Precise neutron diffraction study of hcp and bcc $^{4}$He

Precise lattice parameter measurements are reported for $^{4}$He in both bcc and hcp phases at low density and low temperature. The results can be used to set limits on a proposed incommensurate equilibrium state of solid $^{4}$He near $T$ = 0. ``Incommensurate" means a net difference between atomic sites and atoms. The relative difference is defined as $\epsilon$. Present measurements were made by carefully calibrated neutron diffraction. The value established at melting, by comparison with published bulk density values, is $\epsilon = 0.4\pm 0.4 \%$. Much of the uncertainty comes from uncertainties in the bulk values. These neutron results on hcp $^{4}$He are also consistent with previous precise x-ray diffraction work on bcc $^{4}$He and, at higher densities, on both $^{4}$He and $^{3}$He. Published isochoric measurements of changes in x-ray lattice parameters as $T$ is reduced from melting can be used to extrapolate $\epsilon$ toward zero $T$, where its most probable value is zero, with the same uncertainty. The present neutron work on hcp phase agrees with published high-resolution synchrotron x-ray work in showing that the $(c/a)$ ratio is slightly smaller than that corresponding to ideal close-packing.