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

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640 Principles of Radiation Interaction in Matter and Detection

polyethylene foils with a thickness of the order of 1 cm. Studies, carried out by

Monte-Carlo simulations, have conﬁrmed this possible use of low-A materials as

neutron moderators [Russ (1990); Russ et al. (1991)].

9.4 Energy Resolution in Sampling Electromagnetic Calorimetry

9.4.1 Visible Energy Fluctuations

The energy deposited in a sampling calorimeter is sampled and measured in a num-

ber of readout layers, which interspace the passive samplers. The energy deposition

is dominated by collision losses of positrons and electrons generated in the process

of energy degradation during the cascade development.

The energy deposited in active readout detectors ﬂuctuates on an event-to-event

basis and is aﬀected by the number and energy distribution of charged particles,

which do not usually cross the same amount of active samplers. At any depth in

a calorimeter, the ﬂuctuation of the number of electrons and positrons is intrinsi-

cally related to the statistical nature of the cascade process. Therefore, the average

numb er of electrons and positrons traversing active samplers will undergo statisti-

cal ﬂuctuations. Moreover, the collision loss stopping power of soft-energy electrons

(and positrons) is energy dependent. These particles deposit more energy than fast

electrons (or positrons) moving closer to the shower axis and are also more spread

laterally. Thus, the direction of average motion of soft charged particles makes large

angles with respect to the shower axis. Furthermore, because of the statistical nature

of the shower process, the cascade energy and spatial distributions are characterized

by ﬂuctuations, i.e., deviations from their average behavior.

In sampling calorimeters, these ﬂuctuations result in ﬂuctuations of the sam-

pled visible-energy ²

vis

, i.e., in widening the distribution of the measured ²

vis

. They

are usually called energy sampling ﬂuctuations [Amaldi (1981); Fabjan (1985a,

1986)]. As discussed above, this is related to the ﬂuctuations in the number of par-

ticles and in the energy deposited by electrons and positrons traversing (or being

absorbed in) the active layers of the calorimeter.

The overall physical process of energy deposition in a medium is determined by

the mean energy deposition per charged particle traversing the active readout layers

multiplied by the average numb ers of particles N

. To a ﬁrst approximation, the

value of N

can be estimated as:

vis

el−loss

, (9.20)

where E

el−loss

is the collision energy-loss by electrons (or positrons, whose diﬀerence

can be neglected within our approximation) in a single active readout layer.

As discussed at page 200 and in Sect. 2.1.6, the electron collision energy-losses

are ≈ 10% larger than the collision energy-losses of a massive mip. In addition, in a

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Principles of Particle Energy Determination 641

Fig. 9.12 Energy resolution σ(E)/E in (%) versus E, data from the Si/W electromagnetic sam-

pling calorimeter of SICAPO Collaboration (Fig. 2.78) with τ = 2 (reprinted from Nucl. Instr.

and Meth. in Phys. Res. A 235, Barbiellini, G., Cecchet, G., Hemery, J.Y., Lemeilleur, F., Leroy,

C., Levman, G., Rancoita, P.G. and Seidman, A., Energy Resolution and Longitudinal Shower

Development in a Si/W Electromagnetic Calorimeter, 55–60, Copyright (1985), with permission

from Elsevier; see also [Leroy and Rancoita (2000)]).

shower generator medium, namely the passive sampler of the calorimeter, the mean

cosine value of the charged-particle opening-angle hcos θi is given by Eq. (2.253). In

the active readout layer of thickness X

, the average value E

el−loss

is given by

el−loss

≈ 1.1

hcos θi

. (9.21)

In a sampling calorimeter with a dominant passive sampler of thickness L

and

radiation length X

, ²

vis

is given by Eq. (9.9) and using Eq. (9.21) one obtains

= (e/mip)

hcos θi

1.1

. (9.22)

As seen on page 200, the ratio ²

is ≈ 1.1 (dE/dx)

, where ²

is the critical

energy of the passive sampler. The previous equation can be ﬁnally written as:

= (e/mip)

hcos θi

1.1

= (e/mip)

hcos θi

−3

, (9.23)

where the incoming particle energy E is in units of GeV, the critical energy of

the passive sampler, ²

, is units of MeV and τ is the sampling frequency in units

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642 Principles of Radiation Interaction in Matter and Detection

of radiation lengths of the passive material. Assuming that the charged particles

are statistically independent [Amaldi (1981); Fabjan (1985a, 1986)], N

is also the

value of the variance, i.e., the square value of the root mean squared (r.m.s.) error,

of the distribution of the number of charged particles.

