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I believe that nobody who has a reasonably reliable sense for
the experimental test of a theory will be able to contemplate
these results without becoming convinced of the mighty
logical power of the quantum theory.
W. Nernst, Z. fur Elektrochem. 17, p.265 (1911).
Chapter 10
Lattice Heat Capacity
10.1
Heat Capacity of Solids
The Dulong-Petit (1819) “rule” for molar heat capacities of crystalline matter cv ,
predicts the constant value
3
cv = NA kB
2
=24.94J/mole ,
(10.1)
(10.2)
where NA is Avagadro’s number. Although the Dulong-Petit rule, which assumes
solids to be dense, classical, ideal gases [see Eq.8.29] is in amazingly good agreement
with the high temperature (∼ 300K ○ ) molar heat capacities of many solids, it fails
to account for the observed rapid fall in cv at low temperature. An especially large
effect in diamond caught Einstein’s (1907) attention and with extraordinary insight
he applied Plank’s “quanta” to an oscillator model of an atomic lattice to predict a
universal decline in cv as T → 0K ○ . Several years later, when low temperature molar
heat capacities could be accurately measured, they were indeed found to behave in
approximate agreement with Einstein’s prediction.1 It was this result that ultimately
succeeded in making the case for quantum theory and the need to radically reform
physics to accommodate it.
1
cv in metals has an additional very low temperature contribution from conduction electrons
which, of course, Einstein could not account for.
1
2
CHAPTER 10. LATTICE HEAT CAPACITY
10.1.1
Einstein’s Model
Einstein’s model assumes a solid composed of N atoms, each of mass M , bound to
equilibrium sites within a unit cell by simple harmonic forces. The potential energy
of each atom is
1
V (Rα ) = M ω02 δRα ⋅ δRα
2
(10.3)
with a clasical equation of motion
δ R̈α + ω02 δRα = 0
(10.4)
where δ Rα = Rα − Rα,0 is the displacement vector of the αth ion from its origin Rα,0 .
Einstein’s independent oscillator model ignores any interactions between ions so there
is only a single mode with oscillator frequency ω0 . From a modern perspective
Einstein’s intuitive harmonic assumption is correct, since atoms in a solid are bound
by a total potential energy V (R) consisting of:
1. A short range repulsive component arising from the screened coulomb interaction between positively charged ion cores
Vα,β ≈
Zα Zβ exp−γ∣Rα −Rβ ∣
e2
,
∑
2 α,β
∣Rα − Rβ ∣
(10.5)
α≠β
2. A long range attractive component arising from quantum mechanical electronelectron correlations and ion-electron interactions.
The two potential energy components are shown in Figure 10.1 together with their
sum, V (R), which has a nearly harmonic minimum near R0 .2
10.1.2
Einstein in 1 − D
Einstein’s landmark calculation of the heat capacity of a crystal lattice – the first
application of a quantum theory to solids – is based on an independent oscillator
2
With increasing displacement from R0 departures from pure harmonicity (anharmonicity) become important with significant physical consequences.
10.1. HEAT CAPACITY OF SOLIDS
3
screened coulomb repulsion
Figure 10.1: Long dashed line is the screened (short-range) coulomb repulsion between ion cores. Short dashed line is the effective ion-ion attraction due to quantum
mechanical electron correlations and ion-electron interactions. Solid line, V (r), is
the sum of the two contributions, displaying a nearly harmonic potential minimum
at R0 .
model. In its simplest form, consider a one-dimensional lattice with the αth independent oscillator having the potential
1
V (Rα ) = M ω02 δRα 2
2
(10.6)
̵ 0 (nα + 1 )
E(nα ) = hω
2
(10.7)
and quantum energy levels
where α = 1, 2, ⋯N and ω0 is the natural oscillator frequency. For the 1 − D oscillator
the quantum number nα = 0, 1, 2, ⋯∞.
4
CHAPTER 10. LATTICE HEAT CAPACITY
With all N atoms contributing, the macroscopic eigen-energies are
N
̵ 0
N hω
̵ 0 ∑ nα
+ hω
2
α=1
̵
N hω0 ̵
+ hω0 n
=
2
E (n) =
(10.8)
(10.9)
with
N
∑ nα = n
(10.10)
α=1
Figure 10.2: Cartoon array of 1-D Einstein harmonic oscillator potentials showing
the equally spaced energy levels of Eq.10.7.
