## 6 An Equivalent Gas Conductivity for RIC Phenomenon  
At the beginning of the EM analyses aimed at evaluating the insulation performances of SF\({}_{6}\) for the TL of ITER NBI, an equivalent \(\sigma_{\text{G}}\) accounting for the RIC to be used in the EQS model, was defined from Equation (4) [37]:  
\[\sigma_{\text{G}}=\frac{L}{A}\frac{I_{sat}}{\Delta V}, \tag{17}\]  
where \(L\) is the electrodes gap, \(A\) is the cross-section of TL, and \(\Delta V=1\) MV is the voltage. This expression is only valid for a simple electrode configuration; thus, a generalization is required. In this section, an attempt to obtain a relation linking gas conductivity to ion-pair generation rate \(S\) is carried out by noticing that if current density data are taken from a DDR simulation and plotted versus the corresponding electric field intensity; a recurrent pattern can be found to describe the link between these two quantities. In Figure 3a, an example of this kind of plot is reported, taking current density and electric field values at each node of the tetrahedral mesh implemented for the DDR simulation. As can be observed, two leading trends emerge in the points' distribution, described by two main data populations clustering around straight lines with different slopes. From a qualitative point of view, the distinctive feature of these two populations is ionic concentration in the region of the computational domain where data are collected: points around the lower slope line in each dataset belong to regions with lower concentration than points around the higher slope line. This can be confirmed by the fact that for very low values of \(S\), the higher slope pattern tends to vanish, being completely absent if \(S\) is ideally set to zero. In Figure 3b, a comparison of the population for two different values of \(S\) is given. The red dataset corresponds to \(S=3\times 10^{7}\) m\({}^{-3}\) s\({}^{-1}\), while the blue data are taken at \(S=10^{8}\) m\({}^{-3}\) s\({}^{-1}\). As can be observed by comparing the two datasets, the slope of both lines increases with \(S\). In particular, the growth is linear in the case of the lower slope, while the higher slope shows a nonlinear behavior that tends to stabilize for high values of ion-pairs' generation rate. All the previous considerations motivate the idea to exploit this behavior of gas conductivity to obtain an alternative simulation approach by extrapolating an expression to be enforced in place of Equation (7), which has the non-trivial drawback of requiring to fit experimental data to a functional form defined by nine different parameters, which can be extremely difficult and finally lead to large confidence intervals on the identified parameters. Furthermore, the computational burden of such an alternative approach is still comparable to a standard EQS, since the only change in the implementation is the imposition of a gas conductivity linked to \(S\), but constant inside the gas medium. This last aspect constitutes a fundamental advantage with respect to the DDR approach: indeed, despite its several  
Figure 3: **(a)** Example of \(J\)-\(E\) population clustering. **(b)** Example of \(J\)-\(E\) populations for different values of \(S\). interesting features and accuracy, the solution of two additional nonlinear drift-diffusion equations strongly affects its computational burden, possibly preventing a wide application in the field of nuclear fusion where the simulation of large-scale devices, e.g., the HVB inside the DTT Tokamak hall (see Figure 2), is often an unavoidable necessity [1]. To extract an explicit relation between conductivity and the generation rate, several DDR simulations were obtained for different values of \(S\).