Direct Methods for Modulated Structures

Abstract

Modulated structures belong to that kind of crystal structure, in which the atoms 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. Modulated phases have been found in many important inorganic and organic solids. In many cases, the transition to the modulated structure corresponds to a change of certain physical properties. Hence it is important to know the structure of modulated phases in order to understand the mechanism of the transition and properties in the modulated state. Up to the present most procedures used for solving modulated structures, especially those for solving the incommensurate ones, depend on some preliminary assumption to the form of modulation. It is worthwhile to find a straightforward way to solve this important kind of structures. Both commensurate modulated structure (superstructure) and incommensurate modulated structure possess pseudo-translational symmetry.