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

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680 Principles of Radiation Interaction in Matter and Detection

errors, for incoming hadron energies ≥ 10 GeV.

9.9.1 Compensation Condition by Detecting Neutron Energy

The role of neutron energy detection in compensating or almost compensating ca-

lorimetry and, in particular, the role of the nuclear ﬁssion in the building of the

signal of uranium calorimeters (see e.g., [Leroy, Sirois and Wigmans (1986)]) have

intensively investigated experimentally and, also, by using Monte-Carlo simulations

(e.g., see [Fabjan et al. (1977); Abramowicz et al. (1981); De Vincenzi et al. (1986);

Bernardi et al. (1987); HELIOS Collab. (1987); Drews et al. (1990); Ros (1991)],

and also [Wigmans (1987); Br¨uckmann et al. (1988)], Section 3.3.3.2 in [Wigmans

(2000)]).

In hadronic cascades, neutrons (n) are produced either by spallation

or, mostly,

are evaporated from highly excited nuclear fragments (see [Fraser et al. (1965); Pat-

terson and Thomas (1981); Brau and Gabriel (1985); Leroy, Sirois and Wigmans

(1986); Wigmans (1987); Br¨uckmann et al. (1988)] and references therein). The

energy spectrum has a peak below 10 MeV, corresponding to evaporation neu-

trons. For heavy elements, the evaporation peak is slightly shifted to lower ener-

gies. For ﬁssionable nuclei (

238

U), a large amount of fast neutrons are generated ow-

ing to subsequent ﬁssion processes. Those neutrons with energies larger than ab out

1 MeV are likely to dissipate their energy through inelastic (n,γ) reactions on heavy

nuclei. At lower energies (below 100 keV), neutrons are captured by nuclei. Neutrons

emerging with energies ab ove 10 MeV are mainly produced by spallation processes

and have an energy spectrum almost independent of the atomic number of the ab-

sorber element. The few very highly energetic neutrons, that are generated, travel

some distance inside the calorimeter before interacting. They behave similarly to

highly energetic charged hadrons. Furthermore, the longitudinal proﬁle of neutron

generated in hadronic cascades was measured in iron and lead absorbers for in-

coming hadrons of 24 and 200 GeV ([Russ et al. (1991)] and references therein).

The proton-induced neutron yield, Y (E, A), was measured from light to heavy

nuclei with incoming protons above ≈ 0.5 GeV [Fraser et al. (1965)]. To a ﬁrst

approximation, it is given by [Patterson and Thomas (1981)]:

Y (E, A) = 0.1 × (E − 0.12)(A + 20), for non-ﬁssionable nuclei (9.63)

and

Y (E, A) = 50 × (E − 0.12), for ﬁssionable nuclei (like

238

U), (9.64)

where E (in GeV) is the incoming hadron energy and A is the target mass.

From Eqs. (9.63, 9.64), we expect ≈ 45 n/GeV per incoming proton on depleted

uranium absorbers and ≈ 20 n/GeV on lead absorber. These values agree with the

The relative probability of neutron-to-proton spallation increases as A increases, because the

Z/A ratio decreases.

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Principles of Particle Energy Determination 681

Fig. 9.22 The measured e/π ratio with the compensating uranium/scintillator calorimeter real-

ized by the HELIOS Collaboration (reprinted from Nucl. Instr. and Meth. in Phys. Res. A 262,

Akesson, T. et al., Performance of the uranium/plastic scintillator calorimeter for the HELIOS

experiment at CERN, 243–263, Copyright (1987), with permission from Elsevier, e.g., for the list

of the authors see [HELIOS Collab. (1987)]; see also [Leroy and Rancoita (2000)]).

experimental data from [Fraser et al. (1965); Leroy, Sirois and Wigmans (1986)]

(see also [Russ et al. (1991)] and references therein). Therefore, when uranium

absorbers are employed, the neutron yield is more than doubled with respect to a

high-A nucleus, such as lead.

