Quartz sync: Mar 15, 2024, 5:48 PM
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Ein [[Neutron]] wird zu [[Proton]] und [[Elektron]] + [[Neutrino]] umgewandelt. Für eine genauere Erläuterung siehe [[Kernprozesse#Der Beta-Zerfall]].
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Ein [[Neutron]] wird zu [[Proton]] und [[Elektron]] + [[Neutrino]] umgewandelt. Für eine genauere Erläuterung siehe [[Kernprozesse#Der Beta-Zerfall]].
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![[Feynman_Beta_Negative_Decay.svg]]
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![[Feynman_Beta_Negative_Decay.svg]]
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Im $\beta^{+}$-Zerfall kommt es zur Umwandlung von einem [[Proton]] zu einem [[Neutron]] mit Positron und Neutrino. Der Zerfall ist energetisch nur möglich wenn die Bindungsenergie im Kern sich danach stark verbessert, da sonst energetisch gehindert. (Kerne mit sehr vielen Protonen).
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Im $\beta^{+}$-Zerfall kommt es zur Umwandlung von einem [[Proton]] zu einem [[Neutron]] mit Positron und Neutrino. Der Zerfall ist energetisch nur möglich wenn die Bindungsenergie im Kern sich danach stark verbessert, da sonst energetisch gehindert. (Kerne mit sehr vielen Protonen -> [[Bethe-Weizsäcker-Formel]]).
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Siehe auch [[Kernmodelle]].
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Siehe auch [[Kernmodelle]].
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Neutrinoloser Beta-Zerfall im Doppel-Beta-Zerfall postuliert. Kein Nachweis bisher. Verletzung der [[Leptonenzahl]].
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Neutrinoloser Beta-Zerfall im Doppel-Beta-Zerfall postuliert. Kein Nachweis bisher. Verletzung der [[Leptonenzahl]].
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#question #todo
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#question #kristall
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> [!question] Frage
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> [!question] Frage
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> Erläutern Sie die Grundlagen eines Lasers.
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> Erläutern Sie die Grundlagen eines Lasers.
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# Antwort
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# Antwort
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# Vorkommen
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# Vorkommen
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1. *Insert link here*
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1. *Insert link here*
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2. *And more*
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2. *And more*
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---
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annotation-target: "[[FragenEx.pdf]]"
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---
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>%%
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>```annotation-json
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>{"text":"Eine Übersicht über die Verteilungen","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":720,"end":775},{"type":"TextQuoteSelector","exact":"Boltzmann Verteilung, Fermi-Verteilung, Bose-Verteilung","prefix":"die Teilchenphysik begonnen? •","suffix":"o Graphen zeichnen, diskutieren"}]}],"created":"2024-03-15T13:22:40.779Z","updated":"2024-03-15T13:22:40.779Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf"}
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>```
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>%%
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>*%%PREFIX%%die Teilchenphysik begonnen? •%%HIGHLIGHT%% ==Boltzmann Verteilung, Fermi-Verteilung, Bose-Verteilung== %%POSTFIX%%o Graphen zeichnen, diskutieren*
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>%%LINK%%[[#^gffr7bb3t8t|show annotation]]
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>%%COMMENT%%
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>Eine Übersicht über die Verteilungen
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>%%TAGS%%
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>
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^gffr7bb3t8t
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>%%
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>```annotation-json
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>{"text":"Einbindung mit Bezug auf \\beta^+","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":956,"end":978},{"type":"TextQuoteSelector","exact":"Bethe-Weizäcker-Formel","prefix":"/- Zerfall o Feynman-Diagramm o","suffix":"• Radioaktiver Zerfall, der kei"}]}],"created":"2024-03-15T13:23:05.874Z","updated":"2024-03-15T13:23:05.874Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf"}
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>```
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>%%
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>*%%PREFIX%%/- Zerfall o Feynman-Diagramm o%%HIGHLIGHT%% ==Bethe-Weizäcker-Formel== %%POSTFIX%%• Radioaktiver Zerfall, der kei*
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>%%LINK%%[[#^1nmzolm5e2s|show annotation]]
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>%%COMMENT%%
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>Einbindung mit Bezug auf \beta^+
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>%%TAGS%%
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>
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^1nmzolm5e2s
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>%%
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>```annotation-json
