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There are many different radiation vectors from a nuclear reactor, so there are many different kinds of instrumentation.

Direct radiation is usually gamma and the best instrument is an ion chamber. In the case of an in-containment monitor, they make high range instrumentation that can be used to assess core damage. One that I am familiar with had a range of 1 x 107 rads per hour (1 x 105 grays per hour), and was a post Three Mile Island mandated requirement.

Direct radiation can also be neutron, but you need some kind of neutron detector to measure that, usually, one that detects neutron interaction with something else, such as by absorbtive reaction, activation reaction, or elastic scattering.

Indirect radiation falls into several categories, but generally involves activation of some material, such as the oxygen in the water to radioactive nitrogen, or leakage of mixed fission byproducts, such as krypton, xenon, iodine, cesium, strontium, barium, etc.

Often, there is a process or effluent radiation monitor that takes a sample of the stream and analyzes it for beta or gamma radiation. Depending on the application, a paper filter may be used to accumulate particulates, and a charcoal cartridge may be used to accumulate iodines.

There can also be area radiation monitors that look at areas within the plant. They are used to protect personnel. They are generally some kind of gamma detecting ion chamber, though of a lower range than the high range in-containment monitors.

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