《THE OPTICAL PRINCIPLES OF THE DIFFRACTION OF X-RAYS》求取 ⇩

CHAPTER ⅠTHE GEOMETRICAL THEORY OF DIFFRACTION BY SPACE-LATTICES1

1．DIFFRACTION BY A SIMPLE SPACE-LATTICE1

(a)The Laue theory1

(b)The interference conditions in termsof the reciprocal lattice6

(c)The width of a diffraction maximum and its dependence on crystal size．The interference function8

2．SOME APPLICATIONS OF THE PRINCIPLES TO PARTICULAR PROBLEMS14

(a)Introductory14

(b)The method of the rotating or oscillating crystal15

(c)The method of the powdered crystal18

(d)The Laue method19

3．THE EFFECT OF RANDOM IRREGULARITIES OF THE LATTICE ON THE DIFFRACTION PHENOMENA．THE TEMPERATURE EFFECT20

(a)Introductory20

(b)Diffraction by a lattice whose points are subject to random displacements21

(c)The general scattering when the lattice-points vibrate independently24

4．SOME EFFECTS DUE TO MULTIPLE REFLECTION25

CHAPTER ⅡTHE INTENSITY OF REFLECTION OF χ-RAYS BY CRYSTALS27

1．DIFFRACTION BY A COMPOSITE LATTICE27

(a)The structure amplitude27

(b)The atomic scattering factor29

(c)The structure factor31

(d)Phase changes on scattering and their effect on the structure factor32

(e)The temperature effect and the structure factor34

2．REFLECTION FROM CRYSTALS WHEN DYNAMICAL INTERACTION MAY BB NEGLECTED34

(a)Introductory34

(b)The amplitude reflected by a piane sheet of atoms35

(c)Reflection by a crystal slab of infinite lateral extent,composed of a number of planes36

(d)Reflection by a small crystal of any form39

(e)The integrated reflection41

(f)The integrated reflection derived from an integration in the reciprocal-lattice space41

(g)The crystal mosaic43

(h)Intensity formulae for mosaic Crystals44

(ⅰ)Reflection from a face44

(ⅱ)Reflection through a crystal plate45

(i)Reflection from powdered crystals46

(j)Secondary extinction49

(k)Summary of formulae for intensity of reflection by mosaic crystals50

3．REFLECTION FROM PERFECT CRYSTALS52

(a)Introductory52

(b)The refractive index of the crystal53

(c)Deviations from Bragg＇s law54

(d)Reflection from a set of crystal planes,taking into account multiple reflections55

(e)Crystal with negligible absorption57

(f)The integrated reflection from the face of a perfect crystal59

(g)Primary extinction60

(h)The perfect crystal with absorption—Prins＇s method62

(i)The composite lattice in the perfect crystal65

4．EWALD＇S DYNAMICAL THEORY66

(a)Introductory66

(b)The electromagnetic field due to a dipole-wave in a crystal lattice67

(c)The alternative methods of description of a dipole-wave in a lattice70

(d)The relationship of the interference wave-field to the dipole-waves72

(e)The dynamical problem73

(f)The dispersion equation and the dispersion surface75

(g)The dispersion surface for the case of two waves76

(h)The transition from the case of a single wave to that of two waves79

(i)The field due to a dipole-wave in a semi-infinite crystal The boundary waves80

(j)The relation between the external and internal waves considered in terms of the boundary conditions83

(k)One primary and one secondary wave in the finite crystal84

(l)Unsymmetrical reflection88

(m)Conclusion90

CHAPTER ⅢTHE ATOMIC SCATTERING FACTOR93

1．INTRODUCTORY93

2．THE CLASSICAL CALCULATION OF SCATTERING FACTORS94

(a)General considerations94

(b)Scattering by atoms arranged in a crystal lattice95

(c)Scattering by atoms distributed at random98

(d)Coherent and incoherent radiation100

(e)Numerical calculation of scattering factors on classical basis101

3．THE TREATMENT OF SCATTERING BY THE METHODS OF WAVE-MECHANICS101

(a)Introductory101

(b)The wave-equation103

(c)The calculation of the perturbed wave-function105

(d)The calculation of the current-density and of the corres-ponding scattered radiation107

