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

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130 Principles of Radiation Interaction in Matter and Detection

2.2.3 Emission of Transition Radiation

Transition radiation is emitted when a fast charged particle passes through the

boundary between media with diﬀerent indices of refraction (r

and r

), as shown

in Fig. (2.45) ([Ginzburg and Frank (1946)], see also [Goldsmith and Jelley (1959)]

for the ﬁrst experimental observation of the transition radiation).

The reorganization of the ﬁeld associated with the incoming particle occurs be-

cause a sudden change of the dielectric property of matter. The emitted radiation

comes from a coherent superposition of radiation ﬁelds generated by the molecular

polarization. The coherence is kept in a small volume surrounding the particle path,

whose length extension is referred to as the coherent length or formation zone. It can

result in an observable amount of energy in the X-ray region when a high enough

energy particle (i.e., when its Lorentz factor γ is À 1) traverses the boundary of

a macroscopically thick medium. Like

Cerenkov emission, the process depends on

the particle velocity and is a collective response of the matter close to the particle

path. Like bremsstrahlung, it is sharply peaked in the forward direction [Garibyan

and Barsukov (1959)]. The intensity of the pro cess, i.e., the overall number of emit-

ted photons, can be enhanced by radiators consisting of several boundaries [Frank

(1964)]. The transition-radiation emission depends on a few parameters such as, for

instance, the radiator conﬁguration and the Lorentz factor γ of the particle.

In the X-ray frequency region, a material behaves as an electron gas (see [Artru,

Yodh and Mennessier (1975)]), whose plasma frequency [Eq. (2.25)] is

2π

πm

where n is the electron density and m is the electron mass. The corresponding

plasma photon energy is

hν

= ~ω

= h

ZρNe

πmA

4πNr

Zρ

≈ 28.8

Zρ

[eV]. (2.142)

The plasma photon energies are ≈ 0.7 eV for normal air, 0.27 eV for He, 20 eV

for polypropylene and styrene, 13.8 eV for Li and 24.4 eV for mylar. The dielectric

constant of the medium is given by

²(ω) = 1 −

= 1 − Υ

, (2.143)

where Υ is ¿ 1 in the X-ray region. The formation length D is of the order of

≈ (γc)/ω

and represents the largest value of the frequency dependent depth over

which the coherent superposition can occur [Jackson (1975)].

The complete expression for the energy radiated is rather complicated. But at

large γ, most of the energy is in the forward direction, i.e., for θ < π/2, where θ

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Electromagnetic Interaction of Radiation in Matter 131

Fig. 2.45 Transition-radiation emission at the boundary between two media with diﬀerent indices

of refraction (adapted and republished with permission from Artru, X., Yodh, G.B. and Mennessier,

G., Phys. Rev. D 12, 1289 (1975); Copyright (1975) by the American Physical Society).

is the angle between the direction of the emitted photon and the trajectory of the

incoming particle. For the forward direction (see Fig. 2.45), the energy radiated by

a particle of charge ze at the b oundary between two media, per unit of solid angle

and per unit of frequency interval can be approximated by

dν dΩ

' z

hα

−2

+ θ

+ Υ

−

−2

+ θ

+ Υ

(2.144)

with Υ

p,1

, and Υ

p,2

; where ν

p,1

= ω

p,1

/(2π) and ν

p,2

= ω

p,2

/(2π) are the

plasma frequencies of the two media (see for instance [Artru, Yodh and Mennessier

(1975); Fayard (1988)] and references therein). Equation (2.144) can be rewritten

per unit of solid angle and unit of energy as:

d(~ω) dΩ

d(hν) dΩ

' z

−2

+ θ

+ Υ

−

−2

+ θ

+ Υ

. (2.145)

The radiation is concentrated in a narrow forward cone, deﬁned by θ

and of an

order ranging from

−2

+ Υ

−2

+ Υ

. Therefore, the angular distribution

is conﬁned to a forward cone for which γθ is of the order of the unity. As long as

the cone is completely contained in the second medium, there is no dependence on

the particle incidence angle relative to the boundary. If ω

p,2

¿ ω

p,1

, as it is the case

for a dense material and a gas, the most probable emission angle θ

and the root

mean square angle θ

rms

are given by [Fayard (1988)]:

