Nuclear charge radii of the even sulphur isotopesS32,S34, andS36and ofP31using muonic atoms

Energies of muonic x rays of the Lyman series of the even sulphur isotopes ${}_{}{}^{32}{}_{}{}^{}\mathrm{S}$, ${}_{}{}^{34}{}_{}{}^{}\mathrm{S}$, and ${}_{}{}^{36}{}_{}{}^{}\mathrm{S}$ and of natural phosphorus have been determined with absolute precisions up to 23 parts/${10}^6$. Equivalent Barrett charge radii $R_{k,\alpha }$ have been deduced. Their differences between the sulphur isotopes amount to $\Delta R_{k,\alpha }({}_{}{}^{34}{}_{}{}^{}\mathrm{S}-{}_{}{}^{32}{}_{}{}^{}\mathrm{S})=29.7\mathrm{}\left(1.4\right)\mathrm{}$ am and $\Delta R_{k,\alpha }({}_{}{}^{36}{}_{}{}^{}\mathrm{S}-{}_{}{}^{34}{}_{}{}^{}\mathrm{S})=18.7\mathrm{}\left(1.5\right)\mathrm{}$ am. Combining these results with recent elastic electron scattering data, we obtain in a model-independent way $\Delta {〈r^2〉}^{\frac{1}{2}}({}_{}{}^{34}{}_{}{}^{}\mathrm{S}-{}_{}{}^{32}{}_{}{}^{}\mathrm{S})=23.1\mathrm{}\left(1.2\right)\mathrm{}$ am and $\Delta {〈r^2〉}^{\frac{1}{2}}({}_{}{}^{36}{}_{}{}^{}\mathrm{S}-{}_{}{}^{34}{}_{}{}^{}\mathrm{S})=13.0\mathrm{}\left(1.2\right)\mathrm{}$ am.