Shell-model calculation with density-dependent Gogny interaction

Shell model has been an important and powerful tool for describing the structures of finite nuclei. In shell-model calculations, one of the most important questions is to construct the effective Hamiltonian in the truncated model space. One way is to start from realistic nuclear forces and then use perturbation theory to get the effective interaction. However, these calculations are complicated. If one wants to gain better quantitative descriptions one will have to take the high-order correlations (e.g., three-nucleon force, 3NF) into consideration. High-order correlations make calculations even more complicated and give rise to a significant growth of the model dimension. Another way is to start form an empirical TBMEs which will provide more quantitative results with simpler shell-model calculations. In empirical methods, the TBMEs are usually derived from realistic interactions originally and then are modified by fitting the experimental data coming from nuclear structures(including nuclear binding energies, excitation spectra and transition properties). Effects of the missing high-order correlations (e.g., 3NFs) are assume to be equivalently taken into account by fitting procedure. Yet the fitting can be a troubled process while there are too many TBMEs to be fitted. Meanwhile there will be large uncertainties in evaluating the TBMEs since some of matrix elements (especially for off-diagonal matrix elements) are insensitive to the data fitting. When it come to cross-shell cases, i.e. the model space including two or more major shells, the fitting process will be even more intricate. Not only because the large number of TBMEs to be fitted, but also that it is difficult to obtain the SPEs in such cases. We cannot distinguish the collective core excitations and single-particle excitations because they can get really close for cross-shell cases while the energy for the system is high. To deal with the above problems and to discuss the cross-shell situation properly, in the present work, we introduce the Gogny density-dependent force to study the structure of finite nuclei. The nature of density-dependent interactions is concerned and its application in shell-model is carefully discussed. The Gogny interaction contains totally 14 free parameters. There are several mature sets of the Gogny parameters which are fitted to nuclear structure data determined by mean-field calculations (Hartree-Fock-Bogolyubov). In the present shell-model calculations, we apply the existing mean-field Gogny parameters without refitting. Several Gogny force are tested in which D1S interaction gives the best result. In practice, we need to deal with the density-dependent term of Gogny force in shell-model frame using iteration methods. It is worth mention that, in our method, both two-body matrix elements and single-particle energies are reached by unified Gogny interaction and so are the binding energies. In this paper calculated the spectra, binding energies etc. for p-shell and sd-shell nuclei and the results are in good agreement with experimental data. Particularly, in the calculation of binding energies for oxygen chain isotopes, the present model can reproduce its neutron drip line which is an important task for nuclear structure study. It is due to the fact that Gogny interaction have consider reasonable three-body effects by its density-dependent term. In conclusion, the Gongy-like phenomenological interactions can describe the properties of p-shell and sd-shell nuclei and have the potential for further shell-model calculation.