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

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650 Principles of Radiation Interaction in Matter and Detection

Fig. 9.15 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)].

9.4.2 Eﬀect of Limited Containment on Energy Resolution

So far in the previous sections, the energy resolution of the calorimeter was con-

sidered for negligible lateral and longitudinal visible-energy losses, namely for the

case where the calorimeter longitudinal depth and lateral width are large enough

to contain the shower cascade completely. However, the energy resolution expressed

by Eq. (9.24) is degraded by the eﬀects of the ﬁnite dimensions of the calorimeter.

The energy resolution is increased by leakage ﬂuctuations caused by longitudinal,

, and lateral, E

, energy-losses. The size of these losses depends on the dimen-

sions of the calorimeter. The longitudinal energy-losses display a weak dependence

on the incoming energy and an even weaker dependence on the material. The lon-

gitudinal shower development can be described, as seen for instance in Eq. (2.236),

by a function, whose parameters depend on the incoming energy E as ln E. Thus,

ﬁnite dimensions of a calorimeter result in losses of the average dep osited energy

depending on ln E [Engler (1985); Grupen (1996)]. Moreover, because of the statis-

tical nature of the cascade process (including the location of the ﬁrst interaction of

the primary particle along the shower axis), the ﬁnite longitudinal dimension will

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Principles of Particle Energy Determination 651

aﬀect the energy resolution on an event-by-event basis.

The lateral energy-losses do not critically depend on the incoming energy, on the

type of incident particles (electrons or photons), or on the absorber material at a

depth where the cascade is extended over a small transverse distance in units of the

Moli`ere radius. For large transverse distances in units of Moli`ere radius, a slight de-

pendence on the above parameters is related to soft interactions contributing to the

energy deposition process. Fluctuations coming from the transverse leakage come

out to be less eﬀective compared to those caused by the longitudinal leakage [Amaldi

(1981)]. Thus, the longitudinal energy-loss contribution to the degradation of the

energy resolution is expected to be larger than the contribution depending on lateral

energy-losses (see the discussion on the lateral shower development on page 207).

The longitudinal energy leakage can be calculated by integrating one of the equa-

tions given in Sect. 2.4.2.2. Similarly, the lateral energy leakage can be calculated

by integrating one of the equations given in Sect. 2.4.2.3. However, these equa-

tions give the average behavior of the shower development only approximately. In

a sampling calorimeter, because the visible energy is proportional to the incoming

energy [see Eq. (9.1)], the fraction of visible-energy lost laterally or longitudinally

is given by the fraction of the lateral or longitudinal leakage of the incoming energy,

respectively.

As a rule of thumb [Engler (1985)], the deterioration of the energy resolution is

expected to increase proportionally to the longitudinal leakage. These eﬀects were

experimentally investigated. The marble ﬁne-grained calorimeter of the CHARM

Collaboration [Diddens et al. (1980)] was used to determine the eﬀect of lateral

and longitudinal losses. The experimental data allowed Amaldi (1981) to deter-

mine that the longitudinal losses are aﬀecting more the energy resolution than

the lateral losses, as expected (Fig. 9.15). For energy losses lower than 15%, the

calorimeter energy-resolution is almost linearly related to the average fraction in

% of the lateral energy lost (lt = 100 × [E

/E]) and of the longitudinal energy lost

(Lg = 100 × [E

/E]), according to

σ(E)

≈

σ(E)

(1 + 6.6 × 10

−2

Lg), (9.30)

and

σ(E)

≈

σ(E)

(1 + 1.4 × 10

−2

lt), (9.31)

where (from Fig. 9.15)

σ(E)

≈ 5.58% is the measured calorimeter resolution in

the absence of energy losses.

For highly segmented readout planes such as mosaic planes made of silicon de-

tectors, dead zones between the detectors in the mosaics result in losses of visible

energy, indicated by E

. These dead zones can aﬀect the energy resolution [SICAPO

Collab. (1994a)]. In the following, the fraction in % of visible-energy lost by the

mosaic dead zones is indicated by De (i.e., De = 100 × [E

/E]). These losses are

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652 Principles of Radiation Interaction in Matter and Detection

subjected to event-by-event ﬂuctuations which are more similar to the longitudinal

energy-losses than to the lateral energy-losses. The dead area energy-losses start

much closer to the impinging point than the lateral energy-losses which appear af-

ter some transverse distance from the impinging point and are controlled by the

Moli`ere radius.

