Slow-Neutron Scattering by Liquids: a Hindered-Translator Model

Abstract

A hindered-translator model including damping is constructed with the aim of accounting for the one-particle motions in a liquid, in connection with slow-neutron scattering. Liquids which exhibit the Arrhenius law for self-diffusion are considered and the activation energy ${U}_{0}$ is assumed to represent the energy barrier against translations for a single particle. In the absence of dissipation and for energies below the barrier, the hindered translator is allowed to carry out harmonic oscillations with frequency ${\ensuremath{\omega}}_{0}$ given by the mean field of force experienced by the particle in the liquid. Dissipation is taken into account using methods of non-equilibrium statistical mechanics. Dissipation is considered, first, on the weak-collision hypothesis, and it is assumed that no transitions between vibrational and translational states occur during the time the dynamic correlation persists. By looking into the phase space of the hindered translator at thermal equilibrium, its velocity autocorrelation function is shown to be a weighted superposition of the autocorrelation functions for vibrations and translations, respectively. Explicit expressions are found for their correlation times ${\ensuremath{\tau}}_{\mathrm{I}}$ and ${\ensuremath{\tau}}_{\mathrm{II}}$, in terms of the interatomic potential $\ensuremath{\phi}(r)$, the radial distribution function $g(r)$, the activation energy ${U}_{0}$, and the self-diffusion coefficient $D$. Next, the effect that hard collisions have on dissipation is included by simply superimposing hard on weak collisions; terms which contribute to Gaussian and non-Gaussian parts of Van Hove's ${G}_{s}(r,t)$ are considered. Finally, the expression for Van Hove's scattering function ${S}_{s}(\ensuremath{\kappa},\ensuremath{\epsilon})$ is given in terms of the physical quantities entering our model. The numerical computation concerns quantities related to the Gaussian part of ${G}_{s}(r,t)$. The width of the quasi-elastic peak at different scattering angles, the mean-square displacement versus time, the velocity autocorrelation function, and the Egelstaff $p(\ensuremath{\beta})$ are compared with the experimental data for water at 25\ifmmode^\circ\else\textdegree\fi{}C and 75\ifmmode^\circ\else\textdegree\fi{}C and for liquid argon. Fairly good agreement is obtained without using any adjustable parameter. A significant role is played by ${\ensuremath{\omega}}_{0}$ and ${U}_{0}$ in accounting for the different behavior that liquids with similar self-diffusion coefficient may exhibit with respect to slow-neutron scattering.