Conduction and Ionization in Gases

Begin with a fact that is easy to take for granted: at ordinary pressure, a gas is one of the best insulators we own. Its molecules are electrically neutral, spaced far apart, and carry no loose charge to ferry a current across a gap. Left entirely to itself, an air gap is not perfectly empty of carriers — cosmic rays and the natural radioactivity of the surroundings knock loose on the order of ten ion pairs in every cubic centimetre each second — but those few are swept up or recombine almost as fast as they appear, and the current that crosses even a strongly charged gap is vanishingly small, a matter of microamperes or less.

So the interesting question is not why a gas insulates, but how it ever stops. The answer, and the subject of this entire part of the course, is ionization: the wholesale manufacture of free electrons and positive ions inside the gas itself. Once the gas begins to breed its own carriers faster than they are lost, the trickle of current turns into a flood, and what was an insulator becomes, abruptly, a conductor. Townsend's avalanche, the streamer, the spark, the arc — all the dramatic chapters ahead — are just successive stages of that runaway. This chapter lays the groundwork by examining the elementary events one at a time, before the next chapters assemble them into a criterion for breakdown.

To see how ionization gets started, follow a single free electron through the gap. It does not glide smoothly to the anode; it ricochets, colliding with gas molecules and travelling on average only a short distance — the mean free path \(\lambda\) — between one impact and the next. The denser the gas, the more often it collides, so the free path shrinks as pressure rises:

Superimposed on this random buffeting, the applied field \(E\) drags the electron steadily toward the anode at a drift velocity \(v_d=\mu E\), and along the way it does work on the electron. Between two collisions the energy gained is roughly the force times the distance travelled along the field:

Now the crucial deduction. If that energy \(eE\lambda\) reaches the molecule's ionization energy, the next collision can knock a second electron free; if it falls short, the energy is mostly surrendered as heat or excitation and the electron must build up again. But \(\lambda\propto 1/p\), so the energy won per free path scales as \(eE\lambda \propto E/p\). It is therefore the ratio \(E/p\) — the field per unit pressure, the reduced field — and not the field alone that decides whether ionization can happen. Double the pressure and you must double the field to keep the electron equally energetic. This single grouping, \(E/p\), will turn out to be the master variable of every gas-discharge law in the chapters ahead, from Townsend's coefficient to Paschen's curve.

An electron can be torn from a molecule in several distinct ways, and a high-voltage engineer should know all four — but should also know which one does the heavy lifting. By far the most important for breakdown is ionization by electron impact, also called collision ionization. A drifting electron that has gathered at least the ionization energy \(eV_i\) strikes a neutral molecule and ejects one of its bound electrons, leaving behind a positive ion and now two free electrons where there was one:

Because every freed electron can repeat the act, impact ionization is the one process capable of multiplying, and so it sits at the heart of the avalanche. The other three matter at the margins:

Photoionization. A sufficiently energetic photon absorbed by a molecule can itself eject an electron, \(h\nu + A \to A^{+} + e^{-}\), provided \(h\nu \ge eV_i\). This demands short-wavelength ultraviolet light, and it is the mechanism by which a streamer leaps ahead of its own avalanche by seeding fresh electrons in the gas in front of it — central to Chapter 6.

Thermal ionization. At the temperatures of an arc — several thousand kelvin — the sheer violence of thermal agitation, a blend of energetic collisions, radiation and excitation, ionizes the gas in bulk. Its equilibrium degree of ionization is set by the Saha equation, and it is what keeps an established arc conducting.

Ionization by metastables (the Penning effect). Some excited atoms cannot easily shed their energy by radiating; they are metastable and may live long enough to collide with a second species and ionize it — but only if the energy they carry exceeds that species' ionization energy. A trace of argon in neon is the textbook case: neon's metastables (≈ 16.6 eV) comfortably ionize argon (≈ 15.7 eV), and the breakdown voltage of the mixture collapses well below that of either pure gas.

