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

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278 7 Coordinate Measurement and Fundamental Limits of Accuracy

Fig. 7.16 Schematic view of the experimental setup of [BOY 95]

thousands of these tiny bits of momentum can displace the wire and make it vibrate.

The measurement accuracy is limited to the extent that the wire position is unknown.

Boyko et al. [BOY 95] used a long drift tube and observed wire vibrations

through a window with a microscope (see Fig. 7.16). A radioactive source could

be brought close to the tube to produce counts of the frequency and pulseheight that

were registered. Wire vibration amplitudes up to 30

m were observed at count rates

up to 4.5 kHz.

7.5.1 Linear Harmonic Oscillator Driven by Random Pulses

Before we treat the motion of the wire we look at the simpler case of the linear

harmonic oscillator under the inﬂuence of a driving force consisting of an irregular

(random) sequence of short impact pulses (see Fig. 7.17).

The equation of motion for the one-dimensional damped harmonic oscillator that

is excited by a force of very short duration at time t

x(t)

dx(t)

+ kx(t)= f (t) f (t)= f

(t −t

), (7.53)

where f

characterizes the size of the short impact and has the dimension of

momentum. The solution of this equation is

x(t)=

−

(t−t

)

sin

(t −t

)



1−

4km



. (7.54)

In the case of several consecutive forces at times t

and with amplitudes f

, f (t)=

(t −t

), the solution takes the form

Fig. 7.17 Harmonic

oscillator, characterized by

spring constant k, damping

constant

, and mass m,

driven by a random sequence

of short impact pulses

k , Γ

7.5 Accuracy Limitation Owing to Wire Vibrations 279

x(t)=

∑

−

(t−t

)

sin

(t −t

). (7.55)

For such a superposition of pulses of equal shape with amplitudes f

from a

probability distribution P( f ), separated by random time intervals with an average

frequency

, Campbell’s theorem [CAM 09][RIC 44] applies. This theorem tells us

that the average

x and the variance

of x(t) are given by

x =



∞



−

sin



dt =

(7.56)



∞



−

sin



dt =

, (7.57)

where

f =



fP( f )df and f



P( f )df. In a case in which all pulses have the

same amplitude f

,wehaveP( f )=

( f − f

) and thus f = f

and f

= f

.The

r.m.s. of the amplitude then becomes

= f



. (7.58)

We see that the ﬂuctuation of the position increases with the rate

but decreases

for increased damping

and spring constant k. The fact that

is independent of m

is due to a balance of primary excitation and damping. For larger m, a single pulse

produces a smaller amplitude, a smaller damping factor

/2m, and a decreased

frequency

, but increases the duration of the inﬂuence of f

7.5.2 Wire Excited by Avalanche Ions

The avalanche ions moving from the wire surface to the tube wall exert a force on the

wire. Since the electric ﬁeld in the tube at radius R is given by E(r)=V /(r lnR/a),

the resulting force on the wire by a charge q at position r is given by F(r)=qE(r).

The movement of the ions from the wire surface to the tube wall with the velocity

v(r)=

E(r) transfers the total momentum of

p =



F(t)dt =



F(r)

dr =



F(r)

v(r)

dr =

(7.59)

to the wire. Since the maximum drift time of the ions is typically much shorter

than any time associated with the mechanical movement of the wire (i.e., the period

of the ﬁrst harmonic), one can assume the action of the repulsive force to be f =

qR/

μδ

(t).

The equation determining the wire displacement y(x,t) in a cylindrical drift tube

of radius R and wire radius a in the presence of an external force density f (x,t) is

given by

280 7 Coordinate Measurement and Fundamental Limits of Accuracy

∂

−T

∂

−

y = −g

+ f (x,t), (7.60)

where

is the linear wire density,

the friction coefﬁcient and T the wire tension.

The term

= 2

πε

/(RlnR/a)

represents the electrostatic force, V is the wire

voltage and g

the gravitational force. The expression f (x,t) due to the avalanches

of charge q

at time t

, position x

and angular position

f (x,t)=

∑

cos

(t −t

)

(x −x

). (7.61)

The solution of Eq. (7.60) and the statistical analysis can be performed in analogy

to the simple example from the previous section. The solution obviously depends

on the way the excitations are distributed in the azimuthal direction and along the

wire, and whether the avalanches are created from a point source or from tracks.

