Breakdown voltage — Paschen's law

At what voltage does this gap break down and sustain its own discharge?

The formula

V = \frac{B pd}{\ln (A pd) - \ln (\ln (1 + \frac{1}{γ}))}

LaTeX

V = \frac{B \cdot pd}{\ln(A \cdot pd) - \ln\!\left(\ln\!\left(1 + \frac{1}{γ}\right)\right)}

Work it out

g Gas

The fill gas. Sets the empirical A and B coefficients, the same ones the ionisation coefficient uses.

p Pressure

Torr

The gas pressure in the gap.

d Gap

The electrode separation.

γ Secondary emission coefficient

Secondary electrons released at the cathode per incoming ion. Typically 0.01–0.1 for a clean metal surface; lower for a contaminated or oxidised one.

or see a worked example with typical numbers

Multiply pressure by gap to get the pd product, in torr·centimetres. Paschen's law is a function of this one combined variable, not of pressure and gap separately — the same pd gives the same breakdown voltage whether it comes from a wide gap at low pressure or a narrow one at high pressure.
Take the natural log of A times pd, and separately the natural log of the log of (1 + 1/γ). Subtract the second from the first.
Divide B times pd by that difference. The result is the breakdown voltage — the point where each avalanche produces, on average, exactly one secondary electron to replace the one that started it.
For the minimum: setting the derivative to zero gives (pd)ₘ = (e ÷ A) × ln(1 + 1/γ), and the voltage there is Vₘ = (e × B ÷ A) × ln(1 + 1/γ), with e Euler's number. This is the floor for the gas — no pressure or gap combination breaks down below it.

A and B are the same constants, and the same validity range, as the ionisation coefficient calculation — fitted over a limited pd range for each gas.
The constants set the minimum. With A = 12, B = 180 and γ = 0.01, argon's minimum comes out near 190 V at about 1 torr·cm; the commonly tabulated measured figure is lower, around 137 V. The gap is the coefficients and γ, not the law — treat the minimum from this page as the right neighbourhood, not the measured value.
γ is a single number for the whole cathode. Real secondary emission depends on the ion's energy and species and the surface's exact condition, and varies well outside the 0.01–0.1 range this defaults toward.
This is DC or slowly-varying breakdown. A gap driven by a fast pulse can hold off a higher voltage than this figure — there is not always time for an avalanche to fully develop before the pulse ends.
The formula is undefined, or gives a nonsensical (negative or infinite) answer, for a pd product far enough to the left of the Paschen minimum that A·pd ≤ 1. The warning on this page says so when it happens rather than returning a number.