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.

The description of interactions of particles in terms of the standard model is proving remarkably robust. In this scheme, the electroweak interactions between the leptons and quarks are described by the gauge field theory with broken SU(2) × U(1) symmetry with the associated gauge bosons of the photon, W and Z, while the strong interactions of quarks are governed by quantum chromodynamics (QCD) with SU(3) symmetry where the gauge boson is the gluon. QCD provides a theoretical framework for formulating the structure of hadrons, in particular that of the proton, in terms of quarks and gluons. This structure is revealed when the proton is probed by a virtual photon or weak current at high energy. This is known as deep inelastic scattering by leptons off a nucleon target and is the subject of this book.

I am indebted to many people for educating me in the subtleties of theory and experiment. Especially I thank Graham Ross, Frank Close, Alan Martin, James Stirling and Roger Phillips on the theoretical and phenomenological front. I have benefitted from discussions with Erwin Gabathuler, Peter Norton and Terry Sloan who with their colleagues provided much of recent excitement on the experimental front. Correspondence with Jan Kwiecinski and Louis Miramontes was a great help. I am grateful to Greg Moley for preparing the text in TEX. Finally I thank Peter Landshoff for continual encouragement.