Doctoral Dissertations

Date of Award


Degree Type


Degree Name

Doctor of Philosophy



Major Professor

Robert Hinde

Committee Members

Ben Xue, David Keffer, Tessa Calhoun


When placed under high pressure, and cooled to low temperatures, He-4 will crystalize into a solid with a hexagonal close packed arrangement of atoms. Lattices of this type will exhibit phonons with E2g symmetry. In the He-4 solid, the frequency of this phonon can be measured to a high degree of accuracy using Raman spectroscopy or neutron scattering. It has been shown that the frequency is dependent upon the density of the solid. This data has been used in this work as a highly accurate benchmark to evaluate the accuracy of modelling the solid via the path integral Monte Carlo method, using two-body and three-body potential energy surfaces that are known to be highly accurate. The path integral method was chosen to see if the inclusion of zero-point motion of the atoms in calculations of the phonon frequency would be an improvement on previous theoretical treatments which do not include zero-point motion. Not incorporating zero-point motion causes theoretical calulations to underestimate the phonon frequency, esepecially at high molar volumes. Because He-4 is a quantum solid, the individual atoms within the solid experience large amplitude zero-point motion that can substantially change the average two-body and three-body interactions in the crystal. In calculating the phonon frequency, this work showed that incorporating zero-point motion does increase the phonon frequency in comparison to previous theoretical work. However, the results presented in this thesis overestimate the value of the phonon frequency. It was theorized that this was due to the construction of an external potential that exerted too much force on the Monte Carlo system, which prevented the accurate sampling of two-body and three-body energies. Gaussian quadrature calculations were then done to examine the relationship between the radial distribution of atomic position and the phonon frequency. The minimum energy of the solid systems as a function of the root mean square displacement of atomic position was found. The phonon frequency at low molar volumes were the most affected by three-body corrections, which was not true of the path integral simulations.

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