Scintillation light and photodetector gain

In a limited linear-response regime, a light yield converts deposited energy into an average number of scintillation photons. The yield depends on material, wavelength spectrum, temperature and particle ionization density. Dense tracks can quench light production, so equal deposited energy does not always produce equal light across particle types. A measured electron-equivalent calibration therefore must not be transferred to a heavily ionizing recoil without checking the response model.

Photons can be absorbed, reflected, wavelength shifted or lost before reaching the active surface. Define an optical transport fraction and then a quantum efficiency for primary photoelectron production in the selected spectral band. A separate collection factor accounts for transporting those primary electrons into the multiplier when relevant. In a real calculation these factors can depend on wavelength and position; the scalar product below is an intentionally averaged benchmark, not a universal factorization for arbitrary optical geometry.

In a photomultiplier tube, a primary electron is accelerated toward a dynode and releases secondary electrons, which are multiplied through successive stages. Mean gain is approximately the product of the mean stage gains. It is powered by the applied bias; the large electrical output is not energy extracted solely from the incident optical photon. The anode charge provides a calibrated proxy for the initial number of collected photoelectrons within the multiplier’s linear range.

Mean gain and gain fluctuations are different properties. Statistical multiplication introduces an excess variance, and dark counts, afterpulses and electronic noise can add signals unrelated to a new scintillation photon. Silicon photomultipliers introduce further complications such as finite cell occupancy, recovery, optical crosstalk and saturation. Increasing voltage may change several of these quantities together. A gain number alone is therefore not enough to predict detection threshold, dynamic range, energy resolution or timing precision.

If collected photoelectrons follow a Poisson distribution and each receives identical noiseless gain, their fractional rms fluctuation is the inverse square root of the mean count. Constant gain cancels from that ratio. The assumption can be useful after independent thinning of a Poisson light source, but it is not a consequence of fixing a deterministic photon number. Real scintillation production, optical nonuniformity and multiplication fluctuations can produce different variances.

One can summarize multiplication noise with a defined excess-noise factor and add an independent output-referred electronic charge noise term. Better gain can suppress that electronic term relative to signal, yet it cannot lower the irreducible photon-count term at fixed collected light. Gaussian full width at half maximum is approximately 2.355 times rms width only when the peak is adequately Gaussian. At very small photoelectron counts, threshold losses and discrete charge peaks require a different likelihood rather than a Gaussian percentage.