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Homework Help: Alpha decay vs 12C emission

  1. Sep 30, 2012 #1
    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
  2. jcsd
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