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Purdue University
School of Mechanical Engineering
ME 501: Statistical Thermodynamics
Lecture 24: Statistical Thermodynamic
Analysis of Lattice Vibrations in
Crystals
Prof. Robert P. Lucht
Room 2204, Mechanical Engineering Building
School of Mechanical Engineering
Purdue University
West Lafayette, Indiana
[email protected]
,
765-494-5623 (Phone)
November 7, 2011

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Purdue University
School of Mechanical Engineering
Lecture Topics
•
The crystal lattice.
•
The Einstein theory of the heat capacity of crystalline
solids.
•
The Debye theory of the heat capacity of crystalline
solids.

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Purdue University
School of Mechanical Engineering
The Crystal Lattice
•
Consider the crystal lattice of a solid as a large number N atoms hooked
together by springs:
McQuarrie,
Statistical Mechanics
, 1976

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Purdue University
School of Mechanical Engineering
The Crystal Lattice
•
The N atoms in the crystal lattice are inherently distinguishable rather than
indistinguishable because we can associate each atom with a particular
lattice site.
Therefore Maxwell-Boltzmann (MB) statistics rather than
corrected Maxwell-Boltzmann statistics will apply to the lattice.
From our
previous discussion of MB and CMB statistics we found:
,
,
!
!
ln
ln
ln
j
N
j
m MB
j
j
j
m MB
j
j
j
g
W
N
N
g
W
N
N
N
N
(ln
)
(ln
)
ln
1
ln
0
j
j
mMB
mCMB
j
j
j
j
j
j
g
g
d
W
d
W
dN
dN
N
N
•
Differentiating these expressions to find d(ln
W
m
) gives us:
2
ln
ln
ln
ln
j
j
j
j
mMB
mCMB
j
j
j
j
j
j
j
g
N
g
g
W
W
N
N
N
N
g
N
N
,
ln
ln
j
m CMB
j
j
j
g
W
N
N
N