Tessellations and plane symmetry groups as applied to the derivation of close-packed binary layers

Abstract

A method is described for the derivation of binary closest-packed layers of a given composition with the condition that the atoms of the minor component should be crystallographically equivalent. The method is based on the theory of derivatives of space groups developed by Buerger, which is here applied to those multiple cells that may be regarded as tessellations drawn over a basic triangular net. "Ihe results are presented for a set of compositions, in order to illustrate the method of derivation. IntroductionIn preceeding work concerning the systematic derivation of close-packed structure types (Lima-de-Faria & Figueiredo, 1969), a method was developed in which the rules for derivation followed directly from the conditions for structural stability already discussed (Limade-Faria, 1965). Appropriate layers were so deduced for the generation of ordered binary close-packed structures, under the previous statement of crystallographic equivalency for the atoms of the minor component.The fact that this method applied to the majority of known structures in the area investigated proved its efficacy in deriving the most probable patterns. However, it was not established that all the appropriate layers had been derived. On the other hand, a small proportion were found to have no corresponding appropriate layer; if proved, and not merely a consequence of the simplifications involved in that derivation technique, this circumstance has interesting structural implications. The aim of the present work is to verify these results by applying an exhaustive method to the derivation of these binary close-packed layers. M. O. FIGUEIREDO 235 Statement of the problemThe essential problem is to find all the two-dimensional patterns built up with symmetrically equivalent points as a result of partially filling the nodes of a triangular net.The two designations, triangular net and hexagonal lattice, will be used below to refer to the regular tessellation we are dealing with, whose Schl/~fli symbol is {3,6}; the geometrical aspect is implied with the first designation while the second applies to the crystallographic point of view.The proportion of the triangular net filled will be given by the ratio of the number of occupied nodes in a multiple cell considered over the fundamental net to the total number of nodes covered by this cell. The occupied nodes must be symmetrically equivalent, as stated above, and thus they will belong to a single set of equivalent positions, either general or special, of the plane symmetry group or groups, compatible with the two-dimensional cell used. The groups analysed are restricted to those involving symmetry elements consistent with those already found in the original lattice, and consequently the groups containing fourfold rotation points are excluded. Also, when considering the distribution of one sole component, as is the case here, the general position in pl corresponds to a monoclinic primitive lattice (Burzlaff, Fischer & Hellner, 1968). Therefore, only 13 of the 17 two-dimensional space groups ...