A look at the curve of binding energy shows that if you break up a heavy nucleus (heavier than iron) into smaller pieces you have more nuclear binding energy per nucleon. Of course, because the strong nuclear force is very strong indeed, nuclear fission of most atomic nuclei requires more energy than the excess nuclear binding energy that is released by the fission. For some unstable radiocative isotopes, however, such as uranium-235 and plutonium-239, only a small amount of energy is required for fission to occur, and large quantities of energy are released.

The first explanation of nuclear fission was given by Lise Meitner and Otto Frisch in 1939, based on experiments performed by Meitner in collaboration with Otto Hahn, who, oddly enough, got sole credit for the discovery by being awarded the 1944 Nobel Prize in Chemistry. The Frisch-Meitner theory uses Niels Bohr's liquid drop model of nuclear structure to explain the phenomenon. Atomic nuclei are thought of as being similar to drops of a liquid, the nucleus being held together by surface tension provided by the strong nuclear force, just as a droplet of water is held together by hydrogen bonds in the water molecules. When the nucleus is externally excited, such as by absorption of a neutron, the extra energy causes the nucleus to deform, but the strong nuclear force tries to make the nucleus return to its former equilibrium shape. The inertia of the nucleons, however, causes the nuclear shape to overshoot equilibrium, causing it to oscillate. The excited nucleus usually radiates away the energy when it does this, and the oscillations die down. However, if the energy supplied is sufficient, the deformation may cause portions of the nucleus to become further apart from each other, far enough that the strong force attracting the protons in the nucleus to each other is no longer that effective in keeping them together, and the electromagnetic repulsion between the positively charged protons becomes stronger. The nucleus then breaks apart.

For U-235, when even a slow thermal neutron is absorbed, it turns into U-236, which is so unstable that it almost instantly explodes in the manner described above into two fission fragments and 2-3 extra neutrons, releasing total energy in the vicinity of 188 MeV. This is a truly astounding amount of energy to be released in a single atomic-scale event; a typical combustion reaction such as the burning of gasoline releases only a few eV per molecule. A similar process occurs in the fission of plutonium and similar elements. For the far more common uranium-238 however, the neutrons to be used must have an energy of at least 1 MeV for fission to occur; neutrons released in nuclear fusion reactions have energies in this range, which is why most high-yield hydrogen bombs have natural uranium casings.

Once it was seen that nuclear fission normally liberates extra neutrons, it became clear that it could be possible to make a self-sustaining reaction if one could ensure that at the neutrons liberated by one fission would in turn cause at least one other fission afterwards, causing a chain reaction. If in a mass of fissionable material the fission doesn't do this, the reactions will die down and eventually stop; this mass is called subcritical. If one fission produces yet another fission on the average, the reaction proceeds in a controlled manner and constant energy is released; such a mass is called critical, and it is the process that powers nuclear reactors. If more than one fission is caused on the average by another fission, that mass is called supercritical, and this generally results in an explosion such as that of an atomic bomb. Fortunately, the conditions for such a runaway nuclear chain reaction to occur are only present in highly enriched uranium or plutonium; even uranium fuel rods are only sufficiently enriched to provide at most a critical sustained chain reaction.

dido's write-up is correct except in saying that:

"If more than one fission is caused on the average by another fission, that mass is called supercritical, and this generally results in an explosion such as that of an atomic bomb. Fortunately, the conditions for such a runaway nuclear chain reaction to occur are only present in highly enriched uranium or plutonium; even uranium fuel rods are only sufficiently enriched to provide at most a critical sustained chain reaction."

In accordance with the rules of neutron multiplication, the number of neutrons in each successive neutron generation will increase only when the reactor is super-critical. Thus, one cannot increase power in a nuclear reactor without being super-critical. (Well, technically, one can due to subcritical multiplication, but not very much.) If nuclear reactors were as described above, then they could only ever remain at a constant power or decrease in power, never being able to increase. It is a common misconception that being super-critical is a terrifically bad idea, when in fact the ability to be super-critical is fundamentally necessary to the operation of nuclear reactors. Super-critical does not mean run-away.

A Civilization advance.
The development of a reasonably accurate model of the structure of the atom led physicists to attempt to intentionally split one in two. When a large, unstable atomic nucleus splits--fissions--the result os two or more smaller, more stable nuclei. Other by-products of this fission are a tremendous amount of energy and fast neutrons, plus a lingering, deadly radioactivity. The first intentional use of nuclear fission was in warfare, where the incredible power of an unconfined fission reaction produced an explosion rivaled only by the largest known volcanic eruptions.
Prerequisites: Mass Production and Atomic Theory.
Allows for: Nuclear Power.

One of the things that I find the most interesting about nuclear fission is that a full-fledged nuclear fission chain reaction probably could never occur naturally, and it could be argued that before July 16th, 1945, an explosive nuclear fission reaction never occurred anywhere else in the cosmos---unless there is alien life as intelligent and foolish as our own.

It should be qualified that nuclear fission happens all the time, and that as you are reading this, somewhere in your home, some uranium atom inside of steel, or wood, or your own body is splitting in two. And it would most likely follow that somewhere somewhat richer in uranium, some neutrons from such a fissioning bump into some other atoms and cause them to divide. Apparently, it is already established that some deposits of uranium have had enough radioactivity in them to change their isotope fractions and to create a large amount of heat. But that is not quite the same as the type of reaction you get from having a critical mass of pure, enriched isotope coming together in an atomic device.

The conditions with which nuclear fission can occur are fairly limited, and seem to be only possible by intelligent manipulation. Although elements usually occur somewhat close together, they are usually not in the pure, concentrated form that would be necessary to start a chain reaction. In other words, all that uranium, or thorium, is going to be mixed up in chemical compounds that absorb the neutrons, or they are just going to be physically scattered. Of course, this might not be the case in some alien planet's chemistry, but it would be hard to think up a chemical or physical system that would concentrate a rare element like uranium or thorium close together. Beyond the chemical improbability, there seems to be no plausible way to concentrate one isotope, such as Uranium-235, away from its chemically identitical fellow isotopes.

The only scenario I can think of for such a chain reaction to occur naturally would happen right after the explosion of a supernova. In such an environment, there would be plenty of very radioactive, short lived heavy elements, including transuranium elements. If such elements were to form into a large sphere, they could be joined close together, especially if there was also a large radial momentum that would work as a centrifuge. In these circumstances, it may be possible that a single type of radioactive isotope could be drawn to each other, leading to a chain reaction.

And if this improbable situation is not the case (and how would we know if it was), we can take something like pride in being the first creatures in the cosmos to discover and unleash the force of nuclear fission.

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