Plane nets

The division of an infinite plane into polygons is obviously related to the enumeration of polyhedra, which can be represented as tessellations of polygons on a simple closed surface such as a sphere. In fact we find equations somewhat similar to those for polyhedra except that instead of the number / of n-gon faces we have 4 n as the fraction of the polygons which are n-gons, since we are now dealing with an infinite repeating pattern. These equations, which are derived in Appendbc 2, MSIC, are  

There are three special solutions corresponding to plane nets in which all the polygons have the same number of edges (and the same number of lines meet at every point), namely  

They are illustrated in Fig. 3.9(a). The last is the only plane 6-connected net, and evidently plane nets with more than six lines meeting at every point are not 

In order to derive the general equation for nets containing both 3- and 4-connected points we must allow for variation in the proportions of the two kinds of points. If the ratio of 3- to 4-connected points is R it is readily shown that in a system of N points the number of links is N(3R + 4)/2(/ + 1). Using the same method as for deriving equations (4)-(6) it is found that = 2(3/ + 4)j(R + 2). The value of ranges from 6, when R = ° , to 4, when / = 0, and has the special value 5 if the ratio of 3- to 4-connected points is 2 1. The special solution of the equation 2n0 = 5 is 0 = 1, corresponding to the 5-gon nets of Fig. 3.10. Although (3,4)-connected plane nets are not of much interest in structural chemistry the 3D nets of this type form the bases of a number of crystal structures (p. 77). 

Plane nets in which points of two kinds (p- and -connected) alternate are of interest in connection with layer structures of compounds AmX in which the coordination numbers of both A and X are 3 or more. For example, the simple Cdl2 layer may be represented as the plane (3, 6)-connected net. It is shown in Appendix 2 of MSIC that the only plane nets composed of alternate p- and (/-connected points are those in which the values of p and q are 3 and 4, 3 and 5, or 3 and 6. The impossibility of constructing a plane net with alternate 4- and 6-connected points implies that a simple layer structure is not possible for a compound A2X3 if A is to be 6-coordinated and X 4-coordinated. The nonexistence of an octahedral layer structure for a sesquioxide is, therefore, not a matter of crystal chemistry in the sense in which this term is normally understood but receives a very simple topological explanation. 

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The complex CUCN.NH3 provides an unusual example of CN aeting as a bridging ligand at C, a mode which is common in p,-CO complexes (p. 928) indeed, the complex is unique in featuring tridentate CN groups which link the metal atoms into plane nets via the Cu... 

A considerable body of scientific work has been accomplished in the past to define and characterize point defects. One major reason is that sometimes, the energy of a point defect can be calculated. In others, the charge-compensation within the solid becomes apparent. In many cases, if one deliberately adds an Impurity to a compound to modify its physical properties, the charge-compensation, intrinsic to the defect formed, can be predicted. We are now ready to describe these defects in terms of their energy and to present equations describing their equilibria. One way to do this is to use a "Plane-Net". This is simply a two-dimensional representation which uses symbols to replace the spherical images that we used above to represent the atoms (ions) in the structure. 

It is also easy to see that we can stack a series of these "NETS" to form a three-dimensional solid. We can also suppose that the same type of defects wiU arise in our Plane Net as in either the homogeneous or heterogeneous soM and so proceed to label such defects as Mi, meaning an interstitial In the same way, we label a cation vacancy as Vm,... 

These, then, cire the set of possible defects for the Plane Net, and the following summarizes the types of intrinsic defects expected. Note that we have used the labelling V = vacancy i = interstitial M = cation site X = anion site and s = surface site. We have already stated that surface sites are special. Hence, they are included in our listing of intrinsic defects. 

Returning to the subject of lattice defect formation, we can now proceed to write a series of defect reactions for the defects which we found for our plane net ... 

Whether you recdize it or not, we have already developed our own symbolism for defects and defect reactions based on the Plane Net. It might be well to compeu e our system to those of other authors, who have also considered the same problem in the past. It was Rees (1930) who wrote the first monograph on defects in solids. Rees used a box to represent the cation vacancy, as did Libowitz (1974). This has certain advantages since we can write equation 3.3.5. as shown in the following ... 

