On the fast computing of three-dimensional Fourier transforms of crystallographic dataviaexternal storage

Abstract

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.

Treatment of negative indices in crystallographic fast Fourier transforms

“…There has been some interest in applying the fast Fourier transform (FFT) to crystallography since its discovery by Cooley & Tukey (1965). Methods have been described for incorporating the space-group symmetry (Ten Eyck, 1973), and for using backing store to accommodate the array of * Present address: Chemistry Department, The University, Olas$ow G1_2 8QQ, 8¢0tland, data (Singleton, 1967;Lange, Stolle & Huttner, 1973). The general formula is h, k, l are the indices of the Fourier coefficient Fhkt; (ha, he), (k,,,ke), (l,,,le) are the limits of the indices; Q(x,y,z) is the value of the transform at discrete coordinates x, y, and z.…”

“…On the kth pass data points 2 "-k apart are transformed and the results stored in an output array in locations 2 "-1 apart. The transform is (4) Lange, Stolle & Huttner (1973) deal with the crystallographic case where ha, ka and la may be negative. They modify each row of the array of coefficients by an additional phase factor depending on the row index.…”