Kinematics of Excitation Pair Contributions to Dynamic Structure

The joint density-of-states integral for production of a pair of elementary excitations in an isotropic fluid is studied as a function of energy and momentum. Isotropy permits one to take advantage of the essentially two-dimensional character of the system. Specializing to liquid ${\mathrm{He}}^4$ at $T=0\mathrm{}$ °K, we show that certain abrupt variations in the density of states are associated with thresholds and saddle points in the energy function. The expected influence of these phenomena on $S(\overrightarrow{\mathrm{k}},\omega )$ is described. Resonant pairs of excitations, as judged by their linesharpening effects, are shown to exist at special values of $\overrightarrow{\mathrm{k}}$ and $\omega$.