# Direct nuclear Reactions

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Direct Nuclear Reactions deals with the theory of direct nuclear reactions, their microscopic aspects, and their effect on the motions of the individual nucleons. The principal results of the theory are described, with emphasis on the approximations involved to understand how well the theory can be expected to hold under specific experimental conditions. Applications to the analysis of experiments are also considered. This book consists of 19 chapters and begins by explaining the difference between direct and compound nuclear reactions. The reader is then introduced to the theory of plane waves, some results of scattering theory, and the phenomenological optical potential. The following chapters focus on form factors and their nuclear structure content; the basis of the optical potential as an effective interaction; reactions such as inelastic single- and two-nucleon transfer reactions; the effect of nuclear correlations; and the role of multiple-step reactions. The theory of inelastic scattering and the relationship between the effective and free interactions are also discussed, along with reactions between heavy ions and the polarizability of nuclear wave functions during a heavy-ion reaction. This monograph will be of interest to nuclear physicists.

## Table of Contents

Preface

Acknowledgments

Notational Conventions

1. Introduction: Direct and Compound Nuclear Reactions

A. The Observables

B. Direct and Compound Nuclear Reactions

C. Competition between Direct and Compound Nuclear Reactions

D. Historical Note

2. The Plane-Wave Theory

Notes to Chapter 2

3. Scattering Theory and General Results

A. Motivation

B. The Nuclear Shell Model

C. Reaction Channels or Partitions

D. Integral Equations and the Scattering Amplitude

E. Asymptotic Form of the Complete Wave Function

F. The First Born Approximation

G. Cross Section

Notes to Chapter 3

4. The Phenomenological Optical Potential

A. Rationale for the Optical Potential

B. Partial Wave Expansion, the Radial Wave Function, Its Asymptotic Behavior

C. Elastic Scattering Amplitude

D. Coulomb and Nuclear Potentials

E. Parametrization of the Optical Potential

F. Elastic Scattering of Alpha Particles

G. Spin-Orbit Interaction and Nucleon Elastic Scattering

H. Elastic Scattering of Heavy Ions

I. The Imaginary Potential and Mean Free Path

J. Systematics of the Parameters

K. Non-locality of the Optical Potential

L. Convergence of the Partial Wave Sum

Notes to Chapter 4

5. Distorted-Wave Born Approximation

A. Introduction

B. Distorted-Wave Green's Functions

C. The Gell-Mann-Goldberger Transformation

D. Two-Potential Formula

E. The DWBA Transition Amplitude

F. Discussion of the Approximations

G. Anti-symmetrization

H. Multi-pole Expansion of the Transition Amplitude

Notes to Chapter 5

6. Operator Formalism

A. Introduction

B. Lippmann-Schwinger Equation

C. Formal Solution

D. Transition Amplitude

E. Transition Operator T

F. Gell-Mann-Goldberger Transformation

G. Distorted-Wave Green's Function

H. Distorted-Wave Born Approximation

I. Second Distorted-Born Approximation

J. Multiple-Scattering Series of Watson

K. Green's Theorem and the Hermitian H

Notes to Chapter 6

7. Calculation of the DWBA Amplitude

A. Introduction

B. The (d,p) Stripping Reaction

C. Zero-Range Approximation

D. Examples

E. Improvements within the Framework of the DWBA

F. Spectroscopic Factors for One-Nucleon Transfer

G. Pairing Theory in Nuclear Structure

H. Inelastic Excitation of Surface Vibrations

I. Inelastic Excitation of Nuclear Rotational Levels

J. Inelastic Excitation of Single-Nucleon States

K. Charge-Exchange Reactions

L. Distorted-Wave Impulse Approximation

M. Coulomb Excitation

Notes to Chapter 7

8. Coupled Equations and the Effective Interaction

A. Coupled Equations for Inelastic Scattering

B. Truncation and Effective Interactions

C. The Effective Interaction

D. Non-locality

E. Multiple-Scattering Series for the Effective Interaction

F. Partial-Wave Expansion

G. Boundary Conditions

H. Distorted-Wave Born Approximation

I. Cross Sections

9. Microscopic Theory of Inelastic Nucleon Scattering from Nuclei

