《International Tables For X-Ray Crystallography Volume IV Revised and Supplementary Tables》

INTRODUCTION (James A．Ibers and Walter C．Hamilton)1

Purpose and Scope of the Tables1

Acknowledgments1

1．X-RAY WAVELENGTHS (J．A．Bearden)3

1．1．Comments on the Tables (James A．Ibers and Walter C．Hamilton)5

Table 1．1A．X-ray Wavelengths in A Units and in keV Arranged by Atomic Number6

Table 1．1B．X-ray Wavelengths and Absorption Edges in A Units and in keV in Order of Increasing Wavelength20

2．ABSORPTION AND SCATTERING45

2．1．X-ray Cross Sections and Attenuation Coefficients (J．H．Hubbell，W．H．McMaster，N．Kerr Del Grande，and J．H．Mallett)47

2．1．1．Introduction47

2．1．2．Sources of Information47

2．1．2．1．Experimental Attenuation Coefficient Data47

2．1．2．2．Experimental Photoeffect Data48

2．1．2．3．Theoretical Photoeffect Data48

2．1．2．4．Theoretical Coherent Scattering Data48

2．1．2．5．Theoretical Incoherent Scattering Data48

2．1．3．Compilation Procedure48

2．1．3．1．Synthesis of Photoeffect Cross Sections from th Experimental and Theoretical Input Information48

2．1．3．2．Photoeffect Data Fitting49

2．1．4．Uncertainty of the Compilation Values50

Table 2．1A．Wavelengths and Energies Used for Tables 2．1B and 2．1C54

Table 2．1B．Total Cross Section (σ) in 10-24 cm2 per Atom for Selected X-ray Wavelengths55

Table 2．1C．Mass Attenuation Coefficients (μ/ρ) in cm2 g-161

2．1References67

2．2 Atomic Scattering Factors for X-rays (Don T．Cromer and J．T．Waber)71

Table 2．2A．Mean Atomic Scattering Factors in Electrons for Free Atoms and Chemically Significant Ions72

Table 2．2B．Coefficients for Analytical Approximation to the Scattering Factors of Table 2．2A99

Table 2．2C．Spherical Bonded Hydrogen Atom Scattering Factor102

Table 2．2D．Orbital Scattering Factors for Unfilled p，d，and f Orbitals for Atoms and Ions103

2．2．References147

2．3．Dispersion Effects148

2．3．1．Dispersion Corrections for X-ray Atomic Scattering Factors (Don T．Cromer)148

Table 2．3．1．Real and Imaginary Dispersion Corrections for Atomic Scattering Factors149

2．3．2．Dispersion Corrections and Crystal Structure Analyses (James A．Ibers)148

Table 2．3．2．Reciprocal Lattice Points Equivalent under the Operations of a Given Noncentrosymmetric Point Group151

2．3．References151

2．4．Scattering Factors for the Diffraction of Electrons by Crystalline Solids (P．A．Doyle and J．M．Cowley)152

2．4．1．Elastic Scattering from a Perfect Crystal152

2．4．2．Atomic Scattering Factors152

2．4．3．Approximations of Restricted Validity153

2．4．4．Relativistic Effects154

2．4．5．Absorption Effects154

2．4．6．Tables of Atomic Scattering Amplitudes for Electrons154

Table 2．4．6A．Atomic Scattering Amplitudes in A for Electrons for Neutral Atoms155

Table 2．4．6B．Atomic Scattering Amplitudes in A for Electrons for Ionized Atoms164

Table 2．4．6C．Parameters Useful in Electron Diffraction as a Function of Accelerating Voltage，E174

2．4．References175

2．5．Complex Scattering Factors for the Diffraction of Electrons by Gases (R．A．Bonham and Lothar Schafer)176

2．5．1．Introduction176

2．5．2．Complex Atomic Scattering Factors for Electrons176

2．5．2．1．Elastic Scattering Factors for Atoms176

2．5．2．2．Elastic Scattering Factors for Ions177

2．5．2．3．Total Inelastic Scattering Factors178

2．5．2．4．Corrections for Defects in the Theory of Atomic Scattering178

2．5．3．Molecular Scattering Factors for Electrons178

Table 2．5A．Partial Wave Elastic Scattering Factors for Neutral Atoms181

Table 2．5B．Partial Wave Ionic Scattering Factors257

Table 2．5C．Inelastic Scattering Factors258

Table 2．5D．Inelastic Scattering Factors for He and Ne268

2．5．References269

2．6．neutron Scattering Amplitudes for Elements and Isotopes (G．E．Bacon)270

Table 2．6．Neutron Nuclear Scattering Data for Elements and Isotopes270

2．6．References271

3．ANGLE SETTINGS FOR FOUR-CIRCLE DIFFRACTOMETERS (Walter C．Hamilton)273

3．1．Introduction275

3．2．Definitions276

3．2．1．The Diffractometer276

3．2．2．Coordinate Systems276

3．2．2．1．Diffractometer-Fixed Axis System277

3．2．2．2．Crystal-Fixed Orthonormal System277

3．2．2．3．The Direct Crystal Axes277

3．2．2．4．The Reciprocal Crystal Axes277

3．2．2．5．Coordinates of a Reciprocal Vector277

3．2．2．6．Crystal Rotation277

3．2．2．7．The Orientation Matrix277

3．3．Calculation of Setting Angles278

3．3．1．General Solution278

Table 3．3．1．Various Settings for Observing a Reflection on a Four-Circle Diffractometer278

