Frequency Spectrum of Crystalline Solids. III. Body-Centered Cubic Lattices

This is a study of the dynamics of a Born-Karman atomic model of a body-centered cubic lattice developed with the assumption that only interactions between nearest and next nearest atomic neighbors are significant. The equations of motion of these systems are derived and the secular equations are given from which one can calculate the frequencies of the crystal's normal modes of vibration. From the secular equations the moments of the frequency spectrum are determined and by solving the ``moment problem'' the frequency spectrum itself is obtained as a function of the force constants of the lattice. There are two maxima of approximately equal height in the frequency spectrum. Asymptotic expressions are derived for the specific heat of a body-centered cubic lattice which are valid at high and low temperatures.