PHYS 5012 Radiation Physics and Dosimetry

Transcription

1 PHYS 5012 Radiation Physics and Dosimetry Tuesday 12 March 2013

2 What are the dominant photon interactions?

3 (cont.) Compton scattering, photoelectric absorption and pair production are the three main energy transfer mechanisms in photon interactions with matter. Each of these processes can dominate under specific conditions determined chiefly by the incident photon energy hν and atomic number Z of the absorber.

4 kinematics (cont.) When hν becomes comparable to m e c 2, the simple classical description of electromagnetic waves scattering off an electron in terms of harmonic oscillators breaks down. Both quantum and relativistic effects need to be considered in the kinematics and dynamics of the interaction. The quantization of the electromagnetic field is equivalent to breaking it up into particles (photons). Photon energies are comparable to the electron rest-mass energy, so the centre-of-mass of the scattering is moving in the observer frame, which results in a Doppler shift in the photon energy. In the observer frame, the decrease in photon energy appears as electron recoil energy. This results in the following kinematic relation derived in the last lecture (c.f. eqn. 23): hν = hν 1 + ε(1 cos θ)

5 dynamics Relativistic and quantum corrections to the dynamics of the scattering process take into consideration that the interaction is electromagnetic, and the magnetic moment associated with the electron spin cannot be ignored. Dirac s relativistic theory of the electron posits that for each value of momentum p e, there are two corresponding energy states: E = ± (p e c) 2 + (m e c 2 ) 2 (1) The normally unobserved negative energy states are capable of playing a role in the electromagnetic interaction between a photon and an electron. In quantum electrodynamics, these interactions involve an initial state, an intermediate state and a final state. Energy does not need to be conserved in the intermediate state.

6 Klein-Nishina differential cross section for Compton scattering measures probabilty of photon re-emission into solid angle dω = d cos θ dφ as a result of a Compton interaction between an incident photon and a free electron exact expression is derived from quantum electrodynamics: d e σ KN c dω = 1 ( ) ν 2 ( ν 2 r2 e ν ν + ν ) ν sin2 θ (2) can be written in terms of Thomson differential cross section, d e σ T /dω = 1 2 r2 e(1 + cos 2 θ), and a form factor: d e σ KN c dω = d eσ T dω F KN (3)

7 Klein-Nishina form factor: F KN = [1 + ε (1 cos θ)] 2 (4) { ε 2 (1 cos θ) 2 } 1 + [1 + ε(1 cos θ)](1 + cos 2 θ) The Klein-Nishina form factor plotted against scattering angle for different incident photon energies ε = hν/m ec 2. (Fig in Podgoršak).

8 Klein-Nishina differential cross section forward scattering (θ 0) and backward scattering (θ π) have equal probability in Thomson limit (ε 0) probability of backscattering decreases with increasing ε forward beaming of photon re-emission

9 Energy distribution of recoil electrons The differential electronic Klein-Nishina cross section can also be expressed as a function of the recoil electron kinetic energy, E K, rather than scattering angle θ since E K = E K (θ) (c.f. eqn. 24 in previous lecture): d e σc KN = d eσc KN de K dω = πr2 e εhν dω dθ [ 2 where ξ K = E K /hν. dθ (5) de K 2ξ K ε(1 ξ K ) + ξk 2 ε 2 (1 ξ K ) 2 + ξk 2 ] (1 ξ K )

10 The Klein-Nishina differential electronic cross section for Compton scattering plotted as a function of recoil electron kinetic energy E K. For a given photon energy, the maximum recoil energy is indicated. (Fig in Podgoršak). energy distribution peaks sharply near (E K ) max = 2ε 1 + 2ε hν c.f. eqn. (27) in last lecture

11 Total electronic Klein-Nishina cross section eσ KN c = de σc KN dω { 1 + ε = 2πr 2 e + dω = 2π ε 2 ln(1 + 2ε) 2ε +1 1 [ 2(1 + ε) 1 + 2ε 1 + 3ε (1 + 2ε) 2 d e σc KN dω d cos θ ] ln(1 + 2ε) ε } (6)

12 Limiting solutions: ε 1: eσ KN c 8π 3 r2 e ( 1 2ε ε ε ) 35 ε4... ε 0 : e σ KN c 8π 3 r2 e = e σ T = m 2 = b =Thomson limit ε 1 : e σ KN c πr 2 e(1 + 2 ln ε)/(2ε) (hν) 1 at high photon energies, Klein-Nishina cross section declines rapidly with respect to the Thomson cross section

