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Solve the system of equations using the addition (elimination) method. If the answer is a unique solution, present it as an ordered pair: (x, y). If not, specify whether the answer is "no solution" or "infinitely many solutions." 4x – 3y = 1 -12x + 9y = 5 | 677.169 | 1 |
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0.23 MB | 4 pages
PRODUCT DESCRIPTION
I designed these lessons to teach my students about geometric figures (the 5th in a series of 9). This lesson can be purchased as a complete bundled unit at a discounted price under the listing Features of Functions Complete Bundled Unit.
This lesson focuses on: given a graph of a function--finding where x and y equal specific numbers, given an arithmetic or geometric story problem--finding the rate of change, an explicit and recursive equation and telling what the input and output would mean, interpreting discrete and continuous functions from graphs | 677.169 | 1 |
Intro to Fourier series and how to calculate them
Published on 06 Oct 2010 | over 6 years ago
Download the free PDF from tinyurl.com/EngMathYT
This is a basic introduction to Fourier series and how to calculate them. An example is presented that illustrates the computations involved. Such ideas are seen in university mathematics. | 677.169 | 1 |
Advanced Mathematics Suite, or AMS for short, is a program which makes up for where TI left off when they developed the TI-83 Plus graphing calculator. It includes various useful utilities for work with Algebra I, II, Geometry, and other subjectsThis project is an easy to use Java application framework, providing enough functionalities to be fully usable. It has been developed for a scientific articles library software, but I hope it's sufficiently generic to be usable on different contexts | 677.169 | 1 |
...
Show More mathematical models with user-friendly interactive computer programs, written in the powerful and popular MATLAB. This unique merger of technical referencing and interactive computing allows instant solution of a variety of engineering problems, and in-depth exploration of the physics of deformation, stress and motion by analysis, simulation, graphics, and animation. This book is ideal for both professionals and students dealing with aerospace, mechanical, and civil engineering, as well as naval architecture, biomechanics, robotics, and mechtronics. For engineers and specialists, the book is a valuable resource and handy design tool in research and development. For engineering students at both undergraduate and graduate levels, the book serves as a useful study guide and powerful learning aid in many courses. And for instructors, the book offers an easy and efficient approach to curriculum development and teaching innovation. * Combines knowledge of solid mechanics--including both statics and dynamics, with relevant mathematical physics and offers a viable solution scheme. * Will help the reader better integrate and understand the physical principles of classical mechanics, the applied mathematics of solid mechanics, and computer methods. * The Matlab programs will allow professional engineers to develop a wider range of complex engineering analytical problems, using closed- solution methods to test against numerical and other open-ended methods. * Allows for solution of higher order problems at earlier engineering level than traditional textbook approaches | 677.169 | 1 |
They start with real numbers and their basic properties, then turn to equations and inequalities, graphing and solving systems of equations and inequalities, polynomials, factoring polynomials, proportions and rational expressions, writing equations of lines along with functions and variations, radicals and rational exponents, quadratic functions, inequalities, algebra of functions, exponential and logarithmic functions, conic sections and a set of miscellaneous topics such as geometric sequences | 677.169 | 1 |
Resource Added!
Type:
Interactive, Graphic Organizer/Worksheet
Description:This lesson should take one day of class time. The purpose of this lesson is to
develop an understanding of quadratic functions. We use the linear relation between
distance, constant speed and time and the quadratic relation between the vertical distance
of a falling object and time. From these, students will develop two new quadratic
functions. The graph of one of these provides a picture of the physical phenomenon they
have viewed.
Hands-on activity. The purpose is to help students learn to use variables and to write equations to model a problem situation. Helps students learn how to work simple "word problems" by actually doing what is said.
A hands-on activity to help students understand four basic binary operations in algebraic expressions, solving linear equations in 1 variable, and solving a formula for an indicated variable, using value of collections of coins.
The second part of the Money Investigation activities. A hands-on activity to help students understand four basic binary operations in algebraic expressions, solving linear equations in 1 variable, and solving a formula for an indicated variable, using value of collections of coins.
This investigation of the genetics of the Sickle Cell trait via a mathematical model
uses probability and teaches properties of quadratic functions and the concept of
optimization of a function.
The properties of quadratic functions brought out by this investigation are
The functions that model the process of the elimination of alcohol from the body
serve as an introduction to a study of rational functions at an intermediate algebra level.
The lesson focuses on graphs of the functions with an emphasis on interpretation of the
horizontal and vertical asymptotes in the context of elimination of alcohol from the
body. Other mathematics involved is algebraic manipulation of the rational functions,
solution of equations with rational expressions, realistic domain of a function, inverse
functions, and equilibrium state of a dynamic process.
"Reading This Could Help You Sleep: Caffeine in Your Body" is an introduction to
exponential functions of the form ab^t at the Intermediate Algebra level, with emphasis on the meaning of these functions and their graphs. The concept of half-life is
introduced. A conditional function (a piecewise-defined function) is used.
"Get the Lead Out" extends the study of exponential functions and can be used to
introduce the use of logarithms to "un-do" exponential expressions in solving equations.2 days. Hands-on activity using the actual path of light through your container of water. Students compute the speed of light in water and develop an understanding of why light reflects back if below a critical angle. It provides an opportunity to build a fairly complicated function from simpler ones in a physical setting, and then investigate the function to learn something about the real world. It involves investigation of function with a parameter. This unit is appropriate for a precalculus course in a unit on functions or or in a unit on functions involving radicals. It has also been used in elementary calculus courses as a conceptual introduction to optimization.
2 days. Hands-on simulation activity. Used to introduce solving linear systems of 2 equations in two unknowns, with follow-up involving 3 equations and 3 unkowns. This unit studies interconversion of two drugs in the blood, that is, the case where the body metabolizes each of two drugs into the other, which is what happens for vitamin K and another chemical. This requires one day, with homework given. On the following day, you discuss solving systems of equations and can use a second set of homework that comes with this unit. The second set of homework studies drugs which are absorbed into different compartments of the body, such as vitamin A which is in the blood and in the liver. Intermediate algebra or precalculus. | 677.169 | 1 |
took Linear Algebra at M.I.T. The book was Gilbert Strang's "Linear Algebra and Its Applications." As a mathematician, once you learn linear algebra, you never stop using it. However, for the finer points, I have ready access to the book as a reference | 677.169 | 1 |
Fractions, Decimals, and Percents GMAT Strategy GuideMore...
The practice implementing strategic shortcuts. Each chapter builds comprehensive content understanding by providing rules, strategies, and in-depth examples of how the GMAT tests a given topic and how you can respond accurately and quickly. The Guide contains #x1C;In-Action#x1D; problems of increasing difficulty with detailed answer explanations. The content of the book is aligned to the latest Official Guides from GMAC (12th edition). Purchase of this book includes one year of access to Manhattan GMAT#x19;s online practice exams and Fractions, Decimals, & Percents question bank | 677.169 | 1 |
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7.97 MB
PRODUCT DESCRIPTION
This is two 8th Grade Common Core guided, color-coded notebook pages for the Interactive Math Notebook on Special Types of Solutions to Systems of Equations.
Included is a graphic organizer with the special types of solutions, examples of each graphically and algebraically and hints for each type of solution. (One solution, No Solution and Infinite Many Solutions) And explicit notes on determining the type of solution using the elimination method.
Blackline master and color-coded answer key included.
** My Interactive Note Pages include all or some of the following: step by step color-coded notes, diagrams, academic vocabulary, graphic organizers and example problems.
My Interactive Note Pages were designed to use in my IMN. The students keep the color-coded notes in a 3-pronged folder, and the notes are set up to print back to back | 677.169 | 1 |
Powers, Roots and Absolutes!
40 Downloads
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0.58 MB | 4 pages
PRODUCT DESCRIPTION
This worksheet reviews powers, roots and absolute values. I like to combine these and discuss grouping symbols prior to order of operations. This challenging worksheet includes negatives inside and outside of grouping symbols. Also included are practice problems for comparing values, encouraging students to work on their estimation skills. I don't allow students to use a calculator. Appropriate for introduction in Pre-Algebra, Algebra, and review for Algebra 2 | 677.169 | 1 |
Secondary Calculus
Secondary calculus is the result of a natural evolution of the
classical geometrical theory of partial differential equations (PDE)
originated by Sophus Lie. In particular, it allows the construction
of a general theory of PDE, in the same manner as algebraic geometry
does with respect to algebraic equations. There are strong indications that
secondary calculus may become a natural language for quantum field
theory, just in the same way as standard calculus is for classical physics.
From the mathematical point of view, secondary calculus is a complex
mathematical construction putting into a natural interrelation many
parts of modern mathematics such as commutative and homological
algebra, algebraic and differential topology, differential geometry,
etc. The strategic goal of the Diffiety Schools to involve interested
participants into a series of large scale research programs the Levi-Civita Institute is launching.
Initial
ideas of the area can be got from the books. Advance topics can be found in the various papers. | 677.169 | 1 |
new edition of the precalculus text developed by the Consortium based at Harvard University and funded by a National Science Foundation Grant. The text is thought-provoking for well-prepared students while still accessible to students withMore...
This is a new edition of the precalculus text developed by the Consortium based at Harvard University and funded by a National Science Foundation Grant. The text is thought-provoking for well-prepared students while still accessible to students with weaker backgrounds. It provides numerical and graphical approaches as well as algebraic approaches to give students another way of mastering the material. This approach encourages students to persist, thereby lowering failure rates. A large number of real-world examples and problems enable students to create mathematical models that will help them understand the world in which they live. The focus is on those topics that are essential to the study of calculus and these topics are treated in depth. Linear, exponential, power, and periodic functions are introduced before polynomial and rational functions to take advantage of their use to model physical phenomena. Building on the Consortium's Rule of Four: Each function is represented symbolically, numerically, graphically, and verbally where | 677.169 | 1 |
Business & Everyday Math Course
Get comfortable with numbers in business and life.
The business world is founded on numbers and math. Mathematical mindfulness can help eliminate costly errors, drive profit and generally lead to a greater confidence in business (and everyday) discussions. It's therefore vital for business people to be comfortable with the most important and frequently-occurring features of the discipline. We can help you with that.
Whether you haven't touched a calculator since high school, or you just don't find simple interest simple, our refreshingly practical course will tailor itself closely to your knowledge and ambitions using our signature diagnosis test. Using guided examples, videos and constant application to realistic business scenarios, our course carefully teaches you all the skills required to transform math from a hazard to be feared into a weapon to be utilized. Allow Dr Clare Morris, professor at the University of Gloucestershire and published author, to make you feel more at home in numberland.
Preview
Course Content
Course Content Alt
Syllabus Explorer
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Syllabus
Section 1: Introducing the basics
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Unit 1.1 - Getting Started
Introduces you to the purpose and potential of the course, includes the diagnostic test that will tailor the course to your requirements, and advises you on how to approach both the course and math in general in order to get the most out of your learning.
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Unit 1.2 - Be Confident with Numbers
Teaches you the fundamentals of the decimal number system, outlines the theory behind and the operation of the key arithmetic operations, and explains the essential terminology and anatomy of mathematical calculations.
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Unit 1.3 - Fractions, Decimals, Percentages and Ratios
Moves the course firmly into the world of business application by explaining in depth the four critical mechanisms through which numbers interact with one another, giving constant and explicit examples of how each is central to the business world.
Section 2: Quirks in Math
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Unit 2.1 - Notation
Explains the operation of powers and roots, which are central to the decimal system, and teaches you how to handle all numbers - huge or minute - with ease and clarity.
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Unit 2.2 - Estimation and Accuracy
Teaches you how to remain in control of math even without a calculator or a computer by outlining the ways in which degrees of accuracy, rounding, approximations and calculation checks can all be used to apply math to business in real time, such as in meetings or calls.
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Unit 2.3 - Working with Letters
Guiding you gently through the often-daunting world of algebra, Clare explains the reasons for algebra, its applicability to the business world, and its operation – from the underlying theory to the solution of linear equations.
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Unit 2.4 - Debts and Debits
Outlines how negative numbers operate, utilising a number of business examples to teach you how negative numbers are both critical to the business world, and easy to work with once you know a few simple rules.
Section 3: Working with data and tools
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Unit 3.1 - Tools for Arithmetic
Connects math firmly with the real world by demonstrating how to get maximum use from your calculator and Excel spreadsheets in dealing with business scenarios.
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Unit 3.2 - Graphs
Teaches you how to plot, read, make sense of and utilize the vast potential of different types of graphs as visual representations of business data.
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Unit 3.3 - Business Applications and Handling Data
Applies what you have learnt throughout the course to the very heart of business, demonstrating why math is so critical to interest, depreciation, value, data handling and statistics: concepts unavoidably critical to the everyday running of a successful business.
Section 4: Advanced techniques
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Unit 4.1 - Handling Data
Building on the utilisation of graphs and business applications of the previous two units, this unit also teaches you how to collect, display, interpret and benefit from data collections – a process at the heart of all world-leading businesses and organisations.
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Unit 4.2 - Dealing with Chance
Demonstrates how math does not simply allow you to make sense of the present, but allows you to speculate into the future, by teaching you how to use the concept of probability to make real life business, projections, predictions and decisions.
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Unit 4.3 - Conclusion
Summarizes what you have learnt, how to apply what you have learnt, and how to build upon what you have learnt to be more successful in business.
Pricing
*The AUD price shown above is an estimate. Actual charges are based on the current exchange rate from GBP
Requirements
Software:
Required: none
Completion Time: 10 hours (average)
About the Author
Clare Morris
Professor Clare Morris has more than 30 years' experience of teaching mathematics and statistics to business students, in some of the UK's leading business schools including Warwick Business School and the Cardiff University Business School. She is the author of a number of successful texts including 'Quantitative Approaches in Business Studies', now in its 8th edition (Pearson, 2011) and 'Essential maths for business and management' (Palgrave, 2007), as well as many journal and conference papers.
Clare holds an MA degree from the University of Oxford, and MSc and PhD from the University of Bristol. She is a Chartered Statistician and Fellow of the Royal Statistical Society, and has served as a member of the Council and chair of the Quality Improvement Committee. She has worked as a statistical consultant for major companies including Hewlett- Packard and Land Rover, as well as for a number of UK Government departments. She has a particular interest in distance and online learning, and is a tutor for the UK's Open University. Clare believes strongly that everyone can learn and enjoy mathematics, as long as it is presented in a de-mystified and accessible way. | 677.169 | 1 |
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Free 'Help With GCSE Maths' Udemy Course
"A series of tips and tricks to make the basics of GCSE Maths easier, followed by a whole bunch of past paper questions, with step by step video solutions. This course is designed for anyone looking for GCSE Maths Help and to consolidate at the GCSE C grade, or perhaps looking to improve from a high E or solid D to a grade C within a few weeks with GCSE Maths Online."
So, if you would like to take a step towards help or helping with some GCSE Maths then why not sigh up to this FREE course. To go grab it simply click the GRAB FREEBIE button below to be taken to the Udemy course. If you've not already signed up to Udemy then you need to do that; but that is also FREE.. | 677.169 | 1 |
PRODUCT DESCRIPTION
This unit activity on Radicals is designed for Algebra 1 or Algebra 2 students. It can also be used as a review in PreCalculus or for SAT and ACT skills.
The resource includes:
• 2 versions of a test/study guide/review, each with 40 questions. One version is free response and the other one is multiple choice.
• 21 Task Cards
• Student recording sheet
• Blank Cards for you to customize
Note: The task cards have different questions than the worksheet/tests.
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This worksheet allows students to work on their own or with a partner to investigate, using a graphing calculator, what all the different numbers in a sine or cosine equation do to the graph. It looks at it from a transformations approach, drawing from their prior knowledge of other function transformations. It asks them to look for patterns and explain their reasoning | 677.169 | 1 |
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Ganit Mathematics
Author(s) :
C Sailaja, Lata Thergaonkar & Smita Ratish
Ganit Mathematics series consists of seven textbooks; two textbooks for Primer A, B and five textbooks for classes 1–5. The textbooks meet the requirements of the latest ICSE syllabus classes 6-8 About the Author Lata Wishram is a renowned academician with more than two decades of experience in teaching mathematics. She has worked as Principal at Naval Public School, Visakhapatnam. She has also taught mathematics at various reputed institutions, like Army School, Jhansi; Army School, Kalimpong; St. Miras School, Pune and Sadhu Vaswani International School, Delhi.
For Primer A, B and classes 1-5 C. Sailaja
C. Sailaja is a reputed author having over 15 years of experience in teaching mathematics. She is presently working as Mathematics Teacher(Primary Section), at NASR Girls School, Hyderabad.
Smita Ratish
Smita Ratish is a well-known author having teaching experience of more than a decade. She is presently working as Primary Teacher at New Horizon School, Bengaluru.
GANIT MATHEMATICS series consists of ten textbooks; two textbooks for Primer A and B, eight textbooks for classes 1-8. This series is strictly bases on the syllabus prescribed by the Council for the Indian School Certificate Primer A, B and classes 1-5
For Primer A, B and classes 1-5
Let us Recall given at the beginning of each chapter to brush up the prior knowledge of learner relevant to the forthcoming chapter
Maths and our Environment exercises to associate and blend mathematics with art, language and environmental science
Remember to emphasize on tips and tricks for better retention of concepts
Hands-on Activities based on key concepts to make learning an enjoyable experience
Exercises given at the end of each concept to assess the understanding of the concept | 677.169 | 1 |
Forum for Science, Industry and Business
The Aftermath of Calculator Use in College Classrooms
13.11.2012
Students may rely on calculators to bypass a more holistic understanding of mathematics, says Pitt researcher
Math instructors promoting calculator usage in college classrooms may want to rethink their teaching strategies, says Samuel King, postdoctoral student in the University of Pittsburgh's Learning Research & Development Center.
King has proposed the need for further research regarding calculators' role in the classroom after conducting a limited study with undergraduate engineering students published in the British Journal of Educational Technology.
"We really can't assume that calculators are helping students," said King. "The goal is to understand the core concepts during the lecture. What we found is that use of calculators isn't necessarily helping in that regard."
Together with Carol Robinson, coauthor and director of the Mathematics Education Centre at Loughborough University in England, King examined whether the inherent characteristics of the mathematics questions presented to students facilitated a deep or surface approach to learning. Using a limited sample size, they interviewed 10 second-year undergraduate students enrolled in a competitive engineering program. The students were given a number of mathematical questions related to sine waves—a mathematical function that describes a smooth repetitive oscillation—and were allowed to use calculators to answer them. More than half of the students adopted the option of using the calculators to solve the problem.
"Instead of being able to accurately represent or visualize a sine wave, these students adopted a trial-and-error method by entering values into a calculator to determine which of the four answers provided was correct," said King. "It was apparent that the students who adopted this approach had limited understanding of the concept, as none of them attempted to sketch the sine wave after they worked out one or two values."
After completing the problems, the students were interviewed about their process. A student who had used a calculator noted that she struggled with the answer because she couldn't remember the "rules" regarding sine and it was "easier" to use a calculator. In contrast, a student who did not use a calculator was asked why someone might have a problem answering this question. The student said he didn't see a reason for a problem. However, he noted that one may have trouble visualizing a sine wave if he/she is told not to use a calculator.
"The limited evidence we collected about the largely procedural use of calculators as a substitute for the mathematical thinking presented indicates that there might be a need to rethink how and when calculators may be used in classes—especially at the undergraduate level," said King. "Are these tools really helping to prepare students or are the students using the tools as a way to bypass information that is difficult to understand? Our evidence suggests the latter, and we encourage more research be done in this area."
King also suggests that relevant research should be done investigating the correlation between how and why students use calculators to evaluate the types of learning approaches that students adopt toward problem solving in | 677.169 | 1 |
Tagged: math tools
Microsoft Mathematics is an application that provides a set of mathematical tools which will help students get their school work done quickly and easily. With Microsoft Mathematics, students can easily learn to solve equations step-by-step while gaining a better understanding of fundamental concepts in pre-algebra, algebra, trigonometry, physics, chemistry, and... | 677.169 | 1 |
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$26 thorough revision of Teaching Secondary Mathematics: Techniques and Enrichment Unitsincludes the most practical, step-by-step techniques for teaching mathematics in today#x19;s assessment and standards driven environment.#xA0; Part One on methods discusses all aspects of being a math teacher, from an overview of the discipline, to challenges of teaching today, the role of problem-solving, the importance of planning, assessment strategies, teaching more effective lessons, enriching mathematics instruction, and extracurricular activities for mathematics.#xA0; Part Two supplies 125 enrichment units that teachers can use right away in their own classrooms that are suitable for the entire secondary school curriculum spectrum.#xA0; From methods to hands on activities, this highly successful book takes students through the whole process of what teaching mathematics in the secondary schools will be like.#xA0; #xA0; New To This Edition: #xA0; NEW: Includes the latest information on changes to student assessment that have been made possible due to innovations in technology. #xA0; NEW!#xA0; Features new content on data driven planning. #xA0; NEW!#xA0; Completely revised technology chapter reflects the rapidly changing technology environment of today. #xA0; NEW!#xA0; No chapters remain untouched!
Author Biography
Alfred S. Posamentier is Professor of Mathematics Education and Dean of the School of Education of The City College of the City University of New York. He is the author and co-author of many mathematics books for teachers and secondary school students.
After completing his A.B. degree in mathematics at Hunter College of the City University of New York, he took a position as a teacher of mathematics at Theodore Roosevelt High School in the Bronx (New York), where he focused his attention on the teaching process in general and the improvement of students' problem-solving skills in particular. He developed the school's first mathematics teams and established a special class whose primary focus was enrichment topics in mathematics and problem solving. After six years as a high school teacher, Dr. Posamentier joined the faculty of The City College where he also received his master's degree. He began to develop in-service courses for secondary school mathematics teachers, focusing on practical classroom applications of educational research. These courses addressed such topics as the uses of new technology in mathematics instruction, efficient ways to teach weaker students, problem-solving strategies, and the enrichment of mathematics through a variety of ways including, but not limited to, recreational mathematics.
Dr. Posamentier received his Ph.D. from Fordham University (New York) in mathematics education. He is an Honorary Fellow at the South Bank University (London, England). He has been visiting professor at the Technical University of Vienna and the Humboldt University at Berlin, and a Fulbright Professor at the University of Vienna. Dr. Posamentier is often cited for his outstanding teaching. The City College Alumni Association named him Educator of the Year (1993) and he also on May 1, 1993 had a "Day" named in his honor by the City Council President of New York City. He was awarded the Grand Medal of Honor from the Federal Republic of Austria and the Medal of Distinction from the city of Vienna. In 1999 he was awarded the title of University Professor for Austrian Universities.
Now, after more than 35 years on the faculty of CCNY, he still exudes an ever-increasing energy and enthusiasm for mathematics and mathematics education. With his penchant for mathematics instruction, he has been especially concerned that during the recent years of mathematics teacher shortages, those who enter the classroom are as well prepared as possible. He enthusiastically believes that providing mathematics teachers with an appropriate repertoire of teaching strategies enables them to fulfill an essential role in society: empowering our nation's youngsters to engage in the critical study of mathematics.
Dr. Beverly Smith Beverly Smith holds a, MA and Ed.D. in mathematics education from Teachers College — Columbia University and an M.S. degree in computer science from Union College. Prior to becoming a mathematics-teacher-educator, Dr. Smith taught mathematics and computer science at the secondary school and college level in New York State and Massachusetts. She is currently an Associate Professor in the Secondary Education Department at The City College of New York.
Dr. Smith's research interests are in the area of teacher professional development. As part of her responsibilities as a faculty member of MetroMath: The Center for Mathematics in America's Cities, she is studying the development of alternatively certified mathematics teachers who are participating in the New York City Teaching Fellows Program. In addition, Dr. Smith is working with mathematics teachers to better understand how technology can support formative assessment in urban mathematics classrooms.
Table of Contents
Methods of Teaching Secondary Mathematics
The Challenge of Teaching
Planning for Instruction
Teaching More Effective Lessons
The Role of Problem-Solving
Using Technology to Enhance Mathematics Instruction
Assessment
Enriching Mathematics Instruction
Extracurricular Activities in Mathematics
Enrichment Units for the Secondary School Classroom
Cross-Catalogue of Enrichment Units
Constructing Odd-Order Magic Squares
Constructing Even-Order Magic Squares
Introduction to Alphametics
A Checkerboard Calculator
The Game of Nim
The Tower of Hanoi
What Day of the Week Was It? Palindromic Numbers
The Fascinating Number
Nine Unusual Number
Properties Enrichment with a Handheld
Calculator
Symmetric Multiplication
Variations on a Themendash;Multiplication
Ancient Egyptian Arithmetic Napier's Rods
Unit Pricing Successive Discounts and Increases
Prime and Composite Factors of a Whole Number
Prime Numeration System
Repeating Decimal Expansions Peculiarities of Perfect
Repeating Decimals Patterns in Mathematics
Googol and Googolplex
Mathematics of Life
Insurance Geometric Dissections
The Klein Bottle
The Four-Color Map Problem
Mathematics on a Bicycle
Mathematics and Music
Mathematics in Nature
The Birthday Problem
The Structure of the Number
System Excursions in Number Bases Raising Interest
Reflexive, Symmetric, and Transitive Relations
Bypassing an Inaccessible Region
The Inaccessible Angle Triangle Constructions
The Criterion of Constructibility
Constructing Radical Lengths Constructing a Pentagon
Investigating the Isosceles Triangle Fallacy
The Equiangular Point
The Minimum-Distance Point of a Triangle
The Isosceles Triangle Revisited Reflective
Properties of the Plane
Finding the Length of a Cevian of a Triangle
A Surprising Challenge Making Discoveries in Mathematics Tessellations | 677.169 | 1 |
putational Science75.23Encompasses the full range of computational science and engineering from modelling to solution, both analytical and numerical. It develops a framework for the equations and numerical methods of applied mathematics. Gilbert Strang has taught this material to thousands of engineers and scientists (and many more on MIT's OpenCourseWare 18.085-6). His experience is seen in his clear explanations, wide range of examples, and teaching method. The book is solution-based and not formula-based: it integrates analysis and algorithms and MATLAB codes to explain each topic as effectively as possible. The topics include applied linear algebra and fast solvers, differential equations with finite differences and finite elements, Fourier analysis and optimization. This book also serves as a reference for the whole community of computational scientists and engineers. Supporting resources, including MATLAB codes, problem solutions and video lectures from Gilbert Strang's 18.085 courses at MIT, are provided at math.mit.edu/cse. | 677.169 | 1 |
Pre-Calculus B Course
Registration Code: MA883B Credits: 1 Price: $299.00
Two-semester course with explores in great detail the relationship that exists between advanced algebra topics and trigonometry objectives. An exploration into the nature of graphs (including the twelve basic functions) is provided along with nonlinear systems, polynomial and rational functions. Logarithmic knowledge and application is developed. A large portion of the class focuses on trigonometric graphs and identities. Other topics include vectors, parametric equations and sequences and series. The students are introduced into the world of calculus, by exploring topics of limits, continuity, derivatives and the Fundamental Theorem of Calculus. | 677.169 | 1 |
Need to keep your rental past your due date? At any time before your due date you can extend or purchase your rental through your account.
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Summary
This text, intended for a graphing calculator required precalculus course, shows students when and how to use concepts, and promotes real understanding not just rote memorization. In addition, the graphing calculator is used as a tool to help explain ideas rather than merely to find answers. The book reflects AMATYC, MAA, and NCTM guidelines, and makes use of real world data in presenting a balanced algebraic and graphical approach to understanding precalculus concepts. The result is a thorough preparation for the calculus course. | 677.169 | 1 |
Elementary Teachers is an unbound, binder-ready edition. When students truly understand the mathematical concepts, it's magic. Students who use this text are motivated to learn mathematics. They become more confident and are better able to appreciate the beauty and excitement of the mathematical world. That's why the new Ninth Edition of Musser, Burger, and Peterson's best-selling textbook focuses on one primary goal: helping students develop a true understanding of central concepts using solid mathematical content in an accessible and appealing format. The components in this complete learning program--from the textbook, to the eManipulative activities, to the online problem-solving tools and the resource-rich website--work in harmony to help achieve this goal. This edition can also be accompanied with WileyPlus, an online teaching and learning environment that integrates the entire digital textbook with the most effective resources to fit every learning style. WileyPLUS sold separately from text. | 677.169 | 1 |
Essays on statistics mathematics
Welcome. The Utah State Board of Education (USBE) has a core mission of ensuring students are well prepared for their future by providing high quality instruction in. Related Resources Essays. Each month James Tanton writes math essays for all to enjoy. Here are the latest: September Cool Math Essay (pdf) September Curriculum. On Mathematics, Mathematical Physics, Truth and Reality. NOTE: These pages deal with the Philosophy and Metaphysics of Mathematics and the Mathematical.
Online homework and grading tools for instructors and students that reinforce student learning through practice and instant feedback. Free Career papers, essays, and research papers These results are sorted by most relevant first (ranked search). You may also sort these by color rating or essay. The CLEP College Mathematics exam covers material generally taught in a college course for non-mathematics majors. Mathematics is the study of numbers, quantity, space, structure, and change. Mathematics is used throughout the world as an essential tool in many fields. College study is the process of acquainting students with values and procedures central to scholarship. All students are expected to do their own work. College study is the process of acquainting students with values and procedures central to scholarship. All students are expected to do their own work.
Essays on statistics mathematics
On Mathematics, Mathematical Physics, Truth and Reality. NOTE: These pages deal with the Philosophy and Metaphysics of Mathematics and the Mathematical. Free Career papers, essays, and research papers These results are sorted by most relevant first (ranked search). You may also sort these by color rating or essay. Introduction to Use of statistics in daily life:- Statistics deal with frequency distribution. It is used to compare twoor more frequency distribution. Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of topics such as quantity , structure, space, and change.
Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials. Introduction to Use of statistics in daily life:- Statistics deal with frequency distribution. It is used to compare twoor more frequency distribution.
The CLEP College Mathematics exam covers material generally taught in a college course for non-mathematics majors. Product information and search tools. Find information, locate, learn how to order, and browse the content of NCES publications or download data files. Online homework and grading tools for instructors and students that reinforce student learning through practice and instant feedback.
Ivy League writers provided Admission Essay, Personal statement & Letter of Recommendation writing services for college, grad, mba, med & law school students. Welcome. The Utah State Board of Education (USBE) has a core mission of ensuring students are well prepared for their future by providing high quality instruction in. Product information and search tools. Find information, locate, learn how to order, and browse the content of NCES publications or download data files.
Free descriptive Free descriptive papers, essays, and research papers These results are sorted by most relevant first (ranked search). You may also sort these by color rating or. Mathematics Standards Download the standards Print this page. For more than a decade, research studies of mathematics education in high-performing countries. | 677.169 | 1 |
This new approach to real analysis stresses the use of the subject in applications, by showing how the principles and theory of real analysis can be applied in a variety of settings in subjects ranging from Fourier series and polynomial approximation to discrete dynamical systems and nonlinear optimization. Users will be prepared for more intensive work in each topic through these applications and their accompanying exercises. Chapter topics under the abstract analysis heading include: the real numbers, series, the topology of R^n, functions, normed vector spaces, differentiation and integration, and limits of functions. Applications cover approximation by polynomials, discrete dynamical systems, differential equations, Fourier series and physics, Fourier series and approximation, wavelets, and convexity and optimization. This book is appropriate for math enthusiasts with a prior knowledge of both calculus and linear algebra.
Review of the first edition, Real Analysis with Real Applications:
"A well balanced book! The first solid analysis course, with proofs, is central in the offerings of any math.-dept.; — and yet, the new books that hit the market don't always hit the mark: The balance between theory and applications, — between technical proofs and intuitive ideas, — between classical and modern subjects, and between real life exercises vs. the ones that drill a new concept. The Davidson-Donsig book is outstanding, and it does hit the mark. The writing is both systematic and engaged.- Refreshing! Novel: includes wavelets, approximation theory, discrete dynamics, differential equations, Fourier analysis, and wave mechanics." (Palle E. T. Jorgenson, Review from Amazon.com) | 677.169 | 1 |
Geometry for College Students challenges many mathematics students face occurs after they complete their study of basic calculus and linear algebra, and they start taking courses where they are expected to write proofs. Historically, students have been learning toMore...
One of the challenges many mathematics students face occurs after they complete their study of basic calculus and linear algebra, and they start taking courses where they are expected to write proofs. Historically, students have been learning to think mathematically and to write proofs by studying Euclidean geometry. In the author's opinion, geometry is still the best way to make the transition from elementary to advanced mathematics. The book begins with a thorough review of high school geometry, then goes on to discuss special points associated with triangles, circles and certain associated lines, Ceva's theorem, vector techniques of proof, and compass-and-straightedge constructions. There is also some emphasis on proving numerical formulas like the laws of sines, cosines, and tangents, Stewart's theorem, Ptolemy's theorem, and the area formula of Heron. An important difference of this book from the majority of modern college geometry texts is that it avoids axiomatics. The students using this book have had very little experience with formal mathematics. Instead, the focus of the course and the book is on interesting theorems and on the techniques that can be used to prove them. This makes the book suitable to second- or third-year mathematics majors and also to secondary mathematics education majors, allowing the students to learn how to write proofs of mathematical results and, at the end, showing them what mathematics is really all about | 677.169 | 1 |
Hall and Mercer's text is intended for schools that want a single book covering the standard topics from elementary algebra through intermediate algebra. The text is fully integrated, rather than being simply the joining of two, separate texts. Topics are organized not following the historical pattern, but by using as the guiding principles, the AMATYC standards as outlined in Crossroads in Mathematics. BEGINNING AND INTERMEDIATE ALGEBRA: THE LANGUAGE AND SYMBOLISM OF MATHEMATICS is oriented toward recent reforms in college level mathematics curricula.
"synopsis" may belong to another edition of this title.
From the Publisher:
This book has been written from the ground up as an integrated, combined book. The authors have used the AMATYC Standards as a guiding document in writing the book. The use of the graphing calculator is assumed, and the authors use the graphing calculator to explain a large number of real-data applications. Concepts are presented using "Rule of Four" (multiple representations of mathematical solutions to problems, including graphical, algebraic, numerical, and verbal approaches). "Multiple Perspectives" text boxes feature two or more of the Rule of Four approaches (numerical, algebraic, graphical, verbal) to solving a given problem. | 677.169 | 1 |
Synopses & Reviews
Publisher Comments
"Recommended with confidence" by The Times Literary Supplement, this lively survey starts with simple arithmetic and algebra and proceeds by gradual steps through graphs, logarithms, and trigonometry to calculus and the world of numbers. Generations of readers have found it the ideal introduction to mathematics, offeringand#160;accessible explanations of how theory arises from real-life applications.
"The main object of this book is to dispel the fear of mathematics," declares author W. W. Sawyer, adding that "Many people regard mathematicians as a race apart, possessed of almost supernatural powers. While this is very flattering for successful mathematicians, it is very bad for those who, for one reason or another, are attempting to learn the subject." Now retired, Sawyer won international renown for his innovative teaching methods, which heand#160;used at colleges in England and Scotland as well as Africa, New Zealand, and North America. His insights into the pleasures and practicalities of mathematics will appeal to readers of all backgrounds.
Synopsis
"Recommended with confidence" by The Times Literary Supplement, this lively survey starts with arithmetic and algebra and gradually proceeds to trigonometry and calculus. The author, who is internationally renowned for his innovative teaching methods, offers insights into the pleasures of mathematics that will appeal to readers of all backgrounds. 1943 edition.
Synopsis
"Recommended with confidence" by The Times Literary Supplement, this lively survey was written by a renowned teacher. It starts with arithmetic and algebra, gradually proceeding to trigonometry and calculus. 1943 edition.
Table of Contents
I. The Approach to Mathematics1. The Dread of Mathematics2. Geometryand#151;The Science of Furniture and Walls3. The Nature of Reasoning4. The Strategy and Tactics of StudyII. On Certain Parts of Mathematics5. Arithmetic6. How to Forget the Multiplication Table7. Algebraand#151;the Shorthand of Mathematics8. Ways of Growing9. Graphs, or Thinking in Pictures10. Differential Calculusand#151;the Study of Speed11. From Speed to Curves12. Other Problems of Calculus13. Trigonometry, or How to Make Tunnels and Maps14. On Backgrounds15. The Square Root of Minus One | 677.169 | 1 |
The brand new edition of this classic text--with more exercises and easier to use than ever Like the first edition, this new version of Lamperti's classic text succeeds in making this fascinating area of mathematics accessible to readers who have limited knowledge of measure theory and only some familiarity with elementary probability.
Streamlined for even greater clarity and with more exercises to help develop and reinforce skills, Probability is ideal for graduate and advanced undergraduate students--both in and out of the classroom. Probability covers:
* Probability spaces, random variables, and other fundamental concepts
* Laws of large numbers and random series, including the Law of the Iterated Logarithm
* Characteristic functions, limiting distributions for sums and maxima, and the "Central Limit Problem"
* The Brownian Motion process Buy Premium From My Links To Get Resumable Support,Max Speed & Support Me | 677.169 | 1 |
Think Maths - Matt Parker
This group of mathematics speakers visits schools to perform maths talks and workshops in the UK. Sessions present math in ways that "grab the attention of students and remind them that maths is something to be enjoyed." Free downloads include domino
...more>>
ThinkNum - Gregory Ugwi
A web-based financial data analysis engine. Use regression, curve trades, and other quantitative trading algorithms from scipy, numpy, and similar open source libraries to analyze economic information from hundreds of sources ranging from the ADO National
...more>>
ThinkQuest
An international contest designed to encourage students from different schools and different backgrounds to work together in teams toward creating valuable educational tools on the Internet while enhancing their ability to communicate and cooperate in
...more>>
ThinkWave.com
A free web community that provides secure internet-based communication for parents, students, and teachers: real-time information about attendance, assignments, grades, and school curriculum.
...more>>
Third Apex to Fractovia - Juan Luis Martínez
Several galleries, each with up to 18 fractal images, and discussions on fractals as art. Also an up-to-date list of freeware fractal generators, screen savers, and fractal music software, with links and reviews.
...more>>
This Is Statistics - American Statistical Association
Why study statistics? For students as well as their parents, teachers, and counselors, the American Statistical Association (ASA) provides a host of motives: to make a difference, have fun, satisfy curiosity, and make money, among other reasons. "WhichThree Skills For Algebra - Alan Selby
This book describes three skills key to the algebraic way of writing and thinking, offering a first image of mathematics beyond arithmetic. It also describes the first elements of logic or rule-based reason, needed in all disciplines for writing or a
...more>>
TI-83 / TI-84 Games and Programs - Bill Paetzke
Bill Paetzke offers TI-83/TI-84 Plus programs to help students and teachers be more efficient in using their calculators. The programs are organized under these categories: Algebra, Finance, Geometry, Physics, Programs, Statistics, Trigonometry, and Tutorials.
...more>>
Ti 84 Plus Calculator
Instructional videos include using the parametric function to construct a pentagram, hypothesis testing, sketching polynomial functions, finding critical points of a function, and using the TVM (Time Value of Money) Solver method. The site also offers
...more>>
TI89Prog - Mickaël Nicotera
A French database of programs for the TI-89, TI-92+ and V200 graphing calculators, organized into categories such as math, games, ebooks, pictures, and cours prépa. With an online discussion forum, links to other resources, and more. Available
...more>>
ticalc.org
The unofficial TI-calculator home page. A non-profit, collaborative effort geared toward fans of Texas Instruments' graphing calculators, acting as a headquarters for all available information on the subject. It is also the home of ZShell and Fargo, assemblyiesforteachers
A company in the UK selling neckties specifically with the school curriculum in mind. The growing collection includes ties for teachers of maths, science, English, music, ICT and other general-purpose school subjects.
...more>>
Tiling with Polyominoes - Ivars Peterson (MathTrek)
"Mathematicians have proved that the general question of whether it's possible to cover the plane with identical copies of a given finite set of tiles is, in principle, computationally undecidable. In other words, there's no cookbook recipe or handbook
...more>> | 677.169 | 1 |
6th Grade Math Volume 4
PDF (Acrobat) Document File
Be sure that you have an application to open this file type before downloading and/or purchasing.
2 MB | 34 pages
PRODUCT DESCRIPTION
eWorkbooks are specifically designed to be used as printed workbooks or as on screen instruction. Each page offers focused, engaging exercises.Students master topics with proficiency allowing them to move on to the next level.
Finding the Percent of a Number
Proportion
Algebraic Expressions
Graphing Linear Functions | 677.169 | 1 |
This book would certainly be my choice for any future Maths Studies classes that I have to teach. The style is very student friendly. Explanations and examples are easy to follow, and GDC details are for the modern and versatile TI-Nspire. This book also offers a good number of investigations, there are plenty of exercises, with all answers provided, and there are many exam style questions too. At the back of the book are three chapters that students will find very useful: the internally accessed project, the use of the TI-Nspire, and prior learning. Also worth a mention is the CD which contains an enhanced version of the text and PDFs for those who still use a Casio or TI-84. ( David Getling, Mathematics Teacher)
The committee thinks the books are well written, easy to follow, and they like the TI-NSpire technology throughout. ( Education Ministry, Canada)
We have ordered the Mathematics SL and Mathematical Studies books, and we love them. ( Padmini Nadar-Japal, IB Coordinator, Windhoek International School, Namibia)
The Oxford IB course books are the best ever resource for both teachers and students. They are practical, insightful and fully in line with the IB Course outcomes. ( Pat Hanson, IB Coordinator, Academy of the Holy Cross, USA)
We have adopted most of the Oxford course books for our school. We find them well written, well linked to TOK issues and age appropriate. ( Sheta Saha, IB Coordinator, Chatsworth International School, Singapore)
Vom Verlag:
Part of a completely new offering for IB Mathematics, this text provides extensive practice, detailed examination support, the latest GDC support and a free eBook, in addition to offering the most thorough syllabus coverage, which is crucial for the IB student. Uniquely developed with the IB, you can trust it takes the best approach. With carefully stepped activities with extensive practice, students will gain confidence in their skills. Activities make cross-curricular and real-world connections, while emphasising the historical and cultural aspects of the theory, in line with the Learner Profile. An eBook gives students ultimate flexibility in their study, including animations to simplify challenging concepts, interactive worked solutions and full and up-to-date GDC instructions for the most commonly used calculators. *Full syllabus coverage - the truest match to the IB syllabus, developed with the IB to exactly match IB specifications *Free eBook - a complete interactive eBook is included on CD for free, for the most flexible learning *Complete worked solutions - a full set of worked solutions is included online, in addition to interactive worked solutions on CD, which take learners through problems step-by-step *The most practice - more practice than any other resource, with over 600 pages and an eBook *Up-to-date GDC support - take the confusion out of GDC use and help students focus on the theory *Definitive assessment preparation - exam-style papers and questions will build confidence *The Exploration - supported by a full chapter, to guide you through this new component *Real world approach - connect mathematics with human behaviour, language, morality and more About the series: The only DP resources developed directly with the IB, the Oxford IB Course Books are the most comprehensive core resources to support learners through their study. Fully incorporating the learner profile, resources are assessed by consulting experts in international-mindedness and TOK to ensure these crucial components are deeply embedded into learning. | 677.169 | 1 |
A 21st Century Math & Science Education
with a Distinctly Biblical Perspective
Expert Video Instruction
Our courses feature video instruction by Dr. Shormann. This ensures understanding of complex concepts, leading to higher levels of achievement.
Biblical Perspective
Teaching from a Biblical perspective, Dr. Shormann teaches the purpose and pattern God puts in
math and science.
Academic Excellence
Our courses are designed to help students achieve the highest levels of math and science. Students
who choose to, can earn up to 34 college credits in math and science via CLEP and AP exams.
However, our courses can also be used as standard high school courses.
The
Meet
Teacher
David Shormann, PhD
With degrees in aerospace engineering and science, Dr. Shormann has spent
the past 17 years teaching Saxon Math 5/4 through AP Calculus, as well as
AP Biology, Chemistry, and Physics, to thousands of homeschoolers around
the world. He teaches math as the language of science, a tool to better
understand God and the world He created.
I am majoring in Chemistry at the University of Nevada. I attribute my
knowledge of Chemistry to the fact that I was able to understand the levels
of math presented within the course. And that is all thanks to you. I appreciate all the time and effort you put in to helping me understand the concepts that were a little difficult for me. You are an amazing professor and
such an outstanding person. I cannot wait to start this next chapter in my
educational career and am so thankful that I understand the math needed
to continue it. Your help really made the difference between having a basic
understanding of a concept and comprehending the topic fully. You truly
are the best professor ever :) ~Brittany
2
CD design-front.pdf
1
13/02/14
Journey to Novarupta
11:33 AM
From the creators of Jonathan Park, this 3-part audio adventure is
based on Dr. Shormann's research expeditions to Novarupta volcano, site of the largest eruption in over 100 years! Join the team as
they face grizzly bears, raging rivers, and pumice storms in search
of evidence of Creation, while building their faith in God and in
each other..
C
M
Y
CM
MY
CY
CMY
K
Math
Lesson 27
Lesson 32
x+3=9
- 3 -3
x=6
c2 - b2 = a2
+ b 2 + b2
c2 = a2+ b2
Expert video instruction for every lesson in the corresponding
Saxon textbook, taught from a Biblical foundation. Because the
lesson in the book is not the complete lesson, John Saxon recommended an experienced Saxon instructor who could connect the
incremental lessons, making complex concepts easy to understand.
Earn up to 34 college credits using CLEP and AP exams. These
courses include a diagnostic test, video lectures, practice problems,
video solutions to practice problems, practice exams, and email
support with Dr. Shormann.
936.372.9216 | sales@diveintomath.com
24
4
22
30
3
Saxon + DIVE = Success!
Lesson 27
c2 - b2 = a2
+ b 2 + b2
c2 = a2+ b2
Lesson 32
x+3=9
- 3 -3
x=6
Expert Video Instruction for every Saxon Math Lesson
Math is the language of science. Like learning a
language, the original Saxon curriculum begins with
the fundamentals and provides students ample practice
before gently introducing more advanced material. The
only math program basesd on incremental development,
continual review, and cummulative assesments, Saxon
Math increases long-term retention and recall speed by
continually reviewing previously learned concepts over a
long period of time. Long-term retention increases recall
speed, raising standardized test scores and making it
easier to apply to other subjects, like science.
The lesson in the Saxon textbook is not the complete
lesson. John Saxon designed his program to be taught
by a trained Saxon instructor. The DIVE course provides
this instruction!
Teaching 3-5 practice problems, Dr. Shormann connects
the new concept to previous ones, making complex
concepts easy to understand. The lesson in the book
does not make these connections. Without DIVE, the
concepts can appear disorganized and confusing.
DIVE into Saxon Math and reach higher levels of
achievement.
Student Textbook
• 120 lessons inlcude practice over new and previous concepts
• Homework questions have a lesson reference number directing
students to the lesson where the concept was originally taught
• Step-by-step solutions to all lesson, investigation, and test
questions
Dr. Shormann teaches the new
concept section of every lesson
in the corresponding Saxon
text. He provides important
information, not included in the
Saxon lesson, that makes it
easier to learn. Lectures are 1015 minutes long.
Students take notes and work
problems with Dr. Shormann,
pausing and rewinding as
needed.
The lecture has slightly different
practice problems than the Saxon
lesson. This provides additional
practice and builds profieciency
on the new concept.
This section provides practice on
previous concepts.
While essential to building longterm retention, this continual
practice often requires re-learning
recent concepts until they are
stored in long-term memory.
DIVE makes re-learning quick
and easy. Simply click on the
DIVE lesson (in parentheses,
next to each question) and watch
the lesson. Then try to solve the
problem again.
If you can't, skip it and go to the
next problem. You will correct it in
the next step.
Parents should grade work for
younger students but in Algebra
1 and up should grade their own
work using the Answer Key.
Missed problems should be
corrected by referring to the
DIVE lecture for that problem
(printed in parentheses next to
the problem).
If needed, the Saxon Solutions
Manual has step by-stepsolutions to every homework and
test question.
If more help is needed students
can email Dr. Shormann.
Typically there is one test per
week. The tests are cumulative,
meaning they test over previous
concepts.
Tests should be graded by a
parent then corrected by the
student.
A lesson reference number next
to each question makes it easy to
"plug any holes" by re-watching
the corresponding DIVE lecture.
If more help is needed students
can email Dr. Shormann.
Which Format is Right for Me?
All formats have the same content.
CD-ROM Format
System Requirements: All Macs and Windows XP and higher
Internet Access: Not Required
This format can be used with members of your immediate family. EULA prohibits the CD being
loaned, sold, or given away.
Digital Download
System Requirements: All Macs and Windows XP and higher
Internet Access: Only required for initial download
The files can be downloaded to all of your computers, stored in the cloud, and copied to a USB drive.
There is no CD to lose or damage. It can be used with all the children in your immediate family. However, it cannot be sold, loaned, or given away.
Lesson 32
x+3=9
- 3 -3
x=6
iTunes U Course
System Requirements: iOS 6.0+ (iPad, iPod Touch, iPhone) and iTunes U app (free in App Store)
Internet Access: Required for downloading and streaming
Featuring updated graphics and the power of iTunes U, this is our latest and greatest format. The
course can be used with all the children in your immediate family but cannot be shared, sold, or given
away.
936.372.9216 | sales@diveintomath.com
5
DIVE for
Saxon
K-12
Math Math
Course
Sequence
Advanced (AP)
Honors
Standard
K
Saxon 1
Saxon K
Saxon K
1st
Saxon 2
Saxon 1
Saxon 1
2nd
Saxon 3
Saxon 2
Saxon 2
3rd
Math 5/4
Saxon 3
Saxon 3
4th
Math 6/5
Math 5/4
Math 5/4
5th
Math 7/6
Math 6/5
Math 6/5
6th
Math 8/7
Math 7/6
Math 7/6
7th
Algebra 1
Math 8/7
Math 8/7
8th
Algebra 2
Algebra 1
Algebra 1/2
9th
Advanced Math 1
Algebra 2
Algebra 1
10th
Advanced Math 2/
Calculus 1
Advanced Math 1
Algebra 2
11
th
12
AP Calculus
th
Advanced Math 2
Advanced Math 1
& Calculus 1*
Calculus 2
Advanced Math 2
3 Semester Schedule
While an advanced student can complete them in a year, Saxon
Advanced Math and Calculus are recommended as three
semester courses. This schedule shows you what to do each
semester and how many credits are earned. The DIVE course
includes a weekly assignment chart indicating which lessons are
to be completed each week.
FALL
SPRING
CREDITS
First 1/2 of
ALGEBRA 2
Second 1/2 of
ALGEBRA 2
1 ALGEBRA 2
1/2 GEOMETRY
First 1/3 of
ADVANCED MATH
(Includes Geometry)
Second 1/3 of
ADVANCED MATH
(Includes Geometry)
1/2 PRECALCULUS
1/2 GEOMETRY
Last 1/3 of
ADVANCED MATH
FirstT 1/3 of
CALCULUS
Second 1/3 of
CALCULUS
Last 1/3
CALCULUS
TOTAL
CREDITS
6
HIGH SCHOOL TRANSCRIPT
Putting all these credits on a transcript
can be a little confusing. It's important
to remember to put the name of the
requirement the course fulfilled, not the
name of the textbook.
A subject transcript lists credits by
subject instead of year. This is a
format admissions personnel can
easily understand. Visit our website to
learn more about making high school
transcripts.
SUBJECT
Like the European and Asian countries that outperform the US on
national math exams, Saxon integrates the teaching of algebra
and geometry. This makes geometry easier to understand,
increases long-term retention, and raises college entrance exam
scores.
One-half credit of geometry is earned in Algebra 2 and onehalf credit is earned in the first 40 lessons of Advanced Math.
Therefore it is redundant and unneccessary to take a separate
year of geometry.
VS.
Saxon Teacher
In 2000, Dr. Shormann created the DIVE into Math series to provide daily instruction for the students in his weekly co-op
classes. Shortly thereafter, Saxon Publishers offered Dr. Shormann a job to create a secular version of DIVE. The offer
was declined due to the restrictions on selling our DIVE CDs with Christian content. In 2004, Saxon Publishers was
purchased by Harcourt Achieve. A year later Harcourt Achieve released Saxon Teacher, a whiteboard lecture very similar
to the DIVE into Math courses. Following is our opinion of the differences between the two products.
