R.J. Eden

Publisher Summary

This chapter discusses the main features of different nuclear models and to considers how far each model can be regarded as representing a particular aspect of an integrated picture of the nucleus. The nuclear models throw light on the qualitative behavior of a nucleus under various experimental conditions. They help in understanding the theoretical basis for nuclear structure. The chapter describes nuclear shell model in its simplest form of independent particle motion. This form is applicable to certain ground-state properties of most nuclei, and provides the basis of the shell characteristics of nuclei. The principal refinements of the shell model are presented in the chapter. These include: (1) the consideration of residual two-body interactions, which leads to a description of the properties of low-lying excited states of nuclei near closed shells, (2) the spheroidal shell model, which extends the model to describe nuclear quadrupole moments, and (3) the use of momentum-dependent single-particle potentials, which permits the model to provide a true representation of total nuclear energy.

CONTENTS

1 INTRODUCTION

THE objects of this survey are firstly to describe the main features of different nuclear models and secondly to consider how far each model can be regarded as representing a particular aspect of an integrated picture of the nucleus. The choice of nuclear models to be discussed has been made partly on the basis of their success in classifying a range of experimental results, partly on whether they throw light on the qualitative behaviour of a nucleus under various experimental conditions, and partly on their contribution to understanding the theoretical basis for nuclear structure and for nuclear models.

In describing a nuclear model the following topics will be considered: (1) the assumptions on which the model is based and the resulting physical picture of the model, (2) the experimental evidence in favour of the model and the experimental domain in which the model is useful, (3) refinements of the model to extend its usefulness and its essential limitations which cannot be remedied by refinements, (4) the implications of the model for nuclear structure or nuclear behaviour, (5) the theoretical basis for the model and its relation to a detailed theory of the nucleus. These considerations cannot altogether be kept separate in view of the relations between different models, and some will be postponed to Section 10 of the survey which is concerned with the nuclear many-body problem.

The nuclear shell model is described in Section 2 in its simplest form of independent particle motion. This form is applicable to certain ground-state properties of most nuclei, and provides the basis of the shell characteristics of nuclei. In Section 3 the principal refinements of the shell model are described. These include (i) the consideration of residual two-body interactions which leads to a description of the properties of low-lying excited states of nuclei near closed shells, (ii) the spheroidal shell model which extends the model to describe nuclear quadrupole moments, (iii) the use of momentum-dependent single-particle potentials which permits the model to give a true representation of total nuclear energy.

The first method of investigating collective motion in the nucleus was based on the liquid-drop model. It is possible that this model gives a useful representation of nuclear fission, but it is now known that it gives an incorrect picture of the low-energy spectra of nuclei. The most important consequence of low-energy collective motion is the rotational spectrum and the strong E2 transitions of distorted nuclei. These are described by the rotational model of the nucleus which is considered in Section 4. This contains many of the characteristics of the liquid drop model but the picture is different. The motion can be pictured as a rotating distorted-shell model but not as oscillations of a liquid drop. The moment of inertia estimated from the spheroidal shell model with energy-dependent potential (and possibly also residual interactions) appears to be of the right magnitude to fit observed rotational spectra. Section 5 contains some concluding remarks on models for nuclear structure at low energies.

The compound-nucleus model for nuclear reactions is described in Sections 6 and 7. Section 6 is concerned with the generalizations of the theory which lead to a framework for describing nuclear reactions rather than to detailed predictions. Section 7 describes the extra assumptions which lead to the statistical theory of nuclear reactions.

The optical model in which a target nucleus is represented by a complex potential is described in Section 8. Particular attention is given to the relation of this model to the compound nucleus theory, and it is seen that the detailed mechanism by which a colliding nucleon interacts with a target nucleus to form a compound nucleus will be of importance in most energy ranges. The need for special models for very-high-energy nuclear reactions (greater than 80 MeV) is discussed in Section 9. This section also describes the deuteron model which provides a method of estimating certain types of correlation in the nucleus.

Finally in Section 10 a brief account is given of the theory of the nucleus when it is considered as a system of many nucleons interacting through two-body forces. It is shown how this theory leads to a model similar to the improved shell model with momentum-dependent potentials. The relation of this model to the actual nuclear wave function is not described in detail, but it is noted that the many-body theory provides a general basis for models for nuclear structure and nuclear reactions and provides an integrated picture of nuclear behaviour.

A complete list of references would be disproportionate to an article of this length; the references are therefore limited to a somewhat arbitrary selection of typical or important papers.

2 THE SHELL MODEL

In its simplest form the shell model assumes independent particle motion by nucleons in the nucleus subject only to the requirements of the exclusion principle which must be satisfied by both neutrons and protons. The corresponding wave function is a determinant

(2.1)

The 1, 2, … A denote the co-ordinates, spin, and isotopic spin of the corresponding particle.

The basic assumptions of the present form of the shell model were given by Mrs. M. G. MAYER (1948, 1949) and by HAXEL, JENSEN, and SUESS (1948, 1950). A general account of the model is given by MAYER and JENSEN (1955) who also list detailed references. The assumptions are:

The potential which determines the single-particle wave functions ϕi in (2.1) has the form

(2.2)

where VA(r) and fA(r) depend only on the radial distance r and the size of the nucleus, and (l. s) denotes the coupling of the nucleon spin s and the orbital angular momentum l. There is also a difference between the potentials for protons and for neutrons which is attributed to Coulomb effects. The magnitude of the potential Vs.p. is not a significant feature of the shell model in its present form for reasons to be discussed below. Its order of magnitude will be about 40 MeV and the ratio of VA(r) to fA(r) is about 10 to 1, but both may vary for different shells. The sign of the spin-orbit term is such that the level having angular...