Acta Crystallographica Section A

Foundations of Crystallography

Volume 57, Part 4 (July 2001)

research papers

Acta Cryst. (2001). A57, 461-472 [ doi:10.1107/S0108767301003762 ]

The joint probability distributions of structure-factor doublets in displacive incommensurately modulated structures and their applicability to direct methods

R. Peschar, R. Israël and P. T. Beurskens

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, Israël, 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 h',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.

Keywords: joint probability distributions; incommensurately modulated structures; direct methods.