# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Penetration of the Microwave Field into the Plasma  
As the approach to investigate the linearity of the resonance frequency shift--with that the perturbation of the resonant field--treated in the previous section is laborious, a more practical limit related to the conductivity of the plasma is investigated here. Previously the skin depth has been used as a parameter to limit the range of plasmas parameters which can be probed by MCRS [18; 25]. Figure 15: Investigation of the linearity of the relative resonance frequency shift \(\frac{\Delta f}{f_{1}}\) as a function of the change of the relative permittivity \(\Delta\varepsilon_{\rm r}\) of the plasma (\(r_{\rm p}=10^{-4}\) m) for eight resonant modes. The vertical dashed line at \(-\Delta\varepsilon_{\rm r}=1\) indicates the expanded putative limit of the MCRS diagnostic method and the circles indicate where the response is no longer linear. According to this limit, the plasma radius must be smaller than the skin depth for successful MCRS measurements. Assuming an unmagnetized plasma, Equation (22) for the skin depth \(\delta\) reduces to,  
\[\delta=\sqrt{\frac{\rho}{\pi f_{\mathrm{res}}\mu_{0}}}. \tag{32}\]  
For calculating \(\rho\), the following definition for the conductivity \(\sigma\) (\(\rho=1/\sigma\)) is used [55],  
\[\sigma=\
u_{\mathrm{m}}\varepsilon_{0}\frac{\omega_{\mathrm{pe}}^{2}}{\
u_{ \mathrm{m}}^{2}+4\pi^{2}f_{\mathrm{res}}^{2}}. \tag{33}\]  
Using Equations (4) and (7) while assuming \(\bar{\varepsilon}_{\mathrm{r,gas,1}}=\bar{\varepsilon}_{\mathrm{r,gas,2}}\) and \(n_{\mathrm{e}}=0\) m\({}^{-3}\) for the unperturbed state, the following equation can be obtained for the real part of the change in the permittivity \(\Re\big{(}\Delta\bar{\varepsilon}\big{)}\),  
\[\Re\big{(}\Delta\bar{\varepsilon}\big{)}=\varepsilon_{0}\frac{\omega_{ \mathrm{pe}}^{2}}{\
u_{\mathrm{m}}^{2}+4\pi^{2}f_{\mathrm{res}}^{2}}. \tag{34}\]  
This equation can be used to rewrite Equation (33) to,  
\[\sigma=\
u_{\mathrm{m}}\Re\big{(}\Delta\bar{\varepsilon}\big{)}. \tag{35}\]  
The obtained equation can subsequently be used to express the skin depth as a function of the electron collision frequency \(\
u_{\mathrm{m}}\) and the change in permittivity \(\Delta\bar{\varepsilon}\) with respect to \(\varepsilon_{0}\),  
\[\delta=\sqrt{\frac{1}{\pi f_{\mathrm{res}}\mu_{0}\
u_{\mathrm{m}}\Re\big{(} \Delta\bar{\varepsilon}\big{)}}}. \tag{36}\]  
The motivation to rewrite the skin depth in terms of a change in permittivity instead of the conductivity becomes evident when later in this paragraph the skin depth is compared with the results from the previous section. Figure 16 presents a graph of \(\delta\) as a function of \(-\Delta\varepsilon_{\mathrm{r}}\) for a variety of electron collision frequencies (\(\
u_{\mathrm{m}}~{}=~{}10^{10}\), \(10^{11}\), and \(10^{12}\) Hz). The data show that the skin depth decreases with an increase in \(\
u_{\mathrm{m}}\) and increasing \(-\Delta\varepsilon_{\mathrm{r}}\), i.e., plasma density. According to the limit described by the skin depth and the plasma radius \(r_{\mathrm{p}}\), the plasma parameter space represented by the area below a line does not suffer from a significant perturbation of the microwave field. Figure 16: The skin depth of a plasma for three collision frequencies as a function of the change in permittivity \(\Delta\bar{\varepsilon}\) with respect to \(\varepsilon_{0}\). Furthermore, the parameters in which the shift of the resonance frequency no longer scales linearly with a change in permittivity are indicated by circles for a variety of modes—red circles for plasma at an antinode and black circles for at a node—and plasma radii. In the previous section of this article, the change in relative permittivity \(\Delta\varepsilon_{\rm r}\) for which the shift of the resonance frequency starts to deviate more than 10 percent was determined for a range of plasma radii \(r_{\rm p}\) and several resonant modes.