Direct phasing in crystallography: fundamentals and applications
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Oxford
Oxford Univ. Press
1998
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| Schriftenreihe: | International Union of Crystallography: International Union of Crystallography monographs on crystallography
8 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 767 S. Ill., graph. Darst. |
| ISBN: | 0198500726 |
Internformat
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| 245 | 1 | 0 | |a Direct phasing in crystallography |b fundamentals and applications |c Carmelo Giacovazzo |
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| 490 | 1 | |a International Union of Crystallography: International Union of Crystallography monographs on crystallography |v 8 | |
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Datensatz im Suchindex
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| adam_text | DIRECT PHASING IN CRYSTALLOGRAPHY FUNDAMENTALS AND APPLICATIONS CARMELO
GIACOVAZZO PROFESSOR OF CRYSTALLOGRAPHY, UNIVERSITY OF BARI, ITALY
INTERNATIONAL UNION OF CRYSTALLOGRAPHY OXFORD UNIVERSITY PRESS 1998
CONTENTS ABBREVIATIONS XXIV 1. WILSON STATISTICS 1 1.1 INTRODUCTION 1
1.2 THE STRUCTURE FACTOR 1 1.3 THE ALGEBRAIC FORM OF THE STRUCTURE
FACTOR 2 1.4 ELEMENTS OF STRUCTURE FACTOR ALGEBRA 5 FRIEDEL LAW 6
EFFECTS OF SYMMETRY OPERATORS IN RECIPROCAL SPACE 6 DETERMINATION OF
REFLECTIONS WITH RESTRICTED PHASE VALUES 7 SYSTEMATIC ABSENCES 9 1.5
STRUCTURE FACTOR STATISTICS: FIXED INDICES, RANDOM POSITIONS 11 1.6
STRUCTURE FACTOR STATISTICS IN PI 12 1.7 THE INFLUENCE OF SYMMETRY
ELEMENTS ON STRUCTURE FACTOR STATISTICS 14 STRUCTURE FACTOR STATISTICS
IN PI 14 STRUCTURE FACTOR STATISTICS FOR SPACE GROUPS WITH A PRIMITIVE
UNIT CELL 15 STRUCTURE FACTOR STATISTICS FOR SPACE GROUPS WITH A CENTRED
UNIT CELL 17 1.8 THE DISTRIBUTIONS P(I) : FL 1.9 THE NORMALIZED
STRUCTURE FACTOR AND THE UNITARY STRUCTURE FACTOR 18 1.10 THE
DISTRIBUTIONS P(||) AND P(Z) 22 1.11 STATISTICS OF STRUCTURE FACTORS:
FIXED STRUCTURE, RANDOM INDICES 24 1.12 CUMULATIVE DISTRIBUTIONS 27 1.13
STATISTICAL CRITERIA FOR DISCRIMINATING BETWEEN CENTROSYMMETRIC AND
NON-CENTROSYMMETRIC STRUCTURES 28 1.14 NON-IDEAL DISTRIBUTIONS OF THE
STRUCTURE FACTOR 31 1.15 ABSOLUTE SCALING OF THE INTENSITIES. THE WILSON
METHOD 33 1.16 A LIST OF TEST STRUCTURES AND OF DIRECT METHODS PROGRAMS
38 1.17 ABOUT THE SHAPE OF THE WILSON PLOT 39 CIV CONTENTS 1.18 ABOUT
THE RELIABILITY OF THE WILSON PLOT SCALING METHOD 44 1.19 THE
NORMALIZATION PROCESS IN THE PRESENCE OF PRIOR INFORMATION 49 1.20 THE
