.
The Straggling function. Energy Loss Distribution of charged particles for thin silicon layers
The
collisions are casual, of course, but their
number per macroscopic path length is
generally large and this is the reason why
average quantities are generally used. One
of the most important quantity is the mean
energy loss per units length, often called
stopping power. Many theories have been
developed during the first half of the
twentieth century in order to characterize
this quantity. The correct
quantum‑mechanical calculation was first
described, around 1932, by Hans Bethe,
Bloch and other authors who gave the
formula:
with:
2π N_{a
}r_{e^}^{2 }
m_{e}c^{^2}= 0.1535 MeVcm^{2}/g 
β: v/c of
incident particle 
r_{e}:
electron radius (2.817 x 10^{13} cm) 
ρ:
density of absorbing material 
M_{e}:
electron mass 
γ: 
N_{a}:
Avogadro’s number 
δ:
density correction 
Z:
atomic number of absorbing
material 
C:
shell correction 
A:
atomic weight of absorbing
material 
I:
mean excitation potential 
z:
charge of incident particle 
W_{max}:
maximum energy transferable in a
single collision 
W_{max} can
be calculated using the equation:
where: s=m_{e}/M and η=βγ.
The mean excitation energy I depends by the orbital frequency of the absorbing material and there is not a precise formula to calculate that value. However values of I for several material have been deduced from measurements [3]. The last two terms in the parentheses of (2.3) are the density and the shell corrections and they have been inserted in the original formulation of Bethe‑Bloch in order to enhance the prediction of the formula at certain range compared to the experimental results. The density correction takes into account the effect of the electric field produced by incoming particles and is more evident at high velocity. Instead, the shell correction is noticeable when the velocity of incident particle is comparable to the orbital velocity of the bound electrons of the target material.
At this low energy some other
complicated effects come into play and
the BetheBloch formula breaks down.
When the velocity is comparable with the
speed of orbital electrons of the target
material the energy loss reach a maximum
depending on the sign of the charge
(Barkas effect) and for lower energy
drops sharply. At higher energy (that
means higher velocity) dE/dx is
dominated by the 1/β^{2} factor
and decreases until β ≅ 0.96c where
a minimum is reached. Particle with this
energy is usually indicated with the
name of minimum ionizing particle (MIP).
Increasing the energy the losses do not
increase so much due to the density
effect (Fermi plateau) until the
radiative components, such as the
Cherenkov radiation and Bremsstrahlung,
start to be relevant. The Cherenkov
radiation arises when a charged particle
in a medium moves faster than the speed
of light in that same medium (βc>c/n,
with n:
index of refraction): in such case an
electromagnetic shock wave is created,
just as an aircraft that moves faster
than sound.
Especially for light particles, such as
electrons or positrons at very high
energy, the Bremsstrahlung emission
represents the main energy loss
mechanism. The deflection and the
deceleration of the particle due to the
interaction with the nuclei of the
target cause the emission of photons;
this effect is much greater as lighter
is the particle (in fact the emission
probability by Bremsstrahlung varies as
the inverse square of the particle mass)
and higher is the atomic number of
target material. While ionization loss
rates rise logarithmically with energy ,
Bremsstrahlung losses rise linearly and
dominate at high energy (just only above
few tens of MeV in
most material for electrons). In Figure
2.1 it
is shown the mean energy loss (also
known as stopping power) for muons that
traverse a cupper target in the range of
few hundreds of keV to
tens of TeV.
Figure 2.1 Stopping
power for positive muons in Copper [4]
Correction to
BetheBloch for electrons and positrons
Electrons or positrons needs particular
consideration. First, their small mass
implies the possibility of a large
deflection due to a single collision
too; moreover the collisions are between
identical particles, so that the
calculation must take into account their
indistinguishability. As result the
maximum transferable energy in a single
collision becomes:
with T_{e}:
kinetic energy of the incident particle,
and the BetheBloch formula can be
rearranged as:
with:
where suffix “+”
means positrons and “”
means electrons.
References
 William R. Leo, Techniques for Nuclear and Particle Physics Experiments. Berlin and Heidelberg: Springer, 1987.
 Particle
Data Group PDG,
Passage of
particles through matter, Nuclear
and Particle Physics, vol. 33, no.
27, pp. 258270, July 2006.

P.V.Vavilov,
Ionization losses
of high energy heavy particles,
Soviet Physics JETP, 5:749, 1957.
 S. Meroli
et al.,
Energy loss
measurement for charged particles in
very thin silicon layers
, JINST,
6 P06013 doi:
10.1088/17480221/6/06/P06013
 Claude
Leroy, Pier Giorgio Rancoita,
Principles of radiation interaction
in matter end detection. Singapore:
World Scientific Publishing, 2004.
 International Commission on Radiation Units and Measurements. [Online]. http://www.icru.org/
 H.
Bichsel,
Straggling of
Heavy Charged Particles: Comparison
of Born HydrogenicWaveFunction
Approximation with FreeElectron
Approximation,
Phys. Rev. B1 (1970)
2854
 S. M. Sze, Kwok Kwok Ng, Physics of Semiconductor Devices. John Wiley & Sons, 2007.
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