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

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240 Principles of Radiation Interaction in Matter and Detection

ing a pair of bound nucleons, i.e., it requires ≈ (1–2) MeV. Even-even nuclei with

> 20 may require more than 2 MeV. Even-odd and odd-odd nuclei have many

lower excited states of about 100 keV. The electromagnetic radiation emitted by the

decay of lower excited energy states can be interpreted as given by a superposition

of electric dipole, quadrupole, octopole and other multi-pole transitions (see [Segre

(1977); Povh, Rith, Scholz and Zetsche (1995)]), which have diﬀerent angular di-

stributions and are indicated by E

, E

, etc.. The corresponding multipole

magnetic transitions are called M

, M

, etc.. The angular momentum and

parity conservations determine angular momentum and parity selection rules. The

treatment of the multipole transition probability is rather complicated and still

rather approximative (see, for instance, Chapter 9 in [Marmier and Sheldon (1969)]

and references therein).

When γ-decay is so much inhibited by forbidden transitions that the mean life-

time of the excited state exceeds 0.1 s, this long-lived nuclear state is called isomeric

state, as already mentioned at page 226. Isomeric states have typically large spin

diﬀerences and small energy diﬀerences from lower-lying energy levels. They are

labeled with a small m following the mass number value. For example, we have the

99m

Tc nuclide which has a lifetime of ≈ 6 h and decays by emitting a photon of

140.5 keV. This nuclide is largely used in nuclear medicine.

Some nuclear γ-ray and intensity standards from [IAEA (1991, 1998)], recom-

mended by the IAEA Coordinate Research Programme (CRP) for calibration of

γ-ray measurements, are given in Appendix A.8. Further X- and γ-ray data are

available from [IAEA (1991, 1998); ToI (1996, 1998, 1999); Helmer (1999); Helmer

and van der Leun (1999, 2000)].

3.2 Phenomenology of Interactions on Nuclei at High Energy

In Sect. 3.1, we treated the general properties and models of nuclei without consider-

ing any speciﬁc reaction on nuclei. These nuclear properties are due to short-ranged

nuclear forces, which, in turn, have their origin in strong interactions among nu-

cleons. Moreover, in addition to the electromagnetic interactions discussed in the

chapter on Electromagnetic Interaction of Radiation in Matter, hadronic particles

like protons (p) and pions (π) also undergo strong interactions on their passage in

matter. At high energies, among the resulting eﬀects of strong interactions, there

are particle creation and nuclear breakup.

Hadronic interactions on protons were widely studied in high-energy physics

experiments, during many decades, in order to investigate the fundamental con-

stituents of the so-called elementary particles, like the proton itself and the neu-

tron. Because the nucleus is made of nucleons, it has become standard to express the

relevant characteristics of the hadronic production on nuclei, in terms of ratios to

the corresponding quantities observed in hadronic production on nucleons, i.e., on

protons.

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Nuclear Interactions in Matter 241

Among the salient features of the hadroproduction (and photoproduction) on

nuclei from a few GeV up to very high energies, there is the coherent production [≈

(8–12)% of the total cross section]. The coherent reactions are interactions where

the nucleus remains in its ground state after the collision has occurred. Furthermore,

as discussed in Sect. 3.2.2, there is no evidence of a large intranuclear cascade on

multiparticle incoherent production on nuclei (see for instance [Feinberg (1972);

Busza (1977); Otterlund (1977); Halliwell (1978); Otterlund (1980)] and references

therein). For many years, experimental data on hadro- and photo-production on

nuclei were obtained using photographic emulsions exposed to high, very high and

ultra-high energy cosmic rays. In the last few decades, accelerator experiments were

carried out on both coherent and incoherent production on nuclei, as well.

It has to be noted that experiments on relativistic heavy-ion collisions were also

performed, but this topic

∗∗

is beyond the purpose of the present book.

3.2.1 Energy and A-Dependence of Cross Sections

We already discussed, in the chapter on Electromagnetic Interaction of Radiation

in Matter, how most of the electromagnetic diﬀerential cross sections are strongly

peaked in the forward direction and fall oﬀ as the energy increases. Away from the

forward direction, the high-energy behavior is mostly dominated by the hadronic

strong-interaction properties. However, in the forward direction, strong-interaction

processes like quasi-elastic and coherent production are relevant at high energies.

