Blum W., Riegler W., Rolandi L. Particle Detection with Drift Chambers

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6.6 Three Examples of Modern Drift Chamber Electronics 247

baseline is 30 ns for a v

/g = 2-fC pulse and 60 ns for a v

/g = 200-fC pulse. The

achieved ENC with attached drift tube is approximately 2300 electrons. The sen-

sitivity of g = 20 mV/fC provides a full-scale linear range for pulse heights from

0tov

/g = 200 fC. The last circuit element is a discriminator with two thresholds

providing the leading edge time (low threshold) and the identiﬁcation of an X-ray

photon hit (high threshold). A single-wire resolution of 100

m and efﬁciency of

80% at low rate is achieved. At a rate of 17 MHz, the resolution is 120

m and the

efﬁciency is 50% [AKE 04].

6.6.2 The ATLAS CSC Front-end Electronics

The ATLAS CSC fronted chip [OCO 99][JUN 05] is designed for readout of the

ATLAS cathode strip chambers [BEN 95]. These chambers measure muon tracks

with an accuracy of ≈ 50

m by using charge interpolation of cathode strips with a

pitch of 5.08 mm. The chambers are multiwire proportional chambers with a wire

pitch and wire-to-cathode distance of 2.54 mm. The gas used is an Ar/CO

mixture

operated at a gas gain of 2 ×10

. For a pitch of 5 mm and the speciﬁed position

resolution of 50

m, a signal-to-noise ratio of about 150 is required. This leads to a

speciﬁcation for an ENC of less than 2500 electrons for the chosen operating point.

The cathode strip capacitance ranges from 50 to 100 pF.

The front-end electronics developed for this detector consists of a bipolar shaper

with a peaking time of 75 ns, resulting in a total pulse length of about 300 ns. This

number is a compromise between noise performance and the inefﬁciency at the

highest expected rates of about 600 kHz. This bipolar shaper is not the one described

in the previous sections, but a shaper with one real pole and six complex poles,

called a seventh- order complex Gaussian ﬁlter [OHK 76]. It has the advantage of a

more symmetric response compared to the bipolar shaper realized by an equivalent

number of real coincident poles and having the same pulse-width. The achieved

noise performance is 1140+17.6e

−

/pF. The sensitivity is 3.8 mV/fC, which allows

a linear range of 450 fC. A single chip of 3.29×5.79 mm size contains 25 channels

and the power dissipation amounts to 32.5 mW/channel. The output signals are then

sampled at a 20 MHz rate and readout on a level 1 trigger to a 12 bit analog to

digital converter (ADC).

6.6.3 The PASA and ALTRO Electronics for the ALICE TPC

The PASA (preampliﬁer shaper) chip together with ALTRO (alice TPC readout)

chip [MUS 03] form the front-end of the time projection chamber for the ALICE

[ALI 95] [APP 99] experiment. This ALICE TPC consists of a cylindrical gas vol-

ume of about 90 m

under a uniform electrostatic ﬁeld and a drift distance of up

to 2.5 m. At the endplates, multiwire proportional chambers provide a readout by

248 6 Electronics for Drift Chambers

means of a cathode plane segmented into about 6 ×10

pads. These pads are con-

nected to the PASA chip, a 16-channel ampliﬁer chip providing a delta response

with a peaking time of t

≈150 ns and a full width at half maximum of about 200 ns.

The peaking time is realized by two so-called T-bridge second-order shapers result-

ing in two pairs of real poles. The transfer function is, however, very similar to an

n = 4 unipolar shaper. The diffusion of the gas used is quoted as 220

√

cm, so a

diffusion of

= 2.2 mm at a drift distance of 100 cm and

= 3.5 mm at a drift dis-

tance at 250 cm is expected. With a drift velocity of 28.3

m/ns this corresponds to

a FWHM of the arriving electron cloud of 180 and 290 ns. The choice of 200 ns for

the FWHM of the shaper is therefore a compromise in terms of optimum signal-to-

noise ratio and pulse width. The ENC of the PASA chip is < 1000 electrons in order

to allow a signal-to-noise ratio of 30 at the speciﬁed working point. The sensitivity

is g = 12 mV/fC with a linear range of 2 V.

