1. Lattice Vibrational Contribution to the Heat Capacity of the Solid

2. Lattice Vibrational Contribution to the Heat Capacity of the Solid
The Thermal Energy is the dominant contribution to the
heat capacity in most solids. In non-magnetic insulators,
it is the only contribution. Other contributions may be:
Conduction Electrons in metals & semiconductors.
Magnetic ordering in magnetic materials.
The calculation of the vibrational contribution to the
thermal energy & heat capacity of a solid has 2 parts:
1. Evaluation of the contribution of
a single vibrational mode.
2. Summation over the frequency
distribution of the modes.

3. Vibrational Specific Heat of Solids cp Data at T = 298 K

4. Historical Background
In 1819, using room temperature data, Dulong
& Petit empirically found that the molar
heat capacity at constant volume for solids
is approximately (per mole):
R is the gas constant.
This relationship is now known as the
Dulong-Petit “Law”

5. The Molar Heat Capacity
Assume that the heat supplied to a solid is
transformed into the vibrational kinetic &
potential energies of the lattice.
To explain the Dulong-Petit
“Law” theoretically, using
Classical, Maxwell-
Boltzmann Statistical
Mechanics, a knowledge of how the
heat is divided up among the degrees
of freedom of the solid is needed.

6. The Molar Heat Capacity
The Molar Thermal Energy of a Solid
The Dulong-Petit “Law” can be explained using
Classical Maxwell-Boltzmann
Statistical Physics. Specifically,
The Equipartition Theorem can be used.
This theorem states that, for a system in thermal
equilibrium with a heat reservoir at temperature T,
The thermal average energy per
degree of freedom is (½)kT
If each atom has 6 degrees of
freedom: 3 translational &
3 vibrational, then
R = NAk

7. The Molar Heat Capacity Heat Capacity at Constant Volume
By definition, the heat capacity of a
substance at constant volume is
Classical physics therefore predicts:
A value independent of temperature!

8. The Molar Heat Capacity
Experimentally, however, the Dulong-Petit Law
is found to be valid only at high temperatures.

9. Einstein Model of a Vibrating Solid: I
In 1907, Einstein extended Planck’s ideas to matter:
He proposed that the energy values of
vibrating atoms are quantized & proposed the
following simple model of a vibrating solid:
Assumptions:
Each atom is independent.
Each vibrates in 3-dimensions.
Each vibrational normal mode
has a quantum energy:
  ħ

10. In effect, Einstein modeled one mole of a solid as
an assembly of 3NA distinguishable oscillators.
He then used the Canonical Ensemble to calculate
the average energy of an oscillator in this model.

11. The Partition Function
To compute the average, note that it can be
written as
Z has the form:
  ħ
Here, b  [1/(kT)]
Z is called
The Partition Function

12. With b  [1/(kT)] and En = ne, , the partition
function Z for the Einstein Model is
  ħ
This follows from the
geometric series result

13. Differentiating Z with respect to b gives:
  ħ
Multiplying by –1/Z gives:

14.   ħ
This is the Einstein Model Result for the
average thermal energy of a single oscillator
of energy ħ. The total vibrational energy of
the solid is the sum over the 3NA normal
modes, each of which has a different normal
mode frequency .

15. The Heat Capacity of one normal mode in
the Einstein Model is given by:
  ħ
Do the derivative & define TE  e/k. TE is
called “The Einstein Temperature”
Finally, in the Einstein Model, CV has the
form (for each independent mode):

16. The Einstein Model’s last approximation: All modes have the
same frequency E !
This is clearly unphysical!
Einstein treated E as
an empirical parameter
 the Einstein Frequency

17. The Einstein Model of a Vibrating Solid
Einstein, Annalen der Physik 22 (4), 180 (1907)
CV for Diamond
TE = 1320 K

18. Thermal Energy & Heat Capacity: Einstein Model II: Another Derivation
The following assumes that you know enough
statistical physics to have seen the Cannonical
Ensemble & the Boltzmann Distribution!
The Quantized Energy of a Single Oscillator has
the form:
If the oscillator interacts with a heat reservoir at
absolute temperature T, the probability Pn of it
being in quantum level n is proportional to the
Boltzmann Factor: Pn 

19. Quantized Energy of a Single Oscillator:
In the Cannonical Ensemble, a formal
expression for the average energy of the
harmonic oscillator & therefore of a lattice
normal mode of angular frequency ω at
temperature T is given by:

20. QM energy of one oscillator: Thermal average energy of one oscillator:
The cannonical ensemble probability Pn of the oscillator
being in quantum level n has the form: Pn  [exp (-β)/Z]
where the partition function Z is given by:

