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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Defektkonzentration</span></h1>
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<p>Als <b>Defektkonzentration</b> <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 c}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"></span> bezeichnet man das Verhältnis der Anzahl der <a href="Gitterfehler" title="Gitterfehler">Fehlstellen</a> in der <a href="Gitterstruktur" class="mw-redirect" title="Gitterstruktur">Gitterstruktur</a> eines <a href="Festk%C3%B6rper" title="Festkörper">Festkörpers</a> zur Gesamtzahl der Gitterplätze:
</p>
<dl><dd><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 c=\mathrm {\frac {Anzahl\;Fehlstellen}{Anzahl\;Gitterplaetze}} }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>c</mi>
<mo>=</mo>
<mrow class="MJX-TeXAtom-ORD">
<mfrac>
<mrow>
<mi mathvariant="normal">A</mi>
<mi mathvariant="normal">n</mi>
<mi mathvariant="normal">z</mi>
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<mi mathvariant="normal">l</mi>
<mspace width="thickmathspace"></mspace>
<mi mathvariant="normal">F</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">h</mi>
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">s</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">n</mi>
</mrow>
<mrow>
<mi mathvariant="normal">A</mi>
<mi mathvariant="normal">n</mi>
<mi mathvariant="normal">z</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">h</mi>
<mi mathvariant="normal">l</mi>
<mspace width="thickmathspace"></mspace>
<mi mathvariant="normal">G</mi>
<mi mathvariant="normal">i</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">r</mi>
<mi mathvariant="normal">p</mi>
<mi mathvariant="normal">l</mi>
<mi mathvariant="normal">a</mi>
<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">z</mi>
<mi mathvariant="normal">e</mi>
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</mfrac>
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<annotation encoding="application/x-tex">{\displaystyle c=\mathrm {\frac {Anzahl\;Fehlstellen}{Anzahl\;Gitterplaetze}} }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1a9c938abe7060310d9b55e17275aa9df085e720.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:26.085ex; height:5.843ex;" alt="{\displaystyle c=\mathrm {\frac {Anzahl\;Fehlstellen}{Anzahl\;Gitterplaetze}} }" loading="lazy"></span></dd></dl>
<p>Gitterfehler in einem Festkörper treten auf, weil sie einen <a href="Entropie_(Thermodynamik)" class="mw-redirect" title="Entropie (Thermodynamik)">Entropie</a>gewinn durch Unordnung bringen.
</p><p>Für eine gegebene Defektkonzentration <i>c</i> kann man die <a href="Freie_Enthalpie" class="mw-redirect" title="Freie Enthalpie">freie Enthalpie</a> <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 \Delta G}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>G</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta G}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c2c9bc9950e25c28607799cdd9cb0af1373721fa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.763ex; height:2.176ex;" alt="{\displaystyle \Delta G}" loading="lazy"></span> über die <a href="Gibbs-Helmholtz-Gleichung" title="Gibbs-Helmholtz-Gleichung">Gibbs-Helmholtz-Gleichung</a> bestimmen, vgl. Abb. rechts:
</p>
<dl><dd><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 \Delta G(c)=\Delta H(c)-T\cdot \Delta S(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
<mo>=</mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
<mo>−<!-- − --></mo>
<mi>T</mi>
<mo>⋅<!-- ⋅ --></mo>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta G(c)=\Delta H(c)-T\cdot \Delta S(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b47a3c719bc9aa47c5cc31a5159107d211b37ef7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.9ex; height:2.843ex;" alt="{\displaystyle \Delta G(c)=\Delta H(c)-T\cdot \Delta S(c)}" loading="lazy"></span></dd></dl>
<p>Der Einbau von Fehlstellen kostet <a href="Reaktionsenergie" class="mw-redirect" title="Reaktionsenergie">Reaktionsenergie</a> <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 \Delta H(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta H(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b14d1831406798de98956ea1c1d4de404a32ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.816ex; height:2.843ex;" alt="{\displaystyle \Delta H(c)}" loading="lazy"></span> und erzeugt dafür Entropie <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 \Delta S(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta S(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1e1e20c4e5cee1508f4b64dcfc865718cd0fe329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.251ex; height:2.843ex;" alt="{\displaystyle \Delta S(c)}" 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 T}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" loading="lazy"></span> ist die <a href="Absolute_Temperatur" class="mw-redirect" title="Absolute Temperatur">absolute Temperatur</a>.
