Charged-Particle Interactions in Matter
Introduction Charged-Particle • Charged particles have surrounding Coulomb field • Always interact with electrons or nuclei of atoms in Interactions in Matter matter • In each interaction typically only a small amount of particle’s kinetic energy is lost (“continuous slowing- Chapter 8 down approximation” – CSDA) • Typically undergo very large number of interactions, F.A. Attix, Introduction to Radiological therefore can be roughly characterized by a common Physics and Radiation Dosimetry path length in a specific medium (range)
Charged-particle interactions in Types of charged-particle matter interactions in matter Impact parameter b • “Soft” collisions (b>>a) • “Soft” collisions (b>>a) – The influence of the particle’s Coulomb force • Hard (“Knock-on” field affects the atom as a whole collisions (b~a) – Atom can be excited to a higher energy level, or • Coulomb interactions ionized by ejection of a valence electron with nuclear field (b< Types of charged-particle Types of charged-particle interactions in matter interactions in matter • Hard (“Knock-on”) collisions (b~a) • Coulomb interactions with nuclear field (b< 1 Types of charged-particle Stopping Power interactions in matter dT • Nuclear interactions by heavy charged particles dx – A heavy charged particle with kinetic energy ~ 100 Y ,T ,Z MeV and bnucleons may be driven out of loss per unit of path length x the nucleus in an intranuclear cascade process – Charged particle of type Y – The highly excited nucleus decays by emission of so- – Having kinetic energy T called evaporation particles (mostly nucleons of – Traveling in a medium of atomic number Z relatively low energy) and -rays • Units: MeV/cm or J/m – Dose may not be deposited locally, the effect is <1-2% Mass Stopping Power Mass Collision Stopping Power dT • Only collision stopping power contributes to the energy deposition (dose to medium) dx Y ,T ,Z • Can be further subdivided into soft and hard • - density of the absorbing medium collision contributions 2 2 • Units: MeVcm J m dT dT dT or s h g kg dxc dxc dxc • May be subdivided into two terms: – collision - contributes to local energy deposition • Separately calculated for electrons and heavy particles – radiative - energy is carried away by photons Mass Collision Stopping Power Mass Collision Stopping Power H T dT s max h TQc dT TQc dT T H min dxc 1. T´ is the energy transferred to the atom or electron 4. T´max is related to T´min by 2. H is the somewhat arbitrary energy boundary between 2 2 2 6 2 2 Tmax 2m0c 1.02210 eV soft and hard collisions, in terms of T´ Tmin I I 3. T´max is the maximum energy that can be transferred in a head-on collision with an atomic electron (unbound) where I is the mean excitation potential of the atom 2 s h – For a heavy particle with kinetic energy < than its M0c 5. Q c and Q c are the respective differential mass 2 2 T 2m c2 1.022 MeV, v/c max 0 2 2 collision coefficients for soft and hard collisions, 1 1 2 2 – For positrons incident, T´max = T if annihilation does not occur typically in units of cm /g MeV or m /kg J – For electrons T´max T/2 2 Soft-Collision Term Soft-Collision Term dT 2Cm c2 z 2 2m c2 2 H s 0 ln 0 2 • The mean excitation potential I is the geometric-mean 2 2 2 dxc I 1 value of all the ionization and excitation potentials of an atom of the absorbing medium here C (N Z/A)r 2 = 0.150Z/A cm2/g; in which N Z/A is the A 0 A • In general I for elements cannot be calculated number of electrons per gram of the stopping medium, and r0 = 2 2 -13 e /m0c = 2.818 10 cm is the classical electron radius • Must instead be derived from stopping-power or range measurements • For either electrons or heavy particles (z- elem. charges) – Experiments with cyclotron-accelerated protons, due to their • Based on Born approximation: particle velocity is much availability with high -values and the relatively small effect greater than that of the atomic electrons (v = c>>u) of scattering as they pass through layers of material • Verified with cyclotron-accelerated protons • Appendices B.1 and B.2 list some I-values Mass Collision Stopping Power Hard-Collision Term for Heavy Particles • The form of the hard-collision term depends on dT Zz 2 2 0.3071 13.8373 ln 2 ln I whether the charged particle is an electron, positron, 2 2 dx A 1 or heavy particle c • For heavy particles, having masses much greater • Combines both soft and hard collision contributions than that of an electron, and assuming that H << • Depends on Z - stopping medium, z - particle charge, T´max, the hard-collision term may be written as particle velocity through =v/c (not valid for very low ) • The term –ln I provides even stronger variation with Z dTh Tmax 2 (the