We investigate tile absorption of infra-red radiation by an impure ion
ic crystal, due to coupling with phonon modes. By means of a standard
diagrammatic perturbation theory, the phonon Green's function and henc
e the infra-red spectrum is calculated. The two absorption peaks corre
sponding to longitudinal and transverse modes are broadened by disorde
r into bands of finite width. This behaviour is very similar to that o
f electrons in a dirty metal with a magnetic field, with the phonon mo
des playing a similar role to the Landau levels of the electrons. This
suggests that the diagrammatic formalism can be used to study the tra
nsport properties of phonons in a crystal with impurities and long-ran
ged dipole-dipole interactions, by analogy with the methods used in th
e transport theory of electrons in disordered metals.