Path Integral Monte Carlo study of phonons in the bcc phase of $^4$He
  Using Path Integral Monte Carlo and the Maximum Entropy method, we calculate the dynamic structure factor of solid $^4$He in the bcc phase at a finite temperature of T = 1.6 K and a molar volume of 21 cm$^3$. Both the single-phonon contribution to the dynamic structure factor and the total dynamic structure factor are evaluated. From the dynamic structure factor, we obtain the phonon dispersion relations along the main crystalline directions, [001], [011] and [111]. We calculate both the longitudinal and transverse phonon branches. For the latter, no previous simulations exist. We discuss the differences between dispersion relations resulting from the single-phonon part vs. the total dynamic structure factor. In addition, we evaluate the formation energy of a vacancy. 
V.Sorkin, E. Polturak and Joan Adler