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border-width: 1px; clear: left; margin-bottom:1em; margin-top:1em; padding: 0.25em; overflow: hidden; word-break: break-word; word-wrap: break-word;"><div class="noviewer noresize" style="display: table-cell; padding-bottom: 0.2em; padding-left: 0.25em; padding-right: 1em; padding-top: 0.2em; vertical-align: middle;" aria-hidden="true" role="presentation"><span typeof="mw:File"><a href="/wiki/Wikipedia:Belege" title="Belege"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/24px-Qsicon_Quelle.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/36px-Qsicon_Quelle.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b7/Qsicon_Quelle.svg/48px-Qsicon_Quelle.svg.png 2x" data-file-width="24" data-file-height="24" /></a></span></div> <div style="display: table-cell; vertical-align: middle; width: 100%;"> <div> Dieser Artikel oder nachfolgende Abschnitt ist nicht hinreichend mit <a href="/wiki/Wikipedia:Belege" title="Wikipedia:Belege">Belegen</a> (beispielsweise <a href="/wiki/Hilfe:Einzelnachweise" title="Hilfe:Einzelnachweise">Einzelnachweisen</a>) ausgestattet. Angaben ohne ausreichenden Beleg könnten demnächst entfernt werden. Bitte hilf Wikipedia, indem du die Angaben recherchierst und <span style="white-space:nowrap">gute Belege einfügst.</span><br /> <span class="editoronly" style="display:none;"></span></div> </div></div> <p>In der <a href="/wiki/Physik" title="Physik">Physik</a> beschreibt die <b>Dispersionsrelation</b> (lat. dispergere ‚verteilen', ‚ausbreiten', ‚zerstreuen') den Zusammenhang zwischen dem <i>Ablauf</i> eines physikalischen Prozesses (<a href="/wiki/Frequenz" title="Frequenz">Frequenz</a>, <a href="/wiki/Energie" title="Energie">Energie</a>) und den <i>Eigenschaften</i> der ihn beschreibenden Größen (<a href="/wiki/Wellenzahl" title="Wellenzahl">Wellenzahl</a>, <a href="/wiki/Brechungsindex" title="Brechungsindex">Brechungsindex</a>, <a href="/wiki/Ausbreitungsgeschwindigkeit" class="mw-redirect" title="Ausbreitungsgeschwindigkeit">Ausbreitungsgeschwindigkeit</a>, <a href="/wiki/Impuls_(Physik)" class="mw-redirect" title="Impuls (Physik)">Impuls</a>). </p><p>Mathematisch ist die Dispersionsrelation die Beziehung zwischen der <a href="/wiki/Kreisfrequenz" title="Kreisfrequenz">Kreisfrequenz</a> <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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"></span> und der <a href="/wiki/Kreiswellenzahl" class="mw-redirect" title="Kreiswellenzahl">Kreiswellenzahl</a> <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 k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"></span>. Sie wird aus der linearen <a href="/wiki/Wellengleichung" title="Wellengleichung">Wellengleichung</a> durch eine <a href="/wiki/Fouriertransformation" class="mw-redirect" title="Fouriertransformation">Fouriertransformation</a> in Raum und Zeit gewonnen und hat die Form </p> <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 \omega =f(k)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =f(k)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd65a42c5784bc96b0008b3fee77ca7d68d3d8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.843ex; height:2.843ex;" alt="{\displaystyle \omega =f(k)}"></span>.</dd></dl> <p>Im einfachsten Fall sind Kreisfrequenz und Kreiswellenzahl stets proportional </p> <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 \omega =v_{\text{Phase}}\cdot k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Phase</mtext> </mrow> </msub> <mo>⋅</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =v_{\text{Phase}}\cdot k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/badf289b44cd961815154aaaa1d7e66e596c703f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.028ex; height:2.509ex;" alt="{\displaystyle \omega =v_{\text{Phase}}\cdot k}"></span>,</dd></dl> <p>mit der konstanten <a href="/wiki/Phasengeschwindigkeit" title="Phasengeschwindigkeit">Phasengeschwindigkeit</a> <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 v_{\text{Phase}}={\frac {\omega }{k}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Phase</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0d638ce457abd83c831df472925f2dbe602faa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.973ex; height:4.843ex;" alt="{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}"></span>. In diesem Fall gibt es <i>keine</i> <a href="/wiki/Dispersion_(Physik)" title="Dispersion (Physik)">Dispersion</a>. </p><p>Die Geschwindigkeit eines <a href="/wiki/Wellenpaket" title="Wellenpaket">Wellenpakets</a> ist dagegen die <a href="/wiki/Gruppengeschwindigkeit" title="Gruppengeschwindigkeit">Gruppengeschwindigkeit</a> <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 v_{\text{Gruppe}}={\frac {\mathrm {d} \omega }{\mathrm {d} k}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Gruppe</mtext> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>ω</mi> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>k</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \omega }{\mathrm {d} k}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cefefe292611bcc7dec16c6fa0ed8305961af2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.086ex; height:5.509ex;" alt="{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \omega }{\mathrm {d} k}}.