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

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66 2 The Drift of Electrons and Ions in Gases

Case of E Orthogonal to B. The second case concerns the drift volume of the axial

wire drift chamber, where cos

= 0; here













1/2

−





. (2.49)

We note that the random velocity c is reduced in the presence of a magnetic ﬁeld

orthogonal to E. The same is true for the drift velocity because of (2.47). Any two

values of E and B that produce the same solution for c in (2.49) lead to the same

drift velocity. In particular, if E

is the electric ﬁeld that produces certain values of

c and u in the absence of magnetic ﬁeld (B

= 0), then there is a corresponding E

for the same values of c and u at some non-zero B

. In order to ﬁnd an expression

for E

in terms of E

and B

, we rewrite (2.49) for the two pairs of ﬁelds in the

following form:

−





+ c

= 0,



ecB



−





+ c

= 0.

(2.50)

Note that λ and

are the same in both cases because they are functions of c alone.

From (2.50) we deduce that

= E





cmN





= E

(1+

), (2.51)

which is the microscopic justiﬁcation for (2.12) and (2.13). One cannot understand

(2.12) from a derivation using a constant friction term

because

does not remain

constant when E is varied (see (2.7) and Fig. 2.17). Rather, as we have seen,

also

depends on B in such a way that it comes back to its old value for the appropriate

combination of B and E.

Next, we wish to evaluate (2.49) for large values of the B ﬁeld. In the limit

ωτ

 1 the second term in the square root of (2.49) becomes much larger com-

pared to the ﬁrst term (using (2.46) and (2.47) it can be shown that the two terms

are equal when

ωτ

 2.2), and we have to ﬁrst order in 1/

ωτ





, (2.52)





, u =

, (2.53)

which is remarkable because c no longer depends on

Equation (2.53) has the following signiﬁcance. In the presence of a magnetic

ﬁeld B, orthogonal to E and strong enough for

ωτ

to be large, the drift velocity u

approaches a universal value E/B, which is the same for all gases.

2.2 The Microscopic Picture 67

In order to study the deviations from the universal expression (2.53), we may

carry the expansion of the square root in (2.49) to the second order. Using (2.47)

and (2.53) we ﬁnd

u =



1−



(

ωτ

 1). (2.54)

This means that the universal limit (2.53) is approached rapidly from below as B is

increased.

This physical content of (2.53) can best be understood by comparison with the

motionofanelectronin vacuo: under the inﬂuence of crossed electric and magnetic

ﬁelds, the electron is accelerated by the E ﬁeld and bent by the B ﬁeld; it performs

a cycloidal motion in the plane orthogonal to B (see (2.43) and Fig. 2.7). When

averaging over the periodic part, one ﬁnds that the electron drifts with the constant

speed E/B in the direction of −

E ×

B. This motion in vacuo is the limit of the

gas drift for large values of

ωτ

, the mean number of turns (expressed in radians)

between collisions. Experimentally it can be reached either by increasing the B ﬁeld

or by reducing the gas density, which is roughly proportional to 1/

2.2.4 Diffusion

As the drifting electrons or ions are scattered on the gas molecules, their drift ve-

locity deviates from the average owing to the random nature of the collisions. In the

simplest case the deviation is the same in all directions, and a point-like cloud of

Fig. 2.7 Motion of an elec-

tron in vacuo under the

inﬂuence of orthogonal E

and B ﬁelds: B goes out of the

paper, the electron stays in the

plane of the paper

68 2 The Drift of Electrons and Ions in Gases

electrons which then begins to drift at time t = 0 from the origin in the z direction

will, after some time t, assume the following Gaussian density distribution:

n =



√



exp



−r

4Dt



, (2.55)

where r

= x

+ y

+(z −ut)

; D is the diffusion constant because n satisﬁes the

continuity equation for the conserved electron current Γ :

Γ = nu −D∇n, (2.56)

∂

+ ∇·Γ = 0, (2.57)

∂

+ n∇·u −D∇

n = 0. (2.58)

The diffusion constant D, deﬁned by (2.56), makes the mean squared deviation of

the electrons equal to 2Dt in any one direction from their centre. (This is a special

case of (2.75), which deals with the case of anisotropic diffusion.)

