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<title>Thomson-Problem</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Thomson-Problem</span></h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr"><p>Beim <b>Thomson-Problem</b> sollen <i>n</i> Elektronen so auf der Oberfläche einer <a href="Einheitskugel" title="Einheitskugel">Einheitskugel</a> verteilt werden, dass das gesamte elektrostatische Potential, das sich durch die <a href="Coulombsches_Gesetz" title="Coulombsches Gesetz">Coulombkraft</a> einstellt, sein Minimum annimmt. Der Physiker <a href="Joseph_John_Thomson" title="Joseph John Thomson">Joseph John Thomson</a> formulierte dieses Problem 1904<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>, nachdem er sein <a href="Thomsonsches_Atommodell" title="Thomsonsches Atommodell">Atommodell</a> entwickelte.

</p><p>Mathematisch ist es eines der <a href="Smale-Probleme" title="Smale-Probleme">Smale-Probleme</a>.

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<div class="mw-heading mw-heading2"><h2 id="Mathematische_Beschreibung">Mathematische Beschreibung</h2></div>

<p>Das elektrostatische Potential <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(N)}">

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<annotation encoding="application/x-tex">{\displaystyle U(N)}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75bc0531863b76e4cb6239ccfc14804a5a817d1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.656ex; height:2.843ex;" alt="{\displaystyle U(N)}" loading="lazy"></span>, das zwischen zwei Elektronen entsteht, kann mit dem <a href="Coulombsches_Gesetz" title="Coulombsches Gesetz">coulombschen Gesetz</a> beschrieben werden.

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<annotation encoding="application/x-tex">{\displaystyle U_{ij}(N)=k_{e}{e_{i}e_{j} \over r_{ij}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4e76c2e6908f08f50170bce4a706790fd2e4b3cf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.958ex; height:5.676ex;" alt="{\displaystyle U_{ij}(N)=k_{e}{e_{i}e_{j} \over r_{ij}}}" loading="lazy"></span>.</dd></dl>

<p>Dabei sind <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle e_{i}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ebdc3a9cb1583d3204eff8918b558c293e0d2cf3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.883ex; height:2.009ex;" alt="{\displaystyle e_{i}}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{j}}">

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/64afd228cf00e5024b9cdd277462d24ab97b6d3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:1.993ex; height:2.343ex;" alt="{\displaystyle e_{j}}" loading="lazy"></span> die <a href="Elementarladung" title="Elementarladung">Ladungen</a> der Elektronen, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{e}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{e}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ad7233b322184c1269207c976f301658c163de10.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.21ex; height:2.509ex;" alt="{\displaystyle k_{e}}" loading="lazy"></span> ist die Coulombkonstante (gegeben durch <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e66755e48a90db06e3ffcb12f11156e018e76a85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.468ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{4\pi \varepsilon _{0}}}}" loading="lazy"></span>; <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"></span> ist die <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante">elektrische Feldkonstante</a>) und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}">

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<annotation encoding="application/x-tex">{\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99a35090c1a93faeeef34005b5e4b1305df74c4e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.672ex; height:3.009ex;" alt="{\displaystyle r_{ij}=|\mathbf {r} _{i}-\mathbf {r} _{j}|}" loading="lazy"></span> ist der Abstand der beiden Elektronen zueinander. Zur Vereinfachung des Problems können <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{i}=e_{j}=1}">

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<annotation encoding="application/x-tex">{\displaystyle e_{i}=e_{j}=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f4e7df75ad98383591bc772afe72819c712e3f2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.236ex; height:2.843ex;" alt="{\displaystyle e_{i}=e_{j}=1}" loading="lazy"></span> und <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{e}=1}">

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<annotation encoding="application/x-tex">{\displaystyle k_{e}=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9032bf1f74dd530918736083681dbd2f71620527.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.471ex; height:2.509ex;" alt="{\displaystyle k_{e}=1}" loading="lazy"></span> gesetzt werden.

</p><p>Bei einer Konfiguration von <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

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<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"></span> Elektronen stellt sich dann das Potential

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<annotation encoding="application/x-tex">{\displaystyle U(N)=\sum _{i<j}{\frac {1}{r_{ij}}}}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ef119570dc65b6fc61b8756d807496c9f78bd074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:15.858ex; height:6.676ex;" alt="{\displaystyle U(N)=\sum _{i<j}{\frac {1}{r_{ij}}}}" loading="lazy"></span>.</dd></dl>

<p>ein. Ziel ist es nun, diejenige Form zu finden, bei der dieses Gesamtpotential ein Minimum annimmt. Das Finden einer Lösung geschieht meist durch <a href="Numerische_Verfahren" class="mw-redirect" title="Numerische Verfahren">numerische Verfahren</a>.

