Calculation of Phonon Frequencies and Thermodynamic Properties of Crystalline bcc Helium

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 $\Theta \mathrm{}\left(T\right)\mathrm{}$. It is found that the reduced $\Theta -T$ 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.