1 Introduction ................................................. 1
1.1 Refractive Index for Neutrons and X-rays ............... 13
1.2 CRLs - Thin-Lens Approximation: Focal Length, Ray
Path Lengths, and Attenuation .......................... 23
1.3 CRL Arrays ............................................. 47
1.3.1 One-to-One Imaging .............................. 49
1.3.2 Magnified Imaging ............................... 51
1.4 Integration on the Complex Plane-Cauchy-Riemann
Theorem, Cauchy Integration, and Residues .............. 54
1.5 Derivation of the Complex Refractive Index of
Material Medium (e.g., Lenses) Based on the Rayleigh
Scatter of X-rays and Gammas ........................... 67
1.5.1 The Electromagnetic Wave Equation in a Vacuum
or Dielectric Medium ............................ 67
1.5.2 Electromagnetic Field Produced by an
Accelerated Charge .............................. 80
1.5.3 Acceleration of a Bound Atomic Electron by an
Imposed Electromagnetic Field ................... 96
1.5.4 Extraction of the Complex Refractive Index
from the Electromagnetic Wave Equation ......... 103
1.5.5 Scatter, Absorption, Total Cross Section for
Electromagnetic Waves (X-rays) ................. 114
1.5.6 Derivation of the Optical Theorem .............. 122
1.5.7 Derivation of the Kramers - Kronig Relation
and Calculation of the Refractive Decrement
from the Measured Attenuation Cross Section .... 127
1.6 Refractive of Gammas via Rayleigh and Delbriick
Scatter ............................................... 141
1.7 Historical Introduction to Gamma Lenses-The Dirac
Equation and the Delbriick Effect ..................... 143
1.7.1 Refractive Index and Attenuation Cross
Section for the Delbriick Refraction of
Gammas ......................................... 144
1.7.2 Gamma Refractive Optics-Experimental Results ... 150
References ................................................. 155
2 Neutron Refractive Index in Materials and Fields ........... 161
2.1 Calculation of General Refractive Decrement for
Material or Magnetic Media ............................ 161
2.2 Comparison of the Electron, Neutron, X-ray, and Light
Refractive Index ...................................... 163
2.3 Neutron Decrement for Composite Materials, and
Neutron Refraction Due to Decrement Gradient .......... 168
2.4 Neutron Decrement and Refractive Index in
a Gravitational Field ................................. 169
2.5 Neutron Spin and Magnetic Dipole Moment Vectors in
Applied Magnetic Fields ............................... 171
2.6 Potential Energy, Force, and Decrement for Neutrons
in Applied Magnetic Fields ............................ 175
2.7 The Bloch Equation and Neutron Precession in an
Applied Magnetic Field ................................ 179
2.8 Temperature Effect on Neutron Spin and Magnetic
Dipole Moment Orientation in an Applied Magnetic
Field ................................................. 181
2.9 The Bloch Equation and the Lorentz Force Equation ..... 182
2.10 Average Spin Polarization of a Neutron in an Applied
Magnetic Field ........................................ 184
2.11 Equation of Motion of the Expected Value of the
Neutron Spin Vector in an Applied Magnetic Field ...... 186
2.12 Expected Values of Quantum Mechanical Quantities
Follow Classical Trajectories ......................... 190
2.13 Average Spin Polarization of a Beam of Neutrons in
an Applied Magnetic Field ............................. 196
2.14 Adiabatic and Nonadiabatic Polarization Rotation
About Magnetic Field Lines That Change Direction ...... 197
2.15 Magnetic Resonance .................................... 199
2.16 Ferromagnetic Materials-Domains, Magnetization,
Permeability, Susceptibility .......................... 205
2.17 Law of Refraction of Magnetic Field Lines ............. 207
2.18 Ferromagnetic Materials with Applied Magnetic Fields
and the Hysteresis Loop ............................... 210
2.19 Calculation of the Magnetization Vector from
Unpaired Atomic Electron Magnetic Dipole Moments ...... 211
2.20 Calculation of the Tangential Component of the
Magnetization Vector from Magnetic Field Boundary
Conditions ............................................ 214
