Calculating the half-life of decays of Radium-224

[Demon117]

Homework Statement

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.

Homework 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.

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.

 

[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

 

[ghaith harahsheh]

 

[ghaith harahsheh]

Homework Statement

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.

Homework 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.

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.

Hi, i just answer you of how to calculate the frequency to measure the half life

 

[berkeman]

Hi, i just answer you of how to calculate the frequency to measure the half life

Welcome to PF.

We don't normally allow posting solutions to homework problems at PF (the student must do the bulk of the work), but since the thread is so old, it's fine at this point.

 

[kuruman]

Welcome to PF.

We don't normally allow posting solutions to homework problems at PF (the student must do the bulk of the work), but since the thread is so old, it's fine at this point.

This thread has a looooong half-life.