NE334: Lecture 30
July 22, 2014
Simulations of nano-scale
systems
What to do when you
dont know Z!
1
Fundamentals
In order to perform molecular simulations, we need:
Statistical mechanics (because we cannot
specify everything!)
Quantum Mechanics (thats
NE 334 Lecture 34, July 30 2014
Reading: This and next lectures: Chapter IX
FermiDirac Statistics and Intrinsic Semiconductors.
Crystalline Solids:
Each atom/ion brings electrons to assembly of atoms making up solid: some are called atomic
core, some ato
NE 334 Lecture 33, July 29 2014
Reading: This and next lectures: Chapter IX
The StronglyDegenerate FermiDirac Gas.
Low temperatures and/or high density: consider, for example, the freeelectron model of
metals
for electrons, as we have only translational
NE 334 Lecture 31, July 23 2014
Reading: This and next lectures: Chapter IX
FermiDirac and BoseEinstein Statistics.
FermiDirac distribution
Two cases :
BoseEinstein distribution
(for fermions)
(for bosons)
These are the only exact distributions.
N molecu
NE 334 Lecture 32, July 24 2014
Reading: This and next lectures: Chapter IX
A dierent limiting case is of importance for BoseEinstein particles. We know that nk cannot
be negative; moreover, we may write nBE as
k
nBE
k
e k
=
1 e
k
1
=
1
e
k
1
.
We may al
NE 334 Lecture 28, July 18 2014
Reading for this lecture: Chapter VIII Next lectures: Bose-Einstein and Fermi-Dirac Statistics;
Secs. IX.1-IX.4
nal exam info
problem set 4 (simulations)
Examples:
(a) Dissociation of I2 .
Let us consider the gas phase diss
NE 334 Lecture 29, July 22 2014
Reading: This and next lectures: Bose-Einstein and Fermi-Dirac Statistics; Secs. IX.1-IX.4
Quantum statistics: setting the stage.
Until now, quantum restrictions upon wavefunctions have not had to be considered except
when
NE 334 Lecture 27, July 11 2014
Reading for this lecture: Chapter VIII Next lecture: Bose-Einstein and Fermi-Dirac Statistics;
Secs. IX.1-IX.4
KP =
(zC /N0 )C (zD /N0 )D U /RT
0
.
e
(zA /N0 )A (zB /N0 )B
(1)
Aside : Some practical matters:
We shall examin
NE 334 Lecture 21, June 25, 2015
Reading for this lecture: Section IV.5,Sections V.1-V.2 Reading for upcoming lecture: Chapter
V
Debye model (continued)
We are now set to calculate all thermodynamic properties of a Debye crystal. Let us, however,
simply f
NE 334 Lecture 24, July 2, 2014
Reading for this lecture and upcoming one: Chapter V
Symmetry Requirements for Homonuclear Diatomics.
In the same way that total (1, 2) for a He atom must be antisymmetric under the exchange of
the two He 1s electrons (beca
NE 334 Lecture 25, July 3 2014
Reading for this lecture and upcoming ones: Chapter V, Chapter VIII
Nuclear spin contribution to properties.
More generally, for a homonuclear diatomic molecule made up of atoms with nuclear spin Ia :
Ia MIa +Ia
pair of nucl
NE 334 Lecture 26, July 10 2014
Reading for this lecture and upcoming ones: Chapter VIII
Statistical Mechanical Expression for KP
General equilibrium reaction
A A + B B
C C + D D ,
Setting the Stage.
Gibbs energy for a pure gas: start from canonical parti
NE 334 Lecture 23, July 2 2014
Reading for this lecture and upcoming one: Chapter V
Reminder:
Molecules: = trans + int
translations and internal molecular motions are considered to be independent of one another
z(T, V ) = ztrans (V, T )zrot (T )
additive
NE 334 Lecture 22, June 26 2014
Reading for this lecture and upcoming lecture: Chapter V
Introduce two further approximations:
1) Replace V (R) by parabola tted to minimum
2) Separation of vibrational and rotational motions: assume that diatomic has xed i
NE 334 Lecture 18 and, June 11, 2014
Reading for this lecture: Section IV.5 Next lecture: bring the old midterms and pick questions.
