Determination of proton and neutron contributions to the $0_{g\mathrm{.}s\mathrm{.}}^{+}\to 2_1^{+}$ excitations in ${}^{42}$Si and ${}^{44}$S using inelastic proton scattering in inverse kinematics and intermediate energy Coulomb excitation

We have measured the $0_{g\mathrm{.}s\mathrm{.}}^{+}\to 2_1^{+}$ transition in the neutron rich $N=28$ isotope ${}^{42}$Si using the probes of intermediate energy Coulomb excitation and inelastic proton scattering in inverse kinematics at the Facility for Rare Isotope Beams with beam particle rates of $\approx 5$ particles/s. The results of these two measurements allowed us to determine $M_n\mathrm{/}{M}_p$, the ratio of the neutron and proton transition matrix elements for the $0_{g\mathrm{.}s\mathrm{.}}^{+}\to 2_1^{+}$ transition. In addition, we have measured the $0_{\mathrm{g}\mathrm{.}\mathrm{s}\mathrm{.}}^{+}\to 2_1^{+}$ transition in the isotone ${}^{44}$S using inverse kinematics inelastic proton scattering. By comparing the ${}^{44}$S proton scattering result with a recent intermediate energy Coulomb excitation result on the same transition, we were able to determine $M_n\mathrm{/}{M}_p$ for the $0_{g\mathrm{.}s\mathrm{.}}^{+}\to 2_1^{+}$ transition in this nucleus as well. This work strengthens the evidence that ${}^{42}$Si has a stable quadrupole deformation in its ground state and that ${}^{44}$S does not. Both conclusions are further supported by shell model calculations carried out with the FSU interaction.