Compton Scattering

1. Compton Scattering • There are three related processes • Thomson scattering (classical) • Photon-electron • Compton scattering (QED) • Photon-electron • Rayleigh scattering (coherent) • Photon-atom • Thomson and Rayleigh scattering are elastic-only the direction of the photon changes, not its energy • Plus Thomson and Rayleigh scattering are only important at low energies where the photoelectric effect dominates

2. Thomson Scattering • In Thomson scattering an electromagnetic (EM) wave of frequency f is incident on an electron • What happens to the electron? • Thus the electron will emit EM waves of the same frequency and in phase with the incident wave • The electron absorbs energy from the EM wave and scatters it in a different direction • In particular, the wavelength of the scattered wave is the same as that of the incident wave

3. Thomson Scattering

4. Rayleigh Scattering • Rayleigh scattering is scattering of light from a harmonically bound electron • You may recall the probability for Rayleigh scattering goes as 1/λ4 • Why is the sky blue?

5. Compton Scattering • Compton scattering is the scattering of light (photons) from free electrons

6. Compton Scattering • Calculations • The change in wavelength can be found by applying • Energy conservation • Momentum conservation

7. Compton Scattering • From energy conservation • From momentum conservation • Eliminating pe2

8. Compton Scattering • Continuing on • And using v=c/λ we arrive at the Compton effect • And h/mc is called the Compton wavelength

9. Compton Scattering • Summarizing and adding a few other useful results are

10. Compton Scattering • The differential and total cross sections are calculated in a straightforward manner using QED • Called the Klein-Nishina formula

11. Compton Scattering • On the previous slide • At low energies • At high energies

12. Compton Scattering • Thus at high energies, the Compton scattering cross section sC goes as

13. Compton Scattering • Graphically, ds/dW

14. Compton Scattering • In polar form, assume a photon incident from the left

15. Compton Scattering • At high energies, say > 10 MeV, most of the photons are scattered in the forward direction • Because of the high forward momentum of the incident photons, most of the electrons will also be scattered in the forward direction

16. Compton Scattering • Concerning kerma and absorbed dose, we are particularly interested in the scattered electron because it is ionizing • We can split the Compton cross section into two parts: one giving the fraction of energy transferred to the electron and the other the fraction of energy contained in the scattered photon

17. Compton Scattering

18. Compton Scattering Heresen=str

19. Compton Scattering • Another useful form of the differential cross section is ds/dT, which gives the energy distribution of the electron

20. Compton Scattering • The maximum electron kinetic energy is given by

21. Compton Scattering • In cases where the scattered photon leaves a detector without interaction one would observe

22. Compton Scattering

23. Compton Scattering

24. Pair Production • Pair production is the dominant photon interaction at high energies (> 10 MeV) • In order to create a pair, the photon must have > 2me = 1.022 MeV • In order to conserve energy and momentum, pair production must take place in the Coulomb field of a nucleus or electron • For nuclear field, Ethreshold > 2 x me • For atomic electron field, Ethreshold> 4 x me

25. Pair Production

26. Pair Production • Energy and momentum conservation give • Energy conservation can be re-written • But momentum conservation (x) shows • Thus energy and momentum are not simultaneously conserved

27. Pair Production • The processes of pair production and bremsstrahlung are related (crossed processes) • Thus we’d expect the cross section to depend on the screening of atomic electrons surrounding the nucleus • Does the photon see nuclear charge Ze or 0 or something in between? • The relevant screening parameter is

28. Pair Production • In the Born approximation (which is not very accurate for low energy or high Z) one finds

29. Pair Production • Notes • spair ~ Z2 • Above some photon energy (say > 1 GeV), spair becomes a constant • In order to account for pair production from the Coulomb field of atomic electrons, Z2 is replaced by Z(Z+1) approximately since the cross section is smaller by a factor of Z • Usually we don’t distinguish between the source of the field

30. Pair Production • Notes • In the case of the nuclear field and for large photon energies, the mean scattering angle of the electron and positron is

31. Pair Production • The probability for pair production

32. Pair Production • 2me (1.022 MeV) of the photon’s energy goes into creating the electron and positron • The electron will typically be absorbed in a detector • The positron will typically annihilate with an electron producing two annihilation photons of energy me (0.511 MeV) each • If these photons are not absorbed in the detector than the pair production energy spectrum will look like

33. Pair Production

34. Pair Production • Similar to the photoelectric effect and Compton scattering we define the mass attenuation and mass energy transfer coefficients as

35. Photonuclear Interactions • Here a nucleus is excited by the absorption of a photon, subsequently emitting a neutron or proton • Most important when the energy of the photon is approximately the binding energy of nucleons (5-15 MeV) • Called giant nuclear dipole resonance • Still a small fraction compared to pair production however

36. Photonuclear Interactions • Giant dipole resonance

37. Photonuclear Interactions • These interactions would be observed with higher energy x-ray machines • A 25 MV x-ray beam will contain neutron contamination from photonuclear interactions • Small effect compared to the photon beam itself • Also important in designing shielding since ~MeV neutrons are difficult to contain

38. Photon Interactions • Typical photon cross sections

39. Photon Interactions • Typical photon cross sections

40. Photon Interactions • Notes • Of course different interactions can occur at a given photon energy • A polyenergetic beam such as an x-ray beam is not attenuated exponentially • Lower energy x-rays have higher attenuation coefficients than higher energy x-rays • Thus the attenuation coefficient changes as the beam proceeds through material • An effective attenuation length meff can be estimated as

41. Beam Hardening

42. Photon Interactions • Let’s return to our first slide • As we’ve seen in the different photon interactions • Secondary charged particles are produced • Photons can lose energy through Compton • We define • Narrow beam geometry and attenuation • Only primaries strike the detector or are recorded • Broad beam geometry and attenuation • All or some of the secondary or scattered photons strike the detector or are recorded • Effective attenuation coefficient m’ < m

43. Photon Interactions

44. Photon Interactions • In ideal broad beam geometry all surviving primary, secondary, and scattered photons (from primaries aimed at the detector) is recorded • In this case m’ = men

45. Photon Interactions • There are three relevant mass coefficients

46. Photon Interactions • Tables of photon cross sections, mass attenuation, and mass-energy absorption coefficients can be found in numerous places • http://physics.nist.gov/PhysRefData/contents.html • NIST also gives material constants and composition • Useful since

47. Photon Interactions l=1/(m/r)

48. Photon Interactions • Sometimes easy to loose sight of real thickness of material involved

49. Photon Interactions • X-ray contrast depends on differing attenuation lengths

50. Photon Interactions • What is a cross section? • What is the relation of m to the cross section s for the physical process?