2.21 Calculation of the Neutron Potential Energy and
Magnetic Scatter Length from the Tangential
Component of the Magnetization Vector ................. 215
2.22 Refractive Decrement and Index for a Neutron in a
Ferromagnetic Material ................................ 217
References ................................................. 220
3 Magnetic Neutron Scatter from Magnetic Materials ........... 221
3.1 Partial Differential Cross Section for Neutron
Scatter in Magnetic Materials ......................... 221
3.2 The Transition Matrix Element for Neutron Magnetic
Scatter ............................................... 225
3.3 Boltzmann Thermal Distribution of Initial Scatter
System States ......................................... 226
3.4 Magnetic Fields of Unpaired Atomic Electrons in
Magnetic Materials .................................... 229
3.5 Neutron Magnetic Potential Energy due to the Total
Electron Magnetic Dipole Moment ....................... 231
3.6 Neutron Magnetic Potential Energy Due to the
Electron Spin Magnetic Dipole Moment .................. 233
3.7 Neutron Magnetic Potential Energy Due to the
Electron Orbital Magnetic Dipole Moment ............... 236
3.8 Evaluation of the Matrix Element for the Neutron
Magnetic Potential Energy ............................. 237
3.9 Electron Magnetic Dipole Moment Operator for
Unpaired Atomic Electrons ............................. 240
3.10 Magnetic Dipole Moment Operator and Magnetization
Vector- the Spin Component ............................ 243
3.11 Magnetic Dipole Moment Operator and Magnetization
Vector- the Orbital Component ......................... 245
3.12 Magnetic Dipole Moment Operator Relation with
Magnetization Vector .................................. 251
3.13 Evaluation of the Neutron Magnetic Potential Energy
Operator .............................................. 253
3.14 Evaluation of Transition Matrix Element with Neutron
Spin Eigenstates ...................................... 256
3.15 Coherent, Elastic Differential Cross Section
Expressed by a Magnetization Vector ................... 258
3.16 Coherent, Elastic Differential Cross Section
Expressed by Electron Spin Density .................... 260
3.17 Magnetization Determined from Measuring Bragg Peak
Intensity ............................................. 261
References ................................................. 267
4 LS Coupling Basis for Magnetic Neutron Scatter ............. 269
4.1 Summation and Coupling of Atomic Electron Spin and
Orbital Angular Momentum .............................. 269
4.2 Spin and Orbital Angular Momentum in Two- and Three-
Electron Atoms ........................................ 271
4.3 Spin and Orbital Angular Momentum for an JV-Electron
Atom .................................................. 273
4.4 LS Coupling and the Pauli Exclusion Principle ......... 275
4.5 Eigenfunctions and the Schrodinger Equation for a
Two-Electron Atom ..................................... 276
4.6 Antisymmetric and Symmetric Eigenfunctions Describe
an Identical Electron Pair ............................ 278
4.7 Two-Electron Atom-Symmetric Spatial and
Antisymmetric Spin Components ......................... 280
4.8 Two-Electron Atom - Antisymmetric Spatial and
Symmetric Spin Components ............................. 280
4.9 N-Electron System Described by an Antisymmetric
Total Eigenfunction ................................... 282
4.10 The Physical Basis of LS Coupling of Electron Spin
and Orbital Motion .................................... 283
4.11 Derivation of Thomas Precession Factor for LS
Coupling .............................................. 287
4.12 An Alternative Derivation of the Thomas Precession
Factor ................................................ 292
4.13 Quenching of an Electron Orbital Momentum in a
Crystal ............................................... 294
4.14 Paramagnetic and Ferromagnetic Materials .............. 296
References ................................................. 298
5 LS-Coupled, Localized Electron, Magnetic Scatter of
Neutrons ................................................... 299
5.1 Heitler-London Model for Neutron Scatter by Magnetic