In hadron calorimeters employing hydrogenous material as active devices, like

scintillators or scintillating ﬁbers, the compensation condition can be achieved by

increasing the relative contribution of the pure hadronic signal, namely by increasing

the h/mip ratio. Indeed, the neutrons, generated in the last stages of the hadronic

cascade development where processes at the nuclear level occur, transfer part of

their energy (E

) to the protons of the scintillator. The maximum recoil energy

R,max

) transferred from an incoming non-relativistic neutron to a recoiling nucleus

of atomic weight A is:

R,max

(1 + A)

. (9.65)

For neutron–hydrogen collisions, the full energy of the neutron can be transferred to

the recoiling proton (E

p,R,max

), namely E

p,R,max

= E

. This results in an increase of

the hadronic signal, whose amount can be varied by modifying either the sampling

fraction or the ratio of the thicknesses of the passive sampler and the active medium,

i.e., by increasing the number of protons available as targets for neutrons [Leroy,

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682 Principles of Radiation Interaction in Matter and Detection

Sirois and Wigmans (1986)].

The increase of the hadronic signal is limited by the saturation of the scintillator

response, which occurs in the presence of dense ionization loss, i.e., for fast pro-

tons. In scintillators, this phenomenon was studied for many years and is described

to a ﬁrst approximation

by Birks’ law [Birks (1951)] expressed in Eq. (5.4), where

dL/dx is the light yield per unit path length, dE/dx is the speciﬁc energy-loss by

collisions for the charged particle, S is the normal scintillation eﬃciency, the para-

meter kB (see [Knoll (1999)]) describes the quenching which occurs for high-density

ionization. Extended calculations regarding the saturation eﬀect of the scintillator

response in hadronic calorimetry can be found in the literature, e.g., see [Brau and

Gabriel (1985); Brau et al. (1985); Brau and Gabriel (1989)].

The calorimeter performance also depends on the relative time response for

the various components of the calorimeter signal. The energy loss by collisions of

both charged relativistic particles and spallation protons occurs at a very short

time after the arrival of the incoming primary hadron. The temporal dependence of

ﬁssion and neutron capture was studied for uranium samplers (see e.g. [Brau and

Gabriel (1989)]). It lasts a few hundreds ns. The majority of the hadronic responses

of uranium calorimeters occurs in the ﬁrst (50–100) ns. As consequence, the h/mip

ratio will depend on the integration time of the signal. The typical integration time

is about 100 ns.

A comp ensating uranium/plastic scintillator calorimeter was built by the HE-

LIOS Collaboration [HELIOS Collab. (1987)]. In Fig. 9.22, the e/π ratio, measured

up to 200 GeV, is shown. Their calorimeter modules consisted of uranium and scin-

tillator layers 3.0 mm and 2.5 mm thick, respectively. This calorimeter had a ratio

p/a

of ≈ 1.2, for the thicknesses of the passive relative to active samplers.

The compensation condition or an almost compensating condition were also

achieved by the ZEUS Collaboration (see [Klanner et al. (1988); d’Agostini et al.

(1989); Tiecke (1989); Behrens (1990); Drews et al. (1990); Ros (1991)] and referen-

ces therein), with devices for which the R

p/a

ratios were between 1.07 and 1.26. In

Fig. 9.23, the e/π ratio, measured up to 100 GeV with the ZEUS prototype calorime-

ter, is shown [Behrens (1990); Ros (1991)]. This calorimeter consisted in uranium

and scintillator layers 3.3 mm and 2.6 mm thick, respectively. The ZEUS Collabora-

tion measurements show the calorimeter is compensating or almost compensating

above (2–3) GeV, i.e., the e/π value is close to 1 indep endently of the incoming

hadron energy. However, as discussed in Sect. 9.8.1 and in [Wigmans (1987)], below

≈ 2 GeV the calorimeter properties, like the e/π value, diﬀer from the intrinsic ones

at higher energies (see Fig. 9.23).

