Evidence of an enhanced nuclear radius of the $\alpha$-halo state via $\alpha +{}^{12}\mathrm{C}$ inelastic scattering

Evidence of the enhanced nuclear radius in the Hoyle rotational state, $2_2^{+}$, is derived from the differential cross sections in $\alpha +{}^{12}\mathrm{C}$ inelastic scattering. The prominent shrinkage is observed in the differential cross section of the $2_2^{+}$ state in comparison to the yrast $2_1^{+}$ state, and this shrinkage is the first evidence of the enhanced nuclear radius which originates from the $3\alpha$ structure in the $2_2^{+}$ state. A diffraction formula, that is, Blair's phase rule, is applied to the differential cross sections, and the present analysis predicts an enhancement of 0.6 to 1.0 fm in the nuclear radius of the $2_2^{+}$ state in comparison to the radius of the yrast $2_1^{+}$, which is considered to have a normal nuclear radius. Constraint on the recent ab initio calculation for $3\alpha$ states in ${}^{12}\mathrm{C}$ is also discussed.

Figure 1

Comparison of the MCC calculation with the observed differential cross section in $\alpha +{}^{12}\mathrm{C}$ scattering at $E_{\alpha }=386$ MeV [11]. Filled curves represent theoretical calculations; filled circles, experimental data. From top to bottom, results represent the elastic, $2_1^{+}, 3_1^{-}$, and $0_2^{+}$ channels, respectively.

Figure 2

Comparison of the theoretical calculation with the observed cross section of the excited state at $E_x\approx 10$ MeV. Open circles show experimental data, while the thick solid curve represents the total cross section in theory, which is given by the incoherent summation of the $0_3^{+}$ (dotted curve) and $2_2^{+}$ (thin solid curve) channels.

Figure 3

Theoretical differential cross sections of the final $2_1^{+}$ (dotted curve) and $2_2^{+}$ (solid curve) channels in $\alpha +{}^{12}\mathrm{C}$ inelastic scattering. The former and latter cross sections are same as the cross sections shown in Figs. 1 and 2, respectively. The magnitudes of both cross sections are normalized by the maximum value around ${\theta }_{\mathrm{c}.\mathrm{m}.}\approx 5^{\circ }$.

Figure 4

Theoretical differential cross sections of the final channels of $2_2^{+}$ with excitation energy $E_x=10.3$ MeV (solid curve) and $E_x=4.44$ MeV (dashed curve). The former cross section is the same as the cross sections shown in Fig. 2 (thin solid curve) and Fig. 3 (solid curve). The magnitudes of both cross sections are normalized by the maximum value around ${\theta }_{\mathrm{c}.\mathrm{m}.}\approx 5^{\circ }$.

Figure 5

Partial wave decomposition of angle-integrated cross sections calculated as a function of the incident orbital spin $L$ ($L$ distribution). The dotted curve with open circles shows the partial cross section of the $2_1^{+}$ final channel; the solid curve with filled circles, that of the $2_2^{+}$ final channel. The magnitudes of all the partial cross sections are normalized by the total cross section, which is the summation of the partial cross sections.

Figure 6

Same as Fig. 3 except that the incident energy is $E_{\alpha }=240$ MeV. The magnitudes of the cross sections are normalized to the first peak at ${\theta }_{\mathrm{c}.\mathrm{m}.}= 0^{\circ }$.