Plane Waves and the Brillouin Zone

We emphasized in Chapter 2 that we are interested in applying DFT calculations to arrangements of atoms that are periodic in space. We defined the 

It turns out that many parts of the mathematical problems posed by DFT are much more convenient to solve in terms of k than they are to solve in terms of 

Just as we defined positions in real space in terms of the lattice vectors 

A simple example of this calculation is the simple cubic lattice we discussed in Chapter 2. In that case, the natural choice for the real space lattice vectors has a, = a for all i. You should verify that this means that the reciprocal lattice vectors satisfy b, = 2rr/a for all i. For this system, the lattice vectors and the reciprocal lattice vectors both define cubes, the former with a side length of a and the latter with a side length of 2tt/g. 

In Chapter 2 we mentioned that a simple cubic supercell can be defined with lattice vectors a, = a or alternatively with lattice vectors a = 2a. The first choice uses one atom per supercell and is the primitive cell for the simple cubic material, while the second choice uses eight atoms per supercell. Both choices define the same material. If we made the second choice, then 

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