THE OPTICAL PRINCIPLES OF THE DIFFRACTION OF X-RAYSPDF电子书下载

CHAPTER Ⅰ THE GEOMETRICAL THEORY OF DIFFRACTION BY SPACE-LATTICES 1

1．DIFFRACTION BY A SIMPLE SPACE-LATTICE 1

(a)The Laue theory 1

(b)The interference conditions in termsof the reciprocal lattice 6

(c)The width of a diffraction maximum and its dependence on crystal size．The interference function 8

2．SOME APPLICATIONS OF THE PRINCIPLES TO PARTICULAR PROBLEMS 14

(a)Introductory 14

(b)The method of the rotating or oscillating crystal 15

(c)The method of the powdered crystal 18

(d)The Laue method 19

3．THE EFFECT OF RANDOM IRREGULARITIES OF THE LATTICE ON THE DIFFRACTION PHENOMENA．THE TEMPERATURE EFFECT 20

(a)Introductory 20

(b)Diffraction by a lattice whose points are subject to random displacements 21

(c)The general scattering when the lattice-points vibrate independently 24

4．SOME EFFECTS DUE TO MULTIPLE REFLECTION 25

CHAPTER Ⅱ THE INTENSITY OF REFLECTION OF χ-RAYS BY CRYSTALS 27

1．DIFFRACTION BY A COMPOSITE LATTICE 27

(a)The structure amplitude 27

(b)The atomic scattering factor 29

(c)The structure factor 31

(d)Phase changes on scattering and their effect on the structure factor 32

(e)The temperature effect and the structure factor 34

2．REFLECTION FROM CRYSTALS WHEN DYNAMICAL INTERACTION MAY BB NEGLECTED 34

(a)Introductory 34

(b)The amplitude reflected by a piane sheet of atoms 35

(c)Reflection by a crystal slab of infinite lateral extent,composed of a number of planes 36

(d)Reflection by a small crystal of any form 39

(e)The integrated reflection 41

(f)The integrated reflection derived from an integration in the reciprocal-lattice space 41

(g)The crystal mosaic 43

(h)Intensity formulae for mosaic Crystals 44

(ⅰ)Reflection from a face 44

(ⅱ)Reflection through a crystal plate 45

(i)Reflection from powdered crystals 46

(j)Secondary extinction 49

(k)Summary of formulae for intensity of reflection by mosaic crystals 50

3．REFLECTION FROM PERFECT CRYSTALS 52

(a)Introductory 52

(b)The refractive index of the crystal 53

(c)Deviations from Bragg＇s law 54

(d)Reflection from a set of crystal planes,taking into account multiple reflections 55

(e)Crystal with negligible absorption 57

(f)The integrated reflection from the face of a perfect crystal 59

(g)Primary extinction 60

(h)The perfect crystal with absorption—Prins＇s method 62

(i)The composite lattice in the perfect crystal 65

4．EWALD＇S DYNAMICAL THEORY 66

(a)Introductory 66

(b)The electromagnetic field due to a dipole-wave in a crystal lattice 67

(c)The alternative methods of description of a dipole-wave in a lattice 70

(d)The relationship of the interference wave-field to the dipole-waves 72

(e)The dynamical problem 73

(f)The dispersion equation and the dispersion surface 75

(g)The dispersion surface for the case of two waves 76

(h)The transition from the case of a single wave to that of two waves 79

(i)The field due to a dipole-wave in a semi-infinite crystal The boundary waves 80

(j)The relation between the external and internal waves considered in terms of the boundary conditions 83

