Phonon - Crystal Momentum

Crystal Momentum

It is tempting to treat a phonon with wave vector as though it has a momentum, by analogy to photons and matter waves. This is not entirely correct, for is not actually a physical momentum; it is called the crystal momentum or pseudomomentum. This is because is only determined up to multiples of constant vectors, known as reciprocal lattice vectors. For example, in our one-dimensional model, the normal coordinates and are defined so that

where

for any integer . A phonon with wave number is thus equivalent to an infinite "family" of phonons with wave numbers, 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.

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

It is interesting that similar consideration is needed in analog-to-digital conversion where aliasing may occur under certain conditions.