Supplementary XRD Theory

The structure of a crystalline material may be described by a crystal lattice structure and the locations of atoms in this lattice. The crystal lattice is a 3-dimensional periodic array that defines the symmetries, i.e. shape, of the crystal. There are seven possible crystal lattices: cubic, hexagonal, tetragonal, rombohedral, orthorhombic, monoclinic and triclinic. When we define the possible arrangements of atoms at a point in the crystal lattice, for each lattice, we arrive at the Bravais lattices, sometimes called bases, of which there are 14 possible configurations, divided among the seven crystal lattices. Consider the cubic crystal lattice. It is like a stack of equally sized cubes where the corners of the cubes represent the lattice points. Inside each of these cubes we may arrange equally sized spheres, representing possible atomic sites. The set of possible arrangements that do not repeat within the cube, and cannot be represented by sums of other configurations, are the bases. The six other crystal lattices are represented by various distortions (reductions of the symmetry) of the cubes, or in the case of the hexagonal lattice, the use of hexagonal prisms. A particular crystal can then be defined by placing the appropriate atoms in the appropriate locations in a basis, thus defining the unit cell for the crystal. This unit cell may then be translated in three dimensions, according to the symmetries of the crystal lattice, to generate the entire crystal. In this lab, all the crystals you will investigate have cubic symmetry. For the cubic lattice there are three distinct basis (Fig. 1) . The length of the side of the unit cell, a or a0, is known as the lattice constant.