Isoscalar monopole and dipole excitations of cluster states and giant resonances in ${}^{12}\mathrm{C}$

The isoscalar monopole (ISM) and dipole (ISD) excitations in ${}^{12}\mathrm{C}$ are investigated theoretically with the shifted antisymmetrized molecular dynamics (AMD) plus $3\alpha$-cluster generator coordinate method (GCM). The small-amplitude vibration modes are described by coherent one-particle one-hole excitations expressed by a small shift of single-nucleon Gaussian wave functions within the AMD framework, whereas the large-amplitude cluster modes are incorporated by superposing $3\alpha$-cluster wave functions in the GCM. The coupling of the excitations in the intrinsic frame with the rotation and parity transformation is taken into account microscopically by the angular-momentum and parity projections. The present a calculation that describes the ISM and ISD excitations over a wide energy region covering cluster modes in the low-energy region and the giant resonances in the high-energy region, although the quantitative description of the high-energy part is not satisfactory. The low-energy ISM and ISD strengths of the cluster modes are enhanced by the distance motion between $\alpha$ clusters, and they split into a couple of states because of the angular motion of $\alpha$ clusters. The low-energy ISM strengths exhaust 26% of the energy-weighted sum rule, which is consistent with the experimental data for the ${}^{12}\mathrm{C}$($0_2^{+}$; 7.65 MeV) and ${}^{12}\mathrm{C}$($0_3^{+}$; 10.3 MeV) measured by $(e,e^{\prime }),(\alpha ,{\alpha }^{\prime })$, and (${}^6\mathrm{Li},\stackrel{\prime }{{}^6\mathrm{Li}}$) scatterings. In the calculated low-energy ISD strengths, two $1^{-}$ states (the $1_1^{-}$ and $1_2^{-}$ states) with the significant strengths are obtained over $E=10\text{–}15$ MeV. The results indicate that the ISD excitations can be a good probe to experimentally search for new cluster states such as the ${}^{12}\mathrm{C}\left(1_2^{-}\right)$ obtained in the present calculation.

Figure 1

Schematic figures of $3\alpha$ configurations for (a) the $3\alpha \mathrm{GCM}$ (full) with the generator coordinates $d,D$, and $\theta$, and (b) the $3\alpha \mathrm{GCM}$ (distance) with the generator coordinates $d$ and $D$, and the fixed $\theta =\pi /2$.

Figure 2

(a) ISM and (b) ISD strength distributions of ${}^{12}\mathrm{C}$ calculated by the sAMD with ${\mathit{e}}_{\sigma =x,y,z},{\mathit{e}}_{\sigma =1,...,6},{\mathit{e}}_{\sigma =1,...,8}$, and ${\mathit{e}}_{\sigma =1,...,14}$. The strengths are smeared by a Gaussian with the width $\gamma =2$ MeV.

Figure 3

Density distribution of the intrinsic wave function of the ${}^{12}\mathrm{C}$ ground state obtained by $\text{AMD}+\text{VAP}$. The density is projected onto the $xy$ (left), $yz$ (middle), and $zx$ (right) planes.

Figure 4

The ISM strengths calculated by the sAMD, $\text{sAMD}+3\alpha \text{GCM}$ (distance), $\text{sAMD}+3\alpha \text{GCM}$ (full), and $\text{VAP}+3\alpha \text{GCM}$ (full). The strengths $B\left(\text{IS0}\right)$ are shown by dashed lines and those smeared by a Gaussian with the width $\gamma =1/\sqrt{\pi }$ MeV are shown by solid curves.

Figure 5

The ISD strengths calculated by the sAMD, $\text{sAMD}+3\alpha \text{GCM}$ (distance), $\text{sAMD}+3\alpha \text{GCM}$ (full), and $\text{VAP}+3\alpha \text{GCM}$ (full). The strengths $B\left(\text{IS1}\right)$ are shown by dashed lines and those smeared by a Gaussian with the width $\gamma =1/\sqrt{\pi }$ MeV are shown by solid curves.

Figure 6

The energy-weighted ISM and ISD strength distributions calculated by the $\text{sAMD}+3\alpha \text{GCM}$ (full) and those measured by $(\alpha ,{\alpha }^{\prime })$ scatterings. For the calculated result, the ratio of the strengths in each energy bin to the TEWS is shown. The bin width is chosen to be 1 MeV. The experimental data are the EWSR ratios from Ref. [9] of $(\alpha ,{\alpha }^{\prime })$ scatterings, in which about 40% of the ISM EWSR and 80% of the ISD EWSR were observed in $E\le 45$ MeV. Panel (a) also shows the experimental values of the energy-weighted ISM strengths in the unit of EWSR: the data for the $0_2^{+}$ and $0_3^{+}$ states measured by $(\alpha ,{\alpha }^{\prime })$ [9] and $\left({}^6\mathrm{Li},\stackrel{\prime }{{}^6\mathrm{Li}}\right)$ [17] scatterings, and that evaluated by the $E0$ strength measured by $(e,e^{\prime })$ scatterings [40] assuming mirror symmetry.