Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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16 2 Particle Interactions with Matter and Detectors
•Therelativistic rise of dE=dx in the logarithmic term is due to the relativistic
increase in the laboratory system of the transverse component of the electric field,
E
?
, by a factor .SinceE
k
remains unchanged, the field flattens and extends so
that distant collisions increase as ln.ˇ/. In dense materials, the relativistic slope
is reduced because of the effects of density.
•Thedensity effect. The relativistic effect implies that at high ˇ, target electrons
with large values of b are involved in the interaction; however, these electrons
are screened by electrons closer to the projectile particle trajectory. This effect is
larger in materials of high density (solids and liquids) than in gases. In solids and
for high values of ˇ, the density effect reduces the dE=dx relativistic rise by
about a factor of two.
•Theshell correction takes into account the effects that occur when the speed of
the incident particle is comparable or smaller than the atomic electron orbital
speed. In this case, the approximation of a stationary electron with respect to the
incident particle is not as good. However, the correction that is derived from it is
relatively small.
Figure 2.2a illustrates the behavior of the energy loss as a function of ˇ of the
incident particle; some definitions are given (for instance, the energy loss at the
minimum ionization and the relativistic rise). Note that from (2.10), the energy loss
only depends on ˇ, or better, since relativistically p D M v D Mˇc, from
ˇ D p=Mc.FromEq.2.10, it is then possible to formulate the so-called “scaling
laws” which allow one to calculate the energy loss of a particle with mass m
1
,energy
E
1
and electric charge z
1
starting from m
2
, E
2
, z
2
:
dE
2
dx
.E
2
/ '
z
2
2
z
2
1
dE
1
dx
E
2
m
1
m
2
: (2.11)
Figure 2.2b shows the difference in energy loss in liquid hydrogen .Z=A D 1/,
gaseous materials (He, Z=A D 0:5) and in solid materials with Z=A ' 0:5.
It can be observed that the specific energy loss at the minimum ionization is
.dE=dx/
min
' 1:5 MeV g
1
cm
2
, while it can be considered approximately
constant at high energies, .dE=dx/ ' 2 MeV g
1
cm
2
.
The formula (2.10) can be integrated in order to determine the range,thatis,the
total path length of a particle that loses energy only through ionization. The result is
shown in Fig. 2.3.
I
Fig. 2.2 (a) Energy loss through ionization for
˙
mesons in copper. The general behavior is
shown together with some definitions and corrections due to the density effect (responsible for
the smaller relativistic rise) and two different approximations at low energies. (b) Mean energy
loss in liquid hydrogen (bubble chamber), gaseous helium, carbon, aluminum, iron, tin, and lead.
The horizontal scale is in ˇ units, which is independent of the incident particle type. The
underlying scales indicate the momentum for ; and p, respectively. The energy loss curves
show a minimum for ˇ D 3,thatis,pc ' 3Mc
2
[P08]
2.2 Passage of Charged Particles Through Matter 17
18 2 Particle Interactions with Matter and Detectors
0.05 0.10.2 0.50.2 1.0 5.02.0 10.0
Pion momentum (GeV/c)
0.1 0.50.2 1.0 5.02.0 10.0 50.020.0
Proton momentum (GeV/c)
0.050.02 0.1 0.50.2 1.0 5.02.0 10.0
Muon momentum (GeV/c)
βγ= p / Mc
1
2
5
10
20
50
100
200
500
1000
2000
5000
10000
20000
50000
R/M (g cm
−2
GeV
−1
)
0.1
25
1.0
25
10.0
25
100.0
H
2
liquid
He gas
Pb
Fe
C
Fig. 2.3 “Range” of charged particles, normalized to the mass M of the particle (relationship valid
for >0:4GeV) in liquid hydrogen (bubble chamber), helium gas, carbon, iron and lead [P08] as
a function of the ˇ of the particle. For example, for a proton of 200 MeV energy, ˇ ' 0:2 (see
Fig. 2.2b) and we read R=M M ' 1 gcm
2
, equivalent to 1 cm of water. For 1 GeV protons,
the range is R ' 100 gcm
2
Hadron therapy with nuclei. The concept of the range of a particle has a very
important application in the medical field. Hadron therapy is the most recent
relative of conventional radiotherapy, which uses X-rays. Hadron therapy
instead uses beams of protons, carbon ions or neutrons. For instance, protons
accelerated to 200 MeV or carbon ions accelerated to 4,700 MeV may be
used to irradiate deep tumors by following the tumor contour with millimetric
precision, allowing one to preserve the surrounding healthy tissue.
