Expansion of the Vibrational Spectrum at Low Frequencies

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Abstract
The branches of the vibrational spectrum of a crystal lattice are expanded at low frequencies in a form that enables the coefficients to be evaluated by a digital computer. Here the expansion is applied to the fcc lattice with nearest-neighbor central-force interactions. Thirty terms are derived for each branch of the spectrum. Previously only two coefficients were available. The two-point Padé approximant is applied to the expansion of the thermodynamic functions at low and high temperatures. These approximants provide closed-form approximations which give the correct low- and high-temperature expansions for the number of terms available. It also provides the most accurate approximation at intermediate temperatures. Improved closed-form approximations are derived for the frequency spectrum.