Gas TPC Drift and Gain

What is modelled

A charged particle crossing the gas strips electrons from atoms along its path, about one electron pair per 26 eV it loses. A uniform electric field drags the electrons to the readout plane at a speed that grows with the field and then saturates. On the way they random-walk, which blurs the track in proportion to the square root of the distance. At the readout plane a strong field makes each electron start an avalanche. The arrival time of each electron, times the drift speed, gives its original height; the position on the plane gives the other coordinate.

Equations

• Number of electrons N = Edep / W with W = 26 eV and variance F N, F = 0.2.
• Drift speed vd = vmax (E/E0) / (1 + E/E0) (a simple saturating curve), drift time t = L / vd.
• Diffusion σ = D* √L, time spread σ/vd.
• Townsend avalanche dn/dx = α n, α = A p exp(-B p / E), so G = exp(α d).
• Track fit slope error σ / √Σ(x - x̄)².

Checks run in the page

• A random walk of 3000 electrons against L / v and the diffusion spread.
• Primary ionisation counts and their Fano-factor variance.
• The avalanche differential equation integrated across the gap against the set gain.
• Least-squares track fits against the slope-error formula.

Honest simplifications

The drift speed curve and the diffusion coefficient are stand-ins for a real gas mixture (a real one is computed from electron-atom cross sections), the field is perfectly uniform, there is no attachment to impurities, no ion back-flow, no space charge, no transverse diffusion different from longitudinal, no delta rays, and the avalanche is a single-gap parallel plate with a fixed gain rather than a statistical (Polya) distribution. The drawn drift is slowed by a factor of about 100,000. The 3D cutaway is illustrative and its slider labels carry no units.

• Choose Long drift, big diffusion: the track blurs and the height resolution gets worse as the square root of the distance.
• Choose Low field: the electrons crawl and the drift takes much longer.
• Choose Heavy ionising track: many more electrons, so the gain needed is lower.