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

Подождите немного. Документ загружается.

36 1 Gas Ionization by Charged Particles and by Laser Rays

Table 1.7 Va lu es of N

eff

calculated for various average numbers m of clusters, using the cluster-

size distribution of Fig. 1.7

Average number m 1 5 10 20 50 100 200 500 1000

eff

1 2.4 3.7 5.2 8.0 12 15 26 41

This is because the x

are created independently of each other so that the mixed

terms x

in the square drop out. The statistical factor that multiplies L

/12 has

been denoted 1/N

eff

in (1.73); its value depends on the partitioning of the charge

over the m clusters. If all clusters were equally large, then N

eff

would take on the

value m. It is equal to the equivalent number of independently ﬂuctuating entities

of equal sizes that create a ﬂuctuation that is the same as the one created by the

clustered charges. As these are different in size, N

eff

has to be smaller than m.

For a calculation of N

eff

, which governs the localizability of any cluster ionization

charge, we have to integrate over the cluster-size distribution. Starting from the one

plotted in Fig. 1.7 we calculate, with the Monte Carlo method, the values listed in

Table 1.7.

There are two properties of N

eff

that are remarkable: (i) it is quite small compared

with the average number m of clusters (let alone the total ionization); (ii) it does

not grow as fast as m (see the graph in Fig. 1.22). The reason for the latter is

that the relative contribution of the rare large clusters grows with the increase of

the number of clusters as the tail of the cluster-size distribution is sampled more

often.

The variation of N

eff

with m seen in Fig. 1.22 may be described roughly as

eff

= m

0.54

. (1.74)

There is a statistical correlation between N

eff

and the total charge deposited in

L. It can be used to increase N

eff

by cutting on the pulse height, but the gain is not

overwhelmingly large.

If diffusion is taken into account, N

eff

may become considerably larger in time,

owing to the declustering effect. This is treated in Sect. 1.2.

Fig. 1.22 Dependence of N

eff

the effective number of inde-

pendently ﬂuctuating entities

of charge, as a function of the

mean number m of clusters.

Result of a calculation using

the cluster-size distribution

in Ar

1.2 Calculation of Energy Loss 37

1.2.10 A Measurement of N

eff

The effective number of independently ﬂuctuating charges along the length L of a

track can be measured. The r.m.s. ﬂuctuation of the centre of gravity of the charge

deposited along L is, by deﬁnition of N

eff

, equal to

√

12N

eff

A particle track whose ionization is sampled over a length L by n wires with

pulse heights P

(i = 1,...,n) has the centre of its charge at

∑

where the y

are the wire positions along the track. A measurement of y

of many

tracks allows a determination of

, and hence of N

eff

, to be made.

In an experiment using a model of the ALEPH TPC, cosmic-ray tracks were

collected in a small interval of momentum and angle of incidence. The chamber

contained an Ar (90%)+CH

(10%) gas mixture at atmospheric pressure and had a

sense-wire spacing of 0.4 cm. Track lengths L between 0.8 and 20 cm were speciﬁed

and N

eff

was calculated for every L. The result is shown in Fig. 1.23. The choice

of L is limited on the low side by the wire spacing and on the high side by the

dynamic range of the pulse-height recording. In the case under consideration, the

overﬂow bin contained 1.5% of the pulses. For higher values of L this bin is sampled

more often, and the calculated number N

eff

begins to depend on the details of an

extrapolation of the overﬂow bin. A power-law ﬁt to Fig. 1.23 shows that N

eff

varied

with L according to

eff

∝ (L/1cm)

0.45±0.1

Comparing Figs. 1.22 and 1.23, we notice that the exponent is identical, within

errors, to that obtained from the theoretical cluster-size distribution. Using the power

law, which is only an empirical relation, we may obtain an estimate of the primary

ionization density, because it can be expected that the function N

eff

(L) becomes

equal to 1 at L = λ, the average distance between clusters.

Fig. 1.23 N

eff

measured with

cosmic rays as a function

of the track length L.The

last data point has the largest

dependence on the treatment

of the overﬂow bin (see text).

