Technical Physics, Vol. 46, No. 2, 2001, pp. 133–138. Translated from Zhurnal Tekhnicheskoœ Fiziki, Vol. 71, No. 2, 2001, pp. 1–7. Original Russian Text Copyright © 2001 by Bukina, Dubovik. THEORETICAL AND MATHEMATICAL PHYSICS The Contributions of Polarizabilities to Four Basis Polarizations of Electromagnetic Media E. N. Bukina and V. M. Dubovik Joint Institute for Nuclear Research, Dubna, Moscow oblast, 141980 Russia e-mail: [email protected] Received February 8, 2000 Abstract—The contributions to four basis densities of polarization distribution for an arbitrary electromagnetic medium are determined. Mixed polarizabilities given by tensors of up to the fourth rank are considered. Specific physical examples are discussed. © 2001 MAIK “Nauka/Interperiodica”. INTRODUCTION of a cyclic fragment involved in a complex molecule of a benzene ring C6 (Fig. 1). This has made it possible to Electromagnetic properties of some substances give insight into the interaction between microcrystals (e.g., crystals [1]) are described predominantly by the of aromatic compounds and an alternating magnetic distributions of dipoles. Multipole expansion of the field [14]. Such an extraordinary type of electromag- density of such a distribution differs from expansion of netic induction is known as aromagnetism. the current of charges and leads to unusual interaction of dipole systems with external fields. Comprehensive These advances, as well as the inadequacy of the analysis of such expansions is not only of academic theory of the Aharonov–Bohm effect [15] and the dis- interest. It is known [2, 3] that multipole expansion of covery of similar effects (see, for example, [16]), current (charge) density within the framework of the necessitate a more systematic approach to multipole Maxwell–Lorentz electrodynamics involves three representation of various configurations that can arise, types of basis dipoles: electric Q, magnetic M, and for example, in dipole media and to selecting parame- (polar) toroidal T. An electromagnetic medium without ters for the description of electromagnetic properties of free and bound charges (ρ ≡ 0), as well as unclosed cur- quantum objects like complicated molecules. rents (divj = 0), is described only by magnetic and tor- In this work, we found complete sets of basis oidal polarizations [4]. When describing the distribu- dipoles for dipole media of most physical importance tion of magnetization in a nucleus, Blatt and Weisskopf [9]. Based on the mathematical fact of their existence, (see [5], where they cite earlier works) were apparently we demonstrate possible unusual responses of con- the first to introduce magnetic contributions to toroidal densed matter to uniform and/or dynamic electric and polarizations by the name “induced electric transverse magnetic fields. This paper is an extension of [12, 17]. moments.” In classifying point groups of magnetic symmetry of crystals, several authors [6–9] have pointed to the fact that a toroidal dipole exhibits sym- CONSTRUCTION OF MULTIPOLE MOMENTS metry of a special type. Magnetic substances allowing FOR SPIN (DIPOLE) MEDIA AND THEIR toroidal ordering have been referred to as toroics. The INTERACTION WITH EXTERNAL FIELDS references cited give examples of such magnetic crys- Let us consider a system of continuously distributed tals. Recent experimental studies of toroidal polariza- elementary electric dipoles that can be described by a tion have been made on magnetic piezoelectric function d : r ∈ V limited in space and, in the general Ga2 − xFexO3 [10] and magnetoelectric Cr2O3 [11]. case, time-dependent. Assume that internal interactions In 1986, a family of axial toroidal moments was in this system or applied external fields lead to the for- included in the apparatus of the electrodynamics of mation and ordering of variously configured aggre- continuous media [12].1 A short time later, these gates, each consisting of a finite number of elementary moments found prominent application in molecular dipoles. We will describe the properties of the aggre- physics. In 1991, it was demonstrated [13] that crystal- gates by multipole parameters. The first three terms in lization of some aromatic compounds (e.g., anthracene, the expansion of the energy of their interaction with phenanthrene, or pentacene) may produce an axial tor- external fields are oidal microdipole by ordering electric dipoles of atoms 3 12 ()d W ==–dr()Er()dr --- r divEQET– – curlE 1 Historically [8], the basis dipole of this family is sometimes des- ∫ 60 ignated as G. V 1063-7842/01/4602-0133 $21.00 © 2001 MAIK “Nauka/Interperiodica” 134 BUKINA, DUBOVIK (a) Q (b) electric moment of the system, T(d) = (1/2)∫(r × d)d3r T(d) is the dipole axial toroidal moment, and P(d) = (1/10)∫[ r(rd) – 2r2d]d3r is the dipole polar poloidal moment (see Figs. 1a–1c). These three vectors form a basis allowing one to represent the density