Physics 489 Problem Set 4
Due Tuesday, Oct. 6
(1) Kittel 5.1.
(2) Specific heat in the Debye model in 1D:
(a) Find the density of modes vs. ω for the case that the speed of sound is c and the lattice
constant is a.
(b) Calculate the cutoff frequency, ωD.
(c) Write down the integral expression for the energy, and by change of variables change the
integral to dimensionless units.
(d) Evaluate the integral for the low temperature regime, and then take the derivative to find the
specific heat for this temperature range. In this case we are looking for the leading term; you will
have to find the integral in a table or perhaps in Mathematica. What is the temperature exponent
for the specific heat in this case?
(e) Evaluate the integral in the limit of the high temperature regime, and again take the
derivative to get CV for high temperatures. However, in this case find the leading term and also
the next term in the expansion. Compare to the expectation for a classical harmonic lattice.
(3) Group velocity in 2D: Consider the simple case of a 2-D lattice, the same as was treated in
the last homework set in problem 3.4 (with the same solution for lattice vibration frequencies).
(a) Find the group velocity (as a 2D vector, as a function of k) for this case.
(b) Find the points in the first Brillouin zone where the group velocity magnitude is zero.
(c) Show that the component of the velocity in the direction perpendicular to the zone
boundary is zero, at all points on the boundary.
(d) Find the speed of sound, and show that it is isotropic. (Note, this is often the case but not a
general rule for general crystal symmetries.)
(d) Determine the Debye cutoff wavevector and the cutoff frequency for the Debye model in
this case. Compare the Debye wavevector to the dimensions of the first Brillouin zone.
(4) Thermal conductivity of germanium: See fig. 5.19 in your text showing a plot. At 200 K,
normal Ge has k ~1 W/(cm.K). The decrease with increasing T is due mostly to Umklapp
scattering as noted in class.
(a) Ge has a lattice constant of 5.7 Å (conventional cubic cell, for Ge in the diamond structure).
From this determine the molar density of Ge, moles/cm3.
(b) Assume that at 200 K the phonon contribution to the specific heat for Ge has already reached
the classical limit, and determine the specific heat per cm3. [Alternatively, you can make a
slightly better estimate from the plot earlier in chapter 5, where you can see that C is still ~10%
below the classical limit at this temperature.]
(c) Ge has TA and LA phonon velocities of about 5000 and 3500 m/s, however the optical
branches have considerably smaller velocities (see the slide I showed in class). Assume for a
rough approximation a mean phonon velocity of 2000 m/s, and then based on your above results
determine the phonon Umklapp mean free path at 200 K. Compare this to the period of the
highest-frequency lattice vibration mode for Ge (see again the slide I showed in class; vertical
units are 1012 Hz).
(5) Zero-point motion, 1D crystal:
Consider the state |0>, having phonon number nk = 0 in all of the N phonon states for the simple
1-D lattice considered in appendix C. The expectation value of us2 (or qs2 in the notation of the
appendix) can be calculated for such a state using the raising and lowering operators defined in
(28) and (29). Equation (32) is particularly helpful.
(a) For a given lattice site s, determine the expectation value of qs2 in the zero-phonon
ground state, as a sum over k. Your sum will contain the frequency ωk which depends on
k. Note that the squared operator contains a sum of many pairs of raising-lowering
operator pairs, but most of these pairs evaluate to zero in the expectation value, greatly
simplifying the sum.
(b) Write down the simple expression for the 1-D density of modes in k-space, and turn your
summation from (a) into an integral over k. With the form (15) for the frequency, or its
equivalent half-angle form, evaluate the integral. (The definite integral can be evaluated
numerically, for example one way would be on the Wolfram integral site.) The final
expression should not have factors of N, etc.: only M, C, and Planck’s constant. (C is the
spring constant, which I called K in class.)