interaction between ionizing radiation and matter, part 2 charged-particles audun sanderud...

Interaction Between Ionizing Radiation And Matter,

Part 2 Charged-Particles

Audun Sanderud

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• Incoming charged particle interact with atom/molecule:

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loExcitation / ionization

Ionization

Excitation

• Ion pair created from ionization

• Interaction between two particles with conservation of kinetic energy ( and momentum):D

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Elastic collision

m1, v m2 m1, v1

m2, v2

• Classic mechanics give:2 2 2

0 1 1 1 2 2

1 1 1 2 2

1 1 2 2

1 1 1T m v m v m v

2 2 2m v m v cos m v cos

0 m v sin m v sin

q c

q c

= = +

= +

= +

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loElastic collision(2)

( )

21 1 2

2 1 21 2 1 2

1

2

2m vcos 4m m cosv , v v 1

m m m m

sin 2tan

mcos 2

m

c c

cq

c

Þ = = -+ +

=-

• These equations gives the maximum transferred energy:

( )2 1 2

max 2 2,max 02

1 2

m m1E m v 4 T

2 m m= =

+

• Proton(#1)-electron(#2):max=0.03o, Emax=0.2 % T0

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loElastic collisions(3)

a) m1>>m2 a) m1=m2 a) m1<<m2

1 2

1

2max 0

1

0 2

m0 tan sin 2

m

mE 4 T

m

pc

q c-

£ £

æ ö÷ç ÷£ £ ç ÷ç ÷çè ø

= max 0

0 2

0 2

E T

pc

pq

£ £

£ £

= 1max 0

2

0 2

0

mE 4 T

m

pc

q p

£ £

£ £

=

• Electron(#1)-electron(#2):max=90o, Emax=100 % T0

• Rutherford proved that the cross section of elastic scattering is:

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loElastic collisions-cross section

( )4

d 1

d sin 2

sq

µW

Small scattering angels most probable

• Differentiated by the energy

2

d 1

dE E

sµ

Small energy transferred most probable

• Stopping power, (dT/dx): the expectation value of the rate of energy loss per unit of pathlength. Dependent on: -type of charged particle

-its kinetic energy-the atomic number of the

medium

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loStopping power

T0 T0-dT

dx

nv targets per volume unit max max

min min

max

min

E E

AV v

E E

E

A

E

N Zd ddT En dx n dx EdT dx EdT

dT A dT

N ZS dT dEdT

dx A dT

s ss r

sr r

æ ö÷ç= = = ÷ç ÷çè ø

æ ö æ ö÷ç ÷ç= ÷= ÷ç ç÷ ÷çç ÷ç è øè ø

ò ò

ò

• The charged particle collision is a Coulomb-force interaction

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loImpact parameter

• The impact parameter b useful versus the classic atomic radius a

• Most important: the interaction with electrons

• b>>a: particle passes an atom in a large distance

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loSoft collisions

• The result is excitations (dominant) and ionization;amount energy transferred range from Emin to a certain energy H

• Small energy transitions to the atom

• Hans Bethe did quantum mechanical calculations on the stopping power of soft collision in the 1930• We shall look at the results from particles with much larger mass then the electron

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loSoft collisions(2)

r0: classic electron radius = e2/40mec2

I: mean excitation potential: v/cz: charge of the incoming particle: Density of the medium

NAZ/A: Number of electrons per gram in medium

H: Maximum transferred energy at soft collision

( )2 2 2 2 2

c,soft 2soft 0 e eA2 2 2

c

S dT 2 r m c z 2m c HN Zln

dx A I 1

p bb

r r b b

é ùæ öæ ö ÷çê ú÷ ÷ç ç= ÷= -÷ç ê úç÷ ÷ç ÷ç ç ÷è ø - ÷ê úçè øë û

• The quantum mechanic effects are specially seen in the excitation potential I

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loSoft collisions(3)

• High Z – small transferred energy less likely

Atomic number, Z

Mean e

xci

tati

on

pote

nti

al, I/Z

[eV

]

• b<<a: particle passes trough the atom

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loHard collisions

• Amount energy transferred range from H to Emax

• Large (but few) energy transactions to single electron

• Can be seen as an elastic collision between free particles (bonding energy nelectable)

