Toroidal ring model

<h2 id="theory">Theory</h2>
Instead of a single <a href="/facts/Orbit/nD2TPySG">orbiting</a> <a href="/facts/Electric_charge/xxsBfmlB">charge</a>, the toroidal ring was conceived as a collection of <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> charge elements, which orbited or circulated along a common continuous path or "<a href="/facts/Circuit_theory/IjmVqbG8">loop</a>". In general, this path of charge could assume any shape, but tended toward a circular form due to internal repulsive <a href="/facts/Electromagnetic_force/oAkmALHT">electromagnetic forces</a>. In this configuration the charge elements circulated, but the ring as a whole did not <a href="/facts/Radiation/RwMocyeq">radiate</a> due to changes in <a href="/facts/Electric_field/tt4MooBs">electric</a> or <a href="/facts/Magnetic_field/Ta8Ev588">magnetic</a> <a href="/facts/Field_(physics)/FJdAKeRA">fields</a> since it remained stationary. The ring produced an overall magnetic field ("<a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>") due to the <a href="/facts/Electric_current/Dc1xdnUX">current</a> of the moving charge elements. These elements circulated around the ring at the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> c, but at <a href="/facts/Frequency/QUSbPaW8">frequency</a> ν = c/2πR, which depended <a href="/facts/Inversely_proportional/xDQmfh6t">inversely</a> on the <a href="/facts/Radius/9Gr9deYD">radius</a> R. The ring's <a href="/facts/Inertia/bzK9uzI2">inertial</a> <a href="/facts/Energy/wd8PrQM7">energy</a> increased when <a href="/facts/Compression_(physical)/2wwcVmM0">compressed</a>, like a <a href="/facts/Spring_(device)/jknGXzTq">spring</a>, and was also inversely proportional to its radius, and therefore <a href="/facts/Proportionality_(mathematics)/xDQmfh6t">proportional</a> to its frequency ν. The theory claimed that the <a href="/facts/Proportionality_constant/xDQmfh6t">proportionality constant</a> was the <a href="/facts/Planck_constant/qGHAe3qq">Planck constant</a> h, the conserved <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> of the ring.
According to the model, electrons or protons could be viewed as bundles of "<a href="/facts/Fiber/ttkxAFBu">fibers</a>" or "<a href="/facts/Plasmoid/LKSICN9H">plasmoids</a>" with total charge ±<a href="/facts/Elementary_charge/lPcj6CoQ">e</a>. The <a href="/facts/Electric_force/V0lR5NKv">electrostatic repulsion force</a> between charge elements of the same sign was balanced by the <a href="/facts/Magnetic_force/aq1Rc3SS">magnetic attraction force</a> between the <a href="/facts/Parallel_(geometry)/16r5yq7D">parallel</a> <a href="/facts/Current_(mathematics)/gwhmhzqJ">currents</a> in the fibers of a bundle, per <a href="/facts/Amp%C3%A8re%27s_circuital_law/B6WVCpc6">Ampère's law</a>. These fibers twisted around the <a href="/facts/Torus/J9SkaV1W">torus</a> of the ring as they progressed around its radius, forming a <a href="/facts/Slinky/Xc1Brzrd">Slinky</a>-like <a href="/facts/Helix/BqC4AKwL">helix</a>. Circuit completion demanded that each helical plasmoid fiber twisted around the ring an <a href="/facts/Integer/PUwwrawx">integer</a> number of times as it proceeded around the ring. This requirement was thought to account for "<a href="/facts/Quantum/4x5QTxGB">quantum</a>" values of <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a> and <a href="/facts/Radiation/RwMocyeq">radiation</a>. <a href="/facts/Chirality_(physics)/H8Xbcvzt">Chirality</a> demanded the number of fibers to be <a href="/facts/Odd_number/7zq2UM4V">odd</a>, probably three, like a rope. The <a href="/facts/Helicity_(particle_physics)/vvNpFcFE">helicity</a> of the twist, was thought to distinguish the electron from the proton.
