Relation between Inelastic Neutron Scattering and Thermodynamic Functions of Liquid Helium

A model of liquid helium is analyzed, in which the liquid is regarded as a collection of excitations ("rotons" only with energy>~Δ) with an arbitrary pairwise number-conserving interaction. The entropy and normal fluid density of the liquid, and the energy distribution of scattered neutrons are computed as power series in the density of excitations exp(-ΔkT). The first terms containing effects of the interactions are studied. When the interactions are weak, the entropy [through order exp(-2ΔkT)] is simply related to the neutron scattering, the connection being correctly given by the formula of Bendt, Cowan, and Yarnell. For strong interactions there appears to be no simple connection. Even when interactions are weak, the first correction to the normal fluid density involves information which is not contained in the neutron scattering.A method due to Bloch and de Dominicis is used in the analysis, and leads to a new form for the second virial coefficient. This is closely related to a curious new form for the level shift of a particle in a large spherical box, under the influence of a central potential.