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8/2/2019 14 - Charged Particles I (1)
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Charged-Particle Interactions in
Matter I
Types of Charged-Particle Coulomb-
Force InteractionsStopping Power
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Introduction
Charged particles lose their energy in a manner
that is distinctly different from that of uncharged
radiations (x- or -rays and neutrons) An individual photon or neutron incident upon a
slab of matter may pass through it with no
interactions at all, and consequently no loss of
energy
Or it may interact and thus lose its energy in one
or a few catastrophic events
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Introduction (cont.)
By contrast, a charged particle, being surroundedby its Coulomb electric force field, interacts withone or more electrons or with the nucleus ofpractically every atom it passes
Most of these interactions individually transferonly minute fractions of the incident particleskinetic energy, and it is convenient to think of theparticle as losing its kinetic energy gradually in africtionlike process, often referred to as thecontinuous slowing-down approximation(CSDA)
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Introduction (cont.)
Charged particles can be roughly characterized bya commonpathlength, traced out by most suchparticles of a given type and energy in a specificmedium
Because of the multitude of interactionsundergone by each charged particle in slowingdown, its pathlength tends to approach theexpectation value that would be observed as amean for a very large population of identicalparticles
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Introduction (cont.)
That expectation value, called the range,
will be discussed in a later lecture
Note that because of scattering, all identical
charged particles do not follow the same
path, nor are the paths straight, especially
those of electrons because of their smallmass
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Types of Charged-Particle
Coulomb-Force Interactions Charged-particle Coulomb-force interactions can
be simply characterized in terms of the relativesize of the classical impact parameterb vs. theatomic radius a, as shown in the following figure
The following three types of interactions becomedominant for b >> a, b ~ a, and b
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Important parameters in charged-particle collisions with atoms:
a is the classical atomic radius; b is the classical impact
parameter
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Soft Collisions (b >> a)
When a charged particle passes an atom at a
considerable distance, the influence of the
particles Coulomb force field affects the atom asa whole, thereby distorting it, exciting it to a
higher energy level, and sometimes ionizing it by
ejecting a valence electron
The net effect is the transfer of a very smallamount of energy (a few eV) to an atom of the
absorbing medium
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Soft Collisions (cont.)
Because large values ofb are clearly moreprobable than are near hits on individual atoms,soft collisions are by far the most numerous typeof charged-particle interaction, and they accountfor roughly half of the energy transferred to theabsorbing medium
In condensed media (liquids and solids) the atomicdistortion mentioned above also gives rise to thepolarization (or density) effect, which will bediscussed later
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Hard (or Knock-On) Collisions
(b ~ a) When the impact parameter b is of the order of the
atomic dimensions, it becomes more likely that theincident particle will interact primarily with asingle atomic electron, which is then ejected fromthe atom with considerable kinetic energy and iscalled a delta () ray
In the theoretical treatment of the knock-onprocess, atomic binding energies have beenneglected and the atomic electrons treated asfree
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Hard Collisions (cont.)
-rays are of course energetic enough to
undergo additional Coulomb-force
interactions on their own
Thus a -ray dissipates its kinetic energy
along a separate track (called a spur) from
that of the primary charged particle
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Hard Collisions (cont.)
The probability for hard collisions depends uponquantum-mechanical spin and exchange effects,thus involving the nature of the incident particle
Hence, as will be seen, the form of stopping-
power equations that include the effect of hardcollisions depend on the particle type, beingdifferent especially for electrons vs. heavyparticles
Although hard collisions are few in numbercompared to soft collisions, the fractions of the
primary particles energy that are spent by thesetwo processes are generally comparable
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Hard Collisions (cont.)