The signal from the PASA is then fed into the ALTRO chip [BOS 03], which

integrates 16 channels, each consisting of a 10-bit, 10 MS/s ADC, followed by a

programmable digital baseline subtraction and tail cancellation ﬁlter, a zero suppres-

sion unit, and a multi-event buffer. All this functionality is contained in an 8×8mm

CMOS chip. The tail cancellation ﬁlter can be programmed to represent any fourth-

order ﬁlter, i.e., any transfer function consisting of a fourth-order polynomial in the

numerator and the denominator. The advantage of such an approach is evident: The

signal tail cancellation time constants do not have to be hardwired into the front-end

electronic chip but are fully programmable, which provides maximum ﬂexibility.

Clearly, the integration of the front-end chip in the ALTRO is a desirable next step.

We conclude this section by noting that the extreme miniaturization of micro-

electronic components, together with the reduced power consumption, has allowed

the placement of a large number of electronic channels very close to the detector,

which makes the systems very compact and economic.

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Chapter 7

Coordinate Measurement and Fundamental

Limits of Accuracy

7.1 Methods of Coordinate Measurement

After the passage of a charged particle through the sensitive gas volume, the elec-

trons produced in the ionization process along a trajectory segment drift toward the

sense wire, where they are collected and ampliﬁed in avalanches. There are essen-

tially four different methods to determine particle coordinates in drift chambers:

1. Measurement of the drift time. Using the known drift velocity of the ionization

electrons along their drift trajectory, this determines the distance along the drift

trajectory between wire and track.

2. Measurement of the pulse-height ratios on pick-up electrodes (strips, “pads”,

wires) near the sense-wire avalanche. With electrode response known this de-

termines the coordinate of the avalanche between the pick-up electrodes. If the

drift trajectory is known, the corresponding coordinate is determined. In this way

the coordinate along the sense wire is measurable.

3. Measurement of the pulse-height ratios at the two ends of a sense wire (charge

division). The position of the avalanche along the wire is determined using the

known damping of the pulse as it propagates along the wire.

4. Measurement of the difference of arrival times at the two ends of a sense wire

(time difference). This has been used for the same purpose, applying the known

propagation time of the signal on the wire.

Of these, methods 3 and 4 measure the coordinate along the wire direction and

method 1 measures the coordinate in the drift direction. The coordinate direction

of method 2, ﬁnally, is given by the arrangement of the pick-up electrodes. A com-

bination of method 1 with one of the others allows a three-dimensional position

measurement of the track segment to be made, provided the drift trajectories are

known.

Methods 1 and 2 are the most powerful in terms of the achievable accuracy be-

cause it turns out that the electronic measurement of the time and of the pulse-height

ratio on neighbouring electrodes can often be made better than the fundamental lim-

its imposed by the properties of the ionization that arrives on the wire.

W. Blum et al., Particle Detection with Drift Chambers, 251

doi: 10.1007/978-3-540-76684-1

 Springer-Verlag Berlin Heidelberg 2008

252 7 Coordinate Measurement and Fundamental Limits of Accuracy

Comparing methods 2 and 3 we notice that in both cases the coordinate is ob-

tained from a known function of a pulse-height ratio times a length: the distance D

between the pick-up electrodes in method 2 and the wire length L in the other case.

It is clear that the response (i.e. the change in pulse-height ratio per unit length of

avalanche displacement) with method 2 is better than the response with method 3

by the order of magnitude of the ratio L/D. The designer of the chamber has to pay

for this advantage with a correspondingly larger number of electronic channels.

