Crystal Structure Determination

1. Front Matter

   Pages i-xii

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2. The Theoretical Basis

   1. Front Matter

      Pages 1-2

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   2. The Phase Problem

      • Herbert A. Hauptman

      Pages 3-65

   3. Explicit Formulas for the Cosine Seminvariants: The Algebraic Approach

      • Herbert A. Hauptman

      Pages 67-124

   4. The Joint Probability Distribution of Two Structure Factors and Related Conditional Distributions and Expectation Values

      • Herbert A. Hauptman

      Pages 125-157

   5. Probability Distribution of Three Structure Factors and Applications

      • Herbert A. Hauptman

      Pages 159-183

3. Techniques of Implementation

   1. Front Matter

      Pages 185-186

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   2. The Calculation of the Cosine Seminvariants

      • Herbert A. Hauptman

      Pages 187-209

   3. Using Calculated Cosines to Determine Values of Initial Phases

      • Herbert A. Hauptman

      Pages 211-225

   4. The Tangent Techniques

      • Herbert A. Hauptman

      Pages 227-249

4. The Applications

   1. Front Matter

      Pages 251-251

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   2. Two Androstane Derivatives; Least-Squares Analysis of the Cosine Invariants

      • Herbert A. Hauptman

      Pages 253-282

   3. The Estradiol·Urea Complex and 6α-Fluorocortisol; Two-Dimensional Invariants

      • Herbert A. Hauptman

      Pages 283-306

   4. Hexestrol; Quadruples

      • Charles Weeks

      Pages 307-326

   5. 9α-Fluorocortisol; Pairs

      • Herbert A. Hauptman

      Pages 327-359

   6. Valinomycin; Strong Enantiomorph Discrimination

      • Herbert A. Hauptman

      Pages 361-384

   7. 9-t-Butyl-9,10-Dihydroanthracene; Modified Tangent Procedure

      • Herbert A. Hauptman

      Pages 385-395

5. Back Matter

   Pages 397-407

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The central theme of this monograph is that the cosine seminvariants are the key to crystal structures. The cosine seminvariants are the cosines of those linear combinations of the phases (the so-called structure seminvariants) whose values, for a given functional form for the geometric structure factor, are uniquely determined by the crystal structure alone and are there­ fore independent of the choice of permissible origin. It follows that the cosine seminvariants themselves are uniquely determined, in general, by the observed magnitudes of the normalized structure factors. The values of the cosine seminvariants in turn lead unambiguously to the values of the individual phases and thus to the crystal structure by means of the E-map (Fourier synthesis). It is this property of the cosine seminvariants, that they serve to link the observed magnitudes with the desired phases of the normalized structure factors, which accounts for their importance and explains the emphasis which is here placed on their role.

• crystal
• crystal structure
• invariant
• lead
• phase
• structure
• synthesis

• Mathematical Biophysics Department, Medical Foundation of Buffalo, Buffalo, USA

  Herbert A. Hauptman

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