2 2 2c,hard 2hard 0 e maxA

2c

S dT 2 r m c z EN Zln

dx A H

pb

r r b

æ ö é ùæ ö÷ç ÷çê ú= ÷= -÷ç ç÷ ÷çç ÷ ê úç è øè ø ë û

• The total collision stopping power is then (soft + hard):

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loCollisions stopping power

• Important: increase with z2, decrease with v2, not dependent on particle mass

( )2 2 2 2 2

c,soft c,hard 2c 0 e eA2 2

S SS 4 r m c z 2m cN Zln

A 1 I

p bb

r r r b b

é ùæ ö÷çê ú÷ç= + = -÷ê úç ÷ç ÷- ÷ê úçè øë û

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loSc/ in different media

• I and electron density (ZNA/A) gives the variation

• Electron-electron scattering more complicated;interaction between identical particles D

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SC for electrons/positrons

• Sc,hard/electron-elektron; Møller cross section

positron-electron; Bhabha cross section

The characteristics similar to that of heavy particles

• Sc,soft/Bethe’s soft coll. formula

( )

( )( )

22 2 22c 0 eA

e22 2e

2S 2 r m c zN Z Cln F 2 , T / m c

A Z2 I / m c

t tpt d t

r b±

é ùæ ö÷ç +ê ú÷ç ÷= + - - ºê úç ÷ç ÷ê úç ÷÷çè øê úë û

( )( )

( )

22

2

/ 8 2 1 ln2F 1

1- t - t +

t = - b +t +

( )( ) ( )

2

2 3

14 10 4F 2ln 2 23

12 2 2 2+

ì üï ïb ï ïï ït = - + + +í ýï ït + t + t +ï ïï ïî þ

• The approximation used in the calculations of SC assume v>>vatomic electron

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loShell correction

• C/Z depend on particle velocity and medium

• When v~vatomic electron no ionizations will occur

• Shell correction C/Z handles this, and reduce SC/

• Occur first in the K-shell - highest atomic electron speed

• Charged particles polarizes the medium

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loDensity-effect correction

•

• Weaker interaction with distant atoms because of the reduction of the Coulomb force field

Charged (+z) particle

eff eff pol

eff pol

E E E

E E

= +

<

ur ur ur

ur ur

• Polarization increase with (relativistic) speed

• Most important for electrons / positrons

• But: polarization not important at low

• Density-effect correction reduces Sc/ in solid and liquid elements

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loDensity-effect correction(2)

• Sc/water vapor) > Sc/water

Dashed curves: Sc without

• When charged particles are accelerated by the Coulomb force from atomic electrons or nucleus photons can be emitted; Bremsstrahlung

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loRadiative stopping power

• The Lamor equation (classic el.mag.) denote the radiation power from an acceleration, a, of a charged particle:

0: Permittivity of a vacuum

Charged particle

atomic electron

• The case of a particle accelerated in nucleus field:

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• Comparison of proton and electron as incoming:

• Bremsstrahlung not important for heavy charged particles

• The maximum energy loss to bremsstrahlung is the total kinetic energy of the electron D

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Radiative stopping power(3)

•Br(T,Z) weak dependence of T and Z

• Energy transferred to radiation per pathlength unit: radiative stopping power:

( )2

2 2A0 e r

r r

N ZS dTr T m c B (T, Z)

dx A

æö æ ö÷ ÷ç ç÷= ÷=a +ç ç÷ ÷ç ç÷ ÷ç çr rè ø è ø

• Radiative energy loss increase with T and Z

• Total stopping power, electrons:

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loTotal stopping power, electrons

• Comparison:tot c r

dT dT dT

dx dx dx

æ ö æ ö æ ö÷ ÷ ÷ç ç ç÷ = ÷+ ÷ç ç ç÷ ÷ ÷ç ç ç÷ ÷ ÷ç ç çr r rè ø è ø è ø

r

c

S TZ

S n

n 750MeV

»

=

• Estimated fraction of the electron energy that is emitted as bremsstrahlung:

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loRadiation yield

( )( )

( ) ( )r r

c r

dT / dx SY T

dT / dx dT / dx S

r= =

r + rR

adia

tion y

ield

, Y(T

)

Kinetic energy, T (MeV)

WaterTungsten

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loComparison of Sc

Kinetic energy, T [MeV]

