# Advances in Imaging and Electron Physics, Volume 173

## 1st Edition

### Part B

**Serial Volume Editors:**Jay Theodore Cremer, Jr.

**eBook ISBN:**9780123972767

**Hardcover ISBN:**9780123969699

**Imprint:**Academic Press

**Published Date:**27th July 2012

**Page Count:**540

**View all volumes in this series:**Advances in Imaging and Electron Physics

## Table of Contents

Editor-in-Chief

Edited by

Foreword

Preface

Contributor

Future Contributions

Chapter 1 Fast and Thermal Neutrons for Radiography and Isotope Detection

1 Inelastic and Elastic Fast Neutron Scatter

2 Moderation of Fast Neutrons to Thermal Neutrons

3 Determination of Neutron Moderator Thickness

4 Inelastic Fast Neutron Scatter—Determination of Scattered Target Nucleus Energy

5 Center of Mass to Lab Frame Transformation of a Differential Cross Section

6 Reduction of Gamma Noise in Moderated Fast Neutron Sources of Thermal Neutrons

7 The DD Fusion Reaction and Generator—DD Neutron Yield

8 DD Fusion Neutron Source—Threshold Deuteron Projectile Energy and Neutron Angular Energy Distribution

9 Nuclear-Resonant Fluorescence (NRF) Detection of Isotopes of Interest in Gamma or Fast-Neutron-Interrogated Targets

10 Nuclear-Resonant Detection of Excited State Gammas Emitted from Targets Interrogated by Fast Neutron Inelastic Scatter

11 Criteria for Nuclear Resonant Absorption of Target Isotope-Emitted Gammas

12 Resonant Detection of Target-Emitted Gammas Depends on Angular Position of Resonant Detector

13 Probability of Nuclear Resonant Detection of Target-Emitted Isotope Gammas Verses Detector Angular Position

14 Range of Angles for Nuclear Resonant Detection of Target-Emitted Gammas

Chapter 2 Neutron Scatter Amplitude, Cross Section, Scatter Length, and Refractive Index

1 Wave and Particle Aspects of Neutrons

2 Coherent and Incoherent, Elastic and Inelastic Neutron Scatter

3 The Eikonal Equation for Neutron Optics

4 The Eikonal Equation for a Stationary Wave Function Yields a Refractive Index

5 Scatter Cross Section Expressed by Neutron Scatter Amplitude

6 The Absorption Cross Section Expressed by the Neutron Probability Density and Probability Current Density via the Continuity Equation

7 The Absorption Cross Section Expressed by the Total Neutron Wave Amplitude and Scatter Cross Section by the Scattered Neutron Wave Amplitude

8 Overview—Neutron Wave Amplitude and Cross Sections for Nuclear Scatter

9 The Schrödinger Equation for Neutron Scatter by a Spherical Nuclear Potential

10 The General Solution of the Schrödinger Equation for Nuclear Scatter of Neutrons

11 Boundary Conditions Applied to the Neutron Total Wave Amplitude Solution

12 Angular Momentum of Partial Wave Components of the Asymptotic Solution of the Total Neutron Wave Amplitude

13 Neutron Scatter from the Nucleus—The Total Neutron Wave Amplitude Solution

14 Spherical Wave Representation of the Incident Neutron Plane Wave

15 The Alternative, Total Wave Amplitude Solution of Assumed Form Uses Scatter Amplitude to Describe the Neutron Scatter from Nucleus

16 Neutron Scatter Amplitude Obtained by Equating the Schrödinger Wave Equation Solution and the Alternative, Assumed Solution for the Total Neutron Wave Amplitude

17 The Neutron Scatter Cross Section and Total (Net) Neutron Wave Amplitude Solution Obtained from the Neutron Scatter Amplitude

18 The Neutron Absorption Cross Section Obtained from the Neutron Total Wave Amplitude

19 Total Neutron Cross Section and Optical Theorem

20 Total Neutron Wave Amplitude, Cross Sections, and Complex Phase-Shift Angle

21 Relation of Neutron Scatter Length to the Phase-Shift Angle and to the Spherical Nuclear Potential and Radius

22 Neutron Scatter Length and the Phase-Shift Angle, Scatter Amplitude, and Cross Sections

23 Resonant Neutron Scatter and Absorption—The Breit-Wigner Neutron Cross Section

24 Relation of the Refractive Decrement and Phase-Shift Angle

25 Neutron Scatter Time Corresponding the Scatter Length and Phase-Shift Angle

26 Incident Total Scattered Waves are the Sum of Diverging and Converging Waves via Scatter Coefficient—Alternative Derivations of Cross Sections

