Townsend breakdown condition with secondary ionization

In Townsend's theory, breakdown of a gas in a uniform field is the point where the discharge becomes self-sustaining: each avalanche produces, through secondary processes, at least one new electron at the cathode to start the next avalanche, so the current continues even without the external source of initial electrons.

1. Primary ionization

Let $n_0$ electrons per second leave the cathode due to an external agency (UV light). Each electron makes $\alpha$ ionizing collisions per cm in the field direction ($\alpha$ = Townsend's first ionization coefficient, a function of $E/p$). In distance $dx$:

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

Each electron leaving the cathode produces $e^{\alpha d} - 1$ new electrons and the same number of positive ions in the gap. With primary ionization alone the current is finite for any $d$; it stops if $I_0$ is removed. This cannot explain breakdown.

2. Secondary ionization processes

Experiments show the current rising faster than $e^{\alpha d}$ at higher voltages. This is due to secondary electrons from:

• Positive ions striking the cathode and releasing electrons ($\gamma_i$)
• Photons from excited atoms falling on the cathode (photo-emission, $\gamma_{ph}$)
• Metastable atoms diffusing to the cathode ($\gamma_m$)
• Photo-ionization in the gas (minor in Townsend's range)

All are combined into one coefficient $\gamma = \gamma_i + \gamma_{ph} + \gamma_m$ = number of secondary electrons released from the cathode per positive ion produced (Townsend's second coefficient).

3. Current growth with secondary ionization

Let $n_0'$ = total electrons leaving the cathode, $n_+$ = secondary electrons, $n$ = electrons reaching the anode:

$$\begin{aligned} n_0' &= n_0 + n_+ \\ n &= n_0'\, e^{\alpha d} \\ n_+ &= \gamma\,(n - n_0') \end{aligned}$$

Eliminating $n_0'$ and $n_+$:

$$\begin{aligned} n_0' &= \frac{n_0 + \gamma n}{1 + \gamma} \\ n(1 + \gamma) &= (n_0 + \gamma n)\, e^{\alpha d} \\ n &= \frac{n_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)} \end{aligned}$$ $$I = \frac{I_0\, e^{\alpha d}}{1 - \gamma\left(e^{\alpha d} - 1\right)}$$

4. Condition for breakdown

As the voltage (or gap) is raised, $\alpha$ and $e^{\alpha d}$ increase, and the denominator decreases. When

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

the current becomes theoretically infinite; in practice it is limited only by the external circuit, and a spark forms. Since normally $e^{\alpha d} \gg 1$:

$$\gamma\, e^{\alpha d} \approx 1 \qquad \text{or} \qquad \alpha d = \ln\left(1 + \frac{1}{\gamma}\right)$$

Physical meaning: $\gamma(e^{\alpha d} - 1)$ is the number of secondary electrons produced at the cathode by one primary avalanche.

 ln I
  |                       |  breakdown
  |                      /   (gamma term)
  |                   _/
  |              ___/
  |         ___/  slope = alpha
  |    ___/
  +-------------------------> d (or V)

5. Consequences

• Sparking voltage: the voltage at which the criterion is just satisfied. Combining it with $\alpha/p = Ae^{-Bp/E}$ gives Paschen's law:

$$V_b = \frac{B\,pd}{\ln\left[A\,pd/\ln(1 + 1/\gamma)\right]} = f(pd)$$

• Effect of attachment: in electronegative gases (SF₆, O₂) electrons are captured with coefficient $\eta$; the criterion becomes

$$\frac{\alpha}{\alpha - \eta}\left[e^{(\alpha - \eta)d} - 1\right]\gamma = 1$$

so breakdown needs a higher field, explaining the high strength of SF₆.

• Validity: the Townsend criterion holds for low $pd$ (below about 1000 torr·cm); for long gaps at high pressure the streamer criterion applies.