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Theory and Practice of Direct Methods in Crystallography

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概要 Direct methods of crystal structure determination are usually associated with techniques in which phases for a set of structure factors are determined from the corresponding experimental amplitudes by... probabilistic calcula tions. It is thus implied that such ab initio phase calculations do not require a knowledge of atomic positions, and this basis distinguishes direct methods from other techniques for structure determination. An acceptably wider interpretation of the term direct methods leads to other important applica tions involving, inter alia, the use of heavy atoms, resolution-limited phase data for large molecules, rotation functions, and Fourier series. These topics are discussed in the later chapters of this book. Although some earlier theoretical investigations were made by Harker and Kaspar, direct methods may be considered to have begun around the year 1950. Important landmarks in the development of the subject include the book by Hauptmann and Karle, The Centrosymmetric Crystal (1953), the definitive paper by Karle and Karle in Acta Crystallographica (1966), and the recent (1978) sophisticated program package MULTAN 78 produced mainly by Germain, Main, and Woolfson. Woolfson's book, Direct Methods in Crystallography, was published in 1961, but because of the rapid progress in direct methods, much of it soon became outmoded. It is interesting to note that direct methods nearly came into being many years earlier. Certainly the E2 relationship was used implicitly by Lonsdale in 1928 in determining the crystal structure of hexamethylbenzene.続きを見る
目次 1. Principles of Direct Methods of Phase Determination in Crystal Structure Analysis
1.1. Introduction
1.2. Spherical Symmetry of Atoms: Sayre's Equation
1.3. Unitary and Normalized Structure Factors
1.4. Karle-Hauptman Determinants
1.5. Structure Invariants and Seminvariants
1.6. Probability Theory
1.7. Solving the Phase Problem for a Real Structure
1.8. Refinement of Phases
1.9. Possible Future Developments of Direct Methods
2. Definition of Origin and Enantiomorph and Calculation of |E| Values
I: Definition of Origin and Enantiomorph
2.1. Introduction
2.2. Some Preliminaries
2.3. Invariance
2.4. Variation of Phase among the Laue-Related Reflections
2.5. Defining the Origin
2.6. Some Unusual Requirements of S/I Selectors
2.7. In Conclusion
II: Calculation of |E| Values
3. Symbolic Addition and Multisolution Methods
3.1. Introduction
3.2. Symbolic Addition: Centrosymmetric Case
3.3. Symbolic Addition: Noncentrosymmetric Case
3.4. Advantages and Disadvantages of Symbolic Addition
3.5. Multisolution Methods
3.6. Success Is Not Guaranteed
3.7. More Recent Developments of the Multisolution Method: Magic Integers
4. Probabilistic Theory of the Structure Seminvariants
4.1. Major Goal
4.2. Introduction
4.3. Structure Invariants
4.4. Structure Seminvariants
4.5. The Structure Seminvariants Link the Observed Magnitudes |E| with the Desired Phases ?
4.6. Probabilistic Background
4.7. Three-Phase Structure Invariant
4.8. Four-Phase Structure Invariant (Quartet)
4.9. The Neighborhood Principle
4.10. More on Quartets: Higher Neighborhoods
4.11. Two-Phase Structure Seminvariants (Pairs)
4.12. Concluding Remarks
5. Application of Calculated Cosine Invariants in Phase Determination
5.1. Introduction
5.2. Accuracy of Cosine Calculations
5.3. Quartets, Quintets, and Triplets
5.4. ?1 Cosines
5.5. Pair Relationships
5.6. Strong Enantiomorph Selection
5.7. NQEST
5.8. Automated Procedures
5.9. Exercises
6. Phase Correlation with Calculated Cosine Invariants for Routine Structure Analysis
6.1. Phase Correlation Procedure
6.2. Simple Cosine-Invariant Calculations
6.3. Phase Correlation with Calculated Triple Invariants
6.3.1. Centrosymmetric Space Groups
7. Application of Direct Methods to Difference Structure Factors
7.1. Introduction
7.2. Difference Structure Factors
7.3. Description of the Procedure
7.4. Some Observations
8. Phase Extension and Refinement Using Convolutional and Related Equation Systems
8.1. Introduction
8.2. Outline of the Convolutional Equation Systems
8.3. Outlines of the Phasing Methods Based on the Convolutional Equation Systems
8.4. Comments on the Phasing Methods Based on the Convolutional Equation Systems
8.5. Computational and Practical Aspects
8.6. Summary
9. Maximum Determinant Method
9.1. Introduction
9.2. Inequalities and Algebraic Properties of Determinants
9.3. Maximum Determinant Rule
9.4. Practical Applications
9.5. New Gram Determinants
10. Molecular Replacement Method
10.1. Introduction
10.2. Preliminary Theoretical Considerations
10.3. Rotation Function
10.4. Translation Problem
10.5. Phase Determination
10.6. Noncrystallographic Symmetry and Heavy-Atom Searches
10.7. Applications of Molecular Replacement
10.8. Conclusions.
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登録日 2020.06.27
更新日 2020.06.28