Materials ............................................. 299
5.2 Evaluation of the Unpaired, Atomic Electron,
Magnetic Dipole Moment Transition Matrix Element
(Qs(k,0)) ............................................. 305
5.3 The Magnetic Form Factor .............................. 307
5.4 Partial and Differential Cross Section Expressions
for a Quenched Magnetic Crystal ....................... 308
5.5 Evaluation of Partial and Differential Cross Section
Expressions for a Quenched Magnetic Crystal-
Separation of Unpaired, Atomic Electron Spatial and
Spin Components ....................................... 311
5.6 Evaluation of the Thermal Average of Initial State,
Unpaired, Atomic Electron Positions ................... 314
5.7 Partial Differential Cross Section for the
LS-Coupled, Heitler - London Model in a Quenched
Crystal with Unpaired, Localized Atomic Electron
Spin .................................................. 317
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 .............................. 321
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 .............................. 325
5.10 Expression of Partial Differential Cross Section by
Intermediate Correlation Function ..................... 327
References ................................................. 331
6 Magnetic Scatter of Neutrons in Paramagnetic Materials ..... 333
6.1 General Expression for Coherent, Elastic
Differential Cross Section for Paramagnetic Material
in an Applied Magnetic Field .......................... 333
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 ..................................... 337
6.3 Coherent, Differential Cross Section for Elastic
Neutron Scatter in Paramagnetic Material-With an
Applied Magnetic Field at Low and High Temperatures ... 340
6.4 Coherent, Differential Cross Section for Elastic
Neutron Scatter in Paramagnetic Material-No Applied
Magnetic Field ........................................ 342
6.5 Coherent, Elastic Differential Cross Section for
Scatter of Neutron Spin States from Localized
Electrons in Paramagnetic Materials- No Applied
Magnetic Field ........................................ 343
References ................................................. 351
7 Neutron Scatter in Ferromagnetic, Antiferromagnetic,
and Helical Magnetic Materials ............................. 353
7.1 Coherent, Elastic Differential Cross Section for
Magnetic Neutron Scatter in Ferromagnetic Materials-
Localized Unpaired Electrons .......................... 353
7.2 Antiferromagnetic Materials - Coherent, Elastic
Differential Cross Section for Neutron Scatter from
Localized Unpaired Electrons .......................... 358
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 .......................... 362
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 .......................... 365
7.5 Production and Measurement of Polarized Neutrons by
Ferromagnetic Materials ............................... 372
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 ............................................. 375
7.7 Polarized Neutrons by Grazing Incidence Reflection
via Nuclear Scatter of Neutron Spin States in
Ferromagnetic Materials ............................... 377
7.8 Coherent, Elastic Differential Cross Section for
Scatter of Neutron Spin States from Magnetic
Materials with Helical-Oriented, Localized, Unpaired
Electron Spins ........................................ 379
References ................................................. 391
8 Coherent, Inelastic, Magnetic Neutron Scatter, Spin Waves,
and Magnons ................................................ 393
8.1 Electron Spin, Magnetic Dipole Moment, and
Precession in Applied Magnetic Field .................. 393
8.2 No Magnetic Field-Unpaired Electron Spins Tend to
Align in the Same Direction ........................... 396
8.3 Heisenberg Model of Unpaired Electron Spin in
Magnetic Materials .................................... 397
8.4 Physical Basis of Exchange Integral in the
Heisenberg Model ...................................... 399
8.5 Expression of the Heisenberg Hamiltonian by Spin
Operators ............................................. 404
