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

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190 Principles of Radiation Interaction in Matter and Detection

Fig. 2.72 Mass attenuation coeﬃcients for Ge as a function of the incoming photon energy (data from [Berger, Hubbell, Seltzer, Chang, Coursey,

Sukumar and Zucker (2005)]).

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Electromagnetic Interaction of Radiation in Matter 191

Fig. 2.73 Mass attenuation coeﬃcients for W as a function of the incoming photon energy (data from [Berger, Hubbell, Seltzer, Chang, Coursey,

Sukumar and Zucker (2005)]).

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192 Principles of Radiation Interaction in Matter and Detection

Fig. 2.74 Mass attenuation coeﬃcients for Pb as a function of the incoming photon energy (data from [Berger, Hubbell, Seltzer, Chang, Coursey,

Sukumar and Zucker (2005)]).

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Electromagnetic Interaction of Radiation in Matter 193

tial and total mass interaction coeﬃcients. Total attenuation coeﬃcients without

the contribution from coherent scattering are also given. Interaction coeﬃcients and

total attenuation coeﬃcients for compounds or mixtures are obtained as sums of

the corresponding quantities for atomic constituents. For example, in Figs. 2.67–

2.74, mass attenuation coeﬃcients from XCOM database [Berger, Hubbell, Seltzer,

Chang, Coursey, Sukumar and Zucker (2005)] are shown as a function of the in-

coming photon energy for air

, water (H

O), brain

Al, Si, Ge, W, Pb for photon

energies between 1 keV and 10 GeV.

The eﬀect of γ-ray irradiation on matter is mainly indirect, i.e., via charged

particles (electrons and positrons) generated in the interaction. It is the dissipation-

energy process (mostly by collision energy-losses) of these secondary charged and

ionizing particles, which determines the energy deposition of γ-rays in matter. The

relationship between the deposited energy and the various physical, chemical and

biological eﬀects is usually complex and not fully understood. However, it is com-

monly assumed that a signiﬁcant parameter in radiation eﬀects is the absorbed dose,

which is deﬁned as the mean energy imparted by ionizing radiation (d¯ε) per unit

mass ( dm):

D ≡

d¯ε

The mean energy imparted in a volume is given by [ICRUM (1980a)]

¯ε ≡ R

− R

out

where R

is the radiant energy incident on the volume, i.e., the sum of the ener-

gies (without taking into account the rest energies) of all (charged and uncharged)

ionizing particles entering the volume. R

out

is the radiant energy emerging from the

volume, i.e., the sum of the energies (without taking into account the rest energies)

of all (charged and uncharged) ionizing particles leaving the volume.

Q is the

sum of all changes (which occur in the volume) of the rest mass energy of nuclei

and elementary particles in any nuclear transformation. The SI-unit

∗

of absorbed

dose is designated as the gray

†

(Gy), i.e., in J kg

−1

At energies larger than the critical energy ²

[Eq. (2.121)] of the medium, the cal-

culation of the energy deposition requires the cascade-development treatment for the

transport of electrons and photons. However, at lower energies the energies deposited

by γ-rays can be treated in terms of the photon ﬂux and the mass energy absorption

coeﬃcient µ

att,m,en

(or one of its approximation like the mass absorption coeﬃcient

att,m,ab

or the mass energy transfer coeﬃcient µ

att,m,tr

). The mass energy absorp-

tion coeﬃcient takes into account all possible modes of energy transfer

‡

to electrons

The dry air composition near the sea level is that speciﬁed in the web site given for [ICRUM

(1989)].

For brain, the composition of grey/white matter is from [ICRUM (1989)].

∗

The International System of Units is usually indicated as SI.

†

It is sometimes expressed in rad: 1 rad = 0.01 Gy = 1 cGy.

‡

The reader can see relatively recent calculations by Seltzer (1993) and, for instance, [ICRUM

(1964); Hubbell (1969); Hubbell and Seltzer (2004)].

