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

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700 Principles of Radiation Interaction in Matter and Detection

Fig. 9.30 Intrinsic resolutions (σ

intr

) in [%] calculated by means of Monte-Carlo simulations for

diﬀerent passive and active samplers (namely liquid argon [LAr], scintillator [PMMA] and silicon

detector [Si]) as a function of the passive sampler thickness in mm (from [Leroy and Rancoita

(2000)], see also [Wigmans (1987)] for details). The curves are for: (1) Pb/0.4 mm Si ∼ Pb/2.5 mm

LAr and no neutron capture, (2) Pb/0.4 mm Si ∼ Pb/2.5 mm LAr for 20% neutron capture, (3)

U/0.4 mm Si for 20% neutron capture, (4) U/2.5 mm LAr for 20% neutron capture, (5) Pb/2.5 mm

PMMA, (6) U/2.5 mm PMMA, (7) Fe/0.4 mm Si ∼ Fe/2.5 mm LAr for 20% neutron capture, (8)

Fe/5 mm PMMA.

For an Fe (Pb) sampler with a thickness of 6 mm, one ﬁnds ≈

0.30

√

0.32

√

The intrinsic resolution was also studied with Monte-Carlo simulations

††

and

measured, usually, with compensating or quasi-compensating calorimeters

‡‡

In Fig. 9.30, intrinsic resolutions (σ

intr

) from [Leroy and Rancoita (2000)] (see

also [Wigmans (1987)]) are shown for low-A (Fe) and high-A (Pb and U) passive

samplers, some active media, a small percentage of neutron capture and no neutron

capture (see [Wigmans (1987); Leroy and Rancoita (2000)] for details). These results

show that, for non-hydrogeneous active media, the intrinsic resolution is larger

than for hydrogeneous detectors, since the amount of invisible energy is decreased

††

One can see [Amaldi (1981); Brau and Gabriel (1985); Wigmans (1987); Br¨uckmann et al.

(1988); Fesefelt (1988); Wigmans (1988); Brau and Gabriel (1989); Brau, Gabriel and Rancoita

(1989); Wigmans (1991)] and references therein.

‡‡

The reader can see, for instance, [Fabjan et al. (1977); HELIOS Collab. (1987); Tiecke (1989);

Drews et al. (1990); SICAPO Collab. (1995a,b, 1996)] and references therein.

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Principles of Particle Energy Determination 701

when its ﬂuctuation is reduced. As a consequence, compensating calorimeters with

hydrogeneous readout are expected to minimize the contribution from the intrinsic

resolution. However these latter ones (Sect. 9.9.1) require an almost ﬁxed ratio

between passive and active sampler thicknesses.

For non-hydrogeneous readout, the neutron contribution to the overall visible-

energy is small and the estimated intrinsic energy resolutions are

intr

√

≈











0.2

√

for low-A (like Fe) absorbers,

0.4

√

high-A (like Pb or U) absorbers.

These values depend weakly on the passive sampler thickness. The intrinsic resolu-

tions, σ

intr

, were determined for low-A and high-A passive samplers (Table 9.9) and

are in agreement with calculated ones shown in Fig. 9.30.

For example, in compensating calorimeters with hydrogeneous readout, i.e., with

a ratio R

p/a

of ≈ 1.2 for U samplers (Sect. 9.9.1), the intrinsic energy resolution is

≈ 0.20/

√

E. An overall energy resolution [HELIOS Collab. (1987)] of ≈ 0.34/

√

(Fig. 9.28) was achieved with U samplers, each with a thickness of 2.9% of an intera-

ction length. Also the tuning of the mip ratio allows one to reach the compensating

condition in Pb+Fe calorimeters (Sect. 9.9.2). In this case, the thickness ratio of

