## 1 Introduction  
In low-pressure, low-temperature plasmas, the study of the positive ion current collected by the Langmuir probe is very important, as the smallness of the positive ion current collected by the Langmuir probe when it is polarized negatively with respect to the plasma potential allows local diagnosis of the plasma parameters with very low disturbance to the plasma. In other words, the positive ion sheath that is formed around the probe shields out the influence of the probe, which in plasmas with low plasma density is crucial [1; 2; 3; 4; 5; 6; 7]. On the other hand, many surface technological processes that use plasmas depend on the ion current that reaches the surface, and thus the control of the ion current is essential in this kind of technology. Among these processes, we have plasma-assisted chemical vapor deposition (PACVD), ion implantation, etching, surface coating, thin films, nanotechnology, etc. [8; 9; 10; 11; 12; 13]. In the semiconductor industry, which is a major application of PACVD, the properties of the plasma must be closely examined in order to control the energy and the frequency of the ion impacts against the surface, so that the optimal conditions for ion implantation are obtained [8; 10]. Therefore, both theoretical analysis and experimental studies of the ion sheath surrounding the surface to be treated are important. The ion current collected by a Langmuir probe has been extensively studied from a theoretical point of view. There are two main groups of theories to explain the fall of the ions towards the probe. The orbital theories, of which orbital motion limited (OML) is the most frequently used, study the movement of the ions in orbits around the probe that are calculated using the applicable laws of conservation. Some ions have a trajectory that does not intersect with the probe surface and orbit back to the plasma, so that not all of the ions are collected by the probe [14; 15; 16]. The radial theories, the first being the Allen-Boyd-Reynolds (ABR) theory for a spherical Langmuir probe, study the plasma as a fluid, so that all the ions fall radially towards the probe and therefore all the ions that enter the sheath are collected by the probe [17]. The Allen-Boyd-Reynolds theory, which was soon adapted to cylindrical Langmuir probes by [PERSON] [18], is valid for ions that have an ion temperature that is negligible when compared to the electron temperature, so that the parameter \(\beta=T_{+}/T_{e}\), with \(T_{+}\) and \(T_{e}\) as the positive ion and electron temperature, respectively, can be given a value of \(\beta=0\). The cylindrical radial model has been extended by the authors to \(\beta\
eq 0\)[1; 2; 19; 20; 21; 22; 23; 24]. Both orbital and radial theories are used to diagnose plasmas to obtain plasma parameters such as plasma density. However, the values obtained using the two theories can be very different, the one predicted by the radial theory being up to an order of magnitude higher than the one predicted by the orbital theory. When the values of the plasma parameters deduced from these theories are compared to the values of the plasma parameters obtained using the much higher electron current in the electron saturation zone, depending on the plasma conditions, it is found that either the radial or the orbital theories are consistent with the well-established electron saturation zone theory. This implies a paradox when the ion current is used in plasma diagnosis, given that the appropriate theory that should be used is not known a priori before it is used in plasma diagnosis [1; 4; 25]. Actually, in many situations which depend on the plasma discharge power and the pressure, values for the ion current collected by the probe between the two theories are measured in experiments [5; 26; 27; 28]. In two previous papers [26; 28], the authors showed an experimentally observed transition in the positive ion current values that are derived from the radial theories and the orbital ones, as a function of the \(\beta\) parameter.