Resonances and scattering in microscopic cluster models with the complex-scaled generator coordinate method

The generator coordinate method of a microscopic cluster model is developed to treat the resonance and scattering of nuclear clusters with complex scaling. We consistently derive the formulation of the complex scaling for the microscopic cluster model, in which only the relative motions between clusters are transformed in the generator coordinate wave function. We also reveal the applicability of this method to the cluster wave function. Furthermore, we demonstrate this framework in the $2\alpha$ system of ${}^8\mathrm{Be}$ and obtain the solutions of resonance and nonresonant continuum states. Using these solutions, we calculate the level density, which brings the phase shifts of the cluster-cluster scattering. This work becomes the foundation in the description of the multicluster scattering states of nuclei in a microscopic framework with complex scaling.

Figure 1

Energy eigenvalues of ${}^8\mathrm{Be}$ ($0^{+}, 2^{+}, 4^{+}$, and $6^{+}$) for the full Hamiltonian ${\overline{H}}^{\theta }$ (solid symbols) and the asymptotic one ${\overline{H}}_0^{\theta }$ (open symbols) in the complex energy plane with $\theta ={27}^{\circ }$, measured from the $\alpha +\alpha$ threshold energy.

Figure 2

Two kinds of level densities, ${\rho }^{\theta }\left(E\right)$ and ${\rho }_0^{\theta }\left(E\right)$, and the continuum level density $\mathrm{\Delta }\left(E\right)$ of ${}^8\mathrm{Be}$ ($0^{+}, 2^{+}, 4^{+}$, and $6^{+}$), measured from the $\alpha +\alpha$ threshold energy. The upper arrows indicate the resonance energies of $2^{+}, 4^{+}$, and $6^{+}$ in Table 1.

Figure 3

Phase shifts of the $\alpha -\alpha$ scattering ($0^{+}, 2^{+}, 4^{+}$, and $6^{+}$) in the center-of-mass frame. The open symbols indicate the experi- mental data [42]. The upper arrows indicate the resonance energies of $2^{+}, 4^{+}$, and $6^{+}$ in Table 1.