The toroidal or "helicon" model did not demand a constant radius or inertial energy for a particle. In general its shape, size, and motion adjusted according to the external electromagnetic fields from its environment. These adjustments or reactions to external field changes constituted the <a href="/facts/Emission_(electromagnetic_radiation)/fxumR717">emission</a> or <a href="/facts/Absorption_(electromagnetic_radiation)/xBYKt7yS">absorption</a> of <a href="/facts/Radiation/RwMocyeq">radiation</a> for the particle. The model, then, claimed to explain how particles linked together to form <a href="/facts/Atom/FoQ4kGN5">atoms</a>.

<h2 id="history">History</h2>
<h3>Beginnings</h3>
The development of the helicon or toroidal ring began with <a href="/facts/Andr%C3%A9-Marie_Amp%C3%A8re/qk01uas1">André-Marie Ampère</a>, who in 1823 proposed tiny magnetic "loops of charge" to explain the attractive force between current elements.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a> In that same era <a href="/facts/Carl_Friedrich_Gauss/f4eSMBD7">Carl Friedrich Gauss</a> and <a href="/facts/Michael_Faraday/NDcc3lyt">Michael Faraday</a> also uncovered foundational laws of <a href="/facts/Classical_electrodynamics/vK0YI6dj">classical electrodynamics</a>, later collected by <a href="/facts/James_Clerk_Maxwell/jtYyze61">James Maxwell</a> as <a href="/facts/Maxwell%27s_equations/LfJ1drHS">Maxwell's equations</a>. When Maxwell expressed the laws of <a href="/facts/Gauss%27s_law/Am2X21DU">Gauss</a>, <a href="/facts/Faraday%27s_law_of_induction/PFBreB0F">Faraday</a>, and <a href="/facts/Amp%C3%A8re%27s_circuital_law/B6WVCpc6">Ampère</a> in <a href="/facts/Differential_(calculus)/YmSDWCol">differential form</a>, he assumed <a href="/facts/Point_particle/4MPToN8W">point particles</a>, an assumption that remains foundational to <a href="/facts/Theory_of_relativity/qGHj3eTC">relativity theory</a> and <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a> today. In 1867 <a href="/facts/William_Thomson,_1st_Baron_Kelvin/LTIxG1jV">Lord Kelvin</a> suggested that the <a href="/facts/Vortex/K6oeqcuX">vortex</a> rings of a <a href="/facts/Perfect_fluid/wWpblQcg">perfect fluid</a> discovered by <a href="/facts/Hermann_von_Helmholtz/oFISedae">Hermann von Helmholtz</a> represented "the only true <a href="/facts/Atom/FoQ4kGN5">atoms</a>".<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> Then shortly before 1900, as scientists still debated over the very existence of atoms, <a href="/facts/J._J._Thomson/Y4jyndJF">J. J. Thomson</a><a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> and <a href="/facts/Ernest_Rutherford/hjjMEskl">Ernest Rutherford</a><a class="footnote-ref" id="fnref:5" href="#fn:5">5</a> sparked a revolution with experiments<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a> confirming the existence and properties of electrons, protons, and <a href="/facts/Atomic_nucleus/orpIm5Xl">nuclei</a>. <a href="/facts/Max_Planck/KKk0698y">Max Planck</a> added to the fire when he solved the <a href="/facts/Ultraviolet_catastrophe/rRM4PTQU">blackbody radiation problem</a> by assuming not only <a href="/facts/Quantum/4x5QTxGB">discrete</a> particles, but discrete <a href="/facts/Frequency/QUSbPaW8">frequencies</a> of radiation emanating from these "particles" or "<a href="/facts/Resonator/NjC413Gn">resonators</a>". Planck's famous paper,<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a> which incidentally calculated both the <a href="/facts/Planck_constant/qGHAe3qq">Planck constant</a> h and the <a href="/facts/Boltzmann_constant/5DIejyR3">Boltzmann constant</a> kB, suggested that something in the "resonators" themselves provided these discrete frequencies.
