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Calculation of Photon Attenuation Coefficients of Elements And 732 ISSN 0214-087X Calculation of photon attenuation coeffrcients of elements and compound Roteta, M.1 Baró, 2 Fernández-Varea, J.M.3 Salvat, F.3 1 CIEMAT. Avenida Complutense 22. 28040 Madrid, Spain. 2 Servéis Científico-Técnics, Universitat de Barcelona. Martí i Franqués s/n. 08028 Barcelona, Spain. 3 Facultat de Física (ECM), Universitat de Barcelona. Diagonal 647. 08028 Barcelona, Spain. CENTRO DE INVESTIGACIONES ENERGÉTICAS, MEDIOAMBIENTALES Y TECNOLÓGICAS MADRID, 1994 CLASIFICACIÓN DOE Y DESCRIPTORES: 990200 662300 COMPUTER LODES COMPUTER CALCULATIONS FORTRAN PROGRAMMING LANGUAGES CROSS SECTIONS PHOTONS Toda correspondencia en relación con este trabajo debe dirigirse al Servicio de Información y Documentación, Centro de Investigaciones Energéticas, Medioam- bientales y Tecnológicas, Ciudad Universitaria, 28040-MADRID, ESPAÑA. Las solicitudes de ejemplares deben dirigirse a este mismo Servicio. Los descriptores se han seleccionado del Thesauro del DOE para describir las materias que contiene este informe con vistas a su recuperación. La catalogación se ha hecho utilizando el documento DOE/TIC-4602 (Rev. 1) Descriptive Cataloguing On- Line, y la clasificación de acuerdo con el documento DOE/TIC.4584-R7 Subject Cate- gories and Scope publicados por el Office of Scientific and Technical Information del Departamento de Energía de los Estados Unidos. Se autoriza la reproducción de los resúmenes analíticos que aparecen en esta publicación. Este trabajo se ha recibido para su impresión en Abril de 1993 Depósito Legal n° M-14874-1994 ISBN 84-7834-235-4 ISSN 0214-087-X ÑIPO 238-94-013-4 IMPRIME CIEMAT Calculation of photon attenuation coefíicients of elements and compounds from approximate semi-analytical formulae M. Roteta1, J. Baró2, J.M. Fernández-Varea3 and F. Salvat3 1 CIEMAT. Avenida Complutense 22. 28040 Madrid, