Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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620 Principles of Radiation Interaction in Matter and Detection

From Eqs. (9.4, 9.6), we can write

vis

= (e/mip)E

= (e/mip) F (S) E. (9.7)

Equation (9.7) is the way Equation (9.1) is rewritten in terms of the e/mip ratio.

In the previous section we have seen how the experimental data show a re-

markable linear dependence of the measured ²

vis

values on the incoming particle

energy. These data and Eq. (9.7) indicate that the e/mip ratio is energy indepen-

dent. Furthermore, both experimental data and cascade simulations (see page 623)

support the fact that the e/mip ratio is almost independent of the thickness of read-

out layers for practical sampling calorimeters. The e/mip ratio is a fundamental

characteristic of the structure of sampling calorimeters, essentially related to the

diﬀerence between the readout and absorber Z values (see Sect. 9.2.2.1).

If the sampling calorimeter has a dominant (in units of radiation length) passive

absorber a, with thickness L

and radiation length X

, and if collision energy-losses

in readout active detectors are small compared with those in passive absorbers (these

conditions are usually fulﬁlled in most sampling calorimeters), from Eq. (9.4) one

ﬁnds:

G(S) =

£¡

≈

, (9.8)

where τ = L

. Considering that F (S) = G(S) X

and using Eq. (9.8), Eq. (9.7)

can be rewritten as:

vis

(e/mip)

E. (9.9)

This equation [see Eq. (9.3)] shows the visible energy is inversely proportional to the

sampling frequency τ, as long as the e/mip ratio is independent of τ . This occurs

for τ larger than ≈ 0.8, as discussed at page 626. Since

≈ ²

to better

than 10%, Eq. (9.9) can b e approximated by

vis

≈

(e/mip)

τ²

·µ

E, (9.10)

where ²

is the critical energy (see Sect. 2.1.7.4) of the passive absorber a. By

knowing the e/mip ratio from Eq. (9.7) [or Eq. (9.9)], it is possible to determine

the expected visible-energy for any electromagnetic calorimeter.

In this section, we have seen how the e/mip ratio can be almost considered a

characteristic or intrinsic property of the calorimeter structure, namely of the type of

readout detectors and the kind of passive absorbers used, and is almost independent

of the sampling frequency τ. In addition and within ≈ 10% as shown in Table 9.1,

since readout active detectors (like scintillators, liquid argon or silicon detectors,

etc.) usually have low-Z values, e/mip values (see [Fabjan (1985a); Wigmans (1987);

SICAPO Collab. (1989c)] and references therein) are ≈ 0.85 for medium-Z passive

absorbers (like Fe or Cu) and ≈ 0.65 for high-Z passive absorbers (like W, Pb or

U).

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Principles of Particle Energy Determination 621

Table 9.1 Selection of measured e/mip ratios from [Leroy and Rancoita (2000)]. L

and S are the passive sampler and

active readout thicknesses. Errors are quoted when available in the original reference.

Passive Readout type L

S e/mip Reference

sampler (mm) (mm)

Al Liquid Ar 1.0 3.0 1.00 [Cerri et al. (1983)]

Fe Liquid Ar 1.5 2.0 0.90 [Fabjan (1985a)]

Fe Si det.+G10 plates 35.0 0.2 0.78±0.02 [SICAPO Collab. (1989c)]

Fe Si det. 35.0 0.2 0.82±0.02 [SICAPO Collab. (1989c)]

Cu Si det. 15.0 0.3 0.85 [SICAPO Collab. (1989c); Lindstro em et al. (1987)]

Cu Scintillator 5.0 2.5 0.84±0.05 [Botner et al. (1981)]

Pb Liquid Ar 3.0 2.4 0.54 [Dubois et al. (1985)]

Pb Si det.+G10 plates 11.0 0.2 0.69±0.04 [SICAPO Collab. (1989c)]

Pb Si det. 11.0 0.2 0.76±0.04 [SICAPO Collab. (1989c)]

Pb Si det. 6.0 0.3 0.71 [SICAPO Collab. (1989c); Lindstroem et al. (1987)]

