The dielectric tensor and spatial dispersion

Glazer, A. M.; Cox, K. G.

doi:10.1107/97809553602060000633

International
Tables for
Crystallography
Volume D
Physical properties of crystals
Edited by A. Authier

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International Tables for Crystallography (2006). Vol. D. ch. 1.6, pp. 167-168

Section 1.6.5.2. The dielectric tensor and spatial dispersion

A. M. Glazer^a ^* and K. G. Cox^b ^‡

^a Department of Physics, University of Oxford, Parks Roads, Oxford OX1 3PU, England, and ^bDepartment of Earth Sciences, University of Oxford, Parks Roads, Oxford OX1 3PR, England
Correspondence e-mail: glazer@physics.ox.ac.uk

1.6.5.2. The dielectric tensor and spatial dispersion

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The relevant polarization term to consider here is $[ P_i^\omega = \omega_0\chi_{ij\ell}\nabla_\ell E_j^\omega. \eqno (1.6.5.3)]$

The important part of this expression is the use of the field gradient, which implies a variation of the electric field across the unit cell of the crystal rather than the assumption that $[\bf E]$ is everywhere constant. This variation in $[\bf E]$ is known as spatial dispersion (Agranovich & Ginzburg, 1984).

Assume propagation of a plane wave given by $[{\bf E}=]$ $[{\bf E_o}\exp (i{\bf k}\cdot {\bf r})]$ through an optically active crystal. Substituting into the expression for the polarization gives $[ P_i^\omega = i\omega_0\chi_{ij\ell}E_j^\omega k_\ell .\eqno (1.6.5.4)]$ This term can now be treated as a perturbation to the dielectric tensor $[\varepsilon_{ij}(\omega)]$ to form the effective dielectric tensor $[\varepsilon_{ij}(\omega, {\bf k})]$ : $[\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i\gamma_{ij\ell}k_\ell , \eqno (1.6.5.5)]$ where $[\gamma_{ij\ell}]$ has been written for the susceptibility $[\chi_{ij\ell}]$ in order to distinguish it from the use of $[\chi]$ elsewhere. Note that this can be expressed more generally as a power-series expansion in the vector $[\bf k]$ (Agranovich & Ginzburg, 1984) to allow for a generalization to include all possible spatial dispersion effects: $[\varepsilon_{ij}(\omega, {\bf k}) = \varepsilon_{ij}(\omega) + i\gamma_{ij\ell}(\omega)k_\ell + \alpha_{ij\ell m}(\omega)k_\ell k_m, \eqno (1.6.5.6)]$ where the susceptibilities are in general themselves dependent on frequency.