10.1.2.1
Partition Function for 1 − D Einstein Model
Initially, in the spirit of Boltzmann we emphasize the role of “degeneracy” and reproduce results from Chapter 6 [see Eqs.?? and ??] where the partition function was
written
(N − 1 + n)!
̵ 0 ( N + n)]
exp [−β hω
2
n=0 (N − 1)! n!
̵
N β hω0 ∞ (N − 1 + n)!
−
̵ 0) .
2
=e
exp (−nβ hω
∑
(N
−
1)!
n!
n=0
∞
Z=∑
(10.11)
(10.12)
The result of this sum is not entirely obvious, but with a little fancy mathematics
using the Γ − f unction integral representation for the factorial
∞
Γ (n) = (n − 1)! = ∫ dttn−1 e−t
0
(10.13)
10.1. HEAT CAPACITY OF SOLIDS
5
we can make the replacement
∞
(N + n − 1)! = ∫ dtt(N +n−1) e−t
(10.14)
0
so the partition function becomes
̵ 0 ∞
N β hω
tN −1 e−t ∞ 1 n −nβ hω
−
̵ 0
2
Z =e
dt
.
∑ t e
∫
(N − 1)! n=0 n!
(10.15)
0
Now summing over n (to get an exponential) and then integrating over t (again using
the Γ − f unction)
̵ 0
N β hω
e 2
.
Z=
N
̵ 0
β
hω
(e
− 1)
(10.16)
Evaluation of this 1 − D partition function is not difficult, but it is not extensible to
higher dimensionality or to more physically interesting models.
10.1.3
Quasi-particles and the 1 − D Oscillator
A more useful route is through the occupation number (phonon quasi-particle) method
of Appendix D.
Using Eq.10.9 for the eigen-energies, the 1 − D oscillator partition function is
∞
∞
∞
Z = ∑ ∑ . . . ∑ exp [−β (
n1 =0 n2 =0
nN =0
∞
N
̵ 0
N hω
̵ 0 ∑ nα )]
+ hω
2
α=1
∞
N
nN =0
α=1
∞
̵
̵ 0 ∑ nα ] ,
=e−βN hω0 /2 ∑ ∑ . . . ∑ exp [−β hω
n1 =0 n2 =0
(10.17)
(10.18)
where the sum over states is equivalent to the sum over all phonon quasi-particle
occupation numbers nα . Explicitly summing over α we get a product of N identical
geometrical series
̵ 0 ∞
βN hω
∞
∞
−
̵ 0 n1 ]) ( ∑ exp [−β hω
̵ 0 n2 ]) ⋯ ( ∑ exp [−β hω
̵ 0 nN ])
2
( ∑ exp [−β hω
Z =e
n1 =0
n2 =0
nN =0
(10.19)
6
CHAPTER 10. LATTICE HEAT CAPACITY
or
N
∞
̵
̵
Z =e−βN hω0 /2 ( ∑ e−β hω0 n )
(10.20)
n=0
which is summed to give
̵
eβN hω0 /2
=
N
̵
(eβ hω0 − 1)
(10.21)
the same result as in Eq.10.16.
