Photon Interactions in Matter

Transcription

1 Radiation Dosimetry Attix 7 Photon Interactions in Matter Ho Kyung Kim hokyung@pusan.ac.kr Pusan National University

2 References F. H. Attix, Introduction to Radiological Physics and Radiation Dosimetry, John Wiley and Sons, Inc.,

3 1. Compton effect (σσ) 2. Photoelectric effect (ττ) 3. Pair production (κκ) 4. Rayleigh (coherent) scattering Elastic small-angle scattering with no energy loss 5. Photonuclear interaction EE γγ = hνν > a few MeV (γγ, nn) reaction Resulting in the transfer of energy to electrons (this chapter), which then impart that energy to matter (next chapter) in many small Coulomb-force interactions along their tracks EE γγ (= hνν) of the interaction photon ZZ of the interaction medium 3

4 Attix Fig

5 COMPTON EFFECT Interaction with electrons being unbound and stationary (assumptions) Zero-binding free electrons Kinematics Relationship between energies and angles Cross section Probability that an interaction will occur Attix Fig

6 Kinematics Energy conservation TT = hνν hνν Momentum conservation xx-direction: hνν = hνν cos φφ + pppp cos θθ yy-direction: hνν sin φφ = pppp sin θθ Law of invariance: pppp = TT(TT + 2mm 0 cc 2 ) mm = mm 0 1 vv/cc 2 TT = mmcc 2 2mm 0 cc 2 pp = mmmm Solutions to the kinematics of Compton interactions: hνν hνν = 1+ hνν mm0cc2(1 cos φφ) TT = hνν hνν cot θθ = 1 + hνν mm 0 cc 2 tan φφ 2 6

7 Straight-ahead scattering, φφ = 0 KE of recoiling ee Side scattering Back scattering For hνν < 0.01 MeV, hνν = hνν regardless of φφ No KE transfer to ee Elastic Thomson scattering Attix Fig

8 For hνν << 1, cot θθ tan φφ 2 or θθ ππ 2 φφ 2 For hνν >> 1, θθ = φφ at 2.59 e.g., All ee scattered at θθ btwn 2.59 & 90 are likewise related to the photons scattered forward btwn 0 & 2.59 Attix Fig

9 Thomson scattering The earliest theoretical description of photon scattering with a "free" electron Elastic scattering Only valid to hνν 0.01 MeV Same as the Compton scattering when hνν 0 because of no relativistics Differential Thomson scattering x-sec. per electron per unit solid angle for a photon scattered at φφ: cm 2 sr -1 per electron 2 d ee σσ 0 = rr 0 dω φφ 2 (1 + cos2 φφ) rr 0 = ee2 mm 0 cc 2 = cm, called the "classical electron radius" Front-back symmetrical angular distribution Cylindrical symmetry around the beam axis dω φφ = 2ππ sin φφ dφφ 9

10 Total Thomson scattering x-sec. per electron: ππ ππ d ee σσ 0 eeσσ 0 = dω φφ=0 dω φφ = ππrr 2 0 (1 + cos 2 φφ) sin φφ dφφ = 8ππrr 0 2 φφ φφ=0 3 Note: dω φφ = 2ππ sin φφ dφφ, the annular element of solid angle In units of (cm 2 per electron) Independent of hνν = X-sec. Can be thought of as an effective target area Probability that a Thomson scattering event occurs when a single photon passes through a layer containing on electron per cm 2 Fraction of a large number of incident photons that scatter in passing through the layer (e.g., ~665 events for photons) So long as the fraction of photons interacting in a layer of matter by all processes combined remains less than 0.05, the fraction may be assumed to be proportional to absorber thickness (i.e., linear approximation); otherwise the exponential law must be used 10

11 Klein-Nishina x-sec. Relativistic treatment to unbound electrons using the Dirac's relativistic theory Differential KN scattering x-sec. per electron per unit solid angle for a photon scattered at φφ: 2 d ee σσ = rr 0 dω φφ 2 hνν hνν 2 hνν hνν + hνν hνν sin2 φφ See Attix Fig. 7.5 Forward scattering at high energies hνν hνν for low energies; d ee σσ rr 2 0 dω φφ 2 2 sin2 φφ = rr 0 (1 + 2 cos2 φφ): 2 Identical to the Thomson scattering! 11

