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

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270 Principles of Radiation Interaction in Matter and Detection

Fig. 3.23 Parameter a as a function of the mass number A. The parameter a appearing in the

Fermi-gas level density formula was determined by comparison with the average spacings observed

in slow neutron resonances (◦) and evaporation spectra (×) (from [Rancoita and Seidman (1982)],

were used to describe the general trend of the nuclear process. The excitation energy,

stored by a nucleus, can be compared with the heat energy of a solid body or a

liquid. The subsequent emission of particles (nucleons or nucleon aggregates) is

similar to an evaporation process occurring, for instance, in a liquid. Following such

an analogy, a general statistical treatment was ﬁrst proposed by Weisskopf (1937) for

the evaporation of particles in highly excited heavy nuclei. The nuclear evaporation

temperature, T , of the excited nucleus is again deﬁned, in analogy to statistical

thermodynamics, as the derivative of the entropy S(E), which, in turn, dep ends on

the excitation energy E:

dS(E)

, (3.64)

where the excitation energy is measured from the nucleus ground state. The tem-

perature, deﬁned in this way, has the dimension of an energy and is equal to the

ordinary temperature times the Boltzmann constant k (Appendix A.2). For instance,

the room temperature of 300 K corresponds to ' 0.026 eV. Referring to nuclear re-

actions, the temperature is usually expressed in MeV. The entropy of a nucleus,

with an excitation energy between E and E + dE [Weisskopf (1937)], is deﬁned as

S(E) = ln g(E), (3.65)

where g(E) is the level density of the excited nucleus. The number of levels between

E and E + dE is given by g(E) dE. Although in statistical thermodynamics the

entropy is deﬁned using the logarithm of the number of states available to the

system, thermodynamic transitions are characterized by the diﬀerence between the

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Nuclear Interactions in Matter 271

entropies of the ﬁnal and initial states. In the case of nuclear interactions, the

entropy diﬀerence between two states with excitation energies E

and E

is given

by the quantity ln [g(E

)/g(E

)], which is dimensionless and independent of the

inﬁnitesimal energy-diﬀerence

∗

, dE. The energy-level density available for nucleons

can be computed by approximating the nucleus to a nuclear Fermi-gas discussed

in Sect. 3.1.5. However, it has to be noted that, once a particle is emitted, the

overall excitation energy varies so that a lower temperature must be ascribed to the

remaining nucleus. A detailed treatment of the evaporation process, based on the

continuum theory, is beyond the scope of this book and can be found in [Blatt and

Weisskopf (1952)]. In the following, we will present a simpliﬁed introduction to this

process as given in [Rancoita and Seidman (1982)].

Let us consider the emission of a neutron by an excited nucleus (A) with mass

numb er M

, excitation energy E and neutron binding energy B

< E. In the

continuum theory, the probability P (W, n) dW , that a neutron is emitted with a

kinetic energy between W and W + dW , is evaluated by considering the reverse

process:

+ n → M

where R

is the mass number of the nucleus after the neutron ejection. The proba-

bility is given by:

P (W, n) dW = k

(2S + 1) W

g (E

)

g (E

)

dW, (3.66)

where S and m

are the spin and the rest mass of the neutron, respectively; g (E

)

and g (E

) are the densities of energy-levels computed for the corresponding energies

= E − B

− W and E

= E; and, ﬁnally, k

is an appropriate constant. For

an excited nucleus approximated to a Fermi gas of nucleons with total angular

momentum L = 0 and equal amount of protons and neutrons, the density of energy

levels is:

g (E, L = 0) ≡ g (E) ∝

exp

√

, (3.67)

under the condition that E < E

1/3

and E > E

−1

, where E

is the Fermi

energy (see page 238). From Eqs. (3.64, 3.65, 3.67), we have:

g (E)

∂g (E)

∂E

= E

exp

−2

√





−2 exp

√

exp

√





= −

. (3.68)

∗

It was previously introduced to deﬁne the number of energy levels.