Boyko et al. have shown that these different conditions can be collected into one

dimensionless constant. They ﬁnd the resulting variance of the position of the wire,

averaged over the wire length, to be given by



12T

√

, (7.62)

where

q is the average total charge per event,

is the total particle rate on the

tube, and K, K

are dimensionless geometry factors which together make up

one constant. This solution assumes that the avalanche ions return to the tube wall

along the trajectories of the primary ionization electrons. For point ionization we

have K = 1 and K

= 0.5. For particle tracks through the tube we have K = 0.416

and K

= 0.5–1 depending on the angular distribution of the tracks. K

is another

geometry factor being ≈1 for uniform illumination along the tube.

Figure 7.18 shows results of a calculation assuming a drift tube of R = 1cm,

mobility

= 1.7cm

/Vs and a damping factor

= 7 × 10

−5

kg/sm. The wire

Fig. 7.18 Wire vibration

amplitudes calculated by

[BOY 95] for a practical case

(see text). The uncertainty of

the wire position is shown as

a function of the count rate.

The three bands belong to the

three different avalanche sizes

indicated; in each band the

upper edge is valid for a wire

tension of 50 g and the lower

for one of 500 g

7.6 Accuracy Limitation Owing to Space Charge Fluctuations 281

tension is varied from 50 to 500 g. The diagram makes it clear that wire vibrations

can be a decisive factor for detectors operating in limited streamer mode (charges of

10–100 pC) at ﬂuxes >10

particles/s. In proportional mode (charges around 1 pC)

the vibration makes a signiﬁcant contribution (>10

m) only in the case of lower

ion mobility (larger pressure), lower wire tension, smaller wire damping, and rates

beyond 1 MHz.

7.6 Accuracy Limitation Owing to Space Charge Fluctuations

There are situations in which the drift time measurement becomes inaccurate owing

to ﬂuctuations in the drift ﬁeld. As an example we discuss the ATLAS MDT drift

tubes [ATL 97]. It is expected that their momentum measuring accuracy will be

reduced when the photon background rate goes beyond the design level. In the

tubes, the background

-radiation will produce a space charge whose density de-

pends on the counting rate, the ion charge per count, the ion drift velocity, and

the tube diameter. The space charge density is homogeneous – which produces a

ﬁeld (increasing linearly from the wire to the tube wall) in addition to the main

ﬁeld (proportional to 1/r). This was explained in Sect. 4.5.3 in the context of

the reduction of ampliﬁcation. Using the method outlined there, we can estimate

that at a background count rate of 1.4 kHz per cm of wire length (roughly ﬁve

times the design value), the drift ﬁeld is distorted to an extent that an apparent

shift of track positions of the order of a millimetre takes place in part of the drift

space.

This effect can in principle be corrected by carefully monitoring the history of

the previous few milliseconds (the ion travel time to the tube wall). If the tube works

perfectly, and if the charge associated with each previous pulse is known, it might be

possible to assess the total ion charge present in the tube at the moment of recording

the track in question. Then the drift ﬁeld distortion may be calculated and the track

position corrected.

However, this procedure is limited by the inhomogeneity of the charge density

along the tube. Even if there is some knowledge about the average density of counts

along the tube, there is the statistical ﬂuctuation in the immediate vicinity of the

recorded track. Before we report a measurement quantifying this effect, let us es-

timate what we can expect. At the ﬂux rate of 1400 Hz/cm the charge that will

contribute to the disturbing ﬁeld is the one produced in roughly 3 cm of tube length

(approximately the tube diameter) in 5 ms (the ion travel time to the wall); this cor-

responds to 1400Hz/cm ×5ms×3cm = 21 avalanches on average. As these are

occurring at random we expect a ﬂuctuation of 1/

√

21 or 0.22 of the total shift.

Thus if the total shift goes up to the order of 1 mm it will ﬂuctuate on the order of

0.2 mm.

