A Natural Nuclear Reactor — Question 9

Question 9

Common radioactive aging preserves a normalized isotope deficit

Immediately after an ancient event, a sample’s $^{235}$U/$^{238}$U ratio was a fraction b of a matched unfissioned reference ratio. Both subsequently undergo the same radioactive decay for time t and remain closed. Their measured present ratio quotient is $b=.920\pm.010$.

1. Show that the quotient of isotope ratios remains b at every later time.

2. Infer the depletion fraction and its standard uncertainty, and explain why this comparison does not determine the event age.

Question 9 - Solution

Common radioactive aging preserves a normalized isotope deficit

(a) Method and calculation

$$r_s(t)=br_r(0)e^{-(\lambda_{235}-\lambda_{238})t},\quad r_r(t)=r_r(0)e^{-(\lambda_{235}-\lambda_{238})t}.$$$$\boxed{r_s(t)/r_r(t)=b}\quad\text{the common decay factor cancels}.$$

(b) Interpretation and further calculation

$$\boxed{d=1-b=.080\pm.010}\quad\text{or }(8.0\pm1.0)\%.$$$$\boxed{\partial b/\partial t=0}\quad\text{under matched-history assumptions}.$$

Physical check

The cancellation fails if samples have different gains, losses or isotope histories. Depletion evidence and age evidence are distinct constraints.