It should be noted that whenever an inner-shell
electron is ejected from an atom by a hard
collision, characteristic x rays and/or Augerelectrons will be emitted just as if the same
electron had been removed by a photon interaction
Thus some of the energy transferred to the
medium may be transported some distance awayfrom the primary particle track by these carriers as
well as by the -rays
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Coulomb-Force Interactions with the
External Nuclear Field (b
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Interactions with the External
Nuclear Field (cont.) In all but 23% of such encounters, the electron
is scattered elastically and does not emit an x-ray
photon or excite the nucleus It loses just the insignificant amount of kinetic
energy necessary to satisfy conservation of
momentum for the collision
Hence this is not a mechanism for the transfer of
energy to the absorbing medium, but it is an
important means ofdeflecting electrons
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Interactions with the External
Nuclear Field (cont.) It is the principle reason why electrons follow very
tortuous paths, especially in high-Zmedia, and
why electron backscattering increases withZ In doing Monte Carlo calculations of electron
transport through matter, it is often assumed for
simplicity that the energy-loss interactions may be
treated separately from the scattering (i.e., change-of-direction) interactions
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Interactions with the External
Nuclear Field (cont.) The differential elastic-scattering cross
section per atom is proportional toZ
This means that a thin foil of high-Zmaterial may be used as a scatterer to
spread out an electron beam while
minimizing the energy lost by thetransmitted electrons in traversing a given
mass thickness of foil
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Interactions with the External
Nuclear Field (cont.) In the other 23% of the cases in which the
electron passes near the nucleus, an inelasticradiative interaction occurs in which an x-rayphoton is emitted
The electron is not only deflected in this process,but gives a significant fraction (up to 100%) of itskinetic energy to the photon, slowing down in theprocess
Such x-rays are referred to as bremsstrahlung, theGerman word for braking radiation
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Interactions with the External
Nuclear Field (cont.) This interaction also has a differential atomic
cross section proportional toZ, as was the case
for nuclear elastic scattering Moreover, it depends on the inverse square of the
mass of the particle, for a given particle velocity
Thus bremsstrahlung generation by charged
particles other than electrons is totally
insignificant
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Interactions with the External
Nuclear Field (cont.) Although bremsstrahlung production is an important
means of energy dissipation by energetic electrons in high-Zmedia, it is relatively insignificant in low-Z(tissue-like)
materials for electrons below 10 MeV Not only is the production cross section low in that case,
but the resulting photons are penetrating enough so thatmost of them can escape from objects several centimetersin size
Thus they usually carry away their quantum energy ratherthan expending it in the medium through a furtherinteraction
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Interactions with the External
Nuclear Field (cont.) In addition to the foregoing three modes of kinetic
energy dissipation (soft, hard, and bremsstrahlung
interactions), a fourth channel is available only toantimatter (i.e., positrons): in-flight annihilation
The average fraction of a positrons kinetic energy
that is spent in this type of radiative loss is said to
be comparable to the fraction going intobremsstrahlung production
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Nuclear Interactions by Heavy
Charged Particles A heavy charged particle having sufficiently high
kinetic energy (~ 100 MeV) and an impact
parameter less than the nuclear radius may interactinelastically with the nucleus
When one or more individual nucleons (protons or
neutrons) are struck, they may be driven out of the
nucleus in an intranuclear cascade process,collimated strongly in the forward direction
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Nuclear Interactions by Heavy
Charged Particles (cont.) The highly excited nucleus decays from its excited
state by emission of so-called evaporation
particles (mostly nucleons of relatively lowenergy) and -rays
Thus the spatial distribution of absorbed dose is
changed when nuclear interactions are present,
since some of the kinetic energy that wouldotherwise be deposited as local excitation and
ionization is carried away by neutrons and -rays
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Nuclear Interactions by Heavy
Charged Particles (cont.) One special case where nuclear interactions by
heavy charged particles attain first-orderimportance relative to Coulomb-force interactions
is that of-mesons (negative pions)
These particles have a mass 273 times that of theelectron, or 15% of the proton mass
They interact by Coulomb forces to produceexcitation and ionization along their track in thesame way as any other charged particle, but theyalso display some special characteristics
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Nuclear Interactions by Heavy
Charged Particles (cont.) The effect of nuclear interactions is conventionally
not included in defining the stopping poweror
range of charged particles Nuclear interactions by heavy charged particles
are usually ignored in the context of radiological
physics and dosimetry
Internal nuclear interactions by electrons are
negligible in comparison with the production of
bremsstrahlung
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Stopping Power
The expectation value of the rate of energy loss per unit ofpath lengthx by a charged particle of type Yand kineticenergy T, in a medium of atomic numberZ, is called its
stopping power, (dT/dx)Y,T,Z The subscripts need not be explicitly stated where that
information is clear from the context
Stopping power is typically given in units of MeV/cm orJ/m
Dividing the stopping power by the density of theabsorbing medium results in a quantity called the massstopping power(dT/dx), typically in MeV cm2/g or Jm2/kg
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Stopping Power (cont.)
