Solutions of the Secular Determinant on the Brillouin Zone Faces for Face-Centered Cubic Lattice Vibrations

Abstract

The contours of constant frequency of the longitudinal branch have been plotted on the hexagonal Brillouin zone face to aid in the study of the vibration spectra of the two-force constant model for facecentered cubic lattices. These contours are based on machine solutions of the secular determinant as well as on closed-form solutions along the edge lines of the $\frac{1}{6}\mathrm{th}$ portion of the hexagonal face cut by the planes of the irreducible trihedral angle ($\frac{1}{48}$ zone). Newly found closed-form solutions along the bisectrix of the $\frac{1}{6}\mathrm{th}$ portion provide greater accuracy of the contours. The graphical study has disclosed new types of critical points on the hexagonal zone faces.