12 Same as the Thomson scattering Unlikely backscattering at high hνν Attix Fig

13 Total KN scattering x-sec. per electron: eeσσ = 2ππrr αα αα 2 ππ Note: ee σσ = 2ππ φφ=0 αα = hνν mm 0 cc2 with hνν in MeV See Attix Fig (1 + αα) 1 + 2αα ln(1 + 2αα) αα d ee σσ sin φφ dφφ = ππrr 2 ππ hνν dω 0 φφ 0 hνν ee σσ (hνν) 1 for higher photon energies Independent of ZZ or ee σσ ZZ ln(1 + 2αα) 2αα 1 + 3αα (1 + 2αα) 2 hνν hνν + hνν hνν sin2 φφ sin φφ dφφ Because the electron binding energy has been assumed to be zero (unbound electrons) KN x-sec. per atom: aa σσ = ZZ ee σσ (cm 2 /atom) aa σσ ZZ Compton mass attenuation coefficient σσ ρρ = NN AAZZ AA NN AA = mole -1 NN AAZZ AA ee σσ σσ ρρ ZZ0 = number of electrons per gram of material 13

14 eeσσ ss, x-sec for the energy carried by the scattered photon eeσσ = ee σσ tttt + ee σσ ss eeσσ 0 = cm 2 /e at hνν = 0.01 MeV eeσσ (hνν) 1 eeσσ becomes constant & eeσσ tttt diminishes with decreasing hνν below 0.5 MeV Attix Fig

15 Energy-transfer x-sec. In each Compton scattering; incident (hνν) = scattered (hνν ) + recoiled electron (TT) Related to the "kerma," it is interesting to know TT/hνν or TT averaged over all φφ d ee σσ tttt dω φφ = d eeσσ TT = d eeσσ hνν hνν dω φφ hνν dω φφ hνν = rr hνν hνν 2 hνν hνν + hνν hνν sin2 φφ hνν hνν hνν ππ eeσσ tttt = 2ππ φφ=0 eeσσ tttt dω φφ sin φφ dφφ eeσσ tttt = 2ππrr 0 2 2(1+αα)2 1+3αα 1+αα 2αα2 2αα 1 4αα2 1+αα αα 2 (1+2αα) 1+2αα 2 αα 2 1+2αα 2 3(1+2αα) 3 αα 3 2αα 2αα 3 ln(1 + 2αα) See Attix Fig. 7.6 ee σσ = ee σσ tttt + ee σσ ss TT = ee σσ tttt or TT = hνν ee σσ tttt hνν eeσσ eeσσ See Attix Fig

16 Compton mass energy-transfer coefficient σσ tttt ρρ = NN AAZZ AA ee σσ tttt = σσ TT ρρ hνν Attix Fig

17 Other differential KN x-sec's Differential KN x-sec. for electron scattering at angle θθ per unit solid angle & per electron: d ee σσ = d eeσσ (1 + αα) 2 (1 cos φφ) 2 dω θθ dω φφ cos 3 θθ ππ/2 d eeσσ = ee σσ θθ=0 dω dω θθ θθ High forward momentum in the collision causes most of the electrons and most of the scattered photons to be strongly forward-directed when hνν is large Strong forward scattering at high hνν (Refer to Attix Fig. 7.5) Attix Fig

18 d ee σσ dtt = ππrr 0 2 mm 0 cc 2 hνν 2 In units of (cm 2 MeV -1 e -1 ) mm 0 cc 2 TT hνν hνν hνν 2 + hνν hνν 3 (TT mm 0cc 2 ) 2 (mm 0 cc 2 ) 2 Probability that a single photon will have a Compton interaction in traversing a layer containing 1 e/cm 2, transferring to that electron a KE between TT & TT + dtt Energy distribution of the electrons averaged over all scattering angles θθ TT mmmmmm θθ=0 (head on collision) = 2 hνν 2 = 2hνν+mm 0 cc 2 hνν 1+ mm 0cc 2 2hνν = hνν 1 + mm 0cc 2 2hνν 1 hνν 1 hνν MeV Attix Fig