When one is interested in the fate of the energylost by the charged particle, stopping power may
be subdivided into collision stopping power and
radiative stopping power
The former is the rate of energy loss resultingfrom the sum of the soft and hard collisions, whichare conventionally referred to as collisioninteractions
Radiative stopping power is that owing toradiative interactions
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Stopping Power (cont.)
Unless otherwise specified, radiative stoppingpower may be assumed to be based onbremsstrahlung alone
The effect of in-flight annihilation, which is onlyrelevant for positrons, is accounted for separately
Energy spend in radiative collisions is carriedaway from the charged particle track by the
photons, while that spent in collision interactionsproduces ionization and excitation contributing tothe dose near the track
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Stopping Power (cont.)
The mass collision stopping powercan be writtenas
where subscripts c indicate collision interactions, sbeing soft and h hard
The terms on the right can be rewritten as
c
h
c
s
cdx
dT
dx
dT
dx
dT
max
min
T
H
h
c
H
T
s
c
c
TdQTTdQTdx
dT
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Stopping Power (cont.)
1. T is the energy transferred to the atom orelectron in the interaction
2. His the somewhat arbitrary energyboundary between soft and hard collisions,in terms ofT
3. Tmax
is the maximum energy that can betransferred in a head-on collision with anatomic electron, assumed unbound
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Stopping Power (cont.)
For a heavy particle with kinetic energy less thanits rest-mass energyM0c
2,
which for protons equals 20 keV for T= 10 MeV,or 0.2 MeV for T= 100 MeV
For positrons incident, Tmax = Tif annihilationdoes not occur
For electrons, TmaxT/2
MeV1
022.11
2 2
2
2
2
20max
cmT
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Stopping Power (cont.)
4. Tmax is related to Tmin by
in whichIis the mean excitation potential of the
struck atom, to be discussed later
5. Qs
c and Qh
c are the respective differential masscollision coefficients for soft and hard collisions,
typically in units of cm2/g MeV or m2/kg J
2
262
22
0
min
max eV10022.12
II
cm
T
T
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The Soft-Collision Term
The soft-collision term was derived by Bethe, for eitherelectrons or heavy charged particles withz elementarycharges, on the basis of theBorn approximation which
assumes that the particle velocity (v = c) is much greaterthan the maximum Bohr-orbit velocity (u) of the atomicelectrons
The fractional error in the assumption is of the order of(u/v)2, and Bethes formula is valid for (u/v)2 ~ (Z/137)2
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The Soft-Collision Term (cont.)
The Bethe soft-collision formula can be written as
where C(NAZ/A)r02 = 0.150Z/A cm2/g, in
whichNAZ/A is the number of electrons per gram
of the stopping medium, and r0
= e2/m0
c2 = 2.818
10-13 cm is the classical electron radius
222
22
0
2
22
0
1
2ln
2
I
HcmzcCm
dx
dT
c
s
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The Soft-Collision Term (cont.)
We can further simplify the factor outside thebracket by defining it as
where m0c2 = 0.511 MeV, the rest-mass energy of
an electron
The bracket factor is dimensionless, thus requiringthe quantities m0c
2,H, andIoccurring within it tobe expressed in the same energy units, usually eV
22
2
2
22
0
g/cm
MeV1535.0
2
A
ZzzcCmk
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The Soft-Collision Term (cont.)
The mean excitation potentialIis the geometric-
mean value of all the ionization and excitation
potentials of an atom of the absorbing medium In generalIfor elements cannot be calculated
from atomic theory with useful accuracy, but must
instead be derived from stopping-power or range
measurements Appendices B.1 and B.2 list someI-values
according to Berger and Seltzer
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The Soft-Collision Term (cont.)
