Effective number of electrons (N_eff) in MPGD-readout TPCs: quantitative attribution of the prototype-size discrepancy and experimental test of the de-clustering drift-distance dependence

In a time projection chamber (TPC) with micro-pattern gaseous detector (MPGD) readout, the azimuthal coordinate resolution follows $\sigma_X^2=\sigma_{X_0}^2+D^2z/N_{\mathrm{eff}}$ , where $N_{\mathrm{eff}}$ is the effective number of electrons contributing to the charge-centroid measurement. Kobayashi's 2013 study (paper II) showed by analytic argument and numerical simulation that transverse diffusion perpendicular to the pad rows gradually de-clusters the primary ionization, making $N_{\mathrm{eff}}$ a slowly increasing function of drift distance governed by the scaling parameter $\sigma/h$ — but that this effect is too small to explain the substantially larger $N_{\mathrm{eff}}$ measured with the larger Micromegas-readout LC-TPC prototype ( $\sim56$ per 1-cm pad height) than with the small GEM prototype ( $\approx22.6$ per 6.3-mm pad row, i.e. $\approx36$ per 1-cm pad height under the $N_{\mathrm{eff}}\propto h$ scaling). The paper left open the quantitative attribution of this discrepancy (avalanche fluctuation $f$ , signal-to-noise ratio, readout calibration, electron capture by oxygen) and the experimental verification of the predicted de-clustering drift dependence at negligible magnetic field. Later measurements substantially settled the single-electron avalanche statistics ( $f\approx0.6$ – $0.8$ for GEM stacks and for a Micromegas in argon-based gases, close to the assumed $2/3$ ) and quantified oxygen attachment in the large prototype ( $A\approx0.5~\mathrm{m}^{-1}$ at $\sim50$ ppm O $_2$ ), but no published work quantitatively reconciles the prototype-dependent $N_{\mathrm{eff}}$ values, and the dedicated de-clustering test has not been performed; these two items form the surviving open core.

Background

A time projection chamber (TPC) is a gas-filled tracking detector in which ionization electrons liberated by a charged particle drift in a uniform electric field toward a segmented readout plane; the projected positions and arrival times of the electrons reconstruct the track in three dimensions. For a cylindrical collider TPC the crucial coordinate is the azimuthal one, measured along rows of rectangular pads by the charge-centroid method: the hit position is the pad-position average weighted by collected charge. For tracks perpendicular to the pad rows the accuracy is conventionally parametrized as

\sigma_X^2 = \sigma_{X_0}^2 + \frac{D^2}{N_{\mathrm{eff}}}\,z,

where $\sigma_{X_0}$ is an intrinsic (zero-drift) term, $D$ the transverse diffusion constant, $z$ the drift distance, and $N_{\mathrm{eff}}$ the effective number of electrons contributing to the centroid (a standard parametrization in the TPC literature; see Blum, Riegler and Rolandi, Particle Detection with Drift Chambers). Because diffusion smears each arriving electron over $\sigma = D\sqrt{z}$ , the centroid error improves as $1/\sqrt{N_{\mathrm{eff}}}$ .

Naively $N_{\mathrm{eff}}$ should equal $\langle N\rangle$ , the mean number of drift electrons collected by a pad row (of order 70 per 6-mm row in the gases considered here). Two fluctuation sources reduce it. First, primary ionization is produced in clusters along the track with a very broad (Landau-like) size distribution; a single large cluster can dominate a pad-row signal, so what enters the centroid precision is $\langle 1/N\rangle$ rather than $1/\langle N\rangle$ . Second, each electron is gas-amplified in an avalanche whose size fluctuates. Modern TPCs amplify with micro-pattern gaseous detectors (MPGDs): GEMs (Gas Electron Multipliers — perforated foil electrodes with multiplication inside the holes) or Micromegas (a parallel mesh–anode amplification structure). The avalanche fluctuation is characterized by the relative variance $f = \sigma_q^2/\langle q\rangle^2$ of the charge induced by a single drift electron, often described by a Polya (gamma) distribution with shape parameter $\theta$ , for which $f = 1/(1+\theta)$ . For right-angle tracks and practical pad pitches, Kobayashi's simulation study (paper I) established the approximation

\frac{1}{N_{\mathrm{eff}}} = \left\langle\frac{1}{N}\right\rangle(1+f),

so with $f = 2/3$ and a realistic cluster-size distribution, $N_{\mathrm{eff}}$ can be as small as $\sim30\%$ of $\langle N\rangle$ — about 22 for 6.3-mm pad rows in Ar-CF $_4$ (3%)-isobutane(2%).

