On the Theory of Slow Neutron Scattering by Liquid Helium

Using the extreme Bose-Einstein (B.E.) and Fermi-Dirac (F.D.) models of liquid ${\mathrm{He}}_{4}$ and ${\mathrm{He}}_{3}$, respectively, the theory of inelastic or incoherent slow neutron scattering by these liquids has been studied in this paper. It is, of course, fully realized that these liquids are not ideal symmetric or antisymmetric collections of atoms. Nevertheless, they might exhibit, in their behavior with respect to slow neutron scattering, such trends which could be interpreted as resulting from their respective statistical properties.In liquid ${\mathrm{He}}_{4}$, for neutrons of kinetic energy considerably larger than $\mathrm{kT}$, $k$ being Boltzmann's constant and $T$ the liquid temperature, the incoherent differential cross section per atom is always larger, or at least equal to the free atom cross section exhibited by these atoms in the classical limit of vanishing degeneracy. At constant liquid temperatures, larger than the lambda-point temperature, i.e., in the ${\mathrm{He}}_{4}$I region, the scattering has a maximum finite limit at vanishing scattering angles. This limit depends, at these small angles, on the liquid temperature only. The liquid cross section decreases with increasing scattering angle and tends asymptotically toward its free atom value. The latter is approached faster the larger the kinetic energy of the incident neutrons. At liquid temperatures near the lambda-point temperature, the vanishing small angle scattering cross section becomes very large and becomes some ${N}^{\frac{1}{2}}$ times the free atom value at the lambda-point.In the liquid ${\mathrm{He}}_{4}$II region the very small angle scattering cross section increases as ${\ensuremath{\Theta}}^{\ensuremath{-}2}$, $\ensuremath{\Theta}$ denoting the scattering angle.At constant scattering angles the scattering cross section varies as a function of the liquid temperature, in the following way. In the limit of high temperatures, or small degeneracy, the scattering cross section tends toward its asymptotic free atom value. As the liquid temperature decreases, or the degeneracy increases, the cross section increases. At the limit of the absolute zero temperature, perfect degeneracy, the distribution of the atoms being narrowed down to practically a single state, the scattering cross section is also classical, the fundamental symmetrical correlation having vanished because of the emptiness of the higher energy levels. The constant angle cross section curves thus exhibit a maximum as a function of the temperature.In liquid ${\mathrm{He}}_{3}$, treated here as a limiting ideal antisymmetric fluid, the differential cross section is always smaller, at small scattering angles, than the free atom cross section. The latter is reached asymptotically at large scattering angles from below, in contrast with the symmetric fluid. At sufficiently large scattering angles, the constant angle cross sections vary with temperature by exhibiting a minimum between the temperature of absolute zero and the limit of high temperatures or vanishing degeneracy. Again, in contrast with the symmetric fluid, the fluid cross section remains always finite and has no anomalies.The preceding results correspond to those collision processes in which the incident neutron loses momentum and energy. Since however, in all practical cases, even the total loss of neutron kinetic energy to the target liquid ${\mathrm{He}}_{4}$, over reasonable radiation times, cannot perturb the thermal equilibrium of the liquid, the inverse scattering processes have also been studied by using the principle of detailed balance. The resulting, direct plus inverse, liquid cross sections have also been evaluated both in ${\mathrm{He}}_{4}$ and ${\mathrm{He}}_{3}$. In the latter, however, the thermal equilibrium is far from being as well preserved as in liquid ${\mathrm{He}}_{4}$, because of the large thermal neutron reaction cross section, $n({\mathrm{He}}_{3},p){\mathrm{H}}_{3}$. This reaction appears to present serious difficulties for the eventual experimental investigation of the incoherent slow neutron scattering.