High-Energy Electron Scattering and the Charge Distributions of Selected Nuclei

TLDR

The nuclei are modeled as smoothed uniform charge distributions with a constant central density and a gradual surface. Electron scattering data at 183 MeV (and 153 MeV for selected nuclei) were analyzed with a partial‑wave Dirac equation approach to extract two parameters describing the nuclear charge radius and surface thickness. The radius parameter scales as 1.07 × 10⁻¹³ A¹⁄³ cm, while the surface thickness is constant at about 2.4 × 10⁻¹³ cm across the nuclei studied, indicating a largely flat central charge density and insensitivity to surface shape.

Abstract

Experimental results are presented of electron scattering by Ca, V, Co, In, Sb, Hf, Ta, W, Au, Bi, Th, and U, at 183 Mev and (for some of the elements) at 153 Mev. For those nuclei for which asphericity and inelastic scattering are absent or unimportant, i.e., Ca, V, Co, In, Sb, Au, and Bi, a partial wave analysis of the Dirac equation has been performed in which the nuclei are represented by static, spherically symmetric charge distributions. Smoothed uniform charge distributions have been assumed; these are characterized by a constant charge density in the central region of the nucleus, with a smoothed-out surface. Essentially two parameters can be determined, related to the radius and to the surface thickness. An examination of the Au experiments shows that the functional forms of the surface are not important, and that the charge density in the central regions is probably fairly flat, although it cannot be determined very accurately. An analysis of the experiments on the nuclei Ca, V, Co, In, Sb, Au, and Bi, assuming for convenience the Fermi smoothed uniform shape (1), then leads to the following results: the radial parameter $c$ (the distance to the midpoint of the surface) scales as ${A}^{\frac{1}{3}}$ for the nuclei we have examined and is $(1.07\ifmmode\pm\else\textpm\fi{}0.02){A}^{\frac{1}{3}}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}$ cm; the surface thickness $t$ (the $0.9{\ensuremath{\rho}}_{0}$ to $0.1{\ensuremath{\rho}}_{0}$ distance) is constant for all of these nuclei, to within the estimated error, and is (2.4\ifmmode\pm\else\textpm\fi{}0.3)\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}13}$ cm.