In an experiment using a strong radioactive

-source the ATLAS muon team

[ATL 00] irradiated the tubes in the drift chambers, creating background condi-

tions ﬁve times as severe as expected for the most heavily irradiated parts of the

282 7 Coordinate Measurement and Fundamental Limits of Accuracy

Fig. 7.19 Resolution as a

function of distance from the

wire, comparison of

measurements and

simulations. Full squares:

measurement without

background; open circles:

measurement with a

background rate of

1400 Hz/cm (4.5 kHz per

tube); lines: simulations

without and with background

(36 keV per photon)

spectrometer. The coordinates of high-energy muon tracks were measured at right

angles to the wires, and the resolution was determined. The result of the experiment

is shown in Fig. 7.19. The resolution deteriorates enormously in the outer region of

the tube owing to the presence of the background, and assumes values up to 140

A detailed simulation along the ideas outlined above was carried out and compared

to the measurements. The curves in Fig. 7.19 match the data and show that the effect

is understood.

Finally, let us be aware that it is the problem of ageing to which one owes this

critical dependence of measuring accuracy on background. Had not the hydrocar-

bons been outlawed one could ﬁnd gas mixtures with a much reduced dependence

of the drift velocity on the electric ﬁeld (cf. the remarks at the end of Sect. 12.6.4).

Appendix to Chapter 7. Inﬂuence of Cluster Fluctuations

on the Measurement Accuracy of a Single Wire

An analytical form of Eq. (7.15) can be derived under certain assumptions as fol-

lows. The symbols used here were deﬁned in Sect. 7.2.

We consider the N

clusters produced along the track segment R. N

represents

the clusters of cluster size i, where N

= N

P(i) and P(i) is the cluster-size distribu-

tion deﬁned in Sect. 1.2.2 [Eq. (1.34)].

The N

clusters produce a total of iN

electrons and on average m

of them,

= iN

P(i)

Rcos

= iN

Rcos

, (7.63)

are collected by the wire. The total number N of electrons collected by the wire is

7.6 Accuracy Limitation Owing to Space Charge Fluctuations 283

N =

∑

where the sum on i is performed on the cluster size i.

As we showed in Sect. 7.2.3, N does not depend on diffusion, and we assume in

the following that neither does each m

Deﬁning ¯x

to be the average position of the m

electrons collected by the wire

and produced in clusters of original cluster size i, the quantity X

deﬁned by (7.10)

can be rewritten as

∑

¯x

∑

¯x

. (7.64)

Its variance is given simply by

= Var (X

∑

Var( ¯x

)

, (7.65)

since the various ¯x

are not correlated because they are by deﬁnition averages of

arrival positions of electrons produced in different clusters.

The variance of ¯x

,Var( ¯x

), depends on the distribution of the number k(k =

0,...,i) of collected electrons when i are produced in the same cluster. Referring to

Eqs. (7.1) and (7.3) and assuming for simplicity x

= 0, we ﬁnd that the probability

that k electrons among i are collected is given by

+R/2



−R/2

⎡

⎣

+∞



−∞

dxG(x|ssin

)

⎤

⎦

⎡

⎢

⎣

+b/2



−b/2

dyG(y|scos

)

⎤

⎥

⎦

⎡

⎢

⎣

1−

+b/2



−b/2

dyG(y|scos

)

⎤

⎥

⎦

i−k





, (7.66)

where the term





takes into account the different combinations of k electrons

among i.

Since we have assumed that x

= 0, we also have that

x = ¯x

 = X

 = 0 . (7.67)

The integrals in dx of (7.66) are equal to 1. Deﬁning

(scos

+b/2



−b/2

dyG(y|scos

) (7.68)

as the probability that one electron is collected by the wire (0 <

< 1), we rewrite

the probability that k among i electrons produced in the same cluster are collected

284 7 Coordinate Measurement and Fundamental Limits of Accuracy

by the wire as

+R/2



−R/2

(1−

)

i−k





ds . (7.69)

We note that

∑

k=0

= 1 .

If along the track segment R we produce N

clusters of i electrons, on the wire we

collect on average

• N

single electrons

• N

pairs of electrons

• N

triplets of electrons

• ...

• N

ith of electrons

and in total we have

∑

k=0

= m

electrons. This last equality can be proved using (7.69) and (7.68) and the properties

of the binomial distribution.