10.1.4
The 3 − D Einstein Model
The 3−D Einstein partition function although approaching a realistic model, still falls
short of what is physical observed. [See the Debye model discussion later in the chapter.] It uses the same method as above, except that the coordinate components x, y, z,
of the αth independent oscillator displacements are taken into account. The 3 − D
oscillator model has the eigen-energies
̵ 0 (nα,x + nα,y + nα,z + 3/2)
E (nα,x , nα,y , nα,z ) = hω
(10.22)
where again α = 1, 2, 3, ⋯N , nα,x , nα,y , nα,z = 0, 1, 2, ⋯, ∞ and ω0 is the oscillator
frequency. The N oscillator lattice has the eigen-energies
Ex,y,z =
10.1.4.1
N
̵ 0
3N hω
̵ 0 ∑ (nα,x + nα,y + nα,z )
+ hω
2
α=1
(10.23)
Partition Function for the 3 − D Einstein Model
Using the result of Eq.10.23 the partition function is written
̵ 0
3N β hω
2
Z =e
−
∞
∑
∞
∞
N
nN,x =0
nN,y =0
nN,z =0
α=1
̵ 0 ∑ (nα,x + nα,y + nα,z )]} (10.24)
∑ . . . ∑ exp {−β [hω
n1,x =0 n2,x =0
n1,y =0 n2,y =0
n1,z =0 n3,z =0
10.1. HEAT CAPACITY OF SOLIDS
7
The sum is managed, as in section 10.1.3, by first explicitly taking the sum over α in
the exponential. Then, because the three coordinate sum sets (nα,x , nα,y , nα,z ) are
identical, what remains is
̵ 0 ∞
3
3
3
3βN hω
∞
∞
̵
̵
̵
2
( ∑ exp [−β hω0 n1 ]) ( ∑ exp [−β hω0 n2 ]) ⋯ ( ∑ exp [−β hω0 nN ]) .
Z =e
−
n1 =0
n2 =0
nN =0
(10.25)
Finally,
̵ 0 ∞
3N
3N β hω
̵ 0n
−
β
hω
2
Z =e
(∑ e
) ,
−
(10.26)
n=0
where the remaining sum gives
̵ 0
⎤3N
⎡ β hω
⎥
⎢ −
1
)⎥⎥ .
Z = ⎢⎢e 2 (
̵
⎢
1 − e−β hω0 ⎥⎦
⎣
10.1.4.2
(10.27)
Thermodynamics of the 3 − D Einstein Model
Following the steps from previous chapters, the internal energy is
∂
ln Z
∂β
̵ 0 [ 1 + ⟨n⟩]
=3N hω
2
U =−
(10.28)
(10.29)
where3
⟨n⟩ =
3
1
.
̵
(eβ hω0 − 1)
(10.30)
⟨n⟩ is called a Bose-Einstein “function” or, for the case of phonons, the average Bose-Einstein
quasi-particle occupation number.
There is a distinction between the Bose-Einstein function for real, massive, boson particles, e.g.
He4 [see Chapter 17], and boson quasi-particles, e.g. phonons and photons, which are merely energy
excitations.
8
CHAPTER 10. LATTICE HEAT CAPACITY
Einstein’s constant volume heat capacity is therefore
CN = (
∂U
)
∂T N
∂U
)
∂β N
̵ 0
̵ 0 )2 eβ hω
(β hω
= − kB β 2 (
= 3N kB
(10.31)
2
̵ 0
(eβ hω
− 1)
It is conventional to replace the harmonic force ω0 with an Einstein temperature
θE
̵ 0.
kB θE = hω
(10.32)
Then
2
CN = 3N kB
(θE /T ) eθE /T
(eθE /T − 1)
(10.33)
2
which can characterize specific materials by fitting to experimental data.
3
θ
Figure 10.3: 3 − D Einstein model heat capacity CN vs. T . Note the sharp
3N kB
θE
exponential drop as T → 0
In the low temperature limit Eq.10.33 becomes
2
lim CN → 3N kB (θE /T ) e−θE /T
T →0
(10.34)
10.2. DEBYE MODEL
9
as shown in Figure 10.3. Sadly, this steep exponential decline is never observed.
What is universally observed is C ∼ T 3 .
Einstein was aware that a single oscillator frequency model was bound to be inadequate and he did try to improve upon it, without success. His primary objective,
however, was to apply quantum theory and show that it explained several poorly
understood phenomena. This he achieved.
At high temperature the Einstein result is lim CN = 3N kB , in accord with DulongT →∞
Petit.
10.2
Debye Model
The effect of atom-atom interactions were added to Einstein’s theory by Debye.4
Their consequence is to introduce dispersion into the oscillator frequencies, which is
precisely the correction Einstein sought but never achieved.
