# Homework Help: Alpha decay vs 12C emission

1. Sep 30, 2012

### xiMy

1. The problem statement, all variables and given/known data
Radium 226 usually decays via three consecutive alpha decays into Pb 214. Show that energetically possibly for radium 226 to decay into 214|86 Pb and 12|6 C but tell why it is highly unlikely.
Calculate the lifetime of the direct transition as a function of the possibility $\omega$ for the 12C to be assembled in the nuclear parent.
Hint: lifetime universe : 1.4 10^10 a
2. Relevant equations
gamow factor - alpha decay
fermis golden rule - decay width
we calculated the one particle into two decay which is
$\Gamma=\frac{\vec{p_2}}{32 \pi^2 m_a^2} \int{|M^2| d\Omega}$
but I have absolutely no idea how to get the M
3. The attempt at a solution
I have no idea how to approach this. I can show via the Weizsäcker mass formula that the reaction is energetically possible (funny fact: the alpha decay isn't because the formula calculates the mass for the alpha particle too heavy 4.0071 u instead of 4.0026; 12C is also 12.0026 instead of just 12 - so while its pretty good formula it is actually not good enough) but from then on I am clueless.

The gamov factor is usually derived for the alpha decay. Can I just take the same $G=\int_R^{r_E}dr\frac{\sqrt{2m_{\alpha}(E-V(r))}d}{\hbar}$ for 12C and then compare the values because they are proportionally to squareroot E ?
That would be 15 MeV for 12C vs 4 MeV and the formula is $\lambda \propto e^{-2G}$
Problem is that the radii for the alpha particle and C are different and therefore the potentials V too. Is the problem really that complicated or am I missing something?
Anyway, I really would appreciate the help