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Would Cherenkov radiation occur at the explosion of a nuclear bomb? Suppose it would not be occluded by smoke or anything else for that matter.

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Of course, you could observe it in the water where the speed of light is much smaller than $c$ and electrons (beta-radiation) from the nuclear blast easily beat the light.

The air has index of refraction $1.0003$ so the speed of light is $0.9997c$. Note that $v^2/c^2$ is about $0.9994$ and when subtracted from 1, we get $0.0006$. The square root of it is about $0.025$ so charged particles get about 40 times heavier at this speed.

Clearly, this is about 40 GeV for protons or neutrons (and even more for nuclei), so you can't get this fast hadrons or nuclei in a nuclear blast. Even for electrons, 40 times the rest mass gives about 20 MeV which is, I believe, much more than the energy that an electron can ever get in any process of this kind. Moreover, fission really produces two smaller nuclei that carry the energy away, and the energy of a neutron that is created is just 2 MeV or so.

So you can't get Cherenkov radiation in the air by nuclear blast. Cherenkov radiation in the air is a strange animal, anyway: we usually talk about water or similar dense environments. How air usually enters these discussions is only via the supersonic analogy of the Cherenkov radiation - but when dealing with sound, it's not the real Cherenkov radiation. ;-)

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The index of refraction of air is 1.0003 at STP. Depending on the exact construction of the bomb in question there may be multiple wave fronts allowing for the possibility of much denser regions still relatively near the blast center. However, I doubt that this changes the conclusion. That said gas Cerenkov detectors with thresholds similar to that for dry air are standard tools in medium energy physics: tune the pressure or the gas mixture and dial in a threshold for particle ID. – dmckee♦ Jan 20 '11 at 22:48
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this was nonsense – Georg Jan 20 '11 at 23:37
Dear dmckee, the pressure waves in the air are spreading approximately by the speed of sound which is a million times slower than the speed of light. So long before the pressure increases, the gamma radiation - and right after that, the beta radiation and shortly afterwards other radiation - will get out of the blast at normal pressure (and temperature). – Luboš Motl Jan 21 '11 at 6:51
Only the beta particles could possibly cause Cherenkov because the particles need to be charged and alpha are too heavy. The energy required for beta in air is about 20 MeV, but the upper limit for beta rays from nuclear reactions is about 8 MeV. Possibly a theoretical chance if a particle was accelerated by the EM pulse? Also, the beta rays can be secondary radiation so may still be around when the denser shock wave passes, but they wont travel far in air. – Philip Gibbs Jan 21 '11 at 8:44
I thought the Cherenkov radiation observable in nuclear power stations was also in air? Is this water instead? Or is there a big difference between the reactions in a power station and a bomb? – Ruud v A Jan 21 '11 at 15:03
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