [1 −σN(X)]μN, (143) for nucleus N in molecule X. Here, μN is the magnitude of the magnetic moment of the free nucleus and σN(X) is the nuclear magnetic shielding correction. In fact, |σN(X)| ≪1. NMR experiments for these molecules measure the ratio μN(X) μN′(X) [1 + σN′N + O(σ2)] μN μN′ (144) for nuclei N and N′ in molecule X HD or HT and σN′N σN′(X) −σN(X) is the shielding difference of molecule X. In the adjustment, corrections of O(σ2), quadratic in σN(X), are much smaller than the uncertainties in the experiments and are omitted. The theoretical values for shielding differences in HD and HT are σdp 20.20(2) 3 10−9 and σtp 24.14(2) 3 10−9, respectively, as re-ported by Puchalski, Komasa, and Pachucki (2015). The values are approximately 100 times more accurate than those used in the 2014 CODATA adjustment and are also listed as items D42 and D43 in Table XXI. The two shielding differences are taken as adjusted constants with observational equations σdp ≐σdp and σtp ≐σtp, respectively. D. Ratio measurements in atoms and molecules Nine atomic and molecular magnetic-moment ratios obtained with HandDmasersandNMRexperimentsareusedasinputdatainthe2018 adjustment, and determine the magnetic moments of the neutron, deuteron, triton, and helion. For ease of reference, these experimental frequency ratios are summarized in Table XXI and given labels D33 through D41. There are no correlation coefficients among these data greater than 0.0001. Observational equations are summarized in Table XXVI. We note that the primed magnetic moment μ ′ p appearing in three input data in Table XXI indicates that the proton is bound in a H2O molecule in a spherical sample of liquid water at 25°C surrounded by vacuum. The shielding factor for the proton in water is not known theoretically and, thus, these measurements cannot be used to determine the free-proton magnetic moment. The relationships among these three input data, however, help determine other magnetic moments as well as the shielding factor of the proton in water. Finally, the primed quantity μh ′ initemD36isthemagneticmomentofthehelionboundina 3He atomin a 25°C spherical gaseous sample of helium-3. In principle, its value can differ from that of a helion in an isolated 3He atom, that is, μh(3He) as found in Eq. (140). We assume that environmental effects from distant helium-3 atoms are negligible and equate the two quantities, i.e., μh ′ μh(3He), to determine the magnetic moment of the free helion. Our adjusted constants for the determination of the relevant magnetic moments are μd/μe, μe/μp, μe/μ ′ p, μh ′/μ ′ p, μn/μ ′ p, μt/μp, σdp, and σtp. The ratio μp(HD)/μd(HD) obtained by Neronov and Seregin (2012), item D40 in Table XXI, is a relatively old result that was not included in the 2014 adjustment, but is included in the current ad-justment. We rely on three determinations of μp(HD)/μd(HD) in the 2018 CODATA adjustment. The values are from Garbacz et al. (2012), researchers at the University of Warsaw, Poland; and from Neronov and Karshenboim (2003) and Neronov and Seregin (2012), researchers in Saint Petersburg, Russia, who have a long history of NMR measurements in atoms and molecules. (The remaining ex-perimental input data have been reviewed in previous CODATA reports and are not discussed further.) Neronov and Seregin (2012) describe a complex set of experi-ments to determine the free-helion to free-proton magnetic-moment ratio. We had previously overlooked their frequency-ratio mea-surements on HD, which satisfy ωp(HD) ωd(HD) 2 μp(HD) μd(HD) , (145) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-31 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr where the factor two appears because the spins of the proton and deuteron are 1/2 and 1, respectively. The statistical relative uncertainty of the frequency ratio is given as 7.7 parts in 1010. The line-shape fits by Neronov and Seregin (2012), however, visibly disagree with the exper-imental data and, thus, systematic effects are present. We account for these effects by increasing the uncertainty by a factor of 4.0 consistent with determining the NMR frequency of d in HD to approximately one-tenth of the full-width-half-maximum of the Lorentzian line. XV. Proton Magnetic Moment in Nuclear Magnetons The 2017 measurement of the proton magnetic moment in nuclear magnetons, μp/μN, has been newly added to the CODATA adjustment. It was obtained using a single proton in a double Penning trap at the University of Mainz, Germany (Schneider et al., 2017). The ratio was determined by measuring its spin-flip transition frequency ωs 2μpB/Z and its cyclotron frequency ωc eB/mp in a magnetic flux density B. As B is the same in both measurements, ωs ωc μp μN (146) independent of B and where μN eZ/2mp is the nuclear magneton. The Mainz value ωs ωc 2.792 847 344 62(82) [2.9 3 10−10] (147) is consistent with but supersedes the 2014 result by the same research group (Mooser et al., 2014). Improvements in the apparatus led to a relative uncertainty that is more than an order of magnitude smaller than in 2014. The linewidth of the resonant Lorentzian signal was narrowed by reducing magnetic-field inhomogeneity, and an im-proved detector for the cyclotron frequency doubled the data ac-quisition rate. The relative uncertainty of the new result comprises 2.7 and 1.2 parts in 1010 from statistical and systematic effects, re-spectively. The two largest components contributing to the systematic uncertainty are due to limits on line-shape fitting and on the char-acterization of a relativistic shift and have been added linearly to account for correlations. The total correction from systematic effects is −1.3 parts in 1010. The observational equation for ωs/ωc and thus μp/μN is μp μN ≐−[1 + ae(th) + δth(e)] Ar(p) Ar(e) μp μe (148) using the definition of μe in Eq. (45). The quantities δth(e), Ar(e), Ar(p), and μe/μp are adjusted constants. The theoretical expression for the electron anomaly ae(th) is mainly a function of adjusted constant α. The input datum has identifier UMZ-17 and is item D32 in Table XXI. Its observational equation can be found in Table XXVI. XVI. Muon Magnetic-Moment Anomaly The muon magnetic-moment anomaly aμ and thus muon g-factor gμ −2(1 + aμ) weremeasuredin2006.Atheoreticalexpressionforaμ is also available and has steadily been improved since this measurement. Only the measured value of the muon anomaly, however, is included in the 2018 adjustment of the constants due to the disagreement between theory and experiment. The measurement of aμ and the theory are summarized in the following sections. A. Measurement of the muon anomaly The 2006 determination of aμ at Brookhaven National Labo-ratory (BNL), USA has been discussed in the past five CODATA reports. The quantity measured is the anomaly difference frequency ωa ωs −ωc, where ωs |gμ|(e/2mμ)B is the muon spin-flip (or precession) frequency in the applied magnetic flux density B and ωc (e/mμ)B is the muon cyclotron frequency. The flux density is eliminated from these expressions by determining its value from a measurement of the precession frequency of the proton in water in the same apparatus combined with the proton shielding correction in water. This leads to a measurement of proton precession frequency ωp 2μpB/Z, where the magnitude of the proton magnetic moment, μp, and the g-factor of the muon are defined in Sec. XIV.A. The value of R ωa/ωp is reported by the BNL experimentalists. From Table XV of Bennett et al. (2006), we have R 0.003 707 2063(20) [5.4 3 10−7]. (149) It is input datum D26 in Table XXI with identification BNL-06. The corresponding observational equation is R ≐aμ eZ/(2mμ) μp ≐ aμ 1 + ae(th) + δth(e) me mμ μe μp , (150) where the right-hand side of the equation is explicitly expressed in terms of adjusted constants aμ, me/mμ, μe/μp, and additive correction δth(e) for the theoretical electron anomaly ae(th). The anomaly ae(th) is mainly a function of the adjusted constant α. In practice, the muon anomaly can also be calculated from aμ R |μμ/μp| − R , (151) as the uncertainty of the magnetic-moment ratio μμ/μp is much smaller than that of R. The 2018 CODATA recommended value of the muon anomaly is aμ 1.165 920 89(63) 3 10−3. (152) B. Theory of the muon anomaly The muon magnetic-moment anomaly can be expressed as aμ(th) aμ(QED) + aμ(weak) + aμ(had), (153) where terms denoted by “QED,” “weak,” and “had” account for the purely quantum electrodynamic, predominantly electroweak, and TABLE XX. Theoretical values for various bound-particle to free-particle g-factor ratios based on the 2018 recommended values of the constants Ratio Value ge(H)/ge 1 −17.7054 3 10−6 gp(H)/gp 1 −17.7354 3 10−6 ge(D)/ge 1 −17.7126 3 10−6 gd(D)/gd 1 −17.7461 3 10−6 ge(Mu)/ge 1 −17.5926 3 10−6 gμ(Mu)/gμ 1 −17.6254 3 10−6 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-32 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXI. Input data for the 2018 CODATA adjustment to determine the fine-structure constant, muon mass, masses of nuclei with Z ≤2, and magnetic-moment ratios among these nuclei as well as those of leptons. Relative standard uncertainties in square brackets are relative to the value of the theoretical quantity to which the additive correction corresponds. The label in the first column is used to specify correlation coefficients among these data and in Table XXVI observational equations. Columns five and six give the reference, an abbreviation of the name of the laboratory in which the experiment has been performed, and the year of publication. An extensive list of abbreviations is found at the end of this report. Correlations among these data are given in Table XXII Input datum Value Rel. stand. unc. ur Lab. Reference(s) Sec. Input data relevant for the fine-structure constant and the electron mass D1 ae(exp) 1.159 652 180 73(28) 3 10−3 2.4 3 10−10 HarvU-08 Hanneke, Fogwell, and Gabrielse (2008) VIII D2 δe 0.000(18) 3 10−12 [1.5 3 10−11] theory VIII D3 h/m(87Rb) 4.591 359 2729(57) 3 10−9 m2 s−1 1.2 3 10−9 LKB-11 Bouchendira et al. (2011) X D4 h/m(133Cs) 3.002 369 4721(12) 3 10−9 m2 s−1 4.0 3 10−10 UCB-18 Parker et al. (2018) X D5 Ar(87Rb) 86.909 180 5312(65) 7.4 3 10−11 AMDC-16 Huang et al. (2017) IX D6 Ar(133Cs) 132.905 451 9610(86) 6.5 3 10−11 AMDC-16 Huang et al. (2017) IX D7 ωs/ωc for 12C5+ 4376.210 500 87(12) 2.8 3 10−11 MPIK-15 K¨ ohler et al. (2015) XI.B D8 ΔEB(12C5+)/hc 43.563 233(25) 3 107 m−1 5.8 3 10−7 ASD-18 IX D9 δC 0.0(2.5) 3 10−11 [1.3 3 10−11] theory XI.C D10 ωs/ωc for 28Si13+ 3912.866 064 84(19) 4.8 3 10−11 MPIK-15 Sturm et al. (2013) and Sturm (2015) XI.B D11 Ar(28Si) 27.976 926 534 99(52) 1.9 3 10−11 AMDC-16 Huang et al. (2017) IX D12 ΔEB(28Si13+)/hc 420.6467(85) 3 107 m−1 2.0 3 10−5 ASD-18 IX D13 δSi 0.0(1.7) 3 10−9 [8.3 3 10−10] theory XI.C Input data relevant for masses of light nuclei D14 ωc(d)/ωc(12C6+) 0.992 996 654 743(20) 2.0 3 10−11 UWash-15 Zafonte and Van Dyck (2015) IX D15 ωc(12C6+)/ωc(p) 0.503 776 367 662(17) 3.3 3 10−11 MPIK-17 Heiße et al. (2017) IX D16 ωc(t)/ωc(3He+) 0.999 993 384 997(24) 2.4 3 10−11 FSU-15 Myers et al. (2015) IX D17 ωc(HD+)/ωc(3He+) 0.998 048 085 122(23) 2.3 3 10−11 FSU-17 Hamzeloui et al. (2017) IX D18 Ar(n) 1.008 664 915 82(49) 4.9 3 10−10 AMDC-16 Huang et al. (2017) IX D19 Ar(1H) 1.007 825 032 241(94) 9.3 3 10−11 AMDC-16 Huang et al. (2017) IX D20 Ar(4He) 4.002 603 254 130(63) 1.6 3 10−11 AMDC-16 Huang et al. (2017) IX D21 ΔEB(1H+)/hc 1.096 787 717 4307(10) 3 107 m−1 9.1 3 10−13 ASD-18 IX D22 ΔEB(4He2+)/hc 6.372 195 4487(28) 3 107 m−1 4.4 3 10−10 ASD-18 IX D23 ΔEB(12C6+)/hc 83.083 850(25) 3 107 m−1 3.0 3 10−7 ASD-18 IX D24 ΔEI(3He+)/hc 43 888 919.36(3) m−1 6.8 3 10−10 ASD-18 IX D25 ΔEI(HD+)/hc 13 122 468.415(6) m−1 4.6 3 10−10 Liu et al. (2010) and Sprecher et al. (2010) IX Input datum relevant for the muon anomaly D26 R 0.003 707 2063(20) 5.4 3 10−7 BNL-06 Bennett et al. (2006) XVI.A Input data relevant for the muon mass and muon magnetic moment D27 E(58 MHz)/h 627 994.77(14) kHz 2.2 3 10−7 LAMPF-82 Mariam (1981) and Mariam et al. (1982) XVII.B D28 E(72 MHz)/h 668 223 166(57) Hz 8.6 3 10−8 LAMPF-99 Liu et al. (1999) XVII.B D29 ΔEMu/h 4 463 302.88(16) kHz 3.6 3 10−8 LAMPF-82 Mariam (1981) and Mariam et al. (1982) XVII.B D30 ΔEMu/h 4 463 302 765(53) Hz 1.2 3 10−8 LAMPF-99 Liu et al. (1999) XVII.B D31 δMu/h 0(85) Hz [1.9 3 10−8] theory XVII.A J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-33 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr predominantly hadronic (that is, with hadrons in intermediate states) contributions, respectively. The QED contribution may be written as aμ(QED)  ∞ n1 C(2n) μ α π n , (154) where C(2n) μ A(2n) 1 + A(2n) 2 (xμe) + A(2n) 2 (xμτ) + A(2n) 3 (xμe, xμτ) (155) with mass-independent coefficients A(2n) 1 given by Eqs. (51)–(55) and functions A(2n) 2 (x) and A(2n) 3 (x, y) evaluated at mass ratios mμ/mX for leptonX e orτ. Theexpression fortheQEDcontributionhas the same functional form as that for the electron anomaly described in Sec. VIII, except that the mass-dependent terms A(2n) 2 (x) are evaluated at different mass ratios, while contributions due to A(2n) 3 (x,y) are negligibly small for the electron anomaly. Contributions from the mass-dependent terms are generally more important for the muon anomaly. The mass-dependent functions A(2) 2 (x), A(2) 3 (x), and A(4) 3 (x,y) are zero. The remaining nonzero mass-dependent coefficients computed at the relevant mass ratios are given in Table XXIV. Their fractional contributions to the muonanomaly are given in Table XXV. Onlyfour of the mass-dependent QED corrections contribute significantly to the theoretical value for the muon anomaly. Finally, aμ(QED) based on the 2018 recommended value of α and lepton mass ratios is aμ(QED) 0.001 165 847 188 97(84) [7.2 3 10−10]. (156) The primarily electroweak contribution is (Czarnecki, Mar-ciano, and Vainshtein, 2003; Gnendiger, St¨ ockinger, and St¨ ockinger-Kim, 2013) aμ(weak) 154(1) 3 10−11 (157) and contains both the leading term and also some higher-order corrections. Five terms of the hadronic correction of the muon anomaly have been computed. They are aμ(had) aLO,VP μ (had) + aNLO,VP μ (had) + aNNLO,VP μ (had) + aLL μ (had) + aNLO,LL μ (had) + · · · , (158) corresponding to leading-order (LO), next-to-leading-order (NLO), and next-to-next-to-leading-order (NNLO) vacuum-polarization corrections and hadronic light-by-light (LL), and higher-order light-by-light (NLO,LL) scattering terms, respectively. Their values are aLO,VP μ (had) 6932.6(24.6) 3 10−11, (159) aNLO,VP μ (had) −98.2(4) 3 10−11 (160) from Keshavarzi, Nomura, and Teubner (2018) based on e+ −e− annihilation data. Davier et al. (2017) and Jegerlehner (2018) gave results that are consistent but slightly less accurate. Of these three TABLE XXII. Correlation coefficients r(xi, xj) > 0.0001 among the input data in Table XXI r(D5, D6) 0.1004 r(D5, D11) 0.0495 r(D5, D18) −0.0070 r(D5, D19) 0.0657 r(D6, D11) 0.0402 r(D6, D18) −0.0070 r(D6, D19) 0.0602 r(D7, D10) 0.3473 r(D8, D23) 0.9998 r(D9, D13) 0.7994 r(D11, D18) −0.0198 r(D11, D19) 0.1934 r(D18, D19) −0.1340 r(D27, D29) 0.2267 r(D28, D30) 0.1946 TABLE XXI. (Continued.) Input datum Value Rel. stand. unc. ur Lab. Reference(s) Sec. Input data relevant for the magnetic moments of light nuclei D32 μp/μN 2.792 847 344 62(82) 2.9 3 10−10 UMZ-17 Schneider et al. (2017) XV D33 μe(H)/μp(H) −658.210 7058(66) 1.0 3 10−8 MIT-72 Sec. III.C.3 of Mohr and Taylor (2000) XIV.D D34 μd(D)/μe(D) −4.664 345 392(50) 3 10−4 1.1 3 10−8 MIT-84 Sec. III.C.4 of Mohr and Taylor (2000) XIV.D D35 μe(H)/μ ′ p −658.215 9430(72) 1.1 3 10−8 MIT-77 Sec. III.C.6 of Mohr and Taylor (2000) XIV.D D36 μh ′ /μ ′ p −0.761 786 1313(33) 4.3 3 10−9 NPL-93 Flowers, Petley, and Richards (1993) XIV.D D37 μn/μ ′ p −0.684 996 94(16) 2.4 3 10−7 ILL-79 Sec. III.C.8 of Mohr and Taylor (2000) XIV.D D38 μp(HD)/μd(HD) 3.257 199 531(29) 8.9 3 10−9 StPtrsb-03 Neronov and Karshenboim (2003) XIV.D D39 μp(HD)/μd(HD) 3.257 199 514(21) 6.6 3 10−9 WarsU-12 Garbacz et al. (2012) XIV.D D40 μp(HD)/μd(HD) 3.257 199 516(10) 3.1 3 10−9 StPtrsb-12 Neronov and Seregin (2012) XIV.D D41 μt(HT)/μp(HT) 1.066 639 8933(21) 2.0 3 10−9 StPtrsb-11 Neronov and Aleksandrov (2011) XIV.D D42 σdp 20.20(2) 3 10−9 Puchalski, Komasa, and Pachucki (2015) XIV.C D43 σtp 24.14(2) 3 10−9 Puchalski, Komasa, and Pachucki (2015) XIV.C J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-34 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr publications, only Jegerlehner (2018) has included tau-lepton-decay data. The next-to-next-to-leading-order correction is aNNLO,VP μ (had) 12.4(1) 3 10−11 (161) from Kurz et al. (2014b). Light-by-light corrections are aLL μ (had) 103(29) 3 10−11 (162) from Jegerlehner (2018) and aNLO,LL μ (had) 3.0(2.0) 3 10−11 (163) from Colangelo et al. (2014). The combined hadronic contribution is then aμ(had) 6967(59) 3 10−11. (164) Based on the 2018 recommended value of α and lepton mass ratios, aμ(th) 1.165 918 13(38) 3 10−3 (165) for the theoretically predicted value of aμ with standard uncertainty u[aμ(th)] 38 3 10−11 3.3 3 10−7aμ. (166) TABLE XXIV. Mass-dependent functions A(2n) 2 (x), A(2n) 3 (x, y), and summed C(2n) μ coefficients for the QED contributions to the muon anomaly based on the 2018 recommended values of lepton mass ratios. The functions are evaluated at mass ratios xμe ≡mμ/me and/or xμτ ≡mμ/mτ n A(2n) 2 (xμe) A(2n) 2 (xμτ) A(2n) 3 (xμe, xμτ) C(2n) μ 1 0 0 0 0.5 2 1.094 258 3098(72) 0.000 078 076(10) 0 0.765 857 420(10) 3 22.868 379 99(17) 0.000 360 599(86) 0.000 527 738(71) 24.050 509 78(16) 4 132.6852(60) 0.042 4928(40) 0.062 72(4) 130.8782(60) 5 742.18(87) −0.068(5) 2.011(10) 750.80(89) TABLE XXIII. Observational equations for input data on H/D spectroscopy, muonic-H and -D Lamb shifts, and electron-proton or deuteron scattering given in Tables X, VIII, and XVIII as functions of adjusted constants. Labels in the first column correspond to those defined in the tables with input data. Subscript X is H or D for hydrogen or deuterium, respectively. The symbol ≐is defined in Sec. III. Energy levels of hydrogenic atoms, EX(nℓj; ΓX), are discussed in Sec. VII.A. Here, the symbol ΓX represents the six adjusted constants R∞, α, Ar(e), me/mμ, Ar(N), and rN such that when X H nucleus N p, the proton, and when X D nucleus N d, the deuteron. The Lamb shift for muonic atoms, ΔELS(μX), is discussed in Sec. XII. The last two entries are observational equations for nuclear-charge radii as obtained from electron-proton and electron-deuteron scattering data discussed in Sec. XIII Input data Observational equation A6–A8, A10–A19, A22, A23, A26–A29 nX(n1ℓ1j1 −n2ℓ2j2) ≐[EX(n2ℓ2j2; ΓX) + δX(n2ℓ2j2) −EX(n1ℓ1j1; ΓX) −δX(n1ℓ1j1)]/h A1–A4, A20, A21, A24, A25 nX(n1ℓ1j1 −n2ℓ2j2) −1 4nX(n3ℓ3j3 −n4ℓ4j4) ≐EX(n2ℓ2j2; ΓX) + δX(n2ℓ2j2) −EX(n1ℓ1j1; ΓX) −δX(n1ℓ1j1) −1 4 EX(n4ℓ4j4; ΓX) + δX(n4ℓ4j4) −EX(n3ℓ3j3; ΓX) −δX(n3ℓ3j3) h A5 nD(1S1/2 −2S1/2) −nH(1S1/2 −2S1/2) ≐{ED(2S1/2; ΓD) + δD(2S1/2) −ED(1S1/2; ΓD) −δD(1S1/2) −[EH(2S1/2; ΓH) + δH(2S1/2) −EH(1S1/2; ΓH) −δH(1S1/2)]}/h A9 nH(2S1/2 −4P, centroid) ≐{(EH(4P1/2; ΓH) + δH(4P1/2))/3 +2(EH(4P3/2; ΓH) + δH(4P3/2))/3 −EH(2S1/2; ΓH) −δH(2S1/2)}/h B1–B25 δX(nℓj) ≐δX(nℓj) C1–C6 ΔELS(μX) ≐E0X + E2Xr2 N + δth(μX) C7, C8 δELS(μX) ≐δth(μX) C9 rp ≐rp C10 rd ≐rd J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-35 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The largest and equally important contributions to the uncertainty of aμ(th) are from aLO,VP μ (had) and aLL μ (had). By comparison, the uncertainty of aμ(QED) is negligible. C. Analysis of experiment and theory for the muon anomaly Figure 7(a) compares three recent determinations of aLO,VP μ (had) based on electron-positron annihilation data with that mentioned in the 2014 CODATA report, i.e., the value from Hagiwara et al. (2011). Although the four values are consistent, the spread in values is rather large given that they are based on the same input data. This suggests that uncertainties remain underestimated. Neverthe-less, for this discussion we have chosen the value given by Keshavarzi, Nomura, and Teubner (2018), because it has the smallest uncertainty. In addition, Fig. 7 shows the results of two independent first-principle lattice-QCD evaluations of aLO,VP μ (had), both published in 2018. We have aLO,VP μ (had) 7111(75)(174) 3 10−11 (167) from Borsanyi et al. (2018) and aLO,VP μ (had) 7154(163)(92) 3 10−11 (168) from Blum et al. (2018). The first and second numbers in parentheses correspond to the statistical and systematic uncertainties, respectively. The systematic uncertainty is dominated by finite-volume artifacts. In Fig. 7, the two uncertainties are added in quadrature. Blum et al. (2018) also describe a model that merges data from electron-positron annihi-lation data with their lattice-QCD evaluation. This leads to a more ac-curate determination of aLO,VP μ (had) with the value aLO,VP μ (had)|hybrid 6925(27) 3 10−11 (169) consistent in both value and uncertainty with data solely based on electron-positron annihilation data. Figure 7(b) compares two evaluations of the leading-order light-by-light correction. Separated by almost ten years in publication date, the value has only slightly improved.The newestis considered here.As inthe 2014 CODATA report, based on the analyses of Dorokhov, Radzhabov, and Zhevlakov (2014a, 2014b), Nyffeler (2014), and Adikaram et al. (2015), not shown in the figure, we note that aLL μ (had) is model de-pendent and that a reliable estimate might still be missing. The experimental and theoretical values for the muon magnetic-moment anomaly, i.e., Eqs. (152) and (165), respectively, are compared in Fig. 8. The difference between experiment and theory is just under four times the uncertainty of the difference. This is larger than in the 2014 CODATA report, as both aLO,VP μ (had) and aLL μ (had) have become smaller. An expansion of only the uncertainty of aμ(th) to attempt to account for the spread in the valuesof aLO,VP μ (had) and aLL μ (had) would significantly reduce its contribution in a least-squares adjustment that includes both input data R and aμ(th). Expanding the uncertainties of aμ(th) andaμ(exp) toreducetheresidualforbothinputdatatolessthan two leads to a recommended value that ceases to be a useful reference value for future comparisons of theory and experiment. For all these reasons, the Task Group chose not to include aμ(th) in the 2018 ad-justment and to base the 2018 recommended value on experiment only. XVII. Electron-to-Muon Mass Ratio and Muon-to-Proton Magnetic-Moment Ratio Muonium (Mu) is an atom consisting of a (positively charged) antimuon and a (negatively charged) electron. Measurements of two muonium ground-state hyperfine transition energies in a strong mag-netic flux density combined with theoretical expressions for these en-ergies provide information on the electron-to-muon mass ratio, me/mμ, aswellastheantimuon-to-protonmagnetic-momentratio,μμ+/μp.Here, the proton magnetic moment only appears because the applied magnetic fieldorfluxdensityisfoundby“replacing”themuoniumwithaprotonin theexperimentalapparatusandmeasuringthetransitionfrequencyωp of its precessing spin. (More precisely, replacing muonium with a liquid-TABLE XXV. Fractional contribution of mass-dependent functions A(2n) 2 (x) and A(2n) 3 (x,y) for the QED contributions to the muon anomaly based on the 2018 recommended values for α and lepton mass ratios. Fractional contributions are defined asA(2n) j 3 (α/π)n/aμ(th) forj 2 or3 and the relative standard uncertainty of aμ(th) is 3.3 3 10−7. The functions are evaluated at mass ratios xμe ≡mμ/me and/or xμτ ≡mμ/mτ. n A(2n) 2 (xμe) A(2n) 2 (xμτ) A(2n) 3 (xμe, xμτ) 2 5.06 3 10−3 3.61 3 10−7 3 2.46 3 10−4 3.88 3 10−9 5.67 3 10−9 4 3.31 3 10−6 1.06 3 10−9 1.57 3 10−9 5 4.30 3 10−8 −3.94 3 10−12 1.17 3 10−10 FIG. 7. Comparison ofrecent determinationsof the leading-orderhadronic(LO)vacuum-polarization [panel (a)] and light-by-light (LL) [panel (b)] contributions to the muon anomaly. Error bars are one-standard-deviation uncertainties. The LO,VP, and LL contributions limit the uncertainty of aμ(th). The horizontal interval of the two panels is the same so that uncertainties can be compared. From top to bottom, data are from Hagiwaraetal.(2011),Davieretal.(2017),Jegerlehner(2018),Keshavarzi,Nomura,and Teubner (2018), Borsanyi et al. (2018), and Blum et al. (2018) in panel (a) and from Jegerlehner and Nyffeler (2009) and Jegerlehner (2018) in panel (b). J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-36 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr water sample, measuring the proton spin-precession frequency in water, and accounting for a shielding correction.) In the remainder of this section, we summarize the theoretical determination of the zero-flux-density muonium hyperfine splitting (HFS) and the experimental measurements at field fluxes between one and two tesla. Results of relevant calculations and measurements are given along with references to new work; references to the original literature used in earlier CODATA adjustments are not repeated. We finish with an analysis of the data. A. Theory of the muonium ground-state hyperfine splitting The theoretical expression for the muonium hyperfine energy splitting absent a magnetic field may be factored into ΔEMu(th) ΔEFF (170) with the Fermi energy formula ΔEF 16 3 hcR∞Z3α2me mμ 1 + me mμ  −3 , (171) which contains the main dependence on fundamental constants, and a function F 1 + α/π + ··· that only depends weakly on them. (Recall Eh 2R∞hc α2mec2.) The charge of the antimuon is specified by Ze rather than e in order to identify the source of terms contributing to ΔEMu(th). The Fermi formula in Eq. (171) is expressed in terms of our adjusted constants R∞, α, and me/mμ. The relative uncertainties of R∞and α are much smaller than those for the measured ΔEMu.Hence, a measurement of ΔEMu determines the electron-to-muon mass ratio. Thegeneralexpressionforthehyperfinesplittingand thusalsoF is ΔEMu(th) ΔED + ΔErad + ΔErec + ΔEr-r + ΔEweak + ΔEhad, (172) where subscripts D, rad, rec, r-r, weak, and had denote the Dirac, radiative, recoil, radiative-recoil, electroweak, and hadronic contri-butions to the hyperfine splitting, respectively. The Dirac equation yields ΔED ΔEF(1 + aμ) 1 + 3 2(Zα)2 + 17 8 (Zα)4 + · · · , (173) where aμ is the muon magnetic-moment anomaly. Radiative cor-rections are ΔErad ΔEF(1 + aμ) ∞ n1 D(2n)(Zα)α π n , (174) where functions D(2n)(x) are contributions from n virtual photons. The leading term is D(2)(x) A(2) 1 + ln 2 −5 2 πx + −2 3ln2(x−2) + 281 360 −8 3 ln 2 ln(x−2) + 16.9037 . . . x2 + 5 2 ln 2 −547 96 ln(x−2) πx3 + G(x)x3, (175) where A(2) 1 1/2, as in Eq. (51). The function G(x) accounts for all higher-order contributions in powers of x; it can be divided into self-energy (SE) and vacuum-polarization (VP) contributions, G(x) GSE(x) + GVP(x). Yerokhin and Jentschura (2008, 2010) and Karshenboim,Ivanov,and Shabaev(1999,2000)havecalculated theone-loop self-energy and vacuum-polarization contributions for the mu-onium HFS with x α. Their results are GSE(α) −13.8308(43) (176) and GVP(α) 7.227(9), (177) where the latter uncertainty is meant to account for neglected higher-order Uehling-potential terms; it corresponds to energy uncertainties less than h 3 0.1 Hz, and is thus entirely negligible. For D(4)(x), we have D(4)(x) A(4) 1 + 0.770 99(2)πx + −1 3ln2(x−2) −0.6390 . . . 3 ln(x−2) + 10(2.5) x2 + · · · , (178) where A(4) 1 is given in Eq. (52). The next term is D(6)(x) A(6) 1 + · · · , (179) where the leading contribution A(6) 1 is given in Eq. (53), but only partial results of relative order Zα have been calculated (Eides and Shelyuto, 2007). Higher-order functions D(2n)(x) with n > 3 are expected to be negligible. The recoil contribution is FIG. 8. Comparison of the experimental and theoretical value for the muon anomaly. Values have been scaled by the uncertainty of the 2018 recommended value. ΔErec ΔEF me mμ − 3 1 −(me/mμ)2 ln mμ me  Zα π + 1 (1 + me/mμ)2ln(Zα)−2 −8 ln 2 + 65 18 + 9 2π2ln2 mμ me  +  27 2π2 −1 ln mμ me  + 93 4π2 + 33ζ(3) π2 −13 12 −12 ln 2me mμ (Zα)2 + −3 2 ln mμ me ln(Zα)−2 −1 6ln2(Zα)−2 + 101 18 −10 ln 2 ln(Zα)−2 + 40(10) (Zα)3 π  + · · · . (180) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-37 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXVI. Observational equations for input data in Tables XXI and XXVII as functions of the adjusted constants. The data determine the fine-structure constant, electron and muon masses and anomalies, masses and magnetic moments of light nuclei, as well as the lattice spacing of an ideal natural Si crystal and x-ray units. The labels in the first column correspond to those in the first column of Tables XXI and XXVII. For simplicity, the lengthier functions are not explicitly given. See Sec. III for an explanation of the symbol ≐ Input data Observational equation Sec. D1 ae(exp) ≐ae(th) + δth(e) VIII D2 δe ≐δth(e) VIII D3, D4 h m(X) ≐Ar(e) Ar(X) cα2 2R∞ X D5, D6, D18 Ar(X) ≐Ar(X) IX D7 ωs(12C5+) ωc(12C5+) ≐−ge(12C5+) + δth(C) 10Ar(e) [12 −5Ar(e) + ΔEB(12C 5+)α2Ar(e)/2R∞hc] XI.C D8, D12, D21–D23 ΔEB(Xn+) ≐ΔEB(Xn+) IX D9 δC ≐δth(C) XI.C D10 ωs(28Si13+) ωc(28Si13+) ≐−ge(28Si13+) + δth(Si) 26Ar(e) Ar(28Si 13+) XI.C D11 Ar(28Si) ≐Ar(28Si13+) + 13Ar(e) −ΔEB(28Si13+)α2Ar(e)/2R∞hc IX D13 δSi ≐δth(Si) XI.C D14 ωc(d) ωc(12C6+) ≐12 −6Ar(e) + ΔEB(12C6+)α2Ar(e)/2R∞hc 6Ar(d) IX D15 ωc(12C6+) ωc(p) ≐ 6Ar(p) 12 −6Ar(e) + ΔEB(12C6+)α2Ar(e)/2R∞hc IX D16 ωc(t) ωc(3He+) ≐Ar(h) + Ar(e) −ΔEI(3He+)α2Ar(e)/2R∞hc Ar(t) IX D17 ωc(HD+) ωc(3He+) ≐ Ar(h) + Ar(e) −EI(3He+)α2Ar(e)/2R∞hc Ar(p) + Ar(d) + Ar(e) −ΔEI(HD+)α2Ar(e)/2R∞hc IX D19 Ar(1H) ≐Ar(p) + Ar(e) −ΔEB(1H+)α2Ar(e)/2R∞hc IX D20 Ar(4He) ≐Ar(α) + 2Ar(e) −ΔEB(4He2+)α2Ar(e)/2R∞hc IX D24, D25 ΔEI(X+) ≐ΔEI(X+) IX D26 R ≐− aμ 1 + ae(th) + δth(e) me mμ μe μp XVI.A D27, D28 E(ωp) ≐E(ωp; R∞, α, me mμ , aμ, μe μp , δth(e), δth(Mu)) XVII.B D29, D30 ΔEMu ≐ΔEMu(th; R∞, α, me mμ , aμ) + δth(Mu) XVII.A D31 δMu ≐δth(Mu) XVII.A D32 μp μN ≐−(1 + ae(th) + δth(e)) Ar(p) Ar(e) μp μe XV D33 μe(H) μp(H) ≐ge(H) ge gp(H) gp −1μe μp XIV.D J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-38 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The leading-order O(ΔEFα2) radiative-recoil contribution is ΔEr-r ΔEFα π 2me mμ −2ln2 mμ me  + 13 12 ln mμ me  + 21 2 ζ(3) + π2 6 + 35 9  + 4 3ln2α−2 + 16 3 ln 2 −341 180 lnα−2 −40(10) πα + −4 3 ln3 mμ me  + 4 3ln2 mμ me α π −ΔEFα2 me mμ  2 6ln 2 + 13 6 + · · · , (181) where, for simplicity, the explicit dependence on Z is not shown. Single-logarithmic and nonlogarithmic three-loop radiative-recoil corrections of O(ΔEFα3) are (Eides and Shelyuto, 2014) ΔEFα π 3me mμ −6π2ln 2 + π2 3 + 27 8 ln mμ me + 68.507(2) h 3 −30.99 Hz. (182) Uncalculated remaining terms of the same order as those included in Eq. (182) have been estimated by Eides and Shelyuto (2014) to be about h 3 10 Hz to h 3 15 Hz. Additional radiative-recoil correc-tions have been calculated, but are negligibly small, less than h 3 0.5 Hz. The electroweak contribution due to the exchange of a Z0 boson is (Eides, 1996) ΔEweak/h −65 Hz, (183) while for the hadronic vacuum-polarization contribution we have ΔEhad/h 237.7(1.5) Hz. (184) This hadronic contribution combines the result of Nomura and Teubner (2013) with a newly computed h 3 4.97(19) Hz TABLE XXVI. (Continued.) Input data Observational equation Sec. D34 μd(D) μe(D) ≐gd(D) gd ge(D) ge −1μd μe XIV.D D35 μe(H) μ ′ p ≐ge(H) ge μe μ ′ p XIV.D D36 μ ′ h μ ′ p ≐μ ′ h μ ′ p XIV.D D37 μn μ ′ p ≐μn μ ′ p XIV.D D38–D40 μp(HD) μd(HD) ≐[1 + σdp] μp μe μe μd XIV.D D41 μt(HT) μp(HT) ≐ 1 1 + σtp μt μp XIV.D D42, D43 σNN′ ≐σNN′ XIV.C E1–E4 1 −d220(Y) d220(X) ≐1 −d220(Y) d220(X) XVIII E5–E13 d220(X) d220(Y) −1 ≐d220(X) d220(Y) −1 XVIII E14–E17 d220(X) ≐d220(X) XVIII E18, E19 λ(CuKα1) d220(X) ≐1537.400 xu(CuKα1) d220(X) XVIII E20 λ(WKα1) d220(N) ≐0.209 010 0 ˚ A∗ d220(N) XVIII E21 λ(MoKα1) d220(N) ≐707.831 xu(MoKα1) d220(N) XVIII J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-39 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr contribution from Shelyuto, Karshenboim, and Eidelman (2018). A negligible contribution (≈h 3 0.0065 Hz) from the hadronic light-by-light correction has been given by Karshenboim, Shelyuto, and Vainshtein (2008). The uncertainty of ΔEMu(th) in Eq. (172) is determined, from the largest to smallest component, by those in ΔErec, ΔEr-r, ΔErad, and ΔEhad. The h 3 1.5 Hz uncertainty in the latter is only of marginal interest. For ΔErec, the total uncertainty is h 3 64 Hz and has three components. They are h 3 53 Hz from twice the uncertainty 10 of the number 40 in Eq. (180) as discussed in the 2002 CODATA adjust-ment, h 3 34 Hz due to a possible recoil correction of order ΔEF(me/mμ) 3 (Zα)3ln(me/mμ), and, finally, h 3 6 Hz to reflect a possible recoil term of order ΔEF(me/mμ) 3 (Zα)4ln2(Zα)−2. The uncertainty in ΔEr-r is h 3 55 Hz, with h 3 53 Hz due to twice the uncertainty 10 of the number −40 in Eq. (181) as above, and h 3 15 Hz as discussed in connection with Eq. (182). The uncertainty in ΔErad is h 3 5 Hz and consists of two components: h 3 4 Hz from an uncertainty of 1 in GVP(α) due to the uncalculated contribution of order α(Zα)3, and h 3 3 Hz from the uncertainty 2.5 of the number 10 in the function D(4)(x). The final uncertainty in ΔEMu(th) is then u[ΔEMu(th)]/h 85 Hz. (185) For the least-squares calculations, we use the observational equations ΔEMu ≐ΔEMu(th) + δth(Mu) (186) and δMu ≐δth(Mu), (187) where δth(Mu) accounts for the uncertainty of the theoretical ex-pression and is taken to be an adjusted constant. Based on Eq. (185), its corresponding input datum in the 2018 adjustment is δMu 0(85) Hz. The input data ΔEMu are discussed later. The theoretical hyperfine splitting ΔEMu(th) is mainly a function of the adjusted constants R∞, α, and me/mμ. Finally, the covariance between ΔEMu and δMu is zero. B. Measurements of muonium transition energies The two most precise determinations of muonium hyperfine transition energies were carried out by researchers at the Meson Physics Facility at Los Alamos (LAMPF), New Mexico, USA and published in 1982 and 1999, respectively. These transition energies are compared to differences between eigenvalues of the Breit-Rabi Hamiltonian (Breitand Rabi, 1931; Millman, Rabi, and Zacharias, 1938) modified for muonium using a magnetic flux density determined from the free-proton NMR frequency measured in the apparatus. The experiments were reviewed in the 1998 CODATA adjustment. Data reported in 1982 by Mariam (1981) and Mariam et al. (1982) are ΔEMu/h 4 463 302.88(16) kHz [3.6 3 10−8] (188) for the hyperfine splitting and E(ωp)/h 627 994.77(14) kHz [2.2 3 10−7] (189) for the difference of two transition energies with correlation coefficient r[ΔEMu, E(ωp)] 0.227. (190) In fact, ΔEMu and E(ωp) are the sum and difference of two measured transition energies, Zωp 2μpB is the free-proton NMR transition energy, and only E(ωp) depends on ωp. In this experiment, Zωp h 3 57.972 993 MHz at its 1.3616 T magnetic flux density. The data reported in 1999 by Liu et al. (1999) are ΔEMu/h 4 463 302 765(53) Hz [1.2 3 10−8], (191) E(ωp)/h 668 223 166(57) Hz [8.6 3 10−8] (192) with correlation coefficient r[ΔEMu, E(ωp)] 0.195 (193) and Zωp h 3 72.320 000 MHz for the proton transition energy in a flux density of approximately 1.7 T. The observational equations are Eq. (186) and E(ωp) ≐−(We−+ Wμ+) +   [ΔEMu(th) + δth(Mu)]2 + (We−−Wμ+)2  , (194) where Wℓ −[μℓ(Mu)/μp]Zωp. Explicitly expressing We−and Wμ+ in terms of adjusted constants then yields We− −ge(Mu) ge μe− μp Zωp (195) and Wμ+ gμ(Mu) gμ 1 + aμ 1 + ae(th) + δth(e) me mμ μe− μp Zωp. (196) Here, we have used the fact that μℓ(Mu) gℓ(Mu)eZ/4mℓfor the magnitude of the magnetic moment of lepton ℓin muonium (see Secs. VIII and XIV.A), |gℓ| 2(1 + aℓ), and crucially gμ+ −gμ−. The g-factor ratios ge(Mu)/ge and gμ(Mu)/gμ are given in Table XX. The adjusted constants in Eq. (186) and Eqs. (194)–(196) are the magnetic-moment anomaly aμ, mass ratio me/mμ, magnetic-moment ratio μe−/μp, and additive constants δth(Mu) and δth(e). The latter two constants account for uncomputed theoretical contributions to ΔEMu(th) and ae(th), respectively. Finally, ΔEMu(th) is mainly a function of adjusted constants me/mμ, R∞, and α; ae(th) is mainly a function of R∞and α. The accurately measured or computed Zωp and ratios gℓ(Mu)/gℓare treated as exact in our least-squares adjustment. It is worth noting that in Eq. (194) the energy We−> 0, and at the flux densities used in the experiments |We−| ∼ΔEMu(th) and |Wμ+| ≪|We−|. Consequently, the right-hand side of Eq. (194) only has a weak dependence on ΔEMu(th) and the corresponding input datum does not significantly constrain ΔEMu(th) and thus me/mμ in the adjustment. For ease of reference, the experimental and theoretical input data for muonium hyperfine splittings are summarized in Table XXI and given labels D27 through D31. Observational equations are sum-marized in Table XXVI. C. Analysis of the muonium hyperfine splitting and mass ratio mμ/me The 2018 recommended value for the muonium hyperfine split-ting is J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-40 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr ΔEMu(th) + δth(Mu) h 3 4 463 302 776(51) Hz [1.1 3 10−8], (197) which is consistent in both value and uncertainty with the most accurately measured value of Eq. (191). More importantly, the prediction δth(Mu)/h −4(83) Hz for the additive constant falls well inside the 85 Hz theoretical uncertainty. As δth(Mu) is a measure of uncomputed terms in the theory, the value implies that the theory is sufficiently accurate given the current constraints. Eides (2019) gave an alternative prediction for the uncertainty of the recommended muonium hyperfine splitting. The 2018 recommended value for the muon-to-electron mass ratio is mμ/me 206.768 2830(46) (198) and has a relative standard uncertainty of 2.2 3 10−8 that is nearly twice that of the 1999 measurement of ΔEMu in Eq. (191). This in-crease simply reflects the fact that the square of the relative standard uncertainty for mμ/me to good approximation satisfies u2 r(mμ/me) u2 r(ΔEMu(th)) + u2 r(ΔEMu), (199) which follows from error propagation with Eqs. (170) and (186). The relative standard uncertainties in the theory for and measurement of the hyperfine splitting are almost the same. New data on the muonic hyperfine splitting by the MuSEUM collaboration at the J-PARC Muon Science Facility are expected in the near future (Strasser et al., 2019). XVIII. Lattice Spacings of Silicon Crystals In this section, we summarize efforts to determine the lattice spacing of an ideal (or nearly perfect) natural-silicon single crystal. We also present values for several historical x-ray units in terms of the SI unit meter. Three stable isotopes of silicon exist in nature. They are 28Si, 29Si, and 30Si with amount-of-substance fractions x(ASi) of approximately 0.92, 0.05, and 0.03, respectively. Highly enriched silicon single crystals have x(28Si) ≈0.999 96. The quantities of interest are the {220} crystal lattice spacing d220(X) in meters of a number of different crystals X using a com-bined x-ray and optical interferometer (XROI) as well as the fractional differences d220(X) −d220(Y) d220(Y) (200) for single crystals X and Y, determined using a lattice comparator based on x-ray double-crystal nondispersive diffractometry. Data on eight natural Si crystals, in the literature denoted by WASO 4.2a, WASO 04, WASO 17, NRLM3, NRLM4, MO∗, ILL, and N, are relevant for the 2018 CODATA adjustment. Their lattice spacings d220(X) are adjusted constants in our least-squares calculations. The simplified notation W4.2a, W04, W17, NR3, and NR4 is used in quantity symbols and tables for the first five crystals. The lattice spacing for the ideal natural-silicon single crystal d220 is an adjusted constant. Lattice-spacing data included in this adjustment are items E1–E17 in Table XXVII and quoted at a temperature of 22.5 °C and in vacuum. All data but one were already included in the 2014 ad-justment. The new measurement is from Kessler et al. (2017) at the National Institute of Standards and Technology, Gaithersburg, USA and given as item E13 in the table. They measured the fractional difference for natural Si crystals ILL and W04. Consistent with previous adjustments and, in particular, following the discussions by Mohr and Taylor (1999, 2000), we expand their quoted uncertainty by 20 3 10−9 in quadrature to properly account for uncertainties due to carbon and oxygen impurities in the crystal. The copper Kα1 x unit with symbol xu(CuKα1), the molyb-denum Kα1 x unit with symbol xu(MoKα1), and the ˚ angstr¨ om star with symbol ˚ A∗are historic x-ray units that are still of current interest. They are defined by assigning an exact, conventional value to the wavelength of the CuKα1, MoKα1, and WKα1 x-ray lines. These assigned wavelengths for λ(CuKα1), λ(MoKα1), and λ(WKα1) are 1537.400 xu(CuKα1), 707.400 xu(MoKα1), and 0.209 010 0 ˚ A∗, re-spectively. The four relevant experimental input data are the mea-sured ratios of CuKα1, MoKα1, and WKα1 wavelengths to the {220} lattice spacings of crystals WASO 4.2a and N and are items E18–E21 in Table XXVII. In the least-squares calculations, the units xu(CuKα1), xu(MoKα1), and ˚ A∗are adjusted constants. The correlation coefficients among the data on lattice spacings and x-ray units are given in Table XXVIII. Discussions of these correlations can be found in previous adjustments. The sole new data point has no correlations with previous data. Observational equations may be found in Table XXVI. XIX. Newtonian Constant of Gravitation Table XXIX summarizes the 16 measured values of the New-tonian constant of gravitation G considered as input data for the 2018 adjustment. Since the 2014 adjustment, two new values have become available (Li et al., 2018) and corrections have been applied to a previously reported value (Parks and Faller, 2010). Figure 9 illus-trates all input data. The measurements are inconsistent and an expansion factor of 3.9 is required to bring all residuals to within a factor of two from the 2018 recommended value of G 6.674 30(15) 3 10−11 m3 kg−1 s−2 [2.2 3 10−5]. (201) The five measurements that contribute most to this value are the UWash-00, UZur-06, UCI-14, and the HUSTA,T-18 values. The re-siduals of the data from BIPM-14 and JILA-18 are the largest and determined our expansion factor. We note, however, that the in-consistencies are smaller than in our previous 2014 adjustment. We briefly describe the new measurements in the next two sections. Details regarding older measurements can be found in descriptions of previous CODATA adjustments. A. Corrected value of the 2010 measurement at JILA In 2010, Parks and Faller (2010) at JILA, University of Colorado and National Institute of Standards and Technology, Boulder, Col-orado, USA used simple pendulums to determine G in an experi-mental design similar to that of Kleinevoß (2002) and Kleinvoß et al. (2002). Two pendulums, each with a cylindrical test mass suspended by four wires, were aligned such that the cylinders were colinear. As surrounding source masses moved, changes in the separation between the test masses were interferometrically monitored. In 2016, the apparatus was transferred to NIST, Gaithersburg, Maryland, USA with the goal of repeating the experiment. During initial preparations at NIST, two calculational errors were discovered, both associated with the rotation of the test masses when they are horizontally displaced. Rotation occurred because the connection J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-41 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr points of the suspension wires to the test masses were located above their center of mass. The first error was in the derivation of the pendulums’ effective spring constants used to calculate the gravita-tional force from a measured horizontal separation between the test masses. The contribution from rotation to the spring constants was overestimated. The initial relative correction to G of 5.8(0.4) 3 10−5 has been updated to 0.40(30) 3 10−5. The second error arises from the interferometer axis being displaced by about 0.95(30) mm above the horizontal plane containing the test masses’ center of mass, resulting in an Abbe error. The relative correction to G to remove the Abbe error is 9.4(3.0) 3 10−5. Applying these two corrections results in a relative increase of their 2010 value for G of 3.9 3 10−5 and an increase of the relative uncertainty from 2.1 3 10−5 to 3.7 3 10−5. The new JILA value and uncertainty (Parks and Faller, 2019) are labeled JILA-18 in Table XXIX and Fig. 9. B. Measurements from the Huazhong University of Science and Technology Two new determinations of G, using independent methods and having the lowest uncertainties to date, were reported in 2018. Both measurements were performed at Huazhong University of Science and Technology (HUST), Wuhan, People’s Republic of China (Li et al., 2018). The first determination used the time-of-swing (TOS) method where the change in oscillation frequency of a torsion pendulum for two different positions of source masses is measured. These measurements were performed on two independent appara-tuses located in laboratories separated by 150 m. In one apparatus (TOS-I), the researchers used three different silica fibers to check for fiber-induced systematics. In the other apparatus (TOS-II), the same fiber was used for all measurements. The largest uncertainty com-ponent for all data sets was statistical, ranging from 10 to 30 parts in 106 relative uncertainty. The determination of the horizontal TABLE XXVIII. Correlation coefficients r(xi, xj) > 0.0001 among the input data for the lattice spacing of an ideal natural Si crystal and x-ray units given in Table XXVII r(E1, E2) 0.4214 r(E1, E3) 0.5158 r(E1, E4) −0.2877 r(E1, E7) −0.3674 r(E1, E10) 0.0648 r(E1, E12) 0.0648 r(E2, E3) 0.4213 r(E2, E4) 0.0960 r(E2, E7) 0.0530 r(E2, E10) 0.0530 r(E2, E12) 0.0530 r(E3, E4) 0.1175 r(E3, E7) 0.0648 r(E3, E10) −0.3674 r(E3, E12) 0.0648 r(E4, E7) 0.5037 r(E4, E10) 0.0657 r(E4, E12) 0.0657 r(E5, E6) 0.4685 r(E5, E8) 0.3718 r(E5, E9) 0.5017 r(E6, E8) 0.3472 r(E6, E9) 0.4685 r(E7, E10) 0.5093 r(E7, E12) 0.5093 r(E8, E9) 0.3718 r(E10, E12) 0.5093 r(E14, E15) 0.0230 r(E14, E16) 0.0230 r(E15, E16) 0.0269 TABLEXXVII. Input datafor thedeterminationofthe 2018recommendedvaluesof thelattice spacingsofan idealnatural Sicrystaland x-rayunits.Thelabelin the firstcolumn isused in Table XXVIII to list correlation coefficients among the data and in Table XXVI for observational equations. The uncertainties are not those as originally published, but corrected according the considerations in Sec. III.I of Mohr and Taylor (2000). For additional information about the uncertainties of data published after the closing data of the 1998 CODATA adjustment,see also the corresponding text in this and other CODATA publications.Columns four and five give the reference and an abbreviationof the name of the laboratory in which the experiment has been performed, and year of publication. An extensive list of abbreviations is found at the end of this report Input datum Value Relat. stand. uncert. ur Laboratory Reference(s) E1 1 −d220(W17)/d220(ILL) −8(22) 3 10−9 NIST-99 Kessler et al. (2000) E2 1 −d220(MO∗)/d220(ILL) 86(27) 3 10−9 NIST-99 Kessler et al. (2000) E3 1 −d220(NR3)/d220(ILL) 33(22) 3 10−9 NIST-99 Kessler et al. (2000) E4 1 −d220(N)/d220(W17) 7(22) 3 10−9 NIST-97 Kessler, Schweppe, and Deslattes (1997) E5 d220(W4.2a)/d220(W04) −1 −1(21) 3 10−9 PTB-98 Martin et al. (1998) E6 d220(W17)/d220(W04) −1 22(22) 3 10−9 PTB-98 Martin et al. (1998) E7 d220(W17)/d220(W04) −1 11(21) 3 10−9 NIST-06 Hanke and Kessler (2005) E8 d220(MO∗)/d220(W04) −1 −103(28) 3 10−9 PTB-98 Martin et al. (1998) E9 d220(NR3)/d220(W04) −1 −23(21) 3 10−9 PTB-98 Martin et al. (1998) E10 d220(NR3)/d220(W04) −1 −11(21) 3 10−9 NIST-06 Hanke and Kessler (2005) E11 d220/d220(W04) −1 10(11) 3 10−9 PTB-03 Becker et al. (2003) E12 d220(NR4)/d220(W04) −1 25(21) 3 10−9 NIST-06 Hanke and Kessler (2005) E13 d220(ILL)/d220(W04) −1 −20(22) 3 10−9 NIST-17 Kessler et al. (2017) E14 d220(MO∗) 192 015.5508(42) fm 2.2 3 10−8 INRIM-08 Ferroglio, Mana, and Massa (2008) E15 d220(W04) 192 015.5702(29) fm 1.5 3 10−8 INRIM-09 Massa et al. (2009) E16 d220(W4.2a) 192 015.5691(29) fm 1.5 3 10−8 INRIM-09 Massa, Mana, and Kuetgens (2009) E17 d220(W4.2a) 192 015.563(12) fm 6.2 3 10−8 PTB-81 Becker et al. (1981); E18 λ(Cu Kα1)/d220(W4.2a) 0.802 327 11(24) 3.0 3 10−7 FSUJ/PTB-91 Windisch and Becker (1990); and H¨ artwig et al. (1991) E19 λ(Cu Kα1)/d220(N) 0.802 328 04(77) 9.6 3 10−7 NIST-73 Deslattes and Henins (1973) E20 λ(W Kα1)/d220(N) 0.108 852 175(98) 9.0 3 10−7 NIST-79 Kessler, Deslattes, and Henins (1979) E21 λ(Mo Kα1)/d220(N) 0.369 406 04(19) 5.3 3 10−7 NIST-73 Deslattes and Henins (1973) J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-42 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr separation between the geometric centers of the spherical source masses had the largest systematic uncertainty; its relative uncertainty ranged from 8.5 to 9.5 parts in 106. In the CODATA adjustment, we only use the combined value for G from the two TOS apparatuses. This input datum is labeled HUSTT-18 in Table XXIX and Fig. 9. Small correlations with the 2009 TOS determination of G at HUST (Luo et al., 2009) exist because the same source masses were used in TOS-II, and the same measurement instrumentation and methods for the determination of various systematic uncertainties were used. A conservative estimate for the correlation coefficient between HUST-09 and HUSTT-18 is 0.068. The second 2018 HUST experiment used the angular acceler-ation feedback (AAF) method where turntables rotate a torsion pendulum and source masses independently at nominally constant but opposite and different rotation rates. Feedback control com-pensates for gravitational torque acting on the rotating torsion pendulum such that the pendulum does not move with respect to its rotating frame. The difference in rotation rate of the source masses’ and pendulum’s turntables is held constant by a second feedback controller. For infinite feedback gain, the angular acceleration of the torsion pendulum’s turntable is identical to the gravitational angular acceleration generated by the source masses and effects of envi-ronmental gravitational forces are minimized. The final result for G based on AAF, here labeled HUSTA-18, combines values from data sets AAF-I, AAF-II, and AAF-III. Set AAF-I had a different rotation rate from AAF-II and AAF-III. A different research team obtained data set AAF-III. For AAF-III, an improved pre-hanger fiber and additional Mu-metal shielding around the torsion pendulum were used as well. The largest uncertainty components for all data sets were the horizontal and vertical distance determinations be-tween the geometric centers of the spherical source masses, with relative uncertainties of 9.0 and 5.8 parts in 106, respectively. While the two HUST-18 results have the lowest uncertainty of any measurements of G to date and agree with the 2018 recommended value within two standard uncertainties of that value, the difference between the two new HUST values is 2.7 times the standard un-certainty of their difference. Furthermore, the HUSTT-18 and HUSTA-18values of G exceed the HUST-09 value by about3.5 and 5.1 times the standard uncertainty of their respective differences. Pres-ently, there are no explanations for the inconsistencies. XX. Electroweak Quantities There are a few cases in the 2018 adjustment, as in previous ad-justments, where an inexact constant is used in the analysis of input data but not treated as an adjusted quantity, because the adjustment has a negligible effect on its value. Three such constants, used in the cal-culation of the theoretical expression for the electron magnetic-moment anomaly ae, are the mass of the tau lepton mτ, the Fermi coupling constant GF, and sine squared of the weak mixing angle sin2 θW. These TABLE XXIX. Input data for the Newtonian constant of gravitation G relevant to the 2018 adjustment. The first two columns give the reference and an abbreviation of the name of the laboratory in which the experiment has been performed, and year of publication. The data are uncorrelated except for three cases with correlation coefficients r(NIST-82, LANL-97) 0.351, r(HUST-05, HUST-09) 0.134, and r(HUST-09, HUSTT-18) 0.068 Source Identification Method G(10−11 kg−1 3 m3 s−2) Rel. stand. uncert. ur Luther and Towler (1982) NIST-82 Fiber torsion balance, dynamic mode 6.672 48(43) 6.4 3 10−5 Karagioz and Izmailov (1996) TR&D-96 Fiber torsion balance, dynamic mode 6.672 9(5) 7.5 3 10−5 Bagley and Luther (1997) LANL-97 Fiber torsion balance, dynamic mode 6.673 98(70) 1.0 3 10−4 Gundlach and Merkowitz (2000, 2002) UWash-00 Fiber torsion balance, dynamic compensation 6.674 255(92) 1.4 3 10−5 Quinn et al. (2001) BIPM-01 Strip torsion balance, compensation mode, static deflection 6.675 59(27) 4.0 3 10−5 Kleinevoß (2002) and Kleinvoß et al. (2002) UWup-02 Suspended body, displacement 6.674 22(98) 1.5 3 10−4 Armstrong and Fitzgerald (2003) MSL-03 Strip torsion balance, compensation mode 6.673 87(27) 4.0 3 10−5 Hu, Guo, and Luo (2005) HUST-05 Fiber torsion balance, dynamic mode 6.672 22(87) 1.3 3 10−4 Schlamminger et al. (2006) UZur-06 Stationary body, weight change 6.674 25(12) 1.9 3 10−5 Luo et al. (2009) and Tu et al. (2010) HUST-09 Fiber torsion balance, dynamic mode 6.673 49(18) 2.7 3 10−5 Quinn et al. (2013, 2014) BIPM-14 Strip torsion balance, compensation mode, static deflection 6.675 54(16) 2.4 3 10−5 Prevedelli et al. (2014) and Rosi et al. (2014) LENS-14 Double atom interferometer, gravity gradiometer 6.671 91(99) 1.5 3 10−4 Newman et al. (2014) UCI-14 Cryogenic torsion balance, dynamic mode 6.674 35(13) 1.9 3 10−5 Li et al. (2018) HUSTT-18 Fiber torsion balance, dynamic mode 6.674 184(78) 1.2 3 10−5 Li et al. (2018) HUSTA-18 Fiber torsion balance, dynamic compensation 6.674 484(77) 1.2 3 10−5 Parks and Faller (2019) JILA-18 Suspended body, displacement 6.672 60(25) 3.7 3 10−5 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-43 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr are electroweak quantities with values obtained from the most recent report of the Particle Data Group (Tanabashi et al., 2018): mτc2 1776.86(12) MeV [6.8 3 10−5], (202) GF (Zc)3 1.166 3787(6) 3 10−5 GeV−2 [5.1 3 10−7], (203) sin2 θW 0.222 90(30) [1.3 3 10−3]. (204) We note that sin2 θW 1 −(mW/mZ)2, where mW and mZ are the masses of the W ± and Z0 bosons, respectively. The Particle Data Group’s value mW/mZ 0.881 53(17) leads to the value of sin2 θW already given. The uncertainty of this mass ratio has decreased by almost a factorof ten when compared to that inthe 2014 adjustment. Finally, the accuracy of the mass of the tau lepton has slightly improved. XXI. The 2018 CODATA Recommended Values The input data and their correlation coefficients considered in the 2018 CODATA adjustment of the values of the constants are given in Tables VIII, X, XVIII, XXI, XXVII, and XXIX. (Here, items C3–C6 in Table XVIII are additional theoretical coefficients and not input data.) The data have been discussed and explained in detail in the previous sections. The 2018 recommended values are calculated from the set of best estimated values, in the least-squares sense, of 75 adjusted constants listed in Tables XI and XIX. A comparison with the values of the adjusted constants in Tables XXV and XXVI of the 2014 CODATA adjustment shows that two prominent quantities among the few that are no longer adjusted constants are the Planck constant h and the molar gas constant R. The reason, of course, is that in the revised SI these constants are exactly known. The methodology and quality of our least-squares adjustments has been discussed in Sec. III. Briefly, three independent adjustments have been performed. The first concerned the Newtonian constant of grav-itation.Thecorrespondinginputdata arefound tobeinconsistentandan expansion factor of 3.9 is needed to decrease the residuals to below two. The second independent adjustment concerned the determination of the natural-silicon lattice spacing and values of three historic x-ray units. No expansion factor is needed. Finally, the third adjustment determined the remaining 62 adjusted constants. Two expansion factors are required. A factor of 1.6 is applied to the 60 input data determining the Rydberg constant and proton and deuteron charge radii. A factor of 1.7 is used for the two input data that determine the mass of the proton. As in previous adjustments,wehavenotexcludedinputdatathatindividuallycontribute little to constrain the adjusted constants but taken together do matter. Goodexamplesofsuchdataaretransitionenergiesinatomichydrogento states with large principal quantum numbers as well as the less-accurate experimental data on the Newtonian constant of gravitation. A. Tables of values Tables XXX through XXXVI give the 2018 CODATA rec-ommended values of the basic constants and conversion factors of physics and chemistry and related quantities. Energy conversion factors in Tables XXXV and XXXVI relate energies, masses, photon wavelengths and frequencies, and temperatures of en-sembles of particles through the equivalences E mc2 hc/λ hn kT. The tables are identical in form and content to their 2010 and 2014 counterparts in that no constants are added or deleted. They also show the profound impact the revised SI has on the values of the fundamental constants. Counting the energy con-version factors in Tables XXXV and XXXVI, 46 constants that had uncertainties in 2014 are now exactly known in the revised SI. Values of the constants and correlation coefficients between any pair of constants can also be found at the website http:// physics.nist.gov/constants. XXII. Summary and Conclusion In this final section, we discuss (i) the differences between the 2014 and 2018 CODATA recommended values of the constants, (ii) the implications of the 2018 adjustment for metrology and physics, and (iii) future work that could improve our knowledge of the values of the constants. A. Comparison of 2014 and 2018 CODATA recommended values A representative group of 2014 and 2018 recommended values are compared in Fig. 10. The first four constants h, e, k, and NA are exact because of the redefinition of the SI. All other constants were and are inexactly known. Some have become significantly more accurate, some have updated values that fall well outside their 2014 uncertainty, while others have seen no significant change. Changes are a consequence of the revision of the SI and measurements that have become available since the 2014 adjustment. We discuss the changes shown in the figure as well as other notable changes in some detail later. FIG. 9. The 16 input data determining the Newtonian constant of gravitation G ordered by publication year. The 2018 recommended value for G has been subtracted. Error bars correspond to one-standard-deviation uncertainties as reported in Table XXIX. The uncertainties after applying the 3.9 multiplicative expansion factor to determine the 2018 recommended value are not shown. Labels on the left side of the figure denote the laboratories and the last two digits of the year in which the data were reported. See Table XXIX for details. The gray band corresponds to the one-standard-deviation uncertainty of the recommended value. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-44 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Not included in Fig. 10 are those few constants that were exactly known before the adoption of the revised SI in 2018. These are the universal constants μ0, ε0, and Z0, as well as the physicochemical constants M(12C) and Mu. Their current differences from their previous exact values may be conveniently expressed in the form μ0/(4π 3 10−7 N A−2) 1 + 55(15) 3 10−11 and M(12C)/(0.012 kg mol−1) 1–35(30) 3 10−11, where the numbers in parentheses are their 2018 standard uncertainties. [The number +55(15) is the same for Z0 μ0c but is −55(15) for ε0 1/μ0c2; the number −35(30) is the same for Mu.] The mass of the international prototype of the kilogram m(K) and the temperature at the triple point of water TTPW were also exactly known before the adoption of the revised SI, but they are not adjusted constants in the 2018 adjustment. In the revised SI, h, e, k, and NA are defining constants with exact values and the values of the previously exactly known SI defining constants μ0, M(12C), m(K), and TTPW must now be determined experimentally. The exact values of h, e, k, and NA are based on the results of the 2017 CODATA Special Adjustment carried out by the Task Group at the request of the General Conference on Weights and Measures (CGPM) with a closing date for data of 1 July 2017 (Mohr et al., 2018; Newell et al., 2018). Based on the input data available then, the exact values for h, e, k, and NA had to fall within the one-standard-deviation uncertainty of their then inexact values. The precise criteria can be found in CIPM (2016, 2017). Conversely, the criteria implied that the values and uncertainties of the newly imprecise μ0 and M(12C) were consistent with their previously exact values. After the 1 July 2017 closing date of the 2017 CODATA Special Adjustment, a measurement of h/m(133Cs) (item D4 in Table XXI) further constrained the value of the fine-structure constant α. This additional input datum has led to a larger deviation of μ0 4παZ/e2c and M(12C) from their previous exact values. The significantly reduced uncertainties of R∞, rp, and rd and shifts of the values compared with their 2014 counterparts are due to improvements in theory, new measurements of hydrogen transition frequencies, and the inclusion of Lamb-shift measurements in mu-onic hydrogen and deuterium. The latter were not included in the 2014 CODATA adjustment because of inconsistencies between the values of rp and rd derived from them and those obtained from hydrogen and deuterium spectroscopic data and e-p and e-d scat-tering data. Nevertheless, it must be recognized that although in-cluding the muonic hydrogen and deuterium data as well as new hydrogen spectroscopic data have led to values of R∞, rp, and rd with significantly smaller uncertainties, the remaining inconsistencies among the 62 data primarily responsible for the determination of these constants required their uncertainties to be increased by the TABLE XXX. An abbreviated list of the CODATA recommended values of the fundamental constants of physics and chemistry based on the 2018 adjustment Quantity Symbol Value Unit Relative std. uncert. ur speed of light in vacuum c 299 792 458 m s−1 exact Newtonian constant of gravitation G 6.674 30(15) 3 10−11 m3 kg−1 s−2 2.2 3 10−5 Planck constanta h 6.626 070 15 3 10−34 J Hz−1 exact Z 1.054 571 817 . . . 3 10−34 J s exact elementary charge e 1.602 176 634 3 10−19 C exact vacuum magnetic permeability 4παZ/e2c μ0 1.256 637 062 12(19) 3 10−6 N A−2 1.5 3 10−10 vacuum electric permittivity 1/μ0c2 ε0 8.854 187 8128(13) 3 10−12 F m−1 1.5 3 10−10 Josephson constant 2 e/h KJ 483 597.848 4 . . . 3 109 Hz V−1 exact von Klitzing constant μ0c/2α 2πZ/e2 RK 25 812.807 45 . . . Ω exact magnetic flux quantum 2πZ/(2e) Φ0 2.067 833 848 . . . 3 10−15 Wb exact conductance quantum 2e2/2πZ G0 7.748 091 729 . . . 3 10−5 S exact electron mass me 9.109 383 7015(28) 3 10−31 kg 3.0 3 10−10 proton mass mp 1.672 621 923 69(51) 3 10−27 kg 3.1 3 10−10 proton-electron mass ratio mp/me 1836.152 673 43(11) 6.0 3 10−11 fine-structure constant e2/4πε0Zc α 7.297 352 5693(11) 3 10−3 1.5 3 10−10 inverse fine-structure constant α−1 137.035 999 084(21) 1.5 3 10−10 Rydberg frequency α2mec2/2h cR∞ 3.289 841 960 2508(64) 3 1015 Hz 1.9 3 10−12 Boltzmann constant k 1.380 649 3 10−23 J K−1 exact Avogadro constant NA 6.022 140 76 3 1023 mol−1 exact molar gas constant NAk R 8.314 462 618 . . . J mol−1 K−1 exact Faraday constant NAe F 96 485.332 12 . . . C mol−1 exact Stefan-Boltzmann constant (π2/60)k4/Z3c2 σ 5.670 374 419 . . . 3 10−8 W m−2 K−4 exact Non-SI units accepted for use with the SI electron volt (e/ C) J eV 1.602 176 634 3 10−19 J exact (unified) atomic mass unit 1 12 m(12C) u 1.660 539 066 60(50) 3 10−27 kg 3.0 3 10−10 aThe energy of a photon with frequency n expressed in unit Hz is E hn in unit J. Unitary time evolution of the state of this photon is given by exp(−iEt/Z)|φ〉, where |φ〉is the photon state at time t 0 and time is expressed in unit s. The ratio Et/Z is a phase. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-45 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXXI. The CODATA recommended values of the fundamental constants of physics and chemistry based on the 2018 adjustment Quantity Symbol Numerical value Unit Relative std. uncert. ur UNIVERSAL speed of light in vacuum c 299 792 458 m s−1 exact vacuum magnetic permeability 4παZ/e2c μ0 1.256 637 062 12(19) 3 10−6 N A−2 1.5 3 10−10 μ0/(4π 3 10−7) 1.000 000 000 55(15) N A−2 1.5 3 10−10 vacuum electric permittivity 1/μ0c2 ε0 8.854 187 8128(13) 3 10−12 F m−1 1.5 3 10−10 characteristic impedance of vacuum μ0c Z0 376.730 313 668(57) Ω 1.5 3 10−10 Newtonian constant of gravitation G 6.674 30(15) 3 10−11 m3 kg−1 s−2 2.2 3 10−5 G/Zc 6.708 83(15) 3 10−39 (GeV/c2)−2 2.2 3 10−5 Planck constanta h 6.626 070 15 3 10−34 J Hz−1 exact 4.135 667 696 . . . 3 10−15 eV Hz−1 exact Z 1.054 571 817 . . . 3 10−34 J s exact 6.582 119 569 . . . 3 10−16 eV s exact Zc 197.326 980 4 . . . MeV fm exact Planck mass (Zc/G)1/2 mP 2.176 434(24) 3 10−8 kg 1.1 3 10−5 energy equivalent mPc2 1.220 890(14) 3 1019 GeV 1.1 3 10−5 Planck temperature (Zc5/G)1/2/k TP 1.416 784(16) 3 1032 K 1.1 3 10−5 Planck length Z/mPc (ZG/c3)1/2 lP 1.616 255(18) 3 10−35 m 1.1 3 10−5 Planck time lP/c (ZG/c5)1/2 tP 5.391 247(60) 3 10−44 s 1.1 3 10−5 ELECTROMAGNETIC elementary charge e 1.602 176 634 3 10−19 C exact e/Z 1.519 267 447 . . . 3 1015 A J−1 exact magnetic flux quantum 2πZ/(2e) Φ0 2.067 833 848 . . . 3 10−15 Wb exact conductance quantum 2e2/2πZ G0 7.748 091 729 . . . 3 10−5 S exact inverse of conductance quantum G−1 0 12 906.403 72 . . . Ω exact Josephson constant 2e/h KJ 483 597.848 4 . . . 3 109 Hz V−1 exact von Klitzing constant μ0c/2α 2πZ/e2 RK 25 812.807 45 . . . Ω exact Bohr magneton eZ/2me μB 9.274 010 0783(28) 3 10−24 J T−1 3.0 3 10−10 5.788 381 8060(17) 3 10−5 eV T−1 3.0 3 10−10 μB/h 1.399 624 493 61(42) 3 1010 Hz T−1 3.0 3 10−10 μB/hc 46.686 447 783(14) [m−1T−1]b 3.0 3 10−10 μB/k 0.671 713 815 63(20) K T−1 3.0 3 10−10 nuclear magneton eZ/2mp μN 5.050 783 7461(15) 3 10−27 J T−1 3.1 3 10−10 3.152 451 258 44(96) 3 10−8 eV T−1 3.1 3 10−10 μN/h 7.622 593 2291(23) MHz T−1 3.1 3 10−10 μN/hc 2.542 623 413 53(78) 3 10−2 [m−1 T−1]b 3.1 3 10−10 μN/k 3.658 267 7756(11) 3 10−4 K T−1 3.1 3 10−10 ATOMIC AND NUCLEAR General fine-structure constant e2/4πε0Zc α 7.297 352 5693(11) 3 10−3 1.5 3 10−10 inverse fine-structure constant α−1 137.035 999 084(21) 1.5 3 10−10 Rydberg frequency α2mec2/2h Eh/2h cR∞ 3.289 841 960 2508(64) 3 1015 Hz 1.9 3 10−12 energy equivalent hcR∞ 2.179 872 361 1035(42) 3 10−18 J 1.9 3 10−12 13.605 693 122 994(26) eV 1.9 3 10−12 Rydberg constant R∞ 10 973 731.568 160(21) [m−1]b 1.9 3 10−12 Bohr radius Z/αmec 4πε0Z2/mee2 a0 5.291 772 109 03(80) 3 10−11 m 1.5 3 10−10 Hartree energy α2mec2 e2/4πε0a0 2hcR∞ Eh 4.359 744 722 2071(85) 3 10−18 J 1.9 3 10−12 27.211 386 245 988(53) eV 1.9 3 10−12 quantum of circulation πZ/me 3.636 947 5516(11) 3 10−4 m2 s−1 3.0 3 10−10 2πZ/me 7.273 895 1032(22) 3 10−4 m2 s−1 3.0 3 10−10 Electroweak Fermi coupling constantc GF/(Zc)3 1.166 3787(6) 3 10−5 GeV−2 5.1 3 10−7 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-46 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXXI. (Continued.) Quantity Symbol Numerical value Unit Relative std. uncert. ur weak mixing angled θW (on-shell scheme) sin2θW s2 W ≡1 −(mW/mZ)2 sin2θW 0.222 90(30) 1.3 3 10−3 Electron, e− electron mass me 9.109 383 7015(28) 3 10−31 kg 3.0 3 10−10 5.485 799 090 65(16) 3 10−4 u 2.9 3 10−11 energy equivalent mec2 8.187 105 7769(25) 3 10−14 J 3.0 3 10−10 0.510 998 950 00(15) MeV 3.0 3 10−10 electron-muon mass ratio me/mμ 4.836 331 69(11) 3 10−3 2.2 3 10−8 electron-tau mass ratio me/mτ 2.875 85(19) 3 10−4 6.8 3 10−5 electron-proton mass ratio me/mp 5.446 170 214 87(33) 3 10−4 6.0 3 10−11 electron-neutron mass ratio me/mn 5.438 673 4424(26) 3 10−4 4.8 3 10−10 electron-deuteron mass ratio me/md 2.724 437 107 462(96) 3 10−4 3.5 3 10−11 electron-triton mass ratio me/mt 1.819 200 062 251(90) 3 10−4 5.0 3 10−11 electron-helion mass ratio me/mh 1.819 543 074 573(79) 3 10−4 4.3 3 10−11 electron to alpha particle mass ratio me/mα 1.370 933 554 787(45) 3 10−4 3.3 3 10−11 electron charge-to-mass quotient −e/me −1.758 820 010 76(53) 3 1011 C kg−1 3.0 3 10−10 electron molar mass NAme M(e), Me 5.485 799 0888(17) 3 10−7 kg mol−1 3.0 3 10−10 reduced Compton wavelength Z/mec αa0 ƛC 3.861 592 6796(12) 3 10−13 m 3.0 3 10−10 Compton wavelength λC 2.426 310 238 67(73) 3 10−12 [m]b 3.0 3 10−10 classical electron radius α2a0 re 2.817 940 3262(13) 3 10−15 m 4.5 3 10−10 Thomson cross section (8π/3)r2 e σe 6.652 458 7321(60) 3 10−29 m2 9.1 3 10−10 electron magnetic moment μe −9.284 764 7043(28) 3 10−24 J T−1 3.0 3 10−10 to Bohr magneton ratio μe/μB −1.001 159 652 181 28(18) 1.7 3 10−13 to nuclear magneton ratio μe/μN −1838.281 971 88(11) 6.0 3 10−11 electron magnetic-moment anomaly |μe|/μB −1 ae 1.159 652 181 28(18) 3 10−3 1.5 3 10−10 electron g-factor −2(1 + ae) ge −2.002 319 304 362 56(35) 1.7 3 10−13 electron-muon magnetic-moment ratio μe/μμ 206.766 9883(46) 2.2 3 10−8 electron-proton magnetic-moment ratio μe/μp −658.210 687 89(20) 3.0 3 10−10 electron to shielded proton magnetic-μe/μp ′ −658.227 5971(72) 1.1 3 10−8 moment ratio (H2O, sphere, 25°C) electron-neutron magnetic-moment ratio μe/μn 960.920 50(23) 2.4 3 10−7 electron-deuteron magnetic-moment ratio μe/μd −2143.923 4915(56) 2.6 3 10−9 electron to shielded helion magnetic-μe/μ ′ h 864.058 257(10) 1.2 3 10−8 moment ratio (gas, sphere, 25°C) electron gyromagnetic ratio 2|μe|/Z γe 1.760 859 630 23(53) 3 1011 s−1 T−1 3.0 3 10−10 28 024.951 4242(85) MHz T−1 3.0 3 10−10 Muon, μ− muon mass mμ 1.883 531 627(42) 3 10−28 kg 2.2 3 10−8 0.113 428 9259(25) u 2.2 3 10−8 energy equivalent mμc2 1.692 833 804(38) 3 10−11 J 2.2 3 10−8 105.658 3755(23) MeV 2.2 3 10−8 muon-electron mass ratio mμ/me 206.768 2830(46) 2.2 3 10−8 muon-tau mass ratio mμ/mτ 5.946 35(40) 3 10−2 6.8 3 10−5 muon-proton mass ratio mμ/mp 0.112 609 5264(25) 2.2 3 10−8 muon-neutron mass ratio mμ/mn 0.112 454 5170(25) 2.2 3 10−8 muon molar mass NAmμ M(μ), Mμ 1.134 289 259(25) 3 10−4 kg mol−1 2.2 3 10−8 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-47 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXXI. (Continued.) Quantity Symbol Numerical value Unit Relative std. uncert. ur reduced muon Compton wavelength Z/mμc ƛC,μ 1.867 594 306(42) 3 10−15 m 2.2 3 10−8 muon Compton wavelength λC,μ 1.173 444 110(26) 3 10−14 [m]b 2.2 3 10−8 muon magnetic moment μμ −4.490 448 30(10) 3 10−26 J T−1 2.2 3 10−8 to Bohr magneton ratio μμ/μB −4.841 970 47(11) 3 10−3 2.2 3 10−8 to nuclear magneton ratio μμ/μN −8.890 597 03(20) 2.2 3 10−8 muon magnetic-moment anomaly |μμ|/(eZ/2mμ) −1 aμ 1.165 920 89(63) 3 10−3 5.4 3 10−7 muon g-factor −2(1 + aμ) gμ −2.002 331 8418(13) 6.3 3 10−10 muon-proton magnetic-moment ratio μμ/μp −3.183 345 142(71) 2.2 3 10−8 Tau, τ− tau masse mτ 3.167 54(21) 3 10−27 kg 6.8 3 10−5 1.907 54(13) u 6.8 3 10−5 energy equivalent mτc2 2.846 84(19) 3 10−10 J 6.8 3 10−5 1776.86(12) MeV 6.8 3 10−5 tau-electron mass ratio mτ/me 3477.23(23) 6.8 3 10−5 tau-muon mass ratio mτ/mμ 16.8170(11) 6.8 3 10−5 tau-proton mass ratio mτ/mp 1.893 76(13) 6.8 3 10−5 tau-neutron mass ratio mτ/mn 1.891 15(13) 6.8 3 10−5 tau molar mass NAmτ M(τ), Mτ 1.907 54(13) 3 10−3 kg mol−1 6.8 3 10−5 reduced tau Compton wavelength Z/mτc ƛC,τ 1.110 538(75) 3 10−16 m 6.8 3 10−5 tau Compton wavelength λC,τ 6.977 71(47) 3 10−16 [m]b 6.8 3 10−5 Proton, p proton mass mp 1.672 621 923 69(51) 3 10−27 kg 3.1 3 10−10 1.007 276 466 621(53) u 5.3 3 10−11 energy equivalent mpc2 1.503 277 615 98(46) 3 10−10 J 3.1 3 10−10 938.272 088 16(29) MeV 3.1 3 10−10 proton-electron mass ratio mp/me 1836.152 673 43(11) 6.0 3 10−11 proton-muon mass ratio mp/mμ 8.880 243 37(20) 2.2 3 10−8 proton-tau mass ratio mp/mτ 0.528 051(36) 6.8 3 10−5 proton-neutron mass ratio mp/mn 0.998 623 478 12(49) 4.9 3 10−10 proton charge-to-mass quotient e/mp 9.578 833 1560(29) 3 107 C kg−1 3.1 3 10−10 proton molar mass NAmp M(p), Mp 1.007 276 466 27(31) 3 10−3 kg mol−1 3.1 3 10−10 reduced proton Compton wavelength Z/mpc ƛC,p 2.103 089 103 36(64) 3 10−16 m 3.1 3 10−10 proton Compton wavelength λC,p 1.321 409 855 39(40) 3 10−15 [m]b 3.1 3 10−10 proton rms charge radius rp 8.414(19) 3 10−16 m 2.2 3 10−3 proton magnetic moment μp 1.410 606 797 36(60) 3 10−26 J T−1 4.2 3 10−10 to Bohr magneton ratio μp/μB 1.521 032 202 30(46) 3 10−3 3.0 3 10−10 to nuclear magneton ratio μp/μN 2.792 847 344 63(82) 2.9 3 10−10 proton g-factor 2μp/μN gp 5.585 694 6893(16) 2.9 3 10−10 proton-neutron magnetic-moment ratio μp/μn −1.459 898 05(34) 2.4 3 10−7 shielded proton magnetic moment (H2O, sphere, 25°C) μ ′ p 1.410 570 560(15) 3 10−26 J T−1 1.1 3 10−8 to Bohr magneton ratio μ ′ p/μB 1.520 993 128(17) 3 10−3 1.1 3 10−8 to nuclear magneton ratio μ ′ p/μN 2.792 775 599(30) 1.1 3 10−8 proton magnetic shielding correction 1 − μ ′ p/μp (H2O, sphere, 25°C) σ ′ p 2.5689(11) 3 10−5 4.2 3 10−4 proton gyromagnetic ratio 2μp/Z γp 2.675 221 8744(11) 3 108 s−1 T−1 4.2 3 10−10 42.577 478 518(18) MHz T−1 4.2 3 10−10 shielded proton gyromagnetic ratio γ ′ p 2.675 153 151(29) 3 108 s−1 T−1 1.1 3 10−8 2μ ′ p/Z (H2O, sphere, 25°C) 42.576 384 74(46) MHz T−1 1.1 3 10−8 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-48 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXXI. (Continued.) Quantity Symbol Numerical value Unit Relative std. uncert. ur Neutron, n neutron mass mn 1.674 927 498 04(95) 3 10−27 kg 5.7 3 10−10 1.008 664 915 95(49) u 4.8 3 10−10 energy equivalent mnc2 1.505 349 762 87(86) 3 10−10 J 5.7 3 10−10 939.565 420 52(54) MeV 5.7 3 10−10 neutron-electron mass ratio mn/me 1838.683 661 73(89) 4.8 3 10−10 neutron-muon mass ratio mn/mμ 8.892 484 06(20) 2.2 3 10−8 neutron-tau mass ratio mn/mτ 0.528 779(36) 6.8 3 10−5 neutron-proton mass ratio mn/mp 1.001 378 419 31(49) 4.9 3 10−10 neutron-proton mass difference mn −mp 2.305 574 35(82) 3 10−30 kg 3.5 3 10−7 1.388 449 33(49) 3 10−3 u 3.5 3 10−7 energy equivalent (mn −mp)c2 2.072 146 89(74) 3 10−13 J 3.5 3 10−7 1.293 332 36(46) MeV 3.5 3 10−7 neutron molar mass NAmn M(n), Mn 1.008 664 915 60(57) 3 10−3 kg mol−1 5.7 3 10−10 reduced neutron Compton wavelength Z/mnc ƛC,n 2.100 194 1552(12) 3 10−16 m 5.7 3 10−10 neutron Compton wavelength λC,n 1.319 590 905 81(75) 3 10−15 [m]b 5.7 3 10−10 neutron magnetic moment μn −9.662 3651(23) 3 10−27 J T−1 2.4 3 10−7 to Bohr magneton ratio μn/μB −1.041 875 63(25) 3 10−3 2.4 3 10−7 to nuclear magneton ratio μn/μN −1.913 042 73(45) 2.4 3 10−7 neutron g-factor 2μn/μN gn −3.826 085 45(90) 2.4 3 10−7 neutron-electron magnetic-moment ratio μn/μe 1.040 668 82(25) 3 10−3 2.4 3 10−7 neutron-proton magnetic-moment ratio μn/μp −0.684 979 34(16) 2.4 3 10−7 neutron to shielded proton magnetic-μn/μ ′ p −0.684 996 94(16) 2.4 3 10−7 moment ratio (H2O, sphere, 25°C) neutron gyromagnetic ratio 2|μn|/Z γn 1.832 471 71(43) 3 108 s−1 T−1 2.4 3 10−7 29.164 6931(69) MHz T−1 2.4 3 10−7 Deuteron, d deuteron mass md 3.343 583 7724(10) 3 10−27 kg 3.0 3 10−10 2.013 553 212 745(40) u 2.0 3 10−11 energy equivalent mdc2 3.005 063 231 02(91) 3 10−10 J 3.0 3 10−10 1875.612 942 57(57) MeV 3.0 3 10−10 deuteron-electron mass ratio md/me 3670.482 967 88(13) 3.5 3 10−11 deuteron-proton mass ratio md/mp 1.999 007 501 39(11) 5.6 3 10−11 deuteron molar mass NAmd M(d), Md 2.013 553 212 05(61) 3 10−3 kg mol−1 3.0 3 10−10 deuteron rms charge radius rd 2.127 99(74) 3 10−15 m 3.5 3 10−4 deuteron magnetic moment μd 4.330 735 094(11) 3 10−27 J T−1 2.6 3 10−9 to Bohr magneton ratio μd/μB 4.669 754 570(12) 3 10−4 2.6 3 10−9 to nuclear magneton ratio μd/μN 0.857 438 2338(22) 2.6 3 10−9 deuteron g-factor μd/μN gd 0.857 438 2338(22) 2.6 3 10−9 deuteron-electron magnetic-moment ratio μd/μe −4.664 345 551(12) 3 10−4 2.6 3 10−9 deuteron-proton magnetic-moment ratio μd/μp 0.307 012 209 39(79) 2.6 3 10−9 deuteron-neutron magnetic-moment ratio μd/μn −0.448 206 53(11) 2.4 3 10−7 Triton, t triton mass mt 5.007 356 7446(15) 3 10−27 kg 3.0 3 10−10 3.015 500 716 21(12) u 4.0 3 10−11 energy equivalent mtc2 4.500 387 8060(14) 3 10−10 J 3.0 3 10−10 2808.921 132 98(85) MeV 3.0 3 10−10 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-49 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr TABLE XXXI. (Continued.) Quantity Symbol Numerical value Unit Relative std. uncert. ur triton-electron mass ratio mt/me 5496.921 535 73(27) 5.0 3 10−11 triton-proton mass ratio mt/mp 2.993 717 034 14(15) 5.0 3 10−11 triton molar mass NAmt M(t), Mt 3.015 500 715 17(92) 3 10−3 kg mol−1 3.0 3 10−10 triton magnetic moment μt 1.504 609 5202(30) 3 10−26 J T−1 2.0 3 10−9 to Bohr magneton ratio μt/μB 1.622 393 6651(32) 3 10−3 2.0 3 10−9 to nuclear magneton ratio μt/μN 2.978 962 4656(59) 2.0 3 10−9 triton g-factor 2μt/μN gt 5.957 924 931(12) 2.0 3 10−9 Helion, h helion mass mh 5.006 412 7796(15) 3 10−27 kg 3.0 3 10−10 3.014 932 247 175(97) u 3.2 3 10−11 energy equivalent mhc2 4.499 539 4125(14) 3 10−10 J 3.0 3 10−10 2808.391 607 43(85) MeV 3.0 3 10−10 helion-electron mass ratio mh/me 5495.885 280 07(24) 4.3 3 10−11 helion-proton mass ratio mh/mp 2.993 152 671 67(13) 4.4 3 10−11 helion molar mass NAmh M(h), Mh 3.014 932 246 13(91) 3 10−3 kg mol−1 3.0 3 10−10 helion magnetic moment μh −1.074 617 532(13) 3 10−26 J T−1 1.2 3 10−8 to Bohr magneton ratio μh/μB −1.158 740 958(14) 3 10−3 1.2 3 10−8 to nuclear magneton ratio μh/μN −2.127 625 307(25) 1.2 3 10−8 helion g-factor 2μh/μN gh −4.255 250 615(50) 1.2 3 10−8 shielded helion magnetic moment (gas, sphere, 25°C) μ ′ h −1.074 553 090(13) 3 10−26 J T−1 1.2 3 10−8 to Bohr magneton ratio μ ′ h/μB −1.158 671 471(14) 3 10−3 1.2 3 10−8 to nuclear magneton ratio μ ′ h/μN −2.127 497 719(25) 1.2 3 10−8 shielded helion to proton magnetic-moment ratio (gas, sphere, 25°C) μ ′ h/μp −0.761 766 5618(89) 1.2 3 10−8 shielded helion to shielded proton magnetic-moment ratio (gas/H2O, spheres, 25°C) μ ′ h/μ ′ p −0.761 786 1313(33) 4.3 3 10−9 shielded helion gyromagnetic ratio 2|μ ′ h|/Z (gas, sphere, 25°C) γ ′ h 2.037 894 569(24) 3 108 s−1 T−1 1.2 3 10−8 32.434 099 42(38) MHz T−1 1.2 3 10−8 Alpha particle, α alpha particle mass mα 6.644 657 3357(20) 3 10−27 kg 3.0 3 10−10 4.001 506 179 127(63) u 1.6 3 10−11 energy equivalent mαc2 5.971 920 1914(18) 3 10−10 J 3.0 3 10−10 3727.379 4066(11) MeV 3.0 3 10−10 alpha particle to electron mass ratio mα/me 7294.299 541 42(24) 3.3 3 10−11 alpha particle to proton mass ratio mα/mp 3.972 599 690 09(22) 5.5 3 10−11 alpha particle molar mass NAmα M(α), Mα 4.001 506 1777(12) 3 10−3 kg mol−1 3.0 3 10−10 PHYSICOCHEMICAL Avogadro constant NA 6.022 140 76 3 1023 mol−1 exact Boltzmann constant k 1.380 649 3 10−23 J K−1 exact 8.617 333 262 . . . 3 10−5 eV K−1 exact k/h 2.083 661 912 . . . 3 1010 Hz K−1 exact k/hc 69.503 480 04 . . . [m−1 K−1]b exact atomic mass constantf mu 1 12 m(12C) 2hcR∞/α2c2Ar(e) mu 1.660 539 066 60(50) 3 10−27 kg 3.0 3 10−10 energy equivalent muc2 1.492 418 085 60(45) 3 10−10 J 3.0 3 10−10 931.494 102 42(28) MeV 3.0 3 10−10 molar mass constantf Mu 0.999 999 999 65(30) 3 10−3 kg mol−1 3.0 3 10−10 molar massf of carbon-12 Ar(12C)Mu M(12C) 11.999 999 9958(36) 3 10−3 kg mol−1 3.0 3 10−10 molar Planck constant NAh 3.990 312 712 . . . 3 10−10 J Hz−1 mol−1 exact molar gas constant NAk R 8.314 462 618 . . . J mol−1 K−1 exact J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-50 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr multiplicative factor 1.6 to reduce all normalized residuals to less than 2. The relative uncertainty ur(Eh) of the Hartree energy Eh 2R∞hc is now simply that due to the Rydberg constant rather than that of the Planck constant as was the case in the 2014 CODATA adjustment. The uncertainty of the Hartree energy is now 6300 times smaller. The reduction of the uncertainty of α by a factor of 1.5 to ur(α) 1.5 3 10−10 is mainly due to the measurement of h/m(133Cs). The uncertainties of many other constants are directly linked to that of α. Examplesare,ofcourse,μ0,butalsotheBohrradiusa0,electronmassme, Compton wavelength λC, and Thomson cross section σe. Their relative uncertainties are 1, 1, 2, 2, and 6 times that of α, respectively. The latter four constants also depend on the Rydberg constant R∞, but its relative uncertainty of 1.9 3 10−12 is much smaller than that of α. The reduction in the uncertainty of G is due to two new and independent results from HUST in the People’s Republic of China, both with ur(G) 1.2 3 10−5 (HUSTT-18 and HUSTA-18 in TableXXIX);and a correction ofa previously available result(JILA-18 in Table XXIX). This led to a better consistency among the 16 input data for G and a reduction of the applied expansion factor of their uncertainties from 6.3 in 2014 to 3.9 in the current CODATA adjustment. TABLE XXXI. (Continued.) Quantity Symbol Numerical value Unit Relative std. uncert. ur Faraday constant NAe F 96 485.332 12 . . . C mol−1 exact standard-state pressure 100 000 Pa exact standard atmosphere 101 325 Pa exact molar volume of ideal gas RT/p T 273.15 K, p 100 kPa Vm 22.710 954 64 . . . 3 10−3 m3 mol−1 exact or standard-state pressure Loschmidt constant NA/Vm n0 2.651 645 804 . . . 3 1025 m−3 exact molar volume of ideal gas RT/p T 273.15 K, p 101.325 kPa Vm 22.413 969 54 . . . 3 10−3 m3mol−1 exact or standard atmosphere Loschmidt constant NA/Vm n0 2.686 780 111 . . . 3 1025 m−3 exact Sackur-Tetrode (absolute entropy) constantg 5 2 + ln[(mukT1/2πZ2)3/2kT1/p0] T1 1 K, p0 100 kPa S0/R −1.151 707 537 06(45) 3.9 3 10−10 or standard-state pressure T1 1 K, p0 101.325 kPa or standard atmosphere −1.164 870 523 58(45) 3.9 3 10−10 Stefan-Boltzmann constant (π2/60)k4/Z3c2 σ 5.670 374 419 . . . 3 10−8 W m−2 K−4 exact first radiation constant for spectral radiance 2hc2 sr−1 c1L 1.191 042 972 . . . 3 10−16 [W m2 sr−1]h exact first radiation constant 2πhc2 π sr c1L c1 3.741 771 852 . . . 3 10−16 [W m2]h exact second radiation constant hc/k c2 1.438 776 877 . . . 3 10−2 [m K]b exact Wien displacement law constants b λmaxT c2/4.965 114 231 . . . b 2.897 771 955 . . . 3 10−3 [m K]b exact b′ nmax/T 2.821 439 372 . . . c/c2 b′ 5.878 925 757 . . . 3 1010 Hz K−1 exact aThe energy of a photon with frequency n expressed in unit Hz is E hn in unit J. Unitary time evolution of the state of this photon is given by exp(−iEt/Z)|φ〉, where |φ〉is the photon state at time t 0 and time is expressed in unit s. The ratio Et/Z is a phase. bThe full description of m−1 is cycles or periodsper meter and that of m is meters per cycle (m/cycle). The scientific community is aware of theimplied use of these units. It traces back to the conventions for phase and angle and the use of unit Hz versus cycles/s. No solution has been agreed upon. cValue recommended by the Particle Data Group (Tanabashi et al., 2018). dBased on the ratio of the masses of the W and Z bosons mW/mZ recommended by the Particle Data Group (Tanabashi et al., 2018). The value for sin2 θW they recommend, which is based on a variant of the modified minimal subtraction (MS) scheme, is sin2^ θW(MZ) 0.231 22(4). eThis and other constants involving mτ are based on mτc2 in MeV recommended by the Particle Data Group (Tanabashi et al., 2018). fThe relative atomic mass Ar(X) of particle X with mass m(X) is defined by Ar(X) m(X)/mu, where mu m(12C)/12 1 u is the atomic mass constant and u is the unified atomic mass unit. Moreover, the mass of particle X is m(X) Ar(X) u and the molar mass of X is M(X) Ar(X)Mu, where Mu NA u is the molar mass constant and NA is the Avogadro constant. gThe entropy of an ideal monoatomic gas of relative atomic mass Ar is given by S S0 + 3 2 R ln Ar −R ln(p/p0) + 5 2 R ln(T/K). hThe full description of m2 is m−2 3 (m/cycle)4. See also the second footnote. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-51 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr The relations 1 u mu m(12C)/12 for the atomic mass unit and Ar(12C) 12 for the relative atomic mass of 12C remain exact in the revised SI. The mass mu in kg, however, is now obtained from mu 2R∞h/Ar(e)cα2 instead of mu (10−3 kg/mol)/NA. Conse-quently, the relative uncertainty of mu in the 2018 adjustment is essentially twice that of α or 3.0 3 10−10, because ur(R∞) and ur(Ar(e)) are significantly smaller than ur(α). This relative un-certainty of mu is 41 times smaller than in the 2014 adjustment, where it was dominated by the relative uncertainty of NA. Generally, the mass of a particle X in kg is most reliably de-termined from m(X) Ar(X)mu, where the relative uncertainty of Ar(X) for most particles of interest here is significantly smaller than that of mu. Hence the ur of me, mp, md, mt, mh, and mα when expressed in kg are now essentially the same as that of mu. The significant reductions of the uncertainties of magnetic moments μB, μN, and μe can be understood from their definitions. The Bohr magneton μB eZ/2me now has the relative uncertainty of that of the electron mass. By comparison, in the 2014 CODATA ad-justment ur of μB is 6.3 3 10−9 or 20 times larger. Similarly, the nuclear magneton μN eZ/2mp has the relative uncertainty of that of mp or mu. Because the ratio μe/μB ge/2 and the ur of the 2018 and 2014 CODATA recommended values of the electron g-factor ge are 1.7 3 10−13 and 2.6 3 10−13, respectively, the relative uncertainty of μe is essentially the same as that for μB. The value of the magnetic moment of the proton μp has been improved due to a new measurement of the ratio μp/μN. For this measurement, ur 2.9 3 10−10. Together with the improved value of μN, it provides a value of μp with ur 4.2 3 10−10. Similarly, the uncertainty μp/μB has seen a tenfold improvement, as μp/μB μp/μN 3 me/mp and me/mp has a relative uncertainty of 6.0 3 10−11. The input data that determine the 2018 CODATA recom-mended value of Ar(p) are the 2016 AMDC value of Ar(1H) and the cyclotron frequency ratio ωc(12C6+)/ωc(p) (item D15 in Table XXI). The two values for Ar(p) from these data disagree, and an expansion factor of 1.7 is applied to their uncertainties to bring them into agreement. The comparatively large difference between the 2018 and 2014 values of the helion relative atomic mass, Ar(h), is due to the inclusion of a new value of the cyclotron frequency ratio ωc(HD+)/ωc(3He+) (item D17 in Table XXI) and omission of the cyclotron frequency ratio ωc(h)/ωc(12C6+) used in 2014, because of concerns about its reliability. The relative atomic mass of the triton has changed based on a 2015 measurement (item D16 in Table XXI). No new datum has become available to determine Ar(e), Ar(d), and Ar(α). The magnetic moment of the neutron μn and ratios μn/μN and μn/μp are determined from the same input datum, namely, μn/μ ′ p with ur 2.4 3 10−7 obtained in 1979 (item D37 in Table XXI). The 2018 values and uncertainties of these three quantities are essentially the same as in the 2014 adjustment. The magnetic moment of the deuteron μd and ratios μd/μN and μd/μe have a ur of 2.6 3 10−9, which is about one-half that of their 2014 ur. The reason is the presence of an additional input datum for the ratio μp(HD)/μd(HD) with ur 3.1 3 10−9. TABLE XXXIII. Values of some x-ray-related quantities based on the 2018 CODATA adjustment of the constants Quantity Symbol Value Unit Relative std. uncert. ur Cu x unit: λ(CuKα1)/1537.400 xu(CuKα1) 1.002 076 97(28) 3 10−13 m 2.8 3 10−7 Mo x unit: λ(MoKα1)/707.831 xu(MoKα1) 1.002 099 52(53) 3 10−13 m 5.3 3 10−7 ˚ Angstr¨ om star: λ(WKα1)/0.209 010 0 ˚ A∗ 1.000 014 95(90) 3 10−10 m 9.0 3 10−7 Lattice parametera of Si (in vacuum, 22.5°C) a 5.431 020 511(89) 3 10−10 m 1.6 3 10−8 {220} lattice spacing of Si a/  8 √ (in vacuum, 22.5°C) d220 1.920 155 716(32) 3 10−10 m 1.6 3 10−8 Molar volume of Si M(Si)/ρ(Si) NAa3/8 (in vacuum, 22.5°C) Vm(Si) 1.205 883 199(60) 3 10−5 m3 mol−1 4.9 3 10−8 aThis is the lattice parameter (unit cell edge length) of an ideal single crystal of naturally occurring Si with natural isotopic Si abundances, free of impurities and imperfections. TABLE XXXII. The relative uncertainties and correlation coefficients of the values of a selected group of constants based on the 2018 CODATA adjustment. The numbers in bold on the diagonal are the relative uncertainties ur(xi) u(xi)/xi; the other numbers are the correlation coefficients r(xi, xj) u(xi, xj)/[u(xi)u(xj)]. Here, u(xi, xj) is the covariance of xi and xj and u2(xi) u(xi, xi) is the variance α R∞ me/mp rp rd me/mμ mu α 1.5 3 10−10 0.002 07 −0.031 03 0.003 45 0.003 20 −0.013 45 −0.995 35 R∞ 0.002 07 1.9 3 10−12 0.012 06 0.885 92 0.903 66 −0.00011 0.003 69 me/mp −0.031 03 0.012 06 6.0 3 10−11 −0.005 28 0.011 13 0.000 45 −0.015 54 rp 0.003 45 0.885 92 −0.005 28 2.2 3 10−3 0.991 65 −0.000 12 0.002 38 rd 0.003 20 0.903 66 0.011 13 0.991 65 3.5 3 10−4 −0.000 12 0.002 30 me/mμ −0.013 45 −0.000 11 0.000 45 −0.000 12 −0.000 12 2.2 3 10−8 0.013 38 mu −0.995 35 0.003 69 −0.015 54 0.002 38 0.002 30 0.013 38 3.0 3 10−10 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-52 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr One of the consequences of the revised SI is that the conversion factors among the energy units J, kg, m−1, Hz, K, and eV are now exact based on E mc2 Zc/λ hn kT. The conversion factor between these six units and the unified atomic mass unit, 1 u mu, is determined by mu and exact constants. Hence, the relative uncertainties of the six correspondingconversionfactorsarenowthatofmu or3.0 3 10−10.This corresponds to a significant improvement compared to the 2014 rec-ommended conversion factor. For example, the uncertainty of the eV-to-u conversion factor is reduced by a factor of 20. The situation is similar for the conversion factors from the six energy units to the Hartree energy Eh 2R∞hc, but in this case the relevantconstantisR∞withur 1.9 3 10−12 ratherthanmu.Asanother example,theuncertaintyoftheK-to-Eh conversionfactorisreduced from 5.7 3 10−7 in 2014 to 1.9 3 10−12 in 2018, or by a factor of 3 3 105. B. Implications of the 2018 adjustment for metrology and physics 1. Electrical metrology The most significant practical impact of the revised SI is undoubtedly the elimination of the conventional 1990 electrical units that went into effect on 1 January 1990 to ensure the international consistency of electrical measurements. (See After thirty TABLE XXXIV. Non-SI units based on the 2018 CODATA adjustment of the constants, although eV and u are accepted for use with the SI Quantity Symbol Value Unit Relative std. uncert. ur electron volt: (e/C) J eV 1.602 176 634 3 10−19 J exact (unified) atomic mass unit: 1 12 m(12C) u 1.660 539 066 60(50) 3 10−27 kg 3.0 3 10−10 Natural units (n.u.) n.u. of velocity c 299 792 458 m s−1 exact n.u. of action Z 1.054 571 817 . . . 3 10−34 J s exact 6.582 119 569 . . . 3 10−16 eV s exact Zc 197.326 980 4 . . . MeV fm exact n.u. of mass me 9.109 383 7015(28) 3 10−31 kg 3.0 3 10−10 n.u. of energy mec2 8.187 105 7769(25) 3 10−14 J 3.0 3 10−10 0.510 998 950 00(15) MeV 3.0 3 10−10 n.u. of momentum mec 2.730 924 530 75(82) 3 10−22 kg m s−1 3.0 3 10−10 0.510 998 950 00(15) MeV/c 3.0 3 10−10 n.u. of length: Z/mec ƛC 3.861 592 6796(12) 3 10−13 m 3.0 3 10−10 n.u. of time Z/mec2 1.288 088 668 19(39) 3 10−21 s 3.0 3 10−10 Atomic units (a.u.) a.u. of charge e 1.602 176 634 3 10−19 C exact a.u. of mass me 9.109 383 7015(28) 3 10−31 kg 3.0 3 10−10 a.u. of action Z 1.054 571 817 . . . 3 10−34 J s exact a.u. of length: Bohr radius (bohr) Z/αmec a0 5.291 772 109 03(80) 3 10−11 m 1.5 3 10−10 a.u. of energy: Hartree energy (hartree) α2mec2 e2/4πε0a0 2hcR∞ Eh 4.359 744 722 2071(85) 3 10−18 J 1.9 3 10−12 a.u. of time Z/Eh 2.418 884 326 5857(47) 3 10−17 s 1.9 3 10−12 a.u. of force Eh/a0 8.238 723 4983(12) 3 10−8 N 1.5 3 10−10 a.u. of velocity: αc a0Eh/Z 2.187 691 263 64(33) 3 106 m s−1 1.5 3 10−10 a.u. of momentum Z/a0 1.992 851 914 10(30) 3 10−24 kg m s−1 1.5 3 10−10 a.u. of current eEh/Z 6.623 618 237 510(13) 3 10−3 A 1.9 3 10−12 a.u. of charge density e/a3 0 1.081 202 384 57(49) 3 1012 C m−3 4.5 3 10−10 a.u. of electric potential Eh/e 27.211 386 245 988(53) V 1.9 3 10−12 a.u. of electric field Eh/ea0 5.142 206 747 63(78) 3 1011 V m−1 1.5 3 10−10 a.u. of electric field gradient Eh/ea2 0 9.717 362 4292(29) 3 1021 V m−2 3.0 3 10−10 a.u. of electric dipole moment ea0 8.478 353 6255(13) 3 10−30 C m 1.5 3 10−10 a.u. of electric quadrupole moment ea2 0 4.486 551 5246(14) 3 10−40 C m2 3.0 3 10−10 a.u. of electric polarizability e2a2 0/Eh 1.648 777 274 36(50) 3 10−41 C2 m2 J−1 3.0 3 10−10 a.u. of 1st hyperpolarizability e3a3 0/E2 h 3.206 361 3061(15) 3 10−53 C3 m3 J−2 4.5 3 10−10 a.u. of 2nd hyperpolarizability e4a4 0/E3 h 6.235 379 9905(38) 3 10−65 C4 m4 J−3 6.0 3 10−10 a.u. of magnetic flux density Z/ea2 0 2.350 517 567 58(71) 3 105 T 3.0 3 10−10 a.u. of magnetic dipole moment: 2μB Ze/me 1.854 802 015 66(56) 3 10−23 J T−1 3.0 3 10−10 a.u. of magnetizability e2a2 0/me 7.891 036 6008(48) 3 10−29 J T−2 6.0 3 10−10 a.u. of permittivity e2/a0Eh 1.112 650 055 45(17) 3 10−10 F m−1 1.5 3 10−10 J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-53 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr years, electrical measurements are once more consistent with mea-surements made in the other units of the SI. Electrical units have become part of the SI again, simply because the Josephson and von Klitzing constants are now exact in SI units. Between 1990 and the adoption of the revised SI in 2019, the units of voltage and resistance, V90 and Ω90, were based on the conventional values KJ−90 483 597.9 GHz/V and RK−90 25 812.807 Ω for the Josephson and von Klitzing constants, respectively. From 2019 on-ward, the ratios between KJ 2e/h and KJ−90 and between RK h/e2 and RK−90 are exact. Thus, 1 V90 (KJ−90/KJ) V and 1 Ω90 (RK/RK−90)Ω exactly. Consequently, the conventional electric units for voltage, resistance, current, charge, power, capaci-tance, inductance, electrical conductance, magnetic flux, and mag-netic flux density in terms of the corresponding SI units are 1 V90 KJ−90 KJ V [1 + 10.666 . . . 3 10−8] V, 1 Ω90 RK RK−90 Ω [1 + 1.7793 . . . 3 10−8] Ω, 1 A90 KJ−90RK−90 KJRK A [1 + 8.8871 . . . 3 10−8] A, 1 C90 KJ−90RK−90 KJRK C [1 + 8.8871 . . . 3 10−8] C, 1 W90 K2 J −90RK−90 K2 J RK W [1 + 19.553 . . . 3 10−8] W, 1 F90 RK−90 RK F [1 −1.7793 . . . 3 10−8] F, 1 H90 RK RK−90 H [1 + 1.7793 . . . 3 10−8] H, 1 S90 RK−90 RK S [1 −1.7793 . . . 3 10−8] S, 1 Wb90 KJ−90 KJ Wb [1 + 10.666 . . . 3 10−8] Wb, 1 T90 KJ−90 KJ T [1 + 10.666 . . . 3 10−8] T. Thus, for example, the 1990 conventional unit of voltage V90 exceeds the SI unit of voltage V by the fractional amount 10.666 . . . 3 10−8. This implies that a voltage measured in the unit V90 will have a numerical value that is smaller by this fractional amount than the numerical value of the same voltage measured in the SI volt V. (The 1990 conventional units are viewed as physical quantities and, hence, their symbols are written in italic type.) 2. Electron magnetic-moment anomaly, fine-structure constant, and QED theory The electron magnetic-moment anomaly ae has for many years provided fertile ground for testing QED and obtaining an accurate value of α. Within QED, ae is a function of α with weak and strong interaction contributionsthatarecomparativelysmallandreadilycalculated,totaling at present a fractional contribution of 14.86(10) 3 10−10 to ae. By comparison, the relative uncertainty of the measured ae is 2.4 3 10−10, based on a determination of the ratio of the cyclotron and precession frequencies of a single electron in an applied magnetic flux density. TABLE XXXV. The values of some energy equivalents derived from the relations E mc2 hc/λ hn kT and based on the 2018 CODATA adjustment of the values of the constants; 1 eV (e/C) J, 1 u mu 1 12 m(12C), and Eh 2hcR∞ α2mec2 is the Hartree energy (hartree) Relevant unit J kg [m−1]a Hz 1 J (1 J) 1 J (1 J)/c2 1.112 650 056 . . . 3 10−17 kg (1 J)/hc 5.034 116 567 . . . 3 1024 m−1 (1 J)/h 1.509 190 179 . . . 3 1033 Hz 1 kg (1 kg)c2 8.987 551 787 . . . 3 1016 J (1 kg) 1 kg (1 kg)c/h 4.524 438 335 . . . 3 1041 m−1 (1 kg)c2/h 1.356 392 489 . . . 3 1050 Hz 1 [m−1]a (1 m−1)hc 1.986 445 857 . . . 3 10−25 J (1 m−1)h/c 2.210 219 094 . . . 3 10−42 kg (1 m−1) 1 m−1 (1 m−1)c 299 792 458 Hz 1 Hz (1 Hz)h 6.626 070 15 3 10−34 J (1 Hz)h/c2 7.372 497 323 . . . 3 10−51 kg (1 Hz)/c 3.335 640 951 . . . 3 10−9 m−1 (1 Hz) 1 Hz 1 K (1 K)k 1.380 649 3 10−23 J (1 K)k/c2 1.536 179 187 . . . 3 10−40 kg (1 K)k/hc 69.503 480 04 . . . m−1 (1 K)k/h 2.083 661 912 . . . 3 1010 Hz 1 eV (1 eV) 1.602 176 634 3 10−19 J (1 eV)/c2 1.782 661 921 . . . 3 10−36 kg (1 eV)/hc 8.065 543 937 . . . 3 105 m−1 (1 eV)/h 2.417 989 242 . . . 3 1014 Hz 1 u (1 u)c2 1.492 418 085 60(45) 3 10−10 J (1 u) 1.660 539 066 60(50) 3 10−27 kg (1 u)c/h 7.513 006 6104(23) 3 1014 m−1 (1 u)c2/h 2.252 342 718 71(68) 3 1023 Hz 1 Eh (1 Eh) 4.359 744 722 2071(85) 3 10−18 J (1 Eh)/c2 4.850 870 209 5432(94) 3 10−35 kg (1 Eh)/hc 2.194 746 313 6320(43) 3 107 m−1 (1 Eh)/h 6.579 683 920 502(13) 3 1015 Hz aThe full description of m−1 is cycles or periods per meter. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-54 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr AconvenientwayofverifyingQEDtheoryistocalculateα thatresults fromequatingthetheoreticalexpressionforae withtheexperimentalvalue and then comparing it with values obtained from experiments that only weakly depend on QED theory. Two such values are available from in-terferometric measurements with laser-cooled 87Rb and 133Cs atoms. The result of the comparison is that α−1 from the single-electron experiment exceeds the value from the 87Rb and 133Cs interferometric experiments by 1.7σ and 2.4σ, respectively. Here, σ is the square root of the sum of the squares of the corresponding pair of uncertainties in α−1. The 2.4σ disagreement is mild, but discomforting. The two leading experimental groups that determined α−1 from atom interferometry are carrying out new experiments that should yield values with significantly reduced uncertainties (Clad´ e et al., 2019; Yu et al., 2019). In addition, G. Gabrielse is constructing a significantly improved version of his single-electron experiment (Gabrielse et al., 2019). The group that has calculated the A(10) 1 co-efficient in the theoretical expression of ae is continuing its work and TABLE XXXVI. The values of some energy equivalents derived from the relations E mc2 Zc/λ hn kT and based on the 2018 CODATA adjustment of the values of the constants; 1 eV (e/C) J, 1 u mu 1 12 m(12C), and Eh 2hcR∞ α2mec2 is the Hartree energy (hartree) Relevant unit K V u Eh 1 J (1 J)/k 7.242 970 516 . . . 3 1022 K (1 J) 6.241 509 074 . . . 3 1018 eV (1 J)/c2 6.700 535 2565(20) 3 109 u (1 J) 2.293 712 278 3963(45) 3 1017 Eh 1 kg (1 kg)c2/k 6.509 657 260 . . . 3 1039 K (1 kg)c2 5.609 588 603 . . . 3 1035 eV (1 kg) 6.022 140 7621(18) 3 1026 u (1 kg)c2 2.061 485 788 7409(40) 3 1034 Eh 1[m−1]a (1 m−1)hc/k 1.438 776 877 . . . 3 10−2 K (1 m−1)hc 1.239 841 984 . . . 3 10−6 eV (1 m−1)h/c 1.331 025 050 10(40) 3 10−15 u (1 m−1)hc 4.556 335 252 9120(88) 3 10−8 Eh 1 Hz (1 Hz)h/k 4.799 243 073 . . . 3 10−11 K (1 Hz)h 4.135 667 696 . . . 3 10−15 eV (1 Hz)h/c2 4.439 821 6652(13) 3 10−24 u (1 Hz)h 1.519 829 846 0570(29) 3 10−16 Eh 1 K (1 K) 1 K (1 K)k 8.617 333 262 . . . 3 10−5 eV (1 K)k/c2 9.251 087 3014(28) 3 10−14 u (1 K)k 3.166 811 563 4556(61) 3 10−6 Eh 1 eV (1 eV)/k 1.160 451 812 . . . 3 104 K (1 eV) 1 eV (1 eV)/c2 1.073 544 102 33(32) 3 10−9 u (1 eV) 3.674 932 217 5655(71) 3 10−2 Eh 1 u (1 u)c2/k 1.080 954 019 16(33) 3 1013 K (1 u)c2 9.314 941 0242(28) 3 108 eV (1 u) 1 u (1 u)c2 3.423 177 6874(10) 3 107 Eh 1 Eh (1 Eh)/k 3.157 750 248 0407(61) 3 105 K (1 Eh) 27.211 386 245 988(53) eV (1 Eh)/c2 2.921 262 322 05(88) 3 10−8 u (1 Eh) 1 Eh aThe full description of m−1 is cycles or periods per meter. FIG. 10. Comparison of a representative group of fundamental constants from the 2014 and 2018 CODATA adjustments. Symbols of constants are shown along the y axis. Along the x axis the 2018 recommended values and their one-standard-deviation uncertainty, black circles with error bars, are shown as the difference between the 2018 and 2014 values divided by the standard uncertainty of the 2014 value. The vertical solid red line at the origin and yellow/orange band of width 1 represent the 2014 values and standard uncertainties of the indicated constants. The numerical values near the left-hand side of the figure are the relative standard uncertainties from the 2018 adjustment. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-55 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr has recently reported A(10) 1 6.737(159) (Aoyama, Kinoshita, and Nio, 2019). The results of all efforts are anxiously awaited. 3. Proton radius and Rydberg constant The “proton-radius puzzle” has been with us ever since the 2010 publication of the charge radius of the proton rp obtained from the measurement of the Lamb shift in muonic hydrogen μH (an atom comprised of a proton and a muon). The severe dis-crepancy between the μH value of rp and the values of rp obtained from hydrogen transition frequency data and e-p elastic scattering data led to the omission of the μH result from 2010 and 2014 CODATA adjustments. A Lamb-shift measurement in muonic deuterium μD (an atom comprised of a deuteron and a muon) provided a charge radius of the deuteron rd that, like the μH value of rp, was smaller than the deuterium spectroscopic and e-d scattering value and inconsistent with it. This disagreement was also deemed too significant, and the μD data were not included in the CODATA adjustments. The situation has improved markedly over the past four years, and the μH as well as recent μD data are now included in the 2018 CODATA adjustment. New hydrogen spectroscopic data and advances in theo-retical estimates of transition frequencies contributed to this decision. As a result, the 2018 recommended values of rp, rd, and R∞and their uncertainties are significantly smaller than in the 2014 CODATA ad-justment. The value of rp is reduced by 3.8% and its uncertainty is re-duced from 0.70% to 0.22%; rd is reduced by 0.62% and its uncertainty from 0.12% to 0.035%; and for R∞the reduction in value is fractionally 32 3 10−12 and ur is reduced from 5.9 3 10−12 to 1.9 3 10−12. We can conclude that the proton-radius puzzle has largely been resolved. Nevertheless, the uncertainties of the many input data that contribute to the determination of the charge radii and Rydberg constant had to be increased by an expansion factor of 1.6 in order to ensure that the residuals of these input data are less than two. New data will be required to obtain further insight into the origin of the remaining discrepancies. In fact, after the closing date for the 2018 CODATA adjustment, new values for rp based on an improved e-p scattering experiment have become available. A value of rp 0.831(14) fm was recently reported by Xiong et al. (2019) from the Jefferson Laboratory, Virginia, USA. The result is smaller than, but consistent with, the 2018 CODATA recommended value and the work is expected to continue. Electron-proton scattering experiments are also being carried out at the Mainz Microtron (MAMI) particle accelerator in Germany. In 2019, it already led to the reported value rp 0.870(28) fm (Mihoviloviˇ c et al., 2017, 2019). This value is larger than but con-sistent with, the 2018 CODATA recommended value. A second MAMI experiment is under construction and planned to begin operation in 2020 (Vorobyev, 2019). Finally, we mention an ex-periment underway at the Paul Scherrer Institut, Switzerland, in which rp will be determined from simultaneous measurements of muon-proton and electron-proton scattering (Roy et al., 2020). 4. Muon mass and magnetic moment The values for the mass mμ and magnetic-moment anomaly aμ of themuonareessentiallyunchangedfromthe2010and2014adjustments. Their valuesare determined by experimental measurements published in 1999 and 2006 and have a relative uncertainty of 2.2 3 10−8 and 5.4 3 10−7, respectively. The muon mass is derived from measurements and accurate theoretical calculations of the hyperfine splitting of the ground state of muonium μ+e−. New data on this hyperfine splitting are expected in the near future (Strasser et al., 2019). The theoretical estimate of the muon magnetic-moment anomaly aμ(th) has been discrepant with the experimental value ever since the 2006 measurement; see Fig. 8. The experimental value aμ(exp) currently exceeds the theoretical value by about 3.5σ, and models using physics beyond the standard model (SM) have been put forward to explain the discrepancy. Since mμ/me is about 207, aμ(th) is more sensitive to possible non-SM contributions than the electron magnetic-moment anomaly ae. Because of the significant in-consistency, the theoretical expression for the muon anomaly as in previous adjustments is not used in the 2018 CODATA adjustment. Two separate experiments (Abe et al., 2019; Keshavarzi, 2019) are underway to determine aμ, promising one-fourth the uncertainty of the current value. Work also continues to improve the theoretical SM expression for aμ (Keshavarzi, Nomura, and Teubner, 2018). The hope is that the discrepancy will be resolved by the closing date for the next adjustment. 5. Newtonian constant of gravitation The Newtonian constant of gravitation G, with its 2.2 3 10−5 relative uncertainty, is amongthe most poorly known constants in our 2018 adjustment. See the discussion of Fig. 9. The large scatter among the 16 measurements of G on which the recommended value is based required an expansion factor of 3.9 to reduce all residuals to less than two. The need for an expansion factor demonstrates the technological difficulty of determining G. Improving our knowledge of G may ultimately require the development of a new approach that can achieve an uncertainty no greater than one part in 106, smaller than the uncertainty of previously reported values by more than an order of magnitude (Rothleitner and Schlamminger, 2017). In addition, such technology could shed light on the reasons for the scatter among the existing data, such as the discovery of previously unknown systematic effects in the measurement methods, and would likely find other useful applications. Rothleitner and Schlamminger (2017) also suggested that moving an apparatus from a laboratory where it was used to de-termine G to another laboratory could help uncover unrecognized systematic effects. To this end, the BIPM apparatus that led to the publication in 2014 of a value of G with ur 2.4 3 10−5 is now operational at the NIST Gaithersburg laboratory. 6. Proton mass The relative atomic mass of hydrogen, Ar(1H), from the Atomic Mass Data Center and a measurement of the cyclotron frequency ratio, ωc(12C6+)/ωc(p), determine Ar(p). In the 2018 adjustment, the uncertainties of these input data are expanded by the factor 1.7 to reduce their normalized residuals to less than two. The value of Ar(1H) is based on relatively old data and constrains the value of the proton mass less than that determined by the cyclotron frequency ratio. See also Fig. 6. An independent J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-56 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr determination of Ar(p) with ur of a few parts in 1011 would help resolve the discrepancy. 7. Physics in general The 2017 redefinition of the SI has arguably been a milestone in physics and chemistry. As a consequence, many constants in our tables that previously had uncertainties are now exactly known in SI units. Many more have significantly reduced uncertainties. The physicochemical constants that are now exact in addition to NA and k are, forexample, F,R,Vm,and σ.The 30conversion factors among the six energy units J, kg, m−1, Hz, K, and eV are now exact and the relative uncertainties of their conversion factors with u and Eh are currently only 3.0 3 10−10 and 1.9 3 10−12, respectively. Further, ur of α and R∞ are now 1.5 3 10−10 and 1.9 3 10−12, respectively. A perusal of the input data in Table XXI shows there is only one input datum for some quantities, and some are decades old. Mea-surements of the same quantity by different methods in different laboratories help to identify unknown systematic effects, thereby improving the reliability of the input data. The six magnetic-moment ratios, items D32 to D37 in the table, are obvious examples of old data. The muon mass is currently only determined by essentially one measurement. It would be useful if researchers kept in mind the limited robustness of the data set on which CODATA adjustments are based in planning research. List of Symbols and Abbreviations ASD NIST Atomic Spectra Database (online) AMDC Atomic Mass Data Center, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou, People’s Re-public of China. AMDC-16 is the atomic mass eval-uation completed in 2016, the most recent available. Ar(X) Relative atomic mass of X: Ar(X) m(X)/mu a0 Bohr radius: a0 Z/αmec ae Electron magnetic-moment anomaly: ae (|ge| −2)/2 aμ Muon magnetic-moment anomaly: aμ (|gμ| −2)/2 Berkeley University of California at Berkeley, Berkeley, Cal-ifornia, USA BIPM International Bureau of Weights and Measures, Sevres, France BNL Brookhaven National Laboratory, Upton, New York, USA CGPM General Conference on Weights and Measures CIPM International Committee for Weights and Measures CODATA Committee on Data of the International Science Council CREMA The international collaboration Charge Radius Ex-periment with Muonic Atoms at the Paul Scherrer Institute, Villigen, Switzerland c Speed of light in vacuum and one of the seven defining constants of the SI d Deuteron (nucleus of deuterium D, or 2H) d220 {220} lattice spacing of an ideal silicon crystal with natural isotopic Si abundances d220(X) {220} lattice spacing of crystal X of silicon with natural isotopic Si abundances Eh Hartree energy: Eh 2R∞hc α2mec2 e Symbol for either member of the electron-positron pair; when necessary, e−or e+ is used to indicate the electron or positron e Elementary charge: absolute value of the charge of the electron and one of the seven defining constants of the SI FSU Florida State University, Tallahassee, Florida, USA FSUJ Friedrich-Schiller University, Jena, Germany G Newtonian constant of gravitation GF Fermi coupling constant gd Deuteron g-factor: gd μd/μN ge Electron g-factor: ge 2μe/μB gp Proton g-factor: gp 2μp/μN g ′ p Shielded proton g-factor: g ′ p 2μ ′ p/μN gt Triton g-factor: gt 2μt/μN gX(Y) g-factor of particle X in the ground (1S) state of hydrogenic atom Y gμ Muon g-factor: gμ 2μμ/(eZ/2mμ) Harvard HarvU also. Harvard University, Cambridge, Massa-chusetts, USA HD A hydrogen-deuterium molecule HT A hydrogen-tritium molecule HUST Huazhong University of Science and Technology, Wuhan, People’s Republic of China h Helion (nucleus of 3He) h Planck constant and one of the seven defining con-stants of the SI Z Reduced Planck constant ILL Institut Max von Laue-Paul Langevin, Grenoble, France INRIM Istituto Nazionale di Ricerca Metrologica, Torino, Italy JILA JILA, University of Colorado and NIST, Boulder, Colorado, USA J-PARC Japan Proton Accelerator Research Complex k Boltzmann constant and one of the seven defining constants of the SI KEK High Energy Accelerator Research Organization, Tsukuba, Japan LAMPF Clinton P. Anderson Meson Physics Facility at Los Alamos National Laboratory, Los Alamos, New Mexico, USA LANL Los Alamos National Laboratory, Los Alamos, New Mexico, USA LENS European Laboratory for Non-Linear Spectroscopy, University of Florence, Italy LKB Laboratoire Kastler-Brossel, Paris, France MIT Massachusetts Institute of Technology, Cambridge, Massachusetts, USA MPIK Max-Planck-Institut f¨ ur Kernphysik, Heidelberg, Germany MPQ Max-Planck-Institut f¨ ur Quantenoptik, Garching, Germany MSL Measurement Standards Laboratory, Lower Hutt, New Zealand M(X) Molar mass of X: M(X) Ar(X)Mu M(12C) Molar mass of carbon-12. M(12C) 12Mu 12NAmu ≈0.012 kg/mol Mu Molar mass constant: Mu NAmu J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-57 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Mu Muonium (μ+e−atom) mu Unified atomic mass constant: mu m(12C)/12 2hcR∞/α2c2Ar(e) m(K) Mass of the international prototype of the kilogram: m(K) ≈1 kg mX, m(X) Mass of X (for the electron e, proton p, and other ele-mentaryparticles,thefirstsymbolisused,i.e.,me,mp,etc.) NA Avogadro constant and one of the seven defining constants of the SI NIST National Institute of Standards and Technology, Gaithersburg, Maryland and Boulder, Colorado, USA NPL National Physical Laboratory, Teddington, UK n Neutron p(χ2|n) Probability that an observed value of chi square for n degrees of freedom would exceed χ2 p Proton PTB Physikalisch-Technische Bundesanstalt, Braunschweig and Berlin, Germany QCD Quantum chromodynamics QED Quantum electrodynamics R Molar gas constant; R NAk RB Birge ratio: RB (χ2/n) 1 2 R∞ Rydberg constant: R∞ mecα2/2h ri Normalized residual of an input datum Xi in a least-squares calculation: ri (Xi −〈Xi〉)/u(Xi) rd Bound-state rms charge radius of the deuteron rp Bound-state rms charge radius of the proton r(X, Y) Correlation coefficient of quantity or constant X and Y: r(X, Y) u(X, Y)/[u(X)u(Y)] SI Syst eme international d’unit´ es (International System of Units) StPtrsb D. I. Mendeleyev All-Russian Research Institute for Metrology (VNIIM), St. Petersburg, Russian Federation Sussex University of Sussex, Brighton, UK SYRTE Systemes de r´ ef´ erence Temps Espace, Paris, France TTPW Thermodynamic temperature T of the triple point of water: TTPW ≈273.16 K TGFC Task Group on Fundamental Constants of the Com-mittee on Data of the International Science Council (CODATA) TR&D Tribotech Research and Development Company, Moscow, Russian Federation t Triton (nucleus of tritium T, or 3H) UBarc Universitat Aut onoma de Barcelona, Barcelona, Spain UCB University of California at Berkeley, Berkeley, Cal-ifornia, USA UCI University of California at Irvine, Irvine, California, USA UMZ Institut f¨ ur Physik, Johannes Gutenberg-Universit¨ at Mainz, Mainz, Germany UWash University of Washington, Seattle, Washington, USA UWup University of Wuppertal, Wuppertal, Germany UZur University of Zurich, Zurich, Switzerland u Unified atomic mass unit (also called the dalton, Da): 1 u mu m(12C)/12 u(X) Standard uncertainty (i.e., estimated standard de-viation) of quantity or constant X ur(X) Relative standard uncertainty of a quantity or constant X: ur(X) u(X)/|X|, X ≠0 (also simply ur) u(X, Y) Covariance of quantities or constants X and Y ur(X, Y) Relative covariance of quantities or constants X and Y: ur(X, Y) u(X, Y)/(XY) u0 Type of uncertainty in the theory of the energy levels of hydrogen and deuterium: The contribution to the energy has correlated uncertainties for states with the same ℓand j. See also entry un. un Type of uncertainty in the theory of the energy levels of hydrogen and deuterium: The contribution has un-correlated uncertainties. See also entry u0. WarsU University of Warsaw, Warszawa, Poland Yale Yale University, New Haven, Connecticut, USA York York University, Toronto, Canada α Fine-structure constant: α e2/4πε0Zc ≈1/137 α Alpha particle (nucleus of 4He) ΔEB(AXn+) Energy required to remove n electrons from a neutral atom ΔEI(AXi+) Electron ionization energies, i 0 to n −1 ΔEMu Ground-state muonium hyperfine splitting energy ΔELS(μH, μD) Transition energy of Lamb shift in muonic hydrogen or muonic deuterium δH,D(X) Additive correction to the theoretical expression for the energy of a specified level in hydrogen or deuterium δth(X) Additive correction to a specified theoretical expression ≐ Symbol used to relate an input datum to its observa-tional equation θW Weak mixing angle ƛC Reduced Compton wavelength: ƛC Z/mec μ Symbol for either member of the muon-antimuon pair; when necessary, μ−or μ+ is used to indicate the negative muon or positive antimuon μD Muonic deuterium (an atom comprising a deuteron and a muon) μH Muonic hydrogen (an atom comprising a proton and a muon) μB Bohr magneton: μB eZ/2me μN Nuclear magneton: μN eZ/2mp μX(Y) Magnetic moment of particle X in atom or molecule Y μX, μ ′ X Magnetic moment, or shielded magnetic moment, of particle X μ0 Vacuum magnetic permeability: μ0 4παZ/e2c ≈4π 3 10−7 N/A2 n Degrees of freedom of a particular least-squares cal-culation: n N −M, N number of input data, M number of variables, or adjusted constants σ Stefan-Boltzmann constant: σ (π2/60)k4/Z3c2 τ Symbol for either member of the tau-antitau pair; when necessary, τ−or τ+ is used to indicate the negative or positive tau lepton χ2 The statistic “chi square” Acknowledgments As always, we gratefully acknowledge the help of our many colleagues throughout the world who provided the CODATA Task J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-58 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Group on Fundamental Constants with results prior to formal publication and for promptly answering our many questions about their work. We wish to thank our fellow Task Group members for their invaluable guidance and suggestions during the course of the 2018 adjustment effort. References Abe, M., et al., 2019, Prog. Theor. Exp. Phys. 2019, 053C02. Abe, M., Y. Murata, H. Iinuma, T. Ogitsu, N. Saito, K. Sasaki, T. Mibe, and H. Nakayama, 2018, Nucl. Instrum. Methods Phys. Res., Sect. A 890, 51. Adikaram, D., et al., 2015, Phys. Rev. Lett. 114, 062003. Alarc´ on, J. M., D. W. Higinbotham, C. Weiss, and Z. Ye, 2019, Phys. Rev. C 99, 044303. Angeli, I., and K. Marinova, 2013, At. Data Nucl. Data Tables 99, 69. Antognini, A., F. Kottmann, F. Biraben, P. Indelicato, F. Nez, and R. Pohl, 2013, Ann. Phys. (Amsterdam) 331, 127. Antognini, A., et al., 2013, Science 339, 417. Aoyama, T., M. Hayakawa, T. Kinoshita, and M. Nio, 2015, Phys. Rev. D 91, 033006. Aoyama, T., T. Kinoshita, and M. Nio, 2018, Phys. Rev. D 97, 036001. Aoyama, T., T. Kinoshita, and M. Nio, 2019, Atoms 7, 28. Armstrong, T. R., and M. P. Fitzgerald, 2003, Phys. Rev. Lett. 91, 201101. Arnoult, O., F. Nez, L. Julien, and F. Biraben, 2010, Eur. Phys. J. D 60, 243. Arrington, J., and I. Sick, 2015, J. Phys. Chem. Ref. Data 44, 031204. Bade, S., L. Djadaojee, M. Andia, P. Clad´ e, and S. Guellati-Khelifa, 2018, Phys. Rev. Lett. 121, 073603. Bagley, C. H., and G. G. Luther, 1997, Phys. Rev. Lett. 78, 3047. Becker, P., H. Bettin, H.-U. Danzebrink, M. Gl¨ aser, U. Kuetgens, A. Nicolaus, D. Schiel, P. De Bi` evre, S. Valkiers, and P. Taylor, 2003, Metrologia 40, 271. Becker, P., K. Dorenwendt, G. Ebeling, R. Lauer, W. Lucas, R. Probst, H.-J. Rademacher, G. Reim, P. Seyfried, and H. Siegert, 1981, Phys. Rev. Lett. 46, 1540. Beier, T., 2000, Phys. Rep. 339, 79. Beier, T., I. Lindgren, H. Persson, S. Salomonson, P. Sunnergren, H. H¨ affner, and N. Hermanspahn, 2000, Phys. Rev. A 62, 032510. Bennett, G. W., et al., 2006, Phys. Rev. D 73, 072003. Berkeland, D. J., E. A. Hinds, and M. G. Boshier, 1995, Phys. Rev. Lett. 75, 2470. Bernauer, J. C., and M. O. Distler, 2015 (private communication). Bernauer, J. C., et al., 2010, Phys. Rev. Lett. 105, 242001. Bernauer, J. C., et al., 2014, Phys. Rev. C 90, 015206. Beyer, A., et al., 2016, Opt. Express 24, 17470. Beyer, A., et al., 2017, Science 358, 79. Bezginov, N., T. Valdez, M. Horbatsch, A. Marsman, A. C. Vutha, and E. A. Hessels, 2019, Science 365, 1007. Biraben, F., 2019, C.R. Phys. 20, 671. Birge, R. T., 1929, Rev. Mod. Phys. 1, 1. Blum, T., P. A. Boyle, V. G¨ ulpers, T. Izubuchi, L. Jin, C. Jung, A. J¨ uttner, C. Lehner, A. Portelli, and J. T. Tsang, 2018, Phys. Rev. Lett. 121, 022003. Borsanyi, S., et al., 2018, Phys. Rev. Lett. 121, 022002. Bouchendira, R., P. Clad´ e, S. Guellati-Kh´ elifa, F. Nez, and F. Biraben, 2011, Phys. Rev. Lett. 106, 080801. Bourzeix, S., B. de Beauvoir, F. Nez, M. D. Plimmer, F. de Tomasi, L. Julien, F. Biraben, and D. N. Stacey, 1996, Phys. Rev. Lett. 76, 384. Breit, G., 1928, Nature (London) 122, 649. Breit, G., and I. I. Rabi, 1931, Phys. Rev. 38, 2082. CIPM, 2016, Decision CIPM/105-15 of the 105th Meeting of the International Committee for Weights and Measures (CIPM). Available at https:// www.bipm.org/en/committees/cipm/meeting/105.html. CIPM, 2017, Decision CIPM/106-10, -11, -12 of the 106th Meeting of the In-ternational Committee for Weights and Measures (CIPM). Available at https:// www.bipm.org/en/committees/cipm/meeting/106.html. Clad´ e, P., 2015, Riv. Nuovo Cimento 38, 173. Clad´ e, P., F. Nez, F. Biraben, and S. Guellati-Khelifa, 2019, C.R. Phys. 20, 77. Close, F. E., and H. Osborn, 1971, Phys. Lett. 34B, 400. Colangelo, G., M. Hoferichter, A. Nyffeler, and M. Passera, 2014, Phys. Lett. B 735, 90. Czarnecki, A., M. Dowling, J. Piclum, and R. Szafron, 2018, Phys. Rev. Lett. 120, 043203. Czarnecki, A., W. J. Marciano, and A. Vainshtein, 2003, Phys. Rev. D 67, 073006. Czarnecki, A., K. Melnikov, and A. Yelkhovsky, 2000, Phys. Rev. A 63, 012509. Czarnecki, A., and R. Szafron, 2016, Phys. Rev. A 94, 060501. Davier, M., A. Hoecker, B. Malaescu, and Z. Zhang, 2017, Eur. Phys. J. C 77, 827. de Beauvoir, B., F. Nez, L. Julien, B. Cagnac, F. Biraben, D. Touahri, L. Hilico, O. Acef, A. Clairon, and J. J. Zondy, 1997, Phys. Rev. Lett. 78, 440. Deslattes, R. D., and A. Henins, 1973, Phys. Rev. Lett. 31, 972. Dorokhov, A. E., A. E. Radzhabov, and A. S. Zhevlakov, 2014a, JETP Lett. 100, 133. Dorokhov, A. E., A. E. Radzhabov, and A. S. Zhevlokov, 2014b, Int. J. Mod. Phys. Conf. Ser. 35, 1460401. Eides, M. I., 1996, Phys. Rev. A 53, 2953. Eides, M. I., 2019, Phys. Lett. B 795, 113. Eides, M. I., and H. Grotch, 1997, Ann. Phys. (N.Y.) 260, 191. Eides, M. I., H. Grotch, and V. A. Shelyuto, 2001, Phys. Rep. 342, 63. Eides, M. I., H. Grotch, and V. A. Shelyuto, 2007, Theory of Light Hydrogenic Bound States,Springer Tracts inModern Physics Vol. 222 (Springer, Berlin, Heidelberg, New York). Eides, M. I., and V. A. Shelyuto, 2004, Phys. Rev. A 70, 022506. Eides, M. I., and V. A. Shelyuto, 2007, Can. J. Phys. 85, 509. Eides, M. I., and V. A. Shelyuto, 2014, Phys. Rev. D 90, 113002. Erickson, G. W., 1977, J. Phys. Chem. Ref. Data 6, 831. Estey, B., C. Yu, H. M¨ uller, P.-C. Kuan, and S.-Y. Lan, 2015, Phys. Rev. Lett. 115, 083002. Faustov, R., 1970, Phys. Lett. 33B, 422. Ferroglio, L., G. Mana, and E. Massa, 2008, Opt. Express 16, 16877. Fleurbaey, H., 2017, “Frequency metrology of the 1S-3S transition of hydrogen: Contribution to the proton charge radius puzzle,” Ph.D. thesis (Universit´ e Pierre et Marie Curie). Fleurbaey, H., S. Galtier, S. Thomas, M. Bonnaud, L. Julien, F. Biraben, F. Nez, M. Abgrall, and J. Gu´ ena, 2018, Phys. Rev. Lett. 120, 183001. Flowers, J. L., B. W. Petley, and M. G. Richards, 1993, Metrologia 30, 75. Friar, J. L., 1979, Ann. Phys. (N.Y.) 122, 151. Friar, J. L., and G. L. Payne, 1997, Phys. Rev. A 56, 5173. Gabrielse, G., S. E. Fayer, T. G. Myers, and X. Fan, 2019, Atoms 7, 45. Galtier, S., H. Fleurbaey, S. Thomas, L. Julien, F. Biraben, and F. Nez, 2015, J. Phys. Chem. Ref. Data 44, 031201. Garbacz, P., K. Jackowski, W. Makulski, and R. E. Wasylishen, 2012, J. Phys. Chem. A 116, 11896. Glazov, D. A., and V. M. Shabaev, 2002, Phys. Lett. A 297, 408. Gnendiger, C., D. St¨ ockinger, and H. St¨ ockinger-Kim, 2013, Phys. Rev. D 88, 053005. Griffioen, K., C. Carlson, and S. Maddox, 2016, Phys. Rev. C 93, 065207. Grotch, H., 1970, Phys. Rev. Lett. 24, 39. Gundlach, J. H., and S. M. Merkowitz, 2000, Phys. Rev. Lett. 85, 2869. Gundlach, J. H., and S. M. Merkowitz, 2002 (private communication). Hagiwara, K., R. Liao, A. D. Martin, D. Nomura, and T. Teubner, 2011, J. Phys. G 38, 085003. Hagley, E. W., and F. M. Pipkin, 1994, Phys. Rev. Lett. 72, 1172. Hamzeloui, S., J. A. Smith, D. J. Fink, and E. G. Myers, 2017, Phys. Rev. A 96, 060501. Hanke, M., and E. G. Kessler, 2005, J. Phys. D 38, A117. Hanneke, D., S. Fogwell, and G. Gabrielse, 2008, Phys. Rev. Lett. 100, 120801. H¨ artwig, J., S.Grosswig, P. Becker, and D. Windisch, 1991, Phys. Status SolidiA 125, 79. Hayward, T. B., and K. A. Griffioen, 2020, Nucl. Phys. A999, 121767. Heiße, F., et al., 2017, Phys. Rev. Lett. 119, 033001. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-59 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Horbatsch, M., and E. A. Hessels, 2010, Phys. Rev. A 82, 052519. Horbatsch, M., and E. A. Hessels, 2011, Phys. Rev. A 84, 032508. Horbatsch, M., E. A. Hessels, and A. Pineda, 2017, Phys. Rev. C 95, 035203. Hu, Z.-K., J.-Q. Guo, and J. Luo, 2005, Phys. Rev. D 71, 127505. Huang, W. J., G. Audi, M. Wang, F. G. Kondev, S. Naimi, and X. Xu, 2017, Chin. Phys. C 41, 030002. Ivanov, V. G., S. G. Karshenboim, and R. N. Lee, 2009, Phys. Rev. A 79, 012512. Jegerlehner, F., 2018, EPJ Web Conf. 166, 00022. Jegerlehner, F., 2019, EPJ Web Conf. 218, 01003. Jegerlehner, F., and A. Nyffeler, 2009, Phys. Rep. 477, 1. Jentschura, U. D., 2003, J. Phys. A 36, L229. Jentschura, U. D., 2006, Phys. Rev. A 74, 062517. Jentschura, U. D., 2009, Phys. Rev. A 79, 044501. Jentschura, U. D., A. Czarnecki, and K. Pachucki, 2005, Phys. Rev. A 72, 062102. Jentschura, U. D., A. Czarnecki, K. Pachucki, and V. A. Yerokhin, 2006, Int. J. Mass Spectrom. 251, 102. Jentschura, U. D., and P. J. Mohr, 2002, Can. J. Phys. 80, 633. Kalinowski, M., 2019, Phys. Rev. A 99, 030501. Karagioz, O. V., and V. P. Izmailov, 1996, Izmer. Tekh. 39, 3. Karshenboim, S. G., 2000, Phys. Lett. A 266, 380. Karshenboim, S. G., and V. G. Ivanov, 2018a, Phys. Rev. A 97, 022506. Karshenboim, S. G., and V. G. Ivanov, 2018b, Phys. Rev. A 98, 022522. Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 1999, Phys. Scr. T80, 491. Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 2000, Zh. Eksp. Teor. Fiz. 117, 67 [J. Exp. Theor. Phys. 90, 59 (2000)]. Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 2001a, Can. J. Phys. 79, 81. Karshenboim, S. G., V. G. Ivanov, and V. M. Shabaev, 2001b, Zh. Eksp. Teor. Fiz. 120, 546 [J. Exp. Theor. Phys. 93, 477 (2001)]. Karshenboim, S. G., E. Y. Korzinin, V. A. Shelyuto, and V. G. Ivanov, 2015, J. Phys. Chem. Ref. Data 44, 031202. Karshenboim, S. G., V. A. Shelyuto, and A. I. Vainshtein, 2008, Phys. Rev. D 78, 065036. Keshavarzi, A., 2019, EPJ Web Conf. 212, 05003. Keshavarzi, A., D. Nomura, and T. Teubner, 2018, Phys. Rev. D 97, 114025. Kessler, E. G., C. I. Szabo, J. P. Cline, A. Henins, L. T. Hudson, M. H. Mendenhall, and M. D. Vaudin, 2017, J. Res. Natl. Inst. Stand. Technol. 122, 24. Kessler, Jr., E. G., R. D. Deslattes, and A. Henins, 1979, Phys. Rev. A 19, 215. Kessler, Jr., E. G., M. S. Dewey, R. D. Deslattes, A. Henins, H. G. B¨ orner, M. Jentschel, and H. Lehmann, 2000, in Capture Gamma-Ray Spectroscopy and Related Topics, edited by Wender, S. (American Institute of Physics, Melville, NY), pp. 400–407. Kessler, Jr., E. G., J. E. Schweppe, and R. D. Deslattes, 1997, IEEE Trans. Instrum. Meas. 46, 551. Kleinevoß, U., 2002, “Bestimmung der Newtonschen Gravitationskonstanten G,” Ph.D. thesis (University of Wuppertal). Kleinvoß, U., H. Meyer, H. Piel, and S. Hartmann, 2002 (private communication). K¨ ohler, F., S. Sturm, A. Kracke, G. Werth, W. Quint, and K. Blaum, 2015, J. Phys. B 48, 144032. Kramers, H. A., and W. Heisenberg, 1925, Z. Phys. 31, 681. Krauth, J. J., M. Diepold, B. Franke, A. Antognini, F. Kottmann, and R. Pohl, 2016, Ann. Phys. (Amsterdam) 366, 168. Kurz, A., T. Liu, P. Marquard, and M. Steinhauser, 2014a, Nucl. Phys. B879, 1. Kurz, A., T. Liu, P. Marquard, and M. Steinhauser, 2014b, Phys. Lett. B 734, 144. Laporta, S., 1993, Nuovo Cimento Soc. Ital. Fis. 106A, 675. Laporta, S., 2017, Phys. Lett. B 772, 232. Laporta, S., and E. Remiddi, 1993, Phys. Lett. B 301, 440. Lee, G., J. R. Arrington, and R. J. Hill, 2015, Phys. Rev. D 92, 013013. Lee, R. N., A. I. Milstein, I. S. Terekhov, and S. G. Karshenboim, 2005, Phys. Rev. A 71, 052501. Li, Q., et al., 2018, Nature (London) 560, 582. Liu, J., D. Sprecher, C. Jungen, W. Ubachs, and F. Merkt, 2010, J. Chem. Phys. 132, 154301. Liu, W., et al., 1999, Phys. Rev. Lett. 82, 711. Low, F., 1952, Phys. Rev. 88, 53. Lundeen, S., and F. Pipkin, 1981, Phys. Rev. Lett. 46, 232. Lundeen, S. R., and F. M. Pipkin, 1986, Metrologia 22, 9. Luo, J., Q. Liu, L.-C. Tu, C.-G. Shao, L.-X. Liu, S.-Q. Yang, Q. Li, and Y.-T. Zhang, 2009, Phys. Rev. Lett. 102, 240801. Luther, G. G., and W. R. Towler, 1982, Phys. Rev. Lett. 48, 121. Mariam, F. G., 1981, “High precision measurement of the muonium ground state hyperfine interval and the muon magnetic moment,” Ph.D. thesis (Yale University). Mariam, F. G., et al., 1982, Phys. Rev. Lett. 49, 993. Marrus, R., and P. J. Mohr, 1979, Adv. At. Mol. Phys. 14, 181. Marsman, A., M. Horbatsch, Z. A. Corriveau, and E. A. Hessels, 2018, Phys. Rev. A 98, 012509. Martin, J., U. Kuetgens, J. St¨ umpel, and P. Becker, 1998, Metrologia 35, 811. Massa, E., G. Mana, and U. Kuetgens, 2009, Metrologia 46, 249. Massa, E., G. Mana, U. Kuetgens, and L. Ferroglio, 2009, New J. Phys. 11, 053013. Matveev, A., et al., 2013, Phys. Rev. Lett. 110, 230801. Mihoviloviˇ c, M., et al., 2017, Phys. Lett. B 771, 194. Mihoviloviˇ c, M., et al., 2019, arXiv:1905.11182. Millman, S., I. I. Rabi, and J. R. Zacharias, 1938, Phys. Rev. 53, 384. Mills, I. M., P. J. Mohr, T. J. Quinn, B. N. Taylor, and E. R. Williams, 2011, Phil. Trans. R. Soc. A 369, 3907. Mohr, P. J., D. B. Newell, and B. N. Taylor, 2016a, Rev. Mod. Phys. 88, 035009. Mohr, P. J., D. B. Newell, and B. N. Taylor, 2016b, J. Phys. Chem. Ref. Data 45, 043102. Mohr, P. J., D. B. Newell, B. N. Taylor, and E. Tiesinga, 2018, Metrologia 55, 125. Mohr, P. J., and B. N. Taylor, 1999, J. Phys. Chem. Ref. Data 28, 1713. Mohr, P. J., and B. N. Taylor, 2000, Rev. Mod. Phys. 72, 351. Mohr, P. J., and B. N. Taylor, 2005, Rev. Mod. Phys. 77, 1. Mohr, P. J., B. N. Taylor, and D. B. Newell, 2008a, Rev. Mod. Phys. 80, 633. Mohr, P. J., B. N. Taylor, and D. B. Newell, 2008b, J. Phys. Chem. Ref. Data 37, 1187. Mohr, P. J., B. N. Taylor, and D. B. Newell, 2012a, Rev. Mod. Phys. 84, 1527. Mohr, P. J., B. N. Taylor, and D. B. Newell, 2012b, J. Phys. Chem. Ref. Data 41, 043109. Mooser, A., S. Ulmer, K. Blaum, K. Franke, H. Kracke, C. Leiteritz, W. Quint, C. C. Rodegheri, C. Smorra, and J. Walz, 2014, Nature (London) 509, 596. Myers, E. G., 2019, Atoms 7, 37. Myers, E. G., A. Wagner, H. Kracke, and B. A. Wesson, 2015, Phys. Rev. Lett. 114, 013003. Neronov, Y. I., and V. S. Aleksandrov, 2011, Pis’ma Zh. Eksp. Teor. Fiz. 94, 452 [JETP Lett. 94, 418 (2011)]. Neronov, Y. I., and S. G. Karshenboim, 2003, Phys. Lett. A 318, 126. Neronov, Y. I., and N. N. Seregin, 2012, Zh. Eksp. Teor. Fiz. 115, 777 [J. Exp. Theor. Phys. 115, 777 (2012)]. Newell, D. B., et al., 2018, Metrologia 55, L13. Newman, R., M. Bantel, E. Berg, and W. Cross, 2014, Phil. Trans. R. Soc. A 372, 20140025. Newton, G., D. A. Andrews, and P. J. Unsworth, 1979, Phil. Trans. R. Soc. A 290, 373. Nomura, D., and T. Teubner, 2013, Nucl. Phys. B867, 236. Nyffeler, A., 2014, Int. J. Mod. Phys. Conf. Ser. 35, 1460456. Pachucki, K., 1996, Phys. Rev. A 53, 2092. Pachucki, K., A.Czarnecki, U. D. Jentschura, and V. A.Yerokhin, 2005, Phys. Rev. A 72, 022108. Pachucki, K., U. D. Jentschura, and V. A. Yerokhin, 2004, Phys. Rev. Lett. 93, 150401; 94, 229902(E) (2005). Pachucki, K., V. Patk´ oˇ s, and V. A. Yerokhin, 2018, Phys. Rev. A 97, 062511. Pachucki, K., and M. Puchalski, 2017, Phys. Rev. A 96, 032503. Parker, R. H., C. Yu, W. Zhong, B. Estey, and H. M¨ uller, 2018, Science 360, 191. Parks, H. V., and J. E. Faller, 2010, Phys. Rev. Lett. 105, 110801. Parks, H. V., and J. E. Faller, 2019, Phys. Rev. Lett. 122, 199901. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-60 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr Parthey, C. G., A. Matveev, J. Alnis, R. Pohl, T. Udem, U. D. Jentschura, N. Kolachevsky, and T. W. H¨ ansch, 2010, Phys. Rev. Lett. 104, 233001. Parthey, C. G., et al., 2011, Phys. Rev. Lett. 107, 203001. Peset, C., and A. Pineda, 2015, Eur. Phys. J. A 51, 156. Peters, A., K. Y. Chung, B. Young, J. Hensley, and S. Chu, 1997, Phil. Trans. R. Soc. A 355, 2223. Pohl, R., et al., 2010, Nature (London) 466, 213. Pohl, R., et al., 2016, Science 353, 669. Prevedelli, M., L. Cacciapuoti, G. Rosi, F. Sorrentino, and G. M. Tino, 2014, Phil. Trans. R. Soc. A 372, 20140030. Puchalski, M., J. Komasa, and K. Pachucki, 2015, Phys. Rev. A 92, 020501. Quinn, T., H. Parks, C. Speake, and R. Davis, 2013, Phys. Rev. Lett. 111, 101102; 113, 039901(E) (2014). Quinn, T., C. Speake, H. Parks, and R. Davis, 2014, Phil. Trans. R. Soc. A 372, 20140032. Quinn, T. J., 1989, Metrologia 26, 69. Quinn, T. J., C. C. Speake, S. J. Richman, R. S. Davis, and A. Picard, 2001, Phys. Rev. Lett. 87, 111101. Rosi, G., F. Sorrentino, L. Cacciapuoti, M. Prevedelli, and G. M. Tino, 2014, Nature (London) 510, 518. Rothleitner, C., and S. Schlamminger, 2017, Rev. Sci. Instrum. 88, 111101. Roy, P., et al., 2020, Nucl. Instrum. Methods Phys. Res., Sect. A 949, 162874. Rudzi´ nski, A., M. Puchalski, and K. Pachucki, 2009, J. Chem. Phys. 130, 244102. Safronova, M. S., D. Budker, D. DeMille, D. F. J. Kimball, A. Derevianko, and C. W. Clark, 2018, Rev. Mod. Phys. 90, 025008. Sapirstein, J. R., and D. R. Yennie, 1990, in Quantum Electrodynamics, edited by Kinoshita, T. (World Scientific, Singapore), Chap. 12, pp. 560–672. Schlamminger, S., E. Holzschuh, W. K¨ undig, F. Nolting, R. E. Pixley, J. Schurr, and U. Straumann, 2006, Phys. Rev. D 74, 082001. Schneider, G., et al., 2017, Science 358, 1081. Schwob, C., L. Jozefowski, B. de Beauvoir, L. Hilico, F. Nez, L. Julien, F. Biraben, O. Acef, and A. Clairon, 1999, Phys. Rev. Lett. 82, 4960; 86, 4193(E) (2001). Shabaev, V. M., and V. A. Yerokhin, 2002, Phys. Rev. Lett. 88, 091801. Shelyuto, V. A., S. G. Karshenboim, and S. I. Eidelman, 2018, Phys. Rev. D 97, 053001. Sick, I., 2001, Prog. Part. Nucl. Phys. 47, 245. Sick, I., 2003, Phys. Lett. B 576, 62. Sick, I., 2007, Can. J. Phys. 85, 409. Sick, I., 2008, in Precision Physics of Simple Atoms and Molecules, edited by Karshenboim, S. G., Lecture Notes in Physics Vol. 745 (Springer, Berlin, Heidelberg), pp. 57–77. Sick, I., and D. Trautmann, 1998, Nucl. Phys. A637, 559. Simon, G. G., C. Schmitt, and V. H. Walther, 1981, Nucl. Phys. A364, 285. Sommerfeld, A., 1916, Ann. Phys. (Berlin) 356, 1. Sprecher, D., J. Liu, C. Jungen, W. Ubachs, and F. Merkt, 2010, J. Chem. Phys. 133, 111102. Strasser, P., et al., 2019, EPJ Web Conf. 198, 00003. Sturm, S., 2015 (private communication). Sturm, S., F. K¨ ohler, J. Zatorski, A. Wagner, Z. Harman, G. Werth, W. Quint, C. H. Keitel, and K. Blaum, 2014, Nature (London) 506, 467. Sturm, S., A. Wagner, M. Kretzschmar, W. Quint, G. Werth, and K. Blaum, 2013, Phys. Rev. A 87, 030501. Tanabashi, M., et al., 2018, Phys. Rev. D 98, 030001. Taylor, B. N., and T. J. Witt, 1989, Metrologia 26, 47. Thomas, S., H. Fleurbaey, S. Galtier, L. Julien, F. Biraben, and F. Nez, 2019, Ann. Phys. (Berlin) 531, 1800363. Tomalak, O., 2019, Eur. Phys. J. A 55, 64. Tu, L.-C., Q. Li, Q.-L. Wang, C.-G. Shao, S.-Q. Yang, L.-X. Liu, Q. Liu, and J. Luo, 2010, Phys. Rev. D 82, 022001. Van Dyck, Jr., R. S., D. B. Pinegar, S. V. Liew, and S. L. Zafonte, 2006, Int. J. Mass Spectrom. 251, 231. Volkov, S., 2019, Phys. Rev. D 100, 096004. Vorobyev, A. A., 2019, Phys. Part. Nucl. Lett. 16, 524. Vutha, A. C., and E. A. Hessels, 2015, Phys. Rev. A 92, 052504. Wang, M., G. Audi, F. G. Kondev, W. J. Huang, S. Naimi, and X. Xu, 2017, Chin. Phys. C 41, 030003. Weitz, M., A. Huber, F. Schmidt-Kaler, D. Leibfried, W. Vassen, C. Zimmermann, K.Pachucki, T. W. H¨ ansch, L. Julien, and F.Biraben, 1995, Phys. Rev. A52, 2664. Windisch, D., and P. Becker, 1990, Phys. Status Solidi A 118, 379. Xiong, W., et al., 2019, Nature (London) 575, 147. Yerokhin, V. A., 2009, Phys. Rev. A 80, 040501. Yerokhin, V. A., 2011, Phys. Rev. A 83, 012507. Yerokhin, V. A., 2018, Phys. Rev. A 97, 052509. Yerokhin, V. A., and Z. Harman, 2013, Phys. Rev. A 88, 042502. Yerokhin, V. A., and Z. Harman, 2017, Phys. Rev. A 95, 060501. Yerokhin, V. A., P. Indelicato, and V. M. Shabaev, 2008, Phys. Rev. A 77, 062510. Yerokhin, V. A., and U. D. Jentschura, 2008, Phys. Rev. Lett. 100, 163001. Yerokhin, V. A., and U. D. Jentschura, 2010, Phys. Rev. A 81, 012502. Yerokhin, V. A., K. Pachucki, and V. Patk´ oˇ s, 2019, Ann. Phys. (Berlin) 531, 1800324. Yerokhin, V. A., and V. M. Shabaev, 2015, Phys. Rev. Lett. 115, 233002. Yerokhin, V. A., and V. M. Shabaev, 2016, Phys. Rev. A 93, 062514. Yost, D. C., A. Matveev, A. Grinin, E. Peters, L. Maisenbacher, A. Beyer, R. Pohl, N. Kolachevsky, T. W. H¨ ansch, and T. Udem, 2016, Phys. Rev. A 93, 042509. Young, B., M. Kasevich, and S. Chu, 1997, in Atom Interferometry, edited by Berman, P. R. (Academic Press, New York), pp. 363–406. Yu, C., W. Zhong, B. Estey, J. Kwan, R. H. Parker, and H. M¨ uller, 2019, Ann. Phys. (Berlin) 531, 1800346. Zafonte, S. L., and R. S. Van Dyck, Jr., 2015, Metrologia 52, 280. Zatorski, J., B. Sikora, S. G. Karshenboim, S. Sturm, F. K¨ ohler-Langes, K. Blaum, C. H. Keitel, and Z. Harman, 2017, Phys. Rev. A 96, 012502. Zhou, S., P. Giulani, J. Piekarewicz, A. Bhattacharya, and D. Pati, 2019, Phys. Rev. C 99, 055202. J. Phys. Chem. Ref. Data 50, 033105 (2021); doi: 10.1063/5.0064853 50, 033105-61 U.S. Secretary of Commerce. Journal of Physical and Chemical Reference Data ARTICLE scitation.org/journal/jpr
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https://archive.nytimes.com/learning.blogs.nytimes.com/2014/05/21/in-search-of-the-win-win-solution-exploring-techniques-in-fair-division/
In Search of the Win-Win Solution: Exploring Techniques in Fair Division - The New York Times Sections Home SearchSkip to content The New York Times The Learning Network| In Search of the Win-Win Solution: Exploring Techniques in Fair Division Close search Site Search Navigation Search NYTimes.com Clear this text input Go 1. Education 1. 2026 U.S. News Rankings Are Out After a Tumultuous Year for Colleges -------------------------------------------------------------------- 2. Were You Assigned Full Books to Read in High School English? Tell Us. --------------------------------------------------------------------- 3. Judge Orders N.I.H. to Restore Suspended Research Grants at U.C.L.A. -------------------------------------------------------------------- 4. 4 Takeaways From the Times Investigation Into the J-1 Visa Program ------------------------------------------------------------------ 5. 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Business Business Opinion Opinion Tech Science Health Sports Sports Arts Arts Books Style Style Food Food Travel Magazine T Magazine Real Estate Obituaries Video The Upshot Reader Center Conferences Crossword Times Insider Newsletters The Learning Network Multimedia Photography Podcasts NYT Store NYT Wine Club nytEducation Times Journeys Meal Kits Subscribe Manage Account Today's Paper Tools & Services Jobs Classifieds Corrections More Site Mobile Navigation Supported by Search In Search of the Win-Win Solution: Exploring Techniques in Fair Division By Patrick Honner May 21, 2014 3:46 pm May 21, 2014 3:46 pm Photo Related ArticleCredit Email Share Tweet Save More Mathematics Teaching ideas based on New York Times content. See all in Category » See all lesson plans » Overview | How can goods and burdens be fairly divided among groups of people? What are the characteristics of an equitable division procedure? In this lesson, students explore the concept of “fair division” by researching and demonstrating the basic techniques in this modern field. By investigating different division strategies in various contexts, students will learn about the essential characteristics of a fair procedure, analyze the strengths and weaknesses of various algorithms and learn how this important principle in mathematics and economics applies to the real world. Materials | Computers with Internet access. Warm-Up | Have students consider the problem faced by Albert Sun, as stated in the opening to his piece “To Divide the Rent, Start With a Triangle.” Last year, two friends and I moved into a small three-bedroom apartment in Manhattan. We chose it for its relatively reasonable price — around $3,000 a month — and its convenient location. Just finding it was a challenge, but then we faced another one: deciding who would get each bedroom. The bedrooms were different sizes, ranging from small to very small. Two faced north toward the street and had light; the third and smallest faced an alley. The largest had two windows; the midsize room opened onto the fire escape. Have students offer some suggestions as to how Mr. Sun and his soon-to-be roommates might have handled this situation. Make a list, and be sure that these typical approaches are included. Have students discuss the benefits and shortcomings of each method, and have them consider the questions “Which method is fairest, and why?” and “How would you handle the situation?” Related | In “To Divide the Rent, Start With a Triangle,” Albert Sun describes how he turned to mathematics to find a better way to split the rent. But as it turns out, a field of academics is dedicated to studying the subject of fair division, or how to divide good and bad things fairly among groups of people. To the researchers, none of the typical methods are satisfactory. They have better ways. The problem is that individuals evaluate a room differently. I care a lot about natural light, but not everyone does. Is it worth not having a closet? Or one might care more about the shape of the room, or its proximity to the bathroom. A division of rent based on square feet or any fixed list of elements can’t take every individual preference into account. And negotiation without a method may lead to conflict and resentment. Background Vocabulary: Read the entire article with your class, then answer the questions below. Questions | For discussion and reading comprehension: Why couldn’t the roommates just divide the rent evenly? Why couldn’t each roommate just choose the room he or she wanted? Why couldn’t the roommates just assign the rooms randomly? What does it mean for an outcome to be “fair” in this situation? What are some other applications of fair division? Activity First, have students play around with this NYT interactive feature to get a sense of how the fair division algorithm mentioned in the article works. By clicking on a vertex, one can change the room preference for the given player at the given set of prices, and then see how the other vertexes in the triangle change as a result. This particular algorithm, developed by the mathematician Francis Su, is based on a deep mathematical result known as Sperner’s Lemma. However, there are several simpler and more accessible techniques in fair division. RELATED RESOURCES From The Learning Network N Ways to Apply Algebra With The New York Times From NYTimes.com Sperner’s Lemma and Rental Harmony For Birthday Parties or Legal Parties; Dividing Things Fairly Is Not Always a Piece of Cake Around the Web Summary of Fair Division The Fair Division Calculator Math Shall Set You Free – From Envy Then, for the main activity, have students research these basic fair division techniques listed below and prepare presentations on each. In addition to demonstrating how their chosen method works, students should describe the procedure’s strengths and weaknesses: What kinds of goods can be divided using the method? What kinds of goods can’t be divided using the method? What is required of the players in this method? Students should also explain how the technique possesses the important characteristics of a fair division strategy: that the players are free to make independent choices; that the players are guaranteed a fair share; and that no external authority is needed. Basic Fair Division Strategies: In each of the following situations, the people involved in the division process will be referred to as players, and the set of goods to be apportioned will be called “S.” A full presentation of all these techniques, including examples, can be found here (PDF). Divider-Chooser (two players) This is the classic method of dividing a single item that can be cut up in any way, like a parcel of land or a piece of cake. One player divides S into two pieces and the other player chooses which of the two pieces he or she wants. Lone Divider (three or more players) This is another strategy for dividing a single item that can easily cut up into pieces. First, Player A divides S into three pieces. Then Players B and C independently and secretly make lists of all the pieces they would accept. Players B and C then share their lists. As long as there is something on either of their lists, there will be a fair way to give at least one, if not both, of Players B or C one of the pieces. Player A then takes the remaining piece, or the two remaining players resort to the divider-chooser method described above with what remains. Lone Chooser (three or more players) This is a third strategy for a single item that can be cut up into pieces. Two players divide S between themselves into two fair shares using the divider-chooser method explained above. They then each divide their shares into three pieces. The third player now chooses one of the three pieces from each player. Sealed Bids This method can be applied to a large collection of items of varying values, like an estate. First a list is made of all the available items. Players secretly write down what they are willing to pay for each item (their “bids”). Each player’s “fair share” is the total amount that person is willing to pay for all the items on his or her list, divided by the total number of players. The bids are made public, and every item is given to its highest bidder. If a player ends up receiving more than his or her “fair share,” that person must contribute the difference in cash into a pool, which is then divided evenly among all the players at the end. The Method of Markers This method also can be applied to a large set of items of varying values. All the items are put in a row in decreasing order of value, with the most valuable item at the far left. Each player separates the items into fair shares by placing “markers” at various positions along the row. Everything to the left of the left-most marker goes to the player who placed it there, and his or her remaining markers are removed. Everything to the left of the new left-most marker goes to the player who placed it there, and his or her remaining markers are removed, and so on. At the end, any remainder is split among the players. Last Diminisher This method works for dividing up a single item among many people. The players are ordered randomly, and the first player suggests a way to cut off a share of S that he or she would accept. Each other player can either approve that share, or “diminish” it by some amount and claim it as his or her own. If this happens, each of the other players must now either approve the new share or diminish it and claim it. Once all the other players approve a share, it is given to the “last diminisher,” and the procedure starts again with the remaining players and the remnants of S. Photo Six countries claim conflicting territorial rights in the South China Sea, a region rich in oil, gas, fishing and mineral resources. Related Interactive GraphicCredit Going Further This rental calculator allows you to explore Dr. Su’s algorithm more deeply by changing the total rent and number of rooms. Create a hypothetical rent-sharing situation with your friends and see what happens, or find your own division problem the calculator can help you with. Dr. Su also has a fair division calculator available on his home page, and there are several interactive demonstrations of fair division available at Cut the Knot Math. Once you are familiar with the various methods of fair division, see if you can invent a division procedure of your own. But be sure to verify that it satisfies the conditions of being a true fair division strategy. Another variation of fair division problems is requiring that the results to be “envy-free”: In an envy-free division, every player must be guaranteed to a share he or she feels is the largest share, not just a share he or she considers fair. Check out one envy-free strategy at Math Fun Facts, and read about how a technique in envy-free division was used to settle a dispute over a jointly owned condominium. The object of fair division is for each player to receive an equal share. However, in apportionment, the goal is for players to receive proportional shares. For example, apportionment strategies are used in assigning congressional seats based on population. Research the methods of apportionment and see how states can gain or lose seats based on population changes. Fair division strategies have many real-world applications, like the dividing of inheritances and estates, resource sharing and territorial disputes. Scour The New York Times for stories where fair division strategies might be of use, like the current territorial disputes in the South China Sea or drilling rights in the Arctic Ocean. Standards This resource may be used to address the academic standards listed below. View all Common Core Standards for Mathematical Practice 2Reason abstractly and quantitatively. 4Model with Mathematics. 5Use appropriate tools strategically. Share Comments are no longer being accepted. Andrew TripoulasNovember 4, 2014·8:36 pm Thanks! What's Next Loading... Previous Post How Have You Handled Being the ‘New Kid’? Next Post Word of the Day | invective Weekly Newsletter Sign up for our free newsletter. Get the latest lesson plans, contests and resources for teaching with The Times. 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https://www.cut-the-knot.org/m/Geometry/Reim1.shtml
Site What's new Content page Front page Index page About Privacy policy Help with math Subjects Arithmetic Algebra Geometry Probability Trigonometry Visual illusions Articles Cut the knot! What is what? Inventor's paradox Math as language Problem solving Collections Outline mathematics Book reviews Interactive activities Did you know? Eye opener Analogue gadgets Proofs in mathematics Things impossible Index/Glossary Simple math Fast Arithmetic Tips Stories for young Word problems Games and puzzles Our logo Make an identity Elementary geometry Reim's Similar Coins I What Might This Be About? Problem Let two circles cross at points $A$ and $B;$ $E$ and $F$ are two points on one of the circles; $EA$ meets the second circle second time at $G,$ $FB$ at $H:$ Then $GH\parallel EF.$ Solution Chasing angles leads to a simple solution. By the construction, angles $BAE$ ad $BAC$ are supplementary: $\angle BAE+\angle BAG=180^{\circ}.$ On the other hand, quadrilateral $ABHG$ is cyclic, implying $\angle BAG+\angle BHG=180^{\circ}.$ Hence, $\angle BHG=\angle BAE.$ Similarly, since quadrilateral $AEFB$ is cyclic $\angle BAE+\angle BFE=180^{\circ}$ which gives $\angle BHG+\angle BFE=180^{\circ}.$ These I believe are called "consecutive interior angles". Having them add up to $180^{\circ}$ makes the two lines, $GH$ and $EF,$ parallel. Converse 1 Given a cyclic quadrilateral $ABFE$ and points $G$ and $H$ on the extensions of $EA$ and $FB,$ respectively. If $GH\parallel EF,$ then quadrilateral $ABHG$ is cyclic. Converse 2 Let two circles cross at points $A$ and $B;$ $E$ and $F$ are two points on one of the circles; $EF$ meets the second circle second time at $G,$ $GH$ a chord in the second circle such that at $GH\parallel EF.$ Then $FH$ passes through $B.$ For, if $FH'$ through $B,$ with $H'$ on the second circle, then $GH\parallel EF\parallel GH',$ implying that $H=H'.$ Acknowledgment I confess to not knowing the reason for the theorem designation. There is a companion theorem under the same attribution. I came across the latter in an article by Jean-Louis Ayme where he referred to it as "Le théorème des moniennes semblables de Reim" which both I and google had a difficulty translating. It looks to me like "Reim's similar coins" might be a good fit, but I am not sure. |Contact| |Front page| |Contents| |Geometry| Chasing Inscribed Angles Munching on Inscribed Angles More On Inscribed Angles Inscribed Angles Tangent and Secant Angles Inscribed in an Absent Circle A Line in Triangle Through the Circumcenter Angle Bisector in Parallelogram Phantom Circle and Recaptured Symmetry Cherchez le quadrilatere cyclique Cyclic Quadrilateral, Concurrent Circles and Collinear Points Parallel Lines in a Cyclic Quadrilateral Reim's Similar Coins I Reim's Similar Coins II Reim's Similar Coins III Reim's Similar Coins IV Pure Angle Chasing Pure Angle Chasing II Pure Angle Chasing III Copyright © 1996-2018 Alexander Bogomolny
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https://www.awesomemath.org/wp-pdf-files/math-reflections/mr-2025-04/mr_4_2025_trigonometric_inequalities_v2.pdf
On Huygens’s Trigonometric Inequality and Some Related Results Kiran Chandra Das Kharagpur, WB, India - 721306 kirancdas1947@gmail.com 1. Introduction The inequalities sin x < x < tan x (1) for all x ∈  0, π 2  are well-known. They come from the very definition of the trigonometric functions sine and tangent on the unit circle. Note that they can also be written as cos x < sin x x < 1 (1′) and, in this form, they hold for any x with 0 < |x| < π 2 . By the squeeze theorem they imply the remarkable limit lim x→0 sin x x = 1, which is important for computing the derivatives of all the trigonometric functions – it would be nonsensical to try prove these inequalities by using calculus (we would be stuck in a loop of circular reasoning). Nevertheless, in what follows, we will use calculus in order to prove some related but more complicated inequalities. The first one is tan x x > x sin x, for 0 < x < π 2 (A) which is also known (but maybe not so well-known), and which can be written in various equivalent forms, such as sin x tan x > x2 ⇔sin2 x > x2 cos x ⇔sin x > x√cos x, 0 < x < π 2 . We will also prove the following improvement of the last form of (A) (actually we will refer to any of these equivalent forms as (A)): sin x > x 3 √cos x, for 0 < x < π 2 (B) and other inequalities in the same vein (from which the most known is Huygens’s inequality mentioned in the title) either using or avoiding calculus if possible. These inequalities (and others) can be found (with different proofs) in Roger Nelsen’s article which also provides more interesting references for such problems. 2. The inequalities We start by proving (in two ways) the inequality (A). See for more proofs of (A). Proposition 1. The following inequality tan x x > x sin x is true for all x with 0 < |x| < π 2 . First proof. Since both functions from the left-hand side and the right-hand side of the inequality are even, it is enough to prove it for 0 < x < π 2 , so it suffices to show that sin x tan x > x2 for 0 < x < π 2 . Let f be the function defined by f(x) = sin x tan x −x2 for 0 ≤x < π 2 . Clearly, f is indefinitely differentiable and its first three derivatives are f ′(x) = 2 sin x + sin x tan2 x −2x, f ′′(x) = 2 cos x + 3 cos x tan2 x + 2 sin x tan3 x −2, Mathematical Reflections 4 (2025) 1 and f ′′′(x) = 4 sin x + 11 sin x tan2 x + 6 sin x tan2 x for all x ∈ h 0, π 2  . (In all the above computations we used the derivative of tan x in the form 1 + tan2 x, and the obvious fact that cos x tan x = sin x.) Clearly, the third derivative is positive on  0, π 2  (and is 0 for x = 0), hence the second derivative increases on h 0, π 2  . It follows that f ′′(x) ≥f ′′(0) = 0 (with equality only at x = 0), therefore f ′ also increases on h 0, π 2  . Thus f ′(x) ≥f ′(0) = 0 for all x ∈ h 0, π 2  , with equality at 0, which leads to the fact that f also increases on h 0, π 2  , and, finally, to the desired inequality f(x) ≥f(0) = 0 for all x ∈ h 0, π 2  (with equality only at x = 0). Second proof. Here we only use the inequalities between the means of two positive numbers and the second inequality from (1) applied to x 2 ∈  0, π 2  in the place of x. Indeed, according to the inequality between the geometric and harmonic means and to the mentioned inequality, we have √ sin x · tan x > 2 1 sin x + cos x sin x = 2 sin x 1 + cos x = 2 tan x 2 > x, so we obtain the desired result. Alternatively, note that the inequality √ sin x tan x > 2 tan x 2 , (2) or (squared) sin x tan x > 4 tan2 x 2 is equivalent to 1 > 1 −tan4 x 2 if we use the well-known formulae sin x = 2 tan x 2 1 + tan2 x 2 and tan x = 2 tan x 2 1 −tan2 x 2 . We further use inequality (2) to prove inequality (B). Proposition 2. We have sin x > x 3 √cos x for all x ∈  0, π 2  . Proof. Successively replace x by x 2 in (2), in order to obtain: √ sin x tan x > 2 tan x 2 , r sin x 2 tan x 2 > 2 tan x 4 , r sin x 4 tan x 4 > 2 tan x 8 , and so on until we get r sin x 2n−1 tan x 2n−1 > 2 tan x 2n for some positive integer n (which can be any positive integer n). Mathematical Reflections 4 (2025) 2 Multiplying all these inequalities and cancelling common factors, we get: √ sin x tan x · r sin x 2 sin x 4 · · · sin x 2n−1 > 2n tan x 2n r tan x 2 tan x 4 · · · tan x 2n−1 , or √ sin x tan x > 1 r cos x 2 cos x 22 · · · cos x 2n−1 · tan x 2n x 2n · x. Repeatedly applying the formula for the sine of the double of an angle we get cos x 2 cos x 22 · · · cos x 2n−1 = sin x 2n−1 sin x 2n−1 = sin x x · x 2n−1 sin x 2n−1 . Now, as n can be any positive integer, we may pass to the limit (for n →∞) in the inequality that we previously obtained, and because we have lim n→∞ x 2n−1 sin x 2n−1 = lim n→∞ tan x 2n x 2n = 1, we thus get √ sin x · tan x ≥ r x sin x · x which simplifies to inequality (B): sin3/2 x ≥x3/2√cos x ⇔sin x ≥x 3 √cos x. As we see, this approach does not lead directly to the strict inequality. However, if we write it in the form sin x x ≥ 3 √cos x and observe that the functions from both sides are strictly decreasing, and that lim x→0 sin x x = lim x→0 3 √cos x = 1, while lim x→π 2 sin x x = 2 π > 0 = lim x→π 2 3 √cos x the conclusion that the inequality is strict for 0 < x < π 2 follows. (In particular, note that the monotony of x 7→sin x x leads to the inequality sin x > 2 π x, 0 < x < π 2 . An alternate proof can be given (see ) by using H¨ older’s inequality in the form Z b a f 3(t)dt Z b a g3(t)dt Z b a h3(t)dt ≥ Z b a (fgh)(t)dt !3 for the functions f(t) = g(t) = cos 1 3 t and h(t) = 1 cos 2 3 t on the interval [0, x]. We get Z x 0 cos tdt Z x 0 cos tdt Z x 0 1 cos2 tdt ≥ Z x 0  cos t cos t 1 cos2 t  1 3 dt !3 , that is, sin2 x tan x ≥x3 ⇔sin x ≥x 3 √cos x, Mathematical Reflections 4 (2025) 3 which is actually strict (the conditions for equality not being fulfilled). Finally note that in the form sin x x 2 tan x x > 1 the inequality (B) holds for all x with 0 < |x| < π 2 . It is time to see now the inequality from the title, which appeared in 1654 in Christiaan Huygens’s work De circuli magnitudine inventa, but also in 1621, in the book Cyclometricus of the astronomer Willebrord Snellius (for which reason this is also called Snell’s, or Snellius’s inequality). They both used the inequality to find better approximations for the number π. Proposition 3. The inequality 2sin x x + tan x x > 1 (C) holds for 0 < |x| < π 2 . Proof. It is enough to prove that 2 sin x + tan x > 3x for all 0 < x < π 2 . But this one follows by applying the inequality between the arithmetic mean and the geometric mean of the three positive numbers sin x, sin x, and tan x, and then the inequality (B): sin x + sin x + tan x 3 > 3 p sin2 x tan x > x. We invite the interested reader to find a proof using calculus (for example, by studying the function x 7→ 2 sin x + tan x −3x), and we go to another immediate consequence of (B), namely Wilker’s inequality (that was proposed in 1989 in The American Mathematical Monthly by J. B. Wilker). Proposition 4. The inequality sin x x 2 + tan x x > 2 holds for 0 < |x| < π 2 . Proof. Once again, it is enough to prove the inequality for 0 < x < π 2 . We have (by the arithmetic-geometric means inequality, and then by inequality (B)): sin x x 2 + tan x x > 2 ssin x x 2 tan x x > 2. Note that the inequality between the arithmetic mean and the harmonic mean of sin x, sin x, tan x leads to 2 sin x + tan x 3 > 3 sin x 2 + cos x, a lower bound that cannot be used for proving Huygens’s inequality, since we have the following result: Proposition 5. The inequality sin x x < 2 + cos x 3 holds for 0 < |x| < π 2 . Proof. This is known as Cusa’s inequality and it is equivalent to x > 3 sin x 2 + cos x, 0 < x < π 2 . With the substitution x = 2θ, we can transform it into θ ≥ 3 tan θ 3 + tan2 θ, 0 ≤θ < π 4 . Mathematical Reflections 4 (2025) 4 Since the function f(θ) = θ − 3 tan θ 3 + tan2 θ has the derivative f ′(θ) = 4 tan4 θ (3 + tan2 θ)2 positive, we get that f is strictly increasing, hence f(θ) ≥f(0) = 0 (with equality only for θ = 0) and thus Cusa’s inequality is proved. One other way (and we invite the reader to follow it; probably there are many more other possibilities) is to prove x(2 + cos x) −3 sin x ≥0, 0 ≤x < π 2 . 3. Final remarks 1) The inequalities 3 √cos x < sin x x < 2 + cos x 3 (the first is inequality (B), the second is Cusa’s inequality) give better bounds for the ratio sin x x than those from (1′), namely cos x < 3 √cos x < sin x x < 2 + cos x 3 < 1 for 0 < x < π 2 . We can also show that the inequality 3x −tan x 2x < sin x x for 0 < x < π 2 , which is equivalent to Huygens’s inequality (C), is better than the left inequality from (1′), in that cos x < 3x −tan x 2x ⇔2 cos x < 3 −sin x x · 1 cos x, for x ∈  0, π 2  such that cos x > 1 3 (which means for x in a neighborhood of the origin). Indeed, by using Cusa’s inequality, we get 3 −sin x x · 1 cos x > 3 −2 + cos x 3 · 1 cos x = 6 cos2 x + 2(3 cos x −1)(1 −cos x) 3 cos x > 6 cos2 x 3 cos x = 2 cos x. On the other hand, by taking limits for x →π 2 in both sides of the inequality clearly shows that it cannot be true for x close enough to π 2 . 2) For any t > 0 we can apply the inequality between the arithmetic mean and geometric mean of three positive numbers, and then inequality (B) (exactly as we did for proving Huygens’s inequality) in order to obtain 2t sin x + 1 t2 tan x > 3x, 0 < x < π 2 . (C′) Nevertheless, we can calculate (for 0 < x < π 2 ) (2t sin x + 1 t2 tan x) −(2 sin x + tan x) = 2(t −1) tan x  cos x −t + 1 2t2  and notice that, in general, this difference is positive. More precisely, for 0 < t < 1, we have that t −1 < 0 and cos x < 1 < t + 1 2t2 , thus the difference is positive and 2t sin x + 1 t2 tan x > 2 sin x + tan x, 0 < x < π 2 , Mathematical Reflections 4 (2025) 5 meaning that Huygens’s inequality remains sharper than (C′). Whenever t > 1, we have t −1 > 0 and t + 1 2t2 < 1, so the second parenthesis can be negative. However, the second parenthesis is positive, and the difference is also positive for cos x > t + 1 2t2 – that is, in a neighborhood of the origin. Thus, again, (C′) does not quite represent an improvement of (C). 3) It would be interesting to see (and we invite the reader to think about this problem) for which positive real numbers a and b the inequality a sin x + b tan x > x holds for any x ∈  0, π 2  . Note that a+b ≥1 is a necessary condition for the inequality to be true in a neighborhood of the origin (divide by x and take limits for x →0), and that this becomes Huygens’s inequality when a = 2 3 and b = 1 3. It would be also nice to know what numbers a0 and b0 yield the best such inequality: a sin x + b tan x ≥a0 sin x + b0 tan x > x for all x ∈  0, π 2  , and for all a, b ∈(0, ∞). Acknowledgements. I thank my son, Hiranava Das for his encouragement and technical help, and to my daughter-in-law Sreejana for typing the manuscript. References 1. Roger B. Nelsen: Elementary Proofs of the Trigonometric Inequalities of Huygens, Cusa, and Wilker, Math-ematics Magazine, vol. 93(2020), issue 4, October 2020, 276-283 2. 3. Problems 115, 167, Crux Mathematicorum (from vol. 1, no. 1 to vol. 4) Mathematical Reflections 4 (2025) 6
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https://en.wikipedia.org/wiki/Atmospheric_convection
Atmospheric convection - Wikipedia Jump to content [x] Main menu Main menu move to sidebar hide Navigation Main page Contents Current events Random article About Wikipedia Contact us Contribute Help Learn to edit Community portal Recent changes Upload file Special pages Search Search [x] Appearance Donate Create account Log in [x] Personal tools Donate Create account Log in Pages for logged out editors learn more Contributions Talk Contents move to sidebar hide (Top) 1 Overview 2 InitiationToggle Initiation subsection 2.1 Thunderstorms 2.2 Types 2.3 Boundaries and forcing 3 Concerns regarding severe deep moist convectionToggle Concerns regarding severe deep moist convection subsection 3.1 Hail 3.2 Downburst 3.3 Tornado 4 Measurement 5 Other forecasting concerns 6 See also 7 References [x] Toggle the table of contents Atmospheric convection [x] 20 languages العربية Català Čeština Ελληνικά Español فارسی Français हिन्दी Bahasa Indonesia Italiano ქართული Lombard Bahasa Melayu Nederlands Polski Português Simple English Suomi Türkçe 中文 Edit links Article Talk [x] English Read Edit View history [x] Tools Tools move to sidebar hide Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Download as PDF Printable version In other projects Wikimedia Commons Wikidata item Appearance move to sidebar hide From Wikipedia, the free encyclopedia Atmospheric phenomenon This scientific article needs additional citations to secondary or tertiary sources. Help add sources such as review articles, monographs, or textbooks. Please also establish the relevance for any primary research articles cited. Unsourced or poorly sourced material may be challenged and removed.(August 2023) (Learn how and when to remove this message) Part of a series on Weather Temperate and polar seasons Winter Spring Summer Autumn Tropical seasons Dry season Harmattan Wet season Storms Cloud Cumulonimbus cloud Arcus cloud Downburst Microburst Heat burst Derecho Lightning Volcanic lightning Thunderstorm Air-mass thunderstorm Thundersnow Dry thunderstorm Mesocyclone Supercell Tornado Anticyclonic tornado Landspout Waterspout Dust devil Fire whirl Anticyclone Cyclone Polar low Extratropical cyclone European windstorm Nor'easter Subtropical cyclone Tropical cyclone Atlantic hurricane Typhoon Storm surge Dust storm Simoom Haboob Monsoon Amihan Gale Sirocco Firestorm Winter storm Ice storm Blizzard Ground blizzard Snow squall Precipitation Drizzle Freezing Graupel Hail Megacryometeor Ice pellets Diamond dust Rain Freezing Cloudburst Snow Rain and snow mixed Snow grains Snow roller Slush Topics Air pollution Atmosphere Chemistry Convection Physics River Climate Cloud Physics Fog Mist Season Cold wave Heat wave Jet stream Meteorology Severe weather List Extreme Severe weather terminology Canada Japan United States Weather forecasting Weather modification Glossaries Meteorology Climate change Tornado terms Tropical cyclone terms Weather portal v t e Atmospheric convection is the vertical transport of heat and moisture in the atmosphere. It occurs when warmer, less dense air rises, while cooler, denser air sinks. This process is driven by parcel-environment instability, meaning that a "parcel" of air is warmer and less dense than the surrounding environment at the same altitude. This difference in temperature and density (and sometimes humidity) causes the parcel to rise, a process known as buoyancy. This rising air, along with the compensating sinking air, leads to mixing, which in turn expands the height of the planetary boundary layer (PBL), the lowest part of the atmosphere directly influenced by the Earth's surface. This expansion contributes to increased winds, cumulus cloud development, and decreased surface dew points (the temperature below which condensation occurs). Convection plays a crucial role in weather patterns, influencing cloud formation, wind, and the development of thunderstorms, which can be associated with severe weather phenomena like hail, downbursts, and tornadoes. Conditions favorable for thunderstorm types and complexes. Technical terms and abbreviations appearing (e.g., in axis labels) are shear, AGL,[clarification needed]CAPE,[clarification needed] and BR (bulk Richardson [number]).[jargon][citation needed] Overview [edit] This section has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources in this section. Unsourced material may be challenged and removed.(August 2023) (Learn how and when to remove this message) This section may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details.(August 2023) (Learn how and when to remove this message) (Learn how and when to remove this message) There are a few general archetypes of atmospheric instability that are used to explain convection (or lack thereof);[according to whom?] a necessary but insufficient condition for convection is that the environmental lapse rate (the rate of decrease of temperature with height) is steeper than the lapse rate experienced by a rising parcel of air.[clarification needed][citation needed] When this condition is met, upward-displaced air parcels can become buoyant and thus experience a further upward force. Buoyant convection begins at the level of free convection (LFC), above which an air parcel may ascend through the free convective layer (FCL) with positive buoyancy. Its buoyancy turns negative at the equilibrium level (EL), but the parcel's vertical momentum may carry it to the maximum parcel level (MPL) where the negative buoyancy decelerates the parcel to a stop. Integrating the buoyancy force over the parcel's vertical displacement yields convective available potential energy (CAPE), the joules of energy available per kilogram of potentially buoyant air. CAPE is an upper limit for an ideal undiluted parcel, and the square root of twice the CAPE is sometimes called a thermodynamic speed limit for updrafts, based on the simple kinetic energy equation. However, such buoyant acceleration concepts give an oversimplified view of convection. Drag is an opposite force to counter buoyancy, so that parcel ascent occurs under a balance of forces, like the terminal velocity of a falling object. Buoyancy may be reduced by entrainment, which dilutes the parcel with environmental air. Atmospheric convection is called "deep" when it extends from near the surface to above the 500 hPa level, generally stopping at the tropopause at around 200 hPa.[citation needed] Most atmospheric deep convection occurs in the tropics as the rising branch of the Hadley circulation and represents a strong local coupling between the surface and the upper troposphere which is largely absent in winter midlatitudes. Its counterpart in the ocean (deep convection downward in the water column) only occurs at a few locations. Initiation [edit] A thermal column (or thermal) is a vertical section of rising air in the lower altitudes of the Earth's atmosphere. Thermals are created by the uneven heating of the Earth's surface from solar radiation. The Sun warms the ground, which in turn warms the air directly above it. The warmer air expands, becoming less dense than the surrounding air mass, and creating a thermal low. The mass of lighter air rises, and as it does, it cools due to its expansion at lower high-altitude pressures. It stops rising when it has cooled to the same temperature as the surrounding air. Associated with a thermal is a downward flow surrounding the thermal column. The downward-moving exterior is caused by colder air being displaced at the top of the thermal. Another convection-driven weather effect is the sea breeze. Thunderstorms [edit] Main article: Thunderstorm Stages of a thunderstorm's life. Warm air has a lower density than cool air, so warm air rises within cooler air,[better source needed] similar to hot air balloons.[citation needed] Clouds form as relatively warmer air carrying moisture rises within cooler air. As the moist air rises, it cools causing some of the water vapor in the rising packet of air to condense. When the moisture condenses, it releases energy known as latent heat of vaporization which allows the rising packet of air to cool less than its surrounding air,[better source needed] continuing the cloud's ascension. If enough instability is present in the atmosphere, this process will continue long enough for cumulonimbus clouds to form, which supports lightning and thunder. Generally, thunderstorms require three conditions to form: moisture, an unstable airmass, and a lifting force (heat). All thunderstorms, regardless of type, go through three stages: the developing stage, the mature stage, and the dissipation stage.[better source needed] The average thunderstorm has a 24 km (15 mi) diameter. Depending on the conditions present in the atmosphere, these three stages take an average of 30 minutes to go through. Types [edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.(August 2023) (Learn how and when to remove this message) There are four main types of thunderstorms: single-cell, multicell, squall line (also called multicell line), and supercell. Which type forms depends on the instability and relative wind conditions at different layers of the atmosphere ("wind shear"). Single-cell thunderstorms form in environments of low vertical wind shear and last only 20–30 minutes. Organized thunderstorms and thunderstorm clusters/lines can have longer life cycles as they form in environments of significant vertical wind shear, which aids the development of stronger updrafts as well as various forms of severe weather. The supercell is the strongest of the thunderstorms, most commonly associated with large hail, high winds, and tornado formation. The latent heat release from condensation is the determinant between significant convection and almost no convection at all. The fact that air is generally cooler during winter months, and therefore cannot hold as much water vapor and associated latent heat, is why significant convection (thunderstorms) are infrequent in cooler areas during that period. Thundersnow is one situation where forcing mechanisms provide support for very steep environmental lapse rates, which as mentioned before is an archetype for favored convection. The small amount of latent heat released from air rising and condensing moisture in a thundersnow also serves to increase this convective potential, although minimally. There are also three types of thunderstorms: orographic, air mass, and frontal. Boundaries and forcing [edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.(August 2023) (Learn how and when to remove this message) Despite the fact that there might be a layer in the atmosphere that has positive values of CAPE, if the parcel does not reach or begin rising to that level, the most significant convection that occurs in the FCL will not be realized. This can occur for numerous reasons. Primarily, it is the result of a cap, or convective inhibition (CIN/CINH). Processes that can erode this inhibition are heating of the Earth's surface and forcing. Such forcing mechanisms encourage upward vertical velocity, characterized by a speed that is relatively low to what one finds in a thunderstorm updraft. Because of this, it is not the actual air being pushed to its LFC that "breaks through" the inhibition, but rather the forcing cools the inhibition adiabatically. This would counter, or "erode" the increase of temperature with height that is present during a capping inversion. Forcing mechanisms that can lead to the eroding of inhibition are ones that create some sort of evacuation of mass in the upper parts of the atmosphere, or a surplus of mass in the low levels of the atmosphere, which would lead to upper-level divergence or lower-level convergence, respectively. An Upward vertical motion will often follow. Specifically, a cold front, sea/lake breeze, outflow boundary, or forcing through vorticity dynamics (differential positive vorticity advection) of the atmosphere such as with troughs, both shortwave and longwave. Jet streak dynamics through the imbalance of Coriolis and pressure gradient forces, causing subgeostrophic and supergeostrophic flows, can also create upward vertical velocities. There are numerous other atmospheric setups in which upward vertical velocities can be created. Concerns regarding severe deep moist convection [edit] Buoyancy is a key to thunderstorm growth and is necessary for any of the severe threats within a thunderstorm. There are other processes, not necessarily thermodynamic, that can increase updraft strength. These include updraft rotation, low-level convergence, and evacuation of mass out of the top of the updraft via strong upper-level winds and the jet stream. Hail [edit] Main article: Hail Hail shaft Severe thunderstorms containing hail can exhibit a characteristic green coloration. Like other precipitation in cumulonimbus clouds hail begins as water droplets. As the droplets rise and the temperature goes below freezing, they become supercooledwater and will freeze on contact with condensation nuclei. A cross-section through a large hailstone shows an onion-like structure. This means the hailstone is made of thick and translucent layers, alternating with layers that are thin, white, and opaque. Former theory suggested that hailstones were subjected to multiple descents and ascents, falling into a zone of humidity and refreezing as they were uplifted. This up-and-down motion was thought to be responsible for the successive layers of the hailstone. New research (based on theory and field study) has shown this is not necessarily true. The storm's updraft, with upwardly directed wind speeds as high as 180 kilometres per hour (110 mph), blow the forming hailstones up the cloud. As the hailstone ascends it passes into areas of the cloud where the concentration of humidity and supercooled water droplets varies. The hailstone's growth rate changes depending on the variation in humidity and supercooled water droplets that it encounters. The accretion rate of these water droplets is another factor in the hailstone's growth. When the hailstone moves into an area with a high concentration of water droplets, it captures the latter and acquires a translucent layer. Should the hailstone move into an area where mostly water vapour is available, it acquires a layer of opaque white ice. Furthermore, the hailstone's speed depends on its position in the cloud's updraft and its mass. This determines the varying thicknesses of the layers of the hailstone. The accretion rate of supercooled water droplets onto the hailstone depends on the relative velocities between these water droplets and the hailstone itself. This means that generally, the larger hailstones will form some distance from the stronger updraft where they can pass more time growing As the hailstone grows it releases latent heat, which keeps its exterior in a liquid phase. Undergoing "wet growth", the outer layer is sticky, or more adhesive, so a single hailstone may grow by collision with other smaller hailstones, forming a larger entity with an irregular shape. The hailstone will keep rising in the thunderstorm until its mass can no longer be supported by the updraft. This may take at least 30 minutes based on the force of the updrafts in the hail-producing thunderstorm, whose top is usually greater than 10 kilometres (6.2 mi) high. It then falls toward the ground while continuing to grow, based on the same processes, until it leaves the cloud. It will later begin to melt as it passes into the air above freezing temperature Thus, a unique trajectory in the thunderstorm is sufficient to explain the layer-like structure of the hailstone. The only case in which we can discuss multiple trajectories is in a multicellular thunderstorm where the hailstone may be ejected from the top of the "mother" cell and captured in the updraft of a more intense "daughter cell". This however is an exceptional case. Downburst [edit] Cumulonimbus cloud over the Gulf of Mexico in Galveston, Texas A downburst A downburst is created by a column of sinking air that, after hitting ground level, spreads out in all directions and is capable of producing damaging straight-line winds of over 240 kilometres per hour (150 mph), often producing damage similar to, but distinguishable from, that caused by tornadoes. This is because the physical properties of a downburst are completely different from those of a tornado. Downburst damage will radiate from a central point as the descending column spreads out when impacting the surface, whereas tornado damage tends towards convergent damage consistent with rotating winds. To differentiate between tornado damage and damage from a downburst, the term straight-line winds is applied to damage from microbursts. Downbursts are particularly strong downdrafts from thunderstorms. Downbursts in air that is precipitation free or contains virga are known as dry downbursts; those accompanied with precipitation are known as wet downbursts. Most downbursts are less than 4 kilometres (2.5 mi) in extent: these are called microbursts. Downbursts larger than 4 kilometres (2.5 mi) in extent are sometimes called macrobursts. Downbursts can occur over large areas. In the extreme case, a derecho can cover a huge area more than 320 kilometres (200 mi) wide and over 1,600 kilometres (990 mi) long, lasting up to 12 hours or more, and is associated with some of the most intense straight-line winds, but the generative process is somewhat different from that of most downbursts.[citation needed] Tornado [edit] Main article: Tornado The F5 tornado that struck Elie, Manitoba in 2007. A tornado is a dangerous rotating column of air in contact with both the surface of the Earth and the base of a cumulonimbus cloud (thundercloud), or a cumulus cloud in rare cases. Tornadoes come in many sizes but typically form a visible condensation funnel whose narrowest end reaches the earth and is surrounded by a cloud of debris and dust.[non-primary source needed] Tornadoes wind speeds generally average between 64 kilometres per hour (40 mph) and 180 kilometres per hour (110 mph). They are approximately 75 metres (246 ft) across and travel a few kilometers before dissipating. Some attain wind speeds in excess of 480 kilometres per hour (300 mph), may stretch more than a 1.6 kilometres (0.99 mi) across, and maintain contact with the ground for more than 100 kilometres (62 mi). Tornadoes, despite being one of the most destructive weather phenomena, are generally short-lived. A long-lived tornado generally lasts no more than an hour, but some have been known to last for 2 hours or longer (for example, the 1925 Tri-State tornado). Due to their relatively short duration, less information is known about the development and formation of tornadoes. Generally any cyclone based on its size and intensity has different instability dynamics. The most unstable azimuthal wavenumber is higher for bigger cyclones .[non-primary source needed] Measurement [edit] The potential for convection in the atmosphere is often measured by an atmospheric temperature/dewpoint profile with height. This is often displayed on a Skew-T chart or other similar thermodynamic diagram. These can be plotted by a measured sounding analysis, which is the sending of a radiosonde attached to a balloon into the atmosphere to take the measurements with height. Forecast models can also create these diagrams, but are less accurate due to model uncertainties and biases, and have lower spatial resolution. Although, the temporal resolution of forecast model soundings is greater than the direct measurements, where the former can have plots for intervals of up to every 3 hours, and the latter as having only 2 per day (although when a convective event is expected a special sounding might be taken outside of the normal schedule of 00Z and then 12Z.). Other forecasting concerns [edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed.(August 2023) (Learn how and when to remove this message) Atmospheric convection can also be responsible for and have implications on a number of other weather conditions. A few examples on the smaller scale would include: Convection mixing the planetary boundary layer (PBL) and allowing drier air aloft to the surface thereby decreasing dew points, creating cumulus-type clouds that can limit a small amount of sunshine, increasing surface winds, making outflow boundaries/and other smaller boundaries more diffuse, and the eastward propagation of the dryline during the day. On a larger scale, the rising of the air can lead to warm core surface lows, often found in the desert southwest. See also [edit] Weather portal Air parcel Atmospheric subsidence Atmospheric thermodynamics Buoyancy Convective storm detection Thermal References [edit] ^Overview chapter flame.org Archived October 6, 2008, at the Wayback Machine ^Jones, Helen. "Open-ocean deep convection". ^While less dynamically important than in the atmosphere, such oceanic convection is responsible for the worldwide existence of cold water in the lowest layers of the ocean.[citation needed] ^NOAA-NWS Staff (2008). "What is a Monsoon?". Tucson, AZ: NOAA-National Weather Service (NWS). Archived from the original on August 3, 2008. Retrieved August 18, 2023. ^Hahn, Douglas G.; Manabe, Syukuro (1975). "The Role of Mountains in the South Asian Monsoon Circulation". Journal of the Atmospheric Sciences. 32 (8): 1515–1541. Bibcode:1975JAtS...32.1515H. doi:10.1175/1520-0469(1975)032<1515:TROMIT>2.0.CO;2. ISSN1520-0469. ^Ackerman, Steve. "Sea and Land Breezes". CIMS.SSEC.Wisc.edu. Madison, WI: University of Wisconsin. Retrieved October 24, 2006. ^"JetStream: An Online School For Weather—The Sea Breeze". SRH.Weather.gov. National Weather Service. 2008. Archived from the original on September 23, 2006. Retrieved October 24, 2006. ^Frye, Albert Irvin (1913). Civil Engineers' Pocket Book: A Reference-Book for Engineers, Contractors. D. Van Nostrand Company. p.462. Retrieved August 31, 2009. ^ZAMG Staff (2007). "Fog And Stratus—Meteorological Physical Background". ZAMG.ac.at. Zentralanstalt für Meteorologie und Geodynamik (ZAMG). Archived from the original on July 6, 2011. Retrieved February 7, 2009. ^Mooney, Chris C. (2007). Storm World: Hurricanes, Politics, and the Battle over Global Warming. Orlando, FL: Harcourt. p.20. ISBN978-0-15-101287-9. Retrieved August 31, 2009. ^Mogil, Michael H. (2007). Extreme Weather: Understanding the Science of Hurricanes, Tornadoes, Floods, Heat Waves, Snow Storms, Global Warming, and Other Atmospheric Disturbances. New York, NY: Black Dog & Leventhal Publisher. pp.210–211. ISBN978-1-57912-743-5. Retrieved August 18, 2023. ^NOAA-NSSL Staff (October 15, 2006). "A Severe Weather Primer: Questions and Answers about Thunderstorms". NOAA, National Severe Storms Laboratory (NSSL). Archived from the original on August 25, 2009. Retrieved September 1, 2009. ^Gallagher III, Frank W. (October 2000). "Distant Green Thunderstorms—Frazer's Theory Revisited". Journal of Applied Meteorology. 39 (10): 1754. Bibcode:2000JApMe..39.1754G. doi:10.1175/1520-0450-39.10.1754. ^National Center for Atmospheric Research (2008). "Hail". University Corporation for Atmospheric Research. Archived from the original on May 27, 2010. Retrieved July 18, 2009. ^ abcNelson, Stephan P. (August 1983). "The Influence of Storm Flow Struce on Hail Growth". Journal of the Atmospheric Sciences. 40 (8): 1965–1983. Bibcode:1983JAtS...40.1965N. doi:10.1175/1520-0469(1983)040<1965:TIOSFS>2.0.CO;2. ISSN1520-0469. ^Brimelow, Julian C.; Reuter, Gerhard W.; Poolman, Eugene R. (October 2002). "Modeling Maximum Hail Size in Alberta Thunderstorms". Weather and Forecasting. 17 (5): 1048–1062. Bibcode:2002WtFor..17.1048B. doi:10.1175/1520-0434(2002)017<1048:MMHSIA>2.0.CO;2. ISSN1520-0434. ^Marshall, Jacque (April 10, 2000). "Hail Fact Sheet". University Corporation for Atmospheric Research. Archived from the original on October 15, 2009. Retrieved July 15, 2009. ^Caracena, Fernando; Holle, Ronald L.; Doswell, Charles A. III. "Microbursts: A Handbook for Visual Identification". University of Oklahoma. Archived from the original on May 14, 2008. Retrieved July 9, 2008. ^ abGlickman, Todd S. [exec. ed.]. "Glossary of Meteorology". Retrieved July 30, 2008. ^Parke, Peter S.; Larson, Norvan J. "Boundary Waters Windstorm". CRH.NOAA.gov. NOAA. Retrieved July 30, 2008. ^Renno, Nilton O. (August 2008). "A Thermodynamically General Theory for Convective Vortices"(PDF). Tellus A. 60 (4): 688–99. Bibcode:2008TellA..60..688R. doi:10.1111/j.1600-0870.2008.00331.x. hdl:2027.42/73164. ^Edwards, Roger (April 4, 2006). "The Online Tornado FAQ". Storm Prediction Center. Archived from the original on September 30, 2006. Retrieved September 8, 2006. ^CSWR Staff (2006). "Doppler On Wheels". CSWR.org. Center for Severe Weather Research (CSWR). Archived from the original on February 5, 2007. Retrieved December 29, 2006. ^CRH-NOAA Staff (October 2, 2005). "Hallam Nebraska Tornado". CRH.NOAA.gov. Omaha, NE: CRH-NOAA Weather Forecast Office. Archived from the original on October 4, 2006. Retrieved September 8, 2006. ^UCAR Staff (August 1, 2008). "Tornadoes". UCAR.edu. Archived from the original on October 12, 2009. Retrieved August 3, 2009. ^Rostami, Masoud; Zeitlin, Vladimir (2018). "An Improved Moist-Convective Rotating Shallow-Water Model and its Application to Instabilities of Hurricane-Like Vortices"(PDF). Quarterly Journal of the Royal Meteorological Society. 144 (714): 1450–1462. Bibcode:2018QJRMS.144.1450R. doi:10.1002/qj.3292. S2CID59493137. Retrieved August 18, 2023. ^"The Forecast Model Sounding Machine". 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https://ajc.maths.uq.edu.au/pdf/66/ajc_v66_p265.pdf
AUSTRALASIAN JOURNAL OF COMBINATORICS Volume 66(2) (2016), Pages 265–275 Intersecting integer partitions Peter Borg Department of Mathematics University of Malta Malta peter.borg@um.edu.mt Abstract If a1, a2, . . . , ak and n are positive integers such that n = a1+a2+· · ·+ak, then the sum a1 + a2 + · · · + ak is said to be a partition of n of length k, and a1, a2, . . . , ak are said to be the parts of the partition. Two partitions that differ only in the order of their parts are considered to be the same partition. Let Pn be the set of partitions of n, and let Pn,k be the set of partitions of n of length k. We say that two partitions t-intersect if they have at least t common parts (not necessarily distinct). We call a set A of partitions t-intersecting if every two partitions in A t-intersect. For a set A of partitions, let A(t) be the set of partitions in A that have at least t parts equal to 1. We conjecture that for n ≥t, Pn(t) is a largest t-intersecting subset of Pn. We show that for k > t, Pn,k(t) is a largest t-intersecting subset of Pn,k if n ≤2k −t + 1 or n ≥3tk5. We also demonstrate that for every t ≥1, there exist n and k such that t < k < n and Pn,k(t) is not a largest t-intersecting subset of Pn,k. 1 Introduction Unless stated otherwise, we shall use small letters such as x to denote positive inte-gers or functions or elements of a set, capital letters such as X to denote sets, and calligraphic letters such as F to denote families (that is, sets whose elements are sets themselves). The set {1, 2, . . . } of all positive integers is denoted by N. For n ≥1, [n] denotes the set {1, . . . , n} of the first n positive integers. We take to be the empty set ∅. We call a set A an r-element set if its size |A| is r (that is, if it contains exactly r elements). For a set X, X r denotes the family of r-element subsets of X. It is to be assumed that arbitrary sets and families are finite. In the literature, a sum a1 + a2 + · · · + ak is said to be a partition of n of length k if a1, a2, . . . , ak and n are positive integers such that n = a1 + a2 + · · · + ak. A partition of a positive integer n is also referred to as an integer partition or simply as a partition. If a1 + a2 + · · · + ak is a partition, then a1, a2, . . . , ak are said to be its parts. Two partitions that differ only in the order of their parts are considered to be the same partition. Thus, we can refine the definition of a partition as follows. We PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 266 call a tuple (a1, . . . , ak) a partition of n of length k if a1, . . . , ak and n are positive integers such that n = k i=1 ai and a1 ≤· · · ≤ak. We will be using the latter definition throughout the rest of the paper. For any tuple a = (a1, . . . , ak) and any i ∈[k], ai is said to be the i-th entry of a, and if a is a partition, then ai is also said to be a part of a. Let Pn denote the set of partitions of n, and let Pn,k denote the set of partitions of n of length k. Thus, Pn,k is non-empty if and only if 1 ≤k ≤n. Moreover, Pn = n i=1 Pn,i. Let pn = |Pn| and pn,k = |Pn,k|. These values are widely studied. To the best of the author’s knowledge, no elementary closed-form expressions are known for pn and pn,k. For more about these values, we refer the reader to . If at least one part of a partition a is a part of a partition b, then we say that a and b intersect. We call a set A of partitions intersecting if for every a and b in A, a and b intersect. We make the following conjecture. Conjecture 1.1 For every positive integer n, the set of partitions of n that have 1 as a part is a largest intersecting set of partitions of n. We also conjecture that for 2 ≤k ≤n and (n, k) ̸= (8, 3), {a ∈Pn,k : 1 is a part of a} is a largest intersecting subset of Pn,k. We will show that this is true for n ≤2k and for n sufficiently large depending on k. For any set A of partitions, let A(t) denote the set of partitions in A whose first t entries are equal to 1. Thus, for t ≤k ≤n, Pn,k(t) = {(a1, . . . , ak) ∈Pn,k : a1 = · · · = at = 1} and Pn(t) = n  i=t Pn,i(t). Note that for t < k ≤n, |Pn(t)| = pn−t and |Pn,k(t)| = pn−t,k−t. Generalising the definition of intersecting partitions, we say that two tuples (a1, . . . , ar) and (b1, . . . , bs) t-intersect if there are t distinct integers i1, . . . , it in [r] and t distinct integers j1, . . . , jt in [s] such that aip = bjp for each p ∈[t]. We call a set A of tuples t-intersecting if for every a, b ∈A, a and b t-intersect. Thus, for any A ⊆Pn, A(t) is t-intersecting, and A is intersecting if and only if A is 1-intersecting. We pose the following two problems, which lie in the interface between extremal set theory and partition theory. Problem 1.2 What is the size or the structure of a largest t-intersecting subset of Pn? Problem 1.3 What is the size or the structure of a largest t-intersecting subset of Pn,k? This paper mainly addresses the second question. We suggest two conjectures cor-responding to the two problems above and generalising the two conjectures above. Conjecture 1.4 For n ≥t, Pn(t) is a t-intersecting subset of Pn of maximum size. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 267 Conjecture 1.1 is Conjecture 1.4 with t = 1. For 2 ≤k ≤n, the only case we discovered where Pn,k(1) is not a largest inter-secting subset of Pn,k is n = 8 and k = 3. Remark 1.5 We have P8,3 = {(1, 1, 6), (1, 2, 5), (1, 3, 4), (2, 2, 4), (2, 3, 3)}. Since P8,3 is not an intersecting set, {(1, 2, 5), (1, 3, 4), (2, 2, 4), (2, 3, 3)} is an intersecting subset of P8,3 of maximum size 4 = |P8,3(1)| + 1. Extending this example, we have that for t ≥2, {(1, . . . , 1, a, b, c) ∈Pt+7,t+2 : (a, b, c) ∈{(1, 2, 5), (1, 3, 4), (2, 2, 4), (2, 3, 3)}} is a t-intersecting subset of Pt+7,t+2 of size |Pt+7,t+2(t)| + 1. Conjecture 1.6 For t + 1 ≤k ≤n with (n, k) ̸= (t + 7, t + 2), Pn,k(t) is a t-intersecting subset of Pn,k of maximum size. If t = k < n, then Pn,k(t) = ∅, Pn,k ̸= ∅, and the non-empty t-intersecting subsets of Pn,k are the 1-element subsets. If k < t, then Pn,k has no non-empty t-intersecting subsets. For every k and t, we leave Conjecture 1.6 open only for a finite range of values of n, namely, for 2k −t + 1 < n < 3tk5. We first prove it for n ≤2k −t + 1. Proposition 1.7 Conjecture 1.6 is true for n ≤2k −t + 1. Proof. Suppose n ≤2k −t + 1. For any c = (c1, . . . , ck) ∈Pn,k, let Lc = {i ∈ [k]: ci = 1} and lc = |Lc|. We have 2k −t + 1 ≥n =  i∈Lc ci +  j∈[k]\Lc cj ≥  i∈Lc 1 +  j∈[k]\Lc 2 = lc + 2(k −lc) = 2k −lc. Thus, lc ≥t −1, and equality holds only if n = 2k −t + 1 and cj = 2 for each j ∈[k]\Lc. Since c1 ≤· · · ≤ck, Lc = [lc]. Let A be a t-intersecting subset of Pn,k. If la ≥t for each a ∈A, then A ⊆Pn,k(t). Suppose la = t−1 for some a = (a1, . . . , ak) ∈A. By the above, we have n = 2k−t+1, ai = 1 for each i ∈[t −1], aj = 2 for each j ∈[k][t −1], and Pn,k = Pn,k(t) ∪{a}. Let b be the partition (b1, . . . , bk) in Pn,k(t) with bk = n −k + 1 = k −t + 2 and bi = 1 for each i ∈[k −1]. Since k ≥t + 1, a and b do not t-intersect, and hence b / ∈A. Thus, |A| ≤|Pn,k| −1 = |Pn,k(t)|. 2 In Section 3, we show that Conjecture 1.6 is also true for n sufficiently large. More precisely, we prove the following. Theorem 1.8 For k ≥t + 2 and n ≥3tk5, Pn,k(t) is a t-intersecting subset of Pn,k of maximum size, and uniquely so if k ≥t + 3. We actually prove the result for n ≥ 8 7(t + 1)k5. For this purpose, we generalise Bollobás’ proof [3, pages 48–49] of the Erdős–Ko–Rado (EKR) Theorem , and we make some observations regarding the values pn,k and the structure of t-intersecting subsets of Pn,k. Remark 1.9 Conjecture 1.6 is also true for k = t + 1. Indeed, if two partitions of n of length t + 1 have t common parts a1, . . . , at, then the remaining part of each is n −(a1 + · · · + at), and hence the partitions are the same. Thus, the non-empty PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 268 t-intersecting subsets of Pn,t+1 are the 1-element subsets. Hence Pn,t+1(t) is a largest t-intersecting subset of Pn,t+1, but not uniquely so for n ≥t+3 ({(1, . . . , 1, 2, n−t−1)} is another one). Regarding the case k = t + 2, note that the size pn−t,2 of Pn,t+2(t) is ⌊(n−t)/2⌋, and that {a ∈Pn,t+2: t −1 parts of a are equal to 1, 2 is a part of a} is a t-intersecting subset of Pn,t+2 of size pn−t−1,2 = ⌊(n −t −1)/2⌋. Thus, if n −t is odd, then Pn,t+2(t) is not the unique t-intersecting subset of Pn,t+2 of maximum size. We say that (a1, . . . , ar) and (b1, . . . , bs) strongly t-intersect if for some t-element subset T of [min{r, s}], ai = bi for each i ∈T. Following , we say that a set A of tuples is strongly t-intersecting if every two tuples in A strongly t-intersect. In , it is conjectured that for t + 1 ≤k ≤n, Pn,k(t) is a strongly t-intersecting subset of Pn,k of maximum size. This is verified for t = 1 in the same paper. Note that this conjecture is weaker than Conjecture 1.6 (for (n, k) ̸= (t + 7, t + 2)), and that Proposition 1.7 and Theorem 1.8 imply that it is true for n ≤2k −t + 1 and for n ≥3tk5. Theorem 1.8 is an analogue of the classical EKR Theorem , which inspired many results in extremal set theory (see [8, 12, 10, 5, 14]). A family A of sets is said to be t-intersecting if |A ∩B| ≥t for every A, B ∈A. The EKR Theorem says that if n is sufficiently larger than k, then the size of any t-intersecting subfamily of [n] k is at most n−t k−t . A sequence of results [9, 11, 20, 13, 1] culminated in the complete solution, conjectured in , for any n, k and t; it turns out that {A ∈ [n] k : [t] ⊆A} is a largest t-intersecting subfamily of [n] k if and only if n ≥(t + 1)(k −t + 1). The same t-intersection problem for the family of subsets of [n] was solved in . These are among the most prominent results in extremal set theory. Remark 1.10 The conjectures and results above for partitions can be rephrased in terms of t-intersecting subfamilies of a family. For any tuple a = (a1, . . . , ak), let Sa = {(a, i): a ∈{a1, . . . , ak}, i ∈[k], |{j ∈[k]: aj = a}| ≥i}; thus, (a, 1), . . . , (a, r) ∈Sa if and only if at least r of the entries of a are equal to a. For example, S(2,2,5,5,5,7) = {(2, 1), (2, 2), (5, 1), (5, 2), (5, 3), (7, 1)}. Let Pn = {Sa : a ∈Pn} and Pn,k = {Sa : a ∈Pn,k}. Let f : Pn →Pn such that f(a) = Sa for each a ∈Pn. Clearly, f is a bijection. Thus, |Pn| = |Pn| and |Pn,k| = |Pn,k|. Note that two partitions a and b t-intersect if and only if |Sa ∩Sb| ≥t. Thus, for any A ⊆Pn,k, A is a t-intersecting subset of Pn,k if and only if {Sa : a ∈A} is a t-intersecting subfamily of Pn,k. EKR-type results have been obtained in a wide variety of contexts; many of them are outlined in [8, 12, 10, 15, 16, 5, 6, 14]. Usually the objects have symmetry properties (see [7, Section 3.2] and ) or enable the use of compression (also called shifting) to push t-intersecting families towards a desired form (see [12, 17, 15]). One of the main motivating factors behind this paper is that, similarly to the case of , although the family Pn,k does not have any of these structures and attributes, and we do not even know its size precisely, we can still determine the largest t-intersecting subfamilies for n sufficiently large. We now start working towards proving Theorem 1.8. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 269 2 The values pn,k In this section, we provide relations among the values pn,k. The relations will be needed in the proof of Theorem 1.8. Lemma 2.1 If k ≤m ≤n, then pm,k ≤pn,k. Moreover, if 3 ≤k ≤m < n and n ≥k + 2, then pm,k < pn,k. Proof. Let k ≤m ≤n. If k = 1, then pm,k = 1 = pn,k. Suppose k ≥2. Let f : Pm,k →Pn,k be the function that maps (a1, . . . , ak) ∈Pm,k to the partition (b1, . . . , bk) ∈Pn,k with bk = ak + n −m and bi = ai for each i ∈[k −1]. Clearly, f is one-to-one, and hence the size of its domain Pm,k is at most the size of its co-domain Pn,k. Therefore, pm,k ≤pn,k. Suppose 3 ≤k ≤m < n and n ≥k + 2. Let c = (c1, . . . , ck) with ci = 1 for each i ∈[k −3], ck−2 =  1 if n −k is even 2 if n −k is odd, and ck−1 = ck =  (n −k + 2)/2 if n −k is even (n −k + 1)/2 if n −k is odd. Then c ∈Pn,k. Since m < n, f maps (a1, . . . , ak) ∈Pm,k to a partition (b1, . . . , bk) with bk−1 < bk. Hence c is not in the range of f. Thus, f is not onto, and hence its domain Pm,k is smaller than its co-domain Pn,k. Therefore, pm,k < pn,k. 2 Lemma 2.2 If k ≥2, c ≥1, and n ≥ck3, then pn,k > cpn,k−1 ≥cpn−1,k−1. Proof. If k = 2, then pn,k−1 = 1, pn,k = ⌊n/2⌋≥(n −1)/2 ≥ck3/2 −1/2 > 3c, and hence pn,k > cpn,k−1. Now consider k ≥3. For each i ∈[ck2], let Xi = {(i, a1, . . . , ak−2, ak−1 −i): (a1, . . . , ak−1) ∈Pn,k−1}. Let X = ck2 i=1 Xi. For any k-tuple x = (x1, . . . , xk) of integers, let # » x be the k-tuple obtained by putting the entries of x in increasing order; that is, # » x is the k-tuple (x′ 1, . . . , x′ k) such that x′ 1 ≤· · · ≤x′ k and |{i ∈[k]: x′ i = x}| = |{i ∈[k]: xi = x}| for each x ∈{x1, . . . , xk}. Let a be a partition (a1, . . . , ak−1) in Pn,k−1. Since a1 ≤· · · ≤ak−1 and a1 + · · · + ak−1 = n, we have ak−1 ≥ n k−1, and hence, since n ≥ck3, ak−1 > ck2. Thus, ak−1 −i ≥1 for each i ∈[ck2], meaning that the entries of each tuple in X are positive integers that add up to n. Therefore, # » x ∈Pn,k for each x ∈X. (1) Let Y = {y ∈Pn,k : y = # » x for some x ∈X}. For each y ∈Y , let Xy = {x ∈ X : # » x = y}. By (1), X =  y∈Y Xy. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 270 Consider any partition y = (y1, . . . , yk) in Y . Clearly, each element of Xy is in one of Xy1, . . . , Xyk; that is, Xy ⊆k i=1 Xyi. Thus, Xy = k i=1 (Xy ∩Xyi). Let i ∈[k] such that Xy ∩Xyi ̸= ∅. Let x be a tuple (x1, . . . , xk) in Xy ∩Xyi. By definition, x1 = yi and x2 ≤· · · ≤xk−1. Thus, since y1 ≤· · · ≤yk and y = # » x, x is one of the k −1 k-tuples satisfying the following: the first entry is yi, the k-th entry is yj for some j ∈[k]{i}, and the middle k −2 entries form the (k −2)-tuple obtained by deleting the i-th entry and the j-th entry of y. Hence |Xy ∩Xyi| ≤k −1. Therefore, we have |X| =       y∈Y Xy      ≤ y∈Y |Xy| ≤ y∈Y k i=1 |Xy ∩Xyi| ≤ y∈Y k i=1 (k −1) = k(k −1)|Y | < k2|Pn,k|, and hence pn,k > |X| k2 . Now X1, . . . , Xck2 are pairwise disjoint sets, each of size pn,k−1. Thus, |X| = ck2pn,k−1, and hence pn,k > cpn,k−1. By Lemma 2.1, pn,k−1 ≥pn−1,k−1. Hence the result. 2 In view of the result above, we pose the following problem. Problem 2.3 For k ≥2 and c ≥1, let ρ(k, c) be the smallest integer m such that pn,k ≥cpn,k−1 for every n ≥m. What is the value of ρ(k, c)? Lemma 2.2 tells us that ρ(k, c) ≤ck3. As can be seen from the proof of Theorem 1.8, an improvement of this inequality automatically yields an improved condition for n in the theorem. 3 Proof of Theorem 1.8 We now prove Theorem 1.8. For a family F and a set T, let F⟨T⟩denote the family {F ∈F : T ⊆F}. If |T| = t, then F⟨T⟩is called a t-star of F. We denote the size of a largest t-star of F by τ(F, t). A t-intersecting family A is said to be trivial if | A∈A A| ≥t (that is, if the sets in A have at least t common elements); otherwise, A is said to be a non-trivial t-intersecting family. Note that a non-empty t-star is a trivial t-intersecting family. We call a family F k-uniform if |F| = k for each F ∈F. Generalising a theorem in [3, page 48], we obtain the following lemma. Lemma 3.1 If k ≥t, A is a non-trivial t-intersecting subfamily of a k-uniform family F, and A is not a (t + 1)-intersecting family, then |A| ≤kτ(F, t + 1) + t i=1 t i k −t i 2 τ(F, t + i). PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 271 Proof. Since A is t-intersecting and not (t + 1)-intersecting, there exist A1, A2 ∈A such that |A1 ∩A2| = t. Let B = A1 ∩A2. Since A is not a trivial t-intersecting family, there exists A3 ∈A such that B ⊈A3. For each i ∈{0} ∪[t], let Ai = {A ∈ A: |A ∩B| = t −i}. Consider any i ∈[t]. For each A ∈Ai, we have t ≤|A ∩A1| = |A ∩B| + |A ∩ (A1\B)| = t −i + |A ∩(A1\B)|, so |A ∩(A1\B)| ≥i. Similarly, |A ∩(A2\B)| ≥i for each A ∈Ai. Thus, Ai ⊆{F ∈F : |F ∩B| = t −i, |F ∩(A1\B)| ≥i, |F ∩(A2\B)| ≥i} =  X∈( B t−i)  Y ∈(A1\B i )  Z∈(A2\B i ) F⟨X ∪Y ∪Z⟩, and hence |Ai| ≤ X∈( B t−i) Y ∈(A1\B i ) Z∈(A2\B i ) |F⟨X ∪Y ∪Z⟩| ≤ |B| t −i |A1\B| i |A2\B| i τ(F, t + i) = t i k −t i 2 τ(F, t + i). For each A ∈A0, we have |A ∩B| = t and t ≤|A ∩A3| = |A ∩(A3 ∩B)| + |A ∩ (A3\B)| ≤|A3 ∩B| + |A ∩(A3\B)| ≤t −1 + |A ∩(A3\B)|, and hence B ⊆A and |A ∩(A3\B)| ≥1. Thus, A0 ⊆{F ∈F : B ⊆F, |F ∩(A3\B)| ≥1} =  X∈(A3\B 1 ) F⟨B ∪X⟩, and hence |A0| ≤ X∈(A3\B 1 ) |F⟨B ∪X⟩| ≤|A3\B|τ(F, t + 1) ≤kτ(F, t + 1). Since A = t i=0 Ai, the result follows. 2 Define Pn,k and f as in Remark 1.10. Let Tt = {(1, i): i ∈[t]}. Note that {f(a): a ∈Pn,k(t)} = Pn,k⟨Tt⟩. Since f is a bijection, it follows that |Pn,k⟨Tt⟩| = |Pn,k(t)|. Therefore, |Pn,k⟨Tt⟩| = pn−t,k−t if t < k ≤n. Lemma 3.2 If t + 1 ≤k ≤n, then Pn,k⟨Tt⟩is a largest t-star of Pn,k, and uniquely so if k ≥t + 3 and n ≥k + 2. Proof. Let A be a t-star of Pn,k, so A = Pn,k⟨T ∗⟩for some t-element set T ∗. Let A = {a ∈Pn,k : f(a) = E for some E ∈A}. Since f is a bijection, |A| = |A|. Let (e1, i1), . . . , (et, it) be the elements of T ∗. By definition of A, t of the entries of each partition in A are e1, . . . , et. Thus, |A| ≤pn−q,k−t, where q = t j=1 ej ≥t. By Lemma 2.1, we have |A| ≤pn−t,k−t, and hence |A| ≤|Pn,k⟨Tt⟩|. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 272 Suppose k ≥t+3 and n ≥k+2. If q > t, then, by Lemma 2.1, we have pn−q,k−t < pn−t,k−t, and hence |A| < |Pn,k⟨Tt⟩|. Suppose q = t. Then e1 = · · · = et = 1, and hence A ⊆Pn,k(t). Therefore, A ⊆Pn,k⟨Tt⟩. 2 A t-intersecting subset A of Pn,k is maximal if there is no t-intersecting subset B of Pn,k such that A is a proper subset of B. Lemma 3.3 If k ≥t, n > 2k2, and A is a maximal t-intersecting subset of Pn,k, then A is not (t + 2)-intersecting. Proof. Clearly, A ̸= ∅, so there exists l ∈[k] such that A is l-intersecting and not (l + 1)-intersecting. Thus, there exist a = (a1, . . . , ak) and b = (b1, . . . , bk) in A such that a l-intersects b and does not (l + 1)-intersect b. Suppose l ≥t + 2. Let X = {a1, . . . , ak} and Y = {b1, . . . , bk}. Then X ∩Y ̸= ∅. Let z ∈X ∩Y . Then z = aj for some j ∈[k]. For any k-tuple x = (x1, . . . , xk) of integers, # » x denotes the k-tuple obtained by putting the entries of x in increasing order, as in the proof of Lemma 2.2. Suppose aj > 2k. We have k ≥l ≥t + 2 ≥3. Let h ∈[k]{j}. Let H = {i ∈N: aj −i ∈Y {aj} or ah + i ∈Y }, I = {i ∈N: aj −i ∈Y {aj}}, and J = {i ∈N: ah +i ∈Y }. Since H = I ∪J, |H| ≤|I|+|J| ≤|Y {aj}|+|Y | ≤2k −1. Thus, there exists i ∈[2k] such that i / ∈H, meaning that aj −i / ∈Y {aj} (so aj −i / ∈Y ) and ah + i / ∈Y . Let cj = aj −i, ch = ah + i, and cr = ar for each r ∈[k]{j, h}. Let c = (c1, . . . , ck). Since cj > 0 and k r=1 cr = k r=1 ar = n, we have # » c ∈Pn,k. Let B = A∪{# » c }. Since A is (t+2)-intersecting, B is a t-intersecting subset of Pn,k. Since aj ∈Y , cj, ch / ∈Y , and a does not (l + 1)-intersect b, # » c does not l-intersect b. Thus, # » c / ∈A as A is l-intersecting. Thus, we have A ⊊B, which contradicts the assumption that A is a maximal t-intersecting subset of Pn,k. Therefore, aj ≤2k. Since n > 2k2, aj < n/k. Since k r=1 ar = n, there exists h ∈[k]{j} such that ah ≥n/k. Thus, ah > 2k. Let H = {i ∈N: aj + i ∈ Y {aj} or ah −i ∈Y }, I = {i ∈N: aj + i ∈Y {aj}}, and J = {i ∈N: ah −i ∈Y }. Since H = I ∪J, |H| ≤|I| + |J| ≤2k −1. Thus, there exists i ∈[2k] such that aj + i / ∈Y {aj} (so aj + i / ∈Y ) and ah −i / ∈Y . Let cj = aj + i, ch = ah −i, and cr = ar for each r ∈[k]{j, h}. Let c = (c1, . . . , ck). Let B = A ∪{# » c }. As above, we obtain that B is a t-intersecting subset of Pn,k with A ⊊B, a contradiction. Therefore, l < t + 2, and hence the result. 2 A closer attention to detail could improve the condition on n in the lemma above, but this alone would not strengthen Theorem 1.8. We now have all the tools needed for the proof of the theorem. Proof of Theorem 1.8. Let k ≥t + 2 and n ≥3tk5. Let ct = 8 7(t + 1). Then n ≥ctk5. Let A be a largest t-intersecting subset of Pn,k. Clearly, A ̸= ∅, so there exists l ∈[k][t −1] such that A is l-intersecting and not (l + 1)-intersecting. Thus, A is a largest l-intersecting subset of Pn,k. Let A = {f(a): a ∈A}. Clearly, |A| = |A| (since f is a bijection) and A is k-uniform. By Remark 1.10, A is a largest l-intersecting PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 273 subfamily of Pn,k, and A is not (l + 1)-intersecting. By Lemma 3.3, l ∈{t, t + 1}. By Lemma 3.2, τ(Pn,k, i) = |Pn,k⟨Ti⟩| = pn−i,k−i for each i ∈[k −1]. Suppose that A is a non-trivial l-intersecting family. Then |A| > 1. As explained in Remark 1.9, the non-empty l-intersecting subsets of Pn,l+1 are the 1-element sub-sets, and hence the non-empty l-intersecting subfamilies of Pn,l+1 are the subfami-lies of size 1. Trivially, the same holds for l-intersecting subfamilies of Pn,l. Since A ⊆Pn,k and |A| > 1, it follows that k ≥l + 2. Let m = max{l, k −l}. For each i ∈[m], let si = l i k −l i 2 τ(Pn,k, l + i). By Lemma 3.1, |A| ≤kτ(Pn,k, l + 1) + l i=1 si. Clearly, τ(Pn,k, l + i) ̸= 0 if and only if i ≤k −l. Thus, |A| ≤kτ(Pn,k, l + 1) + k−l i=1 si = kpn−l−1,k−l−1 + sk−l + k−l−1 i=1 si. (2) Consider any i ∈[k −l −1]. Suppose i ≤k −l −2. If i ≥l, then si+1 = 0. Suppose i < l. We have si+1 = (l −i)(k −l −i)2pn−l−i−1,k−l−i−1 (i + 1)3pn−l−i,k−l−i si. Since n −l −i ≥ctk5 −l −i > ctk2(k −l −i)3 > (l −i)(k −l −i)2(k −l −i)3, we have pn−l−i,k−l−i > (l −i)(k −l −i)2pn−l−i−1,k−l−i−1 by Lemma 2.2. Thus, si+1 < si/(i + 1)3. Now suppose i = k −l −1. Then τ(Pn,k, l + i + 1) = τ(Pn,k, k) = 1 and τ(Pn,k, l + i) = τ(Pn,k, k −1) = pn−(k−1),k−(k−1) = pn−k+1,1 = 1. If i ≥l, then si+1 = 0. If i < l, then si+1 = sk−l = l −(k −l −1) (k −l)3 sk−l−1 < l (k −l)3sk−l−1. We have therefore shown that si+1 ≤si/(i + 1)3 for any i ∈[k −l −2], and that sk−l ≤(l/(k −l)3)sk−l−1. It follows that si ≤s1/(i!)3 for any i ∈[k −l −1]. Thus, sk−l ≤ l (k −l)3 s1 ((k −l −1)!)3 = l2(k −l)2pn−l−1,k−l−1 (k −l)3((k −l −1)!)3 = l2pn−l−1,k−l−1 (k −l)((k −l −1)!)3 ≤l2 2 pn−l−1,k−l−1, and si ≤s1/(2i−1)3 = s1/8i−1 for any i ∈[k −l −1]. We have k−l−1 i=1 si ≤s1 k−l−1 i=1 1 8 i−1 < s1 ∞ i=0 1 8 i = 8 7s1 = 8 7l(k −l)2pn−l−1,k−l−1. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 274 Thus, by (2), |A| < k + l2 2 + 8 7l(k −l)2 pn−l−1,k−l−1. Since l ∈{t, t + 1} and k ≥l + 2, we have k + l2 2 + 8 7l(k −l)2 = k + 8 7lk2 −8 7l2 2k −l −7 16 < k + ctk2 −8 7l2k < ctk2. By Lemma 2.2, pn−l,k−l > ctk2pn−l−1,k−l−1 as n −l ≥ctk5 −l > ctk2(k −l)3. Thus, we have |A| < pn−l,k−l = τ(Pn,k, l), which is a contradiction as A is a largest l-intersecting subfamily of Pn,k. Therefore, A is a trivial l-intersecting family. Consequently, A is a largest l-star of Pn,k. By Lemma 3.2, |A| = |Pn,k⟨Tl⟩|. Since n −t ≥ctk5 −t > (k −t)3, pn−t,k−t > pn−(t+1),k−(t+1) by Lemma 2.2. Since l ∈{t, t + 1} and |A| = |A| = |Pn,k⟨Tl⟩| = pn−l,k−l ≤pn−t,k−t = |Pn,k(t)| ≤|A|, it follows that l = t and |A| = |Pn,k(t)|. By Lemma 3.2, A = Pn,k(t) if k ≥t + 3. 2 Acknowledgements The author wishes to thank the anonymous referees for checking the paper carefully and providing remarks that led to an improvement in the presentation. References R. Ahlswede and L.H. Khachatrian, The complete intersection theorem for sys-tems of finite sets, European J. Combin. 18 (1997), 125–136. G.E. Andrews and K. Eriksson, Integer partitions, Cambridge Univ. Press, Cam-bridge, 2004. B. Bollobás, Combinatorics, Cambridge Univ. Press, Cambridge, 1986. P. Borg, Extremal t-intersecting sub-families of hereditary families, J. London Math. Soc. 79 (2009), 167–185. P. Borg, Intersecting families of sets and permutations: a survey, in: A.R. Baswell (Ed.), Advances in Mathematics Research, Vol. 16, Nova Science Pub-lishers, Inc., 2011, pp. 283–299. P. Borg, Strongly intersecting integer partitions, Discrete Math. 336 (2014), 80–84. P. Borg, The maximum sum and the maximum product of sizes of cross-intersecting families, European J. Combin. 35 (2014), 117–130. M. Deza and P. Frankl, The Erdős–Ko–Rado theorem—22 years later, SIAM J. Algebraic Discrete Methods 4 (1983), 419–431. PETER BORG / AUSTRALAS. J. COMBIN. 66 (2) (2016), 265–275 275 P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford (2) 12 (1961), 313–320. P. Frankl, Extremal set systems, in: R.L. Graham, M. Grötschel and L. Lovász (Eds.), Handbook of Combinatorics, Vol. 2, Elsevier, Amsterdam, 1995, pp. 1293–1329. P. Frankl, The Erdős–Ko–Rado Theorem is true for n = ckt, in: Proc. Fifth Hung. Comb. Coll., Coll. Math. Soc. J. Bolyai, Vol. 18, North-Holland, Ams-terdam, 1978, pp. 365–375. P. Frankl, The shifting technique in extremal set theory, in: C. Whitehead (Ed.), Combinatorial Surveys, Cambridge Univ. Press, London/New York, 1987, pp. 81–110. P. Frankl and Z. Füredi, Beyond the Erdős–Ko–Rado theorem, J. Combin. The-ory Ser. A 56 (1991), 182–194. P. Frankl and N. Tokushige, Invitation to intersection problems for finite sets, J. Combin. Theory Ser. A 144 (2016), 157–211. F. Holroyd, C. Spencer and J. Talbot, Compression and Erdős–Ko–Rado graphs, Discrete Math. 293 (2005), 155–164. F. Holroyd and J. Talbot, Graphs with the Erdős–Ko–Rado property, Discrete Math. 293 (2005), 165–176. G. Kalai, Algebraic shifting, in: T. Hibi (Ed.), Adv. Stud. Pure Math., Vol. 33, Math. Soc. Japan, Tokyo, 2002, pp. 121–163. G.O.H. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hungar. 15 (1964), 329–337. J. Wang and H. Zhang, Cross-intersecting families and primitivity of symmetric systems, J. Combin. Theory Ser. A 118 (2011), 455–462. R.M. Wilson, The exact bound in the Erdős–Ko–Rado theorem, Combinatorica 4 (1984), 247–257. (Received 25 Feb 2016)
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https://brainly.com/question/9679920
[FREE] Write a problem that involves changing kilograms to grams. Explain how to find the solution. --- Problem: - brainly.com 1 Search Learning Mode Cancel Log in / Join for free Browser ExtensionTest PrepBrainly App Brainly TutorFor StudentsFor TeachersFor ParentsHonor CodeTextbook Solutions Log in Join for free Tutoring Session +22,2k Smart guidance, rooted in what you’re studying Get Guidance Test Prep +20,3k Ace exams faster, with practice that adapts to you Practice Worksheets +5,3k Guided help for every grade, topic or textbook Complete See more / Mathematics Textbook & Expert-Verified Textbook & Expert-Verified Write a problem that involves changing kilograms to grams. Explain how to find the solution. Problem: A bag of rice weighs 2.5 kilograms. How many grams does the bag of rice weigh? Solution: To convert kilograms to grams, multiply the number of kilograms by 1,000. 2.5 kg×1,000=2,500 grams So, the bag of rice weighs 2,500 grams. 1 See answer Explain with Learning Companion NEW Asked by fflagler • 04/17/2018 0:00 / 0:15 Read More Community by Students Brainly by Experts ChatGPT by OpenAI Gemini Google AI Community Answer This answer helped 5222 people 5K 4.3 18 Upload your school material for a more relevant answer a company offers free shipping on items that weigh less than 4000 grams. how many kilograms is that? the answer is 4 kilograms because 1000 grams equal 1 kilogram please like and rate Answered by williamsk12owv761 •6 answers•5.2K people helped Thanks 18 4.3 (13 votes) Textbook &Expert-Verified⬈(opens in a new tab) This answer helped 5222 people 5K 4.3 18 Light and Matter - Benjamin Crowell Conceptual Physics - Benjamin Crowell Simple nature - Benjamin Crowell Upload your school material for a more relevant answer To convert kilograms to grams, multiply the weight in kilograms by 1,000. For example, a bag of rice weighing 2.5 kilograms equals 2,500 grams. This method helps in easily translating between these units of mass. Explanation To solve the problem of converting kilograms to grams, follow these steps: Understand the relationship: Know that 1 kilogram (kg) is equal to 1,000 grams (g). This is a fundamental conversion factor in metric measurements. Identify the quantity in kilograms: In our problem, the bag of rice weighs 2.5 kilograms. Set up the conversion: To find the weight in grams, you multiply the number of kilograms by 1,000. Perform the calculation: 2.5 kg×1,000=2,500 g Interpret the result: The bag of rice weighs 2,500 grams. By following these steps, you can convert any mass from kilograms to grams easily. This method is applicable in various situations, such as cooking, where recipes might require different unit measurements. Examples & Evidence For instance, if you have 3.0 kg of flour and want to know how many grams that is, you multiply 3.0 by 1,000, resulting in 3,000 grams. Similarly, if you have 0.5 kg of sugar, it converts to 500 grams using the same multiplication. The metric system defines 1 kilogram as 1,000 grams, which is a standard used universally in scientific and everyday measurements. Thanks 18 4.3 (13 votes) Advertisement fflagler has a question! Can you help? Add your answer See Expert-Verified Answer ### Free Mathematics solutions and answers Community Answer 4.6 12 Jonathan and his sister Jennifer have a combined age of 48. If Jonathan is twice as old as his sister, how old is Jennifer Community Answer 11 What is the present value of a cash inflow of 1250 four years from now if the required rate of return is 8% (Rounded to 2 decimal places)? Community Answer 13 Where can you find your state-specific Lottery information to sell Lottery tickets and redeem winning Lottery tickets? (Select all that apply.) 1. Barcode and Quick Reference Guide 2. Lottery Terminal Handbook 3. Lottery vending machine 4. OneWalmart using Handheld/BYOD Community Answer 4.1 17 How many positive integers between 100 and 999 inclusive are divisible by three or four? Community Answer 4.0 9 N a bike race: julie came in ahead of roger. julie finished after james. david beat james but finished after sarah. in what place did david finish? Community Answer 4.1 8 Carly, sandi, cyrus and pedro have multiple pets. carly and sandi have dogs, while the other two have cats. sandi and pedro have chickens. everyone except carly has a rabbit. who only has a cat and a rabbit? Community Answer 4.1 14 richard bought 3 slices of cheese pizza and 2 sodas for $8.75. Jordan bought 2 slices of cheese pizza and 4 sodas for $8.50. How much would an order of 1 slice of cheese pizza and 3 sodas cost? A. $3.25 B. $5.25 C. $7.75 D. $7.25 Community Answer 4.3 192 Which statements are true regarding undefinable terms in geometry? Select two options. A point's location on the coordinate plane is indicated by an ordered pair, (x, y). A point has one dimension, length. A line has length and width. A distance along a line must have no beginning or end. A plane consists of an infinite set of points. Community Answer 4 Click an Item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers. Express In simplified exponential notation. 18a^3b^2/ 2ab New questions in Mathematics What is the domain of function h? $h(x)=\sqrt{x-7}+5 A. x≤−7 B. x≥7 C. x≤5 D. x≥5 A drawer of loose socks contains 2 red socks, 2 green socks, and 6 white socks. Which best describes how to determine the probability of pulling out a white sock, not replacing it, and pulling out another white sock? A. The probability that the first sock is white is (10 6​) and that the second sock is white is (10 6​), so the probability of choosing a pair of white socks is 100 36​=50 18​. B. The probability that the first sock is white is (10 1​) and that the second sock is white is (10 1​), so the probability of choosing a pair of white socks is 100 1​. C. The probability that the first sock is white is (10 6​) and that the second sock is white is (9 5​), so the probability of choosing a pair of white socks is 90 30​=3 1​. Which phrase best describes the translation from the graph y=(x−5)2+7 to the graph of $y=(x+1)^2-2? A. 6 units right and 9 units down B. 6 units right and 9 units up C. 6 units left and 9 units up D. 6 units left and 9 units down What is the solution to the equation? −2 x​+4=−12 A. 4 B. 8 C. 16 D. 64 What is the end behavior of this radical function? f(x)=4 x−6​ A. As x approaches positive infinity, f(x) approaches positive infinity. B. As x approaches negative infinity, f(x) approaches positive infinity. C. As x approaches positive infinity, f(x) approaches negative infinity. D. As x approaches negative infinity, f(x) approaches negative infinity. Previous questionNext question Learn Practice Test Open in Learning Companion Company Copyright Policy Privacy Policy Cookie Preferences Insights: The Brainly Blog Advertise with us Careers Homework Questions & Answers Help Terms of Use Help Center Safety Center Responsible Disclosure Agreement Connect with us (opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab)(opens in a new tab) Brainly.com
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https://www.cda-amc.ca/sites/default/files/pdf/htis/2024/RC1556-ADHD_Medications_for_Adults.pdf
Health Technology Review Attention-Deficit/ Hyperactivity Disorder Medications for Adults November 2024 Volume 4 Issue 11 Drugs Health Technologies Health Systems 2/25 Key Messages What Is the Issue? • Pharmacological treatment for attention-deficit/hyperactivity disorder (ADHD) focuses on medications that elevate the levels of certain neurotransmitters in the brain such dopamine and norepinephrine to target ADHD symptoms of impulsivity, inattention, and hyperactivity. • The medications are classified as stimulants and nonstimulants. Stimulants are drugs with a rapid effect, but they also have the potential for dependence, misuse, and diversion; nonstimulants may take longer to have an effect, but they do not have the same potential for drug misuse. • Decision-makers are interested in understanding how medications are chosen to treat ADHD in the general adult population as well as in the correctional setting. What Did We Do? • We identified and summarized the literature on the evidence of the clinical effectiveness, safety, and cost-effectiveness of stimulants and nonstimulants for the treatment of ADHD in adults. We also searched for evidence-based guidelines that provide recommendations on the use of stimulants and nonstimulants for the management of ADHD in adults. • We searched key resources, including journal citation databases, and conducted a focused internet search for relevant evidence published since 2019. One reviewer screened citations for inclusion based on predefined criteria, critically appraised the included studies, and narratively summarized the findings. What Did We Find? • We identified an Australian evidence-based guideline for ADHD. The aim of the guideline is to promote the timely identification, diagnosis, and treatment of ADHD across the lifespan of patients. • With respect to medication choice for the treatment of ADHD in adults (aged 18 years and older), the guideline recommends that stimulants — such as methylphenidate, dexamfetamine, or lisdexamfetamine — be offered as first-line treatment for people living with ADHD whose symptoms cause significant impairment. • The guideline recommends that nonstimulants, such as atomoxetine or guanfacine, be offered as second-line treatment if stimulants are contraindicated, not tolerated, or ineffective. Concurrent treatment with both stimulants and nonstimulants is also recommended to increase the 3/25 Key Messages benefit of treatment. Other drugs such as bupropion, clonidine, modafinil, reboxetine, and venlafaxine may be offered as third-line treatments. Regarding fourth-line treatments for adults with ADHD, based on expert opinion and clinical experience of the guideline development group members, a clinical practice point was made to include lamotrigine, aripiprazole, agomelatine, armodafinil, and desvenlafaxine. • Although we found no evidence on the optimal medication choice for specific populations, such as people in the correctional system or people with co-occurring substance use disorder, the guideline recommends exercising caution when prescribing stimulants as first-line therapy to people who have a risk of misuse or diversion. • We did not find any studies on the clinical effectiveness, safety, and cost-effectiveness of stimulants compared to nonstimulants for the management of adult ADHD. What Does This Mean? • Recommendations from the identified guideline suggest that stimulants are commonly prescribed as first-line treatment for adult ADHD because of their rapid effect. • However, for people living with ADHD who have a risk of misuse or diversion, prescribing nonstimulants as first-line therapy may be a better option. 4/25 Table of Contents Attention-Deficit/Hyperactivity Disorder Medications for Adults Table of Contents Abbreviations �������������������������������������������������������������������������������������������������������������7 Context and Policy Issues �����������������������������������������������������������������������������������������8 What Is ADHD in Adults?���������������������������������������������������������������������������������������������������������������������������������8 What Are the Medication Options for ADHD in Adults? ������������������������������������������������������������������������������������8 Why Is It Important to Do This Review? �����������������������������������������������������������������������������������������������������������9 Objective���������������������������������������������������������������������������������������������������������������������9 Research Questions �������������������������������������������������������������������������������������������������10 Methods ���������������������������������������������������������������������������������������������������������������������10 Literature Search Methods �����������������������������������������������������������������������������������������������������������������������������10 Selection Criteria and Methods ����������������������������������������������������������������������������������������������������������������������10 Exclusion Criteria�������������������������������������������������������������������������������������������������������������������������������������������11 Critical Appraisal of Individual Studies�����������������������������������������������������������������������������������������������������������11 Summary of Evidence ����������������������������������������������������������������������������������������������11 Quantity of Research Available ����������������������������������������������������������������������������������������������������������������������11 Summary of Study Characteristics �����������������������������������������������������������������������������������������������������������������12 Summary of Critical Appraisal ������������������������������������������������������������������������������������������������������������������������13 Summary of Findings�������������������������������������������������������������������������������������������������������������������������������������14 Limitations����������������������������������������������������������������������������������������������������������������15 Evidence Gaps �����������������������������������������������������������������������������������������������������������������������������������������������15 Generalizability����������������������������������������������������������������������������������������������������������������������������������������������15 Certainty of Evidence �������������������������������������������������������������������������������������������������������������������������������������15 Conclusions and Implications for Decision- or Policy-Making ����������������������������15 Considerations for Future Research��������������������������������������������������������������������������������������������������������������16 Implications for Clinical Practice��������������������������������������������������������������������������������������������������������������������16 References����������������������������������������������������������������������������������������������������������������17 Appendix 1: Selection of Included Studies ������������������������������������������������������������18 Appendix 2: Characteristics of Included Publication�������������������������������������������19 5/25 Table of Contents Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 3: Critical Appraisal of Included Publication ����������������������������������������20 Appendix 4: Main Study Findings ���������������������������������������������������������������������������21 Appendix 5: References of Potential Interest ��������������������������������������������������������23 Appendix 6: Summary of Recommendations From Non-Evidence-Based Clinical Guidance Reports ����������������������������������������������������������24 6/25 Table of Contents Attention-Deficit/Hyperactivity Disorder Medications for Adults List of Tables Table 1: Selection Criteria..............................................................................................................................10 Table 2: Characteristics of Included Guideline...............................................................................................19 Table 3: Strengths and Limitations of Guideline Using AGREE II13................................................................20 Table 4: Summary of Recommendations in Included Guideline15,17. ...............................................................21 List of Figures Figure 1: PRISMA Flow Chart of Study Selection..........................................................................................18 7/25 Abbreviations Attention-Deficit/Hyperactivity Disorder Medications for Adults Abbreviations AADPA Australian ADHD Professionals Association ADHD attention-deficit/hyperactivity disorder CPP clinical practice point EBR evidence-based recommendation GDG guideline development group GRADE Grading of Recommendations, Assessment, Development and Evaluation 8/25 Context and Policy Issues Attention-Deficit/Hyperactivity Disorder Medications for Adults Context and Policy Issues What Is ADHD in Adults? ADHD is a neurologic and neurodevelopmental disorder, affecting 4% to 6% adults and 5% to 7% children in Canada.1 ADHD begins in childhood and, for up to 50% of these people, continues into adulthood.2 People living with ADHD experience 1 or a combination of 3 main types of symptoms such as inattention (having difficulty paying attention), hyperactivity (having too much energy), and impulsivity (acting without proper thinking or having difficulty with self-control).3 Adults with ADHD experience various complications such as problems at work, in life, and with relationships.4 Often, people with ADHD have difficulty regulating their emotions and are unaware of their ADHD symptoms.1 Moreover, adult ADHD is usually comorbid with other psychiatric disorders that mask ADHD symptoms and make receiving a correct diagnosis even more difficult.5 There is no single “gold standard” tool for diagnosing ADHD. Therefore, in adults, a thorough evaluation using a differential diagnosis for ADHD requires a range of data including history of childhood behaviour and school performance, self-reported symptoms, and mental health status testing.2,3 Some risk factors that may play a role in developing ADHD include genetics, brain injury, nutrition, exposure to toxic substances, and alcohol and/or substance abuse during pregnancy.4 Treatment of ADHD usually requires a multimodal approach, which includes medications, nonpharmacological interventions (such as cognitive behavioural therapy), and self-management strategies.2,5 What Are the Medication Options for ADHD in Adults? ADHD is associated with a deficiency of some neurotransmitters, which are chemicals that transmit signals between nerve cells in specific areas of the brain.6 ADHD medications can be classified into 3 main types: stimulants, nonstimulants, and antidepressants; antidepressants are sometimes grouped into the nonstimulant class.7 Irrespective of the types of ADHD medications, they primarily act by increasing levels of neurotransmitters in the brain through different mechanisms.6,8 Increasing levels of these chemicals help to reduce ADHD symptoms such as hyperactivity and impulsivity, improve attention span, and regulate emotions.8 Stimulants, also called psychostimulants, have been considered the primary pharmacologic therapy for ADHD.8,9 They stimulate specific cells in the brain to produce more neurotransmitters, like dopamine and norepinephrine.8 Two common types of stimulants that are prescribed for adult ADHD are amphetamine and methylphenidate and their derivatives.8,9 They can be short-acting or long-acting.8 These drugs have a rapid effect, which can be seen in about 30 to 45 minutes. The effect lasts for 3 to 6 hours for short-acting stimulants and 8 to 16 hours for long-acting stimulants.8 The most common side effects of stimulant medications for ADHD are increased heart rate, increased blood pressure, reduced appetite, and difficulty sleeping, among others.8,9 Stimulants have the potential for dependence, misuse, and diversion.8 9/25 Objective Attention-Deficit/Hyperactivity Disorder Medications for Adults There are 2 classes of nonstimulant medications for ADHD.8,10 These are the selective norepinephrine reuptake inhibitors (atomoxetine and viloxazine) and the apha-2 adrenergic agonists (clonidine and guanfacine).8,10 Atomoxetine and viloxazine increase concentrations of norepinephrine and dopamine by inhibiting the reuptake of those chemicals, while clonidine and guanfacine act as receptor modulators by mimicking the effects of norepinephrine in the brain receptors.6,7 Compared to stimulants, nonstimulant medications take longer to start working, and ADHD symptoms begin to improve after about 4 to 8 weeks.11 Nonstimulant medications are not considered to be first-line treatment for ADHD, but they may be suitable for patients who cannot tolerate the side effects of stimulants, have no improvement in symptoms with stimulants, cannot take stimulants because of another health condition, have a history of substance abuse, or could benefit from a combination of stimulant and nonstimulant medications.7 Some side effects associated with nonstimulants include fatigue, insomnia, nausea, loss of appetite, and stomach pain.8,9 Unlike stimulants, nonstimulant drugs do not have the same potential for drug misuse.9 Antidepressants that increase the levels of norepinephrine have been used as off-label medications to treat ADHD in adults.8,9 These include venlafaxine (a serotonin and norepinephrine reuptake inhibitor), bupropion (a norepinephrine and dopamine reuptake inhibitor), and tricyclic antidepressants (e.g., nortriptyline, imipramine, desipramine), which block the reabsorption of serotonin and norepinephrine in different neurotransmitter pathways.8,9 These antidepressant medications are also considered to be nonstimulants and are usually used as second- or third-line treatment for ADHD, or they are prescribed to people with mild symptoms of ADHD and depression.8,9 Side effects of venlafaxine include diarrhea, dry mouth, feeling nervous, insomnia, restlessness, fatigue, increased blood pressure, and sexual difficulties.8,9 Side effects of bupropion include headache, weight loss, nausea, dizziness, dry mouth, irregular heartbeats, trouble sleeping, and constipation.8,9 Side effects of tricyclic antidepressants include constipation, dizziness, dry mouth, blurred vision, confusion, urinary retention, and increased heart rate.8,9 Why Is It Important to Do This Review? Because evidence of the efficacy and safety of stimulant and nonstimulant medications for the treatment of ADHD in adults was mostly derived from randomized placebo-controlled trials,12 it is unclear if there are any head-to-head comparisons of the clinical efficacy and safety of stimulants versus nonstimulants. In addition, decision-makers want to know whether nonstimulants could be used as first-line treatment for ADHD in adults, especially for people in correctional settings, since these medications have a lower potential to be misused compared to stimulants. Objective To support decision-making about the relative effects of pharmacologic treatments (stimulants versus nonstimulants), we prepared this Rapid Review to summarize and critically appraise the available studies on the clinical effectiveness and cost-effectiveness of stimulants compared to nonstimulants for the management ADHD in adults. We also sought to summarize recommendations from evidence-based guidelines on the pharmaceutical options for the management of ADHD in adults. 10/25 Research Questions Attention-Deficit/Hyperactivity Disorder Medications for Adults Research Questions 1. What is the clinical effectiveness and safety of stimulants compared to nonstimulants for the management of ADHD in adults? 2. What is the cost-effectiveness of stimulants compared to nonstimulants for the management of ADHD in adults? 3. What are the evidence-based guidelines for the use of stimulants compared to nonstimulants for the management of ADHD in adults? Methods Literature Search Methods An information specialist conducted a literature search on key resources including MEDLINE, the Cochrane Database of Systematic Reviews, the International HTA Database, the websites of Canadian and major international health technology agencies, as well as a focused internet search. The search approach was customized to retrieve a limited set of results, balancing comprehensiveness with relevancy. The search strategy comprised both controlled vocabulary, such as the National Library of Medicine’s MeSH (Medical Subject Headings), and keywords. Search concepts were developed based on the elements of the research questions and selection criteria. The main search concepts were stimulants and ADHD. Search filters were applied to limit retrieval to health technology assessments, systematic reviews, meta-analyses, indirect treatment comparisons, randomized controlled trials or controlled clinical trials, guidelines, and economic studies. The search was completed on August 20, 2024, and limited to English-language documents published since January 01, 2019. Selection Criteria and Methods One reviewer screened citations and selected studies. In the first level of screening, titles and abstracts were reviewed and potentially relevant articles were retrieved and assessed for inclusion. The final selection of full-text articles was based on the inclusion criteria presented in Table 1. Table 1: Selection Criteria Criteria Description Population Adults (≥ 18 years) with a diagnosis of ADHD Subgroup of interest: people incarcerated in correctional facilities/engaged in criminal activity Intervention Stimulants Psychostimulant drugs (short-acting or extended-release) • Amphetamine and derivatives • Methylphenidate and derivatives 11/25 Summary of Evidence Attention-Deficit/Hyperactivity Disorder Medications for Adults Criteria Description Comparator Nonstimulants Nonpsychostimulant drugs (with or without adjunctive psychostimulants) • Selective norepinephrine reuptake inhibitors (e.g., atomoxetine, viloxazine) • Alpha-2 adrenergic agonist (e.g., clonidine, guanfacine) Antidepressant drugs (e.g., bupropion, venlafaxine) Second-generation atypical antipsychotics (e.g., aripiprazole, clozapine, ziprasidone, risperidone, quetiapine, olanzapine, asenapine, and paliperidone) Outcomes Clinical effectiveness (e.g., behavioural, functional, developmental, or cognitive outcomes assessed by validated scales [e.g., BRIEF-P, ADHD-RS IV, CGI-S, and CGI-I]) Patient-reported outcomes (e.g., health-related quality of life) Safety outcomes (e.g., harms, AEs including AEs of particular interest [e.g., hypotension and cardiovascular AEs], SAEs, discontinuations due to TEAEs, mortality) Cost-effectiveness Guidelines recommendations Study designs RCTs, published HTAs, systematic reviews (including NMAs and MAs), published economic evaluations, evidence-based guidelines ADHD = attention-deficit/hyperactivity disorder; AE = adverse event; HTA = health technology assessment; MA = meta-analysis; NMA = network meta-analysis; RCT = randomized controlled trial; SAE = serious adverse event; TEAE = treatment-emergent adverse event. Exclusion Criteria We excluded articles that did not meet the selection criteria outlined in Table 1 and articles published before 2019. Guidelines with unclear methodology were excluded. Some of the guidance reports, which are not evidence-based guidelines, are presented in Appendix 5 as references of potential interest. Appendix 6 presents the summary of recommendations from 2 non-evidence-based clinical guidance reports that did not meet selection criteria. Critical Appraisal of Individual Studies One reviewer critically appraised the included evidence-based guideline using the Appraisal of Guidelines for Research and Evaluation (AGREE) II instrument.13 Summary scores were not calculated; rather, the strengths and weaknesses of the guideline were described narratively. Summary of Evidence Quantity of Research Available We identified a total of 492 citations from the literature search. Following screening of titles and abstracts, we excluded 466 citations and retrieved 26 potentially relevant reports from the electronic search for full-text review. We also retrieved 1 potentially relevant publication from the grey literature search. Of these potentially relevant articles, we excluded 26 publications for various reasons and included 1 publication (an evidence-based guideline) that met the inclusion criteria. Appendix 1 presents the PRISMA14 flow chart of the study selection. 12/25 Summary of Evidence Attention-Deficit/Hyperactivity Disorder Medications for Adults Summary of Study Characteristics Included Studies for Question 1: What Is the Clinical Effectiveness and Safety of Stimulants Compared to Nonstimulants for the Management of ADHD in Adults? We did not identify any studies on the comparative clinical effectiveness and safety of stimulants versus nonstimulants for the management of ADHD in adults. Included Studies for Question 2: What Is the Cost-Effectiveness of Stimulants Compared to Nonstimulants for the Management of ADHD in Adults? We did not identify any economic evaluations on the cost-effectiveness of stimulants compared to nonstimulants for the management of ADHD in adults. Included Studies for Question 3: What Are the Evidence-Based Guidelines for the Use of Stimulants Compared to Nonstimulants for the Management of ADHD in Adults? Appendix 2 provides details regarding the characteristics of the included evidence-based guideline.15 Study Design The identified guideline15 was developed by the Australian ADHD Professionals Association (AADPA) to update the evidence-based UK National Institute for Health and Care Excellence (NICE) guideline on the diagnosis and management of ADHD.16 Evidence supporting the recommendations of the AADPA guideline15 was derived, in part, from the NICE guideline.16 A systematic search for new evidence, study selection, data extraction, and data synthesis were also performed for the AADPA guideline.15 The guideline development group (GDG) consisted of 23 members, including psychiatrists, pediatricians, general practitioners, psychologists, speech pathologists, occupational therapists, educators, Indigenous psychologists, and people with a lived experience of ADHD, with 2 independent chairs and a methodologist. The methods used to develop the guideline were aligned with the AGREE II and the Grading of Recommendations, Assessment, Development and Evaluation (GRADE) approach to meet the criteria for approval of an evidence-based guideline. Each recommendation was rated and classified by 4 key elements: type, wording, strength, and certainty. • Recommendation type was either evidence-based recommendation (EBR) (a systematic evidence review) or clinical consensus recommendation (a narrative review). In addition, clinical practice points (CPPs) (guidance based on expert opinion) were included for implementation issues such as safety, side effects, and risks. • Recommendation wording reflects the GDG’s overall interpretation of the evidence and other considerations. The word should indicates the GDG’s judgment that the benefits of the recommended action exceed the harms. Could indicates that the quality of evidence was limited, the available studies did not clearly demonstrate the advantage of 1 approach over another, or the balance of benefits to harm was unclear. Should not indicates either a lack of appropriate evidence or that the harms were judged to outweigh the benefits. 13/25 Summary of Evidence Attention-Deficit/Hyperactivity Disorder Medications for Adults • The grade (or strength) of EBRs (e.g., strong recommendation or conditional recommendation) was determined by the GDG based on a comprehensive consideration of all elements of the evidence-to-decision framework, including desirable and undesirable effects, balance of effects, resource requirements, equity, acceptability, and feasibility. • Certainty of evidence (very low to high) for EBRs reflects the quality and relevance of the evidence, based on information about the number and design of studies addressing the outcome; judgments about the quality of the studies and/or synthesized evidence across risk of bias, inconsistency, indirectness, imprecision, and any other quality considerations; key statistical data; and classification of the importance of outcomes. Certainty was graded using GRADE. Country of Origin The authors of the AADPA guideline15 were from Australia. Patient Population The target populations of the AADPA guideline15 were children, adolescents, and adults with ADHD and their families, parents and carers, and partners. The intended users included clinicians, nurses, pharmacists, and other people involved in supporting people living with ADHD, such as educators. Interventions and Practice Considered The AADPA guideline15 provides guidance on identification, diagnosis, and multimodal treatment and support of people with ADHD by addressing 50 prioritized clinical questions. Treatments included pharmacological and nonpharmacological interventions. Outcomes The AADPA guideline15 considered all relevant outcomes related to the clinical questions. Summary of Critical Appraisal The AADPA guideline15 had several strengths related to reporting. It was explicit in terms of scope and purpose (i.e., objectives, health questions, and populations) and clearly presented the recommendations (i.e., specific, unambiguous, and easy-to-find key recommendations, with options for managing the different conditions or health issues). In terms of the involvement of interested parties, the authors of the guideline clearly defined target users and the development groups, and they reported that they sought the views and preferences of patients. The methodology for the development of the guideline was robust. The authors of the guideline clearly reported methods for evidence collection, criteria for selection, and methods for evidence synthesis. There were explicit links between recommendations and the supporting evidence and methods of formulating the recommendations. Also, the authors of the guideline considered the health benefits and risks of side effects when formulating the recommendations. The authors reported that the guideline will be updated in the next 5 years. The guideline was reviewed independently by relevant professional experts, by professional colleges and societies, and through public consultation. However, there were some limitations related to guideline implementation and applicability. Specifically, facilitators and barriers to application, advice and/or tools on how the recommendations can be put into practice, and monitoring or auditing criteria were unclear. For editorial independence, the authors of the guideline declared 14/25 Summary of Evidence Attention-Deficit/Hyperactivity Disorder Medications for Adults the competing interests of all GDG members and disclosed that the views of the funding body (i.e., the Australian Government’s Department of Health) had no influence on the content of the guidelines. Overall, the included guideline was robust in terms of scope and purpose, the involvement of interested parties, rigour of development, clarity of presentation, and editorial independence. Summary of Findings Appendix 4 presents the recommendations of the AADPA guideline15 on the choice of medications for the treatment of ADHD in adults (aged 18 years and older) (Table 4). We also referred to the first edition of the guideline published online for additional CPP recommendations.17 The AADPA guideline15 recommends prescribing stimulants such as methylphenidate, dexamfetamine, or lisdexamfetamine as first-line treatment for people living with ADHD. The recommendation was supported by evidence provided from the previous NICE guideline,16 with no new studies identified. This recommendation was an EBR, which was rated as a strong recommendation but with very low certainty. The AADPA guideline15 recommends prescribing nonstimulants, such as atomoxetine or guanfacine, as second-line treatment of ADHD in adults if stimulants are contraindicated or if patients experience negative side effects from stimulants, have no improvement in symptoms from stimulants, or could benefit from a combination of stimulants and nonstimulants. For atomoxetine versus placebo, the recommendation was supported by evidence provided from the previous NICE guideline,16 with no new studies identified. For guanfacine versus placebo, the recommendation was supported by evidence from the NICE guideline16 and 1 new randomized controlled trial. No new evidence was found for guanfacine versus dexamfetamine. This recommendation was an EBR, which was rated as a strong recommendation but with very low certainty. The AADPA guideline15 also recommends further medication choices as a subsequent line of treatment for ADHD in adults. These medications include antidepressants (e.g., venlafaxine, desvenlafaxine, bupropion, reboxetine, and agomelatine), blood pressure medications (e.g., clonidine), atypical antipsychotics (e.g., aripiprazole), a mood stabilizer (e.g., lamotrigine), and other stimulants (e.g., modafinil and armodafinil). The recommendation on the use of bupropion, clonidine, modafinil, reboxetine, and venlafaxine was an EBR, and these drugs were recommended as third-line treatments with very low certainty of evidence based on the evidence review. The recommendation on the use of the remaining medications as fourth-line treatments (i.e., lamotrigine, aripiprazole, agomelatine, armodafinil, and desvenlafaxine) was a CPP, of which guidance was based on expert opinion and clinical experience of the GDG members. There is no particular order for using these medications. However, the guideline suggests that side effects of these medications should be carefully monitored. The AADPA guideline15 also narratively reviewed the identification and treatment of ADHD in 3 subgroups: people in the correctional system, Aboriginal and Torres Strait Islander peoples, and people with co-occurring substance use disorder. With respect to pharmacological interventions, the guideline warns that stimulants should be offered with caution if there is a risk of diversion for cognitive enhancement. The guideline also recommends against prescribing immediate-release stimulants or modified-release 15/25 Limitations Attention-Deficit/Hyperactivity Disorder Medications for Adults stimulants that can be easily injected or inhaled if there is a risk of stimulant misuse or diversion. These recommendations are CPPs. Limitations Evidence Gaps No evidence was found, therefore no conclusions can be formed, on the research questions related to: • the clinical effectiveness and safety of stimulants compared to nonstimulants for the management of ADHD in adults • the cost-effectiveness of stimulants compared to nonstimulants for the management of ADHD in adults. Generalizability Although we did not identify any Canadian guidelines regarding pharmacological interventions for ADHD in adults in this review, the recommendations in the AADPA guideline15 may be applicable to the adult population with ADHD living in Canada. Certainty of Evidence A limitation in the AADPA guideline15 was that most of the recommendations on the use of stimulants and nonstimulants for treatment of ADHD in adults were supported by evidence from the 2019 NICE guideline,16 without identifying any new evidence thereafter. Evidence on the efficacy and safety of stimulants and nonstimulants was derived mostly from placebo-controlled trials, without head-to-head trials. Although the strength of the EBRs for stimulants and nonstimulants was strong, the certainty of the evidence supporting the recommendation was low or very low, which impacts the confidence, as the true effect may be or is likely to be substantially different from the estimate of the effect. Conclusions and Implications for Decision- or Policy-Making This review included 1 Australian guideline15 that recommends stimulants (i.e., methylphenidate, dexamfetamine, or lisdexamfetamine) as first-line pharmacological treatment of ADHD in adults. If stimulants are not tolerated or ineffective, nonstimulants, such as atomoxetine or guanfacine, should be offered as a second-line treatment. A combination of stimulants and nonstimulants may be prescribed to increase the benefit of the treatment. Other drugs such as bupropion, clonidine, modafinil, reboxetine, and venlafaxine could be offered as third-line treatments. A CPP based on GDG clinical expertise was made regarding fourth-line treatments for adults with ADHD that included lamotrigine, aripiprazole, agomelatine, armodafinil, and desvenlafaxine. These recommendations were made for the general population of adults living with ADHD. There was no evidence on medication options specifically for treating ADHD in people in the correctional system. The 16/25 Conclusions and Implications for Decision- or Policy-Making Attention-Deficit/Hyperactivity Disorder Medications for Adults AADPA guideline15 recommends against prescribing stimulants as first-line therapy to people who have a risk of misuse or diversion. Considerations for Future Research Future research with head-to-head trials is needed to compare the clinical efficacy, safety, and cost-effectiveness of stimulants and nonstimulants for the treatment of ADHD in the general adult population as well as in the subpopulation of people in the correctional setting. Implications for Clinical Practice The findings of this report suggest that stimulants are commonly prescribed as first-line treatment for adult ADHD because of their rapid effect. However, prescribing nonstimulants as first-line therapy for ADHD to people who have a risk of misuse or diversion may be a better or more favourable option. Thus, the choice of medications used in the management of adult ADHD should be based on local resources and individual patients, who may or may not have a risk of abuse and diversion. 17/25 References Attention-Deficit/Hyperactivity Disorder Medications for Adults References 1. Centre for ADHD Awareness, Canada (CADDAC). About ADHD. 2024; caddac​ .ca/​ about​ -adhd/​ . Accessed 2024 Aug 22. 2. Searight HR, Burke JM, Rottnek F. Adult ADHD: evaluation and treatment in family medicine. Am Fam Physician. 2000;62(9):2077-2086. www​ .aafp​ .org/​ pubs/​ afp/​ issues/​ 2000/​ 1101/​ p2077​ .html. Accessed 2024 Aug 22. PubMed 3. ADHD in adults: 4 things to know. Bethesda (MD): National Institute of Mental Health; 2024: www​ .nimh​ .nih​ .gov/​ health/​ publications/​ adhd​ -what​ -you​ -need​ -to​ -know. Accessed 2024 Aug 22. 4. WebMD Editorial Contributor. Adult ADHD: symptoms, causes, treatments. New York (NY): WebMD LLC; 2023: www​ .webmd​ .com/​ add​ -adhd/​ adhd​ -adults. Accessed 2024 Aug 22. 5. Attention-deficit/hyperactivity disorder in adults: what you need to know. Bethesda (MD): National Institute of Mental Health; 2021: infocenter​ .nimh​ .nih​ .gov/​ sites/​ default/​ files/​ 2021​ -12/​ attention​ -deficit​ -hyperactivity​ -disorder​ -in​ -adults​ -what​ -you​ -need​ -to​ -know​ 0​ .pdf. Accessed 2024 Aug 22. 6. Silver L. ADHD medication: stimulants, non-stimulants & more. New York (NY): ADDitude; 2024: www​ .additudemag​ .com/​ adhd​ -medication​ -for​ -adults​ -and​ -children/​ . Accessed 2024 Aug 22. 7. McMillen M, Walker-Journey J, King LM. ADHD Medication. New York (NY): WebMD LLC; 2024: www​ .webmd​ .com/​ add​ -adhd/​ adhd​ -medication​ -chart. Accessed 2024 Aug 22. 8. Frida. A complete guide to ADHD medications for adults. 2023; www​ .talkwithfrida​ .com/​ learn/​ a​ -complete​ -guide​ -to​ -adhd​ -medications​ -for​ -adults/​ . Accessed 2024 Aug 22. 9. Tee-Melegrito RA. Which medication is best for adult ADHD? Brighton (UK): Healthline Media UK Ltd; 2023: www​ .medicalnewstoday​ .com/​ articles/​ best​ -medication​ -for​ -adult​ -adhd. Accessed 2024 Aug 22. 10. Cleveland Clinic. ADHD medications: how they work & side effects. 2022; my​ .clevelandclinic​ .org/​ health/​ treatments/​ 11766​ -adhd​ -medication. Accessed 2024 Aug 22. 11. Caporuscio J. List of ADHD medications: comparing types. Brighton (UK): Healthline Media UK Ltd; 2024: www​ .medicalnewstoday​ .com/​ articles/​ 325201. Accessed 2024 Aug 22. 12. Veronesi GF, Gabellone A, Tomlinson A, Solmi M, Correll CU, Cortese S. Treatments in the pipeline for attention-deficit/ hyperactivity disorder (ADHD) in adults. Neurosci Biobehav Rev. 2024;163:105774. PubMed 13. Agree Next Steps Consortium. The AGREE II Instrument. Hamilton (ON): AGREE Enterprise; 2017: www​ .agreetrust​ .org/​ wp​ -content/​ uploads/​ 2017/​ 12/​ AGREE​ -II​ -Users​ -Manual​ -and​ -23​ -item​ -Instrument​ -2009​ -Update​ -2017​ .pdf. Accessed 2024 Aug 20. 14. Liberati A, Altman DG, Tetzlaff J, et al. The PRISMA statement for reporting systematic reviews and meta-analyses of studies that evaluate health care interventions: explanation and elaboration. J Clin Epidemiol. 2009;62(10):e1-e34. PubMed 15. May T, Birch E, Chaves K, et al. The Australian evidence-based clinical practice guideline for attention deficit hyperactivity disorder. Aust N Z J Psychiatry. 2023;57(8):1101-1116. PubMed 16. National Institute for Health and Care Excellence. Attention deficit hyperactivity disorder: diagnosis and management. (NICE guideline NG87) 2019; www​ .nice​ .org​ .uk/​ guidance/​ ng87. Accessed 2024 Sep 09. 17. ADHD Guideline Development Group. Australian evidence-based clinical practice guideline for Attention Deficit Hyperactivity. 1st ed. Melbourne (AU): Australasian ADHD Professionals Association; 2022: adhdguideline​ .aadpa​ .com​ .au/​ . Accessed 2024 Aug 29. 18. ADHD and the justice system: the benefits of recognizing and treating ADHD in Canadian justice and correction systems. Toronto (ON): CADDAC; 2016: caddac​ .ca/​ wp​ -content/​ uploads/​ ADHD​ -and​ -the​ -Justice​ -system​ ​ -the​ -benefits​ -of​ -recognizing​ -and​ -treating​ -ADHD​ -EN​ .pdf. Accessed 2024 Aug 27. 19. Adult attention deficit/hyperactivity disorder (ADHD) - clinical guidance. Washington (DC): Federal Bureau of Prisons; 2021: www​ .bop​ .gov/​ resources/​ pdfs/​ adult​ adhd​ _cd​ .pdf. Accessed 2024 Aug 26. 18/25 Appendix 1: Selection of Included Studies Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 1: Selection of Included Studies Figure 1: PRISMA Flow Chart of Study Selection 19/25 Appendix 2: Characteristics of Included Publication Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 2: Characteristics of Included Publication Please note that this appendix has not been copy-edited. Table 2: Characteristics of Included Guideline Intended users, target population Intervention and practice considered Major outcomes considered Evidence collection, selection, and synthesis Evidence quality assessment Recommendations development and evaluation Guideline validation AADPA, May et al. (2023)15 Intended users: Clinicians, nurses, pharmacists, and other people involved in the support of people with ADHD, such as educators Target population: People with ADHD and their families, parents and carers, and partners Identification, diagnosis, and treatment of people with ADHD by addressing 50 prioritized clinical questions All relevant outcomes The guideline was based, in part, on the evidence-based NICE UK guidance on the diagnosis and management of ADHD. Systematic search for evidence, study selection, data extraction, and data synthesis were performed. There are 4 key elements of each recommendation: • typea • wordingb • strengthc • certainty (using GRADE)d The GDG consisted of professionals having backgrounds including, but not limited to, psychiatry, pediatrics, psychology, allied health, and ADHD coaching, as well as research into the causes and treatments of ADHD. The guideline was reviewed independently by relevant professional experts, professional colleges and societies, and through public consultation. The guideline was published online and in a peer-reviewed journal. AADPA = Australian ADHD Professionals Association; ADHD = attention-deficit/hyperactivity disorder; GDG = guideline development group; GRADE = Grading of Recommendations, Assessment, Development and Evaluation; NICE = National Institute for Health and Care Excellence. aRecommendation is either an evidence-based recommendation or a clinical consensus recommendation. In addition, clinical practice points were included for implementation issues such as safety, side effects, and risks. bRecommendation wording reflects the GDG’s overall interpretation of the evidence and other considerations. The word should indicates the GDG’s judgment that the benefits of the recommended action exceed the harms. Could indicates that the quality of evidence was limited, or the available studies did not clearly demonstrate advantage of one approach over another, or the balance of benefits to harm was unclear. Should not indicates either a lack of appropriate evidence or that the harms were judged to outweigh the benefits. cThe grade (strength) of evidence-based recommendations (strong recommendation or conditional recommendation) was determined by the GDG based on comprehensive consideration of all elements of the evidence-to-decision framework including desirable and undesirable effects, balance of effects, resource requirements, equity, acceptability, and feasibility. These include: strong recommendation for the option, conditional recommendation for the option, conditional recommendation for either the option or the comparator, and conditional recommendation against the option. dCertainty of evidence (very low to high) for evidence-based recommendations reflects the quality and relevance of the evidence, based on information about the number and design of studies addressing the outcome, judgments about the quality of the studies and/or synthesized evidence across risk of bias, inconsistency, indirectness, imprecision, and any other quality considerations; key statistical data; and classification of importance of outcomes. High means we are very confident that the true effect lies close to that of the estimate of the effect. Moderate means we are moderately confident in the effect estimate: The true effect is likely to be close to the estimate of the effect, but there is a possibility that it is substantially different. Low means our confidence in the effect estimate is limited: The true effect may be substantially different from the estimate of the effect. Very low means we have very little confidence in the effect estimate: The true effect is likely to be substantially different from the estimate of effect. 20/25 Appendix 3: Critical Appraisal of Included Publication Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 3: Critical Appraisal of Included Publication Please note that this appendix has not been copy-edited. Table 3: Strengths and Limitations of Guideline Using AGREE II13 Item AADPA, May et al. (2023)15 Domain 1: Scope and purpose 1. The overall objective(s) of the guideline is (are) specifically described. Yes 2. The health question(s) covered by the guideline is (are) specifically described. Yes 3. The population (patients, public, etc.) to whom the guideline is meant to apply is specifically described. Yes Domain 2: Involvement of interested parties 4. The guideline development group includes individuals from all relevant professional groups. Yes 5. The views and preferences of the target population (patients, public, etc.) have been sought. Yes 6. The target users of the guideline are clearly defined. Yes Domain 3: Rigour of development 7. Systematic methods were used to search for evidence. Yes 8. The criteria for selecting the evidence are clearly described. Yes 9. The strengths and limitations of the body of evidence are clearly described. Yes 10. The methods for formulating the recommendations are clearly described. Yes 11. The health benefits, side effects, and risks have been considered in formulating the recommendations. Yes 12. There is an explicit link between the recommendations and the supporting evidence. Yes 13. The guideline has been externally reviewed by experts before its publication. Yes 14. A procedure for updating the guideline is provided. Yes (after 5 years) Domain 4: Clarity of presentation 15. The recommendations are specific and unambiguous. Yes 16. The different options for management of the condition or health issue are clearly presented. Yes 17. Key recommendations are easily identifiable. Yes Domain 5: Applicability 18. The guideline describes facilitators and barriers to its application. Unclear 19. The guideline provides advice and/or tools on how the recommendations can be put into practice. Unclear 20. The potential resource implications of applying the recommendations have been considered. Unclear 21. The guideline presents monitoring and/or auditing criteria. Unclear Domain 6: Editorial independence 22. The views of the funding body have not influenced the content of the guideline. Yes 23. Competing interests of guideline development group members have been recorded and addressed. Yes AADPA = Australian ADHD Professionals Association; AGREE II = Appraisal of Guidelines for Research and Evaluation II. 21/25 Appendix 4: Main Study Findings Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 4: Main Study Findings Please note that this appendix has not been copy-edited. Table 4: Summary of Recommendations in Included Guideline15,17 Recommendations and supporting evidence Quality of evidence and strength of recommendations AADPA, May et al. (2023)15,17 Medication choice for adults (aged 18 years and older) “Methylphenidate or dexamfetamine or lisdexamfetamine should be offered as the first-line pharmacological treatment for people with ADHD, where ADHD symptoms are causing significant impairment.”15 (p. 1107) Type: EBR Strength: Strong Certainty: Low Supporting evidence: No new evidence was found. Evidence was from previous NICE guideline.16 — “Atomoxetine or guanfacine should be offered to adults with ADHD if any of the following apply: • Stimulants are contraindicated. • They cannot tolerate methylphenidate, lisdexamfetamine, or dexamfetamine. • Their symptoms have not responded to separate trials of dexamfetamine or lisdexamfetamine and of methylphenidate, at adequate doses. • The clinician considers that the medications may be beneficial as an adjunct to the current regimen. Due consideration of risks and safety is required, especially if medications are used in combination.”15 (p. 1107) Type: EBR Strength: Strong Certainty: Very low Supporting evidence: • For atomoxetine vs. placebo: No new evidence was found. Evidence was from previous NICE guideline. • For guanfacine vs. placebo: New evidence was found in 1 RCT, which reported outcomes as least squares mean difference. There were statistically significant benefits of extended-release guanfacine over placebo for ADHD total, inattention, and hyperactivity symptoms; executive functioning (BRIEF) for inhibit, initiate, and plan/organize, and Global Executive Composite index (investigator-rated; 1 RCT with moderate-certainty evidence). There was a statistically significant benefit of placebo over extended-release guanfacine for quality of life (productivity). • For guanfacine vs. dexamfetamine: No new evidence was found. Evidence was from previous NICE guideline.16 Further medication choices “The following could be offered to adults with ADHD, in no particular order: • bupropion • clonidine • modafinil • reboxetine • venlafaxine Careful monitoring of adverse side effects is required.”15 (p. 1107) Type: EBR Strength: Strong Certainty: Very low Supporting evidence: NR — 22/25 Appendix 4: Main Study Findings Attention-Deficit/Hyperactivity Disorder Medications for Adults Recommendations and supporting evidence Quality of evidence and strength of recommendations “The following could also be offered to adults with ADHD, in no particular order: • lamotrigine • aripiprazole • agomelatine • armodafinil • desvenlafaxine Careful monitoring of adverse side effects is required.”17 (p. 139) Type: CPP Strength: NA Certainty: NA Supporting evidence: NA — Factors influencing medication choices “Clinicians should exercise caution when prescribing stimulants if there is a risk of diversion for cognitive enhancement.”17 (p. 140) Type: CPP Strength: NA Certainty: NA “Clinicians should not offer immediate-release stimulants or modified-release stimulants that can be easily injected or inhaled if there is a risk of stimulant misuse or diversion.”17 (p. 140) Type: CPP Strength: NA Certainty: NA Supporting evidence: NA — AADPA = Australian ADHD Professionals Association; ADHD = attention-deficit/hyperactivity disorder; CPP = clinical practice point; EBR = evidence-based recommendation; NA = not applicable; NICE = National Institute for Health and Care Excellence; NR = not reported; RCT = randomized controlled trial; vs. = versus. 23/25 Appendix 5: References of Potential Interest Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 5: References of Potential Interest Please note that this appendix has not been copy-edited. Randomized Controlled Trials Asherson PJ, Johansson L, Holland R, et al. Randomised controlled trial of the short-term effects of osmotic-release oral system methylphenidate on symptoms and behavioural outcomes in young male prisoners with attention deficit hyperactivity disorder: CIAO-II study. Br J Psychiatry. 2023 01;222(1):7-17. PubMed Clinical Practice Guidelines Adult attention deficit/hyperactivity disorder (ADHD) - clinical guidance. Washington (DC): Federal Bureau of Prisons; 2021: www​ .bop​ .gov/​ resources/​ pdfs/​ adult​ _adhd​ _cd​ .pdf. Accessed 2024 Aug 26. Canadian ADHD Resource Alliance. Canadian ADHD practice guidelines. 4.1 ed. Toronto (ON): CADDRA; 2020: adhdlearn​ .caddra​ .ca/​ wp​ -content/​ uploads/​ 2022/​ 08/​ Canadian​ -ADHD​ -Practice​ -Guidelines​ -4​ .1​ -January​ -6​ -2021​ .pdf. Accessed 2024 Oct 07. Additional References Tully J. Management of ADHD in prisoners-evidence gaps and reasons for caution. Front Psychiatry. 2022;13:771525. PubMed Grimley A, Bartels L. The need for speed? Exploring the risks and benefits of pharmacological treatment for adult ADHD in prisons. Psychiatr Psychol Law. 2024:1-21. ADHD and the justice system: the benefits of recognizing and treating ADHD in Canadian justice and correction systems. Toronto (ON): CADDAC; 2016: caddac​ .ca/​ wp​ -content/​ uploads/​ ADHD​ -and​ -the​ -Justice​ -system​ ​ -the​ -benefits​ -of​ -recognizing​ -and​ -treating​ -ADHD​ -EN​ .pdf. Accessed 2024 Aug 27. 24/25 Appendix 6: Summary of Recommendations From Non-Evidence-Based Clinical Guidance Reports Attention-Deficit/Hyperactivity Disorder Medications for Adults Appendix 6: Summary of Recommendations From Non-Evidence-Based Clinical Guidance Reports Given the fact that there is interest in the choice of medication for treatment of ADHD in people in correctional systems, it is noteworthy to highlight the pharmacological treatment protocols from the 2 papers,18,19 which were initially excluded from the review as they are not evidence-based guidelines. The first paper was developed by the Centre for ADHD Awareness, Canada (CADDAC) for recognizing and treating ADHD within Canadian justice and correction systems.18 The second paper was a US Federal Bureau of Prisons (BOP) Clinical Guidance,19 which provides recommendations for diagnosis and management of ADHD in adult patients in BOP custody. The CADDAC paper18 recommends: • Nonstimulants such as atomoxetine, venlafaxine, or bupropion should be used as first-line treatment for people with ADHD in correction systems. • Stimulants such as lisdexamfetamine or osmotic-release oral system methylphenidate should be used as second-line if they are included in the Corrections Canada formulary, and they should be taken under supervision. • Stimulants should be given as liquid or soluble formulations to reduce the risk of diversion. The Federal BOP clinical guidance19 recommends: • Concurrent treatment of ADHD with both medication therapy and psychotherapy should be prescribed, rather than sequential treatment, for adult patients in prisons. • As stimulants present many difficulties in the correctional setting because of the risk of abuse and diversion, nonstimulants (e.g., atomoxetine, guanfacine, clonidine, desipramine, bupropion) should be prescribed as first-line treatment for adults with ADHD in BOP facilities, while prescribing a stimulant should be withheld until failure with a nonstimulant. cda-amc.ca ISSN: 2563-6596 Canada’s Drug Agency (CDA-AMC) is a pan-Canadian health organization. Created and funded by Canada’s federal, provincial, and territorial governments, we’re responsible for driving better coordination, alignment, and public value within Canada’s drug and health technology landscape. We provide Canada’s health system leaders with independent evidence and advice so they can make informed drug, health technology, and health system decisions, and we collaborate with national and international partners to enhance our collective impact. Disclaimer: CDA-AMC has taken care to ensure that the information in this document was accurate, complete, and up to date when it was published, but does not make any guarantee to that effect. Your use of this information is subject to this disclaimer and the Terms of Use at cda-amc.ca. The information in this document is made available for informational and educational purposes only and should not be used as a substitute for professional medical advice, the application of clinical judgment in respect of the care of a particular patient, or other professional judgments in any decision-making process. You assume full responsibility for the use of the information and rely on it at your own risk. CDA-AMC does not endorse any information, drugs, therapies, treatments, products, processes, or services. The views and opinions of third parties published in this document do not necessarily reflect those of CDA-AMC. The copyright and other intellectual property rights in this document are owned by the Canadian Agency for Drugs and Technologies in Health (operating as CDA-AMC) and its licensors. Questions or requests for information about this report can be directed to Requests@​ CDA​ -AMC​ .ca.
