Electric Octopole Magnetic Quadrupole Research Articles

Multipole theory and the Hehl–Obukhov decomposition of the electromagnetic constitutive tensor

The Hehl–Obukhov decomposition expresses the 36 independent components of the electromagnetic constitutive tensor for a local linear anisotropic medium in a useful general form comprising seven macroscopic property tensors: four of second rank, two vectors, and a four-dimensional (pseudo)scalar. We consider homogeneous media and show that in semi-classical multipole theory, the first full realization of this formulation is obtained (in terms of molecular polarizability tensors) at third order (electric octopole–magnetic quadrupole order). The calculations are an extension of a direct method previously used at second order (electric quadrupole–magnetic dipole order). We consider in what sense this theory is independent of the choice of molecular coordinate origins relative to which polarizabilities are evaluated. The pseudoscalar (axion) observable is expressed relative to the crystallographic origin. The other six property tensors are invariant (with respect to an arbitrary choice of each molecular coordinate origin), or zero, at first and second orders. At third order, this invariance has to be imposed (by transformation of the response fields)—an aspect that is required by consideration of isotropic fluids and is consistent with the invariance of transmission phenomena in dielectrics. Alternative derivations of the property tensors are reviewed, with emphasis on the pseudoscalar, constraint-breaking, translational invariance, and uniqueness.

Electromagnetic boundary conditions in multipole theory

Multipole expansions for the macroscopic charge and current densities in a dielectric half-space involve a hierarchy of singular functions comprising the Dirac delta function and its derivatives. For these, Maxwell's differential equations yield corresponding singular expansions of the macroscopic electromagnetic fields E and B, and the response fields D and H, together with their boundary conditions (in terms of macroscopic multipole moment densities) at a dielectric–vacuum (or dielectric–dielectric) interface. Explicit results are obtained up to electric octopole–magnetic quadrupole order. These show that published expressions for boundary conditions are incomplete beyond electric dipole order, due to an invalid assumption concerning two-dimensional behaviour at the interface. The effect of this on studies of certain reflection effects for anisotropic media is detailed. Comparison of the differential theory with the standard integral formulation shows that, beyond electric dipole order, the latter is incomplete and redundant.

Translationally invariant semi-classical electrodynamics of magnetic media to electric octopole-magnetic quadrupole order

We consider semi-classical macroscopic electrodynamics that is translationally invariant (independent of the choice of an arbitrary, implicit set of coordinate origins for molecule-fixed axes) for linear, homogeneous, anisotropic media interacting with harmonic, plane electromagnetic waves. We extend a previous formulation at electric octopole-magnetic quadrupole order to include media comprising magnetic molecules (those possessing both time-even and time-odd properties). This requires two additional invariant, time-odd molecular polarizabilities. Overall, the electrodynamics depends on 10 invariant polarizabilities—5 time even (one each of electric dipole and electric quadrupole–magnetic dipole order, and three of electric octopole-magnetic quadrupole order) and 5 time odd (one, two, and two, respectively)—that are required for the description of linear transmission and reflection phenomena, and material constants. The two additional time-odd polarizabilities account for certain predicted effects, and one of them contributes to the inverse ac permeability of magnetic media. The results are presented in a form that is suitable for numerical computation.

Translational invariance, the Post constraint and uniqueness in macroscopic electrodynamics

We consider semi-classical multipole theory for non-magnetic molecules interacting with harmonic plane electromagnetic waves, to electric octopole–magnetic quadrupole order and relative to an arbitrary set of molecular coordinate origins {On}. Spatial averaging of expectation values of induced molecular multipole moments produces a macroscopic theory for linear, homogeneous, anisotropic media that has three shortcomings: it is only partially invariant with respect to {On}, it is ambivalent on the Post constraint (equality of the traces of the magnetoelectric tensors), and it yields non-unique dynamic response fields D and H. To remedy these, we present a fully invariant theory that is consistent (affirmative) on the Post constraint, and is based on five time-even, invariant molecular polarizability tensors (one each of electric dipole and electric quadrupole–magnetic dipole order, and three of electric octopole–magnetic quadrupole order). As in previous work on linear phenomena, translational invariance is achieved through the Van Vleck–Buckingham condition. Uniqueness of the invariant response fields is demonstrated, based on linear independence of molecular polarizability tensors at each multipole order above electric dipole. Our results are compared with previously published expressions for two invariant polarizabilities.

The quadrupole–quadrupole polarizability of a non-magnetic molecule

The quadrupole–quadrupole polarizability of a molecule introduced by Buckingham is not a unique molecular property, in the sense that it depends on an arbitrary choice of the origin of the molecular coordinates. To obtain a physically acceptable form of this property for a non-magnetic molecule, a linear combination is taken of molecular polarizabilities of the relevant multipole order (electric octopole-magnetic quadrupole). By requiring this combination to possess the desired space–time and symmetry properties, as well as origin independence, an acceptable quadrupole–quadrupole polarizability is obtained. Expressions are presented for the components of this polarizability for molecules of certain symmetries.

On the ac magnetizability of a molecule

We present a solution to a long-standing problem in molecular physics, namely a physically acceptable expression for the ac magnetizability of a molecule. To this end we apply general arguments based on spatial transformation properties, intrinsic symmetries, linearity, and the dc limit, to express the magnetizability in terms of a linear combination of molecular polarizability tensors of the relevant (electric octopole–magnetic quadrupole) multipole order. Explicit formulae are given in terms of components of these polarizabilities for molecules of various symmetries.

