An approximate method for analysis of double probe characteristic in the absence of ion saturation

Ion current to a double probe (DP) with cylindrical leads is subject to sheath expansion effects, so probe's volt-ampere characteristic (VAC) differs significantly from a simple \textit{tanh}-like shape, which is valid only in the limit of an infinitely large ratio of probe radius, $r_{p}$, to Debye length, \textit{$\lambda $}$_{D}$: \textit{$\xi $}$_{p}=r_{p}$/\textit{$\lambda $}$_{D}$. Thus, a commonly employed simple method, in which straight lines are fitted to VAC at zero and large positive (or negative) bias voltages, and the ordinate of the intersection point is used as $I_{sat}$\textit{ = 2$\pi $ r}$_{p }L_{p} e^{-1/2 }N_{is}^{ }$\textit{$\surd $T}$_{e}/M_{i}^{+}$, often results in overestimation of the calculated ion density, $N_{is}$, by a factor \textit{$\eta $}$_{i}^{-1}=N_{is}$ / $N_{i}^{+ }\sim $ 2 -- 3, where $N_{i}^{+}$ is the true positive ion density. The shape of the VAC and therefore value of \textit{$\eta $}$_{i}$ are also strongly affected by plasma's electronegativity, \textit{$\beta $} = $N_{neg}$ / $N_{e}$. In this work, for the first time, we present an approximate analytical expression for \textit{$\eta $}$_{i }$as a function of \textit{$\beta $} and\textit{ $\xi $}$_{ps}=r_{p}$\textit{ / $\lambda $}$_{D}$ ($N_{is})$, which is valid in the wide range of parameters: \textit{$\beta $} = 0 -- 40, and \textit{$\xi $}$_{pi}=r_{p}$\textit{ / $\lambda $}$_{D}$ ($N_{i}^{+})$ = 0.1 -- 10. We obtained this expression in O$_{2}$ and CF$_{4}$, for $T_{e}/T_{neg}$ = 30, by solving ``radial motion'' equation in the presence of singly-charged electronegative ions [H. Amemiya et al., Plasma Sources Sci. Technol. \textbf{8}, 179 (1999)], and numerically calculating a family of DP VAC. We also obtained a formula for \textit{$\eta $}$_{i}$ ($M_{i}^{+}$, \textit{$\xi $}$_{ps})$ in electropositive plasma for \textit{$\xi $}$_{pi}$ = 0.1 -- 50 and a wide range of ions, from H to Xe.