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272 Principles of Radiation Interaction in Matter and Detection

The values of the constant a (in MeV

−1

) appearing in Eq. (3.68) were experimentally

determined for various nuclei [Erba, Facchini and Saetta Menichella (1961)] and are

shown in Fig. 3.23 as a function of the mass number. For temperatures of a few

MeV’s (easily reached in nuclear interactions) and for a values given in Fig. 3.23,

the second term of Eq. (3.68) dominates the ﬁrst one. The former term can also

be neglected for nuclei with low mass number (M

), such as silicon. Therefore, the

excitation energy can be simply related to the nuclear temperature by:

E ≈ aT

Equation (3.66) can be rewritten by introducing the entropy as:

P (W, n) dW = k

(2S + 1) W exp[S (E

) − S (E

)] dW. (3.69)

The entropy S (E

) can be derived from the entropy at the energy E

via the Taylor

expansion:

S (E

) = S (E

) − (B

+ W )

dS (E)

E=E

+ ···

' S (E

) − (B

+ W )

where T is the temperature of the excited state with excitation energy E =

. Thus, Eq. (3.69) becomes:

P (W, n) dW = k

(2S + 1) W exp

−

+ W

dW. (3.70)

In Eq. (3.70), the kinetic energy distribution of emitted neutrons is an exponential

function of the emitted particle energy, which goes as W exp(−W /T ) and is peaked

at the neutron kinetic energy W

n,mp

= T .

So far, we have considered the evaporation of a neutron. In general, charged

particles of mass M can also be ejected in the de-excitation process. The previous

considerations are still valid if the Coulomb barrier potential-energy, V

, is added

to the binding energy, B

, and if no particle is emitted with kinetic energy W <

. Therefore, by introducing the potential barrier V

, Eqs. (3.69, 3.70) can be

approximated by:

P (W, M) dW = k

M (2S

+ 1) (W −V

) exp

−

+ V

×exp

−

W − V

dW, (3.71)

where S

and M are the spin and the rest mass of the emitted particle, respec-

tively, and k

is an appropriate constant. The Coulomb barrier shifts the most

probable kinetic-energy of the emitted charged particles from the nuclear excitation

temperature, i.e.,

M,mp

= T + V

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Nuclear Interactions in Matter 273

Fig. 3.24 Range distribution in emulsion data for black-tracks in events for which N

> 8

(from [Rancoita and Seidman (1982)], Copyright of the Societ`a Italiana di Fisica; see also [Ciok et

al. (1963)]). Both curves are for 50% p, 25% d and 25% He. In a) T = 7 MeV, V

= V

= 0;

in b) T = 7 MeV, V

= 4 MeV, V

= 5 MeV, V

= 8 MeV. The data with many heavy prongs

are consistent with high nuclear temperature and negligible Coulomb barrier.

It has to be noted that Eq. (3.71) reduces to Eq. (3.70) in case of neutron emis-

sion. We can integrate Eq. (3.71) to obtain the probability of emitting a particle of

mass M:

P (M) =

∞

P (W, M) dW

= k

M (2S

+ 1) exp

−

+ V

∞

(W − V

) exp

−

W − V

= k

M (2S

+ 1) exp

−

+ V

. (3.72)

Experimentally, the nuclear evaporation was also studied in high energy emul-

sion experiments, in order to investigate both nuclear temperatures and Coulomb

potential-barriers. It was observed that the number of black tracks is related to

the level of nuclear excitation. In [Powell, Fowler and Perkins (1959)], the data

regarding Z =1 prongs (i.e., p, d and t) with 2 ≤ N

≤ 12 and Z = 2 prongs

(i.e.,

He and

He) with 2 ≤ N

≤ 6 seem to indicate a temperature T ≈ 3 MeV

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274 Principles of Radiation Interaction in Matter and Detection

Fig. 3.25 Average number of π

◦

’s computed using Eq.(3.73) as a function of the incoming-particle

energy in GeV and > 2.5 GeV.

and Coulomb barriers ' 4 (for Z = 1) and ' 10 MeV (for Z = 2). While for Z = 1

prongs with N

≥ 13 and Z = 2 prongs with N

≥ 7, the experimental data seem to

indicate a larger temperature and lower Coulomb-barriers. In Fig. 3.24 [Ciok et al.

