Photon interactions in matter

Gamma- and X-Ray • Compton effect • Photoelectric effect Interactions in Matter • Pair production • Rayleigh (coherent) scattering Chapter 7 • Photonuclear interactions

F.A. Attix, Introduction to Radiological Kinematics Physics and Radiation Dosimetry Interaction cross sections Energy-transfer cross sections Mass attenuation coefficients

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Compton interaction A.H. Compton

• Inelastic photon scattering by an electron • Arthur Holly Compton (September 10, 1892 – March 15, 1962) • Main assumption: the electron struck by the • Received Nobel prize in physics 1927 for incoming photon is unbound and stationary his discovery of the Compton effect – The largest contribution from binding is under • Was a key figure in the Manhattan Project, condition of high Z, low energy and creation of first nuclear reactor, which went critical in December 1942 – Under these conditions photoelectric effect is dominant Born and buried in • Consider two aspects: kinematics and cross Wooster, OH http://en.wikipedia.org/wiki/Arthur_Compton sections http://www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=22551

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Compton interaction: Kinematics Compton interaction: Kinematics

• An earlier theory of -ray scattering by Thomson, based on observations only at low energies, predicted that the scattered photon should always have the same energy as the incident one, regardless of h or  • The failure of the Thomson theory to describe high-energy photon scattering necessitated the • Inelastic collision • After the collision the electron departs at angle , with development of Compton’s theory kinetic energy T and momentum p • The photon scatters at angle  with a new, lower quantum energy h and momentum h/c

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1 Compton interaction: Kinematics Compton interaction: Kinematics

• The collision kinetics is based upon conservation of 2 • pc can be written in terms of T : pc = T(T + 2 m0c ) both energy and momentum where m0 is the electron’s rest mass • Energy conservation requires • We get a set of three simultaneous equations in T = h − h these five parameters: h, h, T, , and  : • Conservation of momentum along the (0°) direction h h =  2 h = h cos + pc cos 1+ (h / m0c )(1− cos) • Conservation of momentum perpendicular to the T = h − h direction of incidence:  h     cot = 1+  tan h sin  = pc sin   2     m0c   2 

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Compton interaction: Kinematics Compton interaction: Kinematics

energy transferred to the recoiling electron

hv’=0.49 Dependence of on  Photon energy  photon is backscattered – increasing is a strong function of - max energy transferred to e photon energy: (never ALL photon energy) • Low photon energies   90 − / 2 • High photon energies: electrons are mostly forward scattered 2 When photon energy is lower than electron rest mass energy m0c , relativistic effects do not contribute => pure elastic scattering

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Compton interaction: Cross sections Compton interaction: Cross sections Interaction cross section Interaction cross section

Cross section describes the probability of interaction Thomson: elastic scattering on free electron • Thomson: elastic scattering on a free electron, no energy is - total cross section (integrated over all directions ) transferred to electron • Differential cross section (per electron for a photon 2 8r0 −25 2 scattered at angle , per unit solid angle)  = = 6.6510 cm /electron e T 3 d  r 2 max at  = 0, 180o e T 0 2 o = (1+ cos ) ½ max at  = 90 r0 d 2 - classical radius of electron e2 r = - classical radius of electron 0 2 2 m0c Works well for low photon energies, << m0c Overestimates for photon energies > 0.01MeV (factor of 2 for 0.4MeV)

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2 Compton interaction: Cross sections Compton interaction: Cross sections Interaction cross section Interaction cross section

• This cross section (can be thought of as an effective target • Klein-Nishina: Compton scattering on free electron but area) is equal to the probability of a Thomson-scattering event includes Dirac’s quantum relativistic theory occurring when a single photon passes through a layer • Differential cross section: containing one electron per cm2 • It is also the fraction of a large number of incident photons that scatter in passing through the same layer, e.g., approximately 2 ' ' d  r  h  h h 2  665 events for 1027 photons e K−N = 0   + − sin     '  • For such a small fraction of photons interacting in a layer of d 2  h  h h  matter (by all processes combined remains less than about 0.05), the fraction may be assumed to be proportional to absorber thickness; for greater thicknesses the exponential • For elastic scattering (h=h) – reduces to Thomson's expression relation must be used • Needed at high photon energy

