2 Categorizing and description of crystal structures

Crystals, i.e., chemical entities that show a strictly periodic order of their constituents in two or three dimensions, can be described and categorized in many different ways, all of which emphasize a certain aspect of the crystal structure and complement each other. The most important ones are the following: (a) to describe the symmetry of the crystal, (b) to look at the structure if it can be derived from a rather geometrical aspect, if we assume that atoms or ions behave like hard spheres, i.e., to emphasize the way the spheres (or rods) are packed together, (c) to explore the neighbouring species around a certain atom in the form of their coordination polyhedra, (d) to categorize crystals according to prototypical compounds of a certain type of structure (structure types), and (e) to specify the way in which the atoms are connected (bonded) to other atoms, i.e., to describe the network that a structure forms.

2.1 Symmetry

In crystallography, the most important aspect of a crystal structure is its symmetry. The presence (or absence) of certain symmetry elements leads to one of the 17 possible plane groups (for two-periodic systems) or to one of the 230 possible space groups (for three-periodic structures). A plane/space group is, in principle, nothing more than stating which symmetry elements are present in a structure. While the symmetry aspect is completely covered with this specification, it ignores all other aspects, like the nature of the constituents of a structure and how they are held together. Therefore, also other objects, even everyday objects, can be classified according to their symmetry: from a symmetry point of view, a hexagonal tiling in your bathroom is identical to a single graphene sheet of graphite. Disparate substances like the pure metal copper, held together by metallic bonds, the prototypical ionic compound NaCl (i.e., simple rock salt), held together by electrostatic interactions, noble gases (except helium) at very low temperatures, and the solid Buckminsterfullerene (at least at room temperature [2]), consisting of discrete C60 molecules (the famous football molecule), which are interacting only via dispersion forces, all have the same cubic symmetry: they all belong to the space group Fm3¯m (space group no. 225; for an explanation of the nomenclature of space groups, see below). What is different is, of course, (i) the base (also called motif) and (ii) the dimension of the unit cell; but the symmetry elements that are acting on this base are the same. The crystallographic description then will be completed by specifying the (i) lattice parameters and (ii) the nature and position of the atomic species of the base, i.e., their atomic (fractional) coordinates and sometimes their Wyckoff letter, this means their multiplicity and site symmetry. We will recap the most important aspects of the symmetry framework of crystals in Chapter 3.

2.2 Sphere (and rod) packings

Nature generally tends to avoid structures with pronounced voids. The reason for this is that the creation of surfaces or interfaces requires energy. Therefore, there is a trend to realize structures that are as compact as possible by minimizing the surface area and thus reducing energy. In particular, for systems composed of spheres (or rods, i.e., infinite cylinders; for an early overview of crystal structures based on rod packings, see, for instance, Ref. [3]) structures are realized by the guiding principle of how efficient and to which degree space can be filled, so that a minimum of unfilled space is left. In the case of only one type of spheres there are two equivalent densest sphere packings: the cubic closest packing (ccp) and the hexagonal closest packing (hcp).

A large number of metals crystallize in one of these two structures (ccp, for instance, by Rh, Ir, Ni, Pd, Pt, Cu, Ag, Au, and hcp, for instance, by Mg, Ti, Zr, Re, Ru, Os, Co, Cd). In both cases the volume is filled by approx. 74%. And in both arrangements, there are tetrahedral and octahedral voids, which can potentially be occupied by some other, smaller atoms or ions; depending on the composition of the (binary/ternary.) compound and the size and charge of the counterions, a fraction or all of these voids are filled. The structure of many ionic compounds can be described by sphere packings, in which usually the smaller cations are located in the voids of a ccp or hcp arrangement of the larger anions. However, interestingly, there are also a significant number of (simple) structures - although based on spheres - where not densest packings are formed and the principle of realizing structures that are as compact as possible seems to be violated: in elements with a body-centred cubic structure (bcc), for instance, V, Nb, Ta, Cr, Mo, W, a-Fe, the packing density is only 68%, and in primitive cubic (pcu) arrangements, like in a-Po, only 52% of the space is filled. And this means, of course, that atoms are not hard spheres that behave like billiard balls.

The packing density and specifications of the coordination of different kinds of sphere packings are gathered in Table 2.1. A more detailed view on densest sphere packings and their crystallographic descriptions are given in Chapter 4.

Table 2.1:Packing density and coordination in different sphere packings.

Sphere packing Packing density Coordination number of shell Relative distance of the coordination shells 1 2 3 4 d2/d1 d3/d1 d4/d1 Primitive cubic (pcu) 0.52 6 12 8 6 1.41 1.73 2   Body-centred cubic (bcc) 0.68 8 6 12 8 1.15 1.63 1.91 Cubic closest packing (ccp) 0.74 12 6 24 12 1.41 1.73 2   Hexagonal closest packing (hcp) 0.74 12 6 8 24 1.41 1.63 1.73

2.3 Coordination polygons/polyhedra and the five Pauling rules

A slightly extended and complementary approach to categorizing crystal structures, mostly for compact and ionic crystalline compounds, is based on the specification of coordination polyhedra. Here, the coordination environments of the cations and/or anions are specified together with information on how these regular or also slightly distorted polyhedra are connected to each other (i.e., if they are corner- or/and edge- or/and face-connected). The advantage is that structures, which are not based on a densest packing of one of the components, can also be categorized and that polyhedra of any kind can be considered (not only tetrahedral or octahedral ones as in the case of sphere packings). Furthermore, this concept also allows, at least partly, predicting and rationalizing the crystal structure of ionic compounds. As early as 1929, Linus Carl Pauling (1901-1994, American chemist, awarded the Nobel Prize in chemistry in 1954 and the Nobel Peace Prize in 1962) published his seminal work on the principles on which the structure of complex ionic crystals is based, known as the five Pauling rules (see below).

The most important coordination environments, their names, and short symbols are shown in Figure 2.1.

Figure 2.1: Important coordination polyhedra; l?=?collinear/coplanar, n?=?non-collinear/non-coplanar (redrawn and adapted from Ref. [4]).

2.3.1 The five Pauling rules

In 1929, Pauling published five rules for predicting and rationalizing the crystal...