X-ray Debye-Waller factor measurements of solid3Heand4He

Abstract

X-ray synchrotron radiation was used to measure Debye-Waller factors of helium crystals for both ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ in both hcp and fcc phases. To our knowledge, there are no previous measurements for ${}^{3}\mathrm{He}.$ The ranges studied for ${}^{3}\mathrm{He}$ and ${}^{4}\mathrm{He}$ crystals were 11.52--12.82 and $10.95--12.13{\mathrm{cm}}^{3},$ respectively, and 11.5--18.2 and 12.0--20.3 K. With small uncertainty, only a Gaussian dependence upon momentum transfer Q was found, and no anisotropy was detected in the hcp phase. Mean square atomic deviations, $〈{u}^{2}〉,$ and Lindemann ratios were obtained. Large Lindemann ratios confirm that these solids are highly anharmonic. The $〈{u}^{2}〉$ values agree within an average 1% with computations of Draeger and Ceperley from path integral Monte Carlo methods including unusual extrapolations to the thermodynamic limit. Because the path-integral Monte Carlo (PIMC) computations exhibit a ${T}^{3}$ dependence for $〈{u}^{2}〉,$ which also depends upon molar volume, an empirical analysis was made of the present data as well as of published x-ray and neutron data on hcp ${}^{4}\mathrm{He}.$ The volume dependencies are similar to those found from calorimetry, over a large volume range, and the temperature dependencies show similar systematic variations with molar volume both in x-ray data and PIMC results.