Help! I’m trying to figure out how the math that Enrico Fermi used when building the world’s first nuclear reactor worked. As the reactor grew, Fermi’s team tracked the neutron intensity A at the center of the reactor, and computed the effective radius of the reactor squared divided by A. Fermi was able to accurately predict when the reactor would go critical by extrapolating this metric to zero. The pictures below are from Fermi’s collected papers - you can find a full pdf version of this paper here.
Fermi’s math, part 1.
Fermi’s Math, part 2.
What I don’t understand is why this works! Sure, we expect A to get really big as our reactor reaches criticality, but why include the R squared term? Fermi gives a clue in his report:
“In a spherical structure having the reproduction factor 1 for infinite dimensions the activate of a detector placed at the center due to the natural neutrons is proportional to the square of the radius.”
I dug deeper here but can’t quite see how how these pieces fit together. I did find one interesting clue in Principles of Nuclear Engineering by Glasstone, and a similar idea was echoed by a Professor of Nuclear engineering I emailed.
Finally, I posted on physics stack exchange, and there’s been some interesting conversation there.
So, can you help me? Maybe the explanation is really simple - or maybe there’s more going on here - either way, I want to find out! Please post in the comments section here or on physics stack exchange if you have helpful clues or a complete answer!