Extended theorems for signal induction in particle detectors VCI 2004

W Riegler

doi:10.1016/j.nima.2004.07.129

Abstract

Most particle detectors are based on the principle that charged particles leave a trail of ionization in the detector and that the movement of these charges in an electric field induces signals on the detector electrodes. Assuming detector elements that are insulating and electrodes with infinite conductivity one can calculate the signals with an electrostatic approximation using the so-called ‘Ramo theorem’. This is the standard way for the calculation of signals e.g. in wire chambers and silicon detectors. In case the detectors contain resistive elements, which is, e.g. the case in resistive plate chambers or underdepleted silicon detectors, the time dependence of the signals is not only given by the movement of the charges but also by the time-dependent reaction of the detector materials. Using the quasistatic approximation of Maxwell's equations we present an extended formalism that allows the calculation of induced signals for detectors with general materials by time dependent weighting fields. As examples, we will discuss the signals in resistive plate chambers and underdepleted silicon detectors.

Highlights

The currents induced on grounded electrodes by moving charges can be calculated with static weighting fields using Ramo’s theorem [1]
In case the volume between the electrodes contains conductive material in addition, which is the case for e.g. resistive plate chambers or underdepleted silicon detectors, we need a further extension of the theorem, which is the subject of this paper
Ε(x,s) σ (x,s) x2(t) x1(t) q -q where eeff ð~x; sÞ 1⁄4 eð~x; sÞ þ sð~x; sÞ=s and rextð~x; sÞ is an ‘externally impressed’ charge density. This equation is equal to the Poisson equation for electrostatic problems with an effective permittivity, so knowing electrostatic fields for a given permittivity we find the time-depended fields for a scenario with conductive materials by replacing e by e þ s=s and performing the inverse Laplace transform

Summary

Introduction

Be calculated by time-dependent weighting fields as shown in Ref. In case the volume between the electrodes contains conductive material in addition, which is the case for e.g. resistive plate chambers or underdepleted silicon detectors, we need a further extension of the theorem, which is the subject of this paper. This problem was already discussed in Ref. [3], for this conference a more practical ‘recipe’ formulation of the theorems is presented. W. Riegler / Nuclear Instruments and Methods in Physics Research A 535 (2004) 287–293

Formulation of the problem

Induced signals

Extended theorems

Example

Summary