Cluster structures in ${}^{12}\mathrm{C}$ from global energy density functionals

Spectroscopic properties of low-lying states and cluster structures in ${}^{12}\mathrm{C}$ are analyzed in a beyond mean-field framework based on global energy density functionals (EDFs). To build symmetry-conserving collective states, axially symmetric and reflection-asymmetric solutions of the relativistic Hartree-Bogoliubov equations are first projected onto good values of angular momentum, particle number, and parity. Configuration mixing is implemented using the generator coordinate method formalism. It is shown that such a global microscopic approach, based on a relativistic EDF, can account for the main spectroscopic features of ${}^{12}\mathrm{C}$, including the ground-state and linear-chain bands as well as, to a certain approximation, the excitation energy of the Hoyle state. The calculated form factors reproduce reasonably well the available experimental values, and display an accuracy comparable to that of dedicated microscopic cluster models.

Figure 1

Deformation energy surfaces of ${}^{12}\mathrm{C}$ in the ${\beta }_2–{\beta }_3$ plane. In addition to the self-consistent mean-field RHB binding energies (top left panel), the angular momentum-, particle number-, and parity-projected energy surfaces are shown for spin-parity values $J^{\pi }=0^{+},2^{+},4^{+}$. For each surface, energies are normalized with respect to the corresponding absolute minimum. Contour lines are separated by $2.5\mathrm{MeV}$ (dashed lines) and $0.5\mathrm{MeV}$ (dotted lines).

Figure 2

Energy curves of ${}^{12}\mathrm{C}$ as functions of the axial quadrupole deformation ${\beta }_2$ for parity-conserving (${\beta }_3=0$) configurations. In addition to the self-consistent mean-field RHB binding energies (squares), we display the angular momentum- and particle number-projected curves for spin-parity values $J^{\pi }=0^{+},2^{+},4^{+}$.

Figure 3

The calculated low-energy positive-parity excitation spectrum of ${}^{12}\mathrm{C}$ compared to the available data [43]. Intraband $B\left(E2\right)$ transition strengths (red color, in $e^2{\mathrm{fm}}^4$) and spectroscopic quadrupole moments (green color, in $e$ ${\mathrm{fm}}^2$) are also shown. See text for more details.

Figure 4

Amplitudes of collective wave functions squared $|g({\beta }_2,{\beta }_3){|}^2$ for the low-energy levels of ${}^{12}\mathrm{C}$. Dashed contours in the ${\beta }_2\text{−}{\beta }_3$ plane successively denote a $10\%$ decrease starting from the largest value of the amplitude.

Figure 5

Characteristic intrinsic nucleon densities of the first three $0^{+}$ and $2^{+}$ collective states in ${}^{12}\mathrm{C}$. The corresponding average deformation parameters $({\beta }_2,{\beta }_3)$, as well as the respective contributions of prolate and/or oblate configurations, are shown. The bottom panel displays states that exhibit significant contributions from both prolate and oblate configurations, whereas the states shown in the top panel are predominantly characterized by either prolate or oblate configurations. See text for more details.

Figure 6

Form factors for electron scattering on ${}^{12}\mathrm{C}$ for the $0_1^{+}\to 0_1^{+}$ [left panel (a)] and $0_1^{+}\to 0_2^{+}$ [right panel (b)] transitions. The results obtained in the present study (red triangles) are compared to available data for the elastic [47] and inelastic [48] form factors, as well as with predictions of the AMD [6] and THSR [49] models. In addition, the insets show the corresponding charge density [left panel (a)] and the transition charge density [right panel (b)].