Paschen Curve and Gas Breakdown

What is modelled

A uniform electric field in a gas between two plates. Electrons gain energy from the field and ionise atoms, so each electron released at the cathode starts an avalanche. Positive ions striking the cathode release new electrons with probability γ. When one avalanche regenerates itself the discharge becomes self-sustaining and the gas glows.

Equations

• Townsend ionisation coefficient α = A p exp(-B p d / V) (per cm, p in Torr).
• Self-sustaining condition γ (eα d - 1) = 1, which gives the Paschen law Vb = B p d / [ ln(A p d) - ln ln(1 + 1/γ) ].
• Minimum at p d = (e/A) ln(1+1/γ), Vmin = (e B / A) ln(1+1/γ).
• After ignition a ballast resistor limits the current, I = (Vsupply - Vhold) / R, with a phenomenological holding voltage.

Checks run in the page

• Numerical scan of the Paschen minimum against the closed form for all three gases.
• The Townsend condition holds exactly at the computed breakdown voltage.
• The p d scaling law.
• The steady current against (V - V_hold)/R from a fresh engine run.

Honest simplifications

The constants A, B and γ are round illustrative values, not measured data for the named gases. The model is for a uniform field and a fixed γ; it does not cover streamers, the left branch at very small p d in real tubes, field emission or the structure of a real glow discharge. The post-breakdown glow and current are a simple phenomenological model.

• Choose Neon glow lamp and lower the supply below the breakdown line: the avalanche stays dark.
• Double the pressure and halve the gap: the operating point does not move on the Paschen curve.
• Choose Too thin to break down: the gas is too thin at this p d and no voltage helps within the slider range.