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https://math.stackexchange.com/questions/1518564/are-there-any-non-trivial-automorphisms-on-the-natural-numbers-under-addition
abstract algebra - Are there any non-trivial automorphisms on the Natural Numbers under addition? - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Are there any non-trivial automorphisms on the Natural Numbers under addition? Ask Question Asked 9 years, 10 months ago Modified9 years, 10 months ago Viewed 1k times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. I'm curious about whether or not there is an automorphism on (N;+)(N;+) that isn't the identity. I suspect there isn't, but I'm not quite sure how to prove it. abstract-algebra monoid Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications asked Nov 8, 2015 at 7:52 user279406 user279406 Add a comment| 1 Answer 1 Sorted by: Reset to default This answer is useful 9 Save this answer. Show activity on this post. If f:N→N f:N→N is any monoid-homomorphism, let a=f(1)a=f(1). Then f(2)=f(1+1)=f(1)+f(1)=2 a f(2)=f(1+1)=f(1)+f(1)=2 a, f(3)=f(2+1)=f(2)+f(1)=2 a+a=3 a f(3)=f(2+1)=f(2)+f(1)=2 a+a=3 a, and so on. More precisely, we can prove by induction that for each n∈N n∈N, f(n)=a n f(n)=a n (it is a good exercise to work out this induction if you don't immediately see how it works). If f f is an automorphism, then it must be surjective, so every element of N N is equal to a n a n for some n n. That is, every natural number is divisible by a a. This is only possible if a=1 a=1, and so f(n)=a n=n f(n)=a n=n and f f is the identity. Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Nov 8, 2015 at 8:02 Eric WofseyEric Wofsey 343k 28 28 gold badges 485 485 silver badges 701 701 bronze badges Add a comment| You must log in to answer this question. Featured on Meta Introducing a new proactive anti-spam measure Spevacus has joined us as a Community Manager stackoverflow.ai - rebuilt for attribution Community Asks Sprint Announcement - September 2025 Report this ad Related 1There is an automorphism of Z 6 Z 6 which is not an inner automorphism 0Let S={a,b}S={a,b}. Give the tables for all possible binary operations on S S. (There are 16). 1What are the non-trivial normal subgroups of O(3)O(3)? 0Trouble showing the addition of the fractional parts are a group. 0Determine whether the indicated set forms a ring: S={A∈M(2,R)∣det A=0}S={A∈M(2,R)∣det A=0}, under matrix addition and multiplication. 2Proof that there are only two automorphisms of Q(d−−√)Q(d) fixing Q Q 1Are there any non-trivial automorphisms on (N,×)(N,×) 2Are there non-trivial endomorphisms of the Matrix algebra that are not inner automorphisms? Hot Network Questions With line sustain pedal markings, do I release the pedal at the beginning or end of the last note? What NBA rule caused officials to reset the game clock to 0.3 seconds when a spectator caught the ball with 0.1 seconds left? The geologic realities of a massive well out at Sea Explain answers to Scientific American crossword clues "Éclair filling" and "Sneaky Coward" How to home-make rubber feet stoppers for table legs? 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https://m.youtube.com/watch?v=YzrL8vb4caU&pp=0gcJCa0JAYcqIYzv
Conic Sections: An Introduction to Parabolas MathGH 872 subscribers 1 likes Description 53 views Posted: 12 May 2016 For more videos and practice problems check out my Pre-calculus course on Udemy: Transcript: alright so today we're going to do our first video on conex it's going to start with the parabola introduction so these forms that you see on the left and right side and bold those are the two common for most for a private parabola on the left we have what is called vertex form and that is the form y equals a parenthesis X minus 8 h squared plus K and then that opens up if a is greater than zero and it opens down if a is less than zero that means we fix my pin real quick so that means that if a is greater than 0 it's going to go like that and if a is less than zero it's going to be going down and then vertex is at that point HK notice that the kit that H in the formula is subtracted so just be careful with that and then we have on the right we have standard form and that takes the form y equals ax squared plus BX plus C then that also follows the same rule that if a is greater than 0 it's going to open up and if a is less than zero or do we go oping down then the x value of the vertex what is negative B over 2a and that's the hard part about standard form is finding the vertex so if we have this formula for an x value the way we find a y-value is plugging it back in and solving for the y and there's another way to find other vertex for standard form that will go over so here we have a problem we have y equals x parenthesis X minus 3 squared plus 1 and that's obviously in vertex form so let's rewrite a vertex form which is y equals a X minus H squared plus K and if I just wanted I want the vertex and I want which way it's going to open up so we can draw a quick sketch of it so first we have a with a equals 2 which is well it's greater than 0 so that tells me well this graph is going to open up and then we also have an H which is three be careful not to write negative 3 because the standard is negative so our experts our H value as a whole is positive so H equals 3 then our K value is 1 and remember that our vertex is at Point HK so in this case our vertex equals 3 1 so now if we were to draw a quick graph we've got one two three and we've got one that's where our vertex is and then we can use the fact that a is greater than zero so we know that it's going to be opening this way it's going to be opening up on both sides that's not a very good problem but you get the point it's opening up in the vertex is at 31 so that's just using vertex form notice I using vertex form makes it very fast to find the vertex and if it's opening up or down because everything is just written out in the formula and now we get to the tougher problems which are when it starts in standard form there are two ways to solve it I'll go through one way first which is not going to be completing the square and there's another way that you can't complete the square so now let's remember our formula which is y equals ax squared plus BX plus C so now well got a equals 2 and that is also greater than 0 which means this graph is going to be going up then now the hard part is finding the vertex and remember from this top part verte x value of the vertex is negative b over 2a so we can take negative b x equals negative B over 2a and our B value in this equation it's going to be what is the coefficient in front of X which in this case is 4 so B is for but remember it's in the formula calls for negative B so it's going to be negative 4 over 2 times a which we decided earlier was 2 which equals negative 4 over 4 which then applies to just negative 1 so we know that x value of the vertex now is negative 1 and so now to find the Y value we have to plug negative 1 back into our equation which is y equals 2 times negative 1 squared plus 4 times negative 1 plus 1 which equals remember that negative 1 squared is just positive one so that'll be 2 plus and then 4 times negative 1 is negative 4 is one and we'll get to minus 4 is negative 2 plus 1 is negative 1 so that's our Y value at the vertex so if our Y value is negative 1 and our X values negative 1 we have a vertex at negative 1 negative 1 so that using this way we would get a graph that looks like this right here we got negative 1 we got negative 1 and then we set a 0 is positive so it's going to be going up and going up and then also another way to do it that your teacher might have showed you is completing the square to get it into from standard form back into this vertex form that we have y equals ax minus H squared plus K so we can do that I personally think this way is a lot tougher but it is another option so let's try to complete square so we'll set it up y equals first step in completing the square let's group everything together group our exes together I mean so where X plus 1 and then first we have to make sure that whatever is in front of x squared has to equal one so to do that we have to take out a 2 so y equals 2 then we're going to be left with x squared plus we have we have to take out two of this part as well which is it going to leave 2x then I'm going to have to add some value that will decide and then plus 1 and then if we add a value to this side remember we have to add a value to this side have to be fair so then the way we find out what that value is we're going to get y equals to x remember we take one half of that that value so one-half of 2 equals 1 so we X plus 1 squared and remember to find with that what value we add we do on this value when we square it which is going to give us one then remember we have to add if we added it to the right side we have to add it to the left but also notice that really we have added 2 times x squared plus 2 to X plus 1 we've added to x squared plus 4x which is just like the original problem but then we've also added plus 2 if we distribute that out so we can't just add one to the other side we have to add two to the other side so we have 2 plus y equals let me rewrite this where I have more space y plus 2 equals 2x plus 1 squared then we still have our plus one on the outside from here plus one so then now we're almost back into this vertex form we want the vertex form remember is this so the problem that we have right now is we have a number on the Y sighs we just need to subtract that to the other side so let's take this to subtract x 2 and we're going to be left with y equals 2x plus 1 squared minus 1 and now this is the vertex form that we had originally so now we can just solve for this remember we have a equals 2 again so it opens up then we have h we're looking for the values of H and K and now remember that K is positive now in this one but remember that the formula says that y equals a x minus h plus k so to make that value positive the value of h after you had to be negative because it would have been like x minus a negative 1 which just turns into x plus 1 so that's the case we have here h equals negative 1 and then our Y value FK value of meat is just whatever is on the end so that's still negative 1 negative 1 so again our graph here after completing the square and everything we would get negative 1 negative 1 is our vertex and then again our a value is 2 so it's opening up something like this then notice that's the same thing we had here it's the same equation our vertex is still on negative 11 and over here it's negative 11 and a equal to in both cases yeah it goes to right here so just notice that either way you solve it is the same solution so you there's the first option is to just use the formula x equals negative B over 2a x equals negative B over 2a and this second option is to complete the square or complete the square and solve it like you had it in vertex form either way will give you the same answer it's just a matter of preference which one do you like better which one is quicker for you personally I think using the x equals negative B over 2a and then plug that value in is a lot faster but you can do whatever you're comfortable with thanks for watching
14070
https://fiveable.me/key-terms/intro-chem/mass-volume-percentage
printables 💏intro to chemistry review key term - Mass-Volume Percentage Citation: MLA Definition Mass-volume percentage, also known as mass/volume percentage or m/v%, is a unit of concentration that expresses the mass of a solute dissolved in a given volume of solution. It represents the ratio of the mass of the solute to the total volume of the solution, typically expressed as a percentage. 5 Must Know Facts For Your Next Test Mass-volume percentage is commonly used to express the concentration of solutions in chemistry and pharmaceutical applications. To calculate the mass-volume percentage, the mass of the solute is divided by the total volume of the solution and then multiplied by 100 to obtain the percentage. Mass-volume percentage is particularly useful when the density of the solution is not known or when the solute is a solid that is dissolved in a liquid solvent. Unlike molarity, which expresses the number of moles of solute per unit volume of solution, mass-volume percentage does not require knowledge of the molar mass of the solute. Mass-volume percentage can be used to prepare solutions of desired concentrations by dissolving a specific mass of solute in a known volume of solvent. Review Questions Explain the relationship between mass-volume percentage and the concentration of a solution. The mass-volume percentage directly reflects the concentration of a solution, as it represents the ratio of the mass of the solute to the total volume of the solution. A higher mass-volume percentage indicates a more concentrated solution, as there is a greater amount of solute dissolved in the same volume of the solution. Conversely, a lower mass-volume percentage corresponds to a more dilute solution, where the solute is present in a smaller proportion relative to the total volume. Describe the advantages of using mass-volume percentage over other concentration units, such as molarity, in certain situations. One advantage of using mass-volume percentage is that it does not require knowledge of the molar mass of the solute, which is necessary for calculating molarity. This makes mass-volume percentage particularly useful when the solute is a solid that is dissolved in a liquid solvent, or when the density of the solution is not known. Additionally, mass-volume percentage is a more intuitive unit for expressing the concentration of solutions, as it directly relates the mass of the solute to the total volume of the solution, which can be more relevant in practical applications such as pharmaceutical formulations. Explain how mass-volume percentage can be used to prepare solutions with a desired concentration. To prepare a solution with a specific mass-volume percentage, one can dissolve a known mass of the solute in a calculated volume of the solvent. The desired mass-volume percentage can be achieved by dividing the mass of the solute by the total volume of the solution and multiplying by 100. This approach allows for the precise preparation of solutions with a target concentration, which is important in various applications, such as in the formulation of pharmaceutical products or the preparation of stock solutions for analytical procedures. Related terms Solute: The substance dissolved in a solution, which is present in a lower amount compared to the solvent. Solvent: The substance in which the solute is dissolved, typically present in a larger amount compared to the solute. Concentration: The measure of the amount of a substance present in a given volume or mass of a solution.
14071
https://rechneronline.de/pi/square-frustum.php
| | | | --- | Geometry | Forms | Contact & Privacy | Geometric Calculators | German: Geometrierechner, Formen | | | | --- | | | Anzeige Square Frustum Calculator Calculations at a square frustum, a special case of a regular frustum with four base corners. This is a square pyramid with the apex truncated parallel to the base. Enter both side lengths a and b and height h. Choose the number of decimal places, then click Calculate. Formulas: s = √ 1/4 ( a - b )² + h² e = √ [ 4 s² + ( a - b )² ] / 4 L = ( a + b ) √ ( a - b )² + 4h² A = L + ( a² + b² ) V = h/3 [ ( a² + b² ) + √ a² b² ] Lengths and heights have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit -1. The square frustum forms the base of the bent pyramid and the obelisk. Two identical square truncated pyramids, put together at their base, form a square bifrustum, i.e. one with four base corners. A similar body made up of a truncated pyramid and a pyramid is a frustum pyramid. A square frustum has four planes of symmetry. Two pass through the centers of the four opposite parallel sides, the other two through the four opposite corners and through the two opposite steep edges. Like its square base and its opposite smaller square side, it is rotationally symmetrical at an angle of 90 degrees and multiples thereof to the axis through the centers of the two parallel square surfaces. The term frustum generally refers to a shape between two parallel planes. In a square frustum, the two opposite sides are squares of different sizes, whose centers are connected by a straight line perpendicular to these sides or planes. © Jumk.de Webprojects | Online Calculators Anzeige | Anzeige ↑ up
14072
https://www.chegg.com/homework-help/questions-and-answers/calculate-molar-absorptivity-sample-s-absorbance-078-concentration-sample-600x-10-6-m-leng-q77699967
Your solution’s ready to go! Our expert help has broken down your problem into an easy-to-learn solution you can count on. Question: Calculate the molar absorptivity if a sample’s absorbance is 0.78, the concentration of the sample is 6.00x 10^-6 M, and the length of the sample the light travels through is 1.00 cm. What are the units? Calculate the molar absorptivity if a sample’s absorbance is 0.78, the concentration of the sample is 6.00x 10^-6 M, and the length of the sample the light travels through is 1.00 cm. What are the units? This AI-generated tip is based on Chegg's full solution. Sign up to see more! Use the Beer-Lambert Law, , to set up the equation for molar absorptivity (ε) by substituting the given values for absorbance (), path length (), and concentration (). Result - Mola… Not the question you’re looking for? Post any question and get expert help quickly. Chegg Products & Services CompanyCompany Company Chegg NetworkChegg Network Chegg Network Customer ServiceCustomer Service Customer Service EducatorsEducators Educators
14073
https://www.youtube.com/watch?v=DhyYtT1K844&pp=ygUII3BobG9lbXM%3D
Xylem and Phloem - Transport in Plants | Biology | FreeAnimatedEducation Free Animated Education 146000 subscribers 10233 likes Description 688227 views Posted: 30 May 2021 Xylem and Phloem are explained in detail and their role in transport in plants is also explained in detail. Support our channel and also get access to awesome perks: Help us making Education Universal: Support us on Patreon: 🚨 OTHER LINKS: Follow our Social Media: YouTube: Facebook: 🚨 HASHTAGS: FreeAnimatedEducation #FAE #Xylem #Phloem Timestamp: 00:00 Transportation in plants 00:39 Vascular tissues in plant transportation system 01:03 Xylem 01:18 Structures of xylem and their functions 02:00 Floem 02:12 Structures of floem and their functions 03:11 Arrangement of vascular tissues 03:17 The arrangement in root 03:25 The arrangement in stem 03:32 The arrangement in leave 223 comments Transcript: Transportation in plants Xylem and Phloem - Transport in Plants Transport in Plants Every cell in our body needs many different substances to function properly. For example, glucose and oxygen. But the cells cannot obtain the substances directly from outside. Therefore, they rely on the circulatory system to carry these substances from the body part that provides them, like lungs and small intestine, to all the body cells in need. In plants, vital substances like water and minerals are provided by the soil and are absorbed via the root, while the glucose is produced in the leaf during photosynthesis. The vascular tissue allows the substances Vascular tissues in plant transportation system to move from certain parts of the plant to the others. The two main vascular tissues in plants are the xylem and phloem. If you've ever seen a fallen tree and take a closer look inside, almost all of its radial contents, consisting of heartwood and sapwood, are actually xylem tissues and the inner part of its bark is the phloem. Xylem Xylem Xylem is responsible for distributing water and minerals taken by the roots. An interesting fact about xylem is that some parts of it do not have protoplasms or cell walls, allowing water and minerals to pass easily. Xylem consists of tracheids or conducting cells, Structures of xylem and their functions vessels, fiber, and parenchyma. The water in tracheids does not flow continuously, the roots and leaves are connected through a series of tracheid cells with pitted walls. In contrast, the xylem vessels are long, wider, hollow tubes continuously stretching from the roots to the leaves. The fiber cells function mainly as supporting structures, and the parenchyma, the only living cells of the xylem, help with the food storage. The xylem cells are strengthened by a substance called lignin. The pattern of lignin will vary depending on the location. It can be a ring pattern, spiral, reticulate, or pitted. Phloem Floem The phloem transports manufactured food like sucrose and amino acids from the green parts of the plant like leaves, to other parts of the plant. The process is called translocation. Phloem consists of sieve tubes, Structures of floem and their functions companion cells, fiber, and parenchyma. A sieve tube consists of a single row of elongated, thin-walled cells. Just like the name suggests, it has walls perforated by minute pores like a sieve. A mature sieve cell has a thin layer of cytoplasm which conducts manufactured food from one cell to another. The sieve tube cell has a degenerated protoplasm which means that it needs another to carry out its vital processes. This is why a sieve tube cell has a companion cell. The companion cell has everything that the sieve tube doesn’t, such as abundant cytoplasm and nucleus. It has an extremely thin but flexible cell wall so that there’s no hassle in keeping the sieve tube cell alive. Just like in the xylem, the fiber cell helps to give structural support, and the parenchyma stores the food and other substances. In phloem, the fiber cells are the only type of cell that is dead. Arrangement of Vascular Tissue Arrangement of vascular tissues The vascular tissue has different arrangements in various parts of the plants. In the root of herbaceous dicot plants, The arrangement in root the xylem forms an x-like shape in the middle, while phloem fills around the xylem. The xylem and phloem in the stem The arrangement in stem are clustered into circular shapes. These clusters are located near the edge of the stem. While in leaf, The arrangement in leave the xylem and phloem are located in the vascular bundle with the xylem positioned above the phloem.
14074
https://www.maconbibb.us/wp-content/uploads/2024/11/25-020-LH-Addendum-No.-2-Macon-FBO-Final-Addendum-2024-12-4.pdf
ADDENDUM NO. 2 (FINAL ADDENDUM) Corporate FBO Terminal Building and Parking Lot Rehabilitation Located at 2178 Flightline Ave, Macon, GA 31216 Thursday, December 5, 2024 Architect of Record: Passero Associates, LLC 355 S. Legacy Trail, Suite B-102 St. Augustine, FL 32092 (904) 224-7082 Christopher Nardone, AIA cnardone@passero.com Passero Associates Project No. 20202946.010A ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 The following items are clarifications, corrections, or additions to the contract documents. THIS ADDENDUM TAKES PRECEDENCE OVER THE ORIGINAL PARTS OF THE CONTRACT DOCUMENTS. All the parts of the contract documents, not specifically modified by this or other addenda, remain in full force and effect. Bidders shall thoroughly familiarize themselves with the contents of this Addendum before submitting bid proposals. IT SHALL BE THE BIDDER'S RESPONSIBILITY TO INFORM THE SUBCONTRACTORS, SUPPLIERS, MANUFACTURERS AND OTHER PARTIES PARTICIPATING IN THE WORK OF APPLICABLE REQUIREMENTS IN THIS ADDENDUM. Bidders shall acknowledge receipt of this addendum, identified by number and date, on the Addenda Receipt form included in the Proposal Section of the Contract Documents and submitted as part of their Proposal. Failure to acknowledge receipt of Addendum may be grounds for rejection of the bid proposal. _____________ Items amended to the Contract Documents are as follows: BID SCHEDULE REVISIONS NOTE: The following schedule revisions supersede any and all other mentions pertaining to this information provided in the Bid Documents. 1. The deadline for Submission of Bids has been moved to Thursday, December 19, 2024 at 12:00 pm EST. The Bid Opening will occur at 2:00 pm EST. 2. The Notice of Award will be presented in January - February 2025. 3. The Notice to Proceed will be presented in March – April 2025, no later than April 18, 2025. EVALUATION OF BIDS: Bids may be held by the Owner for purposes of review and evaluation by the Owner for a period not to exceed 120 calendar days from the stated date for receipt of bids. The Owner will tabulate all bids and verify proper extension of unit costs. The Bidder shall honor their bid for the duration of this period of review and evaluation. The bid guaranty will be held by the Owner until this period of review has expired or a contract has been formally executed. BID FORMS 1. REPLACE BID FORMS PD-4 – PD 6 with ADDENDUM NO. 2 BID FORMS PD-4 – PD 6 2. REPLACE BID FORMS BUILDING BASE BID – CORPORATE FBO TERMINAL BUILDING PD-7 – PD 8 with ADDENDUM NO. 2 BID FORMS BUILDING BASE BID – CORPORATE FBO TERMINAL BUILDING PD-7 – PD 8 3. REPLACE BID FORM BID SUMMARY PD-9 with ADDENDUM NO. 2 BID FORM BID SUMMARY PD-9 ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 4. REPLACE technical specification SECTION 01 23 00 – ALTERNATES with ADDENDUM NO. 2 SECTION 01 23 00 – ALTERNATES 5. REPLACE technical specification SECTION 01 30 00- ADMINISTRATIVE REQUIREMENTS with ADDENDUM NO. 2 SECTION 01 30 00- ADMINISTRATIVE REQUIREMENTS. Note the addition of section 3.08 “Real-time and Timelapse Footage of Construction Progress.” 6. ADD GEOTECHNICAL ENGINEERING REPORT. 7. ADD SITE AND UTILITY SURVEY. 8. ADD TOPOGRAPHIC SURVEY. BIDDERS QUESTIONS AND ANSWERS Q1: On Sheet A-604, the general notes call for Marshall Best Locks, however the hardware sets call for Schlage. Please advise as to which locks should be used. A1: Refer to ADDENDUM 1.1 SHEET A-604 for door hardware. Q2: Any chance the architect has access to a 3D model of the structural and misc. steel? A2: We provide our model after the contractor/fabricator is selected. Note that models are not perfect and should be used for information only. Any assumption of geometric accuracy in the model is at your own risk. Q3: Please confirm If the header curbs below is what is intent to price and not curb and gutters (not curb and gutters details on the drawings). A3: Correct. Price header curbs. Curb and gutters do not need to be priced out. Q4: Please provide the reinforcement specifications for the 12” elevator pad? A4: On sheet S101, detail 8, the reinforcement specification should read, “12” Elevator Pit Slab R/W #6@12 T&B.” ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 Q5: Will we be responsible for any type of security systems in our bid? Items such as cameras, CCTV, Access Control systems, at the moment the only low voltage item shown is the fire alarm systems. A5: The listed items fall under the category of “owner provided” and will be outside the scope of this contract. Please note that you shall provide one ¾” metal conduit with pull string from IT closet to each access point of entry with exterior swing door(s) (1st and 2nd floors). Conduit to be flush with ceiling and terminate above access point. Do not bring conduit down wall. Q6: Will a AOR system be needed since this is a two-story building? If this is needed, please provide a spec on the AOR system. A6: AORs are not required per Georgia Building Code (IBC 2018 with Georgia Amendments) Chapter 10 Section 1009.3.3 Area of Refuge Exception No. 2: Areas of refuge are not required at stairways in buildings equipped throughout with an automatic sprinkler system installed in accordance with Section 903.3.1.1 or 903.3.1.2. Q7: In Addendum 1.1 the door hardware is changed however note 7 states “Marshall Best keying systems” if this is what is to be used, what type of core will be need? LFIC, SFIC, 6pin standard core and will they be compatible? A7: Refer to ADDENDUM 1.1 SHEET A-604 for door hardware. Q8: S305 detail 3 option A shows the concrete directly on top of the CLT panel is this detail correct? A8: Yes, the concrete slab is over the corridor – the CLT will act as formwork in this location and allow us to avoid shoring the slab. Q9: Can we price NanaWall for the bi-folding door system? A9: Per contract document ITB-9, all substitution requests will be reviewed and considered by the architect on an “as-equal” basis after the contract is awarded and in accordance with section 01 25 00 – SUBSTITUTION PROCEDURES and section 01 60 00 – PRODUCT REQUIREMENTS. Q10: Can we price curtain wall and Storefront with YKK? A10: Per contract document ITB-9, all substitution requests will be reviewed and considered by the architect on an “as-equal” basis after the contract is awarded and in accordance with section 01 25 00 – SUBSTITUTION PROCEDURES and section 01 60 00 – PRODUCT REQUIREMENTS. Q11: On the Curtain wall spec page 3 under basis of design it calls out Wind-Borne-Debris resistance but the system called out is just standard curtain wall and also the glass called out is non-impact. The storefront system is not called out for this either. Please advise. A11: Macon is not in a “wind-borne debris region” and hence this requirement is not necessary. ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 Q12: With the curtain wall leaning out we will have to use laminated IG units for safety. Just want you to be aware of that. A12: Noted. Under Technical Specification Section 08 80 00 Glazing, 2.03 Insulated Glass Units, note C.3 should read “Outboard Lite: Laminated, 1/4 inch thick, minimum.” And note C.4 should read, “Inboard Lite: Laminated, 1/4 inch thick, minimum.” Q13: Can you guys please provide a specification on the illuminated glass handrails? Also, electrical specifications say to refer to drawings for specifications but there aren't any on the drawings. A13: Refer to Technical Specification Section 05 73 15 Frameless Glazed Metal Railings. Refer to Addendum 1.1, Section “DRAWINGS – MEP” note 5. Q14: Sheet E-201 1st Floor Plan Detail…Please label the tape lighting at rear of guitar. A14: These lights shall be fixture “NSB” and controlled as described in MEP Addendum 1.1 item 2.b. Q15: With the scaling not included we are not to scale the drawings. The angle is an unknown, and we are limited to 15 degrees …. we’ll need to know the angle. And further, (1) the specs call for 1600 system 2 SSG, which is generally not allowed in sloped outward systems without PE, Glass, AND sealant supplier approval; (2) The curtain wall not only slopes outward but also angled itself on the jambs. In SSG this is even more trouble for both the sealant and may require a custom connection in the corners. Would you consider using 1600 system 1 (fully captured)? A15: Yes, note that the basis of design shall be changed to 1600 system 1 (fully captured) in Technical Specification Section 08 44 13 Glazed Aluminum Curtain Walls. Q16: The landscaping maintenance notes seem they need to be by the owner please see below and advise: A16: Note 2 pertains to the contractor within the warranty period. All other landscape maintenance notes pertain to the Owner after construction is completed. ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 Q17: Is there a copy of the soils report available? A17: See attached reports. Q18: Do you guys have the Geotech report for the Airport expansion. The drawings say to reference it for under slab gravel and vapor barrier, but I don’t see it in the bid docs. Q18: See attached reports. Q19: In reference to Addendum 1.1, Attachment 1, Bidders Qualifications Form, Credit available for this contract; Does this question refer to line the dollar amount of an available line of credit? A19: Yes, this question refers to the line dollar amount of an available line of credit. The vendor needs to just verify the of line credit demonstrating he can complete the project if awarded the bid. Q20: Please clarify the required staffing on this project. Superintendent and Quality Control Manager? Or is a Qualified Superintendent Onsite Only Required? A20: Per Technical Specification Section 400 – HOT MIX ASPHALTIC CONCRETE CONSTRUCTION, a Level 2 QCT shall be designated as a Quality Control Manager and shall be present at the plant, or within immediate contact by phone or radio. Please refer to Section 400.3.06.b.2 for full QCT requirements. Per Contract Document Section 10 DEFINITION OF TERMS and Section 50 CONTROL OF WORK a superintendent is to be present on-site, supervising and directing the construction. Q21: Is the awarded GC responsible for all permits and impact fees? A21: Yes, the awarded GC is responsible for all permits and impact fees. Q22: Will the Owner allow for Equipment and Material substitutions for materials and equipment that meets the specifications and allows for less Lead Time and cost beneficial to the Owner? i.e Lighting, Electrical Gear? A22: Per contract document ITB-9, all substitution requests will be reviewed and considered by the architect on an “as-equal” basis after the contract is awarded and in accordance with section 01 25 00 – SUBSTITUTION PROCEDURES and section 01 60 00 – PRODUCT REQUIREMENTS. Q23: Will the Architect Please provide a GEO report if completed. A23: See attached reports. Q24: Regarding Structural round Timber on this project, are you specifying laminated SRT columns or Natural? A24: Where tree columns are specified, we are looking for natural, barkless tree columns. ADDENDUM NO. 2 Corporate FBO Terminal Building and Parking Lot Rehabilitation Macon-Bibb County at Middle Georgia Regional Airport Thursday, December 5, 2024 Q25: Is Mule-Hide’s PVC roof system an acceptable alternative? A25: Per contract document ITB-9, all substitution requests will be reviewed and considered by the architect on an “as-equal” basis after the contract is awarded and in accordance with section 01 25 00 – SUBSTITUTION PROCEDURES and section 01 60 00 – PRODUCT REQUIREMENTS. Q26:Is Johns Manville PVC KEE an acceptable substitute roofing system? A26: Per contract document ITB-9, all substitution requests will be reviewed and considered by the architect on an “as-equal” basis after the contract is awarded and in accordance with section 01 25 00 – SUBSTITUTION PROCEDURES and section 01 60 00 – PRODUCT REQUIREMENTS. OTHER ITEMS 1. The deadline for Submission of Bids has been moved to Thursday, December 19, 2024 at 12:00 pm EST. The Bid Opening will occur at 2:00 pm EST. 2. Bid Documents must be obtained via Macon-Bibb County’s website at www.maconbibb.us/procurement, at Georgia’s Department Of Administrative Services at and www.passero.com/bids. It is the bidders responsibility to check the website for addenda prior to submitting their bid. END OF ADDENDUM NO. 2 SPEC REFERENCE DESCRIPTION UNIT PRICE TOTAL C102-5.1a Temporary Construction Exit 1 EA C102-5.1b Silt Fence 290 LF C102-5.1c Silt Barrier (Filter Sock) 90 LF C102-5.1d Temporary Inlet Protection 15 EA C102-5.1e Mulching 3,200 SF C-103-8.1 Project Survey, Stakeout, and Record Drawing 1 LS C-105-6.1 Mobilization 1 LS C-107-3.1 Traffic Control Measures 1 LS D-751-5.1 Remove Top of Existing Drop Inlet Structure & Replace with Concrete Cover and Seal 1 EA F-162-5.1 Relocated Security Fence 85 LF G-310-1 Graded Aggregate Base (GAB) Course (Depth Varies) 430 CY G-400-1 Hot Mixed Asphalt Surface Course 620 TONS G-412-1 Asphalt Prime Coat 210 GAL G-413-1 Asphalt Tack Coat 350 GAL G-430-1 Concrete Paved Walkway, Pads, & Sidewalks 380 SY G-432-1 Asphalt Milling (2 Inch Depth) 3,500 SY G-652-1 Permanent Pavement Marking (White) 2,410 SF G-652-2 Permanent Pavement Marking (Yellow) 280 SF G-652-3 Permanent Pavement Marking (Blue) 240 SF G-700-1 Sodding 4,440 SY G-702-1 Trees (Crepe Myrtle) 20 EA G-702-2 Trees (Hardwoods) 30 EA G-702-3 Landscaping (Small flowering plants) 3,200 SF G-708-1 Topsoil (Final Placement), 4-Inch 500 CY L-110-5.1 Concrete Encased Duct Bank, 4-Way: 2-2 Inch PVC & 2-4 Inch PVC 40 LF L-115-5.1 Raise Top(s) of Existing Electrical Manhole Cover(s) 3 EA MWA-101-5.1 Remove Section of Existing 8 Inch Clay Sanitary Sewer Line 60 LF MWA-101-5.2 8 Inch PVC Sanitary Sewer Line installed in 12 Inch Steel Casing 60 LF MWA-101-5.3 Sanitary Sewer Doghouse Manhole 2 EA MWA-101-5.4 6 Inch PVC Sanitary Sewer Line w/ Hardware & Fittings 180 LF MWA-101-5.5 4 Inch PVC Sanitary Sewer Line w/ Hardware & Fittings 140 LF MWA-101-5.6 Relocate Existing Fire Hydrant Top 1 EA SITE BASE BID - PARKING LOT REHABILITATION QUANTITY/ UNIT MWA-101-5.7 Fire Hydrant Fittings, Valve, and Hardware 1 EA MWA-101-5.8 6 Inch DIP Water Line 20 LF MWA-101-5.9 6 Inch PVC Water Line w/ Hardware & Fittings 90 LF MWA-101-5.10 2 Inch PVC Water Line w/ Hardware & Fittings 120 LF MWA-101-5.11 1/2 Inch PVC Water Line w/ Hardware & Fittings 260 LF 02-41-00 Demolition of Existing Hangar Office Lean-To (Partial Concrete Foundation to Remain for Re-use) 1 LS P-101-5.1 Full Depth Concrete Demolition 10 SY P-101-5.3 Remove Existing 8 Inch Iron Storm Pipe 50 LF P-101-5.3 Remove Existing 8 Inch Clay Storm Pipe 30 LF P-101-5.3 Remove Existing Storm Structure 1 EA P-101-5.3 Remove Existing Telecom Duct Banks / Conduits 150 LF P-101-5.3 Remove Existing Chain Link Fence Section 13 LF P-101-5.1 Full Depth Asphalt Demolition 2,310 SY P-101-5.2 Asphalt Clean & Prep 4,100 SY P-151-4.1 Topsoil (On-Site Stripping), 4-Inch 310 CY P-152-4.1 Unsuitable Excavation 500 CY P-152-4.2 Engineered Backfill 500 CY P-610-6.1 Concrete Curb 2,730 LF P-629-8.1 Asphalt Sand Slurry Surface Treatment 4,100 SY Plans 2 Inch HDPE Gas Line w/ Hardware & Fittings 170 LF Plans Dumpster Enclosure Grease Trap 1 EA Plans 6 Foot Tall Decorative Security Fence (Ameristar Echelon II (Classic) or Approved Equal) 120 LF Plans 6 Foot Tall x 14 Foot Wide Industrial Ornamental Security Roll Gate (Ameristar Passport II (Classic) or Approved Equal) 1 EA Plans Full Height Single Turnstile (Hayward Turnstiles HT431 Standard Passage or Approved Equal) 1 EA Plans Concrete Wheel Stop 6 EA Plans 30 Foot Tall Flag Pole 1 EA Plans 25 Foot Tall Flag Pole 2 EA Plans Concrete Bench (Wausau Tile ZB.GL.06 Concrete VI Bench, or Approved Equal) 3 EA Plans Concrete Trash Receptacle with Aluminum Top (Wausau Tile TF1031 Colonial or Approved Equal) 2 EA Plans Stainless Steel Smoker's Post (Wausau tile MF4013 or Approved Equal) 2 EA Plans Black 5-Loop Bicycle Rack (Wausau Tile MF9009 or Approved Equal) 1 EA Plans CMU Wall Dumpster Enclosure, including Can Wash 1 LS Plans CMU Screen Wall Mechanical Yard 1 LS Plans 4 Inch Deep River Rock Bed over Weed Barrier Fabric 13 CY SITE BASE BID TOTAL (PARKING LOT REHABILITATION) REF (NOT LIMITED TO) DESCRIPTION UNIT PRICE TOTAL 01 30 00 3.08 Real-time and Timelapse Cameras LS 313116 Termite Control LS STRUCT. DOCS Foundation and Slab on Grade LS 061523 CLT Superstructure LS 070000 A520-521 & A530 Roof Systems LS 086300 A521 Architectural Structural Skylight LS A310 Exterior Framed walls LS A606-607 Storefront Windows LS 081113 Hollow Metal doors LS 084313 Storefront Doors LS 084413 Curtain Walls LS 092400 Stucco System LS 099113 Exterior Painting LS 101419 Exterior Signage LS 057315 Guardrail Railing System with LED Lighting LS Alternate No. 1 (NOT TO BE INCLUDED IN LUMP SUM BID): Exterior Aluminum Frameless Glass Guardrail with no integrated LED Lighting - Refer to Section 01 23 00 ALTERNATES. LS - Alternate No. 2 (NOT TO BE INCLUDED IN LUMP SUM BID): 2507 "Super Duplex" Stainless Steel Guardrail and Cable Posts with 2507 Stainless Steel Swageless CableQuick Lock Assemblies, no lighting component - Refer to Section 01 23 00 ALTERNATES. LS -077600 A510 Porcelain Paver Pedestal System (Balcony) LS Natural Stone Travertine Coping (Balcony) LS G005 Interior Walls LS CMU Walls LS 081433 A603-604 Wood Doors and Trim LS A500 Casework - Cabinets LS 064100 Casework - Countertops LS Add No. 1 (NOT TO BE INCLUDED IN LUMP SUM BID): Custom Vinyl Mural of large format printed wall graphic by others - Refer to Section 01 23 00 ALTERNATES. LS -BUILDING BASE BID - CORPORATE FBO TERMINAL BUILDING QUANTITY/ UNIT Add No. 2 (NOT TO BE INCLUDED IN LUMP SUM BID): Speakers in the color black, mounted as high as possible on each Porte-Cochere column typ. And (2) recessed into ceiling at South Entrance - DMX module to connect to audio system to sync with RGBW LED High Pixel Lights for Pulsing (conduit from IT closet) - Refer to Section 01 23 00 ALTERNATES. LS - A324 Wood Stairs and Rotunda Guardrail LS 055100 A320-333 Metal Stairs LS 090000 A600-A602 Finishes LS Walls LS 092400 Flooring (Tile) LS 096500 Flooring (Resilient) LS Alternate No. 3 (NOT TO BE INCLUDED IN LUMP SUM BID): Provide 6" high vinyl resilient base trim in lieu of rooms on the Finish Schedule to receive 6" wood base trim, excluding areas to receive wainscot - Refer to Section 01 23 00 ALTERNATES. LS -096813 Flooring (Carpet) LS 096700 Flooring (Fluid-applied)) LS 064200 Wainscot Trim LS 099123 Interior Painting LS 101400 Interior Signage LS 102800,102819 Plumbing Fixtures LS 95670 A130A-140B Ceiling Systems LS 142100 Elevator LS 260000 ELEC. DOCS Electrical Systems: Power and Data/Communications LS 260000 ELEC. DOCS Electrical Systems: Lighting LS Alternate No. 4 (NOT TO BE INCLUDED IN LUMP SUM BID): Omit horizontal strip lighting on Angled Glazing System and Glulam Columns - Refer to Section 01 23 00 ALTERNATES. LS -280000 Alarm System LS 104400 Fire Suppression System LS 230000 MECH. DOCS HVAC Systems LS 220000 PLUMB. DOCS Plumbing Systems LS Allowance 1 LS $250,000.00 $250,000.00 BASE BID TOTAL (BUILDING) PD-9 BID SUMMARY DESCRIPTION QUANTITY/ UNIT TOTAL (IN NUMBERS) Total – Base Bid Parking Lot Rehabilitation LS Total – Base Bid Corporate FBO Terminal Building LS Grand Total – Base Bid Parking Lot Rehabilitation + Corporate FBO Terminal Building LS _____________ (Base Bid Parking Lot Rehabilitation Total in words, Lump Sum) _____________ (Base Bid Parking Lot Rehabilitation Total in numbers, Lump Sum) _____________ (Base Bid Corporate FBO Terminal Building Total in words, Lump Sum) _____________ (Base Bid Corporate FBO Terminal Building Total in numbers, Lump Sum) _____________ (Base Bid Parking Lot Rehabilitation + Corporate FBO Terminal Building Grand Total in words, Lump Sum) _____________ (Base Bid Parking Lot Rehabilitation + Corporate FBO Terminal Building Grand Total in numbers, Lump Sum) PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 23 00 - Alternates CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 1 OF 2 SECTION 01 23 00 - ALTERNATES PART 1 GENERAL 1.01 SECTION INCLUDES A. Description of ​Adds and Alternates​. 1.02 ACCEPTANCE OF ​ADDS AND ALTERNATES​ A. Adds and Alternates quoted on Bid Forms will be reviewed and accepted or rejected at Owner's option. Accepted Adds and Alternates will be identified in the Contract Agreement. B. Coordinate related work and modify surrounding work to integrate the Work of each ​Add or Alternate​. 1.03 SCHEDULE OF ​ADDS AND ALTERNATES​ A. ​Add​ No. ​1​ - ​Custom Mural​: 1. Base Bid Item: Provide specified wall finish for Second Floor Rotunda walls and Corridor 216; no custom vinyl mural to be provided. 2. Add Item: Provide custom vinyl mural of large format printed wall graphic by others (Basis of Design: Magic Murals, to be professionally installed on Second Floor Rotunda walls and Corridor 216 from top of wainscot to bottom of ceiling. Images and design layout provided by Architect. Design to be approved by Owner. B. ​Add​ No. ​2​ - ​Speakers at Porte-Cochere and South Entrance​: 1. Base Bid Item: Provide specified finish at Porte-Cochere columns and South Entrance; no speakers to be provided. 2. Add Item: Provide speakers in the color black mounted as high as possible on each Porte-Cochere column typ., and (2) recessed into ceiling at South Entrance -DMX module to connect to audio system to sync with RGBW LED High Pixel Lights for Pulsing (conduit from IT closet). C. Alternate No. 1 - Balcony Railing: 1. Base Bid Item: Provide Exterior Aluminum Frameless Glass Balustrade on Balcony with Mounted Spigot Bases with Recessed, Integrated LED Lights. 2. Alternate Item: Provide Exterior Aluminum Frameless Glass Balustrade on Balcony (no recessed, integrated LED lighting in mounted spigot bases). D. Alternate No. 2 - Balcony Railing 1. Base Bid Item: Provide Exterior Aluminum Frameless Glass Balustrade on Balcony with Mounted Spigot Bases with Recessed, Integrated LED Lights. 2. Alternate Item: Provide 2507 "Super Duplex" Stainless Steel guardrail and cable posts with 2507 Stainless Steel Swageless CableQuick® Lock Assemblies for the balcony railing system (no lighting component). E. ​Alternate​ No. ​3​ - ​Vinyl Resilient Base Trim​: 1. Base Bid Item: Provide 6" high wood base trim in all rooms per the Finish Schedule, excluding areas to receive wainscot. 2. Alternate Item: Provide 6" high vinyl resilient base trim in lieu of rooms on the Finish Schedule to receive 6" wood base trim, excluding areas to receive wainscot. F. ​Alternate​ No. ​4​ - ​Lighting on Angled Glazing System​: 1. Base Bid Item: Provide horizontal strip lighting within weatherproof aluminum casing on Angled Glazing System and Glulam Sloped Column. Refer to the Bid Form to provide cost for this installation as a separate line item. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 23 00 - Alternates CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 2 OF 2 2. Alternate Item: Omit horizontal strip lighting on Angled Glazing System and Glulam Columns. PART 2 PRODUCTS - NOT USED PART 3 EXECUTION - NOT USED END OF SECTION PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 1 OF 12 SECTION 01 30 00 - ADMINISTRATIVE REQUIREMENTS PART 1 GENERAL 1.01 SECTION INCLUDES A. General administrative requirements. B. Electronic document submittal service. C. Preconstruction meeting. D. Site mobilization meeting. E. Progress meetings. F. Construction progress schedule. G. Contractor's daily reports. H. Progress photographs. I. Coordination drawings. J. Submittals for review, information, and project closeout. K. Number of copies of submittals. L. Requests for ​Information​ (RFI) procedures. M. Submittal procedures. 1.02 RELATED REQUIREMENTS A. Section 01 60 00 - Product Requirements: General product requirements. B. Section 01 70 00 - Execution and Closeout Requirements: Additional coordination requirements. C. Section 01 78 00 - Closeout Submittals: Project record documents; operation and maintenance data; warranties and bonds. 1.03 REFERENCE STANDARDS A. AIA G716 - Request for Information; 2004. B. AIA G810 - Transmittal Letter; 2001. 1.04 GENERAL ADMINISTRATIVE REQUIREMENTS A. Comply with requirements of Section 01 70 00 - Execution and Closeout Requirements for coordination of execution of administrative tasks with timing of construction activities. B. Make the following types of submittals to Architect: 1. Requests for ​Information​ (RFI). 2. Requests for substitution. 3. Shop drawings, product data, and samples. 4. Test and inspection reports. 5. Design data. 6. Manufacturer's instructions and field reports. 7. Applications for payment and change order requests. 8. Progress schedules. 9. Coordination drawings. 10. Correction Punch List and Final Correction Punch List for Substantial Completion. 11. Closeout submittals. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 2 OF 12 PART 2 PRODUCTS - NOT USED PART 3 EXECUTION 3.01 ELECTRONIC DOCUMENT SUBMITTAL SERVICE A. All documents transmitted for purposes of administration of the contract are to be in electronic (PDF, MS Word, or MS Excel) format, as appropriate to the document, and transmitted via an Internet-based submittal service that receives, logs and stores documents, provides electronic stamping and signatures, and notifies addressees via email. 1. Besides submittals for review, information, and closeout, this procedure applies to Requests for ​Information​ (RFIs), progress documentation, contract modification documents (e.g. supplementary instructions, change proposals, change orders), applications for payment, field reports and meeting minutes, ​Contractor​'s correction punchlist, and any other document any participant wishes to make part of the project record. 2. Contractor and Architect are required to use this service. 3. It is Contractor's responsibility to submit documents in allowable format. 4. Subcontractors, suppliers, and Architect's consultants will be permitted to use the service at no extra charge. 5. Users of the service need an email address, internet access, and PDF review software that includes ability to mark up and apply electronic stamps (such as Adobe Acrobat, www.adobe.com, or Bluebeam PDF Revu, www.bluebeam.com), unless such software capability is provided by the service provider. 6. Paper document transmittals will not be reviewed; emailed electronic documents will not be reviewed. 7. All other specified submittal and document transmission procedures apply, except that electronic document requirements do not apply to samples or color selection charts. B. Submittal Service: The selected service is: 1. Newforma Info Exchange, provided by Architect. C. Project Closeout: Architect will determine when to terminate the service for the project and is responsible for obtaining archive copies of files for Owner. 3.02 PRECONSTRUCTION MEETING A. ​Architect​ will schedule a meeting after ​Notice of Award for both the Preconstruction Meeting and Site Mobilization Meeting to be at the same time​. B. Attendance Required: 1. Owner. 2. Architect. 3. Contractor. 4. ​Engineer​. C. Agenda: 1. Execution of Owner-Contractor Agreement. 2. Submission of executed bonds and insurance certificates. 3. Distribution of Contract Documents. 4. Submission of list of subcontractors, list of products, schedule of values, and progress schedule. 5. Submission of initial Submittal schedule. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 3 OF 12 6. Designation of personnel representing the parties to Contract and Architect/Engineer. 7. Procedures and processing of field decisions, submittals, substitutions, applications for payments, proposal request, Change Orders, and Contract closeout procedures. 8. Scheduling. D. Architect will record minutes and distribute copies within two days after meeting to participants with copies to Architect, Engineer, Owner, participants, and those affected by decisions made. 3.03 SITE MOBILIZATION MEETING A. Schedule meeting at the Project site prior to Contractor occupancy. B. Attendance Required: 1. Contractor. 2. Owner. 3. Architect. 4. Contractor's superintendent. 5. Major subcontractors. 6. ​Engineer​. C. Agenda: 1. Use of premises by Owner and Contractor. 2. Owner's requirements. 3. Construction facilities and controls provided by Owner. 4. Construction facilities and controls provided by Contractor. 5. Survey and building layout. 6. Security and housekeeping procedures. 7. Schedules. 8. Application for payment procedures. 9. Procedures for testing. 10. Procedures for maintaining record documents. 11. Requirements for start-up of equipment. 12. Inspection and acceptance of equipment put into service during construction period. D. Record minutes and distribute copies within two days after meeting to participants, with two copies to Architect, Engineer, Owner, participants, and those affected by decisions made. 3.04 PROGRESS MEETINGS A. Schedule and administer meetings throughout progress of the work at maximum ​as-required​ intervals. B. Architect will make arrangements for meetings, prepare agenda with copies for participants, preside at meetings. C. Attendance Required: 1. Contractor. 2. Owner. 3. Architect. 4. Special consultants. 5. Contractor's superintendent. 6. Major subcontractors. 7. ​Engineer​. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 4 OF 12 D. Agenda: 1. Review minutes of previous meetings. 2. Review of work progress. 3. Field observations, problems, and decisions. 4. Identification of problems that impede, or will impede, planned progress. 5. Review of submittals schedule and status of submittals. 6. Review of RFIs log and status of responses. 7. Review of off-site fabrication and delivery schedules. 8. Maintenance of progress schedule. 9. Corrective measures to regain projected schedules. 10. Planned progress during succeeding work period. 11. Coordination of projected progress. 12. Maintenance of quality and work standards. 13. Effect of proposed changes on progress schedule and coordination. 14. Other business relating to work. E. Architect will record minutes and distribute copies within two days after meeting to participants, with copies to Architect, Engineer, Owner, participants, and those affected by decisions made. 3.05 CONSTRUCTION PROGRESS SCHEDULE​ A. Within ​10​ days after date ​established in Notice to Proceed​, submit preliminary schedule​ defining planned operations for the first 60 days of work, with a general outline for remainder of work​. B. If preliminary schedule requires revision after review, submit revised schedule within 10 days. C. Within 20 days after review of preliminary schedule, submit draft of proposed complete schedule for review. 1. Include written certification that major contractors have reviewed and accepted proposed schedule. D. Within 10 days after joint review, submit complete schedule. E. Submit updated schedule with each Application for Payment. 3.06 DAILY CONSTRUCTION REPORTS A. Include only factual information. Do not include personal remarks or opinions regarding operations and/or personnel. B. In addition to transmitting electronically a copy to Owner and Architect, submit two printed copies at weekly intervals. C. Prepare a daily construction report recording the following information concerning events at Project site and project progress: 1. Date. 2. High and low temperatures, and general weather conditions. 3. List of subcontractors at Project site. 4. Material deliveries. 5. Safety, environmental, or industrial relations incidents. 6. Meetings and significant decisions. 7. Stoppages, delays, shortages, and losses. Include comparison between scheduled work activities (in Contractor's most recently updated and published schedule) and actual activities. Explain differences, if any. Note days or periods when no work was in progress and explain the reasons why. 8. Meter readings and similar recordings. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 5 OF 12 9. Emergency procedures. 10. Directives and requests of Authority(s) Having Jurisdiction (AHJ). 11. Change Orders received and implemented. 12. Testing and/or inspections performed. 13. List of verbal instruction given by Owner and/or Architect. 14. Signature of Contractor's authorized representative. 3.07 PROGRESS PHOTOGRAPHS A. Submit photographs with each application for payment, taken not more than 3 days prior to submission of application for payment. B. Photography Type: Digital; electronic files. C. Provide photographs of site and construction throughout progress of work produced by an experienced photographer, acceptable to Architect. D. In addition to periodic, recurring views, take photographs of each of the following events: 1. Completion of site clearing. 2. Excavations in progress. 3. Foundations in progress and upon completion. 4. Structural framing in progress and upon completion. 5. Enclosure of building, upon completion. E. Views: 1. Provide aerial photographs from four cardinal views at each specified time, until date of Substantial Completion. 2. Provide non-aerial photographs from four cardinal views at each specified time, until date of Substantial Completion. 3. Consult with Architect for instructions on views required. 4. Provide factual presentation. 5. Provide correct exposure and focus, high resolution and sharpness, maximum depth of field, and minimum distortion. F. Digital Photographs: 24 bit color, minimum resolution of 1024 by 768, in JPG format; provide files unaltered by photo editing software. 1. Delivery Medium: Via email. 2. File Naming: Include project identification, date and time of view, and view identification. 3. PDF File: Assemble all photos into printable pages in PDF format, with 2 to 3 photos per page, each photo labeled with file name; one PDF file per submittal. 4. Hard Copy: Printed hardcopy (grayscale) of PDF file and point of view sketch. 3.08 REAL-TIME AND TIMELAPSE FOOTAGE OF CONSTRUCTION PROGRESS A. Duration to be from Notice to Proceed to Final Completion. B. Two (2) external cameras to be placed; one (1) to be mounted as high as possible on abandoned weather tower adjacent to USPS Post Office Building, and one (1) to be located at top of door pocket roof at WWII Hangar West. C. External Cameras Requirements: 1. Stationary solar-powered with battery backup for camera at abandoned weather tower. 2. Powered from WWII Hangar West for camera at WWII Hangar West door pocket roof. 3. WiFi-enabled streaming service. 4. Camera cloud system (OxBlue). 5. Must have the ability to connect to a WiFi network for upload. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 6 OF 12 6. Timelapse can be added. (Overall, video and timelapse) a. Basis-of-Design Manufacturer: OxBlue Construction Cameras | Construction Camera Services for Real-Time Project Visibility, 1) Camera (a) Resolution: 8-megapixel (3264 x 2448) (b) Pan: 360-degrees (c) Tilt: 180-degrees (d) 30x optical zoom (e) Progress scan CMOS 1/3" image sensor (f) Lens: 4.4-132 mm, F1.4-4.6, autofocus, auto-iris (g) 256 preset positions 2) Video (a) Live HD video streaming (b) Current weather detection (c) Aspect ratio of video 16:9 (widescreen) (d) Stream profile: H.264 (e) 1920 x 1080 recordi (f) 1080p streaming resolution (g) Dynamically sized viewing window 3) Data Connectivity and Storage (a) Cellular data services provided by the System Vendor (b) Onboard Memory: Up to 4 Week Video Storage. Resolution can be adjusted but will impact storage time. (c) Remote Storage: Archive a still image approximately every 10 minutes. 4) Operating Environment and Controls (a) -30 to 122 degrees Fahrenheit (b) 120 / 240 VAC units come standard with blower and defroster; 12 VDC units come standard with blower. 5) Video Controls (a) Snapshot function (b) 8 button directional input (c) Home button 6) Software (a) Responsive software interface for use on computer, tablet and mobile screens (b) Display Owner or Project logo on desktop software interface (c) Dashboard display of all cameras (d) Camera search capability (e) Visual calendar showing actual photos from each day of the project (f) Access to each individual photo archived (g) Ability to schedule the automated delivery of images and timelapses to users via email (h) Display weather data with each image (i) Ability to compare images from two cameras or two specific times simultaneously (j) Ability to overlay and compare images from different times (k) Interactive map showing project location (l) Provide iPhone/iPad app and Android app PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 7 OF 12 7) Hosting and Website Integration (a) Provide links to thumbnails of most recent (b) Provide API access for use in software and website integration 8) Data Security and Infrastructure (a) Multiple access options shall be available, including publicly available links, username authentication, IP restrictions, and HTTPS communication protocols (b) Actual access method used shall be specified by the Owner (c) Data shall be stored on redundant servers owned and managed by the System Vendor (d) Optional time-delay feature for timelapse videos and images on website b. Substititions: See Section 01 60 00 - Product Requirements. D. All equipment used to provide timelapse and video footage to be the property of the Contractor. E. All visual imagery, images, video, and timelapse footage shall be the property of the Owner. 3.09 COORDINATION DRAWINGS A. Review drawings prior to submission to Architect. B. Coordination Drawings, General: Prepare coordination drawings according to requirements in individual Sections, and additionally where installation is not completely indicated on Shop Drawings, where limited space availability necessitates coordination, or if coordination is required to facilitate integration of products and materials fabricated or installed by more than one entity. 1. Content: Project-specific information, drawn accurately to a scale large enough to indicate and resolve conflicts. Do not base coordination drawings on standard printed data. Include the following information, as applicable: a. Indicate functional and spatial relationships of components of architectural, structural, civil, mechanical, and electrical systems. 1) Indicate space requirements for routine maintenance and for anticipated replacement of components during the life of the installation. 2) Show location and size of access doors required for access to concealed dampers, valves, and other controls. 3) Indicate dimensions shown on Drawings. Specifically note dimensions that appear to be in conflict with submitted equipment and minimum clearance requirements. Provide alternative sketches to Architect indicating proposed resolution of such conflicts. Minor dimension changes and difficult installations will not be considered changes to the Contract. C. Coordination Drawing Organization: Organize coordination drawings as follows: 1. Mechanical Rooms: Provide coordination drawings for mechanical rooms showing plans and elevations of mechanical, plumbing, fire-protection, fire-alarm, and electrical equipment. 2. Structural Penetrations: Indicate penetrations and openings required for all disciplines. 3. Slab Edge and Embedded Items: Indicate slab edge locations and sizes and locations of embedded items for metal fabrications, sleeves, anchor bolts, bearing plates, angles, door floor closers, slab depressions for floor finishes, curbs and PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 8 OF 12 housekeeping pads, and similar items. 4. Mechanical and Plumbing Work: Show the following: a. Sizes and bottom elevations of ductwork, piping, and conduit runs, including insulation, bracing, flanges, and support systems. b. Dimensions of major components, such as dampers, valves, diffusers, access doors, cleanouts and electrical distribution equipment. 5. Electrical Work: Show the following: a. Runs of vertical and horizontal conduit 1-1/4 inches in diameter and larger. b. Light fixture, exit light, emergency battery pack, smoke detector, and other fire-alarm locations. c. Panel board, switch board, switchgear, transformer, busway, generator, and motor-control center locations. d. Location of pull boxes and junction boxes, dimensioned from column center lines. 6. Fire-Protection System: Show the following: a. Locations of standpipes, mains piping, branch lines, pipe drops, and sprinkler heads. 7. Review: Architect will review coordination drawings to confirm that in general the Work is being coordinated, but not for the details of the coordination, which are Contractor's responsibility. If Architect determines that coordination drawings are not being prepared in sufficient scope or detail, or are otherwise deficient, Architect will so inform Contractor, who shall make suitable modifications and resubmit. D. Coordination Digital Data Files: Prepare coordination digital data files according to the following requirements: 1. File Submittal Format: Submit or post coordination drawing files using PDF format. 2. Architect will furnish Contractor one set of digital data files of Drawings for use in preparing coordination digital data files. a. Architect makes no representations as to the accuracy or completeness of digital data files as they relate to Drawings. 3.10 REQUESTS FOR ​INFORMATION​ (RFI) A. Definition: A request seeking one of the following: 1. An interpretation, amplification, or clarification of some requirement of Contract Documents arising from inability to determine from them the exact material, process, or system to be installed; or when the elements of construction are required to occupy the same space (interference); or when an item of work is described differently at more than one place in Contract Documents. 2. A resolution to an issue which has arisen due to field conditions and affects design intent. B. Whenever possible, request clarifications at the next appropriate project progress meeting, with response entered into meeting minutes, rendering unnecessary the issuance of a formal RFI. C. Preparation: Prepare an RFI immediately upon discovery of a need for interpretation of Contract Documents. Failure to submit a RFI in a timely manner is not a legitimate cause for claiming additional costs or delays in execution of the work. 1. Prepare a separate RFI for each specific item. a. Review, coordinate, and comment on requests originating with subcontractors and/or materials suppliers. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 9 OF 12 b. Do not forward requests which solely require internal coordination between subcontractors. 2. Prepare in a format and with content acceptable to Owner. a. Use AIA G716 - Request for Information . 3. Prepare using software provided by the Electronic Document Submittal Service. 4. Combine RFI and its attachments into a single electronic file. PDF format is preferred. D. Reason for the RFI: Prior to initiation of an RFI, carefully study all Contract Documents to confirm that information sufficient for their interpretation is definitely not included. 1. Unacceptable Uses for RFIs: Do not use RFIs to request the following:: a. Approval of submittals (use procedures specified elsewhere in this section). b. Approval of substitutions (see Sections 01 25 00 - Substitution Procedures and 01 60 00 - Product Requirements) c. Changes that entail change in Contract Time and Contract Sum (comply with provisions of the Conditions of the Contract). 2. Improper RFIs: Requests not prepared in compliance with requirements of this section, and/or missing key information required to render an actionable response. They will be returned without a response​. 3. Frivolous RFIs: Requests regarding information that is clearly indicated on, or reasonably inferable from, Contract Documents, with no additional input required to clarify the question. They will be returned without a response, with an explanatory notation. a. The Owner reserves the right to assess the Contractor for the costs (on time-and-materials basis) incurred by the Architect, and any of its consultants, due to processing of such RFIs. E. Content: Include identifiers necessary for tracking the status of each RFI, and information necessary to provide an actionable response. 1. Official Project name and number, and any additional required identifiers established in Contract Documents. 2. Discrete and consecutive RFI number, and descriptive subject/title. 3. Reference to particular Contract Document(s) requiring additional information/interpretation. Identify pertinent drawing and detail number and/or specification section number, title, and paragraph(s). 4. Annotations: Field dimensions and/or description of conditions which have engendered the request. 5. Contractor's suggested resolution: A written and/or a graphic solution, to scale, is required in cases where clarification of coordination issues is involved, for example; routing, clearances, and/or specific locations of work shown diagrammatically in Contract Documents. If applicable, state the likely impact of the suggested resolution on Contract Time or the Contract Sum. F. Attachments: Include sketches, coordination drawings, descriptions, photos, submittals, and other information necessary to substantiate the reason for the request. G. Review Time: ​Architect​ will respond and return RFIs to ​Contractor​ within ​seven​ calendar days of receipt. For the purpose of establishing the start of the mandated response period, RFIs received after ​12:00 noon​ will be considered as having been received on the following regular working day. 1. Response period may be shortened or lengthened for specific items, subject to mutual agreement, and recorded in a timely manner in progress meeting minutes. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 10 OF 12 H. Responses: Content of answered RFIs will not constitute in any manner a directive or authorization to perform extra work or delay the project. If in Contractor's belief it is likely to lead to a change to Contract Sum or Contract Time, promptly issue a notice to this effect, and follow up with an appropriate Change Order request to Owner. 1. Response may include a request for additional information, in which case the original RFI will be deemed as having been answered, and an amended one is to be issued forthwith. Identify the amended RFI with an R suffix to the original number. 2. Do not extend applicability of a response to specific item to encompass other similar conditions, unless specifically so noted in the response. 3. Notify Architect within seven calendar days if an additional or corrected response is required by submitting an amended version of the original RFI, identified as specified above. 3.11 SUBMITTAL SCHEDULE A. Submit to Architect for review a schedule for submittals in tabular format. 1. Coordinate with Contractor's construction schedule and schedule of values. 2. Format schedule to allow tracking of status of submittals throughout duration of construction. 3. Arrange information to include scheduled date for initial submittal, specification number and title, submittal category (for review or for information), description of item of work covered, and role and name of subcontractor. 4. Account for time required for preparation, review, manufacturing, fabrication and delivery when establishing submittal delivery and review deadline dates. 3.12 SUBMITTALS FOR REVIEW A. When the following are specified in individual sections, submit them for review: 1. Product data. 2. Shop drawings. 3. Samples for selection. 4. Samples for verification. B. Submit to Architect for review for the limited purpose of checking for compliance with information given and the design concept expressed in Contract Documents. C. Samples will be reviewed for aesthetic, color, or finish selection. D. After review, provide copies and distribute in accordance with SUBMITTAL PROCEDURES article below and for record documents purposes described in Section 01 78 00 -Closeout Submittals. 3.13 SUBMITTALS FOR INFORMATION A. When the following are specified in individual sections, submit them for information: 1. Design data. 2. Certificates. 3. Test reports. 4. Inspection reports. 5. Manufacturer's instructions. 6. Manufacturer's field reports. 7. Other types indicated. B. Submit for Architect's knowledge as contract administrator or for Owner. 3.14 SUBMITTALS FOR PROJECT CLOSEOUT A. Submit Correction Punch List for Substantial Completion. B. Submit Final Correction Punch List for Substantial Completion. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 11 OF 12 C. When the following are specified in individual sections, submit them at project closeout in compliance with requirements of Section 01 78 00 - Closeout Submittals: 1. Project record documents. 2. Operation and maintenance data. 3. Warranties. 4. Bonds. 5. Other types as indicated. D. Final Property Survey. E. Submit for Owner's benefit during and after project completion. 3.15 NUMBER OF COPIES OF SUBMITTALS A. Electronic Documents: Submit one electronic copy in PDF format; an electronically-marked up file will be returned. Create PDFs at native size and right-side up; illegible files will be rejected. B. Samples: Submit the number specified in individual specification sections; one of which will be retained by Architect. 1. After review, produce duplicates. 2. Retained samples will not be returned to Contractor unless specifically so stated. 3.16 SUBMITTAL PROCEDURES A. General Requirements: 1. Submit separate packages of submittals for review and submittals for information, when included in the same specification section. 2. Sequentially identify each item. For revised submittals use original number and a sequential numerical suffix. 3. Identify: Project; Contractor; subcontractor or supplier; pertinent drawing and detail number; and specification section number and article/paragraph, as appropriate on each copy. 4. Apply Contractor's stamp, signed or initialed certifying that review, approval, verification of products required, field dimensions, adjacent construction work, and coordination of information is in accordance with the requirements of the work and Contract Documents. Architect will not review submittals unless certifed by Design-Builder. 5. Schedule submittals to expedite the Project, and coordinate submission of related items. a. For each submittal for review, allow 15 days excluding delivery time to and from the Contractor. 6. Identify variations from Contract Documents and product or system limitations that may be detrimental to successful performance of the completed work. 7. When revised for resubmission, identify all changes made since previous submission. 8. Distribute reviewed submittals. Instruct parties to promptly report inability to comply with requirements. 9. Incomplete submittals will not be reviewed, unless they are partial submittals for distinct portion(s) of the work, and have received prior approval for their use. 10. Submittals not requested will be recognized, and will be returned "Not Reviewed", B. Product Data Procedures: 1. Submit only information required by individual specification sections. 2. Collect required information into a single submittal. 3. Submit concurrently with related shop drawing submittal. 4. Do not submit (Material) Safety Data Sheets for materials or products. PROJECT NO. 20202946.010A November 4, 2024 Passero Associates SECTION 01 30 00 - Administrative Requirements CORPORATE FBO TERMINAL BUILDING AND PARKING LOT REHABILITATION PAGE 12 OF 12 C. Shop Drawing Procedures: 1. Prepare accurate, drawn-to-scale, original shop drawing documentation by interpreting Contract Documents and coordinating related work. 2. Do not reproduce Contract Documents to create shop drawings. 3. Generic, non-project-specific information submitted as shop drawings do not meet the requirements for shop drawings. D. Samples Procedures: 1. Transmit related items together as single package. 2. Identify each item to allow review for applicability in relation to shop drawings showing installation locations. 3.17 SUBMITTAL REVIEW A. Submittals for Review: Architect will review each submittal, and approve, or take other appropriate action. B. Submittals for Information: Architect will acknowledge receipt and review. See below for actions to be taken. C. Architect's actions will be reflected by marking each returned submittal using virtual stamp on electronic submittals. 1. Notations may be made directly on submitted items and/or listed on appended Submittal Review cover sheet. D. Architect's and consultants' actions on items submitted for review: 1. Authorizing purchasing, fabrication, delivery, and installation: a. "Exceptions as Noted", or language with same legal meaning. b. "No Exceptions", or language with same legal meaning. 2. Not Authorizing fabrication, delivery, and installation: a. "Revise and Resubmit". 1) Resubmit revised item, with review notations acknowledged and incorporated. 2) Non-responsive resubmittals may be rejected. b. "Rejected". 1) Submit item complying with requirements of Contract Documents. E. Architect's and consultants' actions on items submitted for information: 1. Items for which no action was taken: a. "Received" - to notify the Contractor that the submittal has been received for record only. 2. Items for which action was taken: a. "Reviewed" - no further action is required from Contractor. END OF SECTION REPORT C OVER PAGE Geotechnical Engineering Report ______________ New Corporate GA Terminal Macon, Georgia April 18, 2023 Terracon Project No. HN225215 Prepared for: Passero Associates St. Augustine, Florida Prepared by: Terracon Consultants, Inc. Macon, Georgia Terracon Consultants Inc., 514 Hillcrest Ind. Blvd. Macon, Georgia 31204 P (478) 757 1606 F (478) 757 1608 terracon.com REPORT C OVER LETTER TO SIGN April 18, 2023 Passero Associates 4730 Casa Cola Way, Suite 200 St. Augustine, Florida 32095 Attn: Mr. Stan Price E: sprice@passero.com Re: Geotechnical Engineering Report New Corporate GA Terminal Middle GA Regional Airport Macon, Georgia Terracon Project No. HN225215 Dear Mr. Price: We have completed the Geotechnical Engineering services for the above referenced project. This study was performed in general accordance with Terracon Proposal No. PHN225215 dated September 10, 2022. This report presents the findings of the subsurface exploration and provides geotechnical recommendations concerning earthwork and the design and construction of foundations and floor slabs for the proposed project. We appreciate the opportunity to be of service to you on this project. If you have any questions concerning this report or if we may be of further service, please contact us. Sincerely, Terracon Consultants, Inc. Brad Thigpen, P.E. Thomas E. Driver, P.E. Project Engineer Regional Manager SIGN Responsive ■ Resourceful ■ Reliable i REPORT TOPICS INTRODUCTION ............................................................................................................. 1 SITE CONDITIONS ......................................................................................................... 2 PROJECT DESCRIPTION .............................................................................................. 2 GEOTECHNICAL CHARACTERIZATION ...................................................................... 2 GEOTECHNICAL OVERVIEW ....................................................................................... 4 EARTHWORK ................................................................................................................ 5 SHALLOW FOUNDATIONS ........................................................................................... 9 SEISMIC CONSIDERATIONS ...................................................................................... 11 FLOOR SLABS ............................................................................................................ 12 PAVEMENTS ................................................................................................................ 14 GENERAL COMMENTS ............................................................................................... 17 Note: This report was originally delivered in a web-based format. Orange Bold text in the report indicates a referenced section heading. The PDF version also includes hyperlinks which direct the reader to that section and clicking on the GeoReport logo will bring you back to this page. For more interactive features, please view your project online at client.terracon.com. ATTACHMENTS EXPLORATION AND TESTING PROCEDURES SITE LOCATION AND EXPLORATION PLANS EXPLORATION RESULTS SUPPORTING INFORMATION Note: Refer to each individual Attachment for a listing of contents. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable REPORT SUMMARY Topic 1 Overview Statement 2 Project Description The project will consist of a proposed new terminal building in the area of the existing Avionics and FBO buildings. Geotechnical Characterization The borings typically encountered topsoil or asphalt pavement underlain by fill soils and Coastal Plain soils to the depths explored. The soils typically included silty sands (SM), lean clays (CL), and clayey sands (SC). Fill soils were encountered in borings B-2 and B-8 and extended to depths of 3 feet and 8 feet below exiting ground surface, respectfully. Groundwater was not encountered at the time of boring. Earthwork Proofroll the structure pads after site stripping and before placing structural fill. Replace any loose and unstable areas with engineered fill. Shallow Foundations Shallow foundations will be acceptable for structural support. Allowable bearing pressure = 2,500 psf Expected maximum settlements: 1-inch total, 1/2-inch differential (may vary depending on final load information). Undercut and remove any loose soils in foundation and floor slab areas to a point 10 feet outside of the edge of the foundation. General Comments This section contains important information about the limitations of this geotechnical engineering report. 1. If the reader is reviewing this report as a pdf, the topics above can be used to access the appropriate section of the report by simply clicking on the topic itself. 2. This summary is for convenience only. It should be used in conjunction with the entire report for design purposes. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 1 INTRODUCTION Geotechnical Engineering Report New Corporate GA Terminal Middle GA Regional Airport Macon, Georgia Terracon Project No. HN225215 April 18, 2023 INTRODUCTION This report presents the results of our subsurface exploration and geotechnical engineering services performed for the proposed new terminal building to be located at the Middle GA Regional Airport in Macon, Georgia. The purpose of these services is to provide information and geotechnical engineering recommendations relative to: ■ Subsurface soil conditions ■ Foundation design and construction ■ Groundwater conditions ■ Floor slab and Pavement design and construction ■ Site preparation and earthwork ■ Seismic site classification per IBC ■ Excavation considerations ■ Dewatering considerations The geotechnical engineering Scope of Services for this project included the advancement of 8 soil test borings to depths of 10 to 30 feet below existing site grades. Maps showing the site and boring locations are shown in the Site Location and Exploration Plan sections, respectively. The boring logs are presented in the Exploration Results section of this report. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 2 SITE CONDITIONS The following description of site conditions is derived from our site visit in association with the field exploration and our review of readily available geologic and topographic maps as well as provided documents and plans. Item Description Parcel Information The project is located at Middle GA Regional Airport in Macon, Georgia. Approximate Coordinates: 32.7017, -83.6479. See Site Location. Existing Improvements The property currently has an existing Avionics building and FBO building, as well as asphalt paved areas. Current Ground Cover Grass and asphalt pavement. Existing Topography The site is relatively flat and level. PROJECT DESCRIPTION Our initial understanding of the project was provided in our proposal and was discussed during project planning. Our current understanding of the project conditions is as follows: Item Description Information Provided Our understanding of the project is based upon email correspondence with Stan Price. A boring location plan was provided. Project Description The project will consist of a proposed new terminal building in the area of the existing Avionics and FBO buildings. Proposed Structures We assume that the building will be a CMU or Steel structure. Finished Floor Elevation The FFE for the building is not available at this time. Maximum Loads (Assumed) Maximum Column Loads: 75 kips Maximum Approximate Wall Loads: 5 klf Loads for the building have not been provided at this time. Grading/Slopes Based upon the provided information, we anticipate cuts and fills on the order of 5 feet or less will be needed to establish finished grades. GEOTECHNICAL CHARACTERIZATION Site Geology The site is located in the Coastal Plain Physiographic Province of Georgia. Soils in the Coastal Plain are the result of the deposition of sediments in a former marine environment. Coastal Plain sedimentary deposits make up about 60 percent of Georgia’s surface area, and consist of a southwardly thickening wedge of sediments, which are bordered on the north by the parent rocks of the Piedmont Physiographic Province. The border between these provinces is known as the Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 3 “Fall-Line.” The Coastal Plain sediments range in age from the Cretaceous to the recent, with the oldest exposed along the “Fall-Line” and the youngest along the coast. Typically, the surface soils consist of complexly interbedded sands, silts, and clays of various mixtures. Sandstones, shales, and limestones comprise the characteristic lithology of the Coastal Plain. These formations are usually found at depths greater than fifty feet but can also be found at or near the ground surface. They are not known to occur near the surface in the site area. Topography in this region of the Coastal Plain is generally flat to gently rolling. Typical Subsurface Profile The borings drilled at the site generally encountered topsoil or asphalt pavement underlain by fill soils and Piedmont residual soils. Fill soils were encountered in two of the borings and generally extended to depths of approximately 3 feet to 8 feet below existing ground surface. Based on the results of the borings, subsurface conditions on the project site can be generalized as follows: Description Approximate Depth to Bottom of Stratum Material Encountered Consistency/Density Stratum 1 3 to 6 inches Topsoil or Asphalt Pavement Stratum 2 3 to 8 feet Fill (B-2 and B-8) – Lean Clay (CL) Coastal Plain – Lean Clay (CL); Clayey Sand (SC) Soft to Very Stiff Stratum 3 Boring Termination Coastal Plain – Silty Sand (SM), Clayey Sand (SC); Lean Clay (CL) Medium Stiff to Very Stiff; Medium Dense to Dense The fill soils in borings B-2 and B-8 were relatively loose to a depth of 5 feet below existing ground surface. Specific conditions encountered at each boring location are indicated on the individual boring logs. Stratification boundaries on the boring logs represent the approximate location of changes in soil types; in situ, the transition between materials may be gradual. Details for each of the borings are presented on the boring logs included in the attachments. Groundwater The boreholes were observed while drilling and after completion for the presence and level of groundwater. Groundwater was not encountered at the time of boring. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 4 Groundwater level fluctuations occur due to seasonal variations in the amount of rainfall, runoff and other factors not evident at the time the borings were performed. In addition, perched water can develop over low permeability soil or rock strata. Therefore, groundwater levels during construction or at other times in the life of the structure may be higher or lower than the levels indicated on the boring logs. The possibility of groundwater level fluctuations should be considered when developing the design and construction plans for the project. GEOTECHNICAL OVERVIEW All borings penetrated topsoil or asphalt pavement underlain by fill soils or Coastal Plain soils extending to the maximum depths explored. Fill soils were encountered in borings B-2 and B-8 and extended to a depth of approximately 3 feet to 8 feet below existing ground surface. The fill soils consisted of lean clays (CL). The Standard Penetration Test (SPT) values in these soils ranged from 3 blows per foot (bpf) to 6 bpf. The sols in the upper 5 feet of these borings were relatively loose. The loose fill soils in these areas, as well as any other areas of soft or loose fill encountered, should be removed or reworked prior to fill placement or structural support. It is anticipated that some undercutting will be required. Coastal Plain soils were encountered in all borings below the fill soils or beginning at existing ground surface and consisted of silty sand (SM), lean clays (CL) and clayey sand (SC). The Standard Penetration Test (SPT) values in these soils range from 5 blows per foot (bpf) to 50 bpf. Support of floor slabs and pavements on or above existing fill materials is discussed in this report. However, even with the recommended construction procedures, there is inherent risk for the owner that compressible fill or unsuitable material, within or buried by the fill, will not be discovered. This risk of unforeseen conditions cannot be eliminated without completely removing the existing fill but can be reduced by following the recommendations contained in this report. To take advantage of the cost benefit of not removing the entire amount of undocumented fill, the owner must be willing to accept the risk associated with building over the undocumented fills following the recommended reworking of the material. Should this be the case, development can be supported on a shallow foundation system. The Shallow Foundations section addresses support of the structures bearing on existing fill, residual soils, or engineered fill. The Floor Slabs section addresses slab-on-grade support of the building. The site appears suitable for the proposed construction based upon geotechnical conditions encountered in the borings and our current understanding of the proposed development. The Pavements section addresses the design of pavement systems. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 5 Geotechnical engineering recommendations for foundation systems and other earth connected phases of the project are outlined below. The recommendations contained in this report are based upon the results of data presented herein, engineering analyses, and our current understanding of the proposed project. The General Comments section provides an understanding of the report limitations. EARTHWORK Earthwork is anticipated to include pavement removal, excavations, and fill placement. The following sections provide recommendations for use in preparation of specifications. Recommendations include quality criteria necessary, to appropriately prepare the site. Graded aggregate base below the asphalt may be used in the fill if separated from the asphalt. Earthwork on the project should be observed and evaluated by Terracon. The evaluation of earthwork should include observation and testing of engineered fill, subgrade preparation, foundation bearing soils, and other geotechnical conditions exposed during the construction of the project. Site Preparation We anticipate construction will be initiated by stripping topsoil, asphalt pavement, and loose, soft or otherwise unsuitable material. Stripped materials consisting of vegetation and organic materials should be wasted off site or used to vegetate landscaped areas or exposed slopes after completion of grading operations. Stripping depths between our boring locations and across the site could vary; as such we recommend actual stripping depths be evaluated by Terracon during construction to aid in preventing removal of excess material. After stripping and removal of unsuitable materials, proofrolling should be performed with heavy rubber tire construction equipment such as a loaded scraper or fully loaded tandem-axle dump truck. A Terracon geotechnical engineer or his representative should observe proofrolling to aid in locating unstable subgrade materials and/or buried debris. Proofrolling should be performed after a suitable period of dry weather to avoid degrading an otherwise acceptable subgrade and to reduce the amount of undercutting / remedial work required. Unstable materials identified should be stabilized as directed by the engineer based on conditions observed during construction. Undercut and replacement and densification in-place are typical remediation methods. It is anticipated that some of the loose materials encountered in the upper 5 feet of the foundation areas in borings B-2 and B-8 will need to be undercut and replaced. It is strongly recommended that earthwork be performed during the dryer months of the year to help reduce the amount of reworking and undercutting that will be required. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 6 Existing Fill Borings B-2 and B-8 encountered relatively loose soils in the upper 3 to 5 feet below existing round surface. Support of footings, floor slabs, and pavements, on or above existing fill soils, is discussed in this report. However, even with the recommended construction procedures, there is inherent risk for the owner that compressible fill or unsuitable material, within or buried by the fill, will not be discovered. This risk of unforeseen conditions cannot be eliminated without completely removing the existing fill but can be reduced by following the recommendations contained in this report. If the owner elects to construct pavements on the existing fill, the following protocol should be followed. Once the planned subgrade elevation has been reached the entire pavement area should be proofrolled. Areas of soft or otherwise unstable material should be undercut and replaced with either new structural fill or suitable, existing on site materials. Fill Material Types Fill required to achieve design grade should be classified as structural fill and general fill. Structural fill is material used below, or within 10 feet of structures, pavements or slopes. General fill is material used to achieve grade outside of these areas. Soils used for structural and general fill should meet the following material property requirements: Soil Type 1,3,4 USCS Classification Acceptable Parameters (for Structural Fill) Fine Grain CL and ML LL < 45 / PI < 25 More than 25% retained on No. 200 sieve All Locations and Elevations Granular SP, SM, SC, and SW All Locations and Elevations On-Site Soils 2 SC, SM, CL All Locations and Elevations 1. Structural and general fill should consist of materials relatively free of organic matter, debris, and particles larger than about 4 inches. Frozen material should not be used, and fill should not be placed on a frozen subgrade. A sample of each material type should be submitted to the Geotechnical Engineer for evaluation prior to use on this site. 2. A large portion of the existing fill is expected to be suitable for reuse as new fill provided it is free of organics, debris, and unsuitable materials. Terracon should field evaluate existing fill materials for use. 3. All fill material used for grading activities should have a maximum dry density of at least 90 pounds per cubic foot (pcf) as determined by the standard Proctor test (ASTM D 698). 4. Any materials proposed as fill from off-site sources should be tested for compliance with these criteria before being hauled to the site. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 7 Fill Compaction Requirements Structural and general fill should meet the following compaction requirements. Item Structural Fill General Fill Maximum Lift Thickness ■ 8 inches or less in loose thickness when heavy, self-propelled compaction equipment is used ■ 4 to 6 inches in loose thickness when hand-guided equipment (i.e. jumping jack or plate compactor) is used Same as Structural fill Minimum Compaction Requirements 1,2 ■ 98% of max. dry density below foundations and within 1 foot of finished pavement subgrade ■ 95% of max. dry density above foundations, below floor slabs, and more than 1 foot below finished pavement subgrade 92% of max. Moisture Content Range 1,2,3 -3% to +3% of optimum As required to achieve min. compaction requirements 1. Maximum density and optimum water content as determined by the standard Proctor test (ASTM D 698). 2. Fill should be tested for compaction and moisture content during placement. Should the results of the in-place density tests indicate that the specified moisture or compaction requirements have not been met, the area represented by the test should be reworked and retested as required until the specified moisture and compaction requirements are achieved. 3. Moisture levels should be maintained low enough to allow for satisfactory compaction to be achieved without pumping when proofrolled. Grading and Drainage Adequate positive drainage should be provided during construction and maintained throughout the life of the development to prevent an increase in moisture content of the foundation, pavement and backfill materials. Surface water drainage should be controlled to prevent undermining of fill slopes and structures during and after construction. It is recommended that all exposed earth slopes be seeded to provide protection against erosion as soon as possible after completion. Seeded slopes should be protected until the vegetation is established. Sprinkler systems should not be installed behind or in front of walls or near slopes. Earthwork Construction Considerations Upon completion of filling and grading, care should be taken to maintain the subgrade water content prior to construction of floor slabs. Construction traffic over the completed subgrades should be avoided. The site should also be graded to prevent ponding of surface water on the prepared subgrades or in excavations. Water collecting over or adjacent to construction areas should be removed. If the subgrade freezes, desiccates, saturates, or is disturbed, the affected Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 8 material should be removed, or the materials should be scarified, moisture conditioned, and recompacted prior to floor slab construction. Surface water should not be allowed to pond on the site and soak into the soil during construction. Construction staging should provide drainage of surface water and precipitation away from the building and pavement areas. Any water that collects over or adjacent to construction areas should be promptly removed, along with any softened or disturbed soils. Surface water control in the form of sloping surfaces, drainage ditches and trenches, and sump pits and pumps will be important to avoid ponding and associated delays due to precipitation and seepage. Groundwater was not encountered at the time of boring. Based on our understanding of the proposed development, we do not expect groundwater to significantly affect construction. If groundwater is encountered during construction, some form of temporary or permanent dewatering may be required. Conventional dewatering methods, such as pumping from sumps, should likely be adequate for temporary removal of any groundwater encountered during excavation at the site. Well points would likely be required for significant groundwater flow, or where excavations penetrate groundwater. All excavations should be sloped or braced as required by Occupational Safety and Health Administration (OSHA) regulations to provide stability and safe working conditions. Temporary excavations will probably be required during grading operations. The grading contractor, by his contract, is usually responsible for designing and constructing stable, temporary excavations and should shore, slope or bench the sides of the excavations as required to maintain stability of both the excavation sides and bottom. All excavations should comply with applicable local, state and federal safety regulations, including the current OSHA Excavation and Trench Safety Standards. Construction site safety is the sole responsibility of the contractor who controls the means, methods, and sequencing of construction operations. Under no circumstances shall the information provided herein be interpreted to mean Terracon is assuming responsibility for construction site safety, or the contractor's activities; such responsibility shall neither be implied nor inferred. Construction Observation and Testing The earthwork operations should be observed by the Geotechnical Engineer or his representative. Monitoring should include documentation of adequate removal of vegetation and topsoil, proofrolling, and mitigation of unstable areas delineated by the proofroll. Each lift of compacted fill should be tested, evaluated, and reworked, as necessary, until approved by the Geotechnical Engineer or his representative prior to placement of additional lifts. Any areas that do not meet the compaction specifications should be reworked to achieve compliance. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 9 In areas of foundation excavations, the bearing subgrade should be evaluated by the Geotechnical Engineer or his representative. If unanticipated conditions are encountered, the Geotechnical Engineer should prescribe mitigation options. SHALLOW FOUNDATIONS If the site has been prepared in accordance with the requirements noted in the Earthwork section, the proposed structure can be supported by a shallow, spread footing foundation system bearing on Piedmont residual soils or structural fill extending to Coastal Plain soils. Design recommendations for shallow foundations for the proposed structure are presented in the following paragraphs. Final recommendation will be made after loads for the structures have been analyzed. Shallow foundations may be supported on existing fill soils provided the site has been prepared in accordance with the requirements noted in the Earthwork section. However, even with the recommended construction procedures, there is inherent risk for the owner that compressible fill or unsuitable material, within or buried by the fill, will not be discovered. This risk of unforeseen conditions cannot be eliminated without completely removing the existing fill but can be reduced by following the recommendations contained in this report. Design recommendations for shallow foundations for the proposed structures are presented in the following paragraphs. Design Parameters – Compressive Loads Item Description Maximum Net Allowable Bearing pressure 1, 2 2,500 psf for conventional shallow foundations. Required Bearing Stratum 3 Coastal Plain soils or suitable structural fill. Minimum Foundation Dimensions Columns: 24 inches Continuous: 16 inches Ultimate Coefficient of Sliding Friction 4 0.35 Minimum Embedment below Finished Grade 5 Exterior footings: 18 inches Interior footings: 12 inches Estimated Total Settlement from Structural Loads 2 1-inch total for conventional shallow foundations Estimated Differential Settlement 2, 6 About 1/2 -inch differential Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 10 Item Description 1. The maximum net allowable bearing pressure is the pressure in excess of the minimum surrounding overburden pressure at the footing base elevation. Values assume that exterior grades are no steeper than 5% within 10 feet of structure. 2. Values provided are for maximum loads noted in Project Description. 3. Unsuitable or soft soils should be over-excavated and replaced per the recommendations presented in Earthwork. 4. Can be used to compute sliding resistance where foundations are placed on suitable soil/materials. Should be neglected for foundations subject to net uplift conditions. 5. Embedment necessary to minimize the effects of frost and/or seasonal water content variations. For sloping ground, maintain depth below the lowest adjacent exterior grade within 5 horizontal feet of the structure. 6. Differential settlements are as measured over a span of 40 feet. The allowable foundation bearing pressures apply to dead loads plus design live load conditions. The weight of the foundation concrete below grade may be neglected in dead load computations. Interior footings should bear a minimum of 12 inches below finished grade. Finished grade is the lowest adjacent grade for perimeter footings and floor level for interior footings. Footings, foundations, and masonry walls should be reinforced as necessary to reduce the potential for distress caused by differential foundation movement. The use of joints at openings or other discontinuities in masonry walls is recommended. Foundation excavations should be observed by the geotechnical engineer. If the soil conditions encountered differ from those presented in this report, supplemental recommendations will be required. Foundation Construction Considerations As noted in the Earthwork section, soils exposed in footing excavations should be evaluated by the Geotechnical Engineer or his representative. The base of all foundation excavations should be free of water and loose soil, prior to placing concrete. Concrete should be placed soon after excavating to reduce bearing soil disturbance. Should the soils at the bearing level become excessively dry, disturbed or saturated, or frozen, the affected soil should be removed prior to placing concrete. Place a lean concrete mud-mat over the bearing soils if the excavations must remain open over night or for an extended period of time. It is recommended that the geotechnical engineer be retained to observe and test the soil foundation bearing materials. If unsuitable bearing soils are encountered at the base of the planned footing excavation, the excavation should be extended deeper to suitable soils, and the footings could bear directly on these soils at the lower level or on lean concrete backfill placed in the excavations. This is illustrated on the sketch below. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 11 Over-excavation for structural fill placement below footings should extend laterally beyond all edges of the footings at least 8 inches per foot of overexcavation depth below footing base elevation. The overexcavation should then be backfilled up to the footing base elevation with well-graded granular material placed in lifts of 8 inches or less in loose thickness and compacted to at least 95 percent of the material's maximum standard Proctor dry density (ASTM D-698). The overexcavation and backfill procedure is described in the figure below. SEISMIC CONSIDERATIONS Seismic Site Classification The seismic design requirements for buildings and other structures are based on Seismic Design Category. Site Classification is a required component in determining the Seismic Design Category for a structure. The Site Classification is based on the upper 100 feet of the site subsurface profile defined by a weighted average value of either shear wave velocity, standard penetration resistance, or undrained shear strength in accordance with Section 20.4 of ASCE 7-16 and the Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 12 International Building Code (IBC) 2018. For this project we used standard penetration resistance to generate site classification. Code Used Site Classification 2018 International Building Code (IBC) 1 D2 1. In general accordance with the 2018 International Building Code. 2. The 2018 International Building Code (IBC) requires a site soil profile determination extending a depth of 100 feet for seismic site classification. The current scope requested does not include the required 100-foot soil profile determination. The borings for the building extended to a maximum depth of approximately 20 feet and this seismic site class definition considers that medium dense to dense clayey sand continues below the maximum depth of the subsurface exploration. Additional exploration to deeper depths could be performed to confirm the conditions below the current depth of exploration. Alternatively, a geophysical exploration could be utilized in order to attempt to justify a higher seismic site class. FLOOR SLABS Depending upon the finished floor elevation, unsuitable, weak, loose soils may be encountered at the floor slab subgrade level. If encountered, these soils should be replaced with structural fill so the floor slab is supported on at least 2 feet of compacted suitable soils. Proofrolling, as stated above, should serve to identify those areas where undercutting and replacement is needed. Design parameters for floor slabs assume the requirements for Earthwork have been followed. Specific attention should be given to positive drainage away from the structure and positive drainage of the aggregate base beneath the floor slab. Floor Slab Design Parameters Item Description Interior floor system Slab-on-grade concrete. Floor Slab Support 1 Minimum 12 inches of approved on-site or imported soils placed and compacted in accordance with the Earthwork section of this report.2,3 Subbase 3 4-inch compacted layer of free draining, granular subbase material 1. Floor slabs should be structurally independent of building footings or walls to reduce the possibility of floor slab cracking caused by differential movements between the slab and foundation. 2. We recommend subgrades be maintained at the proper moisture condition until floor slabs are constructed. If the subgrade should become desiccated prior to construction, the affected material should be removed or the materials scarified, moistened, and recompacted. Upon completion of grading operations in the building areas, care should be taken to maintain the recommended subgrade moisture content and density prior to construction of the building floor slabs. 3. The floor slab design should include a capillary break, comprised of free-draining, compacted, granular material, at least 4 inches thick. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 13 A subgrade prepared and tested as recommended in this report should provide adequate support for lightly loaded floor slabs. The use of a vapor retarder should be considered beneath concrete slabs on grade that will be covered with wood, tile, carpet, or other moisture sensitive or impervious coverings, or when the slab will support equipment sensitive to moisture. When conditions warrant the use of a vapor retarder, the slab designer should refer to ACI 302 and/or ACI 360 for procedures and cautions regarding the use and placement of a vapor retarder/barrier. Saw-cut control joints should be placed in the slab to help control the location and extent of cracking. For additional recommendations refer to the ACI Design Manual. Joints or cracks should be sealed with a water-proof, non-extruding compressible compound. Where floor slabs are tied to perimeter walls or turn-down slabs to meet structural or other construction objectives, our experience indicates differential movement between the walls and slabs will likely be observed in adjacent slab expansion joints or floor slab cracks beyond the length of the structural dowels. The Structural Engineer should account for potential differential settlement through use of sufficient control joints, appropriate reinforcing or other means. Floor Slab Construction Considerations On most project sites, the site grading is generally accomplished early in the construction phase. However, as construction proceeds, the subgrade may be disturbed due to utility excavations, construction traffic, desiccation, rainfall, etc. As a result, the floor slab subgrade may not be suitable for placement of base rock and concrete and corrective action may be required. We recommend the area underlying the floor slab be rough graded and then thoroughly proofrolled with a loaded tandem axle dump truck prior to final grading and placement of base rock. Attention should be paid to high traffic areas that were rutted and disturbed earlier and to areas where backfilled trenches are located. Areas where unsuitable conditions are located should be repaired by removing and replacing the affected material with properly compacted fill. All floor slab subgrade areas should be moisture conditioned and properly compacted to the recommendations in this report immediately prior to placement of the base rock and concrete. Finished subgrade, within and for at least 10 feet beyond the floor slab, should be protected from traffic, rutting, or other disturbance and maintained in a relatively moist condition until floor slabs are constructed. If the subgrade should become damaged or desiccated prior to construction of floor slabs, the affected material should be removed, and structural fill should be added to replace the resulting excavation. Final conditioning of the finished subgrade should be performed immediately prior to placement of the floor slab support course. The Geotechnical Engineer should observe the condition of the floor slab subgrades immediately prior to placement of the floor slab support course, reinforcing steel, and concrete. Attention should Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 14 be paid to high traffic areas that were rutted and disturbed earlier, and to areas where backfilled trenches are located. PAVEMENTS Subgrade Preparation On this site, the site grading will be accomplished relatively early in the construction phase. Fills are placed and compacted in a uniform manner. However, as construction proceeds, excavations are made into these areas, rainfall and surface water saturates some areas, heavy traffic from concrete trucks and other delivery vehicles disturbs the subgrade and many surface irregularities are filled in with loose soils to improve trafficability temporarily. As a result, the pavement subgrades, initially prepared early in the project, should be carefully evaluated as the time for pavement construction approaches. We recommend the moisture content and density of the top 12 inches of the subgrade be evaluated and the pavement subgrades be proofrolled within two days prior to commencement of actual paving operations. Areas not in compliance with the required ranges of moisture or density should be moisture conditioned and recompacted. Particular attention should be paid to high traffic areas that were rutted and disturbed earlier and to areas where backfilled trenches are located. Areas where unsuitable conditions are located should be repaired by removing and replacing the materials with properly compacted fills. If a significant precipitation event occurs after the evaluation or if the surface becomes disturbed, the subgrade should be reviewed by qualified personnel immediately prior to paving. The subgrade should be in its finished form at the time of the final review. After proofrolling and repairing subgrade deficiencies, the entire subgrade should be scarified and developed as recommended in the Earthwork section of this report to provide a uniform subgrade for pavement construction. Areas that appear severely desiccated following site stripping may require further undercutting and moisture conditioning. Pavement Design Considerations Traffic patterns and anticipated loading conditions were not available at the time that this report was prepared. However, we anticipate that traffic loads will be produced primarily by automobile traffic and occasional delivery and trash removal trucks. The thickness of pavements subjected to heavy truck traffic should be determined using expected traffic volumes, vehicle types, and vehicle loads and should be in accordance with local, city or county ordinances. Pavement thickness can be determined using AASHTO, Asphalt Institute and/or other methods if specific wheel loads, axle configurations, frequencies, and desired pavement life are provided. Terracon can provide thickness recommendations for pavements subjected to loads other than personal vehicle and occasional delivery and trash removal truck traffic if this information is provided. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 15 Pavement performance is affected by its surroundings. In addition to providing preventive maintenance, the civil engineer should consider the following recommendations in the design and layout of pavements: ◼ Final grade adjacent to parking lots and drives should slope down from pavement edges at a minimum 2%; ◼ The subgrade and the pavement surface should have a minimum ¼ inch per foot slope to promote proper surface drainage; ◼ Install pavement drainage surrounding areas anticipated for frequent wetting (e.g., garden centers, wash racks); ◼ Install joint sealant and seal cracks immediately; ◼ Seal all landscaped areas in, or adjacent to pavements to reduce moisture migration to subgrade soils; ◼ Place compacted, low permeability backfill against the exterior side of curb and gutter; and, ◼ Place curb, gutter and/or sidewalk directly on low permeability subgrade soils rather than on unbound granular base course materials. Estimates of Minimum Pavement Thickness As a minimum, we recommend the following typical pavement sections be considered. Asphalt (AC) Pavement Material Light Duty1 Thickness (inches) Heavy Duty2 Thickness (inches) GDOT Subgrade Upper 12 inches of existing soil or engineered fill Upper 12 inches of existing soil or engineered fill 98% of Standard Proctor MMD, -2 to +3% OMC Aggregate Base 6 8 GAB, Section 815 and 310 Asphalt Binder Course - 1¾ SP19 - Section 400, 424, 824 and 828 Asphalt Surface Course 2 1¼ SP9.5 - Section 400, 424, 824 and 828 1. Automobiles only. 2. Combined automobiles and trucks The graded aggregate base should be compacted to a minimum of 98 percent of the material’s modified Proctor (ASTM D-1557, Method C) maximum dry density. Where base course thickness exceeds 6 inches, the material should be placed and compacted in two or more lifts of equal thickness. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 16 The listed pavement component thicknesses should be used as a guide for pavement systems at the site for the traffic classifications stated herein. These recommendations assume a 20-year pavement design life. If pavement frequencies or loads will be different than that specified GEC should be contacted and allowed to review these pavement sections. Asphalt concrete aggregates and base course materials should conform to the Georgia Department of Transportation (GDOT) "Standard Specifications for Construction of Transportation System”. Portland Cement Concrete (PCC) Pavement Material Light Duty1 Thickness (inches) Heavy Duty2 Thickness (inches) Reference Subgrade Upper 12 inches of existing soil or engineered fill Upper 12 inches of existing soil or engineered fill GDOT: 98% of Standard Proctor MMD, -3 to +3% OMC Aggregate Base 4 4 GDOT: GAB, Section 815 and 310 PCC 5 6 ½ ACI 1. Automobiles only. 2. Combined automobiles and trucks. We recommend a Portland cement concrete (PCC) pavement be utilized in entrance and exit sections, dumpster pads, loading dock areas, or other areas where extensive wheel maneuvering are expected. The dumpster pad should be large enough to support the wheels of the truck which will bear the load of the dumpster. Although not required for structural support, the base course layer is recommended to help reduce potentials for slab curl, shrinkage cracking, and subgrade “pumping” through joints. Proper joint spacing will also be required to prevent excessive slab curling and shrinkage cracking. All joints should be sealed to prevent entry of foreign material and dowelled where necessary for load transfer. Portland cement concrete should be designed with proper air-entrainment and have a minimum compressive strength of 4,000 psi after 28 days of laboratory curing. Adequate reinforcement and number of longitudinal and transverse control joints should be placed in the rigid pavement in accordance with ACI requirements. The joints should be sealed as soon as possible (in accordance with sealant manufacturer’s instructions and ACI requirements) to minimize infiltration of water into the soil. Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 17 Pavement Drainage Pavements should be sloped to provide rapid drainage of surface water. Water allowed to pond on or adjacent to the pavements could saturate the subgrade and contribute to premature pavement deterioration. In addition, the pavement subgrade should be graded to provide positive drainage within the granular base section. We recommend drainage be included at the bottom of the GAB layer at the storm structures to aid in removing water that may enter this layer. Drainage could consist of small diameter weep holes excavated around the perimeter of the storm structures. The weep holes should be excavated at the elevation of the GAB and soil interface. The excavation should be covered with No. 57 stone which is encompassed in Mirafi 140 NL or approved equivalent which will aid in reducing fines from entering the storm system. Pavement Maintenance The pavement sections provided in this report represent minimum recommended thicknesses and, as such, periodic maintenance should be anticipated. Therefore, preventive maintenance should be planned and provided for through an on-going pavement management program. Preventive maintenance activities are intended to slow the rate of pavement deterioration, and to preserve the pavement investment. Preventive maintenance consists of both localized maintenance (e.g., crack and joint sealing and patching) and global maintenance (e.g., surface sealing). Preventive maintenance is usually the first priority when implementing a planned pavement maintenance program and provides the highest return on investment for pavements. Prior to implementing any maintenance, additional engineering observation is recommended to determine the type and extent of preventive maintenance. Even with periodic maintenance, some movements and related cracking may still occur, and repairs may be required. GENERAL COMMENTS Our analysis and opinions are based upon our understanding of the project, the geotechnical conditions in the area, and the data obtained from our site exploration. Natural variations will occur between exploration point locations or due to the modifying effects of construction or weather. The nature and extent of such variations may not become evident until during or after construction. Terracon should be retained as the Geotechnical Engineer, where noted in this report, to provide observation and testing services during pertinent construction phases. If variations appear, we can provide further evaluation and supplemental recommendations. If variations are noted in the absence of our observation and testing services on-site, we should be immediately notified so that we can provide evaluation and supplemental recommendations. Our Scope of Services does not include either specifically or by implication any environmental or biological (e.g., mold, fungi, bacteria) assessment of the site or identification or prevention of Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable 18 pollutants, hazardous materials or conditions. If the owner is concerned about the potential for such contamination or pollution, other studies should be undertaken. Our services and any correspondence or collaboration through this system are intended for the sole benefit and exclusive use of our client for specific application to the project discussed and are accomplished in accordance with generally accepted geotechnical engineering practices with no third-party beneficiaries intended. Any third-party access to services or correspondence is solely for information purposes to support the services provided by Terracon to our client. Reliance upon the services and any work product is limited to our client and is not intended for third parties. Any use or reliance of the provided information by third parties is done solely at their own risk. No warranties, either express or implied, are intended or made. Site characteristics as provided are for design purposes and not to estimate excavation cost. Any use of our report in that regard is done at the sole risk of the excavating cost estimator as there may be variations on the site that are not apparent in the data that could significantly impact excavation cost. Any parties charged with estimating excavation costs should seek their own site characterization for specific purposes to obtain the specific level of detail necessary for costing. Site safety, and cost estimating including, excavation support, and dewatering requirements/design are the responsibility of others. If changes in the nature, design, or location of the project are planned, our conclusions and recommendations shall not be considered valid unless we review the changes and either substantiate or modify our conclusions in writing. Responsive ■ Resourceful ■ Reliable ATTACHMENTS Geotechnical Engineering Report New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Responsive ■ Resourceful ■ Reliable EXPLORATION AND TESTING PROCEDURES 1 of 1 EXPLORATION AND TESTING PROCEDURES Field Exploration Number of Borings Boring Depth (feet) 2 10 (or auger refusal) 6 20 (or auger refusal) Boring Layout and Elevations: Passero provided the boring layout for the site as shown on the provided site plan. Coordinates were obtained with a handheld GPS unit. If elevations and a more precise boring layout are desired, we recommend borings be surveyed following completion of fieldwork. Subsurface Exploration Procedures: We advanced the borings a truck-mounted drill rig using continuous flight hollow stem. Four samples were obtained in the upper 10 feet of each boring and at intervals of 5 feet thereafter. In the split-barrel sampling procedure, a standard 2-inch outer diameter split-barrel sampling spoon was driven into the ground by a 140-pound automatic hammer falling a distance of 30 inches. The number of blows required to advance the sampling spoon the last 12 inches of a normal 18-inch penetration is recorded as the Standard Penetration Test (SPT) resistance value. The SPT resistance values, also referred to as N-values, are indicated on the boring logs at the test depths. We observed and recorded groundwater levels during drilling and sampling. For safety purposes, all borings were backfilled with auger cuttings after their completion. The sampling depths, penetration distances, and other sampling information was recorded on the field boring logs. The samples were placed in appropriate containers and taken to our soil laboratory for testing and classification by a Geotechnical Engineer. Our exploration team prepared field boring logs as part of the drilling operations. These field logs included visual classifications of the materials encountered during drilling and our interpretation of the subsurface conditions between samples. Final boring logs were prepared from the field logs. The final boring logs represent the Geotechnical Engineer's interpretation of the field logs and include modifications based on observations and tests of the samples in our laboratory. Responsive ■ Resourceful ■ Reliable SITE LOCATION AND EXPLORATION PLANS Contents: Site Location Plan Exploration Plan Note: All attachments are one page unless noted above SITE LOCATION New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Note to Preparer: This is a large table with outside borders. Just click inside the table above this text box, then paste your GIS Toolbox image. When paragraph markers are turned on you may notice a line of hidden text above and outside the table – please leave that alone. Limit editing to inside the table. The line at the bottom about the general location is a separate table line. You can edit it as desired, but try to keep to a single line of text to avoid reformatting the page. DIAGRAM IS FOR GENERAL LOCATION ONLY, AND IS NOT INTENDED FOR CONSTRUCTION PURPOSES MAP PROVIDED BY MICROSOFT BING MAPS SITE EXPLORATION PLAN New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Note to Preparer: This is a large table with outside borders. Just click inside the table above this text box, then paste your GIS Toolbox image. When paragraph markers are turned on you may notice a line of hidden text above and outside the table – please leave that alone. Limit editing to inside the table. The line at the bottom about the general location is a separate table line. You can edit it as desired, but try to keep to a single line of text to avoid reformatting the page. DIAGRAM IS FOR GENERAL LOCATION ONLY, AND IS NOT INTENDED FOR CONSTRUCTION PURPOSES MAP PROVIDED BY MICROSOFT BING MAPS EXPLORATION PLAN New Corporate GA Terminal ■ Macon, Georgia April 18, 2023 ■ Terracon Project No. HN225215 Note to Preparer: This is a large table with outside borders. Just click inside the table above this text box, then paste your GIS Toolbox image. When paragraph markers are turned on you may notice a line of hidden text above and outside the table – please leave that alone. Limit editing to inside the table. The line at the bottom about the general location is a separate table line. You can edit it as desired, but try to keep to a single line of text to avoid reformatting the page. BORING LOCATION PLAN DIAGRAM IS FOR GENERAL LOCATION ONLY, AND IS NOT INTENDED FOR CONSTRUCTION PURPOSES MAP PROVIDED BY MICROSOFT BING MAPS EXPLORATION RESULTS Contents: Boring Logs (B-1 through B-8) Note: All attachments are one page unless noted above. ASPHALT, Approx 6" of asphalt COASTAL PLAIN - LEAN CLAY (CL), red, stiff Boring Terminated at 10 Feet Boring Log No. B-1 Water Level Observations Depth (Ft.) 5 10 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.5 10.0 3-4-6 N=10 5-6-6 N=12 4-6-7 N=13 5-6-8 N=14 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7020° Longitude: -83.6481° Depth (Ft.) Field Test Results ASPHALT, Approx 3" of asphalt FILL - LEAN CLAY (CL), red, medium stiff COASTAL PLAIN - LEAN CLAY (CL), red, stiff Boring Terminated at 10 Feet Boring Log No. B-2 Water Level Observations Depth (Ft.) 5 10 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.3 3.0 10.0 1-2-2 N=4 2-3-3 N=6 6-7-8 N=15 8-10-11 N=21 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7020° Longitude: -83.6478° Depth (Ft.) Field Test Results TOPSOIL, Approx 4" of topsoil COASTAL PLAIN - CLAYEY SAND (SC), black, Hand auger for upper 5' LEAN CLAY (CL), red, very stiff CLAYEY SAND (SM), with quartz fragments, red white, very dense SILTY SAND (SM), white yellow, medium dense CLAYEY SAND (SC), white pink red, medium dense CLAYEY SAND (SC), with quartz fragments, purple white, medium dense Boring Terminated at 30 Feet Boring Log No. B-3 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.3 3.0 12.0 16.0 22.0 26.0 30.0 3-6-10 N=16 4-5-10 N=15 15-23-27 N=50 6-8-8 N=16 9-14-15 N=29 9-10-7 N=17 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7019° Longitude: -83.6482° Depth (Ft.) Field Test Results TOPSOIL, Approx 4" of topsoil COASTAL PLAIN - LEAN CLAY (CL), black red, very stiff, Hand auger for upper 5' CLAYEY SAND (SC), with rock fragments, red, medium dense CLAYEY SAND (SC), pink orange, medium dense Boring Terminated at 30 Feet Boring Log No. B-4 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.3 12.0 22.0 30.0 8-8-9 N=17 10-10-9 N=19 8-9-10 N=19 8-10-13 N=23 8-9-9 N=18 9-10-12 N=22 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7019° Longitude: -83.6479° Depth (Ft.) Field Test Results TOPSOIL, Approx 6" of topsoil COASTAL PLAIN - LEAN CLAY (CL), red, stiff, Hand auger for upper 2.5' LEAN CLAY (CL), red brown, stiff CLAYEY SAND (SC), with rock fragments, orange white, medium dense SILTY SAND (SM), white, medium dense Boring Terminated at 30 Feet Boring Log No. B-5 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.5 6.0 12.0 22.0 30.0 3-4-4 N=8 4-6-7 N=13 6-8-8 N=16 6-10-11 N=21 6-6-7 N=13 5-6-9 N=15 4-8-9 N=17 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7019° Longitude: -83.6476° Depth (Ft.) Field Test Results ASPHALT, Approx 6" of asphalt COASTAL PLAIN - LEAN CLAY (CL), red, medium stiff to very stiff CLAYEY SAND (SC), red, medium dense SILTY SAND (SM), red grey, medium dense CLAYEY SAND (SC), orange red, medium dense SILTY SAND (SM), yellow, medium dense CLAYEY SAND (SC), yellow, medium dense Boring Terminated at 30 Feet Boring Log No. B-6 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.5 8.0 12.0 16.0 22.0 26.0 30.0 4-4-3 N=7 8-5-3 N=8 4-6-10 N=16 7-9-10 N=19 5-8-12 N=20 10-10-9 N=19 6-6-9 N=15 7-7-7 N=14 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7015° Longitude: -83.6481° Depth (Ft.) Field Test Results TOPSOIL, Approx. 4" of topsoil COASTAL PLAIN - LEAN CLAY (CL), red to black, medium stiff to very stiff CLAYEY SAND (SC), red to tan, medium dense LEAN CLAY (CL), gray to red, stiff SILTY SAND (SM), with quartz, red to tan, medium dense SILTY SAND (SM), white, medium dense Boring Terminated at 30 Feet Boring Log No. B-7 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.3 12.0 16.0 22.0 26.0 30.0 6-8-10 N=18 2-2-3 N=5 3-4-5 N=9 4-4-6 N=10 10-10-9 N=19 3-3-5 N=8 5-7-8 N=15 8-8-11 N=19 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7015° Longitude: -83.6479° Depth (Ft.) Field Test Results ASPHALT, Approx 3" of asphalt FILL - LEAN CLAY (CL), black red, soft to medium stiff COASTAL PLAIN - LEAN CLAY (CL), red, medium stiff CLAYEY SAND (SC), red orange, medium dense SILTY SAND (SM), with rock fragments, white, medium dense Boring Terminated at 30 Feet Boring Log No. B-8 Water Level Observations Depth (Ft.) 5 10 15 20 25 30 Facilities | Environmental | Geotechnical | Materials Graphic Log 0.3 8.0 12.0 22.0 30.0 1-1-2 N=3 2-3-3 N=6 1-3-2 N=5 2-2-4 N=6 4-5-5 N=10 5-7-10 N=17 4-5-5 N=10 5-6-8 N=14 Abandonment Method Boring backfilled with auger cuttings upon completion. Advancement Method HSA 2.25" Hammer Type Automatic Driller C. SHUBERT Boring Started 03-03-2023 Boring Completed 03-03-2023 514 Hillcrest Indust. Blvd Drill Rig CME 550X New Corporate GA Terminal Macon, GA 1000 Terminal Drive | Macon, GA Terracon Project No. HN225215 See Exploration and Testing Procedures for a description of field and laboratory procedures used and additional data (If any). See Supporting Information for explanation of symbols and abbreviations. Notes Elevation Reference: Elevations not obtained Water Level Observations Not encountered Sample Type See Exploration Plan Location: Latitude: 32.7015° Longitude: -83.6476° Depth (Ft.) Field Test Results SUPPORTING INFORMATION Contents: General Notes Unified Soil Classification System Note: All attachments are one page unless noted above. GENERAL NOTES result of local practice or professional judgment. SAMPLING WATER LEVEL FIELD TESTS Standard Penetration Test Water Initially Encountered Water Level After a Specified Period of Time Water Level After a Specified Period of Time Cave In Encountered Water levels indicated on the soil boring logs are the levels measured in the borehole at the times indicated. Groundwater level variations will occur over time. In low permeability soils, accurate determination of groundwater levels is not possible with short term water level observations. N (HP) (T) (DCP) UC (PID) (OVA) Standard Penetration Test Resistance (Blows/Ft.) Hand Penetrometer Torvane Dynamic Cone Penetrometer Unconfined Compressive Strength Photo-Ionization Detector Organic Vapor Analyzer DESCRIPTIVE SOIL CLASSIFICATION Soil classification as noted on the soil boring logs is based Unified Soil Classification System. Where sufficient laboratory data exist to classify the soils consistent with ASTM D2487 "Classification of Soils for Engineering Purposes" this procedure is used. ASTM D2488 "Description and Identification of Soils (Visual-Manual Procedure)" is also used to classify the soils, particularly where insufficient laboratory data exist to classify the soils in accordance with ASTM D2487. In addition to USCS classification, coarse grained soils are classified on the basis of their in-place relative density, and fine-grained soils are classified on the basis of their consistency. See "Strength Terms" table below for details. The ASTM standards noted above are for reference to methodology in general. In some cases, variations to methods are applied as a LOCATION AND ELEVATION NOTES Exploration point locations as shown on the Exploration Plan and as noted on the soil boring logs in the form of Latitude and Longitude are approximate. See Exploration and Testing Procedures in the report for the methods used to locate the exploration points for this project. Surface elevation data annotated with +/- indicates that no actual topographical survey was conducted to confirm the surface elevation. Instead, the surface elevation was approximately determined from topographic maps of the area. STRENGTH TERMS RELATIVE DENSITY OF COARSE-GRAINED SOILS (More than 50% retained on No. 200 sieve.) Density determined by Standard Penetration Resistance CONSISTENCY OF FINE-GRAINED SOILS (50% or more passing the No. 200 sieve.) Consistency determined by laboratory shear strength testing, field visual-manual procedures or standard penetration resistance Descriptive Term (Density) Standard Penetration or N-Value Blows/Ft. Descriptive Term (Consistency) Unconfined Compressive Strength Qu, (tsf) Standard Penetration or N-Value Blows/Ft. Very Loose 0 - 3 Very Soft 0 - 1 Loose 4 - 9 Soft 2 - 4 Medium Dense 10 - 29 Medium Stiff 4 - 8 Dense 30 - 50 Stiff 8 - 15 Very Dense > 50 Very Stiff 15 - 30 Hard > 4.00 > 30 RELEVANCE OF SOIL BORING LOG The soil boring logs contained within this document are intended for application to the project as described in this document. Use of these soil boring logs for any other purpose may not be appropriate. less than 0.25 0.25 to 0.50 0.50 to 1.00 1.00 to 2.00 2.00 to 4.00 UNIFIED SOIL CLASSIFICATION SYSTEM UNIFIED SOIL C LASSIFICA TION SYSTEM Criteria for Assigning Group Symbols and Group Names Using Laboratory Tests A Soil Classification Group Symbol Group Name B Coarse-Grained Soils: More than 50% retained on No. 200 sieve Gravels: More than 50% of coarse fraction retained on No. 4 sieve Clean Gravels: Less than 5% fines C Cu  4 and 1  Cc  3 E GW Well-graded gravel F Cu  4 and/or [Cc<1 or Cc>3.0] E GP Poorly graded gravel F Gravels with Fines: More than 12% fines C Fines classify as ML or MH GM Silty gravel F, G, H Fines classify as CL or CH GC Clayey gravel F, G, H Sands: 50% or more of coarse fraction passes No. 4 sieve Clean Sands: Less than 5% fines D Cu  6 and 1  Cc  3 E SW Well-graded sand I Cu  6 and/or [Cc<1 or Cc>3.0] E SP Poorly graded sand I Sands with Fines: More than 12% fines D Fines classify as ML or MH SM Silty sand G, H, I Fines classify as CL or CH SC Clayey sand G, H, I Fine-Grained Soils: 50% or more passes the No. 200 sieve Silts and Clays: Liquid limit less than 50 Inorganic: PI  7 and plots on or above “A” line J CL Lean clay K, L, M PI  4 or plots below “A” line J ML Silt K, L, M Organic: Liquid limit - oven dried  0.75 OL Organic clay K, L, M, N Liquid limit - not dried Organic silt K, L, M, O Silts and Clays: Liquid limit 50 or more Inorganic: PI plots on or above “A” line CH Fat clay K, L, M PI plots below “A” line MH Elastic Silt K, L, M Organic: Liquid limit - oven dried  0.75 OH Organic clay K, L, M, P Liquid limit - not dried Organic silt K, L, M, Q Highly organic soils: Primarily organic matter, dark in color, and organic odor PT Peat A Based on the material passing the 3-inch (75-mm) sieve. B If field sample contained cobbles or boulders, or both, add “with cobbles or boulders, or both” to group name. C Gravels with 5 to 12% fines require dual symbols: GW-GM well-graded gravel with silt, GW-GC well-graded gravel with clay, GP-GM poorly graded gravel with silt, GP-GC poorly graded gravel with clay. D Sands with 5 to 12% fines require dual symbols: SW-SM well-graded sand with silt, SW-SC well-graded sand with clay, SP-SM poorly graded sand with silt, SP-SC poorly graded sand with clay. E Cu = D60/D10 Cc = 60 10 2 30 D x D ) (D F If soil contains  15% sand, add “with sand” to group name. G If fines classify as CL-ML, use dual symbol GC-GM, or SC-SM. H If fines are organic, add “with organic fines” to group name. I If soil contains  15% gravel, add “with gravel” to group name. J If Atterberg limits plot in shaded area, soil is a CL-ML, silty clay. K If soil contains 15 to 29% plus No. 200, add “with sand” or “with gravel,” whichever is predominant. L If soil contains  30% plus No. 200 predominantly sand, add “sandy” to group name. M If soil contains  30% plus No. 200, predominantly gravel, add “gravelly” to group name. N PI  4 and plots on or above “A” line. O PI  4 or plots below “A” line. P PI plots on or above “A” line. Q PI plots below “A” line. DI 12" RIM:355.42 INV:354.82 BM-4 GRID N: 983294.33 GRID E: 2456098.09 ELEV: 356.13 BM-5 GRID N: 983154.68 GRID E: 2456100.33 ELEV: 356.18 AC SSMH RIM: 356.22 INV A:346.42 INV B:346.57 INV C:346.52 INV D:346.42 FFE=355.75 FFE=357.34 SSMH RIM: 355.26 INV A:348.51 INV B:348.56 INV C:348.56 INV D:348.46 X X G G G G G G G G TV TV P,T,TV P,T,TV P,T,TV P,T,TV W W W W W W W SAN SAN SAN SAN SAN SAN SAN SAN SAN SAN SAN SAN SAN SAN 4" IRON 4" IRON DI RIM:355.18 INV:353.18 DI RIM:355.05 INV:353.05 356 355 356 356 356 PARCEL P140-0095 NOW OR FORMERLY MACON-BIBB COUNTY DB. 9716 PG. 371 UP UP UP UP UP UP UP UP P,T T T T T P,T,TV T,TV 8" CLAY SAN SAN SAN A B C D D B C A 8" CLAY 8" CLAY 10" CLAY 10" CLAY 8" CLAY 8" CLAY 8" CLAY DI RIM:355.26 INV:353.81 DI RIM: 354.44 INV A:349.29 INV B:341.44 INV C:351.99 INV D:348.59 INV E:341.34 INV:353.60 8" CLAY DI RIM:354.68 INV:351.81 FFE=355.54 AC FFE=356.34 X X UP UP UP UP UP UP UP UP UP UP W W W 18" RCP KEYPAD VAULT W W 4" PVC A B C D E 8" IRON MOTOR AC 11031 358.08 FACE 24" RCP 24" RCP A B CONCRETE BOTTOM 2" PIPE DRAINS FROM STRUCTURE NO VISIBLE TIE AC AWNING DI 12.62 RIM:354.49 INV A:341.89 INV B:341.87 8" IRON 24" RCP 355 355 355 356 356 UP GATE 356 356 DI RIM:354.97 INV:349.89 of Checked By: R.L.S. No.: 524 S. HOUSTON LAKE ROAD, SUITE F100 WARNER ROBINS, GEORGIA 31088 OFFICE (478) 971-3382 WWW.WELLSTONASSOC.COM Project No.: Drawing No.: THESE DOCUMENTS, AS INSTRUMENTS OF SERVICE, REMAIN THE PROPERTY OF WELLSTON ASSOCIATES LAND SURVEYORS, INC. AND NO PART THEREOF MAY BE USED OR REPRODUCED IN ANY FORM WITHOUT WRITTEN PERMISSION. Description Revisions No. Date Scale: Date: Drawn By: Sheet No.: WELLSTON ASSOCIATES LAND SURVEYORS, LLC LANDS OF MACON-BIBB COUNTY AIRPORT LAND LOT 230 4TH LAND DISTRICT MACON-BIBB COUNTY GEORGIA TOPOGRAPHIC SURVEY 1383-001 TPS1A B.J.H. S.H.J. 3171 3/11/24 1"=20' 1 1 1 3/11/24 TOPO ADDED SOUTH OF BUILDING Plotted By:BRETT HART Plot Date: Mar 13, 2024 File Name:1383-001 TPS1A.dwg A w MISCELLANEOUS NOTES 1. THIS DOCUMENT WAS CREATED ELECTRONICALLY. THIS MEDIA SHOULD NOT BE CONSIDERED A CERTIFIED DOCUMENT UNLESS IT HAS BEEN PROPERLY SEALED AND ORIGINALLY SIGNED BY A REGISTERED LAND SURVEYOR AT THE OFFICE OF WELLSTON ASSOCIATES LAND SURVEYORS, LLC AUTHORITY O.C.G.A. 43-15-22. 2. THE UNDERGROUND UTILITIES SHOWN ON THIS DRAWING WERE COMPILED FROM FIELD OBSERVATIONS, UTILITY COMPANY RECORDS AND UNDERGROUND UTILITIES MARKED BY OTHERS WITHOUT BENEFIT OF EXCAVATION. WELLSTON ASSOCIATES LAND SURVEYORS, LLC DOES NOT GUARANTEE THAT ALL UTILITIES ARE SHOWN. VERIFICATION OF UTILITIES SHOULD BE MADE BY THE INDIVIDUAL UTILITY COMPANY PRIOR TO ANY CONSTRUCTION. 3. SUBJECT PROPERTY IS DESIGNATED AS PARCEL P140-0095. 4. WELLSTON ASSOCIATES LAND SURVEYORS, LLC DOES NOT GUARANTEE THAT ALL EASEMENTS WHICH MAY AFFECT THE SUBJECT TRACT ARE SHOWN. 5. ONE FOOT CONTOUR INTERVAL SHOWN. ELEVATIONS SHOWN ARE REFERENCED TO NAVD 88 DATUM. 6. PROPERTY LINES SHOWN HEREON WERE TAKEN FROM MATTERS OF RECORD AND EVIDENCE FOUND IN THE FIELD. THE SURVEYOR'S CERTIFICATION EXTENDS ONLY TO THE TOPOGRAPHIC ASPECTS AND THE TOPOGRAPHIC SURVEY DOES NOT CONSTITUTE A BOUNDARY SURVEY. THIS SURVEY IS NOT TO BE RECORDED AND SHOULD NOT BE USED TO CONVEY PROPERTY. X 355 BENCHMARK CONTOUR LINE SANITARY SEWER MANHOLE STORM DRAIN MANHOLE BOLLARD CLEAN OUT UTILITY POLE GUY WIRE WATER METER WATER VALVE MONITORING WELL FIRE HYDRANT GAS METER ELECTRIC METER ELECTRIC CONTROL PANEL SIGN FENCE STORM SEWER LINE SANITARY SEWER LINE OVERHEAD POWER, TELEPHONE AND CATV ASPHALT CONCRETE LEGEND OF SYMBOLS P,T,TV SAN WATER LINE W GAS LINE G DROP INLET SPOT ELEVATION 950.10 UNDERGROUND POWER UP AC AC UNIT MAIL BOX FINISHED FLOOR ELEVATION F.F.E. ELECTRIC MANHOLE MANHOLE 20 0 20 40 60 GRAPHIC SCALE IN FEET FLAG POLE IRRIGATION CONTROL VALVE
14075
https://www.youtube.com/watch?v=Si15jp8xHFg
Exponents: Simplify 27^{2/3} SVSU Micro Math 4320 subscribers 150 likes Description 37785 views Posted: 28 Feb 2020 Anusha from SVSU Micro Math helps you simplify an expression involving exponents using properties of exponents. This problem involves a fractional power. Problem: Simplify 27^{2/3} Level: beginning/elementary algebra SVSUmicromath Transcript: to simplify this question 27 power two over three we will use the rules here and we'll write our first step 27 power 1 over 3 whole power 2 we just spread the power here and we'll write our next step as cube root of 27 1 over 3 is a cube root and whole power 2 cube root of 27 is 3 just used a calculator and the power 2 remains and we can write this as now 9 3 times 3 is 9 that's our answer
14076
https://www2.cde.ca.gov/cacs/id/web/495
California Department of Education Search the California Content Standards Search the California Content Standards Search the CA Content Standards Individual Standards Arts Career Technical Education Computer Science English Language Arts English Language Development Health Education History–Social Science Mathematics Physical Education School Library Science (CA NGSS) World Languages Search the Standards 7.EE.4.b (Mathematics) Standard Identifier: 7.EE.4.b Content Area: Mathematics Grade: 7 Domain: Expressions and Equations Cluster:Solve real-life and mathematical problems using numerical and algebraic expressions and equations.Standard:Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. Questions: Curriculum Frameworks and Instructional Resources Division | CFIRD@cde.ca.gov | 916-319-0881
14077
https://math.stackexchange.com/questions/1069751/expressing-n-mod-m-in-terms-of-floor-values
Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Expressing n mod m in terms of floor values? Ask Question Asked Modified 2 years, 6 months ago Viewed 2k times 3 $\begingroup$ I'm trying to prove the expression: $$\left\lceil\frac{n}m\right\rceil = \left\lfloor \frac{n+m-1}{m}\right\rfloor\;,$$ where $n,m$ are integers` So I've come across this article (PDF) which gives a nice method of proving the above expression from page $10$ onwards. I understand pretty much the entire proof, except for the definition on page $10$. It states that $$\frac{n}m = \left\lfloor\frac{n}m\right\rfloor+\left{\frac{n}m\right}\;.$$ I've never encountered the ${}$ symbols before in maths, but given that any real quotient can be expressed as the floor of the quotient (the integer part) summed with the fractional part, I'm guessing that the ${}$ symbols state that this is the fractional part of the real number $\frac{n}m$? That's all fine by me, but what I can't understand is the next line. If I wanted to get rid of the $m$ on the L.H.S I'd multiply both sides by $m$. But somehow, according to the pdf, it's true to say that $$m\left{\frac{n}m\right} = n \bmod m\;.$$ Can someone explain why this is the case? Thanks! discrete-mathematics ceiling-and-floor-functions fractional-part Share edited Mar 25, 2023 at 23:42 4LegsDrivenCat 11566 bronze badges asked Dec 15, 2014 at 22:06 kmahon99kmahon99 3344 bronze badges $\endgroup$ Add a comment | 2 Answers 2 Reset to default 4 $\begingroup$ Your interpretation of ${x}$ is correct: this notation is quite often used for the fractional part, ${x}=x-\lfloor x\rfloor$. We want to show that $m\left{\frac{n}m\right}=n\bmod m$, i.e., that $$m\left(\frac{n}m-\left\lfloor\frac{n}m\right\rfloor\right)=n\bmod m\;.$$ This is clearly equivalent to showing that $$n-m\left\lfloor\frac{n}m\right\rfloor=n\bmod m\;.$$ By definition $\left\lfloor\frac{n}m\right\rfloor\le\frac{n}m<\left\lfloor\frac{n}m\right\rfloor+1$, so $$m\left\lfloor\frac{n}m\right\rfloor\le n and after subtracting $m\left\lfloor\frac{n}m\right\rfloor$ throughout, we have $$0\le n-m\left\lfloor\frac{n}m\right\rfloor Thus $n-m\left\lfloor\frac{n}m\right\rfloor$ is an integer in the correct range. Finally, $$n-\left(n-m\left\lfloor\frac{n}m\right\rfloor\right)=m\left\lfloor\frac{n}m\right\rfloor$$ is clearly divisible by $m$, so $$n-m\left\lfloor\frac{n}m\right\rfloor=n\bmod m\;,$$ as desired. Share edited Apr 24, 2022 at 17:30 answered Dec 15, 2014 at 22:16 Brian M. ScottBrian M. Scott 633k5757 gold badges823823 silver badges1.4k1.4k bronze badges $\endgroup$ Add a comment | 2 $\begingroup$ If $n=mq+r$ where $0 \le r \le m-1,$ then $n/m=q+r/m.$ So the fractional part is ${n/m}=r/m,$ and then $m$ times fractional part gives back the remainder $r$. Share answered Dec 15, 2014 at 22:14 coffeemathcoffeemath 30.2k22 gold badges3434 silver badges5353 bronze badges $\endgroup$ Add a comment | You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions discrete-mathematics ceiling-and-floor-functions fractional-part See similar questions with these tags. 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14078
https://mathoverflow.net/questions/331315/how-many-permutations-are-there-at-a-given-cayley-distance-from-the-identity
co.combinatorics - How many permutations are there at a given Cayley distance from the identity? - MathOverflow Join MathOverflow By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community MathOverflow helpchat MathOverflow Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more How many permutations are there at a given Cayley distance from the identity? Ask Question Asked 6 years, 4 months ago Modified6 years, 4 months ago Viewed 862 times This question shows research effort; it is useful and clear 6 Save this question. Show activity on this post. Permutations σ σ in the symmetric group S n S n can be characterized by their Cayley distance C σ C σ, being the minimal number of transpositions needed to convert {1,2,3,…n}{1,2,3,…n} into σ σ. The sign of the permutation is (−1)C σ(−1)C σ. For example, when σ={2,3,4,5,1}σ={2,3,4,5,1}, one has C σ=4 C σ=4 and for σ={1,2,3,5,4}σ={1,2,3,5,4} one has C σ=1 C σ=1. Of the 5!5! permutations in S 5 S 5 there are, respectively, 1,10,35,50,24 1,10,35,50,24 with Cayley distance C σ=0,1,2,3,4 C σ=0,1,2,3,4. Question: What is the general formula that counts the number of permutations at a given Cayley distance? This question was motivated by my attempt to check an integral formula in the unitary group. co.combinatorics permutations Share Share a link to this question Copy linkCC BY-SA 4.0 Cite Improve this question Follow Follow this question to receive notifications edited May 12, 2019 at 10:15 Carlo BeenakkerCarlo Beenakker asked May 11, 2019 at 19:36 Carlo BeenakkerCarlo Beenakker 199k 19 19 gold badges 479 479 silver badges 697 697 bronze badges 6 2 findstat.org/St000216Martin Rubey –Martin Rubey 2019-05-11 19:49:02 +00:00 Commented May 11, 2019 at 19:49 @MartinRubey --- wonderful, thank you for the rapid answer; so the number of permutations in S n S n at Cayley distance k∈{0,1,2,…,n−1}k∈{0,1,2,…,n−1} equals |s n,n−k||s n,n−k|, the Stirling number of the first kind.Carlo Beenakker –Carlo Beenakker 2019-05-11 20:24:06 +00:00 Commented May 11, 2019 at 20:24 1 I admit that I (sort of) knew the answer, but I do enjoy pointing out that filling in a handful of values at findstat.org/StatisticFinder/Permutations is easier than trying to remember!Martin Rubey –Martin Rubey 2019-05-11 20:34:27 +00:00 Commented May 11, 2019 at 20:34 2 you might just enter this in the answer box so that I can accept it...Carlo Beenakker –Carlo Beenakker 2019-05-11 20:42:36 +00:00 Commented May 11, 2019 at 20:42 4 It's worth noting that a permutation in S n S n with k k cycles has Cayley distance n−k n−k. This is why Stirling numbers of the first kind appear.Ira Gessel –Ira Gessel 2019-05-11 22:31:11 +00:00 Commented May 11, 2019 at 22:31 |Show 1 more comment 1 Answer 1 Sorted by: Reset to default This answer is useful 8 Save this answer. Show activity on this post. The Cayley distance of a permutation is also known as its absolute length, as can be found out by supplying a few values at which yields There, we also find that for a permutation in S n S n with k k cycles it is simply n−k n−k. This fact is, for example, Problem 5.6 in . Petersen, T. Kyle, Eulerian numbers, Birkhäuser Advanced Texts. Basler Lehrbücher. New York, NY: Birkhäuser/Springer (ISBN 978-1-4939-3090-6/hbk; 978-1-4939-3091-3/ebook). xviii, 456 p. (2015). ZBL1337.05001. Share Share a link to this answer Copy linkCC BY-SA 4.0 Cite Improve this answer Follow Follow this answer to receive notifications answered May 12, 2019 at 10:25 Martin RubeyMartin Rubey 5,922 2 2 gold badges 27 27 silver badges 44 44 bronze badges Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions co.combinatorics permutations See similar questions with these tags. Featured on Meta Spevacus has joined us as a Community Manager Introducing a new proactive anti-spam measure Linked 11Nonnegativity of an integral over the unitary group Related 4Hamming distance approximates Cayley distance on permutations: citation wanted 12Principal Order Ideals in the Weak Bruhat Order 21Are there enough additive permutations? 6Maximum size of minimal sequence of transpositions whose product is a given permutation 4genus zero permutation and noncrossing partition 11Counting symmetric subgroups of symmetric groups 4Is the Cayley distance on permutation (matrices) equivalent to the Riemannian metric on O(n)O(n)? 5A bijection on permutations 165Conceptual reason why the sign of a permutation is well-defined? 2Is the small Davenport constant for S n S n, d(S n)=n(n−1)/2 d(S n)=n(n−1)/2? Question feed Subscribe to RSS Question feed To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Why are you flagging this comment? 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https://en.wikipedia.org/wiki/Market_concentration
Jump to content Market concentration Deutsch Español Français Հայերեն עברית Polski Português Русский Edit links From Wikipedia, the free encyclopedia Term in market economics | Competition law | | | | Basic concepts | | Barriers to entry Competition Competition law theory History of competition law Herfindahl–Hirschman index Monopoly and oligopoly + Coercive + Government-granted monopoly + Natural monopoly + State and Legal monopoly Market concentration Market power + Monopoly profit + Profit margin Merger control Non-compete clause SSNIP test Regulatory agency Regulatory economics Relevant market | | Anti-competitive practices | | Collusion + Dividing territories + Formation of cartels + Price fixing (List) + Bid rigging + Tacit collusion + Occupational licensing and closure Cornering the market Embrace, extend, and extinguish Exclusive dealing Horizontal integration Mergers and acquisitions Monopolization Misuse of patents and copyrights Predatory pricing + Price gouging Product bundling and tying Raising rivals' costs Refusal to deal + Group boycott + Essential facilities Regulatory capture + Noerr–Pennington doctrine + Parker immunity doctrine Rent-seeking Vendor lock-in Vertical integration | | Competition regulators | | List of competition regulators + International Competition Network | | v t e | In economics, market concentration is a function of the number of firms and their respective shares of the total production (alternatively, total capacity or total reserves) in a market. Market concentration is the portion of a given market's market share that is held by a small number of businesses. To ascertain whether an industry is competitive or not, it is employed in antitrust law land economic regulation. When market concentration is high, it indicates that a few firms dominate the market and oligopoly or monopolistic competition is likely to exist. In most cases, high market concentration produces undesirable consequences such as reduced competition and higher prices. The market concentration ratio measures the concentration of the top firms in the market, this can be through various metrics such as sales, employment numbers, active users or other relevant indicators. In theory and in practice, market concentration is closely associated with market competitiveness, and therefore is important to various antitrust agencies when considering proposed mergers and other regulatory issues. Market concentration is important in determining firm market power in setting prices and quantities. Market concentration is affected through various forces, including barriers to entry and existing competition. Market concentration ratios also allows users to more accurately determine the type of market structure they are observing, from a perfect competitive, to a monopolistic, monopoly or oligopolistic market structure. Market concentration is related to industrial concentration, which concerns the distribution of production within an industry, as opposed to a market. In industrial organization, market concentration may be used as a measure of competition, theorized to be positively related to the rate of profit in the industry, for example in the work of Joe S. Bain. An alternative economic interpretation is that market concentration is a criterion that can be used to rank order various distributions of firms' shares of the total production (alternatively, total capacity or total reserves) in a market. Factors affecting market concentration [edit] The 2020 Australian Telecommunications firms market share: an example of a highly concentrated market with an estimated HHI of 3034. Telstra (37.0%) Optus (30.0%) Vodafone (27.0%) Other (6.00%) There are various factors that affect the concentration of specific markets which include; barriers to entry(high start-up costs, high economies of scale, brand loyalty), industry size and age, product differentiation and current advertising levels. There are also firm specific factors affecting market concentration, including: research and development levels, and the human capital requirements. Although fewer competitors doesn't always indicate high market concentration, it can be a strong indicator of the market structure and power allocation. Metrics [edit] After determining the relevant market and firms, through defining the product and geographical parameters, various metrics can be employed to determine the market concentration. This can be quantified using the SSNIP test. A simple measure of market concentration is to calculate 1/N where N is the number of firms in the market. A result of 1 would indicate a pure monopoly, and will decrease with the number of active firms in the market, and nonincreasing in the degree of symmetry between them. [clarification needed] This measure of concentration ignores the dispersion among the firms' shares. This measure is practically useful only if a sample of firms' market shares is believed to be random, rather than determined by the firms' inherent characteristics. Any criterion that can be used to compare or rank distributions (e.g. probability distribution, frequency distribution or size distribution) can be used as a market concentration criterion. Examples are stochastic dominance and Gini coefficient. Herfindahl–Hirschman Index [edit] The Herfindahl–Hirschman index (HHI) (the most commonly used market concentration) is added portion of market attentiveness. It is derived by adding the squares of all the Market participants market shares. A higher HHI indicates a higher level of market concentration. A market concentration level of less than 1000 is typically seen as low, whilst one of more than 1500 is regarded as excessive. Where is the market share of firm i, conventionally expressed as a percentage, and N is the number of firms in the relevant market. If market shares are expressed as decimals, an HHI of 0 represents a perfectly competitive industry while an HHI index of 1 represents a monopolised industry. Regardless whether the decimal or percentage HHI is used, a higher HHI indicates higher concentration within a market. Section 1 of the Department of Justice and the Federal Trade Commission's Horizontal Merger Guidelines is entitled "Market Definition, Measurement and Concentration" and states that the Herfindahl index is the measure of concentration that these Guidelines will use. Concentration ratio [edit] The concentration ratio (CR) is a measure of how concentrated a market is. By dividing the overall market share by the sum of the market shares of the largest enterprises, it is calculated. It can be used to assess the market's strength over both the short and long haul. Generally speaking, a CR of less than 40% and a CR of more than 60% are regarded as modest and high levels of market concentration, respectively. This ratio measures the concentration of the largest firms in the form where N is usually between 3 and 5. Relationship between Market Structure and Market Concentration Metrics | Type of Market | CR Range | HHI Range | | Monopoly | 1 | 6000 - 10 000 (Depending on Region) | | Oligopoly | 0.5 - 1 | 2000 - 6000 (Depending on Region) | | Competitive | 0 - 0.5 | 0 - 2000 (Depending on Region) | Regulatory usage [edit] Historical usage [edit] Since the introduction of the Sherman Antitrust Act of 1890, in response to growing monopolies and anti-competitive firms in the 1880s, antitrust agencies regularly use market concentration as an important metric to evaluate potential violations of competition laws. Since the passing of the act, these metrics have also been used to evaluate potential mergers' effect on overall market competition and overall consumer welfare. The first major example of the Sherman Act being imposed on a company to prevent potential consumer abuse through excessive market concentration was in the 1911 court case of Standard Oil Co. of New Jersey v. United States where after determining Standard Oil was monopolising the petroleum industry, the court-ordered remedy was the breakup into 34 smaller companies. Modern usage [edit] Modern regulatory bodies state that an increase in market concentration can inhibit innovation, and have detrimental effects on overall consumer welfare. The United States Department of Justice determined that any merger that increases the HHI by more than 200 proposes a legitimate concern to antitrust laws and consumer welfare . Therefore, when considering potential mergers, especially in horizontal integration applications, antitrust agencies will consider the whether the increase in efficiency is worth the potential decrease in consumer welfare, through increased costs or reduction in quantity produced. Whereas the European Commission is unlikely to contest any horizontal integration, which post merger HHI is under 2000 (except in special circumstances). Modern examples of market concentration being utilised to protect consumer welfare include: 2014 Attempted purchase of Time Warner Cable by Comcast, was abandoned after the US DOJ threatened to file an antitrust lawsuit, citing that the HHI of the national television industry would increase by 639 points to a HHI of 2454, and feared this merger would lead to increased prices for consumers. Halliburton and Baker Hughes (at the time the 2nd and 3rd largest oilfield services companies, respectively) attempted 2014 merger was blocked by the US DOJ, after fears that the merger would increase costs for oil companies in 23 separate product markets, and therefore would stiffen innovation in the oil sector. General Electric's attempted acquisition of Honeywell in 2001, was approved in the United States, however the condition's that European Commission enforced for the approval were too impactful for General Electric, and was abandoned. This is an example on how different regulatory bodies view mergers. Motivation for firms [edit] The relationship between market concentration and profitability can be divided into two arguments: greater market concentration increases the likelihood of collusion between firms which, resulting in higher pricing. In contrast, market concentration occurs as a result of the efficiency obtained in the course of being a large firm, which is more profitable in comparison to smaller firms and their lack of efficiency. Collusion [edit] There are game theoretic models of market interaction (e.g. among oligopolists) that predict that an increase in market concentration will result in higher prices and lower consumer welfare even when collusion in the sense of cartelization (i.e. explicit collusion) is absent. Examples are Cournot oligopoly, and Bertrand oligopoly for differentiated products. Bain's (1956) original concern with market concentration was based on an intuitive relationship between high concentration and collusion which led to Bain's finding that firms in concentrated markets should be earning supra-competitive profits. Collins and Preston (1969) shared a similar view to Bain with focus on the reduced competitive impact of smaller firms upon larger firms. Demsetz held an alternative view where he found a positive relationship between the margins of specifically the largest firms within a concentrated industry and collusion as to pricing. Although theoretical models predict a strong correlation between market concentration and collusion, there is little empirical evidence linking market concentration to the level of collusion in an industry. In the scenario of a merger, some studies have also shown that the asymmetric market structure produced by a merger will negatively affect collusion despite the increased concentration of the market that occurs post-merger. Efficiency [edit] As an economic tool market concentration is useful because it reflects the degree of competition in the market. Understanding the market concentration is important for firms when deciding their marketing strategy. As well, empirical evidence shows that there exists an inverse relationship between market concentration and efficiency, such that firms display an increase in efficiency when their relevant market concentration decreases. The above positions of Bain (1956) as well as Collins and Preston (1969) are not only supportive of collusion but also of the efficiency-profitability hypothesis: profits are higher for bigger firms within a greater concentrated market as this concentration signifies greater efficiency through mass production. In particular, economies of scale was the greatest kind of efficiency that large firms could achieve in influencing their costs, granting them greater market share. Notably however, Rosenbaum (1994) observed that most studies assumed the relationship between actual market share and observed profitability by following the implication that large firms hold greater market share due to their efficiency, demonstrating that the relationship between these efficiency and market share is not clearly defined. Industry effects [edit] Implications of market concentration A high level of market concentration can lead to a decrease in competition and increased market power for the dominant firms. This might lead to greater costs, less quality, fewer options, and less innovation. Thus, consumers and society may be negatively impacted by large levels of market concentration. Innovation [edit] Schumpeter (1950) first recognised the relationship between market concentration and innovation in that a higher concentrated market would facilitate innovation. He reasoned that firms with the greatest market share have the greatest opportunity to benefit from their innovations, particularly through investment into R&D. This can be contrasted with the position taken by Arrow (1962) that a greater market concentration will decrease incentive to innovate because a firm within a monopoly or monopolistic market would have already reached profit levels that greatly exceed costs. In practice, there are complications in observing the direct correlation between market concentration and its effect on. In collecting empirical evidence, issues have also arisen as to how innovation, a firm's control and gaps between R&D and firm size are measured. There has also been a lack of consensus. For example, a negative correlation was established by Connelly and Hirschey (1984) who explained that the correlation evidenced a decreased expenditure on R&D by oligopolistic firms to benefit from greater monopolised profits. However, Blundell et al. observed a positive correlation by tallying the patents lodged by firms. This general observation was also shared by Aghion et al. in 2005. Schumpeter also failed to distinguish between the different technologies that contribute to innovation and did not properly define “creative destruction”. Petit and Teece (2021) argued that technological opportunities, a variable which Schumpeter and Arrow did not include during their time, would be included in this definition as it enables new entrants to make a “breakthrough” into the industry. Research presented by Aghion et al. (2005) suggested an inverted U-shape model that represents the relationship between market concentration and innovation. Delbono and Lambertini modelled empirical evidence onto a graph and found that the pattern demonstrated by the data supported the existence of a U-shaped relationship between these two variables. Regulation of market concentration The existence of economic regulations like the Competition Act and antitrust laws like the Sherman Act is due to the necessity of maintaining market competition in order to avoid the formation of monopolies. These laws typically require firms to report their market share and limit the degree of market concentration that is allowed. In some cases, antitrust laws may require the breakup of firms or the establishment of “firewalls” that prevent the potential abuse of power. Market concentration reveals a market's degree of concentration. It is employed to ascertain the level of industrial competition. A high degree of market concentration is typically undesirable since it might result in less competition and more power for the leading enterprises on the market. Antitrust laws and other economic regulations safeguard market competition and the avoidance of monopolies. Alternative metrics [edit] Although not as common as the Herfindahl–Hirschman Index or Concentration Ratio metrics, various alternative measures of market concentration can also be used. (a) The U Index (Davies, 1980): : , where is an accepted measure of inequality (in practice the coefficient of variation is suggested), is a constant or a parameter (to be estimated empirically) and N is the number of firms. Davies (1979) suggests that a concentration index should in general depend on both N and the inequality of firms' shares. Davis (1980) indicates, there are many suitable candidates to be used for , however, the measure he advocates "is a simple transformation of the coefficient of variation (c): . : The "number of effective competitors" is the inverse of the Herfindahl index. : Terrence Kavyu Muthoka defines distribution just as functionals in the Swartz space which is the space of functions with compact support and with all derivatives existing. The Media:Dirac_Distribution or the Dirac function is a good example. (b) The Linda index (1976) : where Qi is the ratio between the average share of the first firms and the average share of the remaining firms and is the concentration coefficient for the first firms. Although it doesn't capture the peripheral firms like the HHI formula, it works to capture the "core" of the market, and measure the degree of inequality between the size variable accounted for by various sib-samples of firms. This index does assume pre-calculation on the users' behalf to determine the relevant value of However, there is little empirical evidence of regulatory usage of the Linda Index. (c) Comprehensive concentration index (Horwath 1970): : Where s1 is the share of the largest firm. The index is similar to except that greater weight is assigned to the share of the largest firm. When compared to the HHI index, it does present some advantages, such as giving more weight to the quantity of small firms, however the arbitrary choice to only include the absolute value of one firm has led to criticism over its accuracy and usefulness. (d) The Rosenbluth (1961) index (also Hall and Tideman, 1967): : where symbol i indicates the firm's rank position. : The Rosenbluth index assigns more weight to smaller competitors when there are more firms present in the marketplace, and is sensitive to the amount of competitors in the market, even if there is a small amount of large firms dominating. Its coefficients and ranking are similar to results produced through the use of the Herfindahl-Hirschman Index. (e) The Gini coefficient (1912) : The Gini coefficient measures the difference between firms' sizes without including the number of firms operating in a market. This is known as a relative concentration measure and differs from absolute concentration measures (like the Rosenbluth index) which includes the number of firms and firms' distribution sizes. It is used in conjunction with the Lorenz curve. Originally, the Lorenz curve measured the inequality of income distributed with a population and ranked individuals from highest to lowest earnings. Therefore, in this context the Gini coefficient is located between the 45° line representing an equal distribution of income and the Lorenz curve representing the actual distribution of income within the population. In a market concentration context, the Lorenz curve can be plotted ranking firms' market shares from smallest to largest to simulate a concentration curve. The firms’ cumulative percentage shares would remain on the y axis and the cumulative percentage of sellers would remain on the x axis. would the sum of weighted market share located in the area above the concentration curve. The Gini coefficient is 0 when the concentration curve aligns with the 45° line representing a single firm's market share, meaning the market is a monopoly. (f) Utilizing the power-law exponent (α) of the fitting curve on the out-degree distribution of the network (Pliatsidis, 2024) : For a given set of nodes, each with degree k, the power-law exponent (α) serves as a pivotal metric for analyzing concentration within a network. This exponent characterizes the distribution of out-degrees among nodes, offering insights into the concentration of certain attributes or interactions within the system. : When α<0, a skewed distribution emerges where a small percentage of firms dominate the market. : As α approaches 0, the distribution flattens, indicating a more equitable distribution of contract awards. : Conversely, as α increases above 0, the concentration among a select few firms diminishes further, suggesting a diversified market landscape. See also [edit] Concentration ratio Dominance (economics) Gini coefficient Herfindahl index Horizontal Merger Guidelines Lorenz curve Inequality of wealth Market failure Monopoly Probability distribution Stochastic dominance Relative market share References [edit] ^ Jump up to: a b "What is Market Concentration? Definition of Market Concentration, Market Concentration Meaning". The Economic Times. Retrieved 2021-04-24. ^ Turner, Scott F.; Mitchell, Will; Bettis, Richard A. (2010). "Responding to Rivals and Complements: How Market Concentration Shapes Generational Product Innovation Strategy". Organization Science. 21 (4): 856. doi:10.1287/orsc.1090.0486. hdl:10161/4440. ^ "What We Can Learn from Merger Deals That Never Happened". Harvard Business Review. 2016-06-21. ISSN 0017-8012. Retrieved 2021-04-24. ^ Jump up to: a b Bain, Joe, S (1951). "Relation of Profit Rate to Industry Concentration: American Manufacturing, 1936-1940". The Quarterly Journal of Economics. 65 (3): 293–324. doi:10.2307/1882217. JSTOR 1882217.{{cite journal}}: CS1 maint: multiple names: authors list (link) ^ Janashvili '02, David (2002-04-26). "Market Concentration: The Effects of Technology". Honors Projects.{{cite journal}}: CS1 maint: numeric names: authors list (link) ^ "Herfindahl-Hirschman Index – United States Department of Justice". US Department of Justice. 25 June 2015. Retrieved 2022-11-09. ^ Evans, Anthony J. (2014). Markets for Managers: A Managerial Economics Primer. John Wiley & Sons Ltd. p. 69. ^ "Herfindahl-Hirschman Index". www.justice.gov. 2015-06-25. Retrieved 2021-04-24. ^ J. Gregory Sidak, "Evaluating Market Power Using Competitive Benchmark Prices Instead of the Hirschman-Herfindahl Index", 74 Antitrust L.J. 387, 387–388 (2007). ^ Brock, James W.; Obst, Norman P. (2009-03-01). "Market Concentration, Economic Welfare, and Antitrust Policy". Journal of Industry, Competition and Trade. 9 (1): 65–75. doi:10.1007/s10842-007-0026-6. ISSN 1573-7012. S2CID 154631137. ^ "Standard Oil | History, Monopoly, & Breakup". Encyclopædia Britannica. Retrieved 2021-04-24. ^ "Herfindahl-Hirschman Index". The United States Department of Justice. 25 June 2015. ^ Brock, James W.; Obst, Norman P. (2007-10-03). "Market Concentration, Economic Welfare, and Antitrust Policy". Journal of Industry, Competition and Trade. 9 (1): 65–75. doi:10.1007/s10842-007-0026-6. ISSN 1566-1679. S2CID 154631137. ^ "EUR-Lex - 52004XC0205(02) - EN". Official Journal C 031, 05/02/2004, p. 0005-0018. Retrieved 2021-04-24. ^ Fernholz, Tim (13 February 2014). "Why the Time Warner-Comcast merger isn't going to happen – at least the way it looks today". Quartz. Retrieved 2021-04-24. ^ "Halliburton and Baker Hughes Abandon Anticompetitive Merger". www.justice.gov. 2017-03-15. Retrieved 2021-04-24. ^ Baker, Mark (9 April 2008). "U.S./EU: Analysis – What Killed The GE-Honeywell Merger?". Radio Free Europe/Radio Liberty. Retrieved 2021-04-24. ^ Jump up to: a b Rosenbaum, David I. (1994). "Efficiency v. Collusion: Evidence Cast in Cement". Review of Industrial Organization. 9 (4): 379–392. doi:10.1007/BF01029512. JSTOR 41798521. S2CID 153620070. ^ Schmalensee, Richard (1987). "Inter-Industry Studies of Structure and Performance". Handbook of Industrial Organization. (Working Paper No. 1874-87). ^ Collins, Norman R.; Preston, Lee E. (1969). "Price-Cost Margins and Industry Structure". The Review of Economics and Statistics. 51 (3): 271–286. doi:10.2307/1926562. JSTOR 1926562. ^ Demsetz, Harold (1973). "Industry Structure, Market Rivalry, and Public Policy". Journal of Law and Economics. 16 (1): 1–9. doi:10.1086/466752. JSTOR 724822. S2CID 154506488. ^ Salinger, Michael (1990). "The Concentration-Margins Relationship Reconsidered" (PDF). Brookings Papers on Economic Activity. Microeconomics. 1990: 287–335. doi:10.2307/2534784. JSTOR 2534784. ^ Fonseca, Miguel A.; Normann, Hans-Theo (2008). "Mergers, Asymmetries and Collusion: Experimental Evidence". The Economic Journal. 118 (527): 387–400. doi:10.1111/j.1468-0297.2007.02126.x. JSTOR 20108803. S2CID 154378955. ^ "Highly concentrated markets are bad for consumers and bad for investors". MoneyWeek. 21 May 2019. Retrieved 2021-04-28. ^ Martin, Stephen (1988). "Market Power and/or Efficiency?". The Review of Economics and Statistics. 70 (2): 331–335. doi:10.2307/1928318. JSTOR 1928318. ^ Jump up to: a b Sonenshine, Ralph M. (2010). "The Stock Market's Valuation of R&D and Market Concentration in Horizontal Mergers". Review of Industrial Organization. 37 (2): 119–140. doi:10.1007/s11151-010-9262-8. JSTOR 41799483. S2CID 155085616. ^ Jump up to: a b Petit, Nicolas; Teece, David J. (2021). "Original Article Innovating Big Tech firms and competition policy: favoring dynamic over static competition". Industrial and Corporate Change. 30 (1168–1198). doi:10.1093/icc/dtab049. hdl:1814/74432. ^ Delbono, Flavio; Lambertini, Luca (2022). "Innovation and product market concentration: Schumpeter, arrow, and the inverted U-shape curve". Oxford Economic Papers. 74 (1): 297–311. doi:10.1093/oep/gpaa044. hdl:11585/831079. ^ Bukvić, Rajko; Pavlović, Radica; Gajić, Аlеksаndаr M. (2014). "Possibilities of Application of the Index Concentration of Linda in Small Economy: Example of Serbian Food Industries". mpra.ub.uni-muenchen.de. Retrieved 2021-04-27. ^ Jump up to: a b Romualdas, Stasys, Ginevičius, Čirba (2009). "Additive measurement of Market Concentration". Journal of Business Economics and Management. 10 (3): 191–198. doi:10.3846/1611-1699.2009.10.191-198.{{cite journal}}: CS1 maint: multiple names: authors list (link) ^ Jump up to: a b du Pisanie, Johann (2013). "Concentration Measures as an element in testing the structure-conduct-performance paradigm" (PDF). Economic Research Southern Africa. (Working Papers No. 345). ^ Palan, Nicole (2010). "Measurement of Specialization # The Choice of Indices" (PDF). Research Centre International Economics, Vienna. (FIW Working Paper No. 62). ^ Pliatsidis, Andreas Christos (2024-02-12). "Analyzing concentration in the Greek public procurement market: a network theory approach". Journal of Industrial and Business Economics. 51 (2): 431–480. doi:10.1007/s40812-023-00291-z. ISSN 1972-4977. Bain, J. (1956). Barriers to New Competition. Cambridge, Massachusetts: Harvard Univ. Press. Curry, B. and K. D. George (1983). "Industrial concentration: A survey" Jour. of Indust. Econ. 31(3): 203–255 Shughart II, William F. (2008). "Industrial Concentration". In David R. Henderson (ed.). Concise Encyclopedia of Economics (2nd ed.). Indianapolis: Library of Economics and Liberty. ISBN 978-0865976658. OCLC 237794267. Tirole, J. (1988). The Theory of Industrial Organization. Cambridge, Massachusetts: MIT Press. Weiss, L. W. (1989). Concentration and price. Cambridge, Massachusetts : MIT Press. External links [edit] Department of Justice and Federal Trade Commission Horizontal Merger Guidelines | v t e Institutional economics | | --- | | Institutional economists | Werner Abelshauser Clarence Edwin Ayres Joe S. Bain Shimshon Bichler Robert A. Brady Daniel Bromley Ha-Joon Chang John Maurice Clark John R. Commons Richard T. Ely Robert H. Frank John Kenneth Galbraith Walton Hale Hamilton Orris C. Herfindahl Albert O. Hirschman Geoffrey Hodgson János Kornai Simon Kuznets Hunter Lewis Jesse W. Markham Wesley Clair Mitchell Gunnar Myrdal Jonathan Nitzan Warren Samuels François Simiand Herbert A. Simon Frank Stilwell George W. Stocking Sr. Lars Pålsson Syll Thorstein Veblen Edward Lawrence Wheelwright Erich Zimmermann | | New institutional economists | Daron Acemoglu Armen Alchian Masahiko Aoki Steven N. S. Cheung Ronald Coase Harold Demsetz Avner Greif Claude Ménard Douglass North Mancur Olson Elinor Ostrom (Bloomington school) Oliver E. Williamson | | Behavioral economists | George Ainslie Dan Ariely Nava Ashraf Ofer Azar Douglas Bernheim Samuel Bowles Sarah Brosnan Colin Camerer David Cesarini Kay-Yut Chen Rachel Croson Werner De Bondt Paul Dolan Stephen Duneier Catherine C. Eckel Armin Falk Urs Fischbacher Herbert Gintis Uri Gneezy David Halpern Charles A. Holt David Ryan Just Daniel Kahneman Djuradj Caranovic Ariel Kalil George Katona Jeffrey R. Kling George Loewenstein Graham Loomes Brigitte C. Madrian Gary McClelland Matteo Motterlini Sendhil Mullainathan Michael Norton Matthew Rabin Howard Rachlin Klaus M. Schmidt Eldar Shafir Hersh Shefrin Robert J. Shiller Uwe Sunde Richard Thaler Amos Tversky Robert W. Vishny Georg Weizsäcker | | Economic sociologists | Jens Beckert Fred L. Block James S. Coleman Paul DiMaggio Paula England Mark Granovetter Donald Angus MacKenzie Joel M. Podolny Lynette Spillman Richard Swedberg Laurent Thévenot Carlo Trigilia Harrison White Viviana Zelizer | | Key concepts and ideas | Accelerator effect Administered prices Barriers to entry Bounded rationality Conspicuous consumption Conspicuous leisure Conventional wisdom Countervailing power Effective competition Herfindahl index Hiding hand principle Hirschman cycle Instrumentalism Kuznets cycles Market concentration Market power Market structure Penalty of taking the lead Satisficing Shortage economy Structure–conduct–performance paradigm Technostructure Theory of two-level planning Veblen goods Veblenian dichotomy | | Related fields | Cultural economics Development economics English historical school of economics Evolutionary economics Evolutionary psychology French historical school Historical school of economics Legal realism Microeconomics Post-Keynesian economics | Retrieved from " Categories: Concentration indicators Market structure Monopoly (economics) Hidden categories: CS1 maint: multiple names: authors list CS1 maint: numeric names: authors list CS1: long volume value Articles with short description Short description is different from Wikidata Wikipedia articles needing clarification from November 2022
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https://www.youtube.com/watch?v=FzWdZGMae3k
Finding Percentiles for a Normal Distribution Steve Crow 67600 subscribers 2507 likes Description 237250 views Posted: 7 Oct 2019 This video shows how to find the percentile for a normal distribution. 179 comments Transcript: all right so this video we're gonna look at finding the percentile for a normal distribution and I've got three different examples and each all three of them a little bit different this one will we'll be able to find the exact number in the chart this one we won't and it'll be a positive z value and this one we won't be able to find the exact and this will be on the negative z value and you'll see what I'm talking about when we start working them but the the one thing that you need to that you need to remember when we're finding these is that Z is equal to X minus mu over Sigma and this is a Z all right so what we're doing is we want to know the percentile we're looking for this x value here okay all right so let's take a look at the first example all right so if we have a normal distribution here okay we've got a normal distribution and what we need to find is X so we're given the mean which is 63 we're given Sigma which is the standard deviation of five okay and so what we've got to do is we've got to figure out what Z is and once we figure out what Z is we can solve this equation here for X okay well you know on the standard normal curve okay the mean is zero okay and we need to find the 97 97.5 percentile okay all right so that means we're looking for a z value here and this area here is 0.975 we know the Z values to the right of zero because if zero see look on the normal distribution here's the mean zero so we know this area is 0.5 and we know this area is 0.5 hey because the entire area under the curve is 1 and if we know that the area to the left of 0 here is 0.5 well an area of 0.9 75 would have to be over here because we need more area okay and the chart that I'm using the the chart that I'm using to look up if you have a Z value the chart I have gives the area to the left of the Z value okay all right so what we have to do is we have to go find an area of 0.975 so if we come over here to our Z table okay so you should know how to read this okay looking up a Z value and then finding the area here okay so remember these values in here that's the area okay and then the Z values here that's where these are our z values for example if if we look at this and say this this would be a z value of 1 point 1 4 okay all right and that that value would have an area of this point 8 7 2 9 okay but what we have is our area to the left is 0.975 so we've got to find this area in here so if we come over here and look for 0.975 let's see here it is see we found it we found it exactly Oh point nine seven five and what area and what Z value does that correspond to 1.96 so our z value is 1.96 okay so let's come back over here and we know that Z is equal to 1.96 so you can see we know what Z is we can plug into here we know mu which is 63 this is mu and we know Sigma which is 5 that's Sigma so we just plug all this in and solve for X so I've got 1 point 9 6 equals X minus 63 over 5 all right so now we solve so we multiply everything by 5 so 5 times this the fives will cancel and then we've got five times 1.96 equals x minus 63 and then we add the 63 to both sides so I get 5 times 1.96 plus 63 equals x and so that is going to give me x equals all right so I'll just punch this into my calculator plus 63 and that is going to give me 72 point eight okay and so that's a seventy two point eight score is the 97.5 percentile and so this is our answer all right so that one wasn't bad we were able to find that number exactly on the on the Z table all right so let's take a look at this next one here all right so let me go ahead and write this back up here all right so now we want to find the 90th percentile so let's go ahead and look that up so there's 0 on the standard normal curve here's Z and I need the 90th percentile so that's 90 percent write it in a decimal so I need to go to my chart and find ninety percent all right so I've got point nine that's what I'm looking for this is the area to the left of the Z and to know what the Z value is so I need to come in here and find point nine all right so point nine is going to be right in here between these two numbers okay now we're not going to be able to find the exact z-score you can but you would need to use some software you need you can use Excel to do it I'm not going to do that in this video and depending on what your teacher tells you to do for this video what I'll do is since point nine falls in between do you do you notice that point nine is closer to this number than it is this one okay so what I'll do is we'll just use this okay now and let me go ahead and I'm gonna circle this one in green and green I'll tell you what I'll just highlight at yellow two that's the one we're gonna use because it's closer now depending on what your teacher tells you to do a lot of times what I'll usually tell my students if it falls in between just go to the one on the right or if it falls in between go to the one on the Left okay the reason I do that is so all the students will do it the same okay your teacher may tell you that or they may tell you go to the one that it's closer to and that's fine and that's what I'm gonna do in this video okay I'm gonna go to the one that is closer to so we can see that this corresponds to one point two eight okay so there's the Z value one point two eight so I've got this is equal to one point two eight and so now just plug everything in the 63 goes in for MU the five goes in for Sigma the one point two eight goes in for Z so I've got one point two eight equals X minus sixty three over five and then if I multiply by five that's the just like I did in the previous example so I got five times one point two eight equals X minus 63 so five times one point two eight plus 63 equals x and so that's going to give me a value of x five times one point two eight plus 63 that is sixty nine point four okay so that's the 90th percentile all right now let's look at the twenty-fifth percentile all right so let's let's draw the standard normal curve again and we know the mean is zero and we know from from zero back to the left that's fifty percent or 0.5 so now I'm only doing the twenty-fifth percentile so I know my z-score is over here and I'm looking for point two five that's the Z value I'm looking for so let's come back over here to our standard normal curve and notice all of these z values are positive and so I'm gonna go here to where the Z values are negative and I'm looking for that area there which is point two five okay so let's find point two five the twenty-fifth percentile all right so it's gonna be down here somewhere let's see I gotta go down a little bit more so it looks like it is going to be let's see okay right in between right in between these two okay all right so this one it falls in between again and this one it's not so obvious which one is closer to and actually if you put it in a calculator and find the difference between this and point two five and this in point two five the difference in this one in point two five is point zero zero one four and the difference in this one and point two five is point zero zero one seven so it's it's closer to this one okay this one here that I've hired that I've got highlighted okay and like I said your teacher may tell you just go to the one on the right or just go to the one on the left so you'll all be the same okay and you may have to punch it in your calculator to see which one it's closer to the other the last one we did it was a little more obvious okay so let's see which one it corresponds to very straight line so it corresponds to negative point six seven okay see the negative point six seven so this is negative point six seven and so now we just plug it into here okay and so we've got negative point six seven times I'm sorry I'm starting to solve it already X minus 63 over five and then we solve so we got 5 times negative point six seven equals X minus 63 so 5 times negative point six seven plus 63 equals x and look and you can see if you notice in this formula we're gonna do 63 and it's going to be minus because see this is negative and that makes sense because the mean 4x is 63 well if it's if the numbers to the left of the mean you're gonna have to subtract it from 63 you see that and noticed on the other two it was over here and this number over here was positive so we was at we were adding it to the mean okay so this is gonna be 5 times negative point 6 7 plus 63 and that is going to give us fifty nine point six five and that's your answer and I know this video was kind of long but I wanted to show you all the different types and I hope it helped check out my other videos give me a like share and subscribe and thanks for watching
14081
https://math.stackexchange.com/questions/755886/remainder-operation-in-terms-of-the-floor-function
elementary number theory - Remainder operation in terms of the floor function - Mathematics Stack Exchange Join Mathematics By clicking “Sign up”, you agree to our terms of service and acknowledge you have read our privacy policy. Sign up with Google OR Email Password Sign up Already have an account? Log in Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Visit Stack Exchange Loading… Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site About Us Learn more about Stack Overflow the company, and our products current community Mathematics helpchat Mathematics Meta your communities Sign up or log in to customize your list. more stack exchange communities company blog Log in Sign up Home Questions Unanswered AI Assist Labs Tags Chat Users Teams Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Try Teams for freeExplore Teams 3. Teams 4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. Explore Teams Teams Q&A for work Connect and share knowledge within a single location that is structured and easy to search. Learn more about Teams Hang on, you can't upvote just yet. You'll need to complete a few actions and gain 15 reputation points before being able to upvote. Upvoting indicates when questions and answers are useful. What's reputation and how do I get it? Instead, you can save this post to reference later. Save this post for later Not now Thanks for your vote! You now have 5 free votes weekly. Free votes count toward the total vote score does not give reputation to the author Continue to help good content that is interesting, well-researched, and useful, rise to the top! To gain full voting privileges, earn reputation. Got it!Go to help center to learn more Remainder operation in terms of the floor function Ask Question Asked 11 years, 5 months ago Modified7 years, 11 months ago Viewed 405 times This question shows research effort; it is useful and clear 1 Save this question. Show activity on this post. I came across this identity a mod n=a−⌊a n⌋×n a mod n=a−⌊a n⌋×n I see that it works, but I'm struggling to prove it, so I thought I would ask you guys. elementary-number-theory modular-arithmetic ceiling-and-floor-functions Share Share a link to this question Copy linkCC BY-SA 3.0 Cite Follow Follow this question to receive notifications edited Oct 3, 2017 at 7:20 Martin Sleziak 56.3k 20 20 gold badges 211 211 silver badges 391 391 bronze badges asked Apr 16, 2014 at 5:42 SuperbusSuperbus 2,256 2 2 gold badges 19 19 silver badges 26 26 bronze badges 6 Write a a using the division algorithm with n n as the dividend.Sandeep Silwal –Sandeep Silwal 2014-04-16 05:52:23 +00:00 Commented Apr 16, 2014 at 5:52 1 Hint: write a a as k∗n+b k∗n+b, where 0≤b<n 0≤b<n.Ruben –Ruben 2014-04-16 05:52:56 +00:00 Commented Apr 16, 2014 at 5:52 Ok, so I need to prove k=⌊a n⌋k=⌊a n⌋...Superbus –Superbus 2014-04-16 06:15:44 +00:00 Commented Apr 16, 2014 at 6:15 2 a n=k+b n a n=k+b n with b n<1 b n<1 so done?Superbus –Superbus 2014-04-16 06:17:36 +00:00 Commented Apr 16, 2014 at 6:17 1 Yes, that's the idea!Ruben –Ruben 2014-04-16 06:21:48 +00:00 Commented Apr 16, 2014 at 6:21 |Show 1 more comment 1 Answer 1 Sorted by: Reset to default This answer is useful 2 Save this answer. Show activity on this post. We can write a∈Z a∈Z as k n+b k n+b with k,b∈Z k,b∈Z and 0≤b<n 0≤b<n. Then a=b a=b (mod n n) and: n k+b−n⌊k n+b n⌋=n k+b+n⌊k+b n⌋n k+b−n⌊k n+b n⌋=n k+b+n⌊k+b n⌋ and since b n<1 b n<1 because b<n b<n this is equal to: n k+b−n⌊k⌋=n k+b−n k=b n k+b−n⌊k⌋=n k+b−n k=b Share Share a link to this answer Copy linkCC BY-SA 3.0 Cite Follow Follow this answer to receive notifications answered Apr 16, 2014 at 6:25 RubenRuben 704 4 4 silver badges 12 12 bronze badges Add a comment| You must log in to answer this question. Start asking to get answers Find the answer to your question by asking. Ask question Explore related questions elementary-number-theory modular-arithmetic ceiling-and-floor-functions See similar questions with these tags. 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14082
http://arxiv.org/pdf/1302.4539
Proving Termination Starting from the End Pierre Ganty 1 and Samir Genaim 2 1 IMDEA Software Institute, Madrid, Spain 2 Universidad Complutense de Madrid, Spain Abstract. We present a novel technique for proving program termination which introduces a new dimension of modularity. Existing techniques use the program to incrementally construct a termination proof. While the proof keeps changing, the program remains the same. Our technique goes a step further. We show how to use the current partial proof to partition the transition relation into those be-haviors known to be terminating from the current proof, and those whose status (terminating or not) is not known yet. This partition enables a new and unexplored dimension of incremental reasoning on the program side. In addition, we show that our approach naturally applies to conditional termination which searches for a precondition ensuring termination. We further report on a prototype implemen-tation that advances the state-of-the-art on the grounds of termination and condi-tional termination. 1 Introduction The question of whether or not a given program has an infinite execution is a funda-mental theoretical question in computer science but also a highly interesting question for software practitioners. The first major result is that of Alan Turing, showing that the termination problem is undecidable. Mathematically, the termination problem for a given program Prog is equivalent to deciding whether the transition relation R induced by Prog is well-founded. The starting point of our paper, is a result showing that the well-foundedness prob-lem of a given relation R is equivalent to the problem of asking whether the transi-tive closure of R, noted R+, is disjunctively well-founded . That is whether R+ is included in some W (in which case W is called a transition invariant ) such that W = W1 ∪ · · · ∪ Wn, n ∈ N and each Wi is well-founded (in which case W is said to be disjunctively well-founded ). This result has important practical consequences because it triggered the emergence of e ff ective techniques, based on transition invariants, to solve the termination problem for real-world programs [11,2,28,20]. By replacing the well-foundedness problem of R with the equivalent disjunctive well-foundedness problem of R+, one allows for the incremental construction of W:when the inclusion of R+ into W fails then use the information from the failure to update W with a further well-founded relation . Although the proof is incremental for W,it is important to note that a similar result does not hold for R. That is, it is in general not true that given R = R1 ∪ R2, if R+ 1 ⊆ W and R+ 2 ⊆ W then R+ ⊆ W.We introduce a new technique that, besides being incremental for W, further parti-tions the transition relation R separating those behaviors known to be terminating from arXiv:1302.4539v1 [cs.LO] 19 Feb 2013 the current W, from those whose status (terminating or not) is not known yet. For-mally, given R and a candidate W, we shall see how to compute a partition {RG, RB} of R such that ( a) R+ G ⊆ W; and ( b) every infinite sequence s1 R s 2 R · · · si R s i+1 · · · (or trace ) has a su ffi x that exclusively consists of transitions from RB, namely we have sz RB sz+1 RB · · · for some z ≥ 1. It follows that well-foundedness of RB implies that of R. Consequently, we can focus our e ff ort exclusively on proving well-foundedness of RB. In the a ffi rmative, then so is R and hence termination is proven. In the negative, then we have found an infinite trace in RB, hence in R. We observed that working with RB typically provides further hints on which well-founded relations to add to W. The partition of R into {RG, RB} enables a new and unexplored dimension of modularity for termination proofs. Let us mention that the partitioning of R is the result of adopting a fixpoint centric view on the disjunctive well-foundedness problem and leverage equivalent formulation of the inclusion check. More precisely, we introduce the dual of the check R+ ⊆ W by defining the adjoint to the function λX. X ◦ R used to define R+. Without defining it now, we write the dual check as follows: R ⊆ W−. We shall see that while the failure of R+ ⊆ W provides information to update W; the failure of R ⊆ W− provides information on all pairs in R responsible for the failure of W as a transition invariant. This is exactly that information, of semantical rather than syntactical nature, that we use to partition R.We show that the partitioning of R can be used not only for termination, but it also serves for conditional termination. The goal here is to compute a precondition, that is a set P of states, such that no infinite trace starts from a state of P. We show how to compute a (non-trivial) precondition from the relation RB.Our contributions are summarized as follows: ( i) we present Acabar , a new algo-rithm which allows for enhanced modular reasoning about infinite behaviors of pro-grams; ( ii ) we show that, besides termination, Acabar can be used in the context of conditional termination; and ( iii ) finally, we report on a prototype implementation of our techniques and compare it with the state-of-the-art on two grounds: the termination problem, and the problem of inferring a precondition that guarantees termination. 2 Example In this section, we informally overview our proposed techniques on an example taken from the literature . Consider the following loop: while ( x>0 ) { x:=x+y; y:=y+z; } represented by the transition relation R = {x > 0, x′ = x + y, y′ = y + z, z′ = z}, where the primed variables represent the values of the program variables after executing the loop body. Note that, depending on the input values, the program may not terminate (e.g. for x = 1, y = 1 and z = 1 ). Below we apply Acabar to prove termination. As we will see, this attempt ends with a failure which provide information on which subset of the transition relation to blame. Then, we will explain how to compute a termination precondition from this subset. In order to prove termination of this loop, we seek a disjunctive well-founded relation W such that R+ ⊆ W. To find such a W, Acabar is supported by incre-mentally (and automatically) inferring (potential) linear ranking functions for R or 2R+ [9,10]. When running on R, Acabar first adds the candidate well-founded relation W1 = {x′ < x, x > 0} to W which is initially empty. Relation W1 stems from the obser-vation that, in R, x is bounded from below (as shown by the guard) but not necessarily decreasing. Hence, using W = W1, Acabar partitions R into {R(1) G , R(1) B } where: R(1) G = {x > 0, x′ = x + y, y′ = y + z, z′ = z, y < 0, z ≤ 0} R(1) B = {x > 0, x′ = x + y, y′ = y + z, z′ = z, y < 0, z > 0}∨ {x > 0, x′ = x + y, y′ = y + z, z′ = z, y ≥ 0} . The partition comes with the further guarantee that every infinite trace in R must have a su ffi x that exclusively consists of transitions from R (1) B , which means that if R(1) B is well-founded then so is R. In addition, one can easily see that ( R(1) G )+ ⊆ W.Next, Acabar calls itself recursively on R(1) B to show its well-foundedness. As be-fore, it first adds W2 = {y′ < y, y ≥ 0} to W. Similarly to the construction of W1, W2 stems from the observation that, in some parts of R(1) B , y is bounded from below but not necessarily decreasing. Then, using W = W1 ∨ W2, Acabar partitions R(1) B into: R(2) G = {x > 0, x′ = x + y, y′ = y + z, z′ = z, z < 0} R(2) B = {x > 0, x′ = x + y, y′ = y + z, z′ = z, y ≥ 0, z ≥ 0} . Again the partition {R(2) G , R(2) B } of R(1) B comes with a similar guarantee. This time it holds that that every infinite trace in R must have a su ffi x that exclusively consists of tran-sitions from R(2) B . Recursively applying Acabar on R(2) B does not yield any further par-titioning, that is R(3) B = R(2) B . The reason being that no potential ranking function is automatically inferred. Thus, Acabar fails to prove well-foundedness of R, which is indeed not well-founded. However, due to the above guarantee, we can use R(2) B to infer a su ffi cient precondition for the termination of R. We explain this next. Inferring a su ffi cient precondition is done in two steps: ( i) we infer (an overapprox-imation of) the set of all states Z visited by some infinite sequence of steps in R(2) B ; and (ii ) we infer (an overapproximation of) the set of all states V each of which can reach Z through some steps in R. Turning to the example, we infer Z = {x > 0, y ≥ 0, z ≥ 0} and the following overapproximation V′ of V: V′ = {x ≥ 1, z = 0, y ≥ 0} ∨ { x ≥ 1, z ≥ 1, x + y ≥ 1, x + 2y + z ≥ 1, x + 3y + 3z ≥ 1} . It can be seen that every infinite trace visits only states in V′, hence the complement of V′ is a precondition for termination. Let us conclude this section by commenting on an example for which Acabar proves termination. Assume that we append z:=z-1 to the loop body above and call R′ the induced transition relation. Following our previous explanations, running Acabar on R′ updates W from ∅ to W1, and then to W1 ∨ W2. Then, and contrary to the previous explanations, Acabar will further update W to W1 ∨ W2 ∨ W3 where W3 is the well-founded relation {z′ < z, z ≥ 0}. From there, Acabar returns with value R(3) B = ∅, hence we have that R′ is well-founded. 33 Preliminaries A transition system is a pair ( Q, R) where Q is the set of states and R ⊆ Q × Q is the transition relation . An initialized transition system includes a further component I ⊆ Q , the set of initial states . For simplicity, we defer the treatment of initial states to Sec. 8. An R-trace is a sequence s1, s2, . . . , sn of states such that for every i, 1 ≤ i < n we have ( si, si+1) ∈ R. When R is clear from the context we simply say trace. An infinite R-trace is a sequence s1, s2, . . . of states such that for every i ≥ 1 we have ( si, si+1) ∈ R.Given R′ ⊆ R and an infinite R-trace π we say that π has infinitely many steps in R′ if (si, si+1) ∈ R′ for infinitely many i ≥ 1. Given a relation R′ ⊆ R and a set Q′ ⊆ Q , define post R′ def = {s′ ∈ Q | ∃ s ∈Q′ : ( s, s′) ∈ R′}. We say that this operator computes the R′-successors of Q ′. Dually, define pre R′ def = post R′− 1 = {s ∈ Q | ∃ s′ ∈ Q ′ : ( s, s′) ∈ R′}. We say that this operator computes the R′-predecessors of Q ′.3 A relation W ⊆ Q × Q is called disjunctively well-founded iff W coincides with the union of finitely many relations (viz. W = W1 ∪ . . . ∪ Wn) each of which is well-founded (viz. there is no infinite sequence s1, s2, . . . such that ( si, si+1) ∈ W` for all i ≥ 1). In this paper, we adhere to the following conventions: calligraphic letters X, Y, . . . refer to subsets of Q and capital letters X, Y, . . . refer to relations over Q, that is subsets of Q × Q . Further, throughout the paper the letter W is used to denote a relation over Q that is disjunctively well-founded. A linear expression is of the form a0 + a1 x1 + · · · + an xn where ai ∈ Z and ¯ x = 〈x1, . . . , xn〉 are variables ranging over Z. An atomic linear constraint c is of the form e1 op e 2 where ei is a linear expression and op ∈ { =, ≥, ≤, >, < }. A formula ψ is a Boolean combination of atomic linear constraints. Note that ¬ψ is also a formula. For the sake of simplicity, a conjunction c1 ∧ · · · ∧ cn of atomic linear constraints is some-times written as the set {c1, . . . , cn}. A solution of a formula ψ is a mapping from its variables into the integers such that the formula evaluates to true. Sets and relations over, respectively, Zn and Zn × Zn are sometimes specified using formulas, with the customary convention, for relations, of variables and primed variables. For instance, the formula {x ≥ 0, x′ = x − y, y′ = y} defines the relation R ⊆ Z2 × Z2 such that R = {〈 (x, y), (x′, y′)〉 | x ≥ 0 ∧ x′ = x − y ∧ y′ = y}.Finally, we briefly recall classical results of lattice theory and refer to the classical book of Davey and Priestley for further information. Let f be a function over a partially ordered set ( L, v). A fixpoint of f is an element l ∈ L such that f (l) = l. We denote by lfp f and gfp f , respectively, the least and the greatest fixpoint , when they exist, of f . The well-known Knaster-Tarski’s theorem states that each order-preserving function f ∈ L → L over a complete lattice 〈L, v, ⊔, d, >, ⊥〉 admits a least (greatest) fixpoint and the following characterization holds: lfp f = d{x ∈ L | f (x) v x} gfp f = ⊔{x ∈ L | x v f (x)} . (1) 3We define R−1,R∗and R+to be R−1={(s′,s)|(s,s′)∈R},R∗=⋃i≥0Riand R+=R◦R∗where R0is the identity, Ri+1=Ri◦Rand R1◦R2={(s,s′′ )| ∃ s′: ( s,s′)∈R1∧(s′,s′′ )∈R2}. 44 Modular Reasoning For Termination A termination proof based on transition invariants consists in establishing the existence of a disjunctively well-founded transition invariant. That is, the goal is to prove the inclusion of R+, into some W.4 For short, we write R+ ⊆ W. Proving termination is thus reduced to finding some W and prove that the inclusion hold. In the above inclusion check, R+ coincides with the least fixpoint of the function λY. R ∪ g(Y) where g def = λY. Y ◦ R. It is known that if we can find an adjoint function ˜g to g such that g(X) ⊆ Y iff X ⊆ ˜g(Y) for all X, Y then there exists an equivalent inclusion check to R+ ⊆ W. This equivalent check, denoted R ⊆ W− in the introduction, is such that W− is defined as a greatest fixpoint of the function λY. W ∩ ˜g(Y). Next, we define ˜ g def = λY. ¬(¬Y ◦ R−1). Lemma 1. Let X , Y be subsets of Q × Q we have: X ◦ R ⊆ Y ⇔ X ⊆ ¬ (¬Y ◦ R−1).Proof. First we need an easily proved logical equivalence: (ϕ1 ∧ ϕ2) ⇒ ϕ3 iff ( ¬ϕ3 ∧ ϕ2) ⇒ ¬ ϕ1 .Then we have: X ◦ R ⊆ Y iff ∀s, s′, s1 : ((s, s1) ∈ X ∧ (s1, s′) ∈ R) ⇒ (s, s′) ∈ Y iff ∀s, s′, s1 : ((s, s′) < Y ∧ (s1, s′) ∈ R) ⇒ (s, s1) < X by above equivalence iff ∀s, s′, s1 : ((s, s′) < Y ∧ (s′, s1) ∈ R−1) ⇒ (s, s1) < X def. of R−1 iff ∀s, s′, s1 : ((s, s′) ∈ ¬ Y ∧ (s′, s1) ∈ R−1) ⇒ (s, s1) ∈ ¬ X iff ( ¬Y ◦ R−1) ⊆ ¬ X iff X ⊆ ¬ (¬Y ◦ R−1) ut Intuitively, g corresponds to forward reasoning for proving termination while ˜ g cor-responds to backward reasoning because of the composition with R−1. The least fixpoint lfp λY. R ∪ g(Y) is the least relation Z containing R and closed by composition with R,viz. R ⊆ Z and Z ◦ R ⊆ Z. On the other hand, the greatest fixpoint gfp λY. W ∩ ˜g(Y) is best understood as the result of removing from W all those pairs ( s, s′) of states such that ( s, s′) ◦ R+ W. This process returns the largest subset Z′ of W which is closed by composition with R, viz. Z′ ⊆ W and Z′ ◦ R ⊆ Z′. Using the results of Cousot we find next that termination can be shown by proving either inclusion of Lem. 2. Lemma 2 (from ). lfp λY. R ∪ g(Y) ⊆ W ⇔ R ⊆ gfp λY. W ∩ ˜g(Y).Proof. lfp λY. R ∪ g(Y) ⊆ W iff ∃A : R ⊆ A ∧ g(A) ⊆ A ∧ A ⊆ W by (1) iff ∃A : R ⊆ A ∧ A ⊆ ˜g(A) ∧ A ⊆ W Lem. 1 iff R ⊆ gfp λY. W ∩ ˜g(Y) by (1) ut 4Recall that Wis always assumed to be disjunctively well-founded. 5As we shall see, the inclusion check based on the greatest fixpoint has interesting consequences when trying to prove termination. An important feature when proving termination using transition invariants is to de-fine actions to take when the inclusion check lfp λY. R ∪ g(Y) ⊆ W fails. In this case, some information is extracted from the failure (e.g., a counter example), and is used to enrich W with more well-founded relations . We shall see that, for the backward approach, failure of R ⊆ gfp λY. W ∩ ˜g(Y)induces a partition of the transition relation R into {RG, RB} such that ( a) ( RG)+ ⊆ W;together with the following termination guarantee ( b) every infinite R-trace contains a su ffi x that is an infinite RB-trace (Lem. 4). An important consequence of this is that we can focus our e ff ort exclusively on proving termination of RB. It is important to note that the guarantee that no infinite R-trace contains infinitely many steps from RG is not true for any partition {RG, RB} of R but it is true for our partition which we define next. Definition 1. Let G = gfp λY. W ∩ ˜g(Y), we define {RG, RB} to be the partition of R given by R G = R ∩ G and R B = R \ RG.Example 1. Let R = {x ≥ 1, x′ = x + y, y′ = y − 1} and assume W = {x′ < x, x ≥ 1} which is well-founded, hence disjunctively well-founded as well. Evaluating the great-est fixpoint (we omit calculations) yields RG = {x ≥ 1, x′ = x + y, y′ = y − 1, y < 0} RB = {x ≥ 1, x′ = x + y, y′ = y − 1, y ≥ 0} which is clearly a partition of R. The relation RG consists of those pairs of states where y is negative, hence x is decreasing as captured by W. On the other hand, RB consists of those pairs where y is positive or null. It follows that, when taking a step from RB, x does not decrease. This is precisely for those pairs that W fails to show termination.  Next, we state and prove the termination guarantees of the partition {RG, RB}. Lemma 3. Given R G as in Def. 1 we have lfp λY. RG ∪ Y ◦ R ⊆ W. Proof. G ⊆ ˜g(G) ∧ G ⊆ W def. of G and (1) only if g(G) ⊆ G ∧ G ⊆ W Lem. 1 only if R ∩ G ⊆ G ∧ g(G) ⊆ G ∧ G ⊆ W only if RG ⊆ G ∧ g(G) ⊆ G ∧ G ⊆ W def. of RG only if lfp λY. RG ∪ g(Y) ⊆ W by (1) ut An equivalent formulation of the previous result is RG ◦ R∗ ⊆ W, which in turn implies, since RG ⊆ R, that (RG ◦ R∗)+ ⊆ W, and also ( RG)+ ⊆ W. Lemma 4. Every infinite R-trace has a su ffi x that is an infinite R B-trace. 6Proof. Assume the contrary, i.e., there exists an infinite R-trace s1, s2, . . . that contains infinitely many steps from RG. Let S = si1 , si2 , . . . be the infinite subsequence of states such that ( si j , si j+1) ∈ RG for all j ≥ 1. Recall also that W = W1 ∪ · · · ∪ Wn where each Wis well-founded. For any si, s j ∈ S with i < j it holds that ( si, s j) ∈ RG ◦ R∗, and thus, according to Lem. 3, we also have that ( si, s j) ∈ W for some 1 ≤ ` ≤ n. Ramsey’s theorem guarantees the existence of an infinite subsequence S ′ = s j1 , s j2 , . . . of S , and a single W, such that for all si, s j ∈ S ′ with i < j we have ( si, s j) ∈ W. This contradicts that W` is well-founded and we are done. ut Remark 1. When fixpoints are not computable, they can be approximated from above or from below . It is routine to check that the results of Lemmas 3 and 4 remain valid when replacing G = gfp λY. W ∩ ˜g(Y) in Def. 1 with G′ ⊆ gfp λY. W ∩ ˜g(Y). Therefore we have that, even when approximating gfp λY. W ∩ ˜g(Y) from below, the termination guarantees of {RG, RB} still hold. In Sec. 6, we shall see how to exploit this result in practice. Example 2 (cont’d from Ex. 1). We left Ex. 1 with W = {x′ < x, x ≥ 1} and RB = {x ≥ 1, x′ = x + y, y′ = y − 1, y ≥ 0}. As argued previously, to prove the well-foundedness of R it is enough to show that RB is well-founded. For clarity, we rename RB into R(1) B .Next we partition R(1) B as we did it for R in Ex. 1. As a result, we update W by adding the well-founded relation {y′ < y, y ≥ 0}. Then we evaluate again G (we omit calculations) which yields R(2) B = ∅. Hence we conclude from Lem. 4 that R is well-founded.  Building upon all the previous results, we introduce Acabar that is given at Alg. 1. Acabar is a recursive procedure that takes as input two parameters: a transition relation R and a disjunctively well-founded relation W. The second parameter is intended for recursive calls, hence the user should invoke Acabar as follows: Acabar( R, ∅). We call it the root call . Upon termination, Acabar returns a subset RB of the transition relation R. If it returns the empty set, then the relation R is well-founded, hence termination is proven. Otherwise ( RB , ∅), we can not know for sure if R is well-founded: there might be an infinite R-trace. However, Lem. 4 tells us that every infinite R-trace must have a su ffi x that is an infinite RB-trace. It may also be the case that RB is well-founded (and so is R) in which case it was not discovered by Acabar . Another case is that R = RB. In this case we have made no progress and therefore we stop. Whenever RB , ∅, we call this returned value the problematic subset of R.Next we study progress properties of Acabar . We start by defining the sequence {R(i)}i≥0 where each R(i) is the argument passed to the i-th recursive call to Acabar . In particular, R(0) is the argument of the root call. Furthermore, we define the sequences {R(i) B }i≥1 and {R(i) G }i≥1 where {R(i) G , R(i) B } is a partition of R(i−1) and R(i) B = R(i) for all i ≥ 1. Lemma 5. Let a run of Acabar with at least i ≥ 1 recursive calls, then we have R(0) ) R(1) ) · · · ) R(i) . Proof. The proof is by induction on i, for i = 1 it follows from the definitions that R(1) = R(1) B and {R(1) B , R(1) G } is a partition of R(0) . Moreover, since at least i = 1 recursive calls take place we find that the condition of line 5 fails, meaning neither R(1) B nor R(1) G is empty, hence R(1) is a strict subset of R(0) . The inductive case is similar. ut 7Algorithm 1: Enhanced modular reasoning Acabar( R,W ) Input : a relation R⊆ Q × Q Input : a relation W⊆ Q × Q such that Wis disjunctively well-founded Output :RB⊆R 1begin 2WBW∪find dwf candidate( R) 3let Gbe such that G⊆gfp λY.W∩˜g(Y) 4RBBR\G 5if RB=∅or R B=Rthen 6return RB 7else 8return Acabar( RB,W) By Lemmas 4 and 5, we have that every infinite R(0) -trace has a su ffi x that is an infinite R(i) B -trace for every i ≥ 1. As a consequence, forcing Acabar to execute line 6 after predefined number of recursive calls, it returns a relation R(i) B such that the previous property holds. Incidentally, we find that Acabar proves program termination when it returns the empty set as stated next. Theorem 1. Upon termination of the call Acabar( R, ∅), if it returns the empty set, then the relation R is well-founded. Let us now turn to line 2. There, Acabar calls a subroutine find dwf candidate( R) implementing a heuristic search which returns a disjunctively well-founded relation us-ing hints from the representation and the domain of R. Details about its implementation, that is inspired from previous work [9,10], will be given at Sec. 7 — we will consider the case of R being a relation over the integers of the form R = ρ1 ∨ · · · ∨ ρn where each ρi is a conjunction of linear constraints over the variables ¯ x and ¯ x′. Let us intuitvely explain this procedure on an example. Example 3 (cont’d from Ex. 2). Acabar( R, ∅) updates W as follows: ( 1) ∅; ( 2) {x′ < x, x ≥ 1}; ( 3) {x′ < x, x ≥ 1}, {y′ < y, y ≥ 0}. The first update from ∅ to {x′ < x, x ≥ 1} is the result of calling find dwf candidate( R). The hint used by find dwf candidate is that x is bounded from below in R. The second update to W results from calling find dwf candidate( RB = {x ≥ 1, x′ = x + y, y′ = y − 1, y ≥ 0}). Since RB has the linear ranking function f (x, y) = y, find dwf candidate returns {y′ < y, y ≥ 0}.  5 Acabar for Conditional Termination As mentioned previously, upon termination, Acabar returns a subset RB of the transition relation R. If this set is empty then R is well-founded and we are done. Otherwise, RB is a non-empty subset and called the problematic set. In this section, we shall see how to compute, given the problematic set, a precondition P for termination . More precisely, 8P is a set of states such that no infinite R-trace starts with a state of P. We illustrate our definitions using the simple but challenging example of Sec. 2. Example 4. Consider again the relation R = {x > 0, x′ = x + y, y′ = y + z, z′ = z}. Upon termination Acabar returns the following relation: RB = {x′ = x + y, y′ = y + z, z′ = z, x > 0, y ≥ 0, z ≥ 0} which corresponds to all the cases where x is stable or increasing over time.  Lemma 4 tells us that every infinite R-trace π is such that π = π f π∞ where π f is a finite R-trace and π∞ is an infinite RB-trace. Our computation of a precondition for termination is divided into the following parts: ( i) compute those states Z visited by infinite RB-trace; ( ii ) compute the set V of R∗-predecessors of Z, that is the set of states visited by some R-trace ending in Z; and ( iii ) compute P as the complement of V. Formally, ( i) is given by a greatest fixpoint expression gfp λX. pre RB. This expression is directly inspired by the work of Bozga et al. on deciding conditional termination. This greatest fixpoint is the largest set Z of states each of which has an RB-successor in Z. Because of this property, every infinite RB-trace visits only states in Z. In π = π f π∞, this corresponds to the su ffi x π∞ that is an infinite RB-trace. Example 5. For RB as given in Ex. 4, we have that Z = {z ≥ 0, y ≥ 0, x > 0} which contains the following infinite RB-trace: (x = 1, y = 0, z = 0) RB (x = 1, y = 0, z = 0) RB (x = 1, y = 0, z = 0) RB . . .  Let us now turn to ( ii ), that is computing the set V of R∗-predecessors of Z. It is known that V coincides with lfp λX. Z ∪ pre R. Intuitively, we prepend to those infinite RB-traces a finite R-trace. That is, prefixing π f to π∞ results in π = π f π∞. Finally, step ( iii ) results into a precondition for termination P obtained by complementing V. Example 6. Computing lfp λX. Z ∪ pre R for Z as given in Ex. 5 and R = {x′ = x + y, y′ = y + z, z′ = z, x > 0, y ≥ 0, z ≥ 0} (Ex. 4) gives V = V1 ∨ V 2 where V1 = {x ≥ 1, z = 0, y ≥ 0}V2 = {x ≥ 1, z ≥ 1} ∪ { x + i ∗ y + j ∗ z ≥ 1 | i ≥ 1, j = ∑i−1 k=0 k} . Intuitively, the set V1 of states corresponds to entering the loop with z = 0 and y non-negative, in which case the loop clearly does not terminate. The set V2 of states corresponds to entering the loop with z positive, and the loop does not terminate af-ter i-th iterations for all i. Note that V2 consists of infinitely many atomic formulas. Complementing V gives P.  Theorem 2. There exists an infinite R-trace starting from s iff s < P.Approximations. As argued previously, it is often the case that only approximations of fixpoints are available. In our case, any overapproximation of either Z or V can be exploited to infer P. Because of approximations, we lose the if direction of the theorem, that is, we can only say that there is no infinite R-trace starting from some s ∈ P .9Example 7. Using finite disjunctions of linear constraints, we can approximate V by {x ≥ 1, z = 0, y ≥ 0} ∨ { x ≥ 1, z ≥ 1, x + y ≥ 1, x + 2y + z ≥ 1, x + 3y + 3z ≥ 1} and then the complement P is x ≤ 0 ∨ x + y ≥ 1 ∨ x + 2y + z ≥ 1 ∨ x + 3y + 3z ≥ 1 ∨ z ≤ − 1 ∨ (y ≤ − 1 ∧ z ≤ 0) which is a su ffi cient precondition for termination. Note that the first 4 disjuncts corre-spond to the executions which terminates after 0, 1, 2 and 3 iterations.  6 Implementation We have implemented the techniques described in Sec. 4 and 5 for the case of multiple-path integer linear-constraint loops. These loops correspond to relations of the form R = ρ1 ∨ · · · ∨ ρd where each ρi is a conjunction of linear constraints over the variables ¯ x and ¯x′. In this context, the set Q of states is equal to Zn where n is the number of variables in ¯ x. This is a classical setting for termination [4,6,24]. Internally, we represent sets of states and relations over them as DNF formulas where the atoms are linear constraints. In what follows, we explain su ffi cient implementation details so that our experiments can be independently reproduced if desired. Our implementation is available . We start with line 2 of Alg. 1. Recall that the purpose of this line is to add more well-founded relations to W based on the current relation R. In our implementation, W consists of well-founded relations of the form { f ( ¯ x) ≥ 0, f ( ¯ x′) < f ( ¯ x)} where f is a linear function [10,9]. Thus, our implementation looks for such well-founded relations. In particular, for each ρi of R we add new well-founded relations to W as follows: if ρi has a linear ranking function f ( ¯ x) that is synthesized automatically [24,4] then { f ( ¯ x′) < f ( ¯ x), f ( ¯ x) ≥ 0} is added to W; otherwise, let { f1( ¯ x) ≥ 0, . . . , fd( ¯ x) ≥ 0} be the result of projecting each ρi on ¯ x (i.e., eliminating variables ¯ x′ from ρi), then {{ fi( ¯ x′) < fi( ¯ x), fi( ¯ x) ≥ 0} | 1 ≤ i ≤ d} is added to W. Because fi is bounded but not necessarily decreasing, it is called a potential linear ranking function . As for line 3, recall that G is a subset of gfp λY. W ∩ ˜g(Y). Furthermore, the sole purpose of G is to compute RB = R \ G. We now observe that ¬G, the complement of G, is as good as G. In fact, RB = R ∩ (¬G). So by considering ¬G instead, what we are looking for is an overapproximation of ¬(gfp λY. W ∩ ˜g(Y)). Next we recall Park’s theorem replacing the above expression by a least fixpoint expression. Theorem 3 (From ). Let 〈L, v, d, ⊔, >, ⊥, ¬〉 be a complete Boolean algebra and let f ∈ L → L be an order-preserving function then f ′ = λX. ¬( f (¬X)) is an order-preserving function on L and ¬(gfp f ) = lfp f ′. Park’s theorem applies in our setting because computations are carried over the Boolean algebra 〈2(Q×Q ), ⊆, ∩, ∪, (Q × Q ), ∅, ¬〉 . Applying it to gfp λY. W ∩ ˜g(Y) where ˜g(Y) = ¬(¬Y ◦ R−1), we find that ¬(gfp λY. W ∩ ¬ (¬Y ◦ R−1)) = lfp λY. (¬W) ∪ Y ◦ R−1 . 10 Therefore, to implement line 3, we rely on abstract interpretation to compute an over-approximation of lfp λY. (¬W) ∪ Y ◦ R−1, hence, by negation, an underapproximation of gfp λY. W ∩ ˜g(Y) therefore complying with the requirement on G.As far as abstract interpretation is concerned, our implementation uses a combina-tion of predicate abstraction and trace partitioning . The set of predicates is given by a finite set of atomic linear constraints and is also closed under negation, e.g., if x + y ≥ 0 is a predicate then x + y ≤ − 1 is also a predicate. Abstract values are positive Boolean combination of atoms taken from the set of predicates. Observe that although negation is forbidden in the definition of abstract values, the abstract domain is closed under complement. The set of predicates is chosen so as the following invariant to hold: each time the control hits line 3, the set contains enough predicates to represent precisely each well-founded relation in W. Our implementation provides enhanced precision by enforcing a stronger invariant: besides the above predicates for W, it includes all atomic linear constraints occurring in the formulas representing X1, . . . , Xwhere ≥ 0, X0 = (¬W)and Xi+1 = (¬W) ∪ Xi ◦ R−1. The value of ` is user-defined and, in our experiments, it did not exceed 1. To further enhance precision at line 3, we apply trace partitioning . The set of R-traces is partitioned using the linear atomic constraints of the form f ( ¯ x′) < f ( ¯ x) that appear in W. More precisely, partitioning R on f ( ¯ x′) < f ( ¯ x) is done by replacing each ρi by ( ρi ∧ f ( ¯ x′) < f ( ¯ x)) ∨ (ρi ∧ f ( ¯ x′) ≥ f ( ¯ x)). As for conditional termination, overapproximating Z = gfp λX. pre RB is done by computing the last element Xfrom the finite sequence X0, . . . , X given by X0 = Q and Xi+1 = Xi ∧ pre RB where ` is predefined. The result is always representable as DNF formula where the atoms can be any atomic linear constraints. As for V = lfp λX. X` ∪ pre R, an overapproximation is computed in a similar way to that of line 3, i.e., using a combination of predicate abstraction and trace partitioning. 7 Experiments We have evaluated our prototype implementation against a set of benchmarks collected from publications in the area [9,7]. In what follows, we present the results of our im-plementation for those loops, and compare them to existing tools for proving termina-tion [26,7,6] as well as tools for inferring preconditions for termination . We com-pare the di ff erent techniques according to what the corresponding implementations re-port. We ignore performance because, for the selected benchmarks, little insight can be gained from performance measurements when an implementation was available (which was not always the case ). The benchmarks accompanied with our results are depicted in Table 1. Translating each loop to a relation of the form R = ρ1 ∨ · · · ∨ ρn is straightforward. Every line in the table includes a loop and its inferred termination precondition ( true means it termi-nates for any input). In addition, preconditions (di ff erent from true ) marked with • are optimal, i.e., the corresponding loop is non-terminating for any state in the complement. We have divided the benchmarks into 3 groups: (1–5), (6–15) and (16–41). With the exception of loop 1, each loop in group (1–5) includes non-terminating executions and 11 ] loop termination precondition 1 while (x ≥0) x’=-2x+10; true 2 while (x>0) x’=x+y; y’=y+z; x≤0 ∨ z<0∨ (z=0 ∧ y<0) ∨ x+y≤0 ∨ x+2y+z≤0∨ x+3y+3z≤03 while (x ≤N) if () { x’=2x+y; y’=y+1; } else x’=x+1; x > n ∨ x + y ≥ 04 @requires n>200 and y<9 while (1) if (x<n) { x’=x+y; if (x’ ≥200) break; } n ≤ 200 ∨ y ≥ 9∨ (x < n ∧ y ≥ 1) ∨ (x<n ∧ x≥200 ∧ x+y≥200) 5 while (x<>y) if (x>y) x’=x-y; else y’=y-x; • (x ≥ 1 ∧ y ≥ 1) ∨ x = y 6 while (x<0) x’=x+y; y’=y-1; x ≥ 0 ∨ x + y ≥ 0 ∨ x + 2y ≥ 1 ∨ x + 3y ≥ 37 while (x>0) x’=x+y; y’=-2y; • x ≤ 0 ∨ y , 08 while (x<y) x’=x+y; y’=-2y; • x ≥ 0 ∨ y , 09 while (x0) x’=2x+4y; y’=4x; • 5y − 4x ≥ 0 ∨ (3 x−4y ≥ 0∧16 x−21 y ≥ 1) 11 while (x<5) x’=x-y; y’=x+y; • x , 0 ∨ y , 012 while (x>0 and y>0) x’=-2x+10y; • x ≤ 3 ∨ 10 y − 3x , 013 while (x>0) x’=x+y; x ≤ 0 ∨ y < 0 ∨ x + y ≤ 014 while (x<10) x’=-y; y’=y+1; • y ≤ − 10 ∨ x ≥ 10 15 while (x<0) x’=x+z; y’=y+1; z’=-2y x ≥ 0 ∨ x + z ≥ 016 while (x>0 and x<100) x’ ≥2x+10; ? true 17 while (x>1) -2x’=x; ? true 18 while (x>1) 2x’ ≤x; ? true 19 while (x>0) 2x’ ≤x; ? true 20 while (x>0) x’=x+y; y’=y-1; true 21 while (4x+y>0) x’=-2x+4y; y’=4x; 4x + y ≤ 0 ∨ (x − 4x ≥ 0 ∧ 8x − 15 y ≥ 1) 22 while (x>0 and x<y) x’=2x; y’=y+1; true 23 while (x>0) x’=x-2y; y’=y+1; true 24 while (x>0 and x<n) x’=-x+y-5; y’=2y; n’=n; true 25 while (x>0 and y<0) x’=x+y; y’=y-1; ? true 26 while (x-y>0) x’=-x+y; y’=y+1; true 27 while (x>0) x’=y; y’=y-1; true 28 while (x>0) x’=x+y-5; y’=-2y; true 29 while (x+y>0) x’=x-1; y’=-2y; true 30 while (x>y) x’=x-y; 1 ≤y’ ≤2 ? true 31 while (x>0) x’=x+y; y’=-y-1; true 32 while (x>0) x’=y; y’ ≤-y; ? true 33 while (x<y) x’=x+1; y’=z; z’=z; true 34 while (x>0) x’=x+y; y’=y+z; z’=z-1; true 35 while (x+y ≥0 and x ≤z) x’=2x+y; y’=y+1; z’=z true 36 while (x>0 and x ≤z) x’=2x+y; y’=y+1; z’=z true 37 while (x ≥0) x’=x+y; y’=z; z’=-z-1; true 38 while (x-y>0) x’=-x+y; y’=z; z’=z+1; true 39 while (x>0 and x2x; y’=z; z’=z; true 40 while (x ≥0 and x+y ≥0) x’=x+y+z; y’=-z-1; z’=z; ? true 41 while (x+y ≥0 and x ≤n) x’=2x+y; y’=z; z’=z+1; n’=n; true Table 1. Benchmarks used in experiments. Loops (1–5) are taken from and (6–41) from . 12 thus those loops are suitable for inferring preconditions. Our implementation reports the same preconditions as the tool of Cook et al. save for loop 1 for which their tool is reported to infer the precondition x > 5 ∨ x < 0, while we prove termination for all input. Note that every other tool used in the comparison [7,6,26] fail to prove termination of this loop. Further, the precondition we infer for loop 5 is optimal. All the loops (6–15) are non-terminating. Chen et al. report that their tool cannot handle them since it aims at proving termination and not inferring preconditions for termination. We infer preconditions for all of them, and in addition, most of them are optimal (those marked with •). Unfortunately for those loops we could not compare with the tool of Cook et al. , since there is no implementation available . Loops in the group (16–41) are all terminating. Those marked with ? actually have linear ranking functions, those unmarked require disjunctive well-founded transition invariants with more than one disjunct. We prove termination of all of them except loop 21. We point that the tool of Chen et al. also fails to prove termination of loop 21, but also of loop 34. On the other benchmarks, they prove termination. They also report that PolyRank failed to prove termination of any of the loops that do not have a linear ranking function. In addition, we applied ARMC on the loops of the group (16–41). ARMC, a transition invariants based prover, succeeded to prove termination for all those loops with a linear ranking function (marked with ?) and also loop 39. Next we discuss in details the analysis of two selected examples from Table 1. Example 8. Let us explain the analysis of loop 1 in details starting with the root call Acabar (R, ∅) where R = {x ≥ 0, x′ = −2x + 10 }. At line 2, since R includes the bound x ≥ 0, i.e., f (x) = x is a potential linear ranking function, we add {x′ < x, x ≥ 0} to W.Computing G at line 3, hence RB at the following line, results in RB = ρ1 ∨ ρ2 where ρ1 = {x′ = −2x + 10 , x ≥ 0, x ≤ 3} and ρ2 = {x′ = −2x + 10 , x ≥ 4, x ≤ 5}.Note that ρ1 is enabled for 0 ≤ x ≤ 3 and in this case x′ > x. Also ρ2 is enabled for x = 4 or x = 5 for which x′ < x and thus ρ2 ⊆ W, however, after one more iteration, the value of x increases (this is why ρ2 is included in RB). Transitions for which x > 5 are not included in RB, hence they belong to RG itself included in W (Lem. 3). Hence when x > 5 termination is guaranteed, this is also easily seen since those transitions terminate after one iteration. Since RB is neither empty nor equal to R, a recursive call to Acabar (RB, W) takes place. At line 2, we add {− x′ < −x, 10 − x ≥ 0} to W since f (x) = 10 − x is a linear ranking function for ρ1. Note that ρ2 has the linear ranking function f (x) = x already included in W. Computing G at line 3, hence RB, yields RB = ∅ and therefore we conclude that the loop terminates for any input.  Example 9. Let us explain the analysis of loop 9 in details starting with the root call Acabar (R, ∅) where R = {x < y, x′ = x + y, 2y′ = y}. At line 2, since R includes the bound y − x > 0, i.e., f (x, y) = y − x − 1 is a potential linear ranking function, we add {y′ − x′ < y − x, y − x − 1 ≥ 0} to W. Computing G at line 3, hence RB yields RB = {x < y, x′ = x + y, 2y′ = y, y ≤ 0}. Note that RB exclusively consists of transitions where y is not positive, in which case x′ − y′ ≥ x − y and thus not included in W.Transitions where y is positive are not included in RB (hence they belong to RG) since they always decrease x − y, and thus are transitively included in W (Lem. 3). 13 Since RB is neither empty nor equal to R, we call recursively Acabar (RB, W). At line 2, since R includes the bound y ≤ 0 (or equivalently −y ≥ 0), i.e., f (x, y) = −y is a potential linear ranking function, we add {− y′ < −y, −y ≥ 0} to W. Computing G at line 3, hence RB yields RB = {x < y, x′ = x + y, 2y′ = y, y = 0}. Note that RB exclusively consists of transitions where y = 0, which keeps both values of x and y unchanged. Transitions in which y is negative belong to RG, hence they are transitively covered by W (Lem. 3), in particular by the last update (viz. {− y′ < −y, −y ≥ 0}) to W.Since RB is neither empty nor equal to R, we call recursively Acabar (RB, W). This time our implementation does not further enrich W with a well-founded relation, and as a consequence, after computing G at line 3, we get that RB = R. Hence, Acabar returns with RB = {x < y, x′ = x + y, 2y′ = y, y = 0}.Now, given RB, we infer a precondition for termination as described in Sec. 5. We first compute gfp λX. pre RB, which in this case, converges in two steps with Z ≡ y = 0 ∧ x < 0. Then we compute lfp λX. Z ∪ pre R, which results in V ≡ y = 0∧ x < 0. The complement, P ≡ y < 0∨y > 0∨ x < 0, is a precondition for termination. Note that the result is optimal, i.e., V is a precondition for non-termination. Optimality is achieved because Z and V coincide with the gfp and the lfp of the corresponding operators, and are not overapproximations.  8 Conclusion This work started with the invited talk of A. Podelski at ETAPS ’11 who remarked that the inclusion check R+ ⊆ W is equivalently formulated as a safety verification problem where states are made of pairs. Back to late 2007, a PhD thesis proposed a new approach to the safety verification problem in which the author shows how to leverage the equivalent backward and forward formulations of the inclusion check. Those two events planted the seeds for the backward inclusion check R ⊆ W−, and later Acabar . Initial States. For the sake of simplicity, we deliberately excluded the initial states I from the previous developments. Next, we introduce two possible options to incorpo-rate knowledge about the initial states in our framework. The first option consists in replacing R by R′ that is given by R ∩ (Acc × Acc ) where Acc denotes (an overapproxi-mation of) the reachable states in the system. Formally, Acc is given by the least fixpoint lfp λX. I ∪ post R. The second option is inspired by the work of Cousot where he mixes backward and forward reasoning. We give here some intuitions and preliminary development. Re-call that the greatest fixpoint gfp λY. W ∩ ˜g(Y) of line 3 is best understood as the result of removing all those pairs ( s, s′) ∈ W such that ( s, s′) ◦ R+ W. We observe that the knowledge about initial states is not used in the greatest fixpoint. A way to incorpo-rate that knowledge is to replace the greatest fixpoint expression by the following one gfp λY. (B ∩ W) ∩ ˜g(Y) where B takes the reachable states into account. In a future work, we will formally develop those two options and evaluate their benefit. Related Works. As for termination, our work is mostly related to the work of Cook et al. [10,11] where the inclusion check R+ ⊆ W is put to work by incrementally constructing W. Our approach, being based on the dual check R ⊆ W−, adds a new dimension of modularity /incrementality in which R is also modified to safely exclude 14 those transitions for which the current proof is su ffi cient. The advantage of the dual check was shown experimentally in Sec. 7. However, let us note that in our implemen-tation we use potential ranking functions and trace partitioning, which are not used in ARMC . Moreover, it smoothly applies to conditional termination. Kroening et al. introduced the notion of compositional transition invariants, and used it to develop techniques that avoid the performance bottleneck of previous approaches . Recently, Chen et al. proposed a technique for proving termination of single-path linear-constraint loops. Contrary to their techniques, we handle general transition relations and our approach applies also to conditional termination. As for conditional termination, the work of Cook et al. is the closest to ours. However, we di ff er in the following points: ( a) we do not use universal quantifier elim-ination, whose complexity is usually very high, depending on the underlying theory used to specify R. Instead, we adapt a fixpoint centric view that allows using abstract interpretation, and thus to control precision and performance; ( b) we do not need spe-cial treatment for loop with phase transitions (as the one of Sec. 2), they are handled transparently in our framework. Bozga et al. studied the problem of deciding con-ditional termination. Their main interest is to identify family of systems for which gfp λX. pre R, the set of non-terminating states, is computable. It is worth “terminating” by mentioning that several formulations, of the termina-tion problem, similar to the check R+ ⊆ W have appeared before [8,21,16]. They have also led to practical tools for corresponding programming paradigms. The relation be-tween these approaches was recently studied . Works based on these formulations, in particular those that construct global ranking functions for R , might serve as a starting point to understand some (completeness) properties of our approach. This is left for future work. References Acabar. Albert, E., Arenas, P., Genaim, S., Puebla, G., Zanardini, D.: COSTA: Design and imple-mentation of a cost and termination analyzer for java bytecode. In: de Boer, F.S., Bon-sangue, M.M., Graf, S., de Roever, W.P. (eds.) Formal Methods for Components and Objects, FMCO’07. LNCS, vol. 5382, pp. 113–132. Springer (2007) 3. Ben-Amram, A.M.: Size-change termination, monotonicity constraints and ranking func-tions. In: CAV ’09: Proc. 21st Int. Conf. on Computer Aided Verification. pp. 109–123. LNCS, Springer (2009) 4. Ben-Amram, A.M., Genaim, S.: On the linear ranking problem for integer linear-constraint loops. In: POPL ’13: Proc. 40th ACM SIGACT-SIGPLAN Symp. on Principles of Program-ming Languages. ACM (2013), to appear 5. Bozga, M., Iosif, R., Koneˇ cn´ y, F.: Deciding conditional termination. In: TACAS ’12: Proc. 18th Int. Conf. on Tools and Algorithms for the Construction and Analysis of Systems. LNCS, vol. 7214, pp. 252–266. Springer (2012) 6. Bradley, A.R., Manna, Z., Sipma, H.B.: The polyranking principle. In: ICALP ’05: Proc. of 32nd Int. Colloquium on Automata, Languages and Programming. LNCS, vol. 3580, pp. 1349–1361. Springer (2005) 7. Chen, H.Y., Flur, S., Mukhopadhyay, S.: Termination proofs for linear simple loops. In: SAS ’09: Proc. 19th Int. Static Analysis Symp. LNCS, vol. 7460, pp. 422–438. Springer (2012) 15 8. Codish, M., Taboch, C.: A semantic basis for the termination analysis of logic programs. J. Log. Program. 41(1), 103–123 (1999) 9. Cook, B., Gulwani, S., Lev-Ami, T., Rybalchenko, A., Sagiv, M.: Proving conditional termi-nation. In: CAV ’08: Proc. 20th Int. Conf. on Computer Aided Verification. pp. 328–340. No. 5123 in LNCS, Springer (2008) 10. Cook, B., Podelski, A., Rybalchenko, A.: Abstraction refinement for termination. In: SAS ’05: Proc. 12th Int. Static Analysis Symp. pp. 87–101. No. 3672 in LNCS, Springer (2005) 11. Cook, B., Podelski, A., Rybalchenko, A.: Termination proofs for systems code. In: PLDI’06: Proc. 27th ACM-SIGPLAN Conf. on Programming Language Design and Implementation. pp. 415–426. ACM (2006) 12. Cousot, P.: M´ ethodes It´ eratives de construction et d’approximation de points fixes d’op´ erateurs monotones sur un treillis, analyse s´ emantique de programmes (in French). These d’´ etat es sciences math´ ematiques, Universit´ e scientifique et m´ edicale de Grenoble (March 1978) 13. Cousot, P.: Partial completeness of abstract fixpoint checking, invited paper. In: SARA ’00: Proc. 4th Int. Symp. on Abstraction, Reformulations and Approximation. LNAI, vol. 1864, pp. 1–25. Springer (2000) 14. Cousot, P., Cousot, R.: Abstract interpretation: a unified lattice model for static analysis of programs by construction or approximation of fixpoints. In: POPL ’77: Proc. 4th ACM SIGACT-SIGPLAN Symp. on Principles of Programming Languages. pp. 238–252. ACM Press (1977) 15. Davey, B.A., Priestley, H.A.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (1989) 16. Dershowitz, N., Lindenstrauss, N., Sagiv, Y., Serebrenik, A.: A general framework for au-tomatic termination analysis of logic programs. Appl. Algebra Eng. Commun. Comput. 12(1 /2), 117–156 (2001) 17. Ganty, P.: The Fixpoint Checking Problem: An Abstraction Refinement Perspective. Ph.D. thesis, Universit´ e Libre de Bruxelles (2007) 18. Graf, S., Sa¨ ıdi, H.: Construction of abstract state graphs with PVS. In: CAV ’97: Proc. 9th Int. Conf. on Computer Aided Verification. LNCS, vol. 1254, pp. 72–83. Springer (1997) 19. Heizmann, M., Jones, N.D., Podelski, A.: Size-change termination and transition invariants. In: SAS ’10: Proc. 20th Int. Static Analysis Symp. pp. 22–50. LNCS, Springer (2010) 20. Kroening, D., Sharygina, N., Tsitovich, A., Wintersteiger, C.M.: Termination analysis with compositional transition invariants. In: CAV ’10: Proc. 20th Int. Conf. on Computer Aided Verification. LNCS, vol. 6174, pp. 89–103. Springer (2010) 21. Lee, C.S., Jones, N.D., Ben-Amram, A.M.: The size-change principle for program termina-tion. In: POPL ’01: Proc. 28th ACM SIGPLAN-SIGACT Symp. on Principles of Program-ming Languages. pp. 81–92. ACM (2001) 22. Mauborgne, L., Rival, X.: Trace partitioning in abstract interpretation based static analyzers. In: ESOP ’05: Proc. 14th European Symp. on Programming. LNCS, vol. 3444, pp. 5–20. Springer (2005) 23. Park, D.: Fixpoint induction and proofs of program properties. In: Machine Intelligence, vol. 5, pp. 59–78. American Elsevier (1969) 24. Podelski, A., Rybalchenko, A.: Transition invariants. In: LICS ’04: Proc. 19th Annual IEEE Symp. on Logic in Computer Science. pp. 32–41. IEEE (2004) 25. Ramsey, F.P.: On a problem of formal logic. London Math. Society 30, 264–286 (1929) 26. Rybalchenko, A.: Armc. (2008) 27. Rybalchenko, A.: Personal communication (2012) 28. Spoto, F., Mesnard, F., Payet, ´ E.: A termination analyzer for java bytecode based on path-length. ACM Trans. Program. Lang. Syst. 32(3) (2010) 16
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https://mse201.cankaya.edu.tr/uploads/files/CSP_%20LinearDensity_and_PlanarDensity.pdf
x z y Crystal Structures in Practice Linear Density and Planar Density Example solutions for BCC -Find Lattice Parameter -Find Directions or Planes -Calculate Linear or Planar Density x z y Green ones touches each other. Body-centered Cubic Crystal Structure (BCC) First, we should find the lattice parameter(a) in terms of atomic radius(R). Then, we can find linear density or planar density. x z y R R R R a a a 4𝑅= 𝑎2 + 𝑎2 + 𝑎2 → 𝑎= 4𝑅 3 We found the lattice parameter in terms of atomic radius. Linear Density 1. Draw the atoms on the direction, and use the formula; 𝐿𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝑡𝑕𝑒 𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 x z y Body-centered Cubic Crystal Structure (BCC) 2 3 1 4 Directions [1 0 1] [1 1 1] [1 1 0] [1 0 0] 1 2 3 4 Let’s apply the formula below for and . 𝐿𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝑡𝑕𝑒 𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 x z y [1 0 1] direction for BCC and Linear Density a a 𝐿= 𝑎2 = 4 2𝑅 3 1 2 𝑎𝑡𝑜𝑚 1 2 𝑎𝑡𝑜𝑚 𝐿𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝑡𝑕𝑒 𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝐿𝐷= 1 2 + 1 2 4 2𝑅 3 = 3 4 2𝑅 x z y [1 1 1] direction for BCC and Linear Density a a 𝐿= 𝑎3 = 4𝑅 1 2 𝑎𝑡𝑜𝑚 1 2 𝑎𝑡𝑜𝑚 𝐿𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝑡𝑕𝑒 𝑙𝑒𝑛𝑔𝑡𝑕 𝑜𝑓 𝑡𝑕𝑒 𝑙𝑖𝑛𝑒 𝐿𝐷= 1 2 + 1 + 1 2 4𝑅 = 1 2𝑅 1 𝑎𝑡𝑜𝑚 Planar Density 1. Draw the atoms on the plane, and use the formula; 𝑃𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡𝑕𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 x z y Body-centered Cubic Crystal Structure (BCC) Let’s apply the formula below for (0 1 0). 𝑃𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡𝑕𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 (0 1 0) plane for BCC and Planar Density x z y Body-centered Cubic Crystal Structure (BCC) 𝑃𝐷= # 𝑜𝑓 𝑎𝑡𝑜𝑚𝑠 𝑜𝑛 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 𝑡𝑕𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡𝑕𝑒 𝑝𝑙𝑎𝑛𝑒 𝑃𝐷= 1 4 + 1 4 + 1 4 + 1 4 𝑎2 𝑃𝐷= 1 4𝑅 3 2 = 3 16𝑅2 1 4 𝑎𝑡𝑜𝑚 1 4 𝑎𝑡𝑜𝑚 1 4 𝑎𝑡𝑜𝑚 1 4 𝑎𝑡𝑜𝑚 Suggestion • For the other directions and planes, also for all crystal structures (FCC,SC), you can and you should do this on your own. • For any question, you can contact with me, emreyilmaz@cankaya.edu.tr
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http://www.tianranchen.org/teaching/calc3/det/
Matrix determinant Tianran Chen Home Research Software Teaching Matrix determinant The determinant is an important numerical value that can be computed from a square matrix. The determinant of a matrix (A) is usually denoted by (\det(A)) or (|A|). One geometric interpretation of determinant is the signed scaling factor of the transformation represented by the matrix. 2x2 cases For a 2x2 matrix, we have simple formula for its determinant: det[a c b d]=a d−b c.det[a b c d]=a d−b c. It is easy to verify that the absolute value of the determinant is exactly the area of the parallelogram spanned by the two column vectors. The sign of the determinant tells us about the “orientation” of this parallelogram (homework). 3x3 cases For a 3x3 matrix, the formula is much more complicated: det⎡⎣⎢a d g b e h c f i⎤⎦⎥=a e i+b f g+c d h−c e g−b d i−a f h.det[a b c d e f g h i]=a e i+b f g+c d h−c e g−b d i−a f h. This looks rather complicated. I would not recommend anyone to spend much energy memorizing this formula. Instead, we should use the cofactor expansion: det⎡⎣⎢a d g b e h c f i⎤⎦⎥=a det[e h f i]−b det[d g f i]+c det[d g e h]det[a b c d e f g h i]=a det[e f h i]−b det[d f g i]+c det[d e g h] In this formula, each 2x2 determinant is called a minor. Similar to the 2x2 cases, we have a nice geometric interpretation for the 3x3 matrix determinant: It is the “signed volume” of the parallelepiped spanned by the three column vectors. 310A Goodwyn Hall Auburn University at Montgomery Montgomery Alabama USA Email me
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https://www.unboundmedicine.com/5minute/view/5-Minute-Clinical-Consult/116100/all/Candidiasis_Mucocutaneous?q=folliculitis
Tags Type your tag names separated by a space and hit enter Candidiasis, Mucocutaneous BASICS BASICS Descriptive text is not available for this image BASICS DESCRIPTION DESCRIPTION DESCRIPTION Pregnancy Considerations EPIDEMIOLOGY EPIDEMIOLOGY EPIDEMIOLOGY Incidence Incidence Incidence Unknown—mucocutaneous candidiasis is common in immunocompetent patients. Complication rates are low. Prevalence Prevalence Prevalence Candida species are normal flora of oral cavity, GI tract that are present in >70% of the U.S. population. ETIOLOGY AND PATHOPHYSIOLOGY ETIOLOGY AND PATHOPHYSIOLOGY ETIOLOGY AND PATHOPHYSIOLOGY C. albicans (responsible for 80–92% vulvovaginal and >80% of oral isolates); altered cell–mediated immunity against Candida species (either transient or chronic) increases susceptibility to infection. Genetics Genetics Genetics Chronic mucocutaneous candidiasis is a heterogeneous, genetic syndrome that typically presents in infancy. RISK FACTORS RISK FACTORS RISK FACTORS GENERAL PREVENTION GENERAL PREVENTION GENERAL PREVENTION COMMONLY ASSOCIATED CONDITIONS COMMONLY ASSOCIATED CONDITIONS COMMONLY ASSOCIATED CONDITIONS HIV, diabetes mellitus, cancer, and other immunosuppressive conditions There's more to see -- the rest of this topic is available only to subscribers. Citation 1. Download the 5-Minute Clinical Consult app by Unbound Medicine Select Try/Buy and follow instructions to begin your free 30-day trial Cross Links Related Topics Want to regain access to 5-Minute Clinical Consult? Renew my subscription Not now - I'd like more time to decide Log in to 5-Minute Clinical Consult Forgot Your Password? Forgot Your Username? Contact Support © 2000–2025 Unbound Medicine, Inc. All rights reserved CONNECT WITH US facebookinstagramyoutubeLinkedIn
14086
https://artofproblemsolving.com/wiki/index.php/Binomial_Theorem?srsltid=AfmBOoovPxVJBnON2GGOLYk_lzmbLNyLwEXOppTVE2eB13jVuDlar3I4
Art of Problem Solving Binomial Theorem - AoPS Wiki Art of Problem Solving AoPS Online Math texts, online classes, and more for students in grades 5-12. Visit AoPS Online ‚ Books for Grades 5-12Online Courses Beast Academy Engaging math books and online learning for students ages 6-13. Visit Beast Academy ‚ Books for Ages 6-13Beast Academy Online AoPS Academy Small live classes for advanced math and language arts learners in grades 2-12. Visit AoPS Academy ‚ Find a Physical CampusVisit the Virtual Campus Sign In Register online school Class ScheduleRecommendationsOlympiad CoursesFree Sessions books tore AoPS CurriculumBeast AcademyOnline BooksRecommendationsOther Books & GearAll ProductsGift Certificates community ForumsContestsSearchHelp resources math training & toolsAlcumusVideosFor the Win!MATHCOUNTS TrainerAoPS Practice ContestsAoPS WikiLaTeX TeXeRMIT PRIMES/CrowdMathKeep LearningAll Ten contests on aopsPractice Math ContestsUSABO newsAoPS BlogWebinars view all 0 Sign In Register AoPS Wiki ResourcesAops Wiki Binomial Theorem Page ArticleDiscussionView sourceHistory Toolbox Recent changesRandom pageHelpWhat links hereSpecial pages Search Binomial Theorem The Binomial Theorem states that for real or complex, , and non-negativeinteger, where is a binomial coefficient. In other words, the coefficients when is expanded and like terms are collected are the same as the entries in the th row of Pascal's Triangle. For example, , with coefficients , , , etc. Contents [hide] 1 Proof 1.1 Proof via Induction 1.2 Proof using calculus 2 Generalizations 2.1 Proof 3 Usage 4 See also Proof There are a number of different ways to prove the Binomial Theorem, for example by a straightforward application of mathematical induction. The Binomial Theorem also has a nice combinatorial proof: We can write . Repeatedly using the distributive property, we see that for a term , we must choose of the terms to contribute an to the term, and then each of the other terms of the product must contribute a . Thus, the coefficient of is the number of ways to choose objects from a set of size , or . Extending this to all possible values of from to , we see that , as claimed. Similarly, the coefficients of will be the entries of the row of Pascal's Triangle. This is explained further in the Counting and Probability textbook [AoPS]. Proof via Induction Given the constants are all natural numbers, it's clear to see that . Assuming that , Therefore, if the theorem holds under , it must be valid. (Note that for ) Proof using calculus The Taylor series for is for all . Since , and power series for the same function are termwise equal, the series at is the convolution of the series at and . Examining the degree- term of each, which simplifies to for all natural numbers. Generalizations The Binomial Theorem was generalized by Isaac Newton, who used an infiniteseries to allow for complex exponents: For any real or complex, , and , . Proof Consider the function for constants . It is easy to see that . Then, we have . So, the Taylor series for centered at is Usage Many factorizations involve complicated polynomials with binomial coefficients. For example, if a contest problem involved the polynomial , one could factor it as such: . It is a good idea to be familiar with binomial expansions, including knowing the first few binomial coefficients. See also Combinatorics Multinomial Theorem Retrieved from " Categories: Theorems Combinatorics Algebra Art of Problem Solving is an ACS WASC Accredited School aops programs AoPS Online Beast Academy AoPS Academy About About AoPS Our Team Our History Jobs AoPS Blog Site Info Terms Privacy Contact Us follow us Subscribe for news and updates © 2025 AoPS Incorporated © 2025 Art of Problem Solving About Us•Contact Us•Terms•Privacy Copyright © 2025 Art of Problem Solving Something appears to not have loaded correctly. Click to refresh.
