Calculation of Phonon Frequencies and Thermodynamic Properties of Crystalline bcc Helium

Abstract

The lattice dynamics of crystalline bcc helium is treated by using the time-dependent Hartree approximation together with the results of variational calculations of the ground-state energy using correlated trial wave functions. The phonon spectrum has been calculated for various densities of bcc ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$. We present dispersion curves, density-of-states histograms, and sound velocities for selected densities. In addition we have calculated (for these densities) the mean-square displacement and the specific heat ${C}_{V}$ which is expressed as a temperature-dependent Debye temperature $\ensuremath{\Theta}(T)$. It is found that the reduced $\ensuremath{\Theta}\ensuremath{-}T$ relationships follow a universal curve for all densities considered. This is a theoretical confirmation of the experimental fact that the bcc phase of crystalline helium (like the other crystalline phases of ${\mathrm{He}}^{3}$ and ${\mathrm{He}}^{4}$) shows a thermodynamic behavior very similar to that of ordinary crystals. The calculated Debye temperatures are about 15-20% higher than those recently measured by Sample and Swenson. A possible cause for this discrepancy is indicated.