Mobility of Electrons in the Noble Gases

This paper presents a theoretical study which leads to formulas that relate the cross section for inelastic collisions near the threshold energy to the drift velocity $v_d$ of electrons in noble gases. Derivation of general formulas for $v_d$ follows a qualitative interpretation of the main features of the experimental drift-velocity curves. The formulas for $v_d$ are derived for the case in which the elastic cross section for momentum transfer is given by $N_0{Q}_m\mathrm{}\left(v\right)={\mathrm{av}}^{j-1\mathrm{}}$. Closed-form expressions for $v_d$ are obtained by integrating only over the distribution function for electrons that have energy less than the excitation energy $u_1$. By this procedure, integration over the high-energy ($u>\mathrm{}{u}_1$) distribution function is avoided and the use of analytical methods is made possible. A partial correction to this approximation is obtained by extending the low-energy ($u<\mathrm{}{u}_1$) distribution function to pass through zero at an energy $u_0$ that is greater than $u_1$. From the drift-velocity formula and the experimental values of $v_d$, the cutoff energy $u_0$ can, in principle, be evaluated as a function of $\frac{E}{p}$. The high-energy distribution function is used to derive an expression for the overshoot ($u_0-u_1$) which is shown to be proportional to ${\left(\frac{E}{p}\right)}^{\frac{2}{(\gamma +2)}}$ and to depend upon the constants $h$ and $\gamma$ when the inelastic-collision cross section has the form $Q_1\mathrm{}\left(u\right)=h{(u-u_1)}^{\gamma }$. Possible applications of these theoretical results to drift-velocity data for gas mixtures are also briefly discussed.