Showing papers in "Acta Crystallographica in 1974"

The pseudo-symmetry of modulated crystal structures

Abstract: 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⊕∊. 1, where 1 is the unit 2 × 2 matrix, and ∊ = ± 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.

Structural study by high-resolution electron microscopy: Tetragonal tungsten bronze-type structures in the system Nb2O5-WO3

Abstract: The contrast in many-beam lattice images from very thin crystals of 4Nb2O5. 9WO3 is shown to be directly related to its known structure. On the basis of this correlation, the structure of 2Nb2O5.7WO3 is derived from the observed image contrast; it is an ordered intergrowth of the ReO3 and tetragonal tungsten bronze structural types. The structures of typical fault boundaries and disordered intergrowths are described, and those of the reported compounds 6Nb2O3. 11WO3 and 3Nb2O5.8WO3 are discussed.