Structure and decays of nuclear three-body systems: The Gamow coupled-channel method in Jacobi coordinates

Background: Weakly bound and unbound nuclear states appearing around particle thresholds are prototypical open quantum systems. Theories of such states must take into account configuration mixing effects in the presence of strong coupling to the particle continuum space.

Purpose: To describe structure and decays of three-body systems, we developed a Gamow coupled-channel (GCC) approach in Jacobi coordinates by employing the complex-momentum formalism. We benchmarked the complex-energy Gamow shell model (GSM) against the new framework.

Methods: The GCC formalism is expressed in Jacobi coordinates, so that the center-of-mass motion is automatically eliminated. To solve the coupled-channel equations, we use hyperspherical harmonics to describe the angular wave functions while the radial wave functions are expanded in the Berggren ensemble, which includes bound, scattering, and Gamow states.

Results: We show that the GCC method is both accurate and robust. Its results for energies, decay widths, and nucleon-nucleon angular correlations are in good agreement with the GSM results.

Conclusions: We have demonstrated that a three-body GSM formalism explicitly constructed in the cluster-orbital shell model coordinates provides results similar to those with a GCC framework expressed in Jacobi coordinates, provided that a large configuration space is employed. Our calculations for $A=6$ systems and ${}^{26}\mathrm{O}$ show that nucleon-nucleon angular correlations are sensitive to the valence-neutron interaction. The new GCC technique has many attractive features when applied to bound and unbound states of three-body systems: it is precise, is efficient, and can be extended by introducing a microscopic model of the core.

Figure 1

Jacobi coordinates in a three-body system.

Figure 2

Comparison between GSM and GCC results for the two-nucleon separation energies of ${}^6\mathrm{Be},{}^6\mathrm{Li}$, and ${}^6\mathrm{He}$ obtained in different model spaces defined by ${\ell }_{\max }$. Vertical bars in (a) represent decay widths.

Figure 3

Similar to Fig. 2, but for the two-neutron separation energy of ${}^6\mathrm{He}$ obtained with the angular-momentum dependent Hamiltonian; see text for details.

Figure 4

Comparison between GSM and GCC results for the two-neutron angular correlation in ${}^6\mathrm{He}$ for different model spaces defined by ${\ell }_{\max }$.

Figure 5

Two-nucleon angular densities (total and in the $S=1$ channel) in the g.s. configurations of (a) ${}^6\mathrm{He}$, (b) ${}^6\mathrm{Li}$, and (c) ${}^6\mathrm{Be}$ obtained in the GSM and GCC approach with ${\ell }_{\max }=10$.

Figure 6

Two-neutron separation energy of the g.s. of ${}^{26}\mathrm{O}$ computed with the GSM and GCC method for different values of ${\ell }_{\max }$.

Figure 7

GCC wave function of the g.s. of ${}^{26}\mathrm{O}$ in the Jacobi coordinates $nn$ and ${}^{24}\mathrm{O}-2n$.

Figure 8

Two-neutron angular correlation for (a) the $0^{+}$ g.s. and (b) the $2_1^{+}$-state configurations of ${}^{26}\mathrm{O}$ computed with the GCC approach (solid line) and GSM (dashed line) with ${\ell }_{\max }=10$. The dash-dotted curve labeled ${\mathrm{GCC}}^{\prime }$ in (a) shows GCC results obtained with the strength of the neutron-neutron interaction reduced by 50%.