For R

p/a

values largely diﬀering from ≈ 1.2, the uranium/plastic scintillator

calorimeter is no longer a compensating calorimeter. The measured e/π ratio, at

10 GeV, was ≈ 1.12 for R

p/a

≈ 0.8 [

Akesson et al. (1985)] and ≈ 0.81 for R

p/a

≈

2.0 [Catanesi et al. (1987)]. Furthermore ([Br¨uckmann and Kowalski (1986); Ros

Other more reﬁned approximations are given in [Knoll (1999)] and in references therein.

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Principles of Particle Energy Determination 683

Fig. 9.23 The measured e/π ratio with the compensating uranium/scintillator calorimeter real-

ized by the ZEUS Collaboration (reprinted from Nucl. Phys. B (Proc. Supp.) 23 (Issue 1), Ros,

E., Inﬂuence of instrumental eﬀects on the performance of the ZEUS calorimeter, 51–61, Copyright

(1991), with permission from Elsevier; see also [Leroy and Rancoita (2000)] and [Behrens (1990)]).

(1991)] and references therein), the experimental data show a decrease (as expected)

of ≈ 7% in the e/π ratio when the integration times are increased from ≈ 100 ns to

≈ 1 µs.

An almost compensating lead/plastic scintillator calorimeter was built by the

ZEUS Collaboration [Bernardi et al. (1987)]. Their calorimeter modules consisted of

lead and scintillator layers 10.0 mm and 2.5 mm thick, respectively. This calorime-

ter had a ratio R

p/a

of ≈ 4 between the thicknesses of the passive to active sam-

plers. The measured e/π ratio was ≈ 1.05, above 10 GeV.

A consequence of these results is that the use of uranium is not an absolute

requirement to achieve compensation, when hydrogeneus detectors are employed as

readout media. However, it is fair to say that uranium still possesses the advantage

of easy calibration with the noise of its natural radioactivity. Furthermore, an almost

non-variable R

p/a

ratio of about 1.2 for passive uranium samplers and of about 4

for passive lead samplers is needed to achieve compensation with hydrogeneus active

media.

Lead and scintillating ﬁbres (or strips) were used in calorimetry, for instance by

the SPACAL Collaboration ([Acosta et al. (1991); DeSalvo (1995)] and references

therein) and the CHORUS ([Buontempo et al. (1995)] and references therein). They

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684 Principles of Radiation Interaction in Matter and Detection

have achieved an e/π ratio ≈ 1.1 (or larger) at 10 GeV.

Data from Fe/scintillator calorimeter with a longitudinal tile conﬁguration [Ariz-

tizabal et al. (1994)] show that such a calorimeter was not compensating (see

also [Botner (1981)]. The e/π ratio was larger than ≈ 1.2 at an incoming parti-

cle energy of 20 GeV.

Other readout media using hydrogeneous liquids were investigated ([Pripstein

(1991)] and references therein) for low-A (e.g., see [Aubert et al. (1993)]) and high-

A passive sampler. A detailed study of their almost compensating properties (as

a result of Monte-Carlo simulation predictions) is presented in [Brau and Gabriel

(1989)].

Hadronic calorimeters employing liquid argon

∗∗

as the active medium were and

are presently used in high-energy physics experiments (e.g., see [Fabjan et al. (1977);

H1 Collab. (1988); D0 Collab. (1989); SSC Detector R&D at BNL (1990); D0 Col-

lab. (1993); Axen et al. (1993); H1 Collab. (1993a); Gingrich et al. (1994); Guida

(1995); ATLAS Collab. (1997)] and references therein). These calorimeters are

also non-compensating, when uranium absorbers constitute the passive samplers

(e.g., see [Kondo et al. (1984)]).

