Incoherent Neutron Scattering by Simple Classical Liquids for Large Momentum Transfer

The general form for the moments of the Van Hove incoherent scattering function is derived for a simple classical liquid with velocity-independent interatomic forces. With the additional assumption of additive central forces, explicit expressions are obtained for the coefficients which determine the asymptotic behavior of the moments at large momentum transfer $\kappa$. This enables one to obtain an exact asymptotic expansion of the scattering function in inverse powers of $\kappa$. The theory is applied in detail to the Lennard-Jones liquid at the triple point and to the hard-sphere fluid at arbitrary density. The results for these two cases are qualitatively different as a consequence of the fact that the atomic velocities are continuous functions of time in the former case and discontinuous in the latter. For example, the leading correction to the impulse approximation due to final-state interactions is proportional to ${\kappa }^{-2\mathrm{}}$ for the Lennard-Jones potential and to ${\kappa }^{-1\mathrm{}}$ for hard spheres. With decreasing $\kappa$ the scattering function changes from the Gaussian shape, characteristic of the impulse approximation at large $\kappa$, to the Lorentzian shape characteristic of simple diffusion at small $\kappa$. This change occurs in the hardsphere fluid when $\kappa l\sim 1$, where $l$ is the mean free path (including the Enskog factor).