A few elements of crystallography

A crystal is a given atomic pattern repeated in three dimensions. The unit cell is defined as constituting a basis for the crystal lattice. This basis, which generally is not orthonormal, is defined by the triplet (a,b,c) and the angles between these different vectors are a = (b, c) P = (a, c) y = (a,b). Inside this cell, the position of each atom in the crystal pattern is specified by the vector r. The position of each cell is specified by the vector R = ua -t vb -I- wc. The crystal is completely described by the translations along the cell s three vectors (a,b, c). 

The lattice points can be arranged in atomic rows. If we go from the point (u,v,w) to the origin of the lattice, the line we draw will go through the points (0,0,0), (u,v,w), (2u,2v,2w), (3u,3v,3w), etc. For each lattice point there is another line, parallel to the first, and containing the same number of points. Thus, the lattice is decomposed in a series of regularly arranged, parallel rows. They are referred to as rows with indices [u,v,w]. 

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