STANDARD REFLECTIONS AND THE UNIQUE LIST 52 1.21 INTENSITY STATISTICS OF
SOME SPECIAL CRYSTALS 53 TWINNING 55 INCOMMENSURATELY MODULATED
STRUCTURES 55 APPENDIX 1.A ABOUT PSEUDOSYMMETRIES 56 1.A.1
PSEUDOSYMMETRY 56 1.A.2 HYPERSYMMETRY 58 APPENDIX L.B FOURIER SERIES
REPRESENTATION OF P(E) 69 L.B.L THE CENTROSYMMETRIC CASE 70 1.B.2 THE
NON-CENTROSYMMETRIC CASE 71 APPENDIX L.C A MODIFIED WILSON PLOT
PROCEDURE 73 2. STRUCTURE INVARIANTS, SEMINVARIANTS AND THE ORIGIN
PROBLEM 75 2.1 INTRODUCTION 75 2.2 THE CONCEPT OF STRUCTURE INVARIANTS
75 2.3 ORIGIN SHIFT AND SYMMETRY OPERATORS 78 2.4 ALLOWED OR PERMISSIBLE
ORIGINS AND TRANSLATIONS IN PRIMITIVE SPACE GROUPS 80 2.5 THE
SEMINVARIANCE CONCEPT 86 2.6 STRUCTURE SEMINVARIANTS 92 2.7 ORIGIN
DEFINITION IN PRIMITIVE UNIT CELLS 93 2.8 ORIGIN DEFINITION IN CENTRED
CELLS 103 2.9 ENANTIOMORPH DEFINITION 107 2.10 FURTHER CONSIDERATIONS ON
ORIGIN AND ENANTIOMORPH DEFINITION 109 3. TRIPLET INVARIANT ESTIMATION
AND CLASSICAL DIRECT PHASING PROCEDURES 113 3.1 INTRODUCTION 113 3.2 THE
POSITIVITY OF A TRIPLET INVARIANT: ALGEBRAIC CONSIDERATIONS 113 3.3 THE
PROBABILITY DENSITY FUNCTION P( H E H , E K , E H _ K ,CT K , *_,)
115 3.4 TRIPLET PHASE ESTIMATION 120 3.5 THE TANGENT FORMULA 123 3.6 THE
DISTRIBUTION OF A H 127 3.7 TRIPLET ESTIMATES FOR CENTROSYMMETRIC SPACE
GROUPS 129 3.8 ABOUT TRIPLET INVARIANT RELIABILITY AND BASIC POSTULATES
131 CONTENTS XV 3.9 PSI-ZERO TRIPLETS 134 3.10 SCHEME OF PROCEDURE FOR
PHASE DETERMINATION 135 3.11 SET-UP OF PHASE RELATIONSHIPS - 135 3.12
FIXING THE REFLECTIONS OF THE STARTING SET 139 3.13 ASSIGNMENT OF THE
STARTING PHASES 142 3.14 PHASE DETERMINATION 146 3.15 FINDING OUT THE
CORRECT SOLUTION 151 3.16 MAP INTERPRETATION 156 3.17 THE EFFICIENCY OF
THE WEIGHTED TANGENT FORMULA BASED ON COCHRAN ESTIMATES 159 3.18
STRUCTURE COMPLETION AND REFINEMENT 162 APPENDIX 3.A OTHER MULTISOLUTION
METHODS 167 3.A.1 THE SYMBOLIC ADDITION PROCEDURE 167 3.A.2 THE
PRIMARY-SECONDARY METHOD 168 3.A.3 LINEAR EQUATIONS METHOD 170 3.A.4
COSINE LEAST SQUARES METHOD 171 3.A.5 THE X-Y METHOD 172 4. DIRECT
METHODS AND REAL SPACE PROPERTIES 173 4.1 INTRODUCTION 173 4.2 WILSON
STATISTICS AND REAL SPACE PROPERTIES 173 4.3 THE SAYRE EQUATION AND ITS
MODIFIED FORM 175 4.4 THE GENERALIZATION OF THE SAYRE EQUATION AND THE
POLYNOMIAL METHOD 178 4.5 THE INTEGRAL CRITERIA V 180 4.6 DIRECT METHODS
AND PATTERSON MAP PROPERTIES 183 4.7 POSITIVENESS OF ELECTRON DENSITY
AND KARLE-HAUPTMAN DETERMINANTS 186 4.8 THE ALGEBRA OF KARLE-HAUPTMAN
DETERMINANTS 193 4.9 INEQUALITIES AND PROBABILITY THEORY 196 4.10
ELECTRON DENSITY MODIFICATION PROCEDURES 198 APPENDIX 4.A DIRECT METHODS
