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

  1. Apr 19, 2012 #1
    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 [itex]\alpha[/itex] 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 [itex]\alpha[/itex] emitter with a half-life of 3.66 days. What I am having trouble with is the Barrier height B given by

    [itex]B = \frac{1}{4\pi \varepsilon_{0}}\frac{zZ'e^{2}}{a}[/itex]

    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 [itex]\alpha[/itex] particle? And finally, is [itex]a[/itex] 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.

    [itex]\lambda=f P[/itex]

    where [itex]P = exp(-2 k_{2}(1/2)(b-a))[/itex] 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 [itex]t_{1/2} = \frac{ln(2)}{\lambda}[/itex]. Any help and suggestions would be appreciated.
  2. jcsd
  3. Mar 6, 2017 #2
    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
    half life equation.jpg
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