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Why is the Brillouin zone of a honeycomb lattice is a hexagon and not a parallelogram? I mean, how can you get the hexagonal BZ using the reciprocal lattice vectors?

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  • $\begingroup$ The brilluoin zone is a parallelogram. But because of the periodicity, you can cut the Brillouin zone apart and patch it back together in many shapes. Convention is a hexagon. $\endgroup$ Commented May 25, 2020 at 0:40

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Brillouin zone is a symmetrized cell, so called Wigner–Seitz cell. Parallelogram doesn't reflect point symmetry of your reciprocal lattice, while hexagon does, thus hexagon is chosen as the Brillouin zone instead of parallelogram.

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