False detail in three-dimensional Fourier representations of crystal structures

Abstract

An investigation is made of the diffraction rings to be expected in a three-dimensional Fourier representation of a crystal strueture when the Fourier series germinates while the coefficients are appreciable. The method followed is an extension of that used by Bragg and West in discussing the analogous problem for two-dimensional projections. The diffraetion rings for one-, two-and three-dimensional representations are compared. The maxima representing heavy atoms in a two-dimensional Fourier projection of a crystal structure are often surrounded by one or more nearly circular regions of negative density, the origin of which is well understood. They are diffraction rings, closely analogous to those which surround the image of a star viewed with a teleseope of small aperture, and are due t.o the fact that in summing the Fourier series from which the pro-jeetion is obtained it has been terminated while its coefficients are still appreciable. We may think of the spectra that can be given by the crystal, the amplitudes of which give the coefficients of the series, as associated with the points of the reciprocal lattice. For a two-dimensional projection, the reciprocal-lattice points concerned are those lying in a plane passing through the origin and perpendicular to a zone axis of the crystal. Terms corresponding to all points in the plane up to a certain distance from the origin are included in the summation, and we may think of the circle drawn on the plane, having the origin as centre, and including these points, as defining an equivalent optical aperture for the projection, which is, formally, an optical image. The problem has been discussed from this point of view by Bragg & West (1930), who calculated, with certain approximations, the form of the rings for a crystal consisting of point atoms, and found that their results agreed faMv closely, as regards the positions of the maxima and minima, with the rings obtained by making a projection of a fictitious rock-salt: crystal, in which the atoms scattered a.s Hartree atoms at rest. Three-dimensional Fourier series are being increasingly used in crystal analysis, and it seems desirable to con-sider the analogous problem when tim volume density is determined by means of such a series for points in a plane passing through the centre of the atom. Let b= be the vector from the origin to the reciprocal-lattice point m, and let F(m) be the strueture amplitude of the speetrum corresponding to this point. The electron density at a point in the structure at a vector distanee r from tile origin is given bv l