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

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710 Principles of Radiation Interaction in Matter and Detection

Fig. 9.35 Estimates (from [Leroy and Rancoita (2000)]) of interstellar antiproton ﬂuxes (Φ

)

from production models as a function of the kinetic energy T in GeV: Webber and Potgieter

(dashed line), leaky box [Webber and Potgieter (1989)]; Simon and Heinbach (dotted line), diﬀusive

reacceleration [Simon and Heinbach (1996)]; Simon and Heinbach (dot-dashed line), leaky box

[Simon and Heinbach (1996)]; Gaisser and Shaefer (solid line), leaky box [Gaisser and Schaefer

(1997)].

Fig. 9.36 Estimates (from [Leroy and Rancoita (2000)]) of eﬀect of the solar modulation as a

function of the kinetic energy T in GeV at minimum activity (Sect. 4.1.2.3) and at the Earth orbit

for the interstellar antiproton ﬂuxes (Φ

) shown in Fig. 9.35.

A knee is observed in the spectrum at E ≈ 3 × 10

GeV. The knee is

usually attributed to a change in the nature of the propagation of cosmic rays

and/or a decrease in the eﬃciency of conﬁning particles at very high energies in

the Galaxy [Greisen (1966); Berezinskii, Bulanov, Dogiel, Ginzburg and Ptuskin

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Principles of Particle Energy Determination 711

(1990)]. The particle spectrum presents an ankle between 10

and 10

GeV. This

ankle is usually interpreted as the result of the injection of a high energy ﬂux of

extragalactic origin [Greisen (1966)].

At low energies, where intensities are high, it is possible to study cosmic rays

through direct measurements via detectors operated in balloons, satellites, space

Shuttles and on board of the International Space Station Alpha. At energies beyond

eV, the intensities are very low and the study of the cosmic rays is performed

through cascade showers resulting from the degradation of the incident cosmic ray

energy in the atmosphere. The extent of these showers is determined by the size of

the incident particle (electrons, positrons, photons or hadrons) energy. The radiator

volume is now very large since it consists of the whole Earth atmosphere from the

ground level up to about an altitude of 30 km. Since the air shower is spread over a

large area with a low ﬂux of particles, detectors with large collection areas have to

be used. Their array sensitive area cover up to several 10

[Bott-Bodenhausen et

al. (1992)] and eﬀective detection over a volume of up to 100 km

[Cassiday (1985)].

9.11.2 Air Showers (AS) and Extensive Air Showers (EAS)

The cosmic rays interact with the terrestrial atmosphere creating air showers

(AS). These interactions produce secondaries through showering processes over long

distances in the atmosphere. If the cosmic ray energy is high enough, the seconda-

ries will reach eventually the ground level where they are detected. The atmosphere,

which acts as a very thick and low density radiator, forces the spread of secondaries

over a large area into so-called extensive air showers (EAS). This spread disperses

the secondaries far enough apart to resolve and count them. In principle, these

secondaries bring information about incident time, direction and energy.

The ratio of extragalactic cosmic rays to local cosmic ray ﬂuxes is thought to be

−4

–10

−5

. If the universe were baryon symmetric, about half of the extragalactic

cosmic rays should be antimatter. Fragmentation processes in the galaxy of ori-

gin and modulation in our galaxy will reduce the ﬂux of antimatter. Therefore,

experiments with an antiproton ﬂux sensitivity at the level of (or better than)

−3

−2

−1

GeV

−1

with an antiproton–proton separation capability at the

−7

level or better are required to study primordial antimatter.

A component from solar origin is present in primary cosmic rays. Variations in

the magnitude of this component are clearly correlated with solar activity [Berezin-

skii, Bulanov, Dogiel, Ginzburg and Ptuskin (1990)]. As mentioned above, the sig-

niﬁcance of this solar comp onent is small for kinetic energies above 1 GeV and

consequently these particles are not generally catalogued as cosmic rays. Thus, the

main part of the primary cosmic rays reach the Earth’s atmosphere from interstellar

space, produced in our Galaxy with the exception perhaps of particles with ultra

high energies (10

–10

) GeV which are presumably of extragalactic origin, where

they were accelerated by very large magnetic ﬁelds or produced by large size objects

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712 Principles of Radiation Interaction in Matter and Detection

or regions.

