Neutron Scattering from Liquid Helium II at Large Momentum Transfer and the Condensate Fraction

A simple theory of neutron scattering from liquid He II at zero temperature for large momentum transfers is described. The theory is based on a generalized mean-field approximation involving the polarization potential and the screened response function. The latter, instead of having a free-particle form, is assumed to be a sum of Gaussian functions, weighted by the momentum distribution function appropriate for liquid helium. The zero and third moments of the scattering law determine the polarization potential and the width of the Gaussians. Numerical calculations have been made for the dynamical structure function $S(q,\omega )$ for different values of the condensate fraction. Both the width and the peak position of $S(q,\omega )$ in the range of momentum transfers 2.5-9 ${\mathrm{\AA }}^{-1\mathrm{}}$ are in fair agreement with experiment for a 6% condensate fraction. Calculations have also been done for $q$ values as large as 20 ${\mathrm{\AA }}^{-1\mathrm{}}$ for three different values of the condensate fraction, including 0. For a condensate fraction of 6% or larger, the contribution of the condensate to $S(q,\omega )$ becomes separated from that of the noncondensed particles at these large -$q$ values. The separation is distinct enough that it should be experimentally observable.