the Fe absorber

∗

to the Pb absorber is ≈ 3. The measured intrinsic energy reso-

lution is ≈ 0. 25/

√

E. For an almost-compensating Pb+Fe calorimeter (see Table 1

of [SICAPO Collab. (1996)]), the overall energy resolution achieved is ≈ 0.66/

√

(Fig. 9.29) with passive samplers, each with a thickness of 14.3% of an interaction

length. For this type of calorimeter the energy resolution can be improved by de-

creasing the passive sampler thickness (while keeping the Fe to Pb sampler thickness

ratio ﬁxed, see Sect. 9.9.2) to reduce the contribution of the sampling ﬂuctuations

to the overall energy resolution [Eq. (9.78)]. As already discussed, for these com-

pensating (or almost compensating) calorimeters the φ(e/π) term is negligible and

the calorimeter energy resolution scales as 1/

√

E (see Figs. 9.28 and 9.29).

9.10.1

Non-Compensation Eﬀects and the

(

e/π

) Term

The function φ(e/π) in Eq. (9.77) takes into account eﬀects of non-compensation. It

should be emphasized that there exists no analytical expression for the function

φ(e/π), since it depends on several complex processes at the nuclear level, which

are at the basis of the e/π ratio value. This ratio (Sect. 9.8) depends on both the

active and passive samplers. Measured e/π ratios, mostly at the reference energy

of 10 GeV, were given in Sect. 9.9 for diﬀerent types of calorimeters (for estimated

values based on Monte-Carlo simulations see references cited on page 670). Their

values cover a wide range: ≈ (0.8–1.3).

∗

The contribution of Fe absorbers to the overall intrinsic resolution is lower than the one due to

Pb absorbers.

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702 Principles of Radiation Interaction in Matter and Detection

Table 9.9 Intrinsic resolutions σ

intr

in [%], measured with hydrogeneous media (plastic scintillators) and

non-hydrogeneous media (silicon detectors) from [Leroy and Rancoita (2000)].

Passive Hydrogeneous Non-hydrogeneous Reference

sampler readout [see(1)] readout

Fe (16.2−17.5) [SICAPO Collab. (1995b)]

Fe 19.8±2.4 [SICAPO Collab. (1996)]

Fe+Pb [see(2)] 24.5 [SICAPO Collab. (1995b)]

Fe+Pb [see(2)] 25.9±2.1 [SICAPO Collab. (1996)]

Pb (44.7−48.5) [SICAPO Collab. (1995b)]

Pb 49.9±14.1 [SICAPO Collab. (1996)]

Pb 13.4±4.7 [Drews et al. (1990)]

Pb 11±5 [Tiecke (1989)]

U 47.8±1.9 [SICAPO Collab. (1995a)]

U 20.4±2.4 [Drews et al. (1990)]

U 19±2 [Tiecke (1989)]

U 22 [Fabjan et al. (1977); Fabjan (1985a)]

(1): measured with compensating or almost compensating calorimeters (Sect. 9.9.1).

(2): the ratio of thicknesses between Fe and Pb absorbers makes the calorimeters almost compensating

(Sect. 9.9.2).

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Principles of Particle Energy Determination 703

The function φ(e/π) vanishes for e/π ∼ 1, i.e., for compensating (or almost

compensating) calorimeters when the equalization of the calorimeter response to

electromagnetic and hadronic cascades is achieved (Sects. 9.8, 9.9). Furthermore, as

discussed so far, φ(e/π) mainly depends on the e/π ratio, but weakly (if at all) on

both the sampling ﬂuctuations and intrinsic resolution of the calorimeter.

