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

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126 4 Ampliﬁcation of Ionization

Fig. 4.1 Schematic

development of the

proportional wire avalanche

develops in the longitudinal direction over as many free paths as there are gener-

ations, say typically over 50 to 100μm. This part ends after a fraction of a nanosec-

ond, under normal gas conditions. The physical processes inside the avalanche

are quite complicated, as they involve single and multiple ionization, optical and

metastable excitations, perhaps recombinations, and energy transfer by collisions

between atoms, much the same as discussed in Sect. 1.1. The de-excitation of

metastable states may in principle increase the duration of the avalanche up to

the collision time between the molecules, but not much is known about

this.

The proportional wire owes its name to the fact that the signal is proportional

to the number of electrons collected. This proportionality is possible to the extent

that the avalanche-induced changes of the electric ﬁeld remain negligible compared

to the ﬁeld of the wire. Referring to Sect. 4.5.1, we may characterize the regime

of proportionality as that in which the charge density in the avalanche is negligi-

ble compared with the linear charge density of the wire. The charge density of the

avalanche is given by the product of the gain factor and the number of electrons that

start the avalanche, divided by its width.

A very interesting role is played by the photons, which are as abundant as elec-

trons because the relevant cross-sections are of the same order of magnitude. A

small fraction of them will be energetic enough to ionize the gas, in all probability

ionizing a gas component with a low ionization potential. If it now happens that

these ionizing photons travel further, on the average, than the longitudinal size of

Fig. 4.2 Mechanism of

breakdown by

photoionization of the gas

outside the cylinder that

contains the full avalanche

4.1 The Proportional Wire 127

the avalanche, then the electrons they produce will each give rise to another full

avalanche and the counter may break down. In order to formulate a stability crite-

rion, we refer to the avalanche of Fig. 4.2, created by one electron e

. Let us call n

the number of electrons in the avalanche and n

the number of energetic photons,

and use q to denote the average probability that one of them ionizes the gas outside

the radius r

min

. Breakdown occurs if

q > 1, xn

q > 1,

where x is the ratio n

. This ratio may be expected to stay roughly constant as

is changed by a variation of the wire voltage.

Fig. 4.3a,b Optical properties of chamber gases: (a) Outstanding spectral emission lines of the

neutral Ar atom [HAN 81]. The intensities are approximate and are plotted on a logarithmic scale.

(b) Absorption cross-sections for UV photons of the normal alkanes [SCH 62]

128 4 Ampliﬁcation of Ionization

Table 4.1 Common quench gases

Methane CH

Ethane C

Propane C

Butane C

Pentane C

Isobutane (CH

)

CHCH

Carbon dioxide CO

Ethylene (C

)

Methylal CH

(CH

OH)

A similar consideration applies also to photons from the avalanche that reach the

conducting surfaces of the cathodes, where they may create free electrons by the

photoelectric effect.

It is because of these far-travelling photons that an organic ‘quench gas’ needs

to be present. Its effect is to reduce q allowing larger values of n

, and hence larger

gain. Organic molecules, with their many degrees of freedom, have large photoab-

sorption coefﬁcients over a range of wavelengths that is wider than that for noble

gas atoms. Some examples are given in Fig. 4.3a,b, which also contains a picture of

the most prominent emission lines of the argon. The photoabsorption spectrum of

argon was shown as a graph in Fig. 1.4 in the context of a calculation of particle ion-

ization. Table 4.1 contains a list of common quench gases. Inorganic quench gases

have been studied by Dwurazny, Jelen, and Rulikowska–Zabrebska [DWU 83].

4.2 Beyond the Proportional Mode

If the avalanche ampliﬁcation is increased beyond the region of proportionality, the

space charge of the positive ions reduces appreciably the ﬁeld near the head of the

avalanche, that is the ﬁeld experienced by the electrons between the wire and the

positive cloud. The ampliﬁcation is smaller for any subsequent increment and we

are in the regime of ‘limited proportionality’.

