2

1 2

1 1 10 0

1, ,..., (0,0,...,0) ...

2

N N N

N j i j

j i jj i j

U UU Ux x x x x x

x x x

Statistical thermodynamics of crystals

Monoatomic crystal „Ideal crystal“:

Regulary ordered point masses

connected via harmonic springs

Interatomic interactions – Represented by the lattice force-constant

Equivalent atom positions – minima on PES

Every atom moves around its equilibrium position

Example: one-dimensional crystal – displacement from equilibrium { ξi }

2

1 2

1 1 10 0

1, ,..., (0,0,...,0) ...

2

N N N

N j i j

j i jj i j

U UU Ux x x x x x

x x x

1 2

1 1

1, ,..., (0,0,...,0)

2

N N

N ij i j

i j

U U kx x x x x

Harmonic approximation – U(ξi) is a quadratic function – „reasonable“ appraoximation

Force constants – kij

U(0,0,...,0) – depends on the lattice parameter → function of ρ = V/N :

( , )U 0 r

1 2, ,..., NU x x x

ijkDepends on ρ

≠0

„Coupled harmonic osc.“

3N-6 independent vibrational modes

~ 3N

1/ 2

1

2

j

j

j

kn

p m

kj and μj stands for effective force constant and effective reduced mass

Solving the variational problem of atom cyrstal: transformation into 3N independent harmonic

oscilators.

Frequency of individual oscilators – depends on masses, force constants and type of the

crystal (complicated equation)

j j ij

Vk k

Nn Frequency of normal modes

depends on density !

Partition function of monoatomic crystal:

3 6( ; ) /

,

1

,N

U kT

vib j

j

VQ T e q

N

0 r (no rotational and translational degrees of freedom)

(atoms are distinguishable !)

Vibrational partition function – harmonic oscilator

1

2n h ne n

Vibrational level degeneracy 1nw

1/2

1

2

kn

p m

Zero energy defined as –De

/2/2

0

( )1

n

hh h n

vib hn n

eq T e e e

e

b nbe b n b n

b n

2 2

/

,

ln ln 1 1

2 1v

v vv v T

N V

Q d qE kT NkT Nk

T dT eQQ

/v hv kQ Vibrational temperature – typically 103 K – just first term needs to be considered

Population of vibrational levels: ( 1/ 2)h n

n

vib

ef T

q

b n

Fraction of molecule in vibrationally excited states:

( 1/ 2)/ 2/ 2

0 0

1 1

1 v

h nTh

n n

n n vib

ef T f T f e e

q

b nQb n

j j ij

Vk k

Nn

3 6( ; ) /

,

1

,N

U kT

vib j

j

VQ T e q

N

0 r

/ 2

/1

h kT

vib h kT

eq

e

n

n

/ 23( ; ) /

/1

,1

j

j

h kTNU kT

h kTj

V eQ T e

N e

0

nr

n

Solving the variational problem of atom cyrstal: transformation into 3N independent harmonic

oscilators.

Frequency of individual oscilators – depends on masses, force constants and type of the

crystal (complicated equation)

Frequency of normal modes

depends on density !

Partition function of monoatomic crystal:

(no rotational and translational degrees of freedom)

(atoms are distinguishable !)

Large number of vibrational modes (3N) – continuous distribution from 0 to νmax

Define „frequency density“ g(ν)dν – number of normal vibrational models in an

interval (ν,ν+dν)

/ 23( ; ) /

/1

,1

j

j

h kTNU kT

h kTj

V eQ T e

N e

0

nr

n

/

0

( ; )ln ln 1 ( )

2

h kTU hQ e g d

kT kT

0 nr nn n

0

( ) 3g d Nn n„Normalization“ condition:

We need a suitable approximation for g(ν); TD properties can be obtained

2

,

ln

N V

QE kT

T

/

/

0

( ; ) ( )21

h kT

h kT

h e hE U g d

e0

n

n

n nr n n

Almost exact (harmonic approximation only) –g(ν) is missing

=> Various approaches to find g(ν)

2 /

2/

0

/ ( )

1

h kT

Vh kT

h kT e g dC k

e

n

n

n n n

,

V

N V

EC

T

I. Classical thermodynamics

Dulong-Petit law

Each vibrational degree of freedom contributes based on equipartition theorem

3 3 6 / deg.VC Nk R cal mol

Works for numerous crystals at high temperatures

Fails at low temperatures

Qualitative failure at very low temperatures (CV approaches 0 K as T3 – experimentally)