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194 Principles of Radiation Interaction in Matter and Detection

in the medium, while the mass energy transfer and the mass absorption co eﬃ-

cient neglect some of the modes. The coeﬃcient µ

att,m,en

accounts for the escape

of ﬂuorescence, Compton scattered, annihilation, and bremsstrahlung photons. The

coeﬃcient µ

att,m,tr

accounts for the escape of these photons, bremsstrahlung pho-

tons excepted. The coeﬃcient µ

att,m,tr

accounts for the escape of Compton scattered

photons only. The mass energy transfer coeﬃcient is given by:

att,m,tr

+ f

incoh

+ f

pair,n

+ f

pair,e

) , (2.224)

where the cross sections σ

refer to the processes (previously discussed) of photoelec-

tric absorption, incoherent Compton scattering (see page 163) and pair production

in the Coulomb ﬁeld of nuclei and atomic electrons; the factors f

represent the

average fractions of the photon energy E

which were transferred as kinetic ener-

gies of charged particles as a result of the interactions. In Eq. (2.224), the coherent

scattering cross section was omitted because the associated energy transfer is negli-

gible. The factors f

in Eq. (2.224) are given by:

= 1 −

where E

is the average energy of the ﬂuorescence radiation emitted per absorbed

photon;

incoh

= 1 −

+ E

where

is the average energy of the photon scattered by Compton eﬀect;

pair,n

= 1 −

2mc

and

pair,e

= 1 −

2mc

+ E

The ﬂuorescence radiation emitted per absorbed photon E

depends on the distribu-

tion of atomic electron vacancies generated in the interaction. Thus, it is diﬀerently

evaluated for various types of interactions and includes the emission of cascade

ﬂuorescence X-rays associated with the atomic relaxation process initiated by the

primary vacancy [Carlsson (1971)].

The dosimetric quantity kerma (K) is deﬁned as the sum of the kinetic energies

of all those charged particles released by uncharged particles (in this case photons)

per unit mass; it can be calculated by multiplying the mass energy transfer coeﬃ-

cient by the photon energy ﬂuence

Ψ = ΦE

where Φ is the ﬂuence of photons of energy E

. At charged particles equilibrium

(i.e., when energy, number and direction are constant throughout the volume of

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Electromagnetic Interaction of Radiation in Matter 195

interest) and for negligible bremsstrahlung production, the kerma approaches the

absorbed dose. The SI-unit of kerma is the gray.

Further emission of radiation, produced by secondary charged particles, is taken

into account by the mass energy absorption coeﬃcient, which is given by

att,m,en

(1 − g

incoh

(1 − g

pair,n

(1 − g

pair,e

= (1 − g) µ

att,m,tr

, (2.225)

where g

is the average fraction of the kinetic energy of secondary charged particles

produced in a speciﬁc type of interaction, that is lost in radiative processes, and g

is the weighted average of g

In Fig. 2.75, mass energy absorption coeﬃcients by Hubbell and Seltzer (2004)

are shown as function of the incoming photon energy for Al, Si, Pb, brain

(grey/white matter), (liquid) water and dry air (near sea level) media

∗∗

for photon

energies between 1 keV and 20 MeV. It has to be noted that at low photon ener-

gies, where g is small, µ

att,m,tr

≈ µ

att,m,en

. At 1.5 MeV, their diﬀerence increases

as the atomic number increases from less than 1% for low-Z media up to ≈ 7%

for high-Z media. At 10 MeV and for high-Z absorb ers, they diﬀer by larger and

energy dependent values; but for air and water, they diﬀer by ≤ 1%.

Furthermore, let us introduce the dosimetric quantity exposure X, which is

related to the mass energy absorption coeﬃcient in air by:

X = Ψµ

air

att,m,en

where W

is the mean energy expended in air per ion-pair and e is the electron

charge, while the kerma in air, K

air

, is related to the energy ﬂuence by

air

= µ

air

att,m,tr

Ψ.

The SI-unit of exposure is expressed in C kg

−1

The equivalent dose

‡‡

) in a organ or tissue (T) depends on the absorbed

dose (D

) caused by diﬀerent radiation types R and averaged over the volume of

that organ or tissue (e.g., see Section 4.5.2.1 of [Dietze (2005)] and Section 29 of

[PDB (2008)]):

× D

T,R

where the weighting factors w

are charecterizing the biological eﬀectiveness of the

speciﬁc radiation (R) and can b e found, for instance, in Section 29 of [PDB (2008)]

∗∗

The dry air and brain composition are the same as for the mass attenuation coeﬃcients from

XCOM database (see page 193 and also [ICRUM (1989)]).

‡‡

In 1977 the the tisse (or organ) dose equivalent was introduced by [ICRP (1977)]. Although

the concept was not changed in [ICRP (1991)], important modiﬁcations where introduced, e.g.,

replacing dose equivalent quantities by equivalent dose quantities. The reader can see Section 4.5.2

of [Dietze (2005)].