There are hadron showers initiated by hadrons, and electromagnetic showers

generated by electrons, positrons and photons. The EAS’s generated by hadrons

develop an electromagnetic component, since an important fraction of the seconda-

ries produced in the process is made of π

and η mesons, which decay via electro-

magnetic interaction and generate their own electromagnetic showers (as explained

in Sect. 3.3). Electromagnetic showers or electromagnetic components contain pho-

tons, electrons and positrons. The mechanism of energy loss for these particles is

pair production for photons and bremsstrahlung for electrons and positrons. Thus,

the number of photons, electrons and positrons rapidly increases with the atmo-

sphere depth until the electron and positron energy is down to the critical energy

(²

= 81 MeV in air), afterwards which they will lose the remaining energy through

ionization and the number of particles will decrease.

Muons are also involved in the generation of hadron and electromagnetic show-

ers. The charged mesons (π

, K

) produced in the hadron showers decay into

muons and are responsible for the presence of a large muon component in hadron

initiated showers. This component has a broad distribution since the muons are pro-

duced high in the atmosphere at the start of the hadron shower. Muons can also be

produced in electromagnetic induced showers through the decay of photoproduced

and K

, but less copiously with respect to hadronic induced shower, since

σ(γ-air)/σ(nucleon-air) ≈ 1.4 mb/300 mb. Another source of muons in an electro-

magnetic induced shower is hadroproduction by electrons, but this contributes even

less than photoproduction. Thus, the lateral spread of hadron initiated showers is

much wider than that of electromagnetic showers, especially at low altitude. As was

the case for calorimeters operated with accelerators (Sect. 9.8), a large fraction of

the hadron shower energy (20–30)% goes into nuclear excitation or is carried away

by neutrinos. These observations have consequences for the choice of techniques

used in the study of high energy cosmic rays.

The high energy cascade, produced by the interaction of ultra high energy parti-

cles entering the top of the atmosphere, can be detected by either the observation of

the electromagnetic radiation emitted in the atmosphere by the shower particles via

Cerenkov radiation or visible nitrogen ﬂuorescence, or by the direct observation of

the particles in the cascade. The combination of several techniques of measurement

allows the identiﬁcation of particles necessary for the search of point sources of high

energy cosmic radiation. The separation between γ’s and hadrons is essential to

this search since γ (like ν), as already stressed, can possibly be traced back to their

sources. In contrast, the sources of charged particles escape retracing since they

reach the Earth surface uniformly after being deﬂected by the galactic magnetic

ﬁeld. The γ-hadron separation requires high angular resolution and the possibility

of measuring the muon component in the electromagnetic induced showers.

Air

Cerenkov telescopes are used for the observation of the fast

Cerenkov light

ﬂash emitted by a shower generated in the atmosphere [Weekes (1988)]. The charged

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Principles of Particle Energy Determination 713

Table 9.10 The value of the refractive index,

Cerenkov angle (θ

)

and

Cerenkov threshold momenta for electrons [p

(e)], pions [p

(π)]

and protons [p

(p)], as a function of altitude from [Leroy and Ran-

coita (2000)].

Altitude Refractive θ

(e) p

(

π) p

(p)

(km) index (degree) (MeV) (GeV) (GeV)

30 1.00000424 0.17 175 48 322

20 1.00001734 0.34 87 24 159

15 1.00003506 0.48 61 17 112

10 1.00007091 0.68 43 12 79

8 1.00009398 0.78 37 10 68

6 1.0001245 0.90 32 9 59

4 1.0001651 1.04 28 8 52

2 1.0002188 1.20 24 7 45

particles traveling at velocities greater than the speed of light in a medium emit

electromagnetic radiation via the

Cerenkov eﬀect (Sects. 2.2.2, 9.5). EAS’s con-

tain relativistic charged particles with momenta above the

Cerenkov threshold. The

Cerenkov angle of the emitted radiation and the

Cerenkov threshold particle veloc-

ity are given by Eqs. (2.132, 2.134), respectively. Since the

Cerenkov eﬀect is related

to the refractive index of the traversed medium, the altitude will be a factor aﬀect-

ing the detection of EAS’s by their

Cerenkov light. At very high altitude i.e., in

the upper layers of the Earth atmosphere, the atmospheric pressure is low and the

refractive index is close to unity. As the altitude decreases, the refractive index

and the

Cerenkov angle increase, while the particle threshold momentum decreases

(Table 9.10).

