Oscillations and line shapes ofS(Q, ω)in quantum fluids

The dynamic form factor $S(Q, \omega )$ of the quantum liquids ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ and ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ is evaluated in the range $3\le Q\le 10$ ${\mathrm{\AA }}^{-1\mathrm{}}$ within the random-phase approximation (RPA) beginning from the He-He pair potential. In ${}_{}{}^4{}_{}{}^{}\mathrm{He}$, the width $W\left(Q\right)\mathrm{}$ of $S(Q, \omega )$ is found to oscillate with $Q$ as observed. These oscillations originate in the He-He interaction in the RPA. In ${}_{}{}^3{}_{}{}^{}\mathrm{He}$, $W\left(Q\right)\mathrm{}$ agrees in magnitude with experiment but does not oscillate. This suggests $W\left(Q\right)\mathrm{}$ is not simply related to the total He-He scattering cross section $\sigma \mathrm{}\left(Q\right)\mathrm{}$. The calculated $S(Q, \omega )$ has high-frequency tails which make the kinetic energy greater than expected from $W\left(Q\right)\mathrm{}$ and a Gaussian $S(Q, \omega )$.