AND PATTERSON MAP PROPERTIES: ADDITIONAL APPROACHES 201 APPENDIX 4.B
ESTIMATING STRUCTURE SEMINVARIANTS FROM THE HARKER SECTIONS 207 APPENDIX
4.C PATTERSON DECONVOLUTION METHODS 208 4.C.1 SUPERPOSITION TECHNIQUES
208 4.C.2 THE MULTIPLE IMPLICATION FUNCTION 212 WI CONTENTS 5. THE
METHOD OF JOINT PROBABILITY DISTRIBUTIONS 214 5.1 INTRODUCTION 214 5.2
MULTIVARIATE DISTRIBUTIONS IN CENTROSYMMETRIC STRUCTURES: THE CASE OF
INDEPENDENT RANDOM VARIABLES 216 5.3 MULTIVARIATE DISTRIBUTIONS IN
NON-CENTROSYMMETRIC STRUCTURES: THE CASE OF INDEPENDENT RANDOM VARIABLES
219 5.4 SIMPLIFIED JOINT PROBABILITY DENSITY FUNCTIONS IN THE ABSENCE OF
PRIOR INFORMATION 222 5.5 THE JOINT PROBABILITY DENSITY FUNCTION WHEN
SOME PRIOR INFORMATION IS AVAILABLE 225 5.6 THE CALCULATION OF P(E) IN
PI IN THE ABSENCE OF PRIOR INFORMATION 227 5.7 THE CALCULATION OF P(R,
4 ) IN PI IN THE ABSENCE OF PRIOR INFORMATION 230 5.8 THE CALCULATION OF
P(R, ) IN ANY SPACE GROUP IN THE ABSENCE OF PRIOR INFORMATION 233
APPENDIX 5.A P{E) AND THE SADDLE POINT METHOD 234 5.A.1 ABOUT THE SADDLE
POINT METHOD 234 5.A.2 P(E) IN PI VIA THE SADDLE POINT METHOD 236 5.A.3
P(E T , E 2 , ...,*) IN PI VIA THE SADDLE POINT METHOD 239 5.A.4 P(E 1
,E 2 ,***,E N ) IN ANY SPACE GROUP VIA THE SADDLE POINT METHOD . 240 6.
REPRESENTATIONS OF STRUCTURE INVARIANTS AND SEMINVARIANTS 243 6.1
INTRODUCTION 243 6.2 THE FIRST REPRESENTATION OF A STRUCTURE INVARIANT
243 6.3 THE UPPER REPRESENTATIONS OF A STRUCTURE INVARIANT 247 6.4
REPRESENTATIONS AND MULTIPOLES 248 6.5 REPRESENTATION THEORY EXTENDED TO
ISOMORPHOUS STRUCTURES 252 6.6 THE FIRST REPRESENTATION OF A STRUCTURE
SEMINVARIANT 256 6.7 ALGEBRAIC PROPERTIES RELATED TO THE RANK OF THE
S.S.S 260 6.8 FINDING THE FIRST REPRESENTAITON OF AN S.S. APPLICATIONS
263 6.9 UPPER REPRESENTATIONS OF STRUCTURES SEMINVARIANTS 270 6.10 A
BRANCH: THE METHOD OF COMPLEMENTARY SEMINVARIANTS 271 APPENDIX 6.A TWO
ALGEBRAIC PROPOSITIONS 272 7. THE PROBABILISTIC ESTIMATION OF TRIPLET
INVARIANTS 275 7.1 INTRODUCTION 275 CONTENTS XVII 7.2 THE ESTIMATION OF
IN PI IN THE ABSENCE OF PRIOR INFORMATION 275 7.3 THE ESTIMATION OF
IN PI IN THE ABSENCE OF PRIOR INFORMATION 279 7.4 THE ESTIMATION OF
IN ANY SPACE GROUP VIA THE FIRST REPRESENTATION: NO PRIOR INFORMATION
AVAILABLE 280 7.5 THE ESTIMATION OF 3 FROM ITS FIRST REPRESENTATION
WHEN PRIOR INFORMATION IS AVAILABLE 284 7.6 RECOVERING OF THE COMPLETE
FROM A PARTIAL STRUCTURE (FIRST REPRESENTATION FORMULA) 287 7.7 THE
ESTIMATE OF A TRIPLET PHASE WHEN PSEUDOTRANSLATIONAL SYMMETRY IS PRESENT
(FIRST REPRESENTATION FORMULA) 292 7.8 THE ESTIMATE OF THE TIPLET PHASE
VIA ITS SECOND REPRESENTATION 294 7.9 THE ESTIMATION OF 3 FROM ITS