Comparing methods 1 and 4, both based on a measurement of time, method 4

is less favourable because the response (i.e. the change in measured time per unit

length of coordinate displacement) is inversely proportional to the velocity involved,

this being the drift velocity u in the gas for method 1 and the signal propagation

velocity v (a good fraction of the speed of the light) along the sense wire for method

4. Therefore the response with method 1 is better than the response with method 4

by the order of magnitude of the ratio v/u, which has typical values of 10

to 10

It is not our goal to review here the various technical aspects of these methods,

such as pulse propagation, ampliﬁer noise or discriminator behaviour. Our interest is

focussed on the fundamental processes that create limitations which cannot be over-

come by technical improvements. In methods 1 and 2 we have techniques to measure

the coordinate such that the error resulting from imperfections of the electronics can

be made much smaller than that deriving from the fundamental processes.

If all the technical problems have been solved, which means that the wire po-

sitions are under control, the drift-velocity ﬁeld is known well enough (implying

the absence of unknown ﬁeld distortions of the electric and magnetic ﬁelds) and if

the electronics is capable of measuring the drift time or the pulse-height ratio of the

induced signals with sufﬁcient accuracy, then we are ultimately faced with the statis-

tical ﬂuctuations of the ﬁnite number of electrons involved in the measuring process.

These ﬂuctuations are strongly inﬂuenced by the clustering of the ionization pro-

cess itself. They are the fundamental limitation of any coordinate measurement in

drift chambers and become effective because of the presence of mechanisms which

spread the ionization cloud as it reaches the wire. In this chapter we will study the

most important mechanisms of spread: the diffusion in the drift toward the sense

wire, the drift path, and time variations due to the inhomogeneous electric ﬁeld near

the wire and those due to an angle the track might have with the wire. In the pres-

ence of a magnetic ﬁeld, the cylindrical electric ﬁeld near the wire can combine with

the magnetic ﬁeld and create a spread of ionization as if the track had an angle with

the wire.

The problem of accuracy can be described in the following terms. The ionization

electrons from the track element in question arrive at the wires; they are spread

both in arrival time and along the wires. What are the best estimates for the two

coordinates and what is their statistical variation?

In order to understand this better, we will ﬁrst analyse the spreading on a single

wire and how it depends on the track angles, the wire geometry, the magnetic and

electric ﬁelds and the diffusion. Then we have to discuss how much we gain from

the fact that there is not one electron but many electrons arriving. Here the nature of

the ionization, which is clustered, plays a role.

7.2 Basic Formulae for a Single Wire 253

Anticipating the result of this chapter, we summarize the beneﬁt of having many

electrons as follows. When diffusion is the main source of spreading, the increase in

accuracy is a function of the total number N of electrons, in the sense that the r.m.s.

measurement error decreases as 1/

√

N. When any of the other mechanisms is the

main source of spreading, the accuracy is a function only of N

eff

, a number which is

usually much smaller than N and depends on the length of the track segment as well

as on diffusion.

Recently another basic problem has been discovered which, in principle, is ca-

pable of limiting the measuring accuracy of drift chambers. Long wires may be

excited to vibrations by the unavoidable momentum transfer from the avalanches to

the wire. Coordinate measurements become less accurate to the extent that the wire

position is unknown. Although it is only in special circumstances large enough to

bother, the phenomenon is a fundamental limitation of coordinate measurements. It

will be treated in Sect. 7.5.

7.2 Basic Formulae for a Single Wire

For the coordinate measurements along the wire direction and along the drift direc-

tion there are some common aspects that we want to underline. In both cases the

accuracy is limited by a track angle which spreads the ionization at the wire and by

diffusion which ampliﬁes this spread.

The most general orientation a track might have with the wire plane is given

by two angles

and

. For the measurement along the wire, only the angle

between the track projected onto the wire plane and the wire is relevant, and for the

measurement along the drift direction it is the angle

between the track projected

onto the plane orthogonal to the wire and the wire plane.