≈

−2

+ Υ

and

rms

≈

−2

+ Υ

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132 Principles of Radiation Interaction in Matter and Detection

It has to be noted that the distributions given by Eqs. (2.144, 2.145) for an incoming

particle with a Lorentz factor γ and two media with ω

p,1

and ω

p,2

are the same as

for an incoming particle with a Lorentz factor γ

and, as media, the vacuum and a

material with plasma frequency ν

p,1

= ω

p,1

/(2π) where:

p,1

− ω

p,2

= 0 (i.e., the vacuum, for which we have: Υ

= 0),

−2

+ Υ

We have always γ

< γ and ω

p,1

< ω

p,1

. Integrating Eq. (2.145) over Ω, we obtain:

d(~ω)

' z

·µ

+ Υ

+ 2γ

−2

− Υ

+ γ

−2

+ γ

−2

− 2

. (2.146)

In Eq. (2.146), dW /d(~ω) depends on γ/ω only. It can be rewritten introducing γ

and Υ

= ω

p,1

/ω, i.e.,

d(~ω)

' z

½·

(ω

p,1

/ω)

+ (ω

p,2

/ω)

+ 2γ

−2

(ω

p,1

/ω)

− (ω

p,2

/ω)

(ω

p,1

/ω)

+ γ

−2

(ω

p,2

/ω)

+ γ

−2

− 2

= z







(ω

p,1

/ω)

+ 2γ

0−2

(ω

p,1

/ω)





p,1

−ω

p,2

+ γ

−2

0−2





− 2







= z

(ω

p,1

/ω)

+ 2γ

0−2

(ω

p,1

/ω)

(ω

p,1

/ω)

+ γ

0−2

− 2

)

= z











1 + 2

p,1





1 +

p,1

− 2







= z

·µ

1 + 2

1 + γ

− 2

= z

, (2.147)

where the function G

is given by

·µ

1 + 2

1 + γ

− 2

(2.148)

and it is shown in Fig. 2.46. For instance, in the case of ω

p,2

¿ ω

p,1

, we can

distinguish three regimes as a function of γ

(i) γ ¿ ω/ω

p,1

= 1/Υ

, i.e., a very low yield for which

d(~ω)

' z

6π

(γΥ

)

= z

6π

γω

p,1

and, in order to have enough yield, it must be noted that there is a frequency cutoﬀ

given by ω ≤ γω

p,1

;

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Electromagnetic Interaction of Radiation in Matter 133

(ii) 1/Υ

= ω/ω

p,1

¿ γ ¿ ω/ω

p,2

= 1/Υ

, where there is a logarithmic increase of

the yield with γ and we have

d(~ω)

' 2z

[ln (γΥ

) − 1] = 2z

γω

p,1

− 1

;

(iii) γ À ω /ω

p,2

= 1/Υ

, in which the yield is almost constant (saturation).

The total energy emitted in forward direction by transition radiation per bound-

ary [Fayard (1988)] is calculated by integrating Eq. (2.146):

W =

∞

d(~ω)

= z

α~ω

p,1

1 −

p,2

p,1

1 +

p,2

p,1

= z

α~

(ω

p,1

− ω

p,2

)

p,1

+ ω

p,2

. (2.149)

It has to be noted that the energy emitted by transition radiation has a linear

dependence on the Lorentz factor γ of the incoming particle. This property can be

exploited for detector applications [Dolgoshein (1993)]. In the case of medium to

vacuum transition for which ω

p,1

= ω

and ω

p,2

= 0, Eq. (2.146) reduces to:

W = z

γα

~ω

Furthermore (see Section 27.7 in [PDB (2008)]), the average number of emitted

photons by transition radiation above a minimal threshold ~ω

is given by:

~ω>~ω

∞

~ω

d(~ω)

= z

γ~ω

~ω

− 1

. (2.150)

Thus, the overall number of photons grows as (ln γ)

, but it is constant above a ﬁxed

fraction of γ~ω

. For instance, over the typical ionization cluster energy between

(2–3) keV in gas detectors, at a lithium to vacuum boundary the number of emitted

transition-radiation photons by an electron

∗

of 5 GeV is hN

i ≈ 5.6 α ≈ 2.6 × 10

−2

with ~ω > 2 keV. The quantum yield is of the order α, whence the necessity of

having a large number of boundary crossings to increase it.

