Phonon magnification in cubic crystals

A method has been suggested to evaluate the phonon magnification for any general direction inside a cubic crystal. The method consists in establishing the functional relationship between the polar variables (${\theta }_s,{\phi }_s$) of the group-velocity vector $\overrightarrow{\mathrm{S}}$ and the polar variables ($\theta ,\phi$) corresponding to the wave vector $\overrightarrow{\mathrm{k}}$, and evaluating the Jacobian of the transformation between these variables. A computer program was written for the phonon amplification factor $A$ and numerical values for the phonon focusing or defocusing have been given for over fifteen crystals for a large number of directions in three dimensions. It is found that the phonon magnification factor varies from values nearly equal to zero to values as high as 300 or 400 for certain directions of some crystals. An analytical expression has been given for the phonon magnification along a cubic axis. For crystals exhibiting a cuspidal edge along a cubic axis, a large number of wave vectors $\overrightarrow{\mathrm{k}}$ correspond to the same group-velocity vector along the $x$ axis. For waves propagating along the $x-y$ plane, these directions are shown to be given by the roots of a quadratic equation. By using a computer-based numerical method, these directions which correspond to a group-velocity vector along the cubic axis have been evaluated numerically for metallic copper.