Polarization of arbitrary charge distributions: The classical electrodynamics perspective

Abstract

The conventional definition of electric polarization as the “dipole moment per unit volume” is valid only for the special case of well-separated dipoles. An alternative general approach is to view polarization as a characteristic not of a single charge distribution but rather of an adiabatic transition between two nearby states. Polarization can then be rigorously defined as the integral of the current density over that transition. In contrast with the “Modern Theory of Polarization,” which is fully quantum-mechanical, in this paper polarization is considered from the classical perspective. Such treatment is less fundamental but simpler, has pedagogical advantages and, importantly, is subject to fewer constraints. Polarization can be rigorously and unambiguously defined for periodic or nonperiodic charge distributions, finite or infinite, microscale or macroscale, electrically neutral or non-neutral, continuous or discrete, at any temperature; polarization can be spontaneous or induced. A previous classical (non-quantum) analysis by Russakoff (Am J Phys 1970) was (i) limited to the Clausius–Mossotti/Lorenz–Lorentz model of molecular dipoles, and (ii) involves multipole expansions, which the analysis of this paper shows to be redundant. The traditional dipole model of polarization is a straightforward special case of the proposed definition. A number of illustrative examples are presented. • Review of models of electric polarization, with an emphasis on generality and rigor. • The definition of polarization as “the dipole moment per unit volume” is flawed. • Polarization is rigorously defined for a transition between two states, not for a single state. • Trade-offs between classical and quantum-mechanical theories are examined. • The classical theory is less fundamental but is subject to much fewer constraints. • A variety of examples are presented.