# Nuclear chain reaction

File:Einstein.jpg
Albert Einstein's letter to President Roosevelt in 1939 about his concern, about (Nuclear chain reactions)
Click for closeup of letter

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).

The only known natural self-sustaining nuclear chain reaction was at Oklo.

The first artificial self-sustaining nuclear chain reaction was initiated by the Metallurgical Laboratory, led by Enrico Fermi, in a racquets court below the bleachers of Stagg Field at the University of Chicago on December 2, 1942 during the Manhattan Project.

The equation for U-235 fission may be summarized as follows: U-235 + 0 N1 = fission fragments + 2.5 0N1 + 180 MeV. The equation for Pu-239 fission is generally as follows: Pu-239 + 0 N1 = fission fragments + 3 0N1 + 200 MeV. 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 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.

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.
• k = 1 (critical mass): starting with one fission, we have on average one fission per unit time; this unit of time is the time it takes for the neutron to hit another nucleus; the distance is something like the diameter of a critical mass; e.g. the speed may be 10 000 km/s and the distance 10 cm, then the time is 10 ns.
• k > 1 (super-critical mass): starting with one fission, we have on average in every generation of the chain reaction, say again once every 10 ns, a multiplication of the number of fissions by a factor a. This cannot continue, of course: k decreases when the amount of fission material that is left decreases, or when the remaining fission material is torn apart. In a nuclear weapon this may e.g. happen after 80 generations of 10 ns, is 800 ns, when there have been ${\displaystyle 1+k+\cdots +k^{80}={{k^{81}-1} \over {k-1}}}$ fissions.

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 percent 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.