combined effect results in (dT/dx) for Pb less than kln c dxc H that for C by 40-60 % within the -range 0.85-0.1) • No dependence on particle mass Mass Collision Stopping Power for Heavy Particles Shell Correction =v/c • When the velocity of the passing particle ceases to be much greater than that of the atomic electrons in the Accounts for stopping medium, the mass-collision stopping power is Bragg peak 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 kinetic energy • The so-called “shell correction” is intended to account required by any particle to reach a for the resulting error in the stopping-power equation given velocity is proportional to its • The correction term C/Z is the same for all charged 2 rest energy, M0c particles of the same , and is a function of the medium 3 Mass Collision Stopping Power for Electrons and Positrons Polarization Effect • Atoms near the particle track get polarized, decreasing the Coulomb dT 2 2 2C kln F force field and corresponding interaction dx 2 2 Z c 2I / m0c • Introduce density-effect correction influencing soft collisions T m c2 • The correction term, , is a function of the composition and density 0 of the stopping medium, and of the parameter 2 • Combines both soft and hard collision contributions log10 p / m0c log10 / 1 • F() term – depends on and for the particle, in which p is its relativistic momentum mv, and m0 • Includes two corrections: is its rest mass - shell correction 2C/Z • Mass collision stopping power decreases in condensed media - correction for polarization effect • Relevant in measurements with ion chambers at energies > 2 MeV Polarization Effect Polarization Effect Appendix E contains tables of electron stopping powers, ranges, radiation yields, and density-effect corrections • increases almost linearly as a function of above 1 for a 2 variety of condensed media The steep rise in collision stopping power for < m0c is not shown, but the 2 • It is somewhat larger for low-Z than for high-Z media minimum at 3 m0c is evident, as is the continuing rise at still higher energy Mass Radiative Stopping Power Mass Radiative Stopping Power • Only electrons and positrons are light enough to • The mass radiative stopping power is generate significant bremsstrahlung (1/m2 dependence 2 proportional to NAZ /A, while the mass for particles of equal velocities) collision stopping power is proportional to • The rate of bremsstrahlung production by electrons or positrons is expressed by the mass radiative stopping NAZ/A, the electron density power (in units of MeV cm2/g) • Ratio of radiative to collision stopping power 2 dT N AZ 2 dT dx TZ T m c B r dx 0 A 0 r r dT dxc n 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, T – kinetic energy, Z – atomic number, n ~700 or 800 and ̅Br is a slowly varying function of Z and T MeV 4 Mass Stopping Powers vs. Energy and Z Restricted Stopping Power • Energy cutoff allows to account for escaping Relatively delta-rays independent of Z • Linear Energy Transfer Range Radiation Yield • The range of a charged particle of a given type • The radiation yield Y(T0) of a charged particle of initial and energy in a given medium is the expectation kinetic energy T0 is the total fraction of that energy that value of the pathlength p that it follows until it is emitted as electromagnetic radiation while the particle comes to rest (discounting thermal motion) slows and comes to rest • The projected range Range CSDA Range: Protons T 1.77 1 ( ) 0 CSDA 415 670 • T0 – starting energy of the particle • Units: g/cm2 • Appendix E Greater for higher Z due to decrease in stopping power 5 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 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 P P CSDA range values for protons at energy T0 T0M0 M0 P CSDAM 0 CSDA P 2 M 0 z Projected Range Electron Range • Electrons typically undergo multiple scatterings • 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 6 Calculation of Absorbed Dose Dose from Heavy Particles Energy lost in collision interactions (energy imparted) Based on range can find the residual kinetic energy of exiting particle T T0 Tex Absorbed dose E T T cos D 1.6021010 mass per unit t area of foil If beam is not perpendicular – accounts for angle Dose in the foil is independent of its thickness 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: T (T0 Tex )c • Average dose: T D 1.6021010 c t The Bragg Curve Dose vs. Depth for Electron Beams • No Bragg peak • Diffused peak at ~half of tmax 7 Calculation of Absorbed Dose at Depth Summary • Types of charged particle interactions • At any point P at depth x in a medium w for • Stopping power know fluence spectrum • Range • For x below particle range • Calculation of absorbed dose 8