Lecture 10: Photon Interactions PDF

Lecture 10 November 7, 2023 BP8103 What we talked about last time Basics of photon interactions Attenuation coefficients Compton effect Photoelectric effect 2 What we’ll talk about today Photoelectric effect cont’d Pair production Putting it a...

Lecture 10 November 7, 2023 BP8103 What we talked about last time Basics of photon interactions Attenuation coefficients Compton effect Photoelectric effect 2 What we’ll talk about today Photoelectric effect cont’d Pair production Putting it all together (photon interactions) Light charged particle interactions 3 Review Number of atoms (Na per volume, V), = to (rho*Na/A) where A is the atomic mass in g/mol 4 Compton effect review 5 Derivation of Compton scattered photon energy 6 Derivation of Compton scattered photon energy ℎ𝑣 + 𝑚𝑒 𝑐 2 = ℎ𝑣 ′ + 𝐸𝑒 Conservation of energy ℎ𝑣 + 𝑚𝑒 𝑐 2 = ℎ𝑣 ′ + 𝑚𝑒 𝑐 2 + 𝐾𝐸 Total energy of electron post collision ℎ𝑣 + 𝑚𝑒 𝑐 2 = ℎ𝑣 ′ + 𝑚𝑒 𝑐 2 + 𝐾𝐸 ℎ𝑣 = ℎ𝑣 ′ + 𝐾𝐸 Simplifies to KE is the kinetic energy of the recoil e- 𝐸𝑒 = 𝑚𝑒 𝑐2 2 + 𝑝𝑒2 𝑐 2 ℎ𝑣 + 𝑚𝑒 𝑐 2 = ℎ𝑣 ′ + 𝐸𝑒 ℎ𝑣 + 𝑚𝑒 𝑐 2 = ℎ𝑣 ′ + 𝑚𝑒 𝑐 2 2 + 𝑝𝑒2 𝑐 2 7 Derivation of Compton scattered photon energy Conservation of momentum 𝑝𝛾 = 𝑝𝛾′ cos θ + pe 𝑐𝑜𝑠𝜑 Along the x-direction 0 = 𝑝𝛾′ sin θ − pe 𝑠𝑖𝑛𝜑 Along the y-direction 𝑝𝑒2 = 𝑝𝛾2 − 2𝑝𝛾 𝑝𝛾′ 𝑐𝑜𝑠𝜃 + 𝑝𝑣2′ Let’s plug in equations 2 and 3 Set them equal to each other ℎ𝑣 After some rearranging and cancelling ℎ𝑣 ′ = ℎ𝑣 1 − 𝑐𝑜𝑠𝜃 + 1 𝑚𝑒 𝑐 2 ℎ𝑣 Set ∈=hv/mec^2 ℎ𝑣 ′ = 8 1 +∈ 1 − 𝑐𝑜𝑠𝜃  Thomson scattering Coherent scattering What happens Photon interacts with atom Photon is elastically scattered (i.e. no energy is transferred) Using a classical model (non-relativistic) model of photon/electron scattering developed by J.J. Thompson Incoming photon is an EM wave Oscillating EM field causes electron to oscillate e- emits EM radiation at an angle theta relative to incident photon 9 Thomson scattering cross section 𝑑𝑒 σ𝑡ℎ 𝑟𝑒2 = 1 + cos 𝜃 𝑑Ω 2 Full derivation in section 7.2.1 Differential electronic attenuation coefficient as a function of solid angle from the interaction point Re=classical radius of e- (2.82 fm) 10 Compton attenuation coefficient Low energy limit is the Thompson cross-section At higher energies, corrections must be applied Klein Nishina Coefficient! (ranges from 0 to 1) 𝑑𝑒 𝜎𝑐𝐾𝑁 𝑑𝑒 𝜎𝑇ℎ = 𝐹𝐾𝑁 ℎ 𝑣 𝜃 𝑑Ω 𝑑Ω 11 Total Compton Cross-section At low energies 𝜎𝑐𝐾𝑁 ≈ 𝜎𝑡ℎ = 0.665 𝑏𝑎𝑟𝑛𝑠 At higher energies Dependent on hv 12 13 14 Scattering angles of Compton photon At low energies, there is an equal likelihood of backscattering and forward scattering Side scattering is half of forward/back scattering As energy increases, the scattering becomes more forward peaked and backscattering probability goes down 15  Binding energy and Compton effect Recall – we assumed that the electron was ‘free and stationary’ this is why the Compton energy transfer fraction (that special graph) only depends on energy but not Z This assumption breaks down at low photon energies Negligible because at those low energies, photoelectric effect dominates 16 Compton atomic cross section 17 Compton cross section electronic  atomic cross section Mass attenuation Based on this final equation, coefficient what is the Z dependence of the Compton attenuation coefficient? Expanded mass attenuation coefficient 