13 Energy Transfer Mean fraction of incident photon energy hν transferred to kinetic energy E K of the recoil electron is an average of the fractional kinetic energy E K /hν weighted over the probability distribution P(θ) for Compton scattering in direction θ, integrated over all scattering angles: E C K hν = EK hν P(θ) d cos θ (7) P(θ) d cos θ where P(θ) = 1 eσ KN c de σ KN c dω 2π dφ = eσc KN d e σ KN c dω and E K = E K (θ) is given by eqn. (24) in the last lecture. Note that P(θ)d cos θ = 1. (8)

14 We can now write where E C K hν = 2π eσc KN = 2π eσc KN = ( eσc KN eσ KN c ) tr E K hν d e σ KN c dω d cos θ (d e σc KN ) tr dω d cos θ ( e σc KN ) tr = electronic energy transfer cross section (d e σc KN ) tr /dω = differential energy transfer cross section (9)

15 Differential energy transfer cross section (d e σc KN ) tr dω E K = d eσc KN (10) dω hν = 1 ( ) ν 2 ( ν 2 r2 e ν ν + ν ) ( ) ν ν ν sin2 θ ν where E K = hν hν is the kinetic energy imparted to the recoil electron. From eqn. (24) in the last lecture, the θ dependence is: E K /(hν) = ε(1 cos θ)[1 + ε(1 cos θ)] 1 Total energy transfer cross section: ( e σ KN c ) tr = d(e σc KN ) +1 tr dω = 2π dω 1 d( e σc KN ) tr d cos θ dω see eqn. (7.108) in Podgoršak for full solution.

16 Recall from eqn. (9) that ( e σc KN ) tr eσc KN = EC K hν

17 Atomic cross section e σ KN c is for free electrons independent of Z at high photon energies (hν E B, where E B = electron binding energy), total Compton cross section for entire atom is aσ KN c = Z ( e σ KN c ) (11)

18 Binding energy effects assumption of free electrons breaks down for photon energies hν E B = electron binding energy overestimates effective Compton atomic cross section a σ C at low hν, especially for high Z material a σ KN c The Compton atomic cross section aσ C (solid curves) compared to the Klein- Nishina atomic cross section aσc KN = Z eσc KN (dashed curves), demonstrating electron binding effects at low hν. (Fig in Podgoršak).

19 binding energy correction to Compton atomic cross section usually has negligible effect on overall attenuation because other processes (e.g. Rayleigh scattering, photoelectric effect) are usually more important than Compton scattering at low hν and high Z The atomic cross section for Rayleigh scattering (solid curves) compared to that for Compton scattering (dotted curves) plotted against incident photon energy for varying Z atoms. (Fig in Podgoršak)

20 Atomic attenuation coefficient and atomic energy transfer coefficient for lead. (Fig in Podgoršak) Radiation Physics Compton mass attenuation coefficient σ c ρ = N A A a σ C (12) Compton mass energy transfer coefficient ( ) σc = σ c EK C ρ ρ hν tr (13)

21 Example: For hν = 1 MeV photons incident on lead (Z = 82, A = kg), the Compton atomic cross section is aσ C = m 2. This can be calculated directly from the expression for e σc KN given by (6) and then using a σ C = Z e σc KN (since binding energy corrections are negligible at this hν). The Compton mass attenuation coefficient is σ c ρ = N A A a σ C = kg m 2 = m 2 kg 1 The values can be checked by going to the NIST/Xcom database: The average fractional recoil energy is obtained by inserting ε = 1.96 into the full solution given by eqn. (7.112) in Podgoršak: E C K hν 0.440

22 interaction between photon and tightly bound orbital electron photon completely absorbed, electron ejected momentum transfer to atom, but recoil negligible due to relatively large nuclear mass, so energy conservation is: E K = hν E B photoelectron kinetic energy (14) E B = binding energy of electron orbital approx. 80% occur with K-shell electrons resulting shell vacancy quickly filled by a higher shell electron; resulting transition energy released either as: characteristic (fluorescent) photon Auger electron probability determined by fluorescent yield

23 Atomic cross section a τ = atomic cross section for photoelectric effect function of hν, exhibits characteristic "sawtooth" behaviour: sharp discontinuities coinciding with E B of a particular shell absorption edges Atomic cross section for photoabsorption. indicated. (Fig in Podgoršak.) Energies of K-shell ionisation are

24 e.g. Lead has prominent edges at the following ionisation energies for respective shells: K edge 88.0 kev L 1 edge 15.9 kev L 2 edge 15.2 kev L 3 edge 13.0 kev M edge 3.9 kev

