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Toroidal moment
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<h2 id="classical-toroidal-dipole-moment">Classical toroidal dipole moment</h2> <p>A complex expression allows the <a href="/facts/Current_density/IlUKSe2B">current density</a> J to be written as a sum of electric, magnetic, and toroidal moments using Cartesian<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> or spherical<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> differential operators. The lowest order toroidal term is the toroidal dipole. Its magnitude along direction <i>i</i> is given by </p> T i = 1 10 c ∫ [ r i ( r ⋅ J ) − 2 r 2 J i ] d 3 x . {\displaystyle T_{i}={\frac {1}{10c}}\int [r_{i}(\mathbf {r} \cdot \mathbf {J} )-2r^{2}J_{i}]\mathrm {d} ^{3}x.} <p>Since this term arises only in an expansion of the current density to second order, it generally vanishes in a long-wavelength approximation. </p><p>However, a recent study comes to the result that the toroidal multipole moments are not a separate multipole family, but rather higher order terms of the electric multipole moments.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p> <h2 id="quantum-toroidal-dipole-moment">Quantum toroidal dipole moment</h2> <p>In 1957, <a href="/facts/Yakov_Zel%2527dovich/JEpdtxIM">Yakov Zel'dovich</a> found that because the <a href="/facts/Weak_interaction/f6upmKMs">weak interaction</a> <a href="/facts/Weak_interaction/f6upmKMs">violates parity symmetry</a>, a spin-1/2 <a href="/facts/Dirac_particle/DzFaXS4p">Dirac particle</a> must have a toroidal dipole moment, also known as an anapole moment, in addition to the usual electric and magnetic dipoles.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> The interaction of this term is most easily understood in the non-relativistic limit, where the Hamiltonian is H ∝ − d ( σ ⋅ E ) − μ ( σ ⋅ B ) − a ( σ ⋅ ∇ × B ) , {\displaystyle {\mathcal {H}}\propto -d(\mathbf {\sigma } \cdot \mathbf {E} )-\mu (\mathbf {\sigma } \cdot \mathbf {B} )-a(\mathbf {\sigma } \cdot \nabla \times \mathbf {B} ),} where <i>d</i>, <i>μ</i>, and <i>a</i> are the electric, magnetic, and anapole moments, respectively, and <i>σ</i> is the vector of <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>The nuclear toroidal moment of <a href="/facts/Cesium/dXKN1S5k">cesium</a> was measured in 1997 by Wood <i>et al.</i>.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h2 id="symmetry-properties-of-dipole-moments">Symmetry properties of dipole moments</h2> <p>All dipole moments are vectors which can be distinguished by their differing symmetries under <a href="/facts/Parity_(physics)/emSbJ4hD">spatial inversion</a> (P : r ↦ −r) and <a href="/facts/T-symmetry/Tgf2LJMc">time reversal</a> (T : <i>t</i> ↦ −<i>t</i>). Either the dipole moment stays invariant under the symmetry transformation ("+1") or it changes its direction ("−1"): </p> <table><tbody><tr><th>Dipole moment</th><th>P</th><th>T</th></tr><tr><td>axial toroidal dipole moment</td><td>+1</td><td>+1</td></tr><tr><td><a href="/facts/Electric_dipole_moment/ou2LDUVg">electric dipole moment</a></td><td>−1</td><td>+1</td></tr><tr><td><a href="/facts/Magnetic_dipole_moment/VpAQKJtr">magnetic dipole moment</a></td><td>+1</td><td>−1</td></tr><tr><td>polar toroidal dipole moment</td><td>−1</td><td>−1</td></tr></tbody></table> <h2 id="magnetic-toroidal-moments-in-condensed-matter-physics">Magnetic toroidal moments in condensed matter physics</h2> <p>In <a href="/facts/Condensed_matter/EBVI5nmL">condensed matter</a> magnetic toroidal order can be induced by different mechanisms:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> <ul><li>Order of localized <a href="/facts/Spin_(physics)/4Jvp7e8P">spins</a> breaking spatial inversion and time reversal. The resulting toroidal moment is described by a sum of cross products of the spins S<i>i</i> of the magnetic ions and their positions r<i>i</i> within the magnetic unit cell:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> T = Σ<i>i</i> r<i>i</i> × S<i>i</i></li> <li>Formation of vortices by delocalized magnetic moments.</li> <li>On-site <a href="/facts/Atomic_orbital/tKhgl6mL">orbital</a> currents (as found in <a href="/facts/Multiferroic/AmlHcVAF">multiferroic</a> <a href="/facts/CuO/62mUJlpm">CuO</a>).<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a></li> <li>Orbital loop currents have been proposed in copper oxides superconductors<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> that might be important to understand <a href="/facts/High-temperature_superconductivity/MXtx2gvV">high-temperature superconductivity</a>. Experimental verification of symmetry-breaking by such orbital currents has been claimed in <a href="/facts/Cuprates/rS6vV6RS">cuprates</a> through polarized neutron-scattering.