 
Summary: Physics 607 Exam 2
Please show all significant steps clearly in all problems.
1. Consider spin waves at the surface of a ferromagnetic solid, with a single polarization and
the dispersion relation = c k2
, where c is a constant. Since a surface is 2dimensional, you
may consider this to be a 2dimensional system with area A.
(a) (5) Show that the number of wavevectors with magnitude in the interval dk at k is
given by
(k) dk = constant × A k dk (1)
and determine the constant.
(b) (15) Show that the heat capacity at low temperature T is given by
CV = constant × T
. (2)
At the same time, determine the constant prefactor and the value of .
1
2. In three dimensions, the number of wavevectors whose magnitude lies in the interval
dk at k is given by
(k) dk =
4k2
dk