(k)One primary and one secondary wave in the finite crystal 84

(l)Unsymmetrical reflection 88

(m)Conclusion 90

CHAPTER Ⅲ THE ATOMIC SCATTERING FACTOR 93

1．INTRODUCTORY 93

2．THE CLASSICAL CALCULATION OF SCATTERING FACTORS 94

(a)General considerations 94

(b)Scattering by atoms arranged in a crystal lattice 95

(c)Scattering by atoms distributed at random 98

(d)Coherent and incoherent radiation 100

(e)Numerical calculation of scattering factors on classical basis 101

3．THE TREATMENT OF SCATTERING BY THE METHODS OF WAVE-MECHANICS 101

(a)Introductory 101

(b)The wave-equation 103

(c)The calculation of the perturbed wave-function 105

(d)The calculation of the current-density and of the corres-ponding scattered radiation 107

(e)The scattering of radiation of frequency large compared with the atomic absorption frequencies 108

(ⅰ)Coherent scattering 108

(ⅱ)Incoherent scattering 110

(f)The dispersion terms 112

(g)The general form of the scattering factor for the many-electron atom 113

(h)Approximate forms of the expressions for coherent and total scattering for a many-electron atom．The exclusion principle 114

4．NUMERICAL METHODS 117

(a)The method of the self-consistent field 117

(b)The self-consistent field,including exchange—Fock＇s equation 121

(c)The Pauling-Sherman method 123

(d)The Thomas-Fermi method 123

(e)The calculation of f-curves from the charge distributions 125

(f)Comparison of f-curves based on different atomic models．A discussion of the available tables of f-values 129

(g)Limitations of applicability of the tabulated f-curves 132

CHAPTER Ⅳ THE ANOMALOUS SCATTERING AND DISPERSION OF χ-RAYS 135

1．THEORETICAL 135

(a)Introductory 135

(b)Scattering by a classical dipole-oscillator 135

(c)The relation between the scattering and the absorption and refractive index in a medium containing dipoles 137

(d)The breadth of the absorption lines 138

(e)Comparison between the classical and quantum-theory expressions for the scattering factor．The oscillator strength 140

(f)Extension to the many-electron atom 143

(g)The oscillator-density for the continuum of positive energy states 144

(h)The determination of the oscillator-density from the photo-electric absorption 146

(i)The imaginary part of the scattering factor 148

(j)The total scattering factor f 149

(k)The variation of the scattering factor with frequency 149

(l)The effect of the imaginary component of f on the amplitude and phase of the scattered wave 151

(m)The influence of damping of the oscillators on f 154

(n)The calculation of oscillator strengths from the atomic wave-functions 157

(o)The quadrupole terms in the scattering factor 161

(p)The total scattering factor,including quadrupole terms 164

(q)The dependence of δf on the angle of scattering 166

2．EXPERIMENTAL VERIFICATION OF THE DISPERSION FORMULAE 167

(a)Introductory 167

(b)Measurement of the refractive index from the deviations from Bragg＇s law 168

(c)Comparison with theory 171

(d)Refractive indices by total reflection 171

(e)Refractive indices from the deviations produced by prisms 177

(f)Tests of the dispersion formulae from the measurement of atomic scattering factors 180

(g)General conclusions 188

CHAPTER Ⅴ THE INFLUENCE OF TEMPERATURE ON THE DIFFRACTION OF χ-rayS BY CRYSTALS 193

1．THEORETICAL 193

(a)Introductory 193

(b)The normal co-ordinates of the lattice vibrations 195

(c)Plane waves in a lattice 196

(d)Waves in a finite lattice 197

(e)The calculation of {kS.(un-un＇)}2 in terms of the normal co-ordinates 198

(f)The relation between the mean-square amplitude and the temperature 199

(g)The calculation of the average scattering from a lattice in thermal movement 201

(h)The scattering function considered with the aid of the reciprocal lattice 202

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

(j)The distribution of intensity in the diffuse maxima 207

(k)The surfaces of equal diffusion for a cubic crystal 210

(l)The shapes of the surfaces of equal diffusion in some par-ticular cases 213

(m)The numerical evaluation of the temperature factor e-2M for a cubic crystal by Debye＇s method 215

(n)Determination of the characteristic temperature 220

(o)The contribution of the different frequency ranges of the elastic spectrum to M 223

