Bravais Lattices, Symmetry and Crystals

3D space can be filled without voids or overlapping by identical prismatic cells with well-defined symmetries, and their types are limited to seven. These units cells can be defined by the lengths of three nonplanar primitive vectors ai, a2 and a3 and by the angles a, (3 and 7 between these vectors. They generate the seven simple crystal systems or classes, defined by the sets of all points taken from a given origin of these cells, that are defined by vectors 

The other crystal lattices can be generated by adding to some of the above-defined cells extra high-symmetry points by the so-called centering method. TableB.2 shows the new systems added to the simple crystal lattices (noted s, or P, for primitive) and the numbers of lattice points in each conventional unit cell. The body-centred lattices are noted be or I (for German Innenzentrierte), the face-centred, fc or F, and the side-centred or base-centred lattices are noted C (an extra atom at the Centre of the base). These 14 lattice systems are known as the Bravais lattices (noted here BLs). A representation of their unit cells can be found in the textbook by Kittel [7]. 

A primitive cell of a BL is a cell of minimum volume that contains only one lattice point, so that the whole lattice can be generated by all the translations of this cell. This definition allows for different primitive cells for the same BL, but their volumes must be the same. The parallelepiped defined by the three primitive vectors ai, a2, and a3 of a simple BL is a primitive cell of this lattice. 

The unit cells of the simple P systems are primitive cells. Primitive cells are not unique and most don t have the BL symmetry, but it is possible 

System Simple (P) Body-centred Face-centred Base-centred 

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