Shell model embedded in the continuum

Theoretical description of properties of weakly bound nuclei requires new methods that take into account the continuum phase space explicitly. Traditional shell-model approaches use well-bound single-particle states, most frequently the harmonic oscillator (HO) basis, to define the Fock space of many-nucleon configurations. Within such a space, particle emission threshold is completely absent, and a large fraction of obtained excited states (even all of them in case of weakly bound nuclei) have excitation energies above the threshold and, at the same time, localized wave functions. This inconsistency precludes by construction any possibility to describe decay widths of excited states, and moreover, introduces an unphysical cut-off in the physically important low-energy phase space.

Improvements upon this unsatisfactory situation have been proposed and defined since many years, see, e.g., Ref. [13], however, only very recently practical applications were realized. The shell model embedded in the continuum (SMEC) [14,15] combines the state-of-the-art shell model methods and interactions with the exact treatment of the one-body continuum. It means that the standard space of shell-model states of an A-particle system is enlarged by the similar shell-model states of the (A-1)-body system coupled with single-particle continuum states. The one-body continuum is appropriately orthogonalized with respect to the space of bound single-particle states. In the SMEC, the biggest unknown factor is the interaction between the shell-model and continuum states (the simplest contact forces have been used up to now), and this aspect certainly requires further analysis. The known nucleon-nucleus potentials may not be directly applicable, because interactions with all states of an (A-1)-body system have, in principle, to be taken into account.

  

Figure 1: The astrophysical S-factors (left) and phase-shifts (right) for the p+¹⁶O reactions, calculated within the SMEC and compared with experimental data. (From Ref. [15]. Reprinted from Physics Letters B, Vol 488, 2000, Page 75, K. Bennaceur et al, ``Shell model description of...'', Copyright (2000), with permission from Elsevier Science.)

Results of calculations performed for the p+¹⁶O reactions are presented in Fig. 1. The obtained quality of description is indeed very satisfactory. One is able to correctly reproduce the total and differential cross-sections as well as the widths of low-excited unbound states. The model may fail in predicting the widths of states for which the two- or three-body continuum becomes available, and the inclusion of these channels is probably one of the most challenging future extensions of the model.