So if I increase the volume, then the reciprocal lattice vectors will shrink in size.

The direction of the reciprocal lattice vector corresponds to the normal to the real space planes.

Due to the definition of , when is the direct lattice vector , we have the same relationship.

The are the reciprocal lattice vectors to which the bands belong.

This is because the number of reciprocal lattice vectors that lie in an interval increases.

Another helpful ingredient in the proof is the reciprocal lattice vectors.

Because the coordinates are integers, this normal is itself always a reciprocal lattice vector.

The requirement of lowest terms means that it is the shortest reciprocal lattice vector in the given direction.

It is even, meaning that the norm of any lattice vector is even.

If is the reciprocal lattice vector, we know .