An Estimation of the Effective Number of Electrons Contributing to the Coordinate Measurement with a TPC

TL;DR

This paper estimates the effective number of electrons contributing to coordinate measurement in a TPC, revealing it can be significantly less than the average due to statistical fluctuations, impacting measurement accuracy.

Contribution

The study introduces a simple numerical simulation to estimate the effective electron number, accounting for fluctuations affecting TPC coordinate accuracy.

Findings

$N_{eff}$ can be as low as 30% of $ar{N}$ due to fluctuations.

Measurement accuracy deteriorates with drift distance, influenced by $N_{eff}$.

Simulation provides a practical way to estimate $N_{eff}$ in TPCs.

Abstract

For time projection chambers (TPCs) the accuracy in measurement of track coordinates along the pad row direction deteriorates with the drift distance ($z$): $\sigma_{\rm X}^2 \sim D_{\rm X}^2 \cdot z / N_{\rm eff}$, where $D_{\rm X}$ is the diffusion constant and $N_{\rm eff}$ is the effective number of electrons. Experimentally it has been shown that $N_{\rm eff}$ is smaller than the average number of drift electrons per pad row ($\overline{N}$). We estimated $N_{\rm eff}$ by means of a simple numerical simulation. The simulation shows that $N_{\rm eff}$ can be as small as $\sim$ 30\% of $\overline{N}$ due to the combined effect of statistical fluctuations in the number of drift electrons ($N$) and in their multiplication in avalanches.

Equations

$${\sigma }_{\mathrm{X}}^2={\sigma }_{0\mathrm{X}}^2+D_{\mathrm{X}}^2\cdot z,$$

$${\sigma }_{\mathrm{PR}}^2={\sigma }_{0\mathrm{PR}}^2+D_{\mathrm{T}}^2\cdot z,$$

$$D_{\mathrm{X}}^2=\frac{D_{\mathrm{T}}^2}{N_{\mathrm{eff}}},$$

$$N_{\mathrm{eff}}=\frac{\overline{N}}{R},$$

$$P(q_{i})=\frac{(1+\theta)^{1+\theta}}{\Gamma(1+\theta)}\cdot\Bigl{(}\frac{q_{i}}{\overline{q}}\Bigr{)}^{\theta}\cdot{\rm exp}\displaystyle\Bigl{(}-(1+\theta)\frac{q_{i}}{\overline{q}}\Bigr{)}\;,$$

$$X=\frac{\sum _{i=1}^N{q}_i\cdot x_i}{\sum _{i=1}^N{q}_i}.$$

$${\sigma }_X^2$$

$${\sigma }_{\mathrm{X}}^2=\frac{{\sigma }_x^2}{N}\cdot \frac{\overline{q^2}}{{\overline{q}}^2}\equiv \frac{{\sigma }_x^2}{N}\cdot (1+f),$$

$${\sigma }_{\mathrm{X}}^2$$

Full text

††journal: Nucl. Instr. and Meth. A

1 Introduction

The spatial resolution of TPCs along the pad row direction is expressed as

[TABLE]

where $\sigma_{0\rm X}$ is the intrinsic resolution, $D_{\rm X}$ takes into account the diffusion, and $z$ is the drift distance. On the other hand the width of pad response is given by

[TABLE]

with $D_{\rm T}$ being the transverse diffusion constant111In presence of axial magnetic field ($B$) the transverse diffusion constant is given by $D_{\rm T}(B=0)/\sqrt{1+\omega^{2}\tau^{2}}$, where $\omega\equiv eB/m$, the electron cyclotron frequency and $\tau$ is the mean free time of drift electrons. $D_{\rm T}$ is related to the diffusion coefficient ($D$) thorough $D_{\rm T}^{2}=2D/W$, where $W$ is the electron drift velocity. . Let us forget about $\sigma_{0\rm X}$ and $\sigma_{0\rm PR}$ here and focus on the $z$-dependent (diffusion) terms although the origins of the intrinsic terms themselves are of great interest.