Their eﬀect on the calorimeter energy resolution was investigated as a func-

tion of the incoming electron energy with Fe and Pb passive samplers and em-

ploying the same readout detectors (namely silicon mosaic planes) [SICAPO Col-

lab. (1994a)]. These passive materials have similar Moli`ere radii (≈ 1.8 and 1.6 cm

respectively), and consequently similar values of De. The data were collected for

sampling frequencies τ = 1.99 and 3.98 for Fe absorbers, and τ = 1.96, 2.36, 2.50 and

3.39 for Pb absorbers [SICAPO Collab. (1994a); Furetta et al. (1995)]. In Figs. 9.16

and 9.17, the measured energy-resolutions for the two calorimeters with Fe (and τ

= 1.99) and Pb (and τ = 3.39) absorbers are shown as a function of 1/

√

E. The

experimental data show (in both cases) a linear dependence on 1/

√

E and are in

agreement with

σ(E)

= K

. (9.32)

Similarly to the cases of longitudinal and lateral energy-losses, these experimentally

measured energy-resolutions [SICAPO Collab. (1994a)] are consistent with

σ(E)

[1 + (6.8 ± 1.7) × 10

−2

De], (9.33)

Fig. 9.16 Energy resolution σ(E)/E in % measured as a function of 1/

√

E (E is the incoming

electron energy) for ﬁxed visible-energy losses due to the mosaic structure of the active readout

silicon plane and Fe passive sampler (reprinted from Nucl. Instr. and Meth. in Phys. Res. A 345,

Bosetti, M. et al., Systematic investigation of the electromagnetic energy resolution on sampling

frequency using silicon calorimeters, 244–249, Copyright (1994), with permission from Elsevier,

e.g., for the list of the authors see [SICAPO Collab. (1994a)]; see also [Leroy and Rancoita (2000)]).

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Principles of Particle Energy Determination 653

where

σ( E)

is the calorimeter resolution in the absence of energy-losses [e.g., see

Eq. (9.25)], and, thus, can be expressed in a general form by

σ(E)

= K[1 + (6.8 ± 1.7) × 10

−2

De]

. (9.34)

As a consequence, from Eqs. (9.32–9.34) we have that

= K[1 + (6.8 ± 1.7) × 10

−2

De],

where K is the parameter of the energy resolution, in the absence of dead-area

energy-losses [see Eqs. (9.25, 9.27)]. As expected from Eq. (9.34), the ratio R

the measured K

value for the Fe and Pb calorimeters is R

= 28.3/22.2 ≈ 1.27

and is consistent with the value calculated from Table 9.4, i.e., ≈ 1.31, without

energy losses.

The visible-energy losses, due to the leakage of the incoming particle energy,

degrade the calorimeter resolution. In fact, the K

values are larger than those

discussed in previous sections on energy resolutions for no dead-area energy-losses.

The multiplicative factor De, appearing in Eqs. (9.33, 9.34), is similar (within the

experimental errors) to the one for longitudinal energy-loss (Lg) given in Eq. (9.30)

since, in both cases, energy losses are subjected to event-to-event ﬂuctuations.

Fig. 9.17 Energy resolution σ(E)/E in % measured as a function of 1/

√

E (E is the incoming

electron energy) for ﬁxed visible-energy losses due to the mosaic structure of the active readout

silicon plane and Pb passive sampler (reprinted from Nucl. Instr. and Meth. in Phys. Res.

A 345, Bosetti, M. et al., Systematic investigation of the electromagnetic energy resolution on

sampling frequency using silicon calorimeters, 244–249, Copyright (1994), with permission from

Elsevier, e.g., for the list of the authors see [SICAPO Collab. (1994a)]; see also [Leroy and Rancoita

(2000)]).

