### K K K K m m m M M

```PHYS 7363 - Homework #1. Due Sept. 9, 2014.
1. Consider a variation on the simple lattice vibration problem we analyzed in class. Let’s assume that we have a
chain of atoms separated by distance a, connected by spring constants K,but of two diﬀerent masses, so that
masses m and M alternate.
m
K
K
K
M
m
K
M
m
a) Find the frequencies of lattice vibrations as a function of the wave number k. Specify the limits for
physically distinct solutions, and give allowed values of k.
b) Check that you recover the results of the discussion in class for m = M .
c) Determine the contribution of these vibrations to the heat capacity of the crystal at low temperatures.
2. In ferromagnetic insulators there are excitations called spin waves whose spectrum depends on the wave vector
as ω(k) = ak 2 . What is the contribution of these excitations to the heat capacity of three-dimensional magnetic
insulators at low temperatures? Is it greater or smaller than the contribution of phonons?
3. What are the density of states, the connection between the Fermi momentum and the Fermi energy and the