Near the tail of the avalanche, however, we have an increase of the electric

ﬁeld owing to the positive ions, especially once the fast-moving electrons have

disappeared into the wire. This situation leads to two different kinds of multiplica-

tive processes, depending on the behaviour of the photons:

• If the UV absorption of the quench gas is very strong, the photons in the

avalanche produce ionization near their creation point, including the region near

the tail of the avalanche where the electric ﬁeld is particularly large. This starts

the phenomenon of the ‘limited streamer’, which is a backward-moving multipli-

cation process in the sense of a series of avalanches whose starting points move

further and further away from the wire. In a paper by Atac, Tollestrup, and Potter

[ATA 82], it is suggested that energetic photons are created by the recombination

process with cool electrons when the ﬁeld at the avalanche head is sufﬁciently

4.2 Beyond the Proportional Mode 129

reduced. Under the inﬂuence of the growing charge, the starting points of suc-

cessive new avalanches move further outwards. Once the streamer has reached

a certain length that depends on the wire voltage, typically 1 to 3 mm, it comes

to an end. The total charge is almost independent of the amount of charge that

originally started the process. Figure 4.4a,b shows a typical pulse height distribu-

tion of the limited streamer for individual electrons (b) and for ∼100 electrons (a)

[BAT 85]: they give approximately the same total charge. The self-quench occurs

because the electric ﬁeld becomes weaker as the avalanches are produced further

away from the wire. But the exact mechanism is not well understood. More de-

tails can be found in the articles by Alekseev et al. [ALE 80], Iarocci [IAR 83]

and Atac et al. [ATA 82] and in the references quoted therein. A summary and

overview has been given by Alexeev, Kruglov, and Khazins [ALE 82].

As the high voltage of a wire tube with the correct quenching gas is increased

(Fig. 4.5), the collected charge at ﬁrst follows a nearly exponential mode (pro-

portional mode), then ﬂattens off as the avalanche charge density approaches the

wire charge density (limited proportional mode), and then suddenly jumps to the

limited streamer mode where the charge multiplication is one and a half orders

of magnitude larger. This behaviour is reported in Fig. 4.5. The collected charge

continues to rise more slowly up to the general breakdown of the counter. F.E.

Fig. 4.4a,b Streamer charge

distribution for (a)

-rays

and (b) photoelectrons from

[BAT 85]. Aluminium tube

1 ×1cm

cell size, 50μm

wire, gas mixture Ar(65%) +

Isobutane(65%), wire poten-

tial 3.7 kV

130 4 Ampliﬁcation of Ionization

Fig. 4.5 Collected charge

as a function of the high

voltage, measured by Atac

et al. [ATA 82] on a 100μm

diameter wire in a tube

12 ×12 mm

,ﬁlledwith

Ar(49.3%)+C

(49.3%)+

OH(1.4%)

Taylor argues that the size of the limited streamer is essentially given by electro-

static considerations [TAY 90].

• If the UV absorption of the gas is so low that ionizing photons can travel distances

that are comparable to the counter dimension, then multiplicative processes may

develop along the full length of the wire. The wire is said to be in Geiger mode

if the external electric circuit terminates the discharge with the help of a large

HV feed resistor. The size of the signal is therefore also independent of the

original charge. The Geiger counter is described, for example, in [KNO 79] and

[KOR 55].

4.3 Lateral Extent of the Avalanche

The electrons from an ionization track, which are collected on a wire and start an

avalanche there, do not usually arrive at the same point in space, but there is some

spread between them. This happens not only because the track was originally ex-

tended in space, and every electron is guided on its drift path to a different place

near the wire, but also because of the diffusion in the gas during the drift. The pro-

portional avalanche has a lateral extent that is at least as large as this spread.