(1963)], the range distribution of black tracks, obtained in high energy interactions

on emulsion nuclei, is shown for the case N

> 8. In Fig. 3.24, both continuous

curves assume the black tracks are constituted by 50% p, 25% d and 25% He and

regard two diﬀerent sets of values for Coulomb barriers: the curve (a) was calcu-

lated for T = 7 MeV and negligible Coulomb-barriers for all the particles, while the

curve (b) was calculated for T = 7 MeV and Coulomb barriers of 4, 5, and 8 MeV

for p, d and

He, respectively. These multiple black prongs data are consistent with

a negligible Coulomb-barrier and T = 7 MeV.

3.3 Hadronic Shower Development and Propagation in Matter

The absorption in matter of high energy hadrons, which undergo strong intera-

ctions, develops as a cascade process. The interactions of incoming particles in the

absorber volume produce a wide spectrum of secondary particles, which have a vari-

ety of subsequent interactions with the nuclei of the medium. This process is similar,

in many respects, to the one of electromagnetic showers, which we have examined

in Sect. 2.4. However, the production mechanism is more complex. Analytical solu-

tions, even following simpliﬁed approaches, are not available [Amaldi (1981); Wig-

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Nuclear Interactions in Matter 275

mans (1987); Br¨uckmann et al. (1988); Wigmans (1988); Brau and Gabriel (1989);

Wigmans (1991)]. Furthermore, the nucleon binding energy and the fraction of pro-

tons among nucleons depend on the mass number (e.g., see [Blatt and Weisskopf

(1952); Born (1969)]). Thus, part of the incoming energy will not be deposited (and

not be detectable) by collision losses and become invisible energy, because of the

nuclear break-up. Its fraction, in turn, depends on the mass number.

At present, quantitative approaches to a detailed description of hadronic showers

use Monte-Carlo codes, written to simulate both the hadronic cascade development

and the performance of the hadron absorbing device, taking into account the physi-

cal pro cesses at work during the intra-nuclear cascading and the various many-body

ﬁnal states interactions (e.g., see [Fesefeld (1985); Aarnio et al. (1987); Br¨uckmann

et al. (1988); Anders et al. (1989); Brau and Gabriel (1989); Alsmiller et al. (1990);

Brun et al. (1992); Giani (1993); Gabriel et al. (1994)] and references therein).

Note that, in Sects. 3.3-3.3.3, the notations are the same as those used in

Sects. 2.4-2.4.3; for instance, A, Z, and N are the atomic weight (Sect. 1.4.1 and

discussion at page 220), the atomic number (Sect. 3.1) and the Avogadro constant

(see Appendix A.2), respectively.

3.3.1 Phenomenology of the Hadronic Cascade in Matter

The hadronic cascade is propagated through a succession of various inelastic intera-

ctions leading to particles production characterized by a multiplicity of secondaries

increasing logarithmically with the available energy, namely increasing as ≈ ln(s),

where s is the square of the energy available in the center-of-mass system (see page

12) divided by c

(c is the speed of light).

In a hadronic interaction, about half of the incoming energy is carried away by

leading particles and the remaining part is absorbed in the production of secon-

daries. Neutral pions amount, on average, to a third of the produced pions. The

nuclear processes

‡

, involved in the generation of the hadronic cascade, produce rel-

ativistic hadrons (mainly pions), nucleons and nucleon aggregates from spallation

(including those ones from the evaporation process), break-up and recoiling nu-

clear fragments. As discussed in Sect. 3.2.6, the energy distribution and the relative

abundance of the evaporated prongs, namely p, n, d, t,

He,

He, may be estima-

ted by following the continuum theory for nuclear reactions (e.g., see Chapter 8

in [Blatt and Weisskopf (1952)], and [Weisskopf (1937)]). For instance, an estimate

of the nuclear break-up as a result of a high-energy interaction, in particular on

silicon, can be found in [Rancoita and Seidman (1982)]. The experimental data in-

dicate large values for nuclear temperatures and a lowering of the Coulomb barrier

(e.g., see [Rancoita and Seidman (1982)] and references therein; see, also, [Powell,

Fowler and Perkins (1959)]). The spallation processes were widely investigated and

‡

A further discussion on these nuclear processes can be found, for instance, in Sections 2.3.2

and 2.3.3 of [Wigmans (2000)]

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276 Principles of Radiation Interaction in Matter and Detection

Table 3.3 Energy deposition in 5 GeV proton showers neglecting the π

com-

ponent (from [Leroy and Rancoita (2000)], see also [Wigmans (1987)]).

