LVIII. Some vibrational properties of solid helium

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Abstract
A series of experimental measurements on the specific heat and other vibrational properties of solid helium, due to J. S. Dugdale and F. E. Simon, are here given a theoretical basis in terms of a new form of harmonic lattice dynamics, due originally to Born, and further developed in two previous papers by the present author. The relative magnitude of the zero-point energy in solid helium is such that the customary lattice dynamics, which takes no account of the anharmonic terms in the actual lattice motion, breaks down (the proper frequencies becoming negative); the new theory provides a harmonic approximation with frequencies anharmonically denned, and can be used over the full range of molar volumes which can be achieved in solid helium. The results are expressed in terms of a Debye approximation with characteristic temperature θ; this θ differs in absolute value from the purely empirical Debye-type parameter introduced by Dugdale and Simon, but otherwise has similar properties. The significance according to this theory of the Grüneisen equation of state found by these authors is fully discussed. Some vibrational properties on the melting curve, treated by C. Domb, are reconsidered from a properly anharmonic point of view.

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