# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 3 Results and Discussion
### Contributors to Changes in the Resonant Behavior
#### 3.2.2 Influence of the Cavity Walls' Temperature  
Temperature-corrected apparent frequencies are introduced to compensate for thermal expansion of the cavity [19]. This apparent frequency \(f_{\text{ap}}\), which is derived in Appendix A, is given by:  
\[f_{\text{ap}}(T_{\text{cav}})=\frac{f}{1+\alpha(T_{\text{cav}}-T_{0})}, \tag{20}\]  
where \(f\) is the applied microwave frequency, \(\alpha\) the thermal expansion coefficient of the cavity's wall material, \(T_{\text{cav}}\) the temperature of the walls during the measurement set with the same probing frequency \(f\), and \(T_{0}\) a fixed temperature for the whole measurement set. In reference [19], the temperature of the cavity was measured with \(\sim\)1 mK resolution for each set of temporal responses with the same probing frequency. In the analysis of the measurement data, the probing frequency is corrected for temperature fluctuations. The correction in frequency for thermal expansion of copper (\(\alpha=16.5\times 10^{-6}\) K\({}^{-1}\)[50]) walls--as used in reference [19]--is approximately 58 kHz/K. For a similar cavity surrounded by aluminium (\(\alpha=23.1\times 10^{-6}\) K\({}^{-1}\)[50]) as used in reference [18] the size of the correction is \(\sim\)81 kHz/K. Although the introduction of temperature-corrected apparent frequencies is a step forward, the approach does not correct for changes in the conductance \(\sigma\) of the cavity's walls due to variations in \(T_{\rm cav}\) which affects the width as well as the position of the peak. The two responsible mechanisms for this dependency are treated below. The first mechanism is related to the Ohmic losses in the cavity's walls and is explored by an experiment. A vector network analyzer and the metal structure surrounding the cavity described in reference [18] were used to investigate the influence of its temperature on the quality factor of the TM\({}_{010}\) resonant mode. A temperature-stabilized box was used to control the temperature of the cavity. Note that in this box the temperature of the gas was used to control \(T_{\rm cav}\) and consequently also the permittivity of the gas was unintentionally changed [17]. Figure 11 presents a graph of the obtained \(Q\) as a function of \(T_{\rm cav}\) and a linear fit through the measurement data. The error bars in the figure represent the standard deviation in the determined \(Q\) in the set of measurements for the same temperature. The derivative of the fit \(\frac{\Delta Q}{\Delta T_{\rm cav}}=-0.8\) translates into a \(\Delta(1/Q)\) of \(\sim\)\(3\times 10^{-7}\) for a temperature difference of 1 K. This is one order of magnitude below the best lower-detection limit for \(\Delta(1/Q)\) as presented in Section 3.1. Although this mechanism is below the detection limit for the determination of the changes in the imaginary part of the permittivity of our previous studies [18; 19] for small temperature variations, it might still affect the determination of \(f_{\rm res}\). To estimate the extent of this effect, the change in the distance from the resonance frequency to the spectral position of the frequency corresponding to the Full-Width-at-Half-Maximum (FWHM) is investigated. This spectral distance equals the difference in  
Figure 11: The quality factor \(Q\) as a function of the temperature of the metal structure surrounding the cavity \(T_{\rm cav}\). The circles represent the measurement data and the dashed line is obtained by a linear fit procedure using the experimental data. half of the FWHM. The change in the FWHM \(\Delta\)FWHM due to a change in \(T_{\rm cav}\) from 300 to 301 K is expressed via Equation (1) as,  
\[\Delta\text{FWHM}=f_{\rm res}\big{(}\frac{1}{Q_{\rm 301K}}-\frac{1}{Q_{\rm 300K }}\big{)}, \tag{21}\]  
where \(Q_{\rm 301K}\) and \(Q_{\rm 300K}\) are the fitted values of the quality factor for the temperatures 301 and 300 K, respectively. In this example and for this position in the peak, the correction term of the spectral position \(\frac{\Delta\text{FWHM}}{2\Delta T_{\rm cav}}\) should be \(\sim\)0.5 kHz/K.