Interaxial angle

Crystal Systems. The cubic crystal system is composed of three space lattices, or unit cells, one of which we have already studied simple cubic (SC), body-centered cubic (BCC), anA face-centered cubic (FCC). The conditions for a crystal to be considered part of the cubic system are that the lattice parameters be the same (so there is really only one lattice parameter, a) and that the interaxial angles all be 90°. 

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. 

There is only one space lattice in the rhombohedral crystal system. This crystal is sometimes called hexagonal R or trigonal R, so don t confnse it with the other two similarly-named crystal systems. The rhombohedral crystal has nniform lattice parameters in all directions and has equivalent interaxial angles, bnt the angles are nonorthogonal and are less than 120°. 

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. 

Mathematically, this is a triple scalar product and can be used to calculate the volume of any cell, with only a knowledge of the lattice translation vectors. If the lattice parameters and interaxial angles are known, the following expression for V can be derived from the vector expression ... 

The general expression for d in terms of lattice parameters and interaxial angles is somewhat complicated... 

Figure 6.7 Calculated diffraction spot configuration in reciprocal space. <5 indicates interaxial angle between c and a crystallographic axes.

The fourteen Bravais lattices are divided into seven crystal systems. The term system indicates reference to a suitable set of axes that bear specific relationships, as illustrated in Table 9.2.1. For example, if the axial lengths take arbitrary values and the interaxial angles are all right angles, the crystal system... 

Interference figures are useful in determining the orientation of a mineral particle and whether the mineral is uniaxial or biaxial, provided the particle is large enough to produce useful interference figures. A consideration of interference figures and interaxial angles is beyond the scope of this discussion. 

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. 

Each unit cell is a parallelepiped whose three axes may or may not be equal in length and whose interaxial angles may or may not be 90 degrees. Relationships between the values of the angles and between the lengths of the axes form the basis for classification into seven types of crystal... 

The crystal axes for a given crystal may be chosen in many different ways however, they are conventionally chosen to yield a coordinate system of the highest possible symmetry. It has been found that crystals can be divided into six possible systems on the basis of the highest possible symmetry that the coordinate system may possess as a result of the symmetry of the crystal. This symmetry is best described in terms of symmetry restrictions governing the values of the axial lengths a, b, and c and the interaxial angles a, (3, and y. 

Over 75% of inorganics and 95% of intermetallics crystallize in crystal structures having 90° and 120° interaxial angles of the unit cell. The residnal compounds for inorganics are mainly oxides, halides, and snlfides, while those for the intermetallics are Se, Te, and S compounds. These compounds represent the boundary compounds between the inorganics and the intermetalhcs. 

Certain trigonal crystals may also be referred to rhombohedral axes, the unit cell being a rhombohedron defined by cell edge a and interaxial angle a ji90P)... 

---