Townsend Breakdown Mechanism

At the close of Chapter 4 we had a single electron leaving the cathode and arriving at the anode as \(e^{\alpha d}\) electrons — a genuine avalanche. It is tempting to call that breakdown, but it is not. The avalanche sweeps across the gap, deposits its electrons on the anode, and is over; if no fresh electron is launched from the cathode, the current simply stops. The gap has conducted a pulse, not struck a spark. To break down, the discharge must become self-sustained — able to keep itself going after the external agent that supplied the first electron is removed.

Townsend's insight was that the avalanche does not only deliver electrons to the anode; it also leaves behind a trail of positive ions and emits a burst of photons, and these can travel back to the cathode and knock loose new electrons. If, on average, each avalanche manages to liberate at least one replacement electron at the cathode, the process becomes a closed loop: avalanche → secondary electron → fresh avalanche → and so on, no longer needing any outside help. That single condition — one avalanche begetting at least one successor — is the Townsend criterion, and the rest of this chapter makes it precise.

Recall the multiplication law from Chapter 4. With the cathode supplying \(n_0\) electrons per second by some external means (irradiation, say), each multiplies by collision ionization as it drifts, so the number arriving at the anode of a gap \(d\) wide is \(n_0 e^{\alpha d}\). In terms of measurable current — the current that would flow if there were no secondary effects — this is the primary Townsend relation:

Two things to notice set up everything that follows. First, the electrons reach the anode in nanoseconds, but the positive ions they leave behind are thousands of times heavier and crawl back toward the cathode far more slowly — so the ions are the messengers that carry the avalanche's influence back to where it began. Second, \(i = i_0 e^{\alpha d}\) alone can never run away: switch off \(i_0\) and the current vanishes. Something must regenerate \(i_0\) from within. That something is the secondary emission of the next section.

An avalanche feeds the cathode in several ways at once, and each can liberate a fresh electron there. Townsend lumped them all into a single number, the second ionization coefficient \(\gamma\): the average number of secondary electrons released at the cathode for every primary ionizing collision made in the gap. There are three contributing mechanisms.

Positive-ion bombardment (γ_i). The slow positive ions left in the avalanche's wake drift back and strike the cathode; if an ion arrives with enough energy it ejects an electron from the metal surface. This is usually the dominant secondary process at the low \(pd\) where Townsend's mechanism reigns.

Photoemission (γ_p). Excited molecules in the avalanche relax by emitting photons; those that strike the cathode can liberate electrons by the photoelectric effect.

Metastable and other surface effects (γ_m). Long-lived excited atoms reaching the cathode, and the neutralisation of ions at the surface, contribute further electrons.

The single most important consequence of \(\gamma\) is that breakdown becomes cathode-dependent. Because the secondary electrons come from the metal surface, the breakdown voltage of a gap depends on what the cathode is made of and how clean it is — a feature that sharply distinguishes the Townsend mechanism from the streamer mechanism of Chapter 6, which needs no cathode help at all. Typical values of \(\gamma\) are small, from about \(10^{-2}\) down to \(10^{-4}\), which is exactly why the primary avalanche must grow enormously before so feeble a feedback can sustain it.

Now assemble the loop into an equation. Suppose the cathode emits \(n_0\) primary electrons per second. Each grows by \(e^{\alpha d}\), so \(n_0(e^{\alpha d}-1)\) new ion pairs are created in the gap. A fraction \(\gamma\) of these returns to liberate further electrons at the cathode, which themselves multiply, and so on. Summing the geometric series of successive generations gives the steady-state current:

Read this carefully, because the whole physics hides in the denominator. When \(\gamma(e^{\alpha d}-1)\) is small, the denominator is close to one and the current is essentially the primary avalanche \(i_0 e^{\alpha d}\) — a non-self-sustained discharge that still needs \(i_0\). But as the voltage rises, \(\alpha\) grows, the term \(\gamma(e^{\alpha d}-1)\) climbs toward unity, and the denominator collapses toward zero. The current then grows without bound, limited in reality only by the external circuit. That run-away is breakdown.

The threshold of run-away is the value of voltage at which the denominator first reaches zero. Setting it to zero gives the celebrated Townsend breakdown criterion:

Its meaning is beautifully physical. The quantity \(e^{\alpha d}-1\) is the number of new ion pairs one avalanche creates; multiplying by \(\gamma\) gives the number of replacement electrons it produces at the cathode. The criterion says simply: each avalanche must produce exactly one successor. Below this, every generation is weaker than the last and the discharge dies; above it, every generation is stronger and the current explodes. Because \(\gamma\) is small, \(1/\gamma\) is large and the criterion is often written in the approximate form \(\gamma e^{\alpha d}\approx 1\), i.e. \(\alpha d \approx \ln(1/\gamma)\) — telling us the avalanche must reach a size of order \(1/\gamma\), typically \(10^{2}\)–\(10^{4}\), before it can sustain itself.

The criterion tells us whether a gap will break down, but not when. Under a slowly rising voltage the distinction hardly matters; under an impulse — a lightning or switching surge lasting microseconds — it is everything, because a gap that needs longer to break down than the surge lasts will simply survive a voltage well above its static threshold. The delay between applying an over-voltage and the collapse of the gap is the time lag, and it has two distinct parts.

The statistical time lag \(t_s\) is the wait for a suitably placed initiating electron to appear in the gap. Until one does, no avalanche can start, however high the field; \(t_s\) is random and is shortened by irradiating the gap or otherwise supplying seed electrons. The formative time lag \(t_f\) is the time the sequence of avalanches and secondary emissions then takes to build up to the full breakdown current once that first electron exists. The total time lag is their sum:

This is why a gap withstands a brief impulse of greater amplitude than it could endure as a steady voltage — the basis of the impulse ratio and of the volt–time curves used in insulation coordination later in the course.

The Townsend picture is not universal. It rests on the assumption that many successive avalanches, linked through slow ion feedback at the cathode, are needed to build the discharge — and that assumption holds only at modest values of the product \(pd\), roughly up to a few hundred \(\mathrm{torr\cdot cm}\) (low pressures, short gaps). In this range the measured breakdown voltage depends on the cathode material, the time lags are relatively long, and everything the equations predict is borne out.

At large \(pd\) — atmospheric pressure across centimetre gaps, the everyday case of a power-system spark — three observations break the Townsend story. Breakdown is far faster than the ion-transit time the mechanism requires; it is essentially independent of the cathode; and it often follows a thin, branched, luminous channel rather than a diffuse glow. The reason is that a single avalanche has by then grown so large that the space charge of its own ions distorts the applied field, and the discharge propagates by photoionization in the gas ahead of it, with no need to wait for the cathode. That is the streamer mechanism, and together with Paschen's law it is the subject of the next chapter.

For each item, first name the idea it tests — the primary avalanche, the coefficient \(\gamma\), the complete current, the breakdown criterion, the time lags, or the limits of the mechanism — then work it through. Difficulty rises down the list.