Neutron and X-ray Optics

1. Introduction

1.1 Refractive Index for Neutrons and X-rays

1.2 CRLs—Thin-Lens Approximation: Focal Length, Ray Path Lengths, and Attenuation

1.3 CRL Arrays

1.4 Integration on the Complex Plane—Cauchy–Riemann Theorem, Cauchy Integration, and Residues

1.5 Derivation of the Complex Refractive Index of Material Medium (e.g., Lenses) Based on the Rayleigh Scatter of X-rays and Gammas

1.6 Refractive of Gammas via Rayleigh and Delbrück Scatter

1.7 Historical Introduction to Gamma Lenses—The Dirac Equation and the Delbrück Effect

References

2. Neutron Refractive Index in Materials and Fields

2.1 Calculation of General Refractive Decrement for Material or Magnetic Media

2.2 Comparison of the Electron, Neutron, X-ray, and Light Refractive Index

2.3 Neutron Decrement for Composite Materials, and Neutron Refraction Due to Decrement Gradient

2.4 Neutron Decrement and Refractive Index in a Gravitational Field

2.5 Neutron Spin and Magnetic Dipole Moment Vectors in Applied Magnetic Fields

2.6 Potential Energy, Force, and Decrement for Neutrons in Applied Magnetic Fields

2.7 The Bloch Equation and Neutron Precession in an Applied Magnetic Field

2.8 Temperature Effect on Neutron Spin and Magnetic Dipole Moment Orientation in an Applied Magnetic Field

2.9 The Bloch Equation and the Lorentz Force Equation

2.10 Average Spin Polarization of a Neutron in an Applied Magnetic Field

2.11 Equation of Motion of the Expected Value of the Neutron Spin Vector in an Applied Magnetic Field

2.12 Expected Values of Quantum Mechanical Quantities Follow Classical Trajectories

2.13 Average Spin Polarization of a Beam of Neutrons in an Applied Magnetic Field

2.14 Adiabatic and Nonadiabatic Polarization Rotation About Magnetic Field Lines That Change Direction

2.15 Magnetic Resonance

2.16 Ferromagnetic Materials—Domains, Magnetization, Permeability, Susceptibility

2.17 Law of Refraction of Magnetic Field Lines

2.18 Ferromagnetic Materials with Applied Magnetic Fields and the Hysteresis Loop

2.19 Calculation of the Magnetization Vector from Unpaired Atomic Electron Magnetic Dipole Moments

2.20 Calculation of the Tangential Component of the Magnetization Vector from Magnetic Field Boundary Conditions

2.21 Calculation of the Neutron Potential Energy and Magnetic Scatter Length from the Tangential Component of the Magnetization Vector