The three values correspond

to extrapolation according

to (a) a1/PH

law, (b) the

Landau distribution or (c) no

extrapolation, respectively

38 1 Gas Ionization by Charged Particles and by Laser Rays

1.3 Gas Ionization by Laser Rays

The light of a narrow pulsed laser beam that traverses a volume of gas, under

favourable conditions, is capable of ionizing the gas in the beam so that it imitates

a straight particle track. For this to occur, there must be some ionizable molecules

in the gas, and the energy density must be sufﬁciently high, depending on the wave-

length. In practice, a small nitrogen laser, emitting pulses of 10

W on a few square

millimetres at λ = 337nm, may be sufﬁcient in an ordinary chamber gas that has not

been especially cleansed and therefore contains suitable molecules in some low con-

centration. This technique has the obvious advantage of producing identical tracks

in the same place. Therefore, by taking repeated measurements of the same coor-

dinate, one may form the average, which can be made almost free from statistical

variations, if only the number of repeated measurements is made large enough. In

practice 100 shots are usually sufﬁcient. The technique has been widely used ever

since its ﬁrst application in 1979 [AND 79].

Chemical compounds used for laser ionization are discussed in Sect. 12.4.

1.3.1 The nth Order Cross-Section Equivalent

The quantum energy of laser light in the visible and the near ultraviolet is much

lower than the ionization energies of molecules. It takes two or more such laser

photons to ionize the organic molecules present in the chamber gas. Multiphoton

ionization processes involving 11 photons have been observed in xenon (see the

review by Lambropoulos [LAM 76]). For the ionization of a molecule to occur, the

n photons have to be incident on the molecule during the lifetime of the intermediate

states. Then the ionization rate varies the nth power of the photon ﬂux because

the photons act incoherently in the gas. The probability of n photons arriving in

a given time interval is equal to the nth power of the probability of each one of

them, and is therefore proportional to the nth power of the photon ﬂux

.Justasthe

concept of ‘cross-section’

describes one-particle collision rates in units of cm

we have an ‘nth-order cross-section equivalent’

(n)

for n-particle collisions in units

of cm

n−1

.InavolumeV containing a density N of molecules, the ionization rate

R is therefore given by the expression

R =

(n)

. (1.75)

In fact, n-photon ionization is most easily identiﬁed by a measurement of R as a

function of

A light pulse with cross-sectional area A and duration T contains m photons if

the ﬂux

and the energy E are given by

= m/(AT ),

E = mh

= mhc/λ.

1.3 Gas Ionization by Laser Rays 39

Considering a traversed volume V = AL, the speciﬁc ionization per unit track length

for the n-photon process is

(AT )

n−1

(n)





AT N





(n)

(1.76)

In (1.76), (E/AT ) is the power density of the beam.

So far, we have implicitly assumed the duration of the light pulse to be short

compared with the inverse transition frequencies between the states involved. The

more general situation develops towards a dynamic equilibrium between excitation

and de-excitation. The case of two-photon ionization is treated in more detail in

Sect. 1.3.2.

The molecular ions created in the ionization do not as a rule resemble the par-

ent molecules very much, because they are cracked in the process. Fragmentation

patterns are complicated and usually not understood [REI 85].

1.3.2 Rate Equations for Two-Photon Ionization

For the electrons in the molecules, we distinguish the ground level (0), the inter-

mediate level (1), and the continuum ionization level (2). We denote the population

densities in the gas by P

, P

, and P

. The incident radiation stimulates transitions

0 →1, 1 → 2, and 1 →0, but there are also spontaneous transitions 1 →0. Depend-

ing on the circumstances, there may be losses from level 1 into other channels. The

transition rates, denoted by k

to k

, are determined by the incident ﬂux and by the

internal transition mechanism.

The rate equations are written in the following form (the primes denote the time

derivatives):



(t)=−k

(t)+(k

+ k

(t),



(t)=+k

(t) −(k

+ k

(t),



(t)=k

(1.77)

They are symbolized in Fig. 1.24.

The rate per molecule k

is taken to be proportional to the incoming ﬂux

of pho-

tons; the constant of proportionality is the cross-section

for the process 0 → 1:

. (1.78)

We want to calculate the transition rate P



(t) with which electrons appear in the

continuum state. It is proportional to the density P

(t) of the intermediate state, and

the corresponding rate per molecule is

. (1.79)

40 1 Gas Ionization by Charged Particles and by Laser Rays

Fig. 1.24 Scheme of the

ﬁve transition rates used in

(1.77a–c)

The ﬁrst two of the rate equations are a system of homogeneous linear differential

equations. Before we derive the general mathematical solution, we shall consider

some limiting cases in order to better understand the physical content of (1.77).

