Thermodynamics of anharmonic crystals with application to Nb

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Abstract
A self-consistent variational method is used to approximate the thermodynamic functions of anharmonic crystals. It is shown that for an arbitrary lattice potential, the entropy is given correctly within the variational approximation by the harmonic theory (SH) with the harmonic frequencies replaced by the self-consistent renormalized frequencies Omega q(T). This simple replacement is not valid for other thermodynamic functions. The specific heats at constant pressure (Cp) or constant volume (Cv) are shown to be simple functions depending only on Omega q(P,T) or Omega q(V,T), which can be calculated from a model of the lattice potential. Several one-dimensional model potentials are used to illustrate the calculations of specific heats from this approach. A test is also presented of the proposition that the measured phonon frequencies give a good approximation to the exact thermodynamics when used to calculate the entropy via the harmonic formula for Nb.