Classification of crystal structures

Symmetry is the fundamental basis for descriptions and classification of crystal structures. The use of symmetry made it possible for early investigators to derive the classification of crystals in the seven systems, 14 Bravais lattices, 32 crystal classes, and the 230 space groups before the discovery of X-ray crystallography. Here we examine symmetry elements needed for the point groups used for discrete molecules or objects. Then we examine additional operations needed for space groups used for crystal structures. 

Here we use bond-valence theory (Brown 1981) and its developments (Hawthorne 1985a, 1994, 1997) to consider structure topology and hierarchical classification of crystal structures, and we point out that bond-valence theory can be considered as a simple form of molecular-orbital theory (Burdett and Hawthorne 1993 Hawthorne 1994, 1997). 

A chirality classification of crystal structures that distinguishes between homochiral (type A), heterochiral (type B), and achiral (type C) lattice types has been provided by Zorkii, Razumaeva, and Belsky [11] and expounded by Mason [12], In the type A structure, the molecules occupy a homochiral system, or a system of equivalent lattice positions. Secondary symmetry elements (e.g., inversion centers, mirror or glide planes, or higher-order inversion axes) are precluded in type A lattices. In the racemic type B lattice, the molecules occupy heterochiral systems of equivalent positions, and opposite enantiomers are related by secondary lattice symmetry operations. In type C structures, the molecules occupy achiral systems of equivalent positions, and each molecule is located on an inversion center, on a mirror plane, or on a special position of a higher-order inversion axis. If there are two or more independent sets of equivalent positions in a crystal lattice, the type D lattice becomes feasible. This structure consists of one set of type B and another of type C, but it is rare. Of the 5,000 crystal structures studied, 28.4% belong to type A, 55.6% are of type B, 15.7% belong to type C, and only 0.3% are considered as type D. 

Fig. 3. Classification of crystal structures according to Hume-Rothery. Also shown are heats of atomization in kcal/g-atom at 300°K or at the melting point, whichever is lower. The three classes into which the elements are divided are discussed in the text. The cohesive energies are taken from compilations of thermodynamic properties in NBS reports (D. D. Wagman et al).

Much surface work is concerned with the local atomic structure associated with a single domain. Some surfaces are essentially bulk-temiinated, i.e. the atomic positions are basically unchanged from those of the bulk as if the atomic bonds in the crystal were simply cut. More coimnon, however, are deviations from the bulk atomic structure. These structural adjustments can be classified as either relaxations or reconstructions. To illustrate the various classifications of surface structures, figure A1.7.3(a ) shows a side-view of a bulk-temiinated surface, figure A1.7.3(b) shows an oscillatory relaxation and figure A1.7.3(c) shows a reconstructed surface. 

The given discussion shows that rather universal and simple classification of porous materials equivalent to classification of crystals is absent. However, one can consider a system of interrelating classifications that take into account order, morphology and sizes at different hierarchical levels, degrees of integrity, structure, heterogeneity of a various type, etc. Such a systematic approach can be used as well for adequate modeling of various hierarchical levels of a porous material structure. 

Our description of atomic packing leads naturally into crystal structures. While some of the simpler structures are used by metals, these structures can be employed by heteronuclear structures, as well. We have already discussed FCC and HCP, but there are 12 other types of crystal structures, for a total of 14 space lattices or Bravais lattices. These 14 space lattices belong to more general classifications called crystal systems, of which there are seven. 

The structures of many inorganic crystal structures can be discussed in terms of the simple packing of spheres, so we will consider this first, before moving on to the more formal classification of crystals. 

A classification of crystals based on bonding is useful in understanding structure-property relations in solids. Five types of solids are readily defined on bonding considerations ionic, covalent, metallic and molecular (van der Waals) and hydrogen-bonded. In Table 1.2, the important characteristics of the five types of solids are presented. In real situations, however, solids may exhibit features of more than one type of bonding. 

In 1970, Heller suggested a classification of borates based on the number of boron atoms in the fundamental building block . In 1971, J. R. Clark added, in an article on crystal chemistry of borates , a further principle as the fifth rule, namely that "the boric acid group, B(OH)3, may exist in isolated form in the presence of more complex polyanions, or such insular groups may themselves polymerize and attach to side-chains of more complex polyanions , as first observed in the crystal structures of veatchite and paraveatchite. In 1977, Christ and Clark reviewed the various principles and classifications in their article on a crystal-chemical classification of borate structures with emphasis on hydrated borates . In addition to a sixth rule. 

Christ, C. L., and J. R. Clark (1977). A crystal-chemical classification of borate structures with emphasis on hydrated borates. Phys. Chem. Minerals 2, 59-88. 

For the Classification of Minerals , crystal-structure determinations improved the definitions of mineral species and varieties, assisted in the development of the concept of crystal structure types, helped to establish isotypic series and homeotypic and heterotypic groups, and pointed to the recognition of much broader crystallochemical relationships. The X-ray method appreciably simplified the generally unique characterization of a mineral species and led to a reduction in varieties and the discreditation of many minerals accepted up to that time, thereby eliminating countless superfluous mineral names . 

Fig.ll. Classification of hydrogen storage alloys in terms of crystal structural evolution in the course of hydrogenation. 

The systematic description of crystal structures is presented primarily in the well-known Structurbericht. The classification of crystals by the Structurbericht does not reflect their crystal class, the Bravais lattice, but is based on the crystaUochemical type. This makes it inconvenient to use the Structurbericht categories for comparison of some individual crystals. Thus, there have been several attempts to provide a more convenient classification of crystals. Table 5 presents a compilation of different classifications which allows the reader to correlate the Structurbericht type with the international and Schoenflies point and space groups and with Pearsons symbols, based on the Bravais lattice and chemical composition of the class prototype. The information included in Table 5 has been chosen as an introduction to a more detailed crystal-lophysical and crystaUochemical description of solids. 

Per formula by Leonhard Euler F + V - E = 2 Strukturbericht Structure triangle TABLE 5 Classification of Crystals Symmetry group Pearson Standard ASTM E157-82a... 

Traditionally, the classification of crystals includes an important category the ionic crystals. These crystals are composed of ions, and the cohesion arises from the balance between the attractive coulombian forces and short-distance repulsive interactions, which prevents the collapse of the crystal This category includes the alkali halides, all crystals possessing a similar structure (MgO, CaO), the oxy-salts (carbonates, nitrates, sulfates, silicates). Sometimes even corundum is included in this categoryI... 

Because the present work is based on the structural analogies that exist between crystals and molecules, some general results for molecules will first be briefly reviewed. After showing how these notions can be transposed to crystals, a scheme of structural classification of crystals will be presented. This will permit a subsequent logical treatment of the problem of crystallographic arrangements in infinite networks. 

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