Interactions of Photons with Matter Compton Scatter (Part 2)

Transcription

1 Interactions of Photons with Matter Compton Scatter (Part 2) George Starkschall, Ph.D. Summary: Compton scatter cross sections Equal to classical scatter at all angles at zero energy Equal to classical scatter at = 0 Peaked in the forward direction as photon energy increases Compton scatter at = is lower than that at = 0 by About 1 order of magnitude at 1 MeV ( =2) About 2 orders of magnitude at 10 MeV ( =20) Total cross section We have differential scattering cross section per unit solid angle. Convert to differential scattering cross section per unit plane angle by using d = 2 sin d 1

2 Total cross section The sin term suppresses this cross section at = 0 and = as it had done for the expression for coherent scatter Integrate over to obtain Total cross section 0 is the classical cross section Term in brackets is the integral F KN (Klein- Nishina coefficient) The cross-section is directly related to the electronic attenuation coefficient (Total Compton coefficient) Cross sections are tabulated in J & C Table A-2a 2 nd column Total cross section At 1 kev, e = m 2 /electron, comparing to classical value of At 7 MeV, down one order of magnitude ( ) At 100 MeV, down two orders of magnitude ( ) Compton scatter coefficient drops two orders of magnitude over 5 orders of magnitude of energy (nearly independent of energy) 2

3 Energy dependence Energy dependence e constant nearly out to 100 kev Falls less than 2 orders of magnitude out to 100 MeV Energy dependence e independent of Z free electron gas model e roughly independent of photon energy 3

4 Energy transfer coefficient e tells us fraction of photons scattered per unit path length Determine how much energy transferred to electrons in target in interaction Follow the energy Energy transfer coefficient We know that energy transferred to electron given by Fraction of energy transferred given by Energy transfer coefficient Same exercise as before integrate over angles 4

5 Energy transfer coefficient Energy transfer coefficient Approaches classical coefficient for high-energy photons (> 1 MeV) Energy transfer coefficient Passes through maximum near MeV 5

6 Energy transfer coefficient Strongly suppressed at low energies where Compton is poor energy transfer mechanism Scatter coefficient Evaluate as difference between total attenuation coefficient and energy transfer coefficient s = - tr At low energies, s, since tr small This is also consistent with classical scatter At high energies, s 0, since tr Mean energy transferred Mean energy transferred to charged particles given by 6

7 Mean energy transferred Energy distribution Look at energy distribution of Compton electrons generated by interaction of photons of a given energy Probability distribution of electrons generated from a single photon of a given energy Energy distribution We can write We have already calculated d /d We can get de/d from 7

8 We find Energy distribution Combining this with d /d we get Energy distribution J & C, fig 6-5 Energy distribution All curves show peaks at maximum energy and at zero energy Low energy and high energy electrons more likely than medium energy electrons 8

9 Energy distribution The maximum energy of the electrons is less than the photon energy Compton edge Total cross section Total cross section (electronic attenuation coefficient) can be obtained by integrating the cross section over energies Total cross section Because distribution relatively constant with electron energy, easy to estimate electronic attenuation coefficient This coefficient is directly proportional to /, the mass attenuation coefficient 9

10 Example Estimate the electronic attenuation coefficient for an 0.8 MeV photon in a free electron gas Maximum energy of electrons is MeV Example d /de is approximately constant with a value of approximately 40 x m 2 /e MeV. So approximately 40 x m 2 /e MeV x 0.6 MeV = 24 x m 2 /e, which is close to the value given in J & C Table A-2a of x m 2 /e Example Follow the energy Determine the number of Compton interactions and the number of electrons set in motion with energies between 0.15 and 0.25 MeV when a bone slab, 0.6 cm thick, is bombarded with MeV photons 10

11 Example Recall that attenuation coefficient is fraction attenuated per unit absorber thickness Number attenuated (thin absorber) = number of incident photons x Compton coefficient x absorber thickness Number of incident photons 10 4 Compton coefficient for 0.5 MeV photons (Table A-2a) x m 2 /electron Absorber thickness 0.6 cm x 1650 kg/m 3 (Table 5-3) x 10-2 m/cm x 3.19 x e/kg = 3.16 x electron/m 2 Example Number scattered = 10 4 x x m 2 /e x 3.16 x e/m 2 = 9.14 x 10 2 photons Assumption of thin absorber reasonable Example How many electrons have energy between 0.15 and 0.25 MeV? From the graph, the differential cross section for 0.5 MeV photons is estimated to be approximately 70 x m 2 electron -1 MeV -1 11

12 Example From the graph, the differential cross section is estimated to be approximately 70 x m 2 electron -1 MeV -1 The energy width is 0.1 MeV so the cross section is 70 x m 2 electron -1 MeV -1 x 0.1 MeV = 7.0 x m 2 electron -1 # electrons generated in energy range = 10 4 x 7.0 x m 2 electron -1 x 3.16 x electron/m 2 = 220 electrons Effects of binding energy So far, we have assumed free electron gas no binding energy To account for binding energy must do quantum mechanical treatment Need wave functions for atoms Only available for simple atoms Effects of binding energy important only for low-energy incident photons Effects of binding energy We can write S(x,Z) represents probability that an electron actually leaves the atom 12

13 Effects of binding energy Values of S(x,Z) tabulated by Hubbell [J Phys Chem Ref Data 4:471 (1975)] Integrate to obtain Compton coefficient for real atoms and molecules Effects of binding energy Integral cannot be evaluated in closed form Information tabulated in J & C Table A-4 Effects of binding energy 13

14 Summary of Compton scatter Based on the free electron model, the attenuation coefficient is nearly constant to 100 kev, falling less than 2 orders of magnitude out to 100 MeV / h <-1 At low energies (<100 kev) binding energy of electrons affects Compton scatter coefficient, suppressing attenuation for higher Z materials Summary of Compton scatter At higher energies (>500 kev) there is no difference between any of the Compton attenuation coefficients, so / is essentially independent of Z In summary / Z 0 h <-1 Summary of Compton scatter 14

15 Summary of Compton scatter This plot is at low Z and low energy Coherent/Rayleigh significant and stronger than Compton Binding energy for Compton significant Compton scatter approaches classical limit Summary of Compton scatter Involves interaction between photon and single electron Almost independent of Z Decreases slightly with increasing energy Some energy scattered and some transferred to electron Amount depends on angle of emission of scattered photon and energy of photon Summary of Compton scatter Fraction of energy transferred to kinetic energy of charged particles per collision increases with increase in photon energy Low energy photons: tr << - inefficient energy transfer High energy photons: tr - more efficient energy transfer In soft tissue, Compton most important process in energy range 30 kev 30 MeV 15