Radioactive Decay — Question 9

Question 9

An equal-lifetime decay chain requires a limiting form

A parent and its radioactive daughter have the same decay constant lambda. Initially there are $N_0$ parents and no daughters. The parent decays entirely to that daughter.

1. Solve the daughter-population equation without dividing by a difference of decay constants.

2. Find the maximum daughter population and the fraction of all original nuclei already in the stable final state at that time.

Question 9 - Solution

An equal-lifetime decay chain requires a limiting form

(a) Method and calculation

$$\dot N_D+\lambda N_D=\lambda N_0e^{-\lambda t}\Rightarrow\frac{d}{dt}(N_De^{\lambda t})=\lambda N_0.$$$$\boxed{N_D=N_0\lambda t e^{-\lambda t}}.$$

(b) Interpretation and further calculation

$$\dot N_D=N_0\lambda e^{-\lambda t}(1-\lambda t)=0\Rightarrow\boxed{t_{\max}=1/\lambda,\ N_D=N_0/e}.$$$$\boxed{N_S/N_0=1-N_P/N_0-N_D/N_0=1-2/e=.2642}.$$

Physical check

The usual unequal-rate chain formula has a removable singularity when the two constants coincide. Direct integration gives the finite limit.