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<button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">déplacer vers la barre latérale</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">masquer</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">Un article de Wikipédia, l'encyclopédie libre.</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="fr" dir="ltr"><p>En <a href="/wiki/Physique_du_solide" title="Physique du solide">physique du solide</a>, la <b>règle de Matthiessen</b> donne le temps caractéristique de relaxation des <a href="/wiki/Phonons" class="mw-redirect" title="Phonons">phonons</a> se propageant dans un cristal lorsqu'il existe plusieurs phénomènes se superposant. Elle a été établie par <a href="/wiki/Augustus_Matthiessen" title="Augustus Matthiessen">Augustus Matthiessen</a> en 1862<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite_crochet">[</span>1<span class="cite_crochet">]</span></a></sup>. </p><p>Cette règle s'exprime de la manière suivante<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite_crochet">[</span>2<span class="cite_crochet">]</span></a></sup> : lorsque plusieurs phénomènes se superposent, le temps caractéristique de relaxation <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }"></span> est obtenu à partir des divers temps caractéristiques <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/814ca8b33360ac3b7db8e9435271b5654175c853" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.816ex; height:2.009ex;" alt="{\displaystyle \tau _{i}}"></span> correspondant aux divers phénomènes de diffusion possibles (vibrations du réseau cristallin, imperfections, limites physiques du milieu) en appliquant la règle suivante </p> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau ^{-1}=\sum _{i}\tau _{i}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </munder> <msubsup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau ^{-1}=\sum _{i}\tau _{i}^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b9c4687381ba427e211068594088aa43510a8c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.022ex; height:5.509ex;" alt="{\displaystyle \tau ^{-1}=\sum _{i}\tau _{i}^{-1}}"></span></dd></dl></dd></dl> <p>Cette règle exprime l'additivité des probabilités de transition d'un état à un autre dans un <a href="/wiki/Syst%C3%A8me_thermodynamique" title="Système thermodynamique">système thermodynamique</a> constitué de phonons lorsque les phénomènes qui en sont à l'origine sont indépendants. Elle n'est pas rigoureuse mais constitue dans la plupart des cas une approximation raisonnable<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite_crochet">[</span>3<span class="cite_crochet">]</span></a></sup>. </p> <div class="mw-heading mw-heading2"><h2 id="Références"><span id="R.C3.A9f.C3.A9rences"></span>Références</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A8gle_de_Matthiessen&veaction=edit&section=1" title="Modifier la section : Références" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A8gle_de_Matthiessen&action=edit&section=1" title="Modifier le code source de la section : Références"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="references-small decimal" style=""><div class="mw-references-wrap"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink noprint"><a href="#cite_ref-1">↑</a> </span><span class="reference-text"><a href="/wiki/Augustus_Matthiessen" title="Augustus Matthiessen">Augustus Matthiessen</a>, Report of the British Association for the Advancement of Science, no.32, 1862, p.144</span> </li> <li id="cite_note-2"><span class="mw-cite-backlink noprint"><a href="#cite_ref-2">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Ziman1960"><span class="ouvrage" id="J._M._Ziman1960"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/John_Ziman" title="John Ziman">J. M. Ziman</a>, <cite class="italique" lang="en">Electrons and Phonons</cite>, <a href="/wiki/Clarendon_Press" class="mw-redirect" title="Clarendon Press">Clarendon Press</a>, <time>1960</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Electrons+and+Phonons&rft.pub=Clarendon+Press&rft.aulast=Ziman&rft.aufirst=J.+M.&rft.date=1960&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AR%C3%A8gle+de+Matthiessen"></span></span></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink noprint"><a href="#cite_ref-3">↑</a> </span><span class="reference-text"><span class="ouvrage" id="Callaway1958"><span class="ouvrage" id="Joseph_Callaway1958"><abbr class="abbr indicateur-langue" title="Langue : anglais">(en)</abbr> <a href="/wiki/Joseph_Callaway" title="Joseph Callaway">Joseph Callaway</a>, « <cite style="font-style:normal" lang="en">Model for Lattice Thermal Conductivity at Low Temperature</cite> », <i><span class="lang-en" lang="en"><a href="/wiki/Physical_Review" title="Physical Review">Physical Review</a></span></i>, <abbr class="abbr" title="volume">vol.</abbr> 113, <abbr class="abbr" title="numéro">n<sup>o</sup></abbr> 4,‎ <time>1958</time><span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Model+for+Lattice+Thermal+Conductivity+at+Low+Temperature&rft.jtitle=Physical+Review&rft.issue=4&rft.aulast=Callaway&rft.aufirst=Joseph&rft.date=1958&rft.volume=113&rfr_id=info%3Asid%2Ffr.wikipedia.org%3AR%C3%A8gle+de+Matthiessen"></span></span></span></span> </li> </ol></div> </div> <div class="mw-heading mw-heading2"><h2 id="Voir_aussi">Voir aussi</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=R%C3%A8gle_de_Matthiessen&veaction=edit&section=2" title="Modifier la section : Voir aussi" class="mw-editsection-visualeditor"><span>modifier</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=R%C3%A8gle_de_Matthiessen&action=edit&section=2" title="Modifier le code source de la section : Voir aussi"><span>modifier le code</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%C3%89quation_de_Boltzmann-Peierls" title="Équation de Boltzmann-Peierls">Équation de Boltzmann-Peierls</a></li></ul> <ul id="bandeau-portail" class="bandeau-portail"><li><span class="bandeau-portail-element"><span class="bandeau-portail-icone"><span class="noviewer" typeof="mw:File"><a href="/wiki/Portail:Physique" title="Portail de la physique"><img alt="icône décorative" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/24px-Circle-icons-physics-logo.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/36px-Circle-icons-physics-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/22/Circle-icons-physics-logo.svg/48px-Circle-icons-physics-logo.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span></span> <span class="bandeau-portail-texte"><a href="/wiki/Portail:Physique" title="Portail:Physique">Portail de la physique</a></span> </span></li> </ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; 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