A Natural Nuclear Reactor — Question 2

Question 2

An isotope deficit must be normalized to the unchanged reference isotope

In a hypothetical fossil sample, present $^{235}$U fraction is .00660, compared with .00720 in a matched unfissioned reference. Suppose $^{238}$U is unchanged by the event, only $^{235}$U was removed, and both samples subsequently remained closed. Neglect other isotopes.

1. Find the fraction of $^{235}$U removed using ratios to $^{238}$U.

2. Compare it with the shortcut $1-f_s/f_r$ and explain the discrepancy.

Question 2 - Solution

An isotope deficit must be normalized to the unchanged reference isotope

(a) Method and calculation

$$r_s=\frac{.00660}{.99340},\quad r_r=\frac{.00720}{.99280},\quad r_s=(1-d)r_r.$$$$\boxed{d=1-\frac{r_s}{r_r}=.0838870=8.38870\%}.$$

(b) Interpretation and further calculation

$$\boxed{d_{\mathrm{shortcut}}=1-.00660/.00720=.0833333=8.33333\%}.$$$$\boxed{d-d_{\mathrm{shortcut}}=.00055365}\quad\text{or .055365 percentage point}.$$

Physical check

Removing atoms changes both the numerator and total in an atom fraction. Ratios to an unchanged isotope provide the appropriate normalization.