# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 2 Theory
### Perturbations  
As mentioned before, the properties of the material inside the cavity volume \(V_{\text{cav}}\) affect the resonant behavior of the excited standing wave. The change in the resonance frequency \(\Delta f\) (\(\Delta f=f_{2}-f_{1}\)) and the change in the reciprocal \(Q\)-factor \(\Delta(\frac{1}{Q})\) (\(\Delta(\frac{1}{Q})=\frac{1}{Q_{2}}-\frac{1}{Q_{1}}\)) are described by the following relation [18; 33],  
\[\frac{\Delta f}{f_{1}}+i\frac{1}{2}\Delta\left(\frac{1}{Q}\right)=-\frac{ \iiint_{V_{\text{cav}}}\left(\Delta\tilde{\varepsilon}(\mathbf{r})|\mathbf{E} _{1}(\mathbf{r})|^{2}+\Delta\tilde{\mu}(\mathbf{r})|\mathbf{H}_{1}(\mathbf{r} )|^{2}\right)d^{3}\mathbf{r}}{2\epsilon_{0}\iiint_{V_{\text{cav}}}|\mathbf{E} _{1}(\mathbf{r})|^{2}d^{3}\mathbf{r}}, \tag{3}\]  
where the subscript 1 is used for the unperturbed cavity and 2 for the perturbed state, \(i\) the imaginary unit number, \(\mathbf{E}_{1}\) the local electric field of the resonant mode of the unperturbed cavity, \(\mathbf{H}_{1}\) the local  
Figure 2: Spectral response of the TM\({}_{010}\) resonant mode of the unperturbed cavity measured in reflection configuration (red marks) and a Lorentzian fit (blue line).  
magnetic field of the resonant mode of the unperturbed cavity, \(\mathbf{r}\) the position vector, and \(\varepsilon_{0}\) the vacuum permittivity. The local change of the complex permittivity \(\Delta\tilde{\varepsilon}\) is given by:  
\[\Delta\tilde{\varepsilon}(\mathbf{r})=\tilde{\varepsilon}_{2}(\mathbf{r})- \tilde{\varepsilon}_{1}(\mathbf{r}), \tag{4}\]  
where \(\tilde{\varepsilon}_{1}\) and \(\tilde{\varepsilon}_{2}\) are the complex permittivity of the unperturbed and perturbed cavity, respectively. The derivation of this equation can be found in reference [18]. In this derivation, it is assumed that the resonant field profile is not disturbed by the perturbation and that \(f_{1}\approx f_{2}\).  
In this work, it is assumed that the plasma is unmagnetized, i.e., \(\tilde{\mu}=\mu_{0}\) ([34], p. 109), where \(\mu_{0}\) is the vacuum permeability. Hence, Equation (3) can be simplified by \(\Delta\tilde{\mu}=0\).  
By separating the real and imaginary parts of Equation (3), the contributors to changes in \(f\) and \(\frac{1}{Q}\) are separated,  
\[\frac{\Delta f}{f_{1}}=-\frac{\Re\bigg{(}\iiint_{V_{\mathrm{cav}}}\Delta \tilde{\varepsilon}(\mathbf{r})|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3}\mathbf{r }\bigg{)}}{2\varepsilon_{0}\iiint_{V_{\mathrm{cav}}}|\mathbf{E}_{1}(\mathbf{ r})|^{2}d^{3}\mathbf{r}}, \tag{5}\]  
\[\Delta\left(\frac{1}{Q}\right)=-\frac{\Im\bigg{(}\iiint_{V_{\mathrm{cav}}} \Delta\tilde{\varepsilon}(\mathbf{r})|\mathbf{E}_{1}(\mathbf{r})|^{2}d^{3} \mathbf{r}\bigg{)}}{\varepsilon_{0}\iiint_{V_{\mathrm{cav}}}|\mathbf{E}_{1}( \mathbf{r})|^{2}d^{3}\mathbf{r}}. \tag{6}\]  
These equations show that the method is most sensitive in the regions where the strength of the resonant field is largest. To be more exact, the material properties obtained by this integral from MCRS are cavity volume averaged and weighted with the square of the local electric field.  
Another type of perturbation is a change in the dimensions or shape of the cavity, e.g., due to thermal expansion of the metal structure which surrounds the cavity. For low-pressure plasmas, this is not of major importance because the effect is relatively small. However, the investigated atmospheric-pressure plasmas [18; 19] induce a change in resonance frequency which can be similar to that caused by a temperature increase of \(\sim\)10 mK and therefore this mechanism can no longer be neglected.  
A third type of perturbation is related to changes in the impedance of the conducting walls surrounding the cavity. The change in the surface reactance leads to, among other things, a 'virtual' change in the dimensions of the cavity due to a change in the skin depth. The importance of this effect when probing plasmas is investigated in Section 3.2.2. A more detailed description of the wall-impedance perturbation can be found in ([35], p. 255).