Interaction BetweenIonizing Radiation And Matter, Part 2

Charged-Particles

Audun Sanderud

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• Incoming charged particle interact with atom/molecule:

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oExcitation / ionization

Ionization

Excitation

• Ion pair created from ionization

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

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oElastic collision

m1, v m2 m1, v1

m2, v2 χ

θ

• 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 v2 2 2

m v m v cos m v cos0 m v sin m v sin

θ χθ χ

= = +

= += +

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oElastic collision(2)

( )

21 1 2

2 1 21 2 1 2

1

2

2m v cos 4m m cosv , v v 1m m m m

sin 2tan m cos 2m

χ χ

χθχ

⇒ = = −+ +

=−

• These equations gives the maximum transferred energy:

( )2 1 2

max 2 2,max 021 2

m m1E m v 4 T2 m m

= =+

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

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oElastic collisions(3)

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

1 2

1

2max 0

1

0 2m0 tan sin 2mmE 4 Tm

πχ

θ χ−

≤ ≤

⎛ ⎞⎟⎜ ⎟≤ ≤ ⎜ ⎟⎜ ⎟⎜⎝ ⎠

= max 0

0 2

0 2

E T

πχ

πθ

≤ ≤

≤ ≤

= 1max 0

2

0 2

0

mE 4 Tm

πχ

θ π

≤ ≤

≤ ≤

=

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

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

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oElastic collisions-cross section

( )4

d 1d sin 2

σθ

∝Ω

→ Small scattering angels most probable • Differentiated by the energy

2

d 1dE Eσ ∝

→ 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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oStopping power

T0 T0-dT

dx

nv targets per volume unitmax max

min min

max

min

E EA

V vE E

EA

E

N Zd ddT En dx n dx EdT dx EdTdT A dT

N ZS dT d EdTdx A dT

σ σσ ρ

σρ ρ

⎛ ⎞⎟⎜= = = ⎟⎜ ⎟⎜⎝ ⎠

⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜= ⎟= ⎟⎜ ⎜⎟ ⎟⎜⎜ ⎟⎜ ⎝ ⎠⎝ ⎠

∫ ∫

∫

• The charged particle collision is a Coulomb-force interaction

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oImpact 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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oSoft 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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oSoft collisions(2)

r0: classic electron radius = e2/4πε0mec2

I: mean excitation potentialβ: v/cz: charge of the incoming particleρ: Density of the medium

NAZ/A: Number of electrons per gram in mediumH: 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 Z lndx A I 1

π β βρ ρ β β

⎡ ⎤⎛ ⎞⎛ ⎞ ⎟⎜⎢ ⎥⎟ ⎟⎜ ⎜= ⎟ = −⎟⎜ ⎢ ⎥⎜⎟ ⎟⎜ ⎟⎜ ⎜ ⎟⎝ ⎠ − ⎟⎢ ⎥⎜⎝ ⎠⎣ ⎦

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

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oSoft collisions(3)

• High Z – small transferred energy less likely

Atomic number, Z

Mea

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[eV

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• b<<a: particle passes trough the atom

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oHard 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 Z lndx A H

π βρ ρ β

⎛ ⎞ ⎡ ⎤⎛ ⎞⎟⎜ ⎟⎜⎢ ⎥= ⎟ = −⎟⎜ ⎜⎟ ⎟⎜⎜ ⎟ ⎢ ⎥⎜ ⎝ ⎠⎝ ⎠ ⎣ ⎦

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

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oCollisions 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 Z lnA 1 I

π β βρ ρ ρ β β

⎡ ⎤⎛ ⎞⎟⎜⎢ ⎥⎟⎜= + = −⎟⎢ ⎥⎜ ⎟⎜ ⎟− ⎟⎢ ⎥⎜⎝ ⎠⎣ ⎦

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oSc/ρ in different media

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

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

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oSC for electrons/positrons

• Sc,hard/ρ: electron-elektron; Møller cross sectionpositron-electron; Bhabha cross section

The characteristics similar to that of heavy particles

• Sc,soft/ρ: Bethe’s soft coll. formula

( )( )

( )22 2 2

2c 0 eAe22 2

e

2S 2 r m c zN Z Cln F 2 , T / m cA Z2 I / m c

τ τπ τ δ τρ β

±

⎡ ⎤⎛ ⎞⎟⎜ +⎢ ⎥⎟⎜ ⎟= + − − ≡⎢ ⎥⎜ ⎟⎜ ⎟⎢ ⎥⎜ ⎟⎟⎜⎝ ⎠⎢ ⎥⎣ ⎦

( ) ( )( )

22

2

/ 8 2 1 ln2F 1

1− τ − τ+

τ = −β +τ+

( )( ) ( )

2

2 314 10 4F 2ln 2 23

12 2 2 2+

⎧ ⎫⎪ ⎪β ⎪ ⎪⎪ ⎪τ = − + + +⎨ ⎬⎪ ⎪τ+ τ+ τ+⎪ ⎪⎪ ⎪⎩ ⎭

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

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oShell 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/ρ

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

• Charged particles polarizes the medium

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oDensity-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

= +

<

• Polarization increase with (relativistic) speed

• Most important for electrons / positrons • But: polarization not important at low ρ

