230 6 Interactions of Charged Particles with Matter
Stopping power is the parameter used to describe the gradual loss of energy
of the charged particle, as it penetrates into an absorbing medium. Two classes
of stopping power are known: collision stopping power that results from charged
particle interaction with orbital electrons of the absorber and radiation stopping
power that results from charged particle interaction with nuclei of the absorber.
Stopping powers play an important role in radiation dosimetry. They depend on the
properties of the charged particle such as its mass, charge, velocity and energy as well
as on the properties of the absorbing medium such as its density and atomic number.
In addition to stopping powers, other parameters of charged particle interaction with
matter, such as the range, energy transfer, mean ionization potential, and radiation
yield, are also discussed in this chapter.
6.1 General Aspects of Energy Transfer from Charged
Particle to Medium
The discovery of energetic charged particle emission from radioactive materials in
1896 stimulated interest not only in the origin of the emitted particles but also in
how they were slowed down as they traversed matter. The theory of stopping power
played an important role in the development of atomic and nuclear models starting
with the α-particle scattering studies of Hans Geiger, Ernest Marsden and Ernest
Rutherford in 1908 and the classical stopping power theory developed by Niels Bohr
in 1913, and culminating with the quantum mechanical and relativistic theory of
stopping power proposed by Hans Bethe in the 1930s and refined by Ugo Fano in the
1960s. More recent developments introduced several additional secondary correction
factors to increase the accuracy of theoretical stopping power expressions; however,
the main theoretical foundations that early workers enunciated decades ago are still
valid today.
As a charged particle travels through an absorber, it experiences Coulomb inter-
actions with the nuclei and orbital electrons of absorber atoms. These interactions
can be divided into three categories depending on the size of the classical impact
parameter b of the charged particle trajectory compared to the classical atomic radius
a of the absorber atom with which the charged particle interacts:
1. Coulomb force interaction of the charged particle with the external nuclear field
of the absorber atom for b a (bremsstrahlung production).
2. Coulomb force interaction of the charged particle with orbital electron of the
absorber atom for b ≈ a (hard collision).
3. Coulomb force interaction of the charged particle with orbital electron of the
absorber atom for b a (soft collision).
Radiation collision, hard collision, and soft collision are shown schematically in
Fig. 6.1, with b the impact parameter of the particle trajectory and a the atomic
radius of the absorber atom.
6.1 General Aspects of Energy Transfer from Charged Particle to Medium 231
Fig. 6.1 Three different types of collision of a charged particle with an atom, depending on the
relative size of the impact parameter b and atomic radius a. Hard (close) collision for b ≈ a; soft
(distant) collision for b a; and radiation collision for b a
6.1.1 Charged Particle Interaction with Coulomb
Field of the Nucleus (Radiation Collision)
When the impact parameter b of a charged particle is much smaller than the radius
a of the absorber atom (i.e., b a), the charged particle interacts mainly with the
nucleus and undergoes either elastic or inelastic scattering possibly accompanied
with a change in direction of motion.
The vast majority of these interactions are elastic so that the particle is scattered
by the nucleus but loses only an insignificant amount of its kinetic energy to satisfy
the conservation of momentum requirement. However, a small percentage of the
scattering interactions are inelastic and may result in significant energy loss for the
charged particle accompanied by emission of x-ray photons. This type of interaction
is called bremsstrahlung collision. At a given particle acceleration, the probability
for this type of interaction is inversely proportional to the square of the mass of the
charged particle, making the bremsstrahlung production for charged particles other
then electrons and positrons essentially negligible.
6.1.2 Hard (Close) Collision
When the impact parameter b of a charged particle trajectory is of the order of the
radius a of the absorber atom (i.e., b ≈ a), the charged particle may have a direct
Coulomb impact interaction with a single atomic orbital electron and transfer to it a
significant amount of energy. The interaction is referred to as hard or close collision.
The orbital electron leaves the atom as a δ-ray, and is energetic enough to undergo
its own Coulomb interactions with absorber atoms. The maximum possible energy
transfer from a charged particle to an orbital electron (δ-ray) is discussed in detail in
232 6 Interactions of Charged Particles with Matter
Sect. 5.3. The number of hard collisions experienced by a charged particle moving
in an absorber is generally small; however, the energy transfers associated with hard
collisions are relatively large, so that the particle loses roughly 50% of its kinetic
energy through hard collisions.
The theories that govern hard collisions depend strongly on the characteristics
of charged particles and generally assume that the orbital electron (δ-ray) released
through a hard collision is free before and after the interaction, since the kinetic
energy transferred to it from the charged particle is much larger than its atomic
binding energy.
6.1.3 Soft (Distant) Collision
When the impact parameter b of the charged particle trajectory is much larger than
the radius a of the absorber atom (i.e., b a), the charged particle interacts with the
whole atom and the whole atomic complement of bound electrons. The interaction
is called a soft or distant collision. The energy transfer from the charged particle
to a given bound electron is very small; however, the number of these interactions
is large, so that approximately 50% of energy loss by a charged particle occurs
through these small-energy-transfer interactions that may cause atomic polarization,
excitation or ionization through removal of a valence electron. In the energy region
of soft collisions the expressions derived with a given theory are valid for all types
of charged particles including electrons and positrons.