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>{"text":" ","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":1735,"end":1749},{"type":"TextQuoteSelector","exact":"Fermigasmodell","prefix":"halenmodell o Tröpfchenmodell o","suffix":"• Entartetes Fermigas (-> Fermi"}]}],"created":"2024-03-15T13:24:01.359Z","updated":"2024-03-15T13:24:01.359Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf"}
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>```
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>%%
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>*%%PREFIX%%halenmodell o Tröpfchenmodell o%%HIGHLIGHT%% ==Fermigasmodell== %%POSTFIX%%• Entartetes Fermigas (-> Fermi*
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>%%LINK%%[[#^fleykmyps57|show annotation]]
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>%%COMMENT%%
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>
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>%%TAGS%%
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>
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^fleykmyps57
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>%%
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>```annotation-json
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>{"created":"2024-03-15T13:24:30.238Z","text":"?","updated":"2024-03-15T13:24:30.238Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":2113,"end":2125},{"type":"TextQuoteSelector","exact":"Mott Sreuung","prefix":"wendet? • Wirkungsquerschnitt • ","suffix":" • Formfaktor • Frank Hertz Vers"}]}]}
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>```
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>%%
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>*%%PREFIX%%wendet? • Wirkungsquerschnitt •%%HIGHLIGHT%% ==Mott Sreuung== %%POSTFIX%%• Formfaktor • Frank Hertz Vers*
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>%%LINK%%[[#^6m3grghsn8|show annotation]]
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>%%COMMENT%%
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>?
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>%%TAGS%%
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>
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^6m3grghsn8
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>%%
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>```annotation-json
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>{"created":"2024-03-15T13:25:44.004Z","updated":"2024-03-15T13:25:44.004Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":3318,"end":3382},{"type":"TextQuoteSelector","exact":"Warum ist der Zerfall in e- gegen mu unterdrückt? (-> Helizität)","prefix":"antenzahlen • Was sind Pionen o ","suffix":" • Neutrino o Wie ist man darauf"}]}]}
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>```
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>%%
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>*%%PREFIX%%antenzahlen • Was sind Pionen o%%HIGHLIGHT%% ==Warum ist der Zerfall in e- gegen mu unterdrückt? (-> Helizität)== %%POSTFIX%%• Neutrino o Wie ist man darauf*
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>%%LINK%%[[#^lhkp9jntm9|show annotation]]
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>%%COMMENT%%
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>
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>%%TAGS%%
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>
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^lhkp9jntm9
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>%%
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>```annotation-json
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>{"created":"2024-03-15T13:25:57.912Z","updated":"2024-03-15T13:25:57.912Z","document":{"title":"FragenEx.pdf","link":[{"href":"urn:x-pdf:9930f0d91509a34eaa86422fe82698c9"},{"href":"vault:/FragenEx.pdf"}],"documentFingerprint":"9930f0d91509a34eaa86422fe82698c9"},"uri":"vault:/FragenEx.pdf","target":[{"source":"vault:/FragenEx.pdf","selector":[{"type":"TextPositionSelector","start":3459,"end":3502},{"type":"TextQuoteSelector","exact":"Erzeugung aus 2 Gluonen -> Feynman-Diagramm","prefix":"ss es das gibt? • Higgs-Boson o ","suffix":" o Zerfall o ... • Austauschtei"}]}]}
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>```
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>%%
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>*%%PREFIX%%ss es das gibt? • Higgs-Boson o%%HIGHLIGHT%% ==Erzeugung aus 2 Gluonen -> Feynman-Diagramm== %%POSTFIX%%o Zerfall o ... • Austauschtei*
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>%%LINK%%[[#^gm3v014ag7|show annotation]]
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>%%COMMENT%%
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>
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>%%TAGS%%
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>
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^gm3v014ag7
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*Folder tags:*
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#folder-Altprotokolle
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#class/Ex-IV #kerne
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Ein empirisches Modell für die Bindungsenergie, nach dem [[Tröpchenmodell]], das die Anziehung zwischen den nächsten Nachbarn berücksichtigt, führt zu einer Gleichung für die Bindungsenergie:
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$$E = a_v A - a_o A^\frac{2}{3} - a_c Z (Z-1) A^{-\frac{1}{3}}$$
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Hierbei repräsentiert der Term mit $a_v$ die paarweise Anziehung der Teilchen, die Oberflächenenergie wird durch $a_o$ beschrieben, und die Proton-Proton-Abstoßung durch den Term mit $a_c$. Diese Werte sind angenähert $a_v \approx \pu{{15.67} {meV}}$, $a_o \approx \pu{{17.23} {meV}}$, und $a_c \approx \pu{{.714} {meV}}$.
