Rate of underground formation of carbon-14 by neutron capture

Question

Carbon-14 constantly forms in the atmosphere due to neutron capture by nitrogen-14 and subsequent emission of a proton. Other less frequent mechanisms involving neutron capture by carbon-13 or oxygen isotopes also exist, as well as radium decay, as the link explains.

My question is: given an old (compared to the half-life of $\ce{^{14}C}$, more than 100 million years old let's say), sample of coal with for example 2% nitrogen and 80% carbon, how much carbon-14 would be present due to the neutron flux underground?

Obviously this is going to be at least 1000 times less than in modern plant materials. But what order of magnitude would be expected?

In case it helps, I'll add these references:

• Background neutron flux determination at a depth of 3200 mwe underground Nuclear Instruments and Methods in Physics Research A 357 (1995), p 524-534.
• For shallower data: Variation of Environmental Neutron Flux with the Depth of Water and Soil Journal of Nuclear and Radiochemical Sciences 9, p 45-47.

Answer 1

I do not have access to the article you cite for the deep subsurface neutron flux but I question whether either article will be particularly useful. The shallow water and soil article is concerned with neutrons in the atmosphere and these attenuate so won't be significant at depth.

The major neutron source at depth are spontaneous fission of U-238 and α-n reactions from U and Th series alpha decay. So your flux will be highly dependent on the amount of uranium and thorium and the presence of light elements that will undergo the α-n reaction. This means that the flux will likely be very different from the Broken Hill ore deposit. You may want to look at the literature on subsurface Cl-36 production for flux estimates, but I'm unaware of any dealing with coal deposits.

In coal the C-14 production will be dominated by neutron capture on C-13. Nitrogen n-p should be insignificant because of it's low abundance. I think the n-p reaction depends on higher energy neutrons produced in the atmosphere (but check that).

Assuming you know or can bound the neutron flux then you can estimate the C-14 from the neutron capture cross section of C-13 - 1.4 mb (Chart of the nuclides 16th ed) and the half life of C-14 - 5715 years, assuming steady-state.