The accelerated hadrons are able to destroy sick tissues mostly at the end
of their range in the body of the patient, where the tumor is situated. This is
evident from Fig. 2.2, where it can be seen that the energy loss for slowing
2.2 Passage of Charged Particles Through Matter 19
down particles (that is, at the end of range) is very high. A beam of charged
hadrons therefore releases the greatest part of its destructive energy on the
target tumor. The dose received at the tumor can therefore be very high while
the healthy tissue is saved. Hadron therapy was initially specially relevant
for tumors located in the cranial base or along the vertebral column and for
ocular cancers. Recently, hadron therapy has successfully treated pediatric
tumors, tumors residing in the central nervous system, prostate, liver, and of
the gastroenterological apparatus.
Many countries are investing in this anticancer tool. The clinics which
use proton accelerators largely dominate, and the number of new clinics
has rapidly increased in the last few years. In 2008, there was nearly the
same number of clinics in Europe (11), North America (8) and Japan (8),
and more than 16 clinics will enter into operation within the next 5 years
[2L07]. The total number of patients treated with protons was about 10,000 in
1993 and 50,000 in 2006, that is, the progression appears to be exponential.
Many excellent clinics exist or are in construction in Europe. A dedicated
heavy ion therapy clinic is being constructed in Heidelberg (Germany). The
construction of ETOILE, the French National Hadron Therapy Centre, was
approved by the French Ministries of Health and Research in 2006. This
carbon therapy clinic will be built within the framework of a Public – Private
Partnership near Lyon. The TERA Foundation [2w1] aims at the development
of the techniques based on the use of hadrons and, more generally, of the
applications of physics and computer science for medicine and biology. In
particular, in Italy, since 2001, the South National Laboratories of INFN
are operating a beam of 62 MeV protons used to treat ocular melanoma
and other non-deep tumors. In 2003, the TERA Foundation completed the
specifications and the technical design of the CNAO, the Italian National
Centre for Oncologic Hadron Therapy aiming at the treatment of deep tumors
with protons and carbon ions, which will be built in Pavia.
An appreciable fraction of the energy lost through ionization can be delivered
more “violently” to some electrons which consequently acquire a relatively high
energy and therefore have a relatively longer path length: they are the so-called
knock-on electrons and they have enough energy to ionize (they are the so-called ı
rays along the path of a charged particle). The number of ı rays increases with the
energy of the primary particle. Fluctuations in the energy loss through ionization are
mainly due to the small number of energetic knock-on electrons. Some detectors, for
example, the Nuclear Track detectors, are sensitive to the Restricted Energy Loss
(REL), that is, the energy deposited in a cylinder, having for an axis the direction of
the particle of about 100
˚
A diameter, corresponding to ı rays with less than 200 eV
energy.
20 2 Particle Interactions with Matter and Detectors
2.2.3 Bremsstrahlung
The bremsstrahlung is the emission of a photon from an electron deflected by
a nucleus; it is a process due to the electromagnetic interaction that produces a
large energy loss. The bremsstrahlung dominates the energy loss with respect to the
energy losses through ionization and excitation for high ˇ. Given the small mass,
this already happens for a few tens of MeV in the case of electrons in lead and
hundreds of MeV for lighter materials; for muons, it only becomes important for
energies larger than 0.5 TeV.
