A Natural Nuclear Reactor — Question 1

Question 1

Different uranium half-lives increase the ancient fissile fraction

Ancient natural fission at Oklo, Gabon, motivates this reference calculation. For an unfissioned closed reference sample, use present $^{235}$U atom fraction $f_0=.00720$ among $^{235}$U and $^{238}$U only. Their half-lives are .704 and 4.47 billion years. Ignore other isotopes. Let t be positive lookback time.

1. Derive the ancient isotope ratio $r(t)=N_{235}/N_{238}$ and atom fraction f(t).

2. Evaluate f at t=2.00 billion years and compare it with the present fraction.

Question 1 - Solution

Different uranium half-lives increase the ancient fissile fraction

(a) Method and calculation

$$r_0=\frac{f_0}{1-f_0},\quad\boxed{r(t)=r_0e^{(\lambda_{235}-\lambda_{238})t}}.$$$$\boxed{f(t)=\frac{r(t)}{1+r(t)},\quad\lambda_i=\ln2/T_i}.$$

(b) Interpretation and further calculation

$$r(2)=\frac{.00720}{.99280}\exp\!\left[2\ln2\left(\frac1{.704}-\frac1{4.47}\right)\right]=.0381049.$$$$\boxed{f(2)=.0367062=3.67062\%,\quad f(2)/f_0=5.0981}.$$

Physical check

The shorter-lived isotope was relatively more abundant in the past. Abundance alone does not establish whether a natural chain reaction occurred.