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654 Principles of Radiation Interaction in Matter and Detection

9.5 Homogeneous Calorimeters

9.5.1 General Considerations

A homogeneous calorimeter is a block or a matrix of blocks of sensitive absorbing

material where the energy of an incoming particle is degraded and measured via

the collection of visible photons. The absorbing material or active medium of the

calorimeter is either scintillator, where ﬂuorescence is produced by the passage of

an ionizing particle or material with a high refractive index transparent to the

Cerenkov photons produced by charged particles, which move with a velocity

v = c/n

larger than that of light through the medium of refractive index n > 1 (for a treat-

ment of scintillator detectors see the chapter on Scintillating Media and Scintillator

Detectors, while for a treatment of the

Cerenkov eﬀect see Sect. 2.2.2).

A variety of scintillating and

Cerenkov materials (see the chapter on Scintillating

Media and Scintillator Detectors) can be used as the active medium of homogeneous

calorimeters. The choice of a particular active medium is made with the goal to

achieve the best possible energy and position resolution over an energy range as

large as possible (from MeV up to TeV energies) taking into account practical and

physical requirements.

The energy resolution is determined by the ﬂuctuations in the number of pho-

tons or ions pairs produced by the cascade. The light yield, i.e, the number of

photons produced per MeV of energy deposition, should be large enough to mini-

mize the contribution of the statistical ﬂuctuations to the energy resolution. Typi-

cally, the amount of collected light should exceed 100 γ/MeV and in fact can exceed

4×10

γ/MeV for NaI(Tl) (Table 5.2).

The size of the sensitive material length and lateral dimensions are important

constraints. The total length should be suﬃciently large to ensure that energy leak-

age of the electromagnetic cascades at the back of the calorimeter is small enough

not to degrade the energy resolution, even for very energetic electrons and pho-

tons. High density sensitive materials have short radiation length and small Moli`ere

radius, so favoring good compactness of the calorimeter.

The length of the sensitive material has also to be compatible with light trans-

mission properties and light attenuation length. The light output along the length

of the crystal should not vary (uniform light collection).

The lateral size of the sensitive material is chosen in such a way that incoming

particles ﬁred centrally into a block at normal incidence deposit a large fraction

between 70 and 80% of their energy in this particular block, while energy deposited

in the neighboring blocks is large enough to measure the coordinates of the center of

gravity and, consequently, of the impact point. This can be achieved by selecting a

material with a small Moli`ere radius (typically ≤ a few cm’s, see Table 5.2) resulting

in narrow cascades and a matrix of sensitive material blocks with the lateral size

of a cell of the order of the Moli`ere radius. However, transverse dimensions of a

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Principles of Particle Energy Determination 655

sensitive block in a matrix result most generally from practical considerations, as

a compromise between the spatial resolution and the number of readout channels

whose number increases fast with the segmentation.

The decay time of scintillation should be as short as possible. This becomes

an important constraint in experiments at the LHC, where the light has to be

collected in less than ∼ 10 ns since the expected LHC interbunch crossing time is

25 ns, requiring the use of new types of crystals such as PbWO

The readout of the homogeneous calorimeters is realized through photomulti-

plier tubes or photodiodes of various types collecting the produced light. Factors

such as the noise, the quantum eﬃciency and the gain of the photosensitive device

contribute to the energy resolution of the calorimeter.

In Sect. 9.9, the mechanisms for achieving the compensation in (hadronic)

calorimeters will be described. These mechanisms do not apply to homogeneous

calorimeters. These mechanisms are based on the fact that only a small fraction of

the cascade energy is deposited in the active medium of the calorimeter. Depen-

ding on the nature of the active and passive medium, there are ways to equalize the

calorimeter response to electromagnetic and non-electromagnetic components of the

cascade. In the case of homogeneous calorimeters, there is no way to compensate

for binding energy-losses occurring in the non-electromagnetic component of the

cascade. This is supported by the results of measurements performed with homoge-

neous hadronic calorimeters of e/h ratio not equal to 1. Then, the non-linearity of

the calorimeter response and the deviation from a Gaussian response are observed

(Sect. 9.8.1).

