Quantum scattering by a disordered target -- The mean cross section

TL;DR

This paper investigates how the average scattering cross section varies with sample density in quantum scattering by disordered pointlike targets, revealing two scattering regimes influenced by double and multiple scattering effects.

Contribution

It provides a theoretical and numerical analysis of the mean cross section behavior in disordered quantum scatterers, including a derivation for an infinite series of scattering contributions.

Findings

Mean cross section exhibits two stages: initial increase or decrease, then uniform decrease with density.

Double scattering causes initial variation depending on phase shift ${oldsymbol{ heta}}$.

Multiple scattering leads to a monotonic decrease in the mean cross section at high densities.

Abstract

We study the variation of the mean cross section with the density of the samples in the quantum scattering of a particle by a disordered target. The target consists of a set of pointlike scatterers, each having an equal probability of being anywhere inside a sphere whose radius may be modified. We first prove that scattering by a pointlike scatterer is characterized by a single phase shift ${\delta}$ which takes on its values in $]0 \, , {\pi}[$ and that the scattering by ${\rm N}$ pointlike scatterers is described by a system of only ${\rm N}$ equations. We then show with the help of numerical calculations that there are two stages in the variation of the mean cross section as the density of the samples (the radius of the target) increases (decreases). Depending on the value of ${\delta}$, the mean cross section first either increases or decreases, each one of the two behaviours being originated by double scattering; it decreases uniformly for any value of ${\delta}$ as the density increases further on, a behaviour which results from multiple scattering and which follows that of the cross section for diffusion by a hard sphere potential of decreasing radius. The expression of the mean cross section is derived in the particular case of an unlimited number of contributions of successive scatterings.