The experimental data obtained from his experiments are described by the following formula
Where is the current flowing in the device, is the photoelectric current generated at the cathode surface, is the Euler number, is the first Townsend ionization coefficient, expressing the number of ion pairs generated per unit length (e.g. meter) by a negative ion (anion) moving from cathode to anode, is the distance between the plates of the device.
The almost constant voltage between the plates is equal to the breakdown voltage needed to create a self-sustaining avalanche: it decreases when the current reaches the glow discharge regime. Subsequent experiments revealed that the current rises faster than predicted by the above formula as the distance increases: two different effects were considered in order to explain the physics of the phenomenon and to be able to do a precise quantitative calculation.
Townsend put forward the hypothesis that positive ions also produce ion pairs, introducing a coefficient expressing the number of ion pairs generated per unit length by a positive ion (cation) moving from anode to cathode. The following formula was found
since , in very good agreement with experiments.
The first Townsend coefficient (α), also known as first Townsend avalanche coefficient is a term used where secondary ionization occurs because the primary ionization electrons gain sufficient energy from the accelerating electric field, or from the original ionizing particle. The coefficient gives the number of secondary electrons produced by primary electron per unit path length [2,4].
Cathode emission caused by impact of ions. Townsend, Holst and Oosterhuis also put forward an alternative hypothesis, considering the augmented emission of electrons by the cathode caused by impact of positive ions. This introduced Townsend's second ionization coefficient ; the average number of electrons released from a surface by an incident positive ion, according to the following formula:
These two formulas may be thought as describing limiting cases of the effective behavior of the process: note that either can be used to describe the same experimental results. Other formulas describing various intermediate behaviors are found in the literature, particularly in reference 1 and citations therein.
A Townsend discharge can be sustained only over a limited range of gas pressure and electric field intensity. The accompanying plot shows the variation of voltage drop and the different operating regions for a gas-filled tube with a constant pressure, but a varying current between its electrodes. The Townsend avalanche phenomena occurs on the sloping plateau B-D. Beyond D the ionisation is sustained.
At higher pressures, discharges occur more rapidly than the calculated time for ions to traverse the gap between electrodes, and the streamer theory of spark discharge of Raether, Meek and Loeb is applicable. In highly non-uniform electric fields, the corona discharge process is applicable. Discharges in vacuum require vaporization and ionization of electrode atoms. An arc can be initiated without a preliminary Townsend discharge; for example when electrodes touch and are then separated [2, 3, 7].
The starting of Townsend discharge sets the upper limit to the blocking voltage a glow discharge gas filled tube can withstand this limit is the Townsend discharge breakdown voltage also called ignition voltage of the tube.
The occurrence of Townsend discharge, leading to glow discharge breakdown shapes the current-voltage characteristic of a gas discharge tube such as a neon lamp in a way such that it has a negative differential resistance region of the S-type. The negative resistance can be used to generate electrical oscillations and waveforms, as in the relaxation oscillator whose schematic is shown in the picture on the right. The sawtooth shaped oscillation generated has frequency
where – is the glow discharge breakdown voltage, is the Townsend discharge breakdown voltage, , and are respectively the capacitance, the resistance and the supply voltage of the circuit.
Since temperature and time stability of the characteristics of gas diodes and neon lamps is low, and also the statistical dispersion of breakdown voltages is high, the above formula can only give a qualitative indication of what the real frequency of oscillation is.
Gas phototubes. Avalanche multiplication during Townsend discharge is naturally used in gas phototubes, to amplify the photoelectric charge generated by incident radiation (visible light or not) on the cathode: achievable current is typically 10~20 times greater respect to that generated by vacuum phototubes.