25 3 distinct regions characterise a τ: 1. in immediate vicinty of absorption edge: a τ poorly known; for K-shell electrons, a τ K ε 3 assumed 2. away from absorption edge: cross section for K-shell electrons is aτ K 32 α 4 eσ T Z n ε 7/2 (15) where α = 1/137 = fine structure constant, n is a power index for Z dependence 3. ε 1: where n where ε = hν/m e c 2. aτ K 1.5 α 4 eσ T Z n ε 1 (16)

26 Energy transfer Auger electrons sometimes produced mean energy transfer to electrons as a result of photoelectric effect can be in range (no Auger electron) hν E B < E PE tr < hν (Auger electron) fluorescent yield for K-shell, ω K : ω K = 1 characteristic emission only, no Auger electrons ω K = 0 no characteristic emission, Auger electrons only in general, E PE tr hν P K ω K hν K (17) where P K = probability of photoelectric interactions that occur in K-shell for photons with hν > E B (K)

27 hν K = K-shell weighted mean of all possible fluorescent transitions (L K, M K); K α usually most probable, giving hν K 0.86E B (K) K-shell binding energy, E B(K), weighted mean characteristic X-ray energy for all transitions to K-shell, hν K, and mean energy of K-shell characteristic emission, P K ω K hν K. (Fig in Podgoršak.)

28 Example: Consider hν = 0.5 MeV photons incident on lead, which has E B (K) = 88 kev. Suppose the photoelectric effect occurs and is immediately followed by a forbidden K α3 (L 1 K) transition, with ejection of an Auger electron from the L 2 shell. The total energy transferred to electrons = photoelectron energy + Auger electron energy: E tr = [ hν E B (K) ] + [ E Kα3 E B (L 2 ) ] But E Kα3 = E B (K) E B (L 1 ), so this simplifies to E tr = hν E B (L 1 ) E B (L 2 ) MeV 0.94hν For all K-shell transitions, the average energy transfer is Etr PE = hν P K ω K hν K 500 kev 60 kev = MeV 0.88hν (where P K ω K hν K 60 kev was obtained from the previous figure.)

29 Mass coefficients for the photoelectric effect mass attenuation coefficient: τ ρ = N A A a τ (18) mass energy transfer coefficient: ( τk ρ ) tr = τ ρ Etr PE hν = τ ( 1 P ) Kω K hν K = τ ρ hν ρ f PE (19) where f PE = mean fraction of energy hν transferred to electrons, for hν > E B (K)

30 f PE 1 for low-z absorbers because Auger effect is more prevalent (i.e. fluorescent yield ω K 0) Mean fraction of photon energy hν transferred to electrons in a K-shell photoelectric interaction.

31 production of electron-positron (e e + ) pair resulting from photon interaction with atomic nucleus incident photon energy must exceed threshold 2m e c 2 = 1.02 MeV triplet production (e e e + ) results when incident photon interacts with orbital electron; higher threshold energy required: 4m e c 2 = MeV (see next lecture for derivation)

32 Relativistic kinematics Particle 4-momenta: photon: p γ = hν c (1, ˆk) where ˆk = unit vector in direction of photon 3-momentum (i.e. propagation direction). ( ) ( ) E E electron and positron: p e = c, p e, p e + = c, p e + where p e and p e + are the electron and positron 3-momenta, with p e = γβm e c = p e + (must have same magnitude, but can have different direction). Must also consider 4-momentum of atom, p a = (E a /c, p a ), which can gain recoil energy.

33 Conservation of 4-momentum: before: p γ = p e + p e + + p a after Note that the modulus of a 4-vector A = (A 0, A) is: A 2 = A µ A µ = (A 0 ) 2 + A A which implies that (p γ ) 2 = 0 always and (p e ) 2 = m 2 ec 2. If we square the conservation equation above, then (p γ ) 2 = (p e + p e + + p a ) 2 = 0 Now consider the case where p a = 0. We have (p e ) 2 + 2p e p e + + (p e +) 2 = 0 ( ) E = 2m 2 ec c 2 + p e p e + = 0

34 = 2(γ 2 1)m 2 ec 2 (1 + cos θ e ) = 0 which can only be satisfied if the electron and positron are emitted in exactly opposite directions, with separation angle θ e = π. In general, the atom must gain some recoil energy from the collision with its nucleus. Because of its relatively large mass, however, the recoil gained by the atom will be small.