8.6 Ferromagnetic Materials-Spin Waves, Dispersion
Relation, and Magnons ................................. 407
8.7 Antiferromagnetic Materials-Spin Wave Dispersion
Relation .............................................. 411
8.8 Exchange and Anisotropy Energy and Domain Formation
in Magnetic Materials ................................. 414
8.9 Hamiltonian Eigenequation for 1-D Ferromagnetic Spin
Lattice ............................................... 419
8.10 Spin and Spin Deviation Operators, Creation and
Annihilation Operators, Holstein-Primakoff
Transformations, and Linear Approximation of
Heisenberg Hamiltonian ................................ 423
8.11 Application of the Bloch Theorem to Express Creation
and Annihilation Operators ............................ 426
8.12 The Heisenberg Hamiltonian Expressed as a Sum of
Harmonic Oscillators .................................. 429
8.13 Coherent, Inelastic Partial Differential Cross
Section for One-Magnon Absorption or Emission for
Neutron Scatter in a Ferromagnetic Crystal ............ 432
8.14 Coherent Inelastic Neutron Scatter - One-Magnon
Exchange in Ferromagnetic Material .................... 442
8.15 Integral Expression for Temperature Dependence of
Spin and Magnetization in a Ferromagnetic Crystal
Based on the Planck Distribution ...................... 445
8.16 Evaluation of Low-Temperature Spin and Magnetization
of the Integral Expression for a Cubic Ferromagnetic
Crystal- T3/2 Dependence .............................. 449
8.17 Magnon Population Low-Temperature Dependence in a
Ferromagnetic Cubic Crystal ........................... 451
References ................................................. 454
9 Coherent, Elastic Scatter of Neutrons by Atomic Electric
Field ...................................................... 455
9.1 Spin-Orbit Electric Field Scatter of Neutrons ......... 455
9.2 Foldy Electric Field Scatter of Neutrons .............. 461
9.3 Scatter of Neutrons by Nuclear and Electric Field
Interactions in a Non-Bravais Lattice Crystal ......... 463
Reference .................................................. 472
10 Diffractive X-ray and Neutron Optics ....................... 473
10.1 Derivation of Helmholtz - Kirchhoff Integral
Theorem ............................................. 473
10.2 Derivation of the Kirchhoff - Fresnel Diffraction
Equation .............................................. 476
10.3 The Obliquity Factor in the Kirchhoff-Fresnel
Diffraction Equation .................................. 479
10.4 The Paraxial Approximation Applied to the Kirchhoff-
Fresnel Diffraction Equation .......................... 480
10.5 Fraunhofer Diffraction of X-rays or Neutrons from a
Rectangular Aperture .................................. 487
10.6 Fraunhofer Diffraction of X-ray or Neutron Line
Source by a Parallel Single Slit ...................... 488
10.7 Fraunhofer Diffraction of X-ray or Neutron Line
Source by a Parallel Slit Pair ........................ 491
10.8 Fraunhofer Diffraction of an X-ray or a Neutron Line
Source from N Parallel Slits .......................... 493
10.9 Fraunhofer Diffraction from Gratings Is Archetype
for Coherent, Elastic Scatter of X-ray or Neutrons
from Material Lattices ................................ 499
10.10 Abbe Theory of Imaging Applied to X-rays or
Neutrons ............................................. 504
10.11 Fraunhofer Diffraction of X-rays or Neutrons from
a Circular Aperture .................................. 506
10.12 Huygens-Fresnel Approach: The Kirchhoff Equation
for a Compound Refractive Lens with X-rays or
Neutrons ............................................. 509
10.13 Compound Refractive Fresnel Lens for X-rays and
Neutrons ............................................. 515
10.14 Fresnel Diffraction of X-rays or Neutrons from
a Circular Aperture .................................. 525
10.15 Fresnel Diffraction of X-rays or Neutrons from
a Rectangular Aperture ............................... 528
10.16 Fresnel Diffraction of X-rays or Neutrons from
a Knife Edge ......................................... 532
10.17 Fresnel Zone Plates (FZP) for X-rays or Neutrons ..... 533
10.18 X-ray or Neutron Achromat Fabricated from FZPs and
