On the number of ambiguities in direct methods - anomalous scattering estimates of the two- and three-phase structure invariants

The theoretical basis for the integration of direct methods into the single-wavelength anomalous dispersion technique is reexamined. The analysis shows that the approximations responsible for the ability to obtain unique estimates of the two- and three-phase structure invariants [Hauptman (1982). Acta Cryst. A38, 632-641; Giacovazzo (1983). Acta Cryst. A39, 585-592] or twofold estimates of the three-phase structure invariants [Kroon, Spek & Krabbendam (1977). Acta Cryst. A33, 382-385] are also responsible for the substantial errors observed in the applications. It is shown that, in the general case, the method of joint probability distributions leads to twofold estimates of the two-phase invariants and eightfold estimates of the three-phase invariants. Finally, it is shown that more accurately determined three-phase invariant estimates can be obtained by the use of anomalous scatterer substructure information, when available, and the use of a strategy that recognizes cases in which the eight estimates are clustered around one or two values. These cases are then distinguished from those where the eight estimates are widely scattered by a weighting function.