Paschen Avalanche Minimum

The Townsend/Paschen law describes the breakdown voltage of a gas between parallel electrodes. By defining reduced units, the relationship illustrates the classical gas-breakdown behavior where a minimum voltage exists to trigger an avalanche.

Worked Calculation

Given a secondary-emission yield γ = 0.1 and a reduced pd value x = 8, we can calculate the breakdown voltage and other properties using the reduced unit model. The auxiliary parameter is k = ln(1 + 1/γ).

Calculate k = ln(1 + 1/0.1) = ln(11) ≈ 2.3978952727983707.
This is the positive-voltage threshold. The minimum occurs at x_min = e * k ≈ e * 2.3978952727983707 ≈ 6.518155146595656.
The minimum voltage is V_min = e * k ≈ 6.518155146595656.
For x = 8, the breakdown voltage is V = x / ln(x/k) = 8 / ln(8/2.397895) ≈ 6.639829809425408.
The voltage slope is (ln(x/k) - 1) / (ln(x/k))² ≈ 0.1411140402698283.

Interactive Experiment

Explore the Paschen curve minimum in the interactive module at https://riano.app/module/paschen-avalanche-minimum.

Select the Secondary yield from the available values. Observe how the minimum point shifts.
Adjust the Reduced pd value. Note how it calculates the voltage and slope when in the finite domain.
Set Reduced pd to "Threshold" to observe the divergent boundary behavior.
Set Reduced pd below the threshold to see that the model status indicates it is outside the positive-voltage domain.

Model Limitations

This reduced-unit model illustrates the classical Paschen law. It does not predict real gas or electrode voltages directly. It assumes an ideal condition and does not account for microscopic behaviors or field-emission departures that occur at very small gaps or high pressures. The threshold itself represents a divergent boundary, not a finite physical state.

Primary References

The underlying physics of Paschen's Law can be derived from gas breakdown models.

NPTEL Course on Paschen's Law - Explains the formulation and derivation of the breakdown voltage curve.

Paschen Avalanche Minimum