Chapter 16: The Heat Capacity of a Solid

16.1 Introduction 1. It is important in the study of condensed matter2. 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.

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.

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!

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

The heat capacity:

Te

NkNk

T

UC

T

EE

vv

E

)

1

321

3(

Case 1: when T >> θE

This result is the same as Dulong & Petit’s

Case 2:

As discussed earlier, the increase of is out powered by the increase of

As a result, when

T << θE

If an element has a large θE , the ratio will be large even for temperatures well above absolute zero

When is large, is small

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

hvE

u

kv

2

1

• 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

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:

For quantum waves in a one dimensional box, the wave function is

with

Since where is the speed

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.

Since

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

is determined by the average inter atomic spacing

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!

• Because the total number of phonons is not an independent variable

The internal energy of the assembly

To get

Debye Temperature

Let

High temperature,

• 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 .

Solution (a)Follow the definition

The value of U is known as

To solve F, we need to know S (as discussed in class)

For distinguishable oscillators

therefore, for distinguishable oscillators (or particles)

since we have 3N oscillators

(this is the solution for b)

a)

c) We have the solution for S

When

For T is high