# Multigap Resistive Plate Chambers for Time of Flight Applications
## 2 Description of MRPC Detector
For example, the commonly used gas mixture consists of a large part (even up to 90%) of tetrafluoroethane (C\({}_{2}\)H\({}_{2}\)F\({}_{4}\)) with a few percentages of isobutane (i-C\({}_{4}\)H\({}_{10}\)) and sulfur hexafluoride (SF\({}_{6}\)). Even though the detector's geometry is simple, as shown above, the physics in MRPCs is quite complicated. The incident particles will interact with the gas molecules in the MRPC detector. Each interaction will deposit a certain amount of energy, which will cause the ionization of extranuclear electrons and generate clusters of primary electron-ion pairs. Each cluster contains one or more electrons, the number of which depends on how much energy is deposited during the collision of the incident particles. The electrons generated by ionization will drift towards the anode under the high electric field, while the positive ions will drift towards the cathode. In the process of drifting towards the anode, the electrons will obtain energy from the electric field and trigger secondary ionization. Similarly, the secondary ionized particles are accelerated again and trigger new ionization. Here is the cascade of electron multiplications known as Townsend avalanches [34]. When the number of electrons in a cluster is large enough, the aggregation of electrons will create a considerable local field, whose magnitude is comparable to the applied field but in the opposite direction. At this time, the avalanche will saturate and stop growing, which is called the \"space charge effect\" [35]. The diffusion effect [36], due to collisions between electrons/ions and the gas molecules, should be discussed for the real conditions. More details about the detector physics and simulation can be found in [32; 37; 38]. As ionized electrons and ions move in the gas gap, a time-varying current signal will be induced on the PCB's readout strips or pads. The induced current signal [39] can be derived by [PERSON]'s theorem [40] and given by:  
\[i(t)=\frac{E_{w}\cdot v}{V_{w}}e_{0}N(t) \tag{6}\]  
where \(N(t)\) is the number of electrons at time \(t\), \(e_{0}\) is the electron charge, \(v\) is the drift velocity which can be simulated by Magboltz, \(E_{w}\) (weighting field) is the electric field obtained by setting the selected electrode to potential \(V_{w}\) and others to 0. Once the currents induced on the electrodes have been calculated, and all the resistive, capacitive and inductive elements of the MRPC system have been introduced, the final currents can be determined [41], and detailed discussions about the signal propagation and signal integrity can be found in [42; 43; 44]. For the multi-conductor transmission lines readout structure in MRPCs, the impedance [45; 46] has been well studied through experimental tests and simulations to help achieve impedance matching with the electronics and optimize the detector structure. An approximate formula for the impedance of a multi-stack MRPC [46] is given by:  
\[Z_{0}=\frac{\frac{386.73}{\sqrt{\epsilon_{eq}+7.35}}\log\frac{6.03 nh_{2}+0.67 7(n+1)h_{1}}{w-0.554}+60\sqrt{\frac{w}{nh_{2}+(n+1)h_{1}}}+37.8\log\frac{h_{1 }}{h_{2}}}{ns+0.895} \tag{7}\]  
where \(n\) is the number of gas gaps in a stack, \(ns\) the number of stacks, \(h_{1}\) the glass thickness, \(h_{2}\) the gap thickness and \(w\) the strip width. Considering the inhomogeneous dielectric mediums between the paired strips of the MRPC, the equivalent dielectric constant can be estimated as follows:  
\[\varepsilon_{eq}=\left(\frac{nh_{2}\sqrt{\varepsilon_{gas}}+(n+1)h_{1}\sqrt{ \varepsilon_{g}}+2h_{Mylar}\sqrt{\varepsilon_{Mylar}}}{nh_{2}+(n+1)h_{1}+2h_{ Mylar}}\right)^{2} \tag{8}\]  
Figure 1: Schematic sketch of a multigap resistive plate chamber (MRPC) detector. The avalanches (red triangles in the gaps) induce negative signals (blue pulse) on the top anode electrodes and positive signals (red pulse) on the bottom cathode electrodes. where the relative dielectric constant of the gas (\(\varepsilon_{gas}\)) is set to 1, and \(\varepsilon_{g}\) is the relative dielectric constant of the glass.