Microscopic events are vivid but hard to compute with. Townsend's contribution was to wrap them in a single bookkeeping quantity, the first ionization coefficient \(\alpha\): the average number of ionizing collisions one electron makes per unit distance as it drifts along the field. If a slab of thickness \(dx\) contains \(n\) electrons, they create \(\alpha\,n\,dx\) new ones in crossing it, so the population obeys a simple growth law that integrates to an exponential:

A single electron leaving the cathode therefore becomes \(e^{\alpha d}\) electrons by the time it reaches the anode of a gap \(d\) wide — an avalanche. And because \(\alpha\) counts ionizing collisions, and those depend on the energy harvested per free path, \(\alpha\) inherits the same dependence on the reduced field we met in §4-2. Experiment is well fitted by a relation of the form

where \(A\) and \(B\) are gas constants. The full consequences of this exponential growth — the breakdown criterion itself, and the indispensable role of secondary electrons at the cathode — are the business of Chapter 5. Here it is enough to see where \(\alpha\) comes from and why it lives or dies on \(E/p\).

Creation does not go unopposed. Three processes continually drain free charge from the gas, and each one raises the field required to sustain a discharge.

Recombination. A positive ion meeting an electron — or a negative ion — can neutralise it, \(e^{-} + A^{+} \to A\), frequently releasing a photon in the bargain. Because it needs two carriers to meet, its rate grows with the product of the carrier densities, so recombination bites hardest where the discharge is most intense.

Attachment. In certain gases a neutral molecule will readily capture a passing electron to form a negative ion, \(e^{-} + A \to A^{-}\). This is the most consequential of the three, because the negative ion it creates is thousands of times heavier than an electron; it drifts sluggishly, gains almost no energy between collisions, and is hopeless at ionizing. An electron lost to attachment is, for the purposes of multiplication, simply removed from the game.

Diffusion. Quite apart from any reaction, carriers wander down their own density gradient and stray out of the high-field region altogether — a steady leak that matters most at low pressure, where free paths are long.

Gases whose molecules snatch electrons eagerly are called electronegative. Oxygen is mildly so; the undisputed champion is sulphur hexafluoride, SF₆, whose fluorine atoms seize free electrons almost on contact. To weigh creation against this kind of loss, we define an attachment coefficient \(\eta\) — attachments per electron per unit length, the mirror image of \(\alpha\) — and fold the two into one number, the effective ionization coefficient:

Net growth — and so the very possibility of breakdown — requires \(\bar{\alpha} > 0\), that is \(\alpha > \eta\). At low reduced field \(\eta\) dominates: electrons are mopped up as fast as they are bred, and the gas holds off. Only when \(E/p\) climbs past a critical value \((E/p)_{\text{lim}}\), at which \(\alpha = \eta\), does multiplication finally overtake attachment and an avalanche become possible.

This crossover is the whole reason SF₆ withstands roughly three times the field of air at the same pressure: its \(\eta\) is so large that \(\alpha\) only overtakes it at a much higher reduced field. The limiting field for SF₆ is about \(89~\mathrm{kV/(cm\cdot bar)}\), against roughly \(30~\mathrm{kV/cm}\) for air at one bar — which is exactly why SF₆ is the insulating gas of choice for gas-insulated switchgear and compact high-voltage apparatus.

Remarkably, everything above can be read off a single experiment. Take a gap between two plane electrodes, irradiate it weakly so that a steady trickle of seed electrons is always present, and raise the voltage slowly while watching the current. Three regions appear in turn.

Region I — the ohmic rise. At low voltage the current grows in step with the voltage, because a stronger field sweeps an ever-larger fraction of the background-produced ions to the electrodes before they can recombine.

Region II — saturation. Once every ion that is created is also collected, raising the voltage cannot increase the count, and the current flattens into a plateau \(i_0\). This saturation current depends only on the rate of background ionization, not on the voltage — a direct measurement of how many seed carriers the source supplies.

Region III — the Townsend rise. When the voltage is high enough that drifting electrons begin to ionize by collision, each seed electron multiplies, and the current turns sharply upward, climbing as \(i = i_0\,e^{\alpha d}\). Throughout all three regions the discharge is non-self-sustained: cut off the external irradiation and the current dies, because the gas cannot yet replace its own seed electrons. At a definite voltage \(V_s\), multiplication together with the secondary processes at the cathode becomes able to feed itself; the current runs away and the gap breaks down into a self-sustained discharge. Pinpointing that voltage is the entire object of the next chapter.

For each item, first name the idea it tests — a loss or creation process, the energy \(eE\lambda\), the reduced field, the coefficient \(\alpha\), effective ionization, or the I–V curve — then work it through. Difficulty rises down the list.