513.001 Molecular and Solid State Physics

Fourier series in 3 dimensions

Every perioic function is associated with a Bravais lattice. You can think of the function as being defined in a primitive unit cell and then repeating the primitive unit cell at every point of the Bravais lattice.

A periodic function can be written as a Fourier series in the form,

where G are the reciprocal lattice vectors of the Bravais lattice and c_G are complex coefficients. For real functions, c_G^* = c_-G.

Using the definition of a reciprocal lattice vector,

the Fourier series can be rewritten in terms of the primitive reciprocal lattice vectors, b_i.

As an example, a real periodic function that has an orthorhombic Bravais lattice can be constructed using just the reciprocal lattice vectors 100, -100, 010, 0-10,001, 00-1. If for all of these reciprocal lattice points c_G^* = 1, then the periodic function is,

Give an example of a real periodic function that has a body centered cubic Bravais lattice.