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I am taking a course in nuclear physics using the bookAn Intro to the Physics of Nuclei & Particlesby Dunlap. I am a little confused by an explanation related to the critical radius for nuclear reactions.

The author first defines a value ##vq## as the average number of neutrons produced by a free neutron in a sample of U-235. Here ##v## is the number of neutrons produced in a fission reaction and ##q## is a factor < 1 to account for the neutron loss in the sample. To obtain a controlled reaction in which the number of neutrons remains constant ##vq## should equal ##1##. This I understand. The book derives that for a typical value ##v = 2.5## one should aim for ##q = 0.4##.

My confusion starts when the author explains how one might obtain such a q value. The author derives the average path length before a neutron fission reaction occurs to be equal to ##0.079## m. Then he states that a sample sphere with this radius will be just the right size to keep the reaction controlled (##vq = 1##), also called the critical radius. However in my understanding this means that every neutron produced at the center will undergo on average 1 fission reaction, and thus ##q \approx 1##. Then the reaction would obviously be unstable.

Thus I would think the critical radius would be some fraction of ##0.079## m, related to the q value one would aim for. However this is apparently not the case, so what am I missing? I feel like I am misunderstanding the probabilities involved, but some pointers would be greatly appreciated.

Thanks!

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# I Critical radius for nuclear reactions

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