The dependence of the total and elastic cross sections of proton (p) and anti-

proton (¯p) on proton is shown in Fig. 3.9 versus the incoming laboratory momentum

in GeV/c [PDB (2002)]. The main diﬀerences on the cross-section behavior appear

at incoming particle momenta lower than 2–3 GeV/c. For incoming momenta <

1 GeV/c, the p–p scattering is dominated by the elastic scattering, which is still

a relevant fraction, ≈ (15–20)%, of the total cross section at higher energies. At

incoming energies of ≈ 200–300 GeV, the total p–p cross section is about 38–40 mb

and increases almost logarithmically with s, where s (see page 12) is the square

of the total energy in the center-of-mass system divided by c

. At high-energies,

the elastic scattering is a relevant fraction of the total cross section for p–n and

¯p–n (see Fig. 3.10), and, also, in π

–p (see Fig. 3.11) scatterings. Furthermore, the

interaction of p and ¯p on the lightest bound nuclide, the deuterium (d), indicates

again similar general features (Fig. 3.10).

Total cross sections on nuclei were measured by various experiments (see for

instance [Murthy et al. (1975); Busza (1977); Carroll et al. (1979); Roberts et al.

(1979)]). The experimental data show that, above 20 GeV, cross sections exhibit

a weak energy-dependence [Fernow (1986)]. Complex multi-step processes, asso-

ciated with an increased nuclear-matter transparency

‡‡

(Sect. 3.2.2) to particles

∗∗

The reader can see Section 16.4 of [Henley and Garcia (2007)], as a short introduction to this

topic, and recent reviews like, for instance, [Ludlam (2005); M¨uller and Nagle (2006)].

‡‡

The reader can see [Rancoita and Seidman (1982)] and references therein.

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242 Principles of Radiation Interaction in Matter and Detection

produced in the primary interaction on nucleons, can be partially responsible for

the weakening of the cross-section energy-dependence. When propagation in matter

is considered, the more relevant cross section is the so-called inelastic cross section,

which is deﬁned as:

A(inel)

= σ

A,tot

− σ

A,el

− σ

A,q−el

, (3.32)

where σ

A,tot

refers to the total cross section on the nucleus with atomic weight

A, σ

A,el

refers to coherent elastic scattering oﬀ the whole nucleus (Sect. 3.2.2)

and σ

A,q−el

refers to the quasi elastic process on individual nucleons, in which no

additional fast secondary particle is produced but the nucleus might be disrupted. It

was determined in neutron–nucleus (n–A) interactions that for A ≥ 9 [Rob erts et

al. (1979)]:

nA(inel)

' 41.2A

0.711

[mb], (3.33)

while the total cross section goes as ∼ A

0.77

. It has to be noted that, for the

nuclear radius value given by Eq. (3.12), we expect that the geometrical cross sec-

tion is approximatively proportional to r

(Sect. 1.4), i.e., proportional to A

2/3

(Sects. 3.1, 3.1.1). Furthermore, the proton–nucleus (p–A) and pion–nucleus (π–A)

inelastic cross sections are given by ([Busza (1977)] and references therein):

pA(inel)

' 46A

0.69

[mb], (3.34)

π A(inel)

' 28.5A

0.75

[mb]. (3.35)

Finally, in interactions on nuclei, the absorption cross section is deﬁned as

([Roberts et al. (1979)] and references therein):

A(absorption)

= σ

A(inel)

+ σ

A,q−el

= σ

A,tot

− σ

A,el

. (3.36)

3.2.1.1 Collision and Inelastic Length

In Sect. 1.4, we introduced the collision length λ

col

as the mean free path between

successive interactions occurring for processes whose total cross section is σ

tot

. In

the case of strong interactions on nuclei, we deﬁne the nuclear inelastic length (λ

inel

)

as the mean free path between successive inelastic collisions whose cross section is

given by Eq. (3.32). By rewriting Eq. (1.41), we have:

inel

Nρ σ

A(inel)

[cm] (3.37)

where A is the atomic weight, N is the Avogadro constant (see Appendix A.2),

A(inel)

is in cm

and ρ in g/cm

As discussed at page 220, to a ﬁrst approximation, the atomic weight, A (also referred to as

relative atomic mass, see page 15), of a nuclide is expressed by its mass number.

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Nuclear Interactions in Matter 243

Fig. 3.9 Total and elastic cross sections of p and ¯p on p as a function of the incoming beam momentum (experimental data and references are

available, also via the web, from [PDB (2002)]). The lines are to guide the eye.

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244 Principles of Radiation Interaction in Matter and Detection

Fig. 3.10 Total and elastic cross sections of p and ¯p on d and n as a function of the incoming beam momentum (experimental data and references

are available, also via the web, from [PDB (2002)]). The lines are to guide the eye.

