Tensor correlations in ${}^4\mathrm{He}$ and ${}^8\mathrm{Be}$ within an antisymmetrized quasicluster model

In this paper, we extend the framework of the improved version of a simplified method to take into account the tensor contribution $\left(i\mathrm{SMT}\right)$ and propose the AQCM-T, a tensor version of the antisymmetrized quasicluster model (AQCM). Although the AQCM-T is phenomenological, we can treat the ${}^3\mathrm{S}\text{−}{}^3\mathrm{D}$ coupling in the deuteronlike $T=0NN$ pair induced by the tensor interaction in a very simplified way, which allows us to proceed to heavier nuclei. Using the AQCM-T and the V2m interaction, where the triplet-even channel of the Volkov no. 2 interaction is weakened to 60% so as to reproduce the binding energy of ${}^4\mathrm{He}$ after including the tensor term of a realistic interaction, the significant tensor contribution in ${}^4\mathrm{He}$ is shown, which is almost comparable to the central interaction, where the $D$ state mixes by 8% to the major $S$ state. The AQCM-T with the new interaction is also applied to ${}^8\mathrm{Be}$. It is found that the tensor suppression gives a significant contribution to the short-range repulsion between two $\alpha$ clusters.

Figure 1

The comparison of the radial part of the G3RS tensor term (solid line), Furutani tensor (dotted line), and three-range fit (dash-dotted line). (a) Triplet even part, (b) triplet odd part.

Figure 2

Matter density distribution of ${}^4\mathrm{He}_{\mathrm{gs}}$ obtained with AQCM-T and V2m-3R. The single Gaussian shape for the ${\left(0s\right)}^4$ configuration with $\nu =0.264{\mathrm{fm}}^{-2}$ that gives the equivalent radius 1.46 fm is also drawn.

Figure 3

The squared overlaps ${\mathcal{O}}_{\beta }\left(k\right)$ and pair wave functions ${\varphi }_{NN}\left(r\right)$ in ${}^4\mathrm{He}_{\mathrm{gs}}$ calculated with AQCM-T and the V2m-3R interaction. (a) Squared overlaps obtained by the full calculation (three-channel calculation with full $k$ configurations, $k=\{0.5,1.0,...,5.5\}{\mathrm{fm}}^{-1})$; (b) pair wave functions in the ${}^3\mathrm{D},{}^3\mathrm{S}$, and ${}^1\mathrm{S}$, and $0s$ components obtained by the full calculation; (c) those obtained by the two-channel calculation with $k=\{1,2,3\}{\mathrm{fm}}^{-1}$; (d) those obtained by the single-channel calculation with $k=2{\mathrm{fm}}^{-1}$. In (b), the ${}^3\mathrm{S}$ pair wave functions in the ${\left(0s\right)}^4$ configuration and that $\left({{}^3\mathrm{S}}_{\mathrm{ortho}}\right)$ in the orthogonal configuration $(1-|0s\rangle \langle 0s\left|\right)|{\mathrm{\Psi }}_{{}^4\mathrm{He},\mathrm{gs}}\rangle$ are also shown.

Figure 4

Energy and the probabilities of $0s$ and $D$ states in ${}^8\mathrm{Be}$ calculated with AQCM-T and the V2m-3R interaction, as a function of the relative distances $d_{\alpha }$ between the two $\alpha$ clusters, (a) total energy, (b) $0s$ and ${}^3\mathrm{D}$ probabilities, and (c) energy of each component of the Hamiltonian. The total energy calculated using the V2m central interaction without the odd part combined with the three-range fit tensor interaction is also shown in (a). The results calculated with the BB cluster model and V2 are shown in (a) and (d). In panel (a), the asymptotic energy for AQCM-T and BB are shown by arrows. The energy calculated with AQCM-T using the V2m central interaction without the odd part combined with the three-range fit tensor interaction is also shown in (a). In (c) and (d), the relative energies measured from $d_{\alpha }=7$ fm are plotted.