The observation of

Cerenkov light, produced in the atmosphere by highly rela-

tivistic particles in EAS’s, provides a tool for investigating the longitudinal structure

of the EAS. The intensity of the

Cerenkov light is proportional to the total energy

dissipated in the atmosphere. This method of measurement provides a good angular

resolution. When the shower size is very large, the eﬀective detecting area will be

extended by observing the scintillation light, produced by EAS particles at high

altitudes or the scattering of electromagnetic waves by an ionized column produced

by the EAS. The application of such methods allows the study of EAS’s of very

large size that can be due to primary cosmic rays of energies greater than 10

eV.

The main shower development at high altitudes can be observed by its radial

Cerenkov light pattern at the Earth ground level. At high altitudes, the lateral

dispersions of electromagnetic and hadron induced showers are quite diﬀerent and

the measurement of the

Cerenkov light provides a way to separate photons from

hadrons. The interactions of the charged particles radiating

Cerenkov light with the

atmosphere modify their trajectories and, in particular, the multiple Coulomb scat-

tering spreads the electron paths. The lateral spread of the shower and the fact that

many particles in the shower have a momentum close to the threshold momentum

i) create a situation of overlap for photons generated at diﬀerent altitudes and ii)

complicates in practice the measurement of the γ/hadron ratio. Therefore, there is a

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714 Principles of Radiation Interaction in Matter and Detection

need to combine

Cerenkov measurements with scintillators or tracking chamber ar-

rays that sample the shower tail, when it reaches the Earth ground level. Signiﬁcant

progress in the understanding of the cascading mechanisms, γ-hadron separation,

and improved sensitivity in the search for point-like sources or diﬀuse γ radiation

is expected from the combination of Earth based scintillator and muon detector ar-

rays with a matrix of air

Cerenkov counters recording the shower parameters from

ground level up to the higher layers of the atmosphere [Bott-Bodenhausen et al.

(1992)].

The passage of EAS’s through the atmosphere can also be detected via the mea-

surement of the nitrogen ﬂuorescence light given oﬀ by relativistic charged particles

in the shower as performed in the Fly’s Eye detector [Cassiday (1985)]. The main

diﬀerence between ﬂuorescence and

Cerenkov light lies in the angular distribution

as the

Cerenkov light is distributed along the shower direction while the ﬂuores-

cence light has an isotropic distribution. The distribution of the number of photons

of ﬂuorescence is approximated by [Chiavassa and Ghia (1996)]:

dl d Ω

4π

, (9.88)

where N

is the number of electrons in the EAS (see Sect. 9.11.3.1 below) and

is the ﬂuorescent yield ≈ 4 γ/electron/m. The fact that ﬂuorescence light is

emitted isotropically from the EAS permits experiments such as the Fly’s Eye de-

tector [Cassiday (1985)] to detect at large distances. The experiments carried out

with this detector include: i) a direct measurement of the proton–air cross section

(at

√

s = 30 TeV), ii) an analysis of the primary cosmic ray spectrum in the energy

range (10

–10

) eV, iii) an extraction of the composition of the high energy cos-

mic ray primaries, iv) a search for anisotropies in arrival directions, v) a search for

deeply penetrating showers indicative of primary neutrinos, possible heavy-lepton

production and quark matter in the primary ﬂux and, ﬁnally, vi) a search for sources

of γ rays near 10

eV.

9.11.3 Electromagnetic Air Showers

9.11.3.1 Longitudinal Development

The longitudinal development of the electromagnetic cascades is parameterized as

a function of the age parameter s. For each sub-shower induced by photon (from

and η decays) or by electrons of energy E, the age s of this sub-shower at depth

t is given by

s =

t + 2y

, (9.89)

where t is the depth measured in radiation length units (∼ 37.0 g/cm

in air) and

y = ln (E

/²

); E

is the energy of the primary photon or electron, and ²

is the

critical energy (as given in Sect. 2.1.7.4, ²

= 81 MeV for air).

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Principles of Particle Energy Determination 715

The age parameter evolves from s = 0 at the point of the ﬁrst interaction to

s = 1 at the shower maximum, and continues to increase (s > 1) beyond the

shower maximum. The longitudinal development of an electromagnetic cascade can

be described by a parametrization of the number of charged particles (electrons and

positrons), N

, in the shower as a function of the depth [Hillas (1982)]:

, E, t) =

0.31

√

exp {t[1.0 − 1.5 ln(s)]}. (9.90)

Here E is the threshold energy of electrons. N

is often called the size of the

shower. As an example of an application, Eq. (9.90) was used in [Fenyves et al.