3．3．2．Rotation Around the Diffraction Vector278

3．3．3．Computational Aspects279

3．4．Determination of U，the Orientation Matrix280

3．4．1．Determination of U from Three Reflections280

3．4．2．Determination of U from Two Reflections280

3．4．3．Least-Squares Refinement of Cell Constants and Orientation Angles280

3．5．Measurement Procedures282

Table 3．5A．Translational Errors for Various Crystal Settings282

Table 3．5B．Equivalent Crystal Setting Angles for Crystal Alignment on a Four-Circle Diffractometer283

3．References284

4．TESTS FOR STATISTICAL SIGNIFICANCE (Walter C．Hamilton)285

4．1．Introduction287

4．2．Tests of the R-Factor Ratio，R288

Table 4．2．Significance Points for the R-Factor Ratio，R289

4．3．Tests for the Consistency of Weights293

4．3．1．Tests on S，the Standard Deviation of the Observation of Unit Weight293

4．3．2．Normal Probability Plots293

Table 4．3．2A．Percentage Points xi of the Cumulative Normal Distribution Function295

Table 4．3．2B．Expected Values ξ(i/n) of Ranked Normal Deviates296

Table 4．3．2C．Expected Values ξ(i/n) of Ranked Normal Deviates Compared with Approximate Values301

Table 4．3．2D．Expected Values of ith Ranked Moduli of Normal Deviates in Samples of Size n302

Table 4．3．2E．Expected Values of Extreme Ranked Moduli of Normal Deviates306

4．References310

5．THERMAL-MOTION ANALYSIS USING BRAGG DIFFRACTION DATA (C．K．Johnson and H．A．Levy)311

5．1．Introduction313

5．2．Thermal Motion of Independent Atoms314

5．2．1．The Gaussian Density Function Model314

5．2．2．Properties of the Dispersion Matrix314

5．2．2．1．Mean-Square Displacement314

5．2．2．2．Probability Ellipsoids314

5．2．2．3．Positive-Definite Matrix315

5．2．2．4．Transformation Properties315

5．2．2．5．Principal Axis Transformation315

5．2．2．6．Cartesian Coordinate Systems and the Reporting of Results315

5．2．2．7．Contraction and Expansion316

5．2．3．Expansions based on the Gaussian Model316

5．2．3．1．Gram-Charlier Series Expansion316

5．2．3．2．Cumulant Expansion317

5．2．4．Models based on Curvilinear Density Functions317

5．2．4．1．Density Functions on a Circle317

5．2．4．2．Density Functions on a Sphere318

5．2．4．3．The Quasi-Gaussian Approximation319

5．3．Thermal Motion of a Rigid Body320

5．3．1．Rigid Motions320

5．3．2．The 6-Space of rotation and Translation320

5．3．3．The Center of Reaction321

5．3．4．Trace-of-S and Geometrical-Arrangement Singularities321

5．3．5．Non-Intersecting Screw Axes321

5．4．Correction of Interatomic Distances and Angles323

5．4．1．The Rigid-Body Model323

5．4．2．The Riding Model323

5．5．Site Symmetry Restrictions on Thermal-Motion Tensor Coefficients324

Table 5．5A．Site Symmetry Table325

Table 5．5B．Symmetry Restrictions of Coefficients in Second-Order Tensors328

Table 5．5C．Symmetry Restrictions on Coefficients in Third-Rank Symmetric Polar Tensors329

Table 5．5D．Symmetry Restrictions on Coefficients in Fourth-Rank Symmetric Polar Tensors331

Table 5．5E．Symmetry Restrictions on Coefficients in Second-Rank Axial Tensors332

5．References335

6．DIRECT METHODS FOR STRUCTURE DETERMINATION：ORIGIN SPECIFICATION，NORMALIZED STRUCTURE FACTORS，FORMULAS，AND THE SYMBOLIC-ADDITION PROCEDURE FOR PHASE DETERMINATION (J．Karle)337

6．1．Origin Specification339

6．1．1．Centrosymmetric Space Groups339

6．1．2．Noncentrosymmetric Space Groups341

6．1．3．Application344

Table 6．1A．Seminvariant Vector，Modulus，and Phases for the Conventionally Primitive Centrosymmetric Space Groups340

Table 6．1B．Seminvariant Vector，Modulus，and Phases for the Conventionally Centered Centrosymmetric Space Groups341

Table 6．1C．Seminvariant Vector，Modulus，and Phases for the Conventionally Primitive Noncentrosymmetric Space Groups342

Table 6．1D．Seminvariant Vector，Modulus，and Phases for the Conventionally Centered Noncentrosymmetric Space Groups345

Table 6．1E．Examples of Sets of Phases and Values for Origin Specification in Noncentrosymmetric Space Groups347

6．2．Normalized Structure Factors350

Table 6．2．Selected Statistical Properties of Normalized Structure-Factor Magnitudes351

6．3．Phase-Determining Formulas352

6．3．1．Introduction352

6．3．2．Centrosymmetric Crystals352

6．3．3．Noncentrosymmetric Crystals353

6．3．4．Auxiliary Phase-Determining Formulas353

6．4．Symbolic-Addition Procedure for Phase Determination：X-ray and Neutron Diffraction355

6．4．1．Introduction355

6．4．2．Centrosymmetric Crystals355

6．4．3．Noncentrosymmetric Crystals356

6．References358

7．SUBJECT INDEX FOR VOLUMES Ⅰ，Ⅱ，Ⅲ，AND Ⅳ359