DIVE
Saxon Teacher
Teaches from a Biblical Worldview, Scripture is used to
inspire godly character.
Reads the exact lessons in the book which do not include
connections to previous concepts, lessons can appear
disorganized or confusing
Students work interactively, taking notes and working
problems with Dr. Shormann.
Dr. Shormann teaches different practice problems than the
Saxon lesson, providing extra practice on new concepts.
This is not confusing since the book should not be open
during the lecture.
No additional practice problems
All the lectures are in 1 file, making it easy to re-learn
previous concepts instead of seeing the answer. This builds
retention and fluency, making math easier. A quick reminder
is usually all that is needed. The Saxon solutions manual in
the kit provides step-by-step solutions, if needed.
Saxon Teacher is a multi-disk program, which makes it
difficult to go back and review previous lessons. Students
typically just watch the solution. Each time they do this,
they lose the opportunity to build rentention and fluency,
making math harder.
A Q&A email service with Dr. Shormann, is included with
each course. Questions are answered by Dr. Shormann
within one business day. Dr. Shormann has taught Saxon
Math and Saxon Physics courses since 1997.
Earn up to 14 college math credits with the free CLEP and
AP preparation program included with DIVE Algebra 2,
Advanced Math, and Calculus.
No CLEP or AP prep provided
936.372.9216 | sales@diveintomath.com
7
half
Math
Math 5/4
76
DIVE
DIVEfor
forSaxon
SaxonMath
Math
Grade
This course teaches numerical operations, functions, geometry,
fractions, patterns, word problems, and data manipulation through
charts and graphs. Critical thinking skills and long-term retention
are built with daily continual review of previously learned concepts.
Weekly tests are cumulative with lesson references for easy relearning of missed concepts. A solutions manual provides stepby-step solutions to all lesson, test, and investigation problems.
The DIVE course includes Q&A email support with Dr. Shormann.
Advanced: 3rd
Honors: 4th
Standard: 5th 6/5
Grade
This course continues to build a firm foundation for upper math
courses with word problems, division, integers, order of operations,
functions all
lesson, test, and investigation problems. The DIVE course includes
a Q&A email support with Dr. Shormann. Read more on page 4
Advanced: 4th
Honors: 5th
Standard: 6This course continues building on the concepts required for upper
level math courses like word problems, division, integers, order of
operations, functions
all lesson, test, and investigation problems. The DIVE course
includes a Q&A email support with Dr. Shormann. Read more on page 4
Grade
Advanced: 5th
Honors: 6th
Standard: 7 8/7 with Pre-Algebra
Grade
This course provides excellent preparation for Algebra 1 by building
fluency in the skills students struggle with most, like fractions, decimals,
percents, and ratios. This course teaches students to quickly simplify
equations, convert between fractions, decimals, percents, and ratios,
and much more. We recommend this pre-algebra course instead of
Saxon Algebra 1/2. A grade of 80% or better indicates the student can
skip Saxon Algebra 1/2 and go directly to Algebra 1.The DIVE course
includes a Q&A email support with Dr. Shormann. Read more on page 4
Advanced: 6th
Honors: 7th
Standard: 8th
Prerequisite
6th Grade Math
Credit
1 Pre-Algebra11
Math
Algebra 1/2
Grade
We prefer students take Math 8/7 for pre-algebra instead of this course
Optional
because it builds fluency in skills that are essential for success in Algebra
1. Algebra 1/2 is for students who struggle in Math 8/7. Otherwise,
Credit
1
Pre-Algebra
students should skip this course and go directly to Algebra 1. Daily
lessons include continual review over previous concepts Weekly tests
are cumulative with lesson references for easy re-learning of missed
concepts. A solutions manual provides step-by-step solutions to all
lesson, test, and investigation problems. The DIVE course includes a
Q&A email support with Dr. Shormann. Readd
is
e
Math
Algebra 1
Grade
This course teaches all the concepts required in a first year algebra
course like proofs, statistics, probablility and real world, algebra based
word problems. Daily lessons include continual review over previous
concepts Weekly tests are cumulative with lesson references for easy
re-learning of missed concepts. A solutions manual provides stepby-step solutions to all lesson, test, and investigation problems. The
DIVE course includes a Q&A email support with Dr. Shormann.
Advanced: 7th
Honors: 8th
Standard: 9th
Prerequisite
Pre-algebra
Credit
1 Algebra13
Math
Geometry
Grade
This curriculum was published by Houghton Mifflin, not John Saxon.
Saxon was a firm believer in the integration of geometry and algebra
because it makes geometry easier to learn and raises college entrance
exam scores. Saxon's original Algebra 2, 3rd edition and Advanced Math,
2nd edition have all the geometry required for a high school geometry
course while providing excellent preparation for the PSAT, SAT, and ACT.
This course is only recommened in rare circumstances. Read more
about the new HMH editions of Saxon on the opposite page. Read more
about
Optional
Prerequisite
Algebra 1 & 2
Credit
1 Geometry
Saxon MathL
c
o
i
a
g
i
a
s
d
s
NEW SAXON 4TH EDITIONS
ARE THEY WORTHY OF THE SAXON NAME?
John Saxon died in 1996. In
200Saxon Publishers was sold to
Harcourt Achieve Incorporated. In
2007 Houghton Mifflin Publishers
purchased Harcourt, acquiring
Saxon in the process. Recently,
Houghton Mifflin Harcourt (HMH)
released their newly revised 4th
editions of Saxon Algebra 1 and 2
as well as a Geometry text. But are
these new editions worthy of the
Saxon name? Or do they go against
the very principles on which John
founded Saxon Publishers?
John Saxon was an Air Force
test pilot with three engineering
degrees. After retiring, he began
teaching algebra at the local junior
college. Appalled at the skills his
students possessed, Saxon began
writing lessons to bring his students
up to speed. His methodology
produced great results so he wrote
and published his first algebra book
in 1981. When he died in 1996,
Saxon Publishers annual sales were
at $27 million.
Math is the language of science.
Just like learning a language, the
original Saxon curriculum begins
with the fundamentals and provides
students ample time to practice
these before gently introducing
more advanced material. John
Saxon created a unique system of
incremental development (small
bite-sized lessons), continual
review and assessment, and the
integration of geometry and
algebra. Researchers have found
these common-sense methods
increase long-term retention and
recall speed, thereby improving
standardized test scores. More
importantly, students are able to use
these skills and apply them to new
situations, like science.
John Saxon's systematic integration
of algebra and geometry is
essentially gone. The European and
Asian countries that consistently
outperform the United States
on international math exams
integrate the teaching of algebra
and geometry. A student who is
learning algebra and geometry
together will understand all math
better , enabling them to apply math
in science and engineering fields.
These students will also generally
outperform their peers on college
admissions exams because algebra
and geometry are tested on these
exams.
The new editions of published
by HMH significantly reduce the
amount of review over previously
learned concepts. The lessons do
no not make the proper connections
between the incremental concepts,
making the lessons appear
disorganized and confusing.
The most disappointing change
found in the new editions is that
Since Saxon Publishers was first
sold in 2004, I've feared that any
new editions might lose their
original methodology that strives to
teach mathematics like the language
of science that it is. I have often
prayed that if that happens, Lord
willing, I would be able to stand
on the shoulders of giants like
John Saxon, Leonard Euler, Isaac
Newton, Euclid, and others to build
an even better math curriculum.
These new editions confirm that
time has come. Coming soon from
DIVE: Shormann Math.
Do I Need a Separate Geometry Course?
Like the European and Asian
countries that outperform the US
on national math exams, Saxon
integrates the teaching of algebra
and geometry. This makes
geometry easier to understand,
increases long-term retention,
and raises college entrance exam
scores.
One-half credit of geometry is earned
in Algebra 2 and one-half credit
is earned in the first 40 lessons of
Advanced Math. Between these
two texts, all the geometry required
on the PSAT, SAT, and ACT are
taught. Therefore it is redundant and
unneccessary to take a separate year
of geometry.
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15
Algebra 2 with Geometry
Grade
John Saxon believed in integrating algebra and geometry because it makes
ithem asier to learn and apply on college entrance exams. This unique
course, with 1 credit of Algebra 2 and 1/2 credit of geometry, teaches all
the topics necessary to excel on the PSAT, SAT, and ACT, all in one year.
With short, bite-sized lessons, daily review over previous concepts, lesson
reference numbers to re-learn missed concepts, and a solutions manual
with step-by-step solutions, this course will build fluency and long-term
retention, thereby raising exam scores. Read more on page 4
Advanced: 8th
Honors: 9th
Standard: 10th
Prerequisite
Algebra 1
Credit
1 Algebra 2
1/2 GeometryWith numerous applications to physics, chemistry, engineering, and
Prerequisite
Precalculus
business, this course meets the requirement for AP Calculus AB and
(Advanced
Math)
BC exams. Most of the advances in math and science over the past 300
Credit
years have been based on Calculus, making it an important goal for
1
Calculus
1
all students. With short, bite-sized lessons, daily review over previous
1 Calculus 2
concepts, lesson reference numbers to re-learn missed concepts, and a
solutions manual with step-by-step solutions, this course will build fluency
and long-term retention, thereby raising exam scores. and APMany students can finish calculus in high school. Setting a
high standard for your homeschool will increase the risk of
failure. But, shouldn't we be training our kids to try hard, and
possibly fail, than to set a low standard? How will they know
what they are capable of if we simply plan their K-12 curriculum based on government school standards, which never
include calculus? In the following paragraphs, Dr. Shormann
presents some thoughts on why Christian home schools
around the world should plan to complete calculus in high
school.
What is calculus?
Answer this question: If you drove 60 miles in one hour, how
fast were you going? If you answered "60 miles per hour",
then congratulations, you just did calculus! Calculus is nothing more than a tool for studying rates of change. And rate
stems from the word ratio, or fraction, so calculus can even be
thought of as a study of fractions. Rates of change in position,
speed, temperature, volume, etc. can all be studied with calculus.
Why is it important to know calculus?
Understand History
Calculus began with the ancient Greeks, but their cyclical view
of time caused their progress to stagnate. Because their worldview was flawed, they were not able to grasp the reality of infinity. But Christians know from Scripture that God is infinite
(eternal). It is no surprise then, that in the 1600's, Newton and
Leibniz (both Christians) discovered calculus. For the next 300
years, practically every development in science and mathematics was connected with calculus. The discovery of calculus is a
powerful lesson that worldview matters!
Enjoy Life
Understanding how things move and change, from motion
pictures to engine pistons, has been one of the most intellectually satisfying experiences in my life. The fundamental idea of
change is at the basis of our whole perception of phenomena.
Life is full of change and predictable patterns of change. Jesus
said that He came to give us an abundant life (John 10:10), and
the study of calculus allows us to enjoy God and his works in a
more abundant way.
Know God
It helps us understand the Trinity One of the biggest differences between Christianity and other religions is the theme of
unity and diversity. Some religions have one god, and others
have many gods, but the only true God is both one and many.
He is our Father, Son and Holy Spirit, a triune God. God's
invisible attributes have been clearly seen by humans from the
beginning (Romans 1:20). The attribute of unity and diversity
is quite visible. Tides would cease without the sun and moon,
dolphins would starve without their sonar, and calculation of
speed would be impossible without a measure of time elapsed.
One cannot study calculus for long without noticing that unity
and diversity abounds, and realizing this will help a student
understand that a biblically-based math and science education
makes the world so much easier to figure out!
Why complete calculus in high school?
Easier Than You Think
What usually intimidates most students is the whole host of
strange symbols like ∂y/∂x and ∫, and confusing words like
function, limit and continuity. However, with a little patience
and perseverance, coupled with skill in working with fractions
and basic algebra, students will find that difficult things readily
become easy things.
A Liberal Education
Not that kind of liberal, the "free to think," "free to pursue"
kind! Up to 80% of undergraduate degrees offered at 4-year
universities require at least one semester of calculus. Calculus
knowledge frees a child up to pursue any college degree. But,
calculus should not be considered just by the college crowd.
The problem solving skills and understanding of God's creation that naturally flow from a study of calculus are invaluable
to anyone who loves the Lord and loves to learn.
A Liberal Education
Many children are capable of completing calculus in high
school, and it is up to parents to provide the love, encouragement and godly counsel that will motivate them to succeed.
Parents must teach their children well, demand excellence, and
set high standards, remembering also that knowledge puffs up,
but love builds up(I Cor. 8:1), and that God considers as "nothing" those with lots of knowledge but little love (I Cor. 13:1-3).
All parents want their children to have a better education, better job, and a better understanding of His Word and His works
than they had, and planning for your children to finish calculus in high school will help them succeed. Humble application
of this great knowledge will result in a child who is better prepared for restoring His kingdom on Earth, as it is in Heaven.
936.372.9216 | sales@diveintomath.com
19
DIVE Science
L
IVE
ONLINE
MATH CLASSES
Classes meet online with Dr. Shormann once per week. During this time Dr.
Shormann will review new concepts, answer questions, and administer a quiz.
During the week students watch DIVE Math video lectures in our online eLearning
Campus and complete daily homework from their Saxon textbook. Homework is
uploaded to Dr. Shormann before each class. Classes are limited to 12 students.
What's Included?
• SYLLABUS WITH WEEKLY ASSIGNMENTS
Tuition: $400 Per Course
This price includes a $95 non-refundable
deposit which is due with registration.
Registration Opens April 1
Register online or by phone at
936.372.9216
Access your DIVE course anytime,
anywhere, on any computer or device
with high speed internet.
DIVE courses are now available in
iTunes U. After purchasing your course
from our website you will be emailed an
enrollment link.
Instantly download DIVE courses from
our website. Burn the course to a CD,
save it to a USB or portable hard drive,
and transfer them to any mobile device
that supports Adobe FlashPlayer.
Digital download versions contain all
the same content as the corresponding
CD-ROM but there is no CD to lose or
damage.
*High speed internet access is required
to download the course. However, after
downloading, internet access is not
required.
The eCourses include all the content
found on the corresponding DIVE
CD-ROM. Science eCourses include
automated grading and grade tracking.
eLearning subscriptions give you
access to your course for 1 year.
Subscriptions can be extended,
renewed, and/or additional family
members added at reduced rates.
936.372.9216 | sales@diveintomath.com
After enrollment you will have access
to all the DIVE lectures and material
for that course, just like any song you
purchase from iTunes.
*HIgh speed internet is required to
download the material to your device.
After downloading you can view your
course material anytime, anywhere!
"I really appreciating your ministry. I have two sons taking your courses. I finally found
what I was looking for in Science and Math. I especially like that Dr. Shormann teaches the
students how to learn: define terms, lecture, take notes
and practice." Esmy L.
12 Month Subscription
eLearning subscriptions give you access to
your course for 1 year. Subscriptions can
be renewed for a sibling in later years for
$20 and a sibling taking the same course at
the same time is $10
Based on the true story of Dr. Shormann's research
expeditions to a powerful volcano in the
remote Alaskan wilderness!
On June 6, 1912, the biggest eruption in over 100 years occurred in what is now Katmai National Park, and The
Valley of Ten Thousand Smokes in Alaska. Ejecting over 30 times more material than Mount St. Helen's, Novarupta
exploded for more than 60 hours. Catastrophic processes consistent with Biblical history are clearly revealed in the
story of Novarupta. The volcano deposited ash and tephra at rates exceeding 10 feet per hour, dropped worldwide
temperatures up to 2 °F, caused large parts of mountains to disappear in the blink of an eye, plus a host of other
cataclysmic events. Novarupta provides many challenges to the uniformitarian assumption that normal everyday
processes are responsible for Earth's surface features. It is a powerful testimony to the young and active planet we
live on, a planet shaped by the hand of God as revealed in Scripture.
JourneyToNovarupta.com
24
EARTH SCIENCE
DIVE Earth Science will change the way you teach science. Giving your child a 21st Century education, this
course will strengthen their faith in the absolute truth of Scripture as a God-inspired work that is a true account
of the history of Earth and the Universe.
Beautiful, engaging video lectures
and labs do all the teaching for you.
Students work interactively, taking
notes and working along with Dr.
Shormann.
GRADES 7+
PREREQUISITE: Prealgebra (or Saxon 8/7) Completed or Concurrently
Biblical Foundation
Taught from a Biblical foundation, this course trains students to enjoy God's
great Earth while equipping them to stand against evolutionism. It covers four
major areas: Earth Science Basics, Flood Geology, Limnology (Freshwater
Ecosytems) and Oceanography, and Astronomy and Meteorology.
Video Lectures Do The Teaching for You
Set up on a 32 week schedule, students complete two lectures, two worksheets,
20 definitions, and one lab each week. Every 8 weeks there is a quarterly
review, which can be used as an exam for high school students. The video
lectures and labs average 20 minutes each. While there are no required reading
assignments, a list of supplemental resources is provided.
Video Labs Spark Interest
The weekly video labs teach students everything from how to use Google Earth
to how to build and fly a model rocket. Many lab activities include experiments
that teach the student to use the scientific method. A hands-on lab kit is
available but is not required.
Middle School or High School Course
Typically recommended for 7th or 8th grade, this course can be modified for use
as a high school course by administering the quarterly reviews as exams and
adding quarterly research papers on a related topic.
Optional: Lab Manual $24
A printable PDF of this lab is included with
your DIVE course. We sell it spiral bound
with heavy covers as a convenience for
those who do not want to print the 118
pages at home.
A complete college preparatory curriculum taught from a Biblical foundation 138 pages
• Spiral bound with heavy covers
An introductory chemistry and physics course designed to spark interest while
building a firm foundation for advanced science courses. Topics include,
electron configuration, chemical bonding, oxidation, balancing equations, nuclear
chemistry, chemical bonds and reactions, fluid dynamics, Ohm's law, gases, and
thermodynamics. Based on the scientific method, the video labs have many fun,
engaging activities that draw students in and inspire them to learn more.
An Advanced or Standard Course
Designed as an honors course, it can also be used as a standard high school
course by administering the quarterly exams as "open note".
Internet Textbook
The DIVE Internet Textbook contains links to complete weekly reading
assignments online. If you prefer a traditional hard copy textbook, we recommend
either Bob Jones Physical World, Bob Jones Physical Science, or Abeka Science
and Matter, 3rd Edition. A reading syllabus that lists exactly what to read each
week for these texts and many others are posted on our website under Support.
Taught From A Biblical Foundation
Upon completionExellent Preparation For Biology
Due to the heavy emphasis on chemistry in today's biology courses, ICP is
recommended before taking Biology.
Physical Science Renamed
Colleges have changed the name of the Physical Science requirement to
Integrated Physics and Chemistry (also known as Integrated Chemistry and
Physics depending on which is taught first). Instead of an earth and space
course, they now require one semester of chemistry and one semester of
physics.
DIVE Biology is a complete, college preparatory biology course. Topics include
Science and Christianity, biochemistry, cells, genetics, epigenetics, Creation/
evolution, bacteria, protozoans, fungi, plants, animals, ecology and human anatomy.
The 28 video labs include all the required AP labs and emphasize important
laboratory techniques used by biologists, including chromatography and gel
electrophoresis.
Advanced or Standard Course
Designed to be an honors course, DIVE Biology can also be used as a standard high
school course Textbook contains links to complete weekly reading assignments
online. Easy to update, this text has the most recent scientific information. If you
prefer a traditional textbook, we recommend either Apologia, Bob Jones, or Abeka.
We have a reading syllabus for each text that lists exactly which pages to read each
week. If we don't have one for your text, email us the table of contents and we will
make one for you.
Earn 8 College Credits
After DIVE Biology, we recommend our CLEP Professor Biology, a three week
course that prepares students for the CLEP and AP Biology exam.See page 18 for
details.
Apologia Users: Five weeks of human body concepts typically found in high school
biology are not in this Biology text, they are in the Human Body. To complete
these reading assignments , you can use the internet links provided in the course.
Alternatively, you can purchase the Human Body text. We have a syllabus that lists
exactly what to read from the Human Body text.
191 pages
• Spiral bound with heavy covers
A complete college preparatory curriculum taught from a Biblical foundation
Complete One Year Course
This course is available in three formats.
All formats include the same components.
See pages 21 & 23 for more information.
CD-ROM
eLearning Course
Digital Download
Optional Workbook: $29
A printable PDF of this workbook is included
with your DIVE course. We sell it as a
convenience for those who do not want to
print the workbook at home.
• 171 pages
• Spiral bound with heavy covers
Designed as an honors level course, DIVE Chemistry includes all the required
AP Chemistry labs. These labs emphasize important lab techniques used by
chemists, including chromotography, colorimetry, spectroscopy, electroplating,
and titrations. However, this course can also be used as a standard high school
course textbook contains links to complete weekly reading
assignments online. Easy to update, this text has the most recent scientific
information. If you prefer a traditional hard copy textbook, we recommend either
Apologia, Bob Jones, or Abeka. We have a reading syllabus for each text that lists
exactly which pages to read each week. If we don't have one for your text, email
us the table of contents and we will make one for you.
Earn 8 College Credits
After DIVE Chemistry, we recommend our CLEP Professor Chemistry, a three
week course that prepares students for the CLEP and AP Chemistry exam. See
page 18 for details on how to earn up to 34 college credits.
Apologia Users: Six weeks of concepts found in a high school chemistry course are
not in the Chemistry text, they are in Advanced Chemistry. To complete these reading
assignments , you can use the internet links provided in the course. Alternatively, you
can purchase the Advanced Chemistry text. We have a syllabus that lists exactly what
to read from textbook.
DIVE Physics teaches every lesson in the Saxon Physics curriculum. It covers
all the topics normally found in an AP Physics B level course including speed
and velocity, torque, impulse/momentum, relativity theory, work, gases and
thermodynamics, potential and kinetic energy, electricity and magnetism, circuit
theory and Ohm's law, and reflection and refraction. It is also contains 23 video
labs and a printable lab manual developed by Dr. Shormann to supplement the
Saxon course.
Advanced Placement or Standard Course
This is an Advanced Placement course. However, it can be used as a standard
high school course
Taught From A Biblical Foundation
This course also teaches science hitsotry, with a special emphasis on the
distinctly Christian foundation of modern science. This foundation is lacking from
secular courses and even many Christian courses, but is essential education for
the informed Christian.
Saxon Physics Bookset Required
Because physics is math based, the Saxon method of incremental development
and continual practice has worked best for our students. Therefore this course
was designed to teach Saxon Physics and does not work well with other texts.
The complete Saxon Physics bookset, including student text, homeschool packet
with test forms, and solutions manual, are required.
Earn 8 College Credits
Upon completion of DIVE Physics, we recommend our CLEP Professor Physics,
a three week course that prepares students for the AP Physics B exam.See page
30 for details on how students can earn up to 34 college credits for their high
school courses.
DIVE Physics Course
This course is available in three formats. All
formats include the same components. See
pages 21 & 23 for more information.
CD-ROM
Digital Download
Optional Lab Manual $27
A printable PDF of this workbook is included
with your DIVE course. We sell it as a
convenience for those who do not want to
print the workbook at home.
• 74 pages
• Spiral bound with heavy covers
936.372.9216 | sales@diveintomath.com
29
CLEP Professor
Earn up to 34 College Credits!
Taught by Dr. Shormann from a Biblical foundation, each 3 week
course prepares students to earn 3-8 college credit by exam.
For questions or help, email sales@diveintomath.com
16 points to spare! Thank you
Dr. Shormann for your great
CDs. We have used them for
all Nick's HS Maths, Biology,
Chemistry and Physics with
great success. We recommend
these to all users out there. As
always, Dr. S., you have done a
great job!"
Peter S.
*This is the typical number of credits awarded. However, each college determines
how much crediit is awarded and the course the credit is applied to.
HOW DOES IT WORK?
DIAGNOSTIC TEST
Each CLEP Professor title has a diagnostic test that determines
which of the 20 lessons to complete.
VIDEO LECTURES
Twenty video lectures teach every topic on the corresponding CLEP
or AP Exam. Students work interactively, taking notes and working
problems along with Dr. Shormann.
What are CLEP and AP Exams?
CLEP and AP exams, administered by the College Board and
accepted by over 2900 colleges and universities nationwide, allow
students to earn college credit by passing an exam instead of
taking the college course.
What is the difference between CLEP and AP exams?
CLEP exams are amultiple choice and last 90 minutes. AP exams
PRACTICE PROBLEMS WITH VIDEO SOLUTIONS
To reinforce understanding, each lesson has 10-15 practice problems. have a 90-minute multiple choice section and a 90-minute free
response section. Because AP exams are more rigorous, they are
Video solutions are provided for each question.
accepted by most colleges.
COMPUTER BASED PRACTICE EXAMS
After completing the prescribed lessons, take the first practice exam.
Each question on the practice exam is referenced to the CLEP
Professor lesson the topic was taught in, making it easy to re-learn
missed concepts. Each title contains two or three practice exams.
TAKE THE CLEP OR AP EXAM
While AP exams are only offered in the spring each year, CLEP
exams can be taken anytime at your local university or community
college. You do not have to be enrolled in college nor are there any
age requirements. However, a picture ID is required.
EXAM SCORES HELD BY THE COLLEGE BOARD
Exam scores are stored at The College Board. Upon request, exam
scores are sent to prospective colleges. Colleges determine the
number of credits awarded for each exam.
FREE Q&A EMAIL SERVICE
Dr. Shormann will personally answer any questions about your
course via email..
When and where can I take CLEP and AP exams?
AP exams are only administered once a year, usually in May. We
recommend talking to your local Christian or public school in the
fall to register. CLEP exams can be taken anytime at your local
community college or university testing center.
Which exams will my college accept?
Visit and click on the CLEP or AP link.
Since most students don't know which college they will attend
until their senior year, we recommend taking each CLEP or AP
exam after completing the corresponding high school course and
CLEP Professor course. Even if your college doesn't award credit,
good scores on these exams make your college admissions and
scholarship application more competitive.
936.372.9216 | sales@diveintomath.com
31
Got Questions?
We can help!
Each of our sales
team members have
more than 20 years
of experience
homeschooling.
Special Conference Hours
Call 936-372-9216
Thursday
Friday
Satruday
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Normal Business Hours
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Saturday
10-2
Shormann
Math
Algebra 1
TM
32
Coming Fall 2014TM
Anchored to a Biblical,
Christian foundtion, Shormann Math
brings John Saxon's provenformula of
incremental development with continual
review, into the 21st Century! | 677.169 | 1 |
FAQ
Algebra In A Flash is perfect for anyone who needs help with Algebra. On a typical American school day some six million high school students and over two million college freshmen who are placed in a Remedial Algebra course, can be found struggling with Algebra.
Yes. The first time you run Algebra In A Flash, it asks for your name and a date and will use both entries for the name of your grade report file. For example, if you enter Uncle Sam, April 15 then the name of your grade file will be US0415.
First check for an update. Math911 is a huge program and sometimes when a new topic is added it inadvertently introduces an error message. If the error persists email the Professor sending a screen shot if possible. Alternatively, call Tech Support at 347-528-7837
Feel Free to Get in Touch with Professor Weissman
Tech Support
Contact Info
Disclaimer
Professor Weissman's Mathematics Software contains NO gamelike graphics and NO animation. This Software is intended for students of all ages who want to learn mathematics, not play games. The real fun is learning. | 677.169 | 1 |
Synopsis
So You Really Want to Learn Maths Book 2 A Textbook for Key Stage 3 and Common Entrance by Serena Alexander
This book consolidates the material covered in So You Really Want To Learn Book 1 and completes the topics required for Level 2 of the Common Entrance Maths syllabus at 13+. Challenging and rigorous throughout, this book features detailed explanations and an impressive bank of practice exercises, to ensure that pupils have fully grasped all the topics covered. - Endorsed by ISEB - Clear explanations are followed by extensive practice material to ensure pupils have understood the material covered - Provides a strong emphasis on numerical work, including fractions, decimals, and algebra to give a strong grounding in mathematical knowledge An Answer Book containing answers and a mark scheme is also available to save you time marking work.
Reviews
Those who appreciate Book 1 will find in Book 2 an equally challenging and enjoyable experience. There is more than enough here to meet the needs of the most able pupils. David E. Hanson, Leader of the ISEB 11+ Mathematics setting team, member of the ISEB 13+ Mathematics setting team, Member of the ISEB Editorial Board These books provide a thorough grounding in maths. Detailed explanations are given through graded exercises, practical work, investigations and puzzles. The material is neatly laid-out, well organised with clear instructions and excellent diagrams, and very informative. The author even brings in fascinating snippets of history throughout the books: how the Egyptians wrote fractions, the history of the penny and of percentages, the origin of imperial units and much more. For instance, did you know that it was the Greeks who established the 'foot
foot
About the Author
Serena Alexander has taught Mathematics since 1987, originally in both maintained and independent senior schools. From 1999 she taught at St Paul's School for Boys, where she was Head of Mathematics at their preparatory school, Colet Court, before moving to Newton Preparatory School in London. She has been a member of the ISEB setting team for mathematics, is an ISI inspector and helps to run regular mathematics conferences for prep school teachers. She has a passion for maths and expects her pupils to feel the same way. After a lesson or two, they normally do! | 677.169 | 1 |
Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context.
Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.
This is an excellent course on the pronunciation of English and very fun too. You can learn the exact sound for each letter and keep a smile throughout the entire work, something quite important dealing with a subject usually related to boring task… | 677.169 | 1 |
Students who do not meet the prerequisite may enrol with MAST10006 Calculus 2 taken as a corequisite.
Recommended Background Knowledge:
None
Non Allowed Subjects:
Students who have completed a level-2 actuarial studies subject will not normally be permitted to enrol in this subject Shuanming Li
Contact
This subject is an introduction to compound interest functions and operations; valuation of annuities, bonds and loans; demography, and factors affecting population growth and size; construction and use of the life table; applications of these in life insurance; types of insurance products; the role of the actuary; and the significance of financial institutions utilising actuarial management.
Learning Outcomes:
Apply relevant pre-requisite mathematical knowledge in the solution of a range of practical problems
Calculate the accumulation or present value of money under compound interest
Calculate the amount or present value of payments under fixed interest contracts such as loans,annuities and bonds;
Solve equations of value for the effective rate of interest;
Describe the factors that affect the growth and structure of populations and explain their impact on these populations; | 677.169 | 1 |
EGR 102Lecture 37Matrix Math Review±The horizontal lines in a matrix are called rows±The vertical lines are called columns. ±A matrix with mrows & ncolumns is called an -by-matrix ±and are called its dimensions. ±The dimensions of a matrix are always given with the number of rows first, then the number of columns. It is commonly said that an -by-matrix has an order(size) of ×. ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=121110987654321AEGR 102Lecture 38Matrix Math Review±Almost always capital letters denote matrices with the corresponding lower-case letters with two indices representing the entries. ±For example, the entry of a matrix Athat lies in the i-th row and the j-thcolumn is written as ai,jand called the entry or (i,j)-th entry of A. ±Alternative notations for that entry are A[] or A. The row is always noted first, then the column. ⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧=121110987654321Aa2,3= ?
EGR 102Lecture 39Vectors±A matrix where one of the dimensions equals 1 is often called a vector±An m× 1 matrix (1 column & rows) is called a columnvector±A 1 ×nmatrix (1 row & columns) is called a rowvector { }4321=AEGR 102Lecture 310Special Matrices•Matrices where =are called square matrices.
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Math 417 Midterm 2 Study Guide Winter 2008
Content
The material will be drawn from Sections 3.3 and 3.4, and from Sections 5.1, 5.2 and 5.3. You will want
to review homework problems as well as the material below. You will find solutions to the odd problems (including True/False problems) on Otto Bretscher's website:
Computational Problems
Computing coordinates for a vector in a given basis. Exercises 1 - 18 of Section 3.4
Computing a matrix for a transformation relative to a given basis.
Exercises 19-30 of Section 3.4
Computing the orthogonal projection, Exercises 26-28 of Section 5.1. You will need to know
how to use Fact 5.3.10. | 677.169 | 1 |
This book is an ideal resource for extra classwork, homework and for use in catch-up or Summer classes. Each practice exercise delivers progression through questions which revisit and extend ideas covered in Year 8 Pupil Book 2.
Maths Frameworking offers you the most comprehensive and engaging route to Framework success.
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Review:58718 Damaged book. Please note this is a damaged book. It is actually new but has been damaged and therefore we are selling it considerably cheaper than the new price - so please grab a bargain. Thanks. Bookseller Inventory # CHL1630556
Book Description Collins Educational 207138821
Book Description Collins Educational 207138821
Book Description Collins Educational 207138821
Book Description Collins Educational, 2002. Paperback. Book Condition: Very Good. In VERY GOOD general condition, with some signs of previous use. Dispatched from the UK daily Another Croaking bargain from the Frog !!!. Bookseller Inventory # mon0001465169 | 677.169 | 1 |
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This activity reviews the mathematics that you may encounter in this course. This exercise will help
you with tools such as working with logarithms, the small-angle formu
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Overview of Astronomy 101A
Autumn Quarter 2016
1
BACKYARD ASTRONOMY ALERT: When the sun goes down on Monday,
Oct. 3rd, step outside and look west. If you have a clear view of the
horizon, you'll see Venus and the exquisitely slender crescent Moon side by
side in the sunset sky. Visit Spaceweather.com fo
Rotational Motion Whiteboarding Exercises (Honors)
1) A large sphere rolls without slipping across a horizontal surface. The sphere has a constant
translational speed of 10 meters per second, a mass m of 25 kilograms, and a radius r of 0.2 meter.
The mome
Math 327 Midterm Solutions Summer 2011
1 By a homework problem, the fraction inside the cosine converges to 5/7,
since the numerator and denominator are polynomials of the same degree.
By the continuity lemma (since the cosine function is continuous), the
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Unformatted text preview: Math 105: Problem Solving in Mathematics Course Description This course introduces students to the true nature mathematics, what mathematicians really do, how they think, and what they try to accomplish. The focus is on using quantitative reasoning and intuitive logical thought techniques to solve problems rather than formal rigid processes. Course Objectives The student will be able to: Be able to recognize and produce a precise and formal statement of a problem. Explore various parts of a problem, including any necessary background information, basic examples, what sort of solution is required, and what techniques might help to solve it. Demonstrate a logical reasoning process in solving problems. Be able to precisely present their ideas to others. Demonstrate an ability to understand and critique non-technical scientific writing. Course Objectives The student will be able to: Be able to recognize and produce a precise and formal statement of a problem. Explore various parts of a problem , including any necessary background information, basic examples, what sort of solution is required, and what techniques might help to solve it. Demonstrate a logical reasoning process in solving problems. Be able to precisely present their ideas to others . Demonstrate an ability to understand and critique non-technical scientific writing. First Assignment What is Mathematics? Describe what you think Mathematicians do. How do you feel about Math? Describe your experience thus far in previous Math classes. Why Solve Problems? What major world problems need solving? Why Solve Problems? What major world problems need solving? Are all of these problems going to be solved by scientists? Why Solve Problems? What major world problems need solving? Are all of these problems going to be solved by scientists? What major problems will you need to solve in your major field of study? Why Solve Problems? What major world problems need solving? Are all of these problems going to be solved by scientists? What major problems will you need to solve in your major field of study?...
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This note was uploaded on 11/11/2011 for the course MATH 112 taught by Professor Jarvis during the Winter '08 term at BYU. | 677.169 | 1 |
In this chapter we hope to present a review of basic skills in the areas of mathematics,
unit conversions, use of the metric system, an introduction to the physical nature of
matter, measurement techniques, and techniques for the presentation of data. At the
end of each section we will include a bibliography for those of you wishing to explore
any of these areas more fully.
REVIEW of MATHEMATICS
Physical measurements you will perform will require accurate recording, calculation, and
reporting of numbers. Numbers that we will use may be broken down into three
categories: integers, real numbers, and rational numbers. We will use the symbols x, /
or ÷, +, and - to denote the mathematical functions multiply, divide, add, and subtract.
Equations: Just as the name suggests, equations are mathematical statements in
which the equality of two statements is expressed:
Example: π x 5 = 15.7 is a mathematical equation which reads Pi multiplied by five is
equal to fifteen point seven (or fifteen and seven tenths).
Variables: Often we do not know the values for all of the expressions in an equation, so
we substitute letters or symbols in place of the numbers. Variables are also used in the
writing of generic equations (formulas) into which numbers are substituted later.
Example: the circumference of a circle C having diameter D is given
by the equation:
circumference = π x D
or
C D π =
In this example, D is a variable and can have any positive value.
When variable are not separated by an operator, it is assumed to
be multiplication.
Las Positas College Vacuum Technology 60A & 60B
Page 7 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Formulas: Formulas are mathematical equations that have been worked out for you; all
that is required to solve a formula is to substitute appropriate numerical values in place
of the variables.
Example: the mass flow rate of a vacuum system is given by the
formula:
Q = S x P
Where Q= mass flow in Torr- liters per second, S= volumetric flow rate (or pumping
speed) in liters per second, and P = pressure, expressed in Torr. Formulas used in
simple vacuum calculations are given in appendix X, and formulas for areas and
volumes of simple geometric shapes are given in the appendix Y.
Exponential Notation: Often in the process of performing calculations, we are
interested in multiplying a number by itself several times.
Example: The area of a circle is equal to a constant (π) times the
square of the radius (the radius multiplied by itself). In this example,
one could write the square of the radius as: r x r, but it is more
frequently written as r
2
. Here, r is the base and 2 is the called the
exponent.
base
exponent
Addition of Numbers Expressed in Exponential Notation: In order to add numbers
such as 4
3
and 3
6
it will be necessary to find the value of each and then add in the usual
manner.
Examples:
4
3=
4x4x4 = 64
3
6
= 3x3x3x3x3x3 = 729
4
3
+ 3
6 =
64 + 729 = 793
Subtraction of Numbers Expressed in Exponential Notation: Same as for addition;
evaluate each exponent, then subtract.
Multiplication of numbers expressed in exponential notation: Now things really get
to be interesting! Numbers expressed in exponential notation that have the same base
may be multiplied by simply adding the exponent.
Example: 2
4
x 2
5
=
2
(4 + 5)
= 2
9
Division of Numbers Expressed in Exponential Notation: In a manner similar to
multiplication of values expressed in exponential notation, division of numbers having
the same base may be accomplished by subtracting the exponents.
5 3 (5 3) 2
4 4 4 4 16
−
÷ = = =
What about any number raised to the zero power (a zero exponent)?
Examples:
3
0
= 1
76
0
= 1
Rule: Any number raised to the zero power is equal to 1.
In all of the examples above both the base and the exponent are integers. It is possible
that either or both could be real numbers (see the table below).
Examples:
5.3
4
= 5.3 x 5.3 x 5.3 x 5.3 = 789
6
2.8
= 6 multiplied by itself 2.8 times = 151
Scientific Notation: Writing and calculating with very large or very small numbers can
result in a great deal of tedium and often create opportunities for mathematical errors.
One can express all real numbers in terms of a number between 1 and 10, multiplied by
10, and raised to some power.
Converting from Scientific Notation: Occasionally you may wish to change a number
expressed in scientific notation back to the normal form of expression. This is
accomplished by moving the decimal point to the right the number of times expressed in
the power of 10 for positive exponents and similarly moving the decimal point left for
negative powers of 10.
Examples:
5.67 x 10
3
= 5670
3.40 x 10
-4
= 0.00034
Addition of Numbers Expressed in Scientific Notation: In order to add numbers
expressed in scientific notation, one must first make the power of 10 for each of the
numbers to be added equal.
Example: 2 x 10
3
+ 5 x10
5
= 0.02 x10
5
+ 5 x10
5
= 5.02 x10
5
Multiplication of Numbers Expressed in Scientific Notation: To multiply numbers
expressed in scientific notation, the following rules are used:
(Ax10
x
) x (B x10
y
) = (AxB) x 10
(x+y)
Example: (4 x 10
4
)
x (7 x10
3
) = (4 x 7) x 10
(4+3)
= 28 x 10
7
= 2.8 x 10
8
Division of Numbers Expressed in Scientific Notation: The rules are similar to
multiplication.
Example: (8 x 10
4
)
÷ (2 x 10
3
) = (8 ÷ 2) x 10
(4-3)
= 4 x 10
1
= 40
Rounding of Data and Significant Figures: A measurement was made of the length
and diameter of a tube in order to calculate its volume. The diameter was measured to
be 4.05 cm and its length was 83.7 cm. The geometric volume of the tube may be
calculated using the formula
2
V r l π =
where V is the volume of the tube, r is the radius, and L the length.
Example:V = π ×(4.05cm / 2)
2
× 83.7cm = 343.222313cm
3
A comment on Significant Digits
Las Positas College Vacuum Technology 60A & 60B
Page 10 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Reporting the calculated volume as 343.222313cm
3
is not truthful, as it suggests that
the volume is known to nine significant figures, when in fact the measurements are only
known to three significant figures. The result should be rounded to 343. If the last figure
to be dropped in a rounding operation is less than five, round down, otherwise, round
up. A good practice to follow is to round the result of a calculation to the lowest number
of significant figures used in the calculation of that result.
Example: If we multiply 5.03 × 6. 7 the result is 33.701, but should be reported as
33, as there are only two significant digits in 6.7.
Logarithms: Every positive number may be expressed as a power of 10.
We can always find a number "p" such that the number N = 10
p
. We call p the
logarithm of N to the base 10 or the common log of N.
Alternatively, we may write p = log
10
(N)
Fairly complex mathematical expressions may be evaluated simply using logarithms.
Log( A
Z
x B
W
÷ C
Y
) = Z x Log(A) + W x Log(B) - Y x Log(C)
Sample Problems:
1.4
Log(2
5
× 3
2
÷4
3
)
1.5 Log(4
3
)
1.6
Log(56 ×12 )
SYSTEM of UNITS
While it is generally agreed that use of the International System of Units (SI units)
is the best practice, you will soon find that the majority of people who work in vacuum
technology use some non-SI units. Torr, for example, is much more frequently used in
the USA than pascal (Pa) or millibar (mbar) as the unit for pressure. We have chosen to
use SI units whenever possible, but to also follow the current conventions in the United
States.
Before moving on to the derived SI units, some explanation of the base units may be
appropriate.
Meter: The length of the path traveled by light in vacuum in the time interval
1/299,792,458 of a second. A meter is approximately 39.4".
Kilogram: Equal to the mass of the international kilogram prototype. The mass of an
object is related to its weight by the force of gravity given by the equation weight =
mass x gravity. A kilogram is about 2.2 pounds.
Second: The duration of 9,192,631,770 periods of radiation emitted by a specific
electronic transition in the cesium-133 atom.
Las Positas College Vacuum Technology 60A & 60B
Page 12 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Ampere: The constant current which if maintained in two straight parallel conductors of
infinite length and 1 meter apart in vacuum, would produce a force between these
conductors of 2 x 10
-7
newtons per meter of conductor length. If one applies 10 volts
across a 10Ω resistor, 1 ampere of current will flow through the circuit. Ω is the capital
greek symbol usually denoting ohms, the standard measure of electrical resistance.
Voltage E, Current I, and Resistance R are related by:
E IR =
Kelvin: The kelvin is the unit of thermodynamic temperature and is equal to 1/276.13 of
the temperature of the triple point of water (temperature at which water can exist as a
solid, liquid, or vapor depending upon the pressure). The melting point of ice is 273K,
room temperature is 298K, and water boils at 373K. To convert from degrees
Centigrade to Kelvin, add 273.15.
Example: 100 °C + 273.15= 373.15K
(the boiling point of water expressed in kelvin).
Mole: A mole of material contains 6.023 x 10
23
particles. A mole of carbon atoms
weighs 12.011g (the atomic weight of carbon). One can think of a mole as a number of
objects. Just as one can have 5 pencils, one can have a mole (6.023 x 10
23
) of pencils.
Dimensional Analysis: Regardless of units nearly all physical measures in the
universe have dimensions
a b c
M L T
where M is mass, L is length, and T is time, and a, b, and c are integers. It is evident
from Table 1.2 that for voltage, a=1, b=2, and c=-3. Equations can be checked for
validity by ensuring these three numbers are the same on both sides of the equal sign.
This is called dimensional analysis and is a useful tool
The System International set of unites, abbreviated SI, has conveniently defined the
internation unit of Mass to be the Kilogram, the international unit of Length to be the
Meter, and the international unit of Time to be the Second.
Force: That which changes the state of rest (or motion) of matter. The rate of change of
momentum is a measure of force.
Force = mass x acceleration
In SI units, one newton is the force that will accelerate a one kilogram mass one meter
per second, per second.
Pressure (force per unit area). Popular units include pounds per square inch and
(PSI), Torr, Bar,, atmospheres, and Pascals (Newtons per square meter). Pressure is a
force distributed over an area. Absolute pressure is measured with respect to zero
pressure (denoted PSIA), and gauge pressure is measured with respect to atmospheric
pressure (denoted PSIG). If your car tire has about 30 PSIG, then it has about 45 PSIA
(atmospheric pressure is about 15 PSI).
Power: The rate at which work is done. Power in watts will be obtained if work in joules
is divided by time in seconds.
Power = work / time
Electrical Potential: The work expended moving a charged body from point A to point
B in an electric field.
Electrical Resistance: For a conductor of electricity, resistance is the relationship of
applied electric potential to voltage. Ohm's Law states that:
1.8 If the meter is the SI unit for length, what would be the SI unit for area and
volume?
1.9 Given that Ohm's Law states that Voltage = Current times resistance, what
current would you expect in a circuit when a 10mV potential is applied across a
50MΩ resistor?
UNIT CONVERSIONS
Often it will be necessary to change from one system of units to another. A technique
for performing unit conversions is given below and tables of conversion factors, grouped
by function are given in Appendix B.
Let's work a simple example first, then examine the technique.
If you are driving at 30 miles per hour, how many feet per second are you traveling?
This problem requires that we change two sets of units; miles to feet and hours to
seconds. Lets do the miles to feet conversion first. We know that there are 5,280 feet in
a mile, therefore we can write:
Now, let's change feet per hour to feet per second. Since there are 60 minutes in an
hour and 60 seconds in a minute, we can calculate that there are 3600 seconds in an
hour. This conversion factor (3600 sec/hour) will allow us to write:
158, 400 feet
hour
×
1 hour
60 minutes
×
1 minute
60 seconds
=
44 feet
second
Las Positas College Vacuum Technology 60A & 60B
Page 16 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Now, you notice that in the first conversion (miles to feet) we multiplied the conversion
factor (5,280 feet/mile) by the original value of 30 MPH, but in the second step
(converting feet per hour to feet per second) we divided. The trick here is to multiply
always by 1, and organize the units to cancel algebraically.
Sample Problems:
1.10 Express the values in the left column in terms of the units in the right
column.
BASIC PROPERTIES of MATTER
For the purposes of our study of vacuum technology, matter may be divided into three
physical categories or states: solid, liquid and gas.
Solids: The most ordered state of matter. Atoms and molecules of solids tend to remain
in fixed positions with respect to one another. Solids have a definite shape and volume.
Solid material may be crystalline or amorphous. Examples of crystalline solids include
natural crystals, such as gemstones, and metals. Metals are typically composed of
many micro-crystallites (grains) that usually require a powerful microscope to observe.
Glasses have solid-like behavior (they have definite shape and volume), but on an
atomic scale, there is no long-range atomic or molecular order, as exists in crystals.
Glasses have been described as "super cooled" liquids.
Liquids: The state of matter in which atoms and molecules are relatively free to move
about with respect to one another. Liquids have a definite volume, but the shape of a
liquid is defined by the walls of its container.
Gas: The state of matter in which atoms or molecules move about freely with respect to
one another, and tend to distribute themselves to fill any container, regardless of size.
About 400 BC, Greek philosophers argued that indivisible units of matter, called "atoms"
existed, and they were the building blocks from which everything was made. Current
understanding of the nature of matter includes a portion of this classical atomic theory.
The indivisible building blocks that our physical world is made up of are called Elements.
Familiar materials such as iron, carbon, oxygen and mercury are examples of elements.
Compounds, on the other hand are materials that are formed from elements through a
chemical reaction. Table salt, water and methane gas are all compounds. One very
interesting feature of compounds is that they are composed of elements in definite
ratios. For example, water molecules are always composed of two atoms of hydrogen
and one atom of oxygen. Usually the physical properties of compounds are radically
Las Positas College Vacuum Technology 60A & 60B
Page 17 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
different than those of their constituent elements. Water, for example, is a liquid at room
temperature, and yet both of its elements, oxygen and hydrogen are gases at room
temperature. Not all material we are exposed to is in the form of compounds. Mixtures
are simply physical combinations of materials (no chemical reaction involved). The air
that we breathe is a mixture of approximately 79% nitrogen (a gaseous element at room
temperature and atmospheric pressure) and 19% oxygen (another gas under the same
conditions).
INTRODUCTION to MEASUREMENT TECHNIQUES
There is little point in setting up an experiment, and observing some physical (or
chemical) phenomena unless one is prepared to record and later report meaningful
information (data). In this section we hope to provide guidelines for use in this endeavor.
First of all, one must think through the entire experiment before it is started to ensure
that the procedure to be followed will result in a meaningful observation. Let me give
you an example. I was working on a project, the goal of which was to use the
characteristic emission lines of iron to determine when a sputter-etching process had
gone to completion. I was using an existing vacuum chamber with a pyrex window view
port for my spectrometer. Several days of data collection and analysis provided
inconclusive results, so I began to wonder what I was doing wrong. As it turned out, the
pyrex window was strongly absorbing the light I was hoping to observe in the vacuum
chamber. After replacing the pyrex window with one made of sapphire, the results were
in good agreement with what I had expected.
Before beginning an experiment, you should at least have a guess (scientists call these
theories) as to what will happen. In many of the vacuum pump experiments you will
perform you will be measuring the pumping speed as a function of pressure. How do
you expect the pump speed to change with pressure? Will the rate of change be
constant? These are the kind of questions you would do well to consider before the
experimental measurement.
Another aspect of good data collection technique is to repeat the measurement
enough times so that you are confident in the results.
Statistics: The science of the meaningful interpretation of data. Let's assume you're
performing a set of experiments to determine the length of time required to pump a
vacuum vessel from atmospheric pressure to 50 microns. You make four runs and the
values recorded are: 124, 136, 118 and 144 seconds respectively.
Average: The sum divided by the number of measurements.
Average = (124+136+118+144)/4 = 522/4 = 130.5, which rounded to the correct
number of significant digits is 131 seconds.
Range = 144 - 118 = 26 seconds.
For this very limited data set one could say that the time required to achieve a pressure
of 50 microns is the average value plus or minus half the range.
An acceptable way to present this data would be:
Las Positas College Vacuum Technology 60A & 60B
Page 18 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Average time to achieve a pressure of 50 microns starting at atmospheric
pressure in a series of four measurements was observed to be 131 seconds ± 13
seconds.
Notice that the measurement conditions (beginning and ending pressure), number of
measurements (four), and the units (seconds) are all clearly stated.
Clarity of data presentation is very important. Your work, both in this laboratory course,
and in your vocation, will be judged not only on its correctness, but on the manner in
which it is presented. Prior to collection of a set of measurements, set up a table to
enter your data in as the experiment progresses. Columns should be provided for the
variables such as time, pressure, temperature, etc. It is good practice to record along
with your table of data the time and date, your name, what experiment is being
performed.
Graphs: Pictorial representation of data that allows one to view the relationships
between variables. In this laboratory you will be constructing graphs of time versus
pressure and pumping speed versus pressure. Typically, the horizontal (X) axis is used
to plot the independent variable (such as time), and the vertical (Y) axis is used to plot
the dependent variable (whose value depends on or is a function of the independent
variable). We have created below a data table using the integers from 1 to 50 as the
independent variable (X) and have calculated the dependent variable values (Y) for
several simple functions.
Las Positas College Vacuum Technology 60A & 60B
Page 19 Rights Reserved, Biltoft, Benapfl, and Swain Fall 2002
Chapter 2: Safety
Our concern for the safety of everyone participating in this laboratory course is
paramount. To achieve this goal, the Vacuum Technology Laboratory has been
equipped with such safety features as smoke detectors, automatic sprinklers, fire
extinguishers, and a first aid kit. Exposure to liquid chemicals has been minimized by
the use of OSHA approved containers and exhaust gases are removed through a
special ventilation system. Please use these physical safety measures that have been
provided as they were intended; if you do not understand their function or proper use,
please ask an instructor. The information presented in this chapter is only a summary of
the material we felt was most important to provide for student safety awareness. In the
final assessment, each individual is responsible for the safety of everyone in the
laboratory.
