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Kerndrehimpulse und elektromagnetische Kernmomente
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==Elektrisches Kernquadrupolmoment Q== Q gibt Abweichung von der Kugelgestalt wieder Potential <math>\phi</math> für p im Außenraum <math>\Delta \phi = 0</math> :<math>\phi(r,\theta) = \frac{1}{4 \pi \epsilon_0} \sum\limits_{n=0}^{\infty} a_n \frac{1}{r^{n+1}} P_n(\cos \theta)</math> {{FB|Legendre Polynome}} <math>P_0 = 1</math> :<math>P_1 = cos \theta \quad P_n(\theta = 0) = 1 \quad P_2 = \frac{1}{2} + \frac{3}{2} \cos^ 2 \theta</math> [[Datei:KernQuadrupolmoment20.png|miniatur|Kugelgestalt des Kerns]] Die Bedeutung der Entwicklungskoeffizienten <math>a_n</math> erkennt man durch direkte Berechnung des Potentials auf der z-Achse, also für <math>e = 0</math> und Koeffizientenvergleich: :<math>\phi(r,\theta=0) = \frac{1}{4 \pi \epsilon_0} \sum\limits_{n=0}^{\infty} a_n \frac{1}{r^{n+1}} 1</math> oder direkt berechnet :<math>\phi(r,\theta=0)=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(r')d\tau}{|r-r'|}=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(r')r'^{n}}{r^{n+1}}\frac{1}{r^{n+1}}P_{n}\cos(\alpha)d\tau</math> mit <math>\frac{1}{|r-r'|}=\sum\limits _{n=0}^{\infty}a_{n}\frac{1}{r^{n+1}}P_{n}\cos(\alpha)</math>. :<math>a_n=\int {\rho(r')r'^{n}}P_{n}\cos(\alpha) d \tau</math> ;n=0:<math>a_0=\int \rho(r') d \tau= Z e</math> Punktladung ;n=1:<math>a_1=\int \rho(r')r'\cos(\alpha) d \tau =0 </math> elektrisches Dipolmoment in <math>z= r'\cos(\alpha)</math>-Richtung (=0 da Kernkräfte die Parität erhalten) ;n=2:<math>a_{2}=\int\rho(r')r'^{2}\left(-\frac{1}{2}+\frac{3}{2}\cos^{2}\alpha\right)=\frac{1}{2}\int\rho(r')(3z^{2}-r'^{2})d\tau\equiv\frac{1}{2}eQ</math> Bei konstanter Ladungsverteilung <math>\rho = \frac{Ze}{V}</math> ist deshalb <math>Q=\frac{Z}{V}\int(3z^2-r'^{2})d \tau</math>. Größenordnung: <math>Q \approx \pi R^2 \approx 10^{-28} m^2</math> (lb) Vorzeichen: [[Datei:KernQuadrupolmoment-Geometry21.png|gerahmt|Formen des Kernquadupolmoments]]
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