Energetic Particle Interactions with Matter: EM Interaction, Bethe-Bloch Equation, Photon Interactions

1. Elementary Particles • Radiation Technology Jan 10, 2019 • Interactions of energetic particles with matter • Electro-Magnetic (EM) Interaction • Bethe-Bloch equation • Landau Distribution • Bremstrahlung • Photon Interactions • Photo Effect • Compton Scattering • PairProduction • Cherenkov Radiation

2. μ+μ - π+π- p e charged particles n π0 γ υ neutral particles

3. Energetic charged particles through matter: Electro-Magnetic interaction special for electrons: Brehmstrahlung Interaction of photons (X-rays – gammas) with matter Photo-effect Compton Effect Pair Production

4. Particle detection is based on the interactions of particles with matter Electro-magnetic (EM) interactions cause detectable phenomena, e.g. in the form of electric signals, production of light, chemical changes, local phase transitions (e.g. bubble chamber), Acoustics The signals observed may be due to secondary particles, e.g. due to particles produced by the interaction of a hadron with a nucleus or by a neutrino with a nucleon. But invariably the effect of the e.m. interactions of charged particles with the matter through which they pass is seen. These interactions manifest themselves by excitation of molecules and/or atoms and ionization. Ionisation

5. charged particle Energy E Energy E’ dX Calculate energy transfer to all charged particles within material Realise that there are large statistical fluctuations Essential energy loss: E – E’ = dE Essential process: dE/dX

7. EM interactions can take place with • electrons • nuclei An important fact: me = 511 keV /c2, mp = 940 MeV/c2, so it is much easier to give an electron a "kick" than a nucleus, i.e. energy loss due to collisions will be dominated by interactions with the electrons.

8. Energyloss due to e.m. interactions, also referred to as "collisions", In the laboratory system S' particle 1 is moving, particle 2 is in rest. Consider a system S in which particle 1 is in rest, so particle 2 is moving in a static electric field. 2 3 E Z e / r = Z e b / r = cos q 1 1 ^ (energy particle is assumed to be high, -> b is assumed to be constant) Movement particle 2 in S 2 q r b Z e 1 1 x = - v t 1 t = 0: distance between 1 and 2 is minimal In the laboratory system S' particle 2 is in rest, so there is no effect from the (time dependent) magnetic field caused by particle 1. In S': time in S' ! electric field strength Beyond the scope of this course

9. (Particle 2 is in rest -> non-relativistic calculation of energy loss possible) energy loss Interactions with nucleus with mass number A and atomic number Z: m = Amp ≈ 2Zmp, Z2 = Z Interactions with electron: m = me, Z2 = 1 -> Energy loss due to collisions is dominated by interactions with the electrons (NB: we are comparing interactions with 1 electron to interactions with the nucleus of an atom, a non-ionized atom has Z electrons) Beyond the scope of this course

10. e b r Z e 1 dx db We have looked at the interaction between a charged particle and a single isolated electron or nucleus. In reality we are dealing with the combined effect of interactions with many electrons (and nuclei). The effect of this was estimated by N. Bohr. He considered interactions with atoms, each with Z2e electrons. Particle 1 is passed through the centre of a thin shell of atoms and the net energy loss is calculated. Then one integrates over b (the "impact parameter") to obtain dE/dx Number of electrons in shell: ne 2pbdb dx with ne = number of atoms per cm3 Beyond the scope of this course

11. To find bmax we note that determines the size of the transversal field strength, from this it follows that only during a time of the order of b/(v1g) the "collision" takes place. This time should be short relative to the time characterizing the orbital frequencies of the electrons in the atoms, otherwise the effect of the binding of the electrons cannot be neglected (Bohr considered harmonically bound charges for distant collisions, but numerically the results are about the same as for the approach discussed here). Define w to be the characterizing orbital frequency, then: bmax = gv1/ w. The energy transfer cannot exceed a certain maximum, as seen earlier. This maximum is for m2 >> me: 2meb2g2 With: so: one obtains The result shows for low energies decreasing dE/dx for increasing E and for higher energies an increase in dE/dx, due to the increase in g and v approaching c. The properties of the material enter via ne, i.e. only the electron density matters. Beyond the scope of this course

12. Maximum energy transfer → "head-on collision", i.e. in CM frame the particles move along the same line in opposite directions. Conservation of momentum: in CM frame total momentum = 0, after collision and if the collision is elastic (i.e. no energy loss in collision) the particles move opposite to the original directions, with speeds equal to their original speeds. The momentum change for the target particle therefore is : 2m bCMgCM Beyond the scope of this course