27 Bound and Free Neutron Scatter Lengths

28 Neutron Scatter Length Accounting for Neutron Spin and Nuclear Spin

Chapter 3 Neutron Scatter Physics and Differential Cross Sections

1 Inverse Relation Between Nuclear Excited State Linewidth and Lifetime

2 Derivation of the Breit-Wigner Equation for Neutron Scatter and Absorption

3 The Dirac Delta Function

4 Basis and State Wave Functions—Operators and Eigenequations

5 Commutation of Operators and the Heisenberg Uncertainty Principle

6 The Hamiltonian Operator, the Schrödinger Equation, and Hermitian Operators

7 Properties of the Hermitian Operator and Eigenvalue as Measured Physical Quantities

8 Hermitian Operators and the Orthogonality of Basis Eigenfunctions

9 Examples of Hermitian Operators—Position and Momentum Operators

10 Commutating Operators Share the Same Eignenfunctions

11 Dirac Notation and Eigenequations

12 Hilbert Space Basis of Quantum Mechanics and Neutron Scatter

13 Hilbert Space—Unitary Transformation between Differing Basis Representations of a State Wave Function

14 Hilbert Space—Unitary-Similarity Transformation Between Operators of Differing Basis Representations of a State Wave Function

15 Evolution of State Functions in Time via the Unitary Operator

16 Ehrenfest’s Principle—Trajectories of Physically Measurable Expectation Values

17 The Schrödinger and Heisenberg Pictures of Quantum Mechanics

18 Obtaining the Heisenberg Operator from the Schrödinger Operator

19 Equation of Motion of the Expectation Value of the Schrödinger Operator

20 Equation of Motion of the Expectation Value of the Heisenberg Operator

21 The Interaction Picture—Time-Dependent Perturbation

22 The Interaction Picture—The Neutron Wave Function Equation of Motion in Scatter

23 The Interaction Picture—Operator Equation of Motion

24 The Interaction Picture—General Solution of Schrödinger Wave Equation for Neutron Scatter

25 Schrödinger Wave Equation Solution—The Lippmann-Schwinger Equation in Dirac Form

26 Dirac Form of the Lippmann-Schwinger Equation Derived from the Interaction Picture

27 Integral Lippmann-Schwinger Equation Derived from the Green’s Function Solution of the Schrödinger Equation

28 The Born Approximation Applied to the Lippmann-Schwinger Integral Solution of the Schrödinger Wave Equation for Neutron Scatter

29 The Born Approximation, Fermi Pseudopotential, and Neutron Scatter Length

30 Derivation of Fermi’s Golden Rule

31 Introduction—Differential and Partial Differential Neutron Scatter Cross Sections

32 Application of Fermi’s Golden Rule to Neutron Scatter

33 Neutron Scatter System in a Box

34 General Neutron Differential Cross Section for Single Transitions

35 Fermi Pseudopotential for Nuclear Scatter and Neutron Scatter Length

36 Differential Cross Section for Single Transitions—Evaluation of the Transition Integral

37 General Neutron Partial Differential Cross Section for Single Transitions

38 The Squared Magnitude of Neutron Scatter Transition Amplitude—Correlation of Scatter of the Neutron Plane Wave at Different Times by Different Nuclei or the Same Nucleus

39 Boltzmann Distribution of Nuclei—The Temperature Effect Initial State Distribution

40 The Density Operator

41 General Neutron Partial Differential Cross Section for All Transitions

42 Application of the Schrödinger and the Heisenberg Pictures to General Neutron Partial Differential Cross Section for All Transitions

43 General Neutron Partial Differential Cross Section—Coherent and Incoherent Scatter

Chapter 4 Kinematic Neutron Scatter in Crystals without Lattice Vibrations

1 Introduction: Neutron Scatter from Crystals—The Kinematical Theory

2 The Simple or Bravais Crystal Lattice—The Unit Cell and Direct Lattice Vectors

3 The Non-Bravais Crystal Lattice—Two or More Atoms per Unit Cell

4 The Reciprocal Lattice and Reciprocal Lattice Vectors

5 The Diffraction Condition

6 The Bragg and Laue Scattering Condition for Diffraction and their Equivalence

7 The Ewald Sphere to Visualize Diffraction in the Reciprocal Lattice Space, First Brillouin Zone, and Wigner-Seitz Cell

8 Elastic Differential Cross Section for Neutron Scatter from a Bravais Lattice Crystal

9 Diffraction Patterns and Images

Chapter 5 Thermal-based Oscillations of Nuclei in a Crystal Lattice

1 Schrödinger Wave Solution of a Crystal Lattice of Simple Harmonic Oscillators

2 The Hamiltonian Operator for a Crystal Lattice of Simple Harmonic Oscillators

3 The Hamiltonian Operator for a Crystal Lattice Represented by Ladder Operators

4 The Average Displacement Amplitude of the Vibrating Nuclei of the Crystal Lattice

5 Derivation of the Equivalent Exponential and Kronecker Delta Function Expressions for the Crystal Lattice Vibration Wave Amplitude

6 1D Crystal Lattice Vibration Modes

7 3D Crystal Lattice Vibration Modes

8 The Debye Model to Characterize Thermal Vibrations in the Crystal Lattice

Chapter 6 Kinematic Neutron Scatter in Crystals with Lattice Nuclei Vibration

1 Non-Bravais Crystal with Nuclei Vibrations—Partial Differential Cross Section

2 The Nucleus Displacement Vector in a Non-Bravais Crystal with Nuclei Vibrations

3 Kinematic, General, Elastic and Inelastic, and Coherent and Incoherent Partial Differential Cross Section for Non-Bravais Crystals