The properties of scintillators, mechanisms of light emission and readout tech-

niques are reviewed in [Suﬀert (1988); Cushman (1992); Bourgeois (1994)] and are

treated in Sect. 5.1. These elements are necessary in order to understand the charac-

teristics of both homogeneous calorimeters and hydrogeneous readouts in hadronic

sampling calorimetry (Sect. 9.9.1).

The inorganic scintillators were used in small-scale homogeneous calorimeters,

since their relative high cost makes their use prohibitive in large-scale calorime-

ters. However, BGO was used in the large electromagnetic calorimeter of the L3

experiment at LEP machine (at CERN). BaF

was also used in smaller arrays [Wis-

shak et al. (1989)]. The decay constants of BGO, NaI(Tl) and CsI(Tl) are in the

hundreds of ns and more. It precludes the use of these crystals for high-rate appli-

cations such as those to be encountered in experiments at the LHC machine.

Organic scintillators are aromatic compounds produced from a benzenic cy-

cle. In organic scintillators, the mechanism of light emission is not a lattice eﬀect,

but rather a molecular eﬀect. It proceeds through excitation of molecular levels in

a primary ﬂuorescent material, which emits bands of ultraviolet (UV) light during

de-excitation. This UV light is readily absorbed in most organic materials transpar-

ent in the visible wavelength region, with an absorption length of a few mm. The

extraction of a light signal becomes possible only by introducing a second ﬂuo-

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656 Principles of Radiation Interaction in Matter and Detection

rescent material in which the UV light is converted into visible light (wavelength

shifter). This second substance is chosen in such a way that its absorption spectrum

is matched to the emission spectrum of the primary ﬂour, and its emission should

be adapted to the spectral dependence of the quantum eﬃciency of the photocath-

ode. These two active components of a scintillator are either dissolved in suitable

organic liquids or mixed with the monomer of a material capable of polymeriza-

tion. The polymer can then be cast into any shape for practical applications. Most

frequently used are rectangular plates with thicknesses from 0.2 up to 5.0 mm and

area from a few mm

up to several m

and are of current use as active medium

of sampling calorimeters. Organic scintillators are very rarely utilized in homoge-

neous calorimeters with the exception of liquid scintillators. In a liquid scintillator,

an organic crystal (solute) is dissolved in a solvent (with typical concentrations of

≈ 3 g/l). The scintillation eﬃciency depends on the solvent used. Xylene and toluene

give good relative pulse height with a PPO solute [Schram (1963)]. The scintillator

BIBUQ dissolved in toluene has the best time resolution (≈ 85 ps) of any scintillator

currently available [Bengston and Moszynski (1982)].

Cerenkov detectors, used as calorimeters, are composed of liquids (with refrac-

tive index n ≈ between 1.2 and 1.5) or solids (n ≈ between 1.4 and 1.8).

Cerenkov

material used in practice are represented by lead glass of various types such as SF-

5, SF-6, PbF

, and their properties are shown in Table 9.6. The

Cerenkov light is

emitted in a forward cone with aperture cos θ = c/(vn) = 1/(βn). The number of

photons, N

, emitted per unit length of track of the incident particle of charge

numb er z through the medium in the dλ is given by Eq. (5.16) [see also Eq. (2.138)

on page 128]. In the wavelength interval, dλ, the radiation intensity depends only on

the charge number z of the incident particle and on the emission angle, and there-

fore on the particle velocity. The presence of λ

in the denominator of Eq. (5.16)

[see also Eq. (2.140) on page 128] indicates an emission concentrated at shorter

wavelength. The emission is taking place in the wavelength domains of visible and

UV light. This is one reason why low light yields are often obtained, since only part

Table 9.6 Properties of

Cerenkov materials of current use (see for in-

stance [Leroy and Rancoita (2000)]).

