text stringlengths 30 4k | source stringlengths 60 201 |
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Color
Wojciech Matusik MIT EECS
Many slides courtesy of Victor Ostromoukhov, Leonard McMillan, Bill Freeman, Fredo Durand
Image courtesy of Chevre on Wikimedia Commons. License: CC-BY-SA. This content is excluded from
our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.
1
... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
light source
spectra
Image courtesy of Cmglee on Wikimedia Commons. License:
CC-BY-SA. This content is excluded from our Creative
Commons license. For more information,
see http://ocw.mit.edu/help/faq-fair-use/.
This image is in the public domain. Source: Wikimedia Commons.
9
Questions?
So far, physical side of co... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
16
Cone spectral sensitivity
• Short, Medium and Long wavelength
• Response for a cone
= ∫ λ stimulus(λ) * response(λ) dλ
17
Cone response
Start from infinite
number of values
(one per
wavelength)
Stimulus
Cone responses
Multiply wavelength by wavelength
End up with 3
values (one per
cone type)
Integ... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
different intensity
• But different response for different cones
24
von Helmholtz 1859: Trichromatic theory
• Colors as relative responses
(ratios)
Violet
Blue
Green
Yellow
Orange
Red
Short wavelength receptors
Medium wavelength receptors
Long wavelength receptors
t
l
e
o
V
i
e
u
B
l
n
e
e
r
G
w
o
l
l
e... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
information, see http://ocw.mit.edu/help/faq-fair-use/.
30
Color blindness test
• Maze in subtle intensity contrast
• Visible only to color blinds
• Color contrast overrides intensity otherwise
© source unknown. All rights reserved. This content is excluded from our Creative
Commons license. For more information,... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
Daylight
Scan (neon)
© source unknown. All rights reserved. This content is
excluded from our Creative Commons license. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
Hallogen
37
Bad consequence: cloth matching
• Clothes appear to match in store (e.g. under neon)
• Don’t match outdoor
38
Rec... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
. For more
information, see http://ocw.mit.edu/help/faq-fair-use/.
43
Additive Synthesis - wrong way
• Take a given stimulus and the corresponding
responses s, m, l (here 0.5, 0, 0)
44
Additive Synthesis - wrong way
• Use it to scale the cone spectra (here 0.5 * S)
• You don’t get the same cone response!
(her... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
not measure cone sensitivity?
Spectral
L
Sensibility
of the
L, M and S
Cones
• Less directly measurable
M
S
– electrode in photoreceptor?
– not available when color spaces were defined
• Most directly available measurement:
– notion of metamers & color matching
– directly in terms of color reproduction:
... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
ceived by a human
• Careful: this is not your usual rgb
58
CIE color matching
• Meaning of these curves: a monochromatic
wavelength λ can be reproduced with
b(λ) amount of the 435.8nm primary,
+g(λ) amount of the 546.1 primary,
+r(λ) amount of the 700 nm primary
• This fully specifies the color
perceived... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
– measure 3 numbers
• But for those primaries, we need negative spectra
64
CIE’s problem
• Obvious solution:
use cone response!
– but unknown at the time
• =>new set of tristimulus curves
– linear combinations of b, g, r
– pretty much add enough b and g
until r is positive
65
Chromaticity diagrams
• 3D... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
0.412424 0.212656 0.0193324
0.357579 0.715158 0.119193
0.180464 0.0721856 0.950444
3.24071 -0.969258 0.0556352
-1.53726 1.87599 -0.203996
0.498571 0.0415557 1.05707
• Adobe RGB to XYZ XYZ to Adobe RGB
2.04148 -0.969258 0.0134455
0.576700 0.297361 0.0270328
-0.564977 1.87599 -0.118373
0.185556 0.627355 0.07... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
0661196
-0.532592 1.99845 -0.118518
0.200112 0.114350 1.11491
-0.288284 -0.0282980 0.898611
71
Color gamut
• Given 3 primaries
• The realizable
chromaticities lay in the
triangle in xy
chromaticity diagram
• Because we can only
add light, no negative
light
© source unknown. All rights reserved. T... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
is more sensitive to ratios
– Is a grey twice as bright as another one?
• If we use linear encoding, we have tons of information
between 128 and 255, but very little between 1 and 2!
• Ideal encoding?
Log
• Problems with log?
