IV. Vibrational Properties of the Lattice
A. Heat Capacity—Einstein Model
B. The Debye Model — Introduction
C. A Continuous Elastic Solid
D. 1-D Monatomic Lattice
E. Counting Modes and Finding N()
F. The Debye Model — Calculation
Having studied the structural arrangements of atoms in solids,
we now turn to properties of solids that arise from collective
vibrations of the atoms about their equilibrium positions.
A. Heat Capacity—Einstein Model (1907)
For a vibrating atom:
kz
E1  K  U
m
kx
 12 mvx2  12 mvy2  12 mvz2  12 k x x 2  12 k y y 2  12 k z z 2
ky
Classical statistical mechanics — equipartition theorem: in thermal
equilibrium each quadratic term in the E has an average energy 12 k BT , so:
E1  6(12 k BT )  3k BT
Classical Heat Capacity
E  NE1  3NkBT
For a solid with N such atomic oscillators:
Total energy per mole:
Heat capacity at constant
volume per mole is:
E 3NkBT

 3N Ak BT  3RT
n
n
d
CV 
dT
E
   3R  25 molJ K
 n V
This law of Dulong and Petit (1819) is approximately obeyed
by most solids at high T ( > 300 K). But by the middle of the
19th century it was clear that CV  0 as T  0 for solids.
So…what was happening?
Einstein Uses Planck’s Work
Planck (1900): vibrating oscillators (atoms) in a solid have quantized
energies En  n n  0, 1, 2, ...
[later QM showed En  n  12  
is actually correct]
Einstein (1907): model solid as collection of 3N independent 1-D
oscillators, all with same , and use Planck’s equation for energy levels
occupation of energy level n:
(probability of oscillator
being in level n)
f ( En ) 
e  En / kT

e
classical physics
(Boltzmann factor)
 En / kT
n 0

Average total
energy of solid:

E  U  3 N  f ( En ) En  3 N
n 0
 En / kT
E
e
 n
n 0

e
n 0
 En / kT
Some Nifty Summing

Using Planck’s equation:
U  3N
 n e
n 0

e
 n / kT
Now let x 
 n / kT

kT
n 0

U  3N
ne
n 0

e
 nx
Which can
be rewritten: U  3N
 nx
n 0
Now we can use
the infinite sum:
So we obtain:
d  nx
 e
dx n 0

e
 3N
 nx
n 0

1
x


1 x
n 0
n
for x  1 To give:
 
d  x
 e
dx n 0
n
 e 

x n
n 0
 e 

x n
n 0
d  ex 
  x 
dx  e  1  3N
3N
U  3N


e x  1 e  / kT  1
 ex 
 x 
 e 1 
1
ex

 x
x
1 e
e 1
At last…the Heat Capacity!
d U 
d  3N A 
CV 
  
  / kT

dT  n V dT  e
1 
Using our previous definition:
Differentiating:
CV 

   3R   e
e 1
 1
 3N A e / kT
Now it is traditional to define
an “Einstein temperature”:
So we obtain the prediction:
e
 / kT
E 
 
kT 2
 2
kT
 / kT
 / kT
2
2

k
CV (T ) 
3R
 e
E 2
e
T
E /T
 E /T

1
2
Limiting Behavior of CV(T)
High T limit:
E
T
 1
CV



1 
(T ) 
 3R
1   1
3R   e
(T ) 
 3R   e
e 
3R
E 2
E
T
T
2
E
T
T
 1
E 2
CV
These predictions are
qualitatively correct: CV  3R
for large T and CV  0 as T  0:
 E /T
T
E 2
E /T 2
T
3R
CV
Low T limit:
E
T/E
 E / T
But Let’s Take a Closer Look:
High T behavior:
Reasonable
agreement with
experiment
Low T behavior:
CV  0 too quickly
as T  0 !
B. The Debye Model (1912)
Despite its success in reproducing the approach of CV  0 as T  0, the
Einstein model is clearly deficient at very low T. What might be wrong with
the assumptions it makes?
• 3N independent oscillators, all with frequency 
• Discrete allowed energies: En  n
n  0, 1, 2, ...
Details of the Debye Model
Pieter Debye succeeded Einstein as professor of physics in Zürich, and soon
developed a more sophisticated (but still approximate) treatment of atomic
vibrations in solids.
Debye’s model of a solid:
• 3N normal modes (patterns) of oscillations
• Spectrum of frequencies from  = 0 to max
• Treat solid as continuous elastic medium (ignore details of atomic structure)
This changes the expression for CV
because each mode of oscillation
contributes a frequency-dependent
heat capacity and we now have to
integrate over all :
CV (T ) 
max
N ( ) C


