Three-Dimensional Lattices and Their Symmetries

For one who has fully understood the preceding development of lattices, symmetry, and symmetry groups in 2D, the same fundamental concepts in 3D should be easy to understand. The principles are the same only the dimensionality in which they are to be implemented will change. 

We next turn to monoclinic lattices, of which there are two types. A monoclinic lattice is one in which we require one vector to be perpendicular to the plane of the other two. The lattice then has twofold rotational symmetry about this unique vector and planes of symmetry perpendicular to it. A monoclinic lattice (so-called because there is only one nonorthogonal pair of 

Note that there would be nothing to gain in defining a C-centered monoclinic cell since the resulting lattice would still be a primitive monoclinic lattice with shorter a and b translation vectors, a smaller volume, and the same symmetry. 

We can continue to apply restrictions to the defining vector set so as to obtain an orthorhombic lattice in which all three vectors are of different lengths, but are required to be orthogonal. The lattice now has considerable symmetry, namely, three mutually perpendicular sets of twofold axes, and three sets of mutually perpendicular reflection planes. 

If in addition to orthogonality of the translation vectors we also require two vectors to be of equal length, say a = b, we have a tetragonal lattice. This now has the same mirror planes and twofold axes as an orthorhombic lattice but has fourfold axes parallel to the c direction. In this case there is only one form of centering possible, namely, / centering. 

---