Inverse-kinematics proton scattering on ${}^{50}$Ca: Determining effective charges using complementary probes

We have performed measurements of the $0_{\mathrm{g}\mathrm{.}\mathrm{s}\mathrm{.}}^{+}\to 2_1^{+}$ excitations in the neutron-rich isotopes ${}^{48,50}$Ca via inelastic proton scattering on a liquid hydrogen target, using the GRETINA $\gamma$-ray tracking array. A comparison of the present results with those from previous measurements of the lifetimes of the $2_1^{+}$ states provides us the ratio of the neutron and proton matrix elements for the $0_$\mathrm{g}\mathrm{.}\mathrm{s}\mathrm{.}$^+ \rightarrow 2_1^+$ transitions. These results allow the determination of the ratio of the proton and neutron effective charges to be used in shell model calculations of neutron-rich isotopes in the vicinity of ${}^{48}$Ca.