E
(, T ) d
0
# of oscillators per
unit 
Einstein function
for one oscillator
C. The Continuous Elastic Solid
We can describe a propagating vibration of amplitude u along a rod of
material with Young’s modulus E and density  with the wave equation:
 2u
E  2u

2
t
 x 2
for wave propagation along the x-direction
By comparison to the general form of the 1-D wave equation:
2
 2u
2  u
v
2
t
x 2
  2f  2
we find that
v

v
E

So the wave speed is independent of
wavelength for an elastic medium!
 (k ) is called the
dispersion relation of
the solid, and here it is
linear (no dispersion!)

 kv
group velocity v g 
d
dk
k
D. 1-D Monatomic Lattice
By contrast to a continuous solid, a real solid is not uniform on an atomic
scale, and thus it will exhibit dispersion. Consider a 1-D chain of atoms:
M
a
In equilibrium:
s p
s
s 1
s 1
Longitudinal wave:
u s 1
For atom s,
Fs   c p u s  p  u s 
p
us
u s 1
p = atom label
p =  1 nearest neighbors
p =  2 next nearest neighbors
cp = force constant for atom p
us p
1-D Monatomic Lattice: Equation of Motion
 2u s
Fs  M 2   c p us  p  us 
t
p
Now we use Newton’s second law:
For the expected harmonic
traveling waves, we can write
Thus:
u s  uei ( kxs t )
xs = sa = position of atom s

Mu (i ) 2 ei ( ksat )   c p uei ( k ( s  p ) a t )  uei ( ksat )

p
Or:


 M 2 e i ( ksat )  e i ( ksat )  c p eikpa  1
p
So:


 M 2   c p e ikpa  1
Now since c-p = cp by symmetry,
p


 M 2   c p eikpa  e  ikpa  2   2c p cos( kpa)  1
p 0
p 0
1-D Monatomic Lattice: Solution!
2 
The result is:
2
M
 c p (1  cos(kpa)) 
p 0
4
M
c
p 0
2 1
sin
( 2 kpa)
p
The dispersion relation of the monatomic 1-D lattice!
Often it is reasonable to make the
nearest-neighbor approximation (p = 1):
4c1
 
sin 2 ( 12 ka)
M
2

The result is periodic in k
and the only unique
solutions that are
physically meaningful
correspond to values in
the range: 


a
k
a
4c1
M
k

2
a


a
0

a
2
a
Dispersion Relations: Theory vs. Experiment
In a 3-D atomic lattice we
expect to observe 3 different
branches of the dispersion
relation, since there are two
mutually perpendicular
transverse wave patterns in
addition to the longitudinal
pattern we have considered.
Along different directions in
the reciprocal lattice the
shape of the dispersion
relation is different. But
note the resemblance to the
simple 1-D result we found.
E. Counting Modes and Finding N()

