Contents

Introduction

1.1 Nuclear Structure Physics

1.2 The Basic Equation

1.3 Microscopic versus Collective Models

1.4 The Role of Symmetries

Symmetries

2.1 General Remarks

2.2 Translation

2.2.1 The Operator for Translation

2.2.2 Translational Invariance

2.2.3 Many-Particle Systems

2.3 Rotation

2.3.1 The Angular Momentum Operators

2.3.2 Representations of the Rotation Group

2.3.3 The Rotation Matrices

2.3.4 SU(2) and Spin

2.3.5 Coupling of Angular Momenta

2.3.6 Intrinsic Angular Momentum

2.3.7 Tensor Operators

2.3.8 The Wigner-Eckart Theorem

2.3.9 6j and 9j Symbols

2.4 Isospin

2.5 Parity

2.5.1 Definition

2.5.2 Vector Fields

2.6 Time Reversal

Second Quantization

3.1 General Formalism

3.1.1 Motivation

3.1.2 Second Quantization for Bosons

3.1.3 Second Quantization for Fermions

3.2 Representation of Operators

3.2.1 One-Particle Operators

3.2.2 Two-Particle Operators

3.3 Evaluation of Matrix Elements for Fermions

3.4 The Particle-Hole Picture

Group Theory in Nuclear Physics

4.1 Lie Groups and Lie Algebras

4.2 Group Chains

4.3 Lie Algebras in Second Quantization

Electromagnetic Moments and Transitions

5.1 Introduction

5.2 The Quantized Electromagnetic Field

5.3 Radiation Fields of Good Angular Momentum

5.3.1 Solutions of the Scalar Helmholtz Equation

5.3.2 Solutions of the Vector Helmholtz Equation

5.3.3 Properties of the Multipole Fields

5.3.4 Multipole Expansion of Plane Waves

5.4 Coupling of Radiation and Matter

5.4.1 Basic Matrix Elements

5.4.2 Multipole Expansion of the Matrix Elements

and Selection Rules

5.4.3 Siegert's Theorem

5.4.4 Matrix Elements for Emission

in the Long-Wavelength Limit

5.4.5 Relative Importance of Transitions

and Weisskopf Estimates

5.4.6 Electric Multipole Moments

5.4.7 Effective Charges

Collective Models

6.1 Nuclear Matter

6.1.1 Mass Formulas

6.1.2 The Fermi-Gas Model

6.1.3 Density-Functional Models

6.2 Nuclear Surface Deformations

6.2.1 General Parametrization

6.2.2 Types of Multipole Deformations

6.2.3 Quadrupole Deformations

6.2.4 Symmetries in Collective Space

6.3 Surface Vibrations

6.3.1 Vibrations of a Classical Liquid Drop

6.3.2 The Harmonic Quadrupole Oscillator

6.3.3 The Collective Angular-Momentum Operator

6.3.4 The Collective Quadrupole Operator

6.3.5 The Quadrupole Vibrational Spectrum

6.4 Rotating Nuclei

6.4.1 The Rigid Rotor

6.4.2 The Symmetric Rotor

6.4.3 The Asymmetric Rotor

6.5 The Rotation-Vibration Model

6.5.1 Classical Energy

6.5.2 Quantal Hamiltonian

6.5.3 Spectrum and Eigenfunctions

6.5.4 Moments and Transition Probabilities

6.6 -Unstable Nuclei

6.7 More General Collective Models for Surface Vibrations

6.7.1 The Generalized Collective Model

6.7.2 Proton-Neutron Vibrations

6.7.3 Higher Multipoles

6.8 The Interacting Boson Model

6.8.1 Introduction

6.8.2 The Hamiltonian

6.8.3 Group Chains

6.8.4 The Casimir Operators

6.8.5 The Dynamical Symmetries

6.8.6 Transition Operators

6.8.7 Extended Versions of the IBA

6.8.8 Comparison to the Geometric Model

6.9 Giant Resonances

6.9.1 Introduction

6.9.2 The Goldhaber-Teller Model

6.9.3 The Steinwedel-Jensen Model

6.9.4 Applications

Microscopic Models

7.1 The Nucleon-Nucleon Interaction

7.1.1 General Properties

7.1.2 Functional Form

7.1.3 Interactions from Nucleon-Nucleon Scattering

7.1.4 Effective Interactions

7.2 The Hartree-Fock Approximation

7.2.1 Introduction

7.2.2 The Variational Principle

7.2.3 The Slater-Determinant Approximation

7.2.4 The Hartree-Fock Equations

7.2.5 Applications

