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
n ex
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pote
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l, I/Z
[eV
]
• 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