# Brillouin Zone Sampling of a Periodic Chain with *N* Sites

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This Demonstration shows the sampling of the -points in the first Brillouin zone (BZ) of a virtually infinite linear crystal as a function of the number of sites in the unit cell. By choosing a periodic chain with sites one can sample -points in the reciprocal space of the first BZ, whose spacing is inversely proportional to and the lattice parameter . Then , where * *is the allowed quantum number for the chain ( or equivalently, ). There is also cyclic periodicity in . The -points thus obtained are mapped onto the analytical form of the tight-binding electronic dispersion relation for the chain. Diagonalizing the associated Bloch Hamiltonian gives the electronic energy eigenvalues . These are calculated and plotted as a function of the tight-binding hopping parameter and the on-site energy parameter expressed in electron volts.

Contributed by: Jessica Alfonsi (University of Padova, Italy) (March 2011)

Open content licensed under CC BY-NC-SA

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## Details

C. Kittel, *Introduction to **Solid State Physics*, 7th ed., Hoboken, New Jersey: J. Wiley and Sons, 1996.

S. L. Altmann, *Band Theory of Solids: An Introduction from the Point of View of Symmetry*, Oxford: Clarendon Press, 1991.

J. Alfonsi, Small Crystal Models for the Electronic Properties of Carbon Nanotubes, PhD thesis, University of Padova, 2009, Chap. 4 and references therein.

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