$D_{\rm X}^{2}$ is expected to be proportional to $D_{\rm T}^{2}$ and expressed as

[TABLE]

where $N_{\rm eff}$ is the effective number of electrons. One may naively expect $N_{\rm eff}$ to be the average number of drift electrons per pad row ($\overline{N}$). In reality, however, $N_{\rm eff}$ is significantly smaller than $\overline{N}$ [1, 2, 3]:

[TABLE]

where $R$ is the reduction factor ($>1$).

In what follows an attempt is made to estimate $N_{\rm eff}$ (or equivalently, $R$) for a typical TPC operated in argon-based gases, by means of a numerical simulation.

2 Expectations

Let us assume:

Readout pads are aligned along the $x$ axis and charged particles traverse the drift volume of the TPC, in parallel with the readout plane ($z$ = constant) and in perpendicular to the pad rows ($x$ = constant = $x_{0}$).

• 2.

A charged particle leaves $N$ electrons along its path, to be detected by a single pad row. The initial electron clusters are considered as point-like. The $x$ coordinate of each electron then deviates from $x_{0}$ during the drift towards a detection gap222 The detection gap may be equipped with a multi-wire proportional chamber (MWPC) or a micro-pattern gas detector (MPGD), along with readout pads.

because of transverse diffusion. The standard deviation along the pad row direction at the entrance of the detection gap is denoted by $\sigma_{x}(\equiv D_{\rm T}\sqrt{z})$333 Let us ignore displacement of the coordinates caused by the $E$ $\times$ $B$ effect near the entrance of the detection gap and possible “quantization effects” on the coordinates due to finite granularity of amplifying elements in the case of MPGD readout. .

• 3.

Each drift electron gets amplified in the detection gap and the charge ($q$) is collected by (induced on) the pads. The charge multiplication process (avalanche) of each electron develops independently of those initiated by other electrons and the fluctuation of avalanche size for the $i$-th electron ($1\leq i\leq N$) is given by the Polya distribution [4]:

[TABLE]

where $\overline{q}$ is the average charge and $\theta$ is a free parameter determining the shape of distribution.

• 4.

The width of the pads is small enough so that the center of gravity of charge distribution on the pads caused by the $i$-th electron be equal to its arrival position $x_{i}$ at the entrance of the detection gap.

• 5.

The “measured” $x$ coordinate of the track ($X$) is defined as the charge centroid of the pad response, which is the superposition of the contribution from each of the drift electrons.

Under these assumptions and definitions the track coordinate measured with the pad row is given by a weighted mean of $x_{i}$:

[TABLE]

The expected spatial resolution is readily obtained as follows444 From now on, both angle brackets and an over-line denote the average of the quantity in-between or beneath. .

[TABLE]

where the symbol $\left<\;\cdot\cdot\cdot\;\right>_{x\;(q)}$ stands for the average taken over a variable (variables) $x\;(q_{i})$. Hereafter we assume $Q\equiv\sum\nolimits_{i=1}^{N}q_{i}={\rm constant}=N\cdot\overline{q}$, expecting $N\gg 1$. Then

[TABLE]

with $f\equiv\sigma_{\rm q}^{2}/\overline{q}^{2}$, the relative variance of $q$.

In fact, $N$ is not a constant and fluctuates according to its probability density function $P(N)$. The expected variance in this case is given by

[TABLE]

where $N_{\rm eff}\equiv\overline{N}/R$, with $R\equiv(1+f)\cdot\overline{N}\cdot\overline{N^{-1}}$. It is worth noting that one needs to evaluate $\overline{N^{-1}}$ instead of $\overline{N}$ for the estimation of $N_{\rm eff}$. The reduction factor ($R$) is a product of statistically independent quantities: $R=R_{\rm q}\cdot R_{\rm N}$, where $R_{\rm q}\equiv 1+f$ and $R_{\rm N}\equiv\overline{N}\cdot\overline{N^{-1}}$. The relative variance of Polya distribution is given by $f=1/(1+\theta)$. If we assume $\theta=0.5$, the frequently used value in wire-chamber simulations, $f=2/3$, yielding $\sim$ 1.67 for $R_{\rm q}$555 When $\theta=0$ the Polya function gives the Furry (exponential) distribution, which is sometimes used to describe avalanche fluctuations. In that case, $R_{\rm q}$ is expected to be $\sim$ 2. .

