Corrections to the Impulse Approximation for High-Energy Neutron Scattering from Liquid Helium

Corrections to the impulse approximation for scattering from a many-body system are derived in the form of a series in inverse powers of the momentum transfer $k$. The coefficients in this series are written in terms of the two-body interaction potential and the particle-density matrices. Application is made to the case of high-energy neutron scattering from liquid helium. Evaluation of correction terms corresponding to a single helium-helium collision during the neutron-helium interaction time indicates (a) a shift in the peak of the incoherent-scattering cross section toward lower energy by a constant amount, and (b) an asymmetry of the cross section with respect to neutron energy loss about its peak value. Numerical estimates are given for these two effects for $k=14.3\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$. Evaluation of condensate broadening due to multiple He-He collisions shows that the condensate-scattering contribution has a width proportional to $k^{\frac{1}{2}}$, and is thus distinguishable from the main noncondensate peak whose width is proportional to $k$. Estimates are given for requirements on experimental-resolution functions necessary to preserve this evidence of a distinct condensate contribution.