Engineering & Science
Townsend Electron Avalanche Current & Spark Breakdown Criterion Formula
Calculate pre-breakdown Townsend gas-discharge current I = (I0 * EXP(alpha * d)) / (1 - gamma * (EXP(alpha * d) - 1)) and test the Townsend self-sustaining spark criterion gamma * (EXP(alpha * d) - 1) >= 1.
Track primary electron avalanches and secondary cathode feedback up to sparkover!
In proportional radiation detectors, Geiger-Müller tubes, and high-voltage gas gaps, UV photons liberate a tiny seed photocurrent I0 (say 10 pA) at the cathode. As each electron drifts across gap d (cm), it creates alpha new electron-ion pairs per centimeter (Townsend's first ionization coefficient), multiplying current exponentially to I0 * EXP(alpha * d)!
Positive ions then drift back and strike the cathode, ejecting gamma secondary electrons per incident ion (Townsend's second coefficient)! Summing the infinite feedback loop yields John Sealy Townsend's complete current multiplication equation: I = (I0 * EXP(alpha * d)) / (1 - gamma * (EXP(alpha * d) - 1))! At I0 = 10 pA, alpha = 8 cm^-1, d = 0.50 cm (primary gain EXP(4) = 54.60 -> 545.98 pA without secondary feedback), adding just gamma = 0.01 shrinks the denominator to 0.4640 and more than doubles anode current to 1,176.64 pA; raising alpha to 10 cm^-1 drives gamma*(EXP(5)-1) = 1.474 >= 1, triggering self-sustained Townsend breakdown!
| Discharge Regime (I0 = 10 pA, d = 0.50 cm) | Alpha (cm^-1) & Gamma | Feedback Term gamma*(e^(ad)-1) | Anode Current I (pA) |
|---|---|---|---|
| Pure Primary Avalanche (No Secondary) | alpha = 8, gamma = 0.00 | 0.0000 (< 1) | 545.98 pA (54.6x gain) |
| Sub-Breakdown Proportional Gain | alpha = 8, gamma = 0.01 | 0.5360 (< 1) | 1,176.64 pA (117.7x gain) |
| Self-Sustained Spark Breakdown! | alpha = 10, gamma = 0.01 | 1.4741 (>= 1!) | "Self-Sustained Breakdown!" |
Townsend Pre-Breakdown Anode Current I (Same Units as I0)
Checks whether secondary feedback gamma * (EXP(alpha * d) - 1) >= 1 (spark breakdown) before dividing.
=IF(OR(B2 < 0, B3 < 0, B4 <= 0, B5 < 0), "Invalid Discharge Inputs", IF(B5 * (EXP(B3 * B4) - 1) >= 1, "Self-Sustained Breakdown!", ROUND((B2 * EXP(B3 * B4)) / (1 - B5 * (EXP(B3 * B4) - 1)), 2)))Townsend Feedback Loop Gain L_loop = gamma * (EXP(alpha * d) - 1)
Measures proximity to the Townsend spark threshold L_loop = 1.0000.
=IF(OR(B3 < 0, B4 <= 0, B5 < 0), "Invalid Discharge Inputs", ROUND(B5 * (EXP(B3 * B4) - 1), 4))Try the example
Use the original values or change them to test your understanding.