Gets crazy around zero
Solution: gamma
77
Color quantization gamma
• The hum... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
: 047119459X
304 pages 3 edition (March 31, 2000)
Vision and Art : The Biology of Seeing by Margaret Livingstone, David H. Hubel Harry N
Abrams; ISBN: 0810904063
208 pages (May 2002)
The Reproduction of Color by R. W. G. Hunt
Fountain Press, 1995
Color Appearance Models by Mark Fairchild
Addison Wesley, 1998
C... | https://ocw.mit.edu/courses/6-837-computer-graphics-fall-2012/2d7476b0ce37e364ac728cd4c8ff41f4_MIT6_837F12_Lec05.pdf |
Outline
Review
Standard Library
<stdio.h>
<ctype.h>
<stdlib.h>
<assert.h>
<stdarg.h>
<time.h>
1
6.087 Lecture 10 – January 25, 2010
Review
Standard Library
<stdio.h>
<ctype.h>
<stdlib.h>
<assert.h>
<stdarg.h>
<time.h>
2
Review: Libraries
•
linking: binds symbols to addresses.
• static linkage: occ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
on error.
FILE∗ freopen(const char∗ filename,const char∗ mode,FILE∗ stream)
• redirects the stream to the file.
• returns NULL on error.
• Where can this be used? (redirecting stdin,stdout,stderr)
int fflush (FILE∗ stream)
•
flushes any unwritten data.
•
if stream is NULL flushes all outputs streams.
•
returns EOF... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
file position in the stream. Subsequent read/write
begins at this location
origin can be SEEK_SET, SEEK_CUR, SEEK_END.
returns non-zero on error.
long ftell (FILE∗ stream)
•
returns the current position within the file. (limitation? long
data type).
•
returns -1L on error.
int rewind(FILE∗ stream)
•
sets the file ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
n)
•
•
fills the first n bytes of dst with the value c.
returns a pointer to dst
12
<stdlib.h>:Utility
double atof(const char∗ s)
int atoi (const char∗ s)
long atol(const char∗ s)
•
converts character to float,integer and long respectively.
int rand()
•
returns a pseduo-random numbers between 0 and
RAND_MAX
... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
∗ base , s i z e _ t n ,
s i z e _ t sz ,
i n t ( ∗ cmp ) ( const void ∗ , const void ∗ ) ) !
• sorts base[0] through base[n-1] in
ascending/descending order.
• function cmp() is used to perform comparison.
16
<assert.h>:Diagnostics
void assert(int expression)
•
used to check for invariants/code consistency dur... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
i s t ap ; i n t
v a _ s t a r t ( ap , num ) ;
while ( num>0)
{
t o t a l +=va_arg ( ap , i n t ) ;
num−−;
}
va_end ( ap ) ;
r e t u r n
t o t a l ;
}
i n t suma=sum ( 4 , 1 , 2 , 3 , 4 ) ; / ∗ c a l l e d w i t h
i n t sumb=sum ( 2 , 1 , 2 ) ;
/ ∗ c a l l e d w i t h
f i v e args ∗ /
t h r e e args ∗ /
20... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
<time.h>
char∗ asctime(const struct tm∗ tp)
•
returns string representation of the form "Sun Jan 3
15:14:13 1988".
•
returns static reference (can be overwritten by other calls).
struct tm∗ localtime(const time_t ∗ tp)
•
converts calendar time to local time".
char∗ ctime(const time_t ∗ tp)
•
converts calendar ... | https://ocw.mit.edu/courses/6-087-practical-programming-in-c-january-iap-2010/2d7ce9e81c1ce03c8c93a12c9d21d0dd_MIT6_087IAP10_lec10.pdf |
18.336 spring 2009
lecture 1
02/03/09
18.336 Numerical Methods for Partial Differential Equations
Fundamental Concepts
Domain Ω ⊂ Rn with boundary ∂Ω
�
PDE
b.c.
in Ω
on Γ ⊂ ∂Ω
�
PDE = “partial differential equation”
b.c. = “boundary conditions”
(if time involved, also i.c. = “initial conditions”)
Def.: An e... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
1u, ..., Du, u, x) = 0
·
|α|≤k
(iv) fully nonlinear, if neither (i), (ii) nor (iii).
Def.: An expression of the form
F (Dk u(x), Dk−1 u(x), ..., Du(x), u(x), x) = 0,
x ∈ Ω ⊂ Rn
is called kth order system of PDE,
where F : Rmnk × Rmnk−1 × ... × Rmn × Rm × Ω Rm
and u : Ω → Rm , u = (u1, ..., um).
