Statistical properties of normalized difference-structure factors for non-centrosymmetric structures

Abstract

The probability density of phase angles for structures with one or more atoms on known positions [Sim (1959). A cta Cryst. 12,[813][814][815][816][817][818] is expressed in terms of normalized difference-structure factors, and used to calculate standard deviations of phases and expectation values for amplitudes (IE~ I) of the normalized difference-structure factors. Numerical results are tabulated for various values of I Ell and I E 21, i.e. the minimum and maximum I Erl value a given reflection can have. Applications to the DIRDIF procedures [van den Hark, Prick & Beurskens (1976). Acta Cryst. A32, 816-821] are described. New applications are the calculation of the a priori probability density function for levi values, and a statistical rriethod for the detection of a centre of symmetry in the remaining part of the structure. For the statistical method, a centricity parameter, X c, which is unity for centric and zero for acentric distributions, is defined; for the difference structure, X c is calculated by an iterative procedure, extrapolating towards zero contribution of the known part of the structure. Numerical results for 13 test structures are given.

On direct-methods phase information from differences between isomorphous structure factors

“…Techniques that have proved to be successful are the tangent recycling methods (Karle, 1970;Hull & Irwin, 1978), the Fourier recycling methods (K_inneging & de Graaff, 1984) and the tangent recycling methods applied to the difference s.f.s (Beurskens, Prick, Doesburg & Gould, 1979). This last technique, upon which the DIRDIF system of programs is based, has been very successful in completing heavy-atom models .…”

Integration of Patterson information into direct methods. I. The theoretical background

“…Such a two-atom fragment may be considered as a well located molecular fragment: recovery of the complete crystal structure may then be accomplished by techniques such as those described by Beurskens, Prick, Doesburg & Gould (1979) or by Burla, Cascarano, Fares, Giacovazzo, Polidori & Spagna (1989).…”

Direct methods and structures showing superstructure effects. IV. A new approach for phase solution

“…Statistical analysis of diffraction data gives in favourable cases approximate information on the scattering power of the average structure ~p(r), so that it seems more useful to study triplet invariants in terms of the electron distribution p(r) = pp(r) + pp_#(r)+ pq (r) ( 1) where p#(r) = t3p(r) and pp_~(r) = pp(r)-t3p(r). It will be seen that probabilistic estimation of triplet invariants benefits from the Fourier transform of the pseudotranslational symmetry just as other methods (Main, 1976;Beurskens, Prick, Doesburg & Gould, 1979;Giacovazzo, 1983;Camalli, Giacovazzo & Spagna, 1985) take advantage of the Fourier transform of a molecular fragment when its orientation or position is known a priori. Though the Fourier transform of the pseudotranslational symmetry does not give information on single triplets but only on classes of triplets, its use as prior information in probabilistic approaches will prove quite useful.…”