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2D Brillouin Zones

Three steps are needed to construct successive Brillouin zones for 2D lattices.
1. Select one reciprocal lattice point as the origin.
2. Draw the bisectors perpendicular to the line segments connecting the origin with the other reciprocal lattice points (the Bragg lines).
3. The Brillouin zone consists of those points in the plane such that a line segment from the point to the origin intersects exactly Bragg lines.

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The dispersion relation for the excitations spectra in a periodic medium is periodic in reciprocal space, that is, in the space of pseudo-momentum (-space). Due to this periodicity, it is possible to find the disjoint regions (called Brillouin zones) in which the dispersion relation is fully characterized.
Reference
[1] N .W. Ashcroft and N. D. Mermin, Solid State Physics, Philadelphia: Saunders College, 1976.
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