Interactions of Photons with Matter

Transcription

1 Interactions of Photons with Matter Photons are elecomagnetic radiation with zero mass, zero charge, and a velocity that is always c, the speed of light. Because they are elecically neual, they do not steadily lose energy via coulombic interactions with atomic elecons, as do charged particles. Photons avel some considerable distance before undergoing a more catasophic interaction leading to partial or total ansfer of the photon energy to elecon energy. These elecons will ultimately deposit their energy in the medium. Photons are far more peneating than charged particles of similar energy. Energy Loss Mechanisms photoelecic effect Compton scattering pair production Interaction probability linear attenuation coefficient, µ, The probability of an interaction per unit distance aveled Dimensions of inverse length (eg. cm - ). µ x N = N 0 e The coefficient µ depends on photon energy and on the material being aversed. µ mass attenuation coefficient, ρ The probability of an interaction per g cm -2 of material aversed. µ ( ρ x) Units of cm 2 g - N = N 0 e ρ

2 Mechanisms of Energy Loss: Photoelecic Effect In the photoelecic absorption process, a photon undergoes an interaction with an absorber atom in which the photon completely disappears. In its place, an energetic photoelecon is ejected from one of the bound shells of the atom. For gamma rays of sufficient energy, the most probable origin of the photoelecon is the most tightly bound or K shell of the atom. The photoelecon appears with an energy given by E e- = hv E b (E b represents the binding energy of the photoelecon in its original shell) Thus for gamma-ray energies of more than a few hundred kev, the photoelecon carries off the majority of the original photon energy. Filling of the inner shell vacancy can produce fluorescence radiation, or x ray photon(s). M L Flourescent radiation E = b.e. K Incoming photon E = hυ Photoelecon leaving E = hυ- b.e. Events in the photoelecic scattering process (after Alpen, E. L. Radiation Biophysics, 2 nd ed. San Diego, CA: Academic Press, 990. Fig. 4.3) 2

3 The photoelecic process is the predominant mode of photon interaction at o relatively low photon energies o high atomic number Z The probability of photoelecic absorption, symbolized τ (tau), is roughly proportional to τ Z n ( hν ) 3 where the exponent n varies between 3 and 4 over the gamma-ray energy region of interest. This severe dependence of the photoelecic absorption probability on the atomic number of the absorber is a primary reason for the preponderance of high-z materials (such as lead) in gamma-ray shields. The photoelecic interaction is most likely to occur if the energy of the incident photon is just greater than the binding energy of the elecon with which it interacts. Image removed due to copyright resictions. Fig 4.4 in Alpen, E. L. Radiation Biophysics, 2 nd ed. San Diego, CA: Academic Press,

4 Compton Scattering Compton scattering takes place between the incident gamma-ray photon and an elecon in the absorbing material. It is most often the predominant interaction mechanism for gamma-ray energies typical of radioisotope sources. It is the most dominant interaction mechanism in tissue. Scattered photon E = hu'; momentum, ρ = hu'/c Incoming photon E = hu; momentum, ρ = hu/c q = photon scattering angle f = elecon scattering angle Recoil elecon momentum, ρ = q; energy = E; velocity = v Events in the Compton (incoherent scattering process) (after Alpen, E. L. Radiation Biophysics, 2nded. San Diego, CA: Academic Press, 990. Fig. 4.5) In Compton scattering, the incoming gamma-ray photon is deflected through an angle θ with respect to its original direction. The photon ansfers a portion of its energy to the elecon (assumed to be initially at rest), which is then known as a recoil elecon, or a Compton elecon. All angles of scattering are possible. The energy ansferred to the elecon can vary from zero to a large fraction of the gamma-ray energy. The Compton process is most important for energy absorption for soft tissues in the range from 00 kev to 0MeV. 4

5 The Compton scattering probability is is symbolized σ (sigma): almost independent of atomic number Z; decreases as the photon energy increases; directly proportional to the number of elecons per gram, which only varies by 20% from the lightest to the heaviest elements (except for hydrogen). Compton Scattering Energetics The energies of the scattered photon hν ' and the Compton elecon E e, are given by h ν ' = hν + α( cosθ ) E e = hν α( cosθ ) + α( cosθ ) hν where α = 2 m 0 c 2 [ m0c is the elecon rest energy, 0.5 MeV, hν is the incoming photon energy] 5

6 Limits of Energy Loss Maximum energy ansfer to recoil elecon: angle of elecon recoil is forward at 0, φ = 0, the scattered photon will be scattered saight back, θ = 80 With θ = 80, cos θ = - the expressions above simplify to: 2α E e(max) = hv + 2α and ' hv min = hv + 2α The Table below illusates how the amount of energy ansferred to the elecon varies with photon energy. Energy ansfer is not large until the incident photon is in excess of approximately 00 kev. Photon Energy, 5. kev Photon Energy, 5. MeV α 5. kev = 0.5 MeV = E e(max) = 5. kev * 2 *.02 = 0.0 kev hv' (min) = 5. kev *.02 = 5.0 kev Energy ansferred: 2% 5. MeV α = 0.5 MeV = 0 0 E e(max) = 5. MeV * 2 * 2 = 4.87 MeV hv' (min) = 5. MeV* 2 = 0.24 MeV Energy ansferred: 95% ( ( ( Figure by MIT OCW. ( For low-energy photons, when the scattering interaction takes place, little energy is ansferred, regardless of the probability of such an interaction. As the energy increases, the fractional ansfer increases, approaching.0 for photons at energies above 0 to 20 MeV. 6