Pb Scintillator 2.1 6.3 0.51 [Stone et al. (1978)]

Pb Scintillator 5.0 5.0 0.65±0.10 [Bernardi (1987)]

Pb Scintillator 6.4 6.0 0.77 [Duﬀy et al. (1984)]

Pb Scintillator 10.0 2.5 0.67±0.03 [Bernardi et al. (1987)]

U Gas 4.5 6.0 0.65±0.04 [Areﬁev et al. (1989)]

U Liquid Ar 1.6 4.4 0.51 [Dubois et al. (1985)]

U Liquid Ar 1.7 2.0 0.61 [Fabjan (1985a)]

U Liquid Ar 4.0 3.2 0.50±0.05 [Aronson et al. (1988)]

U Scintillator 3.0 2.5 0.70±0.05 [HELIOS Collab. (1987)]

U Scintillator 3.0 2.5 0.65±0.03 [Anders et al. (1988)]

U Scintillator 3.2 3.0 0.74±0.03 [Bernardi (1987)]

U Scintillator 3.2 5.0 0.68±0.04 [Bernardi (1987)]

U Scintillator 10.0 5.0 0.61 [Catanesi et al. (1987)]

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622 Principles of Radiation Interaction in Matter and Detection

Finally, it has to be noted [Eq. (9.7)] that the visible energy in a calorimeter is

usually only a few percent of the overall incoming particle energy even for high-Z

passive samplers. It may be reduced to a fraction of a percent when low-Z passive

samplers are used (or for large sampling frequencies).

9.2.2.1 e/mip Dependence on Z-Values of Readout and Passive Absorbers

As discussed in Sect. 2.4, the development of the electromagnetic cascade inside a

medium can be understood within the well established and complete framework of

the QED theory.

Initial analytical treatments and interpretations have dealt with the shower de-

velopment and propagation in a single medium. They have disregarded physical

processes of signal generation in additional media, which are present in sampling

calorimeters. For instance basic properties, like the energy dependence of the depth

of the shower maximum and the longitudinal proﬁle, were predicted in “Approxi-

mation B” (discussed in Sect. 2.4.2.1) considering the fast cascade multiplication and

neglecting low-energy processes (namely those induced by soft electrons, positrons

and photons), by which almost all the energy is deposited in matter. The main

reason for neglecting these soft processes was that they were not important for

the determination of the properties of shower transport inside the absorber: very

soft electrons and positrons are ranging out locally in few tens (hundreds) of mi-

crons. Only the more energetic charged particles undergo approximately constant

collision energy-losses.

It has already been noted (see page 207) that some cascade characteristics de-

pend on the soft-component of the shower: soft photons, electrons and positrons

are mainly responsible for the transverse development of the shower. Furthermore,

low-energy photons below ≈ 1 MeV undergo diﬀerential absorption in matter.

In Fig. 9.4, mass attenuation coeﬃcients from [Leroy and Rancoita (2000)] (see

also [Hubbell (1969); Hubbell, Gimm and Øverbø (1980)]) in units of cm

/g are

shown for materials typically used in electromagnetic calorimeters as readout de-

tectors or passive samplers. Above an energy between '2 and 3 MeV, the pair-

production eﬀect dominates and the multiplication process is still important. For

photon energies between ' 1 and 2 MeV, the Compton scattering dominates and,

because of its dependence on the electron density of the medium, the mass attenua-

tion coeﬃcients depend on the ratio Z/A, which varies only slightly from one mate-

rial to another. As the photon energy decreases below ' 1 MeV for high-Z materials

and even lower energy for lower-Z materials (≈ 100 keV in Al), the photoelectric

eﬀect becomes more and more important. As a consequence, the attenuation mass

coeﬃcient is strongly dependent on the atomic number Z of the material. Further-

more in the photoelectric eﬀect, the interaction essentially occurs with the entire

atomic electron cloud rather than with individual corpuscular electrons. At low pho-

ton energies (for E

< a few hundreds keV), the emerging photoelectrons are very

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Principles of Particle Energy Determination 623

markedly emitted in the transverse direction with respect to the direction of the

incoming photon and enlarge the shower size.

