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 cG are complex coefficients. For real functions, cG* = c-G.
Using the definition of a reciprocal lattice vector,
the Fourier series can be rewritten in terms of the primitive reciprocal lattice vectors, bi.
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 cG* = 1, then the periodic function is,
Give an example of a real periodic function that has a body centered cubic Bravais lattice.