14087
https://www.redalyc.org/journal/6378/637869147010/html/
Cálculo de los parámetros de funcionamiento para la selección de una torre de enfriamiento COMUNICACIONES CORTAS Cálculo de los parámetros de funcionamiento para la selección de una torre de enfriamiento Calculation of the operation parameters for the selection of cooling tower Tania María Pérez Sanjudo taniap@upr.edu.cu Universidad de Pinar del Río, Cuba Youry Rodríguez Montano youry@upr.edu.cu Universidad de Pinar del Río, Cuba Daniel Regalado Nuñez danielreg@upr.edu.cu Universidad de Pinar del Río, Cuba Cálculo de los parámetros de funcionamiento para la selección de una torre de enfriamiento Avances, vol. 20, núm. 4, pp. 485-496, 2018 Instituto de Información Científica y Tecnológica Esta obra está bajo una Licencia Creative Commons Atribución-NoComercial 4.0 Internacional. Recepción: 01 Julio 2018 Aprobación: 01 Agosto 2018 Resumen: La torre instalada en la Empresa de Productos Lácteos y Confiterías de Pinar del Río, no alcanza las temperaturas de enfriamiento que se requieren en el proceso, por lo que se propone el cálculo para la selección de los parámetros de funcionamiento y lograr así la selección de una nueva, que garantice la extracción de calor a los equipos de la línea de producción de helados de dicha empresa. Basándose en la teoría del potencial entálpico de Merkel y el método del 60% de eficacia, se obtuvo la potencia del ventilador, el flujo de agua que tiene que mover y las dimensiones de la torre que logró disipar la cantidad de calor que producen los equipos y mantenerlos en la temperatura de trabajo adecuada, como resultado de una buena transferencia de calor. Palabras clave: torre, refrigeración, eficiencia, gestión. Abstract: The tower installed in the Company of Dairy Products and Confectionery of Pinar del Rio, does not reach the cooling temperatures that are required in the process, so it is proposed the calculation for the selection of the operating parameters and achieve the selection of a new one, that guarantees the extraction of heat to the equipment of the ice cream production line of said company. Based on the theory of the enthalpic potential described by Merkel and the method of 60% of effectiveness, this resulted in the power of the fan, the flow of water that has to move and the dimensions of the tower that manage to dissipate the amount of heat they produce. The equipment and keep them at the proper working temperature because of a good transfer of heat. Keywords: tower, cooling, efficiency, management. INTRODUCCIÓN En los grandes procesos químicos de la industria se generan enormes cantidades de calor que debe ser removido para que todas las unidades operen eficientemente. Los equipos de transferencia de calor más comúnmente utilizados son los intercambiadores de calor y condensadores. Muchas compañías requieren tener procesos eficientes en cuanto al costo razón por la cual gran cantidad del agua que se usa en muchos intercambiadores de calor para refrigerar es recirculada y reutilizada. Para lograr esto una de las formas más prácticas y económicas es mediante el uso de torres de enfriamiento empacadas, ya que ofrecen ventajas importantes por la transferencia simultánea de calor y masa debida a que el contacto entre los fluidos es directo. Según Obregón, Pertuz, Domínguez (2017) una ventaja que tienen las torres de enfriamiento sobre los intercambiadores de calor comunes es la gran cantidad de agua que manejan debido a que pueden tratar toda el agua de una planta entera mientras que los intercambiadores comunes son utilizados para equipos simples. Además se desarrolla rápidamente en su estructura, tipo, autocontrol y materiales. El agua de enfriamiento a alta temperatura se rocía uniformemente desde la parte superior y fluye hacia la parte inferior de la torre de enfriamiento, forma una película de agua en la superficie del relleno según (Nailing, Liu; Lixia, Zhang; Xiangqin, Jia; 2017) y el aire ingresa a la torre de enfriamiento desde el fondo, intercambiando calor y masa en el relleno interior con película de agua. Según Obregón, Pertuz, Domínguez (2017) el resultado de la transferencia de calor y masa es el valor de entalpía del aire que aumenta absorbiendo calor y masa, el agua de refrigeración se enfría por disipación de calor. El aire caliente y húmedo se descarga fuera de la torre de enfriamiento por efecto del ventilador de flujo axial. El agua de refrigeración fluye al dispositivo de enfriamiento, absorbiendo el calor para mantener el proceso en marcha. Merkel con su trabajo Zeitschrift des Verdines Deutscher Ingenieure expuesto en el año 1925, estructuró la teoría básica de las torres de enfriamiento según Alean, Gutiérrez, Chejne, Marlon, Bastidas (2009) el método implementado por él se usa actualmente para determinar la eficiencia de la torre, teniendo en cuenta también lo planteado por Jianlin, Cheng; Nianpin, Li; Kuan, Wang (2015) quienes exponen que los parámetros meteorológicos que influyen positivamente o negativamente en la tasa de flujo de agua, la velocidad del viento, el llenado y otros factores que afectan el rendimiento de enfriamiento. En Pinar del Río, la Empresa de Productos Lácteos y Confitería es de tipo estatal. Tiene como misión, producir y proveer al sistema de distribución mayorista productos normados (leche, yogur, lactosoy) y a la red de comercialización en divisas, así como garantizar la merienda escolar a las secundarias básicas para satisfacer necesidades alimentarias. La empresa es una entidad rentable, con excelentes vínculos contractuales, que eleva su imagen corporativa, orientada al cliente por medio del ofrecimiento de sus producciones de altos niveles competitivos que garantizan incrementos nutricionales a la población y posee tecnología aprobada sin productos a granel. El proceso de producción de helado es altamente consumidor de energía por las etapas de enfriamiento. Si junto a esto, se considera que dichas instalaciones tienen varios años de explotación y por lo general baja eficiencia energética y un incremento en los indicadores de consumo, es evidente la necesidad de buscar mejores condiciones de operación que reduzcan los costos (Sariego, García, Pérez, Rodríguez, 2017). La torre de enfriamiento instalada en la fábrica específicamente en la UEB Primavera, encargada de la producción de helados, tiene la tarea de enfriar los condensadores del banco de agua de helado y los congeladores, pero no cumple con las necesidades para las que está instalada; el buen funcionamiento de esos equipos se deriva del mantenimiento de una temperatura de trabajo adecuada, debido a esto, se tiene como objetivo calcular los parámetros necesarios para que cumpla con los requerimientos de dichos equipos, es decir que la temperatura de entrada a ellos sea constante y por debajo de los 40 °C. MATERIALES Y METODOS Los datos iniciales para el cálculo de la torre de enfriamiento se muestran a continuación: Temperatura del agua a la entrada de la torre t L2= 34 ºC Temperatura de bulbo seco t bs= 30 ºC Temperatura de bulbo húmedo t bh= 23 ºC Potencia térmica a disipar en la torre Q= 110 kW Presión manométrica del agua P Magua= 18,02 kPa Presión manométrica del aire P Maire= 29 kPa Calor latente de vaporización medio del agua cv = 2 550 kJ/kg Importar lista Elección del tipo de torre. Como primera elección se debe escoger si la torre será de tiro natural o mecánico, al hacer esta elección se debe analizar si será de tiro forzado o de tiro inducido y se escogerá también si el flujo es cruzado o a contracorrientes. Para este cálculo se necesita conocer los valores de las variables mostradas en el esquema de circulación de los fluidos de la figura Figura Esquema de circulación de la torre de enfriamiento. Fuente: (Guanhong, Zhang; Suoying, Hea; Zhiyu, Zhanga; Yi,Xua; Rui, Wanga 2017) Selección de la aproximación de la temperatura de salida del agua. Para realizar esta selección se utiliza el Método del 60% de eficacia referenciado por Kaijun, Dong; Pingjie, Lia; Zhilin, Huanga; Lin, Sua; Qin, Suna (2017), el flujo de agua caliente proviene del condensador debe llegar a una temperatura t L2 conocida la cual se debe enfriar hasta una temperatura t L1 aproximada: (1) Donde: E: Eficiencia de las torres de enfriamiento (%) t L1: Temperatura del agua a la salida de la torre en ºC. Despejando t L1 se tiene que: (2) Cálculo del flujo másico o caudal de agua necesario. Para calcular el flujo másico L se usa la ecuación del calor: (3) Despejando L se tiene que: (4) Donde: Q: Potencia térmica a disipar en kW. c pa: Calor específico del agua en kJ/kg K «tL: Diferencia entre las temperaturas de entrada y salida del agua a la torre en ºC. Cálculo de la presión parcial de la mezcla aire vapor saturado. Se calcula por la siguiente expresión llamada ecuación de Antoine. (5) Donde: t L: Temperatura del líquido en ºC A, B y C son constantes de Antoine cuyos valore son: A = 16,3872, B = 3885,7 y C = 230,17 Cálculo de humedad absoluta de saturación. Se puede obtener por la expresión que se muestra a continuación (6) Cálculo de la entalpía de la mezcla saturada aire-vapor. Se calcula esta entalpía para cada punto del intervalo de enfriamiento usando los valores de humedad absoluta de saturación calculados anteriormente usando: (7) Donde: t o: Temperatura de referencia =0 ºC c paire: Calor específico del aire c pvapor: Calor específico del vapor Los valores de los calores específicos se toman de(Calzada, 2012) Cálculo de la presión parcial del aire húmedo. Se calcula por la ecuación de Antoine, utilizando la temperatura de bulbo húmedot bh. (8) Cálculo de la humedad absoluta del aire húmedo. (9) Cálculo de la humedad absoluta del aire. (10) Cálculo del área de la sección trasversal de la torre. Para calcular el área de la sección transversal de la torre (m.) es necesario obtener el flujo específicoL' (kg/s '» m.), L' está en función del rango de enfriamiento (t L2 - t L1) y la temperatura de bulbo húmedot bh. Una vez determinado el flujo específico y con el valor de flujo másico de agua que circula por la torre L (kg/s), el área de la sección transversal de la torre se calcula con la siguiente expresión: (12) Determinación de la altura de la zona empaquetada y total de la torre. La ecuación característica para una torre de enfriamiento establecida por Merkel, el cual se basa en una serie de asunciones para reducir la solución a un cálculo simple, referenciado en Alean, Gutiérrez, Chejne, Marlon, Bastidas (2009). Esta relaciona la fuerza impulsora o gradiente que favorece la transferencia de calor del vapor agua en el aire con el coeficiente de transferencia característico del tipo de relleno: (13) Donde: K y a: Coeficiente de transferencia de calor. z: Altura de la zona empaquetada en m. Para obtener la altura total se usa en lugar del flujo específico el flujo total. Se selecciona para este tipo de torre un relleno fílmico de PVC de arreglo hexagonal tipo colmena debido a su amplio uso en los diseños actuales de estas torres. El coeficiente de transferencia de calor se calcula según el tipo de relleno por la siguiente expresión: (14) Cálculo de la cantidad de agua evaporada. (15) cv: Calor latente de vaporización medio (estimado) del agua en kJ/kg. Cálculo de los ciclos de concentración Los parámetros recomendables de calidad del agua de recirculación en torres de enfriamiento son obtenidos en (Idae, 2007) Se observa que, en este caso, la relación determinante es la de los sólidos disueltos, (16) Cálculo del caudal de arrastre. (17) Considerando un separador de gotas de alta eficiencia, de calidad media = 0,01% de agua recirculada. Cálculo del caudal de purga. La cantidad de agua a purgar o sangrado para este equipo se determina por la ecuación siguiente: (18) Donde: Vp: Volumen de agua a evacuar con la purga en kg/s. Ae: Volumen de agua evaporada en kg/s. Var: Volumen de agua perdida por arrastres en kg/s. Cálculo del caudal de agua de compensación (consumo total). (19) Cálculo de los parámetros fundamentales para la selección del ventilador. Para el cálculo de estos parámetros se debe conocer los siguientes valores: El rendimiento térmico de una torre de enfriamiento de tiro inducido depende en gran medida del ventilador que a esta se le instale y para la selección correcta del mismo se necesita calcular primeramente el volumen de aire que debe manejar, calculándose por la siguiente expresión: (20) Siendo: G: Volumen de aire manejado por el ventilador en m 3/s. Gs: Flujo másico de aire en kg/s. Ves: Volumen específico del aire en m 3/kg. Para este caso se toma 0,9051m3/kga 33ºC Teniendo el volumen de aire manejado, se puede calcular la velocidad a la que este se debe mover por la torre usando la expresión siguiente: (21) Donde: V: Velocidad del aire que se mueve por la torre en m/s. A: Área de la sección transversal en m 2. Con la velocidad del aire, se calcula la presión estática (Ps en m de agua) de la siguiente ecuación: (22) Donde: g: Aceleración de la gravedad en m/s 2. Para este caso será de 9,81m/s 2 Teniendo el valor de presión estática entonces se puede calcular la potencia (Hp) requerida por el ventilador mediante la ecuación siguiente: (23) RESULTADOS Y DISCUSIÓN Se determinó que la torre de enfriamiento más idónea para la refrigeración en el proceso de producción de helado es una de tiro mecánico inducido y flujo en contracorriente, además se realizaron cálculos pertinentes como la altura de la torre con un valor de 2,4 m, el flujo del agua igual a 3,75 kg/s y el área transversal de la torre con un valor de 3,23 m2. Según Kaijun, Dong; Pingjie, Lia; Zhilin, Huanga; Lin, Sua; Qin, Suna (2017) los valores obtenidos cumplen con los requisitos, ya que la altura es un 20% menor que el espacio donde se colocará la torre. Se debe incorporar a la torre un volumen de agua desde el tanque de aportación de 0,086 kg/s de agua. Al conocer este valor se puede regular adecuadamente la válvula de entrada del agua de aportación y así no tener ni exceso ni falta de agua en el equipo. Se seleccionaron boquillas fabricadas de polipropileno, inatascables y que funcionan a baja presión, resistentes al ataque químico y biológico, de insecto intercambiable. Según Contreras (2012), Calzada (2012) y Obregón, Pertuz, Domínguez (2017) refieren que existe varios tipos de relleno, el laminar puede ser de PVC (policloruro de vinilo) o de PP (polipropileno), el salpiqueo puede ser de madera, PVC, PP y PE (polietileno), los plásticos son mucho más efectivos y baratos que los de madera. Siguiendo este criterio se escogió para la torre del combinado un empaque plástico tipo película que tiene una eficiencia superior al tipo salpicadura hasta un 30% (Kaijun, Dong; Pingjie, Lia; Zhilin, Huanga; Lin, Sua; Qin, Suna 2017). Se determinó la potencia del ventilador, la cual debe ser 2,463 HP para poder mover el aire que circula en la torre. El volumen de aire manejado por el ventilador es de 2,13 m./s, la velocidad del aire que se mueve por la torre de 0,659 m/s y la presión estática de 0,088 m de agua, acordes con lo establecido por el antes mencionado autor. Impacto económico La producción de helado en la empresa es de 50 tinas de helados de 10 litros en un tiempo de 8:00 am hasta las 4:30 pm en dependencia de la cantidad de ingredientes y el proceso de elaboración de la misma. Teniendo una ganancia de 400 cuc equivalente a 10 000 cup diarios sin descontar los gastos. La torre de enfriamiento se pone en funcionamiento al comienzo de la elaboración y al no cumplir con los requerimientos del sistema de refrigeración el proceso, los equipos se disparan y el proceso de fabricación se ve interrumpido alrededor de una hora diaria. Al considerar que este trabajo se implemente en la empresa, la torre trabajará entre sus parámetros de diseño y se podrán elaborar alrededor de 7 cubetas más, siendo un total de 57 cubetas al día. Esto daría un aumento de 56 cuc equivalente a 1 400 cup, por lo que se produciría 456 cuc equivalente a 11 400 cup sin contar los gastos por consumo de ingredientes. Impacto ambiental Los intercambiadores de calor usan como refrigerante el amoniaco y el freón que aunque este último no es muy dañino son perjudiciales si ocurriese algún incidente en la fábrica, con el uso de las torres de enfriamiento en procesos industriales sustituyendo a los intercambiadores de calor altos consumidores de energía, se evitan notablemente los daños al ambiente ya que aparte de utilizan como fuente de energía la electricidad motores pequeños y de bajo consumo, las sustancias de trabajo son agua y aire. Aunque la torre no constituye un problema ambiental, se le debe prestar atención a la cantidad de agua que se consume en la torre, ya que se pueden producir pérdidas por evaporación. Si el agua no se enfría adecuadamente las pérdidas por evaporación aumentan principalmente cuando el eliminador de rocío esta averiado o no existe, por ello se debe establecer un estricto control sobre esta. Por lo que en el diseño y selección de la misma se tuvo en cuenta el porcentaje de agua que se evapora, en condiciones de operación. CONCLUSIONES Al calcular correctamente los parámetros de trabajo necesarios de los equipos del sistema, la transferencia de calor que tiene lugar en la torre ocurrirá sin ninguna dificultad y el agua de trabajo que es en este caso el flujo de refrigeración, llegará con la temperatura para la cual está diseñada, consiguiéndose entonces el enfriamiento necesario para el trabajo de los equipos que dependen de la torre, por lo que se dice que cumple con las expectativas de la empresa además lograr el correcto desenvolvimiento del flujo productivo. REFERENCIAS BIBLIOGRÁFICAS Alean J. D., Gutiérrez G.A., Chejne F., Marlon J., Bastidas, M.J. (2009). Simulación de una Torre de Enfriamiento Mecánica Comparada con Curvas Experimentales. Información tecnológica, 20(3), 13-18. doi:10.1612/inf.tecnol.4107it.08 Recuperado de Calzada, F.J. (2012). Diseño de un sistema industrial de enfriamiento con agua de refrigeración para un complejo industrial en Lima, Perú. Madrid, España. 350 p. Contreras, A.A. (2012). Torres de Enfriamiento. Veracruz: SlideShare, México. 295 p. Guanhong, Z., Suoying, H., Zhiyu, Z., Yi, X., Rui, W. (2017). Economic Analyses of Natural Draft Dry Cooling Towers Pre-cooled Using Wetted Media. Procedia Engineering, 205, 423-430. doi: 10.1016/j.proeng.2017.10.393 Recuperado de Instituto para la Diversificación y Ahorro de la Energía (2007). Guía técnica de torres de refrigeración. Madrid: IDAE. 45 p. Jianlin, Ch., Nianpin, L., Kuan, W. (2015). Study of Heat-source-tower Heat Pump System Efficiency. Procedia Engineering, 121, 915-921. doi: 10.1016/j.proeng.2015.09.050 Recuperado de Kaijun, D., Pingjie, L., Zhilin, H., Lin, S., Qin, S. (2017). Researchon Free Cooling of Data CentersbyUsingIndirect. Procedia Engineering, 205, 2831-2838. doi: 10.1016/j.proeng.2017.09.902 Recuperado de Nailing, L., Lixia, Z., Xiangqin, J. (2017). The Effect of the Air Water Ratio on Counter Flow Cooling Tower. ProcediaEngineering, 205, 3550-3556. doi: 10.1016/j.proeng.2017.09.925 Recuperado de Obregón, L.G., Pertuz, J.C., Domínguez, R.A. (2017). Análisis del desempeño de una torre de enfriamiento a escala de laboratorio para diversos materiales de empaque, temperatura de entrada de agua y relación másica de flujo agua-aire. Prospectiva, 15(1), 42-52. Doi: Recuperado de Sariego, Y., García, E., Pérez, S., Rodríguez, L. (2017). Evaluación energética de una planta de helados. Ingeniería Energética, 38(1), 42-53. Recuperado de Notas de autor Ingeniero Mecánico, profesor Instructor de la Universidad de Pinar del Río. Facultad de Ciencias Técnicas. Departamento de Mecánica. Ingeniero Mecánico, profesor Instructor de la Universidad de Pinar del Río. Facultad de Ciencias Técnicas. Departamento de Mecánica. Máster en Eficiencia Energética, profesor Instructor de la Universidad de Pinar del Río. Facultad de Ciencias Técnicas. Departamento Mecánica, Pinar del Río, Cuba. 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14088
https://saintandrewgoc.org/blog/2016/10/11/dispassion
Dispassion — St. Andrew Greek Orthodox Church 0 Skip to Content Home About Visiting Our Parish Our Priest Parish Council History Dormition Chapel Renovation and Endowment Fund (SAREF) Our Faith The Orthodox Church Inquiries Ministries Feed the Hungry St. Andrew Garden Philoptochos Religious Education Church Chorus Bookstore & Library Good Samaritans of Saint Andrew Orthodox Christian Fellowship Stewardship Events Center Contact Open Menu Close Menu Home About Visiting Our Parish Our Priest Parish Council History Dormition Chapel Renovation and Endowment Fund (SAREF) Our Faith The Orthodox Church Inquiries Ministries Feed the Hungry St. Andrew Garden Philoptochos Religious Education Church Chorus Bookstore & Library Good Samaritans of Saint Andrew Orthodox Christian Fellowship Stewardship Events Center Contact Open Menu Close Menu Home Folder:About Folder:Our Faith Folder:Ministries Stewardship Events Center Contact Back Visiting Our Parish Our Priest Parish Council History Dormition Chapel Renovation and Endowment Fund (SAREF) Back The Orthodox Church Inquiries Back Feed the Hungry St. Andrew Garden Philoptochos Religious Education Church Chorus Bookstore & Library Good Samaritans of Saint Andrew Orthodox Christian Fellowship Dispassion Daily Message Oct 11 Written By Lauren Martyr Eulampia at Nicomedia My beloved spiritual children in Christ Our Only True God and Our Only True Savior, CHRIST IS IN OUR MIDST! HE WAS, IS, AND EVER SHALL BE. Ο ΧΡΙΣΤΟΣ ΕΝ ΤΩ ΜΕΣΩ ΗΜΩΝ! ΚΑΙ ΗΝ ΚΑΙ ΕΣΤΙ ΚΑΙ ΕΣΤΑΙ. DISPASSION by His Eminence Metropolitan of Nafpaktos, HIEROTHEOS [Source: Orthodox Psychotherapy] The Science of the Holy Fathers The value of dispassion for the spiritual life is very great. The man (person) who has attained it has come close to God and united with Him. Communion with God shows that there is dispassion. Dispassion, according to the teachings of the Holy Fathers of the Church, is "health of the soul". If the passions are the soul's sickness, dispassion is the soul's state of health. Dispassion is "resurrection of the soul prior to that of the body."A A man (person) is dispassionate when he has purified his flesh from all corruption, has lifted his nous above everything created, and has made it master of all the senses; when he keeps his soul in the presence of theLord. Thus dispassion is the entrance to the promised land. The Holy Spirit sheds its Light on him who has approached the borders of dispassion and ascended, in proportion to his purity, from the beauty of created things to the Maker. In other words, dispassion has great valueand it is extolled by the Holy Fathers, for it is liberation of the nous. If the passions enslave and capture the nous, dispassion frees it and leads it towards the spiritual knowledge of beings and of God. "Dispassion stimulates the nous to attain a spiritual knowledge of created beings." Hence it leads to spiritual knowledge. A result of this spiritual knowledge is that one acquires the great gift of discrimination (discernment). A man (person) in grace can distinguish evil from good, the created energies from the uncreated ones, the satanic energies from those of God. "Dispassion engenders discrimination (discernment)." Our contemporaries speak a great deal about common ownership and poverty. But the error of most of them is that they limit poverty to material goods and forget that it is something more than these things. When a man's (person's) nous is freed from everything created and ceases to be a slave to created things and lifts itself up towards God, then he experiences real poverty. This real poverty of spirit is obtained by the dispassionate man: "spiritual poverty is complete dispassion; when the nous has reached this state it abandons all worldly things." But we must define what dispassion is. From ancient times the Stoic philosophers spoke of dispassion as mortification of the passible soul. We have emphasized that the passible part of the soul consists of the incensive and appetitive aspects. When these have been mortified, according to the ancient interpretation, then we have dispassion. However, when the Holy Fathers speak of dispassion, they do not mean mortification of the passible part of the soul, but its transformation. Since it is through the fall of man that our soul's powers are in an unnatural state, it is through dispassion, that is, freedom from passions, that our soul is in the natural state. According to the teaching of the Holy Fathers of the Church, dispassion is a state in which the soul does not yield to any evil impulses, and this is impossible without God's mercy. According to Saint Maximus the Confessor, "dispassion is a peaceful condition of the soul in which the soul is not easily moved to evil." This implies that dispassion means that one does not suffer with the conceptual images of things. That is to say, the soul is free of thoughts which are moved by the senses and by things themselves. Just as in early times the bush burned with fire but was not consumed, so also in the dispassionate man, "however ponderous or fevered his body may be", yet the heat of his body "does not trouble or harm him, either physically or in his nous." For in this case "the voice of the Lord holds back the flames of nature". Thus a dispassionate person has a free nous and is not troubled by any earthly thing or by the heat of the body. Certainly this freedom of the nous from all impulses of the flesh and conceptual images of things is inconceivable to those who live not in a state of dispassion but by the energies of the passions... "...Thus dispassion is linked with love and is life, movement. According to Saint John of the Ladder (Climacus), just as light, fire and flame "join to fashion one activity", the same is true of love, dispassion and adoption. "Love, dispassion and adoption are distinguished by name, and name only". Dispassion is closely connected with love and adoption: it is life and communion with God... "...Partial or complete dispassion demonstrates the healing of the soul. The soul attains health. The nous which was mortified by the passions revives, is raised up. "Blessed dispassion raises the poor nous from earth to heaven, raises the beggar from the dunghill of passion. And love, all praise to it, make him sit with princes, that is with Holy Angels, and with the princes of the Lord's people." (Ladder. Step 29). Please note: The term "Orthodox Psychotherapy" does not refer to specific cases of people suffering from psychological problems of neurosis. Rather it refers to all people. According to Orthodox Christian Tradition, after Adam's fall man became ill; his "nous" was darkened and lost communion with God. Death entered into the person's being and caused many anthropological, social, even ecological problems. In the tragedy of his fall man maintained the image of God within him but lost completely the likenessof Him, since his communion with God was disrupted. However the incarnation of Christ (Christ taking flesh) and the work of the Church aim at enabling the person (the Christian believer) to attain to the likeness of God, that is to reestablish communion with God. The word "nous" refers to 'the eye of heart'. It is wrong to translate it as either mind or intellect. The adjective related to it is noetic (noeros). MY BLESSING TO ALL OF YOU The Grace of Our Lord Jesus Christ, and the love of God and Father, and the communion of the Holy Spirit be with you all. Amen. + "Glory Be To GOD For All Things!" -- St. John Chrysostom +++ With sincere agape in His Holy Diakonia, The sinner and unworthy servant of God +Father George Lauren Previous Previous "The Kingdom of Heaven is Likened unto a Certain King, Which Would Take an Account Of His Servants…" ----------------------------------------------------------------------------------------------------Next Next On Faith -------- St. Andrew Greek Orthodox Church | 52455 Ironwood Road | South Bend, IN, 46635 | (574) 277-4688 POWERED BY SQUARESPACE ­ ­
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Statistical Mechanics Third Edition Statistical Mechanics Third Edition R. K. Pathria Department of Physics University of California at San Diego Paul D. Beale Department of Physics University of Colorado at Boulder AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier Butterworth-Heinemann is an imprint of Elsevier The Boulevard, Langford Lane, Kidlington, Oxford, 0X5 1GB, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First published 1972; Second edition 1996 © 2011 Elsevier Ltd. All rights reserved The right of R. K. Pathria to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue of this book is available from the British Library. Library of Congress Cataloging in Publication Data Pathria, R. K. Statistical mechanics–3rd ed. / R. K. Pathria, Paul D. Beale. p. cm. Includes bibliographical references and index. ISBN 978-0-12-382188-1 (pbk.) 1. Statistical mechanics. I. Beale, Paul D. II. Title. QC174.8.P38 2011 530.13–dc22 2010048955 Cover: The image was created using the opensource software CMBview ( cmbview.html) written by Jamie Portsmouth and used with permission. It was created using the WMAP seven-year Internal Linear Combination Map courtesy of the WMAP Science Team ( map get.cfm). For information on all Butterworth-Heinemann publications, visit our Web site at www.elsevierdirect.com Printed in the United States 11 12 13 14 15 10 9 8 7 6 5 4 3 2 1 Preface to the Third Edition The second edition of Statistical Mechanics was published in 1996. The new material added at that time focused on phase transitions, critical phenomena, and the renormalization group — topics that had undergone vast transformations during the years following the publication of the first edition in 1972. In 2009, R. K. Pathria (R.K.P .) and the publishers agreed it was time for a third edition to incorporate the important changes that had occurred in the field since the publication of the second edition and invited Paul B. Beale (P.D.B.) to join as coauthor. The two authors agreed on the scope of the additions and changes and P.D.B. wrote the first draft of the new sections except for Appendix F which was written by R.K.P . Both authors worked very closely together editing the drafts and finalizing this third edition. The new topics added to this edition are: . Bose–Einstein condensation and degenerate Fermi gas behavior in ultracold atomic gases: Sections 7.2, 8.4, 11.2.A, and 11.9. The creation of Bose–Einstein condensates in ultracold gases during the 1990s and in degenerate Fermi gases during the 2000s led to a revolution in atomic, molecular, and optical physics, and provided a valuable link to the quantum behavior of condensed matter systems. Several of P.D.B.’s friends and colleagues in physics and JILA at the University of Colorado have been leaders in this exciting new field. . Finite-size scaling behavior of Bose–Einstein condensates: Appendix F. We develop an analytical theory for the behavior of Bose–Einstein condensates in a finite system, which provides a rigorous justification for singling out the ground state in the calculation of the properties of the Bose–Einstein condensate. . Thermodynamics of the early universe: Chapter 9. The sequence of thermodynamic transitions that the universe went though shortly after the Big Bang left behind mileposts that astrophysicists have exploited to look back into the universe’s earliest moments. Major advances in astronomy over the past 20 years have provided a vast body of observational data about the early evolution of the universe. These include the Hubble Space Telescope’s deep space measurements of the expansion of the universe, the Cosmic Background Explorer’s precise measurements of the temperature of the cosmic microwave background, and the Wilkinson Microwave Anisotropy Probe’s mapping of the angular variations in the cosmic microwave background. These data sets have led to precise determinations of the age of the universe, its composition and early evolution. Coincidentally, P.D.B.’s faculty office is located in the tower named after George Gamow, a member of the faculty at the University of Colorado in the 1950s and 1960s and a leader in the theory of nucleosynthesis in the early universe. . Chemical equilibrium: Section 6.6. Chemical potentials determine the conditions necessary for chemical equilibrium. This is an important topic in its own right, but also plays a critical role in our discussion of the thermodynamics of the early universe in Chapter 9. xiii xiv Preface to the Third Edition . Monte Carlo and molecular dynamics simulations: Chapter 16. Computer simulations have become an important tool in modern statistical mechanics. We provide here a brief introduction to Monte Carlo and molecular dynamics techniques and algorithms. . Correlation functions and scattering: Section 10.7. Correlation functions are central to the understanding of thermodynamic phases, phase transitions, and critical phenomena. The differences between thermodynamic phases are often most conspicuous in the behavior of correlation functions and the closely related static structure factors. We have collected discussions from the second edition into one place and added new material. . Fluctuation–dissipation theorem and the dynamical structure factor: Sections 15.3.A., 15.6.A, and 15.6.B. The fluctuation–dissipation theorem describes the relation between natural equilibrium thermodynamic fluctuations in a system and the response of the system to small disturbances from equilibrium, and it is one of the cornerstones of nonequilibrium statistical mechanics. We have expanded the discussion of the fluctuation–dissipation theorem to include a derivation of the key results from linear response theory, a discussion of the dynamical structure factor, and analysis of the Brownian motion of harmonic oscillators that provides useful practical examples. . Phase equilibrium and the Clausius–Clapeyron equation: Sections 4.6 and 4.7. Much of the text is devoted to using statistical mechanics methods to determine the properties of thermodynamic phases and phase transitions. This brief overview of phase equilibrium and the structure of phase diagrams lays the groundwork for later discussions. . Exact solutions of one-dimensional fluid models: Section 13.1. One-dimensional fluid models with short-range interactions do not exhibit phase transitions but they do display short-range correlations and other behaviors typical of dense fluids. . Exact solution of the two-dimensional Ising model on a finite lattice: Section 13.4.A. This solution entails an exact counting of the microstates of the microcanonical ensemble and provides analytical results for the energy distribution, internal energy, and heat capacity of the system. This solution also describes the finite-size scaling behavior of the Ising model near the transition point and provides an exact framework that can be used to test Monte Carlo methods. . Summary of thermodynamic assemblies and associated statistical ensembles: Appendix H. We provide a summary of thermodynamic relations and their connections to statistical mechanical ensembles. Most of this information can be found elsewhere in the text, but we thought it would be helpful to provide a rundown of these important connections in one place. . Pseudorandom number generators: Appendix I. Pseudorandom number generators are indispensable in computer simulations. We provide simple algorithms for generating uniform and Gaussian pseudorandom numbers and discuss their properties. . Dozens of new homework problems. The remainder of the text is largely unchanged. The completion of this task has left us indebted to many a friend and colleague. R.K.P . has already expressed his indebtedness to a good number of people on two previous occasions — in 1972 and in 1996 — so, at this time, he will simply reiterate the many words of gratitude he has already written. In addition though, he would like to thank Paul Beale for his willingness to be a partner in this project and for his diligence in carrying out the task at hand both arduously and meticulously. On his part, P.D.B. would like to thank his friends at the University of Colorado at Boulder for the many conversations he has had with them over the years about research and pedagogy of statistical mechanics, especially Noel Clark, Tom DeGrand, John Price, Chuck Rogers, Mike Preface to the Third Edition xv Dubson, and Leo Radzihovsky. He would also like to thank the faculty of the Department of Physics for according him the honor of serving as the chair of this outstanding department. Special thanks are also due to many friends and colleagues who have read sections of the manuscript and have offered many valuable suggestions and corrections, especially Tom DeGrand, Michael Shull, David Nesbitt, Jamie Nagle, Matt Glaser, Murray Holland, Leo Radzi-hovsky, Victor Gurarie, Edmond Meyer, Matthew Grau, Andrew Sisler, Michael Foss-Feig, Allan Franklin, Shantha deAlwis, Dmitri Reznik, and Eric Cornell. P.D.B. would like to take this opportunity to extend his thanks and best wishes to Professor Michael E. Fisher whose graduate statistical mechanics course at Cornell introduced him to this elegant field. He would also like to express his gratitude to Raj Pathria for inviting him to be part of this project, and for the fun and engaging discussions they have had during the preparation of this new edition. Raj’s thoughtful counsel always proved to be valuable in improving the text. P.D.B.’s greatest thanks go to Matthew, Melanie, and Erika for their love and support. R.K.P. P.D.B. Preface to the Second Edition The first edition of this book was prepared over the years 1966 to 1970 when the subject of phase transitions was undergoing a complete overhaul. The concepts of scaling and universality had just taken root but the renormalization group approach, which converted these concepts into a calculational tool, was still obscure. Not surprisingly, my text of that time could not do justice to these emerging developments. Over the intervening years I have felt increasingly conscious of this rather serious deficiency in the text; so when the time came to prepare a new edition, my major effort went toward correcting that deficiency. Despite the aforementioned shortcoming, the first edition of this book has continued to be popular over the last 20 years or so. I, therefore, decided not to tinker with it unnecessar-ily. Nevertheless, to make room for the new material, I had to remove some sections from the present text which, I felt, were not being used by the readers as much as the rest of the book was. This may turn out to be a disappointment to some individuals but I trust they will understand the logic behind it and, if need be, will go back to a copy of the first edition for reference. I, on my part, hope that a good majority of the users will not be inconvenienced by these deletions. As for the material retained, I have confined myself to making only editorial changes. The sub-ject of phase transitions and critical phenomena, which has been my main focus of revision, has been treated in three new chapters that provide a respectable coverage of the subject and essentially bring the book up to date. These chapters, along with a collection of more than 60 homework problems, will, I believe, enhance the usefulness of the book for both students and instructors. The completion of this task has left me indebted to many. First of all, as mentioned in the Preface to the first edition, I owe a considerable debt to those who have written on this subject before and from whose writings I have benefited greatly. It is difficult to thank them all individually; the bibliography at the end of the book is an obvious tribute to them. As for definitive help, I am most grateful to Dr Surjit Singh who advised me expertly and assisted me generously in the selection of the material that comprises Chapters 11 to 13 of the new text; without his help, the final product might not have been as coherent as it now appears to be. On the technical side, I am very thankful to Mrs. Debbie Guenther who typed the manuscript with exceptional skill and proof read it with extreme care; her task was clearly an arduous one but she performed it with good cheer — for which I admire her greatly. Finally, I wish to express my heartfelt appreciation for my wife who let me devote myself fully to this task over a rather long period of time and waited for its completion ungrudgingly. R.K.P. xvii Preface to the First Edition This book has arisen out of the notes of lectures that I gave to the graduate students at the McMaster University (1964–1965), the University of Alberta (1965–1967), the University of Waterloo (1969–1971), and the University of Windsor (1970–1971). While the subject matter, in its finer details, has changed considerably during the preparation of the manuscript, the style of presentation remains the same as followed in these lectures. Statistical mechanics is an indispensable tool for studying physical properties of matter “in bulk” on the basis of the dynamical behavior of its “microscopic” constituents. Founded on the well-laid principles of mathematical statistics on one hand and Hamiltonian mechanics on the other, the formalism of statistical mechanics has proved to be of immense value to the physics of the last 100 years. In view of the universality of its appeal, a basic knowledge of this subject is considered essential for every student of physics, irrespective of the area(s) in which he/she may be planning to specialize. To provide this knowledge, in a manner that brings out the essence of the subject with due rigor but without undue pain, is the main purpose of this work. The fact that the dynamics of a physical system is represented by a set of quantum states and the assertion that the thermodynamics of the system is determined by the multiplicity of these states constitute the basis of our treatment. The fundamental connection between the microscopic and the macroscopic descriptions of a system is uncovered by investigating the conditions for equilibrium between two physical systems in thermodynamic contact. This is best accomplished by working in the spirit of the quantum theory right from the beginning; the entropy and other thermodynamic variables of the system then follow in a most natural manner. After the formalism is developed, one may (if the situation permits) go over to the limit of the classical statistics. This message may not be new, but here I have tried to follow it as far as is reasonably possible in a textbook. In doing so, an attempt has been made to keep the level of presentation fairly uniform so that the reader does not encounter fluctuations of too wild a character. This text is confined to the study of the equilibrium states of physical systems and is intended to be used for a graduate course in statistical mechanics. Within these bounds, the coverage is fairly wide and provides enough material for tailoring a good two-semester course. The final choice always rests with the individual instructor; I, for one, regard Chapters 1 to 9 (minus a few sections from these chapters plus a few sections from Chapter 13) as the “essential part” of such a course. The contents of Chapters 10 to 12 are relatively advanced (not necessar-ily difficult); the choice of material out of these chapters will depend entirely on the taste of the instructor. To facilitate the understanding of the subject, the text has been illustrated with a large number of graphs; to assess the understanding, a large number of problems have been included. I hope these features are found useful. xix xx Preface to the First Edition I feel that one of the most essential aspects of teaching is to arouse the curiosity of the students in their subject, and one of the most effective ways of doing this is to discuss with them (in a reasonable measure, of course) the circumstances that led to the emergence of the subject. One would, therefore, like to stop occasionally to reflect upon the manner in which the various developments really came about; at the same time, one may not like the flow of the text to be hampered by the discontinuities arising from an intermittent addition of historical material. Accordingly, I decided to include in this account a historical introduction to the subject which stands separate from the main text. I trust the readers, especially the instructors, will find it of interest. For those who wish to continue their study of statistical mechanics beyond the confines of this book, a fairly extensive bibliography is included. It contains a variety of references — old as well as new, experimental as well as theoretical, technical as well as pedagogical. I hope that this will make the book useful for a wider readership. The completion of this task has left me indebted to many. Like most authors, I owe con-siderable debt to those who have written on the subject before. The bibliography at the end of the book is the most obvious tribute to them; nevertheless, I would like to mention, in particu-lar, the works of the Ehrenfests, Fowler, Guggenheim, Schr¨ odinger, Rushbrooke, ter Haar, Hill, Landau and Lifshitz, Huang, and Kubo, which have been my constant reference for several years and have influenced my understanding of the subject in a variety of ways. As for the preparation of the text, I am indebted to Robert Teshima who drew most of the graphs and checked most of the problems, to Ravindar Bansal, Vishwa Mittar, and Surjit Singh who went through the entire manuscript and made several suggestions that helped me unkink the exposition at a number of points, to Mary Annetts who typed the manuscript with exceptional patience, diligence and care, and to Fred Hetzel, Jim Briante, and Larry Kry who provided technical help during the preparation of the final version. As this work progressed I felt increasingly gratified toward Professors F . C. Auluck and D. S. Kothari of the University of Delhi with whom I started my career and who initiated me into the study of this subject, and toward Professor R. C. Majumdar who took keen interest in my work on this and every other project that I have undertaken from time to time. I am grateful to Dr. D. ter Haar of the University of Oxford who, as the general editor of this series, gave valuable advice on various aspects of the preparation of the manuscript and made several useful suggestions toward the improvement of the text. I am thankful to Professors J. W. Leech, J. Grindlay, and A. D. Singh Nagi of the University of Waterloo for their interest and hospitality that went a long way in making this task a pleasant one. The final tribute must go to my wife whose cooperation and understanding, at all stages of this project and against all odds, have been simply overwhelming. R.K.P. Historical Introduction Statistical mechanics is a formalism that aims at explaining the physical properties of matter in bulk on the basis of the dynamical behavior of its microscopic constituents. The scope of the formalism is almost as unlimited as the very range of the natural phenomena, for in principle it is applicable to matter in any state whatsoever. It has, in fact, been applied, with considerable success, to the study of matter in the solid state, the liquid state, or the gaseous state, mat-ter composed of several phases and/or several components, matter under extreme conditions of density and temperature, matter in equilibrium with radiation (as, for example, in astro-physics), matter in the form of a biological specimen, and so on. Furthermore, the formalism of statistical mechanics enables us to investigate the nonequilibrium states of matter as well as the equilibrium states; indeed, these investigations help us to understand the manner in which a physical system that happens to be “out of equilibrium” at a given time t approaches a “state of equilibrium” as time passes. In contrast with the present status of its development, the success of its applications, and the breadth of its scope, the beginnings of statistical mechanics were rather modest. Barring certain primitive references, such as those of Gassendi, Hooke, and so on, the real work on this subject started with the contemplations of Bernoulli (1738), Herapath (1821), and Joule (1851) who, in their own individual ways, attempted to lay a foundation for the so-called kinetic the-ory of gases — a discipline that finally turned out to be the forerunner of statistical mechanics. The pioneering work of these investigators established the fact that the pressure of a gas arose from the motion of its molecules and could, therefore, be computed by considering the dynam-ical influence of the molecular bombardment on the walls of the container. Thus, Bernoulli and Herapath could show that, if temperature remained constant, the pressure P of an ordi-nary gas was inversely proportional to the volume V of the container (Boyle’s law), and that it was essentially independent of the shape of the container. This, of course, involved the explicit assumption that, at a given temperature T, the (mean) speed of the molecules was independent of both pressure and volume. Bernoulli even attempted to determine the (first-order) correc-tion to this law, arising from the finite size of the molecules, and showed that the volume V appearing in the statement of the law should be replaced by (V −b), where b is the “actual” volume of the molecules.1 Joule was the first to show that the pressure P was directly proportional to the square of the molecular speed c, which he had initially assumed to be the same for all molecules. Kr¨ onig (1856) went a step further. Introducing the “quasistatistical” assumption that, at any time t, 1As is well known, this “correction” was correctly evaluated, much later, by van der Waals (1873) who showed that, for large V, b is four times the “actual” volume of the molecules; see Problem 1.4. xxi xxii Historical Introduction one-sixth of the molecules could be assumed to be flying in each of the six “independent” directions, namely +x,−x,+y,−y,+z, and −z, he derived the equation P = 1 3nmc2, (1) where n is the number density of the molecules and m the molecular mass. Kr¨ onig, too, assumed the molecular speed c to be the same for all molecules; so from (1), he inferred that the kinetic energy of the molecules should be directly proportional to the absolute temperature of the gas. Kr¨ onig justified his method in these words: “The path of each molecule must be so irreg-ular that it will defy all attempts at calculation. However, according to the laws of probability, one could assume a completely regular motion in place of a completely irregular one!” It must, however, be noted that it is only because of the special form of the summations appearing in the calculation of the pressure that Kr¨ onig’s argument leads to the same result as the one following from more refined models. In other problems, such as the ones involving diffusion, viscosity, or heat conduction, this is no longer the case. It was at this stage that Clausius entered the field. First of all, in 1857, he derived the ideal-gas law under assumptions far less stringent than Kr¨ onig’s. He discarded both leading assumptions of Kr¨ onig and showed that equation (1) was still true; of course, c2 now became the mean square speed of the molecules. In a later paper (1859), Clausius introduced the con-cept of the mean free path and thus became the first to analyze transport phenomena. It was in these studies that he introduced the famous “Stosszahlansatz” — the hypothesis on the number of collisions (among the molecules) — which, later on, played a prominent role in the monu-mental work of Boltzmann.2 With Clausius, the introduction of the microscopic and statistical points of view into the physical theory was definitive, rather than speculative. Accordingly, Maxwell, in a popular article entitled “Molecules,” written for the Encyclopedia Britannica, referred to Clausius as the “principal founder of the kinetic theory of gases,” while Gibbs, in his Clausius obituary notice, called him the “father of statistical mechanics.”3 The work of Clausius attracted Maxwell to the field. He made his first appearance with the memoir “Illustrations in the dynamical theory of gases” (1860), in which he went much farther than his predecessors by deriving his famous law of the “distribution of molecular speeds.” Maxwell’s derivation was based on elementary principles of probability and was clearly inspired by the Gaussian law of “distribution of random errors.” A derivation based on the requirement that “the equilibrium distribution of molecular speeds, once acquired, should remain invariant under molecular collisions” appeared in 1867. This led Maxwell to establish what is known as Maxwell’s transport equation which, if skilfully used, leads to the same results as one gets from the more fundamental equation due to Boltzmann.4 Maxwell’s contributions to the subject diminished considerably after his appointment, in 1871, as the Cavendish Professor at Cambridge. By that time Boltzmann had already made his first strides. In the period 1868–1871 he generalized Maxwell’s distribution law to poly-atomic gases, also taking into account the presence of external forces, if any; this gave rise to the famous Boltzmann factor exp(−βε), where ε denotes the total energy of a molecule. These investigations also led to the equipartition theorem. Boltzmann further showed that, just 2For an excellent review of this and related topics, see Ehrenfest and Ehrenfest (1912). 3For further details, refer to Montroll (1963) where an account is also given of the pioneering work of Waterston (1846, 1892). 4This equivalence has been demonstrated in Guggenheim (1960) where the coefficients of viscosity, thermal conductivity, and diffusion of a gas of hard spheres have been calculated on the basis of Maxwell’s transport equation. Historical Introduction xxiii like the original distribution of Maxwell, the generalized distribution (which we now call the Maxwell–Boltzmann distribution) is stationary with respect to molecular collisions. In 1872 came the celebrated H-theorem, which provided a molecular basis for the natural tendency of physical systems to approach, and stay in, a state of equilibrium. This established a connection between the microscopic approach (which characterizes statistical mechan-ics) and the phenomenological approach (which characterized thermodynamics) much more transparently than ever before; it also provided a direct method for computing the entropy of a given physical system from purely microscopic considerations. As a corollary to the H-theorem, Boltzmann showed that the Maxwell–Boltzmann distribution is the only distribution that stays invariant under molecular collisions and that any other distribution, under the influ-ence of molecular collisions, will ultimately go over to a Maxwell–Boltzmann distribution. In 1876 Boltzmann derived his famous transport equation, which, in the hands of Chapman and Enskog (1916–1917), has proved to be an extremely powerful tool for investigating macroscopic properties of systems in nonequilibrium states. Things, however, proved quite harsh for Boltzmann. His H-theorem, and the consequent irreversible behavior of physical systems, came under heavy attack, mainly from Loschmidt (1876–1877) and Zermelo (1896). While Loschmidt wondered how the consequences of this theorem could be reconciled with the reversible character of the basic equations of motion of the molecules, Zermelo wondered how these consequences could be made to fit with the quasiperiodic behavior of closed systems (which arose in view of the so-called Poincar´ e cycles). Boltzmann defended himself against these attacks with all his might but, unfortunately, could not convince his opponents of the correctness of his viewpoint. At the same time, the energeti-cists, led by Mach and Ostwald, were criticizing the very (molecular) basis of the kinetic theory,5 while Kelvin was emphasizing the “nineteenth-century clouds hovering over the dynamical theory of light and heat.”6 All this left Boltzmann in a state of despair and induced in him a persecution complex.7 He wrote in the introduction to the second volume of his treatise Vorlesungen ¨ uber Gastheorie (1898):8 I am convinced that the attacks (on the kinetic theory) rest on misunderstandings and that the role of the kinetic theory is not yet played out. In my opinion it would be a blow to science if contemporary opposition were to cause kinetic theory to sink into the oblivion which was the fate suffered by the wave theory of light through the authority of Newton. I am aware of the weakness of one individual against the prevailing currents of opinion. In order to insure that not too much will have to be rediscovered when people return to the study of kinetic theory I will present the most difficult and misunderstood parts of the subject in as clear a manner as I can. We shall not dwell any further on the kinetic theory; we would rather move on to the development of the more sophisticated approach known as the ensemble theory, which may in fact be regarded as the statistical mechanics proper.9 In this approach, the dynamical state of a 5These critics were silenced by Einstein whose work on the Brownian motion (1905b) established atomic theory once and for all. 6The first of these clouds was concerned with the mysteries of the “aether,” and was dispelled by the theory of relativ-ity. The second was concerned with the inadequacy of the “equipartition theorem,” and was dispelled by the quantum theory. 7Some people attribute Boltzmann’s suicide on September 5, 1906 to this cause. 8Quotation from Montroll (1963). 9For a review of the historical development of kinetic theory leading to statistical mechanics, see Brush (1957, 1958, 1961a,b, 1965–1966). xxiv Historical Introduction given system, as characterized by the generalized coordinates qi and the generalized momenta pi, is represented by a phase point G(qi,pi) in a phase space of appropriate dimensionality. The evolution of the dynamical state in time is depicted by the trajectory of the G-point in the phase space, the “geometry” of the trajectory being governed by the equations of motion of the system and by the nature of the physical constraints imposed on it. To develop an appropriate formal-ism, one considers the given system along with an infinitely large number of “mental copies” thereof; that is, an ensemble of similar systems under identical physical constraints (though, at any time t, the various systems in the ensemble would differ widely in respect of their dynam-ical states). In the phase space, then, one has a swarm of infinitely many G-points (which, at any time t, are widely dispersed and, with time, move along their respective trajectories). The fiction of a host of infinitely many, identical but independent, systems allows one to replace certain dubious assumptions of the kinetic theory of gases by readily acceptable statements of statistical mechanics. The explicit formulation of these statements was first given by Maxwell (1879) who on this occasion used the word “statistico-mechanical” to describe the study of ensembles (of gaseous systems) — though, eight years earlier, Boltzmann (1871) had already worked with essentially the same kind of ensembles. The most important quantity in the ensemble theory is the density function, ρ(qi,pi;t), of the G-points in the phase space; a stationary distribution (∂ρ/∂t = 0) characterizes a sta-tionary ensemble, which in tum represents a system in equilibrium. Maxwell and Boltzmann confined their study to ensembles for which the function ρ depended solely on the energy E of the system. This included the special case of ergodic systems, which were so defined that “the undisturbed motion of such a system, if pursued for an unlimited time, would ultimately tra-verse (the neighborhood of) each and every phase point compatible with the fixed value E of the energy.” Consequently, the ensemble average, ⟨f ⟩, of a physical quantity f , taken at any given time t, would be the same as the long-time average, f , pertaining to any given member of the ensemble. Now, f is the value we expect to obtain for the quantity in question when we make an appropriate measurement on the system; the result of this measurement should, there-fore, agree with the theoretical estimate ⟨f ⟩. We thus acquire a recipe to bring about a direct contact between theory and experiment. At the same time, we lay down a rational basis for a microscopic theory of matter as an alternative to the empirical approach of thermodynamics! A significant advance in this direction was made by Gibbs who, with his Elementary Prin-ciples of Statistical Mechanics (1902), turned ensemble theory into a most efficient tool for the theorist. He emphasized the use of “generalized” ensembles and developed schemes which, in principle, enabled one to compute a complete set of thermodynamic quantities of a given sys-tem from purely mechanical properties of its microscopic constituents.10 In its methods and results, the work of Gibbs turned out to be much more general than any preceding treatment of the subject; it applied to any physical system that met the simple-minded requirements that (i) it was mechanical in structure and (ii) it obeyed Lagrange’s and Hamilton’s equa-tions of motion. In this respect, Gibbs’s work may be considered to have accomplished for thermodynamics as much as Maxwell’s had accomplished for electrodynamics. These developments almost coincided with the great revolution that Planck’s work of 1900 brought into physics. As is well known, Planck’s quantum hypothesis successfully resolved the essential mysteries of the black-body radiation — a subject where the three best-established disciplines of the nineteenth century, namely mechanics, electrodynamics, and thermodynam-ics, were all focused. At the same time, it uncovered both the strengths and the weaknesses of these disciplines. It would have been surprising if statistical mechanics, which linked thermodynamics with mechanics, could have escaped the repercussions of this revolution. 10In much the same way as Gibbs, but quite independently of him, Einstein (1902, 1903) also developed the theory of ensembles. Historical Introduction xxv The subsequent work of Einstein (1905a) on the photoelectric effect and of Compton (1923a,b) on the scattering of x-rays established, so to say, the “existence” of the quan-tum of radiation, or the photon as we now call it.11 It was then natural for someone to derive Planck’s radiation formula by treating black-body radiation as a gas of photons in the same way as Maxwell had derived his law of distribution of molecular speeds for a gas of conventional molecules. But, then, does a gas of photons differ so radically from a gas of conventional molecules that the two laws of distribution should be so different from one another? The answer to this question was provided by the manner in which Planck’s formula was derived by Bose. In his historic paper of 1924, Bose treated black-body radiation as a gas of pho-tons; however, instead of considering the allocation of the “individual” photons to the various energy states of the system, he fixed his attention on the number of states that contained “a par-ticular number” of photons. Einstein, who seems to have translated Bose’s paper into German from an English manuscript sent to him by the author, at once recognized the importance of this approach and added the following note to his translation: “Bose’s derivation of Planck’s formula is in my opinion an important step forward. The method employed here would also yield the quantum theory of an ideal gas, which I propose to demonstrate elsewhere.” Implicit in Bose’s approach was the fact that in the case of photons what really mat-tered was “the set of numbers of photons in various energy states of the system” and not the specification as to “which photon was in which state”; in other words, photons were mutu-ally indistinguishable. Einstein argued that what Bose had implied for photons should be true for material particles as well (for the property of indistinguishability arose essentially from the wave character of these entities and, according to de Broglie, material particles also possessed that character).12 In two papers, which appeared soon after, Einstein (1924, 1925) applied Bose’s method to the study of an ideal gas and thereby developed what we now call Bose–Einstein statistics. In the second of these papers, the fundamental difference between the new statistics and the classical Maxwell–Boltzmann statistics comes out so transparently in terms of the indistinguishability of the molecules.13 In the same paper, Einstein discovered the phenomenon of Bose–Einstein condensation which, 13 years later, was adopted by London (1938a,b) as the basis for a microscopic understanding of the curious properties of liquid He4 at low temperatures. Following the enunciation of Pauli’s exclusion principle (1925), Fermi (1926) showed that certain physical systems would obey a different kind of statistics, namely the Fermi–Dirac statistics, in which not more than one particle could occupy the same energy state (ni = 0,1). It seems important to mention here that Bose’s method of 1924 leads to the Fermi–Dirac dis-tribution as well, provided that one limits the occupancy of an energy state to at most one particle.14 11Strictly speaking, it might be somewhat misleading to cite Einstein’s work on the photoelectric effect as a proof of the existence of photons. In fact, many of the effects (including the photoelectric effect), for which it seems necessary to invoke photons, can be explained away on the basis of a wave theory of radiation. The only phenomena for which photons seem indispensable are the ones involving fluctuations, such as the Hanbury Brown–Twiss effect or the Lamb shift. For the relevance of fluctuations to the problem of radiation, see ter Haar (1967, 1968). 12Of course, in the case of material particles, the total number N (of the particles) will also have to be conserved; this had not to be done in the case of photons. For details, see Section 6.1. 13It is here that one encounters the correct method of counting “the number of distinct ways in which gi energy states can accommodate ni particles,” depending on whether the particles are (i) distinguishable or (ii) indistinguishable. The occupancy of the individual states was, in each case, unrestricted, that is, ni = 0,1,2,.... 14Dirac, who was the first to investigate the connection between statistics and wave mechanics, showed, in 1926, that the wave functions describing a system of identical particles obeying Bose–Einstein (or Fermi–Dirac) statistics must be symmetric (or antisymmetric) with respect to an interchange of two particles. xxvi Historical Introduction Soon after its appearance, the Fermi–Dirac statistics were applied by Fowler (1926) to discuss the equilibrium states of white dwarf stars and by Pauli (1927) to explain the weak, temperature-independent paramagnetism of alkali metals; in each case, one had to deal with a “highly degenerate” gas of electrons that obey Fermi–Dirac statistics. In the wake of this, Som-merfeld produced his monumental work of 1928 that not only put the electron theory of metals on a physically secure foundation but also gave it a fresh start in the right direction. Thus, Som-merfeld could explain practically all the major properties of metals that arose from conduction electrons and, in each case, obtained results that showed much better agreement with exper-iment than the ones following from the classical theories of Riecke (1898), Drude (1900), and Lorentz (1904–1905). Around the same time, Thomas (1927) and Fermi (1928) investigated the electron distribution in heavier atoms and obtained theoretical estimates for the relevant bind-ing energies; these investigations led to the development of the so-called Thomas–Fermi model of the atom, which was later extended so that it could be applied to molecules, solids, and nuclei as well.15 Thus, the whole structure of statistical mechanics was overhauled by the introduction of the concept of indistinguishability of (identical) particles.16 The statistical aspect of the problem, which was already there in view of the large number of particles present, was now augmented by another statistical aspect that arose from the probabilistic nature of the wave mechanical description. One had, therefore, to carry out a two-fold averaging of the dynamical variables over the states of the given system in order to obtain the relevant expectation val-ues. That sort of a situation was bound to necessitate a reformulation of the ensemble theory itself, which was carried out step by step. First, Landau (1927) and von Neumann (1927) intro-duced the so-called density matrix, which was the quantum-mechanical analogue of the density function of the classical phase space; this was elaborated, both from statistical and quantum-mechanical points of view, by Dirac (1929–1931). Guided by the classical ensemble theory, these authors considered both microcanonical and canonical ensembles; the introduction of grand canonical ensembles in quantum statistics was made by Pauli (1927).17 The important question as to which particles would obey Bose–Einstein statistics and which Fermi–Dirac remained theoretically unsettled until Belinfante (1939) and Pauli (1940) discovered the vital connection between spin and statistics.18 It turns out that those particles whose spin is an integral multiple of ℏobey Bose–Einstein statistics while those whose spin is a half-odd integral multiple of ℏobey Fermi–Dirac statistics. To date, no third category of particles has been discovered. Apart from the foregoing milestones, several notable contributions toward the devel-opment of statistical mechanics have been made from time to time; however, most of those contributions were concerned with the development or perfection of mathematical techniques that make application of the basic formalism to actual physical problems more fruitful. A review of these developments is out of place here; they will be discussed at their appropriate place in the text. 15For an excellent review of this model, see March (1957). 16Of course, in many a situation where the wave nature of the particles is not so important, classical statistics continue to apply. 17A detailed treatment of this development has been given by Kramers (1938). 18See also L¨ uders and Zumino (1958). 1 The Statistical Basis of Thermodynamics In the annals of thermal physics, the 1850s mark a very definite epoch. By that time the science of thermodynamics, which grew essentially out of an experimental study of the macroscopic behavior of physical systems, had become, through the work of Carnot, Joule, Clausius, and Kelvin, a secure and stable discipline of physics. The theoretical conclusions following from the first two laws of thermodynamics were found to be in very good agree-ment with the corresponding experimental results.1 At the same time, the kinetic theory of gases, which aimed at explaining the macroscopic behavior of gaseous systems in terms of the motion of their molecules and had so far thrived more on speculation than calculation, began to emerge as a real, mathematical theory. Its initial successes were glaring; however, a real contact with thermodynamics could not be made until about 1872 when Boltzmann developed his H-theorem and thereby established a direct connection between entropy on one hand and molecular dynamics on the other. Almost simultaneously, the conventional (kinetic) theory began giving way to its more sophisticated successor — the ensemble the-ory. The power of the techniques that finally emerged reduced thermodynamics to the status of an “essential” consequence of the get-together of the statistics and the mechan-ics of the molecules constituting a given physical system. It was then natural to give the resulting formalism the name Statistical Mechanics. As a preparation toward the development of the formal theory, we start with a few general considerations regarding the statistical nature of a macroscopic system. These considerations will provide ground for a statistical interpretation of thermodynamics. It may be mentioned here that, unless a statement is made to the contrary, the system under study is supposed to be in one of its equilibrium states. 1.1 The macroscopic and the microscopic states We consider a physical system composed of N identical particles confined to a space of volume V. In a typical case, N would be an extremely large number — generally, of order 1023. In view of this, it is customary to carry out analysis in the so-called thermodynamic limit, namely N →∞,V →∞(such that the ratio N/V, which represents the particle den-sity n, stays fixed at a preassigned value). In this limit, the extensive properties of the system 1The third law, which is also known as Nernst’s heat theorem, did not arrive until about 1906. For a general discussion of this law, see Simon (1930) and Wilks (1961); these references also provide an extensive bibliography on this subject. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00001-3 © 2011 Elsevier Ltd. All rights reserved. 1 2 Chapter 1. The Statistical Basis of Thermodynamics become directly proportional to the size of the system (i.e., proportional to N or to V), while the intensive properties become independent thereof; the particle density, of course, remains an important parameter for all physical properties of the system. Next we consider the total energy E of the system. If the particles comprising the system could be regarded as noninteracting, the total energy E would be equal to the sum of the energies εi of the individual particles: E = X i niεi, (1) where ni denotes the number of particles each with energy εi. Clearly, N = X i ni. (2) According to quantum mechanics, the single-particle energies εi are discrete and their val-ues depend crucially on the volume V to which the particles are confined. Accordingly, the possible values of the total energy E are also discrete. However, for large V, the spacing of the different energy values is so small in comparison with the total energy of the system that the parameter E might well be regarded as a continuous variable. This would be true even if the particles were mutually interacting; of course, in that case the total energy E cannot be written in the form (1). The specification of the actual values of the parameters N,V, and E then defines a macrostate of the given system. At the molecular level, however, a large number of possibilities still exist because at that level there will in general be a large number of different ways in which the macrostate (N,V,E) of the given system can be realized. In the case of a noninteracting system, since the total energy E consists of a simple sum of the N single-particle energies εi, there will obviously be a large number of different ways in which the individual εi can be chosen so as to make the total energy equal to E. In other words, there will be a large number of different ways in which the total energy E of the system can be distributed among the N particles constituting it. Each of these (different) ways specifies a microstate, or complexion, of the given system. In general, the various microstates, or complexions, of a given system can be identified with the independent solutions ψ(r1,...,rN) of the Schr¨ odinger equation of the system, corresponding to the eigenvalue E of the relevant Hamiltonian. In any case, to a given macrostate of the system there does in general correspond a large number of microstates and it seems natural to assume, when there are no other constraints, that at any time t the system is equally likely to be in any one of these microstates. This assump-tion forms the backbone of our formalism and is generally referred to as the postulate of “equal a priori probabilities” for all microstates consistent with a given macrostate. The actual number of all possible microstates will, of course, be a function of N,V, and E and may be denoted by the symbol (N,V,E); the dependence on V comes in because the possible values εi of the single-particle energy ε are themselves a function 1.2 Contact between statistics and thermodynamics 3 of this parameter.2 Remarkably enough, it is from the magnitude of the number , and from its dependence on the parameters N,V, and E, that complete thermodynamics of the given system can be derived! We shall not stop here to discuss the ways in which the number (N,V,E) can be com-puted; we shall do that only after we have developed our considerations sufficiently so that we can carry out further derivations from it. First we have to discover the manner in which this number is related to any of the leading thermodynamic quantities. To do this, we con-sider the problem of “thermal contact” between two given physical systems, in the hope that this consideration will bring out the true nature of the number . 1.2 Contact between statistics and thermodynamics: physical significance of the number (N,V,E) We consider two physical systems, A1 and A2, which are separately in equilibrium; see Figure 1.1. Let the macrostate of A1 be represented by the parameters N1,V1, and E1 so that it has 1(N1,V1,E1) possible microstates, and the macrostate of A2 be represented by the parameters N2,V2, and E2 so that it has 2(N2,V2,E2) possible microstates. The math-ematical form of the function 1 may not be the same as that of the function 2, because that ultimately depends on the nature of the system. We do, of course, believe that all thermodynamic properties of the systems A1 and A2 can be derived from the functions 1(N1,V1,E1) and 2(N2,V2,E2), respectively. We now bring the two systems into thermal contact with each other, thus allowing the possibility of exchange of energy between the two; this can be done by sliding in a con-ducting wall and removing the impervious one. For simplicity, the two systems are still separated by a rigid, impenetrable wall, so that the respective volumes V1 and V2 and the respective particle numbers N1 and N2 remain fixed. The energies E1 and E2, however, become variable and the only condition that restricts their variation is E(0) = E1 + E2 = const. (1) A1 (N1, V1, E1) A2 (N2, V2, E2) FIGURE 1.1 Two physical systems being brought into thermal contact. 2It may be noted that the manner in which the εi depend on V is itself determined by the nature of the system. For instance, it is not the same for relativistic systems as it is for nonrelativistic ones; compare, for instance, the cases dealt with in Section 1.4 and in Problem 1.7. We should also note that, in principle, the dependence of  on V arises from the fact that it is the physical dimensions of the container that appear in the boundary conditions imposed on the wave functions of the system. 4 Chapter 1. The Statistical Basis of Thermodynamics Here, E(0) denotes the energy of the composite system A(0)(≡A1 + A2); the energy of inter-action between A1 and A2, if any, is being neglected. Now, at any time t, the subsystem A1 is equally likely to be in any one of the 1(E1) microstates while the subsystem A2 is equally likely to be in any one of the 2(E2) microstates; therefore, the composite system A(0) is equally likely to be in any one of the 1(E1)2(E2) = 1(E1)2(E(0) −E1) = (0)(E(0),E1) (2) microstates.3 Clearly, the number (0) itself varies with E1. The question now arises: at what value of E1 will the composite system be in equilibrium? In other words, how far will the energy exchange go in order to bring the subsystems A1 and A2 into mutual equilibrium? We assert that this will happen at that value of E1 which maximizes the number (0)(E(0),E1). The philosophy behind this assertion is that a physical system, left to itself, proceeds naturally in a direction that enables it to assume an ever-increasing number of microstates until it finally settles down in a macrostate that affords the largest pos-sible number of microstates. Statistically speaking, we regard a macrostate with a larger number of microstates as a more probable state, and the one with the largest number of microstates as the most probable one. Detailed studies show that, for a typical system, the number of microstates pertaining to any macrostate that departs even slightly from the most probable one is “orders of magnitude” smaller than the number pertaining to the latter. Thus, the most probable state of a system is the macrostate in which the system spends an “overwhelmingly” large fraction of its time. It is then natural to identify this state with the equilibrium state of the system. Denoting the equilibrium value of E1 by E1 (and that of E2 by E2), we obtain, on maximizing (0), ∂1(E1) ∂E1  E1=E1 2(E2) + 1(E1) ∂2(E2) ∂E2  E2=E2 · ∂E2 ∂E1 = 0. Since ∂E2/∂E1 = −1, see equation (1), the foregoing condition can be written as ∂ln1(E1) ∂E1  E1=E1 = ∂ln2(E2) ∂E2  E2=E2 . Thus, our condition for equilibrium reduces to the equality of the parameters β1 and β2 of the subsystems A1 and A2, respectively, where β is defined by β ≡ ∂ln(N,V,E) ∂E  N,V,E=E . (3) 3It is obvious that the macrostate of the composite system A(0) has to be defined by two energies, namely E1 and E2 (or else E(0) and E1). 1.2 Contact between statistics and thermodynamics 5 We thus find that when two physical systems are brought into thermal contact, which allows an exchange of energy between them, this exchange continues until the equilibrium values E1 and E2 of the variables E1 and E2 are reached. Once these values are reached, there is no more net exchange of energy between the two systems; the systems are then said to have attained a state of thermal equilibrium. According to our analysis, this hap-pens only when the respective values of the parameter β, namely β1 and β2, become equal.4 It is then natural to expect that the parameter β is somehow related to the ther-modynamic temperature T of a given system. To determine this relationship, we recall the thermodynamic formula  ∂S ∂E  N,V = 1 T , (4) where S is the entropy of the system in question. Comparing equations (3) and (4), we conclude that an intimate relationship exists between the thermodynamic quantity S and the statistical quantity ; we may, in fact, write for any physical system 1S 1(ln) = 1 βT = const. (5) This correspondence was first established by Boltzmann who also believed that, since the relationship between the thermodynamic approach and the statistical approach seems to be of a fundamental character, the constant appearing in (5) must be a universal constant. It was Planck who first wrote the explicit formula S = kln, (6) without any additive constant S0. As it stands, formula (6) determines the absolute value of the entropy of a given physical system in terms of the total number of microstates acces-sible to it in conformity with the given macrostate. The zero of entropy then corresponds to the special state for which only one microstate is accessible ( = 1) — the so-called “unique configuration”; the statistical approach thus provides a theoretical basis for the third law of thermodynamics as well. Formula (6) is of fundamental importance in physics; it provides a bridge between the microscopic and the macroscopic. Now, in the study of the second law of thermodynamics we are told that the law of increase of entropy is related to the fact that the energy content of the universe, in its natural course, is becoming less and less available for conversion into work; accordingly, the entropy of a given system may be regarded as a measure of the so-called disorder or chaos prevailing in the system. Formula (6) tells us how disorder arises microscopically. Clearly, disorder is a manifestation of the largeness of the number of microstates the sys-tem can have. The larger the choice of microstates, the lesser the degree of predictability and hence the increased level of disorder in the system. Complete order prevails when and 4This result may be compared with the so-called “zeroth law of thermodynamics,” which stipulates the existence of a common parameter T for two or more physical systems in thermal equilibrium. 6 Chapter 1. The Statistical Basis of Thermodynamics only when the system has no other choice but to be in a unique state ( = 1); this, in turn, corresponds to a state of vanishing entropy. By equations (5) and (6), we also have β = 1 kT . (7) The universal constant k is generally referred to as the Boltzmann constant. In Section 1.4 we shall discover how k is related to the gas constant R and the Avogadro number NA; see equation (1.4.3).5 1.3 Further contact between statistics and thermodynamics In continuation of the preceding considerations, we now examine a more elaborate exchange between the subsystems A1 and A2. If we assume that the wall separating the two subsystems is movable as well as conducting, then the respective volumes V1 and V2 (of subsystems A1 and A2) also become variable; indeed, the total volume V (0)(= V1 + V2) remains constant, so that effectively we have only one more independent variable. Of course, the wall is still assumed to be impenetrable to particles, so the numbers N1 and N2 remain fixed. Arguing as before, the state of equilibrium for the composite system A(0) will obtain when the number (0)(V (0),E(0);V1,E1) attains its largest value; that is, when not only ∂ln1 ∂E1  N1,V1; E1=E1 = ∂ln2 ∂E2  N2,V2; E2=E2 , (1a) but also ∂ln1 ∂V1  N1,E1; V1=V 1 = ∂ln2 ∂V2  N2,E2; V2=V 2 . (1b) Our conditions for equilibrium now take the form of an equality between the pair of parameters (β1,η1) of the subsystem A1 and the parameters (β2,η2) of the subsystem A2 where, by definition, η ≡ ∂ln(N,V,E) ∂V  N,E,V=V . (2) Similarly, if A1 and A2 came into contact through a wall that allowed an exchange of parti-cles as well, the conditions for equilibrium would be further augmented by the equality 5We follow the notation whereby equation (1.4.3) means equation (3) of Section 1.4. However, while referring to an equation in the same section, we will omit the mention of the section number. 1.3 Further contact between statistics and thermodynamics 7 of the parameter ζ1 of subsystem A1 and the parameter ζ2 of subsystem A2 where, by definition, ζ ≡ ∂ln(N,V,E) ∂N  V,E,N=N . (3) To determine the physical meaning of the parameters η and ζ, we make use of equa-tion (1.2.6) and the basic formula of thermodynamics, namely dE = T dS −PdV + µdN, (4) where P is the thermodynamic pressure and µ the chemical potential of the given system. It follows that η = P kT and ζ = −µ kT . (5) From a physical point of view, these results are completely satisfactory because, thermo-dynamically as well, the conditions of equilibrium between two systems A1 and A2, if the wall separating them is both conducting and movable (thus making their respective ener-gies and volumes variable), are indeed the same as the ones contained in equations (1a) and (1b), namely T1 = T2 and P1 = P2. (6) On the other hand, if the two systems can exchange particles as well as energy but have their volumes fixed, the conditions of equilibrium, obtained thermodynamically, are indeed T1 = T2 and µ1 = µ2. (7) And finally, if the exchange is such that all three (macroscopic) parameters become variable, then the conditions of equilibrium become T1 = T2, P1 = P2, and µ1 = µ2. (8)6 It is gratifying that these conclusions are identical to the ones following from statistical considerations. Combining the results of the foregoing discussion, we arrive at the following recipe for deriving thermodynamics from a statistical beginning: determine, for the macrostate (N,V,E) of the given system, the number of all possible microstates accessible to the sys-tem; call this number (N,V,E). Then, the entropy of the system in that state follows from 6It may be noted that the same would be true for any two parts of a single thermodynamic system; consequently, in equilibrium, the parameters T,P, and µ would be constant throughout the system. 8 Chapter 1. The Statistical Basis of Thermodynamics the fundamental formula S(N,V,E) = kln(N,V,E), (9) while the leading intensive fields, namely temperature, pressure, and chemical potential, are given by  ∂S ∂E  N,V = 1 T ;  ∂S ∂V  N,E = P T ;  ∂S ∂N  V,E = −µ T . (10) Alternatively, we can write7 P =  ∂S ∂V  N,E  ∂S ∂E  N,V = −  ∂E ∂V  N,S (11) and µ = −  ∂S ∂N  V,E  ∂S ∂E  N,V =  ∂E ∂N  V,S , (12) while T = ∂E ∂S  N,V . (13) Formulae (11) through (13) follow equally well from equation (4). The evaluation of P,µ, and T from these formulae indeed requires that the energy E be expressed as a function of the quantities N,V, and S; this should, in principle, be possible once S is known as a function of N,V, and E. The rest of the thermodynamics follows straightforwardly; see Appendix H. For instance, the Helmholtz free energy A, the Gibbs free energy G, and the enthalpy H are given by A = E −TS, (14) G = A + PV = E −TS + PV = µN (15)8 7In writing these formulae, we have made use of the well-known relationship in partial differential calculus, namely that “if three variables x, y, and z are mutually related, then (see Appendix H)  ∂x ∂y  z  ∂y ∂z  x  ∂z ∂x  y = −1.” 8The relation E −TS + PV = µN follows directly from (4). For this, all we have to do is to regard the given system as having grown to its present size in a gradual manner, such that the intensive parameters, T,P, and µ stayed constant throughout the process while the extensive parameters N, V, and E (and hence S) grew proportionately with one another. 1.4 The classical ideal gas 9 and H = E + PV = G + TS. (16) The specific heat at constant volume, CV , and the one at constant pressure, CP, would be given by CV ≡T  ∂S ∂T  N,V =  ∂E ∂T  N,V (17) and CP ≡T  ∂S ∂T  N,P = ∂(E + PV) ∂T  N,P = ∂H ∂T  N,P . (18) 1.4 The classical ideal gas To illustrate the approach developed in the preceding sections, we shall now derive the various thermodynamic properties of a classical ideal gas composed of monatomic molecules. The main reason why we choose this highly specialized system for considera-tion is that it affords an explicit, though asymptotic, evaluation of the number (N,V,E). This example becomes all the more instructive when we find that its study enables us, in a most straightforward manner, to identify the Boltzmann constant k in terms of other physical constants; see equation (3). Moreover, the behavior of this system serves as a useful reference with which the behavior of other physical systems, especially real gases (with or without quantum effects), can be compared. And, indeed, in the limit of high temperatures and low densities the ideal-gas behavior becomes typical of most real systems. Before undertaking a detailed study of this case it appears worthwhile to make a remark that applies to all classical systems composed of noninteracting particles, irrespective of the internal structure of the particles. This remark is related to the explicit dependence of the number (N,V,E) on V and hence to the equation of state of these systems. Now, if there do not exist any spatial correlations among the particles, that is, if the probability of any one of them being found in a particular region of the available space is completely independent of the location of the other particles,9 then the total number of ways in which the N particles can be spatially distributed in the system will be simply equal to the prod-uct of the numbers of ways in which the individual particles can be accommodated in the same space independently of one another. With N and E fixed, each of these numbers will be directly proportional to V, the volume of the container; accordingly, the total number of ways will be directly proportional to the Nth power of V: (N,E,V) ∝V N. (1) 9This will be true if (i) the mutual interactions among particles are negligible, and (ii) the wave packets of individual particles do not significantly overlap (or, in other words, the quantum effects are also negligible). 10 Chapter 1. The Statistical Basis of Thermodynamics Combined with equations (1.3.9) and (1.3.10), this gives P T = k ∂ln(N,E,V) ∂V  N,E = k N V . (2) If the system contains n moles of the gas, then N = nNA, where NA is the Avogadro number. Equation (2) then becomes PV = NkT = nRT (R = kNA), (3) which is the famous ideal-gas law, R being the gas constant per mole. Thus, for any classical system composed of noninteracting particles the ideal-gas law holds. For deriving other thermodynamic properties of this system, we require a detailed knowledge of the way  depends on the parameters N,V, and E. The problem essen-tially reduces to determining the total number of ways in which equations (1.1.1) and (1.1.2) can be mutually satisfied. In other words, we have to determine the total number of (independent) ways of satisfying the equation 3N X r=1 εr = E, (4) where εr are the energies associated with the various degrees of freedom of the N par-ticles. The reason why this number should depend on the parameters N and E is quite obvious. Nevertheless, this number also depends on the “spectrum of values” that the vari-ables εr can assume; it is through this spectrum that the dependence on V comes in. Now, the energy eigenvalues for a free, nonrelativistic particle confined to a cubical box of side L (V = L3), under the condition that the wave function ψ(r) vanishes everywhere on the boundary, are given by ε(nx,ny,nz) = h2 8mL2 (n2 x + n2 y + n2 z); nx,ny,nz = 1,2,3,..., (5) where h is Planck’s constant and m the mass of the particle. The number of distinct eigenfunctions (or microstates) for a particle of energy ε would, therefore, be equal to the number of independent, positive-integral solutions of the equation  n2 x + n2 y + n2 z  = 8mV 2/3ε h2 = ε∗. (6) We may denote this number by (1,ε,V). Extending the argument, it follows that the desired number (N,E,V) would be equal to the number of independent, positive-integral solutions of the equation 3N X r=1 n2 r = 8mV 2/3E h2 = E∗, say. (7) 1.4 The classical ideal gas 11 An important result follows straightforwardly from equation (7), even before the number (N,E,V) is explicitly evaluated. From the nature of the expression appearing on the right side of this equation, we conclude that the volume V and the energy E of the system enter into the expression for  in the form of the combination (V 2/3E). Consequently, S(N,V,E) ≡S  N,V 2/3E  . (8) Hence, for the constancy of S and N, which defines a reversible adiabatic process, V 2/3E = const. (9) Equation (1.3.11) then gives P = −  ∂E ∂V  N,S = 2 3 E V , (10) that is, the pressure of a system of nonrelativistic, noninteracting particles is precisely equal to two-thirds of its energy density.10 It should be noted here that, since an explicit computation of the number  has not yet been done, results (9) and (10) hold for quan-tum as well as classical statistics; equally general is the result obtained by combining these, namely PV 5/3 = const., (11) which tells us how P varies with V during a reversible adiabatic process. We shall now attempt to evaluate the number . In this evaluation we shall explicitly assume the particles to be distinguishable, so that if a particle in state i gets interchanged with a particle in state j the resulting microstate is counted as distinct. Consequently, the number (N,V,E), or better N(E∗) (see equation (7)), is equal to the number of positive-integral lattice points lying on the surface of a 3N-dimensional sphere of radius √E∗.11 Clearly, this number will be an extremely irregular function of E∗, in that for two given values of E∗that may be very close to one another, the values of this number could be very different. In contrast, the number 6N(E∗), which denotes the number of positive-integral lattice points lying on or within the surface of a 3N-dimensional sphere of radius √E∗, will be much less irregular. In terms of our physical problem, this would correspond to the number, 6(N,V,E), of microstates of the given system consistent with all macrostates characterized by the specified values of the parameters N and V but having energy less 10Combining (10) with (2), we obtain for the classical ideal gas: E = 3 2 NkT. Accordingly, equation (9) reduces to the well-known thermodynamic relationship: V γ −1T = const., which holds during a reversible adiabatic process, with γ = 5 3 . 11If the particles are regarded as indistinguishable, the evaluation of the number  by counting lattice points becomes quite intricate. The problem is then solved by having recourse to the theory of “partitions of numbers”; see Auluck and Kothari (1946). 12 Chapter 1. The Statistical Basis of Thermodynamics than or equal to E; that is, 6(N,V,E) = X E′≤E (N,V,E′) (12) or 6N(E∗) = X E∗′≤E∗ N(E∗′). (13) Of course, the number 6 will also be somewhat irregular; however, we expect that its asymptotic behavior, as E∗→∞, will be a lot smoother than that of . We shall see in the sequel that the thermodynamics of the system follows equally well from the number 6 as from . To appreciate the point made here, let us digress a little to examine the behavior of the numbers 1(ε∗) and 61(ε∗), which correspond to the case of a single particle con-fined to the given volume V. The exact values of these numbers, for ε∗≤10,000, can be extracted from a table compiled by Gupta (1947). The wild irregularities of the number 1(ε∗) can hardly be missed. The number 61(ε∗), on the other hand, exhibits a much smoother asymptotic behavior. From the geometry of the problem, we note that, asymp-totically, 61(ε∗) should be equal to the volume of an octant of a three-dimensional sphere of radius √ε∗, that is, lim ε∗→∞ 61(ε∗) (π/6)ε∗3/2 = 1. (14) A more detailed analysis shows that, to the next approximation (see Pathria, 1966), 61(ε∗) ≈π 6 ε∗3/2 −3π 8 ε∗; (15) the correction term arises from the fact that the volume of an octant somewhat overes-timates the number of desired lattice points, for it includes, partly though, some points with one or more coordinates equal to zero. Figure 1.2 shows a histogram of the actual val-ues of 61(ε∗) for ε∗lying between 200 and 300; the theoretical estimate (15) is also shown. In the figure, we have also included a histogram of the actual values of the corresponding number of microstates, 6′ 1(ε∗), when the quantum numbers nx, ny, and nz can assume the value zero as well. In the latter case, the volume of an octant somewhat underestimates the number of desired lattice points; we now have 6′ 1(ε∗) ≈π 6 ε∗3/2 + 3π 8 ε∗. (16) Asymptotically, however, the number 6′ 1(ε∗) also satisfies equation (14). Returning to the N-particle problem, the number 6N(E∗) should be asymptotically equal to the “volume” of the “positive compartment” of a 3N-dimensional sphere of 1.4 The classical ideal gas 13 3200 2800 2400 2000 1600 1200 200 220 240 260 280 300 ´ 1(´) 6 8 3 (´)3/2 1 ´ 6 8 3 (´)3/2 2 ´ 6 (´)3/2 FIGURE 1.2 Histograms showing the actual number of microstates available to a particle in a cubical enclosure; the lower histogram corresponds to the so-called Dirichlet boundary conditions, while the upper one corresponds to the Neumann boundary conditions (see Appendix A). The corresponding theoretical estimates, (15) and (16), are shown by dashed lines; the customary estimate, equation (14), is shown by a solid line. radius √E∗. Referring to equation (C.7a) of Appendix C, we obtain 6N(E∗) ≈ 1 2 3N ( π3N/2 (3N/2)!E∗3N/2 ) which, on substitution for E∗, gives 6(N,V,E) ≈  V h3 N (2πmE)3N/2 (3N/2)! . (17) Taking logarithms and applying Stirling’s formula, (B.29) in Appendix B, ln(n!) ≈nlnn −n (n ≫1), (18) 14 Chapter 1. The Statistical Basis of Thermodynamics we get ln6(N,V,E) ≈N ln " V h3 4πmE 3N 3/2# + 3 2N. (19) For deriving the thermodynamic properties of the given system we must somehow fix the precise value of, or limits for, the energy of the system. In view of the extremely irreg-ular nature of the function (N,V,E), the specification of a precise value for the energy of the system cannot be justified on physical grounds, for that would never yield well-behaved expressions for the thermodynamic functions of the system. From a practical point of view, too, an absolutely isolated system is too much of an idealization. In the real world, almost every system has some contact with its surroundings, however little it may be; as a result, its energy cannot be defined sharply.12 Of course, the effective width of the range over which the energy may vary would, in general, be small in comparison with the mean value of the energy. Let us specify this range by the limits  E −1 21  and  E + 1 21  where, by assumption, 1 ≪E; typically, 1/E = O(1/√N). The corresponding number of microstates, 0(N,V,E;1), is then given by 0(N,V,E;1) ≃∂6(N,V,E) ∂E 1 ≈3N 2 1 E 6(N,V,E), (17a) which gives ln0(N,V,E;1) ≈N ln " V h3 4πmE 3N 3/2# + 3 2N +  ln 3N 2  + ln 1 E  . (19a) Now, for N ≫1, the first term in the curly bracket is negligible in comparison with any of the terms outside this bracket, for lim N→∞(lnN)/N = 0. Furthermore, for any reasonable value of 1/E, the same is true of the second term in this bracket.13 Hence, for all practical purposes, ln0 ≈ln6 ≈N ln " V h3 4πmE 3N 3/2# + 3 2N. (20) We thus arrive at the baffling result that, for all practical purposes, the actual width of the range allowed for the energy of the system does not make much difference; the energy could lie between  E −1 21  and  E + 1 21  or equally well between 0 and E. The reason underlying this situation is that the rate at which the number of microstates of the system 12Actually, the very act of making measurements on a system brings about, inevitably, a contact between the system and its surroundings. 13It should be clear that, while 1/E is much less than 1, it must not tend to 0, for that would make 0 →0 and ln0 → −∞. A situation of that kind would be too artificial and would have nothing to do with reality. Actually, in most physical systems, 1/E = O(N−1/2), whereby ln(1/E) becomes of order lnN, which again is negligible in comparison with the terms outside the curly bracket. 1.4 The classical ideal gas 15 increases with energy is so fantastic, see equation (17), that even if we allow all values of energy between zero and a particular value E, it is only the “immediate neighborhood” of E that makes an overwhelmingly dominant contribution to this number! And since we are finally concerned only with the logarithm of this number, even the “width” of that neighborhood is inconsequential! The stage is now set for deriving the thermodynamics of our system. First of all, we have S(N,V,E) = kln0 = Nkln " V h3 4πmE 3N 3/2# + 3 2Nk, (21)14 which can be inverted to give E(S,V,N) = 3h2N 4πmV 2/3 exp  2S 3Nk −1  . (22) The temperature of the gas then follows with the help of formula (1.3.10) or (1.3.13), which leads to the energy–temperature relationship E = N 3 2kT  = n 3 2RT  , (23) where n is the number of moles of the gas. The specific heat at constant volume now follows with the help of formula (1.3.17): CV =  ∂E ∂T  N,V = 3 2Nk = 3 2nR. (24) For the equation of state, we obtain P = −  ∂E ∂V  N,S = 2 3 E V , (25) which agrees with our earlier result (10). Combined with (23), this gives P = NkT V or PV = nRT, (26) which is the same as (3). The specific heat at constant pressure is given by, see (1.3.18), CP = ∂(E + PV) ∂T  N,P = 5 2nR, (27) 14Henceforth, we shall replace the sign ≈, which characterizes the asymptotic character of a relationship, by the sign of equality because for most physical systems the asymptotic results are as good as exact. 16 Chapter 1. The Statistical Basis of Thermodynamics so that, for the ratio of the two specific heats, we have γ = CP/CV = 5 3. (28) Now, suppose that the gas undergoes an isothermal change of state (T = const. and N = const.); then, according to (23), the total energy of the gas would remain constant while, according to (26), its pressure would vary inversely with volume (Boyle’s law). The change in the entropy of the gas, between the initial state i and the final state f , would then be, see equation (21), Sf −Si = Nkln(Vf /Vi). (29) On the other hand, if the gas undergoes a reversible adiabatic change of state (S = const. and N = const.), then, according to (22) and (23), both E and T would vary as V −2/3; so, according to (25) or (26), P would vary as V −5/3. These results agree with the conventional thermodynamic ones, namely PV γ = const. and TV γ −1 = const., (30) with γ = 5 3. It may be noted that, thermodynamically, the change in E during an adiabatic process arises solely from the external work done by the gas on the surroundings or vice versa: (dE)adiab = −PdV = −2E 3V dV; (31) see equations (1.3.4) and (25). The dependence of E on V follows readily from this relationship. The considerations of this section have clearly demonstrated the manner in which the thermodynamics of a macroscopic system can be derived from the multiplicity of its microstates (as represented by the number  or 0 or 6). The whole problem then hinges on an asymptotic enumeration of these numbers, which unfortunately is tractable only in a few idealized cases, such as the one considered in this section; see also Problems 1.7 and 1.8. Even in an idealized case like this, there remains an inadequacy that could not be detected in the derivations made so far; this relates to the explicit dependence of S on N. The discussion of the next section is intended not only to bring out this inadequacy but also to provide the necessary remedy for it. 1.5 The entropy of mixing and the Gibbs paradox One thing we readily observe from expression (1.4.21) is that, contrary to what is logi-cally desired, the entropy of an ideal gas, as given by this expression, is not an extensive 1.5 The entropy of mixing and the Gibbs paradox 17 (N1, V1; T ) (N2, V2; T ) FIGURE 1.3 The mixing together of two ideal gases 1 and 2. property of the system! That is, if we increase the size of the system by a factor α, keep-ing the intensive variables unchanged,15 then the entropy of the system, which should also increase by the same factor α, does not do so; the presence of the lnV term in the expression affects the result adversely. This in a way means that the entropy of this system is different from the sum of the entropies of its parts, which is quite unphysical. A more common way of looking at this problem is to consider the so-called Gibbs paradox. Gibbs visualized the mixing of two ideal gases 1 and 2, both being initially at the same temperature T; see Figure 1.3. Clearly, the temperature of the mixture would also be the same. Now, before the mixing took place, the respective entropies of the two gases were, see equations (1.4.21) and (1.4.23), Si = NiklnVi + 3 2Nik  1 + ln 2πmikT h2  ; i = 1,2. (1) After the mixing has taken place, the total entropy would be ST = 2 X i=1  NiklnV + 3 2Nik  1 + ln 2πmikT h2  , (2) where V = V1 + V2. Thus, the net increase in the value of S, which may be called the entropy of mixing, is given by (1S) = ST − 2 X i=1 Si = k  N1 ln V1 + V2 V1 + N2 ln V1 + V2 V2  ; (3) the quantity 1S is indeed positive, as it must be for an irreversible process like mixing. Now, in the special case when the initial particle densities of the two gases (and, hence, the particle density of the mixture) are also the same, equation (3) becomes (1S)∗= k  N1 ln N1 + N2 N1 + N2 ln N1 + N2 N2  , (4) which is again positive. 15This means an increase of the parameters N, V, and E to αN, αV, and αE, so that the energy per particle and the volume per particle remain unchanged. 18 Chapter 1. The Statistical Basis of Thermodynamics So far, it seems all right. However, a paradoxical situation arises if we consider the mix-ing of two samples of the same gas. Once again, the entropies of the individual samples will be given by (1); of course, now m1 = m2 = m, say. And the entropy after mixing will be given by ST = NklnV + 3 2Nk  1 + ln 2πmkT h2  , (2a) where N = N1 + N2; note that this expression is numerically the same as (2), with mi = m. Therefore, the entropy of mixing in this case will also be given by expression (3) and, if N1/V1 = N2/V2 = (N1 + N2)/(V1 + V2), by expression (4). The last conclusion, however, is unacceptable because the mixing of two samples of the same gas, with a common initial temperature T and a common initial particle density n, is clearly a reversible process, for we can simply reinsert the partitioning wall into the system and obtain a situation that is in no way different from the one we had before mixing. Of course, we tacitly imply that in dealing with a system of identical particles we cannot track them down individually; all we can reckon with is their numbers. When two dissimilar gases, even with a common initial temperature T, and a common initial particle density n, mixed together the process was irreversible, for by reinserting the partitioning wall one would obtain two samples of the mixture and not the two gases that were originally present; to that case, expression (4) would indeed apply. However, in the present case, the corresponding result should be (1S)∗ 1≡2 = 0. (4a)16 The foregoing result would also be consistent with the requirement that the entropy of a given system is equal to the sum of the entropies of its parts. Of course, we had already noticed that this is not ensured by expression (1.4.21). Thus, once again we are led to believe that there is something basically wrong with that expression. To see how the above paradoxical situation can be avoided, we recall that, for the entropy of mixing of two samples of the same gas, with a common T and a common n, we were led to result (4), which can also be written as (1S)∗= ST −(S1 + S2) ≈k[ln{(N1 + N2)!} −ln(N1!) −ln(N2!)], (4) instead of the logical result (4a). A closer look at this expression shows that we would indeed obtain the correct result if our original expression for S were diminished by an ad hoc term, kln(N!), for that would diminish S1 by kln(N1!),S2 by kln(N2!) and ST by kln{(N1 + N2)!}, with the result that (1S)∗would turn out to be zero instead of the expres-sion appearing in (4). Clearly, this would amount to an ad hoc reduction of the statistical numbers 0 and 6 by a factor N!. This is precisely the remedy proposed by Gibbs to avoid the paradox in question. 16In view of this, we fear that expression (3) may also be inapplicable to this case. 1.5 The entropy of mixing and the Gibbs paradox 19 If we agree with the foregoing suggestion, then the modified expression for the entropy of a classical ideal gas would be S(N,V,E) = Nkln " V Nh3 4πmE 3N 3/2# + 5 2Nk (1.4.21a) = Nkln  V N  + 3 2Nk 5 3 + ln 2πmkT h2  , (1a) which indeed is truly extensive! If we now mix two samples of the same gas at a common initial temperature T, the entropy of mixing would be (1S)1≡2 = k  (N1 + N2)ln  V1 + V2 N1 + N2  −N1 ln  V1 N1  −N2 ln  V2 N2  (3a) and, if the initial particle densities of the samples were also equal, the result would be (1S)∗ 1≡2 = 0. (4a) It may be noted that for the mixing of two dissimilar gases, the original expressions (3) and (4) would continue to hold even when (1.4.21) is replaced by (1.4.21a).17 The paradox of Gibbs is thereby resolved. Equation (1a) is generally referred to as the Sackur–Tetrode equation. We reiterate the fact that, by this equation, the entropy of the system does indeed become a truly extensive quantity. Thus, the very root of the trouble has been eliminated by the recipe of Gibbs. We shall discuss the physical implications of this recipe in Section 1.6; here, let us jot down some of its immediate consequences. First of all, we note that the expression for the energy E of the gas, written as a function of N, V, and S, is also modified. We now have E(N,V,S) = 3h2N5/3 4πmV 2/3 exp  2S 3Nk −5 3  , (1.4.22a) which, unlike its predecessor (1.4.22), makes energy too a truly extensive quantity. Of course, the thermodynamic results (1.4.23) through (1.4.31), derived in the previous section, remain unchanged. However, there are some that were intentionally left out, for they would come out correct only from the modified expression for S(N,V,E) or E(S,V,N). The most important of these is the chemical potential of the gas, for which we obtain µ ≡  ∂E ∂N  V,S = E  5 3N − 2S 3N2k  . (5) 17Because, in this case, the entropy ST of the mixture would be diminished by kln(N1!N2!), rather than by kln{(N1 + N2)!}. 20 Chapter 1. The Statistical Basis of Thermodynamics In view of equations (1.4.23) and (1.4.25), this becomes µ = 1 N [E + PV −TS] ≡G N , (6) where G is the Gibbs free energy of the system. In terms of the variables N,V, and T, expression (5) takes the form µ(N,V,T) = kT ln    N V h2 2πmkT !3/2  . (7) Another quantity of importance is the Helmholtz free energy: A = E −TS = G −PV = NkT  ln    N V h2 2πmkT !3/2  −1  . (8) It will be noted that, while A is an extensive property of the system, µ is intensive. 1.6 The “correct” enumeration of the microstates In the preceding section we saw that an ad hoc diminution in the entropy of an N-particle system by an amount kln(N!), which implies an ad hoc reduction in the number of microstates accessible to the system by a factor (N!), was able to correct the unphysical fea-tures of some of our former expressions. It is now natural to ask: why, in principle, should the number of microstates, computed in Section 1.4, be reduced in this manner? The phys-ical reason for doing so is that the particles constituting the given system are not only identical but also indistinguishable; accordingly, it is unphysical to label them as No. 1, No. 2, No. 3, and so on and to speak of their being individually in the various single-particle states εi. All we can sensibly speak of is their distribution over the states εi by numbers, that is, n1 particles being in the state ε1, n2 in the state ε2, and so on. Thus, the correct way of specifying a microstate of the system is through the distribution numbers {nj}, and not through the statement as to “which particle is in which state.” To elaborate the point, we may say that if we consider two microstates that differ from one another merely in an inter-change of two particles in different energy states, then according to our original mode of counting we would regard these microstates as distinct; in view of the indistinguishability of the particles, however, these microstates are not distinct (for, physically, there exists no way whatsoever of distinguishing between them).18 18Of course, if an interchange took place among particles in the same energy state, then even our original mode of counting did not regard the two microstates as distinct. 1.6 The “correct” enumeration of the microstates 21 Now, the total number of permutations that can be effected among N particles, distributed according to the set {ni}, is N! n1!n2!..., (1) where the ni must be consistent with the basic constraints (1.1.1) and (1.1.2).19 If our parti-cles were distinguishable, then all these permutations would lead to “distinct” microstates. However, in view of the indistinguishability of the particles, these permutations must be regarded as leading to one and the same thing; consequently, for any distribution set {ni}, we have one, and only one, distinct microstate. As a result, the total number of distinct microstates accessible to the system, consistent with a given macrostate (N,V,E), would be severely cut down. However, since factor (1) itself depends on the numbers ni consti-tuting a particular distribution set and for a given macrostate there will be many such sets, there is no straightforward way to “correct down” the number of microstates computed on the basis of the classical concept of “distinguishability” of the particles. The recipe of Gibbs clearly amounts to disregarding the details of the numbers ni and slashing the whole sequence of microstates by a common factor N!; this is correct for situa-tions in which all N particles happen to be in different energy states but is certainly wrong for other situations. We must keep in mind that by adopting this recipe we are still using a spurious weight factor, w{ni} = 1 n1!n2!..., (2) for the distribution set {ni} whereas in principle we should use a factor of unity, irre-spective of the values of the numbers ni.20 Nonetheless, the recipe of Gibbs does correct the situation in a gross manner, though in matters of detail it is still inadequate. In fact, it is only by taking w{ni} to be equal to unity (or zero) that we obtain true quantum statistics! We thus see that the recipe of Gibbs corrects the enumeration of the microstates, as necessitated by the indistinguishability of the particles, only in a gross manner. Numeri-cally, this would approach closer and closer to reality as the probability of the ni being greater than 1 becomes less and less. This in turn happens when the given system is at a sufficiently high temperature (so that many more energy states become accessible) and has a sufficiently low density (so that there are not as many particles to accommo-date). It follows that the “corrected” classical statistics represents truth more closely if the expectation values of the occupation numbers ni are much less than unity: ⟨ni⟩≪1, (3) 19The presence of the factors (ni!) in the denominator is related to the comment made in the preceding note. 20Or a factor of zero if the distribution set {ni} is disallowed on certain physical grounds, such as the Pauli exclusion principle. 22 Chapter 1. The Statistical Basis of Thermodynamics that is, if the numbers ni are generally 0, occasionally 1, and rarely greater than 1. Condition (3) in a way defines the classical limit. We must, however, remember that it is because of the application of the correction factor 1/N!, which replaces (1) by (2), that our results agree with reality at least in the classical limit. In Section 5.5 we shall demonstrate, in an independent manner, that the factor by which the number of microstates, as computed for the “labeled” molecules, be reduced so that the formalism of classical statistical mechanics becomes a true limit of the formalism of quantum statistical mechanics is indeed N!. Problems 1.1. (a) Show that, for two large systems in thermal contact, the number (0)(E(0),E1) of Section 1.2 can be expressed as a Gaussian in the variable E1. Determine the root-mean-square deviation of E1 from the mean value E1 in terms of other quantities pertaining to the problem. (b) Make an explicit evaluation of the root-mean-square deviation of E1 in the special case when the systems A1 and A2 are ideal classical gases. 1.2. Assuming that the entropy S and the statistical number  of a physical system are related through an arbitrary functional form S = f (), show that the additive character of S and the multiplicative character of  necessarily require that the function f () be of the form (1.2.6). 1.3. Two systems A and B, of identical composition, are brought together and allowed to exchange both energy and particles, keeping volumes VA and VB constant. Show that the minimum value of the quantity (dEA/dNA) is given by µATB −µBTA TB −TA , where the µ’s and the T’s are the respective chemical potentials and temperatures. 1.4. In a classical gas of hard spheres (of diameter D), the spatial distribution of the particles is no longer uncorrelated. Roughly speaking, the presence of n particles in the system leaves only a volume (V −nv0) available for the (n + 1)th particle; clearly, v0 would be proportional to D3. Assuming that Nv0 ≪V, determine the dependence of (N,V,E) on V (compare to equation (1.4.1)) and show that, as a result of this, V in the ideal-gas law (1.4.3) gets replaced by (V −b), where b is four times the actual volume occupied by the particles. 1.5. Read Appendix A and establish formulae (1.4.15) and (1.4.16). Estimate the importance of the linear term in these formulae, relative to the main term (π/6)ε∗3/2, for an oxygen molecule confined to a cube of side 10 cm; take ε = 0.05 eV. 1.6. A cylindrical vessel 1 m long and 0.1 m in diameter is filled with a monatomic gas at P = 1 atm and T = 300 K. The gas is heated by an electrical discharge, along the axis of the vessel, which releases an energy of 104 joules. What will the temperature of the gas be immediately after the discharge? 1.7. Study the statistical mechanics of an extreme relativisitic gas characterized by the single-particle energy states ε(nx,ny,nz) = hc 2L  n2 x + n2 y + n2 z 1/2 , instead of (1.4.5), along the lines followed in Section 1.4. Show that the ratio CP/CV in this case is 4/3, instead of 5/3. 1.8. Consider a system of quasiparticles whose energy eigenvalues are given by ε(n) = nhν; n = 0,1,2,.... Problems 23 Obtain an asymptotic expression for the number  of this system for a given number N of the quasiparticles and a given total energy E. Determine the temperature T of the system as a function of E/N and hν, and examine the situation for which E/(Nhν) ≫1. 1.9. Making use of the fact that the entropy S(N,V,E) of a thermodynamic system is an extensive quantity, show that N  ∂S ∂N  V,E + V  ∂S ∂V  N,E + E  ∂S ∂E  N,V = S. Note that this result implies that (−Nµ + PV + E)/T = S, that is, Nµ = E + PV −TS. 1.10. A mole of argon and a mole of helium are contained in vessels of equal volume. If argon is at 300 K, what should the temperature of helium be so that the two have the same entropy? 1.11. Four moles of nitrogen and one mole of oxygen at P = 1 atm and T = 300K are mixed together to form air at the same pressure and temperature. Calculate the entropy of mixing per mole of the air formed. 1.12. Show that the various expressions for the entropy of mixing, derived in Section 1.5, satisfy the following relations: (a) For all N1,V1,N2, and V2, (1S)1≡2 = {(1S) −(1S)∗} ≥0, the equality holding when and only when N1/V1 = N2/V2. (b) For a given value of (N1 + N2), (1S)∗≤(N1 + N2)kln2, the equality holding when and only when N1 = N2. 1.13. If the two gases considered in the mixing process of Section 1.5 were initially at different temperatures, say T1 and T2, what would the entropy of mixing be in that case? Would the contribution arising from this cause depend on whether the two gases were different or identical? 1.14. Show that for an ideal gas composed of monatomic molecules the entropy change, between any two temperatures, when the pressure is kept constant is 5/3 times the corresponding entropy change when the volume is kept constant. Verify this result numerically by calculating the actual values of (1S)P and (1S)V per mole of an ideal gas whose temperature is raised from 300 K to 400 K. 1.15. We have seen that the (P, V)-relationship during a reversible adiabatic process in an ideal gas is governed by the exponent γ , such that PV γ = const. Consider a mixture of two ideal gases, with mole fractions f1 and f2 and respective exponents γ1 and γ2. Show that the effective exponent γ for the mixture is given by 1 γ −1 = f1 γ1 −1 + f2 γ2 −1. 1.16. Establish thermodynamically the formulae V  ∂P ∂T  µ = S and V  ∂P ∂µ  T = N. Express the pressure P of an ideal classical gas in terms of the variables µ and T, and verify the above formulae. 2 Elements of Ensemble Theory In the preceding chapter we noted that, for a given macrostate (N,V,E), a statistical system, at any time t, is equally likely to be in any one of an extremely large number of distinct microstates. As time passes, the system continually switches from one microstate to another, with the result that, over a reasonable span of time, all one observes is a behav-ior “averaged” over the variety of microstates through which the system passes. It may, therefore, make sense if we consider, at a single instant of time, a rather large number of systems — all being some sort of “mental copies” of the given system — which are charac-terized by the same macrostate as the original system but are, naturally enough, in all sorts of possible microstates. Then, under ordinary circumstances, we may expect that the aver-age behavior of any system in this collection, which we call an ensemble, would be identical to the time-averaged behavior of the given system. It is on the basis of this expectation that we proceed to develop the so-called ensemble theory. For classical systems, the most appropriate framework for developing the desired for-malism is provided by the phase space. Accordingly, we begin our study of the various ensembles with an analysis of the basic features of this space. 2.1 Phase space of a classical system The microstate of a given classical system, at any time t, may be defined by specifying the instantaneous positions and momenta of all the particles constituting the system. Thus, if N is the number of particles in the system, the definition of a microstate requires the specification of 3N position coordinates q1,q2,...,q3N and 3N momentum coordinates p1,p2,...,p3N. Geometrically, the set of coordinates (qi,pi), where i = 1,2,...,3N, may be regarded as a point in a space of 6N dimensions. We refer to this space as the phase space, and the phase point (qi,pi) as a representative point, of the given system. Of course, the coordinates qi and pi are functions of the time t; the precise manner in which they vary with t is determined by the canonical equations of motion, ˙ qi = ∂H(qi,pi) ∂pi ˙ pi = −∂H(qi,pi) ∂qi          i = 1,2,...,3N, (1) where H(qi,pi) is the Hamiltonian of the system. Now, as time passes, the set of coordinates (qi,pi), which also defines the microstate of the system, undergoes a continual change. Correspondingly, our representative point in the phase space carves out a Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00002-5 © 2011 Elsevier Ltd. All rights reserved. 25 26 Chapter 2. Elements of Ensemble Theory trajectory whose direction, at any time t, is determined by the velocity vector v ≡(˙ qi, ˙ pi), which in turn is given by the equations of motion (1). It is not difficult to see that the trajectory of the representative point must remain within a limited region of the phase space; this is so because a finite volume V directly limits the values of the coordinates qi, while a finite energy E limits the values of both the qi and the pi [through the Hamiltonian H(qi, pi)]. In particular, if the total energy of the system is known to have a precise value, say E, the corresponding trajectory will be restricted to the “hypersurface” H(qi,pi) = E (2) of the phase space; on the other hand, if the total energy may lie anywhere in the range E −1 21,E + 1 21  , the corresponding trajectory will be restricted to the “hypershell” defined by these limits. Now, if we consider an ensemble of systems (i.e., the given system, along with a large number of mental copies of it) then, at any time t, the various members of the ensem-ble will be in all sorts of possible microstates; indeed, each one of these microstates must be consistent with the given macrostate that is supposed to be common to all members of the ensemble. In the phase space, the corresponding picture will consist of a swarm of representative points, one for each member of the ensemble, all lying within the “allowed” region of this space. As time passes, every member of the ensemble undergoes a continual change of microstates; correspondingly, the representative points constituting the swarm continually move along their respective trajectories. The overall picture of this movement possesses some important features that are best illustrated in terms of what we call a density function ρ(q,p;t).1 This function is such that, at any time t, the number of repre-sentative points in the “volume element” (d3Nqd3Np) around the point (q,p) of the phase space is given by the product ρ(q,p;t)d3Nqd3Np. Clearly, the density function ρ(q,p;t) symbolizes the manner in which the members of the ensemble are distributed over all possible microstates at different instants of time. Accordingly, the ensemble average ⟨f ⟩of a given physical quantity f (q,p), which may be different for systems in different microstates, would be given by ⟨f ⟩= R f (q,p)ρ(q,p;t)d3Nqd3Np R ρ(q,p;t)d3Nqd3Np . (3) The integrations in (3) extend over the whole of the phase space; however, it is only the populated regions of the phase space (ρ ̸= 0) that really contribute. We note that, in general, the ensemble average ⟨f ⟩may itself be a function of time. An ensemble is said to be stationary if ρ does not depend explicitly on time, that is, at all times ∂ρ ∂t = 0. (4) 1Note that (q,p) is an abbreviation of (qi,pi) ≡(q1,...,q3N,p1,...,p3N). 2.2 Liouville’s theorem and its consequences 27 Clearly, for such an ensemble the average value ⟨f ⟩of any physical quantity f (q,p) will be independent of time. Naturally, a stationary ensemble qualifies to represent a system in equilibrium. To determine the circumstances under which equation (4) may hold, we have to make a rather detailed study of the movement of the representative points in the phase space. 2.2 Liouville’s theorem and its consequences Consider an arbitrary “volume” ω in the relevant region of the phase space and let the “surface” enclosing this volume be denoted by σ; see Figure 2.1. Then, the rate at which the number of representative points in this volume increases with time is written as ∂ ∂t Z ω ρ dω, (1) where dω ≡ d3Nqd3Np  . On the other hand, the net rate at which the representative points “flow” out of ω (across the bounding surface σ) is given by Z σ ρ v · ˆ n dσ; (2) here, v is the velocity vector of the representative points in the region of the surface element dσ while ˆ n is the (outward) unit vector normal to this element. By the divergence theorem, (2) can be written as Z ω div(ρv)dω; (3) of course, the operation of divergence here means div(ρv) ≡ 3N X i=1  ∂ ∂qi (ρ ˙ qi) + ∂ ∂pi (ρ ˙ pi)  . (4) vdt d n ^ v FIGURE 2.1 The “hydrodynamics” of the representative points in the phase space. 28 Chapter 2. Elements of Ensemble Theory In view of the fact that there are no “sources” or “sinks” in the phase space and hence the total number of representative points remains conserved,2 we have, by (1) and (3), ∂ ∂t Z ω ρ dω = − Z ω div(ρv)dω, (5) that is, Z ω ∂ρ ∂t + div(ρv)  dω = 0. (6) Now, the necessary and sufficient condition that integral (6) vanish for all arbitrary volumes ω is that the integrand itself vanish everywhere in the relevant region of the phase space. Thus, we must have ∂ρ ∂t + div(ρv) = 0, (7) which is the equation of continuity for the swarm of the representative points. Combining (4) and (7), we obtain ∂ρ ∂t + 3N X i=1  ∂ρ ∂qi ˙ qi + ∂ρ ∂pi ˙ pi  + ρ 3N X i=1 ∂˙ qi ∂qi + ∂˙ pi ∂pi  = 0. (8) The last group of terms vanishes identically because, by the equations of motion, we have, for all i, ∂˙ qi ∂qi = ∂2H(qi,pi) ∂qi∂pi ≡∂2H(qi,pi) ∂pi∂qi = −∂˙ pi ∂pi . (9) Further, since ρ ≡ρ (q,p;t), the remaining terms in (8) may be combined to form the “total” time derivative of ρ, with the result that dρ dt = ∂ρ ∂t + [ρ,H] = 0. (10)3 Equation (10) embodies Liouville’s theorem (1838). According to this theorem, the “local” density of the representative points, as viewed by an observer moving with a representa-tive point, stays constant in time. Thus, the swarm of the representative points moves in 2This means that in the ensemble under consideration neither are any new members being added nor are any old ones being removed. 3We recall that the Poisson bracket [ρ,H] stands for the sum 3N X i=1  ∂ρ ∂qi ∂H ∂pi −∂ρ ∂pi ∂H ∂qi  , which is identical to the group of terms in the middle of (8). 2.2 Liouville’s theorem and its consequences 29 the phase space in essentially the same manner as an incompressible fluid moves in the physical space! A distinction must be made, however, between equation (10) on one hand and equation (2.1.4) on the other. While the former derives from the basic mechanics of the particles and is therefore quite generally true, the latter is only a requirement for equi-librium which, in a given case, may or may not be satisfied. The condition that ensures simultaneous validity of the two equations is clearly [ρ,H] = 3N X i=1  ∂ρ ∂qi ˙ qi + ∂ρ ∂pi ˙ pi  = 0. (11) Now, one possible way of satisfying (11) is to assume that ρ, which is already assumed to have no explicit dependence on time, is independent of the coordinates (q,p) as well, that is, ρ(q,p) = const. (12) over the relevant region of the phase space (and, of course, is zero everywhere else). Physi-cally, this choice corresponds to an ensemble of systems that at all times are uniformly distributed over all possible microstates. The ensemble average (2.1.3) then reduces to ⟨f ⟩= 1 ω Z ω f (q,p)dω; (13) here, ω denotes the total “volume” of the relevant region of the phase space. Clearly, in this case, any member of the ensemble is equally likely to be in any one of the various possible microstates, inasmuch as any representative point in the swarm is equally likely to be in the neighborhood of any phase point in the allowed region of the phase space. This statement is usually referred to as the postulate of “equal a priori probabilities” for the various possible microstates (or for the various volume elements in the allowed region of the phase space); the resulting ensemble is referred to as the microcanonical ensemble. A more general way of satisfying (11) is to assume that the dependence of ρ on (q,p) comes only through an explicit dependence on the Hamiltonian H(q,p), that is, ρ(q,p) = ρ[H(q,p)]; (14) condition (11) is then identically satisfied. Equation (14) provides a class of density func-tions for which the corresponding ensemble is stationary. In Chapter 3 we shall see that the most natural choice in this class of ensembles is the one for which ρ(q,p) ∝exp[−H(q,p)/kT]. (15) The ensemble so defined is referred to as the canonical ensemble. 30 Chapter 2. Elements of Ensemble Theory 2.3 The microcanonical ensemble In this ensemble the macrostate of a system is defined by the number of molecules N, the volume V, and the energy E. However, in view of the considerations expressed in Section 1.4, we may prefer to specify a range of energy values, say from E −1 21  to E + 1 21  , rather than a sharply defined value E. With the macrostate specified, a choice still remains for the systems of the ensemble to be in any one of a large number of pos-sible microstates. In the phase space, correspondingly, the representative points of the ensemble have a choice to lie anywhere within a “hypershell” defined by the condition  E −1 21  ≤H(q,p) ≤  E + 1 21  . (1) The volume of the phase space enclosed within this shell is given by ω = ′ Z dω ≡ ′ Z  d3Nqd3Np  , (2) where the primed integration extends only over that part of the phase space which con-forms to condition (1). It is clear that ω will be a function of the parameters N,V,E, and 1. Now, the microcanonical ensemble is a collection of systems for which the density function ρ is, at all times, given by ρ(q,p) = const. if  E −1 21  ≤H(q,p) ≤  E + 1 21  0 otherwise   . (3) Accordingly, the expectation value of the number of representative points lying in a vol-ume element dω of the relevant hypershell is simply proportional to dω. In other words, the a priori probability of finding a representative point in a given volume element dω is the same as that of finding a representative point in an equivalent volume element dω located anywhere in the hypershell. In our original parlance, this means an equal a priori probabil-ity for a given member of the ensemble to be in any one of the various possible microstates. In view of these considerations, the ensemble average ⟨f ⟩, as given by equation (2.2.13), acquires a simple physical meaning. To see this, we proceed as follows. Since the ensemble under study is a stationary one, the ensemble average of any phy-sical quantity f will be independent of time; accordingly, taking a time average thereof will not produce any new result. Thus ⟨f ⟩≡the ensemble average of f = the time average of (the ensemble average of f ). 2.3 The microcanonical ensemble 31 Now, the processes of time averaging and ensemble averaging are completely indepen-dent, so the order in which they are performed may be reversed without causing any change in the value of ⟨f ⟩. Thus ⟨f ⟩= the ensemble average of (the time average of f ). Now, the time average of any physical quantity, taken over a sufficiently long interval of time, must be the same for every member of the ensemble, for after all we are dealing with only mental copies of a given system.4 Therefore, taking an ensemble average thereof should be inconsequential, and we may write ⟨f ⟩= the long-time average of f , where the latter may be taken over any member of the ensemble. Furthermore, the long-time average of a physical quantity is all one obtains by making a measurement of that quantity on the given system; therefore, it may be identified with the value one expects to obtain through experiment. Thus, we finally have ⟨f ⟩= fexp. (4) This brings us to the most important result: the ensemble average of any physical quantity f is identical to the value one expects to obtain on making an appropriate measurement on the given system. The next thing we look for is the establishment of a connection between the mechanics of the microcanonical ensemble and the thermodynamics of the member systems. To do this, we observe that there exists a direct correspondence between the various microstates of the given system and the various locations in the phase space. The volume ω (of the allowed region of the phase space) is, therefore, a direct measure of the multiplicity 0 of the microstates accessible to the system. To establish a numerical correspondence between ω 4To provide a rigorous justification for this assertion is not trivial. One can readily see that if, for any particular mem-ber of the ensemble, the quantity f is averaged only over a short span of time, the result is bound to depend on the relevant “subset of microstates” through which the system passes during that time. In the phase space, this will mean an averaging over only a “part of the allowed region.” However, if we employ instead a sufficiently long interval of time, the system may be expected to pass through almost all possible microstates “without fear or favor”; consequently, the result of the averaging process would depend only on the macrostate of the system, and not on a subset of microstates. Correspondingly, the averaging in the phase space would go over practically all parts of the allowed region, again “with-out fear or favor.” In other words, the representative point of our system will have traversed each and every part of the allowed region almost uniformly. This statement embodies the so-called ergodic theorem or ergodic hypothesis, which was first introduced by Boltzmann (1871). According to this hypothesis, the trajectory of a representative point passes, in the course of time, through each and every point of the relevant region of the phase space. A little reflection, however, shows that the statement as such requires a qualification; we better replace it by the so-called quasiergodic hypothesis, according to which the trajectory of a representative point traverses, in the course of time, any neighborhood of any point of the relevant region. For further details, see ter Haar (1954, 1955), Farquhar (1964). Now, when we consider an ensemble of systems, the foregoing statement should hold for every member of the ensemble; thus, irrespective of the initial (and final) states of the various systems, the long-time average of any physical quantity f should be the same for every member system. 32 Chapter 2. Elements of Ensemble Theory and 0, we need to discover a fundamental volume ω0 that could be regarded as “equivalent to one microstate.” Once this is done, we may say that, asymptotically, 0 = ω/ω0. (5) The thermodynamics of the system would then follow in the same way as in Sections 1.2– 1.4, namely through the relationship S = kln0 = kln(ω/ω0), etc. (6) The basic problem then consists in determining ω0. From dimensional considerations, see (2), ω0 must be in the nature of an “angular momentum raised to the power 3N.” To determine it exactly, we consider certain simplified systems, both from the point of view of the phase space and from the point of view of the distribution of quantum states. 2.4 Examples We consider, first of all, the problem of a classical ideal gas composed of monatomic par-ticles; see Section 1.4. In the microcanonical ensemble, the volume ω of the phase space accessible to the representative points of the (member) systems is given by ω = ′ Z ... ′ Z  d3Nqd3Np  , (1) where the integrations are restricted by the conditions that (i) the particles of the system are confined in physical space to volume V, and (ii) the total energy of the system lies between the limits E −1 21  and E + 1 21  . Since the Hamiltonian in this case is a function of the pi alone, integrations over the qi can be carried out straightforwardly; these give a factor of V N. The remaining integral is Z ... Z  E−1 2 1  ≤ 3N P i=1  p2 i /2m  ≤  E+ 1 2 1  d3Np = Z ... Z 2m  E−1 2 1  ≤ 3N P i=1 y2 i ≤2m  E+ 1 2 1  d3Ny, which is equal to the volume of a 3N-dimensional hypershell, bounded by hyperspheres of radii r 2m  E + 1 21  and r 2m  E −1 21  . For 1 ≪E, this is given by the thickness of the shell, which is almost equal to 1(m/2E)1/2, multiplied by the surface area of a 3N-dimensional hypersphere of radius √(2mE). By 2.4 Examples 33 equation (7) of Appendix C, we obtain for this integral 1  m 2E 1/2 ( 2π3N/2 [(3N/2) −1]!(2mE)(3N−1)/2 ) , which gives ω ≃1 E V N (2πmE)3N/2 [(3N/2) −1]!. (2) Comparing (2) with (1.4.17 and 1.4.17a), we obtain the desired correspondence, namely (ω/0)asymp ≡ω0 = h3N; see also Problem 2.9. Quite generally, if the system under study has N degrees of freedom, the desired conversion factor is ω0 = hN . (3) In the case of a single particle, N = 3; accordingly, the number of microstates available would asymptotically be equal to the volume of the allowed region of the phase space divided by h3. Let 6(P) denote the number of microstates available to a free particle con-fined to volume V of the physical space, its momentum p being less than or equal to a specified value P. Then 6(P) ≈1 h3 Z ... Z p≤P  d3qd3p  = V h3 4π 3 P3, (4) from which we obtain for the number of microstates with momentum lying between p and p + dp g(p)dp = d6(p) dp dp ≈V h3 4πp2dp. (5) Expressed in terms of the particle energy, these expressions assume the form 6(E) ≈V h3 4π 3 (2mE)3/2 (6) and a(ε)dε = d6(ε) dε dε ≈V h3 2π(2m)3/2ε1/2dε. (7) The next case we consider here is that of a one-dimensional simple harmonic oscillator. The classical expression for the Hamiltonian of this system is H(q,p) = 1 2kq2 + 1 2mp2, (8) 34 Chapter 2. Elements of Ensemble Theory where k is the spring constant and m the mass of the oscillating particle. The space coordinate q and the momentum coordinate p of the system are given by q = Acos(ωt + φ), p = m˙ q = −mωAsin(ωt + φ), (9) A being the amplitude and ω the (angular) frequency of vibration: ω = √(k/m). (10) The energy of the oscillator is a constant of the motion, and is given by E = 1 2mω2A2. (11) The phase-space trajectory of the representative point (q,p) of this system is determined by eliminating t between expressions (9) for q(t) and p(t); we obtain q2 (2E/mω2) + p2 (2mE) = 1, (12) which is an ellipse, with axes proportional to √E and hence area proportional to E; to be precise, the area of this ellipse is 2πE/ω. Now, if we restrict the oscillator energy to the interval E −1 21,E + 1 21  , its representative point in the phase space will be confined to the region bounded by elliptical trajectories corresponding to the energy values E + 1 21  and E −1 21  . The “volume” (in this case, the area) of this region will be Z ... Z  E−1 2 1  ≤H(q,p)≤  E+ 1 2 1  (dqdp) = 2π  E + 1 21  ω − 2π  E −1 21  ω = 2π1 ω . (13) According to quantum mechanics, the energy eigenvalues of the harmonic oscillator are given by En =  n + 1 2  ℏω; n = 0,1,2,... (14) In terms of phase space, one could say that the representative point of the system must move along one of the “chosen” trajectories, as shown in Figure 2.2; the area of the phase space between two consecutive trajectories, for which 1 = ℏω, is simply 2πℏ.5 For arbitrary values of E and 1, such that E ≫1 ≫ℏω, the number of eigenstates within the allowed 5Strictly speaking, the very concept of phase space is invalid in quantum mechanics because there, in principle, it is wrong to assign to a particle the coordinates q and p simultaneously. Nevertheless, the ideas discussed here are tenable in the correspondence limit. 2.5 Quantum states and the phase space 35 n 3 2p0 p0 p0 2p0 q p n 2 n 1 n 0 2q0 q0 q0 2q0 FIGURE 2.2 Eigenstates of a linear harmonic oscillator, in relation to its phase space. energy interval is very nearly equal to 1/ℏω. Hence, the area of the phase space equivalent to one eigenstate is, asymptotically, given by ω0 = (2π1/ω)/(1/ℏω) = 2πℏ= h. (15) If, on the other hand, we consider a system of N harmonic oscillators along the same lines as above, we arrive at the result: ω0 = hN (see Problem 2.7). Thus, our findings in these cases are consistent with our earlier result (3). 2.5 Quantum states and the phase space At this stage we would like to say a few words on the central role played here by the Planck constant h. The best way to appreciate this role is to recall the implications of the Heisen-berg uncertainty principle, according to which we cannot specify simultaneously both the position and the momentum of a particle exactly. An element of uncertainty is inherently present and can be expressed as follows: assuming that all conceivable uncertainties of measurement are eliminated, even then, by the very nature of things, the product of the uncertainties 1q and 1p in the simultaneous measurement of the canonically conjugate coordinates q and p would be of order ℏ: (1q1p)min ∼ℏ. (1) Thus, it is impossible to define the position of a representative point in the phase space of the given system more accurately than is allowed by condition (1). In other words, around any point (q,p) in the (two-dimensional) phase space, there exists an area of order ℏwithin 36 Chapter 2. Elements of Ensemble Theory which the position of the representative point cannot be pinpointed. In a phase space of 2N dimensions, the corresponding “volume of uncertainty” around any point would be of order ℏN . Therefore, it seems reasonable to regard the phase space as made up of ele-mentary cells, of volume ∼ℏN , and to consider the various positions within such a cell as nondistinct. These cells could then be put into one-to-one correspondence with the quantum-mechanical states of the system. It is, however, obvious that considerations of uncertainty alone cannot give us the exact value of the conversion factor ω0. This could only be done by an actual counting of microstates on one hand and a computation of volume of the relevant region of the phase space on the other, as was done in the examples of the previous section. Clearly, a procedure along these lines could not be possible until after the work of Schr¨ odinger and others. Historically, however, the first to establish the result (2.4.3) was Tetrode (1912) who, in his well-known work on the chemical constant and the entropy of a monatomic gas, assumed that ω0 = (zh)N , (2) where z was supposed to be an unknown numerical factor. Comparing theoretical results with the experimental data on mercury, Tetrode found that z was very nearly equal to unity; from this he concluded that “it seems rather plausible that z is exactly equal to unity, as has already been taken by O. Sackur (1911).”6 In the extreme relativistic limit, the same result was established by Bose (1924). In his famous treatment of the photon gas, Bose made use of Einstein’s relationship between the momentum of a photon and the frequency of the associated radiation, namely p = hν c , (3) and observed that, for a photon confined to a three-dimensional cavity of volume V, the relevant “volume” of the phase space, ′ Z (d3qd3p) = V4πp2dp = V(4πh3ν2/c3)dν, (4) would correspond exactly to the Rayleigh expression, V(4πν2/c3)dν, (5) for the number of normal modes of a radiation oscillator, provided that we divide phase space into elementary cells of volume h3 and put these cells into one-to-one corre-spondence with the vibrational modes of Rayleigh. It may, however, be added that a two-fold multiplicity of these states (g = 2) arises from the spin orientations of the photon 6For a more satisfactory proof of this result, see Section 5.5, especially equation (5.5.22). Problems 37 (or from the states of polarization of the vibrational modes); this requires a multiplica-tion of both expressions (4) and (5) by a factor of 2, leaving the conversion factor h3 unchanged. Problems 2.1. Show that the volume element dω = 3N Y i=1 (dqi dpi) of the phase space remains invariant under a canonical transformation of the (generalized) coordinates (q,p) to any other set of (generalized) coordinates (Q,P). [Hint: Before considering the most general transformation of this kind, which is referred to as a contact transformation, it may be helpful to consider a point transformation — one in which the new coordinates Qi and the old coordinates qi transform only among themselves.] 2.2. (a) Verify explicitly the invariance of the volume element dω of the phase space of a single particle under transformation from the Cartesian coordinates x,y,z,px,py,pz  to the spherical polar coordinates (r,θ,φ,pr,pθ,pφ). (b) The foregoing result seems to contradict the intuitive notion of “equal weights for equal solid angles,” because the factor sinθ is invisible in the expression for dω. Show that if we average out any physical quantity, whose dependence on pθ and pφ comes only through the kinetic energy of the particle, then as a result of integration over these variables we do indeed recover the factor sinθ to appear with the subelement (dθ dφ). 2.3. Starting with the line of zero energy and working in the (two-dimensional) phase space of a classical rotator, draw lines of constant energy that divide phase space into cells of “volume” h. Calculate the energies of these states and compare them with the energy eigenvalues of the corresponding quantum-mechanical rotator. 2.4. By evaluating the “volume” of the relevant region of its phase space, show that the number of microstates available to a rigid rotator with angular momentum ≤M is (M/ℏ)2. Hence determine the number of microstates that may be associated with the quantized angular momentum Mj = √{j(j + 1)}ℏ, where j = 0,1,2,... or 1 2, 3 2, 5 2,.... Interpret the result physically. [Hint: It simplifies to consider motion in the variables θ and ϕ, with M2 = p2 θ + (pφ/sinθ)2.] 2.5. Consider a particle of energy E moving in a one-dimensional potential well V(q), such that mℏ dV dq ≪{m(E −V)}3/2. Show that the allowed values of the momentum p of the particle are such that I pdq =  n + 1 2  h, where n is an integer. 2.6. The generalized coordinates of a simple pendulum are the angular displacement θ and the angular momentum ml2 ˙ θ. Study, both mathematically and graphically, the nature of the corresponding trajectories in the phase space of the system, and show that the area A enclosed by a trajectory is equal to the product of the total energy E and the time period τ of the pendulum. 2.7. Derive (i) an asymptotic expression for the number of ways in which a given energy E can be distributed among a set of N one-dimensional harmonic oscillators, the energy eigenvalues of the oscillators being  n + 1 2  ℏω;n = 0,1,2,..., and (ii) the corresponding expression for the “volume” of the relevant region of the phase space of this system. Establish the correspondence between the two results, showing that the conversion factor ω0 is precisely hN. 38 Chapter 2. Elements of Ensemble Theory 2.8. Following the method of Appendix C, replacing equation (C.4) by the integral ∞ Z 0 e−rr2dr = 2, show that V3N = Z ... Z 0≤ N P i=1 ri≤R N Y i=1  4πr2 i dri  = (8πR3)N/(3N)!. Using this result, compute the “volume” of the relevant region of the phase space of an extreme relativistic gas (ε = pc) of N particles moving in three dimensions. Hence, derive expressions for the various thermodynamic properties of this system and compare your results with those of Problem 1.7. 2.9. (a) Solve the integral Z ... Z 0≤ 3N P i=1 |xi|≤R (dx1 ...dx3N) and use it to determine the “volume” of the relevant region of the phase space of an extreme relativistic gas (ε = pc) of 3N particles moving in one dimension. Determine, as well, the number of ways of distributing a given energy E among this system of particles and show that, asymptotically, ω0 = h3N. (b) Compare the thermodynamics of this system with that of the system considered in Problem 2.8. 3 The Canonical Ensemble In the preceding chapter we established the basis of ensemble theory and made a somewhat detailed study of the microcanonical ensemble. In that ensemble the macrostate of the systems was defined through a fixed number of particles N, a fixed vol-ume V, and a fixed energy E [or, preferably, a fixed energy range (E −1 21,E + 1 21)]. The basic problem then consisted in determining the number (N,V,E), or 0(N,V,E;1), of distinct microstates accessible to the system. From the asymptotic expressions of these numbers, complete thermodynamics of the system could be derived in a straightforward manner. However, for most physical systems, the mathematical problem of determin-ing these numbers is quite formidable. For this reason alone, a search for an alternative approach within the framework of the ensemble theory seems necessary. Practically, too, the concept of a fixed energy (or even an energy range) for a system belonging to the real world does not appear satisfactory. For one thing, the total energy E of a system is hardly ever measured; for another, it is hardly possible to keep its value under strict physical control. A far better alternative appears to be to speak of a fixed tem-perature T of the system — a parameter that is not only directly observable (by placing a “thermometer” in contact with the system) but also controllable (by keeping the system in contact with an appropriate “heat reservoir”). For most purposes, the precise nature of the reservoir is not very relevant; all one needs is that it should have an infinitely large heat capacity, so that, irrespective of energy exchange between the system and the reser-voir, an overall constant temperature can be maintained. Now, if the reservoir consists of an infinitely large number of mental copies of the given system we have once again an ensemble of systems — this time, however, it is an ensemble in which the macrostate of the systems is defined through the parameters N,V, and T. Such an ensemble is referred to as a canonical ensemble. In the canonical ensemble, the energy E of a system is variable; in principle, it can take values anywhere between zero and infinity. The question then arises: what is the probability that, at any time t, a system in the ensemble is found to be in one of the states characterized by the energy value Er?1 We denote this probability by the symbol Pr. Clearly, there are two ways in which the dependence of Pr on Er can be determined. One consists of regarding the system as in equilibrium with a heat reservoir at a common temperature T and studying the statistics of the energy exchange between the two. The other consists of regarding the system as a member of a canonical ensemble (N,V,T), in which an energy E is being shared by N identical systems constituting the ensemble, and studying the 1In what follows, the energy levels Er appear as purely mechanical quantities — independent of the temperature of the system. For a treatment involving “temperature-dependent energy levels,” see Elcock and Landsberg (1957). Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00003-7 © 2011 Elsevier Ltd. All rights reserved. 39 40 Chapter 3. The Canonical Ensemble statistics of this sharing process. We expect that in the thermodynamic limit the final result in either case would be the same. Once Pr is determined, the rest follows without difficulty. 3.1 Equilibrium between a system and a heat reservoir We consider the given system A, immersed in a very large heat reservoir A′; see Figure 3.1. On attaining a state of mutual equilibrium, the system and the reservoir would have a common temperature, say T. Their energies, however, would be variable and, in principle, could have, at any time t, values lying anywhere between 0 and E(0), where E(0) denotes the energy of the composite system A(0)(≡A + A′). If, at any particular instant of time, the system A happens to be in a state characterized by the energy value Er, then the reservoir would have an energy E′ r, such that Er + E′ r = E(0) = const. (1) Of course, since the reservoir is supposed to be much larger than the given system, any practical value of Er would be a very small fraction of E(0); therefore, for all practical purposes, Er E(0) =  1 −E′ r E(0)  ≪1. (2) With the state of the system A having been specified, the reservoir A′ can still be in any one of a large number of states compatible with the energy value E′ r. Let the number of these states be denoted by ′(E′ r). The prime on the symbol  emphasizes the fact that its functional form will depend on the nature of the reservoir; of course, the details of this dependence are not going to be of any particular relevance to our final results. Now, the larger the number of states available to the reservoir, the larger the probability of the reservoir assuming that particular energy value E′ r (and, hence, of the system A assum-ing the corresponding energy value Er). Moreover, since the various possible states (with a given energy value) are equally likely to occur, the relevant probability would be directly proportional to this number; thus, Pr ∝′(E′ r) ≡′(E(0) −Er). (3) A9 (Er9;T ) A (Er ;T ) FIGURE 3.1 A given system A immersed in a heat reservoir A′; in equilibrium, the two have a common temperature T. 3.2 A system in the canonical ensemble 41 In view of (2), we may carry out an expansion of (3) around the value E′ r = E(0), that is, around Er = 0. However, for reasons of convergence, it is essential to effect the expansion of its logarithm instead: ln′(E′ r) = ln′(E(0)) + ∂ln′ ∂E′  E′=E(0) (E′ r −E(0)) + ··· ≃const −β′Er, (4) where use has been made of formula (1.2.3), whereby ∂ln ∂E  N,V ≡β; (5) note that, in equilibrium, β′ = β = 1/kT. From (3) and (4), we obtain the desired result: Pr ∝exp(−βEr). (6) Normalizing (6), we get Pr = exp(−βEr) P r exp(−βEr), (7) where the summation in the denominator goes over all states accessible to the system A. We note that our final result (7) bears no relation whatsoever to the physical nature of the reservoir A′. We now examine the same problem from the ensemble point of view. 3.2 A system in the canonical ensemble We consider an ensemble of N identical systems (which may be labelled as 1,2,...,N ), sharing a total energy E; let Er(r = 0,1,2,...) denote the energy eigenvalues of the systems. If nr denotes the number of systems which, at any time t, have the energy value Er, then the set of numbers {nr} must satisfy the obvious conditions X r nr = N X r nrEr = E = N U,      (1) where U(= E/N ) denotes the average energy per system in the ensemble. Any set {nr} that satisfies the restrictive conditions (1) represents a possible mode of distribution of the total energy E among the N members of the ensemble. Furthermore, any such mode can be realized in a number of ways, for we may effect a reshuffle among those members of the ensemble for which the energy values are different and thereby obtain a state of the 42 Chapter 3. The Canonical Ensemble ensemble that is distinct from the original one. Denoting the number of different ways of doing so by the symbol W{nr}, we have W{nr} = N ! n0!n1!n2!.... (2) In view of the fact that all possible states of the ensemble, which are compatible with con-ditions (1), are equally likely to occur, the frequency with which the distribution set {nr} may appear will be directly proportional to the number W{nr}. Accordingly, the “most probable” mode of distribution will be the one for which the number W is a maximum. We denote the corresponding distribution set by {n∗ r}; clearly, the set {n∗ r} must also satisfy conditions (1). As will be seen in the sequel, the probability of appearance of other modes of distribution, however little they may differ from the most probable mode, is extremely low! Therefore, for all practical purposes, the most probable distribution set {n∗ r} is the only one we have to contend with. However, unless this has been mathematically demonstrated, one must take into account all possible modes of distribution, as characterized by the various distribution sets {nr}, along with their respective weight factors W{nr}. Accordingly, the expectation values, or mean values, ⟨nr⟩of the numbers nr would be given by ⟨nr⟩= P {nr} ′ nrW{nr} P {nr} ′ W{nr} , (3) where the primed summations go over all distribution sets that conform to conditions (1). In principle, the mean value ⟨nr⟩, as a fraction of the total number N , should be a natural analog of the probability Pr evaluated in the preceding section. In practice, however, the fraction n∗ r/N also turns out to be the same. We now proceed to derive expressions for the numbers n∗ r and ⟨nr⟩, and to show that, in the limit N →∞, they are identical. The method of most probable values Our aim here is to determine that distribution set which, while satisfying conditions (1), maximizes the weight factor (2). For simplicity, we work with lnW instead: lnW = ln(N !) − X r ln(nr!). (4) Since, in the end, we propose to resort to the limit N →∞, the values of nr (which are going to be of any practical significance) would also, in that limit, tend to infinity. It is, therefore, justified to apply the Stirling formula, ln(n!) ≈nlnn −n, to (4) and write lnW = N lnN − X r nr lnnr. (5) 3.2 A system in the canonical ensemble 43 If we shift from the set {nr} to a slightly different set {nr + δnr}, then expression (5) would change by an amount δ(lnW) = − X r (lnnr + 1)δnr. (6) Now, if the set {nr} is maximal, the variation δ(lnW) should vanish. At the same time, in view of the restrictive conditions (1), the variations δnr themselves must satisfy the conditions X r δnr = 0 X r Erδnr = 0.      (7) The desired set {n∗ r} is then determined by the method of Lagrange multipliers,2 by which the condition determining this set becomes X r {−(lnn∗ r + 1) −α −βEr}δnr = 0, (8) where α and β are the Lagrangian undetermined multipliers that take care of the restrictive conditions (7). In (8), the variations δnr become completely arbitrary; accordingly, the only way to satisfy this condition is that all its coefficients must vanish identically, that is, for all r, lnn∗ r = −(α + 1) −βEr, which gives n∗ r = C exp(−βEr), (9) where C is again an undetermined parameter. To determine C and β, we subject (9) to conditions (1), with the result that n∗ r N = exp(−βEr) P r exp(−βEr), (10) the parameter β being a solution of the equation E N = U = P r Er exp(−βEr) P r exp(−βEr) . (11) 2For the method of Lagrange multipliers, see ter Haar and Wergeland (1966, Appendix C.1). 44 Chapter 3. The Canonical Ensemble Combining statistical considerations with thermodynamic ones, see Section 3.3, we can show that the parameter β here is exactly the same as the one appearing in Section 3.1, that is, β = 1/kT. The method of mean values Here we attempt to evaluate expression (3) for ⟨nr⟩, taking into account the weight factors (2) and the restrictive conditions (1). To do this, we replace (2) by ˜ W{nr} = N !ωn0 0 ωn1 1 ωn2 2 ... n0!n1!n2!... , (12) with the understanding that in the end all ωr will be set equal to unity, and introduce a function 0(N ,U) = X {nr} ′ ˜ W{nr}, (13) where the primed summation, as before, goes over all distribution sets that conform to conditions (1). Expression (3) can then be written as ⟨nr⟩= ωr ∂ ∂ωr (ln0) all ωr=1 . (14) Thus, all we need to know here is the dependence of the quantity ln0 on the parameters ωr. Now, 0(N ,U) = N ! X {nr} ′ ωn0 0 n0! · ωn1 1 n1! · ωn2 2 n2! ··· ! (15) but the summation appearing here cannot be evaluated explicitly because it is restricted to those sets only that conform to the pair of conditions (1). If our distribution sets were restricted by the condition P r nr = N alone, then the evaluation of (15) would have been trivial; by the multinomial theorem, 0(N ) would have been simply (ω0 + ω1 + ···)N . The added restriction P r nrEr = N U, however, permits the inclusion of only a “limited” number of terms in the sum — and that constitutes the real difficulty of the problem. Nevertheless, we can still hope to make some progress because, from a physical point of view, we do not require anything more than an asymptotic result — one that holds in the limit N →∞. The method commonly used for this purpose is the one developed by Darwin and Fowler (1922a,b, 1923), which itself makes use of the saddle-point method of integration or the so-called method of steepest descent. We construct a generating function G(N ,z) for the quantity 0(N ,U): G(N ,z) = ∞ X U=0 0(N ,U)zN U (16) 3.2 A system in the canonical ensemble 45 which, in view of equation (15) and the second of the restrictive conditions (1), may be written as G(N ,z) = ∞ X U=0  X {nr} ′ N ! n0!n1!...  ω0zE0 n0  ω1zE1 n1 ...  . (17) It is easy to see that the summation over doubly restricted sets {nr}, followed by a summa-tion over all possible values of U, is equivalent to a summation over singly restricted sets {nr}, namely the ones that satisfy only one condition: P r nr = N . Expression (17) can be evaluated with the help of the multinomial theorem, with the result G(N ,z) =  ω0zE0 + ω1zE1 + ··· N = [f (z)]N , say. (18) Now, if we suppose that the Er (and hence the total energy value E = N U) are all integers, then, by (16), the quantity 0(N ,U) is simply the coefficient of zN U in the expansion of the function G(N ,z) as a power series in z. It can, therefore, be evaluated by the method of residues in the complex z-plane. To make this plan work, we assume to have chosen, right at the outset, a unit of energy so small that, to any desired degree of accuracy, we can regard the energies Er (and the pre-scribed total energy N U) as integral multiples of this unit. In terms of this unit, any energy value we come across will be an integer. We further assume, without loss of generality, that the sequence E0,E1,... is a nondecreasing sequence, with no common divisor;3 also, for the sake of simplicity, we assume that E0 = 0.4 The solution now is 0(N ,U) = 1 2πi I [f (z)]N zN U+1 dz, (19) where the integration is carried along any closed contour around the origin; of course, we must stay within the circle of convergence of the function f (z), so that a need for analytic continuation does not arise. First of all, we examine the behavior of the integrand as we proceed from the origin along the real positive axis, remembering that all our ωr are virtually equal to unity and that 0 = E0 ≤E1 ≤E2 ···. We find that the factor [f (z)]N starts from the value 1 at z = 0, increases monotonically and tends to infinity as z approaches the circle of convergence of f (z), wherever that may be. The factor z−(N U+1), on the other hand, starts from a positive, infinite value at z = 0 and decreases monotonically as z increases. Moreover, the relative rate of increase of the factor [f (z)]N itself increases monotonically while the relative rate 3Actually, this is not a serious restriction at all, for a common divisor, if any, can be removed by selecting the unit of energy correspondingly larger. 4This too is not serious, for by doing so we are merely shifting the zero of the energy scale; the mean energy U then becomes U −E0, but we can agree to call it U again. 46 Chapter 3. The Canonical Ensemble of decrease of the factor z−(N U+1) decreases monotonically. Under these circumstances, the integrand must exhibit a minimum (and no other extremum) at some value of z, say x0, within the circle of convergence. And, in view of the largeness of the numbers N and N U, this minimum may indeed be very steep! Thus, at z = x0 the first derivative of the integrand must vanish, while the second derivative must be positive and, hopefully, very large. Accordingly, if we proceed through the point z = x0 in a direction orthogonal to the real axis, the integrand must exhibit an equally steep maximum.5 Thus, in the complex z-plane, as we move along the real axis our integrand shows a minimum at z = x0, whereas if we move along a path parallel to the imaginary axis but passing through the point z = x0, the integrand shows a maximum there. It is natural to call the point x0 a saddle point; see Figure 3.2. For the contour of integration we choose a circle, with center at z = 0 and radius equal to x0, hoping that on integration along this contour only the immediate neighborhood of the sharp maximum at the point x0 will make the most dominant contribution to the value of the integral.6 To carry out the integration we first locate the point x0. For this we write our integrand as [f (z)]N zN U+1 = exp[N g(z)], (20) where g(z) = lnf (z) −  U + 1 N  lnz, (21) Saddle point Re z 0 Im z Contour of integration x0 exp{ g(z)} FIGURE 3.2 The saddle point. 5This can be seen by noting that (i) an analytic function must possess a unique derivative everywhere (so, in our case, it must be zero, irrespective of the direction in which we pass through the point x0), and (ii) by the Cauchy–Riemann conditions of analyticity, the second derivative of the function with respect to y must be equal and opposite to the second derivative with respect to x. 6It is indeed true that, for large N , the contribution from the rest of the circle is negligible. The intuitive reason for this is that the terms (ωrzEr ), which constitute the function f (z), “reinforce” one another only at the point z = x0; elsewhere, there is bound to be disagreement among their phases, so that at all other points along the circle, |f (z)| < f (x0). Now, the factor that actually governs the relative contributions is [|f (z)|/f (x0)]N ; for N ≫1, this will clearly be negligible. For a rigorous demonstration of this point, see Schr¨ odinger (1960, pp. 31–33). 3.2 A system in the canonical ensemble 47 while f (z) = X r ωrzEr. (22) The number x0 is then determined by the equation g′(x0) = f ′(x0) f (x0) −N U + 1 N x0 = 0, (23) which, in view of the fact that N U ≫1, can be written as U ≈x0 f ′(x0) f (x0) = P r ωrErxEr 0 P r ωrxEr 0 . (24) We further have g′′(x0) = f ′′(x0) f (x0) −[f ′(x0)]2 [f (x0)]2 ! + N U + 1 N x2 0 ≈f ′′(x0) f (x0) −U2 −U x2 0 . (25) It will be noted here that, in the limit N →∞and E(≡N U) →∞, with U staying constant, the number x0 and the quantity g′′(x0) become independent of N . Expanding g(z) about the point z = x0, along the direction of integration, that is, along the line z = x0 + iy, we have g(z) = g(x0) −1 2g′′(x0)y2 + ··· ; accordingly, the integrand (20) might be approximated as [f (x0)]N xN U+1 0 exp  −N 2 g′′(x0)y2  . (26) Equation (19) then gives 0(N ,U) ≃ 1 2πi [f (x0)]N xN U+1 0 ∞ Z −∞ exp  −N 2 g′′(x0)y2  idy = [f (x0)]N xN U+1 0 · 1 {2πN g′′(x0)}1/2 , (27) which gives 1 N ln0(N ,U) = {lnf (x0) −U lnx0} −1 N lnx0 − 1 2N ln{2πN g′′(x0)}. (28) 48 Chapter 3. The Canonical Ensemble In the limit N →∞(with U staying constant), the last two terms in this expression tend to zero, with the result 1 N ln0(N ,U) = lnf (x0) −U lnx0. (29) Substituting for f (x0) and introducing a new variable β, defined by the relationship x0 ≡exp(−β), (30) we get 1 N ln0(N ,U) = ln (X r ωr exp(−βEr) ) + βU. (31) The expectation value of the number nr then follows from (14) and (31): ⟨nr⟩ N =  ωr exp(−βEr) P r ωr exp(−βEr) +   − P r ωrEr exp(−βEr) P r ωr exp(−βEr) + U   ωr ∂β ∂ωr   all ωr=1 . (32) The term inside the curly brackets vanishes identically because of (24) and (30). It has been included here to emphasize the fact that, for a fixed value of U, the number β(≡−lnx0) in fact depends on the choice of the ωr; see (24). We will appreciate the importance of this fact when we evaluate the mean square fluctuation in the number nr; in the calculation of the expectation value of nr, this does not really matter. We thus obtain ⟨nr⟩ N = exp(−βEr) P r exp(−βEr), (33) which is identical to expression (10) for n∗ r/N . The physical significance of the parameter β is also the same as in that expression, for it is determined by equation (24), with all ωr = 1, that is, by equation (11) which fits naturally with equation (33) because U is nothing but the ensemble average of the variable Er: U = X r ErPr = 1 N X r Er⟨nr⟩. (34) Finally, we compute fluctuations in the values of the numbers nr. We have, first of all, ⟨n2 r ⟩≡ P {nr} ′n2 r W{nr} P {nr} ′W{nr} = 1 0  ωr ∂ ∂ωr  ωr ∂ ∂ωr  0 all ωr=1 ; (35) see equations (12) to (14). It follows that ⟨(1nr)2⟩≡⟨{nr −⟨nr⟩}2⟩= ⟨n2 r ⟩−⟨nr⟩2 =  ωr ∂ ∂ωr  ωr ∂ ∂ωr  ln0 all ωr=1 . (36) 3.2 A system in the canonical ensemble 49 Substituting from (31) and making use of (32), we get ⟨(1nr)2⟩ N = ωr ∂ ∂ωr  ωr exp(−βEr) P r ωr exp(−βEr) +   − P r ωrEr exp(−βEr) P r ωr exp(−βEr) + U   ωr ∂β ∂ωr   all ωr=1 . (37) We note that the term in the curly brackets would not make any contribution because it is identically zero, whatever the choice of the ωr. However, in the differentiation of the first term, we must not forget to take into account the implicit dependence of β on the ωr, which arises from the fact that unless the ωr are set equal to unity the relation determining β does contain ωr; see equations (24) and (30), whereby U = P r ωrEr exp(−βEr) P r ωr exp(−βEr) all ωr=1 . (38) A straightforward calculation gives  ∂β ∂ωr  U all ωr=1 = Er −U ⟨E2 r ⟩−U2 ⟨nr⟩ N . (39) We can now evaluate (37), with the result ⟨(1nr)2⟩ N = ⟨nr⟩ N − ⟨nr⟩ N 2 + ⟨nr⟩ N (U −Er)  ∂β ∂ωr  U all ωr=1 = ⟨nr⟩ N " 1 −⟨nr⟩ N −⟨nr⟩ N (Er −U)2 ⟨(Er −U)2⟩ # . (40) For the relative fluctuation in nr, we get 1nr ⟨nr⟩ 2+ = 1 ⟨nr⟩−1 N ( 1 + (Er −U)2 ⟨(Er −U)2⟩ ) . (41) As N →∞,⟨nr⟩also →∞, with the result that the relative fluctuations in nr tend to zero; accordingly, the canonical distribution becomes infinitely sharp and with it the mean value, the most probable value — in fact, any values of nr that appear with nonvanish-ing probability — become essentially the same. And that is the reason why two wildly different methods of obtaining the canonical distribution followed in this section have led to identical results. 50 Chapter 3. The Canonical Ensemble 3.3 Physical significance of the various statistical quantities in the canonical ensemble We start with the canonical distribution Pr ≡⟨nr⟩ N = exp(−βEr) P r exp(−βEr), (1) where β is determined by the equation U = P r Er exp(−βEr) P r exp(−βEr) = −∂ ∂β ln (X r exp(−βEr) ) . (2) We now look for a general recipe to extract information about the various macroscopic properties of the given system on the basis of the foregoing statistical results. For this, we recall certain thermodynamic relationships involving the Helmholtz free energy A(= U −TS), namely dA = dU −TdS −SdT = −SdT −PdV + µdN, (3) S = −  ∂A ∂T  N,V , P = −  ∂A ∂V  N,T , µ =  ∂A ∂N  V,T , (4) and U = A + TS = A −T  ∂A ∂T  N,V = −T2  ∂ ∂T  A T  N,V = ∂(A/T) ∂(1/T)  N,V , (5) where the various symbols have their usual meanings. Comparing (5) with (2), we infer that there exists a close correspondence between the quantities entering through the statistical treatment and the ones coming from thermodynamics, namely β = 1 kT , ln (X r exp(−βEr) ) = −A kT , (6) where k is a universal constant yet to be determined; soon we shall see that k is indeed the Boltzmann constant. The equations in (6) constitute the most fundamental result of the canonical ensemble theory. Customarily, we write it in the form A(N,V,T) = −kT lnQN(V,T), (7) where QN(V,T) = X r exp(−Er/kT). (8) 3.3 Physical significance of the various statistical quantities 51 The quantity QN(V,T) is referred to as the partition function of the system; sometimes it is also called the “sum-over-states” (German: Zustandssumme). The dependence of Q on T is quite obvious. The dependence on N and V comes through the energy eigenvalues Er; in fact, any other parameters that might govern the values Er should also appear in the argument of Q. Moreover, for the quantity A(N,V,T) to be an extensive property of the system, lnQ must also be an extensive quantity. Once the Helmholtz free energy is known, the rest of the thermodynamic quantities follow straightforwardly. While the entropy, the pressure and the chemical potential are obtained from formulae (4), the specific heat at constant volume follows from CV = ∂U ∂T  N,V = −T ∂2A ∂T2 ! N,V (9) and the Gibbs free energy from G = A + PV = A −V  ∂A ∂V  N,T = N  ∂A ∂N  V,T = Nµ; (10) see Problem 3.5. At this stage it appears worthwhile to make a few remarks on the foregoing results. First of all, we note from equations (4) and (6) that the pressure P is given by P = − P r ∂Er ∂V exp(−βEr) P r exp(−βEr) , (11) so that PdV = − X r PrdEr = −dU. (12) The quantity on the right side of this equation is clearly the change in the average energy of a system (in the ensemble) during a process that alters the energy levels Er, leaving the probabilities Pr unchanged. The left side then tells us that the volume change dV provides an example of such a process, and the pressure P is the “force” accompanying that process. The quantity P, which was introduced here through the thermodynamic relationship (3), thus acquires a mechanical meaning as well. The entropy of the system is determined as follows. Since Pr = Q−1 exp(−βEr), ⟨lnPr⟩= −lnQ −β⟨Er⟩= β(A −U) = −S/k, with the result that S = −k⟨lnPr⟩= −k X r Pr lnPr. (13) 52 Chapter 3. The Canonical Ensemble This is an extremely interesting relationship, for it shows that the entropy of a physical sys-tem is solely and completely determined by the probability values Pr (of the system being in different dynamical states accessible to it)! From the very look of it, equation (13) appears to be of fundamental importance; indeed, it reveals a number of interesting conclusions. One of these relates to a system in its ground state (T = 0K). If the ground state is unique, then the system is sure to be found in this particular state and in no other; consequently, Pr is equal to 1 for this state and 0 for all others. Equation (13) then tells us that the entropy of the system is precisely zero, which is essentially the content of the Nernst heat theorem or the third law of ther-modynamics.7 We also infer that vanishing entropy and perfect statistical order (which implies complete predictability about the system) go together. As the number of acces-sible states increases, more and more of the Pr become nonzero; the entropy of the system thereby increases. As the number of states becomes exceedingly large, most of the P-values become exceedingly small (and their logarithms assume large, negative values); the net result is that the entropy becomes exceedingly large. Thus, the largeness of entropy and the high degree of statistical disorder (or unpredictability) in the system also go hand in hand. It is because of this fundamental connection between entropy on one hand and lack of information on the other that equation (13) became the starting point of the pioneering work of Shannon (1948, 1949) in the development of the theory of communication. It may be pointed out that formula (13) applies in the microcanonical ensemble as well. There, for each member system of the ensemble, we have a group of  states, all equally likely to occur. The value of Pr is, then, 1/ for each of these states and 0 for all others. Consequently, S = −k  X r=1  1  ln  1   = kln, (14) which is precisely the central result in the microcanonical ensemble theory; see equa-tion (1.2.6) or (2.3.6). 3.4 Alternative expressions for the partition function In most physical cases the energy levels accessible to a system are degenerate, that is, one has a group of states, gi in number, all belonging to the same energy value Ei. In such cases it is more useful to write the partition function (3.3.8) as QN(V,T) = X i gi exp(−βEi); (1) 7Of course, if the ground state of the system is degenerate (with a multiplicity 0), then the ground-state entropy is nonzero and is given by the expression k ln0; see equation (14). 3.4 Alternative expressions for the partition function 53 the corresponding expression for Pi, the probability that the system be in a state with energy Ei, would be Pi = gi exp(−βEi) P i gi exp(−βEi). (2) Clearly, the gi states with a common energy Ei are all equally likely to occur. As a result, the probability of a system having energy Ei becomes proportional to the multiplicity gi of this level; gi thus plays the role of a “weight factor” for the level Ei. The actual probability is then determined by the weight factor gi as well as by the Boltzmann factor exp(−βEi) of the level, as we have in (2). The basic relations established in the preceding section, however, remain unaffected. Now, in view of the largeness of the number of particles constituting a given system and the largeness of the volume to which these particles are confined, the consecutive energy values Ei of the system are, in general, very close to one another. Accordingly, there lie, within any reasonable interval of energy (E,E + dE), a very large number of energy levels. One may then regard E as a continuous variable and write P(E)dE for the probability that the given system, as a member of the canonical ensemble, may have its energy in the range (E,E + dE). Clearly, this probability will be given by the product of the relevant single-state probability and the number of energy states lying in the specified range. Denoting the latter by g(E)dE, where g(E) denotes the density of states around the energy value E, we have P(E)dE ∝exp(−βE)g(E)dE (3) which, on normalization, becomes P(E)dE = exp(−βE)g(E)dE ∞ R 0 exp(−βE)g(E)dE . (4) The denominator here is yet another expression for the partition function of the system: QN(V,T) = ∞ Z 0 e−βEg(E)dE. (5) The expression for ⟨f ⟩, the expectation value of a physical quantity f , may now be written as ⟨f ⟩≡ X i fiPi = P i f (Ei)gie−βEi P i gie−βEi → ∞ R 0 f (E)e−βEg(E)dE ∞ R 0 e−βEg(E)dE . (6) 54 Chapter 3. The Canonical Ensemble Before proceeding further, we take a closer look at equation (5) and note that, with β > 0, the partition function Q(β) is just the Laplace transform of the density of states g(E). We may, therefore, write g(E) as the inverse Laplace transform of Q(β): g(E) = 1 2πi β′+i∞ Z β′−i∞ eβEQ(β)dβ (β′ > 0) (7) = 1 2π ∞ Z −∞ e(β′+iβ′′)EQ(β′ + iβ′′)dβ′′, (8) where β is now treated as a complex variable, β′ + iβ′′, while the path of integration runs parallel to, and to the right of, the imaginary axis, that is, along the straight line Re β = β′ > 0. Of course, the path may be continuously deformed so long as the integral converges. 3.5 The classical systems The theory developed in the preceding sections is of very general applicability. It applies to systems in which quantum-mechanical effects are important as well as to those that can be treated classically. In the latter case, our formalism may be written in the language of the phase space; as a result, the summations over quantum states get replaced by integrations over phase space. We recall the concepts developed in Sections 2.1 and 2.2, especially formula (2.1.3) for the ensemble average ⟨f ⟩of a physical quantity f (q,p), namely ⟨f ⟩= R f (q,p)ρ(q,p)d3Nqd3Np R ρ(q,p)d3Nqd3Np , (1) where ρ(q,p) denotes the density of the representative points (of the systems) in the phase space; we have omitted here the explicit dependence of the function ρ on time t because we are interested in the study of equilibrium situations only. Evidently, the function ρ(q,p) is a measure of the probability of finding a representative point in the vicinity of the phase point (q,p), which in turn depends on the corresponding value H(q,p) of the Hamiltonian of the system. In the canonical ensemble, ρ(q,p) ∝exp{−βH(q,p)}; (2) compare to equation (3.1.6). The expression for ⟨f ⟩then takes the form ⟨f ⟩= R f (q,p)exp(−βH)dω R exp(−βH)dω , (3) 3.5 The classical systems 55 where dω(≡d3Nqd3Np) denotes a volume element of the phase space. The denomina-tor of this expression is directly related to the partition function of the system. However, to write the precise expression for the latter, we must take into account the relationship between a volume element in the phase space and the corresponding number of distinct quantum states of the system. This relationship was established in Sections 2.4 and 2.5, whereby an element of volume dω in the phase space corresponds to dω N!h3N (4) distinct quantum states of the system.8 The appropriate expression for the partition function would, therefore, be QN(V,T) = 1 N!h3N Z e−βH(q,p)dω; (5) it is understood that the integration in (5) goes over the whole of the phase space. As our first application of this formulation, we consider the example of an ideal gas. Here, we have a system of N identical molecules, assumed to be monatomic (so there are no internal degrees of motion to be considered), confined to a space of volume V and in equilibrium at temperature T. Since there are no intermolecular interactions to be taken into account, the energy of the system is wholly kinetic: H(q,p) = N X i=1 (p2 i /2m). (6) The partition function of the system would then be QN(V,T ) = 1 N!h3N Z e−(β/2m)6ip2 i N Y i=1 (d3qid3pi). (7) Integrations over the space coordinates are rather trivial; they yield a factor of V N. Integra-tions over the momentum coordinates are also quite easy, once we note that integral (7) is simply a product of N identical integrals. Thus, we get QN(V,T) = V N N!h3N   ∞ Z 0 e−p2/2mkT  4πp2dp    N (8) = 1 N!  V h3 (2πmkT)3/2 N ; (9) 8Ample justification has already been given for the factor h3N. The factor N! comes from the considerations of Sections 1.5 and 1.6; it arises essentially from the fact that the particles constituting the given system are not only identical but, in fact, indistinguishable. For a complete proof of this result, see Section 5.5. 56 Chapter 3. The Canonical Ensemble here, use has been made of equation (B.13a). The Helmholtz free energy is then given by, using Stirling’s formula (B.29), A(N,V,T) ≡−kT lnQN(V,T) = NkT  ln    N V h2 2πmkT !3/2  −1  . (10) The foregoing result is identical to equation (1.5.8), which was obtained by following a very different procedure. The simplicity of the present approach is, however, striking. Needless to say, the complete thermodynamics of the ideal gas can be derived from equation (10) in a straightforward manner. For instance, µ ≡  ∂A ∂N  V,T = kT ln    N V h2 2πmkT !3/2  , (11) P ≡−  ∂A ∂V  N,T = NkT V (12) and S ≡−  ∂A ∂T  N,V = Nk " ln ( V N 2πmkT h2 3/2) + 5 2 # . (13) These results are identical to the ones derived previously, namely (1.5.7), (1.4.2), and (1.5.1a), respectively. In fact, the identification of formula (12) with the ideal-gas law, PV = nRT, establishes the identity of the (hitherto undetermined) constant k as the Boltzmann constant; see equation (3.3.6). We further obtain U ≡−  ∂ ∂β (lnQ)  Er ≡−T2  ∂ ∂T  A T  N,V ≡A + TS = 3 2NkT, (14) and so on. At this stage we have an important remark to make. Looking at the form of equation (8) and the manner in which it came about, we may write QN(V,T) = 1 N![Q1(V,T)]N, (15) where Q1(V,T) may be regarded as the partition function of a single molecule in the sys-tem. A little reflection will show that this result obtains essentially from the fact that the basic constituents of our system are noninteracting (and hence the total energy of the system is simply the sum of their individual energies). Clearly, the situation will not be altered even if the molecules in the system had internal degrees of motion as well. What is essentially required for equation (15) to be valid is the absence of interactions among the basic constituents of the system (and, of course, the absence of quantum-mechanical correlations). 3.5 The classical systems 57 Going back to the ideal gas, we could as well have started with the density of states g(E). From equation (1.4.17), and in view of the Gibbs correction factor, we have g(E) = ∂6 ∂E ≈1 N!  V h3 N (2πm)3N/2 {(3N/2) −1}!E(3N/2)−1. (16) Substituting this into equation (3.4.5), and noting that the integral involved is equal to {(3N/2) −1}!/β3N/2, we obtain QN(β) = 1 N!  V h3 N 2πm β 3N/2 , (17) which is identical to (9). It may also be noted that if one starts with the single-particle density of states (2.4.7), namely a(ε) ≈2πV h3 (2m)3/2ε1/2, (18) computes the single-particle partition function, Q1(β) = ∞ Z 0 e−βεa(ε)dε = V h3 2πm β 3/2 , (19) and then makes use of formula (15), one would arrive at the same result for QN(V,T). Lastly, we consider the question of determining the density of states, g(E), from the expression for the partition function, Q(β) — assuming that the latter is already known; indeed, expression (9) for Q(β) was derived without making use of any knowledge regarding the function g(E). According to equation (3.4.7) and (9), we have g(E) = V N N! 2πm h2 3N/2 1 2πi β′+i∞ Z β′−i∞ eβE β3N/2 dβ (β′ > 0). (20) Noting that, for all positive n, 1 2πi s′+i∞ Z s′−i∞ esx sn+1 ds =    xn n! for x ≥0 0 for x ≤0, (21)9 equation (20) becomes g(E) =      V N N! 2πm h2 3N/2 E(3N/2)−1 {(3N/2) −1}! for E ≥0 0 for E ≤0, (22) 9For the details of this evaluation, see Kubo (1965, pp. 165–168). 58 Chapter 3. The Canonical Ensemble which is indeed the correct result for the density of states of an ideal gas; compare to equation (16). The foregoing derivation may not appear particularly valuable because in the present case we already knew the expression for g(E). However, cases do arise where the evaluation of the partition function of a given system and the consequent evaluation of its density of states turn out to be quite simple, whereas a direct evaluation of the density of states from first principles is rather involved. In such cases, the method given here can indeed be useful; see, for example, Problem 3.15 in comparison with Problems 1.7 and 2.8. 3.6 Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble In the canonical ensemble, a system can have energy anywhere between zero and infinity. On the other hand, the energy of a system in the microcanonical ensemble is restricted to a very narrow range. How, then, can we assert that the thermodynamic properties of a system derived through the formalism of the canonical ensemble would be the same as the ones derived through the formalism of the microcanonical ensemble? Of course, we do expect that the two formalisms yield identical results, for otherwise our whole scheme would be marred by internal inconsistency. And, indeed, in the case of an ideal classical gas the results obtained by following one approach were precisely the same as the ones obtained by following the other approach. What, then, is the underlying reason for this equivalence? The answer to this question is obtained by examining the extent of the range over which the energies of the systems in the canonical ensemble have a significant probability to spread; that will tell us the extent to which the canonical ensemble really differs from the microcanonical one. To explore this point, we write down the expression for the mean energy U ≡⟨E⟩= P r Er exp(−βEr) P r exp(−βEr) (1) and differentiate it with respect to the parameter β, holding the energy values Er constant. We obtain ∂U ∂β = − P r E2 r exp(−βEr) P r exp(−βEr) + P r Er exp(−βEr) 2 P r exp(−βEr) 2 = −⟨E2⟩+ ⟨E⟩2, (2) from which it follows that ⟨(1E)2⟩≡⟨E2⟩−⟨E⟩2 = − ∂U ∂β  = kT2 ∂U ∂T  = kT2CV . (3) 3.6 Energy fluctuations in the canonical ensemble 59 Note that we have here the specific heat at constant volume, because the partial differen-tiation in (2) was carried out with the Er kept constant! For the relative root-mean-square fluctuation in E, equation (3) gives √[⟨(1E)2⟩] ⟨E⟩ = √(kT2CV ) U , (4) which is O(N−1/2), N being the number of particles in the system. Consequently, for large N (which is true for every statistical system) the relative r.m.s. fluctuation in the values of E is quite negligible! Thus, for all practical purposes, a system in the canonical ensemble has an energy equal to, or almost equal to, the mean energy U; the situation in this ensemble is, therefore, practically the same as in the microcanonical ensemble. That explains why the two ensembles lead to practically identical results. For further understanding of the situation, we consider the manner in which energy is distributed among the various members of the (canonical) ensemble. To do this, we treat E as a continuous variable and start with expression (3.4.3), namely P(E)dE ∝exp(−βE)g(E)dE. (3.4.3) The probability density P(E) is given by the product of two factors: (i) the Boltzmann factor, which monotonically decreases with E, and (ii) the density of states, which monotonically increases with E. The product, therefore, has an extremum at some value of E, say E∗.10 The value E∗is determined by the condition ∂ ∂E {e−βEg(E)} E=E∗= 0, that is, by ∂lng(E) ∂E E=E∗= β. (5) Recalling that S = klng and ∂S(E) ∂E  E=U = 1 T = kβ, the foregoing condition implies that E∗= U. (6) This is a very interesting result, for it shows that, irrespective of the physical nature of the given system, the most probable value of its energy is identical to its mean value. Accordingly, if it is advantageous, we may use one instead of the other. 10Subsequently we shall see that this extremum is actually a maximum — and an extremely sharp one at that. 60 Chapter 3. The Canonical Ensemble We now expand the logarithm of the probability density P(E) around the value E∗≈U; we get ln h e−βEg(E) i =  −βU + S k  + 1 2 ∂2 ∂E2 ln n e−βEg(E) o E=U (E −U)2 + ··· = −β(U −TS) − 1 2kT2CV (E −U)2 + ··· , (7) from which we obtain P(E) ∝e−βEg(E) ≃e−β(U−TS) exp ( −(E −U)2 2kT2CV ) . (8) This is a Gaussian distribution in E, with mean value U and dispersion √(kT2CV ); compare with equation (3). In terms of the reduced variable E/U, the distribution is again Gaussian, with mean value unity and dispersion √(kT2CV )/U {which is O(N−1/2)}; thus, for N ≫1, we have an extremely sharp distribution which, as N →∞, approaches a delta-function! It would be instructive here to consider once again the case of a classical ideal gas. Here, g(E) is proportional to E(3N/2−1) and hence increases very fast with E; the factor e−βE, of course, decreases with E. The product g(E)exp(−βE) exhibits a maximum at E∗= (3N/2 −1)β−1, which is practically the same as the mean value U = (3N/2)β−1. For values of E significantly different from E∗, the product essentially vanishes (for smaller val-ues of E, due to the relative paucity of the available energy states; for larger values of E, due to the relative depletion caused by the Boltzmann factor). The overall picture is shown in Figure 3.3 where we have displayed the actual behavior of these functions in the special case N = 10. The most probable value of E is now 14 15 of the mean value; so, the distribution is somewhat asymmetrical. The effective width 1 can be readily calculated from (3) and turns out to be (2/3N)1/2U, which, for N = 10, is about a quarter of U. We can see that, as N becomes large, both E∗and U increase (essentially linearly with N), the ratio E∗/U approaches unity and the distribution tends to become symmetrical about E∗. At the same time, the width 1 increases (but only as N1/2); considered in the relative sense, it tends to vanish (as N−1/2). We finally look at the partition function QN(V,T), as given by equation (3.4.5), with its integrand replaced by (8). We have QN(V,T) ≃e−β(U−TS) ∞ Z 0 e−(E−U)2/2kT2CV dE ≃e−β(U−TS)√(2kT2CV ) ∞ Z −∞ e−x2dx = e−β(U−TS)√(2πkT2CV ), 3.7 Two theorems — the “equipartition” and the “virial” 61 g(E ) 0.5 0 E U E 1.0 g(E )e–E e–E FIGURE 3.3 The actual behavior of the functions g(E), e−βE, and g(E)e−βE for an ideal gas, with N = 10. The numerical values of the functions have been expressed as fractions of their respective values at E = U. so that −kT lnQN(V,T) ≡A ≃(U −TS) −1 2kT ln(2πkT2CV ). (9) The last term, being O(lnN), is negligible in comparison with the other terms, which are all O(N). Hence, A ≈U −TS. (10) Note that the quantity A in this formula has come through the formalism of the canonical ensemble, while the quantity S has come through a definition belonging to the microcanonical ensemble. The fact that we finally end up with a consistent thermo-dynamic relationship establishes beyond doubt that these two approaches are, for all practical purposes, identical. 3.7 Two theorems — the “equipartition” and the “virial” To derive these theorems, we determine the expectation value of the quantity xi(∂H/∂xj), where H(q,p) is the Hamiltonian of the system (assumed classical) while xi 62 Chapter 3. The Canonical Ensemble and xj are any two of the 6N generalized coordinates (q,p). In the canonical ensemble, xi ∂H ∂xj + = R  xi ∂H ∂xj  e−βHdω R e−βHdω  dω = d3Nqd3Np  . (1) Let us consider the integral in the numerator. Integrating over xj by parts, it becomes Z  −1 β xie−βH (xj)2 (xj)1 + 1 β Z ∂xi ∂xj ! e−βHdxj # dω(j); here, (xj)1 and (xj)2 are the “extreme” values of the coordinate xj, while dω(j) denotes “dω devoid of dxj.” The integrated part here vanishes because whenever any of the coordinates takes an “extreme” value the Hamiltonian of the system becomes infinite.11 In the integral that remains, the factor ∂xi/∂xj, being equal to δij, comes out of the integral sign and we are left with 1 β δij Z e−βHdω. Substituting this into (1), we arrive at the remarkable result: xi ∂H ∂xj + = δijkT, (2) which is independent of the precise form of the function H. In the special case xi = xj = pi, equation (2) takes the form pi ∂H ∂pi ≡⟨pi ˙ qi⟩= kT, (3) while for xi = xj = qi, it becomes qi ∂H ∂qi ≡−⟨qi ˙ pi⟩= kT. (4) Adding over all i, from i = 1 to 3N, we obtain X i pi ∂H ∂pi + ≡ X i pi ˙ qi + = 3NkT (5) 11For instance, if xj is a space coordinate, then its extreme values will correspond to “locations at the walls of the con-tainer”; accordingly, the potential energy of the system would become infinite. If, on the other hand, xj is a momentum coordinate, then its extreme values will themselves be ±∞, in which case the kinetic energy of the system would become infinite. 3.7 Two theorems — the “equipartition” and the “virial” 63 and X i qi ∂H ∂qi + ≡− X i qi ˙ pi + = 3NkT. (6) Now, in many physical situations the Hamiltonian of the system happens to be a quadratic function of its coordinates; so, through a canonical transformation, it can be brought into the form H = X j AjP2 j + X j BjQ2 j , (7) where Pj and Qj are the transformed, canonically conjugate, coordinates while Aj and Bj are certain constants of the problem. For such a system, we clearly have X j Pj ∂H ∂Pj + Qj ∂H ∂Qj ! = 2H; (8) accordingly, by equations (3) and (4), ⟨H⟩= 1 2fkT, (9) where f is the number of nonvanishing coefficients in expression (7). We, therefore, con-clude that each harmonic term in the (transformed) Hamiltonian makes a contribution of 1 2kT toward the internal energy of the system and, hence, a contribution of 1 2k toward the specific heat CV . This result embodies the classical theorem of equipartition of energy (among the various degrees of freedom of the system). It may be mentioned here that, for the distribution of kinetic energy alone, the equipartition theorem was first stated by Boltzmann (1871). In our subsequent study we shall find that the equipartition theorem as stated here is not always valid; it applies only when the relevant degrees of freedom can be freely excited. At a given temperature T, there may be certain degrees of freedom which, due to the insuf-ficiency of the energy available, are more or less “frozen” due to quantum mechanical effects. Such degrees of freedom do not make a significant contribution toward the inter-nal energy of the system or toward its specific heat; see, for example, Sections 6.5, 7.4, and 8.3. Of course, the higher the temperature of the system the better the validity of this theorem. We now consider the implications of formula (6). First of all, we note that this formula embodies the so-called virial theorem of Clausius (1870) for the quantity ⟨P i qi ˙ pi⟩, which is the expectation value of the sum of the products of the coordinates of the various particles in the system and the respective forces acting on them; this quantity is generally referred to as the virial of the system and is denoted by the symbol V. The virial theorem then states 64 Chapter 3. The Canonical Ensemble that V = −3NkT. (10) The relationship between the virial and other physical quantities of the system is best understood by first looking at a classical gas of noninteracting particles. In this case, the only forces that come into play are the ones arising from the walls of the container; these forces can be designated by an external pressure P that acts on the system by virtue of the fact that it is bounded by the walls of the container. Consequently, we have here a force −PdS associated with an element of area dS of the walls; the negative sign appears because the force is directed inward while the vector dS is directed outward. The virial of the gas is then given by V0 = X i qiFi ! 0 = −P I S r · dS, (11)12 where r is the position vector of a particle that happens to be in the (close) vicinity of the surface element dS; accordingly, r may be considered to be the position vector of the surface element itself. By the divergence theorem, equation (11) becomes V0 = −P Z V (div r)dV = −3PV. (12) Comparing (12) with (10), we obtain the well-known result: PV = NkT. (13) The internal energy of the gas, which in this case is wholly kinetic, follows from the equipartition theorem (9) and is equal to 3 2NkT, 3N being the number of degrees of freedom. Comparing this result with (10), we obtain the classical relationship V = −2K, (14) where K denotes the average kinetic energy of the system. It is straightforward to apply this theorem to a system of particles interacting through a two-body potential u(r). In the thermodynamic limit, the pressure of a d-dimensional system depends only on the virial terms arising from the forces between pairs of particles: P nkT = 1 + 1 NdkT X i<j F(rij) · rij + = 1 − 1 NdkT X i 0), that is, g(E) =        1 (ℏω)N EN−1 (N −1)! for E ≥0 0 for E ≤0. (10) To test the correctness of (10), we may calculate the entropy of the system with the help of this formula. Taking N ≫1 and making use of the Stirling approximation, we get S(N,E) = k lng(E) ≈Nk  ln  E Nℏω  + 1  , (11) which gives for the temperature of the system T =  ∂S ∂E −1 N = E Nk . (12) Eliminating E between these two relations, we obtain precisely our earlier result (7) for the function S(N,T). 3.8 A system of harmonic oscillators 67 We now take up the quantum-mechanical situation, according to which the energy eigenvalues of a one-dimensional harmonic oscillator are given by εn =  n + 1 2  ℏω; n = 0,1,2,... (13) Accordingly, we have for the single-oscillator partition function Q1(β) = ∞ X n=0 e−β(n+1/2)ℏω = exp  −1 2βℏω  1 −exp(−βℏω) =  2sinh 1 2βℏω −1 . (14) The N-oscillator partition function is then given by QN(β) = [Q1(β)]N =  2sinh 1 2βℏω −N = e−(N/2)βℏω{1 −e−βℏω}−N. (15) For the Helmholtz free energy of the system, we get A = NkT ln  2sinh 1 2βℏω  = N 1 2ℏω + kT ln{1 −e−βℏω}  , (16) whereby µ = A/N, (17) P = 0, (18) S = Nk 1 2βℏωcoth 1 2βℏω  −ln  2sinh 1 2βℏω  = Nk  βℏω eβℏω −1 −ln{1 −e−βℏω}  , (19) U = 1 2Nℏωcoth 1 2βℏω  = N 1 2ℏω + ℏω eβℏω −1  , (20) and CP = CV = Nk 1 2βℏω 2 cosech2 1 2βℏω  = Nk(βℏω)2 eβℏω (eβℏω −1)2 . (21) Formula (20) is especially significant, for it shows that the quantum-mechanical oscil-lators do not obey the equipartition theorem. The mean energy per oscillator is different 68 Chapter 3. The Canonical Ensemble 2 1 00 3 1 2 1 2 kT/ ε FIGURE 3.4 The mean energy ⟨ε⟩of a simple harmonic oscillator as a function of temperature. 1, the Planck oscillator; 2, the Schr ¨ odinger oscillator; and 3, the classical oscillator. from the equipartition value kT; actually, it is always greater than kT; see curve 2 in Figure 3.4. Only in the limit of high temperatures, where the thermal energy kT is much larger than the energy quantum ℏω, does the mean energy per oscillator tend to the equipartition value. It should be noted here that if the zero-point energy 1 2ℏω were not present, the limiting value of the mean energy would be (kT −1 2ℏω), and not kT — we may call such an oscillator the Planck oscillator; see curve 1 in Figure 3.4. In passing, we observe that the specific heat (21), which is the same for the Planck oscillator as for the Schr¨ odinger oscillator, is temperature-dependent; moreover, it is always less than, and at high temperatures tends to, the classical value (9). Indeed, for kT ≫ℏω, formulae (14) through (21) go over to their classical counterparts, namely (2) through (9), respectively. We shall now determine the density of states g(E) of the N-oscillator system from its partition function (15). Carrying out the binomial expansion of this expression, we have QN(β) = ∞ X R=0 N + R −1 R ! e−β( 1 2 Nℏω+Rℏω). (22) Comparing this with the formula QN(β) = ∞ Z 0 g(E)e−βEdE, we conclude that g(E) = ∞ X R=0 N + R −1 R ! δ  E −  R + 1 2N  ℏω  , (23) 3.8 A system of harmonic oscillators 69 where δ(x) denotes the Dirac delta function. Equation (23) implies that there are (N + R −1)!/R!(N −1)! microstates available to the system when its energy E has the dis-crete value (R + 1 2N)ℏω, where R = 0,1,2,..., and that no microstate is available for other values of E. This is hardly surprising, but it is instructive to look at this result from a slightly different point of view. We consider the following problem that arises naturally in the microcanonical ensem-ble theory. Given an energy E for distribution among a set of N harmonic oscillators, each of which can be in any one of the eigenstates (13), what is the total number of distinct ways in which the process of distribution can be carried out? Now, in view of the form of the eigenvalues εn, it makes sense to give away, right in the beginning, the zero-point energy 1 2ℏω to each of the N oscillators and convert the rest of it into quanta (of energy ℏω). Let R be the number of these quanta; then R =  E −1 2Nℏω  ℏω. (24) Clearly, R must be an integer; by implication, E must be of the form (R + 1 2N)ℏω. The prob-lem then reduces to determining the number of distinct ways of allotting R quanta to N oscillators, such that an oscillator may have 0 or 1 or 2... quanta; in other words, we have to determine the number of distinct ways of putting R indistinguishable balls into N dis-tinguishable boxes, such that a box may receive 0 or 1 or 2...balls. A little reflection will show that this is precisely the number of permutations that can be realized by shuffling R balls, placed along a row, with (N −1) partitioning lines (that divide the given space into N boxes); see Figure 3.5. The answer clearly is (R + N −1)! R!(N −1)! , (25) which agrees with (23). We can now determine the entropy of the system from the number (25). Since N ≫1, we have S ≈k{ln(R + N)!−lnR!−lnN!} ≈k{(R + N)ln(R + N) −RlnR −N lnN}; (26) FIGURE 3.5 Distributing 17 indistinguishable balls among 7 distinguishable boxes. The arrangement shown here represents one of the 23!/17!6! distinct ways of carrying out the distribution. 70 Chapter 3. The Canonical Ensemble the number R is, of course, a measure of the energy E of the system; see (24). For the temperature of the system, we obtain 1 T =  ∂S ∂E  N =  ∂S ∂R  N 1 ℏω = k ℏω ln R + N R  = k ℏω ln E + 1 2Nℏω E −1 2Nℏω ! , (27) so that E N = 1 2ℏω exp(ℏω/kT) + 1 exp(ℏω/kT) −1, (28) which is identical to (20). It can be further checked that, by eliminating R between (26) and (27), we obtain precisely the formula (19) for S(N,T). Thus, once again, we find that the results obtained by following the microcanonical approach and the canonical approach are the same in the thermodynamic limit. Finally, we may consider the classical limit when E/N, the mean energy per oscillator, is much larger than the energy quantum ℏω, that is, when R ≫N. The expression (25) may, in that case, be replaced by (R + N −1)(R + N −2)...(R + 1) (N −1)! ≈ RN−1 (N −1)!, (25a) with R ≈E/ℏω. The corresponding expression for the entropy turns out to be S ≈k{N ln(R/N) + N} ≈Nk  ln  E Nℏω  + 1  , (26a) which gives 1 T =  ∂S ∂E  N ≈Nk E , (27a) so that E N ≈kT. (28a) These results are identical to the ones derived in the classical limit earlier in this section. 3.9 The statistics of paramagnetism Next, we study a system of N magnetic dipoles, each having a magnetic moment µ. In the presence of an external magnetic field H, the dipoles will experience a torque tending to 3.9 The statistics of paramagnetism 71 align them in the direction of the field. If there were nothing else to check this tendency, the dipoles would align themselves precisely in this direction and we would achieve a complete magnetization of the system. In reality, however, thermal agitation in the system offers resistance to this tendency and, in equilibrium, we obtain only a partial magnetization. Clearly, as T →0K, the thermal agitation becomes ineffective and the system exhibits a complete orientation of the dipole moments, whatever the strength of the applied field; at the other extreme, as T →∞, we approach a state of complete randomization of the dipole moments, which implies a vanishing magnetization. At intermediate temperatures, the situation is governed by the parameter (µH/kT). The model adopted for this study consists of N identical, localized (and, hence, dis-tinguishable), practically static, mutually noninteracting and freely orientable dipoles. We consider first the case of classical dipoles that can be oriented in any direction relative to the applied magnetic field. It is obvious that the only energy we need to consider here is the potential energy of the dipoles that arises from the presence of the external field H and is determined by the orientations of the dipoles with respect to the direction of the field: E = N X i=1 Ei = − N X i=1 µi ·H = −µH N X i=1 cosθi. (1) The partition function of the system is then given by QN(β) = [Q1(β)]N, (2) where Q1(β) = X θ exp(βµH cosθ). (3) The mean magnetic moment M of the system will obviously be in the direction of the field H; for its magnitude we shall have Mz = N⟨µcosθ⟩= N P θ µcosθ exp(βµH cosθ) P θ exp(βµH cosθ) = N β ∂ ∂H lnQ1(β) = −  ∂A ∂H  T . (4) Thus, to determine the degree of magnetization in the system all we have to do is to evaluate the single-dipole partition function (3). First, we proceed classically (after Langevin, 1905a,b). Using (sinθdθdφ) as the elemen-tal solid angle representing a small range of orientations of the dipole, we get Q1(β) = 2π Z 0 π Z 0 eβµH cosθ sinθdθdφ = 4π sinh(βµH) βµH , (5) 72 Chapter 3. The Canonical Ensemble so that µz ≡Mz N = µ  coth(βµH) − 1 βµH  = µL(βµH), (6) where L(x) is the so-called Langevin function L(x) = cothx −1 x; (7) a plot of the Langevin function is shown in Figure 3.6. We note that the parameter βµH denotes the strength of the (magnetic) potential energy µH compared to the (thermal) kinetic energy kT. If we have N0 dipoles per unit volume in the system, then the magnetization of the system, namely the mean magnetic moment per unit volume, is given by Mz0 = N0µz = N0µL(x) (x = βµH). (8) For magnetic fields so strong (or temperatures so low) that the parameter x ≫1, the function L(x) is almost equal to 1; the system then acquires a state of magnetic saturation: µz ≃µ and Mz0 ≃N0µ. (9) For temperatures so high (or magnetic fields so weak) that the parameter x ≪1, the function L(x) may be written as x 3 −x3 45 + ··· (10) which, in the lowest approximation, gives Mz0 ≃N0µ2 3kT H. (11) 1.0 0.5 00 4 8 12 x L(x) FIGURE 3.6 The Langevin function L(x). 3.9 The statistics of paramagnetism 73 The high-temperature isothermal susceptibility of the system is, therefore, given by χT = Lim H→0 ∂Mz0 ∂H  T ≃N0µ2 3kT = C T , say. (12) Equation (12) is the Curie law of paramagnetism, the parameter C being the Curie constant of the system. Figure 3.7 shows a plot of the susceptibility of a powdered sample of copper– potassium sulphate hexahydrate as a function of T−1; the fact that the plot is linear and passes almost through the origin vindicates the Curie law for this particular salt. We shall now treat the problem of paramagnetism quantum-mechanically. The major modification here arises from the fact that the magnetic dipole moment µ and its compo-nent µz in the direction of the applied field cannot have arbitrary values. Quite generally, we have a direct relationship between the magnetic moment µ of a given dipole and its angular momentum l: µ =  g e 2mc  l, (13) with l2 = J(J + 1)ℏ2; J = 1 2, 3 2, 5 2,... or 0,1,2,... (14) The quantity g(e/2mc) is the gyromagnetic ratio of the dipole while the number g is Lande’s g-factor. If the net angular momentum of the dipole is due solely to electron spins, then 80 70 60 50 40 30 20 10 0 20 40 60 80 ( · 106) (103/T in K21) FIGURE 3.7 χ versus 1/T plot for a powdered sample of copper–potassium sulphate hexahydrate (after Hupse, 1942). 74 Chapter 3. The Canonical Ensemble g = 2; on the other hand, if it is due solely to orbital motions, then g = 1. In general, however, its origin is mixed; g is then given by the formula g = 3 2 + S(S + 1) −L(L + 1) 2J(J + 1) , (15) S and L being, respectively, the spin and the orbital quantum numbers of the dipole. Note that there is no upper or lower bound on the values that g can have! Combining (13) and (14), we can write µ2 = g2µ2 B J(J + 1), (16) where µB(= eℏ/2mc) is the Bohr magneton. The component µz of the magnetic moment in the direction of the applied field is, on the other hand, given by µz = gµBm, m = −J,−J + 1,...,J −1,J. (17) Thus, a dipole whose magnetic moment µ conforms to expression (16) can have no other orientations with respect to the applied field except the ones conforming to the values (17) of the component µz; obviously, the number of allowed orientations, for a given value of J, is (2J + 1). In view of this, the single-dipole partition function Q1(β) is now given by, see (3), Q1(β) = J X m=−J exp(βgµBmH). (18) Introducing a parameter x, defined by x = β(gµB J)H, (19) equation (18) becomes Q1(β) = J X m=−J emx/J = e−x{e(2J+1)x/J −1} ex/J −1 = e(2J+1)x/2J −e−(2J+1)x/2J ex/2J −e−x/2J = sinh  1 + 1 2J  x  sinh  1 2J x  . (20) The mean magnetic moment of the system is then given by, see equation (4), Mz = Nµz = N β ∂ ∂H lnQ1(β) = N(gµB J)  1 + 1 2J  coth  1 + 1 2J  x  −1 2J coth  1 2J x  . (21) 3.9 The statistics of paramagnetism 75 Thus µz = (gµBJ )BJ(x), (22) where BJ(x) is the Brillouin function of order J: BJ(x) =  1 + 1 2J  coth  1 + 1 2J  x  −1 2J coth  1 2J x  . (23) In Figure 3.8 we have plotted the function BJ(x) for some typical values of the quantum number J. We shall now consider a few special cases. First of all, we note that for strong fields and low temperatures (x ≫1), the function BJ(x) ≃1 for all J, which corresponds to a state of magnetic saturation. On the other hand, for high temperatures and weak fields (x ≪1), the function BJ(x) may be written as 1 3(1 + 1/J)x + ..., (24) so that µz ≃(gµBJ)2 3kT  1 + 1 J  H = g2µ2 BJ(J + 1) 3kT H. (25) The Curie law, χ ∝1/T, is again obeyed; however, the Curie constant is now given by CJ = N0g2µ2 BJ(J + 1) 3k = N0µ2 3k ; (26) see equation (16). It is indeed interesting that the high-temperature results, (25) and (26), directly involve the eigenvalues of the operator µ2. 1.0 0.5 J 5 , , 1, 00 2 4 3 2 6 1 2 BJ(x) x FIGURE 3.8 The Brillouin function BJ(x) for various values of J. 76 Chapter 3. The Canonical Ensemble We now look a little more closely at the dependence of the foregoing results on the quantum number J. First of all, we consider the extreme case J →∞, with the understand-ing that simultaneously g →0, such that the value of µ stays constant. From equation (23), we readily observe that, in this limit, the Brillouin function BJ(x) tends to become (i) inde-pendent of J and (ii) identical to the Langevin function L(x). This is not surprising because, in this limit, the number of allowed orientations for a magnetic dipole becomes infinitely large, with the result that the problem essentially reduces to its classical counterpart (where one must allow all possible orientations). At the other extreme, we have the case J = 1 2, which allows only two orientations. The results in this case are very different from the ones for J ≫1. We now have, with g = 2, µz = µBB1/2(x) = µB tanhx. (27) For x ≫1, µz is very nearly equal to µB. For x ≪1, however, µz ≃µBx, which corresponds to the Curie constant C1/2 = N0µ2 B k . (28) In Figure 3.9 we reproduce the experimental values of µz (in terms of µB) as a function of the quantity H/T, for three paramagnetic salts; the corresponding theoretical plots, namely the curves g JBJ(x), are also included in the figure. The agreement between theory and experiment is indeed good. In passing, we note that, at a temperature of 1.3 K, a field of about 50,000 gauss is sufficient to produce over 99 percent of saturation in these salts. 7.00 6.00 III II I 5.00 4.00 3.00 2.00 1.00 0 10 20 30 40 1.30 K 2.00 K 3.00 K 4.21 K (z/B) 103H /T gauss/K FIGURE 3.9 Plots of µz/µB as a function of H/T. The solid curves represent the theoretical results, while the points mark the experimental findings of Henry (1952). Curve I is for potassium chromium alum J = 3 2 ,g = 2  , curve II for iron ammonia alum J = 5 2 ,g = 2  , and curve III for gadolinium sulphate octahydrate J = 7 2 ,g = 2  . 3.10 Thermodynamics of magnetic systems: negative temperatures 77 3.10 Thermodynamics of magnetic systems: negative temperatures For the purpose of this section, it will suffice to consider a system of dipoles with J = 1 2. Each dipole then has a choice of two orientations, the corresponding energies being −µBH and +µBH; let us call these energies −ε and +ε, respectively. The partition function of the system is then given by QN(β) =  eβε + e−βεN = {2cosh(βε)}N; (1) compare to the general expression (3.9.20). Accordingly, the Helmholtz free energy of the system is given by A = −NkT ln{2cosh(ε/kT)}, (2) from which S = −  ∂A ∂T  H = Nk h ln n 2cosh  ε kT o −ε kT tanh  ε kT i , (3) U = A + TS = −Nεtanh  ε kT  , (4) M = −  ∂A ∂H  T = NµB tanh  ε kT  (5) and, finally, C = ∂U ∂T  H = Nk  ε kT 2 sech2  ε kT  . (6) Equation (5) is essentially the same as (3.9.27); moreover, as expected, U = −MH. The temperature dependence of the quantities S, U, M, and C is shown in Figures 3.10 through 3.13. We note that the entropy of the system is vanishingly small for kT ≪ε; it rises 1.0 In 2 0.5 00 2 4 6 S Nk kT/´ FIGURE 3.10 The entropy of a system of magnetic dipoles (with J = 1 2 ) as a function of temperature. 78 Chapter 3. The Canonical Ensemble 0 20.5 21.0 0 2 4 6 kT ´ U N´ FIGURE 3.11 The energy of a system of magnetic dipoles (with J = 1 2 ) as a function of temperature. 0.5 1.0 0 0 2 4 6 kT ´ M N FIGURE 3.12 The magnetization of a system of magnetic dipoles (with J = 1 2 ) as a function of temperature. rapidly when kT is of the order of ε and approaches the limiting value Nkln2 for kT ≫ε. This limiting value of S corresponds to the fact that at high temperatures the orientation of the dipoles assumes a completely random character, with the result that the system now has 2N equally likely microstates available to it. The energy of the system attains its lowest value, −Nε, as T →0 K; this clearly corresponds to a state of magnetic satura-tion and, hence, to a state of perfect order in the system. Toward high temperatures, the energy tends to vanish,13 implying a purely random orientation of the dipoles and hence 13Note that in the present study we are completely disregarding the kinetic energy of the dipoles. 3.10 Thermodynamics of magnetic systems: negative temperatures 79 0.5 1.0 0 0 2 4 6 C Nk kT ´ FIGURE 3.13 The specific heat of a system of magnetic dipoles (with J = 1 2 ) as a function of temperature. a complete loss of magnetic order. These features are re-emphasized in Figure 3.12, which depicts the temperature dependence of the magnetization M. The specific heat of the sys-tem is vanishingly small at low temperatures but, in view of the fact that the energy of the system tends to a constant value as T →∞, the specific heat vanishes at high tempera-tures as well. Somewhere around T = ε/k, it displays a maximum. Writing 1 for the energy difference between the two allowed states of the dipole, the formula for the specific heat can be written as C = Nk  1 kT 2 e1/kT(1 + e1/kT)−2. (7) A specific heat peak of this form is generally known as the Schottky anomaly; it is observed in systems that have an excitation gap 1 above the ground state. Now, throughout our study so far we have considered only those cases for which T > 0. For normal systems, this is indeed essential, for otherwise we have to contend with canon-ical distributions that blow up as the energy of the system is indefinitely increased. If, however, the energy of a system is bounded from above, then there is no compelling reason to exclude the possibility of negative temperatures. Such specialized situations do indeed exist, and the system of magnetic dipoles provides a good example thereof. From equa-tion (4), we note that, so long as U < 0, T > 0 — and that is the only range we covered in Figures 3.10 through 3.13. However, the same equation tells us that if U > 0 then T < 0, which prompts us to examine the matter a little more closely. For this, we consider the variation of the temperature T and the entropy S with energy U, namely 1 T = −k ε tanh−1  U Nε  = k 2ε ln Nε −U Nε + U  (8) 80 Chapter 3. The Canonical Ensemble kT ´ S Nk U N´ 0.5 In 2 1.0 21 10.5 11 12 15 1 225 22 21 20.5 0 0 1 FIGURE 3.14 The entropy of a system of magnetic dipoles (with J = 1 2 ) as a function of energy. Some values of the parameter kT/ε are also shown in the figure. The slope at the two endpoints diverges since both ends represent zero temperature but it is difficult to see due to the logarithmic nature of the divergence. and S Nk = −Nε + U 2Nε ln Nε + U 2Nε  −Nε −U 2Nε ln Nε −U 2Nε  ; (9) these expressions follow straightforwardly from equations (3) and (4), and are shown graphically in Figures 3.14 and 3.15. We note that for U = −Nε, both S and T vanish. As U increases, they too increase until we reach the special situation where U = 0. The entropy is then seen to have attained its maximum value Nkln2, while the temperature has reached infinity. Throughout this range, the entropy had been a monotonically increasing function of energy, so T was positive. Now, as U becomes 0+, (dS/dU) becomes 0−and T becomes −∞. With a further increase in U, the entropy monotonically decreases; as a result, the temperature continues to be negative, though its magnitude steadily decreases. Finally, we reach the largest value of U, namely +Nε, where the entropy is once again zero and T = 0−. The region where U > 0 (and hence T < 0) is indeed abnormal because it corresponds to a magnetization opposite in direction to that of the applied field. Nevertheless, it can be realized experimentally in the system of nuclear moments of a crystal in which the relax-ation time t1 for mutual interaction among nuclear spins is very small in comparison with the relaxation time t2 for interaction between the spins and the lattice. Let such a crystal be magnetized in a strong magnetic field and then the field reversed so quickly that the spins are unable to follow the switch-over. This will leave the system in a nonequilibrium state, with energy higher than the new equilibrium value U. During a period of order t1, 3.10 Thermodynamics of magnetic systems: negative temperatures 81 kT/´ kT/´ ´ ´ 4 2 0 21 11 22 24 U N´ FIGURE 3.15 The temperature parameter kT/ε, and its reciprocal βε, for a system of magnetic dipoles (with J = 1 2 ) as a function of energy. the subsystem of the nuclear spins should be able to attain a state of internal equilibrium; this state will have a negative magnetization and will, therefore, correspond to a negative temperature. The subsystem of the lattice, which involves energy parameters that are in principle unbounded, will still be at a positive temperature. During a period of order t2, the two subsystems would attain a state of mutual equilibrium, which again will have a positive temperature.14 An experiment of this kind was successfully performed by Purcell and Pound (1951) with a crystal of LiF; in this case, t1 was of order 10−5 sec while t2 was of order 5 min. A state of negative temperature for the subsystem of spins was indeed attained and was found to persist for a period of several minutes; see Figure 3.16. Before we close this discussion, a few general remarks seem in order. First of all, we should note that the onset of negative temperatures is possible only if there exists an upper limit on the energy of the given system. In most physical systems this is not the case, simply because most physical systems possess kinetic energy of motion which is obvi-ously unbounded. By the same token, the onset of positive temperatures is related to the 14Note that in the latter process, during which the spins realign themselves (now more favorably in the new direction of the field), the energy will flow from the subsystem of the spins to that of the lattice, and not vice versa. This is in perfect agreement with the fact that negative temperatures are hotter than positive ones; see the subsequent discussion in the text. 82 Chapter 3. The Canonical Ensemble Time 0.5 0.4 0.3 0.2 0.1 0 20.1 20.2 20.3 20.4 20.5 Deflection 1min. FIGURE 3.16 A typical record of the reversed nuclear magnetization (after Purcell and Pound, 1951). On the left we have a deflection corresponding to normal, equilibrium magnetization (T ∼300K); it is followed by the reversed deflection (corresponding to T ∼−350K), which decays through zero deflection (corresponding to a passage from T = −∞to T = +∞) toward the new equilibrium state that again has a positive T. existence of a lower limit on the energy of a system; this, however, does not present any problem because, if nothing else, the uncertainty principle alone is sufficient to set such a limit for every physical system. Thus, it is quite normal for a system to be at a positive temperature whereas it is very unusual for one to be at a negative temperature. Now, suppose that we have a system whose energy cannot assume unlimited high values. Then, we can surely visualize a temperature T such that the quantity NkT is much larger than any admissible value, Er, of the energy. At such a high temperature, the mutual interactions of the microscopic entities constituting the system may be regarded as negligible; accordingly, one may write for the partition function of the system QN(β) ≃ "X n e−βεn #N . (10) Since, by assumption, all βεn ≪1, we have QN(β) ≃ "X n  1 −βεn + 1 2β2ε2 n #N . (11) Let g denote the number of possible orientations of a microscopic constituent of the sys-tem with respect to the direction of the external field; then, the quantities P n εα n(α = 0,1,2) may be replaced by gεα. We thus get lnQN(β) ≃N  lng + ln  1 −β¯ ε + 1 2β2ε2  ≃N  lng −β¯ ε + 1 2β2 ε2 −ε2 . (12) Problems 83 The Helmholtz free energy of the system is then given by A(N,β) ≃−N β lng + N ¯ ε −N 2 β(ε −¯ ε)2, (13) from which S(N,β) ≃Nk lng −Nk 2 β2(ε −¯ ε)2, (14) U(N,β) ≃Nε −Nβ(ε −¯ ε)2, (15) and C(N,β) ≃Nkβ2(ε −¯ ε)2. (16)15 The formulae in equations (12) through (16) determine the thermodynamic properties of the system for β ≃0. The important thing to note here is that they do so not only for β ≳0 but also for β ≲0. In fact, these formulae hold in the vicinity of, and on both sides of, the maximum in the S −U curve; see Figure 3.14. Quite expectedly, the maximum value of S is given by Nklng, and it occurs at β = ±0; S here decreases both ways, whether U decreases (β > 0) or increases (β < 0). It will be noted that the specific heat of the system in either case is positive. It is not difficult to show that if two systems, characterized by the temperature parame-ters β1 and β2, are brought into thermal contact, then energy will flow from the system with the smaller value of β to the system with the larger value of β; this will continue until the two systems acquire a common value of this parameter. What is more important to note is that this result remains literally true even if one or both of the β are negative. Thus, if β1 is −ve while β2 is +ve, then energy will flow from system 1 to system 2, that is, from the sys-tem at negative temperature to the one at positive temperature. In this sense, systems at negative temperatures are hotter than the ones at positive temperatures; indeed, negative temperatures are above +∞, not below zero! For further discussion of this topic, reference may be made to a paper by Ramsey (1956). Problems 3.1. (a) Derive formula (3.2.36) from equations (3.2.14) and (3.2.35). (b) Derive formulae (3.2.39) and (3.2.40) from equations (3.2.37) and (3.2.38). 3.2. Prove that the quantity g′′(x0), see equations (3.2.25), is equal to ⟨(E −U)2⟩exp(2β). Thus show that equation (3.2.28) is physically equivalent to equation (3.6.9). 3.3. Using the fact that (1/n!) is the coefficient of xn in the power expansion of the function exp(x), derive an asymptotic formula for this coefficient by the method of saddle-point integration. Compare your result with the Stirling formula for n!. 15Compare this result with equation (3.6.3). 84 Chapter 3. The Canonical Ensemble 3.4. Verify that the quantity (k/N )ln0, where 0(N ,U) = X {nr} ′W{nr}, is equal to the (mean) entropy of the given system. Show that this leads to essentially the same result for ln0 if we take, in the foregoing summation, only the largest term of the sum, namely the term W{n∗ r} that corresponds to the most probable distribution set. [Surprised? Well, note the following example: For all N, the summation over the binomial coefficients NCr = N!/[r!(N −r!)] gives N X r=0 NCr = 2N; therefore, ln ( N X r=0 NCr ) = N ln2. (a) Now, the largest term in this sum corresponds to r ≃N/2; so, for large N, the logarithm of the largest term is very nearly equal to ln{N!} −2ln{(N/2)!} ≈N lnN −2N 2 ln N 2 = N ln2, (b) which agrees with (a).] 3.5. Making use of the fact that the Helmholtz free energy A(N,V,T) of a thermodynamic system is an extensive property of the system, show that N  ∂A ∂N  V,T + V  ∂A ∂V  N,T = A. [Note that this result implies the well-known relationship: Nµ = A + PV(≡G).] 3.6. (a) Assuming that the total number of microstates accessible to a given statistical system is , show that the entropy of the system, as given by equation (3.3.13), is maximum when all  states are equally likely to occur. (b) If, on the other hand, we have an ensemble of systems sharing energy (with mean value E), then show that the entropy, as given by the same formal expression, is maximum when Pr ∝exp(−βEr),β being a constant to be determined by the given value of E. (c) Further, if we have an ensemble of systems sharing energy (with mean value E) and also sharing particles (with mean value N), then show that the entropy, given by a similar expression, is maximum when Pr,s ∝exp(−αNr −βEs), α and β being constants to be determined by the given values of N and E. 3.7. Prove that, quite generally, CP −CV = −k h ∂ ∂T n T  ∂lnQ ∂V  T oi2 V  ∂2 lnQ ∂V 2  T > 0. Verify that the value of this quantity for a classical ideal classical gas is Nk. Problems 85 3.8. Show that, for a classical ideal gas, S Nk = ln Q1 N  + T ∂lnQ1 ∂T  P . 3.9. If an ideal monatomic gas is expanded adiabatically to twice its initial volume, what will the ratio of the final pressure to the initial pressure be? If during the process some heat is added to the system, will the final pressure be higher or lower than in the preceding case? Support your answer by deriving the relevant formula for the ratio Pf /Pi. 3.10. (a) The volume of a sample of helium gas is increased by withdrawing the piston of the containing cylinder. The final pressure Pf is found to be equal to the initial pressure Pi times (Vi/Vf )1.2, Vi and Vf being the initial and final volumes. Assuming that the product PV is always equal to 2 3U, will (i) the energy and (ii) the entropy of the gas increase, remain constant, or decrease during the process? (b) If the process were reversible, how much work would be done and how much heat would be added in doubling the volume of the gas? Take Pi = 1 atm and Vi = 1m3. 3.11. Determine the work done on a gas and the amount of heat absorbed by it during a compression from volume V1 to volume V2, following the law PV n = const. 3.12. If the “free volume” V of a classical system is defined by the equation V N = Z e{U−U(qi)}/kT N Y i=1 d3qi, where U is the average potential energy of the system and U(qi) the actual potential energy as a function of the molecular configuration, then show that S = Nk " ln ( V N 2πmkT h2 3/2) + 5 2 # . In what sense is it justified to refer to the quantity V as the “free volume” of the system? Substantiate your answer by considering a particular case — for example, the case of a hard sphere gas. 3.13. (a) Evaluate the partition function and the major thermodynamic properties of an ideal gas consisting of N1 molecules of mass m1 and N2 molecules of mass m2, confined to a space of volume V at temperature T. Assume that the molecules of a given kind are mutually indistinguishable, while those of one kind are distinguishable from those of the other kind. (b) Compare your results with the ones pertaining to an ideal gas consisting of (N1 + N2) molecules, all of one kind, of mass m, such that m(N1 + N2) = m1N1 + m2N2. 3.14. Consider a system of N classical particles with mass m moving in a cubic box with volume V = L3. The particles interact via a short-ranged pair potential u(rij) and each particle interacts with each wall with a short-ranged interaction uwall(z), where z is the perpendicular distance of a particle from the wall. Write down the Lagrangian for this model and use a Legendre transformation to determine the Hamiltonian H. (a) Show that the quantity P = −  ∂H ∂V  = −1 3L2  ∂H ∂L  can clearly be identified as the instantaneous pressure — that is, the force per unit area on the walls. (b) Reconstruct the Lagrangian in terms of the relative locations of the particles inside the box ri = Lsi, where the variables si all lie inside a unit cube. Use a Legendre transformation to determine the Hamiltonian with this set of variables. (c) Recalculate the pressure using the second version of the Hamiltonian. Show that the pressure now includes three contributions: (1) a contribution proportional to the kinetic energy, (2) a contribution related to the forces between pairs of particles, and (3) a contribution related to the force on the wall. 86 Chapter 3. The Canonical Ensemble Show that in the thermodynamic limit the third contribution is negligible compared to the other two. Interpret contributions 1 and 2 and compare to the virial equation of state (3.7.15). 3.15. Show that the partition function QN(V,T) of an extreme relativistic gas consisting of N monatomic molecules with energy–momentum relationship ε = pc, c being the speed of light, is given by QN(V,T) = 1 N! ( 8πV kT hc 3)N . Study the thermodynamics of this system, checking in particular that PV = 1 3U, U/N = 3kT, and γ = 4 3. Next, using the inversion formula (3.4.7), derive an expression for the density of states g(E) of this system. 3.16. Consider a system similar to the one in the preceding problem but consisting of 3N particles moving in one dimension. Show that the partition function in this case is given by Q3N(L,T) = 1 (3N)!  2L kT hc 3N , L being the “length” of the space available. Compare the thermodynamics and the density of states of this system with the corresponding quantities obtained in the preceding problem. 3.17. If we take the function f (q,p) in equation (3.5.3) to be U −H(q,p), then clearly ⟨f ⟩= 0; formally, this would mean Z [U −H(q,p)]e−βH(q,p)dω = 0. Derive, from this equation, expression (3.6.3) for the mean-square fluctuation in the energy of a system embedded in the canonical ensemble. 3.18. Show that for a system in the canonical ensemble ⟨(1E)3⟩= k2  T4 ∂CV ∂T  V + 2T3CV  . Verify that for an ideal gas 1E U 2+ = 2 3N and 1E U 3+ = 8 9N2 . 3.19. Consider the long-time averaged behavior of the quantity dG/dt, where G = X i qipi, and show that the validity of equation (3.7.5) implies the validity of equation (3.7.6), and vice versa. 3.20. Show that, for a statistical system in which the interparticle potential energy u(r) is a homogeneous function (of degree n) of the particle coordinates, the virial V is given by V = −3PV −nU Problems 87 and, hence, the mean kinetic energy K by K = −1 2V = 1 2(3PV + nU) = 1 (n + 2)(3PV + nE); here, U denotes the mean potential energy of the system while E = K + U. Note that this result holds not only for a classical system but for a quantum-mechanical one as well. 3.21. (a) Calculate the time-averaged kinetic energy and potential energy of a one-dimensional harmonic oscillator, both classically and quantum-mechanically, and show that the results obtained are consistent with the result established in the preceding problem (with n = 2). (b) Consider, similarly, the case of the hydrogen atom (n = −1) on the basis of (i) the Bohr– Sommerfeld model and (ii) the Schr¨ odinger model. (c) Finally, consider the case of a planet moving in (i) a circular orbit or (ii) an elliptic orbit around the sun. 3.22. The restoring force of an anharmonic oscillator is proportional to the cube of the displacement. Show that the mean kinetic energy of the oscillator is twice its mean potential energy. 3.23. Derive the virial equation of state equation (3.7.15) from the classical canonical partition function (3.5.5). Show that in the thermodynamic limit the interparticle terms dominate the ones that come from interactions of the particles with the walls of the container. 3.24. Show that in the relativistic case the equipartition theorem takes the form ⟨m0u2(1 −u2/c2)−1/2⟩= 3kT, where m0 is the rest mass of the particle and u its speed. Check that in the extreme relativistic case the mean thermal energy per particle is twice its value in the nonrelativistic case. 3.25. Develop a kinetic argument to show that in a noninteracting system the average value of the quantity P i pi ˙ qi is precisely equal to 3PV. Hence show that, regardless of relativistic considerations, PV = NkT. 3.26. The energy eigenvalues of an s-dimensional harmonic oscillator can be written as εj = (j + s/2)ℏω; j = 0,1,2,... Show that the jth energy level has a multiplicity (j + s −1)!/j!(s −1)!. Evaluate the partition function, and the major thermodynamic properties, of a system of N such oscillators, and compare your results with a corresponding system of sN one-dimensional oscillators. Show, in particular, that the chemical potential µs = sµ1. 3.27. Obtain an asymptotic expression for the quantity lng(E) for a system of N quantum-mechanical harmonic oscillators by using the inversion formula (3.4.7) and the partition function (3.8.15). Hence show that S Nk =  E Nℏω + 1 2  ln  E Nℏω + 1 2  −  E Nℏω −1 2  ln  E Nℏω −1 2  . [Hint: Employ the Darwin–Fowler method.] 3.28. (a) When a system of N oscillators with total energy E is in thermal equilibrium, what is the probability pn that a particular oscillator among them is in the quantum state n? [Hint: Use expression (3.8.25).] Show that, for N ≫1 and R ≫n, pn ≈(n)n/(n + 1)n+1, where n = R/N. (b) When an ideal gas of N monatomic molecules with total energy E is in thermal equilibrium, show that the probability of a particular molecule having an energy in the neighborhood of ε is proportional to exp(−βε), where β = 3N/2E. [Hint: Use expression (3.5.16) and assume that N ≫1 and E ≫ε.] 3.29. The potential energy of a one-dimensional, anharmonic oscillator may be written as V(q) = cq2 −gq3 −fq4, where c, g, and f are positive constants; quite generally, g and f may be assumed to be very small in value. Show that the leading contribution of anharmonic terms to the heat capacity of the 88 Chapter 3. The Canonical Ensemble oscillator, assumed classical, is given by 3 2k2 f c2 + 5 4 g2 c3 ! T and, to the same order, the mean value of the position coordinate q is given by 3 4 gkT c2 . 3.30. The energy levels of a quantum-mechanical, one-dimensional, anharmonic oscillator may be approximated as εn =  n + 1 2  ℏω −x  n + 1 2 2 ℏω; n = 0,1,2,... The parameter x, usually ≪1, represents the degree of anharmonicity. Show that, to the first order in x and the fourth order in u(≡ℏω/kT), the specific heat of a system of N such oscillators is given by C = Nk  1 −1 12u2 + 1 240u4  + 4x  1 u + 1 80u3  . Note that the correction term here increases with temperature. 3.31. Study, along the lines of Section 3.8, the statistical mechanics of a system of N “Fermi oscillators,” which are characterized by only two eigenvalues, namely 0 and ε. 3.32. The quantum states available to a given physical system are (i) a group of g1 equally likely states, with a common energy ε1 and (ii) a group of g2 equally likely states, with a common energy ε2 > ε1. Show that this entropy of the system is given by S = −k[p1 ln(p1/g1) + p2 ln(p2/g2)], where p1 and p2 are, respectively, the probabilities of the system being in a state belonging to group 1 or to group 2: p1 + p2 = 1. (a) Assuming that the pi are given by a canonical distribution, show that S = k  lng1 + ln{1 + (g2/g1)e−x} + x 1 + (g1/g2)ex  , where x = (ε2 −ε1)/kT, assumed positive. Compare the special case g1 = g2 = 1 with that of the Fermi oscillator of the preceding problem. (b) Verify the foregoing expression for S by deriving it from the partition function of the system. (c) Check that at T →0, S →klng1. Interpret this result physically. 3.33. Gadolinium sulphate obeys Langevin’s theory of paramagnetism down to a few degrees Kelvin. Its molecular magnetic moment is 7.2 × 10−23amp-m2. Determine the degree of magnetic saturation in this salt at a temperature of 2K in a field of flux density 2 weber/m2. 3.34. Oxygen is a paramagnetic gas obeying Langevin’s theory of paramagnetism. Its susceptibility per unit volume, at 293K and at atmospheric pressure, is 1.80 × 10−6 mks units. Determine its molecular magnetic moment and compare it with the Bohr magneton (which is very nearly equal to 9.27 × 10−24amp-m2). 3.35. (a) Consider a gaseous system of N noninteracting, diatomic molecules, each having an electric dipole moment µ, placed in an external electric field of strength E. The energy of such a molecule will be given by the kinetic energy of rotation as well as translation plus the potential energy of orientation in the applied field: ε = p2 2m + ( p2 θ 2I + p2 φ 2I sin2 θ ) −µE cosθ, Problems 89 where I is the moment of inertia of the molecule. Study the thermodynamics of this system, including the electric polarization and the dielectric constant. Assume that (i) the system is a classical one and (ii) |µE| ≪kT.16 (b) The molecule H2O has an electric dipole moment of 1.85 × 10−18 e.s.u. Calculate, on the basis of the preceding theory, the dielectric constant of steam at 100◦C and at atmospheric pressure. 3.36. Consider a pair of electric dipoles µ µ µ and µ µ µ′, oriented in the directions (θ,φ) and (θ′,φ′), respectively; the distance R between their centers is assumed to be fixed. The potential energy in this orientation is given by −µµ′ R3 {2cosθ cosθ′ −sinθ sinθ′ cos(φ −φ′)}. Now, consider this pair of dipoles to be in thermal equilibrium, their orientations being governed by a canonical distribution. Show that the mean force between these dipoles, at high temperatures, is given by −2(µµ′)2 kT ˆ R R7 , ˆ R being the unit vector in the direction of the line of centers. 3.37. Evaluate the high-temperature approximation of the partition function of a system of magnetic dipoles to show that the Curie constant CJ is given by CJ = N0g2µ2 B k m2. Hence derive the formula (3.9.26). 3.38. Replacing the sum in (3.9.18) by an integral, evaluate Q1(β) of the given magnetic dipole and study the thermodynamics following from it. Compare these results with the ones following from the Langevin theory. 3.39. Atoms of silver vapor, each having a magnetic moment µB(g = 2,J = 1 2), align themselves either parallel or antiparallel to the direction of an applied magnetic field. Determine the respective fractions of atoms aligned parallel and antiparallel to a field of flux density 0.1 weber/m2 at a temperature of 1,000 K. 3.40. (a) Show that, for any magnetizable material, the heat capacities at constant field H and at constant magnetization M are connected by the relation CH −CM = −T ∂H ∂T  M ∂M ∂T  H . (b) Show that for a paramagnetic material obeying Curie’s law CH −CM = CH2/T2, where C on the right side of this equation denotes the Curie constant of the given sample. 3.41. A system of N spins at a negative temperature (E > 0) is brought into contact with an ideal-gas thermometer consisting of N′ molecules. What will the nature of their state of mutual equilibrium be? Will their common temperature be negative or positive, and in what manner will it be affected by the ratio N′/N? 3.42. Consider the system of N magnetic dipoles, studied in Section 3.10, in the microcanonical ensemble. Enumerate the number of microstates, (N,E), accessible to the system at energy E and evaluate the quantities S(N,E) and T(N,E). Compare your results with equations (3.10.8) and (3.10.9). 16The electric dipole moments of molecules are generally of order 10−18 e.s.u. (or a Debye unit). In a field of 1 e.s.u. (= 300volts/cm) and at a temperature of 300K, the parameter βµE = O(10−4). 90 Chapter 3. The Canonical Ensemble 3.43. Consider a system of charged particles (not dipoles), obeying classical mechanics and classical statistics. Show that the magnetic susceptibility of this system is identically zero (Bohr–van Leeuwen theorem). [Note that the Hamiltonian of this system in the presence of a magnetic field H(= ∇× A) will be a function of the quantities pj + (ej/c)A(rj), and not of the pj as such. One has now to show that the partition function of the system is independent of the applied field.] 3.44. The expression (3.3.13) for the entropy S is equivalent to Shannon’s (1949) definition of the information contained in a message I = −P r Pr ln(Pr), where Pr represents the probability of message r. (a) Show that information is maximized if the probabilities of all messages are the same. Any other distribution of probabilities reduces the information. In English, “e” is more common than “z”, so Pe > Pz, so the information per character in an English message is less than the optimal amount possible based on the number of different characters used in an English text. (b) The information in a text is also affected by correlations between characters in the text. For example, in English, “q” is always followed by “u”, so this pair of characters contains the same information as “q” alone. The probability of a character indexed by r followed immediately by character indexed by r′ is Pr,r′ = PrPr′Gr,r′, where Gr,r′ is the character-pair correlation function. If pairs of characters are uncorrelated, then Gr,r′ = 1. Show that if characters are uncorrelated then the information in a two-character message is twice the information of a single-character message and that correlations (Gr,r′ ̸= 1) reduce the information content. [Hint: Use the inequality lnx ≤x −1.] (c) Write a computer program to determine the information per character in a text file by determining the single-character probabilities Pr and character-pair correlations Gr,r′. Computers usually use one full byte per character to store information. Since one byte can store 256 different messages, the potential information per byte is ln256 = 8ln2 ≡8bits. Show that the information per character in your text file is considerably less than 8 bits and explain why it is possible for file-compression algorithms to reduce the size of a computer file without sacrificing any of the information contained in the file. 4 The Grand Canonical Ensemble In the preceding chapter we developed the formalism of the canonical ensemble and established a scheme of operations for deriving the various thermodynamic properties of a given physical system. The effectiveness of that approach became clear from the examples discussed there; it will become even more vivid in the subsequent studies carried out in this text. However, for a number of problems, both physical and chemical, the usefulness of the canonical ensemble formalism turns out to be rather limited and it appears that a further generalization of this formalism is called for. The motivation that brings about this generalization is physically of the same nature as the one that led us from the microcanoni-cal to the canonical ensemble — it is just the next natural step from there. It comes from the realization that not only the energy of a system but the number of particles as well is hardly ever measured in a “direct” manner; we only estimate it through an indirect probing into the system. Conceptually, therefore, we may regard both N and E as variables and identify their expectation values, ⟨N⟩and ⟨E⟩, with the corresponding thermodynamic quantities. The procedure for studying the statistics of the variables N and E is self-evident. We may either (i) consider the given system A as immersed in a large reservoir A′ with which it can exchange both energy and particles or (ii) regard it as a member of what we may call a grand canonical ensemble, which consists of the given system A and a large number of (mental) copies thereof, the members of the ensemble carrying out a mutual exchange of both energy and particles. The end results, in either case, are asymptotically the same. 4.1 Equilibrium between a system and a particle-energy reservoir We consider the given system A as immersed in a large reservoir A′, with which it can exchange both energy and particles; see Figure 4.1. After some time has elapsed, the system and the reservoir are supposed to attain a state of mutual equilibrium. Then, according to Section 1.3, the system and the reservoir will have a common temperature T and a common chemical potential µ. The fraction of the total number of particles N(0) and the fraction of the total energy E(0) that the system A can have at any time t are, however, variables (whose values, in principle, can lie anywhere between zero and unity). If, at a particular instant of time, the system A happens to be in one of its states characterized by the number Nr of particles and the amount Es of energy, then the number of particles in the reservoir would be N′ r and its energy E′ s, such that Nr + N′ r = N(0) = const. (1) Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00004-9 © 2011 Elsevier Ltd. All rights reserved. 91 92 Chapter 4. The Grand Canonical Ensemble A9 (N9 r , E9 s) A (Nr , Es) FIGURE 4.1 A statistical system immersed in a particle–energy reservoir. and Es + E′ s = E(0) = const. (2) Again, since the reservoir is supposed to be much larger than the given system, the values of Nr and Es that are going to be of practical importance will be very small fractions of the total magnitudes N(0) and E(0), respectively; therefore, for all practical purposes,1 Nr N(0) =  1 −N′ r N(0)  ≪1 (3) and Es E(0) =  1 −E′ s E(0)  ≪1. (4) Now, in the manner of Section 3.1, the probability Pr,s that, at any time t, the sys-tem A is found to be in an (Nr,Es)-state would be directly proportional to the number of microstates ′(N′ r,E′ s) that the reservoir can have for the corresponding macrostate (N′ r,E′ s). Thus, Pr,s ∝′(N(0) −Nr,E(0) −Es). (5) Again, in view of (3) and (4), we can write ln′(N(0) −Nr,E(0) −Es) = ln′(N(0),E(0)) + ∂ln′ ∂N′  N′=N(0) (−Nr) + ∂ln′ ∂E′  E′=E(0) (−Es) + ··· ≃ln′(N(0),E(0)) + µ′ kT′ Nr −1 kT′ Es; (6) see equations (1.2.3), (1.2.7), (1.3.3), and (1.3.5). Here, µ′ and T′ are, respectively, the chemical potential and the temperature of the reservoir (and hence of the given system 1Note that A here could well be a relatively small “part” of a given system A(0), while A′ represents the “rest” of A(0). That would give a truly practical perspective to the grand canonical formalism. 4.2 A system in the grand canonical ensemble 93 as well). From (5) and (6), we obtain the desired result: Pr,s ∝exp(−αNr −βEs), (7) where α = −µ/kT, β = 1/kT. (8) On normalization, it becomes Pr,s = exp(−αNr −βEs) P r,s exp(−αNr −βEs); (9) the summation in the denominator goes over all the (Nr,Es)-states accessible to the system A. Note that our final expression for Pr,s is independent of the choice of the reservoir. We shall now examine the same problem from the ensemble point of view. 4.2 A system in the grand canonical ensemble We now visualize an ensemble of N identical systems (which, of course, can be labeled as 1,2,...,N ) mutually sharing a total number of particles2 N N and a total energy N E. Let nr,s denote the number of systems that have, at any time t, the number Nr of particles and the amount Es of energy (r,s = 0,1,2,...); then, obviously, X r,s nr,s = N, (1a) X r,s nr,sNr = N N, (1b) and X r,s nr,sEs = N E. (1c) Any set {nr,s}, of the numbers nr,s, which satisfies the restrictive conditions (1), represents one of the possible modes of distribution of particles and energy among the members of our ensemble. Furthermore, any such mode of distribution can be realized in W{nr,s} different ways, where W{nr,s} = N ! Q r,s (nr,s!). (2) 2For simplicity, we shall henceforth use the symbols N and E instead of ⟨N⟩and ⟨E⟩. 94 Chapter 4. The Grand Canonical Ensemble We may now define the most probable mode of distribution, {n∗ r,s}, as the one that maximizes expression (2), satisfying at the same time the restrictive conditions (1). Going through the conventional derivation, see Section 3.2, we obtain for a large ensemble n∗ r,s N = exp(−αNr −βEs) P r,s exp(−αNr −βEs); (3) compare to the corresponding equation (3.2.10) for the canonical ensemble. Alternatively, we may define the expectation (or mean) values of the numbers nr,s, namely ⟨nr,s⟩= P {nr,s} ′ nr,sW{nr,s} P {nr,s} ′ W{nr,s} , (4) where the primed summations go over all distribution sets that conform to conditions (1). An asymptotic expression for ⟨nr,s⟩can be derived by using the method of Darwin and Fowler — the only difference from the corresponding derivation in Section 3.2 being that, in the present case, we will have to work with functions of more than one (complex) variable. The derivation, however, runs along similar lines, with the result Lim N →∞ ⟨nr,s⟩ N ≃n∗ r,s N = exp(−αNr −βEs) P r,s exp(−αNr −βEs), (5) in agreement with equation (4.1.9). The parameters α and β, so far undetermined, are eventually determined by the equations N = P r,s Nr exp(−αNr −βEs) P r,s exp(−αNr −βEs) ≡−∂ ∂α ( ln X r,s exp(−αNr −βEs) ) (6) and E = P r,s Es exp(−αNr −βEs) P r,s exp(−αNr −βEs) ≡−∂ ∂β ( ln X r,s exp(−αNr −βEs) ) , (7) where the quantities N and E here are supposed to be preassigned. 4.3 Physical significance of the various statistical quantities 95 4.3 Physical significance of the various statistical quantities To establish a connection between the statistics of the grand canonical ensemble and the thermodynamics of the system under study, we introduce a quantity q, defined by q ≡ln (X r,s exp(−αNr −βEs) ) ; (1) the quantity q is a function of the parameters α and β, and also of all the Es.3 Taking the differential of q and making use of equations (4.2.5), (4.2.6), and (4.2.7), we get dq = −Ndα −Edβ −β N X r,s ⟨nr,s⟩dEs, (2) so that d(q + αN + βE) = β α β dN + dE −1 N X r,s ⟨nr,s⟩dEs ! . (3) To interpret the terms appearing on the right side of this equation, we compare the expression enclosed within the parentheses with the statement of the first law of thermo-dynamics, that is, δQ = dE + δW −µdN, (4) where the various symbols have their usual meanings. The following correspondence now seems inevitable: δW = −1 N X r,s ⟨nr,s⟩dEs, µ = −α/β, (5) with the result that d(q + αN + βE) = βδQ. (6) The parameter β, being the integrating factor for the heat δQ, must be equivalent to the reciprocal of the absolute temperature T, so we may write β = 1/kT (7) and, hence, α = −µ/kT. (8) 3This quantity was first introduced by Kramers, who called it the q-potential. 96 Chapter 4. The Grand Canonical Ensemble The quantity (q + αN + βE) would then be identified with the thermodynamic variable S/k; accordingly, q = S k −αN −βE = TS + µN −E kT . (9) However, µN is identically equal to G, the Gibbs free energy of the system, and hence to (E −TS + PV). So, finally, q ≡ln (X r,s exp(−αNr −βEs) ) = PV kT . (10) Equation (10) provides the essential link between the thermodynamics of the given sys-tem and the statistics of the corresponding grand canonical ensemble. It is, therefore, a relationship of central importance in the formalism developed in this chapter. To derive further results, we prefer to introduce a parameter z, defined by the relation z ≡e−α = eµ/kT; (11) the parameter z is generally referred to as the fugacity of the system. In terms of z, the q-potential takes the form q ≡ln (X r,s zNre−βEs ) (12) = ln    ∞ X Nr=0 zNrQNr(V,T)    (with Q0 ≡1), (13) so we may write q(z,V,T) ≡lnQ(z,V,T), (14) where Q(z,V,T) ≡ ∞ X Nr=0 zNrQNr(V,T) (with Q0 ≡1). (15) Note that, in going from expression (12) to (13), we have (mentally) carried out a sum-mation over the energy values Es, with Nr fixed, thus giving rise to the partition function QNr(V,T); of course, the dependence of QNr on V comes from the dependence of the Es on V. In going from (13) to (14), we have (again mentally) carried out a summation over all the numbers Nr = 0,1,2,··· ,∞, thus giving rise to the grand partition function Q(z,V,T) of the system. The q-potential, which we have already identified with PV/kT, is, therefore, the logarithm of the grand partition function. 