Lattice vibrations and elastic constants of crystalline He-4 and He-3 near low-temperature melting

The phonon dispersion curves and the elastic constants are evaluated for crystalline He-4 and He-3 near melting at low temperature in the deep quantal regime. The structures considered are the hexagonal close-packed and the body-centred and face-centred cubic ones for He-4 and the body-centred cubic one for He-3. The calculations use a density functional approach to construct a held of effective force constants in the strongly anharmonic solid near melting from its Debye-Waller factor and from the linear density response function of the fluid phase at freezing. The results of the calculations are compared with the available experimental evidence: good agreement is generally found for transverse phonon frequencies and shear elastic constants, and phenomenological adjustments of the force-constant field are proposed for a quantitative description of the longitudinal modes. A relationship between the dispersion curves of phonons in crystalline He-4 and that of elementary collective excitations in superfluid He-4 is displayed and discussed in the light of current interpretations of atomic dynamics in the superfluid phase, with main emphasis on the region of the roton minimum.