$0_{\mathrm{gs}}^{+}\to 2_1^{+}$ Excitations in the Mirror Nuclei ${}^{32}\mathrm{Ar}$ and ${}^{32}\mathrm{Si}$

We measured the strength of the $0_{\mathrm{gs}}^{+}\to 2_1^{+}$ excitations in the radioactive mirror nuclei ${}^{32}\mathrm{Ar}$ and ${}^{32}\mathrm{Si}$ using the techniques of intermediate-energy Coulomb excitation for ${}^{32}\mathrm{Ar}$ and inelastic proton scattering in inverse kinematics for ${}^{32}\mathrm{Si}$. The ${}^{32}\mathrm{Ar}$ measurement, taken together with previously existing Coulomb excitation data for ${}^{32}\mathrm{Si}$, yields the isoscalar and isovector multipole matrix elements for the $0_1^{+}\to 2_1^{+}$ transition between $T=2\mathrm{}$ states in the $A=32\mathrm{}$ system. The proton scattering measurement for ${}^{32}\mathrm{Si}$, when combined with the Coulomb excitation data for this nucleus, yields a ratio of neutron and proton matrix elements, $M_n{/M}_p$, for ${}^{32}\mathrm{Si}$.