H. Hauptman et F. Han, PHASING MACROMOLECULAR STRUCTURES VIA STRUCTURE-INVARIANT ALGEBRA, Acta crystallographica. Section D, Biological crystallography, 49, 1993, pp. 3-8
Owing to the breakdown of Friedel's law when anomalous scatterers are
present, unique values of the three-phase structure invariants in the
whole range from 0 to 2pi are determined by measured values of diffrac
tion intensities alone. Two methods are described for going from presu
med known values of these invariants to the values of the individual p
hases. The first, dependent on a scheme for resolving the 2pi ambiguit
y in the estimate omega(HK) of the triplet (phi(H) + phi(K) + phi(-H-K
), solves by least squares the resulting redundant system of linear eq
uations phi(H) + phi(K) + phi(-H-K) = omega(HK). The second attempts t
o minimize the weighted sum of squares of differences between the true
values of the cosine and sine invariants and their estimates. The lat
ter method is closely related to one based on the 'minimal principle'
which determines the values of a large set of phases as the constraine
d global minimum of a function of all the phases in the set. Both meth
ods work in the sense that they yield values of the individual phases
substantially better than the values of the initial estimates of the t
riplets. However, the second method proves to be superior to the first
but requires, in addition to estimates of the triplets, initial estim
ates of the values of the individual phases.