2.22 Refractive Decrement and Index for a Neutron in a Ferromagnetic Material

References

3. Magnetic Neutron Scatter from Magnetic Materials

3.1 Partial Differential Cross Section for Neutron Scatter in Magnetic Materials

3.2 The Transition Matrix Element for Neutron Magnetic Scatter

3.3 Boltzmann Thermal Distribution of Initial Scatter System States

3.4 Magnetic Fields of Unpaired Atomic Electrons in Magnetic Materials

3.5 Neutron Magnetic Potential Energy due to the Total Electron Magnetic Dipole Moment

3.6 Neutron Magnetic Potential Energy Due to the Electron Spin Magnetic Dipole Moment

3.7 Neutron Magnetic Potential Energy Due to the Electron Orbital Magnetic Dipole Moment

3.8 Evaluation of the Matrix Element for the Neutron Magnetic Potential Energy

3.9 Electron Magnetic Dipole Moment Operator for Unpaired Atomic Electrons

3.10 Magnetic Dipole Moment Operator and Magnetization Vector—the Spin Component

3.11 Magnetic Dipole Moment Operator and Magnetization Vector—the Orbital Component

3.12 Magnetic Dipole Moment Operator Relation with Magnetization Vector

3.13 Evaluation of the Neutron Magnetic Potential Energy Operator

3.14 Evaluation of Transition Matrix Element with Neutron Spin Eigenstates

3.15 Coherent, Elastic Differential Cross Section Expressed by a Magnetization Vector

3.16 Coherent, Elastic Differential Cross Section Expressed by Electron Spin Density

3.17 Magnetization Determined from Measuring Bragg Peak Intensity

References

4. LS Coupling Basis for Magnetic Neutron Scatter

4.1 Summation and Coupling of Atomic Electron Spin and Orbital Angular Momentum

4.2 Spin and Orbital Angular Momentum in Two- and Three-Electron Atoms

4.3 Spin and Orbital Angular Momentum for an N-Electron Atom

4.4 LS Coupling and the Pauli Exclusion Principle

4.5 Eigenfunctions and the Schrödinger Equation for a Two-Electron Atom

4.6 Antisymmetric and Symmetric Eigenfunctions Describe an Identical Electron Pair

4.7 Two-Electron Atom—Symmetric Spatial and Antisymmetric Spin Components

4.8 Two-Electron Atom—Antisymmetric Spatial and Symmetric Spin Components

4.9 N-Electron System Described by an Antisymmetric Total Eigenfunction

4.10 The Physical Basis of LS Coupling of Electron Spin and Orbital Motion

4.11 Derivation of Thomas Precession Factor for LS Coupling

4.12 An Alternative Derivation of the Thomas Precession Factor

4.13 Quenching of an Electron Orbital Momentum in a Crystal

4.14 Paramagnetic and Ferromagnetic Materials

References

5. LS-Coupled, Localized Electron, Magnetic Scatter of Neutrons

5.1 Heitler–London Model for Neutron Scatter by Magnetic Materials

5.2 Evaluation of the Unpaired, Atomic Electron, Magnetic Dipole Moment Transition Matrix Element

5.3 The Magnetic Form Factor

5.4 Partial and Differential Cross Section Expressions for a Quenched Magnetic Crystal

5.5 Evaluation of Partial and Differential Cross Section Expressions for a Quenched Magnetic Crystal—Separation of Unpaired, Atomic Electron Spatial and Spin Components

5.6 Evaluation of the Thermal Average of Initial State, Unpaired, Atomic Electron Positions

5.7 Partial Differential Cross Section for the LS-Coupled, Heitler–London Model in a Quenched Crystal with Unpaired, Localized Atomic Electron Spin

5.8 Partial Differential Cross Section for an LS-Coupled, Heitler–London Model in a Quenched Crystal with Unpaired, Localized Atomic Electron Spin and Orbital Current

5.9 Coherent, Elastic Differential Cross Section for an LS-Coupled, Heitler–London Model in a Quenched Crystal with Unpaired, Localized Atomic Electron Spin and Orbital Current

5.10 Expression of Partial Differential Cross Section by Intermediate Correlation Function

References

6. Magnetic Scatter of Neutrons in Paramagnetic Materials

6.1 General Expression for Coherent, Elastic Differential Cross Section for Paramagnetic Material in an Applied Magnetic Field

6.2 Coherent, Elastic Differential Cross Section for Paramagnetic Material in an Applied Magnetic Field Expressed by the Total Spin Quantum Number for a Paramagnetic Atom

6.3 Coherent, Differential Cross Section for Elastic Neutron Scatter in Paramagnetic Material—With an Applied Magnetic Field at Low and High Temperatures

6.4 Coherent, Differential Cross Section for Elastic Neutron Scatter in Paramagnetic Material—No Applied Magnetic Field

6.5 Coherent, Elastic Differential Cross Section for Scatter of Neutron Spin States from Localized Electrons in Paramagnetic Materials—No Applied Magnetic Field

References

7. Neutron Scatter in Ferromagnetic, Antiferromagnetic, and Helical Magnetic Materials

7.1 Coherent, Elastic Differential Cross Section for Magnetic Neutron Scatter in Ferromagnetic Materials—Localized Unpaired Electrons

7.2 Antiferromagnetic Materials—Coherent, Elastic Differential Cross Section for Neutron Scatter from Localized Unpaired Electrons

7.3 Coherent, Elastic Differential Cross Section for Magnetic and Nuclear Scatter of Neutron Spin States in a Bravais-Lattice Ferromagnetic Crystal—Localized Unpaired Electrons

7.4 Coherent, Elastic Differential Cross Section for Magnetic and Nuclear Scatter of Neutron Spin States in a Non-Bravais-Lattice Ferromagnetic Crystal—Localized Unpaired Electrons

7.5 Production and Measurement of Polarized Neutrons by Ferromagnetic Materials

7.6 General Expression for the Coherent Differential Cross Section for Nuclear and Magnetic Elastic Scatter of Neutron Spin States in Ferromagnetic Materials—Localized or Delocalized Unpaired Electrons