SECOND REPRESENTATION IN THE PRESENCE OF PSEUDOTRANSLATIONAL SYMMETRY
302 7.10 THE PROBABILISTIC ESTIMATION OF NON-MEASURED DIFFRACTION
MAGNITUDES 306 APPENDIX 7.A COSINE INVARIANT COMPUTATION TECHNIQUES AND
THE P 6 FORMULA 309 8. THE PROBABILISTIC ESTIMATION OF QUARTET
INVARIANTS 311 8.1 INTRODUCTION 311 8.2 THE ESTIMATION OF $ IN PI FROM
ITS FIRST REPRESENTATION: THE HAUPTMAN FORMULA 311 8.3 THE ESTIMATION OF
IN PI FROM ITS FIRST REPRESENTATION: THE HAUPTMAN FORMULA 315 8.4 THE
ESTIMATION OF & IN PI AND PI FROM ITS FIRST REPRESENTATION: THE
GIACOVAZZO FORMULA 316 8.5 FINDING QUARTETS 318 8.6 ABOUT QUARTET
RELIABILITY 319 8.7 THE ESTIMATION OF $ FROM ITS FIRST REPRESENTATION IN
ANY SPACE GROUP 325 8.8 THE ROLE OF THE QUARTETS IN SAYTAN AND SHELX 327
8.9 THE INTEGRATION OF TRIPLET AND QUARTET RELATIONSHIPS IN SIR92 332
8.10 THE INTEGRATION OF STRONG TRIPLETS, NEGATIVE QUARTETS, AND PSI-ZERO
TRIPLETS IN THE SIR92 PHASING PROCESS 337 8.11 IMPROVING QUARTET
ESTIMATES VIA THEIR SECOND REPRESENTATION 340 8.12 QUARTETS AS A FIGURE
OF MERIT 340 APPENDIX 8.A MATHEMATICAL DERIVATION OF THE PHASE
RELATIONSHIP (8.2) 342 APPENDIX 8.B MATHEMATICAL DERIVATION OF THE SIGN
RELATIONSHIP (8.9) 346 APPENDIX 8.C SPECIAL QUARTETS 348 .XVIII CONTENTS
9. THE PROBABILISTIC ESTIMATION OF QUINTET INVARIANTS 352 9.1
INTRODUCTION 352 9.2 THE ESTIMATION OF 3 = H + K + , + *- H+K+L+M IN PI
AND PI VIA ITS FIRST REPRESENTATION 353 9.3 ESTIMATION OF 4 IN ANY
SPACE GROUP FROM ITS FIRST REPRESENTATION: SOME GENERAL CRITERIA 355 9.4
THE ROLE OF THE QUINTETS IN DIRECT PROCEDURES 356 10. THE PROBABILISTIC
ESTIMATION OF ONE- AND TWO-PHASE STRUCTURE SEMINVARIANTS 360 10.1
INTRODUCTION 360 10.2 ESTIMATION OF ONE-PHASE STRUCTURE SEMINVARIANTS OF
FIRST RANK VIA THEIR FIRST REPRESENTATION 360 10.3 ESTIMATION OF
ONE-PHASE STRUCTURE SEMINVARIANTS OF FIRST RANK VIA THEIR SECOND
REPRESENTATION 362 10.4 THE ESTIMATION OF 4 H VIA THE METHOD OF QUARTET
COMPLEMENTARY INVARIANTS 365 10.5 THE ESTIMATION OF TWO-PHASE STRUCTURE
SEMINVARIANTS OF FIRST RANK VIA THEIR FIRST REPRESENTATION 366 10.6
ABOUT THE ROLE OF SEMINVARIANTS IN DIRECT PROCEDURES 368 APPENDIX 10.A
SOME MATHEMATICAL DETAILS ABOUT T X ESTIMATES 370 11. PROBABILISTIC
DETERMINANTAL APPROACHES 372 11.1 INTRODUCTION 372 11.2 THE MAXIMUM
DETERMINANT RULE 372 11.3 THE REGRESSION EQUATION 375 11.4 ABOUT
DETERMINANTAL APPROACHES 377 11.5 PRACTICAL ASPECTS AND APPLICATIONS TO
SMALL MOLECULES 378 12. PHASING VIA NEUTRON AND ELECTRON DATA 381 12.1
INTRODUCTION 381 12.2 ELECTRON SCATTERING 381 12.3 COLLECTING ELECTRON
DIFFRACTION DATA 384 12.4 PERTURBATIONS TO ELECTRON DIFFRACTION
