Why Is Carbon-12 Emission from Radium-226 Highly Unlikely?

[xiMy]

Homework Statement

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

Homework 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

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

 

[mark726]

. Thank you!

Thank you for your post. It is indeed possible for radium 226 to decay into 214|86 Pb and 12|6 C, but as you mentioned, it is highly unlikely. This is due to the difference in energy between the two decay paths.

To calculate the lifetime of the direct transition, we can use the Fermi's golden rule for decay width. This formula is given by:

Γ = 2π|M|^2ρ(E)

where Γ is the decay width, M is the transition matrix element, and ρ(E) is the density of final states.

In this case, we are interested in the transition from radium 226 to 12|6 C and 214|86 Pb. The transition matrix element can be calculated using the formula you provided in your post, but it will be different for each decay path due to the difference in radii and potentials.

The density of final states, ρ(E), can be calculated using the formula:

ρ(E) = \frac{1}{2π^2\hbar^3}\int_0^{r_E}\sqrt{2m(E-V(r))}r^2dr

where r_E is the radius of the nucleus and V(r) is the potential at that radius.

Once you have calculated the decay widths for both decay paths, you can use the relation:

\omega = \frac{\Gamma_{12C}}{\Gamma_{214 Pb + 12C}}

to determine the probability of the direct transition to occur. The lifetime of the direct transition can then be calculated using the relation:

τ = \frac{1}{\Gamma} = \frac{1}{\Gamma_{12C} + \Gamma_{214 Pb + 12C}}

Using these calculations, you can determine the lifetime of the direct transition as a function of the possibility ω for the 12C to be assembled in the nuclear parent.

I hope this helps. Please let me know if you have any further questions or need clarification on any of the steps. Best of luck with your research!