(e)The scattering of radiation of frequency large compared with the atomic absorption frequencies108

(ⅰ)Coherent scattering108

(ⅱ)Incoherent scattering110

(f)The dispersion terms112

(g)The general form of the scattering factor for the many-electron atom113

(h)Approximate forms of the expressions for coherent and total scattering for a many-electron atom．The exclusion principle114

4．NUMERICAL METHODS117

(a)The method of the self-consistent field117

(b)The self-consistent field,including exchange—Fock＇s equation121

(c)The Pauling-Sherman method123

(d)The Thomas-Fermi method123

(e)The calculation of f-curves from the charge distributions125

(f)Comparison of f-curves based on different atomic models．A discussion of the available tables of f-values129

(g)Limitations of applicability of the tabulated f-curves132

CHAPTER ⅣTHE ANOMALOUS SCATTERING AND DISPERSION OF χ-RAYS135

1．THEORETICAL135

(a)Introductory135

(b)Scattering by a classical dipole-oscillator135

(c)The relation between the scattering and the absorption and refractive index in a medium containing dipoles137

(d)The breadth of the absorption lines138

(e)Comparison between the classical and quantum-theory expressions for the scattering factor．The oscillator strength140

(f)Extension to the many-electron atom143

(g)The oscillator-density for the continuum of positive energy states144

(h)The determination of the oscillator-density from the photo-electric absorption146

(i)The imaginary part of the scattering factor148

(j)The total scattering factor f149

(k)The variation of the scattering factor with frequency149

(l)The effect of the imaginary component of f on the amplitude and phase of the scattered wave151

(m)The influence of damping of the oscillators on f154

(n)The calculation of oscillator strengths from the atomic wave-functions157

(o)The quadrupole terms in the scattering factor161

(p)The total scattering factor,including quadrupole terms164

(q)The dependence of δf on the angle of scattering166

2．EXPERIMENTAL VERIFICATION OF THE DISPERSION FORMULAE167

(a)Introductory167

(b)Measurement of the refractive index from the deviations from Bragg＇s law168

(c)Comparison with theory171

(d)Refractive indices by total reflection171

(e)Refractive indices from the deviations produced by prisms177

(f)Tests of the dispersion formulae from the measurement of atomic scattering factors180

(g)General conclusions188

CHAPTER ⅤTHE INFLUENCE OF TEMPERATURE ON THE DIFFRACTION OF χ-rayS BY CRYSTALS193

1．THEORETICAL193

(a)Introductory193

(b)The normal co-ordinates of the lattice vibrations195

(c)Plane waves in a lattice196

(d)Waves in a finite lattice197

(e)The calculation of {kS.(un-un＇)}2 in terms of the normal co-ordinates198

(f)The relation between the mean-square amplitude and the temperature199

(g)The calculation of the average scattering from a lattice in thermal movement201

(h)The scattering function considered with the aid of the reciprocal lattice202

(i)The diffuse maxima considered as optical＇ghosts＇205

(j)The distribution of intensity in the diffuse maxima207

(k)The surfaces of equal diffusion for a cubic crystal210

(l)The shapes of the surfaces of equal diffusion in some par-ticular cases213

(m)The numerical evaluation of the temperature factor e-2M for a cubic crystal by Debye＇s method215

(n)Determination of the characteristic temperature220

(o)The contribution of the different frequency ranges of the elastic spectrum to M223