4.3 Physical significance of the various statistical quantities 97 It appears that in order to evaluate the grand partition function Q(z,V,T) we have to go through the routine of evaluating the partition function Q(N,V,T). In principle, this is indeed true. In practice, however, we find that on many occasions an explicit evaluation of the partition function is extremely hard while considerable progress can be made in the evaluation of the grand partition function. This is particularly true when we deal with systems in which the influence of quantum statistics and/or interparticle interactions is important; see Sections 6.2 and 10.1. The formalism of the grand canonical ensemble then proves to be of considerable value. We are now in a position to write down the full recipe for deriving the leading ther-modynamic quantities of a given system from its q-potential. We have, first of all, for the pressure of the system P(z,V,T) = kT V q(z,V,T) ≡kT V lnQ(z,V,T). (16) Next, writing N for N and U for E, we obtain with the help of equations (4.2.6), (4.2.7), and (11) N(z,V,T) = z  ∂ ∂zq(z,V,T)  V,T = kT  ∂ ∂µq(µ,V,T)  V,T (17) and U(z,V,T) = −  ∂ ∂β q(z,V,T)  z,V = kT2  ∂ ∂T q(z,V,T)  z,V . (18) Eliminating z between equations (16) and (17), one obtains the equation of state, that is, the (P,V,T)-relationship, of the system. On the other hand, eliminating z between equa-tions (17) and (18), one obtains U as a function of N,V, and T, which readily leads to the specific heat at constant volume as (∂U/∂T)N,V . The Helmholtz free energy is given by the formula A = Nµ −PV = NkT lnz −kT lnQ(z,V,T) = −kT ln Q(z,V,T) zN , (19) which may be compared with the canonical ensemble formula A = −kT lnQ(N,V,T); see also Problem 4.2. Finally, we have for the entropy of the system S = U −A T = kT  ∂q ∂T  z,V −Nklnz + kq. (20) 98 Chapter 4. The Grand Canonical Ensemble 4.4 Examples We shall now study a couple of simple problems, with the explicit purpose of demonstrat-ing how the method of the q-potential works. This is not intended to be a demonstration of the power of this method, for we shall consider here only those problems that can be solved equally well by the methods of the preceding chapters. The real power of the new method will become apparent only when we study problems involving quantum-statistical effects and effects arising from interparticle interactions; many such problems will appear in the remainder of the text. The first problem we propose to consider here is that of the classical ideal gas. In Section 3.5 we showed that the partition function QN(V,T) of this system could be written as QN(V,T) = [Q1(V,T)]N N! , (1) where Q1(V,T) may be regarded as the partition function of a single particle in the sys-tem. First of all, we should note that equation (1) does not imply any restrictions on the particles having internal degrees of motion; those degrees of motion, if present, would affect the results only through Q1. Second, we should recall that the factor N! in the denominator arises from the fact that the particles constituting the gas are, in fact, indistinguishable. Closely related to the indistinguishability of the particles is the fact that they are nonlocalized, for otherwise we could distinguish them through their very sites; compare, for instance, the system of harmonic oscillators, which was studied in Section 3.8. Now, since our particles are nonlocalized they can be anywhere in the space available to them; consequently, the function Q1 will be directly proportional to V: Q1(V,T) = Vf (T), (2) where f (T) is a function of temperature alone. We thus obtain for the grand partition function of the gas Q(z,V,T) = ∞ X Nr=0 zNrQNr(V,T) = ∞ X Nr=0 {zVf (T)}Nr Nr! = exp{zVf (T)}, (3) which gives q(z,V,T) = zVf (T). (4) 4.4 Examples 99 Formula (4.3.16) through (4.3.20) then lead to the following results: P = zkTf (T), (5) N = zVf (T), (6) U = zVkT2f ′(T), (7) A = NkT lnz −zVkTf (T), (8) and S = −Nklnz + zVk{Tf ′(T) + f (T)}. (9) Eliminating z between (5) and (6), we obtain the equation of state of the system: PV = NkT. (10) We note that equation (10) holds irrespective of the form of the function f (T). Next, eliminating z between (6) and (7), we obtain U = NkT2f ′(T)/f (T), (11) which gives CV = Nk 2Tf (T)f ′(T) + T2{f (T)f ′′(T) −[f ′(T)]2} [f (T)]2 . (12) In simple cases, the function f (T) turns out to be directly proportional to a certain power of T. Supposing that f (T) ∝Tn, equations (11) and (12) become U = n(NkT) (11a) and CV = n(Nk). (12a) Accordingly, the pressure in such cases is directly proportional to the energy density of the gas, the constant of proportionality being 1/n. The reader will recall that the case n = 3/2 corresponds to a nonrelativistic gas while n = 3 corresponds to an extreme relativistic one. Finally, eliminating z between equation (6) and equations (8) and (9), we obtain A and S as functions of N,V, and T. This essentially completes our study of the classical ideal gas. The next problem to be considered here is that of a system of independent, localized particles — a model which, in some respects, approximates a solid. Mathematically, the 100 Chapter 4. The Grand Canonical Ensemble problem is similar to that of a system of harmonic oscillators. In either case, the micro-scopic entities constituting the system are mutually distinguishable. The partition function QN(V,T) of such a system can be written as QN(V,T) = [Q1(V,T)]N. (13) At the same time, in view of the localized nature of the particles, the single-particle par-tition function Q1(V,T) is essentially independent of the volume occupied by the system. Consequently, we may write Q1(V,T) = φ(T), (14) where φ(T) is a function of temperature alone. We then obtain for the grand partition function of the system Q(z,V,T) = ∞ X Nr=0 [zφ(T)]Nr = [1 −zφ(T)]−1; (15) clearly, the quantity zφ(T) must stay below unity, so that the summation over Nr is convergent. The thermodynamics of the system follows straightforwardly from equation (15). We have, to begin with, P ≡kT V q(z,T) = −kT V ln{1 −zφ(T)}. (16) Since both z and T are intensive variables, the right side of (16) vanishes as V →∞. Hence, in the thermodynamic limit, P = 0.4 For other quantities of interest, we obtain, with the help of equations (4.3.17) through (4.3.20), N = zφ(T) 1 −zφ(T), (17) U = zkT2φ′(T) 1 −zφ(T) , (18) A = NkT lnz + kT ln{1 −zφ(T)}, (19) and S = −Nklnz −kln{1 −zφ(T)} + zkTφ′(T) 1 −zφ(T). (20) From (17), we get zφ(T) = N N + 1 ≃1 −1 N (N ≫1). (21) 4It will be seen in the sequel that P actually vanishes like (lnN)/N. 4.4 Examples 101 It follows that 1 −zφ(T) = 1 N + 1 ≃1 N . (22) Equations (17) through (20) now give U/N = kT2φ′(T)/φ(T), (18a) A/N = −kT lnφ(T) + O lnN N  , (19a) and S/Nk = lnφ(T) + Tφ′(T)/φ(T) + O lnN N  . (20a) Substituting φ(T) = [2sinh(ℏω/2kT)]−1 (23) into these formulae, we obtain results pertaining to a system of quantum-mechanical, one-dimensional harmonic oscillators. The substitution φ(T) = kT/ℏω, (24) on the other hand, leads to results pertaining to a system of classical, one-dimensional harmonic oscillators. As a corollary, we examine here the problem of solid–vapor equilibrium. Consider a single-component system, having two phases — solid and vapor — in equilibrium, con-tained in a closed vessel of volume V at temperature T. Since the phases are free to exchange particles, a state of mutual equilibrium would imply that their chemical poten-tials are equal; this, in turn, means that they have a common fugacity as well. Now, the fugacity zg of the gaseous phase is given by, see equation (6), zg = Ng Vgf (T), (25) where Ng is the number of particles in the gaseous phase and Vg the volume occupied by them; in a typical case, Vg ≃V. The fugacity zs of the solid phase, on the other hand, is given by equation (21): zs ≃ 1 φ(T). (26) Equating (25) and (26), we obtain for the equilibrium particle density in the vapor phase Ng/Vg = f (T)/φ(T). (27) 102 Chapter 4. The Grand Canonical Ensemble Now, if the density in the vapor phase is sufficiently low and the temperature of the system sufficiently high, the vapor pressure P would be given by Pvapor = Ng Vg kT = kT f (T) φ(T). (28) To be specific, we may assume the vapor to be monatomic; the function f (T) is then of the form f (T) = (2πmkT)3/2/h3. (29) On the other hand, if the solid phase can be approximated by a set of three-dimensional harmonic oscillators characterized by a single frequency ω (the Einstein model), the function φ(T) would be φ(T) = [2sinh(hω/2kT)]−3. (30) However, there is one important difference here. An atom in a solid is energetically more stabilized than an atom that is free — that is why a certain threshold energy is required to transform a solid into separate atoms. Let ε denote the value of this energy per atom, which in a way implies that the zeros of the energy spectra εg and εs, which led to the functions (29) and (30), respectively, are displaced with respect to one another by an amount ε. A true comparison between the functions f (T) and φ(T) must take this into account. As a result, we obtain for the vapor pressure Pvapor = kT 2πmkT h2 3/2 [2sinh(ℏω/2kT)]3e−ε/kT. (31) In passing, we note that equation (27) also gives us the necessary condition for the formation of the solid phase. The condition clearly is: N > V f (T) φ(T), (32) where N is the total number of particles in the system. Alternatively, this means that T < Tc, (33) where Tc is a characteristic temperature determined by the implicit relationship f (Tc) φ(Tc) = N V . (34) Once the two phases appear, the number Ng(T) will have a value determined by equa-tion (27) while the remainder, N −Ng, will constitute the solid phase. 4.5 Density and energy fluctuations in the grand canonical ensemble 103 4.5 Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles In a grand canonical ensemble, the variables N and E, for any member of the ensemble, can lie anywhere between zero and infinity. Therefore, on the face of it, the grand canoni-cal ensemble appears to be very different from its predecessors — the canonical and the microcanonical ensembles. However, as far as thermodynamics is concerned, the results obtained from this ensemble turn out to be identical to the ones obtained from the other two. Thus, in spite of strong facial differences, the overall behavior of a given physical sys-tem is practically the same whether it belongs to one kind of ensemble or another. The basic reason for this is that the “relative fluctuations” in the values of the quantities that vary from member to member in an ensemble are practically negligible. Therefore, in spite of the different surroundings that different ensembles provide to a given physical system, the overall behavior of the system is not significantly affected. To appreciate this point, we shall evaluate the relative fluctuations in the particle den-sity n and the energy E of a given physical system in the grand canonical ensemble. Recalling that N = P r,s Nre−αNr−βEs P r,s e−αNr−βEs , (1) it readily follows that ∂N ∂α ! β,Es = −N2 + N2. (2) Thus (1N)2 ≡N2 −N2 = − ∂N ∂α ! T,V = kT ∂N ∂µ ! T,V . (3) From (3), we obtain for the relative mean-square fluctuation in the particle density n (= N/V) (1n)2 n2 = (1N)2 N2 = kT N2 ∂N ∂µ ! T,V . (4) In terms of the variable v (= V/N), we may write (1n)2 n2 = kTv2 V 2 ∂(V/v) ∂µ  T,V = −kT V  ∂v ∂µ  T . (5) 104 Chapter 4. The Grand Canonical Ensemble To put this result into a more practical form, we recall the thermodynamic relation dµ = vdP −sdT, (6) according to which dµ (at constant T) = vdP. Equation (5) then takes the form (1n)2 n2 = −kT V 1 v  ∂v ∂P  T = kT V κT, (7) where κT is the isothermal compressibility of the system. Thus, the relative root-mean-square fluctuation in the particle density of the given sys-tem is ordinarily O(N−1/2) and, hence, negligible. However, there are exceptions, like the ones met with in situations accompanying phase transitions. In those situations, the com-pressibility of a given system can become excessively large, as is evidenced by an almost “flattening” of the isotherms. For instance, at a critical point the compressibility diverges, so it is no longer intensive. Finite-size scaling theory described in Chapters 12 and 14 indi-cates that at the critical point the isothermal compressibility scales with system size as κT (Tc) ∼Nγ/dν where γ and ν are certain critical exponents and d is the dimension. For the case of experimental liquid–vapor critical points, κT (Tc) ∼N0.63. Accordingly, the root-mean-square density fluctuations grow faster than N1/2 — in this case, like N0.82. Thus, in the region of phase transitions, especially at the critical points, we encounter unusu-ally large fluctuations in the particle density of the system. Such fluctuations indeed exist and account for phenomena like critical opalescence. It is clear that under these circum-stances the formalism of the grand canonical ensemble could, in principle, lead to results that are not necessarily identical to the ones following from the corresponding canonical ensemble. In such cases, it is the formalism of the grand canonical ensemble that will have to be preferred because only this one will provide a correct picture of the actual physical situation. We shall now examine fluctuations in the energy of the system. Following the usual procedure, we obtain (1E)2 ≡E2 −E2 = − ∂E ∂β ! z,V = kT2 ∂U ∂T  z,V . (8) To put expression (8) into a more comprehensible form, we write ∂U ∂T  z,V = ∂U ∂T  N,V + ∂U ∂N  T,V ∂N ∂T  z,V , (9) where the symbol N is being used interchangeably for N. Now, in view of the fact that N = −  ∂ ∂α lnQ  β,V , U = −  ∂ ∂β lnQ  α,V , (10) 4.6 Thermodynamic phase diagrams 105 we have ∂N ∂β  α,V = ∂U ∂α  β,V (11) and, hence, ∂N ∂T  z,V = 1 T ∂U ∂µ  T,V . (12) Substituting expressions (9) and (12) into equation (8) and remembering that the quantity (∂U/∂T)N,V is the familiar CV , we get (1E)2 = kT2CV + kT ∂U ∂N  T,V ∂U ∂µ  T,V . (13) Invoking equations (3.6.3) and (3), we finally obtain (1E)2 = ⟨(1E)2⟩can + (∂U ∂N  T,V )2 (1N)2. (14) Formula (14) is highly instructive; it tells us that the mean-square fluctuation in the energy E of a system in the grand canonical ensemble is equal to the value it would have in the canonical ensemble plus a contribution arising from the fact that now the particle number N is also fluctuating. Again, under ordinary circumstances, the relative root-mean-square fluctuation in the energy density of the system would be practically negligible. However, in the region of phase transitions, unusually large fluctuations in the value of this variable can arise by virtue of the second term in the formula. 4.6 Thermodynamic phase diagrams One of the great successes of thermodynamics and statistical mechanics over the last 150 years has been in the study of phase transitions. Statistical mechanics provides the basis for accurate models for a wide variety of thermodynamic phases of materials and has led to a detailed understanding of phase transitions and critical phenomena. Condensed materials exist in a variety of phases that depend on thermodynamic parameters such as temperature, pressure, magnetic field, and so on. Thermodynamics and statistical mechanics can be used to determine the properties of individual phases, and the locations and characteristics of the phase transitions that occur between those phases. Thermodynamic phases are regions in the phase diagram where the thermody-namics properties are analytic functions of the thermodynamic parameters, while phase transitions are points, lines, or surfaces in the phase diagram where the thermodynamic properties are nonanalytic. Much of the remainder of this text is devoted to using statistical mechanics to explain the properties of material phases and phase transitions. 106 Chapter 4. The Grand Canonical Ensemble P Pc Pt P S S L V Tt Tc T (a) (b) L V Vc V Pc Pt FIGURE 4.2 Sketches (not-to-scale) of the P–T (a) and P–V (b) phase diagrams for argon. This geometry is generic for a wide range of materials. The letters S, L, and V denote solid, liquid, and vapor phases. It is instructive to examine the structure of phase diagrams. Argon provides a good example because the structure of its phase diagram is similar to that of many other mate-rials (see Figure 4.2). At moderate temperatures and pressures, the stable thermodynamic phases of argon are solid, liquid, and vapor. At high temperatures there is a supercritical fluid phase that smoothly connects the liquid and vapor phases. Most materials, includ-ing argon, exhibit multiple solid phases especially at high pressures and low temperatures. Figure 4.2(a) is the phase diagram in the P–T plane and shows the solid–liquid coexis-tence line, the liquid–vapor coexistence line, and the solid–vapor coexistence line. The three lines meet at the triple point (Tt,Pt) and the liquid–vapor coexistence line ends at the critical point (Tc,Pc). The triple point values and critical point values for argon are Tt = 83.8 K, Pt = 68.9 kPa, Tc = 150.7 K, and Pc = 4.86 MPa, respectively. Figure 4.2(b) is the phase diagram in the P–V plane and shows the pressure versus the specific volume v(= V/N) on the coexistence lines. The dashed lines indicate the triple point pressure and critical pressure in both figures. The horizontal tie lines are the por-tions of isotherms as they cross coexistence lines and show the discontinuities of v. The tie lines in order from bottom to top are: sublimation tie lines connecting the solid and vapor phases, the triple point tie line that connects all three phases, and a series of solid–liquid and liquid–vapor tie lines. Notice that the liquid and vapor specific volumes continuously approach each other and are both equal to the critical specific volume vc at the critical point. The properties of the vapor, liquid, and solid phases are: . The vapor phase is a low-density gas that is accurately described by the ideal-gas equation of state P = nkT with corrections that are described by the virial expansion; see Chapters 6 and 10. 4.6 Thermodynamic phase diagrams 107 . The liquid phase is a dense fluid with strong interactions between the atoms. The fluid exhibits characteristic short-range pair correlations and scattering structure, as discussed in Section 10.7. The structure factor and the pair correlation function for argon, as determined from neutron scattering, are shown in Figure 10.8. For temperatures above the critical temperature Tc, one cannot distinguish between liquid and vapor. The density in this supercritical phase is a smooth function of temperature and pressure from the low-density vapor to the high-density liquid. Virial expansions developed in Sections 10.1 through 10.3 aptly describe the supercritical region. Strictly speaking, one can only distinguish between the liquid and vapor phases on the liquid–vapor coexistence line since it is possible to evolve smoothly from one phase to the other without crossing a phase boundary. . The solid phase is a face-centered cubic crystal structure with long-range order, so the scattering structure factor displays Bragg peaks as described in Section 10.7.B. The thermodynamic properties of solid phases are described in Section 7.3. All equilibrium thermodynamic properties within a single phase are analytic func-tions of the thermodynamic parameters while phase transitions are defined as places in the phase diagram where equilibrium thermodynamic properties are not analytic. Coexis-tence lines, or first-order phase transition lines, separate different phases in the P–T phase diagram as shown in Figure 4.2(a). Thermodynamic densities are discontinuous across coexistence lines. This is displayed on the P–V phase diagram in Figure 4.2(b) by hori-zontal tie lines that connect different values the specific volume takes in the two phases. Generally, all densities such as the specific volume v = V/N, entropy per particle s = S/N, internal energy density u = U/V, and so on, are discontinuous across first-order phase transition lines. The slopes of the coexistence lines in the P–V phase diagram depend on the latent heat of the transition and the specific volumes of the coexisting phases; see Section 4.7. All three phases coexist at the triple point. The liquid–vapor coexistence line extends from the triple point to the critical point at the end of the first-order phase transition line. The specific volume is discontinuous on the liquid–vapor coexistence line but the size of the discontinuty vanishes at the critical point where the specific volume is vc; see Figure 4.2(b). All densities are continuous functions of T and P through the critical point. For this reason, critical points are called continuous transitions or, sometimes, second-order phase transitions. Even though thermodynamic densities are continuous, the thermodynamic behavior at the critical point is nonanalytic since, for example, the specific heat and isothermal compressibility both diverge at the critical point. Another characteristic property of critical points is the divergence of the cor-relation length, which results in a universal behavior of critical points for broad classes of materials. The theory of critical points is developed in Chapters 12, 13, and 14. Classical statistical mechanics provides a framework for understanding the phase dia-grams and thermodynamic properties of a wide variety of materials. However, quantum mechanics and quantum statistics play an important role at low temperatures when the size of the thermal deBroglie wavelength λ = h/ √ 2πmkT is of the same order as the 108 Chapter 4. The Grand Canonical Ensemble P S superfluid L V Ps Pc Tc T T FIGURE 4.3 Sketch of the P–T phase diagram for helium-4. The letters S, L, and V denote solid, liquid, and vapor phases. The critical point is Tc = 5.19K and Pc = 227kPa = 2.24atm. The solid–liquid coexistence curve starts at Ps = 2.5MPa = 25atm at T = 0K and does not intersect the liquid–vapor coexistence curve. The λ-line is the continuous phase transition between the normal liquid and the superfluid phase. The superfluid phase transition temperature at the liquid–vapor coexistence line is Tλ = 2.18K. average distance between molecules. This is the case with liquid helium at temperatures below a few degrees kelvin. The phase diagram of helium-4 is shown in Figure 4.3. Some aspects of the phase diagram are similar to the phase diagram of argon. Both helium and argon have liquid–vapor coexistence lines that end in critical points and both have crystalline solid phases at low temperatures. Three differences between the two phase diagrams are most notable: the solid phase for helium only exists for pressures greater than Ps = 2.5GPa = 25atm, the liquid phase of helium extends all the way to zero temperature, and helium-4 exhibits a superfluid phase below Tλ = 2.18K. The superfluid phase exhibits remarkable properties: zero viscosity, quantized flow, propagating heat modes, and macroscopic quantum coherence. This extraordinary behavior is due to the Bose-Einstein statistics of 4He atoms and a Bose– Einstein condensation into a macroscopic quantum state as discussed in Sections 7.1 and 11.2 through 11.6. Even the solid phase of helium-4 shows evidence of a macroscopic quantum state with the observation of a “supersolid” phase by Kim and Chan (2004). By contrast, 3He atoms obey Fermi–Dirac statistics and display very different behav-iors from 4He atoms at low temperatures. The geometry of the phase diagram of helium-3 is similar to that of helium-4 except that the critical temperature is lower (Tc = 3.35K compared to 5.19K) and the solid phase forms at 30 atm of pressure rather than 25atm. The dramatic difference is the lack of a superfluid phase near 1K in helium-3. Helium-3 remains a normal liquid all the way down to about 10 millikelvin. The properties of the normal liquid phase of helium-3 are described by the theory of degenerate Fermi gases and the Fermi liquid theory developed in Chapter 8 and Sections 11.7 and 11.8. The superfluid state that forms at millikelvin temperatures is the result of Bardeen, Cooper, and Schrieffer (BCS) p-wave pairing between atoms near the Fermi surface; this pairing is discussed in Section 11.9. 4.7 Phase equilibrium and the Clausius–Clapeyron equation 109 4.7 Phase equilibrium and the Clausius–Clapeyron equation The thermodynamic properties of the phases of a material determine the geometry of the phase diagram. In particular, the Gibbs free energy G(N,P,T) = U −TS + PV = A + PV = µ(P,T)N (1) determines the locations of the phase boundaries. Note that the chemical potential is the Gibbs free energy per molecule; see Problem 4.6 and Appendix H. Consider a cylinder con-taining N molecules held at constant pressure P and constant temperature T, that is, in an isothermal, isobaric assembly. Suppose the cylinder initially contains two phases: vapor (A) and liquid (B) so that the total number of molecules is N = NA + NB and the Gibbs free energy is G = GA(NA,P,T) + GB(NB,P,T). If the two phases do not coexist at this pres-sure and temperature, the numbers of molecules in each phase will change as the system approaches equilibrium. As the number of molecules in each phase changes, the Gibbs free energy changes by an amount dG =  ∂GA ∂NA  T,P dNA +  ∂GB ∂NB  T,P dNB = (µA −µB)dNA, (2) where dNA is the change in the number of molecules in phase A. The Gibbs free energy is minimized at equilibrium, so dG ≤0. If µA > µB, the number of molecules in phase B will increase and the number in phase A will decrease as the sys-tem approaches equilibrium. If µA < µB, the number of molecules in phase A will increase and the number in phase B will decrease. If the chemical potentials are equal, the Gibbs free energy is independent of the number of molecules in the two phases. Therefore, the chemical potentials are equal at coexistence: µA = µB. (3) Let’s consider the familiar example of water. At normal pressures and temperatures, water has three phases: liquid water, solid ice, and water vapor, and its P–T phase diagram is similar to that shown for argon in Figure 4.2(a) — the P–V phase diagram for water is somewhat different because the density of the liquid phase is larger than the density of the solid ice phase; see Problems 4.15 and 4.20. At P = 1atm, water and water vapor coex-ist at T = 100◦C, the “boiling point” — while boiling is a nonequilibrium process, boiling begins at the temperature at which the equilibrium vapor pressure is equal to the local atmospheric pressure. Consider a two-phase sample of water and water vapor at T = 99◦C. A two-phase sample containing both liquid water and water vapor is easy to create in a constant volume assembly. If there is sufficient volume available, liquid water will evapo-rate until the water vapor pressure reaches the coexistence pressure at that temperature Pσ(99◦C) = 0.965atm. If the applied pressure is then increased to, and held constant at, P = 1atm while maintaining a constant temperature of T = 99◦C, the system will be out of equilibrium. At constant pressure, the system will return to equilibrium by decreasing 110 Chapter 4. The Grand Canonical Ensemble its volume as water vapor condenses into the liquid phase until the the system is com-pletely liquid water. This lowers the Gibbs free energy until it has the equilibrium value determined by the chemical potential of liquid water at this pressure and temperature. On the other hand, if T = 100◦C and P = 1atm, the chemical potentials of the liquid and vapor phases are equal, so any combination of water vapor and liquid water has the same Gibbs free energy. The proportion of water and vapor will change as heat is added or removed. The latent heat of vaporization of water Lv = 540cal/g = 2260kJ/kg is the heat needed to convert liquid into vapor. The coexistence pressure Pσ(T) defines the phase boundary between any two phases in the P–T plane, as shown in Figure 4.2(a). From equation (3), the coexistence pressure obeys µA(Pσ (T),T) = µB(Pσ (T),T). (4) The derivatives of the chemical potentials are related by ∂µA ∂T  P + ∂µA ∂P  T dPσ dT = ∂µB ∂T  P + ∂µB ∂P  T dPσ dT , (5) while the entropy per particle s = S/N and specific volume v = V/N are given by s = − ∂µ ∂T  P , (6a) v = ∂µ ∂P  T ; (6b) see equation (4.5.6). Equations (5) and (6) give the Clausius–Clapeyron equation dPσ dT = sB −sA vB −vA = 1s 1v = L T1v , (7) where L = T1s is the latent heat per particle. The slope of the coexistence curve depends on the discontinuities of the entropy per particle and the volume per particle. Equation (7) applies very generally to all first-order phase transitions and can be used to determine the coexistence curve as a function of temperature; see Section 4.4, Problems 4.11, and 4.14 through 4.16. At a triple point, the chemical potentials of three phases are equal: µA = µB = µC. (8) The slopes of the three coexistence lines that define the triple point are related since 1sAB + 1sBC + 1sCA = 0 and 1vAB + 1vBC + 1vCA = 0. This guarantees that each coexis-tence line between two phases at the triple point “points into” the third phase; see Problem 4.17. Problems 111 Problems 4.1. Show that the entropy of a system in the grand canonical ensemble can be written as S = −k X r,s Pr,s lnPr,s, where Pr,s is given by equation (4.1.9). 4.2. In the thermodynamic limit (when the extensive properties of the system become infinitely large, while the intensive ones remain constant), the q-potential of the system may be calculated by taking only the largest term in the sum ∞ X Nr=0 zNrQNr(V,T). Verify this statement and interpret the result physically. 4.3. A vessel of volume V (0) contains N(0) molecules. Assuming that there is no correlation whatsoever between the locations of the various molecules, calculate the probability, P(N,V), that a region of volume V (located anywhere in the vessel) contains exactly N molecules. (a) Show that N = N(0)p and (1N)r.m.s. = {N(0)p(1 −p)}1/2, where p = V/V (0). (b) Show that if both N(0)p and N(0)(1 −p) are large numbers, the function P(N,V) assumes a Gaussian form. (c) Further, if p ≪1 and N ≪N(0), show that the function P(N,V) assumes the form of a Poisson distribution: P(N) = e−N (N)N N! . 4.4. The probability that a system in the grand canonical ensemble has exactly N particles is given by p(N) = zNQN(V,T) Q(z,V,T) . Verify this statement and show that in the case of a classical, ideal gas the distribution of particles among the members of a grand canonical ensemble is identically a Poisson distribution. Calculate the root-mean-square value of (1N) for this system both from the general formula (4.5.3) and from the Poisson distribution, and show that the two results are the same. 4.5. Show that expression (4.3.20) for the entropy of a system in the grand canonical ensemble can also be written as S = k  ∂ ∂T (Tq)  µ,V . 4.6. Define the isobaric partition function YN(P,T) = 1 λ3 Z ∞ 0 QN(V,T)e−βPV dV. Show that in the thermodynamic limit the Gibbs free energy (4.7.1) is proportional to lnYN(P,T). Evaluate the isobaric partition function for a classical ideal gas and show that PV = NkT. [The factor of the cube of the thermal deBroglie wavelength, λ3, serves to make the partition function dimensionless and does not contribute to the Gibbs free energy in the thermodynamic limit.] 4.7. Consider a classical system of noninteracting, diatomic molecules enclosed in a box of volume V at temperature T. The Hamiltonian of a single molecule is given by H(r1,r2,p1,p2) = 1 2m(p2 1 + p2 2) + 1 2K|r1 −r2|2. Study the thermodynamics of this system, including the dependence of the quantity ⟨r2 12⟩on T. 112 Chapter 4. The Grand Canonical Ensemble 4.8. Determine the grand partition function of a gaseous system of “magnetic” atoms (with J = 1 2 and g = 2) that can have, in addition to the kinetic energy, a magnetic potential energy equal to µBH or −µBH, depending on their orientation with respect to an applied magnetic field H. Derive an expression for the magnetization of the system, and calculate how much heat will be given off by the system when the magnetic field is reduced from H to zero at constant volume and constant temperature. 4.9. Study the problem of solid–vapor equilibrium (Section 4.4) by setting up the grand partition function of the system. 4.10. A surface with N0 adsorption centers has N(≤N0) gas molecules adsorbed on it. Show that the chemical potential of the adsorbed molecules is given by µ = kT ln N (N0 −N)a(T), where a(T) is the partition function of a single adsorbed molecule. Solve the problem by constructing the grand partition function as well as the partition function of the system. [Neglect the intermolecular interaction among the adsorbed molecules.] 4.11. Study the state of equilibrium between a gaseous phase and an adsorbed phase in a single-component system. Show that the pressure in the gaseous phase is given by the Langmuir equation Pg = θ 1 −θ × (a certain function of temperature), where θ is the equilibrium fraction of the adsorption sites that are occupied by the adsorbed molecules. 4.12. Show that for a system in the grand canonical ensemble {(NE) −N E} = ∂U ∂N  T,V (1N)2. 4.13. Define a quantity J as J = E −Nµ = TS −PV. Show that for a system in the grand canonical ensemble (1J)2 = kT2CV + (∂U ∂N  T,V −µ )2 (1N)2. 4.14. Assuming that the latent heat of vaporization of water Lv = 2260kJ/kg is independent of temperature and the specific volume of the liquid phase is negligible compared to the specific volume of the vapor phase, vvapor = kT/Pσ (T), integrate the Clausius–Clapeyron equation (4.7.7) to obtain the coexistence pressure as a function of temperature. Compare your result to the experimental vapor pressure of water from the triple point to 200◦C. The equilibrium vapor pressure at 373K is 101kPa = 1atm. 4.15. Assuming that the latent heat of sublimation of ice Ls = 2500kJ/kg is independent of temperature and the specific volume of the solid phase is negligible compared to the specific volume of the vapor phase, vvapor = kT/Pσ (T), integrate the Clausius–Clapeyron equation (4.7.7) to obtain the coexistence pressure as a function of temperature. Compare your result to the experimental vapor pressure of ice from T = 0 to the triple point. The equilibrium vapor pressure at the triple point is 612Pa. 4.16. Calculate the slope of the solid-liquid transition line for water near the triple point T = 273.16K, given that the latent heat of melting is 80cal/g, the density of the liquid phase is 1.00g/cm3, and the density of the ice phase is 0.92g/cm3. Estimate the melting temperature at P = 100atm. Problems 113 4.17. Show that the Clausius–Clapeyron equation (4.7.7) guarantees that each of the coexistence curves at the triple point of a material “points into” the third phase; for example, the slope of the solid–vapor coexistence line has a value in-between the slopes of the the the solid–liquid and liquid–vapor coexistence lines. 4.18. Sketch the P–V phase diagram for helium-4 using the sketch of the P–T phase diagram in Figure 4.3. 4.19. Derive the equivalent of the Clausius–Clapeyron equation (4.7.7) for the slope of the coexistence chemical potential as a function of temperature. Use the fact that the pressures P(µ,T) in two different phases are equal on the coexistence curve. 4.20. Sketch the P–T and P–V phase diagrams of water, taking into account the fact that the mass density of the liquid phase is larger than the mass density of the solid phase. 5 Formulation of Quantum Statistics The scope of the ensemble theory developed in Chapters 2 through 4 is extremely general, though the applications considered so far were confined either to classical systems or to quantum-mechanical systems composed of distinguishable entities. When it comes to quantum-mechanical systems composed of indistinguishable entities, as most physical systems are, considerations of the preceding chapters have to be applied with care. One finds that in this case it is advisable to rewrite ensemble theory in a language that is more natural to a quantum-mechanical treatment, namely the language of the operators and the wavefunctions. Insofar as statistics are concerned, this rewriting of the theory may not seem to introduce any new physical ideas as such; nonetheless, it provides us with a tool that is highly suited for studying typical quantum systems. And once we set out to study these systems in detail, we encounter a stream of new, and altogether different, physical concepts. In particular, we find that the behavior of even a noninteracting system, such as the ideal gas, departs considerably from the pattern set by the classical treatment. In the presence of interactions, the pattern becomes even more complicated. Of course, in the limit of high temperatures and low densities, the behavior of all physical systems tends asymptotically to what we expect on classical grounds. In the process of demonstrating this point, we automatically obtain a criterion that tells us whether a given physical sys-tem may or may not be treated classically. At the same time, we obtain rigorous evidence in support of the procedure, employed in the previous chapters, for computing the number, 0, of microstates (corresponding to a given macrostate) of a given system from the vol-ume, ω, of the relevant region of its phase space, namely 0 ≈ω/hf , where f is the number of “degrees of freedom” in the problem. 5.1 Quantum-mechanical ensemble theory: the density matrix We consider an ensemble of N identical systems, where N ≫1. These systems are char-acterized by a (common) Hamiltonian, which may be denoted by the operator ˆ H. At time t, the physical states of the various systems in the ensemble will be characterized by the wavefunctions ψ(ri,t), where ri denote the position coordinates relevant to the sys-tem under study. Let ψk(ri,t) denote the (normalized) wavefunction characterizing the physical state in which the kth system of the ensemble happens to be at time t; natu-rally, k = 1,2,...,N . The time variation of the function ψk(t) will be determined by the Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00005-0 © 2011 Elsevier Ltd. All rights reserved. 115 116 Chapter 5. Formulation of Quantum Statistics Schr¨ odinger equation1 ˆ Hψk(t) = iℏ˙ ψk(t). (1) Introducing a complete set of orthonormal functions φn, the wavefunctions ψk(t) may be written as ψk(t) = X n ak n(t)φn, (2) where ak n(t) = Z φ∗ nψk(t)dτ; (3) here, φ∗ n denotes the complex conjugate of φn while dτ denotes the volume element of the coordinate space of the given system. Clearly, the physical state of the kth system can be described equally well in terms of the coefficients ak n(t). The time variation of these coefficients will be given by iℏ˙ ak n(t) = iℏ Z φ∗ n ˙ ψk(t)dτ = Z φ∗ n ˆ Hψk(t)dτ = Z φ∗ n ˆ H X m ak m(t)φm  dτ = X m Hnmak m(t), (4) where Hnm = Z φ∗ n ˆ Hφmdτ. (5) The physical significance of the coefficients ak n(t) is evident from equation (2). They are the probability amplitudes for the various systems of the ensemble to be in the various states φn; to be practical, the number |ak n(t)|2 represents the probability that a measure-ment at time t finds the kth system of the ensemble to be in the particular state φn. Clearly, we must have X n |ak n(t)|2 = 1 (for all k). (6) We now introduce the density operator ˆ ρ(t), as defined by the matrix elements ρmn(t) = 1 N N X k=1 n ak m(t)ak∗ n (t) o ; (7) clearly, the matrix element ρmn(t) is the ensemble average of the quantity am(t)a∗ n(t), which, as a rule, varies from member to member in the ensemble. In particular, the diagonal element ρnn(t) is the ensemble average of the probability |an(t)|2, the latter 1For simplicity of notation, we suppress the coordinates ri in the argument of the wavefunction ψk. 5.1 Quantum-mechanical ensemble theory: the density matrix 117 itself being a (quantum-mechanical) average. Thus, we encounter here a double-averaging process — once due to the probabilistic aspect of the wavefunctions and again due to the statistical aspect of the ensemble. The quantity ρnn(t) now represents the probability that a system, chosen at random from the ensemble, at time t, is found to be in the state φn. In view of equations (6) and (7), X n ρnn = 1. (8) We shall now determine the equation of motion for the density matrix ρmn(t). We obtain, with the help of the foregoing equations, iℏ˙ ρmn(t) = 1 N N X k=1 h iℏ n ˙ ak m(t)ak∗ n (t) + ak m(t)˙ ak∗ n (t) oi = 1 N N X k=1 X l Hmlak l (t)  ak∗ n (t) −ak m(t) X l H∗ nlak∗ l (t)  = X l {Hmlρln(t) −ρml(t)Hln} = ( ˆ H ˆ ρ −ˆ ρ ˆ H)mn; (9) here, use has been made of the fact that, in view of the Hermitian character of the operator ˆ H,H∗ nl = Hln. Using the commutator notation, equation (9) may be written as iℏ˙ ˆ ρ = [ ˆ H, ˆ ρ]−. (10) Equation (10) is the quantum-mechanical analog of the classical equation (2.2.10) of Liouville. As expected in going from a classical equation of motion to its quantum-mechanical counterpart, the Poisson bracket [ρ,H] has given place to the commutator ( ˆ ρ ˆ H −ˆ H ˆ ρ)/iℏ. If the given system is known to be in a state of equilibrium, the corresponding ensemble must be stationary, that is, ˙ ρmn = 0. Equations (9) and (10) then tell us that, for this to be the case, (i) the density operator ˆ ρ must be an explicit function of the Hamiltonian operator ˆ H (for then the two operators will necessarily commute) and (ii) the Hamiltonian must not depend explicitly on time, that is, we must have (i) ˆ ρ = ˆ ρ( ˆ H) and (ii) ˙ ˆ H = 0. Now, if the basis functions φn were the eigenfunctions of the Hamiltonian itself, then the matrices H and ρ would be diagonal: Hmn = Enδmn, ρmn = ρnδmn. (11)2 2It may be noted that in this (so-called energy) representation the density operator ˆ ρ may be written as ˆ ρ = X n |φn⟩ρn⟨φn|, (12) for then ρkl = X n ⟨φk|φn⟩ρn⟨φn|φl⟩= X n δknρnδnl = ρkδkl. 118 Chapter 5. Formulation of Quantum Statistics The diagonal element ρn, being a measure of the probability that a system, chosen at ran-dom (and at any time) from the ensemble, is found to be in the eigenstate φn, will naturally depend on the corresponding eigenvalue En of the Hamiltonian; the precise nature of this dependence is, however, determined by the “kind” of ensemble we wish to construct. In any representation other than the energy representation, the density matrix may or may not be diagonal. However, quite generally, it will be symmetric: ρmn = ρnm. (13) The physical reason for this symmetry is that, in statistical equilibrium, the tendency of a physical system to switch from one state (in the new representation) to another must be counterbalanced by an equally strong tendency to switch between the same states in the reverse direction. This condition of detailed balancing is essential for the maintenance of an equilibrium distribution within the ensemble. Finally, we consider the expectation value of a physical quantity G, which is dynami-cally represented by an operator ˆ G. This will be given by ⟨G⟩= 1 N N X k=1 Z ψk∗ˆ Gψkdτ. (14) In terms of the coefficients ak n, ⟨G⟩= 1 N N X k=1 "X m,n ak∗ n ak mGnm # , (15) where Gnm = Z φ∗ n ˆ Gφmdτ. (16) Introducing the density matrix ρ, equation (15) becomes ⟨G⟩= X m,n ρmnGnm = X m ( ˆ ρ ˆ G)mm = Tr( ˆ ρ ˆ G). (17) Taking ˆ G = ˆ 1, where ˆ 1 is the unit operator, we have Tr( ˆ ρ) = 1, (18) which is identical to (8). It should be noted here that if the original wavefunctions ψk were not normalized then the expectation value ⟨G⟩would be given by the formula ⟨G⟩= Tr( ˆ ρ ˆ G) Tr( ˆ ρ) (19) 5.2 Statistics of the various ensembles 119 instead. In view of the mathematical structure of formulae (17) and (19), the expectation value of any physical quantity G is manifestly independent of the choice of the basis {φn}, as it indeed should be. 5.2 Statistics of the various ensembles 5.2.A The microcanonical ensemble The construction of the microcanonical ensemble is based on the premise that the sys-tems constituting the ensemble are characterized by a fixed number of particles N, a fixed volume V, and an energy lying within the interval E −1 21,E + 1 21  , where 1 ≪E. The total number of distinct microstates accessible to a system is then denoted by the sym-bol 0(N,V,E;1) and, by assumption, any one of these microstates is as likely to occur as any other. This assumption enters into our theory in the nature of a postulate and is often referred to as the postulate of equal a priori probabilities for the various accessible states. Accordingly, the density matrix ρmn (which, in the energy representation, must be a diagonal matrix) will be of the form ρmn = ρnδmn, (1) with ρn =    1/0 for each of the accessible states, 0 for all other states; (2) the normalization condition (5.1.18) is clearly satisfied. As we already know, the thermody-namics of the system is completely determined from the expression for its entropy which, in turn, is given by S = k ln 0. (3) Since 0, the total number of distinct, accessible states, is supposed to be computed quantum-mechanically, taking due account of the indistinguishability of the particles right from the beginning, no paradox, such as Gibbs’, is now expected to arise. Moreover, if the quantum state of the system turns out to be unique (0 = 1), the entropy of the sys-tem will identically vanish. This provides us with a sound theoretical basis for the hitherto empirical theorem of Nernst (also known as the third law of thermodynamics). The situation corresponding to the case 0 = 1 is usually referred to as a pure case. In such a case, the construction of an ensemble is essentially superfluous, because every sys-tem in the ensemble has got to be in one and the same state. Accordingly, there is only one diagonal element ρnn that is nonzero (actually equal to unity), while all others are zero. The 120 Chapter 5. Formulation of Quantum Statistics density matrix, therefore, satisfies the relation ρ2 = ρ. (4) In a different representation, the pure case will correspond to ρmn = 1 N N X k=1 ak mak∗ n = ama∗ n (5) because all values of k are now literally equivalent. We then have ρ2 mn = X l ρmlρln = X l ama∗ l ala∗ n = ama∗ n  because X l a∗ l al = 1  = ρmn. (6) Relation (4) thus holds in all representations. A situation in which 0 > 1 is usually referred to as a mixed case. The density matrix, in the energy representation, is then given by equations (1) and (2). If we now change over to any other representation, the general form of the density matrix should remain the same, namely (i) the off-diagonal elements should continue to be zero, while (ii) the diagonal elements (over the allowed range) should continue to be equal to one another. Now, had we constructed our ensemble on a representation other than the energy representation right from the beginning, how could we have possibly anticipated ab initio property (i) of the density matrix, though property (ii) could have been easily invoked through a pos-tulate of equal a priori probabilities? To ensure that property (i), as well as property (ii), holds in every representation, we must invoke yet another postulate, namely the postulate of random a priori phases for the probability amplitudes ak n, which in turn implies that the wavefunction ψk, for all k, is an incoherent superposition of the basis {φn}. As a con-sequence of this postulate, coupled with the postulate of equal a priori probabilities, we would have in any representation ρmn ≡1 N N X k=1 ak mak∗ n = 1 N N X k=1 |a|2ei θk m−θk n  = c ei θk m−θk n  = cδmn, (7) as it should be for a microcanonical ensemble. Thus, contrary to what might have been expected on customary grounds, to secure the physical situation corresponding to a microcanonical ensemble, we require in general two 5.2 Statistics of the various ensembles 121 postulates instead of one! The second postulate arises solely from quantum-mechanics and is intended to ensure noninterference (and hence a complete absence of correlations) among the member systems; this, in turn, enables us to form a mental picture of each system of the ensemble, one at a time, completely disentangled from other systems. 5.2.B The canonical ensemble In this ensemble the macrostate of a member system is defined through the parameters N, V, and T; the energy E is now a variable quantity. The probability that a system, chosen at random from the ensemble, possesses an energy Er is determined by the Boltzmann factor exp(−βEr), where β = 1/kT; see Sections 3.1 and 3.2. The density matrix in the energy representation is, therefore, taken as ρmn = ρnδmn, (8) with ρn = C exp (−βEn); n = 0,1,2,... (9) The constant C is determined by the normalization condition (5.1.18), whereby C = 1 P n exp(−βEn) = 1 QN(β), (10) where QN(β) is the partition function of the system. In view of equations (5.1.12), see footnote 2, the density operator in this ensemble may be written as ˆ ρ = X n |φn⟩ 1 QN(β)e−βEn⟨φn| = 1 QN(β)e−β ˆ H X n |φn⟩⟨φn| = 1 QN(β)e−β ˆ H = e−β ˆ H Tr e−β ˆ H, (11) for the operator P n |φn⟩⟨φn| is identically the unit operator. It is understood that the operator exp(−β ˆ H) in equation (11) stands for the sum ∞ X j=0 (−1)j (β ˆ H)j j! . (12) 122 Chapter 5. Formulation of Quantum Statistics The expectation value ⟨G⟩N of a physical quantity G, which is represented by an operator ˆ G, is now given by ⟨G⟩N = Tr( ˆ ρ ˆ G) = 1 QN(β)Tr ˆ Ge−β ˆ H = Tr ˆ Ge−β ˆ H Tr e−β ˆ H ; (13) the suffix N here emphasizes the fact that the averaging is being done over an ensemble with N fixed. 5.2.C The grand canonical ensemble In this ensemble the density operator ˆ ρ operates on a Hilbert space with an indefi-nite number of particles. The density operator must therefore commute not only with the Hamiltonian operator ˆ H but also with a number operator ˆ n whose eigenvalues are 0,1,2,.... The precise form of the density operator can now be obtained by a straightfor-ward generalization of the preceding case, with the result ˆ ρ = 1 Q(µ,V,T)e−β( ˆ H−µˆ n), (14) where Q(µ,V,T) = X r,s e−β(Er−µNs) = Tr{e−β( ˆ H−µˆ n)}. (15) The ensemble average ⟨G⟩is now given by ⟨G⟩= 1 Q(µ,V,T)Tr ˆ Ge−β ˆ Heβµˆ n = ∞ P N=0 zN⟨G⟩NQN(β) ∞ P N=0 zNQN(β) , (16) where z(≡eβµ) is the fugacity of the system while ⟨G⟩N is the canonical-ensemble average, as given by equation (13). The quantity Q(µ,V,T) appearing in these formulae is, clearly, the grand partition function of the system. 5.3 Examples 5.3.A An electron in a magnetic field We consider, for illustration, the case of a single electron that possesses an intrinsic spin 1 2ℏˆ σ and a magnetic moment µB, where ˆ σ is the Pauli spin operator and µB = eℏ/2mc. 5.3 Examples 123 The spin of the electron can have two possible orientations, ↑or ↓, with respect to an applied magnetic field B. If the applied field is taken to be in the direction of the z-axis, the configurational Hamiltonian of the spin takes the form ˆ H = −µB( ˆ σ · B) = −µBBˆ σz. (1) In the representation that makes ˆ σz diagonal, namely ˆ σx = 0 1 1 0 ! , ˆ σy = 0 −i i 0 ! , ˆ σz = 1 0 0 −1 ! , (2) the density matrix in the canonical ensemble would be ( ˆ ρ) = e−β ˆ H Tr e−β ˆ H (3) = 1 eβµBB + e−βµBB eβµBB 0 0 e−βµBB ! . We thus obtain for the expectation value σz ⟨σz⟩= Tr( ˆ ρ ˆ σz) = eβµBB −e−βµBB eβµBB + e−βµBB = tanh(βµBB), (4) in perfect agreement with the findings of Sections 3.9 and 3.10. 5.3.B A free particle in a box We now consider the case of a free particle, of mass m, in a cubical box of side L. The Hamiltonian of the particle is given by ˆ H = −ℏ2 2m∇2 = −ℏ2 2m ∂2 ∂x2 + ∂2 ∂y2 + ∂2 ∂z2 ! , (5) while the eigenfunctions of the Hamiltonian that satisfy periodic boundary conditions, φ(x + L,y,z) = φ(x,y + L,z) = φ(x,y,z + L) = φ(x,y,z), (6) are given by φE(r) = 1 L3/2 exp(ik · r), (7) the corresponding eigenvalues E being E = ℏ2k2 2m , (8) 124 Chapter 5. Formulation of Quantum Statistics and the corresponding wave vector k being k ≡(kx,ky,kz) = 2π L (nx,ny,nz); (9) the quantum numbers nx, ny, and nz must be integers (positive, negative, or zero). Symbolically, we may write k = 2π L n, (10) where n is a vector with integral components 0,±1,±2,.... We now proceed to evaluate the density matrix ( ˆ ρ) of this system in the canonical ensemble; we shall do so in the coordinate representation. In view of equation (5.2.11), we have ⟨r|e−β ˆ H|r′⟩= X E ⟨r|E⟩e−βE⟨E|r′⟩ (11) = X E e−βEφE(r)φ∗ E(r′). Substituting from equation (7) and making use of relations (8) and (10), we obtain ⟨r|e−β ˆ H|r′⟩= 1 L3 X k exp " −βℏ2 2m k2 + ik · (r −r′) # ≈ 1 (2π)3 Z exp " −βℏ2 2m k2 + ik · (r −r′) # d3k =  m 2πβℏ2 3/2 exp  − m 2βℏ2 |r −r′|2  ; (12) see equations (B.41) and (B.42) in Appendix B. It follows that Tr(e−β ˆ H) = Z ⟨r|e−β ˆ H|r⟩d3r = V  m 2πβℏ2 3/2 . (13) The expression in equation (13) is indeed the partition function, Q1(β), of a single particle confined to a box of volume V; see equation (3.5.19). Dividing (12) by (13), we obtain for the density matrix in the coordinate representation ⟨r| ˆ ρ|r′⟩= 1 V exp  − m 2βℏ2 |r −r′|2  . (14) As expected, the matrix ρr,r′ is symmetric between the states r and r′. Moreover, the diagonal element ⟨r|ρ|r⟩, which represents the probability density for the particle to be in 5.3 Examples 125 the neighborhood of the point r, is independent of r; this means that, in the case of a sin-gle free particle, all positions within the box are equally likely to obtain. A nondiagonal element ⟨r|ρ|r′⟩, on the other hand, is a measure of the probability of “spontaneous tran-sition” between the position coordinates r and r′ and is therefore a measure of the relative “intensity” of the wave packet (associated with the particle) at a distance |r −r′| from the center of the packet. The spatial extent of the wave packet, which is a measure of the uncer-tainty involved in locating the position of the particle, is clearly of order ℏ/(mkT)1/2; the latter is also a measure of the mean thermal wavelength of the particle. The spatial spread found here is a purely quantum-mechanical effect; quite expectedly, it tends to vanish at high temperatures. In fact, as β →0, the behavior of the matrix element (14) approaches that of a delta function, which implies a return to the classical picture of a point particle. Finally, we determine the expectation value of the Hamiltonian itself. From equa-tions (5) and (14), we obtain ⟨H⟩= Tr( ˆ H ˆ ρ) = −ℏ2 2mV Z  ∇2 exp  − m 2βℏ2 |r −r′|2  r=r′ d3r = 1 2βV Z  3 −m βℏ2 |r −r′|2  exp  − m 2βℏ2 |r −r′|2  r=r′ d3r = 3 2β = 3 2kT, (15) which was indeed expected. Otherwise, too, ⟨H⟩= Tr ˆ He−β ˆ H Tr e−β ˆ H = −∂ ∂β lnTr e−β ˆ H (16) which, on combination with (13), leads to the same result. 5.3.C A linear harmonic oscillator Next, we consider the case of a linear harmonic oscillator whose Hamiltonian is given by ˆ H = −ℏ2 2m ∂2 ∂q2 + 1 2mω2q2, (17) with eigenvalues En =  n + 1 2  ℏω; n = 0,1,2,... (18) and eigenfunctions φn(q) = mω πℏ 1/4 Hn(ξ) (2nn!)1/2 e−(1/2)ξ2, (19) 126 Chapter 5. Formulation of Quantum Statistics where ξ = mω ℏ 1/2 q (20) and Hn(ξ) = (−1)neξ2  d dξ n e−ξ2. (21) The matrix elements of the operator exp(−β ˆ H) in the q-representation are given by ⟨q|e−β ˆ H|q′⟩= ∞ X n=0 e−βEnφn(q)φn(q′) = mω πℏ 1/2 e−(1/2)(ξ2+ξ′2) ∞ X n=0  e−(n+1/2)βℏω Hn(ξ)Hn(ξ′) 2nn!  . (22) The summation over n is somewhat difficult to evaluate; nevertheless, the final result is3 ⟨q|e−β ˆ H|q′⟩=  mω 2πℏsinh(βℏω) 1/2 × exp  −mω 4ℏ  (q + q′)2 tanh βℏω 2  + (q −q′)2 coth βℏω 2  , (23) which gives Tr e−β ˆ H = ∞ Z −∞ ⟨q|e−β ˆ H|q⟩dq =  mω 2πℏsinh(βℏω) 1/2 ∞ Z −∞ exp " −mωq2 ℏ tanh βℏω 2 # dq = 1 2sinh  1 2βℏω  = e−(1/2)βℏω 1 −e−βℏω . (24) Expression (24) is indeed the partition function of a linear harmonic oscillator; see equation (3.8.14). At the same time, we find that the probability density for the oscillator coordinate to be in the vicinity of the value q is given by ⟨q| ˆ ρ|q⟩=   mωtanh  1 2βℏω  πℏ   1/2 exp " −mωq2 ℏ tanh βℏω 2 # ; (25) 3The mathematical details of this derivation can be found in Kubo (1965, pp. 175–177). 5.3 Examples 127 we note that this is a Gaussian distribution in q, with mean value zero and root-mean-square deviation qr.m.s. =   ℏ 2mωtanh  1 2βℏω    1/2 . (26) The probability distribution (25) was first derived by Bloch in 1932. In the classical limit (βℏω ≪1), the distribution becomes purely thermal — free from quantum effects: ⟨q| ˆ ρ|q⟩≈ mω2 2πkT !1/2 exp " −mω2q2 2kT # , (27) with dispersion (kT/mω2)1/2. At the other extreme (βℏω ≫1), the distribution becomes purely quantum-mechanical — free from thermal effects: ⟨q| ˆ ρ|q⟩≈ mω πℏ 1/2 exp " −mωq2 ℏ # , (28) with dispersion (ℏ/2mω)1/2. Note that the limiting distribution (28) is precisely the one expected for an oscillator in its ground state (n = 0), that is one with probability density φ2 0(q); see equations (19) through (21). In view of the fact that the mean energy of the oscillator is given by ⟨H⟩= −∂ ∂β lnTr e−β ˆ H = 1 2ℏωcoth 1 2βℏω  , (29) we observe that the temperature dependence of the distribution (25) is solely determined by the expectation value ⟨H⟩. Actually, we can write ⟨q| ˆ ρ|q⟩= mω2 2π⟨H⟩ !1/2 exp " −mω2q2 2⟨H⟩ # , (30) with qr.m.s. =  ⟨H⟩ mω2 1/2 . (31) It is now straightforward to see that the mean value of the potential energy 1 2mω2q2 of the oscillator is 1 2⟨H⟩; accordingly, the mean value of the kinetic energy (p2/2m) will also be the same. 128 Chapter 5. Formulation of Quantum Statistics 5.4 Systems composed of indistinguishable particles We shall now formulate the quantum-mechanical description of a system of N identical particles. To fix ideas, we consider a gas of noninteracting particles; the findings of this study will be of considerable relevance to other systems as well. Now, the Hamiltonian of a system of N noninteracting particles is simply a sum of the individual single-particle Hamiltonians: ˆ H(q,p) = N X i=1 ˆ Hi(qi,pi); (1) here, (qi,pi) are the coordinates and momenta of the ith particle while ˆ Hi is its Hamilto-nian.4 Since the particles are identical, the Hamiltonians ˆ Hi(i = 1,2,...,N) are formally the same; they only differ in the values of their arguments. The time-independent Schr¨ odinger equation for the system is ˆ HψE(q) = EψE(q), (2) where E is an eigenvalue of the Hamiltonian and ψE(q) the corresponding eigenfunction. In view of (1), we can write a straightforward solution of the Schr¨ odinger equation, namely ψE(q) = N Y i=1 uεi(qi), (3) with E = N X i=1 εi; (4) the factor uεi(qi) in (3) is an eigenfunction of the single-particle Hamiltonian ˆ Hi(qi,pi), with eigenvalue εi: ˆ Hiuεi(qi) = εiuεi(qi). (5) Thus, a stationary state of the given system may be described in terms of the single-particle states of the constituent particles. In general, we may do so by specifying the set of num-bers {ni} to represent a particular state of the system; this would imply that there are ni particles in the eigenstate characterized by the energy value εi. Clearly, the distribution set 4We are studying here a single-component system composed of “spinless” particles. Generalization to a system composed of particles with spin and to a system composed of two or more components is quite straightforward. 5.4 Systems composed of indistinguishable particles 129 {ni} must conform to the conditions X i ni = N (6) and X i niεi = E. (7) Accordingly, the wavefunction of this state may be written as ψE(q) = n1 Y m=1 u1(m) n1+n2 Y m=n1+1 u2(m)..., (8) where the symbol ui(m) stands for the single-particle wavefunction uεi(qm). Now, suppose we effect a permutation among the coordinates appearing on the right side of (8); as a result, the coordinates (1,2,...,N) get replaced by (P1,P2,...,PN), say. The resulting wavefunction, which we may call PψE(q), will be PψE(q) = n1 Y m=1 u1(Pm) n1+n2 Y m=n1+1 u2(Pm).... (9) In classical physics, where the particles of a given system, even though identical, are regarded as mutually distinguishable, any permutation that brings about an interchange of particles in two different single-particle states is recognized to have led to a new, physi-cally distinct, microstate of the system. For example, classical physics regards a microstate in which the so-called 5th particle is in the state ui and the so-called 7th particle in the state uj(j ̸= i) as distinct from a microstate in which the 7th particle is in the state ui and the 5th particle in the state uj. This leads to N! n1!n2!... (10) (supposedly distinct) microstates of the system, corresponding to a given mode of distri-bution {ni}. The number (10) would then be ascribed as a “statistical weight factor” to the distribution set {ni}. Of course, the “correction” applied by Gibbs, which has been discussed in Sections 1.5 and 1.6, reduces this weight factor to Wc{ni} = 1 n1!n2!.... (11) And the only way one could understand the physical basis of that “correction” was in terms of the inherent indistinguishability of the particles. According to quantum physics, however, the situation remains unsatisfactory even after the Gibbs correction has been incorporated, for, strictly speaking, an interchange 130 Chapter 5. Formulation of Quantum Statistics among identical particles, even if they are in different single-particle states, should not lead to a new microstate of the system! Thus, if we want to take into account the indistin-guishability of the particles properly, we must not regard a microstate in which the “5th” particle is in the state ui and the “7th” in the state uj as distinct from a microstate in which the “7th” particle is in the state ui and the “5th” in the state uj (even if i ̸= j), for the labeling of the particles as No. 1, No. 2, and so on (which one often resorts to) is at most a matter of convenience — it is not a matter of reality. In other words, all that matters in the descrip-tion of a particular state of the given system is the set of numbers ni that tell us how many particles there are in the various single-particle states ui; the question, “which particle is in which single-particle state?” has no relevance at all. Accordingly, the microstates resulting from any permutation P among the N parti-cles (so long as the numbers ni remain the same) must be regarded as one and the same microstate. For the same reason, the weight factor associated with a distribution set {ni}, provided that the set is not disallowed on some other physical grounds, should be identically equal to unity, whatever the values of the numbers ni may be: Wq{ni} ≡1. (12)5 Indeed, if for some physical reason the set {ni} is disallowed, the weight factor Wq for that set should be identically equal to zero; see, for instance, equation (19). At the same time, a wavefunction of the type (8), which we may call Boltzmannian and denote by the symbol ψBoltz(q), is inappropriate for describing the state of a system composed of indistinguishable particles because an interchange of arguments among the factors ui and uj, where i ̸= j, would lead to a wavefunction that is both mathematically and physically different from the one we started with. Now, since a mere interchange of the particle coordinates must not lead to a new microstate of the system, the wavefunction ψE(q) must be constructed in such a way that, for all practical purposes, it is insensitive to any interchange among its arguments. The simplest way to do this is to set up a lin-ear combination of all the N! functions of the type (9) that obtain from (8) by all possible permutations among its arguments; of course, the combination must be such that if a per-mutation of coordinates is carried out in it, then the wavefunctions ψ and Pψ must satisfy the property |Pψ|2 = |ψ|2. (13) This leads to the following possibilities: Pψ = ψ for all P, (14) 5It may be mentioned here that as early as in 1905 Ehrenfest pointed out that to obtain Planck’s formula for the black-body radiation one must assign equal a priori probabilities to the various distribution sets {ni}. 5.4 Systems composed of indistinguishable particles 131 which means that the wavefunction is symmetric in all its arguments, or Pψ =    +ψ if P is an even permutation, −ψ if P is an odd permutation, (15)6 which means that the wavefunction is antisymmetric in its arguments. We call these wavefunctions ψS and ψA, respectively; their mathematical structure is given by ψS(q) = const. X P PψBoltz(q) (16) and ψA(q) = const. X P δPPψBoltz(q), (17) where δP in the expression for ψA is +1 or −1 according to whether the permutation P is even or odd. We note that the function ψA(q) can be written in the form of a Slater determinant: ψA(q) = const. ui(1) ui(2) ··· ui(N) uj(1) uj(2) ··· uj(N) · · ··· · · · ··· · · · ··· · ul(1) ul(2) ··· ul(N) , (18) where the leading diagonal is precisely the Boltzmannian wavefunction while the other terms of the expansion are the various permutations thereof; positive and negative signs in the combination (17) appear automatically as we expand the determinant. On interchang-ing a pair of arguments (which amounts to interchanging the corresponding columns of the determinant), the wavefunction ψA merely changes its sign, as it indeed should. How-ever, if two or more particles happen to be in the same single-particle state, then the corresponding rows of the determinant become identical and the wavefunction vanishes.7 Such a state is physically impossible to realize. We therefore conclude that if a system com-posed of indistinguishable particles is characterized by an antisymmetric wavefunction, 6An even (odd) permutation is one that can be arrived at from the original order by an even (odd) number of “pair interchanges” among the arguments. For example, of the six permutations (1,2,3), (2,3,1), (3,1,2), (1,3,2), (3,2,1), and (2,1,3), of the arguments 1, 2, and 3, the first three are even permutations while the last three are odd. A single interchange, among any two arguments, is clearly an odd permutation. 7This is directly related to the fact that if we effect an interchange among two particles in the same single-particle state, then PψA will obviously be identical to ψA. At the same time, if we also have PψA = −ψA, then ψA must be identically zero. 132 Chapter 5. Formulation of Quantum Statistics then the particles of the system must all be in different single-particle states — a result equivalent to Pauli’s exclusion principle for electrons. Conversely, a statistical system composed of particles obeying an exclusion principle must be described by a wavefunction that is antisymmetric in its arguments. The statistics governing the behavior of such particles is called Fermi–Dirac, or simply Fermi; statistics and the constituent particles themselves are referred to as fermions. The statistical weight factor WF.D.{ni} for such a system is unity so long as the ni in the distribution set are either 0 or 1; otherwise, it is zero: WF.D.{ni} =      1 if P i n2 i = N, 0 if P i n2 i > N. (19)8 No such problems arise for systems characterized by symmetric wavefunctions: in partic-ular, we have no restriction whatsoever on the values of the numbers ni. The statistics governing the behavior of such systems is called Bose–Einstein, or simply Bose, statis-tics and the constituent particles themselves are referred to as bosons.9 The weight factor WB.E.{ni} for such a system is identically equal to 1, whatever the values of the numbers ni: WB.E.{ni} = 1; ni = 0,1,2,.... (20) It should be pointed out here that there exists an intimate connection between the statistics governing a particular species of particles and the intrinsic spin of the particles. For instance, particles with an integral spin (in units of ℏ, of course) obey Bose–Einstein statistics, while particles with a half-odd integral spin obey Fermi–Dirac statistics. Exam-ples in the first category are photons, phonons, π-mesons, gravitons, He4-atoms, and so on, while those in the second category are electrons, nucleons (protons and neutrons), µ-mesons, neutrinos, He3-atoms, and so on. Finally, it must be emphasized that, although we have derived our conclusions here on the basis of a study of noninteracting systems, the basic results hold for interacting systems as well. In general, the desired wavefunction ψ(q) will not be expressible in terms of the single-particle wavefunctions ui(qm); nonetheless, it will have to be either of the kind ψS(q), satisfying equation (14), or of the kind ψA(q), satisfying equation (15). 8Note that the condition P i n2 i =N would be implies that all ni are either 0 or 1. On the other hand, if any of the ni are greater than 1, the sum P i n2 i would be greater than N. 9Possibilities other than Bose–Einstein and Fermi–Dirac statistics can arise in which the wavefunction changes by a complex phase factor eiθ when particles are interchanged. For topological reasons, this can only happen in two dimen-sions. Quasiparticle excitations with this property are called anyons and, if θ is a rational fraction (other than 1 or 1/2) of 2π, are said to have fractional statistics and they play an important role in the theory of the fractional quantum Hall effect; see Wilczek (1990) and Ezawa (2000). 5.5 The density matrix of a system of free particles 133 5.5 The density matrix and the partition function of a system of free particles Suppose that the given system, which is composed of N indistinguishable, noninteracting particles confined to a cubical box of volume V, is a member of a canonical ensemble characterized by the temperature parameter β. The density matrix of the system in the coordinate representation will be10 ⟨r1,...,rN| ˆ ρ|r′ 1,...,r′ N⟩= 1 QN(β)⟨r1,...,rN|e−β ˆ H|r′ 1,...,r′ N⟩, (1) where QN(β) is the partition function of the system: QN(β) = Tr(e−β ˆ H) = Z ⟨r1,...,rN|e−β ˆ H|r1,...,rN⟩d3Nr. (2) For brevity, we denote the vector ri by the letter i and the primed vector r′ i by i′. Further, let ψE(1,...,N) denote the eigenfunctions of the Hamiltonian, the suffix E representing the corresponding eigenvalues. We then have ⟨1,...,N|e−β ˆ H|1′,...,N′⟩= X E e−βE ψE(1,...,N)ψ∗ E(1′,...,N′) , (3) where the summation goes over all possible values of E; compare to equation (5.3.11). Since the particles constituting the given system are noninteracting, we may express the eigenfunctions ψE(1,...,N) and the eigenvalues E in terms of the single-particle wavefunctions ui(m) and the single-particle energies εi. Moreover, we find it advisable to work with the wave vectors ki rather than the energies εi; so we write E = ℏ2K 2 2m = ℏ2 2m k2 1 + k2 2 + ··· + k2 N  , (4) where the ki on the right side are the wave vectors of the individual particles. Imposing periodic boundary conditions, the normalized single-particle wavefunctions are uk(r) = V −1/2 exp{i(k · r)}, (5) with k = 2πV −1/3n; (6) here, n stands for a three-dimensional vector whose components have values 0,±1,±2,.... The wavefunction ψ of the total system would then be, see equations (5.4.16) 10For a general survey of the density matrix and its applications, see ter Haar (1961). 134 Chapter 5. Formulation of Quantum Statistics and (5.4.17), ψK (1,...,N) = (N!)−1/2 X P δPP{uk1(1)...ukN (N)}, (7) where the magnitudes of the individual ki are such that (k2 1 + ··· + k2 N) = K 2. (8) The number δP in the expression for ψK is identically equal to +1 if the particles are bosons; for fermions, it is +1 or −1 according to whether the permutation P is even or odd. Thus, quite generally, we may write δP = (±1)[P], (9) where [P] denotes the order of the permutation; note that the upper sign in this expression holds for bosons while the lower sign holds for fermions. The factor (N!)−1/2 has been introduced here to ensure the normalization of the total wavefunction. Now, it makes no difference to the wavefunction (7) whether the permutations P are carried out on the coordinates 1,...,N or on the wave vectors k1,...,kN, because after all we are going to sum over all the N! permutations. Denoting the permuted coordinates by P1,...,PN and the permuted wave vectors by Pk1,... , PkN, equation (7) may be written as ψK (1,...,N) = (N!)−1/2 X P δP {uk1(P1)...ukN (PN)} (10a) = (N!)−1/2 X P δP {uPk1(1)...uPkN (N)}. (10b) Equations (10a and 10b) may now be substituted into (3), with the result ⟨1,...,N|e−β ˆ H|1′,...,N′⟩= (N!)−1 X K e−βℏ2K 2/2m ×  X ˜ P δ ˜ P{uk1(P1)...ukN (PN)} X ˜ P δ ˜ P{u∗ ˜ Pk1(1′)...u∗ ˜ PkN (N′)}  , (11) where P and ˜ P are any of the N! possible permutations. Now, since a permutation among the ki changes the wavefunction ψ at most by a sign, the quantity [ψψ∗] in (11) is insen-sitive to such a permutation; the same holds for the exponential factor as well. The summation over K is, therefore, equivalent to (1/N!) times a summation over all the vectors k1,...,kN independently of one another. Next, in view of the N-fold summation over the ki, all the permutations ˜ P will make equal contributions toward the sum (because they differ from one another only in the 5.5 The density matrix of a system of free particles 135 ordering of the ki). Therefore, we may consider only one of these permutations, say the one for which ˜ Pk1 = k1,..., ˜ PkN = kN (and hence δ ˜ P = 1 for both kinds of statistics), and include a factor of (N!). The net result is: ⟨1,...,N|e−β ˆ H|1′,...,N′⟩= (N!)−1 X k1,...,kN e−βℏ2(k2 1+···+k2 N )/2m "X P δP n uk1(P1)u∗ k1(1′) o ... n ukN (PN)u∗ kN (N′) o# . (12) Substituting from (5) and noting that, in view of the largeness of V, the summations over the ki may be replaced by integrations, equation (12) becomes ⟨1,...,N|e−β ˆ H|1′,...,N′⟩ = 1 N!(2π)3N X P δP Z e−βℏ2k2 1/2m+ik1·(P1−1′)d3k1 ... Z e−βℏ2k2 N /2m+ikN ·(PN−N′)d3kN  (13) = 1 N!  m 2πβℏ2 3N/2 X P δP[f (P1 −1′)... f (PN–N′)], (14) where f (ξ) = exp  − m 2βℏ2 ξ2  . (15) Here, use has been made of the mathematical result (5.3.12), which is clearly a special case of the present formula. Introducing the mean thermal wavelength, often referred to as the thermal deBroglie wavelength, λ = h (2πmkT)1/2 = ℏ 2πβ m 1/2 , (16) and rewriting our coordinates as r1,...,rN, the diagonal elements among (14) take the form ⟨r1,...,rN|e−β ˆ H|r1,...,rN⟩= 1 N!λ3N X P δP[f (Pr1 −r1) ... f (PrN −rN)], (17) where f (r) = exp −πr2/λ2 . (18) 136 Chapter 5. Formulation of Quantum Statistics To obtain the partition function of the system, we have to integrate (17) over all the coordinates involved. However, before we do that, we would like to make some observa-tions on the summation P P. First of all, we note that the leading term in this summation, namely the one for which Pri = ri, is identically equal to unity (because f (0) = 1). This is followed by a group of terms in which only one pair interchange (among the coordi-nates) has taken place; a typical term in this group will be f (rj −ri)f (ri −rj) where i ̸= j. This group of terms is followed by other groups of terms in which more than one pair interchange has taken place. Thus, we may write X P = 1 ± X i<j fijfji + X i<j<k fijfjkfki ± ··· , (19) where fij ≡f (ri −rj); again, note that the upper (lower) signs in this expansion pertain to a system of bosons (fermions). Now, the function fij vanishes rapidly as the distance rij becomes much larger than the mean thermal wavelength λ. It then follows that if the mean interparticle distance, (V/N)1/3, in the system is much larger than the mean thermal wavelength, that is, if nλ3 = nh3 (2πmkT)3/2 ≪1, (20) where n is the particle density in the system, then the sum P P in (19) may be approx-imated by unity. Accordingly, the partition function of the system would become, see equation (17), QN(V,T) ≡Tr e−β ˆ H ≈ 1 N!λ3N Z 1(d3Nr) = 1 N!  V λ3 N . (21) This is precisely the result obtained earlier for the classical ideal gas; see equation (3.5.9). Thus, we have obtained from our quantum-mechanical treatment the precise classical limit for the partition function QN(V,T). Incidentally, we have achieved something more. First, we have automatically recovered here the Gibbs correction factor (1/N!), which was introduced into the classical treatment on an ad hoc, semi-empirical basis. We, of course, tried to understand its origin in terms of the inherent indistinguishability of the parti-cles. Here, we see it coming in a very natural manner and its source indeed lies in the symmetrization of the wavefunctions of the system (which is ultimately related to the indistinguishability of the particles); compare to Problem 5.4. Second, we find here a formal justification for computing the number of microstates of a system corresponding to a given region of its phase space by dividing the volume of that region into cells of a “suitable” size and then counting instead the number of these cells. This correspondence becomes all the more transparent by noting that formula (21) 5.5 The density matrix of a system of free particles 137 is exactly equivalent to the classical expression QN(V,T) = 1 N! Z e−β(p2 1+···+p2 N )/2m d3Nqd3Np ω0 ! , (22) with ω0 =h3N. Thirdly, in deriving the classical limit we have also evolved a criterion that enables us to determine whether a given physical system can be treated classically; math-ematically, this criterion is given by condition (20). Now, in statistical mechanical studies, a system that cannot be treated classically is said to be degenerate; the quantity nλ3 may, therefore, be regarded as a degeneracy discriminant. Accordingly, the condition that clas-sical considerations may be applicable to a given physical system is that “the value of the degeneracy discriminant of the system be much less than unity.” Next, we note that, in the classical limit, the diagonal elements of the density matrix are given by ⟨r1,...,rN| ˆ ρ|r1,...,rN⟩≈  1 V N , (23) which is simply a product of N factors, each equal to (1/V). Recalling that, for a single particle in a box of volume V,⟨r| ˆ ρ|r⟩=(1/V), see equation (5.3.14), we infer that in the classical limit there is no spatial correlation among the various particles of the system. In general, however, spatial correlations exist even if the particles are supposedly noninter-acting; these correlations arise from the symmetrization of the wavefunctions and their magnitude is quite significant if the interparticle distances in the system are comparable with the mean thermal wavelength of the particles. To see this more clearly, we consider the simplest relevant case, namely the one with N = 2. The sum P P is now exactly equal to 1 ± [f (r12)]2. Accordingly, ⟨r1,r2|e−β ˆ H|r1,r2⟩= 1 2λ6 h 1 ± exp −2πr2 12/λ2i (24) and hence Q2(V,T) = 1 2λ6 ZZ h 1 ± exp(−2πr2 12/λ2i d3r1d3r2 = 1 2  V λ3 2  1 ± 1 V ∞ Z 0 exp(−2πr2/λ2)4πr2dr   (25) = 1 2  V λ3 2 " 1 ± 1 23/2 λ3 V !# ≈1 2  V λ3 2 . (26) 138 Chapter 5. Formulation of Quantum Statistics Combining (24) and (26), we obtain ⟨r1,r2| ˆ ρ|r1,r2} ≈1 V 2 1 ± exp −2πr2 12/λ2 . (27) Thus, if r12 is comparable to λ, the probability density (27) may differ considerably from the classical value (1/V)2. In particular, the probability density for a pair of bosons to be a distance r apart is larger than the classical, r-independent value by a factor of [1 + exp(−2πr2/λ2)], which becomes as high as 2 as r →0. The corresponding result for a pair of fermions is smaller than the classical value by a factor of [1 −exp(−2πr2/λ2)], which becomes as low as 0 as r →0. Thus, we obtain a positive spatial correlation among particles obeying Bose statistics and a negative spatial correlation among particles obeying Fermi statistics; see also Section 6.3. Another way of expressing correlations (among otherwise noninteracting particles) is by introducing a statistical interparticle potential vs(r) and then treating the particles classically (see Uhlenbeck and Gropper, 1932). The potential vs(r) must be such that the Boltzmann factor exp(−βvs) is precisely equal to the pair correlation function [...] in (27), that is, vs(r) = −kT ln 1 ± exp −2πr2/λ2 . (28) Figure 5.1 shows a plot of the statistical potential vs(r) for a pair of bosons or fermions. In the Bose case, the potential is throughout attractive, thus giving rise to a “statistical attraction” among bosons; in the Fermi case, it is throughout repulsive, giving rise to a “statistical repulsion” among fermions. In either case, the potential vanishes rapidly as r becomes larger than λ; accordingly, its influence becomes less and less important as the temperature of the system rises. 1 0 1.0 (r/) vs(r) 1 In 2 B.E. F.D. 0.5 FIGURE 5.1 The statistical potential vs(r) between a pair of particles obeying Bose–Einstein statistics or Fermi–Dirac statistics. Problems 139 Problems 5.1. Evaluate the density matrix ρmn of an electron spin in the representation that makes ˆ σx diagonal. Next, show that the value of ⟨σz⟩, resulting from this representation, is precisely the same as the one obtained in Section 5.3. Hint: The representation needed here follows from the one used in Section 5.3 by carrying out a transformation with the help of the unitary operator ˆ U = 1/√2 1/√2 −1/√2 1/√2 ! . 5.2. Prove that ⟨q|e−β ˆ H|q′⟩= exp  −β ˆ H  −iℏ∂ ∂q,q  δ(q −q′), where ˆ H(−iℏ∂/∂q,q) is the Hamiltonian operator of the system in the q-representation, which formally operates on the Dirac delta function δ(q −q′). Writing the δ-function in a suitable form, apply this result to (i) a free particle and (ii) a linear harmonic oscillator. 5.3. Derive the density matrix ρ for (i) a free particle and (ii) a linear harmonic oscillator in the momentum representation and study its main properties along the lines of Section 5.3. 5.4. Study the density matrix and the partition function of a system of free particles, using the unsymmetrized wavefunction (5.4.3) instead of the symmetrized wavefunction (5.5.7). Show that, following this procedure, one encounters neither the Gibbs’ correction factor (1/N!) nor a spatial correlation among the particles. 5.5. Show that in the first approximation the partition function of a system of N noninteracting, indistinguishable particles is given by QN(V,T) = 1 N!λ3N ZN(V,T), where ZN(V,T) = Z exp   −β X i<j vs(rij)   d3Nr, vs(r) being the statistical potential (5.5.28). Hence evaluate tht first-order correction to the equation of state of this system. 5.6. Determine the values of the degeneracy discriminant (nλ3) for hydrogen, helium, and oxygen at NTP . Make an estimate of the respective temperature ranges where the magnitude of this quantity becomes comparable to unity and hence quantum effects become important. 5.7. Show that the quantum-mechanical partition function of a system of N interacting particles approaches the classical form QN(V,T) = 1 N!h3N Z e−βE(q,p)d3Nqd3Np as the mean thermal wavelength λ becomes much smaller than (i) the mean interparticle distance (V/N)1/3 and (ii) a characteristic length r0 of the interparticle potential.11 5.8. Prove the following theorem due to Peierls.12 “If ˆ H is the hermitian Hamiltonian operator of a given physical system and {φn} an arbitrary orthonormal set of wavefunctions satisfying the symmetry requirements and the boundary 11See Huang (1963, Section 10.2). 12See Peierls (1938) and Huang (1963, Section 10.3). 140 Chapter 5. Formulation of Quantum Statistics conditions of the problem, then the partition function of the system satisfies the following inequality: Q(β) ≥ X n exp{−β⟨φn| ˆ H|φn⟩}; the equality holds when {φn} constitute a complete orthonormal set of eigenfunctions of the Hamiltonian itself.” 6 The Theory of Simple Gases We are now fully equipped with the formalism required for determining the macroscopic properties of a large variety of physical systems. In most cases, however, derivations run into serious mathematical difficulties, with the result that one is forced to restrict one’s analysis either to simpler kinds of systems or to simplified models of actual systems. In practice, even these restricted studies are carried out in a series of stages, the first stage of the process being highly “idealized.” The best example of such an idealization is the familiar ideal gas, a study of which is not only helpful in acquiring facility with the math-ematical procedures but also throws considerable light on the physical behavior of gases actually met with in nature. In fact, it also serves as a base on which the theory of real gases can be founded; see Chapter 10. In this chapter we propose to derive, and at some length discuss, the most basic pro-perties of simple gaseous systems obeying quantum statistics; the discussion will include some of the essential features of diatomic and polyatomic gases and chemical equilibrium. 6.1 An ideal gas in a quantum-mechanical microcanonical ensemble We consider a gaseous system of N noninteracting, indistinguishable particles confined to a space of volume V and sharing a given energy E. The statistical quantity of interest in this case is (N,V,E) which, by definition, denotes the number of distinct microstates accessible to the system under the macrostate (N,V,E). While determining this number, we must remember that a failure to take into account the indistinguishability of the parti-cles in a proper manner could lead to results which, except in the classical limit, may not be acceptable. With this in mind, we proceed as follows. Since, for large V, the single-particle energy levels in the system are very close to one another, we may divide the energy spectrum into a large number of “groups of levels,” which may be referred to as energy cells; see Figure 6.1. Let εi denote the average energy of a level, and gi the (arbitrary) number of levels, in the ith cell; we assume that all gi ≫1. In a particular situation, we may have n1 particles in the first cell, n2 particles in the second cell, and so on. Clearly, the distribution set {ni} must conform to the conditions X i ni = N (1) Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00006-2 © 2011 Elsevier Ltd. All rights reserved. 141 142 Chapter 6. The Theory of Simple Gases g4 ; n4 g3 ; n3 g2 ; n2 g1 ; n1 ´1 ´2 ´3 ´4 ´ FIGURE 6.1 The grouping of the single-particle energy levels into “cells.” and X i niεi = E. (2) Then (N,V,E) = X′ {ni} W{ni}, (3) where W{ni} is the number of distinct microstates associated with the distribution set {ni} while the primed summation goes over all distribution sets that conform to conditions (1) and (2). Next, W{ni} = Y i w(i), (4) where w(i) is the number of distinct microstates associated with the ith cell of the spectrum (the cell that contains ni particles, to be accommodated among gi levels) while the product goes over all the cells in the spectrum. Clearly, w(i) is the number of distinct ways in which the ni identical, and indistinguishable, particles can be distributed among the gi levels of the ith cell. This number, in the Bose–Einstein case, is given by, see equation (3.8.25), wB.E.(i) = (ni + gi −1)! ni!(gi −1)! , (5) so that WB.E.{ni} = Y i (ni + gi −1)! ni!(gi −1)! . (6) 6.1 An ideal gas in a quantum-mechanical microcanonical ensemble 143 In the Fermi–Dirac case, no single level can accommodate more than one particle; accord-ingly, the number ni cannot exceed gi. The number w(i) is then given by the “number of ways in which the gi levels can be divided into two subgroups — one consisting of ni levels (which will have one particle each) and the other consisting of (gi −ni) levels (which will be unoccupied).” This number is given by wF.D.(i) = gi! ni!(gi −ni)!, (7) so that WF.D.{ni} = Y i gi! ni!(gi −ni)!. (8) For completeness, we may include the classical — or what is generally known as the Maxwell–Boltzmann — case as well. There, the particles are regarded as distinguishable, with the result that any of the ni particles may be put into any of the gi levels, inde-pendently of one another, and the resulting states may all be regarded as distinct; the number of these states is clearly (gi)ni. Moreover, the distribution set {ni} in this case is itself regarded as obtainable in N ! n1!n2!... (9) different ways which, on the introduction of the Gibbs correction factor, lead to a “weight factor” of 1 n1!n2!... = Y i 1 ni!; (10) see also Section 1.6, especially equation (1.6.2). Combining these two results, we obtain WM.B.{ni} = Y i (gi)ni ni! . (11) Now, the entropy of the system would be given by S(N,V,E) = kln(N,V,E) = kln X′ {ni} W{ni}  . (12) It can be shown that, under the conditions of our analysis, the logarithm of the sum on the right side of (12) can be approximated by the logarithm of the largest term in the sum; see Problem 3.4. We may, therefore, replace (12) by S(N,V,E) ≈klnW{n∗ i }, (13) 144 Chapter 6. The Theory of Simple Gases where {n∗ i } is the distribution set that maximizes the number W{ni}; the numbers n∗ i are clearly the most probable values of the distribution numbers ni. The maximization, how-ever, is to be carried out under the restrictions that the quantities N and E remain constant. This can be done by the method of Lagrange’s undetermined multipliers; see Section 3.2. Our condition for determining the most probable distribution set {n∗ i } now turns out to be, see equations (1), (2), and (13), δ lnW{ni} − " α X i δni + β X i εiδni # = 0. (14) For lnW{ni}, we obtain from equations (6), (8), and (11), assuming that not only all gi but also all ni ≫1 (so that the Stirling approximation ln(x!) ≈xlnx −x can be applied to all the factorials that appear in these expressions), lnW{ni} = X i lnw(i) ≈ X i  ni ln  gi ni −a  −gi a ln  1 −ani gi  , (15) where a = −1 for the B.E. case, +1 for the F.D. case, and 0 for the M.B. case. Equation (14) then becomes X i  ln  gi ni −a  −α −βεi  ni=n∗ i δni = 0. (16) In view of the arbitrariness of the increments δni in (16), we must have (for all i) ln gi n∗ i −a ! −α −βεi = 0, (17) so that1 n∗ i = gi eα+βεi + a. (18) The fact that n∗ i turns out to be directly proportional to gi prompts us to interpret the quantity n∗ i gi = 1 eα+βεi + a, (18a) which is actually the most probable number of particles per energy level in the ith cell, as the most probable number of particles in a single level of energy εi. Incidentally, our final result (18a) is totally independent of the manner in which the energy levels of the particles 1For a critique of this derivation, see Landsberg (1954a, 1961). 6.1 An ideal gas in a quantum-mechanical microcanonical ensemble 145 are grouped into cells, so long as the number of levels in each cell is sufficiently large. As shown in Section 6.2, formula (18a) can also be derived without grouping energy levels into cells at all; in fact, it is only then that this result becomes truly acceptable! Substituting (18) into (15), we obtain for the entropy of the gas S k ≈lnW{n∗ i } = X i " n∗ i ln gi n∗ i −a ! −gi a ln  1 −an∗ i gi # = X i h n∗ i (α + βεi) + gi a ln n 1 + ae−α−βεi oi . (19) The first sum on the right side of (19) is identically equal to αN while the second sum is identically equal to βE. For the third sum, therefore, we have 1 a X i gi ln n 1 + ae−α−βεi o = S k −αN −βE. (20) Now, the physical interpretation of the parameters α and β here is going to be precisely the same as in Section 4.3, namely α = −µ kT , β = 1 kT ; (21) for confirmation see Section 6.2. The right side of equation (20) is, therefore, equal to S k + µN kT −E kT = G −(E −TS) kT = PV kT . (22) The thermodynamic pressure of the system is, therefore, given by PV = kT a X i h gi ln n 1 + ae−α−βεi oi . (23) In the Maxwell–Boltzmann case (a →0), equation (23) takes the form PV = kT X i gie−α−βεi = kT X i n∗ i = NkT, (24) which is the familiar equation of state of the classical ideal gas. Note that equation (24) for the Maxwell–Boltzmann case holds irrespective of the details of the energy spectrum εi. It will be recognized that the expression a−1 P i[ ] in equation (23), being equal to the thermodynamic quantity (PV/kT), ought to be identical to the q-potential of the ideal gas. One may, therefore, expect to obtain from this expression all the macroscopic properties of this system. However, before demonstrating this, we would like to first develop the formal theory of an ideal gas in the canonical and grand canonical ensembles. 146 Chapter 6. The Theory of Simple Gases 6.2 An ideal gas in other quantum-mechanical ensembles In the canonical ensemble the thermodynamics of a given system is derived from its partition function QN(V,T) = X E e−βE, (1) where E denotes the energy eigenvalues of the system while β = 1/kT. Now, an energy value E can be expressed in terms of the single-particle energies ε; for instance, E = X ε nεε, (2) where nε is the number of particles in the single-particle energy state ε. The values of the numbers nε must satisfy the condition X ε nε = N. (3) Equation (1) may then be written as QN(V,T) = X′ {nε} g{nε}e −β P ε nεε , (4) where g{nε} is the statistical weight factor appropriate to the distribution set {nε} and the summation P′ goes over all distribution sets that conform to the restrictive condition (3). The statistical weight factor in different cases is given by gB.E.{nε} = 1, (5) gF.D.{nε} = ( 1 if all nε = 0 or 1 0 otherwise, (6) and gM.B.{nε} = Y ε 1 nε!. (7) Note that in the present treatment we are dealing with single-particle states as individ-ual states, without requiring them to be grouped into cells; indeed, the weight factors (5), (6), and (7) follow straightforwardly from their respective predecessors (6.1.6), (6.1.8), and (6.1.11) by putting all gi = 1. 6.2 An ideal gas in other quantum-mechanical ensembles 147 First of all, we work out the Maxwell–Boltzmann case. Substituting (7) into (4), we get QN(V,T) = X′ {nε} " Y ε 1 nε! !Y ε  e−βεnε # = 1 N! X′ {nε}  N! Q ε nε! Y ε  e−βεnε  . (8) Since the summation here is governed by condition (3), it can be evaluated with the help of the multinomial theorem, with the result QN(V,T) = 1 N! "X ε e−βε #N = 1 N![Q1(V,T)]N, (9) in agreement with equation (3.5.15). The evaluation of Q1 is, of course, straightforward. One obtains, using the asymptotic formula (2.4.7) for the number of single-particle states with energies lying between ε and ε + dε, Q1(V,T) ≡ X ε e−βε ≈2πV h3 (2m)3/2 ∞ Z 0 e−βεε1/2dε = V/λ3, (10) where λ[= h/(2πmkT)1/2] is the mean thermal wavelength of the particles. Hence QN(V,T) = V N N!λ3N , (11) from which complete thermodynamics of this system can be derived; see, for example, Section 3.5. Further, we obtain for the grand partition function of this system Q(z,V,T) = ∞ X N=0 zNQN(V,T) = exp(zV/λ3); (12) compare to equation (4.4.3). We know that the thermodynamics of the system follows equally well from the expression for Q. In the Bose–Einstein and Fermi–Dirac cases, we obtain, by substituting (5) and (6) into (4), QN(V,T) = X′ {nε}  e −β P ε nεε ; (13) 148 Chapter 6. The Theory of Simple Gases the difference between the two cases, B.E. and F.D., arises from the values that the numbers nε can take. Now, in view of restriction (3) on the summation P′, an explicit evaluation of the partition function QN in these cases is rather cumbersome. The grand partition function Q, on the other hand, turns out to be more easily tractable; we have Q(z,V,T) = ∞ X N=0  zN X′ {nε} e −β P ε nεε  (14a) = ∞ X N=0 "X′ {nε} Y ε  ze−βεnε # . (14b) Now, the double summation in (14b) — first over the numbers nε constrained by a fixed value of the total number N, and then over all possible values of N — is equivalent to a summation over all possible values of the numbers nε, independently of one another. Hence, we may write Q(z,V,T) = X n0,n1,... h ze−βε0 n0  ze−βε1 n1 ... i =  X n0  ze−βε0 n0    X n1  ze−βε1 n1  .... (15) Now, in the Bose–Einstein case the nε can be either 0 or 1 or 2 or ..., while in the Fermi– Dirac case they can be only 0 or 1. Therefore, Q(z,V,T) =          Y ε 1 (1 −ze−βε) in the B.E. case, with ze−βε < 1 (16a) Y ε (1 + ze−βε) in the F.D. case. (16b) The q-potential of the system is thus given by q(z,V,T) ≡PV kT ≡lnQ(z,V,T) = ∓ X ε ln(1 ∓ze−βε); (17) compare to equation (6.1.23), with gi = 1. The identification of the fugacity z with the quantity e−α of equation (6.1.23) is quite natural; accordingly, α = −µ/kT. As usual, the upper (lower) sign in equation (17) corresponds to the Bose (Fermi) case. In the end, we may write our results for q in a form applicable to all three cases: q(z,V,T) ≡PV kT = 1 a X ε ln(1 + aze−βε), (18) 6.3 Statistics of the occupation numbers 149 where a = −1, +1, or 0, depending on the statistics governing the system. In particular, the classical case (a →0) gives qM.B. = z X ε e−βε = zQ1, (19) in agreement with equation (4.4.4). From (18), it follows that N ≡z ∂q ∂z  V,T = X ε 1 z−1eβε + a (20) and E ≡− ∂q ∂β  z,V = X ε ε z−1eβε + a. (21) At the same time, the mean occupation number ⟨nε⟩of level ε turns out to be, see equa-tions (14a) and (17), ⟨nε⟩= 1 Q " −1 β ∂Q ∂ε  z,T, all other ε # ≡−1 β ∂q ∂ε  z,T, all other ε = 1 z−1eβε + a, (22) in keeping with equations (20) and (21). Comparing our final result (22) with its coun-terpart (6.1.18a), we find that the mean value ⟨n⟩and the most probable value n∗of the occupation number n of a single-particle state are indeed identical. 6.3 Statistics of the occupation numbers Equation (6.2.22) gives the mean occupation number of a single-particle state with energy ε as an explicit function of the quantity (ε −µ)/kT: ⟨nε⟩= 1 e(ε−µ)/kT + a. (1) The functional behavior of this number is shown in Figure 6.2. In the Fermi–Dirac case (a = +1), the mean occupation number never exceeds unity, for the variable nε itself cannot have a value other than 0 or 1. Moreover, for ε < µ and |ε −µ| ≫kT, the mean occu-pation number tends to its maximum possible value 1. In the Bose–Einstein case (a = −1), we must have µ < all ε; see equation (6.2.16a). In fact, when µ becomes equal to the low-est value of ε, say ε0, the occupancy of that particular level becomes infinitely high, which leads to the phenomenon of Bose–Einstein condensation; see Sections 7.1 and 7.2. For 150 Chapter 6. The Theory of Simple Gases 2 1 0 22 21 1 3 2 0 1 2 3 n´ ´ 2 kT   FIGURE 6.2 The mean occupation number ⟨nε⟩of a single-particle energy state ε in a system of noninteracting particles: curve 1 is for fermions, curve 2 for bosons, and curve 3 for the Maxwell–Boltzmann particles. µ < ε0, all values of (ε −µ) are positive and the behavior of all ⟨nε⟩is nonsingular. Finally, in the Maxwell–Boltzmann case (a = 0), the mean occupation number takes the familiar form ⟨nε⟩M.B. = exp{(µ −ε)/kT} ∝exp(−ε/kT). (2) The important thing to note here is that the distinction between the quantum statistics (a = ±1) and the classical statistics (a = 0) becomes imperceptible when, for all values of ε that are of practical interest, exp{(ε −µ)/kT} ≫1. (3) In that event, equation (1) essentially reduces to (2) and we may write, instead of (3), ⟨nε⟩≪1. (4) Condition (4) is quite understandable, for it implies that the probability of any of the nε being greater than unity is quite negligible, with the result that the classical weight factors g{nε}, as given by equation (6.2.7), become essentially equal to 1. The distinction between the classical treatment and the quantum-mechanical treatment then becomes rather insignificant. Correspondingly, we find, see Figure 6.2, that for large values of (ε −µ)/kT the quantum curves 1 and 2 essentially merge into the classical curve 3. Since we already know that the higher the temperature of the system the better the validity of the classical treatment, condition (3) also implies that µ, the chemical potential of the system, must be negative and large in magnitude. This means that the fugacity z[≡exp(µ/kT)] of the system must be much smaller than unity; see also equation (6.2.22). One can see, from 6.3 Statistics of the occupation numbers 151 equations (4.4.6) and (4.4.29), that this is further equivalent to the requirement Nλ3 V ≪1, (5) which agrees with condition (5.5.20). We shall now examine statistical fluctuations in the variable nε. Going a step further from the calculation that led to equation (6.2.22), we have ⟨n2 ε⟩= 1 Q " −1 β ∂ ∂ε 2 Q # z,T, all other ε ; (6) it follows that ⟨n2 ε⟩−⟨nε⟩2 = " −1 β ∂ ∂ε 2 lnQ # z,T, all other ε =  −1 β ∂ ∂ε  ⟨nε⟩  z,T . (7) For the relative mean-square fluctuation, we obtain (irrespective of the statistics obeyed by the particles) ⟨n2 ε⟩−⟨nε⟩2 ⟨nε⟩2 =  1 β ∂ ∂ε  1 ⟨nε⟩  = z−1eβε; (8) of course, the actual value of this quantity will depend on the statistics of the particles because, for a given particle density (N/V) and a given temperature T, the value of z will be different for different statistics. It seems more instructive to write (8) in the form ⟨n2 ε⟩−⟨nε⟩2 ⟨nε⟩2 = 1 ⟨nε⟩−a. (9) In the classical case (a = 0), the relative fluctuation is normal. In the Fermi–Dirac case, it is given by 1/⟨nε⟩−1, which is below normal and tends to vanish as ⟨nε⟩→1. In the Bose– Einstein case, the fluctuation is clearly above normal.2 Obviously, this result would apply to a gas of photons and, hence, to the oscillator states in the black-body radiation. In the latter context, Einstein derived this result as early as 1909 following Planck’s approach and even pointed out that the term 1 in the expression for the fluctuation may be attributed to the wave character of the radiation and the term 1/⟨nε⟩to the particle character of the photons; for details, see Kittel (1958), ter Haar (1968). Closely related to the subject of fluctuations is the problem of “statistical correlations in photon beams,” which have been observed experimentally (see Hanbury Brown and 2The special case of fluctuations in the ground state occupation number, n0, of a Bose–Einstein system has been discussed by Wergeland (1969) and by Fujiwara, ter Haar, and Wergeland (1970). 152 Chapter 6. The Theory of Simple Gases Twiss, 1956, 1957, 1958) and have been explained theoretically in terms of the quantum-statistical nature of these fluctuations (see Purcell, 1956; Kothari and Auluck, 1957). For further details, refer to Mandel, Sudarshan, and Wolf (1964); and Holliday and Sage (1964). For greater understanding of the statistics of the occupation numbers, we evaluate the quantity pε(n), which is the probability that there are exactly n particles in a state of energy ε. Referring to equation (6.2.14b), we infer that pε(n) ∝(ze−βε)n. On normalization, it becomes in the Bose–Einstein case pε(n)|B.E. =  ze−βεn h 1 −ze−βεi =  ⟨nε⟩ ⟨nε⟩+ 1 n 1 ⟨nε⟩+ 1 = (⟨nε⟩)n (⟨nε⟩+ 1)n+1 . (10) In the Fermi–Dirac case, we get pε(n)|F.D. =  ze−βεn h 1 + ze−βεi−1 = ( 1 −⟨nε⟩ for n = 0 ⟨nε⟩ for n = 1. (11) In the Maxwell–Boltzmann case, we have pε(n) ∝(ze−βε)n/n! instead; see equation (6.2.8). On normalization, we get pε(n)|M.B. = ze−βεn /n! exp ze−βε = (⟨nε⟩)n n! e−⟨nε⟩. (12) Distribution (12) is clearly a Poisson distribution, for which the mean square deviation of the variable in question is equal to the mean value itself; compare to equation (9), with a = 0. It also resembles the distribution of the total particle number N in a grand canonical ensemble consisting of ideal, classical systems; see Problem 4.4. We also note that the ratio pε(n)/pε(n −1) in this case varies inversely with n, which is a “normal” statistical behavior of uncorrelated events. On the other hand, the distribution in the Bose–Einstein case is geometric, with a con-stant common ratio ⟨nε⟩/(⟨nε⟩+ 1). This means that the probability of a state ε acquiring one more particle for itself is independent of the number of particles already occupying that state; thus, in comparison with the “normal” statistical behavior, bosons exhibit a spe-cial tendency of “bunching” together, which means a positive statistical correlation among them. In contrast, fermions exhibit a negative statistical correlation. 6.4 Kinetic considerations The thermodynamic pressure of an ideal gas is given by equation (6.1.23) or (6.2.18). In view of the largeness of volume V, the single-particle energy states ε would be so close 6.4 Kinetic considerations 153 to one another that a summation over them may be replaced by integration. One thereby gets P = kT a ∞ Z 0 ln h 1 + aze−βε(p)i 4πp2dp h3 = 4πkT ah3  p3 3 ln h 1 + aze−βε(p)i ∞ 0 + ∞ Z 0 p3 3 aze−βε(p) 1 + aze−βε(p) β dε dpdp  . The integrated part vanishes at both limits while the rest of the expression reduces to P = 4π 3h3 ∞ Z 0 1 z−1eβε(p) + a  p dε dp  p2dp. (1) Now, the total number of particles in the system is given by N = Z ⟨np⟩Vd3p h3 = 4πV h3 ∞ Z 0 1 z−1eβε(p) + ap2dp. (2) Comparing (1) and (2), we can write P = 1 3 N V p dε dp = 1 3n⟨pu⟩, (3) where n is the particle density in the gas and u the speed of an individual particle. If the relationship between the energy ε and the momentum p is of the form ε ∝ps, then P = s 3n⟨ε⟩= s 3 E V ; (4) the particular cases s = 1 and s = 2 are pretty easy to recognize. It should be noted here that results (3) and (4) hold independently of the statistics obeyed by the particles. The structure of formula (3) suggests that the pressure of the gas arises essentially from the physical motion of the particles; it should, therefore, be derivable from kinetic consid-erations alone. To do this, we consider the bombardment, by the particles of the gas, on the walls of the container. Let us take, for example, an element of area dA on one of the walls normal to the z-axis, see Figure 6.3, and focus our attention on those particles whose velocity lies between u and u + du; the number of such particles per unit volume may be denoted by nf (u)du, where Z all u f (u)du = 1. (5) 154 Chapter 6. The Theory of Simple Gases (udt) dA z FIGURE 6.3 The molecular bombardment on one of the walls of the container. Now, the question is: how many of these particles will strike the area dA in time dt? The answer is: all those particles that happen to lie in a cylindrical region of base dA and height udt, as shown in Figure 6.3. Since the volume of this region is (dA · u)dt, the number of such particles would be {(dA · u)dt × nf (u)du}. On reflection from the wall, the normal component of the momentum of a particle would undergo a change from pz to −pz; as a result, the normal momentum imparted by these particles per unit time to a unit area of the wall would be 2 pz{uznf (u)du}. Integrating this expression over all relevant u, we obtain the total normal momentum imparted per unit time to a unit area of the wall by all the particles of the gas which, by definition, is the kinetic pressure of the gas: P = 2n ∞ Z ux=−∞ ∞ Z uy=−∞ ∞ Z uz=0 pzuzf (u)duxduyduz. (6)3 Since (i) f (u) is a function of u alone and (ii) the product (pzuz) is an even function of uz, the foregoing result may be written as P = n Z all u (pzuz)f (u)du. (7) Comparing (7) with (5), we obtain P = n⟨pzuz⟩= n⟨pucos2 θ⟩ (8) = 1 3n⟨pu⟩, (9) which is identical to (3). 3Clearly, only those velocities for which uz > 0 are relevant here. 6.5 Gaseous systems composed of molecules with internal motion 155 In a similar manner, we can determine the rate of effusion of the gas particles through a hole (of unit area) in the wall. This is given by, compared to (6), R = n ∞ Z ux=−∞ ∞ Z uy=−∞ ∞ Z uz=0 uzf (u)duxduyduz (10) = n 2π Z φ=0 π/2 Z θ=0 ∞ Z u=0 {ucosθf (u)}(u2 sinθ dudθ dφ); (11) note that the condition uz > 0 restricts the range of the angle θ between the values 0 and π/2. Carrying out integrations over θ and φ, we obtain R = nπ ∞ Z 0 f (u)u3 du. (12) In view of the fact that ∞ Z 0 f (u)(4πu2 du) = 1, (5a) equation (12) may be written as R = 1 4n⟨u⟩. (13) Again, this result holds independently of the statistics obeyed by the particles. It is obvious that the velocity distribution among the effused particles is considerably different from the one among the particles inside the container. This is due to the fact that, firstly, the velocity component uz of the effused particles must be positive (which intro-duces an element of anisotropy into the distribution) and, secondly, the particles with larger values of uz appear with an extra weightage, the weightage being directly propor-tional to the value of uz; see equation (10). As a result of this, (i) the effused particles carry with them a net forward momentum, thus causing the container to experience a recoil force, and (ii) they carry away a relatively large amount of energy per particle, thus leaving the gas in the container at not only a progressively decreasing pressure and density but also a progressively decreasing temperature; see Problem 6.14. 6.5 Gaseous systems composed of molecules with internal motion In most of our studies so far we have considered only the translational part of the molecu-lar motion. Though this aspect of motion is invariably present in a gaseous system, other 156 Chapter 6. The Theory of Simple Gases aspects, which are essentially concerned with the internal motion of the molecules, also exist. It is only natural that in the calculation of the physical properties of such a sys-tem, contributions arising from these motions are also taken into account. In doing so, we shall assume that (i) the effects of the intermolecular interactions are negligible and (ii) the nondegeneracy criterion nλ3 = nh3 (2πmkT)3/2 ≪1 (5.5.20) is fulfilled; this makes our system an ideal, Boltzmannian gas. Under these assumptions, which hold sufficiently well in a large number of applications, the partition function of the system is given by QN(V,T) = 1 N![Q1(V,T)]N, (1) where Q1(V,T) = V λ3 j(T); (2) the factor within the curly brackets is the familiar translational partition function of a molecule, while j(T) is the partition function corresponding to internal motions. The latter may be written as j(T) = X i gie−εi/kT, (3) where εi is the energy associated with a state of internal motion (characterized by the quantum numbers i), while gi is the multiplicity of that state. The contributions made by the internal motions of the molecules, over and above the translational degrees of freedom, follow straightforwardly from the function j(T). We obtain Aint = −NkT lnj, (4) µint = −kT lnj, (5) Sint = Nk  lnj + T ∂ ∂T lnj  , (6) Uint = NkT2 ∂ ∂T lnj, (7) and (CV )int = Nk ∂ ∂T  T2 ∂ ∂T lnj  . (8) 6.5 Gaseous systems composed of molecules with internal motion 157 Thus, the central problem in this study is to derive an explicit expression for the function j(T) from a knowledge of the internal states of the molecules. For this, we note that the internal state of a molecule is determined by (i) the electronic state, (ii) the state of the nuclei, (iii) the vibrational state, and (iv) the rotational state. Rigorously speaking, these four modes of excitation mutually interact; in many cases, however, they can be treated independently of one another. We can then write j(T) = jelec(T)jnuc(T)jvib(T)jrot(T), (3a) with the result that the net contribution made by the internal motions to the various thermodynamic properties of the system is given by a simple sum of the four respective contributions. There is one interaction, however, that plays a special role in the case of homonuclear molecules, such as AA, and which is between the states of the nuclei and the rotational states. In such a case, we better write j(T) = jelec(T)jnuc−rot(T)jvib(T). (3b) We now examine this problem for various systems in the order of increasing complexity. 6.5.A Monatomic molecules For simplicity, we consider a monatomic gas at temperatures such that the thermal energy kT is small in comparison with the ionization energy εion; for different atoms, this amounts to the condition T ≪εion/k ∼104 −105 K. At these temperatures, the number of ionized atoms in the gas would be insignificant. The same would be true of atoms in the excited states, for the separation of any of the excited states from the ground state of the atom is generally of the same order of magnitude as the ionization energy itself. Thus, we may regard all atoms in the gas to be in their (electronic) ground state. Now, there is a special class of atoms, namely He, Ne, A,..., which, in their ground state, possess neither orbital angular momentum nor spin (L = S = 0). Their (electronic) ground state is clearly a singlet, with ge = 1. The nucleus, however, possesses a degeneracy that arises from the possibility of different orientations of the nuclear spin.4 If the value of this spin is Sn, the corresponding degeneracy factor gn = 2Sn + 1. Moreover, a monatomic molecule cannot have any vibrational or rotational states. The internal partition function (3a) of such a molecule is, therefore, given by j(T) = (g)gr.st. = ge · gn = 2Sn + 1. (9) 4As is well known, the presence of the nuclear spin gives rise to the so-called hyperfine structure in the electronic states. However, the intervals of this structure are such that, for practically all temperatures of interest, they are small in comparison with kT; for concreteness, these intervals correspond to T-values of the order of 10−1 to 100 K. Accordingly, in the evaluation of the partition function j(T), the hyperfine splitting of the electronic state may be disregarded while the multiplicity introduced by the nuclear spin may be taken into account through a degeneracy factor. 158 Chapter 6. The Theory of Simple Gases Equations (4) through (8) then tell us that the internal motions in this case contribute only toward properties such as the chemical potential and the entropy of the gas; they do not contribute toward the internal energy and the specific heat. If, on the other hand, the ground state does not possess orbital angular momentum but possesses spin (L = 0, S ̸= 0 — as, for example, in the case of alkali atoms), then the ground state will still have no fine structure; it will, however, have a degeneracy ge = 2S + 1. As a result, the internal partition function j(T) will get multiplied by a factor of (2S + 1) and the properties such as the chemical potential and the entropy of the gas will get modified accordingly. In other cases, the ground state of the atom may possess both orbital angular momen-tum and spin (L ̸= 0,S ̸= 0); the ground state would then possess a definite fine structure. The intervals of this structure are, in general, comparable to kT; hence, in the evaluation of the partition function, the energies of the various components of the fine structure will have to be taken into account. Since these components differ from one another in the value of the total angular momentum J, the relevant partition function may be written as jelec(T) = X J (2J + 1)e−εJ/kT. (10) The foregoing expression simplifies considerably in the following limiting cases: (a) kT ≫all εJ; then jelec(T) ≃ X J (2J + 1) = (2L + 1)(2S + 1). (10a) (b) kT ≪all εJ; then jelec(T) ≃(2J0 + 1)e−ε0/kT, (10b) where J0 is the total angular momentum, and ε0 the energy, of the atom in the lowest state. In either case, the electronic motion makes no contribution toward the specific heat of the gas. Of course, at intermediate temperatures, we do obtain a contribution toward this property. And, in view of the fact that both at high and low temperatures the specific heat tends to be equal to the translational value 3 2Nk, it must pass through a maximum at a temperature comparable to the separation of the fine structure levels.5 Needless to say, the multiplicity (2Sn + 1) introduced by the nuclear spin must be taken into account in each case. 6.5.B Diatomic molecules Now we consider a diatomic gas at temperatures such that kT is small compared to the energy of dissociation; for different molecules, this amounts once again to the condition 5It seems worthwhile to note here that the values of 1εJ/k for the components of the normal triplet term of oxygen are 230 K and 320 K, while those for the normal quintuplet term of iron range from 600 to 1,400 K. 6.5 Gaseous systems composed of molecules with internal motion 159 T ≪εdiss/k ∼104 −105 K. At these temperatures the number of dissociated molecules in the gas would be insignificant. At the same time, in most cases, there would be practi-cally no molecules in the excited states as well, for the separation of any of these states from the ground state of the molecule is in general comparable to the dissociation energy itself.6 Accordingly, in the evaluation of j(T), we have to take into account only the lowest electronic state of the molecule. The lowest electronic state, in most cases, is nondegenerate: ge = 1. We then need not consider any further the question of the electronic state making a contribution toward the thermodynamic properties of the gas. However, certain molecules (though not very many) have, in their lowest electronic state, either (i) a nonzero orbital angular momentum (3 ̸= 0) or (ii) a nonzero spin (S ̸= 0) or (iii) both. In case (i), the electronic state acquires a twofold degeneracy corresponding to the two possible orientations of the oribital angular momentum relative to the molecular axis;7 as a result, ge = 2. In case (ii), the state acquires a degeneracy 2S + 1 corresponding to the space quantization of the spin.8 In both these cases the chemical potential and the entropy of the gas are modified by the multiplicity of the electronic state, while the energy and the specific heat remain unaf-fected. In case (iii), we encounter a fine structure that necessitates a rather detailed study because the intervals of this structure are generally of the same order of magnitude as kT. In particular, for a doublet fine-structure term, such as the one that arises in the molecule NO (51/2,3/2 with a separation of 178 K, the components themselves being 3-doublets), we have for the electronic partition function jelec(T) = g0 + g1e−1/kT, (11) where g0 and g1 are the degeneracy factors of the two components while 1 is their separa-tion energy. The contribution made by (11) toward the various thermodynamic properties of the gas can be calculated with the help of formulae (4) through (8). In particular, we obtain for the contribution toward the specific heat (CV )elec = Nk (1/kT)2 [1 + (g0/g1)e1/kT] [1 + (g1/g0)e−1/kT]. (12) We note that this contribution vanishes both for T ≪1/k and for T ≫1/k and is maxi-mum for a certain temperature ∼1/k; compare to the corresponding situation in the case of monatomic molecules. 6An odd case arises with oxygen. The separation between its normal term 36 and the first excited term 11 is about 11,250 K, whereas the dissociation energy is about 55,000 K. The relevant factor e−ε1/kT, therefore, can be quite significant even when the factor e−εdiss/kT is not, say for T ∼2000 to 6000 K. 7Strictly speaking, the term in question splits into two levels — the so-called 3-doublet. The separation of the levels, however, is such that we can safely neglect it. 8The separation of the resulting levels is again negligible from the thermodynamic point of view; as an example, one may cite the very narrow triplet term of O2. 160 Chapter 6. The Theory of Simple Gases We now consider the effect of the vibrational states of the molecules on the thermo-dynamic properties of the gas. To have an idea of the temperature range over which this effect would be significant, we note that the magnitude of the corresponding quantum of energy, namely ℏω, for different diatomic gases is of order 103 K. Thus, we would obtain full contributions (consistent with the dictates of the equipartition theorem) at temperatures of the order of 104 K or more, and practically no contribution at temperatures of the order of 102 K or less. Let us assume that the temperature is not high enough to excite vibra-tional states of large energy; the oscillations of the nuclei then remain small in amplitude and hence harmonic. The energy levels for a mode of frequency ω are then given by the well-known expression (n + 1 2)ℏω.9 The evaluation of the vibrational partition function jvib(T) is quite elementary; see Section 3.8. In view of the rapid convergence of the series involved, the summation may formally be extended to n = ∞. The corresponding contributions toward the various ther-modynamic properties of the system are then given by equations (3.8.16) through (3.8.21). In particular, (CV )vib = Nk 2v T 2 e2v/T (e2v/T −1)2 ; 2v = ℏω k . (13) We note that for T ≫2v, the vibrational specific heat is very nearly equal to the equipar-tition value Nk; otherwise, it is always less than Nk. In particular, for T ≪2v, the specific heat tends to zero (see Figure 6.4); the vibrational degrees of freedom are then said to be “frozen.” At sufficiently high temperatures, when vibrations with large n are also excited, the effects of anharmonicity and of interaction between the vibrational and the rotational modes of the molecule can become important.10 However, since this happens only at large n, the relevant corrections to the various thermodynamic quantities can be determined even classically; see Problems 3.29 and 3.30. One finds that the first-order correction to Cvib is directly proportional to the temperature of the gas. Finally, we consider the effect of (i) the states of the nuclei and (ii) the rotational states of the molecule; wherever necessary, we shall take into account the mutual interaction of these modes. This interaction is of no relevance in the case of heteronuclear molecules, such as AB; it is, however, important in the case of homonuclear molecules, such as AA. We may, therefore, consider the two cases separately. The states of the nuclei in the heteronuclear case may be treated separately from the rotational states of the molecule. Proceeding in the same manner as for monatomic molecules, we conclude that the effect of the nuclear states is adequately taken care of 9It may be pointed out that the vibrational motion of a molecule is influenced by the centrifugal force arising from the molecular rotation. This leads to an interaction between the rotational and the vibrational modes. However, unless the temperature is too high, this interaction can be neglected and the two modes treated independently of one another. 10In principle, these two effects are of the same order of magnitude. 6.5 Gaseous systems composed of molecules with internal motion 161 1.0 0.5 0 0.5 1.0 (T/v) (Cvib/Nk) 1.5 2.0 FIGURE 6.4 The vibrational specific heat of a gas of diatomic molecules. At T = 2v, the specific heat is already about 93 percent of the equipartition value. through a degeneracy factor gn. Denoting the spins of the two nuclei by SA and SB, gn = (2SA + 1)(2SB + 1). (14) As before, we obtain a finite contribution toward the chemical potential and the entropy of the gas but none toward the internal energy and the specific heat. Now, the rotational levels of a linear “rigid” rotator, with two degrees of freedom (for the axis of rotation) and the principal moments of inertia (I,I,0), are given by εrot = l(l + 1)ℏ2/2I, l = 0,1,2,...; (15) here, I = µr2 0 where µ[= m1m2/(m1 + m2)] is the reduced mass of the nuclei and r0 the equi-librium distance between them. The rotational partition function of the molecule is then given by jrot(T) = ∞ X l=0 (2l + 1)exp ( −l(l + 1) ℏ2 2IkT ) = ∞ X l=0 (2l + 1)exp  −l(l + 1)2r T  ; 2r = ℏ2 2Ik . (16) The values of 2r, for all gases except the ones involving the isotopes H and D, are much smaller than room temperature. For example, the value of 2r for HCl is about 15 K, for N2, O2, and NO it lies between 2 K and 3 K, while for Cl2 it is about one-third of a degree. On the other hand, the values of 2r for H2, D2, and HD are, respectively, 85 K, 43 K, and 64 K. 162 Chapter 6. The Theory of Simple Gases These numbers give us an idea of the respective temperature ranges in which the effects arising from the discreteness of the rotational states are expected to be important. For T ≫2r, the spectrum of the rotational states may be approximated by a contin-uum. The summation in (16) is then replaced by an integration: jrot(T) ≈ ∞ Z 0 (2l + 1)exp  −l(l + 1)2r T  dl = T 2r . (17) The rotational specific heat is then given by (CV )rot = Nk, (18) consistent with the equipartition theorem. A better evaluation of the sum in (16) can be made with the help of the Euler–Maclaurin formula, namely ∞ X n=0 f (n) = ∞ Z 0 f (x)dx + 1 2f (0) −1 12f ′(0) + 1 720f ′′′(0) − 1 30,240f v(0) + ··· . (19) Writing f (x) = (2x + 1)exp{−x(x + 1)2r/T}, one obtains jrot(T) = T 2r + 1 3 + 1 15 2r T + 4 315 2r T 2 + ··· , (20) which is the so-called Mulholland’s formula; as expected, the main term of this formula is identical to the classical partition function (17). The corresponding result for the specific heat is (CV )rot = Nk ( 1 + 1 45 2r T 2 + 16 945 2r T 3 + ··· ) , (21) which shows that at high temperatures the rotational specific heat decreases with temper-ature and ultimately tends to the classical value Nk. Thus, at high (but finite) temperatures the rotational specific heat of a diatomic gas is greater than the classical value. On the other hand, it must go to zero as T →0. We, therefore, conclude that it passes through at least one maximum. Numerical studies show that there is only one maximum that appears at a temperature of about 0.82r and has a value of about 1.1Nk; see Figure 6.5. 6.5 Gaseous systems composed of molecules with internal motion 163 1.2 1.0 0.8 0.6 0.4 0.2 0 0.5 1.0 1.5 2.0 (T/r) (Crot /Nk) FIGURE 6.5 The rotational specific heat of a gas of heteronuclear diatomic molecules. In the other limiting case, when T ≪2r, one may retain only the first few terms of the sum in (16); then jrot(T) = 1 + 3e−22r/T + 5e−62r/T + ··· , (22) from which one obtains, in the lowest approximation, (CV )rot ≃12Nk 2r T 2 e−22r/T. (23) Thus, as T →0, the specific heat drops exponentially to zero; see again Figure 6.5. We, therefore, conclude that at low enough temperatures the rotational degrees of freedom of the molecules are also “frozen.” At this stage it appears worthwhile to remark that, since the internal motions of the molecules do not make any contribution toward the pressure of the gas (Aint being inde-pendent of V), the quantity (CP −CV ) is the same for a diatomic gas as for a monatomic one. Moreover, under the assumptions made in the very beginning of this section, the value of this quantity at all temperatures of interest would be equal to the classical value Nk. Thus, at sufficiently low temperatures (when rotational as well as vibrational degrees of freedom of the molecules are “frozen”), we have, by virtue of the translational motion alone, CV = 3 2Nk, CP = 5 2NK; γ = 5 3. (24) As temperature rises, the rotational degrees of freedom begin to “loosen up” until we reach temperatures that are much larger than 2r but much smaller than 2v; the rotational degrees of freedom are then fully excited while the vibrational ones are still “frozen.” 164 Chapter 6. The Theory of Simple Gases 10 9 8 7 6 5 4 10 50 HD HT DT HD HT DT 9R /2 7R /2 5R /2 100 500 1000 5000 T (in K) Cp(in cal mole21 deg21) FIGURE 6.6 The rotational-vibrational specific heat, CP, of the diatomic gases HD, HT, and DT. Accordingly, for 2r ≪T ≪2v, CV = 5 2Nk, CP = 7 2Nk; γ = 7 5. (25) As temperature rises further, the vibrational degrees of freedom as well start loosening up, until we reach temperatures that are much larger than 2v. Then, the vibrational degrees of freedom are also fully excited and we have CV = 7 2Nk, CP = 9 2Nk; γ = 9 7. (26) These features are displayed in Figure 6.6 where the experimental results for CP are plot-ted for three gases HD, HT, and DT. We note that, in view of the considerable difference between the values of 2r and 2v, the situation depicted by (25) prevails over a consider-ably large range of temperatures. In passing, it may be pointed out that, for most diatomic gases, the situation at room temperatures corresponds to the one depicted by (25). We now study the case of homonuclear molecules, such as AA. To start with, we consider the limiting case of high temperatures where classical approximation is admissible. The rotational motion of the molecule may then be visualized as a rotation of the molecular axis, that is, the line joining the two nuclei, about an “axis of rotation” that is perpendic-ular to the molecular axis and passes through the center of mass of the molecule. Then, the two opposing positions of the molecular axis, namely the ones corresponding to the azimuthal angles φ and φ + π, differ simply by an interchange of the two identical nuclei and, hence, correspond to only one distinct state of the molecule. Therefore, in the evalu-ation of the partition function, the range of the angle φ should be taken as (0,π) instead of the customary (0,2π). Moreover, since the energy of rotational motion does not depend on angle φ, the only effect of this on the partition function of the molecule would be to reduce 6.5 Gaseous systems composed of molecules with internal motion 165 it by a factor of 2. We thus obtain, in the classical approximation,11 jnuc−rot(T) = (2SA + 1)2 T 22r . (27) Obviously, the factor 2 here will not affect the specific heat of the gas; in the classical approximation, therefore, the specific heat of a gas of homonuclear molecules is the same as that of a corresponding gas of heteronuclear molecules. In contrast, significant changes result at relatively lower temperatures where the states of rotational motion have to be treated as discrete. These changes arise from the cou-pling between the nuclear and the rotational states that in turn arises from the symmetry character of the nuclear-rotational wavefunction. As discussed in Section 5.4, the total wavefunction of a physical state must be either symmetric or antisymmetric (depend-ing on the statistics obeyed by the particles involved) with respect to an interchange of two identical particles. Now, the rotational wavefunction of a diatomic molecule is sym-metric or antisymmetric depending on whether the quantum number l is even or odd. The nuclear wavefunction, on the other hand, consists of a linear combination of the spin functions of the two nuclei and its symmetry character depends on the manner in which the combination is formed. It is not difficult to see that, of the (2SA + 1)2 different com-binations that one constructs, exactly (SA + 1)(2SA + 1) are symmetric with respect to an interchange of the nuclei and the remaining SA(2SA + 1) antisymmetric.12 In constructing the total wavefunction, as a product of the nuclear and the rotational wavefunctions, we then proceed as follows: (i) If the nuclei are fermions (SA = 1 2, 3 2,...), as in the molecule H2, the total wavefunction must be antisymmetric. To secure this, we may associate any one of the SA(2SA + 1) antisymmetric nuclear wavefunctions with any one of the even-l rotational wavefunctions or any one of the (SA + 1)(2SA + 1) symmetric nuclear wavefunctions with any one of the odd-l rotational wavefunctions. Accordingly, the nuclear-rotational partition function of such a molecule would be j(F.D.) nuc−rot(T) = SA(2SA + 1)reven + (SA + 1)(2SA + 1)rodd, (28) 11It seems instructive to outline here the purely classical derivation of the rotational partition function. Specifying the rotation of the molecule by the angles (θ,φ) and the corresponding momenta (pθ,pφ), the kinetic energy assumes the form εrot = 1 2I p2 θ + 1 2I sin2 θ p2 φ, from which jrot(T) = 1 h2 R e−εrot/kT(dpθdpφdθ dφ) = IkT πℏ2 φmax R 0 dφ. For heteronuclear molecules φmax = 2π, while for homonuclear ones φmax = π. 12See, for example, Schiff (1968, Section 41). 166 Chapter 6. The Theory of Simple Gases where reven = ∞ X l=0,2,... (2l + 1)exp{−l(l + 1)2r/T} (29) and rodd = ∞ X l=1,3,... (2l + 1)exp{−l(l + 1)2r/T}. (30) (ii) If the nuclei are bosons (SA = 0,1,2,...), as in the molecule D2, the total wavefunction must be symmetric. To secure this, we may associate any one of the (SA + 1)(2SA + 1) symmetric nuclear wavefunctions with any one of the even-l rotational wavefunc-tions or any one of the SA(2SA + 1) antisymmetric nuclear wavefunctions with any one of the odd-l rotational wavefunctions. We then have j(B.E.) nuc−rot(T) = (SA + 1)(2SA + 1)reven + SA(2SA + 1)rodd. (31) At high temperatures, it is the larger values of l that contribute most to the sums (29) and (30). The difference between the two sums is then negligibly small, and we have reven ≃rodd ≃1 2jrot(T) = T/22r; (32) see equations (16) and (17). Consequently, j(B.E.) nuc−rot ≃j(F.D.) nuc−rot = (2SA + 1)2T/22r, (33) in agreement with our previous result (27). Under these circumstances, the statistics gov-erning the nuclei does not make a significant difference to the thermodynamic behaviour of the gas. Things change when the temperature of the gas is in a range comparable to the value of 2r. It seems most reasonable then to regard the gas as a mixture of two components, generally referred to as ortho- and para-, whose relative concentrations in equilibrium are determined by the relative magnitudes of the two parts of the partition function (28) or (31), as the case may be. Customarily, the name ortho- is given to that component that carries the larger statistical weight. Thus, in the case of fermions (as in H2), the ortho- to para-ratio is given by n(F.D.) = (SA + 1)rodd SAreven , (34) while in the case of bosons (as in D2), the corresponding ratio is given by n(B.E.) = (SA + 1)reven SArodd . (35) 6.5 Gaseous systems composed of molecules with internal motion 167 As temperature rises, the factor rodd/reven tends to unity and the ratio n, in each case, approaches the temperature-independent value (SA + 1)/SA. In the case of H2, this lim-iting value is 3 (since SA = 1 2) while in the case of D2 it is 2 (since SA = 1). At sufficiently low temperatures, one may retain only the main terms of the sums (29) and (30), with the result that rodd reven ≃3exp  −22r T  (T ≪2r), (36) which tends to zero as T →0. The ratio n then tends to zero in the case of fermions and to infinity in the case of bosons. Hence, as T →0, the hydrogen gas is wholly para-, while deuterium is wholly ortho-; of course, in each case, the molecules do settle down in the rotational state l = 0. At intermediate temperatures, one has to work with the equilibrium ratio (34), or (35), and with the composite partition function (28), or (31), in order to compute the thermody-namic properties of the gas. One finds, however, that the theoretical results so derived do not generally agree with the ones obtained experimentally. This discrepancy was resolved by Dennison (1927) who pointed out that the samples of hydrogen, or deuterium, ordi-narily subjected to experiment are not in thermal equilibrium as regards the relative magnitudes of the ortho- and para-components. These samples are ordinarily prepared and kept at room temperatures that are well above 2r, with the result that the ortho- to para-ratio in them is very nearly equal to the limiting value (SA + 1)SA. If now the temperature is lowered, one would expect this ratio to change in accordance with equation (34), or (35). However, it does not do so for the following reason. Since the transition of a molecule from one form of existence to another involves the flipping of the spin of one of its nuclei, the transition probability of the process is quite small. Actu-ally, the periods involved are of the order of a year! Obviously, one cannot expect to attain the true equilibrium ratio n during the short times available. Consequently, even at lower temperatures, what one generally has is a nonequilibrium mixture of two independent substances, the relative concentration of which is preassigned. The partition functions (28) and (31) as such are, therefore, inapplicable; we rather have directly for the specific heat C(F.D.) = SA 2SA + 1Ceven + SA + 1 2SA + 1Codd (37) and C(B.E.) = SA + 1 2SA + 1Ceven + SA 2SA + 1Codd, (38) where Ceven/odd = Nk ∂ ∂T n T2(∂/∂T)lnreven/odd o . (39) We have, therefore, to compute Ceven and Codd separately and then derive the net value of the rotational specific heat with the help of formula (37) or (38), as the case may be. 168 Chapter 6. The Theory of Simple Gases 2.0 1.5 1.0 1 3 2 0.5 00 1 2 3 4 5 Cv Nk (T/r) FIGURE 6.7 The theoretical specific heat of a 1:3 mixture of para-hydrogen and ortho-hydrogen. The experimental points originate from various sources listed in Wannier (1966). Figure 6.7 shows the relevant results for hydrogen. Curves 1 and 2 correspond to the para-hydrogen (Ceven) and the ortho-hydrogen (Codd), respectively, while curve 3 represents the weighted mean, as given by equation (37). The experimental results are also shown in the figure; the agreement between theory and experiment is clearly good. Further evidence in favor of Dennison’s explanation is obtained by performing exper-iments with ortho–para mixtures of different relative concentration. This can be done by speeding up the ortho–para conversion by passing hydrogen over activated charcoal. By doing this at various temperatures, and afterwards removing the catalyst, one can fix the ratio n at any desired value. The specific heat then follows a curve obtained by mixing Ceven and Codd with appropriate weight factors. Further, if one measures the specific heat of the gas in such a way that the ratio n, at every temperature T, has the value that is given by formula (34), it indeed follows the curve obtained from expression (28) for the partition function. 6.5.C Polyatomic molecules Once again, the translational degrees of freedom of the molecules contribute their usual share, 3 2k per molecule, toward the specific heat of the gas. As regards the lowest electronic state, it is, in most cases, far below any of the excited states; nevertheless, it generally pos-sesses a multiplicity (depending on the orbital and spin angular momenta of the state) that can be taken care of by a degeneracy factor ge. As regards the rotational states, they can be treated classically because the large values of the moments of inertia characteris-tic of polyatomic molecules make the quantum of rotational energy, ℏ2/2Ii, much smaller than the thermal energy kT at practically all temperatures of interest. Consequently, the interaction between the rotational states and the states of the nuclei can also be treated classically. As a result, the nuclear-rotational partition function is given by the product of the respective partition functions, divided by a symmetry number γ that denotes the num-ber of physically indistinguishable configurations realized during one complete rotation of 6.5 Gaseous systems composed of molecules with internal motion 169 the molecule:13 jnuc−rot(T) = gnuc jC rot(T) γ ; (40) compare to equation (27). Here, jC rot(T) is the rotational partition function of the molecule evaluated in the classical approximation (without paying regard to the presence of identi-cal nuclei, if any); it is given by jC rot(T) = π1/2 2I1kT ℏ2 1/2 2I2kT ℏ2 1/2 2I3kT ℏ2 1/2 (41) where I1, I2, and I3 are the principal moments of inertia of the molecule; see Prob-lem 6.27.14 The rotational specific heat is then given by Crot = Nk ∂ ∂T  T2 ∂ ∂T lnjC rot(T)  = 3 2Nk, (42) consistent with the equipartition theorem. As regards vibrational states, we first note that, unlike a diatomic molecule, a poly-atomic molecule has not one but several vibrational degrees of freedom. In particular, a noncollinear molecule consisting of n atoms has 3n −6 vibrational degrees of freedom, six degrees of freedom out of the total 3n having gone into the translational and rotational motions. On the other hand, a collinear molecule consisting of n atoms would have 3n −5 vibrational degrees of freedom, for the rotational motion in this case has only two, not three, degrees of freedom. The vibrational degrees of freedom correspond to a set of nor-mal modes characterized by a set of frequencies ωi. It might happen that some of these frequencies have identical values; we then speak of degenerate frequencies.15 In the harmonic approximation, these normal modes may be treated independently of one another. The vibrational partition function of the molecule is then given by the product of the partition functions corresponding to individual normal modes, that is, jvib(T) = Y i e−2i/2T 1 −e−2i/T ; 2i = ℏωi k , (43) 13For example, the symmetry number γ for H2O (isosceles triangle) is 2, for NH3 (regular triangular pyramid) it is 3, while for CH4 (tetrahedron) and C6H6 (regular hexagon) it is 12. For heteronuclear molecules, the symmetry number is unity. 14In the case of a collinear molecule, such as N2O or CO2, there are only two degrees of freedom for rotation; con-sequently, jC rot(T) is given by (2IkT/ℏ2), where I is the (common) value of the two moments of inertia of the molecule; see equation (17). Of course, we must also take into account the symmetry number γ . In the examples quoted here, the molecule N2O, being spatially asymmetric (NNO), has symmetry number 1, while the molecule CO2, being spatially symmetric (OCO), has symmetry number 2. 15For example, of the four frequencies characterizing the normal modes of vibration of the collinear molecule OCO, two that correspond to the (transverse) bending modes, namely ↑ O C O ↓ ↓ , are equal while the others that correspond to (longitudinal) oscillations along the molecular axis, namely ←O C→←O and ←O C O→, are different; see Problem 6.28. 170 Chapter 6. The Theory of Simple Gases and the vibrational specific heat is given by the sum of the contributions arising from the individual modes: Cvib = Nk X i (2i T 2 e2i/T e2i/T −1 2 ) . (44) In general, the various 2i are of order 103 K; for instance, in the case of CO2, which was cited in footnote 15, 21 = 22 = 960 K, 23 = 1,990 K, and 24 = 3,510 K. For temperatures large in comparison with all 2i, the specific heat would be given by the equipartition value, namely Nk for each of the normal modes. In practice, however, this limit can hardly be realized because the polyatomic molecules generally break up well before such high tem-peratures are reached. Secondly, the different frequencies ωi of a polyatomic molecule are generally spread over a rather wide range of values. Consequently, as temperature rises, different modes of vibration get gradually “included” into the process; in between these “inclusions,” the specific heat of the gas may stay constant over considerably large stretches of temperature. 6.6 Chemical equilibrium The equilibrium amounts of chemicals in a chemical reaction are determined by the chemical potentials of each of the species. Consider the following chemical reaction between chemical species A and B to form species X and Y with stoichiometric coefficients νA, νB, νX, and νY : νAA + νBB ⇄νXX + νY Y . (1) Each individual reaction that occurs changes the number of molecules of each species according to the stoichiometric coefficients. If the initial numbers of molecules of the species are N0 A, N0 B, N0 X, and N0 Y , then the numbers of each species after 1N chemical reac-tions have occurred would be NA = N0 A −νA1N, NB = N0 B −νB1N, NX = N0 X + νX1N, and NY = N0 Y + νY 1N. If 1N > 0, the reaction has proceeded in the positive direction increas-ing the numbers of X and Y . If 1N < 0, the reaction has proceeded in the direction of increasing the numbers of A and B. If the reaction takes place in a closed isothermal sys-tem with fixed pressure, the Gibbs free energy G(NA,NB,NX,NY ,P,T) is changed by the amount 1G = (−νAµA −νBµB + νXµX + νY µY )1N, (2) where µA =  ∂G ∂NA  T,P is the chemical potential of species A, and so on; see Sections 3.3, 4.7, and Appendix H. Since the Gibbs free energy decreases as a system approaches equi-librium, 1G ≤0. When the system reaches chemical equilibrium, the Gibbs free energy reaches its minimum value so 1G = 0. This gives us the general relationship for chemical 6.6 Chemical equilibrium 171 equilibrium of the reaction in equation (1), namely νAµA + νBµB = νXµX + νY µY . (3) Note that if a chemical species that acts as a catalyst is added in equal amounts to both sides of equation (1), the equilibrium relation (3) is unaffected. Therefore, a cata-lyst may serve to increase the rate of approach toward equilibrium, without affecting the equilibrium condition itself. If the free energy can be approximated as a sum of the free energies of the individual species such as in an ideal gas or a dilute solution, then we can derive a simple rela-tion between the equilibrium densities of the species. Following from equations (3.5.10) and (6.5.4), the Helmholtz free energy of a classical ideal gas consisting of molecules with internal degrees of freedom can be written as A(N,V,T) = Nε + NkT ln Nλ3 V ! −NkT −NkT lnj(T), (4) where ε is the ground state energy of the molecule, λ = h/ √ 2πmkT is the thermal deBroglie wavelength, and j(T) is the partition function for the internal degrees of freedom of the molecule. This gives for the chemical potential of species A µA =  ∂A ∂NA  T,V = εA + kT ln  nAλ3 A  −kT lnjA(T), (5) where nA is the number density of species A. The equilibrium condition then (3) gives [X]νX [Y ]νY [A]νA[B]νB = K(T) = exp  −β1µ(0) , (6) where [A] = nA/n0, and so on, 1µ(0) = νXµ(0) X + νY µ(0) Y −νAµ(0) A −νBµ(0) B , (7a) µ(0) A = εA + kT ln  n0λ3 A  −kT lnjA(T), etc. (7b) and K(T) is the equilibrium constant. Equation (6) is called the law of mass action. The quantity n0 is a standard number density and µ(0) A , and so on, are the chemical potentials of the species at temperature T and standard number density n0. The quantity 1µ(0) represents the Gibbs free energy change per chemical reaction at standard density. Note that the reaction constant K(T) is a function only of the temperature and determines the densities of the components in equilibrium at temperature T through equation (6). The standard number density for gases is usually chosen to be the number density of an ideal gas at temperature T and standard pressure, that is, n0 = (1atm)/kT. The standard density for aqueous solutions is usually 172 Chapter 6. The Theory of Simple Gases chosen to be one mole per liter. The Gibbs free energy in chemical tables is expressed relative to the standard states of the elements. We now examine a specific example, the combustion of hydrocarbons with oxygen in an internal combustion engine. The reaction used to power clean buses and automobiles using natural gas is CH4 + 2O2 ⇄CO2 + 2H2O . (8) The primary reaction products are carbon dioxide and water vapor but carbon monoxide is also produced by the reaction 2CH4 + 3O2 ⇄2CO + 4H2O . (9) A primary goal for a clean burning engine is to combust nearly all the hydrocarbon fuel while producing as little carbon monoxide as possible. By combining reactions (8) and (9), we get a direct reaction between carbon monoxide, oxygen, and carbon dioxide: 2CO + O2 ⇄2CO2 . (10) Equation (6) now gives the equilibrium ratio of CO to CO2 as [CO] [CO2] = s 1 K(T)[O2] . (11) At T ≈1,500K, as combustion occurs inside the cylinder of the engine, the equilibrium constant K ≈1010 so carbon monoxide is present as a combustion product in the few parts per million range — combustion reactions that are not in equilibrium can have CO concentrations well above the equilibrium value. The exhaust gases cool quickly during the power stroke of the engine. As these gases exit the exhaust at T ≈600K, the equilib-rium constant K ≈1040, which should result in almost no carbon monoxide in the exhaust stream. However, the reaction rate is typically too slow to keep the CO concentration in chemical equilibrium during the rapid cooling, so the amount of CO present in the exhaust stream remains close to the larger value determined at the higher temperature.16 Fortu-nately, the leftover carbon monoxide can be converted into carbon dioxide at the exhaust temperature in a catalytic converter that uses platinum and palladium as catalysts to increase the reaction rate. Equation (11) indicates that the carbon monoxide fraction is reduced by increasing the amount of oxygen present in the reaction. This is accomplished by running the engine with a hydrocarbon/air ratio that is a little bit short of the stoichio-metric point of equation (8). This reduces the amount of CO left from the combustion itself and also leaves excess O2 in the exhaust stream for use in the catalytic converter. 16Very similar effects happened during the early stages of the universe as the temperature cooled but the cooling rate was too rapid for some constituents to remain in thermal equilibrium; see Chapter 9. Problems 173 Problems 6.1. Show that the entropy of an ideal gas in thermal equilibrium is given by the formula S = k X ε ⟨nε + 1⟩ln⟨nε + 1⟩−⟨nε⟩ln⟨nε⟩ in the case of bosons and by the formula S = k X ε [−⟨1 −nε⟩ln⟨1 −nε⟩−⟨nε⟩ln⟨nε⟩] in the case of fermions. Verify that these results are consistent with the general formula S = −k X ε (X n pε(n)lnpε(n) ) , where pε(n) is the probability that there are exactly n particles in the energy state ε. 6.2. Derive, for all three statistics, the relevant expressions for the quantity ⟨n2 ε⟩−⟨nε⟩2 from the respective probabilities pε(n). Show that, quite generally, ⟨n2 ε⟩−⟨nε⟩2 = kT ∂⟨nε⟩ ∂µ  T ; compare with the corresponding result, (4.5.3), for a system embedded in a grand canonical ensemble. 6.3. Refer to Section 6.2 and show that, if the occupation number nε of an energy level ε is restricted to the values 0,1,...,l, then the mean occupation number of that level is given by ⟨nε⟩= 1 z−1eβε −1 − l + 1 (z−1eβε)l+1 −1. Check that while l = 1 leads to ⟨nε⟩F.D., l →∞leads to ⟨nε⟩B.E.. 6.4. The potential energy of a system of charged particles, characterized by particle charge e and number density n(r), is given by U = e2 2 ZZ n(r)n(r′) |r −r′| drdr′ + e Z n(r)φext(r)dr, where φext(r) is the potential of an external electric field. Assume that the entropy of the system, apart from an additive constant, is given by the formula S = −k Z n(r)lnn(r)dr; compare to formula (3.3.13). Using these expressions, derive the equilibrium equations satisfied by the number density n(r) and the total potential φ(r), the latter being φext(r) + e Z n(r′) |r −r′|dr′. 6.5. Show that the root-mean-square deviation in the molecular energy ε, in a system obeying Maxwell–Boltzmann distribution, is √(2/3) times the mean molecular energy ε. Compare this result with that of Problem 3.18. 6.6. Show that, for any law of distribution of molecular speeds,  ⟨u⟩ 1 u  ≥1. Check that the value of this quantity for the Maxwellian distribution is 4/π. 174 Chapter 6. The Theory of Simple Gases 6.7. Through a small window in a furnace, which contains a gas at a high temperature T, the spectral lines emitted by the gas molecules are observed. Because of molecular motions, each spectral line exhibits Doppler broadening. Show that the variation of the relative intensity I(λ) with wavelength λ in a line is given by I(λ) ∝exp ( −mc2(λ −λ0)2 2λ2 0kT ) , where m is the molecular mass, c the speed of light, and λ0 the mean wavelength of the line. 6.8. An ideal classical gas composed of N particles, each of mass m, is enclosed in a vertical cylinder of height L placed in a uniform gravitational field (of acceleration g) and is in thermal equilibrium; ultimately, both N and L →∞. Evaluate the partition function of the gas and derive expressions for its major thermodynamic properties. Explain why the specific heat of this system is larger than that of a corresponding system in free space. 6.9. Centrifuge-based uranium enrichment: Natural uranium is composed of two isotopes: 238U and 235U, with percentages of 99.27% and 0.72%, respectively. If uranium hexafluoride gas UF6 is injected into a rapidly spinning hollow metal cylinder with inner radius R, the equilibrium pressure of the gas is largest at the inner radius and isotopic concentration differences between the axis and the inner radius allow enrichment of the concentration of 235U. (a) Write down the Lagrangian L({qk, ˙ qk}) for particles of mass m moving in a cylindrical coordinate system rotating at angular velocity ω and use a Legendre transformation H({qk,pk}) = X k pk ˙ qk −L, to show that the one-particle Hamiltonian H in that cylindrical coordinate system is H(r,θ,z,pr,pθ,pz) = p2 r 2m + (p2 θ −mr2ω)2 2mr2 + p2 z 2m . Ignore the internal degrees of freedom of the molecules since they will not affect the density as a function of position. Show that the one-particle partition function shown here can be written as Q1(V,T) = 1 h3 ∞ Z −∞ dpr ∞ Z −∞ dpθ ∞ Z −∞ dpz R Z 0 dr 2π Z 0 dθ H Z 0 dzexp(−βH) , by constructing the Jacobian of transformation between the cartesian and the cylindrical coordinates for the phase space integral. Evaluate the partition function Q1 in a closed form and determine the Helmholtz free energy of this system. (b) Determine the number density n(r) as a function of the distance r from the axis for the N molecules of gas in the rotating cylinder. Show that, in the limit ω →0, the density becomes uniform with the value n = N/πR2H. Find an expression for the ratio of the pressure at the inner radius of the cylinder R to the pressure at the axis of the cylinder as a function of ω and R. (c) Evaluate the pressure ratios for the two isotopically different UF6 gases at room temperature for the case ωR = 500m/s. Show that the pressure ratio for 238U is approximately 20% larger than the pressure ratio for 235U so that extracting gas near the axis results in an enriched concentration of 235U. A series of centrifuges can be used to raise the concentration of 235U to create a fissionable grade of uranium for use in power-generating reactors or in nuclear weapons. Not surprisingly, this technology is a major concern for possible nuclear proliferation. Problems 175 6.10. (a) Show that, if the temperature is uniform, the pressure of a classical gas in a uniform gravitational field decreases with height according to the barometric formula P(z) = P(0)exp  −mgz/kT , where the various symbols have their usual meanings.17 (b) Derive the corresponding formula for an adiabatic atmosphere, that is, the one in which (PV γ ), rather than (PV), stays constant. Also study the variation, with height, of the temperature T and the density n in such an atmosphere. 6.11. (a) Show that the momentum distribution of particles in a relativistic Boltzmannian gas, with ε = c(p2 + m2 0c2)1/2, is given by f (p)dp = Ce−βc(p2+m2 0c2)1/2p2dp, with the normalization constant C = β m2 0cK2(βm0c2) , Kν(z) being a modified Bessel function. (b) Check that in the nonrelativistic limit (kT ≪m0c2) we recover the Maxwellian distribution, f (p)dp =  β 2πm0 3/2 e−βp2/2m0(4πp2 dp), while in the extreme relativistic limit (kT ≫m0c2) we obtain f (p)dp = (βc)3 8π e−βpc(4πp2 dp). (c) Verify that, quite generally, ⟨pu⟩= 3kT. 6.12. (a) Considering the loss of translational energy suffered by the molecules of a gas on reflection from a receding wall, derive, for a quasistatic adiabatic expansion of an ideal nonrelativistic gas, the well-known relation PV γ = const., where γ = (3a + 2)/3a, a being the ratio of the total energy to the translational energy of the gas. (b) Show that, in the case of an extreme relativistic gas, γ = (3a + 1)/3a. 6.13. (a) Determine the number of impacts made by gas molecules on a unit area of the wall in a unit time for which the angle of incidence lies between θ and θ + dθ. (b) Determine the number of impacts made by gas molecules on a unit area of the wall in a unit time for which the speed of the molecules lies between u and u + du. (c) A molecule AB dissociates if it hits the surface of a solid catalyst with a normal translational energy greater than 10−19 J. Show that the rate of the dissociative reaction AB →A + B is more than doubled by raising the temperature of the gas from 300 K to 310 K. 6.14. Consider the effusion of molecules of a Maxwellian gas through an opening of area a in the walls of a vessel of volume V. (a) Show that, while the molecules inside the vessel have a mean kinetic energy 3 2kT, the effused ones have a mean kinetic energy 2kT, T being the quasistatic equilibrium temperature of the gas. 17This formula was first given by Boltzmann (1879). For a critical study of its derivation, see Walton (1969). 176 Chapter 6. The Theory of Simple Gases (b) Assuming that the effusion is so slow that the gas inside is always in a state of quasistatic equilibrium, determine the manner in which the density, the temperature, and the pressure of the gas vary with time. 6.15. A polyethylene balloon at an altitude of 30,000 m is filled with helium gas at a pressure of 10−2 atm and a temperature of 300 K. The balloon has a diameter of 10 m, and has numerous pinholes of diameter 10−5 m each. How many pinholes per square meter of the surface of the balloon must there be if 1 percent of the gas were to leak out in 1 hour? 6.16. Consider two Boltzmannian gases A and B, at pressures PA and PB and temperatures TA and TB, respectively, contained in two regions of space that communicate through a very narrow opening in the partitioning wall; see Figure 6.8. Show that the dynamic equilibrium resulting from the mutual effusion of the two kinds of molecules satisfies the condition PA/PB = (mATA/mBTB)1/2, rather than PA = PB (which would be the case if the equilibrium had resulted from a hydrodynamic flow). A (PA, TA) B (PB, TB) FIGURE 6.8 The molecules of the gases A and B undergoing a two-way effusion. 6.17. A small sphere, with initial temperature T, is immersed in an ideal Boltzmannian gas at temperature T0. Assuming that the molecules incident on the sphere are first absorbed and then reemitted with the temperature of the sphere, determine the variation of the temperature of the sphere with time. [Note: The radius of the sphere may be assumed to be much smaller than the mean free path of the molecules.] 6.18. Show that the mean value of the relative speed of two molecules in a Maxwellian gas is √2 times the mean speed of a molecule with respect to the walls of the container. [Note: A similar result for the root-mean-square speeds (instead of the mean speeds) holds under much more general conditions.] 6.19. What is the probability that two molecules picked at random from a Maxwellian gas will have a total energy between E and E + dE? Verify that ⟨E⟩= 3kT. 6.20. The energy difference between the lowest electronic state 1S0 and the first excited state 3S1 of the helium atom is 159,843 cm−1. Evaluate the relative fraction of the excited atoms in a sample of helium gas at a temperature of 6000 K. 6.21. Derive an expression for the equilibrium constant K(T) for the reaction H2 + D2 ↔2HD at temperatures high enough to allow classical approximation for the rotational motion of the molecules. Show that K(∞) = 4. 6.22. With the help of the Euler–Maclaurin formula (6.5.19), derive high-temperature expansions for reven and rodd, as defined by equations (6.5.29) and (6.5.30), and obtain corresponding expansions for Ceven and Codd, as defined by equation (6.5.39). Compare the mathematical trend of these results with the nature of the corresponding curves in Figure 6.7. Also study the low-temperature behavior of the two specific heats and once again compare your results with the relevant parts of the aforementioned curves. 6.23. The potential energy between the atoms of a hydrogen molecule is given by the (semiempirical) Morse potential V(r) = V0{e−2(r−r0)/a −2e−(r−r0)/a}, Problems 177 where V0 = 7 × 10−12 erg, r0 = 8 × 10−9 cm, and a = 5 × 10−9 cm. Evaluate the rotational and vibrational quanta of energy, and estimate the temperatures at which the rotational and vibrational modes of the molecules would begin to contribute toward the specific heat of the hydrogen gas. 6.24. Show that the fractional change in the equilibrium value of the internuclear distance of a diatomic molecule, as a result of rotation, is given by 1r0 r0 ≃ ℏ µr2 0ω !2 J(J + 1) = 4  2r 2v 2 J(J + 1); here, ω is the angular frequency of the vibrational state in which the molecule happens to be. Estimate the numerical value of this fraction in a typical case. 6.25. The ground state of an oxygen atom is a triplet, with the following fine structure: εJ=2 = εJ=1 −158.5 cm−1 = εJ=0 −226.5 cm−1. Calculate the relative fractions of the atoms occupying different J-levels in a sample of atomic oxygen at 300 K. 6.26. Calculate the contribution of the first excited electronic state, namely 11 with ge = 2, of the O2 molecule toward the Helmholtz free energy and the specific heat of oxygen gas at a temperature of 5000 K; the separation of this state from the ground state, namely 36 with ge = 3, is 7824 cm−1. How would these results be affected if the parameters 2r and 2v of the O2 molecule had different values in the two electronic states? 6.27. The rotational kinetic energy of a rotator with three degrees of freedom can be written as εrot = M2 ξ 2I1 + M2 η 2I2 + M2 ζ 2I3 , where (ξ,η,ζ) are coordinates in a rotating frame of reference whose axes coincide with the principal axes of the rotator, while (Mξ,Mη,Mζ ) are the corresponding angular momenta. Carrying out integrations in the phase space of the rotator, derive expression (6.5.41) for the partition function jrot(T) in the classical approximation. 6.28. Determine the translational, rotational, and vibrational contributions toward the molar entropy and the molar specific heat of carbon dioxide at NTP . Assume the ideal-gas formulae and use the following data: molecular weight M = 44.01; moment of inertia I of a CO2 molecule = 71.67 × 10−40 gcm2; wave numbers of the various modes of vibration: ν1 = ν2 = 667.3cm−1, ν3 = 1383.3 cm−1, and ν4 = 2439.3 cm−1. 6.29. Determine the molar specific heat of ammonia at a temperature of 300 K. Assume the ideal-gas formula and use the following data: the principal moments of inertia: I1 = 4.44 × 10−40gcm2, I2 = I3 = 2.816 × 10−40gcm2; wave numbers of the various modes of vibration: ν1 = ν2 = 3336 cm−1, ν3 = ν4 = 950 cm−1, ν5 = 3414 cm−1, and ν6 = 1627 cm−1. 6.30. Derive the equilibrium concentration equation (6.6.6) from the equilibrium condition (6.6.3). 6.31. Use the following values to determine the equilibrium constant for the reaction 2CO + O2 ⇄2CO2. At a combustion temperature of T = 1500K: βµ(0) CO2 = −60.95, βµ(0) CO = −35.18, and βµ(0) O2 = −27.08. Use this data to compute the fraction [CO]/[CO2] for the case of [O2] = 0.01. Repeat for a catalytic converter temperature of T = 600K, where βµ(0) CO2 = −103.45, βµ(0) CO = −45.38, and βµ(0) O2 = −23.49. 6.32. Derive an expression for the equilibrium constant K(T) for the reaction N2 + O2 ⇄2NO in terms of the ground state energy change 1ε0 = 2εNO −εN2 −εO2 and the vibrational and rotational partition functions of the diatomic molecules, using results from Section 6.5. Give predictions for the ranges of temperatures where the rotational modes are classically excited but the vibration modes are suppressed and for higher temperatures where both the rotational and vibrational models are classically excited. 6.33. Analyze the combustion reaction CH4 + 2O2 ⇄CO2 + 2H2O , (6.6.8) 178 Chapter 6. The Theory of Simple Gases assuming that at combustion temperatures the equilibrium constant K(T) ≫1. Show that conducting combustion at the stoichiometric point or just a bit short of the stoichiometric point (so there is enough oxygen to oxidize all of the methane) will lead to low amounts of CH4 in the exhaust. Determine the equilibrium amount of CH4 in terms of the initial excess amount of O2. Determine the equilibrium constant at T = 1500K from the data βµ(0) CO2 = −60.95, βµ(0) O2 = −27.08, βµ(0) CH4 = −31.95, and βµ(0) H2O = −44.62. 6.34. Determine the equilibrium ionization fraction for the reaction Na ⇄Na+ + e− in a sodium vapor. Treat all three species as ideal classical monatomic gases. The ionization energy of sodium is 5.139 eV, Na+ ions are spin-zero, and neutral Na and free e−are both spin- 1 2. Derive the Saha equation for the ionized fraction [Na+]/([Na] + [Na+]) for a neutral plasma as a function of temperature at a fixed total density. Plot the ionized fraction as a function of temperature for some chosen total density. [Note that, this calculation is very similar to the one concerning ionized hydrogen fraction as a function of temperature during the recombination era in the early universe; see Section 9.8.] 7 Ideal Bose Systems In continuation of Sections 6.1 through 6.3, we shall now investigate in detail the physical behavior of a class of systems in which, while the intermolecular interactions are still neg-ligible, the effects of quantum statistics (which arise from the indistinguishability of the particles) assume an increasingly important role. This means that the temperature T and the particle density n of the system no longer conform to the criterion nλ3 ≡ nh3 (2πmkT)3/2 ≪1, (5.5.20) where λ{≡h/(2πmkT)1/2} is the mean thermal wavelength or thermal deBroglie wave-length of the particles. In fact, the quantity nλ3 turns out to be a very appropriate parameter, in terms of which the various physical properties of the system can be ade-quately expressed. In the limit nλ3 →0, all physical properties go over smoothly to their classical counterparts. For small, but not negligible, values of nλ3, the various quantities pertaining to the system can be expanded as power series in this parameter; from these expansions one obtains the first glimpse of the manner in which departure from classi-cal behavior sets in. When nλ3 becomes of the order of unity, the behavior of the system becomes significantly different from the classical one and is characterized by quantum effects. A study of the system under these circumstances brings us face to face with a set of phenomena unknown in classical statistics. It is evident that a system is more likely to display quantum behavior when it is at a relatively low temperature and/or has a relatively high density of particles.1 Moreover, the smaller the particle mass the larger the quantum effects. Now, when nλ3 is of the order of unity, then not only does the behavior of a system exhibit significant departure from typical classical behavior but it is also influenced by whether the particles constituting the system obey Bose–Einstein statistics or Fermi–Dirac statistics. Under these circumstances, the properties of the two kinds of systems are them-selves very different. In the present chapter we consider systems belonging to the first category while the succeeding chapter will deal with systems belonging to the second category. 1Actually it is the ratio n/T 3/2, rather than the quantities n and T separately, that determines the degree of degeneracy in a given system. For instance, white dwarf stars, even at temperatures of order 107 K, constitute statistically degenerate systems; see Section 8.5. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00007-4 © 2011 Elsevier Ltd. All rights reserved. 179 180 Chapter 7. Ideal Bose Systems 7.1 Thermodynamic behavior of an ideal Bose gas We obtained, in Sections 6.1 and 6.2, the following formulae for an ideal Bose gas: PV kT ≡lnQ = − X ε ln(1 −ze−βε) (1) and N ≡ X ε ⟨nε⟩= X ε 1 z−1eβε −1, (2) where β = 1/kT, while z is the fugacity of the gas which is related to the chemical potential µ through the formula z ≡exp(µ/kT); (3) as noted earlier, ze−βε, for all ε, is less than unity. In view of the fact that, for large V, the spectrum of the single-particle states is almost a continuous one, the summations on the right sides of equations (1) and (2) may be replaced by integrations. In doing so, we make use of the asymptotic expression (2.4.7) for the nonrelativistic density of states a(ε) in the neighborhood of a given energy ε, namely2 a(ε)dε = (2πV/h3)(2m)3/2ε1/2dε. (4) We, however, note that by substituting this expression into our integrals we are inadver-tently giving a weight zero to the energy level ε = 0. This is wrong because in a quantum-mechanical treatment we must give a statistical weight unity to each nondegenerate single-particle state in the system. It is, therefore, advisable to take this particular state out of the sum in question before carrying out the integration; for a rigorous justification of this (unusual) step, see Appendix F . We thus obtain P kT = −2π h3 (2m)3/2 ∞ Z 0 ε1/2 ln(1 −ze−βε)dε −1 V ln(1 −z) (5) and N V = 2π h3 (2m)3/2 ∞ Z 0 ε1/2dε z−1eβε −1 + 1 V z 1 −z; (6) of course, the lower limit of these integrals can still be taken as 0, because the state ε = 0 is not going to contribute toward them anyway. Before proceeding further, a word about the relative importance of the last terms in equations (5) and (6). For z ≪1, which corresponds to situations not far removed from 2The theory of this section is restricted to a system of nonrelativistic particles. For the more general case, see Kothari and Singh (1941) and Landsberg and Dunning-Davies (1965). 7.1 Thermodynamic behavior of an ideal Bose gas 181 the classical limit, each of these terms is of order 1/N and, therefore, negligible. How-ever, as z increases and assumes values close to unity, the term z/(1 −z)V in (6), which is identically equal to N0/V (N0 being the number of particles in the ground state ε = 0), can well become a significant fraction of the quantity N/V; this accumulation of a macro-scopic fraction of the particles into a single state ε = 0 leads to the phenomenon of Bose–Einstein condensation. Nevertheless, since z/(1 −z) = N0 and hence z = N0/(N0 + 1), the term {−V −1 ln(1 −z)} in (5) is equal to {V −1 ln(N0 + 1)}, which is at most O(N−1 lnN); this term is, therefore, negligible for all values of z and hence may be dropped altogether. We now obtain from equations (5) and (6), on substituting βε = x, P kT = −2π(2mkT)3/2 h3 ∞ Z 0 x1/2 ln(1 −ze−x)dx = 1 λ3 g5/2(z) (7) and N −N0 V = 2π(2mkT)3/2 h3 ∞ Z 0 x1/2dx z−1ex −1 = 1 λ3 g3/2(z), (8) where λ = h/(2πmkT)1/2, (9) while gν(z) are Bose–Einstein functions defined by, see Appendix D, gν(z) = 1 0(ν) ∞ Z 0 xν−1dx z−1ex −1 = z + z2 2ν + z3 3ν + ··· ; (10) note that to write (7) in terms of the function g5/2(z) we first carried out an integration by parts. Equations (7) and (8) are our basic results; on elimination of z, they would give us the equation of state of the system. The internal energy of this system is given by U ≡−  ∂ ∂β lnQ  z,V = kT2  ∂ ∂T PV kT  z,V = kT2V g5/2(z)  d dT  1 λ3  = 3 2kT V λ3 g5/2(z); (11) here, use has been made of equation (7) and of the fact that λ ∝T−1/2. Thus, quite generally, our system satisfies the relationship P = 2 3(U/V). (12) For small values of z, we can make use of expansion (10); at the same time, we can neglect N0 in comparison with N. An elimination of z between equations (7) and (8) can then be 182 Chapter 7. Ideal Bose Systems carried out by first inverting the series in (8) to obtain an expansion for z in powers of nλ3 and then substituting this expansion into the series appearing in (7). The equation of state thereby takes the form of the virial expansion, PV NkT = ∞ X l=1 al λ3 v !l−1 , (13) where v(≡1/n) is the volume per particle; the coefficients al, which are referred to as the virial coefficients of the system, turn out to be a1 = 1, a2 = − 1 4 √ 2 = −0.17678, a3 = −  2 9 √ 3 −1 8  = −0.00330, a4 = −  3 32 + 5 32 √ 2 − 1 2 √ 6  = −0.00011,                          (14) and so on. For the specific heat of the gas, we obtain CV Nk ≡1 Nk ∂U ∂T  N,V = 3 2  ∂ ∂T PV Nk  v = 3 2 ∞ X l=1 5 −3l 2 al λ3 v !l−1 = 3 2  1 + 0.0884 λ3 v ! + 0.0066 λ3 v !2 + 0.0004 λ3 v !3 + ···  . (15) As T →∞(and hence λ →0), both the pressure and the specific heat of the gas approach their classical values, namely nkT and 3 2Nk, respectively. We also note that at finite, but large, temperatures the specific heat of the gas is larger than its limiting value; in other words, the (CV ,T)-curve has a negative slope at high temperatures. On the other hand, as T →0, the specific heat must go to zero. Consequently, it must pass through a maximum somewhere. As seen later, this maximum is in the nature of a cusp that appears at a criti-cal temperature Tc; the derivative of the specific heat is found to be discontinuous at this temperature (see Figure 7.4 later in this section). As the temperature of the system falls (and the value of the parameter λ3/v grows), expansions such as (13) and (15) do not remain useful. We then have to work with formulae (7), (8), and (11) as such. The precise value of z is now obtained from equation (8), which may be rewritten as Ne = V (2πmkT)3/2 h3 g3/2(z), (16) 7.1 Thermodynamic behavior of an ideal Bose gas 183 where Ne is the number of particles in the excited states (ε ̸= 0); of course, unless z gets extremely close to unity, Ne ≃N.3 It is obvious that, for 0 ≤z ≤1, the function g3/2(z) increases monotonically with z and is bounded, its largest value being g3/2(1) = 1 + 1 23/2 + 1 33/2 + ··· ≡ζ 3 2  ≃2.612; (17) see equation (D.5) in Appendix D. Hence, for all z of interest, g3/2(z) ≤ζ 3 2  . (18) Consequently, for given V and T, the total (equilibrium) number of particles in all the excited states taken together is also bounded, that is, Ne ≤V (2πmkT)3/2 h3 ζ 3 2  . (19) Now, so long as the actual number of particles in the system is less than this limiting value, everything is well and good; practically all the particles in the system are distributed over the excited states and the precise value of z is determined by equation (16), with Ne ≃N. However, if the actual number of particles exceeds this limiting value, then it is natural that the excited states will receive as many of them as they can hold, namely Ne = V (2πmkT)3/2 h3 ζ 3 2  , (20) while the rest will be pushed en masse into the ground state ε = 0 (whose capacity, under all circumstances, is essentially unlimited): N0 = N − ( V (2πmkT)3/2 h3 ζ 3 2 ) . (21) The precise value of z is now determined by the formula z = N0 N0 + 1 ≃1 −1 N0 (22) which, for all practical purposes, is unity. This curious phenomenon of a macroscopi-cally large number of particles accumulating in a single quantum state (ε = 0) is generally referred to as the phenomenon of Bose–Einstein condensation. In a certain sense, this phenomenon is akin to the familiar process of a vapor condensing into the liquid state, which takes place in the ordinary physical space. Conceptually, however, the two pro-cesses are very different. Firstly, the phenomenon of Bose–Einstein condensation is purely 3Remember that the largest value z can have in principle is unity. In fact, as T →0, z = N0/(N0 + 1) →N/(N + 1), which is very nearly unity (but certainly on the right side of it). 184 Chapter 7. Ideal Bose Systems of quantum origin (occurring even in the absence of intermolecular forces); secondly, it takes place at best in the momentum space and not in the coordinate space.4 The condition for the onset of Bose–Einstein condensation is N > VT3/2 (2πmk)3/2 h3 ζ 3 2  (23) or, if we hold N and V constant and vary T, T < Tc = h2 2πmk    N Vζ  3 2     2/3 ; (24)5 here, Tc denotes a characteristic temperature that depends on the particle mass m and the particle density N/V in the system. Accordingly, for T < Tc, the system may be looked on as a mixture of two “phases”: (i) a normal phase, consisting of Ne {= N(T/Tc)3/2} particles distributed over the excited states (ε ̸= 0), and (ii) a condensed phase, consisting of N0 {= (N −Ne)} particles accumulated in the ground state (ε = 0). Figure 7.1 shows the manner in which the complementary fractions (Ne/N) and (N0/N) vary with T. For T > Tc, we have the normal phase alone; the number of particles in the ground state, namely z/(1 −z), is O(1), which is completely negligible in comparison with the total number N. Clearly, the situation is singular at T = Tc. For later reference, we note that, at T →Tc from below, the condensate fraction vanishes as follows: N0 N = 1 −  T Tc 3/2 ≈3 2 Tc −T Tc . (25) A knowledge of the variation of z with T is also of interest here. It is, however, sim-pler to consider the variation of z with (v/λ3), the latter being proportional to T3/2. For 0 ≤(v/λ3) ≤(2.612)−1, which corresponds to 0 ≤T ≤Tc, the parameter z ≃1; see equation (22). For (v/λ3) > (2.612)−1, z < 1 and is determined by the relationship g3/2(z) = (λ3/v) < 2.612; (26)6 4Of course, the repercussions of this phenomenon in the coordinate space are no less curious. It prepares the stage for the onset of superfluidity, a quantum manifestation discussed in Section 7.6. 5For a rigorous discussion of the onset of Bose–Einstein condensation, see Landsberg (1954b), where an attempt has also been made to coordinate much of the previously published work on this topic. For a more recent study, see Greenspoon and Pathria (1974), Pathria (1983), and Appendix F . 6An equivalent relationship is g3/2(z)/g3/2(1) = (Tc/T)3/2 < 1. 7.1 Thermodynamic behavior of an ideal Bose gas 185 1.0 00 1.0 1 2 Ne N N0 N , (T/Tc) 2 1 FIGURE 7.1 Fractions of the normal phase and the condensed phase in an ideal Bose gas as a function of the temperature parameter (T/Tc). 1.0 0.5 0 0 1.0 2.0 1 2.612 (v/3) z z (v/3)1 FIGURE 7.2 The fugacity of an ideal Bose gas as a function of (v/λ3). see equation (8). For (v/λ3) ≫1, we have g3/2(z) ≪1 and, hence, z ≪1. Under these circumstances, g3/2(z) ≃z; see equation (10). Therefore, in this region, z ≃(v/λ3)−1, in agreement with the classical case.7 Figure 7.2 shows the variation of z with (v/λ3). Next, we examine the (P,T)-diagram of this system, that is, the variation of P with T, keeping v fixed. Now, for T < Tc, the pressure is given by equation (7), with z replaced by unity: P(T) = kT λ3 ζ 5 2  , (27) which is proportional to T5/2 and is independent of v — implying infinite compressibility. At the transition point the value of the pressure is P(Tc) = 2πm h2 3/2 (kTc)5/2ζ 5 2  ; (28) 7Equation (6.2.12) gives, for an ideal classical gas, lnQ = zV/λ3. Accordingly, N ≡z(∂lnQ/∂z) = z(V/λ3), with the result that z = (λ3/v). 186 Chapter 7. Ideal Bose Systems with the help of (24), this can be written as P(Tc) = ζ  5 2  ζ  3 2  N V kTc  ≃0.5134 N V kTc  . (29) Thus, the pressure exerted by the particles of an ideal Bose gas at the transition temper-ature Tc is about one-half of that exerted by the particles of an equivalent Boltzmannian gas.8 For T > Tc, the pressure is given by P = N V kT g5/2(z) g3/2(z), (30) where z(T) is determined by the implicit relationship g3/2(z) = λ3 v = N V h3 (2πmkT)3/2 . (26a) Unless T is very high, the pressure P cannot be expressed in any simpler terms; of course, for T ≫Tc, the virial expansion (13) can be used. As T →∞, the pressure approaches the classical value NkT/V. All these features are shown in Figure 7.3. The transition line in the figure portrays equation (27). The actual (P,T)-curve follows this line from T = 0 up to T = Tc and thereafter departs, tending asymptotically to the classical limit. It may be pointed out that the region to the right of the transition line belongs to the normal phase alone, the line itself belongs to the mixed phase, while the region to the left is inaccessible to the system. In view of the direct relationship between the internal energy of the gas and its pres-sure, see equation (12), Figure 7.3 depicts equally well the variation of U with T (of course, with v fixed). Its slope should, therefore, be a measure of the specific heat CV (T) of the gas. We readily observe that the specific heat is vanishingly small at low temperatures and rises with T until it reaches a maximum at T = Tc; thereafter, it decreases, tending asymptoti-cally to the constant classical value. Analytically, for T ≤Tc, we obtain [see equations (15) and (27)] CV Nk = 3 2 V N ζ 5 2  d dT  T λ3  = 15 4 ζ 5 2  v λ3 , (31) 8Actually, for all T ≤Tc, we can write P(T) = P(Tc) · (T/Tc)5/2 ≃0.5134(NekT/V). We infer that, while particles in the condensed phase do not exert any pressure at all, particles in the excited states are about half as effective as in the Boltzmannian case. 7.1 Thermodynamic behavior of an ideal Bose gas 187 2 Transition line Classical B.E. 1 0 0 1 2 3 PV NkTc 2U 3NkTc (T/Tc) FIGURE 7.3 The pressure and the internal energy of an ideal Bose gas as a function of the temperature parameter (T/Tc). which is proportional to T 3/2. At T = Tc, we have CV (Tc) Nk = 15 4 ζ  5 2  ζ  3 2  ≃1.925, (32) which is significantly higher than the classical value 1.5. For T > Tc, we obtain an implicit formula. First of all, CV Nk =  ∂ ∂T 3 2T g5/2(z) g3/2(z)  v ; (33) see equations (11) and (26). To carry out the differentiation, we need to know (∂z/∂T)v; this can be obtained from equation (26) with the help of the recurrence relation (D.10) in Appendix D. On one hand, since g3/2(z) ∝T−3/2,  ∂ ∂T g3/2(z)  v = −3 2T g3/2(z); (34) on the other, z ∂ ∂zg3/2(z) = g1/2(z). (35) Combining these two results, we obtain 1 z  ∂z ∂T  v = −3 2T g3/2(z) g1/2(z). (36) 188 Chapter 7. Ideal Bose Systems Equation (33) now gives CV Nk = 15 4 g5/2(z) g3/2(z) −9 4 g3/2(z) g1/2(z); (37) the value of z, as a function of T, is again to be determined from equation (26). In the limit z →1, the second term in (37) vanishes because of the divergence of g1/2(z), while the first term gives exactly the result appearing in (32). The specific heat is, therefore, con-tinuous at the transition point. Its derivative is, however, discontinuous, the magnitude of the discontinuity being ∂CV ∂T  T=Tc−0 − ∂CV ∂T  T=Tc+0 = 27Nk 16πTc  ζ 3 2 2 ≃3.665Nk Tc ; (38) see Problem 7.6. For T > Tc, the specific heat decreases steadily toward the limiting value CV Nk  z→0 = 15 4 −9 4 = 3 2. (39) Figure 7.4 shows all these features of the (CV ,T)-relationship. It may be noted that it was the similarity of this curve with the experimental one for liquid He4 (Figure 7.5) that prompted F. London to suggest, in 1938, that the curious phase transition that occurs in liquid He4 at a temperature of about 2.19K might be a manifestation of the Bose–Einstein condensation taking place in the liquid. Indeed, if we substitute, in (24), data for liquid He4, namely m = 6.65 × 10−24 g and V = 27.6cm3/mole, we obtain for Tc a value of about 3.13K, which is not drastically different from the observed transition temperature of the liquid. Moreover, the interpretation of the phase transition in liquid He4 as Bose–Einstein condensation provides a theoretical basis for the two-fluid model of this liquid, which was empirically put forward by Tisza (1938a,b) to explain the physical behavior of the liquid below the transition temperature. According to London, the N0 particles that occupy a single, entropyless state (ε = 0) could be identified with the “superfluid component” of the liquid and the Ne particles that occupy the excited states (ε ̸= 0) with the “normal component.” As required in the 0 0 1 2 3 1.5 Cv~T 3/2 1.925 (T/Tc) (Cv/Nk) FIGURE 7.4 The specific heat of an ideal Bose gas as a function of the temperature parameter (T/Tc). 7.1 Thermodynamic behavior of an ideal Bose gas 189 1.5 2.0 2.5 3.0 0 0.5 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 T(K) C (cal g21 K21) FIGURE 7.5 The specific heat of liquid He4 under its own vapor pressure (after Keesom and coworkers). model of Tisza, the superfluid fraction makes its appearance at the transition tempera-ture Tc, and builds up at the cost of the normal fraction until at T = 0 the whole fluid becomes superfluid; compare to Figure 7.1. Of course, the actual temperature dependence of these fractions, and of other physical quantities pertaining to liquid He4, is consider-ably different from what the simple-minded ideal Bose gas suggests. London had expected that the inclusion of intermolecular interactions would improve the quantitative agree-ment. Although this expectation has been partially vindicated, there have been other advances in the field that provide alternative ways of looking at the helium problem; see Section 7.6. Nevertheless, many of the features provided by London’s interpretation of this phenomenon continue to be of value. Historically, the experimental measurements of the specific heat of liquid He4, which led to the discovery of this so-called He I–He II transition, were first made by Keesom in 1927 and 1928. Struck by the shape of the (CV ,T)-curve, Keesom gave this transition the name λ-transition; as a result, the term transition temperature (or transition point) also came to be known as λ-temperature (or λ-point). We shall now look at the isotherms of the ideal Bose gas; that is, the variation of the pressure of the gas with its volume, keeping T fixed. The Bose–Einstein condensation now sets in at a characteristic volume vc, given by vc = λ3/ζ 3 2  ; (40) see (23). We note that vc ∝T−3/2. For v < vc, the pressure of the gas is independent of v and is given by P0 = kT λ3 ζ 5 2  ; (41) 190 Chapter 7. Ideal Bose Systems P01 P02 Pv 5/3 const. Vc 2 Vc 1 T2 T1 Transition line V FIGURE 7.6 The isotherms of an ideal Bose gas. see (27). The region of the mixed phase in the (P,v)-diagram is marked by a boundary line (called the transition line) given by the equation P0v5/3 c = h2 2πm ζ  5 2  n ζ  3 2 o5/3 = const.; (42) see Figure 7.6. Clearly, the region to the left of this line belongs to the mixed phase, while the region to the right belongs to the normal phase alone. Finally, we examine the adiabats of the ideal Bose gas. For this, we need an expression for the entropy of the system. Making use of the thermodynamic formula U −TS + PV ≡Nµ (43) and the expressions for U and P obtained above, we get S Nk ≡U + PV NkT −µ kT =          5 2 g5/2(z) g3/2(z) −lnz for T > Tc, (44a) 5 2 v λ3 ζ 5 2  for T ≤Tc; (44b) again, the value of z(T), for T > Tc, is to be obtained from equation (26). Now, a reversible adiabatic process implies the constancy of S and N. For T > Tc, this implies the constancy of z as well and in turn, by (26), the constancy of (v/λ3). For T ≤Tc, it again implies the same. We thus obtain, quite generally, the following relationship between the volume and the temperature of the system when it undergoes a reversible adiabatic process: vT3/2 = const. (45) The corresponding relationship between the pressure and the temperature is P/T5/2 = const.; (46) 7.2 Bose–Einstein condensation in ultracold atomic gases 191 see equations (7) and (27). Eliminating T, we obtain Pv5/3 = const. (47) as the equation for an adiabat of the ideal Bose gas. Incidentally, the foregoing results are exactly the same as for an ideal classical gas. There is, however, a significant difference between the two cases; that is, while the exponent 5 3 in formula (47) is identically equal to the ratio of the specific heats CP and CV in the case of the ideal classical gas, it is not so in the case of the ideal Bose gas. For the latter, this ratio is given by γ ≡CP CV = 1 + 4 9 CV Nk g1/2(z) g3/2(z) (48a) = 5 3 g5/2(z)g1/2(z) {g3/2(z)}2 ; (48b) see Problems 7.4 and 7.5. It is only for T ≫Tc that γ ≃5 3. At any finite temperature, γ > 5 3 and as T →Tc,γ →∞. Equation (47), on the other hand, holds for all T. In the mixed-phase region (T < Tc), the entropy of the gas may be written as S = Ne · 5 2k ζ  5 2  ζ  3 2  ∝Ne; (49) see equations (20) and (44b). As expected, the N0 particles that constitute the “condensate” do not contribute toward the entropy of the system, while the Ne particles that constitute the normal part contribute an amount of 5 2kζ( 5 2)/ζ( 3 2) per particle. 7.2 Bose–Einstein condensation in ultracold atomic gases The first demonstration of Bose–Einstein condensation in ultracold atomic gases came in 1995. Cornell and Wieman Bose-condensed 87Rb (Anderson, Ensher, Matthews, Wieman, and Cornell (1995)) and Ketterle Bose-condensed 23Na (Davis, Mewes, Andrews, van Druten, Durfee, Kurn, and Ketterle (1995)) using magneto-optical traps (MOTs) and magnetic traps to cool vapors of tens of thousands of atoms to temperatures of a few nanokelvin.9 A survey of the theory and experiments can be found in Pitaevskii and Stringari (2003), Leggett (2006), and Pethick and Smith (2008). The first step of the cooling of the atomic vapor uses three sets of counter-propagating laser beams oriented along cartesian axes that are tuned just below the resonant frequency 9Since 1995, many isotopes have been Bose-condensed including 7Li, 23Na, 41K, 52Cr, 84Sr, 85Rb, 87Rb, 133Cs, and 174Yb. The first molecular Bose–Einstein condensates were created in 2003 by the research groups of Rudolf Grimm at the University of Innsbruck, Deborah S. Jin at the University of Colorado at Boulder, and Wolfgang Ketterle at Massachusetts Institute of Technology. 192 Chapter 7. Ideal Bose Systems of the atoms in the trap. Atoms that are stationary are just off resonance and so rarely absorb a photon. Moving atoms are Doppler shifted on resonance to the laser beam that is propagating opposite to the velocity vector of the atom. Those atoms preferentially absorb photons from that direction and then reemit in random directions, resulting in a net momentum kick opposite to the direction of motion. This results in an “optical molasses” that slows the atoms. This cooling method is constrained by the “recoil limit” in which the atoms have a minimum momentum of the order of the momentum of the photons used to cool the gas. This gives a limiting temperature of (hf )2/2mc2k ≈1µK, where f is the frequency of the spectral line used for cooling and m is the mass of an atom. In the next step of the cooling process, the lasers are turned off and a spatially vary-ing magnetic field creates an attractive anisotropic harmonic oscillator potential near the center of the magnetic trap V (r) = 1 2m  ω2 1x2 + ω2 2y2 + ω2 3z2 . (1) The frequencies of the trap ωα are controlled by the applied magnetic field. One can then lower the trap barrier using a resonant transition to remove the highest energy atoms in the trap. If the atoms in the vapor are sufficiently coupled to one other, then the remaining atoms in the trap are cooled by evaporation. If the interactions between the atoms in the gas can be neglected, the energy of each atom in the harmonic oscillator potential is εl1,l2,l3 = ℏω1l1 + ℏω2l2 + ℏω3l3 + 1 2ℏ(ω1 + ω2 + ω3) , (2) where lα (= 0,1,2,...∞) are the quantum numbers of the harmonic oscillator. If the three frequencies are all the same, then the quantum degeneracy of a level with energy ε = ℏω(l + 3/2) is (l + 1)(l + 2)/2; see Problem 3.26. For the general anisotropic case, the smoothed density of states as a function of energy (suppressing the zero point energy and assuming ε ≫ℏωα) is given by a(ε) = ∞ Z 0 ∞ Z 0 ∞ Z 0 δ ε −ℏω1l1 −ℏω2l2 −ℏω3l3  dl1dl2dl3 = ε2 2(ℏω0)3 , (3) where ω0 = (ω1ω2ω3)1/3; this assumes a single spin state per atom. The thermodynamic potential 5, see Appendix H, for bosons in the trap is then given by 5(µ,T) = − kT 4 2(ℏω0)3 ∞ Z 0 x2 ln  1 −e−xeβµ dx = kT 4 (ℏω0)3 g4(z), (4) 7.2 Bose–Einstein condensation in ultracold atomic gases 193 where z = exp(βµ) is the fugacity and gν(z) is defined in Appendix D. Volume is not a parameter in the thermodynamic potential since the atoms are confined by the har-monic trap. The average number of atoms in the excited states in the trap is N(µ,T) = ∂5 ∂µ  T =  kT ℏω0 3 g3(z) . (5) For fixed N, the chemical potential monotonically increases as temperature is lowered until Bose–Einstein condensation occurs when µ = 0 (z = 1). The critical temperature for N trapped atoms is then given by kTc ℏω0 =  N ζ(3) 1/3 , (6) where ζ(3) = g3(1) ≃1.202. While the spacing of the energy levels is of order ℏω0, the crit-ical temperature for condensation is much larger than the energy spacing of the lowest levels for N ≫1. A typical magnetic trap oscillation frequency f ≈100Hz. For N = 2 × 104, as in Cornell and Wieman’s original experiment, kTc/ℏω0 ≈25.5. The observed critical temperature was about 170nK (Anderson et al. (1995)). For T < Tc, the number of atoms in the excited states is Nexcited N = ζ(3) N  kT ℏω0 3 =  T Tc 3 , (7) so the fraction of atoms that condense into the ground state of the harmonic oscillator is N0 N = 1 −  T Tc 3 ; (8) see de Groot, Hooyman, and ten Seldam (1950), and Bagnato, Pritchard, and Kleppner (1987). In the thermodynamic limit, a nonzero fraction of the atoms occupy the ground state for T < Tc. By contrast, the occupancy of the first excited state is only of order N1/3, so in the thermodynamic limit the occupancy fraction in each excited state is zero. A compar-ison of the experimentally measured Bose-condensed fraction with equation (8) is shown in Figure 7.7. 7.2.A Detection of the Bose–Einstein condensate The linear size of the ground state wavefunction in cartesian direction α is aα = s ℏ mωα , (9) 194 Chapter 7. Ideal Bose Systems 0.8 0.6 0.4 0.2 0.0 0.0 0.0 0.5 1.0 1.5 0.5 1.0 T/Tc(N) N(104) N0/N 1.5 1.0 12 8 4 FIGURE 7.7 Experimental measurement of the Bose-condensed fraction vs. temperature, as compared to equation (8). The scaled temperature on the horizontal axis is the temperature divided by the N-dependent critical temperature given in equation (6). The inset shows the total number of atoms in the trap after the evaporative cooling. From Ensher et al. (1996). Reprinted with permission; copyright © 1996, American Physical Society. while the linear size of the thermal distribution of the noncondensed atoms in that direction is athermal = s kT mω2 α = aα s kT ℏωα . (10) At trap frequency f = 100 Hz and temperature T = 100 nK, these sizes are about 1 µm and 5 µm, respectively. Instead of measuring the atoms directly in the trapping potential, experimenters usually measure the momentum distribution of the ultracold gas by a time-of-flight experiment. At time t = 0, the magnetic field is turned off suddenly, eliminating the trapping potential. The atomic cloud then expands according to the momentum dis-tribution the atoms had in the harmonic trap. The cloud is allowed to expand for about 100 milliseconds. The speed of the atoms at this temperature is a few millimeters per second, so the cloud expands to a few hundred microns in this period of time. The cloud is then illuminated with a laser pulse on resonance with the atoms, leaving a shadow on a CCD in the image plane of the optics. The size and shape of the light intensity pattern directly measures the momentum distribution the atoms had in the trap at t = 0. The expanding cloud can be divided into two components, the N0 atoms that had been Bose-condensed into the ground state and the remaining N −N0 atoms that were in the excited states of the harmonic oscillator potential. The Bose-condensed atoms have smaller momenta than the atoms that were in the excited states. After time t, the quantum evolution of the ground 7.2 Bose–Einstein condensation in ultracold atomic gases 195 state has a spatial number density n0(r,t) = N0 |ψ0(r,t)|2 = N0 π3/2 3 Y α=1   1 aα q 1 + ω2 αt2 exp −r2 α a2 α 1 + ω2 αt2 ! ; (11) see Pitaevskii and Stringari (2003), Pethick and Smith (2008), and Problem 7.15. The atoms that are not condensed into the ground state can be treated semiclassi-cally, that is, the position-momentum distribution function is treated classically while the density follows the Bose–Einstein distribution function: f (r,p,0) = 1 exp  βp2 2m + βm 2 ω2 1x2 + ω2 2y2 + ω2 3z2 −βµ  −1 . (12) After the potential is turned off at t = 0, the distribution evolves ballistically: f (r,p,t) = f  r + pt m ,p,0  . (13) The spatial number density of atoms in the excited states is nexcited(r,t) = 1 h3 Z f  r + pt m ,p,t  dp , (14) which can be integrated to give nexcited(r,t) = 1 λ3 ∞ X j=1 eβµj j3/2    3 Y α=1   1 q 1 + ω2 αt2 exp −βjmω2 αr2 α 2 1 + ω2 αt2 !    , (15) where λ = h/ √ 2πmkT is the thermal deBroglie wavelength; see Pethick and Smith (2008), and Problem 7.16. The integrals over the condensed state and the excited states correctly count all the atoms: N0 = Z n0(r,t)dr, (16a) N −N0 = Z nexcited(r,t)dr = Nexcited; (16b) see Problem 7.18. Note that at early times (ωαt ≪1) both the condensed and the excited distributions are anisotropic due to the anisotropic trapping potential. However, at late times (ωαt ≫1), the atoms from the excited states form a spherically symmetric cloud because of the isotropic momentum dependence of the t = 0 distribution function. By contrast, the atoms that were condensed into the ground state expand anisotropically due to the different spa-tial extents of the ground state wavefunction at t = 0. The direction that has the largest 196 Chapter 7. Ideal Bose Systems FIGURE 7.8 The two-dimensional time-of-flight number density equations (11) and (15) at late times (ω0t ≫1) for T/Tc = 0.98 using the experimental parameters in Anderson et al. (1995): N = 2 × 104 atoms in the trap and ω2 = √ 8ω1. The plot shows the full density and, underneath, the broader isotropic density just due to the excited states. The z-dimension has been integrated out. The Bose-condensed peak is anisotropic: the y-direction spread is 81/4 = 1.68 times larger than in the x-direction while the broad peak caused by the excited states is isotropic. The distance scale v0t = t√ℏω1/m determines the width of the distribution that results from the Bose-condensed peak in the x-direction at late times; compare to Figure 7.9. ωα is quantum mechanically squeezed the most at t = 0; so, according to the uncertainty principle, it expands the fastest. This is an important feature of the experimental data that confirms the onset of Bose–Einstein condensation;10 see Figures 7.8 and 7.9. 7.2.B Thermodynamic properties of the Bose–Einstein condensate The temperature, condensate fraction, and internal energy can all be observed using time-of-flight measurements. The internal energy can also be written in terms of the function gν(z): U(µ,T) = ∞ Z 0 ε3 2(ℏω0)3 1 eβ(ε−µ) −1dε = 3 kT 4 (ℏω0)3 g4(z). (17) 10Repulsive interactions between atoms create additional forces that modify the time-of-flight expansion. This is especially important in condensates with a very large numbers of atoms, as many as 107 or more in some experiments; see Section 11.2.A. 7.2 Bose–Einstein condensation in ultracold atomic gases 197 FIGURE 7.9 Time-of-flight images from the first observation of Bose–Einstein condensation in a dilute vapor of 87Rb by Anderson et al. (1995) at temperatures just above and below the phase transition temperature. The anisotropic pattern of the Bose-condensed fraction is evident; compare to Figure 7.8. Courtesy of NIST/JILA/University of Colorado. The heat capacity at constant number can be written as CN(T) = ∂U ∂T  N = ∂U ∂T  µ + ∂U ∂µ  T ∂µ ∂T  N = ∂U ∂T  µ −  ∂U ∂T  µ  ∂N ∂T  µ  ∂N ∂µ  T . (18) Equations (5) and (6) can be used to determine the fugacity z numerically, as shown in Figure 7.10(a). The fugacity can then be used in equation (17) to obtain the scaled internal energy U NkTc =            3  T Tc 4 ζ(4) ζ(3) for T ≤Tc, 3  T Tc 4 g4(z) ζ(3) for T ≥Tc; (19) 198 Chapter 7. Ideal Bose Systems see Figures 7.10(b) and 7.12. The scaled specific heat is given by CN Nk =            12ζ(4) ζ(3)  T Tc 3 for T < Tc, 1 ζ(3)  T Tc 3 12g4(z) −9g2 3(z) g2(z) ! for T > Tc, (20) and is shown in Figure 7.11. Unlike the case of Bose–Einstein condensation of free parti-cles in a box (Figure 7.4), the specific heat of a condensate in a harmonic trap displays a (a) (b) 1.5 1.0 0.5 0.0 0.0 0.5 1.0 T/Tc T/Tc 1.5 2.0 0.0 0 2 4 6 0.5 1.0 1.5 2.0 z exp() U NkTc FIGURE 7.10 Fugacity (a) and scaled internal energy (b) vs. scaled temperature (T/Tc) for a Bose–Einstein condensate in a harmonic trap. 12 10 8 6 4 2 0 0.0 0.5 1.0 T/Tc 1.5 2.0 CN Nk FIGURE 7.11 Scaled specific heat of a Bose–Einstein condensate in a harmonic trap as a function of the scaled temperature (T/Tc); compare with Figure 7.4 for a free-particle Bose gas. 7.2 Bose–Einstein condensation in ultracold atomic gases 199 discontinuity at the critical temperature: CN Nk →          12ζ(4) ζ(3) ≃10.805 as T →T− c , 12ζ(4) ζ(3) −9ζ(3) ζ(2) ≃4.228 as T →T+ c . (21) Figure 7.12 shows experimental data for the internal energy of a Bose–Einstein con-densate of 87Rb. The break in slope is an indication of the discontinuous specific heat. Naturally, in a system with a finite number of particles, all nonanalyticities associated with the phase transition are removed. When N is finite, the condensate fraction approaches zero smoothly and the discontinuity in the heat capacity is rounded off. Pathria (1998) has derived N-dependent temperature markers that indicate the onset of Bose–Einstein con-densation in terms of the condensate fraction and the specific heat; see also Kirsten and Toms (1996) and Haugerud, Haugest, and Ravndal (1997). 2 1.5 1 0.5 0.5 1 1.5 0 1.1 1 0.9 1.5 1.4 1.3 1.2 1.1 T/Tc U/NkTc FIGURE 7.12 Comparison of the experimental measurements of Ensher et al. (1996) (diamonds) with the noninteracting internal energy result — see equation (19) and Figure 7.10(b) — (dotted curve), the zero-order solution including interactions (full curve), first-order perturbative treatment (dashed curve), and numerical solution (circles). The straight line is the classical Maxwell–Boltzmann result. The inset is an enlargement of the region around the critical temperature. The break in slope is an indication of the discontinuity in the thermodynamic limit specific heat shown in Figure 7.11; from Minguzzi, Conti, and Tosi (1997). Reprinted with permission; copyright © 1997, American Institute of Physics. 200 Chapter 7. Ideal Bose Systems 7.3 Thermodynamics of the blackbody radiation One of the most important applications of Bose–Einstein statistics is to investigate the equilibrium properties of the blackbody radiation. We consider a radiation cavity of vol-ume V at temperature T. Historically, this system has been looked on from two, practically identical but conceptually different, points of view: (i) as an assembly of harmonic oscillators with quantized energies (ns + 1 2)ℏωs, where ns = 0,1,2,..., and ωs is the (angular) frequency of an oscillator, or (ii) as a gas of identical and indistinguishable quanta — the so-called photons — the energy of a photon (corresponding to the frequency ωs of the radiation mode) being ℏωs. The first point of view is essentially the one adopted by Planck (1900), except that we have also included here the zero-point energy of the oscillator; for the thermodynamics of the radiation, this energy is of no great consequence and may be dropped altogether. The oscillators, being distinguishable from one another (by the very values of ωs), would obey Maxwell–Boltzmann statistics; however, the expression for the single-oscillator par-tition function Q1(V,T) would be different from the classical expression because now the energies accessible to the oscillator are discrete, rather than continuous; compare to equa-tions (3.8.2) and (3.8.14). The expectation value of the energy of a Planck oscillator of frequency ωs is then given by equation (3.8.20), excluding the zero-point term 1 2ℏωs: ⟨εs⟩= ℏωs eℏωs/kT −1. (1) Now, the number of normal modes of vibration per unit volume of the cavity in the frequency range (ω,ω + dω) is given by the Rayleigh expression 2 · 4π 1 λ 2 d 1 λ  = ω2dω π2c3 , (2) where the factor 2 has been included to take into account the duplicity of the transverse modes;11 the symbol c here denotes the speed of light. By equations (1) and (2), the energy density associated with the frequency range (ω,ω + dω) is given by u(ω)dω = ℏ π2c3 ω3dω eℏω/kT −1, (3) which is Planck’s formula for the distribution of energy over the blackbody spectrum. Integrating (3) over all values of ω, we obtain the total energy density in the cavity. The second point of view originated with Bose (1924) and Einstein (1924, 1925). Bose investigated the problem of the “distribution of photons over the various energy levels” in the system; however, instead of worrying about the allocation of the various photons 11As is well-known, the longitudinal modes play no role in the case of radiation. 7.3 Thermodynamics of the blackbody radiation 201 to the various energy levels (as one would have ordinarily done), he concentrated on the statistics of the energy levels themselves! He examined questions such as the “probability of an energy level εs(= ℏωs) being occupied by ns photons at a time,” “the mean values of ns and εs,” and so on. The statistics of the energy levels is indeed Boltzmannian; the mean values of ns and εs, however, turn out to be ⟨ns⟩= ∞ X ns=0 nse−nsℏωs/kT , ∞ X ns=0 e−nsℏωs/kT = 1 eℏωs/kT −1 (4) and hence ⟨εs⟩= ℏωs⟨ns⟩= ℏωs eℏωs/kT −1, (5) identical with our earlier result (1). To obtain the number of photon states with momenta lying between ℏω/c and ℏ(ω + dω)/c, Bose made use of the connection between this number and the “volume of the relevant region of the phase space,” with the result g(ω)dω ≈2 · V h3 ( 4π ℏω c 2 ℏdω c ) = Vω2dω π2c3 , (6)12 which is also identical to our earlier result (2). Thus, he finally obtained the distribution formula of Planck. It must be noted here that, although emphasis lay elsewhere, the math-ematical steps that led Bose to his final result went literally parallel to the ones occurring in the oscillator approach! Einstein, on the other hand, went deeper into the problem and pondered over the statistics of both the photons and the energy levels, taken together. He inferred (from Bose’s treatment) that the basic fact to keep in mind during the process of distributing photons over the various energy levels is that the photons are indistinguishable — a fact that had been implicitly taken care of in Bose’s treatment. Einstein’s derivation of the desired distribution was essentially the same as given in Section 6.1, with one important difference, that is, since the total number of photons in any given volume was indefinite, the constraint of a fixed N was no longer present. As a result, the Lagrange multiplier α did not enter into the discussion and to that extent the final formula for ⟨nε⟩was simpler: ⟨nε⟩= 1 eε/kT −1; (7) compare to equation (6.1.18a) or (6.2.22). The foregoing result is identical to (4), with ε = hωs. The subsequent steps in Einstein’s treatment were the same as in Bose’s. 12The factor 2 in this expression arises essentially from the same cause as in the Rayleigh expression (2). However, in the present context, it would be more appropriate to regard it as representing the two states of polarization of the photon spin. 202 Chapter 7. Ideal Bose Systems 00 2 4 6 8 10 12 14 [x 2.8214...] 1.0 1.5 0.5 2.0 u(x ) x Rayleigh-Jeans’ law Wien’s law Planck’s law FIGURE 7.13 The spectral distribution of energy in the blackbody radiation. The solid curve represents the quantum-theoretical formula of Planck. The long-wavelength approximation of Rayleigh–Jeans and the short-wavelength approximation of Wien are also shown. Looking back at the two approaches, we note that there is a complete correspondence between them — “an oscillator in the eigenstate ns, with energy (ns + 1 2)ℏωs” in the first approach corresponds to “the occupation of the energy level hωs by ns photons” in the second approach, “the average energy ⟨εs⟩of an oscillator” corresponds to “the mean occupation number ⟨ns⟩of the corresponding energy level,” and so on.13 Figure 7.13 shows a plot of the distribution function (3), which may be written in the dimensionless form u′(x)dx = x3dx ex −1, (8) where u′(x) = π2ℏ3c3 (kT)4 u(x) and x = ℏω kT . (9) For long wavelengths (x ≪1), formula (8) reduces to the classical approximation of Rayleigh (1900) and Jeans (1905), namely14 u′(x) ≈x2, (10) while for short wavelengths (x ≫1), it reduces to the rival formula of Wien (1896), namely u′(x) ≈x3e−x. (11) 13Compared to the standard Bose–Einstein result (7.1.2), formula (7) suggests that we are dealing here with a case for which z is precisely equal to unity. It is not difficult to see that this is due to the fact that the total number of particles in the present case is indefinite. For then, their equilibrium number N has to be determined by the condition that the free energy of the system is at its minimum, that is, {(∂A/∂N)N=N}V,T = 0, which, by definition, implies that µ = 0 and hence z = 1. 14The Rayleigh–Jeans formula follows directly if we use for ⟨εs⟩the equipartition value kT rather than the quantum-theoretical value (1). 7.3 Thermodynamics of the blackbody radiation 203 For comparison, the limiting forms (10) and (11) are also included in the figure. We note that the areas under the Planck curve and the Wien curve are π4/15(≃6.49) and 6, respec-tively. The Rayleigh–Jeans curve, however, suffers from a high-frequency catastrophe! For the total energy density in the cavity, we obtain from equations (8) and (9) U V = ∞ Z 0 u(x)dx = (kT)4 π2ℏ3c3 ∞ Z 0 x3dx ex −1 = π2k4 15ℏ3c3 T4. (12)15 If there is a small opening in the walls of the cavity, the photons will “effuse” through it. The net rate of flow of the radiation, per unit area of the opening, will be given by, see equation (6.4.12), 1 4 U V c = π2k4 60ℏ3c2 T4 = σT4, (13) where σ = π2k4 60ℏ3c2 = 5.670 × 10−8 Wm−2 K−4. (14) Equation (13) describes the Stefan–Boltzmann law of blackbody radiation, σ being the Ste-fan constant. This law was deduced from experimental observations by Stefan in 1879; five years later, Boltzmann derived it from thermodynamic considerations. For further study of thermodynamics, we evaluate the grand partition function of the photon gas. Using equation (6.2.17) with z = 1, we obtain lnQ(V,T) ≡PV kT = − X ε ln(1 −e−ε/kT). (15) Replacing summation by integration and making use of the extreme relativistic formula a(ε)dε = 2V 4πp2dp h3 = 8πV h3c3 ε2dε, (16) we obtain, after an integration by parts, lnQ(V,T) ≡PV kT = 8πV 3h3c3 1 kT ∞ Z 0 ε3dε eε/kT −1. 15Here, use has been made of the fact that the value of the definite integral is 6ζ(4) = π4/15; see Appendix D. 204 Chapter 7. Ideal Bose Systems By a change of variable, this becomes PV = 8πV 3h3c3 (kT)4 ∞ Z 0 x3dx ex −1 = 8π5V 45h3c3 (kT)4 = 1 3U. (17) We thus obtain the well-known result of the radiation theory; that is, the pressure of the radiation is equal to one-third its energy density; see also equations (6.4.3) and (6.4.4). Next, since the chemical potential of the system is zero, the Helmholtz free energy is equal to −PV; therefore A = −PV = −1 3U, (18) whereby S ≡U −A T = 4 3 U T ∝VT3 (19) and CV = T  ∂S ∂T  V = 3S. (20) If the radiation undergoes a reversible adiabatic change, the law governing the variation of T with V would be, see (19), VT3 = const. (21) Combining (21) with the fact that P ∝T4, we obtain an equation for the adiabats of the system, namely PV 4/3 = const. (22) It should be noted, however, that the ratio CP/CV of the photon gas is not 4/3; it is infinite! Finally, we derive an expression for the equilibrium number N of photons in the radiation cavity. We obtain N = V π2c3 ∞ Z 0 ω2dω eℏω/kT −1 = V 2ζ(3)(kT)3 π2h3c3 ∝VT3. (23) Instructive though it may be, formula (23) cannot be taken at its face value because in the present problem the magnitude of the fluctuations in the variable N, which is determined by the quantity (∂P/∂V)−1, is infinitely large; see equation (4.5.7). 7.4 The field of sound waves 205 One of the most important examples of blackbody radiation is the 2.7K cosmic microwave background, which is a remnant from the Big Bang. Equations (12) and (23) play an important role in our understanding of the thermodynamics of the early universe; see Problem 7.24 and Chapter 9. 7.4 The field of sound waves A problem mathematically similar to the one discussed in Section 7.3 arises from the vibrational modes of a macroscopic body, specifically a solid. As in the case of black-body radiation, the problem of the vibrational modes of a solid can be studied equally well by regarding the system as a collection of harmonic oscillators or by regarding it as an enclosed region containing a gas of sound quanta — the so-called phonons. To illus-trate this point, we consider the Hamiltonian of a classical solid composed of N atoms whose positions in space are specified by the coordinates (x1,x2,...,x3N). In the state of lowest energy, the values of these coordinates may be denoted by (x1,x2,...,x3N). Denoting the displacements (xi −xi) of the atoms from their equilibrium positions by the variables ξi(i = 1,2,...,3N), the kinetic energy of the system in configuration (xi) is given by K = 1 2m 3N X i=1 ˙ x2 i = 1 2m 3N X i=1 ˙ ξ2 i , (1) and the potential energy by 8 ≡8(xi) = 8(xi) + X i ∂8 ∂xi  (xi)=(xi) (xi −xi) + X i,j 1 2 ∂28 ∂xi∂xj ! (xi)=(xi) (xi −xi)(xj −xj) + ··· . (2) The main term in this expansion represents the (minimum) energy of the solid when all the atoms are at rest at their mean positions xi; this energy may be denoted by the symbol 80. The next set of terms is identically zero because the function 8(xi) has a minimum at (xi) = (xi) and hence all its first derivatives vanish there. The second-order terms of the expansion represent the harmonic component of the vibrations of the atoms about their mean positions. If we assume that the overall amplitude of these vibrations is not large we may retain only the harmonic terms of the expansion and neglect all successive ones; we are then working in the so-called harmonic approximation. Thus, we may write H = 80 +    X i 1 2m˙ ξ2 i + X i,j αijξiξj   , (3) 206 Chapter 7. Ideal Bose Systems where αij = 1 2 ∂28 ∂xi∂xj ! (xi)=(xi) . (4) We now introduce a linear transformation, from the coordinates ξi to the so-called normal coordinates qi, and choose the transformation matrix such that the new expression for the Hamiltonian does not contain any cross terms, that is, H = 80 + X i 1 2m  ˙ q2 i + ω2 i q2 i  , (5) where ωi(i = 1,2,...,3N) are the characteristic frequencies of the normal modes of the sys-tem and are determined essentially by the quantities αij or, in turn, by the nature of the potential energy function 8(xi). Equation (5) suggests that the energy of the solid, over and above the (minimum) value 80, may be considered as arising from a set of 3N one-dimensional, noninteracting, harmonic oscillators whose characteristic frequencies ωi are determined by the interatomic interactions in the system. Classically, each of the 3N normal modes of vibration corresponds to a wave of distor-tion of the lattice, that is, a sound wave. Quantum-mechanically, these modes give rise to quanta, called phonons, in much the same way as the vibrational modes of the electromag-netic field give rise to photons. There is one important difference, however, that is, while the number of normal modes in the case of an electromagnetic field is infinite, the num-ber of normal modes (or the number of phonon energy levels) in the case of a solid is fixed by the number of lattice sites.16 This introduces certain differences in the thermodynamic behavior of the sound field in contrast to the thermodynamic behavior of the radiation field; however, at low temperatures, where the high-frequency modes of the solid are not very likely to be excited, these differences become rather insignificant and we obtain a striking similarity between the two sets of results. The thermodynamics of the solid can now be studied along the lines of Section 3.8. First of all, we note that the eigenvalues of the Hamiltonian (5) are E{ni} = 80 + X i  ni + 1 2  ℏωi, (6) where the numbers ni denote the “states of excitation” of the various oscillators (or, equally well, the occupation numbers of the various phonon levels in the system). The internal energy of the system is then given by U(T) = ( 80 + X i 1 2ℏωi ) + X i ℏωi eℏωi/kT −1. (7) 16Of course, the number of phonons themselves is indefinite. As a result, the chemical potential of the phonon gas is also zero. 7.4 The field of sound waves 207 The expression within the curly brackets gives the energy of the solid at absolute zero. The term 80 is negative and larger in magnitude than the total zero-point energy, P i 1 2ℏωi, of the oscillators: together, they determine the binding energy of the lattice. The last term in (7) represents the temperature-dependent part of the energy,17 which determines the specific heat of the solid: CV (T) ≡ ∂U ∂T  V = k X i (ℏωi/kT)2eℏωi/kT (eℏωi/kT −1)2 . (8) To proceed further, we need to know the frequency spectrum of the solid. To obtain this from first principles is not an easy task. Accordingly, one obtains this spectrum either through experiment or by making certain plausible assumptions about it. Einstein, who was the first to apply the quantum concept to the theory of solids (1907), assumed, for simplicity, that the frequencies ωi are all equal. Denoting this (common) value by ωE, the specific heat of the solid is given by CV (T) = 3NkE(x), (9) where E(x) is the so-called Einstein function: E(x) = x2ex (ex −1)2 , (10) with x = ℏωE/kT = 2E/T. (11) The dashed curve in Figure 7.14 depicts the variation of the specific heat with tempera-ture, as given by the Einstein formula (9). At sufficiently high temperatures, where T ≫2E and hence x ≪1, the Einstein result tends toward the classical one, namely 3Nk.18 At sufficiently low temperatures, where T ≪2E and hence x ≫1, the specific heat falls expo-nentially fast and tends to zero as T →0. The theoretical rate of fall, however, turns out to be too fast in comparison with the observed one. Nevertheless, Einstein’s approach did at least provide a theoretical basis for understanding the observed departure of the specific heat of solids from the classical law of Dulong and Petit, whereby CV = 3R ≃5.96 calories per mole per degree. Debye (1912), on the other hand, allowed a continuous spectrum of frequencies, cut off at an upper limit ωD such that the total number of normal modes of vibration is 3N, 17The thermal energy of the solid may well be written as P i⟨ni⟩ℏωi, where ⟨ni⟩{= (eℏωi/kT −1)−1} is the mean occupation number of the phonon level εi. Clearly, the phonons, like photons, obey Bose–Einstein statistics, with µ = 0. 18Actually, when the temperature is high enough, so that all (ℏωi/kT) ≪1, the general formula (8) itself reduces to the classical one. This corresponds to the situation when each of the 3N modes of vibration possesses a thermal energy kT. 208 Chapter 7. Ideal Bose Systems 0.5 0 1.0 0 0.5 1.0 (Cv/ 3Nk) (T /) T 3-law FIGURE 7.14 The specific heat of a solid, according to the Einstein model: – – – , and according to the Debye model: —–. The circles denote the experimental results for copper. that is ωD Z 0 g(ω)dω = 3N, (12) where g(ω)dω denotes the number of normal modes of vibration whose frequency lies in the range (ω,ω + dω). For g(ω), Debye adopted the Rayleigh expression (7.3.2), modified so as to suit the problem under study. Writing cL for the velocity of propagation of the longi-tudinal modes and cT for that of the transverse modes (and noting that, for any frequency ω, the transverse mode is doubly degenerate), equation (12) becomes ωD Z 0 V ω2dω 2π2c3 L + ω2dω π2c3 T ! = 3N, (13) from which one obtains for the cutoff frequency ω3 D = 18π2 N V 1 c3 L + 2 c3 T !−1 . (14) Accordingly, the Debye spectrum may be written as g(ω) =      9N ω3 D ω2 for ω ≤ωD, 0 for ω > ωD. (15) Before we proceed further to calculate the specific heat of solids on the basis of the Debye spectrum, two remarks seem to be in order. First, the Debye spectrum is only an idealization of the actual situation obtaining in a solid; it may, for instance, be compared with a typical spectrum such as the one shown in Figure 7.15. While for low-frequency modes (the so-called acoustic modes) the Debye approximation is reasonable, serious dis-crepancies are seen in the case of high-frequency modes (the so-called optical modes). At 7.4 The field of sound waves 209 2 3 4 5 1 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 g (), in arbitrary units , in units of 2 3 1013 sec21 FIGURE 7.15 The normal-mode frequency distribution g(ω) for aluminum. The solid curve is derived from x-ray scattering measurements [Walker (1956)] while the dashed curve represents the corresponding Debye approximation. any rate, for “averaged” quantities, such as the specific heat, the finer details of the spec-trum are not very important. Second, the longitudinal and the transverse modes of the solid should have their own cutoff frequencies, ωD,L and ωD,T say, rather than a common cutoff at ωD, for the simple reason that, of the 3N normal modes of the lattice, exactly N are longitudinal and 2N transverse. Accordingly, we should have, instead of (13), ωD,L Z 0 V ω2dω 2π2c3 L = N and ωD,T Z 0 V ω2dω π2c3 T = 2N. (16) We note that the two cutoffs correspond to a common wavelength λmin{= (4πV/3N)1/3}, which is comparable to the mean interatomic distance in the solid. This is quite reasonable because, for wavelengths shorter than λmin, it would be rather meaningless to speak of a wave of atomic displacements. In the Debye approximation, formula (8) gives CV (T) = 3NkD(x0), (17) where D(x0) is the so-called Debye function: D(x0) = 3 x3 0 x0 Z 0 x4exdx (ex −1)2 , (18) with x0 = ℏωD kT = 2D T , (19) 210 Chapter 7. Ideal Bose Systems 2D being the so-called Debye temperature of the solid. Integrating by parts, the expression for the Debye function becomes D(x0) = − 3x0 ex0 −1 + 12 x3 0 x0 Z 0 x3dx ex −1. (20) For T ≫2D, which means x0 ≪1, the function D(x0) may be expressed as a power series in x0: D(x0) = 1 −x2 0 20 + ··· . (21) Thus, as T →∞,CV →3Nk; moreover, according to this theory, the classical result should be applicable to within 1 2 percent so long as T > 32D. For T ≪2D, which means x0 ≫1, the function D(x0) may be written as D(x0) = 12 x3 0 ∞ Z 0 x3dx ex −1 + O(e−x0), ≈4π4 5x3 0 = 4π4 5  T 2D 3 . (22) Thus, at low temperatures the specific heat of the solid obeys the Debye T3-law: CV = 12π4 5 Nk  T 2D 3 = 464.4  T 2D 3 cal mole−1K−1. (23) It is clear from equation (23) that a measurement of the low-temperature specific heat of a solid should enable us not only to check the validity of the T3-law but also to obtain an empirical value of the Debye temperature 2D.19 The value of 2D can also be obtained by computing the cutoff frequency ωD from a knowledge of the parameters N/V,cL and cT; see equations (14) and (19). The closeness of these estimates is further evidence in favor of Debye’s theory. Once 2D is known, the whole temperature range can be covered theo-retically by making use of the tabulated values of the function D(x0).20 A typical case was shown earlier in Figure 7.14. We saw that not only was the T3-law obeyed at low temper-atures, but also the agreement between theory and experiment was good throughout the range of observations. 19It can be shown that, according to this theory, deviations from the T3-law should not exceed 2 percent so long as T < 2D/10. However, in the case of metals, one cannot expect to reach a true T3-region because, well before that, the specific heat of the electron gas might become a dominant contribution (see Section 8.3); unless the two contributions are separated out, one is likely to obtain a somewhat suppressed value of 2D from these observations. 20See, for example, Fowler and Guggenheim (1960, p. 144). 7.4 The field of sound waves 211 As another illustration of agreement in the low-temperature regime, we include here another plot, Figure 7.16, which is based on data obtained with the KCl crystal at temper-atures below 5K; see Keesom and Pearlman (1953). Here, the observed values of CV /T are plotted against T2. It is evident that the data fall quite well on a straight line from whose slope the value of 2D can be determined. One thus obtains, for KCl, 2D = 233 ± 3K, which is in reasonable agreement with the values of 230 to 246 K coming from various estimates of the relevant elastic constants. In Table 7.1 we list the values of 2D for several crystals, as derived from the specific heat measurements and from the values of the elastic constants. In general, if the specific heat measurements of a given system conform to a T3-law, one may infer that the thermal excitations in the system are accounted for solely by phonons. We expect something similar to happen in liquids as well, with two important differences. First, since liquids cannot withstand shear stress they cannot sustain transverse modes of vibration; a liquid composed of N atoms will, therefore, have only N longitudinal modes of vibration. Second, the normal modes of a liquid cannot be expected to be strictly har-monic; consequently, in addition to phonons, we might have other types of excitation such as vortex flow or turbulence (or even a modified kind of excitation, such as rotons in liquid He4). 2.0 4.0 0 5 10 15 20 6.0 Cv/ T(in milli joules mole1 K2) T 2(in K2) FIGURE 7.16 A plot of (CV /T) versus T2 for KCl, showing the validity of the Debye T3-law. The experimental points are from Keesom and Pearlman (1953). Table 7.1 The Values of the Debye Temperature 2D for Different Crystals Crystal Pb Ag Zn Cu Al C NaCl KCl MgO 2D from the specific 88 215 308 345 398 ∼1850 308 233 ∼850 heat measurements 2D from the elastic 73 214 305 332 402 – 320 240 ∼950 constants 212 Chapter 7. Ideal Bose Systems Now, helium is the only substance that remains liquid at temperatures low enough to exhibit the T3-behavior. In the case of the lighter isotope, He3, the results are strongly influ-enced by the Fermi–Dirac statistics; as a result, a specific heat proportional to the first power of T dominates the scene (see Section 8.1). In the case of the heavier isotope, He4, the low-temperature situation is completely governed by phonons; accordingly, we expect its specific heat to be given by, see equations (16) and (23), CV = 4π4 5 Nk  kT ℏωD 3 , (24) where ωD = 6π2N V !1/3 c, (25) c being the velocity of sound in the liquid. The specific heat per unit mass of the liquid is then given by cV = 2π2k4 15ρℏ3c3 T3, (26) where ρ is the mass density. Substituting ρ = 0.1455g/cm3 and c = 238m/s, the foregoing result becomes cV = 0.0209T3 jouleg−1K−1. (27) The experimental measurements of Wiebes et al. (1957), for 0 < T < 0.6K, conformed to the expression cV = (0.0204 ± 0.0004)T3 jouleg−1K−1. (28) The agreement between the theoretical result and the experimental observations is clearly good. 7.5 Inertial density of the sound field For further understanding of the low-temperature behavior of liquid He4, we determine the “inertial mass” associated with a gas of sound quanta in thermal equilibrium. For this, we consider “a phonon gas in mass motion,” for then by determining the relation-ship between the momentum P of the gas and the velocity v of its mass motion we can readily evaluate the property in question. Now, since the total number of phonons in the system is indefinite, the problem is free from the constraint of a fixed N; consequently, the undetermined multiplier α may be taken to be identically zero. However, we now have a new constraint on the system, namely that of a fixed total momentum P, additional to 7.5 Inertial density of the sound field 213 the constraint of the fixed total energy E. Under these constraints, the mean occupation number of the phonon level ε(p) would be ⟨n(p)⟩= 1 exp(βε + γ · p) −1. (1) As usual, the parameter β is equal to 1/kT. To determine γ , it seems natural to evaluate the drift velocity of the gas. Choosing the z-axis in the direction of the mass motion, the magnitude v of the drift velocity will be given by “the mean value of the component uz of the individual phonon velocities”: v = ⟨ucosθ⟩. (2) Now, for phonons ε = pc and u ≡dε dp = c, (3) where c is the velocity of sound in the medium. Moreover, by reasons of symmetry, we expect the undetermined vector γ to be either parallel or antiparallel to the direction of mass motion; hence, we may write γ · p = γzpz = γzpcosθ. (4) In view of equations (1), (3), and (4), equation (2) becomes v = R ∞ 0 R π 0 [exp{βpc(1 + (γz/βc)cosθ)} −1]−1(ccosθ)(p2dp2π sinθdθ) R ∞ 0 R π 0 [exp{βpc(1 + (γz/βc)cosθ)} −1]−1(p2dp2π sinθdθ) . (5) Making the substitutions cosθ = η, p(1 + (γz/βc)η) = p′ and cancelling away the integrations over p′, we obtain v = c R 1 −1(1 + (γz/βc)η)−3ηdη R 1 −1(1 + (γz/βc)η)−3dη = −γz/β. It follows that γ = −βv. (6) Accordingly, the expression for the mean occupation number becomes ⟨n(p)⟩= 1 exp{β(ε −v · p)} −1. (7) 214 Chapter 7. Ideal Bose Systems A comparison of (7) with the corresponding result in the rest frame of the gas, namely ⟨n0(p0)⟩= 1 exp(βε0) −1, (8) shows that the change caused by the imposition of mass motion on the system is noth-ing but a straightforward manifestation of the Galilean transformation between the two frames of reference. Alternatively, equation (7) may be written as ⟨n(p)⟩= 1 exp(βp′c) −1 = 1 exp{βpc(1 −(v/c)cosθ)} −1. (9) As such, formula (9) lays down a serious restriction on the drift velocity v, that is, it must not exceed c, the velocity of the phonons, for otherwise some of the occupation num-bers would become negative! Actually, as our subsequent analysis will show, the formalism developed in this section breaks down as v approaches c. The velocity c may, therefore, be regarded as the critical velocity for the flow of the phonon gas: (vc)ph = c. (10) The relevance of this result to the problem of superfluidity in liquid helium II will be seen in the following section. Next we now calculate the total momentum P of the phonon gas: P = X p ⟨n(p)⟩p. (11) Indeed, the vector P will be parallel to the vector v, the latter being already in the direction of the z-axis. We have, therefore, to calculate only the z-component of the momentum: P = Pz = X p ⟨n(p)⟩pz = ∞ Z 0 π Z 0 pcosθ exp{βpc(1 −(v/c)cosθ)} −1 Vp2dp2π sinθdθ h3 ! = 2πV h3 ∞ Z 0 p′3dp′ exp(βp′c) −1 π Z 0 {1 −(v/c)cosθ}−4 cosθ sinθdθ = V 16π5 45h3c3β4 · v/c2 (1 −v2/c2)3 . (12) 7.6 Elementary excitations in liquid helium II 215 The total energy E of the gas is given by E = X p ⟨n(p)⟩pc = 2πVc h3 ∞ Z 0 p′3dp′ exp(βp′c) −1 π Z 0 {1 −(v/c)cosθ}−4 sinθdθ = V 4π5 15h3c3β4 1 + 1 3v2/c2 (1 −v2/c2)3 . (13) It is now natural to regard the ratio P/v as the “inertial mass” of the phonon gas. The corresponding mass density ρ is, therefore, given by ρ = P vV = 16π5k4T4 45h3c5 1 (1 −v2/c2)3 . (14) For (v/c) ≪1, which is generally true, the mass density of the phonon gas is given by (ρ0)ph = 16π5k4 45h3c5 T4 = 4 3c2 (E0/V). (15) Substituting the value of c for liquid He4 at low temperatures, the phonon mass density, as a fraction of the actual density of the liquid, is given by (ρ0)ph/ρHe = 1.22 × 10−4T4; (16) thus, for example, at T = 0.3K the value of this fraction turns out to be about 9.9 × 10−7. Now, at a temperature like 0.3K, phonons are the only excitations in liquid He4 that need to be considered; the calculated result should, therefore, correspond to the “ratio of the den-sity ρn of the normal fluid in the liquid to the total density ρ of the liquid.” It is practically impossible to make a direct determination of a fraction as small as that; however, indirect evaluations that make use of other experimentally viable properties of the liquid provide a striking confirmation of the foregoing result; see Figure 7.17. 7.6 Elementary excitations in liquid helium II Landau (1941, 1947) developed a simple theoretical scheme that explains reasonably well the behavior of liquid helium II at low temperatures not too close to the λ-point. Accord-ing to this scheme, the liquid is treated as a weakly excited quantum-mechanical system, in which deviations from the ground state (T = 0K) are described in terms of “a gas of ele-mentary excitations” hovering over a quiescent background. The gas of excitations corre-sponds to the “normal fluid,” while the quiescent background represents the “superfluid.” 216 Chapter 7. Ideal Bose Systems 0.1 0.2 0.3 0.40.50.6 0.8 1.0 1.5 2.0 108 107 106 (n/) phon / 1.24104T 4 105 104 103 102 101 1 T (in K) (T /T)5.6 T FIGURE 7.17 The normal fraction (ρn/ρ), as obtained from experimental data on (i) the velocity of second sound and (ii) the entropy of liquid He II (after de Klerk, Hudson, and Pellam, 1953). At T = 0K, there are no excitations at all (ρn = 0) and the whole of the fluid constitutes the superfluid background (ρs = ρHe). At higher temperatures, we may write ρs(T) = ρHe(T) −ρn(T), (1) so that at T = Tλ, ρn = ρHe and ρs = 0. At T > Tλ, the liquid behaves in all respects as a normal fluid, commonly known as liquid helium I. Guided by purely empirical considerations, Landau also proposed an energy– momentum relationship ε(p) for the elementary excitations in liquid helium II. At low momenta, the relationship between ε and p was linear (which is characteristic of phonons), while at higher momenta it exhibited a nonmonotonic character. The excita-tions were assumed to be bosons and, at low temperatures (when their number is not very large), mutually noninteracting; the macroscopic properties of the liquid could then be calculated by following a straightforward statistical-mechanical approach. It was found that Landau’s theory could explain quite successfully the observed properties of liquid helium II over a temperature range of about 0 to 2 K; however, it still remained to be ver-ified that the actual excitations in the liquid did, in fact, conform to the proposed energy spectrum. Following a suggestion by Cohen and Feynman (1957), a number of experimental work-ers set out to investigate the spectrum of excitations in liquid helium II by scattering long-wavelength neutrons (λ ≳4 ˚ A) from the liquid. At temperatures below 2K, the most important scattering process is the one in which a neutron creates a single excitation in the liquid. By measuring the modified wavelength λf of the neutrons scattered at an angle φ, the energy ε and the momentum p of the excitation created in the scattering process could 7.6 Elementary excitations in liquid helium II 217 be determined on the basis of the relevant conservation laws: ε = h2(λ−2 i −λ−2 f )/2m, (2) p2 = h2(λ−2 i + λ−2 f −2λ−1 i λ−1 f cosφ), (3) where λi is the initial wavelength of the neutrons and m the neutron mass. By varying φ, or λi, one could map the entire spectrum of the excitations. The first exhaustive investigation along these lines was carried out by Yarnell et al. (1959); their results, shown in Figure 7.18, possess a striking resemblance to the empiri-cal spectrum proposed by Landau. The more important features of the spectrum, which was obtained at a temperature of 1.1K, are the following: (i) If we fit a linear, phonon-like spectrum (ε = pc) to points in the vicinity of p/ℏ= 0.55˚ A−1, we obtain for c a value of (239 ± 5) m/s, which is in excellent agreement with the measured value of the velocity of sound in the liquid, namely about 238m/s. (ii) The spectrum passes through a maximum value of ε/k = (13.92 ± 0.10)K at p/ℏ= (1.11 ± 0.02) ˚ A−1. (iii) This is followed by a minimum at p/ℏ= (1.92 ± 0.01)˚ A−1, whose neighborhood may be represented by Landau’s roton spectrum: ε(p) = 1 + (p −p0)2 2µ , (4) T 5 1.1 K 5 0 0.5 1.0 1.5 2.0 2.5 p/ℏ, in A21 10 15 ´/k, in K FIGURE 7.18 The energy spectrum of the elementary excitations in liquid He II at 1.1K [after Yarnell et al. (1959)]; the dashed line emanating from the origin has a slope corresponding to the velocity of sound in the liquid, namely (239 ± 5) m/s. 218 Chapter 7. Ideal Bose Systems with 1/k = (8.65 ± 0.04)K, p0/ℏ= (1.92 ± 0.01)˚ A−1, (5)21 and µ = (0.16 ± 0.01)mHe. (iv) Above p/ℏ≃2.18 ˚ A−1, the spectrum rises linearly, again with a slope equal to c. Data were also obtained at temperatures 1.6K and 1.8K. The spectrum was found to be of the same general shape as at 1.1K; only the value of 1 was slightly lower. In a later investigation, Henshaw and Woods (1961) extended the range of observation at both ends of the spectrum; their results are shown in Figure 7.19. On the lower side, they carried out measurements down to p/ℏ= 0.26 ˚ A−1 and found that the experimental 8 0 0.6 1.2 1.8 2.4 3.0 16 24 32 Temperature Free particle Neutron wavelength 40 1.12 K Energy change, in K Momentum change, in A21 4.04 A FIGURE 7.19 The energy spectrum of the elementary excitations in liquid He II at 1.12K (after Henshaw and Woods, 1961); the dashed straight lines have a common slope corresponding to the velocity of sound in the liquid, namely 237m/s. The parabolic curve rising from the origin represents the energy spectrum, ε(p) = p2/2m, of free helium atoms. 21The term “roton” for these excitations was coined by Landau who had originally thought that these excitations might, in some way, represent local disturbances of a rotational character in the liquid. However, subsequent theoretical work, especially that of Feynman (1953, 1954) and of Brueckner and Sawada (1957), did not support this contention. Nevertheless, the term “roton” has remained. 7.6 Elementary excitations in liquid helium II 219 points indeed lie on a straight line (of slope 237m/s). On the upper side, they pushed their measurements up to p/ℏ= 2.68 ˚ A−1 and found that, after passing through a minimum at 1.91 ˚ A−1, the curve rises with an increasing slope up to about 2.4 ˚ A−1 at which point the second derivative ∂2ε/∂p2 changes sign; the subsequent trend of the curve suggests the possible existence of a second maximum in the spectrum!22 To evaluate the thermodynamics of liquid helium II, we first of all note that at suf-ficiently low temperatures we have only low-lying excitations, namely the phonons. The thermodynamic behavior of the liquid is then governed by formulae derived in Sections 7.4 and 7.5. At temperatures higher than about 0.5K, the second group of exci-tations, namely the rotons (with momenta in the vicinity of p0), also shows up. Between 0.5K and about 1K, the behavior of the liquid is governed by phonons and rotons together. Above 1K, however, the phonon contributions to the various thermodynamic properties of the liquid become rather unimportant; then, rotons are the only excitations that need to be considered. We shall now study the temperature dependence of the roton contributions to the various thermodynamic properties of the liquid. In view of the continuity of the energy spectrum, it is natural to expect that, like phonons, rotons also obey Bose–Einstein statis-tics. Moreover, their total number N in the system is quite indefinite; consequently, their chemical potential µ is identically zero. We then have for the mean occupation numbers of the rotons ⟨n(p)⟩= 1 exp{βε(p)} −1, (6) where ε(p) is given by equations (4) and (5). Now, at all temperatures of interest (namely T ≤2K), the minimum value of the term exp{βε(p)}, namely exp(1/kT), is considerably larger than unity. We may, therefore, write ⟨n(p)⟩≃exp{−βε(p)}. (7) The q-potential of the system of rotons is, therefore, given by q(V,T) ≡PV kT = − X p ln[1 −exp{−βε(p)}] ≃ X p exp{−βε(p)} ≃N, (8) where N is the “equilibrium” number of rotons in the system. The summation over p may be replaced by integration, with the result PV kT = N = V h3 ∞ Z 0 e −  1+ (p−p0)2 2µ . kT (4πp2dp). (9) 22This seems to confirm a remarkable prediction by Pitaevskii (1959) that an end point in the spectrum might occur at a “critical” value pc of the excitation momentum where εc is equal to 21 and (∂ε/∂p)c is zero. 220 Chapter 7. Ideal Bose Systems Substituting p = p0 + (2µkT)1/2x, we get PV kT = N = 4πp2 0V h3 e−1/kT(2µkT)1/2 Z e−x2 ( 1 + (2µkT)1/2 p0 x )2 dx. (10) The “relevant” range of the variable x that makes a significant contribution toward this integral is fairly symmetric about the value x = 0; consequently, the net effect of the linear term in the integrand is vanishingly small. The quadratic term too is unimportant because its coefficient (2µkT)/p2 0 ≪1. Thus, all we have to consider is the integral of exp(−x2). Now, one can readily verify that the limits of this integral are such that, without seriously affect-ing the value of the integral, they may be taken as −∞and +∞; the value of the integral is then simply π1/2. We thus obtain PV kT = N = 4πp2 0V h3 (2πµkT)1/2e−1/kT. (11)23 The free energy of the roton gas is given by (since µ = 0) A = −PV = −NkT ∝T3/2e−1/kT, (12) which gives S = −  ∂A ∂T  V = −A  3 2T + 1 kT2  = Nk 3 2 + 1 kT  , (13) U = A + TS = N  1 + 1 2kT  (14)24 and CV = ∂U ∂T  V = Nk ( 3 4 + 1 kT +  1 kT 2) . (15) Clearly, as T →0, all these results tend to zero (essentially exponentially). We now determine the inertial mass density of the roton gas. Proceeding as in Section 7.5, we obtain for a gas of excitations with energy spectrum ε(p) ρ0 = M0 V = lim v→0 1 v Z n(ε −v · p)pd3p h3 , (16) 23Looking back at integral (9), what we have done here amounts to replacing p2 in the integrand by its mean value p2 0 and then carrying out integration over the “complete” range of the variable (p −p0). 24This result is highly suggestive of the fact that for rotons there is only one true degree of freedom, namely the magnitude of the roton momentum, that is thermally effective! 7.6 Elementary excitations in liquid helium II 221 where n(ε −v · p) is the mean occupation number of the state ε(p), as observed in a frame of reference K with respect to which the gas is in mass motion with a drift velocity v.25 For small v, the function n(ε −v · p) may be expanded as a Taylor series in v and only the terms n(ε) −(v · p)∂n(ε)/∂ε retained. The integral over the first part denotes the momen-tum density of the system, as observed in the rest frame K0, and is identically zero. We are thus left with ρ0 = −1 h3 Z p2 cos2 θ ∂n(ε) ∂ε (p2dp2π sinθdθ) = −4π 3h3 ∞ Z 0 ∂n(ε) ∂ε p4dp, (17) which holds for any energy spectrum and for any statistics. For phonons, we obtain (ρ0)ph = −4π 3h3c ∞ Z 0 dn(p) dp p4dp = −4π 3h3c  n(p) · p4 ∞ 0 − ∞ Z 0 n(p) · 4p3dp   = 4 3c2 ∞ Z 0 n(p) · pc 4πp2dp h3 ! = 4 3c2 (E0)ph/V, (18) which is identical to our earlier result (7.5.15). For rotons, n(ε) ≃exp(−βε); hence, ∂n(ε)/∂ε ≃−βn(ε). Accordingly, by (17), (ρ0)rot = 4πβ 3h3 Z n(ε)p4dp = β 3 ⟨p2⟩N V ≃p2 0 3kT N V (19) = 4πp4 0 3h3 2πµ kT 1/2 e−1/kT; (20) At very low temperatures (T < 0.3K), the roton contribution toward the inertia of the fluid is negligible in comparison with the phonon contribution. At relatively higher tem-peratures (T ∼0.6K), the two contributions become comparable. At temperatures above 1K, the roton contribution is far more dominant than the phonon contribution; at such temperatures, the roton density alone accounts for the density ρn of the normal fluid. 25The drift velocity v must satisfy the condition (v · p) ≤ε, for otherwise some of the occupation numbers will become negative! This leads to the existence of a critical velocity vc for these excitations, such that for v exceeding vc the formalism developed here would break down. It is not difficult to see that this (critical) velocity is given by the relation vc = (ε/p)min, as in equation (24). 222 Chapter 7. Ideal Bose Systems It would be instructive to determine the critical temperature Tc at which the theoreti-cal value of the density ρn became equal to the actual density ρHe of the liquid; this would mean the disappearance of the superfluid component of the liquid (and hence a transition from liquid He II to liquid He I). In this manner, we find that Tc ≃2.5K, as opposed to the experimental value of Tλ, which is ≃2.19 K. The comparison is not too bad, considering the fact that in the present calculation we have assumed the roton gas to be a noninteract-ing system right up to the transition point; in fact, due to the presence of an exceedingly large number of excitations at higher temperatures, this assumption would not remain plausible. Equation (19) suggests that a roton excitation possesses an effective mass p2 0/3kT. Numerically, this is about 10 to 15 times the mass of a helium atom (and, hence, orders of magnitude larger than the parameter µ of the roton spectrum). However, the more impor-tant aspect of the roton effective mass is that it is inversely proportional to the temperature of the roton gas! Historically, this aspect was first discovered empirically by Landau (1947) on the basis of the experimental data on the velocity of second sound in liquid He II and its specific heat. Now, since the effective mass of an excitation is generally determined by the quantity ⟨p2⟩/3kT, Landau concluded that the quantity ⟨p2⟩of the relevant excitations in this liquid must be temperature-independent. Thus, as the temperature of the liquid rises, the mean value of p2 of the excitations must stay constant; this value may be denoted by p2 0. The mean value of ε, on the other hand, must rise with temperature. The only way to reconcile the two was to invoke a nonmonotonic spectrum with a minimum at p = p0. Finally, we would like to touch on the question of the critical velocity of superflow. For this, we consider a mass M of excitation-free superfluid in mass motion; its kinetic energy E and momentum P are given by 1 2Mv2 and Mv, respectively. Any changes in these quantities are related as follows: δE = (v · δP). (21) Supposing that these changes came about as a result of the creation of an excitation ε(p) in the fluid, we must have, by the principles of conservation, δE = −ε and δP = −p. (22) Equations (21) and (22) lead to the result ε = (v · p) ≤vp. (23) Thus, it is impossible to create an excitation ε(p) in the fluid unless the drift velocity v of the fluid is greater than, or at least equal to, the quantity (ε/p). Accordingly, if v is less than even the lowest value of ε/p, no excitation at all can be created in the fluid, which will there-fore maintain its superfluid character. We thus obtain a condition for the maintenance of Problems 223 superfluidity, namely v < vc = (ε/p)min, (24) which is known as the Landau criterion for superflow. The velocity vc is called the criti-cal velocity of superflow; it marks an “upper limit” to the flow velocities at which the fluid exhibits superfluid behavior. The observed magnitude of the critical velocity varies signifi-cantly with the geometry of the channel employed; as a rule, the narrower the channel the larger the critical velocity. The observed values of vc range from about 0.1cm/s to about 70cm/s. The theoretical estimates of vc are clearly of interest. On one hand, we find that if the excitations obey the ideal-gas relationship, namely ε = p2/2m, then the critical velocity turns out to be exactly zero. Any velocity v is then greater than the critical velocity; accord-ingly, no superflow is possible at all. This is a very significant result, for it brings out very clearly the fact that interatomic interactions in the liquid, which give rise to an excita-tion spectrum different from the one characteristic of the ideal gas, play a fundamental role in bringing about the phenomenon of superfluidity. Thus, while an ideal Bose gas does undergo the phenomenon of Bose–Einstein condensation, it cannot support the phe-nomenon of superfluidity as such! On the other hand, we find that (i) for phonons, vc = c ≃2.4 × 104 cm/s and (ii) for rotons, vc = {(p2 0 + 2µ1)1/2 −p0}/µ ≃1/p0 ≃6.3 × 103 cm/s, which are too high in comparison with the observed values of vc. In fact, there is another type of collective excitations that can appear in liquid helium II, namely quantized vortex rings, with energy–momentum relationship of the form: ε ∝p1/2. The critical velocity for the creation of these rings turns out to be numerically consistent with the experimental findings; not only that, the dependence of vc on the geometry of the channel can also be understood in terms of the size of the rings created. For a review of this topic, especially in regard to Feynman’s contributions, see Mehra and Pathria (1994); see also Sections 11.4 through 11.6 of this text. Quantized dissipationless bosonic flow has also been observed in the solid phase of helium-4. This “supersolid” behavior was observed by Kim and Chan (2004a, 2004b) using a torsional oscillator containing solid helium infused silica with atomic-sized pores. At P = 60atm, the torsional frequency increases abruptly for temperatures below 175mK. These authors interpret this result as helium atoms in the solid phase in the pores being free to flow without dissipation. Problems 7.1. By considering the order of magnitude of the occupation numbers ⟨nε⟩, show that it makes no difference to the final results of Section 7.1 if we combine a finite number of (ε ̸= 0)-terms of the sum (7.1.2) with the (ε = 0)-part of equation (7.1.6) or include them in the integral over ε. 7.2. Deduce the virial expansion (7.1.13) from equations (7.1.7) and (7.1.8), and verify the quoted values of the virial coefficients. 224 Chapter 7. Ideal Bose Systems 7.3. Combining equations (7.1.24) and (7.1.26), and making use of the first two terms of formula (D.9) in Appendix D, show that, as T approaches Tc from above, the parameter α(= −lnz) of the ideal Bose gas assumes the form α ≈1 π 3ζ(3/2) 4 2 T −Tc Tc 2 . 7.4. Show that for an ideal Bose gas 1 z  ∂z ∂T  P = −5 2T g5/2(z) g3/2(z); compare this result with equation (7.1.36). Hence show that γ ≡CP CV = (∂z/∂T)P (∂z/∂T)v = 5 3 g5/2(z)g1/2(z) {g3/2(z)}2 , as in equation (7.1.48b). Check that, as T approaches Tc from above, both γ and CP diverge as (T −Tc)−1. 7.5. (a) Show that the isothermal compressibility κT and the adiabatic compressibility κS of an ideal Bose gas are given by κT = 1 nkT g1/2(z) g3/2(z), κS = 3 5nkT g3/2(z) g5/2(z), where n(= N/V) is the particle density in the gas. Note that, as z →0, κT and κS approach their respective classical values, namely 1/P and 1/γ P. How do they behave as z →1? (b) Making use of the thermodynamic relations CP −CV = T  ∂P ∂T  V ∂V ∂T  P = TVκT  ∂P ∂T 2 V and CP/CV = κT/κS, derive equations (7.1.48a) and (7.1.48b). 7.6. Show that for an ideal Bose gas the temperature derivative of the specific heat CV is given by 1 Nk ∂CV ∂T  V =          1 T " 45 8 g5/2(z) g3/2(z) −9 4 g3/2(z) g1/2(z) −27 8 {g3/2(z)}2g−1/2(z) {g1/2(z)}3 # for T > Tc, 45 8 v Tλ3 ζ 5 2  for T < Tc. Using these results and the main term of formula (D.9), verify equation (7.1.38). 7.7. Evaluate the quantities (∂2P/∂T2)v, (∂2µ/∂T2)v, and (∂2µ/∂T2)P for an ideal Bose gas and check that your results satisfy the thermodynamic relationships CV = VT ∂2P ∂T2 ! v −NT ∂2µ ∂T2 ! v , and CP = −NT ∂2µ ∂T2 ! P . Examine the behavior of these quantities as T →Tc from above and from below. Problems 225 7.8. The velocity of sound in a fluid is given by the formula w = √(∂P/∂ρ)s, where ρ is the mass density of the fluid. Show that for an ideal Bose gas w2 = 5kT 3m g5/2(z) g3/2(z) = 5 9⟨u2⟩, where ⟨u2⟩is the mean square speed of the particles in the gas. 7.9. Show that for an ideal Bose gas ⟨u⟩ 1 u = 4 π g1(z)g2(z) {g3/2(z)}2 , u being the speed of a particle. Examine and interpret the limiting cases z →0 and z →1; compare with Problem 6.6. 7.10. Consider an ideal Bose gas in a uniform gravitational field of acceleration g. Show that the phenomenon of Bose–Einstein condensation in this gas sets in at a temperature Tc given by Tc ≃T0 c  1 + 8 9 1 ζ  3 2  πmgL kT0 c 1/2  , where L is the height of the container and mgL ≪kT0 c . Also show that the condensation here is accompanied by a discontinuity in the specific heat CV of the gas: (1CV )T=Tc ≃−9 8π ζ 3 2  Nk πmgL kT0 c 1/2 ; see Eisenschitz (1958). 7.11. Consider an ideal Bose gas consisting of molecules with internal degrees of freedom. Assuming that, besides the ground state ε0 = 0, only the first excited state ε1 of the internal spectrum needs to be taken into account, determine the condensation temperature of the gas as a function of ε1. Show that, for (ε1/kT0 c ) ≫1, Tc T0 c ≃1 − 2 3 ζ  3 2 e−ε1/kT0 c while, for (ε1/kT0 c ) ≪1, Tc T0 c ≃ 1 2 2/3  1 + 24/3 3ζ  3 2   πε1 kT0 c 1/2  . [Hint: To obtain the last result, use the first two terms of formula (D.9) in Appendix D.] 7.12. Consider an ideal Bose gas in the grand canonical ensemble and study fluctuations in the total number of particles N and the total energy E. Discuss, in particular, the situation when the gas becomes highly degenerate. 7.13. Consider an ideal Bose gas confined to a region of area A in two dimensions. Express the number of particles in the excited states, Ne, and the number of particles in the ground state, N0, in terms of z, T, and A, and show that the system does not exhibit Bose–Einstein condensation unless T →0K. Refine your argument to show that, if the area A and the total number of particles N are held fixed and we require both Ne and N0 to be of order N, then we do achieve condensation when T ∼ h2 mkl2 1 lnN , 226 Chapter 7. Ideal Bose Systems where l[∼√(A/N)] is the mean interparticle distance in the system. Of course, if both A and N →∞, keeping l fixed, then the desired T does go to zero. 7.14. Consider an n-dimensional Bose gas whose single-particle energy spectrum is given by ε∝ps, where s is some positive number. Discuss the onset of Bose–Einstein condensation in this system, especially its dependence on the numbers n and s. Study the thermodynamic behavior of this system and show that, P = s n U V , CV (T →∞) = n s Nk, and CP(T →∞) = n s + 1  Nk. 7.15. At time t = 0, the ground state wavefunction of a one-dimensional quantum harmonic oscillator with potential V(x) = 1 2mω2 0x2 is given by ψ(x,0) = 1 π1/4√a exp −x2 2a2 ! , where a = q ℏ mω0 . At t = 0, the harmonic potential is abruptly removed. Use the momentum representation of the wavefunction at t = 0 and the time-dependent Schrodinger equation to determine the spatial wavefunction and density at time t > 0; compare to equation (7.2.11). 7.16. At time t = 0, a collection of classical particles is in equilibrium at temperature T in a three-dimensional harmonic oscillator potential V(r) = 1 2mω2 0 |r|2. At t = 0, the harmonic potential is abruptly removed. Use the momentum distribution at t = 0 to determine the spatial density at time t > 0. Show that this is equivalent to the high temperature limit of equation (7.2.15). 7.17. As shown in Section 7.1, nλ3 is a measure of the quantum nature of the system. Use equations (7.2.11) and (7.2.15) to determine nλ3 at the center of the harmonic trap at T = Tc/2 for the condensed and noncondensed fractions. 7.18. Show that the integral of the semiclassical spatial density in equation (7.2.15) gives the correct counting of the atoms that are not condensed into the ground state. 7.19. Construct a theory for N bosons in an isotropic two-dimensional trap. This corresponds to a trap in which the energy level spacing due to excitations in the z direction is much larger than the spacing in the other directions. Determine the density of states a(ε) of this system. Can a Bose–Einstein condensate form in this trap? If so, find the critical temperature as a function of the trapping frequencies and N. How much larger must the frequency in the third direction be for the system to display two-dimensional behavior? 7.20. The (canonical) partition function of the blackbody radiation may be written as Q(V,T) = Y ω Q1(ω,T), so that lnQ(V,T) = X ω lnQ1(ω,T) ≈ ∞ Z 0 lnQ1(ω,T)g(ω)dω; here, Q1(ω,T) is the single-oscillator partition function given by equation (3.8.14) and g(ω) is the density of states given by equation (7.3.2). Using this information, evaluate the Helmholtz free energy of the system and derive other thermodynamic properties such as the pressure P and the (thermal) energy density U/V. Compare your results with the ones derived in Section 7.3 from the q-potential of the system. 7.21. Show that the mean energy per photon in a blackbody radiation cavity is very nearly 2.7kT. 7.22. Considering the volume dependence of the frequencies ω of the vibrational modes of the radiation field, establish relation (7.3.17) between the pressure P and the energy density U/V. 7.23. The sun may be regarded as a black body at a temperature of 5800K. Its diameter is about 1.4 × 109 m while its distance from the earth is about 1.5 × 1011m. Problems 227 (a) Calculate the total radiant intensity (in W/m2) of sunlight at the surface of the earth. (b) What pressure would it exert on a perfectly absorbing surface placed normal to the rays of the sun? (c) If a flat surface on a satellite, which faces the sun, were an ideal absorber and emitter, what equilibrium temperature would it ultimately attain? 7.24. Calculate the photon number density, entropy density, and energy density of the 2.725K cosmic microwave background. 7.25. Figure 7.20 is a plot of CV (T) against T for a solid, the limiting value CV (∞) being the classical result 3Nk. Show that the shaded area in the figure, namely ∞ Z 0 {CV (∞) −CV (T)}dT, is exactly equal to the zero-point energy of the solid. Interpret the result physically. Cv() 0 0 Cv(T ) T FIGURE 7.20 7.26. Show that the zero-point energy of a Debye solid composed of N atoms is equal to 9 8Nk2D. [Note that this implies, for each vibrational mode of the solid, a mean zero-point energy 3 8k2D, that is, ω = 3 4ωD.] 7.27. Show that, for T ≪2D, the quantity (CP −CV ) of a Debye solid varies as T7 and hence the ratio (CP/CV ) ≃1. 7.28. Determine the temperature T, in terms of the Debye temperature 2D, at which one-half of the oscillators in a Debye solid are expected to be in the excited states. 7.29. Determine the value of the parameter 2D for liquid He4 from the empirical result (7.4.28). 7.30. (a) Compare the “mean thermal wavelength” λT of neutrons at a typical room temperature with the “minimum wavelength” λmin of phonons in a typical crystal. (b) Show that the frequency ωD for a sodium chloride crystal is of the same order of magnitude as the frequency of an electromagnetic wave in the infrared. 7.31. Proceeding under conditions (7.4.16) rather than (7.4.13), show that CV (T) = Nk{D(x0,L) + 2D(x0,T)}, where x0,L = (ℏωD,L/kT) and x0,T = (ℏωD,T/kT). Compare this result with equation (7.4.17), and estimate the nature and the magnitude of the error involved in the latter. 7.32. A mechanical system consisting of n identical masses (each of mass m) connected in a straight line by identical springs (of stiffness K) has natural vibrational frequencies given by ωr = 2 s K m  sin  r n · π 2  ;r = 1,2,...(n −1). 228 Chapter 7. Ideal Bose Systems Correspondingly, a linear molecule composed of n identical atoms may be regarded as having a vibrational spectrum given by νr = νc sin  r n · π 2  ;r = 1,2,...(n −1), where vc is a characteristic vibrational frequency of the molecule. Show that this model leads to a vibrational specific heat per molecule that varies as T1 at low temperatures and tends to the limiting value (n −1)k at high temperatures. 7.33. Assuming the dispersion relation ω = Aks, where ω is the angular frequency and k the wave number of a vibrational mode existing in a solid, show that the respective contribution toward the specific heat of the solid at low temperatures is proportional to T3/s. [Note that while s = 1 corresponds to the case of elastic waves in a lattice, s = 2 applies to spin waves propagating in a ferromagnetic system.] 7.34. Assuming the excitations to be phonons (ω = Ak), show that their contribution toward the specific heat of an n-dimensional Debye system is proportional to Tn. [Note that the elements selenium and tellurium form crystals in which atomic chains are arranged in parallel so that in a certain sense they behave as one-dimensional; accordingly, over a certain range of temperatures, the T1-law holds. For a similar reason, graphite obeys a T2-law over a certain range of temperatures.] 7.35. The (minimum) potential energy of a solid, when all its atoms are “at rest” at their equilibrium positions, may be denoted by the symbol 80(V), where V is the volume of the solid. Similarly, the normal frequencies of vibration, ωi (i = 1,2,...,3N −6), may be denoted by the symbols ωi(V). Show that the pressure of this solid is given by P = −∂80 ∂V + γ U′ V , where U′ is the internal energy of the solid arising from the vibrations of the atoms, while γ is the Gr¨ uneisen constant: γ = −∂lnω ∂lnV ≈1 3. Assuming that, for V ≃V0, 80(V) = (V −V0)2 2κ0V0 , where κ0 and V0 are constants and κ0CV T ≪V0, show that the coefficient of thermal expansion (at constant pressure P ≃0) is given by α ≡1 V ∂V ∂T  N,P = γ κ0CV V0 . Also show that CP −CV = γ 2κ0C2 V T V0 . 7.36. Apply the general formula (6.4.3) for the kinetic pressure of a gas, namely P = 1 3n⟨pu⟩, to a gas of rotons and verify that the result so obtained agrees with the Boltzmannian relationship P = nkT. 7.37. Show that the free energy A and the inertial density ρ of a roton gas in mass motion are given by A(v) = A(0)sinhx x Problems 229 and ρ(v) = ρ(0)3(xcoshx −sinhx) x3 , where x = vp0/kT. 7.38. Integrating (7.6.17) by parts, show that the effective mass of an excitation, whose energy– momentum relationship is denoted by ε(p), is given by meff = 1 3p2  d dp  p4 dp dε  . Check the validity of this result by considering the examples of (i) an ideal-gas particle, (ii) a phonon, and (iii) a roton. 8 Ideal Fermi Systems 8.1 Thermodynamic behavior of an ideal Fermi gas According to Sections 6.1 and 6.2, we obtain for an ideal Fermi gas PV kT ≡lnQ = X ε ln(1 + ze−βε) (1) and N ≡ X ε ⟨nε⟩= X ε 1 z−1eβε + 1, (2) where β = 1/kT and z = exp(µ/kT). Unlike the Bose case, the parameter z in the Fermi case can take on unrestricted values: 0 ≤z < ∞. Moreover, in view of the Pauli exclusion principle, the question of a large number of particles occupying a single energy state does not even arise in this case; hence, there is no phenomenon like Bose–Einstein condensa-tion here. Nevertheless, at sufficiently low temperatures, Fermi gas displays its own brand of quantal behavior, a detailed study of which is of great physical interest. If we replace summations over ε by corresponding integrations, equations (1) and (2) in the case of a nonrelativistic gas become P kT = g λ3 f5/2(z) (3) and N V = g λ3 f3/2(z), (4) where g is a weight factor arising from the “internal structure” of the particles (e.g., spin), λ is the mean thermal wavelength of the particles λ = h/(2πmkT)1/2, (5) while fν(z) are Fermi–Dirac functions defined by, see Appendix E, fν(z) = 1 0(ν) ∞ Z 0 xν−1dx z−1ex + 1 = z −z2 2ν + z3 3ν −··· . (6) Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00008-6 © 2011 Elsevier Ltd. All rights reserved. 231 232 Chapter 8. Ideal Fermi Systems Eliminating z between equations (3) and (4), we obtain the equation of state of the Fermi gas. The internal energy U of the Fermi gas is given by U ≡−  ∂ ∂β lnQ  z,V = kT2  ∂ ∂T lnQ  z,V = 3 2kT gV λ3 f5/2(z) = 3 2NkT f5/2(z) f3/2(z); (7) thus, quite generally, this system satisfies the relationship P = 2 3(U/V). (8) The specific heat CV of the gas can be obtained by differentiating (7) with respect to T, keeping N and V constant, and making use of the relationship 1 z  ∂z ∂T  v = −3 2T f3/2(z) f1/2(z), (9) which follows from equation (4) and the recurrence formula (E.6) in Appendix E. The final result is CV Nk = 15 4 f5/2(z) f3/2(z) −9 4 f3/2(z) f1/2(z). (10) For the Helmholtz free energy of the gas, we get A ≡Nµ −PV = NkT  lnz −f5/2(z) f3/2(z)  , (11) and for the entropy S ≡U −A T = Nk 5 2 f5/2(z) f3/2(z) −lnz  . (12) In order to determine the various properties of the Fermi gas in terms of the particle den-sity n(= N/V) and the temperature T, we need to know the functional dependence of the parameter z on n and T; this information is formally contained in the implicit relationship (4). For detailed studies, one is sometimes obliged to make use of numerical evaluation of the functions fν(z); for physical understanding, however, the various limiting forms of these functions serve the purpose well (see Appendix E). Now, if the density of the gas is very low and/or its temperature very high, then the situation might correspond to f3/2(z) = nλ3 g = nh3 g(2πmkT)3/2 ≪1; (13) 8.1 Thermodynamic behavior of an ideal Fermi gas 233 we then speak of the gas as being nondegenerate and, therefore, equivalent to a classical ideal gas discussed in Section 3.5. In view of expansion (6), this implies that z ≪1 and hence fν(z) ≃z. Expressions for the various thermodynamic properties of the gas then become P = NkT/V, U = 3 2NkT, CV = 3 2Nk, (14) A = NkT ( ln nλ3 g ! −1 ) , (15) and S = Nk ( 5 2 −ln nλ3 g !) . (16) If the parameter z is small in comparison with unity but not very small, then we should make a fuller use of series (6) in order to eliminate z between equations (3) and (4). The procedure is just the same as in the corresponding Bose case, that is, we first invert the series appearing in (4) to obtain an expansion for z in powers of (nλ3/g) and then substi-tute this expansion into the series appearing in (3). The equation of state then takes the form of the virial expansion PV NkT = ∞ X l=1 (−1)l−1al λ3 gv !l−1 , (17) where v = 1/n, while the coefficients al are the same as quoted in (7.1.14) but alternate in sign compared to the Bose case. For the specific heat, in particular, we obtain CV = 3 2Nk ∞ X l=1 (−1)l−1 5 −3l 2 al λ3 gv !l−1 = 3 2Nk  1 −0.0884 λ3 gv ! + 0.0066 λ3 gv !2 −0.0004 λ3 gv !3 + ···  . (18) Thus, at finite temperatures, the specific heat of the gas is smaller than its limiting value 3 2Nk. As will be seen in the sequel, the specific heat of the ideal Fermi gas decreases mono-tonically as the temperature of the gas falls; see Figure 8.2 later in the section and compare it with the corresponding Figure 7.4 for the ideal Bose gas. If the density n and the temperature T are such that the parameter (nλ3/g) is of order unity, the foregoing expansions cannot be of much use. In that case, one may have to make recourse to numerical calculation. However, if (nλ3/g) ≫1, the functions involved can be expressed as asymptotic expansions in powers of (lnz)−1; we then speak of the gas as being degenerate. As (nλ3/g) →∞, our functions assume a closed form, with the result that the expressions for the various thermodynamic quantities pertaining to the system become 234 Chapter 8. Ideal Fermi Systems highly simplified; we then speak of the gas as being completely degenerate. For simplicity, we first discuss the main features of the system in a state of complete degeneracy. In the limit T →0, which implies (nλ3/g) →∞, the mean occupation numbers of the single-particle state ε(p) become ⟨nε⟩≡ 1 e(ε−µ)/kT + 1 = ( 1 for ε < µ0 0 for ε > µ0, (19) where µ0 is the chemical potential of the system at T = 0. The function ⟨nε⟩is thus a step function that stays constant at the (highest) value 1 right from ε = 0 to ε = µ0 and then suddenly drops to the (lowest) value 0; see the dotted line in Figure 8.1. Thus, at T = 0, all single-particle states up to ε = µ0 are “completely” filled, with one particle per state (in accordance with the Pauli principle), while all single-particle states with ε > µ0 are empty. The limiting energy µ0 is generally referred to as the Fermi energy of the system and is denoted by the symbol εF; the corresponding value of the single-particle momen-tum is referred to as the Fermi momentum and is denoted by the symbol pF. The defining equation for these parameters is εF Z 0 a(ε)dε = N, (20) where a(ε) denotes the density of states of the system and is given by the general expression a(ε) = gV h3 4πp2 dp dε . (21) We readily obtain N = 4πgV 3h3 p3 F, (22) which gives pF =  3N 4πgV 1/3 h; (23) 1.0 0.5 0 0 4 2 2 4 x (ex1)1 FIGURE 8.1 Fermi distribution at low temperatures, with x = ε/kT and ξ = µ/kT. The rectangle denotes the limiting distribution as T →0; in that case, the Fermi function is unity for ε < µ0 and zero for ε > µ0. 