In Sect. 4.6, the Feynman diagrams will be illustrated; we shall obtain a qual-
itative and intuitive dependence of the probability (or better, of a quantity that we
shall call the cross-section) of the bremsstrahlung process, resulting in Z
2
˛
3
EM
,
where Z is the atomic number of the medium nuclei. The process can be seen as
a slowing down of the incident electron caused by the nuclear Coulomb field: the
amplitude of the emitted radiation is inversely proportional to the electron mass. The
cross-section is proportional to 1=m
2
e
. Therefore, one has Z
2
˛
3
EM
=m
2
e
c
4
.Fora
higher mass particle, is smaller since the squared mass appears in the denominator.
It can be derived that the energy loss per path length unit is
dE
dx
rad
'
4N
a
Z
2
˛
3
EM
.„c/
2
m
2
e
c
4
E ln
183
Z
1=3
; (2.12)
where N
a
D number of atoms cm
3
D N
A
=A and N
A
is the Avogadro’s number.
The logarithmic term has its origin in the “screening” of the nucleus from the atomic
electrons and therefore the cross-section is limited. The mathematical demonstration
and more precise formulas are given in [J99].
Figure 2.4a shows how for electrons the energy loss through radiation linearly
increases with the electron energy. Besides, note that for E 20 MeV, the energy
loss through radiation is higher than that through ionization. The critical energy is
defined as the value for which the energy loss through radiation is equal to that of
the energy loss through ionization (the definition given by Rossi [L87] is slightly
different). An approximate formula for the electron critical energy in materials with
a different Z is that of Bethe–Heitler, that is,
E
c
' 1600 m
e
c
2
=Z: (2.13)
Critical energy values for some material elements are given in Table 2.1.
For energies much higher than the critical energy, energy loss through radiation
is practically the only one to be considered. In this situation, the integration of (2.12)
gives
E D E
0
e
x=L
rad
(2.14)
where E
0
is the initial energy, E is the energy after a thickness x of material.
The radiation length, L
rad
, is the length after which the energy E
0
of the incident
2.2 Passage of Charged Particles Through Matter 21
ab
Electron energy (MeV)
Muon energy (GeV)
Fig. 2.4 (a) Electron energy loss in copper as a function of the electron energy. The excitation and
ionization contribution (ionization) remains roughly constant with increasing energy. Instead, the
term due to the energy loss through radiation (bremsstrahlung) increases. The intersecting point of
the two curves defines the critical energy. In this figure, it corresponds to about 20 MeV. (b) Muon
energy loss in hydrogen, iron and uranium as a function of the muon energy. Energy loss through
excitation and ionization is indicated only for the iron (Fe ion). The value of the critical energy is
found at a few hundred GeV. The different behavior of the muons in comparison to the electrons is
due to the mass difference: m
' 200 m
e
[P08]
Table 2.1 Radiation length,
path length (radiation length
divided by the medium
density) and critical energy in
various material absorbers
Medium L
rad
(g cm
2
)
L
rad
(cm) E
C
(MeV)
Air 36:20 30050 83
H
2
O 36:08 36:1 93
Pb 6:37 0:56 9:5
Cu 12:86 1:43 25
Al 24:01 8:9 51
Fe 13:84 1:76 27:4
electron is reduced to E
0
=e,wheree is Nepero’s constant. An approximate formula
for the radiation length L
rad
(often indicated as X
0
)is
X
0
D L
rad
'
716:4 Œgcm
2
A
Z.Z C1/ ln.287=
p
Z/
: (2.15)
It must be emphasized that the energy loss through radiation may suffer strong
fluctuations from the mean value given in (2.12). The number of photons and their
energy fluctuate as well. The probability to emit a photon of high energy is much
smaller than that to emit one of low energy.
The bremsstrahlung may also happen on target electrons. The considerations
made on the electron energy loss through radiation are also valid for other higher
mass charged particles with an equivalent value of ˇ.
22 2 Particle Interactions with Matter and Detectors
2.3 Photon Interactions
The behavior of photons in matter is very different from that of charged particles.
