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<title>Bethe-Heitler-Formel</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Bethe-Heitler-Formel</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">Die Bethe-Heitler-Formel ist eine Formel zur Berechnung des <a href="Wirkungsquerschnitt" title="Wirkungsquerschnitt">Wirkungsquerschnitt</a> für den Streuprozess eines Elektrons an einem Atom verbunden mit der Emission eines Lichtquants. Sie wurde von <a href="Hans_Bethe" title="Hans Bethe">Hans Bethe</a> und <a href="Walter_Heitler" title="Walter Heitler">Walter Heitler</a> 1934 gefunden.<a href="#cite_note-1">[1]</a><a href="#cite_note-2">[2]</a>

Die Bethe-Heitler-Formel lautet für den differentiellen Wirkungsquerschnitt:

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<annotation encoding="application/x-tex">{\displaystyle {\frac {\mathrm {d} ^{4}\sigma }{\mathrm {d} k\,\mathrm {d} \theta \,\mathrm {d} \theta _{0}\,\mathrm {d} \phi }}=8\pi \alpha ^{3}{\frac {\sin \theta \sin \theta _{0}}{k}}{\frac {p}{p_{0}}}{\frac {(Z-F(q))^{2}}{q^{4}}}\Gamma }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e8c4825cd206843410378f8e5d2b27d66bd40011.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.431ex; height:6.509ex;" alt="{\displaystyle {\frac {\mathrm {d} ^{4}\sigma }{\mathrm {d} k\,\mathrm {d} \theta \,\mathrm {d} \theta _{0}\,\mathrm {d} \phi }}=8\pi \alpha ^{3}{\frac {\sin \theta \sin \theta _{0}}{k}}{\frac {p}{p_{0}}}{\frac {(Z-F(q))^{2}}{q^{4}}}\Gamma }" loading="lazy"></dd></dl>

Hierbei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {q}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a4e063f8ee7dae2488859c45a4e645db5148085.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.309ex; height:2.676ex;" alt="{\displaystyle {\vec {q}}}" loading="lazy"> der <a href="Impuls%C3%BCbertrag" class="mw-redirect" title="Impulsübertrag">Impulsübertrag</a> auf das Atom, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}_{0},{\vec {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b597b96b19ec0c2e8ab14fe7a93259a4f68e8a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.828ex; height:2.843ex;" alt="{\displaystyle {\vec {p}}_{0},{\vec {p}}}" loading="lazy"> sind die Impulse des ein- und auslaufendes Elektrons, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ccd4b98d198d6538010ae815ee1199baabd3493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}" loading="lazy"> der Impuls des auslaufenden Photons. Es gilt hierbei:

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<annotation encoding="application/x-tex">{\displaystyle {\vec {q}}={\vec {p}}_{0}-{\vec {p}}-{\vec {k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ac0a47ab85f2241c240bb6a9f1a69fe6d7f90faf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.004ex; height:3.343ex;" alt="{\displaystyle {\vec {q}}={\vec {p}}_{0}-{\vec {p}}-{\vec {k}}}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle F({\vec {q}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ff76028230fc9fb2fc4775eaa43734b487af41c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.859ex; height:2.843ex;" alt="{\displaystyle F({\vec {q}})}" loading="lazy"> ist der <a href="Formfaktor_(Physik)" title="Formfaktor (Physik)">Formfaktor</a> der Ladungsverteilung, welche auch als Screening-Funktion in dem Zusammenhang bezeichnet wird:

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<annotation encoding="application/x-tex">{\displaystyle F({\vec {q}})={\frac {1}{Q}}\int \mathrm {d} ^{3}{\vec {r}}\rho (r)\mathrm {e} ^{\mathrm {i} {\vec {q}}\cdot {\vec {r}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ec9869bd104d62ee3ed680c066fdbc4be0cf5f60.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.293ex; height:5.676ex;" alt="{\displaystyle F({\vec {q}})={\frac {1}{Q}}\int \mathrm {d} ^{3}{\vec {r}}\rho (r)\mathrm {e} ^{\mathrm {i} {\vec {q}}\cdot {\vec {r}}}}" loading="lazy"></dd></dl>

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8752c7023b4b3286800fe3238271bbca681219ed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.838ex; height:2.509ex;" alt="{\displaystyle Q}" loading="lazy"> ist dabei die Gesamtladung des Atoms und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

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

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</math><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"> seine Ladungsverteilung.

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4cfde86a3f7ec967af9955d0988592f0693d2b19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.453ex; height:2.176ex;" alt="{\displaystyle \Gamma }" loading="lazy"> ist definiert durch:

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<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\Gamma =&{\frac {p^{2}\sin ^{2}\theta }{\epsilon ^{2}}}(4E_{0}^{2}-q^{2})+{\frac {p_{0}^{2}\sin ^{2}\theta _{0}}{\epsilon _{0}^{2}}}(4E^{2}-q^{2})-{\frac {2pp_{0}\sin \theta \sin \theta _{0}\cos \phi }{\epsilon \epsilon _{0}}}(4EE_{0}-q^{2})\\&+{\frac {2k^{2}}{\epsilon \epsilon _{0}}}(p^{2}\sin ^{2}\theta +p_{0}^{2}\sin ^{2}\theta _{0}-2pp_{0}\sin \theta \sin \theta _{0}\cos \phi )\end{aligned}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/142fc9381f6a3f8afa9bf6b5a597a9ae301da7fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.171ex; width:84.297ex; height:13.509ex;" alt="{\displaystyle {\begin{aligned}\Gamma =&{\frac {p^{2}\sin ^{2}\theta }{\epsilon ^{2}}}(4E_{0}^{2}-q^{2})+{\frac {p_{0}^{2}\sin ^{2}\theta _{0}}{\epsilon _{0}^{2}}}(4E^{2}-q^{2})-{\frac {2pp_{0}\sin \theta \sin \theta _{0}\cos \phi }{\epsilon \epsilon _{0}}}(4EE_{0}-q^{2})\\&+{\frac {2k^{2}}{\epsilon \epsilon _{0}}}(p^{2}\sin ^{2}\theta +p_{0}^{2}\sin ^{2}\theta _{0}-2pp_{0}\sin \theta \sin \theta _{0}\cos \phi )\end{aligned}}}" loading="lazy"></dd></dl>

mit:

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<annotation encoding="application/x-tex">{\displaystyle \epsilon =E-p\cos \theta }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3abefe16bdf51468c0e24149fb7d4941926f995.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.804ex; height:2.509ex;" alt="{\displaystyle \epsilon =E-p\cos \theta }" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle \epsilon _{0}=E_{0}-p_{0}\cos \theta _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1ff0031142bdc7da40428952c95b5e6a494d6e22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.96ex; height:2.509ex;" alt="{\displaystyle \epsilon _{0}=E_{0}-p_{0}\cos \theta _{0}}" loading="lazy"></li></ul>

Hierbei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{0}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/411d268de7b1cf300d7481e3fe59f3b20887e0d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{0}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> die Energie des ein- und auslaufenden Elektrons <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E={\sqrt {p^{2}+m^{2}}}}">

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<annotation encoding="application/x-tex">{\displaystyle E={\sqrt {p^{2}+m^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0e4539fc5792e0e7e92e9b86abdb5e8e3695fe4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:15.357ex; height:4.843ex;" alt="{\displaystyle E={\sqrt {p^{2}+m^{2}}}}" loading="lazy">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/18b67de6bf25dba7a24e66967ff6319858798734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.145ex; height:2.509ex;" alt="{\displaystyle \theta _{0}}" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" loading="lazy"> sind die Winkel zwischen dem Photonenvektor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {k}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5ccd4b98d198d6538010ae815ee1199baabd3493.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.843ex;" alt="{\displaystyle {\vec {k}}}" loading="lazy"> und den Impulsvektoren <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}_{0}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/11c50d52e622a5d927697119371be2d0fdf9fed7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:2.469ex; height:2.843ex;" alt="{\displaystyle {\vec {p}}_{0}}" loading="lazy"> bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {p}}}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/84fee53c81592db54e0fe6c6f9eba002bb1dc74b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.415ex; height:2.676ex;" alt="{\displaystyle {\vec {p}}}" loading="lazy"> des ein- und auslaufenden Elektrons.</li>

<li>Der Winkel <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72b1f30316670aee6270a28334bdf4f5072cdde4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.385ex; height:2.509ex;" alt="{\displaystyle \phi }" loading="lazy"> ist der Winkel zwischen der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {p}},{\vec {k}})}">

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<mover>

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<annotation encoding="application/x-tex">{\displaystyle ({\vec {p}},{\vec {k}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/99be37757a4767476df3459c43619639a253536c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.38ex; height:3.343ex;" alt="{\displaystyle ({\vec {p}},{\vec {k}})}" loading="lazy"> und der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({\vec {p}}_{0},{\vec {k}})}">

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<annotation encoding="application/x-tex">{\displaystyle ({\vec {p}}_{0},{\vec {k}})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b86a3e6cb67dcb19ad68b4855f1559ecddf538b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.434ex; height:3.343ex;" alt="{\displaystyle ({\vec {p}}_{0},{\vec {k}})}" loading="lazy">-Ebene</li></ul>

Die Behandlung der Streuung von Elektronen an Atomen mit Strahlungsemission durch Bethe und Heitler beruht auf der komplett relativistischen Behandlung des Problems mit der <a href="Dirac-Gleichung" title="Dirac-Gleichung">Dirac-Gleichung</a> mit der <a href="Bornsche_N%C3%A4herung" title="Bornsche Näherung">Bornschen Näherung</a> in der 2. Ordnung <a href="St%C3%B6rungstheorie_(Quantenmechanik)" title="Störungstheorie (Quantenmechanik)">Störungstheorie</a>. Voraussetzung hierfür sind, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\alpha \ll 1}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2709618cc84dc8c71b44106fe226fd855069ba9d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.945ex; height:2.176ex;" alt="{\displaystyle Z\alpha \ll 1}" loading="lazy"> ist, also Ladungszahl des Atom multipliziert mit der <a href="Feinstrukturkonstante" title="Feinstrukturkonstante">Feinstrukturkonstante</a>. Außerdem müssen die Impulse des ein- und auslaufenden Elektrons wesentlich größer sein als <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle mZ\alpha }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7e12c1dbe38d1ac762970341805834b448c21b16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.208ex; height:2.176ex;" alt="{\displaystyle mZ\alpha }" loading="lazy">.

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

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Hans Bethe, Walter Heitler: <cite style="font-style:italic">On the stopping of fast particles and on the creation of positive electrons</cite>. In: <cite style="font-style:italic">Proc. Roy. Soc. A</cite>. Band 146, Nr. 856, 1934, S. 83–112.<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:Bethe-Heitler-Formel&rft.atitle=On+the+stopping+of+fast+particles+and+on+the+creation+of+positive+electrons&rft.au=Hans+Bethe%2C+Walter+Heitler&rft.date=1934&rft.genre=journal&rft.issue=856&rft.jtitle=Proc.+Roy.+Soc.+A&rft.pages=83-112&rft.volume=146" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Hans. A. Bethe, Edwin E. Salpeter: Quantum Mechanics of One- and Two-Electron-Atoms. Academic Press, New York 1957.

</li>

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