In the avalanche process itself there is a lateral development associated with the

multiplication of charges, and this is mainly due to diffusion of electrons, the elec-

trostatic repulsion of charges, and the propagation of ionizing photons. The intrinsic

lateral size of an avalanche therefore depends on the gas (collision cross-section and

4.3 Lateral Extent of the Avalanche 131

UV absorption), on the number of charges in the avalanche and their density and on

the electron energy that is obtained in the various parts of the multiplication process.

If one wants to know whether the charges go fully around the wire or stay on one

side, then the wire diameter also plays an important role.

The spread of avalanches on a wire has been studied experimentally by Okuno

et al. [OKU 79], by observing the signals from the positive ions in a segmented

cathode tube surrounding the wire. In a mixture of Ar(90%)+CH

(10%),the

avalanches, created by a

Fe source on a wire of 25μm diameter, occupied only

100

◦

in azimuth (FWHM) in the proportional region (total charge below 10

elec-

trons). When the voltage was raised, the avalanche started to surround the anode

wire. Figure 4.6 shows how the azimuthal width increased with the total charge of

the avalanche. The increase came at smaller total charges when the concentration

of the quenching gas was smaller. The inﬂuence of the UV photons was more pro-

nounced at higher charge multiplication. The X-ray photon of the

Fe source has

an energy of 5.9 keV and creates approximately 227 electrons. The ﬁnite size of the

electron cloud produced by the X-ray photon can be neglected.

In a detailed Monte Carlo simulation of the scattering processes involved in

the multiplication, Matoba et al. have shown how a small avalanche develops

in three dimensions [MAT 85]. In Fig. 4.7 we reproduce a picture of their elec-

tron density. In recent years the computational techniques for the simulation of

such processes have been considerably advanced; see for example [GRO 89]. We

may expect more detailed insight into the dynamics of the avalanche as com-

puter codes are developed that describe not only the various collision phenomena

between the electrons, ions and gas molecules but also the important effect of the

photons.

Fig. 4.6 Angular spread (FWHM) of the avalanche from

Fe X-rays in various Ar + CH

mix-

tures as a function of the total charge in the avalanche. The anode wire had a diameter of 25μm

[OKU 79]

132 4 Ampliﬁcation of Ionization

Fig. 4.7 Two-dimensional

displays of the electron den-

sity in a small avalanche

created by a Monte Carlo

simulation from a single elec-

tron. Photon ionization was

neglected [MAT 85]

4.4 Ampliﬁcation Factor (Gain) of the Proportional Wire

The multiplication of ionization is described by the ﬁrst Townsend coefﬁcient

.If

multiplication occurs, the increase of the number of electrons per path ds is given

dN = N

ds. (4.1)

The coefﬁcient

is determined by the excitation and ionization cross sections of the

electrons that have acquired sufﬁcient energy in the ﬁeld. It also depends on the vari-

ous transfer mechanism discussed in Chap. 1. Therefore, no fundamental expression

exists for

and it must be measured for every gas mixture. It also depends on the

electric ﬁeld E and increases with the ﬁeld because the ionization cross-section goes

up from threshold as the collision energy

increases. If the gas density

is changed

while keeping the distribution of

ﬁxed (that is, at ﬁxed E/

), then

changes pro-

portionally with the density because all the linear dimensions in the gas scale with

the mean free collision length (cf. Chap. 2). Therefore we may write the functional

dependence of





= f





. (4.2)

4.4 Ampliﬁcation Factor (Gain) of the Proportional Wire 133

Fig. 4.8 First Townsend

ionization coefﬁcient

mea-

sured for some noble gases as

a function of the electric ﬁeld

E, collected by von Engel

[ENG 56]. The plots show

/p versus E/p,wherep is

the gas pressure

In Fig. 4.8 we show measurements with pure noble gases in the form of

over

the gas pressure as a function of the electric ﬁeld over the gas pressure, at normal

temperature, as obtained by von Engel [ENG 56].

