(adsbygoogle = window.adsbygoogle || []).push({}); 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 [itex]\omega[/itex] 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

[itex]\Gamma=\frac{\vec{p_2}}{32 \pi^2 m_a^2} \int{|M^2| d\Omega}[/itex]

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 [itex]G=\int_R^{r_E}dr\frac{\sqrt{2m_{\alpha}(E-V(r))}d}{\hbar}[/itex] 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 [itex]\lambda \propto e^{-2G}[/itex]

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

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# Homework Help: Alpha decay vs 12C emission

Can you offer guidance or do you also need help?

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