18 Compton cross section Z/A  0.5 for all elements except H Attenuation coefficient is indep of Z electronic  atomic cross section Mass attenuation coefficient Expanded mass attenuation coefficient 19 20 Compton mass energy transfer coefficient 21 Photoelectric effect 22 Ejected photoelectron has energy dependent angle of emission 23  Photoelectric atomic cross section As Z increases: Probability of PE increases The atomic cross by Z^(4 to 5) section rises whenever photon energy As E increases: coincides with the binding energy of a Probability of PE decreases particular electron with (1/hv)^3 at low shell energies and 1/(hv) at high energies Absorption edges can be clearly seen, K, L, M edges 24 Atomic attenuation coefficient for photoelectric effect Symbol: 𝑎τ Three distinct regions 1. Around absorption edges (look like cliffs on the graph) Theoretical predictions in this region are uncertain 2. Away from absorption edges 3. Relativistic region far from K absorption edge 25  Atomic attenuation coefficient for photoelectric effect Symbol: 𝑎τ Three distinct regions Based on this information, we can see some general trends: 1. Around absorption edges (look like cliffs on the graph) Theoretical predictions in this region are uncertain 1. Energy dependence goes from 1/hv3 (at low hv) and 2. Away from absorption edges up to 1/hv (at high hv) 2. Atomic number dependence goes from Z4 to Z5 3. Relativistic region far from K absorption edge 26 Based on this information, we can see some general trends: 1. Energy dependence goes from 1/hv3 (at low hv) and up to 1/hv (at high hv) 2. Atomic number dependence goes from Z4 to Z5 27 Photoelectric effect mass attenuation coefficient Atomic photoelectric Number cross section of atoms 28 Energy transferred during photoelectric effect 3 outcomes Energy transferred = incoming photon energy – binding energy In this case, no auger electrons are produced Energy transferred = incoming photon energy No characteristic x-rays produced leaves, all excess energy given to Auger electrons Energy transferred = somewhere in between the above two possibilities 29 Energy transferred to charged particles If only characteristic x-rays are emitted: 𝐸𝑡𝑟 = ℎ𝑣 − 𝐸𝐵 If only Auger e- are emitted: 𝐸𝑡𝑟 ≈ ℎ𝑣 30 What happens if only Auger electrons are emitted (hypothetical) (1) Photoelectron ejected from k-shell with 𝐾𝐸 = ℎ𝑣 − 𝐸𝐵𝑘 (1) Electron from L shell moves to K shell, Auger electron emitted from M shell with the energy: 𝐾𝐸 = 𝐸𝐵𝐾 − 𝐸𝐵𝐿 − 𝐸𝐵𝐾 (1) Two electrons from N shell fill vacancies in M and L shells. 2 Auger electrons emitted from N shell with the following energies: 𝐾𝐸 = 𝐸𝐵𝐿 − 𝐸𝐵𝑀 − 𝐸𝐵𝑁 𝐾𝐸 = 𝐸𝐵𝑀 − 𝐸𝐵𝑁 ෍ 𝐾𝐸 = ℎ𝑣 − 4𝐸𝐵𝑁 ≈ ℎ𝑣 31 How do we determine total energy transferred? Add up Auger, CK, SCK, photoelectron energies Can get cumbersome given all of the cascade effects Smarter way: determine the mean fluorescence emission energy and subtract that from the incident photon energy ത 𝑃𝐸 (𝐸)𝑡𝑟 = ℎ𝑣 − 𝑋𝑃𝐸 Mean fluorescence (char. X-ray) emission energy 32 Photoelectric fluorescence emission energy 33 Fluorescence yield Definition: number of fluorescence (characteristic) photons emitted per vacancy in a given shell or subshell OR  probability (after a vacancy is created) that a fluorescent photon will be emitted instead of a characteristic x-ray