So far, the measured mean value of the deposited energy in a sampling calorime-

ter has b een called visible energy (²

vis

), but ﬂuctuations on its value have not

been considered. Owing to the intrinsic statistical character of the cascade phe-

nomenon (as discussed above), the energy distribution of ²

vis

, for incoming particles

of energy E, is a Gaussian distribution with standard deviation σ(²

vis

). One has

(e.g., see [Amaldi (1981); Fabjan (1985a, 1986)]):

σ(²

vis

)

vis

≈

The energy resolution of the sampling calorimeter, σ(E)/E, on the incoming particle

energy E is

σ(E)

≡

σ(²

vis

)

vis

, (9.24)

and using Eq. (9.23), one obtains:

σ(E)

σ(²

vis

)

vis

≈

= K

. (9.25)

As in Eq. (9.23), τ is the sampling frequency in units of radiation length of the

passive material, the incoming particle energy, E, is in units of GeV and K is given

K =

−3

(e/mip) hcos θi

≈ (3.16%)

(e/mip) hcos θi

, (9.26)

where the passive sampler critical energy ²

is in units of MeV.

Although Eq. (9.26) is similar to Equation 8 given by Amaldi (1981), the e/mip

ratio replaces the function of the minimum detectable energy F (ζ) (deﬁned on page

201) in Eq. (9.26). In this equation, the e/mip ratio takes properly into account

the diﬀerence of the energy deposition per g/cm

between active readout layers and

passive samplers.

9.4.1.1 Calorimeter Energy Resolution for Dense Readout Detectors

Equation (9.26) was obtained under the assumption that electrons (and p ositrons)

depositing their energy by collision losses have similar statistical weights. This as-

sumption is acceptable for suﬃciently thick dense active readout detectors, e.g., a

few cm of scintillator, so that the energy deposited by collision losses (for instance,

≈ 2 MeV/cm in scintillator) by fast charged particles moving forwards and close to

the shower axis are not negligible compared with the energy deposition by collision

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Principles of Particle Energy Determination 643

losses of laterally spread soft electrons. These soft particles have their energies not

exceeding a few MeV’s and can be fully absorbed.

However in thin (although dense) detectors, e.g. a few mm of liquid argon or a

few mm of scintillator or a few hundreds of microns of silicon detectors, particles

passing at large angle deposit much larger energy than those passing close to the

shower axis. As a consequence, N

can be overestimated even by a factor 2, corre-

sponding to particles traversing (without being fully absorbed) at an angle of about

to the shower axis. This factor is reduced, when the mean traversing angle in-

creases. To a rough approximation, for a dense but thin readout detector, we can

assume that the N

value given in Eq. (9.23) has to be multiplied by a reducing

factor 1/(1 + hcos θi). As already mentioned in the previous section, the mean cosine

value of the charged particle opening angle hcos θi is given by Eq. (2.253). Under

these approximations, the calorimeter energy resolution is again given by Eq. (9.25),

but the parameter K becomes:

K =

−3

[1 + P

hcos θi]

(e/mip) hcos θi

≈ (3.16%)

[1 + P

hcos θi]