}"></span> </p><p>Ein Wellenpaket besteht aus Wellen verschiedener Frequenzen, die unterschiedliche Phasengeschwindigkeiten haben können. Daher läuft ein Wellenpaket im Allgemeinen auseinander. Wellenpakete, die aufgrund nichtlinearer Effekte trotz Dispersion <i>nicht</i> auseinanderlaufen, werden als <a href="/wiki/Soliton" title="Soliton">Solitonen</a> bezeichnet. </p> <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"><input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none" /><div class="toctitle" lang="de" dir="ltr"><h2 id="mw-toc-heading">Inhaltsverzeichnis</h2><span class="toctogglespan"><label class="toctogglelabel" for="toctogglecheckbox"></label></span></div> <ul> <li class="toclevel-1 tocsection-1"><a href="#Optik"><span class="tocnumber">1</span> <span class="toctext">Optik</span></a></li> <li class="toclevel-1 tocsection-2"><a href="#Teilchenphysik"><span class="tocnumber">2</span> <span class="toctext">Teilchenphysik</span></a></li> <li class="toclevel-1 tocsection-3"><a href="#Festkörperphysik"><span class="tocnumber">3</span> <span class="toctext">Festkörperphysik</span></a></li> <li class="toclevel-1 tocsection-4"><a href="#Literatur"><span class="tocnumber">4</span> <span class="toctext">Literatur</span></a></li> </ul> </div> <div class="mw-heading mw-heading2"><h2 id="Optik">Optik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dispersionsrelation&veaction=edit&section=1" title="Abschnitt bearbeiten: Optik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Dispersionsrelation&action=edit&section=1" title="Quellcode des Abschnitts bearbeiten: Optik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/Datei:1D_Photonic_Crystal.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/1D_Photonic_Crystal.png/220px-1D_Photonic_Crystal.png" decoding="async" width="220" height="185" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/bf/1D_Photonic_Crystal.png/330px-1D_Photonic_Crystal.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/bf/1D_Photonic_Crystal.png/440px-1D_Photonic_Crystal.png 2x" data-file-width="1318" data-file-height="1111" /></a><figcaption><a href="/wiki/Bandstruktur" title="Bandstruktur">Bandstruktur</a> eines eindimensionalen <a href="/wiki/Photonischer_Kristall" title="Photonischer Kristall">photonischen Kristalls</a>. Die Dispersionsrelation <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 \omega (k_{\mathrm {z} })}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">z</mi> </mrow> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega (k_{\mathrm {z} })}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b5aa4118ef48b9b393b349420c8746c863e338" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.429ex; height:2.843ex;" alt="{\displaystyle \omega (k_{\mathrm {z} })}"></span> lässt sich direkt an der Steigung der Bänder ablesen</figcaption></figure> <p>In der Dispersionsrelation der <a href="/wiki/Optik" title="Optik">Optik</a> taucht der (<a href="/wiki/Komplexe_Zahl" title="Komplexe Zahl">komplexe</a>) <a href="/wiki/Brechungsindex" title="Brechungsindex">Brechungsindex</a> <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 n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" 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}"></span> als Funktion der Kreisfrequenz auf: </p> <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 \omega =c_{\text{M}}\,k={\frac {c}{n(\omega )}}\,k\quad \Rightarrow \quad n={\frac {c}{c_{\text{M}}}}={\frac {c\,k}{\omega }}=f(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>M</mtext> </mrow> </msub> <mspace width="thinmathspace" /> <mi>k</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <mrow> <mi>n</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>k</mi> <mspace width="1em" /> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>c</mi> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>M</mtext> </mrow> </msub> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>c</mi> <mspace width="thinmathspace" /> <mi>k</mi> </mrow> <mi>ω</mi> </mfrac> </mrow> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =c_{\text{M}}\,k={\frac {c}{n(\omega )}}\,k\quad \Rightarrow \quad n={\frac {c}{c_{\text{M}}}}={\frac {c\,k}{\omega }}=f(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0067f69e981ca1740d75745c1d1e641159c39fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:49.578ex; height:6.176ex;" alt="{\displaystyle \omega =c_{\text{M}}\,k={\frac {c}{n(\omega )}}\,k\quad \Rightarrow \quad n={\frac {c}{c_{\text{M}}}}={\frac {c\,k}{\omega }}=f(\omega )}"></span></dd></dl> <p>mit </p> <ul><li>der <a href="/wiki/Phasengeschwindigkeit" title="Phasengeschwindigkeit">Phasengeschwindigkeit</a> <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 c_{\text{M}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>M</mtext> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c_{\text{M}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b59013bc816d30316f116236aaa3f7bbfac1499a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.746ex; height:2.009ex;" alt="{\displaystyle c_{\text{M}}}"></span> von <a href="/wiki/Licht" title="Licht">Licht</a> in einem <a href="/wiki/Ausbreitungsmedium" title="Ausbreitungsmedium">Medium</a></li> <li>der <a href="/wiki/Vakuumlichtgeschwindigkeit" class="mw-redirect" title="Vakuumlichtgeschwindigkeit">Vakuumlichtgeschwindigkeit</a> <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 c}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86a67b81c2de995bd608d5b2df50cd8cd7d92455" 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}"></span>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Teilchenphysik">Teilchenphysik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dispersionsrelation&veaction=edit&section=2" title="Abschnitt bearbeiten: Teilchenphysik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Dispersionsrelation&action=edit&section=2" title="Quellcode des Abschnitts bearbeiten: Teilchenphysik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Da die Frequenz immer in Zusammenhang mit der Energie steht </p> <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 \omega ={\frac {E}{\hbar }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>E</mi> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {E}{\hbar }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9da81c42488056ad3e56a99d73020e3dd3149f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.156ex; height:5.343ex;" alt="{\displaystyle \omega ={\frac {E}{\hbar }}}"></span></dd></dl> <p>und die <a href="/wiki/Wellenzahl" title="Wellenzahl">Wellenzahl</a> (bzw. der <a href="/wiki/Wellenvektor" title="Wellenvektor">Wellenvektor</a>) mit dem <a href="/wiki/Impuls" title="Impuls">Impuls</a> <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 p}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}"></span> </p> <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 {\vec {k}}={\frac {\vec {p}}{\hbar }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>k</mi> <mo stretchy="false">→</mo> </mover> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>p</mi> <mo stretchy="false">→</mo> </mover> </mrow> <mi class="MJX-variant">ℏ</mi> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\vec {k}}={\frac {\vec {p}}{\hbar }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e84a32d0af1fd0ca77c4a0af84d7f1476d82d902" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.118ex; height:5.676ex;" alt="{\displaystyle {\vec {k}}={\frac {\vec {p}}{\hbar }},}"></span></dd></dl> <p>bezeichnet man die Energie-Impuls-Beziehungen der <a href="/wiki/Teilchenphysik" title="Teilchenphysik">Teilchenphysik</a> auch als Dispersionsrelation (oder Dispersionsbeziehung), z. B. bei <a href="/wiki/Freies_Teilchen" title="Freies Teilchen">freien</a> <a href="/wiki/Elektron" title="Elektron">Elektronen</a> im nicht-<a href="/wiki/Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">relativistischen</a> Grenzfall: </p> <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 {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd /> <mtd> <mi>E</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇒</mo> </mtd> <mtd /> <mtd> <mi class="MJX-variant">ℏ</mi> <mspace width="thinmathspace" /> <mi>ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi class="MJX-variant">ℏ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <mo stretchy="false">⇔</mo> </mtd> <mtd /> <mtd> <mi>ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi class="MJX-variant">ℏ</mi> <mrow> <mn>2</mn> <mi>m</mi> </mrow> </mfrac> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8b9daa9955ba8032df2fe4bcbf4c0d9e423b2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:20.91ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2},\end{aligned}}}"></span></dd></dl> <p>wobei <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 \hbar }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi class="MJX-variant">ℏ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de68de3a92517953436c93b5a76461d49160cc41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }"></span> das <a href="/wiki/Reduziertes_Plancksches_Wirkungsquantum" class="mw-redirect" title="Reduziertes Plancksches Wirkungsquantum">reduzierte Plancksche Wirkungsquantum</a> und <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> die <a href="/wiki/Masse_(Physik)" title="Masse (Physik)">Masse</a> des Teilchens bezeichnet. </p> <div class="mw-heading mw-heading2"><h2 id="Festkörperphysik"><span id="Festk.C3.B6rperphysik"></span>Festkörperphysik</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dispersionsrelation&veaction=edit&section=3" title="Abschnitt bearbeiten: Festkörperphysik" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Dispersionsrelation&action=edit&section=3" title="Quellcode des Abschnitts bearbeiten: Festkörperphysik"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In der <a href="/wiki/Festk%C3%B6rperphysik" title="Festkörperphysik">Festkörperphysik</a> wird die Dispersion als Zusammenhang zwischen Energie bzw. Kreisfrequenz und Wellenzahl eines <a href="/wiki/Teilchen" title="Teilchen">Teilchens</a> oder <a href="/wiki/Quasiteilchen" title="Quasiteilchen">Quasiteilchens</a> angegeben. In <a href="/wiki/Festk%C3%B6rper" title="Festkörper">Festkörpern</a> wird dabei einerseits den <a href="/wiki/Phonon" title="Phonon">Phononen</a> (Gitterschwingungen des <a href="/wiki/Atomgitter" title="Atomgitter">Atomgitters</a>) eine <a href="/wiki/Phonon#Dispersion" title="Phonon">Phononen-Dispersionsrelation</a> zugeordnet, andererseits kann den Elektronen eine Elektronen-Dispersionsrelation zugeordnet werden, die mit Hilfe der <a href="/wiki/Bandstruktur" title="Bandstruktur">Bandstruktur</a> beschrieben wird. </p> <div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Dispersionsrelation&veaction=edit&section=4" title="Abschnitt bearbeiten: Literatur" class="mw-editsection-visualeditor"><span>Bearbeiten</span></a><span class="mw-editsection-divider"> | </span><a href="/w/index.php?title=Dispersionsrelation&action=edit&section=4" title="Quellcode des Abschnitts bearbeiten: Literatur"><span>Quelltext bearbeiten</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dieter_Meschede" title="Dieter Meschede">Dieter Meschede</a>: <cite style="font-style:italic">Optik, Licht und Laser</cite>. Springer-Verlag, 2015, <a href="/wiki/Spezial:ISBN-Suche/3663109542" class="internal mw-magiclink-isbn">ISBN 3-663-10954-2</a>, <span style="white-space:nowrap">S.<span style="display:inline-block;width:.2em"> </span>29<span style="display:inline-block;width:.2em"> </span>f</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:Dispersionsrelation&rft.au=Dieter+Meschede&rft.btitle=Optik%2C+Licht+und+Laser&rft.date=2015&rft.genre=book&rft.isbn=3663109542&rft.pages=29f&rft.pub=Springer-Verlag" style="display:none"> </span></li> <li><cite style="font-style:italic">Dispersionsrelation</cite>. In: Ulrich Kilian u. Christine Weber (Hrsg.): <cite style="font-style:italic">Lexikon der Physik</cite>. Spektrum Akademischer Verlag, 2003, <a href="/wiki/Spezial:ISBN-Suche/9783860252963" class="internal mw-magiclink-isbn">ISBN 978-3-86025-296-3</a> (<a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/physik/dispersionsrelation/3190">spektrum.de</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:Dispersionsrelation&rft.atitle=Dispersionsrelation&rft.btitle=Lexikon+der+Physik&rft.date=2003&rft.genre=book&rft.isbn=9783860252963&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></li> <li><cite style="font-style:italic">Dispersionsrelation</cite>. In: <a href="/wiki/Harry_Paul_(Physiker)" title="Harry Paul (Physiker)">Harry Paul</a> (Hrsg.): <cite style="font-style:italic">Lexikon der Optik</cite>. Spektrum Akademischer Verlag, 1999, <a href="/wiki/Spezial:ISBN-Suche/9783827403827" class="internal mw-magiclink-isbn">ISBN 978-3-8274-0382-7</a> (<a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/optik/dispersionsrelation/663">spektrum.de</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:Dispersionsrelation&rft.atitle=Dispersionsrelation&rft.btitle=Lexikon+der+Optik&rft.date=1999&rft.genre=book&rft.isbn=9783827403827&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </span></li></ul></div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1&useformat=desktop" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Abgerufen von „<a dir="ltr" href="https://de.wikipedia.org/w/index.php?title=Dispersionsrelation&oldid=245696251">https://de.wikipedia.org/w/index.php?title=Dispersionsrelation&oldid=245696251</a>“</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Wikipedia:Kategorien" title="Wikipedia:Kategorien">Kategorien</a>: <ul><li><a