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Die Bindungsenergie pro Nukleon zeigt, dass sie für kleine Massenzahlen stark ansteigt und für große Massenzahlen wieder abnimmt. Dies erklärt, warum Kerne mit einer Massenzahl kleiner als Eisen zu Fusionen ([[Kernprozesse#Die Kernfusion]]) neigen und für größere Massenzahlen Spaltungen ([[Kernprozesse#Die Kernspaltung]]) bevorzugt werden.
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Die Gleichung für die Bindungsenergie wird erweitert, um quantenmechanische Korrekturterme einzubeziehen:
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$$E = a_v A - a_o A^\frac{2}{3} - a_c Z (Z-1) A^{-\frac{1}{3}} + a_s \frac{(N-Z)^2}{4A} + \begin{cases}
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a_p A^{-\frac{1}{2}} &\text{für } \mathbf{g,g}\\
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0 &\text{für } \mathbf{g,u} \text{ oder } \mathbf{u,g}\\
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-a_p A^{-\frac{1}{2}} &\text{für } \mathbf{u,u}
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\end{cases}$$
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Diese Korrekturen favorisieren symmetrische Kombinationen von [[Proton]]en und [[Neutron]]en sowie eine gerade Anzahl beider Nukleonenarten. Der Term $a_s$ bevorzugt symmetrische Kerne, während $a_p$ gerade Anzahlen von Protonen und Neutronen begünstigt.
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*Folder tags:*
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#folder-Physikalische-Grundlagen #folder-Atomphysik #folder-Kerne
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#class/Ex-IV #kerne #begriffsdefinition
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Ein weiteres Kernmodell besteht darin den Kern als einen Potentialtopf aufzufassen, welcher mit [[Fermionen]] gefüllt wird. Analog zum [[Elektronengas#Fermi-Gas]] aus Elektronen wie es in der Festkörperphysik verwendet wird, füllt sich der Topf aufgrund des [[Pauli-Prinzip]].
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![[Fermigasmodell_Atomkern.svg]]
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Erklärung der verschiedenen [[Kernprozesse]] durch *tunneln*.