(e/mip) hcos θi

, (9.27)

where the passive sampler critical energy ²

is in units of MeV, and the parame-

ter P

is 0 for thick and dense readout layers [in this case, Eq. (9.27) transforms

into Eq. (9.26)] and 1 for thin and dense readout layers. Within the approxima-

tions used, Eq. (9.27) accounts for eﬀects of diﬀerential collision energy-losses by

charged particles, impinging at diﬀerent angles, traversing (or being absorbed in)

dense readout detectors. These eﬀects are often referred to as path-length ﬂuctu-

ations [Amaldi (1981); Fabjan (1985a, 1986)], and Landau ﬂuctuations [Amaldi

(1981); Fabjan (1985a, 1986); Grupen (1996)]. These eﬀects were introduced in the

sampling calorimetry to take into account the degradation of the energy resolution

because of the diﬀerent path of large-angle particles (path-length ﬂuctuations) and

large energy transfers in ionization process (Landau ﬂuctuations). However, the lat-

ter ﬂuctuations add a minor contribution to the deterioration of the calorimeter

energy resolution [Fabjan (1985a)].

In Eq. (9.27), the e/mip ratio depends on the Z values of the readout and ab-

sorber materials (see Sect. 9.2.2). Furthermore, the parameter K (as it is the case

for the energy and angular distributions of soft electrons) depends on the criti-

cal energy of the passive sampler. The calorimeter resolution σ(E)/E, for a ﬁxed

sampling frequency and incoming particle energy, improves by increasing the visible-

energy sampled as long as sampling ﬂuctuations are the major contribution to the

overall calorimeter resolution and both Landau and path length ﬂuctuations can

be taken into account simply by the multiplicative factor discussed for Eq. (9.27).

Experimental data are in agreement with Eq. (9.25) predicting the energy resolu-

tion dependence on 1/

√

E. In Fig. 9.12, the energy resolution [SICAPO Collab.

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644 Principles of Radiation Interaction in Matter and Detection

Table 9.3 Selection of measured energy resolutions for sampling calorimeters with dense readout detectors from [Leroy and Rancoita (2000)]. Er-

rors are quoted when available in the original reference.

passive readout τ energy resolution K Reference

sampler at 1 GeV

) (%) (%)

Al [see (1)] Scintillator ≈ 1 20.0 20.0 [Amaldi (1981); Fabjan (1985a)]

(marble) (3 cm) [see (1)] [Diddens et al. (1980)]

Fe Scintillator 1.4 23.0 19.4 [Abramowicz et al. (1981)]

Fe liquid Ar 8.5 × 10

−2

7.5 25.7 [Fabjan (1985a); Fabjan et al. (1977)]

Fe Si det. 2.0 32.2±1.1 22.8±0.8 [SICAPO Collab. (1994a)]

W Si det. 2.0 24.6±0.4 17.6±0.3 [SICAPO Collab. (1985a)]

Pb Scintillator 1.8 × 10

−1

7.5 17.7 [Hofmann et al. (1979)]

Pb Scintillator 2.1 × 10

−1

9.0 19.4 [Schneegans et al. (1982)]

Pb Scintillator 1.8 23.5±0.5 17.6±0.4 [Drews et al. (1990)]

Pb liquid Ar 3.1 × 10

−1

[see(2)] >20 GeV 14.4 [Buckardt et al. (1988)]

Pb Si det. 0.9 16.5±0.5 17.4±0.5 [Nakamoto et al. (1985)]

Pb liquid Ar 3.2 × 10

−1

Accordion 9.6 − 10.1 16.9-17.8 [Aubert et al. (1991)]

U Scintillator 0.5 11.0 15.6 [Carosi et al. (1984)]

U Scintillator 1 16.5±0.5 16.5±0.5 [Drews et al. (1990)]

(1): it means that the marble used as passive sampler is equivalent to Al. The scintillator used as readout is 3 cm thick.

(2): the energy resolution is given by:

σ(E)

0.1

0.075

√

+ (0.005)

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Principles of Particle Energy Determination 645

Fig. 9.13 σ(E)/(Eτ

1/2

) (in %) measured for Pb absorbers versus τ at the incoming energy of

4 GeV (data from [SICAPO Collab. (1994a)]). The line is the weighted mean of the data values.

(1985a)] of a sampling calorimeter with tungsten sampler thicknesses of 2 X

and

silicon readout detectors is shown for incoming electron energies between 4 and

49 GeV. The energy resolution is well represented by the ﬁtted curve following

Eq. (9.25) [SICAPO Collab. (1985a)].

