• (Much of the) heat capacity (specific heat) of materials is due to atomic vibrations.
• Boltzmann and Einstein models consider these vibrations as N simple harmonic oscillators.
• Boltzmann classical analysis obtains law of Dulong–Petit C = 3N kB= 3R.
• Einstein quantum analysis shows that at temperatures below the oscillator frequency, degrees of freedom freeze out, and heat capac-ity drops exponentially. Einstein frequency is a fitting parameter.
• Debye Model treats oscillations as sound waves with no fitting parameters.
– ω = v|k|, similar to light (but three polarizations not two) – quantization similar to Planck quantization of light
– maximum frequency cutoff (ωDebye = kBTDebye) necessary to obtain a total of only 3N degrees of freedom
– obtains Dulong–Petit at high T and C∼ T3at low T .
• Metals have an additional (albeit small) linear T term in the heat capacity.
c/T(mJ/mol-K2)
T2 (K2)
0 2 4 12 14 16
1 2 3
Fig. 2.5 Heat capacity divided by temperature of silver at very low tem-perature plotted against temtem-perature squared. At low enough temperature one can see that the heat capacity is actually of the form C = γT + αT3. If the dependence were purely T3, the curve would have a zero intercept. The cubic term is from the Debye theory of specific heat. The linear term is special to metals and will be discussed in Section 4.2. Figure from Corak et al., Phys. Rev. 98 1699 (1955), http://prola.aps.org/abstract/PR/v98/
i6/p1699 1, copyright American Phys-ical Society. Used by permission.
References
Almost every solid state physics book covers the material introduced in this chapter, but frequently it is done late in the book only after the idea of phonons is introduced. We will get to phonons in Chapter 9.
Before we get there the following references cover this material without discussion of phonons:
• Goodstein, sections 3.1–3.2
• Rosenberg, sections 5.1–5.13
• Burns, sections 11.3–11.5
Once we get to phonons, we can look back at this material again. Dis-cussions are then given also by
• Dove, sections 9.1–9.2
• Ashcroft and Mermin, chapter 23
• Hook and Hall, section 2.6
• Kittel, beginning of chapter 5
16 Specific Heat of Solids: Boltzmann, Einstein, and Debye
2.3 Appendix to this Chapter: ζ(4)
The Riemann zeta function is defined as24
24One of the most important un-proven conjectures in all of mathemat-ics is known as the Riemann hypothe-sis and is concerned with determining for which values of p does ζ(p) = 0.
The hypothesis was written down in 1869 by Bernard Riemann (the same guy who invented Riemannian geome-try, crucial to general relativity) and has defied proof ever since. The Clay Mathematics Institute has offered one million dollars for a successful proof.
ζ(p) =
∞ n=1
n−p.
This function occurs frequently in physics, not only in the Debye theory of solids, but also in the Sommerfeld theory of electrons in metals (see Chapter 4), as well as in the study of Bose condensation.
In this appendix we are concerned with the value of ζ(4). To evaluate this we write a Fourier series for the function x2 on the interval [−π, π].
The series is given by
x2= a0
These can be calculated straightforwardly to give an=
2π2/3 n = 0
4(−1)n/n2 n > 0.
We now calculate an integral in two different ways. First we can directly
evaluate π
−πdx(x2)2=2π5 5 .
On the other hand, using the Fourier decomposition of x2 (Eq. 2.9) we can write the same integral as
π
where we have used the orthogonality of Fourier modes to eliminate cross terms in the product. We can do these integrals to obtain
π
Exercises 17
Exercises
(2.1) Einstein Solid
(a) Classical Einstein (or “Boltzmann”) Solid:
Consider a three-dimensional simple harmonic os-cillator with mass m and spring constant k (i.e., the mass is attracted to the origin with the same spring constant in all three directions). The Hamiltonian is given in the usual way by
H = p2 2m+k
2x2.
Calculate the classical partition function Z =
dp
(2π)3
dxe−βH(p,x).
Note: in this exercise p and x are three-dimensional vectors.
Using the partition function, calculate the heat capacity 3kB.
