48 3 Lattice Dynamics: Phonons
The integrand increases with the flatn ess of the dispersion curve and we expect
singular behavior if |∇
q
ω
s
(q)| = 0, which defines a van Hove singularity [
88]
and the corresponding q as critical point (for a classification of the critical
points see [
21, 89]). An example of D(ω)isshowninFig.3.13 for GaAs. It is
clearly seen that the critical points cause pronounced structures in the density
of states. Especially the rather flat dispersion curves of the optical phonons
result in characteristic peaks.
3.4 Acoustic Phonons
Common to the phonon dispersions of all solids is the group of lowest branches
with frequencies starting from the center of the Brillouin zone, with linear
dependence on the wave vector (see the solutions of Problems 3.1 and 3.2).
This particular property b eing connected with sound propagation, as will be
discussed later in this section, has led to the term acoustic phonons. The other
important aspect of these phonons with the smallest quanta of energy is that
they dominate the low-temperature specific heat of the crystal lattice. If the
solid is heated, starting from the ground state at T = 0 K, the macroscopic
change of the temperature is connected microscopically with the creation,
first of all, of acoustic phonons. This addition of energy in a quantized form
is resp onsible fo r a peculiar behavior of the specific heat at low temperatures
to which we turn our attention at the beginning of this section.
The specific heat is the change of the thermal energy E(T )withthetem-
perature T . The thermal energ y of the crystal lattice, connected with the
thermal motion of the ions (or atoms), can be calculated as the thermal
expectation value of the Hamiltonian
ˆ
H (
3.41)
E(T )=E
0
+
sq
n
s
(q,T)¯hω
s
(q). (3.56)
Let us first consider the classical limit of this general expression, which
is valid for sufficiently large temperatu res ¯hω
s
(q) ≪ k
B
T , i.e., for a phonon
energy much smaller than the average thermal energy per degree-of-freedom.
In this case, the phonon o ccupation simplifies according to
n
s
(q,T)=
e
β¯hω
s
(q)
− 1
−1
≃
k
B
T
¯hω
s
(q)
. (3.57)
As we can also neglect the ground state energy E
0
(as compared to k
B
T ), the
thermal energy of the lattice is given by
E(T ) ≃
sq
k
B
T =3rNk
B
T. (3.58)
Tak ing the derivative with respect to T (at constant volume)