Editing Phonon (section)

==Crystal momentum==
{{Main article|Crystal momentum}}
[[File:phonon k 3k.gif|class=skin-invert-image|right|thumb|250px|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 [[photon]]s and [[De Broglie wavelength|matter waves]], phonons have been treated with [[wavevector]] ''k'' as though it has a [[momentum]] ''ħk'';<ref>{{Cite book|title=[[Introduction to Solid State Physics]], 8th Edition|last=Kittel|first=Charles|publisher=Wiley|year=2004|isbn=978-0-471-41526-8|pages=[https://archive.org/details/isbn_9780471415268/page/100 100]}}</ref> 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|reciprocal lattice vector]]s and integer multiples thereof). For example, in the one-dimensional model, the normal coordinates ''Q'' and ''Π'' are defined so that
:<math>Q_k \stackrel{\mathrm{def}}{=} Q_{k+K} ;\quad \Pi_k \stackrel{\mathrm{def}}{=} \Pi_{k + K}</math>

where
:<math>K = \frac{2n\pi}{a}</math>

for any integer ''n''. A phonon with wavenumber ''k'' is thus equivalent to an infinite family of phonons with wavenumbers ''k''&nbsp;±&nbsp;{{sfrac|2{{pi}}|''a''}}, ''k''&nbsp;±&nbsp;{{sfrac|4{{pi}}|''a''}}, and so forth. Physically, the reciprocal lattice vectors act as additional chunks of momentum which the lattice can impart to the phonon. [[Bloch electron]]s obey a similar set of restrictions.

[[File:Brillouin zone.svg|class=skin-invert-image|thumb|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.