English+GermanWikipedia - /page/entry.mu

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<title>Blume-Capel-Modell</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Blume-Capel-Modell</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">Das Blume-Capel-Modell nach <a href="Martin_Blume_(Physiker)" title="Martin Blume (Physiker)">Martin Blume</a><a href="#cite_note-1">[1]</a> und Hans Willem Capel<a href="#cite_note-2">[2]</a> ist eine Verallgemeinerung des <a href="Ising-Modell" title="Ising-Modell">Ising-Modells</a> in der <a href="Festk%C3%B6rperphysik" title="Festkörperphysik">Festkörperphysik</a>. Es beschreibt eine Situation, in der sich die <a href="Spin" title="Spin">Spins</a> mehrerer wechselwirkender Teilchen parallel, antiparallel oder <a href="Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> zu einem externen <a href="Magnetfeld" class="mw-redirect" title="Magnetfeld">Magnetfeld</a> ausrichten können, womit zusätzlich zum Ising-Modell auch der orthogonale Fall abgedeckt ist. Anders ausgedrückt beschreibt das Blume-Capel-Modell Spins mit einem <a href="Vektor#Länge/Betrag_eines_Vektors" title="Vektor">Betrag</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bac386d8f227fb823cede9b3e33d706cad3ed306.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.351ex; height:2.176ex;" alt="{\displaystyle s=1}" loading="lazy">, während das Ising-Modell Spins mit einem Betrag <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s={\tfrac {1}{2}}}">

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<annotation encoding="application/x-tex">{\displaystyle s={\tfrac {1}{2}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84a26f0e8aa20951fc075d0b792ff220d4326802.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.847ex; height:3.509ex;" alt="{\displaystyle s={\tfrac {1}{2}}}" loading="lazy"> beschreibt.

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

Der dem Blume-Capel-Modell zugrunde liegende <a href="Hamilton-Operator" class="mw-redirect" title="Hamilton-Operator">Hamilton-Operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {H}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/19ef4c7b923a5125ac91aa491838a95ee15b804f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.964ex; height:2.176ex;" alt="{\displaystyle {\mathcal {H}}}" loading="lazy">, dessen <a href="Eigenwert" class="mw-redirect" title="Eigenwert">Eigenwerte</a> die möglichen <a href="Energie" title="Energie">Energien</a> des Systems sind, lautet:

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {H}}=-D\sum _{i}(1-s_{i}^{z2})-J\sum _{\langle i,j\rangle }s_{i}^{z}s_{j}^{z}-\mu H\sum _{i}s_{i}^{z}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/43ef0125a7ff083d1fdfd327ff1d6ea1d1900228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:46.415ex; height:6.009ex;" alt="{\displaystyle {\mathcal {H}}=-D\sum _{i}(1-s_{i}^{z2})-J\sum _{\langle i,j\rangle }s_{i}^{z}s_{j}^{z}-\mu H\sum _{i}s_{i}^{z}}" loading="lazy"></dd></dl>

Dabei ist

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> das zero-field splitting, das die Energiedifferenz zwischen dem <a href="Singulett" class="mw-redirect" title="Singulett">Singulett</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{z}=0}">

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<annotation encoding="application/x-tex">{\displaystyle s^{z}=0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cba0d14cd96204afb2a8963e779bc72d91381b86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.353ex; height:2.343ex;" alt="{\displaystyle s^{z}=0}" loading="lazy"> und dem Dublett <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{z}=\pm 1}">

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<annotation encoding="application/x-tex">{\displaystyle s^{z}=\pm 1}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e6169759c117cead1843fb74b87e01a937ac290e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.161ex; height:2.343ex;" alt="{\displaystyle s^{z}=\pm 1}" loading="lazy"> angibt</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/359e4f407b49910e02c27c2f52e87a36cd74c053.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:2.176ex;" alt="{\displaystyle J}" loading="lazy"> die Stärke der Wechselwirkung benachbarter Spins</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" loading="lazy"> das <a href="Magnetisches_Moment" class="mw-redirect" title="Magnetisches Moment">magnetische Moment</a> der Spins</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/75a9edddcca2f782014371f75dca39d7e13a9c1b.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 H}" loading="lazy"> die <a href="Magnetische_Feldst%C3%A4rke" title="Magnetische Feldstärke">Stärke</a> des externen Magnetfeldes</li>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fc1102e3de3633afcebae582196377d78fb408af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.092ex; height:2.843ex;" alt="{\displaystyle s_{i}^{z}}" loading="lazy"> die <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy">-Komponente des <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy">-ten Spins.</li></ul>

Die Notation unter der Summe soll ausdrücken, dass nur über die jeweils nächsten Nachbarn summiert wird.