(p)The limitations of the Debye-Waller formula for M 225

(q)Formal representation of the temperature factor for a com-plex crystal 226

(r)The temperature factor for crystals not having cubic sym-metry 227

(s)Rotations of molecules and groups of atoms in crystals 229

2．EXPERIMENTAL TESTS OF THE FORMULAE FOR THE DEBYE TEMPERA-TURE FACTOR 231

(a)Introductory 231

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

(c)The mean-square amplitude of the atomic vibrations 236

(d)The relation between β and the elastic constants 238

3．THE EXPERIMENTAL STUDY OF DIFFUSE SCATTERING 239

(a)Introductory 239

(b)The study of diffuse patterns on Laue photographs 241

(c)The radial streaks 244

(d)The relation between the characteristic,Laue,and diffuss reflections 246

(e)The qualitative effect of temperature on the diffuse maxima 247

(f)The shape of the diffuse maxima and its dependence on crystal structure 248

(g)The shapes of the diffuse maxima for cubic crystals 250

(h)Primary and secondary extra reflections 252

(i)Laval＇s investigation of diffuse scattering 253

(j)The quantitative dependence of the intensity of the diffuse scattering on temperature 257

(k)Jauncey＇s scattering formula and its investigation 258

CHAPTER Ⅵ EXPERIMENTAL TESTS OF THE INTENSITY FORMULAE 268

1．PRIMARY AND SECONDARY EXTINCTION 268

(a)Introductory 268

(b)The correction for primary extinction 270

(c)Secondary extinction 274

(d)Correction for secondary extinction in a crystal composed of small independent blocks 276

(e)Unsymmetrical reflection from a crystal mosaic 278

(f)The integrated reflection from a mosaic of small blocks 281

(g)Reflection curves from actual crystals 282

(h)Evidence of crystal imperfection from the absolute intensities of χ-ray reflections 285

(i)Experimental determination of secondary extinction 287

(j)Other methods of estimating secondary extinction 292

(k)Effects of simultaneous existence of primary and secondary extinction 294

(l)Artificial alteration of the state of perfection of crystals 295

2．QUANTITATIVE TEST OF THE FORMULA FOR THE MOSAIC CRYSTAL 299

(a)Tests from comparison of observed and calculated structure factors 299

3．EXPERIMENTAL TESTS OF THE REFLECTION FORMULAE FOR PERFECT CRYSTALS 304

(a)Introductory 304

(b)The principle of the double-crystal spectrometer 306

(c)The parallel arrangement for the double-crystal spectrometer 308

(d)Absence of dispersion in the parallel arrangement 312

(e)The reflection curve and the dispersion in the(1,1)arrange-ment of the double-crystal spectrometer 314

(f)The effect of polarisation of the incident radiation on the double-reflection curve 315

(g)Integrated reflections from the double-crystal spectrometer 316

(h)The relation between the widths of the double-reflection and single-reflection curves 317

(i)The diamond as a perfect crystal 318

(j)Tests of the Darwin-Prins formula 322

4．INTENSITY MEASUREMENTS FROM POWDERED CRYSTALS 332

(a)Introductory 332

(b)The transmission method 333

(c)The reflection method and the focusing condition 334

(d)The mixed-powder method,and the Substitution method 337

CHAPTER Ⅶ THE USE OF FOURIER SERIES IN CRYSTAL ANALYSIS 342

1．TRIPLE,DOUBLE,AND SINGLE SERIES AND THEIR APPLICATION 342

(a)Introductory 342

(b)The derivation of the triple series 343

(c)The calculation of the coefficients of the series 345

(d)The physical interpretation of the terms of the series 345

(e)Reality conditions for the density distribution 346

(f)Case of crystal with symmetry centre 347

(g)The Fourier series and the reciprocal lattice 348

(h)Examples of the use of triple series 349

(i)The double series 351

(j)The projection of a slice of the crystal cell 355

(k)Methods of handling the double series 356

(l)Practical application of the double series when the projection is centrosymmetrical 360