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<div class="mw-heading mw-heading2"><h2 id="Bekannte_Lösungen"><span id="Bekannte_L.C3.B6sungen"></span>Bekannte Lösungen</h2></div>

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<annotation encoding="application/x-tex">{\displaystyle N=1}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/85982022b9eb1f295b44de55023687a490db0a39.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=1}" loading="lazy"></span>: Für nur ein einziges Elektron ist die Lösung trivial, da sich, egal wo sich das Elektron auf der Kugeloberfläche befindet, immer dasselbe Potential einstellt.</li></ul>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/405d64b14536deffc3465f1e81b1b7fe9358ad2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=2}" loading="lazy"></span>: Bei zwei Elektronen ist das Potentialminimum dann vorhanden, wenn sie sich diametral gegenüber befinden (z. B. Nord- und Südpol).</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle N=3}</annotation>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b4ab951d779fcdeea3ec188bb5c73518c46b19de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=3}" loading="lazy"></span>: Bei drei Elektronen bildet die Konfiguration ein gleichseitiges Dreieck auf einem <a href="Gro%C3%9Fkreis" title="Großkreis">Großkreis</a> der Kugel.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup></li></ul>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/513c42064b86e0f691571271623451438315767f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=4}" loading="lazy"></span>: Die vier Elektronen bilden ein <a href="Tetraeder" title="Tetraeder">Tetraeder</a>.</li></ul>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4ea204e60e1d27578912bb557d2859e759f4d67c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=5}" loading="lazy"></span>: Für fünf Elektronen wurde 2010 ein computergestützter Beweis erbracht, wonach diese eine dreieckige <a href="Bipyramide" class="mw-redirect" title="Bipyramide">Bipyramide</a> bilden.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup></li></ul>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f87b8c5d4bce3e2d71bb4195b782a629146e9c7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.325ex; height:2.176ex;" alt="{\displaystyle N=6}" loading="lazy"></span>: Die sechs Elektronen bilden ein <a href="Oktaeder" title="Oktaeder">Oktaeder</a>.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup></li></ul>

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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/234a2c9cb6a194372d211edfa84c2706a33d1818.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=12}" loading="lazy"></span>: Diese Konfiguration bildet ein regelmäßiges <a href="Ikosaeder" title="Ikosaeder">Ikosaeder</a>.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Verwandte_wissenschaftliche_Probleme">Verwandte wissenschaftliche Probleme</h2></div>

<p>Das Problem von Thomson spielt eine Rolle in anderen physikalischen Modellen wie zum Beispiel Elektronenblasen oder die Oberflächenbeschaffenheit von flüssigen Metalltropfen in <a href="Paul-Falle" title="Paul-Falle">Paul-Fallen</a>.

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<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li>Carlos Beltrán: <i>The State of the Art in Smale's 7th Problem.</i> In: <a href="Felipe_Cucker" title="Felipe Cucker">Felipe Cucker</a>, Teresa Krick, Allan Pinkus, Agnes Szanto (Hrsg.): <i>Foundations of Computational Mathematics. Budapest 2011</i> (= <i>London Mathematical Society. Lecture Note Series.</i> 403). Cambridge University Press, Cambridge u. a. 2013, ISBN 978-1-107-60407-0, S. 1–15.</li>

<li>L. L. Whyte: <i>Unique arrangements of points on a sphere.</i> In: <i><a href="American_Mathematical_Monthly" title="American Mathematical Monthly">American Mathematical Monthly</a>.</i> Bd. 59, Nr. 9, 1952, S. 606–611, <a href="JSTOR" title="JSTOR">JSTOR</a>:<a rel="nofollow" class="external text" href="http://www.jstor.org/stable/2306764">2306764</a>.</li>

<li>Edward B. Saff, <a href="Arno_Kuijlaars" title="Arno Kuijlaars">Arno B. J. Kuijlaars</a>: <i>Distributing many points on a sphere.</i> In: <i><a href="The_Mathematical_Intelligencer" title="The Mathematical Intelligencer">The Mathematical Intelligencer</a>.</i> Bd. 19, Nr. 1, 1997, S. 5–11, <a href="https://doi.org/10.1007/BF03024331" class="extiw external" title="doi:10.1007/BF03024331">doi:10.1007/BF03024331</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Weblinks">Weblinks</h2></div>

