Ab initio solution of misfit layer structures by automatic Patterson and direct methods

A procedure is presented for the automatic solution of composite (misfit) layer compounds, for the case when the composite crystal structure consists of two types of layer, each of which can be approximately described as a three-dimensional periodic structure with, however, mutually incommensurate lattices and hence mutually induced incommensurate modulations. The composite structure can be described as a periodic structure in four-dimensional superspace [van Smaalen (1992), Mater. Sci. Forum, 100&101, 173-222]. From reflection data indexed with four integer indices HKLM, the phase problem is solved as follows. The basic structures of layers 1 and 2 are solved by routine application of automated Patterson interpretation and Fourier recycling using the main reflections only and ignoring the modulation effects. The two layers are brought to a common origin by a shift function based on correlating F_obs and F_calc using the main reflections common to both layers. All other reflections, which are the (usually weaker) satellite reflections, are phased from the known phases of the main reflections of either layer by application of the Sayre equation in four-dimensional superspace [Hao, Liu & Fan (1987), Acta Cryst. A43, 820-824; Fan, van Smaalen, Lam & Beurskens (1993), Acta Cryst. A49, 704-708]. The procedure is performed by the program MISFIT, which is embedded in the DIRDIF system.