The spark that needs room to start

The field the matter takes away treated an insulator as something that answers a field by polarising: its molecules stretch, the field inside falls, and nothing flows. The constant that depends on how fast it is asked found that the answer depends on how quickly the field changes. The liquid a field turns solid found that a field which pushes on nothing can still arrange everything in a suspension. Every argument along the way assumed that the insulator stays an insulator. None of them asked how much field it would take to stop that being true.

For a gas the answer is surprisingly subtle, and the familiar number — air breaks down at three million volts per metre — turns out to be the value of a function at one point. Gas breakdown is not a property of the gas at a given field. It is a race run across the gap, and the length of the track matters as much as the field driving the runners. Follow that through and the breakdown voltage of every gas gap lies on a single curve with a floor in it, and a gas that is thinned out or squeezed both become better insulators, for opposite reasons.

An avalanche across the gap #

A gas always holds a few free electrons, knocked loose by cosmic rays and natural radioactivity at a rate of some ten pairs per cubic centimetre per second at ground level. Put a field across the gas and each free electron accelerates, collides, loses its direction, and accelerates again. Most collisions just bounce it. But an electron that happens to travel far enough between collisions to pick up the gas’s ionisation energy — about fifteen electronvolts for nitrogen — knocks a second electron loose. Now there are two, and both accelerate.

John Townsend measured the rate around 1900. The number of ionising collisions an electron makes per metre of travel, now written $\alpha$, depends on how often the electron gets a long enough free run, and that is exponentially rare:

$$\alpha = A\,p\,\exp\!\left(-\frac{B\,p}{E}\right).$$

The pressure $p$ appears because how far a molecule gets between collisions is inversely proportional to the pressure — the same fact that makes a gas’s viscosity independent of how much gas there is — and electrons obey the same rule. The ratio $Bp/E$ is, up to a constant, the ionisation energy divided by the energy an electron gains from the field over one mean free path; the exponential is the chance that a free flight is long enough.

The coefficient is steep. Between two and five megavolts per metre it rises four thousandfold. An electron crossing a centimetre at three megavolts per metre makes about one ionising collision and arrives with one companion; at five megavolts per metre it makes forty-four, and its avalanche has $e^{44}$ members, more electrons than there are in the gap’s gas. Somewhere in between, the gap stops insulating.

One generation must replace itself #

Where exactly is decided not by the avalanche but by what comes after it. An avalanche sweeps out of the gap into the anode and is gone. If that were all, every avalanche would end and the current would stop as soon as the background ionisation stopped supplying new seed electrons. The gap would carry a tiny current and remain an insulator.

What sustains a discharge is the positive ions the avalanche leaves behind. They drift back to the cathode, and when they strike it each has a small chance $\gamma$ — typically a hundredth, depending on the metal and its surface — of knocking out a fresh electron, which starts a new avalanche. An avalanche of $e^{\alpha d}$ electrons leaves about $e^{\alpha d}$ ions behind, and these return about $\gamma e^{\alpha d}$ new electrons. If that number is less than one, each generation is smaller than the last and the current dies with the seeding. If it reaches one, the discharge sustains itself with no outside help:

$$\gamma\left(e^{\alpha d} - 1\right) = 1 \quad\Longleftrightarrow\quad \alpha d = \ln\!\left(1 + \frac{1}{\gamma}\right).$$

This is the same condition that decides whether a pile of uranium is a reactor: each generation must, on average, produce exactly one successor. It does not care how many electrons are in the gap, only whether their lineage is self-sustaining. With $\gamma = 0.01$ the right-hand side is 4.62, so an electron must make about four and a half ionising collisions on its way across.

Below the threshold the avalanche is not useless; it is an amplifier. A thin wire held at a high voltage inside a gas-filled tube concentrates the field into a region a fraction of a millimetre across, so that an electron set free anywhere in the tube drifts gently to the wire and only multiplies in the last stretch. Each original electron arrives as an avalanche of thousands to a million, and as long as the ions’ return to the cathode cannot sustain a second generation the output pulse is proportional to the ionisation that started it. That is the proportional counter, which weighs the energy of an X-ray by the size of its pulse. Raise the voltage until the avalanche spreads along the whole wire and every event, large or small, gives the same saturated pulse: the Geiger counter, which counts rather than weighs. Both live just below the line where the gap would carry a current with no ionising event at all.

The horizontal lines on the figure are that requirement for three gaps. A wider gap needs a smaller $\alpha$, because the electron has more room in which to multiply, and the steepness of the curve means the field needed falls only slowly as the gap grows: from five megavolts per metre for a millimetre to 2.7 for ten centimetres. “Three megavolts per metre” is the answer for about a centimetre.