AMPLITUDES 385 12.5 A TYPICAL EXPERIMENTAL PROCEDURE FOR ELECTRON
DIFFRACTION STUDIES 388 12.6 ELECTRON MICROSCOPY, IMAGE PROCESSING, AND
DIRECT METHODS 392 12.7 NEUTRON SCATTERING 394 12.8 ABOUT THE VIOLATION
OF THE POSITIVITY POSTULATE 397 CONTENTS XIX APPENDIX 12.A ABOUT THE
ELASTIC SCATTERING OF ELECTRONS: THE KINEMATIC APPROXIMATION 403
APPENDIX 12.B ABOUT HREM IMAGE FORMATION 407 13. DIRECT METHODS VERSUS
POWDER DATA 410 13.1 INTRODUCTION 410 13.2 ABOUT THE EXPERIMENTAL POWDER
PATTERNS 412 13.3 ABOUT POWDER-PATTERN INDEXING 417 13.4 THE RECOVERY OF
THE F 2 419 13.5 PITFALLS IN PHASING POWDER DATA 423 13.6 PRIOR
INFORMATION VERSUS THE FULL PATTERN DECOMPOSITION PROGRAMS 433 13.7 A
PRACTICAL CASE 440 13.8 POWDER DATA AND ANOMALOUS DISPERSION EFFECTS 441
14. MACROMOLECULAR CRYSTALLOGRAPHY TECHNIQUES AND TRADITIONAL DIRECT
METHODS 445 14.1 INTRODUCTION 445 14.2 THE ISOMORPHOUS REPLACEMENT
METHOD 446 THE ISOMORPHOUS HEAVY ATOM DERIVATIVE 449 THE DETERMINATION
OF HEAVY ATOM POSITIONS 450 THE SIR METHOD - 453 THE MIR METHOD 455
REFINEMENT OF HEAVY ATOM PARAMETERS 457; 14.3 DENSITY MODIFICATION
TECHNIQUES IN MACROMOLECULAR CRYSTALLOGRAPHY 457 POSITIVITY AND
ATOMICITY CONSTRAINTS TO ELECTRON DENSITY MODIFICATIONS 458 SOLVENT
FLATTENING 459 THE ELECTRON DENSITY HISTOGRAM METHOD 459 SKELETONIZATION
TECHNIQUES 462 MOLECULAR AVERAGING 463 14.4 FINDING THE HEAVY ATOM
POSITIONS BY DIRECT METHODS 464 14.5 EARLY DIRECT METHODS APPLICATIONS:
PHASE EXTENSION AND REFINEMENT 465 TANGENT TECHNIQUES 465 THE SAYRE
EQUATION 466 DETERMINANTAL METHOD 467 XX CONTENTS 14.6 ABOUT THE LIMITS
OF THE TANGENT FORMULA AND OF THE SAYRE EQUATION 468 14.7 TRADITIONAL
DIRECT METHODS AND THE USE OF PRIOR INFORMATION 475 14.8 SHAKE AND
BAKE AND HALF-BAKE : TWO DIRECT METHODS PROGRAMS FOR MACROMOLECULES
477 THE SHAKE AND BAKE METHOD 478 THE HALF-BAKE APPROACH 481 APPENDIX
14.A THE CRICK AND MAGDOFF RELATION 483 APPENDIX 14.B PROTEIN PHASE
ESTIMATE INCORPORATING THE TREATMENT OF ERRORS 485 APPENDIX 14.C SOLVENT
FLATTENING 490 14.C.1 WANG, LESLIE, AND OTHER PROCEDURES 490 APPENDIX
14.D ABOUT FOURIER SYNTHESES OF PARTICULAR USE IN MACROMOLECULAR
CRYSTALLOGRAPHY 495 APPENDIX 14.E SOLVENT CONTENT ACCORDING TO MATTHEWS
500 APPENDIX 14.F THE AB INITIO CALCULATION OF LOW-RESOLUTION ENVELOPES
501 15. THE INTEGRATION OF DIRECT METHODS WITH ISOMORPHOUS REPLACEMENT
TECHNIQUES 509 15.1 INTRODUCTION 509 15.2 THE JOINT PROBABILITY
DISTRIBUTION OF THE TWO NORMALIZED STRUCTURE FACTORS E PH , E DH AND
RELATED DISTRIBUTIONS 510 15.3 THE JOINT PROBABILITY DISTRIBUTION P(E PH
,E PK ,E PL ,E DH ,E DK ,E DL )WITH H + K + L = 0 512 15.4 NATIVE
