Computing Methods in Crystallography (eBook)

Front Cover 1
Computing Methods in Crystallography 4
Copyright Page 5
Table of Contents 6
PREFACE 8
PART I: ALGEBRA 10
CHAPTER 1. MATRIX OPERATIONS 12
MATRIX NOTATION 12
MATRIX ADDITION 13
SPECIAL MATRICES 14
SCALAR MULTIPLICATION 15
MATRIX MULTIPLICATION 15
SCALAR PRODUCT 18
TRANSPOSITION 18
FORMS 19
NORMS 20
PARTITIONED MATRICES 20
CHAPTER 2. MATRIX INVERSION AND SOLUTION OF EQUATIONS 22
THE INVERSE TRANSFORMATION 22
SOLUTION OF SIMULTANEOUS EQUATIONS 25
GAUSS ELIMINATION 25
POSITIVE DEFINITE MATRICES 28
THE CHOLESKI PROCESS 28
MATRIX INVERSION 29
CHAPTER 3. APPLICATION OF MATRIX OPERATIONS 31
NATURAL AND ORTHOGONAL AXES 31
SYMMETRY OPERATIONS 33
DISTANCE AND ANGLE CALCULATIONS 34
DIFFRACTOMETER CALCULATIONS 37
LOCATION OF HYDROGEN ATOMS 40
CHAPTER 4. ALGEBRA OF LEAST SQUARES 41
LINEAR OBSERVATIONAL EQUATIONS 41
NON-LINEAR OBSERVATIONAL EQUATIONS 42
WEIGHT OF FUNCTIONS OF UNKNOWNS 44
ELECTRON DENSITY INTERPOLATION 44
CHAPTER 5. STRUCTURE FACTOR ROUTINES 47
GENERAL CONSIDERATIONS 47
CALCULATION OF TRIGONOMETRIC FUNCTIONS 48
FORM-FACTOR INTERPOLATION 51
A GENERAL PURPOSE STRUCTURE FACTOR ROUTINE 52
SPECIAL PURPOSE ROUTINES 54
CHAPTER 6. LEAST-SQUARES ROUTINES 56
OBSERVATIONAL EQUATIONS FOR REFINEMENT 56
NORMAL EQUATIONS 58
CHOICE OF PARAMETERS 60
SOLUTION OF EQUATIONS 61
REPEATED USE OF THE NORMAL MATRIX 61
A GENERAL PURPOSE LEAST-SQUARES REFINEMENT ROUTINE 61
CHAPTER 7. LATENT ROOTS AND VECTORS 66
PROPERTIES OF LATENT ROOTS AND VECTORS 67
THE MODAL MATRIX 68
SYMMETRIC MATRICES 69
METHODS OF COMPUTING LATENT ROOTS AND VECTORS FOR SYMMETRIC MATRICES 69
CHAPTER 8. APPLICATIONS OF LATENT ROOTS AND VECTORS 75
PRINCIPAL ATOMIC VIBRATION DIRECTIONS 75
PRINCIPAL PLANES FOR SETS OF ATOMS 76
LAYER SCALE-FACTORS 78
RIGID BODY OSCILLATIONS 80
CHAPTER 9. CONVERGENCE OF ITERATIVE PROCESSES 82
DIRECT AND ITERATIVE SOLUTION OF LINEAR EQUATIONS 82
STATIONARY METHODS 83
ACCELERATION OF STATIONARY PROCESSES 85
GRADIENT METHODS 87
APPLICATION TO LEAST-SQUARES REFINEMENT 88
CHAPTER 10. FOURIER SERIES ROUTINES 91
METHOD OF CALCULATION 92
A GENERAL ROUTINE 94
SLANT PLANE FOURIER SERIES 97
CHAPTER 11. DATA REDUCTION ROUTINES 98
INTRODUCTION 98
ABSORPTION CORRECTIONS 99
LORENTZ AND POLARISATION CORRECTIONS 100
SPOT SHAPE CORRECTIONS 102
SORTING ROUTINES 102
SCALING 103
ANALYSIS OF DISCREPANCIES 103
MODIFICATION FUNCTIONS 104
PART II: STATISTICS 106
CHAPTER 12. GENERAL THEORY OF STATISTICS 108
INTRODUCTION 108
ASSESSMENT OF THE ACCURACY OF MEASUREMENTS 108
DISTRIBUTION OF THE SUM OF INDEPENDENT RANDOM VARIABLES 109