CRLs ................................................. 535
10.19 The Helmholtz Differential Equation for X-rays and
Neutrons ............................................. 538
10.20 Derivation of the Helmholtz Paraxial Equation for a
Gaussian, Spherical X-ray Laser Beam ................. 540
10.21 Solution of the Helmholtz Paraxial Equation for a
Gaussian, Spherical X-ray Laser Beam ................. 543
References ................................................. 554
11 Kirchhoff Equation Solution for CRL, Pinhole, and Phase
Contrast Imaging ........................................... 555
11.1 Kirchhoff Equation with a 1-D Biconcave, Parabolic,
or Spherical CRL and Thin-Sample Approximation for
X-rays or Neutrons with Gravity ....................... 555
11.2 Derivation of Kirchhoff Equation with a 2-D
Biconcave Parabolic or Spherical CRL for X-rays or
Neutrons with Gravity - Thick Sample .................. 571
11.3 Biconcave, Parabolic CRL - Image Amplitude
Distribution for Incoherent X-rays or Neutrons with
Gravity ............................................... 587
11.3.1 Amplitude in the y-Direction Parallel to the
Direction of Gravity ........................... 588
11.3.2 Amplitude in the x-Direction Perpendicular to
the Direction of Gravity ....................... 593
11.3.3 Object Transmission Function ................... 594
11.3.4 Integration in the y-Direction in the Object
Plane .......................................... 595
11.3.5 Integration in the x-Direction in the Object
Plane .......................................... 600
11.3.6 Amplitude in the Image Plane Along the x- and
y-Directions ................................... 600
11.4 Point Spread Function (PSF) of the Biconcave,
Parabolic CRL for Incoherent X-rays or Neutrons with
Gravity ............................................... 603
11.5 PSF of a Pinhole for Incoherent X-rays or Neutrons
with Gravity .......................................... 611
11.6 The Modulation Transfer Function ...................... 614
11.7 The Modulation Transfer Function (MTF) with a
Biconcave, Parabolic CRL for Incoherent X-rays or
Neutrons with Gravity ................................. 615
11.8 Field of View (FOV) with a Biconcave, Parabolic CRL
for Incoherent X-rays or Neutrons with Gravity ........ 621
11.9 Image Intensity Distribution of a Biconcave,
Parabolic CRL for Coherent X-rays or Neutrons with
Gravity ............................................... 624
11.10 Biconcave, Parabolic CRL Image Intensity
Distribution for Incoherent X-rays or Neutrons with
Gravity-Coherent Amplitude Cross Terms Set to Zero ... 634
11.11 Without CRL-Phase Contrast Imaging for Incoherent
X-rays or Neutrons with Gravity ...................... 641
11.12 Special Case of a Merged Object Plane and Source
Plane ................................................ 649
11.13 Stationary Phase Approach Applied to an Image
Amplitude Integral of a Spherical, Biconcave CRL
for Incoherent X-rays or Neutrons with Gravity ....... 651
References ................................................. 659
12 Electromagnetic Fields of Moving Charges, Electric and
Magnetic Dipoles ........................................... 661
12.1 Maxwell's Equations and the Lienard-Wiechert
Potentials ............................................ 661
12.2 The Lorentz Gauge and the Helmholtz Theorem ........... 667
12.3 Calculation of Scalar and Vector Potentials of an
Oscillating Electric Dipole ........................... 671
12.4 Calculation of a Magnetic Field of an Oscillating
Electric Dipole via the Vector Potential .............. 674
12.5 Calculation of the Electric Field Emitted from an
Oscillating Electric Dipole ........................... 675
12.6 Conversion of Electric and Magnetic Fields from MKS
Units to CGS Units .................................... 679
12.7 Near- and Far-Zone Electric Fields and Power
Emission from an Oscillating Electric Dipole .......... 681
12.8 Frequency and Wave - Number Domain Expressions for
Electric and Magnetic Fields Emission and Power from
the Oscillating Electric Dipole ....................... 683
12.9 Binomial Expansion of the Vector Potential of a