of distribution of a given type. In 2 3 (2) addition, (1) contains r0 = ∫rEd r, a scalar moment (c) (d) r0 P(d) corresponding to the time derivative of the rms charge radius in the expansion of the current (i.e., to the double 2 2 2 3 spherical layer, see Fig. 1d); r0 = ∫[ 2r(rE) + r E]d r, the rms radius of the longitudinal component of the dipole moment Q, generated by the double cylindrical (d) layer of the dipoles (see Fig. 1e); and Qij and T ij , the electric and toroidal quadrupole moments determined by conventional rules. In (1), all the interactions of the (2) (e) r1 (f) multipole characteristics with the field or its derivatives are local and written for the “center” of the system. For the interaction energy between magnetic dipoles and an external nonuniform magnetic field, mH is sub- stituted for dE and we have (µ) W = –∫m(r)H(r)d3r = – MH – T curlH (µ) 1 – P curlcurlH – --M (∇ H + ∇ H ) 2 ij i j j i (µ)∇ ( ) … (µ) ˙ – Tij i curlH j + – MH – T E (2) Fig. 1. (a) Model in which electric microdipoles add up to total dipole moment Q; (b) model of a dipole axial toroidal (µ) (µ) – P (curlj – H˙˙ ) – MH – T curlH moment T(d) realized in a toroid-shaped dielectric when (µ) electric microdipoles are oriented along the azimuthal direc- + P ∆Hfree + … . tions; (c) model of a dipole poloidal moment P(d) realized in a toroid-shaped dielectric when electric microdipoles are It is seen that, according to electromagnetism dual- oriented along the meridional directions; (d) centrosymmet- ity, the dipole moment Q in Eq. (1) is replaced by the ric polarization of a spherical dielectric layer having the dipole magnetic moment M (characterizing an equiva- (2) (d) (µ) symmetry of a spherical capacitor; scalar moment r0 is lent current loop); T , by the polar toroidal dipole T ; the main multipole characteristic of this object; (e) model of and the vector poloidal moment P(d), by the dipole axial the longitudinal rms radius of the electric dipole moment poloidal moment P(µ). If the magnetic moments are car- (2) ried by spin particles (e.g., neutrons), then the multi- r1 (double cylindrical layer); (f) equatorial section of a thin-walled torus with magnetic microdipoles on the merid- pole expansion of the distribution of their magnetic µ ians; poloidal dipole moment P( ) is the multipole charac- moments (in a nucleus or a neutron star) acquires the teristic of this object upon its contraction to a point. primary sense. Zheludev [6] was the first to predict tor- oidal ordering of spin magnetic moments in perovskites within the theory of condensed matter. For ultrathin (d) 1 (d) films, ordering of magnetic particles into closed config- – P curlcurlE – --Q (∇ E + ∇ E ) – T ∇ (curlE) 2 ij i j j i ij i j urations that can be described by toroidal moments seems to be a rule rather than an exception [18, 19]. 1 (2) (d) + -----r —divE + … –QE – T curlE Note that exchange interactions between coplanar spins 10 d (1) give rise to the Kosterlitz–Thouless lattice of “antitor- oidally” arranged vortexes. (d) (d) + P (—ρ – ∆E) + … –QE + T H˙ It is evident that expansions like (1) and (2) can be ( ) obtained for the remaining two fundamental (toroidal) + P d (j˙ – E˙˙) + …, 2 In the specific case of zero inner radius, such a configuration is where Q = ∫d (r)d3r is the conventional total dipole sometimes called “hedgehog.” TECHNICAL PHYSICS Vol. 46 No. 2 2001 THE CONTRIBUTIONS OF POLARIZABILITIES TO FOUR BASIS POLARIZATIONS 135 dipole distributions t(r) and g(r). It is unlikely that the which is nonlinear in field in this case. Eventually, we resulting high-order vortexes are realized in crystals. come to a qualitatively new property of the system of They may form in exotic media where the elementary dipoles. It may be concluded that the external field Hi carriers initially possess, for example, toroidal dipole induces an additional magnetic dipole polarization of moments. Recent experimental data indicate that neu- α(m) trinos are apparently Majorano particles forming dense the medium Mk = ik Hi, which can interact with the 15 –3 α(m) media (10 cm ). It is known [20, 21] that neutrinos field Hk. Here, ik is the quadratic magnetic suscepti- have only toroidal dipole moments. That coherent exci- bility of the medium. tations in such media, in particular, hadron gravita- tional excitations, are a possibility was first indicated Thus, we introduced a nonlinear interaction in [22]. accounting for the nonlinear response of the medium to a magnetic field. Basically, the tensor α may have an arbitrary number of subscripts. In the expression for VECTOR PARAMETERS energy, it must be convolved with the corresponding AND NONLINEAR MEDIA symmetric tensor for the vector of the external mag- netic field strength. A system consisting of electric The above discussion is related to so-called linear dipoles may naturally experience deformations similar electrodynamics.