The joint probability distributions of structure-factor doublets in displacive incommensurately modulated structures and their applicability to direct methods
Peschar R, Israel R, Beurskens PT
ACTA CRYSTALLOGRAPHICA SECTION A 57: 461-472 Part 4 JUL 2001

Abstract:
In 1993, alternative normalized structure factors for incommensurately modulated structures were defined [Lam, Beurskens & van Smaalen (1993). Acta Cryst. A49, 709-721]. The probability distribution associated with the structure invariants E(-H)E(H')E(H - H') has approximately the same functional form as the Cochran distribution. It was shown, however, that triplet-phase relations are relatively less reliable when satellites are involved [de Gelder, Israel, Lam, Beurskens, van Smaalen, Fu & Fan (1996). Acta Cryst. A52, 947-954]. In the present paper, an alternative approach is presented: instead of studying the distribution of a three-phase invariant, the probability distribution of the phase sum of two first-order satellite reflections (h,k,l,1 and h', k',l', - 1) has been derived under the assumption that the phase of the associated main reflection (h + '0,k + k',l + l',0) can be calculated from the known main (or averaged) structure. Intensive tests with randomly generated artificial structures and one real structure show a significant improvement of direct-methods phase-sum statistics. Functional similarities with conventional direct methods, employing normalized structure factors and the Cochran distribution, are discussed.