It should be noted, however, that $Q\equiv\sum\nolimits_{i=1}^{N}q_{i}$ has been assumed to be constant for a given number $N$. The validity of this approximation is checked by a numerical simulation (see the next section). As for the estimation of $R_{\rm N}$ we rely totally on the simulation.

3 Numerical simulation

3.1 Assumptions

The simulation was carried out taking into account the following:

Primary ionization statistics (Poissonian)

The mean number of primary ionization acts per pad row ($\overline{N_{\rm P}}$) is assumed to be $N_{\rm CL}$ $\times$ 0.63 cm $\times$ 1.2, where $N_{\rm CL}$ is the average primary ionization density for minimum ionizing particles [cm*-1*], 0.63 cm is the pad row pitch ( >$$\sim pad length) and the factor 1.2 takes account of the relativistic rise for 4 GeV/c pions (see, for example, fig. 8 of Ref. [5])666 The pad row pitch and the incident particles chosen here are typical of those in the experiments of Refs. [2, 3]. . We use 29.4 cm*-1* [6] or 24.3 cm*-1* [7] for the value of $N_{\rm CL}$ in argon. 2. 2.

Cluster size distribution

We use the probability density listed in table 2 of Ref. [8] for argon (column (b)). This distribution, combined with the primary ionizaion statistics mentioned above, generates the probability density function $P(N)$ in Eq. (1). 3. 3.

Diffusion: fluctuation of $x_{i}$ (Gaussian)

The standard normal distribution is chosen arbitrarily just for simplicity. This defines a dimensionless transverse distance scale. 4. 4.

Avalanche fluctuation

The Polya type with $\theta$ = 0.5. $\overline{q}$ is set arbitrarily to 1.

See Figs. 1–3 for the fundamental distributions. It should be noted that the chamber gas is assumed to be pure argon at NTP.

3.2 Results

Table 1 summarizes the results given by the simulation. See also Figs. 4–6 for the relevant distributions. The $x$ coordinate of particle tracks ($x_{0}$) is arbitrarily chosen to be 0 in Fig. 4. It should also be recalled that $\sigma_{x}$ is set to 1.0 in the simulation.

Table 1. Summary of the simulation results.

[TABLE]

The simulation results deserve several comments below:

We are not sure if the Polya distribution represents the fluctuation in the avalanche size correctly, especially in the case of (cascaded) GEM [9] or MicroMEGAS [10].

• 2.

Granting that the Polya type is an appropriate distribution, the value of parameter $\theta\/$ should depend on gas mixtures and also on the type of detection device: MWPC, GEMs or MicroMEGAS, and its operating high voltages.

• 3.

It may be a bold assumption to suppose no interference between avalanches initiated by different drift electrons when their distances are small.

• 4.

The simulation assumes pure argon as the chamber gas and possible effects of quenchers, such as increase of $\overline{N}$ due to the Penning effect, are not included.

• 5.

The simulation relies on a measurement of cluster size distribution [8], which is a little bit different from the result of a simulation [5].

• 6.

In reality, electron clusters created along a particle path are not point-like and have finite sizes in space (intrinsic track width). However, this hardly affects the $z$-dependence of spatial resolution ($D_{\rm X}$).

• 7.

Drift electrons experience further diffusion in the multiplication region (detection gap). However, this does not contribute to $D_{\rm X}$ defined in the drift region either.

• 8.

The simulation assumes ideal readout electronics, free from noise, and with a dynamic range wide enough to record tracks containing large cluster(s) and/or large avalanche(s) correctly without saturation of amplifiers or ADC-count overflows.

• 9.

