Primitive and nonprimitive cells

Cubic Tetragonal Orthorhombic Rhombohedral Hexagoral Monoclinic Triclinic Four 3 - fold rotation axes One 4 - fold rotation (or rotation - inversion) axis Three perpendicular 2-fold rotation (or rotation - inversion) axes One 3-fold rotation (or rotation - inversion) axis One 6-fold rotation (or rotation - inversion) axis One 2-fold rotation (or rotation-inversion) axis None 

The minimum number of symmetry elements possessed by each crystal system is listed in Table 2-2. Some crystals may possess more than the minimum symmetry elements required by the system to which they belong, but none may have less. The existence of certain symmetry elements often implies the existence of others. For example, a crystal with three 4-fold rotation axes necessarily has, in addition, four 3-fold axes and falls in the cubic system. 

Symmetry operations apply not only to the unit cells shown in Fig. 2-3, considered merely as geometric shapes, but also to the point lattices associated with them. The latter condition rules out the possibility that the cubic system, for example, could include a base-centered point lattice, since such an array of points would not have the minimum set of symmetry elements required by the cubic system, namely four 3-fold rotation axes. Such a lattice would be classified in the tetragonal system, which has no 3-fold axes and in which accidental equality of the a and c axes is allowed. 

Crystals in the rhombohedral (trigonal) system can be referred to either a rhombohedral or a hexagonal lattice. Appendix 4 gives the relation between these two lattices and the transformation equations which allow the Miller indices of a plane (see Sec. 2-6) to be expressed in terms of either set of axes. 

In any point lattice a unit cell may be chosen in an infinite number of ways and may contain one or more lattice points per cell. It is important to note that unit cells do not exist as such in a lattice they are a mental construct and can 

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