# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 2 Theory
### Plasma's Permittivity  
The Drude-Lorentz model is commonly used to describe the complex permittivity of a plasma \(\tilde{\varepsilon}_{i}\) as [34],  
\[\tilde{\varepsilon}_{i}=\varepsilon_{0}\left[\tilde{\varepsilon}_{\mathrm{r,gas },i}-\frac{\omega_{\mathrm{pe}}^{2}}{v_{\mathrm{m}}^{2}+4\pi^{2}f^{2}}-i\frac {v_{\mathrm{m}}}{\omega_{\mathrm{pe}}^{2}}\frac{\omega_{\mathrm{pe}}^{2}}{v_{ \mathrm{m}}^{2}+4\pi^{2}f^{2}}\right], \tag{7}\]  
where \(\varepsilon_{\mathrm{r,gas},i}\) is the relative permittivity of the gas and is traditionally assumed to be unity, \(v_{\mathrm{m}}\) the momentum transfer collision frequency of the electrons, and \(f\) the frequency of the applied electromagnetic field. The electron plasma frequency \(\omega_{\mathrm{pe}}\) is given by [34]:  
\[\omega_{\mathrm{pe}}=\sqrt{\frac{n_{\mathrm{e}}e^{2}}{\varepsilon_{0}m_{ \mathrm{e}}}}, \tag{8}\]  
where \(n_{\mathrm{e}}\) is the electron density, \(e\) the elementary charge, and \(m_{\mathrm{e}}\) the electron mass. One of the assumptions in the Drude-Lorentz model is that the collision cross section does not depend on the energy of the electrons, and this is generally not true. In two of our works [18; 19] an effective collision frequency is used to circumvent this problem.  
The following two equations for \(\Delta f/f_{1}\) and \(\Delta(\frac{1}{Q})\) are obtained by substituting Equation (7) into Equations (5) and (6) while assuming \(\varepsilon_{\rm{r,gas,1}}=\varepsilon_{\rm{r,gas,2}}\) and \(n_{\rm{e}}=0\) m\({}^{-3}\) for the unperturbed state,  
\[\frac{\Delta f}{f_{1}}=\frac{e^{2}}{2\varepsilon_{0}m_{\rm{e}}}\frac{\iiint_{V _{\rm{cav}}}\frac{n_{\rm{e}}(\mathbf{r})}{v_{\rm{m}}^{2}(\mathbf{r})+4\pi^{2} f_{2}^{2}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}{\iiint_{V_{\rm{cav}}}| \mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}, \tag{9}\]  
\[\Delta\left(\frac{1}{Q}\right)=\frac{e^{2}}{2\pi\varepsilon_{0}m_{\rm{e}}f_{ 1}}\frac{\iiint_{V_{\rm{cav}}}\frac{n_{\rm{e}}(\mathbf{r})v_{\rm{m}}(\mathbf{ r})}{v_{\rm{m}}^{2}(\mathbf{r})+4\pi^{2}f_{2}^{2}}|\mathbf{E}_{1}(\mathbf{r})|^{2 }d^{3}\mathbf{r}}{\iiint_{V_{\rm{cav}}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3} \mathbf{r}}. \tag{10}\]  
As an example, in a radio-frequency driven plasma at 10 Pa, the collision frequency \(v_{\rm{m}}=0.2\) GHz ([36], p. 5) which is much smaller than the probing frequencies and can, therefore, be neglected. A typical collision frequency in an atmospheric-pressure plasma jet is \(\sim\)1 THz [37], which is significant with respect to the resonance frequencies used in MCRS experiments. In this case, the role of the collisions of the electrons is very important and must, therefore, be included in the analysis of MCRS experiments. So far only the temporal and spatial afterglow of atmospheric-pressure plasmas were investigated with MCRS [18; 19]. In these studies, the capability of MCRS to probe highly collisional plasmas with collision frequencies up to 0.5 THz was demonstrated [18]. In the literature, no MCRS studies of the bulk--in which the electron temperature and collision frequency of the electrons are higher than in the afterglows--of an atmospheric-pressure plasma are described. Figure 3 presents the change in the real and imaginary part of the relative permittivity with respect to \(\varepsilon_{\rm{r,gas}}\) as a function of the electron density for \(f=3.5\) GHz and a variety of collision frequencies. For \(v_{\rm{m}}=0\) Hz, the imaginary part of the relative permittivity is zero and is, therefore, not visible in this logarithmic plot in Figure 3b. Both graphs show that for increasing collision frequencies the change in the real part and in the imaginary part decreases.  
Note that in this work collisional plasma denotes a plasma in which the momentum transfer collision frequency of the electrons \(v_{\rm{m}}\) is not much smaller than the angular frequency \(\omega\) of the applied microwave field. This definition might differ from the one used in other works.  
Figure 3: _Cont._