4 Kinematic, Coherent and Incoherent Partial Differential Cross Section for a Non-Bravais Crystal with the Debye-Waller Factor

5 The Cubic Bravais Lattice as Archetype Crystal—The Debye-Waller Factor, Debye Temperature, and Nuclei Displacement

6 Hooke’s Law, the Stress Tensor, and Elastic Waves in Cubic Crystals

7 Neutron-based Residual Stress Analysis

8 Kinematic Neutron Coherent and Incoherent Inelastic Scatter—Phonon Exchange

9 Kinematic Elastic, Coherent and Incoherent, Differential Cross Sections

10 Kinematic Inelastic, Coherent Partial Differential Cross Section—One-Phonon Neutron Scatter

11 Coherent, Inelastic One-Phonon Neutron Scatter in the Lab Frame is Coherent, Elastic Scatter in the Rest Frame of Vibrating Crystal Nuclei

12 Conditions for Coherent, Inelastic One-Phonon Neutron Scatter

13 Kinematic Incoherent, Inelastic, Partial Differential Cross Section—One-Phonon Neutron Scatter

Chapter 7 Dynamic Neutron Scatter

1 The Coupled Ewald Equations

2 Extinction in Crystals

3 Broadening of Neutron Diffraction Peaks

4 Characterization of Lattice Nuclei Vibrations as a Stationary Process

5 Average Nuclear Scatter Potential for Dynamic Neutron Scatter in a Crystal

6 Average Nuclear Scatter Potential for Crystal Lattice: Average of Neutron Scatter Lengths and Boltzmann Thermal-Based Distribution of Initial Nuclear States

7 Time-Independent Schrödinger Equation for Dynamic Neutron Scatter in a Perfect Crystal

8 No-Bragg Condition in Time-Independent Schrödinger Equation Solution for a Perfect Crystal Yields Standard Refractive Index without Regard to Material Structure

9 Near-Bragg (Laue Case) of Solution of Time-Independent Schrödinger Equation For Dynamic Neutron Scatter in a Perfect Crystal

10 Two Amplitude Summation Pairs of Transmitted and Bragg-reflected (Laue Case) Waves in a Perfect Crystal Cause Anomalous Absorption at the Bragg Condition

11 Anomalous Absorption in a Perfect Crystal

12 Neutron Wave Pair in a Perfect Crystal: Transmitted Neutron Fluxes

13 Double-Valued Wave Numbers and Refractive Indices for a Perfect Crystal Wave Pair Created by an Incident Wave that Meets the Bragg Condition (Laue Case)

14 Wave Amplitude and Intensity for the Wave Pair in the Perfect Crystal at the Bragg Condition for the Laue Case

15 Interference of Bragg-reflected Waves in a Perfect Crystal: Pendellösung Fringes and Primary Extinction

16 Derivation of the Neutron Transport Equation for Neutron scatter in Mosaic Crystals

17 Expression of the Neutron Transport Equation by Macroscopic Neutron Scatter and Absorption Cross Sections per Unit Volume in a Mosaic Crystal

18 Neutron Transport Equation Solved to Yield Generalized Darwin Equations

19 Single Bragg Reflection for Generalized Darwin Equations Yields Ordinary Darwin Equation for Neutron Scatter in a Mosaic Slab Crystal

20 Solution of the Darwin Equations for the One-Beam (Off-Bragg) and Two-Beam (Bragg) Cases for a Planar Mosaic Crystal

21 Partial Differential Cross Section for Averaged Nuclear Potential in a Crystal

22 Coherent and Incoherent Partial Differential Cross Section for Averaged Nuclear Potential in a Crystal

23 Coherent and Incoherent Elastic Cross Sections and Differential Cross Sections for Averaged Nuclear Potential in a Crystal

Contents of Volumes 151–172

Index

## Description

This special volume of *Advances in Imaging and Electron Physics *details the current theory, experiments, and applications of neutron and x-ray optics and microscopy for an international readership across varying backgrounds and disciplines. Edited by Dr. Ted Cremer, these volumes attempt to provide rapid assimilation of the presented topics that include neutron and x-ray scatter, refraction, diffraction, and reflection and their potential application.

## Key Features

- Contributions from leading authorities
- Informs and updates on all the latest developments in the field

## Readership

Physicists, electrical engineers and applied mathematicians in all branches of image processing and microscopy as well as electron physics in general

## Details

- No. of pages:
- 540

- Language:
- English

- Copyright:
- © Academic Press 2012

- Published:
- 27th July 2012

- Imprint:
- Academic Press

- eBook ISBN:
- 9780123972767

- Hardcover ISBN:
- 9780123969699

## About the Serial Volume Editors

### Jay Theodore Cremer, Jr. Serial Volume Editor

### Affiliations and Expertise

Chief Scientist, Adelphi Technology, Inc.