Two-body density matrix of a Bose fluid

Final-state effects in deep-inelastic neutron scattering experiments on liquid ${\mathrm{He}}^4$ may be analyzed in terms of certain expectation values n(p,q) and n(k,p,q) constructed from elements of the two-body density matrix in the momentum representation. These quantities, called generalized momentum distributions, may be regarded as transition matrix elements for a set of scattering processes. In the case of a Bose fluid in its ground state, two such processes are the creation of a boson out of the condensate, and the absorption of a boson into the condensate, mediated by a phonon excitation. The generalized momentum distributions are specified by the form factors associated with the allowed scatterings. Exact relations for these form factors are derived from an asymptotic structural analysis of the one- and two-body density matrices ${\mathrm{\rho }}_1$(${\mathbf{r}}_1$,${\mathbf{r}}_1^{\prime }$), ${\mathrm{\rho }}_2$(${\mathbf{r}}_1$,${\mathbf{r}}_2$,${\mathbf{r}}_1^{\prime }$,${\mathbf{r}}_2^{\prime }$) of the Bose fluid. Upon specializing to a Jastrow wave function, established techniques may be adapted to the explicit calculation of the requisite ingredients. Numerical results for form factors contributing to n(p,q) are presented for liquid ${\mathrm{He}}^4$ at the experimental equilibrium density. The calculations are based on the HFDHE2 interaction of Aziz et al. and employ the paired-phonon procedure for optimizing the Jastrow trial function in conjunction with a hypernetted-chain approximation in which elementary diagrams are neglected (HNC/0). The results are compared with those corresponding to earlier proposals of Silver, Gersch et al., and Rinat for simple evaluation of n(p,q) or ${\mathrm{\rho }}_2$(${\mathbf{r}}_1$,${\mathbf{r}}_2$,${\mathbf{r}}_1^{\prime }$,${\mathbf{r}}_2^{\prime }$). The various approximations, including HNC/0, are assessed with regard to the preservation or violation of symmetries, consistency conditions, and geometrical constraints. A new approximation is proposed for the compact three-point function entering ${\mathrm{\rho }}_2$(${\mathbf{r}}_1$,${\mathbf{r}}_2$,${\mathbf{r}}_1^{\prime }$,${\mathbf{r}}_2^{\prime }$), which satisfies all the relevant constraints. The present study calls for a reexamination of the final-state correction to the impulse approximation within Silver's hard-core perturbation scheme, using an improved approximation for n(p,q) derived from variational theory.