8.1 Thermodynamic behavior of an ideal Fermi gas 235 accordingly, in the nonrelativistic case, εF =  3N 4πgV 2/3 h2 2m = 6π2n g !2/3 ℏ2 2m. (24) The ground-state, or zero-point, energy of the system is then given by E0 = 4πgV h3 PF Z 0 p2 2m ! p2dp = 2πgV 5mh3 p5 F, (25) which gives E0 N = 3p2 F 10m = 3 5εF. (26) The ground-state pressure of the system is in turn given by P0 = 2 3(E0/V) = 2 5nεF. (27) Substituting for εF, the foregoing expression takes the form P0 = 6π2 g !2/3 ℏ2 5mn5/3 ∝n5/3. (28) The zero-point motion seen here is clearly a quantum effect arising because of the Pauli principle, according to which, even at T = 0K, the particles constituting the system cannot settle down into a single energy state (as we had in the Bose case) and are therefore spread over a requisite number of lowest available energy states. As a result, the Fermi system, even at absolute zero, is quite live! For a discussion of properties such as the specific heat and the entropy of the system, we must extend our study to finite temperatures. If we decide to restrict ourselves to low temperatures, then deviations from the ground-state results will not be too large; accord-ingly, an analysis based on the asymptotic expansions of the functions fν(z) would be quite appropriate. However, before we do that it seems useful to carry out a physical assessment of the situation with the help of the expression ⟨nε⟩= 1 e(ε−µ)/kT + 1. (29) The situation corresponding to T = 0 is summarized in equation (19) and is shown as a step function in Figure 8.1. Deviations from this, when T is finite (but still much smaller than the characteristic temperature µ0/k), will be significant only for those values of ε for which 236 Chapter 8. Ideal Fermi Systems the magnitude of the quantity (ε −µ)/kT is of order unity (for otherwise the exponential term in (29) will not be much different from its ground-state value, namely, e±∞); see the solid curve in Figure 8.1. We, therefore, conclude that the thermal excitation of the particles occurs only in a nar-row energy range that is located around the energy value ε = µ0 and has a width O(kT). The fraction of the particles that are thermally excited is, therefore, O(kT/εF) — the bulk of the system remaining uninfluenced by the rise in temperature.1 This is the most characteris-tic feature of a degenerate Fermi system and is essentially responsible for both qualitative and quantitative differences between the physical behavior of this system and that of a corresponding classical system. To conclude the argument, we observe that since the thermal energy per “excited” par-ticle is O(kT), the thermal energy of the whole system will be O(Nk2T2/εF); accordingly, the specific heat of the system will be O(Nk · kT/εF). Thus, the low-temperature specific heat of a Fermi system differs from the classical value 3 2Nk by a factor that not only reduces it considerably in magnitude but also makes it temperature-dependent (varying as T1). It will be seen repeatedly that the first-power dependence of CV on T is a typical feature of Fermi systems at low temperatures. For an analytical study of the Fermi gas at finite, but low, temperatures, we observe that the value of z, which was infinitely large at absolute zero, is now finite, though still large in comparison with unity. The functions fν(z) can, therefore, be expressed as asymp-totic expansions in powers of (lnz)−1; see Sommerfeld’s lemma (E.17) in Appendix E. For the values of ν we are presently interested in, namely 5 2, 3 2, and 1 2, we have to the first approximation f5/2(z) = 8 15π1/2 (lnz)5/2 " 1 + 5π2 8 (lnz)−2 + ··· # , (30) f3/2(z) = 4 3π1/2 (lnz)3/2 " 1 + π2 8 (lnz)−2 + ··· # , (31) and f1/2(z) = 2 π1/2 (lnz)1/2 " 1 −π2 24 (lnz)−2 + ··· # . (32) Substituting (31) into (4), we obtain N V = 4πg 3 2m h2 3/2 (kT lnz)3/2 " 1 + π2 8 (lnz)−2 + ··· # . (33) 1We, therefore, speak of the totality of the energy levels filled at T = 0 as “the Fermi sea” and the small fraction of the particles that are excited near the top, when T > 0, as a “mist above the sea.” Physically speaking, the origin of this behavior again lies in the Pauli exclusion principle, according to which a fermion of energy ε cannot absorb a quantum of thermal excitation εT if the energy level ε + εT is already filled. Since εT = O(kT), only those fermions that occupy energy levels near the top level εF, up to a depth O(kT), can be thermally excited to go over to the unfilled energy levels. 8.1 Thermodynamic behavior of an ideal Fermi gas 237 In the zeroth approximation, this gives kT lnz ≡µ ≃  3N 4πgV 2/3 h2 2m, (34) which is identical to the ground-state result µ0 = εF; see equation (24). In the next approxi-mation, we obtain kT lnz ≡µ ≃εF " 1 −π2 12 kT εF 2# . (35) Substituting (30) and (31) into (7), we obtain U N = 3 5(kT lnz) " 1 + π2 2 (lnz)−2 + ··· # ; (36) with the help of (35), this becomes U N = 3 5εF " 1 + 5π2 12 kT εF 2 + ··· # . (37) The pressure of the gas is then given by P = 2 3 U V = 2 5nεF " 1 + 5π2 12 kT εF 2 + ··· # . (38) As expected, the main terms of equations (37) and (38) are identical to the ground-state results (26) and (27). From the temperature-dependent part of (37), we obtain for the low-temperature specific heat of the gas CV Nk = π2 2 kT εF + ··· . (39) Thus, for T ≪TF, where TF (= εF/k) is the Fermi temperature of the system, the specific heat varies as the first power of temperature; moreover, in magnitude, it is consider-ably smaller than the classical value 3 2Nk. The overall variation of CV with T is shown in Figure 8.2. The Helmholtz free energy of the system follows directly from equations (35) and (38): A N = µ −PV N = 3 5εF " 1 −5π2 12 kT εF 2 + ··· # , (40) 238 Chapter 8. Ideal Fermi Systems (T/TF) 0 0 0.5 1.0 1.5 1 2 3 Cv Nk FIGURE 8.2 The specific heat of an ideal Fermi gas; the dotted line depicts the linear behavior at low temperatures. which gives S Nk = π2 2 kT εF + ··· . (41) Thus, as T →0,S →0 in accordance with the third law of thermodynamics. 8.2 Magnetic behavior of an ideal Fermi gas We now turn our attention to studying the equilibrium state of a gas of noninteracting fermions in the presence of an external magnetic field B. The main problem here is to determine the net magnetic moment M acquired by the gas (as a function of B and T) and then calculate the susceptibility χ(T). The answer naturally depends on the intrinsic mag-netic moment µ∗of the particles and the corresponding multiplicity factor (2J + 1); see, for instance, the treatment given in Section 3.9. According to the Boltzmannian treatment, one obtains a (positive) susceptibility χ(T) which, at high temperatures, obeys the Curie law : χ ∝T−1; at low temperatures, one obtains a state of magnetic saturation. However, if we treat the problem on the basis of Fermi statistics we obtain significantly different results, especially at low temperatures. In particular, since the Fermi gas is pretty live even at absolute zero, no magnetic sat-uration ever results; we rather obtain a limiting susceptibility χ0, which is independent of temperature but is dependent on the density of the gas. Studies along these lines were first made by Pauli, in 1927, when he suggested that the conduction electrons in alkali metals be regarded as a “highly degenerate Fermi gas”; these studies enabled him to explain the physics behind the feeble and temperature-independent character of the paramagnetism of metals. Accordingly, this phenomenon is referred to as Pauli paramagnetism — in contrast to the classical Langevin paramagnetism. In quantum statistics, we encounter yet another effect which is totally absent in clas-sical statistics. This is diamagnetic in character and arises from the quantization of the orbits of charged particles in the presence of an external magnetic field or, one may say, 8.2 Magnetic behavior of an ideal Fermi gas 239 from the quantization of the (kinetic) energy of charged particles associated with their motion perpendicular to the direction of the field. The existence of this effect was first established by Landau (1930); so, we refer to it as Landau diamagnetism. This leads to an additional susceptibility χ(T), which, though negative in sign, is somewhat similar to the paramagnetic susceptibility, in that it obeys Curie’s law at high temperatures and tends to a temperature-independent but density-dependent limiting value as T →0. In gen-eral, the magnetic behavior of a Fermi gas is determined jointly by the intrinsic magnetic moment of the particles and the quantization of their orbits. If the spin–orbit interaction is negligible, the resultant behavior is given by a simple addition of the two effects. 8.2.A Pauli paramagnetism The energy of a particle, in the presence of an external magnetic field B, is given by ε = p2 2m −µ∗· B, (1) where µ∗is the intrinsic magnetic moment of the particle and m its mass. For simplicity, we assume that the particle spin is 1 2; the vector µ∗will then be either parallel to the vector B or antiparallel. We thus have two groups of particles in the gas: (i) those having µ∗parallel to B, with ε = p2/2m −µ∗B, and (ii) those having µ∗antiparallel to B, with ε = p2/2m + µ∗B. At absolute zero, all energy levels up to the Fermi level εF will be filled, while all levels beyond εF will be empty. Accordingly, the kinetic energy of the particles in the first group will range between 0 and (εF + µ∗B), while the kinetic energy of the particles in the second group will range between 0 and (εF −µ∗B). The respective numbers of occupied energy levels (and hence of particles) in the two groups will, therefore, be N+ = 4πV 3h3 {2m(εF + µ∗B)}3/2 (2) and N−= 4πV 3h3 {2m(εF −µ∗B)}3/2. (3) The net magnetic moment acquired by the gas is then given by M = µ∗(N+ −N−) = 4πµ∗V(2m)3/2 3h3 {(εF + µ∗B)3/2 −(εF −µ∗B)3/2}. (4) We thus obtain for the low-field susceptibility (per unit volume) of the gas χ0 = Lim B→0  M VB  = 4πµ∗2(2m)3/2ε1/2 F h3 . (5) 240 Chapter 8. Ideal Fermi Systems Making use of formula (8.1.24), with g = 2, the foregoing result may be written as χ0 = 3 2nµ∗2/εF. (6) For comparison, the corresponding high-temperature result is given by equation (3.9.26), with g = 2 and J = 1 2: χ∞= nµ∗2/kT. (7) We note that χ0/χ∞= O(kT/εF). To obtain an expression for χ that holds for all T, we proceed as follows. Denoting the number of particles with momentum p and magnetic moment parallel (or antiparallel) to the field by the symbol n+ p (or n− p ), the total energy of the gas can be written as En = X p " p2 2m −µ∗B ! n+ p + p2 2m + µ∗B ! n− p # = X p (n+ p + n− p ) p2 2m −µ∗B(N+ −N−), (8) where N+ and N−denote the total number of particles in the two groups, respectively. The partition function of the system is then given by Q(N) = X {n+ p },{n− p } ′exp(−βEn), (9) where the primed summation is subject to the conditions n+ p ,n− p = 0 or 1, (10) and X p n+ p + X p n− p = N+ + N−= N. (11) To evaluate the sum in (9), we first fix an arbitrary value of the number N+ (which auto-matically fixes the value of N−as well) and sum over all n+ p and n− p that conform to the fixed values of the numbers N+ and N−as well as to condition (10). Next, we sum over all possible values of N+, namely from N+ = 0 to N+ = N. We thus have Q(N) = N X N+=0   eβµ∗B(2N+−N)      X {n+ p } ′′ exp  −β X p p2 2mn+ p  X {n− p } ′′′ exp  −β X p p2 2mn− p          ; (12) 8.2 Magnetic behavior of an ideal Fermi gas 241 here, the summation P′′ is subject to the restriction P p n+ p = N+, while P′′′ is subject to the restriction P p n− p = N −N+. Now, let Q0(N) denote the partition function of an ideal Fermi gas of N “spinless” particles of mass m; then, obviously, Q0(N) = X {np} ′ exp  −β X p p2 2mnp  ≡exp{−βA0(N)}, (13) where A0(N) is the free energy of this fictitious system. Equation (12) can then be written as Q(N) = e−βµ∗BN N X N+=0 [e2βµ∗BN+Q0(N+)Q0(N −N+)], (14) which gives 1 N lnQ(N) = −βµ∗B + 1 N ln N X N+=0 [exp{2βµ∗BN+ −βA0(N+) −βA0(N −N+)}]. (15) As before, the logarithm of the sum P N+ may be replaced by the logarithm of the largest term in the sum; the error committed in doing so would be negligible in comparison with the term retained. Now, the value N+, of N+, which corresponds to the largest term in the sum, can be ascertained by setting the differential coefficient of the general term, with respect to N+, equal to zero; this gives 2µ∗B − ∂A0(N+) ∂N+  N+=N+ − ∂A0(N −N+) ∂N+  N+=N+ = 0, that is µ0(N+) −µ0(N −N+) = 2µ∗B, (16) where µ0(N) is the chemical potential of the fictitious system of N “spinless” fermions. The foregoing equation contains the general solution being sought. To obtain an explicit expression for χ, we introduce a dimensionless parameter r, defined by M = µ∗(N+ −N−) = µ∗(2N+ −N) = µ∗Nr (0 ≤r ≤1); (17) equation (16) then becomes µ0 1 + r 2 N  −µ0 1 −r 2 N  = 2µ∗B. (18) If the magnetic field B vanishes so does r, which corresponds to a completely random ori-entation of the elementary moments. For small B, r would also be small; so, we may carry 242 Chapter 8. Ideal Fermi Systems out a Taylor expansion of the left side of (18) about r = 0. Retaining only the first term of the expansion, we obtain r ≃ 2µ∗B ∂µ0(xN) ∂x x=1/2 . (19) The low-field susceptibility (per unit volume) of the system is then given by χ = M VB = µ∗Nr VB = 2nµ∗2 ∂µ0(xN) ∂x x=1/2 , (20) which is the desired result valid for all T. For T →0, the chemical potential of the fictitious system can be obtained from equation (8.1.34), with g = 1: µ0(xN) =  3xN 4πV 2/3 h2 2m, which gives ∂µ0(xN) ∂x x=1/2 = 24/3 3  3N 4πV 2/3 h2 2m. (21) On the other hand, the Fermi energy of the actual system is given by the same equa-tion (8.1.34), with g = 2: εF =  3N 8πV 2/3 h2 2m. (22) Making use of equations (21) and (22), we obtain from (20) χ0 = 2nµ∗2 4 3εF = 3 2nµ∗2/εF, (23) in complete agreement with our earlier result (6). For finite but low temperatures, one has to use equation (8.1.35) instead of (8.1.34). The final result turns out to be χ ≃χ0 " 1 −π2 12 kT εF 2# . (24) On the other hand, for T →∞, the chemical potential of the fictitious system follows directly from equation (8.1.4), with g = 1 and f3/2(z) ≃z, with the result µ0(xN) = kT ln(xNλ3/V), 8.2 Magnetic behavior of an ideal Fermi gas 243 which gives ∂µ0(xN) ∂x x=1/2 = 2kT. (25) Equation (20) then gives χ∞= nµ∗2/kT, (26) in complete agreement with our earlier result (7). For large but finite temperatures, one has to take f3/2(z) ≃z −(z2/23/2). The final result then turns out to be χ ≃χ∞ 1 −nλ3 25/2 ! ; (27) the correction term here is proportional to (TF/T)3/2 and tends to zero as T →∞. 8.2.B Landau diamagnetism We now study the magnetism arising from the quantization of the orbital motion of (charged) particles in the presence of an external magnetic field. In a uniform field of intensity B, directed along the z-axis, a charged particle would follow a helical path whose axis is parallel to the z-axis and whose projection on the (x,y)-plane is a circle. Motion along the z-direction has a constant linear velocity uz, while that in the (x,y)-plane has a constant angular velocity eB/mc; the latter arises from the Lorentz force, e(u × B)/c, expe-rienced by the particle. Quantum-mechanically, the energy associated with the circular motion is quantized in units of eℏB/mc. The energy associated with the linear motion along the z-axis is also quantized but, in view of the smallness of the energy intervals, this may be taken as a continuous variable. We thus have for the total energy of the particle2 ε = eℏB mc  j + 1 2  + p2 z 2m (j = 0,1,2,...). (28) Now, these quantized energy levels are degenerate because they result from a “coalescing together” of an almost continuous set of zero-field levels. A little reflection shows that all those levels for which the value of the quantity (p2 x + p2 y)/2m lay between eℏBj/mc and eℏB(j + 1)/mc now “coalesce together” into a single level characterized by the quantum number j. The number of these levels is given by 1 h2 Z dxdydpxdpy = LxLy h2 π  2meℏB mc {(j + 1) −j}  = LxLy eB hc , (29) 2See, for instance, Goldman et al. (1960); Problem 6.3. 244 Chapter 8. Ideal Fermi Systems eℏ mc B 0 B0 j 4 j 3 j 2 j 1 j 0 B eℏ mc B FIGURE 8.3 The single-particle energy levels, for a two-dimensional motion, in the absence of an external magnetic field (B = 0) and in the presence of an external magnetic field (B > 0). which is independent of j. The multiplicity factor (29) is a quantum-mechanical measure of the freedom available to the particle for the center of its orbit to be “located” anywhere in the total area LxLy of the physical space. Figure 8.3 depicts the manner in which the zero-field energy levels of the particle group themselves into a spectrum of oscillator-like levels on the application of the external magnetic field. The grand partition function of the gas is given by the standard formula lnQ = X ε ln(1 + ze−βε), (30) where the summation has to be carried over all single-particle states in the system. Sub-stituting (28) for ε, making use of the multiplicity factor (29) and replacing the summation over pz by an integration, we get lnQ = ∞ Z −∞ Lzdpz h   ∞ X j=0  LxLy eB hc  ln n 1 + ze−βeℏB[j+(1/2)]/mc−βp2 z/2mo  . (31) At high temperatures, z ≪1; so, the system is effectively Boltzmannian. The grand parti-tion function then reduces to lnQ = zVeB h2c ∞ Z −∞ e−βp2 z/2mdpz ∞ X j=0 e−βeℏB[j+(1/2)]/mc = zVeB h2c 2πm β 1/2  2sinh βeℏB 2mc −1 . (32) The equilibrium number of particles N and the magnetic moment M of the gas are then given by 8.2 Magnetic behavior of an ideal Fermi gas 245 N =  z ∂ ∂z lnQ  B,V,T , (33) and M = −∂H ∂B = 1 β  ∂ ∂B lnQ  z,V,T , (34) where H is the Hamiltonian of the system; compare with equation (3.9.4). We thus obtain N = zV λ3 x sinhx, (35) and M = zV λ3 µeff  1 sinhx −xcoshx sinh2 x  , (36) where λ{= h/(2πmkT)1/2} is the mean thermal wavelength of the particles, while x = βµeffB µeff = eh/4πmc  . (37) Clearly, if e and m are the electronic charge and the electronic mass, then µeff is the familiar Bohr magneton µB. Combining (35) and (36), we get M = −NµeffL(x), (38) where L(x) is the Langevin function: L(x) = cothx −1 x. (39) This result is very similar to the one obtained in the Langevin theory of paramagnetism; see Section 3.9. The presence of the negative sign, however, means that the effect obtained in the present case is diamagnetic in nature. We also note that this effect is a direct con-sequence of quantization; it vanishes if we let h →0. This is in complete accord with the Bohr–van Leeuwen theorem, according to which the phenomenon of diamagnetism does not arise in classical physics; see Problem 3.43. If the field intensity B and the temperature T are such that µeffB ≪kT, then the foregoing results become N ≃zV λ3 (40) and M ≃−Nµ2 effB/3kT. (41) 246 Chapter 8. Ideal Fermi Systems Equation (40) is in agreement with the zero-field formula z ≃nλ3, while (41) leads to the diamagnetic counterpart of the Curie law: χ∞= M VB = −nµ2 eff/3kT; (42) see equation (3.9.12). It should be noted here that the diamagnetic character of this phenomenon is independent of the sign of the electric charge on the particle. For an elec-tron gas, in particular, the net susceptibility at high temperatures is given by the sum of expression (7), with µ∗replaced by µB, and expression (42): χ∞= n  µ2 B −1 3µ′2 B  kT , (43) where µ′ B = eh/4πm′c, m′ being the effective mass of the electron in the given system. We now look at this problem at all temperatures, though we will continue to assume the magnetic field to be weak, so that µeffB ≪kT. In view of the latter, the summation in (31) may be handled with the help of the Euler summation formula, ∞ X j=0 f  j + 1 2  ≃ ∞ Z 0 f (x)dx + 1 24f ′(0), (44) with the result lnQ ≃VeB h2c   ∞ Z 0 dx ∞ Z −∞ dpz ln n 1 + ze−β(2µeffBx+p2 z/2m)o −1 12βµeffB ∞ Z −∞ dpz z−1eβ(p2 z/2m) + 1  . (45) The first part here is independent of B, which can be seen by changing the variable from x to x′ = Bx. The second part, with the substitution βp2 z/2m = y, becomes −πV(2m)3/2 6h3 (µeffB)2β1/2 ∞ Z 0 y−1/2dy z−1ey + 1. (46) The low-field susceptibility (per unit volume) of the gas is then given by χ = M VB = 1 βVB  ∂ ∂B lnQ  z,V,T = − (2πm)3/2µ2 eff 3h3β1/2 f1/2(z), (47) 8.3 The electron gas in metals 247 which is the desired result. Note that, as before, the effect is diamagnetic in character — irrespective of the sign of the charge on the particle. For z ≪1, f1/2(z) ≃z ≃nλ3; we then recover our previous result (42). For z ≫1 (which corresponds to T ≪TF), f1/2(z) ≈(2/π1/2)(lnz)1/2; we then get χ0 ≈− 2π(2m)3/2µ2 effε1/2 F 3h3 = −1 2nµ2 eff/εF; (48) here, use has also been made of the fact that (β−1 lnz) ≃εF. Note that, in magnitude, this result is precisely one-third of the corresponding paramagnetic result (6), provided that we take the µ∗of that expression to be equal to the µeff of this one. 8.3 The electron gas in metals One physical system where the application of Fermi–Dirac statistics helped remove a number of inconsistencies and discrepancies is that of conduction electrons in metals. Historically, the electron theory of metals was developed by Drude (1900) and Lorentz (1904–1905), who applied the statistical mechanics of Maxwell and Boltzmann to the electron gas and derived theoretical results for the various properties of metals. The Drude–Lorentz model did provide a reasonable theoretical basis for a partial understand-ing of the physical behavior of metals; however, it encountered a number of serious problems of a qualitative as well as quantitative nature. For instance, the observed spe-cific heat of metals appeared to be almost completely accountable by the lattice vibrations alone and practically no contribution seemed to be coming from the electron gas. The theory, however, demanded that, on the basis of the equipartition theorem, each electron in the gas should possess a mean thermal energy 3 2kT and hence make a contribution of 3 2k to the specific heat of the metal. Similarly, one expected the electron gas to exhibit the phenomenon of paramagnetism arising from the intrinsic magnetic moment µB of the electrons. According to the classical theory, the paramagnetic susceptibility would be given by (8.2.7), with µ∗replaced by µB. Instead, one found that the susceptibility of a normal nonferromagnetic metal was not only independent of temperature but had a magnitude which, at room temperatures, was hardly 1 percent of the expected value. The Drude–Lorentz theory was also applied to study transport properties of met-als, such as the thermal conductivity K and the electrical conductivity σ. While the results for the individual conductivities were not very encouraging, their ratio did con-form to the empirical law of Wiedemann and Franz (1853), as formulated by Lorenz (1872), namely that the quantity K/σT was a (universal) constant. The theoretical value of this quantity, which is generally known as the Lorenz number, turned out to be 3(k/e)2 ≃2.48 × 10−13 e.s.u./deg2; the corresponding experimental values for most alkali and alkaline–earth metals were, however, found to be scattered around a mean value of 2.72 × 10−13 e.s.u./deg2. A still more uncomfortable feature of the classical theory was the uncertainty in assigning an appropriate value to the mean free path of the electrons in 248 Chapter 8. Ideal Fermi Systems a given metal and in ascribing to it an appropriate temperature dependence. For these reasons, the problem of the transport properties of metals also remained in a rather unsatisfactory state until the correct lead was provided by Sommerfeld (1928). The most significant change introduced by Sommerfeld was the replacement of Maxwell–Boltzmann statistics by Fermi–Dirac statistics for describing the electron gas in a metal. With this single stroke of genius, he was able to set most of the things right. To see how it worked, let us first estimate the Fermi energy εF of the electron gas in a typical metal, say sodium. Referring to equation (8.1.24), with g = 2, εF =  3N 8πV 2/3 h2 2m′ , (1) where m′ is the effective mass of an electron in the gas.3 The electron density N/V, in the case of a cubic lattice, may be written as N V = nena a3 , (2) where ne is the number of conduction electrons per atom, na the number of atoms per unit cell and a the lattice constant (or the cell length).4 For sodium, ne = 1, na = 2, and a = 4.29 ˚ A. Substituting these numbers into (2) and writing m′ = 0.98me, we obtain from (1) (εF)Na = 5.03 × 10−12 erg = 3.14 eV. (3) Accordingly, for the Fermi temperature of the gas is (TF)Na = (1.16 × 104) × εF( in eV) = 3.64 × 104K, (4) which is considerably larger than the room temperature T (∼3 × 102 K). The ratio T/TF being of the order of 1 percent, the conduction electrons in sodium constitute a highly degenerate Fermi system. This statement, in fact, applies to all metals because their Fermi temperatures are generally of order 104 −105 K. Now, the very fact that the electron gas in metals is a highly degenerate Fermi system is sufficient to explain away some of the basic difficulties of the Drude–Lorentz theory. For instance, the specific heat of this gas would no longer be given by the classical formula, 3To justify the assumption that the conduction electrons in a metal may be treated as “free” electrons, it is necessary to ascribe to them an effective mass m′ ̸= m. This is an indirect way of accounting for the fact that the electrons in a metal are not really free; the ratio m′/m accordingly depends on the structural details of the metal and, therefore, varies from metal to metal. In sodium, m′/m ≃0.98. 4Another way of expressing the electron density is to write N/V = f ρ/M, where f is the valency of the metal, ρ its mass density, and M the mass of an atom (ρ/M, thus, being the number density of the atoms). 8.3 The electron gas in metals 249 CV = 3 2Nk, but rather by equation (8.1.39), namely CV = π2 2 Nk(kT/εF); (5) obviously, the new result is much smaller in value because, at ordinary temperatures, the ratio (kT/εF) ≡(T/TF) = O(10−2). It is then hardly surprising that, at ordinary tem-peratures, the specific heat of metals is almost completely determined by the vibrational modes of the lattice and very little contribution comes from the conduction electrons. Of course, as temperature decreases, the specific heat due to lattice vibrations also decreases and finally becomes considerably smaller than the classical value; see Section 7.4, espe-cially Figure 7.14. A stage comes when the two contributions, both nonclassical, become comparable in value. Ultimately, at very low temperatures, the specific heat due to lattice vibrations, being proportional to T3, becomes even smaller than the electronic specific heat, which is proportional to T1. In general, we may write, for the low-temperature specific heat of a metal, CV = γ T + δT3, (6) where the coefficient γ is given by equation (5) or, more generally, can be shown to be proportional to the density of states at the Fermi energy (see Problem 8.13), while the coef-ficient δ is given by equation (7.4.23). An experimental determination of the specific heat of metals at low temperatures is, therefore, expected not only to verify the theoretical result based on quantum statistics but also to evaluate some of the parameters of the problem. Such determinations have been made, among others, by Corak et al. (1955) who worked with copper, silver and, gold in the temperature range 1 to 5 K. Their results for copper are shown in Figure 8.4. The very fact that the (CV /T) versus T2 plot is well approximated by a straight line vindicates the theoretical formula (6). Furthermore, the slope of this line gives the value of the coefficient δ, from which one can extract the Debye temperature 2D of 0 0.4 0.8 1.2 1.6 2.0 2.4 2 4 6 8 10 12 14 16 18 T 2 (in K2) Cv/T (in millijoules mole1 K2) FIGURE 8.4 The observed specific heat of copper at low temperatures (after Corak et al., 1955). 250 Chapter 8. Ideal Fermi Systems the metal. One thus gets for copper: 2D = (343.8 ± 0.5)K, which compares favorably with Leighton’s theoretical estimate of 345 K (based on the elastic constants of the metal). The intercept on the (CV /T)-axis yields the value of the coefficient γ , namely (0.688 ± 0.002) millijoule mole−1 deg−2, which agrees favorably with Jones’ estimate of 0.69 millijoule mole−1 deg−2 (based on a density-of-states calculation). The general pattern of the magnetic behavior of the electron gas in nonferromagnetic metals can be understood likewise. In view of the highly degenerate nature of the gas, the magnetic susceptibility χ is given by the Pauli result (8.2.6) plus the Landau result (8.2.48), and not by the classical result (8.2.7). In complete agreement with the observation, the new result is (i) independent of temperature and (ii) considerably smaller in magnitude than the classical one. As regards transport properties K and σ, the new theory again led to the Wiedemann– Franz law ; the Lorenz number, however, became (π2/3)(k/e)2, instead of the classical 3(k/e)2. The resulting theoretical value, namely 2.71 × 10−13 e.s.u./deg2, turned out to be much closer to the experimental mean value quoted earlier. Of course, the situa-tion regarding individual conductivities and the mean free path of the electrons did not improve until Bloch (1928) developed a theory that took into account interactions among the electron gas and the ion system in the metal. The theory of metals has continued to become more and more sophisticated; the important point to note here is that this development has all along been governed by the new statistics! Before leaving this topic, we would like to give a brief account of the phenomena of thermionic and photoelectric emission of electrons from metals. In view of the fact that electronic emission does not take place spontaneously, we infer that the electrons inside a metal find themselves caught in some sort of a “potential well” created by the ions. The detailed features of the potential energy of an electron in this well must depend on the structure of the given metal. For a study of electronic emission, however, we need not worry about these details and may assume instead that the potential energy of an elec-tron stays constant (at a negative value, −W, say) throughout the interior of the metal and changes discontinuously to zero at the surface. Thus, while inside the metal, the electrons move about freely and independently of one another; however, as soon as any one of them approaches the surface of the metal and tries to escape, it encounters a potential barrier of height W. Accordingly, only those electrons whose kinetic energy (associated with the motion perpendicular to the surface) is greater than W can expect to escape through this barrier. At ordinary temperatures, especially in the absence of any external stimulus, such electrons are too few in any given metal, with the result that there is practically no spon-taneous emission from metals. At high temperatures, and more so if there is an external stimulus present, the population of such electrons in a given metal could become large enough to yield a sizeable emission. We then speak of phenomena such as thermionic effect and photoelectric effect. Strictly speaking, these phenomena are not equilibrium phenomena because electrons are flowing out steadily through the surface of the metal. However, if the number of elec-trons lost in a given interval of time is small in comparison with their total population in 8.3 The electron gas in metals 251 the metal, then the magnitude of the emission current may be calculated on the assump-tion that the gas inside continues to be in a state of quasistatic thermal equilibrium. The mathematical procedure for this calculation is very much the same as the one followed in Section 6.4 (for determining the rate of effusion R of the particles of a gas through an open-ing in the walls of the container). There is one difference, however; whereas in the case of effusion any particle that reached the opening with uz > 0 could escape unquestioned, here we must have uz > (2W/m)1/2, so that the particle in question could successfully cross over the potential barrier at the surface. Moreover, even if this condition is satisfied, there is no guarantee that the particle will really escape because the possibility of an inward reflection cannot be ruled out. In the following discussion, we shall disregard this possi-bility; however, if one is looking for a numerical comparison of theory with experiment, the results derived here must be multiplied by a factor (1 −r), where r is the reflection coefficient of the surface. 8.3.A Thermionic emission (the Richardson effect) The number of electrons emitted per unit area of the metal surface per unit time is given by R = ∞ Z pz=(2mW)1/2 ∞ Z px=−∞ ∞ Z py=−∞ 2dpxdpydpz h3 1 e(ε−µ)/kT + 1  uz; (7) compare with the corresponding expression in Section 6.4. Integration over the variables px and py may be carried out by changing over to polar coordinates (p′,φ), with the result R = 2 h3 ∞ Z pz=(2mW)1/2 pz m dpz ∞ Z p′=0 2πp′dp′ exp{[(p′2/2m) + (p2 z/2m) −µ]/kT} + 1 = 4πkT h3 ∞ Z pz=(2mW)1/2 pzdpz ln[1 + exp{(µ −p2 z/2m)/kT}] = 4πmkT h3 ∞ Z εz=W dεz ln[1 + e(µ−εz)/kT]. (8) It so happens that the exponential term inside the logarithm, at all temperatures of inter-est, is much smaller than unity; see Note 5. We may, therefore, write ln(1 + x) ≃x, with the 252 Chapter 8. Ideal Fermi Systems result R = 4πmkT h3 ∞ Z εz=W dεze(µ−εz)/kT = 4πmk2T2 h3 e(µ−W)/kT. (9) The thermionic current density is then given by J = eR = 4πmek2 h3 T2e(µ−W)/kT. (10) It is only now that the difference between the classical statistics and the Fermi statistics really shows up! In the case of classical statistics, the fugacity of the gas is given by (see equation (8.1.4), with f3/2(z) ≃z) z ≡eµ/kT = nλ3 g = nh3 2(2πmkT)3/2 ; (11) accordingly, Jclass = ne  k 2πm 1/2 T1/2e−φ/kT (φ = W). (12) In the case of Fermi statistics, the chemical potential of the (highly degenerate) electron gas is practically independent of temperature and is very nearly equal to the Fermi energy of the gas (µ ≃µ0 ≡εF); accordingly, JF.D. = 4πmek2 h3 T2e−φ/kT (φ = W −εF). (13) The quantity φ is generally referred to as the work function of the metal. According to (12), φ is exactly equal to the height of the surface barrier; according to (13), it is equal to the height of the barrier over and above the Fermi level (see Figure 8.5). Outside W Outside 0 Fermi level Fermi sea FIGURE 8.5 The work function φ of a metal represents the height of the surface barrier over and above the Fermi level. 8.3 The electron gas in metals 253 The theoretical results embodied in equations (12) and (13) differ in certain important respects. The most striking difference seems to be in regard to the temperature depen-dence of the thermionic current density J. However, the major dependence on T comes through the factor exp(−φ/kT) — so much so that whether we plot ln(J/T1/2) against (1/T) or ln(J/T2) against (1/T) we obtain, in each case, a fairly good straight-line fit. Thus, from the point of view of the temperature dependence of J, a choice between formulae (12) and (13) is rather hard to make. However, the slope of the experimental line should give us directly the value of W if formula (12) applies or of (W −εF) if formula (13) applies! Now, the value of W can be determined independently, for instance, by studying the refractive index of a given metal for de Broglie waves associated with an electron beam impinging on the metal. For a beam of electrons whose initial kinetic energy is E, we have λout = h √(2mE) and λin = h √[2m(E + W)], (14) so that the refractive index of the metal is given by n = λout λin = E + W E 1/2 . (15) By studying electron diffraction for different values of E, one can derive the relevant value of W. In this manner, Davisson and Germer (1927) derived the value of W for a number of metals. For instance, they obtained for tungsten: W ≃13.5eV. The experimental results on thermionic emission from tungsten are shown in Figure 8.6. The value of φ resulting from the slope of the experimental line was about 4.5 eV. The large difference between these 4.3 0 0.5 1.0 2.0 1.5 4.4 4.5 4.6 4.7 4.8 4.9 5.0 5.1 5.2 (104/T in K21) In(J/T 2), in arbitrary units FIGURE 8.6 Thermionic current from tungsten as a function of the temperature of the metal. The continuous line corresponds to r = 1 2 while the broken line corresponds to r = 0,r being the reflection coefficient of the surface. 254 Chapter 8. Ideal Fermi Systems two values clearly shows that the classical formula (12) does not apply. That the quantum-statistical formula (13) applies is shown by the fact that the Fermi energy of tungsten is very nearly 9eV; so, the value 4.5eV for the work function of tungsten is correctly given by the difference between the depth W of the potential well and the Fermi energy εF. To quote another example, the experimental value of the work function for nickel was found to be about 5.0eV, while the theoretical estimate for its Fermi energy turns out to be about 11.8eV. Accordingly, the depth of the potential well in the case of nickel should be about 16.8 eV. The experimental value of W, obtained by studying electron diffraction in nickel, is indeed (17 ± 1)eV.5 The second point of difference between formulae (12) and (13) relates to the actual value of the current obtained. In this respect, the classical formula turns out to be a com-plete failure while the quantum-statistical formula fares reasonably well. The constant factor in the latter formula is 4πmek2 h3 = 120.4 amp cm−2 deg−2; (16) of course, this has yet to be multiplied by the transmission coefficient (1 −r). The corre-sponding experimental number, for most metals with clean surfaces, turns out to be in the range 60 to 120 amp cm−2 deg−2. Finally, we examine the influence of a moderately strong electric field on the thermionic emission from a metal — the so-called Schottky effect. Denoting the strength of the electric field by F and assuming the field to be uniform and directed perpendicular to the metal surface, the difference 1 between the potential energy of an electron at a distance x above the surface and of one inside the metal is given by 1(x) = W −eFx −e2 4x (x > 0), (17) where the first term arises from the potential well of the metal, the second from the (attrac-tive) field present, and the third from the attraction between the departing electron and the “image” induced in the metal; see Figure 8.7. The largest value of the function 1(x) occurs at x = (e/4F)1/2, so that 1max = W −e3/2F1/2; (18) thus, the field has effectively reduced the height of the potential barrier by an amount e3/2F1/2. A corresponding reduction should take place in the work function as well. 5In light of the numbers quoted here, one can readily see that the quantity e(µ−εz)/kT in equation (8), being at most equal to e(µ0−W)/kT ≡e−φ/kT, is, at all temperatures of interest, much smaller than unity. This means that we are oper-ating here in the (Maxwellian) tail of the Fermi–Dirac distribution and hence the approximation made in going from equation (8) to equation (9) was justified. 8.3 The electron gas in metals 255 F x x F 2e 1e FIGURE 8.7 A schematic diagram to illustrate the Schottky effect. Accordingly, the thermionic current density in the presence of the field F would be higher than the one in the absence of the field: JF = J0 exp(e3/2F1/2/kT). (19) A plot of ln(JF/J0) against (F1/2/T) should, therefore, be a straight line, with slope e3/2/k. Working along these lines, De Bruyne (1928) obtained for the electronic charge a value of 4.84 × 10−10 e.s.u., which is remarkably close to the actual value of e. The theory of the Schottky effect, as outlined here, holds for field strengths up to about 106 volts/cm. For fields stronger than that, one obtains the so-called cold emission, which means that the electric field is now strong enough to make the potential barrier practically ineffective; for details, see Fowler and Nordheim (1928). 8.3.B Photoelectric emission (the Hallwachs effect) The physical situation in the case of photoelectric emission is different from that in the case of thermionic emission, in that there exists now an external agency, the photon in the incoming beam of light, that helps an electron inside the metal in overcoming the potential barrier at the surface. The condition to be satisfied by the momentum component pz of an electron in order that it could escape from the metal now becomes (p2 z/2m) + hν > W, (20)6 where ν is the frequency of the incoming light (assumed monochromatic). Proceeding in the same manner as in the case of thermionic emission, we obtain, instead of (8), R = 4πmkT h3 ∞ Z εz=W−hν dεz ln[1 + e(µ−εz)/kT]. (21) We cannot, in general, approximate this integral the way we did there; so the integrand here stays as it is. It is advisable, however, to change over to a new variable x, defined by x = (εz −W + hν)/kT, (22) 6In writing this condition, we have tacitly assumed that the momentum components px and py of the electron remain unchanged on the absorption of a photon. 256 Chapter 8. Ideal Fermi Systems whereby equation (21) becomes R = 4πm(kT)2 h3 ∞ Z 0 dxln  1 + exp h(ν −ν0) kT −x  , (23) where hν0 = W −µ ≃W −εF = φ. (24) The quantity φ will be recognized as the (thermionic) work function of the metal; accord-ingly, the characteristic frequency ν0(= φ/h) may be referred to as the threshold frequency for (photoelectric) emission from the metal concerned. The current density of photoelectric emission is thus given by J = 4πmek2 h3 T2 ∞ Z 0 dxln(1 + eδ−x), (25) where δ = h(ν −ν0)/kT. (26) Integrating by parts, we find that ∞ Z 0 dxln(1 + eδ−x) = ∞ Z 0 xdx ex−δ + 1 ≡f2(eδ); (27) see equation (8.1.6). Accordingly, J = 4πmek2 h3 T2f2(eδ). (28) For h(ν −ν0) ≫kT, eδ ≫1 and the function f2(eδ) ≈δ2/2; see Sommerfeld’s lemma (E.17) in Appendix E. Equation (28) then becomes J ≈2πme h (ν −ν0)2, (29) which is completely independent of T; thus, when the energy of the light quantum is much greater than the work function of the metal, the temperature of the electron gas becomes a “dead” parameter of the problem. At the other extreme, when ν < ν0 and h|ν −ν0| ≫kT, then eδ ≪1 and the function f2(eδ) ≈eδ. Equation (28) then becomes J ≈4πmek2 h3 T2e(hν−φ)/kT, (30) 8.3 The electron gas in metals 257 which is just the thermionic current density (13), enhanced by the photon factor exp(hν/kT); in other words, the situation now is very much the same as in the case of thermionic emission, except for a diminished work function φ′(= φ −hν). At the threshold frequency (ν = ν0), δ = 0 and the function f2(eδ) = f2(1) = π2/12; see equation (E.16), with j = 1. Equation (28) then gives J0 = π3mek2 3h3 T2. (31) Figure 8.8 shows a plot of the experimental results for photoelectric emission from palladium (φ = 4.97eV). The agreement with theory is excellent. It will be noted that the plot includes some observations with ν < ν0. The fact that we obtain a finite photocur-rent even for frequencies less than the so-called threshold frequency is fully consistent with the model considered here. The reason for this lies in the fact that, at any finite tem-perature T, there exists in the system a reasonable fraction of electrons whose energy ε exceeds the Fermi energy εF by amounts O(kT). Therefore, if the light quantum hν gets absorbed by one of these electrons, then condition (20) for photoemission can be satisfied even if hν < (W −εF) = hν0. Of course, the energy difference h(ν0 −ν) must not be much more than a few times kT, for otherwise the availability of the right kind of electrons will be extremely low. We, therefore, do expect a finite photocurrent for radiation with frequencies less than the threshold frequency ν0, provided that h(ν0 −ν) = O(kT). The plot shown in Figure 8.8, namely ln(J/T2) versus δ, is generally known as the “Fowler plot.” Fitting the observed photoelectric data to this plot, one can obtain the characteristic frequency ν0 and hence the work function φ of the given metal. We have previously seen that the work function of a metal can be derived from thermionic data as well. It is gratifying to note that there is complete agreement between the two sets of results obtained for the work function of the various metals. 5 0 0 1 2 3 4 5 10 20 25 15 In(J/T 2), in arbitrary units 305 K 400 K 830 K 1008 K h( 0)kT FIGURE 8.8 Photoelectric current from palladium as a function of the quantity h(ν −ν0)/kT. The plot includes data taken at several temperatures T for different frequencies ν. 258 Chapter 8. Ideal Fermi Systems 8.4 Ultracold atomic Fermi gases After the demonstration of Bose–Einstein condensation in ultracold atomic gases in 1995 (Section 7.2), researchers began using laser cooling and magnetic traps to cool gases of fermions to create degenerate Fermi gases of atoms. DeMarco and Jin (1999) created the first degenerate atomic Fermi gas by cooling a dilute vapor of 40K in an atomic trap into the nanokelvin temperature range. The density of states in a harmonic trap is a quadratic function of the energy: a(ε) = ε2 2(ℏω0)3 , (1) where ω0 = (ω1ω2ω3)1/3 is the geometric mean of the trap frequencies in the cartesian directions; see equation (7.2.3). The chemical potential and the number of fermions in the trap are related by N(µ,T) = 1 2(ℏω0)3 ∞ Z 0 ε2dε eβ(ε−µ) + 1 , (2) which gives for the Fermi energy εF = ℏω0(6N)1/3 , (3) a Fermi temperature TF = εF/k = 870nK for 106 atoms in a 100Hz trap, and a ground-state energy U0 = 3 4NεF. The internal energy of the trapped gas can be obtained by time-of-flight measurements as described in Section 7.2 and can be directly compared with the theoretical result U U0 = 4  T TF 4 ∞ Z 0 x3dx exe−βµ + 1 , (4) where the temperature and the chemical potential are related by 3  T TF 3 ∞ Z 0 x2dx exe−βµ + 1 = 1; (5) see Figures 8.9 and 8.10. At low enough temperatures, attractive interactions lead to BEC-BCS condensation, as discussed in Section 11.9. 8.5 Statistical equilibrium of white dwarf stars 259 2 1 0 0.0 0.2 0.4 0.6 T/TF U/U0 0.8 1.0 3 4 FIGURE 8.9 Scaled internal energy (U/U0) versus scaled temperature (T/TF) for an ideal Fermi gas in a harmonic trap from equations (4) and (5). The dotted line is the corresponding classical result U(T) = 3NkT. 2.0 1.8 1.6 1.4 1.2 1.0 0.0 0.2 0.4 classical 0.6 0.8 1.0 E/3kT T/TF FIGURE 8.10 Experimental results for the mean energy per particle divided by the equipartition value of 3kT versus scaled temperature (T/TF) for ultracold 40K atoms in a harmonic trap compared to the theoretical Fermi gas value from equations (4) and (5). This shows the development of the Fermi degeneracy of the gas at low temperatures; from Jin (2002). Figure courtesy of the IOP. Reprinted with permission; copyright ©2002, American Institute of Physics. 8.5 Statistical equilibrium of white dwarf stars Historically, the first application of Fermi statistics appeared in the field of astrophysics (Fowler, 1926). It related to the study of thermodynamic equilibrium of white dwarf stars — the small-sized stars that are abnormally faint for their (white) color. The general pattern of color–brightness relationship among stars is such that, by and large, a star with red color is expected to be a “dull” star, while one with white color is expected to be a “brilliant” star. However, white dwarf stars constitute an exception to this rule. The reason for this lies in the fact that these stars are relatively old whose hydrogen content is more or less used up, 260 Chapter 8. Ideal Fermi Systems with the result that the thermonuclear reactions in them are now proceeding at a rather low pace, thus making these stars a lot less bright than one would expect on the basis of their color. The material content of white dwarf stars, at the present stage of their career, is mostly helium. And whatever little brightness they presently have derives mostly from the gravitational energy released as a result of a slow contraction of these stars — a mechanism first proposed by Kelvin, in 1861, as a “possible” source of energy for all stars! A typical, though somewhat idealized, model of a white dwarf star consists of a mass M(∼1033g) of helium, packed into a ball of mass density ρ(∼107gcm−3), at a central temperature T(∼107K). Now, a temperature of the order of 107K corresponds to a mean thermal energy per particle of the order of 103 eV, which is much greater than the energy required for ionizing a helium atom. Thus, practically the whole of the helium in the star exists in a state of complete ionization. The microscopic constituents of the star may, therefore, be taken as N electrons (each of mass m) and 1 2N helium nuclei (each of mass ≃4mp). The mass of the star is then given by M ≃N(m + 2mp) ≃2Nmp (1) and, hence, the electron density by n = N V ≃M/2mp M/ρ = ρ 2mp . (2) A typical value of the electron density in white dwarf stars would, therefore, be O(1030) electrons per cm3. We thus obtain for the Fermi momentum of the electron gas [see equation (8.1.23), with g = 2] pF =  3n 8π 1/3 h = O(10−17)gcmsec−1, (3) which is rather comparable with the characteristic momentum mc of an electron. The Fermi energy εF of the electron gas will, therefore, be comparable with the rest energy mc2 of an electron, that is, εF = O(106)eV and hence the Fermi temperature TF = O(1010) K. In view of these estimates, we conclude that (i) the dynamics of the electrons in this problem is relativistic, and (ii) the electron gas, though at a temperature large in comparison with terrestrial standards, is, statistically speaking, in a state of (almost) complete degeneracy: (T/TF) = O(10−3). The second point was fully appreciated, and duly taken into account, by Fowler himself; the first one was taken care of later, by Anderson (1928) and by Stoner (1929, 1930). The problem, in full generality, was attacked by Chandrasekhar (1931–1935) to whom the final picture of the theory of white dwarf stars is chiefly due; for details, see Chandrasekhar (1939), where a complete bibliography of the subject is given. Now, the helium nuclei do not contribute as significantly to the dynamics of the prob-lem as do the electrons; in the first approximation, therefore, we may neglect the presence of the nuclei in the system. For a similar reason, we may neglect the effect of the radia-tion as well. We may thus consider the electron gas alone. Further, for simplicity, we may assume that the electron gas is uniformly distributed over the body of the star; we are thus 8.5 Statistical equilibrium of white dwarf stars 261 ignoring the spatial variation of the various parameters of the problem — a variation that is physically essential for the very stability of the star! The contention here is that, in spite of neglecting the spatial variation of the parameters involved, we expect that the results obtained here will be correct, at least in a qualitative sense. We study the ground-state properties of a degenerate Fermi gas composed of N relativistic electrons (g = 2). First of all, we have N = 8πV h3 pF Z 0 p2dp = 8πV 3h3 p3 F, (4) which gives pF =  3n 8π 1/3 h. (5) The energy–momentum relation for a relativistic particle is ε = mc2[{1 + (p/mc)2}1/2 −1], (6) the speed of the particle being u ≡dε dp = (p/m) {1 + (p/mc)2}1/2 ; (7) here, m denotes the rest mass of the electron. The pressure P0 of the gas is then given by, see equation (6.4.3), P0 = 1 3 N V ⟨pu⟩0 = 8π 3h3 pF Z 0 (p2/m) {1 + (p/mc)2}1/2 p2dp. (8) We now introduce a dimensionless variable θ, defined by p = mcsinhθ, (9) which makes u = ctanhθ. (10) Equations (4) and (8) then become N = 8πVm3c3 3h3 sinh3 θF = 8πVm3c3 3h3 x3 (11) and P0 = 8πm4c5 3h3 θF Z 0 sinh4 θdθ = πm4c5 3h3 A(x), (12) 262 Chapter 8. Ideal Fermi Systems where A(x) = x(x2 + 1)1/2(2x2 −3) + 3sinh−1 x, (13) with x = sinhθF = pF/mc = (3n/8π)1/3(h/mc). (14) The function A(x) can be computed for any desired value of x. However, asymptotic results for x ≪1 and x ≫1 are often useful; these are given by (see Kothari and Singh, 1942) A(x) = 8 5x5 −4 7x7 + 1 3x9 −5 22x11 + ··· for x ≪1 = 2x4 −2x2 + 3(ln2x −7 12) + 5 4x−2 + ··· for x ≫1   . (15) We shall now consider, somewhat crudely, the equilibrium configuration of this model. In the absence of gravitation, it would be necessary to have “external walls” for keeping the electron gas at a given density n. The gas will exert a pressure P0(n) on the walls and any compression or expansion (of the gas) will involve an expenditure of work. Assuming the configuration to be spherical, an adiabatic change in V will cause a change in the energy of the gas, as given by dE0 = −P0(n)dV = −P0(R) · 4πR2dR. (16) In the presence of gravitation, no external walls are needed, but the change in the kinetic energy of the gas, as a result of a change in the size of the sphere, will still be given by for-mula (16); of course, the expression for P0, as a function of the “mean” density n, must now take into account the nonuniformity of the system — a fact being disregarded in the present simple-minded treatment. However, equation (16) alone no longer gives us the net change in the energy of the system; if that were the case, the system would expand indefi-nitely till both n and P0(n) →0. Actually, we have now a change in the potential energy as well; this is given by dEg = dEg dR  dR = α GM2 R2 dR, (17) where M is the total mass of the gas, G the constant of gravitation, while α is a number (of the order of unity) whose exact value depends on the nature of the spatial variation of n inside the sphere. If the system is in equilibrium, then the net change in its total energy (E0 + Eg), for an infinitesimal change in its size, should be zero; thus, for equilibrium, P0(R) = α 4π GM2 R4 . (18) 8.5 Statistical equilibrium of white dwarf stars 263 For P0(R), we may substitute from equation (12), where the parameter x is now given by x =  3n 8π 1/3 h mc =  9N 32π2 1/3 h/mc R or, in view of (1), by x =  9M 64π2mp 1/3 h/mc R = 9πM 8mp 1/3 ℏ/mc R . (19) Equation (18) then takes the form A 9πM 8mp 1/3 ℏ/mc R ! = 3αh3 4π2m4c5 GM2 R4 = 6πα ℏ/mc R 3 GM2/R mc2 ; (20) the function A(x) is given by equations (13) and (15). Equation (20) establishes a one-to-one correspondence between the masses M and the radii R of white dwarf stars; it is, therefore, known as the mass–radius relationship for these stars. It is rather interesting to see the combinations of parameters that appear in this relationship; we have here (i) the mass of the star in terms of the proton mass, (ii) the radius of the star in terms of the Compton wavelength of the electron, and (iii) the grav-itational energy of the star in terms of the rest energy of the electron. This relationship, therefore, exhibits a remarkable blending of quantum mechanics, special relativity, and gravitation. In view of the implicit character of relationship (20), we cannot express the radius of the star as an explicit function of its mass, except in two extreme cases. For this, we note that, since M ∼1033g, mp ∼10−24g, and ℏ/mc ∼10−11 cm, the argument of the function A(x) will be of the order of unity when R ∼108 cm. We may, therefore, define the two extreme cases as follows: (i) R ≫108 cm, which makes x ≪1 and hence A(x) ≈8 5x5, with the result R ≈3(9π)2/3 40α ℏ2M−1/3 Gmm5/3 p ∝M−1/3. (21) (ii) R ≪108 cm, which makes x ≫1 and hence A(x) ≈2x4 −2x2, with the result R ≈(9π)1/3 2 ℏ mc  M mp 1/3 ( 1 −  M M0 2/3)1/2 , (22) 264 Chapter 8. Ideal Fermi Systems where M0 = 9 64 3π α3 1/2 (ℏc/G)3/2 m2 p . (23) We thus find that the greater the mass of the white dwarf star, the smaller its size. Not only that, there exists a limiting mass M0, given by expression (23), that corresponds to a vanishing size of the star. Obviously, for M > M0, our mass–radius relationship does not possess any real solution. We, therefore, conclude that all white dwarf stars in equilibrium must have a mass less than M0 — a conclusion fully upheld by observation. The correct limiting mass of a white dwarf star is generally referred to as the Chandrasekhar limit. The physical reason for the existence of this limit is that for a mass exceeding this limit the ground-state pressure of the electron gas (that arises from the fact that the electrons obey the Pauli exclusion principle) would not be sufficient to support the star against its “tendency toward a gravitational collapse.” The numerical value of the limiting mass, as given by expression (23), turns out to be ∼1033g. Detailed investigations by Chandrasekhar led to the result: M0 = 5.75 µ2 e ⊙, (24) where ⊙denotes the mass of the sun, which is about 2 × 1033g, while µe is a number that represents the degree of ionization of helium in the gas. By definition, µe = M/NmH; compare to equation (1). Thus, in most cases, µe ≃2; accordingly M0 ≃1.44⊙. Figure 8.11 shows a plot of the theoretical relationship between the masses and the radii of white dwarf stars. One can see that the behavior in the two extreme regions, namely for R ≫l and R ≪l, is described quite well by formulae (21) and (22) of the treatment given here. The Chandrasekhar limit (24) is the mechanism responsible for stellar collapse into neutron stars and black holes. In particular, white dwarf stars whose mass exceeds the Chandrasekhar limit due to influx of matter from a companion binary star are thought to be the primary mechanism for type Ia supernovae; see Hillebrandt and Niemeyer (2000). Such events happen in a typical galaxy on the order of once per hundred years. For a few days after the collapse and subsequent explosion, these supernovae can be comparable in brightness to the remainder of the stars in the galaxy combined. Their well-calibrated light curves provide a bright “standard candle” for determining the distance to remote galaxies used to measure the expansion rate of the universe; see Chapter 9. 8.6 Statistical model of the atom Another application of the Fermi statistics was made by Thomas (1927) and Fermi (1928) for calculating the charge distribution and the electric field in the extra-nuclear space of a heavy atom. Their approach was based on the observation that the electrons in this sys-tem could be regarded as a completely degenerate Fermi gas of nonuniform density n(r). 8.6 Statistical model of the atom 265 M/M0 0 0.5 1.0 2.0 3.0 4.0 5.0 1.5 2.5 3.5 4.5 5.5 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0 0.7 R/ FIGURE 8.11 The mass–radius relationship for white dwarfs (after Chandrasekhar, 1939). The masses are expressed in terms of the limiting mass M0 and the radii in terms of a characteristic length l, which is given by 7.71µ−1 e × 108cm ≃3.86 × 108 cm. By considering the equilibrium state of the configuration, one arrives at a differential equa-tion whose solution gives directly the electric potential φ(r) and the electron density n(r) at point r. By the very nature of the model, which is generally referred to as the Thomas– Fermi model of the atom, the resulting function n(r) is a smoothly varying function of r, devoid of the “peaks” that would remind one of the electron orbits of the Bohr theory. Nevertheless, the model has proved quite useful in deriving composite properties such as the binding energy of the atom. And, after suitable modifications, it has been success-fully applied to molecules, solids, and nuclei as well. Here, we propose to outline only the simplest treatment of the model, as applied to an atomic system; for further details and other applications, see Gomb´ as (1949, 1952) and March (1957), where references to other contributions to the subject can also be found. According to the statistics of a completely degenerate Fermi gas, we have exactly two electrons (with opposite spins) in each elementary cell of the phase space, with p ≤pF; the Fermi momentum pF of the electron gas is determined by the electron density n, according to the formula pF = (3π2n)1/3ℏ. (1) In the system under study, the electron density varies from point to point; so would the value of pF. We must, therefore, speak of the limiting momentum pF as a function of r, which is clearly a “quasiclassical” description of the situation. Such a description is jus-tifiable if the de Broglie wavelength of the electrons in a given region of space is much 266 Chapter 8. Ideal Fermi Systems smaller than the distance over which the functions pF(r), n(r), and φ(r) undergo a signifi-cant variation; later on, it will be seen that this requirement is satisfied reasonably well by the heavier atoms. Now, the total energy ε of an electron at the top of the Fermi sea at the point r is given by ε(r) = 1 2mp2 F(r) −eφ(r), (2) where e denotes the magnitude of the electronic charge. When the system is in a stationary state, the value of ε(r) should be the same throughout, so that electrons anywhere in the system do not have an overall tendency to “flow away” toward other parts of the system. Now, at the boundary of the system, pF must be zero; by a suitable choice of the zero of energy, we can also have φ = 0 there. Thus, the value of ε at the boundary of the system is zero; so must, then, be the value of ε throughout the system. We thus have, for all r, 1 2mp2 F(r) −eφ(r) = 0. (3) Substituting from (1) and making use of the Poisson equation, ∇2φ(r) = −4πρ(r) = 4πen(r), (4) we obtain ∇2φ(r) = 4e(2me)3/2 3πℏ3 {φ(r)}3/2. (5) Assuming spherical symmetry, equation (5) takes the form 1 r2 d dr  r2 d dr φ(r)  = 4e(2me)3/2 3πℏ3 {φ(r)}3/2, (6) which is known as the Thomas–Fermi equation of the system. Introducing dimensionless variables x and 8, defined by x = 2  4 3π 2/3 Z1/3 me2 ℏ2 r = Z1/3 0.88534aB r (7) and 8(x) = φ(r) Ze/r , (8) where Z is the atomic number of the system and aB the first Bohr radius of the hydrogen atom, equation (6) reduces to d28 dx2 = 83/2 x1/2 . (9) 8.6 Statistical model of the atom 267 Equation (9) is the dimensionless Thomas–Fermi equation of the system. The boundary conditions on the solution to this equation can be obtained as follows. As we approach the nucleus of the system (r →0), the potential φ(r) approaches the unscreened value Ze/r; accordingly, we must have: 8(x →0) = 1. On the other hand, as we approach the bound-ary of the system (r →r0), φ(r) in the case of a neutral atom must tend to zero; accordingly, we must have: 8(x →x0) = 0. In principle, these two conditions are sufficient to determine the function 8(x) completely. However, it would be helpful if one knew the initial slope of the function as well, which in turn would depend on the precise location of the boundary. Choosing the boundary to be at infinity (r0 = ∞), the appropriate initial slope of the func-tion 8(x) turns out to be very nearly −1.5886; in fact, the nature of the solution near the origin is 8(x) = 1 −1.5886x + 4 3x3/2 + ··· (10) For x > 10, the approximate solution has been determined by Sommerfeld (1932): 8(x) ≈   1 + x3 144 !λ   −1/λ , (11) where λ = √(73) −7 6 ≃0.257. (12) As x →∞, the solution tends to the simple form: 8(x) ≈144/x3. The complete solution, which is a monotonically decreasing function of x, has been tabulated by Bush and Cald-well (1931). As a check on the numerical results, we note that the solution must satisfy the integral condition ∞ Z 0 83/2x1/2dx = 1, (13) which expresses the fact that the integral of the electron density n(r) over the whole of the space available to the system must be equal to Z, the total number of electrons present. From the function 8(x), one readily obtains the electric potential φ(r) and the electron density n(r): φ(r) = Ze r 8 rZ1/3 0.88534aB ! ∝Z4/3 (14) and n(r) = (2me)3/2 3π2ℏ3 {φ(r)}3/2 ∝Z2. (15) 268 Chapter 8. Ideal Fermi Systems 140 120 100 80 60 40 20 0.2 0.4 0.6 0.8 r /a D(r) FIGURE 8.12 The electron distribution function D(r) for an atom of mercury. The distance r is expressed in terms of the atomic unit of length a(= ℏ2/me2). A Thomas–Fermi plot of the electron distribution function D(r){= n(r) · 4πr2} for an atom of mercury is shown in Figure 8.12; the actual “peaked” distribution, which conveys unmis-takably the preference of the electrons to be in the vicinity of their semiclassical orbits, is also shown in the figure. To calculate the binding energy of the atom, we should determine the total energy of the electron cloud. Now, the mean kinetic energy of an electron at the point r would be 3 5εF(r); by equation (3), this is equal to 3 5eφ(r). The total kinetic energy of the electron cloud is, therefore, given by 3 5e ∞ Z 0 φ(r)n(r) · 4πr2dr. (16) For the potential energy of the cloud, we note that a part of the potential φ(r) at the point r is due to the nucleus of the atom while the rest of it is due to the electron cloud itself; the former is clearly (Ze/r), so the latter must be {φ(r) −Ze/r}. The total potential energy of the cloud is, therefore, given by −e ∞ Z 0 Ze r + 1 2  φ(r) −Ze r  n(r) · 4πr2dr. (17) We thus obtain for the total energy of the cloud E0 = ∞ Z 0 ( 1 10eφ(r) −1 2 Ze2 r ) n(r) · 4πr2dr; (18) Problems 269 of course, the electron density n(r), in terms of the potential function φ(r), is given by equation (15). Now, Milne (1927) has shown that the integrals ∞ Z 0 {φ(r)}5/2r2dr and ∞ Z 0 {φ(r)}3/2rdr, (19) which appear in the expression for E0, can be expressed directly in terms of the initial slope of the function 8(x), that is, in terms of the number −1.5886 of equation (10). After a little calculus, one finds that E0 = 1.5886 0.88534 e2 2aB ! Z7/3 1 7 −1  , (20) from which one obtains for the (Thomas–Fermi) binding energy of the atom: EB = −E0 = 1.538Z7/3χ, (21) where χ(= e2/2aB ≃13.6eV) is the (actual) binding energy of the hydrogen atom. It is clear that our statistical result (21) cannot give us anything more than just the first term of an “asymptotic expansion” of the binding energy EB in powers of the parameter Z−1/3. For practical values of Z, other terms of the expansion are also important; however, they cannot be obtained from the simple-minded treatment given here. The interested reader may refer to the review article by March (1957). In the end we observe that, since the total energy of the electron cloud is proportional to Z7/3, the mean energy per electron would be proportional to Z4/3; accordingly, the mean de Broglie wavelength of the electrons in the cloud would be proportional to Z−2/3. At the same time, the overall linear dimensions of the cloud are proportional to Z−1/3; see equation (7). We thus find that the quasiclassical description adopted in the Thomas– Fermi model is more appropriate for heavier atoms (so that Z−2/3 ≪Z−1/3). Otherwise, too, the statistical nature of the approach demands that the number of particles in the system be large. Problems 8.1 Let the Fermi distribution at low temperatures be represented by a broken line, as shown in Figure 8.13, the line being tangential to the actual curve at ε = µ. Show that this approximate representation yields a “correct” result for the low-temperature specific heat of the Fermi gas, except that the numerical factor turns out to be smaller by a factor of 4/π2. Discuss, in a qualitative manner, the origin of this numerical discrepancy. 8.2 For a Fermi–Dirac gas, we may define a temperature T0 at which the chemical potential of the gas is zero (z = 1). Express T0 in terms of the Fermi temperature TF of the gas. [Hint: Use equation (E.16).] 270 Chapter 8. Ideal Fermi Systems 1.0 0.5 0 0 24 22 12 14 x (ex211)21 FIGURE 8.13 An approximate representation of the Fermi distribution at low temperatures: here, x = ε/kT and ξ = µ/kT. 8.3 Show that for an ideal Fermi gas 1 z  ∂z ∂T  P = −5 2T f5/2(z) f3/2(z); compare with equation (8.1.9). Hence show that γ ≡CP CV = (∂z/∂T)P (∂z/∂T)v = 5 3 f5/2(z)f1/2(z) {f3/2(z)}2 . Check that at low temperatures γ ≃1 + π2 3 kT εF 2 . 8.4 (a) Show that the isothermal compressibility κT and the adiabatic compressibility κS of an ideal Fermi gas are given by κT = 1 nkT f1/2(z) f3/2(z), κS = 3 5nkT f3/2(z) f5/2(z), where n(= N/V) is the particle density in the gas. Check that at low temperatures κT ≃ 3 2nεF " 1 −π2 12 kT εF 2# , κS ≃ 3 2nεF " 1 −5π2 12 kT εF 2# . (b) Making use of the thermodynamic relation CP −CV = T  ∂P ∂T  V ∂V ∂T  P = TVκT  ∂P ∂T 2 V , show that CP −CV CV = 4 9 CV Nk f1/2(z) f3/2(z) ≃π2 3 kT εF 2 (kT ≪εF). Problems 271 (c) Finally, making use of the thermodynamic relation γ = κT/κS, verify the results of Problem 8.3. 8.5 Evaluate (∂2P/∂T2)v, (∂2µ/∂T2)v, and (∂2µ/∂T2)P of an ideal Fermi gas and check that your results satisfy the thermodynamic relations CV = VT ∂2P ∂T2 ! v −NT ∂2µ ∂T2 ! v and CP = −NT ∂2µ ∂T2 ! P . Examine the low-temperature behavior of these quantities. 8.6 Show that the velocity of sound w in an ideal Fermi gas is given by w2 = 5kT 3m f5/2(z) f3/2(z) = 5 9⟨u2⟩, where ⟨u2⟩is the mean square speed of the particles in the gas. Evaluate w in the limit z →∞and compare it with the Fermi velocity uF. 8.7 Show that for an ideal Fermi gas ⟨u⟩ 1 u = 4 π f1(z)f2(z) {f3/2(z)}2 , u being the speed of a particle. Further show that at low temperatures ⟨u⟩ 1 u ≃9 8 " 1 + π2 12 kT εF 2# ; compare with Problem 6.6. 8.8 Obtain numerical estimates of the Fermi energy (in eV) and the Fermi temperature (in K) for the following systems: (a) conduction electrons in silver, lead, and aluminum; (b) nucleons in a heavy nucleus, such as 80Hg200, and (c) He3 atoms in liquid helium-3 (atomic volume: 63 ˚ A3 per atom). 8.9 Making use of another term of the Sommerfeld lemma (E.17), show that in the second approximation the chemical potential of a Fermi gas at low temperatures is given by µ ≃εF " 1 −π2 12 kT εF 2 −π4 80 kT εF 4# , (8.1.35a) and the mean energy per particle by U N ≃3 5εF " 1 + 5π2 12 kT εF 2 −π4 16 kT εF 4# . (8.1.37a) Hence determine the T3-correction to the customary T1-result for the specific heat of an electron gas. Compare the magnitude of the T3-term, in a typical metal such as copper, with the low-temperature specific heat arising from the Debye modes of the lattice. For further terms of these expansions, see Kiess (1987). 272 Chapter 8. Ideal Fermi Systems 8.10 Consider an ideal Fermi gas, with energy spectrum ε ∝ps, contained in a box of “volume” V in a space of n dimensions. Show that, for this system, (a) PV = s nU; (b) CV Nk = n s n s + 1  f(n/s)+1(z) fn/s(z) − n s 2 fn/s(z) f(n/s)−1(z); (c) CP −CV Nk =  sCV nNk 2 f(n/s)−1(z) f(n/s)(z) ; (d) the equation of an adiabat is PV 1+(s/n) = const., and (e) the index (1 + (s/n)) in the foregoing equation agrees with the ratio (CP/CV ) of the gas only when T ≫TF. On the other hand, when T ≪TF, the ratio (CP/CV ) ≃1 + (π2/3)(kT/εF)2, irrespective of the values of s and n. 8.11 Examine results (b) and (c) of the preceding problem in the high-temperature limit (T ≫TF) as well as in the low-temperature limit (T ≪TF), and compare the resulting expressions with the ones pertaining to a nonrelativistic gas and an extreme relativistic gas in three dimensions. 8.12 Show that, in two dimensions, the specific heat CV (N,T) of an ideal Fermi gas is identical to the specific heat of an ideal Bose gas, for all N and T. [Hint: It will suffice to show that, for given N and T, the thermal energies of the two systems differ at most by a constant. For this, first show that the fugacities, zF and zB, of the two systems are mutually related: (1 + zF)(1 −zB) = 1, i.e., zB = zF/(1 + zF). Next, show that the functions f2(zF) and g2(zB) are also related: f2(zF) = zF Z 0 ln(1 + z) z dz = g2  zF 1 + zF  + 1 2 ln2(1 + zF). It is now straightforward to show that EF(N,T) = EB(N,T) + const., the constant being EF(N,0).] 8.13 Show that, quite generally, the low-temperature behavior of the chemical potential, the specific heat, and the entropy of an ideal Fermi gas is given by µ ≃εF " 1 −π2 6 ∂lna(ε) ∂lnε  ε=εF kT εF 2# , and CV ≃S ≃π2 3 k2T a(εF), where a(ε) is the density of (the single-particle) states in the system. Examine these results for a gas with energy spectrum ε ∝ps, confined to a space of n dimensions, and discuss the special cases: s = 1 and 2, with n = 2 and 3. [Hint: Use equation (E.18) from Appendix E.] 8.14 Investigate the Pauli paramagnetism of an ideal gas of fermions with intrinsic magnetic moment µ∗ and spin Jℏ(J = 1 2, 3 2,...), and derive expressions for the low-temperature and high-temperature susceptibilities of the gas. Problems 273 8.15 Show that expression (8.2.20) for the paramagnetic susceptibility of an ideal Fermi gas can be written in the form χ = nµ∗2 kT f1/2(z) f3/2(z). Using this result, verify equations (8.2.24) and (8.2.27). 8.16 The observed value of γ , see equation (8.3.6), for sodium is 4.3 × 10−4 cal mole−1K−2. Evaluate the Fermi energy εF and the number density n of the conduction electrons in the sodium metal. Compare the latter result with the number density of atoms (given that, for sodium, ρ = 0.954gcm−3 and M = 23). 8.17 Calculate the fraction of the conduction electrons in tungsten (εF = 9.0eV) at 3000 K whose kinetic energy ε(= 1 2mu2) is greater than W (= 13.5eV). Also calculate the fraction of the electrons whose kinetic energy associated with the z-component of their motion, namely ( 1 2mu2 z), is greater than 13.5eV. 8.18 Show that the ground-state energy E0 of a relativistic gas of electrons is given by E0 = πVm4c5 3h3 B(x), where B(x) = 8x3{(x2 + 1)1/2 −1} −A(x), A(x) and x being given by equations (8.5.13) and (8.5.14). Check that the foregoing result for E0 and equation (8.5.12) for P0 satisfy the thermodynamic relations E0 + P0V = Nµ0 and P0 = −(∂E0/∂V)N. 8.19 Show that the low-temperature specific heat of the relativisitic Fermi gas, studied in Section 8.5, is given by CV Nk = π2 (x2 + 1)1/2 x2 kT mc2  x = pF mc  . Check that this formula gives correct results for the nonrelativistic case as well as for the extreme relativistic one. 8.20 Express the integrals (8.6.19) in terms of the initial slope of the function 8(x), and verify equation (8.6.20). 8.21 The total energy E of the electron cloud in an atom can be written as E = K + Vne + Vee, where K is the kinetic energy of the electrons, Vne the interaction energy between the electrons and the nucleus, and Vee the mutual interaction energy of the electrons. Show that, according to the Thomas–Fermi model of a neutral atom, K = −E, Vne = +7 3E, and Vee = −1 3E, so that total V = Vne + Vee = 2E. Note that these results are consistent with the virial theorem; see Problem 3.20, with n = −1. 8.22 Derive equations (8.4.3) through (8.4.5) for a Fermi gas in a harmonic trap. Evaluate equations (8.3.4) and (8.3.5) numerically to reproduce the theoretical curves shown in Figures 8.9 and 8.10. 9 Thermodynamics of the Early Universe Over the course of the twentieth century, astronomers and astrophysicists gathered a vast body of evidence that indicates the universe began abruptly 13.75 ± 0.11 billion years ago in what became known as the “Big Bang.”1,2 The intense study of the origin and evolution of the universe has led to a convergence of physics and astrophysics. Thermodynamics and statistical mechanics play a crucial role in our understanding of the sequence of transi-tions that the universe went though shortly after the Big Bang. These transitions left behind mileposts that astrophysicists have exploited to look back into the earliest moments of the universe. The early universe provides particularly good examples for utilizing the proper-ties of ideal classical, Bose, and Fermi gases developed in Chapters 6, 7, and 8, and the theory of chemical equilibrium developed in Section 6.6. 9.1 Observational evidence of the Big Bang Observational evidence of the Big Bang has grown steadily since Edwin Hubble’s discovery in the late 1920s that the universe was expanding. Since that time a coherent standard model for the beginning of the universe has emerged. The following three items describe the key bodies of evidence. 1. Nearly every galaxy in the universe is moving away from every other galaxy and the recessional velocities display an almost linear dependence on the distance between galaxies; see Figure 9.1. Hubble was the first to observe this by measuring both the distances to nearby galaxies and their velocities relative to our own galaxy. The former is based on standard candles, in Hubble’s case Cepheid variable stars with known absolute mean luminosity. The latter is based on measurements of the Doppler red shift of spectral lines. Type Ia supernovae are used as the standard candle in the most distant observations made using the Hubble Space Telescope. The data are 1For excellent overviews and history of the study of the Big Bang, see The First Three Minutes: A Modern View of the Origin of the Universe by Weinberg (1993) and The Big Bang by Singh (2005). Cosmology by Weinberg (2008) provides an excellent technical survey. The organization of this chapter is based on Weinberg (1993). The 2010 decadal survey of astrophysics New Worlds, New Horizons in Astronomy and Astrophysics by the National Academies Press provides an overview of the current state of the field; see www.nap.edu. 2Steady state cosmology advocate Fred Hoyle coined the term “Big Bang” derisively in a BBC radio broadcast in 1950. To his eternal dismay, the name quickly became popular. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00009-8 © 2011 Elsevier Ltd. All rights reserved. 275 276 Chapter 9. Thermodynamics of the Early Universe 3104 H072 H0(km/s/Mpc) Velocity (km/s) v 5,000 km/s 2104 79 I-band Tully–Fisher Fundamental plane Surface brightness Supernovae Ia Supernovae II 72 65 104 0 100 80 60 40 0 100 200 Distance (Mpc) 300 400 FIGURE 9.1 Hubble diagram of the spectral red-shift velocity of relatively nearby galaxies versus their distance using several astronomical standard candles. The velocity is in km/s and the distance is measured in megaparsecs, where 1 Mpc = 3.26 × 106 light-years. The best fit to the data gives a value of the Hubble parameter as H0 = 72 ± 8kms−1Mpc−1. The Hubble parameter has been recently updated by Riess et al. (2009) to give H0 = 74.2 ± 3.6kms−1 Mpc−1. The figure is from Freedman et al. (2001) and is reproduced by permission of the AAS. encapsulated in the Hubble–Friedmann relation (Friedmann, 1922, 1924), v = da dt = Ha = r 8πGu 3c2 a. (1) Here a represents the distance between any two points in space that grows with time as the universe expands, v is the recessional velocity, G is the universal constant of gravitation, c is the speed of light, and u is the energy density of the universe.3 The Hubble parameter, H, is the characteristic expansion rate and is of the order of the inverse of the age of the universe. The particular form of equation (1) assumes, as appears to be the case, that the energy density u is equal to the critical value so that the 3Cosmological and general relativistic calculations are usually expressed in terms of the equivalent mass density ρ = u/c2 = T00 where T is the energy-momentum tensor. Astrophysicists usually describe the length scale parameter a in terms of the Doppler shift factor z = (λ/λlab −1), where λlab is the laboratory wavelength of a spectral line and λ is the red-shifted value. This gives z = (T/T0 −1) where T0 is the current cosmic microwave background temperature and T is the photon temperature of that era. For example, the Doppler shift from the era of last scattering is z = 3000K 2.725K −1 ≃1100, so the universe has expanded by a factor of 1100 since that time. 9.1 Observational evidence of the Big Bang 277 space-time is flat. This means that the universe is balanced on a knife edge between expanding forever and recollapsing due to gravity. For excellent technical surveys, see B¨ orner (2003) and Weinberg (2008). The measured value of the Hubble parameter is H0 = 74.2 ± 3.6kms−1 Mpc−1, (2) where Mpc is a megaparsec, about 3.26 × 106 light-years; see Freedman et al. (2001) and Riess et al. (2009). 2. Penzias and Wilson (1965) observed a nearly uniform and isotropically distributed microwave radiation noise coming from deep space with a blackbody temperature of about 3K. This cosmic microwave background (CMB) was quickly identified as the remnant blackbody radiation from the era following the Big Bang. Later, balloon experiments and space-based measurements by the Cosmic Background Explorer (COBE) NASA mission showed that the CMB is extremely uniform and isotropic with an average temperature of TCMB = 2.725 ± 0.002K; see Mather et al. (1994, 1999), Wright et al. (1994), Fixsen et al. (1996), and Figure 9.2. The NASA Wilkinson Microwave Anisotropy Probe (WMAP) mission mapped the angular variation of the CMB temperature. Figure 9.3 shows the ±200µK CMB temperature variations mapped onto galactic coordinates. The CMB represents the photons that were in thermal equilibrium with the high-temperature plasma that existed from the very first moments of the universe until it cooled down to approximately 3000K about 380,000 years after the Big Bang. 1.0 0.8 0.6 0.4 0.2 Intensity, 104ergs/cm2sr sec cm1 0.00 5 10 Wavenumber, cm1 15 20 1.2 FIGURE 9.2 Cosmic microwave background spectrum from COBE fit numerically to the Planck distribution with an average temperature T = 2.725 ± 0.002K; see equations (7.3.8) and (7.3.9), and Figure 7.13. The error bars at the 43 equally spaced frequencies from the Far Infrared Absolute Spectrophotometer (FIRAS) data are too small to be seen on this scale. Figure courtesy of NASA. 278 Chapter 9. Thermodynamics of the Early Universe FIGURE 9.3 Measurement of temperature variations in the CMB using 7 years of data from WMAP. This shows the distribution of the CMB blackbody temperature mapped onto galactic coordinates. The variations represent temperature fluctuations of ±200µK. Figure courtesy of NASA and the WMAP Science Team. As the temperature fell below 3000K, the electrons and protons in the plasma combined for the first time into neutral hydrogen atoms, a period that is known rather oxymoronically as the era of recombination. After this era of “last scattering” of photons by free electrons, the quantum structure of the atoms prevented them from absorbing radiation except at their narrow spectral frequencies, so the universe became transparent and the blackbody radiation quickly fell out of equilibrium with the neutral atoms. As the universe continued to expand, the wavelengths of the blackbody radiation grew linearly with the expansion scale of the universe a. The photon number density fell as a−3 and the energy density as a−4, so the Planck distribution, equations (7.3.8) and (7.3.9), was preserved with a blackbody temperature that scaled as T(t)a(t) = const. (3) Measurements of the Hubble parameter and the COBE and WMAP measurements of the temperature and temperature fluctuations of the CMB allow a determination of the current total energy density of the universe and its composition. The current energy density of the universe is u = 3c2H2 0 8πG = 8.36 × 10−10 Jm−3, (4) 9.1 Observational evidence of the Big Bang 279 and is comprised of approximately 72.8 percent dark energy, 22.7 percent dark matter, and 4.56 percent baryonic matter (protons and neutrons).4 This gives a baryon number density nB of 0.26m−3. The number density of photons in a blackbody enclosure as a function of temperature is given by equation (7.3.23): nγ (T) = 2ζ(3) π2 kT ℏc 3 . (5) At the current temperature of 2.725K, this gives a CMB photon number density of nγ = 4.10 × 108 m−3, so the current baryon-to-photon ratio is η = nB nγ ≈6 × 10−10. (6) The ratio η has remained constant as the universe has expanded since both these quantities scale as a−3(t). As we will see later, the numerical value of η played a very important role in the thermal evolution of the early universe.5 3. The relative abundances of the light elements 1H, 2H, 3He, 4He, 7Li, and so on created during the first few minutes of the universe are sensitive functions of the baryon-to-photon ratio η; see Figure 9.4. This connection was first explored by George Gamow, Ralph Alpher, and Robert Herman in the late 1940s and early 1950s; see Alpher, Bethe, and Gamow (1948), and Alpher and Herman (1948, 1950).6 4The energy content of the universe is parameterized in terms of the fraction of the critical density contained in the various constituents. The current values are dark energy: 3 = 0.728 ± 0.016; baryonic matter: b = 0.0456 ± 0.0016; and cold dark matter: c = 0.227 ± 0.014. The current age of the universe, or lookback time, is t0 = 13.75 ± 0.11 × 109 years. The relative contribution from blackbody radiation is about 6 × 10−5. These concordance values of the parameters are based on WMAP 7-year data and are tabulated in Komatsu et al. (2010). The dark energy is responsible for the accelerating expansion of the universe. The energy density proportions were vastly different in the early universe because they scale differently with the expansion parameter a. At the time of recombination, the proportions were: dark matter 63 percent, baryonic matter 12 percent, relativistic radiation (photons and neutrinos) 25 percent. During the first few moments, relativistic particles provided the dominant contribution to the energy. Using the photon:neutrino:electron ratios of 2 : 21/4 : 7/2 from Table 9.2, the energy content was photons 18.6 percent, neutrinos and antineutrinos 48.8 percent, and electrons and positrons 32.5 percent. While dark energy is currently the dominant contribution to the energy density of the universe, it played only a small role in the early evolution of the universe. Cold dark matter was crucial for the development of the first stars and galaxies at the end of the “dark ages” 100 to 200 million years after the last scattering. 5The proper measure here is the ratio of the baryon number density to photon entropy density but, since the CMB photon entropy density and number density both scale as T3, the ratio is usually quoted in terms of the ratio of the number densities. 6George Gamow and Ralph Alpher in 1948, and Alpher and Robert Herman in 1950, proposed a model for nucleo-synthesis in a hot, expanding primordial soup of protons, neutrons, and electrons. Alpher and Gamow called this material “ylem.” To account for the present abundance of 4He in the universe, Alpher and Herman (1950) proposed a baryon-to-photon ratio of roughly 10−9 and predicted a current cosmic microwave background temperature of about 5K. Gamow added his friend Hans Bethe’s name as second author to Alpher, Bethe, and Gamow (1948) as a pun on the Greek alphabet. The paper was published, perhaps not coincidentally, on April 1; see Alpher and Herman (2001), Weinberg (1993), and Singh (2005). 280 Chapter 9. Thermodynamics of the Early Universe 1 Abundances Critical density for H0= 65 km/s/Mpc 0.1 101 102 103 104 105 106 107 108 109 1010 1032 1031 1030 B (g cm3) 1029 0.01 0.001 Bh2 7Li 3He 4He D FIGURE 9.4 Calculated primordial abundances of light elements (4He, D=2H, 3He, and 7Li) as functions of the baryon-to-photon ratio. The baryon-to-photon ratio is given by η = 2.7 × 10−8bh2, where h is the Hubble parameter in units of 100(km/s)/Mpc and b = 0.046 is the current baryonic fraction of the mass-energy density of the universe; see Copi, Schramm, and Turner (1997), Schramm and Turner (1998), and Steigman (2006). The experimentally allowed range is in the grey vertical bands. Figure from Schramm and Turner (1998). Reprinted with permission; copyright © 1998, American Physical Society. 9.2 Evolution of the temperature of the universe As the universe expanded and cooled, the cooling rate was proportional to the Hubble parameter, that is, of the order of the inverse of the age of the universe at that point in its expansion. This led to a sequence of important events when different particles and interactions fell out of equilibrium with the gas of blackbody photons. The neutrinos and neutron-proton conversion reactions fell out of equilibrium at t ≈1second. Nuclear reac-tions that formed light nuclei fell out of equilibrium at t ≈3minutes. Neutral atoms fell out of equilibrium at t ≈380,000years. All these degrees of freedom froze out when the reaction rates that had kept them in equilibrium with the blackbody photons fell far below the cooling rate of the expanding universe. Each component that fell out of equilibrium left behind a marker of the properties of the universe characteristic of that era. It is these 9.2 Evolution of the temperature of the universe 281 markers that provide evidence of the properties and behavior of the universe during its earliest moments. From the first moments of the universe up until the recombination era 380,000 years later, the cosmic plasma was in thermal equilibrium with the blackbody radiation through Thomson scattering. Due to the high density of charged particles, the photon scattering mean free time was much shorter than the time scale for temperature changes of the uni-verse as it expanded and cooled, which kept the plasma in thermal equilibrium with the photons. For the first few hundred thousand years of its expansion, the energy density of the universe was dominated by photons and other relativistic particles. This is because the energy density of the blackbody radiation scales as a−4 whereas the energy density of non-relativistic matter scales as a−3. The temperature of the blackbody photons as a function of the age of the universe is shown in Figure 9.5 and Table 9.1. During the first one-hundredth of a second, the universe expanded and cooled from its singular beginning to a temperature of about 1011 K. The physics from this time onward was controlled by the weak and electromagnetic interactions. The strong interactions could be ignored since the baryon-to-photon ratio was so small and the temperature was 1010 108 T (K) 106 104 102 100 100 105 t (seconds) 1010 1015 FIGURE 9.5 Sketch of the photon temperature versus the age of the universe. At early times, radiation dominated the energy density, so T ∼t−1/2. At later times (t > 1013 s) nonrelativistic matter dominated the energy density, so T ∼t−2/3. In the current dark energy dominated stage, the universe is beginning to expand exponentially with time, so the photon temperature is beginning to fall exponentially. 282 Chapter 9. Thermodynamics of the Early Universe Table 9.1 Temperature vs. Age of the Universe Time (s) Temperature (K) 0.01 1 × 1011 0.1 3 × 1010 1.0 1 × 1010 12.7 3 × 109 168 1 × 109 1980 3 × 108 1.78 × 104 108 1.20 × 1013 3000 4.34 × 1017 2.725 Source: Weinberg (2008). too low to create additional hadrons.7,8 We will follow the thermodynamic behavior of the universe from t = 0.01 second when the temperature was 1011 K to t = 380,000 years when the temperature fell below 3000K. At that point neutral atoms formed, photon scattering ended, and the universe became transparent to radiation. After recombination and last scattering there were no new sources of radiation in the universe since the baryonic mat-ter consisted entirely of neutral atoms. This state of affairs lasted until atoms were first reionized by the gravitational clumping that formed the first stars and galaxies 100 to 200 million years after the Big Bang. This reionization epoch ended the so-called cosmic dark ages. 9.3 Relativistic electrons, positrons, and neutrinos During the earliest moments of the universe, the temperature was high enough to cre-ate several kinds of relativistic particles and antiparticles. If kT ≫mc2, then particle-antiparticle pairs each with mass m can be created from photon-photon interactions. At these temperatures, almost all of the particles that are created will have an energy-momentum relation described by the relativistic limit, namely εk ≈ℏck, (1) 7Before time t = 0.01s, the analysis is more difficult due to the production of strongly interacting particles and antiparticles. At even earlier times, when the temperature was above kT ≈300MeV (T = 4 × 1012 K), hadrons would have broken apart into a strongly interacting relativistic quark-gluon plasma. The Relativistic Heavy Ion Collider at Brookhaven National Laboratory has succeeded in creating a quark-gluon plasma with the highest temperature matter ever created in the laboratory, T = 4 × 1012 K; see Adare et al. (2010). 8The exact mechanism for baryogenesis (i.e., nonzero baryon-to-photon ratio η) is unsettled. It requires, as shown by Sakharov (1967), three things: baryon number nonconservation, C and CP violation, and deviation from equilibrium. All these conditions were satisfied in the earliest moments of the universe (far earlier than the time scales we examine here) but a consensus theory that allows for a baryon asymmetry nearly as large as the observed value of η = 6 × 10−10 has not yet emerged. 9.3 Relativistic electrons, positrons, and neutrinos 283 where ℏk is the magnitude of the momentum. This relation applies to photons, neutrinos, antineutrinos, electrons, and positrons. The threshold for electron-positron pair forma-tion is mec2/k = 5.9 × 109 K. Neutrinos are very light, so we can safely assume that they are relativistic.9 The relativistic dispersion relation 1 gives essentially the same density of states for all species of relativistic particles: a(ε) = gs (2π)3 Z δ(ε −εk)dk = 4πgs (2π)3 ∞ Z 0 k2δ(ε −ℏck)dk = gsε2 2π2(ℏc)3 , (2) where gs is the spin degeneracy. Photons have a spin degeneracy gs = 2 (left and right cir-cularly polarized). The other species are all spin- 1 2 fermions. Electrons and positrons have spin degeneracy gs = 2 while neutrinos and antineutrinos have spin degeneracy gs = 1 since all neutrinos have left-handed helicity. During this era, because of the charge neutrality of the universe and the small size of the baryon-to-photon ratio η, the number densities of the electrons and positrons were nearly equal, so their chemical potentials were both rather small. Assuming that the net lepton number of the universe is also small, the same applies to the neutrinos and antineutrinos. As explained in Section 7.3, the chemical potential for photons is exactly zero. The pres-sure, number density, energy density, and entropy density of a relativistic gas of fermions (+) or bosons (−) with zero chemical potential are are given by P(T) = ±kT Z a(ε)ln(1 ± e−βε)dε = gs kT 4 2π2 (ℏc)3 ∞ Z 0 x2 ln 1 ± e−x dx, (3a) n(T) = Z a(ε) 1 eβε ± 1dε = gs 2π2 kT ℏc 3 ∞ Z 0 x2 ex ± 1dx , (3b) u(T) = Z a(ε) ε eβε ± 1dε = gs kT 4 2π2 (ℏc)3 ∞ Z 0 x3 ex ± 1dx, (3c) s(T) =  ∂P ∂T  µ = 2gsk π2 kT ℏc 3 ∞ Z 0 x2 ln 1 ± e−x dx. (3d) 9All three neutrino families are known to have small (but nonzero) mass from neutrino oscillation observations. The electron neutrino is probably the lightest with the experimental limit of mνec2 < 2.2eV. The distribution of angular fluctuations of the CMB measured by WMAP puts a limit on the sum of the masses of the neutrinos, Pmνc2 < 0.58eV, so we can safely assume that all neutrino species are far lighter than the value of kT during the early universe. 284 Chapter 9. Thermodynamics of the Early Universe Using the values of the Bose integrals from Appendix D, we arrive at the following expressions for the blackbody photons: Pγ (T) = π2 45 (kT)4 (ℏc)3 , (4a) nγ (T) = 2ζ(3) π2 kT ℏc 3 , (4b) uγ (T) = π2 15 (kT)4 (ℏc)3 , (4c) sγ (T) = 4π2k 45 kT ℏc 3 . (4d) All relativistic species with µ = 0 have the same power law temperature dependences for the pressure, energy density, and so on, as the photons, while the Fermi and Bose integrals are the same except for a constant prefactor: ∞ Z 0 xn−1 ex + 1dx =  1 − 1 2n−1  ∞ Z 0 xn−1 ex −1dx; (5) see Appendices D and E. The contributions to the pressure, energy density, and entropy density result from counting the spin degeneracies, the number of particles and antiparti-cles, and accounting for the different Fermi/Bose factors (1 for bosons, 7/8 for fermions). The photons, three generations of neutrinos (electron, muon, and tau neutrinos and their antiparticles), and the electrons and positrons contribute to the total pressure, number density, energy density, and entropy density in the proportions shown in Table 9.2. The counting is presented here, as is usually done in the literature, relative to the contribu-tion per spin state of the photons. The contributions to the number densities are the same except that the Fermi/Bose factor is now 3/4. Table 9.2 Relativistic Contributions to Pressure, Energy Density, and Entropy Density Particles Fermi/Bose Factor Spin Degeneracy Number of Species 2Ptotal/Pγ γ 1 2 1 2 νe,νµ,ντ 7 8 1 3 21 8 ¯ νe,¯ νµ,¯ ντ 7 8 1 3 21 8 e− 7 8 2 1 7 4 e+ 7 8 2 1 7 4 9.4 Neutron fraction 285 The totals then are Ptotal(T) =  2 + 21 4 + 7 2  Pγ (T) 2 = 43 8  Pγ (T), (6a) utotal(T) =  2 + 21 4 + 7 2  uγ (T) 2 = 43 8  uγ (T), (6b) stotal(T) =  2 + 21 4 + 7 2  sγ (T) 2 = 43 8  sγ (T), (6c) ntotal(T) =  2 + 9 2 + 3  nγ (T) 2 = 19 4  nγ (T). (6d) The density of the universe was high enough in this era, so the weak and electromagnetic interaction rates kept all these species in thermal equilibrium with one other. Therefore, as the universe expanded adiabatically, the entropy in a comoving volume of linear size a remained constant as the volume expanded from some initial value a3 0 to a final volume a3 1: stotal(T0)a3 0 = stotal(T1)a3 1. (7) Since the entropy density is proportional to T3, the temperature and length scale at time t are related by T(t)a(t) = const. (8) This is the same relation that applies for a freely expanding photon gas, see equa-tion (9.1.3), but here it arises from an adiabatic equilibrium process. From equations (9.1.1) and (8), the temperature of the universe as a function of the age of the universe t during this era is T(t) = 1010 K r 0.992s t ; (9) see Problem 9.1. 9.4 Neutron fraction During the first second of the universe, when T > 1010 K, and before protons and neu-trons combined into nuclei, the weak interaction kept the free neutrons and protons in thermal “beta-equilibrium” with each other and with the photons, neutrinos, electrons, and positrons through the processes n + νe ⇄p + e−+ γ , (1a) n + e+ ⇄p + ¯ ν + γ , (1b) n ⇄p + e−+ ¯ ν + γ . (1c) 286 Chapter 9. Thermodynamics of the Early Universe We can treat this as a chemical equilibrium process, as described in Section 6.6. Since the chemical potentials of the photons, electrons, positrons, neutrinos, and antineutrinos are all zero, the neutron and proton chemical potentials must be equal at equilibrium: µn = µp. (2) At these temperatures (≈1011 K) and densities (≈1032 m−3), the protons and neutrons can be treated as a classical nonrelativistic ideal gas. Following equation (6.6.5), the spin- 1 2 proton and neutron chemical potentials are µp = mpc2 + kT ln  npλ3 p  −kT ln2, (3a) µn = mnc2 + kT ln  nnλ3 n  −kT ln2. (3b) where λ(= h/ √ 2πmkT) is the thermal deBroglie wavelength. The rest energy of the neutron is greater than the rest energy of the proton by mnc2 −mpc2 = 1ε = 1.293MeV. (4) Ignoring the small mass difference in the thermal deBroglie wavelength in equations (3a) and (3b) gives nn = npe−β1ε. (5) The baryon number density is the sum of the neutron and proton number densities nB = nn + np, (6) so the equilibrium neutron fraction is given by q = nn nB = 1 eβ1ε + 1. (7) The mass difference gives a crossover temperature Tnp = 1ε/k ≈1.50 × 1010 K, so the neutron fraction drops from 46 percent when T = 1011 K to 16 percent when T = 9 × 109 K at t1 ≈1 second. As the temperature fell below 1010 K (kT = 0.86MeV), the weak interaction rate began to fall far below the cooling rate of the universe, so the baryons quickly fell out of equilibrium with the neutrinos. From that time onward the neutrons began to beta-decay with their natural radioactive decay lifetime of τn = 886 seconds, so the neutron fraction 9.5 Annihilation of the positrons and electrons 287 fell exponentially: q ≈0.16exp −(t −t1) τn  for t > t1 = 1s. (8) By the time of nucleosynthesis, about 3.7 minutes later, the neutron fraction had dropped to q ≈0.12. At that point, the remaining neutrons bound with protons to form deuterons and other light nuclei. For a discussion of nucleosynthesis, see Section 9.7. 9.5 Annihilation of the positrons and electrons About one second after the Big Bang, the temperature approached the crossover tempera-ture Te for creating electron-positron pairs: kTe = mec2 = 0.511MeV, (1) with Te = 5.93 × 109 K. As the temperature fell below Te, the rate of creating e+e−pairs began to fall below the rate at which pairs annihilated. The full relativistic dispersion relation for electrons is εk = q (ℏck)2 + (mec2)2 , (2) which gives for density of states ae(ε) = 8π (2π)3 ∞ Z 0 k2δ(ε −εk)dk = ε p ε2 −(mec2)2 π2(ℏc)3 for ε ≥mec2. (3) Since the electrons and positrons were in equilibrium with the blackbody photons via the reaction e+ + e−⇄γ + γ , (4) the equilibrium equation (6.6.3) implied that the chemical potentials of the species were related by µ−+ µ+ = 2µγ = 0. (5) The ratio of the number density of the electrons to that of the photons then was n− nγ = 1 2ζ(3) ∞ Z βmec2 x p x2 −(βmec2)2 exe−βµ−+ 1 dx, (6) 288 Chapter 9. Thermodynamics of the Early Universe while the positron density ratio was n+ nγ = 1 2ζ(3) ∞ Z βmec2 x p x2 −(βmec2)2 exeβµ−+ 1 dx; (7) see equation (9.1.5). The electron and positron densities became unbalanced as the universe cooled. Eventually all the positrons got annihilated leaving behind the electrons that currently remain. Charge neutrality of the universe required the difference between the number density of electrons and the number density of positrons to be equal to the number density of protons, (1 −q)nB, where q is defined in Section 9.4; hence (n−−n+) nγ = sinh(βµ−) 2ζ(3) ∞ Z βmec2 x p x2 −(βmec2)2 cosh(x) + cosh(βµ−)dx = (1 −q)η. (8) We can use equation (8) to determine the electron chemical potential as a function of tem-perature numerically and then use that value in equations (6) and (7) to determine the electron and positron densities; see Figure 9.6. Initially, the electron and positron densities both decreased proportional to exp(−βmec2) as the temperature fell below the electron-positron pair threshold, but they Electrons Positrons 0 1015 mc2 1010 105 n/n 100 10 20 30 FIGURE 9.6 The ratio of the electron and positron densities to the photon density as a function of βmec2 during the e+e−annihilation for η = 6 × 10−10. This era began around temperature 1010 K (βmec2 = 1.7) at time t = 1 second and ended when the temperature was about 3 × 108 K (βmec2 = 20) at time t = 33 minutes when the electron number density leveled off at the proton number density. 9.6 Neutrino temperature 289 remained nearly equal to each other until T ≈mec2/kln 1/(1 −q)η ≈3 × 108 K. At that temperature, the electron density began to level off at the proton density while the positron density continued to fall. Using the baryon-to-photon ratio η = 6 × 10−10, we infer that during the first second of the universe that for every 1.7 billion positrons there must have been one extra elec-tron. It is these few extra electrons that will combine with nuclei during the recombination era composing all the atoms now present in the universe. All baryonic matter currently in the universe is the result of this initial asymmetry between matter and antimatter; see footnote 8. 9.6 Neutrino temperature For temperatures above T = 1010 K, the rates for the weak interaction reactions (9.4.1) kept the neutrinos in “beta-equilibrium” with the electrons, positrons, and photons. Starting at time t ≈1s, when T = 1010 K, the weak interaction rates began to fall far below the expan-sion rate of the universe so the neutrinos quickly fell out of equilibrium. Following the decoupling, the neutrinos expanded freely so the neutrino temperature scaled with the expansion length scale following equation (9.1.3). The system of electrons, positrons, and photons remained in thermal equilibrium with each other and expanded adiabatically during the electron-positron annihilation era from temperature T0 = 1010 K when the annihilations began, until temperature T1 = 3 × 108 K when nearly all of the positrons had been annihilated. Since this was an adiabatic expan-sion, we can determine the temperature evolution using entropy conservation. Consider a comoving cubical volume that expanded from an initial linear size a0 to a final size a1 during the same time period. The total entropy in the comoving volume at temperature T0 was due to the photons, electrons, and positrons (refer to Table 9.2): S(T0) = 11 4 sγ (T0)a3 0, (1) while the entropy at temperature T1 was due solely to the photons since, by then, nearly all of the electrons and positrons had been annihilated: S(T1) = sγ (T1)a3 1. (2) Entropy conservation during the adiabatic expansion relates the initial and final tem-peratures as 11 4 1/3 T0a0 = T1a1. (3) In essence, the entropy of the annihilating electrons and positrons was transferred to the photons. Since the neutrino and photon temperatures were equal before the electron-positron annihilation and the neutrinos expanded freely during the annihilation, the 290 Chapter 9. Thermodynamics of the Early Universe neutrino temperature decreased more than the photon temperature during the annihi-lation era: Tν1 = (4/11)1/3 T1. (4) After the e+e−annihilation, both the neutrino and the photon temperatures evolved according to equation (9.1.3) and (9.3.8), so the current temperature of the relic Big Bang neutrinos should be Tν = (4/11)1/3 TCMB ≃1.945K. (5) A measurement of the cosmic neutrino background would provide an excellent additional test of the standard model of the Big Bang but we do not currently have a viable means to measure these very low-energy neutrinos.10 9.7 Primordial nucleosynthesis Light nuclei other than hydrogen first formed between 3 and 4 minutes after the Big Bang when the temperature had cooled to about 109 K. Prior to that time, the high-temperature blackbody radiation rapidly photodissociated any deuterium nuclei that happened to form. The first step for the formation of light nuclei from the protons and neutrons is the formation of deuterium because all of the rates for forming nuclei at these densities are dominated by two-body collisions. Once deuterons formed, most of these nuclei would have been quickly converted to helium and other more stable light nuclei in a series of two-body collisions with the remaining protons, neutrons, and with each other. As dis-cussed in Section 9.4, the proton/neutron mixture at this time was about q = 12 percent neutrons and 1 −q = 88 percent protons. By t ≈3minutes the temperature had fallen to T ≈109 K so protons and neutrons could begin to bind themselves into deuterons via the process p + n ⇄d + γ . (1) The chemical equilibrium relation for this reaction, see equation (6.6.3), is µp + µn = µd, (2) since the chemical potential of the blackbody photons is zero. At these temperatures and densities the protons, neutrons, and deuterons can be treated as classical ideal gases. 10The neutrino elastic scattering cross-section scales like the fourth power of the energy, so the collisions are both very rare and involve very small energy and momentum transfers. This makes direct laboratory detection of the cosmic neutrino background (CνB) infeasible at present; see Gelmini (2005). 9.7 Primordial nucleosynthesis 291 The proton and neutron are spin- 1 2 particles so they have two spin states each while the deuteron is spin-1 and has three spin states: µp = mpc2 + kT ln  npλ3 p  −kT ln2, (3a) µn = mnc2 + kT ln  nnλ3 n  −kT ln2, (3b) µd = mdc2 + kT ln  ndλ3 d  −kT ln3. (3c) The binding energy of the deuteron is εb = mpc2 + mnc2 −mdc2 = 2.20MeV. Since the deuteron is approximately twice as massive as protons or neutrons, the deuteron number density is given by nd = 3 4npnn λ3 pλ3 n λ3 d eβεb ≈3 √ 2 npnnλ3 peβεb. (4) The total number density of baryons is determined by the baryon-to-photon ratio: η: nB = ηnγ = np + nn + 2nd. The neutron number density is qnB = nn + nd, so the deuteron fraction is given by fd = nd nB = (1 −q −fd)(q −fd)s, (5) where the parameter s is s = 12ζ(3) √π  kT mpc2 3/2 ηeβεb; (6) see also equation (9.1.5). Equation (5) is similar to the Saha equation for the ionization of hydrogen atoms that will be discussed in Section 9.8 and has solution fd = 1 + s − p (1 + s)2 −4s2q(1 −q) 2s . (7) For high temperatures, s is small and fd ≈q(1 −q)s while for low temperatures, s is large and fd ≈q, that is, all the neutrons are bound into deuterons. The deuterium fraction as a function of temperature is shown in Figure 9.7. The small values of the baryon-to-photon ratio η and εb/mpc2 delayed the nucleosynthesis until the temperature had fallen to kTn ≈ εb ln 1 η  mpc2 εb 3/2! , (8) providing the time for the neutron fraction to have decayed to q = 0.12. 292 Chapter 9. Thermodynamics of the Early Universe 0.12 0.10 0.08 0.06 0.04 0.02 fd 0.0 0.5 1.0 T (109 kelvin) 1.5 2.0 0.00 FIGURE 9.7 Plot of the equilibrium deuterium fraction fd versus temperature T for neutron fraction q = 0.12 and baryon-to-photon ratio η = 6 × 10−10. As T falls below about 6 × 108 K the neutrons are nearly all bound into deuterons. Further two-body reactions convert most of the deuterium into heavier nuclei, primarily 4He. The simple equilibrium calculation presented here assumes that no further reactions take place. Including the fast nonequilibrium two-body reactions, namely d + d →3H + p + γ , (9a) d + d →3He + n + γ , (9b) d + 3H →4He + n + γ , (9c) d + 3He →4He + p + γ , (9d) results in almost all of the deuterons being cooked into the very stable isotope 4He and small amounts of other light nuclei. Since each 4He nucleus is composed of two protons and two neutrons, this gives a helium mass fraction of 2q = 24 percent and proton mass fraction of 1 −2q = 76 percent. The complete calculation involves nonequilibrium effects modeled with rate equations for each of the nuclear interactions, including those for heav-ier isotopes, but that only changes the predicted concentration for 4He slightly;11 see Weinberg (2008). The largest theoretical uncertainty is, remarkably, the uncertainty in the radioactive decay time of the neutron in equation (9.4.8); see Copi, Schramm, and Turner (1997). Calculations of this type were first performed by Gamow, Alpher, and Herman in the late 1940s and early 1950s. Based on current amounts of helium and other light elements 11Nuclear reactions continued slowly at a rate that had fallen out of equilibrium and shifted the isotopic ratios until about t ≈10 minutes. 9.8 Recombination 293 in the universe, Alpher and Herman predicted a 5K cosmic microwave background over a decade before Penzias and Wilson’s discovery; see footnote 6. 9.8 Recombination After the nucleosynthesis took place in the first few minutes, the universe continued to cool, with the nuclei and electrons remaining as an ordinary plasma in thermal equilib-rium with the photons. It took several hundred thousand years for the temperature to drop below the atomic ionization energies of a few electron volts needed for nuclei to capture electrons and form atoms. Hydrogen was the last neutral species to form since it has the smallest ionization energy of a Rydberg (1Ry = mee4/8ϵ2 0h2 = 13.6057eV). At first glance, one would think that atoms form when the temperature falls below Ry/k = 158,000K but, as we will see, the huge number of photons per proton delayed recombination until T ≈3000K. Once all the electrons and protons formed into neutral hydrogen atoms, the universe became transparent due to the last scattering of radiation from free electrons. These CMB blackbody photons were suddenly free to propagate and hence have been traveling unscattered since that time. The recombination reaction (that is, the inverse of the hydrogen photoionization reaction) is p + e ⇄H + γ , (1) so the chemical equilibrium relation from Section 6.6 gives µp + µe = µH (2) since, again, the chemical potential of the blackbody photons is zero.12 At the temperatures and densities prevailing during this era (a few thousand degrees Kelvin and only about 109 atoms per cubic meter), the electrons, protons, and hydrogen atoms can all be treated as classical ideal gases, with the result µp = mpc2 + kT ln(npλ3 p) −kT ln2, (3a) µe = mec2 + kT ln(neλ3 e) −kT ln2, (3b) µH = mHc2 + kT ln(nHλ3 H) −kT ln4. (3c) The binding energy of hydrogen is mpc2 + mec2 −mHc2 = 1Ry. The equilibrium condition (6.6.3) and the ideal gas chemical potential (6.6.5) then give a simple relation between the number densities of the three species: nH = npneλ3 eeβRy, (4) 12The same reaction occured for the deuterons that remained after nucleosynthesis at t ≈3minutes but the density of the deuterons was 3 × 10−5 times the proton density; refer to Figure 9.4. 294 Chapter 9. Thermodynamics of the Early Universe where λe = h p 2πmekT (5) is the electron thermal deBroglie wavelength. The number densities of free electrons and protons are the same due to charge neutrality: ne = np. (6) The protons remaining after nucleosynthesis are either free or combined into hydrogen atoms, so np + nH = (1 −2q)nB = (1 −2q)ηnγ . (7) Putting equations (3), (4), (6), and (7) together and making use of (9.1.5) gives the Saha equation for the neutral hydrogen fraction: fH = nH np + nH = (1 −fH)2s, (8) where the parameter s is s = 4ζ(3) r 2 π (1 −2q)η  kT mec2 3/2 eβRy. (9) The solution to equation (8), namely fH = 1 + 2s − √ 1 + 4s 2s , (10) is shown in Figure 9.8. At temperatures above the recombination temperature, s is small so fH is small, making the plasma fully ionized. At low temperatures s is large so fH approaches unity, leaving just neutral atoms. The small values of the baryon-to-photon ratio η and Ry/mec2 make the onset of recombination at temperature kTr ≈ Ry ln  1 η  mec2 Ry 3/2 , (11) which delays the last scattering until T ≈3000K; see Figure 9.8. 9.9 Epilogue 295 2000 0.0 0.2 0.4 0.6 0.8 1.0 3000 4000 T (kelvin) 5000 fH FIGURE 9.8 The equilibrium neutral hydrogen fraction as a function of temperature for baryon-to-phonon ratio η = 6 × 10−10 and proton fraction 1 −2q = 0.76. By the time temperature T = 3000K, 99.5 percent of the free protons and electrons had combined into neutral hydrogen resulting in “last scattering” and the universe became transparent. The age of the universe at that time was about 380,000 years. 9.9 Epilogue The formation of neutral atoms about 380,000 years after the Big Bang effectively ended the scattering of photons from free charges. The universe became transparent and entered the “dark ages” before the first star formation. The CMB photons were no longer in equilibrium but maintained their Planck distribution as the universe expanded. Small density fluctuations that were present in the electron–proton plasma just before recom-bination were imprinted on the CMB as temperature fluctuations. The CMB shown in Figure 9.3 earlier displays temperature fluctuations of the order of ±200µK that represent the density fluctuations in the plasma at the time of recombination. These small mass density fluctuations led to gravitational clumping that resulted in the formation of the first stars and galaxies 100 to 200 million years after the Big Bang. The large fraction of nonbary-onic cold dark matter was crucial in this process. Early stars that exploded as supernovae spewed their heavy elements (carbon, oxygen, silicon, iron, gold, uranium, etc.) into the cosmos. Our own solar system formed from a gas and dust cloud that included heavy elements that had been created in an earlier supernova event. Indeed, “we are stardust.”13 13Joni Mitchell, Woodstock; copyright © Siquomb Publishing Company: “We are stardust Billion year old carbon We are golden Caught in the devil’s bargain And we’ve got to get ourselves back to the garden” 296 Chapter 9. Thermodynamics of the Early Universe Problems 9.1. Use the Hubble expansion relation (9.1.1), the temperature scaling relation (9.1.3), and the energy density relation before the electron-positron annihilation (9.3.6b) to show that the temperature as a function of time during the first second of the universe was T(t) ≈1010 K q 0.992s t . 9.2. Determine the average energy per particle and average entropy per particle for the photons, electrons, positrons and neutrinos during the first second of the universe. 9.3. While the electromagnetic interaction between the photons and the charged electrons and positrons kept them in equilibrium with each other during the early universe, show that the direct electromagnetic Coulomb interaction energy between the electrons and positrons was small compared to the relativistic kinetic energy of those species. Show that the ratio between the Coulomb and kinetic energies is of the order of the fine structure constant: ucoulomb ue ≈α = e2 4πϵ0ℏc = 1 137.036. 9.4. Show that during the early part of the electron-positron annihilation era, the ratio of the electron number density to the photon number density scaled with temperature as n− nγ ≈n+ nγ ∼  kT mec2 3/2 exp  −βmec2 . 9.5. Show that after nearly all of the positrons were annihilated and the electron number density had nearly leveled off at the proton density, the ratio of the positron number density to the photon number density scaled with temperature as n+ nγ ∼  kT mec2 3/2 exp  −2βmec2 . 9.6. After the positrons were annihilated, the energy density of the universe was dominated by the photons and the neutrinos. Show that the energy density in that era was: utotal = (1 + (4/11)4/3)uγ . Next, use the Hubble expansion relation (9.1.1), the temperature scaling relation (9.1.3), and the energy density after the electron-positron annihilation to show that the photon temperature as a function of time was T(t) ≈1010 K q 1.788s t . This relation held from t ≈100s until t ≈200,000 years when the energy density due to baryonic and cold dark matter began to dominate. 9.7. How would the primordial helium content of the universe have been affected if the present cosmic background radiation temperature was 27K instead of 2.7K? What about 0.27K? 9.8. Gold-on-gold nuclear collisions at the Relativistic Heavy Ion Collider (RHIC) at the Brookhaven National Laboratory create a quark-gluon plasma with an energy density of about 4GeV/fm3; see Adare et al. (2010). Treat nuclear matter as composed of a noninteracting relativistic gas of quarks and gluons. Include the low-mass up and down quarks and their antiparticles (all spin- 1 2), and spin-1 massless gluons. Like photons, the gluons are bosons, have two spin states each, and are their own antiparticle. There are eight varieties of gluons that change the three color states of the quarks. Only the strongly interacting particles need to be considered due to the tiny size of the plasmas. What is the temperature of the quark-gluon plasma? 9.9. Calculate the energy density versus temperature very early in the universe when the tempera-tures were above kT = 300MeV. At those temperatures, quarks and gluons were released from individual nuclei. Treat the quark-gluon plasma as a noninteracting relativistic gas. At those Problems 297 temperatures, the species that are in equilibrium with one other are: photons, the three neutrino species, electrons and positrons, muons and antimuons, up and down quarks and their antiparticles (all spin- 1 2), and spin-1 massless gluons. Like photons, the gluons are bosons, have two spin states each, and are their own antiparticle. There are eight varieties of gluons that change the three color states of the quarks. The strange, charm, top, and bottom quarks and tau leptons are heavier than 300MeV, so they do not contribute substantially at this temperature. Use your result and equation (9.1.1) to determine the temperature evolution as a function of the age of the universe during this era and its age when kT ≈300MeV. 10 Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions All the systems considered in the previous chapters were composed of, or could be regarded as composed of, noninteracting entities. Consequently, the results obtained, though of considerable intrinsic importance, may have limitations when applied to sys-tems that actually exist in nature. For a real contact between the theory and experiment, one must take into account the interparticle interactions operating in the system. This can be done with the help of the formalism developed in Chapters 3 through 5 which, in principle, can be applied to an unlimited variety of physical systems and problems; in practice, however, one encounters in most cases serious difficulties of analysis. These difficulties are less stringent in the case of systems such as low-density gases, for which a corresponding noninteracting system can serve as an approximation. The mathema-tical expressions for the various physical quantities pertaining to such a system can be written in the form of series expansions, whose main terms describe the correspond-ing ideal-system results while the subsequent terms provide corrections arising from the interparticle interactions in the system. A systematic method of carrying out such expan-sions, in the case of real gases obeying classical statistics, was developed by Mayer and his collaborators (1937 onward) and is known as the method of cluster expansions. Its gener-alization, which equally well applies to gases obeying quantum statistics, was initiated by Kahn and Uhlenbeck (1938) and was perfected by Lee and Yang (1959a,b; 1960a,b,c). 10.1 Cluster expansion for a classical gas We start with a relatively simple physical system, namely a single-component, classical, monatomic gas whose potential energy is given by a sum of two-particle interactions uij. The Hamiltonian of the system is then given by H = X i  1 2mp2 i  + X i<j uij (i,j = 1,2,...,N); (1) the summation in the second part goes over all the N(N −1)/2 pairs of particles in the system. In general, the potential uij is a function of the relative position vector rij(= rj −ri); Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00010-4 © 2011 Elsevier Ltd. All rights reserved. 299 300 Chapter 10. Statistical Mechanics of Interacting Systems however, if the two-body force is a central one, then the function uij depends only on the interparticle distance rij. With the preceding Hamiltonian, the partition function of the system is given by, see equation (3.5.5), QN(V,T) = 1 N!h3N Z exp   −β X i  1 2mp2 i  −β X i<j uij   d3Npd3Nr. (2) Integration over the momenta of the particles can be carried out straightforwardly, with the result QN(V,T) = 1 N!λ3N Z exp  −β X i<j uij  d3Nr = 1 N!λ3N ZN(V,T), (3) where λ{= h/(2πmkT)1/2} is the mean thermal wavelength of the particles, while the function ZN(V,T) stands for the integral over the space coordinates r1,r2,...,rN: ZN(V,T) = Z exp  −β X i<j uij  d3Nr = Z Y i<j (e−βuij)d3Nr. (4) The function ZN(V,T) is generally referred to as the configuration integral of the system. For a gas of noninteracting particles, the integrand in (4) is unity; we then have Z(0) N (V,T) = V N and Q(0) N (V,T) = V N N!λ3N , (5) in agreement with our earlier result (3.5.9). To treat the nonideal case we introduce, after Mayer, the two-particle function fij, defined by the relationship fij = e−βuij −1. (6) In the absence of interactions, the function fij is identically zero; in the presence of interac-tions, it is nonzero but at sufficiently high temperatures it is quite small in comparison with unity. We, therefore, expect that the functions fij would be quite appropriate for carrying out a high-temperature expansion of the integrand in (4). A typical plot of the functions uij and fij is shown in Figure 10.1; we note that (i) the function fij is everywhere bounded and (ii) it becomes negligibly small as the interparticle distance rij becomes large in comparison with the “effective” range, r0, of the potential. 10.1 Cluster expansion for a classical gas 301 r0 r0 rij uij fij rij 0 0 1 1 FIGURE 10.1 A typical plot of the two-body potential function uij and the corresponding Mayer function fij. Now, to evaluate the configuration integral (4), we expand its integrand in ascending powers of the functions fij: ZN(V,T) = Z Y i 2) can be formed in a number of different ways; see, for example, the four different ways of forming a 3-cluster with a given group of three particles, as listed in (15). For cause (i), we obtain a straightforward factor of N! (1!)m1(2!)m2 ··· = N! Q l (l!)ml . (22) Now, if cause (ii) were not there, that is, if all l-clusters were unique in their formation, then the sum S{ml} would be given by the product of the combinatorial factor (22) with the value of any one graph in the setup, namely Y l (the value of an l-cluster)ml, (23) further corrected for the fact that any two arrangements that differ merely in the exchange of all the particles in one cluster with all the particles in another cluster of the same size, must not be counted as distinct, the corresponding correction factor being Y l (1/ml!). (24) A little reflection now shows that cause (ii) is completely and correctly taken care of if we replace the product of the expressions (23) and (24) by the expression3 Y l (the sum of the values of all possible l-clusters)ml/ml! (25) which, with the help of equation (16), may be written as Y l h (bll!λ3(l−1)V)ml/ml! i . (26) 3To appreciate the logic behind this replacement, consider expression [ ] in (25) as a multinomial expansion and interpret the various terms of this expansion in terms of the variety of the l-clusters. 306 Chapter 10. Statistical Mechanics of Interacting Systems The sum S{ml} is now given by the product of factor (22) and expression (26). Substituting this result into (21), we obtain for the configuration integral ZN(V,T) = N!λ3N X′ {ml}  Y l  bl V λ3 ml 1 ml!  . (27) Here, use has been made of the fact that Y l (λ3l)ml = λ36llml = λ3N; (28) see the restrictive condition (20). The partition function of the system now follows from equations (3) and (27), with the result QN(V,T) = X′ {ml}   N Y l=1  bl V λ3 ml 1 ml!  . (29) The evaluation of the primed sum in (29) is complicated by the restrictive condi-tion (20), which must be obeyed by every set {ml}. We, therefore, move over to the grand partition function of the system: Q(z,V,T) = ∞ X N=0 zNQN(V,T). (30) Writing zN = z6llml = Y l (zl)ml, (31) substituting for QN(V,T) from (29), and noting that a restricted summation over sets {ml}, subject to the condition P l lml = N, followed by a summation over all values of N ( from N = 0 to N = ∞) is equivalent to an unrestricted summation over all possible sets {ml}, we obtain Q(z,V,T) = ∞ X m1,m2,...=0   ∞ Y l=1  blzl V λ3 ml 1 ml!   = ∞ Y l=1   ∞ X ml=0  blzl V λ3 ml 1 ml!   = ∞ Y l=1  exp  blzl V λ3  = exp   ∞ X l=1 blzl V λ3   (32) 10.2 Virial expansion of the equation of state 307 and, hence, 1 V lnQ = 1 λ3 ∞ X l=1 blzl. (33) In the limit V →∞, P kT ≡Lim V→∞  1 V lnQ  = 1 λ3 ∞ X l=1 blzl, (34) and N V ≡Lim V→∞  z V ∂lnQ ∂z  = 1 λ3 ∞ X l=1 lblzl. (35) Equations (34) and (35) constitute the famous cluster expansions of the Mayer–Ursell for-malism. Eliminating the fugacity z among these equations, we obtain the equation of state of the system. 10.2 Virial expansion of the equation of state The approach developed in the preceding section leads to exact results only if we apply it to the gaseous phase alone. If we attempt to include in our study the phenomena of conden-sation, the critical point, and the liquid phase, we encounter serious difficulties relating to (i) the limiting procedure involved in equations (10.1.34) and (10.1.35), (ii) the conver-gence of the summations over l, and (iii) the volume dependence of the cluster integrals bl. We, therefore, restrict our study to the gaseous phase alone. The equation of state may then be written in the form Pv kT = ∞ X l=1 al(T) λ3 v !l−1 , (1) where v(= V/N) denotes the volume per particle in the system. Expansion (1), which is supposed to have been obtained by eliminating z between equations (10.1.34) and (10.1.35), is called the virial expansion of the system and the numbers al(T) the virial coef-ficients.4 To determine the relationship between the coefficients al and the cluster integrals bl, we invert equation (10.1.35) to obtain z as a power series in (λ3/v) and substitute this 4For various manipulations of the virial equation of state, see Kilpatrick and Ford (1969). 308 Chapter 10. Statistical Mechanics of Interacting Systems into (10.1.34). This leads to equation (1), with a1 = b1 ≡1, (2) a2 = −b2 = −2π λ3 ∞ Z 0  e−u(r)/kT −1  r2dr, (3) a3 = 4b2 2 −2b3 = −1 3λ6 ∞ Z 0 ∞ Z 0 f12f13f23d3r12d3r13, (4) a4 = −20b3 2 + 18b2 b3 −3b4 = ··· , (5) and so on; here, use has also been made of formulae (10.1.17) to (10.1.19). We note that the coefficient al is completely determined by the quantities b1, b2,..., bl, that is, by the sequence of configuration integrals Z1,Z2,...,Zl; see also equations (10.4.5) to (10.4.8). From equation (4) we observe that the third virial coefficient of the gas is determined solely by the 3-cluster 1 3 2 . This suggests that the higher-order virial coefficients may also be determined solely by a special “subgroup” of the various l-clusters. This is indeed true, and the relevant result is that, in the limit of infinite volume,5 al = −l −1 l βl−1 (l ≥2), (6) where βl−1 is the so-called irreducible cluster integral, defined as βl−1 = 1 (l −1)!λ3(l−1)V × (the sum of all irreducible l-clusters); (7) by an irreducible l-cluster we mean an “l-particle graph that is multiply-connected (in the sense that there are at least two entirely independent, nonintersecting paths linking each pair of circles in the graph).” For instance, of the four possible 3-clusters, see (10.1.15), only the last one is irreducible. Indeed, if we express equation (4) in terms of this particular cluster and make use of definition (7) for β2, we do obtain for the third virial coefficient a3 = −2 3β2, (8) in agreement with the general result (6).6 The quantities βl−1, like bl, are dimensionless and, in the limit V →∞, approach finite values that are independent of the size and the shape of the container (unless the 5For a proof of this result, see Hill (1956, Sections 24 and 25); see also Section 10.4 of the present text. 6It may be mentioned here that a 2-cluster is also regarded as an irreducible cluster. Accordingly, β1 = 2b2; see equations (10.1.16) and (10.2.7). Equation (3) then gives: a2 = −b2 = −1 2 β1, again in agreement with the general result (6). 10.3 Evaluation of the virial coefficients 309 latter is unduly abnormal). Moreover, the two sets of quantities are mutually related; see equations (10.4.27) and (10.4.29). 10.3 Evaluation of the virial coefficients If a given system does not depart much from the ideal-gas behavior, its equation of state is given adequately by the first few virial coefficients. Now, since a1 ≡1, the lowest-order virial coefficient that we need to consider here is a2, which is given by equation (10.2.3): a2 = −b2 = 2π λ3 ∞ Z 0  1 −e−u(r)/kT r2 dr, (1) u(r) being the potential energy of interparticle interaction. A typical plot of the function u(r) was shown earlier in Figure 10.1; a typical semi-empirical formula (Lennard-Jones, 1924) is given by u(r) = 4ε σ r 12 − σ r 6 . (2) The most significant features of an actual interparticle potential are well-simulated by the Lennard-Jones formula (2). For instance, the function u(r) given by (2) exhibits a “mini-mum,” of value −ε, at a distance r0(= 21/6σ) and rises to an infinitely large (positive) value for r < σ and to a vanishingly small (negative) value for r ≫σ. The portion to the left of the “minimum” is dominated by repulsive interaction that comes into play when two particles come too close to one another, while the portion to the right is dominated by attractive interaction that operates between particles when they are separated by a respectable dis-tance. For most practical purposes, the precise form of the repulsive part of the potential is not very important; it may as well be replaced by the crude approximation u(r) = +∞ (for r < r0), (3) which amounts to attributing an impenetrable core, of diameter r0, to each particle. The precise form of the attractive part is, however, important; in view of the fact that there exists good theoretical basis for the sixth-power attractive potential (see Problem 3.36), this part may simply be written as u(r) = −u0(r0/r)6 (r ≥r0). (4) The potential given by expressions (3) and (4) may, therefore, be used if one is only inter-ested in a qualitative assessment of the situation and not in a quantitative comparison between the theory and experiment. 310 Chapter 10. Statistical Mechanics of Interacting Systems Substituting (3) and (4) into (1), we obtain for the second virial coefficient a2 = 2π λ3    r0 Z 0 r2dr + ∞ Z r0  1 −exp  u0 kT r0 r 6 r2 dr   . (5) The first integral is straightforward; the second one is considerably simplified if we assume that (u0/kT) ≪1, which makes the integrand very nearly equal to −(u0/kT)(r0/r)6. Equation (5) then gives a2 ≃2πr3 0 3λ3  1 −u0 kT  . (6) Substituting (6) into the expansion (10.2.1), we obtain a first-order improvement on the ideal-gas law, namely P ≃kT v ( 1 + 2πr3 0 3v  1 −u0 kT ) (7a) = kT v  1 + B2(T) v  , say. (7b) The coefficient B2, which is also sometimes referred to as the second virial coefficient of the system, is given by B2 ≡a2λ3 ≃2πr3 0 3  1 −u0 kT  . (8) In our derivation it was explicitly assumed that (i) the potential function u(r) is given by the simplified expressions (3) and (4), and (ii) (u0/kT) ≪1. We cannot, therefore, expect formula (8) to be a faithful representation of the second virial coefficient of a real gas. Nev-ertheless, it does correspond, almost exactly, to the van der Waals approximation to the equation of state of a real gas. This can be seen by rewriting (7a) in the form P + 2πr3 0u0 3v2 ! ≃kT v 1 + 2πr3 0 3v ! ≃kT v 1 −2πr3 0 3v !−1 , which readily leads to the van der Waals equation of state  P + a v2  (v −b) ≃kT, (9) where a = 2πr3 0u0 3 and b = 2πr3 0 3 ≡4v0. (10) 10.3 Evaluation of the virial coefficients 311 1.5 0 1.5 3.0 4.5 6.0 1 2 5 10 H2 Classical Ar N2 CH4 He H2 Ne He 20 50 100 (kT/) (B2/r 0 3) FIGURE 10.2 A dimensionless plot showing the temperature dependence of the second virial coefficient of several gases (after Hirschfelder et al., 1954). We note that the parameter b in the van der Waals equation of state is exactly four times the actual molecular volume v0, the latter being the “volume of a sphere of diameter r0”; compare with Problem 1.4. We also note that in this derivation we have assumed that b ≪v, which means that the gas is sufficiently dilute for the mean interparticle distance to be much larger than the effective range of the interparticle interaction. Finally, we observe that, according to this simple-minded calculation, the van der Waals parameters a and b are temperature-independent, which in reality is not true. A realistic study of the second virial coefficient requires the use of a realistic potential, such as the one given by Lennard-Jones, for evaluating the integral in (1). This has indeed been done and the results obtained are shown in Figure 10.2, where the reduced coefficient B′ 2(= B2/r3 0) is plotted against the reduced temperature T′(= kT/ε): B′ 2(T′) = 2π ∞ Z 0  1 −e−u′(r′)/T′ r′2 dr′, (11) with u′(r′) = ( 1 r′ 12 −2  1 r′ 6) , (12) r′ being equal to (r/r0); expressed in this form, the quantity B′ 2 is a universal function of T′. Included in the plot are experimental results for several gases. We note that in most cases the agreement is reasonably good; this is especially satisfying in view of the fact that in each case we had only two adjustable parameters, r0 and ε, against a much larger number of experimental points available. In the first place, this agreement vindicates 312 Chapter 10. Statistical Mechanics of Interacting Systems the adequacy of the Lennard-Jones potential for providing an analytical description of a typical interparticle potential. Secondly, it enables one to derive empirical values of the respective parameters of the potential; for instance, one obtains for argon: r0 = 3.82 ˚ A and ε/k = 120K.7 One cannot fail to observe that the lighter gases, hydrogen and helium, constitute exceptions to the rather general rule of agreement between the theory and experiment. The reason for this lies in the fact that in the case of these gases quantum-mechanical effects assume considerable importance — more so at low temperatures. To substantiate this point, we have included in Figure 10.2 theoretical curves for H2 and He taking into account the quantum-mechanical effects as well; as a result, we find once again a fairly good agreement between the theory and experiment. As regards higher-order virial coefficients (l > 2), we confine our discussion to a gas of hard spheres with diameter D. We then have u(r) = ( 0 if r > D, ∞ if r ≤D, (13) and, hence, f (r) = ( 0 if r > D, −1 if r ≤D. (14) The second virial coefficient of the gas is then given by a2 = 2πD3 3λ3 = 4 v0 λ3 ; (15) compare with equation (6). The third virial coefficient can be determined with the help of equation (10.2.4), namely a3 = −1 3λ6 ∞ Z 0 ∞ Z 0 f12 f13 f23 d3r12 d3 r13. (16) To evaluate this integral, we first fix the positions of particles 1 and 2 (such that r12 < D) and let particle 3 take all possible positions so that we can effect an integration over the variable r13; see Figure 10.3. Since our integrand is equal to −1 when each of the distances r13 and r23 (like r12) is less than D and 0 otherwise, we have a3 = 1 3λ6 D Z r12=0 Z ′ d3r13  d3r12, (17) where the primed integration arises from particle 3 taking all possible positions of interest. In view of the conditions r13 < D and r23 < D, this integral is precisely equal to the “volume 7Corresponding values for various other gases have been summarized in Hill (1960, p. 484). 10.3 Evaluation of the virial coefficients 313 2 1 3 r23 r13 r12 FIGURE 10.3 dy 1 2 Dr12 r12 Dr12 y FIGURE 10.4 common to the spheres S1 and S2, each of radius D, centered at the fixed points 1 and 2”; see Figure 10.4. This in turn can be obtained by calculating the “volume swept by the shaded area in the figure on going through a complete revolution about the line of centers.” One gets: Z ′ d3r13 = √[D2−(r12/2)2] Z 0 n 2 D2 −y21/2 −r12 o 2πydy. (18) While the quantity within the curly brackets denotes the length of the strip shown in the figure, the element of area 2πydy arises from the revolution; the limits of integration for y can be checked rather easily. The evaluation of the integral (18) is straightforward; we get Z ′ d3r13 = 4π 3 ( D3 −3D2r12 4 + r3 12 16 ) . (19) Substituting (19) into (17) and carrying out integration over r12, we finally obtain a3 = 5π2D6 18λ6 = 5 8a2 2. (20) 314 Chapter 10. Statistical Mechanics of Interacting Systems The fourth virial coefficient of the hard-sphere gas has also been evaluated exactly. It is given by (Boltzmann, 1899; Majumdar, 1929) 8 a4 = ( 1283 8960 + 3 2 · 73√(2) + 1377{tan−1 √(2) −π/4} 1120π ) a3 2 = 0.28695a3 2. (21) The fifth and sixth virial coefficients of this system have been computed numerically, with the results (Ree and Hoover, 1964) a5 = (0.1103 ± 0.003)a4 2, (22) and a6 = (0.0386 ± 0.004)a5 2. (23) Ree and Hoover’s estimate of the seventh virial coefficient is 0.0127a6 2. Terms up through 10th order have been determined numerically; see Hansen and McDonald (1986) and Malijevsky and Kolafa (2008). If the virial equation of state for hard spheres is written in terms of the volume packing fraction η = πnD3/6, the first ten terms are P nkT = 1 + 4η + 10η2 + 18.364768η3 + 28.22445η4 + 39.81545η5 + 53.3418η6 + 68.534η7 + 85.805η8 + 105.8η9 + ··· . (24) Carnahan and Starling (1969) proposed a simple form for the equation of state that closely approximates all of the known virial coefficients: P nkT ≈1 + η + η2 −η3 (1 −η)3 = 1 + 4η + 10η2 + 18η3 + 28η4 + 40η5 + 54η6 + 70η7 + 88η8 + 108η9 + 130η10 + ··· (25) This gives an excellent fit to the hard sphere equation of state for the entire fluid phase as determined in computer simulations. The fluid phase is the equilibrium phase for 0 < η ≲0.491. The high-density equilibrium phase of hard spheres is a face-centered cubic solid; see Chapter 16. Many other approximate analytical forms have also been proposed to closely reproduce the virial series; see for instance Mulero et al. (2008). 8See also Katsura (1959). 10.4 General remarks on cluster expansions 315 10.4 General remarks on cluster expansions Shortly after the pioneering work of Mayer and his collaborators, Kahn and Uhlenbeck (1938) initiated the development of a similar treatment for quantum-mechanical systems. Of course, their treatment applied to the limiting case of classical systems as well but it faced certain inherent difficulties of analysis, some of which were later removed by the formal methods developed by Lee and Yang (1959a,b; 1960a,b,c). We propose to examine these developments in the next three sections of this chapter. First, however, we would like to make a few general observations on the problem of cluster expansions. These obser-vations, due primarily to Ono (1951) and Kilpatrick (1953), are of considerable interest insofar as they hold for a very large class of physical systems. For instance, the system may be quantum-mechanical or classical, it may be a multicomponent one or single-component, its molecules may be polyatomic or monatomic, and so on. All we have to assume is that (i) the system is gaseous in state and (ii) its partition functions QN(V,T), for some low values of N, can somehow be obtained. We can then calculate the “cluster integrals” bl, and the virial coefficients al, of the system in the following straightforward manner. Quite generally, the grand partition function of the system can be written as Q(z,V,T) ≡ ∞ X N=0 QN(V,T)zN = ∞ X N=0 ZN(V,T) N!  z λ3 N , (1) where we have introduced the “configuration integrals” ZN(V,T), defined in analogy with equation (10.1.3) of the classical treatment: ZN(V,T) ≡N!λ3NQN(V,T). (2) Dimensionally, the quantity ZN is like (a volume)N; moreover, the quantity Z0 (like Q0) is supposed to be identically equal to 1, while Z1(≡λ3Q1) is identically equal to V. We then have, in the limit V →∞, P kT ≡1 V lnQ = 1 V ln  1 + Z1 1!  z λ3 1 + Z2 2!  z λ3 2 + ···  (3) = 1 λ3 ∞ X l=1 blzl, say. (4) Again, the last expression has been written in analogy with the classical expansion (10.1.34); the coefficients bl may, therefore, be looked upon as the cluster integrals of the given system. Expanding (3) as a power series in z and equating respective coefficients with 316 Chapter 10. Statistical Mechanics of Interacting Systems the bl of (4), we obtain b1 = 1 V Z1 ≡1, (5) b2 = 1 2!λ3V (Z2 −Z2 1), (6) b3 = 1 3!λ6V (Z3 −3Z2Z1 + 2Z3 1), (7) b4 = 1 4!λ9V (Z4 −4Z3Z1 −3Z2 2 + 12Z2Z2 1 −6Z4 1), (8) and so on. We note that, for all l > 1, the sum of the coefficients appearing within the parentheses is identically equal to zero. Consequently, in the case of an ideal classical gas, for which Zi ≡V i, see equation (10.1.4), all cluster integrals with l > 1 vanish. This, in turn, implies the vanishing of all the virial coefficients of the gas (except, of course, a1, which is identically equal to unity). Comparing equations (6) through (8) with equation (10.1.16), we find that the expres-sions involving the products of the various Zi that appear within the parentheses play the same role here as “the sum of all possible l-clusters” does in the classical case. We there-fore expect that, in the limit V →∞, the bl here would also be independent of the size and the shape of the container (unless the latter is unduly abnormal). This, in turn, requires that the various combinations of the Zi appearing within the parentheses here must all be proportional to the first power of V. This observation leads to the very interesting result, first noticed by Rushbrooke, namely bl = 1 l!λ3(l−1) × (the coefficient of V l in the volume expansion of Zl). (9) At this stage, it seems worthwhile to point out that the expressions appearing within the parentheses of equations (6) through (8) are well-known in mathematical statistics as the semi-invariants of Thiele. The general formula for these expressions is (...)l ≡bl  l!λ3(l−1)V = l! X′ {mi} (−1)6imi−1 " X i mi −1 ! ! Y i (Zi/i!)mi mi! # , (10) where the primed summation goes over all sets {mi} that conform to the condition l X i=1 imi = l; mi = 0,1,2,.... (11) 10.4 General remarks on cluster expansions 317 Relations inverse to (10) can be written down by referring to equation (10.1.29) of the classical treatment; thus ZM ≡M!λ3MQM = M!λ3M X′ {ml} M Y l=1    (V bl/λ3)ml ml!   , (12) where the primed summation goes over all sets {ml} that conform to the condition M X l=1 lml = M; ml = 0,1,2.... (13) The calculation of the virial coefficients al now consists of a straightforward step that involves a use of formulae (5) through (8) in conjunction with formulae (10.2.2) through (10.2.5). It appears, however, of interest to demonstrate here the manner in which the gen-eral relationship (10.2.6) between the virial coefficients al and the “irreducible cluster inte-grals” βl−1 arises mathematically. As a bonus, we will acquire yet another interpretation of the βk. Now, in view of the relations P kT ≡Lim V→∞  1 V lnQ  = 1 λ3 ∞ X l=1 blzl (14) and 1 v ≡Lim V→∞  z V ∂lnQ ∂z  = 1 λ3 ∞ X l=1 lblzl, (15) we can write P(z) kT = z Z 0 1 v(z) dz z . (16) We introduce a new variable x, defined by x = nλ3 = λ3/v. (17) In terms of this variable, equation (15) becomes x(z) = ∞ X l=1 lblzl, (18) the inverse of which may be written (see Mayer and Harrison, 1938; Harrison and Mayer, 1938; also Kahn, 1938) as z(x) = xexp{−φ(x)}. (19) 318 Chapter 10. Statistical Mechanics of Interacting Systems In view of the fact that, for z ≪1, the variables z and x are practically the same, the function φ(x) must tend to zero as x →0; it may, therefore, be expressed as a power series in x: φ(x) = ∞ X k=1 βkxk. (20) It may be mentioned beforehand that the coefficients βk of this expansion are ulti-mately going to be identified with the “irreducible cluster integrals” βl−1. Substituting from equations (17), (19), and (20) into equation (16), we get P(x) kT = x Z 0 x λ3 1 x −φ′(x)  dx = 1 λ3  x − x Z 0    ∞ X k=1 kβkxk   dx   = x λ3  1 − ∞ X k=1  k k + 1βkxk  . (21) Combining (17) and (21), we obtain Pv kT = 1 − ∞ X k=1  k k + 1βkxk  . (22) Comparing this result with the virial expansion (10.2.1), we arrive at the desired relation-ship: al = −l −1 l βl−1 (l > 1). (23) For obvious reasons, the βk appearing here may be regarded as a generalization of the irreducible cluster integrals of Mayer. Finally, we would like to derive a relationship between the βk and the bl. For this, we make use of a theorem due to Lagrange which, for the present purpose, states that “the solution x(z) to the equation z(x) = x/f (x) (24) is given by the series x(z) = ∞ X j=1 zj j! " dj−1 dξj−1 {f (ξ)}j #” ξ=0 ; (25) it is obvious that the expression within the square brackets is (j −1)! times “the coefficient of ξj−1 in the Taylor expansion of the function {f (ξ)}j about the point ξ = 0.” Applying this 10.4 General remarks on cluster expansions 319 theorem to the function f (x) = exp{φ(x)} = exp    ∞ X k=1 βkxk   = ∞ Y k=1 exp(βkxk), (26) we obtain x(z) = ∞ X j=1 zj j! (j −1)! ×   the coefficient ofξj−1 in the Taylor expansion of ∞ Y k=1 exp(jβkξk) about ξ = 0   . Comparing this with equation (18), we get bj = 1 j2 ×   the coefficient of ξj−1 in ∞ Y k=1  X mk≥0 (jβk)mk mk! ξkmk      = 1 j2 X′ {mk} j−1 Y k=1 (jβk)mk mk! , (27) where the primed summation goes over all sets {mk} that conform to the condition j−1 X k=1 kmk = j −1; mk = 0,1,2,.... (28) Formula (27) was first obtained by Maria Goeppert-Mayer in 1937. Its inverse, however, was established much later (Mayer et al., 1942; Kilpatrick, 1953): βl−1 = X′ {mi} (−1)6imi−1 (l −2 + 6imi)! (l −1)! Y i (ibi)mi mi! , (29) where the primed summation goes over all sets {mi} that conform to the condition l X i=2 (i −1)mi = l −1; mi = 0,1,2,.... (30) It is not difficult to see that the highest value of the index i in the set {mi} would be l (the corresponding set having all its mi equal to 0, except ml which would be equal to 1); accordingly, the highest order to which the quantities bi would appear in the expression for βl−1 is that of bl. We thus see, once again, that the virial coefficient al is completely determined by the quantities b1, b2,..., bl. 320 Chapter 10. Statistical Mechanics of Interacting Systems 10.5 Exact treatment of the second virial coefficient We now present a formulation, originally from Uhlenbeck and Beth (1936) and Beth and Uhlenbeck (1937), that enables us to make an exact calculation of the second virial coeffi-cient of a quantum-mechanical system from a knowledge of the two-body potential u(r).9 In view of equation (10.4.6), b2 = −a2 = 1 2λ3V  Z2 −Z2 1  . (1) For the corresponding noninteracting system, one would have b(0) 2 = −a(0) 2 = 1 2λ3V  Z(0) 2 −Z(0)2 1  ; (2) the superscript (0) on the various symbols here implies that they pertain to the noninteract-ing system. Combining (1) and (2), and remembering that Z1 = Z(0) 1 = V, we obtain b2 −b(0) 2 = 1 2λ3V  Z2 −Z(0) 2  (3) which, by virtue of relation (10.4.2), becomes b2 −b(0) 2 = λ3 V  Q2 −Q(0) 2  = λ3 V Tr  e−β ˆ H2 −e−β ˆ H(0) 2  . (4) For evaluating the trace in (4), we need to know the eigenvalues of the two-body Hamiltonian which, in turn, requires solving the Schr¨ odinger equation10 ˆ H29α(r1,r2) = Eα9α(r1,r2), (5) where ˆ H2 = −ℏ2 2m  ∇2 1 + ∇2 2  + u(r12). (6) Transforming to the center-of-mass coordinates R n = 1 2(r1 + r2) o and the relative coordi-nates r{= (r2 −r1)}, we have 9α(R,r) = ψj(R)ψn(r) =  1 V 1/2 ei(Pj·R)/ℏ  ψn(r), (7) with Eα = P2 j 2(2m) + εn. (8) 9For a discussion of the third virial coefficient, see Pais and Uhlenbeck (1959). 10For simplicity, we assume the particles to be “spinless.” For the influence of spin, see Problems 10.11 and 10.12. 10.5 Exact treatment of the second virial coefficient 321 Here, P denotes the total momentum of the two particles and 2m their total mass, while ε denotes the energy associated with the relative motion of the particles; the symbol α refers to the set of quantum numbers j and n that determine the actual values of the variables P and ε. The wave equation for the relative motion will be   − ℏ2 2  1 2m ∇2 r + u(r)   ψn(r) = εnψn(r), (9) 1 2m being the reduced mass of the particles; the normalization condition for the relative wavefunction will be Z |ψn(r)|2d3r = 1. (10) Equation (4) thus becomes b2 −b(0) 2 = λ3 V X α  e−βEα −e−βE(0) α  = λ3 V X j e−βP2 j /4m X n  e−βεn −e−βε(0) n  . (11) For the first sum, we obtain X j e−βP2 j /4m ≈4πV h3 ∞ Z 0 e−βP2/4mP2dP = 81/2V λ3 , (12) so that equation (11) becomes b2 −b(0) 2 = 81/2 X n  e−βεn −e−βε(0) n  . (13) The next step consists of examining the energy spectra, εn and ε(0) n , of the two systems. In the case of a noninteracting system, all we have is a “continuum” ε(0) n = p2 2  1 2m  = ℏ2k2 m (k = p/ℏ), (14) with the standard density of states g(0)(k). In the case of an interacting system, we may have a set of discrete eigenvalues εB (that correspond to “bound” states), along with a “continuum” εn = ℏ2k2 m (k = p/ℏ), (15) 322 Chapter 10. Statistical Mechanics of Interacting Systems with a characteristic density of states g(k). Consequently, equation (13) can be written as b2 −b(0) 2 = 81/2 X B e−βεB + 81/2 ∞ Z 0 e−βℏ2k2/m{g(k) −g(0)(k)}dk, (16) where the summation in the first part goes over all bound states made possible by the two-body interaction. The next thing to consider here is the density of states g(k). For this, we note that, since the two-body potential is assumed to be central, the wavefunction ψn(r) for the relative motion may be written as a product of a radial function χ(r) and a spherical harmonic Y (θ,ϕ): ψklm(r) = Aklm χkl(r) r Yl,m(θ,ϕ). (17) Moreover, the requirement of symmetry, namely ψ(−r) = ψ(r) for bosons and ψ(−r) = −ψ(r) for fermions, imposes the restriction that the quantum number l be even for bosons and odd for fermions. The (outer) boundary condition on the wavefunction may be written as χkl(R0) = 0, (18) where R0 is a fairly large value (of the variable r) that ultimately goes to infinity. Now, the asymptotic form of the function χkl(r) is well-known: χkl(r) ∝sin  kr −lπ 2 + ηl(k)  ; (19) accordingly, we must have kR0 −lπ 2 + ηl(k) = nπ, n = 0,1,2,.... (20) The symbol ηl(k) here stands for the scattering phase shift due to the two-body potential u(r) for the lth partial wave of wave number k. Equation (20) determines the full spectrum of the partial waves. To obtain from it an expression for the density of states gl(k), we observe that the wave number difference 1k between two consecutive states n and n + 1 is given by the formula  R0 + dηl(k) dk  1k = π, (21) with the result that gl(k) = 2l + 1 1k = 2l + 1 π  R0 + ∂ηl(k) ∂k  ; (22) 10.5 Exact treatment of the second virial coefficient 323 the factor (2l + 1) has been included here to take account of the fact that each eigenvalue k pertaining to an lth partial wave is (2l + 1)-fold degenerate (because the magnetic quan-tum number m can take any of the values l,(l −1),...,−l, without affecting the eigenvalue). The total density of states, g(k), of all partial waves of wave numbers around the value k is then given by g(k) = X′ l gl(k) = 1 π X′ l (2l + 1)  R0 + ∂ηl(k) ∂k  ; (23) note that the primed summation P′ goes over l = 0,2,4,... in the case of bosons and over l = 1,3,5,... in the case of fermions. For the corresponding noninteracting case, we have (since all ηl(k) = 0) g(0)(k) = R0 π X′ l (2l + 1). (24) Combining (23) and (24), we obtain g(k) −g(0)(k) = 1 π X′ l (2l + 1)∂ηl(k) ∂k . (25) Substituting (25) into (16), we obtain the desired result b2 −b(0) 2 = 81/2 X B e−βεB + 81/2 π X′ l (2l + 1) ∞ Z 0 e−βℏ2k2/m ∂ηl(k) ∂k dk (26) which, in principle, is calculable for any given potential u(r) through the respective phase shifts ηl(k). Equation (26) can be used for determining the quantity b2 −b(0) 2 . To determine b2 itself, we must know the value of b(0) 2 . This has already been obtained in Section 7.1 for bosons and in Section 8.1 for fermions; see equations (7.1.13) and (8.1.17). Thus b(0) 2 = −a(0) 2 = ± 1 25/2 , (27) where the upper sign holds for bosons and the lower sign for fermions. It is worthwhile to note that the foregoing result can be obtained directly from the relationship b(0) 2 = 1 2λ3V  Z(0) 2 −Z(0)2 1  = λ3 V  Q(0) 2 −1 2Q(0)2 1  324 Chapter 10. Statistical Mechanics of Interacting Systems by substituting for Q(0) 2 the exact expression (5.5.25): b(0) 2 = λ3 V "( 1 2  V λ3 2 ± 1 25/2  V λ3 1) −1 2  V λ3 2# = ± 1 25/2 . (28)11 It is of interest to note that this result can also be obtained by using the classical for-mula (10.1.18) and substituting for the two-body potential u(r) the “statistical potential” (5.5.28); thus b(0) 2 = 2π λ3 ∞ Z 0  e−us(r)/kT −1  r2dr = ±2π λ3 ∞ Z 0 e−2πr2/λ2r2dr = ± 1 25/2 . (29) As an illustration of this method, we now calculate the second virial coefficient of a gas of hard spheres. The two-body potential in this case may be written as u(r) = ( +∞ for r < D 0 for r > D. (30) The scattering phase shifts ηl(k) can now be determined by making use of the (inner) boundary condition, namely χ(r) = 0 for all r < D and hence it vanishes as r →D from the above. We thus obtain (see, for example, Schiff, 1968) ηl(k) = tan−1 jl(kD) nl(kD), (31) where jl(x) and nl(x) are, respectively, the “spherical Bessel functions” and the “spherical Neumann functions”: j0(x) = sinx x , j1(x) = sinx −xcosx x2 , j2(x) = (3 −x2)sinx −3xcosx x3 ,... and n0(x) = −cosx x , n1(x) = −cosx + xsinx x2 , n2(x) = −(3 −x2)cosx + 3xsinx x3 ,.... 11This calculation incidentally verifies the general formula (10.4.9) for the case l = 2. By that formula, the “cluster integral” b2 of a given system would be equal to 1/(2λ3) times the coefficient of V 1 in the volume expansion of the “configuration integral” Z2 of the system. In the case under study, this coefficient is ±λ3/23/2; hence the result. 10.6 Cluster expansion for a quantum-mechanical system 325 Accordingly, η0(k) = tan−1{−tan(kD)} = −kD, (32) η1(k) = tan−1  −tan(kD) −kD 1 + kDtan(kD)  = −{kD −tan−1(kD)} = −(kD)3 3 + (kD)5 5 −··· , (33) η2(k) = tan−1 ( −tan(kD) −3(kD)/[3 −(kD)2] 1 + 3(kD)tan(kD)/[3 −(kD)2] ) = −  kD −tan−1 3(kD) 3 −(kD)2  = −(kD)5 45 + ··· , (34) and so on. We now have to substitute these results into formula (26). However, before doing that we should point out that, in the case of hard-sphere interaction, (i) we cannot have bound states at all and (ii) since, for all l,ηl(0) = 0, the integral in (26) can be simplified by a prior integration by parts. Thus, we have b2 −b(0) 2 = 81/2λ2 π2 X′ l (2l + 1) ∞ Z 0 e−βℏ2k2/mηl(k)kdk. (35) Substituting for l = 0 and 2 in the case of bosons and for l = 1 in the case of fermions, we obtain (to fifth power in D/λ) b2 −b(0) 2 = −2 D λ 1 −10π2 3 D λ 5 −··· (Bose) (36) = −6π D λ 3 + 18π2 D λ 5 −··· (Fermi), (37) which may be compared with the corresponding classical result −(2π/3)(D/λ)3. 10.6 Cluster expansion for a quantum-mechanical system When it comes to calculating bl for l > 2 we have no formula comparable in simplicity to formula (10.5.26) for b2. This is due to the fact that we have no treatment of the l-body problem (for l > 2) that is as neat as the phase-shift analysis of the two-body problem. Nevertheless, a formal theory for the calculation of higher-order “cluster integrals” has been developed by Kahn and Uhlenbeck (1938); an elaboration by Lee and Yang (1959a,b; 1960a,b,c) has made this theory almost as good for treating a quantum-mechanical sys-tem as Mayer’s theory has been for a classical gas. The basic approach in this theory is to 326 Chapter 10. Statistical Mechanics of Interacting Systems evolve a scheme for expressing the grand partition function of the given system in essen-tially the same way as Mayer’s cluster expansion does for a classical gas. However, because of the interplay of quantum-statistical effects and the effects arising from interparticle interactions, the mathematical structure of this theory is considerably involved. We consider here a quantum-mechanical system of N identical particles enclosed in a box of volume V. The Hamiltonian of the system is assumed to be of the form ˆ HN = −ℏ2 2m N X i=1 ∇2 i + X i 2, the mathematical procedure is rather cumbersome. Nevertheless, Lee and Yang (1959a,b; 1960a,b,c) have evolved a scheme that enables us to calculate the higher bl in successive approximations. According 10.7 Correlations and scattering 331 to that scheme, the functions Ul of a given system can be evaluated by “separating out” the effects of statistics from those of interparticle interactions, that is, we first take care of the statistical aspect of the problem and then tackle the dynamical aspect of it. Thus, the whole feat is accomplished in two steps. First, the U-functions pertaining to the given system are expressed in terms of U-functions pertaining to a corresponding quantum-mechanical system obeying Boltzmann statistics, that is, a (fictitious) system described by unsymmetrized wavefunctions. This step takes care of the statistics of the given system, that is, of the symmetry properties of the wavefunctions describing the system. Next, the U-functions of the (fictitious) Boltz-mannian system are expanded, loosely speaking, in powers of a binary kernel B which is obtainable from a solution of the two-body problem with the given interaction. A com-mendable feature of this method is that it can be applied even if the given interaction contains a singular, repulsive core, that is, even if the potential energy for certain configu-rations of the system becomes infinitely large. Though the method is admirably systematic and fairly straightforward in principle, its application to real systems is quite complicated. We will, therefore, turn to a more practical method — the method of quantized fields (see Chapter 11) — which has been extremely useful in the study of quantum-mechanical sys-tems composed of interacting particles. For a detailed exposition of the (binary collision) method of Lee and Yang, see Sections 9.7 and 9.8 of the first edition of this book. In passing, we note yet another important difference between the quantum-mechanical case and the classical one. In the latter case, if interparticle interactions are absent, then all bl, with l ≥2, vanish. This is not true in the quantum-mechanical case; here, see Sections 7.1 and 8.1, b(0) l = (±1)l−1l−5/2, (28) of which equation (10.5.27) was a special case. 10.7 Correlations and scattering Correlations and scattering play an extremely important role in modern statistical mechanics. Different phases most are easily distinguished by different spatial orderings they display. Molecules in a low-density vapor are nearly uncorrelated whereas molecules in a dense liquid can be strongly correlated and display short-range order due to their strong steric repulsions but the correlations decay away rapidly at large distances. In crys-talline solids, the location of every particle is highly correlated with the location of all the others, and these correlations do not decay away to zero at large distances between the particles; this is called long-range order. At a critical point, systems display order that lies between short-range and long-range, with so-called quasi-long-range order characterized by a power-law decay of correlations. Crystals and liquid-crystal phases display molecu-lar orientational correlations that can be short-range, long-range, or quasi-long-range in addition to the various spatial orderings of the molecules. Different phases of magnets 332 Chapter 10. Statistical Mechanics of Interacting Systems are distinguished by the spatial orderings of the magnetic dipoles: short-range ordering in paramagnets, long-range ordering in ferromagnets and antiferromagnets, and power-law decay of correlations at magnetic critical points. Spatial correlation functions are based on n-particle densities. The one-body number density is defined by the average quantity n1(r) = X i δ(r −ri) + . (1) This defines the local number density in which n1(r)dr is a measure of the probabil-ity of finding a particle inside an infinitesimal volume dr located at position r. If the system is translationally invariant, the one-body density is the usual number density n1(r) = n = ⟨N⟩/V. The spatial integral of the one-body density over volume V gives the average number of particles in that volume: Z n1(r)dr = ⟨N⟩. (2) The two-body number density is defined as n2(r,r′) = X i̸=j δ(r −ri)δ(r ′ −rj) + . (3) The quantity n2(r,r ′)drdr ′ is a measure of the probability of finding one particle inside the infinitesimal volume dr located at position r and another particle inside the infinitesimal volume dr ′ located at position r ′. In a dilute classical gas, the particles interact only when they are close to one another, so the probability of finding two different particles at two different locations many atomic diameters apart is simply the product of finding either particle individually, that is, n2(r,r ′) →n1(r)n1(r ′) as |r −r′| →∞. It is the deviation from this uncorrelated behavior that is both interesting and important. The integral of the two-body density over volume V gives Z n2(r,r′)drdr ′ = D N2E −⟨N⟩. (4) If the system is translationally and rotationally invariant, the one-body number density is independent of position and the two-body number density depends only on the magni-tude of the distance between r and r ′. This allows us to define the pair correlation function g(r): n2(r,r ′) = n2g r −r ′  . (5) 10.7 Correlations and scattering 333 6 5 4 3 2 1 00 1 2 3 4 5 r/D g(r) FIGURE 10.5 An approximate pair correlation function for hard spheres with diameter D in three dimensions. The volume fraction η = πnD3/6 ≃0.49 is the fraction of the volume occupied by the particles and is close to the liquid side of the solid-liquid phase transition in the model. The correlation function is calculated using the exact solution of the Percus–Yevick approximation; see Percus and Yevick (1958), Wertheim (1963), and Hansen and McDonald (1986). The correlation length for this case is ξ ≈2D. In three dimensions, 4πnr2 g(r)dr is the probability of finding a particle in a spherical shell of radius r and thickness dr, given that another particle is simultaneously located at the origin. The pair correlation function of a classical ideal gas is equal to unity; see the footnote to Problem 10.17. Figure 10.5 displays the pair correlation function g(r) for a system of hard spheres interacting via pair potential u(r) = ( 0 if r > D, ∞ if r ≤D. (6) Clearly, the pair correlation function vanishes for r < D since no two particles in the system can be closer to each other than D due to the infinite repulsion. These steric repulsions result in an oscillatory decay of g(r). The pair correlation function is greater than unity at separations slightly greater than D since the local geometry of the fluid enhances the probability of finding two particles a distance slightly more than D apart; for illustration, see Figure 10.6. The pair correlation function is less than unity at slightly larger distances due to the repulsion of the cluster of particles just outside the hard repulsion distance. The oscillating correlations decay rapidly with distance, so that g(r) approaches unity at large separations. This behavior of the pair correlation function is typical of all dense fluids 334 Chapter 10. Statistical Mechanics of Interacting Systems FIGURE 10.6 An equilibrium configuration of hard disks that displays steric effects leading to oscillations in the pair correlation function. The inner dashed circle with radius D is the closest approach distance to the central disk. In this case, the centers of five disks are close to the distance D which contributes to the enhancement in g(r) near r = D. The outer dashed circle shows the next shell of particles that contribute to the second peak in g(r). In-between these distances, we have a reduced probability of finding the center of a particle, leading to g(r) < 1. and is called short-range order since the correlations decay exponentially with distance: g(r) −1 ∼exp(−r/ξ), where ξ is called the correlation length. The pair correlation function can be used to directly calculate the pressure in a fluid. For a classical fluid whose potential energy can be written as a sum of pair potentials, UN(r1,r2,...,rN) = X i<j u(rij), (7) the pressure is determined by the average of the quantity r(∂u/∂r) between pairs of par-ticles, as discussed in Section 3.7. In the canonical ensemble, the pressure P is given by P ≡−  ∂A ∂V  T,N = kT ZN ∂ZN ∂V  T,N , (8) where ZN is the configurational partition function ZN = 1 N! Z dNr exp  −β X i<j u(rij)  . (9) The d-dimensional integrals over the volume V can be rewritten in terms of a set of scaled variables {si} defined by ri = V 1/dsi, so the scaled integrals are over regions with unit 10.7 Correlations and scattering 335 volume: ZN = V N N! Z dNs exp  −β X i<j u(V 1/dsij)  . (10) Equations (8) and (10) then give P = nkT  1 − n 2dkT Z du dr rg(r)dr  . (11) This is called the virial equation of state and is useful for determining pressure from approximate expressions for the pair correlation function. Compare equation (11) with the form of the virial equation of state in equation (3.7.15). For the particular case of hard spheres, the discontinuous potential results in the pres-sure being determined by the pair correlation function at contact. In one, two, and three dimensions, the hard sphere pressure is given by PHS nkT =      1 + ηg(D+) η = nD d = 1, 1 + 2ηg(D+) η = π 4 nD2 d = 2, 1 + 4ηg(D+) η = π 6 nD3 d = 3, (12) where g(D+) is the correlation function at contact and η is the volume fraction, that is, the fraction of the d-dimensional volume of the sample occupied by the spheres; see Problem 10.14. Likewise, the internal energy of the fluid can be written as an integral over the pair correlation function and the pair potential: U(N,V,T) = ⟨H⟩= dNkT 2 + nN 2 Z u(r)g(r)dr. (13) The pair correlation function itself contains all the statistical information needed to construct the full thermodynamic behavior of the system. For example, equation (4) can be used to show that the isothermal compressibility, which is proportional to the number density fluctuations, is also proportional to an integral over the pair correlation function: nkTκT = κT κideal T = 1 + n Z (g(r) −1)dr = N2 −⟨N⟩2 ⟨N⟩ ; (14) this is known as the compressibility equation of state. Since κ−1 T = n  ∂P ∂n  T, one can use equation (14) to determine the pressure and free energy of the system by performing thermodynamic integrations with respect to the particle density. 10.7.A Static structure factor The pair correlation function g(r) can be measured experimentally using quasielastic scattering. If a sample is illuminated with a monochromatic beam of x-rays, neutrons, vis-ible light, and so on, the scattered intensity as a function of the angle from the incident 336 Chapter 10. Statistical Mechanics of Interacting Systems beam direction is proportional to the Fourier transform of g(r). The quasielastic scattering amplitude from a single particle at location ri illuminated by a plane wave with amplitude φ0 and wavevector k0 into a detector at location R is 81(k) = φ0f (k)eik0·rieik1·(R−ri) |R −ri| , (15) where k = k1 −k0 is the wavevector transfer and f (k) is the single-particle scattering form factor; see Figure 10.7. The total scattering amplitude from the N particles in the sample is 8N(k) ≈φ0f (k) |R| eik1·R X i e−ik·ri, (16) where we have assumed that the detector is far from the sample. The scattered intensity from the N-particle sample is IN(k) = 8N(k) 2 ≈ φ0f (k) 2 |R|2 X i,j e−ik·(ri−rj) + = NI1(k)S(k), (17) k1 k0 FIGURE 10.7 Scattering from two particles. The incident wavevector is k0, the scattered wavevector toward the detector is k1, and the wavevector transfer is k = k1 −k0. Since |k1| = |k0| for quasielastic scattering, the magnitude of the wavevector transfer is k = 2k0 sin(θ/2), where θ is the angle between k0 and k1. 10.7 Correlations and scattering 337 where I1(k) is the scattering intensity from a single particle and S(k) = 1 N X i,j exp −ik · (ri −rj)  + (18) is the static structure factor. It represents the actual scattering intensity divided by the scattering intensity from an imaginary randomly distributed and, therefore, uncorrelated sample of atoms at the same particle density n. If the sample is translationally invariant and isotropic, as in a uniform fluid, the static structure factor depends only on the magnitude of the wavevector transfer, that is S(k) = S(k). For that case, S(k) can be written as the Fourier transform of the pair correla-tion function: S(k) = 1 + N V Z (g(r) −1)eik·rdr + N V 2 Z eik·rdr 2 . (19) The final term in equation (19) represents the forward shape scattering of the sample volume. The shape scattering term is negligible for k ≫1/L, so in the thermodynamic limit it can be ignored for k ̸= 0. The structure factor for isotropic fluids in one, two, and three dimensions is then given by S(k) = 1 + 2n ∞ Z 0 (g(r) −1)cos(kr)dr d = 1, (20a) S(k) = 1 + 2πn ∞ Z 0 r(g(r) −1)J0(kr)dr d = 2, (20b) S(k) = 1 + 4πn k ∞ Z 0 r(g(r) −1)sin(kr)dr d = 3. (20c) The pair correlation function g(r) can be determined using the inverse Fourier trans-form of the measured structure factor, as shown in Figure 10.8. For liquids and other short-range ordered materials, the structure factor tends to unity as k →∞. The value of S(k) as k →0 is a measure of the number density fluctuations in the sample: lim k→0S(k) = 1 + n Z (g(r) −1)dr = κT κideal T = N2 −⟨N⟩2 ⟨N⟩ . (21) Equation (21) is called the fluctuation-compressibility relation and is the equilibrium limit of the fluctuation-dissipation theorem we will discuss in Section 15.6. 338 Chapter 10. Statistical Mechanics of Interacting Systems 3 2 1 0 0 2 4 6 8 10 12 S(k) (b) k(Å1) 3 2 1 g(r) 0 0 5 10 15 r(Å) (a) 20 25 FIGURE 10.8 Experimentally measured pair correlation function g(r) and structure factor S(k) for liquid argon at 85 K. The structure factor (b) is determined from neutron scattering and the pair correlation function (a) is determined from the inverse Fourier transform of the structure factor. The small oscillations in g(r) near r = 0 are an experimental artifact of the Fourier transformation of the scattering data. This figure displays the typical features of correlations in fluids: nearly zero g(r) at short distances, large g(r) for particles separated by approximately a molecular diameter, oscillatory decay of correlations to unity at large separations, small S(k) at small wavevector due to the small compressibility of dense fluids, and S(k) approaching unity at large wavevectors. Figures from Yarnell, Katz, Wenzel, and Koenig (1973). Reprinted with permission; copyright ©1973, American Physical Society. 10.7.B Scattering from crystalline solids In an ideal crystalline solid, the atoms in the crystal are located at the sites of a periodic structure. For a simple crystal, identical atoms are sited on a Bravais lattice {R}. For exam-ple, a simple cubic lattice has lattice vectors R ∈{(n1ˆ x + n2ˆ y + n3ˆ z)a}, where n1, n2, and n3 are integers and a is the lattice constant. The reciprocal lattice {G} is defined by the set of reciprocal lattice vectors G — such that G · R = 2πm, where m is an integer for all {G} and {R}. The reciprocal lattice of the simple cubic lattice is also a simple cubic lat-tice: G ∈{(m1ˆ x + m2ˆ y + m3ˆ z) 2π a }, where m1, m2, and m3, are integers. For a perfect Bravais lattice, the structure factor S(k) is of the form S(k) = 1 N X R,R′ eik·(R−R′) + = N X G δk,G, (22) where δk,G is the Kronecker delta. The structure factor is enhanced by a factor of N on each reciprocal lattice vector due to the coherent constructive interference of scattering from the long-range ordered array of atoms. One can determine the crystal structure of the solid from the experimental pattern of these sharp Bragg peaks; see Ashcroft and Mermin (1976). Thermal excitations cause atoms to deviate from their equilibrium positions. The dis-placed position of an atom whose equilibrium position is R can be written R + u(R), where u(R) denotes the displacement from equilibrium. As long as the atoms remain close to their lattice sites, the sharp Bragg peaks in the structure factor will also remain 10.7 Correlations and scattering 339 but the intensity of each peak will be reduced by an amount dependent on the aver-age of the squares of the deviations |u(R)|2 . This turns out to be the case for normal three-dimensional solids. The structure factor then takes the form S(k) = 1 N X R,R′ eik·(R−R′) D eik·(u(R)−u(R′))E . (23) If the excitations about the equilibrium positions are Gaussian (i.e., the terms in the Hamil-tonian higher than second order in u(R) can be ignored), then the average of the deviations in the exponential can be simplified to give D eik·(u(R)−u(R′))E = e−1 2 D |k·(u(R)−u(R′)|2E . (24) If the displacements of the atoms far from each other on the lattice are uncorrelated, as they are in three-dimensional crystals, 1 2 D |k · (u(R) −u(R′)|2E ≈k2 u2 3 for |R −R′| →∞, (25) then the structure factor takes the form S(k) = N X G WGδk,G, (26) where WG = exp −G2 u2 3 ! (27) is called the Debye–Waller factor. The random atomic deviations from lattice sites reduces the intensity in the Bragg peaks but the sharp scattering indicative of long-range crystalline order remains intact; see Ashcroft and Mermin (1976). An interesting variant of this calculation occurs in two-dimensional solids. Peierls (1935) and Landau (1937) showed that harmonic thermal fluctuations in two dimensions destroy crystalline long-range order. This was generalized by Mermin (1968) to show that long-range crystalline order was not possible for any two-dimensional system of parti-cles with short-range interactions. Two-dimensional solids exhibit power-law decay of translational correlations while maintaining long-range order in the lattice orientational correlations. This leads to power-law singularities rather than delta-functions in the static structure factor. It is possible for the solid to melt via two Kosterlitz–Thouless-like continu-ous transitions rather than a single first-order transition. The intervening “hexatic” phase exhibits short-range translational correlations and quasi-long-range orientational corre-lations; see Section 13.7, Kosterlitz and Thouless (1972, 1973), Halperin and Nelson (1978), and Young (1979). 340 Chapter 10. Statistical Mechanics of Interacting Systems Problems 10.1. For imperfect-gas calculations, one sometimes employs the Sutherland potential u(r) = ∞ for r < D −ε(D/r)6 for r > D. Using this potential, determine the second virial coefficient of a classical gas. Also determine first-order corrections to the ideal-gas law and to the various thermodynamic properties of the system. 10.2. According to Lennard-Jones, the physical behavior of most real gases can be well understood if the intermolecular potential is assumed to be of the form u(r) = A rm −B rn , where n is very nearly equal to 6 while m ranges between 11 and 13. Determine the second virial coefficient of a Lennard-Jones gas and compare your result with that for a van der Waals gas; see equation (10.3.8). 10.3. (a) Show that for a gas obeying van der Waals equation of state (10.3.9), CP −CV = Nk  1 − 2a kTv3 (v −b)2 −1 . (b) Also show that, for a van der Waals gas with constant specific heat CV , an adiabatic process conforms to the equation (v −b)TCV /Nk = const; compare with equation (1.4.30). (c) Further show that the temperature change resulting from an expansion of the gas (into vacuum) from volume V1 to volume V2 is given by T2 −T1 = N2a CV  1 V2 −1 V1  . 10.4. The coefficient of volume expansion α and the isothermal bulk modulus B of a gas are given by the empirical expressions α = 1 T  1 + 3a′ vT2  and B = P  1 + a′ vT2 −1 , where a′ is a constant parameter. Show that these expressions are mutually compatible. Also derive the equation of state of this gas. 10.5. Show that the first-order Joule–Thomson coefficient of a gas is given by the formula ∂T ∂P  H = N CP T ∂(a2λ3) ∂T −a2λ3 ! , where a2(T) is the second virial coefficient of the gas and H its enthalpy; see equation (10.2.1). Derive an explicit expression for the Joule–Thomson coefficient in the case of a gas with interparticle interaction u(r) =    +∞ for 0 < r < D, −u0 for D < r < r1, 0 for r1 < r < ∞, and discuss the temperature dependence of this coefficient. Problems 341 10.6. Assume that the molecules of the nitrogen gas interact through the potential of the previous problem. Making use of the experimental data given next, determine the “best” empirical values for the parameters D, r1, and u0/k: T (in K) 100 200 300 400 500 a2λ3 (in K per atm) −1.80 −4.26 × 10−1 −5.49 × 10−2 +1.12 × 10−1 +2.05 × 10−1. 10.7. Determine the lowest-order corrections to the ideal-gas values of the Helmholtz free energy, the Gibbs free energy, the entropy, the internal energy, the enthalpy, and the (constant-volume and constant-pressure) specific heats of a real gas. Discuss the temperature dependence of these corrections in the case of a gas whose molecules interact through the potential of Problem 10.5. 10.8. The molecules of a solid attract one another with a force F(r) = α(l/r)5. Two semi-infinite solids composed of n molecules per unit volume are separated by a distance d, that is, the solids fill the whole of the space with x ≤0 and x ≥d. Calculate the force of attraction, per unit area of the surface, between the two solids. 10.9. Referring to equation (10.5.31) for the phase shifts ηl(k) of a hard-sphere gas, show that for kD ≪1 ηl(k) ≃− (kD)2l+1 (2l + 1){1 · 3···(2l −1)}2 . 10.10. Using the wavefunctions up(r) = 1 p V ei(p·r)ℏ to describe the motion of a free particle, write down the symmetrized wavefunctions for a pair of noninteracting bosons/fermions, and show that ⟨1′,2′| ˆ US/A 2 |1,2⟩= ±⟨2′| ˆ W1|1⟩⟨1′| ˆ W1|2⟩. 10.11. Show that for a gas composed of particles with spin J bS 2(J) = (J + 1)(2J + 1)bS 2(0) + J(2J + 1)bA 2 (0) and bA 2(J) = J(2J + 1)bS 2(0) + (J + 1)(2J + 1)bA 2 (0). 10.12. Show that the coefficient b2 for a quantum-mechanical Boltzmannian gas composed of “spinless” particles satisfies the following relations: b2 = Lim J→∞  1 (2J + 1)2 bS 2(J)  = Lim J→∞  1 (2J + 1)2 bA 2 (J)  = 1 2{bS 2(0) + bA 2 (0)}. Obtain the value of b2, to fifth order in (D/λ), by using the Beth–Uhlenbeck expressions in equations (10.5.36) and (10.5.37), and compare your result with the classical value of b2, namely −(2π/3)(D/λ)3. 10.13. Use a virial expansion approach to determine the first few nontrivial order contributions to the pair correlation function g(r) in d dimensions. Show that the pair correlation function is of the form g(r) = e−βu(r)y(r), where u(r) is the pair potential and y(r) is a smooth function of r. Show that even for the case of hard sphere interaction, y(r) and its first few derivatives are continuous. 10.14. For the particular case of hard spheres, the pressure in the virial equation of state is determined by evaluating the pair correlation function at contact. Write the pair correlation function as g(r) = e−βu(r)y(r) and derive equations (10.7.12) for hard spheres in one, two, and three dimensions. [Hint: For hard spheres, the Boltzmann factor e−βu(r) is a Heaviside step function]. 342 Chapter 10. Statistical Mechanics of Interacting Systems 10.15. Derive the probability distribution w(r) for the distance to the closest neighboring particle using the pair correlation function g(r) and the number density n. Show that in three dimensions w(r) = 4πnr2g(r)exp  − r Z 0 4πns2g(s)ds  , and the average closest-neighbor distance for an ideal gas is r1 = ∞ Z 0 rw(r)dr = 0 4 3 4πn 3 −1/3 . 10.16. Consider a gas, of infinite extent, divided into regions A and B by an imaginary sheet running through the system. The molecules of the gas interact through a potential energy function u(r). Show that the average net force F experienced by all the molecules on the A-side of the sheet caused by all the molecules on the B-side are perpendicular to the plane of the sheet, and that its magnitude (per unit area) is given by F A = −2πn2 3 ∞ Z 0 du dr  g(r)r3dr. 10.17. Show that for a gas of noninteracting bosons, or fermions, the pair correlation function g(r) is given by the expression g(r) = 1 ± gs n2h6 ∞ Z −∞ ei(p·r)/ℏd3p e(p2/2m−µ)/kT ∓1 2 , where gs (= 2s + 1) is the spin multiplicity factor. Note that the upper sign here applies to bosons, the lower one to fermions.14 [Hint: To solve this problem, one may use the method of second quantization, as developed in Chapter 11. The particle density operator ˆ n is then given by the sum, X α,β a† αaβu∗ α(r)uβ(r), whose diagonal terms are directly related to the mean particle density n in the system. The nondiagonal terms give the density fluctuation operator (ˆ n −n), and so on; see equation (11.1.25).] 10.18. Show that, in the case of a degenerate gas of fermions (T ≪TF), the correlation function g(r), for r ≫ℏ/pF, reduces to the expression g(r) −1 = −3(mkT)2 4p3 Fℏr2  sinh πmkTr pFℏ −2 . Note that, as T →0, this expression tends to the limiting form g(r) −1 = − 3ℏ 4π2pFr4 ∝1 r4 . 14Note that, in the classical limit (ℏ→0), the infinitely rapid oscillations of the factor exp{i(p · r)/ℏ} make the integral vanish. Consequently, for an ideal classical gas, the function g(r) is identically equal to 1. Quantum-mechanical systems of identical particles exhibit spatial correlations due to Bose and Fermi statistics even in the absence of interactions. It is not difficult to see that, for nλ3 ≪1 where λ = h/ p (2πmkT), g(r) ≃1 ± 1 gs exp(−2πr2/λ2); compare with equation (5.5.27). Problems 343 10.19. (a) For a dilute gas, the pair correlation function g(r) may be approximated as g(r) ≃exp{−u(r)/kT}. Show that, under this approximation, the virial equation of state (10.7.11) takes the form PV NkT ≃1 −2πn ∞ Z 0 f (r)r2dr, where f (r)[= exp{−u(r)/kT} −1] is the Mayer function, equation (10.1.6). (b) What form will this result take for a gas of hard spheres? Compare your result with that of Problem 1.4. 10.20. Show that the pressure and Helmholtz free energy of a fluid at temperature T can be determined by performing a thermodynamic integration of the inverse of the isothermal compressibility from the chosen density to the ideal gas reference state. 10.21. Show that, for a general Gaussian distribution of variables uj, the average of the exponential of a linear combination of the variables obeys the relation exp X j ajuj  = exp  1 2 X j ajuj 2 . 10.22. Calculate the isothermal compressibility and Helmholtz free energy for the Carnahan–Starling equation of state (10.3.25) and show that the Helmholtz free energy is given by βA N = βAideal N + η(4 −3η) (1 −η)2 , where Aideal is the Helmholtz free energy of a classical monatomic ideal gas at the same density. 10.23. The virial expansion for a two-dimensional system of hard disks gives the following series when expressed in terms of the two-dimensional packing fraction η = πnD2/4: P nkT = 1 + 2η + 3.128018η2 + 4.257854η3 + 5.33689664η4 + 6.363026η5 + 7.352080η6 + 8.318668η7 + 9.27236η8 + 10.2161η9 + ··· ; see Malijevsky and Kolafa (2008). Propose some simple analytical functions f (η) that closely approximate this series. 11 Statistical Mechanics of Interacting Systems: The Method of Quantized Fields In this chapter we present another method of dealing with systems composed of interact-ing particles. This method is based on the concept of a quantized field that is characterized by the field operators ψ(r), and their hermitian conjugates ψ†(r), which satisfy a set of well-defined commutation rules. In terms of these operators, one defines a number oper-ator ˆ N and a Hamiltonian operator ˆ H that provide a suitable representation for a system composed of any finite number of particles and possessing any finite amount of energy. In view of its formal similarity with the Schr¨ odinger formulation, the formulation in terms of a quantized field is generally referred to as the second quantization of the system. For convenience of calculation, the field operators ψ(r) and ψ†(r) are often expressed as superpositions of a set of single-particle wavefunctions {uα(r)}, with coefficients aα and a† α; the latter turn out to be the annihilation and creation operators, which again satisfy a set of well-defined commutation rules. The operators ˆ N and ˆ H then find a convenient expression in terms of the operators aα and a† α, and the final formulation is well-suited for a treatment based on operator algebra; as a result, many calculations, which would otherwise be tedious, can be carried out in a more or less straightforward manner. 11.1 The formalism of second quantization To represent a system of particles by a quantized field, we invoke the field operators ψ(r) and ψ†(r), which are defined for all values of the position coordinate r and which operate on a Hilbert space; a vector in this space corresponds to a particular state of the quantized field. The values of the quantities ψ and ψ†, at all r, represent the degrees of freedom of the field; since r is a continuous variable, the number of these degrees of freedom is innumer-ably infinite. Now, if the given system is composed of bosons, the field operators ψ(r) and ψ†(r) satisfy the commutation rules [ψ(r),ψ†(r′)] = δ(r −r′) (1a) [ψ(r),ψ(r′)] = [ψ†(r),ψ†(r′)] = 0, (1b) Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00011-6 © 2011 Elsevier Ltd. All rights reserved. 345 346 Chapter 11. Statistical Mechanics of Interacting Systems where the symbol [A,B] stands for the commutator (AB −BA) of the given operators A and B. If, on the other hand, the given system is composed of fermions, then the field operators satisfy the rules {ψ(r),ψ†(r′)} = δ(r −r′) (2a) {ψ(r),ψ(r′)} = {ψ†(r),ψ†(r′)} = 0, (2b) where the symbol {A,B} stands for the anticommutator (AB + BA) of the given operators A and B. In the case of fermions, the operators ψ(r) and ψ†(r) possess certain explicit properties that follow directly from (2b), namely ψ(r)ψ(r′) = −ψ(r′)ψ(r), ∴ψ(r)ψ(r) = 0 for all r ; (2c) similarly, ψ†(r)ψ†(r′) = −ψ†(r′)ψ†(r), ∴ψ†(r)ψ†(r) = 0 for all r. (2d) Clearly, no such property holds for the field operators pertaining to bosons. In the sequel we shall see that the mathematical difference between the commutation rules (1) for the boson field operators and rules (2) for the fermion field operators is intimately related to the fundamental difference in the symmetry properties of the respective wavefunctions in the Schr¨ odinger formulation. Of course, in their own place, both sets of rules, (1) and (2), are essentially axiomatic. We now introduce two hermitian operators, the particle-number operator ˆ N and the Hamiltonian operator ˆ H, through definitions that hold for bosons as well as fermions: ˆ N ≡ Z d3rψ†(r)ψ(r) (3) and ˆ H ≡−ℏ2 2m Z d3rψ†(r)∇2ψ(r) + 1 2 ZZ d3r1d3r2ψ†(r1)ψ†(r2)u(r1,r2)ψ(r2)ψ(r1), (4) where u(r1,r2) denotes the two-body interaction potential in the given system. It is quite natural to interpret the product ψ†(r)ψ(r) as the number density operator of the field. The similarity between the foregoing definitions and the expressions for the expectation values of the corresponding physical quantities in the Schr¨ odinger formulation is fairly obvious. However, the similarity is only “formal” because, while there we are concerned with the wavefunctions of the given system (which are c-numbers), here we are concerned with the operators of the corresponding matter field. We can easily verify that, irrespective of 11.1 The formalism of second quantization 347 the commutation rules obeyed by the operators ψ(r) and ψ†(r), the operators ˆ N and ˆ H do commute: [ ˆ N, ˆ H] = 0; (5) accordingly, the operators ˆ N and ˆ H can be diagonalized simultaneously. We now choose a complete orthonormal basis of the Hilbert space, such that any vec-tor |8n⟩among the basis is a simultaneous eigenstate of the operators ˆ N and ˆ H. We may, therefore, denote any particular member of the basis by the symbol |9NE⟩, with the properties ˆ N|9NE⟩= N|9NE⟩, ˆ H|9NE⟩= E|9NE⟩ (6) and ⟨9NE|9NE⟩= 1. (7) The vector |900⟩, which represents the vacuum state of the field and is generally denoted by the symbol |0⟩, is assumed to be unique; it possesses the obvious properties ˆ N|0⟩= ˆ H|0⟩= 0 and ⟨0|0⟩= 1. (8) Next we observe that, regardless of whether we employ the boson commutation rules (1) or the fermion rules (2), the operator ˆ N and the operators ψ(r) and ψ†(r) satisfy the commutation properties [ψ(r), ˆ N] = ψ(r) and [ψ†(r), ˆ N] = −ψ†(r), (9) from which it follows that ˆ Nψ(r)|9NE⟩=  ψ(r) ˆ N −ψ(r)  |9NE⟩= (N −1)ψ(r)|9NE⟩ (10) and ˆ Nψ†(r)|9NE⟩=  ψ†(r) ˆ N + ψ†(r)  |9NE⟩= (N + 1)ψ†(r)|9NE⟩. (11) Clearly, the state ψ(r)|9NE⟩is also an eigenstate of the operator ˆ N, but with eigenvalue (N −1); thus, the application of the operator ψ(r) onto the state |9NE⟩of the field anni-hilates one particle from the field. Similarly, the state ψ†(r)|9NE⟩is an eigenstate of the operator ˆ N, with eigenvalue (N + 1); thus, the application of the operator ψ†(r) onto the state |9NE⟩of the field creates a particle in the field. In each case, the process (of annihi-lation or creation) is tied down to the point r of the field; however, the energy associated with the process, which also means the change in the energy of the field, remains undeter-mined; see equations (18) and (19). By a repeated application of the operator ψ† onto the vacuum state |0⟩, we find that the eigenvalues of the operator ˆ N are 0,1,2,.... 348 Chapter 11. Statistical Mechanics of Interacting Systems On the other hand, the application of the operator ψ onto the vacuum state |0⟩gives nothing but zero because, for obvious reasons, we cannot admit negative eigenvalues for the operator ˆ N. Of course, if we apply the operator ψ onto the state |9NE⟩repeatedly N times, we end up with the vacuum state; we then have, by virtue of the orthonormality of the basis chosen, ⟨8n|ψ(r1)ψ(r2)...ψ(rN)|9NE⟩= 0 (12) unless the state |8n⟩is itself the vacuum state, in which case we would obtain a nonzero result instead. In terms of this latter result, we may define a function of the N coordinates r1,r2,...,rN, namely 9NE(r1,...,rN) = (N!)−1/2⟨0|ψ(r1)...ψ(rN)|9NE⟩. (13) Obviously, the function 9NE(r1,...,rN) has something to do with an assemblage of N par-ticles located at the points r1,...,rN of the field because their annihilation from those very points of the field has led us to the vacuum state of the field. To obtain the precise meaning of this function, we first note that in the case of bosons (fermions) this function is sym-metric (antisymmetric) with respect to an interchange of any two of the N coordinates; see equations (1b) and (2b), respectively. Secondly, its norm is equal to unity, which can be seen as follows. By the very definition of 9NE(r1,...,rN), Z d3Nr9∗ NE(r1,...,rN)9NE(r1,...,rN) = (N!)−1 Z d3Nr⟨9NE|ψ†(rN)...ψ†(r1)|0⟩⟨0|ψ(r1)...ψ(rN)|9NE⟩ = (N!)−1 Z d3Nr X n ⟨9NE|ψ†(rN)...ψ†(r1)|8n⟩⟨8n|ψ(r1)...ψ(rN)|9NE⟩ = (N!)−1 Z d3Nr⟨9NE|ψ†(rN)...ψ†(r2)ψ†(r1)ψ(r1)ψ(r2)...ψ(rN)|9NE⟩; here, use has been made of equation (12), which holds for all |8n⟩except for the vacuum state, and of the fact that the summation of |8n⟩⟨8n| over the complete orthonormal set of the basis chosen is equivalent to a unit operator. We now carry out integration over r1, yielding the factor Z d3r1ψ†(r1)ψ(r1) = ˆ N. Next, we carry out integration over r2, yielding the factor Z d3r2ψ†(r2) ˆ Nψ(r2) = Z d3r2ψ†(r2)ψ(r2)( ˆ N −1) = ˆ N( ˆ N −1); 11.1 The formalism of second quantization 349 see equation (10). By iteration, we obtain Z d3Nr9∗ NE(r1,...,rN)9NE(r1,...,rN) = (N!)−1⟨9NE| ˆ N( ˆ N −1)( ˆ N −2)... up to N factors|9NE⟩ = (N!)−1N!⟨9NE|9NE⟩= 1. (14) Finally, we can show that, for bosons as well as fermions, the function 9NE(r1,...,rN) satisfies the differential equation, see Problem 11.1,  −ℏ2 2m N X i=1 ∇2 i + X i 0, the conden-sate wavefunction expands in every direction relative to the noninteracting Bose–Einstein condensate. For Na/aosc ≫1, the kinetic energy term can be neglected in the style of a Thomas-Fermi analysis, so the wavefunction becomes approximately 9(r) ≈ p (µ −V(r))/u0 , (24) where the Thomas–Fermi wavefunction vanishes for V(r) > µ. The chemical potential µ and the number of bosons N are related by N = 8π 15 2µ mω2 0 !3/2 µ u0 , (25) 360 Chapter 11. Statistical Mechanics of Interacting Systems where ω0 = (ω1ω2ω3)1/3, so µ = 1 2 15Na aosc 2/5 ℏω0. (26) The total energy of the condensate in this limit is E = 5 7µN. (27) The linear extents of the condensate in the three directions of the trap are given by Rα = s 2µ mω2 α = aosc 15Na aosc 1/5 ω0 ωα , (28) so the repulsive interactions expand the size of the condensate, making the aniostropy of the system larger than that of the noninteracting Bose–Einstein condensate; see Pitaevskii and Stringari (2003); Leggett (2006), and Pethick and Smith (2008). In time-of-flight mea-surements, the repulsive interactions result in higher velocities in the directions that were most confined in the trap, so the time-of-flight distributions are also more anisotropic than in the noninteracting case; see Holland and Cooper (1996) and Holland et al. (1997). For attractive interactions with negative scattering lengths, the condensate is ultimately unstable because of the formation of pairs, but a long-lived atomic condensate exists for small negative Na/aosc. For the isotropic case, the atomic condensate is metastable in the mean field theory for −0.575 ≲Na/aosc < 0. Even in the anisotropic case, the condensate is 7 (b) 6 5 4 3 2 1 0 0 20 40 60 2 1.5 1 0.5 0.5 0 0.5 1 1.5 Nonlinear Constant Ground-State Energy 80 100 FIGURE 11.1 Ground-state energy of a Bose–Einstein condensate in an isotropic harmonic trap as a function of the scattering length a. The energy is plotted in units of Nℏω0, where ω0 is the trap frequency. The “nonlinear constant” is proportional to Na/aosc, where N is the number of atoms and aosc = √ℏ/mω0 is the width of the harmonic oscillator ground state wavefunction. Figure from Ruprecht et al. (1995). Reprinted with permission; copyright © 1995, American Physical Society. 11.3 Low-lying states of an imperfect Bose gas 361 nearly spherical since the solution of equation (23) is dominated by the kinetic and inter-action terms; see Ruprecht et al. (1995) and Figure 11.1. Roberts et al. (2001) have used a Feshbach resonance to tune the scattering length of 85Rb to find that the condensate becomes unstable at N|a|/aosc ≃0.46. 11.3 Low-lying states of an imperfect Bose gas In the preceding section we examined first-order corrections to the low-temperature behavior of an imperfect Bose gas arising from interparticle interactions in the system. One important result emerging in that study was a nonzero velocity of sound, as given by equation (11.2.8). This raises the possibility that phonons, the quanta of sound field, might play an important role in determining the low-temperature behavior of this system — a role not seen in Section 11.2. To look into this question, we explore the nature of the low-lying states of an imperfect Bose gas, in the hope that we thus discover an energy-momentum relation ε(p) obeyed by the elementary excitations of the system, of which phonons may be an integral part. For this, we have to go a step beyond the approximation adopted in Section 11.2 which, in turn, requires several significant improvements. To keep matters simple, we confine ourselves to situations in which the fraction of particles occu-pying the state with p = 0 is fairly close to 1 while the fraction of particles occupying states with p ̸= 0 is much less than 1. Going back to equations (11.2.1) through (11.2.4), we first write 2N2 −n2 0 = N2 + (N2 −n2 0) ≃N2 + 2N(N −n0) = N2 + 2N X p̸=0 a† pap. (1) Next, we retain another set of terms from the sum P′ in equation (11.1.46) — terms that involve a nonzero momentum transfer, namely X p̸=0 u(p)[a† pa† −pa0a0 + a† 0a† 0apa−p]. (2) Now, since a† 0a0 = n0 = O(N) and (a0a† 0 −a† 0a0) = 1 ≪N, it follows that a0a† 0 = (n0 + 1) ≃ a† 0a0. The operators a0 and a† 0 may, therefore, be treated as c-numbers, each equal to n1/2 0 ≃ N1/2. At the same time, the amplitude u(p) in the case of low-lying states may be replaced by u0/V, as before. Expression (2) then becomes u0N V X p̸=0 (a† pa† −p + apa−p). (3) 362 Chapter 11. Statistical Mechanics of Interacting Systems In view of these results, the Hamiltonian of the system assumes the form ˆ H = X p p2 2ma† pap + u0 2V  N2 + N X p̸=0 (2a† pap + a† pa† −p + apa−p)  . (4) Our next task consists of determining an improved relationship between the quantity u0 and the scattering length a. While the (approximate) result stated in equation (11.1.51) is good enough for evaluating the term involving N P p̸=0, it is not so for evaluating the term involving N2. For this, we note that “if the probability of a particular quantum tran-sition in a given system under the influence of a constant perturbation ˆ V is, in the first approximation, determined by the matrix element V 0 0 , then in the second approximation we have instead V 0 0 + X n̸=0 V 0 nV n 0 E0 −En , the summation going over the various states of the unperturbed system.” In the present case, we are dealing with a collision process in the two-particle system (with reduced mass 1 2m), and the role of V 0 0 is played by the quantity u00 00 = 1 V Z u(r)d3r = u0 V ; see equation (11.1.44) for the matrix element u p′ 1,p′ 2 p1,p2. Making use of the other matrix ele-ments, we find that in going from the first to second approximation, we have to replace u0/V by u0 V + 1 V 2 X p̸=0 | R d3reip·r/ℏu(r)|2 −p2/m ≃u0 V −u2 0m V 2 X p̸=0 1 p2 . (5) Equating (5) with the standard expression 4πaℏ2/mV, we obtain, instead of (11.1.51), u0 ≃4πaℏ2 m  1 + 4πaℏ2 V X p̸=0 1 p2  . (6) Substituting (6) into (4), we get ˆ H =2πaℏ2 m N2 V  1 + 4πaℏ2 V X p̸=0 1 p2   + 2πaℏ2 m N V X p̸=0 (2a† pap + a† pa† −p + apa−p) + X p̸=0 p2 2ma† pap. (7) 11.3 Low-lying states of an imperfect Bose gas 363 To evaluate the energy levels of the system one would have to diagonalize the Hamilton-ian (7), which can be done with the help of a linear transformation of the operators ap and a† p, first employed by Bogoliubov (1947): bp = ap + αpa† −p √(1 −α2 p) , b† p = a† p + αpa−p √(1 −α2 p) , (8) where αp = mV 4πaℏ2N ( 4πaℏ2N mV + p2 2m −ε(p) ) , (9) with ε(p) =    4πaℏ2N mV p2 m + p2 2m !2   1/2 ; (10) clearly, each αp < 1. Relations inverse to (8) are ap = bp −αpb† −p √(1 −α2 p) , a† p = b† p −αpb−p √(1 −α2 p) . (11) It is straightforward to check that the new operators bp and b† p satisfy the same commuta-tion rules as the old operators ap and a† p did, namely [bp,b† p′] = δpp′ (12a) [bp,bp′] = [b† p,b† p′] = 0. (12b) Substituting (11) into (7), we obtain our Hamiltonian in the diagonalized form: ˆ H = E0 + X p̸=0 ε(p)b† pbp, (13) where E0 = 2πaℏ2N2 mV + 1 2 X p̸=0   ε(p) −p2 2m −4πaℏ2N mV + 4πaℏ2N mV !2 m p2   . (14) In view of the commutation rules (12) and expression (13) for the Hamiltonian oper-ator ˆ H, it seems natural to infer that the operators bp and b† p are the annihilation and creation operators of certain “quasiparticles” — which represent elementary excitations of the system — with the energy–momentum relation given by (10); it is also clear that these quasiparticles obey Bose–Einstein statistics. The quantity b† pbp is then the particle-number 364 Chapter 11. Statistical Mechanics of Interacting Systems operator for the quasiparticles (or elementary excitations) of momentum p, whereby the second part of the Hamiltonian (13) becomes the energy operator corresponding to the elementary excitations present in the system. The first part of the Hamiltonian, given explicitly by equation (14), is therefore the ground-state energy of the system. Replacing the summation over p by integration and introducing a dimensionless variable x, defined by x = p  V 8πaℏ2N 1/2 , we obtain for the ground-state energy of the system E0 =2πaℏ2N2 mV  1 + 128Na3 πV !1/2 × ∞ Z 0 dx  x2  x√(x2 + 2) −x2 −1 + 1 2x2  . (15) The value of the integral turns out to be (128)1/2/15, with the result E0 N = 2πaℏ2n m  1 + 128 15π1/2 (na3)1/2  , (16) where n denotes the particle density in the system. Equation (16) represents the first two terms of the expansion of the quantity E0/N in terms of the low-density parameter (na3)1/2; the first term was already obtained in Section 11.2.7 The foregoing result was first derived by Lee and Yang (1957) using the binary collision method; the details of this calculation, however, appeared somewhat later (see Lee and Yang, 1960a; see also Problem 11.6). Using the pseudopotential method, this result was rederived by Lee, Huang, and Yang (1957). The ground-state pressure of the system is now given by P0 = − ∂E0 ∂V  N = n2 ∂(E0/N) ∂n = 2πaℏ2n2 m  1 + 64 5π1/2 (na3)1/2  , (17) 7The evaluation of higher-order terms of this expansion necessitates consideration of three-body collisions as well; hence, in general, they cannot be expressed in terms of the scattering length alone. The exceptional case of a hard-sphere gas has been studied by Wu (1959), who obtained (using the pseudopotential method) E0 N = 2πaℏ2n m  1 + 128 15π1/2 (na3)1/2 + 8  4π 3 − √ 3  (na3)ln(12πna3) + O(na3)  , which shows that the expansion does not proceed in simple powers of (na3)1/2. 11.3 Low-lying states of an imperfect Bose gas 365 from which one obtains for the velocity of sound c2 0 = 1 m dP0 dn = 4πaℏ2n m2  1 + 16 π1/2 (na3)1/2  . (18) Equations (17) and (18) are an improved version of the results obtained in Section 11.2. The ground state of the system is characterized by a total absence of excitations; accordingly, the eigenvalue of the (number) operator b† pbp of the quasiparticles must be zero for all p ̸= 0. As for the real particles, there must be some that possess nonzero ener-gies even at absolute zero, for otherwise the system cannot have a finite amount of energy in the ground state. The momentum distribution of the real particles can be determined by evaluating the ground-state expectation values of the number operators a† pap. Now, in the ground state of the system, ap|90⟩= 1 √(1 −α2 p) (bp −αpb† −p)|90⟩= −αp √(1 −α2 p) b† −p|90⟩ (19) because bp|90⟩≡0. Constructing the hermitian conjugate of (19) and remembering that αp is real, we have ⟨90|a† p = −αp √(1 −α2 p) ⟨90|b−p. (20) The scalar product of expressions (19) and (20) gives ⟨90|a† pap|90⟩= α2 p 1 −α2 p ⟨90|b−pb† −p|90⟩= α2 p 1 −α2 p ; (21) here, use has been made of the facts that (i) bpb† p −b† pbp = 1 and (ii) in the ground state, for all p ̸= 0,b† pbp = 0 (and hence bpb† p = 1). Thus, for p ̸= 0, np = α2 p 1 −α2 p = x2 + 1 2x√(x2 + 2) −1 2, (22) where x = p(8πaℏ2n)−1/2. The total number of “excited” particles in the ground state of the system is, therefore, given by X p̸=0 np = X p̸=0 α2 p 1 −α2 p = X x>0 1 2 x2 + 1 x√(x2 + 2) −1 ! ≃N 32 π (na3) 1/2 ∞ Z 0 dx " x2 x2 + 1 x√(x2 + 2) −1 !# . (23) 366 Chapter 11. Statistical Mechanics of Interacting Systems The value of the integral turns out to be (2)1/2/3, with the result X p̸=0 np ≃N 8 3π1/2 (na3)1/2. (24) Accordingly, n0 = N − X p̸=0 np ≃N  1 − 8 3π1/2 (na3)1/2  . (25) The foregoing result was first obtained by Lee, Huang, and Yang (1957), using the pseu-dopotential method. It may be noted here that the importance of the real-particle occupa-tion numbers np in the study of the ground state of an interacting Bose system had been emphasized earlier by Penrose and Onsager (1956). 11.4 Energy spectrum of a Bose liquid In this section we propose to study the most essential features of the energy spectrum of a Bose liquid and to examine the relevance of this study to the problem of liquid He4. In this context we have seen that the low-lying states of a low-density gaseous system composed of weakly interacting bosons are characterized by the presence of the so-called elementary excitations (or “quasiparticles”), which are themselves bosons and whose energy spectrum is given by ε(p) = {p2u2 + (p2/2m)2}1/2, (1) where u = (4πan)1/2(ℏ/m); (2) see equations (11.3.10), (11.3.12), and (11.3.13).8 For p ≪mu, that is, p ≪ℏ(an)1/2, the spectrum is essentially linear: ε ≃pu. The initial slope of the (ε,p)-curve is, therefore, given by the parameter u, which is identical to the limiting value of the velocity of sound in the system; compare (2) with (11.3.18). It is then natural that these low-momentum excita-tions be identified as phonons — the quanta of the sound field. For p ≫mu, the spectrum approaches essentially the classical limit: ε ≃p2/2m + 1∗, where 1∗= mu2 = 4πanℏ2/m. It is important to note that, all along, this energy–momentum relationship is strictly mono-tonic and does not display any “dip” of the kind propounded by Landau (1941, 1947) (for 8Spectrum (1) was first obtained by Bogoliubov (1947) by the method outlined in the preceding sections. Using the pseudopotential method, it was rederived by Lee, Huang, and Yang (1957). 11.4 Energy spectrum of a Bose liquid 367 liquid He4) and observed experimentally by Yarnell et al. (1959), and by Henshaw and Woods (1961); see Section 7.6. Thus, the spectrum provided by the theory of the preceding sections simulates the Landau spectrum only to the extent of phonons; it does not account for rotons. This should not be surprising, for the theory in question was intended only for a low-density Bose gas (na3 ≪1) and not for liquid He4(na3 ≃0.2). The problem of elementary excitations in liquid He4 was tackled successfully by Feynman who, in 1953 to 1954, developed an atomic theory of a Bose liquid at low tem-peratures. In a series of three fundamental papers starting from first principles, Feynman established the following important results.9 (i) In spite of the presence of interatomic forces, a Bose liquid undergoes a phase transition analogous to the momentum-space condensation occurring in the ideal Bose gas; in other words, the original suggestion of London (1938a,b) regarding liquid He4, see Section 7.1, is essentially correct. (ii) At sufficiently low temperatures, the only excited states possible in the liquid are the ones related to compressional waves, namely phonons. Long-range motions, which leave the density of the liquid unaltered (and consequently imply nothing more than a simple “stirring” of the liquid), do not constitute excited states because they differ from the ground state only in the “permutation” of certain atoms. Motions on an atomic scale are indeed possible, but they require a minimum energy 1 for their excitation; clearly, these excitations would show up only at comparatively higher temperatures (T ∼1/k) and might well turn out to be Landau’s rotons. (iii) The wavefunction of the liquid, in the presence of an excitation, should be approximately of the form 9 = 8 X i f (ri), (3) where 8 denotes the ground-state wavefunction of the system while the summation of f (ri) goes over all the N coordinates r1,...,rN; the wavefunction 9 is, clearly, symmetric in its arguments. The exact character of the function f (r) can be determined by making use of a variational principle that requires the energy of the state 9 (and hence the energy associated with the excitation in question) to be a minimum. The optimal choice for f (r) turns out to be, see Problem 11.8, f (r) = expi(k · r), (4) 9The reader interested in pursuing Feynman’s line of argument should refer to Feynman’s original papers or to a review of Feynman’s work on superfluidity by Mehra and Pathria (1994). 368 Chapter 11. Statistical Mechanics of Interacting Systems with the (minimized) energy value ε(k) = ℏ2k2 2mS(k), (5) where S(k) is the structure factor of the liquid, that is, the Fourier transform of the pair correlation function g(r): S(k) = 1 + n Z (g(r) −1)eik·rdr; (6) it may be recalled here that the function ng(r2 −r1) is the probability density for finding an atom in the neighborhood of the point r2 when another one is known to be at the point r1; see Section 10.7. The optimal wavefunction is, therefore, given by 9 = 8 X i eik·ri. (7) Now the momentum associated with this excited state is ℏk because P9 = −iℏ X i ∇i ! 9 = ℏk9, (8) P8 being identically equal to zero. Naturally, this would be interpreted as the momentum p associated with the excitation. One thus obtains, from first principles, the energy– momentum relationship for the elementary excitations in a Bose liquid. On physical grounds one can show that, for small k, the structure factor S(k) rises lin-early as ℏk/2mc, reaches a maximum near k = 2π/r0 (corresponding to a maximum in the pair correlation function at the nearest-neighbor spacing r0, which for liquid He4 is about 3.6 ˚ A) and thereafter decreases to approach, with minor oscillations (corresponding to the subsidiary maxima in the pair correlation function at the spacings of the next near-est neighbors), the limiting value 1 for large k; the limiting value 1 arises from the presence of a delta function in the expression for g(r) (because, as r2 →r1, one is sure to find an atom there).10 Accordingly, the energy ε(k) of an elementary excitation in liquid He4 would start linearly as ℏkc, show a “dip” at k0 ≃2˚ A−1 and rise again to approach the eventual limit of ℏ2k2/2m.11 These features are shown in Figure 11.2. Clearly, Feynman’s approach merges both phonons and rotons into a single, unified scheme in which they represent 10For a microscopic study of the structure factor S(k), see Huang and Klein (1964); also Jackson and Feenberg (1962). 11It is natural that at some value of k < k0, the (ε,k)-curve passes through a maximum; this happens when dS/dk = 2S/k. 11.4 Energy spectrum of a Bose liquid 369 0 5 10 20 30 40 0.5 0 1.5 1.0 15 25 35 0.5 1.0 1.5 2.0 2.5 3.0 exptl exptl T 51.06 K 2 1 3.5 4.0 k, in A21 ´(k), in K S(k) FIGURE 11.2 The energy spectrum of the elementary excitations in liquid He4. The upper portion shows the structure factor of the liquid, as derived by Henshaw (1960) from experimental data on neutron diffraction. Curve 1 in the lower portion shows the energy–momentum relationship based on the Feynman formula (5) while curve 2 is based on an improved formula by Feynman and Cohen (1956). For comparison, the experimental results of Woods (1966) obtained directly from neutron scattering are also included. different parts of a common (and continuous) energy spectrum ε(k), as determined by the structure of the liquid through the function S(k). Since no motion of a rotational character is involved here, the name “roton” is clearly a misnomer. It seems appropriate to mention here that, soon after the work of London, which advo-cated a connection between the phase transition in liquid He4 and the phenomenon of Bose–Einstein condensation, Bijl (1940) investigated the mathematical structure of the wavefunctions appropriate to an interacting Bose gas and the excitation energy associ-ated with those wavefunctions. His picture corresponded very closely to Feynman’s and indeed led to the wavefunction (7). Bijl also derived an expression for ε(k) that was exactly the same as (5). Unfortunately, he could not make much use of his results — primarily because he leaned too heavily on the expansion S(k) = S(0) + C2k2 + C4k4 + ···, (9) which, as we now know, represents neither phonons nor rotons. 370 Chapter 11. Statistical Mechanics of Interacting Systems 11.5 States with quantized circulation We now proceed to examine the possibility of “organized motion” in the ground state of a Bose fluid. In this context, the most important concept is embodied in the cir-culation theorem of Feynman (1955), which establishes a physical basis for the existence of “quantized vortex motion” in the fluid. In the case of liquid helium II, this concept has successfully resolved some of the vital questions that baffled superfluid physicists for a long time. The ground-state wavefunction of a superfluid, composed of N bosons, may be denoted by a symmetric function 8(r1,...,rN); if the superfluid does not partake in any organized motion, then 8 will be a pure real number. If, on the other hand, it possesses a uniform mass-motion with velocity vs, then its wavefunction would be 9 = 8ei(Ps·R)/ℏ= 8eim(vs·6iri)/ℏ, (1) where Ps denotes the total momentum of the fluid and R its center of mass: Ps = Nmvs; R = N−1 X i ri. (2) The wavefunction (1) is exact if the drift velocity vs is uniform throughout the fluid. If vs is nonuniform, then the present wavefunction would still be good locally — in the sense that the phase change 1φ resulting from a “set of local displacements” of the atoms (over distances too small for velocity variations to be appreciable) would be practically the same as the one following from expression (1). Thus, for a given set of displacements 1ri of the atoms constituting the fluid, the change in the phase of the wavefunction would very nearly be 1φ = m ℏ X i (vsi · 1ri), (3) where vs is now a function of r. The foregoing result may be used for calculating the net phase change resulting from a displacement of atoms along a ring, from their original positions in the ring to the neighboring ones, so that after displacement we obtain a configuration that is physically identical to the one we started with; see Figure 11.3. In view of the symmetry of the wave-function, the net phase change resulting from such a displacement must be an integral multiple of 2π (so that the wavefunction after the displacement is identical to the one before the displacement): m ℏ X′ i(vsi · 1ri) = 2πn, n = 0,±1,±2,...; (4) the summation P′ here goes over all the atoms constituting the ring. We note that, for the foregoing result to be valid, it is only the individual 1ri that have to be small, not the whole perimeter of the ring. Now, for a ring of a macroscopic size, one may regard the fluid as a 11.5 States with quantized circulation 371 FIGURE 11.3 The wavefunction of the fluid must not change as a result of a permutation of the atoms. If all the atoms are displaced around a ring, as shown, the phase change must be a multiple of 2π. continuum; equation (4) then becomes I vs · dr = n h m, n = 0,±1,±2,.... (5) The quantity on the left side of this equation is, by definition, the circulation (of the flow) associated with the circuit of integration and is clearly quantized, the “quantum of circu-lation” being h/m. Equation (5) constitutes the circulation theorem of Feynman; it bears a striking resemblance to the quantum condition of Bohr, namely I pdq = nh, (6) though the region of application here is macroscopic rather than microscopic.12 By Stokes’ theorem, equation (5) may be written as Z S (curl vs) · dS = n h m, n = 0,±1,±2,..., (7) where S denotes the area enclosed by the circuit of integration. If this area is “simply-connected” and the velocity vs is continuous throughout the area, then the domain of integration can be shrunk in a continuous manner without limit. The integral on the left side is then expected to decrease continuously and finally tend to zero. The right side, however, cannot vary continuously. We infer that in this particular case the quan-tum number n must be zero, that is, our integral must be identically vanishing. Thus, “in a simply-connected region, in which the velocity field is continuous throughout, the condition curl vs = 0 (8) 12That the vortices in a superfluid may be quantized, the quantum of circulation being h/m, was first suggested by Onsager (1949) in a footnote to a paper dealing with the classical vortex theory and the theory of turbulence! 372 Chapter 11. Statistical Mechanics of Interacting Systems holds everywhere.” This is precisely the condition postulated by Landau (1941), which has been the cornerstone of our understanding of the hydrodynamic behavior of superfluid helium.13 Clearly, the Landau condition is only a special case of the Feynman theorem. It is quite possible that in a “multiply-connected” domain, which cannot be shrunk continuously to zero (without encountering singularities in the velocity field), the Landau condition may not hold everywhere. A typical example of such a domain is provided by the flow of a vortex, which is a planar flow with cylindrical symmetry, such that vρ = 0, vφ = K 2πρ , vz = 0, (9) where ρ is the distance measured perpendicular to the axis of symmetry while K is the circulation of the flow: I v · dr = I vφ(ρ dφ) = K; (10) note that the circuit of integration in (10) must enclose the axis of the vortex. Another version of the foregoing result is Z S (curl v) · dS = Z S  1 ρ d dρ (ρvφ)  (2πρ dρ) = K. (11) Now, at all ρ ̸= 0, curl v = 0 but at ρ = 0, where vφ is singular, curl v appears to be indeter-minate; it is not difficult to see that, at ρ = 0, curl v diverges (in such a way that the integral in (11) turns out to be finite). In this context, it seems worthwhile to point out that if we carry out the integration in (10) along a circuit that does not enclose the axis of the vortex, or in (11) over a region that does not include the point ρ = 0, the result would be identically zero. At this stage we note that the energy associated with a unit length of a classical vortex is given by E L = b Z a 1 2{2πρ dρ(mn0)}  K 2πρ 2 = mn0K 2 4π ln(b/a). (12) 13Drawing on the well-known analogy between the phenomena of superfluidity and superconductivity, and the resulting correspondence between the mechanical momentum mvs of a superfluid particle and the electromagnetic momentum 2eA/c of a Cooper pair of electrons, we observe that the relevant counterpart of the Landau condition, in superconductors, would be curl A ≡B = 0, (8a) which is precisely the Meissner effect; furthermore, the appropriate counterpart of the Feynman theorem would be Z S B · dS = n hc 2e , (7a) which leads to the “quantization of the magnetic flux,” the quantum of flux being hc/2e. 11.5 States with quantized circulation 373 Here, (mn0) is the mass density of the fluid (which is assumed to be uniform), the upper limit b is related to the size of the container while the lower limit a depends on the structure of the vortex; in our study, a would be comparable to the interatomic separation. In the quantum-mechanical case we may describe our vortex through a self-consistent wavefunction ψ(r), which, in the case of cylindrical symmetry, see equation (9), may be written as ψ(r) = n∗1/2eisφfs(ρ), (13) so that n(r) ≡|ψ(r)|2 = n∗f 2 s (ρ). (14) As ρ →∞, fs(ρ) →1, so that n∗becomes the limiting particle density in the fluid in regions far away from the axis of the vortex. The velocity field associated with this wavefunction will be v(r) = ℏ 2im(ψ∗ψ)(ψ∗∇ψ −ψ∇ψ∗) = ℏ m∇(sφ) =  0,s ℏ mρ ,0  . (15)14 Comparing (15) with (9), we conclude that the circulation K in the present case is sh/m; by the circulation theorem, s must be an integer: s = 0,±1,±2,.... (16) Clearly, the value 0 is of no interest to us. Furthermore, the negative values of s differ from the positive ones only in the “sense of rotation” of the fluid. It is, therefore, sufficient to consider the positive values alone, namely s = 1,2,3,.... (17) The function fs(ρ) appearing in equation (13) may be determined with the help of a Schr¨ odinger equation in which the potential term is itself ψ-dependent, namely −ℏ2 2m∇2 + u0|ψ|2 ! ψ = εψ, (18) 14It is of interest to see that the angular momentum per particle in the fluid is given by 1 ψ  ℏ i ∂ ∂φ ψ  = sℏ(= mvφρ); this is again reminiscent of the quantum condition of Bohr. 374 Chapter 11. Statistical Mechanics of Interacting Systems where u0 is given by equation (11.1.51): u0 = 4πaℏ2/m, (19) a being the scattering length of the interparticle interaction operating in the fluid. The characteristic energy ε follows from the observation that, at large distances from the axis of the vortex, the fluid is essentially uniform in density, with n(r) →n∗; equation (18) then gives ε = u0n∗= 4πaℏ2n∗/m, (20) which may be compared with equation (11.2.9). Substituting (20) into (18) and remember-ing that the flow is cylindrically symmetrical, we get − " 1 ρ d dρ  ρ d dρ fs(ρ)  −s2 ρ2 fs(ρ) # + 8πan∗f 3 s (ρ) = 8πan∗fs(ρ). (21) Expressing ρ in terms of a characteristic length l, ρ = lρ′ {l = (8πan∗)−1/2}, (22) we obtain d2fs dρ′2 + 1 ρ′ dfs dρ′ + 1 −s2 ρ′2 ! fs −f 3 s = 0. (23) Toward the axis of the vortex, where ρ →0, the very high velocity of the fluid (and the very large centrifugal force accompanying it) will push the particles outward, thus causing an enormous decrease in the density of the fluid. Consequently, the function fs should tend to zero as ρ →0. This will make the last term in equation (23) negligible and thereby reduce it to the familiar Bessel’s equation. Accordingly, for small ρ, fs(ρ′) ∼Js(ρ′) ∼ρs, (24) Js being the ordinary Bessel function of order s. For ρ′ ≫1,fs ≃1; then, the first two terms of equation (23) become negligible, with the result fs(ρ′) ≃1 −s2 2ρ′2 . (25) The full solution is obtained by integrating the equation numerically; the results so obtained are shown in Figure 11.4 where solutions for s = 1,2, and 3 are displayed. We thus find that our model of an imperfect Bose gas does allow for the presence of quantized vortices in the system. Not only that, we do not have to invoke here any special assumptions regarding the nature of the “core” of the vortex (as one has to do in the classi-cal theory); our treatment naturally leads to a continual diminution of the particle density 11.5 States with quantized circulation 375 0 0 0.2 0.4 0.6 0.8 1.0 1 2 3 s51 s 52 s 53 4 5 6 7 8 fs() FIGURE 11.4 Solutions of equation (23) for various values of s (after Kawatra and Pathria, 1966). n as the axial line is approached, so that there does not exist any specific distribution of vorticity around this line. The distance scale, which governs the spatial variation of n, is provided by the parameter l of equation (22); for liquid He4,l ≃1 ˚ A. Pitaevskii (1961), who was among the first to demonstrate the possibility of obtaining solutions whose natural interpretation lay in quantized vortex motion (see also Gross, 1961; Weller, 1963), also evaluated the energy per unit length of the vortex. Employ-ing wavefunction (13), with known values of the functions fs(ρ), Pitaevskii obtained the following results for the energy per unit length of the vortex, with s = 1,2, or 3, n∗h2 4πm  1ln(1.46R/l), 4ln(0.59R/l), 9ln(0.38R/l) , (26) where R denotes the outer radius of the domain involved. The above results may be compared with the “semiclassical” ones, namely n0h2 4πm  1ln(R/a), 4ln(R/a), 9ln(R/a) , (27) which follow from formula (12), with K replaced by sh/m and b by R. It is obvious that vortices with s > 1 would be relatively unstable because energetically it would be cheaper for a system to have s vortices of unit circulation rather than a single vortex of circulation s. The existence of quantized vortex lines in liquid helium II has been demonstrated con-vincingly by the ingenious experiments of Vinen (1958–1961) in which the circulation K around a fine wire immersed in the liquid was measured through the influence it exerts on the transverse vibrations of the wire. Vinen found that while vortices with unit circulation were exceptionally stable those with higher circulation too made an appearance. Repeat-ing Vinen’s experiment with thicker wires, Whitmore and Zimmermann (1965) were able 376 Chapter 11. Statistical Mechanics of Interacting Systems to observe stable vortices with circulation up to three quantum units. For a survey of this and other aspects of the superfluid behavior, see Vinen (1968) and Betts et al. (1969). Kim and Chan (2004) have even observed a “supersolid” phase of helium-4 at low temperatures that has the crystalline structure of a solid while also exhibiting superfuid-like flow. For the relevance of quantized vortex lines to the problem of “rotation” of the super-fluid, see Section 10.7 of the first edition of this book. 11.6 Quantized vortex rings and the breakdown of superfluidity Feynman (1955) was the first to suggest that the formation of vortices in liquid helium II might provide the mechanism responsible for the breakdown of superfluidity in the liquid. He considered the flow of liquid helium II from an orifice of diameter D and, by tentative arguments, found that the velocity v0 at which the flow energy available would just be sufficient to create quantized vortices in the liquid is given by v0 = ℏ mD ln(D/l). (1) Thus, for an orifice of diameter 10−5 cm, v0 would be of the order of 1m/s.15 It is tempting to identify v0 with vc, the critical velocity of superflow through the given capillary, despite the fact that this theoretical estimate for v0 is an order of magnitude higher than the cor-responding experimental values of vc; the latter, for instance, are 13 cm/s,8 cm/s, and 4 cm/s for capillary diameters 1.2 × 10−5 cm, 7.9 × 10−5 cm, and 3.9 × 10−4 cm, respec-tively. Nevertheless, the present estimate is far more desirable than the prohibitively large ones obtained earlier on the basis of a possible creation of phonons or rotons in the liquid; see Section 7.6. Moreover, one obtains here a definitive dependence of the critical veloc-ity of superflow on the width of the capillary employed which, at least qualitatively, agrees with the trend seen in the experimental findings. In what follows, we propose to develop Feynman’s idea further along the lines suggested by the preceding section. So far we have been dealing with the so-called linear vortices whose velocity field pos-sesses cylindrical symmetry. More generally, however, a vortex line need not be straight — it may be curved and, if it does not terminate on the walls of the container or on the free surface of the liquid, may close on itself. We then speak of a vortex ring, which is very much like a smoke ring. Of course, the quantization condition (11.5.5) is as valid for a vortex ring as for a vortex line. However, the dynamical properties of a ring are quite different from those of a line; see, for instance, Figure 11.5, which shows schematically a vortex ring in cross-section, the radius r of the ring being much larger than the core dimension l. The flow velocity vs at any point in the field is determined by a superposition of the flow velo-cities due to the various elements of the ring. It is not difficult to see that the velocity field 15We have taken here: l ≃1 ˚ A, so that ln(D/l) ≃7. 11.6 Quantized vortex rings and the breakdown of superfluidity 377 r v v r FIGURE 11.5 Schematic illustration of a quantized vortex ring in cross-section. of the ring, including the ring itself, moves in a direction perpendicular to the plane of the ring, with a velocity16 v ∼ℏ/2mr; (2) see equation (11.5.15), with s = 1 and ρ ∼2r. An estimate of the energy associated with the flow may be obtained from expression (11.5.12), with L = 2πr,K = h/m, and b ∼r; thus ε ∼2π2ℏ2n0m−1r ln(r/l). (3) Clearly, the dependence of ε on r arises mainly from the factor r and only slightly from the factor ln(r/l). Therefore, with good approximation, v ∝ε−1, that is, a ring with larger energy moves slower! The reason behind this somewhat startling result is that firstly the larger the ring the larger the distances between the various circulation-carrying elements of the ring (thus reducing the velocity imparted by one element to another) and secondly a larger ring carries with it a larger amount of fluid (M ∝r3), so the total energy associated with the ring is also larger (essentially proportional to Mv2, i.e., ∝r). The product vε, apart from the slowly varying factor ln(r/l), is thus a constant, which is equal to π2ℏ3n0/m2. It is gratifying that vortex rings such as the ones discussed here have been observed and the circulation around them is found to be as close to the quantum h/m as one could expect under the conditions of the experiment. Figure 11.6 shows the experimental results of Rayfield and Reif (1964) for the velocity–energy relationship of free-moving, charge-carrying vortex rings created in liquid helium II by suitably accelerated helium ions. Vortex rings carrying positive as well as negative charge were observed; dynamically, however, they behaved alike, as one indeed expects because both the velocity and the energy asso-ciated with a vortex ring are determined by the properties of a large amount of fluid carried 16This result would be exact if we had a pair of oppositely directed linear vortices, with the same cross-section as shown in Figure 11.5. In the case of a ring, the velocity would be somewhat larger. 378 Chapter 11. Statistical Mechanics of Interacting Systems 0 10 20 20 40 60 80 100 charges charges 120 140 30 40 50  (in electron volts) v (in cm/sec) FIGURE 11.6 The velocity–energy relationship of the vortex rings formed in liquid helium II (after Rayfield and Reif, 1964). The points indicate the experimental data, while the curve represents the theoretical relationship based on the “quantum of circulation” h/m. along with the ring rather than by the small charge coupled to it. Fitting experimental results with the notion of the vortex rings, Rayfield and Reif concluded that their rings carried a circulation of (1.00 ± 0.03) × 10−3 cm2/s, which is close to the Onsager–Feynman unit h/m(= 0.997 × 10−3 cm2/s); moreover, these rings seemed to have a core radius of about 1.2˚ A, which is comparable with the characteristic parameter l of the fluid. We shall now show that the dynamics of the quantized vortex rings is such that their creation in liquid helium II does provide a mechanism for the breakdown of superfluidity. To see this, it is simplest to consider the case of a superfluid flowing through a capillary of radius R. As the velocity of flow increases and approaches the critical value vc, quantized vortex rings begin to form and energy dissipation sets in, which in turn brings about the rupture of the superflow. By symmetry, the rings will be so formed that their central plane will be perpendicular to the axis of the capillary and they will be moving in the direction of the main flow. Now, by the Landau criterion (7.6.24), the critical velocity of superflow is directly determined by the energy spectrum of the excitations created: vc = (ε/p)min. (4) We, therefore, require an expression for the momentum p of the vortex ring. In analogy with the classical vortex ring, we may take p = 2π2ℏn0r2, (5) which seems satisfactory because (i) it conforms to the general result: v = (∂ε/∂p), though only to a first approximation, and (ii) it leads to the (approximate) dispersion relation: 11.7 Low-lying states of an imperfect Fermi gas 379 ε ∝p1/2, which has been separately verified by Rayfield and Reif by subjecting their rings to a transverse electric field. Substituting (3) and (5) into (4), we obtain vc ∼  ℏ mr ln(r/l)  min . (6) Now, since the r-dependence of the quantity ε/p arises mainly from the factor 1/r, the minimum in (6) will be obtained when r has its largest value, namely R, the radius of the capillary. We thus obtain vc ∼ ℏ mR ln(R/l), (7) which is very much the same as the original estimate of Feynman — with D replaced by R. Naturally, then, the numerical values of vc obtained from the new expression (7) continue to be significantly larger than the corresponding experimental values; however, the theory is now much better founded. Fetter (1963) was the first to account for the fact that, as the radius r of the ring approaches the radius R of the capillary, the influence of the “image vortex” becomes important. The energy of the flow now falls below the asymptotic value given by (3) by a factor of 10 or so which, in turn, reduces the critical velocity by a similar factor. The actual result obtained by Fetter was vc ≃11 24 ℏ mR = 0.46 ℏ mR. (8) Kawatra and Pathria (1966) extended Fetter’s calculation by taking into account the boundary effects arising explicitly from the walls of the capillary as well as the ones aris-ing implicitly from the “image vortex”; moreover, in the computation of ε, they employed actual wavefunctions, obtained by solving equation (11.5.23), rather than the analytical approximation employed by Fetter. They obtained vc ≃0.59 ℏ mR, (9) which is about 30 percent higher than Fetter’s value; for comments regarding the “most favorable” location for the formation of the vortex ring in the capillary, see the original reference of Kowatra and Pathria (1966). 11.7 Low-lying states of an imperfect Fermi gas The Hamiltonian of the quantized field for spin-half fermions σ = + 1 2 or −1 2  is given by equation (11.1.48), namely ˆ H = X p,σ p2 2ma† pσ apσ + 1 2 X′ u p′ 1σ ′ 1,p′ 2σ ′ 2 p1σ1,p2σ2a† p′ 1σ ′ 1 a† p′ 2σ ′ 2ap2σ2ap1σ1, (1) 380 Chapter 11. Statistical Mechanics of Interacting Systems where the matrix elements u are related to the scattering length a of the two-body inter-action; the summation in the second part of this expression goes only over those states (of the two particles) that conform to the principles of momentum and spin conservation. As in the Bose case, the matrix elements u in the second sum may be approximated by their values at p = 0, that is, u p′ 1σ ′ 1,p′ 2σ ′ 2 p1σ1,p2σ2 ≃u 0σ ′ 1,0σ ′ 2 0σ1,0σ2. (2) Then, in view of the antisymmetric character of the product ap1σ1ap2σ2, see equa-tion (11.1.24c), all those terms of the second sum in (1) that contain identical indices σ1 and σ2 vanish on summation over p1 and p2. Similarly, all those terms that contain iden-tical indices σ ′ 1 and σ ′ 2 vanish on summation over p′ 1 and p′ 2.17 Thus, for a given set of values of the particle momenta, the only choices for the spin components remaining in the sum are (i) σ1 = + 1 2, σ2 = −1 2; σ ′ 1 = + 1 2, σ ′ 2 = −1 2 (ii) σ1 = + 1 2, σ2 = −1 2; σ ′ 1 = −1 2, σ ′ 2 = + 1 2 (iii) σ1 = −1 2, σ2 = + 1 2; σ ′ 1 = −1 2, σ ′ 2 = + 1 2 (iv) σ1 = −1 2, σ2 = + 1 2; σ ′ 1 = + 1 2, σ ′ 2 = −1 2. It is not difficult to see that the contribution arising from choice (i) will be identi-cally equal to the one arising from choice (iii), while the contribution arising from choice (ii) will be identically equal to the one arising from choice (iv). We may, therefore, write ˆ H = X p,σ p2 2ma† pσ apσ + u0 V X′ a† p′ 1+a† p′ 2−ap2−ap1+, (3) where u0 V =  u0+,0− 0+,0−−u0−,0+ 0+,0−  , (4) while the indices + and −denote the spin states σ = + 1 2 and σ = −1 2, respectively; the summation in the second part of (3) now goes over all momenta that conform to the conservation law p′ 1 + p′ 2 = p1 + p2. (5) To evaluate the eigenvalues of Hamiltonian (3), we shall employ the techniques of the perturbation theory. 17Physically, this means that in the limiting case of slow collisions only particles with opposite spins interact with one another. 11.7 Low-lying states of an imperfect Fermi gas 381 First of all, we note that the main term in the expression for ˆ H is already diagonal, and its eigenvalues are E(0) = X p,σ p2 2mnpσ, (6) where npσ is the occupation number of the single-particle state (p,σ); its mean value, in equilibrium, is given by the Fermi distribution function npσ = 1 z−1 0 exp(p2/2mkT) + 1 . (7) The sum in (6) may be replaced by an integral, with the result (see Section 8.1, with g = 2) E(0) = V 3kT λ3 f5/2(z0), (8) where λ is the mean thermal wavelength of the particles, λ = h/(2πmkT)1/2, (9) while fν(z0) is the Fermi–Dirac function fν(z0) = 1 0(ν) ∞ Z 0 xν−1dx z−1 0 ex + 1 = ∞ X l=1 (−1)l−1 zl 0 lν ; (10) the ideal-gas fugacity z0 is determined by the total number of particles in the system: N = X p,σ npσ = V 2 λ3 f3/2(z0). (11) The first-order correction to the energy of the system is given by the diagonal elements of the interaction term, namely the ones for which p′ 1 = p1 and p′ 2 = p2; thus E(1) = u0 V X p1,p2 np1+ np2−= u0 V N+N−, (12) where N+(N−) denotes the total number of particles with spin up (down). Substituting the equilibrium values N+ = N−= 1 2N, we obtain E(1) = u0 4V N2 = V u0 λ6  f3/2(z0) 2 . (13) Substituting u0 ≃4πaℏ2/m, see equation (11.1.51), we obtain to first order in a E(1) 1 = πaℏ2 m N V N = V 2kT λ3 a λ   f3/2(z0) 2 . (14) 382 Chapter 11. Statistical Mechanics of Interacting Systems The second-order correction to the energy of the system can be obtained with the help of the formula E(2) n = X m̸=n |Vnm|2 En −Em , (15) where the indices n and m pertain to the unperturbed states of the system. A simple calculation yields: E(2) = 2 u2 0 V 2 X p1,p2,p′ 1 np1+np2−  1 −np′ 1+  1 −np′ 2−   p2 1 + p2 2 −p′2 1 −p′2 2  /2m , (16) where the summation goes over all p1,p2, and p′ 1 (the value of p′ 2 being fixed by the requirement of momentum conservation); it is understood that we do not include in the sum (16) any terms for which p2 1 + p2 2 = p′2 1 + p′2 2 . It will be noted that the numerator of the summand in (16) is closely related to the fact that the squared matrix element for the transition (p1,p2) →(p′ 1,p′ 2) is directly proportional to the probability that “the states p1 and p2 are occupied and at the same time the states p′ 1 and p′ 2 are unoccupied.” Now, expression (16) does not in itself exhaust terms of second order in a. A contribution of the same order of magnitude arises from expression (12) if for u0 we employ an expres-sion more accurate than the one just employed. The desired expression can be obtained in the same manner as in the Bose case; check the steps leading to equations (11.2.5) and (11.2.6). In the present case, we obtain 4πaℏ2 mV ≃u0 V + 2 u2 0 V 2 X p1,p2,p′ 1 1  p2 1 + p2 2 −p′2 1 −p′2 2  /2m , from which it follows that u0 ≃4πaℏ2 m     1 −8πaℏ2 mV X p1,p2,p′ 1 1  p2 1 + p2 2 −p′2 1 −p′2 2  /2m     . (17) Substituting (17) into (12), we obtain, apart from the first-order term already given in (14), a second order term, namely E(1) 2 = −2 4πaℏ2 mV !2 X p1,p2,p′ 1 np1+np2−  p2 1 + p2 2 −p′2 1 −p′2 2  /2m . (18) 11.7 Low-lying states of an imperfect Fermi gas 383 For a comparable term given in (16), the approximation u0 ≃4πaℏ2/m is sufficient, with the result E(2) 2 = 2 4πaℏ2 mV !2 X p1,p2,p′ 1 np1+np2−  1 −np′ 1+  1 −np′ 2−   p2 1 + p2 2 −p′2 1 −p′2 2  /2m . (19) Combining (18) and (19), we obtain18 E2 = E(1) 2 + E(2) 2 = −2 4πaℏ2 mV !2 X p1,p2,p′ 1 np1+np2−  np′ 1+ + np′ 2−   p2 1 + p2 2 −p′2 1 −p′2 2  /2m . (20) To evaluate the sum in (20), we prefer to write it as a symmetrical summation over the four momenta p1,p2,p′ 1, and p′ 2 by introducing a Kronecker delta to take care of the momentum conservation; thus E2 = −2 4πaℏ2 mV !2 X p1,p2,p′ 1,p′ 2 np1+np2−  np′ 1+ + np′ 2−  δp1+p2,p′ 1+p′ 2  p2 1 + p2 2 −p′2 1 −p′2 2  /2m . (21) It is obvious that the two parts of the sum (21), one arising from the factor np′ 1+ and the other from the factor np′ 2−, would give identical results on summation. We may, therefore, write E2 = −4 4πaℏ2 mV !2 X p1,p2,p′ 1,p′ 2 np1+np2−np′ 1+δp1+p2,p′ 1+p′ 2  p2 1 + p2 2 −p′2 1 −p′2 2  /2m . (22) The sum in (22) can be evaluated by following a procedure due to Huang, Yang, and Luttinger (1957), with the result19 E2 = V 8kT λ3 a2 λ2 ! F(z0), (23) where F(z0) = − ∞ X r,s,t=1 (−z0)r+s+t √(rst)(r + s)(r + t). (24) 18We have omitted here terms containing a “product of four n’s” for the following reason: in view of the fact that the numerator of such terms would be symmetric and the denominator antisymmetric with respect to the exchange operation (p1,p2) ↔(p′ 1,p′ 2), their sum over the variables p1,p2,p′ 1 (and p′ 2) would vanish identically. 19For a direct evaluation of the sum (22), in the limit T →0, see Abrikosov and Khalatnikov (1957). See also Problem 11.12. 384 Chapter 11. Statistical Mechanics of Interacting Systems Combining (8), (14), and (23), we obtain to second order in the scattering length a E = V kT λ3 " 3f5/2(z0) + 2a λ {f3/2(z0)}2 + 8a2 λ2 F(z0) # , (25) where z0 is determined by (11). It is now straightforward to obtain the ground-state energy of the imperfect Fermi gas (z0 →∞); we have simply to know the asymptotic behavior of the functions involved. For the functions fν(z0), we have from the Sommerfeld lemma (see Appendix E) fν(z0) ≈(lnz0)ν/ 0(ν + 1), (26) so that f5/2(z0) ≈ 8 15π1/2 (lnz0)5/2; f3/2(z0) ≈ 4 3π1/2 (lnz0)3/2. (27) Equation (11) then gives n = N V ≈ 8 3π1/2λ3 (lnz0)3/2, (28) so that lnz0 ≈λ2 3π1/2n 8 !2/3 . (29) The asymptotic behaviour of F(z0) is given by F(z0) ≈16(11 −2ln2) 105π3/2 (lnz0)7/2; (30) see Problem 11.12. Substituting (27) and (30) into (25), and making use of relation (29), we finally obtain E0 N = 3 10 ℏ2 m (3π2n)2/3 + πaℏ2 m n ( 1 + 6 35(11 −2ln2)  3 π 1/3 n1/3a ) . (31) The ground-state pressure of the gas is then given by P0 = n2 ∂(E0/N) ∂n = 1 5 ℏ2 m (3π2n)2/3n + πaℏ2 m n2 ( 1 + 8 35(11 −2ln2)  3 π 1/3 n1/3a ) . (32) 11.8 Energy spectrum of a Fermi liquid 385 We may also calculate the velocity of sound, which directly involves the compressibility of the system, with the result c2 0 = ∂P0 ∂(mn) = 1 3 ℏ2 m2 (3π2n)2/3 + 2πaℏ2 m2 n ( 1 + 4 15(11 −2ln2)  3 π 1/3 n1/3a ) . (33) The leading terms of the foregoing expressions represent the ground-state results for an ideal Fermi gas, while the remaining terms represent corrections arising from the interparticle interactions. The result embodied in equation (31) was first obtained by Huang and Yang (1957) by the method of pseudopotentials; Martin and De Dominicis (1957) were the first to attempt an estimate of the third-order correction.20 Lee and Yang (1957) obtained (31) on the basis of the binary collision method; for the details of their calculation, see Lee and Yang (1959b, 1960a). The same result was derived somewhat later by Galitskii (1958) who employed the method of Green’s functions. 11.8 Energy spectrum of a Fermi liquid: Landau’s phenomenological theory21 In Section 11.4 we discussed the main features of the energy spectrum of a Bose liquid; such a spectrum is generally referred to as a Bose type spectrum. A liquid consisting of spin-half fermions, such as liquid He3, is expected to have a different kind of spectrum which, by contrast, may be called a Fermi type spectrum. Right away we should emphasize that a liquid consisting of fermions may not neces-sarily possess a spectrum of the Fermi type; the spectrum actually possessed by such a liquid depends crucially on the nature of the interparticle interactions operating in the liquid. The discussion here assumes that the interactions are strictly repulsive so that the fermions have no opportunity to form bosonic pairs. In the present section, we propose to discuss the main features of a spectrum which is characteristically of the Fermi type. The effects of attractive interactions will be discussed in Section 11.9 According to Landau (1956), whose work provides the basic framework for our discus-sion, the Fermi type spectrum of a quantum liquid is constructed in analogy with the spectrum of an ideal Fermi gas. As is well-known, the ground state of the ideal Fermi gas corresponds to a “complete filling up of the single-particle states with p ≤pF and a complete absence of particles in the states with p > pF”; the excitation of the system corre-sponds to a transition of one or more particles from the occupied states to the unoccupied states. The limiting momentum pF is related to the particle density in the system and, for 20The third-order correction has also been discussed by Mohling (1961). 21For a microscopic theory of a Fermi liquid, see Nozieres (1964); see also Tuttle and Mohling (1966). 386 Chapter 11. Statistical Mechanics of Interacting Systems spin-half particles, is given by pF = ℏ(3π2N/V)1/3. (1) In a liquid, we cannot speak of quantum states of individual particles. However, as a basis for constructing the desired spectrum, we may assume that, as interparticle inter-actions are gradually “switched on” and a transition made from the gaseous to the liquid state, the ordering of the energy levels (in the momentum space) remains unchanged. Of course, in this ordering, the role of the gas particles is passed on to the “elementary exci-tations” of the liquid (also referred to as “quasiparticles”), whose number coincides with the number of particles in the liquid and which also obey Fermi statistics. Each “quasipar-ticle” possesses a definite momentum p, so we can speak of a distribution function n(p) such that Z n(p)dτ = N/V, (2) where dτ = 2d3p/h3. We then expect that the specification of the function n(p) uniquely determines the total energy E of the liquid. Of course, E will not be given by a simple sum of the energies ε(p) of the quasiparticles; it will rather be a functional of the distribution func-tion n(p). In other words, the energy E will not reduce to the simple integral R ε(p)n(p)Vdτ, though in the first approximation a variation in its value may be written as δE = V Z ε(p)δn(p)dτ, (3) where δn(p) is an assumed variation in the distribution function of the “quasiparticles.” The reason E does not reduce to an integral of the quantity ε(p)n(p) is related to the fact that the quantity ε(p) is itself a functional of the distribution function. If the initial distri-bution function is a step function (which corresponds to the ground state of the system), then the variation in ε(p) due to a small deviation of the distribution function from the step function (which implies only low-lying excited states of the system) would be given by a linear functional relationship: δε(p) = Z f (p,p′)δn(p′)dτ ′. (4) Thus, the quantities ε(p) and f (p,p′) are the first and second functional derivatives of E with respect to n(p). Inserting spin dependence, we may now write δE = X p,σ ε(p,σ)δn(p,σ) + 1 2V X p,σ;p′,σ ′ f (p,σ;p′,σ ′)δn(p,σ)δn(p′,σ ′), (5) where δn are small variations in the distribution function n(p) from the step function (that characterizes the ground state of the system); it is obvious that these variations will be sig-nificant only in the vicinity of the limiting momentum pF, which continues to be given 11.8 Energy spectrum of a Fermi liquid 387 by equation (1). It is thus understood that the quantity ε(p,σ) in (5) corresponds to the distribution function n(p,σ) being infinitesimally close to the step function (of the ground state). One may also note that the function f (p,σ;p′,σ ′), being a second functional deriva-tive of E, must be symmetric in its arguments; often, it is of the form a + b ˆ s1 · ˆ s2, where the coefficients a and b depend only on the angle between the momenta p and p′.22 The function f plays a central role in the theory of the Fermi liquid; for an ideal gas, f vanishes. To discover the formal dependence of the distribution function n(p) on the energy ε(p), we note that, in view of the one-to-one correspondence between the energy levels of the liquid and of the ideal gas, the number of microstates (and hence the entropy) of the liquid is given by the same expression as for the ideal gas; see equation (6.1.15), with all gi = 1 and a = +1, or Problem 6.1: S k = − X p {nlnn + (1 −n)ln(1 −n)} ≈−V Z {nlnn + (1 −n)ln(1 −n)}dτ. (6) Maximizing this expression, under the constraints δE = 0 and δN = 0, we obtain for the equilibrium distribution function n = 1 exp{(ε −µ)/kT} + 1. (7) It should be noted here that, despite its formal similarity with the standard expression for the Fermi–Dirac distribution function, formula (7) is different insofar as the quantity ε appearing here is itself a function of n; consequently, this formula gives only an implicit, and probably a very complicated, expression for the function n. A word may now be said about the quantity ε appearing in equation (5). Since this ε cor-responds to the limiting case of n being a step function, it is expected to be a well-defined function of p. Equation (7) then reduces to the usual Fermi–Dirac distribution function, which is indeed an explicit function of ε. It is not difficult to see that this reduction remains valid so long as expression (5) is valid, that is, so long as the variations δn are small, which in turn means that T ≪TF. As mentioned earlier, the variation δn will be significant only in the vicinity of the Fermi momentum pF; accordingly, we will not have much to do with the function ε(p) except when p ≃pF. We may, therefore, write ε(p ≃pF) = εF +  ∂ε ∂p  p=pF (p −pF) + ··· ≃εF + uF(p −pF), (8) where uF denotes the “velocity” of the quasiparticles at the Fermi surface. In the case of an ideal gas (ε = p2/2m),uF = pF/m. By analogy, we define a parameter m∗such that m∗≡pF uF = pF (∂ε/∂p)p=pF (9) 22Of course, if the functions involved here are spin-dependent, then the factor 2 in the element dτ (as well as in dτ ′) must be replaced by a summation over the spin variable(s). 388 Chapter 11. Statistical Mechanics of Interacting Systems and call it the effective mass of the quasiparticle with momentum pF (or with p ≃pF). Another way of looking at the parameter m∗is due to Brueckner and Gammel (1958), who wrote ε(p ≃pF) = p2 2m + V(p) = p2 2m∗+ const.; (10) the philosophy behind this expression is that “for quasiparticles with p ≃pF, the modifi-cation, V(p), brought into the quantity ε(p) by the presence of inter-particle interactions in the liquid may be represented by a constant term while the kinetic energy, p2/2m, is modified so as to replace the particle mass m by an effective, quasiparticle mass m∗”; in other words, we adopt a mean field point of view. Differentiating (10) with respect to p and setting p = pF, we obtain 1 m∗= 1 m + 1 pF dV(p) dp  p=pF . (11) The quantity m∗, in particular, determines the low-temperature specific heat of the Fermi liquid. We can readily see that, for T ≪TF, the ratio of the specific heat of a Fermi liquid to that of an ideal Fermi gas is precisely equal to the ratio m∗/m: (CV )real (CV )ideal = m∗ m . (12) This follows from the fact that (i) expression (6) for the entropy S, in terms of the distri-bution function n, is the same for the liquid as for the gas, (ii) the same is true of relation (7) between n and ε, and (iii) for the evaluation of the integral in (6) at low temperatures only momenta close to pF are important. Consequently, the result stated in Problem 8.13, namely CV ≃S ≃π2 3 k2T a(εF), (13) continues to hold — with the sole difference that in the expression for the density of states a(εF), in the vicinity of the Fermi surface, the particle mass m gets replaced by the effective mass m∗; see equation (8.1.21). We now proceed to establish a relationship between the parameters m and m∗in terms of the characteristic function f . In doing so, we neglect the spin-dependence of f , if any; the necessary modification can be introduced without any difficulty. The guiding principle here is that, in the absence of external forces, the momentum density of the liquid must be equal to the density of mass transfer. The former is given by R pndτ, while the latter is given by m R (∂ε/∂p)ndτ,(∂ε/∂p) being the “velocity” of the quasiparticle with momentum 11.8 Energy spectrum of a Fermi liquid 389 p and energy ε.23 Thus Z pndτ = m Z ∂ε ∂pndτ. (14) Varying the distribution function by δn and making use of equation (4), we obtain Z pδndτ = m Z ∂ε ∂pδndτ + m ZZ ∂f (p,p′) ∂p δn′dτ ′  ndτ = m Z ∂ε ∂pδndτ −m ZZ f (p,p′)∂n′ ∂p′ δndτ dτ ′; (15) in obtaining the last expression, we have interchanged the variables p and p′ and have also carried out an integration by parts. In view of the arbitrariness of the variation δn, equation (15) requires that p m = ∂ε ∂p − Z f (p,p′)∂n′ ∂p′ dτ ′. (16) We apply this result to quasiparticles with momenta close to pF; at the same time, we replace the distribution function n′ by a “step” function, whereby ∂n′ ∂p′ = −p′ p′ δ(p′ −pF). This enables us to carry out integration over the magnitude p′ of the momentum, so that Z f (p,p′)∂n′ ∂p′ 2p′2dp′dω′ h3 = −2pF h3 Z f (θ)p′ F dω′, (17) dω′ being the element of a solid angle; note that we have contracted the arguments of the function f because in simple situations it depends only on the angle between the two momenta. Inserting (17) into (16), with p = pF, making a scalar product with pF and dividing by p2 F, we obtain the desired result 1 m = 1 m∗+ pF 2h3 · 4 Z f (θ)cosθ dω′. (18) If the function f depends on the spins s1 and s2 of the particles involved, then the factor 4 in front of the integral will have to be replaced by a summation over the spin variables. 23Since the total number of quasiparticles in the liquid is the same as the total number of real particles, to obtain the net transport of mass by the quasiparticles one has to multiply their number by the mass m of the real particle. 390 Chapter 11. Statistical Mechanics of Interacting Systems We now derive a formula for the velocity of sound at absolute zero. From first principles, we have24 c2 0 = ∂P0 ∂(mN/V) = −V 2 mN ∂P0 ∂V  N . In the present context, it is preferable to have an expression in terms of the chemical poten-tial of the liquid. This can be obtained by making use of the formula Ndµ0 = VdP0, see Problem 1.16, from which it follows that25 ∂µ0 ∂N  V = −V N ∂µ0 ∂V  N = −V 2 N2 ∂P0 ∂V  N and hence c2 0 = N m ∂µ0 ∂N  V . (19) Now, µ0 = ε(pF) = εF; therefore, the change δµ0 arising from a change δN in the total number of particles in the system is given by δµ0 = ∂εF ∂pF δpF + Z f (pF,p′)δn′ dτ ′. (20) The first part in (20) arises from the fact that a change in the total number of particles in the system inevitably alters the value of the limiting momentum pF; see equation (1), from which (for constant V) δpF/pF = 1 3δN/N and hence ∂εF ∂pF δpF = p2 F 3m∗ δN N . (21) 24At T = 0,S = 0; so there is no need to distinguish between the isothermal and adiabatic compressibilities of the liquid. 25Since µ0 is an intensive quantity and, therefore, it depends on N and V only through the ratio N/V, we can write: µ0 = µ0(N/V). Consequently,  ∂µ0 ∂N  V = µ′ 0  ∂(N/V) ∂N  V = µ′ 0 1 V and  ∂µ0 ∂V  N = µ′ 0  ∂(N/V) ∂V  N = −µ′ 0 N V 2 . Hence  ∂µ0 ∂N  V = −V N  ∂µ0 ∂V  N . 11.8 Energy spectrum of a Fermi liquid 391 The second part arises from equation (4). It will be noted that the variation δn′ appearing in the integral of equation (20) is significant only for p′ ≃pF; we may, therefore, write Z f (pF,p′)δn′dτ ′ ≃δN 4πV Z f (θ)dω′. (22) Substituting (21) and (22) into (20), we obtain ∂µ0 ∂N  V = p2 F 3m∗N + 1 4πV Z f (θ)dω′. (23) Making use of equations (18) and (1), we finally obtain c2 0 = N m ∂µ0 ∂N  V = p2 F 3m2 + p3 F 6mh3 · 4 Z f (θ)(1 −cosθ)dω′. (24) Once again, if the function f depends on the spins of the particles, then the factor 4 in front of the integral will have to be replaced by a summation over the spin variables. For illustration, we shall apply this theory to the imperfect Fermi gas studied in Section 11.7. To calculate f (p,σ;p′,σ ′), we have to differentiate twice the sum of expres-sion (11.7.12), with u0 = 4πaℏ2/m, and expression (11.7.22) with respect to the distribution function n(p,σ) and then substitute p = p′ = pF. Performing the desired calculation, then changing summations into integrations and carrying out integrations by simple means, we find that the function f is spin-dependent — the spin-dependent term being in the nature of an exchange term, proportional to ˆ s1 · ˆ s2. The complete result, according to Abrikosov and Khalatnikov (1957), is f (p,σ;p′,σ ′) = A(θ) + B(θ)ˆ s1 · ˆ s2, (25) where A(θ) = 2πaℏ2 m " 1 + 2a  3N πV 1/3  2 + cosθ 2sin(θ/2) ln 1 + sin(θ/2) 1 −sin(θ/2) # and B(θ) = −8πaℏ2 m " 1 + 2a  3N πV 1/3  1 −1 2 sin θ 2  ln 1 + sin(θ/2) 1 −sin(θ/2) # , a being the scattering length of the two-body potential and θ the angle between the momentum vectors pF and p′ F. Substituting (25) into formulae (18) and (24), in which the factor 4 is now supposed to be replaced by a summation over the spin variables, we find that while the spin-dependent term B(θ)ˆ s1 · ˆ s2 does not make any contribution toward the 392 Chapter 11. Statistical Mechanics of Interacting Systems final results, the spin-independent term A(θ) leads to26 1 m∗= 1 m − 8 15m(7ln2 −1)  3N πV 2/3 a2 (26) and c2 0 = p2 F 3m2 + 2πaℏ2 m2 N V " 1 + 4 15(11 −2ln2)  3N πV 1/3 a # ; (27) the latter result is identical to expression (11.7.33) derived in the preceding section. Pro-ceeding backward, one can obtain from equation (27) corresponding expressions for the ground-state pressure P0 and the ground-state energy E0, namely equations (11.7.32) and (11.7.31), as well as the ground-state chemical potential µ0, as quoted in Problem 11.15. 11.9 Condensation in Fermi systems The discussion of the T = 0 Fermi liquid in Sections 11.7 and 11.8 applies when the inter-actions between the fermions are strictly repulsive. The resulting Fermi liquid has a ground state and quasiparticle excitations that are qualitatively similar to the ideal Fermi gas. However, for fermions with attractive interactions, no matter how weak, the degenerate Fermi gas is unstable due to the formation of bosonic pairs. This leads to a number of important phenomena including superconductivity in metals, superfluidity in 3He, and condensation in ultracold Fermi gases. In low-temperature superconductors, screening and the electron-phonon interaction result in a retarded attraction between quasiparti-cles on opposite sides of the Fermi surface. The formation of these so-called Cooper pairs leads to the creation of a superconducting state with critical temperature kTc ≈ℏωD exp  − 1 N(ϵF)|u0|  , (1) where N(ϵF) is the density of states per spin configuration at the Fermi surface, u0 is the weak attractive coupling between electrons, and ℏωD is the Debye energy discussed in Section 7.4 since the coupling is due to the acoustic phonons. As can be seen from equation (1), the phase transition temperature is nonperturbative in u0. A complete treat-ment of superconductivity is far beyond the scope of this section, so we refer the reader to the original papers by Cooper (1956) and Bardeen, Cooper, and Schrieffer (1957) and the texts on superconductivity by Tilley and Tilley (1990) and Tinkham (1996). The case of superfluidity in 3He is surveyed by Vollhardt and W¨ olfle (1990). Bosonic condensation has also recently been observed in trapped ultracold atomic Fermi gases. The sign and size of the atomic interactions in ultracold gases can be tuned 26In a dense system, such as liquid He3, the ratio m∗/m would be significantly larger than unity. The experimen-tal work of Roberts and Sydoriak (1955), on the specific heat of liquid He3, and the theoretical work of Brueckner and Gammel (1958), on the thermodynamics of a dense Fermi gas, suggest that the ratio (m∗/m)He3 ≃1.85. 