Traceless multipole moment densities and transformations in macroscopic electromagnetism

We consider whether use of traceless multipole moment densities in macroscopic electromagnetism can yield physically acceptable results. For harmonic plane wave fields it is shown that a traceless electric quadrupole density yields linear constitutive relations for which the dynamical material constants (permittivity and magnetoelectric coefficients) and response fields are unphysical. We further show that, within multipole theory, these constitutive relations cannot be transformed into physically acceptable relations. Specifically, the transformed response field D is unphysical for all orders beyond the electric dipole. This contrasts with use of primitive (traced) moment densities, for which unphysical constitutive relations have been successfully transformed up to electric octopole-magnetic quadrupole order, thereby providing also the leading contribution to the ac permeability.

Transformed multipole theory of the response fields D and H to electric octopole–magnetic quadrupole order

We consider multipole theory of linear constitutive relations for the macroscopic electromagnetic response fields D and H to electric octopole–magnetic quadrupole order. We use a recently developed transformation theory to obtain unique, physically acceptable, constitutive relations from the unphysical relations of conventional multipole theory. Explicit expressions for the transformed material constants and transformed macroscopic multipole moment densities in terms of appropriate macroscopic polarizability tensors are presented for both non–magnetic and magnetic media. The electric octopole–magnetic quadrupole contributions yield a new, physically acceptable multipole expression for the AC magnetizability; they satisfy the Post constraint; and they disagree with published results based on trial–and–error methods.

Completion of multipole theory for the electromagnetic response fields D and H

Multipole theory of linear constitutive relations for the response fields D and H in dielectrics is incomplete in the sense that it yields unphysical (for example, origindependent) results when taken beyond electric dipole order. Because of the inherent non–uniqueness of D and H in Maxwell'equations, we consider the role of transformations of these fields in multipole theory. Specifically, we construct (for a non–dissipative magnetic medium to electric quadrupole–magnetic dipole order, and for E and B fields represented by complex harmonic plane waves) the most general transformation which (i) leaves the inhomogeneous Maxwell equations for D and H unchanged; (ii) preserves the linear, homogeneous dependence of D and H on the macroscopic polarizability tensors and fields E and B ; (iii) is consistent with the requirements of space inversion and time reversal; (iv) maintains the order of the multipole expansion; (v) imposes origin independence on the tensors representing material constants; (vi) preserves necessary symmetries of these tensors. The resulting transformed constitutive relations are unique. They satisfy the Post constraint (equality of traces of the magnetoelectric tensors), yield a Fresnel reflection matrix which satisfies reciprocity, and imply multipole moment densities which are origin independent. The transformation theory can be applied to higher multipole orders, and the results adapted to dissipative media.

Post's constraint for electromagnetic constitutive relations

A brief review is given of the Post symmetries and constraint for covariant electromagnetic constitutive relations which are linear in the fields E and B. Their extension to non-covariant constitutive relations which are linear also in space and time derivatives of E and B is discussed. The Post constraint is compared with published results from multipole theory. It is found that for time-dependent fields in non-magnetic and magnetic substances, and to electric octopole-magnetic quadrupole order, the constraint is satisfied and in agreement with experimental data. For static fields of the magnetoelectric effect, available data do not satisfy this constraint.

Multipole solution for the macroscopic electromagnetic boundary conditions at a vacuum—dielectric interface

Multipole solutions are obtained for the boundary conditions on the macroscopic electromagnetic fields at the surface of a semiinfinite homogeneous medium in a vacuum, which may be anisotropic, magnetic and chiral. The theory is based on the differential forms of Maxwell's equations and is developed within the electric octopolemagnetic quadrupole approximation, since optical effects are known which require this multipole order for their description. To ensure that both spatial invariance and reciprocity (timereversal symmetry) are satisfied, it is necessary to take the constitutive relations for the D and H fields in relativistically covariant form. These forms are derived to the order of electric octopole and magnetic quadrupole. The boundary conditions are shown to reduce to the standard Maxwell forms when the properties of the medium are described within the electric dipole approximation D = 0E P and H = 01B. However, when higher multipole contributions are included in the constitutive relations, vario...

Eigenvector approach to the evaluation of the Jones N matrices of nonabsorbing crystalline media

An eigenvector equation within the electric octopole–magnetic quadrupole approximation is derived, from which it is possible in principle to establish the eigenvectors and the refractive indices of polarized light forms sustained by an anisotropic crystal for any chosen direction of phase propagation. It is shown that, for propagation along the crystal axes when the rays do not separate, a decomposition technique based on eigenvectors and the Jones resolution of a medium into independent differential plates substantially simplifies solution of the general problem. Precise expressions are obtained for the Jones parameters n, (ny − nx), (n−45 − n45), and (nR − nL) in terms of multipole-property tensors of the crystal. The effects of crystal symmetry on these expressions are summarized.

Non-reciprocal optical rotation in cubic antiferrornagnets

Abstract A recent multipole theory of light propagation in non-magnetic crystals is extended to non-absorbing magnetic crystals to the order of electric octopoles and magnetic quadrupoles. The theory distinguishes contributions of property tensors of different multipole order, each of which has both a time-symmetric and time-antisymmetric part. Within the electric octopole-magnetic quadrupole approximation there are six such contributions; five of which are associated with known optical effects. The sixth contains time-antisymmetric tensors of electric octopole-magnetic quadrupole order and is shown to describe a novel non-reciprocal (or Faraday-type) rotation which may occur even in the absence of net magnetization. For propagation along a body diagonal in certain cubic antiferromagnets this rotation is not masked by other effects. The related circular birefringence is comparable in magnitude to the linear birefringence that has been observed in some cubic systems, which is due to time-symmetric tensors of the same multipole order.