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**

- Structure Factors of Crystals and of Distributions in General

- Structure Factors and Transforms

- Transforms of Sets of Points

Weighted points on any lattice

The points comprising a lattice

- 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

- 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

- Transforms of Distributions

A distribution within a cube

Spherical volumes, shells and surfaces

A comparative study

Patterson distributions

- Small Crystals

Appendix. Fourier Series and Fourier's Integral Theorem