Typically: # eq... | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
equation
[dynamic-algebraic system]
shallow water equations
[system of hyberbolic conservation laws]
MIT OpenCourseWare
http://ocw.mit.edu
18.336 Numerical Methods for Partial Differential Equations
Spring 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/18-336-numerical-methods-for-partial-differential-equations-spring-2009/2da291f80e0df5ceca25f76df3c91a5d_MIT18_336S09_lec1.pdf |
Classes &
Interfaces
Java’s Object Oriented System
Justin Mazzola Paluska
Keywords
(cid:122) Class – a template of a data object
(cid:122) Interface – a specification
(cid:122) Instance – an instantiation of a Class or Interface
physically represented in memory
(cid:122) Method – a set sequence of instructions
(cid:1... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
{
//constructors
//member functions
//member variables
}
Simple Example
public class BankAccount {
…
}
Class Members
(cid:122) In class definitions we can define the
following members:
(cid:122) Constructors
(cid:122) Instance and static methods
(cid:122) Instance and static fields
(cid:122) Nested classes
Construc... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
method signature
(cid:122) Example: myClass.foo(7) //calls 2nd method
Fields
(cid:122) A field is like a variable, it stores state
(cid:122) A field has a associated data type which
determines the type of data that this field will
hold
(cid:122) Template:
[access] dataType fieldName [= value];
Example Fields
public... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
int amount);
public void deposit (int amount);
public int getBalance ();
}
Notice that for method
declarations, the method
body is not defined.
How do we use the Interface?
(cid:122) We make classes or other interface
implement or extend the interface.
(cid:122) If a class implements an interface, that class
must ... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
class SavingsAccount extends BankAccount {
private int numberOfWithdrawals;
public SavingsAccount () {
balance = 0;
numberOfWithdrawals = 0;
}
public void withdraw (int amount) {
if (numberOfWithdrawals > 5) {
throw new RuntimeException (“Cannot make >5 withdrawals a month”);
} else {
}
balance = balance – amount;
numb... | https://ocw.mit.edu/courses/6-092-java-preparation-for-6-170-january-iap-2006/2db153242ca1c747fd6cd7c2b497162a_lecture2a.pdf |
Course Organization
Spirit of the Undertaking
6.871: Knowledge-Based Systems
Spring 2005
Randall Davis
Logistics
• Info sheet, syllabus
• Personnel:
– Lecturers: Davis (and friends)
• Course notes:
– 1st installment ready now
• You are responsible for what happens in lecture.
• No open laptops.
6.871 - Lecture 1... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
PECTOR: $100m worth of molybdenum
– R1/XCON: from 85% to 97.5% performance
– American Express: 20% operational savings,
$10M’s in added revenue
– DuPont
– Manufacturer’s Hanover: Inspector
– The clever (?) paper clip
6.871 - Lecture 1
8
In 1995, in Singapore…
Crime Case Closed Infamous Crimes
Nick Leeson and Bar... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
401 million euros from 997 million euros
in the previous year.
6.871 - Lecture 1
10
(Why) is this interesting?
• Applied AI leads to advances in basic
science
– Rule-based systems
– Causal reasoning
– Reasoning at multiple levels: Reasoning
under uncertainty
– Case-based reasoning
6.871 - Lecture 1
11
(Wh... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
for expertise?
• Can human thought be formalized?
• (How) Can we get a machine to be smart?
• And what is AI, anyway?
6.871 - Lecture 1
17
Intellectual Origins
• AI is a great intellectual adventure
– Cf cosmology, physics, biology
• AI is the exploration of the design space of
intelligences
• AI is making mac... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
iz and the “algebra of thought”
• 19th century: Boole’s logic and The Laws of Thought
• 19th century: Babbage and the Analytical Engine
• 20th century: Shannon’s insight about switching circuits
• 20th century: Turing’s ideas about thought and computation
6.871 - Lecture 1
24
Intellectual Origins
• What is it w... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
of
rationality.
• Knowledge is closely linked to rationality.
• Knowledge is competence-like notion.
6.871 - Lecture 1
30
Character of Knowledge
• Most of what we know knowledge is non
numeric.
• Most of what we know is heuristic.
– What’s certain?
– What’s the alternative?
6.871 - Lecture 1
31
Character o... | https://ocw.mit.edu/courses/6-871-knowledge-based-applications-systems-spring-2005/2dfcfa88dc81856d86d21e0d67017e94_lect01_intro.pdf |
Intro
This lecture will review everything you learned in 6.042.
• Basic tools in probability
• Expectations
• High probability events
• Deviations from expectation
Coupon collecting.
• n coupon types. Get a random one each round. How long to get all
coupons?
• general example of waiting for combinations of eve... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
deferred decisions
– used intuitively already
– random choices all made in advance
– same as random choices made when algorithm needs them.