(1988)] with a modiﬁed form in order to describe the longitudinal development of

the electromagnetic component of EAS’s generated by 10

–10

eV proton and iron

nuclei:

, E, t) = α

A(E) N

, E, t

), (9.91)

where α

is the number of primary particles (photons or electrons) generating the

shower; A(E) is the fraction of electrons having energies larger than E compared

with the total number of electrons. The modiﬁed age parameter, s

, is calculated

as a function of the mo diﬁed depth, t

, according to:

+ 2y

. (9.92)

The modiﬁed depth, t

, is given by

= t + a

π ,γ

(E), (9.93)

where a

(E) and a

(E) account for the diﬀerent development and diﬀerent t

max

values of electron- and photon-induced showers, resp ectively, with diﬀerent electron

threshold energies (t

max

is the depth where the shower reaches its maximum). Ex-

amples of values for A(E), a

(E) and a

(E) obtained from ﬁts to the data [Fenyves

et al. (1988)] are given in Table 9.11.

9.11.3.2 Lateral Development

The lateral distribution of particles in an electromagnetic shower is usually described

by a parametrization suggested by Nishimura and Kamata (1958):

f(r/R

, s, E) = C(s)

s−2

1 −

s−4.5

, (9.94)

Table 9.11 Values of the parameters for the longitu-

dinal and lateral development of electromagnetic air

showers from [Leroy and Rancoita (2000)].

E 5 MeV 10 MeV 15 MeV 20 MeV

A(E) 0.67 0.59 0.52 0.48

(E) 0.60 0.80 0.92 1.0

(E) 0.00 0.20 0.32 0.40

(E) 0.20 0.40 0.52 0.60

(E) -0.40 -0.20 -0.08 0.00

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716 Principles of Radiation Interaction in Matter and Detection

where s is given by Eq. (9.89), E is the threshold energy of the charged particle, r

is the perpendicular distance from the shower axis, and R

is the Moli`ere radius

at the level of observation (80 m at sea level and 100 m at 2 km); furthermore, one

imposes the normalization condition [Chiavassa and Ghia (1996)]

2π

∞

(r/R

)f(r/R

, s, E) d(r/R

) = 1. (9.95)

The function f(r/R

, s, E) represents the probability that a charged particle falls

at a distance r from the shower axis with an area of R

. It also means that the

factor C(s) is constrained to be:

C(s) =

2π

Γ(4.5 − s)

Γ(s) Γ(4.5 − 2s)

. (9.96)

If N

is the size of the shower, the charged particle density, ρ

, as a function of the

distance r from the shower axis is given by

(r) =

f(r/R

, s, E). (9.97)

Equation (9.94) was applied in [Fenyves et al. (1988)] to the description of the lateral

distribution of the electromagnetic component of an EAS generated by 10

–10

protons and iron nuclei. Equation (9.94) was modiﬁed for the purpose by replacing

by R

= 0.5 R

. The best ﬁt was obtained for R

= 45 m at 850 g/cm

and

= 37.5 m at sea level. The age parameter was given by

s =

+ 2y

(9.98)

with

= t + b

π,γ

(E). (9.99)

Here b

(E) and b

(E) account for the diﬀerent development and aging of electron-

and photon-induced showers, respectively, with diﬀerent electron threshold ener-

gies. Examples of values for b

(E) and b

(E) are also given in Table 9.11.

9.11.4 Hadronic Extensive Air Showers

A hadron-induced EAS has a development following closely the steps described

in Sect. 3.3. Basically, the interaction of a incident nucleon with a nucleus in the

upper atmosphere produces many hadrons. Each of these secondary hadrons will

further interact with atmospheric nuclei or decay into other hadrons (π, K), leptons

, ν’s) and γ’s. The occurrence of a decay or of an interaction depends on the

atmosphere density. Pions and kaons generated in the upper atmosphere, which also

corresponds to the earlier stage of the shower development, have a higher probability

to decay. In the lower atmosphere, the probability to have an interaction instead

of a decay is signiﬁcantly higher. The interaction length λ

g, air

[Eq. (3.76)] in air

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Principles of Particle Energy Determination 717

Fig. 9.37 Energy spectra of various nuclei from hydrogen to iron up to energies of hundreds of GeV

per nucleon on the Earth as a function of the kinetic energy per nucleon (from [Simpson (1983)],

reprinted, with permission, from the Annual Review of Nuclear and Particle Science, Volume 33

°1983 by Annual Reviews www.annualreviews.org; see also [Leroy and Rancoita (2000)]).

is 80 g/cm

. The depth of the atmosphere is commonly put at about 13 λ

g ,air

and

allows for a complete development of the hadronic shower.