The number N

pairs

of different pairs of electrons produced in the same cluster of

cluster size i and collected by the wire is given by

pairs

= N

∑

k=2







∑

k=0

−

∑

k=0



∑

k=0

−

. (7.70)

We are now in a position to compute the variance Var (¯x

) of the average position

¯x

of the m

electrons produced by clusters of cluster size i and collected by the wire:

we have m

variance terms x

 and 2 ×N

pairs

covariance terms xx. The average

¯x

 is zero because we have assumed that x

= 0 in (7.66) [see also (7.67)]. We

obtain

Var( ¯x

x

+ N

pairs

xx

x

+

∑

k=0

−m

xx



(x

−xx)+

∑

k=0

∑

k=0

xx



. (7.71)

7.6 Accuracy Limitation Owing to Space Charge Fluctuations 285

Using (7.69) and the properties of the binomial distribution, one can show that

∑

k=0

−i

+R/2



−R/2

ds +

+R/2



−R/2

ds ,

∑

k=0

+R/2



−R/2

ds .

From these equations we obtain

∑

k=0

∑

k=0

= 1+(i −1)I

where

+R/2



−R/2

+R/2



−R/2

, (7.72)

and we can rewrite (7.71) as

Var( ¯x

(x

−I

xx+ iI

xx) . (7.73)

Finally we substitute the variance Var(¯x

) from this last equation into (7.65) and

set

Var(X

∑

(x

−I

xx+ iI

xx)

x

−I

xx

∑

xx . (7.74)

Using (7.63) we can show that

∑

P(i)[b/(Rcos

)]

∑

P(i)[b/(Rcos

)]

(

∑

P(i)[b/(Rcos

)])

∑

b/(Rcos

)

eff

, (7.75)

286 7 Coordinate Measurement and Fundamental Limits of Accuracy

where the sum over i is performed on the cluster size i, while the sum over j is

performed on the N

clusters produced by the particle in the track segment R and n

is the number of electrons in the cluster j.

eff

was deﬁned in Sect. 1.2.5 as the effective number of electrons computed

along the track segment R; here it is scaled by the normalization factor b/Rcos

derived in (7.2). In Sect. 1.2.5 we showed that N

eff

does not scale linearly with the

length of track sampled, while (7.75) implies a linear dependence of N

eff

on R.This

is an implicit consequence of the assumption that the m

are constant, since we do

not take into account the effect of the rare and large clusters present in the tails

of the cluster-size distribution, and this effect is a limitation of our derivation. We

comment on this point at the end of this appendix.

The quantity I

deﬁned in (7.72) can be computed using (7.68):

= Erf





−

√



1−exp



−



, (7.76)

where Erf is the error function [ABR 64],

Erf(z)=

√



exp(−t

)dt .

In the limit

 b,I

→ 1, since Erf(∞)=1. When

 b, then I

→ b/

√

πσ

) → 0.

In order to compute explicitly the variance Var(X

) of (7.74) we have to evaluate

the correlation term xx.

The covariance of the arrival position of two electrons produced in the same

cluster is evaluated from the distribution function of their arrival positions. Starting

from (7.1) and (7.3) we write it as

+R/2



−R/2

dsG(x

|ssin

)G(y

|scos

)

× G(x

|ssin

)G(y

|scos

), (7.77)

where the two indices 1 and 2 refer to the two electrons.

This distribution is not normalized. Its normalization is the probability that two

electrons produced in the same cluster are both collected by the wire. It can be

shown that

+∞



−∞

b/2



−b/2

∞



−∞

b/2



−b/2

)=I

cos

, (7.78)

7.6 Accuracy Limitation Owing to Space Charge Fluctuations 287

where I

is deﬁned by (7.72). Since in (7.74) we have the product I

xx, we can

compute directly

xx =

cos

+∞



−∞

b/2



−b/2

+∞



−∞

+b/2



−b/2

) , (7.79)

and after some tedious integration we obtain

xx = tan









Erf





−

√



1−exp



−



+ b

√



1−exp



−



−



−

√



1−exp



−



. (7.80)

Now we have all the information we require and can write down the ﬁnal formula

for the variance Var(X

). Using (7.74), (7.75), (7.5) and (7.80) yields

= Var (X

+ tan

θσ





eff

−









+tan





eff

−









, (7.81)

where the two functions F

and F

are given by

= Erf





−

√



1−exp



−



(7.82)

= Erf





−

√



1−exp



−



1−8



−

√

. (7.83)

These are equal to 1 when

/b is zero and go to zero when

/b →∞. Inspecting

(7.81) we note that this behaviour corresponds to a smooth transition between the

coefﬁcients 1/N

eff

and 1/N in the angular terms. In the absence of diffusion there is

complete correlation and the reduction factor of the angular term in the variance of