Indeed, the photons are not subject to the numerous inelastic collisions with atomic
electrons. The main photon interactions are the photoelectric effect, Compton
scattering (including Thomson and Rayleigh collisions) and pair production. In
general, photons are more penetrating (in matter) than charged particles; a beam
of photons is not degraded in energy, but is attenuated in intensity according to the
formula
I.x/ D I.0/ e
x
(2.16)
where is the photon attenuation coefficient
D N
a
D N
A
=A (2.17)
where N
A
D Avogadro’s number, D medium specific mass, A D molecular or
atomic weight, N
a
D atomic density, and D the total cross-section. The mean
free path D 1= of photons as a function of their energy and for various materials
is given in Fig. 2.5.
2.3.1 Photoelectric Effect
In the photoelectric effect, a photon is absorbed by an atomic electron with the
consequent emission of an electron, e
! e
. To conserve momentum, the
photoelectric effect can only happen with bound electrons; the rest of the atom
Photon energy
100
10
10
–4
10
–5
10
–6
1
0.1
0.01
0.001
10 eV 100 eV 1 keV 10 keV 100 keV 1 MeV 10 MeV 100 MeV 1 GeV 10 GeV 100 Ge
V
Absorption length λ (g/cm
2
)
Si
Fe
Pb
Sn
Fig. 2.5 Photon mean free path (in g cm
2
) as a function of photon energy for various elemental
absorbers [P08]. The mean free path is the inverse of the attenuation coefficient (2.17)
2.3 Photon Interactions 23
Photon Energy
Cross section, barns / atom
Cross section, barns / atom
10 mb
1 b
1 kb
1 Mb
σ
coherent
σ
incoh
σ
nuc
κ
N
κ
e
σ
p.e.
10 eV 1 keV 1 MeV
1 GeV 100 GeV
Photon Energy
1 Mb
1 kb
1 b
10 mb
10 eV 10 keV 1 MeV 1 GeV 100 Ge
V
σ
coherent
σ
incoh
Carbon (Z = 6)
− experimental σ
tot
Lead (Z = 82)
− experimental σ
tot
σ
p.e.
σ
nuc
κ
N
κ
e
ab
Fig. 2.6 Total cross-section (circles at the top) for photons (a) in carbon and (b) in lead as a
function of the photon energy. In the figure are also shown the partial cross-sections for the
following processes:
p:e:
for the photoelectric effect on atomic electrons;
coherent
for elastic
collisions on atoms (Rayleigh scattering);
incoh
for Compton scattering on electrons;
N
for pair
production in a nuclear field;
e
for pair production in the field of an electron;
nuc
for photon
absorption in nuclei [P08]
recoils. The energy of the emitted electron is given by E
e
D h h
0
,whereh
0
is
the electron binding energy. Figure 2.6 shows the cross-section for the photoelectric
effect as a function of photon energy for C and Pb elements. The cross-section
strongly decreases with increasing energy and becomes very small for energies
larger than 100 keV. One should also note the series of peaks corresponding to the
ionization energy of the K-shell electrons
1
and to the higher order shell electrons,
that is, L, M, etc.
The calculation of the photoelectric cross-section is complicated; for energies
greater than the K-edge and smaller than m
e
c
2
, the following approximate formula
is valid, that is,
pe
' 4˛
2
EM
p
2Z
5
0
.m
e
c
2
=h/
7=2
(2.18)
per atom, with
0
D 8r
2
e
=3 D 6:65 10
25
cm
2
D Thomson cross-section,where
r
e
is the electron classical radius and ˛
EM
D 1=137. The Thomson cross-section is
the cross-section for the elastic process e
! e
for energies approaching zero.
One should note the dependence from Z
5
of the traversed medium.
2.3.2 Compton Scattering
The Compton scattering consists of an elastic collision of a photon with an electron,
e
! e
. The electrons in ordinary matter are bound electrons; if the incident
1
The peak due to photons with energy slightly higher than the binding energy of the K-shell atomic
electrons is called the K-edge.