$R_{\rm q}$ is (approximately) given once the value of $\theta\/$ is fixed. However, $R_{\rm N}(\equiv\overline{N}\cdot\overline{N^{-1}}$) is sensitive to the cut on the maximum cluster size ($n_{\rm max}$)777 The cluster size distribution has a long tail in proportion to $1/n^{2}$, with $n$ the cluster size. Therefore a cut has to be applied for large $n$ in order to keep the average total number of electrons ($\overline{N}$) finite. In the simulation $n_{\rm max}$ is set somewhat arbitrarily to reproduce $\overline{N}$ consistent with the total ionization density given in Refs. [6, 7] (94 cm*-1* for minimum ionizing particles).

since $\overline{N}$ inevitably depends on $n_{\rm max}$ while $\overline{N^{-1}}$ is fairly constant over a wide range of assumed $n_{\rm max}$888 This was confirmed by the simulation with several numbers between $10^{2}$ and $10^{6}$ for $n_{\rm max}$. . On the other hand $N_{\rm eff}\sim 1/\{(1+f)\cdot\overline{N^{-1}}\}$ is rather stable against variation of $n_{\rm max}$. It should be noted, however, that $N_{\rm eff}$ as well as $R_{\rm N}$ depends on $\overline{N_{\rm P}}$ ($N_{\rm CL}$) and on the probability densities assigned in the populated region of the cluster size distribution999 This was confirmed by intentionally manipulating the cluster size distribution. .

4 Summary

We may summarize the results as follows:

We have carried out a simple numerical simulation of the spatial resolution given by a TPC operated in argon-based gases, taking into account the fluctuations in the total number of drift electrons and in their multiplication in avalanches. The simulation gives 22–28 for the effective number of electrons ($N_{\rm eff}$) contributing to the coordinate measurement along the pad row, depending on the assumed primary ionization density ($N{\rm{}_{CL}}$). These numbers correspond to 30–40% of the average number of drift electrons per pad row made up of 6-mm-long pads ($\overline{N}$ $\sim$ 71).

• 2.

Contributions of the two fluctuations, i.e. ionization statistics and avalanche multiplication, to the reduction factor (R) are found to be of comparable size.

• 3.

$R_{\rm q}$ is close to the expected value quoted in section 2 ($\sim$ 1.67), justifying, to some extent, the “constant $Q$” approximation and the resultant Eq. (1).

• 4.

In view of the contribution of diffusion to the spatial resolution it is advantageous to use a gas with a small diffusion constant ($D_{\rm T}$), a large electron yield ($\propto\overline{N}$), a high average primary ionization density ($N_{\rm CL}$) and a small avalanche fluctuation ($f$)101010 Notice that $\sigma_{\rm X}^{2}=\sigma_{x}^{2}/N_{\rm eff}$ with $\sigma_{x}^{2}=D_{\rm T}^{2}\cdot z$, and that $N_{\rm eff}=\overline{N}/(R_{\rm q}\cdot R_{\rm N})$, where $R_{\rm q}\sim 1+f$ and $R_{\rm N}$ is a decreasing function of $\overline{N_{\rm p}}$ as is expected. We recall that $f$ should depend on the detection device and its operating conditions as well. , as well as long mean free time ($\tau$) in presence of axial magnetic field. In this respect longer pads are also advantageous in the case of conventional TPCs with discrete pad rows since $\overline{N_{\rm P}}$ (as well as $\overline{N}$) is larger, giving a smaller $R_{\rm N}$111111 It should be noted that $R_{\rm N}$ is greater than unity because of the difference between the mode and the average of $N$ distribution, which is smaller for larger $\overline{N_{\rm P}}$. See also Table 1 and compare $R_{\rm N}$ for the different values of $N_{\rm CL}$. , but at the expense of a larger angular pad effect.

• 5.

An argument similar to that employed in section 2 is applicable also to the coordinate measurement in the drift direction by substituting $Z$, $z_{i}$ and $\sigma_{z}$ respectively for $X$, $x_{i}$ and $\sigma_{x}$.

The presented method for estimation of the effective number of electrons can be applied for a given gas mixture and for a given amplifying and readout scheme including pad size if the relevant parameters are available.

Acknowledgments

The author would like to thank many colleagues for their continuous encouragement and support. He is especially grateful to Dr. Keisuke Fujii of KEK and Dr. Khalil Boudjemline at Carleton university for fruitful discussions.