As discussed above, the low-energy phenomena, which control the energy de-

position via soft photon interactions in matter during the development of electro-

magnetic cascades, depend on the diﬀerence of atomic numbers between the ab-

sorber and the active medium. For a ﬁxed amount of matter in g/cm

, the ﬂux

of low energy photons is less attenuated (see Fig. 9.4) in low-Z than in high-Z

media. Consequently, less soft photoelectrons are generated and locally absorbed

in low-Z materials (like detector active readouts as for instance scintillators, liquid

argon or silicon detectors, etc.) compared with medium-Z materials (like Fe and

Cu passive samplers), and much less compared with high-Z materials (like W, Pb

and U passive samplers). Thus, the e/mip value is expected to depend on the Z

diﬀerence among detector active readout and passive samplers.

Many experimental results have conﬁrmed this predicted behavior of the e/mip

ratio (see [Fabjan (1985a); Wigmans (1987); SICAPO Collab. (1989c)] and referen-

ces therein). Wigmans (1987), using the EGS4 code [Nelson, Hirayama and Rogers

(1985)], has systematically simulated electromagnetic shower cascades in sampling

calorimeters with liquid argon and scintillator readout detectors as a function of

the Z-value of passive samplers. His results are in agreement with experimental

data and other simulations [Flauger (1985); del Peso and Ro (1990)] and have con-

ﬁrmed [Wigmans (1987)] that the e/mip ratio mainly depends on the diﬀerential

absorption of low energy photons.

The e/mip ratios are < 1 for Z

absorber

> Z

readout

, ≈ 1 for Z

absorber

≈ Z

readout

and > 1 for Z

absorber

< Z

readout

. Simulations have shown that the energy deposi-

tion is quite uniform in thin (in units of radiation length) readout layers [Flauger

(1985)]. These simulation results and experimental data indicate that the value of

e/mip is almost independent of the readout layer thickness for practical calorimeters.

Several e/mip values, obtained experimentally, are shown in Table 9.1: the data

are mainly given for liquid argon, scintillator, and silicon as readout detectors and

for Al, Fe, Cu, Pb and U as passive samplers. As pointed out before and within

≈ 10% (see also [Fabjan (1985a); Wigmans (1987); SICAPO Collab. (1989c)] and

references therein), e/mip values are ≈ 0.85 for medium-Z passive absorbers (like

Fe or Cu) and ≈ 0.65 for high-Z passive absorbers (like W, Pb or U). Because

in typical sampling calorimeters e/mip ratios are < 1, the phenomenon was also

referred to as electromagnetic sampling ineﬃciencies [Gabriel (1989)].

Sometimes phenomena leading to “sampling ineﬃciencies”, i.e., to the e/mip

ratio dependence on Z

absorber

and Z

readout

, were interpreted as being due to the

so-called transition eﬀects.

Although the calorimeter response is based on the energy deposited in the read-

out detectors after the shower transits from passive samplers to active planes, it

has to be recalled that the term transition eﬀect was introduced by Pinkau (1965)

in the framework of “Approximation B”. In this formulation, not all soft-photon

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624 Principles of Radiation Interaction in Matter and Detection

Fig. 9.4 Mass attenuation coeﬃcients in units of cm

/g for photons b etween 10 keV and 1 GeV

(from [Leroy and Rancoita (2000)], see also [Hubbell (1969); Hubbell, Gimm and Øverbø (1980)]).

interactions with the medium are taken into account. But, as it has been seen, these

interactions play a major role in the process of energy deposition in matter. Pinkau

argued that, when a cascade propagates through matter, the low-energy electrons

absorbed by ionization loss are replaced by new ones from the cloud of γ rays. How-

ever, if the cascade moves into a medium of higher critical energy the absorption

process is stopped, while the replenishment by γ-rays takes place at the old rate

since it is determined by the radiation length and not by the critical energy. In

his argument, soft γ-ray interactions in matter are neglected. These interactions,

when occurring via the photoelectric process (disregarded in the framework of “Ap-

proximation B”), have a dependence on the atomic numbers of materials stronger

than their dependence on radiation lengths. The latter one occurs at higher pho-

ton energies, when pair-production processes dominate. In addition, soft secondary

