General expression of the calibration curve for the CR-39 radon track detectors

The CR-39 detectors are widely used as passive radon dosimeters, as well as in physics laboratories or for industrial applications. For what concerns radon monitoring, the calibration curve which is usually adopted corresponds to a linear relation between the actual etched track density and the track density counted by an automatic acquisition system. This linear calibration provides very accurate radon exposure assessments in a restricted range of etched track density, however it neglects the effect of the tracks overlapping that becomes as relevant as the track density increases. In the present work the mathematical expression of the area covered by a set of uniformly distributed tracks is deduced. This result allows then to infer the probability that the system acquires the right track density, providing a general calibration curve for a solid state radon track detector. The process of track production and reading routine is also simulated adopting a Monte Carlo approach, showing that the obtained results are in agreement with the function proposed as calibration curve. Moreover, a linear dependence between the track average area and the rate parameter $\beta$ has been observed. Finally a semi-empirical correlation based on the previous results is propose