Soluble Three-Dimensional Model for Townsend'sα

A model gas is considered in which all electron-molecule cross sections are isotropic and depend inversely on the velocity $v$. Collisional energy loss is neglected. The Boltzmann equation for the model is solved for the collision density, where the collision density is the number of collisions that an individual electron makes between $v$ and $v+dv$ over its entire history. The Townsend $\alpha$ is obtained from the collision density, and it is found that $\frac{\alpha }{p}$ is inversely proportional to $\frac{E}{p}$. It is argued that this model furnishes an upper bound to the true $\frac{\alpha }{p}$ for all $\frac{E}{p}$; therefore it is concluded that this model demonstrates that at sufficiently high $\frac{E}{p}$ the observed $\frac{\alpha }{p}$ for any real gas must decrease with increasing $\frac{E}{p}$. The results also shed light on the way electron energy balance or lack of energy balance affects $\frac{\alpha }{p}$ and the drift velocity $v_D$; it is shown that energy balance is not possible at arbitrarily large $\frac{E}{p}$. Numerical applications of these results to ${\mathrm{H}}_2$ are discussed.