Dorothy Wrinch

Fourier Transforms and Structure Factors
The American Society for X-Ray and Electron Diffraction, February, 1946

This monograph was reprinted by the American Crystallographic Association, successor to the American Society for X-Ray and Electron Diffraction, May, 1966.

Preface

This Monograph has two aspects. On the one hand, it is presented as a contribution to the study of structure factors — or Fourier transforms — of atomic groupings which occur frequently in a wide variety of crystals, both organic and inorganic. Thus special attention is given to such cases as tetrahedral, octahedral and hexagonal arrays of like atoms. A section on the structure factors of small crystals is also included.

On the other hand, it is presented as a contribution to the x-ray analysis of megamolecular crystals as yet studied by few but destined, it would seem, to play an important role in the crystallography of the future. Megamolecular crystals confront crystallography with a new problem, since the structure of the molecules and indeed, to some extent, even the composition of the molecules is unknown. It is the belief of the Author that a systematic study of what may be called the language of structure factors is a necessary preliminary to the interpretation of the intensity maps of crystals made up of megamolecules of unknown structure. In the sequel the structure factors of distributions of different structural types are recorded. Such mathematical facts provide material for the study of the relationship between distributions and their structure factors — the fundamental theme throughout this Monograph.

The Marine Biological Laboratory
Woods Hole, Mass., August 1, 1945

Contents

1. Structure Factors of Crystals and of Distributions in General
    
2. Structure Factors and Transforms
    
3. Transforms of Sets of Points
      Weighted points on any lattice
      The points comprising a lattice
      
4. Transforms of Sets of Points on a Cubic Lattice
      The vertices of a cube
      Generalization
      The vertices of a tetrahedron
      The vertices of an octahedron
      The "benzene" point set
      Generalizations
      Patterson sets
      Point sets on a triangular mesh
      Point sets on a hexagonal mesh
      
5. Transforms of Sets of Many Points on a Cubic Lattice
      Patterned octahedral surfaces
      Patterned octahedral volumes
      Octahedral surfaces carrying a sub-pattern
      Patterned surfaces of truncated tetrahedra
      Patterned cube surfaces and volumes
      
6. Transforms of Distributions
      A distribution within a cube
      Spherical volumes, shells and surfaces
      A comparative study
      Patterson distributions
      
7. Small Crystals

Appendix. Fourier Series and Fourier's Integral Theorem