Properties SF-5 SF-6 PbF

Water

Density (g/cm

) 4.08 5.20 7.66 1.00

Radiation

Length (cm) 2.54 1.69 0.95 36.40

cut

(nm) 350 350 260 ∼300

Refractive index 1.67 1.81 1.82 1.33

Light yield

(pe/GeV) 600 900 2000

Light yield [vs NaI(Tl)] 1.5 × 10

−4

2.3 × 10

−4

−3

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Principles of Particle Energy Determination 657

of the spectrum is sampled. In the wavelength domain of the visible and UV light

which corresponds to the domain of spectral sensitivity of the standard photocath-

odes (300 to 600 nm for bialkali photocathodes), the light emitted by a

Cerenkov

radiator is smaller than that emitted by scintillators of the same dimensions. The

Cerenkov light is produced very fast and essentially with no time spread (with the

exception of light propagation eﬀects in crystals of large dimensions). The timing

properties are not governed by the speciﬁc

Cerenkov material itself but rather by

the dimensions, geometry of the block or container (for liquid

Cerenkov), and by

the rise time of the photosensitive device.

The threshold eﬀect linked to

Cerenkov light production (v > c/n) is also an-

other reason for low light yields since the cutoﬀ energy (E

cut

) is high [Sowerby

(1971)]. As an example, in the case of SF-5 (n = 1.7), E

cut

≈ 0.15 and ≈ 0.3 MeV

for electrons and photons, respectively. Table 9.6 shows the properties of

Cerenkov

materials of standard use.

9.5.2 Energy Measurement

Ideally, all the incoming particle energy is deposited in the volume (which is the

active medium) of the homogeneous calorimeters [Nakamoto et al. (1986); Bormann

et al. (1987)], and the number of photons or ions pairs n

produced by the cascade

is given by a formula similar to Eq. (9.25):

, (9.35)

where

E is the mean energy needed to create a photon or an ion pair.

Since this creation process is a random process, the intrinsic energy resolution

is dominated by the statistical ﬂuctuations of the number of detected primary pro-

cesses, and is given by

σ(E)

√

constant

√

. (9.36)

The threshold of detection is usually small enough that the number of primary

processes, that can be detected, is large and gives good intrinsic energy resolu-

tion. However, as already discussed in Sect. 9.2 for the case of sampling calorimeters,

Eq. (9.36) only applies to cascades fully contained in the calorimeter volume. Ide-

ally, the calorimeter must be of suﬃcient thickness to allow the particle to deposit

all its energy inside the detector volume in the subsequent cascade of increasingly

lower energy particles. The total depth of the calorimeter must be large enough to

allow a longitudinal containment of the cascade in order to completely absorb the

incoming energy. The necessary longitudinal depth of a calorimeter varies with the

incoming energy E, as ln E.

The incoming energy being distributed among a large number of secondary par-

ticles, energy can be deposited at large angles with respect to the longitudinal axis

of the calorimeter. Thus, the requirement of transverse containment of the cascade

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658 Principles of Radiation Interaction in Matter and Detection

imposes a minimal radial extension of the calorimeter which must reach, at least,

several times the average diameter of a cascade.

In practice, the dimensions of the homogeneous calorimeters are limited by the

cost of crystals. Thus, some fraction of the cascades escapes through the sides and

the rear of any realistic homogeneous calorimeter. This leakage of energy causes the

energy resolution to dep end on the dimensions of the calorimeter. The measured

value of the constant in Eq. (9.36) accounts for the leakage ﬂuctuations caused by

longitudinal and lateral energy-losses. The increase of the value of this constant

with the energy leakage reﬂects the deterioration of the energy resolution.

The readout of the homogeneous calorimeters is mostly achieved through pho-

tomultiplier tubes or photodiodes of various types collecting the produced light,

and instrumental eﬀects, such as the gain of the photosensitive device, contribute

to the energy resolution. The number of photoelectrons, n

, produced by an elec-

tromagnetic cascade is relatively small and can present large statistical ﬂuctua-

tions. The contribution to the overall energy resolution from these statistical ﬂuc-

tuations is [Akrawy et al. (1990); Cushman (1992)]:

σ(E)

(1 + 1/g)

¯n

, (9.37)

where g is the gain of the photosensitive device, ¯n

is the mean number of photo-

electrons, and E is the energy of the shower in GeV units.