The ampliﬁcation factor on a wire is given by integrating (4.1) between the point

min

where the ﬁeld is just sufﬁcient to start the avalanche and the wire radius a:

N/N

= exp



min

(s)ds = exp

E(a)



min

(E)

dE/ds

dE . (4.3)

Here N and N

are the ﬁnal and initial number of electrons in the avalanche; dE/ds is

the electric ﬁeld gradient. The minimal ﬁeld E

min

needed for ionization to multiply

is given by the energy required to ionize the molecules in question, divided by the

mean free path between collisions; therefore E

min

must be proportional to the gas

density.

The electric ﬁeld near a wire whose radius is small compared with the distance to

other electrodes is given by the charge per unit length,

, as a function of the radius:

E(r)=

πε

, (4.4)

where

= 8.85 ×10

−12

As/(Vm),and

can be derived from the wire voltages

and the capacitance matrix (cf. Chap. 3). Inserting (4.4) into (4.3) we have

N/N

= exp

E(a)



min

λα

(E)

πε

dE . (4.5)

134 4 Ampliﬁcation of Ionization

The ratio N/N

is usually called gain. In the following paragraphs we use G =

N/N

to indicate it.

4.4.1 The Diethorn Formula

Diethorn [DIE 56] derived a useful formula for G by assuming

to be proportional

to E. Looking at Fig. 4.8, this is not unreasonable for heavy noble gases between

and 10

[V/cm Torr], a typical range of ﬁelds near thin wires. Inserting

into (4.5) gives

lnG =

βλ

πε

min

. (4.6)

The value of

can be related in the following way to the average energy e

required to produce one more electron. The potential difference between r = a and

r = s

min

that corresponds to relation (4.3) is

(a) −

min



E(r)dr =

πε

min

πε

min

. (4.7)

It gives rise to a number Z of generations of doubling the electrons in the avalanche:

Z =[

(a) −

min

)]/

V , (4.8)

G = 2

, (4.9)

lnG =

ln2

πε

min

. (4.10)

We recognize that in this model the constant

of (4.6) has the meaning of the

inverse of the average potential required to produce one electron in the avalanche

multiplied by ln 2.

We write E

min

, which is proportional to the gas density

, in the form

min

(

)=E

min

(

)

, (4.11)

where

is the normal gas density. We ﬁnally obtain

lnG =

ln2

πε

min

(

)(

)

. (4.12)

Relation (4.12) was compared with measurements using proportional counter

tubes, and was shown to be better than the older models of Rose–Korff and Curran–

Craggs; see [HEN 72, WOL 74, KIS 60] and the literature quoted therein. In a

proportional counter tube with inner radius b and wire radius a, the charge density

is related to the voltage V by

4.4 Ampliﬁcation Factor (Gain) of the Proportional Wire 135

πε

ln(b/a)

. (4.13)

Therefore, (4.12) can also be expressed in the following form (Diethorn’s formula):

lnG =

ln2

ln(b/a)

ln(b/a)aE

min

(

)(

)

. (4.14)

Experimentally, we vary

,a and V/ ln(b/a), and measure G. A plot of

lnGln(b/a)

versus ln

ln(b/a)a(

)

must be linear and yields the two constants E

min

(

) and

V. Figure 4.9 shows

such a plot made by Hendricks [HEN 72] for two xenon gas mixtures, and Table 4.2

contains a collection of Diethorn parameters measured in this way, typically for

ampliﬁcation factors between 10

and 10

. They are only in moderate agreement

with each other.

We recognize the value of formulae (4.12) and (4.14), discussed above, more

from a practical than from a fundamental point of view. Obviously, for a given gas

mixture, the Townsend coefﬁcient

(E) must ﬁrst be known accurately as a function

of the electric ﬁeld. Then the multiplication factor can be calculated using (4.5). See

also Charles [CHA 72] and Shalev and Hopstone [SHA 78], who compared and

criticized various formulae for the gas gain.

Fig. 4.9 Diethorn plot of

ampliﬁcation measurements

using two Xe gas mixtures

[HEN 72]; p =