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Nuclear Interactions in Matter 245

Fig. 3.11 Total and elastic cross sections of π

on p as a function of the incoming beam momentum (experimental data and references are available,

also via the web, from [PDB (2002)]). The lines are to guide the eye.

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246 Principles of Radiation Interaction in Matter and Detection

3.2.2 Coherent and Incoherent Interactions on Nuclei

In spite of the (large) numb er of nucleons involved, the interaction with nuclei is

more simple to deal with than interactions with a nucleon, when the nucleus remains

in its ground state after the interaction. This kind of interaction is called coherent

interaction on nucleus, i.e.,

h + T

→ h

∗

+ T

A coherent nuclear reaction is an interaction induced by a high energy particle, in

which all nucleons, whose amplitudes have strong interference-eﬀects, are partici-

pating. The theory of coherent reaction on nuclei was derived by Glauber (see, for

instance, [Glauber (1959)] and references therein).

The concept that particle production can occur diﬀractively, namely coherently,

from a nucleus, was ﬁrst introduced by Feinberg and Pomeranchuk (1956), who

pointed out that, at large incoming particle momenta, a minimum momentum

transfer, q

, can be suﬃcient to produce a ﬁnal hadronic state with a mass larger

than the incoming one, when the wavelength h/q

is comparable to or larger than

the nuclear size. Hence, owing to the uncertainty principle [Eq. (3.25)], the eﬀec-

tive dimensions of the region involved can be large compared with the target di-

mensions. This process was called the diﬀraction dissociation process or diﬀractive

reaction. Furthermore, the possibility of nuclear excitation in diﬀractive collisions

can be understood by considering that, if the exchanged momentum is capable of

changing the initial particle mass by several hundred MeV/c

(or a few GeV/c

it should easily allow the target nucleus to acquire a few MeV. Reactions in which

the recoil nucleus is a well-deﬁned excited state are called semi-coherent reactions

(e.g., [Stodolsky (1966)] and references therein), like for instance:

h +

C → h

∗

C(2

)

and, subsequently,

C(2

) →

C + γ(4.4MeV).

3.2.2.1 Kinematics for Coherent Condition

Let us consider the production reaction:

h + T

→ h

∗

+ R

. (3.38)

In this reaction,

h =

E/c,

= (M

c, 0) ,

∗

/c,

∗

= (E

/c, ~q)

are the four-momenta of the incoming particle, the target nucleus at rest, the

diﬀracted particle and the recoil nucleus, whose rest masses are m

, M

, m

∗

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Nuclear Interactions in Matter 247

and whose momenta are

h, 0,

∗

, ~q, respectively. From energy-momentum con-

servation, the four-momenta transfer squared, t (deﬁned at page 12), between h and

∗

and between T

and R

must be equal, namely:

t =

h −

∗

−

= T

+ R

− 2

= M

+ M

− 2M

. (3.39)

For a coherent reaction, i.e., when M

= M

and E

' q

/ (2 M

), we have

= M

and we can rewrite Eq. (3.39) as:

−t ' q

= q

+ q

⊥

, (3.40)

where |q

| and |q

⊥

| are the longitudinal and transverse three-momentum transfer,

respectively. Because the incoming momentum is along the longitudinal direction,

the transverse three-momentum transfer can be arbitrarily small (at the limit to

become zero). Thus, in a coherent reaction, the value of

≡ −t

min

(3.41)

is kinematically the minimum four-momentum transfer from the target to the re-

coiling nucleus to produce the particle h

∗

by an incoming particle h. Furthermore,

in coherent reactions the diﬀerential cross section, which depends on the four mo-

mentum transfer, is usually expressed as a function of the quantity t

deﬁned as:

≡ |t − t

min

| = q

⊥

. (3.42)

In order to estimate t

min

, we need ﬁrst to evaluate the limits on the momentum

transfer imposed by the coherent condition.

Let us consider the diﬀraction scattering in the framework of the optical model

(e.g. see [Roman (1965); Fisher (1968)]), ﬁrst developed for the elastic scattering. In

this model, the complete interacting system is assumed in complete stationary mode

and the time dependence of the wave function is omitted. The incident particle beam

moving in the positive z direction is described by a plane wave

= exp(ik

z),

where k

= p/~ is the wave vector, i.e., the particle momentum p in units of ~.

