Correlations in Space and Time and Born Approximation Scattering in Systems of Interacting Particles

TLDR

The derived expression extends the Zernike‑Prins formula to cases where energy transfers are comparable to the scattered particle’s energy. The paper introduces a time‑dependent generalization of the pair distribution function g(r) for interacting particle systems. By defining G(r,t) and reformulating the Born‑approximation scattering formula in terms of it, the authors obtain a simple, general expression for the angular and energy distribution of scattered particles. The study shows that G captures the qualitative behavior of liquids, dense gases, crystals, and ideal quantum gases, including a long‑range component near the critical point.

Abstract

A natural time-dependent generalization is given for the well-known pair distribution function $g(\mathrm{r})$ of systems of interacting particles. The pair distribution in space and time thus defined, denoted by $G(\mathrm{r}, t)$, gives rise to a very simple and entirely general expression for the angular and energy distribution of Born approximation scattering by the system. This expression is the natural extension of the familiar Zernike-Prins formula to scattering in which the energy transfers are not negligible compared to the energy of the scattered particle. It is therefore of particular interest for scattering of slow neutrons by general systems of interacting particles: $G$ is then the proper function in terms of which to analyze the scattering data.After defining the $G$ function and expressing the Born approximation scattering formula in terms of it, the paper studies its general properties and indicates its role for neutron scattering. The qualitative behavior of $G$ for liquids and dense gases is then described and the long-range part exhibited by the function near the critical point is calculated. The explicit expression of $G$ for crystals and for ideal quantum gases is briefly derived and discussed.