Crystal Structure Determination

The Role of the Cosine Seminvariants

  • Herbert A. Hauptman

Table of contents

  1. Front Matter
    Pages i-xii
  2. The Theoretical Basis

  3. Techniques of Implementation

    1. Front Matter
      Pages 185-186
    2. Herbert A. Hauptman
      Pages 187-209
    3. Herbert A. Hauptman
      Pages 227-249
  4. The Applications

    1. Front Matter
      Pages 251-251
    2. Charles Weeks
      Pages 307-326
    3. Herbert A. Hauptman
      Pages 327-359
    4. Herbert A. Hauptman
      Pages 361-384
  5. Back Matter
    Pages 397-407

About this book


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

Authors and affiliations

  • Herbert A. Hauptman
    • 1
  1. 1.Mathematical Biophysics DepartmentMedical Foundation of BuffaloBuffaloUSA

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