We consider the simpliﬁed scheme of a drift chamber given in Figs. 7.1 and 7.2.

The sense wire is parallel to the x axis and contains the point of origin y = z = 0.

We show two regions: the “drift region”, where the ionization electrons drift and

Fig. 7.1 Scheme of a drift cell. For a study of the ionization spread on the wire, we consider a

track parallel to the x–y plane

254 7 Coordinate Measurement and Fundamental Limits of Accuracy

Fig. 7.2 Scheme of a drift cell. For a study of the spread of arrival times we consider a track

parallel to the z−y plane

diffuse, and a “wire region” where they are collected on the wire. The drift-velocity

vector is assumed constant in the drift region and the diffusion in the wire region is

neglected.

In Fig. 7.1 we have drawn the case relevant for the coordinate along the wire,

where the track may be assumed to be parallel to the x–y plane. In Fig. 7.2 we have

drawn a different track, parallel to the x–z plane, relevant for the coordinate along

the drift direction. In the following we discuss those aspects of the problem that are

common to the determination of both coordinates.

The ionization clusters are randomly distributed along the particle trajectory with

uniform distribution. An electron is collected by the wire if it arrives at −b/2 < y <

b/2 at the entrance of the wire region after its drift and its random diffusion. Because

of diffusion, electrons produced at y < −b/2oraty > b/2 may also be collected by

the wire and contribute to the measurement.

Since we neglect the diffusion in the wire region, the arrival position and time

of one electron on the wire are completely determined by its arrival coordinates and

time at the entrance to the wire region. We discuss ﬁrst the distribution of these

variables in the case of a single electron, and later we treat the statistical problems

connected with the ensemble of collected electrons.

7.2.1 Frequency Distribution of the Coordinates of a Single

Electron at the Entrance to the Wire Region

This distribution is determined by the trajectory of the particle and by the random

diffusion of the drifting electron in the drift region.

We begin with the particular case shown in Fig. 7.1: a charged particle parallel to

the x–y plane produces ionization clusters at coordinates x = x

+ ssin

,y = scos

and z = z

, where s is the coordinate along the track and

is the angle between the

perpendicular to the wire and the projection of the trajectory on the wire plane.

7.2 Basic Formulae for a Single Wire 255

The arrival position of an electron produced at coordinate s along the track is dis-

tributed according to a Gaussian diffusion around the production coordinate, while

s is uniformly distributed along the trajectory. Introducing the symbol G,

G(x|x,

√

πσ

exp



−

(x −x)



to denote the generic Gaussian function of average x and r.m.s.

we can write

down the frequency distribution of the arrival positions and time of the electron at

the entrance of the wire region (z = z

) as

F(x,y,t)dxdydt =

dxdydt

+R/2



−R/2

dsG(x|x

+ ssin

)

× G(y|scos

)G(t|(z

−z

)/u,

) . (7.1)

Here

/u denote the diffusion r.m.s. in the three directions (in the

most general case we should introduce the diffusion tensor) and u is the drift velocity

assumed along z. R is the length of a track arbitrarily chosen so large that Rcos

b

and Rcos



(i = x,y,z). In the following we will always take the limit R → ∞.

This frequency distribution has been written assuming that just one ionization

electron is produced along the track segment R. This electron is collected by the

wire and contributes to the coordinate measurement only if its arrival position at the

entrance to the wire region satisﬁes the condition −b/2 < y < b/2. The probability

that this electron is collected by the wire is

+∞



−∞

+∞



−∞

+b/2



−b/2

dyF(x,y,t)=

Rcos

(7.2)

and is equal to the ratio between the length of the portion of the track between the

planes y = −b/2 and y = b/2 and R.