At large distances from the scattering center, the scattered wave is an outgoing

spherical wave described by

scat

= f(θ)

exp (ik

, (3.43)

where k

is the wave vector of the outgoing particle after elastic scattering, θ is the

scattering angle, r is radial distance from the scattering center, f (θ) is the scattering

amplitude, whose related angular diﬀerential cross section is given by:

dσ

dΩ

= |f(θ)|

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248 Principles of Radiation Interaction in Matter and Detection

The solution of Schr¨odinger’s equation, when the scattered wave is given by Ψ

scat

allows one to establish a relationship (the so-called optical theorem) between the

total cross section and the imaginary part of the scattering amplitude in the forward

direction:

Imf(θ = 0) =

4π

tot

. (3.44)

In addition, it can be shown (see, for instance, [Fisher (1968); Bingham (1970)]

and references therein) that, in order to produce coherently a diﬀracted wave on a

nuclear target, the phase diﬀerence between the outgoing diﬀracted wave and the

incident wave is ' (2r

)/~, where r

is the radius of the nucleus [Eq. (3.12)],

and must be lower than π. Then, we can have a constructive interference between

contributions from all positions, from the front to the back of the nucleus of diameter

. Therefore, the coherent condition for the interaction on a nucleus is satisﬁed

when:

≤ π ⇒ q

≤ ~

. (3.45)

By taking a factor ∼ 2 lower for the right hand side of Eq. (3.45), the coherent

condition is even better satisﬁed. Since numerically we have

~c ' 197.3 MeV fm,

= 139.6 MeV

(where m

is the rest mass of the pion), and

' 1.4 fm,

we can write the practical coherent condition, as:

≤

∼

4.4 m

4.8 M

1/3

∼

1/3

[MeV/c], (3.46)

where m

is in units of MeV/c

and the mass number of the target nucleus (M

)

is numerically approximated by the atomic weight (A) of the nuclide (see Sect. 3.1

and, also, the discussion at page 220)

∼ A.

Let us evaluate the value of t

min

for the reaction shown in Eq. (3.38) for a high

energy coherent scattering, induced by an incoming particle of momentum p. We

have for q

⊥

= 0:

min

= −q

h −

∗

= m

+ m

∗

− 2

−

+ 2p (p − q

), (3.47)

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Nuclear Interactions in Matter 249

where E −(q

/2M

) and p −q

are the energy and longitudinal momentum of the

diﬀracted hadron h

∗

, respectively. In a coherent scattering, q

is small

∗

and, thus,

the term (q

/2M

) can be neglected. Introducing Eq. (1.8), the expression (3.47)

becomes

−q

∼ m

+ m

∗

− 2

+ m

+ 2p

− 2p q

= m

∗

− m

− 2p q

= m

∗

− m

− 2p q

and, ﬁnally, by neglecting q

with respect to 2p q

, we have:

| − t

min

| = q

∼

∗

− m

. (3.48)

For instance, from Eq. (3.46), in interactions on C and Pb nuclei, the coherently

limited q

values are ≈ 61 and 24 MeV/c, respectively, within a factor 2. While from

Eq. (3.48) at an incoming particle momentum of 25 GeV/c, the largest masses of

particles coherently produced are ≈ 1.75 and 1.10 GeV/c

, respectively, within a

factor

√

The coherent condition given in Eq. (3.46) has to be applied also to the transverse

momentum transfer q

⊥

. Thus, the coherent production in a forward cone is limited

by:

coh

∼

1/3

∼

1/3

For instance, the coherent production on C and Pb nuclei at an incoming momentum

of 25 GeV/c is almost limited to the angles θ

coh

of ≈ 2.4 and 1 mrad, respectively.

3.2.2.2 Coherent and Incoherent Scattering

In this section, we will employ relativistic units used for calculations, i.e., all quan-

tities are express in units so that

~ = c = 1.

The masses are measured in GeV, i.e., the mass of the proton is ' 0.938 GeV. Be-

cause in the International System of Units ~c ' 0.1973 GeV fm, the conversion

factor of the unit of length to cm becomes 1.973 × 10

−14

cm. If the unit of mass

were chosen diﬀerently, for instance the proton mass, the conversion factor to cm

would be modiﬁed accordingly. Furthermore, for a compact notation the number of

nucleons in a nucleus is indicated with that for the atomic weight

A, instead of

the usual symbol M

The coherent elastic scattering on nuclei was ﬁrst introduced within the frame-

work of Glauber’s multiple scattering theory. This high-energy approach assumes i)

∗

The largest practical value of q

can be estimated using Eq. (3.46).

As discussed at page 220, to a ﬁrst approximation the atomic weight (also referred to as relative

atomic mass, see page 15) of a nuclide is expressed by its mass number.