Dynamics of quantum liquids at high momentum

The dynamic form factor S(Q,ω) in liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ is evaluated in the wave-vector transfer range 3≤Q≤15 A${̊}^{\mathrm{-}1}$. The input is the pair interatomic potential, developed by Aziz et al. The S(Q,ω) is calculated within the random-phase approximation (RPA) which becomes valid when ħQ is much larger than the average momentum in the liquid. A T-matrix approximation represents the interaction appearing in the RPA. The aim is to explore how well S(Q,ω) can be described for 3≤Q≤15 A${\mathrm{̊}}^{\mathrm{-}1}$ from first principles. In ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, we find S(Q,ω) is a broad, nearly Gaussian function, centered just below the recoil frequency having a width and shape that agrees well with experiment. It does, however, have tails at high frequency which make important contributions to its moments. In ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, S(Q,ω) is a more sharply peaked function which also agrees quite well with experiment. We are able to reproduce the oscillations in the peak position and in the width of S(Q,ω) with Q in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ observed by Martel et al. In the present model, these oscillations originate from oscillations in the magnitude of the T-matrix interaction with Q. The corresponding oscillations are predicted to be very small and probably unobservable in liquid ${}_{}{}^3{}_{}{}^{}\mathrm{He}$.