Proton and neutron correlations in ${}^{10}\mathrm{B}$

We investigate positive-parity states of ${}^{10}\mathrm{B}$ with the calculation of antisymmetrized molecular dynamics focusing on $pn$ pair correlations. We discuss effects of the spin-orbit interaction on energy spectra and $pn$ correlations of the $J^{\pi }T=1_1^{+}0,3_1^{+}0$, and $0_1^{+}1$ states. The $1_1^{+}0$ state has almost no energy gain of the spin-orbit interaction, whereas the $3_1^{+}0$ state gains the spin-orbit interaction energy largely to come down to the ground state. We analyze the spin-orbit interaction dependence of the ${}^{10}\mathrm{B}$ spectra and find that the ordering of the $3_1^{+}0$ and $1_1^{+}0$ states is sensitive to the spin-orbit interaction. We also apply a $2\alpha +pn$ model to discuss effects of the spin-orbit interaction on $T=0$ and $T=1pn$ pairs around the $2\alpha$ core. In the spin-aligned $J^{\pi }T=3^{+}0$ state, the spin-orbit interaction affects the $\left(ST\right)=\left(10\right)$ pair attractively and keeps the pair close to the core, whereas, in the $1^{+}0$ state, it gives a minor effect to the $\left(ST\right)=\left(10\right)$ pair. In the $0^{+}1$ state, the $\left(ST\right)=\left(01\right)$ pair is somewhat dissociated by the spin-orbit interaction.

Figure 1

Energy spectra of ${}^6\mathrm{Li}$ and ${}^9\mathrm{Be}$ obtained by AMD+VAP using the interaction parameter sets (A) and (B) compared with the experimental spectra [40, 41, 42].

Figure 2

Energy spectra of ${}^{10}\mathrm{B}$. The theoretical result of AMD+VAP using the interaction parameter sets (A) and (B), the experimental data [40, 42], and the NCSM calculation with the chiral NN+NNN force [20] are shown.

Figure 3

Energy spectra for ${}^{10}\mathrm{B}$ calculated with AMD+VAP using interactions (B) and $\left({\mathrm{B}}^{\prime }\right)$. The experimental data are taken from Refs. [40, 42].

Figure 4

Dependence of the energy spectra on the strength $u_{\mathrm{ls}}$ of the spin-orbit interaction for ${}^{10}\mathrm{B}$ and ${}^6\mathrm{Li}$ calculated with AMD+VAP. Energy spectra of ${}^{10}\mathrm{B}$ obtained using (B) with the default spin-orbit interaction $u_{\mathrm{ls}}=1300$ MeV and $\left({\mathrm{B}}^{\prime }\right)$ with the weaker one, $u_{\mathrm{ls}}=1000$ MeV, and those of ${}^6\mathrm{Li}$ obtained using (A) with $u_{\mathrm{ls}}=1300$ MeV and $\left({\mathrm{A}}^{\prime }\right)$ with $u_{\mathrm{ls}}=1000$ MeV are shown as well as the experimental energy spectra. The NCSM calculation using the chiral nuclear forces with the NNN force (the chiral NN+NNN force) and without the NNN force (the chiral NN force) from Ref. [20] are also shown.

Figure 5

Distributions of matter densities of ${}^{10}\mathrm{B}\left(3_1^{+}\right),{}^{10}\mathrm{B}\left(1_1^{+}\right)$, and ${}^{10}\mathrm{B}(0_1^{+},T=1)$ calculated with AMD+VAP using the interaction (A). Densities of intrinsic states are integrated with respect to the $z$ axis and plotted on the $x\text{−}y$ plane (the box size is 10 fm $\times$ 10 fm). Here, axes of the intrinsic frame are chosen as $\langle x^2\rangle \ge \langle y^2\rangle \ge \langle z^2\rangle$.

Figure 6

Schematic figures for the $S_z=0pn$ pair (top) and the $S_z=1pn$ pair (bottom) around the $2\alpha$ core in the $2\alpha +pn$ model.

Figure 7

$d_x$ dependence of energy calculated with the $2\alpha +pn$ model. Energies of $J^{\pi }=0^{+},1^{+}$, and $3^{+}$ projected states for $k_y=0$ are calculated using the interaction (A). The $\alpha \text{−}\alpha$ distance is fixed to be $d_{\alpha \alpha }=3$ fm. $S_z=0$ is chosen for the $0^{+}$ state, and $S_z=1$ and $K=1$ $\left(K=3\right)$ are chosen for the $1^{+}$ $\left(3^{+}\right)$ state. The $3^{+}$ energy calculated without the spin-orbit interaction is also shown.

Figure 8

Energies and $\langle S^2\rangle$ calculated with the $2\alpha +pn$ model using the interaction (A). (a) Energy of the nonprojected state for the $S_z=0pn$ pair, (b) that of the $J^{\pi }=0^{+}$ projected state for the $S_z=0pn$ pair, and (c) that of the nonprojected state for the $S_z=1pn$ pair. The energies with and without the spin-orbit force are plotted as functions of $k_y$. $d_{\alpha \alpha }=3$ fm and $d_x=2$ fm are chosen. (d) The spin expectation value $\langle S^2\rangle$ of the $J^{\pi }=0^{+}$ projected state for the $S_z=0pn$ pair.

Figure 9

Energies of the $J^{\pi }=0^{+}$ projected state for the $S_z=0pn$ pair calculated with the $2\alpha +pn$ model using the interaction (A). (a) Energy with the spin-orbit interaction, (b) that without the spin-orbit interaction, and (c) the expectation value of the spin-orbit interaction.