1. Department of Physics University of Oslo
Interaction Between Ionizing Radiation And Matter, Part 2 Charged-Particles Audun Sanderud

2. Excitation / ionization
Department of Physics University of Oslo
Excitation / ionization
• Incoming charged particle interact with atom/molecule:
Ionization
Excitation
• Ion pair created from ionization

3. Department of Physics University of Oslo
Elastic collision
• Interaction between two particles with conservation of kinetic energy ( and momentum):
m1, v m2
m1, v1
m2, v2 c q
• Classic mechanics give:

4. Department of Physics University of Oslo
Elastic collision(2)
• These equations gives the maximum transferred energy:

5. Department of Physics University of Oslo
Elastic collisions(3)
a) m1>>m2
a) m1=m2
a) m1<<m2
• Proton(#1)-electron(#2): qmax=0.03o, Emax=0.2 % T0
• Electron(#1)-electron(#2): qmax=90o, Emax=100 % T0

6. Elastic collisions-cross section
Department of Physics University of Oslo
Elastic collisions-cross section
• Rutherford proved that the cross section of elastic scattering is:
 Small scattering angels most probable
• Differentiated by the energy
 Small energy transferred most probable

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Stopping power
• 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 T0
T0-dT dx
nv targets per volume unit

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Impact parameter
• The charged particle collision is a Coulomb-force interaction
• Most important: the interaction with electrons
• The impact parameter b useful versus the classic atomic radius a

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Soft collisions
• b>>a: particle passes an atom in a large distance
• Small energy transitions to the atom
• The result is excitations (dominant) and ionization; amount energy transferred range from Emin to a certain energy H
• 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

10. Department of Physics University of Oslo
Soft collisions(2)
r0: classic electron radius = e2/4pe0mec2 I: mean excitation potential b: v/c z: charge of the incoming particle r: Density of the medium NAZ/A: Number of electrons per gram in medium H: Maximum transferred energy at soft collision

11. Mean excitation potential, I/Z [eV]
Department of Physics University of Oslo
Soft collisions(3)
• The quantum mechanic effects are specially seen in the excitation potential I
Mean excitation potential, I/Z [eV]
Atomic number, Z
• High Z – small transferred energy less likely

12. Hard collisions • b<<a: particle passes trough the atom
Department of Physics University of Oslo
Hard collisions
• b<<a: particle passes trough the atom
• Large (but few) energy transactions to single electron
• Amount energy transferred range from H to Emax
• Can be seen as an elastic collision between free particles (bonding energy nelectable)

13. Collisions stopping power
Department of Physics University of Oslo
Collisions stopping power
• The total collision stopping power is then (soft + hard):
• Important: increase with z2, decrease with v2, not dependent on particle mass

14. Sc/r in different media
Department of Physics University of Oslo
Sc/r in different media
• I and electron density (ZNA/A) gives the variation

15. SC for electrons/positrons
Department of Physics University of Oslo
SC for electrons/positrons
• Electron-electron scattering more complicated; interaction between identical particles
• Sc,soft/r: Bethe’s soft coll. formula
• Sc,hard/r: electron-elektron; Møller cross section positron-electron; Bhabha cross section
The characteristics similar to that of heavy particles

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Shell correction
• The approximation used in the calculations of SC assume v>>vatomic electron
• When v~vatomic electron no ionizations will occur
• Occur first in the K-shell - highest atomic electron speed
• Shell correction C/Z handles this, and reduce SC/r
• C/Z depend on particle velocity and medium

17. Density-effect correction
Department of Physics University of Oslo
Density-effect correction
• Charged particles polarizes the medium
Charged (+z) particle
• Weaker interaction with distant atoms because of the reduction of the Coulomb force field •
• Polarization increase with (relativistic) speed
• But: polarization not important at low r
• Most important for electrons / positrons

18. Density-effect correction(2)
Department of Physics University of Oslo
Density-effect correction(2)
• Density-effect correction d reduces Sc/r in solid and liquid elements
• Sc/r (water vapor) > Sc/r (water)
Dashed curves: Sc without d

19. Radiative stopping power
Department of Physics University of Oslo
Radiative stopping power
• When charged particles are accelerated by the Coulomb force from atomic electrons or nucleus photons can be emitted; Bremsstrahlung
Charged particle
atomic electron
• The Lamor equation (classic el.mag.) denote the radiation power from an acceleration, a, of a charged particle: e0: Permittivity of a vacuum