As a result of atom-atom interactions:
1. Translational (crystal) symmetry is introduced, with a new wave vector quantum number k, sometimes called crystal momentum, with
2π
kj =
νj j = x, y, z ,
(10.35)
Nj aj
where aj is the j th crystal direction lattice spacing, Nj the number of atoms in
the j th crystal direction and where νj = 1, 2, 3 . . . Nj . 5
2. As shown in Figure 10.4, rather than a single lattice frequency ω0 there is now
a range of frequencies6 which Debye assumed varied linearly with ∣k∣
ω =ω (k)
=cs ∣k∣
(10.36)
(10.37)
where cs is an average speed of sound in the crystal.7
4
P. Debye, “Zur Theorie der spezifischen Waerme,” Annalen der Physik (Leipzig), 39, 789 (1912).
In solid state physics it is conventional to choose − N2 < ν ≤ N2 which, in this example, would
define a 1 − D Brillouin Zone.
6
One might say that the atom-atom interactions have lifted the degeneracy among single atom
oscillator frequencies.
7
This turns out to be an approximation that accurately replicates the small ∣k∣ behavior of lattice
vibrations in 3 − D crystals.
5
10
CHAPTER 10. LATTICE HEAT CAPACITY
3. The infinitely sharp Einstein “phonon” density of states
DE (ω) = N δ (ω − ω0 )
(10.38)
becomes, in the Debye model,8
V ω2
2 π 2 c3s
DD (ω) =
(10.39)
4. Since in a finite crystal the quantum number k is bounded, the range of Debye’s
oscillator frequencies is also bounded, i.e.
0 ≤ ω < ΩD .
(10.40)
so that
ΩD
∞
∫ dωDE (ω) = ∫ dωDD (ω) ,
0
(10.41)
0
i.e. the total number of modes is the same in both models.
Whereas the Einstein internal energy UE is [see Eq.10.29]
∞
̵ ∫ ω dω {N δ (ω − ω0 )} [ 1 + ̵ 1 ]
UE =3h
2 eβ hω − 1
(10.42)
0
1
1
=3N ω0 [ + β hω
]
̵ 0
2 e
−1
(10.43)
the changes introduced by Debye give instead the internal energy UD
ΩD
2
̵ ∫ ω dω { V ω } [ 1 + ̵ 1 ]
UD =3h
2π 2 c3s
2 eβ hω − 1
(10.44)
0
̵ D
β hΩ
3V
= 3̵3 3 4 ∫
2π h cs β
0
8
See Appendix E.
1
1
)
dx x3 ( + x
2 e −1
(10.45)
10.2. DEBYE MODEL
11
Figure 10.4: Mode dependent frequencies. The solid curve represents an approximate
result for a real lattice. The dashed line represents Debye’s linear approximation.
The slope of the dashed line is the average speed of sound in the crystal. The
horizontal dashed grey line at ω = ω0 represents the dispersion of an Einstein lattice
model.
10.2.0.3
Thermodynamics of the Debye Lattice
̵ D << 1, the integral in Eq.10.45 can be approximated by
At high temperature, β hΩ
expanding ex ≅ 1 + x. Then using Eq.10.41 the Debye internal energy is
V ΩD 3
lim UD = 2 (
) kB T
T →∞
2π
cs
=3N kB T
(10.46)
(10.47)
12
CHAPTER 10. LATTICE HEAT CAPACITY
a result also consistent with the Dulong-Petit rule.
̵ D >> 1, the internal energy integral can be approximated
At low temperature, β hΩ
as
∞
1
1
3V
lim UD → 3 ̵ 3 3 4 ∫ dxx3 ( + x
)
T →0
2π h cs β
2 e −1
(10.48)
0
from which follows the low temperature heat capacity
4
6kB
π4 3
×
̵ 3 c3 15 T
π2h
s
12π 4
T 3
=N kB (
)×(
) .
5
ΘD
lim CV →
T →0
(10.49)
(10.50)
where
̵ D = kB ΘD
hΩ
(10.51)
with ΘD the Debye temperature.
The T 3 low temperature heat capacity is almost universally observed in solids. Examples of Debye temperatures are given in Figure 10.5
10.2. DEBYE MODEL
13
Diamond
Figure 10.5: Debye temperatures ΘD in solids. Charles Kittel, Introduction to Solid
State Physics, 7th Ed., Wiley, (1996).