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

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oDensity-effect correction(2)

• Sc/ρ (water vapor) > Sc/ρ (water)

Dashed curves: Sc without δ

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

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Radiative stopping power

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

ε0: Permittivity of a vacuum

Charged particle atomic

electron

2 2

30

(ze) aP6 cπε

=

• The case of a particle accelerated in nucleus field:

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oRadiative stopping power(2)

• Comparison of proton and electron as incoming:

22 2 22 2

2 20 0

zZe zZe ZzF ma a P a z4 r 4 mr m

⎛ ⎞⎟⎜ ⎟= = ⇒ = ⇒ ∝ ∝⎜ ⎟⎜ ⎟⎜πε πε ⎝ ⎠

2

proton electron2

electron proton

P m 1P m 1836

⎛ ⎞⎟⎜ ⎟⎜= ≈⎟⎜ ⎟⎟⎜⎝ ⎠

• Bremsstrahlung not important for heavy charged particles

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

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oRadiative 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 dT r T m c B (T, Z)dx A

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ = ⎟ =α +⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜ρ ρ⎝ ⎠ ⎝ ⎠

• Radiative energy loss increase with T and Z

• Total stopping power, electrons:

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oTotal stopping power, electrons

• Comparison:tot c r

dT dT dTdx dx dx

⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ = ⎟ + ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜⎟ ⎟ ⎟⎜ ⎜ ⎜ρ ρ ρ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

r

c

S TZS nn 750MeV

≈

=

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

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oRadiation yield

( ) ( )( ) ( )

r r

c r

dT / dx SY TdT / dx dT / dx S

ρ= =

ρ + ρR

adia

tion

yiel

d, Y

(T)

Kinetic energy, T (MeV)

WaterTungsten

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oComparison 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)D

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Other 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, γ-quant, α-particleNot important below ~10 MeV (proton)

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oOther 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

,

lnln ,

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 iZ Z

mix mixn n

Z Zi iZ Z

mS Sf fm

Z Zf I fA AI

Z Zf fA A

ρ ρ

δδ

=

=

= =

= =

⎛ ⎞ ⎛ ⎞⎟ ⎟⎜ ⎜⎟ = ⎟ =⎜ ⎜⎟ ⎟⎜ ⎜⎟ ⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

= =

∑∑

∑ ∑

∑ ∑

• LETΔ; also known as restricted stopping power

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oLinear Energy Transfer

• Sc includes energy transitions from Emin to Emax

• Δ, cutoff value; LETΔ includes all the soft and the fraction of the hard collision δ-rays with energy<Δ

δ-electron as a result of ionization

Trace of charged particle

δ-electrons living the volume → energy transferred > Δ

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

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

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oLinear Energy Transfer(2)

• LETΔ given in keV/μm• If Δ = Emax then L∞= Sc ; unrestricted LET

• 30 MeV protons in water: LET100eV/L ∞ = 0.53

min

A

E

2 2 22 2 2eA0 e 2 2

N ZdT dL EdEdx A dE

2m cN Z z2 r m c ln 2A (1 )I

σρ

βρ π ββ β

Δ

ΔΔ

⎛ ⎞⎛ ⎞ ⎟⎟ ⎜⎜= = ⎟⎟ ⎜⎜ ⎟⎟⎜ ⎜⎝ ⎠ ⎝ ⎠

⎡ ⎤⎛ ⎞⎛ ⎞⎛ ⎞ Δ⎟⎟ ⎜⎟⎜ ⎢ ⎥⎜ ⎟= −⎟ ⎜⎟⎜⎜ ⎟⎟⎟ ⎢ ⎥⎜⎜ ⎟⎜ ⎟⎜⎝ ⎠ −⎝ ⎠ ⎝ ⎠⎣ ⎦

∫

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

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oRange

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

Electrons:<t> < ℜ

Heavy particles:<t> ≈ ℜ

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

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oRange(2)

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

T0

Δx

0 0

n n

ii 1 i 1 i

dTT T T xdx

dx dxx T, x TdT dT= =

−Δ = − Δ

⎛ ⎞⎟⎜Δ = Δ ⇒ ℜ= Δ = Δ⎟⎜ ⎟⎜⎝ ⎠∑ ∑0

1T

CSDA0

dT dTdx

−⎛ ⎞⎟⎜⇒ℜ = ⎟⎜ ⎟⎜ ⎟⎜ρ⎝ ⎠∫

• Range is often given multiplied by density

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oRange(3)

• Unit is then [cm][g/cm3]=[g/cm2]

01T

CSDA0

dT dTdx

−⎛ ⎞⎟⎜ℜ = ⎟⎜ ⎟⎜ ⎟⎜ρ⎝ ⎠∫

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

potential of absorbent

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oRange(4)

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

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Straggling and multiple scattering

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

v4v

3v 2v1v

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oMultiple 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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oProjected range <t>

• Protons energy disposal at a given depth:

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oEnergy disposal

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

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oEnergy disposal(2)

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

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oMonte Carlo simulations

• Notice: e- most scattered α has highest S

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

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Hadron therapy

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

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oTables on the web

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

Summary