</p><p>Die Funktion <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 \Delta G(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta G(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d06c55f7cbc434ee913feb95a55adc4fb7f038d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.579ex; height:2.843ex;" alt="{\displaystyle \Delta G(c)}" loading="lazy"></span> durchläuft ein Minimum, weil <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 T\Delta S(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>T</mi>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle T\Delta S(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/04da2ba56bd7b3e7c920a29d7a7e62f9681cbd06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.888ex; height:2.843ex;" alt="{\displaystyle T\Delta S(c)}" loading="lazy"></span> für kleine <i>c</i> stärker wächst als <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 \Delta H(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>H</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta H(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b14d1831406798de98956ea1c1d4de404a32ac6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.816ex; height:2.843ex;" alt="{\displaystyle \Delta H(c)}" loading="lazy"></span>. Wenn die Defekte in beliebiger Menge erzeugt werden können (z. B. <a href="Leerstelle" title="Leerstelle">Leerstellen</a>) bzw. zur Verfügung stehen (z. B. Ein<a href="Diffusion" title="Diffusion">diffusion</a> von <a href="Fremdatom" title="Fremdatom">Fremdatomen</a> aus einer anderen <a href="Phase_(Materie)" title="Phase (Materie)">Phase</a>), wird die Defektkonzentration den Wert anstreben, bei dem <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 \Delta G(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>G</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \Delta G(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1d06c55f7cbc434ee913feb95a55adc4fb7f038d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.579ex; height:2.843ex;" alt="{\displaystyle \Delta G(c)}" loading="lazy"></span> das Minimum annimmt. Ob und wann der Gleichgewichtswert erreicht wird, hängt davon ab, wie schnell die Diffusion abläuft.
</p><p>Bei steigender Temperatur verläuft die entropische Kurve (<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 -T\Delta S(c)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mo>−<!-- − --></mo>
<mi>T</mi>
<mi mathvariant="normal">Δ<!-- Δ --></mi>
<mi>S</mi>
<mo stretchy="false">(</mo>
<mi>c</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle -T\Delta S(c)}</annotation>
</semantics>
</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7aab7f17468b0c5524159b83b055847fe8094e62.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.696ex; height:2.843ex;" alt="{\displaystyle -T\Delta S(c)}" loading="lazy"></span>) steiler, wodurch sich das Minimum der Freien Energie zu höheren Defektkonzentrationen verschiebt.
</p><p>Nur <a href="Punktdefekt" title="Punktdefekt">nulldimensionale Gitterfehler</a> stehen im <a href="Thermodynamisches_Gleichgewicht" title="Thermodynamisches Gleichgewicht">thermodynamischen Gleichgewicht</a>. Andere Gitterfehler (z. B. <a href="Versetzung_(Materialwissenschaft)" title="Versetzung (Materialwissenschaft)">Versetzungen</a> und <a href="Grenzfl%C3%A4che" title="Grenzfläche">Grenzflächen</a>) können aber mechanisch stabil sein, so dass sie sich auch durch längeres <a href="Gl%C3%BChen" title="Glühen">Glühen</a> nicht vollständig beseitigen lassen.
</p>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Lesley Smart, Elaine Moore: <cite style="font-style:italic">Einführung in die Festkörperchemie</cite>. Springer, 1997, ISBN 3-540-67066-1, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>148–151</span>.<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:Defektkonzentration&rft.au=Lesley+Smart%2C+Elaine+Moore&rft.btitle=Einf%C3%BChrung+in+die+Festk%C3%B6rperchemie&rft.date=1997&rft.genre=book&rft.isbn=3540670661&rft.pages=148-151&rft.pub=Springer" style="display:none"> </span></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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