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Criticality is the event of a nuclear reactor reaching a self-sustained chain reaction. This means no external source of neutrons is required to sustain fission in the reactor core. Criticality is a precondition of producing useful amounts of energy from the reactor. The term can also be used in relation to nuclear weapons.
When a reactor is "going critical", it is not about to blow up. This is a popular misconception in fiction. It's actually just starting to produce power and operate normally because the reactor has just begun to support an ongoing chain reaction of fissions. Nuclear detonation is caused by supercriticality under extreme conditions.
There are two kinds of criticality: delayed criticality and prompt criticality. Delayed criticality means that the chain reaction is self-sustained due to delayed neutrons, which come from the decay of short-lived fission products, but would not be if those fission products were neglected. All nuclear reactors operate in the delayed criticality regime. Prompt criticality means the reaction is self-sustaining using only the neutrons released immediately from fission, regardless of the neutrons from fission products. Nuclear bombs operate in the far reaches of the supercritical regime. Nuclear reactors are designed to prevent prompt criticality from happening in a reactor, in that it can lead to core meltdown and/or explosion (which should also be limited by containment facilities).
Accidentally assembling a critical mass of fissile material can lead to sudden bursts of radioactivity and in some cases even explosions. Such events are called criticality accidents. Several such accidents have happened in nuclear weapons programs, as well as in civilian reprocessing facilities. The worst in recent history was an accident in Tokaimura (Japan) in 1999, which killed two workers.
The term explosion, used above, does not refer to nuclear explosions. Nuclear detonation requires a mass far beyond critical of weapons-grade fissionable materials (much higher-grade than are used in power plants), under extremely demanding conditions of pressure and uniformity. These conditions are so difficult to produce through purposeful engineering that the idea of them arising randomly in the course of an accident strains credulity to the breaking point. Accidental assembly of such an extreme supercritical mass is theoretically possible, but vanishingly unlikely, as the effects of criticality would disrupt the assembly long before such a mass could be formed. Criticality, particularly in accidents, may, however, result in chemical explosions (rapid combustion of materials initiated directly or indirectly by the energy release) or physical explosions (failure of pressure containment, such as in steam pipes).