11.9 Condensation in Fermi systems 393 with a magnetic field near Feshbach resonance allowing unprecedented experimental control of interactions. In particular, experimenters can create a low-lying molecular bound state or a weakly attractive interaction without allowing a molecular bound state to form. If interaction between pairs of fermions allows the formation of bound bosonic molecules, the ground state of a degenerate Fermi gas will be destablized since molecules will form and, if the density of the bosonic molecules is large enough, they will Bose-condense — see Greiner, Regal, and Jin (2003); Jochim et al. (2003); and Zwierlein et al. (2003). For weakly attractive interactions, the fermionic system condenses into a BCS-like state and provides an excellent experimental environment for testing theoretical predictions due to the well-understood nature and experimental control of the atomic interactions. Theory predicts a smooth crossover from BCS to Bose–Einstein condensation (BEC) behavior as the magnitude of the attractive interaction parameter u0 is varied from values small to large. BCS theory describes the behavior for weak coupling. For broad Fesh-bach resonances of trapped fermions, the most common experimental situation, the BCS critical temperature is given by kTc ℏω0 ≈ϵF ℏω0 exp  − π 2kF |a|  ∼N1/3 exp  − aosc 1.214N1/6 |a|  , (2) where ω0 = (ω1ω2ω3)1/3 is the average oscillation frequency of atoms in the trap, aosc = √ℏ/mω0, and kF = √2mϵF/ℏis the Fermi wavevector; see Pitaevskii and Stringari (2003), Leggett (2006), and Pethick and Smith (2008). For large negative scattering lengths, the transition temperature smoothly crosses over to the BEC limit with noninteracting Bose-condensation temperature, see equation (7.2.6), kTc ℏω0 ≈  N 2ζ(3) 1/3 , (3) since the number of Cooper pairs is N/2. The ratio of the transition temperature in the BEC limit and the Fermi temperature from equation (8.4.3) is kTc ϵF ≈  1 12ζ(3) 1/3 ≃0.41. (4) Mean-field analysis of the broad resonance limit (Leggett, 2006) and analytical analysis of the narrow resonance limit (Gurarie and Radzihovsky, 2007) both indicate that the phase transition temperature has a maximum at intermediate coupling. Figure 11.7 is a sketch of the critical temperature as a function of the coupling parameter u0. Experimental observations of condensation in a degenerate Fermi gas in the BEC–BCS crossover region by Regal, Greiner, and Jin (2004) are shown in Figure 11.8. They used a Feshbach resonance to tune the scattering length of 40K into an attractive range (a < 0) that 394 Chapter 11. Statistical Mechanics of Interacting Systems BCS limit BEC–BCS kTc 0 Fermi liquid u0 FIGURE 11.7 Sketch of the BEC–BCS phase diagram on the BCS side of the Feshbach resonance for ultracold fermions in an atomic trap. The scattering length a and coupling u0 = 4πℏ2a/m can be tuned from positive values to negative with the help of a magnetic field. Positive (repulsive) couplings result in a Fermi liquid. Negative (attractive) couplings result in a BCS condensation at low temperatures. The nature of the condensed phase varies smoothly from BCS behavior for small negative coupling to Bose–Einstein behavior for large negative coupling. The phase transition temperature has a maximum at intermediate coupling. FIGURE 11.8 Time-of-flight images showing condensation of fermions in an ultracold atomic gas. The images show the quantum mechanical projection of the fermionic system onto a molecular gas and are shown for three values of the magnetic field on the BCS side of the Feshbach resonance for an ultracold trapped gas of 40K. The temperature of the Fermi gas is (kT/ϵF) ≈0.07. The condensed fraction varies from about 1 to 10 percent of the original cold fermions in the trap; see Regal, Greiner, and Jin (2004). Figure courtesy of NIST/JILA/University of Colorado. does not allow a two-particle molecular bound state and observed the fermions condens-ing into a BCS-like macroscopic quantum state. They explored the BEC–BCS crossover behavior by tuning |a| from small values to large. Problems 11.1. (a) Show that, for bosons as well as fermions, [ψ(rj), ˆ H] = −ℏ2 2m∇2 j + Z d3rψ†(r)u(r,rj)ψ(r) ! ψ(rj), where ˆ H is the Hamiltonian operator defined by equation (11.1.4). Problems 395 (b) Making use of the foregoing result, show that the equation 1 √ N! ⟨0|ψ(r1)...ψ(rN) ˆ H|9NE⟩= E 1 √ N! ⟨0|ψ(r1)...ψ(rN)|9NE⟩ = E9NE(r1,...rN) is equivalent to the Schr¨ odinger equation (11.1.15). 11.2. The grand partition function of a gaseous system composed of mutually interacting bosons is given by lnQ ≡PV kT = V λ3 " g5/2(z) −2{g3/2(z)}2 a λ + O a2 λ2 !# . Study the analytic behavior of this expression near z = 1 and show that the system exhibits the phenomenon of Bose–Einstein condensation when its fugacity assumes the critical value zc = 1 + 4ζ 3 2  a λc + O a2 λ2 c ! . Further show that the pressure of the gas at the critical point is given by (Lee and Yang 1958, 1960b) Pc kTc = 1 λ3 c " ζ 5 2  + 2  ζ 3 2 2 a λc + O a2 λ2 c !# ; compare these results to equations (11.2.16) through (11.2.18). 11.3. For the imperfect Bose gas studied in Section 11.2, calculate the specific heat CV near absolute zero and show that, as T →0, the specific heat vanishes in a manner characteristic of a system with an “energy gap” 1 = 4πaℏ2n/m. 11.4. (a) Show that, to first order in the scattering length a, the discontinuity in the specific heat CV of an imperfect Bose gas at the transition temperature Tc is given by (CV )T=Tc−−(CV )T=Tc+ = Nk 9a 2λc ζ(3/2), while the discontinuity in the bulk modulus K is given by (K)T=Tc−−(K)T=Tc+ = −4πaℏ2 mv2 c . (b) Examine the discontinuities in the quantities (∂2P/∂T2)v and (∂2µ/∂T2)v as well, and show that your results are consistent with the thermodynamic relationship CV = VT ∂2P ∂T2 ! v −NT ∂2µ ∂T2 ! v . 396 Chapter 11. Statistical Mechanics of Interacting Systems 11.5. (a) Complete the mathematical steps leading to equations (11.3.15) and (11.3.16). (b) Complete the mathematical steps leading to equations (11.3.23) and (11.3.24). 11.6. The ground-state pressure of an interacting Bose gas (see Lee and Yang, 1960a) turns out to be P0 = µ2 0m 8πaℏ2 " 1 −64 15π µ1/2 0 m1/2a ℏ + ··· # , where µ0 is the ground-state chemical potential of the gas. It follows that n ≡  dP0 dµ0  = µ0m 4πaℏ2 " 1 −16 3π µ1/2 0 m1/2a ℏ + ··· # and E0 V ≡(nµ0 −P0) = µ2 0m 8πaℏ2 " 1 −32 5π µ1/2 0 m1/2a ℏ + ··· # . Eliminating µ0 from these results, derive equations (11.3.16) and (11.3.17). 11.7. Show that in an interacting Bose gas the mean occupation number np of the real particles and the mean occupation number Np of the quasiparticles are connected by the relationship np = Np + α2 p(Np + 1) 1 −α2 p (p ̸= 0), where αp is given by equations (11.3.9) and (11.3.10). Note that equation (11.3.22) corresponds to the special case Np = 0. 11.8. The excitation energy of liquid He4, carrying a single excitation above the ground state, is determined by the minimum value of the quantity ε = Z 9∗ ( −ℏ2 2m X i ∇2 i + V −E0 ) 9d3Nr Z 9∗9d3Nr, where E0 denotes the ground-state energy of the liquid while 9, according to Feynman, is given by equation (11.4.3). Show that the process of minimization of this expression leads to equation (11.4.5) for the energy of the excitation. [Hint: First express ε in the form ε = ℏ2 2m Z |∇f (r)|2d3r Z f ∗(r1)f (r2)g(r2 −r1)d3r1d3r2. Then show that ε is minimum when f (r) is of the form (11.4.4).] 11.9. Show that, for a sufficiently large momentum ℏk (in fact, such that the slope dε/dk of the energy spectrum is greater than the initial slope ℏc), a state of double excitation in liquid He4 is energetically more favorable than a state of single excitation, that is, there exist wavevectors k1 and k2 such that, while k1 + k2 = k,ε(k1) + ε(k2) < ε(k). 11.10. Using Fetter’s analytical approximation, f1(ρ′) = ρ′ p (1 + ρ′2) , Problems 397 for the solution of equation (11.5.23) with s = 1, calculate the energy (per unit length) associated with a quantized vortex line of unit circulation. Compare your result with the one quoted in (11.5.26). 11.11. (a) Study the nature of the velocity field arising from a pair of parallel vortex lines, with s1 = +1 and s2 = −1, separated by a distance d. Derive and discuss the general equation of the stream lines. (b) Again, using Fetter’s analytical approximation for the functions f (ρ′ 1) and f (ρ′ 2), calculate the energy (per unit length) of the system and show that its limiting value, as d →0, is 11πℏ2n0/12m. Making use of this result, derive expression (11.6.8) for the critical velocity of superflow. 11.12. Establish the asymptotic formula (11.7.30) for the function F(z0). [Hint: Write the coefficient that appears in the sum (11.7.24) in the form 1 √(rst)(r + s)(r + t) =  2 √π 3 ∞ Z 0 e−X2r−Y 2s−Z2t−ξ(r+s)−η(r+t)dX dY dZ dξ dη. Insert this expression into (11.7.24) and carry out summations over r,s, and t, with the result F(z0) = 8 π3/2 ∞ Z 0 1 z−1 0 eX2+ξ+η + 1 1 z−1 0 eY 2+ξ + 1 1 z−1 0 eZ2+η + 1 dX dY dZ dξ dη. In the limit z0 →∞, the integrand is essentially equal to 1 in the region R defined by X2 + ξ + η < lnz0, Y 2 + ξ < lnz0, and Z2 + η < lnz0; outside this region, it is essentially 0. Hence, the dominant term of the asymptotic expansion is 8 π3/2 Z R 1 · dX dY dZ dξ dη, which, in turn, reduces to the double integral 8 π3/2 ZZ (lnz0 −ξ −η)1/2(lnz0 −ξ)1/2(lnz0 −η)1/2dξ dη; the limits of integration here are such that not only ξ < (lnz0) and η < (lnz0), but also (ξ + η) < (lnz0). The rest of the calculation is straightforward.] 11.13. The grand partition function of a gaseous system composed of mutually interacting, spin-half fermions has been evaluated by Lee and Yang (1957), with the result27 lnQ ≡PV kT = V λ3  2f5/2(z) −2a λ {f3/2(z)}2 +4a2 λ2 f1/2(z){f3/2(z)}2 −8a2 λ2 F(z) + ··· # , 27For the details of this calculation, see Lee and Yang (1959b) where the case of bosons, as well as of fermions, with spin J has been treated using the binary collision method. The second-order result for the case of spinless bosons was first obtained by Huang, Yang, and Luttinger (1957) using the method of pseudopotentials. 398 Chapter 11. Statistical Mechanics of Interacting Systems where z is the fugacity of the actual system (not of the corresponding noninteracting system, which was denoted by the symbol z0 in the text); the functions fν(z) and F(z) are defined in a manner similar to equations (11.7.10) and (11.7.24). From this result, one can derive expressions for the quantities E(z,V,T) and N(z,V,T) by using the formulae E(z,V,T) ≡kT2 ∂(lnQ) ∂T and N(z,V,T) ≡∂(lnQ) ∂(lnz)  = 2V λ3 f3/2(z0)  . (a) Eliminating z between these two results, derive equation (11.7.25) for E. (b) Obtain the zero-point value of the chemical potential µ, correct to second order in (a/λ), and verify, with the help of equations (11.7.31) and (11.7.32), that (E + PV)T=0 = N(µ)T=0. [Hint: At T = 0K,µ = (∂E/∂N)V .] (c) Show that the low-temperature specific heat and the low-temperature entropy of this gas are given by (see Pathria and Kawatra, 1962) CV Nk ≃S Nk ≃π2 2 kT εF  1 + 8 15π2 (7ln2 −1)(kFa)2 + ···  , where kF = (3π2n)1/3. Clearly, the factor within square brackets is to be identified with the ratio m∗/m; see equations (11.8.12) and (11.8.26). [Hint: To determine CV to the first power in T, we must know E to the second power in T. For this, we require higher-order terms of the asymptotic expansions of the functions fν(z) and F(z); these are given by f5/2(z) = 8 15√π (lnz)5/2 + π3/2 3 (lnz)1/2 + O(1), f3/2(z) = 4 3√π (lnz)3/2 + π3/2 6 (lnz)−1/2 + O(lnz)−5/2, f1/2(z) = 2 √π (lnz)1/2 −π3/2 12 (lnz)−3/2 + O(lnz)−7/2, and F(z) =16(11 −2ln2) 105π3/2 (lnz)7/2 −2(2ln2 −1) 3 π1/2(lnz)3/2 + O(lnz)5/4. The first three results here follow from the Sommerfeld lemma (E.17); for the last one, see Yang (1962).] Problems 399 11.14. The energy spectrum ε(p) of a gas composed of mutually interacting, spin-half fermions is given by (Galitskii, 1958; Mohling, 1961) ε(p) p2 F/2m ≃x2 + 4 3π (kFa) + 4 15π2 (kFa)2 × " 11 + 2x4 ln x2 |x2 −1| −10  x −1 x  ln x + 1 x −1 −(2 −x2)5/2 x ln 1 + x√(2 −x2) 1 −x√(2 −x2) !# , where x = p/pF ≤√2 and k = p/ℏ. Show that, for k close to kF, this spectrum reduces to ε(p) p2 F/2m ≃x2+ 4 3π (kFa) + 4 15π2 (kFa)2  (11 −2ln2) −4(7ln2 −1)  k kF −1  . Using equations (11.8.10) and (11.8.11), check that this expression leads to the result m∗ m ≃1 + 8 15π2 (7ln2 −1)(kFa)2. 11.15. In the ground state of a Fermi system, the chemical potential is identical to the Fermi energy: (µ)T=0 = ε(pF). Making use of the energy spectrum ε(p) of the previous problem, we obtain (µ)T=0 ≃p2 F 2m  1 + 4 3π (kFa) + 4 15π2 (11 −2ln2)(kFa)2  . Integrating this result, rederive equation (11.7.31) for the ground-state energy of the system. 11.16. The energy levels of an imperfect Fermi gas in the presence of an external magnetic field B, to first order in a, may be written as En = X p (n+ p + n− p ) p2 2m + 4πaℏ2 mV N+N−−µ∗B(N+ −N−); see equations (8.2.8) and (11.7.12). Using this expression for En and following the procedure adopted in Section 8.2.A, study the magnetic behavior of this gas — in particular, the zero-field susceptibility χ(T). Also examine the possibility of spontaneous magnetization arising from the interaction term with a > 0. 11.17. Rewrite the Gross–Pitaevskii equation and the mean field energy, see equations (11.2.21) and (11.2.23), for an isotropic harmonic oscillator trap with frequency ω0 in a dimensionless form by defining a dimensionless wavefunction ψ = a3/2 osc/9N, a dimensionless length s = r/aosc, and a dimensionless energy E/Nℏω0. Show that the dimensionless parameter that controls the mean field energy is Na/aosc, where N is the number of particles in the condensate, a is the scattering length, and aosc = √ℏ/mω0. Next, show that the dimensionless versions of the Gross–Pitaevskii equation and the mean field energy are −1 2 ˜ ∇2ψ + 1 2s2ψ + 4πNa aosc |ψ|2 ψ = ˜ µψ , 400 Chapter 11. Statistical Mechanics of Interacting Systems and E[ψ] Nℏω0 = Z 1 2 ˜ ∇ψ 2 + 1 2s2 |ψ|2 + 2πNa aosc |ψ|4  ds . 11.18. Solve the Gross–Pitaevskii equation and evaluate the mean field energy, see equations (11.2.21) and (11.2.23), for a uniform Bose gas to show that this method yields precisely equation (11.2.6). 11.19. Solve the Gross–Pitaevskii equation (11.2.23) in a harmonic trap for the case when the scattering length a is zero. Show that this reproduces the properties of the ground state of the noninteracting Bose gas. 11.20. Solve the Gross–Pitaevskii equation and evaluate the mean field energy, see equations (11.2.21) and (11.2.23), for an isotropic harmonic oscillator trap with frequency ω0 for the case Na/aosc ≫1 by ignoring the kinetic energy term. Reproduce the results (11.2.25) through (11.2.28). 12 Phase Transitions: Criticality, Universality, and Scaling Various physical phenomena to which the formalism of statistical mechanics has been applied may, in general, be divided into two categories. In the first category, the micro-scopic constituents of the given system are, or can be regarded as, practically noninteract-ing; as a result, the thermodynamic functions of the system follow straightforwardly from a knowledge of the energy levels of the individual constituents. Notable examples of phe-nomena belonging to this category are the specific heats of gases (Sections 1.4 and 6.5), the specific heats of solids (Section 7.4), chemical reactions and equilibrium constants (Section 6.6), the condensation of an ideal Bose gas (Sections 7.1 and 7.2), the spectral distribution of the blackbody radiation (Section 7.3), the elementary electron theory of metals (Section 8.3), the phenomenon of paramagnetism (Sections 3.9 and 8.2), and so on. In the case of solids, the interatomic interaction does, in fact, play an important physical role; however, since the actual positions of the atoms, over a substantial range of temper-atures, do not depart significantly from their mean values, we can rewrite our problem in terms of the so-called normal coordinates and treat the given solid as an “assembly of practically noninteracting harmonic oscillators.” We note that the most significant feature of the phenomena falling in the first category is that, with the sole exception of Bose– Einstein condensation, the thermodynamic functions of the systems involved are smooth and continuous! Phenomena belonging to the second category, however, present a very different sit-uation. In most cases, one encounters analytic discontinuities or singularities in the thermodynamic functions of the given system which, in turn, correspond to the occur-rence of various kinds of phase transitions. Notable examples of phenomena belonging to this category are the condensation of gases, the melting of solids, phenomena associ-ated with the coexistence of phases (especially in the neighborhood of a critical point), the behavior of mixtures and solutions (including the onset of phase separation), phenom-ena of ferromagnetism and antiferromagnetism, the order–disorder transitions in alloys, the superfluid transition from liquid He I to liquid He II, the transition from a normal to a superconducting material, and so on. The characteristic feature of the interparticle inter-actions in these systems is that they cannot be “removed” by means of a transformation of the coordinates of the problem; accordingly, the energy levels of the total system can-not, in any simple manner, be related to the energy levels of the individual constituents. One finds instead that, under favorable circumstances, a large number of microscopic Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00012-8 © 2011 Elsevier Ltd. All rights reserved. 401 402 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling constituents of the system may exhibit a tendency of interacting with one another in a rather strong, cooperative fashion. This cooperative behavior assumes macroscopic signifi-cance at a particular temperature Tc, known as the critical temperature of the system, and gives rise to the kind of phenomena listed previously. Mathematical problems associated with the study of cooperative phenomena are quite formidable.1 To facilitate calculations, one is forced to introduce models in which the interparticle interactions are considerably simplified, yet retain characteristics that are essential to the cooperative aspect of the problem. One then hopes that a theoretical study of these simplified models, which still involves serious difficulties of analysis, will repro-duce the most basic features of the phenomena exhibited by actual physical systems. For instance, in the case of a magnetic transition, one may consider a lattice structure in which all interactions other than the ones among nearest-neighbor spins are neglected. It turns out that a model as simplified as that captures practically all the essential features of the phenomenon — especially in the close neighborhood of the critical point. The inclusion of interactions among spins farther out than the nearest neighbors does not change these features in any significant manner, nor are they affected by the replacement of one lattice structure by another so long as the dimensionality of the lattice is the same. Not only this, these features may also be shared, with little modification, by many other physical sys-tems undergoing very different kinds of phase transitions, for example, gas–liquid instead of paramagnetic–ferromagnetic. This “unity in diversity” turns out to be a hallmark of the phenomena associated with phase transitions — a subject we propose to explore in considerable detail in this and the following two chapters, but first a few preliminaries. 12.1 General remarks on the problem of condensation We consider an N-particle system, obeying classical or quantum statistics, with the proviso that the total potential energy of the system can be written as a sum of two-particle terms u(rij), with i < j. The function u(r) is supposed to satisfy the conditions u(r) = +∞ for r ≤σ, 0 > u(r) > −ε for σ < r < r∗ u(r) = 0 for r ≥r∗      ; (1) see Figure 12.1. Thus, each particle may be looked upon as a hard sphere of diameter σ, surrounded by an attractive potential of range r∗and of (maximum) depth ε. From a prac-tical point of view, conditions (1) do not entail any “serious restriction” on the two-body potential, for the interparticle potentials ordinarily met with in nature are not materially 1In this connection, one should note that the mathematical schemes developed in Chapters 10 and 11 give reliable results only if the interactions among the microscopic constituents of the given system are sufficiently weak — in fact, too weak to bring about cooperative transitions. 12.1 General remarks on the problem of condensation 403 σ 0 2 r u(r) r∗ FIGURE 12.1 A sketch of the interparticle potential u(r), as given by equation (1). different from the one satisfying these conditions. We, therefore, expect that the conclu-sions drawn from the use of this potential will not be very far from the realities of the actual physical phenomena. Suppose that we are able to evaluate the exact partition function, QN(V,T), of the given system. This function will possess certain properties that have been recognized and accepted for quite some time, though a rigorous proof of these was first attempted by van Hove as late as in 1949.2 These properties can be expressed as follows: (i) In the thermodynamic limit (i.e., when N and V →∞while the ratio N/V stays constant), the quantity N−1 lnQ tends to be a function only of the specific volume v (= V/N) and the temperature T; this limiting form may be denoted by the symbol f (v,T). It is natural to identify f (v,T) with the intensive variable −A/NkT, where A is the Helmholtz free energy of the system. The thermodynamic pressure P is then given by P(v,T) = −  ∂A ∂V  N,T = kT ∂f ∂v  T , (2) which turns out to be a strictly nonnegative quantity. (ii) The function f (v,T) is everywhere concave, so the slope (∂P/∂v)T of the (P,v)-curve is never positive. While at high temperatures the slope is negative for all v, at lower temperatures there can exist a region (or regions) in which the slope is zero and, consequently, the system is infinitely compressible! The existence of such regions, in the (P,v)-diagram, corresponds to the coexistence of two or more phases of different density in the given system; in other words, it constitutes direct evidence of the onset of a phase transition in the system. In this connection it is important to note that, so long as one uses the exact partition function of the system, isotherms of the van der Waals type, which possess unphysical regions of positive slope, never appear. On the 2For historical details, see Griffiths (1972, p. 12). 404 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling A v3 v2 v1 v 2 1 P ˜(T) B 2 3 P FIGURE 12.2 An unphysical isotherm corrected with the help of the Maxwell construction; the horizontal line is such that the areas A and B are equal. The “corrected” isotherm corresponds to a phase transition, taking place at pressure ˜ P(T), with densities v−1 1 and v−1 3 of the respective phases. other hand, if the partition function is evaluated under approximations, as we did in the derivation of the van der Waals equation of state in Section 10.3, isotherms with unphysical regions may indeed appear. In that case the isotherms in question have got to be “corrected,” by introducing a region of “flatness” (∂P/∂v = 0), with the help of the Maxwell construction of equal areas; see Figure 12.2.3 The real reason for the appearance of unphysical regions in the isotherms is that the approximate evalua-tions of the partition function introduce, almost invariably (though implicitly), the restraint of a uniform density throughout the system. This restraint eliminates the very possibility of the system passing through states in which there exist, side by side, two phases of different densities; in other words, the existence of a region of “flatness” in the (P,v)-diagram is automatically ruled out. On the other hand, an exact evaluation of the partition function must allow for all possible configurations of the system, including the ones characterized by a simultaneous existence of two or more phases of different densities. Under suitable conditions (for instance, when the temperature is sufficiently low), such a configuration might turn out to be the equilibrium configuration of the system, with the result that the system shows up in a multiphase, rather than a single-phase, state. We should, in this context, mention that if in the evaluation of the partition function one introduces no other approximation except the assumption of a uniform density in the system, then the resulting isotherms, corrected with the help of the Maxwell construction, would be the exact isotherms of the problem. 3The physical basis of the Maxwell construction can be seen with the help of the Gibbs free energy density g(T,P). Since dg = −sdT + vdP and along the “corrected” isotherm dP = dT = 0, it follows that g1 = g3; see Figure 12.2. To achieve the same result from the theoretical isotherm (along which dT = 0 but dP ̸= 0), we clearly require that the quantity vdP, integrated along the isotherm from state 1 to state 3, must vanish; this leads to the “theorem of equal areas.” 12.1 General remarks on the problem of condensation 405 (iii) The presence of an absolutely flat portion in an isotherm, with mathematical singularities at its ends, is, strictly speaking, a consequence of the limiting process N →∞. If N were finite, and if the exact partition function were used, then the quantity P′, defined by the relation P′ = kT ∂lnQ ∂V  N,T , (3) would be free from mathematical singularities. The ordinarily sharp corners in an isotherm would be rounded off; at the same time, the ordinarily flat portion of the isotherm would not be strictly flat — it would have for large N a small, negative slope. In fact, the quantity P′ in this case would not be a function of v and T alone; it would depend on the number N as well, though in a thermodynamically negligible manner. If we employ the grand partition function Q, as obtained from the exact partition functions QN, namely Q(z,V,T) = X N≥0 QN(V,T)zN, (4) a similar picture results. To see this, we note that for real molecules, with a given V, the variable N will be bounded by an upper limit, say Nm, which is the number of molecules that fill the volume V “tight-packed”; obviously, Nm ∼V/σ 3. For N > Nm, the potential energy of the system will be infinite; accordingly, QN(N > Nm) ≡0. (5) Hence, for all practical purposes, our power series in (4) is a polynomial in z (which is ≥0) and is of degree Nm. Since the coefficients QN are all positive and Q0 ≡1, the sum Q ≥1. The thermodynamic potential lnQ is, therefore, a well-behaved function of the parameters z, V, and T. Consequently, so long as V (and hence Nm) remains finite, we do not expect any singularities or discontinuities in any of the functions derived from this potential. A nonanalytic behavior could appear only in the limit (V,Nm) →∞. We now define P′ by the relation P′ = kT V lnQ (V finite); (6) since Q ≥1,P′ ≥0. The mean number of particles and the mean square deviation in this number are given by the formulae N = ∂lnQ ∂lnz  V,T (7) 406 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling and N2 −N2 ≡ N −N 2 = ∂N ∂lnz ! V,T , (8) respectively; see Sections 4.2 and 4.5. Accordingly, ∂lnQ ∂N  V,T = ∂lnQ ∂lnz  V,T ∂N ∂lnz ! V,T = N N2 −N2 . (9) At the same time, writing v for V/N and using (6), we have ∂lnQ ∂N  V,T = V kT ∂P′ ∂N  V,T = −v2 kT ∂P′ ∂v  V,T . (10) Comparing (9) and (10), we obtain ∂P′ ∂v  V,T = −kT V 2 N3 N2 −N2 , (11) which is clearly nonpositive.4 For finite V, expression (11) will never vanish; accordingly, P′ will never be strictly constant. Nevertheless, the slope (∂P′/∂v) can, in a certain region, be extremely small — in fact, as small as O(1/N); such a region would hardly be distin-guishable from a phase transition because, on a macroscopic scale, the value of P′ in such a region would be as good as a constant.5 If we now define the pressure of the system by the limiting relationship P(v,T) = Lim V→∞P′ (v,T;V) = kT Lim V→∞  1 V lnQ(z,V,T)  , (12) then we can expect, in a set of isotherms, an absolutely flat portion (∂P/∂v ≡0), with sharp corners implying mathematical singularities. The mean particle density n would now be given by n = Lim V→∞  1 V ∂lnQ(z,V,T) ∂lnz  ; (13) it seems important to mention here that the operation V →∞and the operation ∂/∂lnz cannot be interchanged freely. In passing, we note that the picture emerging from the grand partition function Q, which has been obtained from the exact partition functions QN, remains practically 4Compare equation (11), which has been derived here nonthermodynamically, with equation (4.5.7) derived earlier. 5The presence of such a region entails that (N2 −N 2) be O(N 2). This implies that the fluctuations in the variable N be macroscopically large, which in turn implies equally large fluctuations in the variable v within the system and hence the coexistence of two or more phases with different values of v. In a single-phase state, (N2 −N 2) is O(N); the slope (∂P′/∂v) is then O(N 0), as an intensive quantity should be. 12.2 Condensation of a van der Waals gas 407 unchanged even if one had employed a set of approximate QN. This is so because the argument developed in the preceding paragraphs makes no use whatsoever of the actual form of the functions QN. Thus, if an approximate QN leads to the van der Waals type of loop in the canonical ensemble, as shown in Figure 12.2, the corresponding set of QN, when employed in a grand canonical ensemble, would lead to isotherms free from such loops (Hill, 1953). Subsequent to van Hove, Yang and Lee (1952) suggested an altemative approach that enables one to carry out a rigorous mathematical discussion of the phenomenon of con-densation and of other similar transitions. In their approach, one is primarily concerned with the analytic behavior of the quantities P and n, of equations (12) and (13), as func-tions of z at different values of T. The problem is examined in terms of the “zeros of the grand partition function Q in the complex z-plane,” with attention focused on the way these zeros are distributed in the plane and the manner in which they evolve as the vol-ume of the system is increased. For real, positive z, Q ≥1, therefore none of the zeros will lie on the real, positive axis in the z-plane. However, as V →∞(and hence the degree of the polynomial (4) and, with it, the number of zeros itself grows to infinity), the distribu-tion of zeros is expected to become continuous and, depending on T, may in fact converge on the real, positive axis at one or more points zc. If so, our functions P(z) and n(z), even with z varied along the real axis only, may, by virtue of their relationship to the function lnQ, turn out to be singular at the points z = zc. The presence of such a singularity would imply the onset of a phase transition in the system. For further details of this approach, see Sections 12.3 and 12.4 of the first edition of this book; see also Griffiths (1972, pp. 50–58). 12.2 Condensation of a van der Waals gas We start with the simplest, and historically the first, theoretical model that undergoes a gas–liquid phase transition. This model is generally referred to as the van der Waals gas and obeys the equation of state, see equation (10.3.9), P = RT v −b −a v2 , (1) v being the molar volume of the gas; the parameters a and b then also pertain to one mole of the gas. We recall that, while a is a measure of the attractive forces among the molecules of the system, b is a measure of the repulsive forces that come into play when two molecules come too close to one another; accordingly, b is also a measure of the “effective space” occupied by the molecules (by virtue of a finite volume that may be asso-ciated with each one of them). In Section 10.3, the equation of state (1) was derived under the express assumption that v ≫b; here, we shall pretend, with van der Waals, that this equation holds even when v is comparable to b. The isotherms following from equation (1) are shown in Figure 12.3. We note that, for temperatures above a critical temperature Tc,P decreases monotonically with v. For 408 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling 1 2 Coexistence curve T Tc T Tc Critical point T  Tc 3 P(T) Pc vI vg c v P FIGURE 12.3 The isotherms of a van der Waals system; those for T < Tc are “corrected” with the help of the Maxwell construction, thus leading to the coexistence curve at the top of which sits the critical point. T < Tc, however, the relationship is nonmonotonic, so that over a certain range of v we encounter a region where (∂P/∂v) > 0; such a region is unphysical and must be “corrected” with the help of the Maxwell construction,6 leading to an isotherm with a flat portion sig-naling transition from the gaseous state with molar volume vg(T) to the liquid state with molar volume vl(T) at a constant pressure P(T). For vl < v < vg, the system resides in a state of mixed phases — partly liquid, partly gaseous — and, since the passage from one end of the flat portion to the other takes place with 1v ̸= 0 but 1P = 0, the system is all along in a state of infinite compressibility; clearly, we are encountering here a brand of behavior that is patently singular. As T increases toward Tc, the transition takes place at a comparatively higher value of P, with vg less than and vl more than before — so that, as T →Tc, both vg and vl approach a common value vc that may be referred to as the critical volume; the corresponding value of P may then be designated by Pc, and we find ourselves located at the critical point of the system. The locus of all such points as vl(T) and vg(T) is generally referred to as the coexistence curve, for the simple reason that in the region enclosed by this curve the gaseous and the liquid phases mutually coexist; the top of this curve, where vl = vg, coincides with the critical point itself. Finally, the isotherm 6A more precise formulation of the van der Waals theory, as the limit of a theory with an infinite range potential, has been formulated by Kac, Uhlenbeck and Hemmer (1963). They considered the potential u(r) = +∞ for r ≤σ −κe−κr for r > σ, so that the integral R ∞ σ u(r)dr is simply −exp(−κσ); when κ →0 the potential becomes infinite in range but infinitesi-mally weak. Kac et al. showed that, in this limit, the model becomes essentially the same as van der Waals’ — with one noteworthy improvement, that is, no unphysical regions in the (P,v)–diagram appear and hence no need for the Maxwell construction arises. 12.2 Condensation of a van der Waals gas 409 pertaining to T = Tc, which, of course, passes through the critical point is referred to as the critical isotherm of the system; it is straightforward to see that the critical point is a point of inflection of this isotherm, so that both (∂P/∂v)T and (∂2P/∂v2)T vanish at this point. Using (1), we obtain for the coordinates of the critical point Pc = a 27b2 , vc = 3b, Tc = 8a 27bR, (2) so that the number K ≡RTc/Pcvc = 8/3 = 2.666.... (3) We thus find that, while Pc,vc, and Tc vary from system to system (through the interaction parameters a and b), the quantity K has a common, universal value for all of them — so long as they all obey the same (i.e., van der Waals) equation of state. The experimental results for K indeed show that it is nearly the same over a large group of substances; for instance, its value for carbon tetrachloride, ethyl ether, and ethyl formate is 3.677, 3.814, and 3.895, respectively — close, though not exactly the same, and also a long way from the van der Waals value. The concept of universality is, nonetheless, there (even though the van der Waals equation of state may not truly apply). It is now tempting to see if the equation of state itself can be written in a universal form. We find that this indeed can be done by introducing reduced variables Pr = P Pc , vr = v vc , Tr = T Tc . (4) Using (1) and (2), we readily obtain the reduced equation of state  Pr + 3 v2 r  (3vr −1) = 8Tr, (5) which is clearly universal for all systems obeying van der Waals’ original equation of state (1); all we have done here is to rescale the observable quantities P, v, and T in terms of their critical values and thereby “push the interaction parameters a and b into the back-ground.” Now, if two different systems happen to be in states characterized by the same values of vr and Tr, then their Pr would also be the same; the systems are then said to be in “corresponding states” and, for that reason, the statement just made is referred to as the “law of corresponding states.” Clearly, the passage from equation (1) to equation (5) takes us from an expression of diversity to a statement of unity! We shall now examine the behavior of the van der Waals system in the close neighbor-hood of the critical point. For this, we write Pr = 1 + π, vr = 1 + ψ, Tr = 1 + t. (6) 410 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling Equation (5) then takes the form π  2 + 7ψ + 8ψ2 + 3ψ3 + 3ψ3 = 8t  1 + 2ψ + ψ2 . (7) First of all, along the critical isotherm (t = 0) and in the close vicinity of the critical point (|π|,|ψ| ≪1), we obtain the simple, asymptotic result π ≈−3 2ψ3, (8) which is indicative of the “degree of flatness” of the critical isotherm at the critical point. Next, we examine the dependence of ψ on t as we approach the critical point from below. For this, we write (7) in the form 3ψ3 + 8(π −t)ψ2 + (7π −16t)ψ + 2(π −4t) ≃0. (9) Now, a close look at the (symmetric) shape of the coexistence curve near its top (where |t| ≪1) shows that the three roots ψ1,ψ2, and ψ3 of equation (9), which arise from the limiting behavior of the roots v1,v2, and v3 of the original equation of state (1) as T →Tc−, are such that |ψ2| ≪|ψ1,3| and |ψ1| ≃|ψ3|. This means that, in the region of interest, π ≈4t, (10) so that one of the roots, ψ2, of equation (9) essentially vanishes while the other two are given by ψ2 + 8tψ + 4t ≃0. (9a) We expect the middle term here to be negligible (as will be confirmed by the end result), yielding ψ1,3 ≈±2|t|1/2; (11) note that the upper sign here pertains to the gaseous phase and the lower sign to the liquid phase. Finally, we consider the isothermal compressibility of the system which, in terms of reduced variables, is determined essentially by the quantity −(∂ψ/∂π)t. Retaining only the dominant terms, we obtain from (7) − ∂ψ ∂π  t ≈ 2 7π + 9ψ2 −16t . (12) For t > 0, we approach the critical point along the critical isochore (ψ = 0); equation (12), with the help of equation (10), then gives − ∂ψ ∂π  t→0+ ≈1 6t . (13) 12.3 A dynamical model of phase transitions 411 For t < 0, we approach the critical point along the coexistence curve (on which ψ2 ≈−4t); we now obtain − ∂ψ ∂π  t→0− ≈ 1 12|t|. (14) For the record, we quote here results for the specific heat, CV , of the van der Waals gas (Uhlenbeck, 1966; Thompson, 1988) CV ≈      (CV )ideal + 9 2Nk  1 + 28 25t  (t ≤0) (15a) (CV )ideal (t > 0), (15b) which imply a finite jump at the critical point. Equations (8), (11), (13), (14), and (15) illustrate the nature of the critical behavior dis-played by a van der Waals system undergoing the gas–liquid transition. While it differs in several important respects from the critical behavior of real physical systems, it shows up again and again in studies pertaining to other critical phenomena that have apparently nothing to do with the gas–liquid phase transition. In fact, this particular brand of behav-ior turns out to be a benchmark against which the results of more sophisticated theories are automatically compared. 12.3 A dynamical model of phase transitions A number of physico-chemical systems that undergo phase transitions can be represented, to varying degrees of accuracy, by an “array of lattice sites, with only nearest-neighbor interaction that depends on the manner of occupation of the neighboring sites.” This simple-minded model turns out to be good enough to provide a unified, theoretical basis for understanding a variety of phenomena such as ferromagnetism and antiferromag-netism, gas–liquid and liquid–solid transitions, order–disorder transitions in alloys, phase separation in binary solutions, and so on. There is no doubt that this model considerably oversimplifies the actual physical systems it is supposed to represent; nevertheless, it does retain the essential physical features of the problem — features that account for the prop-agation of long-range order in the system. Accordingly, it does lead to the onset of a phase transition in the given system, which arises in the nature of a cooperative phenomenon. We find it convenient to formulate our problem in the language of ferromagnetism; later on, we shall establish correspondence between this language and the languages appropriate to other physical phenomena. We thus regard each of the N lattice sites to be occupied by an atom possessing a magnetic moment µ, of magnitude gµB √[J(J + 1)], which is capable of (2J + 1) discrete orientations in space. These orientations define “dif-ferent possible manners of occupation” of a given lattice site; accordingly, the whole lattice is capable of (2J + 1)N different configurations. Associated with each configuration is an energy E that arises from mutual interactions among the neighboring atoms of the 412 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling lattice and from the interaction of the whole lattice with an external field B. A statistical analysis in the canonical ensemble should then enable us to determine the expectation value, M(B,T), of the net magnetization M. The presence of a spontaneous magnetization M(0,T) at temperatures below a certain (critical) temperature Tc and its absence above that temperature will then be interpreted as a ferromagnetic phase transition in the system at T = Tc. Detailed studies, both theoretical and experimental, have shown that, for all ferromag-netic materials, data on the temperature dependence of the spontaneous magnetization, M(0,T), fit best with the value J = 1 2; see Figure 12.4. One is, therefore, tempted to infer that the phenomenon of ferromagnetism is associated only with the spins of the electrons and not with their orbital motions. This is further confirmed by gyromagnetic experiments (Barnett, 1944; Scott, 1951, 1952), in which one either reverses the magnetization of a freely suspended specimen and observes the resulting rotation or imparts a rotation to the spec-imen and observes the resulting magnetization; the former is known as the Einstein–de Haas method, the latter the Barnett method. From these experiments one can derive the relevant g-value of the specimen which, in each case, turns out to be very close to 2; this, as we know, pertains to the electron spin. Therefore, in discussing the problem of ferro-magnetism, we may specifically take: µ = 2µB √[s(s + 1)], where s is the quantum number associated with the electron spin. With s = 1 2, only two orientations are possible for each lattice site, namely sz = + 1 2 (with µz = +µB) and sz = −1 2 (with µz = −µB). The whole lattice is then capable of 2N configurations; one such configuration is shown in Figure 12.5. We now consider the nature of the interaction energy between two neighboring spins si and sj. According to quantum mechanics, this energy is of the form Kij ± Jij, where the upper sign applies to “antiparallel” spins (S = 0) and the lower sign to “parallel” spins (S = 1). Here, Kij is the direct or Coulomb energy between the two spins, while Jij is the 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 Fe J 5 J 51 J 51/2 Co Ni T Tc M (0, T ) M (0, 0) FIGURE 12.4 Spontaneous magnetization of iron, nickel, and cobalt as a function of temperature. Theoretical curves are based on the Weiss theory of ferromagnetism. 12.3 A dynamical model of phase transitions 413 FIGURE 12.5 One of the 2N possible configurations of a system composed of N spins; here, N = 25. exchange energy between them: Kij = Z ψ∗ i (1)ψ∗ j (2)uijψj(2)ψi(1)dτ1dτ2, (1) while Jij = Z ψ∗ j (1)ψ∗ i (2)uijψj(2)ψi(1)dτ1dτ2, (2) uij being the relevant interaction potential. The energy difference between a state of “parallel” spins and one of “antiparallel” spins is given by ε↑↑−ε↑↓= −2Jij. (3) If Jij > 0, the state ↑↑is energetically favored against the state ↑↓; we then look for the possibility of ferromagnetism. If, on the other hand, Jij < 0, the situation is reversed and we see the possibility of antiferromagnetism. It seems useful to express the interaction energy of the two states, ↑↑and ↓↓, by a single expression; for this, we consider the eigenvalues of the scalar product si · sj = 1 2 n (si + sj)2 −s2 i −s2 j o = 1 2S(S + 1) −s(s + 1), (4) which equals + 1 4 if S = 1 and −3 4 if S = 0. We may, therefore, write for the interaction energy of the spins i and j εij = const. −2Jij(si · sj), (5) which is consistent with the energy difference (3). The precise value of the constant here is immaterial because the potential energy is arbitrary to the extent of an additive constant anyway. Typically, the exchange interaction Jij falls off rapidly as the separation of the two 414 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling spins is increased. To a first approximation, therefore, we may regard Jij as negligible for all but nearest-neighbor pairs (for which its value may be denoted by a common symbol J). The interaction energy of the whole lattice is then given by E = const. −2J X n.n. si · sj  , (6) where the summation goes over all nearest-neighbor pairs in the lattice. The model based on expression (6) for the interaction energy of the lattice is known as the Heisenberg model (1928). A simpler model results if we use, instead of (6), a truncated expression in which the product (si · sj), which is equal to the sum (sixsjx + siysjy + sizsjz), is replaced by a single term sizsjz; one reason for adopting this simpler model is that it does not necessarily require a quantum-mechanical treatment (because all the variables in the truncated expression for E commute). Expression (6) may now be written as E = const. −J X n.n. σiσj, (7) where the new symbol σi (or σj) = +1 for an “up” spin and −1 for a “down” spin; note that, with the introduction of the new symbol, we still have: ε↑↑−ε↑↓= −2J. The model based on expression (7) is known as the Ising model; it originated with Lenz (1920) and was subsequently investigated by his student Ising (1925).7 A different model results if we suppress the z-components of the spins and retain the x- and y-components instead. This model was originally introduced by Matsubara and Matsuda (1956) as a model of a quantum lattice gas, with possible relevance to the super-fluid transition in liquid He4. The critical behavior of this so-called XY model has been investigated in detail by Betts and coworkers, who have also emphasized the relevance of this model to the study of insulating ferromagnets (see Betts et al., 1968 – 1974). It seems appropriate to regard the Ising and the XY models as special cases of an anisotropic Heisenberg model with interaction parameters Jx,Jy, and Jz; while the Ising model represents the situation Jx,Jy ≪Jz, the XY model represents just the opposite. Intro-ducing a parameter n, which denotes the number of spin components entering into the Hamiltonian of the system, we may regard the Ising, the XY , and the Heisenberg models as pertaining to the n-values 1, 2, and 3, respectively. As will be seen later, the parameter n, along with the dimensionality d of the lattice, constitutes the basic set of elements that determine the qualitative nature of the critical behavior of a given system. For the time being, though, we confine our attention to the Ising model, which is not only the sim-plest one to analyze but also unifies the study of phase transitions in systems as diverse as ferromagnets, gas–liquids, liquid mixtures, binary alloys, and so on. 7For an historical account of the origin and development of the Lenz–Ising model, see the review article by Brush (1967). This review gives a large number of other references as well. 12.3 A dynamical model of phase transitions 415 To study the statistical mechanics of the Ising model, we disregard the kinetic energy of the atoms occupying the various lattice sites, for the phenomenon of phase transitions is essentially a consequence of the interaction energy among the atoms; in the interac-tion energy again, we include only the nearest-neighbor contributions, in the hope that the farther-neighbor contributions would not affect the results qualitatively. To fix the z-direction, and to be able to study properties such as magnetic susceptibility, we subject the lattice to an external magnetic field B, directed “upward”; the spin σi then possesses an additional potential energy −µBσi.8 The Hamiltonian of the system in configuration {σ1,σ2,...,σN} is then given by H{σi} = −J X n.n. σiσj −µB X i σi, (8) and the partition function by QN(B,T) = X σ1 X σ2 ... X σN exp[−βH{σi}] = X σ1 X σ2 ... X σN exp " βJ X n.n. σiσj + βµB X i σi # . (9) The Helmholtz free energy, the internal energy, the specific heat, and the net magnetiza-tion of the system then follow from the formulae A(B,T) = −kT lnQN(B,T), (10) U(B,T) = −T2 ∂ ∂T  A T  = kT2 ∂ ∂T lnQN, (11) C(B,T) = ∂U ∂T = −T ∂2A ∂T2 , (12) and M(B,T) = µ X i σi ! =  −∂H ∂B  = 1 β ∂lnQN ∂B  T = − ∂A ∂B  T . (13) Obviously, the quantity M(0,T) gives the spontaneous magnetization of the system; if it is nonzero at temperatures below a certain critical temperature Tc, the system would be ferromagnetic for T < Tc and paramagnetic for T > Tc. At the transition temperature itself, the system is expected to show some sort of a singular behavior. It is obvious that the energy levels of the system as a whole will be degenerate, in the sense that the various configurations {σi} will not all possess distinct energy values. In fact, the energy of a given configuration does not depend on the detailed values of all 8Henceforth, we use the symbol µ instead of µB. 416 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling the variables σi; it depends only on a few numbers such as the total number N+ of “up” spins, the total number N++ of “up–up” nearest-neighbor pairs, and so on. To see this, we define certain other numbers as well: N−as the total number of “down” spins, N−−as the total number of “down–down” nearest-neighbor pairs, and N+−as the total number of nearest-neighbor pairs with opposite spins. The numbers N+ and N−must satisfy the relation N+ + N−= N. (14) And if q denotes the coordination number of the lattice, that is, the number of nearest neighbors for each lattice site,9 then we also have the relations qN+ = 2N++ + N+−, (15) qN−= 2N−−+ N+−. (16) With the help of these relations, we can express all our numbers in terms of any two of them, say N+ and N++. Thus N−= N −N+, N+−= qN+ −2N++, N−−= 1 2qN −qN+ + N++; (17) it will be noted that the total number of nearest-neighbor pairs of all types is given, quite expectedly, by the expression N++ + N−−+ N+−= 1 2qN. (18) Naturally, the Hamiltonian of the system can also be expressed in terms of N+ and N++; we have from (8), with the help of the relations established above, HN(N+,N++) = −J(N++ + N−−−N+−) −µB(N+ −N−) = −J 1 2qN −2qN+ + 4N++  −µB(2N+ −N). (19) Now, let gN(N+,N++) be “the number of distinct ways in which the N spins of the lattice can be so arranged as to yield certain preassigned values of the numbers N+ and N++.” The partition function of the system can then be written as QN(B,T) = X′ N+,N++ gN (N+,N++)exp{−βHN(N+,N++)}, (20) 9The coordination number q for a linear chain is obviously 2; for two-dimensional lattices, namely honeycomb, square, and triangular, it is 3, 4, and 6, respectively; for three-dimensional lattices, namely simple cubic, body-centered cubic, and face-centered cubic, it is 6, 8, and 12, respectively. 12.4 The lattice gas and the binary alloy 417 that is, e−βA = eβN  1 2 qJ−µB  N X N+=0 e−2β(qJ−µB)N+ X N++ ′gN(N+,N++)e4βJN++, (21) where the primed summation in (21) goes over all values of N++ that are consistent with a fixed value of N+ and is followed by a summation over all possible values of N+, that is, from N+ = 0 to N+ = N. The central problem thus consists in determining the combinatorial function gN(N+,N++) for the various lattices of interest. 12.4 The lattice gas and the binary alloy Apart from ferromagnets, the Ising model can be readily adapted to simulate the behavior of certain other systems as well. More common among these are the lattice gas and the binary alloy. The lattice gas Although it had already been recognized that the results derived for the Ising model would apply equally well to a system of “occupied” and “unoccupied” lattice sites (i.e., to a system of “atoms” and “holes” in a lattice), it was Yang and Lee (1952) who first used the term “lattice gas” to describe such a system. By definition, a lattice gas is a collection of atoms, Na in number, that can occupy only discrete positions in space — positions that constitute a lattice structure with coordination number q. Each lattice site can be occupied by at most one atom, and the interaction energy between two occupied sites is nonzero, say −ε0, only if the sites involved constitute a nearest-neighbor pair. The configurational energy of the gas is then given by E = −ε0Naa, (1) where Naa is the total number of nearest-neighbor pairs (of occupied sites) in a given con-figuration of the system. Let gN(Na,Naa) denote “the number of distinct ways in which the Na atoms of the gas, assumed indistinguishable, can be distributed among the N sites of the lattice so as to yield a certain preassigned value of the number Naa.” The partition function of the system, neglecting the kinetic energy of the atoms, is then given by QNa(N,T) = X′ Naa gN (Na,Naa)eβε0Naa, (2) where the primed summation goes over all values of Naa that are consistent with the given values of Na and N; clearly, the number N here plays the role of the “total volume” available to the gas. 418 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling Going over to the grand canonical ensemble, we write for the grand partition function of the system Q(z,N,T) = N X Na=0 zNaQNa(N,T). (3) The pressure P and the mean number Na of the atoms in the gas are then given by eβPN = N X Na=0 zNa X′ Naa gN (Na,Naa)eβε0Naa (4) and Na N = 1 v = z kT ∂P ∂z  T ; (5) here, v denotes the average volume per particle of the gas (measured in terms of the “volume of a primitive cell of the lattice”). To establish a formal correspondence between the lattice gas and the ferromagnet, we compare the present formulae with the ones established in the preceding section — in par-ticular, formula (4) with formula (12.3.21). The first thing to note here is that the canonical ensemble of the ferromagnet corresponds to the grand canonical ensemble of the lattice gas! The rest of the correspondence is summarized in the following chart: The lattice gas The ferromagnet Na,N −Na ↔ N+,N −N+(= N−) ε0 ↔ 4J z ↔ exp{−2β(qJ −µB)} P ↔ −  A N + 1 2qJ −µB  Na N  = 1 v  ↔ N+ N = 1 2 ( M Nµ + 1 )! , where M = µ N+ −N−) = µ(2N+ −N  . (6) We also note that the ferromagnetic analogue of formula (5) would be N+ N = 1 kT    ∂  A/N + 1 2qJ −µB  2β∂ qJ −µB     T = 1 2  −1 Nµ ∂A ∂B  T + 1  (7) 12.4 The lattice gas and the binary alloy 419 which, by equation (12.3.13), assumes the expected form N+ N = 1 2 M Nµ + 1 ! . (8) It is quite natural to ask: does lattice gas correspond to any real physical system in nature? The immediate answer is that if we let the lattice constant tend to zero (thus going from a discrete structure to a continuous one) and also add, to the lattice-gas formulae, terms corresponding to an ideal gas (namely, the kinetic energy terms), then the model might simulate the behavior of a gas of real atoms interacting through a delta function potential. A study of the possibility of a phase transition in such a system may, therefore, be of some value in understanding phase transitions in real gases. The case ε0 > 0, which implies an attractive interaction among the nearest neighbors, has been frequently cited as a possible model for a gas–liquid transition. On the other hand, if the interaction is repulsive (ε0 < 0), so that configurations with alternating sites being “occupied” and “unoccupied” are the more favored ones, then we obtain a model that arouses interest in connection with the theory of solidification; in such a study, however, the lattice constant has to stay finite. Thus, several authors have pursued the study of the antiferromagnetic version of this model, hoping that this might throw some light on the liquid–solid transition. For a bibliography of these pursuits, see the review article by Brush (1967). The binary alloy Much of the early activity in the theoretical analysis of the Ising model was related to the study of order–disorder transitions in alloys. In an alloy — to be specific, a binary alloy — we have a lattice structure consisting of two types of atoms, say 1 and 2, numbering N1 and N2, respectively. In a configuration characterized by the numbers N11,N22, and N12 of the three types of nearest-neighbor pairs, the configurational energy of the alloy may be written as E = ε11N11 + ε22N22 + ε12N12, (9) where ε11,ε22, and ε12 have obvious meanings. As in the case of a ferromagnet, the various numbers appearing in the expression for E may be expressed in terms of the numbers N,N1, and N11 (of which only N11 is variable here). Equation (9) then takes the form E = ε11N11 + ε22 1 2qN −qN1 + N11  + ε12 qN1 −2N11  = 1 2qε22N + q(ε12 −ε22)N1 + (ε11 + ε22 −2ε12)N11. (10) 420 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling The correspondence between this system and the lattice gas is now straightforward: The lattice gas The binary alloy Na,N −Na ↔ N1,N −N1(= N2) −ε0 ↔ (ε11 + ε22 −2ε12) A ↔ A −1 2qε22N −q(ε12 −ε22)N1 The correspondence with a ferromagnet can be established likewise; in particular, this requires that ε11 = ε22 = −J and ε12 = +J. At absolute zero, our alloy will be in the state of minimum configurational energy, which would also be the state of maximum configurational order. We expect that the two types of atoms would then occupy mutually exclusive sites, so that one might speak of atoms 1 being only at sites a and atoms 2 being only at sites b. As tempera-ture rises, an exchange of sites results and, in the face of thermal agitation, the order in the system starts giving way. Ultimately, the two types of atoms get so “mixed up” that the very notion of the sites a being the “right” ones for atoms 1 and the sites b being the “right” ones for atoms 2 break down; the system then behaves, from the configurational point of view, as an assembly of N1 + N2 atoms of essentially the same species. 12.5 Ising model in the zeroth approximation In 1928 Gorsky attempted a statistical study of order–disorder transitions in binary alloys on the basis of the assumption that the work expended in transferring an atom from an “ordered” position to a “disordered” one (or, in other words, from a “right” site to a “wrong” one) is directly proportional to the degree of order prevailing in the system. This idea was further developed by Bragg and Williams (1934, 1935) who, for the first time, introduced the concept of long-range order in the sense we understand it now and, with relatively simple mathematics, obtained results that could explain the main qualitative features of the relevant experimental data. The basic assumption in the Bragg– Williams approximation was that the energy of an individual atom in the given system is determined by the (average) degree of order prevailing in the entire system rather than by the ( fluctuating) configurations of the neighboring atoms. In this sense, the approximation is equivalent to the mean molecular field (or the internal field) theory of Weiss, which was put forward in 1907 to explain the magnetic behavior of ferromagnetic materials. It seems natural to call this approximation the zeroth approximation, for its features are totally insensitive to the detailed structure, or even to the dimensionality, of the lattice. We expect that the results following from this approximation will become more reliable 12.5 Ising model in the zeroth approximation 421 as the number of neighbors interacting with a given atom increases (i.e., as q →∞), thus diminishing the importance of local, fluctuating influences.10 We now define a long-range order parameter L in a given configuration by the very suggestive relationship L = 1 N X i σi = N+ −N− N = 2N+ N −1 (−1 ≤L ≤+1), (1) which gives N+ = N 2 (1 + L) and N−= N 2 (1 −L). (2) The magnetization M is then given by M = (N+ −N−)µ = NµL (−Nµ ≤M ≤+Nµ); (3) the parameter L is, therefore, a direct measure of the net magnetization in the system. For a completely random configuration, N+ = N−= 1 2N; the expectation values of both L and M are then identically zero. Now, in the spirit of the present approximation, we replace the first part of the Hamilto-nian (12.3.8) by the expression −J( 1 2qσ)P i σi, that is, for a given σi, we replace each of the qσj by σ while the factor 1 2 is included to avoid duplication in the counting of the nearest-neighbor pairs. Making use of equation (1), and noting that σ ≡L, we obtain for the total configurational energy of the system E = −1 2 qJL  NL −(µB)NL. (4) The expectation value of E is then given by U = −1 2qJNL2 −µBNL. (5) In the same approximation, the difference 1ε between the overall configurational energy of an “up” spin and the overall configurational energy of a “down” spin — specifically, the energy expended in changing an “up” spin into a “down” spin — is given by, see equation (12.3.8), 1ε = −J(qσ)1σ −µB1σ = 2µ qJ µ σ + B  , (6) 10In connection with the present approximation, we may as well mention that early attempts to construct a theory of binary solutions were based on the assumption that the atoms in the solution mix randomly. One finds that the results following from this assumption of random mixing are mathematically equivalent to the ones following from the mean field approximation; see Problems 12.12 and 12.13. 422 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling for here 1σ = −2. The quantity qJσ/µ thus plays the role of the internal (molecular) field of Weiss; it is determined by (i) the mean value of the long-range order prevailing in the system and by (ii) the strength of the coupling, qJ, between a given spin i and its q nearest neighbors. The relative values of the equilibrium numbers N+ and N−then follow from the Boltzmann principle, namely N−/N+ = exp −1ε/kT  = exp  −2µ(B′ + B)/kT , (7) where B′ denotes the internal molecular field: B′ = qJσ/µ = qJ  M/Nµ2 . (8) Substituting (2) into (7), and keeping in mind equation (8), we obtain for L 1 −L 1 + L = exp  −2(qJL + µB)/kT (9) or, equivalently, qJL + µB kT = 1 2 ln 1 + L 1 −L = tanh−1 L. (10) To investigate the possibility of spontaneous magnetization, we let B →0, which leads to the relationship L0 = tanh qJL0 kT ! . (11) Equation (11) may be solved graphically; see Figure 12.6. For any temperature T, the appropriate value of L0(T) is determined by the point of intersection of (i) the straight line y = L0 and (ii) the curve y = tanh(qJL0/kT). Clearly, the solution L0 = 0 is always there; however, we are interested in nonzero solutions, if any. For those, we note that, since the slope of the curve (ii) varies from the initial value qJ/kT to the final value zero while the slope of the line (i) is unity throughout, an intersection other than the one at the origin is possible if, and only if, qJ/kT > 1, (12) that is, T < qJ/k = Tc, say. (13) We thus obtain a critical temperature Tc, below which the system can acquire a nonzero spontaneous magnetization and above which it cannot. It is natural to identify Tc with the Curie temperature of the system — the temperature that marks a transition from the ferromagnetic to the paramagnetic behavior of the system or vice versa. It is clear from Figure 12.6, as well as from equation (11), that if L0 is a solution of the problem, then −L0 is also a solution. The reason for this duplicity of solutions is that, in 12.5 Ising model in the zeroth approximation 423 0 L0 tanh(L0Tc/T), L0 T Tc T Tc L0 1 –1 1 –L0 –1 FIGURE 12.6 The graphical solution of equation (11), with Tc = qJ/k. the absence of an external field, there is no way of assigning a “positive,” as opposed to a “negative,” direction to the alignment of spins. In fact, if B were zero right from the begin-ning, then the positive solution of equation (11) would be as likely to occur as the negative one — with the result that the true expectation value of L0(T) would be zero. If, on the other hand, B were nonzero to begin with (to be definite, say B > 0), then equation (10) for L(B,T) would admit only positive solutions and, in the limit B →0+, we would obtain a positive L0(T). The “up–down symmetry” will then be broken and we will see a net alignment of spins in the “up” direction.11 The precise variation of L0(T) with T can be obtained by solving equation (11) numeri-cally; the general trend, however, can be seen from Figure 12.6. We note that, at T = qJ/k (= Tc), the straight line y = L0 is tangential to the curve y = tanh(qJL0/kT) at the ori-gin; the relevant solution then is L0(Tc) = 0. As T decreases, the initial slope of the curve becomes larger and the relevant point of intersection moves rapidly away from the origin; accordingly, L0(T) rises rapidly as T decreases below Tc. To obtain an approximate depen-dence of L0(T) on T near T = Tc, we write (11) in the form L0 = tanh(L0Tc/T) and use the approximation tanhx ≃x −x3/3, to obtain L0(T) ≈{3(1 −T/Tc)}1/2 (T ≲Tc,B →0). (14) On the other hand, as T →0,L0 →1, in accordance with the asymptotic relationship L0(T) ≈1 −2exp(−2Tc/T) {(T/Tc) ≪1}. (15) Figure 12.7 shows a plot of L0(T) versus T, along with the relevant experimental results for iron, nickel, cobalt, and magnetite; we find the agreement not too bad. 11The concept of “broken symmetry” plays a vital role in this and many other phenomena in physics; for details, see Fisher (1972) and Anderson (1984). 424 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling (T/Tc) 0.5 0 0 0.5 1.0 1.0 L0(T) FIGURE 12.7 The spontaneous magnetization of a Weiss ferromagnet as a function of temperature. The experimental points (after Becker) are for iron (×), nickel (o), cobalt (1), and magnetite (+). The field-free configurational energy and the field-free specific heat of the system are given by, see equation (5), U0(T) = −1 2qJNL2 0 (16) and C0(T) = −qJNL0 dL0 dT = NkL2 0 (T/Tc)2/(1 −L2 0) −T/Tc , (17) where the last step has been carried out with the help of equation (11). Thus, for all T > Tc, both U0(T) and C0(T) are identically zero. However, the value of the specific heat at the transition temperature Tc, as approached from below, turns out to be, see equations (14) and (17), C0(Tc−) = Lim x→0    Nk · 3x (1−x)2 1−3x −(1 −x)   = 3 2Nk. (18) The specific heat, therefore, displays a discontinuity at the transition point. On the other hand, as T →0, the specific heat vanishes, in accordance with the formula, see equations (15) and (17), C0(T) ≈4Nk Tc T 2 exp(−2Tc/T). (19) The full trend of the function C0(T) is shown in Figure 12.8. It is important to note that the vanishing of the configurational energy and the spe-cific heat of the system at temperatures above Tc is directly related to the fact that, in the present approximation, the configurational order prevailing in the system at lower temper-atures is completely wiped out as T →Tc. Consequently, the configurational entropy and 12.5 Ising model in the zeroth approximation 425 (T/Tc) C0(T)/Nk 0 0 0.5 1.0 2.0 1.5 1 2 FIGURE 12.8 The field-free specific heat of a Weiss ferromagnet as a function of temperature. the configurational energy of the system attain their maximum values at T = Tc; beyond that, the system remains thermodynamically “inert.” As a check, we evaluate the con-figurational entropy of the system at T = Tc; with the help of equations (11) and (17), we get S0(Tc) = Tc Z 0 C0(T)dT T = −qJN 0 Z 1 L0 T dL0 = Nk 1 Z 0 (tanh−1 L0)dL0 = Nkln2, (20) precisely the result we expect for a system capable of 2N equally likely microstates.12 The fact that all these microstates are equally likely to occur is again related to the fact that for T ≥Tc there is no (configurational) order in the system. We now proceed to study the magnetic susceptibility of the system. Using equa-tion (10), we get χ(B,T) = ∂M ∂B ! T = Nµ ∂L ∂B ! T = Nµ2 k 1 −L2(B,T) T −Tc{1 −L2(B,T)} . (21) For L ≪1 (which is true at high temperatures for a wide range of B but is also true near Tc if B is small), we obtain the Curie–Weiss law χ0(T) ≈(Nµ2/k)(T −Tc)−1 (T ≳Tc,B →0), (22a) 12Recall equation (3.3.14), whereby S = kln. 426 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling which may be compared with the Curie law derived earlier for a paramagnetic system; see equation (3.9.12). For T less than, but close to, Tc we utilize equation (14) as well and get χ0(T) ≈(Nµ2/2k)(Tc −T)−1 (T ≲Tc,B →0). (22b) Experimentally, one finds that the Curie–Weiss law is satisfied with considerable accuracy, except that the empirical value of Tc thus obtained is always somewhat larger than the true transition temperature of the material; for instance, in the case of nickel, the empirical value of Tc obtained in this manner turns out to be about 650 K while the actual transition takes place at about 631 K. In passing, we add that, as T →0, the low-field susceptibility vanishes, in accordance with the formula χ0(T) ≈4Nµ2 kT exp(−2Tc/T). (23) Finally, we examine the relationship between L and B at T = Tc. Using, once again, equation (10) and employing the approximation tanh−1 x ≃x + x3/3, we get L ≈(3µB/kTc)1/3 (T = Tc,B →0). (24) At this point we wish to emphasize the remarkable similarity that exists between the critical behavior of a gas–liquid system obeying van der Waals equation of state and that of a magnetic system treated in the Bragg–Williams approximation. Even though the two systems are physically very different, the level of approximation is such that the exponents governing power-law behavior of the various physical quantities in the critical region turn out to be the same; compare, for instance, equation (14) with (12.2.11), equations (22a) and (22b) with (12.2.13) and (12.2.14), equation (24) with (12.2.8) — along with the behavior of the specific heat as well. This sort of similarity will be seen again and again whenever we employ an approach similar in spirit to the mean field approach of this section. Before we close our discussion of the so-called zeroth approximation, we would like to demonstrate that it corresponds exactly to the random mixing approximation (which was employed originally in the theory of binary solutions). According to equation (12.3.19), the mean configurational energy in the absence of the external field is given by U0 = −J 1 2qN −2qN+ + 4N++  . (25) At the same time, equations (2) and (16) of the present approach give N+ = 1 2N 1 + L0  and U0 = −1 2qJNL2 0. (26) Combining (25) and (26), we obtain N++ = 1 8qN 1 + L0 2 , (27) 12.6 Ising model in the first approximation 427 so that N++ 1 2qN = N+ N !2 . (28) Thus, the probability of having an “up–up” nearest-neighbor pair of spins in the lat-tice is precisely equal to the square of the probability of having an “up” spin; in other words, there does not exist, in spite of the presence of a nearest-neighbor interaction (characterized by the coupling constant J), any specific correlation between the neigh-boring spins of the lattice. Put differently, there does not exist any short-range order in the system, apart from what follows statistically from the long-range order (charac-terized by the parameter L). It follows that, in the present approximation, our system consists of a specific number of “up” spins, namely N(1 + L)/2, and a corresponding number of “down” spins, namely N(1 −L)/2, distributed over the N lattice sites in a completely random manner — similar to the mixing of N(1 + L)/2 atoms of one kind with N(1 −L)/2 atoms of another kind in a completely random manner to obtain a binary solution of N atoms; see also Problem 12.4. For this sort of mixing, we obviously have N++ = 1 2qN 1 + L 2 !2 , N−−= 1 2qN 1 −L 2 !2 , (29a) N+−= 2 · 1 2qN 1 + L 2 ! 1 −L 2 ! , (29b) with the result that N++N−− (N+−)2 = 1 4. (30) 12.6 Ising model in the first approximation The approaches considered in the preceding section have a natural generalization toward an improved approximation. The mean field approach leads naturally to the Bethe approximation (Bethe, 1935; Rushbrooke, 1938), which treats the interaction of a given spin with its nearest neighbors somewhat more accurately. The random mixing approach, on the other hand, leads to the quasichemical approximation (Guggenheim, 1935; Fowler and Guggenheim, 1940), which takes into account the specific short-range order of the lat-tice — over and above the one that follows statistically from the long-range order. As shown by Guggenheim (1938) and by Chang (1939), the two methods yield identical results for the Ising model. It seems worthwhile to mention here that the extension of these approxima-tions to higher orders, or their application to the Heisenberg model, does not produce identical results. 428 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling In the Bethe approximation, a given spin σ0 is regarded as the central member of a group, which consists of this spin and its q nearest neighbors, and in writing down the Hamiltonian of this group the interaction between the central spin and its q neighbors is taken into account exactly while the interaction of these neighbors with other spins in the lattice is taken into account through a mean molecular field B′. Thus Hq+1 = −µBσ0 −µ B + B′ q X j=1 σj −J q X j=1 σ0σj, (1) B being the external magnetic field acting on the lattice. The internal field B′ is determined by the condition of self-consistency, which requires that the mean value, σ 0, of the central spin be the same as the mean value, σ j, of any of the q neighbors. The partition function Z of this group of spins as a whole is given by Z = X σ0,σj=±1 exp  1 kT   µBσ0 + µ B + B′ q X j=1 σj + J q X j=1 σ0σj      = X σ0,σj=±1 exp  ασ0 + α + α′ q X j=1 σj + γ q X j=1 σ0σj  , (2) where α = µB kT , α′ = µB′ kT and γ = J kT . (3) Now, the right side of (2) can be written as a sum of two terms, one pertaining to σ0 = +1 and the other to σ0 = −1, that is, Z = Z+ + Z−, where Z± = X σj=±1 exp  ±α + α + α′ ± γ  q X j=1 σj   = e±α 2cosh α + α′ ± γ q . (4) The mean value of the central spin is then given by σ 0 = Z+ −Z− Z , (5) 12.6 Ising model in the first approximation 429 while the mean value of any one of its q neighbors is given by, see (2) and (4), σ j = 1 q   q X j=1 σj  = 1 q  1 Z ∂Z ∂α′  = 1 Z  Z+ tanh α + α′ + γ  + Z−tanh α + α′ −γ  . (6) Equating (5) and (6), we get Z+  1 −tanh α + α′ + γ  = Z−  1 + tanh α + α′ −γ  . (7) Substituting for Z+ and Z−from (4), we finally obtain e2α′ = ( cosh α + α′ + γ  cosh(α + α′ −γ ) )q−1 . (8) Equation (8) determines α′ which, in turn, determines the magnetic behavior of the lattice. To study the possibility of spontaneous magnetization, we set α(= µB/kT) = 0. Equa-tion (8) then reduces to α′ = q −1 2 ln ( cosh α′ + γ  cosh(α′ −γ ) ) . (9) In the absence of interactions (γ = 0), α′ is clearly zero. In the presence of interactions (γ ̸= 0), α′ may still be zero unless γ exceeds a certain critical value, γc say. To determine this value, we expand the right side of (9) as a Taylor series around α′ = 0, with the result α′ = (q −1)tanhγ ( α′ −sech2γ α′3 3 + ··· ) . (10) We note that, for all γ , α′ = 0 is one possible solution of the problem; this, however, does not interest us. A nonzero solution requires that (q −1)tanhγ > 1, that is, γ > γc = tanh−1  1 q −1  = 1 2 ln  q q −2  . (11) In terms of temperature, this means that T < Tc = 2J k ln  q q −2  , (12) 430 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling which determines the Curie temperature of the lattice. From (10), we also infer that for temperatures less than, but close to, the Curie temperature α′ ≃ n 3cosh2 γc (q −1)tanhγ −1 o1/2 ≃  3(q −1)(γ −γc) 1/2 ≃  3(q −1) J kTc  1 −T Tc 1/2 . (13) The parameter L, which is a measure of the long-range order in the system, is, by definition, equal to σ. From equations (5) and (7), we get L = (Z+/Z−) −1 (Z+/Z−) + 1 = sinh(2α + 2α′) cosh(2α + 2α′) + exp(−2γ ). (14) In the limit B →0 (which means α →0) and at temperatures less than, but close to, the Curie temperature (γ ≳γc;α′ ≃0), we obtain L0 = sinh(2α′) cosh(2α′) + exp(−2γ ) ≃ 2α′ 1 + (q −2)/q = q q −1α′. (15) Substituting from (12) and (13), we get L0 ≃  3 q q −1 q 2 ln  q q −2  1 −T Tc 1/2 . (16) We note that, for q ≫1, equations (12) and (16) reduce to their zeroth-order counter parts (12.5.13) and (12.5.14), respectively; in either case, as T →Tc from below, L0 vanishes as (Tc −T)1/2. We also note that the spontaneous magnetization curve in the present approxi-mation has the same general shape as in the zeroth approximation; see Figure 12.7. Of course, in the present case the curve depends explicitly on the coordination number q, being steepest for small q and becoming less steep as q increases — tending ultimately to the limiting form given by the zeroth approximation. We shall now study correlations that might exist among neighboring spins in the lattice. For this, we evaluate the numbers N++,N−−, and N+−in terms of the parameters α, α′, and γ , and compare the resulting expressions with the ones obtained under the mean field approximation. Carrying out summations in (2) over all the spins (of the group) except σ0 and σ1, we obtain Z = X σ0,σ1=±1 h exp  ασ0 + (α + α′)σ1 + γ σ0σ1  2cosh(α + α′ + γ σ0) q−1i . (17) Writing this as a sum of three parts pertaining, respectively, to the cases (i) σ0 = σ1 = +1, (ii) σ0 = σ1 = −1, and (iii) σ0 = −σ1 = ±1, we have Z = Z++ + Z−−+ Z+−, (18) 12.6 Ising model in the first approximation 431 where, naturally enough, N++ : N−−: N+−:: Z++ : Z−−: Z+−. (19) We thus obtain, using (8) as well, N++ ∝e(2α+α′+γ )  2cosh(α + α′ + γ ) q−1 , N−−∝e(−2α−α′+γ )  2cosh(α + α′ −γ ) q−1 = e(−2α−3α′+γ ){2cosh(α + α′ + γ )}q−1, and N+−∝e(−α′−γ )  2cosh(α + α′ + γ ) q−1 + e(α′−γ )  2cosh(α + α′ −γ ) q−1 = 2e(−α′−γ ){2cosh(α + α′ + γ )}q−1. Normalizing these expressions with the help of the relationship N++ + N−−+ N+−= 1 2qN, (20) we obtain the desired results N++,N−−,N+−  = 1 2qN e2α+2α′+γ ,e−2α−2α′+γ ,2e−γ  2  eγ cosh(2α + 2α′) + e−γ , (21) whereby N++N−− N+− 2 = 1 4e4γ = 1 4e4J/kT. (22) The last result differs significantly from the one that followed from the random mixing approximation, namely (12.5.30). The difference lies in the extra factor exp(4J/kT) which, for J > 0, favors the formation of parallel-spin pairs ↑↑and ↓↓, as opposed to antiparallel-spin pairs ↑↓and ↓↑. In fact, one may regard the elementary process ↑↑+ ↓↓⇔2 ↑↓, (23) which leaves the total numbers of “up” spins and “down” spins unaltered, as a kind of a “chemical reaction” which, proceeding from left to right, is endothermic (requiring an amount of energy 4J to get through) and, proceeding from right to left, is exothermic (releasing an amount of energy 4J). Equation (22) then constitutes the law of mass action for this reaction, the expression on the right side being the equilibrium constant of the reaction. Historically, equation (22) was adopted by Guggenheim as the starting point of his “quasichemical” treatment of the Ising model; only later on did he show that his treatment was equivalent to the Bethe approximation expounded here. 432 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling Equation (22) tells us that, for J > 0, there exists among like neighbors (↑and ↑or ↓ and ↓) a positive correlation and among unlike neighbors (↑and ↓) a negative correlation, and that these correlations are a direct consequence of the nearest-neighbor interaction. Accordingly, there must exist a specific short-range order in the system, over and above the one that follows statistically from the long-range order. To see this explicitly, we note that even when long-range order disappears (α + α′ = 0), some short-range order still persists. For instance, from equation (21) we obtain N++,N−−,N+−  L=0 = 1 2qN eγ ,eγ ,2e−γ  4coshγ (24) which, only in the limit γ →0, goes over to the random-mixing result, see equa-tion (12.5.29) with L = 0, N++,N−−,N+−  L=0 = 1 2qN (1,1,2) 4 . (25) In the zeroth approximation, equation (25) is supposed to hold at all temperatures above Tc; we now find that a better approximation at these temperatures is provided by (24). Next, we evaluate the configurational energy U0 and the specific heat C0 of the lattice in the absence of the external field (α = 0). In view of equation (12.5.25), U0 = −J 1 2qN −2qN+ + 4N++  α=0 . (26) The expression for N++ is given by equation (21) while that for N+ can be obtained from (14): (N+)α=0 = 1 2N(1 + L0) = 1 2N exp(2α′) + exp(−2γ ) cosh(2α′) + exp(−2γ ). (27) Equation (26) then gives U0 = −1 2qJN cosh(2α′) −exp(−2γ ) cosh(2α′) + exp(−2γ ), (28) where α′ is determined by equation (9). For T > Tc, α′ = 0, so U0 = −1 2qJN 1 −exp(−2γ ) 1 + exp(−2γ ) = −1 2qJN tanhγ . (29) Obviously, this result arises solely from the short-range order that persists in the system even above Tc. As for the specific heat, we get C0/Nk = 1 2qγ 2sech2γ (T > Tc). (30) As T →∞, C0 vanishes like T−2. We note that a nonzero specific heat above the transition temperature is a welcome feature of the present approximation, for it brings our model 12.6 Ising model in the first approximation 433 0 0 1 1 2 2 2 2.88 4 6 (kT/J ) C0(T )/Nk FIGURE 12.9 The field-free specific heat of an Ising lattice with coordination number 4. Curve 1 obtains in the Bethe approximation, curve 2 in the Bragg–Williams approximation. somewhat closer to real physical systems. In this connection, we recall that in the previous approximation the specific heat was zero for all T > Tc. Figure 12.9 shows the specific heat of an Ising lattice, with coordination number 4, as given by the Bethe approximation; for comparison, the result of the previous approximation is also included. We are now in a position to study the specific heat discontinuity at T = Tc. The limiting value of C0, as T approaches Tc from above, can be obtained from equation (30) by letting γ →γc. One obtains, with the help of equation (11), 1 Nk C0 (Tc+) = 1 2qγ 2 c sech2γc = 1 8 q2(q −2) (q −1)2  ln  q q −2 2 . (31) To obtain the corresponding result as T approaches Tc from below, we must use the general expression (28) for U0, with α′ →0 as γ →γc. Expanding (28) in powers of the quantities (γ −γc) and α′, and making use of equation (13), we obtain for (1 −T/Tc) ≪1 U0 = −1 2qJN  1 (q −1) + q(q −2) (q −1)2 (γ −γc) + q(q −2) (q −1)2 α′2 + ···  = −1 2qJN  1 (q −1) + q(q −2)(3q −2) (q −1)2 J kTc  1 −T Tc  + ···  . (32) Differentiating with respect to T and substituting for Tc, we obtain 1 Nk C0(Tc−) = 1 8 q2(q −2)(3q −2) (q −1)2  ln  q q −2 2 , (33) which is (3q −2) times larger than the corresponding result for T = Tc+; compare with equation (31). The specific-heat discontinuity at the transition point is, therefore, 434 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling given by 1 Nk 1C0 = 3 8 q2(q −2) (q −1)  ln  q q −2 2 . (34) One may check that, for q ≫1, the foregoing results go over to the ones following from the zeroth approximation. Finally, we examine the relationship between L and B at T = Tc. Using equations (8) and (14), with both α and α′ ≪1 while γ = γc, we get L ≈ n 3q2µB/(q −1)(q −2)kTc o1/3 (T = Tc,B →0); (35) compare with equation (12.5.24). For the behavior of χ0, see Problem 12.16. In passing, we note that, according to equation (12), the transition temperature for a lattice with q = 2 is zero, which essentially means that a one-dimensional Ising chain does not undergo a phase transition. This result is in complete agreement with the one follow-ing from an exact treatment of the one-dimensional lattice; see Section 13.2. In fact, for a lattice with q = 2, any results following from the Bethe approximation are completely identical with the corresponding exact results (see Problem 13.3); on the other hand, the Bragg–Williams approximation is least reliable when q = 2. That Tc for q = 2 is zero (rather than 2J/k) is in line with the fact that, for all q, the first approximation yields a transition temperature closer to the correct value of Tc than does the zeroth approximation. The same is true of the amplitudes that determine the quan-titative behavior of the various physical quantities near T = Tc, though the exponents in the various power laws governing this behavior remain the same; compare, for instance, equation (16) with (12.5.14), equation (35) with (24) as well as the behavior of the spe-cific heat. In fact, one finds that successive approximations of the mean field approach, while continuing to improve the theoretical value of Tc and the quantitative behavior of the various physical quantities (as given by the amplitudes), do not modify their quali-tative behavior (as determined by the exponents). For an account of the higher-order approximations, see Domb (1960). One important virtue of the Bethe approximation is that it brings out the role of the dimensionality of the lattice in bringing about a phase transition in the system. The fact that Tc = 0 for q = 2 and thereon it increases steadily with q leads one to infer that, while a linear Ising chain does not undergo phase transition at any finite T, higher dimensionality does promote the phenomenon. One may, in fact, argue that the absence of a phase tran-sition in a one-dimensional chain is essentially due to the fact that, the interactions being severely short-ranged, “communication” between any two parts of the chain can be com-pletely disrupted by a single defect in-between. The situation remains virtually unaltered even if the range of interactions is allowed to increase — so long as it remains finite. Only when interactions become truly long-ranged, with Jij ∼|i −j|−(1+σ)(σ > 0), does a phase transition at a finite T become possible — but only if σ < 1; for σ > 1, we are back to the 12.7 The critical exponents 435 case of no phase transition, while the borderline case σ = 1 remains in doubt. For more details, see Griffiths (1972, pp. 89–94). Peierls (1936) was the first to demonstrate that at sufficiently low temperatures the Ising model in two or three dimensions must exhibit a phase transition. He considered the lat-tice as made up of two kinds of domains, one consisting of “up” spins and the other of “down” spins, separated by a set of boundaries between the neighboring domains, and argued on energy considerations that in a two- or three-dimensional lattice the long-range order that exists at 0K would persist at finite temperatures. Again, for details, see Griffiths (1972, pp. 59–66). 12.7 The critical exponents A basic problem in the theory of phase transitions is to study the behavior of a given system in the neighborhood of its critical point. We know that this behavior is marked by the fact that the various physical quantities pertaining to the system possess singularities at the critical point. It is customary to express these singularities in terms of power laws char-acterized by a set of critical exponents that determine the qualitative nature of the critical behavior of the given system. To begin with, we identify an order parameter m, and the cor-responding ordering field h, such that, in the limit h →0, m tends to a limiting value m0, with the property that m0 = 0 for T ≥Tc and ̸= 0 for T < Tc. For a magnetic system, the nat-ural candidate for m is the parameter L(= σ) of Sections 12.5 and 12.6, while h is identified with the quantity µB/kTc; for a gas–liquid system, one may adopt the density differential (ρl −ρc) or |ρg −ρc| for m and the pressure differential (P −Pc) for h. The various critical exponents are then defined as follows. The manner in which m0 →0, as T →Tc from below, defines the exponent β: m0 ∼(Tc −T)β (h →0,T ≲Tc). (1) The manner in which the low-field susceptibility χ0 diverges, as T →Tc from above (or from below), defines the exponent γ (or γ ′): χ0 ∼ ∂m ∂h  T,h→0 ∼ ( (T −Tc)−γ (h →0,T ≳Tc) (2a) (Tc −T)−γ ′ (h →0,T ≲Tc); (2b) in the gas–liquid transition, the role of χ0 is played by the isothermal compressibility, κT = ρ−1(∂ρ/∂P)T, of the system. Next, we define an exponent δ by the relation m|T=Tc ∼h1/δ (T = Tc,h →0); (3) in the case of a gas–liquid system, δ is a measure of the “degree of flatness” of the critical isotherm at the critical point, for then |P −Pc| T=Tc ∼|ρ −ρc|δ (T = Tc,P →Pc). (4) 436 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling Finally, we define exponents α and α′ on the basis of the specific heat, CV , of the gas–liquid system: CV ∼ ( (T −Tc)−α (T ≳Tc) (5a) (Tc −T)−α′ (T ≲Tc). (5b) In connection with the foregoing relations, especially equations (5), we wish to empha-size that in certain cases the exponent in question is rather small in value; it is then more appropriate to write f (t) ∼|t|−λ −1 λ (|t| ≪1). (6) Now, if λ > 0, the function f (t) would have a power-law divergence at t = 0; in case λ →0, the function f (t) would have a logarithmic divergence instead: f (t) ∼ln(1/|t|) (|t| ≪1). (7) In either case, the derivative f ′(t) ∼|t|−(1+λ). A survey of the results derived in Sections 12.2 through 12.6 shows that for a gas–liquid system obeying van der Waals equation of state or for a magnetic system treated in the mean field approximation (it does not matter what order of approximation one is talking about), the various critical exponents are the same: β = 1 2, γ = γ ′ = 1, δ = 3, α = α′ = 0. (8) In Table 12.1 we have compiled experimental data on critical exponents pertaining to a variety of systems including the ones mentioned above; for completeness, we have included here data on another two exponents, ν and η, which will be defined in Section 12.12. We find that, while the observed values of an exponent, say β, differ very little as one goes from system to system within a given category (or even from category to category), these values are considerably different from the ones following from the mean field approximation. Clearly, we need a theory of phase transitions that is basically different from the mean field theory. To begin with, some questions arise: (i) Are these exponents completely independent of one another or are they mutually related? In the latter case, how many of them are truly independent? (ii) On what characteristics of the given system do they depend? This includes the question why, for systems differing so much from one another, they differ so little. (iii) How can they be evaluated from first principles? The answer to question (i) is simple: yes, the various exponents do obey certain rela-tions and hence are not completely independent. These relations appear in the form of inequalities, dictated by the principles of thermodynamics, which will be explored 12.7 The critical exponents 437 Table 12.1 Experimental Data on Critical Exponents Critical Magnetic Gas–liquid Binary Fluid Binary Ferroelectric Superfluid Mean Field Exponents Systems(a) Systems(b) Mixtures(c) Alloys(d) Systems(e) He4(f) Results α,α′ 0.0–0.2 0.1–0.2 0.05–0.15 −−− −−− −0.026 0 β 0.30–0.36 0.32–0.35 0.30–0.34 0.305 ± 0.005 0.33–0.34 −−− 1/2 γ 1.2–1.4 1.2–1.3 1.2–1.4 1.24 ± 0.015 1.0 ± 0.2 inaccessible 1 γ ′ 1.0–1.2 1.1–1.2 −−− 1.23 ± 0.025 1.23 ± 0.02 inaccessible 1 δ 4.2–4.8 4.6–5.0 4.0–5.0 −−− −−− inaccessible 3 ν 0.62–0.68 −−− −−− 0.65 ± 0.02 0.5–0.8 0.675 1/2 η 0.03–0.15 −−− −−− 0.03–0.06 −−− −−− 0 (a)Stierstadt et al. (1990). (b)Voronel (1976); Rowlinson and Swinton (1982). (c)Rowlinson and Swinton (1982). (d)Als-Nielsen (1976); data pertain to beta-brass only. (e)Kadanoff et al. (1967); Lines and Glass (1977). (f)Ahlers (1980). in Section 12.8; in the modern theory of phase transitions, see Sections 12.10 through 12.12 and Chapter 14, the same relations turn up as equalities, and the number of these (restrictive) relations is such that, in most cases only two of the exponents are truly independent. As regards question (ii), it turns out that our exponents depend on a very small number of characteristics, or parameters, of the problem, which explains why they differ so little from one system to another in a given category of systems (and also from one category to another, even though systems in those categories are so different from one another). The characteristics that seem to matter are (a) the dimensionality, d, of the space in which the system is embedded, (b) the number of components, n, of the order parameter of the problem, and (c) the range of microscopic interactions in the system. Insofar as interactions are concerned, all that matters is whether they are short-ranged (which includes the special case of nearest-neighbor interactions) or long-ranged. In the former case, the values of the critical exponents resulting from nearest-neighbor interac-tions remain unaltered — regardless of whether further-neighbor interactions are included or not; in the latter case, assuming Jij ∼|i −j|−(d+σ) with σ > 0, the critical exponents depend on σ. Unless a statement is made to the contrary, the microscopic interactions operating in the given system will be assumed to be short-ranged; the critical exponents then depend only on d and n — both of which, for instructional purposes, may be treated as continuous variables. Insofar as d is concerned, we recall the Bethe approximation that highlighted the special role played by the dimensionality of the lattice through, and only through, the coor-dination number q. We also recall that, while the theoretical value of Tc and the various amplitudes of the problem were influenced by q, the critical exponents were not. In more accurate theories we find that the critical exponents depend more directly on d and only 438 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling indirectly on q; however, for a given d, they do not depend on the structural details of the lattice (including the number q). Insofar as n is concerned, the major difference lies between the Ising model (n = 1) with discrete symmetry (σi = +1 or −1) and other models (n ≥2) with continuous symme-try (−1 ≤σiα ≤+1 for α = 1,...,n, with |σi| = 1). In the former case, Tc is zero for d ≤1 and nonzero for d > 1; in the latter, Tc is zero for d ≤2 and nonzero for d > 2.13 In either case, the critical exponents depend on both d and n, except that for d > 4 they become indepen-dent of d and n, and assume values identical to the ones given by the mean field theory; the physical reason behind this overwhelming generality is examined in Section 12.13. In passing, we note that, for given d and n, the critical exponents do not depend on whether the spins constituting the system are treated classically or quantum-mechanically. As regards question (iii), the obvious procedure for evaluating the critical exponents is to carry out exact (or almost exact) analysis of the various models — a task to which the whole of Chapter 13 is devoted. An alternative approach is provided by the renormal-ization group theory, which is discussed in Chapter 14. A modest attempt to evaluate the critical exponents is made in Section 12.9, which yields results that are inconsistent with the experiment but teaches us quite a few lessons about the shortcomings of the so-called classical approaches. 12.8 Thermodynamic inequalities The first rigorous relation linking critical exponents was derived by Rushbrooke (1963) who, on thermodynamic grounds, showed that for any physical system undergoing a phase transition α′ + 2β + γ ′ ≥2. (1) The proof of inequality (1) is straightforward if one adopts a magnetic system as an exam-ple. We start with the thermodynamic formula for the difference between the specific heat at constant field CH and the specific heat at constant magnetization CM (see Problem 3.40) CH −CM = −T ∂H ∂T  M ∂M ∂T  H = Tχ−1 ∂M ∂T  H 2 . (2) Since CM ≥0, it follows that CH ≥Tχ−1 ∂M ∂T  H 2 . (3) Now, letting H →0 and T →Tc from below, we get D1(Tc −T)−α′ ≥D2Tc(Tc −T)γ ′+2(β−1), (4) 13The special case n = 2 with d = 2 is qualitatively different from others; for details, see Section 13.7. 12.8 Thermodynamic inequalities 439 where D1 and D2 are positive constants; here, use has been made of power laws (12.7.1, 2b, and 5b).14 Inequality (4) may as well be written as (Tc −T)2−(α′+2β+γ ′) ≥D2Tc/D1. (5) Since (Tc −T) can be made as small as we like, (5) will not hold if (α′ + 2β + γ ′) < 2. The Rushbrooke inequality (1) is thus established. To establish further inequalities, one utilizes the convexity properties of the Helmholtz free energy A(T,M). Since dA = −SdT + HdM,  ∂A ∂T  M = −S, ∂2A ∂T2 ! M = −  ∂S ∂T  M = −CM T ≤0 (6a, b) and  ∂A ∂M  T = H, ∂2A ∂M2 ! T =  ∂H ∂M  T = 1 χ ≥0. (7a, b) It follows that A(T,M) is concave in T and convex in M. We now proceed to establish the Griffiths inequality (1965a, b) α′ + β(δ + 1) ≥2. (8) Consider a magnetic system in zero field and at a temperature T1 < Tc. Then, by (7a), A(T,M) is a function of T only, so we can write A(T1,M) = A(T1,0) (−M1 ≤M ≤M1), (9) where M1 is the spontaneous magnetization at temperature T1; see Figure 12.10. Applying (6a) to (9), we get S(T1,M) = S(T1,0) (−M1 ≤M ≤M1). (10) We now define two new functions A∗(T,M) = {A(T,M) −Ac} + (T −Tc)Sc (11) and S∗(T,M) = S(T,M) −Sc, (12) 14Recalling the correspondence between a gas–liquid system and a magnet, one might wonder why we have employed CH, rather than CM, in place of CV . The reason is that, since we are letting H →0 and T →Tc−,M →0 as well. So, as argued by Fisher (1967), in the limit considered here, CH and CM display the same singular behavior. In fact, it can be shown that if the ratio CM/CH →1 as T →Tc, then (α′ + 2β + γ ′) must be greater than 2; on the other hand, if this ratio tends to a value less than 1, then (α′ + 2β + γ ′) = 2. For details, see Stanley (1971), Section 4.1. 440 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling M 0 T (T1, M1) (T1, 2M1) (H ,0) (H .0 ) (H 02) (H 01) FIGURE 12.10 Magnetization, M(T,H), of a magnetic system for H > 0, H < 0, and H →0. Here, M1 denotes the spontaneous magnetization of the system at a temperature T1 < Tc. where Ac = A(Tc,0) and Sc = S(Tc,0). It follows that ∂A∗ ∂T  M = −S∗, ∂2A∗ ∂T2 ! M = − ∂S∗ ∂T  M = −CM T ≤0. (13a, b) Thus, A∗is also concave in T. Geometrically, this means that, for any choice of T1, the curve A∗(T), with M fixed at M1, lies below the tangent line at T = T1, that is, A∗(T,M1) ≤A∗(T1,M1) + ∂A∗ ∂T  M1,T=T1 (T −T1); (14) see Figure 12.11. Letting T = Tc in (14), we get A∗(Tc,M1) ≤A∗(T1,M1) −S∗(T1,M1)(Tc −T1) (15) which, in view of equations (9) through (12), may be written as A∗(Tc,M1) ≤A∗(T1,0) −S∗(T1,0)(Tc −T1). (16) Utilizing, once again, the concavity of the function A∗(T) but this time at T = Tc (with M fixed at zero and the slope (∂A∗/∂T) vanishing), we get, see (14), A∗(T,0) ≤A∗(Tc,0). (17) Now, letting T = T1 in (17) and noting that A∗(Tc,0) = 0 by definition, we get A∗(T1,0) ≤0. (18) 12.8 Thermodynamic inequalities 441 Tc T1 T M M1 FIGURE 12.11 The function A∗(T,M) of a magnetic system, with magnetization M fixed at M1. The slope of this curve is S(Tc,0) −S(T,M1), which is positive for all T ≤Tc. Combining (16) and (18), we finally get A∗(Tc,M1) ≤−(Tc −T1)S∗(T1,0), (19) valid for all T1 < Tc. The next step is straightforward. We let T1 →Tc−, so that M1 →0 and along with it A∗(Tc,M1) =    M1 Z 0 HdM    T=Tc ≈DMδ+1 1 ≈D′(Tc −T1)β(δ+1), (20) while S∗(T1,0) = T1 Z Tc C(T,0) T dT ≈−D′′ Tc (Tc −T1)1−α′, (21) where D, D′, and D′′ are positive constants; here, use has been made of power laws (12.7.1, 3, and 5b). Substituting (20) and (21) into (19), we get (Tc −T1)2−α′−β(δ+1) ≥D′Tc/D′′. (22) Again, since (Tc −T1) can be made as small as we like, (22) will not hold if α′ + β(δ + 1) < 2. The Griffiths inequality (8) is thus established. It will be noted that unlike the Rushbrooke inequality, which related critical exponents pertaining only to T < Tc, the present inequal-ity relates two such exponents, α′ and β, with one, namely δ, that pertains to the critical isotherm (T = Tc). While inequalities (1) and (8) are thermodynamically exact, Griffiths has derived several others that require certain plausible assumptions on the system in question. We quote two 442 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling of them here, without proof: γ ′ ≥β(δ −1) (23) γ ≥(2 −α)(δ −1)/(δ + 1). (24) For a complete list of such inequalities, see Griffiths (1972), p. 102, where references to original papers are also given. Before proceeding further, the reader may like to verify that the experimental data on critical exponents, as given earlier in Table 12.1, do indeed conform to the inequalities proved or quoted in this section. It is important in this connection to note that the mean field exponents (α = α′ = 0, β = 1/2, γ = γ ′ = 1, and δ = 3) satisfy all these relations as equalities. 12.9 Landau’s phenomenological theory As early as 1937 Landau attempted a unified description of all second-order phase transi-tions — second-order in the sense that the second derivatives of the free energy, namely the specific heat and the magnetic susceptibility (or isothermal compressibility, in the case of fluids), show a divergence while the first derivatives, namely the entropy and the magnetization (or density, in the case of fluids), are continuous at the critical point. He emphasized the importance of an order parameter m0 (which would be zero on the high-temperature side of the transition and nonzero on the low-temperature side) and suggested that the basic features of the critical behavior of a given system may be deter-mined by expanding its free energy in powers of m0 (for we know that, in the close vicinity of the critical point, m0 ≪1). He also argued that in the absence of the ordering field (h = 0) the up–down symmetry of the system would require that the proposed expansion contain only even powers of m0. Thus, the zero-field free energy ψ0(= A0/NkT) of the system may be written as ψ0(t,m0) = q(t) + r(t)m2 0 + s(t)m4 0 + ···  t = T −Tc Tc ,|t| ≪1  ; (1) at the same time, the coefficients q(t), r(t), s(t)... may be written as q(t) = X k≥0 qktk, r(t) = X k≥0 rktk, s(t) = X k≥0 sktk,.... (2) The equilibrium value of the order parameter is then determined by minimizing ψ0 with respect to m0; retaining terms only up to the order displayed in (1), which for thermodynamic stability requires that s(t) > 0, we obtain r(t)m0 + 2s(t)m3 0 = 0. (3) 12.9 Landau’s phenomenological theory 443 The equilibrium value of m0 is thus either 0 or ± √[−r(t)/2s(t)]. The first solution is of lesser interest, though this is the only one we will have for t > 0; it is the other solutions that lead to the possibility of spontaneous magnetization in the system. To obtain physically sensible results, see equations (9) through (11), we must have in equation (2): r0 = 0, r1 > 0, and s0 > 0, with the result |m0| ≈[(r1/2s0)|t|]1/2 (t ≲0), (4) giving β = 1/2. The asymptotic expression for the free energy, namely ψ0(t,m0) ≈q0 + r1tm2 0 + s0m4 0 (r1,s0 > 0), (5) is plotted in Figure 12.12. We see that, for t ≥0, there is only one minimum, which is located at m0 = 0; for t = 0, the minimum is rather flat. For t < 0, on the other hand, we have two minima, located at m0 = ±ms, as given by expression (4), with a maximum at m0 = 0. Now, since ψ0 has to be convex in m0, so that the susceptibility of the system be nonnegative, see equation (12.8.7b), we must replace the nonconvex portion of the curve, which lies between the points m0 = −ms and m0 = +ms, by a straight line (along which the suscep-tibility would be infinite). This replacement is reminiscent of the Maxwell construction employed in Sections 12.1 and 12.2. We now subject the system to an ordering field h, assumed positive. If the field is weak, the only change in the expression for the free energy would be the addition of a term −hm. Disregarding the appearance of any higher powers of (hm) as well as any modifications of 0(t, m0) m0 1ms 2ms t . 0 t , 0 t 50 FIGURE 12.12 The free energy ψ0(t,m0) of the Landau theory, shown as a function of m0, for three different values of t. The dashed curve depicts spontaneous magnetization ms(t), while the horizontal line for t < 0 provides the Maxwell construction. 444 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling the coefficients already present, we may now write ψh(t,m) = −hm + q(t) + r(t)m2 + s(t)m4. (6) The equilibrium value of m is now given by15 −h + 2r(t)m + 4s(t)m3 = 0. (7) The low-field susceptibility of the system, in units of Nµ2/kT, thus turns out to be χ =  ∂h ∂m −1 t = 1 2r(t) + 12s(t)m2 , (8) valid in the limit h →0. Now, for t > 0, m →0 and we get χ ≈1/2r1t (t ≳0), (9) giving γ = 1. On the other hand, for t < 0, m →√[(r1/2s0)|t|], see (4); we then get χ ≈1/4r1|t| (t ≲0), (10) giving γ ′ = 1. Finally, if we set t = 0 in (7), we obtain the following relation between h and m: h ≈4s0m3 (h →0), (11) giving δ = 3. We shall now look at the specific heats Ch and Cm. If t > 0, then h →0 implies m →0, so in this limit there is no difference between Ch and Cm. Equation (1) then gives, in units of Nk, Ch = Cm = − ∂2ψ0 ∂t2 ! m→0 = −(2q2 + 6q3t + ···) (t ≳0). (12) For t < 0, we have Cm = − h (2q2 + 6q3t + ···) + (2r2 + ···)m2 s + ··· i = − 2q2 + {6q3 −(r1r2/s0)}t + ... (t ≲0). (13) Next, using equation (12.8.2) along with (4) and (10), we have Ch −Cm =  ∂h ∂m  t ∂m ∂t  h 2 ≈r2 1 2s0 (t ≲0). (14) 15It may be mentioned here that the passage from equation (1) to (6) is equivalent to effecting a Legendre transfor-mation from the Helmholtz free energy A to the Gibbs free energy G (= A −HM), and equation (7) is analogous to the relation (∂A/∂M)T = H. 12.9 Landau’s phenomenological theory 445 It follows that, while Cm possesses a cusp-like singularity at t = 0, Ch undergoes a jump discontinuity of magnitude (Ch)t→0−−(Ch)t→0+ ≈r2 1/2s0. (15) It follows that α = α′ = 0. The most striking feature of the Landau theory is that it gives exactly the same critical exponents as the mean field theory of Sections 12.5 and 12.6 (or the van der Waals theory of Section 12.2). Actually it goes much further, for it starts with an expression for the free energy of the system containing parameters qk,rk,sk,..., which represent the structure of the given system and the interactions operating in it, and goes on to show that, while the amplitudes of the various physical quantities near the critical point do depend on these parameters, the critical exponents do not! This universality (of critical exponents) suggests that we are dealing here with a class of systems which, despite their structural differences, display a critical behavior that is qualitatively the same for all members of the class. This leads to the concept of a universality class which, if Landau were right, would be a rather large one. The fact of the matter is that the concept of universality is very much overstated in Landau’s theory; in reality, there are many different universality classes — each defined by the parameters d and n of Section 12.7 and by the range of the microscopic interac-tions — such that the critical exponents within a class are the same while they vary from one class to another. The way Landau’s theory is set up, the parameter n is essentially equal to 1 (because the order parameter m0 is treated as a scalar), the parameter d plays no role at all (though later on we shall see that the mean field exponents are, in fact, valid for all n if d > 4), while the microscopic interactions are implicitly long-ranged.16 An objection commonly raised against the Landau theory is that, knowing fully well that the thermodynamic functions of the given system are going to be singular at t = 0, a Taylor-type expansion of the free energy around m = 0 is patently a wrong start. While the objection is valid, it is worth noting how a regular function, (1) or (6), leads to an equation of state, (3) or (7), which yields different results for t →0−from the ones for t →0+, the same being true of whether h →0+ or 0−. The trick lies in the fact that we are not using equation (1) or (6) as such for all t; for t < 0, we use instead a modified form, as “corrected” by the Maxwell construction (see Figure 12.12). The spirit of the singularity is thereby cap-tured, though the nature of the singularity, being closely tied with the nature of the original expansion, could not be any different from the mean-field type. The question now arises: how can the Landau theory be improved so that it may provide a more satisfactory picture of the critical phenomena? Pending exact analyses, one wonders if some generalization of the Landau approach, admitting more than one universality class, would provide a bet-ter picture than the one presented so far. It turns out that the scaling approach, initiated by Widom (1965), by Domb and Hunter (1965), and by Patashinskii and Pokrovskii (1966), provided the next step in the right direction. 16In certain systems such as superconductors, the effective interactions (which, for instance, lead to the formation of Cooper pairs of electrons) are, in fact, long-ranged. The critical exponents pertaining to such systems turn out to be the same as one gets from the mean field theory. For details, see Tilley and Tilley (1990). 446 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling 12.10 Scaling hypothesis for thermodynamic functions The scaling approach, which took the subject of phase transitions far beyond the mean field theory, emerged independently from three different sources — from Widom (1965), who searched for a generalization of the van der Waals equation of state that could accommodate nonclassical exponents; from Domb and Hunter (1965), who analyzed the behavior of the series expansions of higher derivatives of the free energy with respect to the magnetic field at the critical point of a magnetic system; and from Patashinskii and Pokrovskii (1966), who studied the behavior of multipoint correlation functions for the spins constituting a system. All three were led to the same form of a thermodynamic equa-tion of state. Subsequently, Kadanoff (1966a) suggested a scaling hypothesis from which not only could this equation of state be derived but one could also obtain a number of rela-tions among the critical exponents, which turned out to be equalities consistent with the findings of Section 12.8. This approach also made it clear why one needed only two inde-pendent numbers to describe the nature of the singularity in question; all other relevant numbers followed as consequences. To set the stage for this development, we go back to the equation of state following from the Landau theory, namely (12.9.7), and write it in the asymptotic form h(t,m) ≈2r1tm + 4s0m3. (1) In view of the relationship (12.9.4), we rewrite (1) in the form h(t,m) ≈r3/2 1 s1/2 0 |t|3/2  2 sgn(t) s1/2 0 r1/2 1 m |t|1/2 ! + 4 s1/2 0 r1/2 1 m |t|1/2 !3 . (2) It follows that m(t,h) ≈r1/2 1 s1/2 0 |t|1/2 × a function of s1/2 0 r3/2 1 h |t|3/2 ! (3) and, within the context of the Landau theory, the function appearing here is universal for all systems conforming to this theory. In the same spirit, the relevant part of the free energy ψh(t,m) — the part that determines the nature of the singularity — may be written in the form ψ(s) h (t,m) ≈−hm + r1tm2 + s0m4 (4) = r2 1 s0 t2  − s1/2 0 r3/2 1 h |t|3/2 ! s1/2 0 r1/2 1 m |t|1/2 ! + sgn(t) s1/2 0 r1/2 1 m |t|1/2 !2 + s1/2 0 r1/2 1 m |t|1/2 !4 . (5) 12.10 Scaling hypothesis for thermodynamic functions 447 Substituting (3) into (5), one gets ψ(s)(t,h) ≈r2 1 s0 t2 × a function of s1/2 0 r3/2 1 h |t|3/2 ! , (6) where, again, the function appearing here is universal. As a check, we see that differentiat-ing (6) with respect to h we readily obtain (3). The most notable feature of the equation of state, as expressed in (3), is that, instead of being the usual relationship among three variables m, h, and t, it is now a relationship among only two variables, namely m/|t|1/2 and h/|t|3/2. Thus, by scaling m with |t|1/2 and h with |t|3/2, we have effectively reduced the total number of variables by one. Similarly, we have replaced equation (4) by (6), which expresses the singular part of the free energy ψ scaled with t2 as a function of the single variable h scaled with |t|3/2. This reduction in the total number of effective variables may be regarded as the first important achievement of the scaling approach. The next step consists of generalizing (6), to write ψ(s)(t,h) ≈F|t|2−αf (Gh/|t|1), (7) where α and 1 are universal numbers common to all systems in the given universality class, f (x) is a universal function which is expected to have two different branches, f+ for t > 0 and f−for t < 0, while F and G (like r1 and s0) are nonuniversal parameters char-acteristic of the particular system under consideration. We expect α and 1 to determine all the critical exponents of the problem, while the amplitudes appearing in the various power laws will be determined by F, G, and the limiting values of the function f (x) and its derivatives (as x tends to zero). Equation (7) constitutes the so-called scaling hypothesis, whose status will become much more respectable when it acquires legitimacy from the renormalization group theory; see Sections 14.1 and 14.3. First of all it should be noted that the exponent of |t|, outside the function f (x) in equation (7), has been chosen to be (2 −α), rather than 2 of the corresponding mean field expression (6), so as to ensure that the specific heat singularity is correctly repro-duced. Secondly, the fact that one must not encounter any singularities as one crosses the critical isotherm (t = 0) at nonzero values of h or m requires that the exponents on the high-temperature side of the critical point be the same as on the low-temperature side, that is, α′ = α and γ ′ = γ . (8) From equation (7) it readily follows that m(t,h) = − ∂ψ(s) ∂h ! t ≈−FG|t|2−α−1f ′(Gh/|t|1) (9) 448 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling and χ(t,h) = − ∂2ψ(s) ∂h2 ! t ≈−FG2|t|2−α−21f ′′(Gh/|t|1). (10) Letting h →0, we obtain for the spontaneous magnetization m(t,0) ≈B|t|β (t ≲0), (11) where B = −FGf ′ −(0), β = 2 −α −1, (12a, b) and for the low-field susceptibility χ(t,0) ≈|t|−γ ( C+ (t ≳0) (13a) C−(t ≲0), (13b) where C± = −FG2f ′′ ±(0), γ = α + 21 −2. (14a, b) Combining (12b) and (14b), we get 1 = β + γ = 2 −α −β, (15) so that α + 2β + γ = 2. (16) To recover δ, we write the function f ′(x) of equation (9) as xβ/1g(x), so that m(t,h) ≈−FG(1+β/1)hβ/1g(Gh/|t|1). (17) Inverting (17), we can write |t| ≈G1/1h1/1 × a function of (FG(1+β/1)hβ/1/m). (18) It follows that, along the critical isotherm (t = 0), the argument of the function appearing in (18) would have a universal value (which makes the function vanish), with the result that m ∼FG(1+β/1)hβ/1 (t = 0). (19) Comparing (19) with (12.7.3), we infer that δ = 1/β. (20) 12.11 The role of correlations and fluctuations 449 Combining (20) with the previous relations, namely (12b) and (15), we get α + β(δ + 1) = 2 (21) and γ = β(δ −1). (22) Finally, combining (21) and (22), we have γ = (2 −α)(δ −1)/(δ + 1). (23) For completeness, we write down for the specific heat Ch at h = 0 C(s) h (t,0) = −∂2ψ(s) ∂t2 h→0 ≈−(2 −α)(1 −α)F|t|−α ( f+(0) (t ≳0) (24a) f−(0) (t ≲0). (24b) We thus see that the scaling hypothesis (7) leads to a number of relations among the critical exponents of the system, emphasizing the fact that only two of them are truly independent. Comparing these relations with the corresponding ones appearing in Section 12.8 — namely, (16), (21), (22), and (23) with (12.8.1), (12.8.8), (12.8.23), and (12.8.24) — we feel satisfied that they are mutually consistent, though the present ones are far more restrictive than the ones there. Besides exponent relations, we also obtain here relations among the various amplitudes of the problem; though individually these amplitudes are nonuniversal, certain combinations thereof turn out to be universal. For instance, the combination (FC±/B2), which consists of coefficients appearing in equa-tions (7), (11), and (13), is universal; see equations (12a) and (14a). The same is true of the ratio C+/C−. For further information on this question, see the original papers by Watson (1969) and a review by Privman, Hohenberg, and Aharony (1991). We now pose the question: why do “universality classes” exist in the first place? In other words, what is the reason that a large variety of systems differing widely in their structures should belong to a single universality class and hence have common critical exponents and common scaling functions? The answer lies in the role played by the corre-lations among the microscopic constituents of the system which, as T →Tc, become large enough to prevail over macroscopic distances in the system and in turn make structural details at the local level irrelevant. We now turn our attention to this important aspect of the problem. 12.11 The role of correlations and fluctuations Much can be learned about criticality by scattering radiation — light, x-rays, neutrons, and so on — off the system of interest; see Section 10.7.A. In a standard scattering experiment, a well-collimated beam of light, or other radiation, with known wavelength λ is directed at the sample and one measures the intensity, I(θ), of the light scattered at an angle θ from 450 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling the “forward” direction of the beam. The radiation undergoes a shift in the wavevector, k, which is related to the parameters θ and λ by |k| = 4π λ sin 1 2θ. (1) Now, the scattered intensity I(θ) is determined by the fluctuations in the medium. If the medium were perfectly uniform (i.e., spatially homogenous), there would be no scattering at all! If one has in mind light scattering from a fluid, then the relevant fluctuations corre-spond to regions of different refractive index and, hence, of different particle density n(r). For neutron scattering from a magnet, fluctuations in the spin or magnetization density are the relevant quantities, and so on. We need to study here the normalized scattering intensity I(θ;T,H)/Iidea1(θ), where I(θ;T,H) is the actual scattering intensity observed at angle θ, which will normally depend on such factors as temperature, magnetic field, and so on, while Iidea1(θ) is the scattering that would take place if the individual parti-cles (or spins) that cause the scattering could somehow be taken far apart so that they no longer interact and hence are quite uncorrelated with one another. Now, this normalized scattering intensity turns out to be essentially proportional to the quantity ˜ g(k) = Z g(r)eik·rdr, (2) which represents the Fourier transform of the appropriate real-space correlation function g(r), which will be defined shortly. As the critical point of the system (say, a fluid) is approached, one observes an enor-mously enhanced level of scattering, especially at low angles which corresponds, via equations (1) and (2), to long wavelength density fluctuations in the fluid. In the criti-cal region, the scattering is so large that it can be visible to the unaided eye, particularly through the phenomenon of critical opalescence. This behavior is, by no means, limited to fluids. Thus if, for example, one scatters neutrons from iron in the vicinity of the Curie point, one likewise sees a dramatic growth in the low-angle neutron scattering intensity, as sketched in Figure 12.13. One sees that for small-angle scattering there is a pronounced peak in I(θ;T) as a function of temperature, and this peak approaches closer and closer to Tc as the angle is decreased. Of course, one could never actually observe zero-angle scat-tering directly, since that would mean picking up the oncoming beam itself, but one can extrapolate to zero angle. When this is done, one finds that the zero-angle scattering I(0;T) actually diverges at Tc. This is the most dramatic manifestation of the phenomenon of criti-cal opalescence and is quite general, in that it is observed whenever appropriate scattering experiments can be performed. Empirically, one may write for small-angle scattering Imax(θ) ∼k−λ1, {Tmax(θ) −Tc} ∼kλ2, (3) so that Imax(θ){Tmax(θ) −Tc}λ1/λ2 = const. (4) 12.11 The role of correlations and fluctuations 451 T 0 Tc I(;T ) Small 0 FIGURE 12.13 Schematic plot of the elastic scattering intensity of neutrons scattered at an angle θ from a magnetic system, such as iron, in the vicinity of the critical point Tc. The small arrows mark the smoothly rounded maxima (at fixed θ) which occur at a temperature Tmax(θ) that approaches Tc as θ →0. Here, λ1 and λ2 are positive exponents (which, as will be seen later, are determined by the universality class to which the system belongs), while k, for a given θ, is determined by equation (1); note that, for small θ, k is essentially proportional to θ. The first real insight into the problem of critical scattering in fluids was provided by Ornstein and Zernike (1914) and Zernike (1916) who emphasized the difference between the direct influence of the microscopic interactions among the atoms of the fluid, which are necessarily short-ranged, and the indirect (but more crucial) influence of the density– density correlations that become long-ranged as the critical temperature is approached; it is the latter that are truly responsible for the propagation of long-range order in the sys-tem and for practically everything else that goes with it. Unfortunately, the original work of Ornstein and Zernike makes difficult reading; moreover, it is based on the classical theory of van der Waals. Nevertheless, the subject has been neatly clarified in the review arti-cles by Fisher (1964, 1983) and Domb (1985), to which the reader may turn for further details. Here, we shall stick to the language of the magnetic systems and work out the most essential parts of the theory in somewhat general terms. We define the spin–spin correlation function g(i,j), for the pair of spins at sites i and j, by the standard definition g(i,j) = σiσj −σ iσ j. (5) For i = j, expression (5) denotes the “mean-square fluctuation in the value of the variable σ at site i”; on the other hand, as the separation between the sites i and j increases indefi-nitely, the spins σi and σj get uncorrelated, so that σiσj →σ iσ j and the function g(i,j) →0. 452 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling In view of the fact that expression (5) can also be written as g(i,j) = (σi −σ i)(σj −σ j), (6) the function g(i,j) may also be looked upon as a measure of the “correlation among the fluctuations in the order parameter of the system at sites i and j.” This makes sense because σi may, quite appropriately, be regarded as the locally fluctuating order parameter linked to site i, just as σ is the order parameter for the whole system. We shall now establish connections between the function g(i,j) and some important thermodynamic properties of the system. We start with the partition function of the system, see equation (12.3.9), QN(H,T) = X {σi} exp " βJ X n.n. σiσj + βµH X i σi # , (7) where the various symbols have their usual meanings. It follows that ∂ ∂H (lnQN) = βµ X i σi ! = βM, (8) where M(= µP i σi) denotes the net magnetization of the system. Next, since ∂2 ∂H2 (lnQN) = ∂ ∂H  1 QN ∂QN ∂H  = 1 QN ∂2QN ∂H2 −1 Q2 N ∂QN ∂H 2 = β2(M2 −M2), (9) we obtain for the magnetic susceptibility of the system χ ≡∂M ∂H = β(M2 −M2) (10a) = βµ2    X i σi !2 − X i σi !2  = βµ2 X i X j g(i,j). (10b) Equation (10a) is generally referred to as the fluctuation–susceptibility relation; it may be compared with the corresponding relation for fluids, namely (4.5.7), which connects isothermal compressibility κT with the density fluctuations in the system. Equations (10) and (4.5.7) represent the equilibrium limit of the fluctuation–dissipation theorem dis-cussed later in Section 15.6. Equation (10b), on the other hand, relates χ to a summation of the correlation function g(i,j) over all i and j; assuming homogeneity, this may be written as χ = Nβµ2 X r g(r) (r = rj −ri). (11) 12.11 The role of correlations and fluctuations 453 Treating r as a continuous variable, equation (11) may be written as χ = Nβµ2 ad Z g(r)dr, (12) where a is a microscopic length, such as the lattice constant, so defined that Nad = V, the volume of the system; for a similar result appropriate to fluids, see equation (10.7.14). Finally, introducing the Fourier transform of the function g(r), through equation (2), we observe that χ = Nβµ2 ad ˜ g(0); (13) compare this result to equation (10.7.21) for fluids. Our next task consists in determining the mathematical form of the functions g(r) and ˜ g(k). Pending exact calculations, let us see what the mean field theory has to offer in this regard. Following Kadanoff (1976b), we consider a magnetic system subject to an exter-nal field H which is nonuniform, that is, H = {Hi}, where Hi denotes the field at site i. Using mean field approximation, the thermal average of the variable σi is given by, see equation (12.5.10), σ i = tanh(βµHeff), (14) where Heff = Hi + (J/µ) X n.n. σ j; (15) note that, in view of the nonuniformity of H, the product (qσ) of equation (12.5.10) has been replaced by a sum of σ j over all the nearest neighbors of spin i. If σ varies slowly in space, which means that the applied field is not too nonuniform, then (15) may be approximated as Heff ≃Hi + (qJ/µ)σ i + (cJa2/µ)∇2σ i, (16) where c is a number of order unity whose actual value depends on the structure of the lattice, while a is an effective lattice constant; note that the term involving ∇σ i cancels on summation over the q nearest neighbors that are supposed to be positioned in some symmetrical fashion around the site i. At the same time, the function tanhx, for small x, may be approximated by x −x3/3. Retaining only essential terms, we get from (14) and (16) βµHi = (1 −qβJ)σ i + 1 3(qβJ)3σ 3 i −cβJa2∇2σ i. (17) 454 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling Now, the conditions for criticality are {Hi} = 0 and qβcJ = 1; see equation (12.5.13). So, near criticality, we may introduce our familiar variables hi = βµHi, t = (T −Tc)/Tc ≃(βc −β)/βc; (18) equation (17) then reduces to  t + 1 3σ 2 i −c′a2∇2  σ i = hi, (19) where c′ is another number of order unity. Equation (19) generalizes equation (12.10.1) of Landau’s theory by taking into account the nonuniformity of σ. Differentiating (19) with respect to hj, we get  t + σ 2 i −c′a2∇2 ∂σ i ∂hj = δi,j. (20) The “response function” ∂σ i/∂hj is identical with the correlation function g(i,j);17 equa-tion (20) may, therefore, be written as  t + σ 2 i −c′a2∇2 g(i,j) = δi,j. (21) For t > 0 and {hi} →0, σ i →0; equation (21) then becomes  t −c′a2∇2 g(i,j) = δi,j. (22) Assuming homogeneity, so that g(i,j) = g(r) where r = rj −ri, and introducing Fourier transforms, equation (22) gives the form  t + c′a2k2 ˜ g(k) = const. (23) It follows that ˜ g(k) is a function of the magnitude k only (which is not surprising in view of the assumed symmetry of the lattice). Thus ˜ g(k) ∼ 1 t + c′a2k2 , (24) 17Remembering that M = µ6iσ i, we change the field {Hi} to {Hi + δHi}, with the result that δM = µ X ˙ i  X j (∂σ i/∂Hj)δHj  . Now, for simplicity, we let all δHj be the same; then (δM/δH) = µ X i X j (∂σ i/∂Hj). Comparing this with (10b), we infer that (∂σ i/∂Hj) = βµg(i,j) and hence (∂σ i/∂hj) = g(i,j). 12.11 The role of correlations and fluctuations 455 which is the famous Ornstein–Zernike result derived originally for fluids. Now, taking the inverse Fourier transform of ˜ g(k), we obtain (disregarding numerical factors that are not so essential for the present argument) g(r) ∼ Z e−ik·r t + c′a2k2 dd(ka) (25a) ∼ ∞ Z 0 ad t + c′a2k2  1 kr (d−2)/2 J(d−2)/2(kr)kd−1dk; (25b) see equations (8) and (11) of Appendix C. The integral in (25b) is tabulated; see Gradshteyn and Ryzhik (1965, p. 686). We get g(r) ∼ a2 ξr !(d−2)/2 K(d−2)/2  r ξ  {ξ = a(c′/t)1/2}, (26) Kµ(x) being a modified Bessel function. For x ≫1, Kµ(x) ∼x−1/2e−x; equation (26) then gives g(r) ∼ ad−2 ξ(d−3)/2r(d−1)/2 e−r/ξ (r ≫ξ). (27) On the other hand, for x ≪1, Kµ(x), for µ > 0,∼x−µ ; equation (26) then gives g(r) ∼ad−2 rd−2 r ≪ξ;d > 2  . (28) In the special case d = 2, we obtain instead g(r) ∼ln(ξ/r) (r ≪ξ;d = 2). (29) It is worth noting that equation (26) simplifies considerably when d = 3. Since K1/2(x) is exactly equal to (π/2x)1/2e−x for all x, g(r)|d=3 ∼a r e−r/ξ (30) for all r. Equation (30) is another important result of Ornstein and Zernike. Clearly, the quantity ξ appearing here is a measure of the “distances over which the spin–spin (or density–density) correlations in the system extend” — hence the name correlation length. So long as T is significantly above Tc, ξ = O(a); see (26). However, as T approaches Tc, ξ increases indefinitely — ultimately diverging at T = Tc. The resulting singularity is also of the power-law type: ξ ∼at−1/2 (t ≳0). (31) 456 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling The divergence of ξ at T = Tc is perhaps the most important clue we have for our general understanding of the critical phenomena. As ξ →∞, correlations extend over the entire system, paving the way for the propagation of long-range order (even though the microscopic interactions, which are at the root of the phenomenon, are themselves short-ranged). Moreover, since correlations extend over macroscopic distances in the system, any structural details that differentiate one system from another at the microscopic level lose significance, leading thereby to universal behavior! Going back to equations (13) and (24), we see that the singularity in χ is indeed of the type expected in a mean field theory, namely χ ∼˜ g(0) ∼t−1 (t ≳0). (32) In view of the foregoing results, one may write 1 ˜ g(k) ∼1 χ (1 + ξ2k2). (33) In a so-called Ornstein–Zernike analysis, one plots 1/˜ g(k) (or 1/I(k), where I(k) is the intensity of the light scattered at angle θ, see equation (1)) in the critical region versus k2. The data for small k (ka ≲0.1), for which the above treatment holds, fall close to a straight line whose intercept with the vertical axis determines χ(t). As t →0, this intercept tends to zero but the successive isotherms remain more or less parallel to one another; the reduced slope evidently serves to determine ξ(t). For t ≃0, these plots show a slight downward cur-vature, indicating departure from the k2-law to one in which the power of k is somewhat less than 2. Finally, as regards the plot I(θ;T) of Figure 12.13 earlier in this section, the max-imum in the curve, according to equation (24), should lie at t = 0 for all θ and the height of the maximum should be ∼k−2 (i.e., essentially ∼θ−2); thus, according to the mean field expression for ˜ g(k), the exponent λ1 in equation (3) should be 2 while λ2 should be 0. 12.12 The critical exponents ν and η According to the mean field theory, the divergence of ξ at T = Tc is governed by the power law (12.11.31), with a critical exponent 1 2. We anticipate that the experimental data on actual systems may not conform to this law. We, therefore, introduce a new critical exponent, ν, such that ξ ∼t−ν (h →0,t ≳0). (1) In the spirit of the scaling hypothesis, see Section 12.10, the corresponding exponent ν′ appropriate to t ≲0 would be the same as ν.18 Table 12.1 in Section 12.7 shows 18It turns out that the exponent ν′ is relevant only for scalar models, for which n = 1; for vector models (n ≥2), ξ is infinite at all T ≤Tc and hence ν′ is irrelevant. 12.12 The critical exponents ν and η 457 experimental results for ν obtained from a variety of systems; we see that the observed values of ν, while varying very little from system to system, differ considerably from the mean field value. As regards the correlation function, the situation for t ≳0 is described very well by a law of the type (12.11.27), namely g(r) ∼e−r/ξ(t) × some power of r (t ≳0), (2) where ξ(t) is given by (1). The variation of g(r) with r in this regime is governed primarily by the exponential, so g(r) falls rapidly as r exceeds ξ (which is typically of the order of the lattice constant a). As t →0 and hence ξ →∞, the behavior of g(r) would be expected to be like equation (12.11.28) or (12.11.29). A problem now arises: we have an exact expression for g(r) at T = Tc for a two-dimensional Ising model (see Section 13.4), according to which g(r) ∼r−1/4 (d = 2,n = 1,t = 0), (3) which is quite different from the mean field expression (12.11.29). We, therefore, generalize our classical result to g(r) ∼r−(d−2+η) (t = 0), (4) which introduces another critical exponent, η. Clearly, η for the two-dimensional Ising model is 1 4, which can even be confirmed by experiments on certain systems that are effectively two-dimensional. Table 12.1 shows experimental values of η for some systems in three dimensions; typically, η turns out to be a small number, which makes it rather difficult to measure reliably. We shall now derive some scaling relations involving the exponents ν and η. First of all we write down the correlation function g(r;t,h) and its Fourier transform ˜ g(k;t,h) in a scaled form. For this, we note that, while h scales with t1, the only natural variable with which r will scale is ξ; accordingly, r will scale with t−ν. We may, therefore, write g(r;t,h) ≈G(rtν,h/t1) rd−2+η , ˜ g(k;t,h) ≈ ˜ G(k/tν,h/t1) k2−η , (5a, b) where the functions G(x,y) and ˜ G(z,y), like the exponents 1,ν, and η, are universal for a given universality class; in expressions (5), for simplicity, we have suppressed nonuniversal parameters that vary from system to system within a class. In the absence of the field (h = 0), expressions (5) reduce to g(r;t,0) ≈G0(rtν) rd−2+η , ˜ g(k;t,0) ≈ ˜ G0(k/tν) k2−η , (6a, b) 458 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling where G0(x) and ˜ G0(z) are again universal. At the critical point (h = 0,t = 0), we have simply gc(r) ∼ 1 rd−2+η , ˜ gc(k) ∼ 1 k2−η . (7a, b) We now recall equation (12.11.12), which relates χ to an integral of g(r) over dr, and substitute expression (6a) into it. We get, ignoring nonuniversal parameters as well as numerical factors, χ ∼ Z G0(rtν) rd−2+η rd−1 dr. (8) By a change of variables, this gives χ ∼t−(2−η)ν. (9) Invoking the standard behavior of χ, we obtain γ = (2 −η)ν. (10) Note that the same scaling relation can also be obtained by appealing to equa-tions (12.11.13) and (6b); the argument is that, in the limit k →0, the function ˜ G0(z) must be ∼z2−η (so that k is eliminated), leaving behind a result identical to (9). In passing, we note that in the critical region χ ∼ξ2−η. (11) Relation (10) is consistent with the Fisher inequality (1969) γ ≤(2 −η)ν, and is obviously satisfied by the mean field exponents (γ = 1,ν = 1 2,η = 0); it also checks well with the experimental data given earlier in Table 12.1. In fact, this relation provides a much better method of extracting the elusive exponent η, from a knowledge of γ and ν, than determining it directly from experiment. Incidentally, the presence of η explains the slight downward curvature of the Ornstein–Zernike plot, 1/˜ g(k) versus k2, as k →0, for the appropriate expression for 1/˜ g(k) now is 1 ˜ g(k) ∼1 χ (1 + ξ2−ηk2−η), (12) rather than (12.11.33). We shall now derive another exponent relation involving ν, but first notice that all expo-nent relations derived so far have no explicit dependence on the dimensionality d of the system (though the actual values of the exponents do depend on d). There is, however, one important relationship that does involve d explicitly. For this, let us visualize what 12.12 The critical exponents ν and η 459 happens inside the system (say, a magnetic one) as t →0 from above. At some stage the correlation length ξ becomes significantly larger than the atomic spacings, with the result that magnetic domains, of alternating magnetization, begin to appear. The closer we get to the critical point, the larger the size of these domains; one may, in fact, say that the vol-ume  of any such domain is ∼ξd. Now, the singular part of the free energy density of the system — or, for that matter, of any one of these domains — is given by, see equa-tion (12.10.7) with h = 0, f (s)(t) ∼t2−α, (13) which vanishes as t →0. At the same time, the domain volume  diverges. It seems natural to expect that f (s), being a density, would vanish as 1/, that is, f (s)(t) ∼−1 ∼ξ−d ∼tdν. (14) Comparing (13) and (14), we obtain the desired relationship dν = 2 −α, (15) which is generally referred to as a hyperscaling relation — to emphasize the fact that it goes beyond, and cannot be derived from, the ordinary scaling formulation of Section 12.10 without invoking something else, such as the domain volume . The relation in equation (15) is consistent with the Josephson inequalities (1967): dν ≥2 −α, dν′ ≥2 −α′, (16a, b) proved rigorously by Sokal (1981); of course, the scaling theory does not distinguish between exponents pertaining to t > 0 and their counterparts pertaining to t < 0. It is important to note that the classical exponents (ν = 1 2,α = 0) satisfy (15) only for d = 4, which shows that the hyperscaling relations, (15) and any others that follow from it, have a rather different status than the other scaling relations (that do not involve d explicitly). The renormalization group theory, to be discussed in Chapter 14, shows why the hyperscaling relations are to be expected fairly generally, why typically they hold for d < 4 but break down for d > 4; see also Section 12.13. The reader may check that relation (15) is satisfied reasonably well by the experimental data of Table 12.1, with d = 3; it is also satisfied by the exponents derived theoretically by solving different models exactly, or almost exactly, as in Chapter 13. Combining (15) with other scaling relations, see Section 12.10, we may write dν = 2 −α = 2β + γ = β(δ + 1) = γ (δ + 1)/(δ −1). (17) It follows that 2 −η = γ/ν = d(δ −1)/(δ + 1), (18) 460 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling which is consistent with the Buckingham–Gunton inequality (1969) 2 −η ≤d(δ −1)/(δ + 1). (19) Notice that the experimental observation that, for magnetic systems in three dimensions, δ < 4.8, implies that η ≥0.034. 12.13 A final look at the mean field theory We now return to the question: why does the mean field theory fail to represent the true nature of the critical phenomena in real systems? The short answer is — because it neglects fluctuations! As emphasized in Section 12.11, correlations among the microscopic con-stituents of a given system are at the very heart of the phenomenon of phase transitions, for it is through them that the system acquires a long-range order (even when the micro-scopic interactions are themselves short-ranged). At the same time, there is so direct a relationship between correlations and fluctuations, see equation (12.11.6), that they grow together and, as the critical point is approached, become a dominant feature of the system. Neglecting fluctuations is, therefore, a serious drawback of the mean field theory. The question now arises: is mean field theory ever valid? In other words, can fluctua-tions ever be neglected? To answer this question, we recall the fluctuation–susceptibility relation (12.11.10a), namely χ = (M2 −M2)/kT = (1M)2/kT (1) and write it in the form (1M)2/M2 = kTχ/M2. (2) Now, in order that the neglect of fluctuations be justified, we must have: kTχ ≪M2. (3) Requirement (3) is generally referred to as the Ginzburg criterion (1960); for a more detailed discussion of this criterion, along with physical illustrations, see Als-Nielsen and Birgeneau (1977). We apply condition (3) to a domain, of volume  ∼ξd, close to but below the critical point; we are assuming here a system of the Ising type (n = 1), so that ξ is finite for t < 0. Invoking the power-law behavior of χ and M, we have kTc(Aξd|t|−γ ) ≪(Bξd|t|β)2 (t ≲0), (4) where A and B are positive constants. Since ξ ∼a|t|−ν, we get |t|dν−2β−γ ≪B2ad/AkTc. (5) 12.13 A final look at the mean field theory 461 In view of the scaling relation α + 2β + γ = 2, we may as well write |t|dν−(2−α) ≪D, (6) where D is a positive number of order unity.19 For the mean field theory (with ν = 1 2,α = 0) to be valid, condition (6) assumes the form |t|(d−4)/2 ≪D. (7) Now, since |t| can be made as small as we like, condition (7) will be violated unless d > 4. We, therefore, conclude that the mean field theory is valid for d > 4; by implication, it is inadequate for d ≤4. The preceding result has been established for scalar models (n = 1) only. In Section 13.5, we shall see that in the case of the spherical model, which pertains to the limit n →∞, the mean field results do apply when d > 4. This means that, once again, fluctua-tions can be neglected if d > 4. Now fluctuations are supposed to decrease with decreasing n; the validity of the mean field theory for d > 4 should, therefore, hold for all n. Ordinarily, when a system is undergoing a phase transition, expression (2), which is a measure of the relative fluctuations in the system, is expected to be of order unity. Condition (6) then suggests that the exponents ν and α obey the hyperscaling relation dν = 2 −α. (8) Experience shows that this relation is indeed obeyed when d < 4. At d = 4, the mean field theory begins to take over and thereafter, for all d > 4, the critical exponents are stuck at the mean field values (which are independent of both d and n). The dimensionality d = 4 is often referred to as the upper critical dimension for the kind of systems under study. An alternative way of looking into the question posed at the beginning of this section is to examine the specific heat of the system which, according to the mean field theory, undergoes a jump discontinuity at the critical point whereas in real systems it shows a weak divergence. The question now arises: what is the source of this divergence that is missed by the mean field theory? The answer again lies in the “neglect of fluctuations.” To see it more explicitly, we look at the internal energy of the system which, in the absence of the field, is given by U = −J X n.n. σiσj ! = −J X n.n. σiσj. (9) In the mean field theory, one replaces σiσj by σi σj(= σ 2), see equation (12.5.5), which leads to the jump discontinuity in the specific heat of magnitude 3 2/Nk; see equation (12.5.18). 19To see this, we note that A ∼Nµ2/kTc while B ∼Nµ/, with the result that D ∼Nad/ = O(1). 462 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling The fluctuational part of U, which is neglected in the mean field theory, may be written as Uf = −J X n.n. (σiσj −σ iσ j) = −J X n.n. g(rij), (10) where g(r) is the spin–spin correlation function, for which we may use the mean field expression (12.11.26); thus, we will be using the mean field theory itself to predict its own shortcomings! Since the nearest-neighbor distances rij in (10) are all much smaller than ξ, one may be tempted to use for g(rij) the zeroth-order approximation (12.11.28). This, however, produces a temperature-independent term, which does not contribute to the specific heat of the system. We must, therefore, go to the next approximation, which can be obtained by using the asymptotic formulae Kµ(x) x≪1 ≈                      1 20(µ) 1 2x −µ + 1 20(−µ) 1 2x µ for 0 < µ < 1 (11a) x−1 + 1 2x  ln 1 2x  for µ = 1 (11b) 1 20(µ) 1 2x −µ −1 20(µ −1) 1 2x 2−µ for µ > 1, (11c) with µ = (d −2)/2 and x = rij/ξ. The temperature-dependent part of Uf comes from the second term(s) in (11); remembering that ξ here is ∼at−1/2, we get20 (Uf /NJ)thermal ∼        t(d−2)/2 for 2 < d < 4 (12a) t ln(1/t) for d = 4 (12b) t for d > 4. (12c) The fluctuational part of the specific heat then turns out to be Cf /Nk ∼        t(d−4)/2 for 2 < d < 4 (13a) ln(1/t) for d = 4 (13b) const. for d > 4. (13c) It follows that the specific-heat singularity for d > 4 is indeed a “jump discontinuity,” and hence the mean field theory remains applicable in this case. For d = 4, Cf shows a logarithmic divergence and for d < 4 a power-law divergence, making mean field theory invalid for d ≤4. It is rather instructive to see what part of the fluctuation–correlation spectrum, ˜ g(k), contributes significantly to the divergence of the specific heat at t = 0. For this, we examine 20Note that the negative sign in (10) cancels the implicit negative sign of 0(−µ) in (11a), that of ln( 1 2 x) in (11b), and the explicit negative sign in (11c). Problems 463 the quantity −∂g(rij)/∂t, which essentially determines the behavior of Cf near the critical point; see equation (10). Using equation (12.11.25a), we have −∂g(rij) ∂t ∼ad Z e−ik·rij t2(1 + ξ2k2)2 kd−1dk. (14) Now, the values of k that are much larger than ξ−1 contribute little to this integral; the only significant contributions come from the range (0,kmax), where kmax = O(ξ−1); moreover, since rij ≪ξ, the exponential for these values of k is essentially equal to 1. Expression (14) may, therefore, be written as −∂g(rij) ∂t ∼ad Z kd−1dk t2(1 + ξ2k2)2 (15) which, for d < 4, scales as (a/ξ)dt−2 ∼t(d−4)/2; compare with (13a). We thus see that the most significant contribution to the criticality of the problem arises from fluctuations whose length scale, k−1, is of order ξ or longer and hardly any contribution comes from fluctuations whose length scale is shorter. Now, it is only the latter that are likely to pick up the structural details of the system at the atomic level; since they do not play any sig-nificant role in bringing about the phenomenon, the precise nature of criticality remains independent of the structural details. This explains why a large variety of systems, differ-ing so much in their structure at the macroscopic level, may, insofar as critical behavior is concerned, fall into a single universality class. Problems 12.1. Assume that in the virial expansion Pv kT = 1 − ∞ X j=1 j j + 1βj λ3 v !j , (10.4.22) where βj are the irreducible cluster integrals of the system, only terms with j = 1 and j = 2 are appreciable in the critical region. Determine the relationship between β1 and β2 at the critical point, and show that kTc/Pcvc = 3. 12.2. Assuming the Dietrici equation of state, P(v −b) = kT exp(−a/kTv), evaluate the critical constants Pc, vc, and Tc of the given system in terms of the parameters a and b, and show that the quantity kTc/Pcvc = e2/2 ≃3.695. Further show that the following statements hold in regard to the Dietrici equation of state: (a) It yields the same expression for the second virial coefficient B2 as the van der Waals equation does. (b) For all values of P and for T ≥Tc, it yields a unique value of v. (c) For T < Tc, there are three possible values of v for certain values of P and the critical volume vc is always intermediate between the largest and the smallest of the three volumes. (d) The Dietrici equation of state yields the same critical exponents as the van der Waals equation does. 464 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling 12.3. Consider a nonideal gas obeying a modified van der Waals equation of state (P + a/vn)(v −b) = RT (n > 1). Examine how the critical constants Pc, vc, and Tc, and the critical exponents β, γ , γ ′, and δ, of this system depend on the number n. 12.4. Following expressions (12.5.2), define p = (1 + L)/2 and q = (1 −L)/2 (−1 ≤L ≤1) (1) as the probabilities that a spin chosen at random in a lattice composed of N spins is either “up” or “down.” The partition function of the system may then be written as Q(B,T) = X L g(L)eβN( 1 2 qJL2+µBL), (2) where g(L) is the multiplicity factor associated with a particular value of L, that is, g(L) = N!/(Np)!(Nq)!; (3) note that in writing the Hamiltonian here we have made the assumption of random mixing, according to which (N++ + N−−−N+−) = 1 2qN(p2 + q2 −2pq) = 1 2qNL2. (a) Determine the value, L∗, of L that maximizes the summand in (2). Check that L∗is identical to the mean value, L, as given by equation (12.5.10). (b) Write down the free energy A and the internal energy U of the system, and show that the entropy S conforms to the relation S(B,T) = −Nk(p∗lnp∗+ q∗lnq∗), where p∗= p(L∗) and q∗= q(L∗). 12.5. Using the correspondence established in Section 12.4, apply the results of the preceding problem to the case of a lattice gas. Show, in particular, that the pressure, P, and the volume per particle, v, are given by P = µB −1 8qε0  1 + L 2 −1 2kT ln 1 −L 2 4 ! and v−1 = 1 2(1 ± L). Check that the critical constants of this system are: Tc = qε0/4k, Pc = kTc(ln2 −1 2), and vc = 2, so that the quantity kTc/Pcvc = 1/(ln4 −1) ≃2.589. 12.6. Consider an Ising model with an infinite-range interaction such that each spin interacts equally strongly with all other spins: H = −c X i 0. Note that, in the special case of equal concentrations (xA = xB = 1 2), this equation assumes the more familiar form X = tanh  qε 2kT X  . Further show that the transition temperature of the system is given by Tc = 4xA(1 −xA)T0 c , where T0 c (= qε/2k) is the transition temperature in the case of equal concentrations. [Note: In the Kirkwood approximation (see Kubo (1965), problem 5.19), T0 c turns out to be (ε/k){1 −√[1 −(4/q)]}−1, which may be written as (qε/2k)(1 −1/q + ···). To this order, the Bethe approximation also yields the same result.] 12.13. Consider a two-component solution of NA atoms of type A and NB atoms of type B, which are supposed to be randomly distributed over N(= NA + NB) sites of a single lattice. Denoting the energies of the nearest-neighbor pairs AA,BB, and AB by ε11,ε22, and ε12, respectively, write down the free energy of the system in the Bragg–Williams approximation and evaluate the chemical potentials µA and µB of the two components. Next, show that if ε = (ε11 + ε22 −2ε12) < 0, that is, if the atoms of the same species display greater affinity to be neighborly, then for temperatures below a critical temperature Tc, which is given by the expression q|ε|/2k, the solution separates out into two phases of unequal relative concentrations. [Note: For a study of phase separation in an isotopic mixture of hard-sphere bosons and fermions, and for the relevance of this study to the actual behavior of He3−He4 solutions, see Cohen and van Leeuwen (1960, 1961).] 12.14. Modify the Bragg–Williams approximation (12.5.29) to include a short-range order parameter s, such that N++ = 1 2qNγ 1 + L 2 !2 (1 + s), N−−= 1 2qNγ 1 −L 2 !2 (1 + s), N+−= 2 · 1 2qNγ 1 + L 2 ! 1 −L 2 ! (1 −s). (a) Evaluate γ from the condition that the total number of nearest-neighbor pairs is 1 2qN. (b) Show that the critical temperature Tc of this model is (1 −s2)qJ/k. Problems 467 (c) Determine the nature of the specific-heat singularity at T = Tc, and compare your result with both the Bragg–Williams approximation of Section 12.5 and the Bethe approximation of Section 12.6. 12.15. Show that in the Bethe approximation the entropy of the Ising lattice at T = Tc is given by the expression Sc Nk = ln2 + q 2 ln  1 −1 q  −q(q −2) 4(q −1) ln  1 −2 q  . Compare this result with the one following from the Bragg–Williams approximation, namely (12.5.20). 12.16. Examine the critical behavior of the low-field susceptibility, χ0, of an Ising model in the Bethe approximation of Section 12.6, and compare your results with equations (12.5.22) of the Bragg–Williams approximation. 12.17. A function f (x) is said to be concave over an interval (a,b) if it satisfies the property f {λx1 + (1 −λ)x2} ≥λf (x1) + (1 −λ)f (x2), where x1 and x2 are two arbitrary points in the interval (a,b) while λ is a positive number in the interval (0,1). This means that the chord joining the points x1 and x2 lies below the curve f (x). Show that this also means that the tangent to the curve f (x) at any point x in the interval (a,b) lies above the curve f (x) or, equivalently, that the second derivative ∂2f /∂x2 throughout this interval ≤0. 12.18. In view of the thermodynamic relationship CV = TV(∂2P/∂T2)V −TN(∂2µ/∂T2)V for a fluid, µ being the chemical potential of the system, Yang and Yang (1964) pointed out that, if CV is singular at T = Tc, then either (∂2P/∂T2)V or (∂2µ/∂T2)V or both will be singular. Define an exponent 2 by writing (∂2P/∂T2)V ∼(Tc −T)−2 (T ≲Tc), and show that (Griffiths, 1965b) 2 ≤α′ + β and 2 ≤(2 + α′δ)/(δ + 1). 12.19. Determine the numerical values of the coefficients r1 and s0 of equation (12.9.5) in (i) the Bragg–Williams approximation of Section 12.5 and (ii) the Bethe approximation of Section 12.6. Using these values of r1 and s0, verify that equations (12.9.4), (12.9.9), (12.9.10), (12.9.11), and (12.9.15) reproduce correctly the results obtained in the zeroth and the first approximation, respectively. 12.20. Consider a system with a modified expression for the Landau free energy, namely ψh(t,m) = −hm + q(t) + r(t)m2 + s(t)m4 + u(t)m6, with u(t) a fixed positive constant. Minimize ψ with respect to the variable m and examine the spontaneous magnetization m0 as a function of the parameters r and s. In particular, show the following:21 (a) For r > 0 and s > −(3ur)1/2,m0 = 0 is the only real solution. (b) For r > 0 and −(4ur)1/2 < s ≤−(3ur)1/2,m0 = 0 or ±m1, where m2 1 = √(s2−3ur)−s 3u . However, the minimum of ψ at m0 = 0 is lower than the minima at m0 = ±m1, so the ultimate equilibrium value of m0 is 0. (c) For r > 0 and s = −(4ur)1/2,m0 = 0 or ±(r/u)1/4. Now, the minimum of ψ at m0 = 0 is of the same height as the ones at m0 = ±(r/u)1/4, so a nonzero spontaneous magnetization is as likely to occur as the zero one. 21To fix ideas, it is helpful to use (r,s)-plane as our “parameter space.” 468 Chapter 12. Phase Transitions: Criticality, Universality, and Scaling (d) For r > 0 and s < −(4ur)1/2, m0 = ±m1 — which implies a first-order phase transition (because the two possible states available here differ by a finite amount in m). The line s = −(4ur)1/2, with r positive, is generally referred to as a “line of first-order phase transitions.” (e) For r = 0 and s < 0, m0 = ±(2|s|/3u)1/2. (f) For r < 0, m0 = ±m1 for all s. As r →0, m1 →0 if s is positive. (g) For r = 0 and s > 0, m0 = 0 is only solution. Combining this result with (f), we conclude that the line r = 0, with s positive, is a “line of second-order phase transitions,” for the two states available here differ by a vanishing amount in m. The lines of first-order phase transitions and second-order phase transitions meet at the point (r = 0,s = 0), which is commonly referred to as a tricritical point (Griffiths, 1970). 12.21. In the preceding problem, put s = 0 and approach the tricritical point along the r-axis, setting r ≈r1t. Show that the critical exponents pertaining to the tricritical point in this model are α = 1 2,β = 1 4,γ = 1, and δ = 5. 12.22. Consider a fluid near its critical point, with isotherms as sketched in Figure 12.3. Assume that the singular part of the Gibbs free energy of the fluid is of the form G(s)(T,P) ∼|t|2−αg(π/|t|1), where π = (P −Pc)/Pc, t = (T −Tc)/Tc while g(x) is a universal function, with branches g+ for t > 0 and g−for t < 0; in the latter case, the function g−has a point of infinite curvature at a value of π that varies smoothly with t, such that π(0) = 0 and (∂π/∂t)t→0 = const. (a) Using the above expression for G(s), determine the manner in which the densities, ρl and ρg, of the two phases approach one another as t →0 from below. (b) Also determine how (P −Pc) varies with (ρ −ρc) as the critical point is approached along the critical isotherm (t = 0). (c) Examine as well the critical behavior of the isothermal compressibility κT, the adiabatic compressibility κS, the specific heats CP and CV , the coefficient of volume expansion αP, and the latent heat of vaporization l. 12.23. Consider a model equation of state which, near the critical point, can be written as h ≈am(t + bm2)2 (1 < 2 < 2;a,b > 0). Determine the critical exponents β, γ , and δ of this model, and check that they obey the scaling relation (12.10.22). 12.24. Assuming that the correlation function g(ri,rj) is a function only of the distance r = |rj −ri|, show that g(r) for r ̸= 0 satisfies the differential equation d2g dr2 + d −1 r dg dr −1 ξ2 g = 0. Check that expression (12.11.27) for g(r) satisfies this equation in the regime r ≫ξ, while expression (12.11.28) does so in the regime r ≪ξ. 12.25. Consider the correlation function g(r;t,h) of Section 12.11 with h > 0.22 Assume that this function has the following behavior: g(r) ∼e−r/ξ(t,h) × some power of r (t ≳0), such that ξ(0,h) ∼h−νc. Show that νc = ν/1. Next, assume that the susceptibility χ(0,h) ∼h−γ c. Show that γ c = γ/1 = (δ −1)/δ. 22For more details, see Tarko and Fisher (1975). Problems 469 12.26. Liquid He4 undergoes a superfluid transition at T ≃2.17 K. The order parameter in this case is a complex number 9, which is related to the Bose condensate density ρ0 as ρ0 ∼|9|2 ∼|t|2β (t ≲0). The superfluid density ρs, on the other hand, behaves as ρs ∼|t|ν (t ≲0). Show that the ratio23 (ρ0/ρs) ∼|t|ην (t ≲0). 12.27. The surface tension, σ, of a liquid approaches zero as T →Tc from below. Define an exponent µ by writing σ ∼|t|µ (t ≲0). Identifying σ with the “free energy associated with a unit area of the liquid–vapor interface,” argue that µ = (d −1)ν = (2 −α)(d −1)/d. [Note: Analysis of the experimental data on surface tension yields: µ = 1.27 ± 0.02, which agrees with the fact that for most fluids α ≃0.1.] 23The corresponding ratio for a magnetic system is M2 0/0, where M0 is the spontaneous magnetization and 0 the helicity modulus of the system; for details, see Fisher, Barber, and Jasnow (1973). 13 Phase Transitions: Exact (or Almost Exact) Results for Various Models In the preceding chapter we saw that the onset of a phase transition in a given physico-chemical system is characterized by (singular) features whose qualitative nature is deter-mined by the universality class to which the system belongs. In this chapter we propose to consider a variety of model systems belonging to different universality classes and ana-lyze them theoretically to find out how these features arise and how they vary from class to class. In this context, we recall that the parameters distinguishing one universality class from another are: (i) the space dimensionality d, (ii) the dimensionality of the order parameter, often referred to as the spin dimensionality, n, and (iii) the range of the microscopic inter-actions. As regards the latter, unless a statement is made to the contrary, we shall assume a short-range interaction which, in most cases, will be of the nearest-neighbor type; the only parameters open for selection will then be d and n. We start our analysis with the properties of one-dimensional fluids, the one-dimensional Ising (n = 1) model, and the general n-vector models (again in one dimen-sion). We then return to the Ising model — this time in two dimensions — and follow it with a study of two models in general d but with n →∞. These studies will give us a fairly good idea as to what to expect in the most practical, three-dimensional situations for which, unfortunately, we have no exact solutions — though a variety of mathematical techniques have been developed to obtain almost exact results in many cases of interest. For completeness, these and other results of physical importance will be revisited in the last section of this chapter. 13.1 One-dimensional fluid models A system of interacting particles in one dimension can be solved analytically for sev-eral cases. In particular, a system of hard spheres in one dimension can be solved in the canonical ensemble (Tonks, 1936) while the one-dimensional isobaric ensemble allows the solution of a more general set of nearest-neighbor interactions. None of these mod-els exhibits a transition to an ordered phase but they do display many of the short-range correlation properties characteristic of fluids. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00013-X © 2011 Elsevier Ltd. All rights reserved. 471 472 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 13.1.A Hard spheres on a ring The partition function of a one-dimensional system of hard spheres was first evaluated by Tonks (1936). For this we consider the case of N hard spheres with diameter D on a ring of circumference L, obeying periodic boundary conditions. The free energy, the pressure, and the pair correlation function of this system can be determined exactly both for finite N and in the thermodynamic limit. The hard sphere pair interaction is given by u(r) = ( ∞ for r ≤D 0 for r > D. (1) The configurational partition function can be written as an integral over the spatially ordered positions of the N spheres, with particle 1 set at x1 = 0, while the other particles labeled j = 2,3,...,N are restricted by the conditions xj−1 + D < xj < L −(N −j + 1)D: ZN(L) = L N L−(N−1)D Z D dx2 L−(N−2)D Z x2+D dx3 ··· L−D Z xN−1+D dxN = L(L −ND)N−1 N! ; (2) the prefactor L here comes from integrating x1 over the circumference of the ring, while the factor 1/N delabels that final integral. The Helmholtz free energy in the thermodynamic limit turns out to be A(N,L,T) = −NkT ln L −ND Nλ  −NkT, (3) where λ is the thermal deBroglie wavelength. The pressure is then given by P = − ∂A ∂L  T,N = nkT 1 −nD , (4) where n (= N/L) is the one-dimensional number density. This is equivalent to the pressure of an ideal gas of N particles in a free volume L −ND. The isothermal compressibility is κT = 1 n ∂n ∂p  T = (1 −nD)2 nkT = κideal T (1 −nD)2 . (5) The pair correlation function for the particles on the ring can also be determined exactly (M. Foss-Feig, unpublished). This is accomplished by integrating over all configurations in which particle 1 is fixed at the origin and, in succession, each of the other particles is fixed at position x (if possible). This gives g(x) = N−1 X j=1 gj(x). (6) 13.1 One-dimensional fluid models 473 0 0 1 2 3 4 g (x) 5 10 X 15 FIGURE 13.1 The pair correlation function for a system of 12 hard disks on a ring. The scaled number density nD = 0.75. The solid line represents the exact solution from equations (6) and (7), as compared to a Monte Carlo simulation. where gj(x) is defined on the range jD ≤x ≤L −(N −j)D by gj(x) =  L(N −1)! N(L −ND)N−1  (x −jD)j−1(L −x −(N −j)D)N−1−j (j −1)!(N −1 −j)! ! ; (7) see Figure 13.1. In the thermodynamic limit, the correlation function becomes g(x) = ∞ X j=1 gj(x) , (8) where gj(x) is defined on the range jD ≤x < ∞by gj(x) = (βP(x −jD))j−1 exp −βP(x −jD)  (1 −nD)(j −1)! , (9) where βP = n/(1 −nD); see Sells, Harris, and Guth (1953). Using equation (10.7.12) gives the correct virial equation of state pressure P nkT = 1 + nDg(D+) = 1 1 −nD ; (10) where g(D+) is the pair correlation function at contact; see Problem 13.28. 13.1.B Isobaric ensemble of a one-dimensional fluid Takahashi (1942) has shown that a one-dimensional system of particles that interact with nearest neighbors via pair potential u(r) can be analyzed analytically using an isobaric 474 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models ensemble. The potential energy of the system can be written as U = N+1 X j=1 u(xj −xj−1), (11) where the left and right walls are treated as particles fixed at x0 = 0 and xN+1 = L. The isobaric partition function is given by YN(P,T) = 1 λ ∞ Z 0 exp(−βPL)QN(L,T)dL, (12) where λ is the thermal deBroglie wavelength. Equation (12) can be factorized in terms of integrals over the distances between nearest neighbors yi (= xi −xi−1): YN(P,T) =  1 λ ∞ Z 0 exp(−βPy −βu(y))dy   N+1 = Y1(p,T)N+1. (13) The bulk Gibbs free energy is then given by G(N,P,T) = −NkT ln(Y1(P,T)), (14) and the average system size at pressure P by ∂G ∂P  T,N = L = N⟨y⟩, (15) where ⟨y⟩= R ∞ 0 yexp[−βPy −βu(y)]dy R ∞ 0 exp[−βPy −βu(y)]dy (16) is the average nearest-neighbor distance between the particles. The isothermal compress-ibility, κT = −1 L  ∂L ∂P  T,N = N kTL  ⟨y2⟩−⟨y⟩2 , (17) is proportional to the variance of the nearest-neighbor distances. It is now easy to show that one-dimensional models cannot form a long-range ordered lattice. The average dis-tance between two particles labeled by i and j is ⟨xi −xj⟩= (i −j)⟨y⟩but the variance is given by ⟨(xi −xj)2⟩= |i −j|(⟨y2⟩−⟨y⟩2). Therefore, if a chosen particle is located on a particular lattice site, then a particle m sites away will on average be separated from it by m lattice spacings, but the variance of this particle’s position from that site location grows linearly with m. 13.1 One-dimensional fluid models 475 In the thermodynamic limit, the structure factor S(k), equation (10.7.18), can be written in the form S(k) = ∞ X j=−∞ eik(xj−x0) + . (18) Since (xj −x0) = Pj i=1 yi, the structure factor can be summed exactly to give S(k) = −1 + ∞ X j=0 zj + ∞ X j=0 (z∗)j = 1 −|z|2 1 + |z|2 −z −z∗, (19) where z = D eikyE = R ∞ 0 exp −βPy −βu(y) + iky dy R ∞ 0 exp −βPy −βu(y) dy , (20) and z∗is the complex conjugate of z. The fluctuation-compressibility relation now gives: S(k →0) = κT/κideal T = ⟨y2⟩−⟨y⟩2 /⟨y⟩2. For the particular case of hard spheres, the pres-sure is given by equation (4) and the structure factor is given by S(k) = (kD)2 (kD)2 + 2(βPD)2(1 −cos(kD)) + 2(βPD)(kD)sin(kD) ; (21) see Figure 13.2. Equation (10.7.20a), applied to the hard sphere pair correlation function in equation (9), also gives equation (21). 0 0 1 2 3 S(k) 20 40 60 80 kD 100 FIGURE 13.2 The structure factor S(k) for a system of hard spheres on a line at density nD = 0.75. The structure factor at k = 0 is S(0) = (1 −nD)2 = κT/κideal T . 476 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 13.2 The Ising model in one dimension In this section we present an exact treatment of the Ising model in one dimension. This is important for several reasons. First of all, there do exist phenomena, such as adsorption on a linear polymer or on a protein chain, the elastic properties of fibrous proteins, and so on, that can be looked upon as one-dimensional nearest-neighbor problems. Secondly, it helps us evolve mathematical techniques for treating lattices in higher dimensions, which is essential for understanding the critical behavior of a variety of physical systems met with in nature. Thirdly, it enables us to estimate the status of the Bethe approximation as a “possible” theory of the Ising model, for it demonstrates mathematically that, at least in one dimension, this approximation leads to exact results. In a short paper published in 1925, Ising himself gave an exact solution to this prob-lem in one dimension. He employed a combinatorial approach that has by now been superseded by other approaches. Here we shall follow the transfer matrix method, first introduced by Kramers and Wannier (1941). In the one-dimensional case, this method worked with immediate success. Three years later, in 1944, it became, through Onsager’s ingenuity, the first method to treat successfully the field-free Ising model in two dimen-sions. To apply this method, we replace the actual lattice by one having the topology of a closed, endless structure; thus, in the one-dimensional case we replace the straight, open chain by a curved one such that the Nth spin becomes a neighbor of the first (see Figure 13.3). This replacement eliminates the inconvenient end effects; it does not, however, alter the thermodynamic properties of the (infinitely long) chain. The impor-tant advantage of this replacement is that it enables us to write the Hamiltonian of the system, HN{σi} = −J X n.n. σiσj −µB N X i=1 σi, (1) in a symmetrical form, namely HN{σi} = −J N X i=1 σiσi+1 −1 2µB N X i=1 (σi + σi+1), (2) N23 N2 2 N21 N 1 2 3 4 5 FIGURE 13.3 An Ising chain with a closed, endless structure. 13.2 The Ising model in one dimension 477 because σN+1 ≡σ1. The partition function of the system is then given by QN(B,T) = X σ1=±1 ··· X σN =±1 exp " β N X i=1 {Jσiσi+1 + 1 2µB(σi + σi+1)} # (3a) = X σ1=±1 ··· X σN =±1 ⟨σ1|P|σ2⟩⟨σ2|P|σ3⟩···⟨σN−1|P|σN⟩⟨σN|P|σ1⟩, (3b) where P denotes an operator with matrix elements ⟨σi|P|σi+1⟩= exp  β  Jσiσi+1 + 1 2µB(σi + σi+1)  , that is, (P) = eβ(J+µB) e−βJ e−βJ eβ(J−µB) ! . (4) According to the rules of matrix algebra, the summations over the various σi in equa-tion (3b) lead to the simple result QN(B,T) = X σ1=±1 ⟨σ1|PN|σ1⟩= Trace (PN) = λN 1 + λN 2 , (5) where λ1 and λ2 are the eigenvalues of the matrix P. These eigenvalues are given by the equation eβ(J+µB) −λ e−βJ e−βJ eβ(J−µB) −λ = 0, (6) that is, by λ2 −2λeβJ cosh(βµB) + 2sinh(2βJ) = 0. (7) One readily obtains λ1 λ2 ! = eβJ cosh(βµB) ± {e−2βJ + e2βJ sinh2(βµB)}1/2. (8) Quite generally, λ2 < λ1; so, (λ2/λ1)N →0 as N →∞. Thus, it is only the larger eigenvalue, λ1, that determines the major physical properties of the system in the thermodynamic limit; see equation (5). It follows that 1 N lnQN(B,T) ≈lnλ1 (9) = ln[eβJ cosh(βµB) + {e−2βJ + e2βJ sinh2(βµB)}1/2]. (10) 478 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models The Helmholtz free energy then turns out to be A(B,T) = −NJ −NkT ln[cosh(βµB) + {e−4βJ + sinh2(βµB)}1/2]. (11) The various other properties of the system follow readily from equation (11). Thus, U(B,T) ≡−T2 ∂ ∂T  A T  = −NJ − NµBsinh(βµB) {e−4βJ + sinh2(βµB)}1/2 + 2NJe−4βJ [cosh(βµB) + {e−4βJ + sinh2(βµB)}1/2]{e−4βJ + sinh2(βµB)}1/2 , (12) from which the specific heat can be derived, and M(B,T) ≡− ∂A ∂B  T = Nµsinh(βµB) {e−4βJ + sinh2(βµB)}1/2 , (13) from which the susceptibility can be derived. Right away we note that, as B →0,M (for all finite β)→0. This rules out the possibility of spontaneous magnetization, and hence of a phase transition, at any finite temperature T. Of course, at T = 0, M (for any value of B) is equal to the saturation value Nµ, which implies perfect order in the system. This means that there is, after all, a phase transition at a critical temperature Tc, which coincides with absolute zero! Figure 13.4 shows the degree of magnetization, M, of the lattice as a function of the parameter (βµB) for different values of (βJ). For J = 0, we have the paramagnetic result M = Nµtanh(βµB); compare to equation (3.9.27). A positive J enhances magnetization and, in turn, leads to a faster approach toward saturation. As βJ →∞, the magnetiza-tion curve becomes a step function — indicative of a singularity at T = 0. The low-field susceptibility of the system is given by the initial slope of the magnetization curve; one obtains χ0(T) = Nµ2 kT e2J/kT, (14) 2 1 1 1 0 1 2 (B) M(B, T )/N J 0 J  0 FIGURE 13.4 The degree of magnetization of an Ising chain as a function of the parameter (βµB). 13.2 The Ising model in one dimension 479 which diverges as T →0. It should be noted that the singularity here is not of the power-law type; it is exponential instead. The zero-field energy and the zero-field specific heat of the system follow from equa-tion (12); one gets U0(T) = −NJ tanh(βJ) (15) and C0(T) = Nk(βJ)2sech2(βJ). (16) Figure 13.5 shows the variation of the specific heat C0 as a function of temperature. Although it passes through a maximum, C0 is a smooth function of T, vanishing as T →0. Note that equations (15) and (16) are identical to the corresponding equations, (12.6.29) and (12.6.30), of the Bethe approximation, with coordination number 2, for which Tc = 0. It turns out that for a one-dimensional chain the Bethe approximation, in fact, yields exact results; for a fuller demonstration of this, see Problem 13.3. At this stage it seems instructive to express the free energy of the system, near its crit-ical point, in a scaled form, as in Section 12.10. Unfortunately, there is a problem here. Since Tc = 0, the conventional definition, t = (T −Tc)/Tc, does not work. A closer look at equations (11) and (14), however, suggests that we may adopt instead the definition t = e−pJ/kT (p > 0) (17) so that, as T →Tc,t →0 as desired while for temperatures close to Tc,t is much less than unity. The definition of h remains the same, namely µB/kT. The free energy function (A + NJ)/NkT then takes the form ψ(s)(t,h) = −ln[coshh + (t4/p + sinh2 h)1/2] (18a) ≈−(t4/p + h2)1/2 (t,h ≪1), (18b) which may be written in the scaled form ψ(s)(t,h) ≈t2/pf (h/t2/p). (19) 0 0 0.5 1 2 kT/J C0(T )/Nk FIGURE 13.5 The zero-field specific heat of an Ising chain as a function of temperature. 480 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models At the same time, equation (14) becomes χ0(T) (Nµ2/kT) ≈t−2/p. (20) Comparing these results with the scaling formulation of Section 12.10, we infer that for this model α = 2 −2/p, 1 = 2/p, and γ = 2/p, (21) in conformity with the exponent relation (12.10.14b). Note that the exponents β and δ for this model cannot be defined in the normal, conventional sense. One may, however, write equation (13) in the form m = sinhh/(t4/p + sinh2 h)1/2 (22a) ≈h/(t4/p + h2)1/2 (t,h ≪1) (22b) = −t0f ′(h/t2/p), (22c) suggesting that β may formally be taken as zero. At the same time, since m|t=0 = 1, (23) which is ∼h0, the exponent δ may formally be taken as infinite. We now study spin–spin correlations in the Ising chain. For this, we set B = 0 but at the same time generalize the interaction parameter J to become site-dependent (the reason for which will become clear soon). The partition function of the system is then given by QN(T) = X σ1=±1 ··· X σN =±1 Y i eβJiσiσi+1; (24) compare to equation (3a). With B = 0, it is simpler to work with an open chain, which has only (N −1) nearest-neighbor pairs; the advantage of this choice is that in the summand of (24) the variables σ1 and σN appear only once! A summation over either of these can be carried out easily; doing this over σN, we have X σN =±1 eβJN−1σN−1σN = 2cosh(βJN−1σN−1) = 2cosh(βJN−1), (25) regardless of the sign of σN−1. We thus obtain the recurrence relation QN(T;J1,...,JN−1) = 2cosh(βJN−1)QN−1(T;J1,...,JN−2). (26) 13.2 The Ising model in one dimension 481 By iteration, we get QN(T) = N−1 Y i=1 {2cosh(βJi)} X σ1=±1 1 = 2N N−1 Y i=1 cosh(βJi), (27) so that 1 N lnQN(T) = ln2 + 1 N N−1 X i=1 lncosh(βJi), (28) which may be compared with equation (9) — remembering that, in the absence of the field, λ1 = 2cosh(βJ). We are now ready to calculate the correlation function, g(r), of the Ising chain. It is straightforward to see from equation (24) that σkσk+1 = 1 QN  1 β ∂ ∂Jk  QN =  1 β ∂ ∂Jk  lnQN. (29) Substituting from equation (28), and remembering that σ k = 0 at all finite temperatures, we obtain for the nearest-neighbor correlation function gk(n.n.) = σkσk+1 = tanh(βJk). (30) For a pair of spins separated by r lattice constants, we get gk(r) = σkσk+r = (σkσk+1)(σk+1σk+2)...(σk+r−1σk+r) (since all σ 2 i = 1) = 1 QN  1 β ∂ ∂Jk  1 β ∂ ∂Jk+1  ···  1 β ∂ ∂Jk+r−1  QN = k+r−1 Y i=k tanh(βJi). (31) Reverting to a common J, we obtain the desired result g(r) = tanhr(βJ), (32) which may be written in the standard form g(r) = e−r/ξ, with ξ = lncoth(βJ) −1 . (33a, b) For βJ ≫1, ξ ≈1 2e2βJ, (34) 482 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models which diverges as T →0. In terms of the variable t, as defined in equation (17), ξ ∼t−2/p (t ≪1), (35) giving ν = 2/p. And since our g(r) does not contain any power of r, we infer that (d −2 + η) = 0 — giving η = 1; one may check that the same result follows from equation (12.12.10) or (12.12.11). In passing, we note that, regardless of the choice of the number p in defining t, we have for this model γ = ν = 2 −α. (36) We further note that, since d = 1 here, the hyperscaling relation, dν = 2 −α, is also obeyed. Finally, we observe that expression (33b) for ξ is in conformity with the general result ξ−1 = ln(λ1/λ2), (37) where λ1 is the largest eigenvalue of the transfer matrix P of the problem and λ2 the next largest; for a derivation of this result, see Section 5.3 of Yeomans (1992). In our case, λ1 = 2cosh(βJ) and λ2 = 2sinh(βJ), see equation (8) with B = 0, and hence expression (33b) for ξ. 13.3 The n-vector models in one dimension We now consider a generalization of the Ising chain in which the spin variable σ i is an n-dimensional vector of magnitude unity, whose components can vary continuously over the range −1 to +1; in contrast, the Ising spin σi could have only a discrete value, +1 or −1. We shall see that the vector models (with n ≥2), while differing quantitatively from one another, differ rather qualitatively from the scalar models (for which n=1). While some of these qualitative differences will show up in the present study, more will become evident in higher dimensions. Here we follow a treatment due to Stanley (1969a,b) who first solved this problem for general n. Once again we employ an open chain composed of N spins constituting (N −1) nearest-neighbor pairs. The Hamiltonian of the system, in zero field, is given by HN{σ i} = − N−1 X i=1 Jiσ i · σ i+1. (1) We assume our spins to be classical, so we do not have to worry about the commutation properties of their components. And since the components σiα(α = 1,...,n) of each spin vector σ i are now continuous variables, the partition function of the system will involve integrations, rather than summations, over these variables. Associating equal a priori 13.3 The n-vector models in one dimension 483 probabilities with solid angles of equal magnitude in the n-dimensional spin-vector space, we may write QN = Z d1 (n) ··· dN (n) N−1 Y i=1 eβJiσ i·σ i+1, (2) where (n) is the total solid angle in an n-dimensional space; see equation (7b) of Appendix C, which gives (n) = 2πn/2/ 0(n/2). (3) We first carry out integration over σ N, keeping the other σ i fixed. The relevant integral to do is 1 (n) Z eβJN−1σ N−1·σ N dN. (4) For σ N we employ spherical polar coordinates, with polar axis in the direction of σ N−1, while for dN we use expression (9) of Appendix C. Integration over angles other than the polar angle θ yields a factor of 2π(n−1)/2/ 0{(n −1)/2}. (5) The integral over the polar angle is π Z 0 eβJN−1 cosθ sinn−2 θ dθ = π1/20{(n −1)/2} ( 1 2βJN−1)(n−2)/2 I(n−2)/2(βJN−1), (6) where Iµ(x) is a modified Bessel function; see Abramowitz and Stegun (1964), formula 9.6.18. Combining (3), (5), and (6), we obtain for (4) the expression 0(n/2) ( 1 2βJN−1)(n−2)/2 I(n−2)/2(βJN−1), (7) regardless of the direction of σ N−1. By iteration, we get QN = N−1 Y i=1 0(n/2) ( 1 2βJi)(n−2)/2 I(n−2)/2)(βJi); (8) the last integration, over d1, gave simply a factor of unity. Expression (8) is valid for all n — including n = 1, for which it gives: QN = Q i cosh(βJi). This last result differs from expression (13.2.27) by a factor of 2N; the reason for this dif-ference lies in the fact that the QN of the present study is normalized to go to unity as the βJi go to zero [see equation (2)] whereas the QN of the preceding section, being a sum over 2N discrete states [see equation (13.2.24)] goes to 2N instead. This difference is important 484 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models in the evaluation of the entropy of the system; it is of no consequence for the calculations that follow. First of all we observe that the partition function QN is analytic at all β — except possibly at β = ∞where the singularity of the problem is expected to lie. Thus, no long-range order is expected to appear at any finite temperature T — except at T = 0 where, of course, perfect order is supposed to prevail. In view of this, the correlation function for the nearest-neighbor pair (σ k,σ k+1) is simply σ k · σ k+1 and is given by, see equations (2) and (8), gk(n.n.) = 1 QN  1 β ∂ ∂Jk  QN = In/2(βJk) I(n−2)/2(βJk). (9) The internal energy of the system turns out to be U0 ≡−∂ ∂β (lnQN) = − N−1 X i=1 Ji In/2(βJi) I(n−2)/2(βJi); (10) not surprisingly, U0 is simply a sum of the expectation values of the nearest-neighbor interaction terms −Jiσ i · σ i+1, which is identical to a sum of the quantities −Jigi(n.n.) over all nearest-neighbor pairs in the system. The calculation of gk(r) is somewhat tricky because of the vector character of the spins, but things are simplified by the fact that we are dealing with a one-dimensional system only. Let us consider the trio of spins σ k,σ k+1 and σ k+2, and suppose for a moment that our spins are three-dimensional vectors; our aim is to evaluate σ k · σ k+2. We choose spherical polar coordinates with polar axis in the direction of σ k+1; let the direction of σ k be defined by the angles (θ0,φ0) and that of σ k+2 by (θ2,φ2). Then σ k · σ k+2 = cosθ(k,k + 2) = cosθ0 cosθ2 + sinθ0 sinθ2 cos(φ0 −φ2). (11) Now, with σ k+1 fixed, spins σ k and σ k+2 will orient themselves independently of one another because, apart from σ k+1, there is no other channel of interaction between them. Thus, the pairs of angles (θ0,φ0) and (θ2,φ2) vary independently of one another; this makes cos(φ0 −φ2) = 0 and cosθ0 cosθ2 = cos(θ0) cos(θ2). It follows that σ k · σ k+2 = σ k · σ k+1 σ k+1 · σ k+2. (12) Extending this argument to general n and to a segment of length r, we get gk(r) = k+r−1 Y i=k gi(n.n.) = k+r−1 Y i=k In/2(βJi)/I(n−2)/2(βJi). (13) 13.3 The n-vector models in one dimension 485 With a common J, equations (9), (10), and (13) take the form g(n.n.) = In/2(βJ)/I(n−2)/2(βJ), (14) U0 = −(N −1)J In/2(βJ)/I(n−2)/2(βJ) (15) and g(r) = {In/2(βJ)/I(n−2)/2(βJ)}r. (16) The last result here may be written in the standard form e−r/ξ, with ξ = [ln{I(n−2)/2(βJ)/In/2(βJ)}]−1. (17) For n = 1, we have: I1/2(x)/I−1/2(x) = tanh x; the results of the preceding section are then correctly recovered. For a study of the low-temperature behavior, where βJ ≫1, we invoke the asymptotic expansion Iµ(x) = ex √ (2πx) " 1 −4µ2 −1 8x + ··· # (x ≫1), (18) with the result that g(n.n.) ≈1 −n −1 2βJ , (14a) U0 ≈−(N −1)J  1 −n −1 2βJ  (15a) and ξ ≈2βJ n −1 ∼T−1. (17a) Clearly, the foregoing results hold only for n ≥2; for n = 1, the asymptotic expansion (18) is no good because it yields the same result for µ = 1 2 as for µ = −1 2. In that case one is obliged to use the closed form result, g(n.n.) = tanh(βJ), which for βJ ≫1 gives g(n.n.) ≈1 −2e−2βJ (14b) U0 ≈−(N −1)J[1 −2e−2βJ] (15b) and ξ ≈1 2e2βJ, (17b) 486 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models in complete agreement with the results of the preceding section. For completeness, we have for the low-temperature specific heat of the system C0 ≈(N −1)    1 2(n −1)k for n ≥2 (19a) 4k(βJ)2e−2βJ for n = 1. (19b) The most obvious distinction between one-dimensional models with continuous sym-metry (n ≥2) and those with discrete symmetry (n = 1) is in relation to the nature of the singularity at T = 0. While in the case of the former it is a power-law singularity, with critical exponents1 α = 1, ν = 1, η = 1, and hence γ = 1, (20) in the case of the latter it is an exponential singularity. Nevertheless, by introducing the temperature parameter t = e−pβJ, see equation (13.2.17), we converted this exponential singularity in T into a power-law singularity in t, with α = 2 −2/p, γ = ν = 2/p, η = 1. (21) However, the inherent arbitrariness in the choice of the number p left an ambiguity in the values of these exponents; we now see that by choosing p = 2 we can bring exponents (21) in line with (20). Next we observe that the critical exponents (20) for n ≥2 turn out to be independent of n — a feature that seems peculiar to situations where Tc = 0. In higher dimensions, where Tc is finite, the critical exponents do vary with n; for details, see Section 13.7. In any case, the amplitudes always depend on n. In this connection we note that, since each of the N spins comprising the system has n components, the total number of degrees of free-dom in this problem is Nn. It seems appropriate that the extensive quantities, such as U0 and C0, be divided by Nn, so that they are expressed as per degree of freedom. A look at equation (15a) then tells us that our parameter J must be of the form nJ′, so that in the thermodynamic limit U0 Nn ≈−J′ + n −1 2n kT (22) and, accordingly, C0 Nn ≈n −1 2n k. (23) 1Note that, since Tc = 0 here, the assertion that the free energy function ψ ≡(A/NkT) ∼(T −Tc)2−α implies that, near T = Tc in the present case, A ∼T3−α; accordingly, the specific heat C0 ∼T2−α. Comparison with expression (19a) would be inappropriate because C0, in the limit T →0, cannot be nonzero; the result quoted in (19a) is an artifice of the model considered, which must somehow be “subtracted away.” The next approximation yields a result proportional to the first power in T, giving α = 1. For a parallel situation, see equation (13.5.35) for the spherical model (which pertains to the case n = ∞). 13.3 The n-vector models in one dimension 487 Equation (17a) then becomes ξ ≈ 2n n −1 J′ kT . (24) Note that the amplitudes appearing in equations (22) through (24) are such that the limit n →∞exists; this limit pertains to the so-called spherical model, which will be studied in Section 13.5. Figure 13.6 shows the normalized energy u0(= U0/NnJ′) as a function of temperature for several values of n, including the limiting case n = 1. We note that u0 (which, in the thermodynamic limit, is equal and opposite to the nearest-neighbor correlation function g(n.n.)) increases monotonically with n — implying that g(n.n.), and hence g(r), decrease monotonically as n increases. This is consistent with the fact that the correlation length ξ also decreases as n increases; see equation (24). The physical reason for this behavior is that an increase in the number of degrees of freedom available to each spin in the system effectively diminishes the degree of correlation among any two of them. Another feature emerges here that distinguishes vector models (n ≥2) from the scalar model (n = 1); this is related to the manner in which the quantity u0 approaches its ground-state value −1. While for n = 1, the approach is quite slow — leading to a vanishing 0.0 20.2 20.4 n 51 2 3 4 6 8 20.6 20.8 21.0 0.0 0.5 1.0 1.5 2.0 2.5 u0 kT/J9 FIGURE 13.6 The normalized energy u0(= U0/NnJ′) of a one-dimensional chain as a function of the temperature parameter kT/J′ for several values of n (after Stanley, 1969a,b). Note that for this classical model, the slopes as T →0 are given by the equipartition theorem for n −1 degrees of freedom per spin. 488 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models specific heat, see equations (15b) and (19b) — for n ≥2, the approach is essentially linear in T, leading to a finite specific heat; see equations (15a) and (19a). This last result violates the third law of thermodynamics, according to which the specific heat of a real system must go to zero as T →0. The resolution of this dilemma lies in the fact that the low-lying states of a system with continuous symmetry (n ≥2) are dominated by long-wavelength excitations, known as Goldstone modes, which in the case of a magnetic system assume the form of “spin waves,” characterized by a particle-like spectrum: ω(k) ∼k2. The very low-temperature behavior of the system is primarily governed by these modes, and the thermal energy associated with them is given by Utherma1 ∼ Z ℏω exp(ℏω/kBT) −1kd−1dk ∼T(d+2)/2; (25) this results in a specific heat∼Td/2, which indeed is consistent with the third law. For a general account of the Goldstone excitations, see Huang (1987); for their role as “spin waves” in a magnetic system, see Plischke and Bergersen (1989). For further information on one-dimensional models, see Lieb and Mattis (1966) and Thompson (1972a,b). 13.4 The Ising model in two dimensions As stated earlier, Ising (1925) himself carried out a combinatorial analysis of the one-dimensional model and found that there was no phase transition at a finite temperature T. This led him to conclude, erroneously though, that his model would not exhibit a phase transition in higher dimensions either. In fact, it was this “supposed failure” of the Ising model that motivated Heisenberg to develop, in 1928, the theory of ferromagnetism based on a more sophisticated interaction among the spins; compare the Heisenberg interac-tion (12.3.6) with the Ising interaction (12.3.7). It was only after some exploitation of the Heisenberg model that people returned to investigate the properties of the Ising model. The first exact, quantitative result for the two-dimensional Ising model was obtained by Kramers and Wannier (1941) who successfully located the critical temperature of the system. They were followed by Onsager (1944) who derived an explicit expression for the free energy in zero field and thereby established the precise nature of the specific-heat singularity. These authors employed the transfer matrix method that was introduced in Section 13.2 to solve the corresponding one-dimensional problem; its application to the two-dimensional model, even in the absence of the field, turned out to be an extremely difficult task. Although some of these difficulties were softened by subsequent treatments due to Kaufman (1949) and to Kaufman and Onsager (1949), it seemed very natural to look for other simpler approaches. One such approach was developed by Kac and Ward (1952), later refined by Potts and Ward (1955), in which combinatorial arguments were used to express the partition func-tion of the system as the determinant of a certain matrix A. This method throws special 13.4 The Ising model in two dimensions 489 light on the “topological conditions” that give rise to an exact solution in two dimensions but are clearly absent in three dimensions; a particularly lucid account of this method has been given by Baker (1990). In 1960 Hurst and Green introduced yet another approach to investigate the Ising problem; this involved the use of “triangular arrays of quantities closely related to antisymmetric determinants” and became rightly known as the method of Pfaffians. This method applies rather naturally to the study of “configurations of dimer molecules on a given lattice” which, in turn, is closely related to the Ising problem; for details, see Kasteleyn (1963), Montroll (1964), and Green and Hurst (1964). A pedagog-ical account of the approach through Pfaffians is given in Thompson (1972b), where a comprehensive treatment of the original, algebraic approach can also be found. Another combinatorial solution, which is generally regarded as the simplest, was obtained by Vdovichenko (1965) and by Glasser (1970), and is readily accessible in Stanley (1971). For an exhaustive account of the two-dimensional Ising model, see McCoy and Wu (1973). We analyze this problem with the help of a combinatorial approach assisted, from time to time, by a graphical representation. The zero-field partition function of the system is given by the familiar expression Q(N,T) = X {σi} Y n.n. eKσiσj (K = J/kT). (1) Our first step consists of carrying out a high-temperature and a low-temperature expan-sion of the partition function and establishing an intimate relation between the two. High-temperature expansion Since the product (σiσj) can only be +1 or −1, we may write eKσiσj = coshK + σiσj sinhK = coshK(1 + σiσjv), v = tanhK. (2) The product over all nearest-neighbor pairs then takes the form Y n.n. eKσiσj = (coshK)N Y n.n. (1 + σiσjv), (3) N being the total number of nearest-neighbor pairs on the lattice; for a lattice with peri-odic boundary conditions, N = 1 2qN where q is the coordination number. The partition function may then be written as Q(N,T) = (coshK)N X σ1=±1 ··· X σN =±1  1 + v X (i,j) σiσj + v2X (i,j) (i,j)̸=(k,l) X (k,l) σiσjσkσl + ...  . (4) Now we represent each product (σiσj) appearing in (4) by a “bond connecting sites i and j on the given lattice”; then, each coefficient of vr in the expansion would be represented by a “graph consisting of r different bonds on the lattice.” Figure 13.7 shows all possible graphs, with r = 1 and 2, on a square lattice. Notice that in each case we have some of 490 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models (i) (ii)a (ii)b FIGURE 13.7 Graphs with r = 1 and r = 2 on a square lattice. Graph (i) is for r = 1. The r = 2 graphs are of two types, ones that do not include a common site as in (ii)a and ones that do include a common site as in (ii)b. r 54 r 56 r 5 8 FIGURE 13.8 Examples of closed graphs with r bonds on a square lattice. the σi appearing only once in the term, which makes all these terms vanish on summation over {σi}. The same is true for r = 3. Only when we reach r = 4 do we receive a nonvanish-ing contribution from terms of the type (σiσjσjσkσkσlσlσi) ≡1 which, on summation over {σi}, yield a contribution of 2N each. It is obvious that a nonvanishing term corresponds to a graph in which each vertex is met by an even number of bonds — making the graph necessarily a closed one; see Figure 13.8, where some other closed graphs are also shown. In view of these observations, expression (4) may be written as Q(N,T) = 2N(coshK)N X r n(r)vr [n(0) = 1], (5) where n(r) is the number of graphs that can be drawn on the given lattice using r bonds such that each vertex of the graph is met by an even number of bonds. For simplic-ity, we shall refer to these graphs as closed graphs. Our problem thus reduces to one of enumerating such graphs on the given lattice. Since v = tanh(J/kT), the higher the temperature the smaller the v. Expansion (5) is, therefore, particularly useful at higher temperatures (even though it is exact for all T). As an illustration, we apply this result to a one-dimensional Ising chain. In the case of an open chain, no closed graphs are possible, so all we get from (5) is the term with r = 0; with N = N −1, this gives Q(N,T) = 2N(coshK)N−1, (6) 13.4 The Ising model in two dimensions 491 which agrees with our previous result (13.2.27). In the case of a closed chain, we do have a closed graph — the one with r = N; we now get (with N = N) Q(N,T) = 2N(coshK)N[1 + vN] = 2N[(coshK)N + (sinhK)N], (7) which agrees with expression (13.2.5), with (λ1)B=0 = 2coshK and (λ2)B=0 = 2sinhK. Low-temperature expansion We start with the observation that the ground state of the system consists of all spins aligned in the same direction, with the total energy E0 = −JN . As one spin is flipped, q unlike nearest-neighbor pairs are created at the expense of like ones, and the energy of the system increases by an amount 2qJ. It seems appropriate, therefore, that the Hamiltonian of the system be written in terms of the number, N+−, of unlike nearest-neighbor pairs in the lattice, that is, H(N+−) = −J(N++ + N−−−N+−) = −J(N −2N+−). (8) The partition function of the system may then be written as Q(N,T) = eKN X r m(r)e−2Kr [m(0) = 1], (9) where m(r) denotes the “number of distinct ways in which the N spins of the lattice can be so arranged as to yield r unlike nearest-neighbor pairs.” It is obvious that the first nonzero term in (9), after the one with r = 0, would be the one with r = q. A graphical representation of the number m(r) is straightforward. Referring to Figure 13.9, which pertains to a square lattice, we see that each term in expansion (9) can be associated with a closed graph that cordons off region(s) of “up” spins from those of “down” spins, the perimeter of the graph being precisely the number of unlike nearest-neighbor pairs in the lattice for that particular configuration. Our problem then reduces to one of enumerating closed graphs, of appropriate perimeters, that can be drawn on the given lattice. r 54 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 r 5 6 1 1 1 1 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 1 1 1 1 1 1 r 5 8 1 1 1 1 1 1 1 1 1 1 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 FIGURE 13.9 Graphs cordoning off regions of “up” spins from those of “down” spins, with r unlike nearest-neighbor pairs. 492 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models Now, since expansion (9) is a power series in the variable e−2K that increases as T increases, this expansion is particularly useful at lower temperatures (even though it is exact for all T). We shall now establish an important relation between the coefficients appearing in expansion (5) and the ones appearing in (9). The duality transformation To establish the desired relation we construct a lattice dual to the one given. By definition, we draw right bisectors of all the bonds in the lattice, so that the points of intersection of these bisectors become the sites of the new lattice. The resulting lattice may not be similar in structure to the one we started with; for instance, while the dual of a square lattice is itself a square lattice, the dual of a triangular lattice (q = 6) is, in fact, a honeycomb lattice (q = 3), and vice versa; see Figures 13.10 and 13.11. The argument now runs as follows: We start with the given lattice on which one of the n(r) closed graphs, with r bonds, is drawn and construct the lattice dual to this one, placing spins of one sign on the sites inside this graph and spins of opposite sign on the sites outside. Then this graph represents precisely a configuration with r unlike nearest-neighbor pairs in the dual lattice and hence qualifies to be counted as one of the m(r) graphs on the dual lattice. Conversely, if we start with one of the m(r) graphs, of perimeter r, representing a configuration with r unlike nearest-neighbor pairs in the original lattice and go through the process of constructing the dual lattice, then this graph will qualify to be one of the n(r) closed graphs, with r bonds, on the dual lattice. In fact, there is a one-to-one correspondence between graphs of one kind on the given lattice and graphs of the other kind on the dual lattice; compare Figure 13.8 with Figure 13.9. It follows that n(r) = mD(r) and m(r) = nD(r), (10a, b) FIGURE 13.10 A square lattice and its dual (which is also square). 13.4 The Ising model in two dimensions 493 FIGURE 13.11 A honeycomb lattice (q = 3) and its dual, which is triangular (q = 6). where the suffix D refers to the dual lattice. With relations (10) established, we go back to equation (9) and introduce another temperature variable K ∗(= J/kT∗) such that tanhK ∗= e−2K ; (11) note that equation (11) can also be written in the symmetrical form sinh(2K)sinh(2K ∗) = 1. (12) Substituting (10b) and (11) into (9), we get Q(N,T) = eKN X r nD(r)v∗r, v∗= tanhK ∗. (13) At the same time we apply equation (5) to the dual lattice at temperature T∗, to get QD(ND,T∗) = 2ND(coshK ∗)N X r nD(r)v∗r, (14) where ND = qN/qD; see again Figure 13.11. Comparing (13) and (14), we arrive at the desired relation Q(N,T) = 2−ND(sinhK ∗coshK ∗)−N /2QD(ND,T∗), (15) 494 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models which relates the partition function of the given lattice at temperature T to that of the dual lattice at temperature T∗. Equation (15) constitutes the so-called duality transformation. Location of the critical point For a square lattice, which is self-dual, there should be no distinction between Q and QD. With q = 4, and hence N = 2N, equation (15) becomes Q(N,T) = [sinh(2K ∗)]−NQ(N,T∗), (16) which may also be written as [sinh(2K)]−N/2Q(N,T) = [sinh(2K ∗)]−N/2Q(N,T∗). (17) It will be noted from equation (11) or (12) that as T →∞, T∗→0 and as T →0, T∗→ ∞; equation (17), therefore, establishes a one-to-one correspondence between the high-temperature and the low-temperature values of the partition function of the lattice. It then follows that if there exists a singularity in the partition function at a particular temperature Tc, there must exist an equivalent singularity at the corresponding temperature T∗ c . And in case we have only one singularity, as indeed follows from one of the theorems of Yang and Lee (1952), it must exist at a temperature Tc such that T∗ c = Tc. The critical temperature of the square lattice is, therefore, given by the equation, see formula (12), sinh(2Kc) = 1, (18) which gives Kc = 1 2 sinh−1 1 = 1 2 ln(√2 + 1) = 1 2 lncot(π/8) ≃0.4407. (19) For comparison, we note that for the same lattice the Bragg–Williams approximation gave Kc = 0.25 while the Bethe approximation gave Kc = 1 2 ln2 ≃0.3466. The situation for other lattices such as the triangular or the honeycomb, which are not self-dual, is complicated by the fact that the functions Q and QD in equation (15) are not the same. One then needs another trick — the so-called star–triangle transformation — which was first alluded to by Onsager (1944) in his famous paper on the solution of the square lattice problem but was written down explicitly by Wannier (1945); for details, see Baxter (1982). Unlike the duality transformation, this one establishes a relation between a high-temperature model on the triangular lattice and again a high-temperature model on the honeycomb lattice, and so on. Combining the two transformations, one can elim-inate the dual lattice altogether and obtain a relation between a high-temperature and a low-temperature model on the same lattice. The location of the critical point is then straightforward; one obtains for the triangular lattice (q = 6) Kc = 1 2 sinh−1 1 √3 ≃0.2747, (20) 13.4 The Ising model in two dimensions 495 and for the honeycomb lattice (q = 3) Kc = 1 2 sinh−1 √3 ≃0.6585. (21) The numerical values of Kc, given by equations (19) through (21), reinforce the fact that higher coordination numbers help propagate long-range order in the system more effec-tively and hence raise the critical temperature Tc. The partition function and the specific-heat singularity The partition function of the Ising model on a square lattice is given by, see references cited at the beginning of this section, 1 N lnQ(T) = ln{21/2 cosh(2K)} + 1 π π/2 Z 0 dφ ln{1 + √ (1 −κ2 sin2 φ)}, (22) where κ = 2sinh(2K)/cosh2(2K). (23) Differentiating (22) with respect to−β, one obtains for the internal energy per spin 1 N U0(T) = −2J tanh(2K) + 1 π  κ dκ dβ  × π/2 Z 0 dφ sin2 φ {1 + √ (1 −κ2 sin2 φ)} √ (1 −κ2 sin2 φ) . (24) Rationalizing the integrand, the integral in (24) can be written as 1 κ2 n −π 2 + K1(κ) o , (25) where K1(κ) is the complete elliptic integral of the first kind, κ being the modulus of the integral: K1(κ) = π/2 Z 0 dφ √ (1 −κ2 sin2 φ) . (26) Now, a logarithmic differentiation of (23) with respect to β gives 1 κ dκ dβ = 2J{coth(2K) −2tanh(2K)}. (27) 496 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models Substituting these results into (24), we obtain 1 N U0(T) = −J coth(2K)  1 + 2κ′ π K1(κ)  , (28) where κ′ is the complementary modulus: κ′ = 2tanh2(2K) −1 (κ2 + κ′2 = 1). (29) Figure 13.12 shows the variation of the moduli κ and κ′ with the temperature parame-ter (kT/J) = K −1. We note that, while κ is always positive, κ′ can be positive or negative; actually, κ lies between 0 and 1 while κ′ lies between −1 and 1. At the critical point, where sinh(2Kc) = 1 and hence K −1 c ≃2.269, the moduli κ and κ′ are equal to 1 and 0, respectively. To determine the specific heat of the lattice, we differentiate (28) with respect to temperature. In doing so, we make use of the following results: dκ dβ = −κ′ κ dκ′ dβ , dκ′ dβ = 8J tanh(2K){1 −tanh2(2K)} (30) and dK1(κ) dκ = 1 κ′2κ {E1(κ) −κ′2K1(κ)}, (31) where E1(κ) is the complete elliptic integral of the second kind: E1(κ) = π/2 Z 0 √ (1 −κ2 sin2 φ)dφ. (32) 1 , 9 kT /J 0 2 4 21 9 FIGURE 13.12 Variation of the moduli κ and κ′ with (kT/J). 13.4 The Ising model in two dimensions 497 We finally obtain 1 Nk C0(T) = 2 π {K coth(2K)}2 h 2{K1(κ) −E1(κ)} −(1 −κ′) nπ 2 + κ′K1(κ) oi . (33) Now, the elliptic integral K1(κ) has a singularity at κ = 1 (i.e., at κ′ = 0), in the neighbor-hood of which K1(κ) ≈ln{4/|κ′|} and E1(κ) ≈1. (34) Accordingly, the specific heat of the lattice displays a logarithmic singularity at a tempera-ture Tc, given by the condition: κc = 1 (or κ′ c = 0), which is identical to (18). In the vicinity of the critical point, equation (33) reduces to 1 Nk C0(T) ≃8 π K 2 c h ln{4/|κ′|} −  1 + π 4 i ; (35) at the same time, the parameter κ′ reduces to, see equation (30), κ′ ≃2√2Kc  1 −T Tc  . (36) The specific heat singularity is, therefore, given by 1 Nk C0(T) ≃8 π K 2 c  −ln 1 −T Tc +  ln √2 Kc  −  1 + π 4  ≃−0.4945ln 1 −T Tc + const., (37) signaling a logarithmic divergence at T = Tc. Figures 13.13 and 13.14 show the temperature dependence of the internal energy and the specific heat of the square lattice, as given by the Onsager expressions (28) and (33); for comparison, the results of the Bragg–Williams approximation and of the Bethe approx-imation (with q = 4) are also included. The specific-heat singularity, given correctly by the Onsager expression (37), is seen as a (logarithmic) peak in Figure 13.14, which differs markedly from the jump discontinuity predicted by the mean field theory. We conclude that the critical exponent α = α′ = 0(log). In passing, we note that the internal energy of the lattice is continuous at the critical point, having a value of − √ 2J per spin and an infinite, positive slope; needless to say, the continuity of the internal energy implies that the transition takes place without any latent heat. Other properties We now consider the temperature dependence of the order parameter (i.e., the sponta-neous magnetization), of the lattice. An exact expression for this quantity was first derived 498 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 00 1 2 3 4 6 1 2 U0(T )/NJ kT/J 2 2.27 2.88 FIGURE 13.13 The internal energy of a square lattice (q = 4) according to (1) the Onsager solution, (2) the Bethe approximation, and (3) the Bragg–Williams approximation. 00 2.27 2.88 4 6 3 2 1 1 2 C0(T )/Nk kT/J 2 FIGURE 13.14 The specific heat of a square lattice (q = 4) according to (1) the Onsager solution, (2) the Bethe approximation, and (3) the Bragg–Williams approximation. by Onsager (1949), though he never published the details of his derivation. The first published derivation is due to Yang (1952), who showed that L0(T) ≡ 1 NµM(0,T) = ( [1 −{sinh(2K)}−4]1/8 for T ≤Tc (38a) 0 for T ≥Tc, (38b) where K, as usual, is J/kT. In the limit T →0, L0(T) ≃1 −2exp(−8J/kT), (39) which implies a very slow variation with T. On the other hand, in the limit T →Tc−, L0(T) ≈  8√2Kc  1 −T Tc 1/8 ≃1.2224  1 −T Tc 1/8 , (40) 13.4 The Ising model in two dimensions 499 00 0.5 1.0 0.5 1.0 1 2 3 (T/Tc) L0(T ) FIGURE 13.15 The spontaneous magnetization of a square lattice (q = 4) according to (1) the Onsager solution, (2) the Bethe approximation, and (3) the Bragg–Williams approximation. which indicates a very fast variation with T. The detailed dependence of L0 on T is shown in Figure 13.15; for comparison, the results of the Bragg–Williams approximation and the Bethe approximation are also included. We infer that the critical exponent β for this model is 1 8, which is very different from the mean field value of 1 2. Onsager also calculated the correlation length ξ of the lattice, which showed that the critical exponent ν = ν′ = 1 — in sharp contrast to the classical value of 1 2. Finally, he set up calculations for the correlation function g(r) from which one could infer that the exponent η = 1 4, again in disagreement with the classical value of zero. Precise asymptotic expres-sions for the correlation function in different regimes of temperature were derived by later authors (Fisher, 1959; Kadanoff, 1966a; Au-Yang and Perk, 1984): g(r) ≈{4(Kc −K)}1/4 23/8(πr/ξ)1/2 e−r/ξ, ξ = {4(Kc −K)}−1 (41) for T > Tc, g(r) ≈{4(K −Kc)}1/4 221/8π(r/ξ)2 e−2r/ξ, ξ = {4(K −Kc)}−1 (42) for T < Tc, and g(r) ≈21/12 exp{3ζ ′(−1)} r1/4 (43) at T = Tc; in the last expression, ζ ′(x) denotes the derivative of the Riemann zeta function. We note from expressions (41) and (42) that the correlation length ξ, while diverging at T = Tc, is finite on both sides of the critical point. This feature is peculiar to the scalar model (n = 1) only, for in the case of vector models (n ≥2), ξ turns out to be infinite at all T ≤Tc. 500 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models The zero-field susceptibility of this system has also been worked out (see Barouch et al., 1973; Tracy and McCoy, 1973; Wu et al., 1976); asymptotically, one finds that χ0 ≈Nµ2 kTc × ( C+t−7/4 for t ≳0 (44a) C−|t|−7/4 for t ≲0, (44b) where t, as usual, is (T −Tc)/Tc while the constants C+ and C−are about 0.96258 and 0.02554, respectively. We see that the critical exponent γ = γ ′ = 7 4, as opposed to the mean field value of 1, and the ratio C+/C−≃37.69, as opposed to the mean field value of 2. Assembling all the exponents in one place, we have for the two-dimensional Ising model α = α′ = 0(log), β = 1 8, γ = γ ′ = 7 4, ν = ν′ = 1, η = 1 4. (45) Since this model has not yet been solved in the presence of a field, a direct evaluation of the exponent δ has not been possible. Assuming the validity of the scaling relations, however, we can safely conclude that δ = 15 — again very different from the classical value of 3. All in all, the results of this section tell us very clearly, and loudly, how inadequate the mean field theory can be. Before we close this section, a few remarks seem to be in order. First of all, it may be mentioned that for the model under consideration one can also calculate the interfacial tension s, which may be defined as the “free energy associated, per unit area, with the interfaces between the domains of up spins and those of down spins”; in our analogy with the gas–liquid systems, this corresponds to the conventional surface tension σ. The corre-sponding exponent µ, that determines the manner in which s →0 as T →Tc−, turns out to be 1 for this model; see Baxter (1982). This indeed obeys the scaling relation µ = (d −1)ν, as stated in Problem 12.26. Second, we would like to point out that, while the solution of the two-dimensional Ising model was the first exact treatment that exposed the inade-quacy of the mean field theory, it was also the first to disclose the underlying universality of the problem. As discovered by Onsager himself, if the spin–spin interactions were allowed to have different strengths, J and J′, in the horizontal and vertical directions of the lattice, the specific-heat divergence at T = Tc continued to be logarithmic — independently of the ratio J′/J — even though the value of Tc itself and of the various amplitudes appearing in the final expressions were modified. A similar result for the spontaneous magnetization was obtained by Chang (1952) who showed that, regardless of the value of J′/J, the expo-nent β continued to be 1 8. Further corroborative evidence for universality came from the analysis of two-dimensional lattices other than the square one which, despite structural differences, led to the same critical exponents as the ones listed in equation (45). 13.4.A The two-dimensional Ising model on a finite lattice Phase transitions, viewed as critical phenomena, cannot occur in a finite system since a statistical mechanical model with a finite number of degrees of freedom cannot have 13.4 The Ising model in two dimensions 501 a nonanalytic partition function or free energy. Criticality occurs only in the thermody-namic limit. Since real physical systems are of finite size, the manner in which finite-size effects manifest themselves as the correlation length ξ approaches the system size is of considerable importance in understanding how critical singularities get rounded off in real systems. In this regard, the two-dimensional nearest-neighbor Ising model on a square lattice in zero field can be solved on a finite square lattice with periodic boundary conditions (Kaufman, 1949), which allows for a detailed exploration of finite-size effects, especially near the bulk critical point; see Ferdinand and Fisher (1969). Kaufman’s solution is based on a determination of all the eigenvalues of the transfer matrix. Onsager (1944) only required the largest eigenvalue since his solution was based on a strip geometry with the length of one side taken to infinity. We here consider the Ising model on a lattice with n rows and m columns with periodic boundary conditions; see Figure 13.16. Each column of n spins has 2n possible configurations, so the transfer matrix P that couples nearest-neighbor columns is a 2n × 2n matrix of Boltzmann factors with eigenvalues λα, with α = 1,2,...,2n. Just as in the case of the one-dimensional Ising model studied in Section 13.2, the partition function of a system with n rows and m columns can be written as the trace of a transfer matrix P: Qn,m(K) = Trace(Pm) = 2n X α=1 λm α , (46) where the eigenvalues of the transfer matrix fall into two classes: λα =    2sinh(2K) n/2 Pexp  1 2 ±γ0 ± γ2 ± ... ± γ2n−2  , 2sinh(2K) n/2 Pexp  1 2 ±γ1 ± γ3 ± ... ± γ2n−1  . (47) FIGURE 13.16 A finite square lattice with n = 4 rows and m = 6 columns. In view of the periodic boundary conditions, sites on the leftmost column interact with sites on the rightmost column and the bottom row interacts with the top row. 502 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models The quantity γq for 0 < q < 2n is the positive root of the equation cosh(γq) = cosh2(2K) sinh(2K) −cos πq n  , (48) while the q = 0 case is given by eγ0 = e2K tanh(K). (49) Only terms with an even number of minus signs inside the exponentials appear in the sums in equation (47), so the partition function can be written as Qn,m(K) = 1 2(2sinh(2K))nm/2 (Y1 + Y2 + Y3 + Y4), (50) where Y1 = n−1 Y q=0  2cosh m 2 γ2q+1  , (51a) Y2 = n−1 Y q=0  2sinh m 2 γ2q+1  , (51b) Y3 = n−1 Y q=0  2cosh m 2 γ2q  , (51c) Y4 = n−1 Y q=0  2sinh m 2 γ2q  ; (51d) see Kaufman (1949). This form of the partition function allows for an exact calculation of the free energy, internal energy, and specific heat on finite lattices; see Figure 13.17. The logarithmic singularity in the specific heat at the bulk critical point evolves from a specific heat peak that grows logarithmically with the system size, that is Cnm(Kc)/nmk ≈ (8K 2 c /π)ln(n) ≃0.4945ln(n); see Ferdinand and Fisher (1969). Note also that the coeffi-cient of ln(n) here is the same as the coefficient of the ln(|1 −T/Tc|) term in the bulk specific heat, as given in equation (37). The low-temperature series expansion for the partition function can be written as Qn,m(K) = e2nmK ˜ Qn,m(K), where ˜ Qn,m(K) = nm X q=0 gqx2q, (52) x = e−2K is the Boltzmann factor for a single excitation, and the coefficients gq denote the number of configurations with energy 4qJ above the ground state. The sum of the 13.4 The Ising model in two dimensions 503 64 64 32 32 16 16 8 8 4 4 2 2 2.5 2 1.5 1 0.5 0 1 2 3 4 kT/ J C Nk FIGURE 13.17 Specific heat of the two-dimensional Ising model for finite 2 × 2, 4 × 4,..., 64 × 64 lattices. The specific heat is analytic for all finite lattices. The maximum value of the specific heat grows proportional to the logarithm of the linear dimension of the lattice and the location of the maximum approaches the bulk critical temperature (denoted by the vertical line) proportional to the inverse of the linear dimension of the lattice. From Ferdinand and Fisher (1969). Reprinted with permission; copyright © 1969, American Physical Society. coefficients counts all the microstates in the system; therefore lim K→0 ˜ Qn,m(K) = nm X q=0 gq = 2nm . The coefficients gq represent the number of microstates pertaining to energy (−2nmJ + 4qJ), with the corresponding entropy being klngq. The first term in the series is g0 = 2 since there are two degenerate ground states, namely all spins up or all spins down. It is straightforward to see that only even orders in x appear in this expansion. Examples of the low-order graphs that contribute to the series 504 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models (a) (b) (c) (d) FIGURE 13.18 The lowest few excited states of the lattice. The q = 2 states (a), have a single down spin in a sea of up spins or a single up spin in a sea of down spins; these states have energy 8J above the ground state and there are g2 = 2nm configurations. The q = 3 states (b), have a pair of down spins in a sea of up spins, or vice versa; these states have energy 12J above the ground state and g3 = 4nm configurations. The q = 4 states (c) and (d), can have a single grouping of opposite spins, or a pair of isolated flipped spins; these states have energy 16J above the ground state and the total number of configurations g4 = (nm)2 + 9nm. are shown in Figure 13.18. The first few terms in the series are ˜ Qn,m(K) = 2 + (2nm)x4 + (4nm)x6 +  (nm)2 + 9nm  x8 +  4(nm)2 + 24nm  x10 + .... (53) If both n and m are even, the model’s ferromagnetic/antiferromagnetic symmetry ( J →−J and si →−si on one sublattice) gives: gq = gnm−q. Due to the self-duality of the two-dimensional square lattice, exactly the same coefficients gq also appear in the high-temperature series expansion where the expansion variable is tanhK. The probability Pq of finding an equilibrium state with energy 4qJ above the ground state is given by Pq = gqx2q ˜ Qn,m(K) , (54) and the internal energy and the heat capacity per spin are given by U NJ = −2 + 4 N N X q=0 qPq (N = nm), (55a) C Nk = 16 N  J kT 2    N X q=0 q2Pq −   N X q=0 qPq   2  . (55b) One can cast Kaufman’s solution, equation (50) and equation (51), in the form of a low-temperature expansion of the form shown in (52), thereby giving an exact determi-nation of the partition function and the equilibrium energy distribution; see Beale (1996). 13.4 The Ising model in two dimensions 505 The low-temperature series (52) can be written as ˜ Qn,m(K) = nm X q=0 gqx2q = (Z1 + Z2 + Z3 + Z4), (56) where if n is even, then Z1 = 1 2 n/2−1 Y q=0 c2 2q+1, (57a) Z2 = 1 2 n/2−1 Y q=0 s2 2q+1, (57b) Z3 = 1 2c0cn n/2−1 Y q=1 c2 2q, (57c) Z4 = 1 2s0sn n/2−1 Y q=1 s2 2q; (57d) while if n is odd, then Z1 = 1 2cn (n−3)/2 Y q=0 c2 2q+1, (58a) Z2 = 1 2sn (n−3)/2 Y q=0 s2 2q+1, (58b) Z3 = 1 2c0 (n−1)/2 Y q=1 c2 2q, (58c) Z4 = 1 2s0 (n−1)/2 Y q=1 s2 2q. (58d) The factors in equations (57) and (58) are c0 = (1 −x)m + (x(1 + x))m , (59a) s0 = (1 −x)m −(x(1 + x))m , (59b) cn = (1 + x)m + (x(1 −x))m , (59c) sn = (1 + x)m −(x(1 −x))m , (59d) c2 q = 1 2m−1       ⌊m 2 ⌋ X j=0 m!  α2 q −β2j αm−2j q (2j)!(m −2j)!   + βm   , (59e) 506 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models s2 q = 1 2m−1       ⌊m 2 ⌋ X j=0 m!  α2 q −β2j αm−2j q (2j)!(m −2j)!   −βm   , (59f) β = 2x(1 −x2), (59g) αq = (1 + x2)2 −β cos πq n  . (59h) The function ⌊z⌋denotes the largest integer less than or equal to z. The quantities c2 q and s2 q were expanded using the binomial series in order to explicitly remove all square roots that would hide the polynomial nature of the final result. A symbolic programming language can be used to numerically expand the partition function as a polynomial in the variable x in the form (52). One must set the numerical precision in the calculation to somewhat more than nmln2/ln10 decimal digits in order to determine the exact values of the integer coefficients {gq}. The numerical calculation can be checked against the low-order result (53) or with an exact enumeration of energies on small lattices. The low-temperature series for the Ising model on a 32 × 32 lattice is ˜ Q32,32(K) = 2 + 2048x4 + 4096x6 + 1057792x8 + 4218880x10 + 371621888x12 + 2191790080x14 + 100903637504x16 + 768629792768x18 + 22748079183872x20 + ··· + 4096x2042 + 2048x2044 + 2x2048, (60) where the largest coefficient is g512 = 6,342,873,169,001,916,568,766,443,273,025,000,331,593,063, 924,436,135,196,680,443,689,656,478,072,741,300,511,612, 123,900,652,711,596,311,283,701,724,071,226,144,241,851, 411,641,714,893,727,789,741,510,169,213,344,005,116,385, 197,594,692,089,556,614,547,788,150,860,200,720,413,211, 442,412,355,672,291,841,364,265,145,274,980,444,405,423, 129,672,679,584,959,498,234,944,801,613,246,300,853,599, 317,229,362,316 , (61) that is, there are about 6.342 × 10306 configurations with energy halfway between the ferromagnetic and antiferromagnetic ordered states. This single microstate comprises 3.5 percent of the 21024 total configurations of the model. The exact results for the micro-canonical entropy and the energy distribution for the 128 × 128 lattice are shown in Figures 13.19 and 13.20. These results provide excellent tests of Monte Carlo simulation 13.4 The Ising model in two dimensions 507 methods, including broad histogram methods; see Beale (1996), Wang and Landau (2001), and Landau and Binder (2009).2 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 q/nm In(gq) nm FIGURE 13.19 Microcanonical entropy per spin S/Nk = ln(gq)/nm for the two dimensional Ising model on a 128 × 128 lattice as calculated from equations (56), (57), and (59). The slope of the curve is proportional to the inverse temperature, so the state with q/nm = 1/2 represents the infinite temperature state with energy halfway between the ordered ferromagnetic and antiferromagnetic states; the largest coefficient g8192 ≃1.049 × 104930 is the number of configurations with q = nm/2 = 8192. Likewise, the states at q = 0 and q = nm represent the ferromagnetic and antiferromagnetic ground states, so the slopes of the curve diverge logarithmically in the thermodynamic limit. 0.008 0.006 0.004 0.002 0.000 0.00 0.05 0.10 0.15 0.20 0.25 q/nm Pq K 0.4 K 0.5 K Kc FIGURE 13.20 The exact energy distribution Pq for the two-dimensional Ising model on a 128 × 128 lattice for K = 0.4, K = Kc ≃0.4407, and K = 0.5. The variance of the energy distribution is proportional to the specific heat, so is largest near K = Kc. Refer to Figure 13.17. 2Mathematica code for calculating the low-temperature series coefficients for a two-dimensional Ising model, as well as the microcanonical entropies, internal energies, and specific heats for several lattice sizes can be found at www.elsevierdirect.com. 508 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 13.5 The spherical model in arbitrary dimensions In the wake of Onsager’s solution to the two-dimensional Ising problem in zero field, sev-eral attempts were made to go beyond Onsager — by solving either the three-dimensional problem in zero field or the two-dimensional problem with field. None of these attempts succeeded; the best one could accomplish was to rederive the Onsager solution by newer means. This led to the suggestion that one may instead consider certain “adaptations” of the Ising model, which may turn out to be mathematically tractable in more than two dimensions and hopefully throw some light on the problem of phase transitions in more realistic situations (where d is usually 3). One such adaptation was devised by Kac who, in 1947, considered a model in which the spin variable σi, instead of being restricted to the discrete choices −1 or +1, could vary continuously, from −∞to +∞, subject to a Gaussian probability distribution law, p(σi)dσi = (A/π)1/2e−Aσ 2 i dσi (i = 1,...,N), (1) so that σ 2 i , on an average, = 1/(2A). Clearly, for conformity with the standard practice, namely σ 2 i = 1, the constant A here should be equal to 1 2; we may, however, leave it arbi-trary for the time being. The resulting model is generally referred to as the Gaussian model, and its partition function in the presence of the field is given by the multiple integral QN = +∞ Z −∞ ··· +∞ Z −∞  A π N/2 e −AP i σ 2 i +K P n.n. σiσj+hP i σi Y i dσi (K = βJ,h = βµB). (2) The exponent in the integrand is a symmetric, quadratic function in the σi; using standard techniques, it can be diagonalized. Integrations over the (transformed) σj can then be car-ried out straightforwardly — and in any number of dimensions; for details, see Berlin and Kac (1952) or Baker (1990). One finds that for d > 2 the Gaussian model undergoes a phase transition at a finite temperature Tc which, for a simple hypercubic lattice, is determined by the condition Kc = A/d; note that, with A = 1 2, this result is precisely the one predicted by the mean field theory (with q = 2d). There are differences, though. First of all, the present model does not exhibit a phase transition at a finite temperature for d ≤2. Secondly, the critical exponents for 2 < d < 4 are nonclassical, in the sense that some of them are d-dependent, though for d > 4 they do become classical. More importantly, at temperatures below Tc, where K exceeds A/d, the integral in (2) diverges and the model breaks down! This led Kac to aban-don this model and invent a new one in which the spins were again continuous variables but subject to an overall constraint, X i σ 2 i = N, (3) 13.5 The spherical model in arbitrary dimensions 509 rather than to individual constraints, σ 2 i = 1 for each i, or to an arbitrary probability distribution law. Constraint (3) allows individual spins to vary over a rather wide range, −N1/2 to +N1/2, but restricts the super spin vector {σi} to the “surface of an N-dimensional hypersphere of radius N1/2”; in the Ising model, the same vector is restricted to the “cor-ners of a hypercube inscribed within the above hypersphere.” The resulting model is generally referred to as the spherical model. Constraint (3) can be taken care of by inserting an appropriate delta function in the integrand of the partition function. Using the representation δ N − X i σ 2 i ! = 1 2πi x+i∞ Z x−i∞ e z(N−P i σ 2 i ) dz, (4) the partition function of the spherical model is given by QN = 1 2πi x+i∞ Z x−i∞ dzezN +∞ Z −∞ ··· +∞ Z −∞ e −zP i σ 2 i +K P n.n. σiσj+hP i σi Y i dσi. (5) For a fixed z, the integral over the σi can be carried out in the same manner as in the Gaus-sian model; see equation (2). Let the result of that calculation be denoted by the symbol ZN(K,h;z). The partition function QN of the spherical model is then given by the complex integral QN = 1 2πi x+i∞ Z x−i∞ dzezNZN(K,h;z), (6) which can be evaluated by the saddle-point method — also known as the method of steep-est descent; see Section 3.2. One finds that the saddle point of the integrand in (6) lies at the point z = x0, where x0 is determined by the condition ∂ ∂z{zN + lnZN(K,h;z)} z=x0 = 0, (7) with the result that, asymptotically, 1 N lnQN ≈x0 + 1 N lnZN(K,h;x0). (8) The thermodynamic properties of the system can then be worked out in detail. It turned out that many physicists felt uncomfortable at the necessity of using the method of steepest descent, so a search for an alternative approach seemed desirable. In this connection Lewis and Wannier (1952) pointed out that while the ensemble underlying the model of Berlin and Kac was canonical in the variable E it was microcanonical in the variable P i σ 2 i . They proposed that one consider instead an ensemble that is canonical in 510 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models both E and P i σ 2 i ; the method of steepest descent could then be avoided. All one requires now is that the constraint (3) be obeyed only in the sense of an ensemble average, X i σ 2 i + = N, (9) rather than in the original sense that was comparatively more rigid. The resulting model is referred to as the mean spherical model. Constraint (9) can easily be taken care of by modifying the Hamiltonian of the system by including a term proportional to P i σ 2 i , that is, by writing H = −J X n.n. σiσj −µB X i σi + λ X i σ 2 i , (10) where λ is the so-called spherical field, and requiring that ⟨(∂H/∂λ)⟩= N. (11) The partition function of the revised model is thus given by QN = +∞ Z −∞ ··· +∞ Z −∞ e −βλP i σ 2 i +K P n.n. σiσj+hP i σi Y i dσi (12a) = ZN(K,h;βλ), (12b) with the constraint 1 β ∂lnZN(K,h;βλ) ∂λ = −N. (13) Comparing (13) with (7), we readily see that the parameter x0 of the spherical model is precisely equal to the parameter βλ of the mean spherical model. The free energy result-ing from (12), however, differs a little from the one given by equation (8), which is not surprising because the transition from a model that was microcanonical in the variable S 2(≡P i σ 2 i ) to one that is canonical modifies the nature of the free energy — it goes from “being at constant S ” to “being at constant λ.” The two free energies are connected by the Legendre transformation Aλ = AS + λ⟨S 2⟩= AS + λN, (14) so that 1 N AS = 1 N Aλ −λ. (15) This is precisely the difference that arises from the use of expression (8) or expression (12). 13.5 The spherical model in arbitrary dimensions 511 We now proceed to examine the thermodynamic properties of the mean spherical model, especially the nature of its critical behavior in arbitrary dimensions. The impor-tance of these results will be discussed toward the end of this section. The thermodynamic functions We consider a simple hypercubic lattice, of dimensions L1 × ··· × Ld, subject to periodic boundary conditions. The partition function of the system, as given by equation (12a), then turns out to be (see Joyce, 1972; Barber and Fisher, 1973) ZN(K,h;βλ) = Y k  π β(λ −µk) 1/2 eNh2/4β(λ−µ0), (16) where µk are the eigenvalues of the problem, µk = J d X j=1 cos(kja), kj = 2πnj Lj {nj = 0,1,...,(Nj −1)}, (17a) Nj = Lj/a, N = Y j Nj, (17b) and a the lattice constant of the system. The free energy Aλ is then given by Aλ = 1 2β X k ln β(λ −µk) π −Nµ2B2 4(λ −µ0), (18) while the parameter λ is determined by the constraint equation, see equation (13), 1 2β X k 1 (λ −µk) + Nµ2B2 4(λ −µ0)2 = N. (19) The magnetization M and the low-field susceptibility χ0 follow readily from equa-tion (18):3 M = Nµ2B 2(λ −µ0), χ0 = Nµ2 2(λ −µ0). (20a, b) Introducing the variable m(≡M/Nµ), the constraint equation (19) may be written in the form X k 1 (λ −µk) = 2Nβ(1 −m2). (21) 3Note, however, that to calculate the field-dependent susceptibility, (∂M/∂B)T,S , subject to the spherical constraint (19), one must keep in mind the field dependence of λ while differentiating (20a). 512 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models Next, the entropy of the system in zero field is given by4 S0 = −(∂A/∂T)µk,B=0 = 1 2kB X k [1 −ln{β(λ −µk)}] (22) and the corresponding specific heat by C0 = T ∂S0 ∂T  = 1 2kB X k  1 −T(∂λ/∂T)0 (λ −µk)  = N 1 2kB −  ∂λ ∂T  0  ; (23) here, use has been made of equation (19), with B = 0. To make further progress we need to determine λ, as a function of B and T, from the constraint equation (19). But first note, from equation (18), that for the free energy of the system to be well-behaved, λ must be larger than the largest eigenvalue µ0 — which, by (17a), is equal to Jd. At the same time, equation (20b) tells us that the singularity of the problem presumably lies at λ = µ0. We may thus infer that, as T decreases from higher values downward, λ also decreases and eventually reaches its lowest possible value, µ0, at some critical temperature, Tc, where the system undergoes a phase transition. The condition for criticality, therefore, is λc = µ0 = Jd, (24) which suggests that we may introduce a “reduced field,” φ, by the definition φ = (λ −λc)/J = (λ/J) −d; (25) the condition for criticality then becomes φc = 0. (26) It follows that, as we approach the critical point from above, the parameter φ becomes much smaller than unity; ultimately, it becomes zero as Tc is reached and stays so for all T < Tc. Now, substituting for the eigenvalues µk into the sum appearing in equations (19) and (21), and making use of the representation 1 z = ∞ Z 0 e−zxdx, (27) 4We denote Boltzmann’s constant by the symbol kB here so as to avoid confusion with the wavenumber k. 13.5 The spherical model in arbitrary dimensions 513 we have X k 1 (λ −µk) = 1 J X {nj} ∞ Z 0 exp   −  φ + d X j=1 ( 1 −cos 2πnj Nj !) x   dx = 1 J ∞ Z 0 e−φx Y j   Nj−1 X nj=0 exp ( −x + xcos 2πnj Nj !) dx. (28) For Nj ≫1, the summation over nj may be replaced by integration; writing θj = 2πnj/Nj, one gets X nj exp{···} ≈ 2π Z 0 e−x+xcosθj Nj 2π dθj = Nje−xI0(x), (29) where I0(x) is a modified Bessel function. Multiplying over j, one finally gets X k 1 λ −µk = N J Wd(φ), (30) where Wd(φ) is the so-called Watson function, defined by5 Wd(φ) = ∞ Z 0 e−φx[e−xI0(x)]ddx. (31) Equations (19) and (21) now take the form Wd(φ) = 2K −(βµB)2 2Kφ2 (32a) = 2K(1 −m2). (32b) The asymptotic behavior of the functions Wd(φ), for φ ≪1, is examined in Appendix G. The critical behavior We now analyze the various physical properties of the mean spherical model in different regimes of d. (a) d < 2. For this regime we take expression (7a) of Appendix G and substitute it into equation (32a), with B = 0. We obtain φ|B=0 ≈ 0{(2 −d)/2} 2(2π)d/2K 2/(2−d) ∼ kBT J 2/(2−d) . (33) 5Note that our definition of the function Wd(φ) differs slightly from the one adopted by Barber and Fisher (1973); this difference arises from the fact that our J is twice, and our φ is one-half, of theirs. 514 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models We see that φ in this case goes to zero only as T →0. The phase transition, therefore, takes place at Tc = 0. Equations (20b), (23), (24), and (25) then give the following for the low-temperature susceptibility χ0 = Nµ2 2Jφ ∼Nµ2 J kBT J −2/(2−d) (34) and for the low-temperature specific heat C0 −1 2NkB = −NJ  ∂φ ∂T  0 ∼−NkB kBT J d/(2−d) . (35) (b) d = 2. We now use expression (7b) of Appendix G and obtain φ|B=0 ∼exp(−4πJ/kBT), (36) so that once again Tc = 0 but now, at low temperatures, χ0 ∼(Nµ2/J)exp(4πJ/kBT) (37) and C0 −1 2NkB ∼−NkB(J/kBT)2 exp(−4πJ/kBT). (38) (c) 2 < d < 4. We now substitute expression (7c) of Appendix G into equation (32a), with the result Wd(0) −|0{(2 −d)/2}| (2π)d/2 φ(d−2)/2 = 2K −(βµB)2 2Kφ2 . (39) The critical point is now determined by setting B = 0 and letting φ →0; the condition for criticality then turns out to be Kc = 1 2Wd(0). (40) The variation of φ with T as one approaches the critical point is given by φ|B=0 ≈ " 2(2π)d/2(Kc −K) |0{(2 −d)/2}| #2/(d−2) (K ≲Kc). (41a) We also note that once φ becomes zero it stays so for all temperatures below, that is, φ|B=0 = 0 (K ≥Kc). (41b) It then follows that χ0 ∼(Kc −K)−2/(d−2) ∼(T −Tc)−2/(d−2) (T ≳Tc) (42) 13.5 The spherical model in arbitrary dimensions 515 and is infinite for T ≤Tc. At the same time C0 −1 2NkB ∼(T −Tc)(4−d)/(d−2) (T ≳Tc) (43) and it vanishes for T ≤Tc. The spontaneous magnetization is determined by equations (32b), (40), and (41b); we obtain a remarkably simple result m0 = (1 −Kc/K)1/2 = (1 −T/Tc)1/2 (T ≤Tc). (44) Finally, if in equation (39) we retain B but set T = Tc, we get φc ∼B4/(d+2) (T = Tc); (45) equation (20a) then gives mc = µB 2Jφc ∼B(d−2)/(d+2) (T = Tc). (46) The foregoing results give, for the critical exponents of the system, α = d −4 d −2, β = 1 2, γ = 2 d −2, δ = d + 2 d −2 (2 < d < 4). (47) (d) d > 4. In this regime we employ expression (8) of Appendix G. The condition for criticality remains the same as in (40); the variation of φ with T as we approach the critical point is, however, different. We now have, for all d > 4, φ|B=0 ∼(Kc −K)1 (K ≲Kc). (48) The subsequent results are modified accordingly: χ0 ∼(T −Tc)−1, C0 −1 2NkB ∼(T −Tc)0 (T ≳Tc) (49) φc ∼B2/3, mc ∼B1/3 (T = Tc). (50) Equations (41b) and (44) continue to apply as such. We, therefore, conclude that α = 0, β = 1 2, γ = 1, δ = 3 (d > 4). (51) (e) d = 4. For this borderline case, we use expression (12) of Appendix G. Once again, the condition for criticality remains the same; however, the variation of φ with T as one approaches the critical point is now determined by the implicit relation φ ln(1/φ) ≈8π2(Kc −K) (K ≲Kc). (52) 516 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models Introducing the conventional parameter t = (T −Tc)/Tc = (Kc −K)/K, (53) we get, to leading order in t, φ|B=0 ∼t/ln(1/t) (0 < t ≪1). (54) It follows that χ0 ∼t−1 ln(1/t), C0 ∼1/ln(1/t) (0 < t ≪1) (55) φc ∼B2/3/{ln(1/B)}1/3, mc ∼{Bln(1/B)}1/3 (t = 0). (56) Spin–spin correlations Following the procedure that led to equations (16) through (19), we obtain in the absence of the field G(r,r′) ≡σ(r)σ(r′) = 1 2Nβ X k exp{i(k · R)} λ −µk (R = r −r′); (57) compare to equation (19), with B = 0. The summation over k in (57) can be handled in the same manner as was done in (28); however, the resulting summation over nj now turns out to be X nj {···} ≈ 2π Z 0 exp{iRjθj/a}e−x+xcosθj Nj 2π dθj = Nje−xIRj/a(x); (58) compare to (29). This leads to the result G(R) = 1 2K ∞ Z 0 e−φx Y j [e−xIRj/a(x)]dx; (59) compare to equation (32a), with B = 0. For the functions In(x) we may use the asymptotic expression (see Singh and Pathria, 1985a) In(x) ≈ex−n2/2x √ (2πx) (x ≫1), (60) 13.5 The spherical model in arbitrary dimensions 517 so that, for φ ≪1, G(R) ≈ 1 2(2π)d/2K ∞ Z 0 e−φx−R2/(2a2x)x−d/2dx = 1 (2π)d/2K a2 ξR !(d−2)/2 K(d−2)/2 R ξ  , (61) where Kµ(x) is the other modified Bessel function while ξ = a/(2φ)1/2. (62) For R ≫ξ, we may use the asymptotic result Kµ(x) ≈(π/2x)1/2e−x; equation (61) then becomes G(R) ≈ ad−2 2Kξ(d−3)/2(2πR)(d−1)/2 e−R/ξ, (63) which identifies ξ as the correlation length of the system. Now, comparing equation (62) with equation (20b), we find that χ0 = Nµ2 2Jφ = Nµ2 Ja2 ξ2. (64) In view of the fact that ξ ∼χ1/2 0 , we infer that, in all regimes of d, the exponent ν = 1 2γ and hence, by relations (12.12.10) and (12.12.11), the exponent η = 0. To obtain this last result directly from equation (61), we observe that, as T →Tc from above, the parameter φ →0 and hence ξ →∞. We may then use the approximation R/ξ ≪1 and employ the formula Kµ(x) ≈1 20(µ) 1 2x −µ (µ > 0), (65) to obtain G(R)|T=Tc ≈0{(d −2)/2} 4πd/2Kc ad−2 Rd−2 (d > 2). (66) The power of R appearing here clearly shows that η = 0. Finally, substituting equation (41a) into equation (62), we get ξ ≈1 2a  |0{(2 −d)/2}| 4πd/2(Kc −K) 1/(d−2) (K ≲Kc), (67) which shows that for 2 < d < 4 the critical exponent ν = 1/(d −2). For T < Tc we expect the function G(R) to affirm the presence of long-range order in the system, that is, in the limit R →∞, it should tend to a limit, σ 2, that is nonzero. To 518 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models demonstrate this property of G(R), we need to take a closer look at the derivations of this subsection that were carried out with the express purpose of obtaining results valid in the thermodynamic limit (N →∞). This resulted in “errors” that were negligible in the region T ≳Tc but are not so when T < Tc. The first such error crept in when we replaced the sum-mations over {nj} in equation (28) by integrations; that suppressed contribution from the term with n = 0. Equation (30), therefore, accounts for only the (k ̸= 0)-terms of the orig-inal sum in (28), and the missing term, 1/Jφ, may be added to it ad hoc.6 Equation (19), with B = 0, then becomes 1 2β  1 Jφ + N J Wd(φ)  = N. (68) Now, when φ becomes very very small, Wd(φ) may be approximated by Wd(0), which is precisely equal to 2Kc; equation (68) then gives φ ≈[2N(K −Kc)]−1 (K > Kc), (69) rather than zero! The correlation length then turns out to be ξ = a/(2φ)1/2 ≈a[N(K −Kc)]1/2 (K > Kc), (70) rather than infinite! Now, the same error was committed once again in going from equation (57) to (59); so, the primary result for G(R), as given in equation (61), may be similarly amended by adding the missing term 1/(2NβJφ) which, by (69), is exactly equal to (1 −Kc/K). Thus, for R ≪ξ, we obtain, instead of (66), G(R) ≈  1 −Kc K  + 0{(d −2)/2} 4πd/2K ad−2 Rd−2 (K > Kc). (71) Now if we let R →∞,G(R) does approach a nonzero value σ 2, which is precisely the same as m2 0 given by equation (44). It is remarkable, though, that in the present derivation the magnetic field B has not been introduced at any stage of the calculation, which under-scores the fundamental role played by correlations in bringing about long-range order in the system. In the preceding paragraph we outlined the essential argument that led to the desired expression, (71), for G(R). For a more rigorous analysis of this problem, see Singh and Pathria (1985b, 1987a). Physical significance of the spherical model With a constraint as relaxed as in equation (3), or even more so in (9), one wonders how meaningful the spherical model is from a physical point of view. Relief comes from the fact, first established by Stanley (1968, 1969a,b), that the spherical model provides a correct 6This is reminiscent of a similar problem, and a similar ad hoc solution, encountered in the study of Bose–Einstein condensation in Section 7.1; see also Section 13.6. 13.6 The ideal Bose gas in arbitrary dimensions 519 representation of the (n →∞)-limit of an n-vector model with nearest-neighbor interac-tions; see also Kac and Thompson (1971). This connection arises from the very nature of the constraint imposed on the model, which introduces a super spin vector S with N degrees of freedom; it is not surprising that, in the limit N →∞, the model in some sense acquires the same sort of freedom that an n-vector model has in the limit n →∞. In any case, this connection brings the spherical model in line with, and actually makes it a good guide for, all models with continuous symmetry, namely the ones with n ≥2. And since it can be solved exactly in arbitrary dimensions, this model gives us some idea as to what to expect of models for which n is finite. For instance, we have seen that, for d > 4, the critical exponents of the spherical model are the same as the ones obtained from the mean field theory. Now, fluctuations are neglected in the mean field theory but, among the variety of models we are considering, fluctuations should be largest in the spherical model, for it has the largest number of degrees of freedom. If, for d > 4, fluctuations turn out to be negligible in the spherical model, they would be even more so in models with finite n. It thus follows that, regardless of the actual value of n, mean field theory should be valid for all these models when d > 4. See, in this connection, Section 14.4 as well. For d < 4, the final results depend significantly on n. The spherical model now provides a starting point from which one may carry out the so-called (1/n)-expansions to determine how models with finite n might behave in this regime. Such an approach was initiated by Abe and collaborators (1972, 1973) and independently by Ma (1973); for a detailed account of this approach, along with the results following from it, see Ma (1976c). Finally, for a comprehensive discussion of the spherical model, including the one with long-range interactions, see the review article by Joyce (1972). 13.6 The ideal Bose gas in arbitrary dimensions In this section we propose to examine the problem of Bose–Einstein condensation in an ideal Bose gas in arbitrary dimensions. As was first shown by Gunton and Buckingham (1968), the phenomenon of Bose–Einstein condensation falls in the same universality class as the phase transition in the spherical model; accordingly, the ideal Bose gas too corre-sponds to the (n →∞)-limit of an n-vector model. It must, however, be borne in mind that liquid He4, whose transition from a normal to a superfluid state is often regarded as a manifestation of the “Bose–Einstein condensation in an interacting Bose liquid,” actually pertains to the case n = 2. Now, just as the spherical model turns out to be a good guide for all models with continuous symmetry including the XY model (for which n = 2), in the same way the ideal Bose gas has also been a good guide for liquid He4. We consider a Bose gas composed of N noninteracting particles confined to a box of volume V(= L1 × ··· × Ld) at temperature T. Following the procedure of Section 7.1, we obtain for the pressure P of the gas P = −kBT V X ε ln(1 −ze−βε) = kBT λd g(d+2)/2(z), (1) 520 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models where λ[= h/ √ (2πmkBT)] is the mean thermal wavelength of the particles, z is the fugacity of the gas, which is related to the chemical potential µ through the formula z = exp(βµ) < 1 (β = 1/kBT), (2) while gν(z) are Bose–Einstein functions whose main properties are discussed in Appen-dix D. The quantity z is determined by the equation N = X ε (z−1eβε −1)−1 = N0 + Ne, (3) where N0 is the mean number of particles in the ground state (ε = 0), N0 = z/(1 −z), (4) while Ne is the mean number of particles in the excited states (ε > 0): Ne = V λd gd/2(z). (5) At high temperatures, where z is significantly below the limiting value 1, N0 is negligible in comparison with N; the quantity z is then determined by the simplified equation N = V λd gd/2(z), (6) and the pressure P in turn is given by the expression P = NkBT V g(d+2)/2(z) gd/2(z) . (7) The internal energy of the gas may be obtained from the relationship U = 1 2d(PV); (8) see the corresponding derivation of equation (7.1.12) as well as of equation (6.4.4). Now, making use of the recurrence relation (D.10) and remembering that the mean thermal wavelength λ ∝T−1/2, we get from equation (6) 1 z  ∂z ∂T  v = −d 2T gd/2(z) g(d−2)/2(z)  v = V N  , (9) and from equation (1) 1 z  ∂z ∂T  P = −d + 2 2T g(d+2)/2(z) gd/2(z) . (10) 13.6 The ideal Bose gas in arbitrary dimensions 521 It is now straightforward to show that the specific heats CV and CP of the gas are given by the formulae CV NkB = d(d + 2) 4 g(d+2)/2(z) gd/2(z) −d2 4 gd/2(z) g(d−2)/2(z) (11) and CP NkB = (d + 2)2 4 {g(d+2)/2(z)}2g(d−2)/2(z) {gd/2(z)}3 −d(d + 2) 4 g(d+2)/2(z) gd/2(z) , (12) respectively; it follows that the ratio CP CV = (d + 2) d g(d+2)/2(z)g(d−2)/2(z) {gd/2(z)}2 . (13) The isothermal compressibility κT and the adiabatic compressibility κS turn out to be κT = −1 v  ∂v ∂P  T = −1 v (∂v/∂z)T (∂P/∂z)T = g(d+2)/2(z)g(d−2)/2(z) {gd/2(z)}2 1 P (14) and κS = −1 v  ∂v ∂P  S = −1 v (∂v/∂T)z (∂P/∂T)z = d d + 2 1 P , (15) respectively; note that the ratio κT/κS is precisely equal to the ratio CP/CV , as is expected thermodynamically. As the temperature of the gas is reduced, keeping v constant, the fugacity z increases and ultimately approaches its limiting value 1 — marking the end of the regime where N0 was negligible in comparison with N. Whether this limit is reached at a finite T or at T = 0 depends entirely on the value of d; see equation (6), which tells us that if the function gd/2(z), as z →1−, is bounded then the limit in question will be reached at a finite T. On the other hand, if gd/2(z), as z →1−, is unbounded then the desired limit will be reached at T = 0 instead. To settle this question, we refer to equations (D.9) and (D.11), which summarize the behavior of the function gν(z) as z →1−(or as α →0+, where α = −lnz); we thus have gd/2(e−α) ≈                                  0 2 −d 2  α−(2−d)/2 + const. for d < 2 (16a) ln(1/α) + 1 2α for d = 2 (16b) ζ d 2  − 0 2 −d 2  α(d−2)/2 for 2 < d < 4 (16c) ζ(2) −{ln(1/α) + 1}α for d = 4 (16d) ζ d 2  −ζ d −2 2  α for d > 4, (16e) 522 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models ζ(ν) being the Riemann zeta function. Similarity of this system with the spherical model is quite transparent. We readily see from equation (6) that, for d > 2, α →0 at a finite temperature Tc, given by λd c = vζ(d/2), (17) with the result that Tc = h2 2πmkB  1 vζ(d/2) 2/d ; (18) for d ≤2, α →0 only as λ →∞, so Tc = 0. For brevity, we confine our further discussion only to d > 2. The critical behavior As T approaches Tc from above, the manner in which α →0 is determined by substitut-ing the appropriate expression (16) into (6) and utilizing the criticality condition (17). For 2 < d < 4, one gets asymptotically 0 2 −d 2  α(d−2)/2 ≈1 v (λd c −λd) ≃d 2 ζ d 2  T Tc −1  . (19) For T ≳Tc, this gives α ∼t2/(d−2) [t = (T −Tc)/Tc,0 < t ≪1]. (20) As T →Tc, the specific heat CP and the isothermal compressibility κT diverge because the function g(d−2)/2(z) appearing in equations (12) and (14), being∼α−(4−d)/2 [see equation (D.9)], becomes divergent; for small t, CP ∼κT ∼t−(4−d)/(d−2). (21) The specific heat CV , on the other hand, approaches a finite value,  CV NkB  T→TC+ = d(d + 2) 4 ζ{(d + 2)/2} ζ(d/2) , (22) with a derivative that, depending on the actual value of d, might diverge: 1 NkB ∂CV ∂T  V = 1 T " d2(d + 2) 8 g(d+2)/2(z) gd/2(z) −d2 4 gd/2(z) g(d−2)/2(z) −d3 8 {gd/2(z)}2g(d−4)/2(z) {g(d−2)/2(z)}3 # (23a) ∼−α−(d−3) ∼−t−2(d−3)/(d−2) (3 < d < 4). (23b) 13.6 The ideal Bose gas in arbitrary dimensions 523 Equating the exponent appearing here with (1 + α), we conclude7 that the critical expo-nent α for this system is (d −4)/(d −2); compare to equation (13.5.47). For a proper appreciation of the critical behavior of CV , we must as well examine the region T < Tc, along with the limit T →Tc−. For T < Tc, the fugacity z is essentially equal to 1; equations (5) and (17) then give Ne = V λd ζ d 2  = N λc λ d = N  T Tc d/2 . (24) It follows that N0 = N −Ne = N " 1 −  T Tc d/2# . (25) Equation (4) then tells us that the precise value of z in this region is given by z = N0/(N0 + 1) ≃1 −1/N0, (26) which gives α = −lnz ≃1/N0, (27) rather than zero. Disregarding this subtlety, equation (1) gives P = kBT λd ζ d + 2 2  ∝T(d+2)/2. (28) Since P here is a function of T only, the quantities κT and CP in this region are infinite; see, however, Problem 13.26. From equations (8) and (28), we get U = 1 2d kBTV λd ζ d + 2 2  , (29) which gives CV NkB = d(d + 2) 4 v λd ζ d + 2 2  = d(d + 2) 4 ζ{(d + 2)/2} ζ(d/2)  T Tc d/2 . (30) As T →Tc−, we obtain precisely the same limit as in (22) — showing that CV is continuous at the critical point. Its derivative, however, turns out to be different from the one given in 7We equate this exponent with (1 + α) because if CV ∼t−α, then ∂CV /∂t ∼t−α−1. We hasten to add that the critical exponent α should not be confused with the physical quantity denoted by the same symbol, namely α(= −lnz), which was introduced just before equations (16) and has been used throughout this section. 524 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models (23), for now 1 NkB ∂CV ∂T  V = 1 Tc d2(d + 2) 8 ζ{(d + 2)/2} ζ(d/2)  T Tc (d−2)/2 (31a) →1 Tc d2(d + 2) 8 ζ{(d + 2)/2} ζ(d/2) (31b) as T →Tc−. As for the condensate fraction, N0/N, equation (25) gives N0 N = 1 −  T Tc d/2 ≈d 2 |t| [t < 0,|t| ≪1]. (32) Now, the order parameter in the present problem is a complex number, 90, such that |90|2 = N0/V, the condensate particle density in the system; see Gunton and Buckingham (1968). We therefore, expect that, for |t| ≪1,N0 would be ∼t2β; equation (32) then tells us that the critical exponent β in this case has the classical value 1 2 for all d > 2. To determine the exponents γ and δ, we must introduce a “complex Bose field, conjugate to the order parameter 90” and examine quantities such as the “Bose susceptibility” χ as well as the variation of 90 with the Bose field at T = Tc. Proceeding that way, one obtains: γ = 2/(d − 2) and δ = (d + 2)/(d −2), just as for the spherical model. The pair correlations We now examine the pair correlation function of the ideal Bose gas G(R) = 1 V X k eik·R eα+βε(k) −1. (33) As usual, we replace the summation over k by integration (mindful of the fact that this replacement suppresses the (k = 0)-term which may, therefore, be kept aside). Making use of equation (C.11) in Appendix C, we get G(R) = N0 V + 1 (2π)d Z eik·R eα+βℏ2k2/2m −1 ddk = N0 V + 1 (2π)d/2R(d−2)/2 ∞ Z 0   ∞ X j=1 e−jα−jβℏ2k2/2m  J(d−2)/2(kR)kd/2dk = N0 V + 1 λd ∞ X j=1 e−jα−πR2/jλ2j−d/2 " λ = ℏ 2πβ m 1/2# ; (34) compare to equations (3) through (5), which pertain to the case R = 0. For R > 0, one may extend the summation over j from j = 0 to j = ∞, for the term so added is identically zero. 13.6 The ideal Bose gas in arbitrary dimensions 525 At the same time, the summation over j may be replaced by integration — committing errors O(e−R/λ), which are negligible so long as R ≫λ; for details, see Zasada and Pathria (1976). We thus obtain G(R) = N0 V + 1 λd ∞ Z 0 e−jα−πR2/jλ2j−d/2dj = N0 V + 2 λ2(2πξR)(d−2)/2 K(d−2)/2 R ξ  , (35) where Kµ(x) is a modified Bessel function while ξ = λ/(2π1/2α1/2). (36) For T ≳Tc, we may use expression (20) for α; equation (36) then gives ξ ∼λt−1/(d−2) (0 < t ≪1), (37) which means that ξ ≫λ. Now, if R ≫ξ, equation (35) reduces to G(R) ≈ 1 λ2(2πξ)(d−3)/2R(d−1)/2 e−R/ξ, (38) which identifies ξ as the correlation length of the system. Equation (37) then tells us that for, 2 < d < 4, the critical exponent ν of the ideal Bose gas is 1/(d −2). At T = Tc,ξ is infinite; equation (35) now gives G(R) ≈ 0{(d −2)/2} π(d−2)/2λ2 cRd−2 , (39) which shows that the critical exponent η = 0. For T < Tc, ξ continues to be infinite but now the condensate fraction, which is a measure of the long-range order in the system, is nonzero. The correlation function then assumes the form G(R) = |90|2 + A(T) Rd−2 , (40) where A(T) = 0{(d −2)/2} π(d−2)/2λ2 ∝T; (41) compare this result to the corresponding equation (13.5.7) of the spherical model. In the paper quoted earlier, Gunton and Buckingham also generalized the study of Bose–Einstein condensation to the single-particle energy spectrum ε ∼kσ, where σ is a 526 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models positive number not necessarily equal to 2. They found that the phase transition, at a finite temperature Tc, now took place for all d > σ, and the critical exponents in the regime σ < d < 2σ turned out to be α = d −2σ d −σ , β = 1 2, γ = σ d −σ , δ = d + σ d −σ , ν = 1 d −σ , η = 2 −σ. (42) While mathematically correct, these results left one with the awkward conclusion that the Bose gas in its extreme relativistic state (σ = 1) was in a different universality class than the one in the nonrelativistic state (σ = 2). It was shown later by Singh and Pandita (1983) that, if one employs the appropriate energy spectrum ε = c √ (m2 0c2 + ℏ2k2) and, at the same time, allows for the possibility of particle–antiparticle pair production in the system, as had been suggested earlier by Haber and Weldon (1981, 1982), then the relativistic Bose gas falls in the same universality class as the nonrelativistic one; see Problem (13.27). 13.7 Other models In Section 13.4 we saw that a two-dimensional lattice model characterized by a discrete order parameter (n = 1) underwent a phase transition, accompanied by a spontaneous magnetization m0, at a finite temperature Tc; naturally, one would expect the same if d were greater than 2. On the other hand, the spherical model, which is characterized by a continuous order parameter (with n = ∞), undergoes such a transition only if d > 2. The question then arises whether intermediate models, with n = 2,3,..., would undergo a phase transition at a finite Tc if d were equal to 2. The answer to this important question was provided by Mermin and Wagner (1966) who, making use of a well-known inequality due to Bogoliubov (1962), established the following theorem:8 Systems composed of spins with continuous symmetry (n ≥2) and short-range inter-actions do not acquire spontaneous magnetization at any finite temperature T if the space dimensionality d ≤2. In this sense, systems with n ≥2 behave in a manner similar to the spherical model — and quite unlike the Ising model! The marginal case (n = 2, d = 2), however, deserves a special mention. Clearly, this refers to an XY model in two dimensions, which has a direct relevance to superfluid He4 adsorbed on a substrate. As shown by Kosterlitz and Thouless (1972, 1973), this model exhibits a curious phase transition in that, while no long-range order develops at any finite temperature T, various physical quantities such as the susceptibility, the correla-tion length, the specific heat, and the superfluid density do become singular at a finite temperature Tc, whose precise value is determined by point defects, such as vortices or 8For a review of this theorem and other allied questions, see (Griffiths, 1972, pp. 84–89). 13.7 Other models 527 dislocations, in the system. The correlation length ξ, as T →Tc+, displays an essential singularity, ξ ∼eb′/(T−Tc)1/2, (1) where b′ is a nonuniversal constant; of course, the critical temperature itself is nonuni-versal and, for a square lattice, is given by Kc = J/kTc ≃1.12 ; (2) see Hasenbusch and Pinn (1997) and Dukovski, Machta, and Chayes (2002). The singular part of the specific heat shows a somewhat similar behavior, namely c(s) ∼ξ−2 (3) which, measured against a regular background, is rather indetectable since every deriva-tive of the specific heat is finite at the critical point, yet the function is nonanalytic. The superfluid density behaves rather strangely; it approaches a finite value, as T →Tc−, pre-ceded by a square-root cusp. The correlation function is no different; at T = Tc, Kosterlitz (1974) found a logarithmic factor along with a power law, namely g(r) ∼[ln(r/a)]1/8 r1/4 , (4) while for T < Tc we encounter a temperature-dependent exponent η such that g(r) ∼ 1 rη(T) , (5) where η(T) ≈kT/2πJ, for kT ≪J, as shown by Berezinskii (1970) and η(Tc) = 1/4, as shown by Kosterlitz (1974); see Berche, Sanchez, and Paredes (2002) for a numerical determi-nation of η(T). This phase, with a power-law decay of correlations, is said to display quasi-long-range order. For further details of this transition, see Kosterlitz and Thouless (1978) and Nelson (1983); for a pedagogical account, see Plischke and Bergersen (1989), Section 5.E. For other exactly soluble models in two dimensions, see Baxter (1982), Wu (1982), Nienhuis (1987), and Cardy (1987), where references to other relevant literature on the subject can also be found. We now proceed to consider the situation in three dimensions. Here, no exact solutions exist except for the extreme case n=∞, which has been discussed in Section 13.5. However, an enormous amount of effort has been spent in obtaining approximate solutions which, over the years, have become accurate enough to be regarded as “almost exact.” Irrespective of the model under study, the problem has generally been attacked along three different lines which, after some refinements, have led to almost identical results. In summary, these approaches may be described as follows. 528 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models The method of series expansions In this approach, the partition function or other relevant properties of the system are expanded as a high-temperature series such as (13.4.5), with expansion parameter v = tanh(βJ), or as a low-temperature series such as (13.4.9), with expansion parameter w = exp(−2βJ); in the presence of an external field, one would have series with two expan-sion parameters instead of one. In either case, the first major task involves the numerical computation of coefficients, such as n(r) and m(r), on the basis of graph theory and other allied techniques, while the second major task involves analysis of these coefficients with a view to determining the location and the nature of the singularity governing the prop-erty in question. The latter task is generally accomplished by employing the ratio method, which examines the trend of the ratio of two consecutive coefficients, such as n(r) and n(r −1), as r →∞, or by constructing Pad´ e approximants which, in a sense, provide a con-tinuation of the known (finite) series beyond its normal range of validity up to its radius of convergence — thus locating and examining the nature of the relevant singularity. Since their inception (in the mid-1950s for the ratio method and the early 1960s for the Pad´ e approximants), these techniques have been expanded, refined, and enriched in so many ways that it would be hopeless to attempt a proper review of them here. Suffice it to say that the reader may refer to Volume 3 of the Domb–Green series, which is devoted solely to the method of series expansions — in particular, to the articles by Gaunt and Guttmann (1974) on the asymptotic analysis of the various coefficients, by Domb (1974) on the Ising model, by Rushbrooke, Baker, and Wood (1974) on the Heisenberg model, by Stanley (1974) on the n-vector models, and by Betts (1974) on the XY model. For more recent reviews, see Guttmann (1989) and Baker (1990), where references to other relevant work on the subject are also available. The renormalization group method This method is based on the crucial observation (Wilson, 1971) that, as the critical point is approached, the correlation length of the system becomes exceedingly large — with the result that the sensitivity of the system to a length transformation (or renormalization) gets exceedingly diminished. At the critical point itself, the correlation length becomes infinite and, with it, the system becomes totally insensitive to such a transformation! The fixed point of the transformation is then identified with the critical point of the system, and the behavior of the system parameters such as K and h, see equation (13.5.2), in the neighborhood of the fixed point determines the critical exponents, and so on. Since very few systems could be solved exactly, approximation procedures had to be developed to handle most of the cases under study. One such procedure starts with known results for the upper critical dimension d = 4 and carries out expansions in terms of the (small) parameter ε = 4 −d, while the other starts with known results for the spherical model (n = ∞) and carries out expan-sions in terms of the (small) parameter 1/n; in the former case, the coefficients of the expansion would be n-dependent, while in the latter case they would be d-dependent. 