In order to express the diffusion constant in terms of the microscopic picture, we

suppose that one electron or ion starts at time t = 0 and has a velocity c; hence,

according to (2.16), there is probability distribution of free path l equal to

g(l)dl =

−l/l

dl, (2.59)

where l

= c

is the mean free path. The simplest case is the one where scattering

is isotropic with respect to the drift direction. We consider this case ﬁrst. It applies

to ions at low electric ﬁelds E(E/(N

)  kT) and, to a good ﬁrst approximation,

to electrons.

Consider a time t at which a large number, n, of encounters have already oc-

curred; then n = t/

. The mean square displacement in one direction, say x,is





∑

cos



∏

g(l

)dl

d cos

= n

t. (2.60)

Hence l

cos

is the displacement along x between the (i −1)th and the ith colli-

sion (Fig. 2.8); the cos

are uniformly distributed between −1 and +1 according

to our assumption. The l

are distributed with the probability density (2.59). Note

that the mixed terms in the sum vanish. The part proportional to 2t is the diffusion

coefﬁcient D,

D =

. (2.61)

Recalling the expression for the electron mobility

2.2 The Microscopic Picture 69

Fig. 2.8 Electron paths for

the derivation of (2.60)

we notice that the electron energy can be determined by a measurement of the

ratio D/

. (2.62)

When the diffusing body has thermal energy,

=(3/2)kT, (2.62) takes the form

which is known as the Nernst–Townsend formula,ortheEinstein formula.(Forhis-

torical references, see [HUX 74].)

The energy determines the diffusion width

of an electron cloud which, after

starting point-like, has travelled over a distance L:

= 2Dt =

2DL

3eE

. (2.63)

In drift chambers we therefore require small electron energies at high drift ﬁelds

in order to have

as small as possible. In the literature one ﬁnds the concept of

characteristic energy,

, which is related to our

by the relation

=(2/3)

In Fig. 2.9 we show the variation of

with the electric ﬁeld strength, measured

with electrons drifting in the two gases, argon and carbon dioxide, which repre-

sent somewhat extreme cases concerning the change-over from thermal behaviour

to ﬁeld-dominated behaviour. In argon a ﬁeld strength as low as 1 V/cm produces

electron energies distinctly larger than thermal (‘hot gas’). In carbon dioxide the

same behaviour occurs only at ﬁeld strengths above 2 kV/cm (‘cold gas’). The rea-

son is a large value of the relative energy loss λ in CO

, due to the internal degrees

of freedom of the CO

molecule, which are accessible at low collision energies.

70 2 The Drift of Electrons and Ions in Gases

Fig. 2.9 Characteristic energy

of electrons in Ar and CO

as a function of the reduced

electric ﬁeld. The electric

ﬁeld under normal condi-

tions is also indicated. The

parameters refer to the dif-

ferent temperatures at which

the measurements were made

[SCH 76]

The fractional energy loss λ was introduced in Eq. 2.15 and played a decisive role

throughout Sect. 2.2.

For ions and electrons with thermal energy

=(3/2)kT, (2.63) shows that the

diffusion width of a cloud is independent of the nature of the gas and is proportional

to the square root of the absolute temperature: the ‘thermal limit’ is given by



2kTL



1/2

. (2.64)

2.2.5 Electric Anisotropy

Until 1967 it had always been assumed that the diffusion of drifting electrons in

gases has the isotropic form implied by (2.55). But then Wagner et al. [WAG 67]

discovered experimentally that the value of electron diffusion along the electric ﬁeld

can be quite different from that in the perpendicular direction. The diffusion of drift-

ing ions is also often found to be non-isotropic.

When ions collide with the gas molecules, they retain their direction of motion

to some extent because the masses of the two collision partners are similar, and

therefore the instantaneous velocity has a preferential direction along the electric

ﬁeld (see Sect. 2.2.2). This causes the diffusion to be larger in the drift direction; the

mechanism is at work for ions travelling in high electric ﬁelds E(eE/(N

) ≥ kT),

and we do not treat it here because it plays no role in the detection of particles.

