Supersolid ⁴He Likely Has Nearly Isotropic Superflow

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We extend previous calculations of the zero temperature superfluid fraction fₛ (SFF) vs localization, from the fcc lattice to the experimentally realized (for solid ⁴He) hcp and bcc lattices. The superfluid velocity is assumed to be a one-body function, and dependent only on the local density, taken to be a sum over sites of gaussians of width σ. Localization is defined as σ/d, with d the nearest-neighbor distance. As expected, for fcc and bcc lattices the superfluid density tensor is proportional to the unit tensor. To numerical accuracy of three-places (but no more), the hcp superfluid density tensor is proportional to the unit tensor. This implies that a larger spread in data on fₛ, if measured on pure crystals, is unlikely to be due to crystal orientation. In addition, to three decimal places (but no more) the curves of fₛ vs σ/d are the same for both the hcp and fcc cases. An expected value for the localization gives an fₛ in reasonable agreement with experiment. The bcc lattice has a similar curve of fₛ vs σ/d, but is generally smaller because the lattice is more dilute.