Gain fluctuations in uniform fields

Because of the stochastic nature of the charge amplification process, the size of an electron avalanche in a gas-based detector is subject to fluctuations. These fluctuations can play a key role for the performance of a detector, in particular for the energy (and spatial) resolution and the single electron detection efficiency.

We can describe gas gain fluctuations in terms of probabilities \(P_{n}\) that a single electron creates an avalanche comprising \(n\) electrons. A useful metric is the relative variance

\[f = \frac{\sigma^2}{\overline{n}^2}\]

of the avalanche size distribution, where \(\overline{n}\) is the mean number of electrons.

In this example, we consider avalanches in a constant electric field, and in the proportional regime, i. e. we assume that distortions of the electric field caused by space charge can be neglected. A comparison of calculations and measurements and a discussion of the impact of the gas composition and the electric field on the avalanche size distribution can be found in this paper

Nucl. Instr. Meth. A 624 (2010), 78-84

The following snippet shows the core of a program for calculating the avalanche size distribution in a uniform electric field for a mixture of 90% argon and 10% CO2.

// Electric field [kV / cm].
const double field = 20.;

// Make a gas medium.
MediumMagboltz gas("ar", 90., "co2", 10.);
gas.SetTemperature(293.15);
gas.SetPressure(760.);
gas.SetMaxElectronEnergy(150.);
gas.Initialise();

// Choose the gap to get a reasonable gain.
const double gap = 1. / (3.5 * field - 60.);

// Make a component with constant drift field.
ComponentConstant cmp;
cmp.SetArea(-2., -2., 0., 2., 2., 2 * gap);
cmp.SetMedium(&gas);
cmp.SetElectricField(0, 0, field * 1.e3);

Sensor sensor(&cmp);

AvalancheMicroscopic aval(&sensor);

// Histograms
TH1::StatOverflows(true);
TH1F hElectrons("hElectrons", "Avalanche Size", 200, 0, 2000);
TH1F hIons("hIons", "Avalanche Size", 200, 0, 2000);

constexpr unsigned int nEvents = 1000;
for (unsigned int j = 0; j < nEvents; ++j) {
  aval.AvalancheElectron(0, 0, gap, 0, e0, 0, 0, 0);
  int ne = 0, ni = 0;
  aval.GetAvalancheSize(ne, ni);
  hElectrons.Fill(ne);
  hIons.Fill(ni);
}
const double mean = hElectrons.GetMean();
const double rms = hElectrons.GetRMS();
std::cout << " f = " << rms * rms / (mean * mean) << "\n";

Repeating this calculation for different fields we can make a plot of the relative width f as function of the Townsend coefficient α.

At low electric field (low α) the relative width can be seen to approach unity, and the avalanche size distribution becomes exponential. With increasing field, the distribution becomes more and more rounded and the relative width decreases. The reason for this is that with increasing field, the electron energy distribution shifts towards higher energies, and the relative frequency of ionising collisions increases compared to non-ionising collisions such as excitations.

In addition to Ar/CO₂, the plot also shows the relative width in Ne, Kr, and Xe, all with an admixture of 10% CO₂. Comparing the different noble gases at constant α, one sees that the lighter elements show a lower value of f. This can be understood as a consequence of the higher ionization yield in the lighter gases.