Moving Charge Includes Contributions from Electric
and Magnetic Dipole and Electric Quadrupole Moments ... 687
12.10 Transformation of the Fields and Vector Potential
of the Electric Dipole to the Magnetic Dipole ........ 691
12.11 Electric and Magnetic Fields and Power Emission
from an Oscillating Magnetic Dipole .................. 694
12.12 Derivation of the Electric Field of a Charge in
Arbitrary Motion from Lienard-Wiechert Potentials .... 697
12.13 Derivation of the Magnetic Field of a Charged
Particle in Arbitrary Motion from Lienard-Wiechert
Potentials ........................................... 705
12.14 Poynting Vector and Electromagnetic Energy Radiated
from an Arbitrary Accelerated Charge per Solid
Angle per Frequency Interval ......................... 708
12.15 Calculation of the Electron Trajectory in
Synchrotron Ring Insertion Devices ................... 715
12.16 Simplified Velocity-Dependent Expression for the
per Solid Angle Frequency Spectrum Emitted by
Accelerated Charged Particles ........................ 719
12.17 Bremsstrahlung and Electromagnetic Wave
Polarization ......................................... 727
12.18 Net Neutron Magnetic Dipole Moment Produced in an
Ultracold Neutron Population with Applied Magnetic
Field, and Possibilities for Population Inversion .... 734
12.19 Energy Radiated per Solid Angle for a Moving
Magnetic Dipole Moment of Neutrons and Charged
Particles ............................................ 739
12.20 Relation of a Magnetization Vector of a Moving
Charge with a Polarization Vector of a Stationary
Charge ............................................... 745
References ................................................. 748
13 Special Relativity, Electrodynamics, Least Action, and
Hamiltonians ............................................... 751
13.1 Special Relativity - Minkowski Space and Invariant
Space-Time Distance ................................... 751
13.2 Special Relativity - Length Contraction ............... 754
13.3 Special Relativity - Time Dilation .................... 756
13.4 Special Relativity - Lorentz Transformation of Space -
Time Position ......................................... 757
13.5 Special Relativity - Lorentz Transformation with
Position and Velocity Four-Vectors .................... 760
13.6 Special Relativity - Four-Vector and Lorentz
Transformation Representation in Minkowski Space ...... 762
13.7 Special Relativity - Velocity, Mass, Momentum, and
Energy ................................................ 765
13.8 Radiated Power from an Accelerated Charge Moving at
Nonrelativistic Velocity via the Nonrelativistic
Lamor Formula and Lorentz Transformation at an
Instant of Time ....................................... 769
13.9 Special Relativity - Electromagnetic Four-Vectors ..... 776
13.10 Special Relativity - Electromagnetic Fields and
Potentials as Four-Vectors ........................... 778
13.11 Special Relativity - The Electromagnetic Field
Tensor ............................................... 780
13.12 The Maxwell Stress Tensor ............................ 786
13.13 Force Density and the Maxwell Stress Tensor .......... 789
13.14 The Principle of Least Action and the Lagrangian
Yields Euler-Lagrange Equations ...................... 796
13.15 Derivation of the Hamiltonian from the Lagrangian .... 799
13.16 Derivation of the Lagrangian for Relativistic
Charged Particle in an Electromagnetic Field ......... 802
13.17 Euler-Lagrange Equation for Langrangian Density for
Real Scalar Fields ................................... 809
References ................................................. 812
14 The Klein-Gordon and Dirac Equations ....................... 813
14.1 Relativistic Correct Schrodinger Wave Equation - the
Klein-Gordon Equation ................................. 813
14.2 The Dirac Wave Equation for Spin 1/2 Particles -
Overview .............................................. 815
14.3 Derivation of the Dirac Equation that Predicts
Correct LS Coupling Term for Unpaired Atomic
Electrons in Magnetic Neutron Scatter ................. 817
14.4 Solution of the Dirac Equation for a Free Particle .... 823
14.5 A Useful Vector Property with the Pauli Spin Matrix ... 829