35 Energy transfer to electrons and positrons in pair production interactions: (E K pp ) tr = hν 2m e c 2 (20) is the total kinetic energy gained by the particles (ignoring atom recoil energy). Generally, the electron and positron can be emitted with different kinetic energies, but the average energy transferred to each is E K pp = 1 2 (hν 2m ec 2 ) (21) For triplet production, the amount of energy converted to mass is also 2m e c 2, but the kinetic energy is shared amongst 3 charged particles, so the mean energy transfer is E K tp = 1 3 (hν 2m ec 2 ) (22)

36 Example: For a 2 MeV photon, the average energy of charged particles resulting from pair production in the nuclear field is pp 1 E K = 2 (hν 2m ec 2 ) = 1 (2 MeV MeV) 2 = MeV = hν and in the electron field, the average energy is E K tp = 0 because 2 MeV is less than the threshold energy 4m e c 2 = 2.04 MeV needed for triplet production.

37 Nuclear screening occurs for hν > 20 MeV photons that interact with the nuclear Coloumb field outside the K-shell; effective nuclear field is screened by two K-shell electrons and the interaction cross section is reduced.

38 Atomic cross section General form for pair production atomic cross section in field of nucleus or orbital electron is aκ = αr 2 ez 2 P(ε, Z) (23) P(ε, Z) = complicated function field energy P(ε, Z) 1. nucleus 1 ε (αz 1/3 ) ln(2ε) nucleus ε (αz 1/3 ) ln(183z 1/3 ) nucleus ε > 4 9 ln(2ε) electron ε > 4 Z 1 [ 28 9 ln(2ε) 11.3] nuclear screening only important in case 2 ε > 4 in case 3 must lie outside the limits of cases 1 and 2

39 P(ε, Z) Z 1 for electron field triplet production usually makes negligible contribution to total cross section: aκ = a κ pp + a κ tp = a κ pp [ 1 + (ηz) 1 ) (24) where η 1 as hν Atomic cross sections for pair production (solid curves) and triplet production (dashed curves) for a high-z and low-z absorber. (Fig in Podgoršak.)

40 Total atomic cross sections for pair production (including triplet production) for different Z. (Fig in Podgoršak.)

41 mass attenuation coefficient for pair production: mass energy transfer coefficient: ( ) κ = κ ρ tr ρ κ ρ = N A A a κ (25) E pp tr hν = κ ρ (1 2m ec 2 ) hν (26) Mass energy transfer coefficient (solid curves) and mass attenuation coefficient (dashed curves) for pair production. (Fig in Podgoršak.)

42 of Photon Interactions Tabulation of interactions and symbols used: electronic atomic linear cross cross attenuation section section coefficient [m 2 ] [m 2 ] [m 1 ] Thomson scattering eσ T a σ T σ T Rayleigh scattering aσ R σ R Compton scattering eσ C a σ C σ C Photoelectric effect aτ τ Pair production aκ pp κ pp Triplet production eκ tp a κ tp κ tp

43 Tabulation of attenuation coefficients: Total linear and mass attenuation coefficients are a sum of the partial linear and mass attenuation coefficients for individual photon interactions: µ = τ + σ R + σ C + κ (27) µ m = τ ρ + σ R ρ + σ C ρ + κ ρ Similarly, the total atomic cross section is (28) aµ = µ m A N A = a τ + a σ R + a σ C + a κ = Z e µ (29) where e µ = total electronic cross section.

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45 For compounds or mixtures, the mass attenuation coefficient is a summation of a weighted average of its constituents: µ i µ m = Σ i w i (30) ρ where w i = proportion by weight of i-th constituent. Example: Water, H 2 O, has w H = w O = = = For 1 MeV photons, µ H /ρ = m 2 kg 1 and µ O /ρ = m 2 kg 1 (data from the NIST/Xcom database), so µ ρ m 2 kg 1

46 Mass attenuation coefficient for water (NIST):

47 Mass attenuation coefficient for soft tissue (NIST):

48 Comparison of the three main photon energy transfer processes: photoelectric effect dominates at low hν and high Z pair production dominates at high hν and high Z Compton scattering dominates over a broad range in hν for low/moderate Z (including water and tissue)

49 Energy transfer coefficient is the sum of the partial energy transfer coefficients for the photoelectric effect, Compton scattering and pair production: µ tr = µ E tr hν E C tr = τ EPE tr hν + σ C hν + κepp tr hν = τ f PE + σ C f C + κ f pp (31) where each f is the average fraction of incident photon energy hν transferred to electrons (and positrons) by the corresponding physical process, with f PE = 1 P Kω K hν K hν f C = EC tr hν f pp = 1 2m ec 2 hν (32) (33) (34)

50 What is the dominant photon interaction initially?