7.2.6 The Density Matrix Formulation

7.2.7 Constrained Hartree-Fock

7.2.8 Alternative Formulations and Three-Body Forces

7.2.9 Hartree-Fock with Skyrme Forces

7.3 Phenomenological Single-Particle Models

7.3.1 The Spherical-Shell Model

7.3.2 The Deformed-Shell Model

7.4 The Relativistic Mean-Field Model

7.4.1 Introduction

7.4.2 Formulation of the Model

7.4.3 Applications

7.5 Pairing

7.5.1 Motivation

7.5.2 The Seniority Model

7.5.3 The Quasispin Model

7.5.4 The BCS Model

7.5.5 The Bogolyubov Transformation

7.5.6 Generalized Density Matrices

Interplay of Collective and Single-Particle Motion

8.1 The Core-plus-Particle Models

8.1.1 Basic Considerations

8.1.2 The Weak-Coupling Limit

8.1.3 The Strong-Coupling Approximation

8.1.4 The Interacting Boson-Fermion Model

8.2 Collective Vibrations in Microscopic Models

8.2.1 The Tamm-Dancoff Approximation

8.2.2 The Random-Phase Approximation (RPA)

8.2.3 Time-Dependent Hartree-Fock and Linear Response

Large-Amplitude Collective Motion

9.1 Introduction

9.2 The Macroscopic-Microscopic Method

9.2.1 The Liquid-Drop Model

9.2.2 The Shell-Correction Method

9.2.3 Two-Center Shell Models

9.2.4 Fission in Self-Consistent Models

9.3 Mass Parameters and the Cranking Model

9.3.1 Overview

9.3.2 The Irrotational-Flow Model

9.3.3 The Cranking Formula

9.3.4 Applications of the Cranking Formula

9.4 Time-Dependent Hartree-Fock

9.5 The Generator-Coordinate Method

9.6 High-Spin States

9.6.1 Overview

9.6.2 The Cranked Nilsson Model

Appendix: Some Formulas from Angular-Momentum Theory

References

Subject Index

Contents of Examples and Exercises

2.1 Cartesian Form of the Angular-Momentum Operator

2.2 Cayley-Klein Representation of the Rotation Matrix

2.3 Coupling of Two Vectors to Good Angular Momentum

2.4 The Position Operator as an Irreducible Spherical Tensor

2.5 Commutation Relations of the Position Operator

2.6 The Two-Nucleon System

2.7 The Time-Reversal Operator for Spinors with Spin 1/2

3.1 Two-Body Operators in Second Quantization

4.1 The Lie Algebra of Angular-Momentum Operators

4.2 The Casimir Operator of the Angular-MomentumAlgebra

4.3 The Lie Algebra of SO(n)

5.1 The Vector Spherical Harmonics

5.2 The Weisskopf Estimates for Electric Transitions

6.1 Volume and Center-of-Mass Vector for a Deformed Nucleus

6.2 Quadrupole Deformations

6.3 Angular-Momentum Operator for Quadrupole Phonons

in Second Quantization

6.4 114Cd as aSpherical Vibrator

6.5 Angular-Momentum Operators in the Intrinsic System

6.6 States with Angular Momentum 2

in the Asymmetric Rotor Model

6.7 The Time Derivatives of the

6.8 Transformation of the Quadmpole Operator

6.9 Quadrupole Moments and Transition Probabilities

6.10 238U in the Rotation-Vibration Model

6.11 Casimir Operators of U(N)

6.12 Contributions to the Thomas-Reiche-Kuhn Sum Rule

7.1 The Angular Average of the Tensor Force

7.2 Matrix Elements in the Variational Equation

7.3 The Skyrme Energy Functional

7.4 The Eigenfunctions of the Harmonic Oscillator

7.5 The Quadrupole Moment of a Nucleus

7.6 Single-Particle Energies in the Deformed Oscillator

7.7 The Crossing of Energy Levels

7.8 Pairing in aj = 7/2 Shell

The Spectrum of 183W in the Strong-Coupling Model

Tamm-Dancoff Calculation for 16O

Derivation of the Tamm-Dancoff Equation

The Extended Schematic Model

The Cranking Formula for the BCS Model

The Harmonic Oscillator in the Generator-Coordinate Method

1.1 Nuclear Structure Physics

The nuclear models discussed in this book belong to the realm of nuclear structure

