Theory of the lattice vibrations of germanium

Abstract

Previous attempts to explain the frequency-wave number relations for the normal modes of germanium, which have been determined experimentally by Brockhouse & Iyengar, have required the assumption of force constants between atoms which are relatively far apart. The theory then involves a large number of undetermined parameters which have no obvious physical justification, and the fact that the elastic constants of germanium satisfy an identity which might suggest that only interactions between adjacent atoms are important, has to be dismissed as a coincidence. In this paper we extend the Born-von Kármán theory of lattice dynamics to apply to a simple model of the germanium crystal, in which each atom is regarded as a charged core coupled to an oppositely charged shell. This gives the atom the property of polarizability, not only in an electric field but also under the influence of bonding interactions between adjacent atoms. On the basis of this model, the frequency-wave number relations can be reasonably well accounted for with only two disposable parameters, and a simple explanation is provided of the fact that the elastic constants satisfy Born’s identity. The value deduced for the polarizability of a germanium atom, using the neutron spectroscopy data of Brockhouse & Iyengar, is in good agreement with that determined directly from the dielectric constant. An extension of the theory of Mashkevich & Tolpygo provides some theoretical justification for the use of a shell model for germanium.