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 $M_n/M_p$ for the $2^+_1$ excitations in these nuclei. The present results for $^{41,43}$P$(p,p)$ 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.