Numerous <a href="/facts/Atomic_theory/B2WvuLf6">theories about the structure of the atom</a> developed in the wake of all the new information,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a><a class="footnote-ref" id="fnref:9" href="#fn:9">9</a> of which the 1913 model of <a href="/facts/Niels_Bohr/ydLPvQXT">Niels Bohr</a> came to predominate. The <a href="/facts/Bohr_model/FmTHjaxy">Bohr model</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a> proposed electrons in circular orbit around the <a href="/facts/Atomic_nucleus/orpIm5Xl">nucleus</a> with <a href="/facts/Quantization_(physics)/fvs2Jwpn">quantized</a> values of <a href="/facts/Angular_momentum/7TuSVFxq">angular momentum</a>. Instead of <a href="/facts/Synchrotron_radiation/mTsSKmRz">radiating energy continuously</a>, as <a href="/facts/Classical_electrodynamics/vK0YI6dj">classical electrodynamics</a> demanded from an accelerating charge, Bohr's electron radiated discretely when it "<a href="/facts/Atomic_electron_transition/HeaF4JJk">leaped</a>" from one <a href="/facts/Quantum_state/7jbxd7Dx">state</a> of angular momentum to another.

<h3>Parson magneton</h3>
In 1915, <a href="/facts/Alfred_Lauck_Parson/oVA8wmNK">Alfred Lauck Parson</a> proposed his "<a href="/facts/Parson_magneton/T66VTbto">magneton</a>"<a class="footnote-ref" id="fnref:11" href="#fn:11">11</a> as an improvement over the <a href="/facts/Bohr_model/FmTHjaxy">Bohr model</a>, depicting finite-sized particles with the ability to maintain <a href="/facts/Mechanical_equilibrium/N2XKaK6s">stability</a> and <a href="/facts/Emission_(electromagnetic_radiation)/fxumR717">emit</a> and <a href="/facts/Absorption_(electromagnetic_radiation)/xBYKt7yS">absorb</a> <a href="/facts/Radiation/RwMocyeq">radiation</a> from <a href="/facts/Electromagnetic_wave/Ynx7ujCW">electromagnetic waves</a>. At about the same time <a href="/facts/Leigh_Page/wjCyxTYQ">Leigh Page</a> developed a <a href="/facts/Classical_physics/gGnINVPE">classical</a> theory of <a href="/facts/Blackbody_radiation/K4YKPKXL">blackbody radiation</a> assuming rotating "<a href="/facts/Oscillators/6vSrm84a">oscillators</a>", able to store energy without radiating.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a> <a href="/facts/Gilbert_N._Lewis/wWnV6QXL">Gilbert N. Lewis</a> was inspired in part by Parson's model in developing his theory of <a href="/facts/Chemical_bonding/ArNUMnIB">chemical bonding</a>.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a> Then <a href="/facts/David_L._Webster/XWXxlbNJ">David L. Webster</a> wrote three papers connecting Parson's magneton with Page's oscillator<a class="footnote-ref" id="fnref:14" href="#fn:14">14</a> and explaining <a href="/facts/Mass/HHdc8XmF">mass</a><a class="footnote-ref" id="fnref:15" href="#fn:15">15</a> and <a href="/facts/Alpha_particle/dzNMs9WP">alpha</a> <a href="/facts/Scattering/nMou7lEH">scattering</a><a class="footnote-ref" id="fnref:16" href="#fn:16">16</a> in terms of the magneton. In 1917 Lars O. Grondahl confirmed the model with his experiments on free electrons in <a href="/facts/Iron/QK1JYy8E">iron</a> wires.<a class="footnote-ref" id="fnref:17" href="#fn:17">17</a> Parson's theory next attracted the attention of <a href="/facts/Arthur_Compton/FrNYLBj4">Arthur Compton</a>, who wrote a series of papers on the properties of the electron,<a class="footnote-ref" id="fnref:18" href="#fn:18">18</a><a class="footnote-ref" id="fnref:19" href="#fn:19">19</a><a class="footnote-ref" id="fnref:20" href="#fn:20">20</a><a class="footnote-ref" id="fnref:21" href="#fn:21">21</a> and <a href="/facts/H._Stanley_Allen/bkHkBPCh">H. Stanley Allen</a>, whose papers also argued for a "ring electron".<a class="footnote-ref" id="fnref:22" href="#fn:22">22</a><a class="footnote-ref" id="fnref:23" href="#fn:23">23</a><a class="footnote-ref" id="fnref:24" href="#fn:24">24</a>

