Multi-Dimensional Direct Methods

Abstract

In single crystal structure analysis, it is usually assumed that crystals are ideal 3-dimensional periodic objects. However real crystals are never perfect. What we obtained under this assumption is not the real structure but just an averaged structure over a large number of unit cells. Unfortunately a knowledge on the averaged structure is often not enough for understanding the properties of many solid state materials. Therefore an important task for methods of solving crystal structures is to extend from ideal periodic crystals to real crystals which contain various kinds of defects. Modulated crystal structures belong to a kind of crystal structures containing periodic defects, i.e. the atoms in which suffer from certain occupational and/or positional fluctuation. If the period of fluctuation is commensurate with that of the three-dimensional unit cell then a superstructure results, otherwise an incommensurate modulated structure is obtained. Incommensurate modulated phases can be found in many important solid state materials. In many cases, the transition to the incommensurate modulated structure corresponds to a change of certain physical properties. Hence it is important to know the structure of incommensurate modulated phases in order to understand the mechanism of the transition and properties in the modulated state. Up to the present many incommensurate modulated structures were solved by using some kind of trial-and-error methods. With these methods it is necessary to make assumption on the property of modulation before we can solve the structure. This often causes difficulties and leads easily to errors. In view of diffraction analysis, it is possible to phase the reflections directly and solve the structure objectively without relying on any assumption about the modulation wave. Multidimensional direct methods have been developed for this purpose. The theoretical background and practical applications will be discussed in detail.