- #1
Fluffy86
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Hey
have another problem with one of my exercises
Make a crude estimate for the mean life of an electric dipole transition
in a atom [tex]E_\gamma = 10 eV[/tex]
in a nucleus [tex]E_\gamma = 1 MeV[/tex]
[tex]W_{\alpha \beta} &=& \frac{4}{3} \frac{e^2}{\hbar^4 c^3} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 \
&=& \frac{4}{3} \frac{\alpha}{\hbar^3 c^2} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 [/tex]
with the first \alpha beeing the fine structure constant [tex]\alpha = \frac{e^2}{\hbar c}=\frac{1}{137} [/tex]
I am not quite sure how to estimate the last factor in the equation. Since we just have to do a crude estimate i don't think we have to calculate it with real wavefunctions(dont know if there are even wavefunctions for nuclei)
So my first thought was since [tex]|<\beta|\vec{x}|\alpha>|^2[/tex] has the dimension of a length^2 I inserted the typical lengthscales of an atom, the Bohr radius, and for the nucleus 1fm.
For the atom I get W= 1.1 10^9 1/s and for the nucleus 3.82 *10^14 1/s.
The lifetime is just the inverse of these. But I think the lifetime is then too small, I have something like 10^(-8) in my mind for the atom.
Anyone has a idea how to estimate it in a better way?
Bye
Fluffy
have another problem with one of my exercises
Homework Statement
Make a crude estimate for the mean life of an electric dipole transition
in a atom [tex]E_\gamma = 10 eV[/tex]
in a nucleus [tex]E_\gamma = 1 MeV[/tex]
Homework Equations
[tex]W_{\alpha \beta} &=& \frac{4}{3} \frac{e^2}{\hbar^4 c^3} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 \
&=& \frac{4}{3} \frac{\alpha}{\hbar^3 c^2} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 [/tex]
with the first \alpha beeing the fine structure constant [tex]\alpha = \frac{e^2}{\hbar c}=\frac{1}{137} [/tex]
The Attempt at a Solution
I am not quite sure how to estimate the last factor in the equation. Since we just have to do a crude estimate i don't think we have to calculate it with real wavefunctions(dont know if there are even wavefunctions for nuclei)
So my first thought was since [tex]|<\beta|\vec{x}|\alpha>|^2[/tex] has the dimension of a length^2 I inserted the typical lengthscales of an atom, the Bohr radius, and for the nucleus 1fm.
For the atom I get W= 1.1 10^9 1/s and for the nucleus 3.82 *10^14 1/s.
The lifetime is just the inverse of these. But I think the lifetime is then too small, I have something like 10^(-8) in my mind for the atom.
Anyone has a idea how to estimate it in a better way?
Bye
Fluffy