Report

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 1 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 2 Thomson Scattering S 2 E c 2 B c 8 8 F eE eE0 sin t power/area eE0 sin t a m 2 e2 2 1 2 e4 E02 power emitted P a 3 3 2 3c 23c m 8 e2 2 S P 3 mc thisis thepower subtract edfrom theincomingbeam 8 T 3 2 e 8 2 2 re 0.655 1024 cm2 3 mc 2 3 Rayleigh Scattering Rayleigh scattering is scattering of light from a harmonically bound electron AssumingSHO with frequency0 for an electronin an atom 4 Rayleigh T hom son 2 2 0 2 You may recall the probability for Rayleigh scattering goes as 1/λ4 Why is the sky blue? 4 Compton Scattering Compton scattering is the scattering of light (photons) from free electrons 5 Compton Scattering Calculations The change in wavelength can be found by applying Energy conservation h me c h Ee h p c m c 2 2 2 e 2 4 1/ 2 e Momentum conservation p p pe p p p 2 p p p 2 p2 2 p p cos 2 e 2 2 6 Compton Scattering From energy conservation me2 c 4 (h h ) 2 2me c 2 h h me2 c 4 pe2 c 2 h h 2hh 2 pe 2me h h 2 c c c 2 2 From momentum conservation 2 p p p 2 p p p 2 p2 2 p p cos 2 e 2 h h h h p cos 2 c c c c 2 2 2 e Eliminating pe2 mec2 h h hh 1 cos 7 Compton Scattering Continuing on h (1 cos ) 2 me c And using v=c/λ we arrive at the Compton effect h 1 cos mec And h/mc is called the Compton wavelength h C 2.43 1012 m mec 8 Compton Scattering Summarizing and adding a few other useful results are h 1 cos me c hv h hv 1 cos 1 2 me c Te h h hv t an cot 1 2 me c 2 9 Compton Scattering The differential and total cross sections are calculated in a straightforward manner using QED Called the Klein-Nishina formula 2 2 d r 1 1 cos 2 1 cos 2 d 2 1 1 cos 1 1 cos 2 e Compton 1 21 1 ln1 2 2 1 2 2 2re 1 ln1 2 1 3 2 2 1 2 10 Compton Scattering On the previous slide At low energies Compton T hom son hv 2 mec 8 2 re 3 At high energies Compton 8r 3 1 ln 2 3 8 2 2 e 11 Compton Scattering Thus at high energies, the Compton scattering cross section C goes as Compton Z ~ hv 12 Compton Scattering Graphically, d/d 13 Compton Scattering In polar form, assume a photon incident from the left 14 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 15 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 16 Compton Scattering C tr C sc C T hv hv C C h h hv sc C C h similarlyfor t hemass energy t ransfer at t enuat ion coefficient tr C T C T N Av C h h A tr C 17 Compton Scattering Here en=tr 18 Compton Scattering Another useful form of the differential cross section is d/dT, which gives the energy distribution of the electron 19 Compton Scattering The maximum electron kinetic energy is given by 2 hv and Tmax hv 2 me c 1 2 hvme c 2 2 hv Tmax hv1 2 1 2 m c 2hv e and for hv large me c 2 hv Tmax 0.2555MeV 2 20 Compton Scattering In cases where the scattered photon leaves a detector without interaction one would observe 21 Compton Scattering 22 Compton Scattering me c 2 hv hv | 2 1 2hv / me c 2 hv | 255keV 23 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 24 Pair Production 25 Pair Production Energy and momentum conservation give Energy hf E E hf Momentum(x) p cos p cos c Momentum(y) 0 p sin p sin Energy conservation can be re-written hf p2 c 2 m 2 c 4 p2 c 2 m 2c 4 But momentum conservation (x) shows hfmax pc pc Thus energy and momentum are not simultaneously conserved 26 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 2 100me c hv 1/ 3 E E Z 27 Pair Production In the Born approximation (which is not very accurate for low energy or high Z) one finds No screening 1 and me c 2 h 137me c 2 Z 1/ 3 7 2h 109 pair 4 Z r ln f Z 2 54 9 me c Completescreening 0 and h 137me c 2 Z 1/ 3 2 pair 2 e 7 1 1 / 3 4 Z r ln 183Z f Z 9 54 2 2 e 28 Pair Production Notes pair ~ Z2 Above some photon energy (say > 1 GeV), pair 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 29 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 me c 2 T h 1.022 T 2 For h 5MeV T 2 MeV and 15 30 Pair Production The probability for pair production 31 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 32 Pair Production 33 Pair Production Similar to the photoelectric effect and Compton scattering we define the mass attenuation and mass energy transfer coefficients as pair N Av pair A trpair hv 2me c 2 pair hv 34 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 35 Photonuclear Interactions Giant dipole resonance 36 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 37 Photon Interactions Typical photon cross sections 38 Photon Interactions Typical photon cross sections 39 Photon Interactions Notes Of course different interactions can occur at a given photon energy Z pe Compton pair pe ZCompton pair 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 eff can be estimated as eff 0.693 HVL 40 Beam Hardening 41 Photon Interactions Let’s return to our first slide I I0e x 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 ’ < 42 Photon Interactions 43 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 ’ = en 44 Photon Interactions There are three relevant mass coefficients N Av mass absorptioncoefficient A tr mass energy transfer coefficient en mass energy absorptioncoefficient en tr 1 g where g is theaveragefractionof secondary electronenergy lost toradiativeinteractions (bremsstrahlung and annhilation) 45 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 fA f B ... mixture A B where f i are the weight fractions of separateelements 46 Photon Interactions =1/(/) 47 Photon Interactions Sometimes easy to loose sight of real thickness of material involved 48 Photon Interactions X-ray contrast depends on differing attenuation lengths 49 Photon Interactions What is a cross section? What is the relation of to the cross section for the physical process? has units cm2 and has units1 / cm N where N is thedensityof atoms N Av is thelinear absorptioncoefficient A cm2 in is morecommon g 50