<ul><li><a rel="nofollow" class="external text" href="http://thomson.phy.syr.edu/thomsonapplet.php">Interaktives Java-Applet zum Thomson-Problem</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Joseph J. Thomson: <i>On the structure of the atom: an investigation of the stability and periods of oscillation of a number of corpuscles arranged at equal intervals around the circumference of a circle; with application of the results to the theory of atomic structure.</i> In: <i>The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science.</i> Series 6, Bd. 7, Nr. 39, 1904, <a href="Zeitschriftendatenbank" title="Zeitschriftendatenbank">ZDB</a>-ID <span class="-print"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=5450-1&key=zdb">5450-1</a></span>, S. 237–265, <a href="https://doi.org/10.1080/14786440409463107" class="extiw external" title="doi:10.1080/14786440409463107">doi:10.1080/14786440409463107</a>.</span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><a href="#cite_ref-2">↑</a></span> <span class="reference-text"><a href="Ludwig_F%C3%B6ppl" title="Ludwig Föppl">Ludwig Föppl</a>: <i>Stabile Anordnungen von Elektronen im Atom.</i> In: <i><a href="Journal_f%C3%BCr_die_reine_und_angewandte_Mathematik" title="Journal für die reine und angewandte Mathematik">Journal für die reine und angewandte Mathematik</a>.</i> Bd. 141, 1912, S. 251–302, (<a rel="nofollow" class="external text" href="http://www.digizeitschriften.de/dms/img/?PID=GDZPPN002167611&physid=phys257#navi">Digitalisat</a>).</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><a href="#cite_ref-3">↑</a></span> <span class="reference-text">Richard Evan Schwartz: <cite style="font-style:italic">The 5 Electron Case of Thomson’s Problem</cite>. In: <cite style="font-style:italic">Mathematical Physics</cite>. 21. Januar 2010, <a href="ArXiv" title="ArXiv">arxiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1001.3702">1001.3702</a>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Thomson-Problem&rft.atitle=The+5+Electron+Case+of+Thomson%E2%80%99s+Problem&rft.au=Richard+Evan+Schwartz&rft.btitle=Mathematical+Physics&rft.date=2010-01-21&rft.genre=book" style="display:none"> </span></span>

</li>

<li id="cite_note-4"><span class="mw-cite-backlink"><a href="#cite_ref-4">↑</a></span> <span class="reference-text">V. A. Yudin: <cite style="font-style:italic">The minimum of potential energy of a System of point charges</cite>. In: <cite style="font-style:italic">Discrete Mathematics and Applications</cite>. <span style="white-space:nowrap">Band<span style="display:inline-block;width:.2em"> </span>3</span>, <span style="white-space:nowrap">Nr.<span style="display:inline-block;width:.2em"> </span>1</span>, 2009, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>75–82</span>, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<span class="uri-handle" style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://doi.org/10.1515/dma.1993.3.1.75">10.1515/dma.1993.3.1.75</a></span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rfr_id=info:sid/de.wikipedia.org:Thomson-Problem&rft.atitle=The+minimum+of+potential+energy+of+a+System+of+point+charges&rft.au=V.+A.+Yudin&rft.date=2009&rft.doi=10.1515%2Fdma.1993.3.1.75&rft.genre=journal&rft.issue=1&rft.jtitle=Discrete+Mathematics+and+Applications&rft.pages=75-82&rft.volume=3" style="display:none"> </span></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><a href="#cite_ref-5">↑</a></span> <span class="reference-text">Nikolay N. Andreev: <i>An extremal property of the icosahedron.</i> In: <i>East Journal on Approximations.</i> Bd. 2, Nr. 4, 1996, <span class="-print"><a href="Internationale_Standardnummer_f%C3%BCr_fortlaufende_Sammelwerke" title="Internationale Standardnummer für fortlaufende Sammelwerke">ISSN</a> <span style="white-space:nowrap"><a rel="nofollow" class="external text" href="https://zdb-katalog.de/list.xhtml?t=iss%3D%221310-6236%22&key=cql">1310-6236</a></span></span>, S. 459–462, <a href="Mathematical_Reviews" title="Mathematical Reviews">MR</a> 97m:52022, <a href="Zentralblatt_MATH" class="mw-redirect" title="Zentralblatt MATH">Zbl</a> 0877.51021.</span>

</li>

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