In the case of electrons, there is almost no preferential direction for the instan-

taneous velocity. We will describe the electron diffusion anisotropy following the

2.2 The Microscopic Picture 71

semiquantitative treatment of Partker and Lowke [PAR 69], which is restricted to

energy loss by elastic collisions. The essential point of the argument is that the mo-

bility of the electrons assumes different values in the leading edge and in the centre

of the travelling cloud if the collision rate is a function of electron energy. This

change of mobility inside the cloud is equivalent to a change of diffusion in the

longitudinal direction.

We express the energy balance (2.15) in the drifting cloud more precisely by

making use of the electron current introduced in (2.51); it now contains the diffu-

sion term D∇n. Let the drift be in the z direction (E = E ˆz). Using the collision

frequency

≡ 1/

instead of

and dropping the index from

, the energy balance

takes the form

= eE ·Γ = e

n −eED

∂

, (2.65)

where

, λ and D are functions of

. In principle, we can solve this equation

for

in terms of (1/n)(

∂

z). Here we will put λ constant, as it is in the case of

elastic scattering, and develop

(

) around the equilibrium point, which is given

when

∂

z = 0:

mλ





The functions D and m are approximated by their expressions (2.61) and (2.62).

Putting

Δε

and

∂ν

∂ε

)

Δε

in (2.65), the variation of energy

inside the cloud can be expressed to ﬁrst order as

Δε

= −

3eE[1 + 2(

∂ν

∂ε

)

(

)]

∂

, (2.66)

(

∂ν

∂ε

)

being the derivative of the collision frequency with respect to the energy,

evaluated at

. Equation (2.66) shows that

is larger in the leading edge of the cloud

and smaller in the trailing edge, and from (2.62) it follows that – unless (

∂ν

∂ε

0 – the mobility

also is a function of position in the cloud (see Fig. 2.10):



∂ν

∂ε

Δε



We now rearrange the terms in the expression for the electron current and ﬁnd

Γ =

Enˆz −D



∂

ˆx +

∂

ˆy



−D



1−

1+ 2



∂

ˆz,

where

≡ (

)(

∂ν

∂ε

). Obviously the diffusion in the drift direction has

changed and is no longer equal to the diffusion in the perpendicular direction. We

distinguish the two diffusion coefﬁcients by the indices L and T. The ratio is

1+ 2

. (2.67)

72 2 The Drift of Electrons and Ions in Gases

Fig. 2.10 Mobility variation

inside an electron cloud trav-

elling in the z direction

Instead of the density distribution (2.55) of the diffusing cloud of electrons, we have

in the anisotropic case

n =

√



√



exp



−

+ y

−

(z −ut)



. (2.68)

2.2.6 Magnetic Anisotropy

Let us now consider the effect of a magnetic ﬁeld B along z. The electric ﬁeld is

in the x −z plane and we assume there is no electric anisotropy. The magnetic

ﬁeld causes the electrons to move in helices rather than in straight lines between

collisions. Projection onto the x −y plane yields circles with radii

sin

, (2.69)

where

=(e/m)B is the cyclotron frequency of the electron, c its velocity and

the angle with respect to the z axis. Projection onto the x−z plane yields sinusoidal

curves. Figure 2.11a, in contrast to Fig. 2.8, shows how the random propagation

of the electron is diminished by the magnetic ﬁeld. We must repeat our calculation

(2.60) with the new orbits.

2.2 The Microscopic Picture 73

Fig. 2.11 (a) Electron paths causing the magnetic anisotropy of diffusion. (b) Firstpartofthe

trajectory showing the direction projected into the x−y plane

Looking at Fig. 2.11b we describe the motion of the electron which has suffered

a collision at the origin by the orbit

x(l)=



sin



−



+ sin



y(l)=



cos



−



−cos



z(l)=l cos

(2.70)

where l is the length of the trajectory and

is the starting direction in the x −y

plane. More precisely, the derivatives with respect to l at the origin, giving the initial

direction of the electron, are the following:



(0)=sin

cos



(0)=sin

sin



(0)=cos

The mean square displacement of the electron after the ﬁrst collision is given

by an integration over the solid angle and over the distribution (2.59) of path

lengths:

x

 =



−l/l

(l)d

dcos

dl (2.71)

and similarly for y

 and z

. It is not difﬁcult to integrate (2.71), using (2.70) and

(2.69). The result is

74 2 The Drift of Electrons and Ions in Gases

x

 = y

 =

z

 =

In comparison to (2.60), the magnetic ﬁeld has caused the diffusion along x and y