6.2 General Aspects of Stopping Power
During its motion through an absorbing medium a charged particle experiences a large
number of interactions before its kinetic energy is expended. In each interaction the
charged particle’s path may be altered (elastic or inelastic scattering) and it may lose
some of its kinetic energy that will be transferred to the medium (collision loss) or
to photons (radiation loss). Each of these possible interactions between the charged
particle and orbital electrons or the nucleus of the absorber atoms is characterized by a
specific cross section (probability) σ for the particular interaction. The energy loss of
the charged particle propagating through an absorber depends on the characteristics
of the particle as well as the absorber.
The rate of energy loss (typically expressed in MeV) per unit of path length
(typically expressed in cm) by a charged particle in an absorbing medium is called
the linear stopping power (−dE/dx). Dividing the linear stopping power by the
density ρ of the absorber results in the mass stopping power S given in units of
MeV·cm2 ·g−1 . The stopping power is a property of the material in which a charged
particle propagates.
6.2 General Aspects of Stopping Power 233
In general, the average energy loss per unit path length −dE/d experienced by
the heavy particle is calculated by multiplying the cross section for a given energy loss
σni by the energy loss E ni and a summation over all possible individual collisions i
dE
− = Ni E ni σni , (6.1)
d i n
where Ni is the density of atoms i that can be expressed:
1. Either in number of atoms per unit volume resulting in −dE/d referred to as
the linear stopping power −dE/dx and representing energy loss per unit distance
traversed in the absorber. The typical units of linear stopping power are MeV/cm
or less common keV/µm.
2. Or in number of atoms per unit mass resulting in −dE/d referred to as mass
stopping power S = − (1/ρ) dE/dx and representing energy loss per g/cm2 of
material traversed in the absorber. The typical unit of mass stopping power is
MeV·cm2 ·g−1 .
With regard to charged particle interaction, two types of stopping power are known:
1. Radiation stopping power (also called nuclear stopping power) resulting from
charged particle Coulomb interaction with the nuclei of the absorber. Only light
charged particles (electrons and positrons) experience appreciable energy loss
through these interactions that are usually referred to as bremsstrahlung inter-
actions. For heavy charged particles (protons, α-particles, etc.) the radiation
(bremsstrahlung) loss is negligible in comparison with the collision loss.
2. Collision stopping power (also called ionization or electronic stopping power)
resulting from charged particle Coulomb interactions with orbital electrons of
the absorber. Both heavy and light charged particles experience these interactions
that result in energy transfer from the charged particle to orbital electrons through
impact excitation and ionization of absorber atoms.
The total stopping power Stot for a charged particle of kinetic energy E K traveling
through an absorber of atomic number Z is in general the sum of the radiation
(nuclear) stopping power Srad and collision (electronic) stopping power Scol , i.e.,
Stot = Srad + Scol . (6.2)
The collision stopping power Scol is further subdivided into two components: the soft
soft
(distant) collision stopping power Scol and the hard (close) collision stopping power
hard
Scol
Scol = Scol
soft
+ Scol
hard
(6.3)
The total stopping power is thus in general terms expressed as the following sum
Stot = Srad + Scol = Srad + Scol
soft
+ Scol
hard
(6.4)
234 6 Interactions of Charged Particles with Matter
6.3 Radiation (Nuclear) Stopping Power
As shown by the Larmor relationship (4.18), any time a charged particle is accelerated
or decelerated part of its kinetic energy is emitted in the form of bremsstrahlung
photons. The rate of bremsstrahlung energy dissipation is proportional to a 2 (the
square of the charged particle acceleration a) which in turn is proportional to (z Z /m)2
with z and m the atomic number and mass, respectively, of the radiating charged
particle, and Z the atomic number of the absorber target.
The bremsstrahlung intensity is thus linearly proportional to (z Z )2 and inversely
proportional to m 2 . As a consequence of the relatively large mass of heavy charged
particles, the bremsstrahlung yield produced by heavy charged particles such as
protons and α-particles in comparison with electrons and positrons is insignificant
and generally ignored.
Hans Bethe and Walter Heitler have shown in 1930s that the cross section for
emission of bremsstrahlung σrad has the same form in classical and quantum theory
and is proportional to
σrad ∝ αre2 Z 2 cm2 /nucleus , (6.5)
where
α is the fine structure constant e2 / (4πε
0 c) = 1/137
.
re is the classical electron radius e2 / 4πε0 m e c2 = 2.818 fm .
Z is the atomic number of the absorber target.
Table 6.1 provides expressions for σrad for various regions of incident electron kinetic
energy (E K )0 from the classical region where (E K )0 m e c2 all the way to the
extreme relativistic region where (E K )0 m e c2 /α.
