dispersion relation

When white light passes through a prism it fans out into a rainbow, because different colours — different wavelengths — travel at slightly different speeds in the glass. That speed-depends-on-wavelength rule is the original meaning of dispersion. Physicists borrowed the word for a much broader idea: any rule that ties how fast a wave or particle moves to its wavelength or momentum.

A dispersion relation is the curve linking an electron's energy to its crystal momentum — in everyday terms, how its energy changes as it moves faster or in a different direction. For a free electron the relation is a simple upward bowl. Inside a crystal the lattice bends this curve into the shapes that make up the energy bands, and the slope of the curve gives the electron's speed while its curvature gives its effective mass. The dispersion relation is, in effect, a single band drawn out as a graph.

Dispersion relations matter because nearly everything an electron does in a solid — how fast it drifts, how it responds to a field, whether the material conducts — can be read straight off these curves. The honest subtlety is that crystal momentum is not ordinary momentum: it is defined only up to the repeating pattern of the lattice, which is why dispersion curves are drawn within a special repeating region rather than running off to infinity. The same idea also describes phonons, photons, and many other waves.