Alpha Decay — Question 8

Question 8

A preformation probability multiplies the tunneling decay rate

A model gives $\lambda=S_\alpha\nu P$, where $S_\alpha$ is the alpha preformation probability, nu the barrier encounter frequency and P the penetration probability. Let $\nu=10^{21}\,\mathrm{s^{-1}}$, $P=10^{-25}$ and measured half-life $T_{1/2}=10^5\,\mathrm s$.

1. Infer $S_\alpha$ and verify it is a probability.

2. If a second state has the same nu and P but half the preformation probability, find its half-life and relative activity for equal parent populations.

Question 8 - Solution

A preformation probability multiplies the tunneling decay rate

(a) Method and calculation

$$\lambda=\ln2/T_{1/2}=6.931\times10^{-6}\,\mathrm{s^{-1}}.$$$$\boxed{S_\alpha=\lambda/(\nu P)=.06931}\quad(0<S_\alpha<1).$$

(b) Interpretation and further calculation

$$\lambda\prime=\lambda/2\Rightarrow\boxed{T_{1/2}\prime=2.00\times10^5\,\mathrm s}.$$$$\mathcal A=\lambda N\Rightarrow\boxed{\mathcal A\prime/\mathcal A=1/2}\quad(N\prime=N).$$

Physical check

Penetration probability alone does not fix the lifetime. The nucleus must also contain the appropriate alpha-like configuration and supply an encounter rate.