(m)Some examples of the use of the double series when the pro-jection has a centre of symmetry 361

(n)The use of the double series when the projection has no centre of symmetry 365

(o)One-dimensional series 369

(p)Fourier methods requiring no knowledge of the phases of the spectra:Patterson＇s series 371

(q)The properties of Patterson＇s series 373

(r)Methods of increasing the resolution of the Patterson distri-bution 376

(s)Harker＇s application of the Patterson method 377

(t)An example of the use of the Patterson-Harker method 380

(u)Summary 383

2．THE FOURIER PROJECTION CONSIDERED AS AN OPTICAL IMAGE 385

(a)Introductory 385

(b)The reciprocal relation between the net of a two-dimensional grating and the array of spectra produced by it 386

(c)The relationship between the two-dimensional grating and the projection of a crystal structure on a plane 388

(d)Abbe＇s theory of image formation 390

(e)The relationship between the spectra and the Fourier com-ponents of the image 392

(f)Diffraction effects in the Fourier projection 396

(g)Some examples of false detail due to diffraction 401

3．APPLICATION OF THE METHODS OF FOURIER ANALYSIS TO THE DETERMINATION OF THE ELECTRON DISTRIBUTION IN ATOMS 403

(a)Fourier integrals 403

(b)The determination of the radial electron density,U(r),from the atomic scattering factor,f 404

(c)Series for the radial electron distribution 408

CHAPTER Ⅷ LAUE＇S DEVELOPMENT OF THE DYNAMICAL THEORY—KOSSEL LINES 413

1．THE DYNAMICAL THEORY IN TERMS OF A CONTINUOUS CHARGE DISTRIBUTION 413

(a)Introductory 413

(b)The fictitious dielectric constant and polarisation expressed as a Fourier seres 414

(c)The fundamental equations of the wave-field 416

(d)The case in which two waves only are appreciable 419

(e)Determination of the wave-points when the crystal is bounded by a plane surface and a primary wave enters it from outside 422

(f)The calculation of the ratio Dm/Do 425

(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 Ⅱ 428

(j)Detailed discussion of the values of x for a non-absorbing crystal in Case Ⅱ 429

(k)Extinction within the range of total reflection 430

(l)The intensity of the wave-field in the crystal in the reflection problem(Case Ⅱ),as a function of the angle of incidence 431

(m)The transmission coefficient in Case Ⅰ 435

(n)The intensity of the wave-field in the crystal in Case Ⅰ 436

2．DIFFRACTION PHENOMENA WHEN THE SOURCE OF RADIATION LIES WITHIN THE CRYSTAL 438

(a)Introductory 438

(b)The reciprocity theorem in optics 439

(c)The application of the reciprocity theorem to the problem of the diffraction of radiation excited within the crystal 440

(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 lines 442

(f)The nature of the diffraction cones corresponding to Case Ⅰ(transmission)in the dynamical problem 443

(g)Composite cones 445

(h)Experimental details 446

(i)The geometry of the cones 448

(j)Seemann＇s wide-angle diagrams 452

(k)Divergent-beam photography 452

(l)Laue＇s explanation of Kikuchi lines 455

CHAPTER Ⅸ THE SCATTERING OF χ-rayS BY GASES,LIQUIDS AND AMORPHOUS SOLIDS 458

1．INTRODUCTORY 458

(a)General survey of the subject 458

(b)Formulae for incoherent scattering 461

(c)General formulae for the diffraction of coherent radiation by assemblages of atoms 463

