First, neutrons hitting the nucleus of most atoms will either do very little or will actually use up energy. Only "fissile" materials will generate more energy, such as some forms of uranium and plutonium.

When an explosive chain reaction happens, two things cause it to end quickly:

1. The fissile atom's nucleus changes when it releases energy. The atom becomes a different element. The elements that result are non-fissile. As a result, the fissile material is used up, leaving no fuel for the chain reaction.
2. In order to have enough neutrons hit other atoms in the fissile core, that core has to be pretty dense. When the explosion happens, the fissile material spreads out in a cloud of superhot plasma. This plasma is not dense enough to maintain fission.

During the explosion, each fission event causes >1 additional fission events. Let's say it causes 1.1 more.

As a rough rule of thumb, if something is increasing by X% per step, it doubles in 70/X steps. So in this case, at a 10% increase per step, it doubles every 7. And since each step here takes a tiny, tiny fraction of a second, this doubling can happen many, many times within a slightly less tiny fraction of a second. A rough estimate for the step time here is about 10 nanoseconds (that's 1/100 million of a second), so you're doubling in less than 100 ns, you've doubled more than ten times within 1000 ns = 1 microsecond, and you've doubled more than a hundred times within 10000 ns = 10 microsecond. (It turns out that you don't actually get this far, for reasons we'll see in a second.)

If you naively continue this process) you've doubled more than a thousand times within 100 microsecond (= 0.1 millisecond). 1000 doublings is 2^1000 = 10^300 or so. Since there aren't even 10^300 atoms in the entire Universe, this obviously can't be the case. In fact, even 2^100 = 10^30 or so is beyond the number of atoms in a supercritical chunk of normal nuclear materials.

But nothing about our math is wrong here. Instead, something must be wrong with our assumption that this process continues the way we've modeled it here. And the reason it doesn't - and hence the answer to your question - is that the developing nuclear blast starts to split its fuel apart. To make it explode in the first place, you needed to compress the fuel into a small area, so that the neutrons emitted by each fission event can be captured by other atoms of your fuel. Once the fuel isn't compressed into a small area, the number of fission events caused by each fission event (our 1.1 above) goes down, and we can no longer model the explosion's progress as a raw exponential curve. Once it falls below 1, the reaction starts to slow, and if it's much below 1, it slows down quickly.

In practice, once it is below 1 in a nuclear explosion, you've already got a very violent explosion blasting the fuel apart. So it very quickly drops far below 1, and any energy release past that point isn't caused by the initial chain reaction. This isn't quite the end of the the energy release, since the decay products from the initial chain reaction are also exceptionally radioactive and themselves quickly decay, but even those quick decays are minor-ish contributors to a nuclear explosion because their time frames are much longer than the duration of a nuclear blast (usually, anyway).

Neutrons only trigger more decay when they hit the right kinds of atoms at the right speed. This is reasonably likely when you have a dense clump of enriched uranium. Once that clump starts exploding, it gets way less dense. Neutrons created then miss the other fissile atoms and just fly off, failing to continue the reaction

How close each nuclei is to each other matters. The neutrons get emitted randomly and if everything is packed together tightly that doesn't matter because no matter where they go they're going to hit another fissile nucleus. Once the reaction begins to emit energy it's going to push that core apart and now some of the nuclei being emitted are going to miss a fissile nucleus and just go away into the surrounding environment. Eventually you also convert enough fissile nuclei to non-fissile and they might just absorb the neutron taking it out of the reaction without a new fission. But that won't matter for a bomb, it becomes more of an issue in reactors.

Only a select few materials are fissile, and then, only if they exist in sufficient purity. ^235 U and ^239 Pu are the most common fissile isotopes used in fission weapon cores. To detonate a fission core, usually the core is contained within an outer shell of conventional high explosive specifically shaped into explosive lenses (based on the positions of the detonators and speed of propagation of the detonation wavefront through the explosive). The multiple simultaneous explosions occurring within these explosive lenses create a symmetrical shockwave which acts to evenly implode the fissile material core, rapidly increasing its density and initiating the prompt-critical nuclear chain reaction in as many locations as possible, simultaneously. Neutrons released by every individual fission event at the atomic level may strike the nuclei of nearby fissile atoms, and there is a certain probability that this causes that nucleus to undergo fission itself, releasing more neutrons and thus propagating the chain reaction. While all this is happening, there are two things that are working to arrest the chain reaction: one is that the primary fissile core material is fissioning into child materials which have a much lesser (or even negligible) probability of further fission when struck by the reaction neutrons. Essentially, decay products are being created which are radioactive, but not fissile, so no further chain reaction is possible. The second thing is that the incredible energy release associated with the fissioning core is actively working to blow that core apart, both reducing it below the critical density first established by the conventional explosive trigger, and physically dispersing it and so reducing the probability of neutron collisions which result in fission. A goal of nuclear weapon design is to maintain supercritical density for as long as possible in order to maximize fission yield, but 100% fission of the core is not attainable. Some portion is always just blown apart to become a contribution to the radioactive fallout instead of actively contributing to the destructive yield.

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