Search

In this talk, I will give an overview of the phenomenologic calculations that have been performed in the framework of the I.B.A. model. All the calculations that I will present were done in the neutron-proton I.B.A. model. The main interest in these calculations was to extract from the experimental data the parameters appearing in the Hamiltonian, which could then later be compared with those arising from the microscopic calculations1. For this reason, we did not try to get the best possible fit for each individual nucleus but rather attempted to reproduce the observed trends in excitation energies, binding energies, B(E2) etc. for a chain of isotopes or isotones.

In the IBA model, collective states in even-even nuclei are described in terms of a system of interacting s- and d-bosons. This model can be extended to odd-A nuclei by coupling the degrees of freedom of the odd nucleon to the system of bosons1. In general, the Hamiltonian for this coupled system can be written as $$H\, = \,{H_B}\, + \,{H_F}\, + \,{H_{BF}}\,,$$ (1.1) where HB is the usual IBA Hamiltonian which describes the system of s- and d-bosons, and HF is the Hamiltonian of the odd particle. Since only a single odd particle is coupled to the bosons, it is sufficient to consider only the one-body part of HF $${H_F} = \,\mathop \sum \limits_{jm} \,{\varepsilon _j}\,a_{jm}^\dag \,{a_{jm}}\,.$$ (1.2) Here, as well as in the following, the summation index j runs over.the shell model orbits in the valence shell, and ej represents the single particle energy.

Since it has become increasingly evident that the IBA model and its extension to odd-A nuclei provides at the level of phenomenology an extremely successful description of nuclear collective phenomena, a microscopic understanding of the model is clearly of considerable interest. For the case of even-even nuclei Arima et al.1 have proposed a method to derive the main properties of the boson-boson interaction from a shell model picture. Application of this method by Otsuka2 to realistic cases has been shown to give promising results. In the present contribution the boson-fermion3 interaction, that enters in the description of odd-A nuclei, is derived from the shell model interaction between fermions. The latter has two main characteristics: i) a strong pairing force between the like nucleons, and ii) a quadrupole-quadrupole interaction between neutrons and protons. In terms of boson-boson interactions the former gives rise to attractive binding energies of the s- and d-bosons, whereas the latter leads to the quadrupole-quadrupole force between the neutron and proton bosons. It will be shown in this contribution that this interaction also plays a dominant role in the coupling of the odd-fermion to the bosons in the core4.

In this contribution the extension will be discussed of the Interacting Boson Approximation (IBA) model to odd-A nuclei where the fermion degrees of freedom need to be considered explicitly. The IBA model describes even-even nuclei as a system of interacting s- and d-boson. By coupling the degrees of freedom of a single nucleon to these bosons one can describe also odd-A nuclei. The general Hamiltonian for this coupled system can be written as $$ H = H_B + H_F + V_{BF} $$ (1) The IBA Hamiltonian, HB, describes the system of s- and d-bosons. HF is the Hamiltonian for the single odd particle and thus contains only a one-body term $$ H_F = \sum\limits_{j,m} {\varepsilon _j a_{jm}^\dag a_{jm} } $$ (2) In our convention the summation index j denotes the shell model orbits in the valence shell. The quasi-particle energies are denoted by ϕj.