<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a></li></ul> <h2 id="magnetic-toroidal-moment-and-its-relation-to-the-magnetoelectric-effect">Magnetic toroidal moment and its relation to the magnetoelectric effect</h2> <p>The presence of a magnetic toroidic dipole moment T in condensed matter is due to the presence of a <a href="/facts/Magnetoelectric_effect/gdEMDvYC">magnetoelectric effect</a>: Application of a magnetic field H in the plane of a toroidal solenoid leads via the <a href="/facts/Lorentz_force/aq1Rc3SS">Lorentz force</a> to an accumulation of current loops and thus to an <a href="/facts/Electric_polarization/2oQv9EHx">electric polarization</a> perpendicular to both T and H. The resulting polarization has the form <i>P</i><i>i</i> = <i>ε</i><i>ijk</i><i>T</i><i>j</i><i>H</i><i>k</i> (with <i>ε</i> being the <a href="/facts/Levi-Civita_symbol/WBKUA3Tw">Levi-Civita symbol</a>). The resulting magnetoelectric <a href="/facts/Tensor/pNjFo5tV">tensor</a> describing the cross-correlated response is thus <a href="/facts/Antisymmetric_tensor/atcSaPST">antisymmetric</a>. </p> <h2 id="ferrotoroidicity-in-condensed-matter-physics">Ferrotoroidicity in condensed matter physics</h2> <p>A <a href="/facts/Phase_transition/eLP2BY8R">phase transition</a> to spontaneous <a href="/facts/Long-range_order/VAqeaphv">long-range order</a> of microscopic magnetic toroidal moments has been termed <i>ferrotoroidicity</i>.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> It is expected to fill the symmetry schemes of primary <a href="/facts/Ferroics/zd5Cj66J">ferroics</a> (phase transitions with spontaneous point symmetry breaking) with a space-odd, time-odd macroscopic order parameter. A ferrotoroidic material would exhibit domains which could be switched by an appropriate field, e.g. a magnetic field curl. Both of these hallmark properties of a ferroic state have been demonstrated in an artificial ferrotoroidic model system based on a <a href="/facts/Nanomagnet/FdLmruTk">nanomagnetic</a> array<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p><p>The existence of ferrotoroidicity is still under debate and clear-cut evidence has not been presented yet—mostly due to the difficulty to distinguish ferrotoroidicity from <a href="/facts/Antiferromagnetic/aQ124Kh7">antiferromagnetic</a> order, as both have no net <a href="/facts/Magnetization/zxrtXzTN">magnetization</a> and the <a href="/facts/Order_parameter/eLP2BY8R">order parameter</a> <a href="/facts/Symmetry/hpu7DK8p">symmetry</a> is the same. </p> <h2 id="anapole-dark-matter">Anapole dark matter</h2> <p>All <a href="/facts/CPT_symmetry/JCH2z2j2">CPT self-conjugate</a> particles, in particular the <a href="/facts/Majorana_fermion/39gzL0Nv">Majorana fermion</a>, are forbidden from having any multipole moments other than toroidal moments.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> At tree level (i.e. without allowing loops in <a href="/facts/Feynman_diagram/BRFNvDdr">Feynman diagrams</a>) an anapole-only particle interacts only with external currents, not with free-space electromagnetic fields, and the interaction cross-section diminishes as the particle velocity slows. For this reason, heavy Majorana fermions have been suggested as plausible candidates for <a href="/facts/Cold_dark_matter/vL4134Kd">cold dark matter</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a><a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Spheromak/1nSg4I4R">Spheromak</a></li> <li><a href="/facts/Dynamic_toroidal_dipole/wMfbRXxn">Dynamic toroidal dipole</a></li> <li><a href="/facts/Anapole/RN0bw3da">Anapole</a></li></ul> <h2 id="literature">Literature</h2> <ul><li>Stefan Nanz: <a href="https://link.springer.com/book/10.1007/978-3-658-12549-3"><i>Toroidal Multipole Moments in Classical Electrodynamics</i></a>. Springer 2016. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-3-658-12548-6</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Radescu, E. Jr.; Vaman, G. (2012), "Cartesian multipole expansions and tensorial identities", Progress in Electromagnetics Research B, 36: 89–111, doi:10.2528/PIERB11090702 <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Dubovik, V. M.; Tugushev, V. V. (March 1990), "Toroid moments in electrodynamics and solid-state physics", Physics Reports, 187 (4): 145–202, Bibcode:1990PhR...187..145D, doi:10.1016/0370-1573(90)90042-Z <a href="/wiki/Bibcode_(identifier)" target="_blank">/wiki/Bibcode_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>I. Fernandez-Corbaton et al.: On the dynamic toroidal multipoles from localized electric current distributions. 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