The source paper (paper II) added the effect of transverse diffusion perpendicular to the pad-row direction, ignored in paper I: this diffusion gradually breaks up ("de-clusters") the primary clusters, so the number of electrons reaching a given pad row approaches Poisson statistics as the drift distance grows; since a Poisson mode sits at its mean, $\langle 1/N\rangle$ decreases and $N_{\mathrm{eff}}$ rises slowly toward its asymptote $\langle N\rangle/(1+f)$ . The transition is governed by the scaling variable $\sigma/h$ ( $h$ = pad height), is very slow, and is suppressed by an axial magnetic field (which reduces $D$ ). The work was performed within the R&D of the LCTPC collaboration toward the TPC of the ILD detector at the International Linear Collider (ILC), whose $\sim100~\mu\mathrm{m}$ single-point-resolution requirement over up to 2.35 m of drift motivates knowing $N_{\mathrm{eff}}$ precisely; gas transport coefficients were computed with the Magboltz program, a Monte Carlo program for electron drift and diffusion in counting gases. The paper's two central empirical conclusions were: (i) de-clustering is too small to explain the discrepancy between the $N_{\mathrm{eff}}$ measured with a small GEM-based prototype TPC (cosmic rays and 4-GeV/c pions; $N_{\mathrm{eff}} \approx 22.6$ for 6.3-mm pad rows at $B = 0$ , i.e. $\approx36$ per 1-cm pad height if $N_{\mathrm{eff}} \propto h$ ) and the value from a larger Micromegas-based LC-TPC prototype (the Large Prototype built for linear-collider TPC R&D; 5-GeV/c electron beam, $\sim60$ -cm maximum drift, 7-mm pad-row pitch; $N_{\mathrm{eff}} \sim56$ per 1-cm pad height); and (ii) a visible rise of $N_{\mathrm{eff}}$ with drift distance should nonetheless appear at long drift in a large TPC operated without magnetic field in a gas with large transverse diffusion (for Ar-CF $_4$ (3%)-isobutane(2%) at $B=0$ , $D \approx 315~\mu\mathrm{m}/\sqrt{\mathrm{cm}}$ ).

Problem Statement

The problem, as left open by the source paper, is to bring the measured effective number of electrons $N_{\mathrm{eff}}$ in MPGD-readout TPCs into quantitative agreement with the analytic and simulation-based expectation

\sigma_X^2 = \sigma_{X_0}^2 + \frac{D^2}{N_{\mathrm{eff}}}z,\qquad \frac{1}{N_{\mathrm{eff}}} = \left\langle\frac{1}{N}\right\rangle(1+f),

where $\sigma_{X_0}$ is a zero-drift intrinsic term, $D$ the transverse diffusion constant, $z$ the drift distance, $f$ the relative variance of the single-electron avalanche charge, and $\langle 1/N\rangle$ the mean of the inverse number of drift electrons collected by a pad row — through dedicated experimental measurements, so that the azimuthal-resolution model used for ILC-class TPC design rests on verified effective-electron statistics. Three linked components constitute the question:

Quantitative attribution of the prototype-size discrepancy: determine how much of the gap between the small GEM prototype ( $N_{\mathrm{eff}} \approx 22.6$ for 6.3-mm pad rows at $B = 0$ in Ar-CF $_4$ (3%)-isobutane(2%), i.e. $\approx 36$ per 1-cm pad height under the $N_{\mathrm{eff}} \propto h$ scaling) and the larger Micromegas-readout prototype ( $N_{\mathrm{eff}} \sim 56$ per 1-cm pad height) is due to a smaller avalanche fluctuation $f$ , improved signal-to-noise ratio and readout-electronics calibration, and electron capture by gas contaminants such as oxygen during drift — together with any further cause (e.g. ionization-density or analysis-method differences) — such that the discrepancy is closed within stated uncertainties or shown to be an artifact of measurement or analysis.
Experimental verification or refutation of the de-clustering drift-distance dependence: a dedicated measurement of $N_{\mathrm{eff}}$ as a function of drift distance in a large TPC operated at negligible magnetic field in a gas with a relatively large transverse diffusion constant, of sufficient precision to test the predicted magnitude and drift-length scale (the $\sigma/h$ scaling, where $\sigma = D\sqrt{z}$ is the transverse diffusion width and $h$ the pad height) of the de-clustering-induced rise of $N_{\mathrm{eff}}$ and its approach to the asymptote $\langle N\rangle/(1+f)$ , with $\langle N\rangle$ the mean number of drift electrons collected by a pad row.
Single-electron avalanche statistics in the readout configurations used: measure $f$ (or the Polya shape parameter of the avalanche-charge distribution) for the specific MPGD configurations entering the comparison; configurations for which later literature already provides measured values serve as inputs rather than targets.

Status boundary: component 3 has been substantially settled by measurements after 2015 ( $f \approx 0.6$ – $0.8$ for GEM stacks in the actual TPC gas and for a Micromegas in argon-based mixtures, close to the assumed $2/3$ ), so answers take those values as given; components 1 and 2 are the surviving open core.

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