The geometrical basis of crystal chemistry. XI. Further study of 3-dimensional 3-connected nets

Abstract

In part I (Wells, 1954) the author derived and illustrated two 3-dimensional 3-connected nets having 4 points in the (topological) repeat unit and nineteen such nets with Zt = 6. Since no systematic way of deriving theselnets is known it cannot be claimed that any compilation is exhaustive, but the existence of only two nets with Zt = 4 seems certain. Some further 3connected nets of much greater complexity were added in parts V and VI (Wells, 1955, 1956). It is convenient to illustrate a net in its most symmetrical configuration, that is, with equal links and with equal interbond angles if this is possible. The unit cell of such a configuration of the net contains Zc points where Zc may be, but is not necessarily, a multiple of Zt. The most symmetrical forms of the two nets with Z, = 4 have respectively cubic and tetragonal symmetry, equal links, all interbond angles 120 °, and in both cases Zc = 8. Both nets are of the type (10,3) or (10a), that is, the shortest circuits including any pair of links from any pcint are 10-gons (uniform net). In order to characterize and to distinguish between nets with the same value of Zt further, we may determine the values of x, the number of 10-gons to which each point belongs, and y, the number of 10-gons to which each link belongs. The ratio x]y is equal to ~ for a 3-connected net. It is important to note that the condition for uniformity does not imply that all points are topologically equivalent or that all links are equivalent, as will be evident from the values of x and y in Table 1. For example, in the cubic (10,3) net all the points are equivalent and all the links are equivalent; the net is therefore described by one value of x (15) and one value of y (10). In the tetragonal (10,3) net the points are equivalent but the links are of two kinds, and it is therefore necessary to assign two different values of y. The relation x/y = 3 still holds provided that y is now the weighted mean of the values for the two kinds of link, as noted in part VII (Wells, 1963). For one-third of the links Yl = 8 and for the remainder Y2 = 6, whence Ymean = 6~, consistent with x = 10. In addition to these two uniform nets for which Zt = 4 three other uniform (10,3) nets will be described later which have Z t = 8 , 12, and 20 respectively. The publication of the crystal structure of the normal form of BzO3 (Strong & Kaplow, 1968)* has indicated the existence of a new 3-dimensional 3-connected net with Zt = 6 which was missed in the earlier derivation of the nineteen nets with Z t=6 . Moreover, the new net is a uniform (10,3) net. The structure of B203 consists of planar BO3 groups sharing each 0 atom with one other similar group. It is therefore basically a 3connected net of B atoms linked through O atoms which are simply 2-connected units situated between each bonded pair of B atoms; these O atoms may be omitted when considering the topology of the net. The most symmetrical configuration of the basic net has three equal coplanar links from each point and interbond angles of 120 °. It has trigonal symmetry and 6 equivalent points in the unit cell (Fig. 1), so that for this particular net Zc= Zt. The net is fully described as follows: Hexagonal axes, c/a = (3 I/3)/2; Z = 6; space group