# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion  
### Increased Accuracy in the Permittivity  
The high collisionality of the electrons in atmospheric-pressure plasmas suppresses the change in the real part of the permittivity, which is responsible for the shift in \(f_{\rm res}\). Hence, an increase in the accuracy of the determination of the resonance frequency was required for the transition towards probing atmospheric-pressure plasmas. The typical resolution in the change in the relative permittivity is \(\sim 10^{-3}-10^{-5}\)[9; 13; 39; 40]. However, a resolution of \(\sim 10^{-7}\) is reported in exceptional cases [7; 41; 42]. This value means that the standard deviation in the resonance frequency is less than 1-part-per-5-billion. In this article, we present an increase of the resolution in the real part of the permittivity by more than two orders of magnitude to \(\sim 4\times 10^{-10}\) with respect to these works. This improvement in resolution has been made possible by replacing the trusted transmission scheme [8; 31; 43; 44; 45] by measurements in reflection mode [14], improving the fitting algorithm [18], increasing the number of averages [15; 18; 19], increasing the number of probing frequencies [15; 18; 19], improving the mechanical stability of the cavity [19], introducing temperature-corrected apparent frequencies [19], and temporal averaging [15; 20]. To the best of the authors' knowledge, the resolution in the quality factor has not been specified earlier. In this article, the best accuracy in \(\Delta(1/Q)\) was \(\sim\)\(10^{-6}\). In this section, the influence of the number of averages and the frequency step size on the error of the determined resonance frequency and quality factor is covered to help other users of the diagnostic method to optimize the total duration of the experiment. The cavity--and the rest of the setup--used for this study is described in detail in reference [19] and had a \(f_{\rm res}\) of 3.51 GHz and a Q of 4\(\times 10^{3}\) for the TM\({}_{010}\) mode. The experiment was performed frequency-resolved with 4096 averages of the temporal responses per probing frequency and a frequency step size of 1 kHz over a range of 3.5075 to 3.5175 GHz. A typical spectral response for the TM\({}_{010}\) mode of the cavity is shown in Figure 2.  
The characteristic time \(\tau\) of the used resonant mode was \(\sim\)0.36 \(\upmu\)s. This means that only 0.4 % of the resonant field was still remnant after 2 \(\upmu\)s and that measurements with these intervals were assumed to be uncoupled. Within each temporal response, 100 uncoupled samples were obtained. Due to a technical limitation, the maximum sample interval of 100 ns was shorter than the response time of the cavity. For this reason, the duration of each temporal response for a single frequency was 200 \(\upmu\)s and contained 2000 samples while only 1 in 20 subsequent samples in the temporal response were used in the analyses. During the analyses, subsets of the experimental data were selected and used to emulate experiments for a range of frequency step sizes and a number of averages. The results presented here were obtained while 1 SLM He flow was passing through the cavity and were very similar to the results obtained without flow. In both experimental conditions, no discharges were produced. Note that in the past a larger standard deviation has been observed in experiments with flow compared to cases without flow. However, this problem has been solved by simultaneously improving the mechanical stability of the metal structure forming the cavity and its mounting. For comparative purposes, the coefficient of variation \(\hat{C}_{\mathrm{v}}\) of each emulated experiment is computed,  
\[\hat{C}_{\mathrm{v}}=\frac{s}{\bar{x}}, \tag{18}\]  
where \(s\) is the standard deviation and \(\bar{x}\) the mean of the parameter of interest. Figure 9 presents a graph in which \(\hat{C}_{\mathrm{v}}\) is indicated as a function of the frequency step size and the number of averages.