09-28-2021-2045 - crystal momentum

 

Crystal momentum[edit]

Main article: Crystal momentum

k-vectors exceeding the first Brillouin zone (red) do not carry any more information than their counterparts (black) in the first Brillouin zone.

By analogy to photons and matter waves, phonons have been treated with wavevector k as though it has a momentum ħk,^[12] however, this is not strictly correct, because ħk is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because k is only determined up to addition of constant vectors (the reciprocal lattice vectors and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates Q and Π are defined so that

where

for any integer n. A phonon with wavenumber k is thus equivalent to an infinite family of phonons with wavenumbers k ± 2π/a, k ± 4π/a, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. Bloch electrons obey a similar set of restrictions.

Brillouin zones, (a) in a square lattice, and (b) in a hexagonal lattice

It is usually convenient to consider phonon wavevectors k which have the smallest magnitude |k| in their "family". The set of all such wavevectors defines the first Brillouin zone. Additional Brillouin zones may be defined as copies of the first zone, shifted by some reciprocal lattice vector.

https://en.wikipedia.org/wiki/Phonon#Crystal_momentum