The underlying phenomena that are resp onsible for the signiﬁcant diﬀerence

with respect to the plastic scintillator calorimeters, are mainly explained taking

into account the neutron cross sections and the saturation eﬀects of the active

medium (e.g., see [Brau and Gabriel (1985); Wigmans (1987); Brau and Gabriel

(1989)]). The neutron cross section on hydrogen continues to rise below (1–2) MeV,

while in argon falls. In this way, the neutron cross section is larger on hydrogen

than on argon nucleus. In addition, the maximum recoil energy transferred to an

argon nucleus is [from Eq. (9.65)] ≈ 9.5% of the incoming neutron energy, while it

is the entire neutron energy for an interaction on hydrogen. Therefore, on average,

the recoil nucleus will emerge with a lower velocity for the same incoming neutron

energy.

Following the suggestion of Brau and Gabriel (1985), saturation eﬀects in li-

quid argon can be expressed in the form of Birks’ law [Eq. (5.4)]. In scintillators,

the value of the parameter kB is ≈ (0.01–0.02) g cm

MeV

−1

, while in liquid ar-

gon it is 0.0045 g cm

MeV

−1

[Fabjan et al. (1977); Babaev et al. (1979); Brau and

Gabriel (1985)]. For electrons at all energies, it is assumed that kB = 0. Although

liquid argon is less saturating than plastic scintillator, the kinematic constraint on

the energy transferred in neutron–nucleus collision leads (as mentioned above) to

a greater suppression of the signal, because the highly ionizing recoil nuclei receive

only one-tenth the relative energy of the less highly ionizing recoil protons in scin-

tillator. The e/mip ratio is also slightly larger in the case of liquid argon, whose

Z-value is larger than the one of plastic scintillator (see Sect. 9.2.2).

In uranium/liquid argon calorimetry, the e/π ratio measured at 10 GeV is ≈ 1.09

([D0 Collab. (1993)], e.g., see also [D0 Collab. (1989); Guida (1995)]), or even larger

∗∗

Reviews on liquid-argon calorimetry are given in [Gordon (1991); Fabjan (1995b)].

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Principles of Particle Energy Determination 685

(e.g., see [SSC Detector R&D at BNL (1990)]), depending on sampler thicknesses

and integration times.

Iron cladding of high-Z passive absorbers was investigated for reducing the e/π

ratio (e.g., see [Gordon (1991); Mockett and Boulware (1991)]), but the compensa-

tion is hardly achievable. It was also suggested (e.g. see [D0 Collab. (1989); Cennini

et al. (1990); Fabjan (1995b)]) that, by the addition of dopants containing hydrogen

(like methane) to liquid argon, the e/π ratio might be reduced. The experimental

results [D0 Collab. (1989)] do not conﬁrm that the compensation condition can be

achieved when methane is added to liquid argon.

In large liquid-argon calorimeters, diﬀerent passive samplers have often been

used inside the same device (e.g., [H1 Collab. (1988, 1993a); Gingrich et al. (1994);

ATLAS Collab. (1997)]). The choice of the types and the sequence of absorbers is

related to constraints coming from the experimental apparatus and the physics of

the experiment. In particular, lead absorbers were employed in the ﬁrst part of the

device. This part is able to longitudinally contain almost all the energy deposited by

electromagnetic showers and is called the electromagnetic section. In the second part,

called the hadronic section, iron or copper absorbers were used. These calorime-

ters are intrinsically non-compensating, e.g., the measured e/π ratio is ≈ 1.17 at

20 GeV for the Pb (passive samplers of the electromagnetic section) and Cu (passive

samplers of the hadronic section) calorimeter of the H1 Collaboration [H1 Collab.

(1988)]. However, methods for restoring the calorimeter response almost to linearity

were developed taking into account the response of each active sampler (e.g., [H1

Collab. (1988, 1993a); Gingrich et al. (1994); ATLAS Collab. (1997)]) or the intrinsic

e/h ratios of the two sections of the calorimeter (e.g., [ATLAS Collab. (1997)]).

The method of weighted energies, develop ed by the H1 Collaboration in [H1

Collab. (1988, 1993a)] (see also [Abramowicz et al. (1981); Beretvas et al. (1993)]),

takes into account the ﬂuctuations of the electromagnetic energy deposition in the

electromagnetic and hadronic sections of the calorimeter and requires to determine

an energy-dependent weight for the response of each active sampler.

