A brief overview of existence results and decay time estimates for a mathematical modeling of scintillating crystals

Abstract

Inorganic scintillating crystals can be modelled as continua with microstructure.For rigid and isothermal crystals, the evolution of charge carriers becomes in this way described by a reaction-diffusion-drift equation coupled with the Poisson equation of electrostatic. Here, we give a survey of the available existence and asymptotic decays results for the resulting boundary value problem, the latter being a direct estimate of the scintillation decay time. We also show how to recover various approximated models which encompass also the two most used phenomenological models for scintillators, namely, the kinetic and diffusive ones. Also for these cases, we show, whenever it is possible, which existence and asymptotic decays estimate results are known to date. KEYWORDS entropy methods, existence of solutions of PDE, exponential rate of convergence, reaction-diffusion-drift equations, scintillators MSC CLASSIFICATION 35K57; 35B40; 35B45 INTRODUCTIONA scintillator crystal is a material which converts ionizing radiations into photons in the frequency range of visible light, hence its name. It acts as a true "wavelength shifter" and in such a role is used as radiation sensor into high-energy physics, in medical imaging and in security applications. 1 The physics of scintillation, which is a complex multi-scale phenomenon (see, e.g., Vasil'ev and Getkin 2 ) can be described within a continuum approach at three scales: at a microscopic scale, the incoming energy E generates a population of charged energy carriers which moves in straight directions for few nanometer 3 and whose density N = N(E) can be found by the means of approximated solutions of the Bethe-Bloch equation. 4-7* These energy carriers wander and migrate within a greater region either generating other energy carriers or recombining with emission of photons h𝜈. In the process, some energy is lost, and a scintillator is a material in which