Isoscalar monopole and dipole transitions in ${}^{24}\mathrm{Mg}$, ${}^{26}\mathrm{Mg}$, and ${}^{28}\mathrm{Si}$

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 $\alpha$ 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 $\alpha$ 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 $\alpha$-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 ${}^{28}\mathrm{Si}$.

Figure 1

Fitted ${}^{24}\mathrm{Mg}$ spectra for: $\theta <2$ degrees (top), $2<\theta <3$ degrees (middle), and $5<\theta <6$ degrees (bottom). Some states have been labeled to guide the reader.

Figure 2

Fitted ${}^{26}\mathrm{Mg}$ spectra for $\theta <2$ degrees (top), $2<\theta <3$ degrees (middle), and $5\theta <6$ degrees (bottom). Some states have been labeled to guide the reader.

Figure 3

Differential cross section for the $J^{\pi }=0^{+}$ state at 10.68 MeV in ${}^{24}\mathrm{Mg}$. The data are represented by points with the horizontal error bar delineating the angular range covered. The calculated angle-averaged differential cross section is shown in red.

Figure 4

As Figure 3 but for the $J^{\pi }=1^{-}$ state at 11.86 MeV in ${}^{24}\mathrm{Mg}$.

Figure 5

Experimental transition factors ${\beta }_{R,1}^2$ (upper), QRPA isoscalar dipole compression strength $B\left(IS1K\right)$ for $K=0$ and $K=1$ (middle), and isoscalar octupole strength $B\left(IS3K\right)$ (bottom) in ${}^{24}\mathrm{Mg}$. The $\alpha$-particle threshold energy $S_{\alpha }$ and QRPA equilibrium deformation ${\beta }_2$ are displayed.

Figure 6

QRPA results for isoscalar vortical (black bars) and compression (red filled squares) dipole strengths in $K=0$ (upper) and $K=1$ (bottom) dipole states in ${}^{24}\mathrm{Mg}$.

Figure 7

The same as in Fig. 5 but for ${}^{26}\mathrm{Mg}$.

Figure 8

The same as in Fig. 6 but for ${}^{26}\mathrm{Mg}$.

Figure 9

The same as in Fig. 5 but for ${}^{28}\mathrm{Si}$.

Figure 10

The same as in Fig. 6 but for ${}^{28}\mathrm{Si}$.

Figure 11

Experimental ${\beta }_{R,0}^2$ factors (a)–(c), QRPA $B\left(IS0\right)$ values for $K^{\pi }=0^{+}$ excitations at 0–16 MeV (d)–(f) and 0-30 MeV (g)–(i), QRPA $B\left(IS20\right)$ values at 0-30 MeV (j)–(l).