THE NORMAL OR GAUSSIAN DISTRIBUTION 109
THE CENTRAL LIMIT THEOREM 110
STATISTICAL INFERENCE: CONFIDENCE RANGES 112
ONE PARAMETER SIGNIFICANCE TESTS 113
WEIGHTED MEAN VALUES 113
MULTIVARIATE PROBABILITY DISTRIBUTIONS 114
CHAPTER 13. ERRORS IN FOURIER SERIES 116
ELECTRON DENSITY ERRORS 116
COORDINATE ERRORS 118
ESTIMATES OF s2(F) 120
CHAPTER 14. ERRORS IN LEAST-SQUARES METHODS 121
DERIVATION OF ERROR FORMULAE 121
RELATIONS BETWEEN LEAST-SQUARES AND FOURIER METHODS 124
SIMPLE FORMULA FOR COORDINATE ACCURACY 124
CHAPTER 15. STATISTICAL PROPERTIES OF RECIPROCAL SPACE 126
INTRODUCTION 126
INTENSITY DISTRIBUTIONS 127
ABNORMAL INTENSITY AVERAGES 134
RECOMMENDED PROCEDURE FOR ESTIMATING < 1>
CHAPTER 16. THE SCALING OF INTENSITIES 142
PROGRAM FOR SCALING INTENSITY DATA AND FOR APPLYING STATISTICAL TESTS 147
PATHOLOGICAL CASES 148
PART III: PHASE DETERMINATION 158
CHAPTER 17. PHASE DETERMINATION FOR CENTROSYMMETRIC CRYSTALS BY PROBABILITY METHODS 160
INTRODUCTION 160
NORMALISED STRUCTURE FACTORS 160
PROCEDURE FOR S15 S2, AND S3 162
E-MAPS 169
ALTERNATIVE PROCEDURE 171
TRIPLES FORMULA 172
CONCLUSION 174
CHAPTER 18. THE APPLICATIONS OF THE SAYRE SIGN RELATIONSHIP TO STRUCTURE DETERMINATION 175
THE SAYRE SIGN RELATIONSHIP 175
AN IBM 704 PROGRAM 190
CONCLUSION 191
CHAPTER 19. ISOMORPHOUS REPLACEMENT METHODS 192
NOTATION 192
METHOD IN OUTLINE 192
APPROXIMATE SCALING OF DATA 194
SINGLE ISOMORPHOUS REPLACEMENT METHOD 195
SYNTHESES FOR FINDING HEAVY ATOMS: DELTA F-SQUARED PATTERSON 197
THE C2 SYNTHESIS 201
COMPUTING HAZARDS IN THE ß TYPE SYNTHESIS 202
LEAST-SQUARES REFINEMENT OF HEAVY ATOM PARAMETERS 202
ALTERNATE PHASE DETERMINATION AND HEAVY ATOM REFINEMENT 205
DIFFERENCE FOURIERS 206
COMPUTATION OF PHASE ANGLES 206
THE BEST FOURIER 210
CALCULATING THE CENTROID AND FIGURE OF MERIT 211
INCORPORATION OF ANOMALOUS SCATTERING DATA 213
CIS AND DIS 214
PART IV: PROGRAMMING 216
CHAPTER 20. SOME CRYSTALLOGRAPHIC PROGRAMS IN FORTRAN 218
1. INTRODUCTION 218
2. FORTRAN PROGRAMMING SYSTEM 219
3. FORMAT STATEMENTS 221
4. FUNCTIONS 221
5. BRANCHING OR TRANSFER OF CONTROL 222
6. PROGRAMMING A LOOP 222
7. ARRAYS OF NUMBERS 223
8. TABLE LOOK-UP 223
9. COMBINING OF PROGRAMS 225
10. RECIPROCAL UNIT CELL FROM DIRECT CELL 225
11. CALCULATION OF BRAGG ANGLES 226
12. CALCULATION OF ATOMIC SCATTERING FACTORS 227
13. STRUCTURE FACTOR CALCULATION 228
14. LEAST-SQUARES PROGRAM 228
15. FOURIER SERIES CALCULATION 229
16. PATTERSON SYNTHESIS 230
17. SOME PROGRAMS AVAILABLE 231
18. EXCHANGE OF PROGRAMS 231
APPENDIX 232
EXAMPLES 236
ANSWERS TO EXAMPLES 244
REFERENCES 259
INDEX 262