1. Chapter 16: The Heat Capacity of
a Solid

2. 16.1 Introduction
1. It is important in the study of condensed matter
2. This is another example that classical kinetic theory
cannot provide answers that agree with
experimental observations.
3. Dulong and Petit observed in 1819 that the specific
heat capacity at constant volume of all elementary
solids is approximately 2.49*104
J .kilomole-1
K-1
i.e.
3R.
4. Dulong and Petit’s result can be explained by the
principle of equipartition of energy via treating
every atom of the solid as a linear oscillator with six
degrees of freedom.

3. 5. Extensive studies show that the specific heat
capacity of solid varies with temperature,
becomes zero as the temperature
approaches zero.
6. Specific heat capacities of certain substances
such as boron, carbon and silicon are found
to be much smaller than 3R at room
temperature.
7. The discrepancy between experimental
results and theoretical prediction leads to
the development of new theory.

4. 16.2 Einstein’s Theory of The Heat
Capacity of a Solid
• The crystal lattice structure of a solid
comprising N atoms can be treated as an
assembly of 3N distinguishable one-
dimensional oscillators!
• The assumption is based on that each atom is
free to move in three dimensions!

5. From chapter 15:
the internal energy for N linear oscillators
is
U= Nkθ(1/2 + 1/(eθ/T
-1)) with θ = hv/k
The internal energy of a solid is thus
Here θ is the Einstein temperature and can be
replaced by θE.
)
1
1
2
1
(3
−
+=
T
E E
e
NkU θ
θ

6. The heat capacity:
T
e
Nk
Nk
T
U
C
T
E
E
v
v
E
∂
−
+∂
=





∂
∂
=
)
1
3
2
1
3( θ
θ
θ

7. Case 1: when T >> θE
This result is the same as Dulong & Petit’s

8. Case 2:
As discussed earlier, the increase of is out
powered by the increase of
As a result, when
T << θE

9. If an element has a large θE , the ratio will be
large even for temperatures well above
absolute zero
When is large, is small

10. Since
A large θE value means a bigger
On the other hand
To achieve a larger , we need a large k or a
small u (reduced mass), which corresponds to
lighter element and elements that produce
very hard crystals.
k
hv
E =θ
u
k
v
π2
1
=

11. • The essential behavior of the specific heat capacity
of solid is incorporated in the ratio of θE/T.
• For example, the heat capacity of diamond
approaches 3Nk only at extremely high temperatures
as θE = 1450 k for diamond.
• Different elements at different temperatures will
poses the same specific heat capacity if the ratio θE/T
is the same.
• Careful measurements of heat capacity show that
Einstein’s model gives results which are slightly
below experimental values in the transition range of

12. 16.3 Debye’s theory of the heat capacity of
a solid
• The main problem of Einstein theory lies in the
assumption that a single frequency of vibration
characterizes all 3N oscillators.
• Considering the vibrations of a body as a whole,
regarding it as a continuous elastic solid.
• In Debye’s theory a solid is viewed as a phonon gas.
Vibrational waves are matter waves, each with its
own de Broglie wavelength and associated particle
• De Broglie relationship: any particle travelling with a
linear momentum P should have a wavelength given
by the de Broglie relation:

13. For quantum waves in a one dimensional box,
the wave function is
with
Since where is the speed

14. Considering an elastic solid as a cube of volume
v = L3
where
The quantum numbers are positive integers.
Let f(v)dv be the number of possible frequencies
in the range v to v + dv, since n is proportional
to v, f(v)dv is the number of positive sets of
integers in the interval n to n + dn.

15. Since

16. In a vibrating solid, there are three types of
waves
After considering one longitudinal and two
transverse waves,
Note that: since each oscillator of the assembly
vibrates with its own frequency, and we are
considering an assembly of 3N linear
oscillators, there must be an upper limit to the
frequency, so that

17. is determined by the
average inter atomic
spacing

18. The principle difference between Einstein’s
description and Debye’s model
There is no restriction on the number of
phonons per energy level, therefore phonons
are bosons!

19. • Because the total number of phonons is not
an independent variable
The internal energy of the assembly

20. To get
Debye Temperature

21. Let
High temperature,

22. • Example I: (problem 16.1) The partition function of
an Einstein solid is
where θE is the Einstein temperature. Treat the
crystalline lattice as an assembly of 3N
distinguishable oscillators.
(a) Calculate the Helmholtz function F.
(b) Calculate entropy S.
(c) Show that the entropy approaches zero as the
temperature goes to absolute zero. Show that at
high temperatures, S ≈ 3Nk[1 + ln(T/ θE )]. Sketch
S/3Nk as a function of T/ θE .

23. Solution (a)
Follow the definition
The value of U is known as
To solve F, we need to know S (as discussed in
class)

24. For distinguishable oscillators
therefore, for distinguishable oscillators (or
particles)

25. since we have 3N oscillators
(this is the solution for b)

26. a)
c) We have the solution for S

27. When
For T is high