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196 Principles of Radiation Interaction in Matter and Detection

Fig. 2.75 Mass energy-absorption coeﬃcients for brain (grey/white matter), dry air (near sea

level), Al, Si, Pb and (liquid) water as function of the incoming photon energy (data from [Hubbell

and Seltzer (2004)]). The air and brain densities are 1.205×10

−3

and 1.04 g/cm

, respectively. The

air and brain Z/A ratios are 0.499 and 0.552, respectively.

(see also references therein). The SI-unit of equivalent dose is designated as the

sievert

†

(Sv), i.e., in J kg

−1

A treatment of the physical (stochastic and non-stochastic) quantities and units

used to describe radiological and/or radioactive phenomena can be found in [Dietze

(2005)]. Stochastic quantitities are those used when the phenomena are subject to

inherent ﬂuctuations and follow a probability distribution (e.g., the decay constant

of a radionuclide). For other phenomena, a non-stochastic quantititity (e.g., the

†

It is sometimes expressed in rem: 1 rem = 0.01 Sv = 1 cSv.

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Electromagnetic Interaction of Radiation in Matter 197

Table 2.14 Measured longitudinal attenuation-lengths of elec-

tron-induced cascades (from [Leroy and Rancoita (2000)]).

Incident Attenuation

Material Energy λ

att

References

GeV g/cm

Al 6 64.3 2.7 [Bathow et al. (1970)]

Cu 1 41.7 3.2 [Nelson et al. (1966)]

0.9 34.4 2.7 [Crannell (1967)]

6 39.0 3.0 [Bathow et al. (1970)]

1 35.7 2.8 [Yuda et al. (1970)]

0.6 34.5 2.7 [Yuda et al. (1970)]

Sn 0.9 30.3 3.4 [Crannell (1967)]

Pb 1 22.2 3.5 [Nelson et al. (1966)]

0.9 23.0 3.6 [Crannell (1967)]

6 24.7 3.9 [Bathow et al. (1970)]

1 21.7 3.4 [Yuda et al. (1970)]

0.6 21.3 3.3 [Yuda et al. (1970)]

ﬂuence or the absorbed dose) is deﬁned by averaging in time or over a volume

which results in a single value with no inherent ﬂuctuation.

2.4 Electromagnetic Cascades in Matter

Electromagnetic-cascade showers have been one of the most striking phenomena

observed in high-energy cosmic rays and were discovered by Blackett and Oc-

chialini (1933) (see also [Blackett, Occhialini and Chadwick (1933)]). An extensive

review covering both the electromagnetic (discussed in this section) and hadronic

(discussed in the chapter on Nuclear Interactions in Matter) cascading shower gen-

eration and propagation in matter was provided by Leroy and Rancoita (2000).

The degradation of the incident energy results in a multiplicative process, whose

extent is controlled by the magnitude of the incident-particle energy. High energy

electrons lose most of their energy by radiation via bremsstrahlung eﬀect and, there-

fore, produce high energy photons. These high-energy photons, in turn, mainly un-

dergo materialization

∗

in the Coulomb ﬁeld of a nucleus, or they produce Compton

electrons. These electrons and positrons, in turn, radiate new photons which un-

dergo again pair production or Compton scattering, if their energy is suﬃciently

large. This phenomenon is often referred to as a multiplicative shower or cascade

shower.

At high-(and ultra high-) energies the bremsstrahlung and pair production pro-

cesses are described by the Landau–Pomeranchuk–Migdal (LPM) formulae [Migdal

(1956); Landau and Pomeranchuck (1965)], which, at lower energies [Migdal (1956)],

reduce to the expressions given by Bethe and Heitler (1934). At very high energy,

LPM formulae predict that the production of low-energy photons by high-energy

∗

The pair production process is dominant beyond 100 MeV.

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198 Principles of Radiation Interaction in Matter and Detection

electrons is suppressed in dense media: deviations as large as 30% compared with

the Bethe–Heitler formulae were predicted for lead in [Migdal (1956)]. This sup-

pression was conﬁrmed experimentally [Antony et al. (1995)].

In a cascade shower initiated by an electron or a photon, the generated parti-

cles are electrons, positrons and photons. This kind of cascade shower is called an

electromagnetic shower. Soft photons and electrons are generated in the ﬁnal stage

of the multiplication process. These photons mainly interact via the photoelectric

process while the soft electrons, including the photo electric electrons, dissipate their

energy through collisions. In this way, the incoming particle energy is absorbed by

the medium.

