1. Basic photon interactions
Lecture 3

2. Coherent scattering Photoelectric effect Compton scattering
Relative importance of different X-ray interactions with atoms listed below depends on the incident energy and different for the energy range used for the diagnostic and in radiology.
Coherent scattering
Photoelectric effect
Compton scattering
Pair production
Photodisintegration

3. Kinematics of collisions.
Several features of dynamics can be understood from energy-momentum conservation: sum of initial energies is equal to the sum of final energies, momentum of the initial particle is equal to the sum of the final momenta.
The simplest case: photon-atom scattering at photon energies small enough (< few eV) so that atom cannot be excited:
since

4. Therefore
Energy (wave length) of the photon in the final state is practically the same as in the initial state. Same argument works for elastic scattering off an electron.
To characterize the cross section of the elastic scattering we need to introduce the concept of differential cross section which measures distribution of scattered particles over the spherical angle:

5. For low energy photon - electron scattering
Obviously
satisfies:
For low energy photon - electron scattering
where
is the classical electron radius.

6. Difference between dσ/d Ω and dσ/dθ plots is due to relation sinθ dθ = d Ω

7. is the form factor of electron in the atom q=p - p
Low energy coherent photon - atom scattering - Rayleigh scattering. Photon scatters of individual Z electrons coherently - that is Z amplitudes add coherently. 2 2 γΑ
Z F (q) Α
Thomson
F (q)
is the form factor of electron in the atom Α
q=p - p in
out
is the momentum transfer to the atom. Form γ γ
factor is equal 1 for q=0, and drops when q ~ 1/R A
At large energies q becomes large except for very small angles, and Rayleigh scattering is not important.

8. Visualization of e.m. interactions on quantum level
Existence of maximum speed → no instantaneous interactions. Only way to describe interactions is to introduce carries of different interactions.
Photon is the carrier of the e.m. interaction.
The best graphical way (actually also mathematical tool) to represent the interaction is the Feynman diagrams. Each particle is denoted by a line.

9. Positron - antielectron - has the same mass, opposite charge +1.
Feynman has suggested to treat it as an electron moving back in time. General idea of these diagrams is that inside the interaction volume particles can go both forward and backward in time. Mass in the volume is not conserved.

10. Photon absorption by electron
Photon - charge particle (electron, ...) interaction elementary block one vertex which corresponds to several time sequences.
Photon emission by electron
Photon absorption by electron
Photon conversion to electron-positron pair
Electron-Positron annihilation to a photon

11. Elastic photon- electron scattering - Compton process
In the intermediate state virtual electron that is electron with a mass not equal to the free electron mass. Alternative way to look at it is through energy-time uncertainty relation:
Δ E Δ t>h/2π
Over short times characterizing interaction energy is not conserved.

12. e e e e e e e e e+ e+ e+ e+

13. e γ e+ A A
Production of electron-positron pair off the Coulomb field of the nucleus.

14. Multiple photon exchanges between electron and the nucleus is the space -time picture of the hydrogen-like bound state.
There exist many other more complicated electrodynamic processes with multiple photon emission, virtual photons - photons emitted and absorbed during the interaction, virtual electron-positron pairs, etc. Possible to calculate in excellent agreement with the data.

15. At high energies in most of the processes electrons are knocked out from the atoms.
First, the simplest case - photon scattering off a free electron - Compton scattering (difference from Thomson limit - the photon energy)
E=hν 15

16. From the momentum conservation for the longitudinal and transverse components of the momentum
Combining with energy conservation we obtain
Maximal energy transfer to the electron corresponds to

19. In the Compton scattering photons loose large fraction of their energy.19

20. Graph showing the mean energy and the maximal energy (expressed as a fraction of the incident energy) given to the recoil electron in the Compton collision.

21. Photoelectric effect:
Knock out of the electron from the atom leaving it in an excited state with an electron hole. How to satisfy energy-momentum conservation?
Mass of the atom is large, so that its kinetic energy is negligible as compared to the photon energy, and can be neglected in the eqs. for energy conservation.
Minimal energy to remove an electron from a shell is given by the energy of the shell - sharp increase of the photoelectron cross section with the energy of the shell is reached.

22. Energy dependence of the absorption for lead and water.
Jump of the absorption above the threshold for the interaction with K-shell electrons.
Energy dependence of the absorption for lead and water.

23. After electron is removed, atom de-excites
After electron is removed, atom de-excites. Several channels: characteristic photon emission (single photon emission, multiple photon emission...) given by differences between the energies of the levels, Auger electrons.
Angular distribution of photons is isotropic. Ion ionization has important implications for the oncologic applications.

24. Atomic and energy dependence of photoelectric effect
Energy dependence of cross section
for energies up to 1 MeV. For higher energies for heavy nuclei cross section drops more slowly closer to 1/E.
Z dependence
Look again at the kinematics near say K-shell threshold. Since this is a threshold, the final electron is nearly at rest. So the momentum of the final nucleus is approximately
equal to that of the initial photon. Since the atom is heavy this corresponds to very small final kinetic energy of the atom. Hence 2
Eγ≈ΔE=(energy binding of the K-shell electron)/c = 13.6 eV/c Z
To ensure kinematics corresponding to a low momentum final electron the photon has to hit an electron with momentum equal and opposite to the photon momentum. This momentum is much smaller than the average electron momentum for K-shell. Indeed for the Coulomb case

25. Cross section should strongly increase with Z If
high momenta of electrons are necessary which are present in the heavy nucleus with much higher probability.
Cross section should strongly increase with Z
Cross section per atom increases as Z4 for heavy nuclei and as Z4.8 for light nuclei.
It decreases with increase of as

26. Applications to Diagnostic Radiology: plus of photoelectric effect - large discrimination between bone and water; minus most of the energy is absorbed by the body - bigger exposure.

27. In the case of heavy target its kinetic energy is small and
Pair production
Energy conservation:
In the case of heavy target its kinetic energy is small and

28. In the case of the scattering off a free electron one cannot neglect the kinetic energy of the target electron. Minimal energy in this case is a factor of two larger.
Physics majors - try to test this statement using energy-momentum conservation. Hint - near threshold two electrons and positron have zero relative velocity - that is equal momenta. Relativistic kinematics is crucial to get the correct answer.

29. 2
Production off the nucleus dominates (~Z ) over production off electrons (~Z). Dominant mechanism of interaction for E> few MeV

30. Relative contributions of different mechanisms for Al and Pb
Relative contributions of different mechanisms for Al and Pb. Larger binding in Pb case leads to extention of the energy range where photoelectric effect is important.
For example for E=300 keV for scattering off lead 21% of events are due to Compton.

31. Simulations of the photon interactions in the media using
EGS tool developed at Stanford Linear Acceleration Center.

32. Photoelectric Effect
Compton Scattering
Pair Production

33. Note green dots - these are electrons which stop very close to the production points.33

34. 34

35. EGS Simulation: 300 keV photons/2-cm Pb
From this plot - all photons should be interact. From the plot in p % of interaction should be photoabsorption
However 10 events could give results rather different from average - need larger data sample - on the scale of 100 events.

36. Summary on photon interactions.
In the energy range relevant for the applications we discuss photons interact in three different ways with photoeffect being dominant at the lower end the relevant for medical application, and
pair production at high energies.
The EGS Monte Carlo code seems to be doing a good job of simulating photon interactions.
Next step is electron interaction.