42 3 Lattice Dynamics: Phonons
yields 3 r eigenfrequencies ω
s
(q)=ω
s
(−q),s=1,...,3r, with corresponding
normalized eigenvectors e
s
τ
(q). It is important to note that the solutions of
(
3.19) describe collective modes or excitations , for which all ions of the lattice
move with the same time dependence but phase shifted with respect to each
other according to Bloch’s theorem. Fo r the collective mode sq,themotion
of the individual ion (or mass) at R
nτ
is described by the displacement
u
s
nτ
(q,t) ∼
1
√
M
τ
e
s
τ
(q)e
i
(
q·R
0
n
−ω
s
(q)t
)
. (3.22)
As usual, the eigenmodes form a complete set of solutions that can be used as
the basis for representing an arbitrary motion of the lattice or of the individual
ion.
Essential asp ects of lattice dynamics in the harmonic approximation, such
as the dependence of the frequencies on the force constants a nd masses, the
distinction between acoustic and optical branches (see Sects.
3.4 and 3.5),
and the anisotropy of the dispersion curves, can be studied already in sim-
plified models as those of Problems 3.1 and 3.2. Moreover, even the reduced
dimension assumed in these problems is not hypothetical b ut corresponds to
physical reality: Take the atoms at the surfaces of solids, they move differently
from those in the bulk and give rise to investigate surface phonons [
85–87].
They represent a (quasi) two-dimensional dynamical system with phonon
amplitudes that decay over a few lattice constants away from the surface.
3.2 Normal Coordinates
Having found the eigensolutions of the Hamiltonian (
3.8) in the pr evious
section, we now aim at formulating this Hamiltonian in terms of these eigen-
solutions or normal coordinates. We expect that in this representation the
Hamiltonian will be that of a set of uncoupled harmonic oscillators, each
corresponding to a collective mode.
The displacement of an ion (or atom) can be expressed in terms of the
complete set of eigensolutions (
3.22)
u
nτ i
(t)=
1
√
NM
τ
sq
f
s
(q)e
−iω
s
(q)t
e
s
τ i
(q)e
iq·R
0
n
, (3.23)
with expansion coefficients f
s
(q). The normal co ordinate for the collective
lattice mode sq is defined by
Q
s
(q,t)=f
s
(q)e
−iω
s
(q)t
. (3.24)
The following scheme demonstrates the intended transformation from a sys-
tem with coupled localized motions of the individual masses (or ions) around
their equilibrium position to the uncoupled delocalized collective motions with
all ions moving in phase: