Gas discharge

A gas discharge is electrical conduction through a gas containing mobile charged particles. An applied electric field accelerates electrons; collisions can cause ionization, generating additional electrons and ions. A sustained discharge requires charge-production and electrode-emission processes to offset losses, rather than merely the presence of a few initially charged particles.

In a uniform Townsend avalanche model, $d{n}_e/dx=\alpha n_e$ gives $n_e(d)=n_e(0)e^{\alpha d}$ across a gap $d$. If each arriving positive ion releases on average $\gamma$ electrons from the cathode, the self-sustaining threshold is $\gamma (e^{\alpha d}-1)=1$. Combining that condition with the field dependence of $\alpha$ gives the Paschen law.^[ref]

A motivating measurement fixes the gas, electrode material and gap, varies pressure and records the onset electric potential difference with a current-limited source. A pressure gauge and current measurement distinguish the breakdown threshold from the later conducting regime. At larger currents, space charge and heating can invalidate the uniform-field avalanche approximation.

The Paschen law models gas-breakdown electric potential difference as a function of pressure times electrode spacing. For a uniform gap $d$, fixed gas conditions and effective secondary-emission coefficient $\gamma$, an empirical ionization law $\alpha =Ap\exp (-Bp/E)$ together with $\gamma (e^{\alpha d}-1)=1$ givesHere $E=V_B/d$, $A$ has units inverse to pressure times length, and $Bpd$ has units of potential. The denominator must be positive. These constants and the fitted range are part of the model.^[ref]

Writing $L=\log (1+1/\gamma )$ and $x=pd$ gives $V_B=Bx/\log (Ax/L)$. Differentiation places its minimum at $x=eL/A$, with $V_{B,\min }=BeL/A$. A pressure sweep at fixed gap can therefore first lower and then raise the threshold. Nonuniform fields, changing cathode surfaces, temperature changes or microscopic gaps can break this scaling; it is not a universal law for every electrical spark.