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Principles of Particle Energy Determination 625

Fig. 9.5 The e/mip ratio as a function of the thickness of passive samplers for U/LAr and

U/scintillator calorimeters (reprinted from Nucl. Instr. and Meth. in Phys. Res. A 259, Wig-

mans, R., On the energy resolution of uranium and other hadron calorimeters, 389–429, Copyright

(1987), with permission from Elsevier; see also [Leroy and Rancoita (2000)]).

electrons are not absorbed instantaneously by collision losses. In “Approximation

B” (see Sect. 2.4.2.1), because the analytical treatment is oriented to provide a de-

scription of energetic electrons and positrons propagating in matter, soft secondary

electrons are taken out of the propagating stream of particles. In other words, these

charged particles are supposed to dissipate their energy immediately.

In reality as seen in Sect. 2.4.2.1, the collision energy-loss per unit of length

by soft electrons and positrons, having energies close to the critical energy, is ≈

. They are absorbed in thicknesses which are fractions, not necessarily small,

of radiation length. The readjustment of shower characteristics is expected to take

place gradually. Pinkau’s predictions were found to be in disagreement with the

experimental data [Crannell et al. (1969)].

Furthermore, it has to be noted that the presence of thin, in units of radiation

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626 Principles of Radiation Interaction in Matter and Detection

length, G10 plates (or other low-Z layers, having large ²

values) in the calorimeter

conﬁguration does not introduce any sizable modiﬁcation to the transversal and lon-

gitudinal cascade development and more generally leaves the containment properties

of electromagnetic cascades unchanged, as was experimentally observed [SICAPO

Collab. (1989a, 1991a)] (see page 207 and also Figs. 2.81 and 2.83). This point is

demonstrated by the values of ﬁtted radial and longitudinal parameters of the ca-

scade energy distribution, which are almost the same independently of the fact that

G10 plates are used in between or next to readout detectors or not at all. Only

the normalization coeﬃcient is modiﬁed because of the e/mip variation. Also, Wig-

mans (1987) has drawn the conclusion from his simulations that this transition

eﬀect in crossing the boundary between two layers with diﬀerent Z plays a very

minor role: the shower will adapt itself “adiabatically” to the new situation.

9.2.2.2 e/mip Dependence on Absorber Thickness

Low energy photons can interact via Compton or photoelectric eﬀects in the passive

sampler close to the readout layer. Thus, soft electrons generated in the passive

absorber may enter the readout detector and contribute to the visible energy. The

electromagnetic response is expected to increase slightly, when the passive sampler

is thin. In turn, because this eﬀect is related to electrons generated close to the

readout detectors, the e/mip ratio will not vary, when the absorber thickness is

increased.

Extensive simulations [Wigmans (1987)] have investigated the e/mip ratio de-

pendence on the thickness of the passive sampler. Medium-Z (like iron) and high-Z

(like uranium) absorbers and commonly employed readout detectors (like liquid ar-

gon and scintillator plates), and dependence on the energy of incoming particles,

were considered. It is observed that the e/mip ratio slightly increases for very thin

absorber plates with a thickness smaller than ≈ 0.8X

. For larger passive samplers

and within the simulation uncertainties, the e/mip ratio does not depend on the

absorber thickness. These results cover well the range of sampler thicknesses com-

monly employed in sampling calorimeters.

An example of such simulations is given in Fig. 9.5, where the e/mip ratio

is shown as a function of the thickness of the uranium passive sampler for liquid

argon and plastic scintillator readouts [Wigmans (1987)]. The eﬀect of the e/mip

ratio dependence on the absorber thickness is visible, but for plates thicker than ≈

(2.5–3.0) mm, i.e., above ≈ 0.8 X

, this eﬀect apparently does not play any further

role.

In summary, the e/mip ratio can be considered to be independent of the passive

sampler thickness for thicknesses above ≈ 0.8 X

, i.e., for most practical sampling

calorimeters.

