Analytical description of the time-over-threshold method based on the time properties of plastic scintillators equipped with silicon photomultipliers

A new high-granular compact time-of-flight neutron detector for the identification and energy measurement of neutrons produced in nucleus-nucleus interactions at the BM@N experiment, Dubna, Russia, at energies up to 4 AGeV is under development. The detector consists of approximately 2000 fast plastic scintillators, each with dimensions of 40$\times$40$\times$25 mm$^3$, equiped with SiPM (Silicon Photomultiplier) with an active area of 6$\times$6 mm$^2$. The signal readout from these scintillators will employ a single-threshold multichannel Time-to-Digital Converter (TDC) to measure their response time and amplitude using the time-over-threshold (ToT) method. This article focuses on the analytical description of the signals from the plastic scintillator detectors equipped with silicon photomultipliers. This description is crucial for establishing the ToT-amplitude relationship and implementing slewing correction techniques to improve the time resolution of the detector. The methodology presented in this paper demonstrates that a time resolution at the 70 ps level can be achieved for the fast plastic scintillator coupled with silicon photomultiplier with epitaxial quenching resistors.