In nuclear fission, when a stable nucleus is bombarded by particles, it breaks into two unequal lighter nuclei. Energy is released in the process (exothermic process) and there is an emission of neutrons. 

Neutrons are good projectiles in this reaction, as because they have no charge they are less repelled by the nucleus. Neutrons emitted by fission are fast neutrons and with high energy levels of around 1 MeV. They can become slow neutrons or thermal neutrons, with  an energy level of 1 eV, if they lose part of their energy by colliding with the particles of a moderator.
The energy of a slow neutron is enough to split U-235 into two unequal parts.
The massive nuclei when split produce two new elements.
These nuclei tend to have mass numbers of between 50 and 82.
You can see an animated diagram of a nuclear fission reaction here.
The process can be described in two stages:

Capturing the neutron
The process of fission

The nuclear fission reaction is extremely useful because:
1.- A great deal of energy is released
2.- The process is self-perpetuating. The release of neutrons, two or three per split nucleus (about 2.5 on average), means that these can provoke more fission when they collide with other nuclei giving rise to a chain reaction.

The problems associated with the use of fission are also considerable.

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