Der größte Unterschied zum Ising-Modell ist der zusätzliche, vom Parameter <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> abhängige Term im Hamilton-Operator. Für nur zwei mögliche Ausrichtungen des Spins wie im Ising-Modell wäre dieser Term eine Konstante und, wie ein konstanter Term in einem <a href="Potential_(Physik)" title="Potential (Physik)">Potential</a>, physikalisch bedeutungslos.

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

Abhängig vom Wert von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> nimmt das System verschiedene <a href="Grundzustand" title="Grundzustand">Grundzustände</a> ein und zeigt unterschiedliches Verhalten beim <a href="Phasen%C3%BCbergang" title="Phasenübergang">Phasenübergang</a>.

Bezeichne <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><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> die Anzahl der nächsten Nachbarn und <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><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"> die Anzahl der Spins im System, so gilt:

<ul><li>Für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D>{\tfrac {1}{2}}nJ}">

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<annotation encoding="application/x-tex">{\displaystyle D>{\tfrac {1}{2}}nJ}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b68b180fa244ccbef4388b56fd6f5d659e50a5f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.547ex; height:3.509ex;" alt="{\displaystyle D>{\tfrac {1}{2}}nJ}" loading="lazy"> ist der Grundzustand magnetisch ungeordnet und alle Spins liegen orthogonal zum Magnetfeld, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=0}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9d82e1293328f9760b19c84b9f7f611dbdb0da79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.151ex; height:2.509ex;" alt="{\displaystyle s_{i}=0}" loading="lazy">. Die Grundzustandsenergie liegt bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=-ND}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e9384e2ed6a30e6d7813b8552aecc015817ae7a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.664ex; height:2.509ex;" alt="{\displaystyle E_{0}=-ND}" loading="lazy">.</li>

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<annotation encoding="application/x-tex">{\displaystyle D={\tfrac {1}{2}}nJ}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/143e2106f59f6f722a75459cf542aa0907554fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.547ex; height:3.509ex;" alt="{\displaystyle D={\tfrac {1}{2}}nJ}" loading="lazy"> ist der Grundzustand <a href="Entartet" class="mw-redirect mw-disambig" title="Entartet">entartet</a>.</li>

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<annotation encoding="application/x-tex">{\displaystyle D<{\tfrac {1}{2}}nJ}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/25b56f83abc75bc90060dddb7a787678e6bcb0b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.547ex; height:3.509ex;" alt="{\displaystyle D<{\tfrac {1}{2}}nJ}" loading="lazy"> ist der Grundzustand vollständig <a href="Magnetische_Ordnung" title="Magnetische Ordnung">magnetisch geordnet</a>. Das heißt, alle Spins nehmen den Wert <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{i}=1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8757dc8386d9685bf56c6f638c028507ce3845aa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.151ex; height:2.509ex;" alt="{\displaystyle s_{i}=1}" loading="lazy"> an und die Grundzustandsenergie liegt bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}=-{\tfrac {1}{2}}NnJ}">

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<annotation encoding="application/x-tex">{\displaystyle E_{0}=-{\tfrac {1}{2}}NnJ}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/60a9710b99e47a27da6e214490443b84e037772a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.264ex; height:3.509ex;" alt="{\displaystyle E_{0}=-{\tfrac {1}{2}}NnJ}" loading="lazy">. In diesem Bereich existiert also eine <a href="Curie-Temperatur" title="Curie-Temperatur">Curie-Temperatur</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T_{\mathrm {C} }}">

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<annotation encoding="application/x-tex">{\displaystyle T_{\mathrm {C} }}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f38820828ca2ea1ffd8de70606ae3dd263fc3b5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.776ex; height:2.509ex;" alt="{\displaystyle T_{\mathrm {C} }}" loading="lazy">, bei der das System von einem magnetisch ungeordneten in einen magnetisch geordneten Zustand übergeht. Dieser Phasenübergang ist

