Introduction

Dynamic symmetries and supersymmetries provide a convenient framework within which spectra of nuclei, either individually (symmetries) or in a set (supersymmetries) can be analyzed. However, they usually provide only a first approximation to the observed properties and furthermore there are nuclei for which they cannot be used. In these cases, one needs to do numerical studies. The starting point for these studies is the diagonalization of the Hamiltonian for the combined system of bosons and fermions written in one of its forms. Computer programs are available for the numerical solution of this problem (Scholten, 1979). The structure of the Hamiltonian is as in (1.16). Numerical studies are done by first analyzing the spectra of even-even nuclei as in Volume 1. This analysis determines the parameters appearing in HB. In a second step, the spectra of odd-even nuclei are studied. For odd-even nuclei with one unpaired particle, HF contains only the single-particle energies, ηJ. If states originating from only one single-particle level are studied, there is only one single-particle energy, η, which can be chosen as zero on the energy scale. If states originating from m single-particle orbits are included, the number of input parameters for the calculation is m – 1, since the lowest level can be chosen as zero on the energy scale. The crucial property that determines the structure of spectra of odd-even nuclei is the coupling between the collective degrees of freedom (bosons) and the single-particle degrees of freedom (fermions).

Introduction

The interacting boson-fermion model-1 describes properties of odd-even nuclei by coupling collective and single-particle degrees of freedom much in the same way this is done in the collective model (Bohr and Mottelson, 1975). The collective degrees of freedom are described either by shape variables αμ (μ = 0, ±1, ±2) or by boson operators s, dμ (μ = 0, ±1, ±2), with no direct link to the underlying microscopic structure. A microscopic description of nuclei is provided by the spherical shell model. Collective features in this model can be obtained by introducing the concept of correlated pairs with angular momentum and parity Jp = 0+ and Jp = 2+. A treatment of these pairs as bosons leads to the interacting boson model. However, since there are protons and neutrons, one has the possibility of forming proton and neutron pairs. In heavy nuclei, the neutron excess prevents the formation of correlated proton-neutron pairs and one thus is led to consider only proton-proton and neutron-neutron pairs. The corresponding model is the interacting boson model-2 (Arima et al., 1977; Otsuka et al., 1978). The introduction of fermions in this models leads to the interacting boson-fermion model-2. In addition to a more direct connection with the spherical shell model, the interacting boson-fermion model-2 has features that cannot be obtained in the interacting boson-fermion model-1.

The structure of model-2 is very similar to that of model-1.

Introduction

In addition to low-lying collective modes extensively discussed in Volume 1 and in this book in terms of bosonic degrees of freedom, nuclei also display high-lying collective modes. The microscopic description of these modes is different from that of the low-lying modes, as shown schematically in Fig. 12.1. The latter are built from correlated pairs of nucleons in the valence shell, while the former are built from correlated particle-hole pairs, with one or more particles outside the valence shell. A description of high-lying modes in terms of bosons is also possible, although not particularly useful in itself since only one vibrational state of each mode is observed. It becomes useful only when coupling low-lying and high-lying modes. This coupling leads to the splitting and mixing of the high-lying modes which is often observed.

High-lying collective modes have been introduced in the interacting boson model by Morrison and Weise (1982) and, independently, by Scholtz and Hahne (1983). They proposed a description of the giant dipole resonance via a p boson coupled to a system of interacting s and d bosons and solved the resulting Hamiltonian numerically. Subsequently, Rowe and Iachello (1983) showed that, for deformed nuclei, a class of Hamiltonians exists that correspond to dynamic symmetries and that for such Hamiltonians analytic results can be obtained for energies and transition matrix elements. Since then the model has been applied to several (series of) isotopes (Maino et al., 1984; 1985; Scholtz, 1985; Maino et al., 1986a; Scholtz and Hahne, 1987; Nathan, 1988) and has been extended to include monopole and quadrupole giant reso-nances (Maino et al., 1986b) and dipole resonances in light (Maino et al., 1988) and odd-even nuclei (Maino, 1989).