The method of the benchmark approach, developed by the ATLAS Collabora-

tion [ATLAS Collab. (1997)], consists in a two-step procedure, in which the fractions

of the scaled hadronic energies E

(see the deﬁnition on page 675) deposited in the

two sections of the calorimeter are determined.

A comparison of these two methods is given in [ATLAS Collab. (1997)]. It is

shown that the calorimeter response is readjusted to linearity, within ±2% for inco-

ming hadrons of energy between 20 and 300 GeV, by using both methods. However,

it must be remembered that in hadronic jets (e.g., see [Perkins (1986)]) with the

same energy, the number of particles and their energies vary from event to event. As

a consequence, these procedures have to be carefully studied and tuned when used

in a high energy physics experiments [H1 Collab. (1988); Wigmans (1988, 1991)].

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686 Principles of Radiation Interaction in Matter and Detection

9.9.2 Compensation Condition by Tuning the e/mip Ratio

The compensating (or almost compensating) condition can be achieved by tuning

the electromagnetic response of the calorimeter. In Sects. 9.2.3, 9.3, it was discussed

how an intrinsic calorimetric property, namely the e/mip ratio, can be largely de-

creased by choosing and locating the set of passive samplers in the suited sequence.

Mainly two eﬀects, or their combination ([SICAPO Collab. (1993c)] and refe-

rences therein), were exploited to provide such a tuning. The former is the local

hardening eﬀect

∗

, realized by inserting thin low-Z absorbers next to the readout

detectors when high-Z absorbers constitute the passive samplers. The latter is the

ﬁltering eﬀect

†

, obtained by using a combination of low-Z and high-Z materials

as absorbers, both of them non-negligible in units of radiation length. The local

hardening eﬀect exploits the way the energy deposition occurs in the ﬁnal stage of

the electromagnetic cascade, when soft electrons and photons interact. The ﬁltering

eﬀect makes use of the diﬀerent electron (and positron) energy distributions gener-

ated by electromagnetic showers in passive media, namely combining passive media

whose critical energies have greatly diﬀerent values. The property of these absorbers

is such that the radiation losses by electrons (and positrons) dominate at diﬀerent

values of electron (and positron) energy in the subsequent passive media. Moreover,

these eﬀects were studied by using Monte-Carlo simulations (e.g., see [Wigmans

(1988); Brau and Gabriel (1989); Brau, Gabriel and Rancoita (1989); Mockett and

Boulware (1991); Job et al. (1994)]).

In hadronic cascades, the visible energy of the pure hadronic comp onent

(Sect. 3.3.1) is mainly generated by collision losses by charged particles (spallation

protons and relativistic secondary particles).

The pure hadronic visible-energy is expected to be aﬀected neither by the loca-

tion of passive absorbers nor by eﬀects related to the limited path needed to fully

absorb very soft charged particles. In fact, both secondaries and recoil protons (in-

volved in collision loss processes) undergo negligible (if any) radiation losses and,

being fast, can travel long distances inside the calorimeter before being absorbed.

The overall calorimeter response to the energy deposited by collisions may de-

pend on saturation eﬀects (even in the case of fast protons) occurring in active

samplers, by the passage of densely ionizing particles. Also the neutron contribu-

tion to the hadronic visible-energy depends on the property of the active medium. As

discussed in the previous section, the maximum recoiling energy [Eq. (9.65)] trans-

ferred from an incoming non-relativistic neutron is E

(i.e., the kinetic energy of

the incoming neutron) onto a hydrogen target, like in scintillators, but only 9.5%

of E

on an argon nucleus and ≈ 13.3% of E

on silicon nucleus. For instance, the

saturation aﬀects detectors based more on scintillation media than on liquid argon

∗

The experimental evidence for this eﬀect was given by the SICAPO Collaboration, see [SICAPO

Collab. (1989b)] and references therein, and also Sect. 9.2.3.

