Description of Crystal Structures

Tc or c cubic closest-packing of spheres Th or h hexagonal closest-packing of spheres Ts stacking sequence AA... of hexagonal layers Qs stacking sequence AA... of square layers 

For additional symbols of further packings cf. [38, 156], T (triangular) refers to hexagonal layers, Q to layers with a periodic pattern of squares. The packing Qs yields a primitive cubic lattice (Fig. 2.4), Qf a body-centered cubic lattice (cf. Fig. 14.3, p. 153). Sometimes the symbols are set as superscripts without the angular brackets, for example Ti[Ca03]c. 

Another type of notation, introduced by P. Niggli, uses fractional numbers in the chemical formula. The formula Ti06/, for instance means that every titanium atom is surrounded by 6 O atoms, each of which is coordinated to 3 Ti atoms. Another example is NbOCl3 = NbC C C /i which has coordination number 6 for the niobium atom (= 2 -)- 2 + 2 = sum of the numerators), coordination number 2 for the O atom and coordination numbers 2 and 1 for the two different kinds of Cl atoms (cf. Fig. 16.11, p. 176). 

In a crystal atoms are joined to form a larger network with a periodical order in three dimensions. The spatial order of the atoms is called the crystal structure. When we connect the periodically repeated atoms of one kind in three space directions to a three-dimensional grid, we obtain the crystal lattice. The crystal lattice represents a three-dimensional order of points all points of the lattice are completely equivalent and have the same surroundings. We can think of the crystal lattice as generated by periodically repeating a small parallelepiped in three dimensions without gaps (Fig. 2.4 parallelepiped = body limited by six faces that are parallel in pairs). The parallelepiped is called the unit cell. 

Primitive cubic crystal lattice. One unit cell is marked 

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Aside from the conventions mentioned for the cell choice, further rules have been developed to achieve standardized descriptions of crystal structures [36], They should be followed to assure a systematic and comparable documentation of the data and to facilitate the inclusion in databases. However, contraventions of the standards are rather frequent, not only from negligence or ignorance of the rules, but often for compelling reasons, for example when the relationships between different structures are to be pointed out. 

Pauling, L. (1960). The Nature of the Chemical Bond, 3rd ed. Cornell University Press, Ithaca, NY. A classic book that presents a good description of crystal structures and bonding in solids. 

Although it has often proved useful as a mnemonic, this approach has led to a number of misconceptions about relative atomic sizes and the origin of close-packing geometry, to some of which we allude below. More relevant in the present context is the observation that one natural and simple description of crystal structures has been. overlooked, and an unnecessarily complicated and opaque one used instead. We will provide many examples throughout this article. 

For the description of crystal structures one adds to the preceding molecular parameters an appropriate combination of rotation angles and translations 6 parameters or less according the space group symmetry. 

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. 

For a detailed description of crystal structure.s of Pc complexes, see Refs. [4,24]. 

Ternary two anions compounds - References of the second column are concerned with the description of compounds. References of the third column are concerned with the description of crystal structures. 

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