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

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86 2 The Drift of Electrons and Ions in Gases

Fig. 2.17 Drift velocity u in

an argon–methane mixture

as a function of the electric

ﬁeld E, measured by Jean-

Marie et al. [JEA 77]. The

lowest curve is for 90%

Ar + 10% CH

etc.

up to E

= 900V/cm for pure CH

. For every concentration, the maximum occurs

at the electric-ﬁeld value where the average electron energy is at the Ramsauer min-

imum of the momentum transfer cross-section (e = 0.3eV for CH

and 0.25 eV for

Ar). According to (2.19), the peak behaviour of u

is determined mostly by the

(

)

in the denominator because the factor



λ(

) varies much less.

We can actually measure the values of

for every E and R, and check that the

Ramsauer minimum coincides with the velocity peak. For this we make use of the

systematic measurements of the width

of the single-electron diffusion conducted

by the Berkeley group [PEP 76]. Their results have been rescaled to 1 cm drift length

at 760 Torr, and are plotted in Fig. 2.18. They exhibit a fairly constant behaviour of

the diffusion when E is changed, and an increase for smaller CH

concentrations.

Referring to our equation (2.63), we may calculate the electron energy for every

E and every mixing ratio R:

(E, R)=

(E, R)

1cm

eE, (2.88)

This has been graphed in Fig. 2.19, where we have also marked on the horizontal

scale the velocity peaks for the various values of R. We observe that the measured

electron energies are in the Ramsauer minimum at the velocity peaks (see Fig. 2.2a).

2.4 Applications 87

Fig. 2.18 Width of the diffu-

sion cloud after a drift of 1 cm

for various mixtures of CH

and Ar at 760 Torr. The upper

group of points refer to 10%

+ 90% Ar, etc. Measure-

ments by the Berkeley TPC

group [PEP 76] were done at

600 Torr over a drift length

of 15 cm

Fig. 2.19 Electron energy

deﬁned in (2.63) as a function

of the electric ﬁeld for

various mixtures of CH

+ Ar

at 760 Torr. The data of

Fig. 2.18 were used. The

arrows on the horizontal scale

indicate the position of the

maxima of the drift velocities

according to Fig. 2.17

88 2 The Drift of Electrons and Ions in Gases

2.4.3 Experimental Check of the Universal Drift Velocity

for Large

ωτ

In a drift experiment with crossed electric and magnetic ﬁelds, Merck [MER 89]

measured drift angles and drift velocities at atmospheric pressure in Ar + CH

mixtures that are known to have large values of

under typical drift-chamber con-

ditions. Using the tangent of the drift angle as measure for

ωτ

(2.11), he found the

drift velocities that are plotted as a function of

ωτ

in Fig. 2.20a,b. We observe that

they do approach the limit

u →

quickly from below. However, the approximation (2.54) is not very good, as dif-

ferent combinations of E and B that lead to the same

ωτ

do not yield the same

Fig. 2.20a,b Measured drift

velocity in units of E/B as

a function of the tangent

of the observed drift angle,

for various magnetic ﬁelds.

(a) Ar (50%)+CH

(50%),

(b) Ar (95%)+CH

(5%)

[MER 89]

2.4 Applications 89

u, especially at low concentrations of CH

. We would expect that the more detailed

theory involving the electron energy distribution is required in order to explain these

variations.

2.4.4 A Measurement of Track Displacement as a Function

of Magnetic Field

In a TPC with almost perfect parallelism between B and E, the small angle a of

misalignment causes a displacement of every track element that is proportional to

the drift length L. If we imagine the coordinate system aligned with the E ﬁeld along

z and with small extra B components along x and y, there will be displacements at

the wire plane, equal to

= Lu

and

= Lu

; according to (2.6), these are

= L



−

ωτ



= L



ωτ



(2.89)

with

= B

and

= B

. It is by reversing the magnetic ﬁeld that the

different contributions can be separated. The symmetric part of

[

(B)+

(−B)] = L

, (2.90)

and its antisymmetric part is

[

(B) −

(−B)] = L

−

ωτ

. (2.91)

In Fig. 2.21 we see measured displacements [AME 85] for two different values

of L and a range of values of B. Fitting the symmetric part in Fig. 2.21 with the func-

tional form of (2.90) yielded

= 0.29 ×10

−10

s, and the antisymmetric part ﬁtted

with (2.91) yielded

= 0.35 ×10

−10

s, for the particular conditions of the experi-

ment (argon (91%) + methane (9%), 1 bar, E = 110V/cm, u = 5.05cm/μs). These

values of

are not the same because they represent averages over the distribution

functions, which are written explicitly in (2.84).

2.4.5 A Measurement of the Magnetic Anisotropy of Diffusion

In a TPC with parallel E and B ﬁelds, the transverse width

diff

of the electron

cloud originating from a laser track was measured for various drift distances L and

magnetic ﬁelds B [AME 86]. Since this width is related to the diffusion constant D

and the known drift velocity u by

90 2 The Drift of Electrons and Ions in Gases

Fig. 2.21 Measured track displacements due to a small angle between the electric and magnetic

ﬁelds; (a) symmetric part, (b) antisymmetric part. The curves are ﬁts to (2.90) and (2.91). Curves

A: L = 126cm; curves B: L = 65cm [AME 85]

diff

(L,B)=2D(B)L/u,

we determined D(B). The quantity actually observed,

, was the quadratic sum of

diff

and of a constant,

, given by the width of the laser beam and the electrode

conﬁguration. Figure 2.22 depicts the observations.