7.7 Polarized Neutrons by Grazing Incidence Reflection via Nuclear Scatter of Neutron Spin States in Ferromagnetic Materials

7.8 Coherent, Elastic Differential Cross Section for Scatter of Neutron Spin States from Magnetic Materials with Helical-Oriented, Localized, Unpaired Electron Spins

References

8. Coherent, Inelastic, Magnetic Neutron Scatter, Spin Waves, and Magnons

8.1 Electron Spin, Magnetic Dipole Moment, and Precession in Applied Magnetic Field

8.2 No Magnetic Field—Unpaired Electron Spins Tend to Align in the Same Direction

8.3 Heisenberg Model of Unpaired Electron Spin in Magnetic Materials

8.4 Physical Basis of Exchange Integral in the Heisenberg Model

8.5 Expression of the Heisenberg Hamiltonian by Spin Operators

8.6 Ferromagnetic Materials—Spin Waves, Dispersion Relation, and Magnons

8.7 Antiferromagnetic Materials—Spin Wave Dispersion Relation

8.8 Exchange and Anisotropy Energy and Domain Formation in Magnetic Materials

8.9 Hamiltonian Eigenequation for 1-D Ferromagnetic Spin Lattice

8.10 Spin and Spin Deviation Operators, Creation and Annihilation Operators, Holstein–Primakoff Transformations, and Linear Approximation of Heisenberg Hamiltonian

8.11 Application of the Bloch Theorem to Express Creation and Annihilation Operators

8.12 The Heisenberg Hamiltonian Expressed as a Sum of Harmonic Oscillators

8.13 Coherent, Inelastic Partial Differential Cross Section for One-Magnon Absorption or Emission for Neutron Scatter in a Ferromagnetic Crystal

8.14 Coherent Inelastic Neutron Scatter—One-Magnon Exchange in Ferromagnetic Material

8.15 Integral Expression for Temperature Dependence of Spin and Magnetization in a Ferromagnetic Crystal Based on the Planck Distribution

8.16 Evaluation of Low-Temperature Spin and Magnetization of the Integral Expression for a Cubic Ferromagnetic Crystal— Dependence

8.17 Magnon Population Low-Temperature Dependence in a Ferromagnetic Cubic Crystal

References

9. Coherent, Elastic Scatter of Neutrons by Atomic Electric Field

9.1 Spin–Orbit Electric Field Scatter of Neutrons

9.2 Foldy Electric Field Scatter of Neutrons

9.3 Scatter of Neutrons by Nuclear and Electric Field Interactions in a Non-Bravais Lattice Crystal

References

10. Diffractive X-ray and Neutron Optics

10.1 Derivation of Helmholtz–Kirchhoff Integral Theorem

10.2 Derivation of the Kirchhoff–Fresnel Diffraction Equation

10.3 The Obliquity Factor in the Kirchhoff–Fresnel Diffraction Equation

10.4 The Paraxial Approximation Applied to the Kirchhoff–Fresnel Diffraction Equation

10.5 Fraunhofer Diffraction of X-rays or Neutrons from a Rectangular Aperture

10.6 Fraunhofer Diffraction of X-ray or Neutron Line Source by a Parallel Single Slit

10.7 Fraunhofer Diffraction of X-ray or Neutron Line Source by a Parallel Slit Pair

10.8 Fraunhofer Diffraction of an X-ray or a Neutron Line Source from N Parallel Slits

10.9 Fraunhofer Diffraction from Gratings Is Archetype for Coherent, Elastic Scatter of X-ray or Neutrons from Material Lattices

