Townsend current growth with secondary ionization

Townsend's theory explains breakdown of a gas in a uniform field as the growth of electron avalanches by collision ionization (primary process, coefficient $\alpha$), sustained by the release of new electrons from the cathode by the products of the avalanche (secondary process, coefficient $\gamma$). Breakdown occurs when each avalanche produces enough secondary electrons to start the next one, so the discharge becomes self-sustaining.

1. Current growth with primary ionization only

Let $n_0$ electrons per second leave the cathode (e.g. due to UV light), and let $n_x$ be the number at distance $x$. In length $dx$ each electron makes $\alpha\,dx$ ionizing collisions:

$$\begin{aligned} dn &= \alpha\, n_x\, dx \\ n_x &= n_0\, e^{\alpha x} \\ I &= I_0\, e^{\alpha d} \end{aligned}$$

$\alpha$ (Townsend's first coefficient) = number of electrons produced by one electron per cm of path in the field direction. $\ln I$ vs $d$ is a straight line of slope $\alpha$.

2. Current growth including secondary ionization

At higher voltages the current rises faster than $e^{\alpha d}$. This is due to secondary processes: positive ions striking the cathode, photons from excited atoms, and metastables, which release fresh electrons from the cathode. Let $\gamma$ = number of secondary electrons released from the cathode per positive ion (Townsend's second coefficient).

Let $n_0'$ = total electrons leaving the cathode per second, and $n_+$ = secondary electrons from the cathode:

$$n_0' = n_0 + n_+$$

Electrons reaching the anode: $n = n_0' e^{\alpha d}$. Positive ions created in the gap $= n - n_0'$; these release $\gamma(n - n_0')$ electrons at the cathode:

$$\begin{aligned} n_+ &= \gamma\left(n - n_0'\right) \\ n_0' &= n_0 + \gamma\left(n - n_0'\right) \\ n_0' &= \frac{n_0 + \gamma n}{1 + \gamma} \end{aligned}$$

Substituting into $n = n_0' e^{\alpha d}$:

$$\begin{aligned} n(1 + \gamma) &= (n_0 + \gamma n)\, e^{\alpha d} \\ n &= \frac{n_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)} \end{aligned}$$

In terms of current:

$$\boxed{I = \frac{I_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)}}$$

3. How current grows in presence of the secondary process

 ln I
  |                          /  breakdown
  |                        /    (upturn due
  |                     /        to gamma)
  |              ..../
  |        ..../  straight line, slope = alpha
  |  ..../
  +-------------------------------> d

• For small $d$, $\gamma(e^{\alpha d} - 1) \ll 1$; the denominator is about 1 and current follows $I_0 e^{\alpha d}$.
• As $d$ (or $V$) increases, $e^{\alpha d}$ grows, the denominator decreases and the current rises faster than exponential (upturn in the curve).
• Each avalanche now produces secondary electrons that start new avalanches, so successive avalanches feed one another.
• When the denominator becomes zero, the current becomes theoretically infinite (limited only by the external circuit):

$$\gamma\left(e^{\alpha d} - 1\right) = 1 \quad \text{or} \quad \gamma\, e^{\alpha d} \approx 1$$

This is Townsend's breakdown criterion. At this point the discharge no longer needs the external source ($I_0$) and becomes self-sustained: the gap has broken down. If $\gamma(e^{\alpha d}-1) < 1$ the discharge is non-self-sustaining and stops when the external ionization is removed.