19 PHOTOELECTRIC EFFECT Most important interaction of low-e photons with high-z matter Incident photon can give up all of its hνν in colliding with a tightly bound electron (e.g., inner shell electrons bounded by potential energy EE bb ) especially of high-z atom Attix Fig

20 Kinematics TT = hνν EE bb TT aa = hνν EE bb KE given to the recoiling atom TT aa 0 Can occur only when hνν > EE bb More likely occur as hνν is smaller TT is independent of scattering angle θθ 20

21 Photoelectric interaction x-sec. No simple equation for the differential photoelectric x-sec. X-sec. is available based on experimental results supplemented by theoretically assisted interpolations Directional distribution of photoelectrons per unit solid angle (See Attix Fig. 7.11) Emitted to sideways along the direction of the photon's electric field for low photon energies Emitted toward smaller angles with increasing photon energy Theoretical calculation results Attix Fig

22 Photoelectric x-sec. per atom (integrated over all angles of photoelectron emission) aaττ kk ZZnn (hνν) mm In units of (cm 2 /atom) kk = constant nn 4 at hνν = 0.1 MeV, gradually rising to about 4.6 at 3 MeV mm 3 at hνν = 0.1 MeV, gradually decreasing to about 1 at 3 MeV For hνν 0.1 MeV aaττ ZZ4 (hνν) 3 Mass attenuation coefficient ττ = NN AAZZ ρρ AA eeττ = NN AAZZ AA See Attix Fig aaττ ZZ ττ ρρ ZZ3 (hνν) 3 22

23 ZZ PPPP ZZ CC = ττ/ρρ PPPP ττ/ρρ PPPP (EE bb ) LLL = 15.9, (EE bb ) LL2 = 15.2, (EE bb ) LL3 = 13.0 kev Two K-shell electrons cannot participate in the PE effect when hνν < (EE bb ) KK = 88 kev (hνν) 3 (hνν) 3 Attix Fig

24 Energy-transfer x-sec. TT hνν = hνν EE bb hνν First approximation to the total fraction of hνν that is transferred to "all" electrons Part or all of EE bb is converted to electron KE through the Auger effect Disposal mechanisms of EE bb Fluorescence x-ray Inner shell vacancies due to PE, IC, EC, or charged-particle collision are promptly filled by another electrons falling from less tightly bound shells This transition is sometimes accompanied by the emission of a fluorescence x-ray with an energy equal to the difference in potential energy between the donor and recipient levels (e.g., hνν KK, hνν LL ) with a probability of fluorescence yield (See Attix Fig. 7.14) YY KK > YY LL YY MM 0 24

25 0.42 Cu Attix Fig

26 hνν KK rather than hνν KK because of several energy levels in the L or higher shells hνν KK < (EE bb ) KK ; See Attix Fig PP KK = ττ KK ττ = the fraction of all PE interactions that occur in the K-shell for photons of hνν > (EE bb) KK PP LL = ττ LL ττ for photons where (EE bb) LLL < hνν < (EE bb ) KK PP KK YY KK = the fraction of all PE events in which a K-fluorescence x-ray is emitted by the atom PP LL YY LL = the fraction of all PE events in which an L-fluorescence x-ray is emitted by the atom PP KK YY KK hνν KK = the mean energy carried away from the atom by K-fluorescence x-rays per PE interaction PP LL YY LL hνν LL PP LL YY LL (EE bb ) LLL 26

27 Attix Fig

28 Auger effect EE bb disposal mechanism alternative to the fluorescence x-ray An atom ejects one or more Auger electrons with sufficient KE simultaneously in a kind of chain reaction Contributing to the kerma Exchanging one energetically "deep" inner vacancies for a number of relatively shallow outer-shell vacancies (these vacancies are finally neutralized by conduction-band electrons) Let's consider a possible scenario for a K-shell vacancy (Auger chain reaction or shower) hνν KK = (EE bb ) KK (EE bb ) LL : the remaining (EE bb ) KK hνν KK will become electron KE Auger effect ejecting an M-shell electron for example» TT MM = (EE bb ) KK (EE bb ) LL (EE bb ) MM» Then, two vacancies in the L- & M-shells» Two N-shell electrons fill these vacancies, and the atom emits two more Auger electrons from the N-shell for example Then, the atom has four N-shell vacancies TT NNN = (EE bb ) LL (EE bb ) NN EE bb NN = (EE bb ) LL 2(EE bb ) NN TT NN2 = (EE bb ) MM 2(EE bb ) NN» Thus, the total KE of the 3 Auger electrons: TT AA = TT MM + TT NNN + TT NNN = (EE bb ) KK 4(EE bb ) NN» This process is repeated until all the vacancies are located in the outermost shell» Total KE of all the Auger electrons = (EE bb ) KK sum of BE of all the final electron vacancies» (EE bb ) KK ends up as electron KE, contributing to the kerma 28

