Inelastic Effects in the Study ofNa23andMg23

14 states of ${}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ with excitation energies <5.6 MeV and their mirror states in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Mg}$ have been studied with the ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}(d,{}_{}{}^3{}_{}{}^{}\mathrm{He}){}_{}{}^{23}{}_{}{}^{}\mathrm{Na}$ and ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}\mathrm{}(d,t)\mathrm{}{}_{}{}^{23}{}_{}{}^{}\mathrm{Mg}$ reactions with a bombarding energy of 21.1 MeV. Differential cross-section measurements indicated significant differences between established $l=2\mathrm{}$ transitions and large cross sections for "$j$-forbidden" ${\frac{7}{2}}^{+}$ and ${\frac{9}{2}}^{+}$ states. Failure of a distorted-wave analysis to predict these anomalies suggested that inelastic effects should not be neglected. Calculations with the coupled-channel Born approximation (CCBA) have produced agreement with experiment and have shown that multiple-step processes not only dominate the transitions to the "$j$-forbidden" states, but have large effects upon allowed transitions as well. From the CCBA predictions it was possible to identify mirror states of the nuclei and to establish $J^{\pi }$ assignments of ${\frac{9}{2}}^{+}$, ${\frac{3}{2}}^{+}$, (${\frac{5}{2}}^{+}$), (${\frac{5}{2}}^{-}$), (${\frac{7}{2}}^{+}$), (${\frac{5}{2}}^{+}$), and (${\frac{11}{2}}^{+}$) for the states in ${}_{}{}^{23}{}_{}{}^{}\mathrm{Mg}$ at 2.71, 2.90, 3.86, 3.97, 4.68, 5.29, and 5.45 MeV, respectively.