Silver crystal

II. Einstein model

Quantization of vibrational energy (similar to Planck model of black body)

Each atoms vibrates around its equilibrium position independently of other atoms

3N independent oscillators with the same frequency νE

1907

Using g(ν): ( ) 3 Eg Nn d n n (delta function)

νE ... Frekvency (Einstein’s) 3N independent oscillators

Specifc for each crystal – depends on the PES details

/

/

0

( ; ) ( )21

h kT

h kT

h e hE U g d

e0

n

n

n nr n n

V

V

EC

T

2 /

2/

0

/( )

1

h kT

Vh kT

h kT eC k g d

e

n

n

nn n

2 /

2/

31

E

E

h kT

EV

h kT

h eC Nk

kT e

n

n

nE

E

h

k

nQ

2 /

2/

31

E

E

T

EV

T

eC Nk

T e

Q

Q

Q

EE

h

k

nQEinstein temperature:

A. Einstein, Ann. Physik, 22 (1907) 180.

Heat capacity of diamond

ΘE = 1320o K

Only parameter (Einstein temperature):

Works remarkably except for very low temps.

2

/0 : 3 E TEVT C Nk e

T

QQ

Dependence of CV on reduced temperature (ΘE/T) is

universal for all crystals

III. Debye model

Einstein model – fails at low temps

Oscillator energy depends on frequency T→0 : Low energy modes become

important

Norma mode frequency varies from 0 do 1013 Hz

Below – normal modes in 1-D crystel (high and low energy models depicted below)

A mode having the highest frequency: wavelength ~ 2a – atoms move against each other

A mode with minimal frequency – atoms moves in the same direction

Debye: modes with wavelength » lattice constant – independent of material – crystal

behaves as continuous elastic body

Wave with amplitude A and frequency ω=2πν and moving in the direction k :

( )( , ) i tu t Ae k rr

w

k je wave vector; 2π/λ

v ... Velocity of the wave / ku w nl

Superposition of waves moving in opposite direction:

Standing wave 2 cosiu Ae tk r w

To form a standing wave - its imaginary part must be zero on the border (crystal edge):

x x

y y

z z

k L n

k L n

k L n

p

p

pL

k np

/ ku w nl

Frequency depends on k

2

2 2 2 2

x y zk n n nL

p

Number of vawes with wavenumber in

interval (k, k+dk)

Number of waves having wavevector smaller

than k.

3 3 3 3

2 2( )

6 6 6

Lk L k Vkk

pF

p p p

2

2( )

2

d Vk dkk dk dk

dk

Fw

p

2

3

4( )

Vg d d

p nn n n

u

2

ku un

l pDistinguishing the direction

Of the wave

2

3 3

2 1( ) 4

t l

g d V dn n p n nu u

Vibrational modes in the direction perpendicual (or parallel)

3 3 3

0

3 2 1

t lu u u

2

3

0

12( )

Vg d d

pn n n n

u

Introducing average velocity:

Exact expression for low energy modes

Debye frequency – Maximal frequency of the crystal – follows from 0

( ) 3

D

g d N

n

n n

1/3

0

3

4D

N

Vn u

p

2

3

9( ) 0

0

D

D

D

Ng d dn n n n n n

n

n n

34/

20

91

DxT

Vx

D

T x eC Nk dx

e

Q

QD

D

h

k

nQ Debye temperature

2 /

2/

0

/( )

1

h kT

Vh kT

h kT eC k g d

e

n

n

nn n

3V

D

TC Nk D

Q

34/

20

31

DxT

xD D

T T x eD dx

e

Q

Q QDebye function:

One-parameter equation, numerical solution

For temperature approaching 0 K:

3412

0 :5

V

D

TT K C Nk

pQ

A proper behavior

Even for T goes to 0

Heat capacity as a function of T/ΘD – single universal curve

Aluminium 428 K

Cadmium 209 K

Chromium 630 K

Copper 343.5 K

Gold 165 K

Iron 470 K

Lead 105 K

Manganese 410 K

Nickel 450 K

Platinum 240 K

Silicon 645 K

Silver 225 K

Tantalum 240 K

Tin (white) 200 K

Titanium 420 K

Tungsten 400 K

Zinc 327 K

Carbon 2230 K

Ice 192 K