THE SUPERSPACE APPROACH TO STRUCTURE ANALYSIS OF M0DULATED AND COMPOSITE (INTERGROWTH) CRYSTALS
V. Petricek
Institute of Physics, Academy of Sciences of the Czech
Republic, Na Slovance 2, 180 40 Praha 8, Czech Republic
Incommensurate modulations in the crystal lead to additional diffraction spots, which cannot be indexed by three indices. This fact means that the basic property of the crystal, the 3-dimensional periodicity, is lost. On the other hand the additional, so called satellite, spots are regularly distributed in the reciprocal space in such a way that all diffraction spots can be indexed by adding a few additional (modulation) vectors. The superspace approach as introduced by P.M.deWolff, T.Janssen and A.Janner [1] accounts for these facts and moreover it makes possible the straightforward generalisation of standard methods of crystal structure analysis including collection techniques [2]. Thus the higher dimensional concept affects all steps of the crystal structure analysis.
Parameters as used to describe a regular structure (site occupancy, atomic position, atomic displacement tensors,…) are generally modulated with the periodicity defined by a modulation vector and they are generally different in different cells of the modulated structure. The solution and refinement of modulated structure is equivalent to the problem of finding the basic parameters and their modulations. The modulation functions introduced for different parameters with the periodicity derived from the modulation vector are usually expressed by Fourier series. Such an approach is general but the number of parameters grows very fast with the number of harmonics used. Therefore for some cases special functions such as a step-like (crenel) function [3] for occupational modulation or a linear function for the positional modulation [4] can be applied to reduce the number of parameters.
The technique of solution is strongly dependent on the amplitude of the modulated atoms. The more strongly modulated is the structure the more advanced methods are to be applied. In the case that the modulations are not too strong the solution may be separated into two steps: the solution of the basic structure and the determination of the modulations. The (3+d)-dimensional Fourier synthesis for modulated structures can help to find the modulations of heavy atom(s) ( F2obs Patterson function) and/or modulations of all atoms (Fobs with phases either from direct methods or from already known modulations).
The refined parameters including their modulations allow
calculation of geometrical parameters such as interatomic
distances and angles as function of the internal coordinate t = x4
+ q.r and making the final
interpretation and presentation results in 3-dimensional space.