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

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• Collective levels
• Coulomb excitation
• Direct reactions
• Electromagnetic transitions
• H & He induced nuclear reactions
• Nuclear forces
• 39 ≤ A ≤ 58