Before the pulse arrives, P

(0) is of the order of 10

per cubic centimetre in a

gas having a typical concentration of the ionizing molecules of one part per million,

and P

(0)=0. While P

(t) is slowly built up at the rate P



(t), we may consider

as a constant as long as P

(t) has not developed into a similar order of magni-

tude. The situation of making only a few ionization electrons with the laser shot is

well described by this approximation, which treats the ground state as an inﬁnite

reservoir.

(a) Case of P

as an Inﬁnite Reservoir. We may write the ﬁrst two of (1.77) in the

following form:



(t)=−k

+(k

+ k

(t), (1.80a)



(t)=+k

−(k

+ k

(t). (1.80b)

The solution of (1.80b) is

(t)=

∑

(1−e

−t

∑

), (1.81)

where

∑

k = k

+ k

. Obviously, our approximation is always fulﬁlled if



∑

k or



∑

k (using (1.78)). Let us assume that this is the case. Equa-

tion (1.81) leads to a production rate of ionization equal to



(t)=k

(t)=

∑

(1−e

−t

∑

). (1.82)

Integrated over the duration T of a constant light pulse, the total density reached at

the end of the pulse is given by

1.3 Gas Ionization by Laser Rays 41

(T)=

∑



T −(1 −e

−T

∑

)



∑



. (1.83)

In the limit of short pulses, i.e. T  1/

∑

k, the exponential may be expanded to

second order and gives us

(T) →

. (1.84)

In the limit of long pulses, i.e. T  1/

∑

k, the exponential vanishes, so that

(T) →

∑

T. (1.85)

Expression (1.84) is proportional to

, the square of the photon ﬂux, because of

(1.78) and (1.79). This is also the case for expression (1.85) unless k

or k

domi-

nates

∑

k. In our approximation that treats P

as an inﬁnite reservoir, we assume that



∑

k; then a similar condition holds for k

(except for special circumstances in

which

is orders of magnitude larger than

(T) →

(small T, small

), (1.86)

(T) →

+ k

T (large T, small

). (1.87)

(b) General Case. If we want to know what happens when the photon ﬂux is strong,

(1.77) have to be solved in a rigorous way. This is achieved by a variable

transformation that separates the equations (see any textbook on differential

equations, for example [KAM 59].) One ﬁnds the eigenvalues s

of the ma-

trix of coefﬁcients from the determinant



−k

−sk

+ k

−

∑

k −s



= 0, (1.88)

1,2

= −

∑

k + k





∑

k + k



−k

+ k

)



1/2

. (1.89)

These two eigenvalues are real and negative; we take s

< s

< 0.

The solutions to the differential equations (1.77) satisfying the initial conditions

(0)=0,P

(0),aregivenby

(t)=

(0)

−s

[−(k

+ s

+(k

+ s

], (1.90a)

(t)=

(0)k

−s

−e

], (1.90b)

(t)=

(0)k



−s

−

−s



. (1.90c)

42 1 Gas Ionization by Charged Particles and by Laser Rays

Equation (1.90c) represents the general solution to the problem. It describes the

concentration of ionization electrons as a function of time and of the rate coefﬁcients

to k

. For small t we recover our expressions (1.84) and (1.86) by developing the

exponentials to the second order in s

t and s

(t) →

(0)t

. (1.91)

For large t, using (1.89), we obtain

(t) →

+ k

(0). (1.92)

This enormous concentration of electrons (saturation) would imply that the ground

state has been emptied, a situation that should not concern us when we study ion-

ization tracks that are similar to particle tracks.

Equation (1.91), identical to (1.84) and (1.86), gives rise to the deﬁnition of the

second-order cross-section equivalent described in Sect. 1.3.1. For a laser shot of

duration T, identifying R/V with P

/T,wehave

(2)

T. (1.93)

Under the conditions that lead to (1.87), we have instead

(2)

/(k

+ k

), (1.94)

whereas under the most general conditions, including saturation,

(2)

depends not

only on T but also on

1.3.3 Dependence of Laser Ionization on Wavelength

The ionization yield depends very much on the wavelength. Using tunable dye

lasers, Ledingham and co-workers have found an increase of four orders of mag-

nitude, when going from λ = 330nm to λ = 260nm. In Fig. 1.25 ‘clean’ counter

gas is compared with a gas seeded with a small amount of phenol. The ﬁne

structure visible around 270 nm was resolved using high-resolution techniques

and was compared with the known single-photon UV absorption of phenol

(see Fig. 1.26).