10.10 Abbe Theory of Imaging Applied to X-rays or Neutrons

10.11 Fraunhofer Diffraction of X-rays or Neutrons from a Circular Aperture

10.12 Huygens–Fresnel Approach: The Kirchhoff Equation for a Compound Refractive Lens with X-rays or Neutrons

10.13 Compound Refractive Fresnel Lens for X-rays and Neutrons

10.14 Fresnel Diffraction of X-rays or Neutrons from a Circular Aperture

10.15 Fresnel Diffraction of X-rays or Neutrons from a Rectangular Aperture

10.16 Fresnel Diffraction of X-rays or Neutrons from a Knife Edge

10.17 Fresnel Zone Plates (FZP) for X-rays or Neutrons

10.18 X-ray or Neutron Achromat Fabricated from FZPs and CRLs

10.19 The Helmholtz Differential Equation for X-rays and Neutrons

10.20 Derivation of the Helmholtz Paraxial Equation for a Gaussian, Spherical X-ray Laser Beam

10.21 Solution of the Helmholtz Paraxial Equation for a Gaussian, Spherical X-ray Laser Beam

References

11. Kirchhoff Equation Solution for CRL, Pinhole, and Phase Contrast Imaging

11.1 Kirchhoff Equation with a 1-D Biconcave, Parabolic, or Spherical CRL and Thin-Sample Approximation for X-rays or Neutrons with Gravity

11.2 Derivation of Kirchhoff Equation with a 2-D Biconcave Parabolic or Spherical CRL for X-rays or Neutrons with Gravity—Thick Sample

11.3 Biconcave, Parabolic CRL—Image Amplitude Distribution for Incoherent X-rays or Neutrons with Gravity

11.4 Point Spread Function (PSF) of the Biconcave, Parabolic CRL for Incoherent X-rays or Neutrons with Gravity

11.5 PSF of a Pinhole for Incoherent X-rays or Neutrons with Gravity

11.6 The Modulation Transfer Function

11.7 The Modulation Transfer Function (MTF) with a Biconcave, Parabolic CRL for Incoherent X-rays or Neutrons with Gravity

11.8 Field of View (FOV) with a Biconcave, Parabolic CRL for Incoherent X-rays or Neutrons with Gravity

11.9 Image Intensity Distribution of a Biconcave, Parabolic CRL for Coherent X-rays or Neutrons with Gravity

11.10 Biconcave, Parabolic CRL Image Intensity Distribution for Incoherent X-rays or Neutrons with Gravity—Coherent Amplitude Cross Terms Set to Zero

11.11 Without CRL—Phase Contrast Imaging for Incoherent X-rays or Neutrons with Gravity

11.12 Special Case of a Merged Object Plane and Source Plane

11.13 Stationary Phase Approach Applied to an Image Amplitude Integral of a Spherical, Biconcave CRL for Incoherent X-rays or Neutrons with Gravity

References

12. Electromagnetic Fields of Moving Charges, Electric and Magnetic Dipoles

12.1 Maxwell’s Equations and the Lienard–Wiechert Potentials

12.2 The Lorentz Gauge and the Helmholtz Theorem

12.3 Calculation of Scalar and Vector Potentials of an Oscillating Electric Dipole

12.4 Calculation of a Magnetic Field of an Oscillating Electric Dipole via the Vector Potential

12.5 Calculation of the Electric Field Emitted from an Oscillating Electric Dipole

12.6 Conversion of Electric and Magnetic Fields from MKS Units to CGS Units

12.7 Near- and Far-Zone Electric Fields and Power Emission from an Oscillating Electric Dipole

12.8 Frequency and Wave-Number Domain Expressions for Electric and Magnetic Fields Emission and Power from the Oscillating Electric Dipole

12.9 Binomial Expansion of the Vector Potential of a Moving Charge Includes Contributions from Electric and Magnetic Dipole and Electric Quadrupole Moments

12.10 Transformation of the Fields and Vector Potential of the Electric Dipole to the Magnetic Dipole

12.11 Electric and Magnetic Fields and Power Emission from an Oscillating Magnetic Dipole

12.12 Derivation of the Electric Field of a Charge in Arbitrary Motion from Lienard–Wiechert Potentials

12.13 Derivation of the Magnetic Field of a Charged Particle in Arbitrary Motion from Lienard–Wiechert Potentials

12.14 Poynting Vector and Electromagnetic Energy Radiated from an Arbitrary Accelerated Charge per Solid Angle per Frequency Interval

12.15 Calculation of the Electron Trajectory in Synchrotron Ring Insertion Devices

12.16 Simplified Velocity-Dependent Expression for the per Solid Angle Frequency Spectrum Emitted by Accelerated Charged Particles

12.17 Bremsstrahlung and Electromagnetic Wave Polarization

12.18 Net Neutron Magnetic Dipole Moment Produced in an Ultracold Neutron Population with Applied Magnetic Field, and Possibilities for Population Inversion