24 2 Particle Interactions with Matter and Detectors
Fig. 2.7 Energy distribution
of the recoiling Compton
electrons for various energies
h of the incident photon
hν= 1 .0 MeV
hν= 1.5 MeV
hν= 0.5 MeV
Relative intensity
0 0.5 1.0 1.5
Electron energy (MeV)
photon has an energy much higher than that of the bound electrons, these can
be considered as free. The kinematics of Compton scattering gives the following
expression for the photon energy after the collision, h
0
,thatis,
h
0
D
h
1 C .1 cos /
(2.19)
where Dh=m
e
c
2
. Quantum electrodynamics allow one to compute the Compton
scattering cross-section (Klein–Nishina formula)
d
d˝
D
r
2
e
2
1
Œ1 C .1cos /
2
1 Ccos
2
C
2
.1 cos /
2
1 C.1 cos /
(2.20)
where r
e
is the electron classical radius.
The integration of the Klein–Nishina formula gives the total cross-section of
Compton scattering as shown in Fig. 2.6 (
incoh
). It should be noted that such a
process is dominant in the few tens of keV region for C and a few MeV for Pb.
To evaluate the energy response of some detectors, for example, scintillation
counters, it is important to know the energy distribution of the recoiling electrons in
Compton scattering. This distribution is shown in Fig. 2.7 for different energies of
the incident photons. One should note that the maximum in intensity corresponds to
the maximum energy allowed by the kinematics
T
max
D h
2
1 C 2
(2.21)
(it is referred to as the Compton edge). In the classical low energy limit, the Klein–
Nishina formula reduces to the Thomson formula,
0
D 8r
2
e
=3.
2.4 Electromagnetic Showers 25
The Rayleigh scattering (denoted
coherent
in Fig. 2.6) is the elastic diffusion of a
photon on an atom taken as a whole. It is referred to as a coherent scattering. Given
the high atomic mass, no energy is transferred to the medium.
2.3.3 Pair Production
In the pair production process, a photon converts into an e
C
e
pair: C Z !
Z C e
C
C e
,whereZ is an atomic nucleus or an electron. This reaction has a
threshold energy of 2m
e
c
2
D 1:022 MeV. The lowest order Feynman diagram is
similar to the bremsstrahlung one, as will be explained in Chap. 4.
The differential cross-section has a complicated mathematical form. The more
common practical formula is that of Bethe–Heitler; the integrated cross-section
is shown in Fig. 2.6, and is indicated as
N
and
e
. Note that the pair production
process in the Coulomb field of atomic nuclei .
N
/ dominates for energies of the
incident photon larger than a few MeV. For higher energies, h 137 m
e
c
2
Z
1=3
,
one can use the formula obtained considering a complete screening of the atomic
electrons, that is,
N
' aZ
2
˛
EM
r
2
e
7
9
Œln.183 Z
1=3
/ f.Z/
1
54
: (2.22)
The pair production on the electron (
e
curves of Fig. 2.6) is given by an equation
similar to that with Z D1. To take it into account, it is sufficient to substitute
Z
2
with Z.Z C 1/. From this latest formula, the following mean free path can be
obtained, that is,
1
pair
D N
a
n
'
7
9
Z.Z C1/N
a
r
2
e
˛
EM
Œln.183 Z
1=3
/ f.Z/: (2.23)
Note that the expression for
pair
is quite similar to that for the radiation length.
Indeed, one has
pair
'
9
7
L
rad
: (2.24)
2.4 Electromagnetic Showers
In matter, a high energy photon converts into an electron-positron pair; each
e
.e
C
/ is able to radiate energetic photons through bremsstrahlung. These radiated
photons can convert into pairs that, for example, radiate. In conclusion, one has an
electromagnetic shower (electromagnetic cascade) with a large number of photons,
electrons and positrons. The process continues up to when the energies of the