I. Normal Frequencies of a One-Dimensional Crystal. II. An Approximation to the Lattice Frequency Distribution in Isotropic Solids

Abstract

I. The accurate expression for the individual frequencies of a linear crystal of any number of particles is derived. II. It is shown that, for an isotropic three‐dimensional array of uniform masses, the root‐mean‐square frequency is easily evaluated and the maximum square must always be less, but not much less, than twice the mean square. For two commonly studied simple cubic systems, the maximum square is evaluated and shown to be twice the mean square. The Debye expression for the frequencies in terms of standing wave components is modified empirically to give the correct maximum and mean squares by substituting linear crystal frequencies for the components and introducing a second force constant. The resulting expression, a simplified form of the factored secular equation, should yield a more realistic and probably therefore a more widely useful distribution function than the Debye equation.