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The Basic Interactions Between Photons and Charged Particles With Outline Chapter 6 The Basic Interactions between • Photon interactions Photons and Charged Particles – Photoelectric effect – Compton scattering with Matter – Pair productions Radiation Dosimetry I – Coherent scattering • Charged particle interactions – Stopping power and range Text: H.E Johns and J.R. Cunningham, The – Bremsstrahlung interaction th physics of radiology, 4 ed. – Bragg peak http://www.utoledo.edu/med/depts/radther Photon interactions Photoelectric effect • Collision between a photon and an • With energy deposition atom results in ejection of a bound – Photoelectric effect electron – Compton scattering • The photon disappears and is replaced by an electron ejected from the atom • No energy deposition in classical Thomson treatment with kinetic energy KE = hν − Eb – Pair production (above the threshold of 1.02 MeV) • Highest probability if the photon – Photo-nuclear interactions for higher energies energy is just above the binding energy (above 10 MeV) of the electron (absorption edge) • Additional energy may be deposited • Without energy deposition locally by Auger electrons and/or – Coherent scattering Photoelectric mass attenuation coefficients fluorescence photons of lead and soft tissue as a function of photon energy. K and L-absorption edges are shown for lead Thomson scattering Photoelectric effect (classical treatment) • Electron tends to be ejected • Elastic scattering of photon (EM wave) on free electron o at 90 for low energy • Electron is accelerated by EM wave and radiates a wave photons, and approaching • No energy is given to the electron; wavelength of the scattered 0o with increase in energy photon does not change • Kinetic energy given to • Classical scattering coefficient per electron per unit solid angle: electron KE=hv-E , b d r 2 independent of scattering 0 0 1 cos2 angle d 2 0, 180o 3 3 max at • Interaction probability ~Z /(hv) ½ max at 90o • No universal analytical expression for cross-section e2 r0 2 - classical radius of electron m0c 1 Thomson scattering (classical treatment) Coherent scattering 30 2 • Photon is scattered by combined action of the whole atom • No energy dependence, total 0 66.510 m electron • Photons do not lose energy, just get redirected through a • Works for low photon energies, << m c2 0 small angle • Overestimates for photon energies > 0.01MeV (factor of 2 • No charged particles receive energy, no excitation produced for 0.4MeV) • Scattering coefficient (F - atomic form factor, tabulated): d 2 sind 2 d r coh 0 1 cos2 F(x,Z)2 2 sin 2 d 2 d 0 r0 2 1 cos 2 sin x sin / 2/ d 2 Typical ratios of coherent to total attenuation coefficient Example 1 Compton scattering kinematics • Incoherent scattering – energy is transferred • A 5-keV photon undergoing coherent or to electron (inelastic scattering) classical scatter would be most likely to lose • Energy and momentum are conserved ___% of its energy in the process. 2 1 hv hv'E hv'm0c 1 A. 0 2 2 1 v / c B. 10 hv hv' m0v C. 50 pII cos cos c c 1 v2 / c2 D. 90 E. 100 hv' m0v p 0 sin sin c 1 v2 / c2 Compton scattering kinematics Compton scattering probability 2 • Introducing parameter hv/ m0c • Klein-Nishina coefficient: Compton scattering on free electron, but includes Dirac’s quantum relativistic theory (1 cos ) E hv Energy of electron 1(1 cos) d d r 2 0 F 0 1 cos2 F , where 1 d d KN 2 KN hv' hv Energy of scattered photon 1(1 cos ) 2 1 2 (1 cos)2 F 1 • The maximum energy that electron can acquire in a direct hit KN 2 ( = 180o ): 1(1 cos) 1(1 cos)(1 cos ) 2 1 Emax hv ; photon energy : hv'min hv 0.255MeV 1 2 1 2 • Factor FKN describes the deviation from classical scattering For high-energy photon – FKN <1 for higher energies; o • In a grazing hit ( ~ 0 ) E=0, and for the scattered photon – FKN =1 for small (low energy) or =0 hv' hv 2 Energy distribution of Compton electrons Effect of binding energy • In real situation electrons (bound are bound and in motion • Lower energy and high electron) energy electrons are • Additional factor S(x, Z) (free electron) more likely to be is introduced (free produced electron) d d inc S(x,Z) • Higher energy photons d d are more likely to x=sin(/2)/ transfer more energy to Compton electrons • Makes the scattering coefficient Z-dependent Example 2 Example 3 • Compton scattered electrons can be emitted at: • In Compton scattering original photon has A. Any angle. energy 1.25 MeV, scattered at 30°. B. 0o-90o with respect to the direction of the incident photon Calculate the energy of scattered photon. C. 30o-120o with respect to the direction of the incident