14.6 Dirac Equation for Bound Electron-Proton Interaction
- The Basis of LS Coupling ............................ 830
14.7 Second Term HD2 of the Dirac Equation for Bound
Electron - Proton Interaction that Includes a LS
Coupling Term ......................................... 835
14.8 Third Term HD3 of the Dirac Equation for Bound
Electron - Proton Interaction ......................... 838
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 ............... 844
14.10 Evaluated Dirac Equation for Bound Electron - Proton
Interaction that Predicts LS Coupling and Hyperfine
Interactions ......................................... 848
14.11 Equation of Motion for Electron Spin, Orbital, and
Total Angular Momentum Vectors ....................... 851
14.12 Derivation of the Dirac Equation for the Hydrogen
Atom - Step 1: Evaluation of α • p in the Dirac
Hamiltonian .......................................... 857
14.13 Derivation of the Dirac Equation for the Hydrogen
Atom - Step 2: Evaluation of σ • L in a Dirac
Hamiltonian α • p Term ............................... 860
14.14 Derivation of the Dirac Equation for a Hydrogen
Atom - Step 3: Introduction of the Squared Angular
Momentum Operator K2 in a Dirac Hamiltonian .......... 864
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 .......................................... 865
14.16 Asymptotic Solution - A Coupled, Dirac
Eigenequation Pair for the Hydrogen Atom ............. 869
14.17 Regular Solution of the Coupled, Dirac
Eigenequation Pair and Electron Energy Formula for
the Hydrogen Atom .................................... 873
14.18 Quantum Number Relationships in the Dirac Electron
Energy Formula for a Hydrogen Atom ................... 880
14.19 The Nuclear Electric Quadrupole Potential ............ 886
References ................................................. 888
15 Neutron and X-ray Optics in General Relativity and
Cosmology .................................................. 889
15.1 Special and General Relativity - History and
Relation to Neutron and X-ray Optics .................. 889
15.2 Equivalence Principle, Manifolds, Parallel Vector
Translation, and Covariant Derivatives ................ 891
15.3 Surfaces and Gauss's Remarkable Theorem, Gaussian
Curvature, and Metric Coefficients .................... 893
15.4 Curvilinear Coordinate Systems ........................ 897
15.5 Metric Tensor and Invariant Distance in Coordinate
System Transformations ................................ 901
15.6 Metric Tensor and Invariant Area and Volume in
Coordinate System Transformations ..................... 904
15.7 Curvilinear Coordinate Systems and the Jacobian
Determinant ........................................... 905
15.8 Metric Coefficients, Jacobian Determinant, and
Cartesian-to-Curvilinear Coordinate System
Transformation ........................................ 910
15.9 Variation of Physical Quantity by Displacement-
Scalar and Vector Fields .............................. 914
15.10 Invariance, Summation Convention, and Contravariant
Vectors .............................................. 918
15.11 Natural Basis Vectors are Tangent to Surfaces in a
Contravariant Vector Space ........................... 920
15.12 Geometric View of Contravariant Representation of
a Vector ............................................. 923
15.13 Geometric View of Covariant (Dual) Representation
of a Vector .......................................... 925
15.14 Covariant Vectors Transformation Is Similar to the
Gradient of a Scalar Function ........................ 928
15.15 Dual Basis Vectors are Normal to Surfaces in the
Covariant Vector Space ............................... 931
15.16 Covariant and Contravariant Tensors, Pseudo and
Polar Scalars, Vectors, and Tensors .................. 933
15.17 Natural and Dual Basis Vectors are Reciprocals ....... 937
15.18 Natural and Dual Basis Vectors and the Metric
Tensor ............................................... 939
15.19 Differentials in Curvilinear Coordinates, Curved
Surfaces, and Parallel Transport ..................... 943
15.20 Absolute and Covariant Derivatives of Vectors with
Contravariant Representation ......................... 947