• use for discussing autopartition, quicksort
2
• Proposal algorithm:
– defered decision: each proposal is random among unchosen women
– still hard
– Each proposal among ... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
but at most n
– So, expected max O(ln n/ ln ln n)
• Typical approach: small expectation as small “common case” plus large
“rare case”
Example: coupon collection/stable marriage.
• Probability didn’t get kth coupon after r rounds is (1 − 1/n)r
≤ e−r/n
•
which is n−β for r = βn ln n
• so probability didn’t get som... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
– if neither affirms, run again
– Each iter has probability 1/2 to affirm
– So expected iterations 2:
– So ZP P .
•
If ZP P
– suppose expected time T (n)
– Run for time 2T (n), then stop and give default answer
– Probability of default answer at most 1/2 (Markov)
– So, RP .
– If flip default answer, coRP
On flip sid... | https://ocw.mit.edu/courses/6-856j-randomized-algorithms-fall-2002/2e00cd3ccdd9e9e9aa3f4b8305a6abef_n4.pdf |
Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.341: Discrete-Time Signal Processing
OpenCourseWare 2006
Lecture 6
Quantization and Oversampled Noise Shaping
Reading: Sections 4.8 - 4.9 in Oppenheim, Schafer & Buck (OSB).
While the title of the course is Discrete-... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
B+1 levels, so if x[n] has maximum value XM such that
−XM � x[n] < +XM , these 2B+1 levels must cover the range ±XM . This implies that the
spacing � between adjacent quantization levels is therefore
� = XM 2
−B .
The question then arises of how to analyze the error introduced by the process of quantization.
Since... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
sequence of a stationary random process.
• e[n] is uncorreletaed with x[n].
• e[n] is a white-noise process.
• The probability distribution of e[n] is uniform over the range −�/2 to �/2, as illustrated
in OSB Figure 4.52.
The expected value of e[n] is therefore
and its variance is
E{e[n]} = 0,
�2 =
e
�2
.
12... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
57 is, according to the linear noise model, �ee[0] = �2
12 . When this signal is lowpass
filtered by the ideal LPF with �c = φ/M , the total noise power is divided by M , resulting in a
3
total noise power of �2
after lowpass filtering. As long as no aliasing occurs, the total noise
12M
power at the output of a co... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
a method for further controlling quantization noise which has
the ability to reduce in-band noise to a still greater degree. Let’s first consider the system in
OSB Figure 4.68 and its associated additive noise model (OSB Figure 4.69).
The system in OSB Figure 4.68 implements what might be considered a “first logical s... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
4 sin2(�/2).
The PSD of the noise at the system output then becomes
�eˆeˆ(ej�) =
�2
12
4 sin2(�/2),
as depicted in OSB Figure 4.64. Lowpass filtering the resulting signal by an ideal LPF with
�c = φ/M and compressing by M eliminates out-of-band noise and therefore reduces the overall
noise power.... | https://ocw.mit.edu/courses/6-341-discrete-time-signal-processing-fall-2005/2e04b01fe7072772ba39f0c8dfb58306_lec06.pdf |
Table of Contents
Lec 1
Introduction to Nanotechnology and Nanoscale Transport Phenomena.
Microscopic Pictures of Heat Carriers
Lec 2
Characteristic Time and Length, Simple Kinetic Theory, Characteristic
Lec 3
Schrödinger Equation
Lec 4
Quantum Wells, Harmonic Oscillators, Rigid Rotors, and Hydrogen Atoms
Lec... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Potentials, Electrokinetics, Surface Tension
2
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 1
1. Overview for nano sciences
1.1 Length scale
1.2 Examples in microtechnology
1.3 Examples in nanotechnology
1.4 Nano for energy (phonon, phonon, electron; wavelength, mean free path)
1.5 Na... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
=σ T
4
where:
-8
σ =5.67 × 10 W/m K .
2
4
For real surface, we define “emissivity” as
ε=
e
eb
.
For the two planar walls shown below, the heat flux of radiation is evaluated as
q =σ (T − T2
4
1
4 ) .
T1
ε 1=1
T2
ε 2=1
2.2 Newton shear stress law
The shear stress for the sketched one-dimensional flow is: ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
reducing length scale. Surface effect becomes
dominant at smaller scales.
For a spherical fluid drop, we have
3
γ =
Gravitational force ρg (4πr / 3) 2ρgr
=
3σ
Surface force
σ(2πr)
=
2
.
Substituting ρ= 10 kg / m , σ = 78mN / m into this equation, we get
3
2
r =1 m, γ = 8.4e4; r =1 mm, γ = 8.4e-2.
mg
2.57 Fa... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
long wavelength lattice
waves. Quantum mechanics states that the energy of each lattice wave is discrete and
must be multiples of hν. Based on argument we will discuss in chapter 5, the spring
system can be further simplified as a box of phonon particles.