At ﬁxed energy, Eq. (9.25) predicts that the energy resolution depends on

√

τ. In

Fig. 9.13, the energy resolution divided by

√

τ, i.e., σ(E)/(Eτ

1/2

) in %, is shown

as a function of τ, for electrons of 4 GeV energy and passive samplers made of Pb

plates [SICAPO Collab. (1994a)]. The values of σ(E)/(Eτ

1/2

) do not depend on τ,

as expected from Eq. (9.25).

A summary of experimental data on electromagnetic sampling calorimeters using

dense readout detectors (namely silicon detectors, scintillators and liquid argon

ionization chambers) is presented in Table 9.3. Reported data include the type of

readout detector, the sampling frequency τ, the measured energy resolution ﬁtted

by Eq. (9.25) and evaluated at 1 GeV of incoming electron energy, and the pa-

rameter K. The experimental energy resolutions are measured by employing low-

(Al), medium- (Fe), and high-Z (W, Pb, U) passive samplers, which are the most

frequently used absorbers. The sampling frequency varies between ≈ 9 × 10

−2

and

2. However, in spite of experimental uncertainties and diﬀerent calorimeter para-

meters, for a given type of passive sampler the values of K do not diﬀer by more

than ≈ 10% from their mean values. These averages of experimental K values, for

calorimeters using the same passive samplers, are shown in Table 9.4 together with

the K values calculated by means of Eq. (9.27). In Table 9.4, the e/mip ratios were

estimated by averaging the experimental data

shown in Table 9.1. The agreement

among estimated and averaged experimental K values (shown in Table 9.4) is sati-

The same value of Pb sampler was assumed for W absorber.

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646 Principles of Radiation Interaction in Matter and Detection

sfactory and indicates that Eqs. (9.25, 9.27) take well into account the ﬂuctuations

of the visible energy for dense readout detectors. So far, it was assumed, in the

comparison of experimental data with predictions, that the intrinsic statistical ﬂuc-

tuations are the limiting constraints on the calorimeter energy resolution. In a real

detector, many instrumental eﬀects add to these statistical ﬂuctuations and worsen

the energy resolution. They depend on the type of device (sampling or homogeneous

calorimeter) and readout medium. The instrumental noise, detection statistics and

intercalibration errors aﬀect the overall energy resolution and, particularly, limit

the detector performance at high energies. In [Engler (1985)] and references therein,

their inﬂuence on calorimeter resolution is treated in a general way. Furthermore in

Sect. 9.4.2, the degradation of the calorimeter resolution induced by visible-energy

losses, due to the ﬁnite dimensions of the device, is evaluated.

9.4.1.2 Calorimeter Energy Resolution for Gas Readout Detectors

Sampling calorimeters were also built with gas detectors as active readout de-

vices. As discussed in the previous section, the wide spread of electron angles is

among the causes of degradation of the calorimeter energy resolution. These path-

length ﬂuctuations depend on the thickness of readout detectors, namely they in-

crease as the thickness decreases (as discussed above). In the same way, Landau

ﬂuctuations [as shown by Amaldi (1981) in Equation 21] are larger for calorimeters

in which the deposited energy is measured. However, in calorimeters using streamer-

tubes as readout detectors, tracks are counted at least as long as the particles are

close to the shower axis. For each ionization track one streamer is formed and,

consequently, Landau ﬂuctuations aﬀect less the energy resolution.