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Principles of Particle Energy Determination 627

9.2.3 e/mip Reduction in High-Z Sampling Calorimeters:

The Local Hardening Eﬀect

A considerable fraction of the energy deposition in the calorimeter, which increases

logarithmically with the absorber Z-value, as discussed at page 214, occurs through

low-energy particles. In the case of uranium (similarly in other high-Z materials like

tungsten and lead), particles with an energy below 1 MeV account for about 40 %

of the total energy deposition (Fig. 2.85) and are related to the low-energy photons

generated during the cascade development. In these media, the interaction between

generated soft-photons and matter is dominated by the Compton scattering and the

photoelectric eﬀect for energies below ≈ 4 MeV and, mainly, by the photoelectric

eﬀect below ≈ 1 MeV.

The contribution of the Compton cross section to the total cross section for pho-

ton interactions in a given material is proportional to its Z-number (in fact there

are Z electrons per atom). The dependence of the photoelectric cross section on the

atomic number of the material varies somewhat with the energy of the photon. How-

ever, it approximately goes between Z

and Z

at these energies. Therefore, the

interaction cross section of soft photons is much larger in high-Z absorbers than in

low-Z media.

In high-Z absorbers, the transverse shower size, for which ≈ 95% of the energy is

contained within 2R

[Eq. (2.242)], is larger in units of X

(see Table 2.3) than in

low-Z media because of the way electrons and positrons are generated and scattered,

as discussed in Sect. 2.4.2. The mean cosine of charged-particle angles formed with

the shower axis varies as 1/²

, as given in Eq. (2.253). Consequently, large particle

angles are expected for high-Z materials, since these materials have low ²

values

(see Table 2.3).

The soft electron component was studied by simulations [Fisher (1978)] for

the case of a Pb proportional chamber quantameter. The angular distribution of

shower electrons is p eaked along the shower axis. However, there is a large angle

spread. About 12% of all electrons are backscattered in the sampling gap. For low

energy electrons with energy < 3 MeV, the forward peak vanishes and the angular

distribution (although not fully forward-backward symmetric) behaves like ≈ cos θ,

where θ is the angle with respect the shower axis [Fisher (1978); Amaldi (1981)].

According to Amaldi’s estimates on the Z-dependence of the soft electron com-

ponent [Amaldi (1981)], there is a sizeable fraction of large angle low energy elec-

trons, a part of them moving backwards. The fraction of large angle soft electrons

varies as 1/²

and is about 40% in Pb. Soft electrons are locally absorbed: their

mean range is ab out 2.0 g/cm

for an energy of ≈ 4 MeV, and about 0.4 g/cm

for an energy of ≈ 1 MeV. However in sampling calorimeters, low-energy photons

from the electromagnetic cascade may convert into electrons suﬃciently close to the

surface of the absorber plate to escape and contribute to the measured signal. Al-

though for thin readout layers these electrons can give a substantial contribution to

the overall signal, the e/mip ratio is < 1 for high-Z passive samplers in sampling

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628 Principles of Radiation Interaction in Matter and Detection

calorimeters. To further decrease the e/mip ratio, low-Z passive absorbers have to

be added in b etween high-Z and readout layers, bringing to three the number of

media of the sampling calorimeter.

The additional low-Z absorbers have a lower photoelectric cross section than

high-Z samplers. Thus photons, whose energy distribution has been determined by

the high-Z samplers, will propagate in a medium with a relative lower photoelec-

tric cross section. These additional absorbers have to be thin in units of radiation

lengths in order to absorb soft electrons generated close to the surface in the high-Z

samplers, while at the same time these additional thin passive samplers are not (or

marginally) becoming part of the shower generation process.

Taking into account the mean absorption range of 1 MeV electrons (about

0.4 g/cm

) and Eq. (2.253), thicknesses (along the shower axis) of ≈ 0.3 g/cm

are needed to prevent a large number of soft electrons from entering the readout

detectors and consequently to reduce the e/mip ratio. This eﬀect was referred to as

the local hardening eﬀect [SICAPO Collab. (1989b)], because softer electrons and

positrons are locally taken out of the charged-particle stream, thus hardening the

shower while traversing readout layers.