Physical Safety
Eyes: Approved laboratory safety glasses (available at the bookstore) are required to
be worn whenever any experimental work is being conducted in the laboratory. If you
would like to use some of the laboratory time to perform calculations or plot data, we
suggest moving to the campus library (it will be quieter there anyway).
Clothing: Many of the experiments involve the use of motor driven mechanical pumps.
It is strongly encouraged that no excessively loose fitting clothing (ie: neckties, scarves,
very loose shirt sleeves) be worn while working on this equipment. Long hair that may
be caught in the pulley of a mechanical pump is also a possible hazard; please tie back
or otherwise prevent long hair from being entangled in any motorized device.
Skin Rupture: Sharp objects, including tools and vacuum system components may, if
used or handled incorrectly result in tearing the skin. Beyond the physical discomfort of
such an occurrence, there exists a real danger of injection of chemicals and infection. If
a skin rupture occurs, immediately flush the area with clean water and apply pressure
using a clean cloth or towel, if bleeding is profuse, notify the instructor and if necessary,
go to the school infirmary. An injured person should be accompanied to the infirmary
either by another student or an instructor.
Sample Problem:
2.1 List three possible ways in which equipment (vacuum vessels, pressure vessels,
etc.) could fail causing projectiles to be scattered in the laboratory.
Electrical Safety
Electric shock is a major cause of fatalities at R&D and production facilities.
Surprisingly, the most likely victim of an electrical accident is an experienced person
with the equipment being used. Electricity is used to power some portion of every
experiment that will be performed in this laboratory course. Generally, this electrical
energy is well contained, so we are protected from its effects, and may become
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complacent with electrical safety measures. Almost everyone has experienced a mild
electrical shock. The result is an unreasonable expectation that one will survive future
electrical shocks. Electricity is uniquely dangerous because it is invisible. The danger
that exists is that electrical hazards may surface in unexpected locations, and be
undetected.
Electric Shock: Passage of electrical current through some part of the body. The
current may be alternating (AC) or direct (DC) and vary from being so low in magnitude
to be detected to so high as to cause fatality.
Our bodies may incur damage by two mechanisms: damage to the nervous system and
joule heating.
Nervous System Damage: External mA range current input into the body causes pain
by stimulating nerves in our skin. As applied current increases, control of muscles is lost
and cramping occurs, often preventing an individual from releasing the source of
current. Further damage may result if the electrical signals that control our involuntary
muscles such as the heart, lungs and other vital organs is scrambled so as to prevent
the proper operation of these organs. This may cause the heart to stop beating.
Joule Heating: In a resistive heating element, current passing through live tissue will
meet with resistance, causing heat to be generated. This heat can cause severe third
degree burns along the path of the current, which may include vital organs.
Sample problem:Physiological effects of electrical current passing through the
body.
SAFE DEFINITIONS
1 mA no physical sensation
1 to 8 mA sensation of shock-no muscle spasms (5 mA max safe current)
UNSAFE DEFINITIONS
8 to 15 mA Painful shock. Muscle control is not lost.
20 to 50 mA Painful shock. Local muscle control is lost.
100 to 200mA Normal Heart beat affected. Victim holds onto current source as long
as current flows. Death may result.
> 200 mA Severe burns. Muscular contraction is so severe that chest muscles
clamp the heart and stop it for the duration of the shock. If current
continues, for several minutes, the heart may be too weak to restart
after the flow of electricity is stopped.
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What to do in the case of a severe electrical shock:
1) Call for help immediately.
Call loudly to other people in the immediate area. Have a person
telephone for emergency medical services, specifying the incident
and location.
2) Rescue the victim.
Locate and safely de-energize the source of electrical power. Take extreme caution not
to expose yourself to the electrical hazard. If necessary, use an insulated implement
(dry broom handle) to remove the victim from the current source.
3) Apply CPR.
Apply CPR as soon as possible if the victim is not breathing.
4) Continue to give aid.
Continue CPR Until a medical rescue team arrives. Electrical shock victims have been
revived from up to an hour after the electrical shock occurred.
5) Get the victim medical attention.
Even if the victim appears to have recovered, a professional medical examination is
required to check for invisible internal injuries.
Basics of Electricity
Electrical energy: The flow of electrons in a conductor.
Potential: The ability of an electric field to do work; the ability to cause motion of a
charge. Electrical potential is expressed in volts (V).
Current: The flow of electrons; expressed in amperes (A).
Resistance: The degree to which a material allows the flow of electrons; units: ohms
(Ω).
Power: The time rate of energy transport or transformation; watts (W).
Frequency: Number of periods of a wave form per unit time; hertz (Hz).
DC: Direct Current. A constant (with time) electrical potential; may be positive or
negative.
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0
Time
DC
V
o
l
t
a
g
e
AC: Alternating current. The voltage of an ac current source varies sinusoidally with
time. House current is 60 cycle (60Hz) AC.
Time
Alternating Current
V
o
l
t
a
g
e
RF: radio frequency alternating current; typically kilohertz to gigahertz frequency.
The different current wave forms have different effects on the human body. For
example; AC causes heart fibrillation and muscle spasms. DC causes muscle clamping,
heart seizures and burns at a higher current level than AC. RF alternating current
passes through the skin readily and causes burns at much lower voltages than AC or
DC.
Capacitors: Electrical devices that store electrical energy. Many of the power supplies
and control units used with vacuum equipment have capacitors in their circuits. Severe
injury can result from coming into contact with a charged capacitor.
Always assume that a capacitor is fully charged
Before beginning any work with a circuit that has a capacitor, de-energize the capacitor
using a grounding strap designed for that purpose.
Hazards Related With Electrical Equipment
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Fire: Many fires are initiated by electrical causes. Be familiar with appropriate electrical
fire extinguishing techniques. Fire extinguishers have information on their labels
regarding their proper use, read the label before the emergency.
Toxic gas evolution: Electrical energy may cause the chemical breakdown of
insulating materials and the decomposition of gases (creating ozone for example). In
addition, older capacitors may contain toxic PCB's.
X-rays: High voltage applied under vacuum will almost always result in the generation
of X-rays. Appropriate shielding is necessary to reduce radiation exposure to an
acceptable level.
Bright light: Sparks and arcs can cause severe eye damage.
Radiation: Microwave and RF radiation from electronic devices can be a health hazard,
especially over a long periods of exposure.
Chemical Safety
Some of the materials (solids, liquids, and gases) you may encounter are chemical
health hazards. Examples are chlorinated and fluorinated solvents, and mercury. Care
must be taken to store, use and dispose of chemicals in a safe and environmentally
sound manner. Specific details for the proper handling of chemicals must be researched
using materials safety data sheets (MSDS). All chemical producers are required by
federal law to supply an MSDS for their products upon request.
Organic solvents: In this laboratory organic solvents are stored in OSHA approved fire
safe red metal cabinets. Transfer the minimum amount of the appropriate solvent to a
suitable container (ie: glass beaker) for use near the experiment. Do not expose
flammable solvents to sparks, hot surfaces or open flames. Use these solvents only in a
well ventilated area. Prevent exposure or contact of solvents to the skin. After use,
return the unused (clean) solvent to the red metal storage can. Allow any rags saturated
with solvent to dry thoroughly then dispose of in a fire safe container.
Mercury: The use of mercury in vacuum technology has greatly diminished, but one
should still be aware of the hazards involved with the use of this element. Mercury has
an appreciable equilibrium vapor pressure under laboratory room conditions (2x10
-3
Torr). The toxic effects of mercury are cumulative, and cause irreversible damage to the
brain and kidneys. Mercury should be stored in a tightly sealed non- breakable
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container (polypropylene) and handled so as to minimize contact with skin, ingestion
and inhalation of its odorless vapor.
Mechanical Safety
Pneumatic Lines: Compressed air often provides a robust and compact energy source
for the actuation of vacuum valves and other pneumatic devices. This is often provided
at a pressure between 70 and 120 PSIG. Always wear safety glasses when working
around live pneumatic lines, as plastic tubes carrying this pressure can whip through the
air wildly if they become disconnected. Never attempt to cover the end of a line with
your finger tip, as air can be directly injected through the skin into the body with painful
or even fatal results.
Vacuum Gate valves: These often generate high actuation forces and one should
never reach through a gate valve without first disconnecting the pneumatics. Gate
valves are often actuated with a small electropneumatic pilot valve (frequently referred
to as a solenoid) that frequently require continuous power to remain open. A PG&E
power failure at eactly the wrong moment could crush bone or even cause
dismemberment.
Thermal Safety
Some of the equipment involved in vacuum technology operates at extremely high or
extremely low temperatures and requires some attention to safety.
Equipment operating at high temperature: Diffusion pumps and evaporation
processes. Second and third degree burns may occur if skin comes into contact with
this equipment.
Equipment operating at moderately high temperature: Mechanical pumps, power
supplies and electronic components.
Equipment operating at low temperature: Cold traps(LN2), cryosorption pumps,
cryogenic pumps, liquid helium lines, and helium compressors.
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Answers to Sample Problems:
2.1 A flawed, fractured or misused glass vacuum vessel could implode; a vessel
could be over pressurized, causing it to explode; volatile gases in a vessel could ignite,
causing an explosion; parts of a rotating mechanical device could fail.
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Chapter 3: Review of Basic Vacuum Calculations
Before we go any further, some time should be spent on some of the vocabulary
specific to vacuum technology.
Vacuum technology is based upon the creation of an environment in which a process
(thin film deposition, electron beam welding, etc.) can be carried out. This normally
implies that one remove air from a system to some acceptable sub atmospheric
pressure by the use of some type of vacuum pumping equipment.
Atmosphere: The blanket of gases that surrounds the surface of the earth and extends
outward to a distance of about 25 miles is referred to as "air" or "the atmosphere". This
mixture of gases exerts a pressure that presses uniformly on all objects on the surface
of the earth. This pressure is about 15 pounds per square inch at sea level.
If we remove some amount of atmospheric gas from a leak-free vessel we will have
created an environment that is drastically different in many respects: mechanically,
chemically and physically.
Mechanical Effects of Vacuum: Have you ever placed a half full 2 liter plastic soft
drink container that is at room temperature into a refrigerator, and noticed later after it
has cooled that its sides are distorted and pulled inwards? What you have inadvertently
done is create a condition in which the internal pressure of the plastic container was
reduced, causing its surface to buckle. Vacuum engineers are acutely aware of this
phenomenon, and design vacuum vessels to be sturdy enough to withstand the external
atmospheric pressure of 14.7 pounds per square inch (at sea level) in the absence of
compensating internal pressure. Structures and components that are particularly
susceptible to distortion under vacuum conditions include flat, unsupported surfaces,
thin sections, and flexible lines or bellows.
Sample Problem:
3.1 Calculate the approximate total force that will be exerted on a 4" diameter glass
view port used in a vessel under high vacuum conditions.
Chemical Effects of Vacuum: The removal of gases from a container will reduce the
number of gas atoms that are available to interact with materials in the container. For
this reason many materials that are hydroscopic (have a tendency to absorb water from
the atmosphere) are stored under vacuum. Materials that readily oxidize are also often
stored either under high vacuum, or in an inert atmosphere (nitrogen or argon gas) after
the air has been removed from the storage vessel.
Sample Problem:
3.2 List as many reactive elements or compounds that you know of which you would
consider storing under vacuum or inert gas conditions.
Physical Effects of Vacuum: Many of the physical properties of gases are strongly
affected by the pressure of the gas. Thermal conductivity, electrical conductivity,
propagation of sound, optical transmission, optical absorption are just a few. In addition
to the effect of reduced pressure on the physical properties of gases, under vacuum
solids and liquids also show markedly different behavior. Liquids, such as water, can be
made to boil in a vacuum vessel without the application of heat. This occurs as soon as
the vapor pressure of the water exceeds that of the vacuum environment.. Similarly,
atoms of solid material under vacuum conditions will spontaneously leave the surface of
the solid. The rate at which materials vaporize under vacuum is a function of the
pressure in the system and the vapor pressure of the material. A more in-depth
discussion of vapor pressure will be presented later.
Sample Problem:
3.3 We have suggested that physical changes in the thermal and electrical
conduction of gases are brought about by a decrease in pressure. What are the trends
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you would expect in these two physical characteristics as pressure is decreased from
atmospheric? (Increase or decrease?)
Gas Laws
Gases are composed of independent, randomly moving atoms or molecules that
spontaneously expand to fill any container. The collective behavior of these atoms or
molecules in a contained volume can be described when one knows any three of the
four following quantities:
1. Pressure: The force per unit area a gas exerts on its surroundings. (in our
calculations we will use primarily Torr or atmospheres).
2. Volume: The internal capacity of a container, or vessel. (Liters)
3. Temperature: The temperature of a gas is a function of its kinetic energy, that
is, how vigorously the gas atoms are vibrating. Temperature
must be specified in terms of an absolute temperature scale.
We will use the kelvin scale (K=°C + 273).
4. Amount: The number of gas atoms in a volume (can be in terms of atoms
or moles). {A mole of material is 6.02 x 10
23
particles}.
Boyle's Law: Under conditions of constant temperature, Boyle's Law gives the
relationship between volume and pressure for a fixed quantity of gas.
P
1
× V
1
= P
2
× V
2
Let's do a thought experiment to demonstrate Boyle's Law. Imagine a system of two
leak-free vessels as shown below.
Vacuum Vessel
2
TC2
Vacuum Vessel
1
TC1
Figure 3.1
Assuming that the temperature is constant everywhere in our system, and that we can
accurately measure the pressure in both vessels, we should be able to apply Boyle's
law to calculate the volume of vacuum vessel 2.
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If we know that at the beginning of our experiment the volume of vessel 1 is 120 liters,
and the pressure of gas inside vessel 1 is 760 Torr, and that vacuum vessel 2 has been
rough pumped to about 10 mTorr we can write:
P
1
= 760 Torr
V
1
= 120 Liters
Now, if we open the valve between vessels 1 and 2, and allow sufficient time for the
system to equilibrate, we read pressures at TC1 and TC2 to be 500 Torr.
(760 Torr)(120Liters)=(500 Torr)(V
2
+120 Liters)
Solving for V
2
we find the second vessel has a volume of 62 liters (note that we include
the tubulation to the right of the valve as part of the volume of vessel V
2
.).
Sample Problem:
3.4 What would be the volume of vessel 2 in figure 3.1 if the final pressure read on TC1
and TC2 was 350 Torr rather than 500 Torr?
Charles' Law: Under conditions of fixed volume and amount of gas, Charles' Law
describes the relationship between the temperature and pressure of a gas.
Sample Problem:
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3.5 If the initial pressure and temperature of the leak-free vessel in figure 3.2 were 50
mTorr and 25 °C respectively, and the vessel was heated uniformly to 100 °C
what would be the new pressure reading?
The Ideal Gas Law: The relationship between pressure, volume, amount of gas and
temperature of gas for "ideal" gases is given by the Ideal gas law. Fortunately, most
gases behave "ideally" under subatmospheric conditions.
Sample Problem:
3.6 If a 100 liter vessel at room temperature is evacuated to a pressure of 50 mTorr,
how many moles of gas are in the vessel? How many molecules is this? How
many molecules per cubic centimeter is this?
Kinetic Description of the Behavior of Gases
As the name may suggest, the kinetic theory of gases has to do with describing how
gases behave under the influence of external forces that induce motion. There are four
basic assumptions that provide the foundation of the kinetic theory of gases:
1) Gases are comprised of a large number of extremely small particles
(atoms or molecules).
2) These gas molecules are in constant, rapid motion in a chaotic manner.
3) The distances between individual gas molecules are large compared with
the diameter of the molecules.
4) The molecules exert no force on one another, or on the walls of a
container except during collisions.
Velocity of Gas Molecules: The speed at which gas molecules travel is independent of
pressure, but is a function of the temperature and molecular weight of the gas.
For most clean vacuum systems the majority of the gas load may be assumed to be
nitrogen, and at room temperature the following approximation may be used to calculate
the mean free path for N
2
molecules:
3
5.0 10 x
L
P
−
=
L= mean free path [cm]
P= pressure [Torr]
Collisions of Gaseous Species: Gas molecules travel in straight lines between
collisions and tend to strike all exposed internal surfaces of the vessel in which they are
contained. Pressures that we measure using various types of gauges (more on this in
chapter 5) are the result of the collective impacts of these gas molecules on the inner
surfaces of the containing vessel. The rate of impact (or impingement rate) of gas
molecules per second per square centimeter of surface area is a function of the speed
of the molecules and the gas density
N= molecular density, [cm
-3
]
v= molecular velocity [cm/sec]
I = impingement rate [cm
-2
-sec
-1
]
Usually, the quantities that we can easily measure are pressure and temperature, so,
the same equation expressed in terms of these units is:
22
2
1
3.5 10
sec
m
P
I x
cm
W T
=
−
Sample Problem:
3.9 What is the impingement rate for nitrogen molecules on the inner surface of a
vacuum vessel having a pressure of 5 x 10
-6
Torr and a temperature of 25 °C?
What is I for the same system at 5 x 10
-9
Torr?
Motion of Gas Molecules: As collisions occur between gas molecules and the inner
exposed surfaces of a vessel, the molecules are "diffusely" reflected, that is there is no
relationship between the arrival angle and the departure angle following a collision. The
angle of departure from a planar surface has been studied and was observed to follow a
cosine distribution as shown in figure 3.4
Figure 3.4: Cumulative probability of departure angles of gas molecules departing a
smooth surface. For example, 50% of particles will depart with an angle of 30° or less.
Flow of Gas Through an Orifice: Let's do another thought experiment. Imagine a leak
free vacuum system comprised of two vessels separated by a closed valve. One vessel
contains nitrogen gas at a pressure of 5 x 10
-5
Torr and the other vessel is under
extreme high vacuum (5 x 10
-10
Torr). Both vessels are at room temperature. If we
suddenly open the valve what will happen during the pressure equilibration time? Only
those molecules that randomly impinge {molecular flow, right?} upon the opening
between the vessels will leave the vessel at higher pressure and move into the vessel at
lower pressure. Let me make the point clear by stating the reverse: those molecules in
the vessel at initially higher pressure that don't impinge upon the opening between the
vessels can not leave the vessel they are in. What this suggests is that the flow rate for
gas molecules leaving a vessel is a function of the collision rate of molecules per unit
surface area. The number of gas molecules leaving is:
2
1
4 sec
Nv
I
cm
=
−
The volume of gas leaving may be
calculated by dividing the number of
gas molecules leaving by the number
of molecules per unit volume (N)
The volumetric flow rate of gas
through a hole is independent of the
gas pressure; but depends on the
gas velocity, v, which is a function of
temperature and molecular weight.
For the situation in which the mean free path of gas molecules is greater than the
diameter of the opening in the wall of the chamber, the volumetric flow rate (s) is given
by:
Modes of Gas Flow Under Various Vacuum Conditions
The three modes of gas flow that we will be interested in describing are: turbulent (or
viscous), laminar (or transition) and molecular flow.
The flow regime created when air is induced to move through cylindrical tubes is a
function of the tube diameter and the average pressure.
For these equations, D is the inside
diameter, in inches, and P
bar
is the
average pressure in Torr
Criteria for Viscous Flow Regime
Criteria for Molecular Flow Regime
Criteria for Transition Flow Regime
Upon initiating a pump down, the flow of gas molecules is often turbulent, exhibiting
eddies and currents much like a raging rapid.
Volumetric: Flow rate (S) is the volume amount of gas that passes by a point per unit
time. Examples of units are: liters/second or cubic feet/ minute. The volumetric flow rate
may be considered to be the pumping speed of a system at a specified point in the
conductance path.
Quantitative: Flow rate (Q) is the amount of gas that passes by a point per unit time.
Units are: Torr-liters/second. The quantitative flow rate is also referred to as the
throughput, or mass flow, and is constant everywhere in the vacuum system, unless gas
is leaking or is being captured or condensed along the path.
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foreline
valve
Q
Q
Vacuum Chamber
TC1
TC2
IG1
vent valve chamber
rough valve
head gate
valve
chamber vent
DP vent
IG2
TC3
Figure 3.8 Gas flowing through a vacuum system beginning at the inlet to the vacuum
vessel and exiting at the exhaust of the roughing pump. At all locations in the vacuum
circuit the quantitative flow rate (Q) is the same.
Conductance in a Vacuum System:
Gases moving through conductance elements (pipes, tubes, vessels, and orifices) in a
vacuum system encounter resistance to their motion. At higher pressures, this
resistance is a function pressure differnce and geometry of the conductance element.
1 2
P P
Z
Q
−
=
Z= resistance [sec/liter]
P= pressure [Torr]
Q=flow rate [Torr-L/s]
Conductance is the inverse of resistance and therefore,
1 2
1 Q
C
Z P P
= =
−
C= conductance [liter/sec]
Even very simple vacuum systems are comprised of many conductance elements,
some are connected in series, some in parallel. Let's examine how to calculate the
effect of various components in simple systems.
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Vacuum
chamber
valve trap
pump
90° elbow
C1 C2
C3
Molecular Flow: For the situation in which the mean free path of molecules is much
larger than the diameter of the conductance element, the relationships for throughput, Q
and conductance, C
m
are given by:
3
1 2
80
( )
m
D
Q P P
L
= −
1 2
( )
m
m
Q
C
P P
=
−
3
80
m
D
C
L
=
Now let's apply this information to the calculation of pumping speeds in a simple
vacuum system.
At any location in the vacuum circuit the volumetric flow rate or pumping speed at that
point can be found if we know the pressure at that point, the pressure at the pump, and
the conductance of the path connecting these two. At the chamber side of the system
P
1
= Q/S
t
, and similarly at the pump side, P
2
=Q/S
p
. We will make use of the relationship
between mass flow (Q), pressure drop (P
1
-P
2
), and conductance (C):
Sample Problem:
3.12 For the simple vacuum system pictured in figure 3.11, calculate the pumping speed
at the chamber if the pump has a speed of 200 L/s, the pressure at the vacuum
pump is 5 x 10
-6
, and the conductance element is a tube 30" long by 2" in
diameter.
Conductance of apertures will become important when performing vacuum system
calculations when tube diameters abruptly change diameter by a factor of two or more
(see picture below).
While accurate equations for aperture flow are complex in the viscous flow regime, this
approximation is often reasonably valid:
A = Area of circular or nearly square aperture.
C
v
= Conductance in l/s in viscous regime
In the molecular flow regime the following is valid for Nitrogen near room temperature:
A = Area of circular or nearly square aperture.
C
v
= Conductance in l/s in molecular regime
Before purchasing components for and building an expensive vacuum system,
calculations are generally performed that provide information as to the amount of time
that will be needed to evacuate the vacuum system to a certain pressure. Typically the
calculation is broken (partitioned) into five or more "pressure intervals" and at each
interval the conductance of the tubulation between the vessel and pump is calculated.
The pump speed delivered to the vessel is calculated, and finally, the time to pump from
the upper pressure limit of the interval to the lower pressure limit of the interval is
computed. The calculation is performed in segments because both conductance (in
viscous flow) and pump speed change as pressure decreases. Plots on the following
page give the general behavior of tube conductance and vacuum pump speed as a
function of pressure.
Pressure Interval 1: Time required to pump from 760 Torr to 100 Torr.
A. Calculate the conductance of the tube between the pump and vessel (be sure to
check for viscous or molecular flow conditions).
B. Use a pump efficiency curve to determine the volumetric pumping speed of the
pump for the pressure interval of interest.
C. Calculate the delivered pump speed.
D. Calculate the time to pump from P
1
to P
2
.
Experimental set-up: Select a working rotary vane mechanical pump and the
appropriate gages, tubulation, and fittings to assemble a vacuum system as shown in
figure 3.13 First, assemble the system with a short conductance element between the
chamber and pump. Calculate the conductance of the element using formulas
discussed in this chapter. Evacuate the chamber, allowing the mechanical pump to
achieve its base pressure. Record P
1
and P
2
, and using the calculated value of C, and
the manufacturer's value of pump speed, solve for the pumping speed at the chamber.
Repeat the experiment for a much longer conductance element of the same diameter.
One area of vacuum technology that has evolved rapidly in the last 20 years is the
development of system hardware. Those of us that had been exposed to 1960's vintage
vacuum systems can remember how crude and clumsy they were! Today, vacuum
hardware is generally streamlined, attractive, and functional. We usually attribute the
pleasant appearance of modern vacuum systems to 1) manufacturers who compete for
the business of the user, and 2) users who desire clean, attractive systems to keep their
facilities modern and up-to-date. There are other contributors, of course. And one point
that needs to be mentioned is that as vacuum systems progress, their price tags also
soar!
In this chapter we hope to present information that will allow you to become familiar with
some of the hardware that is currently in use in the field of vacuum technology. We
have included by way of introduction, a set of the symbols used by the American
Vacuum Society (AVS) to schematically diagram vacuum systems. This set of symbols
functions much like the symbols used in the electronics field to represent electric
circuits. It should be noted here, that there exists a European set of symbols that is
completely different from the AVS set. The European symbols are presented in the
technical reference section of the Leybold Hereaus catalog.
Also covered here is material that will introduce the subject of materials selection for
vacuum applications. In particular, the section on valves provides some insight into the
selection of materials based upon their outgassing behavior, permeability to various
gases, and mechanical strength. Since no single component stands alone, some
mention of techniques used to join various components is also given.
There is one laboratory exercise included in this chapter, and there are several
discussion questions provided to provoke your curiosity. As you explore vacuum
technology we encourage you to pay close attention to the way in which hardware is
assembled and maintained. We believe that as your experience in this field increases,
so will your curiosity.
Stopcocks
Feed-throughs
rotating
sliding
bellows sealed
electrical
2 way 2 position
3 way 3 position
3 way 2 position
Vacuum Chambers
Vacuum chambers come in a variety of sizes and configurations, which are generally
specified to accomplish a defined task. Bench top experiments can be carried out in
vessels that have volumes as small as one liter. On the other end of the physical size
spectrum are vessels that are built to contain large physics experiments, such as the
tandem magnet magnetic fusion experiment at Lawrence Livermore National Laboratory
into which several city buses could easily fit. Material selection is also of great
importance in the design of a vacuum chamber or vessel. Early vacuum experiments
(around 1640) were performed in tight wooden casks. As the vacuum technology
became more sophisticated, materials such as lead, copper, brass, glass, steel and
recently stainless steel have been used in the construction of vacuum hardware.
Las Positas College Vacuum Technology 60A & 60B
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Size: In general, there is a strong advantage in designing a vacuum vessel as small as
possible without hampering the process to be run in the vessel. There are several
obvious reasons for this choice. Cost of vacuum hardware generally increases rapidly
as size increases. For example, a simple flange that is to be welded onto a vessel costs
$20 for the 2
3
/
4
" diameter size, and $250 for the 10" diameter size flange of the exact
same design! In addition to the issue of initial cost, operating costs for unnecessarily
large vacuum vessels will be high due to the longer time required to achieve working
pressures (dead time for operators), and increased frequency of repairs on pumping
equipment.
Geometry: Commercial vendors of vacuum vessels offer a wide variety of vessel
shapes which are typically cylindrical, spherical or rectangular. These vacuum vessels
can be grouped into the following categories: bell jars, cylinders, box coaters, "tees" and
"crosses" and custom made vessels.
Bell Jars: May be made from pyrex glass which is selected to resist damage from
thermal shock, mechanical stress, and chemical attack. Bell jars are quite inexpensive
compared to their stainless steel counterparts., and are often selected for experimental
or prototype work. Pyrex glass bell jars have one clear advantage: they can allow direct
observation of the vacuum environment as long as the interior of the bell jar in kept
clean. Glass bell jars range in price from $350 (10" diameter, 12" tall) to $1,000 (18"
diameter, 30" tall). Rubber gaskets are fitted to the base of the bell jar to provide a
vacuum seal with a metal flange. Synthetic rubber materials used for bell jar gaskets
include: Neoprene, Viton, and Buna. The choice of gasket material should be made
according to the expected operating conditions (temperature, pressure, chemical
environment, mechanical wear, etc.). Viton gaskets are the most expensive and cost
approximately $200 for a 24" diameter bell jar. Glass bell jars require a safety guard,
made from expanded sheet metal. The function of the guard is two-fold: to protect the
glass vessel from damage, and to contain the glass in the event the vessel ruptures.
Stainless steel bell jars are also commercially available, and typically come with
an exterior water cooling circuit, and at least one viewport flange. Metal bell jars
typically cost from $2000 (18" diameter, 30 tall) to $4000 (24' diameter, 30" tall). Some
18" diameter and smaller metal bell jars can be obtained with a metal seal "Wheeler
flange" which would allow attainment of lower base pressures than polymeric seals. The
most common grade of stainless steel chosen for vacuum application is 304. This
material has low gas permeability, can be outgassed by heating, and resists chemical
attack.
Cylinders: Most cylindrical vacuum vessels are larger in diameter than they are tall, are
made of 304 stainless steel, and have all-metal Wheeler flanges on their top and
bottom. Quite often many ports are built into cylinders for viewports and feed throughs.
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Box Coaters: Increasingly popular in the optics and micro-electronics industries, box
coaters are often chosen for the ease in which batches of parts can be loaded and
unloaded. Another advantage of box coaters is that with many standard flanges welded
to the chamber's walls, reconfiguration of deposition sources, substrate holders, and
process diagnostics is a simple matter. Box coaters are often made from 304 stainless
steel, and have trace cooling on their exterior surface. Typically, an o-ring sealed hinged
door is provided for interior access. To avoid costly down time for cleaning, inexpensive
sheet metal shields are installed inside the chamber, and are removed and replaced
periodically. External reinforcements are often required to supply sufficient mechanical
strength to the walls of the chamber while evacuated.
"Tees" and "Crosses": Many of the vacuum components vendors carry a variety of
"vacuum plumbing" hardware in the form of tubes, "tees" and "crosses" with standard
flanges welded on that can be easily assembled with valves, pumps and gauges to build
superb vacuum vessels. Most of this hardware is made of 304 stainless steel, and is
available in sizes ranging from 3/4" diameter O.D. to 10" diameter O.D.
Traps, Baffles and Filters
Traps: Broadly defined, a trap is any device placed in a vacuum circuit that serves to
reduce the partial pressure of gas in the circuit. Traps are auxiliary capture pumps that
are generally used to prevent flow of gas in an undesirable direction (as in an oil
diffusion pump, see chapter 7). Traps are often placed in vacuum conductance paths
between vacuum pumps and the vacuum vessel to reduce to a minimum the chance of
oil back streaming from the pump to the vessel. The various mechanisms for trapping
gas vapors include: adsorption, cryo-sorption, and cryo-condensation.
Traps that rely upon adsorption typically are filled with material having a very
large surface area to volume ratio, such as fiberglass, copper wool and micro porous
material such as zeolite.
Cryo-sorption traps take advantage of the fact that vapors, upon striking cold surfaces
(below the boiling point of the gas at the operating pressure) will condense and
effectively be pumped from the system. Liquid nitrogen (LN) is generally chosen as the
coolant for these type traps. The boiling point of liquid nitrogen is -195 °C; gases having
boiling points above this temperature will generally condense upon striking a liquid
nitrogen cooled surface. A point of caution should be made: if the LN cooled trap is
inadvertently allowed to run dry and warm up, those vapors that have been condensed
will evaporate and move about the system. Many modern vacuum systems have LN
traps that are automatically filled through the use of a device having a thermally
actuated solenoid valve to control LN level in the trap. Be aware that these automatic
systems can fail!
Cryo-condensation traps make use of both cryogenically cooled surfaces, as
described above, and the gas trapping effects of micro porous surfaces. Zeolite is a
molecular sieve material that contains pores that are approximately 7 +/- 3 Angstroms in
diameter. This pore size is optimal for trapping gases through the use of Van der Waals'
attraction.
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Maintenance of Traps: Traps do require maintenance, and if this work is carried out
improperly, the effect on the vacuum system is worse than having no trap at all.
Regeneration of traps is necessary because traps have a finite capacity, and will after
some time become saturated with condensed vapors (oil and water, usually).
Adsorption traps that are saturated must have the active element regenerated or
replaced. Many of the commercial traps have quick connect fittings and are designed for
easy replacement. Other designs allow for baking of the trap (150 to 300 °C is typical) to
evaporated condensed vapors, or replacement of the sieve material.
Cryo-condensation (LN trap) maintenance is easy- simply keep a constant level of LN in
the trap. To bake out an LN cold trap, isolate the trap from the system (close the gate
valve), keep the high vacuum pump operating (as well as the fore line pump), and allow
the LN trap to warm to room temperature. Hot air may be blown through the LN
passages to warm the trap above room temperature to speed evaporation and pumping
of gases condensed in the trap. Following this bake-out simply refill the trap with LN,
and allow time for condensation surfaces to cool before opening the gate valve.
Cryo-sorption traps are either entirely replaced, or are isolated from the clean side of
the vacuum vessel, and are baked out while under vacuum, as described for cryo-
condensation traps.
Quick Connect Hardware
Several manufacturers produce quick connect vacuum system hardware that
allows for rapid construction and reconfiguration of vacuum systems. Most of this
hardware makes use of elastomeric materials (o-rings) to provide the vacuum seal. The
hardware, in the form of tubes, elbows, tees and crosses is available either in plastic
(high impact strength polycarbonate), aluminum, brass or stainless steel. Practical base
pressures for polycarbonate and stainless steel quick connect hardware are 10
-4
Torr
and 10
-6
Torr respectively.
weld rings
O-ring
centering ring
Components of a typical
vacuum system quick
connect assembly are
shown to the left. The
stainless steel weld rings are
often welded to stainless
tubing of the appropriate di-
ameter, or are welded
directly to a vacuum vessel
wall. Not shown here is the
clamp that is used to hold
the assembly together, and
provide sufficient
compressive force to the O-
ring to achieve a vacuum
seal.
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This quick connect hardware is available in sizes compatible with Stainless or Aluminum
tubing from 1/2" to 2" outside diameter. This connection type is often designated "NW"
or "KF" with a metric size designation. For example, 1.5" diameter quick connect tubing
is often designated NW-40 or KF-40.
A few of the available configurations for quick connect vacuum hardware.
Another type of mechanical connection frequently used employs a groove or
"gland" machined into the face of one of the flanges and an elastomeric gasket, or o-
ring. Compressive forces exerted on the O-ring when the two flanges are tightened
together squeezes the O-ring providing the seal. An extremely light coating of O-ring
grease is applied to the O-ring, not to fill voids, but to allow the O-ring to move in the
groove under the applied force to achieve the optimal position for a tight seal. A general
rule of thumb with respect to O-ring grease is: if you can feel the grease on the O-ring, it
is too much! Several companies supply special greases that have vapor pressure and
lubricating properties that are engineered for vacuum system application.
An O-ring sealed flange assembly. Note that the gland is a trapezoid, and its cross-
sectional area is larger than that of an undeformed O-ring. Also note that provisions
have been made to pump out the vacuum side of the gland.
UHV Hardware
Vacuum systems designed to operate at pressures below 10
-7
Torr require special "Ultra
High Vacuum" or "UHV" hardware, in the form of flanges, gaskets, hoses, bellows,
tubes, viewports and the body of the vessel itself. There is absolutely no point in putting
money and effort into construction of a UHV vessel if any of the components used will
be inadequate to achieve the desired performance. As with many things, a vacuum
system's performance will be limited by the lowest quality component on the system.
This is not to say that the foreline roughing connections cannot be of lower quality
components; only those components that are part of the UHV circuit need be of the
highest quality.
Knife Edge Flanges: The leak tight seal that is required for UHV system operation is
created by compressing and deforming a soft metal gasket (usually copper) between
two stainless steel flanges having recessed knife edges. The illustration below shows a
typical metal sealed assembly (exploded view on the left). Knife-edged flanges are
welded to tubes with the weld being made on the inside surface only to minimize virtual
leaks. In the assembly of a metal sealed flange it is important to uniformly squeeze the
copper gasket to achieve an adequate seal. To do this, one must carefully seat the
copper gasket in the counter bore where the knife edge resides, and tighten the bolts
sequentially, first tightening (snug) one bolt, then its nearest neighbor, and proceeding
to tighten bolts in the same direction around the flange. Several "laps" around the flange
will be required to compress the copper gasket properly. This technique is perplexing for
most people who have become familiar with mounting automobile wheels. In the case of
a wheel, one tightens bolts in a "star" pattern to minimize warping the rim. The goal with
metal sealed flanges is exactly the opposite- it is very desirable to deform the copper
gasket!
internal
welds
copper
gasket
tube
flange
internal
welds
copper
gasket
Mass production of these UHV flanges has made them cost effective readily available,
and compatible with tubing in a variety of standard sizes: 1
1
/
2
", 2
3
/
4
", 4
1
/
2
", 6" and 8". It
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should be noted here that UHV hardware is sold under a variety of proprietary names,
such as: Conflat, Del-seal, F-style, CF, FC, Vac-U-Flat, and Aflat.
Gaskets used for UHV seals are typically made from high purity oxygen free copper that
has been annealed to make it more readily deformable. For high temperature
applications (vacuum furnaces) copper gaskets are often coated with silver to minimize
oxidation of the copper. The concern here is to prevent copper oxide flakes from being
formed inside the vacuum vessel. Aluminum gaskets are used in vacuum vessels
constructed from aluminum to insure material compatibility.
Bellows: Flexible vacuum tubing, in the form of either welded or hydraulically formed
stainless steel bellows allows for isolation of vibration in vacuum systems, and reduces
the mechanical tolerances to which some portions of a vacuum system must be made,
reducing design, fabrication and assembly costs. Bellows must be supported on both
ends, as they become compressed when evacuated. Allow room for the middle section
of a bellows to move during pump down, so that it will not come into contact with other
components and become damaged. When possible, avoid forcing a bellows to bend at
multiple points or in more than one plane. Use a bellows with an elbow welded to one or
both ends to keep the flexible section of the bellows from suffering too many bends.
Flange Stainless steel welded bellows
Valves
Virtually all of the vacuum processes that are conducted require some means of
controlling the admittance of air or process gases into or out of the vacuum chamber.
Valves designed for vacuum application provide the means to satisfy this important
function. An ideal valve would have the following characteristics: it would allow for the
maximum amount of conductance while open, have zero conductance when closed,
and no leakage of atmospheric gases into the vacuum vessel would occur due to the
presence of the valve in the vacuum circuit. It would have infinite cycle life, never shed
particles, be bakeable to any temperature, be lightweight and cost nothing. An
equipment designer often balances real-world requirements against cost. Valves must
withstand a pressure differential of 1 atmosphere (14.7 PSI) when closed. Valve
designs in use today include: angle, gate, butterfly, pendulum, leak, soft vent, ball and
pressure control (dynamically varying conductance). In addition to the geometry and
throughput, the materials from which a valve is manufactured should be considered
when selecting a valve. Valve bodies are commonly constructed from aluminum (both
cast to shape and machined from bar stock) and stainless steel.
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Aluminum: Cast aluminum valve bodies may be used in high vacuum application, but
for attainment of UHV, aluminum valve bodies must be machined from solid bar stock.
Aluminum valves are common on modern semiconductor process tools. Aluminum
valves are generally always elastomer sealed. Aluminum offers outstanding thermal
conductivity. This is a benefit when operating heated valves in the presence of
condensible by-products where cleaning is periodically required.
Hard-Anodized Aluminum: A popular and economical alternative to Stainless "wetted
surfaces", this coating offers elevated resistance to acids and aggressive by-products.
Deployment will result in a higher leak rate across the seal surface in comparison to
uncoated valves. Surfaces will also outgas more, and so these surfaces are not
recommended for UHV systems.
Stainless Steel: The most popular, though most expensive material, stainless steel
valves provide the means to achieve UHV and also resist most corrosive gases. They
offer higher temperature service (usually to 200°C with elastomer seals, 300°C with
metal seals) and
Valves typically contain the following seals:
• Two flange seals, for the vacuum connections
• A bonnet seal, for service, inspection, and assembly.
• A seal at the actuator mechanism, to allow motion to be passed through to
vacuum
• The main gate seal
All these seals see static service except the gate seal which see dynamic service. All of
these seals can be from Viton, metal, or perfluorelastomers (a class of chemically
resistant, high temperature specialty materials).
Generally, modern commercial Gate and Angle valves can reliably maintain vacuum on
either side of the gate (with atmosphere on the opposite side). It is wise to check with
the manufacturer if it is an older gate valve.
Angle Valves: Commonly used in foreline connections, these valves may have either
O-ring or metal seals. Right angle valves have low conductance for the bore size due to
the 90 degree turn. Below is shown a pneumatically actuated version with spring return
(normally closed)
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Common angle valve construction
Gate Valves: Are usually used for pump isolation or media transfer (such as moving
wafers or other material form one chamber to another). For pump isolation, they offer
highest conductance between chamber and vacuum pump.
Gate valves may be of cast or machined aluminum or stainless steel. They usually
incorporate elastomeric seals, while those designed for UHV applications will have
metal seals. Gate valves may be manually, electrically or pneumatically actuated. Great
care must be taken with these valves when connected to a source of power for
operation. Several people have lost hands working on vacuum systems when a gate
valve was accidentally closed with the individual's hand in the valve.
Feedthroughs
Vacuum processes would be severely limited without the means to bring electrical
power, motion, and cooling water into vacuum systems. Similarly, the ability to move
objects inside an evacuated vessel greatly enhances the capabilities of a vacuum
system (depositing thin films onto all surfaces of a complex shaped part for example).
Feedthroughs provide the mechanism for accomplishing these goals without
compromising the vacuum environment one works so hard to achieve.
Electrical Power Feed Throughs: Electrical power is introduced into a vacuum vessel
from some external source by means of a well designed feed through. These devices
are used when some process, such as electron beam evaporation, sputtering or
vacuum brazing is being conducted. Additionally, most vacuum instrumentation,
including pressure gauges rely upon electrical vacuum Feed throughs. Ceramic
insulators built into the feed through prevent the dissipation of electrical energy through
the walls of the vacuum vessel, and the commensurate danger of electrical shock
When selecting an electrical power feed through, care must be taken to choose a unit
that will be able to safely conduct the voltage and current that will be applied. A
conservative safety factor is recommended. Gauge applications require feed throughs
that will conduct signals of low voltage and/or low current. As with power feed throughs,
correct selection in light of the process requirements is essential for optimal
performance.
Fluid Feed Throughs: Water, liquid nitrogen or process gases are frequently required
to be admitted into a vacuum system in a controlled manner without degrading the
vacuum environment. Feedthroughs designed for this application come in a wide
variety. Common designs include dual line water feed throughs (inlet and outlet for
water used to cool thin film deposition sources, such as sputtering guns, in the vacuum
chamber ), and single line gas inlet (for introduction of a process gas).
Vacuum
side
Atmosphere
side
Vessel wall
O-ring
Motion Feed Throughs: In order to open and close valves, move substrates, and
articulate specimens under vacuum a variety of motion feed throughs are required.
These feed throughs may be categorized according to the type of motion they allow.
Rotary feed throughs are used for actuating the flapper in butterfly valves, linear feed
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throughs provide the motion required by gate valves and "wobblers" or wobble stick
manipulators allow for angular motion. Some complex feed throughs provide
combinations of movements such as: rotary-linear, wobble-linear, etc.
Pictured above is a cross-sectional view of a simple rotary feed through. A pump-out
port is provided to allow for differential pumping between the two elastomeric seals. This
design rotary feed through is suitable for rough and high vacuum, but not UHV. When
choosing a rotary feed through be sure to consider the maximum shaft rotary speed,
maximum torque the shaft will experience, the cantilever loading, and other pertinent
mechanical factors. Also to be kept in mind are the temperature requirements. Will the
feed through be exposed to liquid nitrogen? Will the feed through be subjected to a high
temperature bake-out? for UHV applications Ferrofluidic and bellows sealed feed
throughs are available; see specific vacuum components catalogs for details on these
rotary feed throughs.
Rotary feed throughs are commercially available with a variety of flanges including
quick-connect and all-metal sealed designs. Another consideration for your selection of
motion feed throughs is the motive force that will cause the feed through to rotate,
extend or retract. Most of the available models can be purchased as manual (operator
hand controlled) or motorized. Motor selection is important, and should be made in light
of the process requirements.
Wobble-stick manipulators may be used for changing the position of parts being coated
in a vacuum vessel, positioning sensors, or even pointing deposition or radiation
sources inside a vessel. "Ball and socket" design wobblers (see picture below) with a
stainless steel welded bellows providing the vacuum seal are common.
Other types include cam and wedge designs which also employ welded bellows
for the vacuum seal.
Linear motion manipulators come in several sizes, geometries, and load capacities, and
may be manually controlled (push-pull, rack and pinion, lead-screw) or motor driven.
With all of these manipulators, care should be taken to insure that during pump down
the pressure differential across the sealing surfaces of the manipulator will not cause
damage to it or any other equipment. Remember that bellows if unconstrained will
collapse during evacuation. Pictured on this page is a simple linear motion feed through
of the manual lead-screw design. A scale ruled on the lead-screw cylinder gives an
indication of the extension of the shaft on the vacuum side of the vessel.
Vacuum
side
Atmosphere
side
position scale
welded bellows
Valve Identification and Inspection.
Identify the vacuum valve you have selected.
A. Valve Identification: Who is the manufacturer? What is the valve model number?
Locate the manufacturer's literature from the bookcase, and find the appropriate
reference information. What is the advertised conductance in viscous flow? In molecular
flow? What pressure range is the valve designed for?
B. Physical Inspection of the vacuum valve: Inspect the valve for signs of wear or
misuse. Check the sealing surfaces, are they worn? Is the flapper actuator in good
condition? Record your observations.
Discussion:
What are the primary drivers for the cost of a vacuum valve besides price?
Where would an Angle valve be preferred over a Gate valve? A Gate valve over
an Angle valve?
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Chapter 5: Subatmospheric Total Pressure Gauges
The pressure gauges that will be described in this section are used to monitor the
environment in a vacuum vessel so that processes conducted under subatmospheric
conditions may be understood and made to repeat. As with other topics in vacuum
technology, the subject of pressure gauges can be subdivided several ways. Gauges
could be lumped together based upon the pressure range in which they operate, the
physical principle behind their operation, or by size, cost or complexity. We have chosen
to use the time honored method of grouping gauges to be discussed primarily by the
physical basis of operation. In this scheme, gauges are grouped into the following four
categories: gauges that measure the physical force exerted on a surface, gauges that
measure momentum transfer by gas molecules, gauges that measure heat transfer, and
gauges that measure gas density by ionization of gas molecules.
Examples of each of these four categories are:
Experiments and processes are performed every day in vacuum vessels that have total
pressures ranging from 10
-13
Torr to almost atmospheric pressure (760 Torr). This
pressure range extends almost 16 decades! No one pressure gauge available can
accurately measure the pressure across this enormous range. The gauges that are
sensitive enough to be accurate at extremely low gas densities would be swamped if
not seriously damaged if operated at pressures above 10
-3
Torr. For each of the vacuum
gauges that will be covered, we will make every effort to describe the application that is
appropriate for the gauge and also give useful notes on materials compatibility,
mechanical durability, and susceptibility of gauges to outside influences.
Force Measurement Pressure Gauges:
U-Tube Manometer:
Historically, the liquid level gauge was the first to be used as a means of monitoring
pressure changes. Water was the liquid used initially, but its low density required that a
gauge capable of measuring atmospheric pressure would be over 30 feet tall. Mercury
replaced water as the higher density of mercury ( 13.6 times more dense than water)
allowed for much more compact gauges to be built. The basic principle of operation of
simple liquid level gauges is as follows: a "U" shaped glass tube having a vertical
section one meter tall is filled approximately half way with liquid mercury. A valve on the
"reference" side of the tube is opened to an operating high vacuum pump, the air
pressure above the mercury is reduced to 10
-5
Torr or less, then the reference valve is
closed. At this point the gauge inlet side of the U-tube may be connected to a vacuum
system. If the system is initially at atmospheric pressure, the mercury column height
difference, H, between the reference and gauge legs of the U-tube should be
approximately 760 mm. If the vacuum system connected to the gauge inlet is
evacuated, the difference in height between the two legs of the U-tube will reduce.
Sample Problems:
5.1 If the height difference between the reference and gauge legs of a mercury filled
U-tube manometer connected to a vacuum vessel is 100 mm, what is the
pressure measured in the vacuum vessel?
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5.2 What are the disadvantages of a mercury filled U-tube manometer?
H
Reference
inlet
gage
inlet
Figure 5.2 U-tube manometer.
McLeod Gauge:
The pressure range over which liquid level gauges read can be extended if a sample of
the gas to be measured is isolated from the vacuum vessel and compressed in a well
controlled manner to amplify the force per unit area thus making the pressure easier to
measure accurately. A McLeod gauge accomplishes this through the use of a movable
mercury reservoir, a bulb of known volume, a set of capillary tubes and a tube allowing
for connection to the vacuum vessel (see figure 5.3). Lowering the mercury reservoir will
allow gas from the vacuum vessel to fill the bulb of known volume situated directly
below the closed capillary. This sample of gas is then isolated from the vacuum vessel
by the rising mercury reservoir at the cutoff level. At this same time, the captured gas is
compressed into the sealed capillary tube. It can be shown for a calibration constant k,
2
P kh =
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The Bourdon Tube Pressure Gauge is comprised of a leak tight case with a glass plate
in front to allow a view of the pressure indicator dial and pressure scale. The curved
metal tube elastically deforms and the end deflection is proportional to the differential
pressure across its wall (think of a garden hose trying to straighten out when the water
is turned on). This deflection is mechanically transformed into a rotation of the indicator
dial by a rack and pinion mechanism. There are several variations of the basic design,
some having evacuated cases and reference tubes that protect the mechanism inside
the case from the environment of the vacuum system. Small, inexpensive Bourdon tube
gauges (2" diameter faces) may be accurately read from atmospheric pressure down to
100 Torr. Larger, more sensitive gauges (8-9" diameter faces) can read down to 10
Torr.
Bourdon tube gauges are simple, inexpensive and relatively rugged. As such, they are
often found on high vacuum systems as a means of verifying the gross pressure
conditions in a vacuum system.
Mechanical Diaphragm Gauge
gage inlet
diaphragm
mechanical
linkage
pointer
gage dial
Figure 5.5 Cross-section of a mechanical diaphragm gauge.
The mechanically actuated diaphragm gauge makes use of a thin flexible metal
diaphragm that deflects in proportion to the pressure differential across it. By means of
levers and pulleys this deflection is amplified and transformed into rotational motion of a
pointer in front of a calibrated dial face. Since the gauge side of the diaphragm is
exposed to the environment of the vacuum system, care must be taken to control
exposure of the gauge to oils, water, or reactive gases.
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Capacitance Diaphragm Gauge
gage inlet
to vacuum
system
reference
inlet
electrode diaphragm
Figure 5.6 Capacitance gauge head in cross-section.
Capacitance diaphragm gauges, or capacitance manometers, are another variety of
pressure gauge that rely upon the pressure differential across a flexible diaphragm as a
means of pressure measurement. In this gauge, the flexible diaphragm is made the
variable element in a three-terminal capacitance potentiometer; for a given input
voltage, the change in capacitance as a function of diaphragm deflection is measured,
and translated into pressure units. The absence of mechanisms with backlash and
counter forces means superior accuracy, repeatability over a mechanical diaphram
gauge. Capacitance manometer heads are available in a series of sensitivities; the less
sensitive models being more rugged. Some of the most sensitive units can measure
pressures as low as 10
-5
Torr. These gauges measure presure as an aggregate kinetic
manifestation of the molecules and hence are not gas-species sensitive.
Sample Problem:
5.3 A capacitance manometer is used to measure the pressure in a vacuum vessel
during a sputter deposition operation. If the process gas is changed from argon
to xenon what will be the effect on the pressure reading made using the
capacitance manometer?
