Coulomb Energy and Nuclear Radius

Abstract

The apparent discrepancy between the values ($1.45\times {10}^{-13\mathrm{}}{A}^{\frac{1}{3}}$ cm) for the nuclear radii derived from mirror nuclei and those ($1.2\times {10}^{-13\mathrm{}}{A}^{\frac{1}{3}}$ cm) derived from $\mu$-mesonic atoms was investigated. The conventional calculation of the Coulomb energy difference $\Delta E_c$ between mirror nuclei is improved in two respects: the usual uniform model is replaced by the more elaborate shell model with a finite square well potential, and the exchange terms are taken into account. An equivalent Coulomb radius $R_c$ is defined by $R_c=\left(\frac{6}{5}\right)$ ($\frac{Z{e}^2}{\Delta E_c}$); an equivalent meson radius is defined by $R_M={\left(\frac{5}{3}\right)}^{\frac{1}{2}}{{〈r^2〉}_{\mathrm{Av}}}^{\frac{1}{2}}$, where ${{〈r^2〉}_{\mathrm{Av}}}^{\frac{1}{2}}$ is the mean square radius of the electrical charge distribution. The two extreme cases of the pairs (${\mathrm{F}}^{17}$, ${\mathrm{O}}^{17}$) and (${\mathrm{O}}^{15}$, ${\mathrm{N}}^{15}$) are investigated. The computation gives $\frac{R_c}{R_M}=1.18\mathrm{}$ in ${\mathrm{O}}^{17}$, $\frac{R_c}{R_M}=1.07\mathrm{}$ in ${\mathrm{N}}^{15}$. These results are smaller by about 8 percent than the experimental ratios. However, the experimental discontinuity in $R_c$ at the closure of the $p$ shell is reproduced.