60 3 Lattice Dynamics: Phonons
ε(0)
ε
∞
=
ω
2
L
ω
2
T
(3.114)
known as the Lyddane–Sachs–Teller
5
relation. Making use of this relation, we
also may write S = ω
2
T
(ε(0) − ε
∞
) in terms of the macroscopic quantities
ω
T
,ε(0), and ε
∞
. On the other hand, S =2ω
T
|M
T
|
2
/¯hV ε
0
is determined by
the microscopic parameters of the system (see (
3.104)), the charge η
±
= ±η
and the masses M
±
of the ions. In the long-wavelength limit of the optical
mode the latter move against each other with eigenvectors e
T
±
= ±
#
(M
∓
/M ),
where M = M
+
+ M
−
. This allows us to express
η =
μV ε
0
N
(ε(0) − ε
∞
)
!
1/2
ω
T
(3.115)
in terms of the macroscopic material parameters (here the reduced mass μ =
(1/M
+
+1/M
−
)
−1
appears b ecause o f the relative motion of the two ions
in the unit cell, see Problem 3.1). As this expression for η contains also the
transverse phonon frequency ω
T
, it is called the transverse charge [
89]. It is
related to the strength of the phonon oscillator and can be determined from
the measured spectrum by a line-shape fit.
In crystalline solids with more complex unit cells than the one with two
oppositely charged ions assumed here, there are several triples of optical
phonon branches with different longitudinal-transverse splittings, giving rise
to different transverse effective charges [
95].
3.6 Examples: Phonon Dispersion Curves
Phonon dispersion curves, showing the phonon frequencies ω
s
(q)fordifferent
branches s, are usually plotted versus q along different high symmetry direc-
tions in the Brillouin zone. For the examples to be discussed in this section,
which all have fcc or bcc point lattices, we refer to the Brillouin zones depicted
in Figs.
1.1 and 1.2. Phonon dispersion curves are obtained either experimen-
tally from inelastic scattering preferentially with neutr ons (for a more recent
introduction and examples see [
96, 97]) and also with photons and atoms, or
from model calculations of different sophistication. Both kinds of investigation
have influenced and stimulated each other and are well documented [
77–79],
thus, at present the phonon dispersion curves of solids are well known. For
collections of phonon dispersion curves, together with a compilation of the
original references, we refer to [
76, 81, 94]. A selection will be presented
and discussed in this section to provide the knowledge how to read phonon
dispersion curves and understand their principal material specific features.
5
Russell Hancock Lyddane 1913–2001, Robert Green Sachs 1916–1990, Edward
Teller 1908–2003