Nuclear fission is the process by which a large nucleus is split into two smaller nuclei called “fission fragments”.
Fission occurs when a large nucleus absorbs a neutron. When the neutron is absorbed, the nucleus is deformed (picture a big water balloon getting smacked by a smaller one) and splits in two.
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Unstable fission fragment
Click here for a nice diagram of this process Link
Click here for an animation of this process: Link
When the nucleus splits it forms two fission fragments (Sr90 and Xe143 in the example above). Since these newly-formed nuclei have a randomly determined ratio of neutrons to protons they are almost always very unstable and are therefor highly radioactive. For example, when neutrons split Uranium-235, it can break into 200 different isotopes of 35 different elements, all of which are highly radioactive.
The greatest health threat posed by nuclear energy is “nuclear waste”. Nuclear waste contains high levels of unstable fission fragments, which usually have very unstable neutron to proton ratios, and are extremely radioactive for many, many years. Oddly, the fuel that goes into a nuclear power plant is not all that radioactive, but the “leftovers” are.
A fissionable material is one that contains large nuclei that can be split by neutrons. There are hundreds of fissionable isotopes, mostly of larger nuclei at the bottom of the periodic table.
A fissile material is one that contains nuclei that can be split by neutrons, but also release more neutrons during the fission event, which can then split other nuclei in a sustained chain reaction. There are only three fissile isotopes: U-235, Pu-239, and U-233.
The fission process releases energy because each proton and neutron in the larger nucleus is more massive than it is in the smaller nucleus. As each proton and neutron reduces it mass, the mass is converted to energy via the famous e= mc2 equation.
Some isotopes, such as U-235 and Pu-239 release two or more neutrons when they split. If these newly-released neutrons then split other U-235 or Pu-239 nuclei, then a nuclear chain reaction can be started.

Nuclear
chain reaction
Click here for an animation of a nuclear chain reaction: Link Link
U-235 is a fairly rare isotope of Uranium, the more common isotope U-238 does not release neutrons when is splits, and therefor does not produce chain-reactions. In order to get a chain-reaction going in Uranium, the percentage of U-235 must be increased. Uranium with an artificially elevated percentage of U-235 is called “enriched” Uranium. This is used in nuclear power plants. If the percentage of U-235 is high enough, then it can be used in a bomb and is called “weapons-grade” Uranium.
In order to achieve a sustained nuclear chain-reaction, you must have enough U-235 or Pu-239 to ensure that when a nucleus splits, the extra neutrons that it produces will also split other nuclei.
If there is too little material present, then more neutrons escape into the environment at a great rate than they split other atoms. This is called a sub-critical mass. When this happens the reaction quickly stops.
If there is just enough material present to maintain a constant rate of fission (each atom that splits is able to cause one other atom to split), then the reaction will be self-sustaining. This is called a critical mass.
It the each atom that splits causes more than one atom to split, then the temperature of the fuel skyrockets. This is called a super-critical mass and this is how nuclear bombs work.
There was an unfortunate accident at Los Alamos during the Manhattan Project. A scientist was attempting to determine the critical mass of Plutonium when he made a bad mistake. Click here to learn the weird details: Link
In a fission-type nuclear bomb, two sub-critical masses are forced together by precisely timed explosive charges. In the high-pressure environment created by the explosives, the two smaller masses of nuclear fuel unite to form a super-critical mass, splitting many atoms, and converting several kilograms of mass into energy.

Click here to watch video clips of nuclear bomb tests: Link Link
In the process splitting all those atoms, a large amount of radioactive fallout is produced. Radioactive fallout is comprised of both fission fragments from the blast, and dust particles that have been transformed into unstable nuclei by absorbing neutrons from the blast.
In a nuclear reactor, a critical mass is maintained, and the energy it releases is used to boil water into steam, which is then used to turn turbines.
It is possible to control the rate of the chain-reaction by absorbing neutrons before they can split other atoms. Control rods are pieces of a metal (usually Cadmium) that are efficient at absorbing neutrons. When the control rods are raised, more neutrons find their targets and the reaction rate increases. When the rods are lowered, fewer neutrons find their targets and the reaction rate decreases.
It is not possible for a nuclear reactor to explode in the same way that a nuclear bomb does. A bomb uses explosive charges to keep the reaction highly super-critical.
It is possible for the reaction in a nuclear reactor to get
so hot that it melts the fuel rods (where the nuclear fuel is housed) and
control rods. When this happens it is
called a nuclear meltdown. If the
control rods melt, then there is no way for the operators to control the
reaction and a great deal of heat can be produced. Since the reactions normally occur underwater
(the water slows the neutrons so that they are more likely to be absorbed), a
steam explosion may accompany a meltdown.
Reactors in the
The greatest drawback to nuclear power is the highly radioactive waste products (fission fragments) in the spent fuel rods. These have to be stored in a protected area for tens of thousands of years. There is nothing that can be done to lessen the radioactivity of the waste.