Advanced Solid State Physics

(a) (b)

Figure 18: a)ωas a function ofkin the 11-direction; b)ωas a function ofkx

Plottingωin the 10-direction (kx-direction), we get 2 frequencies, which correspond to one longitudinal
and one transverse mode (fig. 18(a)). If we plotωas a function of the 11-direction however, we only
get 1 frequency, because thex- and they-direction have the same influence (fig. 18(b)).

If we would have calculated this in 3 dimension, we would have ended up with a matrix equation for
everyk, which would look like this:



axx axy axz

ayx ayy ayz

azx azy azz









xk

yk

zk



=mω 2





xk

yk

zk



 (44)

This equation has 3 solutions, which correspond to 1 longitudinal and 2 transverse modes. (Fig.
19(a) shows the phonon dispersion relation for Hafnium, which has bcc-structure and 1 atom per unit
cell). Because there is onekfor every unit cell we have 3 Nnormal modes in our crystal (Nshall be
the number of unit cells in our crystal). If we generalize our assumption as previously and take two
different types of atoms into account, we have 2 atoms per unit cell, which would lead to a 6x6-matrix.
Hence, there would also be 6 eigenvalues which correspond to 3 acoustic and 3 optical branches. (Fig.
19(b) shows the phonon dispersion relation for Silicon, a material with two atoms per unit cell and a
fcc Bravais lattice) If we would have even more atoms per unit cell, the number of optical branches
would be higher, but we still would have 3 acoustic branches.

Although the phonon dispersion relations of some specific materials might look very complicated,
materials with the same crystal structure will have very similar dispersion relations.

6.2.1 Raman Spectroscopy

Raman spectroscopy is a way to determine the occupation of the phonon modes. Typically, a sample
is illuminated with a laser beam. Most of the light scatters elastically (Rayleigh scattering), but the