†

The experimental evidence for this eﬀect was given by the SICAPO Collaboration, see [SICAPO

Collab. (1989c)] and Sect. 9.3.

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Principles of Particle Energy Determination 687

[see Sect. 9.9.1 and the values of kB coeﬃcient in Eq. (5.4)], while saturation is very

small in other devices, like silicon detectors ([Hancock, James, Movchet, Rancoita

and Van Rossum (1983, 1984); Rancoita (1984); Brau and Gabriel (1989); Brau,

Gabriel and Rancoita (1989)] and references therein). In particular, silicon detec-

tors demonstrated to have an extremely linear response for very high densities of

ionization, as for heavily ionizing ions above ≈ a few MeV [Sattler (1965); Wilkins

et al. (1971)]. Some saturation eﬀects are observed for recoils of silicon atoms below

≈ 3 MeV [Sattler (1965)]. However, even for silicon ions of kinetic energy as low as

100 keV nearly a half of the deposited energy is detected as an observable output

signal relatively to an electron of the same energy [Sattler (1965)]. For recoiling

nuclei, the saturation properties of the active medium determine the fraction of

the recoil energy, which can contribute to the overall hadronic visible-energy: this

fraction increases as the detector resp onse approaches linearity. We can conclude

that the purely hadronic visible-energy is expected, to a ﬁrst approximation, to be

marginally aﬀected when tuning the e/mip value (e.g., see [SICAPO Collab. (1991c)]

and references therein). A systematic investigation of the local hardening eﬀect on

hadronic cascades was carried out for incoming electrons and protons of 8, 10, and

12 GeV [SICAPO Collab. (1990a, 1991b)], employing a calorimeter with silicon de-

tectors as active media and passive uranium samplers (each 1.5 cm thick). Additional

low-Z G10 plates were located on the front and at the rear of the active readout

detectors in order to tune the electromagnetic response (Sect. 9.2.3). The G10 plates

(of equal thickness on the front and at the rear of silicon readout planes) were 1,

3, and 5 mm thick. The silicon planes had a support structure made by two G10

sheets 0.2 and 1.0 mm thick, respectively. The visible-energy reduction,

∆²

(G10),

measured (from an average of measurements at the three incoming-particle ener-

gies) for protons and electrons is shown in Fig. 9.24 ([Leroy and Rancoita (2000)],

see also [SICAPO Collab. (1990a, 1991b)]), being

∆²

(G10) =

vis

− ²

vis

(G10)

vis

1 −

vis

(G10)

vis

, (9.66)

where ²

vis

and ²

vis

(G10) are the visible energy without and with additional G10

plates, respectively. In the following, we will indicate with

∆²

(e, G10) the visible-

energy reduction for electromagnetic showers, i.e., for incoming electrons, and with

∆²

(π, G10) the visible-energy reduction for incoming hadrons. The data show that

∆²

(G10) increases as the G10 plate thickness increases, as expected. These re-

sults, for incoming electrons, are in agreement with previous measurements per-

formed using electromagnetic calorimeters (see Sect. 9.2.3, and [SICAPO Collab.

(1988b, 1989b,d, 1992b)]). Equation (9.54) gives the hadronic visible-energy, ²

vis

(π),

in terms of the visible energies of the purely electromagnetic, ²

vis

(e), and purely

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688 Principles of Radiation Interaction in Matter and Detection

hadronic, ²

vis

(h), components and, consequently [see also Eq. (9.55)],

∆²

(π, G10) =

vis

(π) − ²

vis

(π, G10)

vis

(π)

[²

vis

(e) − ²

vis

(e, G10)]

vis

(π)

(1 − f

) [²

vis

(h) − ²

vis

(h, G10)]

vis

(π)

∆²(e, G10) + (1 − f

)∆²(h, G10)

vis

(π)

= f

∆²(e, G10)

vis

(π)

vis

(π)

vis

(e)

(e/π) + (1 − f

)

