A Natural Nuclear Reactor — Question 3

Question 3

Stable isotope mixing separates a fission signature from background

A hypothetical stable element has isotope-X atom fraction .100 in geological background and .250 in a fission-derived endmember. A sample has fraction .160. Let w be the fraction of this element’s atoms contributed by the fission endmember.

1. Infer w by atom-number mixing.

2. If the measured fraction has standard uncertainty .003 and the endmembers are exact, find the uncertainty in w and the admissible measurement interval.

Question 3 - Solution

Stable isotope mixing separates a fission signature from background

(a) Method and calculation

$$f=(1-w)f_b+wf_F\Rightarrow w=\frac{f-f_b}{f_F-f_b}.$$$$\boxed{w=(.160-.100)/(.250-.100)=.400}.$$

(b) Interpretation and further calculation

$$\boxed{\sigma_w=\sigma_f/|f_F-f_b|=.003/.150=.020}.$$$$0\le w\le1\Rightarrow\boxed{.100\le f\le.250}.$$

Physical check

The inferred w is an atom fraction of this element, not a rock mass fraction. Values outside the endmember interval falsify this two-component model.