Time response functions and avalanche fluctuations in resistive plate chambers

The time response function of RPCs is derived. First, primary electron distributions in the RPC gas gap are discussed. Then the exact expression for the fluctuations of an avalanche starting with a fixed number of primary electrons is derived, using Legler's model of avalanche multiplication in electronegative gases. By means of the Z-Transform formalism, the primary electron distributions and avalanche fluctuations are then combined and an analytic expression for the RPC time response function is derived. The solution is further used to discuss signal threshold and attachment effects. Finally, the time response function is evaluated for several primary ionization models.