On the fast computing of three-dimensional Fourier transforms of crystallographic data via external storage

Fast Fourier Transforms (FFT) are widely known as useful tools for the evaluation of spectral data. This article discusses the applicability of FFT methods to crystallographic problems. Formulae are derived which make it possible to use fast Fourier transforms for the general Fourier summation of crystallographic data in all space groups and for the computation of slant planes at arbitrary positions in the unit cell. As even moderate resolutions would produce arrays of data too large to fit within internal computer memory, they must be kept to an external storage device. The organization of data and the resulting time requirements are thoroughly discussed. An ALGOL 60 program has been developed with as many redundancies eliminated as possible. An example shows that for a problem of moderate size this algorithm is faster by an order of magnitude than those which have traditionally been used.