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<<a) • Nuclear interactions by heavy charged particles 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<<a) • Nuclear interactions by heavy charged particles – Most important for electrons – A heavy charged particle with kinetic energy ~ 100 • In all but 2-3% of cases electron is deflected through MeV and b<a may interact inelastically with the nucleus almost elastic scattering, losing almost no energy • In 2-3% of cases electron loses almost all of its energy – One or more individual nucleons may be driven out of through inelastic radiative (bremsstrahlung) interaction the nucleus in an intranuclear cascade process – Important for high Z materials, high energies (MeV) – The highly excited nucleus decays by emission of so- • For 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 dT 2Cm c2 z 2 2m c2 2H s 0 ln 0 2 2 2 2 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: Hard-Collision Term Excitation Potential H T dT s max h • The form of the hardTQ-collisiondT termTQ dependsdT on c c dx Tmin H whether thec 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