Absorber U Pb Fe

Ionization

(fraction due to spallation protons)

(%) 38 (0.70) 43 (0.72) 57 (0.74)

Excitation γ

(%) 2 3 3

Neutrons < 20 MeV

(%) 15 15 8

Invisible energy,

i.e.binding energy and target recoil

(%) 45 42 32

are treated in Sect. 3.2.5. An empirical formula, valid for energies > 50 MeV or mass

numb ers > 20, provides a description of experimental data on spallation reactions

and is given in [Rudstam (1966)] (see also Sect. 3.2.5). An important fraction of the

secondaries produced in the process is comprised of particles, mainly π

mesons,

which decay via electromagnetic interaction and generate their own electromagnetic

cascade. Thus, hadronic cascades always have an electromagnetic shower compo-

nent. The number of π

ﬂuctuates from event to event, depending on the processes

occurring in the various stages of the cascade development.

The average number (n

) of π

’s (Fig. 3.25) was estimated by Monte-Carlo

simulations [Baroncelli (1974)] for E in GeV and > 2.5 GeV (see Equation 3.26

of [Amaldi (1981)] or Equation 88 of [Leroy and Rancoita (2000)] and references

therein) as:

≈ 5 ln(E) − 4.6. (3.73)

The average fraction, f

(Fig. 3.26), of incoming-hadron energy deposited by elec-

tromagnetic cascades of secondary particles was also studied using Monte-Carlo

simulations [Ranft (1972); Baroncelli (1974)] and estimated by Fabjan and Lud-

lam (1982) and Wigmans (1987) as a function of the incoming-hadron energy E in

GeV for E & 1 GeV:

≈ (0.11 − 0.12) ln(E). (3.74)

To prevent the average electromagnetic fraction from becoming larger than 1 for

hadron energies in the ultra high energy range, f

can be parametrized, follow-

ing another approximate expression [Groom (1990); Gabriel et al. (1994); Groom

(2007)], by

' 1 − E(GeV)

−n

, (3.75)

where n is 0.15

. The two parametrizations give similar results over the energy range

between 5 and 200 GeV, where most of the data were collected.

The parameter n is estimated to be ≈ 0.17 in Ref. [Groom (2007)].

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Nuclear Interactions in Matter 277

Fig. 3.26 The average fraction, f

, of incoming-hadron energy (in GeV) deposited by electromagnetic cascades of secondary particles as computed

using Eqs. (3.74, 3.75).

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278 Principles of Radiation Interaction in Matter and Detection

The purely hadronic-component

of the hadronic cascade deposits its energy in

the calorimeter via several mechanisms

, which were studied by means of Monte-

Carlo simulations and, for instance, summarized in Table 3.3 for 5 GeV proton

showers neglecting the π

component [Wigmans (1987)]. This energy deposition

consists of:

• ionization (between 40% and 60%, decreasing with increasing A), the largest

contribution (∼ 70%) comes from spallation protons;

• generated neutrons (between 10% and 15%, decreasing with decreasing A);

• a few percents (∼ 3%) photons (from ﬁssion); and,

• nuclear break-up and recoil of nuclear fragments and target (between 30%

and 45%, decreasing with decreasing A in typically employed materials,

because of nuclear binding energies).

The increase of invisible energy with decreasing A is related to the boil oﬀ of nu-

cleons. For instance, the average binding energy of the last nucleon amounts to

≈ 6.4, 7.6 and 10.5 MeV in U, Pb and Fe nuclei, respectively [Wigmans (1987)]. The

energy, deposited by collision losses, decreases with increasing A, mainly because

the dominant contribution comes from spallation protons, whose fraction (among

the spallation nucleons) depends on the ratio Z/A. The ratio Z/A is larger in low-A

nuclei than in high-A nuclei. For

238

U nuclei, the nuclear ﬁssion mechanism in-

creases the number of neutrons emitted in the nuclear de-excitation process. Their

role in hadronic calorimetry with hydrogeneous readout was investigated by Wig-

mans (1987; 1988; 1991).

