Realistic Effective Interactions and Nuclear Structure Calculations

We address the two main questions relevant to microscopic nuclear structure calculations starting from a free NN potential. These concern the accuracy of these kinds of calculations and the extent to which they depend on the potential used as input. Regarding the first question, we present some results obtained for nuclei around doubly magic Sn and Pb by making use of an effective interaction derived from the Bonn A potential. Comparison shows that our results are in very good agreement with the experimental data. As for the second question, we present the results obtained for the nucleus Te by making use of four different NN potentials. They indicate that nuclear structure calculations may help in understanding the off-shell nature of the NN potential. INTRODUCTION The shell model is the basic framework for nuclear structure calculations in terms of nucleons. Since the early 1950s many hundreds of shell-model calculations have been carried out, most of them being very successful in describing a variety of nuclear structure phenomena. In any standard shell-model calculation one has to start by defining a model space, namely by specifying a set of active single-particle (s.p.) orbits. The choice of the model space is of course conditioned by the size of the matrices to be set up and diagonalized. The rapid increase in computer power and the development of high-quality codes in the last decade has greatly extended the feasibility of large-scale calculations [1]. While these technical improvements add to the practical value of the shell model, much uncertainty still exists for what concerns the model-space effective interaction Veff . In most of the existing calculations to date either empirical effective interaction containing several adjustable parameters have been used or the two-body matrix elements have been treated as free parameters, this latter approach being limited to small model spaces. This uncertainty in shell model work can only be removed by taking a more fundamental approach, namely by deriving the effective interaction from the free nucleon-nucleon (NN) potential. As is well known, the first step in this direction was taken in the mid 1960s by Kuo and Brown [2] who derived an s-d shell effective interaction from the Hamada-Johnston potential [3]. Since that time there has been substantial progress towards a microscopic approach to nuclear structure calculations starting from a free NN potential. On the one hand, high-quality NN potentials have been constructed which reproduce quite accurately all the known NN data. On the other hand, the many-body methods for calculating the matrix elements of the effective interaction have been largely improved. A review of modern NN potentials is given in Ref. [4] while the main aspects of the derivation of Veff are discussed in Ref. [5]. These improvements have brought about renewed interest in shell-model calculations with realistic effective interactions. In this context, the two crucial questions are: i) how accurate is an effective interaction derived from the NN potential? ii) to which extent can nuclear structure calculations distinguish between different NN potentials? Recent calculations for nuclei in the Sn and Sn regions [6–11] have achieved very good agreement with experiment indicating the ability of realistic effective interactions to provide a description of nuclear structure properties at least as accurate as that provided by traditional, empirical interactions. To our knowledge, no systematic investigation concerning the second question has been carried out thus far. The main interest in trying to answer this question stems from the fact that two potentials which fit equally well the NN data up to the inelastic threshold may differ substantially in their off-shell behavior. Thus, from microscopic nuclear structure calculations we may learn something about the off-shell properties of the nuclear potential. The main aim of this paper is to report on some achievements of our current work relevant to both the above questions. We shall first present some results of realistic shell-model calculations for nuclei having either few protons outside doubly magic Sn or few neutron holes in doubly magic Pb. They are I, Xe and Pb. In all of these calculations we have made use of a realistic effective interaction derived from the Bonn A free NN potential [12]. Then we shall present the results obtained for the two proton-nucleus Te by making use of four different potentials, Paris [13], Nijmegen93 [14], Bonn A and CD Bonn [15], which are all based on the meson theory of nuclear force. We shall see that while the former study confirms what was learned from our previous calculations with the Bonn potential, the latter indicates a dependence of nuclear structure results on the kind of potential used as input. OUTLINE OF CALCULATIONS As already mentioned in the Introduction, for all the six nuclei considered in this paper we have employed an effective interaction derived from the Bonn A potential. For Te we have also performed calculations employing three other effective interactions derived from the Paris, Nijmegen93 and CD Bonn potential, respectively. These effective interactions were all obtained using a G-matrix folded-diagram formalism, including renormalizations from both core polarization and folded diagrams. For the N = 82 isotones Te, I and Xe we have considered Sn as an inert core and let the valence protons occupy the five single-particle (s.p.) orbits 0g7/2, 1d5/2, 2s1/2, 1d3/2, and 0h11/2. For the Pb isotopes, we have treated neutrons as valence holes with respect to the Pb closed core and included in the model space the six single-hole (s.h.) orbits 2p1/2, 1f5/2, 2p3/2, 0i13/2, 1f7/2, and 0h9/2. A description of the derivation of our Veff for the N = 82 isotones and for the Pb isotopes can be found in Refs. [16] and [17], respectively. For the shell-model oscillator parameter h̄ω we have used the value 7.88 MeV for the N = 82 isotones and 6.88 MeV for the Pb isotopes, as obtained from the relationship h̄ω = 45A − 25A for A= 132 and A= 208, respectively. As regards the s.p. energies, for the N = 82 isotones we have taken three s.p. spacings from the experimental spectrum of Sb [18,19]. In fact, the g7/2, d5/2, d3/2, and h11/2 states can be associated with the ground state and the 0.962, 2.439 and 2.793 MeV excited levels, respectively. As for the s1/2 state, its position has been determined by reproducing the experimental energy of the 1 2 + level at 2.15 MeV in Cs. This yields the value ǫs1/2 = 2.8 MeV. Regarding the Pb isotopes, the s.h. energies have all been taken from the experimental spectrum of Pb [20]. They are (in MeV) ǫp1/2 = 0, ǫf5/2 = 0.570, ǫp3/2 = 0.898, ǫi13/2 = 1.633, ǫf7/2 = 2.340, and ǫh9/2 = 3.414.