photon. hv 1.25 A. 1.2 MeV 2 2.45 o o m c 0.511 D. 90 -180 with respect to the direction of the incident B. 1.1 MeV 0 1 photon. hv' hv C. 0.94 MeV 1(1 cos ) D. 0.83 MeV 1 1.25 0.941MeV 1 2.45(1 cos 30) Pair production Pair production • Energy in excess of 1.02 MeV is released as kinetic energy • Photon is absorbed giving rise to of the electron and positron electron and positron 2 hv 2m0c T T 1.022 MeVT T • Occurs in Coulomb force field - usually near atomic nucleus; • Differential cross section: threshold energy 1.022 MeV 2 2 d Z r 2 4 • Sometimes occurs in a field of atomic electron (triplet 0 m c F d 137 2 0 pair production); threshold energy 2.044 MeV • For 10-MeV photons in soft tissue, for example, about 10% • Fpair is a function of • The ratio of triplet to pair production increases with E of momentum, energy and incident photons and decreases with Z angle of positron projection 3 Heavy charged particle Charged particle interactions interactions • All charged particles lose kinetic energy through • Energy transferred to the electron of the medium Coulomb field interactions with charged particles z 2r 2m c4 M (electrons or nuclei) of the medium E 0 0 b2 E • In each interaction typically only a small amount of • Parameters: particle’s kinetic energy is lost (“continuous – E – kinetic energy of the particle; ze- its charge; M – slowing-down approximation” – CSDA) mass • Typically undergo very large number of interactions, – b is the impact parameter therefore can be roughly characterized by a common • Slower moving particle (lower KE) transfers more path length in a specific medium (range) energy Interactions of electrons Bremsstrahlung interactions • In any given collision with another electron, one • Fast moving charged particle of mass M, and emerging with higher energy is assumed to be charge ze, passing close to a nucleus of mass MN primary (max energy exchange is limited to half of >>M and charge Ze will experience electric force, its original energy) corresponding to accelerations: F kzZe2 • Due to small mass a – Relativistic effects are important M r 2M – Interactions with nucleus: rapid deceleration results in • Accelerated charge radiates energy at a rate ~a2 bremsstrahlung (breaking) radiation • The rate of energy loss is negligible for particles other than electrons (even protons) due to 1/M2 Stopping power Stopping power • Assuming CSDA, can evaluate the amount of energy • Mass stopping power – energy loss per unit track lost by charged particle per unit track length, dE/dx length in producing ionization in the absorbing • Total mass stopping power – energy loss per unit medium 2 2 1 dE 2 z m0c track length, normalized to the medium density Sion 4r0 Ne ... dx 2 • Parameters: Ne – number of electrons per gram; […] is a slowly increasing function of the particle energy • Energy will be spent on excitations and ionizations • For electrons […] term is more complex (through collisions), can also be emitted in a • Radiation stopping power – energy loss due to radiative process bremsstrahlung 1 dE 2 NeZE for electrons Srad 4r0 ... dx 137 4 Stopping power Bragg peak Never observed • Slower moving charged particle (lower KE) transfers more energy, • Energy loss by radiation increases with atomic number of medium and resulting in Bragg peak at the end of its track electron energy • Never observed for electrons: due to their low mass, electrons constantly change direction as they slow down • Sharp increase in stopping power at low energy leads to the Bragg peak Range of electrons Example 4 • Charge particles are characterized by a range – a finite distance • A 6 MeV electron beam passes through 2 cm of Half-value depth • Can be calculated from stopping tissue, overlying lung (density 0.25 g/cm3). The power in continuous slowing approximate range in the patient is ___cm. Practical range down approximation (CSDA) E0 R ~3 cm dE A. 1 tissue R R ~2 cm (=1) + 1 cm (=0.25) S(E) B. 3 tissue+lung 0 C. 6 =2 cm + 4 cm = 6 cm Rule of thumb: for electron D. 9 energies > 0.5 MeV, the range in unit density material ( in E. 12 g/cm3) is, in cm, about half the energy in MeV Example 5 Mean stopping power • A Cerrobend cutout for a 12 MeV electron beam • If electron beam is not monoenergetic need to should be approximately ___cm thick, to stop all average over all electron energies the electrons and transmit only the E1 dF(E) bremsstrahlung. (Hint: Cerrobend requires 20% S (E)dE ion more thickness than lead.) 0 dE S (Ei ) E1 dF(E) 3 dE A. 7 lead=11 g/cm dE R ~0.5x12/11=0.55 cm 0 B. 4 lead • Similarly, if electrons are produced by C. 1.5 RCerrobend~0.55x(1+0.2)= 0.66 cm D.
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