14 - charged particles i (1)

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 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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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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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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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

14 - charged particles i (1)

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