The identical wavelength dependence is apparent. We understand this behaviour

from (1.93) and (1.94): the wavelength dependence resides almost entirely in the

factor

, implying that the cross-section for ionization from the intermediate level

does not vary so much. The amount of characteristic structure in the spectrum of

laser-induced resonant two-photon ionization (‘R2PI’) makes it a sensitive tool for

the identiﬁcation of molecules [HUR 79].

1.3 Gas Ionization by Laser Rays 43

Fig. 1.25 Ionization induced in untreated counter gas [Ar (90%)+CH

(10%)] by a 1 ×1mm

pulsed laser beam of 1μJ(×), compared with the same gas seeded with a small amount of phenol

(+) [TOW 86]

Fig. 1.26 A comparison of the laser-induced R2PI spectrum of counter gas doped with a trace of

phenol (B) and the single-photon UV absorption spectrum of phenol [TOW 86]

44 1 Gas Ionization by Charged Particles and by Laser Rays

1.3.4 Laser-Beam Optics

A beam of laser light that is to create ionization as if from a particle must have an

approximately constant cross-sectional area along the beam. Relations (1.75) and

(1.76) imply that, for example, in two-photon ionization a light pulse with a given

energy produces half as many ions per unit track length when it has twice the cross-

sectional area. In the interest of a constant high yield and of a narrow deposit of

ionization, one would like to have the beam width as small as possible over the

largest possible track length.

The photons in a beam occupy a volume in the four-dimensional phase space

at each point along the beam. For illustration, we consider the horizontal plane in

Fig. 1.27, where an almost parallel beam is shown to be focused into a narrow

waist by a lens. After the focus, the beam opens up again. Each photon trajectory

is represented by a point in the phase-space diagrams below where the distance

from the axis is plotted against the small angle of the trajectory with the axis. Here

we deal with geometrical optics only; the wave optical aspects are treated later. In

this sense we may regard the optical elements of the beam (lenses, mirrors, light

guides, free space, apertures) as determining a trajectory for each photon in a time-

independent fashion. Now, Liouville’s theorem in statistical mechanics states that

in such a passive system the density of photons in phase space is a constant of

motion, meaning that it does not change along the beam. As long as there are no

photons lost, the occupied volume of cross-sectional area times solid angle is the

same everywhere along the beam and the same as just behind the laser. In the phase-

space diagrams of Fig. 1.27 the envelopes containing the beam do not change their

area. A concentration of the beam in space by a focusing lens is accompanied by a

corresponding extension of the solid angle.

Fig. 1.27 Phase-space diagrams at three points along a light-beam, which is almost parallel at the

left and is focussed into B by a lens at A

1.3 Gas Ionization by Laser Rays 45

There is a principal limit of the size, in phase space, of a photon beam.

Heisenberg’s uncertainty principle of quantum mechanics states that, in each of the

two transverse dimensions separately, the limits are given by

> h/4

, (1.95)

where

x and

are the root mean square widths in space and in momentum

[MES 59] and h is Planck’s constant. Since the r.m.s. opening angle is

Δα

/p =

λ/h  1,

we have

Δα

> λ/4

. (1.96)

An example not very far from the principal limit is provided by the situation where a

plane light-wave is delimited by an aperture, say a slit of width ±A. The diffraction

caused by the slit will open the range of the angles

of propagation behind the slit

so that approximately

|±

|≈

. (1.97)

Many lasers can be tuned to emit beams near the fundamental limit. If their wave-

length is subsequently halved by a frequency doubler, the limit refers to the old

wavelength unless the phase space and, inevitably, the power of the beam are re-

duced afterwards.

For practical purposes we work with approximate full widths W

and W

ignoring the exact distribution of intensity across the beam, and use the order-of-

magnitude relation

> λ. (1.98)

The highest spatial concentration of light is reached in a beam focus; let it have a

width W

. Ideally, the arrival angles of the photons do not depend on the point across

the focus; let the angular width be W

. At a distance D behind the focus, every such

point has reached a width

D. The statistical ensemble of all the photons has a

combined width of W

tot

given by

tot

= W

+(W

. (1.99)

The quadratic addition of these two widths is correct in the approximation that the

light intensity in the focus is a Gaussian function of both the space and the angular

coordinates. For most applications this is not far from reality.

Applying (1.98) and (1.99), the total width of a focused beam, at a distance D

away from the focus, is not smaller than

min

tot





λD





1/2

. (1.100)

This is plotted in Fig. 1.28 as a function of D for various values of W