Cardan's solution for determining the dispersion relations for monatomic crystal lattices

Abstract

It is well known that the phonon dispersion relations for monatomic crystal lattices may be obtained from the dynamical matrix by solving the associated secular equation. For general directions of propagation, Cardan's solution may be applied to avoid the use of numerical techniques (for example those for matrix diagonalisation). The present analysis allows a degeneracy parameter T to be defined in simple terms: if T=0, the transverse branches are degenerate (usually in high symmetry directions): if T0 corresponds to two non-physical (complex) solutions. The analysis shows that only the first two cases will occur in practice (at least for cubic and tetragonal lattices) and that for these cases, the longitudinal modes have higher frequencies than the transverse modes.