Nuclear shell model in natural basis

Shell model calculation is a key tool in investigating the nuclear structure. Traditional shell model calculations deal with the many-body correlations through the configurations interaction, in which one performs large-scale matrix diagonalization to study the excitation spectra for system in many basis states. The crucial starting point in shell model calculations is the derivative of effective interaction, which be either a microscopic approach based on realistic nuclear forces or a phenomenological determined interaction. Conventional shell model calculation always build the many-body wave functions in harmonic oscillator basis, which is always bound and localized. However, nuclei far from stability have strong coupling with the scattering states, low particle emission threshold and the wave functions have large spatial dimensions, which may lead the harmonic oscillator basis fail in describing the weakly bound nuclei. In this paper, we presented a systemically shell model calculation in natural basis, which can treat the bound and scattering states on equal footing. The natural basis was generated by solving the one-body Schrödinger equation with Woods-Saxon potential using the shooting method, and the full natural space is consisting of the bound states and scattering states. The scattering states can well describe the weakly bound properties of nuclei far from stability, but the infinite dimension of continuum states makes it difficult to perform large-scale shell model calculation. By deriving effective interaction, we are able to investigate the influence of scattering states on well bound and weakly bound nuclei in small model space. In our calculation, the model space is spanned by the sd shell with d many-body states. In deriving the effective interactions, we used the folded-diagram method, and toke the core polarization effects into consideration by summing up the two-body irreducible diagrams Q box to the second order. The folded-diagram method can be treated as a Taylor expansion of the energy-dependent interaction which can reproduce the lowest d states. In evaluating the Ĝ box, we used the V_lowk nucleon-nucleon interaction derived from the charge-dependent Bonn potential (CD-Bonn) using the Okamoto-Lee-Suzuki (OLS) transformation in momentum space. The OLS renormalization method is energy independent, which make the low-momentum and high-momentum nuclear interaction decoupled and keep the low-momentum nucleon-nucleon scattering phase shift unchanged. The Woods-Saxon single-particle states are non-degenerated in a major shell, but the conventional folded-diagram method can only deal with the degenerated model space. In order to deal with the non-degenerated model space, the extended Krenciglowa-Kuo (KK) method is adopted. By an energy shift in the decoupling equation and the KK iteration scheme, the EKK approach can deal with degenerated and non-degenerated model space by the same iteration procedure. In our calculation, instead of treating the scattering states explicitly, contributions from the scattering states are included through the effective interactions, which make it possible to perform shell model calculation in a small model space without loss of physics. We calculated the oxygen isotopes in the two sets of base respectively, and analyzed the low-lying states of these nuclei in the two cases. The effects of scattering states in well-bound nucleus are investigated, and the advantages of natural basis shell model calculations for weakly bound nucleus are discussed. From our calculation, we find the natural basis are equivalent to the harmonic oscillator basis in describing well bound nuclei, which is guaranteed by the completeness of this two basis. However, natural basis is much more efficient than the harmonic oscillator basis in describing the weakly bound nuclei as the shell model calculation are always performed in a truncated model space. We can conclude that the scattering states are important in describing weakly bound nuclei, and the natural basis are more efficient in shell model calculation.