Class Bravais

If atoms, molecules, or ions of a unit cell are treated as points, the lattice stmcture of the entire crystal can be shown to be a multiplication ia three dimensions of the unit cell. Only 14 possible lattices (called Bravais lattices) can be drawn in three dimensions. These can be classified into seven groups based on their elements of symmetry. Moreover, examination of the elements of symmetry (about a point, a line, or a plane) for a crystal shows that there are 32 different combinations (classes) that can be grouped into seven systems. The correspondence of these seven systems to the seven lattice groups is shown in Table 1. 

Crystallographic nomenclature (Bravais lattices, crystal classes, space groups) The following information is generally included in a usual crystallographic description ... 

Crystal family Symbol Crystal system Crystallographic point groups (crystal classes) Number of space groups Conventional coordinate system Bravais lattices... 

Orthorhombic Space Groups. There are 59 of these space groups divided among three crystal classes 222(D2), mm2(C2l.), and mmm(DVt). Within each class there is at least one group associated with each of the four types of orthorhombic Bravais lattice, / C (or A), F, /. We shall make no attempt to derive these systematically, but a few examples and some useful observations are warranted. The complete list of the 230 space groups given in Appendix VIII should be consulted at this time. 

When we consider crystal structures we usually think of the pattern and symmetry of the packing of the atoms, ions, or molecules in building the lattice based on X-ray crystallography. However, detailed descriptions of crystals and their classification are much older. The seven systems of crystals and the 32 classes of crystal symmetry were recognized by 1830. The 14 Bravais Lattices were presented by A. Bravais in 1848. 

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. 

FIGURE 7-1 The Seven Crystal Classes and Fourteen Bravais Lattices. The points shown are not necessarily individual atoms, but are included to show the necessary symmetry. 

TABLE 4.3. The 14 Bravais Lattices, 32 Crystallographic Point Groups (Crystal Classes) and Some Space Groups. 

The class symbols can be derived from the space group symbols by deleting the Bravais symbols (P, C, etc.), dropping all subscripts from screw axes (2i, 3i, 4i, etc. -> 2,3,4, etc.) and replacing all glide plane symbols by the mirror plane symbol, m. Thus I4i/acd becomes 4/mmm. A slash means perpendicularity of a rotational element and a reflection element. 

In responding to all five questions, we obtain the crystal class, unit cell dimensions, Bravais lattice, and space group. All are essential crystal properties, but the diffraction pattern contains other useful information as well. 

Mathematics (Hassel, 1830) has shown that there are only 32 combinations of symmetry operations (rotation, inversion, and reflection) that are consistent with a three-dimensional crystal lattice. These 32 point groups, or crystal classes, can be grouped into one of the seven crystal systems given in Table 2.1. There are four types of crystal lattices primitive (P), end-centered (C, B, and A), face-centered (/O, and body-centered (/). The primitive lattice contains a lattice point at each comer of the unit cell, the end-centered lattice has an additional lattice point on one of the lattice faces, the face-centered lattice has an extra lattice on each of the lattice faces, and the body-centered lattice has an extra lattice point at the center of the crystal lattice. By combining the seven crystal systems with the four lattice types (P, C, I, F), 14 unique crystal lattices, also known as Bravais lattices (Bravais, 1849), are produced. 

Crystal system Possible Bravais lattices Crystal classes or point groups Number of space groups... 

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. 

The Bravais lattices (or Bravais classes) represent a classification of the translation lattices according to the symmetry imposed metric. They were originally derived by... 

In Fig. 2.24, m represents a mirror line. Let T be a primitive translation and T the mirror image of T. (A translation T is primitive if 1/2T is not a translation.) T + T and T — T are perpendicular and define a rectangular cell. If T + T and T — T are primitive translations, the rectangular cell is centered because there is a lattice point in the middle. We would thus choose the diamond (T,T) as the primitive cell. In contrast, if 1/2(T + T ) and 1/2(T — T ) are translations, we then obtain a rectangular primitive cell. These two rectangular planar lattices, primitive, on the one hand and centered or diamond, on the other, are representative of two types of lattice that it is important to differentiate. It is impossible to find a primitive diamond-shaped cell for the first, or a primitive rectangular cell for the second. These considerations lead us to an operational definition for the Bravais lattices (or classes). A Bravais lattice is characterized by ... 

The Bravais class of a lattice is given by the metric and the type of the smallest cell that can be obtained by choosing a canonical basis in accord with the crystal system. 

The conventional unit cells for the fourteen three-dimensional Bravais lattices are shown in Fig. 2.25. Each of these cells represents one lattice class. Some important supplementary information is presented in Section 1.4.1 to which may be added the following comments ... 

Each Bravais system has its corresponding minimum and maximum symmetry. Thus the Bravais lattice must be monoclinic (P or C) if the crystal has only one mirror plane or one twofold axis (crystal classes m or 2). However, the monoclinic unit cell will also allow the symmetry 2/m. Thus the symmetry of the contents of the unit cell (the motif) may be less than that of the empty cell. In this case we speak of merohedry. The formation of twins is relatively frequent for merohedral crystals. A twin (Fig. 2.28) is an interpenetration or aggregation of several crystals of the same species whose relative orientations follow well-defined laws. These orientations are related by symmetry operations which do not belong to the crystal class of the untwinned crystal, either by a rotation about a translation... 

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