(p)The limitations of the Debye-Waller formula for M225

(q)Formal representation of the temperature factor for a com-plex crystal226

(r)The temperature factor for crystals not having cubic sym-metry227

(s)Rotations of molecules and groups of atoms in crystals229

2．EXPERIMENTAL TESTS OF THE FORMULAE FOR THE DEBYE TEMPERA-TURE FACTOR231

(a)Introductory231

(b)The temperature factor for sylvine(KCl)233

(c)The mean-square amplitude of the atomic vibrations236

(d)The relation between β and the elastic constants238

3．THE EXPERIMENTAL STUDY OF DIFFUSE SCATTERING239

(a)Introductory239

(b)The study of diffuse patterns on Laue photographs241

(c)The radial streaks244

(d)The relation between the characteristic,Laue,and diffuss reflections246

(e)The qualitative effect of temperature on the diffuse maxima247

(f)The shape of the diffuse maxima and its dependence on crystal structure248

(g)The shapes of the diffuse maxima for cubic crystals250

(h)Primary and secondary extra reflections252

(i)Laval＇s investigation of diffuse scattering253

(j)The quantitative dependence of the intensity of the diffuse scattering on temperature257

(k)Jauncey＇s scattering formula and its investigation258

CHAPTER ⅥEXPERIMENTAL TESTS OF THE INTENSITY FORMULAE268

1．PRIMARY AND SECONDARY EXTINCTION268

(a)Introductory268

(b)The correction for primary extinction270

(c)Secondary extinction274

(d)Correction for secondary extinction in a crystal composed of small independent blocks276

(e)Unsymmetrical reflection from a crystal mosaic278

(f)The integrated reflection from a mosaic of small blocks281

(g)Reflection curves from actual crystals282

(h)Evidence of crystal imperfection from the absolute intensities of χ-ray reflections285

(i)Experimental determination of secondary extinction287

(j)Other methods of estimating secondary extinction292

(k)Effects of simultaneous existence of primary and secondary extinction294

(l)Artificial alteration of the state of perfection of crystals295

2．QUANTITATIVE TEST OF THE FORMULA FOR THE MOSAIC CRYSTAL299

(a)Tests from comparison of observed and calculated structure factors299

3．EXPERIMENTAL TESTS OF THE REFLECTION FORMULAE FOR PERFECT CRYSTALS304

(a)Introductory304

(b)The principle of the double-crystal spectrometer306

(c)The parallel arrangement for the double-crystal spectrometer308

(d)Absence of dispersion in the parallel arrangement312

(e)The reflection curve and the dispersion in the(1,1)arrange-ment of the double-crystal spectrometer314

(f)The effect of polarisation of the incident radiation on the double-reflection curve315

(g)Integrated reflections from the double-crystal spectrometer316

(h)The relation between the widths of the double-reflection and single-reflection curves317

(i)The diamond as a perfect crystal318

(j)Tests of the Darwin-Prins formula322

4．INTENSITY MEASUREMENTS FROM POWDERED CRYSTALS332

(a)Introductory332

(b)The transmission method333

(c)The reflection method and the focusing condition334

(d)The mixed-powder method,and the Substitution method337

CHAPTER ⅦTHE USE OF FOURIER SERIES IN CRYSTAL ANALYSIS342

1．TRIPLE,DOUBLE,AND SINGLE SERIES AND THEIR APPLICATION342

(a)Introductory342

(b)The derivation of the triple series343

(c)The calculation of the coefficients of the series345

(d)The physical interpretation of the terms of the series345

(e)Reality conditions for the density distribution346

(f)Case of crystal with symmetry centre347

(g)The Fourier series and the reciprocal lattice348

(h)Examples of the use of triple series349

(i)The double series351

(j)The projection of a slice of the crystal cell355

(k)Methods of handling the double series356

(l)Practical application of the double series when the projection is centrosymmetrical360

(m)Some examples of the use of the double series when the pro-jection has a centre of symmetry361

(n)The use of the double series when the projection has no centre of symmetry365

(o)One-dimensional series369

(p)Fourier methods requiring no knowledge of the phases of the spectra:Patterson＇s series371

