Set of supplementary notes.

52 CHAPTER 3. FROM ATOMS TO SOLIDS
Figure 3.15: Comparison of Debye density of states (a) with that of a real material (b).
There are 3 acoustic branches, and 3(m − 1) optical branches.
It is convenient to start with a simple description of the optical branch(es), the Einstein
model, which approximates the branch as having a completely flat dispersion ω(k) = ω 0 . In
that case, the density of states in frequency is simply
D E (ω) = Nδ(ω − ω 0 ) . (3.34)
We have a different results for the acoustic modes, which disperse linearly with momentum
as ω → 0. Using a dispersion ω = vk, and following the earlier argument used for electrons, we
get the Debye model
D D (ω) =
4πk2 dk
(2π/L) 3 dω = V ω2
2π 2 v . (3.35)
3
Of course this result cannot apply once the dispersion curves towards, the zone boundary, and
there must be an upper limit to the spectrum. In the Debye model, we cut off the spectrum at a
frequency ω D , which is determined so that the total number of states (N) is correctly counted,
i.e. by choosing
which yields
∫ ωD
0
dωD D (ω) = N (3.36)
ωD 3 = 6π2 v 3 N
. (3.37)
V
Notice that this corresponds to replacing the correct cutoff in momentum space (determined
by intersecting Brillouin zone planes) with a sphere of radius
k D = ω D /v . (3.38)
3.7 Lattice specific heat
Phonons obey Bose-Einstein statistics, but their number is not conserved and so the chemical
potential is zero, leading to the Planck distribution
n(ω) =
1
exp(h¯ω/k B T ) − 1 . (3.39)