Draw one or more "plane-nets" for the "P" eation eombined with a "U" anion. Indicate all of the possible defects that can appear. Write the symbol of each as you proceed. Include pairs of defects as needed. 

Write equations for as many of the thirty (30) defects reactions of your "PU" plane-net as you can. Do not forget the defect-pairs. 

In contrast to these limitations on the structures of molecules and crystals which arise from metrical considerations there are others which may be described as topological in character. For example, the non-existence of compounds A2X3 (e.g. sesquioxides) with simple layer structures in which A is bonded to 6 X and X to 4 A is not a matter of crystal chemistry but of topology it is concerned with the non-existence of the appropriate plane nets, as explained on p. 72. The nonexistence of certain other structures for compounds A X , such as an AX2 structure of 10 5 coordination, may conceivably be due to the (topological)... 

In this very brief survey we have not been concerned with the detailed structure of the layers, various aspects of which are discussed in more detail later, in particular the basic 2D nets and structures based on the simplest 3- and 4-connected plane nets (Chapter 3), and layers formed from tetrahedral and octahedral coordination groups (Chapter 5). The Cdl2 layer and more complex structures derived from this layer are further discussed in Chapter 6. We shall see that there are corrugated as well as plane layers, and also composite layers consisting of two interwoven layers (red P, Ag[C(CN)3]) these are included in the chapters just mentioned. 

We now come to the second point concerning plane patterns. An isolated object (for example, a polygon) can possess any kind of rotational symmetry but there is an important limitation on the types of rotational symmetry that a plane repeating pattern as a whole may possess. The possession of n-fold rotational symmetry would imply a pattern of -fold rotation axes normal to the plane (or strictly a pattern of -fold rotation points in the plane) since the pattern is a repeating one. In Fig. 2.4 let there be an axis of -fold rotation normal to the plane of the paper at /, and at Q one of the nearest other axes of -fold rotation. The rotation through Ivjn about Q transforms P into F and the same kind of rotation about P transforms Q into Q. It may happen that P and Q coincide, in which case n = 6. n all other cases PQ must be equal to, or an integral multiple of, PQ (since Q was chosen as one of the nearest axes), i.e. 4. The permissible values of n are therefore 1, 2, 3, 4, and 6. Since a 3-dimensional lattice may be regarded as built of plane nets the same restriction on kinds of symmetry applies to the 3-dimensional lattices, and hence to the symmetry of crystals. 

Still retaining the condition that all points have the same connectedness we find that there are certain limitations on the types of system that can be formed. For example, there are no polyhedra with all vertices 6-connected, and there is only one 6-connected plane net. These limitations are summarized in Table 3.1, which may be considered in two ways ... 

No equations are known analogous to those for polyhedra and plane nets relating to the proportions of polygons (circuits) of different kinds. A different approach is therefore necessary if we wish to derive the basic 3D nets. 

Table 3.13 includes examples of structures based on 2D and 3D 4-connected nets. Only the simplest of the plane 4-connected nets is of importance in crystals as the basis of layer structures in its most symmetrical configuration it is one of the three regular plane nets. 

We have commented on the absence of structures of ionic compounds A X with coordination numbers of A greater than eight or nine. If we derive 2-dimensional nets in which A has some number (p) of X atoms and X has some number (q) of A atoms as nearest neighbours we find that the only possible (p, <7)-connected nets (p and <7 > 3) in which p- and <7-connected points alternate are the (3,4), (3, 5), and (3,6) nets. Since there are upper limits to the values of p and q in plane nets, it is reasonable to assume that the same is true of 3D nets. We are not aware that this problem has been studied. However, the existence or otherwise of crystal structures... 

Inorganic acids provide a number of examples of layer structures. The layers in H2Se03, H2Se04, and H2SO4 are all of the same topological type, being based on the simplest 4-connected plane net, but whereas in acids H2XO4 (Fig. 8.11(a))... 

Tricyanomethanide ion. The very small deviations from planarity of this ion observed in certain salts have been attributed to crystal interactions. This ion can form interesting polymeric structures with metal ions. In the argentous compound the group acts as a 3-connected unit in layers based on the simplest 3-connected plane net, and these layers occur as interwoven pairs as described on p. 90. 

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