Paschen’s single curve #

Put Townsend’s coefficient into the criterion and solve for the voltage $V = Ed$:

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

The pressure and the gap appear only as their product. Friedrich Paschen found this experimentally in 1889, eleven years before anyone knew why: a gap of one millimetre at one atmosphere and a gap of one centimetre at a tenth of an atmosphere break down at the same voltage. The reason is now plain. The product $pd$ is proportional to the number of mean free paths across the gap, and a given voltage spread over a given number of mean free paths gives the electron the same energy per flight. Two gaps with the same number of collisions in them and the same voltage across them are, as far as an electron can tell, the same gap.

The curve has a minimum, and the reason it must is the heart of the argument. Far to the right, the gap holds a great many mean free paths: the electron collides constantly, gains little energy between collisions, and needs a strong field to ionise at all. The voltage rises nearly in proportion to $pd$. Far to the left, the gap is so short or the gas so thin that an electron crossing it makes only a few collisions of any kind, and however much energy it gains it cannot make four and a half ionising ones; the voltage must climb steeply to make nearly every collision count, and below a value of $pd$ where even every collision ionising would not suffice — about 0.4 pascal-metres in air — the gas cannot break down at any voltage.

Between the two there is a best arrangement, where the electrons make just enough collisions, each with just enough energy. For air it is at about one pascal-metre, and the voltage there is about 330 volts as measured, 305 in the model. No spark can cross a gap of air between flat electrodes at less than about 330 volts, however narrow or wide the gap is. That is why the electronics of a car or a telephone, at twelve volts or less, can be packed with air gaps of a few micrometres and never arc through the gas, while a few hundred volts needs real design.

A centimetre and a micrometre #

At one atmosphere the minimum falls at a gap of about eleven micrometres. Reading the same curve as field against gap shows how strongly the familiar number depends on scale.

Across a metre of uniform field air holds about two and a half megavolts per metre; across a centimetre, a little over three. Below a millimetre the field needed climbs quickly — nine megavolts per metre across a tenth of a millimetre, seventeen across twenty micrometres. Small gaps are strong. That is part of why how much charge a shape will hold could find microscopic actuators running at fields thirty times the “breakdown field of air” without sparking: they are on the far left of Paschen’s curve.

The model stops short of the smallest gaps. Below about five micrometres the field at the metal’s surface becomes strong enough to pull electrons straight out of it, through the barrier that holds them in, by the tunnelling that a wall a factor of two makes impassable describes in a different setting. Measured breakdown in micrometre gaps falls below Paschen’s prediction for that reason, and the left-hand branch of the curve is never reached at atmospheric pressure. Microelectromechanical devices are designed against this modified curve, not the textbook one.

At the other end, long gaps break down below Townsend’s prediction for a different reason. At atmospheric pressure over centimetres, an avalanche grows to some hundred million electrons before reaching the anode, and its own space charge distorts the field enough to launch a self-propagating ionisation front — a streamer — that crosses the gap far faster than ions could return to the cathode. The measured law that the figures draw beside the model is the streamer regime, and over metres, with electrodes that are not flat, sparks advance as stepped leaders at average fields far lower still. Lightning crosses kilometres of air at a few hundred thousand volts per metre on average, because it builds a conducting channel as it goes rather than asking the whole column to break down at once.

Thinner is not always weaker #

The curve’s left branch contains the result that most surprises engineers meeting it for the first time. Lower the pressure in a fixed gap and the breakdown voltage falls, as intuition says — but only down to the minimum. Below that pressure it rises again, and in a good vacuum the gas cannot break down at all.

A one-millimetre gap holds about four and a half kilovolts in air at sea level. At the pressure where its $pd$ reaches one pascal-metre — about eleven hundred pascals, a hundredth of an atmosphere, which the atmosphere reaches about thirty kilometres up — it holds only about three hundred volts. Above that altitude it holds more again. Equipment rated for high voltage at sea level, and perfectly safe in the vacuum of orbit, can arc during the minutes it spends climbing through the middle of the atmosphere, and spacecraft high-voltage supplies are routinely switched off through the launch until the pressure has fallen past the danger region. Aircraft wiring, connectors and radar transmitters carrying a few kilovolts are designed for the reduced breakdown at cruising altitude, where the same connector holds a third of what it holds on the ground.

In a vacuum there are no molecules to ionise, and the gap’s strength is set by the electrodes themselves: field emission from microscopic points, and metal vapour boiled off them. Vacuum gaps hold tens of megavolts per metre, and vacuum interrupters, which open high-voltage circuits by pulling contacts apart in a sealed vacuum, depend on the left branch of a curve whose right branch every spark plug depends on. A petrol engine compresses its mixture tenfold before the spark, which moves the plug’s sub-millimetre gap far to the right on the curve and demands tens of kilovolts from the ignition coil; the same plug at atmospheric pressure would spark at a few kilovolts.