Capacitance manometers can measure pressure very accurately in the pressure range
for which the head was designed. Since the displacement of the diaphragm is very
small in sensitive capacitance manometer heads, the pressure readings may be thrown
off by temperature changes in the environment around the gauge head. Situations to
avoid include placing the manometer head next to an operating hot cathode ion gauge
or a liquid nitrogen cold trap. To decrease the effects of variable room temperature on
the gauge readings, some manufacturers have included heating elements in the gauge
that serve to maintain a constant operating temperature.
Thermal Conductivity Gauges
Thermocouple Gauge:
The most basic of the pressure gauges that measure the change in thermal conductivity
of a gas to infer pressure is the thermocouple gauge. A constant electrical current is
supplied to the filament inside the gauge to which a thermocouple is spot welded. As
pressure is reduced during evacuation, fewer gas molecules impinge upon the heated
filament per unit time, and the filament therefore operates at higher temperatures.
Filament temperature is monitored using the thermocouple, and is transformed into
pressure units at the gauge read-out dial. Since some molecules are better at acquiring
thermal energy than others, these gauges are gas species sensitive.
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gage inlet
power supply
milliammeter
milli-
voltmeter
filament
thermocouple
element
Figure 5.8 Cutaway view of a thermocouple gauge and a schematic of the gauge and
control circuitry.
The operating range of most thermocouple gauges is between atmospheric pressure
and 10
-3
Torr. Thermocouple gauges are very widely used in the vacuum industry due to
their low cost, ease of installation, use, ruggedness, and small size. Common
applications for this type of pressure gauge include measurement of the foreline
pressure of a high vacuum pump. The major disadvantage of the gauge is its inherent
slow response to pressure change. The pressure range of operation of simple TC
gauges is from about 1 Torr to 10
-3
Torr. As rugged and reliable as these gauges are,
the quality of the pressure measurement will be seriously degraded if any foreign fluid,
such as pump oil is allowed into the gauge body where it may become pyrolyzed on the
hot filament. Gauges are often mounted vertically with the gauge inlet pointing
downwards for this reason.
Sample Problem:
5.5 A thermocouple gauge is used to measure the pressure in a vacuum vessel
during a sputter deposition operation. If the process gas is changed from argon
to xenon what will be the effect on the pressure reading made using the
thermocouple gauge?
Pirani Gauge:
In the Pirani gauge, the reference filament (or compensator) is enclosed in a leak tight
glass envelope evacuated to a pressure of less than 1 Torr. In a similar glass envelope
which is open at the gauge inlet end is housed the gauge filament. As gas density
exposed to the gauge filament changes, the gauge filament, which is heated using a
constant electrical current flow, experiences a change in electrical resistivity and this is
measured in the Wheatstone bridge circuit and displayed in pressure units on the read-
out dial. As with other gauges that measure the thermal conductivity of gases, the Pirani
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gauge does not read pressure changes instantaneously. Some time is required for the
heated filament to respond to changes in its environment.
reference
fila
(inside sealed tube)
gage filament
(inside open
tube)
power
supply
reference
filament
gage
filament
meter
Figure 5.9 Pirani gauge drawing (above) and control circuit (below).
Convectron Gauge:
A useful (and patented) modification of the thermal conductivity gauge allows for
measurement of convection currents at higher pressures, increasing the range of this
gauge to atmospheric pressure. Convectron gauges typically include a gold plated
tungsten sensing wire surrounded by a cylinder wound with kovar wire. This cylindrical
temperature compensator helps to reduce the effect changes in ambient temperature
has on the gauge readings. The large volume inside the compensator provides space
for convection currents to develop at higher gas densities (1 Torr to atmospheric
pressure), improving the resolution of the gauge at the high pressure end of its range of
operation. Because this gauge uses convection currents to infer gas pressure,
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orientation of the gauge is critical. The body of a convectron gauge should always be
oriented horizontally (as shown below, in figure 5.8).
Gas Ionization Gauges
Gas density (and pressure) may be deduced from the ability of the gas to undergo
ionization caused by electron impact with the gas molecules. The ionization gauges that
will be discussed here are both designed for use in the medium to ultra-high vacuum
range (10
-3
Torr to 10
-10
Torr).
Hot Cathode Ionization gauge:
Also called the Bayard-Alpert gauge, the hot cathode ion gauge is a simple and
reliable gauge that is widely used in vacuum processing industries. The triode (three
electrode) design is easily understood if we examine each component independently,
become familiar with its function, then see how they work together. The filament is
usually a thin ribbon of tungsten which is in the shape of a coil or inverted "vee". Low
voltage electric current from the gauge power supply is passed through the filament
which heats up much like the filament in an incandescent light bulb (operating
temperature of a gauge filament is almost 1800°C). In addition to heat and light, the
filament emits enormous quantities of electrons which can collide with gas atoms and in
that collision, eject an electron from the gas atom making it an ion. Electrons from the
filament are attracted to a helical "grid" or electron collector which is maintained at a
positive voltage of approximately 150V with respect to the filament. The additional
energy input into the electrons by the electron collector bias is to insure efficient
ionization of gases in the gauge. Finally, the gas ions created are collected on an ion
collector operated at zero volts with respect to the electron collector.
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nucleus
orbiting electrons
+
incident electron
from filament
ejected electron
The operating range of hot cathode ionization gauges is from 10
-3
Torr to 10
-9
Torr.
These gauges are small in size, relatively easy to operate and accurate to +/- 10% of
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the reading in the pressure range in which they are designed to operate. Volatile
contaminants (hydrocarbon oil, process gases, etc.) may impair proper operation of the
ion gauge. If this occurs, one may restore the gauge by performing a "degassing"
operation in which current is supplied to the gauge electrodes to drive off the unwanted
contamination. Most commercial ion gauge control units provide for "degassing"
operation and also prevent operation of the gauge at a pressure at which the gauge
would be damaged. Response time of ion gauges is quite fast, and this attribute is used
for process control and vacuum system Two common configurations of hot cathode
ionization gauge are shown in figure 5.10. The nude gauge is less frequently used, as it
protrudes into the vacuum system and may interfere with the process being conducted.
Cold Cathode Ionization Gauge
Operating in the medium to high vacuum range (10
-3
Torr to 10
-8
Torr), the cold cathode
ionization gauge uses electrons emitted from electrodes maintained at electrical
potentials of 10,000 volts to ionize gas in the gauge body. Ejected electrons are forced
to orbit in a helical path by a strong magnetic field provided by the external permanent
magnet. This increases the probability that gas molecules will be struck by orbiting
electrons and become ionized and subsequently "counted".
Power
supply
milliammeter
cathode
cathode
anode
to
vacuum
system
N
S
Figure 5.13 Cold cathode ionization gauge.
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The accuracy of cold cathode gauges is severely impaired by a dirty environment, as
the number of electrons emitted from the cathodes is strongly dependent upon the
cathode surface condition. Some models of cold cathode gauge may be disassembled
for cleaning, but great care must be used during reassembly, as misalignment of the
electrodes or magnet can cause the gauge to give inaccurate pressure readings.
Momentum Transfer Pressure Gauges
Spinning Rotor Gauge (SRG):
In the spinning rotor gauge, the drag caused by gas molecules hitting the surface of a
magnetically levitated spinning steel sphere is used to infer gas pressure. The control
unit for the SRG brings the levitated ball to a rotational velocity of approximately 400
RPM using a set of electromagnetic coils. Once the rotational speed is constant (as
measured by a set of detector coils), the accelerating coils are turned off, and the steel
sphere is allowed to "coast". The rate at which the rotation of the ball decreases is a
function of the gas density and composition.
levitating magnets
spinning ball
conflat flange
to vacuum
vessel
Figure 5.14 Cutaway view of a spinning rotor gauge.
The pressure range of the SRG is from 10
-2
Torr to 10
-7
Torr. As the gauge is delicate,
expensive, and requires several minutes for each pressure reading, its primary use is
found in calibration of other gauges, and in precise vacuum measurements.
5.2 There are some distinct disadvantages to the U-tube manometer that explains
why they are not widely used. The pressure range of U-tubes is limited; carefully
constructed models can only read pressure from atmospheric down to about 1
Torr. Mercury has the obvious health and safety concerns, but also may
cause problems if the process being measured reacts chemically with mercury
vapor. Other concerns include the substantial equilibrium vapor pressure of
mercury at room temperature and the fragile nature of the glass tubulation.
Laboratory Exercise 5.1:
Pressure Gauge Identification and Inspection
Identify the vacuum gauge you have selected for this exercise:
A. Gauge Identification: What is the principle of operation? Who is the manufacturer?
What is the gauge model number? Locate the manufacturer's literature from the
bookcase and find the appropriate reference information. What is the advertised
pressure range? Is the gauge gas specific? Are there any calibration curves available to
aid in understanding the performance of the gauge as a function of pressure or gas
specie?
B. Physical Inspection of Pressure Gauge: Inspect the gauge for signs of wear or
misuse. What type of vacuum connection is provided? Is this connection appropriate for
the application the gauge was designed for? Locate the gauge control unit and/or power
supply (if applicable). Check electrical cables of the power supply for cracks in
insulation.
Laboratory Exercise 5.2:
Operation of Spinning Rotor Gauge
Before beginning this procedure, read the operating manual carefully.
Procedure:
Assemble an operating vacuum system capable of attaining a pressure of
10
-5
Torr or lower using an ion pump as the high vacuum pump. Operate the SRG
following the instructions in the manual.
Discussion:
• What assumptions did you make in the gauge calibration?
• How did the gauge perform compared to the hot cathode ion gauge?
• Can you explain any inconsistencies you observed?
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Chapter 6: Mechanical Vacuum Pumps
In this chapter we will review the principles of operation of several commonly used
mechanical vacuum pumps, provide information on the performance and typical
applications, and describe appropriate preventative maintenance techniques. This
chapter also includes several laboratory procedures that are designed to aid in your
understanding of mechanical vacuum pumps.
Positive gas displacement pumps of one type or another have been used since 1640!
Almost all of the very early pumps used liquid mercury within glass tubes and vessels to
create a vacuum. For an excellent review of this early technology, read the History of
Vacuum Science and Technology, edited by T.E. Madley and W.C Brown, published for
the American Vacuum Society by the American Institute of Physics.
Modern mechanical pumps may well be considered the workhorses of vacuum
technology; they are simple in design, require little maintenance, are relatively
inexpensive, and can operate for long periods of time without failure. Several
mechanical vacuum pumps that we are aware of have operated continuously for fifteen
years with only occasional oil changes! The range of pumping speeds for commercially
available pumps runs from about 0.5 liters per second to over 300 liters per second.
Mechanical vacuum pumps fall into two basic categories: reciprocating pumps, and
rotary pumps. Further distinctions for mechanical pumps include: the number of stages
(single stage or compound), the use of oil in a pump (pumps may be oil sealed or "dry"),
and the means of driving the mechanics of a pump (direct drive or belt drive). Below is a
brief outline of the types of modern mechanical vacuum pumps.
For this laboratory, we will concentrate on two oil sealed mechanical pumps: the sliding
vane rotary pump, and the rotary piston pump.
Theory of Operation
Mechanical vacuum pumps work by the process of positive gas displacement, that is,
during operation the pump periodically creates increasing and decreasing volumes to
remove gases from the system, and exhaust them to the atmosphere. In most designs a
motor driven rotor spins inside a cylindrical stator of larger diameter. The ratio of the
exhaust pressure (atmospheric) to the base pressure (lowest pressure obtained at the
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vacuum pump inlet) is referred to as the Compression Ratio of the pump. For
example, if a mechanical vacuum pump obtains a base pressure of 15 mTorr, its
compression ratio is:
7 60 Torr
0.0 15 Torr
= 5 1,0 00
Another more common way to state this is to say that the pump has a compression ratio
of 51,000:1. At pressures above 1 Torr, rotary mechanical pumps have a fairly constant
pumping speed. The pumping speed decreases rapidly below this pressure, and
approaches zero at the pump's base pressure. Most manufacturers of mechanical
vacuum pumps will include in their product literature information on the pump's
performance including a pump speed curve.
1000 100 10 1 .1 .01
.1
1
10
100
Pressure [Torr]
P
u
m
p
S
p
e
e
d
[
L
i
t
e
r
s
/
s
e
c
]
Rotary Vane Mechanical Vacuum Pumps
Rotary vane pumps typically have an electric motor driven rotor (either belt or directly
driven) which has one to three sliding vanes that maintain close contact with the inner
wall of the cylindrical stator. The vanes are metal in oil sealed pumps, and carbon in dry
pumps. Centripetal force acts upon the vanes in the spinning rotor so as to force them
against the inner sealing surface of the stator. In some mechanical pumps springs are
used to augment this action. Rotary vane pumps may be of the single or double stage
design. Single stage pumps are simpler, having only one rotor and stator, and are less
expensive. The base pressure one can expect from a good single stage mechanical
pump is about 20 mTorr. In a two stage design, the exhaust port of the first stage is
connected to the inlet port of the second stage which exhausts to atmospheric pressure.
Two stage pumps may attain a base pressure of one to two millitorr, but are more
expensive than single stage pumps.
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1 2
In the figure above are simplified drawings of a single stage oil sealed rotary vane
mechanical pump (left) and a two stage, or compound pump of the same type. In the
compound design the high vacuum side of the pump (stage labeled 1) operates at a
lower pressure due to the lack of exposure to high partial pressures of oxygen in that
stage. It should be noted that supply of very little or no oil to the first stage of a
compound pump in order to achieve even lower pressures can, in practice, lead to
severe difficulties in the reliable operation of a compound pump.
The oil in an oil sealed pump serves three important functions: A) providing a
vacuum seal at the pump exhaust, B) as a lubricant and C) provides cooling for
the pump.
1
2
3 4
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In this figure, and on the following page sequences in a single pump cycle of a rotary
vane pump are shown. Note how the rotor vanes work with the stator to create
increasing and decreasing volumes on each stroke.
7
5
6
8
Also note how the gas discharge valve opens and closes on each cycle.
Belt driven rotary vane pumps typically operate at about 400 to 600 RPM, while the
direct-drive models spin at 1500 to 1725 RPM. Most failures in rotary vane pumps can
be attributed to poor oil maintenance. O'Hanlon states that 95% of all mechanical pump
problems can be resolved by flushing the pump and changing the oil. Because of the
close tolerances between the rotor vanes and the stator, solid particulate matter
entering the pump is likely to cause scoring of the vacuum sealing surfaces, resulting in
a decrease in pump performance. For this reason, precautions should be taken to
minimize intake of particulates. Several manufacturers produce small screens and filters
that fit on the inlet of a pump to accomplish this.
Sample Problems:
6.1 What is the principle by which positive displacement pumps operate?
6.2 If a mechanical pump achieves a base pressure of 30 mTorr, what is the
compression ratio of the pump?
6.3 What are the three functions of the oil in a mechanical vacuum pump?
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Rotary Piston Mechanical Vacuum Pumps
Rotary piston (or rotary plunger)
mechanical pumps like that to the
left also operate on the principle of
positive displacement of gas. On
each cycle the rotating eccentric
piston and the sliding valve work
together to suck gas into the stator,
compress it, and expel the gas to
atmosphere. As with rotary vane
pumps, rotary piston type pumps
may be single stage or compound.
Rotational speed is typically 600 to
800 RPM.
Dimensional tolerances between the stator and piston in pumps of this design are
usually 0.003 to 0.004". Because of this, piston pumps are more tolerant of particulate
contamination that rotary vane pumps. Higher viscosity oil is used in rotary piston
pumps due to the larger dimensional tolerances. Large rotary piston pumps are often
water cooled to increase pump life and performance.
Mechanical Vacuum Pump Fluids
Selecting the appropriate pump fluid is as important as choosing the right pump. In
today's vacuum technology, many processes are not compatible with typical
hydrocarbon pump oil. For example, if you're running a process in which a significant
amount of oxygen is used, a synthetic pump oil that is much less reactive with oxygen is
the preferred choice. In this example, if hydrocarbon oil is chosen, the potential for
creating an explosive mixture of oxygen and hot pump oil vapor exists. Likewise, if a
process involving the use of corrosive gases is being run, you should think about the
chemical reactivity of the process gases being pumped that will be exposed to
mechanical pump oil vapor. Fluorocarbon pump fluids may be chosen for an application
such as this due to their low chemical reactivity. Under certain circumstances, you may
wish to operate a mechanical pump with fluid of higher viscosity. For this purpose, the
clearances between moving parts may need to be increased. Pumps that are modified
for special service should be permanently labeled to let future users know of the
modifications and application.
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One last word on mechanical vacuum pump fluids-research the characteristics of a fluid
carefully before using it. Many of the current commercially available fluids will not
operate well when mixed with one another! For a good review of mechanical pump
fluids, see O'Hanlon's A User's Guide to Vacuum Technology, page 163.
Dry Mechanical Vacuum Pumps
In recent years, the concern over mechanical pump fluids (from both safety and vacuum
system contamination standpoints) has become a great concern. Vacuum pump
manufacturers have responded by developing and marketing oil-free mechanical
roughing pumps. These pumps have, for some applications, very appealing
characteristics, but there are a few drawbacks of which to be aware.
The advantages of dry pumps (usually of the rotary vane design) are that they eliminate
the possibility of backstreaming pump oil into your vacuum vessel. In addition, dry
pumps may be used to safely pump large percentages of oxygen without fear of
explosion. Dry pumps are also well suited for pumping of certain corrosive vapors and
gases.
Disadvantages of dry mechanical vacuum pumps include their initial high cost (as much
as 5 times the cost of a oil-sealed pump of equal capacity), excessive noise, and higher
ultimate pressure.
Identify the mechanical vacuum pump you have selected for the next three exercises: Is the pump of single stage or compound design? What is the rotational speed?
What is the suggested volume of pump fluid?
B. Physical Inspection of Mechanical Pump: Inspect the pump for signs of wear or
misuse. Check electrical cables for cracks in insulation. Are the prongs of the electrical
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plug bent or missing? Check the pump fluid. Is the fluid transparent or milky; is the fluid
level correct? If the pump is a belt-driven model, is the belt tensioned correctly, and is
the belt worn? Is the safety cover in good condition? Locate the gas ballast, inlet and
exhaust ports. Is everything as expected? Once you have carefully inspected the pump
and corrected any problems, cap off the pump inlet and operate the pump briefly.
Record your observations.
{Please prepare a written laboratory report on this and each of the following
exercises using guidelines presented in the section called "How to Use This
Manual"}
Laboratory Exercise 6.2:
Mechanical Pump Ultimate Base Pressure.
The two operational characteristics that define the performance of a mechanical
vacuum pump are: 1) the ultimate (or base) pressure, and 2) the pumping speed. In this
exercise, you will determine the base pressure of your pump, and compare these
results with the manufacturer's specifications.
Procedure:
A. Measurement of ultimate pressure. Place a valve on the inlet of the mechanical
pump. Devise a manifold so that a thermocouple gauge (or pirani gauge) can be
installed somewhere near the pump inlet. Close the valve, and turn the mechanical
pump on. Observe the pump's behavior. Once you're certain the pump is operating
properly, open the valve, and allow the pump to base out (achieve its ultimate
pressure). This may take 15 to 20 minutes. Record the ultimate pressure. How does
your reading compare with the manufacturer's specification? If there is a discrepancy,
what do you attribute it to?
TC1
A schematic of the experimental
set-up for part A of Exercise II is
shown to the left.
Attach a suitable vacuum vessel
having a volume of from 50 to 100
liters to the manifold assembly
used in part A. Place a second
thermocouple gauge on a port of
the vacuum vessel; connect all
required read-outs to the vacuum
gauges.
Before beginning this procedure the vacuum pump should be running, and base
pressure should be read at gauge TC1, the valve to the vacuum vessel should be
closed, and the vessel at atmospheric pressure. In the next step, the pressure as read
at the vacuum vessel (TC2) will be recorded as a function of time. We suggest taking
pressure readings every 30 seconds for the first five minutes, then recording pressure at
one minute intervals until base pressure is achieved in the vacuum vessel. The table to
plot your data is on the following page. This data will allow you to plot vessel pressure
as a function of time on semi-logarithmic graph paper. Label your graph with all
pertinent pump data.
Now vent your system to atmosphere, and leave it open for one minute. Repeat
procedure 6.2-B. Plot the data collected for this second pump down measurement as
you did for the first measurement, and compare the results. Is there a noticeable
difference between the two curves? Would you expect a difference? What would you
attribute this behavior to? The table to plot your data is on the following page.
Remember the first (and easiest) way to test the integrity of a vacuum system is
to check its ultimate pressure, and the time required to reach this pressure. {Hint: after
characterizing the pump down behavior of your clean, dry and empty vacuum system,
plot the data as time vs. pressure and file that information away for future reference.
Your curve becomes an excellent tool for gauging the performance of your vacuum
system}.
The manufacturer's listed pumping speed for any given pump is usually the free
air displacement at STP (standard temperature and pressure). As pressure decreases
from atmospheric, there will be a reduction in the amount of gas pumped per unit time
(mass flow rate). The pumping speed (volumetric flow rate) will decrease only slightly
until a pressure of about 1 Torr is attained. Below this pressure, the decrease in
pumping speed becomes more rapid, depending upon the type of mechanical vacuum
pump, and falls to zero at the ultimate pressure.
We can determine the speed of a pump by measuring either pumping speed
under constant volume or constant pressure conditions. The constant volume technique
is generally used in the pressure range between atmospheric and one Torr. In this
method, you will measure the time required to reduce the pressure in a vessel a
specified amount. The pump speed in that pressure range is then calculated using the
equation:
S
p
=2.3
V
t
2
−t
1
|
\
|
.
| Log
10
P
1
P
2
|
\
|
.
|
|
V = volume of vessel [liters]
t
1
= time at pressure P
1
[seconds]
t
2
= time to reach pressure P
2
from
pressure P
1
[seconds]
In contrast to the constant volume method, the measurement of pumping speed at
constant pressure is typically performed in the pressure range between one Torr and
the mechanical pump's ultimate pressure. To determine pumping speed by the constant
pressure method, a measured amount of gas (Q) is admitted to the vacuum system
being pumped to establish a constant pressure P. Pumping speed is then obtained from
the equation:
6.3-A. Pumping Speed by constant volume method:
For this exercise, you will need a functioning rotary mechanical pump, a vacuum
chamber, a valve, and at least one vacuum gauge capable of reading from atmospheric
pressure to about one Torr.
Vacuum Vessel
TC1
Install the valve between the chamber and
the mechanical pump using the minimum
amount of connecting line to reduce
conductance losses. Begin this exercise
with all valves closed and the vessel at
atmospheric pressure. Start the
mechanical pump, and after it has
warmed up, open the valve to the vacuum
vessel and
Record the time required to achieve a pressure of 100 Torr as read with the pressure
gauge mounted on the vessel. Repeat this measurement until you are confident in the
consistency of your readings. Now record the time required to pump from 100 Torr to 10
Torr, exactly as was done before. And finally, record the time required to pump from 10
Torr to 1 Torr. Table to record your data is on the following page.
Now plot the calculated pumping speed as a function of the average pressure for each
of the four pressure regimes in Table 6.3-A.2.
Following your splendid success in this measurement, replace the vacuum vessel in
your system with another vessel of significantly different volume. Repeat the
measurements performed and plot the data. How do the speed vs. average pressure
curves compare? Is the behavior as you would expect? Why or why not?
Discussion:
Is it possible to make your plots more representative by using shorter time
increments? What are the drawbacks (if any) for this idea?
How do the speeds that you have calculated compare to those listed by the
manufacturer for this pressure range?
Is there any significant difference in speeds calculated for the two vacuum
vessels of differing volumes?
6.3 B: Measurement of pumping speed by the constant pressure method.
For this portion of the exercise, you will need a mechanical vacuum pump, a vacuum
valve, a variable leak valve, an atmosphere valve, a vacuum vessel, a flow indicator and
a pressure gauge capable of reading pressures from one Torr to about one millitorr.
Vacuum Vessel
TC1
TC2
atmosphere
valve
pipette
Install the pump valve at the pump inlet. Place the pressure gauge on the vacuum
vessel, and install the variable leak valve on the chamber also. The flow meter must be
plumbed to the inlet of the leak valve and the atmosphere valve must be plumbed to the
flow meter. Confused? Follow the diagram and have a lab instructor check your setup
before you begin.
Initial conditions should be something like this: mechanical vacuum pump is off,
the valve between the vessel and pump is closed; the vessel is at atmospheric
pressure; the leak valve is closed. Start the mechanical pump, and allow it a few
minutes to warm up to operating temperature. Open the valve between the pump and
vessel, and allow the pressure to be reduced to a stable base pressure (~20 mTorr).
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Once a stable base pressure is achieved, with the atmosphere valve open, slowly open
the calibrated leak valve until you notice a slight rise in vessel pressure. Observe this
pressure (~100mTorr might be a good initial value) for a little time to insure that the
system is stable at this pressure. Close the atmosphere valve, and observe air being
drawn into the vessel through the flow meter. Fluid will rise in the volumetric burette to
replace air being pumped out of the system by the mechanical pump. We now know
that the air being leaked into the chamber is at atmospheric pressure, we know the
volume being leaked in per unit time, and we know the pressure inside the vacuum
chamber. We are now prepared to calculate the rate at which the vacuum pump is
removing air from the chamber using the equation:
Repeat the procedure for various pressure values between one millitorr and one Torr.
Try to get at least five stable readings.
Plot your calculated data as pump speed (S
P
) vs. pressure. Be sure to include all
pertinent data regarding the experiment.
Discussion:
How do the speeds you have calculated compare with those listed in the vacuum
pump manufacturer's literature?
What would be the effect of using a vessel having twice the volume on the
pumping speed?
It wasn't that long ago when you could walk into any vacuum laboratory and find
a vapor diffusion pump on every system. Vapor diffusion pumps were first conceived
about 1915-16, and used mercury as the pumping fluid. A decade later, experimenters
found that some oils had high boiling points and low vapor pressures and were good
pumping fluids. These oils were useful because they remained in the pump indefinitely
and allowed lower pressures to be attained without the use of a cold trap. During
W.W.II, and again during the 1960's for the space effort, diffusion pumps went through
some significant design changes that increased their pumping speed, increased their
ability to produce lower pressures, and oils gave way to synthetic pumping fluids. Due to
it's simplicity, high performance, and low initial cost, the diffusion pump remains the
primary industrial high vacuum pumping mechanism. Applications for this type of pump
are found in R&D labs, coatings facilities, manufacturing, and space simulation. When
diffusion pumps are used with the correct fluid, traps, and baffle, they can produce
pressures to approximately 2*10
-10
Torr.
Theory Of Operation
Diffusion pumps are vapor jet pumps that work on the principle of momentum
transfer. This occurs when a heavy, high speed vapor molecule collides with a gas
molecule and moves it in a preferred direction through the pump. The bottom of the
pump contains an electric heater which is used to heat the pumping fluid to it's boiling
point, thus, producing the vapor. This must be done at a reduced pressure. This means
that before the diffusion pump is started, it must be "rough pumped" down to an
acceptable pressure, typically 100 millitorr. To do otherwise will result in no pumping
action and possible damage to the pumping fluid. Once boiling of the fluid has begun,
the vapor is forced up the central columns of the jet assembly. It then exits at each
downward directed jet in the form of a molecular curtain that impacts the pump body.
The pump body is externally cooled so that the fluid will condense on its inside surface
and run back down into the boiler. Pump bodies are typically water-cooled, but some
are air-cooled. As gas molecules from the system randomly enter the pump (molecular
flow conditions), they encounter the top jet. Some of them are impacted and driven on
to the next jet. Subsequently, they reach the foreline where they are exhausted to the
atmosphere by the mechanical backing pump.
Compression Ratio
The diffusion pump is similar in character to other compression pumps in that it
develops a relatively high exhaust pressure compared to the inlet pressure. For most
gases this compression ratio may be one million to one (or greater). For example; for an
inlet pressure of 2*10
-7
Torr and a foreline pressure of 2.0*10
-1
Torr, the compression
ratio would be one million. As far as compression goes, in a mixture of gases, each
species may be pumped with different effects. It is possible to have different maximum
compression ratios and different flow rates for gases having different molecular weights.
For example, the compression ratio for hydrogen will differ greatly from the compression
ratio for argon simply because their molecular weights are very different. Also, when the
pumped gas has a molecular weight different from air the maximum compression ratio
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will shift, but the tolerable foreline pressure (critical discharge pressure) remains the
same.
Critical Discharge Pressure
The critical discharge pressure of a diffusion pump is the maximum permissible
pressure at the foreline during normal pump operation. The expected pumping action of
a diffusion pump ceases when the critical discharge pressure is exceeded. That is, the
vapor of the discharge stage of the pump does not have sufficient energy and density to
provide a barrier for the air in the foreline, thus, this air will flow through the pump in the
wrong direction carrying with it the pumping fluid vapor. For most modern diffusion
pumps, the maximum allowable foreline pressure is about 0.5 Torr. Diffusion pumps
cannot function at all unless the foreline pressure is held below this limit by the backing
pump. The most important rule of diffusion pump operation is: Do not exceed the critical
discharge pressure! If this single most important rule is observed, then most difficulties
associated with diffusion pump operation can be eliminated.
Backstreaming
Backstreaming can be defined as the passage of the pumping fluid through the inlet port
of the pump and in the direction opposite to the direction of desired gas flow. However,
backstreaming must not be limited to the pump, but must include the trap, baffle, and
plumbing as well because all affect the transfer of pumping fluid vapors from the pump
body to the chamber. There is a multitude of conditions that can cause backstreaming.
The most common are; exceeding the critical discharge pressure in the foreline,
exceeding maximum throughput capacity for long periods of time, and incorrect start-up
or shutdown procedures. Backstreaming of pumping fluids into your work environment
is always considered catastrophic. I know of very few vacuum related processes in
which oil contamination is not a disaster! My suggestion to system operators is to know
their equipment thoroughly and learn proper operating techniques. Ninety-nine percent
of costly backstreaming problems are due to operator error. Finally, equip your system
with the appropriate interlocks that will prohibit valve cycling above a specified pressure.
This will protect your system whenever it is left unattended.
Baffles And Traps
Baffles have one particular purpose: to reduce the backstreaming of pump fluid into the
vacuum chamber. Most baffles are "optically opaque" which implies that their internal
geometry is such that light cannot pass directly through them. This insures that a
molecule will collide at least once with a surface regardless of the incoming direction.
Baffles do impede the flow of pumped gases, but well designed units can retain about
60% of the pumping speed. Baffles are installed directly above the pump inlet and are
often used in conjunction with a trap. Water-cooled baffles can reduce the rate of
reevaporation of condensed fluid thereby reducing the density of vapor in the space
between the baffle and the trap. See illustration for several baffle designs. Cryogenic or
refrigerated traps serve two purposes. They act as barriers against the flow of
condensable vapors from pump to system; and they also serve as cryopumps for
condensable vapors (primarily water vapor) emanating from the system. In typical
unbaked systems, water vapor may constitute about 90% of the remaining gas after
initial evacuation. Chilled traps increase the pumping speed for water vapor and
therefore can in many cases lower the base pressure of your system. I know of two
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distinct varieties of liquid nitrogen traps. One is a trap that is placed anywhere within the
vacuum chamber. This may be a cryopanel, a sphere or cylindrical bottle, or a tubular
arrangement acting as a "cold-finger" on which condensable gases will be trapped. The
other is of the optically opaque design and is placed between the chamber and the
pump inlet. These traps insure that gas molecules collide at least once with a cold
surface.
Vacuum chamber
Liquid nitrogen cold trap
Chevron baffle
Diffusion pump
Figure 7.1 Configuration of traps and baffles used on diffusion pumped systems.
Fluids
Many of the pumping fluids used today have been developed within the last 30
years. Up to about 1960, most fluids had a vapor pressure of 10
-7
Torr or 10
-8
Torr and
the base pressure of the system was limited to that range. The choices of pumping
fluids became greater after Hickman publicized the used of polyphenyl ethers which
offered exceptional thermal and chemical stability. Operational characteristics of
another low vapor pressure silicone fluid (DC705) were also found to be excellent. The
use of either of these fluids will permit base pressures of 10
-9
Torr or 10
-10
Torr to be
achieved. More recently, fluorinated oils have been developed for use in diffusion
pumps. These have the added advantage of compatibility with corrosive gases used in
some processes.
Ultimate Pressure
Two distinct observations can be made regarding the ultimate pressure of a
diffusion pump. Ultimate pressure may be considered to be a gas load or a pressure
ratio limit. The pressure ratio limit is usually associated with light gases (hydrogen,
helium, xenon). The pumping action of the vapor jets does not cease at any pressure,
however low. The ultimate pressure of the pump depends on the ratio of pumped versus
back-diffused molecules, plus the ratio of the gas load to pumping speed. Also, the
pump itself can contribute a gas load either through backstreaming of pump fluid vapor
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and its cracked fractions or the outgassing from its parts. In practice, then, the ultimate
pressure of a pump is a composite of several elements. The first limit of the ultimate
pressure is usually due to the vapor pressure of the pumping fluid, although this limit
may not be observed at pressures below 10
-8
Torr.
Operating Procedures
The operation of high vacuum, diffusion pumped systems requires certain care
and attention to several items. General cleanliness is extremely important, especially in
smaller systems. Remember, if a drop of oil were to be trapped somewhere in your
vacuum system, it may take days or weeks to evaporate that drop from your system.
Humidity and temperature can be important in view of the constant presence of water
vapor in the atmosphere. When your system is opened to the environment, pump down
time is significantly longer if the air is humid. The time of exposure is also significant. If
possible, the backfilling should be done with nitrogen or argon. For short exposures, this
appears to reduce the amount of water vapor adsorption in the vacuum system.
It is extremely important to develop good habits in valve sequencing operations,
especially in systems with manual valves. It is useful to have a "map" or schematic of
your system on your control panel that shows valve locations and functions. A single
wrong operation can result in very costly maintenance to the system. Automatic valve
sequence controllers have been used widely for many years, and they all have built in
interlocks to prevent accidental opening of the wrong valves. During the evacuation of a
vessel, the question arises regarding the proper time to switch from the roughing pump
to the diffusion pump. In other words, when should the high vacuum valve be opened?
There is no general answer to this question because each system is different with
different gas loads and different volumes. In practice, the transfer from roughing to the
diffusion pump is made between 50 and 150 millitorr. Below this pressure region, the
mechanical pump rapidly loses it's pumping effectiveness and the possibility of oil
backstreaming increases. Although the throughput of a diffusion pump is nearly
constant when inlet pressures are in the 1 to 100 millitorr range, the initial surge of air
into the pump when the high vacuum valve is opened will overload the diffusion pump
temporarily. We recommend that the period in which pump inlet pressure is above 150
millitorr be kept as short as possible; i.e., just a few seconds! Without a doubt, you'll
have questions on proper diffusion pump operation. There is literature available to help
you, and one document we suggest is a Varian Corp. publication written by M. H.
Hablanian called "DIFFUSION PUMPS: PERFORMANCE AND OPERATION" which is
part of the AVS Monograph Series.
Sample Problems:
7.1 What is generally regarded as the single most important thing to
remember about operating your diffusion pumped vacuum system?
7.2 Determine the compression ratio of a typical diffusion pump which has an
inlet pressure of 5x10
-7
Torr and a discharge pressure of 1x10
-1
Torr.
7.3 Explain what may happen if an operating diffusion pump is accidentally
vented through the foreline with air.
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Start-Up of Diffusion Pump:
Observation of diffusion pump operation.
Identify the diffusion pump as you did in exercise 6.1 for the mechanical pump
{manufacturer, model, size, capacity, etc.}. Assemble your pumping system so that your
diffusion pump is backed by a small mechanical pump (see figure 7.2).
Place a pyrex view port on the pump inlet and start the mechanical pump. When system
pressure is below 100 millitorr, turn on the diffusion pump and observe the formation of
oil condensation on the pump side of the pyrex view port. How long does it take for a
slight haze to form? a heavy haze?, how long before droplets appear on the view port?
Report your observations. What, in your mind, is the sequence of events that cause
droplet formation on the view port? If you took a heat lamp and aimed it at the view port,
what would you expect to see?
Laboratory Exercise 7.2:
Testing and recording your system's base pressure.
Now add the components to make your system look like the one in figure 7.3.
TC2
TC1 IG
Figure 7.3 Experimental set-up for measurement of DP base pressure.
You'll need a thermocouple gauge at the DP inlet, a pressure gauge on the mechanical
pump, and an ionization gauge at the inlet to the diffusion pump. Start the mechanical
pump and reduce the pressure in the system to below 100 millitorr. Now, turn on the
diffusion pump and allow the pump to reach normal operating temperature. As you
progress through this assignment, record the foreline pressure at frequent intervals so
that it may be graphed later. Once your pump is working, turn on the ionization gauge
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and record the steady drop in pressure. You should give the pump a couple hours to
reach it's base pressure. Now you have two data sets to record on semi-log graph
paper. A table for data entry is provided for you on the following page.
Laboratory Exercise7.3:
Measurement of pump down behavior for DP system.
Assemble your system so that a vacuum chamber is attached to the diffusion pump
through an isolation valve. You'll need pressure gauges on the chamber and also at the
DP inlet. Rough pump the system as you've done previously, and allow the DP to warm
up to normal operating temperature. Open the isolation valve and record time versus
pressure data until the base pressure is achieved. You can monitor foreline pressure as
well, and plot both sets of data on semi-log paper.
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Assemble your system similar to the way you did when testing the speed of a
mechanical pump. That is, attach a variable leak valve and an atmosphere valve to the
chamber and use your burette to find the volume displaced in a specific time period.
Since you're using the constant pressure method, you must select several pressures at
which to conduct the speed tests. Let's recall the vacuum formula used in the previous
exercise:
We suggest you start somewhere in the mid to high 10
-5
Torr range. Then pick higher
pressures at intervals suitable to the apparatus you have assembled. You will be
plotting pumping speed versus pressure. When you graph pumping speed for the
diffusion pumped system, be sure to include all information that may be pertinent to the
outcome of the test. A table for data entry is provided for you on the following page.
Chapter 8: Cryo-sorption pumps
Cryo-sorption pumps offer a clean, quiet, safe, vibration free and inexpensive way to
rough pump a vacuum system. They are often used on vacuum systems that are
sensitive to oil contamination from mechanical roughing pumps (surface science
instruments, for example). Cryo-sorption pumps are a sub category of sorption pumps.
All sorption pumps work by gas-capture. Pumped gases and vapors are bound at the
active surfaces of these pumps by physical means (Van der Waal's Forces), chemical
means ( Chemisorption) or are mechanically embedded in a continuous deposition of
material, as in a sputter ion pump (more on this in Chapter 9). Gas capture pumps of
these types share a few operational characteristics. With use, they will eventually
become "saturated" and will cease to pump- gases effectively. When this occurs, a
sorption pump will either need to be "regenerated" or replaced.
Theory of operation-
Cryo-sorption pumps work by providing a very large surface area of material that is
cooled to below the boiling point of most gases. Gas molecules that strike this cooled
micro-porous surface become attached and are removed from the gas phase, and are
effectively "pumped" from the vacuum system. The active surface area of a cryo-
sorption pump is typically made of zeolite 13X. This alkali alumino-silicate possesses a
very high surface area to mass ratio (about 10
3
m
2
per gram). The diameter of pores in
this material is about 13Å (1.3*10
-9
m) which is approximately the size of a molecule of
water, oil vapor and larger gas molecules (nitrogen and oxygen, for example). The pore
size is appropriate for capture of the gases most predominant in the atmosphere. Low
atomic weight gases, such as hydrogen, helium and neon have molecular diameters
smaller than the 13Å pore size of the zeolite, and are captured by this material less
effectively. Absorption of gases by a given sorbent is a function of gas specie, sorbent
temperature, and gas pressure.
As nitrogen gas is cooled, the amount of gas that can be adsorbed by the zeolite per
gram increases, as is shown in figure 8.1. Also note in this figure that helium, even
when cooled to -195 °C is pumped much less efficiently than nitrogen. Another piece of
information that may be gleaned from the data presented in figure 8.1 is that in general,
as gas pressure decreases, the amount of gas that is adsorbed per gram of sorbent
decreases.
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Nitrogen (-195 °C)
Nitrogen (20 °C)
Helium (-195 °C)
10-7 10-6 10-5 10-4 10-3 10-2 10-1 10 0 10+1 10+2 10+3
10+3
10+2
10+1
10 0
10-1
10-2
10-3
10-4
10-5
Pressure [Torr]
Q
u
a
n
t
i
t
y
o
f
A
d
s
o
r
b
e
d
G
a
s
[
T
o
r
r
-
L
]
p
e
r
g
r
a
m
o
f
s
o
r
b
e
n
t
Figure 8.1 Pumping behavior of Zeolite X-13 as a function of pressure.
Range of operation
Due to the extremely large sorbent surface area, these pumps can begin to trap gases
at atmospheric pressure (no roughing pump required), and can achieve pressures of 20
microns or less depending on the gas being pumped, and ratio of the volume of the
chamber to the capacity of the pumps.
Inspection and First Use
Prior to Operation of cryo-sorption pumps it is probably best to inspect a cryo-sorption
pump before initial use, especially on a critical vacuum system to insure that the pump
contains the correct sorbent, and is filled to the recommended level. If internal hardware
(screens,grids, etc,) are used, is it installed and in good condition? Is the pump body
sound? How about the vacuum flanges and connections? Do they mate with the
vacuum vessel's hardware? Are they in good mechanical condition (no scratches
running across sealing surfaces)? Prior to the first use of a new cryo-sorption pump, it
should be baked out at 250 °C for 24 hours to insure removal of water adsorbed on the
zeolite.
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Viton Stopper
Viton cuff
Zeolite
Liquid nitrogen
Dewar
screen
Pump inlet
pressure relief valve
Figure 8.2 Typical cryo-sorption vacuum pump.
Typical configuration
Sorption pumps are usually connected to vacuum chambers in a valved manifold, such
as shown in figure 8.3.
TC1
Figure 8.3 Cryo-sorption pumps connected to a vacuum vessel.
Operation
With the valve to the cryo-sorption pump closed, attach the liquid nitrogen dewar to the
pump body, and fill the dewar to within 1/2" of the top with liquid nitrogen. Allow 30
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minutes for the sorbent to reach operating temperature. Care should be taken to avoid
splashing liquid nitrogen on the skin. See chapter 2 for more safety details in handling
cryogenic materials.
Regeneration of cryo-sorption pumps-following repeated use, the sorbent material
will become saturated with gas molecules, and the pump's ability to remove gas from
the vacuum system will rapidly deteriorate. When this occurs, regeneration may be
performed by simply valving the pump off from the system, and allowing it to come to
room temperature. Gases will be liberated from the zeolite, and will escape the pump
body through the pressure relief valve. Make sure that the pressure relief valve is in
good operating condition, and is free to operate (no obstructions or blockages. The cork
style relief valve may pose a danger in that if the cork's tether is broken, the cork may
shoot across the room.
In industrial situations it is possible that toxic or explosive combinations of gases
may be released on pump regeneration. Be aware!
In situations where significant amounts of water vapor are pumped with a cryo-sorption
pump, heating at 250 °C for several hours is recommended in the regeneration
sequence.
Performance characteristics-the important quantities for cryo-sorption pumps are the
pump's capacity (expressed in Torr-liters), and its operating temperature (which will
determine which gas species will be pumped and how efficiently).
Pump capacity-
each gram of zeolite cooled to liquid nitrogen temperature (77k, or -195 °C)
approximately 30 Torr-liters of atmospheric gas can be pumped. Remember, at liquid
nitrogen temperature, helium, neon and hydrogen gas are not pumped, as they have
boiling points below that of liquid nitrogen.
Sample problem:
8.1 What temperature would a sorbent material have to be cooled to in order
to pump helium, neon and hydrogen?
A User's Guide to Vacuum Technology, O'Hanlon, John F. John Wiley & Sons
New York, New York. 1980.
Answers to Chapter 8 sample problems
8.1 Below 10k or -263 °C
Laboratory Exercise 8.1: Performance of a single cryo-sorption
vacuum pump.
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A. Pump Identification: Who is the manufacturer? What is the pump model number?
Locate the manufacturer's literature from the bookcase, and find the appropriate
reference information. What is the sorbent? What is the advertised pump capacity?
B. Physical Inspection of Cryo-sorption Pump: Inspect the pump for signs of wear or
misuse. Are the screens in place? Is the correct amount of sorbent in place? Are the
vacuum sealing surfaces in good condition?
C. Bake-out of Cryo-sorption Pump: in a safe area, set up a fire-safe area to bake-out
your cryo-sorption pump. Bake out the pump for 60 minutes.
D. Pumping speed and capacity: once the pump has been regenerated, allow it to
cool to room temperature with the pump isolation and relief valves closed. Attach the
pump to a vacuum vessel of at least 10 liter volume as shown in figure 8.4. Connect a
dewar to the pump body, and fill the dewar with liquid nitrogen. Allow 30 minutes for the
sorbent to cool. With the vessel at atmospheric pressure, and the vent valve closed,
open the cryo-sorption pump isolation valve, and record pressure versus time for 20
minutes. Close the cryo-sorption pump isolation valve and vent the chamber to
atmosphere. Close the vent valve and repeat the experiment. Do this sequence of steps
until a noticeable decrease in pumping speed is noted. Plot your data as pressure vs.
time and pumping speed vs pressure.
TC1
Discussion:
How does the pump capacity that you have calculated compare to those listed by
the manufacturer for this pressure range?
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What was the general trend in pumping speed for the series of pumpdowns for
each of the gases pumped?
Laboratory Exercise 8.2: Performance of multiple cryo-sorption
vacuum pumps.
A. Using the same vacuum vessel as in the previous experiment, connect two similar
cryo-sorption pumps as shown in figure 8.5.
TC1
Figure 8.5 Experimental set-up for experiment 8.2.
As was done in the previous experiment, bake out the pumps if necessary and measure
the pumping speed for two cryo-sorption pumps used simultaneously. Make two plots of
your data: pressure vs time and pumping speed vs. pressure. Calculate the total
amount of gas pumped in each experiment. If bottled nitrogen, helium or argon are
available, repeat the experiment with these gases. A data table is provided on the next
page.
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The invention of the ion pump did not occur until the 1950's when the Varian company
exploited the pumping characteristics of the Penning cold cathode gauge. While it had
been known that the sputtering effect caused by high voltage in the Penning gauge
resulted in burial of ionized gas molecules, and that gettering of gases such as oxygen
by reactive metals (titanium) were both occurring, the concept of using these
mechanisms to remove gas molecules from a system was ignored. Soon after
commercial sputter-ion pumps were made available, they were applied to the then new
field of space environment simulation. Ion pumps were fitted to large carefully
constructed vacuum vessels, and pressures as low as 10
-11
Torr were obtained. This
enabled evaluation of satellite components, space suits and rocket components.
Currently sputter ion pumps are used in a variety of UHV applications including surface
science techniques (study of the first few atomic layers of a surface), and ultra-high
purity thin film deposition processes (e.g. molecular beam epitaxy).
Sputter-ion pumps are gas capture type vacuum pumps that function without pump
fluids or any moving parts. They offer a clean, quiet, and safe way to achieve ultra-high
vacuum (10
-11
Torr). Sputter-ion, or getter-ion pumps are often used on vacuum systems
that are sensitive to oil contamination that is possible from oil diffusion pumps and turbo
pumps. In general, sputter-ion pumps are used in systems in which pumping speed is
less important than cleanliness and achieving an extremely low base pressure.
Sputter-ion pump characteristics The operational characteristics of a sputter-ion
pump may be simply described by the following three factors:
1. Pumping speed
As with any high vacuum pump, the pumping speed will determine the ultimate base
pressure for a given gas load. Ion pumps, however, exhibit pumping speeds that are a
function of the gas specie being pumped. Hydrogen is pumped at a relatively high rate
compared to argon. It is critical to match the ion pump to the application.
2. Starting pressure
Ion pumps must be rough pumped to an acceptable pressure (2x10
-2
Torr or lower)
before being turned on. Typically, this is done with a cryo-sorption pump or dry pump to
eliminate the possiblity of oil backstreaming into the vacuum vessel or ion pump body. If
well-trapped, oil sealed mechanical pumps may be used for roughing the ion pump and
vacuum vessel.
3. Operating principle
Sputter-ion pumps may be single or multiple cell types, and can be of diode or triode
design. For the purposes of this laboratory, we will describe the operation of a single
cell diode type sputter-ion vacuum pump. As with all gas capture pumps, the sputter-ion
pump requires no backing pump, and does have a limited lifetime, based on its
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capacity. The pump reduces pressure in a vacuum vessel by three distinct mechanisms:
chemisorption, ion burial, and physical burial. During operation, titanium metal is
sputtered (energetically liberated by ion bombardment) from the cathode surfaces.
Titanium, being a very reactive metal, will chemically combine with active gas molecules
present (oxygen and hydrogen) to form stable compounds, thus removing the gases
from the vacuum vessel. Additionally, gas molecules and atoms are ionized by electrons
that are constrained to orbit in the anode tube by a strong external magnet. These
ionized gases are accelerated to the cathode by high voltage from the pump power
supply. On impact, gas ions become buried in the titanium cathode, and also sputter (or
knock free by momentum transfer) titanium atoms that act as getters as explained
earlier. On start-up the amount of sputtering that occurs is very high, resulting in an
initially high electrical current in the pump. Sputter-ion pumps will be warm or even hot
to the touch during this phase of operation. After the gas pressure reduces, the pump
will draw much less current from the power supply, and the operating voltage will
increase. The amount of current that a sputter-ion pump draws during operation may be
used, along with conversion charts supplied by the vendor, to determine pressure in the
pump.
Figure 9.1 Cutaway view of a single cell sputter-ion vacuum pump. High voltage applied
between the anode and cathodes generates primary electrons that are constrained to
spiral orbits within the anode. Collisions of these primary electrons with neutral gas
atoms causes the atoms to become ionized. The positive gas ions are accelerated into
the cathodes, resulting in burial of the gas ion and/or sputtering of the cathode material
(titanium).
Titanium atoms
gas molecules or atoms
positive gas ions
electrons
magnetic field
A
B
C
+ HV
Anode tube
cathode cathode
Figure 9.2
Detail of the processes in a sputter-ion pump.
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At location C a primary electron ionizes a gas atom; at B an ion impacts the cathode
ejecting a titanium atom, and at A a gas ion is buried in the cathode. What is the cathode material? Is it a single or multiple cell pump? Is it a diode or
triode design?
B. Physical Inspection of Sputter-Ion Pump: Inspect the pump for signs of damage
or misuse. Check power supply electrical cables for cracks in insulation. Is the power
supply appropriate for the pump? What is the input power requirement of the power
supply? What is the power supply output voltage and current at start-up? What are
theses values during operation at 10
-6
Torr (approximately)?
C. Measurement of ultimate pressure: assemble a system similar to that shown in
figure 9.3. It would be preferable to use cryo-sorption pumps to rough the vacuum
vessel and the sputter-ion pump to a pressure of less than 20 microns. A trapped
mechanical pump will suffice if cryo-sorption pumps are unavailable.
C. Measurement of ultimate pressure (cont.)
Evacuate the sputter-ion pump and the vacuum vessel to a pressure of less than 20
microns (2
x
10
-2
Torr). Valve off the roughing pump, and start the sputter-ion pump.
Record vessel pressure, and sputter-ion pump power supply voltage and current as a
function of time.
Time Pressure Current Voltage Power
[seconds] [Torr] [amps] [volts] [Watts]
Calculate power for each of your readings (power = current*voltage). Plot the data you
have collected as sputter-ion pump current, voltage and power as a function of time.
Also plot vessel pressure vs. time. If the equipment and materials are available, isolate
the sputter-ion pump from the evacuated chamber, and back fill the process chamber
with an inert gas such as helium or argon and repeat the experiment.
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In the previous chapter we covered the operation of diffusion pumps that pump gases
by the mechanism of momentum transfer. Molecular pumps, of which turbo pumps are a
subset, are also momentum transfer pumps. In turbo pumps gases are caused to move
in a preferred direction due to the interaction with high speed surfaces. Gaede
recognized the possibility of pumping gases by this technique as early as 1912, and he
constructed a simple molecular pump that demonstrated his theory. This early
molecular pump was similar in construction to a modern rotary vane mechanical pump,
with the exception that the rotor of Gaede's molecular pump had no moving vanes and
was concentric with the stator (see figure 10.1).
inlet
exhaust
rotor
stator
Figure 10.1 Gaede's molecular drag pump of 1912.