13. Taking into account quantum-mechanical effects and using first-order perturbation theory the Bethe-Bloch equation is obtained: Tmax is the maximum energy transfer to a single electron: , Tmax is often approximated by 2meb2g2. re is the classical electron radius (re = e2 / mec2 = 2.82 x10-13 cm) (radius of a classical distribution of the electron charge with electrostatic self-energy equal to the electron mass). I is the mean ionization energy. NB: for high momentum particles Substituting this and also e2 / mec2 for re gives eq. (2.19) of Fernow Hans Albrecht Bethe Felix Bloch Beyond the scope of this course

14. Bethe- Bloch: Z dE/dX = C1/v2 . ln C2 v3γ2 dE = E-E’ dX

15. dE/dx for pions as computed with Bethe-Bloch equation dE/dx divided by density r (approximately material independent) slope due to 1/v2 high bg: dE/dx independent of bg due to density effect, "Fermi plateau" relativistic rise due to ln g • about proportional to ne, as ne = na Z = NAr Z / A, -> ne ≈ NAr / 2 From PDG, Summer 2002

16. Bubble chamber photograph shows different bubble density along tracks for different particle momenta and particle type. http://physics.hallym.ac.kr/education/hep/adventure/bubble_chamber.html

17. Fluctuations in energy loss The energy transfer for each collision is determined by a probability distribution, i.e. is not fixed. The collision process itself is also a random process determined by a probability distribution. The number of collisions per unit length of material is determined by a Gaussian distribution, the energy loss distribution usually is referred to as a "Landau" distribution. This is a distribution with a long tail for high values of the energy loss. The tail is caused by collisions with a high energy transfer. Collisions with a high-energy transfer produce d-electrons or "d-rays". These electrons also lose energy and may cause further ionization, i.e. "secondary electrons" are produced. In bubble chamber pictures the ionization produced by d-electrons can be seen as short track segments. Lev Davidovich Landau

18. Detail of bubble chamber picture showing delta-ray http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/mbitu/electromag-events1.htm

19. Bubble chamber picture showing delta-rays The red arrows indicate some of the d-electrons, looping in the magnetic field applied CERN photo, http://weblib.cern.ch/Home/Media/Photos/CERN_PhotoLab/?p=

20. From PDG, Summer 2002 Landau Distribution

21. Number of particles with energy loss between W and W+dW in x+dx (I) = number of particles with energy loss between W and W+dW in x (II) - number of particles with energy loss > 0 in dx (III) + number of particles with energy loss between W-e and W-e+dW in x (IV) and energy loss e in dx or: Beyond the scope of this course

22. Essential (in gas): - number of clusters per mm tracklength - number of electrons per cluster specific for gas(and density ρ, thus T, P!)

23. Stopping power charged particle Transmission 90 % absorption detector Thickness

24. Practical Range From PDG, Summer 2002

25. Most of the energy deposited at end of track Fraction of particles surviving 100 % Sir William Henry Bragg Sir William Lawrence Bragg dE/dx Bragg curve Averange range R Depth x in material

26. S.Rossi, Medical applications of accelerators, CERN, Ac. training 1997-1998

27. Energetic charged particles through matter: Electro-Magnetic interaction special for electrons: Brehmstrahlung Cherenkov Radiation Interaction of photons (X-rays – gammas) with matter Photo-effect Compton Effect Pair Production

28. Photon (invisible) Kink in trajectory Curvature smaller after kink: due to lower momentum e Walter Heitler Hans Albrecht Bethe http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/mbitu/electromag-events1.htm

29. Photons can not be radiated in vacuum Ef,pf Before: Ei2 - pi2 = me2 After: e g e Ei,pi not possible Feynman diagram Photons can be radiated in the field of an object with charge Ef,pf e Very heavy pointlike spinless object with charge Z2 (atomic nucleus) Classically: electric field accelerates electron. Quantum-mechanically: virtual photon exchange decreases pf, so that Ef2 - pf2 = me2 is satisfied g e Ei,pi g e This diagram also contributes e Z2 Beyond the scope of this course

30. Electrons with energies in the MeV range can also produce photons up to the MeV range by Bremsstrahlung. This is much more likely for a high Z-material and should be taken into account when shielding a beta source. A combination of a light material (usually acryl or perspex) near the source and a surrounding layer of lead for absorbing photons from the source and/or produced in the light material should be used. The thickness of the light material is typically of the order of a few - 10 cm, see range curves on next slide.

