Triplet Phase Invariants from an Exact Algebraic Analysis of Anomalous Dispersion

Abstract

In a previous investigation, a system of exact algebraic equations was derived for any number and type of anomalous scatterers. Solution of the equations provides information concerning intensities of scattering and certain phase differences. In this paper, it is shown that when appropriate combinations of the phase differences and their values are made, the result is the evaluation of the differences of pairs of triplet phase invariants, one associated with the macromolecular structure and the second associated with the structure of the anomalous scatterers. It is usually easy to satisfy the condition that the values of triplet phase invariants associated with the structures of the anomalous scatterers be close to zero. This permits the evaluation of triplet phase invariants associated with the macromolecular structure. Since the structures of the anomalous scatterers are quite simple in many of the substances of interest, a theoretical and experimental study of the distribution of values for triplet phase invariants associated with simple structures has been carried out. This has provided a quantitative insight into the distribution of values of the cosines of triplet phase invariants for such structures. It has also identified useful functions, based on knowledge of the values of normalized structure factor magnitudes, that permit a reliable prediction of those triplet phase invariants that have values close to zero. In the mathematical sense, the evaluation of the triplet phase invariants for a macromolecular structure, solely from the intensity data, is exact, except for the deviation of the triplet phase invariants for the structure of the anomalous scatterers from zero. No structural information concerning the anomalous scatterers is required. In practice, of course, experimental error will affect the accuracy of the information derived from the algebraic equations. The possibility of overdeterminacy in the equations should be beneficial in reducing the effect of experimental error.