Inverse-kinematics proton scattering from ${}^{42,44}$S, ${}^{41,43}$P and the collapse of the $N=28$ major shell closure

Excited states of the neutron-rich isotopes ${}^{42,44}$S and ${}^{41,43}$P have been studied via inverse-kinematics proton scattering from a liquid hydrogen target, using the GRETINA $\gamma$-ray tracking array to extract inelastic scattering cross sections. Deformation lengths of the $2_1^{+}$ excitations in ${}^{42,44}$S have been determined and, when combined with deformation lengths determined with electromagnetic probes, yield the ratio of neutron-to-proton matrix elements $M_n\mathrm{/}{M}_p$ for the $2_1^{+}$ excitations in these nuclei. The present results for ${}^{41,43}$P$(p,p^{\mathrm{\prime }})$ are used to compare two shell model interactions, SDPF-U and SDPF-MU. As in a recent study of ${}^{42}$Si, the present results on ${}^{41,43}$P favor the SDPF-MU interaction.