12.19 Energy Radiated per Solid Angle for a Moving Magnetic Dipole Moment of Neutrons and Charged Particles

12.20 Relation of a Magnetization Vector of a Moving Charge with a Polarization Vector of a Stationary Charge

References

13. Special Relativity, Electrodynamics, Least Action, and Hamiltonians

13.1 Special Relativity—Minkowski Space and Invariant Space–Time Distance

13.2 Special Relativity—Length Contraction

13.3 Special Relativity—Time Dilation

13.4 Special Relativity—Lorentz Transformation of Space–Time Position

13.5 Special Relativity—Lorentz Transformation with Position and Velocity Four-Vectors

13.6 Special Relativity—Four-Vector and Lorentz Transformation Representation in Minkowski Space

13.7 Special Relativity—Velocity, Mass, Momentum, and Energy

13.8 Radiated Power from an Accelerated Charge Moving at Nonrelativistic Velocity via the Nonrelativistic Larmor Formula and Lorentz Transformation at an Instant of Time

13.9 Special Relativity—Electromagnetic Four-Vectors

13.10 Special Relativity—Electromagnetic Fields and Potentials as Four-Vectors

13.11 Special Relativity—The Electromagnetic Field Tensor

13.12 The Maxwell Stress Tensor

13.13 Force Density and the Maxwell Stress Tensor

13.14 The Principle of Least Action and the Lagrangian Yields Euler–Lagrange Equations

13.15 Derivation of the Hamiltonian from the Lagrangian

13.16 Derivation of the Lagrangian for Relativistic Charged Particle in an Electromagnetic Field

13.17 Euler–Lagrange Equation for Langrangian Density for Real Scalar Fields

References

14. The Klein–Gordon and Dirac Equations

14.1 Relativistic Correct Schrödinger Wave Equation—the Klein–Gordon Equation

14.2 The Dirac Wave Equation for Spin 1/2 Particles—Overview

14.3 Derivation of the Dirac Equation that Predicts Correct LS Coupling Term for Unpaired Atomic Electrons in Magnetic Neutron Scatter

14.4 Solution of the Dirac Equation for a Free Particle

14.5 A Useful Vector Property with the Pauli Spin Matrix

14.6 Dirac Equation for Bound Electron–Proton Interaction—The Basis of LS Coupling

14.7 Second Term of the Dirac Equation for Bound Electron–Proton Interaction that Includes an LS Coupling Term

14.8 Third Term of the Dirac Equation for Bound Electron–Proton Interaction

14.9 Magnetic Field of a Proton Current in an Electron Rest Frame Expressed by the Proton Magnetic Dipole Moment Vector and Electron Orbit Radius

14.10 Evaluated Dirac Equation for Bound Electron–Proton Interaction that Predicts LS Coupling and Hyperfine Interactions

14.11 Equation of Motion for Electron Spin, Orbital, and Total Angular Momentum Vectors

14.12 Derivation of the Dirac Equation for the Hydrogen Atom—Step 1: Evaluation of in the Dirac Hamiltonian

14.13 Derivation of the Dirac Equation for the Hydrogen Atom—Step 2: Evaluation of in a Dirac Hamiltonian Term

14.14 Derivation of the Dirac Equation for a Hydrogen Atom—Step 3: Introduction of the Squared Angular Momentum Operator in a Dirac Hamiltonian

14.15 Derivation of the Dirac Equation for a Hydrogen Atom—Step 4: Obtain a Pair of Coupled First-Order Differential Equations from the Developed Dirac Hamiltonian

