Streamer Theory and Paschen's Law

The Townsend criterion of Chapter 5, \(\gamma(e^{\alpha d}-1)=1\), is a statement about a particular gap at a particular voltage. To make it useful we want a law: a formula that predicts the breakdown voltage of any uniform-field gap from the things we control — the gas, the pressure \(p\), and the spacing \(d\). The bridge is the dependence of \(\alpha\) on the reduced field, which we met in Chapter 4. Since \(\gamma\) is small, the criterion is well approximated by demanding that the avalanche reach a fixed size:

Everything now hinges on writing \(\alpha\) in terms of \(p\) and the field \(E=V/d\) (uniform field), substituting, and solving for the voltage \(V\). When we do, a small miracle occurs: \(p\) and \(d\) appear only ever as the product \(pd\). The breakdown voltage of a uniform gap is not a function of pressure and spacing separately, but of their product alone. This is Paschen's law, discovered experimentally in 1889 and here recovered from first principles.

Start from the experimental fit for the first ionization coefficient and put \(E=V/d\):

Multiply by \(d\) and impose the criterion \(\alpha d = K\):

Take logarithms and rearrange for \(V\). The result is Paschen's law in closed form — the breakdown voltage as an explicit function of the single variable \(pd\):

The constants \(A\) and \(B\) are properties of the gas (for air, roughly \(A\approx 15~(\mathrm{cm\cdot torr})^{-1}\) and \(B\approx 365~\mathrm{V/(cm\cdot torr)}\) over the usual range), and \(\gamma\) carries the cathode. The formula is unwieldy to look at but tells a clean story once plotted, which is the next section.

Plotted against \(pd\), Paschen's law traces a distinctive U-shape: the breakdown voltage is high at very small \(pd\), falls to a minimum, and rises again at large \(pd\). That a curve with a minimum should exist at all is the law's most striking and useful feature.

The minimum is found by differentiating \(V_b\) with respect to \(pd\) and setting the derivative to zero. The algebra yields compact expressions for the optimum product and the minimum sparking voltage:

Here \(e=2.718\) is Euler's number. For air the minimum sits at a few hundred volts — about \(327~\mathrm{V}\) near \(pd \approx 0.57~\mathrm{torr\cdot cm}\) — and remarkably, no uniform air gap, at any pressure or spacing, can be broken down by less than this. It is a hard floor on sparking voltage, and it explains why vacuum and very small gaps can hold off surprisingly little voltage: you cannot get below the Paschen minimum.

The two arms of the curve climb for opposite physical reasons, and understanding them is more valuable than memorising the formula.

The right branch (large \(pd\)). At high pressure or wide gaps there are too many molecules. An electron collides so often that it cannot gather the ionization energy between impacts; each free path is short, so a higher field — and hence a higher voltage — is needed to make each collision count. As \(pd\) grows, \(V_b\) rises roughly in proportion. This is the everyday branch of atmospheric, centimetre-scale gaps.

The left branch (small \(pd\)). At very low pressure or tiny gaps there are too few molecules. Now an electron may cross the whole gap making only a handful of collisions — sometimes none — so even though each electron is highly energetic, there are simply not enough ionizing events to build an avalanche. A higher voltage is required to ensure that the rare collisions that do occur are all ionizing. This is the branch of vacuum gaps and micro-gaps, and its rising trend is why a near-perfect vacuum is an excellent insulator.

Paschen's law and the Townsend mechanism behind it work beautifully at low \(pd\). But at atmospheric pressure across a centimetre gap — \(pd\) of several hundred torr·cm — careful measurement contradicts the theory on three counts, all flagged at the end of Chapter 5: breakdown is far too fast for ion feedback, it ignores the cathode material, and it carves a thin luminous channel rather than a uniform glow. The resolution lies in something Townsend's accounting quietly ignored: the avalanche's own charge.

By the time an avalanche has grown to perhaps \(10^{8}\) electrons, the fast electrons crowd at its head and the slow positive ions trail behind, forming a compact dipole of space charge. The field this dipole produces is no longer negligible — it becomes comparable to the applied field itself. In front of and behind the avalanche head the two fields add; on its flanks they oppose. The applied field is locally distorted, intensely strengthened just ahead of the advancing tip, and the orderly Townsend picture of independent electrons drifting in a uniform field collapses.

The intensified field at the avalanche head does two things. It accelerates ionization locally, and — crucially — the excited and recombining molecules there emit a copious flood of ultraviolet photons. These photons fly out into the gas ahead of the head and, by the photoionization of Chapter 4, liberate fresh electrons in the region of already-enhanced field. Each such electron immediately launches its own secondary avalanche, which is drawn straight back into the head, lengthening and brightening it. The discharge no longer waits for slow ions to crawl to the cathode; it advances by photons at nearly the speed of light, growing a thin, branching, self-propagating filament — a streamer.

Two kinds of streamer are distinguished by the direction of advance. A cathode-directed (positive) streamer develops once the primary avalanche has reached the anode: its positive-ion trail, reinforced by photoelectrons from the gas, extends back toward the cathode, and is the common form in long gaps. An anode-directed (negative) streamer can form before the avalanche reaches the anode, when the head's field is already critical, and advances forward toward the anode. In either case, once a streamer bridges the gap it provides a conducting channel and the spark follows at once. Because the whole process is driven by photons in the gas volume, it needs no help from the cathode — exactly the cathode-independence that the measurements demanded.

The transition from a harmless avalanche to a self-propagating streamer happens when the avalanche's space-charge field grows to roughly the size of the applied field. Raether and, independently, Meek quantified this. The space-charge field at the head depends on how many electrons it carries, \(e^{\alpha x}\), so the condition for transition is reached when the avalanche has travelled a critical distance \(x_c\) such that this number reaches about \(10^{8}\). In its most-used form the criterion is simply:

Meek's fuller statement equates the radial space-charge field of the avalanche head to the applied field \(E\), giving a relation of the form \(E_{r}\approx 5.3\times10^{-7}\,\dfrac{\alpha\,e^{\alpha x}}{\sqrt{x/p}} = E\), but the round number \(\alpha x_c \approx 20\) captures the essential idea for design: breakdown by streamer requires an avalanche to reach a critical multiplication before it crosses the gap. Where \(\alpha d < 18\) the gap tends to break down in the slow, cathode-dependent Townsend way; where \(\alpha d\) comfortably exceeds 20, the fast streamer dominates. The two mechanisms are not rivals so much as the same physics at different \(pd\), and the table below sets the boundary in context.

For each item, first name the idea it tests — the derivation, Paschen's law, the minimum, the branches, space charge, the streamer, or the Raether–Meek criterion — then work it through. Difficulty rises down the list.