The interacting boson model has emerged in the last fifteen years as a unified framework for the description of the collective properties of nuclei. The key ingredients of this model are its algebraic structure based on the powerful methods of group theory, the possibility it gives to perform calculations in all nuclei and its direct connection with the shell model that allows one to derive its properties from microscopic interactions.

The interacting boson model deals with nuclei with an even number of protons and neutrons. However, more than half of the nuclear species have an odd number of protons and/or neutrons. In these nuclei there is an interplay between collective (bosonic) and single-particle (fermionic) degrees of freedom. The interacting boson model was extended to cover these situations by introducing the interacting boson-fermion model. This book, which is the second in a series of three, describes the interacting boson-fermion model and its applications. It has two aspects, an algebraic (group-theoretic) aspect and a numerical one. The algebraic aspect describes the coupling of bosons and fermions. The situation here is by far more complex than in the case of eveneven nuclei and, for this reason, it is described in greater detail. The study of coupled Bose-Fermi systems is a novel application of algebraic methods and as such has a wider scope than that presented here. It has been used recently in other fields of physics, as for example in the coupling of electronic and rotation-vibration degrees of freedom in molecules.

Introduction

Every algebraic structure has associated with it geometric structures. The choice of the geometric structure with which it is most convenient to visualize the situation depends on the physics that one wishes to expose. For boson systems of the type discussed in Volume 1 and also for those used in the description of molecules (Iachello and Levine, 1982), there is a very natural geometric structure provided by the coset space U(n)/U(n – 1)⊗ U(1). This leads in the case of nuclei to a description in terms of five variables, αμ (μ = 0, ±1, ±2), which can then be associated with the shape of a liquid drop with quadrupole deformation (Bohr and Mottelson, 1975). Similarly, in molecules, use of the coset space mentioned above leads to a description in terms of three variables, rμ (μ = 0, ±l), which can be associated with the vector distance between the two atoms in the molecule.

For fermionic systems or when bosons and fermions coexist, the introduction of a geometric space is not so obvious. One can, if one wishes, introduce coset spaces, as briefly discussed in the following section, but even with this introduction, the geometric structure of the problem remains as abstract as before. A simpler situation arises if one is interested only in the case of a single fermion coupled to a system of bosons. In this case one can analyze the algebraic structure in terms of the motion of a single particle in a potential well generated by the bosons.

Introduction

In many cases in physics, one has to deal simultaneously with collective and single-particle excitations of the system. The collective excitations are usually bosonic in nature while the single-particle excitations are often fermionic. One is therefore led to consider a system which includes bosons and fermions. In this book we discuss applications of a general algebraic theory of mixed Bose- Fermi systems to atomic nuclei. The collective degrees of freedom here can be described in terms of a system of interacting bosons as discussed in a previous book (Iachello and Arima, 1987), henceforth referred to as Volume 1. The single-particle degrees of freedom represent the motion of individual nucleons in the average nuclear field. They are described in terms of a system of interacting fermions. The coupling of fermions and bosons leads to the interacting boson-fermion model which has been used extensively in recent years to discuss the properties of nuclei with an odd number of nucleons.

The interacting boson-fermion model was introduced by Arima and one of us in 1975 (Arima and Iachello, 1975). It was subsequently expanded by Iachello and Scholten (1979) and cast into a form more readily amenable to calculations. As in the corresponding case of even-mass systems, the algebra of creation and annihilation operators can be realized in several ways. One of these is the Hoistein-Primakoff realization which leads to a slightly different version of the interacting boson-fermion model called the truncated quadrupole phonon-fermion model (Paar, 1980; Paar and Brant, 1981), based on the boson realization introduced by Janssen, Jolos and Donau in 1974 and discussed in Sect. 1.4.6 of Volume 1.