# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Inhomogenous Plasmas  
For plasmas with a spatial dependency of the plasma parameters, there is a fundamental challenge in disentangling the contributions of \(n_{\rm e}\) and \(\
u_{\rm m}\) to the changes in the cavity volume averaged complex permittivity \(\bar{\epsilon}\). This paragraph discusses the factors that impede the separation of these plasma properties. When low-pressure plasmas are investigated, MCRS yields a cavity volume averaged electric-field-squared-weighted electron density \(\bar{n}_{\rm e}\). This parameter is defined as follows [14; 39],  
\[\bar{n}_{\rm e}=\frac{\iiint_{V_{\rm P}}n_{\rm e}(\mathbf{r})|\mathbf{E}_{1} (\mathbf{r})|^{2}d^{3}\mathbf{r}}{\iiint_{V_{\rm cav}}|\mathbf{E}_{1}( \mathbf{r})|^{2}d^{3}\mathbf{r}}. \tag{23}\]As users of the diagnostic method are familiar with this definition, \(\tilde{v}_{\text{m}}\) is defined here in a similar fashion for the collision frequency of the electrons,  
\[\tilde{v}_{\text{m}}=\frac{\iiint_{V_{\text{P}}}v_{\text{m}}(\mathbf{r})|\mathbf{ E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}{\iiint_{V_{\text{Cav}}}|\mathbf{E}_{1}( \mathbf{r})|^{2}d^{3}\mathbf{r}}. \tag{24}\]  
In the two prior MCRS studies of atmospheric-pressure plasmas [18; 19], it was assumed that the electron dynamics are uniform over the plasma volume. This assumption has been used because only one resonant mode has been used and the spatial information concerning the electron dynamics got lost due to the nature of the MCRS approach. In the analysis of these experiments, the contributions of the electron density and the effective collision frequency to the changes in resonant behavior of the resonant mode have been separated by Equations (15) and (16), which have been derived with the assumption of a uniform plasma volume. After substituting Equations (9) and (10) back into Equations (15) and (16)--purely to assess the error when using Equations (15) and (16) in a non-uniform plasma--Equations (25) and (26) are obtained. In general, this substitution is not valid because Equations (15) and (16) already assume that \(n_{\text{e}}\) and \(v_{\text{m}}\) are homogeneous. The relatively complicated and somewhat arbitrary definition and method of weighing of the collision frequency \(\hat{v}_{\text{m}}\) and the electron density \  
consequence of the division of the two plasma regions in the mode structure instead of a general rule as can be seen in the simplified relation (\(n_{\text{e,1}}=n_{\text{e,2}}\)) based on Equation (25) presented below,  
\[\hat{v}_{\text{m}}=\frac{\
u_{\text{m,1}}+\
u_{\text{m,2}}\frac{\
u_{\text{m,1} }^{2}+4\pi^{2}f_{1}^{2}}{\
u_{\text{m,2}}^{2}+4\pi^{2}f_{1}^{2}}\iiint_{V_{ \text{p,2}}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}{1+\frac{\
u_{\text {m,1}}^{2}+4\pi^{2}f_{1}^{2}}{\
u_{\text{m,2}}^{2}+4\pi^{2}f_{1}^{2}}\iiint_{V_ {\text{p,1}}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}}. \tag{27}\]  
The discrepancies between the results of the two definitions increase when the properties of the two regions become more diverse. For increasing differences between \(\
u_{\text{m,1}}\) and \(\
u_{\text{m,2}}\) the ratio converges to almost 5 orders of magnitude from unity for this arbitrary parameter range. This means that the spatial profile of the collision frequency greatly influences the obtained \(\hat{v}_{\text{m}}\). Even for low-pressure situations where the collision frequency does not exceed the applied microwave frequency [6], this discrepancy in the outcome for the two definitions might be extremely relevant. For this reason, users of MCRS should hedge concerning the obtained electron collision frequency, even more so concerning the electron temperature as it requires an additional assumption: the type of Electron-Energy Distribution Function (EEDF). Secondly, the sensitivity of the outcome in terms of the electron density to spatially dependent plasma parameters is explored. The difference between the two definitions is visualized in Figure 12b.