Outline
Charged-Particle • 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 Physics and Radiation Dosimetry mass (heavy vs. electrons)
Charged-particle interactions in Introduction matter
• Charged particles have surrounding Coulomb field • Always interact with electrons or nuclei of atoms in a - classical matter radius of atom • In each interaction typically only a small amount of particle’s kinetic energy is lost (“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 electron (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<antimatter only: in-flight annihilations called evaporation particles (mostly nucleons of – Two photons are produced relatively low energy) and -rays – Dose may not be deposited locally, the effect is <1-2% 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 MeVcm 2 Jm2 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 TQc dT TQc 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 positrons 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 TQ dT TQ 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 1.022106 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 protons 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, positron, 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 kln 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 kln F t d dx 2 2 Z Accounts for c 2I / 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 log10p / 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 ion 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 the constant 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 dxc 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 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 tdt t dt 0 0 0 0 dt 1 t 0 0 t t tdt P dNt N f CSDAM 0 z – charge of 0 0 t f tdt 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 dxc t dT MeV D t dx g c DT cos D 1.6021010 mass per unit 10 dT area of foil D 1.60210 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 1.6021010 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 1.6021010 (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 1.60210 0 dx c,w Summary • Types of charged particle interactions • Stopping power • Range • Calculation of absorbed dose 9