The Pseudo-Symmetry of Modulated Crystal Structures

A modulated structure can be depicted as a section through a four-dimensional periodic structure. In the latter, each atom is represented by a string continuing endlessly in the overall direction (e₄) of the normal to R₃, R₃ being the hyperplane of the section. The strings have periodic bends or densifications for displacive and substitutional modulation respectively. Formulae for structure factors can be derived from this picture with little effort. The pseudo-symmetry of modulated structures can be described conveniently in this picture. Each four-dimensional space group to which the four-dimensional structure can belong is a possible MS₃ (modulated three-dimensional structure) group of pseudo-symmetry, and is called an MS₃ space group. It is shown that MS₃ point groups are reducible in the form Q⊕[epsilon]. 1, where 1 is the unit 2 × 2 matrix, and [epsilon] = ± 1. A list is presented of these 31 groups written as black-and-white or colourless groups of three-dimensional symmetry. The MS₃ space groups are discussed briefly. As an example of the peculiar differentiations caused by e₄ being a unique direction, the 23 MS₂ space groups are listed explicitly. Finally, it is shown that MS groups are essential for the description of MS symmetry, because very often the latter cannot be represented completely and unambiguously by the normal space group of an approximate superstructure.