In the operation of Gaede's design, gas molecules entering the molecular drag pump's
inlet strike the surface of the moving rotor, and remain on this moving surface for a
period called the "Residence time" (see equation 4.8). Molecules leave the surface of
the rotor, obeying the "Cosine Law" distribution presented in figure 4.1. The molecule
then strikes the inner surface of the stator, remains there for the "Residence time",
desorbs and may again strike the surface of the rotor to again be moved in the preferred
direction.
Chapter 10: Turbomolecular Pumps
Figure 10.2 Cross-section of a molecular drag pump using a spiral channel machined
into the stator and a flared disc as a rotor.
Molecular drag pumps designed in the early 1900's had low pumping speeds, due in
part to the practical limits encountered in machining techniques and bearing designs
which limited rotational velocity of the rotor.
Many of the current molecular drag vacuum pumps are similar in design to that
presented in figure 10.3. The rotor is often fabricated out of a high strength aluminum
alloy and is shaped like an inverted cup. Both inside and outside surfaces of the rotor
are machined to create spiral grooves which work with the surfaces of the stator to
provide the pumping action. Using both the internal and external surfaces of the rotor
creates an elongated pumping path.
The size, shape and tolerances of the grooves change from the inlet side to the exhaust
side of the pump to allow for multiple compression stages. Flush gas is intentionally
admitted to the pump to provide cooling and as an aid to exhausting the compressed
gas. High quality molecular drag pumps can attain compression ratios for nitrogen of
approximately 10
9
:1. Since the pumping action is dependent upon the residence time of
a gas on the stator and rotor, and the average velocity of gases, it should be obvious
that the pumping efficiency for molecular drag pumps decreases with the molecular
weight of the gas being pumped.
Sample Problem:
10.1 For the atmospheric gases listed in table 4.1 arrange the gases in order
according to the pumping speed you would expect for a molecular drag
pump.
Molecular drag pumps in general cannot compress gases to atmospheric pressure, and
must, therefore have a backing pump attached to the exhaust of the drag pump to
accomplish this final stage of compression. The critical foreline pressure range for many
molecular drag pumps is 10 to 40 Torr, which allows these pumps to be backed by
diaphragm or dry pumps, greatly reducing the possibility of oil backstreaming into the
vacuum vessel. Crossover pressures for molecular drag pumps is often as high as 1
Torr, and systems pumped by this means may achieve base pressures as low as 10
-6
Torr.
Modern Turbomolecular Pumps
Some insight into the operational characteristics of turbomolecular pumps may
be gained by comparing and contrasting them to diffusion pumps. Both turbomolecular
and diffusion pumps are high vacuum pumps which cannot compress gases to
atmospheric pressure, and therefore both require backing pumps (typically rotary vane
mechanical pumps). Both turbo pumps and diffusion pumps provide pumping action by
momentum transfer, that is, they induce molecules to flow in a preferred direction
through the use of high speed surfaces or particles. The pumping efficiency of both
types of pumps is a function of the gas specie being pumped, and in general, the
pumping efficiency decreases with decreasing molecular weight of the gas. Unlike
diffusion pumps, turbomolecular vacuum pumps do not require traps and baffles, as the
possibility for backstreaming into the vacuum vessel is very limited in a turbo pump.
Advances in fabrication techniques, high strength/low weight alloys and bearing design
have allowed the development of high performance turbo-molecular vacuum pumps.
Two designs for turbomolecular pumps have been produced commercially: the
horizontal twin rotor design (Sargent-Welch) and the axial flow design (Balzers, Leybold,
Inc., and others). The operating principles are the same for both designs, so we will use
the more currently popular axial flow design to illustrate the operation of turbo pumps. In
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practice, the horizontal twin rotor design has a much more massive rotor assembly
which is more difficult to dynamically balance than the lighter weight rotor of an axial
flow turbo pump. The result is that the rotational velocity (Rpm's) that a horizontal turbo
pump may attain is much lower than for a comparably sized axial flow turbo, and
therefore pumping speed for the horizontal pump is less than for an axial design pump.
In the axial flow design, the compressor is comprised of matched sets of rotors
and stators, which are typically fabricated from aircraft quality aluminum alloys. Typical
rotational velocities for the rotor of turbo pumps of this design are from 20,000 to 60,000
RPM.
Sample Problems:
10.2 Calculate the speed of the tip of a 5 cm radius rotor operating at 60,000
RPM and compare that value to the average velocity of nitrogen and
hydrogen at room temperature. What conclusions can you draw from this
data?
Look again at figure 10.4. Note that the size and aspect ratio (length divided by
width) of the rotor blades at the inlet are different that for the rotor blades at the exhaust.
Most modern axial flow turbo pumps have rotors and stators which are designed in
stages to optimize pumping performance for the pressures at locations throughout the
pump. The inlet stage typically is designed with the goal of achieving high volumetric
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speed with minimal compression. Stages at the exhaust line are designed with the
opposite goal in mind: maximizing compression of the gas at the expense of volumetric
speed. It should be noted that the function of the stators and rotors is slightly different.
The high speed rotors provide a surface on which gas molecules "reside" for some short
time, then desorb, leaving in a preferred direction. The stators serve to improve the
effectiveness of the rotors by providing a baffle effect, directing the gas flow to the next
rotor. For this reason, the stator is often omitted from the final stage of the turbo pump,
as it would serve no purpose, and would impede the flow of gas to the backing pump.
Since the low molecular weight gases are the most difficult to pump using a turbo, the
ultimate pressure one may attain using a turbomolecular pump is often due to the
inefficiency of the pumping of these gases.
Operational Aspects of Turbomolecular Pumps
The maximum crossover pressure for turbomolecular pumps is approximately 1 Torr,
this is a factor of ten times higher pressure than the maximum suggested crossover
pressure for most oil vapor diffusion pumps (100 mTorr). At pressures above 1 Torr the
turbo pump blades will be slowed by collisions with gas molecules such that the motor
will overload and the rotational velocity of the rotor will decrease to a speed that is
ineffective for pumping gas. Unlike diffusion pumps, turbo pumps do have moving parts
that can cause vibration which may adversely affect some precision instruments
including scanning electron microscopes and surface science probes. A 60 or 120 Hz
vibration typically is caused by a mechanical backing pump, while high frequency
vibration is due to imbalances in the turbo pump rotor. Most vacuum applications are
insensitive to this minute amount of vibration, but if vibration must be held to a
minimum, and the pumping characteristics of a turbo pump are desired, a magnetically
levitated rotor design may provide the solution. In this type of turbo pump conventional
(but oil free) bearings are only used on start-up and shut-down of the turbo. During
normal operation the rotor is suspended above the bearings by well matched sets of
strong magnets, virtually eliminating all mechanical vibration. Magnetically levitated
turbo pumps are designed to operate for long periods of time with very few interruptions.
Each time a magnetically levitated turbo pump is started or stopped, the oil-free
mechanical bearings suffer wear and eventually will require replacement. Beyond
reduction of vibration, the magnetically levitated rotor design turbos offer the option of
mounting in any orientation, as there is no oil sump as in most conventional turbo
pumps. Standard sequence of operation of turbomolecular pumped vacuum systems
(see figure 10.5) is as follows:
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Start-up:
1. Close all valves in the vacuum system.
2. Start the mechanical pump.
3. Open the foreline valve and rough pump the turbomolecular pump to a
pressure of less than 1 Torr.
4. Start turbo pump; wait for rotor to attain normal operational velocity (20
minutes for most small to medium size pumps).
5. Close the foreline valve.
6. Open the vessel roughing valve; evacuate the vacuum vessel to a
pressure of less than 500 mTorr*.
7. Close the vessel roughing valve; open the foreline valve.
8. Open the head gate valve; turn on the ion gauge.
Venting the vessel without stopping the turbo pump:
1. Turn off the ion gauge.
2. Close the head gate valve.
3. Open the vessel vent valve.
4. Open the vessel as soon as it reaches an internal pressure equal to
atmospheric.
5. Close the vacuum vessel and the vessel vent valve.
6. Close the foreline valve.
7. Open the vessel roughing valve; evacuate the vacuum vessel to a
pressure of less than 500 mTorr*.
8. Close the vessel roughing valve; open the foreline valve.
9. Open the head gate valve; turn on the ion gauge.
Shut-down:
1. Turn off the ion gauge.
2. Close the head gate valve.
3. Turn off power to the turbo pump, wait for rotation to stop.
4. Close the foreline valve and turn off the mechanical pump.
5. Vent the roughing line.
6. Open the air admittance valve on the turbo pump to gradually bring the
pump to atmospheric pressure.
7. Open the vacuum vessel vent valve.
*Check manufacturer's suggested crossover pressure for the particular pump you
are using.
Sample Problem:
10.3 Why are turbomolecular pump compressors designed in several "stages"? What
are the characteristics of the inlet and exhaust stages?
Maintenance of Turbomolecular Pumps
Normally, turbomolecular pumps operate for years and require little maintenance. Those
pumps which have an oil sump and circulation system should have the oil changed
approximately every six months or when the oil turns from clear to brown. Turbo pumps
that use thick grease should have the lubricant replaced every six months. Bearing life
in turbo pumps is approximately two to three years. Replacement of bearings is usually
performed by trained technicians at the manufacturer's facility due to the precision
balancing required for the high speed rotor.
If a turbomolecular pump inlet becomes contaminated, due to backstreaming of
lubrication oil, occasionally a user may be able to clean the pump inlet and the first few
stages by inverting the pump (oil having previously drained from the sump) in a
container of solvent. Care must be taken to prevent immersion of any electrical
components of the pump. It is wise to check with the pump manufacturer prior to
cleaning a turbo pump by this method.
Applications for Turbomolecular Pumps
Three areas of vacuum technology that take advantage of the pumping characteristics
of turbomolecular pumps are semiconductor equipment manufacturing, thin film
deposition industry and the leak detector manufacturing industry.
Vacuum processes such as sputter deposition, which rely upon the flow of a process
gas, usually at pressures of 3 to 50 milliTorr are often conducted using throttled
turbomolecular pumps. Argon, a common process gas is pumped effectively by turbo
pumps; variable orifice valves are used to control the pressure or throughput of gas in
the vacuum vessel.
Modern vacuum leak detectors also often use turbo pumps as the high vacuum pump.
Portable units typically have turbo pumps with greased bearings or magnetically
levitated rotors so as to eliminate the possibility of oil contamination from the sump were
the unit to be oriented horizontally during shipment. Another desirable characteristic of
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turbo pumps for leak detector application is the relatively high pumping speed for
atmospheric gases (oxygen, nitrogen, carbon dioxide) compared with that for the light
gas, helium. In most instances helium is used for leak detection due to its small
molecular size, rarity in the atmosphere, and low toxicity. Some of the newer "counter-
flow" leak detectors rely upon the low pumping efficiency of turbo pumps for light gases,
such as helium, to permit backwards flow of helium through the operating turbo pump.
This design allows for a much more compact and portable leak detector unit (more on
this in Unit 13, Leak detection).
Sample Problems:
10.4 What are advantages of a turbomolecular pump over a diffusion pump?
10.5 Turbomolecular pump suffer severe mechanical damage when solid
objects fall into the inlet during operation of the pump. Can you suggest
two ways to prevent this occurrence?
10.6 What limits the base pressure one may attain using a turbomolecular
pump?
10.7 Accessories available for turbomolecular pumps include the following:
Flange Heaters: To aid in the removal of residual gases and any contamination that
may be present at the inlet area of the pump. Care must be taken not to exceed the
manufacturer's suggested maximum temperature, as severe bearing wear may result.
Venting Devices: Upon interruption of electrical power these valves admit air into the
inlet area of the pump to achieve pressure equilibrium within the turbo pump. This
action serves to reduce the possibility of mechanical damage to the rotors and to
minimize backstreaming of oil from the foreline.
Vibration Isolation Bellows: Reduce the transmission of high frequency vibration from
the turbo pump to the vacuum vessel.
Water Flow Interlock: Prevents the operation of the turbo pump without proper flow of
cooling water.
Compound Molecular Pumps
Compound molecular pumps are typically of the axial flow design and are essentially a
combination of a turbomolecular and a molecular drag pump built into one unit. The
advantage of a compound pump is that the molecular drag pump at the exhaust stage is
able to compress the gas to a higher pressure than a conventional turbo pump. Most
compound pumps are made to be backed by a diaphragm pump, thereby eliminating
the possibility of backstreaming oil from an oil sealed rotary vane pump. Some of the
newer compound pump can exhaust to atmospheric pressure; these pumps are often
small (less than 150 liters per second pumping speed).
Goals of this experiment: to operate a turbo pumped vacuum system, to observe the
pump down rate for a vessel, to observe the base pressure of the system and to
calculate the total gas load in the vessel.
Equipment required: An axial flow turbomolecular pump.
Procedure: Locate the manufacturer's literature for the model pump you will be working
with. What is the rated speed for air? What is the ultimate pressure the pump can
attain? What is the crossover pressure? What is the critical foreline pressure? What
type bearings does the pump have? How is the pump cooled? What are the utilities
requirements (water and electrical)? What is the recommended routine maintenance
procedure?
Inspect the pump. Is there an air admittance valve on the unit? How is this valve
actuated? Will this valve safely vent the pump during a power failure? If the pump has
an oil sump, inspect the oil through the viewport. Note the appearance, clarity and level
of the oil.
Create a written report of your findings for submission to the Laboratory Instructor.
Laboratory Exercise 10.2: Operation of a vacuum system with a turbomolecular
pump.
Equipment required: small vacuum vessel or bell jar vacuum system to which a
suitably sized turbomolecular pump can be attached, a vent valve for the vessel, two
ionization gauges two thermocouple gauges and controllers, a gate valve to match the
turbo pump inlet, roughing lines, a mechanical pump, a roughing valve, a foreline valve
and a roughing line vent valve.
Procedure: Prior to any experimental work, make certain that the roughing pump
selected will deliver adequate pumping speed to the foreline of the turbo pump.
Assemble the equipment as shown in figure 10.5. Make certain the cooling water
flow is adequate if the pump is a water cooled model. After the Laboratory Instructor has
checked the vacuum system, follow the system start-up procedures provided in this unit.
Measure the vacuum vessel pressure as a function of time during pump down. Graph
the data clearly showing which portion of the curve is due to evacuation by the
mechanical pump, the point of crossover, and the pumping action due to the turbo
pump. Following completion of the measurements safely shut down the turbo pumped
vacuum system as outlined in this unit.
Procedure: Using the vacuum system assembled for experiment 10.1, attach a needle
valve/atmosphere valve assembly to the vacuum vessel as was done for the speed
measurement at constant pressure for diffusion pumps. Bring the system into high
vacuum operation using the set of procedures used in this unit. Open the needle valve
slightly; wait for the pressure in the vessel to stabilize. Close the atmosphere valve by
placing a thumb over the hole, and record the time required to pull 1 ml of water up into
the graduated pipette. Record this time along with the pressure during the
measurement. Open the needle valve another small increment and repeat the
measurement of time required to draw 1 ml of water into the graduated pipette at the
new pressure. Make at least ten measurements in this manner. Using equations 8.1 and
8.2 calculate the speed at each pressure, then plot the data as speed versus pressure.
Following completion of the measurements safely shut down the turbo pumped vacuum
system as outlined in this unit.
Submit a written report for this experiment to the Laboratory Instructor.
Discussion:
Was the pumping speed data you collected over a range of pressures comparable to
the data published by the turbo pump manufacturer?
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Chapter 11: Cryogenic Vacuum Pumps
In the preceding units we have described vacuum pumps that operate by isolating
volumes of gas, compressing the gas and exhausting to atmosphere (mechanical
pumps) and vacuum pumps that move gas through the interaction of high velocity
particles or surfaces (diffusion pumps and molecular pumps). Gases may be removed
from a vacuum vessel by a third pumping mechanism: gas capture. In this scheme, gas
molecules are removed from the gas phase by one of several techniques. Gas capture
may be accomplished by solidifying the gas on extremely cold surfaces. This form of
gas capture is called cryo-condensation. Some gases, such as helium, neon and
hydrogen have such low boiling points that they are not readily condensed. Pumping of
these gases may be accomplished by adsorption. If, through a series of collisions with
cooled surfaces, a helium atom loses kinetic energy, it may become "adsorbed" onto a
surface. In this state, the helium gas molecule is weakly attached to the cold surface
and is, for all purposes, removed from the gas phase. Other mechanisms for gas
capture will be detailed in the following unit on sputter-ion pumps.
One other difference between gas capture pumps and positive displacement or
momentum transfer pumps is that gas capture pumps have a finite capacity; once they
are full, pumping action will cease until the pumping media can be renewed or
regenerated. For this reason gas capture pumps are seldom used on vacuum systems
which are designed for continuous high gas throughput. This fact has an associated
safety issue: gas capture pumps collect and concentrate all the gas species they have
pumped during their service time. If the pumping media is to be regenerated for further
pumping, considerable care should be taken to carefully exhaust the gases which will be
released from the pump during regeneration.
The Effect of Temperature on the Vapor Pressure of Gases
Gas molecules, upon collision with cooled surfaces, lose a significant amount of
their thermal energy to the cooled surface. In general it may be said that the thermal
energy of a gas molecule is determined almost entirely by the temperature of the last
surface the gas molecule desorbed from. If a surface is intentionally cooled to the
temperature of liquid nitrogen (-196°C or 77K), all gas molecules which have a boiling
point higher than -196 °C can be cryo-condensed on this surface. These gas molecules
will literally freeze, transforming from a gas to a solid. As solid material, these
condensed gases are captured and eliminated from the gas load inside the vacuum
vessel.
cryo-surface -196 °C
molecules in
the gas phase
cryo condensed
molecules
desorbed
molecule
Figure 11.1 Molecules in the gas phase,upon contact with a cryo-cooled surface,
condense on that surface. The residence time for molecules is dependent upon the
specie of gas, the temperature of the cryo-surface and the heat of adsorption (see
equation 4.8).
Sample Problems:
11.1 The boiling point of liquid nitrogen is -196 °C. Which of the following gases
can be condensed on a surface cooled by liquid nitrogen?
11.2 For gas molecules which are not cryo-condensed onto a surface cooled by
liquid nitrogen, what would the temperature of the desorbed gas
molecules be after they leave the cold surface? How would the velocity of
these gas molecules be affected?
11.3 Calculate the residence time for common atmospheric gases which have
been condensed onto a surface cooled to liquid nitrogen temperature.
Cryo-sorption Pumps
Cryo-sorption pumps provide a safe, quiet, clean and reasonably inexpensive method
for evacuation of a vacuum vessel to a pressure of 10
-3
Torr. Most commercial cryo
sorption pumps resemble the diagram in figure 11.2. Liquid nitrogen is used to cool the
exterior of the welded aluminum pump body to -196 °C. The interior of the pump body
has radially arrayed heat transfer fins that aid in extracting the heat from the sorbent,
which is usually activated carbon or alumina. A metal screen, often made of stainless
steel creates an open channel that runs vertically through the pump body. This channel
aids in exposing the pumping media, or sorbent to gases entering the pump inlet. Linde
5A is a popular sorbent material, which has micro pores of approximately 5Å diameter,
which are optimal for trapping most atmospheric gases. Other advantages of this
sorbent are that it is chemically inert and will not thermally decompose during a bake-
out at 250 °C.
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Viton Stopper
Viton cuff
sorbent
Liquid nitrogen
Dewar
screen
Pump inlet
pressure relief valve
Figure 11.2 Cutaway drawing of a cryo-sorption vacuum pump.
TC 1
vacuum
vessel
Figure 11.3 Schematic of a vacuum vessel rough pumped by a bank of three cryo-sorption pumps.
Several cryo-sorption pumps may be arranged in a bank as shown in figure 11.3. This
configuration has several advantages over the use of a single cry-sorption pump. A
multiple pump system provides the capability of regenerating one pump while using
others, increasing the capability of the vacuum system for repetitive pump down cycles.
Other advantages include the flexibility of pumping with cryo-sorption pumps in parallel
connection (all roughing valves opened at once) or in series (opening one valve, utilizing
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the first pump until it is saturated, then closing its valve and opening the valve of the
second pump). In general, if one wishes to achieve maximum pumping speed at the
expense of base pressure, the parallel mode of operation is used. Conversely, if a low
base pressure is of utmost importance, the cryo-sorption pumps are operated in series
(sequentially).
Sample Problem:
11.4 Parallel and series operation of cryo-sorption roughing pumps will produce
very different pump down curves. For the vacuum system pictured in
figure 11.3 draw the pump down curves (pressure versus time) that you
would expect for series and parallel arrangement of the cryo-sorption
pumps.
Liquefied Gas Cryogenic Pumps
liquid nitrogen
inlet
nitrogen gas
outlet
TC
IG
vacuum vessel
vessel vent valve
cryo panels
Cryo-panels, placed inside a vacuum vessel, and cooled with liquefied gas (most
commonly liquid nitrogen) are often employed to reduce the base pressure that may be
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attained using a high vacuum pump such as a diffusion pump or a turbomolecular
pump.
Cryo-panels should be designed to allow easy filling and purging of the liquefied
gas, and also must allow for boil-off of the gas during operation. Typically, cryo-panels
are filled after a pressure of less than 10
-5
Torr has been attained. This reduces the
loading of the cryo-panels with atmospheric water vapor, which the diffusion or turbo
pump can normally handle. Prior to venting the vessel to atmosphere, the cryo-panels
should be warmed to room temperature. Cold cryo-panels exposed to air would ice up,
and the ice, upon melting would drip water into the vacuum vessel.
Theory of Operation of Compressed Helium Cryogenic Pumps
Everyone who has pumped up a bicycle tire using a hand pump has experienced the
effect of gas heating upon compression. As the piston in the air pump is forced down,
air is compressed and forced through the inner tube valve stem. At this point the
compression of gas is high and the heat generated is conducted through the valve stem
to the fingers. In just the opposite way, gases may be allowed to expand rapidly to pull
heat from their surroundings. This is why the tip of aerosol cans become cold when the
compressed gas is released. This effect is particularly noticeable for the cans of
compressed freon (microdusters) used to blow dust off of microelectronics parts.
Compressed helium refrigerators take advantage of the cooling effect of expanding
gases to produce extremely cold surface onto which gas molecules may be captured. It
should be noted here that at no point in the operation of the helium compressor is the
helium condensed to a liquid. All helium refrigerators used to produce cold surfaces for
cryo pumping have three basic components: the helium gas compressor, the connecting
lines and the cold head. These components are carefully matched to work together
properly. With very few exceptions, components from different manufacturers cannot be
intermixed and be made to operate properly.
The unit referred to as the "Compressor" actually serves several functions in addition to
compressing the helium gas. Following compression, the gas is forced through a heat
exchanger which is cooled using flowing water. The cooled helium may contain some
residual oil vapor from the compressor. This oil vapor would condense in the cryo-pump
regenerator and severely hamper its ability to produce the cold temperatures required
for cryo-pumping. To remove oil vapor, an oil separator and an oil adsorber are used in
series as shown in figure 11.5. The oil adsorber has a finite service life, and must be
replaced with a new unit periodically. Typically, adsorbers are renewed every 6 months.
The lines which transport the high purity compressed helium between the compressor
and the cryo-pump are specially designed to contain the high pressure helium gas.
These lines have special fittings on each end which allow connection and disconnection
without losing the helium in the lines. Maximum line length varies from manufacturer to
manufacturer, but most models allow the cryo-pump to be at least 20 feet from the
compressor. This permits one to place the compressor outside a clean room to reduce
contamination, or to isolate the vacuum vessel from heat or vibration generated by the
compressor.
The operation of a compressed helium refrigerator is based upon the cooling
cycle as described by Gifford and McMahon in several articles published in 1960. The
following series of diagrams and footnotes are presented to demonstrate the principles
of operation of this type of refrigerator.
Figure 11.7 Cross-section of a compressed helium refrigerator. In the first part of the
refrigeration cycle, the displacer, which is made of a thermally insulating material
(usually micarta) is at the lower end of its stroke. The compressed helium supply valve
is opened, and high pressure (300 PSI) helium gas at room temperature is flowed into
the cylinder in which the displacer oscillates.
supply
valve
helium compressor
return
valve
Figure 11.8 As the high pressure gas is admitted into the cylinder, the displacer moves
upwards, forcing the gas to pass into and through the regenerator. The regenerator is
made up of tightly packed material of high thermal inertia or heat capacity. {Heat
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capacity may be defined as the amount of thermal energy required to raise a specified
amount of material from one temperature to another temperature. A material having a
high heat capacity required more thermal energy input to change its temperature than a
material of low heat capacity}. The materials most often used in the regenerator are
lead or copper spheres. Even though the regenerator is tightly packed with these
spheres, gas flow is not seriously impeded.
supply
valve
helium compressor
return
valve
Figure 11.9 The supply valve admitting compressed gaseous helium into the cylinder is
closed as the displacer moves upwards nearing the top of its stroke. At this point the
helium gas has traveled through the regenerator, and assuming several cycles have
already occurred, the helium gas will lose some of its thermal energy to the cooler
regenerator.
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helium compressor
return
valve
heat heat
Figure 11.10 In the next stage of the cycle, the return valve is opened. The gas in the
cylinder is at 300 PSI while the pressure in the return line is at approximately 80 PSI.
The gas responds to the pressure differential by expanding into the return line. This
expansion is what causes cooling in this type of heat pump. Heat flows from the
external heat load (dark rectangle at the bottom of the drawing through the cylinder
walls to the cold interior of the cylinder. As the helium passes through the regenerator it
also cools the metal spheres.
helium compressor
return
valve
heat heat
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Figure 11.11 In the final stage of the cycle, the displacer is forced downwards to push
any remaining helium gas through the regenerator and into the return line to the helium
compressor. The return valve is closed and the helium is again compressed in the
compressor for the next cycle.
motor
valve disc
low pressure line
high pressure line
seal
first stage
second stage
first stage heat load
second stage heat load
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Figure 11.12 Cross-section of a two-stage compressed helium refrigerator. The motor
serves to rotate the valve disc which is ported to control flow of high pressure gas into
the regenerator and flow of low pressure gas back to the helium compressor.
The majority of commercially available cryogenic pumps are similar to that represented
in figure 11.13. At the pump inlet is the 80 K array, which is thermally connected to the
first stage of the refrigerator by the radiation shield. Indium foil is used at the mechanical
junctions to improve thermal conductivity. Water vapor is the primary gas that is
condensed on the inlet array. Without the optically opaque inlet array, water vapor
would condense on the 15 K array severely limiting its ability to pump oxygen, nitrogen
and the non-condensable gases, helium, hydrogen and neon. The diagonally positioned
plates of the 15 K array serve two functions: the top surfaces are used to pump oxygen,
nitrogen and argon, while the sorbent attached to the underside of each array is used to
cryo-adsorb the three non-condensable gases.
Figure 11.14 Schematic of a vacuum system using a compressed helium refrigerator
cryogenic pump and a mechanical roughing pump
As with all high vacuum pumps, the compressed helium cryogenic pump is
unable to evacuate vessels which are at atmospheric pressure. Unlike diffusion and
turbo pumps, the appropriate conditions for crossover for a cryo pump are a function of
the amount of gas in the vessel rather than simply the pressure in the vessel. This is
best illustrated by example. If a manufacturer's specification for the cross-over of a cryo-
pump is 150 Torr-liters, and the vessel to be pumped has a volume of 100 liters, then
the cross-over pressure is given by:
It has been mentioned that cryogenic pumps, being of the gas capture type, have a
finite capacity. Once these pumps have reached their capacity (become saturated)
pumping action will cease. At this point the pump needs to be warmed up in a controlled
manner to allow the release of the condensed gases in the pump's cryo arrays. This
process is referred to as regeneration. During regeneration, all of the gases which have
been captured by the pump will be released in concentrations much greater than normal
in the atmosphere. It is possible, during regeneration, to release explosive or toxic
gases in dangerous concentrations. For this reason the process of regeneration must
be performed safely, following the pump manufacturer's directions. In general, one
regenerates a compressed helium cryogenic pump following this procedure:
Regeneration Procedure for a Compressed Helium Cryo-pump:
1. Close the head gate valve between the pump inlet and the vacuum vessel.
Turn off any pressure gauges that are exposed to the cryo-pump body.
2. Stop the cold head motor.
3. Check the poppet valve on the cryo-pump body to insure that it is in good
condition and is not physically obstructed. Dangerous over pressurization
of the pump body will occur if the pressure release valve fails to operate
properly.
4. Begin purging the pump body with dry, inert gas such as nitrogen or
argon. (In some cases it is possible to speed regeneration if the purge gas
is heated by flowing it through an electrically heated tube on its way to the
cryo pump body).
5. If the cryo-pump is equipped with a blanket heater, turn this heater on.
6. Allow the pump to be purged with gas for a sufficient amount of time to
allow removal of all trapped gases inside the cryo-pump. This time is a
function of pump size and design; check manufacturer's specifications for
the time duration for this operation.
Following proper regeneration of the cryo-pump, the pump will be ready to resume
service.
Operation of a Cryo-pumped Vacuum System:
Assuming all of the compressed helium lines are connected and properly purged, the
sequence of operations is as follows:
1. Check the compressor to verify cooling water flow to the heat exchanger.
2. Start the compressor, allow it to operate for 30 minutes (the auto bypass
circuit in the compressor will cycle compressed helium from the high
pressure side of the compressor to the low pressure side).
3. Close all valves on the vacuum system.
4. Start the roughing pump; allow time for the roughing pump to warm up.
5. Open the roughing valve to the cryo-pump; pump the cryo-pump body
down to a pressure of less than 50 milliTorr.
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6. Close the roughing valve.
7. Record the rate of pressure rise inside the cryo-pump body. If the rate of
rise is less than 10 millitorr per minute, begin operation of the cold head. If
the rate of rise is greater than 10 milliTorr per minute, the gas load in the
pump is unacceptably high, and the cause must be identified and
eliminated.
8. Observe the temperature of the cooled surfaces inside the pump using the
pump's temperature monitor. Assuming the temp. probe is reading the
second stage, when the temperature is below 10K, the pump is in a stable
operational mode.
9. Refer to the manufacturer's literature for the crossover pressure
specification for the pump model you are using. Calculate the crossover
pressure.
10. Evacuate the vacuum vessel to a pressure below that calculated in step
10.
11. Close the vessel roughing valve and open the head gate valve.
12. Turn on vessel ionization gauge.
Sample Problems:
11.5 Why is the regenerator filled with lead spheres?
11.6 Is the helium in the refrigeration cycle of a compressed helium cryogenic pump
ever in the form of a liquid?
2. What type of sorbent is recommended? What amount of sorbent is
recommended?
3. What are the time and temperatures recommended for regeneration of the
cryo-sorption pump?
4. Identify the pressure relief valve on the cryo-sorption pump. How does it
work? Is the pressure relief valve in good operating condition? What are
the potential dangers associated with regeneration of this type of vacuum
pump?
Procedure: Assemble the equipment as shown in figure 11.3. Make certain the vacuum
connections are secure. If necessary, regenerate the cryo-sorption pumps following the
manufacturer's directions. Be careful to avoid contacting the hot surfaces with the skin
and keep flammable materials at a safe distance from the sorption pumps during
regeneration. Close isolation valves on each pump at the end of the regeneration cycle.
When the pumps have cooled to room temperature, and are ready for use, attach a
dewar to one pump at a time, fill the dewar with liquid nitrogen, and perform the
following experimental measurements:
1. Using only one cryo-sorption pump, measure the time to evacuate the
vessel from atmospheric pressure to 50 milliTorr. Close the isolation valve,
vent the vacuum vessel to room air and repeat the evacuation, again
recording the time to pump to 50 milliTorr. Repeat this process until the
time to achieve the specified pressure is unacceptably long, or the pump
fails to reach 50 milliTorr. Plot the data as pressure versus time for all the
runs performed on a single piece of graph paper, clearly identifying each
plot. Calculate the number of Torr-liters pumped during each
measurement. Calculate the amount of gas (expressed in Torr-liters)
required to saturate the pump. Compare this value to the manufacturer's
specifications.
2. Repeat the steps in the first series of measurements, using a fresh cryo-
sorption pump, this time initially flooding the vessel with helium. Following
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each evacuation, back-fill the vacuum vessel with helium. Plot the data as
before and compare the results with the data for pumping air. What
conclusions can you draw from this comparison?
3. Repeat the procedure in (2) using argon gas instead of helium. Again plot
the data and draw conclusions on the performance of cryo-sorption pumps
used to pump these three gas loads.
Procedure: Assemble the equipment as was done for experiment 11.1. Regenerate the
cryo-sorption pumps if necessary. Evacuate the vessel by opening all isolation valves
simultaneously (parallel pumping). Record pressure as a function of time. Regenerate
the cryo-sorption pumps and repeat the experiment, only this time open the cryo-
sorption pump isolation valves sequentially (series pumping). Allow each pump to
achieve its base pressure before closing its isolation valve and opening the isolation
valve to the next pump. Plot the data for both measurements as pressure versus time,
carefully labeling each set of data. Mark the plot of series pumping to show the point at
which switching from one pump to the next occurred.
How do the two sets of data (parallel versus series) compare? Which configuration
produced the fastest initial pumping speed? Which achieved the lowest base pressure?
Is your data consistent with your understanding of cryo-sorption pump operation?
Laboratory Exercise 11.4:
Inspection of a compressed helium cryogenic pump.
Equipment required: a cryogenic pump, manufacturer's literature.
Procedure: Using manufacturer's literature for the model cryo-sorption pump to be
used in this activity. Answer the following questions:
1. What is the capacity of the pump for atmospheric gases?
2. Which gases are pumped on the inlet (80 K) array?
3. Which gases are pumped on the inlet (15 K) array?
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4. What is the sorbent material used on the underside of the 15 K array?
5. Locate the temperature gauge and probe. Where is the temperature
measured? What is the operating temperature of this component?
6. Locate the radiation shield. Note the color of the shields interior and
exterior. Why would the manufacturer intentionally choose these finishes?
7. How is the radiation shield attached to the inlet array?
8. Identify the pressure relief valve on the cryogenic pump body. How does
the valve work? Is the pressure relief valve in good operating condition?
What are the potential dangers associated with regeneration of this
type of vacuum pump?
9. Find the purge gas inlet. How is flow of purge gas controlled?
10. Is the pump fitted with a blanket heater? How is the temperature controlled
during regeneration? What is the maximum recommended temperature?
What limits the maximum suggested temperature?
Laboratory Exercise 11.5:
Operation of a compressed helium cryogenic vacuum pump.
Equipment required: a compressed helium cryogenic pump with cold head,
compressor and charged helium lines, for cryo-pumps with an O-ring seal at the inlet
flange: a 1" thick pyrex glass plate having a diameter at least 1" larger than the O-ring
diameter; for other flange styles: a pyrex glass viewport to match the inlet flange. An oil
sealed mechanical pump, connecting lines and an in-line pressure gauge capable of
reading from atmospheric pressure to 1 milliTorr.
Procedure: The instructor will assemble the cryogenic pump system. Place the glass
plate or viewport on the inlet flange of the cryo-pump (see figure 11.3). Begin flowing
cooling water into the compressor, and start the compressor.
Rough pump the cryogenic pump body to approximately 10 milliTorr. Isolate the
mechanical pump and record the rate of pressure rise for five minutes. Does the
pressure rise indicate the pump is ready for operation? If the rate of rise test indicates
the need for regeneration, follow the manufacturers recommendations and the
procedure given in this unit to regenerate the pump. If the pump is ready for pumping,
begin operation of the cold-head. Observe through the viewport the operation of the
cryo-surfaces. Upon completion of the experiment, stop the cold head, turn off the
compressor and vent the system (with dry inert gas, if possible).
Discussion:
When in the procedure for this experiment is it acceptable to turn off the
mechanical roughing pump? Do you see any advantages to this?
What temperature did you read on the cryogenic pump's temperature gauge?
What does this suggest about the operation of the pump?
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Chapter 12: Leak Detection
Not available at present
Previously several types of sub-atmospheric pressure gauge were described. All of
these gauges share one common feature: they report the total gas pressure. Partial
pressure analyzers, in contrast, provide more detailed information about the gases that
exist in vacuum systems following evacuation (the so-called "Residual gases"). The
data provided by partial pressure analyzers can be qualitative (specifying the identity of
the gases present), or quantitative (giving the partial pressure of each gas). As one
might expect, instruments that can identify and measure the partial pressures of
individual gases that exist in a working vacuum system are somewhat more complicated
than simple total pressure gauges. Partial pressure analyzers, or residual gas analyzers
(RGA's) as they are commonly known, function by ionizing samples of gas from the
vacuum system, separating the ions into discrete groups based upon their masses, and
then counting the amount of ions in each group. The details of each of these steps will
be discussed in the unit.
Partial pressure analysis is a comparatively recent addition to vacuum technology.
While the principles of mass spectroscopy (analysis of ionized gases on the basis of
mass differences) have been known since 1918, practical application in the field of
vacuum technology was not demonstrated until 1960. In his ground-breaking work, H.L.
Caswell used a mass spectrometer to show the beneficial effect of viton gaskets over
other elastomer seals, and also the effectiveness of Meissner coils and getter pumps.
Today we can select from a wide variety of partial pressure analysis instruments which
conveniently attach to standard vacuum hardware. These instruments can range from
small, simple to operate and reasonably inexpensive units used to monitor specific
gases in a vacuum process chamber, to large, extremely sensitive and very expensive
instruments used to detect minute traces of gases. Although there exist quite a few
methods by which ions may be separated, only two of these methods are used in
current commercial partial pressure analyzers: magnetic field and electric field
separation.
Magnetic fields Magnetic & Electric fields Electric fields
Quadrupole
Monopole
Cycloidial
Cyclotron Resonance
Magnetic sector
SPATIAL SEPARATION
TEMPORAL SEPARATION
Time of flight
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Specifically, we will discuss the principles of operation of quadrupole and magnetic
sector mass spectrometers. Both of these mass spectrometers fall in the category of
"spatial separators" that is, they physically separate beams of ions on the basis of
mass-to-charge ratio. Time of flight mass spectrometers, in contrast, rely upon the
differing velocities of ions having different masses as a means of separation. The
concept of mass separation was introduced in the previous unit on leak detection.
Helium mass spectrometer leak detectors are, in fact, partial pressure analyzers
(usually of the magnetic sector type) which are permanently tuned to detect a test gas
such as helium.
Ionization of Gas
While other methods for ionization of sub-atmospheric pressure gases exist (such as
field emission and chemical ionization), the most widely used technique for partial
pressure analyzers is electron-impact ionization. Electrons emitted from a heated metal
filament are electrostatically attracted to an anode, or electron collector plate, by an
imposed electric field of from 50 to 150 V DC. On route to the anode, some of the
electrons strike neutral gas molecules, stripping off one or more outer-shell electrons,
creating positive ions. Some molecules may be split into fragments during this process,
each fragment being a positive ion which will be mass separated and detected in the
spectrometer. Filaments used in partial pressure analyzers may be made from a variety
of refractory metals and alloys, each having unique characteristics that become
important when performing critical work. Pure tungsten filaments when heated emit
significant amounts of carbon monoxide and carbon dioxide. Iridium filaments which
have been treated with thorium ("thoriated") are selected for use when high partial
pressures of oxygen will be present. It should be noted that thoriated iridium filaments
are susceptible to contamination from hydrocarbons. When this occurs, the electron
emission from this type of filament will be degraded. Other special purpose filaments
may be made of rhenium or lanthanum hexaboride.
Figure 13.1 The components in the ionizer of a partial pressure analyzer.
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Not all of the molecules which enter the ionizer exit the other side as ions. For a fixed
electron accelerating potential (70 V, for example), the probability of ionization is gas
specie dependent.
In addition to having different ionization probabilities for a given electron energy, the
response of each gas to electrons of differing energies is unique.
Fortunately, the ion production by electron impact for each gas specie is directly
proportional to the partial pressure of that gas specie.
Acceleration of Ions
Once positive ions are created in the ionizer, they are accelerated towards the mass
separator by an electric field applied to a set of apertures called the accelerating
aperture or entrance aperture. The degree to which ions are accelerated is a function of
the mass of the ion, the charge on the ion, and the accelerating voltage (V
a
) on the
entrance aperture. For singly charged ions, accelerated by a fixed voltage, V
a,
the
velocity to which the ions are accelerated is greatest for ions of low mass and lowest for
ions of high mass. In some mass spectrometers the accelerating voltage is ramped from
an initial low value to a higher value in order to aid in mass separation. Typical values
for the bias on the acceleration apertures are from 1 to 5 kV DC.
Mass Separation
Of the two mass separation techniques that will be covered in this unit, (quadrupole
electric field separation and magnetic sector separation), the magnetic sector is the
easiest to understand. In this method, ions emerging from the ionizer and accelerated
by the entrance slit enter a strong magnetic field (usually generated by a permanent
magnet). Under the influence of this magnetic field the trajectory of the ions is bent
according to the formula given in 13.1 below.
Sample Problems:
13.1 Calculate the radii of curvature for common atmospheric gases and water vapor
using the following criteria: magnetic field strength = 0.1 Tesla, accelerating
potential = 2000 V, all species are singly ionized.
13.2 Explain why it is the mass-to-charge ratio that determines the trajectory of an ion
in a magnetic sector mass separator rather than simply the mass of the ion.
magnet
M-1
z
M
z
M+1
z
ion
source
slits
1-5 kV
detector
r
Figure 13.2 Simplified drawing of the components in a magnetic sector mass
spectrometer.
As is suggested pictorially in figure 13.2, for a given set of conditions (accelerating
potential and magnetic field strength) only ions of a specific mass-to-charge ratio will
have the correct trajectory to pass through the slits just before the detector. Ions that
have a larger mass-to-charge ratio are less strongly deflected by the magnet, and swing
wide of the exit aperture. Similarly, ions with a lower mass-to-charge ratio have their
trajectories more severely curved by the magnetic field, and also are prevented from
reaching the detector by the exit aperture.
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In practice, to achieve mass selection by varying acceleration potential alone would
require a power supply capable of generating stable sweep voltages across a large
voltage range. The practical solution to this problem is to divide the mass-to-charge
range into two or three segments, and to use multiple permanent magnets to augment
the magnetic field strength. In this method the atomic mass unit (AMU) range of from 2
to 50 is scanned using a 0.1 Tesla permanent magnet, while the 50 to 300 AMU range
is scanned using a 0.25 Tesla magnet. In some expensive mass spectrometers
electromagnets are used instead of permanent magnets. The electromagnets in these
units have variable field strength, based upon the amount of electric current passed
through the coils of the electromagnet.
Quadrupole mass spectrometers use AC and DC electric fields to perform separation of
ions based upon the mass-to-charge ratio.
Figure 13.3 Simplified representation of the electrical circuits supplying AC and DC
voltages to the two pair of rod-shaped electrodes in a quadrupole mass separator.
As the name suggests, there are four "poles" or rod-shaped electrodes in a quadrupole
mass spectrometer that function to separate ions based upon the mass-to-charge ratio
of the ions. The poles of the spectrometer are paired electrically as shown in figure
13.3. One set of opposing electrodes are biased positively using a DC power supply,
while the other two are biased negatively by another DC power supply. A radio
frequency (RF) alternating current is superimposed on the DC voltage applied to both
sets of electrodes. The rods are held in precise position with respect to one another and
the other components of the spectrometer by precision machined ceramic discs. Each
disc has four holes in it to support, align and electrically insulate the four rods. These
ceramic supports allow the rods to be accurately repositioned in the spectrometer
following cleaning.
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The set of rods biased positively by a DC power supply acts as a "high-pass" filter
collecting ions having a mass-to-charge ratio greater than a specified value. The Other
set of rods, biased negatively, by the other DC power supply acts as a "low-pass" filter,
and collects ions having mass-to-charge ratios less than a certain value. Together the
two sets of rods provide an effective means for allowing only the ions having the desired
mass-to-charge ratio to be counted at the detector.
+ +
-
-
U + V(cos t) ω
U - V(cos t) ω
−U
+U
+/- V
+/- V
Figure 13.4 A positive ion of low mass-to-charge ratio oscillating under the influence of
applied AC and DC electric fields.
As shown in figure 13.4 ions having a low value of mass-to-charge are strongly affected
by the radio frequency AC current superimposed upon the positively biased rods. The
amplitude of oscillation for these ions grows rapidly as the ion moves through the mass
separator until the ion strikes one of the rods. Ions which impact a rod lose their charge
and cannot be detected. Ions of high mass-to-charge ratio are "filtered' by the effect of
the rods which have a negative DC bias as shown in figure 13.5. These more massive
ions are much more sluggish in their response to the RF AC electric field than the lighter
ions. The net effect of the negative bias on the more massive ions is to gradually drag
them towards one of the negatively biased rods as the ion passes through the mass
separator. Again, once an ion collides with an electrode, it loses its charge and cannot
be detected.
By choosing appropriate values for the acceleration potential, and the DC and AC bias
potentials, a very effective mass filter can be created. In practice, one parameter
(accelerating voltage, RF or DC potentials) is varied in time, and ion current is recorded
for each mass-to-charge ratio.
Figure 13.5 A positive ion of high mass-to-charge ratio oscillating under the influence of
applied AC and DC electric fields is strongly attracted to the rods having a negativeDC
bias.
Detection of Ions
For either type of mass separator (magnetic sector or quadrupole) the ions which are
not "selected out" impact the ion detector, where they generate an electrical signal. This
signal is amplified electronically and sent on to pulse counting circuitry, and finally
emerges as intensity (ion current) versus mass-to-charge ratio. Several types of ion
detectors are used in commercial mass spectrometers. Simple, inexpensive units often
employ a Faraday cup, while the more sensitive, higher-end units use either a Faraday
cup/secondary electron multiplier combination or a channel electron multiplier. The
sensitivity of ion detectors is typically specified in terms of electrical current per
pressure, such as amps/Torr. Values for the sensitivity of detectors can range from 4 x
10
-6
to 1.0 x 10
-5
Amp/Torr, assuming nitrogen ions. For an operational pressure range
of from 10
-2
to 10
-12
Torr, the current range that a typical detector must be able to register
is from 10
-6
to 10
-17
amps.
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anode
first dynode -V
Faraday cup
ions from
mass-selector
Figure 13.6 .Ion detector for a mass spectrometer. This is a diagram of a combined
Faraday Cup and secondary electron multiplier type detector.
In a compound detector, as pictured in figure 13.6, the broad range of possible ion
currents is handled by operating either the Faraday cup alone or in conjunction with the
secondary electron multiplier (SEM).
Ion currents from 10
-6
to 10
-12
amps may be measured using the Faraday cup alone. Ion
currents of from 10
-12
to 10
-17
are measured by grounding the Faraday cup and applying
a negative bias (-1 to -3 kV) to the resistor chain attached to the dynodes of the
secondary electron multiplier. During operation of the SEM the initial low ion current (10
-
12
to 10
-17
amps) is amplified by electron multiplication. Ions striking the grounded
Faraday cup create secondary electrons upon impact. These secondary electrons are
electrostatically attracted to the first dynode of the SEM. These dynodes are often
fabricated from material which readily emits many electrons during bombardment with
electrons. Copper-beryllium alloys (Cu 2-4 %, Be) which have been heat-treated to
create a beryllium oxide surface film exhibit this favorable electron emission
characteristic. Electrons created at the first dynode are attracted to the second dynode
by the applied electric field, and upon striking the surface of the second dynode, again
generates a cascade of secondary electrons for every arriving electron. In this manner,
signal gains of from 10
5
to 10
6
may be achieved.
Sample Problems:
13.2 Describe the differences in the principle of operation between magnetic sector
and quadrupole mass spectrometers.
13.5 What characteristic of Cu-Be alloys make them a good choice for the
dynodes of a secondary electron multiplier?
Another type of ion detector is the channel electron detector. These detectors achieve
gain by the same mechanism as the SEM previously described: electron multiplication.
electrometer
-HV
positive
ions
Figure 13.7 Channel electron multiplier and associated electronics.
In the channel electron multiplier, an ion incident upon the funnel shaped cathode
creates a cascade of secondary electrons that are electrostatically attracted down the
curved electron multiplier tube. The tube is made of a special glass containing lead
oxide and bismuth oxides. The inherent high resistivity of the glass provides an
electrical resistivity similar to that made by the chain of resistors in an SEM. The
channel multiplier tube is curved for two reasons: it prevents positive ions from traveling
backwards through the tube, and to maximize the effective number of "dynodes" for
electron multiplication. An advantage of this type of electron multiplier over an SEM is
that the channel electron multiplier can withstand repeated exposure to air. Both types
of electron multipliers have a finite useful lifetime, which is generally on the order of one
to two years. Be aware that stray magnetic fields (from ion pumps, for example) can
affect the trajectory of electrons within either type of electron multiplier.
The culmination of this complex series of steps (ionization, acceleration, mass
selection and ion detection) is the representation of the data as signal intensity as a
function of mass-to-charge ratio. Almost universally this data is output to a CRT screen
as a graph which may look something like that presented in figure 13.8.
Operation of Partial Pressure Analyzers
Care should be taken in the use of partial pressure analyzers. Instruments of this type,
even the "low-end" units are quite expensive and easily damaged by misuse. Installation
of an analyzer on a vessel should be made with the following questions in mind:
1) What characteristic of the vacuum environment am I attempting to measure?
(qualitative versus quantitative data).
2) What will the maximum pressure be in the spectrometer?
3) What mass range of gas (AMU) is expected?
4) Will the vessel and the spectrometer need to be baked-out? If so, at what
temperature?
5) Is contamination of the partial pressure analyzer possible? How can the
possibility of contamination be minimized?
6) Will the resolution of the partial pressure analyzer be sufficient for the
application.
Definition of the Peak Resolving Ability of a Mass Spectrometer
Resolution in a mass spectrometer may be broadly defined as the ability of the
instrument to clearly identify signals from ions of two similar mass-to-charge ratios.
There are several accepted means for analytically defining the resolution of a mass
spectrometer. For adjacent peaks M
1
and M
2
in a spectra, if the intensity in the valley (h)
between the peaks is less than 10% of the value of the intensity at peak maxima (H) the
resolution is defined as M
1
/ (M
1
and M
2
) (see figure 13.9).
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H
h
R =
M
1
M
2
− M
1
for H > 10 h
M
1
M
2
Figure 13.9 Definition of resolution for two adjacent peaks observed in a mass spectra.
For a single peak in a mass spectra, the resolution may be defined as the quotient of
the mass-to-charge ratio at maximum peak intensity divided by of peak width at half
maximum intensity, as shown in figure 13.10.
H
M
∆M
H/2
R =
M
∆M
Figure 13.10 Definition of resolution for a single peak observed in a mass spectra.
Differentially Pumped Partial Pressure Analyzers
Some vacuum processes are conducted at pressures above the recommended value
for operation of partial pressure analyzers. Examples of such processes include: sputter
deposition and plasma etching. It may be very useful to diagnose processes such as
these using partial pressure analysis. This is typically done by limiting the flow of
process gases into the spectrometer and by adding a dedicated high vacuum pump to
evacuate the spectrometer. Such a system is referred to as a "differentially pumped"
partial pressure analyzer. A drawing of such an instrument is presented in figure 13.11
The purpose of the aperture between the analyzer and the vessel is to limit flow of gas
through the spectrometer. The second (often variable opening) aperture's function is to
allow control of the pumping speed of the high vacuum pump dedicated to the
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spectrometer. Fixed apertures may conveniently be made by drilling a small hole in a
copper disc which is then substituted for the copper gasket in the flange joint.
Small turbomolecular pumps are often selected for this application, as they generally
produce very little contamination due to backstreaming, do not create a strong magnetic
field (as do ion pumps). Differentially pumped partial pressure analyzers find much use
in the semiconductor industry. It is of much economic importance to that industry to
have the capability to accurately determine the "endpoint" of a plasma etching process.
This is accomplished by monitoring the partial pressures of the gaseous by-products of
the etching process. Similarly, in the process of thin film deposition by sputtering, it is
occasionally very useful to monitor the purity of the process gas and any contaminants
due to outgassing, permeation or leaks.
