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<h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Matthiessensche Regel</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>Die <b>Matthiessensche Regel</b> (benannt nach <a href="Augustus_Matthiessen" title="Augustus Matthiessen">Augustus Matthiessen</a>) ist ein Zusammenhang aus der <a href="Materialwissenschaft" class="mw-redirect" title="Materialwissenschaft">Materialwissenschaft</a>. Sie besagt allgemein, dass sich der <a href="Spezifischer_Widerstand" title="Spezifischer Widerstand">spezifische Widerstand</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 \rho }">
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<mi>ρ<!-- ρ --></mi>
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<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }" loading="lazy"></span> für mehrere voneinander unabhängige <a href="Streuprozess" class="mw-redirect" title="Streuprozess">Streuprozesse</a> zusammensetzt aus der Summe der spezifischen Widerstände der einzelnen Streuprozesse:
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<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 \rho =\rho _{1}+\rho _{2}}">
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<mi>ρ<!-- ρ --></mi>
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<mi>ρ<!-- ρ --></mi>
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<mn>1</mn>
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<annotation encoding="application/x-tex">{\displaystyle \rho =\rho _{1}+\rho _{2}}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5b80a27022fb489b6fabf5a7996b89ab8cc5a85f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.653ex; height:2.509ex;" alt="{\displaystyle \rho =\rho _{1}+\rho _{2}}" loading="lazy"></span></dd></dl>
<p>Beispielsweise setzt sich der spezifische Widerstand eines <a href="Metall" class="mw-redirect" title="Metall">Metalls</a> zusammen aus
</p>
<ul><li>einem temperaturabhängigen Term, der durch <a href="Streuung_(Physik)" title="Streuung (Physik)">Streuung</a> der <a href="Elektron" title="Elektron">Elektronen</a> an den <a href="Gitterschwingung" title="Gitterschwingung">Gitterschwingungen</a> (<a href="Phonon" title="Phonon">Phononen</a>) entsteht</li>
<li>einem weiteren temperaturabhängigen Term, der durch Elektronen-Elektronen-Streuung entsteht, und</li>
<li>einem temperatur<i>un</i>abhängigen Term, der durch Streuung der Elektronen an <a href="Gitterfehler" title="Gitterfehler">Gitterfehlern</a> (Defekten) entsteht:</li></ul>
<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 \rho (T)\ =\ \rho _{\mathrm {Phononen} }(T)+\rho _{\mathrm {Elektronen-Elektronen} }(T)+\rho _{\mathrm {Defekte} }}">
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<mi>ρ<!-- ρ --></mi>
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<mi>T</mi>
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<mi>ρ<!-- ρ --></mi>
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<mi>ρ<!-- ρ --></mi>
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<mi mathvariant="normal">E</mi>
<mi mathvariant="normal">l</mi>
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<mi mathvariant="normal">t</mi>
<mi mathvariant="normal">r</mi>
<mi mathvariant="normal">o</mi>
<mi mathvariant="normal">n</mi>
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<mo>−<!-- − --></mo>
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<mi mathvariant="normal">e</mi>
<mi mathvariant="normal">n</mi>
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<mo stretchy="false">(</mo>
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<mi>ρ<!-- ρ --></mi>
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<mi mathvariant="normal">D</mi>
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<annotation encoding="application/x-tex">{\displaystyle \rho (T)\ =\ \rho _{\mathrm {Phononen} }(T)+\rho _{\mathrm {Elektronen-Elektronen} }(T)+\rho _{\mathrm {Defekte} }}</annotation>
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</math></span><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c793ba67bb1d5c3d490ea26c70d9c4b0b1011c29.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:55.452ex; height:2.843ex;" alt="{\displaystyle \rho (T)\ =\ \rho _{\mathrm {Phononen} }(T)+\rho _{\mathrm {Elektronen-Elektronen} }(T)+\rho _{\mathrm {Defekte} }}" loading="lazy"></span>,</dd></dl>
<p>mit der Temperatur <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}">
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<mi>T</mi>
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<annotation encoding="application/x-tex">{\displaystyle T}</annotation>
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</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>.
</p><p>Bei <a href="Zimmertemperatur" class="mw-redirect" title="Zimmertemperatur">Zimmertemperatur</a>&nbsp;(300&nbsp;<a href="Kelvin" title="Kelvin">K</a>) wird der spezifische Widerstand vorwiegend durch Stöße mit Gitterphononen verursacht, nahe beim <a href="Absoluter_Temperaturnullpunkt" class="mw-redirect" title="Absoluter Temperaturnullpunkt">absoluten Temperaturnullpunkt</a> (4&nbsp;K) sind <a href="Sto%C3%9F_(Physik)" title="Stoß (Physik)">Stöße</a> mit <a href="Fremdatom" title="Fremdatom">Fremdatomen</a> und mechanischen Gitterfehlern relevanter. Diese <a href="Empirisch" class="mw-redirect" title="Empirisch">empirisch</a> gefundene Regel gilt nur näherungsweise und <i>nicht</i> für verschiedene <a href="Einkristall" title="Einkristall">Einkristalle</a> (wie <a href="Wolfram" title="Wolfram">wolfram</a><a href="Dotierung" title="Dotierung">dotierte</a> <a href="Molybd%C3%A4n" title="Molybdän">Molybdän</a>-Einkristalle).<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>
</p>
<div class="mw-heading mw-heading2"><h2 id="Siehe_auch">Siehe auch</h2></div>
<ul><li><a href="Elektronenmobilit%C3%A4t" class="mw-redirect" title="Elektronenmobilität">Elektronenmobilität</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>
<ul><li>Neil W. Ashcroft, N. David Mermin: <cite style="font-style:italic">Solid State Physics</cite>. Saunders College Publishing, New York 1976, ISBN 0-03-083993-9, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em">&nbsp;</span>323–324</span>.<span class="Z3988" title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&amp;rfr_id=info:sid/de.wikipedia.org:Matthiessensche+Regel&amp;rft.au=Neil+W.+Ashcroft%2C+N.+David+Mermin&amp;rft.btitle=Solid+State+Physics&amp;rft.date=1976&amp;rft.genre=book&amp;rft.isbn=0030839939&amp;rft.pages=323-324&amp;rft.place=New+York&amp;rft.pub=Saunders+College+Publishing" style="display:none">&nbsp;</span></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>
<ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><a href="#cite_ref-1">↑</a></span> <span class="reference-text">Wissenschaft-Online Lexika: <i>Eintrag für „Matthiessensche Regel“ im Lexikon der Physik.</i> Abgerufen am 9. März 2011.</span>
</li>
</ol>
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