31. ESRF: European Synchrotron Radiation Facility, Grenoble, France 300 m circumference booster synchrotron, 6 GeV 16 m linac, 200 MeV

32. Energetic charged particles through matter: Electro-Magnetic interaction special for electrons: Brehmstrahlung Interaction of photons (X-rays – gammas) with matter Photo-effect Compton Effect Pair Production

33. (main contributions) dominates below Eg ≈ few 100 keV dominates around Eg ≈ 1 MeV dominates above Eg ≈ 2 MeV (minimum energy = 2 me = 1.02 MeV)

34. N photons N-dN photons dX

36. From PDG, Summer 2002

37. Kleinknecht, p.20 Photoelectric effect A photon causes emission of a "photoelectron' from an atom, the energy of this electron is equal to the energy of the photon minus the binding energy of the electron. -> The electron has a well-defined energy, directly related to the energy of the photon, therefore energy measurement of the electron (by determining the energy deposition when stopping in matter) provides the possibility to measure the energy of the photon Albert Einstein Nobel Prize 1921 "for his services to Theoretical Physics, and especially for his discovery of the law of the photoelectric effect"

38. The cross-section for the photo-electric effect has maxima at photon energies equal to binding energies of electrons and decreases in between for increasing photon energy: "K-edge" or "K-absorption edge" with: and for eK<e<1 Rayleigh scattering, atom neither ionized nor excited Pair production, nuc: nuclear field, e: electron field For e >> 1: Note the dependence of s on the 5th power of Z Beyond the scope of this course re = e2 / mec2 = 2.82 x10-13 cm

39. From PDG, Summer 2002

40. Scintillator Photomultiplier NaI ComptonRidge Photopeak

41. Energetic charged particles through matter: Electro-Magnetic interaction special for electrons: Brehmstrahlung Cherenkov Radiation Interaction of photons (X-rays – gammas) with matter Photo-effect Compton Effect Pair Production

42. Compton effect, scattering of photon from an atomic electron, results in lower energy photon, the energy lost is transferred to the electron Incoming and outgoing photon not seen Arthur Holly Compton Nobel prize 1927 http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/mbitu/electromag-events1.htm

43. 2 2 2 ( ) p m T m = + - 2 T 2 mT = + k k p = + k k k T k k cos p cos = + = q + f 0 k0 0 q k sin p sin q = f 0 f 2 2 2 ( ) p cos k k cos f = - q 0 p, T 2 2 2 2 p sin k sin f = q 2 2 2 p k 2 k k cos k = - q + 0 0 2 2 k k T = - p T 2 mT = + 0 2 2 ( ) k 1 cos ( ) 1 cos - q e - q 0 T m = = ( ) ( ) m k 1 cos 1 1 cos + - q + e - q 0 NB: m = me, mk k 0 0 k k T = - = = 0 m k 1 cos 1 cos q q 1 ( ) ( ) + - + e - 0 Minimum value for k:  = p, Beyond the scope of this course

44. 1 cos ] ( ) - q The emission angle of the electron can be expressed in terms of e and : k k cos k k cos - q - q 0 0 cos f = = p 2 2 k 2 k k cos k - q + 0 0 k cos k 1 1 cos ( ) ( ) q + e - q 0 0 k k cos k - q = - = 0 0 1 1 cos 1 1 cos ( ) ( ) + e - q + e - q 2 (1-cos ) k 2 2 [ ( ) + e e + 2 2 0 k 2 k k cos k - q + = 0 0 2 1 1 cos [ ] ( ) + e - q 1 cos ( ) - q cos 1 ( ) f = + e 2 2 1 cos ( ) ( ) + e e + - q Beyond the scope of this course

45. For high energies and scattering under 180 degrees: i.e. backscattered photons tend to have the same energy, over a considerable angular range, which can give rise to a "backscatter" peak at 256 keV in gamma spectra Beyond the scope of this course, maar figuur moet je echt snappen!

46. Scintillator Photomultiplier NaI ComptonRidge Photopeak

47. Energetic charged particles through matter: Electro-Magnetic interaction special for electrons: Brehmstrahlung Cherenkov Radiation Interaction of photons (X-rays – gammas) with matter Photo-effect Compton Effect Pair Production

48. Pair creation Positron Bremsstrahlung produces photon Electron

49. Pair creation http://teachers.web.cern.ch/teachers/archiv/HST2002/Bubblech/omegaminus.html

50. Bremstrahlung en Pair Production: identiek proces Bremstrahlung Pair Production e- e- e- e+ photon photon