15.21 Absolute and Covariant Derivatives of Vectors and
Tensors with Covariant Representation ................ 950
15.22 Geodesic Curves and Connection Coefficients
Expressed by a Metric Tensor ......................... 953
15.23 Gradient of a Vector in Curvilinear Coordinates ...... 958
15.24 Cylindrical Coordinate System - Covariant and
Contravariant Metric Coefficients and Connection
Coefficients ......................................... 962
15.25 Spherical Coordinate System - Covariant and
Contravariant Metric Coefficients and Connection
Coefficients ......................................... 965
15.26 Newton's Laws and Maxwell's Equations in General
Relativity ........................................... 967
15.27 The Curvature, Ricci, and Einstein Tensors and the
Curvature or Ricci Scalar ............................ 970
15.28 The Stress Tensor, Universe Fluid Model, and the
Equation of Continuity and Motion .................... 975
15.29 Einstein Field Equations ............................. 978
15.30 Hilbert Derived Einstein Field Equations from the
Principle of Least Action ............................ 981
15.31 Gravity Waves Derived from Linearized Einstein
Field Equations for the Weak Gravity Condition ....... 984
15.32 Schwarzschild Solution to Einstein Field Equations
and Experimental Predictions ......................... 994
15.33 The Friedmann Equation and Cosmological Issues ...... 1006
References ................................................ 1013
16 Radiation Imaging Systems and Performance ................. 1015
16.1 Liouville's Theorem, the Vlasov Equation, and the
Continuity Equation .................................. 1015
16.2 Fraction of Source Intensity Intercepted and
Focused by Lens ...................................... 1023
16.3 Depth of Field of a Lens ............................. 1024
16.4 Radiation Dose Rate and Radiation Shielding .......... 1026
16.5 Imaging System Modulation Transfer Function (MTF) .... 1030
16.6 Rose Model, Contrast, Detective Quantum Efficiency
(DQE), and Noise Power Spectrum (NPS) ................ 1033
16.7 Derivation of DQE from NPS, Contrast, Gain, and MTF .. 1036
16.8 Criteria for Sample Feature Imaging .................. 1041
16.9 Imaging System Performance - Receiver Operating
Characteristic (ROC), Positive and Negative
Likelihood, and Positive and Negative Predictive
Value Curves ......................................... 1043
16.10 Reliable Detection of a Signal in the Presence of
Background and Noise ................................ 1047
References ................................................ 1051
17 Neutron and Charged Particle Magnetic Optics .............. 1053
17.1 Charged Particle Motion in Axial Symmetric Magnetic
Field-Busch's Theorem ................................ 1053
17.2 Paraxial Ray Equation and Focusing of Charged
Particles by an Axial Symmetric Magnetic Field ....... 1056
17.3 Paraxial Ray Equation for Azimuthally Symmetric
Magnetic and Electric Fields and Busch's Theorem
Derived from the Least-Action Principle .............. 1060
17.4 Multipole Magnetic Fields and Lenses for Charged
Particles ............................................ 1062
17.5 FODO Quadrupole Magnetic Lens Pair for Charged
Particles ............................................ 1067
17.6 Neutron Trajectory in Magnetic Fields and Magnetic
Field Gradient Focusing of Neutrons .................. 1070
17.7 Neutron Compound Cylindrical Magnetic Lens ........... 1073
17.8 Calculation of Magnetic Fields of Planar Magnets by
Paired Magnetic Charge Sheets ........................ 1076
17.9 1-D Magnetic Gravitational Trap for Ultracold
Neutrons ............................................. 1083
17.10 Electron Trajectory in Wigglers and Undulator
X-ray Sources ....................................... 1087
17.11 Derivation of Emitted X-ray Wavelength, Bandwidth,
and Cone Angle for the Undulator .................... 1091
17.12 X-ray Power and Differential Power Emission per
Solid Angle from an Undulator ....................... 1094
References ................................................... 1103
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