Extraction of the second and fourth radial moments of nuclear charge density from the elastic electron-nucleus scattering

TL;DR

This paper develops a method to extract the second and fourth radial moments of nuclear charge density from elastic electron scattering data using DWBA, improving accuracy over PWBA and considering neutron and spin-orbit effects.

Contribution

It extends previous PWBA-based methods by applying DWBA and relativistic corrections to accurately extract higher-order charge density moments from experimental form factors.

Findings

DWBA form factors fit experimental data well at small q

Extracted moments are consistent with existing experimental data

Relationship between expansion coefficients and proton number Z established

Abstract

The expression for the radial moments $\left\langle r^{n}\right\rangle_{c}$ of the nuclear charge density has been discussed under the plane wave Born approximation (PWBA) method recently, which is significant to investigate the nuclear surface thickness and neutron distribution radius. In this paper, we extend the studies of extracting second-order moment $\left\langle r^{2}\right\rangle_{c}$ and fourth-order moment $\left\langle r^{4}\right\rangle_{c}$ from the Coulomb form factors $|F_{C}(q)|^2$ by the distorted wave Born approximation (DWBA) at the small momentum transfer $ q $ region. Based on the relativistic mean-field (RMF) calculations, the DWBA form factors $F_{C}^{DW}(q)$ are expanded into $ q^4 $, where the corresponding charge distributions are corrected by the contributions of neutron and spin-orbit densities. In the small $ q $ region, it is found that the experimental $|F_{C}(q)|^2$ can be well reproduced by considering the contributions of the $\left\langle r^{4}\right\rangle_{c}$ at the small $ q $ region. Through further analyzing the second-order and fourth-order expansion coefficients of the $F_{C}^{DW}(q)$, the relationship between the expansion coefficients and proton number $ Z $ is obtained. By the relationship, we extract the $\left\langle r^{2}\right\rangle_{c}$ and $\left\langle r^{4}\right\rangle_{c}$ from the limited experimental data of form factors at the small $ q $ region. Within the permissible range of error, the extracted $\left\langle r^{n}\right\rangle_{c}$ are consistent with the experimental data in this paper.