As seen before [i.e., Eq. (2.150)], the photon emission probability from a bound-

ary is quite low, even for low photon detectable energies. Usually, in order to increase

the probability of photon emission, many adjacent thin layers (foils) of material re-

ferred to as radiators are joined together. Thus, the number of surfaces can become

large enough to allow a detectable photon-emission. The limiting factor is the pho-

ton re-absorption inside the radiator itself. The foil thickness cannot be reduced

∗

The Lorentz factor is γ ≈ 10

for an electron of ≈ 5 GeV.

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134 Principles of Radiation Interaction in Matter and Detection

Fig. 2.46 The function G

from Eq. (2.148) as a function of

(adapted and republished

with permission from Artru, X., Yodh, G.B. and Mennessier, G., Phys. Rev. D 12, 1289 (1975);

) − 1].

beyond some minimal value, without compromising the creation of the formation

zone. It can be shown that [Artru, Yodh and Mennessier (1975); Fayard (1988)] the

energy emitted per unit of energy and unit of angle in a foil of thickness l

d(~ω) dΩ

foil

= 4 sin

d(~ω) dΩ

, (2.151)

where d

W /d(~ω)dΩ is given by Eq. (2.145) for the boundary crossing; 4 sin

is the interference term and ϕ

= (γ

−2

+ θ

+ Υ

)ωl

/(2c). In the relevant region

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Electromagnetic Interaction of Radiation in Matter 135

Fig. 2.47 Diﬀerent regions of the ω–γ plane regarding the single foil yield, in the vacuum case

(adapted and republished with permission from Artru, X., Yodh, G.B. and Mennessier, G., Phys.

Rev. D 12, 1289 (1975); Copyright (1975) by the American Physical Society). In this ﬁgure units

are chosen such that c = ~ = 1.

of integration over θ, the values of ϕ

are

≈

−2

+ Υ

ωl

Therefore, for thicknesses much lower than Z

given by

(ω) =

ω(γ

−2

+ Υ

)

, (2.152)

the yield is strongly reduced. For large ω’s, we have

≥ Z

(ω) ∼

2cγ

where Z

(ω) is referred to as the formation zone and can be understood as the

minimal depth inside a foil required by the electromagnetic ﬁeld carried by the

incoming charged-particle to reach a new equilibrium state inside the medium. The

ω–γ plane related to the formation zone is shown in Fig. 2.47 [Artru, Yodh and

Mennessier (1975)]. For Lorentz factors larger than γ

= ω

p,1

/(2 c), the frequency

cutoﬀ is no longer γω

p,1

as in the single surface case, but it is rather determined by

the formation-zone eﬀect. The condition to have enough yield becomes:

ω < min(γω

p,1

, ω

) (2.153)

where ω

= γ

p,1

= (ω

p,1

)/(2c) [Artru, Yodh and Mennessier (1975)]. For practi-

cal calculations with l

in units of µm and the plasma photon energy hν

p,1

in units

of eV, we have:

p,1

2 c

× 10

−4

hν

p,1

2 c~

× 10

−4

≈ 2.5 hν

p,1

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136 Principles of Radiation Interaction in Matter and Detection

Furthermore, from Eq. (2.142) assuming Z

≈ 0.5 for a foil with density ρ

units of g/cm

and l

in units of µm, we have:

~ω

= γ

(~ω

p,1

) =

(~ω

p,1

)

2~c

× 10

−4

≈

28.8

4 × 10

−5

× 10

−4

≈ 10

× (ρ

) [eV].

Experimentally, the formation zone eﬀect was observed by reducing the foil thickness

to the order of some µm’s.