2．THE SCATTERING OF COHERENT RADIATION BY ASSEMBLAGES OF MONATOMIC MOLECULES 465

(a)Scattering by a gas consisting of point atoms 465

(b)Scattering by a gas consisting of monatomic molecules of finite size 469

(c)Experimental tests of the scattering formula for monatomic gases 472

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

(e)Formulae for scattering by monatomic liquids 475

(f)Examples of scattering by monatomic liquids 478

3．THE SCATTERING OF COHERENT RADIATION BY ASSEMBLAGES OF POLY-ATOMIC MOLECULES 480

(a)Introductory 480

(b)Scattering by gases consisting of polyatomic molecules 482

(c)Scattering by gases consisting of diatomic molecules 483

(d)Scattering by molecules with more than two atoms 486

(e)Experimental methods 487

(f)The effect of temperature on scattering by gases 488

(g)Scattering of electrons by gas molecules 493

(h)Scattering by liquids with complex molecules 494

(i)The structure of water 497

4．ANALYSIS OF DIFFRACTION PATTERNS DUE TO POWDERED CRYSTALS AND VITREOUS SOLIDS 501

(a)Introductory 501

(b)Powdered crystals containing only one kind of atom 502

(c)Crystalpowder or solid containing more than one kind of atom 506

(d)Some examples 504

(e)The analysis of vitreous solids 508

CHAPTER Ⅹ DIFFRACTION BY SMALL CRYSTALS AND ITS RELATIONSHIP TO DIFFRACTION BY AMORPHOUS MATERIAL 513

1．A COMPARISON BETWEEN DIFFRACTION BY CRYSTAL POWDERS AND BY POLYATOMIC MOLECULES 513

(a)Introductory 513

(b)The interference function and the structure factor in terms of the reciprocal lattice 513

(c)Scattering by irregular assemblages of crystallites 517

(d)The linear crystallite 518

(e)The transition from the case of the linear crystallite to that of the diatomic molecule 520

(f)The interference function I0(ξ,η,ζ)expressed as a Fourier series 522

(g)Scattering from crystallites in random orientation 525

2．THE EFFECT OF CRYSTAL SIZE ON THE WIDTHS OF THE DIFFRACTION SPECTRA 528

(a)Introductory 528

(b)The breadths of Debye-Scherrer rings 529

(c)The generality of the expression for the integral line-breadth 535

(d)Calculation of the integral line-breadth in some special cases:the Scherrer constant 536

(e)The use of approximation functions:Laue＇s formula 537

(f)Experimental application of the formulae for line-breadth 540

(g)Some examples of the diffraction of electrons by very small single crystals 545

(h)The effect of the external form of the crystal on the inter-ference function 548

(i)The crystal-form factor expressed as a surface integral 550

(j)The crystal-form factor expressed as the sum of a set of integrals along the crystal edges 552

(k)The optical analogy to the interference function 555

3．THE EFFECT OF CERTAIN TYPES OF FAULT AND IMPERFECTION OF THE LATTICE ON THE DIFFRACTION SPECTRA 555

(a)Scattering by crystals with non-identical unit cells based on a regular lattice 555

(b)Diffraction by a distorted lattice 560

(c)Diffraction by lattices with periodic distortions 563

(d)Experimental evidence of structures with periodic faults 568

4．DIFFRACTION BY FIBROUS MATERIALS 571

(a)Introductory 571

(b)Diffraction by elongated crystallites with parailel arrangement and random orientation about the direction of their lengths 572

(c)Effects of lack of parallelism of the fibres 577

(d)The elongated two-dimensional structure with a number of rows 578

(e)The elongated three-dimensional crystallite 580

(f)The external interference terms 582

(g)The structure factor and the atomic scattering factor 585

(h)Summary of conclusions 588

APPENDICES 591

Ⅰ．SUMMARY OF VECTOR FORMULAE 592

Ⅱ．THE RECIPROCAL LATTICE 598

Ⅲ．TABLES FOR ESTIMATING THE CORRECTION TO BE APPLIED TO THE SCATTERING FACTOR ON ACCOUNT OF DISPERSION BY THE K ELECTRONS 608

Ⅳ．DERIVATION OF THE FOURIER INTEGRAL 611

Ⅴ．THE APPLICATION OF THE FOURIER TRANSFORM TO DIFFRACTION PROBLEMS 613

INDEX OF SUBJECTS 632

INDEX OF AUTHORS 640