In general, these ﬂuctuations cannot be neglected, when gas detectors are used

as active readout samplers. The energy resolutions of multiwire proportional quan-

tameters were investigated by means of Monte-Carlo simulations [Fisher (1978)]

and compared with experimental data. Fisher (1978) has shown, see Fig. 9.14, that

path length (also termed track length) and Landau ﬂuctuations worsen the sampling

ﬂuctuations in these quantameters [Fisher (1978)] by a factor ≈ 2. The energy re-

solutions obtained using Monte-Carlo simulations were found to agree with those

measured with Pb [Nordberg (1971)] and Fe [Anderson et al. (1976)] gas quan-

tameters [Fisher (1978)]. Measured energy resolutions of calorimeters employing

gas readout detectors are given for Fe and Pb passive samplers in Table 9.5. The

average measured K values for Fe and Pb absorbers are 32% and 26%, respecti-

vely. Although the gas readout detectors are operated in diﬀerent way, the mea-

sured K values do not diﬀer by more than ≈ 15% from their respective average

values. These average values, in turn, are in agreement with K-parameters derived

from Eq. (9.27) (see Table 9.4) once multiplied by a factor ≈

√

2. This factor can

be taken into account empirically in Eqs. (9.26, 9.27). For a gas calorimeter (within

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Principles of Particle Energy Determination 647

Table 9.4 Comparison of K values from measured energy resolutions and calculated by means of Eq. (9.27)

from [Leroy and Rancoita (2000)].

passive readout (e/mip) hcos θi K K

sampler [see(1)] from Eq. (2.253) from Eq. (9.27) experim. [see(2)]

(%) (%)

Al [see(3)] thick 1.00 0.986 20.2 20.0

(marble)

Fe thin 0.83 0.949 22.8 22.6

W thin 0.66 0.671 17.4 17.6

Pb thin 0.66 0.614 17.2 17.3

U thin 0.63 0.543 17.5 16.1

(1): average values from data of Table 9.1; it was assumed that W and Pb have equal (e/mip) ratios.

(2): average values from data of Table 9.3.

(3): it means that the marble used as passive sampler is equivalent to Al.

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648 Principles of Radiation Interaction in Matter and Detection

Fig. 9.14 Contributions of sampling, track length (also termed path length) and Landau ﬂuc-

tuations to the energy resolution of a Pb gas quantameter as computed by Fisher (1978) (see

also [Amaldi (1981); Leroy and Rancoita (2000)]), using Monte-Carlo simulations (adapted

from Nucl. Instr. and Meth. 156, Fisher, H.G., Multiwire Proportional Quantameters, 81–85,

resolution for a liquid argon calorimeter (for details see [Fisher (1978)]).

an approximation of ≈ 15%), one has

K ≈ (3.16%)

(1 + P

hcos θi)(1 + P

)

(e/mip) hcos θi

, (9.28)

where ²

is the critical energy of the passive sampler in units of MeV, hcos θi is given

by Eq. (2.253); the parameter P

is 0 for thick and dense readout layers [in this case,

Eq. (9.28) transforms into Eq. (9.26)] and 1 for thin and dense readout layers [in

this case Eq. (9.28) transforms into Eq. (9.27)]. In both cases, the parameter P

0. For sampling calorimeters with gas readout detectors, the parameters P

and P

are both equal to 1. Then, the energy resolution of an electromagnetic calorimeter

can be ﬁnally written as

σ(E)

≈ (3.16%)

(1 + P

hcos θi)(1 + P

)

(e/mip) hcos θi

. (9.29)

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Principles of Particle Energy Determination 649

Table 9.5 Selection of measured energy resolutions for sampling calorimeters with gas readout detectors from [Leroy and Rancoita

(2000)]. Errors are quoted when available in the original reference.

passive readout τ energy resolution K Reference

sampler at 1 GeV

) (%) (%)

Fe MWPC 1.6 ≈ 47.2 ≈ 37.4 [Fisher (1978)],[Anderson et al. (1976)]

Fe streamer mode 1.2 28.8 26.4 [Baumgart (1987); Baumgart et al. (1988a)]

Fe streamer mode 2.8 55.0 32.6 [Catanesi et al. (1986)]

Pb MWPC 5.6 × 10

−1

≈ 20.6 ≈ 27.6 [Fisher (1978); Nordberg (1971)]

Pb saturated prop.-mode 0.5 16.2 22.9 [Atac et al. (1983)]

Pb saturated prop.-mode 3.6 × 10

−1

16.5 27.6 [Videau et al. (1984)]