The rate of bremsstrahlung production by light charged particles (electrons and
positrons) traveling through an absorber is generally expressed by the mass radiation
stopping power Srad (in MeV·cm2 ·g−1 ) given as follows
Srad = Na σrad E i , (6.6)
where
Na is the atomic density, i.e., number of atoms per unit mass: Na = N /m =
NA /A.
σrad is the total cross section for bremsstrahlung production given for various
energy regions in Table 6.1.
Ei is the initial total energy of the light charged particle, i.e., E i = (E K )0 +
m e c2 .
(E K )0 is the initial kinetic energy of the light charged particle.
6.3 Radiation (Nuclear) Stopping Power 235
Table 6.1 Total cross section for bremsstrahlung production and parameter Brad for various ranges
of electron kinetic energies
Energy range σrad cm2 /nucleon Brad = σrad / αre2 Z 2
Non-relativistic 16 2 2 16
αr Z (6.8)
(E K )0 m e c2 3 e 3
Relativistic
Complicated power series – (6.9)
(E K )0 ≈ m e c2
High-relativistic Ei 1 Ei 1
8re2 Z 2 ln − 8 ln − (6.10)
me c2 m e c2 6 m e c2 6
me c2 (E K )0
αZ 1/3
Extreme relativistic 183 1 183 1
m e c2 4αre2 Z 2 ln 1/3 + 4 ln 1/3 + (6.11)
(E K )0 αZ 1/3 Z 18 Z 18
Inserting σrad from Table 6.1 into (6.6) we obtain the following expression for Srad
NA
Srad = α re2 Z 2 E i Brad , (6.7)
A
where Brad is a slowly varying function of Z and E i , also given in Table 6.1 and
determined from σrad / α re2 Z 2 . As shown in Table 6.2, the parameter Brad has a value
of 16
3
for light charged particles in the non-relativistic energy range (E K )0 m e c2 ;
about 6 at (E K )0 = 1 MeV; 12 at (E K )0 = 10 MeV; and 15 at (E K )0 = 100 MeV.
Hans Bethe and Walter Heitler derived (6.7) theoretically. Martin Berger and Stephen
Seltzer have provided extensive tables of Srad for a wide range of absorbing materials.
As indicated in (6.7), the mass radiation stopping power Srad is proportional to:
1. NA Z 2 /A indicating a proportionality with the atomic number of the absorber
Z by virtue of Z /A ≈ 0.5 for all elements with the exception of hydrogen. The
higher is the atomic number Z of the absorber, the larger is the radiation stopping
power Srad and the larger is the radiation yield.
2. Total energy E i [(or kinetic energy (E K )0 for (E K )0 m e c2 ] of the light charged
particle.
3. Parameter Brad which is a slowly varying function of light charged particle total
energy E i and absorber atomic number Z , as shown in Table 6.2.
Figure 6.2 shows the mass radiation stopping power Srad for electrons in water, alu-
minum, and lead based on tabulated data obtained from the National Institute of
Standards and Technology (NIST). The Srad data are shown with heavy solid curves,
Table 6.2 Parameter Brad for various initial kinetic energies of light charged particles
Kinetic energy Classical 1 MeV 10 MeV 100 MeV
Brad ∼5.3 ∼6 ∼10 ∼15
236 6 Interactions of Charged Particles with Matter
Fig. 6.2 Mass radiation
stopping power Srad for
electrons in water, aluminum
and lead shown with heavy
solid curves against the
electron kinetic energy E K .
Mass collision stopping
powers Scol , discussed in
Sect. 6.5, for the same
materials are shown with
light solid curves for
comparison. Data were
obtained from the NIST
mass collision stopping powers Scol (discussed in Sect. 6.5) are shown with light
curves for comparison. The radiation stopping power Srad clearly shows an approx-
imate proportionality: (1) to the atomic number Z of the absorber at a given initial
kinetic energy of the light charged particle and (2) to initial kinetic energy (E K )0 of
the light charged particle for a given absorber material.
6.4 Collision (Electronic) Stopping Power for Heavy
Charged Particles
Energy transfer from energetic heavy charged particles to a medium (absorber)
they traverse occurs mainly through Coulomb interactions of the charged particles
with orbital electrons of the absorber atoms (collision or electronic loss); inelastic
Coulomb interactions between heavy charged particles and nuclei of absorber atoms
(radiation loss) are negligible and thus ignored.
Two different approaches were developed to describe a heavy charged particle
energy loss to orbital electrons of absorber atoms:
1. Bohr’s approach (1913) is in the realm of classical physics and is based on the
concept of impact parameter between the particle’s trajectory and the absorber
nucleus.
2. Bethe’s approach (1931) is in the realm of quantum mechanics and relativistic
physics, and assumes that the momentum transfer related to the particle’s energy
loss is quantized.
The basic theories dealing with collision loss of energetic heavy charged particles in
absorbing media make the following assumptions:
1. The energetic charged particle is moving through the absorber much faster than
the orbital electrons of absorber atoms.
2. The energetic charged particle is much heavier than the energy-absorbing orbital
electrons.
