Effective operators in a single-j orbital
Journal of Physics G: Nuclear and Particle Physics
We present an analysis of effective operators in the shell model with up to three-body interactions in the Hamiltonian and two-body terms in electromagnetic transition operators when the nucleons are either neutrons or protons occupying a single-j orbital. We first show that evidence for an effective three-body interaction exists in the N = 50 isotones and in the lead isotopes but that the separate components of such interaction are difficult to obtain empirically. We then determine higherorder
... terms on more microscopic grounds. The starting point is a realistic two-body interaction in a large shell-model space together with a standard one-body transition operator, which, after restriction to the dominant orbital and with use of stationary perturbation theory, are transformed into effective versions with higher-order terms. An application is presented for the lead isotopes with neutrons in the 1g 9/2 orbital. In this paper we report on a study of effective Hamiltonians with up to threebody interactions and effective electromagnetic transition operators with up to twobody terms. For simplicity we limit ourselves to the manifestation of such higher-order corrections in nuclei with one kind of valence nucleon, and study the T = 3/2 three-body component of the nuclear interaction on top of its usual T = 1 two-body component. Therefore, our attention is focused on semi-magic nuclei with only nucleons of one kind, neutrons or protons, in the valence shell. The approach is further simplified by considering nuclei where the valence nucleons are dominantly in a single-j orbital. While these are extreme simplifications of more realistic situations, this approach presents certain advantages. First of all, the shell-model calculations in the full space can be carried out for some of the nuclei that we consider here, thus enabling to check whether an expansion to a given order yields satisfactory results. Also, given the simplifying assumptions made, the perturbation method of Bloch and Horowitz  can be pushed to third order in the Hamiltonian and second order in the transition operators without an unwieldy proliferation of diagrams. Finally, for a single-j orbital the energy matrices are known analytically in terms of the interaction matrix elements, and calculations are easily carried out. In fact, it would be relatively straightforward to extend the current approach to test the performance of effective four-body interactions. An early application of the shell model was the description of Ca isotopes (Z = 20) and N = 28 isotones with a two-body Hamiltonian in the 0f 7/2 orbital [9, 10]. To improve results for binding energies and spectra, several authors considered, already many years ago, the inclusion of three-body interactions [11, 12] . In view of the current interest in three-body forces, the issue was revisited more recently by Volya , who studied the same semi-magic nuclei with a two-plus-three-body Hamiltonian. It was later shown  that the extracted three-body component can to some extent be understood as the result of excluded higher-lying orbitals, in particular 1p 3/2 . It the purpose of this paper (i) to present an application in the same spirit but to different nuclei, (ii) to extend perturbation theory for the effective two-plus-three-body Hamiltonian to third order, and (iii) to determine the effective one-plus-two-body transition operators to second order. This paper is organized as follows. In Sect. 2 we define the Hamiltonian appropriate for identical nucleons in a single-j orbital. An effective single-j Hamiltonian with up to three-body interactions is derived in Sect. 3 and electromagnetic transition operators with up to two-body terms are derived in Sect. 4. With the recurrence relations for scalar and non-scalar k-body operators, as given in Sect. 5, we are then in a position to obtain all results, analytical if needed, in the context of a single-j orbital. In Sect. 6 we present the results of two applications, namely to the N = 50 isotones with protons in 0g 9/2 and to the lead isotopes (Z = 82) with neutrons in 1g 9/2 . Finally, in Sect. 7 the conclusions of this study are summarized.