20. Radiative stopping power(2)
Department of Physics University of Oslo
Radiative stopping power(2)
• The case of a particle accelerated in nucleus field:
• Comparison of proton and electron as incoming:
• Bremsstrahlung not important for heavy charged particles

21. Radiative stopping power(3)
Department of Physics University of Oslo
Radiative stopping power(3)
• The maximum energy loss to bremsstrahlung is the total kinetic energy of the electron
• Energy transferred to radiation per pathlength unit: radiative stopping power:
•Br(T,Z) weak dependence of T and Z
• Radiative energy loss increase with T and Z

22. Total stopping power, electrons
Department of Physics University of Oslo
Total stopping power, electrons
• Total stopping power, electrons:
• Comparison:

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Radiation yield
• Estimated fraction of the electron energy that is emitted as bremsstrahlung:
Radiation yield, Y(T)
Water
Tungsten
Kinetic energy, T (MeV)

24. Comparison of Sc Electrons, total Electrons, collision
Department of Physics University of Oslo
Comparison of Sc
Electrons, total
Electrons, collision
Electrons, radiative
Protons, total
Kinetic energy, T [MeV]

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

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Other interactions(2)
• Nuclear interactions: Inelastic process in which the charged particle cause an excitation of the nucleus. Result: - Scattering of charged particle - Emission of neutron, g-quant, a-particle Not important below ~10 MeV (proton)
• 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

27. Braggs rule • Braggs rule for mixtures of n-atoms/elements:
Department of Physics University of Oslo
Braggs rule
• Braggs rule for mixtures of n-atoms/elements:

28. Linear Energy Transfer
Department of Physics University of Oslo
Linear Energy Transfer
• LETD; also known as restricted stopping power
• D, cutoff value; LETD includes all the soft and the fraction of the hard collision d-rays with energy<D
d-electron as a result of ionization
Trace of charged particle
d-electrons living the volume → energy transferred > D
• Sc includes energy transitions from Emin to Emax
• LETD the amount of energy disposed in a volume defined by the range of an electron with energy D

29. Linear Energy Transfer(2)
Department of Physics University of Oslo
Linear Energy Transfer(2)
• The energy loss per length unit by transitions of energy between Emin < E < D:
• If D = Emax then L= Sc ; unrestricted LET
• LETD given in keV/mm
• 30 MeV protons in water: LET100eV/L  = 0.53

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Range
• The range  of a charge particle in a medium is the expectation value of the pathlength p
• The projected range <t> is the expectation value of the farthest depth of penetration tf in its initial direction
Electrons: <t> <  Heavy particles: <t> ≈ 

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Range(2)
• Range can by approximated by the Continuous Slowing Down Approximation, CSDA
• Energy loss per unit length is given by dT/dx – gives an indirect measure of the range: T0 Dx

32. Range(3) • Range is often given multiplied by density
Department of Physics University of Oslo
Range(3)
• Range is often given multiplied by density
• Unit is then [cm][g/cm3]=[g/cm2]
• Range of a charge particle depend on: - Charge and kinetic energy - Density, electron density and average excitation potential of absorbent

33. Department of Physics University of Oslo
Range(4)

34. Straggling and multiple scattering
Department of Physics University of Oslo
Straggling and multiple scattering
• In a radiation field of charged particles there is: - variations in rate of energy loss - variations in scattering
The initial beam of particle at same speed and direction, are spread as they penetrate a medium

35. Department of Physics University of Oslo
Multiple scattering
• Electrons experience most scattering – characteristic of initially close to monoenergetic beam:
Initial beam
Beam at small depth in absorbent
Beam at large depth in absorbent
Number
Energy [MeV]

36. Projected range <t>
Department of Physics University of Oslo
Projected range <t>
• Characteristic of different type of particles penetrating a medium:

37. Energy disposal • Protons energy disposal at a given depth:
Department of Physics University of Oslo
Energy disposal
• Protons energy disposal at a given depth:

38. Department of Physics University of Oslo
Energy disposal(2)
• Electrons energy disposal at a given depth; multiple scattering decrease with kinetic energy:

39. Monte Carlo simulations
Department of Physics University of Oslo
Monte Carlo simulations
• Monte Carlo simulations of the trace after an electron (0.5 MeV) and an a-particle (4 MeV) in water
• Notice: e- most scattered a has highest S

40. Department of Physics University of Oslo
Hadron therapy
• Heavy charged particles can be used in radiation therapy – gives better dose distribution to tumor than photons/electrons

41. Department of Physics University of Oslo
Tables on the web
• Stopping power
• Attenuation coefficients

42. Summary