Spontaneous fission, decay rates, and critical mass

In "How did the Aborigines get to Australia?" (Message 57): Nonukes quite correctly points out the errors in the preceding, namely, that the phenomenon of induced fission, which proceeds extremely rapidly on the successful absorption of a free neutron, is very different from spontaneous fission, which is an extremely rare phenomenon compared to the typical natural decay rate of a radioactive substance.The statements "You can't affect decay rates without affecting fission decay" and "When you reduce the nuclear binding energy or lower the barrier for radioactive decay to occur, and reduce the decay rate, you would increase the occurrence of all forms of radioactive decay, including fission" and "the critical mass required to reach a sustained reaction is reduced" collectively imply some sort of correlation between decay rates and critical mass. This is arguably not the case. Here is the raw data for fissionable isotopes of plutonium, extracted from Wikipedia (Critical mass and Isotopes of plutonium)

Isotope    Critical mass            Half-life       Proportion of
(bare unbounded sphere)                 spontaneous fission events
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Pu 238        9 kg                    87.7 y       1.9 * 10^ -7 %
Pu 239       10 kg                 24100   y       3.1 * 10^-10 %
Pu 240       40 kg                  6560   y       5.7 * 10^ -6 %
Pu 241       12 kg                    14.3 y       2.4 * 10^-14 %
Pu 242       85 kg                375000   y       5.5 * 10^ -4 %
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Dividing the proportion of spontaneous fission events by the half-life (where half-life is inversely proportional to activity) would, in addition, give an indication of how many spontaneous fission events are to be expected per unit time per unit mass.It turns out that, as far as I can see (at least for plutonium), there seems to be essentially no correlation between the following paired sets of data:1) half-life and proportional spontaneous fission rate
2) half-life and critical mass
3) proportional spontaneous fission rate and critical mass
4) absolute rate of spontaneous fission and critical mass(Feel free to make your own scatter-plots, of course.)As Nonukes points out, the rate of spontaneous fission events has nothing to do with the size of a fissionable isotope's critical mass (all you really need is one fortuitous event in a critical mass to start a chain reaction), and even less to do with the overall decay rate of a fissionable isotope that is not undergoing a chain reaction.An analogy would be what is known as the "percolation threshold" for feedback systems, such as a forest fire, or a field of upright dominoes. It's a given that the probability of a tree spontaneously catching fire is extremely low, but, if that probability isn't precisely zero, a spreading conflagration would be inevitable solely dependent on tree flammability, asymptotic density, and size of the forest.

And yet Zen Deist was apparently arguing for lower critical masses on the basis of lower nuclear stability by causing more induced fission events due to more spontaneous fission events, something which both you and I disagree with. Anyway, he seems to have since changed gears somewhat, claiming that enrichment due to lower nuclear stability would also play a factor. Well, we will see, I hope... I agree that it's not a particularly great example, given that the numbers are all over the board just for those five closely-similar isotopes. You are quite right in that a difference of just one neutron can have profound consequences for many aspects of nuclear behavior; but in the absence of magically increasing overall nuclear decay rates (or equivalently having a complete theory of nuclear stability), perhaps it could be seen as the next best thing for the sake of testing for any such correlations with respect to critical mass.

Well, that goes without saying; it's practically a tautology. A decay rate is (by definition) inversely proportional to stability. You may be right, but how could we directly test that? Given that (1) spontaneous fission decay events tend to be an extremely small proportion of all decay events for a typical isotope in question, and (2) even among the slightly different isotopes which I put forth as a test sampling, the range of variation of the absolute rate of spontaneous fissions varies by a whopping 6 orders of magnitude, even though the overall decay rates vary by no more than 4 orders of magnitude, the available evidence seems to be inconclusive at best. In fact, the half-lives and the absolute rates of spontaneous fission exhibit less correlation than any of the other four pairings of data which I alluded to earlier.For all we know, the spontaneous fission decay rates themselves may be extremely sensitive to changes in nuclear stability, and not always in the same direction. To initially achieve a chain reaction, perhaps. If the spontaneous fission rate was absolutely zero, it would never get started in the first place (barring an external source of thermal neutrons). But to sustain a chain reaction is another thing entirely, and that's what k=1 criticality (as opposed to k<1 PFM-supress all forms of spontaneous decay, including the already-comparatively-rare spontaneous fission events while leaving the specifics of induced fission unchanged, and the chain reaction process itself would be (to an accuracy of many decimal places) completely unaffected. Again, for all we know, the specifics of neutron emission rates via fission (such as average number of neutrons released, and/or the energy spectrum of those neutrons, not even to mention rates of subsequent neutron absorption and/or rates of subsequent induced fissioning) may very well be extremely sensitive to small changes in overall nuclear stability, and not always in the same directions. Edited by DWIII, : re: insufficient evidenceEdited by DWIII, : fixed html-character bug