Th
Tc
Phonon gas
ε = nhν
Now, molecules, electrons, and ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
we use 900 MHz as the frequency. This wavelength is still much larger than that of
the thermal radiation (around 0.5 µm).
2) Generation of thermal radiation
Thermal radiation typically refers to the electromagnetic waves that are generated by
the oscillation charges in the atoms and crystals, while TV and radio si... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
= 3v 2 . From here, you can derive the ideal
2
x
x
2
y
5.5 Charge transport
x
Similarly, we have
G
G
G
=F qε = − eε; J =σε,
G
G
G
G
where ε is electrical field and J is electric current density. An expression of σ can be
derived.
.
5.6 Mass diffusion
2.57 Fall 2004 – Lecture 2
2
9
6. To understand tr... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
we give the
allowed energy levels of a harmonic oscillator, which is a model for the vibrations of a
diatomic molecule such as H2. The dispersion relation (E-k relation) for electrons,
phonons, and photons are sketched in following figures, in which the wavevector k points
to the direction of wave propagation (elec... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
1.
z
y
x
z
-∞
-∞
-∞
Using spherical coordinates can simplify the calculation as
∞
2
∫
4 v f(v)dv = 1.
π
0
Both of above equations yield
A = (
m
2πk TB
)3/ 2 .
Thus
2
3/2
⎛
⎞
m
( ) = ⎜
⎟
f v
⎟
⎜
⎝ 2πκ BT ⎠
(
v m x + v y + vz )
2 ⎤
⎥
,
⎥
⎦
which is also called the Maxwell distribution. The average en... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
X X v , which
is already used in 5.4.
)
(
⋅
2.57 Fall 2004 – Lecture 2
4
11
6.4 How far can they travel?
The effective diameter for two atoms to collide is 2D. . If the number concentration is
n [m-3], then the number of molecules that this particle will collide with is nπD2L. The
average distance L between e... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
.5e-10)2 ×1.01 5 e
= 0.14µm ,
in which 2 comes when we consider relative velocity instead of assuming other
particles are stationary. The collision time (relaxation time) is τ = Λ / ~ 10 10 s .
v
−
For P=1 mTorr, we have Λ = 0.1m . The molecules seldom collide with each other.
Note: This relaxation time is not very... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
.
/ dT
, v 2 = v2 / 3 . The above equation changes into
x
Compared with the Fourier’s law, we know that thermal conductivity k =
2τCv
3
. The
capital C is specific heat per unit volume, C=ρc.
2.57 Fall 2004 – Lecture 2
6
13
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 3
8. Micro & Nanoscale... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
and phonons are also material waves; the
finite size of the system can influence the energy transport by altering the wave
characteristics, such as forming standing waves and creating new modes that do not exist
in bulk materials.
For example, electrons in a thin film can be approximated as standing waves sitting
... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
be as short as a few
femtosecond, we cannot use diffusion theory when the time scale is shorter than the
relaxation time.
Chapter 2 Material Waves & Energy Quantization
2.1 Basic wave characteristics
2.1.1 Traveling wave
First consider a harmonic wave (such as an electric or a magnetic field) represented by a
si... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
= − 2 A cos (ωt )sin (kx
⎦
which has fixed nodes in space such that Φ=0. These nodes are similar to the string ends
of a guitar.
) − sin (ωt + kx
) yˆ ,
t
2
The energy of a wave is typically U ∝ Φ
.
2.2 Wave-particle duality
2.2.1 Electromagnetic (EM) wave
Quantum mechanics started with the explanation for bla... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the Planck constant and has a value h=6.6x10-34 J·s.
2
According to the relativity theory, we have
E = mc .
Thus the momentum is
p = mc = E / c = hν / c = h /λ .
After 1926, people used the term “photon” to name the quantum with E h= ν , p = h / λ .
Einstein used the corpuscular characteristics of electromagnetic... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
A TEM uses thermal excitation or applying a high voltage to draw electrons from the tip
end. The electrons are then accelerated by the strong electrical field to gain a large
momentum p (small λ~1 Å). Since the resolution is normally comparable to wavelength λ,
high resolution is obtained with electron energy as hig... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2
∂ T
∂ t
but the magic imaginary unit
“i” really gives rise to wave behavior.
