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Brillouin zones and the geometry of numbers

Abstract

It is shown in the paper that Brillouin zones (which originally appeared in the quantum theory of solids) possess a number of remarkable, purely geometrical and arithmetical properties. In their terms, problems in the geometry of numbers concerning the best packing and covering of circles and balls by fundamental domains of lattices can be solved. The geometry of Brillouin zones is found to be also closely connected with the classical number-theoretical problems concerning the number of lattice points in a circle and a ball.

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Additional information

Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 134, pp. 206–225, 1984.

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Skriganov, M.M. Brillouin zones and the geometry of numbers. J Math Sci 36, 140–154 (1987). https://doi.org/10.1007/BF01104979

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Keywords

  • Quantum Theory
  • Lattice Point
  • Brillouin Zone
  • Fundamental Domain
  • Good Packing