# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Lower Limit for the Permittivity  
As discussed previously in this article, the shift of the resonance frequency scales linearly with the changes in the real part of the permittivity--being proportional to the electron density--according to the following equation,  
\[\frac{\Delta f}{f_{1}}=\frac{e^{2}}{2\varepsilon_{0}m_{\mathrm{e}}}\frac{ \iiint_{V_{\mathrm{cav}}}\frac{n_{\mathrm{e}}(\mathbf{r})}{v_{\mathrm{m}}^{2} (\mathbf{r})+4\pi^{2}f_{2}^{2}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r }}{\iiint_{V_{\mathrm{cav}}}|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}. \tag{28}\]As in the derivation of the equation, it is assumed that the microwave field is not perturbed by the presence of the plasma, the linearity of this relation holds in a limited range of plasma parameters. The plasma parameter space suitable to be explored by MCRS is investigated and redefined in this section. For this study a uniform (non-)collisional plasma in a coaxial geometry as shown in Figure 13 is assumed. The cylindrical cavity with a height \(L\) and a radius \(r_{2}\) contains a homogenous coaxial plasma column with a radius \(r_{1}\). The permittivity and permeability of region 1, i.e., the core, are \(\varepsilon_{1}\) and \(\mu_{1}\), respectively. The properties of the outer region, also referred to as region 2, are described by \(\varepsilon_{2}\) and \(\mu_{2}\). Based on the boundary conditions of the coaxial geometry, a characteristic equation for the microwave field in the aforementioned geometry is derived from first principles in the Appendix B in order to redefine the limitations of the diagnostic method. The found relation can be used to determine the angular resonance frequency \(\omega_{\text{res}}\). The characteristic equation is expressed as,  
\[ \begin{split}&\left[\frac{J_{11}^{\prime}}{J_{11}}\frac{k_{2}}{k_{ 1}}\frac{\varepsilon_{1}}{\varepsilon_{0}}\big{(}J_{22}N_{21}-N_{22}J_{21} \big{)}-\frac{\varepsilon_{2}}{\varepsilon_{0}}\big{(}J_{22}N_{21}^{\prime}- N_{22}J_{21}^{\prime}\big{)}\right]\\ &\left[\frac{J_{11}^{\prime}}{J_{11}}\frac{k_{2}}{k_{1}}\frac{ \mu_{1}}{\mu_{0}}\big{(}J_{22}^{\prime}N_{21}-N_{22}^{\prime}J_{21}\big{)}- \frac{\mu_{2}}{\mu_{0}}\big{(}J_{22}^{\prime}N_{21}^{\prime}-N_{22}^{\prime}J_ {21}^{\prime}\big{)}\right]\\ &=\bigg{(}\frac{1}{k_{1}^{2}}-\frac{1}{k_{2}^{2}}\bigg{)}^{2} \bigg{(}\frac{mk_{2}k_{2}}{r_{1}k_{0}}\bigg{)}^{2}\big{(}J_{22}N_{21}-N_{22}J_ {21}\big{)}\big{(}J_{22}^{\prime}N_{21}-J_{21}N_{22}^{\prime}\big{)}.\end{split}  \tag{29}\]  
where \(J\) and \(N\) are the Bessel functions of the first and the second kind, respectively, and their derivatives are \(J^{\prime}\) and \(N^{\prime}\). For clarity the following abbreviations,  
\[J_{11}=J_{n}(k_{\rho,1}r_{1}), J_{21}=J_{n}(k_{\rho,2}r_{1}), J_{22}=J_{n}(k_{\rho,2}r_{2}), \tag{30}\]  
are used, equivalently for \(J^{\prime}\), \(N\), and \(N^{\prime}\). The governing wavenumbers \(k_{0}\), \(k_{\text{z}}\), \(k_{1}\), and \(k_{2}\) are given by,  
\[k_{0}=\frac{\omega_{\text{res}}}{c}, k_{z}=\frac{p\pi}{L}, k_{1}=\sqrt{\omega_{\text{res}}^{2}\varepsilon_{1}\mu_{1}-k_{2}^{2}}, k_{2}=\sqrt{\omega_{\text{res}}^{2}\varepsilon_{2}\mu_{2}-k_{2}^{2}}. \tag{31}\]  
where \(p\) is the mode index for the \(z\)-direction, \(m\) determines the number of nodes of the field profile along \(\phi\), while the \(n\)th root of Equation (29) relates to the radial mode index \(n\). Computer code based on the characteristic Equation (29) is written in order to calculate resonance frequencies of the coaxial geometry. This implementation is able to resolve resonance frequencies for \(\varepsilon_{0}=\varepsilon_{1}=\varepsilon_{2}\) and \(\mu_{0}=\mu_{1}=\mu_{2}\) with a difference of only a few Hz (\(\sim 1\) part in 1 billion) from the solutions  
Figure 13: The coaxial geometry where the outer cylinder is the volume occupied by the resonance cavity and the inner cylinder is an homogenous plasma column. for the ideal resonance cavity [54].