Response of a lattice of polarisable points to imposed electric fields

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Abstract
A simple cubic lattice Lambda of unit spacing with a point of polarisability alpha at each lattice vertex is considered. The response of the lattice to an applied constant electric field and to the electric field due to a charged or dipolar impurity is calculated and asymptotic representations for the polarisation at lattice points far from the surface of the lattice sample and far from the impurities are obtained. Polarisation due to a constant applied external field is shown to depend on the shape of the lattice sample and the dielectric constant of the medium exterior to the lattice sample. Polarisation due to charged or dipolar impurities is shown to be independent of shape and external dielectric constant. The asymptotic representations obtained for the polarisation are compared with model calculations treating the lattice sample as a continuum dielectric of dielectric constant epsilon . This dielectric constant is found to be epsilon =(1+8 pi alpha /3)/(1-4 pi alpha /3) so that the standard Lorentz relation between macroscopic and microscopic electric fields is confirmed for all the cases considered. For the dipolar impurity case the continuum picture must use a rescaled dipole moment which is calculated explicitly. The reaction fields giving this rescaled dipole moment are discussed.

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