Coupled-channels analysis of proton inelastic scattering to theγ-vibrational band inMg24

Results are presented of coupled-channels analyses of proton inelastic scattering data for the nucleus ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$, in which the $2_2^{+}$, 4.24 MeV, $3^{+}$, 5.2 MeV, $4_2^{+}$, 6.01 MeV, and $5^{+}$(?), 7.8 MeV states of the $\gamma$-vibrational band are excited, at incident proton energies of 20.3, 40, and 800 MeV. Previous coupled-channels analyses of proton and $\alpha$ particle inelastic scattering data for these states in ${}_{}{}^{24}{}_{}{}^{}\mathrm{Mg}$ have completely failed to account for the shapes and magnitudes of the $3^{+}$, $4_2^{+}$, and $5^{+}$ inelastic cross sections. In the present analysis, the inclusion of an additional nuclear vibrational multipole which permits a direct transition from the ground state to the $4_2^{+}$, 6.01 MeV state is shown to provide a tremendous improvement in the theoretical description of the inelastic cross sections of all the members of the $\gamma$-vibrational band, at each of the three incident proton energies considered. The same nuclear structure parameters are used at all three incident energies, along with phenomenological optical potentials specific to each energy. The new results for the $3^{+}$, 5.2 MeV state also shed light on the energy dependence of the direct spin-flip mechanism in proton inelastic scattering.