Frequency Spectrum of Crystalline Solids. II. General Theory and Applications to Simple Cubic Lattices

Abstract

At temperatures not too close to its melting point, the molecules in a crystalline solid vibrate with simple harmonic motion about fixed equilibrium positions in a space lattice. In order to calculate the thermodynamic properties of the solid, one must know the distribution of frequencies of the normal modes of vibration of the component oscillators. On the basis of the Born-Kármán model, the frequencies of the normal modes are the characteristic roots of a matrix. The moments of the frequency distribution can be found from the traces of powers of the matrix. Methods are developed for calculating the frequency spectrum and thermodynamic functions from these moments. The frequency spectrum of a simple cubic lattice is studied by deriving its polynomial approximation from its moments. The spectrum has two maxima, one near the middle of its frequency range and the other at the high frequency end. As the ratio of the interactions between next nearest neighbors to those of nearest neighbors increases, the height of the low frequency maximum increases and that of the high frequency maximum decreases. Specific heats are calculated from the moments of the frequency distribution.