Isoscalar monopole and dipole transitions in Mg24, Mg26, and Si28

Background: Nuclei in the $sd$ shell demonstrate a remarkable interplay of cluster and mean-field phenomena. The $N=Z$ nuclei, such as $^{24}\mathrm{Mg}$ and $^{28}\mathrm{Si}$, have been the focus of the theoretical study of both phenomena in the past. A variety of different cluster structures in these nuclei are predicted, characterized by isoscalar dipole and monopole transitions. For example, low-energy isoscalar vortical dipole states were predicted in $^{24}\mathrm{Mg}$. The cluster and vortical mean-field phenomena can be probed by excitation of isoscalar monopole and dipole states in scattering of isoscalar particles such as deuterons or $\ensuremath{\alpha}$ particles.Purpose: We investigate, both experimentally and theoretically, the isoscalar dipole $IS1$ and monopole $IS0$ strengths in three essentially different light nuclei with different properties: stiff prolate $^{24}\mathrm{Mg}$, soft prolate $^{26}\mathrm{Mg}$, and soft oblate $^{28}\mathrm{Si}$. We analyze possible manifestations of clustering and vorticity in these nuclei.Methods: Inelastically scattered $\ensuremath{\alpha}$ particles were momentum analyzed in the K600 magnetic spectrometer at iThemba LABS, Cape Town, South Africa. The scattered particles were detected in two multiwire drift chambers and two plastic scintillators placed at the focal plane of the K600. In the theoretical discussion, the Skyrme quasiparticle random-phase approximation (QRPA) and antisymmetrized molecular dynamics $+$ generator coordinate method ($\mathrm{AMD}+\mathrm{GCM}$) were used.Results: A number of isoscalar monopole and dipole transitions were observed in the nuclei studied. Using this information, suggested structural assignments have been made for the various excited states. $IS1$ and $IS0$ strengths obtained within QRPA and $\mathrm{AMD}+\mathrm{GCM}$ are compared with the experimental data. The QRPA calculations lead us to conclude that (i) the mean-field vorticity appears mainly in dipole states with $K=1$, (ii) the dipole (monopole) states should have strong deformation-induced octupole (quadrupole) admixtures, and (iii) near the $\ensuremath{\alpha}$-particle threshold there should exist a collective state with $K=0$ for prolate nuclei and $K=1$ for oblate nuclei, with an impressive octupole strength. The results of the $\mathrm{AMD}+\mathrm{GCM}$ calculations suggest that some observed states may have a mixed (mean-field $+$ cluster) character or correspond to particular cluster configurations.Conclusion: A tentative correspondence between observed states and theoretical states from QRPA and $\mathrm{AMD}+\mathrm{GCM}$ was established. The QRPA and $\mathrm{AMD}+\mathrm{GCM}$ analysis shows that low-energy isoscalar dipole states combine cluster and mean-field properties. The QRPA calculations show that the low-energy vorticity is well localized in $^{24}\mathrm{Mg}$, fragmented in $^{26}\mathrm{Mg}$, and absent in $^{28}\mathrm{Si}$.