First principles pulse pile-up balance equation and fast deterministic solution

Pulse pile-up (PPU) is an always present effect which introduces a distortion into the spectrum measured with radiation detectors and that worsen with the increasing emission rate of the radiation source. It is fully ascribable to the pulse handling circuitry of the detector and it is not comprised in the detector response function which is well explained by a physical model. The PPU changes both the number and the height of the recorded pulses, which are related, respectively, with the number of detected particles and their energy. In the present work, it is derived a first principles balance equation for second order PPU to obtain a post-processing correction to apply to X-ray measurements. The balance equation is solved for the particular case of rectangular pulse shape using a deterministic iterative procedure for which it will be shown the convergence. The proposed method, deterministic rectangular PPU (DRPPU), requires minimum amount of information and, as example, it is applied to a solid state Si detector with active or off-line PPU suppression circuitry. A comparison shows that the results obtained with this fast and simple approach are comparable to those from the more sophisticated procedure using precise detector pulse shapes.