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Outline

Charged- • Impact parameter and types of charged Interactions in Matter particle interactions • Stopping power • Range Chapter 8 • Calculation of absorbed dose F.A. Attix, Introduction to Radiological • Approach typically depends on particle and Radiation Dosimetry mass (heavy vs. )

Charged-particle interactions in Introduction matter

• Charged have surrounding Coulomb field • Always interact with electrons or nuclei of in a - classical matter radius of • In each interaction typically only a small amount of particle’s kinetic energy is (“continuous slowing- down approximation” – CSDA) • Typically undergo very large number of interactions, therefore can be roughly characterized by a common Impact parameter b E. B. Podgorsak, Radiation • “Soft” collisions (b>>a) Physics for Medical Physicists, path length in a specific medium (range) 3rd Ed., Springer 2016, p.231 • Hard (“Knock-on” collisions (b~a) • Coulomb interactions with nuclear field (b<

Types of charged-particle Types of charged-particle interactions in matter interactions in matter • “Soft” collisions (b>>a) • Hard (“Knock-on”) collisions (b~a) – The influence of the particle’s Coulomb force – Interaction with a single atomic (treated as field affects the atom as a whole free), which gets ejected with a considerable K.E. – Atom can be excited to a higher energy level, or – Ejected d-ray dissipates energy along its track ionized by ejection of a valence electron – Interaction probability is different for different – Atom receives a small amount of energy (~eV) particles (much lower than for soft collisions, but – The most probable type of interactions; accounts fraction of energy spent is comparable) for about half of energy transferred to the medium – Characteristic x-ray or Auger electron is also produced

1 Types of charged-particle Types of charged-particle interactions in matter interactions in matter • Coulomb interactions with nuclear field (b<

Stopping Power Mass Stopping Power  dT   dT       dx Y ,T ,Z  dx Y ,T ,Z • The expectation value of the rate of energy •  - density of the absorbing medium MeVcm 2 Jm2 loss per unit of path length x • Units: or – Charged particle of type Y g kg – Having kinetic energy T • May be subdivided into two terms: – Traveling in a medium of atomic number Z • Units: MeV/cm or J/m – collision - contributes to local energy deposition – radiative - energy is carried away by photons

Mass Collision Stopping Power Mass Collision Stopping Power

H T • Only collision stopping power contributes  dT  s max h    TQc dT  TQc dT   T H to the energy deposition (dose to medium) min  dx c • Can be further subdivided into soft and hard 1. T´ is the energy transferred to the atom or electron collision contributions 2. H is the somewhat arbitrary energy boundary between  dT   dT   dT  soft and hard collisions, in terms of T´     s    h   dx   dx   dx  3. T´max is the maximum energy that can be transferred in  c  c  c a head-on collision with an atomic electron (unbound) 2 – For a heavy particle with kinetic energy < than its M0c   2    2  • Separately calculated for electrons and T  2m c2   1.022  MeV,   v/c max 0  2   2  heavy particles 1   1   – For incident, T´max = T if annihilation does not occur – For electrons T´max  T/2

2 Mass Collision Stopping Power Soft-Collision Term

H T  dT  s max h 2 2  2 2     TQ dT  TQ dT  dTs  2Cm0c z  2m0c  H  2    c c    ln    dx Tmin H   2   2 2    c  dx c    I 1    

4. T´max is related to T´min by 2 2 here C  (NAZ/A)r0 = 0.150Z/A cm /g; in which NAZ/A is the 2 2 T  2m c2 2   1.022106 eV 2  number of electrons per gram of the stopping medium, and r0 = max  0    2 2 -13       e /m0c = 2.818  10 cm is the classical electron radius Tmin  I   I  where I is the mean excitation potential of the atom • For either electrons or heavy particles (z - elem. charges) s h 5. Q c and Q c are the respective differential mass • Based on Born approximation: particle velocity is much collision coefficients for soft and hard collisions, greater than that of the atomic electrons (v = c>>u) 2 2 typically in units of cm /g MeV or m /kg J • Verified with cyclotron-accelerated

Soft-Collision Term: Soft-Collision Term Excitation Potential • Can further simplify the expression by introducing • The mean excitation potential I is the geometric-mean value of all the ionization and excitation potentials of an 2Cm c2 z 2 Zz2 MeV atom of the absorbing medium k  0  0.1535  2 A 2 g / cm 2 • In general I for elements cannot be calculated   v c • Must instead be derived from stopping-power or range measurements • The factor k controls the dimensions in which the – Experiments with cyclotron-accelerated protons, due to their stopping power is to be expressed availability with high -values and the relatively small effect of scattering as they pass through layers of material • Appendices B.1 and B.2 list some I-values

