16/02/2005 FYS-KJM4710 - Lection II 1 Interaction Between Ionizing Radiation And Matter, Part 2 Charged-Particles Audun Sanderud Department of Physics University of Oslo 16/02/2005 FYS-KJM4710 - Lection II 2 • Incoming charged particle interact with atom/molecule: Department of Physics University of Oslo Excitation / ionization Ionization Excitation • Ion pair created from ionization
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16/02/2005 FYS-KJM4710 - Lection II 1
Interaction BetweenIonizing Radiation And Matter, Part 2
Charged-Particles
Audun Sanderud
Dep
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• Incoming charged particle interact with atom/molecule:
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Excitation / ionization
Ionization
Excitation
• Ion pair created from ionization
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• 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 v
2 2 2m v m v cos m v cos
0 m v sin m v sin
θ χθ χ
= = +
= += +
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Elastic 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
χ χ
χθχ
⇒ = = −+ +
=−
• These equations gives the maximum transferred energy:
• Rutherford proved that the cross section of elastic scattering is:
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Elastic collisions-cross section
( )4
d 1
d sin 2
σθ
∝Ω
→ Small scattering angels most probable
• Differentiated by the energy
2
d 1
dE E
σ ∝
→ Small energy transferred most probable
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• 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 and the atomic number of the medium
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oStopping power
T0 T0-dT
dx
nv targets per volume unit
max
min
max
min
max
min
E
V v
E
E
A
E
E
A
E
ddT En dx n dx EdE
dE
N Z ddx EdE
A dE
N ZS dT dEdE
dx A dE
σσ
σρ
σρ ρ
= =
⎛ ⎞⎟⎜= ⎟⎜ ⎟⎜⎝ ⎠
⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜= ⎟= ⎟⎜ ⎜⎟ ⎟⎜⎜ ⎟⎜ ⎝ ⎠⎝ ⎠
∫
∫
∫
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• The charged particle collision is a Coulomb-force interaction
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Impact parameter
• The impact parameter b useful versus the classic atomic radius a
• Most important: the interaction with electrons
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• 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 more mass then electrons
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Soft 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
• Estimated fraction of the electron energy that is emitted as bremsstrahlung:
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oRadiation yield
( ) ( )( ) ( )
r r
c r
dT / dx SY T
dT / dx dT / dx S
ρ= =
ρ + ρR
adia
tion
yie
ld, Y
(T)
Kinetic energy, T (MeV)
WaterTungsten
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Comparison of Sc
Kinetic energy, T [MeV]
Electrons, totalElectrons, collisionElectrons, radiativeProtons, total
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• Cerenkov effect: very high energetic electrons (v>c/n) polarize a medium (water) of refractive index n and bluish light is emitted (+UV)D
epar
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Other interactions
• Little energy is emitted
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• 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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Other 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
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• The range ℜ of a charge particle in a medium is the expectation value of the pathlength p
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• 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
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• Range can by approximated by the Continuous Slowing Down Approximation, ℜCSDA
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Range(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 x
dxdx dx
x T, x TdT dT= =
−Δ = − Δ
⎛ ⎞⎟⎜Δ = Δ ⇒ ℜ= Δ = Δ⎟⎜ ⎟⎜⎝ ⎠∑ ∑0
1T
CSDA
0
dTdT
dx
−⎛ ⎞⎟⎜⇒ℜ = ⎟⎜ ⎟⎜ ⎟⎜ρ⎝ ⎠∫
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• Range is often given multiplied by density
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• Unit is then [cm][g/cm3]=[g/cm2]
01T
CSDA
0
dTdT
dx
−⎛ ⎞⎟⎜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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Range(4)
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• In a radiation field of charged particles there is:- variations in rate of energy loss- variations in scattering D
epar
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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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Multiple 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
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• Characteristic of different type of particles penetrating a medium:
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• Protons energy disposal at a given depth:
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Energy disposal
16/02/2005 FYS-KJM4710 - Lection II 37
• Electrons energy disposal at a given depth; multiple scattering decrease with kinetic energy:
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• Monte Carlo simulations of the trace after an electron (0.5 MeV) and an α-particle (4 MeV) in water
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Monte Carlo simulations
• Notice: e- most scattered α has highest S
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• Heavy charged particles can be used in radiation therapy – gives better dose distribution to tumor than photons/electronsD