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Would the effect be different than if the nuclear power plant were bombed with traditional explosives?

Would the atomic bomb neutralise any potential radioactive effect of the obliteration of the nuclear power plant?

Would it make it even worse than the atomic explosion itself?

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The power plant would fall down. Some fraction of its walls, grounds, cooling water, stored fuel, and spent fuel would be volatilized into airborne dust.

The fuel assemblies would almost certainly not melt down, unless you make some very favorable assumptions about the type of power plant and the geometry of the blast damage. Fission occurs when nuclear fuel exceeds a critical density. Power plants generally operate by constructing a supercritical fuel mass with some amount of neutron-absorbing "control rods" interspersed. Meltdowns happen if the control rods are removed (as at Chernobyl) or if the reactor's heat can't be removed while it's operating (as at Fukushima). But if you just knocked the whole assembly down, it would tend to become less dense.

Nuclear weapons work by very carefully engineering the fissionable material to maintain its density for as long as possible during the blast, which has a lot to do with its shape, its surface, the geometry of the chemical explosives which bring the pieces together, and other secret technical details. You can't generate a high-power fission chain reaction by starting from a fuel assembly meant to operate in a controlled way and knocking it over.

Any vaporized fuel or waste would eventually fall back down out of the atmosphere, a phenomenon known imaginatively as "radioactive fallout." The waste would be more dangerous than the fuel.

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The increase in damage -- over and above that caused by the bomb itself -- would be dominated by the dispersion of radioactive isotopes from spent fuel stored onsite, and/or from the reactor core.

Both the core and any spent fuel rods contain fission products: new nuclei formed by the nuclear reactions. The radioactive intensity and long half lives of some of the fission products makes them much more dangerous in aggregate than fallout from the nuclear bomb itself. The available mass of such isotopes further magnifies the potential for deaths and grave health effects from their release.

The extent to which a bomb could release reactor fission products depends on its yield, distance from the reactor building, and the plant's design.

It's a difficult case to analyze, but I judge it likely that even with the most conservative reactor building designs, a fusion bomb exploded at short range would likely rupture the reactor pressure vessel; spent fuel storage will be much more vulnerable.

The bomb need not vaporize the fuel rods: when they are removed from cooling water, their internal heat will cause vaporization and combustion, dispersing the radionuclides through the atmosphere.

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