## 1 Introduction  
Radon (Rn) is a natural radioactive gas present in the environment. Two are the main isotopes: \({}^{222}\)Rn (radon) with a decay half-life of 3.823 days, and \({}^{220}\)Rn (thoron) with a half-life of 55.8 s. Both are carefully studied worldwide by the scientific community especially for radioprotection purposes [1, 2] and for geophysical phenomena tracking [3, 4]. These goals require measurement systems suitable for monitoring the gas isotopes. Various methods for Rn detection have been developed throughout the years, those based on solid-state nuclear track detection and electrostatic collection are the most widespread and used nowadays [5]. The first method utilizes plastic devices sensitive to \(\alpha\)-radiation emitted by Rn and decay products, for mean long-term monitoring [6]. In the second method, naturally ionized Rn daughter, inside a metallic chamber put to a high voltage, are electrostatically driven on to the surface of a silicon detector set to ground potential, enabling the collection of full alpha energies and obtaining high resolution \(\alpha\)-spectrometry. The electrostatic collection systems have optimal detection efficiencies; they allow the continuous detection of the Rn by distinguishing both gas isotopes with their progeny [7]. Currently, three are the main commercial devices for measuring environmental Rn concentrations based on electrostatic collection (by diffusion or pumping the gas in the detection chamber): RAD7 (Durrige Company, Billerica, MA, USA) [8]; RaMonA (University of Naples and University of Campania-Caserta, Italy) [9]; Radim3A (Tesla Company, Pruhonice, Czech Republic) [10]. Rn measurements and the development of new detection systems remain popular topics [11]. Using of tools for the simulation is a required and useful part for the complete evaluation and realization of such kind of Rn measurement systems. In particular, it is necessary to reproduce the geometry of the system, and simulate the \(\alpha\)-spectrum of the Rn with progeny and the correspondent detection efficiency for each involved radionuclides. This leads to being able to study the efficiency versus the different geometries of the system,in order to find the best experimental solution to be realized. Different Rn measurement systems can, hence, be developed for several purposes. Tools for the simulation of the particles passage through matter are based on Monte Carlo method, the major ones are FLUKA and Geant4 [12]. Both can handle complex 3D-geometries using combinatorial geometry algorithms. In literature, papers focused on Rn monitoring only use commercial devices for their purposes, without worrying about simulations [13; 14]; while papers focused on investigation of new Rn measurement systems only deal with the realization and the testing of them without handling the simulation [15; 16; 17]. Within the research framework of simulation of Rn detection systems, in particular those based on electrostatic collection, very few studies are published, e.g., a low-cost monitor for the thoron [18; 19] and radon [20] measurement was designed. These studies obtained results by using FLUKA or Geant4 with restrictions due to the gaps of both simulation tools. In [19] only non-ionized \({}^{220}\)Rn atoms are simulated with FLUKA, while in [18; 20] the efficiency of \({}^{220}\)Rn and the spatial distribution of \({}^{214}\)Po ions were determined using Geant4 with a constant electric field inside the metallic chamber. In fact, FLUKA does not allow to add the electric field, contrary to Geant4 where it is possible but constant or as a function of spatial coordinates. For these reasons, in the present work we introduce the COMSOL software, a powerful tool for the resolution of problems at partial differential equations, which simulates an electric field as gradient of the applied potential difference within metallic chamber and creates the correspondent map useful for Geant4 input [21].