href="/wiki/Kategorie:Wellenlehre" title="Kategorie:Wellenlehre">Wellenlehre</a></li><li><a href="/wiki/Kategorie:Optik" title="Kategorie:Optik">Optik</a></li><li><a href="/wiki/Kategorie:Teilchenphysik" title="Kategorie:Teilchenphysik">Teilchenphysik</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Versteckte Kategorie: <ul><li><a href="/wiki/Kategorie:Wikipedia:Belege_fehlen" title="Kategorie:Wikipedia:Belege fehlen">Wikipedia:Belege fehlen</a></li></ul></div></div> </div> </div> <div id="mw-navigation"> <h2>Navigationsmenü</h2> <div id="mw-head"> <nav id="p-personal" class="mw-portlet mw-portlet-personal vector-user-menu-legacy vector-menu" aria-labelledby="p-personal-label" > <h3 id="p-personal-label" class="vector-menu-heading " > <span class="vector-menu-heading-label">Meine Werkzeuge</span> </h3> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-anonuserpage" class="mw-list-item"><span title="Benutzerseite der IP-Adresse, von der aus du Änderungen durchführst">Nicht angemeldet</span></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Spezial:Meine_Diskussionsseite" title="Diskussion über Änderungen von dieser IP-Adresse [n]" accesskey="n"><span>Diskussionsseite</span></a></li><li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Spezial:Meine_Beitr%C3%A4ge" title="Eine Liste der Bearbeitungen, die von dieser IP-Adresse gemacht wurden [y]" accesskey="y"><span>Beiträge</span></a></li><li id="pt-createaccount" class="mw-list-item"><a href="/w/index.php?title=Spezial:Benutzerkonto_anlegen&returnto=Dispersionsrelation" title="Wir ermutigen dich dazu, ein Benutzerkonto zu erstellen und dich anzumelden. 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class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Relation_de_dispersion" title="Relation de dispersion – Französisch" lang="fr" hreflang="fr" data-title="Relation de dispersion" data-language-autonym="Français" data-language-local-name="Französisch" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%99%D7%97%D7%A1_%D7%A0%D7%A4%D7%99%D7%A6%D7%94" title="יחס נפיצה – Hebräisch" lang="he" hreflang="he" data-title="יחס נפיצה" data-language-autonym="עברית" data-language-local-name="Hebräisch" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D4%B4%D5%AB%D5%BD%D5%BA%D5%A5%D6%80%D5%BD%D5%AB%D5%A1%D5%B5%D5%AB_%D6%85%D6%80%D5%A5%D5%B6%D6%84" title="Դիսպերսիայի օրենք – 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href="https://ko.wikipedia.org/wiki/%EB%B6%84%EC%82%B0_%EA%B4%80%EA%B3%84" title="분산 관계 – Koreanisch" lang="ko" hreflang="ko" data-title="분산 관계" data-language-autonym="한국어" data-language-local-name="Koreanisch" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-ky mw-list-item"><a href="https://ky.wikipedia.org/wiki/%D0%94%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D0%B8%D1%8F_%D0%BC%D1%8B%D0%B9%D0%B7%D0%B0%D0%BC%D1%8B" title="Дисперсия мыйзамы – Kirgisisch" lang="ky" hreflang="ky" data-title="Дисперсия мыйзамы" data-language-autonym="Кыргызча" data-language-local-name="Kirgisisch" class="interlanguage-link-target"><span>Кыргызча</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hubungan_sebaran" title="Hubungan sebaran – Malaiisch" lang="ms" hreflang="ms" data-title="Hubungan sebaran" data-language-autonym="Bahasa Melayu" data-language-local-name="Malaiisch" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Spredningsrelasjon" title="Spredningsrelasjon – Norwegisch (Bokmål)" lang="nb" hreflang="nb" data-title="Spredningsrelasjon" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegisch (Bokmål)" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Dyspersja_fali" title="Dyspersja fali – Polnisch" lang="pl" hreflang="pl" data-title="Dyspersja fali" data-language-autonym="Polski" data-language-local-name="Polnisch" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Rela%C3%A7%C3%A3o_de_dispers%C3%A3o" title="Relação de dispersão – Portugiesisch" lang="pt" hreflang="pt" data-title="Relação de 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mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B0%D0%BA%D0%BE%D0%BD_%D0%B4%D0%B8%D1%81%D0%BF%D0%B5%D1%80%D1%81%D1%96%D1%97" title="Закон дисперсії – Ukrainisch" lang="uk" hreflang="uk" data-title="Закон дисперсії" data-language-autonym="Українська" data-language-local-name="Ukrainisch" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Yorug%CA%BBlik_dispersiyasining_elementar_klassik_nazariyasi" title="Yorugʻlik dispersiyasining elementar klassik nazariyasi – Usbekisch" lang="uz" hreflang="uz" data-title="Yorugʻlik dispersiyasining elementar klassik nazariyasi" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Usbekisch" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%89%B2%E6%95%A3%E5%85%B3%E7%B3%BB" title="色散关系 – 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