(then perpendicular to the magnetic ﬁeld) to be reduced by the factor

(

)

(0)

, (2.72)

whereas the longitudinal diffusion is the same as before:

(

)=D

(0). (2.73)

If there is an E ﬁeld as well as a B ﬁeld in the gas, the electric and magnetic

anisotropies combine. In the most general case of arbitrary ﬁeld directions, the diffu-

sion is described by a 3×3 tensor: let the B ﬁeld be along the z axis of a right-handed

coordinate system S, and let the drift direction ˆu, which is at an angle

with respect

to B, have components along ˆz and ˆx. The electric anisotropy is along ˆu, and the dif-

fusion tensor is diagonal in the system S



which is rotated around ˆy by the angle

In order to describe the two anisotropies in S, we must transform the diagonal

tensor S



to S before we multiply by the diagonal tensor that represents the magnetic

anisotropy. If the electric anisotropy is equal to D

and the magnetic one is

equal to D(0)/D(

)=1/

, we get for the combined tensor S

⎛

⎝

cos

0sin

010

−sin

0 cos

⎞

⎠

⎛

⎝

0 D

00D

⎞

⎠

⎛

⎝

cos

0 −sin

01 0

sin

0 cos

⎞

⎠

⎛

⎝

001

⎞

⎠

⎛

⎝

cos

+ D

sin

) 0 (D

−D

)sin

cos

0 D

−D

)sin

cos

0 D

sin

+ D

cos

⎞

⎠

. (2.74)

The importance of diffusion for drift chambers is in the limitation for the co-

ordinate measurement. Hence, we are interested in the deviation along a given

direction of an electron that has been diffusing for a time t. We treat the gen-

eral case of a diffusion tensor D

and a direction ˆα given by the three cosine

(

= 1), both expressed in the same coordinate system. We make

use of the continuity equation (2.58) for the density n(x

), normalized so that



n dx = 1. For a time-independent and homogeneous ﬁeld the drift velocity is

constant, and we have

2.2 The Microscopic Picture 75

= D

∂

(summation over identical indices is always understood). The rate of the mean

square deviation along x



is given by

dx





+∞



−∞

(

)

dx =

+∞



−∞

∂

dx = 2

In the last integral all the 81 terms vanish except when the powers of the x

match

the powers of the derivatives: one shows by two partial integrations that

+∞



−∞

∂

dx = 2

where

= 1ifi = k, and zero otherwise. We have used the fact that electron density

and its derivatives vanish at inﬁnity

If a point-like ensemble of electrons begins to diffuse at time zero, then after a

time t it has grown so that the mean square width of the cloud in the direction α has

the value

x



 = 2

t. (2.75)

The isotropic case in which x

2

is independent of

is obviously given by a D

which is the unit matrix multiplied with the isotropic diffusion constant. The factor

2 in (2.75) is also present in (2.55) and in the comparison between (2.60) and (2.61).

Equation (2.75) implies that the width of the cloud is calculated only from the sym-

metric part of D

; furthermore, the diffusion tensor must be positive deﬁnite, other-

wise our cloud would shrink in some direction–a thermo-dynamic impossibility.

2.2.7 Electron Attachment

During their drift, electrons may be absorbed in the gas by the formation of negative

ions. Whereas the noble gases and most organic molecules can form only stable

negative ions at collision energies of several electronvolts (which is higher than the

energies reached during the drift in gas chambers), there are some molecules that

are capable of attaching electrons at much lower collision energies. Such molecules

are sometimes present in the chamber gas as impurities. Among all the elements,

the largest electron afﬁnities, i.e. binding energies of an electron to the atom in

question, are found with the halogenides (3.1–3.7 eV) and with oxygen (∼0.5eV).

Therefore we have in mind contaminations due to air, water, and halogen-containing

chemicals.

Our account must necessarily be brief; for a thorough discussion of the atomic

physics of electron attachment, the reader is referred to Massey et al. [MAS 69].