The devices built to measure the energy deposited in their volume from the

degradation of the energy carried by incident electrons and/or photons are called

electromagnetic calorimeters.

2.4.1 Phenomenology and Natural Units of Electromagnetic Ca-

scades

Bremsstrahlung and pair production are the dominant interaction processes for

high-energy electrons and photons, respectively. Their cross sections become almost

energy independent in the case of complete screening, i.e., for incoming electron and

photon energies À mc

/(αZ

1/3

), where Z is the atomic number of the absorber. As

a consequence, the radiation length

†

, X

, emerges as a natural unit of length and

represents the mean-path length of an electron in a material. The absorber depth t

is usually given in units of radiation length.

Radiation lengths were calculated and tabulated (see, for instance, [Tsai (1974);

PDB (2008); PDG (2008)] and references therein). They are reported in Table 2.3,

for various absorbers used for calorimeters, and in Tables 2.7 and 2.8, for elements

up to Z = 92. In practice, the following approximation is often used [Amaldi (1981)]:

g 0

= X

ρ ≈ 180

[g/cm

], (2.226)

where X

is the value of the radiation length in cm; ρ in g/cm

and A are the density

and atomic weight (see Sect. 1.4.1 and page 220) of the medium, respectively, and

with

∆X

g 0

≤ 20% for Z ≥ 13.

When more than one absorber is present in the showering medium, the overall

radiation length can be calculated by means of Eq. (2.119). The corresponding

medium density can be calculated from Eq. (2.120).

There is a depth, indicated as t

max

and called shower maximum, where the

largest number of secondary particles created during the multiplication process is

reached and, correspondingly, where the largest energy dissipation occurs. Beyond

†

It was discussed in Sect. 2.1.7.3 on Radiation Length and Complete Screening Approximation.

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Electromagnetic Interaction of Radiation in Matter 199

that maximum depth, the cascade decays slowly. Each step of the energy degrada-

tion is characterized by an increase of secondary particles produced with a decreas-

ing average energy: low energy photons either transfer their energy to low-energy

electrons via Compton scattering or lead to low energy electron production via the

photoelectric eﬀect. This latter phenomenon

is relevant or dominant for photon

energies below about a few hundred keV’s (Sects. 2.3.1 and 2.3.5). Low energy elec-

trons mostly lose their energy via collisions and, as the cascade evolves along the

calorimeter depth, more and more electrons fall into an energy range where collision

energy-losses dominate radiation energy-losses. In the showering process, the mul-

tiplication process is almost stopped when the electron energy ﬁnally reaches the

critical energy

. The value of the critical energy, calculated using Eq. (2.121), is

given in Table 2.3 for several materials commonly used as absorb er in calorimeters.

2.4.2 Propagation and Diﬀusion of Electromagnetic Cascades in

Matter

The electromagnetic processes, generating an electromagnetic cascade, are well de-

scribed by the quantum-electrodynamics theory (QED), which accounts for the

basic interactions of electrons and photons inside the calorimeter medium (see, for

instance, [Berestetskii, Lifshitz and Pitaevskii (1971); Tsai (1974)] and references

therein). In principle, an analytical description of the cascade behavior is possi-

ble. However, the interaction-mechanism complexity

of cascading particles with

the calorimeter medium imposes treatments in which part of the physical processes

is neglected or taken into account in an approximate way. The theory of cascade

showers originates from two independent papers by Bhabha and Heitler (1936) and

by Carlson and Oppenheimer (1936). Afterwards, improvements and ampliﬁcations

with respect to the physical and mathematical approximations were proposed by

several authors [Arley (1938); Landau and Rumer (1938); Bhabha and Chakrabarty

(1943); Heitler (1954); Rossi (1964)].

2.4.2.1 Rossi’s Approximation B and Cascade Multiplication of Electro-

magnetic Shower

A simpliﬁed analytical model of the cascade development was proposed by

Rossi (1964). In Rossi’s formulation, the shower development theory is supposed

to predict the number and the energy distribution of electrons and photons with

energy large compared with a given energy η

≈ 5 MeV. Collisions, giving rise to

secondary electrons with energy less than η

, represent the process in which the

The photoelectric eﬀect also depends on the absorber atomic-number (see discussion in

Sect. 2.3.1).

The critical energy was deﬁned in Sect. 2.1.7.4.

As it will be seen later, the interaction of cascading particles with low-energy is particularly

complex.