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. The fission fragments consist of atoms more lightweight than the original heavy atom. The sum of their masses do not precisely equal that of the heavy atom, even while accounting for the incident neutron. The difference (mass difference) consists of ejected neutrons and the release of binding energy. The neutrons leave the reaction at high speed, and may collide with other heavy atoms in a phenomenon known as "fission capture". This could result in nuclear fission, forming the basis of a chain reaction.

Average generation time

The average generation time is the average time from neutron emission to fission capture. The neutrons travel only short distances, on the order of 10cm (the diameter of a critical mass). An average neutron's speed varies around ca. 10,000 km/s, resulting in a timescale on the order of 10 ns. This quantity is often referred to as a shake.

Effective neutron multiplication factor

The effective neutron multiplication factor or κ, is the average number of neutrons, of those released in one fission, which cause another fission. The remaining neutrons either fail to induce fission, or are never absorbed and exit the system. The value of κ for a combination of two masses is always greater than that of its components. In some cases its value is equal to the sum of the component κ values. The magnitude of the difference depends on velocity and distance, as well as physical orientation. Passing a small circle through a small round hole produces a particularly large κ: like firing a fissile 'bullet' into a shaped fissile target.

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 beginning 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 it 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 , 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 reaction

A chain reaction can be started by a single neutron or by neutrons from a single spontaneous fission. However, it may well happen that a single such event does not start a chain reaction. It is possible to calculate the probability that it will.

For example, suppose 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.

Predetonation

Detonation 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 (where the bomb blows itself apart before it is ready to produce a large explosion). 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.

History

The concept was first developed by Leó Szilárd in 1933. He supposedly thought of the idea while waiting at a red light. He then patented the concept the following year.

Leo Szilárd attempted to create a chain reaction using beryllium and indium in 1936 but was unsuccessful. In 1939, Leo Szilárd and Enrico Fermi discovered neutron multiplication in Uranium, proving that the chain reaction was possible.

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 Gabon, Africa in September 1972.

See also

• chain reaction
• critical mass
• criticality accident
• nuclear physics
• nuclear reaction
• nuclear weapon design
• Nuclear Criticality Safety
• Nuclear reactor physics
• Nuclear Chain Reaction Animation
• Einstein's Letter to President Roosevelt - 1939
• Annotated bibliography on nuclear chain reactions from the Alsos Digital Library