theory. In present usage, nuclear stmcture physics is devoted to the study of the

properties of nuclei at low excitation energies, where individual energy levels can

be resolved. This means that typically quantum effects are predominant and the

states of the nucleus have a very complicated structure that depends on the intricate

interrelations of all the many nucleons involved.

In contrast, at higher energies and especially for heavy-ion reactions, quantum

mechanics becomes less important and the preeminent place is instead given to

methods of statistical mechanics. Theories then typically employ bulk properties

of nuclear matter such as the equation of state or the dissipation coefficients, or

are even based on purely classical many-body physics like the cascade models.

Of course it is impossible to give an exact energy boundary between these

types of theories. The theories presented here, however, are typically employed for

excitation energies up to 2-3 MeV. Usually only the lowest few energy levels can

be described well by a theoretical model, and the number of levels increases so

rapidly above that energy range that it becomes impossible to make any sensible

comparison with experiment (for nuclei with an odd number of neutrons or protons

or both this is even more dramatic - most nuclear models prefer even-even nuclei

with their relatively simple spectra). Also one should remember that in experimental

spectra only a relatively small number of states can be identified as to spin and

parity, and that to really test a model transitions, i.e, essentially overlaps between

the wave functions, are needed, which again are often not known even for the most

interesting states.

It is thus not surprising that the models presented in this book usually explain a

relatively small number of low-lying states and to a modest accuracy, and even this

is a considerable achievement. To esteem that, remember that we are dealing with

a system of particles whose number is neither small enough to allow direct solution

nor large enough to make statistical methods highly accurate, and which interact

through an interaction that has still not been pinned down to any definite form.

It is this extraordinary difficulty and the freedom with which methods and ideas

from many other branches are applied here that make nuclear structure physics so

fascinating and so much alive.

1.2 The Basic Equation

To find the proper theoretical starting point some more ballpark estimates of the

relevent physical quantitities have to be introduced. Let us first recall a few numbers

from elementary experimental nuclear physics.

The elements known at the time of this writing have nuclei consisting of (at

present) Z ==1,...,111 protons and N = 0,..., 161 neutrons, giving a total

number of A nucleons. The radii of nuclei follow the empirical law

R(A) = roA1/3 (1.1)

with ro =1.2fm. Nuclear radii thus range up to about 7.5 fm. The formula also

implies that the nuclear volume is proportional to the number of particles in the

nucleus, indicating the near incompressibility of nuclear matter (the true density

profile observed by electron scattering is a bit more complicated). The least-bound

nucleon has a binding energy of the order of 8 MeV and a kinetic energy close to

40 MeV.

This information is already sufficient to form some rough ideas about what is

essential in the theories. Since a nucleon has a mass of 938 MeV, the kinetic

energy is quite negligible by comparison, so that a nonrelativistic approach appears

quite sufficient, and this assumption is made in the vast majority ofnuclear structure

models. More recently, however, relativistic approaches have become important -

this theme is taken up in Sect.7.4 in connection with the relativistic mean-field

model, and we also explain there why relativistic effects can be important in spite

of the simple estimate given above.

The velocity of a nucleon with a kinetic energy of T = 40 MeV is given by

v=^^= (1.2)

and the associated de Broglie wavelength by

Here the useful constant hc= w 197.32 MeVfm was used. The result shows that

quantum effects are certainly not negligible, as A is by no means small compared

to the nuclear radii. This is even more pronounced for the more tightly bound

nucleons, which have a smaller kinetic energy.

Taking these considerations into account, the starting point for a theory of

nuclear eigenstates should be a stationary Schrodinger equation very generally

given by

The rest of this book is about what to write for and which degrees of freedom

to use in the wave functions.