The Seven Crystal Systems

Crystal system Synonyms, old names Symbol Geometrical description Symmetry Hermann- Mauguin (Schoenflies- Fedorov) Lattice parameters (lUCr) (edges length, interaxial angles) 

Hexagonal H(h) Upright prism with a regular hexagonal basis 6tmmm (D,T a = b c a = P= nil rad and Y= lnl3 rad 

Rhombohedral trigonal R(h) Prism with each face equal to identical lozenges 3m (D,j) a = b = c a = P= Y nllt Ld 

Orthorhombic orthogonal 0(0) Upright prism with a rectangular basis mmm a bi=c a = P= Y=nHrad 

The crystal symmetry defines the basic shape of the unit cell. There are seven shapes, which correspond to the ways in which symmetrical solid objects can pack. They can be distinguished by their symmetry properties, i.e. whieh symmetry elements they are based on. The resulting unit-cell dimensions can be classified depending on whether their edges are all different or two or three of them are equal in length, and whether their angles are equal or not, or exactly 90° or 120°. These seven crystal systems are hsted alongside other data, which we refer to later, in Table 10.1. 

The identity of the crystal system arises from the internal symmetry of the cell and the minimum symmetry for each crystal system is listed in Table 10.1. For example, if there is a mirror plane of symmetry perpendicular to the b axis of a unit cell, then the a and c axes must lie in this plane and so the angles between a and 6 (i.e. y) and between c and h (i.e. a) must be 90°, but there is no restriction on the angle between a and c (i.e. fi). Or if there is a threefold axis of rotation along c, the axes a and b must enclose an angle of 120°, and 

The 14 Bravais lattices. Adapted from [28]. Copyright 2007 Teubner. 

Within the unit cell we can define any point (termed a lattice point) by its fractional coordinates. Any two lattice points will be related by symmetry if they have the same three-dimensional surrounding of atoms. In the case of C-centering, for example, this means that every lattice point (which could represent an atom) is found again by translation along half the face diagonal defined by a and ft, so the lattice point (x , y, z) can also be found at (V2 + X, % + y, z). 

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Distribution of Crystalline Materials Among the Seven Crystal Systems ... 

Identify the seven crystal systems and 14 Bravais lattices. 

Continuing with our survey of the seven crystal systems, we see that the tetragonal crystal system is similar to the cubic system in that all the interaxial angles are 90°. However, the cell height, characterized by the lattice parameter, c, is not equal to the base, which is square (a = b). There are two types of tetragonal space lattices simple tetragonal, with atoms only at the comers of the unit cell, and body-centered tetragonal, with an additional atom at the center of the unit cell. 

FIGURE 1.23 (a) The unit cells of the seven crystal systems, (h) Assemblies of cubic unit cells in one, two, and three dimensions. 

Diffraction patterns can be described in terms of three-dimensional arrays called lattice points.33 The simplest array of points from which a crystal can be created is called a unit cell. In two dimensions, unit cells may be compared to tiles on a floor. A unit cell will have one of seven basic shapes (the seven crystal systems), all constructed from parallelepipeds with six sides in parallel pairs. They are defined ac-... 

Fig. 3.28 The fourteen Bravais lattices grouped according to the seven crystal systems.

Only fourteen space lattices, called Bravais lattices, are possible for the seven crystal systems (Fig. 328). Designations are P (primitive), / (body-centered), F (face-centered),34 C pace-centered in one set of laces), and R (rhombohedral) Thus our monoclinic structure P2Jc belongs to the monoclinic crystal system and has a primitive Bravais lattice. 

The next step is for a protein crystallographer to mount a small perfect crystal in a closed silica capillary tube and to use an X-ray camera to record diffraction patterns such as that in Fig. 3-20. These patterns indicate how perfectly the crystal is formed and how well it diffracts X-rays. The patterns are also used to calculate the dimensions of the unit cell and to assign the crystal to one of the seven crystal systems and one of the 65 enantiomorphic space groups. This provides important information about the relationship of one molecule to another within the unit cell of the crystal. The unit cell (Fig. 3-21) is a parallelopiped... 

Figure 2.2. The seven Crystal Systems and the 14 Bravais Lattices.

Table 2.3. The seven crystal systems and the 32 crystal classes.

Inorganic compounds are fairly evenly distributed among the seven crystal systems. Orthorhombic, monoclinic, and cubic crystals (in the order of abundance) comprise about 60% of the total number. 

A precise description of the structure of a crystalline compound necessitates prior knowledge of its space group and atomic coordinates in the asymmetric unit. Illustrative examples of compounds belonging to selected space groups in the seven crystal systems are presented in the following sections. 

Figure 1-36 The seven crystal systems corresponding to 14 Bravais lattices. (Reproduced with permission from Ref. 45.)...

In direct analogy with two dimensions, we can define a primitive unit cell that when repeated by translations in space, generates a 3D space lattice. There are only 14 unique ways of connecting lattice points in three dimensions, which define unit cells (Bravais, 1850). These are the 14 three-dimensional Bravais lattices. The unit cells of the Bravais lattices may be described by six parameters three translation vectors (a, b, c) and three interaxial angle (a, (3, y). These six parameters differentiate the seven crystal systems triclinic, monoclinic, orthorhombic, tetragonal, trigonal, hexagonal, and cubic. 

The seven crystal systems in three-dimensional space are listed with their defining symmetry elements in Table 7.4. 

Bravais in 1849 showed that there are only 14 ways that identical points can be arranged in space subject to the condition that each point has the same number of neighbors at the same distances and in the same directions.Moritz Ludwig Frankenheim, in an extension of this study, showed that this number, 14, could also be used to describe the total number of distinct three-dimensional crystal lattices.These are referred to as the 14 Bravais lattices (Figure 4.9), and they represent combinations of the seven crystal systems and the four lattice centering types (P, C, F, I). Rhombohedral and hexagonal lattices are primitive, but the letter R is used for the former. 

Crystal system The seven crystal systems, classified in terms of their symmetry and corresponding to the seven fundamental shapes for unit cells consistent with the 14 Bravais lattices. 

With this background, together with the knowledge about the seven crystal systems (Christian Samuel Weiss, 1815), the recognition of isomorphy and polymorphy (Mit-scherlich, 1819,1824), and the triad rule of chemical elements (Dbbereiner, 1829), Gustav Rose (1798-1873) developed a chemical-morphological mineral system (Berlin, 1838, 1852), which looks quite modern, even today ... 

FIGURE 21.6 Shapes of the unit cells in the seven crystal systems. The... 

In the 19th century, crystallographers could classify crystals into the seven crystal systems only on the basis of their external symmetries. They could not measure the dimensions of unit cells or the positions of atoms within them. Several... 

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