Analytical model for the photomultiplier single photoelectron response including the electron back-scattering contribution

Many models exist to describe the single photoelectron response of single photon counting photomultipliers. Generally to describe the spectral region between the fully amplified primary photoelectron peak and the electronics pedestal an ad hoc function is used (often an exponentially modified gaussian) attributing this region to `noise'. In this paper, following the physical description of back-scattered primary photoelectrons at the first dynode described in the "The Photomultiplier Handbook" by A. G. Wright published by Oxford University Press, we derive an analytical function describing these partially amplified primary photoelectron at the first dynode. This function depends only on intrinsic parameters of the photomultiplier such as the gain at the first dynode and the intrinsic resolution of the dynode chain following the first. Furthermore, analytical descriptions of the fully amplified peak and very low charge signals are derived. The model has been successfully validated with data from two different photomultipliers acquired with a low-noise amplifier.