∆²(h, G10)

vis

(π)

= (e/π)f

∆²

(e, G10) + (1 − f

)

vis

(h) − ²

vis

(h, G10)

vis

(π)

= (e/π)f

∆²

(e, G10)

+[1 − f

]

vis

(h)

vis

(π)

¸·

1 −

vis

(h, G10)

vis

(h)

= (e/π)f

∆²

(e, G10)

vis

(π) − f

vis

(e)

vis

(π)

¸·

1 −

vis

(h, G10)

vis

(h)

= (e/π)f

∆²

(e, G10)

+[1 − (e/π)f

]

1 −

h/mip(G10)

h/mip

(G10)

, (9.67)

where the e/π ratio is the one obtained without additional G10 plates.

For instance (Fig. 9.24) the measured energy reduction,

∆²

(e, G10 = 5 mm),

for incoming electrons is ≈ 40% with the insertion of 5 mm G10 plates, while

for incoming protons the measured value of

∆²

(π, G10 = 5 mm) is ≈ 12%. The

lower reduction of visible energy for incoming protons is consistent with the value

predicted by Eq. (9.67). In fact at these energies, f

is ≈ (25–27)% on average

[Eq. (3.74)]. In addition, the measured e/π ratio (Fig. 9.25), without additional G10

plates, is ≈ 1.22.

In order to estimate

∆²

(π, G10 = 5 mm), we have to take into account the

fact that the local hardening eﬀect acts on the electromagnetic component of the

hadronic shower and that, to a ﬁrst approximation even for the case of 5 mm thick

G10 low-Z additional absorbers, we have ²

vis

(h) ≈ ²

vis

(h, G10 = 5 mm). In fact,

the slight decrease of the purely hadronic visible-energy, due to the decrease of the

shared energy (E

) when the G10 plates are inserted, is mostly compensated by

the slight increase of the h/mip ratio, since the low-Z (i.e., low-A) materials have

a larger A/Z ratio than the uranium absorbers (see discussion in [SICAPO Collab.

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Principles of Particle Energy Determination 689

Fig. 9.24 Percentage (R) of the visible-energy reduction for protons (bottom data) and electrons

(top data) versus the thickness of the added G10 plates (reprinted from Phys. Lett. B 242,

Angelis, A.L.S. et al., Evidence for the compensation condition in Si/U hadronic calorimetry by

the local hardening eﬀect, 293–298, Copyright (1990), with permission from Elsevier, e.g., for the

list of the authors see [SICAPO Collab. (1990a)]; see also [Leroy and Rancoita (2000); SICAPO

Collab. (1991b)]). R(%) is deﬁned as: R[%] = 100

∆²

(G10) = 100

vis

−²

vis

(G10)

vis

, where ²

vis

the visible energy without additional G10 plates and ²

vis

(G10) is the visible energy with additional

G10 plates.

(1990a, 1991b)], and Sect. 3.3.1). Consequently, the second term in Eq. (9.67) can

be neglected, namely we have:

∆²

(π, G10) ≈ (e/π)f

∆²

(e, G10). (9.68)

Thus, from Eq. (9.68) we get

∆²

(π, G10 = 5 mm) ≈ 1.22 × 0.26 × 40% ≈ 12.6% in

agreement with experimental data shown in Fig. 9.24.

The local hardening eﬀect allows the tuning of the e/mip ratio in such a way

that the compensation condition is achieved. In Fig. 9.25, the e/π ratio [SICAPO

Collab. (1990a, 1991b, 1993c)] measured with the Si/U calorimeter, described above,

is shown as a function of the thickness of G10 plates inserted. The experimental data

show that the e/π ratio decreases from ≈ 1.22 up to ≈ 0.87, when 5 mm G10 plates

are inserted. The compensation condition is obtained for a G10 additional thickness

of 1.2 ± 0.2 mm. The calorimeter properties which determine the value of the e/π

ratio are mostly due to the added low-Z material, which causes the occurrence of

the local hardening eﬀect.