(q)The properties of Patterson＇s series373

(r)Methods of increasing the resolution of the Patterson distri-bution376

(s)Harker＇s application of the Patterson method377

(t)An example of the use of the Patterson-Harker method380

(u)Summary383

2．THE FOURIER PROJECTION CONSIDERED AS AN OPTICAL IMAGE385

(a)Introductory385

(b)The reciprocal relation between the net of a two-dimensional grating and the array of spectra produced by it386

(c)The relationship between the two-dimensional grating and the projection of a crystal structure on a plane388

(d)Abbe＇s theory of image formation390

(e)The relationship between the spectra and the Fourier com-ponents of the image392

(f)Diffraction effects in the Fourier projection396

(g)Some examples of false detail due to diffraction401

3．APPLICATION OF THE METHODS OF FOURIER ANALYSIS TO THE DETERMINATION OF THE ELECTRON DISTRIBUTION IN ATOMS403

(a)Fourier integrals403

(b)The determination of the radial electron density,U(r),from the atomic scattering factor,f404

(c)Series for the radial electron distribution408

CHAPTER ⅧLAUE＇S DEVELOPMENT OF THE DYNAMICAL THEORY—KOSSEL LINES413

1．THE DYNAMICAL THEORY IN TERMS OF A CONTINUOUS CHARGE DISTRIBUTION413

(a)Introductory413

(b)The fictitious dielectric constant and polarisation expressed as a Fourier seres414

(c)The fundamental equations of the wave-field416

(d)The case in which two waves only are appreciable419

(e)Determination of the wave-points when the crystal is bounded by a plane surface and a primary wave enters it from outside422

(f)The calculation of the ratio Dm/Do425

(g)Total reflection at a crystal surface(Case Ⅱ)425

(h)The application of the boundary conditions in Case Ⅱ(reflection)427

(i)The reflection coefficient in Case Ⅱ428

(j)Detailed discussion of the values of x for a non-absorbing crystal in Case Ⅱ429

(k)Extinction within the range of total reflection430

(l)The intensity of the wave-field in the crystal in the reflection problem(Case Ⅱ),as a function of the angle of incidence431

(m)The transmission coefficient in Case Ⅰ435

(n)The intensity of the wave-field in the crystal in Case Ⅰ436

2．DIFFRACTION PHENOMENA WHEN THE SOURCE OF RADIATION LIES WITHIN THE CRYSTAL438

(a)Introductory438

(b)The reciprocity theorem in optics439

(c)The application of the reciprocity theorem to the problem of the diffraction of radiation excited within the crystal440

(d)Detailed consideration of the diffraction cones corresponding to the reflection problem(Case Ⅱ)441

(e)The total excess or defect of intensity in the Kossel lines442

(f)The nature of the diffraction cones corresponding to Case Ⅰ(transmission)in the dynamical problem443

(g)Composite cones445

(h)Experimental details446

(i)The geometry of the cones448

(j)Seemann＇s wide-angle diagrams452

(k)Divergent-beam photography452

(l)Laue＇s explanation of Kikuchi lines455

CHAPTER ⅨTHE SCATTERING OF χ-rayS BY GASES,LIQUIDS AND AMORPHOUS SOLIDS458

1．INTRODUCTORY458

(a)General survey of the subject458

(b)Formulae for incoherent scattering461

(c)General formulae for the diffraction of coherent radiation by assemblages of atoms463