<h2 id="current-status">Current status</h2>
The aspect of the Parson magneton with the most experimental relevance (and the aspect investigated by Grondahl and Webster) was the existence of an <a href="/facts/Electron_magnetic_dipole_moment/guAK2JJ2">electron magnetic dipole moment</a>; this dipole moment is indeed present. However, later work by <a href="/facts/Paul_Dirac/6bt5yRdk">Paul Dirac</a> and <a href="/facts/Alfred_Land%C3%A9/axkIBIx8">Alfred Landé</a> showed that a pointlike particle could have an intrinsic quantum <a href="/facts/Spin_(physics)/4Jvp7e8P">spin</a>, and also a magnetic moment. The highly successful modern theory, <a href="/facts/Standard_Model/LhVGf9ow">Standard Model</a> of particle physics describes a pointlike electron with an intrinsic spin and magnetic moment. On the other hand, the usual assertion that an electron is pointlike may be conventionally associated only with a "bare" electron. The pointlike electron would have a diverging electromagnetic field, which should create a strong vacuum polarization. In accordance with QED, deviations from the Coulomb law are predicted at Compton scale distances from the centre of electron, 10−11 cm. Virtual processes in the Compton region determine the spin of electron and renormalization of its charge and mass. It shows that the Compton region of the electron should be considered as a coherent whole with its pointlike core, forming a physical ("dressed") electron. Notice that the Dirac theory of electron also exhibits the peculiar behaviour of the Compton region. In particular, electrons display <a href="/facts/Zitterbewegung/khtLRolH">zitterbewegung</a> at the Compton scale. From this point of view, the ring model does not contradict QED or the Dirac theory and some versions could possibly be used to incorporate gravity in quantum theory.
The question of whether the electron has a substructure of any sort must be decided by experiment. All experiments to date agree with the Standard Model of the electron, with no substructure, ring-like or otherwise. The two major approaches are high-energy electron–positron scattering<a class="footnote-ref" id="fnref:25" href="#fn:25">25</a> and <a href="/facts/Precision_tests_of_QED/ddojHb67">high-precision atomic tests of quantum electrodynamics</a>,<a class="footnote-ref" id="fnref:26" href="#fn:26">26</a> both of which agree that the electron is point-like at resolutions down to 10−20 m. At present, the Compton region of virtual processes, 10−11 cm across, is not exhibited in the high-energy experiments on electron–positron scattering.
<a href="/facts/Nikodem_Pop%C5%82awski/uK54fULE">Nikodem Popławski</a> use the Papapetrou method of multipole expansion to show that torsion modifies Burinskii’s model of the Dirac electron by replacing the Kerr–Newman singular ring of the Compton size with a toroidal structure with the outer radius of the Compton size and the inner radius of the Cartan size (10−27 m) in the <a href="/facts/Einstein%E2%80%93Cartan_theory/GPU36Nsu">Einstein–Cartan theory</a> of gravity.<a class="footnote-ref" id="fnref:27" href="#fn:27">27</a>

<h2 id="further-reading">Further reading</h2>
<ul><li>David L. Bergman, J. Paul Wesley ; Spinning Charged Ring Model of Electron Yielding Anomalous Magnetic Moment, Galilean Electrodynamics. Vol. 1, 63-67 (Sept./Oct. 1990).</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">André-Marie Ampère (1823). "Sur la théorie mathématique des phénomènes électrodynamiques uniquement déduite de l'expérience" [On the mathematical theory of electrodynamic phenomena only deduced from experience]. Mémoires de l'Académie des sciences de l'Institut de France Académie des sciences (in French). 6: 175. <a href="/wiki/Andr%C3%A9-Marie_Amp%C3%A8re" target="_blank">/wiki/Andr%C3%A9-Marie_Amp%C3%A8re</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Thomson, William (1869). "On Vortex Atoms". Proceedings of the Royal Society of Edinburgh. 