Soft-Collision Term: Excitation Potential Hard-Collision Term • The form of the hard-collision term depends on whether the charged particle is an electron, , or heavy particle • For heavy particles, having masses much greater than that of an electron, and assuming that H <<

T´max, the hard-collision term may be written as

Mean ionization/excitation potential I against atomic  dTh   Tmax  2     kln     number Z. Also plotted are dx  H  three empirical relationships  c  

E. B. Podgorsak, Radiation Physics for Medical Physicists, 3rd Ed., Springer 2016, p.240

3 Mass Collision Stopping Power Mass Collision Stopping Power for Heavy Particles for Heavy Particles

 dT  Zz2    2      0.3071 13.8373 ln    2  ln I   2   2    dx c A  1   

• Combines both soft and hard collision contributions • Depends on stopping medium properties through Z/A, • No dependence on particle mass which decreases by ~20% from C to Pb • Depends on particle properties: • The term –ln I provides even stronger variation with Z – z - particle charge (the combined effect results in (dT/dx)c for Pb less than that for C by 40-60 % within the -range 0.85-0.1) – particle velocity through =v/c (not valid for very low ) Overall trend for heavy charged particles: stopping power decreases with Z

Mass Collision Stopping Power Mass Collision Stopping Power for Heavy Particles for Electrons and Positrons =v/c  dT    t 2 t  2  2C     kln   F  t d    dx   2 2  Z Accounts for  c   2I / m0c    Bragg peak 2 t  T m0c

• Combines both soft and hard collision contributions The kinetic energy • F(t) term – depends on  and t required by any particle to reach a • Includes two corrections (introduced by U. Fano): given velocity is proportional to its 2 - shell correction 2C/Z rest energy, M0c - correction for polarization effect d

Shell Correction Shell Correction • When the velocity of the passing particle is not much greater than that of the atomic electrons in the stopping medium, the mass-collision stopping power is over- estimated • Since K-shell electrons have the highest velocities, they are the first to be affected by insufficient particle velocity, the slower L-shell electrons are next, and so on • The so-called “shell correction” is intended to account for the resulting error in the stopping-power equation • The correction term C/Z is a function of the medium and • The correction term C/Z is the same for all charged the particle velocity for all charged particles of the same particles of the same , and is a function of the medium velocity, including electrons

4 Polarization Effect Polarization Effect • Atoms near the particle track get polarized, decreasing the Coulomb force field and corresponding interaction (and energy loss) • Introduce density-effect correction influencing soft collisions • The correction term, d, is a function of the composition and density of the stopping medium, and of the parameter

2   log10p / m0c  log10 / 1   Appendix E contains tables of electron for the particle, in which p is its relativistic momentum mv, and m0 stopping powers, ranges, radiation is its rest mass yields, and density-effect corrections d • Mass collision stopping power decreases in condensed media • d increases almost linearly as a function of  above   1 for a • Relevant in measurements with chambers at energies > 2 MeV variety of condensed media (relativistic effect) • Effect is always present in metals (d ~0.1 even at low energies) • It is somewhat larger for low-Z than for high-Z media

Polarization Effect Mass Radiative Stopping Power • Only electrons and positrons are light enough to generate significant bremsstrahlung (1/m2 dependence for particles of equal velocities) • The rate of bremsstrahlung production by electrons or positrons is expressed by the mass radiative stopping power (in units of MeV cm2/g)

2  dT  N AZ 2     0 T  m0c Br  dx r A 1 2 2 2 -28 here 0 = /137(e /m0c ) = 5.80  10 cm2/atom, T is the particle kinetic energy in MeV, 2 The steep rise in collision stopping power for  < m0c is not shown, but the 2 and ̅Br is a slowly varying function of Z and T minimum at  3 m0c is evident, as is the continuing rise at still higher energy

Mass Radiative Stopping Power Radiation Yield • The mass radiative stopping power is 2 • The radiation yield Y(T0) of a charged particle of proportional to NAZ /A, while the mass initial kinetic energy T0 is the total fraction of that collision stopping power is proportional to energy that is emitted as electromagnetic radiation NAZ/A, the electron density while the particle slows and comes to rest • Ratio of radiative to collision stopping power • For heavy particles Y(T0)  0 • For electrons the production of bremsstrahlung x-rays dT dx TZ r  in radiative collisions is the only significant dT dxc n contributor to Y(T0) • For positrons, in-flight annihilation would be a second T – kinetic energy, Z – atomic number, n ~700 or 800 significant component, but this has typically been MeV omitted in calculating Y(T0)