2.3 Photon Interaction and Absorption in Matter

Ionizing processes embrace ﬁelds like nuclear, atomic, solid state, molecular

physics. They aﬀect the kinetic energy of incoming particles. Usually, these par-

ticles are not removed from the incoming beam, except when their kinetic energy

is fully absorbed in matter or (as discussed later) a shower generation process is

initiated.

In sharp contrast with the behavior of charged particles, any beam of monochro-

matic photons traversing an absorber exhibits a characteristic exponential reduction

of the number of its own photons traveling along the original direction. The reason

is that, in processes of photon scattering or absorption, each photon is individually

removed from the incoming beam by the interaction.

Let us consider a monochromatic photon beam of initial intensity

††

. In ad-

dition, let σ

a,tot

be the total photon atomic cross section for either scattering or

absorbing photons with energy equal to the beam energy. In the passage through a

thickness dx

of a medium, the number of removed photons

‡‡

per unit of time −dI

is proportional to the photon beam intensity I

at depth x

and to the number of

target atoms per unit of volume n

of the traversed material, i.e.,

−dI = I

rem

where P

rem

= n

a,tot

is the probability for a photon removal in the thickness

. In addition, we have

−dI = I

a,tot

= I

att,l

As a consequence, we obtain

= −µ

att,l

⇒

−µ

att,l

⇒ ln

= −µ

att,l

and, ﬁnally,

I = I

exp [−(µ

att,l

x)] . (2.154)

††

The intensity is given by the number of photons per unit of time impinging onto the absorber

surface.

‡‡

These photons can be fully or partially absorbed so that they have no longer their initial energy

and initial incoming direction.

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Electromagnetic Interaction of Radiation in Matter 137

The coeﬃcient

att,l

= n

a,tot

[cm

−1

] (2.155)

is the so-called linear attenuation coeﬃcient with σ

a,tot

in cm

/atom. In Eq. (2.155),

the number of atoms per cm

) of the traversed material is given by (ρN)/A

[Eq. (1.39)], where ρ is the material density in g/cm

, N is the Avogadro number (see

Appendix A.2), A is atomic weight (see page 14 and Sect. 1.4.1) of the material. In

this section as usual, Z indicates the atomic number (Sect. 3.1) of the material. By

introducing the absorber density ρ, we get:

att,m

a,tot

−1

], (2.156)

i.e., the so-called mass attenuation coeﬃcient. If the absorber is a chemical com-

pound or a mixture, its mass attenuation coeﬃcient µ

att,m

can be calculated from

the mass attenuation coeﬃcients of its constituent elements µ

att,m,i

using the

weighted average

att,m

att,m,i

where w

is the proportion by weight of the ith constituent element [Hubbell (1969)].

As mentioned above, the photon interaction on atoms or atomic electrons in

matter results in a change of the incoming photon energy and/or of the scattered-

photon direction. Atomic electrons can be emitted following the full or partial ab-

sorption of the primary photon. Apart resonance eﬀects at frequencies related to

Fig. 2.48 Absorption curve for X-rays in Pb as a function of incident photon wavelength and

energy (see for instance [Marmier and Sheldon (1969)]), showing the characteristic absorption

edges (see Table 2.12).

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138 Principles of Radiation Interaction in Matter and Detection

atomic or nuclear transitions, the main competing and energy dependent processes

contributing to the total cross section are:

• the photoelectric eﬀect, in which the interaction occurs with the entire

atomic electron cloud and results in the complete absorption of the pri-

mary photon energy;

• Thomson and Compton scattering on atomic electrons at photon energies

so that the electron binding energies can be neglected and electrons can be

treated as quasi-free;

• pair production, in which the photon incoming energy is high enough to

allow the creation of an electron–positron pair in the Coulomb ﬁeld of an

electron or a nucleus.