6: 94–105. doi:10.1017/S0370164600045430. <a href="https://zenodo.org/record/2101269" target="_blank">https://zenodo.org/record/2101269</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">William Thomson, "On Vortex Atoms", Proceedings of the Royal Society of Edinburgh, V6, pp. 94–105 (1867) {reprinted in Philosophical Magazine, V34, pp. 15–24 (1867)}. <a href="/wiki/William_Thomson,_1st_Baron_Kelvin" target="_blank">/wiki/William_Thomson,_1st_Baron_Kelvin</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">J. J. Thomson, "Cathode Rays", Philosophical Magazine, S5, V44, p. 293 (1897). <a href="/wiki/J._J._Thomson" target="_blank">/wiki/J._J._Thomson</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Ernest Rutherford, "Uranium Radiation and the Electrical Conduction Archived 2007-09-08 at the Wayback Machine", Philosophical Magazine, S5, V47, pp. 109–163 (Jan 1899). <a href="/wiki/Ernest_Rutherford" target="_blank">/wiki/Ernest_Rutherford</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">See Thomson experiment and Rutherford experiment for details. <a href="/wiki/Thomson_experiment" target="_blank">/wiki/Thomson_experiment</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">Max Planck, "On the Law of Distribution of Energy in the Normal Spectrum”, Annalen der Physik, V4, p. 553 ff (1901). Archived October 25, 2007, at the Wayback Machine <a href="/wiki/Max_Planck" target="_blank">/wiki/Max_Planck</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">J. J. Thomson, "On the Structure of the Atom... Archived 2007-09-09 at the Wayback Machine", Philosophical Magazine, S6, V7, N39, pp. 237–265 (Mar 1904). <a href="/wiki/J._J._Thomson" target="_blank">/wiki/J._J._Thomson</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Ernest Rutherford, "The Scattering of α and β Particles by Matter and the Structure of the Atom Archived 2007-02-05 at the Wayback Machine", Philosophical Magazine, S6, V21, pp. 669–688 (May 1911). <a href="/wiki/Ernest_Rutherford" target="_blank">/wiki/Ernest_Rutherford</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">Niels Bohr, "On the Constitution of Atoms and Molecules Archived 2007-07-04 at the Wayback Machine", Philosophical Magazine, S6, V26, p. 1–25 (July 1913). <a href="/wiki/Niels_Bohr" target="_blank">/wiki/Niels_Bohr</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Alfred L. Parson, "A Magneton Theory of the Structure of the Atom", Smithsonian Miscellaneous Collection, Pub 2371, 80pp (Nov 1915) {Reprinted Pub 2419, V65, N11 (1916)}. <a href="/wiki/Alfred_Lauck_Parson" target="_blank">/wiki/Alfred_Lauck_Parson</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">Leigh Page, "The Distribution of Energy in the Normal Radiation Spectrum", Physical Review, S2, V7, N2, pp. 229–240 (Feb 1916). <a href="/wiki/Leigh_Page" target="_blank">/wiki/Leigh_Page</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Gilbert N. Lewis, "The Atom and the Molecule Archived 2007-09-18 at the Wayback Machine", Journal of the American Chemical Society, V38, pp. 762–786 (1916). <a href="/wiki/Gilbert_N._Lewis" target="_blank">/wiki/Gilbert_N._Lewis</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">David L. Webster, "Notes on Page's Theory of Heat Radiation", Physical Review, S2, V8, N1, pp. 66–69 (Jul 1916). <a href="/wiki/David_L._Webster" target="_blank">/wiki/David_L._Webster</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
<li id="fn:15">David L. Webster, "The Theory of Electromagnetic Mass of the Parson Magneton and other Non-Spherical Systems", Physical Review, S2, V9, N6, pp. 484–499 (Jun 1917). <a href="/wiki/David_L._Webster" target="_blank">/wiki/David_L._Webster</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></li>
