# Electron Impact Cross Sections and Transport Studies of C\({}_{3}\)F\({}_{6}\)[PERSON]
## 2 Calculation Methods
### Buffer Gas Mixtures  
In this section, we describe the approach used to study the effect of adding buffer gases to pure gas. We analyzed the effect on the dielectric strength and the breakdown voltage. The former was examined using the synergistic effect and the latter using the Paschen curve. When two gases are mixed and if the mixture follows the synergistic phenomena, then the dielectric strength of the gas mixture is higher than the value obtained by the linear interpolation of the two pure gases. We used the following synergistic effect coefficient to present this nonlinear characteristic of the gas mixture [7]:  
\[h=\frac{(E/N)_{crit,mix}}{k(E/N)_{crit,1}}-1 \tag{11}\]  
where \((E/N)_{crit,mix}\) is the critical field of the gas mixture. \((E/N)_{crit,1}\) is the critical electric field for the pure gas 1, which was PMVE in the present case, with \(k\) being the molar fraction of PMVE in the gas mixture. There is no synergistic effect if \(h=0\), and the critical field of the gas mixture is simply the sum of the critical field of the two pure gases. For \(h>0\), the critical field of the gas mixture is larger than the sum of the weighted critical field of the pure gases. Thus, the \((E/N)_{crit,mix}\) increases linearly with change in the gas proportions and shows the synergistic effect. The greater the value of \(h\), the more pronounced the synergistic effect. For \(h<0\), the critical field of the gas mixture is lower than the sum of the weighted critical field of the pure gases and is termed a negative synergistic effect.  
We now present the method used to obtain the breakdown characteristics of the gas mixtures. The ionization coefficient can be expressed as [9]  
\[\frac{\alpha}{N}=Ae^{\frac{-B}{kT/N}} \tag{12}\]  
where \(A\) and \(B\) are constants and do not depend on temperature. However, in most of the practical applications, temperature varies; hence, assuming the ideal gas condition \(p=Nk_{B}T\), we can write [10]  
\[\frac{\alpha}{p}=A^{\prime}e^{\frac{-B^{\prime}}{kT/p}} \tag{13}\]  
The temperature-dependent coefficients \(A^{\prime}\) and \(B^{\prime}\) are related to \(A\) and \(B\) as  
\[A^{\prime}=\frac{A}{k_{B}T},B^{\prime}=\frac{B}{k_{B}T} \tag{14}\]  
where \(k_{B}\) is the Boltzmann constant and \(T\) is the temperature. The BOLSIG+ data for the Townsend ionization coefficient can be used to obtain A and B using a fitting Equation (12) on the data. Once \(A\) and \(B\) are determined, \(A^{\prime}\) and \(B^{\prime}\) can be easily derived.  
Now, to predict the Paschen breakdown curve, the breakdown voltage, as a function of the product of pressure (\(p\)) and gap length (\(d\)) [10], is given as  
\[V_{b}=\frac{B^{\prime}(pd)}{ln\big{(}\frac{A^{\prime}(pd)}{ln(1+1/\gamma)} \big{)}} \tag{15}\]  
Moreover, \(\gamma\) is the second Townsend coefficient and is assumed to be 0.1 here. There was no information on its acceptable value and since it appears in the log term, the influence on the final results is not pronounced with its variation. In the following section, we present our results obtained using the methods presented in this section.