# 7.8: Additional Problems

- Page ID
- 18930

*7.6 Spin waves *

For harmonic lattice vibrations at low frequencies, *ω* = *c _{s}k*. There are analogous excitations of ferromagnets called “spin waves” which, at low frequencies, satisfy

*ω*=

*Ak*

^{2}. Find the temperature dependence of the heat capacity of a ferromagnet at low temperatures. (Do not bother to evaluate constants. . . I am only looking for the functional form of the temperature dependence.)

*7.7 Comparison of models *

(This problem is stolen from a GRE Physics test.)

One feature common to both the Debye and Einstein models for the specific heat of a crystal composed of *N* identical atoms is that

a. The average energy of each atom is 3*k _{B}T*.

b. The vibrational energy of the crystal is equivalent to the energy of 3*N* independent harmonic oscillators.

c. The crystal is assumed to be continuous for all elastic waves.

d. The speed of longitudinal elastic waves is less than the speed of transverse elastic waves.

e. The upper cutoff frequency of elastic waves is the same.