Empty lattice approximation

The empty lattice approximation is a theoretical electronic band structure model in which the potential is periodic and weak (close to constant).

One may also consider an empty[clarification needed] irregular lattice, in which the potential is not even periodic.

[1] The empty lattice approximation describes a number of properties of energy dispersion relations of non-interacting free electrons that move through a crystal lattice.

The model is useful because it clearly illustrates a number of the sometimes very complex features of energy dispersion relations in solids which are fundamental to all electronic band structures.

The strength of the scattering mainly depends on the geometry and topology of the system.

For a particle in a one-dimensional lattice, like the Kronig–Penney model, it is possible to calculate the band structure analytically by substituting the values for the potential, the lattice spacing and the size of potential well.

[2] For two and three-dimensional problems it is more difficult to calculate a band structure based on a similar model with a few parameters accurately.

Nevertheless, the properties of the band structure can easily be approximated in most regions by perturbation methods.

In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave.

The scattering process results in the well known Bragg reflections of electrons by the periodic potential of the crystal structure.

This is the origin of the periodicity of the dispersion relation and the division of k-space in Brillouin zones.

The periodic energy dispersion relation is expressed as: The

are the reciprocal lattice vectors to which the bands[clarification needed]

Though the lattice cells are not spherically symmetric, the dispersion relation still has spherical symmetry from the point of view of a fixed central point in a reciprocal lattice cell if the dispersion relation is extended outside the central Brillouin zone.

is; In three-dimensional space the Brillouin zone boundaries are planes.

This results in a very complicated set intersecting of curves when the dispersion relations are calculated because there is a large number of possible angles between evaluation trajectories, first and higher order Brillouin zone boundaries and dispersion parabola intersection cones.

"Free electrons" that move through the lattice of a solid with wave vectors

See the external links section for sites with examples and figures.

In most simple metals, like aluminium, the screening effect strongly reduces the electric field of the ions in the solid.

The electrostatic potential is expressed as where Z is the atomic number, e is the elementary unit charge, r is the distance to the nucleus of the embedded ion and q is a screening parameter that determines the range of the potential.

collapses and the empty lattice approximation is obtained.