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Townsend ionisation coefficient

How fast does ionisation multiply along the field at this pressure and field strength?

The formula
α=A⁢p⁢exp−B⁢pE
α
Ionisation coefficient, /cm
A
Gas constant A
B
Gas constant B
p
Pressure, Torr
E
Electric field, V/cm
LaTeX 
α = A \cdot p \cdot \exp\left(-\frac{B \cdot p}{E}\right)

Work it out

The fill gas. Sets the empirical A and B coefficients.

Torr

The gas pressure in the gap.

V/cm

The field across the gap — from the electric field calculation, in volts per centimetre.

Compare with a variation

Result

α Ionisation coefficient 0.007405 /cm

Ion pairs generated per centimetre of path an electron travels along the field.

With your numbers
0.007405 /cm=12⁢5 Torr⁢exp−180⁢5 Torr100 V/cm
LaTeX 
0.007405\,\mathrm{/cm} = 12 \cdot 5\,\mathrm{Torr} \cdot \exp\left(-\frac{180 \cdot 5\,\mathrm{Torr}}{100\,\mathrm{V/cm}}\right)

This result is a link — the address bar holds your numbers, so it can be pasted into a post and opened to the same answer.

What this looks like

Ionisation coefficient against electric field Electric field swept from 50 V/cm to 150 V/cm with everything else held at your numbers. The dashed lines cross where you are.
Ionisation coefficient against electric fieldIonisation coefficient rises from 9.14e-7 /cm to 0.149 /cm as electric field rises from 50 V/cm to 150 V/cm. At your electric field of 100 V/cm it is 0.0074 /cm.1e-71e-51e-31e-15075100125150100 V/cm0.0074 /cmElectric field (V/cm)Ionisation coefficient (/cm)
The formula behind the curve
α=A⁢p⁢exp−B⁢pE
α
Ionisation coefficient, /cm
A
Gas constant A
B
Gas constant B
p
Pressure, Torr
E
Electric field, V/cm
LaTeX 
α = A \cdot p \cdot \exp\left(-\frac{B \cdot p}{E}\right)
What moves the answer Each input moved 10% either way, with the others held still, and the effect on ionisation coefficient.
What moves the answerIonisation coefficient is most sensitive to Electric field, which moves it by about 127% for a 10% change. It is least sensitive to Pressure, at about 121%.Change in the answer when each input moves by 10%-200%-100%0%100%200%Electric field±127Pressure±121
The formula behind the curve
α=A⁢p⁢exp−B⁢pE
α
Ionisation coefficient, /cm
A
Gas constant A
B
Gas constant B
p
Pressure, Torr
E
Electric field, V/cm
LaTeX 
α = A \cdot p \cdot \exp\left(-\frac{B \cdot p}{E}\right)

Method

  1. Look up the gas's empirical constants A and B — fitted to measured ionisation rates for that gas, in torr and volts per centimetre.
  2. Multiply A by the pressure.
  3. Divide B times the pressure by the field, and take the negative exponential of that ratio. This term collapses toward zero when the field is weak relative to the pressure, which is why a low field at high pressure ionises almost nothing.
  4. Multiply the two together to get α — ion pairs per centimetre of travel.

Assumptions

  • A and B are constants over the pressure and field range entered. The real coefficients drift outside the range they were fitted over — roughly tens to a few hundred V/(cm·torr) for these three gases — and this does not know where that range ends.
  • The gas is pure. A mixture, or contamination from the electrode material or the tube surface, shifts the effective A and B away from the pure-gas figures used here.
  • The field is uniform across the gap, matching the electric field calculation this feeds from.