29 The probability of any other fluorescence x-ray except those from the K-shell being able to carry energy out of an atom is negligible for hνν > (EE bb ) KK Then, all the rest of the (EE bb ) KK and all of the BE involved in PE interactions in other shells may be assumed to be given to Auger electrons The mean energy transferred to charged particles per PE event = hνν PP KK YY KK hνν KK Photoelectric mass energy-transfer coefficient for hνν > (EE bb ) KK ττ tttt ρρ = ττ hνν PP KK YY KK hνν KK (1 PP KK )PP LL YY LL h ρρ hνν νν LL For (EE bb ) LLL < hνν < (EE bb ) KK ττ tttt ρρ = ττ ρρ hνν PP LL YY LL hνν LL hνν ττ ρρ hνν PP LL YY LL (EE bb ) LLL hνν 29

30 The size of K-edge step is less than that in ττ ρρ curve due to the loss of K-fluorescence energy Attix Fig

31 PAIR PRODUCTION Absorption process in which a photon disappears and gives rise to an electron & a positron Only occurs in a Coulomb field Near an atomic nucleus field Dominant process hνν 2mm 0 cc 2 = MeV Near an atomic electron field Called "triplet production" = 2 electrons + 1 positron = host electron (w/ significant KE) + pair production hνν 4mm 0 cc 2 = MeV due to momentum-conservation considerations 31

32 PP in the nuclear coulomb force field Energy conservation hνν = 2mm 0 cc 2 + TT + TT + = MeV + 2 TT Average KE of the products hνν MeV TT = 2 Approximate departure angle of the product relative to the original photon direction θθ mm 0cc 2 TT In units of (radians) e.g., For hνν = 5 MeV, TT = MeV and θθ 0.26 radians = 15 Attix Fig

33 Atomic differential x-sec. (Bethe & Heitler) for the creation of TT + σσ 0 = rr = ee 2 mm 0 cc 2 d aa κκ = σσ 0ZZ 2 PP hνν 2mm 0 cc 2 dtt+ 2 = cm 2 /electron PP is dependent upon hνν & ZZ (See Attix Fig. 7.18) Attix Fig

34 Total nuclear PP x-sec. per atom aaκκ = TT + d aa κκ = σσ 0 ZZ 2 0 hνν 2mm 0 cc 2 PPdTT + hνν 2mm 0 cc 2 = σσ 0ZZ PPd TT + hνν 2mm 0 cc 2 = σσ 0 ZZ 2 PP aaκκ ZZ 2 aaκκ~ log(hνν) See Attix Fig Becomes a constant independent of hνν for very large hνν because of electron screening of the nuclear field Nuclear PP mass attenuation coefficient κκ ρρ = NN AA ρρ aa κκ = NN AAZZ ρρ aaκκ ZZ κκ ρρ ZZ 34

35 PP in the electron field Energy conservation Average KE of the products Threshold (See Attix Fig. 7.19) hνν = MeV + TT + + TT 1 + TT 2 TT = hνν MeV 3 hνν mmmmmm = 4mm 0 cc 2 Attix Fig

36 Momentum conservation in a moving frame RR with a velocity +ββcc relative to the laboratory frame RR hνν hνν mmββcc = cc cc mm 0ββcc 1 ββ = 0 2 Due to the Doppler effect: νν = νν 1 ββ 1+ββ Then we have Energy conservation hνν mmmmmm + TT 1 = hνν mmmmmm ββ = αα 1 + αα + cc 2 mm mm 0 = hνν mmmmmm + mm 0 cc ββ mmmmmm 1 = 2mm 0 cc 2 hνν mmmmmm Then we have 1 ββ = hνν mmmmmm mmmmmm, ββ 1+ββ mmmmmm = αα mmmmmm, and αα mmmmmm 1+αα mmmmmm = hνν mmmmmm mmmmmm mm 0 cc 2 ββ mmmmmm = 4 5 hνν mmmmmm = 4mm 0 cc 2 36