Electrons, totalElectrons, collisionElectrons, radiativeProtons, total

• Cerenkov effect: very high energetic electrons (v>c/n) polarize a medium (water) of refractive index n and bluish light is emitted (+UV)

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loOther interactions

• Little energy is emitted

• Nuclear interactions: Inelastic process in which the charged particle cause an excitation of the nucleus. Result: -Scattering of charged particle

-Emission of neutron, -quant, -particleNot important below ~10 MeV (proton)

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loOther interactions(2)

• Positron annihilation: Positron interact with atomic electron, and a photon pair of energy ≥ 2x0.511MeV is created. The two photons are emitted 180o apart.Probability decrease by ~1/v

Braggs ruleD

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• Braggs rule for mixtures of n-atoms/elements:

( )( ) ( )

( )

( )

( )

1

1

1 1

1 1

,

ln

ln ,

i

i i

i ii

i

i i

i i

i i

i i

nZc c

Z Z nZmix Z

ZZ

n n

Z i Z ii i

Z Zmix mixn n

Z Zi i

Z Z

mS Sf f

m

Z Zf I fA AI

Z Zf fA A

r r

d

d

=

=

= =

= =

æ ö æ ö÷ ÷ç ç÷ = ÷ =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø

é ù é ùê ú ê úë û ë û

= =

åå

å å

å å

• LET; also known as restricted stopping power

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loLinear Energy Transfer

• Sc includes energy transitions from Emin to Emax

•cutoff value; LET includes all the soft and the fraction of the hard collision -rays with energy< -electron as a result of

ionization

Trace of charged particle

-electrons living the volume → energy transferred >

• LET the amount of energy disposed in a volume defined by the range of an electron with energy

• The energy loss per length unit by transitions of energy between Emin < E < :

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• LET given in keV/m

• If = Emax then L= Sc ; unrestricted LET

• 30 MeV protons in water: LET100eV/L

• The range of a charge particle in a medium is the expectation value of the pathlength pD

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Range

• The projected range <t> is the expectation value of the farthest depth of penetration tf in its initial direction

Electrons:<t> <

Heavy particles:<t> ≈

• Range can by approximated by the Continuous Slowing Down Approximation, CSDA

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loRange(2)

• Energy loss per unit length is given by dT/dx – gives an indirect measure of the range:T0

x

0 0

n n

ii 1 i 1 i

dTT T T x

dxdx dx

x T, x TdT dT= =

- D = - D

æ ö÷çD = D Þ Â = D = D÷ç ÷çè øå å

01T

CSDA

0

dTdT

dx

-æ ö÷çÞ Â = ÷ç ÷ç ÷çrè øò

• Range is often given multiplied by density

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loRange(3)

• Unit is then [cm][g/cm3]=[g/cm2]

01T

CSDA

0

dTdT

dx

-æ ö÷ç = ÷ç ÷ç ÷çrè øò

• Range of a charge particle depend on:- Charge and kinetic energy- Density, electron density and average excitation

potential of absorbent

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loRange(4)

• In a radiation field of charged particles there is:- variations in rate of energy loss- variations in scattering

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loStraggling and multiple scattering

The initial beam of particle at same speed and direction, are spread as they penetrate a medium

vr

4vr

3vr 2v

r1vr

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loMultiple scattering

• Electrons experience most scattering – characteristic of initially close to monoenergetic beam:

Energy [MeV]

Num

ber

Initial beamBeam at small depth in absorbentBeam at large depth in absorbent

• Characteristic of different type of particles penetrating a medium:

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loProjected range <t>

• Protons energy disposal at a given depth:

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loEnergy disposal

• Electrons energy disposal at a given depth; multiple scattering decrease with kinetic energy: D

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Energy disposal(2)

• Monte Carlo simulations of the trace after an electron (0.5 MeV) and an -particle (4 MeV) in waterD

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Monte Carlo simulations

• Notice: e- most scattered has highest S

• Heavy charged particles can be used in radiation therapy – gives better dose distribution to tumor than photons/electrons

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loHadron therapy

• Stopping powerhttp://physics.nist.gov/PhysRefData/Star/Text/

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loTables on the web

• Attenuation coefficients http://physics.nist.gov/PhysRefData/XrayMassCoef/cover.html

Summary