Critical points in three dimensions

Abstract

Critical points of the E (κ) function that defines constant energy surfaces in crystals are of importance in the problem of electronic state densities in solids, spin wave theory, lattice dynamics, and other problems. Thus, the critical points of the function defining frequency in wave number space cause singularities in the frequency spectrum of a crystal. In this note, the difference between the number of three dimensional saddle points and the number of three dimensional maxima and minima of such a function of three variables contained in a region is computed from the number of two dimensional saddle points, maxima and minima on the boundary surface of the region. It is shown that this is the maximum information obtainable about three dimensional critical points from such knowledge of the function on the boundary.