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*Folder tags:*
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#folder-Physikalische-Grundlagen #folder-Atomphysik #folder-Kerne
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#class/Ex-IV #kerne #übersicht
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#class/Ex-IV #kerne #übersicht
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# Fermi-Gas-Modell
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![[Fermi-Gas-Modell]]
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# Schalenmodell
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![[Schalenmodell des Kerns]]
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![[Schalenmodell des Kerns]]
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# Tröpchenmodell
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![[Tröpchenmodell]]
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![[Tröpchenmodell]]
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@@ -11,7 +11,10 @@ Der Beta-Zerfall wurde 1899 von *Ernest Rutherford* entdeckt. Dabei wird zwische
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&{\mathbf{n}^0}&\longrightarrow &{\mathbf{p}^+}&+{\mathbf{e}^-}&+\bar{\nu}_e\\
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&{\mathbf{n}^0}&\longrightarrow &{\mathbf{p}^+}&+{\mathbf{e}^-}&+\bar{\nu}_e\\
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q:\quad &0 &\longrightarrow &1 &-1 &+0\\
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q:\quad &0 &\longrightarrow &1 &-1 &+0\\
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s: \quad&\frac{1}{2} &\longrightarrow &\frac{1}{2} &-\frac{1}{2} &+\frac{1}{2}
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s: \quad&\frac{1}{2} &\longrightarrow &\frac{1}{2} &-\frac{1}{2} &+\frac{1}{2}
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\end{aligned}$$ Unter Betrachtung von Ladung und [[Spin des Elektrons]] wird schnell ersichtlich warum es diese Kombination aus [[Elektron]] $\mathbf{e}^-$ und Anti-Elektron-[[Neutrino]] $\bar{\nu}_e$ benötigt bei der Umwandlung. Das Elektron und das Neutrino werden im folgenden aus dem Kern mit hoher Energie emittiert. Die Gesamtreaktion kann geschrieben werden als $$\ce{^A_ZA} \longrightarrow \ce{^A_{Z+1}B} + {\mathbf{e}^-}+ \bar{\nu}_e$$ Außerdem werden kurz angerissen:
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\end{aligned}$$ Unter Betrachtung von Ladung und [[Spin des Elektrons]] wird schnell ersichtlich warum es diese Kombination aus [[Elektron]] $\mathbf{e}^-$ und Anti-Elektron-[[Neutrino]] $\bar{\nu}_e$ benötigt bei der Umwandlung. Das Elektron und das Neutrino werden im folgenden aus dem Kern mit hoher Energie emittiert. Die Gesamtreaktion kann geschrieben werden als $$\ce{^A_ZA} \longrightarrow \ce{^A_{Z+1}B} + {\mathbf{e}^-}+ \bar{\nu}_e$$
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Grundlage für alle Übergänge ist die energetisch günstigere Konfiguration nach der [[Bethe-Weizsäcker-Formel]].
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Außerdem werden kurz angerissen:
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## $\beta^+$-Zerfall
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## $\beta^+$-Zerfall
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# Kernphysik und Bindungsenergie
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# Kernphysik und Bindungsenergie
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In der Kernphysik lassen sich die Nukleonen als eine Art inkompressible Flüssigkeit auffassen. Ein empirisches Modell für die Bindungsenergie, das die Anziehung zwischen den nächsten Nachbarn berücksichtigt, führt zu einer Gleichung für die Bindungsenergie:
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In der Kernphysik lassen sich die Nukleonen als eine Art inkompressible Flüssigkeit auffassen.
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$$E = a_v A - a_o A^\frac{2}{3} - a_c Z (Z-1) A^{-\frac{1}{3}}$$
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![[Bethe-Weizsäcker-Formel]]
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Hierbei repräsentiert der Term mit $a_v$ die paarweise Anziehung der Teilchen, die Oberflächenenergie wird durch $a_o$ beschrieben, und die Proton-Proton-Abstoßung durch den Term mit $a_c$. Diese Werte sind angenähert $a_v \approx \pu{{15.67} {meV}}$, $a_o \approx \pu{{17.23} {meV}}$, und $a_c \approx \pu{{.714} {meV}}$.
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Die Bindungsenergie pro Nukleon zeigt, dass sie für kleine Massenzahlen stark ansteigt und für große Massenzahlen wieder abnimmt. Dies erklärt, warum Kerne mit einer Massenzahl kleiner als Eisen zu Fusionen ([[Kernprozesse#Die Kernfusion]]) neigen und für größere Massenzahlen Spaltungen ([[Kernprozesse#Die Kernspaltung]]) bevorzugt werden.