7 Pair Production If a photon enters matter with an energy in excess of.022 MeV, it may interact by a process called pair production. The photon, passing near the nucleus of an atom, is subjected to song field effects from the nucleus and may disappear as a photon and reappear as a positive and negative elecon pair. The two elecons produced, e- and e+, are not scattered orbital elecons, but are created, de novo, in the energy/mass conversion of the disappearing photon. hυ = 0.5 MeV e + hυ = 0.5 MeV hυ (>.02 MeV) nucleus e (after Alpen, E. L. Radiation Biophysics, 2 nd ed. San Diego, CA: Academic Press, 990. Fig. 4.7) Pair Production Energetics The kinetic energy of the elecons produced will be the difference between the energy of the incoming photon and the energy equivalent of two elecon masses (2 x 0.5, or.022 MeV). E e+ + E e- = hν (MeV) Pair production probability, symbolized κ (kappa), Increases with increasing photon energy Increases with atomic number approximately as Z 2 7

8 N dx (a) N - dn (a) Compton scattering (b) (b) Photoelecic effect (c) + (c) Pair production Z of absorber Photoelecic effect dominant σ = τ Compton effect dominant Pair production dominant σ = κ Photon Energy hν, in Mev Photoelecic effect: produces a scattered photon and an elecon, varies as ~ Z 4 /E 3 Compton effect: produces an elecon, varies as ~ Z Figure by MIT OCW. Pair production: produces an elecon and a posion, varies as ~Z 2 8

9 Bulk Behavior of Photons in an Absorber Attenuation Coefficients Linear attenuation coefficient µ: The probability of an interaction per unit distance aveled. µ has the dimensions of inverse length (eg. cm - ). N = N 0 e µ x The coefficient µ depends on photon energy and on the material being aversed. x R dür narrow beam detector d good scattering geomey for measuring linear attenuation coefficient m (after Turner, J.E. Atoms, Radiation, and Radiation Protection, 2 nd ed. New York, NY: Wiley-Interscience, 995. Fig. 8.7) The interaction probability µ is actually the sum of the three possible photon interaction mechanisms: µ = τ + σ + κ τ is the photoelecic effect interaction probability σ is the Compton scattering interaction probability κ is the pair production interaction probability 9

10 Image removed due to copyright resictions. Flux = photons/cm 2 sec Image removed due to copyright resictions. Intensity = energy x flux hν # photons Intensity ( I) = photon Area time MeV Intensity = 2 cm sec Image removed due to copyright resictions. 0

11 The mass attenuation coefficient, µ /ρ, is obtained by dividing µ by the density ρ of the material, usually expressed in cm 2 g -. Each of the components, τ, σ, and κ can be expressed as mass attenuation µ τ σ κ coefficients. = + + ρ ρ ρ ρ.0 Carbon a 3 energy ansfer processes each depends differently on Z and E. Mass Attenuation Coefficients, m 2 /kg µ/ρ σ/ρ σ R /ρ τ/ρ σ/ρ κ/ρ µ/ρ hν, MeV Figure by MIT OCW. 0-2 L 3 L 2 0 Edges L b Lead Mass Attenuation Coefficients, m 2 /kg σ/ρ K Edge τ/ρ σ R /ρ µ/ρ κ/ρ hν, MeV Mass attenuation coefficients for carbon (a) and lead (b). τ/ρ indicates the conibution of the photoelecic effect, σ/ρ is that of the Compton effect, κ/ρ that of pair production, and σ R /ρ that of Rayleigh (coherent) scattering. µ/ρ is their sum, which is closely approximated in Pb by the τ/ρ curve below hν = 0. MeV. Figure by MIT OCW.

12 Linear Attenuation, Energy Transfer and Energy Absorption Not all of the energy of the incoming photons that interact in the material is necessarily absorbed there. Energy absorbed = Energy ansferred Energy lost Some energy may be lost from the absorber region due to fluorescence or bremssahlung. Linear Attenuation: probability of an interaction Energy Transfer: energy ansferred to short-range elecons Energy absorbed: energy ansferred to short-range elecons minus the bremssahlung photons A photon interaction will, in general, result in ansfer of energy to a short range particle (mostly elecons). The energy ansferred to the elecon may be absorbed within the material, dx, or it may leave the region of interest. Each of the energy ansfer mechanisms has an energy ansfer attenuation coefficient and an energy absorption attenuation coefficient. These coefficients each depend differently on the incoming photon energy and the Z of the absorber. 2

13 Energy ansferred locally i.e., to elecons. τ σ δ = τ + hν photoelecic effect: E avg σ hν δ is the average energy emitted as fluorescence = Compton scattering: E avg /hν is the average photon energy converted into elecon energy κ hν m0c = κ hν 2 pair production: (hν-mc 2 )/hν is the fraction of energy converted to photons by β + β - annihilation. µ = τ + σ + κ µ is the total kinetic energy of all elecons produced by photons µ takes no account of subsequent bremssahlung ( g) µ en = µ g is the average fraction of initial photon energy, hν, ansferred to elecons and subsequently emitted as bremssahlung. g is largest for high-z absorbers 0 Coefficient (cm - ) σ σ t τ σ s µ µ en κ σ σ s Energy (MeV) Linear attenuation and energy-absorption coefficients as functions of energy for photons in water. ** µ en is the most important parameter for calculation of dose** 3 Figure by MIT OCW.

14 I = E ab I 0 = e µ x E ( g) Image removed due to copyright resictions. µ en = µ ( g) Mass Attenuation, Mass Energy-Transfer, Mass Energy-Absorption Coefficients (cm 2 g - ) for photons in Water and Lead 4 Image removed due to copyright resictions. Figs. 8.2, 8.3 in [Turner].