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<annotation encoding="application/x-tex">{\displaystyle -\infty <D<{\tfrac {1}{3}}nJ\ln 4}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fb5d62e130a2d92822065df816aad85acdf53737.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:20.653ex; height:3.676ex;" alt="{\displaystyle -\infty <D<{\tfrac {1}{3}}nJ\ln 4}" loading="lazy"> ein <a href="Phasen%C3%BCbergang#Klassifikation_nach_Ehrenfest" title="Phasenübergang">Phasenübergang zweiter Ordnung</a>. Die Curie-Temperatur sinkt von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {nJ}{k_{\mathrm {B} }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {nJ}{k_{\mathrm {B} }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9b101642df3f14273535479598066ec35c865d81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:3.702ex; height:5.676ex;" alt="{\displaystyle {\frac {nJ}{k_{\mathrm {B} }}}}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=-\infty }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e51997a0d9befa21df0a8c7191998cb39fe6f130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:9.155ex; height:2.343ex;" alt="{\displaystyle D=-\infty }" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93fbf5e413664a7d67baee745cf444264ccb8d72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.701ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}">

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<annotation encoding="application/x-tex">{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/848c1c4997f2c24e1803030ccfab4e0b4abbfacd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.423ex; height:3.676ex;" alt="{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}" loading="lazy"> (darin ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{\mathrm {B} }}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c9b90d21a1c8fb907fddab1caded3b7f1eeffe3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.607ex; height:2.509ex;" alt="{\displaystyle k_{\mathrm {B} }}" loading="lazy"> die <a href="Boltzmann-Konstante" title="Boltzmann-Konstante">Boltzmann-Konstante</a>)</li>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{3}}nJ\ln 4<D<{\tfrac {1}{2}}nJ}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2f9e587ebb30d1f3ebf5b1bbed6dd5648ad3e49b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:21.046ex; height:3.676ex;" alt="{\displaystyle {\tfrac {1}{3}}nJ\ln 4<D<{\tfrac {1}{2}}nJ}" loading="lazy"> ein Phasenübergang erster Ordnung, bei dem die <a href="Magnetisierung" title="Magnetisierung">Magnetisierung</a> abrupt von Null auf einen endlichen Wert springt. Die Curie-Temperatur sinkt weiter von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/93fbf5e413664a7d67baee745cf444264ccb8d72.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:5.701ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{3}}{\frac {nJ}{k_{\mathrm {B} }}}}" loading="lazy"> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}">

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<annotation encoding="application/x-tex">{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/848c1c4997f2c24e1803030ccfab4e0b4abbfacd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:13.423ex; height:3.676ex;" alt="{\displaystyle D={\tfrac {1}{3}}nJ\ln 4}" loading="lazy"> auf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0}">

</mstyle>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2aae8864a3c1fec9585261791a809ddec1489950.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 0}" loading="lazy"> <a href="Kelvin" title="Kelvin">K</a> bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D={\tfrac {1}{2}}nJ}">

<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle D={\tfrac {1}{2}}nJ}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/143e2106f59f6f722a75459cf542aa0907554fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.547ex; height:3.509ex;" alt="{\displaystyle D={\tfrac {1}{2}}nJ}" loading="lazy">.</li></ul></li></ul>

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Martin Blume: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Theory of the First-Order Magnetic Phase Change in UO2</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Physical Review</cite>. Band 141, Nr. 2, 1965, S. 517–524, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103/PhysRev.141.517">10.1103/PhysRev.141.517</a> (englisch).<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:Blume-Capel-Modell&rft.atitle=Theory+of+the+First-Order+Magnetic+Phase+Change+in+UO2&rft.au=Martin+Blume&rft.date=1965&rft.doi=10.1103%2FPhysRev.141.517&rft.genre=journal&rft.issue=2&rft.jtitle=Physical+Review&rft.pages=517-524&rft.volume=141" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Hans Willem Capel: <cite class="lang" lang="en" dir="auto" style="font-style:italic">On the possibility of first-order phase transitions in Ising systems of triplet ions with zero-field splitting</cite>. In: <cite class="lang" lang="en" dir="auto" style="font-style:italic">Physica</cite>. Band 32, Nr. 5, 1966, S. 966–988, <a href="Digital_Object_Identifier" title="Digital Object Identifier">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016/0031-8914%2866%2990027-9">10.1016/0031-8914(66)90027-9</a> (englisch).<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:Blume-Capel-Modell&rft.atitle=On+the+possibility+of+first-order+phase+transitions+in+Ising+systems+of+triplet+ions+with+zero-field+splitting&rft.au=Hans+Willem+Capel&rft.date=1966&rft.doi=10.1016%2F0031-8914%2866%2990027-9&rft.genre=journal&rft.issue=5&rft.jtitle=Physica&rft.pages=966-988&rft.volume=32" style="display:none">

</li>

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