Candidate for the 2${}^{+}$ excited Hoyle state at $E_x\sim 10$ MeV in ${}^{12}$C

Inelastic scattering from ${}^{12}$C has been measured at extremely forward angles including 0 ${}^{\circ }$ using 386 MeV $\alpha$ particles to study the $\alpha$-cluster states around $E_x$ $\sim$ 10 MeV, especially the 2${}^{+}$ state predicted by the $\alpha$-cluster model. We have analyzed ($\alpha$,${\alpha }^{\prime }$) cross-section data using both peak-fitting and multipole decomposition techniques. A 2${}^{+}$ state at $E_x=$ 9.84 $\pm$ 0.06 MeV with a width of 1.01 $\pm$ 0.15 MeV is found to be submerged in the broad 0${}^{+}$ state at $E_x=$ 9.93 $\pm$ 0.03 MeV with a width of 2.71 $\pm$ 0.08 MeV. This 2${}^{+}$ state may be interpreted as the 2${}^{+}$ excitation of the Hoyle state and the $\alpha$-condensate state.

Figure 1

Energy spectra for the ${}^{12}$C($\alpha$,${\alpha }^{\prime }$) reaction at scattering angles (a) ${\theta }_{\mathrm{lab}}=$ 0 ${}^{\circ }$ and (b) ${\theta }_{\mathrm{lab}}=$ 3.7 ${}^{\circ }$. The momentum acceptances of the spectrometer were in the range of $3⩽E_x⩽$ 30 MeV at 2.0 ${}^{\circ }$–15 ${}^{\circ }$ and of 7 $⩽$ $E_x$$⩽$ 30 MeV at 0 ${}^{\circ }$, respectively.

Figure 2

Angular distribution of elastic scattering from ${}^{12}$C at $E_{\alpha }=$ 386 MeV. The solid line shows the result of the DWBA calculation using the single-folding model with the effective N-$\alpha$ interaction.

Figure 3

Angular distributions of inelastic $\alpha$ scattering to (a) the 4.44 MeV 2${}_1^{+}$ state, (b) the 7.65 MeV 0${}_2^{+}$ state, and (c) the 9.64 MeV 3${}_1^{-}$ state at $E_{\alpha }=$ 386 MeV. The solid and dotted lines show the result of the DWBA calculation with the 3$\alpha$ RGM and collective models, respectively. The dashed lines show the 3$\alpha$ RGM calculations normalized to the experimental data.

Figure 4

Excitation-energy spectrum for ${}^{12}$C fitted to the 7.65 MeV 0${}_2^{+}$, 9.64 MeV 3${}_1^{-}$, 10.84 MeV 1${}_1^{-}$, and the 10 MeV broad bump at 0 ${}^{\circ }$. The thick line shows the fitting result. The thin solid lines show the individual peaks. The additional peak around 8.5 MeV was needed to fit at 0 ${}^{\circ }$, 1.9 ${}^{\circ }$, and 2.3 ${}^{\circ }$.

Figure 5

Angular distribution of inelastic $\alpha$ scattering to the broad bump around 10 MeV. The thin solid and dashed lines show the angular distributions calculated using the collective ($L=$ 0) and the $\alpha$-condensed model [30] ($L=$ 2) transition densities, respectively. The thick solid line shows the sum of the $L=$ 0 and 2 angular distributions. The dot-dashed line shows the possible $L=$ 1 component.

Figure 6

Angular distributions of the double-differential cross section at $E_x=$ 8–12 MeV for various excitation-energy bins in ${}^{12}$C. The thick solid lines show the fits to the data from multipole decomposition. In each panel, the contributions from $L=0$ (thin solid), $L=1$ (dot-dashed), $L=2$ (dashed) and $L=3$ (dotted) are also displayed. Contributions from other multipoles are not displayed.

Figure 7

Excitation-energy spectra from ${}^{12}$C($\alpha ,{\alpha }^{\prime }$) at ${\theta }_{\mathrm{lab}}=$ 0 ${}^{\circ }$, 3.7 ${}^{\circ }$. The hatched regions were constructed from the results of the MDA. The cross-hatched region is $L=$ 0, the right-hatched is $L=$ 2, the left-hatched is $L=$ 3, and the vertical-hatched regions represent contributions from other multipoles.

Figure 8

Isoscalar-strength distributions for (a) $L=$ 0, (b) $L=$ 1, (c) $L=$ 2, and (d) $L=$ 3 as obtained in this paper. The vertical axes show the corresponding isoscalar strengths. The solid lines show the conventional fit by a Gaussian function in (a) and (c). The dashed line in (a) shows the fit by two Gaussian functions.