2．THE SCATTERING OF COHERENT RADIATION BY ASSEMBLAGES OF MONATOMIC MOLECULES465

(a)Scattering by a gas consisting of point atoms465

(b)Scattering by a gas consisting of monatomic molecules of finite size469

(c)Experimental tests of the scattering formula for monatomic gases472

(d)The probability function W(r)and the density function ρ(r)474

(e)Formulae for scattering by monatomic liquids475

(f)Examples of scattering by monatomic liquids478

3．THE SCATTERING OF COHERENT RADIATION BY ASSEMBLAGES OF POLY-ATOMIC MOLECULES480

(a)Introductory480

(b)Scattering by gases consisting of polyatomic molecules482

(c)Scattering by gases consisting of diatomic molecules483

(d)Scattering by molecules with more than two atoms486

(e)Experimental methods487

(f)The effect of temperature on scattering by gases488

(g)Scattering of electrons by gas molecules493

(h)Scattering by liquids with complex molecules494

(i)The structure of water497

4．ANALYSIS OF DIFFRACTION PATTERNS DUE TO POWDERED CRYSTALS AND VITREOUS SOLIDS501

(a)Introductory501

(b)Powdered crystals containing only one kind of atom502

(c)Crystalpowder or solid containing more than one kind of atom506

(d)Some examples504

(e)The analysis of vitreous solids508

CHAPTER ⅩDIFFRACTION BY SMALL CRYSTALS AND ITS RELATIONSHIP TO DIFFRACTION BY AMORPHOUS MATERIAL513

1．A COMPARISON BETWEEN DIFFRACTION BY CRYSTAL POWDERS AND BY POLYATOMIC MOLECULES513

(a)Introductory513

(b)The interference function and the structure factor in terms of the reciprocal lattice513

(c)Scattering by irregular assemblages of crystallites517

(d)The linear crystallite518

(e)The transition from the case of the linear crystallite to that of the diatomic molecule520

(f)The interference function I0(ξ,η,ζ)expressed as a Fourier series522

(g)Scattering from crystallites in random orientation525

2．THE EFFECT OF CRYSTAL SIZE ON THE WIDTHS OF THE DIFFRACTION SPECTRA528

(a)Introductory528

(b)The breadths of Debye-Scherrer rings529

(c)The generality of the expression for the integral line-breadth535

(d)Calculation of the integral line-breadth in some special cases:the Scherrer constant536

(e)The use of approximation functions:Laue＇s formula537

(f)Experimental application of the formulae for line-breadth540

(g)Some examples of the diffraction of electrons by very small single crystals545

(h)The effect of the external form of the crystal on the inter-ference function548

(i)The crystal-form factor expressed as a surface integral550

(j)The crystal-form factor expressed as the sum of a set of integrals along the crystal edges552

(k)The optical analogy to the interference function555

3．THE EFFECT OF CERTAIN TYPES OF FAULT AND IMPERFECTION OF THE LATTICE ON THE DIFFRACTION SPECTRA555

(a)Scattering by crystals with non-identical unit cells based on a regular lattice555

(b)Diffraction by a distorted lattice560

(c)Diffraction by lattices with periodic distortions563

(d)Experimental evidence of structures with periodic faults568

4．DIFFRACTION BY FIBROUS MATERIALS571

(a)Introductory571

(b)Diffraction by elongated crystallites with parailel arrangement and random orientation about the direction of their lengths572

(c)Effects of lack of parallelism of the fibres577

(d)The elongated two-dimensional structure with a number of rows578

(e)The elongated three-dimensional crystallite580

(f)The external interference terms582

(g)The structure factor and the atomic scattering factor585

(h)Summary of conclusions588

APPENDICES591

Ⅰ．SUMMARY OF VECTOR FORMULAE592

Ⅱ．THE RECIPROCAL LATTICE598

Ⅲ．TABLES FOR ESTIMATING THE CORRECTION TO BE APPLIED TO THE SCATTERING FACTOR ON ACCOUNT OF DISPERSION BY THE K ELECTRONS608

Ⅳ．DERIVATION OF THE FOURIER INTEGRAL611

Ⅴ．THE APPLICATION OF THE FOURIER TRANSFORM TO DIFFRACTION PROBLEMS613

INDEX OF SUBJECTS632

INDEX OF AUTHORS640