Fourier series in 2-D and 3-D

Every periodic 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 f_G are complex coefficients (called the structure factors). If f(r) is a real function, then f_-G = f_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.

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

In a similar manner, periodic functions with a bcc, fcc, or hexagonal Bravais lattice can be constructed.

bcc:

fcc:

hexagonal:

If the periodic function f(r) is known, the Fourier coefficients f_G can be determined by multiplying both sides of Eq. [1] by exp(-iG'·r) and integrating over a primitive unit cell.

The left-hand side is the Fouier transform of the function f(r) restricted to a unit cell. On the right-hand side, only the term where G = G' contributes and the integral evaluates to f_G times the volume V of the primitive unit cell.

The function f_cell(r) = f(r) within the primitive unit cell and is zero outside the primitive unit cell. The Fourier coefficient f_G is the Fourier transform of the function f_cell(r) divided by the volume.

Example 1: cubes repeated on a bcc lattice

Cubes are arranged on a bcc lattice such that the corners of the cubes just touch.

A three-dimensional periodic function f is defined such that it has a constant value C inside the cubes and is zero outside the cubes. This function can be expressed as a Fourier series,

The Fourier coefficients f_G are given by,

The cube at the origin lies entirely within the Wigner-Seitz cell and the function f is zero outside the cube so we just need to integrate over the cube. The integrals over x, y and z are easily performed.

This result can readily be generalized to any rectangular cuboid with dimensions L_x×L_y×L_z that is repeated on any three-dimensional Bravais lattice. As long as a primitive unit cell can be defined such that the cuboid is entirely within the cell, the Fourier series for a function that has the value C inside the cuboids and zero outside the cuboids is,

Here V is the volume of the primitive unit cell.

Example 2: spheres on an fcc lattice

Spheres of radius R are arranged on a fcc lattice.

A three-dimensional periodic function f is defined such that it has a constant value C inside the spheres and is zero outside the spheres. This function can be expressed as a Fourier series,

The Fourier coefficients f_G are given by,

As long as the spheres do not overlap, a primitive unit cell can be defined so that the sphere at the origin lies entirely within the primitive unit cell. Since the function f is zero outside sphere, we just need to integrate over the sphere.

The integral over φ is easily performed.

Recognizing that,

the integral over θ can be performed.

Finally, performing the integral over r,

As long as the spheres do not overlap, the Fourier series for spheres repeated on any Bravais lattice is,

Example 3: Total electron density

The total electron density in a crystal is a periodic function. Most of the electrons in a solid are core electrons that are very tightly bound to the nuclei. To a good approximation, the total electron density of an atom can be described by a Gaussian function exp(-r²/r₀²) where typically the half-width r₀ of the Gaussian is much smaller than the lattice constant. For a crystal with one atom in the basis, the electron density of a primitive unit cell would be,

This function only depends on r so the angular integration is the same as in example 2. The Fourier coefficents are,

where the limits of integration can be extended to infinity because the Gaussian function goes to zero so quickly. The Fourier series that describes the electron density is,

If there are more atoms in the basis, the Gaussian functions that describe the total electron density of different atoms do not overlap with each other significantly and the Gaussians can just be summed. The Fourier coefficients are,

The index j sums over all of the atoms of the basis. A_j describes the height and r_0j the half-width of the electron density of the atom at position r_j. The quantity in square brackets is n_cell. The Fourier series is,

A muffin tin is used to bake muffins.

In solid state physics, a 2-d muffin tin potential is a potential consisting of a lattice of circular regions. Within the circular regions the potential is a Coulomb potential with the form -Ze²/(4πεr) and between the circular regions the potential is constant. The constant is chosen so that there is no discontinuity in the potential. This potential looks like a muffin tin.

In three dimensions, a muffin tin potential U(r) is a lattice of spherical regions where the potential inside the the spheres is -Ze²/(4πεr) and outside the spheres the potential is constant.

The integral that must be performed to determine the Fourier coefficients is,

The first term is the radial Fourier transform of the Coulomb potential and the second term adds a constant value to the potential in the spherical regions to match the Coulomb potential to the zero potential outside the spherical regions.

The Fourier series for a muffin tin potential is,

Example 6: circles on a 2-d Bravais lattice

Consider a periodic function defined by non-overapping circles arranged on a 2-D Bravais lattice.

A function f is defined such that it has a constant value C inside the circles and is zero outside the circles. As long as the circles do not overlap, a primitive unit cell can be defined so that the circle at the origin lies entirely within the primitive unit cell. Since the function f is zero outside circle, we just need to integrate over the circle. The Fourier coefficients f_G are given by,

Here A is the area of the primitive unit cell and R is the radius of the circles. Performing the integral over θ.

Here J₀ is the zeroth order Bessel function. Integrating over r yields,

Here J₁ is the first order Bessel function. As long as the circles do not overlap, the Fourier series for circles repeated on any 2-D Bravais lattice is,

Example 7: parallelograms repeated on a 2-D Bravais lattice

Consider a periodic function defined by non-overapping parallelograms arranged on a 2-D Bravais lattice.

A function f is defined such that it has a constant value C inside the parallelograms and is zero outside the parallelograms. As long as the parallelograms do not overlap, a primitive unit cell can be defined so that the parallelogram at the origin lies entirely within the primitive unit cell. Since the function f is zero outside parallelogram, we just need to integrate over the parallelogram. The Fourier coefficients f_G are given by,

Here A is the area of the primitive unit cell. b and h are shown in the drawing. Performing the integral over y,

Integrating over x yields,

As long as the parallelograms do not overlap, the Fourier series for parallelograms repeated on any 2-D Bravais lattice is,

This includes the results for squares and rectangles as special cases (tan(0) = 0). e.g. rectangle: