513.001 Molecular and Solid State Physics

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Physical
Constants

Periodic System
of Elements

      

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.