Bloch wave

A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form: A Bloch wave (also called Bloch state or Bloch function or Bloch wavefunction), named after Swiss physicist Felix Bloch, is a type of wavefunction for a particle in a periodically-repeating environment, most commonly an electron in a crystal. A wavefunction ψ is a Bloch wave if it has the form: where r is position, ψ is the Bloch wave, u is a periodic function with the same periodicity as the crystal, k is a vector of real numbers called the crystal wave vector, e is Euler's number, and i is the imaginary unit. In other words, if we multiply a plane wave by a periodic function, we get a Bloch wave. Bloch waves are important because of Bloch's theorem, which states that the energy eigenstates for an electron in a crystal can be written as Bloch waves. (More precisely, it states that the electron wave functions in a crystal have a basis consisting entirely of Bloch wave energy eigenstates.) This fact underlies the concept of electronic band structures. These Bloch wave energy eigenstates are written with subscripts as ψn k, where n is a discrete index, called the band index, which is present because there are many different Bloch waves with the same k (each has a different periodic component u). Within a band (i.e., for fixed n), ψn k varies continuously with k, as does its energy. Also, for any reciprocal lattice vector K, ψn k = ψn,(k+K). Therefore, all distinct Bloch waves occur for k-values within the first Brillouin zone of the reciprocal lattice. The most common example of Bloch's theorem is describing electrons in a crystal. However, a Bloch-wave description applies more generally to any wave-like phenomenon in a periodic medium. For example, a periodic dielectric in electromagnetism leads to photonic crystals, and a periodic acoustic medium leads to phononic crystals. It is generally treated in the various forms of the dynamical theory of diffraction.