Dispersion theorem for the thermodynamic properties of harmonic crystals

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Abstract
The thermodynamic properties of a crystal lattice in the harmonic approximation are expressed in terms of a dispersion function conjugate to the spectral distribution of normal-mode vibrations. Frequency moments of the normal-mode distribution provide the necessary and sufficient information for construction of accurate bounds on the dispersion function and on the associated thermodynamic properties. This clarifies the successful employment of moment expansions and approximate frequency spectra in determinations of thermodynamic properties, and establishes the equivalence of recently devised continued-fraction and moment-theory techniques for investigating crystal lattice vibrations. When sufficient numbers of frequency moments are available, accurate histogram approximations to the normal-mode distribution, which correctly image band gaps and van Hove singularities, are obtained from Stieltjes inversion of the dispersion function, thereby providing an alternative to the more customary root-sampling techniques.