Cluster and toroidal aspects of isoscalar dipole excitations in ${}^{12}\mathrm{C}$

We investigate cluster and toroidal aspects of isoscalar dipole excitations in ${}^{12}\mathrm{C}$ based on the shifted basis antisymmetrized molecular dynamics combined with the generator coordinate method, which can describe 1p-1h excitations and $3\alpha$ dynamics. In the $E=10\text{−}15\mathrm{MeV}$ region, we find two low-energy dipole modes separating from the giant dipole resonance. One is the developed $3\alpha$-cluster state and the other is the toroidal dipole mode. The cluster state is characterized by the high-amplitude cluster motion beyond the 1p-1h model space, whereas the toroidal dipole mode is predominantly described by 1p-1h excitations in the ground state. The low-energy dipole states are remarkably excited by the toroidal dipole operator, which can measure the nuclear vorticity. For compressive dipole transition strengths, a major part is distributed in the 30- to 50-MeV region for the giant dipole resonance, and 5% of the total energy-weighted sum exists in the $E<20$ MeV region.

Figure 1

Schematic of $3\alpha$ configurations in the GCM for ${}^{12}\mathrm{C}$.

Figure 2

Strength functions of the TD, CD, and VD transitions calculated with (a) the full sAMD + GCM, (b) the sAMD with $|{\mathit{S}}_k|\le 4$ fm $3\alpha$ configurations, (c) the sAMD with $|{\mathit{S}}_k|\le 3$ fm $3\alpha$ configurations, and (d) the sAMD without $3\alpha$ configurations. The strengths of discrete states are smeared by a Gaussian with the range $\gamma =1/\sqrt{\pi }$ MeV.

Figure 3

Energy-weighted TD, VD, and CD strengths summed up to $E=20$ MeV obtained with the sAMD, sAMD + GCM ($|{\mathit{R}}_k|\le 3$ fm), sAMD + GCM ($|{\mathit{R}}_k|\le 4$ fm), and full sAMD + GCM calculations. The ratio to the total energy weighted sum value of the full sAMD + GCM (${\mathrm{TEWS}}_{\mathrm{full}}$) is shown. The TEWS value of each calculation relative to the ${\mathrm{TEWS}}_{\mathrm{full}}$ is also shown.

Figure 4

EWSR ratio of the energy-weighted IS1 strengths. (a) Calculated strengths smeared by a Gaussian of width $\gamma =1/\sqrt{\pi }$ MeV. (b) Experimental data measured by $\alpha$ inelastic scattering from Ref. [18]. Figures are from Ref. [29].

Figure 5

Occupation probability of harmonic oscillator quanta $N$ in the shell-model basis expansion of the $0_{1,2}^{+}$ and $1_{1,2}^{-}$ states obtained with the sAMD + GCM calculation. The horizontal axis indicates the difference $\mathrm{\Delta }N$ from the minimum quanta $N_{\min }=8$.

Figure 6

Intrinsic density distributions of the dominant configurations written by the E-$3\alpha$ wave functions for the (a) $0_1^{+}$, (c) $1_1^{-}$, and (d) $1_2^{-}$ states. The intrinsic density of the AMD wave function ${\mathrm{\Phi }}_{\mathrm{AMD}}\left({\mathit{Z}}_{\mathrm{VAP}}^0\right)$ obtained with the VAP for the ground state is also shown. The density projected onto the $X\text{−}Z$, $Z\text{−}Y$, and $X\text{−}Y$ planes by integrating along the axes perpendicular to the planes is shown. Contour lines are drawn with a 0.1-${\mathrm{fm}}^{-2}$ interval.

Figure 7

(a) Transition current density for $0_1^{+}\to 1_2^{-}$ in the intrinsic frame (at $Y=0$ in the $X\text{−}Z$ plane). The vector plot of the transition current density for the parity-projected states of the dominant E-$3\alpha$ configurations is shown. Solid red and dashed magenta lines indicate contours for the matter densities $\rho (X,0,Z)=0.08{\mathrm{fm}}^{-3}$ of the parity-projected initial and final states, respectively. (b) Intrinsic density distribution of the dominant configuration for the initial state ($0_1^{+}$) and (c) that for the final state ($1_2^{-}$). (d) Schematic of the $0_1^{+}\to 1_2^{-}$ excitation in the cluster picture. ${\alpha }_1$ and ${\alpha }_2$ clusters contain cluster-breaking.