The degree of hardening and consequently the size of the reduction of the e/mip

ratio depend both on the thickness of low-Z absorbers and, because of the photo-

electric Z-dependence on the photon cross section, on the diﬀerence between the

absorber

and the low-Z of added thin absorber plates. This allows the ﬁne tuning

of the calorimeter response to electromagnetic cascades. In addition, as it will be

seen in following sections, the exploitation of the local hardening eﬀect is a way to

achieve the compensation condition in hadronic calorimetry.

A reduction of the energy detected in electromagnetic showers, due to the

presence of low-Z absorbers in front of silicon mosaics in a Si/U calorimeter,

was observed in [Pensotti, Rancoita, Simeone, Vismara, Barbiellini and Seidman

(1988); Pensotti, Rancoita, Seidman and Vismara (1988)]. This reduction can be

exploited to equalize the response of a Si/U calorimeter to incoming electrons and

hadrons. The evidence of the local hardening eﬀect was found and systematically

studied for Si/U and Si/W calorimeters [SICAPO Collab. (1989b)]. This eﬀect was

also investigated by means of EGS4 simulations [Br¨uckmann et al. (1988); Wigmans

(1988); Lindstroem et al. (1990); Hirayama (1992)]. In this way, it was possible to

evaluate the reduction of the e/mip ratio in the case of U/scintillator calorimeter

with uranium plates wrapped in stainless steel.

Experimentally, the local hardening eﬀect is found to be almost independent

of the incoming electron energy [SICAPO Collab. (1989b,d)], and of the sampling

frequency [SICAPO Collab. (1992b)]. In other words, the reduction of visible energy

∆²/² depends on the thickness of the low-Z absorb er, but is independent of the

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Principles of Particle Energy Determination 629

thickness of the passive high-Z absorber, where we deﬁne:

∆²

vis

(no low−Z) − ²

vis

(low−Z)

vis

(no low−Z)

= 1 −

vis

(low−Z)

vis

(no low−Z)

. (9.11)

Moreover, the insertion of low-Z material downstream, behind the silicon detectors,

is almost twice as eﬀective as inserting the same thickness in front of the readout de-

tectors [SICAPO Collab. (1988b)]. This occurs because the back-scattered electrons

are softer than the forward electrons.

The downstream insertion (behind the silicon detectors) of G10 plates with

thicknesses varying from 0.5 up to 5.0 mm in a Si/U calorimeter conﬁguration is at

the origin of the local hardening eﬀect. Thus, it is responsible for the reduction of

the visible energy

∆²

vis

(no G10) − ²

vis

(G10)

vis

(no G10)

= 1 −

vis

(G10)

vis

(no G10)

shown in Fig. 9.6 from [SICAPO Collab. (1989b); Leroy and Rancoita (2000)]. The

value of ∆²/² is ≈ 28% for ²

vis

(G10 = 5mm). By means of Eq. (9.4) the change in

the shared energy, due to a 5 mm G10 absorber added to the passive sampler, is

given by

∆E

(G10 = 5mm) =

(no G10) − E

(G10 = 5mm)

(no G10)

= 1 −

(G10 = 5mm)

(no G10)

≈ 10%.

The corresponding e/mip ratio reduction is

e/mip(G10 = 5mm) =

vis

(G10 = 5mm)

vis

(no G10)

1 −

∆²

(G10 = 5mm)

(no G10)

1 −

∆E

(G10 = 5mm)

, (9.12)

therefore,

e/mip(G10 = 5mm) ≈ 0.8

vis

(no G10)

= 0.8 e/mip(no G10). (9.13)

In addition, the e/mip ratio is not modiﬁed by inserting low-Z layers between the

uranium plates. The visible energy was measured to be reduced by about 9.5%,

instead of ≈ 28%, when 5 mm G10 plates were located inside the uranium samplers

and not adjacent to the silicon readout detectors [SICAPO Collab. (1989b)]. This

reduction corresponds to the decrease of E

, as discussed above, and is explained

by the increase of the overall passive sampler (low- and high-Z plates). Thus, the

e/mip ratio is almost constant when no local hardening eﬀect occurs, as expected.