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7. Gamma and X-Ray Interactions in Matter

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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 1 2 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 3 4 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 5 6 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 7 8 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 9 10 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 8r0 −25 2 scattered at angle , per unit solid angle) = = 6.6510 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) 11 12 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 13 14 Compton interaction: Cross sections Compton interaction: Cross sections Interaction cross section Interaction cross section Thomson’s: 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 15 16 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 = 2r2... e K−N 0 T e tr =e d r 2 h ... - depends one incidentT = 0 (1+ cos2 ) Approx. photon energy:d higher 2 ~1/hv Average kinetic energy of recoiling electrons: energy => lower interaction probability; T = h e tr expressed through parameter 2 e = h m0 c 17 18 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) 19 20 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 21 22 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 23 24 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 25 26 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 27 28 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 29 30 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
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