Which gas, which metal #

The two constants $A$ and $B$ belong to the gas, and the minimum voltage is $eB\ln(1+1/\gamma)/A$.

The noble gases break down at markedly lower voltages than the molecular ones. An electron moving through helium or argon bounces off atoms almost elastically, losing almost none of its energy, until it has enough to excite or ionise one; in nitrogen it spends its energy setting molecules vibrating and rotating, a few tenths of an electronvolt at a time, and needs a stronger field to get through to ionising energy. The ratio $B/A$ is a measure of that waste. Gases that grab free electrons and hold them as negative ions are better still as insulators, because an electron captured is an avalanche cut short. Sulphur hexafluoride, which does this voraciously, holds about three times the voltage of air at the same pressure and has insulated most of the world’s high-voltage switchgear for half a century — at the cost that a kilogram of it warms the climate as much as some twenty-three tonnes of carbon dioxide, and replacements are now being installed.

The third constant belongs to the cathode.

A surface that gives up electrons readily when an ion strikes it — a thin oxide, or a coating chosen for it — needs a smaller avalanche to replace each generation, so it lowers the minimum and shifts it to smaller $pd$. Because $\gamma$ enters only through a logarithm, the effect is moderate: a hundredfold change in $\gamma$ halves the minimum. On the right of the curve, where the gas sets everything, the cathode hardly matters. Glow lamps exploit both freedoms, filling a tube with neon mixed with a trace of argon, whose atoms the neon’s long-lived excited states ionise on contact, and coating the electrodes to raise $\gamma$, so that an indicator lamp strikes at under a hundred volts from the mains. Plasma display panels did the same with a layer of magnesium oxide over the electrodes, whose yield under neon ions is unusually high.

What the flat electrodes hide #

Every figure here assumes a uniform field between flat, parallel, clean electrodes, a direct voltage applied slowly, and a gas described by two constants fitted over a limited range of $E/p$. Real gaps break each assumption, and the departures are often larger than anything the curve shows.

A sharp point or a thin wire concentrates the field near itself, as the pressure a charge puts on its own metal found charge crowding onto any tight curve, and the gas there ionises at a voltage far below breakdown, producing a corona — the glow and hiss around power lines in wet weather — without the gap as a whole failing. Rapidly varying voltages change the race: at radio frequencies electrons can oscillate in the gap without reaching either electrode, and the breakdown voltage falls; for very short pulses there is a delay, partly statistical, while the gap waits for a seed electron in the right place, and a gap can briefly hold far more than its steady breakdown voltage. Dust, humidity and surface films change $\gamma$ and the field at the electrodes. The Townsend constants used here come from fits valid for $E/p$ between roughly 100 and 800 volts per centimetre per torr, and at atmospheric pressure over centimetres the model overestimates the breakdown voltage by about a tenth, which is why the measured law is drawn beside it.

The figures also cannot show the transition itself — what happens in the microseconds after the criterion is met, as the space charge of a growing discharge reshapes the field, a sheath forms against each electrode, and the gap passes through a glow into an arc, where the cathode is heated enough to emit electrons thermally and the voltage across the gap collapses to tens of volts. Paschen’s curve says when an insulator stops being one. What the discharge becomes afterwards is a separate subject, and the voltage it needs to keep going is far lower than the voltage it needed to start.

The domain of the argument is a gas between electrodes far enough apart that electrons are not pulled from the metal, at pressures where collisions dominate, under fields that change slowly compared with the time an ion takes to cross. Inside that domain the breakdown voltage depends on the pressure and gap only through their product, and has a floor.

Still open: what an ion knocks loose #

The secondary-emission yield $\gamma$ is the least understood number in the criterion, and it is the one that decides the floor. It depends on the ion’s energy and species, on the metal, on whether the surface carries an oxide, adsorbed gas or a film left by previous discharges, and on contributions from photons and excited atoms reaching the cathode as well as ions; in practice it is fitted to a measured breakdown curve rather than computed, and the same nominal electrodes give different values in different laboratories. For the microscopic gaps of modern devices, where field emission and ion-enhanced emission compete and the surface is a few hundred atoms across, how the left-hand branch of the curve should be drawn is being worked out by simulation and experiment one geometry at a time.

The criterion itself is simple. A gas gap breaks down when one electron’s avalanche returns enough ions to the cathode to replace itself, αd = ln(1 + 1/γ), and because the ionisation rate depends on the pressure only through the mean free path, the breakdown voltage depends on pressure and gap only through their product — with a floor of about 330 V in air at one pascal-metre, below which no gap sparks, and above which a thinner gas and a denser one both insulate better. Three megavolts a metre is where a centimetre of air at sea level sits on that curve, not a property of air.