The additional term φ(e/π) was estimated using Monte-Carlo simulations by

Wigmans (1987) (see references given in Sect. 9.8, and also, e.g., Section 27.9.2

on Hadronic calorimeters in [PDB (2002)]) as a function of the e/π ratio and of

the incoming hadron energy. These simulations show that φ(e/π) depends slightly

on the incoming hadron energy. As an example, the value φ(e/π), when e/π ratio

is between ≈ (1.11–1.26) at the reference energy of 10 GeV, increases by no more

than ≈ 0.003 in the energy range (10–70) GeV ([SICAPO Collab. (1996)] and also

e.g., [Wigmans (1987)]). From [SICAPO Collab. (1996)] for e/π ratio ≈ 1.11 at

10 GeV, we have φ( e/π) ≈ 0.0094, 0.0101, 0.0104, 0.0109, 0.0115, 0.0122 and 0.0128

at 10, 15, 20, 30, 40, 50 and 70 GeV respectively.

Experimental data on φ(e/π) obtained with calorimeters providing a ratio e/π ∼

1.11 at 10 GeV ([Acosta et al. (1991)], see also [SICAPO Collab. (1995a,b, 1996)] and

references therein) are found to be in agreement with the simulations of [Wigmans

(1987)].

Fig. 9.31 φ(e/π) in [%] is shown as a function of the e/π ratio at the reference energy of 10 GeV

(from [Leroy and Rancoita (2000)]).

In Fig. 9.31, φ(e/π) is shown in percentage (namely multiplied by a factor 100)

as a function of the e/π ratio at the reference energy of 10 GeV. This curve was

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704 Principles of Radiation Interaction in Matter and Detection

obtained by interpolating the simulation results of [Wigmans (1987)]. The function

φ(e/π) is given, within a ≈ 10% approximation, by:

φ(e/π) ≈ −0.207 ln(e/π) − 4.5 × 10

−3

for e/π < 0.955, (9.79)

φ(e/π) ≈ 0.281 ln(e/π) − 2.1 × 10

−2

for e/π > 1.115. (9.80)

For 0.95 < e/π < 1.10, the function φ(e/π) is lower than 0.007 (see Fig. 9.31).

9.10.2 Determination of Eﬀective Intrinsic Resolutions

The energy resolution, σ(E)/E, in non-compensating calorimeters does not scale as

√

E, owing to the non-vanishing term φ(e/π) (Sect. 9.10.1). The overall eﬀect is

to make the parameter C of Eq. (9.77) dependent on the incoming energy. We can

rewrite Eq. (9.77) ([SICAPO Collab. (1995a,b, 1996)] and references therein) as:

σ(E)

C(E)

√

eﬀ

+ σ

samp

√

, (9.81)

where σ

eﬀ

C(E)

− σ

samp

and σ

samp

is given by Eq. (9.78).

The eﬀective intrinsic resolution σ

eﬀ

includes both the eﬀect of the intrinsic

resolution and the eﬀect of non-compensation on the calorimeter energy resolution,

the latter being accounted for by the constant term φ(e/π). Using Eqs. (9.78, 9.77),

the eﬀective intrinsic resolution can be expressed as:

eﬀ

C(E)

− σ

samp

(9.82)

intr

+ φ

(e/π) E + 2φ(e/π) C

√

E. (9.83)

For a compensating calorimeter [i.e., e/π ∼ 1 and φ(e/π) ∼ 0], we get

eﬀ

≡ σ

intr

Once the calorimeter energy resolution

σ (E)

was measured, it is possible to eval-

uate C(E) =

σ (E)

√

E and the value of σ

eﬀ

from Eqs. (9.83, 9.77).

The intrinsic resolution [Eq. (9.83)] can be determined from

intr

C(E) − φ(e/π)

√

− σ

samp

, (9.84)

where φ(e/π) can be estimated [see Sect. 9.10.1 and references therein, and for

instance Eqs. (9.79, 9.80)], once the e/π ratio of the calorimeter was measured.

