Three-body model of light nuclei with microscopic nonlocal interactions

A three-body cluster model involving microscopic nonlocal interactions is developed and compared with a fully microscopic cluster model. The energy-independent nonlocal interactions are obtained from a renormalization of the energy-dependent kernels of the resonating-group method. Such interactions are derived for the $\alpha \alpha$ and $\alpha n$ systems. The role and importance of nonlocality are discussed. These interactions are employed in three-body studies of the $\alpha \mathit{nn},\alpha \alpha n$, and $3\alpha$ descriptions of the ${}^6\mathrm{He}$, ${}^9\mathrm{Be}$, and ${}^{12}\mathrm{C}$ nuclei. A comparison with fully microscopic calculations provides a measure of the importance of three-cluster exchanges in those states. The differences between both cluster-model calculations are in general small, except in the densities at short distances.

Figure 1

The potential function $U_{\ell }^{\mathrm{RGM}}(r)$ and the direct potential $V_D$ for the $\alpha \alpha$ system (the MN interaction is used).

Figure 2

Decomposition of the $\alpha \alpha$ $U_{\ell }^{\mathit{NL}}(r)$ potential into the kinetic energy, the nuclear potential and the Coulomb potential contributions (the MN interaction is used).

Figure 3

$\alpha \alpha$ phase shifts. The solid lines are obtained with the standard potentials (a) and the dotted lines with $u=0.9122$ and $M=0.5820$ (thin curves: MN, thick curves: V2). The data are taken from Ref. [34].

Figure 4

$\alpha n$ phase shifts. The solid lines are obtained with the standard potentials (a) and the dotted lines with $u=1.0045$ and $M=0.6573$ (thin curves: MN, thick curves: V2). The data are taken from Ref. [35].

Figure 5

Charge densities of ${}^{12}\mathrm{C}$ with the MN and V2 potentials in the microscopic (solid lines) and semimicroscopic (dashed lines) models. The dotted line corresponds to the MN calculation (a). The inset represents the densities in a logarithmic scale.

Figure 6

Neutron densities of ${}^9\mathrm{Be}$ with the MN and V2 potentials in the microscopic (solid lines) and semimicroscopic (dashed lines) models. The inset represents the densities in a logarithmic scale.

Figure 7

Neutron densities of ${}^6\mathrm{He}$ with the MN and V2 potentials in the microscopic (solid lines) and semimicroscopic (dashed lines) models. The inset represents the densities in a logarithmic scale.

Figure 8

Coordinate system in the three-cluster model.