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