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Lecture Note on Photon Interactions and Cross Sections KEK Internal 2000-10 November 2000 R Lecture Note on Photon interactions and Cross Sections H. Hirayama Lecture Note on Photon Interactions and Cross Sections Hideo Hirayama KEK, High Energy Accelerator Research Organization, 1-1, Oho, Tsukuba, Ibaraki, 305-0801 Japan Abstract This is the lecture note used at the \Tutorials for Electron/Gamma Monte Carlo: building blo cks and applications" as part of the International Conference on the Monte Carlo 2000, Advanced Monte Carlo on Radiation Physics, Particle Transp ort Simulation and Applications (Monte Carlo 2000) held at Lisb on Portugal, 23-26 Octob er, 2000. 1 Intro duction This lecture contains descriptions of the individual atomic cross sections of photons for scat- tering, absorption, and pair pro duction that are necessary for photon transp ort calculations, together with related cross section data. Many parts on this lecture are based on the series of works by J. H. Hubb ell[1, 2, 3, 4, 5, 6] in this eld. 2 Classi cation of Interaction The interaction of photons with matter may b e classi ed according to: 1. The kind of target, e.g., electrons, atoms, or nuclei, with which the photons interact, and 2. The typ e of event, e.g., scattering, absorption, pair pro duction, etc:, whichtakes place. Possible interactions are summarized in Table 1.[7 ] As an example, the cross sections of a photon with Cu is shown in Fig. 1[8 ]. It is apparent from Fig. 1 that for the attenuation of photons the most imp ortantinteractions are: 1. The photo electric e ect, , pe 2. Compton scattering, , and C 3. Electron-p ositron pair pro duction, ( + ). pair tr ip Rayleigh scattering, , is usually of minor imp ortance for the broad b eam conditions typi- R cally found in shielding, but must b e known for the interaction of attenuation co ecient data. The photonuclear e ect, , is mostly restricted to the region of the giant resonance around ph:n 10 to 30 MeV where, at the resonance p eak, it may amount to as much as 10 p ercent of the total \electronic" cross section. Elastic nuclear scattering, inelastic nuclear scattering and Delbruc k scattering are negligible pro cesses in photon interactions. Elastic nuclear scattering is regard as a nuclear analog to very low energy Compton scattering by an electron. In this pro cess, a photon interacts with a nucleon in such a manner that is a photon re-emitted with the same energy. During inelastic nuclear scattering, the nucleus is raised to an excited level by absorbing a photon. The excited nucleus subsequently de-excites by emitting a photon of equal or lower energy. The phenomenon of photon scattering by the Coulomb eldofanucleus is called Delbruc k scattering (also called nuclear p otential scattering). It can b e thought of as virtual pair pro duc- tion in the eld of the nucleus { that is, pair pro duction followed by annihilation of the created pair. 1 4 10 Cu ) Total 2 2 cm 10 -24 Pair production (nucleus) (10 0 10 Compton Rayleigh scattering (γ,n) Pair production (barn/atom) (electron) σ -2 10 Photoelectric Effect Pion Quasi-deutron production disintegration 10-4 0.01 0.1 1 10 100 1000 104 105 Photon Energy (MeV) Figure 1: Cross section of Cu for photons b etween 10 keV and 100 GeV.[8 ]. 2 Table 1. Classi cation of elementary photon interactions. Typ e of Scattering interaction Absorption Elastic Inelastic Interaction (Coherent) (Incoherent) with: Atomic Photo electric Rayleigh Compton electrons e ect scattering scattering ( 2 4 Z Z (L:E :) R Z pe C 5 (L:E :) Z (H:E :) Nucleus Photonuclear Elastic Inelastic reactions nuclear nuclear ( ,n),( ,p), scattering scattering 2 0 photo ssion, etc. ( ; ) Z ( ; ) Z ph:n: (h 10MeV) Electric eld Electron-p ositron Delbruc k surrounding pair pro duction in scattering charged eld of nucleus, 2 particles Z pair (h 1:02MeV) Electron-p ositron pair pro duction in electron eld, 2 Z tr ip (h 2:04MeV) Nucleon-antinucleon pair pro duction (h 3 GeV) Mesons Photomeson Mo di ed pro duction, ( ; ) (h 150 MeV) 3 Photo electric E ect Studies related to the photo electric e ect are reviewed historically by Hubb ell in NSRDS-NBS 29[1 ]. In the atomic photo e ect, a photon disapp ears and an electron is ejected from an atom. The electron carries away all of the energy of the absorb ed photon, minus the energy binding the electron to the atom. The K -shell electrons are the most tightly b ound, and are the most imp ortant contributions to the atomic photo e ect cross-section in most cases. However, if the photon energy drops b elow the binding energy of a given shell, an electron from that shell cannot b e ejected. Hence, particularly for medium- and high-Z elements, a plot of versus the photon pe energy exhibits the characteristic sawto oth absorption edges, since the binding energy of each electron subshell is attained and this pro cess is p ermitted to o ccur. These feature can b e seen well in Fig. 2, which shows the total photo electric cross section, , for Ag together with pe subshell ones[9 ]. 