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In a modern nuclear reactor for example a PWR there are multiple containment systems which prevent the release of radioactive material into the environment and shield the environment from the radiation.

Here is a quote from the wikipedia article about this:

The reactor vessel is the first layer of shielding around the nuclear fuel and usually is designed to trap most of the radiation released during a nuclear reaction. The reactor vessel is also designed to withstand high pressures.

I think that the radiation in this part is shielded partially by the walls of the pressure vessel and partially by the water it contains.

However how much percent of the radiation is shielded by the water alone and how much by the walls? I.e. how do the shielding effects of the walls and the water compare to each other (roughly) and why?

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Julia, your question is not entirely clear as to what radiation you are talking about. For instance, there is no shielding against neutrinos. The worst offenders are actually not radiation but fission products (iodine-131, caesium-137, strontium-90 etc.) that may leak into the environment and decay emitting $\beta^-$ (electrons) and $\gamma$ radiation. –  Deer Hunter May 13 '13 at 13:11
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The primary purpose of the water moderator is not to shield the outside world from radiation (although this is one benefit). The water moderator has two primary tasks in a PWR or BWR:

  1. High energy (~2 MeV) neutrons released in fission are down-scattered (or moderated) into the thermal energy range (~0.25 MeV). The cross section for fission is much higher at this energy level allowing a sustained chain reaction for power production.
  2. The water is used as the working fluid to transfer the thermal energy released in fission for power generation. The energy released from fission cannot be harnessed directly for electricity. In a BWR, fission heats and boils the water which drives a steam turbine to produce electricity. In a PWR, fission super-heats pressurized water which transfers heat to a secondary boiling water loop that drives a steam turbine to produce electricity.

Because of its properties as a moderator, the water tends to scatter most high energy neutrons to lower energies. Generally, any neutrons escaping the reactor will have been moderated down to a low energy and will therefore not have a long mean-free-path in the outside world. This is the shielding benefit provided by the water moderator - but it is not the purpose of the water in a reactor.

The reactor vessel, most likely steel, is also not primarily designed for radiation shielding. The purpose of the reactor vessel is to contain the moderator in the proper geometry to sustain the chain reaction for power generation. In the case of a PWR, the vessel must also contain the moderator at high pressure to prevent boiling of the super-heated water.

Since the steel vessel is sealed, it prevents fission products and activated isotopes from escaping the system and it can also provide some shielding from gamm and x- rays release in fission. This is not the purpose of the reactor vessel though, only an added benefit.

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