Conclude that if you can consider a solid to con-sist of N atoms all in harmonic wells, then the heat capacity should be 3N kB= 3R, in agreement with the law of Dulong and Petit.
(b) Quantum Einstein Solid:
Now consider the same Hamiltonian quantum-mechanically.
Calculate the quantum partition function Z =
j
e−βEj
where the sum over j is a sum over all eigenstates.
Explain the relationship with Bose statistics.
Find an expression for the heat capacity.
Show that the high-temperature limit agrees with the law of Dulong and Petit.
Sketch the heat capacity as a function of tem-perature.
(See also Exercise 2.7 for more on the same topic) (2.2) Debye Theory I
(a)‡ State the assumptions of the Debye model of heat capacity of a solid.
Derive the Debye heat capacity as a function of temperature (you will have to leave the final re-sult in terms of an integral that cannot be done analytically).
From the final result, obtain the high- and low-temperature limits of the heat capacity analyti-cally.
You may find the following integral to be useful
∞
By integrating by parts this can also be written as
∞
0
dx x4ex
(ex− 1)2 =4π4 15 .
(b) The following table gives the heat capacity C for potassium iodide as a function of temperature.
T (K) C(J K−1mol−1)
Discuss, with reference to the Debye theory, and make an estimate of the Debye temperature.
(2.3) Debye Theory II
Use the Debye approximation to determine the heat capacity of a two-dimensional solid as a function of temperature.
State your assumptions.
You will need to leave your answer in terms of an integral that one cannot do analytically.
At high T , show the heat capacity goes to a constant and find that constant.
At low T , show that Cv= KTn Find n. Find K in terms of a definite integral.
If you are brave you can try to evaluate the inte-gral, but you will need to leave your result in terms of the Riemann zeta function.
(2.4) Debye Theory III
Physicists should be good at making educated guesses. Guess the element with the highest Debye
18 Exercises
temperature. The lowest? You might not guess the ones with the absolutely highest or lowest temper-atures, but you should be able to get close.
(2.5) Debye Theory IV
From Fig. 2.3 estimate the Debye temperature of diamond. Why does it not quite match the result listed in Table 2.2?
(2.6) Debye Theory V*
In the text we derived the low-temperature Debye heat capacity assuming that the longitudinal and transverse sound velocities are the same and also that the sound velocity is independent of the direc-tion the sound wave is propagating.
(a) Suppose the transverse velocity is vt and the longitudinal velocity is vl. How does this change the Debye result? State any assumptions you make.
(b) Instead suppose the velocity is anisotropic. For example, suppose in the ˆx, ˆy and ˆz direction, the sound velocity is vx, vy and vz respectively. How might this change the Debye result?
(2.7) Diatomic Einstein Solid*
Having studied Exercise 2.1, consider now a solid made up of diatomic molecules. We can (very crudely) model this as two particles in three di-mensions, connected to each other with a spring, both in the bottom of a harmonic well.
H = p12 2m1 + p22
2m2 +k 2x12+k
2x22+K
2(x1− x2)2 where k is the spring constant holding both par-ticles in the bottom of the well, and K is the
spring constant holding the two particles together.
Assume that the two particles are distinguishable atoms.
(If you find this exercise difficult, for simplicity you may assume that m1= m2.)
(a) Analogous to Exercise 2.1, calculate the classi-cal partition function and show that the heat ca-pacity is again 3kB per particle (i.e., 6kB total).
(b) Analogous to Exercise 2.1, calculate the quan-tum partition function and find an expression for the heat capacity. Sketch the heat capacity as a function of temperature if K k.
(c)** How does the result change if the atoms are indistinguishable?
(2.8) Einstein versus Debye*
In both the Einstein model and the Debye model the high-temperature heat capacity is of the form
C = N kB(1− κ/T2+ . . .).
For the Einstein model calculate κ in terms of the Einstein temperature.
For the Debye model calculate κ in terms of the Debye temperature.
From your results give an approximate ratio TEinstein/TDebye. Compare your result to the val-ues for silver given in Fig. 2.4. (The ratio you cal-culate should be close to the ratio stated in the caption of the figure. It is not exactly the same though. Why might it not be?)