Sample Problem:
13.6 Calculate the resolution of a mass spectrometer if the width of a peak at half
maximum is 0.1 AMU for a peak centered at 50 AMU.
auxilliary
pump
RGA Vacuum
vessel
aperture 1
aperture 2
Figure 13.11 Cross-section of a differentially pumped partial pressure analyzer with two
flow limiting apertures and an isolation valve.
Residual Gases in Vacuum Vessels - Their Characteristics and Probable
Sources
Quite often the technique of partial pressure analysis is applied to a vacuum system
which is exhibiting out of normal performance (high base pressure, frequent filament
burnout for systems with heated filaments, poor film adhesion in deposition systems,
etc.). Interpretation of data from a partial pressure analyzer can be made significantly
more straight-forward if some information about the recent history of the vacuum system
under study is known. As with most fault-finding techniques is it often useful to start with
the most recent occurrences (vessel modifications, significant deviations from normal
operating procedures, etc.) and work backwards. Below are some questions which may
provide insight into the sources of residual gases in a vacuum vessel.
1) Has any fixturing internal to the vacuum vessel been modified or replaced.
Reference Information for Partial Pressure Analysis
Assuming a partial pressure analysis is performed on a system, the first step in turning
the data into meaningful information is to gain a qualitative understanding of the nature
of the gases in the system. As an aid in this process table 13.2 provides some
information relevant to specific mass-to-charge ratios which may show peaks of varying
intensity in a partial pressure analysis.
Mass-to-
charge ratio
Suspected gas
specie
Comments
2 hydrogen hydrogen is often the major gas load in UHV
systems due to permeation through
stainless steel vessel walls. Dissociation of
water and hydrocarbons may also give a
peak at 2.
4 helium May be present following leak checking.
Helium also permeates elastomeric seals.
16 oxygen Singly ionized monatomic oxygen may be
present due to dissociation of water, or from
an air leak.
18 water In the high vacuum range water vapor is the
largest contributor to the gas load. If water is
present peaks should also be seen at 16
and 17.
19 fluorine May indicate the decomposition of
fluorinated hydrocarbons in the vessel.
20 neon May be observed in UHV systems with ion
pumps.
28 nitrogen Diatomic nitrogen, single ionized. If nitrogen
is present, an air leak may be the cause. A
peak at 14 for monatomic nitrogen, singly
ionized should also be present.
32 oxygen Diatomic oxygen, singly ionized. The
presence of this specie may indicate an air
leak, especially if a stronger peak at 28
(nitrogen) is observed.
40 Argon Argon may be present due to an air leak.
Check for oxygen and nitrogen.
44 carbon dioxide May be generated from heated tungsten
filaments, as with CO.
45 isopropyl alcohol May be a residue from a cleaning process
used on a component in the vessel,
especially in tapped holes.
58 acetone See comments for isopropyl alcohol.
95 trichloro-ethylene See comments for isopropyl alcohol.
Another aid in gaining useful information from partial pressure data are the reference
library and spectra search functions that are available on many modern computer-based
instruments. With these features one may compare spectra obtained by the instrument
to known reference spectra that exist in the library. Some of the correlation functions
also provide the means to analytically describe the quality of the match between the
data and the reference spectra. Some of the computer controlled instruments will also
permit automated periodic sampling and will generate a history of the partial pressures
of selected gases as a function of time.
Understanding how to interpret scans from a mass spectrometer is a valuable skill. Use
of the process of elimination will quickly provide a very short list of possible gas
identities. The following simple rules will help in establishing which gases are likely
present in the vacuum system under analysis:
1) Start with the most intense peak in the spectra. Assume that this peak is due to a
singly ionized atom or molecule. Refer to figure 13.12 for this example.
2) Note the mass-to-charge ratio of this most intense peak. The molecular weight (if
the single ionization assumption is correct) cannot be more than the value of the
mass-to-charge ratio for this peak.
3) Refer to the periodic table, using the listed atomic weights, write down the
possible combinations of elements that have atomic weights that sum to equal
the mass-to-charge ratio of the most intense peak in the scan. In our example,
the mass-to-charge ratio is 18. the possible combinations of elements whose
atomic weights sum to 18 are: 2H + O; N + He; 2Be, B + 2He; and C + He
+2H. Of these, the only likely possibility is 2H + O, otherwise known as H
2
O.
4) Look at the peaks associated with the major peak which have lower mass-to-
charge ratios. Determine if it is possible that these peaks may be molecular
fragments of the major peak. In this example, some of the H
2
O has been
dissociated in the ionizer of the partial pressure analyzer to create the fragments
OH and O, which have mass-to-charge ratios of 17 and 16 respectively.
5) Mark those peaks that have tentatively been identified, and repeat steps 1
through 5 for the remaining peaks in the spectra.
As was mentioned in the section describing the ionizer, molecular gases, such as water,
carbon dioxide, and oil vapors will almost certainly become dissociated (fragmented)
during the ionization process. Each of the fragments will become positive ions which will
be accelerated, mass analyzed and detected. The peaks on a mass spectra that are
due to the fragments of a disassociated molecule are often referred to as a "cracking
pattern". Understanding this concept will aid greatly in both qualitative and quantitative
interpretation of mass spectra In the following table are listed the fragments and mass-
to-charge ratios for several common gases.
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Sample Problem:
13.7 The mass spectra for each of the cracking patters listed in equation set 13.2 is
provided in figures 13.12 through 13.14. For each of these mass spectra, identify the
peaks by writing the ion next to the peak it corresponds to.
High resolution mass spectrometers can also discriminate between the isotopes of gas
species. An isotope of an element has a different number of neutrons in its nucleus than
other isotopes of the same element. This difference in nuclear structure creates a
slightly different atomic weight. This effect can be seen in the mass spectra of the noble
gas, Argon(see equation 13.3, and figure 13.15)
Quantitative Analysis of Mass Spectra
As was suggested earlier in this unit, two types of information about the residual gases
in an evacuated vessel may be gained through partial pressure analysis: identification of
species present (qualitative information) and the amount of each species (quantitative
information). Inexpensive mass spectrometers typically do not have the resolving power
required to clearly identify overlapping peaks (carbon monoxide and nitrogen, for
example) and are typically not used for quantitative analysis of mixtures of gases. For
spectra of mixtures of gases which do not have overlapping peaks, one may use the
following steps to perform a rough quantitative measurement:
1) Identify all of the peaks in the mass spectra.
2) For each peak obtain from the instrument's manual the sensitivity of the
instrument for each gas specie (S), as well as the detector gain for each
specie (G).
3) Calculate the partial pressure of each gas using the formula provided in
equation 13.5.
Analysis of mass spectra of gas mixtures in which peaks overlap, such as carbon
monoxide and nitrogen are somewhat more complicated. The measured intensity of a
peak will be the algebraic sum of the intensities of the two peaks which are overlapping.
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One must use reference data to establish the ratios of peak intensities for the peaks
from each of the gases in the mixture. Using the subsidiary peaks which do not overlap,
estimate the partial pressures of the gases in the mixture, using the technique just
described. Once this initial estimate of the partial pressures of each gas is in hand, use
the gain and sensitivity to solve for the total ion current for each component that
contributes to the intensity of an overlapped peak.
Laboratory Exercise 13.1:
Qualitative identification of species in mass spectra.
Equipment required: none.
Procedure: Using the table of cracking patterns for materials commonly used in
vacuum technology (Appendix X), identify the constituents in each of the following mass
spectra
Procedure: Review the installation and operating guidelines for the partial pressure
analyzer you have selected for this experiment. After reading and understanding the
procedures in these instructions, inform the laboratory instructor of your procedure for
installation and operation of the instrument. With his approval, begin the installation of
the spectrometer head onto the vacuum vessel. Attach the mass spectrometer head to
the vacuum vessel with an isolation valve between the two. Attach the reference leaks
to the vessel. Connect the spectrometer to the control unit as suggested by the
manufacturer. Evacuate the vacuum vessel to a pressure of 10
-5
Torr or less. Following
the manufacturer's operating procedures, obtain the partial pressure analysis of the
residual gases in the vessel. Repeat the measurement at five minute intervals for an
hour to see how the partial pressures of gases in the vessel change during operation of
the high vacuum pump. Following this series of measurements, open one reference
leak briefly (1-2 seconds), and observe the mass spectra. Note any changes in the
mass spectra. Wait until the mass spectra returns to a "baseline" reading similar to that
prior to the injection of gas from the reference leak. Repeat the controlled injection of
known gas with the remaining reference leaks that are attached to the vessel. Following
completion of all experimental work, turn off the partial pressure analyzer following the
manufacturer's suggested procedures. Shut down the vacuum system safely, and vent
the pumps and vessel. Write a laboratory report of your procedures and findings,
including the data collected.
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Chapter 14: Thin Film Deposition Processes
Up to this point in the course almost all of the emphasis has been placed on the
techniques involved with certain activities related to achieving and characterizing a
vacuum environment. Now we will turn our attention to the reasons for working so hard
to achieve a vacuum: the processes that are conducted in this environment. The
deposition of thin films has made a tremendous impact on the level of technology we
utilize in our daily lives. Thin film coatings provide enhanced optical performance on
items ranging from camera lenses to sunglasses. Architectural glass is often coated to
reduce the heat load in large office buildings, and provide significant cost savings by
reducing air conditioning requirements. Microelectronics as we know them today would
not be possible without vacuum technology. Microcircuits fabricated in multi-step
vacuum processes are used in devices ranging from wrist watches to microwave ovens
to automobile ignition and monitoring systems. The computer industry would not exist if
it were not for vacuum technology. In 1990 the world market for integrated circuits was
$50 billion; and for the electronic devices which rely on these microcircuits, $0.9 trillion.
Decorative coatings applied to jewelry and plumbing fixtures is another large industry
based upon vacuum technology. Many of the components of plumbing fixtures are
manufactured by depositing thin films of chromium onto injection molded plastic parts.
The useful life of tool bits has also been increased by the application of thin films that
are chemical compounds. Tool steel cutting tools used in lathes and mills are often
coated with the chemical compound titanium nitride to reduce wear of the cutting edges.
The deposition of thin films composed of chemical compounds may be performed in
several ways. Co-deposition is a technique in which vapors of two different materials
are generated simultaneously. These two vapors condense together, forming an alloy or
compound. Other techniques for deposition of compounds include thermal evaporation
of the compound (as is performed for salt coatings), sputtering of the compound, and
reactive sputtering or evaporation. In the reactive processes, atoms of the evaporant
(typically a metal) chemically react with gas species which are intentionally injected into
the process chamber. Each of these processes will be described in detail.
Thin Film Deposition in a Vacuum Environment
Early references to the science of thin film deposition include the research conducted by
Michael Faraday in 1857. In this series of experiments, Faraday created thin metallic
films by exploding metal wires in a vacuum vessel. Historically, the techniques for thin
film deposition have evolved in approximately this order: thermally induced evaporation
(by electrical resistance heating, induction heating, and electron beam heating),
sputtering (diode, triode, magnetron, ion beam), arc processes, and most recently, laser
ablation.
In general, there are three steps in any physical vacuum deposition (PVD) process:
creation of an evaporant from the source material, transport of the evaporant from the
source to the substrate (item to be coated), and condensation of the evaporant onto the
substrate to form the thin film deposit. There are two reasons why this process is best
conducted under vacuum: 1. the process of evaporation involves significant amounts of
heat, if oxygen were present, any reactive metal would form oxides; 2. collisions with
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gas molecules during the transport of evaporant from source to substrate would reduce
the net deposition rate significantly, and would also prevent growth of dense films.
source
evaporant
substrate
vacuum vessel
Figure 14.1 The three basic steps in any physical vapor deposition process:
evaporation from the source, transport of evaporant, and condensation of the
evaporant.
Upon arrival at the substrate, evaporated material condenses on the substrate in a
complex sequence of events that determine many of the physical properties of the
deposited film. The steps in the growth of thin films are generally referred to as
nucleation and growth. In nucleation, the atoms and molecules which are arriving
(called ad atoms) at the surface lose thermal energy to the surface, and the surface
absorbs that energy. Depending on the amount of thermal energy the ad atoms and the
surface have, the ad atoms move about on the surface until they lose the thermal
energy required to move about the surface (referred to as Adam mobility). As nuclei
continue to form, the film grows into a continuous sheet covering the substrate.
Chemical interactions between the ad atoms and the surface determine the strength of
the bond between the film and substrate. Gold, for example, does not form a chemical
bond with silicon dioxide, and therefore, the adherence of gold films on glass are very
weak. Improvement of this adhesion may be made by first depositing a thin (500Å thick)
"Binder" layer of chromium or niobium, then depositing the gold over the binder layer.
Chromium and niobium do form chemical bonds with the silicon dioxide in glass, and
also form metallic bonds with the following gold layer. Once a few monolayers of
evaporant have condensed on the substrate, the film continues to grow in thickness as
if the entire substrate were made of the material being deposited. During film growth the
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microstructure of the deposited film will be developed. This microstructure may be
described in terms of grain size, orientation, porosity, impurity content, and entrained
gases. Normally, vacuum deposition processes are selected over other processes
(electrochemical deposition, flame spraying, etc.) to achieve the following desirable
properties:
1) High chemical purity.
2) Good adhesion between the thin film and substrate.
3) Control over mechanical stress in the film.
4) Deposition of very thin layers, and multiple layers of different materials.
5) Low gas entrapment.
For each of the vacuum deposition process described in this unit, keep in mind
the ultimate goal is to provide a means for depositing a thin film having the required
physical and chemical properties. The parameters one can control to achieve the
specified goals are:
1) Kinetic energy of the ad atoms.
2) Substrate temperature.
3) Deposition rate of the thin film.
4) Augmented energy applied to the film during growth.
5) Gas scattering during transport of the evaporant.
By varying these parameters one can generate thin films of a given material that have
different mechanical strength, adhesion, optical reflectivity, electrical resistivity,
magnetic properties and density.
Thermally Induced Evaporation
In this process, heat is input into the source material (often called the charge) to create
a plume of vapor which travels in straight-line paths to the substrate. Upon arrival at the
substrate, the atoms, molecules, and clusters of molecules condense from the vapor
phase to form a solid film. The heat of condensation is absorbed by the substrate. On a
microscopic scale the localized heating from this process can be enormous. It is
common, in the development of metal coating techniques for thin cross-section plastic
parts, to melt substrates during the initial deposition runs. With experience, one can
select source-to-substrate distances and deposition rates which will allow coating of
temperature sensitive substrates without melting.
There are several methods by which heat can be delivered to the charge to cause
vaporization: electric resistance heating, induction heating, and electron beam heating.
Deposition of thin films by laser ablation and cathodic arc could be grouped in this
section with thermal processes, but there are some unique characteristics of these
techniques which are beyond the simple model of thermally induced evaporation. For
this reason we will cover these two deposition techniques separately.
Resistance Evaporation
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Evaporation of material by electrical resistance evaporation is very likely the easiest of
the thermal evaporation techniques. Quite simply, in a vacuum environment the charge
(which may be an elemental metal, an alloy, a mixture or a compound) is heated to
become a vapor. Low voltage, high current (typically 10 to 40 VDC, 1 to 10 amps)
power is brought into the vacuum vessel using electrical power feed throughs. The
electrical power is passed through a filament which is in intimate contact with the charge
(see figure 13.1). Filaments are often heated to 1000 to 2000° C. A materials
requirement for efficient thermal evaporation is that the charge have an appreciable
vapor pressure at the operating temperature of the filament.
All materials evaporate, even at room temperature. The addition of heat simply
accelerates the process. At a specified temperature the pressure of the vapor emitted
by a material is called the equilibrium vapor pressure.
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Figure 14.2 Equilibrium vapor pressure as a function of temperature
for five metallic elements.
The relationship between the equilibrium vapor pressure of a material and the
temperature generally follows the pattern shown in figure 14.2. Cadmium, for example
has a significantly higher vapor pressure at all temperatures than any of the other
metals shown in this figure. Similarly, Rhenium has the lowest vapor pressures of these
five metals at all temperatures. Vapor pressure curves for many of the metals that are
used as charge material are provided in Appendix Y. Numerical values for the vapor
pressure of a given metal at a specified temperature may be read directly from vapor
pressure curves like that shown in figure 14.2. Cadmium, for example, has a vapor
pressure of approximately 5 x 10
-3
Torr at 500 K (227 °C). By comparison, rhenium does
not achieve a vapor pressure of 5 x 10
-3
Torr until it is heated to almost 3000 K (2730
°C)!
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Sample Problem:
14.1 Use figure 14.2 to determine the vapor pressure of aluminum and silver at 1300
°C.
14.2 Equilibrium vapor pressure curves, such as that presented in figure 14.2 are
useful for thin film evaporation. What other uses for this information can you imagine for
vacuum technology?
Thermal evaporation is typically conducted under high to ultra-high vacuum conditions.
Bell jar type vacuum vessels are often selected to perform this process on a laboratory
scale, as they offer a great deal of design flexibility as well as the advantage of direct
observation of the process at a very reasonable cost.
Filaments are usually made of refractory metals such as Tungsten, Tantalum, or their
alloys. Some of the requirements for a good filament material are:
There exists a great variety of filament configurations ranging from straight and coiled
wires to "boats" and boxes (see figure 14.4). Each type of filament is designed for a
unique application. Coil filaments made of refractory metal strands are loaded with the
charge by hanging small sections of wire made of the charge material on the coil. Upon
heating, the charge melts, and wets the coil. Further heating causes the evaporation of
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the charge from the coil filament. Very rapid heating of a filament with hanging charge
material may cause the charge to melt locally and fall off the filament.
Some skill is required to attain thin film deposits from run to run which have consistent
thicknesses. One technique that helps is to carefully weigh the charge for each run,
keeping the weight the same, and operating the filament so as to completely evaporate
the charge each run.
There are some inherent disadvantages of resistance heated thermal evaporation that
should be kept in mind when selecting a deposition technique:
1) The source may generate impurities which may co-deposit in the condensing thin
film.
2) Accurate control of the deposition rate is difficult.
3) The composition of alloy thin films deposited may differ from that of the charge
material (especially if the elements in the alloy have markedly different vapor
pressures).
4) The amount of material which may be evaporated per run is limited.
5) The substrate will experience heating due to radiant energy from the
source.
helical
filament
conical
basket
flat boat
with dimple
trough style
boat
Molecular beam epitaxy (MBE) is a modern application of electric resistance heated
thermal evaporation. This technique is used to accurately deposit ultra-high purity
semiconductor materials with specified crystallographic orientations between the layers
of different materials. The vacuum environment for MBE is typically in the extreme ultra-
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high vacuum range (10
-9
to 10
-11
Torr). Evaporation of materials in an MBE vacuum
system is conducted using a special thermal evaporation source called a Knudsen cell.
Sample Problem:
14.3 Define "Epitaxy". How is an epitaxial thin film different than other films that are
deposited on a crystalline substrate?. {Hint: see the dictionary}.
Induction Heated Thermal Evaporation
In this technique an electric current is induced to flow through an electrically conductive
charge material by the application of radio-frequency (RF) alternating current. The RF
current is generated by a power supply which may range in output from 1 to 50
kilowatts, depending on the size of the charge. The AC current is flowed through the
copper coil which surrounds a refractory ceramic crucible.
induction coil
crucible
Figure 14.6 A crucible and coil used for induction
heating for thermal evaporation.
For all forms of thermal evaporation which employ ceramic crucibles, selection of the
appropriate material for the crucible is vitally important. If the incorrect selection is
made, the charge may chemically react with the crucible, ruining both and possibly
harming other components of the vacuum deposition system.
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1) The charge must be electrically conductive.
2) RF power supplies and matching networks can be expensive and quite large.
3) Chemical interaction between the charge and crucible can occur.
Sample Problem:
14.4 What property of materials makes thermal evaporation possible? How can
one manipulate this property to deposit thin films of materials?
Electron Beam Evaporation
In this thermal evaporation process, a beam of energetic electrons generated from a
heated filament supplies the thermal energy to evaporate of the charge. There are
several variants of the electron beam evaporation process. All electron beam thermal
evaporation systems have an anode (biased positively) and a cathode (either grounded,
or biased negatively with respect to the anode). The two divisions we will cover are self-
accelerated and work-accelerated electron beam evaporators. In the work-accelerated
scheme, electrons emitted from a heated tungsten filament are attracted to the charge
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material by an applied high voltage bias (10- 40 kV DC). A focusing aperture aids in
minimizing spurious heating of the vessel interior by off-axis electrons.
filament
focussing
apreture
Crucible
charge
electrons
filament
heating
power
supply
accelerating
voltage
vacuum vessel
Figure 14.7 A work-accelerated electron beam evaporator.
Several configurations of the work-accelerated electron beam evaporator have been
designed and used. In figure 14.8 electrons emitted from the heated cathode are
caused to travel in an oval shaped path by the electrostatic negative field applied to the
cylindrical focusing electrode. Water cooling is provided via concentric tubes connected
thermally to the bottom of the charge. This design has a distinct advantage over that
shown in figure 14.7: the evaporant may be directed at a substrate placed above the
source without interference by a focusing aperture or filament. Additionally, the focusing
aperture and filament do not become heavily overcoated with evaporant.
Some work-accelerated electron beam evaporators use electromagnetic coils to steer
and focus the electron beam as shown in figure 14.9. This system provides several
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advantages: source utilization can be controlled and maximized by rastering the
electron beam; and the effective source size can be made large by rastering the beam,
improving the thickness uniformity and coverage of the substrate. Note that in this
design the electrons emitted from the filament impact the backside of the cathode,
heating it so that it will in turn emit electrons. The cathode area which emits electrons is
hemispherical, which improves the three-dimensional uniformity of the electron beam
emitted from it.
Self-accelerated electron beam evaporators use an auxiliary anode to impart kinetic
energy to the electrons emitted by the filament (cathode). This allows the evaporation of
charge materials which are not electrically conductive, such as calcium fluoride, which is
used to coat camera lenses for improved optical performance. A self-accelerated
electron beam gun configuration is presented in figure 14.10.
As was the case with the work accelerated electron beam evaporator in figure 14.9, the
self-accelerated gun has a set of electromagnets which may be used to scan the
electron beam during evaporation. Automated scan controls for both types of guns are
commercially available. These scan controls vary the current in the electromagnets so
as to sweep the beam in a "Lissajous" pattern, which is sinusoidal in two dimensions.
The majority of commercial electron beam evaporators are of the transverse design, as
shown in figure 14.11. These guns use a permanent magnet to steer the electrons
emitted from the cathode around 270°, and a set of electromagnets to raster the beam
on the charge material. Modern electron beam guns of the transverse design are
available with a rotating multiple pocket hearth, which allows deposition of up to five
different materials without venting the vacuum vessel. Some of these units also have
integral shutters which allow the gun to achieve a stable operating temperature before
deposition of the substrate is begun.
Since the process of electron beam evaporation is normally performed under UHV
conditions, the evaporant generally travels in straight-line paths from the source to the
substrate. To aid in attaining uniform thickness coatings by this technique, substrates
are often mounted on "carousels" which rotate, and may wobble or spin individual
substrates each revolution (see figure 14.12). Deposition shielding is often placed inside
the vacuum vessel to facilitate cleaning. Multiple sets of shields allows for continuous
operation of the system: one set may be cleaned while the other set is in operation.
Instead of using a heated filament to generate the electrons for evaporation of a
material, plasma guns utilize the electrons which exist in high temperature gases called
plasmas. Two types of plasma sources exist: cold cathode and hot cathode.
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In the cold cathode design (figure 14.13), the cathode is biased negatively from -5 to -20
kV, the rest of the source is at ground potential. Following evacuation of the vacuum
system to its base pressure, process gas, such as helium is injected into the system. A
partial pressure of from 1 to 100 mTorr is dynamically maintained using flow controllers
at the process gas inlet, and adjustable apertures at the high vacuum stack inlet.
Electrons emitted by the cathode can ionize process gas atoms, creating positive ions.
These ions are accelerated to the cathode, and upon impact, generate a cascade of
secondary electrons. Many of these secondary electrons escape the source and are
accelerated and focused on the work piece as described in the section on work
accelerated electron beam guns.
14.5 A process gas is used in the operation of the cold cathode electron gun. What
effect may this have on the uniformity of the deposited coating?
14.6 List and describe the methods by which electrons can be generated to provide
the heat input to conduct a thermal evaporation under vacuum.
14.7 Describe the difference between work accelerated and self-accelerated
electron beam evaporation systems.
14.8 What would be the consequences of the presence of a partial pressure of
oxygen in an electron beam evaporation deposition system?
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14.9 Calculate the mean-free-path in a vacuum vessel at a pressure of
5 x 10
-7
Torr at room temperature. From this data what could you predict about
the path vapor atoms and molecules would take between the evaporation
source and the substrate? Can you think of situations in which this would be
advantageous or disadvantageous?
The Hot Hollow Cathode Electron Beam Gun
Following evacuation to the vessel's base pressure, a process gas, such as Argon is
flowed in a controlled manner through the cylindrical hollow cathode. Radio frequency
AC electric current is supplied to the cathode of the electron beam source from a low
voltage, high current power supply. Ionization of the process gas occurs as a result of
the applied electrical power. The cathode operating temperature is very high. For this
reason, the cathode is made of a refractory metal, such as tungsten, and must be
actively water cooled. Electrons emitted from the hot cathode and from the plasma may
be steered and accelerated as described for other electron sources.
Very high deposition rates may be achieved using the hot hollow cathode electron beam
source to perform thermal evaporation of materials. The electrons emitted from this
source may be work or self-accelerated, and may be magnetically steered.
Sample Problems:
14.10 The operating pressure in a thermal evaporation deposition chamber using
a hot hollow cathode electron source is approximately 50 mTorr, with Argon
as the process gas. Calculate the mean-free-path at this pressure, and
describe the effect this will have on the deposited thin film.
14.11 If you were interested in depositing a thin film of Zirconium Oxide, which of
the thermal evaporation deposition processes described would you select?
Why?
Figure 14.14 Hot hollow cathode electron beam source.
Safety notes for electron beam evaporation systems.
There are several potential dangers associated with the use of high energy electron
beams that operators should be aware of:
1) Radiation: A significant amount of X-rays are generated whenever high energy
electrons impact materials. The maximum energy of the X-rays is limited by the
electrical potential applied to accelerate the electrons. A radiation survey of the
exterior of the vacuum vessel, especially viewports, should be conducted to
insure a safe working environment.
2) If the beam control systems fail, high energy electrons may be directed to the
vacuum vessel walls or to internal fixturing, including water lines. The electron
beam sweep pattern should be monitored carefully at the beginning of a
deposition run, and periodically during the run. Leaving an electron beam system
to run unattended is not recommended.
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3) High voltage and high current are used in the process. The vacuum vessel and
power supplies should be electrically grounded.
Sputter Deposition of Thin Films
Unlike thermal evaporation, in which a material is heated to increase its equilibrium
vapor pressure, in sputter deposition, material is made to go into the vapor phase by the
physical interaction of particles impacting the source material (often referred to as the
"target"). There is a wide variety of sputtering techniques that are currently used to
deposit thin films for use in magnetic storage media (cassette tapes, computer discs),
optical thin films and microcircuits. The forms of sputtering which will be described in
this section are: diode sputtering, magnetron sputtering, RF sputtering, and ion beam
sputtering.
Diode Sputtering
A simple diode sputtering process system is presented in figure 14.15. Following
evacuation of the vessel to its base pressure to reduce contamination of the film by
atmospheric gases or water vapor, a process gas (most often Argon) is admitted into
the vessel. The pressure of this process gas is maintained at a user-selectable pressure
between 1-100 mTorr using a set of upstream mass flow controllers to regulate the
process gas mix, as well as a downstream throttling gate valve. Such a valve, situated
right at the turbpuump, is equipped with a controller that can perform fast, realtime
pressure control using a capacitance manometer as a sensor element. An electric bias
of from 500 to 5000 V DC is applied to the target. Electrons emitted by the target strike
process gas molecules in the vicinity of the target, and may cause the gas to become
ionized. The positive ions thus created are accelerated towards the cathode by the
applied negative bias. When the positive ions collide with the cathode, the kinetic
energy transferred is sufficient to eject atoms of the cathode material. Secondary
electrons, ions, and light (IR, visible, UV and X-rays) are also emitted during this
collision.
Power
supply
plasma
target
substrate
film
to
vacuum pumps
process gas
-V
The ejected (sputtered) material travels towards surfaces in the vacuum vessel where it
condenses to form films. Since the process gas pressure is in the range of from 1 to 100
mTorr, a significant amount of scattering of the sputtered material by the process gas
occurs. In this scattering, sputtered material loses its directional identity, may become
neutral, if it was ionized, and also loses kinetic energy. The visible glow that surrounds
the cathode during sputtering is called the "glow discharge" or plasma. Visible light of
the plasma has a color which is characteristic of the process gas and the material being
sputtered. This visible light may be used to monitor the chemical composition of the
plasma using a visible light spectrometer (details of this and other deposition monitoring
techniques will be covered later in this unit). One requirement of the diode sputtering
technique is that the cathode be electrically conductive. Elements and compounds
which are insulators can be sputtered by other techniques.
Depending on the composition of the cathode, this set of parameters should yield a
deposition rate of from 60 to 400 Å/minute. It should be noted that the rate of erosion of
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the cathode is expected to be greater than the rate of deposition at the substrate. Some
of the sputtered material condenses on the inner surfaces of the vacuum vessel (a good
reason for removable deposition shields), and some may react chemically with residual
gases to form volatile by-products which are pumped away.
Now that the concept of diode sputtering has been introduced, a few of the relevant
concepts that apply to all sputtering processes will be covered.
Sputtering Yield
The number of target atoms which are ejected from the cathode for every incident ion is
called the "Sputtering yield". The magnitude of the sputtering yield is a function of the
composition of the process gas and target material, the energy of the incident ions, and
the angle of incidence of the ions on the target. In general, the sputter yield is greatest
for the following set of conditions:
The noble gas Argon is the most commonly employed process gas for sputter
deposition processes, as it has a high sputter yield for most metals, is chemically inert
and non-toxic, and is relatively inexpensive (compared with the other noble gases
(Krypton and Xenon).
Figure 14.17 Sputter yields for metals sputtered with Argon as a function of ion energy.
It was shown, in the section on thermal evaporation of materials, that there exists a wide
range of vapor pressures for materials, the vapor pressure at a given temperature being
proportional to the evaporation rate. The magnitude of the variation in sputtering rate is
much smaller. Most metals exhibit sputter yields that are between 1 and 5 atoms per ion
when sputtered with Argon as the process gas. Prediction of the sputtering behavior of
metals is made relatively easy by this fact.
The deposition of thin films of metallic alloys and some chemical compounds may be
accomplished by sputtering. In general, the composition of thin films deposited by
sputtering will have the same overall chemical composition as the source (target) after
an initial equilibration period. In figure 14.18 is presented a simplified representation of a
two component alloy target, made of "A" and "B". Assume that the sputtering yields for
material "A" is higher than for material "B". Initially, as material is sputtered from this
compound cathode the vapor stream will be higher in concentration of "A" due to its
larger sputter yield. After some time, the surface of the cathode will become depleted in
"A" and more concentrated in "B". When this occurs, the thin films being deposited will
have approximately the same composition as the target.
A B
A B
initial
erosion of
target
equilibrium
erosion of
target
Figure 14.18 Sputtering alloy target made of "A" and "B".
Characteristics of Sputtered Material in the Vapor Phase
The vapor emanating from a sputter deposition source cathode contains neutral atoms,
ions (both positive and negative), electrons, neutral clusters of atoms and charged
clusters of atoms. Of these, the vast majority are neutral atoms. These atoms have
kinetic energies approximately 50 to 100 times that of neutral atoms generated from
thermal evaporation sources. This additional energy is thought to be the reason for the
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greater adhesion often observed for sputter deposited films over thermally evaporated
films of the same material. Due to the relatively high pressure in an operating sputter
deposition chamber, the mean-free path of sputtered species is short. The numerous
gas-phase collisions which the sputtered material suffers between the target and
substrate tend to reduce the amount of kinetic energy the depositing species have upon
arrival. This affects the ad atom mobility and therefore, the density and crystal structure
of the thin film. When sputtered atoms lose energy by gas collisions, they are said to be
"thermalized", that is, their kinetic energy is reduced to equal that expected for similar
atoms at the ambient temperature. A plot of the average distance sputtered tantalum
and aluminum atoms can travel before becoming thermalized as a function of process
gas pressure is presented in figure 14.19.
100 10 1 .1
.1
1
10
100
Argon Pressure [mTorr]
D
i
s
t
a
n
c
e
t
o
R
e
a
c
h
T
h
e
r
m
a
l
E
n
e
r
g
y
[
c
m
]
Ta
Al
Figure 14.19 Distance tantalum and aluminum sputtered atoms travel before becoming
"thermalized" as a function of process gas pressure..
Gas scattering has been used to some advantage in the coating of substrates having
complex geometries. By operating at the high end of the pressure range (70-100 mTorr)
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the maximum number of gas phase collisions may be induced, effectively reducing the
"directionality" of the deposition flux from source. Using this technique the insides of
tubes having aspect ratios (length: diameter) of 10:1 have been coated.
Now that some of the features common to all sputtering processes have been
described, we will return our attention to the methods of sputtering.
Triode Sputtering
In this process, an auxiliary set of electrodes are employed to enhance the electron
emission and increase the ionization of the process gas to generate a higher flux of
material sputtered from the cathode.
cathode
anode
vacuum
vessel
wall
mounting
flange
electrical power
feedthrough
insulator
cathode
power
supply
- V
Filament
power
supply
e-
e-
e-
e-
anode
power
supply
substrate
Figure 14.20 Detail of the components of a triode sputter deposition system.
The configuration of electrodes in a triode sputter deposition system is presented in
figure 14.20. Three separate power supplies are used: one to resistively heat a filament
to emit electrons, one to accelerate the electrons into the anode, and a high voltage
power supply to accelerate positive ions towards the cathode. High deposition rates
(>1000 Å/ minute) may be achieved using this configuration. Typical ranges for the
operating parameters are:
Due to the high currents and the intense plasma generated, active water cooling of the
sputter target and electron accelerating anode is required. The power supplies should
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be interlocked with cooling water flow monitors to interrupt power in the event of cooling
water loss. Triode sputtering systems have successfully been operated for deposition of
a wide range of materials, but the complexity of the process, and the difficulty in scaling
the process hardware up in physical size has limited its commercial application.
Magnetron Sputter Deposition
Another technique by which the deposition rate achieved over that of the simple diode
sputtering process may be increased is through the use of magnetic fields to constrain
the plasma close to the sputter target. Magnets situated beside or underneath the target
of a diode sputtering source can be used to constrain the electrons emitted from the
cathode to orbit in close proximity of the cathode. The probability that such an orbiting
electron will strike a process gas molecule, causing an ionization, is greatly increased
without the need to increase process gas pressure. The strength of the magnetic field
and placement of the magnets with respect to the cathode is crucial to the proper
operation of a magnetron sputter deposition source. In figure 14.21 the plasma ring
generated on a planar magnetron as well as a cross section of a magnetron showing
the magnetic structure under the cathode. Since the plasma is very localized, the
current density can be quite high at the cathode surface, generating a significant heat
load, which is very ineffectively dissipated by thermal radiation. Active water cooling of
the cathode provides the means to control temperatures, and to prevent
demagnetization of the permanent magnets under the cathode. A diagram showing
more detail of the design of a magnetron sputter source is presented in figure 14. 22.
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The DC magnetron sputter deposition source has found wide application in industry.
Deposition systems using this source range from desktop units for deposition of thin
films for prototype electronic devices to 150' long architectural glass coating chambers
which use arrays of magnetron sputter guns to coat 4 x 8 foot sheets of glass with
several layers in a single pass. Sputtering of all areas of the gun except the cathode is
prevented by the use of a ground plane shield (see figure 14.22). This shield, along with
the water cooling lines are electrically insulated from the high voltage applied to the
cathode through the use of ceramic insulating spacers. It should be noted that in some
magnetron sputter guns the water lines (often made from formed stainless steel
bellows) are used to conduct high voltage power to the cathode, and are electrically
isolated at the vacuum feed through. Under some circumstances, sputtering of these
biased lines may occur, which has been known to lead to rupture of the water cooling
lines within the vacuum vessel.
N
S
N
S
S
N
S S
water cooling circuit
magnetic field lines
sputtered
material
ground plane
shield
In addition to the planar magnetron configuration, there has been developed a
cylindrical or "Post" magnetron source which may be used to deposit thin films onto the
inner surfaces of tubes (see figures 14.22 and 14.23). Electromagnets may be used
instead of permanent magnets to generate the magnetic field which constrains the
electrons to orbit near the cathode. Using electromagnets it is possible to vary the
position of the plasma with respect to the cathode by changing the field strength in sets
of magnets. If the magnets are mounted outside the vacuum environment, as shown in
figure 14.
14.13 What factors influence the sputtering yield? How can the sputtering yield be
maximized?
14.14 What is the function of the process gas in diode sputter deposition?
magnets
water in
anode
cathode
water out
substrate
Figure 14.22. Detail of cylindrical post magnetron with electromagnets arrayed outside
the tube to be coated.
Sample Problem:
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14.15 What would be the consequences of an interruption in cooling water flow to a
magnetron sputter source?
Power Supplies for DC Sputtering
Electric arcs and the resulting high voltage spikes which may occur during warm-up of
DC sputter deposition sources can cause damage to power supplies not specifically
designed for this application. These arcs are caused by electrical charge buildup on the
thin naturally occurring oxide surface layer on the sputter cathode. As the target is
bombarded with ionized process gas, the thin oxide layer will sputter away, and arcs
should no longer occur. It is strongly recommended that only power supplies designed
for sputtering applications be used. These power supplies have internal arc suppression
circuitry which is intended to protect the power supply from damage.
logic
circuit
saturable
reactor
circuit breakers
-V
+V
Figure 14.24 Schematic of a power supply designed for DC sputtering applications.
One limitation to the deposition of thin films by any of the DC sputtering techniques just
covered is the requirement that the material to be sputtered be electrically conductive.
This restriction is not true for radio frequency AC sputtering. In this process an AC
power supply is used to apply a voltage that varies sinusoidally with time to a set of
electrodes in a vacuum vessel. Attached to one of the electrodes is a target made of the
material which is to be sputter deposited. If the material to be RF sputtered is an
electrical conductor, an electrical insulator is placed between it and the electrode. When
the electrodes are energized, a net negative bias occurs on the sputter target, which is
electrically insulated from the electrode. Substrates to be coated may be placed on the
other electrode, or mounted elsewhere in the vessel.
Materials which can be deposited by RF sputtering include metal oxides
(SiO
2,
Al
2
O
3
, ZnO, TiO
2
, etc.), mixed oxides (Indium-tin oxide, which forms a
transparent, electrically conductive film used to defrost wind shields), plastics, and
glass, such as pyrex. In general, the composition of the deposited thin film is very close
to that of the cathode, after an initial warm-up period.
As with DC sputtering, special power supplies are required for RF sputtering (see figure
14.26). The radio frequency AC power supply operates at a frequency of 13.56 kHz;
many of these supplies output power in the 0.5 to 10 kW range.
Sample Problem:
14.16 Why are special power supplies needed for DC and RF sputtering applications?
Ion-Beam Sputtering
In this sputter deposition process, special ion sources, such as that presented in figure
14.27 are used to generate ions and accelerate these ions towards a sputtering target.
The material sputtered from the target by impact of the energetic ions forms the coating
on the substrate.
process gas inlet
magnets
anode
cathode
accelerator grid
screen grid
Figure 14.27 Kaufman ion source used in ion-beam sputtering.
In the Kaufman ion source, electrons emitted from the heated filament (cathode) are
attracted to the anode, but the strong magnetic field prevents this. Gas molecules
impacted by the oscillating electrons become ionized, and, being positively charged, are
attracted to the negatively biased accelerator grid. By controlling the bias applied to the
screen and accelerator grids, a certain amount of ion beam focusing may be
accomplished.
Through control of the operating parameters, the ion current density and the ion energy
of the beam may be independently varied. Ions generated from a Kaufman source may
be used to clean surfaces or to deposit thin films as shown in figure 14.28.
Figure 14.28. Use of ion an ion source to remove material from a surface (left) or to
deposit a thin film (right).
Note that since the target used in the ion-beam sputter deposition technique is not a
functional electrode, that it need not be electrically conductive. For most commercial
applications the deposition rate one can achieve using ion-beam deposition is too low to
be practical. Ion-beam sources are used in laboratories to produce high purity thin film
coatings for research and development.
Sample Problem:
14.17 What are some of the advantages of RF sputter deposition versus DC
sputter deposition?
Arc Deposition
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High current, low voltage electric arcs, if constrained, may be used to create evaporant
for thin film coatings. There are several techniques used to control the arc, including use
of electrically insulating hearths, electrostatic, and magnetic constraints. Arc deposition
has been demonstrated to produce thin films having high densities and excellent
adherence to the substrate. Normally the arc process is conducted in a UHV
environment, but partial pressures of reactive process gases, such as oxygen and
nitrogen have been used to deposit metal oxide and nitride thin films. High deposition
rates, in the range of from 50 to 500 Å/ second have been reported for the arc
deposition process. One current drawback to this technique is the creation of macro-
particles along with the atomic and ionic evaporant. These macro-particles have
diameters ranging from 0.5 to 50 µm, and are co-deposited in the film, degrading the
film's appearance and physical properties. Significant effort has been expended to
eliminate the macro-particles without reducing the deposition rate. Once this is
accomplished, arc deposition may provide a viable alternative to electroplating.
Laser Ablation
High energy density pulsed laser beams have been used to deposit thin films of a
variety of elements, alloys and compounds. In this process, a laser source, external to
the vacuum vessel generates a beam which is focused, passed through a viewport and
impinges on a target within the vacuum vessel. Sufficient energy is generated to blast
(ablate) material from the surface of the target. This ablated material consists of neutral
atoms, ions, clusters of atoms and macro particles. The amount of material deposited
per laser pulse is very consistent, allowing one to accurately deposit films of a specified
thickness. The deposition rate is low compared to other techniques (electron beam
evaporation and sputtering, for example). The range of commonly used operating
parameters is given below:
Laser ablation, as a deposition technique is currently limited to research and
development laboratories due to the low deposition rate, the additional safety issues
involved with the use of UV lasers and the expense of the equipment. Some of the thin
films that have been deposited using laser ablation include super conducting thin films,
ceramic coatings, and amorphous metallic layers.
Characterization of Thin Film Deposition Processes
There are a variety of means by which one can get information about the thin films
created by physical vapor deposition processes. Of these techniques, some provide
information while the film is being deposited (so-called "in-situ" techniques) while others
give information after the deposition process is completed, and the coated part is
removed from the vacuum vessel ("ex-situ" techniques). The in-situ techniques provide
the means for control of a process during deposition. This real-time information may
simply be collected, and used as quality control data, or can be used as input to the
process parameters to optimize the process (closed loop operation).
In-Situ Characterization Techniques - Quartz Crystal Microbalance
In this technique a small quartz crystal is caused to oscillate by the application of an
electric field. This crystal is placed inside a vacuum vessel, and during deposition of a
film, the crystal gains mass due to the material condensing on it. The frequency of this
oscillation is reduced as the mass of the film on the crystal increases. Through cross-
calibration using a technique such profilometry, the quartz microbalance can be used to
accurately monitor film thickness as the film is deposited, and to provide feedback
information to control the process (close shutters, increase or decrease power to the
deposition source).
Optical monitoring.
Lenses and mirrors, which are to be coated with a thin film material to enhance their
optical performance, are often monitored in the vacuum deposition chamber during the
coating process. In-situ optical monitoring typically falls into two broad categories:
transmission and reflection. If an optic has a performance specification (% transmission
or reflection) at a specific wavelength, it is best to monitor the deposition process with
that same wavelength light.
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light
source
viewport
substrate
deposition
sources
detector
viewport
light
source
viewport
substrate
deposition
sources
detector
Interpretation of in-situ reflectivity and transmission data can become complex. The
effect of absorption and reflection of light by the glass viewport can affect the results. It
is important to choose viewports having good transmission for the wavelength of light
being used for the in-situ monitoring. Both reflection and transmission measurements
can be used to monitor film thickness during the deposition. In the reflection method, the
intensity of the reflected light will resemble a sine curve as the film grows. This is due to
constructive interference between the light reflected from the surface of the film (I
f
) and
light reflected from the film/substrate interface (I
s
).
Figure 14.32 Reflection of light from a substrate coated with a thin film.
Optical transmission data may also be used to monitor the thickness of a film during
deposition, and can yield quantitative data if an independent thickness calibration is
performed.
.
I I
o
substrate thin film
coating
Figure 14.33 Transmission of light through a thin film on a substrate.
I
I
0
= e
−ax
where:
I
0
= intensity of the incident beam
I = intensity of the transmitted beam
a = absorption coefficient of the material
at the wavelength of interest
x = thickness of the material
The effect of absorption from the substrate must be taken into account in the thickness
measurement by optical transmission. This can be accomplished by measuring the
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transmission of the uncoated substrate, and subtracting that value from the
transmission measured during deposition.
Film Stress Measurement
Thin films condensing onto a substrate may induce mechanical stress in the
film/substrate assembly due to the heat of condensation and the dissimilar coefficients
of thermal expansion between the two materials. In-situ thin film stress measurement
devices are commercially available, and generally are of the cantilever beam geometry
shown in figure 14.34.
deposition source
laser photo-detector
film stress
monitor
unstressed
net tensile stress
in thin film
Figure 14.34 In-situ film stress monitor.
If a thin film is in a state of tension, the cantilever film stress monitor will be deflected as
shown in the detail of figure 14.34. Alternatively, if the net mechanical stress in the film
is compressive, the cantilever beam will bend in the opposite direction (upwards in the
detail of figure 14.34).
Many of the thin film deposition techniques described in this unit create a visible plasma
which may be analyzed using spectroscopic techniques to determine the chemical
composition and state of excitation of species in the plasma.
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deposition source
optical
spectrometer
plasma
substrate
Figure 14.35 Optical spectrometer used to monitor light emissions from the
plasma during the deposition of a thin film.
In sputtering processes, for example, the optical emission can be monitored to establish
the purity of the sputtering gas, or to determine when the native oxide layer has been
sputtered off a target. The intensity of peaks in the spectra observed using this
technique are proportional to the concentration of species in the plasma. One can select
a spectral line of interest (the line at 4189 Å for oxygen) to monitor intensity as a
function of sputtering time. In-situ monitors of this type are commercially available, and
can provide feedback to control the deposition process, based on the intensity of a
particular spectral feature.
Ex-situ Thin Film Characterization Techniques
Care must be taken to prevent altering the thin film coating in the performance of ex-situ
characterization techniques. In general, it is good practice to allow a coated substrate to
cool to room temperature before venting the vacuum vessel. This minimizes the
possibility of oxidation of the part. Thermal shock, due to rapid cooling caused by
premature venting may cause films to tear or delaminate from the substrate, making
them virtually useless for physical characterization purposes. Contamination of the
coated substrate due to handling and storage should be kept to a minimum for best
characterization results.
Contact Profilometry
In this ex-situ characterization technique, the sample is prepared by masking off
a portion of a smooth substrate, such as floatglass, then depositing a thin film onto the
sample. This creates coated and uncoated areas ( a so-called "step slide") on the
smooth substrate which can be used to determine film thickness. After the part is
removed from the deposition system, a ruby or diamond tipped miniature stylus is
scrolled across the sample.
Figure 14.36 Stylus of contact profilometer on a "step slide" sample, and the
resulting data.
Contact profilometry is frequently used to establish deposition rates, and as a means of
calibration for in-situ rate monitoring techniques, such as the quartz crystal
microbalance. Films having thicknesses of from 50 to 100,000Å have been accurately
measured using contact profilometry.
Sheet Resistivity
Some thin films are deposited with the goal of attaining a specified electrical
characteristic, such as resistivity. These films may be characterized using a commercial
instrument called a four-point resistivity probe. Current is passed between each of the
points of the probe through the thin film, and is measured. The data is output as sheet
resistivity expressed in ohms per square centimeter (Ω/cm
2
). Other electrical properties
of interest in thin film coatings are the dielectric strength (the ability of a film to prevent
high voltage from dissipating to ground through the film), and the critical current density
(current density below which a material is super conducting at a given temperature).
Optical Transmission
Ex-situ optical transmission measurements often are more detailed and
extensive than the optical monitoring described earlier for in-situ measurements. A
single wavelength, or narrow band of wavelengths are often used, along with low
resolution spectrometers for the in-situ optical measurements. Optics are often carefully
inspected following deposition of a thin film coating to establish the transmission over a
broad range of wavelengths.
Compositional analysis
There are a variety of techniques used to determine the chemical composition of
thin film coatings following a deposition run. These techniques may be grouped into two
broad categories: destructive and non-destructive. Wet chemical analyses, in which the
film is dissolved, the solution being analyzed using techniques such as atomic emission
spectroscopy are obviously harmful to the coating. If destructive analytical techniques
are to be used, it is possible to perform them on "dummy parts" that were coated along
with the part of interest. Non-destructive analytical techniques include x-ray techniques
(x-ray florescence, x-ray diffraction), electron spectroscopy for chemical analysis
(ESCA) and other surface science techniques. Most of these non-destructive analytical
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methods are conducted under vacuum in relatively small chambers, which places size
limitations on the parts to be analyzed.
Mechanical Testing
Frequently the mechanical property of a thin film coating which is of most interest
is its adhesion to a substrate. Simple tests of adhesion include the "scotch tape test" in
which a piece of scotch tape is pressed firmly to the coated side of a part, then peeled
away. Other more sophisticated tests of adhesion include the Sebastian pull test and
the ring shear test. In the Sebastian test, a metal stump is glued to the thin film
deposited onto a substrate. The force required to pull the stump away from the
substrate is recorded, and plotted as a graph of stress (force per unit area) versus strain
(deformation).
thin film
substrate
test stump
Figure 14.36 Sebastian pull test for adhesion of thin films.
In the ring shear test the circumference of a test cylinder is coated with a thin film. The
area coated is subsequently built up by depositing additional material by electroplating.
The "ring" is machined to prescribed dimensions, then tested, as shown in figure 14.37.
thin film
substrate
shaft
test block
Figure 14.37 Ring shear test for thin film adhesion.
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The amount of force required to push the specimen through a hole in the test block is
recorded. The location of failure is also observed. In some cases the thin film is so
tenaciously adhered to the substrate that failure occurs in the bulk material.
Procedure: for each of the following components to be coated with the specified thin
film, describe in detail the deposition technique that would be appropriate. In the
description include drawings of the hardware required, showing locations and relative
sizes of components. Also include in your description the in-situ or ex-situ
characterization techniques you would use to monitor the quality of the thin film
deposits.
1) Deposit magnetic thin film coatings (iron and chromium oxides onto strips of
plastic tape (12" wide by 100' long) for use in the manufacture of cassette
recording media.
Equipment required: small vacuum vessel or bell jar vacuum system with Feed
throughs and internal hardware for deposition of thin films by resistance heated thermal
evaporation (see figure 14.3). A low voltage power supply (5 to 40 VDC, 10 to 50 A)
with power leads compatible with power Feed throughs on the vacuum vessel.
Procedure: Assemble the equipment as shown in figure 14.3. Make certain the
electrical connections are secure, and that the vessel and power supply are both safely
grounded. Prior to any experimental work, have the laboratory instructor inspect the
vacuum vessel and power supply. Before pump down mount at least one substrate
(glass microscope slides work well for this) on the opposite side of the shutter from the
deposition source. Load the source with an appropriate amount of charge material
(aluminum). Evacuate the vessel and conduct the thermal evaporation. Allow the
filament to cool for 30 minutes after the power is turned off prior to venting the vessel.
Upon completion of experimental work shut down the vacuum system safely and write a
lab report describing your experiment. Include any characterization of the thin film you
perform.
Experiment 14.2: Deposition of a thin film by DC diode sputtering.
Equipment required: small vacuum vessel or bell jar vacuum system with electrical
power feed throughs; a diode sputtering power supply with appropriate power leads; a
copper cathode and an aluminum cathode; a bottle of compressed Argon with regulator;
a leak valve.
