A microscopic approach to the response of $^{\bf 3}$He -$^{\bf 4}$He mixtures

Abstract

Correlated Basis Function perturbation theory is used to evaluate the zero temperature response $S(q,ω)$ of $^3$He-$^4$He mixtures for inelastic neutron scattering, at momentum transfers $q$ ranging from $1.1$ to $1.7 Å^{-1}$. We adopt a Jastrow correlated ground state and a basis of correlated particle-hole and phonon states. We insert correlated one particle-one hole and one-phonon states to compute the second order response. The decay of the one-phonon states into two-phonon states is accounted for in boson-boson approximation. The full response is splitted into three partial components $S_{αβ}(q,ω)$, each of them showing a particle-hole bump and a one phonon, delta shaped peak, which stays separated from the multiphonon background. The cross term $S_{34}(q,ω)$ results to be of comparable importance to $S_{33}(q,ω)$ in the particle-hole sector and to $S_{44}(q,ω)$ in the phonon one. Once the one-phonon peak has been convoluted with the experimental broadening, the computed scattering function is in semiquantitative agreement with recent experimental measurements.