Monatomic Solids
A. The Simple Harmonic Oscillator (SHO).
Energy levels:
1
= ( + 2 )hosc ,
= 0, 1, 2,
The canonical part
NE 334 Lecture 20, June 24, 2014
Reading for this lecture: Section IV.5 Reading for upcoming lecture: Sections V.1-V.2
Einstein model (continued)
6
4.5
4
5
CV / cal mol-1K-1
CV / cal mol-1 K-1
3.5
4
C
Si
Cu
Pb
3
2
3
2.5
C
2
Si
Cu
Pb
1.5
1
1
Einstein model
NE 334 Lectures 14 and 15, June 4, 2014
Reading for lectures 14 and 15: Sections III.2 in textbook. Reading for upcoming lectures:
Section IV.1 and IV.2
Midterm info: midterms from previous years are on the UW Learn page.
Formula sheet: one sheet, two sid
NE 334 Lecture 16, June 5, 2014
Reading for this lecture: Section IV.1 and IV.2 Reading for upcoming lecture: Section IV.3
Thermodynamic State Functions for Structureless Particles.
Internal Energy.
We shall now apply our general formulation to the specia
NE 334 Lecture 17, June 10, 2014
Reading for this lecture: Sections IV.3 and IV.4 Reading for upcoming lecture: Section IV.5
The Role of Nuclear States.
The canonical partition function for nuclear energy states has the same structure as that
already disc
NE 334 Lecture 13, May 30, 2014
This lecture: Section III.1 in text.
Next Lecture: Section III.2 in text
Entropy and its statistical mechanical interpretation (continued).
Lets rewrite the expression that we obtained previously for the entropy, i.e.,
S =
NE 334 Lecture 9, May 22 2014
What happens when there are more states at this energy?
(a) If energy Ei has degeneracy i , then we obtain
pr =
r eEr
.(summing over levels)
Ei
i i e
(b) If there are continuum states as accessible states?
Translational motio
NE 334 Lecture 12, May 29, 2014
Section III.1 in textbook (this lecture and upcoming lectures)
Swap Lecture < > Tutorial:
Lecture tomorrow at 16:30 instead of tutorial
Tutorial Tuesday June 2nd at 14:30 instead of lecture
macroscopic result:
W = X dx ,
Er
NE 334 Lecture 7, May 20, 2014
KEY SUPPOSITION: if there are accessible states, it is extremely dicult to see why the
system should preferably be in any specic one of these states. We therefore GUESS that the
1
probability will be p =
for any given state.
NE 334 Lecture 8, May 21, 2014
The Canonical Ensemble.
Let us examine in some detail what happens when two systems can exchange energy between
them, i.e., when two systems are not individually isolated systems.
A
A
A*
Can we make any quantitative statemen
NE 334 Lecture 11, May 28, 2014
Mean Values and Thermodynamics. Section III.1 in textbook (this lecture and upcoming
lectures)
The Canonical Ensemble: Closed Systems.
Mean energy E for a canonical distribution:
E
pr Er =
r
r
Er eEr
Er
r e
or
E =
sum over
NE 334 Lecture 10, May 27, 2014
Reading for next class: Sec. III.1 in textbook.
The Grand (Canonical) Ensemble: Open Systems. Another tool we will need in our toolbox. Our last ensemble. The remainder of the course will be applications of statistical mech
NE 334 Lecture 6, May 15, 2014
Beginning of Aside on derivation
Aside: we can evaluate this integral by recognizing that the integrand is a symmetric function
of y, so that
+
dy y 2 ey
2 /(2N pq)
dy y 2 ey
=2
2 /(2N pq)
,
0
which can be obtained from the
NE 334 Lecture 4, May 13, 2014
Binomial Distribution.
p(N, n) =
N!
pn (1 p)N n
n!(N n)!
What is a better way of representing the binomial distribution when N becomes very large?
Let us try to approximate it by an equivalent continuous distribution, so tha
NE 334 Lecture 3, May 8, 2014
We obtained the ideal gas equation of state from Newtons law.
PV =
2
3
N
1
mv 2
2
.
Let us call the total energy U , and hence write
PV =
2
3
U.
What about the factor 2 ? Is it always that, independent of the nature of the ga
NE 334 Lecture 2, May 7, 2014
Ideal gas microscopic derivation (continued)
MODEL for explaining gas pressure.
Piston
V
F
vacuum
x
dx
We consider an idealized model in order to dispense with irrelevant details, and to bring out
the essential elements. We s