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Compton interaction: Cross sections Compton interaction: Cross sections Interaction cross section Interaction cross section d  r 2 Thomson’s: e T = 0 (1+ cos2 ) d 2

d e K −N shows probability for a photon to be d scattered at an angle 

Photons with high energies tend to scatter in the forward direction ( ~ 0 ) K–N Compton cross section differential in solid angle: (a) data plotted in Cartesian coordinates; (b) the same data plotted in polar coordinates. When the photon energy increases, the scattered photon distribution becomes strongly forward peaked

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Compton interaction: Cross sections Compton interaction: Cross sections Interaction cross section Energy-transfer cross section Klein-Nishina: Compton scattering on free electron (includes Dirac’s quantum relativistic theory) Total cross section –> fraction of energy diverted into Compton Integrating over all photon scattering angles obtain the total cross interactions –> fraction of energy transferred to electrons –> dose section per electron Thomson’s Energy transfer cross section  = 2r2... e K−N 0 T  =   e tr e h . . . - depends on incident Approx. photon energy: higher ~1/hv Average kinetic energy of recoiling electrons: energy => lower interaction  probability; T = h  e tr expressed through parameter  2 e  = h m0 c

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3 Compton interaction: Cross sections Compton interaction: Cross sections Energy-transfer cross section Other cross sections

Differential K-N cross section for electron scattering at angle , per unit solid angle, hv'/ hv T / hv per electron

d e K −N d photon energies:

For high photon energies electrons are preferentially For hv = 1.6 MeV – half of the photon energy is transferred forward scattered ( = 0) to the electron (T = 0.8 MeV)

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Compton interaction: Cross sections Compton interaction: Other cross sections Mass attenuation coefficient

Cross section per electron 0 The distribution of kinetic energies given to the   Z Probability that a single photon (no Z dependence due to free e Compton recoiling electrons is ~ flat from 0 will have Compton interaction in almost up to the max electron energy electron assumption) 2 a layer of 1 e/cm and transfers   Z  energy between T and T+dT Cross section per atom a e

d  e K−N in cm2MeV −1e−1  N Z dT Cross section per unit mass = A   (mass attenuation  A e Energy distribution of coefficient) electrons, averaged over all For head-on collisions: NA – Avogadro’s constant; Z – number of electrons per atom; A – number of grams per mole of material;  – density in g/cm3 Tmax ~ hv - 0.2555 MeV Stronger effect for high hv

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Photoelectric effect: Kinematics Photoelectric effect basics

Most important at low photon energies E  • Interaction with atomic-shell electrons tightly bound with L potential energy E < hv mom=hv/c L b h K • Photon is completely absorbed K • Kinetic energy to electron:

T = h − Eb • Photon transfers its momentum hv/c plus some independent of scattering angle transversal momentum due to the perpendicular • Atom acquires some momentum electric field in the electromagnetic wave • No universal analytical expression for • Final state = free electron + hole in the atomic shell cross sections

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4 Photoelectric effect: Photoelectric effect: Cross sections Directional distribution Interaction cross section For higher photon energies electrons tend to scatter in forward direction ( =0 is forbidden since it is perpendicular to the vector E) Total interaction cross section per atom, in cm2/atom Z n Directional distribution Half of all electrons is ejected a  k m within a forward cone of half (h ) angle equal to bipartition angle k = Const m,n − energy dependent Z 4 n  4, m  3 at hν = 0.1 MeV   (h )3 Mass attenuation coefficient 3   Z       h 

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Photoelectric effect: Mass Photoelectric effect: Cross sections attenuation coefficient Energy-transfer cross section 3   Z       h  Fraction of energy transferred to all electrons T h − E = b E=88keV h h Vacancy created by a photon in the inner shell has to be filled If through Auger process - additional contribution to kerma Final result:

 tr  h − PKYK h K − (1− PK )PLYL h L  =      h 

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Photoelectric effect: Mass energy-transfer coefficients Atom relaxation

h

L L K K

Excited atom relaxes its energy by - fluorescence (emission of photons) or - Auger process (emission of electron) when the higher energy shell electrons move downward

