5.2 Lattice Vibrations
109
N (E) =
V
2π 2
E
3
3 v 3
s
.
(5.36)
Thus the density of states is approximately
3 proportional to E
2 ,
D(E) =
3 V
2π 2
E
2
3 v 3
s
.
(5.37)
As realistic example for the phonon density of states, the DOS of bulk BN is depicted next to the
dispersion in Fig. 5.11.
5.2.6 Phonons
Phonons are the quantized quasi-particles of the lattice vibrations (normal modes). The energy of a
phonon can take the discrete values of a harmonic oscillator
E ph =
n +
1
2
ω ,
(5.38)
where n denotes the quantum number of the state, which corresponds to the number of energy quanta
in the vibration. The amplitude of the vibration can be connected to n via the following discussion.
For the classical oscillation u = u 0 exp i(kx − ωt) the space and time average for the kinetic energy
yields
E kin =
1
2
ρ V
∂u
∂t
2
=
1
8
ρ V ω
2 u
2
0 ,
(5.39)
where ρ is the density and V the volume of the (homogeneous) solid. The energy of the oscillation is
split in half between kinetic and potential energy. From 2E kin = E ph we find
(a)
(b)
Fig. 5.16 a Phonon dispersion for the diatomic linear chain model for γ = 1 (black line) and γ = 0.9 (blue lines). b
Corresponding density of states (in units of N /E m )
3 This dependence is the base for Debye law for the T 3 temperature dependence of the heat capacity at low temperatures
when only states with sufficiently small wave vector are thermally populated.
109
N (E) =
V
2π 2
E
3
3 v 3
s
.
(5.36)
Thus the density of states is approximately
3 proportional to E
2 ,
D(E) =
3 V
2π 2
E
2
3 v 3
s
.
(5.37)
As realistic example for the phonon density of states, the DOS of bulk BN is depicted next to the
dispersion in Fig. 5.11.
5.2.6 Phonons
Phonons are the quantized quasi-particles of the lattice vibrations (normal modes). The energy of a
phonon can take the discrete values of a harmonic oscillator
E ph =
n +
1
2
ω ,
(5.38)
where n denotes the quantum number of the state, which corresponds to the number of energy quanta
in the vibration. The amplitude of the vibration can be connected to n via the following discussion.
For the classical oscillation u = u 0 exp i(kx − ωt) the space and time average for the kinetic energy
yields
E kin =
1
2
ρ V
∂u
∂t
2
=
1
8
ρ V ω
2 u
2
0 ,
(5.39)
where ρ is the density and V the volume of the (homogeneous) solid. The energy of the oscillation is
split in half between kinetic and potential energy. From 2E kin = E ph we find
(a)
(b)
Fig. 5.16 a Phonon dispersion for the diatomic linear chain model for γ = 1 (black line) and γ = 0.9 (blue lines). b
Corresponding density of states (in units of N /E m )
3 This dependence is the base for Debye law for the T 3 temperature dependence of the heat capacity at low temperatures
when only states with sufficiently small wave vector are thermally populated.