2.57 Fall 2004 – Lecture 3
18 Schrödinger himself did not come up with a correct explanation for the meaning of
wavefunction. The right explanation was given by Born, who suggested that Ψt itself is
* is the probability density functio... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
+
∂
∂z 2
+ U
⎞
⎟
⎟
⎠
Note: in equation < Ω >= ∫ Ψ ΩΨ dV , you cannot switch Ψt
gradient operator ∇ and the Laplace operator ∇ 2 .
t
*
t
* and Ψt if Ω contains the
Standard deviation:
Similar to
x
∆ =
n
1
− ∑
(
x
− < >
1
n
x
i
1
=
2
)
,
in quantum mechanics we have
*
q
(
< ∆ >= Ψ − < > Ψ
t
q ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
= E ). Solving for Y leads to
⎤
UΨ⎥ = i=
⎥
⎦
1 dY
Y dt
2
+ Ψ ∇
= E ,
Y C1 exp −i
=
E
⎤
⎡
⎥ = C1 exp [−iωt] .
t
⎢⎣ = ⎦
And the governing equation for Ψ(r) is called the steady-state Schrödinger equation
− ∇ Ψ + (U E ) Ψ = 0 .
−
2
2
=
2m
This is an eigen equation with the eigenvalue E and eigenfunction Ψ d... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the boundary conditions and will not consider the u r t
u r
Here we determine
case.
2.3.1 Free particles in 1D
In this case, there are no constraints for the particles. The potential energy u=0 so that
E
2 − Ψ =
−
=2 d 2Ψ
2m dx
This gives
−ikx
ikx
ψ = Ae + Be
,
0 .
m
−i
(ωt kx )
+
x t = Ae
where k = 2 E... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
=
=
2
mE
=
=
p
=
.
The general boundary conditions are the continuity of the wave functions and their first
derivatives at the boundaries. The latter derives from the continuity of particle flux at the
boundary. For the current problem, the continuity of the first derivatives is not required
because the wavef... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
y
D
O
U=
0
U = ∞
x
D
We establish a coordinate system as shown in the schematic above. Clearly, outside the
potential well, we have Ψ = 0 because U = ∞ . We thus focus on the solution inside the
potential well. The Schrödinger equation inside the well (U=0) is
−
2
2
⎛
=
⎜
⎜
2
2m ⎝ ∂x
∂ Ψ ∂ Ψ
+
2
⎞
⎟
2 ⎟ ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
1, 2,3
") .
Similarly, k y =
lπ
D
Thus
(l = 1, 2,3
") .
2.57 Fall 2004 – Lecture 4
23
(A + n )π =
2 2
2
2
2
2mD
E An =
and
( A ,n=1,2,…)
⎛ Aπy ⎞
⎛ nπx ⎞
.
sin
Ψ = C sin
⎜
⎟
⎜
⎟
⎝ D ⎠
⎝ D ⎠
An
An
We can further determine the constant C A n=2/D.
For quantum ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
cannot distinguish the spin of particles. For electrons,
corresponding to each wavefunction obtained from the Schördinger equation, there are
two quantum states (or two relativistic wavefunctions), which are usually denoted by an
additional quantum number s that can have the following values:
s =
1
2
or −
1
,
2 ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the nucleus and the orbiting electron is governed by
the Coulomb force
F = −
1 e2
4πεo r
2 = −
du
dr
,
where εo=1.124x10-10/4π [C2/(m2N)] is the electrical permittivity of the vacuum. It yields
u r( ) = −
2e
4πε0r
.
Using separation of variables, we assume ΨnAm = RnA (r)YA
energy levels of the electron-nu... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
=3
(-1.5 eV)
p orbitals
n=2
(-3.4 eV)
s orbitals
1s
n=1
(- 13.6 eV)
Note: the energy gap between different n values is much larger than the thermal energy
(kBT~26 meV) and it is almost impossible to thermally excite electrons to a higher n level.
A stable element is obtained only if all the orbitals for the ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
4
.
2
xn = (
Thus
E ∝ Kx =
2
n
nh
4
K
m
,
which is a linear functions of n, similar to what we obtain from solving Schrodinger
equation. Similarly, we can deal with electrons moving around the nucleus but changing
the potential energy to u =
e2
4πε0 xn
. Finally we get E ∝
1
n2
.
Rigid rotation
G
G
I... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the wavelength.
E p2 / 2m =
=
(
2
h
/λ)
2m
=
= k 2
2
2m
; k = 2π/λ, = = h / 2π
2. Quantum well
ENERGY AND
WAVEFUNCTION
U
∞ =
n=3
n=2
n=1
U=0
x
Energy has discrete levels, and we have one quantum number n.