The photoelectric process dominates at low energies, i.e., below 50 keV for alu-

minum and 500 keV for lead absorb er. As the energy increases, between 0.05 and

15 MeV for aluminum and between 0.5 and 5 MeV for lead, the main contribution

to the attenuation coeﬃcient comes from Compton scattering. At larger energies,

pair production becomes the dominant mechanism of photon interaction with mat-

ter. The photon can be scattered or absorbed by the nucleus. The photonuclear cross

section is a measurable eﬀect. However, this kind of process is not easily treated

for systematic calculations due to a number of factors. Among these factors, we

have both A and Z, and sensitivity to the isotopic abundance. Reviews of the γ-

ray interaction processes and practical coeﬃcients tables can be found in Chapter 2

of [Marmier and Sheldon (1969)], and [Hubbell (1969); Messel and Crawford (1970);

Hubbell and Seltzer (2004)] and references therein. At present, the tabulations of

mass attenuation coeﬃcients are also available on the web (see Sect. 2.3.5).

2.3.1 The Photoelectric Eﬀect

When the energy hν is larger than the binding energies (B

) of atomic electrons,

photons can be completely absorbed in the interaction with an atom, which, in

turn, emits an electron raised into a state of the continuous spectrum. This eﬀect

is called photoelectric eﬀect.

The interaction involves the entire electron cloud, rather than the individual

(corpuscular) electron. Furthermore, the atom as a whole takes up the quite small

recoil energy to preserve the momentum and energy conservation. Thus, the kinetic

energy K

of the electron after leaving the atom is determined by the equation:

= hν − B

. (2.157)

Since a free electron cannot absorb a photon, we should expect that the photoelectric

absorption probability is larger for more tightly bound electrons, i.e., for K-shell

electrons. In fact, for incoming photon energies larger than K-shell energies, more

than about 80% of the photoelectric absorption occurs involving the emission of

K-shell electrons (see for instance Chapter V, Section 21 in [Heitler (1954)]). If

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Electromagnetic Interaction of Radiation in Matter 139

the photon energy is lower than the binding energy of a shell (see Tables 2.11 and

2.12), an electron cannot be emitted from that shell. Therefore, the absorption

curve exhibits the characteristic absorption edges, whenever the incoming photon

energy coincides with the ionization energy of electrons of K, L, M, . . . shells. In

addition (see Fig. 2.48), the electron shells (except the K-shell) have substructures

with slightly diﬀerent binding energies, which result in close absorption edges (3 for

the L-shell, 5 for the M-shell, etc).

The binding energy depends on the atomic number Z and the electron shell: it

decreases, as proceeding towards the outer shells, according to these approximate

formulae for the K, L, M shells, respectively:

(K) ≈ Ry(Z − 1)

[eV],

(L) ≈

Ry(Z − 5)

[eV],

(M) ≈

Ry(Z − 13)

[eV],

where Ry = 13.61 eV is the Rydberg energy. These formulae are in agreement with

the values given in Tables 2.11 and 2.12 within ± (3–7)% for K-shell absorption

edges and within ±10% for L-shell absorption edges.

An exact theoretical calculation of the photo electric eﬀect presents diﬃculties

and, thus, empirical formulae are used for computing the total (τ

) and K-shell

cross sections per atom. However, an estimate of the K-shell photoelectric cross

Table 2.11 Energies of the absorption edges above

10 keV for elements with Z up to 68 from [Hubbell

(1969)].

Element Z K-edge Element Z K-edge

keV keV

Ga 31 10.368 Sn 50 29.195

Ge 32 11.104 Sb 51 30.486

As 33 11.865 Te 52 31.811

Se 34 12.654 I 53 33.166

Br 35 13.470 Xe 54 34.590

Kr 36 14.324 Cs 55 35.987

Rb 37 15.202 Ba 56 37.452

Sr 38 16.107 La 57 38.934

Y 39 17.038 Ce 58 40.453

Zr 40 17.999 Pr 59 42.002

Nb 41 18.987 Nd 60 43.574

Mo 42 20.004 Pm 61 45.198

Tc 43 21.047 Sm 62 46.849

Ru 44 22.119 Eu 63 48.519

Rh 45 23.219 Gd 64 50.233

Pd 46 24.348 Tb 65 52.002

Ag 47 25.517 Dy 66 53.793

Cd 48 26.716 Ho 67 55.619

In 49 27.942 Er 68 57.487