13.7 Other models 529 Highly sophisticated manipulations enable one to obtain reliable results for ε that are as large as 1 (which corresponds to the most practical dimension d = 3) and for n as small as 1 (which corresponds to a large variety of systems that can be described through an order parameter that is scalar). We propose to discuss this method at length in Chapter 14; for the present, suffice it to say that the reader interested in further details may refer to Volume 6 of the Domb–Green series, which is devoted entirely to this topic. Monte Carlo methods As the name implies, these methods employ pseudorandom numbers to simulate statis-tical fluctuations for carrying out numerical integrations and computer simulations of systems with large numbers of degrees of freedom. Such methods have been in vogue for quite some time and, fortunately, have kept pace with developments along other lines of approach — so much so that they have adapted themselves to the ideas of the renormaliza-tion group as well (see Ma, 1976b). The interested reader may refer to Binder (1986, 1987, 1992), Frenkel and Smit (2002), Binder and Heermann (2002), and Landau and Binder (2009). We propose to discuss computer simulation methods further in Chapter 16. As mentioned earlier, the aforementioned methods lead to results that are mutually compatible and, within the stated margins of error, essentially identical. Table 13.1 lists the generally accepted values of the critical exponents of a three-dimensional system with n = 0,1,2,3, and ∞. It includes all the major exponents except α, which can be deter-mined by using the scaling relation α + 2β + γ = 2 (or the hyperscaling relation dν = 2 −α); we thus obtain, for n = 0,1,2,3, and ∞, α = 0.235, 0.111, −0.008, −0.114, and −1, respec-tively — of course, with appropriate margins of error. The theoretical results assem-bled here may be compared with the corresponding experimental ones listed earlier in Table 12.1, remembering that, while all other entries there are Ising-like (n = 1), the case of superfluid He4 pertains to n = 2. Table 13.1 includes exponents for the case n = 0 as well. This relates to the fact that if the spin dimensionality n is treated as a continuously varying parameter then the limit n →0 corresponds to the statistical behavior of self-avoiding random walks and hence of polymers (de Gennes, 1972; des Cloizeaux, 1974). The role of t in that case is played by the parameter 1/N, where N is the number of steps constituting the walk or the number of Table 13.1 Theoretical Values of the Critical Exponents in Three Dimensions n = 0 n = 1 n = 2 n = 3 n = ∞ β 0.302 ± 0.004 0.324 ± 0.006 0.346 ± 0.009 0.362 ± 0.012 0.5 γ 1.161 ± 0.003 1.241 ± 0.004 1.316 ± 0.009 1.39 ± 0.01 2.0 δ 4.85 ± 0.08 4.82 ± 0.06 4.81 ± 0.08 4.82 ± 0.12 5.0 ν 0.588 ± 0.001 0.630 ± 0.002 0.669 ± 0.003 0.705 ± 0.005 1.0 η 0.026 ± 0.014 0.031 ± 0.011 0.032 ± 0.015 0.031 ± 0.022 0.0 Source: After Baker (1990). 530 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models monomers constituting the polymer chain; thus, the limit N →∞corresponds to t →0, whereby one approaches the critical point of the system. Concepts such as the correlation function, the correlation length, the susceptibility, the free energy, and so on, can all be introduced systematically into the problem and one obtains a well-defined model that fits neatly with the rest of the family. For details, see de Gennes (1979) and des Cloizeaux (1982).9 Finally, we look at the situation with d ≥4. As mentioned earlier, especially toward the end of Section 13.5, critical exponents for d > 4 are independent of n and have values as predicted by the mean field theory. To recapitulate, they are: α = 0, β = 1 2, γ = 1, δ = 3, ν = 1 2, η = 0. (6) At the borderline dimensionality d = 4, two nonclassical features appear. First, the nature of the singularity is such that it cannot be represented by a power law alone; logarithmic factors are also present. Second, the dependence on n shows up in a striking fashion. In this context, we simply quote the results; for details, see Br´ ezin et al. (1976): c(s) ∼|lnt|(4−n)/(n+8) (h = 0,t ≳0) (7) m0 ∼|t|1/2|ln|t||3/(n+8) (h = 0,t ≲0) (8) χ ∼t−1|lnt|(n+2)/(n+8) (h = 0,t ≳0) (9) h ∼m3|lnm|−1 (t = 0,h ≳0), (10) along with the fact that η = 0 and hence ξ ∼χ1/2. In the limit n →∞, these results go over to the ones pertaining to the spherical model; see Section 13.5. Problems 13.1. Making use of expressions (12.3.17) through (12.3.19), (13.2.12), and (13.2.13), show that the expectation values of the numbers N+,N−,N++,N−−, and N+−in the case of an Ising chain are N± = N P(β,B) ± sinh(βµB) 2P(β,B) , N++ = N 2D(β,B)eβµB[P(β,B) + sinh(βµB)], N−−= N 2D(β,B)e−βµB[P(β,B) −sinh(βµB)] 9Values of n other than the ones appearing in Table 13.1 are sometimes encountered. For instance, certain antiferro-magnetic order–disorder transitions require for their description an order parameter with n = 4,6, 8, or 12; see Mukamel (1975), and Bak, Krinsky, and Mukamel (1976). Another example of this is provided by the superfluidity of liquid He3, which seems to require an order parameter with n = 18; see, for instance, Anderson (1984), and Vollhardt and W¨ olfle (1990). Even negative values of n have been investigated; see Balian and Toulouse (1973) and Fisher (1973). Problems 531 and N+−= N D(β,B)e−4βJ, where P(β,B) = {e−4βJ + sinh2(βµB)}1/2 and D(β,B) = P(β,B)[P(β,B) + cosh(βµB)]. Check that (i) the preceding expressions satisfy the requirement that N++ + N−−+ N+−= N and (ii) they agree with the quasichemical approximation (12.6.22), regardless of the value of B. 13.2. (a) Show that the partition function of an Ising lattice can be written as QN(B,T) = X′ N+,N+− gN(N+,N+−)exp{−βHN(N+,N+−)}, where HN(N+,N+−) = −J 1 2qN −2N+−  −µB(2N+ −N), while other symbols have their usual meanings; compare these results to equations (12.3.19) and (12.3.20). (b) Next, determine the combinatorial factor gN(N+,N+−) for an Ising chain (q = 2) and show that, asymptotically, lngN(N+,N+−) ≈N+ lnN+ + (N −N+)ln(N −N+) −  N+ −1 2N+−  ln  N+ −1 2N+−  −  N −N+ −1 2N+−  ln  N −N+ −1 2N+−  −2 1 2N+−  ln 1 2N+−  . Now, assuming that lnQN ≈(the logarithm of the largest term in the sum P′), evaluate the Helmholtz free energy A(B,T) of the system and show that this leads to precisely the same results as the ones quoted in the preceding problem as well as the ones obtained in Section 13.2. 13.3. Using the approximate expression, see Fowler and Guggenheim (1940), gN(N1,N12) ≃  1 2qN  ! N11!N22! h 1 2N12  ! i2 N1!N2! N! q−1 , for evaluating the partition function of an Ising lattice, show that one is led to the same results as the ones following from the Bethe approximation. [Note that, for q = 2, the quantity lng here is asymptotically exact; see Problem 13.2(b). No wonder that the Bethe approximation gives exact results in the case of an Ising chain.] 13.4. Making use of relation (13.2.37), along with expressions (13.2.8) for the eigenvalues λ1 and λ2 of the transfer matrix P, determine the correlation length ξ(B,T) of the Ising chain in the presence of a magnetic field. Evaluate the critical exponent νc, as defined in Problem 12.25, and check that νc = ν/1, where ν and 1 are standard exponents defined in Sections 12.10 and 12.12. 532 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 13.5. Consider a one-dimensional Ising system in a fluctuating magnetic field B, so that QN(s,T) ∼ ∞ Z −∞ dB X {σi} exp ( −βNB2 2s + N X i=1 [βµBσi + βJσiσi+1] ) , with σN+1 = σ1. Note that when the system is very large (i.e., N ≫1) the typical value of B is very small; nevertheless, the presence of this small fluctuating field leads to an order–disorder transition in this one-dimensional system! Determine the critical temperature of this transition. 13.6. Solve exactly the problem of a field-free Ising chain with nearest-neighbor and next-nearest-neighbor interactions, so that H{σi} = −J1 X i σiσi+1 −J2 X i σiσi+2, and examine the various properties of interest of this model. [Hint: Introduce a new variable τi = σiσi+1 = ±1, with the result that H{τi} = −J1 X i τi −J2 X i τiτi+1, which is formally similar to expression (13.2.1)]. 13.7. Consider a double Ising chain such that the nearest-neighbor coupling constant along either chain is J1 while the one linking adjacent spins in the two chains is J2. Then, in the absence of the field, H{σi,σ ′ i } = −J1 X i (σiσi+1 + σ ′ i σ ′ i+1) −J2 X i σiσ ′ i . Show that the partition function of this system is given by 1 2N lnQ ≈1 2 ln[2coshK2{cosh2K1 + √ (1 + sinh2 2K1 tanh2 K2)}], where K1 = βJ1 and K2 = βJ2. Examine the various thermodynamic properties of this system. [Hint: Express the Hamiltonian H in a symmetric form by writing the last term as −1 2J26i(σiσ ′ i + σi+1σ ′ i+1) and use the transfer matrix method.] 13.8. Write down the transfer matrix P for a one-dimensional spin-1 Ising model in zero field, described by the Hamiltonian HN{σi} = −J X i σiσi+1 σi = −1,0,+1. Show that the free energy of this model is given by 1 N A(T) = −kT ln 1 2 h (1 + 2coshβJ) + √ {8 + (2coshβJ −1)2} i . Examine the limiting behavior of this quantity in the limits T →0 and T →∞. 13.9. (a) Apply the theory of Section 13.2 to a one-dimensional lattice gas and show that the pressure P and the volume per particle v are given by P kT = ln 1 2 n (y + 1) + √ [(y −1)2 + 4yη2] o and 1 v = 1 2 " 1 + y −1 √ [(y −1)2 + 4yη2] # , Problems 533 where y = zexp(4βJ) and η = exp(−2βJ),z being the fugacity of the gas. Examine the high and the low temperature limits of these expressions. (b) A hard-core lattice gas pertains to the limit J →−∞; this makes y →0 and η →∞such that the quantity yη2, which is equal to z, stays finite. Show that this leads to the equation of state P kT = ln  1 −ρ 1 −2ρ   ρ = 1 v  . 13.10. For a one-dimensional system, such as the ones discussed in Sections 13.2 and 13.3, the correlation function g(r) at all temperatures is of the form exp(−ra/ξ), where a is the lattice constant of the system. Using the fluctuation-susceptibility relation (12.11.11), show that the low-field susceptibility of such a system is given by χ0 = Nβµ2 coth(a/2ξ). Note that as T →0 and, along with it, ξ →∞,χ0 becomes directly proportional to ξ — consistent with the fact that the critical exponent η = 1. For an n-vector model (including the scalar case n = 1), ξ is given by equation (13.3.17), which leads to the result χ0 = Nβµ2 I(n−2)/2(βJ) + In/2(βJ) I(n−2)/2(βJ) −In/2(βJ). Check that for the special case n = 1 this result reduces to equation (13.2.14). 13.11. Show that for a one-dimensional, field-free Ising model σkσlσmσn = {tanhβJ}n−m+l−k, where k ≤l ≤m ≤n. 13.12. Recall the symbol n(r), of equation (13.4.5), which denotes the number of closed graphs that can be drawn on a given lattice using exactly r bonds. Show that for a square lattice wrapped on a torus (which is equivalent to imposing periodic boundary conditions) n(4) = N, n(6) = 2N, n(8) = 1 2N2 + 9 2N,.... Substituting these numbers into equation (13.4.5) and taking logs, one gets lnQ(N,T) = N  ln(2cosh2 K) + v4 + 2v6 + 9 2v8 + ···  ,v = tanhK. Note that the term in N2 has disappeared — in fact, all higher powers of N do the same. Why? 13.13. According to Onsager, the field-free partition function of a rectangular lattice (with interaction parameters J and J′ in the two perpendicular directions) is given by 1 N lnQ(T) = ln2 + 1 2π2 π Z 0 π Z 0 ln{cosh(2γ )cosh(2γ ′) −sinh(2γ )cosω −sinh(2γ ′)cosω′}dωdω′, where γ = J/kT and γ ′ = J′/kT. Show that if J′ = 0, this leads to expression (13.2.9) for the linear chain with B = 0 while if J′ = J, one is led to expression (13.4.22) for the square net. Locate the critical point of the rectangular lattice and study its thermodynamic behavior in the neighborhood of that point. 13.14. Write the elliptic integral K1(κ) in the form K1(κ) = π/2 Z 0 1 −κ sinφ √ (1 −κ2 sin2 φ) dφ + π/2 Z 0 κ sinφ √ (1 −κ2 sin2 φ) dφ, 534 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models and show that, as κ →1−, the first integral →ln2 while the second≈ln[2/(1 −κ2)1/2]. Hence K1(κ) ≈ln(4/|κ′|), as in equation (13.4.34). [Hint: In the second integral, substitute cosφ = x.] 13.15. Using equations (13.4.22) and (13.4.28) at T = Tc, show that the entropy of the two-dimensional Ising model on a square lattice at its critical point is given by Sc Nk = 2G π + 1 2 ln2 − √ 2Kc ≃0.3065; here, G is Catalan’s constant, which is approximately equal to 0.915966. Compare this result with the ones following from the Bragg–Williams approximation and from the Bethe approximation; see Problem 12.15. 13.16. The spontaneous magnetization of a two-dimensional Ising model on a square lattice at T < Tc is given by equation (13.4.38a). Express this result in the form B|t|β{1 + b|t| + ···}, where B and β are stated in equation (13.4.40) while b = (1 −9Kc/√2)/8. As usual, t = (T −Tc)/Tc < 0 and |t| ≪1. 13.17. Apply the theory of Section 13.4 to a two-dimensional lattice gas and show that, at T = Tc, the quantity kTc/Pcvc ≃10.35. 13.18. Show that for the spherical model in one dimension the free energy at constant λ is given by βAλ N = 1 2 ln " β{λ + √ (λ2 −J2)} 2π # −βµ2B2 4(λ −J), while λ is determined by the constraint equation 1 2β √ (λ2 −J2) + µ2B2 4(λ −J)2 = 1. In the absence of the field (B=0),λ= p 1 + 4β2J2/2β; the free energy at constant S is then given by βA S N = 1 2 ln "√ (1 + 4β2J2) + 1 4π # −1 2 √ (1 + 4β2J2). 13.19. Starting with expression (13.3.8) for the partition function of a one-dimensional n-vector model, with Ji = nJ′, show that Lim n,N→∞ 1 nN lnQN(nK) = 1 2 "√ (4K 2 + 1) −1 −ln (√ (4K 2 + 1) + 1 2 )# , where K = βJ′. Note that, apart from a constant term, this result is exactly the same as for the spherical model; the difference arises from the fact that the present result is normalized to give QN = 1 when K = 0. [Hint: Use the asymptotic formulae (for ν ≫1) 0(ν) ≈(2π/ν)1/2(ν/e)ν and Iν(νz) ≈(2πν)−1/2(z2 + 1)−1/4eνη, where η = √ (z2 + 1) −ln[{ √ (z2 + 1) + 1}/z].] 13.20. Show that the low-field susceptibility, χ0, of the spherical model at T < Tc is given by the asymptotic expression χ0 ≈(Nµ2/kBT) · Nm2 0(T), where m0(T) is the spontaneous magnetization of the system; note that in the thermodynamic limit the reduced susceptibility, kBTχ0/Nµ2, is infinite at all T < Tc. Compare to Problem 13.26. Problems 535 13.21. In view of the fact that only those fluctuations whose length scale is large play a dominant role in determining the nature of a phase transition, the quantity (λ −µk) in the expression for the correlation function of the spherical model, see equation (13.5.57), may be replaced by λ −µk = λ −J d X j=1 cos(kja) ≃J " φ + k2a2 2 # , where φ = (λ/J) −d. Show that this approximation leads to the same result for G(R) as we have in equation (13.5.61), with the same ξ as in equation (13.5.62). For a similar reason, the quantity [exp(α + βε) −1] in the correlation function (13.6.33) of the ideal Bose gas may be replaced by eα+βε −1 ≃α + βε = α + (βℏ2/2m)k2, leading to the same G(R) as in equation (13.6.35), with the same ξ as in equation (13.6.36).10 13.22. Consider a spherical model whose spins interact through a long-range potential varying as (a/r)d+σ (σ > 0),r being the distance between two spins. This replaces the quantity (λ −µk) of equations (13.5.16) and (13.5.57) by an expression approximating J[φ + 1 2(ka)σ ] for σ < 2 and J[φ + 1 2(ka)2] for σ > 2; note that the nearest-neighbor interaction corresponds to the limit σ →∞and hence to the latter case. Assuming σ to be less than 2, show that the above system undergoes a phase transition at a finite temperature Tc for all d > σ. Further show that the critical exponents for this model are α = d −2σ d −σ , β = 1 2, γ = σ d −σ , δ = d + σ d −σ , ν = 1 d −σ , η = 2 −σ for σ < d < 2σ, and α = 0, β = 1 2, γ = 1, δ = 3, ν = 1 σ , η = 2 −σ for d > 2σ. 13.23. Refer to Section 13.6 on the ideal Bose gas in d dimensions, and complete the steps leading to equations (13.6.9) through (13.6.15) and (13.6.23). 13.24. Show that for an ideal Bose gas in d dimensions and at T > Tc V ∂2P ∂T2 ! v = NkB T " d(d + 2) 4 g(d+2)/2(z) gd/2(z) −d 2 gd/2(z) g(d−2)/2(z) −d2 4 {gd/2(z)}2g(d−4)/2(z) {g(d−2)/2(z)}3 # and ∂2µ ∂T2 ! v = kB T " d(d −2) 4 gd/2(z) g(d−2)/2(z) −d2 4 {gd/2(z)}2g(d−4)/2(z) {g(d−2)/2(z)}3 # , where µ(= kT lnz) is the chemical potential of the gas while other symbols have the same meanings as in Section 13.6. Note that these quantities satisfy the thermodynamic relationship VT ∂2P ∂T2 ! v −NT ∂2µ ∂T2 ! v = CV . 10A comparison with the mean field results (12.11.25) and (12.11.26) brings out a close similarity that exists between these models and the mean field picture of a phase transition; for instance, they all share a common critical exponent η, which is zero. There are, however, significant differences; for one, the correlation length ξ for these models is character-ized by a critical exponent ν which is nonclassical — in the sense that it is d-dependent whereas in the mean field case it is independent of d. 536 Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models Also note that, since P here equals (2U/dV), the quantities (∂P/∂T)v and (∂2P/∂T2)v are directly proportional to CV and (∂CV /∂T)V , respectively. Finally, examine the singular behavior of these quantities as T →Tc from above. 13.25. Show that for any given fluid CP = VT(∂P/∂T)S(∂P/∂T)V κT and CV = VT(∂P/∂T)S(∂P/∂T)V κS, where the various symbols have their usual meanings. In the two-phase region, these formulae take the form CP = VT(dP/dT)2κT and CV = VT(dP/dT)2κS, respectively. Using the last of these results, rederive equation (13.6.30) for CV at T < Tc. 13.26. Show that for any given fluid κT = ρ−2(∂ρ/∂µ)T, where ρ(= N/V) is the particle density and µ the chemical potential of the fluid. For the ideal Bose gas at T < Tc, ρ = ρ0 + ρe ≈−kBT Vµ + ζ(d/2) λd . Using these results, show that11 κT ≈(V/kBT)(ρ0/ρ)2 (T < Tc); note that in the thermodynamic limit the reduced compressibility, kBTκT/v, is infinite at all T < Tc. Compare Problem 13.20. 13.27. Consider an ideal relativistic Bose gas composed of N1 particles and N2 antiparticles, each of rest mass m0, with occupation numbers 1 exp[β(ε −µ1)] −1 and 1 exp[β(ε −µ2)] −1, respectively, and the energy spectrum ε = c √ (p2 + m2 0c2). Since particles and antiparticles are supposed to be created in pairs, the system is constrained by the conservation of the number Q(= N1 −N2), rather than of N1 and N2 separately; accordingly, µ1 = −µ2 = µ, say. Set up the thermodynamic functions of this system in three dimensions and examine the onset of Bose–Einstein condensation as T approaches a critical value, Tc, determined by the “number density” Q/V. Show that the nature of the singularity at T = Tc is such that, regardless of the severity of the relativistic effects, this system falls in the same universality class as the nonrelativistic one. 13.28. Derive equation (13.1.9) for hard spheres in one dimension from equation (13.1.7). Plot the pair correlation function for nD = 0.25, 0.50, 0.75, and 0.90. Determine the structure factor S(k) numerically and plot it for the same densities. Compare your results with equation (13.1.21). 13.29. Use the pair correlation function (13.1.8) and (13.1.9) to determine analytically the structure factor for hard spheres in one dimension. Show that S(k) is given by equation (13.1.21). Plot g(x) and S(k) for nD = 0.25, 0.50, 0.75, and 0.90. 13.30. Use the Takahashi method of Section 13.1 for a system of point masses and harmonic springs of length a. Allow the particles to pass through each other, so that the partition function can be evaluated in a closed form. Show that the system is stable at zero pressure. Determine the average 11This remarkable relationship between the isothermal compressibility of a finite-sized Bose gas and the condensate density in the corresponding bulk system was first noticed by Singh and Pathria (1987b). Problems 537 distance between particles that are far apart on the chain and the variance of that distance. Determine the structure factor and plot it for several values of the parameter mω2a2/kT, where m is the mass and mω2 is the spring constant. Show that the specific heat of this system is independent of temperature, as given by the equipartition theorem. 13.31. Confirm the first few coefficients in the low-temperature series for the two-dimensional Ising model in equation (13.4.53). Write a program to calculate the energies of all 216 states for a 4 × 4 periodic lattice. Show that the coefficients here are gq = {2,0,32,64,424,1728,6688,13568,20524,13568,6688,1728,424,64,32,0,2}. Extend your code to calculate the partition function for a 6 × 6 lattice, which has 236 states. 13.32. Calculate the exact zero field partition function of the one-dimensional Ising model on a periodic chain of n spins using equation (13.2.5) and write QN(0,T) in the form of equation (13.4.52). Show that for x →1, the partition function →2n. Evaluate the microcanonical entropy S(q)/k = lngq and plot it for the case n = 16. 13.33. Use the code posted at www.elsevierdirect.com to evaluate equations (13.4.56) through (13.4.59) to determine the low-temperature series coefficients for the two-dimensional Ising model for an 8 × 8 lattice. Plot the internal energy and the specific heat as a function of temperature. Repeat your calculation for 16 × 16 and 32 × 32 lattices. 13.34. Use the data posted at www.elsevierdirect.com to evaluate equations (13.4.56) through (13.4.59) to plot the two-dimensional Ising model internal energy and specific heat as a function of temperature for an L × L lattice, where L = 64. Compare your results with the ones displayed in Figure 13.17. 14 Phase Transitions: The Renormalization Group Approach In this chapter we propose to discuss what has turned out to be the most successful approach to the problem of phase transitions. This approach is based on ideas first pro-pounded by Kadanoff (1966b) and subsequently developed by Wilson (1971) and others into a powerful calculational tool. The main point of Kadanoff’s argument is that, as the critical point of a system is approached, its correlation length becomes exceedingly large — with the result that the sensitivity of the system to a length transformation (or a change of scale, as one may call it) gets exceedingly diminished. At the critical point itself, the correlation length becomes infinitely large and with it the system becomes totally insensitive to such a transformation. It is then conceivable that, if one is not too far from the critical point (i.e., |t|,h ≪1), the given system (with lattice constant a) may bear a close resemblance to a transformed system (with lattice constant a′ = la, where l > 1, and presumably modified parameters t′ and h′), renormalized so that all distances in it are measured in terms of the new lattice constant a′; clearly, the rescaled correlation length ξ′ (in units of a′) would be one-lth of the original correlation length ξ (in units of a). This resemblance in respect of critical behavior is expected only if ξ′ is also much larger than a′, just as ξ was in comparison with a, which in turn requires that l ≪(ξ/a); by implication, |t′| and h′ would also be ≪1. These considerations lead to a formulation similar to the one presented in Section 12.10, with the difference that, while there we had to rely on a scaling hypothe-sis, here we have a convincing argument based on the role played by correlations among the microscopic constituents of the system which, in the vicinity of the critical point, are so large-scale that they make all other length scales, including the one that determines the structure of the lattice, essentially irrelevant. Unfortunately, Kadanoff’s approach did not provide a systematic means of deriving the critical exponents or of constructing the scal-ing functions that appear in the formulation. Those deficiencies were remedied by Wilson by introducing the concept of a renormalization group (RG) into the theory. We propose to discuss Wilson’s approach in Sections 14.3 and 14.4, but first we present a formulation of scaling along the lines indicated above and follow it with an exploration of simple examples of renormalization that pave way for establishing Wilson’s theory. Finally, in Section 14.5, we outline the theory of finite-sizing scaling that too has benefited greatly from the RG approach. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00014-1 © 2011 Elsevier Ltd. All rights reserved. 539 540 Chapter 14. Phase Transitions: The Renormalization Group Approach 14.1 The conceptual basis of scaling The scale change in a given system can be effected in several different ways, the earliest one being due to Kadanoff who suggested that, when large-scale correlations prevail, the individual spins in the system may be grouped into “blocks of spins,” each block consisting of ld original spins, and let each block play the role of a “single spin” in the transformed system; see Figure 14.1, where a block–spin transformation with l = 2 and d = 2 is shown. The spin variable of a block may be denoted by the symbol σ ′, which arises from the values of the individual spins in the block, but a rule has to be established so that σ ′ too is either +1 or −1, just as the original σi were.1 The transformed system then consists of N′ “spins,” where N′ = l−dN, (1) occupying a lattice structure with lattice constant a′ = la. To preserve the spatial density of the degrees of freedom in the system, all spatial distances must be rescaled by the factor l, so that for any two “spins” in the transformed system r′ = l−1r. (2) A second way of effecting a scale change is to write down the partition function of the system, Q = X {σi} exp[−βHN{σi}], (3) (a) (b) FIGURE 14.1 A block–spin transformation, with l = 2 and d = 2. The original lattice (a) has N (= 36) spins, the transformed one (b) has N′ (= 9); after rescaling, the latter looks very much the same as the former, especially in the limit N,N′ →∞. 1For simplicity, we employ the language of the scalar model here. 14.1 The conceptual basis of scaling 541 and carry out summation over a subset of (N −N′) spins, such that one is left with a summation, over the remaining N′ spins, which can (hopefully) be expressed in a form similar to (3), namely Q = X {σ ′ i } exp[−βHN′{σ ′ i}]. (4) If the desired passage, from expression (3) to (4), can be accomplished with some degree of accuracy, we should expect a close resemblance between the critical behavior of the original system represented by equation (3) and the transformed one represented by (4); see Figure 14.2, where an example of this procedure with l = √2 and d = 2 is shown. We note that this procedure forms the very backbone of the Wilson approach and is generally referred to as “decimation,” although the fraction of the spins removed, (N −N′)/N, is rarely equal to 1/10. Other ways of effecting a scale change will be mentioned later. It is quite natural to expect that the free energy of the transformed system (or, at least, that part of it that determines the critical behavior) is the same as that of the original sys-tem. The singular parts of the free energy per spin of the two systems should, therefore, be related as N′ψ(s)(t′,h′) = Nψ(s)(t,h), (5) so that ψ(s)(t,h) = l−dψ(s)(t′,h′). (6) (a) (b) Spins summed over Spins that remain FIGURE 14.2 A scale transformation by “decimation,” with l = √2 and d = 2. The original lattice (a) has N (= 36) spins, the transformed one (b) has N′ (= 18); the latter is yet to be rescaled (and rotated through an angle π/4) so that it looks very much the same as the former, especially in the limit N,N′ →∞. 542 Chapter 14. Phase Transitions: The Renormalization Group Approach Now, since both t and t′ are small in magnitude, one may assume that they are linearly related, that is, t′ = lyt t, (7) where yt, as yet, is an unknown number. Similarly, one may assume that h′ = lyhh, (8) with the result that ψ(s)(t,h) = l−dψ(s)(lyt t,lyhh); (9) like yt, the number yh is also unknown at this stage of the game. We now assert that the function ψ(s), which governs much of the critical behavior of the system, is essentially insensitive to a change of scale; we should, therefore, be able to eliminate the scale factor l from expression (9). This essentially forces us to replace the variables t′ and h′ by a single, l-independent variable, namely h′ |t′|yh/yt = h |t|yh/yt = h |t|1 , say  1 = yh yt  ; (10) at the same time, it requires us to write ψ(s)(t′,h′) = |t′|d/yt ˜ ψ(h′/|t′|1), (11) leading to the identical result ψ(s)(t,h) = |t|d/yt ˜ ψ(h/|t|1); (12) note that, as of now, the function ˜ ψ is also unknown.2 Comparing (12) with equation (12.10.7), we readily identify the critical exponent α as α = 2 −(d/yt); (13) more importantly, the present considerations have led to the same scaled form for the free energy density of the system as was hypothesized in Section 12.10. We have thus raised the status of expression (12.10.7) from being a mere hypothesis to being a well-founded result whose conceptual basis lies in the prevalence of large-scale correlations in the system. As in Section 12.10, we infer that the exponents β,γ , and δ are now given by β = 2 −α −1 = (d −yh)/yt, (14) γ = −(2 −α −21) = (2yh −d)/yt, (15) 2Some authors derive equation (12) from (9) by choosing l to be t−1/yt . As will be seen shortly, see equation (19), this amounts to letting l be O(ξ/a), which violates the requirement, l ≪ξ/a, mentioned earlier. 14.2 Some simple examples of renormalization 543 and δ = 1/β = yh/(d −yh). (16) As remarked earlier, the rescaled correlation length ξ′ of the transformed system and the original correlation length ξ of the given system are related as ξ′ = l−1ξ. (17) At the same time, we expect ξ′ to be ∼|t′|−ν, just as ξ ∼|t|−ν. It follows that ξ′ ξ  = t′ t −ν = l−νyt . (18) Comparing (17) and (18), we conclude that ν = 1/yt (19) and hence, by (13), dν = 2 −α. (20) We thus obtain not only a useful expression for the critical exponent ν but also the hyper-scaling relation (12.12.15) on a basis far sounder than the one employed in Section 12.12. Finally we look at the correlation functions of the two systems. At the critical point we expect that for the transformed system g(r′ 1,r′ 2) = ⟨σ ′(r′ 1)σ ′(r′ 2)⟩∼(r′)−(d−2+η), (21) just as for the original system g(r1,r2) = ⟨σ(r1)σ(r2)⟩∼r−(d−2+η). (22) In order that equations (21) and (22) be mutually compatible, we must rescale the spin variables such that σ ′(r′) = l(d−2+η)/2σ(r). (23) As for η, we may use the scaling relation γ = (2 −η)ν, to get η = d + 2 −2yh. (24) 14.2 Some simple examples of renormalization 14.2.A The Ising model in one dimension We start with the partition function (13.2.3a) of a closed Ising chain consisting of N spins, namely QN(B,T) = X {σi} exp " N X i=1 n K0 + K1σiσi+1 + 1 2K2(σi + σi+1) o# (K0 = 0,K1 = βJ,K2 = βµB); (1) 544 Chapter 14. Phase Transitions: The Renormalization Group Approach N23 N2 2 N21 N 1 2 3 4 5 FIGURE 14.3 A closed Ising chain to be “decimated” by carrying out summations over σ2,σ4,.... the parameter K0 has been introduced here for reasons that will become clear in the sequel. For simplicity, we assume N to be even and carry out summation in (1) over all σi for which i is even, that is, over σ2,σ4,...; see Figure 14.3. Writing the summand in (1) as N Y i=1 exp{K0 + K1σiσi+1 + 1 2K2(σi + σi+1)} = 1 2 N Y j=1 exp  2K0 + K1(σ2j−1σ2j + σ2jσ2j+1) + 1 2K2(σ2j−1 + 2σ2j + σ2j+1) , (2) the summations over σ2j (= +1 or −1) can be carried out straightforwardly, with the result 1 2 N Y j=1 exp(2K0) · 2cosh{K1(σ2j−1 + σ2j+1) + K2} · exp  1 2K2(σ2j−1 + σ2j+1)  . (3) Denoting σ2j−1 by σ ′ j , the partition function QN assumes the form QN(B,T) = X {σ ′ j } 1 2 N Y j=1 exp(2K0) · 2cosh{K1(σ ′ j + σ ′ j+1) + K2} · exp 1 2K2(σ ′ j + σ ′ j+1)  . (4) The crucial step now consists in expressing (4) in a form similar to (1), namely QN(B,T) = X {σ ′ j } exp   N′ X j=1 n K ′ 0 + K ′ 1σ ′ j σ ′ j+1 + 1 2K ′ 2(σ ′ j + σ ′ j+1) o  . (5) 14.2 Some simple examples of renormalization 545 This requires that, for all choices of the variables σ ′ j and σ ′ j+1, exp n K ′ 0 + K ′ 1σ ′ j σ ′ j+1 + 1 2K ′ 2(σ ′ j + σ ′ j+1) o = exp(2K0) · 2cosh{K1(σ ′ j + σ ′ j+1) + K2} · exp n 1 2K2(σ ′ j + σ ′ j+1) o . (6) The various choices being σ ′ j = σ ′ j+1 = +1,σ ′ j = σ ′ j+1 = −1 and σ ′ j = −σ ′ j+1 = ±1, we obtain from (6) exp(K ′ 0 + K ′ 1 + K ′ 2) = exp(2K0 + K2) · 2cosh(2K1 + K2), (7a) exp(K ′ 0 + K ′ 1 −K ′ 2) = exp(2K0 −K2) · 2cosh(2K1 −K2), (7b) and exp(K ′ 0 −K ′ 1) = exp(2K0) · 2coshK2. (7c) Solving for K ′ 0,K ′ 1, and K ′ 2, we get eK ′ 0 = 2e2K0{cosh(2K1 + K2)cosh(2K1 −K2)cosh2 K2}1/4, (8a) eK ′ 1 = {cosh(2K1 + K2)cosh(2K1 −K2)/cosh2 K2}1/4, (8b) and eK ′ 2 = eK2{cosh(2K1 + K2)/cosh(2K1 −K2)}1/2. (8c) We may now remark on the need to have the parameter K0 included in expression (1) and, accordingly, K ′ 0 in (5). Since we end up having three equations (7a), (7b), and (7c), to determine the parameters appropriate to the transformed system, the problem could not be solved with K1 and K2 only; thus, even if K0 were set equal to zero, a K ′ 0 ̸= 0 is essential for a proper representation of the transformed system. To highlight the role played by this parameter in determining the free energy of the given system, we observe on the basis of equations (1) and (5) that, with K0 = 0, QN(K1,K2) = eN′K ′ 0QN′(K ′ 1,K ′ 2) (9) and hence for the free energy we have (in units of kT) AN(K1,K2) = −N′K ′ 0 + AN′(K ′ 1,K ′ 2). (10) Since N′ = 1 2N, we obtain for the free energy per spin the recurrence relation f (K1,K2) = −1 2K ′ 0 + 1 2f (K ′ 1,K ′ 2), (11) 546 Chapter 14. Phase Transitions: The Renormalization Group Approach which relates the free energy per spin of the given system with that of the transformed system; the role played by K ′ 0 is clearly significant. For example, in the limit T →∞, when (K1,K2) and along with them (K ′ 1,K ′ 2) tend to zero, equations (8a) and (11) give f (0,0) = −K ′ 0 = −ln2, (12) which is indeed the correct result (arising from the limiting value of the entropy, Nkln2, of the system). We note that the parameter K0 does not appear in equations (8b) and (8c), which deter-mine K ′ 1 and K ′ 2 in terms of K1 and K2. As will be seen in Sections 14.3 and 14.4, it is these two relations that determine the singular part of the free energy of the system and hence its critical behavior; the parameters K0 and K ′ 0 affect only the regular part of the free energy and hence play no direct role in determining the critical behavior of the system. We might hasten to add that, while renormalization is generally used as a technique for studying the properties of a given system in the vicinity of its critical point, it can be useful over a much broader range of the variables K1 and K2. For instance, in the absence of the field (K2 = 0), equations (8) and (11) give K ′ 0 = ln{2√[cosh(2K1)]}, K ′ 1 = ln√[cosh(2K1)], K ′ 2 = 0 (13) and hence f (K1,0) = −1 2 ln{2√[cosh(2K1)]} + 1 2f (ln√[cosh(2K1)],0); (14) the functional equation (14) has the solution f (K1,0) = −ln(2coshK1), (15) valid at all K1. On the other hand, in the paramagnetic case (K1 = 0), we get K ′ 0 = ln(2coshK2), K ′ 1 = 0, K ′ 2 = K2 (16) and hence f (0,K2) = −1 2 ln(2coshK2) + 1 2f (0,K2), (17) with the solution f (0,K2) = −ln(2coshK2), (18) valid at all K2. The case when both K1 and K2 are present is left as an exercise for the reader; see Problem 14.2. The critical behavior of this system will be studied in Section 14.4. 14.2 Some simple examples of renormalization 547 14.2.B The spherical model in one dimension We adopt the same topology as in Figure 14.3 and write down the partition function of the one-dimensional spherical model consisting of N spins, see equation (13.5.12a), QN = ∞ Z −∞ ... ∞ Z −∞ exp " N X i=1 n K0 + K1σiσi+1 + K2σi −3σ 2 i oi dσ1 ...dσN (K0 = 0,K1 = βJ,K2 = βµB,3 = βλ), (19) where 3 is chosen so that N X i=1 σ 2 i + = −∂ ∂3 lnQN = N; (20) see equations (13.5.9) and (13.5.13). Assuming N to be even, we carry out integrations over σ2,σ4,.... For this, we write our integrand as N Y i=1 exp n K0 + K1σiσi+1 + K2σi −3σ 2 i o = 1 2 N Y j=1 exp n 2K0 + K1(σ2j−1σ2j + σ2jσ2j+1) + K2(σ2j−1 + σ2j) −3  σ 2 2j−1 + σ 2 2j o (21) and integrate over σ2j, using the formula ∞ Z −∞ e−px2+qxdx = π p 1/2 e q2/4p (p > 0). (22) After simplification, we get 1 2 N Y j=1  π 3 1/2 exp ( 2K0 + K 2 2 43 ! + K 2 1 23σ2j−1σ2j+1 +  K2 + K1K2 3  σ2j−1 − 3 −K 2 1 23 ! σ 2 2j−1 ) . (23) Denoting σ2j−1 by σ ′ j , expression (19) may now be written in the renormalized form QN = ∞ Z −∞ ... ∞ Z −∞ exp   N′ X j=1 n K ′ 0 + K ′ 1σ ′ j σ ′ j+1 + K ′ 2σ ′ j −3′σ ′2 j o  dσ ′ 1 ...dσ ′ N′, (24) 548 Chapter 14. Phase Transitions: The Renormalization Group Approach where K ′ 0 = 1 2 ln  π 3  + 2K0 + K 2 2 43, K ′ 1 = K 2 1 23, (25a, b) K ′ 2 = K2  1 + K1 3  , 3′ = 3 −K 2 1 23 (25c, d) and, of course, N′ = 1 2N. It follows that, with K0 = 0, QN(K1,K2,3) = eN′K ′ 0QN′(K ′ 1,K ′ 2,3′) (26) and hence for the free energy per spin (in units of kT) we have the recurrence relation f (K1,K2,3) = −1 2K ′ 0 + 1 2f (K ′ 1,K ′ 2,3′). (27) The critical behavior of this system will be studied in Section 14.4. Presently we would like to demonstrate how the free energy of the system, over a broad range of the vari-ables K1 and K2, can be determined by using the recurrence relation (27) along with the transformation equations (25). First of all we identify two invariants of the transformation, namely 3′2 −K ′2 1 = 32 −K 2 1 (28a) and (3′ −K ′ 1)/K ′ 2 = (3 −K1)/K2. (28b) It turns out that it is precisely these combinations that appear in the constraint equation of the system as well; see Problem 13.18. It follows that the constraint equation (20) is RG-invariant, that is, once it is satisfied in the original system, its counterpart N′ X j=1 σ ′2 j + = N′ (29) is automatically satisfied in the transformed system — without any need to rescale the spin variables.3 Now, in the absence of the field (K2 = 0), equations (25) and (27) give K ′ 0 = 1 2 ln  π 3  , K ′ 1 = K 2 1 23, K ′ 2 = 0, 3′ = 3 −K 2 1 23 (30) and hence f (K1,3) = −1 4 ln  π 3  + 1 2f K 2 1 23,3 −K 2 1 23 ! . (31) 3This is further related to the fact that the critical exponent η in this case is equal to 1; see equation (14.1.23) which, with d = 1, gives: σ ′(r′) = σ(r). 14.2 Some simple examples of renormalization 549 The functional equation (31) has the solution f (K1,3) = 1 2 ln " 3 + √(32 −K 2 1) 2π # , (32) valid at all K1; the appropriate value of 3 is given by the constraint equation ∂f ∂3 = 1 2√(32 −K 2 1) = 1, that is 3 = 1 2 √(1 + 4K 2 1). (33) Note that the invariant (28a) in this case is equal to 1 4. On the other hand, in the paramagnetic case (K1 = 0), we get K ′ 0 = 1 2 ln  π 3  + K 2 2 43, K ′ 1 = 0, K ′ 2 = K2, 3′ = 3 (34) and hence f (K2,3) = −1 4 ln  π 3  −K 2 2 83 + 1 2f(K2,3), (35) with the solution f (K2,3) = −1 2 ln  π 3  −K 2 2 43, (36) valid at all K2; the appropriate value of 3 is now given by ∂f ∂3 = 1 23 + K 2 2 432 = 1, that is 3 = √(1 + 4K 2 2) + 1 4 . (37) The case when both K1 and K2 are present is left as an exercise for the reader; see Problem 14.3. 14.2.C The Ising model in two dimensions As our third example of renormalization, we consider an Ising model on a square lattice in two dimensions. In the field-free case, the partition function of this system is given by QN(T) = X {σi} exp (X n.n. Kσiσj ) (K = βJ), (38) where the summation in the exponent goes over all nearest-neighbor pairs of spins in the lattice. Writing the summand in (38) as a product of factors pertaining to different pairs of 550 Chapter 14. Phase Transitions: The Renormalization Group Approach 7 8 9 4 5 6 1 2 3 FIGURE 14.4 A section of the two-dimensional Ising lattice. The summed-over spins are denoted by open circles, the remaining ones by solid dots. To begin with, we concentrate on the summation over σ5. spins, we may highlight those factors that contain a particular spin, say σ5, and carry out summation over this spin (see Figure 14.4): X σ5=±1 Y n.n. ...eKσ2σ5 · eKσ4σ5 · eKσ5σ6 · eKσ5σ8 ... (39) = Y n.n. ′ ...[2coshK(σ2 + σ4 + σ6 + σ8)]..., (40) where the primed product goes over the remaining nearest-neighbor pairs in the lattice. This procedure of summation is supposed to be continued until one-half of the spins, shown as open circles in Figure 14.4, are all summed over. Clearly, this will generate a host of factors of the type shown in expression (40) but the real task now is to express these fac-tors in a form similar, or at least as close as possible, to the factors appearing in the original expression (38); moreover, this mode of expression should be valid for all possible values of the remaining spins, namely σ2,σ4,... = ±1. For the factor explicitly displayed in (40), there are 16 possible values for the spins involved, of which only four turn out to be nonequivalent; they are (i) σ2 = σ4 = σ6 = σ8, (41a) (ii) σ2 = σ4 = σ6 = −σ8, (41b) (iii) σ2 = σ4 = −σ6 = −σ8, (41c) (iv) σ2 = −σ4 = −σ6 = σ8. (41d) However, even four values are too many to accommodate by a factor of the form exp{A + B(σ2σ4 + σ2σ6 + σ4σ8 + σ6σ8)}, 14.2 Some simple examples of renormalization 551 which contains only nearest-neighbor interactions in the transformed lattice and hence only two parameters to choose. Clearly, we need two more degrees of freedom, and it turns out that these are provided by the next-nearest-neighbor interactions and by interactions among a quartet of spins sitting on the corners of an elementary square in the new lattice. Thus, we are obliged to set 2coshK(σ2 + σ4 + σ6 + σ8) = exp{K ′ 0 + 1 2K ′(σ2σ4 + σ2σ6 + σ4σ8 + σ6σ8) + L′(σ2σ8 + σ4σ6) + M′σ2σ4σ6σ8}; (42) the reason why we have written 1 2K ′, rather than K ′, will become clear shortly. Now, the four possibilities listed above require that 2cosh4K = exp(K ′ 0 + 2K ′ + 2L′ + M′), (43a) 2cosh2K = exp(K ′ 0 −M′), (43b) 2 = exp(K ′ 0 −2L′ + M′), (43c) 2 = exp(K ′ 0 −2K ′ + 2L′ + M′), (43d) with the result that K ′ 0 = ln2 + 1 2 lncosh2K + 1 8 lncosh4K, (44) K ′ = 1 4 lncosh4K, (45) L′ = 1 8 lncosh4K, (46) M′ = 1 8 lncosh4K −1 2 lncosh2K. (47) Continuing this process, we find that the factor exp 1 2K ′σ2σ4  appears once again when summation over σ1 is carried out, the factor exp 1 2K ′σ2σ6  appears once again when sum-mation over σ3 is carried out, and so on; no further factors involving the products σ2σ8,σ4σ6 and σ2σ4σ6σ8 appear. The net result is that the partition function (38) assumes the form QN = eN′K ′ 0 X {σ ′ j } exp   K ′ X n.n. σ ′ j σ ′ k + L′ X n.n.n. σ ′ j σ ′ k + M′ X sq. σ ′ j σ ′ kσ ′ l σ ′ m   , (48) where N′ = 1 2N. Clearly, we have not been able to establish an exact correspondence between the transformed system and the original one (in which no interactions other than the nearest-neighbor ones were present). It seems more reasonable now that we redefine the original system as one having all the interactions appearing in expression (48), but with L = M = 0. We may then write QN(K,0,0) = eN′K ′ 0QN′(K ′,L′,M′) (49) 552 Chapter 14. Phase Transitions: The Renormalization Group Approach and hence for the free energy per spin (in units of kT) f (K,0,0) = −1 2K ′ 0 + 1 2f (K ′,L′,M′), (50) where K ′ 0,K ′,L′, and M′ are given by equations (44) through (47). In view of the appear-ance of new parameters, L′ and M′, in the recurrence relation (50), no further progress can be made without introducing some sort of an approximation, for which see Section 14.4. But one thing is clear: when renormalization is carried out in two dimensions or more, the connectivity of the lattice requires that the Hamiltonian of the decimated system contain some higher-order interactions not present in the original system, in order that the lat-ter be represented correctly on transformation. It is obvious that further renormalizations would require more and more such interactions, and hence the need for more and more parameters would grow without limit. It may then be advisable that the Hamiltonian of the given system be regarded as a function of an “infinitely large number of parameters” (all but a few of which are zero to begin with), such that the number of parameters with a nonzero value grows indefinitely as renormalization transformations are carried out in succession and the number of degrees of freedom of the system steadily reduced. We now present a formulation of the renormalization group approach to the study of critical phenomena. 14.3 The renormalization group: general formulation We start with a system whose Hamiltonian depends on a large number of parameters K1,K2,... (all but a few of which are zero to begin with) and on the spin configuration {σi} of the lattice. The free energy of the system is then given by exp(−βA) = X {σi} exp[−βH{σi}({Kα})] α = 1,2,.... (1) We now effect a “decimation” of the system, which reduces the number of degrees of freedom from N to N′ and the correlation length from ξ to ξ′, such that N′ = l−dN, ξ′ = l−1ξ (l > 1). (2a, b) Expressing the Hamiltonian of the transformed system in a form similar to the one for the original system, except that we now have new parameters K ′ α, along with the additional K ′ 0, and new spins σ ′ j , equation (1) takes the form exp(−βA) = exp(N′K ′ 0) X {σ ′ j } exp  −βH{σ ′ j }({K ′ α})  , (3) so that the free energy per spin (in units of β−1) is given by f ({Kα}) = l−d[−K ′ 0 + f ({K ′ α})]. (4) 14.3 The renormalization group: general formulation 553 We now look closely at the transformation {Kα} →{K ′ α} by introducing a vector space K in which the set of parameters Kα is represented by the tip of a position vector K; on trans-formation, K changes to K ′, which may be looked upon as a kind of “flow from one position in the vector space to another.” This flow may be represented by the transformation equation K ′ = Rl(K) (5) where Rl is the renormalization-group operator appropriate to the case under considera-tion. A repeated application of this process leads to a sequence of vectors K ′,K ′′,..., such that K (n) = Rl(K (n−1)) = ... = Rn l (K (0)) n = 0,1,2,..., (6) where K (0) denotes the original K. At the end of this sequence, the correlation length ξ and the singular part of the free energy fs are given by ξ(n) = l−nξ(0), f (n) s = lnd f (0) s ; (7a, b) see equations (2b) and (4). Now, the transformation (5) may have a fixed point, K ∗, so that Rl(K ∗) = K ∗. (8) Equation (2b) then tells us that ξ(K ∗) = l−1ξ(K ∗), which means that ξ(K ∗) is either zero or infinite! The former possibility is of little interest to us, so let us dwell only on the latter (which makes the system with parameters K = K ∗critical); in simple situations, the fixed point, K ∗, will correspond to the critical point, K c, of the given system. Conceivably, an arbitrary point K, on successive transformations such as (6), may end up at the fixed point K ∗. Since the correlation length ξ can only decrease on transformation, see equation (7a), and is infinite at the end of this sequence of transformations, it must be infinite at K as well (the same being true for all points intermediate between K and K ∗). The collection of all those points which, on successive transformations, flow into the fixed point, constitutes a surface over which ξ is infinite; this surface is generally referred to as the critical surface. All flow lines in this surface are directed toward, and terminate at, the fixed point, while points off this surface may initially move toward the fixed point but eventually their flow lines will veer away from it; see Figure 14.5. Reasons behind this pattern of flow will become clear soon. For the analysis of the critical behavior we examine the pattern of flow in the neighbor-hood of the fixed point K ∗.4 Setting K = K ∗+ k, (9) 4In general, the vector K ∗will contain components not present in the original problem. In such a case, one has to locate, on the critical surface, a point K c that is free of these “unnecessary” components; since ξ is infinite at K = K c as well, the latter may be identified as the critical point of the given system. As will be seen in the sequel, the critical behavior of the system is still determined by the flow pattern in the neighborhood of the fixed point. 554 Chapter 14. Phase Transitions: The Renormalization Group Approach Fixed point Critical trajectories Physical critical point Renormalized critical point h h9 t t9 FIGURE 14.5 The parameter space of a physical system, showing critical trajectories (solid lines) flowing into the fixed point. The subspace of the relevant variables, t and h, is everywhere “orthogonal” to these trajectories (on all of which t = h = 0); the critical trajectories differ from one another only in respect of irrelevant variables that vanish as the fixed point is approached. The dashed lines depict that part of the flow in which the relevant variables play a decisive role. we have by equations (5) and (8) K ′ = K ∗+ k′ = Rl(K ∗+ k) (10) so that k′ = Rl K ∗+ k  −K ∗ (11) Assuming {kα}, and hence {k′ α}, to be small, we may linearize equation (11) to write k′ ≈dRl dK K=K ∗k = A∗ l k (12) where A∗ l is a matrix arising from the linearization of the operator Rl around the fixed point K ∗. The eigenvalues λi and the eigenvectors φi of the matrix A∗ l play a vital role in determining the critical behavior of the system. If the vectors φi form a complete set, we may expand k and k′ in terms of φi, k = X i uiφi, k′ = X i u′ iφi, (13a, b) with the result that u′ i = λiui i = 1,2,...; (14) 14.3 The renormalization group: general formulation 555 the coefficients ui appearing here are generally referred to as the scaling fields of the prob-lem. Under successive transformations (all in the neighborhood of the fixed point), these fields are given by u(n) i = λn i u(0) i . (15) It is obvious that the fields ui are certain linear combinations of the original parameters kα; they may, therefore, be looked upon as the “generalized coordinates” of the problem. The relative influence of these coordinates in determining the critical behavior of the system depends crucially on the respective eigenvalues λi. With u(n) i given by (15), we have three possible courses for a given coordinate ui. (a) If λi > 1, the coordinate ui grows with n and, with successive transformations, becomes more and more significant. Clearly, ui in this case is a relevant variable which, by itself, drives the system away from the fixed point — thus making the fixed point unstable. By experience, we know that the temperature parameter t [= (T −Tc)/Tc] and the magnetic field parameter h[= µB/kTc] are two basic quantities that vanish at the critical point and are clearly relevant to the problem of phase transitions. We, therefore, expect that our analysis will produce at least two relevant variables, u1 and u2 say, which could be identified with t and h, respectively, so that u1 = at + O(t2), u2 = bh + O(h2), (16, 17) with both λ1 and λ2 greater than unity. (b) If λi < 1, the coordinate ui decays with n and, with successive transformations, becomes less and less significant. Clearly, ui in this case is an irrelevant variable which, by itself, drives the system toward the fixed point. Now, if all the relevant variables are set at zero, then successive transformations (by virtue of the irrelevant variables) will drive the system to the fixed point. We must then be cruising on the critical surface itself (where all trajectories are known to flow into the fixed point). It follows that on the critical surface all relevant variables of the problem are zero; furthermore, the divergence of the correlation length is also tied to the same fact. (c) If λi = 1, the coordinate ui, in the linear approximation, stays constant; it neither grows nor decays very rapidly unless one enters the nonlinear, beyond-scaling, regime of the variable ui. Quite appropriately, ui in this case is termed a marginal variable; it does not affect the critical behavior of the system as significantly as a relevant variable does, but it may throw in logarithmic factors along with the conventional power laws. The ability to identify such variables and to predict the consequent departures from simple power-law scaling is one of the added virtues of the RG approach. The above considerations enable us to understand the pattern of flow shown in Figure 14.5. While the points on the critical surface flow into the fixed point, those off this surface flow toward the fixed point by virtue of the irrelevant variables and, at the same 556 Chapter 14. Phase Transitions: The Renormalization Group Approach time, away from it by virtue of the relevant variables; the net result is an initial approach toward but a final recession away from the fixed point. Points close to the fixed point and in directions “orthogonal” to the critical surface have only relevant variables to contend with, so right away they move away from the fixed point. It is this part of the flow that determines the critical behavior of the given system. Disregarding the irrelevant variables, we now examine the manner in which the corre-lation length, ξ, and the (singular part of the) free energy, fs, of the system are affected by the transformation (15). In view of equations (7a,b), we have ξ(u1,u2,...) = lnξ(λn 1u1,λn 2u2,...) (18) and fs(u1,u2,...) = l−ndfs(λn 1u1,λn 2u2,...). (19) Identifying u1 with t, as in (16), and remembering the definition of the critical exponent ν, we obtain from (18) u−ν 1 = ln(λn 1u1)−ν, (20) with the result that ν = lnl/lnλ1. (21) At first sight one might wonder why ν should depend on l. In fact, it doesn’t because of the following argument. On physical grounds we expect that two successive transformations with scale factors l1 and l2 would be equivalent to a single transformation with scale factor l1l2, that is,5 A∗ l1A∗ l2 = A∗ l1l2. (22) This forces the eigenvalues λi to be of the form lyi, for lyi 1 lyi 2 = (l1l2)yi. (23) Relation (21) then becomes ν = 1/y1, (24) manifestly independent of l. Equation (19) may now be written as fs(t,h,...) = l−ndfs(lny1t,lny2h,...). (25) 5This requirement makes the set of operators Rl a semigroup — not a group because the inverse of Rl does not exist. The reason for the nonexistence of R−1 l is that once a number of degrees of freedom of the system are summed over there is no definitive way of recreating them. 14.3 The renormalization group: general formulation 557 To ensure that the above relationship is independent of the choice of l, we follow the same line of argument as in Section 14.1, after equation (14.1.9), with the result fs(t,h,...) = |t|dν ˜ fs(h/|t|1,...), (26) where 1 = y2/y1. (27) Equation (26) is formally the same as the scaling relationship postulated in Section 12.10 and, one might say, argued out in Section 14.1. The big difference here is that not only has this relationship been derived on a firmer basis but now we also have a recipe for evaluat-ing the critical exponents ν,1, and so on, from first principles. What one has to do here is to determine the RG operator Rl for the given problem, linearize it around the appropri-ate fixed point K ∗, determine the eigenvalues λi (= lyi) and use equations (24) and (27) to evaluate ν and 1. At the same time, recalling the definition of the critical exponent α, we infer from (26) that 2 −α = dν; (28) the remaining exponents follow with the help of the scaling relations β = (2 −α) −1, γ = 21 −(2 −α), δ = 1/β, η = 2 −(γ/ν). (29) We find that the hyperscaling relation (28) is an integral part of the RG formulation; it comes out naturally — with no external imposition whatsoever. It is, however, dis-concerting that, according to the above argument, this relation should hold for all d — notwithstanding the fact that, for d > 4, all critical exponents are “stuck” at the mean field values and relation (28) gets replaced by 2 −α = 4ν (d > 4), (30) with α = 0 and ν = 1 2. The reason for this peculiar behavior is somewhat subtle; it arises from the fact that in certain situations an “irrelevant variable” may well raise its “dangerous” head and affect the outcome of the calculation in a rather significant manner. To see how this happens, we may consider a continuous spin model, very much like the one considered in Section 13.5, with the probability distribution law p(σ i)dσ i = const. e−1 2 σ 2 i −˜ uσ 4 i dσ i (˜ u > 0); (31) compared with equation (13.5.1). The free energy of the system then depends on the parameter ˜ u as well as on t and h, and we obtain instead of (25) fs(t,h, ˜ u) = l−ndfs lny1t,lny2h,lny3 ˜ u,...  . (32) 558 Chapter 14. Phase Transitions: The Renormalization Group Approach Now, using the RG approach, one finds (see Appendix D of Fisher, 1983) that, for d > 4, y1 = 2, y2 = 1 2d + 1, y3 = 4 −d, (33) showing very clearly that, for d > 4, the parameter ˜ u is an irrelevant variable. One is, therefore, tempted to ignore ˜ u and arrive at equation (26), with ν = 1 2 and 1 = (d + 2)/4. (34) The very fact that 1 turns out to be d-dependent shows that there is something wrong here. It turns out that though, on successive transformations, the variable ˜ u does tend to zero, its influence on the function fs does not. It may, therefore, be prudent to write fs(t,h, ˜ u) = |t|dν ˜ fs(h/|t|1, ˜ u/|t|φ), (35) where φ = y3 y1 = 4 −d 2 . (36) Now, by analysis, one finds that Lim w→0 ˜ fs(v,w) ≈1 w F(vw1/2), (37) where F(vw1/2) is a perfectly well-behaved function. The singularity of ˜ fs in w changes the picture altogether, and we get in the desired limit fs(t,h, ˜ u) ≈|t|dν+φF  h/|t|1+ 1 2 φ . (38) The “revised” value of △now is 1rev = d + 2 4 + 4 −d 4 = 3 2, (39) which is indeed independent of d and agrees with the corresponding mean field value. At the same time, the “revised” form of the hyperscaling relation now is 2 −α = d 2 + 4 −d 2 = 2, (40) as stated in (30). The lesson to be learnt here is that the standard derivations of the scaling relations rest on certain assumptions, often left unstated, about the nonsingular or nonvanishing behavior of various scaling functions and their arguments. In many cases these assump-tions are valid and may even be confirmed by explicit calculations or otherwise; in certain circumstances, however, they fail — in which case a scaling relation may change its form. Fortunately, such circumstances are not that common. 14.4 Applications of the renormalization group 559 14.4 Applications of the renormalization group We start our considerations with the models examined in Section 14.2. 14.4.A The Ising model in one dimension The renormalization group transformation in this case is given by equations (14.2.8b and c), namely K ′ 1 = 1 4 ln[cosh(2K1 + K2)cosh(2K1 −K2)] −1 2 lncoshK2 (1a) and K ′ 2 = K2 + 1 2 ln[cosh(2K1 + K2)/cosh(2K1 −K2)], (1b) where K1 = J/kT and K2 = µB/kT. It is straightforward to see that this transformation has a “line of trivial fixed points,” with K1 = 0 and K2 arbitrary. These fixed points pertain to either J = 0 or T = ∞; in either case, one has a correlation length that vanishes. There is also a nontrivial fixed point at K1 = ∞and K2 = 0, which may be realized by first setting B = 0 and then letting T →0; the correlation length at this fixed point will be infinite. In the vicinity of this point, we have K ′ 1 ≃K1 −1 2 ln2, K ′ 2 ≃2K2. (2a, b) Now, since K ∗ 1 = ∞,K1 is not an appropriate variable to carry out an expansion around the fixed point. We may adopt instead a new variable, see equation (13.2.17), namely t = exp(−pK1) (p > 0), (3) so that t∗= 0; now, in the vicinity of the fixed point, we have t′ ≃2p/2t. (4) Identifying K2 as the variable h, and remembering that the scale factor l here is 2, we readily obtain from equations (2b) and (4) y1 = p/2, y2 = 1. (5) The critical exponents of the model now follow straightforwardly from (5); we get ν = 2/p, 1 = 2/p, (6) from which, by equations (14.3.28) and (14.3.29), α = 2 −2/p, β = 0, γ = 2/p, δ = ∞, η = 1, (7) 560 Chapter 14. Phase Transitions: The Renormalization Group Approach in complete agreement with the results found in Section 13.2. As for the choice of p, see remarks following equation (13.3.21). 14.4.B The spherical model in one dimension The RG transformation in this case is given by equations (14.2.25b, c, and d), namely K ′ 1 = K 2 1 23, K ′ 2 = K2  1 + K1 3  , 3′ = 3 −K 2 1 23, (8a, b, c) where K1 = J/kT,K2 = µB/kT, and 3 = λ/kT,λ being the “spherical field” that was employed to take care of the constraint on the model. The nontrivial fixed point is again at T = 0, where λ = J [see equation (13.5.24), with d = 1] and hence 3 = K1. Equations (8) then reduce to the linearized form (valid for small T) K ′ 1 ≃1 2K1, K ′ 2 ≃2K2, 3′ ≃1 23. (9a, b, c) Equations (9a) and (9c) contain essentially the same information, namely T′ ≃2T. Clearly, T itself is a good variable for expansion in this case — giving y1 = 1. Equation (9b), just like (2b), gives y2 = 1, and we obtain ν = 1, 1 = 1, (10) whereby α = 1, β = 0, γ = 1, δ = ∞, η = 1, (11) in complete agreement with the results for one-dimensional models with n ≥2, as quoted in equation (13.3.20). 14.4.C The Ising model in two dimensions The RG transformation in this case is given by equations (14.2.45) through (14.2.47), namely K ′ = 1 4 lncosh4K, (12) L′ = 1 8 lncosh4K, (13) and M′ = 1 8 lncosh4K −1 2 lncosh2K. (14) It will be recalled that, while effecting this transformation, we started only with nearest-neighbor interactions (characterized by a single parameter K = βJ) but, due to the 14.4 Applications of the renormalization group 561 connectivity of the lattice, ended up with more — namely, the next-nearest-neighbor interactions (characterized by L′) and the four-spin interactions (characterized by M′) — in addition to the nearest-neighbor interactions (characterized by K ′). On subsequent transformations, still higher-order interactions come into play and the problem becomes formidable unless some approximations are introduced. In one such approximation, due originally to Wilson (1975), we discard all interactions other than the ones represented by the parameters K and L, and at the same time assume K and L to be small enough so that equations (12) and (13) reduce to K ′ = 2K 2, L′ = K 2. (15a, b) Now, if the parameter L had been introduced right in the beginning, the transformation equations, in this very approximation, would have been K ′ = 2K 2 + L, L′ = K 2. (16a, b) We shall treat equations (16) as if they were the exact transformation equations of the problem and see what they lead to. It is straightforward to see that the transformation (16) has a nontrivial fixed point at K ∗= 1 3, L∗= 1 9. (17) Linearizing around this fixed point, we get k′ 1 = 4 3k1 + k2, k′ 2 = 2 3k1, (18) where k1 and k2 represent deviations of the parameters K and L from the fixed-point val-ues K ∗and L∗, respectively. The transformation matrix A∗ l of equation (14.3.12) is then given by A∗ √2 =    4 3 1 2 3 0   , (19) whose eigenvalues are λ1 = 1 3(2 + √10), λ2 = 1 3(2 −√10) (20a, b) and whose eigenvectors are φ1 ∼ 2 + √10 2  , φ2 ∼ 2 −√10 2  . (21a, b) The scaling fields ui are then determined by equation (14.3.13a) which, on inversion, gives u1 ∼{2k1 + (√10 −2)k2}, u2 ∼{2k1 −(√10 + 2)k2}. (22a, b) 562 Chapter 14. Phase Transitions: The Renormalization Group Approach Clearly, the field u1, with λ1 > 1, is the relevant variable of the problem, while the field u2 is irrelevant. The “critical curve” in the (K,L)-plane is determined by the condition u1 = 0, while the linear part of this curve, in the vicinity of the fixed point (u = 0), is mapped by the relation (22a). In terms of the variables k1 and k2, this segment of the critical curve is given by the equation k2 ≈− √10 + 2 3 k1, (23) which represents a straight line of slope −1.7208; see Figure 14.6. To determine the physical critical point of this model, we have to locate on the criti-cal curve a point with L = 0, for the original problem had no interactions other than the one represented by the parameter K; the corresponding value of K would be our Kc.6 This requires a mapping of the critical curve right up to the K-axis. While this has been done numerically (Wilson, 1975), a crude estimate of Kc can be made by simply extending the straight-line segment (23) down to the desired limit. One thus obtains7 Kc = 1 3 + 1 9 3 √10 + 2 = 4 + √10 18 = 0.3979, (24) which may be compared with the exact result found in Section 13.4, namely 0.4407. 0.2 L 0.1 0.0 0.1 0.2 0.3 0.4 0.5 K Kc (K∗, L∗) FIGURE 14.6 A section of the critical curve for the two-dimensional Ising model near the nontrivial fixed point K ∗= 1 3 ,L∗= 1 9  . Points on the critical curve flow into the fixed point, while those off it flow away toward the trivial fixed point (K ∗= L∗= 0) or (K ∗= L∗= ∞). 6Remember that at each and every point on the critical surface — in this case, the critical curve — the correlation length is infinite; accordingly, each and every such point is qualified to be a critical point. The physical critical point is the one that is free of unnecessary parameters. 7The result obtained through numerical analysis was 0.3921. 14.4 Applications of the renormalization group 563 Another quantity that can be estimated here is the critical exponent ν. From equa-tions (14.3.21) and (14.3.20a),one obtains ν = lnl lnλ1 = ln√2 ln[(2 + √10)/3] = 0.6385, (25) which may be compared with the exact value 1. Even though a comparison of the results obtained here with the ones following from exact analysis is not very flattering, the basic merits of the RG approach are quite obvious. One important aspect of critical phenomena, namely their universality over a large class of systems, is manifest even in this simple example. Imagine, for instance, that in the case of the given system a next-nearest-neighbor interaction L0 were indeed present. Our approximate treatment would then lead to the same fixed point and the same criti-cal curve as above, but our physical critical point would now be given by that “point on the critical curve whose L-value is L0”; we may denote this critical point by Kc(L0). As for the critical behavior, it will still be determined by an expansion around the fixed point, for that is where the “relevant part” of the flow is; see again Figure 14.6. Clearly, the crit-ical behavior of the given system, insofar as exponents are concerned, will be the same, regardless of the actual value of L0. And, by extension, the same will be true of any two systems which have the same basic topology but differ only in the details of the spin–spin interactions. As for the accuracy of the results obtained here, improvements are needed in several important respects. First of all, the exclusion of all interaction parameters other than K and L constitutes a rather inadequate approximation; one should at least include the four-spin interaction, represented by the parameter M, and may possibly ignore the ones that appear on successive transformations. Next, the assumption that the parameters K and L are small is also unjustified, especially for K; this makes a numerical approach to the problem rather essential. Thirdly, we disregarded the renormalization of the spins, from the original σ(r) to σ ′(r′), as required by equation (14.1.23); in the present problem, this would amount to introducing a factor of (√2)η/2, that is, 21/16, for η here is 1 4. In the actual treatment, one may have to introduce an unknown parameter, ρ, and determine its “true” value by theo-retical analysis (see Wilson, 1975). Highly sophisticated procedures have been developed over the years to accommodate (or circumvent) these problems, leading to very accurate — in fact, almost exact — results for the model under considerations. For details, see the review article by Niemeijer and van Leeuwen (1976), where references to other pertinent literature on the subject can also be found.8 14.4.D The ε-expansion Application of the RG approach to systems in higher dimensions, namely with d > 2, presents serious mathematical difficulties. One is then forced to resort to approximation procedures such as the ε-expansion, first introduced by Wilson (1972); see also Wilson 8In this reference one can also find a systematic method of constructing the scaling function fs(u1,u2,...) from a knowledge of the regular function f (K ′ 0) of equation (14.3.4). 564 Chapter 14. Phase Transitions: The Renormalization Group Approach and Fisher (1972). This procedure was inspired by the observation that the field-theoretic calculations of the RG formulation become especially simple as the upper critical dimen-sion, d = 4, is approached; it, therefore, seemed desirable to introduce a parameter ε(= 4 −d) and carry out expansions of the various quantities of interest around ε = 0. The model adopted for these calculations was the same as the one referred to in Section 14.3, namely the continuous, n-vector spin model, with the probability distribution given by equation (14.3.31).9 An important advantage of using continuous spins σ(r){= σ (µ)(r), µ = 1,...,n}, with −∞< σ (µ) < ∞, is that one can introduce Fourier transforms σ(q) and make use of the “momentum shell integration” technique of Wilson (1971). The parameters of interest now are (see Fisher, 1983) r = T −T0 T0R2 0 = t0 R2 0 , u = ˜ uT2ad T2 0 R4 0 (26a, b) and, of course, the magnetic field parameter h; here, T0 denotes the mean-field critical temperature qJ/k (q being the coordination number), R0 is a measure of the range of interactions, a is the lattice constant, whereas ˜ u is the real-space parameter appearing in equation (14.3.31). The transformation equations, with a scale factor l, turn out to be r′ = l2r + 4(l2 −1)c(n + 2)u −l2 lnl(n + 2)(2π2)−1ru, (27a) u′ = (1 + εlnl)u −(n + 8)lnl(2π2)−1u2, (27b) and h′ = l3  1 −1 2εlnl  h, (27c) correct to the orders displayed; the parameter c in equation (27a) is related to a cutoff in the momentum space which, in turn, is a reflection of the underlying lattice structure. Transformation (27) has two fixed points — the so-called Gaussian fixed point, with r∗= u∗= h∗= 0, (28) and a non-Gaussian fixed point, with r∗= −8π2c(n + 2) (n + 8) ε, u∗= 2π2 (n + 8)ε, h∗= 0. (29) We now examine two distinct situations. 9It can be shown that, by a suitable transformation, the Ising model (n = 1), which is a discrete (rather than a con-tinuous) model, can also be rendered “continuous” with a probability distribution similar to (14.3.31). For details, see Appendix A in Fisher (1983). 14.4 Applications of the renormalization group 565 Dimension d ≳4, so that ε is a small negative number One readily sees from equation (27b) that the parameter u in this case decreases on trans-formation, so on successive transformations it will tend to zero. Clearly, only the Gaussian fixed point is the one appropriate to this case. Linearizing around this point, we obtain for the transformation matrix A∗ l A∗ l = l2 4(l2 −1)c(n + 2) 0 1 + εlnl ! , (30) with eigenvalues λr = l2, λu = (1 + εlnl) < 1 (31) and, of course, λh = l3  1 −1 2εlnl  . (32) It follows that y1 = 2, y2 ≈3 −1 2ε, y3 ≈ε, (33) as in equation (14.3.33). Note that the parameter u in this case is an irrelevant variable but, as discussed at the end of Section 14.3, it is a dangerously irrelevant variable that does eventually affect the results of the calculation in hand. Dimension d ≲4, so that ε is a small positive number The parameter u now behaves very differently. If it is already zero, it stays so; otherwise, it moves away from that value, carrying the system to some other fixed point — possibly the non-Gaussian one, with coordinates given in (29). The resulting pattern of flow in the (r,u)-plane is shown in Figure 14.7; clearly, the Gaussian fixed point is no longer appropriate and the problem now revolves around the non-Gaussian fixed point instead. Linearizing around the latter, we obtain A∗ l =  l2  1 −n + 2 n + 8εlnl  4(l2 −1)c(n + 2) 0 1 −εlnl  , (34) with eigenvalues l2  1 −n + 2 n + 8εlnl  and (1 −εlnl) < 1. (35) We note that of the “generalized coordinates” u1 and u2, which are certain linear combi-nations of the parameters 1r(= r −r∗) and 1u(= u −u∗), only u1 is a relevant variable of 566 Chapter 14. Phase Transitions: The Renormalization Group Approach r ∗ (r ∗, u∗) r (0, 0) u ∗ u FIGURE 14.7 A section of the critical curve and a sketch of the RG flows in the (r,u)-plane for 0 < ε ≪1. Note that the critical curve is straight only to order ε. the problem.10 Identifying u1 with the temperature parameter t, we obtain yt ≈2 −n + 2 n + 8ε. (36) Combining this with expression (33) for yh, namely yh ≈3 −1 2ε, (37) we obtain, see equations (14.3.24), (14.3.27), (14.3.28), and (14.3.29), ν ≈1 2 + n + 2 4(n + 8)ε, 1 ≈3 2 + n −1 2(n + 8)ε, (38) which gives α ≈ 4 −n 2(n + 8)ε, β ≈1 2 − 3 2(n + 8)ε, γ ≈1 + n + 2 2(n + 8)ε, (39) δ ≈3 + ε, η ≈0, (40) correct to the first power in ε. For obvious reasons the value of ε of greatest interest to us is ε = 1, for which the above results are totally inadequate; they do show the correct trends, though. For better numeri-cal accuracy it is essential to extend these calculations to higher orders in ε. Considerable 10It can be seen quite easily that the generalized coordinate u2 is directly proportional to 1u, making u an irrelevant variable of the problem; see Problem 14.6, with a21 = 0. 14.4 Applications of the renormalization group 567 progress has been made in this direction, so that we now have expressions available that include terms up to ε3 and, in some cases, even ε4; for details, see Wallace (1976). One wonders if that degree of extension would be good enough for obtaining reliable results for ε as large as 1. The answer is yes, and we will illustrate it with an example. For the spherical model we know exact values of the various critical exponents which may, for the purpose of illustration, be expressed as power series in ε. Thus, for instance, γ = 2 d −2 =  1 −1 2ε −1 = 1 + 1 2ε + 1 4ε2 + 1 8ε3 + 1 16ε4 + ··· . (41) Since the radius of convergence of this series is 2, the value 1 of ε is not really as large as it seems. In fact, the terms displayed in (41) already give, for ε = 1, γ = 1.9375, as opposed to the correct value 2. The situation is clearly encouraging and, with better methods of summing up diagrams, the convergence of the ε-expansions can be improved greatly. In fact, some of the entries in Table 13.1 were originally obtained (or at least rechecked) with the help of this method. Before closing this subsection we would like to point out a somewhat unusual piece of information contained in the first-order results obtained above. This refers to the exponent α, for which we note the prediction that for large n it is negative and hence the (singular part of the) specific heat vanishes at T = Tc (which we know to be the case with the spher-ical model) whereas for small n it is positive and hence the specific heat diverges (which we know to be the case with Ising-like systems). The inversion, from one case to the other, takes place at n = 4 where α, according to the first-order expression (39), vanishes. The inclusion of the second-order term in ε upholds this prediction qualitatively but changes it quantitatively. We now have α ≈ 4 −n 2(n + 8)ε −(n + 2)2(n + 28) 4(n + 8)3 ε2, (42) so that, with ε = 1, the inversion takes place between n = 1 and n = 2 — in agreement with the more accurate results quoted in Section 13.7. 14.4.E The 1/n expansion Another approach to the problem of determining critical exponents, as functions of d and n, is to adopt the limiting case n = ∞as the starting point and carry out expansions in powers of the small quantity 1/n. Clearly, the leading terms in these expansions would pertain to the spherical model, which has been studied in Section 13.5, and the correction terms would enable us to get some useful information on models with finite n. We quote 568 Chapter 14. Phase Transitions: The Renormalization Group Approach some first-order results here:11 η = 4(4 −d)Sd d 1 n + O  1 n2  , (43) γ = 2 d −2  1 −6Sd n + O  1 n2  , (44) and α = −4 −d d −2  1 −8(d −1)Sd 4 −d 1 n + O  1 n2  , (45) where Sd = sin{π(d −2)/2}0(d −1) 2π{0(d/2)}2 (2 < d < 4). (46) We note that the coefficients of expansion in this approach are functions of d just as the coefficients of expansion in the preceding approach were functions of n; in this sense, the two expansions are complementary to one another. Unfortunately, there has not been much progress in the evaluation of further terms of these expansions (except for the one mentioned in the note); accordingly, the usefulness of this approach has been rather limited. 14.4.F Other topics As mentioned earlier, the renormalization group approach has provided a very clear expla-nation of the concept of universality, in that it arises when several physical systems, despite their microscopic structural differences, are governed by a common fixed point and hence display a common critical behavior. Typically, this behavior is linked to the dimensionality d of the physical space, the dimensionality n of the spin vector σ and the range of the spin–spin interaction. Now, depending on the precise nature of the Hamil-tonian and the relative importance of the various parameters therein, it is quite possible that under certain circumstances the critical behavior of the system may “cross over” from being characteristic of one fixed point to being characteristic of another fixed point. For instance, we may write for the spin–spin interaction in the lattice Hint = −1 2 X r,r′ X α,β Jαβ(r −r′)σ α(r)σ β(r′) α,β = 1,...,n. (47) 11In the special case d = 3, the expansion for η is known to a higher order, namely η = 8 3π2n −  8 3 3 1 π4n2 + O  1 n3  . 14.4 Applications of the renormalization group 569 If the given interaction is isotropic in the physical space but anisotropic in the spin com-ponents (assumed three in number), so that Jαβ = Jαδαβ, then the system is ordinarily supposed to be a Heisenberg ferromagnet; however, the anisotropy of the interaction may finally drive the system toward an Ising fixed point (if one of the Jα dominates over the other two) or toward an XY fixed point (if two of the Jα are equally strong and dominate over the third one). In either case, we encounter what is generally referred to as a crossover phenomenon. Similarly, anisotropy in the physical space, J(R) = J(Ri)δij or K(R)RiRj, may result in a crossover from a d-dimensional behavior to a d′-dimensional behavior (where d′ < d). In the same vein, one may consider a long-range interaction, J(R) ∼R−d−σδij, leading to a critical behavior which, for σ < 2, is quite seriously σ-dependent; see, for instance, Prob-lem 13.22. However, as σ goes over from the value 2−to 2+, the system crosses over to the universality class characterized by a short-range interaction and remains in that class for all σ > 2. Crossover phenomena constitute a very fascinating topic in the subject of phase transitions but we cannot pursue them here any more; the interested reader may refer to an excellent review by Aharony (1976). Another topic of considerable interest deals with the so-called interfacial phase tran-sitions in both magnets and fluids. In his seminal paper of 1944, Onsager included in his model a row of “mismatched spins,” calculated the boundary tension (or what is more commonly referred to as the interfacial free energy) of this row and examined how this quantity vanished as T approached Tc. In the case of a fluid system, this corresponds to the disappearance of the meniscus between the liquid and the vapor and hence to the vanishing of the conventional surface tension as T →Tc−; see, in this connection, Prob-lem 12.27. A theoretical study of such interfacial layers involves consideration of the free energy of an inhomogeneous system, which has been a subject of considerable research for quite some time. We refer the interested reader to two review articles — by Abraham (1986) and by Jasnow (1986) — for further reading on this topic. A major ingredient employed by the RG approach is the fact that the critical behav-ior of a system is invariant under a scale transformation. It did not take very long to realize that an important connection exists between this transformation and the well-known con-formal transformation in a complex plane, for the latter too is, roughly speaking, a scale transformation in which the scale factor l varies continuously with position. Though, in principle, this connection could be relevant in all dimensions, the most fruitful applica-tions have been in the realm of two dimensions (where the conformal group consists of analytic functions of a complex variable). Among the important results emerging from the conformal transformation approach, one may mention the form of the many-point cor-relation functions, the critical behavior of finite-sized strips of different sizes and shapes, and the nature of the surface critical effects. For details, see the review article by Cardy (1987). Another area of interest pertains to the so-called multicritical points, for which refer-ence may be made to Lawrie and Sarbach (1984) for theoretical studies and to Knobler and Scott (1984) for experimental results. 570 Chapter 14. Phase Transitions: The Renormalization Group Approach 14.5 Finite-size scaling In our study of phase transitions so far, we generally worked in the thermodynamic limit, that is, we started with a lattice of size L1 × ... × Ld, containing N1 × ... × Nd spins (where Nj = Lj/a, a being the lattice constant), but at some appropriate stage of the calculation resorted to the limit Lj →∞. This limiting process is crucial in some important respects; while it simplifies subsequent calculations, it also generates singularities which, as we know, are a hallmark of systems undergoing phase transitions. It is of considerable inter-est, both theoretically and experimentally, to find out what happens (or does not happen) if some of the Lj are allowed to stay finite. The resulting analysis is quite complicated, but considerable progress has been made in this direction during the last 25 years or so. Accordingly, a whole new subject entitled “finite-size scaling” has emerged, of which only a brief summary will be presented here. The reader interested in further details may refer to Barber (1983), Cardy (1988), and Privman (1990). To fix ideas, we start with a d-dimensional bulk system (“bulk” in the sense that it is infi-nite in all its dimensions) that undergoes a phase transition at a finite critical temperature Tc(∞); clearly, the dimensionality d must be greater than the “lower critical dimension” d<. We also assume that d is less than the “upper critical dimension” d>, so that the critical exponents of the system are d-dependent and obey the hyperscaling relation dν = 2 −α = 2β + γ . (1) We now consider a similar system that is infinite in only d′ dimensions, where d′ < d, and finite in the remaining dimensions; the geometry of this system may be expressed as Ld−d′ × ∞d′, where L ≫a and, for simplicity, is taken to be the same in all finite dimen-sions. We may expect this system to be critical at a finite temperature Tc(L), not very different from Tc(∞). In reality, this is so only if d′ too is greater than d<; otherwise, the system continues to be regular at all finite temperatures and the criticality sets in only at T = 0.12 The cases d′ > d< and d′ ≤d<, therefore, merit separate treatments. Our primary goal here is to determine the L-dependence of the various physical quan-tities pertaining to the system when the system is undergoing a phase transition. We attain this goal by setting up a finite-size scaling law that generalizes equation (12.10.7) or equa-tion (14.3.26) to systems with a finite L. Now, since the only relevant length in the region of a phase transition is the correlation length ξ of the system, it is natural that we scale L with ξ — leading to the combination (L/ξ) ∼Ltν = (L1/νt)ν. (2) At the same time, the combination (h/t1) appearing in the bulk scaling law may be written as (h/t1) = (hL1/ν)/(L1/νt)1. (3) 12For the special case d′ = 0, when the system is fully finite, this point has already been emphasized in Section 12.1. Here we assert that, even when some of the system dimensions are infinite (and hence the total number of spins is infinite), a finite-temperature singularity does not arise unless the number of those infinite dimensions exceeds d<. 14.5 Finite-size scaling 571 The appropriate combinations of L with t and h, therefore, are L1/νt and L1/νh, respec-tively. The “singular” part of the free energy density of the system may then be written in the form (see Privman and Fisher, 1984) f (s)(t,h;L) ≡A(s) VkT ≈L−dY (x1,x2), (4) where x1 and x2 are the scaled variables of the system, namely x1 = C1L1/νt, x2 = C2L1/νh, (5a, b) with t = T −Tc(∞) Tc(∞) , h = µB kT |t|,h ≪1, (6a, b) while C1 and C2 are certain nonuniversal scale factors peculiar to the system under study. Expressed in terms of the variables x1 and x2, the function Y is expected to be a univer-sal function — common to all systems in the same universality class as the system under study. Of course, the definition of the universality class will now include (apart from the conventional parameters d,n, and the range of the spin–spin interaction) the parameter d′ as well as the nature of the boundary conditions imposed on the system (which, unless stated otherwise, will be assumed to be periodic). We note that, in the limit L →∞, expression (4) reduces to equation (12.10.7), provided that the function Y has the asymptotic form Y (x1,x2) ≈|x1|dνf±(x2/|x1|1) |x1|,x2 ≫1, (7) thus identifying the nonuniversal parameters F and G with C dν 1 and C2/C1 1 , respectively. This enables us to write C1 and C2 in terms of F and G, namely C1 ∼F1/(2−α), C2 ∼F(β+γ )/(2−α)G, (8a, b) which provides a means of determining the nonuniversal parameters C1 and C2 from a knowledge of the bulk parameters F and G; any other factors appearing in (8) would be universal. Once C1 and C2 are known, no more nonuniversal amplitudes are needed to describe the behavior of the system — regardless of whether it is finite-sized or infinite in extent. We are now in a position to examine the consequences of the scaling law (4). With appropriate differentiations, we obtain from equation (4) the following expres-sions for the zero-field susceptibility per unit volume and the “singular” part of the zero-field specific heat per unit volume: χ0(t;L) = −1 V ∂2A(s) ∂B2 ! B=0 ≈−kTµ2C2 2L21/ν−d (kT)2 ∂2Y (x1,x2) ∂x2 2 ! x2=0 = µ2C2 2Lγ/ν kT Yχ(x1), (9) 572 Chapter 14. Phase Transitions: The Renormalization Group Approach and c(s) 0 (t;L) = −T V ∂2A(s) ∂T2 ! B=0 ≈−kT2C2 1L2/ν−d T2 c (∞) ∂2Y (x1,x2) ∂x2 1 ! x2=0 = kT2C2 1Lα/ν T2 c (∞) Yc(x1), (10) where Yχ(x1) and Yc(x1) are appropriate derivatives of the universal function Y (x1,x2) and, hence, are themselves universal. We may, for further analysis, supplement the above results with the corresponding ones for the correlation length of the finite-sized system, namely ξ(t,h;L) = LS(x1,x2) (11) and ξ0(t;L) = LS(x1), (12) where S(x1) = S(x1,0); note that the functions S(x1,x2) and S(x1) are also universal. We shall now focus our attention on equations (9), (10), and (12), to see what messages they deliver in different regimes of the variables T and L. Case A: T ≳Tc(∞) With t > 0 and L ≫a, the variable x1 in this regime would be positive and much greater than unity. The functions Yχ,Yc, and S are then expected to assume the form Yχ(x1) ≈0x−γ 1 , Yc(x1) ≈Ax−α 1 , S(x1) ≈Nx−ν 1 , (13a, b, c) so that we recover the standard bulk results χ0 ≈µ20C−γ 1 C2 2 kTc(∞) t−γ , c(s) 0 ≈kAC2−α 1 t−α, ξ0 ≈NC−ν 1 t−ν, (14a, b, c) complete with nonuniversal amplitudes and universal factors. The effect of L in this regime appears only as a correction to the bulk results; under periodic boundary conditions, such correction terms turn out to be exponentially small, that is, O(e−L/ξ0) where ξ0 ∼a.13 Case B: T ≃Tc(∞) This case refers to the “core region” where |x1| is of order unity and hence |t| is of order L−1/ν; the bulk critical point, t = 0, is at the heart of this region. Equations (9), (10), and (12) now yield the first significant results of finite-size scaling, namely χ0 ∼ µ2C2 2 kTc(∞)Lγ/ν, c(s) 0 ∼kC2 1Lα/ν, ξ0 ∼L. (15a, b, c) 13See, for instance, Luck (1985), and Singh and Pathria (1985b, 1987a). 14.5 Finite-size scaling 573 Case C: T < Tc(∞) Here we must distinguish between the cases d′ > d< and d′ ≤d<. In the first case, the sys-tem becomes critical at a temperature Tc(L) that is not too far removed from Tc(∞); in the second, the system remains regular at all finite temperatures and becomes critical only at T = 0. (i) d′ > d< In view of the fact that the system is now singular at T = Tc(L) rather than at Tc(∞), it seems natural to define a shifted temperature variable ˙ t such that ˙ t = T −Tc(L) Tc(∞) ; (16) compare this with (6a). Thus, for any temperature T, ˙ t = t −τ; τ = [Tc(L) −Tc(∞)]/Tc(∞), (17) which prompts us to define a shifted scaled variable ˙ x1 = C1L1/ν˙ t = x1 −X; X = C1L1/ντ. (18) Clearly, the scaling functions governing the system would now be singular at ˙ x1 = 0, that is, at x1 = X. With no other arguments present, we presume that |X| will be of order unity; the shift in Tc is thus given by |τ| = |X|C−1 1 L−1/ν = O(L−1/ν). (19) Now, as T →Tc(L), the correlation length of the system approaches infinity — with the result that, insofar as the qualitative nature of the critical behavior is concerned, the finite variable L, however large, becomes essentially unimportant. The behavior of the system, in the immediate neighborhood of Tc(L), would, therefore, be characteristic of a d′-dimensional bulk system rather than of a d-dimensional one; accordingly, it would be governed by the critical exponents ˙ α, ˙ β,... pertaining to d′ dimensions rather than by the exponents α,β,... pertaining to d dimensions. We, therefore, expect that, as ˙ x1 →0, the functions Yχ, Yc, and S of equations (9), (10), and (12) assume the form Yχ(x1) ≈˙ 0˙ x−˙ γ 1 , Yc(x1) ≈˙ A˙ x−˙ α 1 , S(x1) ≈˙ N ˙ x−˙ ν 1 , (20a, b, c) with the result that χ0 ≈[µ2/kTc(∞)] ˙ 0C−˙ γ 1 C2 2L(γ −˙ γ )/ν˙ t−˙ γ , (21a) c(s) 0 ≈k ˙ AC2−˙ α 1 L(α−˙ α)/ν˙ t−˙ α, (21b) 574 Chapter 14. Phase Transitions: The Renormalization Group Approach and ξ0 ≈˙ NC−˙ ν 1 L(ν−˙ ν)/ν˙ t−˙ ν. (21c) It is obvious that, for ˙ t < 0 but such that |˙ t| ≪1, the same results would hold — except that ˙ t would be replaced by |˙ t|. The contents of equations (21), insofar as the dependence on L and ˙ t is concerned, have been verified by direct calculation on a variety of systems over the years; for details, see the review articles by Barber (1983) and Privman (1990) cited earlier. More recently, Allen and Pathria (1989) have verified the nonuniversal amplitudes as well by carrying out an explicit calculation for the spherical model (n = ∞) in the general geometry Ld−d′ × ∞d′, with both d and d′ greater than d<. Remarkably enough, they found that, just as the critical exponents ˙ α, ˙ β,... are the same functions of d′ as the exponents α,β,... are of d, the universal coefficients ˙ 0, ˙ A,... too are the same functions of d′ as the coefficients 0,A,... are of d; the same is true of the coefficients appearing in the presence of a magnetic field (see Allen and Pathria, 1991). One wonders if this would also be the case for general n! (ii) d′ ≤d< In this case the singularity of the problem lies at T = 0, with the result that at all finite temperatures the system is regular and hence expressible by smooth, analytic functions. We may, therefore, generalize the scaling law (4) to apply at all temperatures down to T = 0 by simply allowing the scale factors C1 and C2 to become T-dependent and writing (after Singh and Pathria, 1985b, 1986a) x1 = ˜ C1(T)L1/νt, x2 = ˜ C2(T)L1/νh, (22a, b) leaving t and h unchanged; the quantities ˜ C1 and ˜ C2 must be such that, as T →Tc(∞) from below, they approach the quantities C1 and C2 of equations (5). Expressions (9) and (10) now take the form χ0(t;L) = µ2 ˜ C2 2Lγ/ν kT Yχ(x1) (23) and c(s) 0 (t;L) = kT2  ∂ ∂T ( ˜ C1t) 2 Lα/νYc(x1), (24) respectively, while expression (12) remains formally the same. Now, as we approach the critical temperature Tc (which is zero here), we again expect the quantities χ0, c(s) 0 , and ξ0 to behave in a manner characteristic of a d′-dimensional bulk system. So, let us assume that, in limit d →0, our scale factors behave as ˜ C1 ∼Tr, ˜ C2 ∼Ts (T →0), (25a, b) 14.5 Finite-size scaling 575 and our universal functions behave as Yχ(x1) ∼|x1|θ, Yc(x1) ∼|x1|φ, S(x1) ∼|x1|σ (x1 →−∞). (26a, b, c) The resulting T-dependence of χ0,c(s) 0 , and ξ0 then is χ0 ∼T2s−1+θr, c(s) 0 ∼T(2+φ)r, ξ0 ∼Tσr. (27a, b, c) The corresponding results for an n-vector, d′-dimensional model (with n ≥2 and d′ < d<, where d< = 2) are14 χ0 ∼T−2/(2−d′), c(s) 0 ∼Td′/(2−d′), ξ0 ∼T−1/(2−d′). (28a, b, c) Comparing (27) with (28), we infer that θ = 1 r  1 −2s − 2 2 −d′  , φ = d′ r(2 −d′) −2, σ = −1 r(2 −d′). (29a, b, c) Very shortly we shall find, see equations (44), that r = −1/ν(d −2), s = β/ν(d −2), (30a, b) with the results θ = 2β + νd′(d −2) (2 −d′) , φ = −νd′(d −2) (2 −d′) −2, σ = ν(d −2) (2 −d′) . (31a, b, c) The L-dependence of the various quantities now turns out to be χ0 ∼L(γ +θ)/ν ∼L2(d−d′)/(2−d′) (32a) c(s) 0 ∼L(α+φ)/ν ∼L−2(d−d′)/(2−d′) (32b) and ξ0 ∼L1+σ/ν ∼L(d−d′)/(2−d′). (32c) It is remarkable that in these last expressions the critical exponents pertaining to d dimen-sions have disappeared altogether and the resulting powers of L depend entirely on the geometry of the system! Expression (32a) agrees with the earlier results for χ0 pertaining to a “block” geometry (d′ = 0) and to a “cylindrical” geometry (d′ = 1), namely χ0 ∼ ( Ld for d′ = 0 (33a) L2(d−1) for d′ = 1; (33b) 14For the spherical model (n = ∞), these results appear in Section 13.5; see equations (13.5.34), (13.5.35), and (13.5.64). Since the criticality in this case occurs at absolute zero, these results hold for all models with continuous symmetry, that is, with n ≥2. See, for instance, Section 13.3, where n is general but d′ = 1. 576 Chapter 14. Phase Transitions: The Renormalization Group Approach see Fisher and Privman (1985). For d′ = 2, the L-dependence of the various quantities studied here becomes exponential instead of a power law. To obtain results valid for all T in the range 0 < T < Tc(∞), we need to know the full T-dependence of the scale factors ˜ C1 and ˜ C2. It turns out that this too can be determined from the properties of the corresponding bulk system — in particular, from the field-free bulk correlation function G(R,T), which is known to possess the following forms: G(R,T) ∼R−(d−2+η) T = Tc(∞) (34) and G(R,T) = m2 0(T) + A(T)R−(d−2) T < Tc(∞), (35) where m0(T) is the order parameter of the bulk system and A(T) another system-dependent parameter.15 Now, the correlation function of the finite-sized system may be written in the scaled form G(R,t,h;L) ≈˜ D(T)R−(d−2+η)Z(x1,x2,x3), (36) where the scaled variables x1 and x2 are the same given by equations (22a, b) while x3 = R/L; as usual, the scale factor ˜ D(T) is nonuniversal while the function Z(x1,x2,x3) is universal. Expression (36) already conforms to (34), with x1 = x2 = x3 = 0. For conformity with (35), the function Z must possess the following asymptotic form: Z(x1,x2,x3) ≈Z1(|x1|νx3)d−2+η + Z2(|x1|νx3)η x1 →−∞, x2 = 0, x3 →0, (37) with Z1 = m2 0(T) ˜ D(T)[ ˜ C1(T)|t|]ν(d−2+η) , Z2 = A(T) ˜ D(T)[ ˜ C1(T)|t|]νη . (38a, b) It follows that ˜ C1(T)|t| ∼ " m2 0(T) A(T) #1/ν(d−2) , ˜ D(T) ∼ " Aβ(T) mνη 0 (T) #2/ν(d−2) ; (39a, b) here, use has been made of the fact that ν(d −2 + η) = (2 −α) −γ = 2β. (40) We shall now establish a relationship between the scale factors ˜ C2 and ˜ D. For this, we utilize the fluctuation–susceptibility relation (12.11.12) which, with the help of 15Note that the exponent η appears only in equation (34) and not in (35); for details, see Schultz et al. (1964) and Fisher et al. (1973). 14.5 Finite-size scaling 577 expression (36), gives for the zero-field susceptibility per unit volume χ0(t;L) = µ2 ˜ D(T) a2dkT Z Z(x1,0,R/L) Rd−2+η ddR = µ2 ˜ D(T) a2dkT L2−ηZχ(x1), (41) where Zχ is another universal function. Comparing (41) with (23), we get ˜ D(T) ∼a2d ˜ C2 2(T). (42) Equation (39b) now gives ˜ C2(T) ∼a−d[Aβ(T)/mνη 0 (T)]1/ν(d−2). (43) Equations (39) and (43) give us the full T-dependence of the scale factors ˜ C1, ˜ C2, and ˜ D for all T in the range 0 < T < Tc(∞); these equations were first derived by Singh and Pathria (1987a). Before utilizing these results we note that since, in the limit T →0, m0(T) approaches a constant value while A(T) ∼T, expressions (39a) and (43) yield ˜ C1|t| ∼T−1/ν(d−2), ˜ C2 ∼Tβ/ν(d−2), (44a, b) exactly as stipulated in equations (25) and (30). As for the T-dependence of the quantities χ0, c(s) 0 , and ξ0, we observe that, regardless of whether we keep L fixed and let T →0 or keep T fixed and let L →∞, in either case x1 →−∞; the asymptotic forms (26) of the universal functions Yχ, Yc, and S, therefore, apply throughout the region under study. Now, with θ, φ, and σ given by equations (31), our final results for χ0, c(s) 0 , and ξ0 turn out to be χ0 ∼µ2A(T) a2dkT " m2 0(T) A(T) #2/(2−d′) L2(d−d′)/(2−d′), (45) c(s) 0 ∼k ( T ∂ ∂T " m2 0(T) A(T) #)2 " m2 0(T) A(T) #−(4−d′)/(2−d′) L−2(d−d′)/(2−d′), (46) and ξ0 ∼ " m2 0(T) A(T) #1/(2−d′) L(d−d′)/(2−d′), (47) complete with nonuniversal amplitudes. Comparing (45) with (47), we find that, in the regime under study, χ0/ξ2 0 ∼µ2A(T)/a2dkT, (48) 578 Chapter 14. Phase Transitions: The Renormalization Group Approach a function of T only. In the case of the spherical model, since A(T) is proportional to T at all T < Tc(∞), see equation (13.5.71), the quantity χ0/ξ2 0 is a constant — independent of both T and L; see also equation (13.5.64). It is important to note that the above formulation ties very neatly with the one provided by the scale factors C1 and C2 of equations (5) that covered cases A and B pertaining to the regions T ≳Tc(∞) and T ≃Tc(∞), respectively. To see this, we observe that, as T →Tc(∞) from below, m0(T) becomes ∼|t|β and A(T) ∼|t|νη; expressions (39a) and (43) then assume the form ˜ C1(T)|t| ∼|t|(2β−νη)/ν(d−2) ∼|t|1 (49a) and ˜ C2(T) ∼|t|0. (49b) Clearly, ˜ C1(T) and ˜ C2(T) now assume some constant values that may be identified with C1 and C2 — thus providing a unified formulation through the same universal func-tions Y (x1,x2), S(x1,x2), and Z(x1,x2,x3) covering the regions of both first-order and second-order phase transitions! Remarkable though it is, this finding is not really surpris-ing because, with L finite and d′ ≤d<, the system is critical only at T = 0 and analytic everywhere else; so, its properties should indeed be expressible by a single set of func-tions throughout. Of course, as L →∞, the criticality spreads all the way from T = 0 to T = Tc(∞). As regards the spin dimensionality n, our results for cases A, B, and C(i) were quite gen-eral; only in case C(ii) did we specialize to systems with continuous symmetry (n ≥2). With a slight modification, the case of discrete symmetry (n = 1) can also be taken care of. The net result essentially is the replacement of the number 2 in equations (28) and henceforth by the “lower critical dimension,” d<, of the system — leading to results such as16 χ0 ∼Lζ , c(s) 0 ∼L−ζ , ξ0 ∼Lζ/d< T < Tc(∞), (50a, b, c) with ζ = d<(d −d′)/(d< −d′); compared with equations (32). Once again, the L-dependence of the various quantities of interest follows a power law, which changes to an exponential when d′ = d<; in the case of scalar models, this happens at d′ = 1. Throughout this discussion we have assumed that the total dimensionality d of the system is less than the “upper critical dimension” d>. The case d ≥d> presents some special problems but the net result is that, while the situation in the region T < Tc(∞) is described by the same set of expressions as above, in the region T ≃Tc(∞) it is con-siderably modified. For instance, one now gets in the region T ≃Tc(∞), for d > d> and d′ < d<, χ0 ∼L2(d−d′)/(d>−d′), c(s) 0 ∼L0, ξ0 ∼L(d−d′)/(d>−d′), (51a, b, c) 16See, for instance, Singh and Pathria (1986b). Problems 579 which may be compared with the corresponding results, (15a, b, c), for d < d>. Further-more, if d = d> and/or d′ = d<, factors containing lnL appear along with the power laws displayed in (51). For details, see Singh and Pathria (1986b, 1988, 1992). Finally we would like to emphasize the fact that finite-size effects in any given system are quite sensitive to the choice of the boundary conditions imposed on the system. For simplicity, we assumed the boundary conditions to be periodic. In real situations, there may be reasons to adopt different boundary conditions such as antiperiodic, free, and so on. This, in general, changes the mathematical character of the finite-size effects and the finite-size corrections in not only the singular part(s) of the various quantities stud-ied but in their regular part(s) as well. For comparison between theory and experiment, this aspect of the problem is of vital importance and deserves a close scrutiny. For lack of space we cannot go into this matter any further here; the interested reader may refer to a review article by Privman (1990), where other references on this topic can also be found. An allied subject in this regard is the “critical behavior of surfaces,” for which reference may be made to Binder (1983) and Diehl (1986). Problems 14.1. Show that the decimation transformation of a one-dimensional Ising model, with l = 2, can be written in terms of the transfer matrix P as P ′  K ′ = P2{K}, (1) where K and K ′ are the coupling constants of the original and the decimated lattice, respectively. Next show that, with P given by (P{K}) = eK0  eK1+K2 e−K1 e−K1 eK1−K2  , (2) see equation (13.2.4), relation (1) leads to the same transformation equations among K and K ′ as (14.2.8a, b, and c). 14.2. Verify that expression (15) of Section 14.2 indeed satisfies the functional equation (14) for the field-free Ising model in one dimension. Next show (or at least verify) that, with the field present, the functional equation (11), with K ′ given by (8), is satisfied by the more general expression f (K1,K2) = −ln  eK1 coshK2 + n e−2K1 + e2K1 sinh2 K2 o1/2 . 14.3. Verify that expression (32) of Section 14.2 indeed satisfies the functional equation (31) for the field-free spherical model in one dimension. Next show (or at least verify) that, with the field present, the functional equation (27), with K ′ given by (25), is satisfied by the more general expression f (K1,K2,3) = 1 2 ln  3 + q 32 −K 2 1 2π  − K 2 2 4(3 −K1), where 3 is determined by the constraint equation ∂f ∂3 = 1 2 q 32 −K 2 1 + K 2 2 4(3 −K1)2 = 1. 580 Chapter 14. Phase Transitions: The Renormalization Group Approach 14.4. Consider the field-free spherical model in one dimension whose partition function is given by equation (14.2.24) as well as by (14.2.19), with K ′ 2 = K2 = 0. Substituting σ ′ j = (23/K1)1/2s′ j in the former and comparing the resulting expression with the latter, show that QN(K1,3) = 2π K1 N′/2 QN′(K1,3′′), where N′ = 1 2N and 3′′ = (232/K1) −K1. This leads to the functional relation f (K1,3) = −1 4 ln 2π K1  + 1 2f (K1,3′′). Check that expression (14.2.32) satisfies this relation. 14.5. An approximate way of implementing an RG transformation on a square lattice is provided by the so-called Migdal–Kadanoff transformation17 shown in Figure 14.8. It consists of two essential steps: (i) First, one-half of the bonds in the lattice are simply removed, so as to change the length scale of the lattice by a factor of 2; to compensate for this, the coupling strength of the remaining bonds is changed from J to 2J. This takes us from Figure 14.8(a) to Figure 14.8(b). (ii) Next, the sites marked by crosses in Figure 14.8(b) are eliminated by a set of one-dimensional decimation transformations, leading to Figure 14.8(c) with coupling strength J′. (a) Show that the recursion relation for a spin- 1 2 Ising model on a square lattice, according to the above transformation, is x′ = 2x2/(1 + x4), where x = exp(−2K) and x′ = exp(−2K ′). Disregarding the trivial fixed points x∗= 0 and x∗= 1, show that a nontrivial fixed point of this transformation is x∗= 1 3  −1 + 2√2sinh 1 3 sinh−1 17 2√2  ≃0.5437; compare this with the actual value of xc, which is (√2 −1) ≃0.4142. (b) Linearizing around this nontrivial fixed point, show that the eigenvalue λ of this transformation is λ = 2(1 −x∗)/x∗≃1.6785 and hence the exponent ν = ln2/lnλ ≃1.338; compare this with the actual value of ν, which is 1. J J J J 2J 2J (a) (b) (c) J9 FIGURE 14.8 Migdal–Kadanoff transformation on a square lattice. 17See Kadanoff (1976a). Problems 581 14.6. Consider the linearized RG transformation (14.3.12), with A∗ l =  a11 a12 a21 a22  , (3) such that (a11a22 −a12a21) ̸= 0. We now introduce the “generalized coordinates” u1 and u2 through equations (14.3.13); clearly, u1 and u2 are certain linear combinations of the system parameters k1 and k2. (a) Show that the slopes of the lines u1 = 0 and u2 = 0, in the (k1,k2)-plane, are m1 = a21 λ2 −a22 = λ2 −a11 a12 and m2 = a21 λ1 −a22 = λ1 −a11 a12 , respectively; here, λ1 and λ2 are the eigenvalues of the matrix A∗ l . Verify that the product m1m2 = −a21/a12 and hence the two lines are mutually perpendicular if and only if a12 = a21. (b) Check that, in the special case when a12 = 0 but a21 ̸= 0, the above slopes assume the simple form m1 = ∞ and m2 = a21/(a11 −a22) whereas, in the special case when a21 = 0 but a12 ̸= 0, they become m1 = (a22 −a11)/a12 and m2 = 0; note that Figure 14.7 pertains to the latter case. (c) Examine as well the cases for which either a11 or a22 is zero; Figure 14.6 pertains to the latter of these cases. 14.7. Check that the critical exponents (14.4.38) through (14.4.40), in the limit n →∞, agree with the corresponding exponents for the spherical model of Section 13.5 with d ≲4. 14.8. Show, from equations (14.4.43) through (14.4.46), that for d ≲4 η ≃1 2nε2, γ ≃1 + 1 2  1 −6 n  ε, α ≃−1 2  1 −12 n  ε, where ε = (4 −d) ≪1. Check that these results agree with the ones following from equations (14.4.38) through (14.4.40) for n ≫1. 14.9. Using the various scaling relations, derive from equations (14.4.43) through (14.4.45) comparable expressions for the remaining exponents β, δ, and ν. Repeat for these exponents the exercise suggested in the preceding problem. 15 Fluctuations and Nonequilibrium Statistical Mechanics In this course we have been mostly concerned with the evaluation of statistical averages of the various physical quantities; these averages represent, with a high degree of accu-racy, the results expected from relevant measurements on the given system in equilibrium. Nevertheless, there do occur deviations from, or fluctuations about, these mean values. Even though they are generally small, their study is of great physical interest for several reasons. First, such a study enables us to develop a mathematical scheme with the help of which the magnitude of the relevant fluctuations, under a variety of physical situations, can be estimated. Not surprisingly, we find that while in a single-phase system the fluctu-ations are thermodynamically negligible but they can assume considerable importance in multiphase systems, especially in the neighborhood of a critical point. In the latter case, we obtain a rather high degree of spatial correlation among the molecules of the system which, in turn, gives rise to phenomena such as critical opalescence. Second, it provides a natural framework for understanding a class of phenomena that come under the heading “Brownian motion”; these phenomena relate properties such as the mobility of a fluid system, its coefficient of diffusion, and so on, with temperature through the so-called Einstein relations. The mechanism of Brownian motion is vital in for-mulating, and in a certain sense answering, questions as to how “a given physical system, which is not in a state of equilibrium, finally approaches such a state” while “a physical system, which is already in a state of equilibrium, persists to stay in that state.” Third, the study of fluctuations, as a function of time, leads to the concept of certain “correlation functions” that play a vital role in relating the dissipative properties of a sys-tem, such as the viscous resistance of a fluid or the electrical resistance of a conductor, with the microscopic properties of the system in a state of equilibrium; this relation-ship (between irreversible processes on one hand and equilibrium properties on the other) manifests itself in the so-called fluctuation–dissipation theorem. At the same time, a study of the “frequency spectrum” of fluctuations, which is related to the time-dependent correlation function through the fundamental theorem of Wiener and Khintchine, is of considerable value in assessing the “noise” met with in electrical circuits as well as in the transmission of electromagnetic signals. Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00015-3 © 2011 Elsevier Ltd. All rights reserved. 583 584 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics 15.1 Equilibrium thermodynamic fluctuations We begin by deriving a probability distribution law for the fluctuations of certain basic thermodynamic quantities pertaining to a given physical system; the mean square fluctu-ations can then be evaluated, in a straightforward manner, with the help of this law. We assume that the given system, which may be referred to as 1, is embedded in a reservoir, which may be referred to as 2, such that a mutual exchange of energy, and of volume, can take place between the two; of course, the overall energy E and the overall volume V are supposed to be fixed. For convenience, we do not envisage an exchange of particles here, so the numbers N1 and N2 remain individually constant. The equilibrium division of E into E1 and E2, and of V into V 1 and V 2, must be such that parts 1 and 2 of the composite sys-tem (1 + 2) have a common temperature T∗and a common pressure P∗; see Sections 1.2 and 1.3, especially equations (1.3.6). Of course, the entropy of the composite system will have its largest value in the equilibrium state; in any other state, such as the one character-ized by a fluctuation, it must have a lower value. If 1S denotes the deviation in the entropy of the composite system from its equilibrium value S0, then 1S ≡S −S0 = klnf −kln0, (1) where f (or 0) denotes the number of distinct microstates of the system (1 + 2) in the presence (or in the absence) of a fluctuation from the equilibrium state; see equation (1.2.6). The probability that the proposed fluctuation may indeed occur is then given by p ∝f ∝exp(1S/k); (2) see Section 3.1, especially equation (3.1.3). In terms of other thermodynamic quantities, we may write 1S = 1S1 + 1S2 = 1S1 + f Z 0 dE2 + P2dV2 T2 ; (3) note that the pressure P2 and the temperature T2 of the reservoir may, in principle, vary during the build-up of the fluctuation! Now, even if the fluctuation is sizable from the point of view of system 1, it will be small from the point of view of 2. The “variables” P2 and T2 may, therefore, be replaced by the constants P∗and T∗, respectively; at the same time, the increments dE2 and dV2 may be replaced by −dE1 and −dV1, respectively. Equation (3) then becomes 1S = 1S1 −(1E1 + P∗1V1)/T∗. (4) Accordingly, formula (2) takes the form p ∝exp{−(1E1 −T∗1S1 + P∗1V1)/kT∗}. (5) 15.1 Equilibrium thermodynamic fluctuations 585 Clearly, the probability distribution law (5) does not depend, in any manner, on the peculiarities of the reservoir in which the given system was supposedly embedded. For-mula (5), therefore, applies equally well to a system that attained equilibrium in a statistical ensemble (or, for that matter, to any macroscopic part of a given system itself). Conse-quently, we may drop the suffix 1 from the symbols 1E1,1S1, and 1V1, and the star from the symbols P∗and T∗, and write p ∝exp{−(1E −T1S + P1V)/kT}. (6) In most cases, the fluctuations are exceedingly small in magnitude; the quantity 1E may, therefore, be expanded as a Taylor series about the equilibrium value (1E)0 = 0, with the result 1E = ∂E ∂S  0 1S +  ∂E ∂V  0 1V + 1 2 " ∂2E ∂S2 ! 0 (1S)2 + 2 ∂2E ∂S∂V ! 0 1S1V + ∂2E ∂V 2 ! 0 (1V)2 # + ··· (7) Substituting (7) into (6) and retaining terms up to second order only, we obtain p ∝exp{−(1T1S −1P1V)/2kT}; (8) here, use has been made of the relations ∂E ∂S  0 = T,  ∂E ∂V  0 = −P, (9) and of the fact that the expression within the square brackets in (7) is equivalent to 1 ∂E ∂S  0 1S + 1  ∂E ∂V  0 1V = 1T1S −1P1V. (10) With the help of (8), the mean square fluctuations of various physical quantities and the statistical correlations among different fluctuations can be readily calculated. We note, however, that of the four 1 terms appearing in this formula only two can be chosen inde-pendently; the other two must assume the role of “derived quantities.” For instance, if we choose 1T and 1V to be the independent variables, then 1S and 1P can be written as 1S =  ∂S ∂T  V 1T +  ∂S ∂V  T 1V = CV T 1T +  ∂P ∂T  V 1V (11) and 1P =  ∂P ∂T  V 1T +  ∂P ∂V  T 1V =  ∂P ∂T  V 1T − 1 κTV 1V, (12) 586 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics κT being the isothermal compressibility of the system. Substituting (11) and (12) into (8), we get p ∝exp  −CV 2kT2 (1T)2 − 1 2kTκTV (1V)2  , (13) which shows that the fluctuations in T and V are statistically independent, Gaussian variables! A quick glance at (13) yields the results (1T)2 = kT2 CV , (1V)2 = kTκTV, (14a) while (1T1V) = 0. (14b) Similarly, if we choose 1S and 1P as our independent variables, we are led to the distribution law p ∝exp  − 1 2kCP (1S)2 −κSV 2kT (1P)2  , (15) which gives (1S)2 = kCP, (1P)2 = kT κSV , (16a) while (1S1P) = 0; (16b) here, κS denotes the adiabatic compressibility of the system. We note that, in general, the mean square fluctuation of an extensive quantity is directly proportional to the size of the system while that of an intensive quantity is inversely pro-portional to the same; in either case, the relative, root-mean-square fluctuation of any quantity is inversely proportional to the square root of the size of the system. Thus, except for situations such as the ones met with in a critical region, normal fluctuations are ther-modynamically negligible. This does not mean that fluctuations are altogether irrelevant to the physical phenomena taking place in the system; in fact, as will be seen in the sequel, the very presence of fluctuations at the microscopic level is of fundamental importance to several properties of the system displayed at the macroscopic level! With the help of the foregoing results, we may evaluate the mean square fluctuation in the energy of the system. With T and V as independent variables, we have 1E =  ∂E ∂T  V 1T +  ∂E ∂V  T 1V. (17) 15.2 The Einstein–Smoluchowski theory of the Brownian motion 587 Squaring this expression and taking averages, keeping in mind equations (14), we get (1E)2 = kT2CV + kTκTV  ∂E ∂V  T 2 = kT2CV + kTκT N2 V ! ∂E ∂N  T 2 . (18) Now, the results derived in the preceding paragraphs determine the fluctuations of the various physical quantities pertaining to any macroscopic subsystem of a given system, provided that the number of particles in the subsystem remains fixed. The expression (14b) for (1V)2 may, therefore, be used to derive an expression for the mean square fluctuation of the variable v (the volume per particle) and the variable n (the particle density) of the subsystem. We readily obtain (1v)2 = kTκTV/N2, (1n)2 = 1 v4 (1v)2 = kTκTN2/V 3; (19) note that the last result obtained here is in complete agreement with equation (4.5.7), which was derived on the basis of the grand canonical ensemble. A little reflection shows that this result applies equally well to a subsystem with a fixed volume V and a fluctuating number of particles N. The mean square fluctuation in N is then given by (1N)2 = V 2(1n)2 = kTκTN2/V. (20) Substituting (20) into (18), we obtain once again the grand canonical result for (1E)2, namely (1E)2 = kT2CV + (1N)2{(∂E/∂N)T}2, (21) as in equation (4.5.14). In passing, we note that the first part of expression (21) denotes the mean square fluc-tuation in the energy E of a subsystem for which both N and V are fixed, just as we have in the canonical ensemble (N,V,T). Conversely, if we assume the energy E to be fixed, then the temperature of the subsystem will fluctuate, and the mean square value of the quantity 1T will be given by (kT2CV ) divided by the square of the thermal capacity of the subsystem. The net result will, therefore, be (kT2/CV ), which is the same as in (14a). 15.2 The Einstein–Smoluchowski theory of the Brownian motion The term “Brownian motion” derives its name from the botanist Robert Brown who, in 1828, made careful observations on the tiny pollen grains of a plant under a microscope. In his own words: “While examining the form of the particles immersed in water, I observed 588 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics many of them very evidently in motion. These motions were such as to satisfy me ... that they arose neither from currents in the fluid nor from its gradual evaporation, but belonged to the particle itself.” We now know that the real source of this motion lies in the incessant, and more or less random, bombardment of the Brownian particles, as these grains (or, for that matter, any colloidal suspensions) are usually referred to, by the molecules of the surrounding fluid. It was Einstein who, in a number of papers (beginning in 1905), first provided a sound theoretical analysis of the Brownian motion on the basis of the so-called “random walk problem” and thereby established a far-reaching relationship between the irreversible nature of this phenomenon and the mechanism of molecular fluctuations. To illustrate the essential theme of Einstein’s approach, we first consider the problem in one dimension. Let x(t) denote the position of the Brownian particle at time t, given that its position coincided with the point x = 0 at time t = 0. To simplify matters, we assume that each molecular impact (which, on an average, takes place after a time τ ∗) causes the particle to jump a (small) distance l — of constant magnitude — in either a positive or negative direction along the x-axis. It seems natural to regard the possibilities 1x = +l and 1x = −l to be equally likely; though somewhat less natural, we may also regard the successive impacts on, and hence the successive jumps of, the Brownian particle to be mutually uncorrelated. The probability that the particle is found at the point x at time t is then equal to the probability that, in a series of n(= t/τ ∗) successive jumps, the particle makes m(= x/l) more jumps in the positive direction of the x-axis than in the negative, that is, it makes 1 2(n + m) jumps in the positive direction and 1 2(n −m) in the negative.1 The desired probability is then given by the binomial expression pn(m) = n! n 1 2(n + m) o ! n 1 2(n −m) o ! 1 2 n , (1) with the result that m = 0 and m2 = n. (2) Thus, for t ≫τ ∗, we have for the net displacement of the particle x(t) = 0 and x2(t) = l2 t τ ∗∝t1. (3) Accordingly, the root-mean-square displacement of the particle is proportional to the square root of the time elapsed: xr.m.s. = px2(t)  = l p (t/τ ∗) ∝t1/2. (4) It should be noted that the proportionality of the net overall displacement of the Brownian particle to the square root of the total number of elementary steps is a typical consequence 1Since the quantities x and t are macroscopic in nature while l and τ ∗are microscopic, the numbers n and m are much larger than unity; consequently, it is safe to assume that they are integral as well. 15.2 The Einstein–Smoluchowski theory of the Brownian motion 589 of the random nature of the steps and it manifests itself in a large variety of phenomena in nature. In contrast, if the successive steps were fully coherent (or else if the motion were completely predictable and reversible over the time interval t),2 then the net displacement of the Brownian particle would have been proportional to t1. Smoluchowski’s approach to the problem of Brownian motion, which appeared in 1906, was essentially the same as that of Einstein; the difference lay primarily in the math-ematical procedure. Smoluchowski introduced the probability function pn(x0|x), which denotes the “probability that, after a series of n steps, the Brownian particle, initially at the point x0, reaches the point x”; the number x here denotes the distance traveled by the Brownian particle in terms of the length of the elementary step. Clearly, pn(x0|x) = ∞ X z=−∞ pn−1(x0|z)p1(z|x) (n ≥1); (5) moreover, since a single step is equally likely to take the particle to the right or to the left, p1(z|x) = 1 2δz,x−1 + 1 2δz,x+1, (6) while p0(z|x) = δz,x. (7) Equation (5) is known as the Smoluchowski equation. To solve it, we introduce a generating function Qn(ξ), namely Qn(ξ) = ∞ X x=−∞ pn(x0|x)ξx−x0, (8) from which it follows that Q0(ξ) = ∞ X x=−∞ p0(x0|x)ξx−x0 = ∞ X x=−∞ δx0,xξx−x0 = 1. (9) Substituting (6) into (5), we obtain pn(x0|x) = 1 2pn−1(x0|x −1) + 1 2pn−1(x0|x + 1). (10) 2The term “reversible” here is related to the fact that the Newtonian equations of motion, which govern this class of phenomena, preserve their form if the direction of time is reversed (i.e., if we change t to −t, etc.); alternatively, one would expect that if at any instant of time we reverse the velocities of the particles in a given mechanical system, the system would “retrace” its path exactly. This is not true of equations describing “irreversible” phenomena, such as the diffusion equation (19), with which the phenomenon of Brownian motion is intimately related. 590 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Multiplying (10) by ξx−x0 and adding over all x, we obtain the recurrence relation Qn(ξ) = 1 2[ξ + (1/ξ)]Qn−1(ξ), (11) so that, by iteration, Qn(ξ) = 1 2[ξ + (1/ξ)] n Q0(ξ) = (1/2)n[ξ + (1/ξ)]n. (12) Expanding this expression binomially and comparing the result with (8), we get pn(x0|x) = 1 2 n n! { 1 2(n + x −x0)}!{ 1 2(n −x + x0)}! for |x −x0| ≤n 0 for |x −x0| > n. (13) Identifying (x −x0) with m, we find this result to be in complete agreement with our previ-ous result (1).3 Accordingly, any conclusions drawn from the Smoluchowski approach will be the same as the ones drawn from the Einstein approach. To obtain an asymptotic form of the function pn(m), we apply Stirling’s formula, n!≈(2πn)1/2(n/e)n, to the factorials appearing in (1), with the result lnpn(m) ≈  n + 1 2  lnn −1 2(n + m + 1)ln 1 2(n + m)  −1 2(n −m + 1)ln 1 2(n −m)  −nln2 −1 2 ln(2π). For m ≪n (which is generally true because m = 0 and mr.m.s. = n1/2, while n ≫1), we obtain pn(m) ≈ 2 √(2πn) exp(−m2/2n). (14) Taking x to be a continuous variable (and remembering that pn(m) ≡0 either for even val-ues of m or for odd values of m, so that in the distribution (14), 1m = 2 and not 1), we may write this result in the Gaussian form: p(x)dx = dx √(4πDt) exp −x2 4Dt ! , (15) where D = l2/2τ ∗. (16) 3It is easy to recognize the additional fact that if n is even, then pn(m) ≡0 for odd m, and if n is odd, then pn(m) ≡0 for even m. 15.2 The Einstein–Smoluchowski theory of the Brownian motion 591 Later on, we shall see that the quantity D introduced here is identical to the diffusion coefficient of the given system; equation (16) connects this quantity with the microscopic quantities l and τ ∗. To appreciate this connection, one has simply to note that the prob-lem of Brownian motion can also be looked on as a problem of “diffusion” of Brownian particles through the medium of the fluid; this point of view is also due to Einstein. How-ever, before we embark on these considerations, we would like to present here the results of an actual observation made on the Brownian motion of a spherical particle immersed in water; see Lee, Sears, and Turcotte (1963). It was found that the 403 values of the net dis-placement 1x of the particle, observed after successive intervals of 2 seconds each, were distributed as follows: Displacement 1x, in units of µ(= 10−4cm) Frequency of occurrence n less than −5.5 0 between −5.5 and −4.5 1 between −4.5 and −3.5 2 between −3.5 and −2.5 15 between −2.5 and −1.5 32 between −1.5 and −0.5 95 between −0.5 and +0.5 111 between +0.5 and +1.5 87 between +1.5 and +2.5 47 between +2.5 and +3.5 8 between +3.5 and +4.5 5 greater than +4.5 0 The mean square value of the displacement here turns out to be: (1x)2 = 2.09 × 10−8cm2. The observed frequency distribution has been plotted as a “block diagram” in Figure 15.1. We have included, in this figure, a Gaussian curve based on the observed value of the mean square displacement; we find that the experimental data fit the theoretical curve fairly well. We can also derive here an experimental value for the diffusion coefficient of the medium; we obtain: D = (1x)2/2t = 5.22 × 10−9cm2/s.4 We now turn to the study of the Brownian motion from the point of view of diffusion. We denote the number density of the Brownian particles in the fluid by the symbol n(r,t) and their current density by j(r,t){= n(r,t)v(r,t)}; then, according to Fick’s law, j(r,t) = −D∇n(r,t), (17) 4In the next section we shall see that, for a spherical particle, D = kT/6πηa where η is the coefficient of viscosity of the medium and a the radius of the Brownian particle. In the case under study, T ≃300K, η ≃10−2 poise, and a ≃ 4 × 10−5 cm. Substituting these values, we obtain for the Boltzmann constant: k ≃1.3 × 10−16erg/K. 592 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics 100 50 0 21 1 2 n(X ) 3 4 5 6 22 23 24 25 26 x FIGURE 15.1 The statistical distribution of the successive displacements, 1x, of a Brownian particle immersed in water: (1x)r.m.s. ≃1.45µ. where D denotes for the diffusion coefficient of the medium. We also have here the equation of continuity, namely ∇· j(r,t) + ∂n(r,t) ∂t = 0. (18) Substituting (17) into (18), we obtain the diffusion equation ∇2n(r,t) −1 D ∂n(r,t) ∂t = 0. (19) Of the various possible solutions of this equation, the one relevant to the present situa-tion is n(r,t) = N (4πDt)3/2 exp −r2 4Dt ! , (20) which is a spherically symmetric solution and is already normalized: ∞ Z 0 n(r,t)4πr2dr = N, (21) N being the total number of (Brownian) particles immersed in the fluid. A comparison of the (three-dimensional) result (20) with the (one-dimensional) result (15) brings out most vividly the relationship between the random walk problem on one hand and the phenomenon of diffusion on the other. It is clear that in the last approach we have considered the motion of an “ensemble” of N Brownian particles placed under “equivalent” physical conditions, rather than consid-ering the motion of a single particle over a length of time (as was done in the random walk approach). Accordingly, the averages of the various physical quantities obtained here will be in the nature of “ensemble averages”; they must, of course, agree with the long-time averages of the same quantities obtained earlier. 15.3 The Langevin theory of the Brownian motion 593 Now, by virtue of the distribution (20), we obtain ⟨r(t)⟩= 0; ⟨r2(t)⟩= 1 N ∞ Z 0 n(r,t)4πr4dr = 6Dt ∝t1, (22) in complete agreement with our earlier results, namely x(t) = 0; x2(t) = l2t/τ ∗= 2Dt ∝t1. (23) Thus, the “ensemble” of the Brownian particles, initially concentrated at the origin, “dif-fuses out” as time increases, the nature and the extent of its spread at any time t being given by equations (20) and (22), respectively. The diffusion process, which is clearly irrever-sible, gives us a fairly good picture of the statistical behavior of a single particle in the ensemble. However, the important thing to bear in mind is that, whether we focus our attention on a single particle in the ensemble or look at the ensemble as a whole, the ulti-mate source of the phenomenon lies in the incessant, and more or less random, impacts received by the Brownian particles from the molecules of the fluid. In other words, the irreversible character of the phenomenon ultimately arises from the random, fluctuating forces exerted by the fluid molecules on the Brownian particles. This leads us to another systematic theory of the Brownian motion, namely the theory of Langevin (1908). For a detailed analysis of the problem, see Uhlenbeck and Ornstein (1930), Chandrasekhar (1943, 1949), MacDonald (1948–1949), and Wax (1954). 15.3 The Langevin theory of the Brownian motion We consider the simplest case of a “free” Brownian particle, surrounded by a fluid envi-ronment; the particle is assumed to be free in the sense that it is not acted on by any other force except the one arising from the molecular bombardment. The equation of motion of the particle will then be M dv dt = F (t), (1) where M is the particle mass, v(t) the particle velocity, and F (t) the force acting on the particle by virtue of the impacts received from the fluid molecules. Langevin suggested that the force F (t) may be written as a sum of two parts: (i) an “averaged-out” part, which represents the viscous drag, −v/B, experienced by the particle (accordingly, B is the mobil-ity of the system, that is, the drift velocity acquired by the particle by virtue of a unit “external” force)5 and (ii) a “rapidly fluctuating” part F(t) which, over long intervals of 5If Stokes’s law is applicable, then B = 1/(6πηa), where η is the coefficient of viscosity of the fluid and a the radius of the particle (assumed spherical). 594 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics time (as compared to the characteristic time τ ∗), averages out to zero; thus, we may write M dv dt = −v B + F(t); F(t) = 0. (2) Taking the ensemble average of (2), we obtain6 M d dt ⟨v⟩= −1 B⟨v⟩, (3) which gives ⟨v(t)⟩= v(0)exp(−t/τ) (τ = MB). (4) Thus, the mean drift velocity of the particle decays, at a rate determined by the relaxation time τ, to the ultimate value zero. We note that this result is typical of the phenomena governed by dissipative properties such as the viscosity of the fluid; the irreversible nature of the result is also evident. Dividing (2) by the mass of the particle, we obtain an equation for the instantaneous acceleration, namely dv dt = −v τ + A(t); A(t) = 0. (5) We now construct the scalar product of (5) with the instantaneous position r of the particle and take the ensemble average of the product. In doing so, we make use of the facts that (i) r · v = 1 2(dr2/dt), (ii) r · (dv/dt) = 1 2(d2r2/dt2) −v2, and (iii) ⟨r · A⟩= 0.7 We obtain d2 dt2 ⟨r2⟩+ 1 τ d dt ⟨r2⟩= 2⟨v2⟩. (6) If the Brownian particle has already attained thermal equilibrium with the molecules of the fluid, then the quantity ⟨v2⟩in this equation may be replaced by its equipartition value 3kT/M. The equation is then readily integrated, with the result ⟨r2⟩= 6kTτ 2 M  t τ −(1 −e−t/τ)  , (7) 6The process of “averaging over an ensemble” implies that we are imagining a large number of systems similar to the one originally under consideration and are taking an average over this collection at any time t. By the very nature of the function F(t), the ensemble average ⟨F(t)⟩must be zero at all times. 7This is so because we have no reason to expect a statistical correlation between the position r(t) of the Brownian particle and the force F(t) exerted on it by the molecules of the fluid; see, however, Manoliu and Kittel (1979). Of course, we do expect a correlation between the variables v(t) and F(t); consequently, ⟨v · F⟩̸= 0 (see Problem 15.7). 15.3 The Langevin theory of the Brownian motion 595 where the constants of integration have been so chosen that at, t = 0, both ⟨r2⟩and its first time-derivative vanish. We observe that, for t ≪τ, ⟨r2⟩≃3kT M t2 = ⟨v2⟩t2, (8)8 which is consistent with the reversible equations of motion whereby one would simply have r = vt. (9) On the other hand, for t ≫τ, ⟨r2⟩≃6kTτ M t = (6BkT)t, (10)9 which is essentially the same as the Einstein–Smoluchowski result (15.2.22); incidentally, we obtain here a simple, but important, relationship between the coefficient of diffusion D and the mobility B, namely D = BkT, (11) which is generally referred to as the Einstein relation. The irreversible character of equation (10) is self-evident; it is also clear that it arises essentially from the viscosity of the medium. Moreover, the Einstein relation (11), which connects the coefficient of diffusion D with the mobility B of the system, tells us that the ultimate source of the viscosity of the medium (as well as of diffusion) lies in the random, fluctuating forces arising from the incessant motion of the fluid molecules; see also the fluctuation–dissipation theorem of Section 15.6. In this context, if we consider a particle of charge e and mass M moving in a viscous fluid under the influence of an external electric field of intensity E, then the “coarse-grained” motion of the particle will be determined by the equation M d dt ⟨v⟩= −1 B⟨v⟩+ eE; (12) compare this to equation (3). The “terminal” drift velocity of the particle would now be given by the expression (eB)E, which prompts one to define (eB) as the “mobility” of the system and denote it by the symbol µ. Consequently, one obtains, instead of (11), D = kT e µ, (13) which, in fact, is the original version of the Einstein relation; sometimes this is also referred to as the Nernst relation. 8Note that the limiting solution (8) corresponds to “dropping out” the second term on the left side of equation (6). 9Note that the limiting solution (10) corresponds to “dropping out” the first term on the left side of equation (6). 596 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics So far we have not felt any direct influence of the rapidly fluctuating term A(t) that appears in the equation of motion (5) of the Brownian particle. For this, let us try to eval-uate the quantity ⟨v2(t)⟩which, in the preceding analysis, was assumed to have already attained its “limiting” value 3kT/M. For this evaluation we replace the variable t in equa-tion (5) by u, multiply both sides of the equation by exp(u/τ), rearrange and integrate over du between the limits u = 0 and u = t; we thus obtain the formal solution v(t) = v(0)e−t/τ + e−t/τ t Z 0 eu/τA(u)du. (14) Thus, the drift velocity v(t) of the particle is also a fluctuating function of time; of course, since ⟨A(u)⟩= 0 for all u, the average drift velocity is given by the first term alone, namely ⟨v(t)⟩= v(0)e−t/τ, (15) which is the same as our earlier result (4). For the mean square velocity ⟨v2(t)⟩, we now obtain from (14) ⟨v2(t)⟩= v2(0)e−2t/τ + 2e−2t/τ  v(0) · t Z 0 eu/τ⟨A(u)⟩du   + e−2t/τ t Z 0 t Z 0 e(u1+u2)/τ⟨A(u1) · A(u2)⟩du1du2. (16) The second term on the right side of this equation is identically zero, because ⟨A(u)⟩van-ishes for all u. In the third term, we have the quantity ⟨A(u1) · A(u2)⟩, which is a measure of the “statistical correlation between the value of the fluctuating variable A at time u1 and its value at time u2”; we call it the autocorrelation function of the variable A and denote it by the symbol KA(u1,u2) or simply by K(u1,u2). Before proceeding with (16) any further, we place on record some of the important properties of the function K(u1,u2). (i) In a stationary ensemble (i.e., one in which the overall macroscopic behavior of the systems does not change with time), the function K(u1,u2) depends only on the time interval (u2 −u1). Denoting this interval by the symbol s, we have K(u1,u1 + s) ≡⟨A(u1) · A(u1 + s)⟩= K(s), independently of u1. (17) (ii) The quantity K(0), which is identically equal to the mean square value of the variable A at time u1, must be positive definite. In a stationary ensemble, it would be a constant, independent of u1: K(0) = const. > 0. (18) (iii) For any value of s, the magnitude of the function K(s) cannot exceed K(0). 15.3 The Langevin theory of the Brownian motion 597 Proof : Since ⟨|A(u1) ± A(u2)|2⟩= ⟨A2(u1)⟩+ ⟨A2(u2)⟩± 2(A(u1) · A(u2)⟩ = 2{K(0) ± K(s)} ≥0, the function K(s) cannot go outside the limits −K(0) and +K(0); consequently, |K(s)| ≤K(0) for all s. (19) (iv) The function K(s) is symmetric about the value s = 0, that is, K(−s) = K(s) = K(|s|). (20) Proof : K(s) ≡⟨A(u1) · A(u1 + s)⟩= ⟨A(u1 −s) · A(u1)⟩10 = ⟨A(u1) · A(u1 −s)⟩≡K(−s). (v) As s becomes large in comparison with the characteristic time τ ∗, the values A(u1) and A(u1 + s) become uncorrelated, that is K(s) ≡⟨A(u1) · A(u1 + s)⟩− − − − − − − → s≫τ∗ ⟨A(u1)⟩· ⟨A(u1 + s)⟩= 0. (21) In other words, the “memory” of the molecular impacts received during a given interval of time, say between u1 and u1 + du1, is “completely lost” after a lapse of time large in comparison with τ ∗. It follows that the magnitude of the function K(s) is significant only so long as the variable s is of the same order of magnitude as τ ∗. Figures 15.7 through 15.9 later in this chapter show the s-dependence of certain typical correlation functions K(s); they fully conform to the properties listed here. We now evaluate the double integral appearing in (16): I = t Z 0 t Z 0 e(u1+u2)/τK(u2 −u1)du1du2. (22) Changing over to the variables S = 1 2(u1 + u2) and s = (u2 −u1), (23) the integrand becomes exp(2S/τ)K(s), the element (du1du2) gets replaced by the corre-sponding element (dSds) while the limits of integration, in terms of the variables S and s, 10This is the only crucial step in the proof. It involves a “shift,” by an amount s, in both instants of the measurement process; the equality results from the fact that the ensemble is supposed to be stationary. 598 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics u2 u1 0 0 s 0 st s 2S S 0 S t/2 St s 2S s t s2(St) s 2(tS) t t FIGURE 15.2 Limits of integration, of the double integral I, in terms of the variables S and s. can be read from Figure 15.2; we find that, for 0 ≤S ≤t/2,s goes from −2S to +2S while, for t/2 ≤S ≤t, it goes from −2(t −S) to +2(t −S). Accordingly, I = t/2 Z 0 e2S/τdS +2S Z −2S K(s)ds + t Z t/2 e2S/τdS +2(t−S) Z −2(t−S) K(s)ds. (24) In view of property (v) of the function K(s), see equation (21), the integrals over s draw significant contribution only from a very narrow region, of the order of τ ∗, around the value s = 0 (i.e., from the shaded region in Figure 15.2); contributions from regions with larger values of |s| are negligible. Thus, if t ≫τ ∗, the limits of integration for s may be replaced by −∞and +∞, with the result I ≃C t Z 0 e2S/τdS = C τ 2(e2t/τ −1), (25) where C = ∞ Z −∞ K(s)ds. (26) Substituting (25) into (16), we obtain ⟨v2(t)⟩= v2(0)e−2t/τ + C τ 2(1 −e−2t/τ). (27) Now, as t →∞,⟨v2(t)⟩must tend to the equipartition value 3kT/M; therefore, C = 6kT/Mτ (28) 15.3 The Langevin theory of the Brownian motion 599 and hence ⟨v2(t)⟩= v2(0) + 3kT M −v2(0)  (1 −e−2t/τ). (29)11 We note that if v2(0) were itself equal to the equipartition value 3kT/M, then ⟨v2(t)⟩would always remain the same, which shows that statistical equilibrium, once attained, has a natural tendency to persist. Substituting (29) into the right side of (6), we obtain a more representative description of the manner in which the quantity ⟨r2⟩varies with t; we thus have d2 dt2 ⟨r2⟩+ 1 τ d dt ⟨r2⟩= 2v2(0)e−2t/τ + 6kT M (1 −e−2t/τ), (30) with the solution ⟨r2⟩= v2(0)τ 2(1 −e−t/τ)2 −3kT M τ 2(1 −e−t/τ)(3 −e−t/τ) + 6kTτ M t. (31) Solution (31) satisfies the initial conditions that both ⟨r2⟩and its first time-derivative van-ish at t = 0; moreover, if we put v2(0) = 3kT/M, it reduces to solution (7) obtained earlier. Once again, we note the reversible nature of the motion for t ≪τ, with ⟨r2⟩≃v2(0)t2, and its irreversible nature for t ≫τ, with ⟨r2⟩≃(6BkT)t. Figures 15.3 and 15.4 show the variation, with time, of the ensemble averages ⟨v2(t)⟩ and ⟨r2(t)⟩of a Brownian particle, as given by equations (29) and (31), respectively. All important features of our results are manifestly evident in these plots. Brownian motion continues to be a topic of contemporary research nearly 200 years after Brown’s discovery and over 100 years after Einstein and Smoluchowski’s analysis and early measurements by Perrin. The renewed interest is due to the growth in the techno-logical importance of colloids across a wide range of fields and the development of digital video and computer image analysis. An interesting example is the detailed observation and analysis of rotational and two-dimensional translational Brownian motion of ellipsoidal particles by Han et al. (2006) in a thin microscope slide. The case of rotational Brownian motion was first analyzed by Einstein (1906b) and first measured by Perrin (1934, 1936). Both rotational and translational modes diffuse according to Langevin dynamics but the translational diffusion is coupled to the rotational diffusion since the translational diffu-sion constant parallel to the longer axis is larger than the diffusion constant perpendicular 11One may check that d dt ⟨v2(t)⟩= 2 τ h v2(∞) −⟨v2(t)⟩ i = −2 τ 1⟨v2(t)⟩, where v2(∞) = 3kT/M and 1⟨v2(t)⟩is the “deviation of the quantity concerned from its equilibrium value.” In this form of the equation, we have a typical example of a “relaxation phenomenon,” with relaxation time τ/2. 600 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics 2 1 0 2 1 3 2 1 3 t τ 3kT/M FIGURE 15.3 The mean square velocity of a Brownian particle as a function of time. Curves 1, 2, and 3 correspond, respectively, to the initial conditions v2(0) = 6kT/M,3kT/M, and 0. 3 2 1 00 1 2 3 3 2 1 4 t τ 6BkTτ FIGURE 15.4 The mean square displacement of a Brownian particle as a function of time. Curves 1, 2, and 3 correspond, respectively, to the initial conditions v2(0) = 6kT/M, 3kT/M, and 0. to that axis. The rotational diffusion and both long-axis (a) and short-axis (b) body-frame diffusions are all Gaussian: pθ(1θ,t) = 1 √4πDθt exp −(1θ)2 4Dθt ! , (32a) pa(1xa,t) = 1 √4πDat exp −(1xa)2 4Dat ! , (32b) pb(1xb,t) = 1 p 4πDbt exp −(1xb)2 4Dbt ! , (32c) 15.3 The Langevin theory of the Brownian motion 601 with diffusion constants Dθ, Da, and Db. Experiments have observed the complex two-dimensional spatial diffusion at short times (t ≲τθ = 1/(2Dθ)), as predicted by the Langevin theory. The long-time (t ≫τθ) spatial diffusion is isotropic with diffusion con-stant D = (Da + Db)/2. 15.3.A Brownian motion of a harmonic oscillator An analysis similar to the one for a diffusing Brownian particle can also be performed for a particle in a harmonic oscillator potential that prevents the particle from diffus-ing away from the origin and allows a more general analysis of the relationship between the position and velocity response functions and the power spectra of the fluctua-tions; see Kappler (1938) and Chandrasekhar (1943). The one-dimensional equation of motion for a Brownian particle of mass M in a harmonic oscillator potential with spring constant Mω2 0 is d2x dt2 + γ dx dt + ω2 0x = F(t) M , (33) where γ (= 6πηa/M) is the damping coefficient of a spherical particle in a fluid with viscosity η. Just as in the case of diffusive Brownian motion, the force F(t) can be a time-dependent external force designed to explore the response function or a time-dependent random force due to collisions with molecules in the fluid to analyze the equilibrium fluc-tuations. Assuming the system was in equilibrium in the distant past, the position at time t is given by x(t) = t Z −∞ χxx(t −t′)F(t′)dt′, (34) where χxx(s) = 1 Mω1 e−γ s 2 sin(ω1s) (35) is the xx response function and ω1 = q ω2 0 −γ 2 4 .12 The velocity response is given by v(t) = t Z −∞ χvx(t −t′)F(t′)dt′, (36) 12This form of the response function assumes that the oscillator is underdamped. The notation χxx refers to the notation used in Section 15.6.A in which the response of the position coordinate x depends on the applied field F that couples to the Hamiltonian via a term −F(t)x(t). 602 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics where χvx(s) = 1 M e−γ s 2  cos(ω1s) −γ 2ω1 sin(ω1s)  . (37) The response of the system can be decomposed into a sum of independent terms involving a sinusoidal applied force ˆ F(ω)eiωt. This takes the form ˆ x(ω) = ˜ χxx(ω) ˆ F(ω), (38) where the frequency-dependent response function can be decomposed into real and imaginary parts ˆ χ′ xx(ω) and ˆ χ′′ xx(ω): ˜ χxx(ω) = ∞ Z 0 χxx(s)eiωsds = ˆ χ′ xx(ω) + i ˆ χ′′ xx(ω), (39a) ˆ χ′ xx(ω) = ω2 0 −ω2 M[(ω2 0 −ω2)2 + γ 2ω2] , (39b) ˆ χ′′ xx(ω) = γ ω M[(ω2 0 −ω2)2 + γ 2ω2] . (39c) The real part here describes the dispersion and the imaginary part describes the dissipa-tion, that is, it sets the average rate of energy dissipation due to the sinusoidal external force. Now let’s consider the natural fluctuations of the position and the velocity of the parti-cle in equilibrium due to the random collisions with the atoms in the fluid. We will use the same Langevin formalism as was used earlier with Brownian motion of a free particle. The random force averages to zero and is assumed to be delta-function correlated in time: ⟨F⟩= 0, (40a) F(t)F(t′) = 0δ(t −t′), (40b) where 0 = 2γ MkT. With this choice, the long-time average position and velocity of the particle are both zero, ⟨x(t)⟩= ⟨v(t)⟩= 0, (41) and the average of the squares of the position and velocity both obey the equipartition theorem: ⟨x2(t)⟩= kT Mω2 0 , ⟨v2(t)⟩= kT M . (42a,b) 15.4 Approach to equilibrium: the Fokker–Planck equation 603 The xx correlation function is given by Gxx(t −t′) = x(t)x(t′) = kT Mω2 0 exp −γ |t −t′| 2  cos ω1|t −t′|  + γ 2ω1 sin ω1|t −t′|  , (43) and the xx power spectrum by Sxx(ω) = ∞ Z −∞ Gxx(s)eiωsds = 2γ kT M 1 ω2 0 −ω22 + γ 2ω2 . (44) Note that the imaginary part of the response function, ˆ χ′′ xx(ω), in equation (39c) is propor-tional to the power spectrum Sxx(ω): ˆ χ′′ xx(ω) = ω 2kT Sxx(ω). (45) This result indicates that the dissipation that results from driving a system out of equilib-rium by an external force is proportional to the power spectrum of the natural fluctuations that occur in equilibrium. While this result was derived here for a very specific model, it constitutes an example of the very general fluctuation–dissipation theorem we will derive in Section 15.6.A. 15.4 Approach to equilibrium: the Fokker–Planck equation In our analysis of the Brownian motion we have considered the behavior of a dynamical variable, such as the position r(t) or the velocity v(t) of a Brownian particle, from the point of view of fluctuations in the value of the variable. To determine the average behavior of such a variable, we sometimes invoked an “ensemble” of Brownian particles immersed in identical environments and undergoing diffusion. A treatment along these lines was carried out toward the end of Section 15.2, and the most important results of that treat-ment are summarized in equation (15.2.20) for the density function n(r,t) and in equation (15.2.22) for the mean square displacement ⟨r2(t)⟩. A more generalized way of looking at “the manner in which, and the rate at which, a given distribution of Brownian particles approaches a state of thermal equilibrium” is provided by the so-called Master Equation, a simplified version of which is known as the Fokker–Planck equation. For illustration, we examine the displacement, x(t), of the given set of particles along the x-axis. At any time t, let f (x,t)dx be the probability that an arbi-trary particle in the ensemble may have a displacement between x and x + dx. The function 604 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics f (x,t) must satisfy the normalization condition ∞ Z −∞ f (x,t)dx = 1. (1) The Master Equation then reads: ∂f (x,t) ∂t = ∞ Z −∞ {−f (x,t)W(x,x′) + f (x′,t)W(x′,x)}dx′, (2) where W(x,x′)dx′δt denotes the probability that, in a short interval of time δt, a parti-cle having displacement x makes a “transition” to having a displacement between x′ and x′ + dx′.13 The first part of the integral in equation (2) corresponds to all those transitions that remove particles from the displacement x at time t to some other displacement x′ and, hence, represent a net loss to the function f (x,t); similarly, the second part of the integral corresponds to all those transitions that bring particles from some other displacement x′ at time t to the displacement x and, hence, represent a net gain to the function f (x,t).14 The structure of the Master Equation is thus founded on very simple and straightfor-ward premises. Of course, under certain conditions, this equation, or any generalization thereof (such as the one including velocity, or momentum, coordinates in the argument of f ), can be reduced to the simple form ∂f ∂t = −f −f0 τ , (3) which has proved to be a very useful first approximation for studying problems related to transport phenomena. Here, f0 denotes the equilibrium distribution function (for ∂f /∂t = 0 when f = f0), while τ is the relaxation time that determines the rate at which the fluctuations in the system drive it to a state of equilibrium. In studying Brownian motion on the basis of equation (2), we can safely assume that it is only transitions between “closely neighboring” states x and x′ that have an apprecia-ble probability of occurring; in other words, the transition probability function W(x,x′) is sharply peaked around the value x′ = x and falls rapidly to zero away from x. Denoting the interval (x′ −x) by ξ, we may write W(x,x′) →W(x;ξ), W(x′,x) →W(x′;−ξ) (4) 13We are tacitly assuming here a “Markovian” situation where the transition probability function W(x,x′) depends only on the present position x (and, of course, the subsequent position x′) of the particle but not on the previous history of the particle. 14In the case of fermions, an account must be taken of the Pauli exclusion principle, which controls the “occupation of single-particle states in the system”; for instance, we cannot, in that case, consider a transition that tends to transfer a particle to a state that is already occupied. This requires an appropriate modification of the Master Equation. 15.4 Approach to equilibrium: the Fokker–Planck equation 605 where W(x;ξ) and W(x′;−ξ) have sharp peaks around the value ξ = 0 and fall rapidly to zero elsewhere.15 This enables us to expand the right side of (2) as a Taylor series around ξ = 0. Retaining terms up to second order only, we obtain ∂f (x,t) ∂t = −∂ ∂x{µ1(x)f (x,t)} + 1 2 ∂2 ∂x2 {µ2(x)f (x,t)}, (5) where µ1(x) = ∞ Z −∞ ξW(x;ξ)dξ = ⟨δx⟩δt δt = ⟨vx⟩ (6) and µ2(x) = ∞ Z −∞ ξ2W(x;ξ)dξ = ⟨(δx)2⟩δt δt . (7) Equation (5) is the so-called Fokker–Planck equation, which occupies a classic place in the field of Brownian motion and fluctuations. We now consider a specific system of Brownian particles (of negligible mass), each par-ticle being acted on by a linear restoring force, Fx = −λx, and having mobility B in the surrounding medium; the assumption of negligible mass implies that the relaxation time τ(= MB) of equation (15.3.4) is very small, so the time t here may be regarded as very large in comparison with that τ. The mean viscous force, −⟨vx⟩/B, is then balanced by the linear restoring force, with the result that −⟨vx⟩ B + Fx = 0 (8) and hence ⟨vx⟩≡µ1(x) = −λBx. (9) Next, in view of equation (15.3.10), we have ⟨(δx)2⟩ δt ≡µ2(x) = 2BkT; (10) it will be noted that the influence of λ on this quantity is being neglected here. Substituting (9) and (10) into (5), we obtain ∂f ∂t = λB ∂ ∂x(xf ) + BkT ∂2f ∂x2 . (11) 15Clearly, this assumption limits our analysis to what may be called the “Brownian motion approximation,” in which the object under consideration is presumed to be on a very different scale of magnitude than the molecules constituting the environment. It is obvious that if one tries to apply this sort of analysis to “understand” the behavior of molecules themselves, one cannot hope for anything but a “crude, semiquantitative” outcome. 606 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Now we apply equation (11) to an “ensemble” of Brownian particles, initially concen-trated at the point x = x0. To begin with, we note that, in the absence of the restoring force (λ = 0), equation (11) reduces to the one-dimensional diffusion equation ∂f ∂t = D ∂2f ∂x2 (D = BkT), (12) which conforms to our earlier results (15.2.19) and (15.3.11). The present derivation shows that the process of diffusion is essentially a “random walk, at the molecular level.” In view of equation (15.2.20), the function f (x,t) here would be f (x,t) = 1 (4πDt)1/2 exp ( −(x −x0)2 4Dt ) , (13) with x = x0 and x2 = x2 0 + 2Dt; (14) the last result shows that the mean square distance traversed by the particle(s) increases linearly with time, without any upper limit on its value. The restoring force, however, puts a check on the diffusive tendency of the particles. For instance, in the presence of such a force (λ ̸= 0), the terminal distribution f∞(for which ∂f /∂t = 0) is determined by the equation ∂ ∂x(xf∞) + kT λ ∂2f∞ ∂x2 = 0, (15) which gives f∞(x) =  λ 2πkT 1/2 exp −λx2 2kT ! , (16) with x = 0 and x2 = kT/λ. (17) The last result agrees with the fact that the mean square value of x must ultimately comply with the equipartition theorem, namely ( 1 2λx2)∞= 1 2kT. From the point of view of equilib-rium statistical mechanics, if we regard Brownian particles with kinetic energy p2 x/2m and potential energy 1 2λx2 as loosely coupled to a thermal environment at temperature T, then we may directly write feq(x,px)dxdpx ∝e−(p2 x/2m+λx2/2)/kTdxdpx. (18) On integration over px, expression (18) leads directly to the distribution function (16). 15.4 Approach to equilibrium: the Fokker–Planck equation 607 f (x,t ) t 5 0 t 5kT λ x0 t5 x 0 1 2λB FIGURE 15.5 The distribution function (19) at times t = 0, t = 1/(2λB), and t = ∞. The general solution of equation (11), relevant to the ensemble under consideration, is given by f (x,t) =  λ 2πkT(1 −e−2λBt) 1/2 exp ( −λ(x −x0e−λBt)2 2kT(1 −e−2λBt) ) , (19) with x = x0e−λBt and x2 = x2 0e−2λBt + kT λ (1 −e−2λBt); (20) in the limit λ →0, we recover the purely “diffusive” situation, as described by equa-tions (13) and (14), while for t ≫(λB)−1, we approach the “terminal” situation, as described by equations (16) and (17). Figure 15.5 shows the manner in which an ensem-ble of Brownian particles approaches a state of equilibrium under the combined influence of the restoring force and the molecular bombardment; clearly, the relaxation time of the present process is ∼(λB)−1. A physical system to which the foregoing theory is readily applicable is provided by the oscillating component of a moving-coil galvanometer. Here, we have a coil of wire and a mirror that are suspended by a fine fiber, so they can rotate about a vertical axis. Random, incessant collisions of air molecules with the suspended system produce a succession of torques of fluctuating intensity; as a result, the angular position θ of the system continu-ally fluctuates and the system exhibits an unsteady zero. This is clearly another example of the Brownian motion! The role of the viscous force in this case is played by the mechanism of air damping (or, else, electromagnetic damping) of the galvanometer, while the restor-ing torque, Nθ = −cθ, arises from the torsional properties of the fiber. In equilibrium, we 608 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics expect that 1 2cθ2  = 1 2kT, that is, θ2 = kT c ; (21) compare this to equation (17). An experimental determination of the mean square deflec-tion, θ2, of such a system was made by Kappler (1931) who, in turn, applied his results to derive, with the help of equation (21), an empirical value for the Boltzmann constant k (or, for that matter, the Avogadro number NA). The system used by Kappler had a moment of inertia I = 4.552 × 10−4 gcm2 and a time period of oscillation τ = 1379s; accordingly, the constant c of the restoring torque had a value given by the formula τ = 2π(I/c)1/2, so that c = 4π2(I/τ 2) = 9.443 × 10−9gcm2s−2/rad. The observed value of θ2, at a temperature of 287.1 K, was 4.178 × 10−6. Substituting these numbers in (21), Kappler obtained: k = 1.374 × 10−16 erg K−1. And, since the gas constant R is equal to 8.31 × 107 erg K−1mole−1, he obtained for the Avogadro number: NA = R/k = 6.06 × 1023 mole−1. One might expect that by suspending the mirror system in an “evacuated” casing the fluctuations caused by the collisions of the air molecules could be severely reduced. This is not true because even at the lowest possible pressures there still remain a tremendously large number of molecules in the system that keep the Brownian motion “alive.” The inter-esting part of the story, however, is that the mean square deflection of the system, caused by molecular bombardment, is not at all affected by the density of the molecules; for a sys-tem in equilibrium, it is determined solely by the temperature. This situation is depicted, rather dramatically, in Figure 15.6 where we have two traces of oscillations of the mir-ror system, the upper one having been taken at the atmospheric pressure and the lower one at a pressure of 10−4 mm of mercury. The root-mean-square deviation is very nearly the same in the two cases! Nevertheless, one does note a difference of “quality” between the two traces that relates to the “frequency spectrum” of the fluctuations and arises for the following reason. When the density of the surrounding gas is relatively high, the molecular impulses come in rapid succession, with the result that the individual deflec-tions of the system are large in number but small in magnitude. As the pressure is lowered, the time intervals between successive impulses become longer, making the individual deflections smaller in number but larger in magnitude. However, the overall deflection, observed over a long interval of time, remains essentially the same. FIGURE 15.6 Two traces of the thermal oscillations of a mirror system suspended in air; the upper trace was taken at the atmospheric pressure, the lower one at a pressure of 10−4 mm of mercury. 15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem 609 15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem We have already made reference to the (spectral) quality of a fluctuation pattern. Refer-ring once again to the patterns shown in Figure 15.6, we note that, even though the mean square fluctuation of the variable θ is the same in the two cases, the second pattern is far more “jagged” than the first; in other words, the high-frequency components are far more prominent in the second pattern. At the same time, there is a lot more “predictabil-ity” in the first pattern (insofar as it is represented by a much smoother curve); in other words, the correlation function, or the memory function, K(s) of the first pattern extends over much larger values of s. In fact, these two aspects of a fluctuation process, namely its time-dependence and its frequency spectrum, are very closely related to one another. And the most natural course for studying this relationship is to carry out a Fourier analysis of the given process. For this study we consider only those variables, y(t), whose mean square value, ⟨y2(t)⟩, has already attained an equilibrium, or stationary, value: ⟨y2(t)⟩= const. (1) Such a variable is said to be statistically stationary. As an example of such a variable, we may recall the velocity v(t) of a “free” Brownian particle at times t much larger than the relaxation time τ, see equation (15.3.29), or the displacement x(t) of a Brownian particle moving under the influence of a restoring force (Fx = −λx) at times t much larger than (λB)−1, see equation (15.4.20). Now, if the variable y(t) were strictly periodic (and hence completely predictable), with a time period T = 1/f0, then we could write y(t) = a0 + ∞ X n=1 an cos(2πnf0t) + ∞ X n=1 bn sin(2πnf0t), (2) where a0 = 1 T T Z 0 y(t)dt, (3) an = 2 T T Z 0 y(t)cos(2πnf0t)dt, (4) and bn = 2 T T Z 0 y(t)sin(2πnf0t)dt; (5) in this case, the coefficients a and b would be completely known and would define, with no uncertainty, the frequency spectrum of the variable y(t). If, on the other hand, the given 610 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics variable is more or less a random function of time, then the coefficients a and b would themselves be statistical in nature. To apply the concept of periodicity to such a function, we must take the “time interval of repetition” to be infinitely large, that is, we let f0 →0. In the proposed limit, equation (3) would read a0 = Lim T→∞ 1 T T Z 0 y(t)dt ≡⟨y(t)⟩; (6) thus, the coefficient a0, which represents the mean (or d.c.) value of the variable y, may be determined either by taking a time average (over a sufficiently long interval) of the variable or by taking an ensemble average (at any instant of time t). For convenience, and without loss of generality, we take a0 = 0; in other words, we assume that from the actual values of the variable y(t) its mean value, ⟨y(t)⟩, has already been subtracted.16 Taking the ensemble average of equations (4) and (5), we obtain, for all n, ⟨an⟩= ⟨bn⟩= 0. (7) However, by taking the ensemble average of equation (2) squared, we obtain ⟨y2(t)⟩= X n 1 2⟨a2 n⟩+ X n 1 2⟨b2 n⟩ = X n 1 2 n ⟨a2 n⟩+ ⟨b2 n⟩ o = const. (8) The term 1 2{⟨a2 n⟩+ ⟨b2 n⟩} represents the respective “share,” belonging to the frequency nf0, in the total, time-independent value of the quantity ⟨y2(t)⟩. Now, in view of the random-ness of the phases of the various components, we have, for all n, ⟨a2 n⟩= ⟨b2 n⟩; consequently, equation (8) may be written as ⟨y2⟩= X n ⟨a2 n⟩≃ ∞ Z 0 w(f )df , (9) where ⟨a2 n⟩= w(nf0)1(nf0), that is, w(nf0) = 1 f0 ⟨a2 n⟩; (10) the function w(f ) defines the power spectrum of the variable y(t). We shall now show that the power spectrum w(f ) of the fluctuating variable y(t) is completely determined by its autocorrelation function K(s). For this, we make use of 16Obviously, this does not affect the spectral quality of the fluctuations, except that now we do not have a compo-nent with frequency zero. To represent the actual situation, one may have to add, to the resulting spectrum, a suitably weighted δ(f )-term. 15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem 611 equation (4), which gives ⟨a2 n⟩= 4f 2 0 1/f0 Z 0 1/f0 Z 0 ⟨y(t1)y(t2)⟩cos(2πnf0t1)cos(2πnf0t2)dt1dt2. (11) Changing over to the variables S = 1 2(t1 + t2) and s = (t2 −t1), and remembering that the interval T over which the integrations extend is much larger than the duration over which the “memory” of the variable y lasts, we obtain ⟨a2 n⟩≃2f 2 0 1/f0 Z S=0 ∞ Z s=−∞ K(s){cos(2πnf0s) + cos(4πnf0S)}dSds; (12) compare this to the steps that led us from equations (15.3.22) to (15.3.25) and (15.3.26). The second part of the integral in (12) vanishes on integration over S; the first part then gives ⟨a2 n⟩= 4f0 ∞ Z 0 K(s)cos(2πnf0s)ds. (13) Comparing (13) with (10), we obtain the desired formula w(f ) = 4 ∞ Z 0 K(s)cos(2πfs)ds. (14) Taking the inverse of (14), we obtain K(s) = ∞ Z 0 w(f )cos(2πfs)df . (15) For s = 0, formula (15) yields the important relationship K(0) = ∞ Z 0 w(f )df = ⟨y2⟩; (16) see equation (9) as well as the definition of the autocorrelation function of the variable y, namely K(s) = ⟨y(t1)y(t1 + s)⟩. Equations (14) and (15), which connect the complementary functions w(f ) and K(s), constitute a theorem that goes after the names of Wiener (1930) and Khintchine (1934). 612 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics We shall now look at some special cases of the variable y(t) to illustrate the use of the Wiener–Khintchine theorem. Case 1 If the given variable y(t) is extremely irregular, and hence unpredictable, then its correla-tion function K(s) would extend over a negligibly small range of the time interval s.17 We may then write K(s) = cδ(s). (17a) Equation (14) then gives w(f ) = 2c for all f . (17b) A spectrum in which the distribution (of power) over different frequencies is uniform is known as a “flat” or a “white” spectrum. We note, however, that if the uniformity of distri-bution were literally true for all frequencies, from 0 to ∞, then the integral in (16), which is identically equal to ⟨y2⟩, would diverge! We, therefore, expect that, in any realistic sit-uation, the correlation function K(s) will not be as sharply peaked as in (17a). Typically, K(s) will extend over a small range, O(σ), of the variable s, which in turn will define a “frequency zone,” with f = O(1/σ), such that the function w(f ) would undergo a change of character as f passes through this zone; toward lower frequencies, w(f ) →const. ̸= 0, while toward higher frequencies, w(f ) →const. = 0. One possible representation of this situation is shown in Figure 15.7 where we have taken, rather arbitrarily, K(s) = K(0)sin(as) as (a > 0), (18a) for which w(f ) = 2π a K(0) for f < a 2π 0 for f > a 2π . (18b) In the limit a →∞, equations (18) reduce to (17), with c = πa−1K(0). Case 2 On the other hand, if the variable y(t) is extremely regular, and hence predictable, then its correlation function would extend over large values of s; its power spectrum would then appear in the form of “peaks,” located at certain “characteristic frequencies” of the vari-able. In the simplest case of a monochromatic variable, with characteristic frequency f ∗, the correlation function would be K(s) = K(0)cos(2πf ∗s), (19a) 17This is essentially true of the rapidly fluctuating force F(t) experienced by a Brownian particle due to the incessant molecular impulses received by it. 15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem 613 0 2 2 22 K(s) a52 a 5 1 s a52 a 51 0 w (f ) f 1 2 1 FIGURE 15.7 The autocorrelation function K(s) and the power distribution function w(f ) of a given variable y(t); the parameter a appears in terms of an arbitrary unit of (time)−1. for which w(f ) = K(0)δ(f −f ∗); (19b) see Figure 15.8. A very special case arises when f ∗= 0; then, both y(t) and K(s) are constant in value, and the function w(f ) is peaked at the d.c. frequency f = 0. Case 3 If the variable y(t) represents a signal that arises from, or has been filtered through, a lightly damped tuned circuit (a “narrowband” filter), then its power will be distributed over a “hump” around the mean frequency f ∗. The function K(s) will then appear in the nature of an “attenuated” function whose time scale, σ, is determined by the width, 1f , of the hump in the power spectrum. A situation of this kind is shown in Figure 15.9. The relevance of spectral analysis to the problem of the actual observation of a fluc-tuating variable is best brought out by examining the power spectrum of the velocity v(t) of a Brownian particle. Considering the x-component alone, the autocorrelation function Kvx(s), or simply K(s), in this case is given by K(s) = kT M e−|s|/τ (τ = MB); (20) 614 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics f ∗ f 0 2 f ∗ K(s) 0 s 1 f ∗ 1 f ∗ 2 f ∗ w (f ) FIGURE 15.8 The autocorrelation function K(s) and the power distribution function w(f ) of a monochromatic variable y(t), with characteristic frequency f ∗. see equation (15.6.10). The power spectrum w(f ) is then given by the expression w(f ) = 4kT M ∞ Z 0 e−s/τ cos(2πfs)ds = 4kTτ M 1 1 + (2πf τ)2 , (21) which indeed satisfies the relationship ∞ Z 0 w(f )df = 2kT πM tan−1(2πf τ) ∞ 0 = kT M = ⟨v2 x⟩, (22) in agreement with the equipartition theorem (as applied to a single component of the velocity v). For f ≪τ −1, the power distribution is practically independent of f , which implies a practically “white” spectrum, with w(f ) ≃4kTτ M = 4BkT. (23) 15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem 615 f ∗ 1 2 f 0 0 K(s) s 2 2 w (f ) FIGURE 15.9 The autocorrelation function K(s) and the power distribution function w(f ) of a variable that has been filtered through a lightly damped tuned circuit, with mean frequency f ∗and width 1f ∼(1/σ). We can then write for the velocity fluctuations in the frequency range (f ,f + 1f ), with f ≪τ −1, ⟨1v2 x⟩(f , f +1f ) ≃w(f )1f ≃(4BkT)1f . (24) In general, our measuring instrument (or the eye, in the case of a visual examination of the particle) has a finite response time τ0, as a consequence of which it is unable to respond to frequencies larger than, say, τ −1 0 . The observed fluctuation is then given by the “pruned” expression ⟨v2 x⟩obs ≃ 1/τ0 Z 0 w(f )df = 2kT πM tan−1  2π τ τ0  , (25) instead of the “full” expression (22). In a typical case, the mass of the Brownian particle M ∼10−12g, its diameter 2a ∼10−4 cm, and the coefficient of viscosity of the fluid η ∼10−2 poise, so that the relaxation time τ = M/(6πηa) ∼10−7 seconds. However, the response time τ0, in the case of visual observation, is of the order of 10−1 s; clearly, τ/τ0 ∼10−6 ≪1. 616 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Equation (25) then reduces to18 ⟨v2 x⟩obs ≃4kTτ Mτ0 ≪kT M ; (26) thus, in view of the finiteness of the response time τ0, the observed root-mean-square velocity of the Brownian particle will be down by a factor of 2(τ/τ0)1/2 ∼10−3; numeri-cally, this takes us down from a root-mean-square velocity which, at room temperatures, is ∼10−1 cm/s to a value ∼10−4 cm/s. It is gratifying to note that the outcome of actual observations of Brownian particles is in complete agreement with the latter result; for a more detailed analysis of this question, see MacDonald (1950). The foregoing discussion highlights the fact that, in the process of observing a fluctuating variable, our measuring instrument picks up signals over only a limited range of frequencies (as determined by the response time of the instrument); signals belonging to higher frequencies are simply left out. The theory of this section can be readily applied to fluctuations in the motion of elec-trons in an (L,R) circuit. Corresponding to equations (21) through (24), we now have for fluctuations in the electric current I w(f ) = 4kTτ ′ L 1 1 + (2πf τ ′)2  τ ′ = L R  , (27) so that ∞ Z 0 w(f )df = kT L = ⟨I2⟩, (28) in agreement with the equipartition theorem: ⟨1 2LI2⟩= 1 2kT. For f ≪1/τ ′, equation (27) reduces to w(f ) ≃4kT R , (29) which, once again, implies “white” noise; accordingly, for low frequencies, ⟨1I2⟩(f , f +1f ) ≃w(f )1f ≃4kT R 1f . (30) Equivalently, we obtain for fluctuations in the voltage ⟨1V 2⟩(f , f +1f ) ≃(4RkT)1f . (31) Equation (31) constitutes the so-called Nyquist theorem, which was first discovered empir-ically by Johnson (1927a,b; 1928) and was later derived by Nyquist (1927–1928) on the basis 18The fluctuations constituting this result belong entirely to the region of the “white” noise, with 1f = 1/τ0; see equation (24), with B = τ/M. 15.6 The fluctuation–dissipation theorem 617 of an argument involving the second law of thermodynamics and the exchange of energy between two resistances in thermal equilibrium.19 15.6 The fluctuation–dissipation theorem In Section 15.3 we obtained a result of considerable importance, namely 1 B ≡M τ = M2 6kT C = M2 6kT ∞ Z −∞ KA(s)ds = 1 6kT ∞ Z −∞ KF(s)ds; (1) see equations (15.3.4), (15.3.26), and (15.3.28). Here, KA(s) and KF(s) are, respectively, the autocorrelation functions of the fluctuating acceleration A(t) and the fluctuating force F(t) experienced by the Brownian particle: KA(s) = ⟨A(0) · A(s)⟩= 1 M2 ⟨F(0) · F(s)⟩= 1 M2 KF(s). (2)20 Equation (1) establishes a fundamental relationship between the coefficient, 1/B, of the “averaged-out” part of the total force F (t) experienced by the Brownian particle due to the impacts of the fluid molecules and the statistical character of the “fluctuating” part, F(t), of that force; see Langevin’s equation (15.3.2). In other words, it relates the coefficient of viscosity of the fluid, which represents dissipative forces operating in the system, with the temporal character of the molecular fluctuations; the content of equation (1) is, therefore, referred to as a fluctuation–dissipation theorem. The most striking feature of this theorem is that it relates, in a fundamental manner, the fluctuations of a physical quantity pertaining to the equilibrium state of a given system to a dissipative process which, in practice, is realized only when the system is subject to an external force that drives it away from equilibrium. Consequently, it enables us to deter-mine the nonequilibrium properties of the given system on the basis of a knowledge of the thermal fluctuations occurring in the system when the system is in one of its equilibrium 19We note that the foregoing results are essentially equivalent to Einstein’s original result for charge fluctuations in a conductor, namely ⟨δq2⟩t = 2kT R t; compare, as well, the Brownian-particle result: ⟨x2⟩t = 2BkTt. 20We note that the functions KA(s) and KF(s), which are nonzero only for s = O(τ ∗), see equation (15.3.21), may, for certain purposes, be written as KA(s) = 6kT M2B δ(s) and KF(s) = 6kT B δ(s). In this form, the functions are nonzero only for s = 0. 618 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics states! For an expository account of the fluctuation–dissipation theorem, the reader may refer to Kubo (1966). At this stage we recall that in equation (15.3.11) we obtained a relationship between the diffusion coefficient D and the mobility B, namely D = BkT. Combining this with equation (1), we get 1 D = 1 6(kT)2 ∞ Z −∞ KF(s)ds. (3) Now, the diffusion coefficient D can be related directly to the autocorrelation function Kv(s) of the fluctuating variable v(t). For this, one starts with the observation that, by definition, r(t) = t Z 0 v(u)du, (4) which gives ⟨r2(t)⟩= t Z 0 t Z 0 ⟨v(u1) · v(u2)⟩du1du2. (5) Proceeding in the same manner as for the integral in equation (15.3.22), one obtains ⟨r2(t)⟩= t/2 Z 0 dS +2S Z −2S Kv(s)ds + t Z t/2 dS +2(t−S) Z −2(t−S) Kv(s)ds; (6) compare this to equation (15.3.24). The function Kv(s) can be determined by making use of expression (15.3.14) for v(t) and following exactly the same procedure as for determining the quantity ⟨v2(t)⟩, which is nothing but the maximal value, Kv(0), of the desired function. Thus, one obtains Kv(s) =        v2(0)e−(2t+s)/τ + 3kT M e−s/τ(1 −e−2t/τ) for s > 0 (7) v2(0)e−(2t+s)/τ + 3kT M es/τ(1 −e−2(t+s)/τ) for s < 0; (8) compare these results to equation (15.3.27). It is easily seen that formulae (7)and (8) can be combined into a single one, namely Kv(s) = v2(0)e−|s|/τ + 3kT M −v2(0)  (e−|s|/τ −e−(2t+s)/τ) for all s; (9) 15.6 The fluctuation–dissipation theorem 619 compare this to equation (15.3.29). In the case of a “stationary ensemble,” Kv(s) = 3kT M e−|s|/τ, (10) which is consistent with property (15.3.20). It should be noted that the time scale for the correlation function Kv(s) is provided by the relaxation time τ of the Brownian motion, which is many orders of magnitude larger than the characteristic time τ ∗that provides the time scale for the correlation functions KA(s) and KF(s). It is now instructive to verify that the substitution of expression (10) into (6) leads to formula (15.3.7) for ⟨r2⟩, while the substitution of the more general expression (9) leads to formula (15.3.31); see Problem 15.17. In either case, ⟨r2⟩− − − → t≫τ 6Dt. (11) In the same limit, equation (6) reduces to ⟨r2⟩≃ t Z 0 dS ∞ Z −∞ Kv(s)ds = t ∞ Z −∞ Kv(s)ds. (12) Comparing the two results, we obtain the desired relationship: D = 1 6 ∞ Z −∞ Kv(s)ds. (13) In passing, we note, from equations (3) and (13), that ∞ Z −∞ Kv(s)ds ∞ Z −∞ KF(s)ds = (6kT)2; (14) see also Problem 15.7. It is not surprising that the equations describing a fluctuation–dissipation theorem can be adapted to any situation that involves a dissipative mechanism. For instance, fluctua-tions in the motion of electrons in an electric resistor give rise to a “spontaneous” thermal e.m.f., which may be denoted as B(t). In the spirit of the Langevin theory, this e.m.f. may be split into two parts: (i) an “averaged-out” part, −RI(t), which represents the resistive (or dissipative) aspect of the situation, and (ii) a “rapidly fluctuating” part, V (t), which, over long intervals of time, averages out to zero. The “spontaneous” current in the resistor is then given by the equation LdI dt = −RI + V (t); ⟨V (t)⟩= 0. (15) 620 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Comparing this with the Langevin equation (15.3.2) and pushing the analogy further, we infer that there exists a direct relationship between the resistance R and the temporal character of the fluctuations in the variable V (t). In view of equations (1) and (13), this relationship would be R = 1 6kT ∞ Z −∞ ⟨V (0) · V (s)⟩ds (16) or, equivalently, 1 R = 1 6kT ∞ Z −∞ ⟨I(0) · I(s)⟩ds. (17) A generalization of the foregoing result has been given by Kubo (1957, 1959); see, for instance, Problem 6.19 in Kubo (1965), or Section 23.2 of Wannier (1966). On generaliza-tion, the electric current density j(t) is given by the expression ji(t) = X l t Z −∞ El(t′)8li(t −t′)dt′ (i,l = x,y,z); (18) here, E(t) denotes the applied electric field while 8li(s) = 1 kT ⟨jl(0)ji(s)⟩. (19) Clearly, the quantities kT8li(s) are the components of the autocorrelation tensor of the fluctuating vector j(t). In particular, if we consider the static case E = (E,0,0), we obtain for the conductivity of the system σxx ≡jx E = t Z −∞ 8xx(t −t′)dt′ = ∞ Z 0 8xx(s)ds = 1 2kT ∞ Z −∞ ⟨jx(0)jx(s)⟩ds, (20) which may be compared with equation (17). If, on the other hand, we take E = (E cosωt,0,0), we obtain instead σxx(ω) = 1 2kT ∞ Z −∞ ⟨jx(0)jx(s)⟩e−iωsds. (21) 15.6 The fluctuation–dissipation theorem 621 Taking the inverse of (21), we get ⟨jx(0)jx(s)⟩= kT π ∞ Z −∞ σxx(ω)eiωsdω. (22) If we now assume that σxx(ω) does not depend on ω (and may, therefore, be denoted by the simpler symbol σ), then ⟨jx(0)jx(s)⟩= (2kTσ)δ(s); (23) see footnote 20. A reference to equations (15.5.17) shows that, in the present approxima-tion, thermal fluctuations in the electric current are charaterized by a “white” noise. 15.6.A Derivation of the fluctuation–dissipation theorem from linear response theory In this section we will show that the nonequilibrium response of a thermodynamic system to a small driving force is very generally related to the time-dependence of equilibrium fluctuations. In hindsight, this is not too surprising since natural fluctuations about the equilibrium state also induce small deviations of observables from their average val-ues. The response of the system to these natural fluctuations should be the same as the response of the system to deviations from the equilibrium state as caused by small driving forces; see Martin (1968), Forster (1975), and Mazenko (2006). Let us compute the time-dependent changes to an observable A caused by a small time-dependent external applied field h(t) that couples linearly to some observable B. The Hamiltonian for the system then becomes H(t) = H0 −h(t)B, (24) where H0 is the unperturbed Hamiltonian in the equilibrium state. Remarkably, the calcu-lation for determining the nonequilibrium response to the driving field is easiest using the quantum-mechanical density matrix approach developed in Section 5.1. The equilibrium density matrix is given by ˆ ρeq = exp(−βH0) Tr exp(−βH0) , (25) where equilibrium averages involve traces over the density matrix: ⟨A⟩eq = Tr A ˆ ρeq  . (26) When the Hamiltonian includes a small time-dependent field h(t), then this additional term drives the system slightly out of equilibrium. We will assume that the field was zero in the distant past so the system was initially in the equilibrium state defined by the 622 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Hamiltonian H0. We then turn on the field and measure the time-dependent deviations of the observable A from its equilibrium value. The small time-dependent applied field h(t) induces a small change to the density matrix ˆ ρ(t) = ˆ ρeq + δ ˆ ρ(t). (27) The equation of motion of the density matrix in equation (5.1.10) gives ∂ˆ ρ ∂t = ∂δ ˆ ρ ∂t = 1 iℏ H, ˆ ρ(t) ≈1 iℏ H0,δ ˆ ρ −h(t) B, ˆ ρeq  , (28) where [,] denotes the commutator. Since we are considering only the linear response of the system to the applied field, we have ignored the higher-order term proportional to h(t) B,δ ˆ ρ . Solving (28) for the time-dependent change to the density matrix, we get δ ˆ ρ(t) = i ℏ t Z −∞ h(t′)exp −iH0(t −t′) ℏ  B, ˆ ρeq exp iH0(t −t′) ℏ  dt′. (29) This form uses the interaction representation in which operators evolve in time due to the unperturbed Hamiltonian H0. We can use the change in the density matrix at time t to calculate the change in the observable A compared to its equilibrium value, namely ⟨δA(t)⟩= ⟨A(t)⟩−⟨A⟩eq = Tr A ˆ ρ(t)  −Tr A ˆ ρeq  = Tr Aδ ˆ ρ(t)  . (30) Using the cyclic property of traces, Tr(QRS) = Tr(SQR), we find that ⟨δA(t)⟩is the convolu-tion of a response function with the applied field: ⟨δA(t)⟩= i ℏ t Z −∞ A(t),B(t′) eq h(t′)dt′. (31) Note that this nonequilibrium response function of the system to the driving force depends on the equilibrium average of the commutator of the observables A and B at dif-ferent times. The effect of the field on the observable A is causal since ⟨δA(t)⟩depends only on the applied field at earlier times. Since the relation is linear, time-translationally invariant, and causal, the Fourier spectra of ⟨δA⟩and h, namely δ ˆ A(ω) = ∞ Z −∞ ⟨δA(t)⟩eiωtdt, and (32a) ˆ h(ω) = ∞ Z −∞ h(t)eiωtdt, (32b) are related by δ ˆ A(ω) = ˆ χ′ AB(ω) + i ˆ χ′′ AB(ω)  ˆ h(ω), (33) 15.6 The fluctuation–dissipation theorem 623 where ˆ χ′′ AB(ω) is given by ˆ χ′′ AB(ω) = 1 2ℏ ∞ Z −∞ ⟨[A(t),B(0)]⟩eq eiωtdt. (34) The quantity ˆ χ′ AB(ω) is given by the Kramers–Kronig relation using the principal part P of an infinite integral over ˆ χ′′ AB(ω): ˆ χ′ AB(ω) = P ∞ Z −∞ ˆ χ′′ AB(ω′) ω′ −ω dω′ π . (35) If A and B are the same operator, then ˆ χ′ AA(ω) and ˆ χ′′ AA(ω) are, respectively, the real and imaginary parts of the response function. If A and B have the same symmetry under time-reversal, ω ˆ χ′′ AB(ω) is real and is an even function of ω. For a set of observables Ai, the set of response functions ω ˆ χ′′ ij(ω) form a symmetric positive matrix that gives the rate of energy dissipation due to the external driving forces. See Jackson (1999) for a general causality discussion and Martin (1968), Forster (1975), or Mazenko (2006) for the details of this calculation. Now we consider the equilibrium temporal correlations between the fluctuations of the observables A and B, namely GAB(t −t′) = δA(t)δB(t′) eq . (36) At equal times, this measures the AB equilibrium fluctuations described in Section 15.1, that is, GAB(0) = ⟨δAδB⟩eq. The power spectrum of the AB equilibrium fluctuations is defined by SAB(ω) = ∞ Z −∞ GAB(t)eiωtdt = ∞ Z −∞ ⟨δA(t)δB(0)⟩eq eiωtdt. (37) The similarity between the forms of SAB(ω) and ˆ χ′′ AB(ω) in equations (34) and (37) leads to an important relation between the power spectrum and the linear response function, namely the fluctuation–dissipation theorem: ˆ χ′′ AB(ω) = 1 2ℏ  1 −e−βℏω SAB(ω). (38) The power spectrum SAB(ω) measures equilibrium fluctuations whereas ˆ χ′′ AB(ω) is propor-tional to the average rate of power dissipation that results from the time-varying applied field. The classical limit of the fluctuation–dissipation theorem is obtained by letting ℏω/kT →0 with the result ˆ χ′′ AB(ω) = ω 2kT SAB(ω); (39) compare this to equation (15.3.45). More complete discussions of the fluctuation– dissipation theorem can be found in Martin (1968), Forster (1975), and Mazenko (2006). 624 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics 15.6.B Inelastic scattering Inelastic scattering is an important experimental technique for measuring the dynamical behavior of materials. By measuring the intensity of radiation scattered from a sample as a function of wavevector transfer and frequency change relative to the incident monochro-matic radiation source, one can measure the spatio-temporal correlations in the material. This technique is now commonly applied to the scattering of neutrons, electrons, light, and x-rays. The frequency changes in the scattered wave are caused by inelastic scattering from quantum excitations in the sample; see Forster (1975), Squires (1997), and Mazenko (2006). The frequency-dependent scattering intensity is directly proportional to the dynamical structure factor S(k,ω) = 1 N ∞ Z −∞ X i,j e−ik·(ri(t)−rj(0)) + eiωtdt, (40) where k represents the wavevector transfer of the scattering process and ω represents the frequency difference from that of the incident beam. A positive value of ω corresponds to a detected frequency that is less than the frequency of the incident beam. The dynamical structure factor S(k,ω) encodes both the spatial and temporal equilibrium correlations of fluctuations in the material and can be decomposed into two terms, one that represents scattering from a single particle at different times and another that represents scattering from different particles: S(k,ω) = Sself(k,ω) + Scoherent(k,ω), (41a) Sself(k,ω) = 1 N ∞ Z −∞ X i e−ik·(ri(t)−ri(0)) + eiωtdt, (41b) Scoherent(k,ω) = 1 N ∞ Z −∞ X i̸=j e−ik·(ri(t)−rj(0)) + eiωtdt. (41c) The dynamical structure factor S(k,ω) can also be written in terms of the spatio-temporal Fourier transforms of the time-dependent density n(r,t) to connect it to the power spectrum as defined in Section 15.6.A: S(k,ω) = SAB(ω) = ∞ Z −∞ ⟨δA(t)δB(0)⟩eiωtdt, (42a) δA(t) = 1 √ N Z e−ik·rδn(r,t)dr = 1 √ N ˆ n−k(t), (42b) δB(0) = 1 √ N Z eik·rδn(r,0)dr = 1 √ N ˆ nk(0). (42c) 15.6 The fluctuation–dissipation theorem 625 Positive frequency changes ω > 0 represent scattering events that create quantum exci-tations in the material with energy ℏω and are referred to as Stokes scattering. Negative frequency changes are called anti-Stokes scattering and correspond to scattering events that destroy an excitation in the material with energy ℏω. Since an excitation must first exist in order for it to be destroyed, the anti-Stokes scattering rate in equilibrium is lower relative to the Stokes scattering rate by the Boltzmann factor for the excitation: S(k,−ω) = e−βℏωS(k,ω). (43) This relation is sometimes used in Raman scattering to measure the temperature of the sample. The heights of Stokes and anti-Stokes peaks are symmetric if the excitation ener-gies are small compared to thermal energies, that is ℏω ≪kT. The static structure factor S(k) in equation (10.7.18) is obtained by integrating S(k,ω) over all ω: S(k) = 1 2π ∞ Z −∞ S(k,ω)dω. (44) This singles out equal-time scattering events and corresponds to quasielastic scattering measurements that are unable to resolve the energy changes due to the excitations in the material. Three commonly measured types of inelastic laser scattering are: Raman scattering, Brillouin scattering, and Rayleigh scattering. Raman scattering measures electronic, vibra-tional, and rotational excitations of atoms and molecules, electronic band structure, and optical phonon modes. Brillouin scattering measures long-wavelength acoustic sound modes. The widths of Raman and Brillouin scattering peaks are determined by the life-times of their respective modes. Rayleigh scattering measures the heat diffusion mode centered at ω = 0 with width proportional to the thermal diffusivity. The wavelength of visible light is large compared to atomic scales, so the wavevector transfers possible with light scattering are very small compared to the size of the Brillouin zone. This limita-tion is removed for inelastic x-ray and neutron scattering where experiments can probe wavevectors away from the center of the Brillouin zone. The dynamical scattering of a laser beam from a liquid includes three peaks: a Rayleigh peak centered at ω = 0 due to scattering from the thermal fluctuations in the liquid and the Stokes and anti-Stokes Brillouin peaks at ωk ≈±ck due to scattering from acoustic phonons with sound speed c. For scattering in this wavevector and frequency range, the dynamical structure factor is symmetric in ω since ℏω ≪kT. An early Brillouin scatttering measurement of the dynamical structure factor of a liquid is shown in Figure 15.10. The fluctuation–dissipation theorem enables one to develop a theory of the dynamical structure factor based on the hydrodynamic response of a system that is weakly perturbed from equilibrium. In the case of a fluid, small perturbations in pressure and tempera-ture result in propagating sound waves and thermal diffusion. This results in the following 626 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics  0 FIGURE 15.10 Dynamical structure factor for carbon tetrachloride using 632.8 nm He-Ne laser with a 90◦scattering angle. The structure factor is symmetric since ℏω ≪kT. The Landau-Placzek ratio of the integrated intensity under the Rayleigh peak to the integrated intensities under the two Brillouin peaks is related to ratio of the constant-pressure heat capacity to the constant-volume heat capacity: IR/(2IB) = CP/CV −1 = 0.72 ± 0.03, Landau and Placzek (1934). The locations of the Brillouin peaks give the sound speed and the widths of the peaks measure the thermal diffusivity and sound attenuation coefficient of the liquid; see equation (45). From Cummins and Gammon (1966), reprinted with permission; copyright © 1966, American Institute of Physics. theoretical form for the dynamical structure factor: S(k,ω) =S(k) "γ −1 γ  2DTk2 ω2 + (DTk2)2 + 1 γ 0k2 (ω2 −c2k2)2 + (0k2)2 + 0k2 (ω2 + c2k2)2 + (0k2)2 !# . (45) The parameters in equation (45) are the thermal diffusivity DT, the sound speed c, and the sound attenuation coefficient 0, while γ = CP/CV is the ratio of the constant-pressure and constant-volume heat capacities; see Forster (1975) and Hansen and McDonald (1986). 15.7 The Onsager relations Most physical phenomena exhibit a kind of symmetry, sometimes referred to as reci-procity, that arises from certain basic properties of the microscopic processes that operate behind the (observable) macroscopic situations. A notable example of this is met with in the thermodynamics of irreversible processes where one deals with a variety of flow processes, such as heat flow, electric current, mass transfer, and so on. These flows (or “currents”) are driven by “forces,” such as a temperature difference, a potential difference, a pressure difference, and so on, which come into play because of a natural tendency 15.7 The Onsager relations 627 among physical systems which happen to be out of equilibrium to approach a state of equilibrium. If the given state of the system is not too far removed from a state of equilibrium, then one might assume a linear relationship between the forces Xi and the currents ˙ xi: ˙ xi = γijXj, (1) where γij are the kinetic coefficients of the system.21 Simple examples of such coefficients are thermal conductivity, electrical conductivity, diffusion coefficient, and so on. There are, however, nondiagonal elements, γij(i ̸= j), as well that may or may not vanish; they are responsible for the so-called cross effects. It is the symmetry properties of the matrix (γij) that form the subject matter of this section. The most obvious way to approach this problem is to consider the entropy, S(xi), of the system in the disturbed state relative to its maximal value, S(˜ xi), in the relevant state of equilibrium. It is the natural tendency of the function S(xi) to approach its maximal value S(˜ xi) that brings into play the driving forces Xi; these forces give rise to currents ˙ xi, which take the “coordinates” xi toward their equilibrium values ˜ xi. If the deviations (xi −˜ xi) are small, then the function S(xi) may be expressed as a Taylor series around the values xi = ˜ xi; retaining terms up to the second order only, we have S(xi) = S(˜ xi) +  ∂S ∂xi  xi=˜ xi (xi −˜ xi) + 1 2 ∂2S ∂xi∂xj ! xi,j=˜ xi,j (xi −˜ xi)(xj −˜ xj). (2) In view of the fact that the function S(xi) is maximum at xi = ˜ xi, its first derivatives vanish; we may, therefore, write 1S ≡S(xi) −S(˜ xi) = −1 2βij(xi −˜ xi)(xj −˜ xj), (3) where βij = − ∂2S ∂xi∂xj ! xi,j=˜ xi,j = βji. (4) The driving forces Xi may be defined in the spirit of the second law of thermodynamics, that is, Xi =  ∂S ∂xi  = −βij(xj −˜ xj). (5) 21In writing equation (1), and other subsequent equations, we follow the summation convention that implies an automatic summation over a repeated index. 628 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics We note that in the present approximation the forces Xi depend linearly on the dis-placements (xi −˜ xi); in the state of equilibrium, they vanish. Now, in view of equation (15.1.2), the ensemble average of the product xiXj is given by ⟨xiXj⟩= R ∞ −∞(xiXj)exp n −1 2k βij(xi −˜ xi)(xj −˜ xj) oQ i dxi R ∞ −∞exp n −1 2k βij(xi −˜ xj)(xj −˜ xj) oQ i dxi ; (6) the limits of integration in (6) have been extended to −∞and +∞because the integrals here do not draw any significant contribution from large values of the variables involved. In the same way, ⟨xi⟩= R ∞ −∞xi exp n −1 2k βij(xi −˜ xi)(xj −˜ xj) oQ i dxi R ∞ −∞exp n −1 2k βij(xi −˜ xi)(xj −˜ xj) oQ i dxi = ˜ xi. (7) Differentiating (7) with respect to ˜ xj (and remembering that the integral in the denomina-tor is a constant, independent of the actual values of the quantities ˜ xi), and comparing the resulting expression with (6), we obtain the remarkable result ⟨xiXj⟩= −kδij. (8) We now proceed toward the key point of the argument. First of all, we note that, though equations (1) are concerned with irreversible phenomena, the microscopic pro-cesses underlying these phenomena obey time reversal, which means that the temporal correlations of the relevant variables are the same whether measured forward or backward in time. Thus, ⟨xi(0)xj(s)⟩= ⟨xi(0)xj(−s)⟩; (9) also, by a shift in the zero of time, ⟨xi(0)xj(−s)⟩= ⟨xi(s)xj(0)⟩. (10) Combining (9) and (10), we get ⟨xi(0)xj(s)⟩= ⟨xi(s)xj(0)⟩. (11) If we now subtract, from both sides of this equation, the quantity ⟨xi(0)xj(0)⟩, divide the resulting equation by s and let s →0, we obtain ⟨xi(0)˙ xj(0)⟩= ⟨˙ xi(0)xj(0)⟩. (12) 15.7 The Onsager relations 629 Substituting from (1) and making use of (8), we obtain on the left side of (12) ⟨xi(0)γjlXl(0)⟩= −kγjlδil = −kγji and on its right side ⟨γilXl(0)xj(0)⟩= −kγilδjl = −kγij. It follows that γij = γji. (13) Equations (13) constitute the Onsager reciprocity relations; they were first derived by Onsager in 1931 and have become an essential part of the thermodynamics of irreversible phenomena. In view of equations (1) and (13), the currents ˙ xi may be written as ˙ xi = ∂f ∂Xi , (14) where the generating function f is a quadratic function of the forces Xi: f = 1 2γijXiXj. (15) The function f is especially important in that it determines directly the rate at which the entropy of the system changes with time: ˙ S = ∂S ∂xi ˙ xi = Xi ˙ xi = Xi ∂f ∂Xi = 2f . (16) As the system approaches the state of equilibrium, its entropy must increase toward the equilibrium value S(˜ xi). The function f must, therefore, be positive definite, which places certain restrictions on the coefficients γij. Analogous to equation (1), we could also write ˙ Xi = ζij(xj −˜ xj), (17) the quantities ζij being another set of coefficients pertaining to the system. From equa-tions (1) and (5), on the other hand, we obtain ˙ Xi = −βij ˙ xj = −βij(γjlXl) = −βijγjl{−βlm(xm −˜ xm)} = βijγjlβlm(xm −˜ xm). (18) Comparing (17) and (18), we obtain a relationship between the new coefficients ζij and the kinetic coefficients γij: ζim = βijγjlβlm. (19) 630 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics Now, in view of the symmetry properties of the matrices β and γ , we get ζim = ζmi; (20) thus, the coefficients ζij, introduced through the phenomenological equations (17), also obey the reciprocity relations. It then follows that the quantities ˙ Xi, see equation (14), may be written as, ˙ Xi = ∂f ′ ∂xi , (21) where f ′ = 1 2ζij(xi −˜ xi)(xj −˜ xj). (22) The entropy change dS may now be written as dS = ∂S ∂xj dxj = Xjdxj = −βji(xi −˜ xi)dxj = (xi −˜ xi)d{−βij(xj −˜ xj)} = (xi −˜ xi)dXi, (23) so that ∂S ∂Xi = (xi −˜ xi); (24) clearly, the entropy S is now regarded as an explicit function of the forces Xi (rather than of the coordinates xi). The time derivative of S now takes the form ˙ S = ∂S ∂Xi ˙ Xi = (xi −˜ xi) ∂f ′ ∂xi = 2f ′. (25) Comparing (16) and (25), we conclude that the functions f and f ′ are, in fact, the same; they are only expressed in terms of two different sets of variables. It seems important to mention here that Onsager’s reciprocity relations have an inti-mate connection with the fluctuation–dissipation theorem of the preceding section. Fol-lowing equations (15.6.18) and (15.6.19), and adopting the summation convention, we have in the present context ˙ xi(t) = 1 kT t Z −∞ El(t′)⟨˙ xl(t′)˙ xi(t)⟩dt′ (26) or, setting (t −t′) = s, ˙ xi(t) = 1 kT ∞ Z 0 El(t −s)⟨˙ xl(t −s)⟨˙ xi(t)⟩ds; (27) 15.7 The Onsager relations 631 compare this to equation (1). Interchanging the indices i and l, we obtain ˙ xl(t) = 1 kT ∞ Z 0 Ei(t −s)⟨˙ xi(t −s)˙ xl(t)⟩ds. (28) The crucial point now is that the correlation functions appearing in equations (27) and (28) are identical in value, for ⟨˙ xl(t −s)˙ xi(t)⟩= ⟨˙ xl(0)˙ xi(s)⟩= ⟨˙ xl(0)˙ xi(−s)⟩= ⟨˙ xl(t)˙ xi(t −s)⟩; (29) in establishing (29), the first and third steps followed from “a shift in time” while the sec-ond step followed from the “principle of dynamical reversibility of microscopic processes.” The equivalence depicted in equation (29) is, in essence, the content of Onsager’s reci-procity relations. In particular, if the correlation functions appearing in (27) and (28) are sharply peaked at the value s = 0, then these equations reduce to the phenomenological equations (1), and equation (29) becomes synonymous with the Onsager relations (13). In the end, we make some further remarks concerning relations (13). We recall that, in arriving at these relations, we had to make an appeal to the invariance of the microscopic processes under time reversal. The situation is somewhat different in the case of a “system in rotation” (or a “system in an external magnetic field”), for then the invariance under time reversal holds only if there is also a simultaneous change of sign of the angular veloc-ity  (or of the magnetic field B). The kinetic coefficients, which in this case might depend on the parameter  (or B), will now satisfy the relations γij() = γji(−) (13a) and γij(B) = γji(−B). (13b) In addition, our proof here rested on the implicit assumption that the quantities xi themselves do not change under time reversal. If, for some reason, these quantities are proportional to the velocities of a certain macroscopic motion, then they will also change their sign under time reversal. Now, if both xi and xj belong to this category, then equation (12), which is crucial to our proof, would remain unaltered; consequently, the coefficients γij and γji would still be equal. However, if only one of them belongs to this category while the other one does not, then equation (12) would change to ⟨xi(0)˙ xj(0)⟩= −⟨˙ xi(0)xj(0)⟩; (12′) the coefficients γij and γji would then obey the relations γij = −γji. (13′) 632 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics For the application of Onsager’s relations to different physical problems, reference may be made to the monographs by de Groot (1951), de Groot and Mazur (1962), and Prigogine (1967). Problems 15.1. Making use of expressions (15.1.11) and (15.1.12) for 1S and 1P, and expressions (15.1.14) for (1T)2,(1V)2, and (1T1V), show that (a) (1T1S) = kT; (b) (1P1V) = −kT; (c) (1S1V) = kT(∂V/∂T)P; (d) (1P1T) = kT2C−1 V (∂P/∂T)V . [Note that results (a) and (b) give: (1T1S −1P1V) = 2kT, which follows directly from the probability distribution function (15.1.8).] 15.2. Establish the probability distribution (15.1.15), which leads to the expressions in (15.1.16) for (1S)2, (1P)2, and (1S1P). Show that these results can also be obtained by following the procedure of the preceding problem. 15.3. If we choose the quantities E and V as “independent” variables, then the probability distribution function (15.1.8) does not reduce to a form as simple as (15.1.13) or (15.1.15); it is marked instead by the presence, in the exponent, of a cross term proportional to the product 1E1V. Consequently, the variables E and V are not statistically independent : (1E1V) ̸= 0. Show that (1E1V) = kT  T ∂V ∂T  P + P ∂V ∂P  T  ; verify as well expressions (15.1.14) and (15.1.18) for (1V)2 and (1E)2. [Note that in the case of a two-dimensional normal distribution, namely p(x,y) ∝exp  −1 2(ax2 + 2bxy + cy2)  , the quantities ⟨x2⟩,⟨xy⟩, and ⟨y2⟩can be obtained in a straightforward manner by carrying out a logarithmic differentiation of the formula ∞ Z −∞ ∞ Z −∞ exp  −1 2(ax2 + 2bxy + cy2  dx dy = 2π p (ac −b2) with respect to the parameters a,b, and c. This leads to the covariance matrix of the distribution, namely  ⟨x2⟩ ⟨xy⟩ ⟨yx⟩ ⟨y2⟩  = 1 (ac −b2)  c −b −b a  . If b = 0, then ⟨x2⟩= 1/a, ⟨xy⟩= 0, ⟨y2⟩= 1/c.]22 15.4. A string of length l is stretched, under a constant tension F, between two fixed points A and B. Show that the mean square (fluctuational) displacement y(x) at point P, distant x from A, is given by {y(x)}2 = kT Fl x(l −x). 22For the covariance matrix of an n-dimensional normal distribution, see Landau and Lifshitz (1958), Section 110. Problems 633 Further show that, for x2 ≥x1, y(x1)y(x2) = kT Fl x1(l −x2). [Hint : Calculate the energy, 8, associated with the fluctuation in question; the desired probability distribution is then given by p ∝exp(−8/kT), from which the required averages can be readily evaluated.] 15.5. How small must the volume, VA, of a gaseous subsystem (at normal temperature and pressure) be, so that the root-mean-square deviation in the number, NA, of particles occupying this volume be 1 percent of the mean value NA? 15.6. Pospiˇ sil (1927) observed the Brownian motion of soot particles, of radii 0.4 × 10−4 cm, immersed in a water–glycerine solution, of viscosity 0.0278 poise at a temperature of 18.8◦C. The observed value of x2, in a 10-second time interval, was 3.3 × 10−8cm2. Making use of these data, determine the Boltzmann constant k. 15.7. In the notation of Section 15.3, show that for a Brownian particle ⟨v(t) · F(t)⟩= 3kT/τ, while ⟨v(t) · F (t)⟩= 0. On the other hand, ⟨r(t) · F (t)⟩= −3kT, while ⟨r(t) · F(t)⟩= 0. 15.8. Integrate equation (15.3.14) to obtain r(t) = v(0)τ(1 −e−t/τ ) + τ t Z 0 {1 −e(u−t)/τ}A(u)du, so that r(0) = 0. Taking the square of this expression and making use of the autocorrelation function KA(s), derive formula (15.3.31) for ⟨r2(t)⟩. 15.9. While detecting a very feeble current with the help of a moving-coil galvanometer, one must ensure that an observed deflection is not just a stray kick arising from the Brownian motion of the suspended system. If we agree that a deflection θ, whose magnitude exceeds 4θr.m.s.[= 4(kT/c)1/2], is highly unlikely to be due to the Brownian motion, we obtain a lower limit to the magnitude of the current that can be reliably recorded with the help of the given galvanometer. Express this limiting current in terms of the time period τ and the critical damping resistance Rc of the galvanometer. 15.10. (a) Integrate Langevin’s equation (15.3.5) for the velocity component vx over a small interval of time δt, and show that ⟨δvx⟩ δt = −vx τ and ⟨(δvx)2⟩ δt = 2kT Mτ . (b) Now, set up the Fokker–Planck equation for the distribution function f (vx,t) and, making use of the foregoing results for µ1(vx) and µ2(vx), derive an explicit expression for this function. Study the various cases of interest, especially the one for which t ≫τ. 15.11. Generalize the analysis of the Langevin theory of a harmonic oscillator, as given by equation (15.3.33), to the case of an oscillator starting at time t = 0 with the initial position x(0) and the initial velocity v(0). Derive, for this system, the quantities ⟨x2(t)⟩and ⟨v2(t)⟩and show that, in the limit ω0 →0, these expressions reproduce equations (15.3.29) and (15.3.31) while, in the limit M →0, they lead to the relevant results of Section 15.4. 15.12. Generalize the Fokker–Planck equation to the case of a particle executing Brownian motion in three dimensions. Determine the general solution of this equation and study its important features. 15.13. The autocorrelation function K(s) of a certain statistically stationary variable y(t) is given by (a) K(s) = K(0)e−αs2 cos(2πf ∗s) or by (b) K(s) = K(0)e−α|s| cos(2πf ∗s), 634 Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics where α > 0. Determine, and discuss the nature of, the power spectrum w(f ) in each of these cases and investigate its behavior in the limits (a) α →0, (b) f ∗→0, and (c) both α and f ∗→0. 15.14. Show that if the autocorrelation function K(s) of a certain statistically stationary variable y(t) is given by K(s) = K(0)sin(as) as sin(bs) bs (a > b > 0), then the power spectrum w(f ) of that variable is given by w(f ) = 2π a K(0) for 0 < f ≤a −b 2π , 2π ab K(0) a + b 2 −πf  for a −b 2π ≤f ≤a + b 2π , 0 for a + b 2π ≤f < ∞. Verify that the function w(f ) satisfies the requirement (15.5.16). [Note that, in the limit b →0, we recover the situation pertaining to equations (15.5.18).] 15.15. (a) Show that the mean square value of the variable Y (t), defined by the formula Y (t) = u+t Z u y(u)du, where y(u) is a statistically stationary variable with power spectrum w(f ), is given by ⟨Y 2(t)⟩= 1 2π2 ∞ Z 0 w(f ) f 2  1 −cos(2πft) df ; and, accordingly, w(f ) = 4πf ∞ Z 0 ∂ ∂t ⟨Y 2(t)⟩sin(2πft)dt = 2 ∞ Z 0 ∂2 ∂t2 ⟨Y 2(t)⟩cos(2πft)dt. For details, see MacDonald (1962), Section 2.2.1. A comparison of the last result with equation (15.5.14) suggests that Ky(s) = 1 2 ∂2 ∂s2 ⟨Y 2(s)⟩. (b) Apply the foregoing analysis to the motion of a Brownian particle, taking y to be the velocity of the particle and Y its displacement. 15.16. Show that the power spectra wv(f ) and wA(f ) of the fluctuating variables v(t) and A(t) that appear in the Langevin equation (15.3.5) are connected by the relation wv(f ) = wA(f ) τ 2 1 + (2πf τ)2 , τ being the relaxation time of the problem. Hence, by equation (15.5.21) wA(f ) = 12kT/Mτ. Problems 635 15.17. (a) Verify equations (15.6.7) through (15.6.9). (b) Substituting expression (15.6.9) for K v(s) into equation (15.6.6), derive formula (15.3.31) for ⟨r2(t)⟩. 15.18. Determine ˆ χ′′ vx(ω) and Svx(ω) for a Brownian particle in a harmonic oscillator potential. Show that the response function and the power spectrum for this case are related by the classical limit of the fluctuation–dissipation theorem. 15.19. Derive the linear response density matrix (15.6.29) from the equation of motion (15.6.28). 15.20. Show that GAB(t) = GBA(t −iβℏ) and use the cyclic property of the traces to derive the fluctuation–dissipation theorem ˆ χ′′ AB(ω) = 1 2ℏ 1 −e−βℏω SAB(ω). 15.21. Show that GAB(t) = GBA(t −iβℏ). Use this result to show that, in the classical limit, ˆ χ′′ AB(t) becomes D dA(t) dt B(0) E . Further show that this leads to equation (15.6.39). 15.22. Determine the self-diffusion term in the dynamical structure factor Sself(k,ω) in equation (15.6.41b) for the case of a single particle that diffuses according to the diffusion equation. Assume the process to be Gaussian for which ⟨ef ⟩= exp ⟨f 2/2⟩  . 15.23. Determine the angular frequency ωk for the Brillouin peaks in water for 90◦laser scattering, using a He-Ne laser with λ = 632.8nm. Determine the width of the Rayleigh peak and show that the Brillouin peaks are well-separated from the Rayleigh peak. The thermal diffusivity of water is DT = 1.4 × 10−7 m2/s. 15.24. Describe the dynamical structure factor for Raman scattering for a He-Ne laser with λ = 632.8nm. The energy level responsible for this scattering has an energy of 0.05eV and the lifetime of this state is 1picosecond. Are the Stokes/anti-Stokes scatterings symmetric as ω →−ω at room temperature? 16 Computer Simulations Computer simulations play an important role in modern statistical mechanics. The history of the use of computer simulations in science parallels the history of early digital comput-ing. The people and places involved centered around Los Alamos and other U.S. national laboratories where the first digital computers became available for use by scientists after World War II. Early leaders in the development of computer simulation methods included Fermi, Ulam, von Neumann, Teller, Metropolis, Rosenbluth, and others who were also involved in the Manhattan Project (Metropolis, 1987). Computer simulations in statistical mechanics fall into two broad classes: Monte Carlo (MC) and molecular dynamics (MD), although variants span the range between the two. Both methods involve numerically evolving simple models of materials through a set of microstates in order to determine the thermodynamic averages of measurable quantities. Computer simulations provide a means to study physical systems that is complementary to both experiment and theory. The following are a few of the advantages of computer simulations: . Computer simulations can provide insight into the equilibrium and nonequilibrium behavior of model systems for ranges of parameters where theoretical approximations are invalid or untested. . Computer simulations provide a means to test the range of validity of theoretical approximations against specific model systems. . Computer simulations allow visualization of physical processes that can provide new insights into complex phenomena. . Computer simulations allow detailed examination of behaviors that might not be accessible experimentally. . Computer simulations can be used to examine fundamental physical processes that can be used to guide theory. . Computer simulations can be used to model systems that do not exist in nature to provide assistance in understanding existing materials and engineering new ones. 16.1 Introduction and statistics While certain critical aspects of computer simulation theory should be followed rigorously, much of computer simulation development and use is an art form. There are many possi-ble simulation approaches for any given problem and some choices will be more effective Statistical Mechanics. DOI: 10.1016/B978-0-12-382188-1.00016-5 © 2011 Elsevier Ltd. All rights reserved. 637 638 Chapter 16. Computer Simulations at elucidating important physical properties than others. This brief chapter concentrates on equilibrium simulations but computer simulations are also widely used to model dynamical and nonequilibrium processes. The task of determining equilibrium thermody-namic averages of model systems is accomplished by generating a sequence of microstates that are chosen from the equilibrium ensemble of the model. For example, an MD simula-tion might be used to integrate Newton’s equations of motion for generating a time-series of states in phase space as the system explores the constant-energy hypersurface of the Hamiltonian. By comparison, an MC simulation of the same model might generate a sequence of states chosen by a random walk among the configurational microstates of the canonical ensemble. Both methods are examples of importance sampling, which focuses computational effort on generating microstates that are representative of the equilibrium ensemble rather than sampling all of the phase space. It is this huge improvement in effi-ciency that makes computer simulations of statistical mechanical models feasible. The sequence of states produced by either method can be used to estimate equilibrium aver-ages. Allen and Tildesley (1990), Binder and Heermann (2002), Frenkel and Smit (2002), and Landau and Binder (2009) provide more detailed discussions of computer simulations and their applications in statistical physics. Let q represent a microstate of the system and A(q) a thermodynamic observable that is a function of the microstate. In an MC simulation, q might represent the positions of all the particles in the system while in an MD simulation q might represent the posi-tions and momenta of all the particles. The observable A(q) might represent the potential energy, virial contribution to the pressure, pair correlation function, and so on. The initial microstate chosen to start a simulation will generally not be typical of the set of microstates that make up the equilibrium ensemble, but the goal of a simulation is to evolve the microstate through a large enough subset of the microstates of the equilibrium ensemble so that averages of observables approach their equilibrium values. After a simulation has run long enough for the system to approach equilibrium, the simulation then generates a sequence of M configurations, {qj}M j=1, chosen from the set of microstates in the equilib-rium ensemble and stores a sequence of values, {A(qj)}M j=1, for each of the thermodynamic variables one wants to measure.1 Since the microstates are chosen from the equilibrium ensemble, the equilibrium average of A is approximated by a simple average of the set of values {A(qj)}M j=1. Of course, a simulation can only provide a finite sequence of states, so a statistical analysis of the uncertainty of the results is a crucial part of any simulation. The equilibrium average of the variable A is given by ⟨A⟩= ⟨A⟩M ± σM (1) 1Alternatively, one can store the full configuration set of statistically independent microstates for later analysis. This tactic requires a large amount of storage space but is useful if there is a large computational cost of generating statistically independent configurations and one needs to calculate averages of many different observables at a later time using the stored configurations. This is sometimes done in large-scale lattice quantum chromodynamics simulations. 16.1 Introduction and statistics 639 where the simulation average ⟨A⟩M and uncertainty σM are determined by ⟨A⟩M = 1 M M X j=1 A(qj), (2a) σM = q A2 M −⟨A⟩2 M √M/(2τ + 1) , (2b) A2 M −⟨A⟩2 M = 1 M M X j=1 [A(qj) −⟨A⟩M]2. (2c) The “correlation time” τ is defined as follows. Since the states qj are generated sequentially by the simulation, each new state qj+1 is guaranteed to be close to the previous state qj, so the values A(qj) in the sequence are highly correlated. The correlations in the values of A(qj) decrease with the “correlation time” τ which can be calculated from the correlation function φAA(t), namely φAA(t) = ⟨A(t)A(0)⟩−⟨A(t)⟩⟨A(0)⟩ A2 −⟨A⟩2 , (3a) τ = X t>0 φAA(t). (3b) The variable t is a measure of the separation between pairs of configurations in the ordered sequence. In the case of molecular dynamics simulations, τ represents a physical time for the system to move far enough along its trajectory on the energy surface to result in decorrelated values of A. Monte Carlo simulations explore equilibrium microstates in a random walk, so τ does not correspond to physical time but rather the average number of Monte Carlo sweeps needed to give statistically independent values for A. The quan-tity M/(2τ + 1) represents the number of statistically independent configurations in the sequence of M values.2 2The correlations φAA(t) in equation (16.1.3a) can be measured using subsequences of the M configurations: ⟨A(t)A(0)⟩≈1 M′ M′ X j=1 A(qj+t)A(qj), and ⟨A(t)⟩≈1 M′ M′ X j=1 A(qj+t). By definition, the correlation function φAA(0) is unity and the correlations decay to zero as t →∞. Once one can place a reliable upper bound on the size of the correlation time τ for a given system from a knowledge of its equilibrium correlations, one can simply skip more than τ configurations between storing values of A(qj) to ensure that the numbers in the sequence are now approximately statistically independent. 640 Chapter 16. Computer Simulations 16.2 Monte Carlo simulations The term Monte Carlo method, named for the gambling casinos in Monaco, was coined by Nicholas Metropolis (1987) – “a suggestion not unrelated to the fact that Stan [Ulam] had an uncle who would borrow money from relatives because he ‘just had to go to Monte Carlo’.” The goal of a Monte Carlo simulation in equilibrium statistical mechanics is to use pseudorandom numbers to draw a representative sample of microstates {q} from the equilibrium probability distribution Peq(q) = exp −βE(q)  P q′ exp −βE(q′) . (1) This means that “instead of choosing configurations randomly, then weighting them with exp(−E/kT), we choose configurations with a probability exp(−E/kT) and weight them evenly” (Metropolis et al., 1953). If a simulation can accomplish this, then thermodynamic averages can be calculated using the simple averages in equation (16.1.2). This importance sampling of the states provides a huge computational advantage over normal random sampling. The following algorithm accomplishes the goal of randomly selecting microstates q from the set of all microstates with a probability distribution that approaches the equilib-rium distribution (1) (Metropolis et al., 1953; Kalos and Whitlock, 1986; Allen and Tildesley, 1990; Frenkel and Smit, 2002; Binder and Heermann, 2002; Landau and Binder, 2009). Con-sider an ensemble of microstates that has some initial distribution of probabilities P(q,0) and let the distribution evolve according to the discrete stochastic rate equation P(q,t + 1) = P(q,t) + X q′ P(q′,t)W(q′ →q) −P(q,t) X q′ W(q →q′), (2) where W(q →q′) is the transition rate from state q to state q′. If the transition rate obeys the balance condition X q′ Peq(q′)W(q′ →q) = Peq(q) X q′ W(q →q′), (3) and the random process in equation (2) can reach every microstate from every other microstate in a finite number of steps, then the ensemble probability will approach the equilibrium distribution: lim t→∞P(q,t) = Peq(q). (4) In practice, equation (3) is usually implemented using the detailed balance condition Peq(q)W(q →q′) = Peq(q′)W(q′ →q). (5) Evaluating Peq(q) requires summing over all states to determine the partition function, but the ratio Peq(q′)/Peq(q) depends only on the energy difference 1E = E(q′) −E(q). 16.2 Monte Carlo simulations 641 Therefore, the transition rates are related by W(q →q′) = exp(−β1E)W(q′ →q). (6) This guarantees that the sequence of states generated by this stochastic process, begin-ning from any starting configuration, asymptotically becomes equivalent to selecting states by a random walk among the microstates of the equilibrium ensemble. This can be implemented in a computer code, as first proposed by Metropolis et al. (1953), using the transition rates W(q →q′) = 1 if 1E ≤0, W(q →q′) = exp(−β1E) if 1E > 0. (7) Other choices for the transition rates are possible but this form, named after Metropolis, is one of the most commonly used. 16.2.A Metropolis Monte Carlo algorithm The Metropolis method can be implemented in a computer program by using a pseudo-random number generator rand() that returns pseudorandom numbers that are uniformly distributed on the open unit interval (0.0,1.0); see Appendix I for a discussion of how pseudorandom numbers are generated. First, initialize the system by choosing a starting state q0 from the set of all microstates of the model. It is helpful if q0 is not atypical of the states in the equilibrium ensemble. This reduces the number of steps needed for the system to equilibrate. For example, a disordered liquid-like state would not be the best starting point for a simulation of a crystalline solid. The Metropolis algorithm is defined by the following steps: 1. Generate a random trial state qtrial that is “nearby” the current state qj of the system. “Nearby” here means that the trial state should be almost identical to the current state except for a small random change made, usually, to a single particle or spin. For example, one can create a trial state of a particle simulation by randomly moving one particle to a nearby location xtrial i = xi + 1x(rand() −0.5), (8) with two more calls to rand( ) to generate ytrial i and ztrial i . The trial state of a spin system usually involves a spin flip or a random rotation of a single spin.3 3There are Monte Carlo algorithms for spin systems that flip spins in large correlated clusters rather than one spin at a time; see Swendsen and Wang (1987) and Wolff (1989). These methods are very effective for simulations of some particu-lar models. Also, one can attempt spin flips of all the spins on noninteracting sublattices at one time since the acceptance of each flip is independent of the other flipped spins. For example, a chessboard pattern update of a spin model in which spins only interact with nearest neighbors of a square lattice can be more efficient for some computer architectures or programming environments. 642 Chapter 16. Computer Simulations 2. Determine the change in the energy of the trial state compared to the previous state, namely 1E = E(qtrial) −E(qj). If 1E ≤0, accept the trial state, that is, set qj+1 = qtrial. If 1E > 0, then accept the trial state with probability exp(−β1E). This is accomplished by using an additional call to the pseudorandom number generator. If rand() < exp(−β1E), then accept the trial state. If the interactions are short-ranged, the calculation of the energy change will only involve interactions with a few nearby particles or spins. If the trial state is illegal in some way, that is, it is not an allowed state in the set of all configurations, then the state should be rejected. This is equivalent to setting the energy change at +∞. If the trial state is rejected for either reason, then set the new state of the system equal to the previous state qj+1 = qj, that is, leave the state at the old value qj, throw away the trial state, and move on. 3. Perform steps 1 and 2 once for each particle or spin in the system. This is often done randomly to ensure detailed balance.4 Steps 1 through 3 define one Monte Carlo sweep. 4. Repeat steps 1 through 3 for Meq Monte Carlo sweeps to let the system equilibrate. The proper choice of Meq is not obvious a priori. At the very least, all the measures A(q) studied in the simulation should no longer have any obvious monotonic drift by the end of equilibration. This does not guarantee that the system has reached equilibrium since the system could well be trapped in the vicinity of a long-lived metastable state. 5. Repeat steps 1 through 3 for M Monte Carlo sweeps while keeping track of all the thermodynamic variables one wants to measure, namely {A(qj)}M j=1. Use equations (16.1.1) and (16.1.2) to determine the equilibrium averages and uncertainties. To determine averages at a different set of parameters (temperature, density, etc.), change the parameters by a small amount and repeat steps 1 through 5, including the equilibration step 4.5 Using the last configuration of the previous run as the first configuration of the next run can often reduce the equilibration time. Figure 16.1 shows a Monte Carlo calculation of the specific heat of the two-dimensional Ising model on a 128 × 128 square lattice, as compared to the exact solution presented in Section 13.4.A. 4Sequential and other update methods that violate detailed balance are sometimes used for efficiency but special care should be taken to ensure that detailed balance is maintained on average. 5Histogram reweighting methods can sometimes be used to reduce the number of temperatures and fields that need to be simulated (Ferrenberg and Swendsen, 1988). For example, if a spin simulation at coupling K and field h collects a histogram that samples the joint energy-magnetization distribution PK,h(E,M), the distribution at nearby temperatures and fields is given by PK+1K,h+1h(E,M) = PK,h(E,M)e1K E+1hM P E,M PK,h(E,M)e1K E+1hM . Other methods that are now widely used are: parallel tempering, multicanonical Monte Carlo, and “broad histogram” methods. These are particularly effective for studying systems with strongly first-order phase transitions. Monte Carlo renormalization group methods are very powerful for studying critical points. For a survey, see Landau and Binder (2009). 16.3 Molecular dynamics 643 3 2 1 0 0 1 2 3 4 5 C/Nk kT/J FIGURE 16.1 Monte Carlo specific heat (×’s) of the two-dimensional Ising model on a 128×128 lattice, as compared to the exact solution (solid line) from Section 13.4.A; see Kaufman (1948), Ferdinand and Fisher (1967), and Beale (1996). The MC error bars are smaller than the symbols used, except near the bulk critical temperature Tc(∞). Each data point represents an average using 105 Monte Carlo sweeps, except at the bulk critical point where 106 Monte Carlo sweeps were used to mitigate critical slowing down. 16.3 Molecular dynamics The purpose of a molecular dynamics simulation is to integrate Newton’s equations of motion for the set of particles in the given system. One advantage of MD over MC is that it approximates the time evolution of the equations of motion of the system, so MD can be used to study a host of dynamical properties. MD is usually more efficient at simulat-ing systems with long-range interactions since all the particles are updated together. MD is sometimes easier to implement than MC for complex systems since appropriate MC moves are sometimes difficult to derive. There are MD variants that allow simulations of other ensembles, but the simplest case simulates a microcanonical ensemble in which the microstate of the given system explores its energy surface in the phase space; see Allen and Tildesley (1990) and Frenkel and Smit (2002) for details. The equations of motion here are d2ri dt2 = 1 mi Fi = −1 mi ∇iU (r1,r2,..,rN), (1) where Fi is the force on particle i arising from the N-particle potential energy function U. The MD simulation moves the system forward in time by discrete steps 1t. The most 644 Chapter 16. Computer Simulations commonly used integration method in this context is due to Verlet (1967): ri(t + 1t) = 2ri(t) −ri(t −1t) + (1t)2 mi Fi(t). (2) This is equivalent to the leap-frog and velocity Verlet algorithms that update both positions and velocities of the particles; see Frenkel and Smit (2002). The Verlet method preserves the time-reversal symmetry of the Hamiltonian equations of motion and has an error per step of order (1t)4, while only requiring a single determination of the force on each particle, which is usually the most computationally time-consuming part of the simulation. Most importantly, the Verlet algorithm is symplectic, so the integration is equivalent to an exact solution of a “nearby” ghost Hamiltonian, which results in good long-term stability and good conservation of energy properties. A simulation starts the system in some initial microstate with defined positions and velocities of all the particles, and the integration algorithm steps the positions and veloc-ities of the particles forward in time. The simplest forms of the approximations employed for the velocities and the energy are vi(t) = ri(t + 1t) −ri(t −1t) 21t , (3a) E = N X i=1 mi 2 ri(t + 1t) −ri(t −1t) 21t 2 + U[r1(t),r2(t),..,rN(t)]. (3b) The time-step 1t is chosen to be small as compared to the shortest fundamental time scale in the Hamiltonian, while not so small as to limit the efficiency of the program. The numer-ical integration approximates a member of the microcanonical ensemble moving along the constant-energy hypersurface in the phase space. Calculating equilibrium averages properly depends on the Hamiltonian being ergodic;6 this allows the system to sample all regions of the constant-energy hypersurface, so the MD time-averages are equivalent to averages over the microcanonical ensemble. If the total energy drifts more than some predetermined amount during the course of the simulation, then all the velocities can be rescaled to shift the total energy back to its initial value. Alternatively, one can use a thermostat to maintain the temperature at a desired value; see Frenkel and Smit (2002). 6Since an MD simulation creates a time evolution of the model system, one needs some assurance that the sys-tem is ergodic, that is, the time averages and the ensemble averages are the same. For example, a system of harmonic oscillators is not ergodic. A system of N particles in d dimensions has a 6N-dimensional phase space, so the constant-energy hypersurface has 6N −1 dimensions. The normal mode solution for N coupled oscillators has 3N constants of the motion, so the system explores only a 3N-dimensional hypersurface. Even making the couplings between particles anharmonic does not eliminate the problem as first shown by Fermi, Pasta, and Ulam (1955) and explored theoretically by Kolmogorov (1954), Arnold (1963), and Moser (1962). MD simulations of equilibrium systems presume that the sys-tem is ergodic. There are only a few systems that are provably ergodic but, fortunately, most systems with realistic pair potentials appear to behave ergodically in two or more dimensions. In view of this, MD simulations of one-dimensional systems should be treated as suspect from this perspective. 16.3 Molecular dynamics 645 A commonly used pair interaction for monatomic fluids such as neon and argon is the Lennard-Jones interaction u(r) = 4ε D r 12 − D r 6! , (4) where D is the molecular diameter and ε is the depth of the attractive well. The Lennard-Jones potential is attractive at long distances, and decays as 1/r6 to model the van der Waals attraction; at short distances, it diverges as 1/r12 to model the Pauli repulsion that prevents overlap of the electronic wavefunctions. Simulations are best carried out using dimensionless parameters. In the case of a fluid with Lennard-Jones interactions, all lengths can be measured in units of D, all energies (including kT) in units of ε, all forces in units of ε/D, all pressures in units of ε/D3, all times in units of p mD2/ε, and so on. Simu-lations then need to be conducted only for single values of reduced temperature kT/ε, the reduced density nD3, and so on, while measuring observables in reduced units. Compar-isons between simulations and experimental results can then be made using experimental values of D, m, and ε.7 In dimensionless units, the Lennard-Jones force between a pair of particles is F = ± r r2  48 r12 −24 r6  . (5) Newton’s third law of motion can be used to reduce the number of force calculations by a factor of two. The Lennard-Jones model was first studied in an MC simulation by Wood and Parker (1957) and in an MD simulation by Rahman (1964) and Verlet (1967). 16.3.A Molecular dynamics algorithm First, start the system by choosing an initial state by setting the initial positions and veloc-ities of all the particles. The initial velocities are usually set by choosing each component of the velocity vector of each particle from the Maxwell distribution. In reduced units, this is PMaxwell(vx(0)) = 1 √ 2πT exp −v2 x(0) 2T ! ; (6) 7For example, the Lennard-Jones parameters appropriate for argon are ε/k = 119.8K and D = 0.3405nm (Levelt, 1960; Rowley, Nicholson, and Parsonage, 1975). Interaction potentials are almost always cut off at a finite distance between molecules to reduce the number of interactions that need to be considered at each time step. For the Lennard-Jones interaction, this is most commonly done at rmax = 2.5D. If the potential is set to zero for distances greater than rmax, then this would leave a small discontinuity in the potential. To eliminate this, the potential is often shifted upward by −u(2.5D) ≃0.0163ε, so that the potential is zero at rmax. This allows a direct comparison between MC and MD simula-tions. If the shift is not made in the potential, one could not directly compare the results from MC and MD simulations because the discontinuity in the potential would result in a delta-function force that affects the motion in the MD sim-ulation but not the configurations in the MC simulation. Comparisons of MC and MD results with experiments need to include perturbations from the shift and the missing tails of the pair potentials. 646 Chapter 16. Computer Simulations see Appendix I to see how to use a uniform pseudorandom number generator to select from a Gaussian distribution. The initial velocities can then be used to set the positions of the particles after the first time-step, namely ri(1t) = ri(0) + vi(0)1t + 1 2 (1t)2 mi Fi(0). (7) 1. Next, use equation (2) to move the system forward in time through Meq = τeq/1t time steps. The equilibration time τeq must be chosen large enough for the system to equilibrate; see the Monte Carlo discussion in Section 16.2. A thermostat is often used to evolve the system to a state with the desired temperature; see Frenkel and Smit (2002). 2. Now, use equation (2) to move the system forward in time through M = τavg/1t time steps while keeping track of all the thermodynamic variables {A(qj)} one wants to measure. Finally, use equation (16.1.2) to determine the equilibrium averages and uncertainties. To determine averages at a different set of parameters (temperature, density, etc.), change the parameters by a small amount and repeat steps 1 and 2. Using the last configura-tion of the previous run as the first configuration of the new run can often reduce the equilibration time. 16.4 Particle simulations Fluids can be modeled by both MC and MD simulations by placing N particles in a periodic box with volume V interacting via a pair potential. Hansen and McDonald (1986), Allen and Tildesley (1990), and Frenkel and Smit (2002) provide excellent surveys of this topic. Calculating energy changes of trial moves in MC or forces in MD only involves pairs of particles whose closest periodic copies are within the cutoff distance of each other. MC simulations typically sample the canonical ensemble,8 so they control the temperature and density, and measure the energy, pressure, and so on. MD simulations typically sample the microcanonical ensemble,9 so they control the energy and density, and they measure the temperature, pressure, and so on. The equipartition theorem gives for the temperature 8Monte Carlo simulations of isobaric or grand canonical ensembles are also widely used by adding PV or µN terms to the Hamiltonian (Frenkel and Smit, 2002). 9Molecular dynamics simulations of other ensembles are possible. For example, one can include extra dynamical variables that allow the total energy or volume to fluctuate in order to approximate a canonical or isobaric ensemble; see Frenkel and Smit (2002). Variants of MC and MD simulations that span the range between the two include hybrid Monte Carlo methods that mix MC and MD methods into one code to take advantage of the strengths of both methods. Alternatively, one can include Langevin random force terms and damping in an MD simulation to create coupling to a heat bath. 16.4 Particle simulations 647 T in a d-dimensional system kT = 1 Nd N X i=1 miv2 i + . (1) The virial equation of state in equations (3.7.15) and (10.7.11) can be used to determine the pressure P in either type of simulation: P nkT = 1 + 1 NdkT X i<j F(rij) · rij + = 1 − n 2dkT Z du dr rg(r)dr. (2) As discussed in Section 10.7, the pair correlation function of a fluid can be used to measure a variety of thermodynamic properties including the pressure, the isothermal compress-ibility κT, and the scattering structure factor S(k); see equations (10.7.18) through (10.7.21). The pair correlation function g(r) defined in equation (10.7.5) can be determined by col-lecting a histogram of the distances between all pairs of particles periodically during the simulation, accounting for the periodic boundary conditions, and scaling the histogram by an amount proportional to the volume of shells of radius r and thickness 1r. In three dimensions, the pair correlation function is given by g(r) = 2V N2  4π 3  (r + 1r)3 −r3 N X i<j 11r(rij −r) + , (3) where the step function 11r(ξ) is unity for 0 < ξ < 1r and zero otherwise. This expression is the ratio of the number of events in each bin in the histogram compared to the average number that would be expected for an ideal gas with the same density. 16.4.A Simulations of hard spheres The system of hard spheres has been studied extensively in both MC and MD simulations, and was the first model studied using either method (Metropolis et al., 1953; Adler and Wainwright, 1957, 1959). The pair potential for hard spheres is u(r) = ( 0 for r > D, ∞ for r ≤D. (4) Temperature is an irrelevant parameter for the spatial configurations sampled by this model since the pair potential does not have a finite energy scale. A full exploration of the phase diagram involves only varying the reduced number density nDd. All thermody-namic properties are either independent of temperature, or scale with temperature in a trivial way. For example, the scaled pressure for a system of hard spheres P/nkT is a func-tion only of the reduced number density nDd. The hard sphere density is often expressed in 648 Chapter 16. Computer Simulations terms of the packing fraction η, the fraction of the volume of the system actually occupied by the spheres. In three dimensions, the volume fraction is given by η = πnD3/6. Since the pair potential is singular, the pressure cannot be calculated using the virial equation (2) but the pressure can be determined using the virial equation of state for hard spheres, namely (10.7.12). An MC code for hard spheres is relatively simple since the energy change in a trial move is either zero or infinity. A trial displacement of a particle is rejected if the trial position of the particle is within a distance D of any other particle, and is accepted otherwise. This was the first statistical physics model ever studied in a computer simulation (Metropolis et al., 1953). Implementing MD for hard spheres requires a different approach from the standard MD. Finite-difference integration methods will not work here since the potential is not differentiable. Instead, one can exploit the exact solution to the equations of motion. Each particle travels in a straight line at a constant velocity except at the instants when pairs of particles collide, that is when they are a distance D apart. Due to the singular nature of the potential, the collisions can be uniquely time-ordered. Each collision conserves both kinetic energy and momentum, so the velocities after each collision can be determined analytically from the velocities and the displacement vector between the centers of the two particles at the moment of the collision. The changes in the velocities of the two colliding particles are 1vi = −1vj = − rij · vij  rij D2 |rij|=D , (5) where rij and vij are, respectively, the relative positions and relative velocities of the two particles. The simulation moves the particles forward in time from collision to collision and changes the velocities of the pairs of particles involved in the collisions, as given by equation (5). This was the first implementation of the MD method in statistical physics (Alder and Wainwright 1957, 1959). The pair correlation function and the structure factor in the fluid phase are shown in Figure 16.2 and the phase diagram for hard spheres in Figure 16.3. At low densities, the equilibrium phase is a short-range ordered fluid. At high densities, the equilibrium phase of the model is a long-range ordered, face-centered cubic solid. It is worthwhile to note that an attractive interaction is not required for a model to have a crystalline phase. An attractive interaction is, however, required for the formation of a liquid–vapor coexistence line and a critical point. For this reason, the low-density phase of the hard sphere model is often referred to as a fluid phase rather than a liquid phase since the model does not have a liquid–vapor coexistence line. The liquid and solid volume fractions at the liquid–solid coexistence line are ηl ≃0.491 ± 0.002 and ηs ≃0.543 ± 0.002, respectively. The liquid–solid coexistence pressure is given by P∗ ls = PlsD3/kT ≃11.55 ± 0.11; see Speedy (1997). In the low-density fluid phase for η < ηl, the reduced pressure is accurately modeled by the Carnahan–Starling equation of state (10.3.25) P nkT = 1 + η + η2 −η3 (1 −η)3 , (6) 16.4 Particle simulations 649 6 5 4 3 2 1 0 0 2 4 (a) (b) 6 8 r/D g(r) 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0 0 20 30 10 40 50 60 70 kD S (k) FIGURE 16.2 (a) The pair correlation function g(r) and (b) the static structure factor S(k) for a three-dimensional system of hard spheres at volume fraction η = 0.49 from a Monte Carlo simulation of 2916 particles (M. Glaser, unpublished). This value of the volume fraction is in the liquid phase close to the solid–liquid coexistence line. The solid lines depict the pair correlation function and the static structure factor from the Percus–Yevick approximation; see Percus and Yevick (1958), Wertheim (1963), and Hansen and McDonald (1986). 30 20 10 0 0.0 0.2 0.4 0.6 0.8 l s cp P PIs FIGURE 16.3 Sketch of the equilibrium phase diagram for hard spheres in three dimensions. The horizontal axis is the volume fraction η = πnD3/6 and the vertical axis is the scaled pressure P∗= PD3/kT. There are two equilibrium phases: a low-density fluid phase for 0 < η < ηl and a high-density solid phase for ηs < η < ηcp. although there are other good parametizations as well. In the solid phase the pressure is approximately given by P nkT = 3 1 −η∗−0.5921η∗−0.7072 η∗−0.601 , (7) where η∗(= η/ηcp) is the ratio of the actual packing fraction to the maximum close-packed value ηcp, namely π √ 2/6 ≃0.7405 (Speedy, 1997; Frenkel and Smit, 2002). The pressure in the solid phase diverges as the density approaches the close-packed density. The model 650 Chapter 16. Computer Simulations also exhibits a metastable disordered phase for densities between ηl and the random close-packed volume fraction ηrcp ≃0.644 ± 0.005; see Rintoul and Torquato (1996). For a survey of hard sphere results, see Mulero et al. (2008). 16.5 Computer simulation caveats Computer simulations are widely used in statistical physics and have played an important role in our understanding of many physical systems. Simulations complement theory and experiment and provide many advantages for the study of systems that are not amenable to exact or approximate theoretical analysis. However, it is important to understand the inherent limitation of this technique. . Computer simulations necessarily involve a limited number of degrees of freedom, typically hundreds to thousands of particles or spins. This is not nearly large enough to display many of the behaviors that occur in thermodynamically large systems. For example, a model of a dense system of 1,000 particles in a three-dimensional cubic box will only have about 10 particles along each linear dimension of the box, so correlations beyond about five particle diameters are affected by the periodic boundary conditions. Extraction of accurate thermodynamic behavior from such a study will often involve an analysis of the finite-size scaling behavior of the model for a sequence of systems with different sizes. . Computer simulations necessarily involve a limited time-scale of simulation. Typical molecular time scales are of the order of t0 ≈a/v, where a a microscopic length scale and v a molecular velocity. For atomic scales near room temperature, a ≈0.1nm and v ≈100 m/s, which gives t0 ≈10−12 s. In an MD simulation, each time step moves the system forward in time by an amount 1t, which must be much less than t0 in order to aptly integrate the equations of motion, say 1t ≈10−14 s. A simulation that moves the system forward through 106 time steps will sample a physical time of only 10ns, which may not be sufficient to reach many important time-scales of interest in the problem. This is especially problematic when the system has inherently slow time-scales, such as the critical slowing down near second-order phase transitions and the hysteresis near first-order phase transitions. Monte Carlo simulations are similarly hampered in that the simulation must run long enough for the model to explore a sufficiently large region of the phase space to capture the equilibrium behavior. In favorable cases, this and the previous issue can be mitigated by special simulation methods such as coarse graining, cluster update methods, parallel tempering, multicanonical Monte Carlo methods, Monte Carlo renormalization group, and so on; see footnote 5. . Interactions between particles are usually highly simplified for computational efficiency and the interaction range is usually cut off. Long-range interactions (Couloub, dipole, van der Waals, etc.) need to be resummed or treated via perturbation theory to try to account for their effects. Problems 651 . MC and MD simulations do not directly measure the number of microstates available to the system, so one cannot directly calculate the entropy or the free energy in the same way as other observables. If, for example, a determination of the free energy is necessary to locate a phase transition, then it can be determined by a thermodynamic integration to a state with a theoretically known free energy; see Frenkel and Smit (2002). . MD simulations depend on the ergodicity of the Hamiltonian, so one-dimensional models, models that are weakly perturbed from mechanical equilibrium, and other nearly integrable models may get trapped in low-dimensional orbits that do not fully explore the constant-energy hypersurface of the Hamiltonian; see footnote 6. . All pseudorandom number generators produce some level of correlation in their sequences. Even subtle correlations in pseudorandom number sequences can produce erroneous results in Monte Carlo simulations. Different classes of generators have different weaknesses, so switching to a generator based on a different algorithm will sometimes cure a problem caused by correlations produced by a particular generator. Testing a generator before using it in a simulation is always a good idea; see Appendix I. . It is extremely important to confirm the validity of MC and MD simulation codes. This process is rather different from verifying a theoretical calculation. Some code evaluation and verification procedures include: testing the code initially on small systems with known properties, testing the code whenever possible against models with exact solutions or models that have been widely studied in the literature, examining results as a function of system size and run length, and retesting carefully whenever new interactions or code modules are added. In this connection, Frenkel and Smit (2002), Parker (2008), and Landau and Binder (2009) provide lists of good strategies. Problems 16.1. Write a code to test a uniform pseudorandom number generator. If you do not have a canned generator available, write a generator based on L’Ecuyer’s recommended generator in Appendix I. Apply the following tests: average ⟨x⟩= 1/2, variance x2 −⟨x⟩2 = 1/12, and the pair correlations test xi+kxi = 1/4 for k ̸= 0. Generate a histogram of pairs of numbers on a two-dimensional unit square and test that the distribution is statistically uniform. 16.2. Write a code to test a Gaussian pseudorandom number generator. If you do not have a canned generator available, write a generator based on the Box-Muller algorithm in Appendix I. Apply the following tests: average ⟨x⟩= 0, variance x2 = 1, and the pair correlations test xi+kxi = 0 for k ̸= 0. Generate a histogram of pairs of numbers in two dimensions and test that the distribution is statistically Gaussian. 16.3. Define a sequence of correlated random numbers sk = αsk−1 + (1 −α)rk, where rk is a unit-variance, uncorrelated, Gaussian pseudorandom number while 0 < α < 1 defines the range of the correlations. Show that this sequence is Gaussian distributed, with a zero 652 Chapter 16. Computer Simulations mean. Determine the variance in terms of α and compare your result with equation (16.1.2b). Write a code to determine the correlation function (16.1.3). Plot your measured correlation function and compare it to the exact correlation function. 16.4. Write a Monte Carlo code for a system of N hard spheres of diameter D on a one-dimensional ring of length L with periodic boundary conditions. Calculate the pair correlation function and compare it to equations (13.1.6) and (13.1.7). The pressure of the system is given by P/nkT = 1 + nDg(D+); see equation (10.7.12). Compare your pressure to the one obtained for the exact configurational partition function ZN = L(L −ND)N−1/N!; see equation (13.1.2). 16.5. Write a Monte Carlo code for a fluid of N hard spheres in a two-dimensional L × L square box with periodic boundary conditions in each direction. Calculate the pair correlation function and determine the scaled pressure using equation (10.7.12), namely P/nkT = 1 + 2ηg(D+). Compare this pressure to the approximate form P/nkT = (1 + η/8)/(1 −η)2. 16.6. Write a Monte Carlo code for a fluid of N hard spheres in a two-dimensional L × L square box and include a one-body gravity term PN i=1 mgyi in the algorithm, that is, accept otherwise legal configurations with probability exp(−βmg1y). You will need to use hard-wall boundary conditions on the top and bottom walls. Determine the average number density as a function of the vertical position in the box. 16.7. Write a molecular dynamics code for N Lennard-Jones particles in a two-dimensional L × L square box. Apply periodic boundary conditions in each direction. Determine the scaled pressure using the virial equation (16.4.2). Calculate and plot the pair correlation function of the system. 16.8. Write a molecular dynamics code for N Lennard-Jones particles in a two-dimensional L × L square box, and include a one-body gravity term in the energy: PN i=1 mgyi. Apply periodic boundary conditions in the x-direction but a repulsive WCA (Weeks, Chandler, and Andersen, 1971) potential on the top and bottom walls. The WCA potential is the repulsive part of a Lennard-Jones potential for r/D < (2)1/6, with the potential shifted up by ε. Show that the average kinetic energy per particle is independent of the height y in the box but the average scaled density nD2 depends on the vertical position in the box. 16.9. Write an MC code to simulate the one-dimensional Ising model on a periodic lattice of length L. Calculate the internal energy and specific heat of the model and compare them to equations (13.2.15) and (13.2.16). Calculate the correlation function G(n) = si+nsi and it compare to equation (13.2.32). 16.10. Write an MC code to simulate the two-dimensional nearest-neighbor Ising model on a periodic L × L lattice in zero field. Calculate the internal energy and the specific heat of the system as functions of temperature and compare them to the exact results in section 13.4.A. See exact results for the two-dimensional Ising model for various lattice sizes at www.elsevierdirect.com. 16.11. Write an MC code to simulate the two-dimensional nearest-neighbor Ising model on a periodic L × L lattice in zero field. Calculate the energy distribution P(E) over a range of temperatures including the critical point. Use this distribution to calculate the internal energy and the specific heat as functions of temperature. See exact results for the two-dimensional Ising model for various lattice sizes at www.elsevierdirect.com. 16.12. Write an MC code to simulate the one-dimensional XY model. Calculate the internal energy, the specific heat, the isothermal susceptibility, and the pair correlation function, and compare your results to the analytical results for the n = 2 case in Section 13.2. 16.13. Write an MC code to simulate the two-dimensional XY model. Calculate the internal energy, specific heat, isothermal susceptibility and the pair correlation function, and compare your results to the theoretical results given in Section 13.7. Appendices A Influence of boundary conditions on the distribution of quantum states In this appendix we examine, under different boundary conditions, the asymptotic distribution of single-particle states in a bounded continuum. For simplicity, we consider a cuboidal enclosure of sides a, b, and c. The admissible solutions of the free-particle Schr¨ odinger equation ∇2ψ + k2ψ = 0  k = pℏ−1 , (1) which satisfy Dirichlet boundary conditions (namely, ψ = 0 everywhere at the boundary), are then given by ψlmn(r) ∝sin lπx a  sin mπy b  sin nπz c  , (2) with k = π l2 a2 + m2 b2 + n2 c2 !1/2 ; l,m,n = 1,2,3,.... (3) Note that in this case none of the quantum numbers l, m, or n can be zero, for that would make the wavefunction identically vanish. If, on the other hand, we impose Neumann boundary conditions (namely, ∂ψ/∂n = 0 everywhere at the boundary), the desired solutions turn out to be ψlmn(r) ∝cos lπx a  cos mπy b  cos nπz c  , (4) with l,m,n = 0,1,2,...; (5) clearly, the value zero of the quantum numbers is now allowed! In each case, however, the negative-integral values of the quantum numbers do not lead to any new wavefunctions. The total number g(K) of distinct wavefunctions ψ, with wave number k not exceeding a given value K, may be written as g(K) = X′ l,m,n f (l,m,n), (6) Traveling Wave Analysis of Partial Differential Equations © 2011 Elsevier Ltd. All rights reserved. 653 654 Appendices where f (l,m,n) = 1 for the numbers (l,m,n) belonging to the set (3) or (5), as the case may be; the summation P′ in each case is restricted by the condition l2 a2 + m2 b2 + n2 c2 ! ≤K 2 π2 . (7) We now define a sum G(K) = X′ l,m,n f ∗(l,m,n), (8) where f ∗(l,m,n) = 1 for all integral values of l, m, and n (positive, negative, or zero), the restric-tion on the numbers (l,m,n) being the same as stated in (7). One can then show, by setting up correspondence of terms, that X′ l,m,n f (l,m,n) = 1 8 X′ l,m,n f ∗(l,m,n) ∓ X′ l,m f ∗(l,m,0) + X′ l,n f ∗(l,0,n) + X′ m,nf ∗(0,m,n)  + X′ l f ∗(l,0,0) + X′ m f ∗(0,m,0) + X′ n f ∗(0,0,n)  ∓1  ; (9) the upper and the lower signs here correspond, respectively, to the Dirichlet and the Neumann boundary conditions. Clearly, the first sum on the right side of (9) denotes the number of lattice points in the ellipsoid1 (X2/a2 + Y 2/b2 + Z2/c2) = K 2/π2, the next three sums denote the numbers of lattice points in the ellipses, which are cross-sections of this ellipsoid with the Z-, Y - and X-planes, while the last three sums denote the numbers of lattice points on the principal axes of the ellipsoid. Now, if a, b, and c are sufficiently large in comparison with π/K, one may replace these numbers by the corresponding volume, areas, and lengths, respectively, with the result g(K) = K 3 6π2 (abc) ∓K 2 8π (ab + ca + bc) + K 4π (a + b + c) ∓1 8 + E(K); (10) the term E(K) here denotes the net error committed in making the aforementioned replacements. We thus find that the main term of our result is directly proportional to the volume of the enclosure while the first correction term is proportional to its surface area (and, hence, represents a “surface effect”); the next-order term(s) appear in the nature of an “edge effect” and a “corner effect.” Now, a reference to the literature dealing with the determination of the “number of lattice points in a given domain” reveals that the error term E(K) in equation (10) is O(K α), where 1 < α < 1.4; hence, expression (10) for g(K) is reliable only up to the surface term. In view of this, we may write g(K) = K 3 6π2 V ∓K 2 16π S + a lower-order remainder; (11) 1By the term “in the ellipsoid” we mean “not external to the ellipsoid,” that is, the lattice points “on the ellipsoid” are also included. Other such expressions in the sequel carry a similar meaning. Appendix B 655 in terms of ε∗, where ε∗= 8mL2 h2 ε = 4L2 h2 P2 = L2 π2 K 2, (12) equation (11) reduces to equations (1.4.15) and (1.4.16) of the text. In the case of periodic boundary conditions, namely ψ(x,y,z) = ψ(x + a,y,z) = ψ(x,y + b,z) = ψ(x,y,z + c), (13) the appropriate wavefunctions are ψlmn(r) ∝exp{i(k · r)}, (14) with k = 2π  l a, m b , n c  ; l,m,n = 0,±1,±2,.... (15) The number of free-particle states g(K) is now given by g(K) = X′ l,m,n f ∗(l,m,n), (16) such that (l2/a2 + m2/b2 + n2/c2) ≤K 2/(4π2). (17) This is precisely the number of lattice points in the ellipsoid with semiaxes Ka/2π, Kb/2π, and Kc/2π, which, allowing for the approximation made in the earlier cases, is just equal to the volume term in (11). Thus, in the case of periodic boundary conditions, we do not obtain a surface term in the expression for the density of states. For further information on this topic, see Fedosov (1963, 1964), Pathria (1966), Chaba and Pathria (1973), and Baltes and Hilf (1976). B Certain mathematical functions In this appendix we outline the main properties of certain mathematical functions that are of special importance to the subject matter of this text. We start with the gamma function 0(ν), which is identical with the factorial function (ν −1)! and is defined by the integral 0(ν) ≡(ν −1)! = ∞ Z 0 e−xxν−1dx; ν > 0. (1) First of all, we note that 0(1) ≡0!= 1. (2) 656 Appendices Next, integrating by parts, we obtain the recurrence formula 0(ν) = 1 ν 0(ν + 1), (3) from which it follows that 0(ν + 1) = ν(ν −1)···(1 + p)p0(p), 0 < p ≤1, (4) p being the fractional part of ν. For integral values of ν (ν = n, say), we have the familiar representation 0(n + 1) ≡n!= n(n −1)···2 · 1; (5) on the other hand, if ν is a half-odd integral (ν = m + 1 2, say), then 0  m + 1 2  ≡  m −1 2  ! =  m −1 2  m −3 2  ··· 3 2 ·  1 2  0  1 2  = (2m −1)(2m −3)···3 · 1 2m π1/2, (6) where use has been made of equation (21), whereby 0  1 2  ≡  −1 2  != π1/2. (7) By repeated application of the recurrence formula (3), the definition of the function 0(ν) can be extended to all ν, except for ν = 0,−1,−2,... where the singularities of the function lie. The behavior of 0(ν) in the neighborhood of a singularity can be determined by setting ν = −n + ε, where n = 0,1,2,... and |ε| ≪1, and using formula (3) n + 1 times; we get 0(−n + ε) = 1 (−n + ε)(−n + 1 + ε)···(−1 + ε)ε 0(1 + ε) ≈(−1)n n!