Coulomb Energy and Nuclear Radius

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The apparent discrepancy between the values ($1.45\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{A}^{\frac{1}{3}}$ cm) for the nuclear radii derived from mirror nuclei and those ($1.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}13}{A}^{\frac{1}{3}}$ cm) derived from $\ensuremath{\mu}$-mesonic atoms was investigated. The conventional calculation of the Coulomb energy difference $\ensuremath{\Delta}{E}_{c}$ 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 ${R}_{c}=(\frac{6}{5})$ ($\frac{Z{e}^{2}}{\ensuremath{\Delta}{E}_{c}}$); an equivalent meson radius is defined by ${R}_{M}={(\frac{5}{3})}^{\frac{1}{2}}{{〈{r}^{2}〉}_{\mathrm{Av}}}^{\frac{1}{2}}$, 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 $\frac{{R}_{c}}{{R}_{M}}=1.18$ in ${\mathrm{O}}^{17}$, $\frac{{R}_{c}}{{R}_{M}}=1.07$ 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.