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We derive the full expression for the shape of the charge spectrum thatresults from the illumination of a photo-multiplier tube. The derivation is forlow intensity illumination with constant gain, a common condition for mostnuclear and particle physics applications. Under these conditions, it is shownthat an analytic expression can be formulated that describes details of thespectrum including the pedestal and dark noise with excellent fidelity to allowstatistical fits to data. The derivation and full formula using either Gaussianor Poisson models for gain, and its limiting forms under various simplifyingassumptions are presented with strategies on their use. The analyticdescription can be used to formulate data acquisition strategies to performprecise absolute calibration of photo-multipliers, the digitizers, and the dataacquisition system.

Statistics of the Charge Spectrum of Photo-Multipliers and Methods for Absolute Calibration