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Die Gleichung für die Bindungsenergie wird erweitert, um quantenmechanische Korrekturterme einzubeziehen:
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$$E = a_v A - a_o A^\frac{2}{3} - a_c Z (Z-1) A^{-\frac{1}{3}} + a_s \frac{(N-Z)^2}{4A} + \begin{cases}
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a_p A^{-\frac{1}{2}} &\text{für } \mathbf{g,g}\\
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0 &\text{für } \mathbf{g,u} \text{ oder } \mathbf{u,g}\\
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-a_p A^{-\frac{1}{2}} &\text{für } \mathbf{u,u}
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\end{cases}$$
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Diese Korrekturen favorisieren symmetrische Kombinationen von [[Proton]]en und [[Neutron]]en sowie eine gerade Anzahl beider Nukleonenarten. Der Term $a_s$ bevorzugt symmetrische Kerne, während $a_p$ gerade Anzahlen von Protonen und Neutronen begünstigt.
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Diese Präferenz für symmetrische Kerne kann zur Umwandlung zwischen Protonen und Neutronen führen, ein Vorgang, der als [[Kernprozesse#Der Beta-Zerfall]] bekannt ist:
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Diese Präferenz für symmetrische Kerne kann zur Umwandlung zwischen Protonen und Neutronen führen, ein Vorgang, der als [[Kernprozesse#Der Beta-Zerfall]] bekannt ist:
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@@ -10,7 +10,7 @@ $$
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p_{F}= h \sqrt[3]{\frac{3n}{8 \pi}}
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p_{F}= h \sqrt[3]{\frac{3n}{8 \pi}}
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$$
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$$
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Die *Fermi-Energie* ist die dem Impuls zugeordnete Energie. Ein solches Gas wird auch als **entartetes Gas** bezeichnet.
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Die *Fermi-Energie* ist die dem Impuls zugeordnete Energie. Ein solches Gas wird auch als **entartetes Gas** bezeichnet.
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Im Extremfall von $T=0$ sind alle Zustände bis $p_F$ besetzt und kein einziger darüber. Mit steigender Temperatur verschwimmt die Grenze.
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Im Extremfall von $T=0$ sind alle Zustände bis $p_F$ besetzt und kein einziger darüber. Mit steigender Temperatur verschwimmt die Grenze. Für die Verteilung siehe [[Stochastische Verteilungen#*Fermi-Dirac*-Verteilung]]
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#class/Ex-IV #QM
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Für die Warscheinlichkeitsdichte der Besetzung thermodynamischer Zustände gibt es unterschiedliche *Verteilungsfunktionen*.
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Insbesondere kommt es dabei auf die Unterschiede in der [[Mathematik der Bosonen und Fermionen]] an.
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# *Boltzmann*-Verteilung
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- **Anwendbarkeit**: Gilt für klassische Teilchen, die unterscheidbar sind und nicht dem quantenmechanischen Prinzip der Ununterscheidbarkeit folgen. Diese Teilchen unterliegen auch nicht dem [[Pauli-Prinzip]], das besagt, dass keine zwei [[Fermionen]] gleichzeitig denselben Quantenzustand besetzen können.
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- **Merkmale**: Die Verteilung wird verwendet, um die Verteilung von Teilchen über verschiedene Energiezustände in Systemen zu beschreiben, in denen die Energieunterschiede deutlich größer sind als die thermische Energie, was die Annahme von unterscheidbaren Teilchen ermöglicht.
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- **Mathematische Form**: Die Wahrscheinlichkeit, ein Teilchen in einem Zustand mit der Energie $E$ zu finden, ist proportional zu $\exp(-E/kT)$, wobei $k$ die Boltzmann-Konstante und $T$ die Temperatur ist.
