A Photons per unit of Energy in Cherenkov radiation?

If I have a beam of protons that go into a block of water, how can I know how many photons will be produced along the beam path?. I'm assuming all the photons have the same energy.

I know that the energy deposition will follow a Bragg Curve, and I think that energy deposition is probably proportional to the amount of photons, is it?.

I was given some experimental data. A beam of 20 MeV produces ~550 photons per proton per MeV per mm^2 in it's first 2mm of water.

I tried using the Frank-Tamm formula to try to reproduce this result, I assumed that v≈c, that permeability was that of vacuum and that the frequency was 789THz (my supervisor told me to assume al photons had this frequency), and then dividing over the energy of a single photon with that energy to get the amount of photons, but... it didn't work.

I was also told that in the Bragg Peak, at approximately 4.25 mm, the amount of photons will have doubled... but I want to estimate it for any energy at any point between the beginning of the Bragg curve and the Bragg Peak, but short of actually learning Geant4 and doing the simulation, I have no idea how to do this.
There are a couple of odd things here. At 20 MeV a proton is slow. The approximation ##v\approx c## doesn’t work and there won’t be Cherenkov radiation. There will be some random photons due to ionization but they will have a broad distribution. Scintillators are made to produce light from ionizing radiation but water is not a good material for that - it leads to photons but if you want to measure them there are better materials.

What is the overall goal of this project?

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