3 6 10 L -edge Ag 3 L -edge 2 L -edge 1 Total 105 K (barn/atom) L1 L2 4 K-edge L3 10 M section 103 cross 102 1 Photoelectric 10 1 10 100 Photon Energy (keV) Figure 2: Total and partial atomic photo e ect of Ag.[9 ]. The order of magnitude of the photo electric atomic-absorption cross section is ( 4 3 Z =(h ) low energy (1) pe 5 Z =h high energy : Dramatic resonance structures of the order of 10% just ab ove the absorption edges are well known and can be easily observed with high-resolution sp ectrometers. Owing to their dep endence on the temp erature, chemical binding and other variable atomic environments, this extended X-ray absorption ne structure (EXAFS) is not included in the photon transp ort calculation. However, EXAFS can b e a ma jor analytical to ol in X-ray di raction sp ectrometry. 3.1 Relaxation pro cesses after the photo electric e ect Avacancy created by the ejection of an electron from an inner shells is lled by an outer electron falling into it (de-excitation); this pro cess may b e accompanied by one of the following mo des: 1. A uorescence X-ray is emitted from the atom, with a photon energy equal to the di erence between the vacancy-site inner-shell energy level and energy level of the particular outer shell which happ ens to supply the electron to ll the vacancy ( uorescence yield, ! ,isthe fraction of uorescence X-ray emission). 2. The excess energy ejects an outer-shell electron from the atom. This electron is known as an Auger electron (Auger yield, a, is the fraction of Auger emission). 3. A vacancy is lled by an electron in a higher subshell, like from the L subshell to the 2 L subshell. This pro cess is called Coster-Kronig. As a result, a new vacancy is created, 1 in which is lled by one of the mo des (Coster-Kronig yield, f , is the fraction of Coster- Kronig). 4 The sum of uorescence yield, ! , Auger yield, a, and Coster-Kronig yield, f , is unity: ! + a + f =1 (2) Exp erimental and theoretical uorescence-yield information have b een reviewed by Fink et al.[10 ], Bambynek et al.[11 ], Krause[12 ] and Hubb ell[2]. The yields of uorescence, Auger electrons and Coster-Kronig after the K-, L -, L - or L - 1 2 3 photo electric e ect[13 ] are shown in Fig. 3. 1 L1-Fluorescence 1.2 L2-L1 Coster-Kronig 0.8 L3-L1 Coster-Kronig L1 Auger 1 0.6 0.8 0.6 K-Fluorescence Yields 0.4 K-Auger Yields 0.4 0.2 0.2 0 0 0 20406080100 0 20406080100 Atomic Number, Z Atomic Number, Z 1 1 0.8 0.8 0.6 0.6 L3 Fluorescence L2 Fluorescence L3 Auger Yields 0.4 L3-L2 Coster-Kronig Yields 0.4 L2 Auger 0.2 0.2 0 0 0 20406080100 0 20406080100 Atomic Number, Z Atomic Number, Z Figure 3: Yields of uorescence, Auger electrons and Coster-Kronig after the K-, L -, L - or 1 2 L -photo electric e ect[13 ]. 3 The uorescence X-ray intensities are imp ortant for low energy photon transp ort. Sco eld published K and L uorescence X-ray intensities calculated with the relativistic Hatree-Slater theory for elements with Z=5 to 104[14 ]. His data are cited in a Table of Isotop es Eighth Edition[13], but are slightly di erent with the exp erimental results, as presented by Salem et al[15 ]. 5 4 Scattering 4.1 Compton scattering, Klein-Nishina Formula In Compton scattering, a photon collides with an electron, loses some of its energy and is de ected from its original direction of travel (Fig. 4). The basic theory of this e ect, assuming the electron to b e initially free and at rest, is that of Klein and Nishina[16]. Y hν θ hν0 O X φ E Figure 4: Compton scattering with a free electron at rest. The relation b etween photon de ection and energy loss for Compton scattering is determined from the conservation of momentum and energy b etween the photon and the recoiling electron. This relation can b e expressed as: h 0 h = ; (3) h 0 1+ (1 cos ) 2 m c e 2 2 2(h ) cos 0 2 E = h h = m c ; (4) 0 e 2 2 2 2 (h + m c ) (h ) cos 0 e 0 1 ; (5) cot tan = h 0 2 1+ 2 m c e where h is the energy of incident photon, h is the energy of scattered photon, E is the energy 0 of recoil electron, m is the rest mass of an electron, and c is the sp eed of light.
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