Quantum Mechanical Methods for Studying the Solid State

8 Quantum Mechanical Methods for Studying the Solid State 

Another concept that is extremely powerful when considering lattice structures is the reciprocal lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar product of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions  

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a., h and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal lattice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2ir of the X-ray crystallographic reciprocal lattice vectors  

A simple illustrative example of reciprocal space is that of a 2D square lattice where the vectors a and b are orthogonal and of length equal to the lattice spacing, a. Here a and b are directed along the same directions as a and b respectively and have a length 1/a 

The periodicity of the lattice means that the values of a function (such as the electron density) will be identical at equivalent points on the lattice. Likewise there is a relationship between the wavefunction at a point (x in our ID lattice) and at an equivalent point elsewhere on the lattice (for the ID lattice this would he x + na, where n is an integer). Bloch s theorem provides the link each allowed lattice wavefunction must satisfy the following relationship  

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