PROTEIN-HEAVY ATOM DERIVATIVE. THE DISTRIBUTIONS FOR TWO-PHASE
INVARIANTS 514 15.5 NATIVE PROTEIN-HEAVY ATOM DERIVATIVE. THE
DISTRIBUTIONS FOR THREE-PHASE INVARIANTS 517 15.6 A PROCEDURE FOR THE
DIRECT PHASING OF PROTEINS 521 15.7 ABOUT THE QUALITY OF THE ELECTRON
DENSITY MAPS 530 15.8 ESTIMATE OF THE QUARTET INVARIANTS 535 15.9 THE
TRIPLET INVARIANT ESTIMATE VIA ITS SECOND REPRESENTATION 540 15.10 THE
RELATION BETWEEN TRADITIONAL SIR AND DIRECT METHODS TECHNIQUES 542 15.11
IMPROVING DIRECT METHODS PHASES WITH HEAVY ATOM INFORMATION 546 APPENDIX
15.A KARLE S ALGEBRAIC RULE R ISO 549 APPENDIX 15.B ABOUT THE LOCAL
SCALING 550 16. THE INTEGRATION OF DIRECT METHODS WITH ANOMALOUS
DISPERSION TECHNIQUES 551 16.1 INTRODUCTION 551 ; CONTENTS XXI 16.2
VIOLATION OF THE FRIEDEL LAW 552 16.3 OAS: FINDING THE POSITIONS OF THE
ANOMALOUS SCATTERERS 556 16.4 OAS: A BASIC ALGEBRAIC FORMULA 559 16.5
OAS: THE RESOLUTION OF PHASE AMBIGUITY VIA ALGEBRAIC METHODS 561 16.6
OAS: THE RESOLUTION OF PHASE AMBIGUITY VIA TRADITIONAL DIRECT METHODS
564 16.7 MAD PHASING METHODS 566 16.8 SIRAS AND MIRAS: THE ALGEBRAIC
METHODS 569 16.9 WILSON STATISTICS FOR THE OAS CASE 577 16.10 THE
DISTRIBUTION P(E H ,E_ H ) AND RELATED DISTRIBUTIONS IN N.CS. CRYSTALS
581 16.11 THE ESTIMATION OF $ = H + _ H IN CS. CRYSTALS OR FOR SYMMETRY
RESTRICTED PHASES IN CS. SPACE GROUPS 585 16.12 DIRECT METHODS FOR THE
OAS CASE: A FIRST STEP 585 16.13 ALGEBRAIC ESTIMATES OF TRIPLET
INVARIANTS: KARLE S RULE 587 16.14 THE JOINT PROBABILITY DISTRIBUTION
P(E H ,E H2 ,E H3 ,E_ HI ,E_ H2 ,E_ HJ ) 591 16.15 THE FIRST
APPLICATIONS OF THE TRIPLET FORMULA 594 16.16 PERSPECTIVES FOR THE
INTEGRATION OF DIRECT METHODS WITH ANOMALOUS DISPERSION TECHNIQUES 595
APPENDIX 16.A PROBABILISTIC TREATMENT OF THE ERRORS IN THE SIRAS CASE
595 APPENDIX 16.B COEFFICIENTS IN THE DISTRIBUTION (16.76) 596 APPENDIX
16.C THE SAS MAXIMAL PRINCIPLE 599 APPENDIX 16.D ANOMALOUS SCATTERING
TERMS AND MAD PERFORMANCES 600 17. MOLECULAR REPLACEMENT TECHNIQUES AND
DIRECT METHODS 602 17.1 INTRODUCTION 602 17.2 ROTATION FUNCTIONS 604
17.3 TRANSLATION FUNCTIONS 608 17.4 DIRECT METHODS AND THE ROTATION
PROBLEM 615 17.5 THE TRANSLATION PROBLEM: THE ROLE OF THE TRIPLET
INVARIANTS 619 17.6 QUARTET INVARIANTS AND THE TRANSLATION PROBLEM 628
17.7 THE TRANSLATION PROBLEM IN POLAR SPACE GROUPS 634 APPENDIX 17.A
CALCULATION OF THE ROTATION FUNCTION IN ORTHOGONALIZED CRYSTAL AXES 635
APPENDIX 17.B ABOUT THE FAST ROTATION FUNCTION 645 APPENDIX 17.C
ROTATION FUNCTIONS USING PHASES 651 APPENDIX 17.D THE ESTIMATION OF |
F | 2 | F MOD | 2 653 XXII CONTENTS 18. MAXIMUM ENTROPY TECHNIQUES FOR