5 Mass Stopping Powers vs. Energy and Z Restricted Stopping Power • Energy cutoff allows to account for escaping delta- Relatively rays with energy ≥D (not depositing energy locally) independent of Z • Linear Energy Transfer (radiobiology and microdosimetry)

  dT  2  LD keV m    MeV cm g 10  dx D 

• If the cut-off energy D is increased to Tmax – use unrestricted LET:  dT   dT       and LD  L  dx D  dx c

Restricted Stopping Power Range • The range  of a charged particle of a given type • Restricted stopping and energy in a given medium is the expectation power LD is smaller than unrestricted value of the pathlength p that it follows until it • For air-filled ion comes to rest (discounting thermal motion) chamber with • The projected range of a charged particle of a electrode separation given type and initial energy in a given medium is 2mm Δ=10keV is the expectation value of the farthest depth of reasonable (the range of a 10 keV electron penetration tf of the particle in its initial direction in air ~2mm) • Both are non-stochastic quantities • Can introduce range based on starting kinetic energy

E. B. Podgorsak, Radiation Physics for Medical Physicists, 3rd Ed., Springer 2016, p.240

CSDA Range CSDA Range: Protons

T 1.77 1 ( )  0  for carbon, 1 to 300MeV CSDA 415 670

1 Continuous T0  dT  CSDA    dT slowing down 0    dx  approximation

• T0 – starting energy of the particle • Units: g/cm2

• Appendix E Range is greater for higher Z due to decrease in stopping power

6 CSDA Range: Other Heavy Particles Projected Range N  1  N0 t T  M c2  1 For particles with the same velocity 0 2  1   t • Kinetic energy of a particle ~ to its rest mass • Count the number of particles that penetrate a slab • Stopping power for singly charged particle is of increasing thickness (disregard nuclear reactions)

independent of mass • N0 number of incident mono-energetic particles in • Consequently, the range is ~ to its rest mass a beam perpendicular to the slab

• Can calculate the range for a heavy particle based on   dN t P P   CSDA range values for protons at energy T  T M M t t f tdt t  dt  0 0 0 0   dt 1 t  0  0   t t tdt P   dNt N  f CSDAM 0 z – charge of 0 0 t f tdt dt CSDA  P 2   heavy particle 0 0 dt M 0 z

Projected Range Electron Range

range straggling – due to • Electrons typically undergo multiple stochastic scatterings variations in rates of energy loss • Range straggling and energy straggling due to stochastic variations in rates of energy loss • Makes range less useful characteristic, except for low-Z materials, where range is comparable to max penetration depth tmax • For high-Z range increases, t is almost independent

Electron Range Calculation of Absorbed Dose

Parallel beam of charged particles of kinetic

energy T0 perpendicularly incident on a foil Z Assumptions: • Collision stopping power is constant and

depends on T0 • Scattering is negligible • Effect of delta rays is negligible

• For low-Z media, tmax is comparable to R, which increases as a function of Z • tmax is almost independent of Z

7 Calculation of Absorbed Dose Dose from Heavy Particles Energy lost in collision interactions (energy imparted) Based on range can find the residual kinetic energy  dT  MeV t – mass thickness E    t    2  of foil, t – mean of exiting particle  dx   cm  c electron pathlength DT  T T Absorbed dose: 0 ex E  DT dT dxc t  dT  MeV D       t  dx   g  c DT cos D 1.6021010 mass per unit 10  dT  area of foil D 1.60210   Gy t  dx c If beam is not perpendicular – t/cos accounts for angle Dose in the foil is independent of its thickness (under our assumptions) Dose in Gray

Dose from Electrons Electron Backscattering

• Need to account for path lengthening due to scatter • Need to account for bremsstrahlung production, consider radiation yield • Energy spent in collisions:

DTc  (T0 Tex )c • Average dose: DT D 1.6021010 c t

The Bragg Curve Dose vs. Depth for Electron Beams

8 Dose vs. Depth for Electron Beams Calculation of Absorbed Dose at Depth • No Bragg peak • Diffused peak at Tmax  dT  ~half of t D 1.6021010  (T)  dT max w  x   0  dx c,w • At any point P at depth x in a medium w for know fluence spectrum • For x below particle range

10  dT  Dx 1.60210 0    dx c,w

Summary

• Types of charged particle interactions • Stopping power • Range • Calculation of absorbed dose

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