a0a0 (r) r effects due to anharmonicity of the lattice potential frequencies become volume...
Post on 18-Dec-2015
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a0
(r)
r
Effects due to anharmonicity of the lattice potential
2 30 0 0( ) ( ) ( ) ...2
fa r a B r a
Frequencies become volume dependentVlnd
lnd kk
Frequency change modifies internal energy
Grueneisen parameter
BV
C
T
V
Vv
p 33
1
linear thermal expansion coefficient
Detailed approach:
Remember differential of Helmholtz free energy dF SdT pdV
T
Fp
V
We consider expansion of the sample in a stress-free state where p=0
0T
F
V
used to calculate expansion coefficient
1V
P
V
V T
Statistical physics provides relation between free energy and partition function
lnBF k T ZLet’s consider a single oscillator and later generalize to 3d sample
nE
n
Z e ( 1/ 2)n
n
e / 2
1
e
e
vibrational contribution to free energy
/ 2
ln1vib B
eF k T
e
1ln 1
2 Bk T e
Total free energy F
vibF F
value of the potential energy in equilibrium
In the anharmonic case time-averaged position of the oscillator no longer given by a0 .
a0
atom longer at positions r>a0
harmonic case: 0ta a
(r)
r
0a an 20
1
2f aa
ta in anharmonic case
vibF F
20 0
1
2a f aa 1
ln 12 Bk T e
For our 1d problem p=0 0T
F
a
where ( )a
a
2
0 0
1 1ln 1
2 2 Ba f a k T ea
0
01
02 1B
eaf a k T
a ea
01
02 1
af aa e
a
01 1
02 1
f aa e
a
0
10f a E
aa
Average thermal energy of the oscillator
Linear expansion coefficient
0
1 a
a T
01
f a Ea
a
T
0
10
EfT a T
a
0
1 E
a f a T
Fromln
ln
a
a a
ln
:ln a
11 ln a
a a
a
a
20
E
Ta f
20
VC
a f
With 1
3VP
V
V T and
20a f 1d ->3d V BT
1
3 3v
p T
CV
V T B V
0
1 a
a T
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