# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Accuracy in \(\mathcal{V}\)  
In the current approach, the determination of the ratio of the obtained effective plasma volume and the effective cavity volume \(\mathcal{V}\) limits the accuracy in the electron density significantly [18,19]. The difficulty lies in determining the plasma occupied volume accurately and this is especially important since the resonant field strength is not uniform in the immediate region of the plasma  
Figure 12: The ratio between the plasma parameters, i.e., (**a**) the collision frequency of the electrons and (**b**) the electron density, obtained by the measurement approach used for collisional plasmas and the electric-field-squared-weighted averaged values as a function of the collision frequency in region 2 \(\
u_{\text{m,2}}\) for several \(n_{\text{e,2}}\). in these investigations. In these two MCRS studies, the field is--due to the perturbation of the resonant field by the concentric holes in the walls which allow the discharge and gas flow to enter and exit the cavity--weaker near the geometry which produces the plasma and becomes stronger towards the heart of the cavity. This means that the error in the volume translates more than linearly into an error in \(\mathcal{V}\) and subsequently in \(n_{\mathrm{e}}\). Possibilities to reduce the error in \(\mathcal{V}\) are conveyed below. Making the cavity appear 'virtually' larger for the microwaves by using a high permittivity material as concentric filling while the plasma remains concentrically on the axis, reduces the leakage of the microwave field out of the concentric holes in the top and bottom faces of the cavity. This approach will reduce the non-idealities in the resonant field. Therefore, the field will be more homogenous along the axial and radial directions near the holes in the metal structure. Hence, for small plasmas \(\mathcal{V}\) will scale more linearly with the \(V_{\mathrm{cav}}\) and this will improve the accuracy in \(\mathcal{V}\). Moreover, the presence of the high permittivity material will also lower \(f_{\mathrm{res}}\), which also reduces the lower detection limit of \(\
u_{\mathrm{m}}\) slightly (Equation (15)). An advantage of using a high permittivity material over increasing the dimensions of the cavity volume is that, as explained and applied in reference [14], \(\mathcal{V}\) remains large. Another method to reduce the non-uniformity of the resonant field caused by the holes in the cavity is the addition of reflecting waveguides to the design. More information relating to this topic can be found in, for example reference [51]. A less elegant approach to reduce the effect of the concentric holes on the resonant field is reducing the size of the holes. By doing so the resonant field is indeed less perturbed from the ideal resonant field profile. However, the smaller holes affect the plasma to a larger extent via the changes in the gas flow and plasma-induced electric fields. Reducing the height of the cavity to the point that plasma is longer than the height of the probing volume helps to increase the accuracy in \(\mathcal{V}\) as one dimension, i.e., the axial, no longer needs to be determined. Further advantages of this approach are: \(\mathcal{V}\) could become larger as the non-plasma occupied volume decreases, and smaller spatial variations of the plasma parameters in the probing volume can be expected. The latter could help to reduce the effect discussed in Section 3.3. Another advantage of this approach is that the plasma can be probed spatially by moving the cavity with respect to the plasma and using multiple positions to probe the discharge. In experiments in which the plasma is confined by, e.g., a glass tube, the volume occupied by the plasma is clearly defined. This translates into a reduced uncertainty in \(\mathcal{V}\). Approaches to correct the resonant field for the presence of the glass tube could be used to improve the accuracy even further [7; 28; 53].