Low-energy cluster modes in $N=Z$ nuclei

Significant transition strength in light $\alpha$-conjugate nuclei at low energy, typically below 10 MeV, has been observed in many experiments. In this work the isoscalar low-energy response of $N=Z$ nuclei is explored using the finite amplitude method based on the microscopic framework of nuclear energy density functionals. Depending on the multipolarity of the excitation and the equilibrium deformation of a particular isotope, the low-energy strength functions display prominent peaks that can be attributed to cluster mode structures: $\alpha +{}^{12}\mathrm{C}+\alpha$ and $\alpha +{}^{16}\mathrm{O}$ in ${}^{20}\mathrm{Ne}, {}^{12}\mathrm{C}+{}^{12}\mathrm{C}$ in ${}^{24}\mathrm{Mg}, 4\alpha +{}^{12}\mathrm{C}$ in ${}^{28}\mathrm{Si}$, etc. Such cluster modes are favored in light nuclei with large deformation.

Figure 1

The self-consistent equilibrium density of ${}^{20}\mathrm{Ne}$ (left panel), and localization function $C_{\tau \sigma }$ (right panel) obtained using the RHB model with the DD-PC1 energy density functional.

Figure 2

Evolution of the monopole strength function in ${}^{20}\mathrm{Ne}$ with the size of the harmonic oscillator basis.

Figure 3

${}^{20}\mathrm{Ne}$ strength distribution functions for the QFAM response to the isoscalar monopole (a), isoscalar dipole (b), isoscalar quadrupole (c), and isoscalar octupole (d) operator. For $J>0$ multipoles, the corresponding projections $K=0$ (solid blue), $K=1$ (dashed red), $K=2$ (dot-dashed green), and $K=3$ (dotted orange) are plotted separately. The thin dashed curves denote the total strength.

Figure 4

Snapshots of the ${}^{20}\mathrm{Ne}$ density and localization function at energy $\hslash \omega =6.75$ MeV induced by monopole perturbation. Time increases from top to bottom then left to right and a full period is shown. The horizontal black lines represent the position of the center of mass of the clusters in the ground state.

Figure 5

Snapshots of the ${}^{20}\mathrm{Ne}$ density and localization function at energy $\hslash \omega =7.65$ MeV induced by octupole perturbation ($K=0$ component). Time increases from top to bottom then left to right and a full period is shown. The horizontal black lines represent the position of the center of mass of the clusters in the ground state.

Figure 6

Radial parts of the angular-momentum projected transition densities that correspond to the low-energy peak of the isoscalar monopole response of ${}^{20}\mathrm{Ne}$. The real and imaginary parts of the transition density are shown in the left and right panels, respectively. The ground state rms radius is indicated by the vertical dashed line.

Figure 7

Same as in the caption to Fig. 6 but for the isoscalar octupole response ($K=0$ component).

Figure 8

Schematic illustration of the most important neutron 2-qp contributions to the isoscalar monopole excitation at $\hslash \omega =6.7$ MeV in ${}^{20}\mathrm{Ne}$. The area and the number below represent the fraction of the total ${\left|X\right|}^2-{\left|Y\right|}^2$ [see Eq. (17)] for this particular excitation. The $\mathrm{\Omega }\pi$ quantum numbers are listed on the right of the figure. The associated partial densities are also plotted for each of the configurations as well as the total density in the background. The Fermi level is shown as a red dash-dotted line.

Figure 9

The low-energy isoscalar monopole strength distribution in ${}^{20}\mathrm{Ne}$ isotope. The QFAM response is calculated for several constrained values of the axial quadrupole deformation ${\beta }_2$, and the dashed curve corresponds to the equilibrium deformation ${\beta }_2=0.525$.

Figure 10

Evolution of the leading neutron 2qp contributions to the low-energy monopole mode with constrained deformation (upper panel). The lower panel shows the evolution of the single-particle energies (left) and occupation number (right) in the canonical basis with deformation. The vertical black lines denote the transitions that correspond to the principal 2-qp contribution shown in the upper panel. The thick black curve denotes the Fermi level.

Figure 11

Snapshots of ${}^{20}\mathrm{Ne}$ total density monopole modes at energy $\hslash \omega$ and constrained initial deformation ${\beta }_2$. The time flows from the top to the bottom and a full period is shown.

Figure 12

Low-energy isoscalar monopole strength distribution in $N=Z$ nuclei: ${}^{24}\mathrm{Mg}, {}^{28}\mathrm{Si}$, and ${}^{32}\mathrm{S}$. The QFAM response is calculated for several values of constrained axial quadrupole deformation ${\beta }_2$, and the dashed curves correspond to the equilibrium deformation for each nucleus.

Figure 13

Upper panel: leading neutron 2-qp contributions to the low-energy monopole modes in ${}^{24}\mathrm{Mg}, {}^{28}\mathrm{Si}$, and ${}^{32}\mathrm{S}$ isotopes (for detailed description see the caption to Fig. 10). Lower panel: snapshots of the corresponding density (see the caption to Fig. 11).