37 KE of each of the products TT = αα2 2αα 2 ± αα αα(αα 4) 2αα + 1 e.g., For hνν = 10 MeV, 3 kev TT 8.7 MeV TT = 2mm 0cc 2 for hνν = 4mm 0 cc 2 3 X-sec. e.g., aa κκ (electrons) 1% in Pb aaκκ (nucleus) CC = 1 for hνν aaκκ (electrons) aaκκ (nucleus) 1 CCCC CC increases slowly to ~2 with decreasing hνν to 5 MeV 37

38 Mass attenuation coefficient κκ = ρρ pppppppp κκ + ρρ nnnnnnnnnnnnnn κκ ρρ eeeeeeeeeeeeeeee PP energy-transfer coefficient κκ tttt ρρ = κκ ρρ hνν 2mm 0 cc 2 hνν 38

39 RAYLEIGH SCATTERING Also called "coherent" scattering because the photon is scattered by the combined action of the whole atom Elastic scattering Contributes to nothing to kerma or dose Small-angle scattering Can only be detected in narrow-beam geometry Dependent upon both ZZ & hνν hνν (MeV) Al Pb X-sec. aaσσ RR ZZ2 (hνν) 2 σσ RR ZZ ρρ (hνν) 2 See Attix Fig hνν (MeV) C σσ RR μμ Cu Pb

40 PHOTONUCLEAR INTERACTIONS An energetic hνν > a few MeV enters and excites a nucleus, which then emits a proton or neutron (γγ, pp) interaction Contributes directly to the kerma < 5% of the kerma due to pair production Negligible in dosimetry considerations (γγ, nn) interaction Neutrons can cause biological consequences to patients Allowable neutron levels should be regulated in radiotherapy x-ray beams (e.g., Linacs) Neutrons can activate accelerator hardware 40

41 TOTAL COEFFICIENTS Mass attenuation coefficient (neglecting photonuclear interactions) μμ ρρ = ττ ρρ + σσ ρρ + κκ ρρ + σσ RR ρρ Mass energy-transfer coefficient (for hνν > (EE bb ) KK ) μμ tttt ρρ = ττ tttt ρρ + σσ tttt ρρ + κκ tttt ρρ = ττ ρρ hνν PP KK YY KK h hνν νν KK + σσ ρρ TT hνν + κκ ρρ hνν 2mm 0 cc 2 hνν Mass energy-absorption coefficient μμ eeee ρρ = μμ tttt ρρ (1 gg) gg = the average fraction of secondary-electron energy that is lost in radiative interactions (bremsstrahlung & in-flight annihilation) e.g., In Pb with hνν = 10 MeV, μμ eeee ρρ For low ZZ & hνν, gg 0 or μμ eeee ρρ = μμ tttt ρρ = 0.74 μμ tttt ρρ 41

42 Coefficients for compounds and mixtures (using the Bragg rule) μμ ρρ mmmmmm = μμ ρρ AA ff AA + μμ ρρ BB ff BB + μμ tttt = μμ tttt ρρ mmmmmm ρρ ff AA + μμ tttt AA ρρ BB ff BB + ff jj = the weight fraction of the separate element jj If radiative losses are small: μμ eeee μμ eeee ρρ mmmmmm ρρ μμ eeee ff AA + μμ eeee ff AA ρρ BB + μμ tttt BB ρρ (1 gg AA )ff AA + μμ tttt AA ρρ BB (1 gg BB )ff BB + = μμ tttt 1 ff ρρ mmmmmm ρρ AA gg AA ff BB gg BB ff AA + μμ tttt 1 ff AA ρρ AA gg AA ff BB gg BB ff BB + = BB μμ tttt 1 ff ρρ AA gg AA ff BB gg BB μμ tttt (1 gg mmmmmm ρρ mmmmmm ) mmmmmm 42