Plane nets in crystal chemistry

Abstract

In the present paper we consider not only the simplest periodic nets (such as arise from the equivalent circle packings of Niggli, Fejes Toth and others) but also less regular ones, ignored by mathematicians but nevertheless of widespread occurrence and usefulness in crystal chemistry. After a general introduction including some mathematical theorems a catalogue of about 30 nets gives, in most cases, the plane group short symbol, and the unit cell parameters and the coordinates of the nodes in terms of unit spacing between nearest nodes. Examples of their occurrence in compounds of established structure are given in each case. The related concepts of the dual of a simple net and primary and secondary nets in less simple cases are then treated briefly. Transformations between nets are discussed, also with crystal structure examples: first in the case that there is no change in the shape of the unit cell, and using a proposed `compatibility' principle. It transpires that compatible nets are simply derivable from one another, and that in most classes the simplest member is a regular net (4$^{4}$, 3$^{6}$, or 6$^{3}$). A few of the transformations are relatively well known, but most are new. Together they emphasise the fact that crystal structures do not constitute a massive collection of unrelated types, but rather a group of patterns largely derivable one from another by a few simple, geometrical-crystallographic operations. Here, as elsewhere in the paper, it frequently occurs that transformations are equivalent to the regular incorporation of `point defects' (missing atoms = `vacancies' or additional atoms = `interstitials'). Hence `point defects' may be readily generated (even in very small concentrations) by cooperative operations, without any need for long-range diffusion of single atoms. This possibility is not generally considered in theories of diffusion in solids. Another type of transformation involves slip, and does result in a change in the shape of the unit cell, sometimes by a homogeneous deformation. It allows transformation between different (compatibility) classes of nets. Section 9 deals with the (hexagon-pentagon-triangle) net description of `tetrahedrally closed packed' alloy structures - Frank-Kasper and Friauf-Laves phases - and transformations relating them. The $\beta $-U$_{3}$O$_{8}$ and related nets discussed in section 10 are some-what similar, but also contain quadrangles. In section 11 a different type of operation is used to relate structures: adjacent planes are combined by collapse to form a composite net on a single plane. This produces further crystal structure relations that were not previously available, e.g. between ReO$_{3}$, HTB and the pyrochlore framework. Finally, in section 12, some conclusions are drawn, and some of the more novel points developed in the paper are summarized and emphasized.