Limitations on the determination of phases by means of inequalities

The inequality method of determining signs of Fourier coefficients is discussed from the point of view of the average sizes of the coefficients. It is shown theoretically and confirmed empirically that for crystals containing atoms of approximately the same atomic factor, the root-mean-square average of || is 1/[square root]N, where N is the number of atoms per cell. It is shown empirically that for N larger than about ten, values are distributed about zero approximately according to the normal error curve. With these results it is first shown that the power of the inequalities varies little with symmetry if the number of atoms in the asymmetric unit is kept constant. It is then shown that the simple inequalities cannot solve crystals of greater than moderate complexity, as judged by the number of atoms in the asymmetric unit, and that to attempt more complicated crystals will require more complicated inequalities. It is suggested that experimental errors may limit the usefulness of these more complicated expressions to the same range of crystals that can be solved by complete three-dimensional, absolute-scale Patterson methods. Attention is called to an older method by Banerjee.