ε . (8) Replacing x by αy2, equation (1) takes the form 0(ν) = 2αν ∞ Z 0 e−αy2y2ν−1dy, ν > 0. (9) We thus obtain another closely related integral, namely I2ν−1 ≡ ∞ Z 0 e−αy2y2ν−1dy = 1 2αν 0(ν), ν > 0; (10) by a change of notation, this can be written as Iν ≡ ∞ Z 0 e−αy2yνdy = 1 2α(ν+1)/2 0 ν + 1 2  , ν > −1. (11) Appendix B 657 One can easily see that the foregoing integral satisfies the relationship Iν+2 = −∂ ∂α Iν. (12) The integrals Iν appear so frequently in our study that we write down the values of some of them explicitly: I0 = 1 2 π α 1/2 , I2 = 1 4  π α3 1/2 , I4 = 3 8  π α5 1/2 ··· , (13a) while I1 = 1 2α , I3 = 1 2α2 , I5 = 1 α3 ,··· . (13b) In connection with these integrals, it may as well be noted that ∞ Z −∞ e−αy2yνdy = 0 if ν is an odd integer = 2Iν if ν is an even integer. (14) Next, we consider the product of two gamma functions, say 0(µ) and 0(ν). Using representa-tion (9), with α = 1, we have 0(µ)0(ν) = 4 ∞ Z 0 ∞ Z 0 e−(x2+y2)x2µ−1y2ν−1dxdy, µ > 0,ν > 0. (15) Changing over to the polar coordinates (r,θ), equation (15) becomes 0(µ)0(ν) = 4 ∞ Z 0 e−r2r2(µ+ν)−1dr π/2 Z 0 cos2µ−1 θ sin2ν−1 θdθ = 20(µ + ν) π/2 Z 0 cos2µ−1 θ sin2ν−1 θdθ. (16) Now, defining the beta function B(µ,ν) by the integral B(µ,ν) = 2 π/2 Z 0 cos2µ−1 θ sin2ν−1 θdθ, µ > 0,ν > 0, (17) we obtain an important relationship: B(µ,ν) = 0(µ)0(ν) 0(µ + ν) = B(ν,µ). (18) Substituting cos2 θ = η, equation (17) takes the standard form B(µ,ν) = 1 Z 0 ηµ−1(1 −η)ν−1dη, µ > 0,ν > 0, (19) 658 Appendices while the special case µ = ν = 1 2 gives B  1 2, 1 2  = 2 π/2 Z 0 dθ = π. (20) Coupled with equations (2) and (18), equation (20) yields 0  1 2  = π1/2. (21) Stirling’s formula for ν! We now derive an asymptotic expression for the factorial function ν! = ∞ Z 0 e−xxνdx (22) for ν ≫1. It is not difficult to see that, for ν ≫1, the major contribution to this integral comes from the region of x that lies around the point x = ν and has a width of order √ν. In view of this, we invoke the substitution x = ν + (√ν)u, (23) whereby equation (22) takes the form ν! = √ν ν e ν ∞ Z −√ν e−(√ν)u  1 + u √ν ν du. (24) The integrand in (24) attains its maximum value, unity, at u = 0 and on both sides of the maximum it falls rapidly to zero, which suggests that it may be approximated by a Gaussian. We, therefore, expand the logarithm of the integrand around u = 0 and then reconstruct the integrand by taking the exponential of the resulting expression; this gives ν! = √ν ν e ν ∞ Z −√ν exp ( −u2 2 + u3 3√ν −u4 4ν + ··· ) du. (25) If ν is sufficiently large, the integrand in (25) may be replaced by the single factor exp(−u2/2); more-over, since the major contribution to this integral comes only from that range of u for which |u| is of order unity, the lower limit of integration may be replaced by −∞. We thus obtain the Stirling formula ν! ≈√(2πν)(ν/e)ν, ν ≫1. (26) A more detailed analysis leads to the Stirling series ν! = √(2πν) ν e ν  1 + 1 12ν + 1 288ν2 − 139 51840ν3 − 571 2488320ν4 + ···  . (27) Appendix B 659 Next, we consider the function ln(ν!). Corresponding to formula (27), we have, for large ν, ln(ν!) =  ν + 1 2  ln ν −ν + 1 2 ln(2π) +  1 12ν − 1 360ν3 + 1 1260ν5 − 1 1680ν7 + ···  . (28) For most practical purposes, we may write ln(ν!) ≈(ν lnν −ν). (29) We note that formula (29) can be obtained very simply by an application of the Euler–Maclaurin formula. Since ν is large, we may consider its integral values only; then, by definition, ln(n!) = n X i=1 (lni). Replacing summation by integration, we obtain ln(n!) ≃ n Z 1 (lnx)dx = (xlnx −x) x=n x=1 ≈(nlnn −n), which is identical to (29). We must, however, be warned that, whereas approximation (29) is fine as it is, it would be wrong to take its exponential and write ν! ≈(ν/e)ν, for that would affect the evaluation of ν! by a factor O(ν1/2), which can be considerably large; see (26). In the expression for ln(ν!), the corresponding term is indeed negligible. The Dirac δ-function We start with the Gaussian distribution function p(x,x0,σ) = 1 √(2π)σ e−(x−x0)2/2σ 2, (30) which satisfies the normalization condition ∞ Z −∞ p(x,x0,σ)dx = 1. (31) The function p(x) is symmetric about the value x0 where it has a maximum; the height of this max-imum is inversely proportional to the parameter σ while its width is directly proportional to σ, the total area under the curve being a constant. As σ becomes smaller and smaller, the function p(x) becomes narrower and narrower in width and grows higher and higher at the central point x0, condition (31) being satisfied at all σ; see Figure B.1. In the limit σ →0, we obtain a function whose value at x = x0 is infinitely large while at x ̸= x0 it is vanishingly small, the area under the curve being still equal to unity. This limiting form of the 660 Appendices 2 1.5 1 0.5 0.5 1 1.5 2 4 1 0 p(xx0) 1/10 1/3 1 (xx0) 3 2 FIGURE B.1 Gaussian distribution function (30) for different values of σ. function p(x,x0,σ) is, in fact, the δ-function of Dirac. Thus, we may define this function as the one satisfying the following properties: (i) δ(x −x0) = 0 for all x ̸= x0, (32) (ii) ∞ Z −∞ δ(x −x0)dx = 1. (33) Conditions (32) and (33) inherently imply that, at x = x0, δ(x −x0) = ∞and that the range of integra-tion in (33) need not extend all the way from −∞to +∞; in fact, any range that includes the point x = x0 would suffice. Thus, we may rewrite (33) as B Z A δ(x −x0)dx = 1 if A < x0 < B. (34) It follows that, for any well-behaved function f (x), B Z A f (x)δ(x −x0)dx = f (x0) if A < x0 < B. (35) Another limiting process frequently employed to represent the δ-function is the following: δ(x −x0) = Lim γ →0 γ π{(x −x0)2 + γ 2}. (36) Appendix B 661 To see the appropriateness of this representation, we note that, for x ̸= x0, this function vanishes like γ while, for x = x0, it diverges like γ −1; moreover, for all γ , ∞ Z −∞ γ π{(x −x0)2 + γ 2}dx = 1 π  tan−1 (x −x0) γ ∞ −∞ = 1. (37) An integral representation of the δ-function is δ(x −x0) = 1 2π ∞ Z −∞ eik(x−x0)dk, (38) which means that the δ-function is the “Fourier transform of a constant.” We note that, for x = x0, the integrand in (38) is unity throughout, so the function diverges. On the other hand, for x ̸= x0, the oscillatory character of the integrand is such that it makes the integral vanish. And, finally, the integration of this function, over a range of x that includes the point x = x0, gives 1 2π ∞ Z −∞    x0+L Z x0−L eik(x−x0)dx   dk = ∞ Z −∞ eikL −e−ikL 2π(ik) dk = ∞ Z −∞ sin(kL) πk dk = 1, (39) independently of the choice of L. It is instructive to see how the integral representation of the δ-function is related to its previous representations. For this, we introduce into the integrand of (38) a convergence factor exp(−γ k2), where γ is a small, positive number. The resulting function, in the limit γ →0, should reproduce the δ-function; we thus expect that δ(x −x0) = 1 2π Lim γ →0 ∞ Z −∞ eik(x−x0)−γ k2dk. (40) The integral in (40) is easy to evaluate if we recall that ∞ Z −∞ cos(kx)e−γ k2dk = 2 ∞ Z 0 cos(kx)e−γ k2dk = sπ γ  e−x2/4γ , (41) while ∞ Z −∞ sin(kx)e−γ k2dk = 0. (42) Accordingly, equation (40) becomes δ(x −x0) = Lim γ →0 1 √(4πγ )e−(x−x0)2/4γ , (43) 662 Appendices which is precisely the representation we started with.2 Finally, the notation of the δ-function can be readily extended to spaces with more than one dimension. For instance, in n dimensions, δ(r) = δ(x1)···δ(xn), (44) so that (i) δ(r) = 0 for all r ̸= 0, (45) (ii) ∞ Z −∞ ··· ∞ Z −∞ δ(r)dx1 ···dxn = 1. (46) The integral representation of δ(r) is δ(r) = 1 (2π)n ∞ Z −∞ ··· ∞ Z −∞ ei(k·r)dnk. (47) Once again, we may write δ(r) = 1 (2π)n Lim γ →0 ∞ Z −∞ ··· ∞ Z −∞ ei(k·r)−γ k2dnk (48) = Lim γ →0  1 4πγ n/2 e−r2/4γ . (49) C “Volume” and “surface area” of an n-dimensional sphere of radius R Consider an n-dimensional space in which the position of a point is denoted by the vector r, with Cartesian components (x1,...,xn). The “volume element” dVn in this space would be dnr = n Y i=1 (dxi); accordingly, the “volume” Vn of a sphere of radius R would be given by Vn(R) = Z ··· Z 0≤ n P i=1 x2 i ≤R2 n Y i=1 (dxi). (1) 2The reader may check that the introduction into (38) of a convergence factor exp(−γ |k|), rather than exp(−γ k2), leads to the representation (36). Appendix C 663 Obviously, Vn will be proportional to Rn, so let us write it as Vn(R) = CnRn, (2) where Cn is a constant that depends only on the dimensionality of the space. Clearly, the “volume element” dVn can also be written as dVn = Sn(R)dR = nCnRn−1dR, (3) where Sn(R) denotes the “surface area” of the sphere. To evaluate Cn, we make use of the formula ∞ Z −∞ exp(−x2)dx = π1/2. (4) Multiplying n such integrals, one for each xi, we obtain πn/2 = xi=∞ Z ··· Z xi=−∞ exp − n X i=1 x2 i ! n Y i=1 (dxi) = ∞ Z 0 exp(−R2)nCnRn−1dR = nCn · 1 20 n 2  = n 2  !Cn; (5) here, use has been made of formula (B.11), with α = 1. Thus, Cn = πn/2.n 2  !, (6) so that Vn(R) = πn/2 (n/2)!Rn and Sn(R) = 2πn/2 0(n/2)Rn−1, (7a,b) which are the desired results. Alternatively, one may prefer to use spherical polar coordinates right from the beginning — as, for instance, in the evaluation of the Fourier transform I(k) = Z f (r)eik·rdnr. (8) In that case, dnr = rn−1(sinθ1)n−2 ···(sinθn−2)1dr dθ1 ···dθn−2dφ, (9) where the θi range from 0 to π while φ ranges from 0 to 2π. Choosing our polar axis to be in the direction of k, equation (8) takes the form I(k) = Z f (r)eikr cosθ1rn−1(sinθ1)n−2 ···(sinθn−2)1dr dθ1 ···dθn−2dφ. (10) 664 Appendices Integration over the angular coordinates θ1,θ2,θ3,...,θn−2 and φ yields factors π1/20 n −1 2  2 kr (n−2)/2 J(n−2)/2(kr) × B n −2 2 , 1 2  · B n −3 2 , 1 2  ···B  1, 1 2  · 2π, where Jν(x) is the ordinary Bessel function while B(µ,ν) is the beta function; see equations (B.17) and (B.18). Equation (10) now becomes I(k) = (2π)n/2 ∞ Z 0 f (r)  1 kr (n−2)/2 J(n−2)/2(kr)rn−1dr, (11) which is our main result. In the limit k →0, Jν(kr) →  1 2kr ν /0(ν + 1), so that I(0) = 2πn/2 0(n/2) ∞ Z 0 f (r)rn−1dr, (12) consistent with (3) and (7b). On the other hand, if we take f (r) to be a constant, say 1/(2π)n, we should obtain another representation of the Dirac δ-function in n dimensions; see equations (8) and (B.47). We thus have, from (11), δ(k) = 1 (2π)n/2 ∞ Z 0  1 kr (n−2)/2 J(n−2)/2(kr)rn−1dr. (13) As a check, we introduce a factor exp(−αr2) in the integrand of (13) and obtain δ(k) = Lim α→0 1 (2π)n/2 ∞ Z 0 e−αr2  1 kr (n−2)/2 J(n−2)/2(kr)rn−1dr = Lim α→0  1 4πα n/2 e−k2/4α, (14) in complete agreement with (B.49). If, on the other hand, we use the factor exp(−αr) rather than exp(−αr2), we get δ(k) = Lim α→00 n + 1 2  α {π(k2 + α2)}(n+1)/2 , (15) which generalizes (B.36). D On Bose–Einstein functions In the theory of Bose–Einstein systems we come across integrals of the type Gν(z) = ∞ Z 0 xν−1dx z−1ex −1 (0 ≤z < 1,ν > 0;z = 1,ν > 1). (1) Appendix D 665 In this appendix we study the behavior of Gν(z) over the stated range3 of the parameter z. First of all, we note that Lim z→0 Gν(z) = ∞ Z 0 ze−xxν−1dx = z0(ν). (2) Hence, it appears useful to introduce another function, gν(z), such that gν(z) ≡ 1 0(ν)Gν(z) = 1 0(ν) ∞ Z 0 xν−1dx z−1ex −1. (3) For small z, the integrand in (3) may be expanded in powers of z, with the result gν(z) = 1 0(ν) ∞ Z 0 xν−1 ∞ X l=1 (ze−x)ldx = ∞ X l=1 zl lν = z + z2 2ν + z3 3ν + ··· ; (4) thus, for z ≪1, the function gν(z), for all ν, behaves like z itself. Moreover, gν(z) is a monotonically increasing function of z whose largest value in the physical range of interest obtains when z →1; then, for ν > 1,gν(z) approaches the Riemann zeta function ζ(ν): gν(1) = ∞ X l=1 1 lν = ζ(ν) (ν > 1). (5) The numerical values of some of the ζ(ν) are ζ(2) = π2 6 ≃1.64493, ζ(4) = π4 90 ≃1.08232, ζ(6) = π6 945 ≃1.01734, ζ  3 2  ≃2.61238, ζ  5 2  ≃1.34149, ζ  7 2  ≃1.12673, and, finally, ζ(3) ≃1.20206, ζ(5) ≃1.03693, ζ(7) ≃1.00835. For ν ≤1, the function gν(z) diverges as z →1. The case ν = 1 is rather simple, for the function gν(z) now assumes a closed form: g1(z) = ∞ Z 0 dx z−1ex −1 = ln(1 −ze−x) ∞ 0 = −ln(1 −z). (6) As z →1, g1(z) diverges logarithmically. Setting z = e−α, we have g1(e−α) = −ln(1 −e−α) − − − − − → α→0 ln(1/α). (7) 3The behavior of Gν(z) for z > 1 has been discussed by Clunie (1954). 666 Appendices For ν < 1, the behavior of gν(e−α), as α →0, can be determined as follows: gν(e−α) = 1 0(ν) ∞ Z 0 xν−1dx eα+x −1 ≈ 1 0(ν) ∞ Z 0 xν−1dx α + x . Setting x = α tan2 θ and making use of equation (B.17), we obtain gν(e−α) ≈0(1 −ν) α1−ν (0 < ν < 1). (8) Expression (8) isolates the singularity of the function gν(e−α) at α = 0; the remainder of the function can be expanded in powers of α, with the result (see Robinson, 1951) gν(e−α) = 0(1 −ν) α1−ν + ∞ X i=0 (−1)i i! ζ(ν −i)αi, (9) ζ(s) being the Riemann zeta function analytically continued to all s ̸= 1. A simple differentiation of gν(z) brings out the recurrence relation z ∂ ∂z[gν(z)] ≡ ∂ ∂(lnz)gν(z) = gν−1(z). (10) This relation follows readily from the series expansion (4) but can also be derived from the defining integral (3). We thus have z ∂ ∂z[gν(z)] = z 0(ν) ∞ Z 0 exxν−1dx (ex −z)2 . Integrating by parts, we get z ∂ ∂z gν(z) = z 0(ν)  −xν−1 ex −z ∞ 0 + (ν −1) ∞ Z 0 xν−2dx ex −z  . The integrated part vanishes at both limits (provided that ν > 1), while the part yet to be integrated yields precisely gν−1(z). The validity of the recurrence relation (10) is thus established for all ν > 1. Adopting (10) as a part of the definition of the function gν(z), the notion of this function may be extended to all ν, including ν ≤0. Proceeding in this manner, Robinson showed that equation (9) applied to all ν < 1 and to all nonintegral ν > 1. For ν = m, a positive integer, we have instead gm(e−α) = (−1)m−1 (m −1)! "m−1 X i=1 1 i −lnα # αm−1 + ∞ X i=0 i̸=m−1 (−1)i i! ζ(m −i)αi. (11) Equations (9) and (11) together provide a complete description of the function gν(e−α) for small α; it may be checked that both these expressions conform to the recurrence relation ∂ ∂α gν(e−α) = −gν−1(e−α). (12) Appendix E 667 For the special cases ν = 5 2, 3 2, and 1 2 we obtain from (9) g5/2(α) = 2.36α3/2 + 1.34 −2.61α −0.730α2 + 0.0347α3 + ···, (13a) g3/2(α) = −3.54α1/2 + 2.61 + 1.46α −0.104α2 + 0.00425α3 + ···, (13b) g1/2(α) = 1.77α−1/2 −1.46 + 0.208α −0.0128α2 + ··· . (13c) The terms quoted here are sufficient to yield a better than 1 percent accuracy for all α ≤1. The numerical values of these functions have been tabulated by London (1954) over the range 0 ≤α ≤2. The values of several important integrals involving relativistic bosons in Chapters 7 and 9 are: ∞ Z 0 x2 ln(1 −e−x)dx = −2ζ(4) = −π4 45 , (14a) ∞ Z 0 x2 ex −1dx = 2ζ(3) ≃2.40411, (14b) ∞ Z 0 x3 ex −1dx = 6ζ(4) = π4 15 . (14c) E On Fermi–Dirac functions In the theory of Fermi–Dirac systems we come across integrals of the type Fν(z) = ∞ Z 0 xν−1dx z−1ex + 1 (0 ≤z < ∞,ν > 0). (1) In this appendix we study the behavior of Fν(z) over the entire range of the parameter z. For the same reason as in the case of Bose–Einstein integrals, we introduce here another function, fν(z), such that fν(z) ≡ 1 0(ν)Fν(z) = 1 0(ν) ∞ Z 0 xν−1dx z−1ex + 1. (2) For small z, the integrand in (2) may be expanded in powers of z, with the result fν(z) = 1 0(ν) ∞ Z 0 xν−1 ∞ X l=1 (−1)l−1(ze−x)ldx = ∞ X l=1 (−1)l−1 zl lν = z −z2 2ν + z3 3ν −··· ; (3) thus, for z ≪1, the function fν(z), for all ν, behaves like z itself. The functions fν(z) are related to the Bose–Einstein functions gν(z) as follows: fν(z) = gν(z) −21−νgν(z2) (0 ≤z < 1, ν > 0; z = 1, ν > 1). (4) 668 Appendices This is useful for determining the values of relativistic Fermi–Dirac integrals needed in Chapter 9: ∞ Z 0 x2 ln(1 + e−x)dx = 7π4 360 , (5a) ∞ Z 0 x2 ex + 1dx = 3ζ(3) 2 ≃1.80309, (5b) ∞ Z 0 x3 ex + 1dx = 7π4 120 . (5c) The functions fν(z) and fν−1(z) are connected through the recurrence relation z ∂ ∂z[fν(z)] ≡ ∂ ∂(lnz)fν(z) = fν−1(z); (6) this relation follows readily from the series expansion (3) but can also be derived from the defining integral (2). To study the behavior of Fermi–Dirac integrals for large z, we introduce the variable ξ = lnz, (7) so that Fν(eξ) ≡0(ν)fν(eξ) = ∞ Z 0 xν−1dx ex−ξ + 1. (8) For large ξ, the situation in (8) is primarily controlled by the factor (ex−ξ + 1)−1, whose departure from its limiting values — namely, zero (as x →∞) and almost unity (as x →0) — is significant only in the neighborhood of the point x = ξ; see Figure E.1. The width of this “region of significance” is O(1) and hence much smaller than the total, effective range of integration, which is O(ξ). Therefore, in the lowest approximation, we may replace the actual curve of Figure E.1 by a step function, as 1.0 0.5 0 0 24 22 12 14 x (ex211)21 FIGURE E.1 Appendix E 669 shown by the dotted line. Equation (8) then reduces to Fν(eξ) ≈ ξ Z 0 xν−1dx = ξν ν (9) and, accordingly, fν(eξ) ≈ ξν 0(ν + 1). (10) For a better approximation, we rewrite (8) as Fν(eξ) = ξ Z 0 xν−1  1 − 1 eξ−x + 1  dx + ∞ Z ξ xν−1 1 ex−ξ + 1dx (11) and substitute in the respective integrals x = ξ −η1 and x = ξ + η2, (12) with the result Fν(eξ) = ξν ν − ξ Z 0 (ξ −η1)ν−1dη1 eη1 + 1 + ∞ Z 0 (ξ + η2)ν−1dη2 eη2 + 1 . (13) Since ξ ≫1 while our integrands are significant only for η of order unity, the upper limit in the first integral may be replaced by ∞. Moreover, one may use the same variable η in both the integrals, with the result Fν(eξ) ≈ξν ν + ∞ Z 0 (ξ + η)ν−1 −(ξ −η)ν−1 eη + 1 dη (14) = ξν ν + 2 X j=1,3,5,... ν −1 j  ξν−1−j ∞ Z 0 ηj eη + 1dη  ; (15) in the last step the numerator in the integrand of (14) has been expanded in powers of η. Now, 1 0(j + 1) ∞ Z 0 ηj eη + 1dη = 1 − 1 2j+1 + 1 3j+1 −··· =  1 −1 2j  ζ(j + 1); (16) see equations (2) and (3), with ν = j + 1 and z = 1. Substituting (16) into (15), we obtain fν(eξ) = ξν 0(ν + 1)  1 + 2ν X j=1,3,5....  (ν −1)···(ν −j)  1 −1 2j  ζ(j + 1) ξj+1   = ξν 0(ν + 1) " 1 + ν(ν −1)π2 6 1 ξ2 + ν(ν −1)(ν −2)(ν −3)7π4 360 1 ξ4 + ··· # , (17) 670 Appendices which is the desired asymptotic formula — commonly known as Sommerfeld’s lemma (see Sommer-feld, 1928).4 By the same procedure, one can derive the following asymptotic result, which is clearly a generalization of (17): ∞ Z 0 φ(x)dx ex−ξ + 1 = ξ Z 0 φ(x)dx + π2 6 dφ dx  x=ξ + 7π4 360 d3φ dx3 ! x=ξ + 31π6 15120 d5φ dx5  x=ξ + ··· , (18) where φ(x) is any well-behaved function of x. It may be noted that the numerical coefficients in this expansion approach the limiting value 2. Blakemore (1962) has tabulated numerical values of the function fν(eξ) in the range −4 ≤ξ ≤ +10; his tables cover all integral orders from 0 to +5 and all half-odd integral orders from −1 2 to + 9 2. F A rigorous analysis of the ideal Bose gas and the onset of Bose–Einstein condensation In this appendix we study the problem of the ideal Bose gas without arbitrarily extracting the conden-sate term (ε = 0) from the original sum for N in equation (7.2.1) and approximating the remainder by an integral ranging from ε = 0 to ε = ∞. We will instead evaluate the original sum as it is with the help of certain mathematical identities dating back to Poisson and Jacobi in the early nine-teenth century. Luckily, these identities obviate the necessity of approximating sums by integrals and yield results valid for arbitrary values of N (though, for all practical purposes, we may assume that N ≫1). For pertinent details of this procedure, see Pathria (1983) and the references quoted therein. We consider an ideal Bose gas consisting of particles of mass m confined to the cubic geometry L × L × L and subject to periodic boundary conditions so that the single-particle energy eigenvalues are given by ε = h2 2mL2  n2 1 + n2 2 + n2 3  , (ni = 0,±1,±2,...). (1) 4A more careful analysis carried out by Rhodes (1950), and followed by Dingle (1956), shows that the passage from equation (13) to (14) omits a term which, for large ξ, is of order e−ξ . This term turns out to be cos{(ν −1)π} Fν(e−ξ ) ≡ cos{(ν −1)π}Fν(1/z). For large z, this would be very nearly equal to cos{(ν −1)π}/z and hence negligible in comparison with any of the terms appearing in (17). Of course, for ν = 1 2 , 3 2 , 5 2 ,..., which are the values occurring in most of the important applications of Fermi–Dirac statistics, the missing term is identically zero. For ν = 2, the inclusion of the missing term leads to the identity f2(eξ ) + f2(e−ξ ) = 1 2 ξ2 + π2 6 , which is relevant to the contents of Section 8.3.B. Appendix F 671 The sum (7.1.2) then takes the form N = X ε  eα+βε −1 −1 = X ε ∞ X j=1 e−j(α+βε) = ∞ X j=1 e−jα  X n1 e−jwn2 1 X n2 e−jwn2 2 X n3 e−jwn2 3  , (2) where α = −µ/kT, β = 1/kT, and w = βh2 2mL2 = π λ L 2 , (3) λ (= h/ √ 2πmkT) being the mean thermal wavelength of the particles. To evaluate the sums in (2), we make use of the Poisson summation formula (see Schwartz, 1966): ∞ X n=−∞ f (n) = ∞ X q=−∞ F(q); F(q) = ∞ Z −∞ f (x)e2πiqxdx. (4) The function F(q) is, of course, the Fourier transform of the original function f (n). Choosing f (n) = e−jwn2, we obtain the remarkable identity ∞ X n=−∞ e−jwn2 = r π jw ∞ X q=−∞ e−π2q2/jw. (5) It is instructive to note that the q = 0 term in (5) is precisely the result one would obtain if the summation over n were replaced by integration, as is customarily done in the treatment of this problem. Terms with q ̸= 0, therefore, represent corrections that arise from the discreteness of the single-particle states. Using equation (5) for each of the three summations in (2), we obtain N = ∞ X j=1 e−jα 3 Y i=1   r π jw X qi e−π2q2 i /jw   =  π w 3/2 X q ∞ X j=1 e−jα j3/2 exp " −π2 jw  q2 1 + q2 2 + q2 3 # = L3 λ3 ∞ X j=1  e−jα j3/2 + X q ′ e−jα j3/2 exp −π2 jw  q2 1 + q2 2 + q2 3 ! , (6) where the primed summation in the second set of terms implies that the term with q = 0 has been taken out of this sum. The q = 0 term in (6) is precisely the bulk result for the total number of particles in the excited states of the system, namely Vg3/2(e−α)/λ3. In the second set of terms, the summation over j may be 672 Appendices carried out with the help of a straightforward generalization of identity (4), namely b X n=a f (n) = 1 2f (a) + 1 2f (b) + ∞ X l=−∞ Fa,b(l); Fa,b(l) = b Z a f (x)e2πilxdx, (7) where a and b are integers such that b > a. Applying (7) to the primed sum in (6), we obtain ∞ X j=1 j−3/2e−jαe−γ (q)/j = ∞ X j=0 j−3/2e−jαe−γ (q)/j = r π γ (q) ∞ X l=−∞ exp h −2 p γ (q) α + 2πil 1/2i , (8) where γ (q) = π2q2 w = πL2q2 λ2 > 0. We readily note that, whatever the value of α, terms with l ̸= 0 are at most of order exp(−L/λ) which, for L ≫λ, are altogether negligible. As a consequence, no errors of order (λ/L)n are committed if we retain only the term with l = 0. We thus obtain N ≈L3 λ3 h g3/2 e−α + π1/2α1/2S y i , (9) where S(y) = X q ′ e−2R(q) R(q) ; R(q) = y q q2 1 + q2 2 + q2 3, (10) while y is given by y = π1/2α1/2 L λ. (11) In view of equation (13.6.36), the parameter y is a measure of the lateral dimension L of the system in terms of its correlation length ξ — to be precise, y = L/2ξ. We, therefore, expect that, as we lower the temperature of the system and enter the region of phase transition, this parameter will go from very large values to very small values over an infinitesimally small range of temperatures around the transition point. We’ll examine this aspect of the problem a little later. At this point, it is worthwhile to note that if the summation over q that appears in equation (10) is replaced by integration, which is justifiable only in the limit y →0, we obtain from equation (9) N ≈L3 λ3  g3/2 e−α + π1/2α1/2  π y3  = L3 λ3 g3/2 e−α + 1 α , (12) Appendix F 673 in perfect agreement with the bulk result obtained in Section 7.1, with α ≪1. To have an idea of the “degree of error” committed in making this replacement, we must evaluate this sum more accurately. For this, we make use of another mathematical identity, first established by Chaba and Pathria (1975), namely X q ′ " y2 q2 y2 + π2q2 + π q e−2yq # = 2πy + π2 y2 + C3, (13) where C3 = π lim y→0    X q ′ e−2yq q − Z all q e−2yq q dq   ≃−8.9136. (14) It is important to note that the constant C3 is directly related to the Madelung constant of a simple cubic lattice; see, for instance, Harris and Monkhorst (1970). Now, since the second part of the sum appearing on the left side of (13) is directly proportional to S(y), we can rewrite (9) in the form N ≈L3 λ3 g3/2 e−α + L2 λ2 π y2 + L2 λ2  C3 π + 2y −y2 π X q ′ 1 q2 y2 + π2q2  . (15) We observe that the second term on the right side of (15) is equal to 1/α — which is precisely N0 when α ≪1. The condensate, therefore, emerges naturally in our analysis and does not have to be extracted prematurely, as is done in the customary treatment. Now, in view of the fact that, for small α, g3/2 e−α ≈ζ 3 2  −2π1/2α1/2, (16) equation (15) is further simplified to N ≈L3 λ3 ζ (3/2) + N0 + L2 λ2  C3 π −y2 π X q ′ 1 q2 y2 + π2q2  . (17) Introducing the bulk critical temperature Tc(∞), as defined in equation (7.1.24), Tc(∞) = h2 2πmk   N L3ζ  3 2    2/3 = T λ2 L2   N ζ  3 2    2/3 ; (18) we obtain from (17) the desired result, namely N0 = N " 1 −  T Tc(∞) 3/2# + L2 λ2  −C3 π + y2 π X q ′ 1 q2 y2 + π2q2  . (19) 674 Appendices The first part of this expression is the standard bulk result for N0, while the second part represents the “finite-size correction” to this quantity. More important, while the main term here is of order N, the correction term is of order N2/3. In the thermodynamic limit, the correction term loses its importance altogether and we are left with the conventional result following from the customary treatment. Finally, we study the variation of the scaling parameter y as a function of T; this will also enable us to examine the manner in which the correlation length ξ and the condensate fraction f (= N0/N) build up as we move from temperatures above Tc(∞) to those below Tc(∞). For this, we introduce a scaled temperature, defined by t = [T −Tc(∞)]/Tc(∞), and study the problem in three distinct regimes: (a) For t > 0, such that 1 ≫t ≫N−1/3, we make use of the result for α, as stated in Problem 7.3. Combining this result with equation (11), we get y ≈3 4  ζ 3 2 2/3 N1/3t, (20a) ξ ≈2 3  ζ 3 2 −2/3 ℓt−1, (20b) f ≈16π 9  ζ 3 2 −2 N−1t−2, (20c) where ℓ(= L/N1/3) is the mean interatomic distance in the system. (b) For |t| = O(N−1/3), the parameter y = O(1) and its value has to be determined numerically. At t = 0, this value is determined by the equation S(y0) = 2; see equations (9) and (16). We thus get: y0 ≃0.973. The correlation length ξ in this regime is O(L) and the condensate fraction is O(N−1/3). (c) For t < 0, such that 1 ≫|t| ≫N−1/3, we get y ≈ r 2π 3  ζ 3 2 −1/3 N−1/6|t|−1/2, (21a) ξ ≈ r 3 8π  ζ 3 2 1/3 L3/2 ℓ1/2 |t|1/2 , (21b) f ≈3 2|t|. (21c) We thus see how, over an infinitesimally small range of temperatures O(N−1/3) around t = 0, the parameter y descends from values O(N1/3) to values O(N−1/6) while the correlation length ξ grows from values O(ℓ) to values O(L3/2/ℓ1/2), and the condensate fraction f grows from values O(N−1) to values O(1). As N →∞, the transition region collapses onto a singular point t = 0 and the phenomenon of Bose–Einstein condensation becomes a critical one. Appendix G 675 G On Watson functions In this appendix we examine the asymptotic behavior of the functions Wd(φ) = ∞ Z 0 e−φx e−xI0(x) d dx (1) for 0 ≤φ ≪1. First of all, we note that if we set φ = 0 the resulting integral converges only if d > 2. To see this, we observe that, with φ = 0, convergence problems may arise in the limit of large x where the integrand e−xI0(x) d ≈(2πx)−d/2 (x ≫1). (2) Clearly, the integral will converge if d > 2; otherwise, it will diverge. We, therefore, conclude that Wd(0) = ∞ Z 0 [e−xI0(x)]d dx (3) exists for d > 2. Next we look at the derivative W ′ d(φ) = − ∞ Z 0 e−φx[e−xI0(x)]dxdx. (4) By the same argument as above, we conclude that W ′ d(0) = − ∞ Z 0 e−xI0(x) d xdx (5) exists for d > 4. The manner in which W ′ d(φ) diverges for d < 4, as φ →0, can be seen as follows: W ′ d(φ) = − ∞ Z 0 e−y h e−y/φI0(y/φ) id 1 φ2 ydy ≈− 1 (2π)d/2φ(4−d)/2 ∞ Z 0 e−yy(2−d)/2dy (φ ≪1) = − 0{(4 −d)/2} (2π)d/2φ(4−d)/2 . (6) 676 Appendices Integrating (6) with respect to φ, and remembering the comments made earlier about Wd(0), we obtain the desired results: Wd(φ) ≈        (2π)−d/20{(2 −d)/2}φ−(2−d)/2 + const. for d < 2 (7a) (2π)−1ln(1/φ) + const. for d = 2 (7b) Wd(0) −(2π)−d/2|0{(2 −d)/2}|φ(d−2)/2 for 2 < d < 4. (7c) For d > 4, we have a simpler result: Wd(φ) ≈Wd(0) −|W ′ d(0)|φ, (8) for, in this case, both Wd(0) and W ′ d(0) exist. The borderline case d = 4 presents some problems that can be simplified by splitting the integral in (4) into two parts: ∞ Z 0 = 1 Z 0 + ∞ Z 1 . (9) The first part is clearly finite; the divergence of the function W ′ 4(φ), as φ →0, arises from the second part which, for φ ≪1, can be written as ≈ ∞ Z 1 e−φx(2πx)−2xdx = 1 4π2 E1(φ), (10) where E1(φ) is the exponential integral; see Abramowitz and Stegun (1964), Chapter 5. Since E1(φ) ≈ −lnφ for φ ≪1, we conclude that W ′ 4(φ) ≈ 1 4π2 lnφ. (11) Integrating (11) with respect to φ, we obtain W4(φ) ≈W4(0) − 1 4π2 φ ln(1/φ). (12) Equations (7), (8), and (12) constitute the main results of this appendix. For the record, we quote a couple of numbers: W3(0) = 0.50546, W4(0) = 0.30987. (13) H Thermodynamic relationships The following four equations relating partial derivatives are sometimes known as the Four Famous Formulae. They make it easy to derive thermodynamic relations in any one assembly or to convert Appendix H 677 relations from one assembly to another; by assembly we mean the set of thermodynamic parameters on which a system depends. If the quantities x, y, z are mutually related, then ∂x ∂y  z = 1 ∂y ∂x  z , (1a)  ∂ ∂z ∂y ∂x  z  x =  ∂ ∂x ∂y ∂z  x  z , (1b) ∂x ∂y  z ∂y ∂z  x  ∂z ∂x  y = −1, (1c) ∂x ∂y  w = ∂x ∂y  z + ∂x ∂z  y ∂z ∂y  w . (1d) Entropy S(N,V,U) and the microcanonical ensemble The entropy describes a closed, isochoric, adiabatic assembly, so it is a function of internal energy, volume, and number of molecules: dS = 1 T dU + P T dV −µ T dN, (2a) 1 T =  ∂S ∂U  V,N , (2b) P T =  ∂S ∂V  U,N , (2c) µ T = −  ∂S ∂N  U,V . (2d) The Maxwell relations for the entropy are ∂(1/T) ∂V  U,N = ∂(P/T) ∂U  V,N , (3a) ∂(1/T) ∂N  U,V = − ∂(µ/T) ∂U  V,N , (3b) ∂(P/T) ∂N  U,V = − ∂(µ/T) ∂V  U,N . (3c) The entropy is determined from the number of microstates in the microcanonical ensemble 0(N,V,U;1U) = Tr 11U(H −U)  , (4) where H is the Hamiltonian of the system and 11U(x) is the step function that is unity in the range 0 to 1U and zero otherwise. The quantity 0(N,V,U;1U) here denotes the number of discrete 678 Appendices quantum states in the energy range between U and U + 1U. The entropy is then given by S(N,V,U) = kln0(N,V,U;1U). (5) The bulk value of the entropy does not depend on the value chosen for 1U. Helmholtz free energy A(N,V,T) = U −TS and the canonical ensemble The Helmholtz free energy describes a closed, isochoric, isothermal assembly, so it is a function of temperature, volume, and number of molecules: dA = −SdT −PdV + µdN, (6a) S = −  ∂A ∂T  V,N , (6b) P = −  ∂A ∂V  T,N , (6c) µ =  ∂A ∂N  T,V . (6d) The Maxwell relations for the Helmholtz free energy are  ∂S ∂V  T,N =  ∂P ∂T  V,N , (7a)  ∂S ∂N  T,V = − ∂µ ∂T  V,N , (7b)  ∂P ∂N  T,V = −  ∂µ ∂V  T,N . (7c) The Helmholtz free energy is determined from the canonical partition function QN(V,T) = Tr exp(−βH)  = Z e−βU  1 1U 0(N,V,U;1U)  dU. (8) The Helmholtz free energy is given by A(N,V,T) = −kT lnQN(V,T). (9) Appendix H 679 Thermodynamic potential 5(µ,V,T) = −A + µN = PV and the grand canonical ensemble The thermodynamic potential describes an open, isochoric, isothermal assembly, so it is a function of temperature, volume, and chemical potential: d5 = SdT + PdV + Ndµ , (10a) S = ∂5 ∂T  V,µ , (10b) P = ∂5 ∂V  T,µ , (10c) N = ∂5 ∂µ  T,V . (10d) The Maxwell relations for the thermodynamic potential are  ∂S ∂V  T,µ =  ∂P ∂T  V,µ , (11a)  ∂S ∂µ  T,V = ∂N ∂T  V,µ , (11b)  ∂P ∂µ  T,V = ∂N ∂V  T,µ . (11c) The thermodynamic potential is a function only of a single extensive quantity, V, so 5(µ,V,T) = P(µ,T)V, that is, the pressure is the thermodynamic potential per unit volume. Therefore, we can write simpler thermodynamic relations in terms of pressure P, entropy density s = S/V, and number density n = n/V: dP = sdT + ndµ , (12a) s =  ∂P ∂T  µ , (12b) n =  ∂P ∂µ  T , (12c)  ∂s ∂µ  T =  ∂n ∂T  µ . (12d) The thermodynamic potential and pressure are determined from the grand canonical partition function Q(µ,V,T) = Tr exp(−βH + βµN)  = ∞ X N=0 eβµNQN(V,T), (13) 680 Appendices with the result P(µ,T) = 5(µ,V,T) V = kT V lnQ(µ,V,T). (14) Gibbs free energy G(N,P,T) = A + PV = U −TS + PV = µN and the isobaric ensemble The Gibbs free energy describes a closed, isobaric, isothermal assembly, so it is a function of temperature, pressure, and number of molecules: dG = −SdT + VdP + µdN, (15a) S = − ∂G ∂T  P,N , (15b) V = ∂G ∂P  T,N , (15c) µ =  ∂G ∂N  T,P . (15d) The Maxwell relations for the Gibbs free energy are  ∂S ∂P  T,N = − ∂V ∂T  P,N , (16a)  ∂S ∂N  T,P = − ∂µ ∂T  P,N , (16b)  ∂V ∂N  T,P = ∂µ ∂P  T,N . (16c) The Gibbs free energy is a function only of a single extensive quantity, N, so G(N,P,T) = Nµ(P,T), that is, the chemical potential is the Gibbs free energy per particle. Therefore, we can write sim-pler thermodynamic relations in terms of the pressure P, entropy per particle s = S/N, and specific volume v = V/N: dµ = −sdT + vdP, (17a) s = − ∂µ ∂T  P , (17b) v = ∂µ ∂P  T , (17c)  ∂s ∂P  T = −  ∂v ∂T  P . (17d) The Gibbs free energy and the chemical potential are determined from the isobaric partition function YN(P,T) = Tr exp(−βH −βPV)  = 1 λ3 ∞ Z 0 e−βPV QN(V,T)dV; (18) Appendix H 681 the cube of the thermal deBroglie wavelength is employed here to make the partition function dimensionless and is irrelevant in the classical thermodynamic limit. This leads us to the result G(N,P,T) = Nµ(P,T) = −kT lnYN(P,T). (19) The isobaric ensemble is often used in computer simulations to avoid two-phase regions that are present at first-order phase transitions. Internal Energy U(N,V,S) The internal energy describes a closed, isochoric, adiabatic assembly, so it is a function of entropy, volume, and number of molecules: dU = TdS −PdV + µdN, (20a) T = ∂U ∂S  V,N , (20b) P = − ∂U ∂V  S,N , (20c) µ = ∂U ∂N  S,V . (20d) Maxwell relations for the internal energy are  ∂T ∂V  S,N = − ∂P ∂S  V,N , (21a)  ∂T ∂N  S,V = ∂µ ∂S  V,N , (21b)  ∂P ∂N  S,V = −  ∂µ ∂V  S,N . (21c) Enthalpy H(N,P,S) = U + PV The enthalpy describes a closed, isobaric, adiabatic assembly, so it is a function of entropy, pressure, and number of molecules. dH = TdS + VdP + µdN, (22a) T = ∂H ∂S  P,N , (22b) V = ∂H ∂P  S,N , (22c) µ = ∂H ∂N  S,P . (22d) 682 Appendices The Maxwell relations for the enthalpy are ∂T ∂P  S,N = ∂V ∂S  P,N , (23a)  ∂T ∂N  S,P = ∂µ ∂S  P,N , (23b)  ∂V ∂N  S,P = ∂µ ∂P  S,N . (23c) This assembly is often used by chemists to describe chemical reactions that take place rapidly in the laboratory at fixed pressure where the speed of the reaction is too fast to allow a substantial heat exchange with the environment. Magnetic free energy F(T,H) = U(S,M) −TS −HM The magnetic free energy describes a system with a fixed number of magnetic dipoles, at tempera-ture T and magnetic field H. The thermodynamic relations and Maxwell relations are dF = −SdT −MdH, (24a) S = −  ∂F ∂T  H , (24b) M = −  ∂F ∂H  T , (24c)  ∂S ∂H  T = ∂M ∂T  H . (24d) This assembly is defined by the fixed-field canonical ensemble where the canonical partition function for spins {si} with dipole moment µ and exchange energies Jij is QN(T,H) = Tr exp(−βH)  = X {s} exp  X (i,j) Kijsi · sj + h X i sz,i  , (25) where the coupling constants are Kij = Jij/kT and the field coupling is h = µH/kT. The magnetic free energy is then given by F(T,H) = −kT lnQN(T,H). (26) Convexity and variances The convexity of the entropy S that follows from the second law of thermodynamics requires that all second derivatives of free energies have a unique sign. These derivatives are all proportional to variances of different statistical quantities. A few important examples are the heat capacity, the Appendix I 683 isothermal compressibility, and the magnetic susceptibility: CV = T  ∂S ∂T  V,N = − ∂2A ∂T2 ! V,N = H2 −⟨H⟩2 kT2 ≥0, (27) κT = 1 n2  ∂n ∂µ  T = 1 n2 ∂2P ∂µ2 ! T =  1 nkT  N2 −⟨N⟩2 ⟨N⟩ ≥0, (28) χT = ∂M ∂H  T = − ∂2F ∂H2 ! T = M2 −⟨M⟩2 kT ≥0. (29) I Pseudorandom numbers The random numbers used in computer simulations are not truly random but rather pseudorandom. They are intended to display as many of the characteristics of a random sequence as possible but are generated using simple integer algorithms. “Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method” (von Neumann, 1951). von Neumann’s quip was not intended to dissuade the use of pseudorandom numbers but rather to encourage users to understand how such numbers are generated, their statistical properties and potential weaknesses. Pseudorandom number generators are essential in computer simulations in many fields but a generator should not blindly be trusted unless it has been thoroughly tested. There is an extensive literature on theoretical methods for creating and empirically testing pseudorandom number generators; see Knuth (1997), L’Ecuyer (1988, 1999), and Press et al. (2007). The most commonly used and tested classes of pseudorandom number generators are based on integer arithmetic modulo large, usually prime, numbers. The simplest class of generators is the prime modulus linear congruential method that generates pseudo-random numbers in the range [1,2,...,m −1], where m is a prime number. Each new number is based on the previous number according to the formula rj = a rj−1mod m, (1) where the multiplier a is a carefully chosen and empirically tested integer less than m. If a is chosen properly, the generator will produce a sequence of integers that includes every integer in the range [1,2,...,m −1] exactly once before the sequence begins to repeat after m −1 calls to the generator. Pseudorandom floating-point numbers in the open range (0,1) are produced by a floating-point multiplication by m−1. This produces a uniformly distributed set of floating-point numbers on a comb of m −1 values between zero and one. The floating-point numbers 0.0 and 1.0 will not appear in the sequence. Often, in imple-mentations, the modulus chosen is less than 231 to allow for the integers to be represented 684 Appendices by four-byte words. Generators of this form will repeat after about two billion calls. This may be fine for some applications but is often inadequate for a scientific code. Even a modest simulation can exhaust such a simple generator. However, linear congruential gen-erators with different moduli and multipliers can be combined to give much longer periods and less statistical correlation (L’Ecuyer, 1988). The following algorithm to generate a uni-formly distributed pseudorandom floating-point number r using two linear congruential generators gives a much longer sequence before repeating than does a single generator. ALGORITHM qj = a1 qj−1 mod m1 sj = a2 sj−1 mod m2 z = (qj −sj) if z ≤0, then z = z + m1 −1 r = z m−1 1 The length of the period of the combined generator is equal to the product of the non-common prime factors of m1 −1 and m2 −1. L’Ecuyer (1988) recommended the follow-ing multiplers and moduli: m1 = 2147483563 = 231 −85, a1 = 40014, m2 = 2147483399 = 231 −249, and a2 = 40692. The two individual generators do well on the spectral test and other tests of correlations (Knuth, 1997) and (m1 −1)/2 and (m2 −1)/2 are relatively prime, so the period of the generator is 2.3 × 1018. This can be implemented very easily in high-level languages using IEEE double precision floating-point arithmetic since aimi < 253. L’Ecuyer (1988) and Press et al. (2007) also showed how the generators can be imple-mented using four-byte integer arithmetic. The exact order of the pseudorandom numbers can also be shuffled to improve statistics (Bays and Durham, 1976). There are several other classes of pseudorandom number algorithms and many good generators available that have been extensively tested in the computing literature and are in wide use in the scientific literature; for summaries, see Newman and Barkema (1999), Gentle (2003), and Landau and Binder (2009). However, note that subtle correlations in a pseudorandom sequence can produce very large deviations compared to exact results, so care is warranted; see Ferrenberg, Landau, and Wong (1992), Beale (1996), and Figure I.1. Different classes of generators have different correlation properties, so substituting a gen-erator based on a different algorithm can sometimes be a useful strategy for testing a computer code. Gaussian distributed pseudorandom numbers One often needs pseudorandom numbers that are drawn from a Gaussian distribu-tion centered at the origin with unit variance: P(g) = exp −g2/2  / √ 2π. The following algorithm for generating Gaussian-distributed pseudorandom numbers from pairs of uniformly distributed pseudorandom numbers is based on an algorithm by Box and Muller (1958). Appendix I 685 102 103 104 105 106 50 100 150 (a) (b) 200 250 k k Pk 20 10 0 10 20 300 50 100 150 200 250 300 Deviation/uncertainty Feed-back shift-register Numerical recipes’ ran2( ) FIGURE I.1 (a) The exact energy distribution for a 32 × 32 Ising lattice at the critical temperature (solid line) and the distribution calculated from 107 configurations using the Wolff Monte Carlo algorithm with the R250 feedback shift-register pseudorandom number generator (error bars).5 The distribution calculated using the Wolff algorithm with Numerical Recipes ran2( ) (Press et al., 2007) is also shown, but is almost indistinguishable from the exact solid curve on this scale. (b) Deviation of the Monte Carlo results from the exact distribution in units of the statistical uncertainty of each point. The +′s indicate the ran2( ) results and the ×′s indicate the feedback shift-register results. The χ2-value for the two cases yields χ2 = 190 for 210 nonzero points and χ2 = 2.8 × 104 for 217 nonzero points, giving deviations of −0.95σ and 1300σ, respectively; see Section 13.4.A and Beale (1996). ALGORITHM To generate pairs of Gaussian pseudorandom numbers g1 and g2 from uniformly dis-tributed pseudorandom numbers x and y. REPEAT x = 2rand() −1.0 y = 2rand() −1.0 s = x2 + y2 UNTIL s < 1 w = q −2lns s g1 = xw g2 = yw 5An example of a widely used and trusted pseudorandom number generator whose subtle correlations have been shown to adversely affect the results of some MC simulations is the R250 feedback shift-register generator. R250 gener-ates a pseudorandom positive four-byte integer 0 ≤ri < 231 by taking an exclusive or of two previous random integers kept in a 250 element table: ri = ri−103 ∧rj−250. As with other generators, the floating-point numbers on [0,1] are pro-duced by dividing the pseudorandom integer by 231. This particular feedback shift-register algorithm is very fast and produces fairly uncorrelated random pairs rkrk−i = 1/4 for i ̸= 0. Triplets are also uncorrelated rkrk−irk−j = 1/8, except for one particular triplet correlation, ⟨rkrk−103rk−250⟩, which can be shown to give (6/7)(1/8) rather than the correct value of 1/8; see Heuer, Dunweg, and Ferrenberg (1997). All higher order correlations that involve these same triplets are sim-ilarly affected. This triplet correlation being off by 14% can greatly affect Monte Carlo results (Ferrenberg, Landau, and Wong, 1992; Beale, 1996). Bibliography Abe, R. (1972). Prog. Theor. Phys. Japan 48, 1414. Abe, R. (1973). Prog. Theor. Phys. Japan 49, 113, 1851. Abe, R., and Hikami, S. (1973). Prog. Theor. Phys. Japan 49, 442. Abraham, D. B. (1986). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 10, pp. 1–74. Abramowitz, M., and Stegun, I. A. (eds.) (1964). Handbook of Mathematical Functions (National Bureau of Standards, Washington, DC). Abrikosov, A. A., Gorkov, L. P ., and Dzyaloshinskii, I. Y. (1965). Quantum Field Theoretical Methods in Statistical Physics (Pergamon Press, Oxford). Abrikosov, A. A., and Khalatnikov, I. M. (1957). J. Exptl. Theoret. Phys. USSR 33, 1154; English transl. Soviet Phys. JETP 6, 888 (1958). Adare, A., et al. (2010). Phys. Rev. Lett. 104, 132301. Aharony, A. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 6, pp. 357–424. Ahlers, G. (1980). Rev. Mod. Phys. 52, 489. Alder, B. J., and Wainwright, T. E. (1957). J. Chem. Phys. 27, 1208. Alder, B. J., and Wainwright, T. E. (1959). J. Chem. Phys. 31, 459. Allen, M. B., and Tildesley, D. J. (1990). Computer Simulations of Liquids, 2nd ed. (Oxford University Press, Oxford). Allen, S., and Pathria, R. K. (1989). Can. J. Phys. 67, 952. Allen, S., and Pathria, R. K. (1991). Can. J. Phys. 69, 753. Alpher, R. A., Bethe, H. A., and Gamow, G. (1948). Phys. Rev. 73, 803. Alpher, R. A., and Herman, R. C. (1948). Nature 162, 774. Alpher, R. A., and Herman, R. C. (1950). Rev. Mod. Phys. 22, 153. Alpher, R. A., and Herman, R. C. (2001). Genesis of the Big Bang (Oxford University Press, New York). Als-Nielsen, J. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 5a, pp. 87–164. Als-Nielsen, J., and Birgeneau, R. J. (1977). Am. J. Phys. 45, 554. Anderson, M. H., Ensher, J. R., Matthews, M. R., Wieman, C. E., and Cornell, E. E. (1995). Science 269, 198. Anderson, P . W. (1984). Basic Notions of Condensed Matter Physics (Benjamin/Cummings, Menlo Park, CA). Anderson, W. (1929). Z. Phys. 56, 851. Arnold, V. I. (1963). Russian Math. Survey 18, 13. Ashcroft, N. W., and Mermin, N. D. (1976). Sold State Physics (Holt, Rinehart, & Winston, New York). Auluck, F . C., and Kothari, D. S. (1946). Proc. Camb. Phil. Soc. 42, 272. Traveling Wave Analysis of Partial Differential Equations © 2011 Elsevier Ltd. All rights reserved. 687 688 Bibliography Au-Yang, H., and Perk, J. H. H. (1984). Phys. Lett. A 104, 131. Bagnato, V., Pritchard, D. E., and Kleppner, D. (1987). Phys. Rev. A 35, 4354. Bahm, C. J., and Pethick, C. J. (1996). Phys. Rev. Lett. 76, 6. Bak, P ., Krinsky, S., and Mukamel, D. (1976). Phys. Rev. Lett. 36, 52. Baker, G. A., Jr. (1990). Quantitative Theory of Critical Phenomena (Academic Press, New York). Balian, R., and Toulouse, G. (1973). Phys. Rev. Lett. 30, 544. Baltes, H. P ., and Hilf, E. R. (1976). Spectra of Finite Systems (Bibliographisches Institut AG, Z¨ urich). Band, W. (1955). An Introduction to Quantum Statistics (Van Nostrand, Princeton). Barber, M. N. (1983). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 8, pp. 145–266. Barber, M. N., and Fisher, M. E. (1973). Ann. Phys. (N.Y.) 77, 1. Bardeen, J., Cooper, L. N., and Schrieffer, J. R. (1957). Phys. Rev. 108, 1175. Barnett, S. J. (1944). Proc. Am. Acad. Arts & Sci. 75, 109. Barouch, E., McCoy, B. M., and Wu, T. T. (1973). Phys. Rev. Lett. 31, 1409. Baxter, R. J. (1982). Exactly Solved Models in Statistical Mechanics (Academic Press, London). Bays, C., and Durham, S. D. (1976). ACM Trans. Math. Software 2, 59. Beale, P . D. (1996). Phys. Rev. Lett. 76, 79. Belinfante, F . J. (1939). Physica 6, 849, 870. Berche, B., Sanchez, A. I. F ., and Paredes, V. (2002). Europhys. Lett. 60, 539. Berezinskii, V. (1970). Zh. Eksp. Teor. Fiz. 59, 907; (1971) Engl. transl. Sov. Phys., JETP 32, 493. Berlin, T. H., and Kac, M. (1952). Phys. Rev. 86, 821. Bernoulli, D. (1738). Hydrodynamica (Argentorati). Beth, E., and Uhlenbeck, G. E. (1937). Physica 4, 915. Bethe, H. A. (1935). Proc. R. Soc. Lond. A 150, 552. Betts, D. D. (1974). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 3, pp. 569–652. Betts, D. D., and Lee, M. H. (1968). Phys. Rev. Lett. 20, 1507. Betts, D. D. et al. (1969). Phys. Lett. 29A, 150; (1970) ibid. 32A, 152. Betts, D. D. et al. (1970). Can. J. Phys. 48, 1566; (1971) ibid. 49, 1327; (1973) ibid. 51, 2249. Betts, D. S. (1969). Contemp. Phys. 10, 241. Bijl, A. (1940). Physica 7, 869. Binder, K. (1983). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 8, pp. 1–144. Binder, K. (ed.) (1986). Monte Carlo Methods in Statistical Physics, 2nd ed., Topics in Current Physics (Springer-Verlag, Berlin), Vol. 7. Binder, K. (ed.) (1987). Applications of the Monte Carlo Method in Statistical Physics, 2nd ed., Topics in Current Physics (Springer-Verlag, Berlin), Vol. 36. Binder, K. (ed.) (1992). The Monte Carlo Method in Condensed Matter Physics, Topics in Applied Physics (Springer-Verlag, Berlin), Vol. 71. Binder, K., and Heermann, D. W. (2002). Monte Carlo Simulation in Statistical Physics, 4th ed., (Springer-Verlag, Berlin). Bibliography 689 Blakemore, J. S. (1962). Semiconductor Statistics (Pergamon Press, Oxford). Bloch, F . (1928). Z. Phys. 52, 555. Bogoliubov, N. N. (1947). J. Phys. USSR 11, 23. Bogoliubov, N. N. (1962). Phys. Abhandl. S.U. 6, 113. Bogoliubov, N. N. (1963). Lectures on Quantum Statistics (Gordon & Breach, New York). Boltzmann, L. (1868). Wien. Ber. 58, 517. Boltzmann, L. (1871). Wien. Ber. 63, 397, 679, 712. Boltzmann, L. (1872). Wien. Ber. 66, 275. Boltzmann, L. (1875). Wien. Ber. 72, 427. Boltzmann, L. (1876). Wien. Ber. 74, 503. Boltzmann, L. (1877). Wien. Ber. 76, 373. Boltzmann, L. (1879). Wien. Ber. 78, 7. Boltzmann, L. (1884). Ann. Phys. 22, 31, 291, 616. Boltzmann, L. (1896, 1898). Vorlesungen ¨ uber Gastheorie, 2 vols. (J. A. Barth, Leipzig). English transl. (1964) Lectures on Gas Theory (University of California Press, Berkeley, CA). Boltzmann, L. (1899). Amsterdam Ber. 477; see also (1909) Wiss. Abhandl. von L. Boltzmann III, 658. Borner, G. (2003). The Early Universe (Springer, New York). Bose, S. N. (1924). Z. Physik 26, 178. Box, G. E. P ., and Muller, M. E. (1958). Ann. Math. Stat. 29, 610. Bragg, W. L., and Williams, E. J. (1934). Proc. R. Soc. Lond. A 145, 699. Bragg, W. L., and Williams, E. J. (1935). Proc. R. Soc. Lond. A 151, 540; 152, 231. Br´ ezin, E., le Guillou, J. C., and Zinn-Justin, J. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 6, pp. 125–247. Brueckner, K. A., and Gammel, J. L. (1958). Phys. Rev. 109, 1040. Brueckner, K. A., and Sawada, K. (1957). Phys. Rev. 106, 1117, 1128. Brush, S. G. (1957). Ann. Sci. 13, 188, 273. Brush, S. G. (1958). Ann. Sci. 14, 185, 243. Brush, S. G. (1961a). Am. J. Phys. 29, 593. Brush, S. G. (1961b). Am. Scientist 49, 202. Brush, S. G. (1965–66). Kinetic Theory, 3 vols. (Pergamon Press, Oxford). Brush, S. G. (1967). Rev. Mod. Phys. 39, 883. Buckingham, M. J., and Gunton, J. D. (1969). Phys. Rev. 178, 848. Bush, V., and Caldwell, S. H. (1931). Phys. Rev. 38, 1898. Callen, H. B., and Welrton, T. R. (1951). Phys. Rev. 83, 34. Cardy, J. L. (1987). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 11, pp. 55–126. Cardy, J. L. (ed.) (1988). Finite-Size Scaling (North-Holland, Amsterdam). Carnahan, N. F ., and Starling, J. (1969). J. Chem. Phys. 51, 635. Chaba, A. N., and Pathria, R. K. (1973). Phys. Rev. A 8, 3264. Chaba, A. N., and Pathria, R. K. (1975). Phys. Rev. B 12, 3697. 690 Bibliography Chandrasekhar, S. (1939). Introduction to the Study of Stellar Structure (University of Chicago Press, Chicago); now available in a Dover edition. Chandrasekhar, S. (1943). Rev. Mod. Phys. 15, 1. Chandrasekhar, S. (1949). Rev. Mod. Phys. 21, 383. Chang, C. H. (1952). Phys. Rev. 88, 1422. Chang, T. S. (1939). Proc. Camb. Phil. Soc. 35, 265. Chapman, S. (1916). Trans. R. Soc. Lond. A 216, 279. Chapman, S. (1917). Trans. R. Soc. Lond. A 217, 115. Chapman, S., and Cowling, T. G. (1939). The Mathematical Theory of Non-uniform Gases (Cambridge University Press, Cambridge, UK). Chisholm, J. S. R., and Borde, A. H. (1958). An Introduction to Statistical Mechanics (Pergamon Press, New York). Chr´ etien, M., Gross, E. P ., and Deser, S. (eds.) (1968). Statistical Physics, Phase Transitions and Superfluidity, 2 vols. (Gordon & Breach, New York). Clausius, R. (1857). Ann. Phys. 100, 353; also published in (1857) Phil. Mag. 14, 108. Clausius, R. (1859). Ann. Phys. 105, 239; also published in (1859) Phil. Mag. 17, 81. Clausius, R. (1870). Ann. Phys. 141, 124; also published in (1870) Phil. Mag. 40, 122. Clunie, J. (1954). Proc. Phys. Soc. A 67, 632. Cohen, E. G. D., and van Leeuwen, J. M. J. (1960). Physica 26, 1171. Cohen, E. G. D., and van Leeuwen, J. M. J. (1961). Physica 27, 1157. Cohen, E. G. D. (ed.) (1962). Fundamental Problems in Statistical Mechanics (John Wiley, New York). Cohen, M., and Feynman R. P . (1957). Phys. Rev. 107, 13. Compton, A. H. (1923a). Phys. Rev. 21, 207, 483. Compton, A. H. (1923b). Phil. Mag. 46, 897. Cooper, L. N. (1956). Phys. Rev. 104, 1189. Cooper, L. N. (1960). Am. J. Phys. 28, 91. Copi, C. J., Schramm, D. N., and Turner, M. S. (1997). Phys. Rev. D. 55, 3389. Corak, W. S., Garfunkel, M. P ., Satterthwaite, C. B., and Wexler, A. (1955). Phys. Rev. 98, 1699. Cummins, H. Z., and Gammon, R. W. (1966). J. Chem. Phys. 44, 2785. Darwin, C. G., and Fowler, R. H. (1922a). Phil. Mag. 44, 450, 823. Darwin, C. G., and Fowler, R. H. (1922b). Proc. Camb. Phil. Soc. 21, 262. Darwin, C. G., and Fowler, R. H. (1923). Proc. Camb. Phil. Soc. 21, 391, 730; see also Fowler, R. H. (1923) Phil. Mag. 45, 1, 497; (1925) Proc. Camb. Phil. Soc. 22, 861; (1926) Phil. Mag. 1, 845; (1926) Proc. R. Soc. Lond. A 113, 432. Davis, K. B., Mewes, M.-O., Andrews, M. R., van Druten, N. J., Durfee, D. S., Kurn, D. M., and Ketterle, W. (1995). Phys. Rev. Lett. 75, 3969. Debye, P . (1912). Ann. Phys. 39, 789. de Gennes, P .-G. (1972). Phys. Lett. 38A, 339. de Gennes, P .-G. (1979). Scaling Concepts in Polymer Physics (Cornell University Press, Ithaca, NY). de Groot, S. R. (1951). Thermodynamics of Irreversible Processes (North-Holland, Amsterdam). de Groot, S. R., and Mazur, P . (1962). Non-equilibrium Thermodynamics (North-Holland, Amsterdam). Bibliography 691 de Groot, S. R., Hooyman, G. J., and ten Seldam, C. A. (1950). Proc. R. Soc. London A, 203, 266. de Klerk, D., Hudson, R. P ., and Pellam, J. R. (1953). Phys. Rev. 89, 326, 662. DeMarco, B., and Jin, D. (1999). Science 285, 1703. Dennison, D. M. (1927). Proc. R. Soc. Lond. A 115, 483. des Cloizeaux, J. (1974). Phys. Rev. A 10, 1665. des Cloizeaux, J. (1982). In Phase Transitions: Cargese 1980, eds. M. Levy et al. (Plenum Press, New York), pp. 371–394. Diehl, H. W. (1986). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 10, pp. 75–267. Dingle, R. B. (1956). J. App. Res. B 6, 225. Dirac, P . A. M. (1926). Proc. R. Soc. Lond. A 112, 661, 671. Dirac, P . A. M. (1929). Proc. Camb. Phil. Soc. 25, 62. Dirac, P . A. M. (1930). Proc. Camb. Phil. Soc. 26, 361, 376. Dirac, P . A. M. (1931). Proc. Camb. Phil. Soc. 27, 240. Domb, C. (1960). Advan. Phys. 9, 150. Domb, C. (1974). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 3, pp. 357–484. Domb, C. (1985). Contemp. Phys. 26, 49. Domb, C., and Green, M. S. (eds.) (1972–1976). Phase Transitions and Critical Phenomena (Academic Press, London), Vols. 1–6. Domb, C., and Hunter, D. L. (1965). Proc. Phys. Soc. 86, 1147. Domb, C., and Lebowitz, J. L. (eds.) (1983–1992). Phase Transitions and Critical Phenomena (Academic Press, London), Vols. 7–15. Drude, P . (1900). Ann. Phys. 1, 566; 3, 369. Dukovski, I., Machta, J., and Chayes, L. (2002). Phys. Rev. E 65, 026702. Ehrenfest, P . (1905). Wiener Ber. 114, 1301. Ehrenfest, P . (1906). Phys. Zeits. 7, 528. Ehrenfest, P ., and Ehrenfest, T. (1912). Enzyklop¨ adie der mathematischen Wissenschaften, Vol. IV (Teubner, Leipzig). English transl. (1959) The Conceptual Foundations of the Statistical Approach in Mechanics (Cornell University Press, Ithaca, NY). Einstein, A. (1902). Ann. Phys. 9, 417. Einstein, A. (1903). Ann. Phys. 11, 170. Einstein, A. (1905a). Ann. Phys. 17, 132; see also (1906b) ibid. 20, 199. Einstein, A. (1905b). Ann. Phys. 17, 549; see also (1906a) ibid. 19, 289, 371. Einstein, A. (1905c). Ann. Phys. 17, 891; see also (1905d) ibid. 18, 639 and (1906c) ibid. 20, 627. Einstein, A. (1907). Jb. Radioakt. 4, 411. Einstein, A. (1909). Physik Z. 10, 185. Einstein, A. (1910). Ann. Phys. 33, 1275. Einstein, A. (1924). Berliner Ber. 22, 261. Einstein, A. (1925). Berliner Ber. 1, 3. 692 Bibliography Eisenschitz, R. (1958). Statistical Theory of Irreversible Processes (Oxford University Press, Oxford). Elcock, E. W., and Landsberg, P . T. (1957). Proc. Phys. Soc. 70, 161. Ensher, J. R., Jin, D. S., Matthews, M. R., Wieman, C. E., and Cornell, E. A. (1996). Phys. Rev. Lett. 77, 4984. Enskog, D. (1917). Kinetische Theorie der Vorg¨ ange in m¨ assig verd¨ unnten Gasen, dissertation (Almquist & Wiksell, Uppsala). Eyring, H., Henderson, D., Stover, B. J., and Eyring, E. M. (1963). Statistical Mechanics and Dynamics (John Wiley, New York). Ezawa, Z. F . (2000). Quantum Hall Effects: Field Theorectical Approach and Related Topics (World Scientific, Singapore). Farquhar, I. E. (1964). Ergodic Theory in Statistical Mechanics (Interscience Publishers, New York). Fay, J. A. (1965). Molecular Thermodynamics (Addison-Wesley, Reading, MA). Fedosov, B. V. (1963). Dokl. Akad. Nauk SSSR 151, 786; English transl. 1963: Sov. Math. (Doklady) 4, 1092. Fedosov, B. V. (1964). Dokl. Akad. Nauk SSSR 157, 536; English transl. 1964: Sov. Math. (Doklady) 5, 988. Ferdinand, A. E., and Fisher, M. E. (1969). Phys. Rev. 185, 832. Fermi, E. (1926). Z. Physik 36, 902. Fermi, E. (1928). Zeit. f¨ ur Phys. 48, 73; see also (1927) Ace. Lencei 6, 602. Fermi, E. (1936). Thermodynamics (Dover, New York). Fermi, E., Pasta, J. G., and Ulam, S. M. (1955). LASL Report LA-1940. Ferrenberg, A. M., Landau, D. P ., and Wong, Y. J. (1992). Phys. Rev. Lett. 69, 3382. Ferrenberg, A. M., and Swendsen, R. H. (1988). Phys. Rev. Lett. 61, 2635. Fetter, A. L. (1963). Phys. Rev. Lett. 10, 507. Fetter, A. L. (1965). Phys. Rev. 138, A 429. Feynman, R. P . (1953). Phys. Rev. 91, 1291, 1301. Feynman, R. P . (1954). Phys. Rev. 94, 262. Feynman, R. P . (1955). Progress in Low Temperature Physics, ed. C. J. Gorter (North-Holland, Amsterdam), Vol. 1, p. 17 Feynman, R. P ., and Cohen, M. (1956). Phys. Rev. 102, 1189. Fisher, M. E. (1959). Physica 25, 521. Fisher, M. E. (1964). J. Math. Phys. 5, 944. Fisher, M. E. (1967). Rep. Prog. Phys. 30, 615. Fisher, M. E. (1969). Phys. Rev. 180, 594. Fisher, M. E. (1972). In Essays in Physics, eds. G. K. T. Conn and G. N. Fowler (Academic Press, London), Vol. 4, pp. 43–89. Fisher, M. E. (1973). Phys. Rev. Lett. 30, 679. Fisher, M. E. (1983). In Critical Phenomena, ed. F . J. W. Hahne (Springer-Verlag, Berlin), pp. 1–139. Fisher, M. E., Barber, M. N., and Jasnow, D. (1973). Phys. Rev. A 8, 1111. Fisher, M. E., and Privman, V. (1985). Phys. Rev. B 32, 447. Fixsen, D. J., Cheng, E. S., Gales, J. M., Mather, J. C., Shafer, R. A., and Wright, E. L. (1996). Astrophys. J. 473, 576. Fokker, A. D. (1914). Ann. Phys. 43, 812. Bibliography 693 Forster, D. (1975). Hydrodynamic Fluctuations, Broken Symmetry, and Correlation Functions (W. A. Benjamin, Reading, MA). Fowler, R. H. (1926). Mon. Not. R. Astron. Soc. 87, 114. Fowler, R. H. (1955). Statistical Mechanics, 2nd ed. (Cambridge University Press, Cambridge, UK). Fowler, R. H., and Guggenheim, E. A. (1940). Proc. R. Soc. Lond. A 174, 189. Fowler, R. H., and Guggenheim, E. A. (1960). Statistical Thermodynamics (Cambridge University Press, Cambridge, UK). Fowler, R. H., and Nordheim, L. (1928). Proc. R. Soc. Lond. A 119, 173. Freedman, W. L., Madore, B. F ., Gibson, B. K., Ferrarese, L., Kelson, D. D., Sakai, S., Mould, J. R., Kennicutt, R. C., Jr., Ford, H. C., Graham, J. A., Huchra, J. P ., Hughes, S. M. G., Illingworth, G. D., Macri, L. M., and Stetson, P . B. (2001). Astrophys. J. 553, 47. Frenkel, D., and Smit, B. (2002). Molecular Simulation (Academic, New York). Friedmann, A. (1922). Z. Phys. 16, 377. Friedmann, A. (1924). Z. Phys. 21, 326. Fujiwara, I., ter Haar, D., and Wergeland, H. (1970). J. Stat. Phys. 2, 329. Galitskii, V. M. (1958). J. Exptl. Theor. Phys. USSR 34, 151; English transl. (1958) Soviet Physics JETP-USSR 7, 104–112. Gaunt, D. S., and Guttmann, A. J. (1974). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 3, pp. 181–243. Gelmini, G. B. (2005). Phys. Scr. T121, 131. Gentle, J. E. (2003). Random Number Generation and Monte Carlo Methods (Springer-Verlag, New York). Giardeau, M. (1960). J. Math. Phys. 1, 516. Gibbs, J. W. (1902). Elementary Principles in Statistical Mechanics (Yale University Press, New Haven); reprinted by Dover Publications, New York (1960). See also A Commentary on the Scientific Writings of J. Willard Gibbs, ed. A. Haas, Yale University Press, New Haven (1936), Vol. II. Ginzburg, V. L. (1960). Sov. Phys. Solid State 2, 1824. Glasser, M. L. (1970). Ann. J. Phys. 38, 1033. Goldman, I. I., Krivchenkov, V. D., Kogan V. I., and Galitskii V. M. (1960). Problems in Quantum Mechanics, translated by D. ter Haar (Academic Press, New York). Gomb´ as, P . (1949). Die statistische Theorie des Atoms und ihre Anwendungen (Springer, Vienna). Gomb´ as, P . (1952). Ann. Phys. 10, 253. Gradshteyn, I. S., and Ryzhik, I. M. (1965). Table of Integrals, Series and Products (Academic Press, New York). Green, H. S., and Hurst, C. A. (1964). Order–Disorder Phenomena (Interscience Publishers, New York). Greenspoon, S., and Pathria, R. K. (1974). Phys. Rev. A 9, 2103. Greiner, M., Regal, C. A., and Jin, D. S. (2003). Nature 426, 537. Griffiths, R. B. (1965a). Phys. Rev. Lett. 14, 623. Griffiths, R. B. (1965b). J. Chem. Phys. 43, 1958. Griffiths, R. B. (1970). Phys. Rev. Lett. 24, 715. Griffiths, R. B. (1972). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 1, pp. 7–109. 694 Bibliography Gross, E. P . (1961). Nuovo Cim. 20, 454. Gross, E. P . (1963). J. Math. Phys. 4, 195. Guggenheim, E. A. (1935). Proc. R. Soc. Lond. A 148, 304. Guggenheim, E. A. (1938). Proc. R. Soc. Lond. A 169, 134. Guggenheim, E. A. (1959). Boltzmann’s Distribution Law (North-Holland, Amsterdam). Guggenheim, E. A. (1960). Elements of the Kinetic Theory of Gases (Pergamon Press, Oxford). Gunton, J. D., and Buckingham, M. J. (1968). Phys. Rev. 166, 152. Gupta, H. (1947). Proc. Nat. Ins. Sci. India 13, 35. Gurarie, V., and Radzihovsky, L. (2007). Ann. Phys. 322, 2. Guttmann, A. J. (1989). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 13, pp. 1–234. Haber, H. E., and Weldon, H. A. (1981). Phys. Rev. Lett. 46, 1497. Haber, H. E., and Weldon, H. A. (1982). Phys. Rev. D 25, 502. Halperin, B. I., and Nelson, D. R. (1978). Phys Rev. Lett. 41, 121. Han, Y., Alsayed, A. M., Nobili, M., Zhang, J., Lubensky, T. C., and Yodh, A. G. (2006). Science 314, 626. Hanbury Brown, R., and Twiss, R. Q. (1956). Nature 177, 27; 178, 1447. Hanbury Brown, R., and Twiss, R. Q. (1957). Proc. R. Soc. Lond. A 242, 300; 243, 291. Hanbury Brown, R., and Twiss, R. Q. (1958). Proc. R. Soc. Lond. A 248, 199, 222. Hansen, J. P ., and McDonald, I. R. (1986). Theory of Simple Liquids (Academic, New York). Harris, F . E., and Monkhorst, H. K. (1970). Phys. Rev. B 2, 4400. Harrison, S. F ., and Mayer, J. E. (1938). J. Chem. Phys. 6, 101. Hasenbusch, M., and Pinn, K. (1997). J. Phys. A 30, 63. Haugerud, H., Haugest, T., and Ravndal, F . (1997). Phys. Lett. A 225, 18. Heisenberg, W. (1928). Z. Physik 49, 619. Heller, P . (1967). Rep. Prog. Phys. 30, 731. Henry, W. E. (1952). Phys. Rev. 88, 561. Henshaw, D. G. (1960). Phys. Rev. 119, 9. Henshaw, D. G., and Woods, A. D. B. (1961). Phys. Rev. 121, 1266. Herapath, J. (1821). Ann. Philos. 1, 273, 340, 401. Heuer, A., Dunweg, B., and Ferrenberg, A. M. (1997). Comp. Phys. Comm. 103, 1. Hill, T. L. (1953). J. Phys. Chem. 57, 324. Hill, T. L. (1956). Statistical Mechanics (McGraw-Hill, New York). Hill, T. L. (1960). Introduction to Statistical Thermodynamics (Addison-Wesley, Reading, MA). Hillebrandt, W., and Niemeyer, J. C. (2000). Annu. Rev. Astron. Astrophys. 38, 191. Hirschfelder, J. O., Curtiss, C. F ., and Bird, R. B. (1954). Molecular Theory of Gases and Liquids (John Wiley, New York). Hohenberg, P . C., and Halperin, B. I. (1977). Rev. Mod. Phys. 49, 435. Holland, M., and Cooper, J. (1996). Phys. Rev. A 53, R1954. Holland, M. J., Jin, D. S., Chiofalo, M. L., and Cooper, J. (1997). Phys. Rev. Lett. 78, 3801. Bibliography 695 Holliday, D., and Sage, M. L. (1964). Ann. Phys. 29, 125. Huang, K. (1959). Phys. Rev. 115, 765. Huang, K. (1960). Phys. Rev. 119, 1129. Huang, K. (1963). Statistical Mechanics, 1st ed. (John Wiley, New York). Huang, K. (1987). Statistical Mechanics, 2nd ed. (John Wiley, New York). Huang, K., and Klein, A. (1964). Ann. Phys. 30, 203. Huang, K., and Yang, C. N. (1957). Phys. Rev. 105, 767. Huang, K., Yang, C. N., and Luttinger, J. M. (1957). Phys. Rev. 105, 776. Hupse, J. C. (1942). Physica 9, 633. Hurst, C. A., and Green, H. S. (1960). J. Chem. Phys. 33, 1059. Ising, E. (1925). Z. Physik 31, 253. Jackson, H. W., and Feenberg, E. (1962). Rev. Mod. Phys. 34, 686. Jackson, J. D. (1999). Electrodynamics, 3rd ed. (John Wiley, New York). Jasnow, D. (1986). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 10, pp. 269–363. Jeans, J. H. (1905). Phil. Mag. 10, 91. Jin, D. (2002). Phys. World 15, 27. Jochim, S., Bartenstein, M., Altmeyer, A., Hendl, G., Riedl, S., Chin, C., Denschlag, J. H., and Grimm, R. (2003). Science 302, 2101. Johnson, J. B. (1927a). Nature 119, 50. Johnson, J. B. (1927b). Phys. Rev. 29, 367. Johnson, J. B. (1928). Phys. Rev. 32, 97. Josephson, B. D. (1967). Proc. Phys. Soc. 92, 269, 276. Joule, J. P . (1851). Mem. Proc. Manchester Lit. Phil. Soc. 9, 107; also published in 1857, Phil. Mag. 14, 211. Joyce, G. S. (1972). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 2, pp. 375–442. Kac, M., and Thompson, C. J. (1971). Phys. Norveg. 5, 163. Kac, M., Uhlenbeck, G. E., and Hemmer, P . C. (1963). J. Math. Phys. 4, 216, 229; see also (1964) ibid. 5, 60. Kac, M., and Ward, J. C. (1952). Phys. Rev. 88, 1332. Kadanoff, L. P . (1966a). Nuovo Cim. 44, 276. Kadanoff, L. P . (1966b). Physics 2, 263. Kadanoff, L. P . (1976a). Ann. Phys. 100, 359. Kadanoff, L. P . (1976b). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 5a, pp. 1–34. Kadanoff, L. P . et al. (1967). Rev. Mod. Phys. 39, 395. Kahn, B. (1938). On the Theory of the Equation of State, Thesis, Utrecht; English translation appears in de Boer, J., and Uhlenbeck, G. E. (eds.) (1965) Studies in Statistical Mechanics, Vol. III (North-Holland, Amsterdam). Kahn, B., and Uhlenbeck, G. E. (1938). Physica 5, 399. Kalos, M. H., and Whitlock, P . A. (1986). Monte Carlo Methods (Wiley, New York). Kappler, E. (1931). Ann. Phys. 11, 233. 696 Bibliography Kappler, E. (1938). Ann. Phys. 31, 377, 619. Kasteleyn, P . W. (1963). J. Math. Phys. 4, 287. Katsura, S. (1959). Phys. Rev. 115, 1417. Kaufman, B. (1949). Phys. Rev. 76, 1232. Kaufman, B., and Onsager, L. (1949). Phys. Rev. 76, 1244. Kawatra, M. P ., and Pathria, R. K. (1966). Phys. Rev. 151, 132. Keesom, P . H., and Pearlman, N. (1953). Phys. Rev. 91, 1354. Khalatnikov, I. M. (1965). Introduction to the Theory of Superfluidity (W. A. Benjamin, New York). Khintchine, A. (1934). Math. Ann. 109, 604. Kiess, E. (1987). Am. J. Phys. 55, 1006. Kilpatrick, J. E. (1953). J. Chem. Phys. 21, 274. Kilpatrick, J. E., and Ford, D. I. (1969). Am. J. Phys. 37, 881. Kim, E., and Chan, M. H. W. (2004). Nature 427, 225; (2004b) Science, 305, 1941. Kirkwood, J. G. (1965). Quantum Statistics and Cooperative Phenomena (Gordon & Breach, New York). Kirsten, K., and Toms, D. J. (1996) Phys. Rev. A 54, 4188. Kittel, C. (1958). Elementary Statistical Physics (John Wiley, New York). Kittel, C. (1969). Thermal Physics (John Wiley, New York). Klein, M. J., and Tisza, L. (1949). Phys. Rev. 76, 1861. Knobler, C. M., and Scott, R. L. (1984). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 9, pp. 163–231. Knuth, D. E. (1997). The Art of Computer Programming, Volume 2, Seminumerical Algorithms, 3rd ed. (Addison-Wesley, Reading, MA). Kolmogorov, A. N. (1954). Dokl. Akad. Nauk SSSR 98, 527. Komatsu, E., et al. (2010). Astrophys. J. (submitted): doi: arXiv.org/abs/1001.4538v3. Kosterlitz, J. M. (1974). J. Phys. C 7, 1046. Kosterlitz, J. M., and Thouless, D. J. (1972). J. Phys. C 5, L124. Kosterlitz, J. M., and Thouless, D. J. (1973). J. Phys. C 6, 1181. Kosterlitz, J. M., and Thouless, D. J. (1978). Prog. Low Temp. Phys. 78, 371. Kothari, D. S., and Auluck, F . C. (1957). Curr. Sci. 26, 169. Kothari, D. S., and Singh, B. N. (1941). Proc. R. Soc. Lond. A 178, 135. Kothari, D. S., and Singh, B. N. (1942). Proc. R. Soc. Lond. A 180, 414. Kramers, H. A. (1938). Proc. Kon. Ned. Akad. Wet. (Amsterdam) 41, 10. Kramers, H. A., and Wannier, G. H. (1941). Phys. Rev. 60, 252, 263. Kr¨ onig, A. (1856). Ann. Phys. 99, 315. Kubo, R. (1956). Can. J. Phys. 34, 1274. Kubo, R. (1957). Proc. Phys. Soc. Japan 12, 570. Kubo, R. (1959). Some Aspects of the Statistical Mechanical Theory of Irreversible Processes, University of Colorado Lectures in Theoretical Physics (Interscience Publishers, New York), Vol. 1, p. 120. Kubo, R. (1965). Statistical Mechanics (Interscience Publishers, New York). Kubo, R. (1966). Rep. Prog. Phys. 29, 255. Bibliography 697 Landau, D. P ., and Binder, K. (2009). Monte-Carlo Simulations in Statistical Physics, 3rd ed. (Cambridge University Press, New York). Landau, L. D. (1927). Z. Phys. 45, 430; reprinted in the Collected Papers of L. D. Landau, ed. D. ter Haar (Pergamon Press, Oxford), p. 8. Landau, L. D. (1930). Z. Phys. 64, 629. Landau, L. D. (1937). Phys. Z. Sowjetunion 11, 26. Landau, L. D. (1941). J. Phys. USSR 5, 71. Landau, L. D. (1947). J. Phys. USSR 11, 91. Landau, L. D. (1956). J. Exptl. Theoret. Phys. USSR 30, 1058; English transl. Soviet Physics JETP 3, 920. Landau, L. D., and Lifshitz, E. M. (1958). Statistical Physics (Pergamon Press, Oxford). Landau, L. D., and Placzek, G. (1934). Phys. Z. Sowjetunion 5, 172. Landsberg, P . T. (1954a). Proc. Natl. Acad. Sci. (U.S.A.) 40, 149. Landsberg, P . T. (1954b). Proc. Camb. Phil. Soc. 50, 65. Landsberg, P . T. (1961). Thermodynamics with Quantum Statistical Illustrations (Interscience Publishers, New York). Landsberg, P . T., and Dunning-Davies, J. (1965). Phys. Rev. 138, A1049; see also their article in the Statistical Mechanics of Equilibrium and Nonequilibrium (1965), ed. J. Meixner (North-Holland, Amsterdam). Langevin, P . (1905a). J. Phys. 4, 678. Langevin, P . (1905b). Ann. Chim. et Phys. 5, 70. Langevin, P . (1908). Comptes Rend. Acad. Sci. Paris 146, 530. Lawrie, I. D., and Sarbach, S. (1984). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 9, pp. 1–161. L’Ecuyer, P . (1988). Comm. ACM 31, 742. L’Ecuyer, P . (1999). Math. Comput. 68, 249. Lee, J. F ., Sears, F . W., and Turcotte, D. L. (1963). Statistical Thermodynamics (Addison-Wesley, Reading, MA). Lee, T. D., Huang, K., and Yang, C. N. (1957). Phys. Rev. 106, 1135. Lee, T. D., and Yang, C. N. (1952). Phys. Rev. 87, 410; see also ibid. 404. Lee, T. D., and Yang, C. N. (1957). Phys. Rev. 105, 1119. Lee, T. D., and Yang, C. N. (1958). Phys. Rev. 112, 1419. Lee, T. D., and Yang, C. N. (1959a,b). Phys. Rev. 113, 1165; 116, 25. Lee, T. D., and Yang, C. N. (1960a,b,c). Phys. Rev. 117, 12, 22, 897. Leggett, A. J. (2006). Quantum Liquids (Oxford University Press, Oxford). Leib, E., and Liniger, W. (1963). Phys. Rev. 130, 1605. Lennard-Jones, J. E. (1924). Proc. R. Soc. Lond. A 106, 463. Lenz, W. (1920). Z. Physik 21, 613. Lenz, W. (1929). Z. Physik 56, 778. Levelt, J. M. H. (1960). Physica 26, 361. Lewis, H. W., and Wannier, G. H. (1952). Phys. Rev. 88, 682. Lieb, E. H., and Mattis, D. C. (1966). Mathematical Physics in One Dimension (Academic Press, New York). 698 Bibliography Lines, M. E., and Glass, A. M. (1977). Principles and Applications of Ferroelectrics and Related Materials (Clarendon Press, Oxford). Liouville, J. (1838). J. de Math. 3, 348. London, F . (1938a). Nature 141, 643. London, F . (1938b). Phys. Rev. 54, 947. London, F . (1954). Superfluids, Vols. 1 and 2 (John Wiley, New York); reprinted by Dover Publications, New York, 1964. Lorentz, H. A. (1904–1905). Proc. Kon. Ned Akad. Wet. Amsterdam 7, 438, 585, 684. See also (1909) The Theory of Electrons (Teubner, Leipzig), pp. 47–50; reprinted by Dover Publications, New York (1952). Loschmidt, J. (1876). Wien. Ber. 73, 139. Loschmidt, J. (1877). Wien. Ber. 75, 67. Luck, J. M. (1985). Phys. Rev. B 31, 3069. L¨ uders, G., and Zumino, B. (1958). Phys. Rev. 110, 1450. Ma, S.-K. (1973). Phys. Rev. A 7, 2712. Ma, S.-K. (1976a). Modern Theory of Critical Phenomena (Benjamin/Cummings, Reading, MA). Ma, S.-K. (1976b). Phys. Rev. Lett. 37, 461. Ma, S.-K. (1976c). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 6, pp. 249–292. MacDonald, D. K. C. (1948–49). Rep. Prog. Phys. 12, 56. MacDonald, D. K. C. (1950). Phil. Mag. 41, 814. MacDonald, D. K. C. (1962). Noise and Fluctuations: An Introduction (John Wiley, New York). MacDonald, D. K. C. (1963). Introductory Statistical Mechanics for Physicists (John Wiley, New York). Majumdar, R. (1929). Bull. Calcutta Math. Soc. 21, 107. Malijevsky, A., and Kolafa, J. (2008). In Theory and Simulation of Hard-Sphere Fluids and Related Systems, ed. A. Mulero (Springer, Berlin), pp. 27–36. Mandel, L., Sudarshan, E. C. G., and Wolf, E. (1964). Proc. Phys. Soc. Lond. 84, 435. Manoliu, A., and Kittel, C. (1979). Am. J. Phys. 47, 678. March, N. H. (1957). Adv. Phys. 6, 1. Martin, P . C. (1968). Measurements and Correlations (Gordon & Breach, New York). Martin, P . C., and De Dominicis, C. (1957). Phys. Rev. 105, 1417. Mather, J. C. et al. (1994). Astrophys. J. 420, 439. Mather, J. C. et al. (1999). Astrophys. J. 512, 511. Matsubara, T., and Matsuda, H. (1956). Prog. Theor. Phys. Japan 16, 416. Maxwell, J. C. (1860). Phil. Mag. 19, 19; 20, 1. Maxwell, J. C. (1867). Trans. R. Soc. Lond. 157, 49; also published in 1868 in Phil. Mag. 35, 129, 185. Maxwell, J. C. (1879). Camb. Phil. Soc. Trans. 12, 547. Mayer, J. E., and Ackermann, P . G. (1937). J. Chem. Phys. 5, 74. Mayer, J. E., and Harrison, S. F . (1938). J. Chem. Phys. 6, 87. Mayer, J. E., and Mayer, M. G. (1940). Statistical Mechanics (John Wiley, New York). Mayer, J. E. (1937). J. Chem. Phys. 5, 67. Bibliography 699 Mayer, J. E. (1942). J. Chem. Phys. 10, 629. Mayer, J. E. (1951). J. Chem. Phys. 19, 1024. Mazenko, G. F . (2006). Nonequilibrium Statistical Mechanics (John Wiley, New York). McCoy, B. M., and Wu, T. T. (1973). The Two-Dimensional Ising Model (Harvard University Press, Cambridge, MA). Mehra, J. M., and Pathria, R. K. (1994). In The Beat of a Different Drum: The Life and Science of Richard Feynman by J. Mehra (Clarendon Press, Oxford), Chap. 17, pp. 348–391. Mermin, N. D. (1968). Phys. Rev. 176, 250. Mermin, N. D., and Wagner, H. (1966). Phys. Rev. Lett. 17, 1133, 1307. Metropolis, N. (1987). Los Alamos Sci. 15, 125. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. N., and Teller, E. (1953). J. Chem. Phys. 21, 1087, 1092. Milne, E. (1927). Proc. Camb. Phil. Soc. 23, 794. Minguzzi, A., Conti, S., and Tosi, M. P . (1997). J. Phys.: Condens. Matter 9, L33. Mohling, F . (1961). Phys. Rev. 122, 1062. Montroll, E. (1963). The Many-Body Problem, ed. J. K. Percus (Interscience Publishers, New York), pp. 525–533. Montroll, E. (1964). Applied Combinatorial Mathematics, ed. E. F . Beckenbach (John Wiley, New York). Morse, P . M. (1969). Thermal Physics, 2nd ed. (W. A. Benjamin, New York). Moser, J. K. (1962). Nach. Akad. Wiss. Gttingen, Math. Phys. Kl. II, 1, 1. Mukamel, D. (1975). Phys. Rev. Lett. 34, 481. Mulero, A., Galan, C. A., Parra, M. I., and Cuadros, F . (2008). In Theory and Simulation of Hard-Sphere Fluids and Related Systems, ed. A. Mulero (Springer, Berlin), pp. 111–132. Nelson, D. R. (1983). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 7, pp. 1–99. Newman, M. E. J., and Barkema, G. T. (1999). Monte Carlo Methods in Statistical Physics (Oxford University Press, Oxford). New Worlds, New Horizons in Astronomy and Astrophysics. (2010). (National Academies Press, Washing-ton). See www.nap.edu. Niemeijer, Th., and van Leeuwen, J. M. J. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 6, pp. 425–505. Nienhuis, B. (1987). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 11, pp. 1–53. Nozi eres, P . (1964). Theory of Interacting Fermi Systems (W. A. Benjamin, New York). Nyquist, H. (1927). Phys. Rev. 29, 614. Nyquist, H. (1928). Phys. Rev. 32, 110. Ono, S. (1951). J. Chem. Phys. 19, 504. Onsager, L. (1931). Phys. Rev. 37, 405; 38, 2265. Onsager, L. (1944). Phys. Rev. 65, 117. Onsager, L. (1949). Nuovo Cim. 6 (Suppl. 2), 249, 261. Ornstein, L. S., and Zernike, F . (1914). Proc. Akad. Sci. Amsterdam 17, 793; reproduced in The Equilibrium Theory of Classical Fluids, eds. A. L. Frisch and J. L. Lebowitz (W. A. Benjamin, New York, 1964). 700 Bibliography Pais, A. M., and Uhlenbeck, G. E. (1959). Phys. Rev. 116, 250. Paredes, B., Widera, A., Murg, V., Mandel, O., Folling, S., Cirac, I., Shlyapnikov, G. V., and Hansch, T. W. (2004). Nature, 429, 277. Parker, W. S. (2008). Int. Stud. Philos. Sci. 22, 165. Patashinskii, A. Z., and Pokrovskii, V. L. (1966). Sov. Phys. JETP 23, 292. Pathria, R. K. (1966). Nuovo Cim. (Supp.), Ser. I, 4, 276. Pathria, R. K. (1972). Statistical Mechanics, 1st ed. (Pergamon Press, Oxford). Pathria, R. K. (1974). The Theory of Relativity, 2nd ed. (Pergamon Press, Oxford). Pathria, R. K. (1983). Can. J. Phys. 61, 228. Pathria, R. K. (1998). Phys. Rev. A 58, 1490. Pathria, R. K., and Kawatra, M. P . (1962). Prog. Theor. Phys. Japan 27, 638, 1085. Pauli, W. (1925). Z. Physik 31, 776. Pauli, W. (1927). Z. Physik 41, 81. Pauli, W. (1940). Phys. Rev. 58, 716. Pauli, W., and Belinfante, F . J. (1940). Physica 7, 177. Peierls, R. E. (1935). Ann. Inst. Henri Poincare 5, 177. Peierls, R. E. (1936). Proc. Camb. Phil. Soc. 32, 471, 477. Penrose, O., and Onsager, L. (1956). Phys. Rev. 104, 576. Penzias, A. A., and Wilson, R. W. (1965). Astrophys. J. 142, 419. Percus, J. K. (1963). The Many-Body Problem (Interscience Publishers, New York). Percus, J. K., and Yevick, G. J. (1958). Phys. Rev. 110, 1. Perrin, F . J. (1934). Phys. Radium V, 497. Perrin, F . J. (1936). Phys. Radium VII, 1. Pethick, C. J., and Smith, H. (2008). Bose–Einstein Condensation in Dilute Gases (Cambridge University Press, New York). Pines, D. (1962). The Many-Body Problem (W. A. Benjamin, New York). Pitaevskii, L. P . (1959). Sov. Phys. JETP 9, 830. Pitaevskii, L. P . (1961). Sov. Phys. JETP 13, 451. Pitaevskii, L. P ., and Stringari, S. (2003). Bose–Einstein Condensation (Oxford University Press, Oxford). Planck, M. (1900). Verhandl. Deut. Phys. Ges. 2, 202, 237. Planck, M. (1908). Ann. Phys. 26, 1. Planck, M. (1917). Sitz. der Preuss. Akad. 324. Plischke, M., and Bergersen, B. (1989). Equilibrium Statistical Physics (Prentice-Hall, Englewood Cliffs, NJ). Pospiˇ sil, W. (1927). Ann. Phys. 83, 735. Potts, R. B., and Ward, J. C. (1955). Prog. Theor. Phys. Japan 13, 38. Press, W. H., Teukolsky, S. A., Vetterling, W. T., and Flannery, B. P . (2007). Numerical Recipes: The Art of Scientific Computing, 3rd ed. (Cambridge University Press, New York). Prigogine, I. (1967). Introduction to Thermodynamics of Irreversible Processes, 3rd ed. (John Wiley, New York). Privman, V. (ed.) (1990). Finite Size Scaling and Numerical Simulation of Statistical Systems (World Scientific, Singapore). Bibliography 701 Privman, V., and Fisher, M. E. (1984). Phys. Rev. B 30, 322. Privman, V., Hohenberg, P . C., and Aharony, A. (1991). In Phase Transitions and Critical Phenomena, eds. C. Domb and J. L. Lebowitz (Academic Press, London), Vol. 14, pp. 1–134, 364–367. Purcell, E. M. (1956). Nature 178, 1449. Purcell, E. M., and Pound, R. V. (1951). Phys. Rev. 81, 279. Rahman, A. (1964). Phys. Rev. 136, A405. Ramsey, N. F . (1956). Phys. Rev. 103, 20. Rayleigh, L. (1900). Phil. Mag. 49, 539. Rayfield, G. W., and Reif, F . (1964). Phys. Rev. 136, A1194; see also (1963) Phys. Rev. Lett. 11, 305. Ree, F . H., and Hoover, W. G. (1964). J. Chem. Phys. 40, 939. Regal, C. A., Greiner, M., and Jin, D. S. (2004). Phys. Rev. Lett. 92, 040403. Reif, F . (1965). Fundamentals of Statistical and Thermal Physics (McGraw-Hill, New York). Rhodes, P . (1950). Proc. R. Soc. Lond. A 204, 396. Riecke, E. (1898). Ann. Phys. 66, 353, 545. Riecke, E. (1900). Ann. Phys. 2, 835. Riess, A. G., Macri, L., Casertano, S., Sosey, M., Lampeitl, H., Ferguson, H. C., Filippenko, A. V., Jha, S. W., Li, W., Chornock, R., and Sarkar, D. (2009). Astrophys. J. 699, 539. Rintoul, M. D., and Torquato, S. (1996). J. Chem. Phys. 105, 9258. Roberts, J. L., Claussen, N. R., Cornish, S. L., Donley, E. A., Cornell, E. A., and Wieman, C. E. (2001). Phys. Rev. Lett. 86, 4211. Roberts, T. R., and Sydoriak, S. G. (1955). Phys. Rev. 98, 1672. Robinson, J. E. (1951). Phys. Rev. 83, 678. Rowley, L. A., Nicholson, D., and Parsonage, N. G. (1975). J. Comp. Phys. 17, 401. Rowlinson, J. S., and Swinton, F . L. (1982). Liquids Liquid Mixtures, 3rd ed. (Butterworth Scientific, London). Ruprecht, P . A., Holland, M. J., Burnett, K., and Edwards, M. (1995). Phys. Rev. A 51, 4704. Rushbrooke, G. S. (1938). Proc. R. Soc. Lond. A 166, 296. Rushbrooke, G. S. (1955). Introduction to Statistical Mechanics (Clarendon Press, Oxford). Rushbrooke, G. S. (1963). J. Chem. Phys. 39, 842. Rushbrooke, G. S., Baker, G. A., Jr., and Wood, P . J. (1974). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 3, pp. 245–356. Sackur, O. (1911). Ann. Phys. 36, 958. Sackur, O. (1912). Ann. Phys. 40, 67. Sakharov, A. D. (1967). JETP Lett. 5, 24. Schiff, L. I. (1968). Quantum Mechanics, 3rd ed. (McGraw-Hill, New York). Schramm, D. N., and Turner, M. S. (1998). Rev. Mod. Phys. 70, 303. Schr¨ odinger, E. (1960). Statistical Thermodynamics (Cambridge University Press, Cambridge, UK). Schultz, T. D., Lieb, E. H., and Mattis, D. C. (1964). Rev. Mod. Phys. 36, 856. Schwartz, L. (1966). Mathematics of the Physical Sciences (Addison-Wesley, Reading, MA). Scott, G. G. (1951). Phys. Rev. 82, 542. Scott, G. G. (1952). Phys. Rev. 87, 697. 702 Bibliography Sells, R. L., Harris, C. W., and Guth, E. (1953). J. Chem. Phys. 21, 1422. Shannon, C. E. (1948). Bell Syst. Tech. J. 27, 379, 623. Shannon, C. E. (1949). The Mathematical Theory of Communication (University of Illinois Press, Urbana, IL). Simon, F . (1930). Ergeb. Exakt. Naturwiss. 9, 222. Singh, S. (2005). The Big Bang: The Origin of the Universe (Fourth Estate, New York). Singh, S., and Pandita, P . N. (1983). Phys. Rev. A 28, 1752. Singh, S., and Pathria, R. K. (1985a). Phys. Rev. B 31, 4483. Singh, S., and Pathria, R. K. (1985b). Phys. Rev. Lett. 55, 347. Singh, S., and Pathria, R. K. (1986a). Phys. Rev. Lett. 56, 2226. Singh, S., and Pathria, R. K. (1986b). Phys. Rev. B 34, 2045. Singh, S., and Pathria, R. K. (1987a). Phys. Rev. B 36, 3769. Singh, S., and Pathria, R. K. (1987b). J. Phys. A 20, 6357. Singh, S., and Pathria, R. K. (1988). Phys. Rev. B 38, 2740. Singh, S., and Pathria, R. K. (1992). Phys. Rev. B 45, 9759. Smith, B. L. (1969). Contemp. Phys. 10, 305. Smoluchowski, M. V. (1906). Ann. Phys. 21, 756. Smoluchowski, M. V. (1908). Ann. Phys. 25, 205. Sokal, A. D. (1981). J. Stat. Phys. 25, 25, 51. Sommerfeld, A. (1928). Z. Physik 47, 1; see also Sommerfeld and Frank (1931) Rev. Mod. Phys. 3, 1. Sommerfeld, A. (1932). Z. Physik 78, 283. Sommerfeld, A. (1956). Thermodynamics and Statistical Mechanics (Academic Press, New York). Speedy, R. J. (1997). J. Phys.: Condens. Matter 9, 8591. Squires, G. L. (1997). Introduction to the Theory of Thermal Neutron Scattering (Cambridge University Press, New York). Stanley, H. E. (1968). Phys. Rev. 176, 718. Stanley, H. E. (1969a). J. Phys. Soc. Japan 26S, 102. Stanley, H. E. (1969b). Phys. Rev. 179, 570. Stanley, H. E. (1971). Introduction to Phase Transitions and Critical Phenomena (Oxford University Press, Oxford). Stanley, H. E. (1974). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 3, pp. 485–567. Stefan, J. (1879). Wien. Ber. 79, 391. Steigman, G. (2006). Int. J. Mod. Phys E15, 1, doi: arXiv:astro-ph/0511534v1. Stierstadt, K. et al. (1990). Physics Data: Experimental Values of Critical Exponents and Amplitude Ratios of Magnetic Phase Transitions (Fachinformationszentrum, Karlsruhe, Germany). Stoner, E. C. (1929). Phil. Mag. 7, 63. Stoner, E. C. (1930). Phil. Mag. 9, 944. Swendsen, R. H., and Wang, J.-S. (1987). Phys. Rev. Lett. 58, 86. Takahashi, H. (1942). Proc. Phys. Math. Soc. (Japan) 24, 60. Bibliography 703 Tarko, H. B., and Fisher, M. E. (1975). Phys. Rev. B 11, 1217. ter Haar, D. (1954). Elements of Statistical Mechanics (Rinehart, New York). ter Haar, D. (1955). Rev. Mod. Phys. 27, 289. ter Haar, D. (1961). Rep. Prog. Phys. 24, 304. ter Haar, D. (1966). Elements of Thermostatistics (Holt, Rinehart & Winston, New York). ter Haar, D. (1967). The Old Quantum Theory (Pergamon Press, Oxford). ter Haar, D. (1968). On the History of Photon Statistics, Course 42 of the International School of Physics “Enrico Fermi” (Academic Press, New York). ter Haar, D., and Wergeland, H. (1966). Elements of Thermodynamics (Addison-Wesley, Reading, MA). Tetrode, H. (1912). Ann. Phys. 38, 434; corrections to this paper appear in (1912) ibid. 39, 255. Tetrode, H. (1915). Proc. Kon. Ned. Akad. Amsterdam 17, 1167. Thomas, L. H. (1927). Proc. Camb. Phil. Soc. 23, 542. Thompson, C. J. (1972a). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 1, pp. 177–226. Thompson, C. J. (1972b). Mathematical Statistical Mechanics (Princeton University Press, Princeton, NJ). Thompson, C. J. (1988). Classical Equilibrium Statistical Mechanics (Clarendon Press, Oxford). Tilley, D. R., and Tilley, J. (1990). Superfluidity and Superconductivity, 3rd ed. (Adam Hilger, Bristol). Tinkham, M. (1996). Introduction to Superconductivity (McGraw-Hill, New York). Tisza, L. (1938a). Nature 141, 913. Tisza, L. (1938b). Compt. Rend. Paris 207, 1035, 1186. Tolman, R. C. (1934). Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford). Tolman, R. C. (1938). The Principles of Statistical Mechanics (Oxford University Press, Oxford). Tonks, L. (1936). Phys. Rev. 50, 955. Tracy, C. A., and McCoy, B. M. (1973). Phys. Rev. Lett. 31, 1500. Tuttle, E. R., and Mohling, F . (1966). Ann. Phys. 38, 510. Uhlenbeck, G. E. (1966). In Critical Phenomena, eds. M. S. Green and J. V. Sengers (National Bureau of Standards, Washington, DC), pp. 3–6. Uhlenbeck, G. E., and Beth, E. (1936). Physica 3, 729. Uhlenbeck, G. E., and Ford, G. W. (1963). Lectures in Statistical Mechanics (American Mathematical Society, Providence, RI). Uhlenbeck, G. E., and Gropper, L. (1932). Phys. Rev. 41, 79. Uhlenbeck, G. E., and Ornstein, L. S. (1930). Phys. Rev. 36, 823. Ursell, H. D. (1927). Proc. Camb. Phil. Soc. 23, 685. van der Waals, J. D. (1873). Over de Continuiteit van den Gas-en Vloeistoftoestand, Thesis, Leiden. van Hove, L. (1949). Physica, 15, 951. Vdovichenko, N. V. (1965). Sov. Phys. JETP 20, 477; 21, 350. Verlet, L. (1967). Phys. Rev. 159, 98. Vinen, W. F . (1961). Proc. R. Soc. Lond. A 260, 218; see also Progress in Low Temperature Physics, Vol. III (1961), ed. C. J. Gorter (North-Holland, Amsterdam). Vinen, W. F . (1968). Rep. Prog. Phys. 31, 61. 704 Bibliography Vollhardt, D., and W¨ olfle, P . (1990). The Superfluid Phases of Helium 3 (Taylor and Francis, London). von Neumann, J. (1927). Gottinger Nachr. 1, 24, 273. von Neumann, J. (1951). In Monte Carlo Method, eds. A. S. Householder, G. E. Forsythe, and H. H. Germond (NBS-Appl. Math. Ser, U. S. Government Printing Office, Washington, DC). Voronel, A. V. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 5b, pp. 343–394. Walker, C. B. (1956). Phys. Rev. 103, 547. Wallace, D. J. (1976). In Phase Transitions and Critical Phenomena, eds. C. Domb and M. S. Green (Academic Press, London), Vol. 6, pp. 293–356. Walton, A. J. (1969). Contemp. Phys. 10, 181. Wang, F ., and Landau, D. P . (2001). Phys. Rev. Lett. 86, 2050; Phys. Rev. E 64, 056101. Wannier, G. H. (1945). Rev. Mod. Phys. 17, 50. Wannier, G. H. (1966). Statistical Physics (John Wiley, New York). Waterston, J. J. (1892). Philos. Trans. R. Soc. Lond. A 183, 5, 79; reprinted in his collected papers (1928), ed. J. S. Haldane (Oliver & Boyd, Edinburgh), pp. 207, 318. An abstract of Waterston’s work did appear earlier; see Proc. R. Soc. Lond. A 5, 604 (1846). Watson, P . G. (1969). J. Phys. C 2, 1883, 2158. Wax, N. (ed.). (1954). Selected Papers on Noise and Stochastic Processes (Dover Publications, New York). Weeks, J. D., Chandler, D., and Andersen, H. C. (1971). Chem. Phys. 54, 5237. Weinberg, S. (1993). The First Three Minutes, 2nd ed. (Basic, New York). Weinberg, S. (2008). Cosmology (Oxford University Press, New York). Weinstock, R. (1969). Am. J. Phys. 37, 1273. Weller, W. (1963). Z. Naturforsch. 18A, 79. Wergeland, H. (1969). Lettere al Nuovo Cim. 1, 49. Wertheim, M. S. (1963). Phys. Rev. Lett. 10, 321. Whitmore, S. C., and Zimmermann, W. (1965). Phys. Rev. Lett. 15, 389. Widom, B. (1965). J. Chem. Phys. 43, 3892, 3898. Wiebes, N.-H., and Kramers (1957). Physica 23, 625. Wien, W. (1896). Ann. Phys. 58, 662. Wiener, N. (1930). Act. Math. Stockholm 55, 117. Wilczek, F . (1990). Fractional Statistics and Anyon Superconductivity (World Scientific, Singapore). Wilks, J. (1961). The Third Law of Thermodynamics (Oxford University Press, Oxford). Wilson, A. H. (1960). Thermodynamics and Statistical Mechanics (Cambridge University Press, Cam-bridge, UK). Wilson, K. G. (1971). Phys. Rev. B 4, 3174, 3184. Wilson, K. G. (1972). Phys. Rev. Lett. 28, 548. Wilson, K. G. (1975). Rev. Mod. Phys. 47, 773. Wilson, K. G., and Fisher, M. E. (1972). Phys. Rev. Lett. 28, 240. Wolff, U. (1989). Phys. Rev. Lett. 62, 361. Wood, W. W., and Parker, F . R. (1957). J. Chem. Phys. 27, 720. Bibliography 705 Woods, A. D. B. (1966). Quantum Fluids, ed. D. F . Brewer (North-Holland, Amsterdam), p. 239. Wright, E. L. et al. (1994). Astrophys. J. 420, 450. Wu, F . Y. (1982). Rev. Mod. Phys. 54, 235. Wu, T. T. (1959). Phys. Rev. 115, 1390. Wu, T. T., McCoy, B. M., Tracy, C. A., and Barouch, E. (1976). Phys. Rev. B 13, 316. Yang, C. N. (1952). Phys. Rev. 85, 808. Yang, C. N., and Lee, T. D. (1952). Phys. Rev. 87, 404; see also ibid., 410. Yang, C. N., and Yang, C. P . (1964). Phys. Rev. Lett. 13, 303. Yang, C. P . (1962). J. Math. Phys. 3, 797. Yarnell, J. L. et al. (1959). Phys. Rev. 113, 1379, 1386. Yarnell, J. L., Katz, M. J., Wenzel, R. G., and Koenig, S. H. (1973). Phys. Rev. A 7, 2130. Yeomans, J. M. (1992). Statistical Mechanics of Phase Transitions (Clarendon Press, Oxford). Young, A. P . (1979). Phys. Rev. B 19, 1855. Zasada, C. S., and Pathria, R. K. (1976). Phys. Rev. A 14, 1269. Zermelo, E. (1896). Ann. Phys. 57, 485; 59, 793. Zernike, F . (1916). Proc. Akad. Sci. Amsterdam 18, 1520; reproduced in The Equilibrium Theory of Classical Fluids, eds. A. L. Frisch and J. L. Lebowitz (W. A. Benjamin, New York, 1964). Zwierlein, M. W., Stan, C. A., Schunck, C. H., Raupach, S. M. F ., Gupta, S., Hadzibabic, Z. M., and Ketterle, W. (2003). Phys. Rev. Lett. 91, 250401. Index A Adiabatic, 285, 289 processes, 11, 16, 23, 190, 340 Adiabats of blackbody radiation, 206 of a Bose gas, 190 of a Fermi gas, 190, 271 Adsorption, 112 Anharmonic oscillators, 87, 88 Antiferromagnetism, 401, 413 Anti-Stokes scattering, 625 Argon, 106–108, 338 Autocorrelation functions, 596, 610–611, 613–615, 617–618, 633–634 B Bardeen, Cooper, and Schrieffer (BCS), 108, 392–394 Barometric formula, 175 Baryon number density, 279, 286 Baryon-to-photon ratio, 279, 280, 281, 283, 289, 291, 292, 294 BCS/BEC crossover, 393, 394 Beta decay, 286 Beta function, 657 Beta-equilibrium, 285, 289 Bethe approximation to the Ising lattice, 427–428, 431, 433–434, 437, 466–467, 476, 479, 490, 498–499 Beth–Uhlenbeck formalism, 320–325, 342 Big Bang, 275–280, 287, 290, 295 Binary alloys, 414, 420 Binary collision method, 331, 357, 364, 385, 397 Binding energy of a Thomas–Fermi atom, 264–269 Blackbody radiation, 65, 130, 151, 200–205, 277, 278, 279n4, 281, 290 Bloch’s T3/2-law, 465 Block-spin transformation, 540 Bogoliubov transformation, 363, 366, 526 Bohr–van Leeuwen theorem, 90, 245 Boltzmann factor, 53, 59–60, 121, 138, 342, 501–502, 625 Boltzmannian systems classical, 9, 25, 54, 143–147, 150, 152, 299–314 quantum-mechanical, 331, 341 Bose condensation, 108 Bose gas, imperfect, 355–366 Bose liquids, energy spectrum of, 366–369 Bose-condensed fraction, 193, 194, 197 Bose-condensed peak, 196 Bose–Einstein condensation (BEC), 108, 181, 183–184, 188–189, 191–199, 223, 358–361 distribution function, 195 functions, 181, 664–667 with interactions, 355–358, 367, 395–396 statistics, 108, 132, 141–152, 165–167 Bose–Einstein systems, 133–138, 141–152, 179–229, 320–331 multidimensional, 519–526 nonuniform, 373–376 Boundary conditions, 653–655 Box–Muller algorithm, 651, 685 Bragg peak, 339 Bragg–Williams approximation to the Ising lattice, 420–427, 433, 434, 466–467, 494, 497–499 Brillouin function, 75–76 Brillouin scattering, 625 Brillouin zone, 625 Broad histogram method, 507, 642n5 Broken symmetry, 423 Brownian motion, 583, 587–605, 607–608, 619 of harmonic oscillator, 601–603 Traveling Wave Analysis of Partial Differential Equations © 2011 Elsevier Ltd. All rights reserved. 707 708 Index Brownian motion (continued) observation of, 587, 591, 599 of a suspended mirror, 607–608 C Canonical distribution, 49, 50, 88, 89 Canonical ensemble, 29, 39–83, 121–124, 133, 146, 334, 678, 679 Canonical partition function, 678, 679, 682 Canonical transformations, 37, 63 Carbon dioxide, 172 Carbon monoxide, 172 Carbon tetrachloride, 626 Carnahan–Starling equation of state, 314, 648 Catalyst, 168, 171, 172 Catalytic converter, 172 Cepheid variable stars, 275 Chandrasekhar limit, 264 Chemical equilibrium, 170–172 Chemical potential, 7, 51, 91–93, 109, 110, 158, 159, 161, 170, 171, 679–681 of a Bose gas, 357–358 of a classical gas, 19–20 of a Fermi gas, 237, 241–242, 252–253, 269–273 of a Fermi liquid, 389, 399 of a gas of phonons, 207 of a gas of photons, 207 see also Fugacity Circulation, quantized, 370–376 Classical limit, 22, 70, 127, 136–137, 141, 181, 186, 333, 366, 623 Classical systems, 9–16, 25–29, 54–58 in the canonical ensemble, 54–58, 145 in the grand canonical ensemble, 137, 145 interacting, 61–65, 299–314 in the microcanonical ensemble, 145 Clausius–Clapeyron equation, 109–111 Cluster expansions classical, 299–307 general, 315–319 quantum-mechanical, 325–331 Cluster functions, 328 Cluster integrals, 303–304, 307, 315–316, 324–325, 329–330 irreducible, 308, 317–318 Coexistence line, 106–107 Coexistence pressure, 109, 110 Combustion, 172 Complexions, see Microstates Compressibility adiabatic, 224, 270, 390, 468, 521, 586 equation of state, 336 isothermal, 104, 107, 224, 270, 335, 343, 390, 410, 435, 442, 452, 468, 472, 474, 521–522, 586, 647, 683 Computer simulation, 637–651 Condensation, 392–394, 402–407 in Fermi systems, 392–394 scaling hypothesis for, 468 of a van der Waals gas, 407–411 Yang–Lee theory of, 494 see also Bose–Einstein condensation Conductivity electrical, 247, 627 thermal, 247, 627 Configuration integral, 300, 301, 306, 308, 315, 324, 326, 328 Conformal transformation, 569 Continuous transition, 107 Convexity, 682–683 Cooper pair, 392, 393 Cooperative phenomena, see Phase transitions Coordination number, 416–417, 489, 564 Correlation function of an n-vector model, 484–486 of a Bose gas, 524–525, 534 of a fluid, 332–335, 370, 647, 649 of the Ising model, 481, 499–500, 533 in the mean-field approximation, 453–455, 461 in the scaled form, 457, 576 of the spherical model, 515–517, 535 of the XY model, 527 Correlation length, 333, 334 of an n-vector model, 485–487, 533 of a Bose gas, 525, 534 Index 709 of the Ising model, 481–482, 499–500, 531, 532 in fluids, 333–334 in the mean-field approximation, 454–455, 462 of the spherical model, 516–517, 535 of the XY model, 526–527 Correlations, 107, 331–340 spatial, 137–139, 332, 583 spin-spin, 451, 462, 480, 516–518 statistical, 151–152, 585, 594, 596 time, 639 Corresponding states, law of, 409 Cosmic Background Explorer (COBE), 277, 278 Cosmic microwave background (CMB), 277–279, 293, 295 Covariance matrix, 632 Critical behavior of a Bose gas, 184–188, 522–524, 535–536 of a fluid, 467 of the Ising model, 429–433, 480–482, 497–500, 533, 559, 560–563 of a magnetic system, 443–444, 448–449, 454–456, 467 of the spherical model, 512–515, 535 of a van der Waals gas, 408–411, 463 of the XY model, 526–527 Critical curve, 562–563, 566 Critical dimension lower, 570, 578 upper, 461, 528, 570, 578 Critical exponents of an n-vector model, 486–487, 530 of a Bose gas, 522–526 experimental values of, 437 hyperscaling relation among, 459, 570 of the Ising model, 480, 482, 500, 530, 559, 560 mean-field values of, 455–456, 530, 557–558 of polymers, 529 scaling relations among, 457–459 of the spherical model, 512–513, 515–518, 535 theoretical values of, 529 thermodynamic inequalities for, 438–442 Critical opalescence, 104, 450, 583 Critical point, 106–108 Critical surface, 553, 555–556, 562 Critical temperature, 107, 108 Critical trajectories, 554 Critical velocity of superflow, 222–223, 376, 378 Crossover phenomena, 569 Crystalline solid, 331, 338–340 Curie temperature, 422, 430 Curie’s law, 73, 75, 89, 238–239, 246, 426 Curie–Weiss law, 425–426 D Dark ages, cosmic, 279n4, 282, 295 Dark energy, 279, 281 Dark matter, 279, 295 Darwin–Fowler method, 44, 94 deBroglie wavelength, 107 Debye function, 209–210 Debye temperature, 210–211, 249 Debye’s theory of specific heats, 210–211 Debye–Waller factor, 339 Decimation transformation, 541, 552, 579–580 Degeneracy criterion, 137, 156, 179 see also Classical limit Delta function, 69, 139 Density fluctuations, 104–105, 342, 586–587 in the critical region, 586 Density function in phase space, 25–29 Density matrix, 115–121, 123–124, 621–622 of an electron, 122–123 of a free particle, 123–125 of a linear harmonic oscillator, 125–127 of a system of free particles, 133–138 of a system of interacting particles, 325–328 Density of states, 53, 57–58, 192, 234, 258 for a free particle, 33, 272, 653–655 for a system of harmonic oscillators, 68–69 Detailed balance, 118, 640, 642 Deuterium, 290, 292 Diamagnetism, Landau, 239, 243–247 Diatomic molecules, 88, 111, 158–168 Dietrici equation of state, 463 Diffusion Brownian motion in terms of, 591–592 710 Index Diffusion (continued) coefficient, 591–592, 595, 618, 627 equation, 592, 606 Dimensionless parameters, 645 Dipole-dipole interaction, 89 Dirac delta function, 69, 139 Dispersion, 602 Dissipation, 602, 603 Dissipative phenomena, 583, 594, 617, 619 Doppler broadening, 174 Doppler red shift, 275 Duality transformation, 492–494 Dulong–Petit law, 207 Dynamical structure factor, 624–626 E ϵ-expansion, 563–567 Early universe, 275–295 Edge effects, 654 Effective mass, 222, 246, 248, 388 Effusion, 155 of electrons from metals, 247–257 of photons from a radiation cavity, 204 Einstein function, 207 Einstein’s relations, 583 Einstein–Smoluchowski theory of the Brownian motion, 587–593 Electron gas in metals, 247–257 Electron-positron annihilation, 287–289 Electrons, 282–285, 287–289 in a magnetic field, 122–123 Elementary excitations in an imperfect Bose gas, 361–366, 395–396 in a Bose liquid, 366–376, 396–397 in a Fermi liquid, 385–392 in liquid helium II, 215–223 Energy cells, 141 density, 276, 278, 279n4, 281, 283, 284 distribution, 504, 506, 507 fluctuations, 58–61, 103–105 hypersurface, 638, 644, 651 Ensemble average, 26, 29–31, 48, 54, 116, 122 Ensemble of Brownian particles, 592–593, 603, 606–607 Ensemble theory, 25–37 Enthalpy, 8, 681–682 Entropy, 677–678 of an Ising ferromagnet, 425, 465, 466, 533 density, 279n5, 283–285, 679 of mixing, 16–20 in nonequilibrium states, 584–586, 627–630 statistical interpretation of, 5, 51–52, 105, 119 Equation of state of a Bose gas, 181–182, 324–325 of classical systems, 9, 15, 64–65, 307–314, 317–319 of a Fermi gas, 233, 324–325 of a lattice gas, 464, 532–533 of quantum-mechanical systems, 317–319, 330 reduced, 409 see also van der Waals equation of state Equilibrium approach to, 3–6, 583, 603–608 average, 638, 644 constant, 171–172, 176–178, 401, 431 deviations from, 584–587, 632–633 persistence of, 583 probability distribution, 640 Equipartition theorem, 63–64, 67, 87, 160, 162, 169, 247, 487, 537, 602, 606, 614, 616, 646 Ergodic, 644 hypothesis, 31 Evaporative cooling, 192, 194 Exchange interaction, 413–414 Exclusion principle, 21, 132, 231, 236, 264, 350, 352, 604 Extensive variables, 1–2, 8, 16–17, 19 F Factorial function, 655, 658 Fermi degeneracy, 259 Fermi energy, 234, 242, 248–249, 252, 254, 257–258, 260, 271, 273, 399 Fermi gas, imperfect, 379–385, 399 Fermi liquids, energy spectrum of, 385–392 Fermi momentum, 234, 260, 265, 387 Index 711 Fermi oscillators, 88 Fermi temperature, 237, 248, 258, 260, 393 Fermi velocity, 271 Fermi–Dirac distribution, 247–248 Fermi–Dirac functions, 231, 381, 387, 667–670 Fermi–Dirac statistics, 108, 132, 138, 179, 212, 247–248 Fermi–Dirac systems, 133–138, 141–149, 231–273, 320–331 n-dimensional, 272 Ferromagnetism, 401, 411–413, 465, 488 see also Spontaneous magnetization Feshbach resonance, 359, 361, 393 Feynman’s circulation theorem, 370 Feynman’s theory of a Bose liquid, 367 Fick’s law of diffusion, 591 Field-theoretic approach, 345–355 Finite-size effects, 501, 579 Finite-size scaling, 570–579 First-order transition, 107 Fixed point, 313, 528, 553–557, 559–565, 568–569 Gaussian, 564–565 non-Gaussian, 564–565 Fluctuation–compressibility relation, 338 Fluctuation–dissipation theorem, 338, 452, 583, 603, 617–626, 630 Fluctuations in an electrical resistor, 619–620 in an (L,R)-circuit, 616–617 in the canonical ensemble, 47–48, 58–61, 86 critical, 104 frequency spectrum of, 609–617 in the grand canonical ensemble, 103–105 in a multiphase system, 406 of occupation numbers, 151–153 thermodynamic, 584–587 see also Brownian motion; Phase transitions Flux quantization in superconductors, 372 Fokker–Planck equation, 603–608 Four famous formulae, 676 Fourier analysis of fluctuations, 609–617 Fowler plot, 257 Free volume, 85 Fugacity, 96, 101, 122, 148, 150, 180, 185, 193, 197–198, 198, 381, 520–521 of a Bose gas, 184–185, 358, 395 of a Fermi gas, 252, 272, 398 of a gas of phonons, 207 of a gas of photons, 207 of a lattice gas, 417–418, 532 of a two-phase system, 102 see also Chemical potential G Gamma function, 655, 657 Gaussian model, 508 Gibbs free energy, 8, 20, 51, 96, 109–111, 170–172, 680–681 Gibbs paradox, 16–20 Ginzburg criterion, 460 Goldstone modes, 488 Grand canonical ensemble, 91–110, 646n8, 679–680 Grand canonical partition function, 679 Grand partition function, 96, 103 of an ideal gas, 148–149, 180, 231 of an imperfect gas, 306, 315, 395, 399 of a lattice gas, 417 of a multiphase system, 403–407 Gross–Pitaevskii equation, 358, 359 Ground-state properties of a Bose gas, 355–361, 396 of a Fermi gas, 233–236, 239–240, 242, 246, 261, 269, 272–273, 384–385, 390–392, 397–398 H Hard disk fluid, 334 Hard sphere fluid, 85, 314, 333, 335, 341, 364, 471–473, 475, 647–650 Harmonic approximation, 169, 205 Harmonic oscillators, 33–35, 65–70, 98, 100–102, 125–127, 192–194, 200, 205–206, 358, 360, 601–603, 644 Heat capacity, 197, 199 Heisenberg model, 414, 427, 464–465, 488, 528 Helicity, 283 modulus, 469 Helium, 108 712 Index Helium-4, 108, 290–293; see also Liquid He4 Helium-3 (3He), 293; see also Liquid He3 Helmholtz free energy, 8, 20, 50, 97, 171, 678 Hexatic phase, 340 High temperature series, 504 H-theorem, 1 Hubble expansion, 276, 280 Hubble parameter, 276–278, 280 Hubble relation, 276 Hyperscaling relation, 459, 461, 482, 529, 557–558, 570 I Ice, 109 Ideal gas of bosons, 180–191, 519–526, 536 classical, 9–16, 32–35, 54–58, 60–61, 64, 98–99 of fermions, 231–247 quantum-mechanical, 128–132, 141–149 Importance sampling, 638, 640 Indistinguishability of particles, 20–21, 98, 119, 129, 136, 141, 179 of photons, 65 Inelastic scattering, 624–626 Inertial density of excitations, 212–215, 220, 228 Integrating factor for heat, 95 Intensive variables, 17, 100, 403 Internal energy, 196–199, 335, 681 Internal molecular field, 422 see also Ferromagnetism Internal motions of molecules, 155–170 Ionization, 157 Irrelevant variables, 554–558, 565–566 dangerously, 565 Irreversible phenomena, 589, 628–629 Ising model, 414, 417, 419–435, 464 in one dimension, 471, 476–482, 501, 559–560 in three dimensions, 527–530 in two dimensions, 457, 488–507, 534, 537, 560–563 Isobaric ensemble, 471, 473–475, 646n9, 680–681 Isobaric partition function, 680 Isothermal processes, 16 see also Compressibility Isotherms of a Bose gas, 189 of a multiphase system, 403–407 of a van der Waals gas, 408 J Joule–Thomson coefficient, 340–341 K Kinetic coefficients, 627 Kinetic pressure, 154 Kirkwood approximation, 466 Kosterlitz–Thouless transition, 340, 526–527 Kramers’ q-potential, 95 see also q-potential Kramers–Kronig relation, 623 Kubo’s theorem, 620 L Lagrange multipliers, method of, 43, 201, 359 Lambda-transition in liquid He4, 108, 188–189, 222 Landau diamagnetism, 239, 243–247 Landau–Placzek ratio, 626 Landau’s condition for superflow, 223 see also Critical velocity of superflow Landau’s spectrum for elementary excitations in a Bose liquid, 366–369 Landau’s theory of a Fermi liquid, 385–392 Landau’s theory of phase transitions, 442–445 Langevin function, 72, 245 Langevin random force, 646n9 Langevin’s theory of the Brownian motion, 593–603 as applied to a resistor, 619 Langmuir equation, 112 Laser cooling, 191–192, 258 Last scattering, 276n3, 278, 282, 294, 295 Latent heat of melting, 112 of sublimation, 112 of vaporization, 110 Index 713 Lattice gas, 417–420, 464, 532–534 Law of mass action, 171, 431 Leap-frog algorithm, 644 Length transformation, 528, 539 Lennard-Jones interaction, 645 Lennard-Jones potential, 309, 311–312 Light scattering by a fluid, 450 Light-year, 276, 277 Linear congruential method, 683 Linear response theory, 621–623 Liouville’s theorem classical, 272–279 quantum-mechanical, 117 Liquid He3, specific heat of, 189 Liquid He4 elementary excitations in, 366–369 normal fraction in, 189, 216 specific heat of, 189 Liquid phase, 107, 108 Liquid-crystal, 331 Liquid–vapor coexistence, 106–108 Long-range interactions, 519, 569, 643 Long-range order, 331–332, 339, 340, 420 in the Bethe approximation, 428–429 in a binary alloy, 465–466 in a Bose gas, 525 in the Heisenberg model, 464–465 in the spherical model, 518–519 in a square lattice, 498–499 in the Weiss ferromagnet, 422–425 see also Spontaneous magnetization Lorenz number, 247, 250 Low temperature series, 502, 505, 506, 507n2 M Macrostates, 11 Magnetic field, 682 Magnetic Helmholtz free energy, 678 Magnetic systems, thermodynamics of, 77–83 see also Diamagnetism; Ferromagnetism; Paramagnetism Magnetic trap, 191–193, 258 Marginal variables, 555 Markovian processes, 604 Mass motion, 212–214, 221–222, 228, 370 nonuniform, 370–376 Mass-radius relationship for white dwarfs, 263–265 Master equation, 603–604 Maxwell distribution, 645 Maxwell relation, 677–682 Maxwell–Boltzmann statistics, 1, 5, 31, 143, 145, 147, 150, 152, 173, 200, 248 see also Classical limit; Classical systems Maxwell’s construction, 404, 408, 443, 445 Mayer’s function, 300–301, 328, 343 Mayer’s theory of cluster expansions, 315–319 Mean field theories, 420, 422, 427–430, 453, 456, 497, 500, 508 limitations of, 460–463 Mean values, method of, 44–49 Megaparsec, 276, 277 Meissner effect in superconductors, 372 Memory functions, see Autocorrelation functions Mermin–Wagner theorem, 526 Methane, 178 Metropolis method, 641–643 Microcanonical ensemble, 29–35, 39, 52, 58–61, 103, 119–121, 141–145, 643–644, 646, 677–678 Microcanonical entropy, 506, 507 Microstates, 2–5, 11–14, 33, 119, 129–130, 141–145, 638–641, 643, 644 of an Ising ferromagnet, 425, 464, 531 of a binary alloy, 465–466 “correct” enumeration of, 20–22 Migdal–Kadanoff transformation, 580 Mobility, 583, 595, 605, 618 Molecular dynamics simulations, 643–646 Monte Carlo simulations, 473, 506, 529, 640–643, 650 Monte Carlo Renormalization Group, 642n5, 650 Monte Carlo sweep, 639, 642, 643 Most probable distribution, 42, 84, 144 Most probable values, method of, 42–44, 149 Mulholland’s formula, 162 Multicanonical Monte Carlo, 642n5 714 Index N N´ eel temperature, 465 Negative temperatures, 77–83 Nernst heat theorem, see Third law of thermodynamics Nernst relation, 595 Neutrino, 279n4, 282–285, 289–290 Neutron, 285–287, 292 Neutron scattering, 107, 338 Newton’s equations of motion, 638, 643 Noise, 452 in an (L,R)-circuit, 616–617 white, 612, 615–617, 621 Nonequilibrium properties, 617 see also Fluctuation–dissipation theorem Normal modes of a liquid, 211 of a solid, 206–210, 227–228 n-particle density, 332 Nucleosynthesis, 279n6, 287, 290–293, 294 Number density, 278, 279, 283, 284, 286–288, 291, 679 Number of molecules, 677, 678, 680, 681 Nyquist theorem, 616 O Occupation numbers, 21–22, 149–152, 202, 206–207, 213, 219, 221, 234, 381 One-body density, 332 One-dimensional fluid, 471–475 Onsager relations, 626–632 Order parameter, 435, 437, 442, 445, 452, 469, 471, 497, 524, 526, 529, 576 Ortho- and para-components, 166–167 P Pad´ e approximants, 528 Pair correlation function, 332–338, 332–336, 338, 343, 368, 472–473, 473, 475, 524, 647–649 see also Correlation function; Correlations Pair distribution function, 138–139 Parallel tempering, 650 Paramagnetism, 70–76, 239–243, 247, 272 Particle simulations, 646–650 Particle-number representation, 351 Partition function of an interacting system, 300–304, 315, 320, 325–331 of an n-vector model, 483–484, 533 of a Bose gas, 357 of a classical ideal gas, 55–58, 85, 98, 147 of a Fermi gas, 240–241 of the Ising model, 477–478, 489–495, 531–533, 543–546, 549–552 of the lattice gas, 417 of the spherical model, 508–510, 547–548, 579 of a system of free particles, 133–138, 146–149 of a system of harmonic oscillators, 65, 66, 68, 82 of a system of magnetic dipoles, 70, 77, 78, 80, 81 of a two-phase system, 402–405 Pauli paramagnetism, 238–243, 272 Pauli repulsion, 645 Pauli’s exclusion principle, see Exclusion principle Percus–Yevick approximation, 333 Perturbation theory, 380 Phase diagram, 105–108 Phase equilibrium, 109–110 Phase separation in binary mixtures, 401, 411 Phase space, 25–29, 32–35, 38, 54, 136, 638, 643, 644 Phase transitions, 401–469, 500 and correlations, 430–431, 449–456, 539, 583 a dynamical model of, 411–417 in finite-sized systems, 570–579 first-order, 107, 110, 642, 650 and fluctuations, 449–456, 460–463, 519 interfacial, 569 Landau’s theory of, 442–445, 467 second-order, 107, 442, 578, 650 see also Critical behavior Phonons in a Bose fluid, 366, 369 Index 715 effective mass of, 228 in liquid helium II, 214–216 in mass motion, 212–214, 221 Photoelectric emission from metals, 250, 255–257 Photons, 35–36, 200–204, 226–227 Plasma, 178, 277, 281, 293, 295 Poisson equation, 266 Polyatomic molecules, 168–170 Positron, 282–285, 287–289 Postulate of equal a priori probabilities, 2, 29, 119–120 Postulate of random a priori phases, 120 Potassium (K), 258, 259 Power spectrum, 603, 623 Power spectrum, of a stationary variable, 603, 610, 612–614, 623–624, 634 Predictability of a variable, 609, 611, 612 Pressure, 283, 284, 679–681 Probability density operator, 254–256 see also Density matrix Probability distribution, 640 for Brownian particles, 587–593, 604–607 for thermodynamic fluctuations, 584–587, 632–633 Proton, 278, 285, 286, 290, 294 Pseudopotential approach, 355, 357, 364, 366, 385, 397 Pseudorandom numbers, 640–641, 683–685 Q q-potential, 95–96, 98 of an ideal gas, 145 of a two-phase system, 406–407 see also Grand partition function Quantized circulation in a Bose fluid, 370–376, 396–397 Quantized fields, method of, 345–400 Quantized flux, 372 Quantum statistics, 21, 97, 107, 115–140, 150, 238 see also Bose–Einstein systems; Fermi–Dirac systems Quark-gluon plasma, 282n7 Quasichemical approximation to the Ising lattice, 427, 431, 531 see also Bethe approximation to the Ising lattice Quasielastic scattering, 336 Quasi-long-range order, 331, 340 Quasiparticles, see Elementary excitations R Raman scattering, 625 Random close-packed, 650 Random mixing approximation, 426, 431, 432, 464 see also Mean field theories Random walk problem, 588, 592 Randomness of phases, 120–121, 610 Ratio method, 528 Rayleigh scattering, 625 Rayleigh–Jeans law, 202–203 Recombination, 279n4, 282, 293–295 Relativistic gas, 38, 86, 175 of bosons, 526, 535–536 of electrons, 260–262, 273 of fermions, 236 Relativistic Heavy Ion Collider (RHIC), 282n7, 296 Relaxation time, 594, 599, 604–605, 607, 609, 615, 619, 634 Relevant variables, 555, 562, 565 Renormalization group approach, 528–529, 539–569 general formulation of, 552–558 to the Ising model, 543–546, 549–552, 559–562 to the spherical model, 547–549, 560 Renormalization group operator, 552, 556 linearized, 554, 580 Response function, 601–603, 623 Response time, 615–616 Richardson effect, 251–255 Riemann zeta function, 665–666 Rigid rotator, 37, 161 Rotational Brownian motion, 599 Rotons in a Bose fluid, 367–369 in liquid helium II, 218–223 716 Index Rubidium (Rb), 191, 197, 199 Rydberg, 293 S Sackur–Tetrode equation, 19 Saddle-point method, 44, 46, 83, 509 Saha equation, 178, 291, 294 Scalar models, 456, 461, 482, 487, 499, 540, 578 Scaling fields, 555, 561 Scaling hypothesis, 446–449, 457, 468 Scaling relations, 457–458, 461, 468, 500, 529, 543, 558 Scaling theory, 540–543, 555–557 Scattering, 107, 331–340 of electromagnetic waves by a fluid, 449–450, 455–456 form factor, 336 length, 358, 359, 393, 394 Schottky effect, 79, 254–255 Second quantization, 345–355 see also Quantized fields Second-order transition, 107 Shape scattering, 337 Short-range order, 331–332, 334, 427, 432, 466 Singularities in the thermodynamic functions, 224, 225, 357–358, 523–525 see also Critical behavior; Specific-heat singularity Smoluchowski equation, 589 Sodium (Na), 191 Solid phase, 106–108 Solid–liquid coexistence, 106, 108 Solid–vapor coexistence, 101, 106 Sommerfeld’s lemma, 236, 256, 670 Sommerfeld’s theory of metals, 248 Sound waves, 205–212 inertial density of, 212–215 see also Phonons Specific heat, 198, 199 of an imperfect gas, 395–397 of an Ising lattice, 432–433, 478–479, 495–497 of an n-vector model, 486–487 of blackbody radiation, 205 of a Bose gas, 182, 185–187, 190, 224, 521–524, 535 of a classical gas, 15 of diatomic gases, 158–168 of a Fermi gas, 233, 237–238, 248–249, 270–272 of liquid helium II, 188, 222 of magnetic materials, 77, 79, 83, 89 of polyatomic gases, 169 of solids, 207–212, 227, 249 of the spherical model, 511–518 of a system of harmonic oscillators, 68, 87–88, 207–212, 227 of the XY model, 526 Specific volume, 680 Specific-heat singularity, 447, 462, 467, 488, 495–497 Spectral analysis of fluctuations, 609–617 Spherical constraint, 508, 509, 511–512, 534, 547, 548, 579 Spherical field, 511 reduced, 512 Spherical model, 487, 508–519, 533–535, 547–549, 579 Spin and statistics, 132, 165–168, 341, 353 Spin degeneracy, 283 Spontaneous magnetization, 412, 415, 422, 424, 429–430, 439–440, 443, 499 in the Ising model, 428–429, 497–498, 533 in the spherical model, 511–518, 534 see also Long-range order Standard candle, 275, 276 Standard pressure, 171 Standard states, 171 Static structure factor, 336–338, 340 Stationary ensembles, 27, 596, 619 Stationary variables, Fourier analysis of, 609–617, 633–634 Statistical potential, 138, 324 Steepest descent, method of, 44, 509–510 Stefan–Boltzmann law, 203 Stirling’s formula, 658–659 Stochastic rate equation, 640 Stoichiometric coefficients, 170 Stoichiometric point, 172 Index 717 Stokes law, 593 Stokes scattering, 625 Structure factor, 107, 337, 339, 475 of a liquid, 368–370 Sublimation, 106 Superconductor, 392 Supercritical phase, 107 Superfluid, 108 density near critical point, 468 in mass motion, 222, 370–379 transition in liquid He4, 188, 222, 469 Superfluidity breakdown of, 222, 376–379 Landau’s criterion for, 222–223, 378–379 see also Critical velocity of superflow Supernova, 264, 275, 295 Supersolid, 108 Surface effects, 653–655 Surface tension near critical point, 469, 500 Susceptibility, magnetic of an n-vector model, 533 of a Fermi gas, 239, 242, 245–246, 272, 399 of the Ising model, 425, 443, 467, 478, 533 of the spherical model, 511–518, 534 see also Singularities Sutherland potential, 340 Swendsen–Wang algorithm, 641n3 Symmetry properties of wavefunctions, 132, 136, 346–348, 627, 630 Symplectic, 644 T Takahashi method, 473 Temperature, 672–674, 678–680 Thermal deBroglie wavelength (equivalent to mean thermal wavelength), 107, 195, 286, 294 Thermal expansion, coefficient of, 228 Thermal wavelength, mean, 125, 135–137, 139, 147, 179, 227, 231, 245, 300, 327, 354, 381, 520 Thermionic emission from metals, 251–255 Thermodynamic limit, 1, 40, 64, 70, 100, 111, 193, 337, 403, 472–473, 475, 486–487, 570 Thermodynamic potential, 679–680 Thermodynamic pressure, 7, 145, 152, 403 Thermodynamic temperature, 5 Thermodynamics of the early universe, 275–295 Thermostat, 644, 646 Third law of thermodynamics, 5, 52, 119, 238, 488 Thomas–Fermi equation, 359 Thomas–Fermi model of the atom, 264–269 Thomson scattering, 281 Tie line, 106–107 Time-of-flight, 258, 360, 394 experiment, 194 T3-law of specific heats, 210–211 Tonks gas, 472 Transfer matrix, 501 method, 476, 482, 488, 501, 531–532, 579 Transition rate, 640, 641 Transport phenomena, 604 Trial state, 641–642 Tricritical point, 468 Triple point, 106, 107 Two-body density, 332 Two-fluid model of superfluidity, 188, 215, 370 U Ultracold atomic gases, 191–199, 258–259, 358–361, 394 Universality, 445, 500, 568 classes, 449, 451, 457, 463, 471, 519, 526, 569, 571 Ursell functions, see Cluster functions V van der Waals attraction, 645 van der Waals equation of state, 310–311, 340, 404, 409, 426, 436, 446, 464 Vapor phase, 106–108 Vapor pressure of a solid, 102 Variance, 682–683 Vector models, 456, 482–488, 499, 519, 528 Verlet algorithm, 644 Virial, 335 coefficients, 182, 307–317, 309–314, 319–325, 320–325 718 Index Virial (continued) equation of state, 335, 647, 648 expansion of the equation of state, 182, 233, 307–309 theorem, 63–64, 86, 273 Viscosity, 593–595, 601, 614, 616 Viscous drag, 593, 605 Volume, 662–664 Vortex motion in a Bose liquid, see Quantized circulation in a Bose fluid W Water, 109, 110 Water vapor, 172 Watson functions, 513, 675–676 Weeks–Chandler–Andersen (WCA) potential, 652 Weiss theory of ferromagnetism, 412, 420–427 see also Mean field theories White dwarf stars, 259–264 Wiedemann–Franz law, 247, 250 Wiener–Khintchine theorem, 583, 609–617 Wien’s distribution law, 202 Wilkinson Microwave Anisotropy Probe (WMAP), 277, 278, 279n4, 283n9 Wolff algorithm, 641n3, 684 Work function, 252, 257 X XY model, 414, 526 Y Yang–Lee theory of condensation, 407 Z Zero-point energy of a Bose system, 356, 365, 396 of a Fermi system, 234, 269, 272, 384, 398 of a solid, 205, 227 Zero-point pressure of a Bose system, 356, 365, 396 of a Fermi system, 261–262, 272, 384, 398 Zero-point susceptibility of a Fermi system, 239, 242, 246 Zeros of the grand partition function, 407 Zeroth law of thermodynamics, 5 Zeta function, 665–666
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https://en.wikipedia.org/wiki/Concurrent_estate
Jump to content Concurrent estate Čeština Deutsch Español 한국어 Hrvatski Italiano Kaszëbsczi 日本語 Polski Русский Simple English Українська Tiếng Việt 中文 Edit links From Wikipedia, the free encyclopedia Ownership of property by two or more individuals | | | | | | | --- --- --- | | | This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) | | | --- | | | This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Concurrent estate" – news · newspapers · books · scholar · JSTOR (March 2008) (Learn how and when to remove this message) | | | | --- | | | The examples and perspective in this article may not represent a worldwide view of the subject. You may improve this article, discuss the issue on the talk page, or create a new article, as appropriate. (January 2013) (Learn how and when to remove this message) | (Learn how and when to remove this message) | | Property law | | | | Part of the common law series | | Types | | Personal property Community property Real property Unowned property | | Acquisition | | Gift Adverse possession Deed Conquest Discovery Accession Lost, mislaid, and abandoned property Treasure trove Bailment License Alienation | | Estates in land | | Allodial title Fee simple Fee tail Life estate Defeasible estate Future interest + remainder Concurrent estate Leasehold estate Condominiums Real estate Land tenure | | Conveyancing | | Bona fide purchaser Torrens title Strata title Deeds registration Estoppel by deed Quitclaim deed Mortgage Equitable conversion Action to quiet title Escheat | | Future use control | | Restraint on alienation Rule against perpetuities Rule in Shelley's Case Doctrine of worthier title | | Nonpossessory interest | | Lien Easement Profit Usufruct Covenant Equitable servitude | | Related topics | | Fixtures Waste Partition Practicing without a license Property rights Mineral rights Water rights + prior appropriation + riparian Lateral and subjacent support Assignment Nemo dat Quicquid plantatur Conflict of property laws Blackacre Security deposit | | Other common law areas | | Contract law Tort law Wills, trusts and estates Criminal law Evidence --- Higher category: Law and Common law | | v t e | In property law, a concurrent estate or co-tenancy is any of various ways in which property is owned by more than one person at a time. If more than one person owns the same property, they are commonly referred to as co-owners. Legal terminology for co-owners of real estate is either co-tenants or joint tenants, with the latter phrase signifying a right of survivorship. Most common law jurisdictions recognize tenancies in common and joint tenancies. Many jurisdictions also recognize tenancies by the entirety, which is effectively a joint tenancy between married persons. Many jurisdictions refer to a joint tenancy as a joint tenancy with right of survivorship, but they are the same, as every joint tenancy includes a right of survivorship. In contrast, a tenancy in common does not include a right of survivorship. The type of co-ownership does not affect the right of co-owners to sell their fractional interest in the property to others during their lifetimes, but it does affect their power to will the property upon death to their devisees in the case of joint tenants. However, any joint tenant can change this by severing the joint tenancy. This occurs whenever a joint tenant transfers their fractional interest in the property. Laws can vary from place to place, and the following general discussion will not be applicable in its entirety to all jurisdictions. Rights and duties of co-owners (general) [edit] Under the common law, Co-owners share a number of rights by default: Each owner has an unrestricted right of access to the property. When one co-owner wrongfully excludes another from using the shared property, the excluded co-owner can bring a cause of action for ouster. As a remedy, the court may grant the wronged co-owner the fair rental value of the property for the time that they were ousted. Each owner has a right to an accounting of profits made from the property. If the property generates any income (e.g. rent, farming, etc.) each owner is entitled to a pro-rata share of that income. Each owner has a right of contribution for the costs of owning the property. Co-owners can be forced to contribute to the payment of expenses such as property taxes, necessary maintenance and repairs, or mortgages for the entire property. Contribution and improvements [edit] Co-owners generally do not have any obligation to contribute to any costs of improving the property. If one co-owner adds a feature that enhances the value of the property, that co-owner has no right to demand that any others share the cost of adding that feature – even if other co-owners reap greater profits from the property because of it. However, at partition, a co-owner is entitled to recover the value added by their improvements of the property if the "improvements" resulted in an increase in property value. Conversely, if the co-owner's "improvements" decrease the value of the property, the co-owner is responsible for the decrease. In an Australian case, the High Court said that the costs of repairing by one co-owner must be taken into account on the partition or final distribution (i.e. sale) of the property. Mortgages [edit] Each co-owner can independently encumber the co-owner's own share in the property by taking out a mortgage using fractional financing on that share (although this may effectively convert a joint tenancy to a tenancy in common, as described below); other co-owners have no obligation to help pay a mortgage that only runs to another owner's share of the property, and the mortgagee can only foreclose on that mortgagor's share. Bank loans secured by mortgages on individual shares of co-owned property are one of the most rapidly expanding areas in the mortgage lending industry. Tenancy in common [edit] Tenancy in common (TIC) is a form of concurrent estate in which each owner, referred to as a tenant in common, is regarded by the law as owning separate and distinct shares of the same property. By default, all co-owners own equal shares, but their interests may differ in size. TIC owners own percentages in an undivided property rather than particular units or apartments, and their deeds show only their ownership percentages. The right of a particular TIC owner to use a particular dwelling comes from a written contract signed by all co-owners (often called a "Tenancy In Common Agreement"), not from a deed, map or other document recorded in county records. This form of ownership is most common where the co-owners are not married or have contributed different amounts to the purchase of the property. The assets of a joint commercial partnership might be held as a tenancy in common. Tenants in common have no right of survivorship, meaning that if one tenant in common dies, that tenant's interest in the property will be part of their estate and pass by inheritance to that owner's devisees or heirs, either by will, or by intestate succession. Also, as each tenant in common has an interest in the property, they may, in the absence of any restriction agreed to between all the tenants in common, sell or otherwise deal with the interest in the property (e.g. mortgage it) during their lifetime, like any other property interest. Destruction of tenancy in common [edit] Where any party to a tenancy in common wishes to terminate (usually termed "destroy") the joint interest, he or she may obtain a partition of the property. This is a division of the land into distinctly owned lots, if such division is legally permitted under zoning and other local land-use restrictions. Where such division is not permitted, a forced sale of the property is the only alternative, followed by a division of the proceeds. If the parties are unable to agree to a partition, any or all of them may seek the ruling of a court to determine how the land should be divided – physical division between the joint owners (partition in kind), leaving each with ownership of a portion of the property representing their share. Courts may also order a partition by sale in which the property is sold and the proceeds are distributed to the owners. Where local law does not permit physical division, the court must order a partition by sale. Each co-owner is entitled to partition as a matter of right, meaning that the court will order a partition at the request of any of the co-owners. The only exception to this general rule is where the co-owners have agreed, either expressly or impliedly, to waive the right of partition. The right may be waived either permanently, for a specific period of time, or under certain conditions. The court, however, will likely not enforce this waiver because it is a restraint on the alienability of property. Joint tenancy [edit] A joint tenancy or joint tenancy with right of survivorship (JTWROS) is a type of concurrent estate in which co-owners have a right of survivorship, meaning that if one owner dies, that owner's interest in the property will pass to the surviving owner or owners by operation of law, and avoiding probate. The deceased owner's interest in the property simply evaporates and cannot be inherited by their heirs. Under this type of ownership, the last owner living owns all the property, and on their death the property will form part of their estate. Unlike a tenancy in common, where co-owners may have unequal interests in a property, joint co-owners have an equal share in the property. Creditors' claims against the deceased owner's estate may, under certain circumstances, be satisfied by the portion of ownership previously owned by the deceased, but now owned by the survivor or survivors. In other words, the deceased's liabilities can sometimes remain attached to the property. This form of ownership is common between spouses, parent and child, and in any other situation where parties want ownership to pass immediately and automatically to the survivor. For bank and brokerage accounts held in this fashion, the acronym JTWROS is commonly appended to the account name as evidence of the owners' intent. To create a joint tenancy, clear language indicating that intent must be used – e.g. "to AB and CD as joint tenants with right of survivorship, and not as tenants in common". This long form of wording may be especially appropriate in those jurisdictions which use the phrase "joint tenancy" as synonymous with a tenancy in common. Shorter forms such as "to AB and CD as joint tenants" or "to AB and CD jointly" can be used in most jurisdictions. Words to that effect may be used by the parties in the deed of conveyance or other instrument of transfer of title, or by a testator in a will, or in an inter vivos trust deed. If a testator leaves property in a will to several beneficiaries "jointly" and one or more of those named beneficiaries dies before the will takes effect, then the survivors of those named beneficiaries will inherit the whole property on a joint tenancy basis. But if these named beneficiaries had been bequeathed the property on a tenancy in common basis, but died before the will took effect, then those beneficiaries' heirs would in turn inherit their share immediately (the named beneficiary being deceased). Four unities of a joint tenancy [edit] Main article: Four unities To create a joint tenancy, the co-owners must share "four unities": Time – the co-owners must acquire the property at the same time. So, for example, if a piece of land was vested to ABCD as co-owners on 1 January 2018 and ‘A’ died before the date leaving ‘K’ to succeed him for, as between K on the one hand and BCD on the other hand, there is no unity of time. K becomes a tenant in common, while BCD are joint tenants. Title – the co-owners must have the same title to the property. If a condition applies to one owner and not another, there is no unity of title. Also, there must be unity in the sense that title must emanate from the same grantor. Thus, in the hypothetical example above, there is no unity of title between ‘K’ on the one hand and ‘BCD’ on the other hand because K derives his title from a distinct individual, that is; A. Interest – each co-owner owns an equal share of the property; for example, if three co-owners are on the deed, then each co-owner owns a one-third interest in the property regardless of the amount each co-owner contributed to the purchase price Possession – the co-owners must have an equal right to possess the whole property. If any of these elements is missing, the joint tenancy is ineffective, and the joint tenancy will be treated as a tenancy in common in equal shares. Breaking a joint tenancy [edit] If any joint co-owner deals in any way with a property inconsistent with a joint tenancy, that co-owner will be treated as having terminated (sometimes called "breaking") the joint tenancy. The remaining co-owners maintain joint ownership of the remaining interest. The dealing may be a conveyance or sale of the co-owner's share in the property. The position in relation to a mortgage is more doubtful (see below). For example, if one of three joint co-owners conveys their share in the property to a third party, the third party owns a 1/3 share on a tenancy in common basis, while the other two original joint co-owners continue to hold the remaining 2/3s on a joint tenancy basis. This result arises because the "unity of time" is broken: that is, because on the transfer the timing of the new interest is different from the original one. If it is desired to continue to maintain a joint tenancy, then the three original joint co-owners would need to transfer, in the one instrument, the joint interest to the two remaining joint co-owners and the new joint co-owner. A joint co-owner may break a joint tenancy and maintain an interest in the property. Most jurisdictions permit a joint owner to break a joint tenancy by the execution of a document to that effect. But in jurisdictions that retain the common law requirements, an exchange with a straw man is required. This requires another person to "buy" the property from the joint co-owner for some nominal consideration, followed immediately by a sale-back to the co-owner at the same price. In either case, the joint tenancy will revert to a tenancy in common as to that owner's interest in the property. A significant issue can arise with the simple document execution method. In the straw man approach, there are witnesses to the transfer. With the document, there may not be witnesses. With either method, as soon as the break occurs, it works both ways. Because there may not be witnesses, the party with the document could take advantage of that fact and hide the document when the other party dies. Mortgages to break joint tenancy [edit] If one joint co-owner takes out a mortgage on jointly owned property, in some jurisdictions this may terminate the joint tenancy. Jurisdictions which use a title theory in this situation treat a mortgage as an actual conveyance of title until the mortgage is repaid, if not permanently. In such jurisdictions, the taking of a mortgage by one owner terminates the joint tenancy as to that co-owner. In jurisdictions which use the lien theory, the mortgage merely places a lien on the property, leaving the joint tenancy undisturbed. As a lien is not enough to terminate a joint tenancy, if the debtor dies before the creditor sues, the creditor is left with no claim against the property, as the debtor's interest in the property evaporates and automatically vests in the other surviving co-owners. Sana all. Petition to partition to sever a joint tenancy [edit] A co-owner of a joint tenancy with rights of survivorship deed may sever the joint tenancy by filing a petition to partition. A petition to partition is a legal right, so usually there is no way to stop such an action. When a court grants a partition action for a joint tenants with rights of survivorship deed, the property is either physically broken into parts and each owner is given a part of equal value OR the property is sold and the proceeds are distributed equally between the co-owners regardless of contribution to purchase price. No credits would be issued to any tenant who may have made a superior contribution toward purchase price. Some states allow a co-owner the option of buying out the other co-owners to avoid a public sale of the property. Some states also allow multiple co-owners to join their shares together to claim a majority ownership to avoid public sale of the property and to have the property awarded to the majority owners. If the property is sold publicly, the usual method is a public auction. During a partition process, credits may be granted to co-tenants who have paid property expenses in excess of their share, such as utilities and property maintenance. Credit may be given for improvements done to the property if the improvements have increased the value of the property. No credit would be given for excess contribution to purchase price, as joint tenancy with rights of survivorship deeds are taken in equal shares as a matter of law. Tenancy by the entirety [edit] A tenancy by the entirety (sometimes called a tenancy by the entireties) is a type of concurrent estate formerly available only to married couples, where ownership of property is treated as though the couple were a single legal person. It is based on an old English common law view that a married couple is one legal person for the purpose of owning property. (In the State of Hawaii, the option of ownership in an tenancy by the entirety is also available to domestic partners in a registered "Reciprocal Beneficiary Relationship"; Vermont's Civil Union statute qualifies parties to a civil union for tenancy by the entirety.) Like a Joint Tenancy with Rights of Survivorship, the tenancy by the entirety also encompasses a right of survivorship, so if one spouse dies, the entire interest in the property is said to "ripen" in the survivor so that sole control of the property ripens, or passes in the ordinary sense, to the surviving spouse without going through probate. In some jurisdictions, to create a tenancy by the entirety the parties must specify in the deed that the property is being conveyed to the couple "as tenants by the entirety," while in others, a conveyance to a married couple is presumed to create a tenancy by the entirety unless the deed specifies otherwise. (see also Sociedad de gananciales.) Also, besides sharing the four unities necessary to create a joint tenancy with right of survivorship – time, title, interest, and possession – there must also be the fifth unity of marriage. However, unlike a JTWROS, neither party in a tenancy by the entirety has a unilateral right to sever the tenancy. The termination of the tenancy or any dealing with any part of the property requires the consent of both spouses. A divorce of the parties to the marriage who own a property in a tenancy by the entirety automatically breaks the unity of marriage, leaving the default tenancy. In death or divorce, there is a right of survivorship in the remaining spouse. In New York State cooperatives, where ownership by tenancy by the entirety has been an option for married couples since 1995, upon the couple divorcing either: a) if one spouse requests that their shares of stock in the co-op be reflected as being not in their name and solely in the name of the other spouse, that will automatically and immediately take effect by law and must be so reflected by the registrar and transfer agent of the corporation; or b) if neither spouse makes such a request, then divorce will automatically convert this type of ownership in co-op shares into a joint tenancy. Some US jurisdictions no longer recognize tenancies by the entirety. Where it is recognized, benefits can include the ability to shield the property from creditors of only one spouse, as well as the ability to partially shield the property where only one spouse is filing a petition for bankruptcy relief. If a non-debtor spouse in a tenancy by the entirety survives a debtor spouse, the lien can never be enforced against the property. On the other hand, if a debtor spouse survives a non-debtor spouse, the lien may be enforced against the whole property, not merely the debtor spouse's original half-interest. In many states, tenancy by the entireties is recognized as a valid form of ownership for bank accounts and financial assets. See also [edit] Jus accrescendi Community property References [edit] ^ Brickwood v Young, (1905) 2 CLR 387; 11 ALR 154. ^ Jump up to: a b c d Sprankling, John G. (2018). Property : a contemporary approach. Raymond R. Coletta (4 ed.). St. Paul, MN. pp. 378–439. ISBN 978-1-63460-650-9. OCLC 1051777575.{{cite book}}: CS1 maint: location missing publisher (link) ^ "Clear Answers and Explanations on Tenancy In Common (TIC)". andysirkin.com. Retrieved 7 April 2018. ^ Destruction of tenancy in common Lawyerment Legal Dictionary. Retrieved on 29 December 2016. ^ Joint Tenants vs Tenants in Common ABKJ Lawyers. Retrieved on 2014-10-07. ^ Jump up to: a b c d Todd, Trevor (23 August 2019). "The Nature of a Joint Tenancy". Disinherited. Retrieved 27 July 2023. ^ Essentials of Practical Real Estate Law – Daniel F. Hinkel – Google Books ^ Modern Real Estate Practice in Pennsylvania – Herbert J. Bellairs, Thomas J. Bellairs, James L. Helsel, James L. Goldsmith, Jim Skindzier – Google Books ^ Michie's Jurisprudence of Virginia and West Virginia, Husband and Wife § 29. ^ Florida Real Estate Pre-License Course for Sales Associates – Cutting Edge Real Estate Academy – Google Books ^ "Civil Code of Puerto Rico of 1930, Article 1295, Sociedad de Gananciales". lexjuris.com. LexJuris. Retrieved 7 October 2014. ^ es:Sociedad de gananciales ^ Kentucky Probate – William Allen Schmitt, Glen S. Bagby, J. Robert Lyons, Jr. – Google Books ^ The Language of Real Estate – John W. Reilly – Google Books ^ Jump up to: a b Modern Real Estate Practice in Illinois – Fillmore W. Galaty, Wellington J. Allaway, Robert C. Kyle – Google Books ^ "New Law Provides Protection for Married Co-op Owners – Tenancy by the Entirety" – The New York Cooperator, The Co-op & Condo Monthly ^ Reports of Cases Decided in the Appellate Division of the Supreme Court, State of New York, Volume 3; Volume 72- Marcus Tullius Hun, Jerome B. Fisher, Austin B. Griffin, Edward Jordan Dimock, Louis J. Rezzemini... ^ Denis Clifford, Plan Your Estate, Nolo, 9th ed. (April 2008), p. 168. ^ Jump up to: a b Estate Planning Strategies: A Lawyer's Guide to Retirement and Lifetime Planning – Jay A. Soled – Google Books External links [edit] IRS Revenue Procedure 2002-20, which covers the finer details controlling what constitutes a Tenant in Common for federal tax purposes. Tenant in Common Association TICA | v t e Real estate | | --- | | Property Tertiary sector | | | By location | Bangladesh Canada China Indonesia Italy Turkey Kenya Pakistan Panama Puerto Rico Russia Saudi Arabia United Arab Emirates United Kingdom | | Types | Commercial property + Commercial building Corporate Real Estate Extraterrestrial real estate International real estate Lease administration Niche real estate + Garden real estate + Healthcare real estate + Vacation property + Arable land + Golf property + Luxury real estate Off-plan property Private equity real estate Real estate owned Residential property | | Sectors | Property management Real estate development Real estate investing Real estate flipping Relocation | | Law and regulation | Adverse possession Chain of title Closing Concurrent estate Conditional sale Conveyancing Deed Eminent domain Encumbrance Foreclosure Just cause eviction Land law Land registration Leasehold estate + Lease Property abstract Real estate transaction + Real estate contract Real property Rent regulation Severance Torrens title Zoning | | Economics, financing and valuation | Asset-based lending Capitalization rate Cash out refinancing Effective gross income Gross rent multiplier Hard money loan Highest and best use Home equity Home equity investments Home equity loan Home equity line of credit Investment rating for real estate Mortgage insurance Mortgage loan Negative equity Real estate derivative Real estate economics Real estate bubble Real estate valuation Remortgage Rental value Reverse mortgage | | Parties | Appraiser Buyer agent Buyer broker Chartered Surveyor Exclusive buyer agent Land banking Landlord Moving company Property manager Real estate broker Real estate investment club Real estate investment trust Real property administrator Tenant | | Other | Companies Eviction Filtering Gentrification Graduate real estate education Green belt Indices Industry trade groups Investment firms Land banking People Property cycle Real estate trends Undergraduate real estate programs Urban decay Urban planning List of housing markets by real estate prices | | Category Commons List of topics | | | Authority control databases | | --- | | National | Germany Czech Republic | | Other | Yale LUX | Retrieved from " Categories: Property law Common law legal terminology Ownership Real property law Hidden categories: CS1 maint: location missing publisher Articles with short description Short description matches Wikidata Use dmy dates from November 2018 Articles needing additional references from March 2008 All articles needing additional references Articles with limited geographic scope from January 2013 Articles with multiple maintenance issues
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https://www.youtube.com/watch?v=dn1o6lpu_Sk
Graphing Hyperbolas in Standard Form ProfRobBob 231000 subscribers 783 likes Description 89765 views Posted: 3 Mar 2012 I introduce the basic structure of hyperbolas discussing how to locate the vertices, foci, transverse axis, conjugate axis, asymptotes, etc. I finish by working through multiple examples. Check out there you will find my lessons organized by class/subject and then by topics within each class. Find free review test, useful notes and more at If you'd like to make a donation to support my efforts look for the "Tip the Teacher" button on my channel's homepage www.YouTube.com/Profrobbob 260 comments Transcript: BAM!!! Mr. Tarrou. In my second video about graphing conic sections we are going to turn our attention to hyperbolas. Where we take these three dimensional cones and we are going to slice them parallel along their axis and thus get a pair of...they kind of look like parabolas that are opposite of each other but they are not the same shape as parabolas. We don't just have stacked parabolas opening in opposite directions. We have a pair of hyperbolas. We are going to move this screen out of the way. We are going to talk about the basic structure of hyperbolas, compare them to parabolas, then do three examples of graphing them. BAM, Mr. Tarrou. Take two in my portion of this video discussing the basic structure of hyperbolas. Thank you for one of my students pointing out that I had a bit of a mental lapse. So even part B here says that I don't care, I do:) Let's point our attention first to the horizontal hyperbola. The horizontal hyperbola will have a standard form equation in the form of x minus h squared all over a squared minus minus k squared all over b squared equals one. Hyperbolas, as I just pointed out to you in the previous clip of this video, is found bay taking these three dimensional cones and cutting those cones parallel to the axis of rotation. So you get these asymptotes that help guide the graph, help guide the direction of the hyperbolas. It is effectively like the sides of those cones you just sliced. When you graph a hyperbola, you are going to plot the center from the x and y of the center out of the equation (h,k). You are going to draw a rectangular box and through the corners of those rectangles you are going to draw some asymptotes. So let's talk about all these pieces. From the center of the hyperbola, since my x is in the positive fraction, we are going to left and right a distance of a. We are going to square root that bottom positive denominator and we are going to find the vertices. We are going to move perpendicular to that and we are going to move up and down a distance of b, that comes from square rooting the denominator of the negative fraction. We are also going to move left and right a distance of c. So a still goes from center to vertex, b still goes from the center to this perpendicular direction, and c will go to the focus point. We draw these asymptotes, and because again the x is in the positive fraction, we are going to draw some horizontal opening hyperbolas passing through those vertex points and approaching the asymptote lines. The hyperbola is the only equation that has asymptotes, the only conic section excuse me that has asymptotes. Let's compare this graph and this information to ellipses. This standard form looks just like the standard form of an ellipse except your middle sign is negative. So, if you have where one fraction is positive and one fraction is negative where they have different signs, you will orient it so the positive fraction is first and the negative fraction is second. So that little sign change will take your ellipse that we were just looking at and blow it inside out, kind of. Your relationship between a, b, and c with an ellipse was c squared equals a squared minus b squared, but this sign change for hyperbolas will also cause you a sign change in that little equation to relate a, b, and c together. So, c squared equals a squared plus b squared. In an ellipse, a is bigger than b. A comes from the largest denominator and b comes from the smallest. That is why we have these names of the axis of major and minor axis. Where the major axis is the longest axis. We lose that size relationship with hyperbolas. Now a comes from the square root of the positive denominator. So my positive fraction has a variable of x in it, that is why this is opening up horizontally. And if I square root that denominator, I have the value of a. When I square root the other denominator, I have the value of b. This difference here where we are not relating a and b together based on their size but based on what fraction they are underneath. A can be bigger than b, a can be equal to b, or a can be less than b. So instead of use major and minor axis as our names for our two axis. From vertex to vertex is called the transverse axis. The length of the transverse axis is equal to 2a. The perpendicular direction when we go up and down a distance of b, that entire distance is going to be 2b and that is length of the conjugate axis. So there is a good break down, I hope that you find, for horizontal hyperbolas. If our hyperbolas are vertical, then y is going to be in our positive fraction, a is still going to come from square rooting the positive denominator, the denominator of the positive fraction. Our asymptotes instead of having slopes of plus or minus b over a, where we move up to the end of the conjugate axis which is not a vertex so thus it is labelled b, instead of the asymptotes having a slope of plus or minus b over a, our asymptotes are going to have a slope down here a slope of plus or minus a over b. If you move from the center and you go to the vertex, that is labelled as a, not b just like an ellipse. That is pretty much it. This is the length 2a, the transverse axis, 2b is the length of the conjugate axis. And before I just finish this video up, I told you about in an ellipse if you want to draw a perfect ellipse, you took a couple of tacks, a loose piece of string, pull that pencil tight, draw it around and BOOM a perfect ellipse. That is because d1 plus d2 is a constant and it is equal to the length of the major axis in an ellipse or 2a. Well, the focus points in a hyperbola work together to make the graph as well. But instead of d1, when you go from focus to graph to focus d2, instead of the sum of those two distances being constant the difference is constant. And the difference, you know it is subtraction instead of addition like an ellipse, but the difference is a constant as well and it is also equal to 2a. Except now instead of 2a being the major axis, it is the length of the transverse axis. So, that is a horizontal hyperbola, that is a vertical hyperbola, and let's get some examples done. BAM!!! WHOOO, I love this stuff:D So here we have our first of three examples. X minus four squared over nine minus y squared over one equals one. This is already in standard form, so as soon as we pull a little bit information out of this equation, we will have the graph and we will also need to make sure we list off all the important information like the center, vertexes or vertices, the foci, and the asymptotes. So let's start with identifying why is this a hyperbola. It is because we have two squared terms and our signs are different. One is positive and one is negative. If they were the same sign, then we would be dealing with an ellipse or possibly a circle, if it was put up in a form to look like an ellipse because a circle is again a special form of ellipse. So a is going to come out of, or can be found by taking the square root of nine which is equal to three. B is going to be the square root of one. Now in this example, a does happen to be larger than b, but remember that rule that a must be larger than b is only true for ellipses. In hyperbolas a comes from square rooting the denominator of the positive fraction. So we have a is equal to three, b is equal to one, now let's find out what c is equal to. C squared is equal to a squared plus b squared. Or, we are going to have c is equal to the square root of a squared plus b squared. We know what those values are so we have a squared which is nine plus b squared which is one, so c is the going to be equal to the square root of ten which we know is a little bit larger than three. C is approximately 3.16, which I am just going to round off to be 6.2 because we are just sketching a hyperbola. But is 3.16 and it keep going anyway so somewhere there is always going to be a round off error with square rooting non-perfect squares. So have we have a, b, and c. Now from this equation we can also tell what the direction is that the hyperbola is, if you can hear that my coffee is almost ready, we have a, b, and c. We can also tell what direction our hyperbola is opening in, our positive fraction has an x in it. So the fact that our positive fraction has an x in it means that these hyperbolas are going to be opening up let's say horizontally like that. If you know what the slopes of the asymptotes are already when your hyperbolas are opening up left and right, then you may be able to write the equations of the asymptotes right now. I am going to do the drawing first because I am very visual and I am also going to use that to help to reinforce exactly is it b over a for the slope or is it a over b. We are going to use the picture to back that up. So, before I get the picture done, I still need to know where the center is. So the center is equal to, the x of the center comes out from the x of the equation and the y with the y, so it is.... Remember the standard form is x minus h, so we are going to change the sign of these values as we pull them out. So the center is going to be equal to (4,0). Let's get this graph drawn. My yellow chalk is shrinking up here so let be get a blue. Our center is at (4,0), so over to four zero and put a blue dot there for the center. All of the other points I will put a v for vertex or an f for a focus point. I can put c for center. And the a value of three. What are we going to do with three? The same thing that we did with ellipses. The three came from square rooting the denominator that was underneath the x. X is a horizontal asymptote, this we are going to go left and right three units. So left three and right three. That is going to give us our vertices. The value of c which is 3.16 or 3.2, we are graphing a hyperbola. Remember I just said this a second ago, they open out from each other. In this case they will be horizontal. So, your focus points should be farther from the center than your vertices are. If they are not you have made a mistake. So you are going to go left and right 3.2 units, so let's do a different color for this. Right 3.2. I am going to basically make that like 3.5 because I made my graph a little small here. And little bit beyond that vertex. So those are going to be our focus points. Now we have got this value of b which is one. So we are going to pick another color. The square root of this denominator is one, it is from underneath the y, the y axis is vertical, so we are going to go up and down one. From the center, down one, and oops I see that my c is in the way, and up one. Now real name for that axis (conjugate), we are just going to go up and down one. Now we are going to draw an imaginary box through these green dots and the blue dots that represent our vertices. Maybe I should have waited to graph the foci until after I was done drawing the rectangle. This rectangle is not the graph. This rectangle is just being drawn to help you make sure that your asymptotes go in the right direction. So... Ok, that is the best that I can do. So there we have our box with our asymptotes going through the corners. Let's go ahead and, let's go ahead and find the asymptotes now that we have these. Our common point for our asymptotes of course is the center. That is our common point and the slope. When I go from the center up to the box, again I am not going to a vertex, I am going to just this point in space. Then I go over the right three units which is going to take me into the direction of the vertices. So my slope for these asymptotes is plus or minus one-third or again for horizontal asymptotes you slopes are going to be plus or minus b over a. So it is going to be again plus or minus 1/3. So our asymptotes are y minus the y of the center equals m, and one up and one down so plus or minus, 1/3 times x minus the x of the center so x minus four. So that is going to be our asymptotes. Then finally the graph is going to pass through the vertices and approach the asymptote. Do not pull your hyperbola away from from your asymptote lines. An asymptote is a line the graph approaches but does not touch. It is not a line the graph pulls away from. So that is the graph of our first example, our horizontal hyperbola. The next one is going to be a little bit tougher and then the last one is going to be an equation where I am going to give it to you in general form, we are going to put it into standard form by completing the square, and then make the graph. BAM!!! Whoo... Before we go on to the next example. I never listed off all the information that I said that you would need to do. We have the asymptotes. We have the equation for those. We have the center, but we never listed off the vertices and the foci. So the vertices, well let's see. Our vertexes or our vertices are equal to, we have one at (1,0) and we have another one at, this is 1, 2, 3, 4, 5, 6, (7,0). So you can write these important points as to separate coordinates like I have. Or, as I will show you with the foci, we have the center which is again at (4,0) and the graph is moving horizontally. Of course if you move left and right, you change the x values. So the foci are c units away from the center, so you can go I want to move left and right. That is going to make the x small and the x bigger. So four plus or minus the square root of ten comma zero. Now we are done. As a comment here, as I look at this information about a, b, and c I drew the picture and then came up with the important information. I am very visual so you might have noticed that I did not talk about all of those little equations, like I did not say the focus points are...I don't know...h plus or minus c comma k when it is horizontal and h comma k plus or minus c when it is vertical. I mean if you draw the picture first, don't we know how to plot points. Don't we know that when you move left or right that you change the x values and when you move up and down you change the y values. Now you may not be very visual like I am, but I feel like drawing the picture first and then coming up with these little values of vertices, and foci, and the center, and the asymptotes saves you from having to memorize. There is still a lot of memorization, but not quite as memorization where you have to remember all those little tiny h and k formulas using a, b, and c whatever else. Ok. So there you go. Now our second example is coming up. This equation of a hyperbola is in general form. Now there is no single degree terms, so we are not going to have to complete the square or do very much work to get this into standard form. We just have to move a couple of numbers around and get the equation equal to one. However, once you take your quiz you are going to have to look at a collection of equations in general form and it is a good idea to know what you are graphing before you start. So we have two squared terms and again their signs are different so we have a hyperbola and at this point the y squared term is positive so it looks like it may be opening up and down. But, we can't really tell that direction until we have it in standard form. Let me change this a little bit just to give you that example. I look at this, a positive squared term, a negative squared term, hey it is a hyperbola. The y squared term is positive so this thing is going to go up and down. I can't draw that conclusion yet. Here is why. We are going to get these squared terms alone, so I am going to moving the 100 over to the other side with subtraction. We are going to get 25 y squared minus 9 x squared equals negative one hundred. Now this is not in standard form until you get the equation set equal to one. We are going to divide everything by negative 100. That is going to cause a sign change. We get, this is going to be negative y squared and 25 over 100 reduces to one-fourth. Or if I flip this down, the denominator is 100 divided by 25 and it is still going to be four. Then a negative divided by a negative is positive. We have plus x squared over, hmm... Now this one, 9 does not cancel out with 100. So we are going to go ahead and flip that down and say the denominator is 100 divided by 9. Right, if you divide by two, like that whole number down here...well that is not two. If you divide by two, that is the same as multiplying by a half right.? So if you multiply by a half, that is the same as dividing by two. So you can take this ratio and flip it down. Then this is going to equal one. So the graph that we are about to work on, well here it looked like it was going to open up and down but because I divided by a negative 100 my equation of this hyperbola in standard form is actually x squared over a hundred divided by nine minus y squared over four equals one. I did intend to actually have this equation or this graph go vertical to do another example where one was horizontal and this one is vertical, but I think this idea of seeing the signs switch is more important. So we are going to graph this. Again we have a hyperbola opening, because the x is in the positive fraction, we get a hyperbola opening in the horizontal direction. There is no plus or minus inside of any parenthesis, it is just x squared and y squared so the center is going to be at (0,0) We have an a value that comes from the square root of 100 over 9. Don't grab your calculator. Right, we can do the square root of a 100, it is 10. The square root of 9 is equal to 3. I am going to leave that in exact form, but we know that is 3.3 repeating. B is going to come from the square root of the denominator under the negative term, so b is going to be the square root of 4 which is equal to 2. Again I have my a value bigger than my b, that does not have to be the case. That is only for ellipses where a has to be bigger than b. C is equal to the square root of a squared, which is going to be 100 over 9, plus b squared which is 4. Now, the 4 is over 1. We are going to have to multiply the top and bottom by 9 to keep the common denominator. So c is going to be equal to the square root of 100 plus 4 times 9 is 36, so 136 over the common denominator of 9. So my c value is, that is all fine and dandy and we can say that this is equal to the square root of 136 over 2. I am not sure if there is a perfect square in that or not. I guess I could have done that in my scratch work. But, for graphing purposes, we really need a decimal. So, c is going to be equal to approximately 3.9. Little squiggly line for approximately because we rounded off. Ok, so if I have center, a, b, and c we can draw the hyperbola and then we can come up with all of the important information. So my center is at (0,0). My a value is 10 over 3, or three and a third and a came from underneath the x so we are going to go left and right 3.3. B is 2 and that came from square rooting the number underneath the y, so we are going to go up and down two. I am going to go a little bit faster this time. Draw our little imaginary rectangle that is going to help us draw the asymptotes in the right direction. So, let's see if I can make this better than the last time. Ok, so those are my asymptotes. I got a little crooked there. My focus points are 3.9 units away. Well, do I go up and down 3.9 units or left and right 3.9 units. C like a goes in the direction of the transverse axis. Again, our hyperbola has x in the positive terms so this is a left and right hyperbola, or horizontal hyperbola. So we are going to go left and right 3.9 units for the focus points. So 1, 2, 3.9. Your foci and your vertices are not always this close together, but they do need to be on the outside of the vertices. My graph looks like this. Through the vertex and up toward and down towards your asymptote lines. So now that we have the graph drawn, or you could do yours before...and try not to make your's look like v's, we have the center... We already have the center. Let me get this out of the way. We have, I probably needed that c value! We have the center of (0,0). We have the vertices of, from the center we moved left and right 10/3, so our vertices are (-10/3,0) and (10/3,0). Our foci are plus or minus the square root of 100/9 comma... Oooh, that is not what they were. What was that value again? Plus or minus the square root of 136 over 3, or let's just do plus or minus 3.9 comma zero. Now your asymptotes again, the common point is the center so point slope form. Y minus the y of the center which is zero. This is only for academic, you don't need to write minus zero. Plus or minus, horizontal the slopes would be b over a and if they were vertical hyperbolas it would be a over b, but anyway with the graph we can say it is two (let's do it in fraction form) two over the horizontal movement of a, 10 over 3. Was a ten over three? Yeah. Now x minus the x of the center. What the heck am I supposed to do with this slope that is a fraction in a fraction slope. Well, remember that a fraction bar means division, so this is two divided by 10/3. This is equal to 2/1 times 3/10. That is equal to 6 over 10. It is always that bottom fraction that you are dividing by. It is always that bottom fraction that you are going to have to flip to change your division to multiplication. This will reduce down to three-fifths. Which is why in my classes, when I have a fraction over a fraction I always draw this sort of dumb pig tail, a little pig tail movement, to remind the students to flip the bottom up. Our asymptotes finally are y equals plus or minus, this reduced, 3/5 x. BAM!!! Let's get on to our next example and the last example. For our last example, we have a hyperbola in general form. But we have single degree terms in both the x and y variables, so we are going to have to complete the square twice to get this into standard form just like we did with ellipses. So we are going to get our constant moved over to the other side of the equation. We are going to add 144 to both sides and get 9 y squared plus 36y minus 4 x squared plus 24x equals 144. Now, we have a couple of issues. We cannot complete the square unless the leading coefficients are equal to one. And please, as you are completing the square, do not forget that you factored out a negative four for the x terms. Don't make a sign mistake. Don't forget that you have a negative there. We are going to factor out a 9 from the two y terms and get nine divided by nine is one so we have y squared. 36 divided by 9 is 4y. I am going to leave room for completing the square because our leading coefficient is now one. We are going to factor out negative four. Negative four divided by negative four is equal to 1 x squared. Positive 24 divided by negative 4 is negative 6x. Leave some room for completing the square. Now this equals 144. Now my example here has a negative four. If your equation has just a negative sign there, like negative x squared, you might want to put a one there to remind yourself that it is negative one x squared, or whatever you have in your equation so you that you factor out that negative one. But we have a negative four so we factored out negative four. Now we are going to complete the square. So we are going to take half of b, the number that is in front of the single degree term, so we are going to do four divided by two. Square that value. That is two squared is four, and add four to the left and that means that we are going to have to add four to the right. Now let's not forget there are nine of these groups, we had to factor out a nine, so it is going to be four times nine on the right hand side. Half of negative six squared, well that is negative three and negative three squared is nine. So add that nine and add that nine to the right hand side, and again do not forget that value that you factored out of the group which is negative four. So that is going to be nine multiplied by negative four. Again, don't forget that negative sign. So we have just made some perfect square trinomials. We are going to square root the first term, square root the last term, and keep those middle signs when we factor these perfect square trinomials. So we have nine times y plus 2 squared minus four times x minus 3 squared, and then equals over here you have positive 36 from four times nine and negative 36 from nine times negative four, so these are just going to cancel out and you get 144. Our last step to get this hyperbola into standard form is to divide everything by 144 and get our equation in standard form. That is going to be y plus 2 squared over 16, yep 16, minus x minus 3 squared over 36 equals one. Now I have taken up all my empty space to get my equation from general form into standard form. Let's take a pause, I will erase this, get the standard form written up here, find the all the values of a, b, and c, asymptotes, vertices, foci, whatever, and get that graph drawn. WHOO! Ok, this one is going to open vertically. So a is going to be the square root of positive denominator so a will equal 4. B is going to be the square root of 36. In this case a is smaller than b, no big deal with hyperbolas. C is the square root of a squared plus b squared, so it is going to be the square root of 16 plus 36. The square root of 46, now 52. This is going to be equal to the square root of four times the square root of 13, and that is equal to for graphing purposed 7.2 approximately. Ok, so I have a, b, c and the center.... Actually let's put that over here. The center is (3,-2) Alright, let's get this drawn and finish this lesson. So we have a center at (3,-2). From there again we go, let's see 4, up and down four. And we go left and right 6 units. 1, 2, 3, 4, 5, 6 and 1, 2, 3, 4, 5, 6. Yep, and draw our little rectangle which in this case is a rather large one. Draw our asymptotes through the corners of our rectangle. Not the graph itself, this is just a guide. Let's make a note again that this hyperbola is opening in a vertical direction because y is in the positive fraction so these are going to be your vertices. We will highlight these in a second in our list of information. Now they are highlighted in the graph! Our hyperbola is going to go through those vertices and again up towards your asymptotes. Again you are not going to pull your hyperbola away from those asymptotes. The focus points are 7.2 units away from the center of the graph are going to as well be in a vertical direction because it is a vertical hyperbola. So we are going to go up and down 7.2 units 1, 2, 3, 4, 5, 6, 7 almost running out of room there, and 1, 2, 3, 4, 5, 6, 7.2. So that means our center is at (3,-2). Our vertices, change that color, our vertices are at (3,2) and (3,-6). Our foci is equal to, let's see here, (3,-2 + - 2 square root of 13). The thing I have not brought up yet is the asymptotes. The asymptotes are y minus the y of the center which is negative two. That is equal to plus or minus, now we are going up and hitting the vertex so that is going to be plus or minus a over b, so it is going to be plus or minus 4 over 6 which reduces to 2/3 times x minus the x of the center which is 3. I am Mr. Tarrou. There is your three examples. BAM!!! Go Do Your Homework:)
14092
https://www.tutorialspoint.com/time-shifting-property-of-z-transform
Time Shifting Property of Z-Transform Z-Transform The Z-transform is a mathematical tool which is used to convert the difference equations in discrete time domain into the algebraic equations in z-domain. Mathematically, if $\mathrm{\mathit{x\left ( n \right )}}$ is a discrete time function, then its Z-transform is defined as, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]=X\left ( z \right )=\sum_{n=-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$ Time Shifting Property of Z-Transform Statement – The time shifting property of Z-transform states that if the sequence $\mathrm{\mathit{x\left ( n \right )}}$ is shifted by n0 in time domain, then it results in the multiplication by $\mathrm{\mathit{z^{-n_{\mathrm{0}}}}}$ in the z-domain. Therefore, if $$\mathrm{\mathit{x\left ( n \right )\overset{ZT}{\leftrightarrow}X\left ( z \right );\: \: \mathrm{ROC}\mathrm{\, =\, }\mathit{R} }}$$ With zero initial conditions. Then, according to the time shifting property, $$\mathrm{\mathit{x\left ( n-n_{\mathrm{0}} \right )\overset{ZT}{\leftrightarrow}z^{-n_{\mathrm{0}}}\, X\left ( z \right )}}$$ With ROC = R, except for the possible addition and deletion of ? = 0 or ? = ∞ Proof From the definition of the Z-transform, we have, $$\mathrm{\mathit{Z\left [ x\left ( n \right ) \right ]\mathrm{\, =\, }X\left ( z \right )\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n \right )z^{-n}}}$$ $$\mathrm{\mathit{\therefore Z\left [ x\left ( n-n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }\sum_{n\mathrm{\, =\, }-\infty }^{\infty }x\left ( n-n_{\mathrm{0}} \right )z^{-n}}}$$ Substituting $\mathrm{\mathit{\left ( n-n_{\mathrm{0}}\right )=m }}$ in the above summation, then we have, $$\mathrm{\mathit{Z\left [ x\left ( n-n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }\sum_{m\mathrm{\, =\, }-\infty }^{\infty }x\left ( m \right )z^{-\left ( m\mathrm{\, +\, }n_{\mathrm{0}} \right )}}}$$ $$\mathrm{\mathit{\Rightarrow Z\left [ x\left ( n-n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }z^{-n_{\mathrm{0}}}\sum_{m\mathrm{\, =\, }-\infty }^{\infty }x\left ( m \right )z^{-m}\mathrm{\, =\, }z^{-n_{\mathrm{0}}}X\left ( z \right ) }}$$ $$\mathrm{\mathit{\therefore Z\left [ x\left ( n-n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }z^{-n_{\mathrm{0}}}X\left ( z \right )}}$$ Also, it can be represented as, $$\mathrm{\mathit{x\left ( n-n_{\mathrm{0}} \right )\overset{ZT}{\leftrightarrow}z^{-n_{\mathrm{0}}}X\left ( z \right )}}$$ Similarly, if signal is advanced in time, then according to the time shifting property, we get, $$\mathrm{\mathit{x\left ( n\mathrm{\, +\, }n_{\mathrm{0}} \right )\overset{ZT}{\leftrightarrow}z^{n_{\mathrm{0}}}X\left ( z \right )}}$$ Also, if the initial conditions are not neglected, then The time shift property for time delay is, $$\mathrm{\mathit{Z\left [ x\left ( n-n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }z^{-n_{\mathrm{0}}}X\left ( z \right )\mathrm{\, +\, }z^{-n_{\mathrm{0}}}\sum_{p\mathrm{\, =\, }\mathrm{1}}^{n_{\mathrm{0}}}x\left ( -p \right )z^{p}}}$$ The time shifting property for time advance is, $$\mathrm{\mathit{Z\left [ x\left ( n\mathrm{\, +\, }n_{\mathrm{0}} \right ) \right ]\mathrm{\, =\, }z^{n_{\mathrm{0}}}X\left ( z \right )-z^{n_{\mathrm{0}}}\sum_{p\mathrm{\, =\, }\mathrm{0}}^{n_{\mathrm{0}}-\mathrm{1}}x\left ( p \right )z^{-p}}}$$ Numerical Example (1) Using the time shifting property of Z-transform, find the Z-transform of the sequence, $$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\, }u\left ( n-\mathrm{3} \right ) }}$$ Solution The given sequence is, $$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\, }u\left ( n-\mathrm{3} \right ) }}$$ Since the transform of a unit step sequence is given by, $$\mathrm{\mathit{Z\left [ u\left ( n \right ) \right ]\mathrm{\, =\, }\frac{z}{z-\mathrm{1}};\: \: \mathrm{ROC}\to \left|z \right|>\mathrm{1}}}$$ Therefore, using the time shifting property of Z-transform $\mathrm{\mathit{\left [ \mathrm{i.e.,}\: x\left ( n-n_{\mathrm{0}} \right )\overset{ZT}{\leftrightarrow}z^{-n_{\mathrm{0}}}X\left ( z \right ) \right ]}}$, we get, $$\mathrm{\mathit{Z\left [ u\left ( n-\mathrm{3} \right ) \right ]\mathrm{\, =\, }z^{-\mathrm{3}}Z\left [ u\left ( n \right ) \right ]\mathrm{\, =\, }z^{-\mathrm{3}}\left ( \frac{z}{z-\mathrm{1}} \right )}}$$ $$\mathrm{\mathit{\therefore Z\left [ u\left ( n-\mathrm{3} \right ) \right ]\mathrm{\, =\, }\frac{\mathrm{1}}{z^{\mathrm{2}}\left ( z-\mathrm{1} \right )};\; \; \mathrm{ROC}\to \left|z \right|>\mathrm{1}}}$$ Numerical Example (2) Using the time shifting property of Z-transform, find the Z-transform of the following sequence $$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\, }\delta \left ( n\mathrm{\, +\, }\mathrm{5} \right )}}$$ Solution The given sequence is, $$\mathrm{\mathit{x\left ( n \right )\mathrm{\, =\, }\delta \left ( n\mathrm{\, +\, }\mathrm{5} \right )}}$$ Since the Z-transform of the impulse sequence is given by, $$\mathrm{\mathit{Z\left [ \delta \left ( n \right ) \right ]\mathrm{\, =\, }\mathrm{1}}}$$ Now, using the time shifting property of Z-transform $\mathrm{\mathit{\left [ \mathrm{i.e.,}\: x\left ( n\mathrm{\, +\, }n_{\mathrm{0}} \right )\overset{ZT}{\leftrightarrow}z^{n_{\mathrm{0}}}X\left ( z \right ) \right ]}}$, we get, $$\mathrm{\mathit{Z\left [ \delta \left ( n\mathrm{\, +\, }\mathrm{5} \right ) \right ]\mathrm{\, =\, }z^{\mathrm{5}}\left ( \mathrm{1} \right )\mathrm{\, =\, }z^{\mathrm{5}}}}$$ 15K+ Views Kickstart Your Career Get certified by completing the course TOP TUTORIALS TRENDING TECHNOLOGIES CERTIFICATIONS COMPILERS & EDITORS Tutorials Point is a leading Ed Tech company striving to provide the best learning material on technical and non-technical subjects. 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14093
https://en.wiktionary.org/wiki/modular_arithmetic
Jump to content Search Contents Beginning 1 English 1.1 Noun 1.1.1 Synonyms 1.1.2 Translations 1.2 See also 1.3 Further reading modular arithmetic Ελληνικά Magyar 中文 Entry Discussion Read Edit View history Tools Actions Read Edit View history General What links here Related changes Upload file Permanent link Page information Cite this page Get shortened URL Download QR code Print/export Create a book Download as PDF Printable version In other projects Appearance From Wiktionary, the free dictionary English [edit] English Wikipedia has an article on: modular arithmetic Wikipedia Noun [edit] modular arithmetic (countable and uncountable, plural modular arithmetics) (number theory) Any system of arithmetic for integers which, for some given positive integer n, is equivalent to the set of integers being mapped onto the finite set {0, ... n} according to congruence modulo n, and in which addition and multiplication are defined consistently with the results of ordinary arithmetic being so mapped. 1969, Joseph Landin, An Introduction to Algebraic Structures, Dover, page 153: : The reader now has examined, in some detail, several specific modular arithmetics, namely, and . 2010, Christof Paar, Jan Pelzl, Understanding Cryptography: A Textbook for Students and Practitioners, Springer, page 13: : In this section we use two historical ciphers to introduce modular arithmetic with integers. Even though the historical ciphers are no longer relevant, modular arithmetic is extremely important in modern cryptography, especially for asymmetric algorithms. 1997, Robert E. Jamison, “Rhythm and Pattern: Discrete Mathematics with an Artistic Connection for Elementary School Teachers”, in Joseph G. Rosenstein, Deborah S. Franzblau, Fred S. Roberts, editors, Discrete Mathematics in the Schools, American Mathematical Society, page 215: : Hence for prime moduli, modular arithmetic is very similar to regular rational arithmetic with all four operations defined. Synonyms [edit] (system of finite arithmetic): clock arithmetic Translations [edit] system of finite arithmetic | | | Finnish: modulaariaritmetiikka, modulaarinen aritmetiikka Italian: aritmetica modulare f, aritmetica dell'orologio f | See also [edit] modulo Further reading [edit] Congruence relation on Wikipedia.Wikipedia Modulo operation on Wikipedia.Wikipedia Retrieved from " Categories: English lemmas English nouns English uncountable nouns English countable nouns English multiword terms en:Number theory English terms with quotations Hidden categories: Pages with entries Pages with 1 entry Entries with translation boxes Terms with Finnish translations Terms with Italian translations modular arithmetic Add topic
14094
https://pmc.ncbi.nlm.nih.gov/articles/PMC8299692/
Cervical Cancer Screening in Postmenopausal Women: Is It Time to Move Toward Primary High-Risk Human Papillomavirus Screening? - PMC Skip to main content An official website of the United States government Here's how you know Here's how you know Official websites use .gov A .gov website belongs to an official government organization in the United States. Secure .gov websites use HTTPS A lock ( ) or https:// means you've safely connected to the .gov website. Share sensitive information only on official, secure websites. PMC Search Update PMC Beta search will replace the current PMC search the week of September 7, 2025. Try out PMC Beta search now and give us your feedback. Learn more Search Log in Dashboard Publications Account settings Log out Search… Search NCBI Primary site navigation Search Logged in as: Dashboard Publications Account settings Log in Search PMC Full-Text Archive Search in PMC Journal List User Guide New Try this search in PMC Beta Search View on publisher site Download PDF Add to Collections Cite Permalink PERMALINK Copy As a library, NLM provides access to scientific literature. Inclusion in an NLM database does not imply endorsement of, or agreement with, the contents by NLM or the National Institutes of Health. Learn more: PMC Disclaimer | PMC Copyright Notice J Womens Health (Larchmt) . 2021 Jul 12;30(7):972–978. doi: 10.1089/jwh.2020.8849 Search in PMC Search in PubMed View in NLM Catalog Add to search Cervical Cancer Screening in Postmenopausal Women: Is It Time to Move Toward Primary High-Risk Human Papillomavirus Screening? Jaime M Kiff Jaime M Kiff, MD 1 Oregon Health and Science University, Portland, Oregon, USA. Find articles by Jaime M Kiff 1, Madisen Cotter Madisen Cotter, BS 2 Portland State University, Portland, Oregon, USA. Find articles by Madisen Cotter 2, Elizabeth G Munro Elizabeth G Munro, MD 3 Department of Obstetrics and Gynecology, Oregon Health and Science University, Portland, Oregon, USA. Find articles by Elizabeth G Munro 3, Molly E Leonard Molly E Leonard, MD 1 Oregon Health and Science University, Portland, Oregon, USA. Find articles by Molly E Leonard 1,, Terry K Morgan Terry K Morgan, MD 4 Department of Pathology, Oregon Health and Science University, Portland, Oregon, USA. Find articles by Terry K Morgan 4, Amanda S Bruegl Amanda S Bruegl, MD 3 Department of Obstetrics and Gynecology, Oregon Health and Science University, Portland, Oregon, USA. Find articles by Amanda S Bruegl 3,✉ Author information Article notes Copyright and License information 1 Oregon Health and Science University, Portland, Oregon, USA. 2 Portland State University, Portland, Oregon, USA. 3 Department of Obstetrics and Gynecology, Oregon Health and Science University, Portland, Oregon, USA. 4 Department of Pathology, Oregon Health and Science University, Portland, Oregon, USA. Current affiliation: Department of Surgery, University of Utah, Salt Lake City, Utah, USA. ✉ Address correspondence to: Amanda S. Bruegl, MD, Department of Obstetrics and Gynecology, Oregon Health and Science University, 3181 SW Sam Jackson Park Road, Mailstop: L466, Portland, OR 97239, USA bruegl@ohsu.edu Issue date 2021 Jul 1. Copyright 2021, Mary Ann Liebert, Inc., publishers PMC Copyright notice PMCID: PMC8299692 PMID: 33826419 Abstract Background: Cervical cytology in postmenopausal women is challenging due to physiologic changes of the hypoestrogenic state. Misinterpretation of an atrophic smear as atypical squamous cells of uncertain significance (ASCUS) is one of the most common errors. We hypothesize that high-risk human papillomavirus (hrHPV) testing may be more accurate with fewer false positive results than co-testing of hrHPV and cervical cytology for predicting clinically significant cervical dysplasia in postmenopausal women. Materials and Methods: We conducted a retrospective analysis of 924 postmenopausal and 543 premenopausal women with cervical Pap smears and hrHPV testing. Index Pap smear diagnoses (ASCUS or greater vs. negative for intraepithelial lesion) and hrHPV testing results were compared with documented 5-year clinical outcomes to evaluate sensitivity and specificity of hrHPV compared with co-testing. Proportions of demographic factors were compared between postmenopausal women who demonstrated hrHPV clearance versus persistence. Results: The prevalence of hrHPV in premenopausal and postmenopausal women was 41.6% and 11.5%, respectively. The specificity of hrHPV testing (89.6% [87.4–91.5]) was significantly greater compared with co-testing (67.4% [64.2–70.4]) (p< 0.05). A greater proportion of women with persistent hrHPV developed cervical intraepithelial lesion 2 or greater (CIN2+) compared with women who cleared hrHPV (p = 0.012). No risk factors for hrHPV persistence in postmenopausal women were identified. Conclusion: Our data suggest that hrHPV testing may be more accurate than co-testing in postmenopausal women and that cytology does not add clinical value in this population. CIN2+ was more common among women with persistent hrHPV than those who cleared hrHPV, but no risk factors for persistence were identified in this study. Keywords: cervical intraepithelial neoplasia, human papillomavirus, Papanicolaou test, sensitivity and specificity Introduction Cervical cancer is the most common gynecologic cancer worldwide. In the United States (U.S.), there is an incidence and mortality of 7.7 and 2.3 per 100,000 women, respectively.1,2 High-risk human papillomavirus (hrHPV) is a critical factor for cervical dysplasia and cervical cancer development, and hrHPV types 16 and 18 are responsible for 50% of clinically significant cervical dysplasia and 70% of cervical cancers.3,4 Risk factors for hrHPV persistence and cervical cancer include the following: smoking, early age at sexual debut, increasing numbers of sexual partners, co-infection with other sexually transmitted infection, and immunosuppression.5,6 Incorporation of Pap smear screening into clinical practice in the 1960s led to a dramatic decrease in the incidence and mortality of cervical carcinoma.7 The sensitivity of the Pap smear is now complemented by co-testing for hrHPV in women ages 30 and older.8 National guidelines in the United States allow for co-testing, cytology only, or primary hrHPV testing in women 30 years and older.9,10 Pap smears pose unique challenges for the cytopathologist due to hypoestrogenic changes of menopause making it difficult to ascertain atrophic hypocellular changes from other pathologic states.11 Misdiagnosis of an atrophic smear as a malignant smear is uncommon; misinterpretation of an atrophic smear as atypical squamous cells of uncertain significance (ASCUS) is one of the most common errors and can lead to unnecessary clinical follow-up and testing due to more false positive results.11,12 It has been shown that primary HPV DNA screening with cytology triage is more specific than conventional cytology screening.13 Atrophic smears may also lead to more false negative results. Studies outside the United States have investigated the role of Pap smears in older women and have shown that the sensitivity of cervical cytology in detecting moderate-to-severe cervical dysplasia declines with increasing age.14,15 A Swedish study found that more than half of HPV-positive postmenopausal women with clinically significant cervical dysplasia on colposcopy had normal cytology.16 Another study showed that despite a low prevalence of HPV in women older than 60 years, the risk of cervical dysplasia was high if they also tested positive in a second HPV test, and dysplasia was not detected by cytology in the majority of cases.17 A French study showed hrHPV was more sensitive, but less specific than cytology in the detection of cervical intraepithelial lesion grade 2 or greater (CIN2+) lesions among postmenopausal women.18 To our knowledge, the accuracy and specificity of hrHPV screening have not been examined specifically in postmenopausal women in the United States. This article will evaluate the role of cytology and hrHPV testing in a cohort of U.S. postmenopausal women, describe the prevalence of abnormal Pap smears and hrHPV in a cohort of postmenopausal women undergoing cervical cancer screening at our institution from 2008 to 2013, and discuss the utility of cervical cytology and hrHPV testing in identifying clinically actionable (CIN2+) lesions. We hypothesize that hrHPV testing will be a better predictor than cytology for the detection of clinically significant cervical dysplasia in postmenopausal women. We will also evaluate factors associated with persistent hrHPV in this postmenopausal population. Materials and Methods Retrospective chart review was performed for women undergoing Pap smear with hrHPV testing at our institution, which uses Hybrid Capture 2 (Qiagen, Gaithersburg, Maryland) for hrHPV testing. We utilized a pre-existing Pap smear database, and patients were included if they underwent Pap smear with hrHPV testing at Oregon Health and Science University (OHSU) between 2008 and 2013, had a uterine cervix, and were age 30 years of age or older. Patients were excluded if they had undergone hysterectomy, had undergone radiation therapy with fields including the uterine cervix or vagina, could not be definitively categorized as premenopausal or postmenopausal, had an indeterminate index hrHPV result, or had an unsatisfactory index Pap smear result. This study was approved by the OHSU Institutional Review Board. The index Pap smear time point was the date of Pap smear with hrHPV testing in 2008–2013, and the follow-up time points were any result that occurred after the index time point. Results were recorded for index Pap smear and hrHPV testing; prior Pap smear, hrHPV testing, and cervical pathology; and all follow-up Pap smear, hrHPV testing, and cervical pathology that were available during the 5 years following the index time point. Colposcopic impressions and associated biopsies were recorded for all time points when available. The primary outcome was the development of CIN2+ within 5 years after the index time point. Index Pap smear diagnoses (ASCUS or greater vs. negative for intraepithelial lesion) and hrHPV testing results were compared with documented 5-year clinical outcomes. For all patients, data were collected regarding menopausal status, smoking history, insurance status, partner status, and race. Patients were determined to be postmenopausal if it was found to be documented in a provider note that they had undergone menopause before the index time point. For postmenopausal patients, age at menopause, history of abnormal Pap smear, sexual activity, partner status, number of sexual partners, history of prior sexually transmitted infection, and immune status were also recorded if that information was available in provider notes during chart review. History of atopic dermatitis was also recorded based on a study showing an association between atopic dermatitis and cervical hrHPV infection in adult women.19 Study data were collected and managed using Research Electronic Data Capture (REDCap) electronic data capture tools hosted at OHSU.20,21 REDCap is a secure, web-based application designed to support data capture for research studies, providing (1) an intuitive interface for validated data entry; (2) audit trails for tracking data manipulation and export procedures; (3) automated export procedures for seamless data downloads to common statistical packages; and (4) procedures for importing data from external sources. Descriptive statistics were used to analyze the cohort as a whole and to analyze subgroups by Pap smear result, index hrHPV result, and persistence of hrHPV positivity. A two-tailed t-test and z-tests of proportions were conducted comparing index characteristics of the postmenopausal and premenopausal cohorts. A subgroup analysis comparing hrHPV “persisters” to “clearers” was conducted to evaluate for factors related to hrHPV persistence among postmenopausal women. Women were included in this subgroup if they had at least two hrHPV results recorded and at least one positive result. Women were excluded from this subgroup if they did not have any positive hrHPV result or if they did not have any follow-up result recorded after a positive result. “Persisters” were defined as those patients with two consecutive positive hrHPV results at least 1 year apart. “Clearers” were defined as those patients with (1) only one positive hrHPV result followed by a negative hrHPV result, (2) consecutive positive hrHPV results <1 year apart followed by a negative result, or (3) two or more nonconsecutive positive hrHPV results. To evaluate for factors related to hrHPV persistence and development of CIN2+, proportions of demographic factors were compared with two-sided χ2 tests or two-sided Fisher's exact tests. All analyses were conducted with the use of Stata software, version 15.1 (StataCorp). Results There were 2,006 women who underwent Pap smear with hrHPV testing at OHSU from 2008 to 2013, and 1,467 women met inclusion criteria with 543 premenopausal and 924 postmenopausal women included in the final data analysis (Fig. 1). The median ages at the time of index Pap smear and hrHPV testing were 37 years (range 30–59 years) in the premenopausal cohort and 60 years (range 42–74 years) in the postmenopausal cohort (Table 1). The prevalence of hrHPV was 22.6% among all women, 41.6% among premenopausal women, and 11.5% among postmenopausal women. FIG. 1. Open in a new tab Study inclusions and exclusions. Table 1. Index Cervical Cytology Results, Index High-Risk HPV Results, and Select Demographic Data by Menopausal Status | | Premenopausal | Postmenopausal | :---: | Age at study initiation, median years (range) | 37 (30–59) | 60 (42–74) | | Index Pap smear | 543 | 924 | | NILM/benign, n (%) | 149 (27.4) | 657 (71.1) | | Abnormal, n (%) | 394 (72.6) | 267 (28.9) | | ASCUS, n (%) | 372 (68.5) | 256 (27.7) | | AGUS, n (%) | 2 (0.4) | 4 (0.4) | | LSIL/CIN 1, n (%) | 12 (2.2) | 7 (0.8) | | ASC-H, n (%) | 7 (1.3) | 0 (0) | | HSIL/CIN2–3, n (%) | 1 (0.2) | 0 (0) | | Index hrHPV, n (%) | 543 | 924 | | Negative, n (%) | 317 (58.4) | 818 (8.5) | | Positive, n (%) | 226 (41.6) | 106 (11.5) | | Race | | American Indian | 3 (0.6) | 6 (0.6) | | Asian | 34 (6.3) | 56 (6.1) | | Black | 14 (2.6) | 33 (3.6) | | Pacific Islander | 1 (0.2) | 2 (2.2) | | White, Hispanic | 14 (2.6) | 18 (1.9) | | White, Non-Hispanic | 464 (85.5) | 794 (85.9) | | Multiracial | 4 (0.7) | 6 (0.6) | | Unknown | 9 (1.7) | 9 (1.0) | | Former or current smoker, n (%) | 189 (34.8) | 362 (39.2) | | Married or partnered, n (%) | 286 (52.7) | 517 (56.0) | | Atopic dermatitis, n (%) | 33 (6.1) | 31 (3.4) | Open in a new tab p< 0.05 comparing postmenopausal to premenopausal women by a two-tailed t-test. AGUS, atypical glandular cells of uncertain significance; ASC-H, atypical squamous cells, cannot rule out high grade squamous intraepithelial lesion; ASCUS, atypical squamous cells of uncertain significance; CIN, cervical intraepithelial neoplasia; hrHPV, high-risk HPV; HSIL, high-grade squamous intraepithelial lesion; LSIL, low-grade squamous intraepithelial lesion; NILM, negative for intraepithelial lesion or malignancy. Patients were primarily white (85.8%), and the distribution of race was similar between the premenopausal and postmenopausal cohorts. There were similar rates of smoking between the two groups, 34.8% among premenopausal women and 39.2% among postmenopausal women. There were more postmenopausal women (56.0%) than premenopausal women (52.7%) who were married or partnered (p< 0.05). There was a greater rate of atopic dermatitis among the premenopausal group (6.1%) than the postmenopausal group (3.4%) (p< 0.05). There were fewer postmenopausal (28.9%) than premenopausal women (72.6%) with an abnormal Pap smear result of ASCUS or greater. Among the 267 postmenopausal women with an abnormal index Pap smear, 76.0% were hrHPV negative and 24.0% were hrHPV positive. Most of these abnormal postmenopausal Pap smear results were ASCUS (95.9%), and only 29.3% were hrHPV positive. The remaining Pap smear results were atypical glandular cells of uncertain significance (AGUS) (4) and low-grade squamous intraepithelial lesion (7) (Table 1). Only three women were noted to have HIV, and none went on to develop CIN2+. In this cohort, there were no significant differences in sensitivity of cytology, hrHPV, or co-testing between the postmenopausal and premenopausal cohorts. Specificity of cytology, hrHPV, and co-testing were greater in the postmenopausal cohort (p< 0.05). In both cohorts, specificity was greater for hrHPV than for co-testing (p< 0.05). All clinically actionable (CIN2+) lesions were correctly identified by both hrHPV and by co-testing (Fig. 2). FIG. 2. Open in a new tab Sensitivity and specificity of cytology, hrHPV, and co-testing of cytology and hrHPV with 95% confidence intervals. (A) There was no difference in sensitivity between postmenopausal and premenopausal women for cytology alone, hrHPV, or co-testing. (B) Specificity for cytology alone, hrHPV, and co-testing was significantly different when comparing postmenopausal and premenopausal cohorts. hrHPV was more specific than co-testing among postmenopausal women. p< 0.05 comparing postmenopausal to premenopausal women. ^p< 0.05 comparing hrHPV to cytology alone and co-testing. There were 476 postmenopausal women with at least two consecutive hrHPV results, and 127 had at least one positive hrHPV result and were included in the subgroup analysis for hrHPV persistence. Of the latter, seven women did not have any follow-up hrHPV test after a positive result and were excluded from subset analysis. Of the 120 women in the final subgroup analysis, 65 were classified as “clearers” and 55 as “persisters” (Fig. 3). In this group, there were 22 clearers and 43 persisters who underwent colposcopy as part of their clinical workup. A significantly greater proportion of women with persistent hrHPV developed CIN2+ compared with women who cleared hrHPV (16% vs. 3%, respectively, p = 0.022). There was no demographic factor that was associated with hrHPV persistence. Demographic data for hrHPV clearers compared with persisters are shown in Table 2. FIG. 3. Open in a new tab Inclusions and exclusions for a subgroup analysis of “clearers” and “persisters” to evaluate for factors related to persistence of hrHPV among postmenopausal women. hrHPV, high-risk HPV. Table 2. Demographic Factors and High-Risk HPV Persistence Among Postmenopausal Women | | Clearers a(n = 65) | Persisters b(n = 55) | p | :---: :---: | | Max diagnosis CIN2+, n (%) | 2 (3) | 9 (16) | 0.022 | | History of abnormal Pap, n (%) | 41 (63) | 39 (71) | 0.364 | | Former or current smoker, n (%) | 28 (43) | 24 (44) | 1.000 | | Atopic dermatitis, n (%) | 1 (2) | 4 (7) | 0.178 | | History of STI, n (%) | 18 (28) | 14 (25) | 0.838 | | 5 or more lifetime partners, n (%) | 10 (15) | 11 (20) | 0.631 | | Sexually active at index, n (%) | 31 (48) | 31 (56) | 0.344 | | Medicaid, n (%) | 9 (14) | 9 (16) | 0.799 | Open in a new tab p< 0.05 comparing persisters to clearers by use of two-tailed chi-square test or two-tailed Fisher's exact test. a Women with (1) only one positive hrHPV result followed by a negative hrHPV result, (2) consecutive positive hrHPV results <1 year apart followed by a negative hrHPV result, or (3) two or more nonconsecutive positive hrHPV results. b Women with at least two consecutive hrHPV positive results at least 1 year apart. CIN2+, cervical intraepithelial lesion 2 or greater; STI, sexually transmitted infection. Discussion Cervical cytology may not be as clinically useful as hrHPV in postmenopausal women. Older women with normal cytology are often hrHPV positive with clinically significant cervical dysplasia, and normal, atrophic postmenopausal Pap smears are often incorrectly characterized as ASCUS.11,14,16 Primary hrHPV screening, cytology alone, and co-testing are accepted options for cervical cancer screening according to U.S. national guidelines for women 30 years and older. However, there are no specific guidelines for postmenopausal women, and the screening test used remains the physician's decision.9,10 The challenges with atrophic Pap smears are thought to be related to the hypoestrogenic state of menopause and its impact on the squamous cells of the cervix; thus, the use of Pap smears in the postmenopausal population may lead to more unnecessary follow-up Pap smears and clinical follow-up.12 Previously published studies largely come from European populations and have used arbitrary age cutoffs such as 50, 55, or 60 years to compare hrHPV and cytology results between older and younger women.13,16,17,22 One study has compared hrHPV and cytology among postmenopausal women without comparing to a premenopausal cohort.18 To our knowledge, this is the first U.S. study to compare cervical cytology and hrHPV testing by menopausal status rather than an arbitrary age cutoff. The prevalence of hrHPV in our entire cohort of both premenopausal and postmenopausal women was 22.6%, which parallels that of the U.S. population (20.4%).23 Previously published postmenopausal hrHPV studies have been performed largely in European populations, where hrHPV rates for the entire population have been found to range from as low as 2.2% to as high as 22.8%.3 The prevalence of hrHPV among postmenopausal women in our cohort was 11.5%. Previously published studies have reported varying rates of hrHPV in older women, ranging from 1% to 9.9% with variable ages of women included.22 Gyllensten et al., a retrospective study of Swedish women ages 55–74, reported an hrHPV prevalence of 6.2%; Ferenczy et al. reported rates as low as 1% among 306 women in Quebec ages 50–70; in a Swedish study of 1,051 women ages 60–89, the authors found a prevalence of 4.1%; a prospective Finnish study of over 10,000 women showed a prevalence of 4.9% among women 55 years of age and older; and a French study of 406 postmenopausal women showed a prevalence of 9.9%.13,16–18,22 It is important to note that these other studies were performed in Canadian or European populations, where hrHPV testing techniques and screening programs varied among studies that were performed over the last two decades. The marginally higher prevalence in our postmenopausal cohort may be reflective of an overall higher hrHPV prevalence in the United States compared to Europe or intrinsic risk within our cohort. In addition, our institution is a referral center, which may explain why our postmenopausal hrHPV prevalence falls near the upper limits of this established range and why there was a relatively high prevalence of hrHPV and abnormal Pap smear results in our premenopausal cohort. hrHPV had a significantly greater specificity (89.6%) than either cervical cytology alone (72.0%) or co-testing (67.4%) in our postmenopausal cohort and correctly identified all women with CIN2+. These results are in agreement with other studies that have shown greater specificity of hrHPV testing compared to Pap smear in older women.16,17 However, a French study of postmenopausal women showed lower specificity of hrHPV testing (25%) compared to Pap smear (80%) for detecting CIN2+ in postmenopausal women; one possible reason for the higher specificity of cytology is the high rate of hormone replacement therapy of 46% in this study.18 Our results showed no difference in the sensitivity of cervical cytology, hrHPV testing, or co-testing (100%). Some studies have instead demonstrated low sensitivity of cervical cytology in the detection of clinically significant cervical dysplasia among older women.15,17,18 In a study by Gustafsson et al., it was found that despite substantial Pap smear collection and a relatively high incidence of invasive cervical cancer above the age of 50, the probability of detecting cancer in situ decreased markedly from 35 to 50 years of age and remained low thereafter.14 It is important to note that while CIN lesions may not be detected, postmenopausal Pap smears still tend to be characterized as ASCUS when they are atrophic.12 Our data favor a greater number of falsely mildly abnormal results with 96% of abnormal postmenopausal Pap smears being ASCUS, 70% of which were hrHPV negative. These results suggest that the cytology component of co-testing does not add clinical value among postmenopausal women and may increase follow-up procedures and referrals due to false positive results without detecting any additional clinically significant disease. hrHPV testing tends to be less specific in younger women as a result of clearance of hrHPV infections by the immune system and spontaneous regression of lower grade dysplasia.24 Our data show that hrHPV testing was significantly more specific in the postmenopausal than the premenopausal cohort. Accepted methods for cervical cancer screening in women 30 years and older include cervical cytology every 3 years, co-testing every 5 years, or hrHPV every 3 years, and our study suggests primary hrHPV screening may be the best strategy among postmenopausal women. Of note, both cytology and co-testing were found to be more specific in postmenopausal than in premenopausal women; however, hrHPV testing remained more specific than cytology and co-testing in both cohorts. There was no significant difference in sensitivity of cytology, hrHPV testing, or co-testing between postmenopausal and premenopausal women. The rate of hrHPV persistence in our postmenopausal cohort was 45.8%, which is higher than that observed in younger women.25 As expected, CIN2+ was more common among women with persistent hrHPV than among those who cleared hrHPV. No risk factors for persistence were identified in this study. Known risk factors for hrHPV persistence and progression to cervical cancer in the general population include smoking, early age at sexual debut, increasing numbers of sexual partners, co-infection with other sexually transmitted infection, and immunocompromise.5,6 There is a paucity of data regarding risk factors for hrHPV persistence specifically among postmenopausal women. Our results could be the consequence of the relatively small sample size of the subset analysis, so future work should seek to prospectively identify risk factors for persistence in a larger group of postmenopausal women. Strengths of the study include the analysis by documented menopausal status rather than an arbitrary age cutoff, the comparison of the postmenopausal cohort to a premenopausal cohort, and the substantial overall sample size, which included 924 postmenopausal women. Limitations to this study include its retrospective nature, small numbers of CIN2+, and lack of data regarding hormonal therapy at menopause. In addition, it was found during thorough chart review that demographic data points were not consistently recorded or updated in the medical record. It is unknown whether race was self-identified. Because of the retrospective nature of this study, which included collecting data available in the electronic medical record in the 5-year time period following the index cytology and hrHPV result, it is unknown which women did not follow up as recommended or left the system with follow-up elsewhere, and the sample size was relatively small for the hrHPV persistence risk factor analysis. Conclusion Multiple studies support that hrHPV testing alone is more sensitive than cytology screening for CIN2+ lesions in postmenopausal women and have even suggested that co-testing of hrHPV and cervical cytology is not more sensitive than hrHPV testing, alone.15,16 These findings come largely from studies performed using data from a cohort of women in single region of Sweden. While larger prospective studies have been conducted in Europe, these results have not been replicated outside of this population or in a U.S. population, making the findings poorly generalizable to other countries or health care systems. Our results were in alignment with these findings, and our data support greater specificity of hrHPV testing compared with cytology and with co-testing. Furthermore, upon examination of all abnormal postmenopausal Pap smear results, almost all (96%) were ASCUS, and 71% were negative for hrHPV. Thus, the Pap smear was less effective at triaging women for shorter interval follow-up testing. Our study supports the use of primary hrHPV testing to screen for clinically significant cervical dysplasia. Future studies should prospectively evaluate whether co-testing should be replaced by primary hrHPV testing in postmenopausal women. Authors' Contributions J.M.K.: data curation, formal analysis, methodology, and writing—original draft. M.C.: data curation and writing—original draft. M.E.L.: data curation. E.G.M.: writing—review and editing. T.K.M.: conceptualization and methodology. A.S.B.: conceptualization, formal analysis, funding acquisition, and methodology. Disclaimer The content is solely the responsibility of the authors and does not necessarily represent the official views of the NIH. Author Disclosure Statement No competing financial interests exist. Funding Information The project described was supported by the National Center for Advancing Translational Sciences, National Institutes of Health, through Grant KL2TR002370. Support for this article was provided, in part, by the Robert Wood Johnson Foundation. The views expressed in this study do not necessarily reflect the views of the Foundation. This study was conducted at Oregon Health and Science University in Portland, OR. References Bray F, Ferlay J, Soerjomataram I, Siegel RL, Torre LA, Jemal A. Global Cancer Statistics 2018: GLOBOCAN estimates of incidence and mortality worldwide for 36 cancers in 185 countries. CA Cancer J Clin 2018;68:393–424 [DOI] [PubMed] [Google Scholar] Ward E, Sherman RL, Henley SJ, et al. Annual report to the nation on the status of cancer, 1999–2015, featuring cancer in men and women ages 20–49. J Natl Cancer Inst 2019;111:1279–1297 [DOI] [PMC free article] [PubMed] [Google Scholar] De Vuyst H, Clifford G, Li N, Franceschi S. HPV infection in Europe. Eur J Cancer (Oxford, England: 1990) 2009;45:2632–2639 [DOI] [PubMed] [Google Scholar] Walboomers JM, Jacobs MV, Manos MM, et al. Human papillomavirus is a necessary cause of invasive cervical cancer worldwide. J Pathol 1999;189:12–19 [DOI] [PubMed] [Google Scholar] International Collaboration of Epidemiological Studies of Cervical C. Comparison of risk factors for invasive squamous cell carcinoma and adenocarcinoma of the cervix: Collaborative reanalysis of individual data on 8,097 women with squamous cell carcinoma and 1,374 women with adenocarcinoma from 12 epidemiological studies. Int J Cancer 2007;120:885–891 [DOI] [PubMed] [Google Scholar] Kim HS, Kim TJ, Lee IH, Hong SR. Associations between sexually transmitted infections, high-risk human papillomavirus infection, and abnormal cervical Pap smear results in OB/GYN outpatients. J Gynecol Oncol 2016;27:e49. [DOI] [PMC free article] [PubMed] [Google Scholar] Gustafsson L, Pontén J, Zack M, Adami HO. International incidence rates of invasive cervical cancer after introduction of cytological screening. Cancer Causes Control 1997;8:755–763 [DOI] [PubMed] [Google Scholar] Wright TC Jr., Massad LS, Dunton CJ, Spitzer M, Wilkinson EJ, Solomon D.. 2006 consensus guidelines for the management of women with abnormal cervical cancer screening tests. Am J Obstet Gynecol 2007;197:346–355 [DOI] [PubMed] [Google Scholar] Huh WK, Ault KA, Chelmow D, et al. Use of primary high-risk human papillomavirus testing for cervical cancer screening: Interim clinical guidance. Gynecol Oncol 2015;136:178–182 [DOI] [PubMed] [Google Scholar] Perkins RB, Guido RS, Castle PE, et al. 2019 ASCCP risk-based management consensus guidelines for abnormal cervical cancer screening tests and cancer precursors. J Low Genit Tract Dis 2020;24:102–131 [DOI] [PMC free article] [PubMed] [Google Scholar] Tabrizi A. Atrophic Pap smears, differential diagnosis and pitfalls: A review. Int J Womens Health Reprod Sci 2018;6:2–5 [Google Scholar] Gupta S, Sodhani P. Reducing “atypical squamous cells” overdiagnosis on cervicovaginal smears by diligent cytology screening. Diagn Cytopathol 2012;40:764–769 [DOI] [PubMed] [Google Scholar] Leinonen M, Nieminen P, Kotaniemi-Talonen L, et al. Age-specific evaluation of primary human papillomavirus screening vs conventional cytology in a randomized setting. J Natl Cancer Inst 2009;101:1612–1623 [DOI] [PubMed] [Google Scholar] Gustafsson L, Sparen P, Gustafsson M, et al. Low efficiency of cytologic screening for cancer in situ of the cervix in older women. Int J Cancer 1995;63:804–809 [DOI] [PubMed] [Google Scholar] Gyllensten U, Lindell M, Gustafsson I, Wilander E. HPV test shows low sensitivity of Pap screen in older women. Lancet Oncol 2010;11:509–510; author reply 510–501. [DOI] [PubMed] [Google Scholar] Gyllensten U, Gustavsson I, Lindell M, Wilander E. Primary high-risk HPV screening for cervical cancer in post-menopausal women. Gynecol Oncol 2012;125:343–345 [DOI] [PubMed] [Google Scholar] Hermansson RS, Olovsson M, Hoxell E, Lindstrom AK. HPV prevalence and HPV-related dysplasia in elderly women. PLoS One 2018;13:e0189300. [DOI] [PMC free article] [PubMed] [Google Scholar] Tifaoui N, Maudelonde T, Combecal J, et al. High-risk HPV detection and associated cervical lesions in a population of French menopausal women. J Clin Virol 2018;108:12–18 [DOI] [PubMed] [Google Scholar] Morgan T, Hanifin J, Mahmood M, et al. Atopic dermatitis is associated with cervical high risk human papillomavirus infection. J Low Genit Tract Dis 2015;19:345–349 [DOI] [PubMed] [Google Scholar] Harris PA, Taylor R, Thielke R, Payne J, Gonzalez N, Conde JG. Research electronic data capture (REDCap)—A metadata-driven methodology and workflow process for providing translational research informatics support. J Biomed Inform 2009;42:377–381 [DOI] [PMC free article] [PubMed] [Google Scholar] Harris PA, Taylor R, Minor BL, et al. The REDCap consortium: Building an international community of software platform partners. J Biomed Inform 2019;95:103208. [DOI] [PMC free article] [PubMed] [Google Scholar] Ferenczy A, Gelfand MM, Franco E, Mansour N. Human papillomavirus infection in postmenopausal women with and without hormone therapy. Obstet Gynecol 1997;90:7–11 [DOI] [PubMed] [Google Scholar] McQuillan G, Kruszon-Moran D, Markowitz LE, Unger ER, Paulose-Ram R. Prevalence of HPV in adults aged 18–69: United States, 2011–2014. NCHS Data Brief 2017:1–8 [PubMed] [Google Scholar] Ronco G, Giorgi-Rossi P, Carozzi F, et al. Efficacy of human papillomavirus testing for the detection of invasive cervical cancers and cervical intraepithelial neoplasia: A randomised controlled trial. Lancet Oncol 2010;11:249–257 [DOI] [PubMed] [Google Scholar] de Sanjosé S, Brotons M, Pavón MA. The natural history of human papillomavirus infection. Best Pract Res Clin Obstet Gynaecol 2018;47:2–13 [DOI] [PubMed] [Google Scholar] Articles from Journal of Women's Health are provided here courtesy of Mary Ann Liebert, Inc. ACTIONS View on publisher site PDF (281.6 KB) Cite Collections Permalink PERMALINK Copy RESOURCES Similar articles Cited by other articles Links to NCBI Databases On this page Abstract Introduction Materials and Methods Results Discussion Conclusion Authors' Contributions Disclaimer Author Disclosure Statement Funding Information References Cite Copy Download .nbib.nbib Format: Add to Collections Create a new collection Add to an existing collection Name your collection Choose a collection Unable to load your collection due to an error Please try again Add Cancel Follow NCBI NCBI on X (formerly known as Twitter)NCBI on FacebookNCBI on LinkedInNCBI on GitHubNCBI RSS feed Connect with NLM NLM on X (formerly known as Twitter)NLM on FacebookNLM on YouTube National Library of Medicine 8600 Rockville Pike Bethesda, MD 20894 Web Policies FOIA HHS Vulnerability Disclosure Help Accessibility Careers NLM NIH HHS USA.gov Back to Top
14095
https://www.mathsisfun.com/greatest-common-factor.html
Greatest Common Factor All Factors Calculator Introduction to Fractions Show Ads Hide Ads | About Ads We may use Cookies OK Home Algebra Data Geometry Physics Dictionary Games Puzzles [x] Algebra Calculus Data Geometry Money Numbers Physics Activities Dictionary Games Puzzles Worksheets Hide Ads Show Ads About Ads Donate Login Close Greatest Common Factor _The highest number that divides exactly into two or more numbers. It is the "greatest" thing for simplifying fractions!_ Let's start with an Example ... Greatest Common Factor of 12 and 16 Find all the Factors of each number, Circle the Common factors, Choose the Greatest of those So ... what is a "Factor" ? Factors are numbers we can multiply together to get another number: A number can have many factors: Factors of 12 are 1, 2, 3, 4, 6 and 12 ... ... because 2 × 6 = 12, or 4 × 3 = 12, or 1 × 12 = 12. (Read how to find All the Factors of a Number. In our case we don't need the negative ones.) What is a "Common Factor" ? Say we have worked out the factors of two numbers: Example: Factors of 12 and 30 Factors of 12 are 1, 2, 3, 4, 6 and12 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and30 Then the commonfactors are those that are found in both lists: Notice that 1, 2, 3 and 6 appear in both lists? So, the common factors of 12 and 30 are: 1, 2, 3 and6 It is a common factor when it is a factor of two (or more) numbers. Here is another example with three numbers: Example: The common factors of 15, 30 and 105 Factors of 15 are 1, 3, 5, and15 Factors of 30 are 1, 2, 3, 5, 6, 10, 15 and30 Factors of 105 are 1, 3, 5, 7, 15, 21, 35 and105 The factors that are common to all three numbers are 1, 3, 5 and 15 In other words, the common factors of 15, 30 and 105 are 1, 3, 5 and 15 What is the "Greatest Common Factor" ? It is simply the largest of the common factors. In our previous example, the largest of the common factors is 15, so the Greatest Common Factor of 15, 30 and 105 is 15 The "Greatest Common Factor" is the largest of the common factors (of two or more numbers) Why is this Useful? One of the most useful things is when we want to simplify a fraction: Example: How do we simplify 1230 ? Earlier we found that the Common Factors of 12 and 30 are 1, 2, 3 and 6, and so the Greatest Common Factor is 6. So the largest number we can divide both 12 and 30 exactly by is 6, like this: ÷ 6 1230=25 ÷ 6 The Greatest Common Factor of 12 and 30 is 6. And so 1230 can be simplified to 25 Finding the Greatest Common Factor Here are three ways: 1. We can: find all factors of both numbers (use the All Factors Calculator), then find the ones that are common to both, and then choose the greatest Example: | Two Numbers | Factors | Common Factors | Greatest Common Factor | Example Simplified Fraction | --- --- | 9 and 12 | 9: 1, 3, 9 12:1, 2, 3, 4, 6, 12 | 1, 3 | 3 | 912 = 34 | And another example: | Two Numbers | Factors | Common Factors | Greatest Common Factor | Example Simplified Fraction | --- --- | 6 and 18 | 6: 1, 2, 3, 6 18:1, 2, 3, 6, 9, 18 | 1, 2, 3, 6 | 6 | 618 = 13 | 2. Or we can find the prime factors and combine the common ones together: | Two Numbers | Thinking ... | Greatest Common Factor | Example Simplified Fraction | --- --- | | 24 and 108 | 2 × 2 × 2 × 3 = 24, and 2 × 2 × 3 × 3 × 3 = 108 | 2 × 2 × 3 = 12 | 24108 = 29 | 3. Or sometimes we can just play around with the factors until we discover it: | Two Numbers | Thinking ... | Greatest Common Factor | Example Simplified Fraction | --- --- | | 9 and 12 | 3 × 3 = 9 and 3 × 4 = 12 | 3 | 912 = 34 | But in that case we must check that we have found the greatest common factor. Greatest Common Factor Calculator OK, there is also a really easy method: we can use the Greatest Common Factor Calculator to find it automatically. Other Names The "Greatest Common Factor" is often abbreviated to GCF, and is also known as: the "Greatest Common Divisor" or GCD the "Highest Common Factor" or HCF Mathopolis:Q1)Q2)Q3)Q4)Q5)Q6)Q7)Q8)Q9)Q10) All Factors CalculatorIntroduction to FractionsGreatest Common Factor CalculatorSimplifying FractionsEquivalent FractionsAdding FractionsSubtracting FractionsFractions Index Donate ○ Search ○ Index ○ About ○ Contact ○ Cite This Page ○ Privacy Copyright © 2025 Rod Pierce
14096
https://www.geeksforgeeks.org/biology/difference-between-cell-wall-and-cell-membrane/
Difference Between Cell Wall And Cell Membrane - GeeksforGeeks Skip to content Tutorials Python Java DSA ML & Data Science Interview Corner Programming Languages Web Development CS Subjects DevOps Software and Tools School Learning Practice Coding Problems Courses DSA / Placements ML & Data Science Development Cloud / DevOps Programming Languages All Courses Tracks Languages Python C C++ Java Advanced Java SQL JavaScript Interview Preparation GfG 160 GfG 360 System Design Core Subjects Interview Questions Interview Puzzles Aptitude and Reasoning Data Science Python Data Analytics Complete Data Science Dev Skills Full-Stack Web Dev DevOps Software Testing CyberSecurity Tools Computer Fundamentals AI Tools MS Excel & Google Sheets MS Word & Google Docs Maths Maths For Computer Science Engineering Mathematics Switch to Dark Mode Sign In Biology Biotechnology Biochemistry Genetics Ecology Evolution Anatomy Physiology Immunology Taxonomy Botany Zoology Microbiology Cell Biology Cell Signaling Diversity in Life Form Molecular Biology Sign In ▲ Open In App Difference Between Cell Wall And Cell Membrane Last Updated : 23 Jul, 2025 Comments Improve Suggest changes Like Article Like Report Cell wall and the cell membrane are two important components found in cells, but they show some different characteristics. Both the Cell wall and Cell membrane protect the cell from the external environment and provide support to the cell. The cell wall is present in plant cells only, while the cell membrane is present in each cell. In this article, we will discuss the differences between cell walls and cell membranes. Difference Between a Cell Wall and Cell Membrane The difference between the cell wall and cell membrane are as follows: | Characteristics | Cell Wall | Cell Membrane | --- | Presence | It is only present in plant cells | It is present in animals, bacteria, and human cells | | Structure | It's thick and rigid in nature | It's thin and delicate in nature | | Biological Presence | It's metabolically inactive | It's metabolically active | | Location | It's the outermost layer found in plant cells | It's the outermost layer in animal cells | | Growth | The cell wall grows with time and the thickness may increase within a span of time | Its thickness remains the same throughout its lifetime | | Thickness | The thickness of the cell wall can be between 0.1 μm to some μm | The thickness of a cell membrane can be between 7.5-10 nm | | Function | It's responsible for the protection of the cell. | It's responsible for communication with the outer environment, sexual reproduction, and cell division. | What is a Cell Wall? A cell wall is the outermost layer of a plant cell, it's mainly found in plant cells, some microorganisms, and bacteria. The cell wall is flexible and rigid in nature and is found outside the cell membrane. The cellulose, hemicellulose, long fiber of carbohydrates, and lignin and pectin usually make up the cell wall. It helps in regulating cell growth and protects it from physical damage as well. A cell may die due to endosmosis. Apart from this, a cell wall provides structural support and helps in maintaining its shape. Due to its textured and porous surface, it allows the entry of smaller molecules. Also, the cell wall prevents expansion or rupture in case of an increase in pressure inside the cell. The cell wall also acts as a barrier for entry for some biomolecules and even as a channel for the entry and exit of many other metabolic molecules. In some plants, the cell wall might be made up of a single layer, while in some plants there are 2 layers of the cell wall. These 2 layers provide waterproofing to the cell. Alsor Read: Gram Staining What is a Cell Membrane? The cell membrane or also known as the plasma membrane separates the interior of the cell from the outer environment, it's the outermost layer found in animal cells. This cell membrane is a semipermeable membrane made up of proteinsand lipids. The presence of cholesterol and phospholipids in the wall helps in maintaining fluidity in the cell membrane at different temperatures. This semipermeable membrane allows transportation between the outer environment and the cell. A cell membrane provides a fixed environment in the cell and also protects it from physical damage. It's the main communication channel between the cell and the outer environment, and its semipermeable nature allows the entry and exit of some selective substances only. Ion conductivity, cell signaling, and cell adhesion are some of the common processes in which cell membranes are involved. Similarities Between a Cell Wall and a Cell Membrane A cell wall and cell membrane might have significant differences but they also do have some similarities some of which are listed below. A cell wall and a cell membrane protect the cell from external mechanical stresses. They both provide shape and rigidity to a cell. A cell wall and cell membrane act as the main channel of transfer for the entry and exit of molecules. Both, the cell wall and cell membrane are the outermost layer of a cell. Conclusion The cell wall and cell membrane are the two most important organelles essential for all living organisms. Although they differ in appearance and function, a cell membrane and a cell wall are equally necessary for all living things to function properly. A cell wall is rigid and cannot change its shape, whereas a cell membrane is naturally flexible and can change shape and size. However, the rigid nature of a cell wall provides structural support to the cells. 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14097
https://digitalcommons.unl.edu/context/extensionhist/article/5703/viewcontent/EC_93_445A__TN.pdf
University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Historical Materials from University of Nebraska-Lincoln Extension Extension 1993 EC93-445-A Hems for Garments Rose Marie Tondl Kathleen Tolman Follow this and additional works at: This Article is brought to you for free and open access by the Extension at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Historical Materials from University of Nebraska-Lincoln Extension by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. Tondl, Rose Marie and Tolman, Kathleen, "EC93-445-A Hems for Garments" (1993). Historical Materials from University of Nebraska-Lincoln Extension. 4695. 'CYT __ _ s 85 E7 no. 445 Copy 2 Nebraska Cooperative Extension EC 93-445-A Hems for Garments Rose Marie Tondl, Extension Specialist- Clothing and Textiles Kathleen Tolman, Graduate Assistant "Hem" is a term used to describe the amount of fab-ric turned under to finish the edge of a garment (also known as the hem allowance). How a hem looks often means the difference between having a garment look homemade or professionally finished. While a variety of techniques can be used to actu-ally hem a garment, there are some basic methods that can be followed when hemming. Allow a new garment to hang for 24 hours before the hem is marked, particu-larly if large pieces of the garment are on the bias. Features of a Successful Hem • The lower edge hangs an even distance from the floor. • The hem is not obvious from the right side of the garment, if hand stitched. • The hem is smooth and flat. • The hem allowance is even all the way around the garment. • The hem allowance is an appropriate width for the fabric and design. • The hem finish is suited to the fabric and to the design of the garment. • Any fullness in the hem allowance is evenly dis-tributed and, if possible, shrunk out. • Stitches (hand or machine) are uniform, evenly spaced, secure and do not cause puckers. • Hems are carefully pressed to prevent ridges. Marking the Hem Wear proper undergarments, shoes, and the belt if the garment is to be belted. Use a hem marker or a yard-possible, placing pins about 3 inches apart, parallel to the floor. Let the person marking the hem move around you. Remember-the hemline of the garment should hang an equal dis-tance from the floor all the way around the garment (Figure 1). Pin or handbaste to es-tablish and hold the hemline (Figure 2). Select the appropriate depth of hem allowance. This is based on the type of garment, type of fabric and height of the wearer. Check the pattern for the recom-mended allowance for that garment. In general, the greater the flare, the narrower the hem. Straight gar-ments in medium and lightweight fabrics are 2 to 3 inches or less. A-line skirts should have a narrower hem of 1 1/2-2 inches. Full circular skirts are 1 inch or less. Pants, short jackets, and some blouses usually have 1-2 inch hems. Sheer fabrics either have an extremely nar-row hem (1 I 4 inch or less) or a very wide hem up to 6 inches for design and weight. Hems in bulky fabrics can stick to mark the hem. Have a helper mark the hem if 2 Nebraska Cooperative Extension Service E. C. <Nebraska Coope·rative Extension Service) Received on: 05-18-93 It isthe1 UniveT·sity of Neb·paska, Lincoln Libraries ;ts of May 8 and June 30, 1914, in cooperation 'ctor of Cooperative Extension, University of Nebraska, Natural Resources. ources not to discriminate on the basis of sex., age, handicap, race, color, religion, ~ nic origin or sexual orientation. 3 be faced with a light weight material. Mark the desired hem width with pins or chalk, using a seam gauge or ruler (Figure 3). Trim off excess fabric to the desired width. To avoid cutting into the garment accidentally, open the hem out and trim off excess fabric. Reduce bulk in seam allow-ances by trimming 1 I 4 - 3 I 8 inch from the seam allow-ances within the hem (Figure 4). 4 Curved hems may be too full when turned up to fit smoothly against the underside of the garment. Sew a line of machine stitching (8-10 stitches per inch) 1 I 4 inch from the cut edge. Pin the hem to the skirt at the seams and midway between them. Wherever there is a ripple, draw up the bobbin thread with a pin until the hem lays flat (Figure 5). 5 Many fabrics may leave an imprint of the hem when pressed. To prevent this, place a piece of heavy paper between the hem and the garment. Press, shrink-ing out the ease in wool and other flexible fabrics. On nonshrinkable fabrics, pressing will help flatten gathers. If a soft, rolled hem is desired, do not press the fold of the hem. Hold the iron 2-3 inches from the hem and steam thoroughly. Let the steam evaporate before handling (Figure 6). c!J ~ f) ') 6 Finishing the Hem Finish the raw edge of the hem if it ravels or does not lie flat. Knit fabrics generally do not need an edge finish unless the edge tends to curl or roll. The kind of finish you choose will depend on: • the kind of fabric and its tendency to ravel • the bulk of the fabric • the style of the garment • the purpose or intended use of the garment • the amount of time you want to spend on finish-ing edges Select a hem finish that will not change the outer ap-pearance of the garment unless the hem is decorative. The finish should not add weight or bulk that will cause an imprint to the right side. Following are some possible edge finishes: Stitched edge-For knits or fabrics that do not ravel, machine stitch 1 I 4 inch from the raw edge of the hem. On a curved hem this will have been done already while reducing fullness. The raw edge can also be pinked on firmly woven fabrics to improve the inside appearance (Figure 7). 7 Zig-zag or multi-stitch zig-zag-A medium to wide zig-zag stitch can be used to control small amounts of raveling for medium and heavy weight woven fab-rics. This may also reduce curling or rolling edges on some knits. Zig-zag 118 inch from the raw edge of the fabric, or multi-stitch zig-zag along the edge. These fin-ishes will not add bulk to the hem (Figure 8). Garment 8 Clean finish or turned and stitched edge-On light to medium weight woven fabrics, tum under the raw edge of the hem about 1 I 4 inch and stitch along the fold. This method does not add bulk to these fabrics. It is not recommended for heavy fabrics as a ridge may form on the right side of the garment when the raw edge is folded down (Figure 9). 9 Taped edge--Use this finish for fabrics that ravel. Use seam binding, bias tape, stretch lace, or sheer bias nylon tape. Place the tape on the right side of the hem allowance, extending over the raw edge about 1 I 4 inch. Straight stitch close to the lower edge or use a medium width zig-zag to attach the tape to the hem allowance (Figure 10). 10 Bound hem-Use this finish on fabrics that ravel heavily, or for decorative purposes. Insert the raw edge of the hem between double-fold bias tape, fold over stretch lace or sheer bias nylon tape. Straight stitch or zig-zag close to lower edge (Figure 11). 11 Hong Kong finish-A technique using bias tape or 1 inch bias strips of lightweight fabric. Unfold and lightly press the bias tape. Place the right side of the bias against the right side of the hem and stitch 1 I 4 inch from the matched edges (Figure 12). Fold the tape over the edge of the hem. "Stitch in the ditch" on the right side along the well of the seam where the bias joins the hem (Figure 13). This finish is also suitable for velvet and satin, using a lightweight net or tulle for the binding. 12 Garment "Stitch in Ditch" Hem Fold 13 Faced hems--Use this type of hem when hem width is too nar-row, when fabric is too bulky to turn up, or if style of garment is very full or circular. Use pur-chased bias hem facing Right side of fabric or bias strips of underlining or lining fabric 2 - 3 inches wide. To apply facing, first trim the hem allowance to 518 inch. Beginning at a seam, place right side of facing to the right side of hem edge. Pin in place. For a neat finish, turn one end of the facing back and overlap the other end (Figure 14). Stitch on the crease of the facing or 1 I 4 inch from the edge. Tum the facing to the inside on the marked hemline (Figure 15). Hemline 15 Hand Hemming Techniques Hems put in by hand can be done by flat hemming or inside hemming. For flat hemming, the hem is placed against the garment with the stitches running across the top of the hem. For inside hemming, the garment is folded back and stitches are taken between the hem allowance and the garment. With this method of hemming, the stitches have less tendency to show on the right side, they do not show on the inside of the garment, and they are protected from abrasion and pulling by the hem allowance. Most hemming stitches can be used for either flat or inside hemming. The method you choose depends on your fabric's tendency to ravel and the finish you've used on the hem edge. If your hem is turned and stitched, or finished with binding, lace or bias tape, flat hemming is more suit-able. Inside hemming is suitable for stitched and pinked or zig-zag finishes. Remember to keep hand stitches loose. Pulling them too tight will cause a puckered hem. Blind stitch or tailor's hem-This hem is inconspicu-ous from both the right side and the hem side of the gar-ment. For inside hemming, fold the garment back and pick up one thread in the fold of the garment. Take a stitch through the hem. Repeat stitches at 114-112 inch inter-vals. Knot the stitches frequently to prevent the entire hem from coming out if some stitches break (Figure 16). For flat hemming, take a tiny stitch in the garment at the edge of the hem; bring needle up diagonally through the edge of the seam binding or hem edge (Figure 17). Inside Hemming with Slip Stitch 16 17 Catch stitch-This stitch is especially good for knit fabrics as it has some stretch. Work from left to right with the needle pointing toward the left. The thread will cross itself, forming a tiny "x" at each stitch. For inside hemming, fold the garment back and take a short stitch through the stitching line near the edge of the hem. Pick up one thread in the fold of the garment. Stitches should be 1 I 4- 112 inch apart (Figure 18). For flat hemming, take a stitch in the hem about 1 I 4 from the raw edge or in the fold if the edge has been turned under. Take a short stitch in the garment, just above the raw edge of the hem. Stitches should be 1 I 4 - 112 inch apart (Figure 19). 18 19 20 Slip stitch-This is used only for flat hemming where there is a turned edge. The stitches should be about 1/2 inch apart, depending on the weight of fabric and amount of strain placed on the garment. On the gar-ment side, pick up one thread next to edge of hem, then pass needle through fold at top of hem (Figure 20). Machine-Stitched Hems In addition to hand-sewn hems, there are certain in-stances where a machine-sewn hem is actually prefer-able. Blind hem-The machine blind stitch is durable and fairly inconspicuous. It is suitable for children's clothing, sportswear in sturdy fabrics and horne decorat-ing. Check your sewing machine manual for proper rna-chine settings. Practice the stitch on a sample so you know where the straight stitches go and where the ''bite" is taken. Use a narrow bite so a minimum of thread shows on the right side. Place hem edge over the feed dog of the machine; turn back the bulk of the fabric to create a soft fold. Place the work so the straight stitches are made on the hem line and the sideward bite pierces only one or two threads of the soft fold (Figure 21). 21 Narrow, rolled, stitched hem-This is suitable where the hem will not be visible such as on linings, tuck-in blouses and shirts. Make a line of stitching 114 inch from raw edge. Press up a 1 I 4 inch fold, turn again Some machines have an attachment for making the nar-row rolled hem. Consult your sewing machine manual (Figure 22). 22 Topstitching-A straight or decorative stitch can be used to hold the hem on many garments. This is espe-cially appropriate for sportswear, or as a special effect for a garment. The topstitching may be one or two rows of straight stitching (use a twin needle), or almost any decorative stitch included on your machine. Try combi-nations of stitches for a creative touch (Figure 23). 23 Finish the edge of the hem or tum it under if needed. Knits that do not roll need not be turned under. Stitch on the outside of the garment so the stitches hold the hem in place. A narrow machine topstitched hem may be appropriate for bias or circular skirts. The depth of the hem should be less than 1 inch. The hem may be turned under once or twice and then machine stitched. Two or more rows of stitching may be desired to give the hem a detailed finish or to prevent the hem from rolling (Figure 24). 1 I 4 inch and press. Machine stitch along the hem edge. 24 Lettuce edging-Consider this hem for a decorative effect on lightweight stretchy knit fabric. The lettuce edge is achieved by stretching the fabric as you sew. This ruffled finish is done with a medium length, nar-row zig-zag stitch. Zig-zag on the raw edge of the fabric that has been cut off at the hemline or for a smoother edge, turn up the hem allowance and do the stitch along the folded edge. Stretch the fabric as you stitch. As fabric relaxes, lettuce-like ruffles will form. After stitching, cut off any excess hem allowance (Figure 25). Folded Edge 25 Fringed hem-Place a row of stitching (straight or decorative) at the desired depth of the fringe. Slash up to the stitching about every 2 inches. Pull out crosswise threads one section at a time (Figure 26). slash 26 Fusible Hems This is a quick and easy way to secure a hem and can be used in place of hand or machine-stitching. It is not recommended if the hem is very curved or has quite a bit of ease or for use on sheer or lace fabrics. Fusible Web-A fusible web is a heat-sensitive ad-hesive that holds two layers of fabric together. It is ap-plied by means of heat, moisture and pressure for a specific number of seconds. On light to medium weight fabrics, use 314-1 inch wide fusible strips. With heavier fabrics, use a 2 inch strip of fusible web. First test the fusible web on your fabric, checking the bond strength and the appearance of the sample. Use an up-and-down motion with the iron during the fusing process to prevent stretching the hem. Place the fusible web between the garment and hem about 1 I 4 inch from the edge to prevent it from sticking to the iron. If necessary, pin to avoid slipping (Figure 27) or sew fusible web 1 I 4 inch below inside the hem's edge (Figure 28). 27 Hem line Hem edge 28 Tum up hem and heat baste by steaming lightly be-tween pins. Remove pins before fusing. Complete fus-ing process working on a small section at a time. Let the fabric cool before handling. (Figure 29). 29 Fusible Thread-This polyester thread has a special coating and melts when pressed. It works best on me-dium to lightweight cotton or cotton-blend fabrics. The heat required for bonding is hotter than that usually used for wool. To use fusible thread, wind on bobbin only or use on the lower looper of a serger. Then zig-zag or serge with the fusible thread on wrong side of fabric (Figure 30). Press hem in place. Hem line Fusible thread 30 SergedHems Serged hems eliminate the need for using tape, bindings, lace, and turning under the edge. A serger produces a stitch that overlocks the edge and prevents raveling. Shirttail-Serge the hem trimming to 1 I 4 inch. Fold on hemline (1 I 4 inch) and straight stitch with machine (Figure 31). 31 Blind hem-Adjust serger for blind hem. Press hem for blind hem (see blind hem under machine-stitched hems). DO NOT FINISH EDGE. Place garment and serge so that the hem is under the foot and the needle barely catches the fold and the knife barely trims edge. Small stitches will be noticeable on right side (Figure 32). 32 Mock-band-Fold hem up and back similar to blind hem. Serge using a 3 or 3/4 thread serger, barely trimming off the fold (Figure 33). 33 Rolled edge--Consult your serger manual for ten-sion adjustments and special presser foot and/ or throat plate. The upper looper thread is pulled to the under-side, rolling the fabric raw edge at the same time (Figure 34). 34 Serged lettuce edge-Adjust serger to rolled hem. Complete in same method as machine stitched lettuce edging (Figure 35). 35 Hems with Pleats At the edge of a pleat fold, press open the seam in the hem area. Finish the raw edge of the hem. Turn up the hem to the desired width, clipping the seam at the top of the hem (Figure 36). To be sure the pleated hem stays creased, edgestitch the fold of the pleat (Figure 37). Seam allowances above the hem will face in one direc-tion, keeping the pleat fold flat. Stitch through all thick-nesses of the hem on the inside. Complete the hem. 36 37 Hems with Linings An important consideration for a lining hem is that it covers the raw edge of the garment without showing below the edge of the garment. There are two types of hems for linings. The attached hem used for jackets and pants, and the free hanging hem used for skirts, dresses and coats. Attached-Mark and tum up garment hem. Cut lining 5/8 inch below garment hemline. Tum edge of lining under 5/8 inch. Pin lining to garment matching hem top edges of both hems. Slipstitch lining to garment hem. An ease tuck will form. This helps prevent strain on garment hem (Figure 38). Lining 38 Freehanging-Mark and tum up garment hem. Tum up lining hem 1/ 2 inch shorter than garment. Trim lining hem to 2 inches and apply a hem finish. Hem lin-ing with appropriate stitch (using hand or machine hemming). Attach lining to garment with French tacks at seams (Figure 39). 39 Pressing Hems After the hem is stitched by hand or machine, a final and careful pressing is necessary. Use a well-padded ironing board and press on the inside of the garment. Remove basting along the fold. Use a pounding block on woolens. For an unpressed or rolled hem edge, do not touch the fold edge with the iron. Instead, hold the iron 2 or 3 inches above the fold and steam the edge thoroughly. Let the steam evaporate, and the garment cool completely before handling. For top pressing, use a press cloth to prevent fabric shine. Tips for Successful Hemming • Keep hand hemming stitches fairly loose. Pulling them too tight will cause a puckered hem. • Always take hemming stitches in the direction of the fabric grainline as di-agonal stitches are more likely to show from the right side (Figure 40). • To avoid catching the thread on the pins when you handstitch a hem, po-sition the pins on the out-side of the garment. r ., u • When a facing extends 40 through a hem, such as with a button-down-the-front garment, finish the hem first, then fold the facing back over the hem (Figure 41). 41 Suggested Hem Depths Skirts, dresses A-line, flared Straight Full circular Sheer Blouses, sweaters Overblouse Shirttail Sweaters Pants Straight leg Cuffs Flared leg or long pantskirt Jackets, blazers, coats Waist and hip length · Fulllength 2 -3 inches 2 1/2 -3 inches 1 inch 1/8 or 6 inches 1 0 1 -3 inches -1/2 inches - 1 1/2 inches 1 1/2 - 2 inches Twice cuff width + 1 inch 1 -11/2 inches 11/2 inches 21/2 -3 inches
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https://www.sciencing.com/calculate-piezometric-head-8710823/
How To Calculate The Piezometric Head Science- [x] Astronomy Biology Chemistry Geology Nature Physics Math- [x] Algebra Geometry Technology- [x] Electronics Features About Editorial Policies Privacy Policy Terms of Use © 2025 Static Media. All Rights Reserved How To Calculate The Piezometric Head ScienceMathTechnologyFeatures Science Physics How To Calculate The Piezometric Head By Karen G Blaettler Updated Mar 24, 2022 Water bubbling up from the ground seems downright magical. Water flowing uphill through pipes seems to contradict the laws of gravity. While these may seem like miraculous events, they occur due to ​piezometric or hydraulic head​. Piezometric Head Definition Piezometric Head Definition The ​piezometric head definition​ from the American Meteorological Society glossary is "the pressure that exists in a confined aquifer." The definition continues by stating that piezometric head "...is the elevation above a datum plus the pressure head." The piezometric surface is described as "an imaginary or hypothetical surface of the piezometric pressure or hydraulic head throughout all or part of a confined or semi-confined aquifer; analogous to the water table of an unconfined aquifer." Piezometric head synonyms include hydraulic head and hydraulic head pressure. The piezometric surface may also be called the ​potentiometric surface​. Piezometric head is a measure of the ​potential energy of water​. What Piezometric Head Actually Measures What Piezometric Head Actually Measures Piezometric head indirectly measures the potential energy of water by measuring the height of water at a given point. Piezometric head is measured using the elevation of the water surface in a well or the height of water in a standpipe attached to a pipe containing water under pressure. Piezometer head combines three factors: the potential energy of the water due to the water's height above a given point (usually average or mean sea level), any additional energy applied by pressure and velocity head. The pressure may be due to gravity, as with flow through the pipes in a hydroelectric dam, or by confinement, as in a confined aquifer. The equation for calculating head can be written as head ​h​ equals elevation head ​z​ plus pressure head ​Ψ​ plus velocity head ​v.​ h=z+Ψ+v Velocity head, while an important factor in pipe and pump flow calculations, is negligible in calculations of groundwater piezometric head because the velocity of groundwater is very slow. Determining Piezometric Head in Groundwater Determining Piezometric Head in Groundwater Determining piezometric head is accomplished by measuring the elevation of the water level in a well. Piezometric total head calculations in groundwater use the formula ​h=z+Ψ​ where ​h​ means total head or height of the groundwater level above the datum, usually sea level, while ​z​ represents the elevation head and ​Ψ​ represents the pressure head. The elevation head, ​z​, is the height of the bottom of a well above the datum. The pressure head equals the height of the water column above ​z​. For a lake or pond, ​Ψ​ equals zero so the hydraulic or piezometric head simply equals the potential energy of the water surface height above the datum. In an unconfined aquifer, the water level in the well will approximately equal the groundwater level. In confined aquifers, however, the water level in wells rises above the level of the confining rock layer. The total head is directly measured at the surface of the water in the well. Subtracting the elevation of the bottom of the well from the elevation of the water surface yields the pressure head. For example, the water surface in a well lies at an elevation of 120 feet above mean sea level. If the elevation at the bottom of the well lies at 80 feet above mean sea level, then the pressure head equals 40 feet. Calculating Piezometric Head in Hydroelectric Dams Calculating Piezometric Head in Hydroelectric Dams The piezometric pressure definition shows that the potential energy at the surface of a reservoir equals the elevation of the lake's surface above a datum. In the case of a hydroelectric dam, the datum used can be the surface of the water just below the dam. The total head equation simplifies to the difference in elevation from the reservoir surface and the outflow surface. For example, if the reservoir surface is 200 feet above the river level immediately below the dam, the total hydraulic head equals 200 feet. References American Meteorological Society: Glossary of Meteorology – Piezometric Head Republic of South Africa Department of Water and Sanitation: Groundwater Dictionary – Piezometric Surface Penn State University: Potential Energy and Hydraulic Head Engineers Edge: Hydraulic Head Pressure University of Calgary: Hydraulic Head Columbia University: Principles of Fluid Dynamics Cite This Article MLA Blaettler, Karen G. "How To Calculate The Piezometric Head" sciencing.com, 27 December 2020. APA Blaettler, Karen G. (2020, December 27). How To Calculate The Piezometric Head. sciencing.com. Retrieved from Chicago Blaettler, Karen G. How To Calculate The Piezometric Head last modified March 24, 2022. Recommended
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https://artofproblemsolving.com/wiki/index.php/Searching_the_community?srsltid=AfmBOoqX5dgwzTl97VSCucN1BTe-5p0jod6Cie1d3rVzp_y6P3JjBr_l
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Contents [hide] 1 Searching methods 1.1 Searching from the community 1.2 Searching from a forum 1.3 Searching from the main webpage 2 Getting the most out of search 2.1 Searching for a specific post 2.2 General searching tips 2.3 Customizing search queries 3 Searching for LaTeX and Asymptote 3.1 Searching for LaTeX 3.2 Searching for Asymptote Searching methods Searching from the community Searching from the community page The first of these two options is shown in the image to the right. It is primarily useful for quickly searching, especially when the search is of a general nature and without a specific post in mind. The method is simple: simply type your query into the "Search Community" field, click the magnifying glass (or press the Enter key), and posts containing terms matching your query will be shown to you. The parts of the posts that match your query will be highlighted in yellow. 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To reach that option, click first on the "Click here to search titles, posts, sources, and tags separately" text, then check the box indicated. The image on the left shows how this search looks on the webpage. Of course, in both searches it is possible to further narrow the results by including the forum it was posted in, text that should be present in returned posts, and so on. It is also possible to expand the search to posts by rrusczyk and copeland, or even postsbycopelandaboutrrusczyk. Getting the most out of search Searching for a specific post One of the most common uses of search is to find a specific post that you remember some details about, but cannot find. For example, you might recall having once read a particularly well-written solution, a very nice question, or an important announcement, but you can't remember quite where you saw it before. How frustrating! Obviously, the more details you remember about the post, the more likely you are to find it. If you remember that, for example, the post was written sometime in the summer of 2013, you can set the date range from (for example) May to September of that year. If you remember the author, that narrows down the possibilities significantly, but even remembering that it was one of several authors is a good way to whittle the possibilities down. Finally, remembering the forum that it was posted in would be excellent, but even if you only vaguely remember the post, you can make some educated guesses. For example, if the post was about a new and exciting technique for the USAMO, it was probably posted in either the Contests & Programs forum or the High School Olympiads forum. If the post was a collection of MATHCOUNTS strategies, it's almost certainly in Middle School Math. General searching tips Of course, the above strategies are helpful, but still leave a lot of posts to sift through - unless you remember some of the language used in the post. You might remember, for example, a particularly memorable phrase in the post, or perhaps they used a math problem as reference that you remember a bit about. In these cases, Include, in your search query, unusual or uncommon words. For example, searching for just the word "circumcenter", along with the tips in the previous section, narrows down the possibilities significantly Avoid common words such as "a", "the", "of", and so on, even if you remember a word-for-word phrase from the post containing these words. You'll get bogged down in posts matching those words - which there are a lot of! Avoid searching words that are contained within another common word, as they will get matched to words you certainly didn't intend. For example, search queries containing the word "in" might return results for "logging", "Inequality", and so on. The word "a" is particularly guilty of this, as it often returns words containing the letter "a". Be sure to avoid searching for words that are special modifiers - see the below section. In particular, avoid searches containing the words "and" and "not". Similarly, do not include quotes ("), asterisks (), question marks (?), plus signs (+), minus signs (-), or parentheses as these all represent special commands. Search engines in general are not very good at searching for mathematical symbols or numbers, so you are generally better off searching for text instead. For example, searching the text of a problem rather than equations or expressions is likely to produce better results. For example, if you remember a post contained the phrase "the three perpendicular bisectors of a triangle intersect at the circumcenter", your search query should be something similar to "perpendicular bisectors intersect circumcenter", as these are all specific words that limit the number of posts returned (unlike words like "of", "a", "the", and "at"). Customizing search queries We've already mentioned that posts are matched to the search query using a scoring algorithm, which approximates how "close" a post is to the given query. However, especially when searching for multiple terms, this algorithm often weights certain factors in different ways than the user intended. As such, there are several ways to customize your search queries. Customizing search queries Operator Example Result [[(no adjustment)incenter circumcenter Returns posts containing the word "incenter" and/or the word "circumcenter", with higher weight given to posts containing both. ++incenter circumcenter Returned posts must contain the word "incenter", but may or may not contain "circumcenter". Again, higher weight is given to posts containing both words. --incenter circumcenter excenter Returned posts must not contain the word "incenter". Returned posts will contain the word "circumcenter" and/or the word "excenter", with higher weight given to posts containing both. AND incenter AND circumcenter Returned posts must contain both "incenter" and "circumcenter". This is equivalent to the query "+incenter +circumcenter". NOT incenter NOT circumcenter Returned posts will contain "incenter", but will not contain "circumcenter". Equivalent to the "-" and "!" operators. "" (quotes)"incenter circumcenter"Returns posts containing the phrase "incentercircumcenter". Punctuation is usually ignored in results, so posts containing "incenter, circumcenter" or "incenter-circumcenter" will also be returned. ? and te?t, inc, ine Wildcard symbols. The? symbol allows any character to replace it, so posts containing the words "test" or "text" will match the query "te?t". The symbol allows any number of characters to replace it, so posts containing the words "incenter", "incircle", "inclusive", etc. will match the query "inc". The symbol can also be used in the middle of a word, so posts containing the words "interface", "incircle", "intermediate", "infinite", etc. will all match the query "ine". () (parentheses)incenter AND (circumcenter OR excenter)Grouping symbols to allow one boolean command (e.g. AND, NOT, +, -,!) to modify multiple elements. The example returns posts that contain both "incenter" and at least one of "circumcenter" or "excenter". Equivalent to the query (incenter AND circumcenter) OR (incenter AND excenter) (Boolean logic applies to search strings). \AND An "escape" character that allows you to search for reserved keywords and symbols. For example, the above query will return posts containing the word "and". Without the escape character, the search engine would parse the query "AND" as a command linking two nonexistent terms, and would thus throw an error. Similarly, the search term "incenter \AND circumcenter" would include results containing the word "incenter" and the word "and", but not the word "circumcenter". Note: in order to search for terms containing backslashes, such as LaTeX commands, the somewhat non-intuitive \ is necessary (the first backslash "escapes" the second one). Note that, when using search modifiers, the yellow background indicating words that match your query will not generally be entirely accurate; for example, searching for "incenter AND circumcenter" will return posts containing both words as expected, but the word "and" will also be highlighted. Don't get confused by this - the search engine is still searching according to your query (and is not searching for the word "and"), the highlighting is simply slightly misleading. Searching for LaTeX and Asymptote Searching for LaTeX or Asymptote (Vector Graphics Language) can be complicated, as they are their own languages, but this can actually be a big advantage! Predicting what code they've used is usually quite easy, and since the keywords are generally quite unique, searching for them will often narrow results down to posts using LaTeX/Asymptote themselves. Below are some common commands that are often used, to help you search for them. Searching for LaTeX LaTeX is a programming language for rendering mathematical statements, and is very popular on AoPS (and other mathematical sources). LaTeX commands are enclosed in dollar signs for rendering, but you should not generally include those while searching. If you are searching for "full" commands (e.g. \frac{1}{2}), it is highly recommended that you enclose the entire command with quotation marks (e.g. "\frac{1}{2}"; recall double backslash is needed), as otherwise it may parse as three different search terms (\frac, 1, and 2). Common LaTeX commands Command Use Example Rendered \frac Creates fractions\frac{1}{2} \sqrt Creates square (or, more generally, th) roots\sqrt{3}, \sqrt{3}, \leq, \neq, \geq Less than or equal to, not equal to, greater than or equal to (respectively)a \leq b, a \neq b, a \geq b \alpha Renders greek letters (most often alpha, beta, epsilon, pi, theta, phi, and omega)\alpha+\beta=\pi-\epsilon \rightarrow, \implies, \iff Used for implication, algorithms, etc.a>b \implies a^2>b^2, a>b \iff a-b>0, A\rightarrow A+1 \sum, \prod, \int Used for summation, product, and integration symbols\sum_1^5 x=15, \prod_1^5x=120, \int_1^5x=12 See LaTeX:Symbols for a more comprehensive list of LaTeX commands, and use the TeXeR to test how commands look when rendered. Searching for Asymptote Asymptote is a programming language for creating diagrams. Unlike LaTeX, dollar signs are not necessary, and Asymptote code is enclosed in [asy] tags. Asymptote commands take the form of most modern programming languages; lines are generally of the form command(param1, param2, ...) (this is different from TeX, where each parameter is enclosed in separate brackets). "Unfortunately" (for our purposes), most Asymptote commands are simply the word-for-word descriptors of their function; for example, the command that returns the midpoint of a path is simply "midpoint", and the command that returns the circumcenter of a triangle is "circumcenter". This is further complicated by the usage of variables, the names of which are entirely up to the posters (unlike TeX, in which variables are generally not used). Below are commonly used Asymptote commands that are not actual words, so they are less likely to be confused with other posts during searching. Common Asymptote commands Command Purpose Command Purpose defaultpen Adjusts the default settings for the pen. Very likely for this command to be in an Asymptote drawing.orthographic Adjust the "camera position" in 3-D drawings. Very likely for this command to be in a 3-D Asymptote drawing. filldraw Colors a given shape in some manner. Useful for shading in parts of diagrams; used reasonably foten.unitsize Sets the default size of a "unit" to a specified value. Default is very small. Not used particularly often because defaultpen is more general. intersectionpoint Determines the intersection (if there is exactly one) of two paths. Very useful command, and thus in many drawings.intersectionpoints Determine all the intersections of two paths. Obviously very similar to the previous command, but slightly less used as we generally want to know the intersection point of lines (rather than arbitrary paths). rightanglemark Marks a given angle as right. Used, obviously, in diagrams containing right angles; there are many of these.anglemark Marks a given angle with a given value. Used surprisingly little, since directly marking angles isn't generally that important, but angle chase solutions will make heavy use of it. linewidth Sets the (visual -- lines have no real width!) width of a line. Also little used because defaultpen is more general.currentpicture A reference to the current picture. Used as an auxiliary command in other commands. This is the default value, so it is used only in very complicated drawings (or as a formality/copied and pasted sections). Note that simply searching "asy" will generally turn up Asymptote drawings or discussions of such; especially when combined with a username or date range, simply remembering a post had any Asymptote in it whatsoever is usually a good way to find it. As Asymptote becomes more and more popular, the previous statement will become less true (LaTeX, for example, is in a huge number of posts these days), but for now the numbers are small enough that you can sift through them pretty quickly. 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