In Figs. 9.32, and 9.33, for instance, both σ

intr

[σ

samp

is obtained from Eq. (9.78)]

and σ

eﬀ

[from Eq. (9.83)] are shown for silicon sampling calorimeters using U

(Fig. 9.32) and Fe (Fig. 9.33) passive samplers. For the uranium calorimeter

(Fig. 9.32), σ

eﬀ

was determined as a function of the G10 plate thickness, which are

interleaved in order to achieve the local hardening eﬀect ([SICAPO Collab. (1995a)]

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Principles of Particle Energy Determination 705

Fig. 9.32 σ

eﬀ

(•) in [%] as a function of the e/π ratio in a Si/U sampling calorimeter (reprinted

from Nucl. Instr. and Meth. in Phys. Res. A 361, Furetta, C., Leroy, C., Pensotti, S., Penzo, A.

and Rancoita, P.G., Experimental determination of the intrinsic ﬂuctuations from binding energy

losses in Si/U hadron calorimeters, 149–156, Copyright (1995), with permission from Elsevier; see

also [Leroy and Rancoita (2000)]). The intrinsic energy resolution (◦) are also shown. The e/π ratio

depends on the amount of G10 plates used for the local hardening eﬀect (see also Sect. 9.9.2).

and Sect. 9.9.2). When the e/mip ratio is tuned to achieve the compensation condi-

tion by the local hardening eﬀect, we can observe the eﬀective intrinsic resolution ap-

proaching the intrinsic resolution. For the iron calorimeter (Fig. 9.33), the eﬀective

intrinsic resolution is given as a function of the incoming hadron energy [SICAPO

Collab. (1996)]. σ

eﬀ

increases [Eq. (9.83)] because the constant term φ(e/π) is non

vanishing for non-compensating calorimeters.

From the experimental data, we can observe that the intrinsic resolution is

independent of the incoming energy, as expected. Measured values of intrinsic re-

solutions are given in Table 9.9 (data are from [Fabjan (1985a); HELIOS Collab.

(1987); Tiecke (1989); Drews et al. (1990); SICAPO Collab. (1995a,b, 1996)]).

9.10.3 Eﬀect of Visible-Energy Losses on Calorimeter Energy

Resolution

In Sects. 9.10.1, 9.10.2, the energy resolution of the calorimeter was investigated for

negligible visible-energy losses, i.e., the calorimeter physical sizes are large enough

to almost fully contain the hadronic cascades. However, the energy resolution (as

discussed in Sect. 9.4.2 for electromagnetic showers) is aﬀected by the ﬁnite dimen-

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706 Principles of Radiation Interaction in Matter and Detection

Fig. 9.33 σ

eﬀ

(◦) in [%] in a Si/Fe sampling calorimeter (reprinted from Nucl. Instr. and Meth.

in Phys. Res. A 368, Furetta, C., Gambirasio, A., Lamarche, F., Leroy, C., Pensotti, S., Penzo,

A., Rattaggi, M. and Rancoita, P.G., Determination of intrinsic ﬂuctuations and energy response

of Si/Fe and Si/Fe + Pb calorimeters up to 70 GeV of incoming hadron energy, 378–384, Copyright

(1996), with permission from Elsevier; see also [Leroy and Rancoita (2000)]) as a function of the

incoming hadron energy. The intrinsic energy resolution (•) are also shown.

sions of calorimeters, which can cause leakage of the deposited energy of hadronic

showers. Furthermore in Sect. 9.4.2, we have seen that the limited shower contain-

ment degrades the energy resolution almost linearly with increasing energy-loss.

In a sampling hadron calorimeter, the measured energy resolution

σ( E)

exp

given by:

σ(E)

exp

≡

σ(²

vis

)

vis

(π)

, (9.85)

where σ(²

vis

) is the standard deviation value of the Gaussian-like distribution of

the hadronic visible-energy, ²

vis

(π), measured in the calorimeter when no correction

for energy losses are included. These losses can be longitudinal, lateral, and dead-

area visible-energy losses (as discussed in Sect. 9.4.2 for electromagnetic calorime-

ters). The latter ones have particularly to be considered for calorimeters with high

granularity, as is the case for silicon readout.

