= Solution
Construct <Bragg planes> by taking the midpoints between the reciprocal origin and the other <reciprocal lattice> points. In one dimension these are points at $k=m\pi/a$, for nonzero integer $m$.
The \b[<first Brillouin zone>] is the interval bounded by the nearest two such points:
$$
\boxed{-\frac{\pi}{a}\leq k\leq\frac{\pi}{a}}.
$$
The \b[<second Brillouin zone>] consists of the next intervals on both sides:
$$
\boxed{-\frac{2\pi}{a}\leq k\leq-\frac{\pi}{a}
\quad\text{or}\quad
\frac{\pi}{a}\leq k\leq\frac{2\pi}{a}}.
$$
Boundary points may be assigned to one adjacent zone to avoid double counting. The first zone is reached from the origin without crossing a Bragg point; the second is reached after crossing one. Each zone has total length $2\pi/a$. In particular, the two separated pieces of the second zone are not the entire translated first-zone interval centered at $2\pi/a$.
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