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Paschen curves

Background

Paschen curves relate the breakdown voltage VB of a gas-filled parallel plate chamber with the product pd of the pressure and the gap thickness. Breakdown is held to occur when an avalanche triggered by a single electron produces on average at least one fully fledged secondary avalanche. Secondary avalanches can be created in a variety of ways, perhaps the simplest of which is electron emission from the cathode hit by avalanche ions. The probability for this to happen is the second Townsend coefficient γ. Writing G for the gain, the breakdown condition reads:

\[\gamma (G - 1) > 1\]

Paschen curves have been measured for numerous gases. In this example, we compare some of the measurements with calculations for Ar-iC4H10 mixtures.

Parametrised Townsend coefficient

A simple formula that provides a reasonable approximation of the measurements can be derived from the Townsend coefficient parametrisation

\[\frac{\alpha}{p} = A\exp(-B p / E)\]

Although sometimes attributed to M.E. Rose and S.A. Korff [Phys. Rev. 59 (1941) 850-859 is sometimes cited, even though this paper does not contain the formula] the expression is probably due to John Sealy Townsend himself [he used similar formulae in Phil. Mag papers published in 1900-1903].

In a constant field the gain is

\[G = \exp(\alpha d).\]

Inserting this in the breakdown condition, gives

\[V_{B} = \frac{a p d}{\ln(pd) + b}\]

where a and b are functions of A, B and γ.

When plotting the breakdown voltage against \(pd\), the Paschen curves for all pressures overlap. This is because the reduced electric field \(E/p=V/(pd)\) remains constant when changing the pressure but keeping \(pd\) constant. Similarly the gain remains constant because \(\alpha/p\) scales in \(E/p\) and \(\alpha d = pd \alpha/p\).

Using Magboltz to calculate the Townsend coefficient

The Magboltz program is capable of computing the Townsend coefficient with considerably better precision than the parametrisation. The first step when calculating the Paschen curve using Magboltz consists in generating the electron transport tables and storing them in a gas file (the full program can be found here):

// Describe the gas: composition, pressure, temperature
MediumMagboltz gas("ar", 90.0, "ic4h10", 10.0);
gas.SetTemperature(293.15);
gas.SetPressure(760.);

// Ask for calculations from 1 kV/cm to 1.5 MV/cm in 40 steps
gas.SetFieldGrid(1000., 1500000., 40, true, 0., 0., 1, 0., 0., 1);

// No Penning transfer
const double lP = 0.0;
const double rP = 0.0;
gas.EnablePenningTransfer(rP, lP, "ar");

// Run Magboltz with moderate precision
gas.GenerateGasTable(3);

// Write the gas file
gas.WriteGasFile("ar-90-ic4h10-10-p1.gas");

Note that Penning transfer is disabled in the above example. Ar/iC4H10 is a Penning mixture and the actual gain is substantially larger as a result.

For this example, we have created the following gas files:

Garfield and Garfield++ use the same gas file format and these files can therefore be visualised by either program.

Plotting the Paschen curve

The core of a Garfield++ program that plots the Paschen curves follows:

// Read the Magboltz transport tables
MediumMagboltz gas;
const double gamma = 1e-2;
const double p = 1.;

gas.LoadGasFile("ar-99-ic4h10-1-p1.gas");

// Retrieve the points at which the Townsend coefficient was computed.
std::vector<double> efields,  bfields, angles;
gas.GetFieldGrid(efields, bfields, angles);

const size_t nE = efields.size();
std::vector<double> dvec(nE, 0.);
std::vector<double> vbvec(nE, 0.);
// Retrieve the Townsend coefficient and prepare plot vectors
for (size_t j = 0; j < nE; ++j) {
  double logalpha;
  gas.GetElectronTownsend(i, 0, 0, logalpha);
  const double alpha = exp(logalpha);
  const double d = log1p(1.0 / gamma) / alpha;
  dvec[j] = d * p;
  vbvec[j] = efields[j] * d;
}

// Plot the graph
TGraph* g = new TGraph(nE, dvec.data(), vbvec.data());
g->Draw();

Paschen curves

The orange, red and brown curves show the Paschen curves calculated by Magboltz for various mixing proportions and γ values, without Penning effect. The blue curves were calculated using a parametrised Townsend coefficient with A = 8.63 Pa-1m-1 and B = 132.0 V Pa-1m-1.

Plotting the Townsend coefficient

The behaviour of the Paschen curve is most easily understood by looking at the Townsend coefficient. With 10% iC4H10, the electron energy is on average lower than with 1% iC4H10. As a result, the low-field Townsend coefficient is larger with 1% iC4H10. For typical operating fields, the Townsend coefficient depends little on the mixture and it is only at the highest fields that the 10% iC4H10 admixture provides a slightly higher gain.

A graph comparing the Townsend coefficients can be prepared with this Garfield++ program or this script for classic Garfield.

Townsend coefficient