CRYSTAL STRUCTURE SOLUTION 655 18.1 INTRODUCTION 655 18.2 ENTROPY AS A
MEASURE OF THE MISSING INFORMATION 656 18.3 JAYNES MAXIMUM ENTROPY
FORMALISM 658 18.4 THE MAXIMUM ENTROPY CRITERION WHEN A SET OF STRUCTURE
FACTORS ARE KNOWN A PRIORI 664 18.5 MAXIMUM ENTROPY AND SADDLE POINT
METHODS 671 18.6 PHASE EXTENSION BY MAXIMUM ENTROPY 673 18.7 ENTROPY AND
LIKELIHOOD: A PHASING PROCEDURE 676 18.8 ABOUT VARIOUS IMPLEMENTATIONS
OF THE ME FORMALISM 678 18.9 SOME APPLICATIONS OF MAXIMUM ENTROPY 684
APPENDIX 18.A BAYESIAN STATISTICS AND MAXIMUM ENTROPY 687 APPENDIX 18.B
A SHORT RECAP OF NEWTON S METHOD 691 18.B.1 THE NEWTON-RAPHSON METHOD
691 18.B.2 MINIMIZING A FUNCTION OF SEVERAL VARIABLES: GRADIENT METHODS
693 18.B.3 CONSTRAINED OPTIMIZATION 694 APPENDIX 18.C ABOUT ENTROPICAL
PROPERTIES OF GAUSSIAN AND VON MISES DISTRIBUTIONS 695 APPENDIX MA SOME
BASIC RESULTS IN PROBABILITY THEORY 699 M.A.1 PROBABILITY DISTRIBUTION
FUNCTIONS 699 M.A.2- . MOMENTS OF A DISTRIBUTION 700 M.A.3 THE
CHARACTERISTIC FUNCTION 700 M.A.4 CUMULANTS OF A DISTRIBUTION 702 M.A.5
THE NORMAL OR GAUSSIAN DISTRIBUTION 703 M.A.6 THE CENTRAL LIMIT THEOREM
705 M.A.7 MULTIVARIATE DISTRIBUTIONS 706 M.A.8 EVALUATION OF THE MOMENTS
IN THE STRUCTURE FACTOR DISTRIBUTIONS 710 M.A.9 JOINT PROBABILITY
DISTRIBUTIONS OF THE SIGNS OF THE STRUCTURE FACTORS 712 M.A.10 SOME
MEASURES OF LOCATION AND DISPERSION IN THE STATISTICS OF DIRECTIONAL
DATA 713 APPENDIX M.B CONGRUENCES 715 APPENDIX M.C THE GAMMA FUNCTION
718 CONTENTS XXIII APPENDIX M.D THE HERMITE AND LAGUERRE POLYNOMIALS 720
APPENDIX M.E SOME RESULTS IN, THE THEORY OF BESSEL FUNCTIONS 723 M.E.I
BESSEL FUNCTIONS 723 M.E.2 GENERALIZED HYPERGEOMETRIC FUNCTIONS 728
APPENDIX M.F SOME DEFINITE INTEGRALS AND FORMULAE OF FREQUENT
APPLICATION 729 REFERENCES 733 INDEX 765
|
| any_adam_object | 1 |
| author | Giacovazzo, Carmelo |
| author_facet | Giacovazzo, Carmelo |
| author_role | aut |
| author_sort | Giacovazzo, Carmelo |
| author_variant | c g cg |
| building | Verbundindex |
| bvnumber | BV012362658 |
| classification_rvk | UQ 5000 |
| ctrlnum | (OCoLC)845047143 (DE-599)BVBBV012362658 |
| dewey-full | 548 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 548 - Crystallography |
| dewey-raw | 548 |
| dewey-search | 548 |
| dewey-sort | 3548 |
| dewey-tens | 540 - Chemistry and allied sciences |
| discipline | Chemie / Pharmazie Physik |
| format | Book |
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| id | DE-604.BV012362658 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:26:14Z |
| institution | BVB |
| isbn | 0198500726 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-008383128 |