E =
2h n2
2
8m D
(n=1,2,…)
For 2D constraints, we have two quantum numbers n and l. In the dis... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
210s,
n=2
Ψ
(-3.4 eV)
Ψ
21(+1)s
8 quantum states
1s
Ψ
100s
n=1
(- 13.6 eV)
2 quantum states
The corresponding Ψnlms are marked in the figure for some energy levels. The
degeneracy follows g(n)=2n2.
Note: As electrons number goes up, the orbit will split. The energy of 3d is lifted up
above 4s because of th... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
energy can be approximated as the summation of translational, vibrational, rotational, and
electronic energies:
tot = E trans
+ E el + E vib + E rot .
E
We talk about the translational energy as a particle in a box. To simplify, for hydrogen
molecules we neglect other effects and only consider the vibrational and ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
solid surface,
because the spring constant of the beam is much lower than that of the atomic vibration,
the beam will deform instead of deforming the atoms on the surface. The beam
deformation can be further measured by a laser and the topography-scanning resolution
can reach nm level and even atomic resolution.
A... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
asymmetric
stretching, and bending. The absorption wavelength is around 10 µm, which is just the
earth emission wavelength at the room temperature.
=
Note: the green housing effects occur when solar emission (~0.5 µm) can pass CO2 and
heat up the earth. Then the emission of the earth will be absorbed by CO2 and th... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
of the reflection waves from different interfaces.
Incoming wave
Reflection wave
Transmission wave
2.57 Fall 2004 – Lecture 5
33 Now we will start solving the Schrödinger equation with periodic u:
(1) First we will consider periodic quantum wells with finite depth, and electrons with
energy larger than the barri... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
one negative-direction
wave in the same region. Two coefficients A and B need to be determined in
Ψ = Ae
+ Be
−iKx
iKx
.
2
1
For [0,a] and [a,a+b] regions, we have totally four unknown coefficients. The continuity
boundary condition at x=a only gives two equations
Ψ1(x = a) = Ψ (x = a) , Ψ '(x = a) = Ψ '(x = a... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
1
2.57 Fall 2004 – Lecture 5
35
1
N+1
(
a N + b )] Ψ =
First we have
Ψ [x +
Using Bloch's theorem, it can be written as
Ψ (x ) Ψ =
(x ) exp[ikN (a + b )] ,
which yields
(x ) .
k =
n 2
π
N(a + b)
2π n
L
where L is the length of the crystal.
(n=0, ± 1, ± 2,...),
=
To get a better idea of the soluti... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
m
m
m
m
8
8
8
8
(
(
(
(
/
/
/
/
2
2
2
2
h
h
h
h
=
=
=
=
0
0
0
0
E
E
E
E
,
,
,
,
0
0
0
0
E
E
E
E
/
/
/
/
E
E
E
E
15
15
15
15
10
10
10
10
5
5
5
5
Eo =
Eo =
2
2
h
h
2
2
8ma
8ma
-4
-4
-4
-3
-3
-3
-2
-2
-2
-1
-1
-1
0
0
0
0
0
0
0
0
1
1
1
1
2
2
2
2
3
3
3
3
4
4
4
4
Normalized Wavevector
Normalized Wavevector
Normalized Wavevec... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
approximate the potential field as rectangular wells in the crystal.
From periodicity, the Bloch theorem gives additional equations
Ψ + (a + b
+ BeiKx . We have also
which is used to determine the coefficients in
determined the value of the wavector k in the Bloch theorem, using the Born-von
Karman periodic bounda... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
3.1.3 Consequences of solid energy levels we just obtained:
(1) Electrons wave function extends through the whole crystal, they belong to all atoms
collectively.
Recall the splitting of waves discussed in last lecture. According to Pauli’s exclusion
principle, the wavefunctions of adjacent wells cannot overlap. A c... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Energy
Energy
Energy
Energy
Energy
Energy
Conduction
Conduction
Conduction
Conduction
Band
Band
Band
Band
Gap Eg
Gap Eg
Gap Eg
Gap Eg
Eg
Eg
Eg
Eg
Electrons
Electrons
Electrons
Electrons
Holes
Holes
Holes
Holes
Donor
Donor
Donor
Donor
Energy Level
Energy Level
Energy Level
Energy Level
Acceptor
Acceptor
Acceptor
Energy ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
exactly fill one or more bands, leaving others empty, the crystal
will be an insulator at zero temperature and can be an insulator or a semiconductor at
other temperatures depending on the value of the energy gap. If a filled band is separated
by a large energy gap (>3 eV) from the next higher band, one cannot chang... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
doping. (3) The lowest band starts from a nonzero energy, which is the
consequence of the uncertainty principle.
