Charge mobility and motion

In the absence of an electric field, free electrons in a gas will move with a random thermal velocity of average magnitude $\bar{v}$. Because the motion is random, the net drift velocity ${\langle{v}\rangle}_t\!=\!0$, where the angle brackets and subscript denote a time average. If an electric field $\vec{E}$ is applied, the electrons will acquire a net drift velocity $\vec{v}_e$ along the electric field lines but in the opposite direction to $\vec{E}$. $\vec{v}_e$ is directly related to $\vec{E}$ by:

$$\vec{v}_e = -\mu_{e}\vec{E}\;,$$ (2.1)

where $\mu_{e}$ is the mobility of electrons.

Electron mobility is a function of both the gas composition and density. It is also dependent on the magnitude of the applied electric field divided by the gas density (see Section 2.4.2). Electron mobility is defined by the following relationship:

$$\mu_{e} = \frac{q_e}{m_{e}\nu_{m}}$$ (2.2)

where $q_e$ is the electronic charge, $m_e$ is the electron mass, and $\nu_m$ is the effective collision frequency of electrons with neutral molecules and atoms. A similar relation also applies for ions, though ion mobility will be very much less than that of electrons since $m_{ion}\!\gg\!m_e$.

The effective collision frequency $\nu_m$ is proportional to the number density of gas molecules ($N$), the thermal velocity of the electrons ($\bar{v}$), and the scattering cross section ($\sigma_c$) of the gas:

$$\nu_m \propto N\bar{v}\sigma_c$$ (2.3)

A rigorous relationship between the above quantities would include the mean scattering angle.