The critical points (points where ∂ω∂xᵢ=0 for all $i=1, 2, 3$) of the frequency ω of the lattice vibrations in wave number space (x₁, x₂, x₃), shown by Van Hove to exist for a very general class of crystals, are located for the monatomic simple, face-centered, and body-centered cubic lattices. The position and nature of the resulting singularities in the frequency distribution are found as a function of the ratio of force constants for second-nearest neighbors and nearest neighbors, and the qualitative features of the frequency distribution are thus determined. A method for using the information so obtained to determine the frequency distribution quantitatively is outlined.
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Herbert B. Rosenstock (1955) studied this question.
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