Homework Help: Calculating the half-life of decays of Radium-224

1. Apr 19, 2012

Demon117

1. The problem statement, all variables and given/known data
Problem 1 – Krane 8.7] (a) compute the Q values for the decays 224Ra → 212Pb + 12C and 224Ra → 210Pb + 14C. (b) Estimate the half-lives for these two possible decay processes. 224Ra is a α emitter with a half-life of 3.66 days.

2. Relevant equations
I am assuming that the entire section 8.4 in Krane (Introduction to Nuclear Physics) on the Theory of $\alpha$ Emission is useful here. This discussion is found on pages 251 - 257.

3. The attempt at a solution
Part (a) is easy. I simply reduced the reactions to their mass excesses and computed the differences between the reactants and products. The results are

224Ra → 212Pb + 12C ====> Q = 26.4 MeV
224Ra → 210Pb + 14C ====> Q = 30.5 MeV

Part (b) is the one I am having trouble with. The 224Ra is known to be an $\alpha$ emitter with a half-life of 3.66 days. What I am having trouble with is the Barrier height B given by

$B = \frac{1}{4\pi \varepsilon_{0}}\frac{zZ'e^{2}}{a}$

I am under the assumption that Z' is for the daughter; which is 212Pb or 12C in the first decay, and 210Pb or 14C in the second decay? Is z for the $\alpha$ particle? And finally, is $a$ the nuclear radius of the 224Ra?

If I can figure out that relationship, then I think I could go through the process and figure out the half-life by the following relationships.

$\lambda=f P$

where $P = exp(-2 k_{2}(1/2)(b-a))$ and f is the frequency with which the alpha presents itself at the barrier. How does one calculate f?

From there it is just simply $t_{1/2} = \frac{ln(2)}{\lambda}$. Any help and suggestions would be appreciated.

2. Mar 6, 2017

llatosz

I know its like 5 years into the future, but here is the equation I think you need for half life. I am also stuck on this too so I am replying to bump this and maybe hijack this thread if you are no longer around