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# *Fermi-Dirac*-Verteilung
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- **Anwendbarkeit**: Gilt speziell für [[Fermionen]], das sind Teilchen wie Elektronen, Protonen und Neutronen, die einen halbzahligen Spin haben und dem [[Pauli-Prinzip]] folgen.
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- **Merkmale**: Sie beschreibt die Belegung von [[Fermionen]] in Energiezuständen bei absoluter Nulltemperatur, wo alle Teilchen die niedrigstmöglichen Energiezustände besetzen, was zu einem gefüllten "Fermi-See" bis zur Fermi-Energie führt (siehe [[Elektronengas#Fermi-Gas]]). Bei endlichen Temperaturen können Teilchen thermisch in höhere Energiezustände angeregt werden.
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- **Mathematische Form**: Die Wahrscheinlichkeit, dass ein Zustand besetzt ist, wird durch $\frac{1}{\exp((E-\mu)/kT) + 1}$ gegeben, wobei $\mu$ das chemische Potential (Fermi-Energie bei absoluter Nulltemperatur) ist.
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# *Bose-Einstein*-Verteilung
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- **Anwendbarkeit**: Gilt für [[Bosonen]], das sind Teilchen wie Photonen und Helium-4-Atome, die einen ganzzahligen Spin haben und nicht dem [[Pauli-Prinzip]] unterliegen. Diese Teilchen können denselben Quantenzustand besetzen, was zu Phänomenen wie der Bose-Einstein-Kondensation führt.
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- **Merkmale**: Bei niedrigen Temperaturen können eine große Anzahl von [[Bosonen]] den niedrigsten Energiezustand besetzen, was zu makroskopischen Quantenphänomenen führt. Im Gegensatz zu [[Fermionen]] gibt es keine Beschränkung für die Anzahl der [[Bosonen]], die einen gegebenen Energiezustand besetzen können.
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- **Mathematische Form**: Die Wahrscheinlichkeit, dass ein Zustand besetzt ist, wird durch $\frac{1}{\exp((E-\mu)/kT) - 1}$ gegeben, wobei $\mu$ das chemische Potential ist, das für [[Bosonen]] negativ oder null sein kann.
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![[Verteilungen.png]]
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*Folder tags:*
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#folder-Physikalische-Grundlagen #folder-Quantenmechanik
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@@ -61,7 +61,14 @@ Für massebehaftete Teilchen ist im Allgemeinen die Helizität ungleich der Chir
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![[chiralität-helizität.png]]
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![[chiralität-helizität.png]]
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## Folgen
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Durch die Erhaltung der *Helizität* kommt es beispielsweise zur [[Streuexperimente zur Strukturanalyse#Mott-Streeung]], da das nötige umklappen des Spins nicht möglich ist.
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Weiterhin ist der Zerfall eines [[Pion⁻]] in ein [[Elektron]] beispielsweise unterdrückt aufgrund des Übergangs von *Chiralität* und *Helizität*.
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Der Zerfall erfolgt
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nach
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$\pi^{-}\to e^{-} + \bar \nu_{e}$ bzw. analog für [[Muon]].
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Aufgrund der Massenverhältnisse ist das Elektron hochrelativistisch), wohingegen das Myon eine geringere Geschwindigkeit hat. Die *Helizität* der Antineutrinos (die hier als masselos betrachtet werden können) ist positiv. Da das Pion keinen Spin trägt und die Zerfallsteilchen sich in entgegengesetzte Richtungen bewegen, müssen aufgrund der Drehimpulserhaltung Elektron bzw. Myon ebenfalls positive *Helizität* haben. Die [[Schwache Wechselwirkung]], die den Zerfall bewirkt, koppelt aber nur an Elektronen und Myonen linkshändiger *Chiralität*. Da das hochrelativistische Elektron nur eine sehr kleine linkshändige Komponente hat, ist der elektronische Zerfall gegenüber dem myonischen Zerfall stark unterdrückt.
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After Width: | Height: | Size: 221 KiB |
Reference in New Issue
Block a user