3.3.2 Natural Units in the Hadronic Cascade

As already discussed in Sect. 3.2.1, above ≈ 5 GeV up to very high energies, all

hadronic cross-sections are observed to rise slowly [Giacomelli (1976)]. The p–p data

show that the increase is consistent with a ln(s) dependence. In p–p collisions at high

energy, the elastic scattering process amounts to about (15–20)% of the total cross

section. Similar behavior and values of cross sections are observed in scattering from

neutrons. Other hadrons, for instance π’s, have similar energy behavior although

with diﬀerent values of cross-sections on proton. When cascading processes in matter

are considered, the most relevant quantity is the inelastic cross section, σ

hA(inel)

[Eq. (3.32)], in which the cross sections for both the coherent elastic scattering on

nucleus and the quasi-elastic scattering on individual nucleons are subtracted from

the total cross section on nucleus (for experimental data on the total cross sections

see [Murthy et al. (1975)]). Measurements of inelastic cross sections showed that

these ones are approximately independent of the particle momentum at high energy.

The hadronic-component is that one in which particles deposit their energy by non-

electromagnetic processes.

The reader can see, for instance, [Amaldi (1981); Wigmans (1987); Br¨uckmann et al. (1988);

Wigmans (1988); Brau and Gabriel (1989)].

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Nuclear Interactions in Matter 279

Table 3.4 Values of atomic number (Z), density (ρ), nuclear collision length [λ

gA(col)

and nuclear interaction length (λ

, λ

) for several materials commonly used to absorb

hadronic showers (data from [PDB (2008); PDG (2008)]).

Material Z ρ [g/cm]

gA(col)

[g/cm

] λ

[g/cm

] λ

[cm]

Be 4 1.848 55.3 77.8 42.10

C (amorphous) 6 2.00 59.2 85.8 42.90

Al 13 2.70 69.7 107.2 39.70

Si 14 2.33 70.2 108.4 46.52

Ti 22 4.54 78.8 126.2 27.80

Fe 26 7.87 81.7 132.1 16.77

Cu 29 8.96 84.2 137.3 15.32

Ge 32 5.323 86.9 143.0 26.86

Sn 50 7.31 98.2 166.7 22.80

W 74 19.30 110.4 191.9 9.95

Pb 82 11.35 114.1 199.6 17.59

U 92 18.95 118.6 209.0 11.03

The development of an hadronic cascade along the direction of motion of the

incoming particle (referred to as the longitudinal direction) is usually described in

units of interaction length (or nuclear interaction length) λ

. This latter is the

nuclear inelastic length for n–A inelastic interaction, i.e.:

Nρ σ

nA(inel)

, (3.76)

where A, N, ρ are the atomic weight, the Avogadro number, the density of the

material (in g cm

−3

), respectively. σ

nA(inel)

(in cm

) is the inelastic cross section

on the nucleus of atomic weight A, measured for incoming neutrons (see Section 6

of [PDB (2004)] and references therein). The weak energy dependence of the inelastic

cross-section and, to a ﬁrst approximation, the similar A-dependence also observed

(see Sect. 3.2.1; [Busza (1977)] and references therein) in p–nucleus and π–nucleus

justify the choice of that unit to measure the hadronic-cascade depths. The nuclear

interaction length λ

g A

in g cm

−2

is given by

g A

= ρλ

The nuclear interaction length is sometimes written using the approximate for-

mula [Amaldi (1981)]

≈ 35

1/3

cm. (3.77)

In a similar way, the nuclear collision length λ

A(col)

is related to the n–A to-

tal cross section. Values of λ

, λ

= ρλ

and λ

g A (col)

= ρλ

A(col)

, for commonly

employed hadronic-absorbers, are listed in Table 3.4; the nuclear collision and inte-

raction lengths for more than 300 substances are available at [PDG (2008)].