<li id="fn:16">David L. Webster, "The Scattering of Alpha Rays as Evidence on the Parson Magnetron Hypothesis", Physical Review, S2 (Feb 1918). <a href="/wiki/David_L._Webster" target="_blank">/wiki/David_L._Webster</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></li>
<li id="fn:17">Lars O. Grondahl, "Proceedings of the American Physical Society: Experimental Evidence for the Parson Magneton", Physical Review, S2, V10, N5, pp. 586–588 (Nov 1917). <a href="/w/index.php?title=Lars_O._Grondahl&action=edit&redlink=1" target="_blank">/w/index.php?title=Lars_O._Grondahl&action=edit&redlink=1</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></li>
<li id="fn:18">Arthur H. Compton, "The Size and Shape of the Electron – American Physical Society address (Dec 1917)", Journal of the Washington Academy of Sciences, pp. 330 (Jan 1918). <a href="/wiki/Arthur_H._Compton" target="_blank">/wiki/Arthur_H._Compton</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></li>
<li id="fn:19">Arthur H. Compton (July 1919). "The Size and Shape of the Electron". Physical Review. 14 (1). doi:10.1103/PhysRev.14.20. <a href="/wiki/Physical_Review" target="_blank">/wiki/Physical_Review</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></li>
<li id="fn:20">Arthur H. Compton, "Possible Magnetic Polarity of Free Electrons", Philosophical Magazine, S6, V41 (Feb 1921). <a href="/wiki/Arthur_H._Compton" target="_blank">/wiki/Arthur_H._Compton</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></li>
<li id="fn:21">Arthur H. Compton, "The Magnetic Electron", Journal of the Franklin Institute, V192, N2, pp. 145–155 (Aug 1921) <a href="/wiki/Arthur_H._Compton" target="_blank">/wiki/Arthur_H._Compton</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></li>
<li id="fn:22">H S Allen (1918). "The case for a ring electron". Proceedings of the Physical Society of London. 31: 49–68. doi:10.1088/1478-7814/31/1/303. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></li>
<li id="fn:23">H. Stanley Allen, "Optical Rotation, Optical Isomerism, and the Ring Electron", Philosophical Magazine, S6, V40, N6, p. 426 (1920). <a href="/wiki/H._Stanley_Allen" target="_blank">/wiki/H._Stanley_Allen</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></li>
<li id="fn:24">H. Stanley Allen, "The Angular Momentum and Some Related Properties of the Ring Electron", Philosophical Magazine, S6, V41, N6, p. 113 (1921). <a href="/wiki/H._Stanley_Allen" target="_blank">/wiki/H._Stanley_Allen</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></li>
<li id="fn:25">D. Bourilkov, "Hint for axial-vector contact interactions in the data on e+e− → e+e−(γ) at center-of-mass energies 192–208 GeV", Phys. Rev. D 64, 071701 (2001), Physical Review Online Archive. <a href="http://prola.aps.org/abstract/PRD/v64/i7/e071701" target="_blank">http://prola.aps.org/abstract/PRD/v64/i7/e071701</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></li>
<li id="fn:26">B. Odom, D. Hanneke, B. D'Urso, and G. Gabrielse, New Measurement of the Electron Magnetic Moment Using a One-Electron Quantum Cyclotron, Phys. Rev. Lett. 97, 030801 (2006), PHYSICAL REVIEW LETTERS. <a href="https://www.uni-ulm.de/fileadmin/website_uni_ulm/nawi.inst.220/lehre/Atomphysik_SS2008/gabrielse_g_Faktor_2006_PRL.pdf" target="_blank">https://www.uni-ulm.de/fileadmin/website_uni_ulm/nawi.inst.220/lehre/Atomphysik_SS2008/gabrielse_g_Faktor_2006_PRL.pdf</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></li>
<li id="fn:27">Popławski, Nikodem J. (2010). "Nonsingular Dirac particles in spacetime with torsion". Physics Letters B. 690 (1): 73–77. arXiv:0910.1181. Bibcode:2010PhLB..690...73P. doi:10.1016/j.physletb.2010.04.073. <a href="/wiki/Physics_Letters_B" target="_blank">/wiki/Physics_Letters_B</a> <a href="#fnref:27" class="footnote-back-ref">↩</a></li>
</ol>

Toroidal ring model open-in-new

Toroidal ring model