The eﬀects of energy losses on the energy resolution were experimentally mea-

sured. In particular, for hadronic cascades, the linear dependence of the hadronic

resolution on lateral and longitudinal energy-losses was observed up to energy losses

of about 15% [Amaldi (1981)] (see Fig. 9.34). To a ﬁrst approximation (for energy

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Principles of Particle Energy Determination 707

Fig. 9.34 Hadronic energy resolution, σ(E)/E in [%], measured as a function of the fraction in

[%] of lateral and longitudinal deposited-energy losses (curves from [Amaldi (1981)], see also [Leroy

and Rancoita (2000)]) for the marble calorimeter of CHARM Collaboration [Diddens et al. (1980)].

losses up to 15%), the combined eﬀect of the overall energy lost on the measured

energy resolution can be expressed, following the discussion in [Amaldi (1981);

SICAPO Collab. (1995b)] (and references therein), according to

σ(E)

exp

≈

σ(E)

1 +

(λ

P )

√

1 +

(λ

P )

, (9.86)

where P is the fraction in % of lateral (E

), longitudinal (E

) and dead area (E

)

energy losses to the total visible-energy; λ

is a coeﬃcient depending on the kind of

energy lost (namely, lateral, longitudinal or dead area energy-loss); and

σ( E)

√

is the expected calorimeter energy resolution in absence of visible-energy losses, i.e.,

the one referred to in Eq. (9.77).

In Fig. 9.34 (see discussion in [Amaldi (1981); Leroy and Rancoita (2000)]), we

can observe that ﬂuctuations produced by lateral energy-losses are less eﬀective,

by a factor ≈ (2–3), than longitudinal energy-losses, as in the case of electromag-

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708 Principles of Radiation Interaction in Matter and Detection

netic showers (Sect. 9.4.2 and references therein), i.e., λ(E

) < λ(E

). Furthermore,

it was also shown [SICAPO Collab. (1994a)] that the longitudinal energy-losses

are as eﬀective as the dead area energy-losses, i.e., λ(E

) ≈ λ(E

), since both

of them are related to event-to-event ﬂuctuations. The experimental data allowed

Amaldi (1981) to determine the curves shown in Fig. 9.34, from which one can

derive that, for hadronic cascades, the values of λ

and λ(E

) ≈ λ(E

) are 0.027

and 0.077, respectively.

9.11 Calorimetry at Very High Energy

9.11.1 General Considerations

The concept of cascade and its propagation in matter were studied in previous sec-

tions at the energy range obtainable at particle accelerators. This section turns to

the study of the cascade behavior in an energy regime beyond accelerator energies,

such as the energies characterizing cosmic rays in a band from about 10

up to about

eV, with a ﬂux of particles decreasing rapidly with the increase of the energy. A

considerable number of excellent articles and books have been published over the

years covering the topic of cosmic ray physics and related subjects (see for in-

stance [Greisen (1960); Berezinskii, Bulanov, Dogiel, Ginzburg and Ptuskin (1990);

Knapp et al. (2003); Allard, Parizot and Olinto (2007); Matthiae (2008)]). The aim

of this section is rather to discuss the physics of cascading processes relevant to

cosmic rays

††

From our point of view, cosmic rays are subatomic particles and nuclei (helium,

carbon, nitrogen, oxygen up to iron), which hit the Earth’s atmosphere. Primary

cosmic rays, which are particles (such as electrons, protons, helium nuclei and nuclei

synthesized in stars) accelerated at astrophysical sources, are distinguished from

secondary cosmic rays produced by the interaction of primaries with the interstellar

matter (the same cosmic rays can be considered as a part of the interstellar medium,

see [McCray and Snow (1981); Berezinskii, Bulanov, Dogiel, Ginzburg and Ptuskin

(1990)] and references therein). This distinction is not strict. While most of the

observed electrons and protons are believed to be of primary origin, positrons and

antiprotons are generally considered to be secondaries. However, there exists the

possibility that a fraction of the observed positrons and antiprotons is of primary

origin. For instance, the measurement of the energy spectrum of cosmic antiprotons

provides important information on cosmic ray propagation models and might be

sensitive to possible novel processes such as the annihilation of supersymmetric

weakly interacting massive neutral particles, called neutralinos, or the evaporation

of primordial black holes (e.g., see [AMS Collab. (1999)]).