Procedure: Assemble the equipment as shown in figure 14.38. Make certain the
electrical connections are secure, and that the vessel and power supply are both safely
grounded. Prior to any experimental work, have the laboratory instructor inspect the
vessel for safety. Evacuate the vessel to a pressure of less than 5 x 10
-5
Torr. Before
proceeding, turn off the ion gauge. The next task is to throttle the high vacuum pump to
control the flow of process gas through the system. If an iris valve is installed between
the inlet of the high vacuum pump and the vessel, close it approximately 95%. In the
absence of an iris valve, close the gate valve almost completely. Use the leak valve to
inject process gas into the vessel. The goal here is to achieve a steady pressure of
between 5 to 50 mTorr of Argon.
Once a stable process gas pressure is established, turn on the sputter source power
supply and slowly increase the negative bias applied to the cathode. A glow discharge
should appear at an applied bias of from 500 to 800 V. Note the color and distribution of
the plasma. Allow the cathode to sputter for ten minutes. Turn the sputter source power
supply down to zero volts and shut it off. Allow the system to cool for 30 minutes. Vent
the vessel and inspect the cathode and substrate. Exchange the copper cathode for the
aluminum cathode and repeat the sputtering experiment. Upon completion of all
experimental work, shut the vacuum system down safely, venting all pumps. Write a
report of your observations.
Discussion questions:
1. What causes the plasma to be a different color when different cathodes are
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• Station Activities to promote collaborative learning and problem-solving skills | 677.169 | 1 |
Number"Number Theory" is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included.
The book is divided into two parts. Part A covers key concepts of number theory and could serve as a first course on the subject. Part B delves into more advanced topics and an exploration of related mathematics. Part B contains, for example, complete proofs of the Hasse–Minkowski theorem and the prime number theorem, as well as self-contained accounts of the character theory of finite groups and the theory of elliptic functions.
TheFrom the reviews:
"This is a book which many mathematicians could enjoy browsing, and one which a good undergraduate could be encouraged to read to learn something of the interconnections, which exist between apparently disparate parts of mathematics."
—Canadian Mathematical Society
"As a source for information on the 'reach' of number theory into other areas of mathematics, it is an excellent work."
—Mathematical Association of America
Contenuti:
The Expanding Universe of Numbers.- Divisibility.- More on Divisibility.- Continued Fractions and Their Uses.- Hadamard's Determinant Problem.- Hensel's -adic Numbers.- The Arithmetic of Quadratic Forms.- The Geometry of Numbers.- The Number of Prime Numbers.- A Character Study.- Uniform Distribution and Ergodic Theory.- Elliptic Functions.- Connections with Number Theory.7894850
Descrizione libro Springer-Verlag New York Inc.,7894850
Descrizione libro Springer, 2009. Paperback. Condizione libro: NEW. 9780387894850 This listing is a new book, a title currently in-print which we order directly and immediately from the publisher. Codice libro della libreria HTANDREE0275039
Descrizione libro Springer-Verlag Gmbh Aug 2009, 2009. Taschenbuch. Condizione libro: Neu. 236x178x38 mm. Neuware - Number Theory is more than a comprehensive treatment of the subject. It is an introduction to topics in higher level mathematics, and unique in its scope; topics from analysis, modern algebra, and discrete mathematics are all included. 610 pp. Englisch. Codice libro della libreria 9780387894850 | 677.169 | 1 |
Latest
Download Introduction to Graph Theory by Richard J. Trudeau ebook free using the link below. This is the best sellling book on Graph Theory in Amazon.
A stimulating excursion into pure mathematics aimed at "the
mathematically traumatized," but great fun for mathematical hobbyists
and serious mathematicians as well. Requiring only high school algebra
as mathematical background, the book leads the reader from simple graphs
through planar graphs, Euler's formula, Platonic graphs, coloring, the
genus of a graph, Euler walks, Hamilton walks, and a discussion of The
Seven Bridges of Konigsberg. Exercises are included at the end of each
chapter. | 677.169 | 1 |
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Summary
While using Intermediate Algebra, Second Edition, you will find that the text focuses on building competence and confidence. The authors present the concepts, show how to do the math, and explain the reasoning behind it in a language you can understand. The text ties concepts together using the Algebra Pyramid, which will help you see the big picture of algebra. The skills Carson presents through both the Learning Strategy boxes and the Study System, introduced in the Preface and incorporated throughout the text, will not only enhance your algebra experience but will also help you succeed in future college courses. Book jacket. | 677.169 | 1 |
Topics in Mathematical Modeling is an introductory textbook on mathematical modeling. The book teaches how simple mathematics can help formulate and solve real problems of current research interest in a wide range of fields, including biology, ecology, computer science, geophysics, engineering, and the social sciences. Yet the prerequisites are minimal: calculus and elementary differential equations. Among the many topics addressed are HIV; plant phyllotaxis; global warming; the World Wide Web; plant and animal vascular networks; social networks; chaos and fractals; marriage and divorce; and El Ni#65533;o. Traditional modeling topics such as predator-prey interaction, harvesting, and wars of attrition are also included. Most chapters begin with the history of a problem, follow with a demonstration of how it can be modeled using various mathematical tools, and close with a discussion of its remaining unsolved aspects.
Designed for a one-semester course, the book progresses from problems that can be solved with relatively simple mathematics to ones that require more sophisticated methods. The math techniques are taught as needed to solve the problem being addressed, and each chapter is designed to be largely independent to give teachers flexibility.
The book, which can be used as an overview and introduction to applied mathematics, is particularly suitable for sophomore, junior, and senior students in math, science, and engineering.
REVIEWS for Topics in | 677.169 | 1 |
Mathematics
This text will be an excellent resource for undergraduate students in physics and a quick reference guide for more advanced students, as well as being appropriate for students in other physical sciences, such as astronomy, chemistry and earth sciences. 20159889 | 677.169 | 1 |
This is
a work in progress so please be patient. As I work on it you will find
more information available to you. If bad links should occur please
e-mail me and let me know.
Home: This
is the home page. It offers a directory of available resources relevant
to this class.
Newsletter:I may occassionaly write a newsletter as need occurs and
this is where you'lll find it.
Syllabus:This is where you will find a copy of the class syllabus.
The syllabus outlines the course, goals and objectives, citizenship
standards, consequences, homework policy, and grading policy.
Standards:This page provides links to the California State Standards
in Algebra which have been adopted by the San Diego City Schools.
Resources:This page provides links to resources which may be helpful
for the successful completion of this class. It also provides links
to sites which may be of general math interest or just plain fun.
Homework:This page gives a listing of the weeks homework. Please
note homework is subject to change in class. You can also find the
homework assignments on the class calendar.
Grades:On this page you can access your grades. To do so you will
need your student number and an identification number.
Other
Stuff:Miscellaneous stuff that didn't seem to fit
anywhere else.
Some
files on this web site may use Adobe Acrobat PDF files and you will
need to install this software program to use with your browser. Click
on the "Get Acrobat Reader" icon to get your free copy of the software. | 677.169 | 1 |
PDF (Acrobat) Document File
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0.97 MB | 16 pages
PRODUCT DESCRIPTION
Algebra that Functions
Graphing Systems of Equations Math and Art Project
Your students will apply their skills of graphing systems of linear equations with art! Students will graph specified types of systems of linear equations and choose how they create an original piece of art. My students love this project because it provides an opportunity for them to be creative in math class!
This project is a great formative assessment because it assesses students' skills and is highly engaging! My students are so proud of the art they create! I created two forms of this project. The second form is more challenging than the first form. This allows you to differentiate based on your students' needs.
I included a rubric and a peer evaluation, which requires students to further apply their skills of systems of equations as they verify that the equations do model the graphs of the lines.
This product is a paid digital download from my TpT store Algebra that Functions and it is for use in one classroom only. This productGo to your My Purchases page (you need to login). Beside each purchase you'll see a Provide Feedback button. Simply
Be the first to know about my sales, freebies and product launches! CLICK ON THE GREEN STAR next to my store logo to become a follower. You will receive | 677.169 | 1 |
The book constitutes a basic, concise, yet rigorous course in complex analysis, for students who have studied calculus in one and several variables, but have not previously been exposed to complex analysis.
Database Systems: A Pragmatic Approach is a classroom textbook providing a comprehensive yet concise introduction to the theory and practice of database systems as they are used in corporate software development | 677.169 | 1 |
Concord, CA Algebra 2 learn how to solve and graph quadratic equations by factoring, completing the square, or using the quadratic formula, including in the complex numbers. Student knows how to apply these techniques in solving word problems. Knowledge and skills in Geometry are essential and useful for students, especially those who are ready to go on to Algebra or Precalculus | 677.169 | 1 |
Algebra and Expressions PowerPoint
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0.09 MB | 4 pages
PRODUCT DESCRIPTION
This PowerPoint in an introduction to algebra and expressions. It helps the students understand what they are, and what the vocabulary means. It discusses how to set up expression from word phrases. It also has built in | 677.169 | 1 |
Mathematics
Our team of Maths teachers is dedicated to ensuring our students fulfil their potential in one of the curriculum's most rewarding subjects. Mathematics is not just about 'doing sums'; it is about challenging the mind to solve problems, and it is these problem solving skills that are so useful as our students move on to the world of work.
A qualification in Mathematics can lead to a number of careers. These include careers in engineering, finance, ICT and computer design, scientific research, medicine and architecture, to name but a few key examples. Most importantly, a good qualification in Mathematics tells a potential employer that you are a good problem solver!
Curriculum Content
Exam Boards:
Level 2 Further Maths – AQA
KS4 – Edexcel GCSE Maths, Algebra and Statistical Methods awards
Key Stage 3
Year 7
Basic skills are the focus for Year 7, ensuring students from different schools are equipped with the skills needed to move forward in their Mathematics learning. These focus on the early parts of the Foundation level GCSE.
Many of these topics will be revisited and developed in each of the subsequent years.
Key Stage 4
The main GCSE syllabus will be covered as a three-year plan, with students sitting their GCSE at the end of Year 11. The examination board is Edexcel. As previously, topics are revisited on a regular basis, with reinforcement and developmental activities to be delivered within this framework. It is intended that all students will take their GCSE at the higher level. Syllabus coverage is heaviest in Years 9 and 10 to give ample time to revise thoroughly for the examination in Year 11.
In 2015-16 all students will also sit the Edexcel Algebra award at level 2, with some EAA students sitting at level 3. Following this course will strengthen their algebraic skills in preparation for their GCSE next year.
Current Year 11 students are following a mixture of courses designed for those who passed their GCSE last year. These are the AQA Further Maths Award and the Edexcel Algebra and Statistical methods Awards
Key Stage 5
Exam Board – AQA
We currently use the AQA examination board for AS/A2 studies. We follow courses for both Mathematics and Further Mathematics in the sixth form.
Calculus in kinematics, velocity at an instant, motion in two and three dimensions, moments, equilibriums, centres of mass, centres of gravity, work, energy and power, elasticity – springs and strings, circular motion, and differential equations. | 677.169 | 1 |
Algebra Task Cards MEGA Bundle
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3.61 MB | 52 pages
PRODUCT DESCRIPTION
This MEGA bundle of 6 sets of Algebra task cards includes task cards for Slope, Functions (identifying and evaluating) Quadratics (word problems with and without C) and Systems of Inequalities. Below is a description of each set you will receive in this download.
Slope Task Cards: Identifying Slope in Different Forms (station activities)
Slope comes in many forms. Students work either independently or together to find slopes given: graphs, right triangles, tables of data, equations, and coordinate pairs (8.F.A.2). On each of the 10 task cards is an additional question, which asks students to find the equations of the lines represented in the data (8.F.B.4). You may choose to have your students work only to find slope or also to find these equations. A student answer sheet is included as well as an answer key.
Functions Task Cards: Identifying Functions in Different Forms (stations)
What makes a function? Students work either independently or together to identify functions represented in: tables, graphs, coordinate pairs, equations, scatter plots, and in verbal word problems (8.F.A.1), to identify which of two functions has a greater rate of change (8.F.A.2), and if a function is linear or nonlinear (8.F.A.3). You may choose to have your students simply identify the functions on the 16 task cards or also explain answers. The answer sheet provides space for both. An answer key is also included.
Systems of Inequalities Task Cards (with answer key)
Students work either independently or together to work through 10 task cards that ask them to:
1: Write systems of inequalities from graphs
2: Determine which points on a graph are solutions
3: Graph systems of inequalities
4: Complete a system and determine if points are solutions
5: Figure out what is wrong ("Find the error") in a system.
A student answer sheet is included as well as an answer key.
Quadratic Word Problem Task Cards: Ax^2+Bx (with optional QR codes)
Students can choose to factor, use the Quadratic Formula or their graphing calculators to solve projectile motion word problems where objects start from the ground (no C). projectile motion task cards [each with 3 questions covering zeros (roots), vertex x and vertex y], a consecutive integer bonus card, a student response sheet, an answer key and a separate (optional) QR code sheet that students can scan to check answers.
Quadratic Word Problem Task Cards: Ax^2+Bx + C (with optional QR codes)
Students can choose to factor, use the Quadratic Formula or their graphing calculators to solve quadratic word problems covering zeros (roots), vertex x, vertex y and y-intercept. task cards [each with 3 questions covering zeros (roots), vertex x, vertex y and y-intercept], an unknown area bonus card, a student response sheet, an answer key and a separate (optional) QR code sheet that students can scan to check answers.
Functions Task Cards: Evaluating Functions in Different Forms (stations)
Students use the given tables, graphs, equations and word problems to evaluate functions written in function notation. There are 10 task cards with an average of 7 questions per card as well as a bonus card. Students find the values of functions given tables, graphs, equations and word problems and solve for x given f(x) values. Cards also include function composition and adding and subtracting questions. A student answer sheet and complete answer key | 677.169 | 1 |
ISBN 13: 9780071222112
Calculus
Students who have used Smith/Minton's Calculus say it was easier to read than any other math book they've used. Smith/Minton wrote the book for the students who will use it, in a language that they understand, and with the expectation that their backgrounds may have some gaps. Smith/Minton provide exceptional, reality-based applications that appeal to students' interests and demonstrate the elegance of math in the world around us | 677.169 | 1 |
Every year, thousands of students in the USA declare mathematics as their major. Many are extremely intelligent and hardworking. However, even the best will encounter challenges, because upper-level mathematics involves not only independent study and learning from lectures, but also a fundamental shift from calculation to proof. This shift is demanding but it need not be mysterious -- research has revealed many insights into the mathematical thinking required, and this book translates these into practical advice for a student audience. It covers every aspect of studying as a mathematics major, from tackling abstract intellectual challenges to interacting with professors and making good use of study time. Part 1 discusses the nature of upper-level mathematics, and explains how students can adapt and extend theirexisting skills in order to develop good understanding. Part 2 covers study skills as these relate to mathematics, and suggests practical approaches to learning effectively while enjoying undergraduate life.As the first mathematics-specific study guide, this friendly, practical text is essential reading for any mathematics major | 677.169 | 1 |
..., measurement, and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity for as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry... Learn about: Mathematics Engineering, Mathematics Teacher, Mathematics Algebra Algebra, Mathematics Series... | 677.169 | 1 |
Here's why students need algebra
Experts are split on the necessity of requiring algebra in high school
In his recent book, "The Math Myth: And Other STEM Delusions," political scientist Andrew Hacker argues, among other things, that we should not require high school students to take algebra.
Part of his argument, based on data some have questioned, is that algebra courses are a major contributor to students dropping out of high school. He also argues that algebra is nothing more than an "enigmatic orbit of abstractions" that most people will never use in their jobs.
There is no doubt that this kind of argument resonates with people who had bad experiences in a math class in their past, and for this reason Hacker's book is getting lots of attention. On the other hand, there are many reasons why I and many others in the mathematical community disagree with Hacker's opinions.
Fundamentally, Hacker has a misunderstanding of what algebra is.
The word "algebra" comes from the Arab word "al-jabr," which means "to balance." Using it in a mathematical context dates back to a Persian manuscript in the ninth century, which introduced the beginnings of what grew into what we now study in high school.
The big idea that distinguishes algebra from the mathematics that had come before is to think of operations taking place simultaneously on whole collections of numbers rather than on a single number. | 677.169 | 1 |
About this product
Description
Description
Deciphering the Proof is for students, parents, and new teachers who need practice solving proofs in Geometry. Specifically, where Geometry is part of the 4e curriculum in a French program, or for American students taking Geometry between Grades 8 and 10. The book shows, step-by-step, how to reason and solve Geometry problems, by writing solutions in a clear, logical, and deductive sequence. This strategy is called, modeling. Students learn, by imitating the method, and eliminating all the n-value adding verbiage that distract graders. By showing the core steps required to solve a problem, students avoid extraneous text, and steps, which make the solution difficult to follow, and difficult for the grader to evaluate with precision. Teachers can use the material, in class, by showing partial solutions (of the reasoning or the proof), and asking the students to complete the other part. The book should be used as a complement to a Geometry textbook. It is especially beneficial for average students with difficulties writing the solution to a problem in a logical deductive process. It is recommended to the user of the book to, first, try to solve the problems entirely, before comparing with the step-by-step solutions following each chapter. | 677.169 | 1 |
2 ME 176 2 Mathematical Modeling
2.
Mathematical Modeling: Introduction
Mathematical Models are representation of a system's schematics,
which in turn is a representation of a system simplified using
assumptions in order to keep the model manageable and still an
approximation of reality.
1. Transfer Functions (Frequency Domain)
2. State Equations (Time Domain)
First step in creating a mathematical model is applying the
fundamental laws of physics and engineering:
Electrical Networks - Ohm's law and Kirchhoff's laws
Mechanical Systems - Newton's laws.
Department of
Mechanical Engineering
6.
Mathematical Modeling: Laplace Transform
Partial Fraction Expansion, where roots of the Denominator of F(s) are:
Note: N(s) must be less order that D(s) .
1. Real and Distinct
where,
Department of
Mechanical Engineering | 677.169 | 1 |
The packet contains 100 questions for the students to solve, as well as worked-out examples and descriptions of how to solve the various types of problems. An answer key is also included.
This math packet would make an excellent summer math packet for pre-algebra students, but it would also be a great August/September back to school review for Algebra I students, as well. It also would be great to leave for emergency sub plans in Algebra or Pre-Algebra classes, or to use as a study guide for a Pre-Algebra final exam or end of course test40. | 677.169 | 1 |
Academic Algebra II
Academic Algebra II follows either Academic Algebra I or Academic Geometry and precedes either Academic Geometry or Pre-Calculus in the academic, college-bound student's mathematical career.
Academic Algebra II is a full year course in which we will explore many facets of advanced algebraic topics. This course is designed to provide highly motivated students with a deep and thorough understanding of functions and the algebraic concepts that are associated with them. Students will use a graphing calculator (TI-83 / TI-84) on almost a daily basis--- if you are pursuing further math/science courses, I strongly encourage you to purchase your own for you to use at home!!
Be sure to check out all of the useful info below such as the Google Calendar containing all assignments/due dates as well as any of the documents I post below such as notes, study guides, etc.
Pre-Requisite: Academic Algebra I
ATTENTION STUDENTS! The textbook for algebra 2 can be found at The login information is as follows:Username: Acad_Alg2 Password: algebra2 | 677.169 | 1 |
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Advanced Algebra Problems Assignment Help
Posted on December 31, 2016 by admin
Best Advanced Algebra Problems Assignment Help Service
Advanced Algebra Problems Assignment Help
Algebra is one of the important section of mathematics along with geometry and analysis. Algebra is also called as mathematical study of symbols and rules for manipulating these symbols. The most basic part of algebra is called elementary algebra. The word algebra is also used in specialized ways. Today algebra has taken its new features and name known as advanced algebra. And our experts of advanced algebra problems assignment help discuss about the problems and solutions of algebra. A mathematician who studies about research of algebra is known as algebraist. In mathematics algebra has its broad part and it began with computation.
There are number of names which falls under mathematics with abstract algebra they are:
There are two parts of algebra that is algebra1 and algebra2. In this algebra1 notes for basic arithmetic operations that is addition, subtraction, division and multiplication. Teaching and learning both are important for a professor. A professor never stops learning. This makes him a good and professional expert. Our Besthomeworkhelpers provide best advanced algebra problems assignment help to our students of mathematics departments.
The root of mathematics goes back to 5000 years ago and today also it is as important and useful as in ancient times. Students of mathematics department take help of advanced algebra problems homework help service, so that they can better improve their algebra. Our experts provide help on advanced algebra. It includes linear and complex equations. Our team of expert holds the degree of Masters and Doctor. They are specialised in the field of mathematics.
We all know that how mathematics is important in our daily life. It had covered a vast area of our life. Today mathematics is applied in variety of subjects like Economics, Physics, Chemistry and Computer science. It had played a dynamic role in the field of Engineering and Information Technology. Along with providing advanced algebra problems homework help, our experts also take quiz and provide practices papers to better understand the concepts and sums of algebra. Mainly, there are two types of mathematics they are applied mathematics and pure mathematics.
Applied mathematics has its implication in different field they are theoretical computer science, computational biology and many more. Pure mathematics has its existence in the field of Geometry, Physical Mathematics and many more. Our experts of algebra problems homework help students to take algebra as an interesting subject. They put their effort best to make mathematics a pure and interesting subject. In today's world mathematics has its leading role. Hence, it is very important to have basic and calculative knowledge of maths. Our experts will help such students who are poor and finds maths a boring subject.
Students need more Advanced Algebra Problems Assignment Help, please click over here. You can read more about our Math Homework Help services here. | 677.169 | 1 |
A Level – Mathematics
This course is for you if you achieved well at GCSE maths, particularly enjoying the algebra work.
An essential part of maths is the challenge of analysing and solving a problem and the satisfaction and confidence gained from achieving a 'correct' answer. However, a big difference between GCSE and A Level is the depth of knowledge and understanding needed. If you like understanding why, rather than just being able to "do" maths then A Level is for you; there are a much smaller number of topics than at GCSE so your depth of knowledge and understanding is tested.
The exams are also marked differently at A Level; if you just put down the correct answer you often only get one of the marks for the question. There are many marks for showing how you got the answer; if you choose maths you will not have to write essays, but you will need to be able to communicate well in written work to explain your solutions.
Entry Criteria
Course Details
The A level course lasts for 2 years with all exams taken at the end of the second year. Students receive 4 lessons each week and additional work is done out of lesson, some of this using "MyMaths", some producing worked solutions to exam-type questions.
The course develops understanding of:
The fundamental theories and concepts of Mathematics
The practical applications of Mathematics to other areas
Logical problem solving
The new A Level Mathematics syllabus is completely prescribed and the content is the same for each examination board. All external examinations are taken at the end of the 2 year course.
The course is made up of a combination of pure and applied Mathematics.
The pure Maths content build on the algebra, graphs and trigonometry from GCSE, and introduce new topics such as calculus and exponentials. While many of the ideas you will meet in pure maths are interesting in their own right, they also serve as an important foundation for other branches of maths, especially mechanics and statistics.
The applied content covers;
Mechanics including work on forces and Newton's laws of motion and applies Mathematical modelling to simple problems.
Statistics builds on work on averages and probabilities from GCSE and introduces topics such as probability distributions and correlation.
How the course is delivered
You will have one teacher for the course each year. Lessons include a variety of group, paired and individual work, resources you will draw on include text books, exam questions and jigsaw/domino/card matching activities as well as web-sites.
Support outside lessons is available, both informally and formally. Regular "maths clinic" sessions are run during the college week and all of the department are available for support at lunchtime and at the end of the day.
Departmental Enrichment
The Maths Department offers a number of enrichment activities:
The Individual Senior Maths Challenge takes place in the autumn term.
The Team Challenge involves our students competing against other maths students both at a regional and national level.
The Advanced Extension Award is designed to challenge the top 10% of students and is offered to all students excelling in their A level Maths course.
We also run an engineering enrichment aimed at students who want to go on to careers using maths and physics, but are not certain what choice to make. It is a series of external speakers talking about a variety of careers. This includes a wide variety of talks, examples include; civil engineering; financial mathematical modelling; operating a production plant; designing and testing artificial body tissues.
We also take a group of students to the Maths Enrichment talks in Leeds each year. These are always very popular and include topics such as "the maths of juggling", "musical maths", and "the maths behind the dam busters".
New for 2016 was the international residential visit to Florence. Highlights will include The Garden of Archimedes, The Galileo Museum and the Tower of Pisa.
Our Department
We're based in five dedicated rooms in the Wilson building. Between us we have over 75 years of teaching experience. We have degrees from Imperial College, York, East Anglia, Sheffield and Durham.
We all enjoy teaching maths, some have always been teachers, and others have taken a less direct route. One of us used to be a Chartered Civil Engineer; one of us used to run a pub!
We're not just mathematicians though: Between us we enjoy climbing trees, rambling, photography, baking, travelling. See if you can find out which of us;
Plays the piano.
Had their photo in the New Zealand Fishing News.
Had an audition to be a presenter on "That's Life".
Has a wardrobe full of Panini Football Stickers.
Only read Harry Potter to spoil it for other people.
Has been on a stunt driving course.
Has spent time wading in sewers.
Our other Courses
Our other Maths courses are;
Further Maths A Level – In combination with A-Level Maths and gives you a broader and deeper understanding of advanced mathematics.
Core Maths level 3 (Mathematical Studies) – Equivalent to half an A level and looks at the applications of mathematics.
Foundation GCSE – For those who need to retake GCSE.
Numeracy level 1 – For those building up to GCSE level.
What can I do now that would help prepare me for this course?
Students who have taken GCSE Maths early may find they need to review what they have learnt at GCSE through the summer, maths skills need to be kept in use. Many Maths students enjoy extending their knowledge of maths and there are many ways of doing this including;
"Maths is always challenging but with the fantastic teachers and support available, every aspect is easier" Brandon Bone "Intriguing challenging, fun" Natalie Wilson "Its great how everything you learn fits together" Andrew Houghton "One does not walk out of maths unhappy" James Carr "Further maths expands your mathematical knowledge and introduces you to a different way of thinking" Elliot Tennison "I've enjoyed the course immensely, Further Maths is the most interesting course I have done and the teachers are amazing" Hash Rehman "It's challenging but very satisfying" Rebecca Brown
This course is vital if you are considering a degree or career in engineering, any of the sciences or maths. Higher Education courses or careers that either require Advanced GCE Mathematics or are strongly related include:
Economics
Medicine
Architecture
Engineering
Accountancy
Teaching
Psychology
Physics
Computing
Information and communication technology
A Level Maths is highly regarded by employers and universities and is therefore a very sought after qualification. Having this qualification identifies you as someone with developed mathematical skills such as logical thinking, problem solving and statistical analysis, all of which are wanted by many employers and Universities.
What does this course lead to?
In recent years students have gone on to a wide variety of careers from their maths A-Level. These include;
Students have gone on to a wide variety of careers from their maths A-Level. These include;
Aerospace Materials at Sheffield University.
Maths and Music at Birmingham University.
Civil and Structural Engineering at Leeds. Geology at Leicester.
Computer Science at University College London. Mathematics at Bath University. | 677.169 | 1 |
Appropriate for one- or two-semester Advanced Engineering Mathematics courses in departments of Mathematics and Engineering.
This clear, pedagogically rich book develops a strong understanding of the mathematical principles and practices that today's engineers and scientists need to know. Equally effective as either a textbook or reference manual, it approaches mathematical concepts from a practical-use perspective making physical applications more vivid and substantial. Its comprehensive instructional framework supports a conversational, down-to-earth narrative style offering easy accessibility and frequent opportunities for application and reinforcement. | 677.169 | 1 |
PRODUCT DESCRIPTION
Three real life situations are presented, each involving a constant (an upfront fee for example) and a variable ($30 per month). The presentation walks students through how to create a function table from a word problem, write a function rule to describe the data, and determine the nth term (find the total cost spent over 12 months) by using the function rule.
Thinking questions are asked before answers are given to prompt students to consider what they may encounter when doing similar problems on their own.
This PowerPoint is an excellent way to introduce the concept, allowing for teacher modeling, guided practice, and work | 677.169 | 1 |
Category Archives for "Resources"Well, it's finally public: I'm working on an eCourse to go alongside Principles of Mathematics: Book 1. The eCourse will feature a short video to go with every lesson in the textbook. The videos will walk through the material covered in the curriculum, making it a perfect supplement for auditory or visual learners, or any […]
Do you have a child who's frustrated in math? Unsure where to go next? I'm excited to announce that, in addition to the math resources in the store, I'm now offering online math tutoring for elementary through Algebra 2. My goal is to quickly get students unstuck, to help them see the concepts as a […]
The second book in Principles of Mathematics series is now complete! I'm super excited, not only because it is done, but also because it's my hope that this material will help students see God's handiwork in mathematics and realize to a deeper level what an amazing, faithful God we serve. Many students (myself included years | 677.169 | 1 |
Description
This book shows students of science and engineering the potential computers have for solving numerical problems and gives them ample opportunities to hone their skills in programming and problem solving. The text also helps students learn about errors that inevitably accompany scientific computations and arms them with methods for detecting, predicting, and controlling these errors. Additional ResourcesCompanion websiteRelated TopicsApplied Mathematics | 677.169 | 1 |
Alg 1 -- Polynomials Review (Survivor)
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43.24 MB | 59 pages
PRODUCT DESCRIPTION
This is a review of operations with polynomials -- adding, subtracting, multiplication and division. It would be for the first half of Chapter 8 in the Prentice Hall CA Algebra 1 textbook. I teach Chapter 8 in two sections: 1. Polynomials, including long division and 2. Factoring. I think each topic is important enough to be separate from the other. A review for factoring can be found at: Factoring Feud
This powerpoint has a Survivor theme, and contains 20 questions. Additional slides are provided so you can add more questions, and all slides can be edited to fit your classes. Video and sound waves are also included.
Common Core Standard A-APR
Perform arithmetic operations on polynomials.
1.Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Previous CA Standard 10.0
Students add, subtract, multiply, and divide monomials and polynomials. Students solve multi-step problems, including word problems, by using these | 677.169 | 1 |
MATH 164
Final
December 17, 2011
NAME (please print legibly):
Your University ID Number:
Circle your Instructors Name along with the Lecture Time:
Mueller (9:00am)
Bailesteanu (10:00am)
Part A of the nal can replace a bad midterm. However, Part A will st
MATH 164
Midterm 1 ANSWERS
October 18, 2011
1. (12 points)
(a) Find the angle between vectors a = (1, 2, 3) and b = ( 1, 2, 2) in terms of an inverse
trig function. Simplify as much as you can without using a calculator.
(b) Find a vector which is perpend
MATH 164
Midterm 1 ANSWERS
October 18, 2011
1. (12 points)
(a) Find the angle between vectors a = (1, 2, 3) and b = (1, 2, 2) in terms of an inverse
trig function. Simplify as much as you can without using a calculator.
(b) Find a vector which is perpendiULTIDIMENSIONAL CALCULUS Advice
Showing 1 to 1 of 1
This professor is great. She makes everything clear and fun. She is beautiful and humous. It is such a pity that she is a visiting professor, so she will not be here in Spring. However, if you can be her student, try all your best to register her course. She will give you a theoretical proof which let you have deep knowledge about the material, she will also give you the approach of how to do the problem which will let you get an easy A. Great lecture and Great teacher.
Course highlights:
MTH 164 is mostly for physical student and math student. DO NOT TAKE IT if you are just interested in math (no one will be interested in math). I am a math student so I have to take it, and due to my physical background, I think this course is pretty helpful and will make you know what exactly happens in the physical field with mathematical approach.
Hours per week:
3-5 hours
Advice for students:
Do the text book questions to get fully familiar with the materials covered. Do the practice midterm and final. Go to class. Go to office hour. Things will get confusing after midterm 2. | 677.169 | 1 |
Introduction to Complex Analysis
Complex analysis is a classic and central area of mathematics, which is studied and exploited in a range of important fields, from number theory to engineering. Introduction to Complex Analysis was first published in 1985, and for this much awaited second edition the text has been considerably expanded, while retaining the style of the original. More detailed presentation is given of elementary topics, to reflect the knowledge base of current students. Exercise setshave been substantially revised and enlarged, with carefully graded exercises at the end of each chapter.This is the latest addition to the growing list of Oxford undergraduate textbooks in mathematics, which includes: Biggs: Discrete Mathematics 2nd Edition, Cameron: Introduction to Algebra, Needham: Visual Complex Analysis, Kaye and Wilson: Linear Algebra, Acheson: Elementary Fluid Dynamics, Jordan and Smith: Nonlinear Ordinary Differential Equations, Smith: Numerical Solution of Partial Differential Equations, Wilson: Graphs, Colourings and the Four-Colour Theorem, Bishop: Neural Networks forPattern Recognition, Gelman and Nolan: Teaching Statistics | 677.169 | 1 |
Course Description
The primary focus of this course is to learn some portion of the
history of mathematics, paying particular attention to the
strands that let to developments that are important to today's
mathematics.While
we will be learning the history of mathematics, our primary
focus will be on the mathematics itself.
Course
Student Learning Outcomes:
1.Demonstrate
knowledge of the historical development of number
systems, algebra,
2.Understand the
contributions of diverse cultures to these developments.
Text: The History of Mathematics - An Introduction (sixth
edition),
David M. Burton, McGraw-Hill, New York 2007.
Grades will be determined on the following basis:
3 mini exams (average counts as 70% of
final grade) Final
presentation (counts as 20% of final grade)
Portfolio (counts as 10% of final grade)
90% guarantees A
80% guarantees B
70% guarantees C
60% guarantees D
Regarding the
presentations, there is a written component (handouts consisting
of notes to accompany your talk are expected) as well as a
verbal one. The presentation should be in Power Point and should
take approximately 40 minutes. You will receive a grade on
a scale of 1-5 in each of the following categories:
Preparation (were you well-prepared?), clarity to audience
(how well did you explain the material?), knowledge (how well
did you understand your topic?) and mathematical content (both
breadth and depth will be considered here). So the total
number of points possible will be 20.
The portfolio will consist
of your class notes and your notes on assigned readings.
Attendance: Students are responsible for all material presented
in class, so it is in your best interest to attend.
Help during office hours is available only to those who
either attended the class in which the material was
presented or whose absence is excused by the Associate Dean
of Students Office at 67 George Street.
Students with disabilities: The
College will make reasonable accommodations for persons with
documented disabilities. Students should apply at the
Center for Disability Services/SNAP, located on the first floor
of the Lightsey Center, Suite 104. Students approved for
accommodations are responsible for notifying the instructor as
soon as possible and for contacting the instructor at least one
week before any accommodation is needed.
Academic Integrity Statement:The Honor Code at the College of
Charleston specifically forbids cheating, attempted cheating,
and plagiarism. Cases of suspected academic dishonesty
will be reported directly to the Dean of Students. A
student found responsible for academic dishonesty will receive
a XF in the course, indicating failure of the course due to
academic dishonesty. This grade will appear on the
student's transcript for two years after which the student may
petition for the X to be expunged. The student may also
be placed on disciplinary probation, suspended (temporary
removal) or expelled (permanent removal) from the College by
the Honor Board.
It
is important for students to remember that unauthorized
collaborations—working together without permission—is a form
of cheating. Unless a professor specifies that students
can work together on an assignment and/or test, no
collaboration is permitted. Other forms of cheating
include possessing or using an unauthorized study aid (such as
a PDA), copying from another's exam, fabricating data, and
giving unauthorized assistance. | 677.169 | 1 |
Critical thinking mathematical reasoning and proof
Read this article to learn how brilliant minds like Elon Musk and Bill Thurston use first principles thinking to simplify and solve difficult problems.
The Critical Thinking Company publishes PreK-12+ books and software to develop critical thinking in core subject areas. The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students.
Critical thinking mathematical reasoning and proof
What are the Importance and Benefits of "Critical Thinking Skills"? S. M. Rayhanul Islam Apr 01, 2003 · 1. Introduction: The Question and the Strategy 1.1 The Nature of the Question. In Book One, the Republic's question first emerges in the figure of Cephalus.
1000_critical_reasoning_questions (1) - GMAT &... This preview shows document page 1. Sign up to view the full document. 3. Dimension 1 SCIENTIFIC AND ENGINEERING PRACTICES. F rom its inception, one of the principal goals of science education has been to cultivate students' scientific. A practical text for building thinking skills. In today's competitive business world, businesses must have an edge to remain competitive and be successful.
Please confirm that you want to add Critical Thinker Academy: Learn to Think Like a Philosopher to your Wishlist. Dr. Steven Novella of the Yale School of Medicine equips you with the knowledge and skills you need to become a savvier, sharper critical thinker in your professional. Here Be Dragons is a video introduction to critical thinking. Most people fully accept paranormal and pseudoscientific … Deductive reasoning, also deductive logic, logical deduction is the process of reasoning from one or more statements (premises) to reach a logically certain.
A comprehensive and coherent set of mathematics standards for each and every student from prekindergarten through grade 12, Principles and Standards is the first … | 677.169 | 1 |
In class we will use the TI-84 plus. Students are not required to purchase their own calculator, as we will provide one that can be used in class. If they prefer to purchase their own, I recommend keeping it at home so it doesn't get lost, stolen, or damaged. In addition, there are a lot of free online calculators that can be used for the same functions we will be using in class. | 677.169 | 1 |
CAPS GRADE 12 TEACHER GUIDE. 1. INTRODUCTION. Assessment is a ... tasks, particularly the investigation and assignment; hence these exemplars were The sum of the first n terms of a sequence is given by: Sn = n(23 3n). 2.1 Write .
ebook.dexcargas.com is a PDF Ebook search engine and unrelated to Adobe System Inc. No pdf files hosted in Our server. All trademarks and copyrights on this website are property of their respective owners. | 677.169 | 1 |
Product Description:
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred categories. There is ample material here for a graduate course in category theory, and the book should also serve as a reference for users.
REVIEWS for Handbook of Categorical | 677.169 | 1 |
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Polynomial Assignment Help
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An introduction of Polynomials:
A mathematical expression which comprises of coefficients and variables is referred to as polynomial. The only operations associated with polynomials include addition, multiplication, non-negative integer exponents and subtraction.
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A trajectory of projectiles can be depicted using polynomials.
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Matrix polynomials - Matrices are variables in the case of a matrix polynomial.
Laurent polynomials - In these forms of polynomials, negative powers of variables are allowed to take place.
Rational functions - Functions through which rational expressions can be deduced are called rational functions. Rational expressions are quotients related to polynomials.
Power Series - In these form of polynomials, several non-zero terms are allowed to occur infinitely.
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This book is a unique work which provides an in-depth exploration into the mathematical expertise, philosophy, and kwledge of H W Gould. It is written in a style that is accessible to the reader with basic mathematical kwledge, and yet contains material that will be of interest to the specialist in enumerative combinatorics. This book begins with exposition on the combinatorial and algebraic techniques that Professor Gould uses for proving bimial identities. These techniques are then applied to develop formulas which relate Stirling numbers of the second kind to Stirling numbers of the first kind. Professor Gould's techniques also provide connections between both types of Stirling numbers and Berulli numbers. Professor Gould believes his research success comes from his intuition on how to discover combinatorial identities.This book will appeal to a wide audience and may be used either as lecture tes for a beginning graduate level combinatorics class, or as a research supplement for the specialist in enumerative combinatorics. | 677.169 | 1 |
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The development of mathematics curriculum in Malaysia
The development of Mathematics Curriculum in Malaysia had started before the Second World War (WW II: 1941). Before that, our mathematics syllabus in Peninsula Malaysia was not standardised, which means that all type of school were freely to choose any text books and topics which considered suitable for the students. The mathematics curriculum become standardised and officially only after the year of 1956 when the Razak Report suggested that there should be a formal curriculum for all government schools. In the respect, an official common syllabus for mathematics curriculum was planed and later was implemented after 1956. However, this common syllabus was only a slightly change in some arithmetic topics and certain topics. It was not until 1970¶s when the Sp ecial Project was implemented. This Special Project was set up by the previous Ministry of Education of Malaysia and headed by En. Abu Hassan bin Ali in the year of 1968. The objective was to improve the standard of mathematics and science in primary level following the trend and development of Modern Mathematics in advanced countries. This project was funded by the Asian Foundation (Yayasan Asia) and some of the American Peace Corps members were invited as advisers of this project. Materials for the teachi ng and learning of mathematics were designed by lecturers and mathematics teachers who had their training overseas. There were only minor changes in the contents of the mathematics syllabus in the Malaysian primary school in this Special Project. However, new approach, strategy and method of presentation were introduced which is the pupil-centered strategy and the inquiry-discovery method. This Special Project was being launched as a pilot project in the year 1970. About 30 schools in Kuala Lumpur were chosen as a trial centers. Eventually, this programme was modified and improved from time being and was adopted in primary schools until it was replaced by KBSR Mathematics syllabus which was fully implemented in 1983. In 1983, the Modern Mathematics Curricul um was modified become the KBSR Mathematics. The difference was the arrangement with reduction of some mathematics contents to suit the pupils¶ ability to master the skill. The mathematics syllabus was divided into two levels. Level 1 (Year 1 ± 3) emphasized the mastering of the basic concepts of numbers and their four basic operations. Level 2 (Year 4 ± 6) emphasized application of the basic skill in solving mathematics problems. This programme was aimed at providing equal opportunity for every pupil to ac quire knowledge, skills, attitudes, rules and desired common social practice in society. The main aim of KBSR was to enable pupils to develop their counting skills. To achieve the aim, pupils must first attempt to master the basic mathematics skills. The Primary School New Curriculum (KBSR) was changed to Primary School Integrated Curriculum in the respect to achieve the noble ambition envisaged in the National Education Philosophy. The aim of KBSR Mathematics could be expressed as follow:
In addition. Decimal and their operations 4. which was listed according to their sequence as follow:
1. The mathematical knowledge gained would also help pupil to manage their daily activities systematically.tutorvista. allowing all pupils to acquire basic mathematics skills and to us the acquired skills in daily situations at all levels. The Syllab us Committee responsible for KBSR Mathematics has grouped all the required learning skills into 9 main topics. pupils would learn to appreciate mathematics. These would help pupils to solve daily problems effectively. thus fulfilling the requirements for our society and nation to progress as well as to help our pupils to further their studies in time to come´ (The Primary School Mathematics Syllabus: April 1993)
The Primary School Mathematics Syllabus which was revised again in 1998. Money 5.com/answers/what -is-the-development-of-mathematicscurriculum-in-malaysia/485674
. Whole numbers and their operations 2.³The main aim of Primary School Mathematics Education was to improve and develop the understanding of number concept and acquiring basic calculation skills. Fractions and their operations 3. Measurement of Lengths and Weights Reference: | 677.169 | 1 |
Solving Exponential and Logarithmic Equations
This book is a study guide for solving exponential and logarithmic equations. As a prelude to this topic, the exponential and logarithmic functions are graphed and discussed. The properties of exponential and logarithmic expressions are also covered and later used to solve different types of equations. The final section has several types of applications for these types of equations. Each topic includes solved examples and practice exercises. A practice test and answers to exercises are included.
This book is ideal for upper level high school students and college students that are enrolled in college level algebra courses or need to review these topics. In order to understand the content of this book, it is assumed that students have basic knowledge of the laws of exponents including negative and rational exponents.
An Introduction to Solving Exponential and Logarithmic Equations
This book explains how the exponential and logarithmic functions are related to each other as inverse functions. The properties of the exponential and logarithmic expressions are consistent with the laws of exponents and these properties can be used to solve a variety of different types of exponential and logarithmic equations. There are many applications for these types of equations including, compound interest, population growth, and depreciation models.
This book is divided into two main sections. Section one introduces the exponential and logarithmic functions and how they are represented graphically. The inverse relationship and other properties of these functions and expressions are also covered in section one. Section two illustrates the strategies used to solve different types of exponential and logarithmic equations. Properties in the first section are put to use in the solving process. Application problems are then used to show how these types of equations solve real life problems. After completing this text, the reader should be proficient with exponential and logarithmic functions, expressions, and equations.
About the Author
My name is John M. Gillis, Ph.D. I have been teaching mathematics since 1994 in Columbus, Georgia. I teach in both high school and college venues and have been an adjunct professor at Columbus State University since 1996. My degrees include a BS in Mathematics from the University of Florida, Med and EdS degrees in Mathematics Education from Columbus State University, a MS in Applied Mathematics from Auburn University, and a Ph.D in Mathematics Education also from Auburn University.
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Before teaching, I spent four years in the U.S. Army. My hobbies include fitness and guitar. To further my career, I have turned to the Internet. I am interested in using my knowledge of mathematics to consult and write. BrainMass has given me the opportunity as an Academic Expert to do both in an online environment.
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Problem-solving strategies algorithms and heuristics
Webmath is designed to help you solve your math problems. Webmath also shows the student how to arrive at the answer. .
Webmath is designed to help you solve your math problems. Webmath also shows the student how to arrive at the answer. . Simple steps to solve math word problems. His emphasis has always been about teaching math skills that will help students outside of school. . Free math problem solver answers your algebra homework questions with step. Help sin. Arcsin. You must allow mathway to access your email. . Solve calculus and algebra problems online with cymath math problem solver with steps to show your work. Get the cymath math solving app on your. Solve equation. . Wolframalpha shows steps to solve math problems, allowing you to learn the basics on your own, check your work, or give you insight on different ways to solve problems. .
In case you think you havent received a comprehensive explanation using our math solver online, you can consider working directly with one of our qualified tutors. Not hard but different lol lol to obtain the logarithm with a fraction as base you have to write the base without the 0 before the decimal point if you try to solve a limit, but the left-handed limit and the right-handed limit are not the same, wolfram alpha gives both one-sided limits, but the show steps button doesnt appear. Youll find hundreds of instant-answer, self-help, math solvers, ready to provide you with instant help on your math problem. I have to re-type my question because ive moved on and then decided to take another look at it. For example with (ddx ax) it doesnt show useful steps.
That would be really sweet, but that would require going through all the textbooks in all the high schools of the country and it recognizing where you are to base your school district and analyzing the textbook that that district uses. Easily one of the most exciting things i have seen recently. I even tried using the identical limit example pictured above and the steps dont show for me. I also did it by hand and i got a different answer (the one given from wolfram is correct). It would be amazing to figure out where i made my mistake no matter what function i put in, it seems as though the show steps part of the derivative doesnt work.
An example ive tried is y (x-3)(4x2) i havent been able to get any of the show steps to work for derivatives. Is there a way to make it come up? Or to request only one of the one-sided limits and have the button appear? How could i input this equation to get a step by step process to solve this difference quotient? Tried it. Now that you understand the word problems purpose, determine the answers unit. Just want to ask are those values fed in database or calculated in real time and then cached. There are many such links on the answers which are easily overlooked. I have also purchased the wolfram alpha mobile application (android os) and this same show steps functionality is as it is on the rest of the site, not working. If you want to be really good at all types of math, you need to practice please enter the email address and well send you an email containing instructions for changing your password. Enjoy other services with math word problem solver at studygeek. Please fix this website, to me this is the alpha and the omega (pun intended), and i cant wait to have full functionality back! Hi all, thank you for bringing this to our attention! Our team is working to get this back up and running. The wolframalpha blog is now part of the wolfram blog.
Five-Step Strategy to Solving Word Problems
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Advances in materials science have given rise to vel materials with unique properties, through the manipulation of structure at the atomic level. Elucidating the shape and form of matter at this scale requires the application of mathematical concepts. This 2006 book presents the geometrical ideas that are being developed and integrated into materials science to provide descriptors and enable visualisation of the atomic arrangements in three-dimensional space. Emphasis is placed on the intuitive understanding of geometrical principles, presented through numerous illustrations. Mathematical complexity is kept to a minimum and only a superficial kwledge of vectors and matrices is required, making this an accessible introduction to the area. With a comprehensive reference list, this book will appeal to those working in crystallography, solid state and materials science. | 677.169 | 1 |
In the mathematics course of secondary schools students get acquainted with the properties of inequalities and methods of their solution in elementary cases (inequalities of the first and the second degree).
In this booklet the author did not pursue the aim of presenting the basic properties of inequalities and made an attempt only to familiarize students of senior classes with some particularly remarkable inequalities playing an important role in various sections of higher mathematics and with their use for finding the greatest and the least values of quantities and for calculating some limits.
The book contains 63 problems, 35 of which are provided with detailed solutions, composing thus its main subject, and 28 others are given in Sections 1.1 and 2.1, 2.3, 2.4 as exercises for individual training. At the end of the book the reader will find the solutions to the' given exercises.
The solution of some difficult problems carried out individually will undoubtedly do the reader more good than the solution of a large number of simple ones.
For this reason we strongly recommend the readers to perform their own solutions before referring to the solutions given by the author at the end of the book. However, one should not be disappointed if the obtained results differ from those of the patterns. The author considers it as a positive factor.
When proving the inequalities and solving the given problems, the author has used only the properties of inequalities and limits actually covered by the curriculum on mathematics in the secondary school.
P. K orookin,
6
CHAPTER 1 Inequalities
The important role of inequalities is determined by their application in different fields of natural science and engineering. The point is that the values of quantities defined from various practical problems (e.g. the distance to the Moon, its speed of rotation, etc.) may be found not exactly, but only approximately. If x is the found value of a quantity, and I'1x is an error of its measurement, then the real value y satisfies the inequalities
x-I I'1x 1 ~ y ~ x + 1 I'1x I.
When solving practical problems, it is necessary to take into account all the errors of the measurements. Moreover, in accordance with the technical progress and the degree of complexity of the problem, it becomes necessary to improve the technique of measurement of quantities. Considerable errors of measurement become inadmissible in solving complicated engineering problems (i.e., landing the mooncar in a specified region of the Moon, landing spaceships on the Venus and so on).
1.1. The Whole Part of a Number
The whole (or integral) part of the number x (denoted by [ x]) is understood to be the greatest integer not exceeding x. It follows from this definition that [z] ~ x, since the integral part does not exceed x. On the other hand, since [xl is the greatest integer, satisfying the latter inequality, then [xl + 1 > x.
Thus, [x] is the integer (whole number) defined by the inequalities
[xl~x <~Jx] +' 1.
7
For example, from the inequalities
3<31<4, 5< ~ <6, -2<-V"2<-1, 5=5<6
it follows that
[31]=3, [1;J=5, [-V2]=-2, [5]=5.
The ability to find the integral part of a quantity is an important factor in approximate calculations. If we have the skill to find an integral part of a quantity x, then taking [xl or [xl + 1 for an approximate value of the quantity x, we shall make an error whose quantity is not greater than 1,
since
o:s;; x - [xl < [xl + 1 - [z ] = 1,
o < [xl + 1 - x:S;; [xl + 1 - [x 1 = 1.
Furthermore, the knowledge of the integral part of a quantity permits to find its value with an accuracy up to -}
The quantity [xl + -} may be taken for this value.
Yet, it is important to note, that the ability to find the whole part of a number will permit to define this number and, with any degree of accuracy. Indeed, since
[Nxl<Nx<[Nxl + 1,
then
[Nx]./ ./ [Nx] +_1
N ~x~ N N •
Thus, the number
[Nx] + 1
----r 2N
differs from the number x not more than by 2~. With large N the error will be small. The integral part of a number is found in the following problems.
Problem 1. Find the integral part of the number
111 1
,1:= 1 + V2 + -Va + V4 + V5 .
8
Solution. Let us use the following inequalitres
1-~ 1<1,
0.7 <V~ <0.8,
0.5<vi <0.6, 0.5<V! <0.5, 0.4<V! <0.5
(which are obtained by extracting roots (evolution) with an' accuracy to 0.1 in excess or deficiency). Combining them we get
1 + 0.7 + 0.5 + 0.5 + 0.4 < x <
< 1 + 0.8 + 0.6 + 0.5 + 0.5, that is, 3.1 < x < 3.4, hence, [xl = 3.
In this relation, it is necessary to note that the number 3.25 differs from x not more than by 0.15.
Problem 2. Find the integral part of the number
1 1 1 1
y=1+ V2 + -vs + V4 + ... + V1000000
Solution. This problem differs from the previous one only by the number of addends (in the first, there were only 5 addends, while in the second, 1000, 000 addends). This circumstance makes it practically impossible to get the solution by the former method.