For phototubes with a gain of 10

, the second term in Eq. (9.37) is negligi-

ble. However, for photodiodes the gain is small compared with phototubes. With

a photodiode gain of about 1, the gain term in Eq. (9.37) will be large and leads

to a degradation of the energy resolution. As an example, the following results are

obtained for the OPAL end cap electromagnetic calorimeter (EMEC), made of a

large array of lead glass (total of 2264 lead glass blocks) [Akrawy et al. (1990)], for

triode (gain ∼ 10) readout (in a magnetic ﬁeld of 0.4 T):

¯n

= 1.8E × 10

and

σ(E)

0.027

√

The importance of electronic (stochastic) noise in the ampliﬁer readout circuit

grows for low-gain photosensitive devices. The contribution of this noise to the

energy resolution can be parameterized as [Akrawy et al. (1990)]:

√

¯n

× 100% , (9.38)

where σ

is the noise equivalent electrons rms of each ampliﬁer contribution [Gatti

and Manfredi (1986)], and m is the total number of channels collecting the energy

deposited by the electromagnetic shower. In the case of low-energy incident particles,

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Principles of Particle Energy Determination 659

this contribution [Eq. (9.38)] can largely aﬀect the energy resolution even for σ

≈

several hundreds of electrons [Akrawy et al. (1990); Cushman (1992)].

The resolution achievable with lead-glass detectors is determined by the ﬂuctu-

ations of the electromagnetic shower development, by the

Cerenkov light absorp-

tion in a radiator σ

(E), and by the photoelectron statistics (∼

√

¯n

). One can

write [Prokoshkin (1980)]:

σ(E)

(E) +

gξ

1/2

, (9.39)

where ξ

is the ratio of the photocathode area to the radiator (counter) exit

area. Here ¯n

= g ξ

E and g is about 1000 photoelectrons per GeV. At ξ = 0.5,

the energy resolution can be parameterized as:

σ(E)

= 0.0064 + 0.042

√

, (9.40)

where σ

(E) can be obtained from Eqs. (9.39, 9.40). The value of σ

(E)/E de-

creases more slowly than the law E

−1/2

when the incoming energy increases, as a

consequence of large ﬂuctuations of

Cerenkov insertion of a light ﬁlter between the

radiator and the photosensitive device, which absorbs the short-wave fraction of the

spectrum [Prokoshkin (1980)]. The contribution from the photoelectron statistics

to the resolution [Eq. (9.40)] decreases with the increase of the incoming energy.

The energy resolution of the OPAL EMEC end cap [Akrawy et al. (1990)] has

an energy dependence well described by ∼ 5%/

√

E at 6 GeV for a array of nine

lead glass blocks with no material in front. The addition of 1.5 X

of lead in front of

the lead glass 20.5 X

deep reduces the cascade leakage from the back and enlarges

the lateral spread since the cascade starts earlier. The energy deposited in the lead

glass is lower, since a fraction of the incoming energy is lost in the lead layer. The

ﬂuctuations of this energy loss degrades the energy resolution. However, at 50 GeV

the improved containment almost exactly compensates for the increased ﬂuctuations

in energy caused by the lead layer, and the energy resolution is found to be ∼ 1.8%

at that energy. The energy resolution is ∼ 3.2% at 6 GeV [Akrawy et al. (1990)].

Deviations from the E

−1/2

law exist for crystal calorimeters. These are at-

tributed to energy leakage [Hughes et al. (1972)] and other instrumental eﬀects

such as the number of photons absorbed or refracted, or the shape of the spectral

overlap between photocathodes and photon energy. For instance, the energy reso-

lution of NaI(Tl) is rather found to obey a law E

−1/4

. The energy resolution of the

Crystal Ball detector is

σ (E)

= (2.7 ±0.2)% E

−1/4

for 16 X

of NaI(Tl) [Oreglia et

al. (1982)]. The energy resolution of CUSB detector at CESR is

σ (E)

= 3.9% E

−1/4

for 8.8 X

of NaI(Tl). Tables 9.7 and 9.8 report the energy resolution found for vari-

ous homogeneous calorimeters. From Table 9.7, NaI(Tl) shows an energy resolution

σ (E)

= 2.8% E

−1/4

and

σ (E)

= 0.9% E

−1/4

for a calorimeter depth of 16 X

and

24 X

[Hughes et al. (1972)], respectively. The improved energy resolution in the

second case is consistent with a larger longitudinal containment.