Can you give us a little more information? Are you literally doing direct electron detection on the photographic emulsion, or, are you using anything else, such as avalanche amplification, scintillation, scintillation with photomultipliers, etc? From the scientific literature, which appears sparse on actual exposure calculation, photographic exposures on emulsions which are specifically not x-ray films from natural sources (lumps of radioisotopes) appear typically in the days to weeks kind of scale. You could do a back of the envelope calculation based upon a known source and a known film and
hope that any scaling towards a modern source and modern emulsion is linear, but this will almost certainly result in a lot of experimentation.
Personally? Had I no access to literature, I would start with a few assumptions and hope that that gets to a good enough ballpark:
Ignoring the spectral dependence of the film response for a moment, we can then assume that one electron absorbed fogs the film similarly to one photon being absorbed - after all, a photon being absorbed excites an electron to a state where it is free to move around (random walk) until captured by something else thus making a silver atom - or something like that. I'm not actually sure of the exact physical process, but it's something like that. I'm sure someone will correct that in due course. Anyway, if you supply the electron, that process can be bypassed. I'd then need to look at the statistics which govern the probability of (for a single silver halid crystal) a single photon being absorbed and ultimately fogging the emulsion. In other words, averaged over the film, what is probability of [electron capture / photons incident]. For a modern emulsion, I'd hazard a wild guess at 0.5? As said, I don't know the exact physical processes here, so for direct electron exposure, I'd start with 0.5. Following this, and given the above, you'd need to know the photon flux (photons per unit area per unit time) required to acceptably expose the film (I mean from a scientific perspective, not some pedantic arty farty eejit stating "the correct exposure is what I saw in my artistic vision"
) and divide your source intensity by this figure to leave you with units of time. Does that help in any way?
Edit: also, I just thought, typically electrons generated from radioactive decay (beta) are stopped by something like a thin layer of aluminium, so your exposure would be heavily affected by the properties of any layer deposited on the film on top of the emulsion. Something to keep in mind perhaps?
