Crystal space lattices, types

The color variations of a nonmetallic mineral are often the result of ionic trace impurities in the crystal space lattice structure. Since the impurities vary from sample to sample, the color may vary. Some nonmetallic minerals have no color and are referred to as colorless. This variability in color, which can sometimes be extreme, means that color is one of the least useful properties for identifying nonmetallic minerals even though it is probably the most obvious. The origin of a mineral s color can be explained by three types of electronic transitions in the crystalline solids. 

Therefore, a space group is a possible combination of all the symmetry elements, macroscopic and microscopic, in space of the Bravais lattice and can be derived. It is found that when all such symmetry elements are combined and applied in the Bravais lattices, 230 different types of crystal space lattices are possible. It is appropriate to mention here that any crystal either naturally free grown or crystallized artificially from the solutions of the synthesized compounds must belong to any of these possible 230 types of space groups [1,2]. 

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. 

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. 

Orthorhombic crystals are similar to both tetragonal and cubic crystals because their coordinate axes are still orthogonal, but now all the lattice parameters are unequal. There are four types of orthorhombic space lattices simple orthorhombic, face-centered orthorhombic, body-centered orthorhombic, and a type we have not yet encountered, base-centered orthorhombic. The first three types are similar to those we have seen for the cubic and tetragonal systems. The base-centered orthorhombic space lattice has a lattice point (atom) at each comer, as well as a lattice point only on the top and bottom faces (called basal faces). All four orthorhombic space lattices are shown in Figure 1.20. 

The crystal descriptions become increasingly more complex as we move to the monoclinic system. Here all lattice parameters are different, and only two of the interaxial angles are orthogonal. The third angle is not 90°. There are two types of monoclinic space lattices simple monoclinic and base-centered monoclinic. The triclinic crystal, of which there is only one type, has three different lattice parameters, and none of its interaxial angles are orthogonal, though they are all equal. 

To derive the 9 space groups in crystal class 222 is no more complicated than was the derivation of the 13 monoclinic space groups. We take, in turn, each lattice type and associate all possible combinations of 2 or 2, axes with it, and then weed out the duplications. For the primitive ones we get 4, with no duplications ... 

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

Cl-. The lattice type is reduced from face-centered cubic to primitive cubic, and the space group of CsCl is 0 - Pm3m. Figure 10.3.3(a) shows a unit cell in the crystal structure of CsCl. 

William B. Pearson (1921-2005) developed a shorthand system for denoting alloy and intermetallic structure types (Pearson, 1967). It is now widely used for ionic and covalent solids, as well. The Pearson symbol consists of a small letter that denotes the crystal system, followed by a capital letter to identify the space lattice. To these a number is added that is equal to the number of atoms in the unit cell. Thus, the Pearson symbol for wurtzite (hexagonal, space group PS mc), which has four atoms in the unit ceU, is hPA. Similarly, the symbol for sodium chloride (cubic, space group Fm3m), with eight atoms in the unit cell, is cF8. 

The angle between two sets of planes in any type of direct-space lattice is equal to the angle between the corresponding reciprocal-space lattice vectors, which are the plane normals. In the cubic system, the [h k /] direction is always perpendicular to the (h k 1) plane with numerically identical indices. For a cubic direct-space lattice, therefore, one merely substimtes the [h k 1] values for [u v w] in Eq. 10.57 to determine the angle between crystal planes with Miller indices h k l ) and (h2 h)- With all other lattice types, this simple... 

Semiconductor crystals add impurities easily. This modifies the properties of the pure crystals so that they do not obey normal valence rules. Zinc oxide is an example of this type of crystal. Pure crystals of this type are characterized by deficient space lattices. 

If the contents of a unit cell have symmetry, containing a number of units of pattern (atoms, molecules), the number of distinct types of space lattice becomes fourteen (Fig. 82) (Bravais, 1818). And when other symmetry operations are recognised (e.g. rotation of the lattice) there are found to be 230 distinct varieties of crystal symmetr ... 

Fig 2. The 14 three-dimensional translation lattice types (Bravais, 1850). P = primitive C = c-face centred, F = all face centred, R = rhombohedral (crystals In the 7 rhombohedral space groups are described in relation to hexagonal axes). 

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