Structure of $A=10–13$ Nuclei with Two- Plus Three-Nucleon Interactions from Chiral Effective Field Theory

Properties of finite nuclei are evaluated with two-nucleon ($NN$) and three-nucleon ($NNN$) interactions derived within chiral effective field theory. The nuclear Hamiltonian is fixed by properties of the $A=2$ system, except for two low-energy constants (LECs) that parametrize the short range $NNN$ interaction, which we constrain with the $A=3$ binding energies. We investigate the sensitivity of ${}^4\mathrm{He}$, ${}^6\mathrm{Li}$, ${}^{10,11}\mathrm{B}$, and ${}^{12,13}\mathrm{C}$ properties to the variation of the constrained LECs. We identify observables that are sensitive to this variation and find preferred values that give the best overall description. We demonstrate that the $NNN$ interaction terms significantly improve the binding energies and spectra of mid-$p$-shell nuclei not just with the preferred choice of the LECs but even within a wide range of the constrained LECs. We find that a very high quality description of these nuclei requires further improvements to the chiral Hamiltonian.

Figure 1

Relations between $c_D$ and $c_E$ for which the binding energy of ${}^3\mathrm{H}$ (8.482 MeV) and ${}^3\mathrm{He}$ (7.718 MeV) are reproduced. (a) ${}^4\mathrm{He}$ ground-state energy along the averaged curve. (b) ${}^4\mathrm{He}$ charge radius $r_c$ along the averaged curve. Dotted lines represent the $r_c$ uncertainty due to the uncertainties in the proton charge radius.

Figure 2

Dependence on the $c_D$ with the $c_E$ constrained by the $A=3$ binding energy fit for different basis sizes for: ${}^6\mathrm{Li}$ quadrupole moment, ${}^{10}\mathrm{B}$ $B(E2;3_1^{+}0\to 1_1^{+}0)/B(E2;3_1^{+}0\to 1_2^{+}0)$ ratio, and the ${}^{12}\mathrm{C}$ $B(M1;0^{+}0\to 1^{+}1)$. The HO frequency of $\hslash \Omega =13$, 14, 15 MeV was employed for ${}^6\mathrm{Li}$, ${}^{10}\mathrm{B}$, ${}^{12}\mathrm{C}$, respectively. In the inset of the ${}^{12}\mathrm{C}$ figure, the convergence of the $B(M1;0^{+}0\to 1^{+}1)$ is presented for calculations with (using $c_D=-1$) and without the $NNN$ interaction.

Figure 3

${}^{11}\mathrm{B}$ excitation spectra as function of the basis space size $N_{\max }$ at $\hslash \Omega =15\mathrm{MeV}$ and comparison with experiment. The isospin of the states depicted is $T=1/2$.

Figure 4

States dominated by $p$-shell configurations for ${}^{10}\mathrm{B}$, ${}^{11}\mathrm{B}$, ${}^{12}\mathrm{C}$, and ${}^{13}\mathrm{C}$ calculated at $N_{\max }=6$ using $\hslash \Omega =15\mathrm{MeV}$ (14 MeV for ${}^{10}\mathrm{B}$). Most of the eigenstates are isospin $T=0$ or $1/2$, the isospin label is explicitly shown only for states with $T=1$ or $3/2$. The excitation energy scales are in MeV.