To solve this problem, let us investigate the sum
1 1 1 1
1+ ,10 + ,r + ,r + ... +--::;-t.=-
v2 v3 v4 vn
and prove that
2Vn+1-2Vn< ~n <2Vn -2Vn-1. (1)
Indeed, since
2V n+ 1-2Vn = 2 eVii+1- Vii) (Vn:tI+ lfri) = Vn+1+ lfri
. 2
::::::::;::
Vn+t+ lfri
e
and
yn+1> Vii,
it follows that
2 Yn+1-2 Vn <_2_=_1_. 2Vn Vn
Thereby proof has been made for the first part of the inequality (1); its second part is proved in a similar way.
Assuming in the inequalities (1) n = 2, ~, 4, ... , n, we get
2y3-2Y2< -0 <2Y2-2, 2Y4-2Y3< 0 <2y3-2Y2, 2 V5-2 Y4 < ,~_ <2 Y4-2y3,
. v 4
2 V n+ 1-2 Vn < -Vn <2 Vn-2Y n-1.
Adding these inequalities, we get 2yn+1-2Y2<
1 1 1 1 «r:
< 112 + lis + 114 +"'+1/n <2 V n-2.
Adding 1 to all parts of the obtained inequalities, we find
2 Yn+1-2 Y2+1 <
1 1 1 I 1 ~/- 2
<1+ ,'" +-/- + ,r + ... T ,r <2 V n-1. ()
v2 1 3 v4 vn
Since 2 Y2 < 3, and V n + 1 > yri, it follows from the inequalities (2) that
~/- 1 1 1
2 V n-2< 1 + V2 + V3 -+- 114 + ...
1 2,r-
+ Vii < V n-:1.
(3)
Using the inequalities (3) we can easily find the integral part of the number
From the inequalities (2) it follows that the number 1998.6 differs from y not more than by 0.4. Thus, we have
calculated the number y with an accuracy up to 19~~.4 % = = 0.02%. The numbers 1998 and 1999 differ from the number y not more than by unity, and the number 1998.5 differs not more than by 0.5.
Now let us examine the next problem of somewhat different pattern.
Problem 3, Prove the inequality
1 3 5 99 1
x='2'"4''6'" 100 < 10'
Solution, Suppose
Since
1 2 3 4 5 6 99 100
'2 < '3' "4 < 5"' '6 < '"1' "" 100 < 101 '
it follows that x<y and, consequently,
2 1 2 3 4 5 6 99 100 1
x < xy='2''3'"4'5"'!f''"1'" 100' 101 = 101 •
Finding the square root of both members of the inequalities yields
1
x< VTIIT <0,1.
if
Exercises
1. Prove the inequalities
V-V-11
2 n+1-2 m «; ,r + Vm+1 + ...
V m m+1
1 V- V... + VB <2 n-2 m-1.
2. Prove the inequalities
1800< 1 1 + 1
, 1110,000 + 1110,001 +... 111,000,000 <
< 1,800.02.
3. Find [50z] , where 1
Z= 1110,000 +
1 1
1110,001 + ... + 111,000,000
Answer. [50z] = 90,000.
4. Prove the following inequality using the method of mathematical induction
1 3 5 2n-1 «; 1
2'4' 6'" 2n ---:::: V3n+1
5. Prove the inequality
1 3 5 99 1
2'4'6'" 100 <1"2'
1.2. The Arithmetic Mean and the Geometric Mean
If Xl' X2, ••• , Xn are positive numbers, then the numbers formed with them
x1+x2+" .+xn
a = --=--'--=---:....----'--::....
n
g=;Y XtX2'" Xn
are called, respectively, the arithmetic mean and the geometric mean of the numbers Xl, X2, ••• , Xn. At the beginning of the last century, the French mathematician O. Cauchy has established for these numbers the inequality
s «; a,
often used in solving problems. Before proving the inequality we have to establish the validity of an auxiliary assertion
12
Theorem 1. 1/ the product n 0/ the positive numbers .tl, X2, ••• , Xn is equal to 1, then the sum of these numbers is not less than n:
XlX2, ••• , Xn = 1 =} Xl + x2 + ... + xn ~ n.
Proof. Use the method of mathematical induction'. First of all check up the validity of the theorem for n = 2, i.e. show that
XIX2 = 1 =} Xl + x2 ~ 2.
Solving the question, examine the two given cases separately:
(1) Xl = X2 = 1.
In this case Xl + X2 = 2, and the theorem is proved.
(2) 0 < Xl < X2•
Here Xl < 1, and X2 > 1, since their product is equal to 1. From the equation
(1 - Xl) (X2 - 1) = X2 + Xl - XlX2 - 1 it follows that
Xl + X2 = XlX2 + 1 + (1 - Xl) (X2 - 1). (4)
The equation (4) has been established without limitations to the numbers Xl and X2• Yet, taking into account, that XIX2 = 1, we get
Xl + X2 = 2 + (1 - Xl) (X2 - 1).
At length, since Xl < 1 < X2, then the last number is positive and Xl + X2 > 2. Thus, for n = 2 the theorem is already proved. Notice, that the equation
Xl + X2 = 2
is realized only when Xl = X2• But if Xl =1= X2, then Xl + X2 > 2.
Now, making use of the method of mathematical induction, assume that the theorem is true for n = k, that is, sup-
1 More detailed information concerning mathematical induction is published in the book by 1. S. Sominsky "The Method of Mathemati cal Induction", Nauka, Moscow, 1974.
First of all, it is necessary to notice that if XIX2X3 ••• Xh,Xl<+1 = 1,
then there may be two cases:
(1) when all the multipliers XII X2, Xa, ••• , xl<, Xh+l are equal, that is
(2) when not all multipliers are equal.
In the first case every multiplier is equal to unity, and their sum equals k + 1, that is
Xl + X2 + Xa + ... + Xl< + Xh+l = k + 1.
In the second case, among the multipliers of the product XIX2 ••• Xh,Xh+lI there may be both numbers greater than unity and numbers less than unity (if all the multipliers were less than unity, then their product as well would be less than unity).
For example, suppose Xl < 1, and Xh+l> 1. We have
(,XIXh+l) X2Xa ••• Xl< = 1.
Assuming YI = xlxh,+ 11 we get
YlX2Xa ••• Xl< = 1.
Since here the product k of positive numbers is equal to uni ty, then (according to the assumption) their sum is not less than k, that is
Raising both parts of the inequality to the (n + 1)th power, we shall obtain
(1 + ~ r < (1 +-n~1 r+1, that is xn< Xn+1' The second inequality is proved in a similar way.
Problem 3. Prove that
_ (1 _1 )"+1
Yn -_ +
n
decreases with the increase of the number n,
( 1) n+2
Yn>Yn+l=1+n+1 •
Solution. We have
_ (1 _1) n+ 1 _ ( n + 1 ) n+ 1 _ 1
Y n - + n - n - ---:(-=---=---n~~"--) n-+-:-:-l
n+1
1
that is
( 1 __ 1_)n+l zn+l
n+1
(see designations of Problem 2). Since Zn increases with the increase of the number n, then Yn decreases.
In Problems 2 and 3 we have proved that
Xi = ( 1 + + ) 1 = 2 < X2 = ( 1 + ~ ) 2 =
=2.25<X3<"· <xn< ... ,
1
u. = ( 1 + ~ ) 2 =, 4 > Yz =
= (1 + ~ ) 3 = 3.375 > v« > ... > Yn > ....
20
On the other hand,
2=XI<Xn= (1 ++)n < (1 ++ )n+l =Yn<YI=4.
Thus, the variable Xn satisfies two conditions:
(1) Xn monotonically increases together with the increase of the number n;
(2) Xn is a limited quantity, 2 < Xn < 4.
It is known, that monotonically increasing and restricted variable has a limit. Hence, there exists a limit of the variable quantity Xn. This limit is marked by the letter e, that is,
e = lim Xn = lim ( 1 + -1-f .
n-e- eo 'n-s oc n
As the quantity xn increases reaching its limit, then Xn is smaller than its limit, that is
Xn = ( 1 + + r < e. (8)
It is not difficult to check that e < 3. Indeed, if the number n is high, then
Xn<Yn<Y5,=(1-t- ~)6=2.985984.
Hence,
e ~ lim Xn ~ 2.985984 < 3.
rt-s-co
In mathematics, the number e together with the number Jt is of great significance. It is used, for instance, as the base of logarithms, known as natural logarithms. The logarithm of the number N at the base e is symbolically denoted by In N (reads: logarithm natural N).
It is common knowledge that the numbers e and Jt are irrational. Each of them is calculated with an accuracy of up to 808 signs after the decimal point, and
e = 2.7182818285490 ....
Now, let us show that the limit of the variable Yn also equals e. Indeed,
lim Yn = lim ( 1 + ! ) n+ 1 =-c lim ( 1 + ! ) n ( 1 + ! ) =
=,e·1=e.
21
Since Yn diminishes coming close to the number e (Problem 2), then
( 1 1 ) n+ 1
+- »:».
, 'n
(9)
Problem 4. Prove the inequality
(10)
Solution. We shall prove the inequality (10) using the method of mathematical induction. The inequality is easily checked for n = 1. Actually,
Assume, that the inequality (10) is true for n = k, that is
Multiplying both members of the last inequality by k + 1, we get
(k+1)k!=(k+1)!>(:r(k+1)=( k~1 )k+1 (1+*r
Since, according to the inequality (8) (1 + + r < e, then
(k+1)!> (k~1 r+1 : = (k~1 r+1,
that is the inequality (9) is proved for n = k + 1. Thus the inequality (9) is proved to be true for all values of n.
Since e < 3, it follows from the inequality (9) that
I (n)n
n.> ""3 .
By means of the last inequality, it is easy to prove that 300! > 100300•
Indeed, setting in it n = 300, we get
300! > ( 3~O ) 300 = 100300•
22
The inequality
I ( n+ 1 )n+l
n.<e -
e
is proved completely the same way as it is done with the inequality of Problem 4.
1.4. The Bernoulli Inequality
In this section, making use of Theorem 2 we shall prove the Bernoulli inequality which is of individual interest and is often used in solving problems.
Theorem 3. It x ;;:: -1 and 0< a < 1, then
(1 +x)ct :::;; 1 + ax. (11)
However if a < ° or a> 1, then
(1 + x)ct ;;:: 1 + ax.
(12)
The sign of equaliiu in (11) and (12) holds only when x = 0.
Proof. Suppose that a is a rational number, bearing in mind that ° < a < 1. Let a = !:!:., where m and n are n
positive integers, 1 :::;; m < n. Since according to the condi-
tion, 1 + x ;;:: 0, then
m
(1+x)ct=(1+xfn =;Y(1+x)m.1n m =
=;Y(1+x)(1+x).- .. (1+x).1.1 ... 1:::;;
-----' ~
m
n-m
«1+x)+(1+x)+ ... +(1+x)+1+1+ ... +1 =
n
The sign of equality occurs only when all multipliers standing under the root sign are identical, i.e., when 1 + x = 1, x ,= O. But if x =1= 0, then
(1 +: x)ct < 1 +- ax.
Thus, we have proved the first part of the theorem considering the case, when a is a rational number.
23
Assume now, that a is an irrational number, 0 < a < 1.
Let r1, r2, ••• , r n ... be the sequence of rational numbers, having for a limit the number a. Bear in mind that ° < < r; < 1. From the inequalities
(1 + xrn<1 + rnx, x ~ -1, n = 1, 2, 3, ... , already proved by us for the case when the exponent is a rational number, it follows that
(1 + x)a = lim (1 + x(n< lim (1 -/- rnx) = 1 + ax.
rn-CX
r ... a n
Thus the inequality (11) is proved for irrational values of a as well. What we still have to prove is that for irrational values of a when x =1= ° and 0 < a < 1
(1 + x)a < 1 + ax,
i.e., that when x =1= 0 in (11), the sign of equality does not hold. For this reason, take a rational number r such t hat a < r < 1. Obviously, we have
a
(1 + x)a = [(1 + xfry.
Since ° < __::_ < 1, then as it has already been proved r
Hence,
(t + x)a< (1 + ~ x r.
1£ x =1= 0, then (1+~x)r<1+r.!!_x~c1+ax, that is
r r
(1 +x)a <1 + ax.
Thus the first part of the theorem is proved completely.
Now, move on to proving the second part of the theorem. If 1 + ax < 0, then the inequality (12) is obvious, since its left part is not negative, and its right part is negative.
If 1 + ax ~ 0, ax ~ -1, then let us consider both cases separately.
Su ppose a > 1; then by virtue of the first part of the theorem proved above we have
~ 1
(1 + ax)a <1 +- ax= 1 +x.
ex
24
Here the sign of equality holds only when x = O. Raising both parts of the last inequality to the power a we get
1 +- ax ~ (1 + .1')a.
Now let us suppose a < O. If 1 + ax < 0, then the inequality' (12) is obvious. But if 1 + ax ;? 0, then select the
positive integer n, so that the inequality - ..::..< 1 would n
be valid. By virtue of the first part of the theorem we get
a
(1+x)n::p 1a ::pi + ~ X
1--;r n
( the latter 'inequality is true, since 1 ;? 1- ~: X2) . Raising both parts of the latter inequality to the nth power we get
(1 + xt::P(1 +..::.. x)n::p1 + n"::" X= 1 + ax.
n' n
Notice, that the equality is possible only when x = O. Thus, the theorem is proved completely.
Problem 1. Prove, that if 0> a> -1, then
(n+ 1)a+l_na+l na+1_(n_1)a+l
a+1 < w «: a+1 (13)
Solution. Since 0 < a + 1 < 1, then accord ing to the inequality (11) we have
Proof. For the case, when the numbers ex and ~ have different signs the theorem has been proved above (refer to Problem 7, Sec. 1.2 and the definition prior to it). Thus, we have to prove the theorem only for the case when ex and ~ have the same signs.
Assume, that 0 < ex < ~, and let
Dividing C(3 by k, we get
Now, supposing
we obtain
(15)
27
Since
1
( dt + d2 ~ .•• + dn ) a =
((~)a (~)a. (.!:!!:...)a)a
k + k + ... + k
, n
1
a+ a+ -t a -
_ 1 (a1 a2 ••. - an ) a _ 1 _ 1 . __
_ - _-ca--ca;.-1,
k n k Ca
then
d1+d2+ .. · +dn n
Suppose
dl = 1 +XI, d2 = 1 +X2, ••• , dn = 1 +xn.
From the equality dl + d2 + + dn = n it follows that
Xl +X2 + -I-xn = o.
On the basis of Theorem 3 (notice, that ! > 1) we have
1
! ! ~
a; = (1 +xn) ::;:>1 +-aXn. J
Adding these inequalities, we get
!! ! ~
d1 +d2 + ... +dn ::;:>n+-(X1+X2+ ... +Xn)=n. (16)
a
(*)
From the inequalities (15) and (16) it follows that 1
C: ::;:> ( : ) ~ = 1, C(3::;:>k = Ca.
28
I t is necessary to note that C fl = k = Ca. only when the signs of equality occur everywhere in (*), that is when
Xl = x2 = = xn = 0 (Theorem 3). In this case d, =
,= d2 = = dn = 1 and, hence, a1 = a2 = ... = an =
= k, But if the numbers aI' a2, ... , an are not identical, then
Thus Theorem 4 is proved regarding the case when ° < <a< ~.
If a < ~ < 0, then ° <1.< 1. Reasoning the same way a
as before, we get in (*) and (16) the opposite signs of inequalities. But. since ~ < 0, then from the inequality
1. 1. _i
df +d~ + ... +d;t --------<;:1
n
it follows that
that is
Thus, Theorem 4 is proved completely.
Further on we shall name the geometric mean by mean power of the order zero, that is, we shall assume g = co.
i.e. the harmonic mean does not exceed the geometric mean, the geometric mean in its turn does not exceed the arithmetic mean, while the arithmetic mean does not exceed the root-mean-square of positive numbers. For example, if
The use of inequalities in finding the greatest and the least function values and in calculating limits of some sequences will be examined in this chapter. Besides that, some important inequalities will be demonstrated here as well.
2.1. The Greatest and the Least Function Values
A great deal of practical problems come to various functions. For example, if x, y, z are the lengths of the edges of a box with a cover (a parallelepiped), then the area of the box surface is
S = 2xy + 2yz + 2zx,
and its volume is
v =, xyz.
If the material from which the box is made is expensive, then, certainly, it is desirable, with the given volume of the box, to manufacture it with the least consumption of the material, i.e., so that the area of the box surface should be the least. We gave a simple example of a problem considering the maximum and the minimum functions of a great number of variables. One may encounter similar problems very often and the most celebrated mathematicians always pay considerable attention to working out methods of their solution.
Here, we shall solve a number of such problems, making use of the inequalities, studied in the first chapter". First of all, we shall prove one theorem.
1 Concerning the application of inequalities of the second degree to solving problems for finding the greatest and the least values see the book by I.P. Natanson "Simplest Problems for Calculating the Maximum and Minimum Values", 2nd edition, Gostekhizdat, Moscow, 1952.
32
Theorem 5. If a>O, a>1, x:>-O, then the junction 1
xu. _ ax takes the least value in the point x = ( : ) 1 -a , a
(a )a=-T""
equal to (1- a) a .
Proof. The theorem is proved very simply for the case when a = 2. Indeed, since
x2 _ ax = ( x _ ~ ) 2 _ a: '
a
the function has the least value when x ="2 > 0, this
a2
value being equal to - T'
In case of arbitrary value of a > 1 the theorem is proved by using the inequality (12), demonstrated in Theorem 3. Since a > 1, then
(1 + z)a ~ 1 + az, z ~ -1,
the equality holding only when z = 0. Assuming hsre, that 1 + Z = y, we get
ya ~ 1 + a (y - 1), ya - ay ~ 1 - a, y ~ 0,
the sign of equality holds only when y = 1. Multiplying both members of the latter inequality by cu., we get
the conclusion, obtained earlier by a different method. The function x3 - 27 x takes the least value in the point
1 3
_ ( 27 )"'3"='1 _ 3 ( 27 )3-"1
x- 3"" -, equal to (1-3) 3 = -54.
Note. Let us mark for the following, that the function ax _XIX = -(xa. - ax),
where a > 1, a> 0, x ~ 0, takes the greatest value in the point
1
(a .)a=-t
X= a f
Fig. 1
equal to
a.
(a )a=-t
(a-i) a .
Problem 1. It is required to saw out a beam of the greatest durability from a round log (the durability of the beam is directly proportional to the product of the width of the beam by the square of its height).
Solution. Suppose AB = x is the width of the beam, BC = y is its height and AC -- d is the diameter of the log (Fig. 1). Denoting the durability of the beam by P, we get
P = kxy2 = kx (d2 - x2) = k (d2x - x3).
The function d2x - x3 takes the greatest value when 1
( d2)3=1 d 2
X= 3 = va ' y2=d2-X2=3d2,
d V- VY= Va 2=x 2.
34
Thus, the beam may have the highest (greatest) durability if the ratio of its height to its width will be equal to y:r ~ 7
the sign of equality occurring only when y = 1. From the last inequality it follows, that
ya _ ay ~ 1 - a, (cy)a - aca-1 (cy) ~ (1 - a) ca.
Assuming a = -aca-\ x = cy, we get
t
(a )a=T
the equality holding only when x = c""'" --=ex .
36
Thus, the function xrx +- ax takes the least value in the point
1
_ (_a )a=T"
x- ,
-(Z
o:
equal to (1- a) ( ~(Z ) rx-1 • For example, the function 1
sr-+-27x, x>O, 11 x
takes the least value in the point 1 -1-
x~ C{ f--' ~ 2~'
This value equals
1 -3
_ 1 -1
(1 +{ ) ( y) 3 = 4.
Problem 5. Find the optimum dimensions of a cylindrical tin having a bottom and a cover (dimensions of a vessel are considered to be the most profitable, if for a given volume the least amount of material is required for its manufacture, that is, the vessel has the least surface area).
Solu tion. Let V = nr2h be the volume of the vessel, where r is the radius, h is the height of the cylinder. Tho total surface area of the cylinder is
S = 2nr2 + Zscrh, V
Since h = --2 , then nr
V 2V
S=2:rtr2+2nr--2 =2nr2+-.
nr r
Assuming
1
X=-, we get r
S=2nx-2+2V;r=~n (X-2+ : x).
37
The function x-2 + ~ x, according to the solution of the 31
previous problem, takes the least value when
1
X- (~) -2-1 - V· 231
- 231 - V'
Returning back to our previous designations, we find
_!_ = J31 231 r3 = ~ = 31r2h r __ .!!_
r V ' 231 231' 2 '
h = 2r = d.
Thus, the vessel has the most profitable dimensions, if the height and diameter of the vessel are equal.
Exercises
6. Find the greatest value of the function x (6 - X)2 when 0 < x < 6.
Indication. Suppose y = 6 - x.
7. From a square sheet whose side is equal to 2a it is required to make a box without a cover by cutting out a square at each vertex and then bending the obtained edges,
20
Fig. 4
Ie--- 2a-2x
so that the box would be produced with the greatest volume (Fig. 4). What should the length of the side of the cut-out squares be?
8. Find the least value of the function
~6, + S,x2 t 5,
9. Find the least value of the function x6 _ 8x2 + 5.
10. Find the greatest value of the function
xcx, - ax when 0< ex < 1, a > 0, x;;? 0.
11. Prove that, when x ;;? 0, the following inequality
is true
Vx-<:; +2x.
12. Prove that, when n ;;? 3, the following inequality is
true
yn>n+Yn+1.
Indication. Make use of the inequality (8). 13. Find the greatest of the numbers
~ f') V- V· - 5 /'"i'; n r:
1, V "" 3, 4, V 5, ... , V n,
14. Prove the inequality
n/- 2
V n<1+ lin .
15. Prove the inequality
(1 + al) (1 + a2) ••• (1 + an) ;;?
? 1 + al + a2 + ... + an, if the numbers ai are of the same sign and are not less than -1.
The series is said to be convergent, if the sequence of its partial sums has a finite limit. In this case the number S = lim Xn is called the sum of the series.
n ... oo
From Problem 3, it follows that the series
11111 1 1
1-2:+"3-"4+"5-6"+'" + 2n---,-1 --2n+'"
converges and its sum equals In 2.
Problem 4. The series
,1 1 1 1
112+"3+4+'" +-n+'"
is called harmonic series. Prove that the harmonic series diverges.
Solution. According to the inequality (23)
.!.>In n+1 •
n n
Assuming n = 1, 2, 3, ... , n, write n inequalities
2
1 >lnT'
1 3
2>ln2:'
1 4
"3>ln"3'
.!.>In n+1 •
n n
Adding them, we get
1 1 1 1 I 2·3·4 ... (n+1) 1 ( 1)
Xn= +2:+"3+'" +-n> n 1.2.3 ... n =;: n n+ .
It follows from this. inequality that
lim xn >-lim In (n + 1) = 00;
hence, the harmonic series diverges. 46
Problem 5. Prove that the series
1+_1 +_1 + ... +~+ ...
20. 30. n
converges at any a> 1.
Solution. The sequence of partial sums of this series
(2())
Xl= 1,
1 X2= 1 +""""'(i""' 2
1 1
x3=1+-+-
2a. 3a. '
1 1 1
x.-1+-+-+-
~- 2a. 3a. 40.'
111 Xn= 1 +-+-+ ... +-
2a. 3a. na.
is monotonically increasing, that is
Xl < x2 < X3 < X4 < ... < Xn < ...
On the other hand, it is known that monotonically increasing limited sequence of numbers has a finite limit. Therefore, if we prove that the sequence of numbers Xn is. limited, then the convergence of the series (26) will be proved as well. Suppose
Y2n=1 __ 1_+_1 1_+_1 1_+ ...
2a. 3a. 4a. 5a. 6a.
1 1
... + (2n-1)a. (2n)a.
Since
Y2n = 1- ( 2~ - 3~ ) - ( 4~ - 5~ ) - .••
'" -C2n~2)o. - (2n~1)a.) - (2:)a.' then (the numbers in each bracket are positive)
Y2n < 1.
On the other hand,
11111 1 1
Y2n=1-~+ -sa-Ta+ 5a.- 6a.+· •. + (2n-1)a. (2n)a. =
= (1+_1_+_1_+_1_+_1_+_1_+ + 1 +_1_)_
2a. 3a. 4a. 5a. 60. . . . (2n-1)a. (2n)a.
47
1 1 1 1
-2 (za-+ 4a +6U"+ ... + (2n)a ) =
= (1+~+~+_1_+_1_+_1_+ .. + 1 +_1_)_
2a 3a 4a 5a 6a . (2n-1)a (2n)a
2 ( 1 1 1 )
-za- 1 + 2a +?+ ... + na •
Since Xn=1+~+~+ ... +_1 , then
2 3~ na
2
Y2n = x2n -za- Xn·
Now, since X2n > Xn, Y2n < 1, then
2 2a_2
1 >Y'l.n >Xn--Xn=-- Xn.
Za Za
Hence, it follows that
za Xn< 2a_2 '
that is, the numbers Xn are limited when a> 1. Thus, it is proved that the series (26) converges and its sum is 2a
not greater than --.
2a_Z
For example, if a = 2, then
1 1 1 22
xn=1+22"+"32+'" +-;i2< 22_2 =2,
S = lim Xn = 1 + Z12 + 3\ + ... +-; + ... <2.
n-s-cc n
In the course of higher mathematics it is proved that
1 1 1 n2
S= 1 +'22+3"2+" . +"/i2+'" ="""6' (27)
Exercises
22. Find the sum of the series
S = 1- ;2 + 12 -12 -I- ... + (_1)n-l ;2 + ...
Indication. Use the equality (27).
n2
Answer. S = 12'
48
1
23. Prove the inequalities
na+1 a a rx (n-;--1f+1
ex. -1-1 < 1 + 2 + 3 + ... + n < ex. + 1 ,a> O.
24. Assuming
xn=1+2a+3eG+ ... _I_neG,
prove that
. xn 1
hm--=-- a>O.
rr-s-co neG + 1 ex.+·1'
25. Prove the inequality (ajbjcj + a2b2c2 + ... + anbncn)3--<
--«a~+a~+ ... +a~) (b~ +b~+ ... +b~)(c~+c~+ ... +c~),
if ak, bk, Ck are positive numbers.
Indication. Use the inequality (7) and the method given in (22).
. 1 1 1 1
26. Assuming Xn =-n+ n+1 + n+f + ... + kn " whe-
re k is a positive integral number, prove that
lim Xn = Ink.
n .... oo
Indication. Use the method of solving Problem 2 of the present section.
I
I
I
I
2.4. The Use of Inequalities
for Approximate Calculation of Quantities
At the very beginning of Chapter 1, we have paid attention to the fact that practical problems require, as a rule, an approximate calculation of quantities and, as well, an ability to treat such approximately calculated quantities. A more accurate estimation of such quantities will certainly permit to decrease errors in solving problems.
In the present section, we are going to return to an approximate calculation of numbers of the form
1/2 4-0866
49
In Sec. 1.1 we have succeeded in finding the number S n, k with an accuracy of up to 0.4 for k = 1, n = 1,000,000
• 1
and ex. = 2" (refer to Problem 2). In the same section (see
Exercises 2 and 3), for n = 106 and k = 10,000, we were able to find the number Sn. k already with an accuracy of up to 0.01. The comparison of these two examples shows, that the indicated method of their solution yields much better results of calculation for greater values of k.
In Sec. 1.4 (Problem 3) we found the integral part of the
number Sn. k for k = 4, n = 106 and ex. = f. Thus, this number was also calculated with an accuracy of up to 0.5. However, we could not find the integral part of the number
1
Sn.I for ex. = 3 and n = 106 because the method of calcu-
lation of such quantities, indicated in Chapter 1, did not permit doing it. In this section, we shall improve the method of calculation of the quantity Sn. l' This improvement will make it possible to find similar quantities with a higher degree of accuracy quite easily.
Lemma 1. If =. > X2 > Xa > > Xn, then
0< A = Xl - X2 + Xa - x4+ + (_1t-l Xn < Xl'
Proof. The number of positive terms in the written algebraic sum is not less than the number of negative terms. Besides this, the preceding positive terms are greater than the following negative term. This proves that their algebraic sum is positive, A > O. On the other hand, since
A = Xl - (X2 - Xa + xCG - ... + (_1)n-2 xn)
and the quantity in brackets is positive too, then A < xl" Thus, the lemma is proved.
- V 2.106+ 1- 0.0003 +,12 = 0.1710 + ,12, Now let us return to the quantity C. We have
1 1 111 1 . 1
C = 1- yz + Y3 - Yl; + V5 - YB + y"7 - ys +D =
1 1 111
= 1 - yz + Y3 - y4: + y5 - y6 +
1 1
+ y"7- VS+0.1710+,12=
1 1 ( 1) 1 1 1 1
= 1 -- 2" - yz 1 +"2 + Y3 + y5 - YB + Y7 +
+0.1710+,12= ~ _ 3yZ + ~3 + 1~5 _ ~ + \"7 + +0.1710 ± ,12'
Thus, for the calculation of the number C with an accuracy of up to 3.10-4 it will be required to find only 5 roots and to produce a number of arithmetic operations. Using the tables and carrying out necessary calculations, we find
C = 0.6035 + ,12'
Taking into consideration the found quantities Band C, and returning to the quantity A, we get
A = (V2 + 1) (B - C) = (V2 + 1) (827.8226+,13) =
= (V2 + 1) ·827.8226 + 2.5,13'
where
1 2.5,13 1 ~ 2.5 (I ,11 1 + 1 ,12 I) < 2.5.5.10-4 < 2.10-3.
Thus, the calculation with an accuracy of up to 2.10-3 will be
A = (V2 + 1) 827.8226 = 1998.539.
55
Problem 2. Calculate the number
1 1 1
A = 1 + y- + 4 r- + ... + 4 ,-
y 2 V 3 y' 1012
with an accuracy of up to unity.
Solution. By virtue of Theorem 8
V2
A = 2-V2 (V 10~2+1 + 11 10~2_f-2 + '" + t 2\012 ) -
V2( 1 1 1)
- 2-V2 1 - V2 + V3 - ... -. V2.1012 .
The first term can be easily found and with a high degree of accuracy by means of the inequalities (28). By virtue of these inequalities the first term can he substituted by the number
3 3
t~_ (2.1012/~_(1012)4 = ~ .109 (y/S-1) V} _ "-"""~-109.
2-12 1-~ 3 2-1/2 3
4
By virtue of Lemma 1 the sum
V2 1 1 1
2-V2 (1 - V2 + V3 - ... - V2.1012 )
is positive and is not greater than the first term. Since the term is less than two, then
4 4
3.109-2 <A < 3.109.
The extreme numbers differ from each other by 2, and from the number A by less than 2. The middle number ; .109 - 1 differs from A by less than unity. Substituting this number, we get
A = 1333333332.3 +~, I ~ I < 1.
Notice that the accuracy of calculating the number A, containing a trillion of addends, is extremely high. The relative error is less than
Notice, that the sign of equality holds only when the polynomial under consideration has a true root, i.e. when
at a2 an
bt=b;"='" =1i;;'
17. U sing the inequality (19), we get 2 _ ( at + a2 + ... + an ) 2 _
C! - n -
(at 1 an 1) 2
= lIii Vii + ... + lin vn -<
:::;;;; ( ai + a~ + . , , + a~ ) (_!_ + _!_ + ' .. + _!_) =
n n n n n n
n
64
Hence, it follows that C1 ~ C2 (the arithmetic mean does not exceed the root-mean-square).
18. From the inequality
(vn+I + V n-1)2=n+1 + 2Vn2--1 +n-1 =
= 2n + 2 V n2 - 1 < 2n + 2 V n2 = 4n
it follows that
V n + 1 + V n -1 < 2 Vn,
_1_< 1 =
2Vn yn+1+yn-1
_ Y~-~ _Y~-Y~
- (Yn +1+ -vn=-T) (Yn +1- Vn -1) - 2
Multiplying by 2, we get
1 11- ll-
yn < v n+1-v n-1.
19. Setting in the inequality of Exercise 18n = 2, 3, ... , Vn 1-
-<V3-1
V2 '
1 1/- 1/-
Y3 < V 4- V 2,
1 1/- 1/-3
y;;;<v5-v,
~<Y6-V4,
;1i <Yn+1-Vn-1.
Combining the written inequalities, we get
1 1, 1 11-' 11- 11-
Vz+ V3""+YIi < V n-r1+v n-v 2-1.
Adding 1 to both parts of the inequality, we finally get
,
1 1 1 1 1
1+ Y2+ Y3+ y7;+ 115+ ... + yli <
< y n+1 + Vn - V2. 6!)
Note. It was proved in Sec. 2.1 that
1 1 1 V- ,(-
1+l/2+Vg+ .. ·+-vn>2 n+1-2v2+1.
The numbers V n + 1 + Vn - V2 and 2 V n + 1 - - 2 V2 + 1 differ from each other less than by 0.42. Each of these numbers could be taken for an approximate value of the sum
1 1 1
1 + V2 -+- 113 + ... + -vn = Zn-
Let us notice without proving, that the number V n + 1 + + lin - V2 differs less from the number Zn, than the number 2 V n + 1 - 2 li2 + 1.
3
20. The function x'~5 takes a negative value when
x < O. Therefore, the greatest value of the function is obtained for positive values of x.
Since
x3 1
x4 + 5 = 5 ( ; x + X-3) ,
then the greatest value of the function is reached in the same point in which the function ; x + X _3 takes the least value. It follows from Problem 4 Sec. 2.1 that the least value of this function is equal to
-3
( 1 )-=3="1 3
(1 + 3)~ = 4 ( 115 f;-.
The greatest value of
s
the function x4x+ 5 is equal to
3
154" 3 =20
5.4.( 115)4
To find the greatest value of the function x6 - O. 6x10, we get y = x6• It is clear that y>O. The function
10 (10 10
y-0.6Ji6 = 0.6 TY- ys)
15
3
66
takes the greatest value (see the note on p. 34) equal to
10
-6-
-1-0-
0.6 ( '~ -1) ( ~ r -, ~ 04.
21 A '~1l ming in this exercise that y = ;2 , we get 1
v J + -..;- = y - 4 -/- ay.
r
1
The least value of the lunct ion y-4+ay, as it follows from Problem 4 See. 2.1, i.'< oq u»l to | 677.169 | 1 |
Mathematical Conversations to Transform Algebra Class
Three topics worthy of classroom discussions help beginning algebra students create meaning and build understanding as a community.
We don't see things as they are. We see things as we are.
—Anaïs Nin
Imagine your beginning algebra class. Together you and your students engage in making sense of notation, representations, and terms. Your students watch you—and one another—covertly to see what makes an acceptable question, what strategies are valued, what pictures and symbols mean and how they are used, how mathematics is written and discussed, and how to justify a solution. These "hidden regularities . . . become the taken-for-granted ways of interacting" that constitute the culture of doing mathematics in your class (Wood 1998, p. 170).
Classroom culture is established through both conversations and practices. Traditionally in mathematics class, we focus primarily on the latter; that is, we show our students what "doing mathematics" looks like and then ask that they try it themselves. In this article, I suggest three mathematical conversations that help bring covert—and often ineffectual—meanings into the light. The process I describe, sometimes called interrogating meaning, allows students to make explicit their assumptions about how, when, and for what purpose a mathematical notation, representation, or term is used (Rosebery et al. 2005). In other words, these conversations can help students recognize the strengths and weaknesses of their own interpretations and give them agency in changing them. Further, they allow us as teachers a window into our students' thinking.
Each mathematical conversation begins with a question for your class to ponder and discuss. I will describe some typical student thinking about each question and suggest some ways to build on students' ideas. Each topic has been addressed in the classroom with students at a variety of levels and makes a powerful point about mathematical culture.
The equals sign, =, signifies that two quantities are the same. It does not mean "write the answer."
During the first week of my algebra classes, I write the problem shown in figure 1 on the board and ask that everyone decide, without talking to one another, what numbers go in the blanks. Then we discuss their ideas.
More than ninety percent of upper elementary school students will interpret = as "write the answer to the preceding computation" and will fill in the blanks with 19 and 25 (Falkner, Levi, and Carpenter 1999). At the university level, a majority of students in introductory mathematics courses, such as college algebra, mathematics for elementary school teachers, and liberal arts mathematics, will also respond with 19 and 25 (Szydlik, Kuennen, and Seaman 2009), so it is a good bet that your beginning algebra students harbor this alternative conception about the meaning of the equals sign. This interpretation is not consistent with that of the mathematical community, but it is not objectively incorrect. We could have decided as a mathematical culture that = means "write the answer." However, we did not, and we had important reasons for defining equality the way that we did. Equality is a fundamental mathematical relationship between quantities signifying that these quantities are exactly the same. Students need to be told all this explicitly and helped to understand that meaning.
Carpenter, Franke, and Levi (2003) recommend that elementary school teachers pose number sentences in ways that better reveal limitations of the "write the answer" interpretation. For example, a teacher might write equations in a variety of forms—
___ = 4 + 7
3 + ___ = 12
8 + 4 = 5 + ___
—and ask students what values make the equations true. Those of us who teach algebra might take a similar tact before formally solving an equation. For example, after making the point that equality is a relationship between two numbers, I will write something like 2x = x2 – 3 and ask, "What values of x make this equation true? Can you find any without writing anything down?" Then, when we discuss algebraic moves that will help find solutions, I can reinforce for my students that the point of algebraic manipulation is to give us exactly those moves that allow us to preserve equality.
I find a conversation about equality particularly valuable when my students use the equals symbol to mean something like "and then I did this . . ." during the course of performing a series of algebraic steps. For example, a student solving for a semicircular area in which she needs to first square the circle's radius, then multiply by π and then divide by 2 might write something like this: A = 42 = 16 = 16 = 16/2= 8.
In a class in which we have had an equality conversation, the student and I might have this exchange:
Teacher: What did you mean here when you wrote 16 = 16?
Audrey: I was showing all my steps. First, I squared the 4, and then I multiplied by .
Teacher: So in this calculation, = means "and then I did this. . . ."
Audrey: [laughs] And then I did that and that.
Teacher: I get it. But looking at this, remember that I would think you meant that 16 is exactly the same quantity as 16. And that might even lead me to solve for and think that = 1.
Audrey: How would you think that? Oh, I see. You'd divide both sides of 16 = 16 by 16.
Teacher: [nods] Algebra is about all the things you can do that keep two sides of an equation balanced, and mathematicians have decided to use the equals sign to show that balance. It is important to write mathematics in a way that is consistent with that meaning. It gives you power in organizing your work, and it allows you to communicate with others who are learning the language of algebra. How could you change what you wrote so that it made sense to me?
The focus here is not on the student being wrong. Rather, I acknowledge that the student did have a meaning for the symbol, that her meaning made some sense, but that she will not be able to effectively communicate her thinking in the language of algebra unless she adopts the mathematical culture.
I ask each student to take a minute to make up a mathematical meaning for the representation shown in figure 2. Then I solicit a variety of ideas. What follows is a typical start of a discussion.
Teacher: What do you see in this picture?
[Fifteen seconds or more of wait time]
Sara: I see a side view of a three-dimensional house.
Teacher: Do you mean the white part?
Sara: Uh-huh. You might get the view from the different sides and have to imagine the whole house.
Richard: I've seen those types of problems before.
Iiona: I see that. But I was picturing it [as] just the fraction 5/12.
Teacher: Okay. How did you see that?
Iiona: Five white squares out of the total of twelve squares.
Teacher: Who else saw that fraction? [Several hands are raised; many students see fraction representations by what is not shaded rather than by what is shaded.] Did anyone see the fraction 7/12? [Hands are raised.] Either one is reasonable, right? So I guess when we make pictures of fractions, we should say whether we are looking at the shaded part or the unshaded part. Other ideas?
Charlotte: Could it be the fraction 5/7?
Iiona: No. That would mean 5 out of 7, and it is 5 out of 12.
Charlotte: I mean 5 white and 7 shaded.
Teacher: So you are thinking of the ratio 5 white to 7 shaded [writing on board: 5 : 7]. Maybe we could write it like this? Does this seem okay to you, Iiona? [Iiona nods.]
Teacher: The big point here might be that this picture has lots of reasonable mathematical interpretations. There is not just one correct way to see it. When we make a representation, we need to talk about what it means for that particular problem or situation. Can anybody think of another possible meaning for this picture?
Diego: It could be an area model for a probability problem.
Teacher: Ah. Can anyone come up with a problem for which that picture would be a model?
I have heard all these responses (depending in part on the current content of the class in which the question was posed) and lots of others too. For example, students have said that the picture shows 12 – 5 = 3 + 4. That it is the number 17 (on a digital clock). That it suggests the expression (3 • 4) – 5. That it is showing that 3/12 + 4/12 = 7/12. That it shows an impossible net for an open box.
The idea that representations do not carry meaning is not new. In the 1980s, researchers published empirical studies showing that even "standard" mathematical representations have many viable interpretations (Schipper 1982; Feller 1983; Radatz 1986). For example, when Schipper asked 109 first-grade children to interpret pictorial representations (like that shown in fig. 2) from standard first-grade mathematics textbooks, he found that about a third of the children gave alternative meanings to pictures that had (or were similar to those that had) been used in their classes; another third declined to attempt a representation.
Thompson (1994) suggested that without awareness of alternative meanings, teachers may assume that students see what we intend for them to see, and he warns that communication can break down when students see something other than what we intend. Conversation 2 gives me the opportunity to see and validate many student conceptions and to acknowledgethat representations require clarification. It also lets me explicitly tell students that if they do not understand what a picture or symbol represents in a particular context, or if they are seeing something different in a representation, they need to bring this to the attention of the class. We talk about the fact that pictures, diagrams, and symbols can have many reasonable meanings and that it is our job as a class to make sure that we discuss and agree on what representations mean. Having an alternative conception does not imply that the student is wrong or bad at mathematics. Pictures do not carry one correct meaning.
Bauersfeld (1995) argued that these types of conversations about alternative conceptions can help students build groundwork for future mathematics. "As soon as we narrow the students' interpretations of pictures and situations toward an unequivocal ascription of mathematical meaning," he warns, "we throw away the opportunity for an early and powerful preparation for later problem solving" (p. 146). So not only does this conversation give students opportunities to describe their thinking, but it also allows them to argue why a certain interpretation may be valuable in one context, whereas another interpretation may be valuable in another, and to create scenarios (such as the probability model problem) that may be useful in future mathematical contexts.
Mathematicians have agreed on precise meanings for words. Pay careful attention to their language.
Many students do not appreciate the subtleties of mathematical talk. As part of a research study (Szydlik, Kuennen, and Seaman 2009), we posed the item shown in figure 3 to a large sample of college students in their beginning mathematics class (such as college algebra, mathematics for elementary teachers, or liberal arts mathematics). Try it with your students as a way to begin a conversation about how mathematicians use language.
Only approximately half the students in our study responded in a way that is consistent with the interpretation of the mathematical community—that neither (a) nor (b) need be true. Students who chose (b), the most common alternative response, argued that if you are talking about people in a pool, there must be someone in the pool. This led us to question the precision with which students attend to language and, specifically, the meanings that students may be giving to statements containing quantifiers (e.g., for all, there exists), quantifying language (e.g., at least, at most, exactly), and words such as unique, distinct, and, and or.
Precise statements are a hallmark of algebra. Algebraic identities are those statements that are true in all cases (e.g., for all x and for all y, (x + y)2 = x2 + 2xy + y2 ). When we solve an equation, we are implicitly thinking, "If there exists a solution, x, then I could do all these moves to find it." We might tell students that two distinct lines in the plane can intersect at most once or that a cubic equation has at least one real root. When talking about solving an inequality, we may explain that x > 7 or x < –7. But what do students make of this language? First, we need to ask them. Second, we need to share explicitly the mathematical culture regarding the precision of our language.
Some researchers have found that revoicing (repeating or rephrasing) student talk can help students clarify ideas, learn mathematical vocabulary, or attend to specific words and their meanings (O'Connor and Michaels 1993). In my algebra classes, I look for opportunities to amplify words by emphasizing their importance when revoicing student talk and by emphasizing them in my own speaking. Revoicing might sound something like this:
Aidos: I got x is bigger than 7 and x is less than negative 7.
Teacher: [to the class] Aidos says x is bigger than 7 and x is less than negative 7. [short pause] Hmm. Give me a number that x could be.
Violet: 10.
Teacher: Okay, then 10 has to satisfy Aidos's statement. Let's read it with 10 in there. Ten is bigger than 7 and 10 is less than negative 7.
[Five seconds of silence]
Aidos: I meant that it just has to be one or the other.
Violet: So it should be x is bigger than 7 or less than negative 7?
Teacher: Yes. Then you are saying that all values of x that are bigger than, greater than, 7 along with all values of x that are less than negative seven make the inequality true.
I also share with students, through stories, mathematical culture regarding language. For example, I explain that if a mathematician has six children and you ask whether she has three children (in the context of mathematics—and probably outside it too), she will answer in the affirmative, because if she has six children, then she also has three children. I tell them that my father (a mathematician too) will respond to and-or questions with either yes or no. I learned quite young that if I asked him if he wanted peas or beans at dinner, he would simply say yes. (I am delighted when students adopt this language and start to answer my questions in that manner. Teacher: "True or false? (x + y)2 = x2 + y2?" Class: "Yes.") Stories like these give us opportunities to share meanings that mathematicians give to language.
The types of conversations described here pay high mathematical dividends for the class time invested. They allow us to hear student thinking about mathematical symbols, representations, and language and share meanings given to these objects by the mathematical community. They provide teachers and students opportunities to lay groundwork for future problem solving and to discuss larger mathematical values and practices. In addition, they specifically address the Common Core Standards for Mathematical Practice regarding attention to precision: "Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equals sign consistently and appropriately" (CCSSI 2010, pp. 7).
Further, the conversations encourage contributions from students who may have been reticent to engage because they allow us to validate their different ways of seeing mathematical objects. In other words, these types of discussion can enhance the culture of participation in our classrooms because they allow us to shift the notion that mathematical rules are based on the teacher's authority to a more inclusive and empowering view in which mathematical understanding is developed by a community of learners.
This is not to say that three conversations are sufficient; changing classroom norms is an ongoing project. The practice of interrogating meaning of symbols, representations, and terms must be ongoing if this type of participation is to become the norm. These conversations are meant to serve as openings to begin that transformation.
Bauersfeld, Heinrich. 1995. "The Structuring of Structures: Development and Function of Mathematizing as a Social Practice." In Constructivism in Education, edited by Leslie Steffe and Jerry Gale, pp. 137–58. Hillsdale, NJ: Lawrence Erlbaum Associates.
Schipper, Wilhelm. 1982. "Selection and Order of Mathematical Content in the Early Grades." Journal fur Mathematik-Didaktik 2: 91–120.
Szydlik, Jennifer E., Eric Kuennen, and Carol E. Seaman. 2009. "Development of an Instrument to Measure Mathematical Sophistication." In Proceedings for the Twelfth Conference of the MAA's Special Interest Group on Research in Undergraduate Mathematics Education (SIGMAA on RUME). | 677.169 | 1 |
Why choose OCR GCSE (9–1) Maths?
100 marks per paper, giving us a large scope for awarding more method marks within questions. This means candidates can be better rewarded for each correct step on the way towards an answer.
A column of required content suitable for 'initial learning' is set out in the specification, ensuring that the basics can be established with learners before moving on to more difficult areas.
Mathematical formulae will be provided directly in each question when relevant, rather than on a formulae sheet at the front of the paper where candidates would then have to identify and choose the correct formula from.
GCSE Maths is designed to be straightforward and accessible so you can tailor how you deliver the course to suit your students' needs and is backed up by high-quality resources to support you.
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GCSE Mathematics for OCR Foundation Homework Book
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This Resource Partner Homework Book is an ideal companion to the OCR Foundation tier Student Book and can be used as a standalone resource. With exercises that correspond to each section of the Student Book, it offers a wealth of additional questions for practice and consolidation.
GCSE Mathematics for OCR FoundationGCSE Mathematics for OCR Higher Homework Book
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GCSE Mathematics for OCR HigherMastering Mathematics for OCR GCSE: Foundation Foundation 2/Higher Higher 2 | 677.169 | 1 |
in its eighth edition, this text masterfully integrates skills, concepts, and activities to motivate learning. It emphasises the relevance of mathematics to help students learn the importance of the information being covered. This approach ensures that they develop a sold mathematics foundation and discover how to apply the content in the real world. | 677.169 | 1 |
The Numbers Guide, now in its fifth edition, is aimed at managers who have budgetary, planning or forecasting responsibilities and is invaluable for everyone who wants to be competent, and able to communicate effectively, with numbers. There are chapters on Key Concepts * Finance and investment * Measures for interpretation and analysisForecasting techniques * Sampling and hypothesis testingIncorporating judgments into decisions * Decision-making Linear programming and networking * How spreadsheet programmes can make it easy
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Trigonometry: An analytic approach
This fifth edition reflects twenty years of classroom experience with the earlier editions. As in the previous editions, the circular functions, which are pairings of real numbers with real numbers, and their properties are introduced in Chapter 1 and are the basis for a mathematical characterization of periodic phenomena. This approach gives early importance to those aspects of trigonometry that students who plan to continue their study of mathematics will need in calculus. This edition is designed for use in a one-semester or one-quarter courses in trigonometry. | 677.169 | 1 |
This course provides an introduction to complex analysis which is the theory of complex functions of a complex variable. We will start by introducing the complex plane, along with the algebra and... more
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Use this textbook help course as a companion to your Amsco geometry textbook to study for exams or clarify information you don't quite understand. Each one of our video lessons aligns with the chapters in your textbook, so you can easily work through challenging topics at your own pace with these extra resources | 677.169 | 1 |
This straightforward guide describes the main methods used to prove mathematical theorems. Shows how and when to use each technique such as the contrapositive, induction and proof by contradiction. Each method is illustrated by step-by-step examples.
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Idea I: Limit Definition of a Derivative
History:
A long, long time ago, in two places half a world away, two
philosophers (Isaac Newton and Gottfried Leibniz) created a
system of mathematics at practically the same time in almost the
same way. Through th
Geography | Webquest | NASA
Webquest
NASA
Directions: Go to and answer the questions.
1. What is the NASA image of the day for today (the day you are completing this exercise)?
Answer:
Tectonically Active Planet Mercury.
2. Who is the Adminis
AP Calculus AB Advice
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Even if math isn't your strong suit, I would highly recommend this class. Mr. Wood's incredible teaching makes all his students feel empowered, and capable. He treats his students like adults, and works with them to find the best path to help them succeed.
Course highlights:
I learned more than just math, I learned how to be a thinker, and a problem solver. Instead of memorizing information, I actually learned it, and used my combined knowledge to solve problems I never knew I could. I was able to grow into a well rounded person, who can think out side of the box, and find solutions to more than just math equations.
Hours per week:
3-5 hours
Advice for students:
Do the work! Ever problem is preparing you for the end of the class, and beyond. If you don't put in the effort to learn, then you won't.
Course Term:Winter 2016
Professor:Brian Wood
Course Tags:Math-heavyBackground Knowledge ExpectedGreat Discussions
Nov 15, 2016
| Would highly recommend.
This class was tough.
Course Overview:
This is a very challenging but very useful course. It's important to begin creating good study habits in High School to prepare you for college. If you like math I would STRONGLY recommend this course. It may be difficult, but it so satisfying when you succeed and begin to understand.
Course highlights:
I took the majority of Calc A last semester and I a currently in the course now. It's a very hard curriculum and you start out with learning things like Limits, and Optimization. It's very tough to wrap your brain around but with a really good teacher and determination to do good you will succeed! Just a reminder that you have resources all around you, take advantage of them!!
Hours per week:
3-5 hours
Advice for students:
I would recommend to take time out of your daily life if you are struggling to go see your teacher. The stronger the relationship is with your teacher the better you will preform. Teachers are here to help you succeed so why not take advantage? One on one help is so valuable in general and it can turn such confusion into something so clear. I would also advise to find someone who can explain hard concepts in everyday ideas, sometimes X's and Y's are just too complicated to wrap your mind around but once you see how to do a similar problem with easy numbers, or something like dog houses in place of X's it seems to make much more sense! | 677.169 | 1 |
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