Nonuniform Nuclear Charge Distributions and Measurements of Nuclear Electrical Radius

The dependence of three long wavelength electronic effects and of the nuclear Coulomb energy on the size and shape of the electrical charge distribution in heavy nuclei is examined. We note that the x-ray fine structure effect and the isotope shift effect measure the volume integral of $r^{2\sigma }$ weighted by the nuclear charge density. [$\sigma ={\{1-{\left(\alpha Z\right)}^2\}}^{\frac{1}{2}}$]. An earlier result of Feshbach—that medium-energy electron scattering measures the similarly weighted mean value of $r^2$—is derived more simply. For these electronic effects, therefore, exact calculations for one shape of charge distribution may easily be scaled to other shapes. The nuclear Coulomb energy has no such simple dependence, but measures very roughly the weighted mean value of $r^{0.8}$—nearly the same quantity as measured by the $2P\to 1S$ mu-mesonic transition in heavy nuclei.