If we express the factor D(0)/D(B) by which the magnetic ﬁeld reduces the

diffusion constant as a function of B

, we see a linear behaviour for small B

as well

Fig. 2.22 Measured dependence of

on the magnetic ﬁeld and the drift length. The slope

decreases with increasing B as the diffusion is reduced [AME 86]

2.4 Applications 91

Fig. 2.23 The factor by which

the magnetic ﬁeld reduces the

transverse diffusion in the

example elaborated in the

text. It is a linear function of

for both small and large B

[AME 86]

as for large B

. The transition region between the two straight lines is around 6kG

(see Fig. 2.23). The interpretation is the following.

Referring to (2.84) and (2.87) we note that the off-diagonal elements of the dif-

fusion tensor do not contribute to

diff

. We write

(

∞



(c)c

dc =



)



This expression can be evaluated in the two limiting cases of

small and

large compared with

, the square of a typical collision frequency

. For this

typical frequency we may take the value derived from the electron mobility

in the

limit in which

is independent of c :

=(e/m)/

. We ﬁnd in lowest order

(0)

(B)

= 1+



(0)

(B)

= C +



(2.92)

The values of

c



c



c



c



C =

c

c



c



(2.93)

are independent of

, because the two ﬁelds are parallel to each other.

Since

is a function of c and is therefore distributed over a wide range of

values, the averages in (2.93) are all different. In fact the straight lines in Fig. 2.23

92 2 The Drift of Electrons and Ions in Gases

are described by

=(0.4 ±0.02) ×10

−10

=(0.266 ±0.006) ×10

−10

s, and

C = 2.8 ±0.2 for the particular conditions investigated, which were argon (91%)

+ methane (9%) at 1 bar, E = 110V/cm, and u = 5.05cm/μs, implying

4.2×10

−1

The two straight lines cross each other at

=(3.6±0.3)×10

−1

, where

is nearly 1. If

were independent of c,orifthedistributionofc were extremely

narrow, we would have had

and C = 1.

2.4.6 Calculated and Measured Electron Drift Velocities

in Crossed Electric and Magnetic Fields

Drift chambers that have to operate in a magnetic ﬁeld B often have their pro-

portional wires parallel to B, hence their electrons drifting in crossed electric and

magnetic ﬁelds.

Daum et al. [DAU 78] have measured the drift-velocity vector for a variety of

suitable gas mixtures as functions of both ﬁelds. Typical velocities |u| are shown in

Fig. 2.24, and typical drift angles

, measured against the direction of E towards

E ×B, are shown in Fig. 2.25. We have selected two examples of gas mixtures,

Fig. 2.24 Magnitude of the

drift velocity in crossed elec-

tric and magnetic ﬁelds for

two different gas mixtures,

plotted as a function of the

electric ﬁeld (at atmospheric

pressure) for four differ-

ent values of B. The curves

represent calculations by Ra-

manantsizhena [RAM 79],

the points measurements by

Daum et al. [DAU 78]

2.4 Applications 93

Fig. 2.25 Drift angle in

crossed electric and magnetic

ﬁelds for the same conditions

as in Fig. 2.24

Fig. 2.26 The magnitude of the drift velocity in two different mixtures of argon–methane,

measured at various magnetic ﬁelds by Merck [MER 89], and plotted as a function of the electric

ﬁeld component along the drift direction. The measured points fall essentially on universal curves

that are independent of the magnetic ﬁeld

94 2 The Drift of Electrons and Ions in Gases

argon + ethane, and xenon + ethane, which exhibit different mobilities and,

therefore, different values of

ωτ

at a given magnetic ﬁeld.

These measurements have been compared with calculations based on the for-

malism explained in Sect. 2.3, by Ramanantsizhena [RAM 79]. He used elastic

cross-sections for the e−C

collision, which are based on the work of Duncan and

Walker [DUN 74], and inelastic cross-sections derived from Palladino and Sadoulet

[PAL 75]. Comparing the measurements with the calculations, we notice that agree-

ment exists, mostly at the level of 5 to 10%, occasionally worse. In addition we

observe the small variation of the calculated maximum of the drift velocity as the

magnetic ﬁeld is changed: according to Tonks’ theorem it should stay the same, but

with a realistic distribution function it decreases by 1 or 2% for the gas mixtures of

Fig. 2.24 as B goes up from 0 to 1.5 T.

In Fig. 2.26 we see drift velocities measured by Merck [MER 89] in crossed

electric and magnetic ﬁelds as functions of the electric ﬁeld component along the

drift direction. We note that the points belonging to different B fall on the same

line characteristic of the gas, within the measurement error, which is 2 or 3% (only

for the small CH

concentration is there a 12% deviation near the peak). Obviously

Tonks’ theorem, (2.13) and (2.51), is quite well conﬁrmed.

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