- #1
Fluffy86
- 4
- 0
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 E_\gamma = 10 eV
in a nucleus E_\gamma = 1 MeV
W_{\alpha \beta} &=& \frac{4}{3} \frac{e^2}{\hbar^4 c^3} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 \<br /> &=& \frac{4}{3} \frac{\alpha}{\hbar^3 c^2} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2
with the first \alpha beeing the fine structure constant \alpha = \frac{e^2}{\hbar c}=\frac{1}{137}
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 |<\beta|\vec{x}|\alpha>|^2 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 E_\gamma = 10 eV
in a nucleus E_\gamma = 1 MeV
Homework Equations
W_{\alpha \beta} &=& \frac{4}{3} \frac{e^2}{\hbar^4 c^3} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2 \<br /> &=& \frac{4}{3} \frac{\alpha}{\hbar^3 c^2} E_\gamma^3 |<\beta|\vec{x}|\alpha>|^2
with the first \alpha beeing the fine structure constant \alpha = \frac{e^2}{\hbar c}=\frac{1}{137}
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 |<\beta|\vec{x}|\alpha>|^2 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
