1. Chapter 16: The Heat Capacity of a Solid

2. 16.1 Introduction It is important in the study of condensed matter
This is another example that classical kinetic theory cannot provide answers that agree with experimental observations.
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.
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.

6. The heat capacity:

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.

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.
Calculate the Helmholtz function F.
Calculate entropy S.
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