12
H. Bichsel and H. Schindler
At high photon energies, the electron-positron pair is emitted preferentially in
the forward direction and the absorption coefficient due to pair production can be
approximated by
μ = Nσ
(pair production)
=
7
9
1
X 0
,
where X 0 is a material-dependent parameter known as the radiation length (see
Sect. 2.4.1). More accurate expressions are given in Ref. [8]. Tabulations of calculated pair-production cross sections can be found in Ref. [36] and are available
online [24].
2.3 Interaction of Heavy Charged Particles with Matter
The main ingredient for computing the energy loss of an incident charged particle
due to interactions with the electrons of the target medium is the single-differential
cross section with respect to the energy transfer E in a collision. In this section, we
discuss the calculation of dσ/dE and its moments for “fast”, point-like particles.
To be precise, we consider particles with a velocity that is large compared to the
velocities of the atomic electrons, corresponding to the domain of validity of the
first-order Born approximation.
In the limit where the energy transfer E is large compared to the atomic binding
energies, dσ/dE approaches the cross section for scattering off a free electron. For
a spin-zero particle with charge ze and speed βc, the asymptotic cross section (per
electron) towards large energy transfers is given by [8]
dσ
dE
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
E 2
Rutherford cross section
1 − β
2 E
E max
=
dσ R
dE
1 − β
2 E
E max
.
(2.11)
Similar expressions have been derived for particles with spin 1 and spin 1/2 [8]. The
maximum energy transfer is given by the kinematics of a head-on collision between
a particle with mass M and an electron (mass m) at rest,
E max = 2mc
2 β
2 γ
2
1 + 2γ
m
M
+
m
M
2
−1
,
(2.12)
which for M m becomes E max ∼ 2mc 2 β 2 γ 2 .
These so-called “close” or “knock-on” collisions, in which the projectile interacts
with a single atomic electron, contribute a significant fraction (roughly half) to the
average energy loss of a charged particle in matter but are rare compared to “distant”
collisions in which the particle interacts with the atom as a whole or with a group of
H. Bichsel and H. Schindler
At high photon energies, the electron-positron pair is emitted preferentially in
the forward direction and the absorption coefficient due to pair production can be
approximated by
μ = Nσ
(pair production)
=
7
9
1
X 0
,
where X 0 is a material-dependent parameter known as the radiation length (see
Sect. 2.4.1). More accurate expressions are given in Ref. [8]. Tabulations of calculated pair-production cross sections can be found in Ref. [36] and are available
online [24].
2.3 Interaction of Heavy Charged Particles with Matter
The main ingredient for computing the energy loss of an incident charged particle
due to interactions with the electrons of the target medium is the single-differential
cross section with respect to the energy transfer E in a collision. In this section, we
discuss the calculation of dσ/dE and its moments for “fast”, point-like particles.
To be precise, we consider particles with a velocity that is large compared to the
velocities of the atomic electrons, corresponding to the domain of validity of the
first-order Born approximation.
In the limit where the energy transfer E is large compared to the atomic binding
energies, dσ/dE approaches the cross section for scattering off a free electron. For
a spin-zero particle with charge ze and speed βc, the asymptotic cross section (per
electron) towards large energy transfers is given by [8]
dσ
dE
=
2πz 2 (α ¯
hc)
2
mc 2 β 2
1
E 2
Rutherford cross section
1 − β
2 E
E max
=
dσ R
dE
1 − β
2 E
E max
.
(2.11)
Similar expressions have been derived for particles with spin 1 and spin 1/2 [8]. The
maximum energy transfer is given by the kinematics of a head-on collision between
a particle with mass M and an electron (mass m) at rest,
E max = 2mc
2 β
2 γ
2
1 + 2γ
m
M
+
m
M
2
−1
,
(2.12)
which for M m becomes E max ∼ 2mc 2 β 2 γ 2 .
These so-called “close” or “knock-on” collisions, in which the projectile interacts
with a single atomic electron, contribute a significant fraction (roughly half) to the
average energy loss of a charged particle in matter but are rare compared to “distant”
collisions in which the particle interacts with the atom as a whole or with a group of