We can express the probability of Auger emission as 34 Photoelectric fluorescence emission energy w = 1  no auger electrons, only the photoelectron is released w=0  no characteristic x- ray [no fluorescence], photoelectron and Auger electrons released w=0-1  both released 35 Probability of photoelectric effect (P) Probability of PE is highest for tightly bound e- in k- shell, Fig 7.29 in Podgorsak 36 Mean photoelectric energy transfer fraction Mean fraction of the incident photon’s energy that is transferred to kinetic energy of secondary charged particles released in a photoelectric event 37 Mean photoelectric energy transfer fraction 38 Pair and Triplet Production Photon – atom interaction Photon interacts with the Coulomb field of either the nucleus (Pair production) or an orbital e- (Triplet production) Photon disappears and an electron-positron pair is created (Pair) In the case of triplet production, the orbital e- is ejected 39 40 Energy Threshold for Pair Production Below threshold energy, pair production cannot occur Rough estimate is 2mec2 (1.022 MeV) More accurate calculation of this threshold energy if calculate momentum transferred to atom 41 before after Calculate using invariant (system that remains unchanged under some transformation) 𝐸 2 − 𝑝2 𝑐 2 𝑏𝑒𝑓𝑜𝑟𝑒 = 𝐸 2 − 𝑝2 𝑐 2 𝑎𝑓𝑡𝑒𝑟 Before (lab reference frame) After (Center of Mass reference frame) 𝐸 = ℎ𝑣 + 𝑚𝐴 𝑐 2 𝐸 = (𝑚𝐴 𝑐 2 + 2𝑚𝑒 𝑐 2 )^2 ℎ𝑣 𝑝=0 𝑝= 𝑐 2 ℎ𝑣 ℎ𝑣 + 𝑚𝐴 𝑐 2 2 − 𝑐 2 = 𝑚𝐴 𝑐 2 + 2𝑚𝑒 𝑐 2 2 𝑐 4𝑚𝑒 𝑐 2 𝑚𝐴 𝑐 2 + 𝑚𝑒 𝑐 2 ℎ𝑣 = 2𝑚𝐴 𝑐 2 𝑚𝑒 ℎ𝑣𝑡ℎ𝑟𝑒𝑠ℎ = 2𝑚𝑒 𝑐2 1+ ≈ 2𝑚𝑒 𝑐 2 42 𝑚𝐴 Triplet Production before after Before After 𝐸 = ℎ𝑣 + 𝑚𝑒 𝑐 2 𝐸𝑓 = 3𝑚𝑒 𝑐 2 2 +0 ℎ𝑣 𝑝𝑓 = 0 𝑝= 𝑐 2 ℎ𝑣 ℎ𝑣 + 𝑚𝑒 𝑐2 2 − 𝑐 2 = 3𝑚𝑒 𝑐 2 2 𝑐 8 𝑚𝑒 𝑐 2 2 2 = 2.044 𝑀𝑒𝑉 ℎ𝑣 = = 4𝑚 𝑒 𝑐 2𝑚𝑒 𝑐 2 43 Pair production 44 Kinetic energy of electron and positron Interaction Kinetic Energy Notes Pair Production ℎ𝑣 − 2𝑚𝑒 𝑐 2 - Not equally shared! 𝐾𝐸 = - Average energy 2 transferred to each e-/positron - Ignore KEatom Triplet Production ℎ𝑣 − 2𝑚𝑒 𝑐 2 - Ignore binding 𝐾𝐸 = energy 3 45 Nuclear screening For high photon energies (> 20 MeV), a large part of the pair production cross section is outside of the K-shell electron orbit K-shell electrons shield the incoming photon from the nucleus’s Coulomb field Screening correction is required to account for this 46  Attenuation Coefficients Function of energy and Z which account for SCREENING of nuclear charge by e- Pair production 2 2 𝜘 𝑎 𝑝𝑝 = 𝛼𝑟𝑒𝑍 𝑃 ∈ 𝑍 When there is no screening, 2𝑍2𝑃 ∈ 𝑍 When there is complete screening (E.g higher Z materials) 𝜘 𝑎 𝑝𝑝 = 𝛼𝑟𝑒 𝑃 ∈ 𝑍 proportional to ln(Z^(-1/3)) a 𝝒𝒑𝒑 is proportional to Z^2 (independent of energy after threshold) 47 48 Energy Transferred to CPs Pair 𝐸𝑡𝑟 = 𝐾𝐸+ + 𝐾𝐸− = 𝒉𝒗 − 𝟐𝒎𝒆 𝒄𝟐 Triplet 𝐸𝑡𝑟 = 𝐾𝐸+ + 𝐾𝐸− + 𝐾𝐸𝑒 = 𝒉𝒗 − 𝟑𝒎𝒆 𝒄𝟐 49 Pair production mean energy transfer fraction 50  Summary of Photon Interactions Interaction Cross-section Z-Dependence Energy Dependence Photoelectric 𝒂𝝉 Z4-5 1/hv3 Rayleigh 𝑎𝜎𝑅 Z2 1/hv2 Compton 𝒂 𝝈𝒄 / 1/hv Pair 𝒂𝜿𝒑𝒑 Z2 Ln(hv) Triplet 𝑎𝜅𝑡𝑝 Z Ln(hv) 51 Total Attenuation Coefficient 52 Questions 1. List the main photon interactions contributing to the mass attenuation coefficient of an x- ray. On a graph of Z vs. log Energy, sketch the photoelectric, Compton, and pair production probabilities 53 Questions 1. List the main photon interactions