21. Some straightforward math manipulation!
Thermal Average Energy:
Putting in the explicit form for n gives:
This sum is a geometric series which can be done exactly:

22. The equation for E can be rewritten:
Finally, the
result is:

23. The thermal average number of phonons n(ω) at
(1)
This is the Mean Phonon Energy. The first term in (1) is
called the Zero-Point Energy. As mentioned before, even
at 0 K the atoms vibrate in the crystal & have a Zero
Point Energy. This is the minimum energy of the system.
The thermal average number of phonons n(ω) at
temperature T is given by The Bose-Einstein (or
Planck) Distribution, & the denominator of the
second term in (1) is often written:

24. <E> = ћω[n() + ½] The number of phonons at temperature
(1)
(2)
By using (2) in (1), (1) can be rewritten:
<E> = ћω[n() + ½]
In this form, the mean energy <E> looks analogous to a
quantum mechanical energy level for a simple harmonic
oscillator. That is, it looks similar to:
So the second term in the mean energy (1) is interpreted as
The number of phonons at temperature
T & frequency ω.

25. ħω << kBT High Temperature Limit: <E>
Temperature dependence of
the mean energy <> of a quantum harmonic oscillator.
Taylor’s series
expansion of ex 
for x << 1
<E>
High Temperature Limit:
ħω << kBT
<E>
At high T, <E> is independent of ω. This high T limit is equivalent to the classical limit, (the energy steps are small compared to the total energy). So, in this case, <E> is the thermal energy of the classical 1D harmonic oscillator (given by the equipartition theorem).
<E>

26. ħω > > kBT “Zero Point Energy”
Temperature dependence of
the mean energy <> of a quantum harmonic oscillator.
Low Temperature Limit:
ħω > > kBT
“Zero Point
Energy”
At low T, the exponential in the denominator of the
2nd term gets larger as T gets smaller. At small
enough T, neglect 1 in the denominator. Then, the
2nd term is e-x, x = (ħω/(kBT). At very small T,
e-x  0. So, in this case, <E> is
independent of T: <E>  (½)ħω
<E>

27. Einstein Heat Capacity CV
The heat capacity CV is found by differentiating the average phonon energy:
Let

28. Einstein Heat Capacity CV
The specific heat CV in
this approximation
vanishes exponentially at
low T & tends to the
classical value at high T.
These features are
common to all quantum
systems; the energy tends
to the zero-point-energy
at low T & to the
where
Area =

29. at high temperatures for all substances. This is the
The specific heat at constant volume Cv depends
qualitatively on temperature T as shown in the
figure below. For high temperatures, Cv (per mole) is
close to 3R (R = universal gas constant. R  2 cal/K mole).
So, at high temperatures Cv  6 cal/K-mole
The figure shows that Cv = 3R
at high temperatures for
all substances. This is the
classical Dulong-Petit law.
This states that specific heat of a
given number of atoms of any solid
is independent of temperature & is
the same for all materials!

30. Einstein Model for Lattice Vibrations in a Solid Cv vs T for Diamond
Einstein, Annalen der Physik 22 (4), 180 (1907)
Points = Experiment. Curve = Einstein Model Prediction
TE = 1320 K

31. Einstein Model of the Heat Capacity of Solids
The Einstein Model was the first quantum theory of lattice vibrations in solids. He made the assumption that all 3N vibrational modes of a 3D solid of N atoms had the same frequency, so that the whole solid had a heat capacity 3N times

32. In the Einstein Model, the whole solid had a heat capacity 3N times
In this model, the atoms are treated as independent oscillators, but the energies of the oscillators are the quantum mechanical energies. This assumes that the atoms are isolated oscillators, which is not at all realistic. In reality, there are a huge number of coupled oscillators. Even this crude model gives the correct limit at high temperatures, where it reproduces the Dulong-Petit law of 3R per mole.

33. At high temperatures, all crystalline solids
have vibrational specific heat Cv = 3R = 6
cal/K per mole; they require 6 calories per
mole to raise the temperature 1 K.
This agreement between observation &
classical theory breaks down if the
temperature is not high.
Observations show that at room
temperatures and below the specific heat of
crystalline solids is not a universal
constant.

34. Cv vs T in Different Materials
In each of these materials (Pb,Al, Si,and Diamond) the specific heat approaches a constant value asymptotically at high T. But at low T, the specific heat decreases towards zero which is in a complete contradiction with the above classical result.

35. The Einstein model also correctly gives a specific heat tending to zero at absolute zero, but the temperature dependence near T= 0 does not agree with experiment.
Taking into account the actual distribution of vibration frequencies in a solid this discrepancy can be approximately accounted for using a model first proposed by Debye.