Antiprotons, which are produced by collisions of cosmic rays with the interstellar

††

For a general introduction to cosmic rays and trapped particle in the Earth magnetosphere, the

reader may see Sects. 4.1.2.4 and 4.1.2.5.

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Principles of Particle Energy Determination 709

medium, suﬀer a kinematical suppression and will show a characteristic energy spec-

trum at kinetic energies below 1 GeV, which falls oﬀ towards lower energies. There-

fore, the antiprotons primarily produced by novel processes are predicted to emerge

at these low energies (see, for example, [Jungman and Kamionkowski (1994)] for

antiproton spectra from neutralino masses of 30 and 60 GeV/c

). In Fig. 9.35, Local

Interstellar energy Spectra (LIS) of antiprotons from diﬀerent production models

from [Webber and Potgieter (1989); Simon and Heinbach (1996); Gaisser and Schae-

fer (1997)] are shown. These models diﬀer by the choice of main parameters and

their estimated values (see, for example, [Boella et al. (1998)] for a discussion on

these production models).

However for observations close to the Earth, it has to be considered that the

Sun emits a plasma wind with an embedded magnetic ﬁeld (Sect. 4.1.2.1), which

may prevent the propagation of low energy cosmic ray particles inside the helio-

sphere. Furthermore in the heliosphere, interactions with particles coming from the

Sun generate an adiabatic process of energy loss for the incoming cosmic rays. Thus,

the LIS spectra are largely modiﬁed, particularly for energies up to a few GeV’s. The

eﬀect of solar modulation depends on solar activity (Sect. 4.1.2.3) and results in a

time dependence of the interstellar energy spectra at the Earth orbit (Fig. 9.36, see

also [Boella et al. (1998)]).

The exact origin of cosmic rays is presently unknown. Charged particles which

dominate (> 99.9%) hadronic cosmic radiation cannot be tracked back to their

origin, since they are deﬂected by the weak galactic magnetic ﬁeld and, then, the

reach Earth uniformly. Therefore, high energy photons serve an important role,

when one tries to ﬁnd the origin of cosmic rays. Since they are uncharged and reach

the Earth undeﬂected by the galactic magnetic ﬁeld, their possible detection can

lead to the identiﬁcation of their source.

At present, data exist (see, for example, [Werber et al. (1991); Simpson (1983)])

on the energy spectra of various particles and nuclei from hydrogen to iron with

kinetic energies of up to hundreds of GeV per nucleon (Fig. 9.37). The fraction of

secondary nuclei in the cosmic rays decreases with the increase of the energy. Direct

measurements of the intensity of protons and helium nuclei give energies

of the

order of 3 × 10

GeV/nucleon. For energies beyond 10

GeV, the spectra consist

almost exclusively of data from the measurements of extended air showers. Between

10 and 10

GeV, the full cosmic ray spectrum is described by a power law as a

function of the particle (kinetic-) energy [Yodh (1987)]:

∝ E

−γ

, (9.87)

where γ

≈ 2.7 (see, for instance, Sect. 4.1.2.4; [Berezinskii, Bulanov, Dogiel,

Ginzburg and Ptuskin (1990); Gaisser (1990)] and references therein). Possible dif-

ferential galactic spectra (i.e., with slightly diﬀerent γ

values

‡‡

) were observed for

protons and helium nuclei [Randall and Van Allen (1986); AMS Collab. (2002)].

The reader can ﬁnd the deﬁnition of kinetic energies per nucleon in Sect. 1.4.1.

‡‡

is the diﬀerential spectral index see page 332.