contributing to the mass attenuation coefficient of an x- ray. On a graph of Z vs. log Energy, sketch the photoelectric, Compton, and pair production probabilities 54 Question 2 Explain + draw the Compton effect. Describe the average kinetic energy of a Compton recoil e- from 10 keV to 3 MeV 55  Question 2 Explain + draw the Compton effect. Describe the average kinetic energy of a Compton recoil e- from 10 keV to 3 MeV 56 Question 3 Interaction Coefficient Energy dependence Z-dependence Photoelectric Compton effect Pair production 57 Question 3 58 Question 4 Describe energy transferred to the medium for all 3 photon interaction types 59 Question 4 Describe energy transferred to the medium for all 3 photon interaction types 60  Photoelectric All photon energy is transferred Ejected electron has the energy hv-Eb Photoelectron will go on to undergo soft and hard collisions Photoelectron can also undergo radiative collisions (Bremstrahlung) Compton Scattered photon and electron are the products of a Compton interaction Photon – continues in the medium and can undergo additional interactions Electron – will undergo soft and hard collisions Can also undergo Brem Pair production Electron and positron travel and impart energy via soft, hard, and radiative interactions. Positron slows down and undergoes annihilation with a resting e-. This results in the release of two 511 keV photons 61 Question 6 A 4 MeV photon is incident upon lead, given the following cross sections, calculate mass energy absorption coefficient 62  A 4 MeV photon is incident upon lead, given the following cross sections, calculate mass energy absorption coefficient 63  For a 2 MeV photon going through lead Question 7 What is the maximum energy lost through each of the three interactions 64  For a 2 MeV photon going through lead Question 7 What is the maximum energy lost through each of the three interactions 65 Charged Particle Interactions 66 Types of CP used in Medical Physics Electrons (e-) Primary Secondary Electrons Linear accelerator Interactions of photons with debris Auger effect Ionization Positrons Beta particle emitter Pair/triplet production Protons Cyclotron, synchrotron, Neutron activation particle therapy Alpha particles Particle therapy Proton activation Heavy ions (6+ C) Particle therapy 67  CP Interactions Charged particles interact via Coulomb interactions with nucleus and orbital e- Also include annihilation – positron + e- interact and disappear, two 511 keV photons released in opposite directions 68 69 Stopped here. 70 Stopping Power Since CPs have a lot of interactions compared to photons, we can visualize them as continuously losing energy Rate of energy loss is quantized by ‘stopping power’, which is the average energy lost per unit path length in MeV/cm Linear stopping power Mass stopping power Ne = density of atoms (atoms/cm^3) E= energy lost per interaction 71 Dsigma/dE = cross section for energy loss Total Stopping Power S𝑟𝑎𝑑 = 𝑆𝑟𝑎𝑑 + 𝑆𝑐𝑜𝑙 Stot = total stopping power Srad = radiative stopping power, energy lost due to radiative collisions (Brem) Scoll = collisional stopping power, energy lost due to soft and hard collisions Tables of Scol and Srad can be found on the NIST estar/pstar database 72 Question 8 What’ s the difference between stopping power and LET? 73