| oclc_num | 845047143 |
| open_access_boolean | |
| owner | DE-703 DE-29T DE-83 |
| owner_facet | DE-703 DE-29T DE-83 |
| physical | XXIII, 767 S. Ill., graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series | International Union of Crystallography: International Union of Crystallography monographs on crystallography |
| series2 | International Union of Crystallography: International Union of Crystallography monographs on crystallography International Union of Cristallography book series |
| spelling | Giacovazzo, Carmelo Verfasser aut Direct phasing in crystallography fundamentals and applications Carmelo Giacovazzo Oxford Oxford Univ. Press 1998 XXIII, 767 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier International Union of Crystallography: International Union of Crystallography monographs on crystallography 8 International Union of Cristallography book series Kristallstrukturanalyse (DE-588)4137204-9 gnd rswk-swf Kristallstrukturanalyse (DE-588)4137204-9 s DE-604 International Union of Crystallography: International Union of Crystallography monographs on crystallography 8 (DE-604)BV005455767 8 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008383128&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Giacovazzo, Carmelo Direct phasing in crystallography fundamentals and applications International Union of Crystallography: International Union of Crystallography monographs on crystallography Kristallstrukturanalyse (DE-588)4137204-9 gnd |
| subject_GND | (DE-588)4137204-9 |
| title | Direct phasing in crystallography fundamentals and applications |
| title_auth | Direct phasing in crystallography fundamentals and applications |
| title_exact_search | Direct phasing in crystallography fundamentals and applications |
| title_full | Direct phasing in crystallography fundamentals and applications Carmelo Giacovazzo |
| title_fullStr | Direct phasing in crystallography fundamentals and applications Carmelo Giacovazzo |
| title_full_unstemmed | Direct phasing in crystallography fundamentals and applications Carmelo Giacovazzo |
| title_short | Direct phasing in crystallography |
| title_sort | direct phasing in crystallography fundamentals and applications |
| title_sub | fundamentals and applications |
| topic | Kristallstrukturanalyse (DE-588)4137204-9 gnd |
| topic_facet | Kristallstrukturanalyse |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008383128&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV005455767 |
| work_keys_str_mv | AT giacovazzocarmelo directphasingincrystallographyfundamentalsandapplications |