SinceIonly depends on the stopping medium, butnot on the type of charged particle, experimentaldeterminations have been done preferentially with
cyclotron-accelerated protons, because of theiravailability with high -values and the relativelysmall effect of scattering as they pass throughlayers of material
The paths of electrons are too crooked to allowtheir use in accurate stopping powerdeterminations
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The Hard-Collision Term for
Heavy Particles The form of the hard-collision term depends on
whether the charged particle is an electron,
positron, or heavy particle We will treat the case of heavy particles first,
having masses much greater than that of an
electron, and will assume thatH
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The Hard-Collision Term for
Heavy Particles (cont.) The mass collision stopping power for combined soft and
hard collisions by heavy particles becomes:
which can be simplified further by substituting for Tmax
222
max
22
0
21
2
ln
I
Tcm
kdx
dT
c
IA
Zz
I
Tcmk
dx
dT
c
ln1
ln8373.133071.0
1
2ln2
2
2
2
2
2
2
2
max
22
0
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Dependence on the Stopping
Medium There are two expressions influencing this
dependence, and both decrease the mass collision
stopping power asZis increased The first is the factorZ/A outside the bracket,
which makes the formula proportional to the
number of electrons per unit mass of the medium
The second is the termlnIin the bracket, whichfurther decreases the stopping power asZis
increased
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Dependence on the Stopping
Medium (cont.) The termlnIprovides the stronger
variation withZ
The combined effect of the twoZ-dependent expressions is to make (dT/dx)c
for Pb less than that for C by 40-60 %
within the -range 0.85-0.1, respectively
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Dependence on Particle Velocity
The strongest dependence on velocity comes fromthe inverse 2 (outside of the bracket), whichrapidly decreases the stopping power as
increases
That term loses its influence as approaches aconstant value at unity, while the sum of the 2terms in the bracket continues to increase
The stopping power gradually flattens to a broadminimum of 1-2 MeV cm2/g at T/M0c
2 3, andthen slowly rises again with further increasing T
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Mass collision stopping power for singly charged heavy
particles, as a function of or of their kinetic energy T
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Dependence on Particle Charge
The factorz2 means that a doubly charged particle
of a given velocity has 4 times the collision
stopping power as a singly charged particle of thesame velocity in the same medium
For example, an -particle with = 0.141 would
have a mass collision stopping power of 200 MeV
cm2/g, compared with the 50 MeV cm2/g shown inthe figure for a singly charged heavy particle in
water
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Dependence on Particle Mass
There is none
All heavy charged particles of a given
velocity andz will have the same collisionstopping power
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Relativistic Scaling
Considerations For any particle, = v/c is related to the kinetic
energy Tby
The kinetic energy required by any particle toreach a given velocity is proportional to its rest
energy, M0c2
The rest energies of some heavy particles arelisted in the following table
2/12
2
02
20
1/
11and1
1
1
cMTcMT
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Shell Correction
The Born approximation assumption, whichunderlies the stopping-power equation, is not wellsatisfied when the velocity of the passing particleceases to be much greater than that of the atomic
electrons in the stopping medium Since K-shell electrons have the highest velocities,
they are the first to be affected by insufficientparticle velocity, the slowerL-shell electrons are
next, and so on The so-called shell correction is intended to
account for the resulting error in the stopping-power equation
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Shell Correction (cont.)
As the particle velocity is decreased toward that ofthe K-shell electrons, those electrons graduallydecrease their participation in the collisionprocess, and the stopping power is thereby
decreased below the value given by the equation When the particle velocity falls below that of the
K-shell electrons, they cease participating in thecollision stopping-power process
The equation underestimates the stopping powerbecause it contains too large anI-value
The properI-value would ignore the K-shellcontribution
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Shell Correction (cont.)
Bichsel estimated the combined effect of all i
shells into a single approximate correction C/Z, to
be subtracted from the bracketed terms
The corrected formula for the mass collision
stopping power for heavy particles then becomes
Z
CI
A
Zz
Z
C
I
Tcm
kdx
dT
c
ln1
ln8373.133071.0
1
2
ln2
2
2
2
2
2
2
2
max
22
0
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Shell Correction (cont.)
The correction term C/Zis the same for allcharged particles of the same velocity , includingelectrons, and its size is a function of the mediumas well as the particle velocity
C/Zis shown in the following figure for protons inseveral elements
A second correction term, , to account for thepolarization or density effectin condensed media,
is sometimes included also
It is negligible for all heavy particles within theenergy range of interest in radiological physics
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Semiempirical shell corrections of Bichsel for selected elements,
as a function of proton energy
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Mass Collision Stopping Power
for Electrons and Positrons The formulae for the mass collision stopping
power for electrons and positrons are gotten bycombining Bethes soft collision formula with a
hard-collision relation based on the Mller crosssection for electrons or the Bhabha cross sectionfor positrons
The resulting formula, common to both particles,
in terms ofT/m0c2, is
Z
CF
cmIk
dx
dT
c
2
/2
2ln
22
0
2
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Mass Collision Stopping Power for
Electrons and Positrons (cont.) For electrons,
and for positrons,
Here C/Zis the previously discussed shellcorrection and is the correction term for thepolarization or density effect
2
22
1
2ln128/1
F
32
2
2
4
2
10
2
1423
122ln2
F