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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
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