It is appetizing to amusement a phonon with beachcomber agent \, k as admitting it has a drive \,\hbar k, by affinity to photons and amount waves. This is not absolutely correct, for \,\hbar k is not in fact a concrete momentum; it is alleged the clear drive or pseudomomentum. This is because \, k is alone bent up to multiples of connected vectors, accepted as alternate filigree vectors. For example, in our apparent model, the accustomed coordinates \, Q and \,\Pi are authentic so that
Q_k \ \stackrel{\mathrm{def}}{=}\ Q_{k+K} \quad;\quad \Pi_k \ \stackrel{\mathrm{def}}{=}\ \Pi_{k + K} \quad
where
\, K = 2n\pi/a
for any accumulation \, n. A phonon with beachcomber amount \, k is appropriately agnate to an absolute "family" of phonons with beachcomber numbers \, k\pm\tfrac{2\,\pi}{a}, \, k\pm\tfrac{4\,\pi}{a}, and so forth. Physically, the alternate filigree vectors act as added "chunks" of drive which the filigree can admit to the phonon. Bloch electrons obey a agnate set of restrictions.
Brillouin zones, a) in a aboveboard lattice, and b) in a hexagonal lattice
It is usually acceptable to accede phonon beachcomber vectors \, k which accept the aboriginal consequence \, (|k|) in their "family". The set of all such beachcomber vectors defines the aboriginal Brillouin zone. Added Brillouin zones may be authentic as copies of the aboriginal zone, confused by some alternate filigree vector.
It is absorbing that agnate application is bare in analog-to-digital about-face area aliasing may action beneath assertive conditions.
Q_k \ \stackrel{\mathrm{def}}{=}\ Q_{k+K} \quad;\quad \Pi_k \ \stackrel{\mathrm{def}}{=}\ \Pi_{k + K} \quad
where
\, K = 2n\pi/a
for any accumulation \, n. A phonon with beachcomber amount \, k is appropriately agnate to an absolute "family" of phonons with beachcomber numbers \, k\pm\tfrac{2\,\pi}{a}, \, k\pm\tfrac{4\,\pi}{a}, and so forth. Physically, the alternate filigree vectors act as added "chunks" of drive which the filigree can admit to the phonon. Bloch electrons obey a agnate set of restrictions.
Brillouin zones, a) in a aboveboard lattice, and b) in a hexagonal lattice
It is usually acceptable to accede phonon beachcomber vectors \, k which accept the aboriginal consequence \, (|k|) in their "family". The set of all such beachcomber vectors defines the aboriginal Brillouin zone. Added Brillouin zones may be authentic as copies of the aboriginal zone, confused by some alternate filigree vector.
It is absorbing that agnate application is bare in analog-to-digital about-face area aliasing may action beneath assertive conditions.
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