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5 Pair production: Third body is Pair production needed • Photon is absorbed giving rise to electron and positron e • Occurs predominantly in Coulomb force field • There exists a reference frame usually near atomic nucleus where the total momentum of sometimes in a field of atomic electron h electron and positron is zero 2 p • Minimum photon energy required 2m c =1.022 MeV 0 • But photon momentum is always 2 − + hv = 2m0c + T + T h/c =1.022 MeV + T − + T + • Third body needed for momentum e conservation: electron or nucleus

h p

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Pair production in Nuclear Pair production in Electron Coulomb Force Field Coulomb Force Field Energy transferred to electron Total cross section per atom Z-dependence 2 Triplet production – higher threshold 4m0c = 2.044 MeV 2 required for conservation of momentum a = 0Z P P – function of hv and Ratio of cross section for all electrons of the atom to Z (at higher E) P nuclear cross section of the same atom is small:

2 r 2 1  e2   = 0 =   = 5.80x10−28 cm2/electron (electron) 1 0 137 137  m c2    0  (nucleus) CZ

Nuclear attraction and repulsion tend to give C – parameter depending on energy, close to 1 the positron slightly more energy than the Example: for Pb (Z=82) triplet cross section is 1% of that of PP; Fraction of energy electron, the difference being less than rising to 5-10% for Z=10 0.0075Z MeV transferred to positron

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Pair production: Cross sections Rayleigh (coherent) scattering

Total cross section for pair production per unit mass: • Photon is scattered by combined action of whole atom • Photons do not lose energy, redirected through only a small          angle   =   +      pair   nuclear   electron • No charged particles receive energy, no excitation produced => No contribution to kerma or dose Pair production energy transfer coefficient:  Z Atomic cross section: R   (h )2  2  tr  hv − 2m0c Typical ratios of Rayleigh to total attenuation coefficient  /  =   R    hv 

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6 Photonuclear Interactions Photonuclear Interactions

• Photonuclear interactions (photodisintegration) are direct • After being excited by a photon, nucleus emits a proton interactions of energetic photons and absorber atom nuclei (,p) or, more likely, a neutron (,n) • The threshold energy is ∼8 MeV for all nuclides with two • (,p) events contribute to kerma and dose exceptions: deuteron at 2.22 MeV and beryllium-9 at 1.67 MeV – Relative amount less that 5% of pair production • Cross sections for photonuclear reactions exhibit a broad – Usually not included in dosimetry consideration “giant resonance” peak at ~23 MeV for low Z and at ~12 MeV • (,n) events are important for shielding design, especially for high Z absorbers. The peak’s FWHM typically ranges from for higher energies ∼3 MeV to ∼9 MeV – Contribution to the total atomic cross section is still only a few – Typical RT threshold >10MeV, resulting in slight percent neutron contamination of the photon beam – Increases by order of magnitude from 10 to 25MeV

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Photonuclear Interactions Photonuclear Interactions:

• High Z absorbers found in the linac head serve as the main site of further reading photonuclear interactions for typical treatment energies; a smaller portion • NCRP Report No. 79, ‘‘Neutron contamination from of interactions occurs in tissue/phantom medical electron accelerators,’’ National Council on • Example: The dose from photon- radiation Protection and Measurements, Bethesda, MD, 1984 induced nuclear particles (neutrons, protons, and alpha particles) • NCRP Report No. 144, Radiation Protection for Particle Accelerator Facilities, National Council on radiation generated by high-energy photon Protection and Measurements, Bethesda, MD, 2003 beams from medical linacs (Siemens 18 MV, Varian 15 MV, • O. Chibani, C.-M. C Ma, Photonuclear dose calculations for high-energy photon beams from Siemens and Varian and Varian 18 MV) linacs, Med. Phys. 30 1990 (2003)

O. Chibani, C.-M. C Ma, Med. Phys. 30 1990 (2003)

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Total coefficients for attenuation, Mass attenuation coefficients energy transfer and absorption Total mass attenuation coefficient for photon interactions - add probabilities for photoelectric effect, Compton effect, pair production and Rayleigh scattering      E=88keV = + + + R     

Total mass energy-transfer coefficient:

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7 Mass energy-absorption Mass energy-transfer coefficients coefficient   en = tr (1− g)  

g - average fraction of secondary electron energy lost in radiative interactions For low Z and high hv, g --> 0

Appendix D

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Summary

• Compton effect • Photoelectric effect the most important • Pair production • Rayleigh (coherent) scattering – no energy transferred to the medium • Photonuclear interactions – relevant at high energies

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