I am thinking if it's possible to detect Cherenkov radiation in water caused by dissolved natural potassium (KOH) in it without using photo-multipliers.
The main question is does 1 beta particle generate 1 photon, or multiple? Specifically, how many photons of visible light should I expect from 1.33 MeV electron in water?
I know that blue light photon has energy of 2.76 eV, but I have no idea about efficiency of this process.
Answer
The Frank-Tamm formula previously mentioned by another user is indeed the correct one to use, however actually implementing it in order to come up with a final estimated number of blue photons is not trivial.
The standard simulation tool preferred by working physicists to perform these sorts of calculations is a library of C++ classes called Geant4: http://geant4.cern.ch/. It is a gargantuan package which simulates far more types of interactions than just Cherenkov radiation.
The teeny tiny little subsection which handles Cherenkov radiation can be found here: http://www-geant4.kek.jp/lxr/source/processes/electromagnetic/xrays/src/G4Cerenkov.cc. The code is written and maintained by and for the professional academic physics community, which means that if you can get it running and actually use it to produce a comprehensible output, the physics results themselves are very likely to be validated and error-free.
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