As described in Sect. 3.3, there is a multiplication of particles accompanied

with a decrease of the average energy of the secondaries along the shower develop-

ment. The number of particles in the shower will reach a maximum at a depth, which

depends on the primary energy (E

), on the type of primary particle, and on the

history of interaction of secondary particles. The interaction of a nucleus with the

atmosphere can be viewed as the interaction of A independent nucleons each with

the energy E

/A. The interaction length of a nucleus in the atmosphere is somewhat

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718 Principles of Radiation Interaction in Matter and Detection

reduced compared to that of a nucleon and estimated to be a few g/cm

. The intera-

ction of a nucleus with the atmosphere can be thought as a superposition of nucleon

interactions with the atmosphere. The propagation of particles through the atmo-

sphere is described by a system of cascade equations, which describe the transport of

particles through the atmosphere, taking into account the particle properties, their

interactions, and the properties of the atmosphere traversed by the particles. Using

the equations from Gaisser’s textbook [Gaisser (1990)], the transport of nucleons in

the atmosphere is described by

(E, X)

= −

(E, X)

gN (E)

+ I

, (9.100)

where

∞

, X)

gN (E

)

(E, E

)

, (9.101)

and where N

(E, X) is the nucleon ﬂux at depth X (expressed in g/cm

) in the

atmosphere, and E

is the primary energy. The nucleon interaction length in air,

gN (E)

in g/cm

, is given at the energy E by means of Eq. (3.76) for A = air, i.e.,

gN (E)

≈ λ

g, air

≈ 80 g/cm

The atmospheric density, ρ, depends on the altitude, as pointed out earlier. The

function

(E, E

) = E

, E

)

(9.102)

is the (dimensionless) inclusive cross section, integrated over transverse momenta,

for an incident nucleon of energy E

to collide with an air nucleus and produce an

outgoing nucleon of energy E; dn

is the number of particle of type c produced on

average in the energy bin, dE

, around E

per collision of an incident particle of

type a [Gaisser (1990)].

If one takes into account the secondary pions, Eq. (9.100) has to b e associated

with the following equations [Sokolsky (1989); Gaisser (1990)]:

dΠ

(E, X)

= −Π

(E, X)

gπ

(E)

EX cos θ

+ I

Nπ

+ I

Nπ

, (9.103)

where

∞

, X)

g N (E

)

Nπ

(E, E

)

(9.104)

and

Nπ

∞

, X)

g π(E

)

Nπ

(E, E

)

, (9.105)

in which Π

is the average number of pions and λ

g π

(E) is the pion interaction

length in air at the energy E; the term

EX cos θ

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Principles of Particle Energy Determination 719

accounts for pion decay at the air shower zenith angle θ;

cτ

(τ

is the pion lifetime) and h

varies with the altitude: h

= 8400 m at sea level and,

for a vertical atmospheric depth X

< 200 g/cm

, h

= 6400 m [Gaisser (1990)]. π

’s

decay (π

→ γγ) before they have the chance to interact and, consequently, do

not feed-back the hadronic cascade. The functions F

Nπ

and F

are deﬁned

analogously to Eq. (9.102) for the processes

N + air → π

+ anything

and

+ air → π

+ anything.

The function representing the process

a + air → b + anything

is [Sokolsky (1989)]:

(E, E

) =

inel

dσ

. (9.106)

Applying Feynman scaling [Feynman (1969); Gaisser and Yodh (1980)], F

depends

only on one variable and can b e rewritten as:

= F

) ,

where

√

is the Feynman variable. p

is the particle momentum component parallel to the

incident particle direction. As a consequence of Feynman scaling hypothesis, the

multiplicity, n, of secondary particles (mostly pions) produced in hadron (nucleon

or π) nucleon collisions depends logarithmically on energy

n = a ln(E). (9.107)

The shower generation is then a process in which a particle almost interacts once

every interaction length, producing a number of secondaries that is roughly constant

per interaction. The maximum of the shower depth increases logarithmically with

. In the case of Feynman scaling violation, the multiplicity increases faster with

energy, from E

1/4

to E

1/2

. Then, the shower develops faster and earlier in the upper

atmosphere. Indeed, violations of the Feynman scaling are increasingly present as

the energy increases due to the increased hard QCD scattering contribution. As

pointed out in [Battistoni and Grillo (1996)], the scaling hypothesis remains a useful

tool for the understanding of hadronic shower behavior in the atmosphere.