= Solution
A <phonon> is a quantum of a <normal mode> of <Bravais lattice> vibration. A harmonic mode of <angular frequency> $\omega$ has <energy> $\hbar\omega(n+1/2)$, and adding one <phonon> increases its <energy> by $\hbar\omega$. The <phonon> carries <crystal momentum> $\hbar\mathbf q$, defined modulo a <reciprocal lattice> vector.
The first <Brillouin zone> is the <Wigner-Seitz cell> of the <reciprocal lattice> about the origin: the set of <wavevectors> at least as close to the origin as to any other <reciprocal lattice> point. Its boundaries are reciprocal-space perpendicular bisectors. It gives one representative of each <wavevector> modulo <reciprocal lattice> translations.
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