On fast charged particle scattering by periodic atomic planes: quadratic potential corrections

TL;DR

This work extends the eikonal (Glauber) framework for fast charged-particle scattering by periodic atomic planes to include quadratic potential corrections, enabling explicit inclusion of incoherent scattering. Using structure-factor decomposition with $D_{N_x}$ and a set of potential-dependent functions $F$, $F'$ and $G$, the authors derive a cross section formula $\frac{d\sigma}{d^2q_\perp}=\frac{1}{4\pi^2}\int d^2\rho d^2\rho' e^{i\vec{q}_\perp(\vec{\rho}-\vec{\rho}')} (1-e^F-e^{F'}+e^F e^{F'} e^G)$ and reduce it to tractable integrals $I_0$–$I_3$, with $I_0$ factoring as a structure-factor times a single-plane term. They compute the potential-dependent functions for a single plane, including $\langle\chi_0^{(1)}\rangle$, $\langle(\chi_0^{(1)})^2\rangle$, and $\langle\chi_0^{(1)}\chi'^{(1)}_0\rangle$, and apply the formalism to $N_x$ planes to obtain $d\sigma/dq_x$, comparing with the Born approximation and highlighting the significance of quadratic corrections and rainbow-like oscillations. The paper also outlines a general method to extend the approach to arbitrary isolated substructures, offering a path to modeling complex targets beyond parallel planes. Overall, the work provides a more complete quantum-mechanical description of coherent and incoherent scattering in layered targets, with implications for crystal diffraction and rainbow scattering phenomena.

Abstract

In this paper, the approach for considering fast charged particles scattering on targets of complex structure, which contains some isolated substructures, was expanded to account quadratic potential terms. Based on this approach, the differential cross section for scattering on the set of parallel planes with uniformly distributed atoms in each plane was obtained. It was shown that for this case the differential scattering cross section splits into coherent and incohent cross sections in the eikonal approximation analogously with the Born approximation.

Figures (8)

• Figure 1: The fast charged particle scattering on the set of parallel planes with uniformly distributed atoms in each plane
• Figure 2: Potential-dependent functions for $A=10$ for 1 and 3 atomic planes
• Figure 3: Dependence of $H$ on $x$ and $x'$ for $A=1$ for 3 atomic planes
• Figure 4: The differential cross section of fast charged particles scattering on 1 atomic plane. $\frac{d\sigma_{lin}}{dq_x}$ denotes the differential scattering cross section calculated with accuracy up to linear potential terms
• Figure 5: The differential cross section of fast charged particles scattering on 2 atomic planes