14.16 Asymptotic Solution—A Coupled, Dirac Eigenequation Pair for the Hydrogen Atom

14.17 Regular Solution of the Coupled, Dirac Eigenequation Pair and Electron Energy Formula for the Hydrogen Atom

14.18 Quantum Number Relationships in the Dirac Electron Energy Formula for a Hydrogen Atom

14.19 The Nuclear Electric Quadrupole Potential

References

15. Neutron and X-ray Optics in General Relativity and Cosmology

15.1 Special and General Relativity—History and Relation to Neutron and X-ray Optics

15.2 Equivalence Principle, Manifolds, Parallel Vector Translation, and Covariant Derivatives

15.3 Surfaces and Gauss’s Remarkable Theorem, Gaussian Curvature, and Metric Coefficients

15.4 Curvilinear Coordinate Systems

15.5 Metric Tensor and Invariant Distance in Coordinate System Transformations

15.6 Metric Tensor and Invariant Area and Volume in Coordinate System Transformations

15.7 Curvilinear Coordinate Systems and the Jacobian Determinant

15.8 Metric Coefficients, Jacobian Determinant, and Cartesian-to-Curvilinear Coordinate System Transformation

15.9 Variation of Physical Quantity by Displacement—Scalar and Vector Fields

15.10 Invariance, Summation Convention, and Contravariant Vectors

15.11 Natural Basis Vectors are Tangent to Surfaces in a Contravariant Vector Space

15.12 Geometric View of Contravariant Representation of a Vector

15.13 Geometric View of Covariant (Dual) Representation of a Vector

15.14 Covariant Vectors Transformation Is Similar to the Gradient of a Scalar Function

15.15 Dual Basis Vectors are Normal to Surfaces in the Covariant Vector Space

15.16 Covariant and Contravariant Tensors, Pseudo and Polar Scalars, Vectors, and Tensors

15.17 Natural and Dual Basis Vectors are Reciprocals

15.18 Natural and Dual Basis Vectors and the Metric Tensor

15.19 Differentials in Curvilinear Coordinates, Curved Surfaces, and Parallel Transport

15.20 Absolute and Covariant Derivatives of Vectors with Contravariant Representation

15.21 Absolute and Covariant Derivatives of Vectors and Tensors with Covariant Representation

15.22 Geodesic Curves and Connection Coefficients Expressed by a Metric Tensor

15.23 Gradient of a Vector in Curvilinear Coordinates

15.24 Cylindrical Coordinate System—Covariant and Contravariant Metric Coefficients and Connection Coefficients

15.25 Spherical Coordinate System—Covariant and Contravariant Metric Coefficients and Connection Coefficients

15.26 Newton’s Laws and Maxwell’s Equations in General Relativity

15.27 The Curvature, Ricci, and Einstein Tensors and the Curvature or Ricci Scalar

15.28 The Stress Tensor, Universe Fluid Model, and the Equation of Continuity and Motion

15.29 Einstein Field Equations

15.30 Hilbert Derived Einstein Field Equations from the Principle of Least Action

15.31 Gravity Waves Derived from Linearized Einstein Field Equations for the Weak Gravity Condition

15.32 Schwarzschild Solution to Einstein Field Equations and Experimental Predictions

15.33 The Friedmann Equation and Cosmological Issues

References

16. Radiation Imaging Systems and Performance

16.1 Liouville’s Theorem, the Vlasov Equation, and the Continuity Equation

16.2 Fraction of Source Intensity Intercepted and Focused by Lens

16.3 Depth of Field of a Lens

16.4 Radiation Dose Rate and Radiation Shielding

16.5 Imaging System Modulation Transfer Function (MTF)

16.6 Rose Model, Contrast, Detective Quantum Efficiency (DQE), and Noise Power Spectrum (NPS)

16.7 Derivation of DQE from NPS, Contrast, Gain, and MTF

16.8 Criteria for Sample Feature Imaging

16.9 Imaging System Performance—Receiver Operating Characteristic (ROC), Positive and Negative Likelihood, and Positive and Negative Predictive Value Curves

16.10 Reliable Detection of a Signal in the Presence of Background and Noise

References

17. Neutron and Charged Particle Magnetic Optics

17.1 Charged Particle Motion in Axial Symmetric Magnetic Field—Busch’s Theorem

17.2 Paraxial Ray Equation and Focusing of Charged Particles by an Axial Symmetric Magnetic Field

17.3 Paraxial Ray Equation for Azimuthally Symmetric Magnetic and Electric Fields and Busch’s Theorem Derived from the Least-Action Principle

17.4 Multipole Magnetic Fields and Lenses for Charged Particles

17.5 FODO Quadrupole Magnetic Lens Pair for Charged Particles

17.6 Neutron Trajectory in Magnetic Fields and Magnetic Field Gradient Focusing of Neutrons

17.7 Neutron Compound Cylindrical Magnetic Lens

17.8 Calculation of Magnetic Fields of Planar Magnets by Paired Magnetic Charge Sheets

17.9 1-D Magnetic Gravitational Trap for Ultracold Neutrons

17.10 Electron Trajectory in Wigglers and Undulator X-ray Sources

17.11 Derivation of Emitted X-ray Wavelength, Bandwidth, and Cone Angle for the Undulator

17.12 X-ray Power and Differential Power Emission per Solid Angle from an Undulator

References