A. The Richness of Nuclear Structure

B. Discussion of the Interactions in the Coupled Equations

C. Matrix Elements of the Effective Interaction

D. Selection Rules

E. Nuclear Form Factor

F. Single-Particle Form Factors

G. Nuclear Structure Amplitudes

H. Shell-Model Configurations

I. Particle Creation and Destruction Operators

J. Particle-Hole Configurations

K. Quasi-Particle Configurations

L. Recapitulation

M. The Direct Interaction and Its Spin Dependence

N. Selection Rules and the Direct Interaction

O. Application of the Theory

Notes to Chapter 9

10. Core Polarization

11. Effective Interactions and the Free Nucleon-Nucleon Force

A. Introduction

B. The Free-Nucleon-Nucleon Interaction

C. The Brueckner G Matrix

D. Bare G Matrix and Core Polarization

E. Effective Interaction in Scattering

F. Low-Energy Domain

G. High-Energy Domain (E>100 MeV)

H. Microscopic Calculation of the Optical Potential

12. Further Developments in the Theory of Inelastic Scattering

A. Introduction

B. General Form of the Two-Body Amplitude

C. Local Coordinate-Space Representation of G or t

D. Energy and Momentum Dependence of the Effective Interaction in the High-Energy Domain

E. Effective Operator for the Exchange Contribution

F. Multi-pole Expansion of the Interaction

G. Transition Densities

H. Cross Sections

I. Applications

13. Scattering from Deformed Rotational Nuclei

A. Wave Functions of Deformed Nuclei

B. Multi-pole Expansion of the Interaction

C. Coupled Equations

D. Relationship between Optical Potential of Spherical and Deformed Nuclei

E. Alpha Scattering from Deformed Rare-Earth Nuclei

Notes to Chapter 13

14. Calculation of Specific Components of the Optical Potential

A. Introduction

B. Potential Components of a Single Level

C. Trivially Equivalent Local Potential

D. Long-Range Absorption Due to Coulomb Excitation

15. Two-Nucleon Transfer Reactions

A. Contrast between One- and Two-Nucleon Transfer Reactions

B. Transfer from a Light-Ion Projectile

C. Spin-Isospin Selection Rules

D. Form Factor

E. Interpretation of the Form Factor

F. Transition Amplitude and Cross Section

G. Correlations

H. Parentage Amplitude for Two Nucleons

16. Finite-Range Interaction in Transfer Reactions

A. Introduction

B. The DWBA Amplitude and Its Coordinate Dependences

C. Parentage Expansions

D. The (t,p) Reaction

E. Two-Nucleon Transfer between Heavy Ions

17. Higher-Order Processes in Particle Transfer Reactions

A. Beyond the DWBA

B. Coupled-Channel Born Approximation (Generalized DWBA)

C. Source-Term Method

D. Derivation of the CCBA

E. Coupled Reaction Channels (CRC)

F. Partial-Wave Expansions and Derivation of the Cross Section

G. Two-Nucleon Transfer between Vibrational Nuclei

H. Two-Nucleon Transfer between Rotational Nuclei

I. Indirect Transitions in Analog Reactions

18. Heavy-Ion Reactions

A. Special Features

B. The Potential

C. Deflection Function, Classical Conditions, Plunging Orbits, Grazing Peak

D. Elastic and Inelastic Scattering and the Optical Potential

E. Indirect Transitions in Two-Nucleon Transfer, Opposite Behavior for Stripping and Pickup

F. Forward-Angle Ripples, Uncertainty Principle

G. Structure of the S Matrix, Speculations on Future Developments

H. Summary

Notes to Chapter 18

19. Polarizability of Nuclear Wave Functions in Heavy-Ion Reactions

A. Introduction

B. The Experimental Facts

C. A Possible Explanation

D. Adiabatic Estimate

E. Dynamical Description of Polarizability

Appendix: Some Useful Reminders

A. Laplace Operator

B. Spherical Harmonics

C. Angular-Momentum Coupling; Clebsch-Gordan Coefficients

D. Angular-Momentum Re-coupling

E. Spherical Tensors

F. Reduced Matrix Elements—Wigner-Eckart Theorem

G. Scalar Product of Two Commuting Tensors

H. Vector Product of Two Commuting Tensors

I. Vector and Vector Product

J. Reduced Matrix Element of Spin Operator

K. Rotation Functions

L. Oscillator Functions

M. Spherical Bessel Functions

Notes to Appendix

References

Index

## Product details

- No. of pages: 396
- Language: English
- Copyright: © Academic Press 1983
- Published: December 1, 1983
- Imprint: Academic Press
- eBook ISBN: 9780323152372