Eur. Phys. J. B 45, 19-31 (2005)
https://doi.org/10.1140/epjb/e2005-00160-1

Microscopic calculation of the constitutive relations

¹ Laboratoire de Minéralogie-Cristallographie, CNRS UMR 7590, Universités Paris 6 et 7, IPGP, 4 place Jussieu, 75252 Paris Cedex 05, France
² BESSY GmbH, Albert-Einstein-Str.15, 12489 Berlin, Germany
³ Laboratoire des Géomatériaux, IFG, Université de Marne la Vallée, Cité Descartes - Champs-sur-Marne, 77454 Marne La Vallée Cedex 2, France

Corresponding author: ^a brouder@lmcp.jussieu.fr

Received: 22 September 2004
Published online: 16 June 2005

Abstract

Homogenization theory is used to calculate the macroscopic dielectric constant from the quantum microscopic dielectric function in a periodic medium. The method can be used to calculate any macroscopic constitutive relation, but it is illustrated here for the case of electrodynamics of matter. The so-called cell problem of homogenization theory is solved and an explicit expression is given for the macroscopic dielectric constant in a form akin to the Clausius-Mossotti or Lorentz-Lorenz relation.
The validity of this expression is checked by showing that the standard formula is recovered for cubic materials and that the average of the microscopic energy density is the macroscopic one. Finally, the general expression is applied to Bloch eigenstates.