Impurities are added to most semiconductors and these impurities have energy levels
somewhere within the band gap, some are close to the bottom edge of the conduction
band or top edge of the valence band... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
+
+
1 ∂ 2 E
2 ∂ k
1 =2
2 m *
2 |km
(k
−
k )2
m
(k
−
k )2 ,
m
where the effective mass is defined as m * =
2=
(∂ 2 E k ) |km
/ ∂ 2
. In differential geometry, the
/ ∂ 2
term 1/ ∂ 2 E k
is just the curvature. Thus, effective mass is proportional to local
curvature at band maxima or minima. For electrons clo... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
2a
a
E
Indirect
E
Direct
Eg
k
Eg
k
In above figures, we demonstrate the idea of direct and indirect band gaps. In a direct
band structure, both the minimum in the conduction band and the maximum in the
valence band occur at the same location of k (=0 for the example given). A good example
for direct band g... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
)a
ja
(j+1)a
We make the following assumptions for the analysis. First, we consider only the
interaction force between the nearest neighbors. Second, the interaction force between
atoms is a harmonic force (which obeys Hook's law). This can be justified as we have
done for harmonic oscillators. Now consider a typ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 7
In the last lecture, we have talked about atoms in a one-dimensional chain. We find the
solution as
u j = A exp[−i(ωt − kja)],
where the frequency is ω= 2
K
m
sin
ka
2
.
Na
(j-1)a ja (j+1)a
Note: When k approaches zero for large wavelength, the fr... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
E = hν (n +1/
n
2);
ν =
1
2π
K
m
,
in which the frequency is consistent with classical mechanics.
Similarly, quantum mechanics gives the energy levels for a chain with N atoms as
En = hν(n +
1
)
2
or
nE
=
nω=
(
+
1
2
)
(n=0,1,2,3,…),
in which ω = 2
K
m
sin
ka
2
.
The basic vibrational energy q... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
the optical branch, because the high frequency phonons in the
optical branch can interact with electromagnetic waves more easily. In general, if there
are m atoms in a basis and N lattice points in the chain, there are one acoustic branch
with N acoustic modes, and (m-1) optical branches with (m-1) N optic modes.
ω... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
Debye
approximation is used, which assumes a linear and isotropic relation between the
frequency and the wavevector. This approximation is valid at low frequencies but is
not a good approximation at high frequencies. In the very low frequency region, the
lattice vibration carries the sound wave.
(2) Einstein appro... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
a2
a1’
1
a2’
A Conventional
Unit Cell
A primitive unit cell is the parallelepiped defined by the primitive lattice vectors. There
is only one lattice point (equivalently speaking) per primitive unit cell. For example,
each of the four lattice points in the two parallelograms formed by the two sets of
primitive ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
systems
There are totally 14 types of Bravais lattices and can be further grouped into 7 types of
point symmetry operations (seven crystal systems). Here we will focus on the cubic
system.
simple cubic (sc) body centered cubic face-centered cubic (fcc)
(g) Cubic system
a
1
=
a
2
=
a
3
γβα
=
=
o90=
A simple c... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
to the three smallest
integers that have the same ratio as the original set. The result is enclosed in parenthesis
(hkl) and this set of numbers is called the Miller indices of the plane. The above example
yields
(1/0.8, 1/0.4, 5/3) (for inner plane) or (1/4,1/2,1/3) (for outer plane) → (364).
3a3
a3
0.6a3
0.4a... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
(Born-Mayer)
−r / ζ
Note: The repulsive forces are normally very strong and the curve is sharp when the
distance approaches zero.
Molecular crystals are characterized by the dipole-dipole interaction between atoms. An
isolated atom is not polarized, but when another atom is close by, the electrical field of
electr... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
required by the second law of thermodynamics for a stable system.
2.57 Fall 2004 – Lecture 7
53
2.57 Nano-to-Macro Transport Processes
Fall 2004
Lecture 8
In the last lecture, we have talked about the primitive unit cell. There is only one lattice
point (equivalently speaking) per primitive unit c... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
as NaCl, the single valence electron in the sodium atom moves to
the chlorine atom such that both Na+ and Cl- have closed electron-shells but meanwhile,
become charged. The Coulomb potential among the ions becomes the major attractive
force. The potential energy of any ion i in the presence of other ions j is then, ... | https://ocw.mit.edu/courses/2-57-nano-to-macro-transport-processes-spring-2012/2e4ecaa5cf55f03bcefbc8ccce79aed6_MIT2_57S12_lec_notes_2004.pdf |
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