Derivation of a simple analytical expression of the gain of pure argon filled CAT proportional counters

In previous publications, we described the intrinsic properties of CAT detectors, formed as a unique or a collection of individual single holes (gas proportional pixel detector). In our last paper, we have shown that the calculated gain of the avalanche along the electric field lines was dependent on the distance of photon absorption with respect to the symmetry axis of a single hole. On the contrary, the measured gain did not follow this dependence and it was suggested that the gain of the detector was similar to the one measured along the hole symmetry axis, irrespective of the shape of the off-axis electric field lines. We present here a simple calculation of the on-axis gain of such structures that corresponds to the measurements and takes into account the detector geometry.