A schematic nuclear fission chain reaction. 1. A uranium-235 atom absorbs a neutron, and fissions in two new atoms (fission fragments), releasing three new neutrons and some binding energy. 2. One of those neutrons is absorbed by an atom of uranium-238, and does not continue the reaction.

Another neutron is simply lost and does not collide with anything, also not continuing the reaction. However one neutron does collide with an atom of uranium-235, which then fissions and releases two neutrons and some binding energy. 3. Both of those neutrons collide with uranium-235 atoms, each of which fission and release between one and three neutrons, which can then continue the reaction.

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A nuclear chain reaction occurs when on average more than one nuclear reaction is caused by another nuclear reaction, thus leading to an exponential increase in the number of nuclear reactions. An uncontrolled chain reaction within a sufficiently large amount of fission fuel (critical mass) can lead to an explosive energy release and is the concept behind nuclear weapons. The chain reaction could also be adequately controlled and used as an energy source (nuclear reactor).

Some fission equations, showing averages:

- U-235 + neutron = fission fragments + 2.52 neutrons + 180 MeV
- Pu-239 + neutron = fission fragments + 2.95 neutrons + 200 MeV.

This excludes 10 MeV for unusable and hardly detectable neutrinos.

When a heavy atom undergoes nuclear fission it breaks into two or more fission fragments. Each of these fission fragments is an atom of a more lightweight element on the periodic table of the elements.

Thus a neutron can cause a nuclear fission reaction which releases ca. 2.5 or 3 neutrons. Crucial is how many of these cause another fission reaction. The effective neutron multiplication factor k is the average number of neutrons from these 2.5 or 3 cause another fission reaction, as opposed to neutrons produced by the fission which are being absorbed without causing a new fission, and those travelling out of the system. The value of k for a combination of two objects is always more than the larger of the two k values of each separately. It may or may not be more than the sum of these two: for two objects far apart it is little more than the larger value, for an object inserted in a hole in the other, as the assembled parts of a gun method weapon, it may well be more than the sum.

The average generation time is the average time from neutron emission to fission capture. This time is very short: the distance is something like the diameter of a critical mass; the speed may be ca. 10 000 km/s and the distance 10 cm, so the time is of the order 10 ns = 1 shake.

We can distinguish the following cases:

- k < 1 (sub-critical mass): starting with one fission, we have on average a total of 1/(1 − k) fissions. Any begin of a chain reaction dies out quickly.
- k = 1 (critical mass): Starting with one free neutron, the expected value of the number of free neutrons resulting from it is 1 at any time; in the course of time there is a decreasing additional probability that the beginning chain reaction has died out, which is compensated by the possibility of multiple neutrons still being present.
- k > 1 (super-critical mass): starting with one free neutron, there is a non-trivial probability that is does not cause a fission or that a beginning chain reaction dies out. However, once the number of free neutrons is more than a few, it is very likely that it will increase exponentially. Both the number of neutrons present in the assembly (and thus the instantaneous rate of the fission reaction), and the number of fissions that have occurred since the reaction began, is proportional to e(k − 1)t / g, where g is the average generation time and t is the elapsed time. This cannot continue, of course: k decreases when the amount of fission material that is left decreases; also the geometry and density can change: the geometry radically changes when the remaining fission material is torn apart, but in other circumstances it can just melt and flow away, etc.

When k is close to 1, this calculation somewhat over-estimates the 'doubling rate'. When a uranium nucleus absorbs a neutron it enters a very-short-lived excited state which then decays by several possible routes. Typically it decays into two fragments, fission products, typically isotopes of Iodine and Cesium, with expulsion of a number of neutrons. The fission products are themselves unstable, with a wide range of lifetimes, but typically several seconds, and decay producing further neutrons.

It is usual to split the population of neutrons which are emitted into two sorts - 'prompt neutrons' and 'delayed neutrons' Typically, the 'delayed neutron fraction' is less than 1 % of the whole. In a nuclear reactor the variable k is typically around 1 to have a steady process. When a value of k = 1 is achieved when all neutrons produced are considered the reaction is said to be 'critical'. This is the situation achieved in a nuclear reactor. The power changes are then slow, and controllable e.g. with control rods. When k = 1 is achieved counting only the 'prompt' neutrons, the reaction is said to be 'prompt critical' - much shorter doubling rates can then occur, depending on the excess criticality (k-1). The change in reactivity needed to go from critical to prompt critical (ie the delayed neutron fraction) is defined as a dollar.

The value of k is increased by a neutron reflector surrounding the fissile material, and also by increasing the density of the fissile material: the probability for a neutron per cm travelled to hit a nucleus is proportional to the density, while the distance travelled before leaving the system is only reduced by the cube root of the density. In the implosion method for nuclear weapons, detonation takes place by increasing the density with a conventional explosive.

The probability of a chain reactionSuppose a fission caused by a neutron hitting a nucleus produces 3 neutrons (i.e. 2 extra). Also suppose k > 1. The probability that a neutron causes a fission is k / 3. The probability that a free neutron does not cause a chain reaction is (1 - k / 3) (no fission at all) plus the probability of at least one fission, while none of the 3 neutrons produced causes a chain reaction. The latter has a probability of k / 3 times the cube of the first-mentioned probability that a free neutron does not cause a chain reaction. This equation can be solved easily, giving a probability of a chain reaction of

which ranges from 0 for k = 1 to 1 for k = 3.

For values of k which are little above 1 we get approximately k-1.

PredetonationDetonation of a nuclear weapon involves bringing fissile material into its optimal supercritical state very rapidly. During part of this process the assembly is supercritical, but not yet in optimal state for a chain reaction. Free neutrons, in particular from spontaneous fissions, can cause predetonation. To keep the probability low, the duration of this period is minimized and fissile and other materials are used for which there are not too many spontaneous fissions. In fact, the combination has to be such that it is unlikely that there is even a single spontaneous fission during the period of assembly. In particular the gun method cannot be used with plutonium, see nuclear weapon design.

HistoryThe concept was first developed by Leó Szilárd in 1933 which he then proceeded to get a patent on the concept the following year. Leo Szilárd attempted to create a chain reaction using beryllium and indium in 1936 but was unsuccessful. The first artificial self-sustaining nuclear chain reaction was initiated by the Metallurgical Laboratory, led by Enrico Fermi and Leó Szilárd, in a racquets court below the bleachers of Stagg Field at the University of Chicago on December 2, 1942 during the Manhattan Project. The only known natural self-sustaining nuclear chain reactions were discovered at Oklo in Septemeber 1972.