Some properties of the vibrational spectrum of a lattice

Abstract

The vibrations of the particles which constitute a crystal can be represented in terms of normal vibrations subject to the condition that the energy of the crystal can be expressed as a homogeneous function of the second order in the displacements. This has been shown by Born and by Waller. It has been proved by Debye and, more generally by Born, that the density of the normal vibrations varies as ν2 when the frequency is very small. The spectrum has recently been worked out for a Born-v. Kármán lattice in the two- and three-dimensional cases. Otherwise practically nothing is known about the spectrum in individual cases, except perhaps in isolated cases where the frequency branches split up and the frequency spectrum shows certain gaps.