Crystal Structure Determination: The Role of the Cosine SeminvariantsThe 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. |
Contents
| 4 | |
| 12 | |
6 | 29 |
CHAPTER II | 64 |
Introduction | 67 |
Formulas for the Cosine Seminvariants Cos 1+2 | 88 |
5 | 112 |
The Joint Probability Distribution of Two Structure | 125 |
CHAPTER XII | 199 |
4 | 208 |
Using Calculated Cosines to Determine Values | 211 |
Valinomycin Strong Enantiomorph Discrimination | 216 |
4 | 218 |
The Tangent Techniques | 227 |
THE APPLICATIONS | 251 |
4 | 288 |
The Conditional Distribution of E Given | 142 |
k hk | 145 |
The Conditional Distribution of E3+ Given | 177 |
Part B TECHNIQUES OF IMPLEMENTATION | 185 |
3 | 197 |
Conclusion | 304 |
CHAPTER XIII | 385 |
| 397 | |
| 404 | |
Other editions - View all
Crystal Structure Determination: The Role of the Cosine Seminvariants H. Hauptman No preview available - 2012 |
Crystal Structure Determination: The Role of Cosine Semivariants H. Hauptman No preview available - 2014 |
Common terms and phrases
6a-fluorocortisol arbitrarily arbitrary assumed atoms average calculated cosine calculated values Chapter chosen components computed conditional distribution cos2 cosine invariants cosine seminvariants cosine values crystal structure cycles defined dependent modulo derived double sum E₁ E₂ enantiomorph Enke 12 epiandrosterone equation equivalence classes estradiol urea complex fixed functional form geometric structure factor given h₁ ĥ₂ hexestrol independent integer k₁ k₂ least-squares linear combinations linearly accessible modulo linearly dependent MDKS modified tangent procedure negative cosines normalized structure factor observed values pair permissible origins phase determination probability distribution quadruple R₁ R₂ random variable ranges uniformly rationally dependent reciprocal space reciprocal vector reflections restricted set of phases space group space group P21 specified structure invariants structure seminvariants Table Theorem triple product two-dimensional uniquely determined unit cell valinomycin vectors in reciprocal X X X zero μ μ μήν μν ΡΙ