If we are interested in the measurement of the coordinate along the wire direction

(x) alone, the readout electronics will integrate electrons arriving at different times:

the probability distribution of the arrival positions at the entrance of the wire region

for the electrons that are collected by the wire is obtained integrating the distribution

F(x,y,t) given by (7.1) in the variable t and normalizing it using (7.2):

(x,y)dxdy = dx dy

cos

+R/2



−R/2

dsG(x|x

+ ssin

)G(y|scos

) . (7.3)

Assuming that the x coordinate on the wire is the same as at the entrance to the

wire region we can compute the average and the variance of the arrival position at

the wire as

256 7 Coordinate Measurement and Fundamental Limits of Accuracy

x =

b/2



−b/2

+∞



−∞

(x,y)dx = x

, (7.4)

x

−x

tan

. (7.5)

The variance of x is the sum of three terms: the ﬁrst two depend on diffusion,

the third is the projection of the track segment on the wire; this latter we call the

angular wire effect. The factor 1/12 that divides the square of the width b of the drift

cell comes from the variance of the uniform distribution of the electrons along the

track.

7.2.2 Frequency Distribution of the Arrival Time of a Single

Electron at the Entrance to the Wire Region

We now turn to the particular case shown in Fig. 7.2: a charged particle parallel to

the y–z plane produces ionization clusters at coordinates x = x

,y = scos

and z =

+ ssin

, where s is the coordinate along the track and

is the angle between the

perpendicular to the drift-velocity direction (z) and the projection of the trajectory

on the wire z–y plane.

The frequency distribution of the arrival positions and time of the electron at the

entrance of the wire region (z = z

) is

F(x,y,t)dxdydt =

dxdydt

+R/2



−R/2

dsG(x|x

)

× G(y|scos

)G(t|(z

+ ssin

−z

)/u,

) . (7.6)

If we are interested in the measurement of the coordinate along the drift direction

z alone, the readout electronics will not distinguish among electrons arriving at dif-

ferent wire positions: the probability distribution of the arrival time at the entrance

to the wire region for the electrons that are collected by the wire is obtained by inte-

grating the distribution given by (7.6) in the variable x and normalizing it using (7.2):

(t, y)dtdy = dtdy

cos

+R/2



−R/2

dsG(t|(z

+ ssin

−z

)/u,

)

× G(y|scos

) . (7.7)

Assuming for the moment that the arrival time on the wire is equal to the time

at the entrance to the wire region, plus the constant shift z

/u, we can compute the

average and the variance of the arrival time at the wire;

7.2 Basic Formulae for a Single Wire 257

t =

b/2



−b/2

+∞



−∞

(t, y)dt = z

/u , (7.8)

t

−t

tan

12u

tan

. (7.9)

Comparing (7.3) and (7.7) we observe that one can obtain one distribution from

the other with the substitutions

↔ f

x ↔ z = tu ,

↔ z

−z

↔

u ,

↔

In the following, we discuss how the cluster ﬂuctuations inﬂuence the resolution

in the coordinate measurement along the wire direction (x). This will be done using

the probability distribution (7.3) and assuming that the x coordinate at the wire is

the same as at the entrance of the wire region. Analogously, the same conclusions

can be drawn with regard to the coordinate measurement in the drift direction, using

the formal substitutions (7.10). Drift-path variations are introduced in Sect. 7.4.

7.2.3 Inﬂuence of the Cluster Fluctuations on the Resolution – the

Effective Number of Electrons

Up to this point we have calculated the average and the variance of the position x

of a single electron (see (7.4) and (7.5)). For the measurement of the coordinate we

average over all electrons that are collected by the wire in a given time interval,

deﬁned by the readout electronics, and the variance of the coordinate measurement

will be reduced. As it turns out, this reduction is not given by a single factor. This is

because of the ionization clustering.

We measure the x coordinate averaging the arrival positions of N electrons:

+ x

+ ...+ x

. (7.10)

The variance of X

depends on the characteristics of the ionization process be-

cause the electrons are produced in clusters and therefore the x are, in general, not

independent.

Assume that the N electrons are grouped in M sets of n electrons, each set con-

taining the electrons that were produced in the same cluster and after drift and