A vibrational mode is a vibration of a given wave vector k (and thus ),
E   . How many
frequency  , and energy
are found in the
 modes


interval between ( , E , k ) and (  d, E  dE, k  dk ) ?
# modes

dN  N ( )d  N ( E )dE  N (k )d k
3
We will first find N(k) by examining allowed values of k. Then we will be
able to calculate N() and evaluate CV in the Debye model.
First step: simplify problem by using periodic boundary conditions for the
linear chain of atoms:
We assume atoms s
and s+N have the
same displacement—
the lattice has periodic
behavior, where N is
very large.
s+N-1
L = Na
s
s+1
x = sa
x = (s+N)a
s+2
First: finding N(k)
Since atoms s and s+N have the same displacement, we can write:
us  us  N
uei ( ksat )  uei ( k ( s  N ) a t )
This sets a condition on
allowed k values:
kNa  2n 
So the separation between
allowed solutions (k values) is:
Thus, in 1-D:
k
1  e ikNa
2n
Na
2
2
k 
n 
Na
Na
n  1, 2, 3, ...
independent of k, so
the density of modes
in k-space is uniform
# of modes
1
Na L



interval of k  space k 2 2
Next: finding N()
Now for a 3-D lattice we can apply periodic boundary
conditions to a sample of N1 x N2 x N3 atoms:
N3c
# of modes
N a N 2b N 3c
V
 1
 3  N (k )
volume of k  space 2 2 2 8
Now we know from before
that we can write the
differential # of modes as:
We carry out the integration
in k-space by using a
“volume” element made up
of a constant  surface with
thickness dk:
N2b
N1 a
 V 3
dN  N ( )d  N (k )d k  3 d k
8
3

d k  ( surface area ) dk 
3
 dS dk

N() at last!
Rewriting the differential
number of modes in an interval:
We get the result:
N ( ) 
dN  N ( )d 
V
dS dk
3 
8
V
dk
V
1
dS

dS

 
8 3 
d 8 3 
k
A very similar result holds for N(E) using constant energy surfaces for the
density of electron states in a periodic lattice!
This equation gives the prescription for calculating the density of modes
N() if we know the dispersion relation (k).
We can now set up the Debye’s calculation of the heat capacity of a solid.
F. The Debye Model Calculation
We know that we need to evaluate an upper limit for the heat capacity integral:
CV (T ) 
max
N ( ) C


E
(, T ) d
0
If the dispersion relation is known, the upper limit will be the maximum  value.
But Debye made several simple assumptions, consistent with a uniform, isotropic,
elastic solid:
• 3 independent polarizations (L, T1, T2) with equal propagation speeds vg
• continuous, elastic solid:  = vgk
• max given by the value that gives the correct number of modes per polarization (N)
N() in the Debye Model
d
vg 
dk
First we can evaluate
the density of modes:
N ( ) 
Since the solid is isotropic, all
directions in k-space are the same, so
the constant  surface is a sphere of
radius k, and the integral reduces to:
Giving:
V 2
N ( )  3 4k 
8 v g
2 2 v g3
V
2
k
V
1
V
dS

dS

3 
3

8
vg 8 vg
2
dS

4

k
 
for one polarization
Next we need to find the upper limit for the integral over the allowed range of
frequencies.
max in the Debye Model
Since there are N atoms in the solid, there are N unique
modes of vibration for each polarization. This requires:
max
N ( )d  N


0
max
3
Vmax
2
 d  2 3  N
Giving:
2 3 
2 vg  0
6 vg
V
1/ 3
 6 N 

 V 
max  v g 
2
 D
The Debye cutoff frequency
Now the pieces are in place to evaluate the heat capacity using the Debye
model! This is the subject of problem 5.2 in Myers’ book. Remember that
there are three polarizations, so you should add a factor of 3 in the expression
for CV. If you follow the instructions in the problem, you should obtain:
T 
CV (T )  9 Nk B  
 D 
3  /T
D

0
4 z
z e dz
(e z  1) 2
And you should evaluate this
expression in the limits of low T
(T << D) and high T (T >> D).
Debye Model:
Theory vs. Expt.
Better agreement
than Einstein
model at low T
Universal behavior
for all solids!
Debye temperature
is related to
“stiffness” of solid,
as expected
Debye Model at
low T: Theory vs.
Expt.
Quite impressive
agreement with
predicted CV  T3
dependence for Ar!
(noble gas solid)
(See SSS program
debye to make a
similar comparison
for Al, Cu and Pb)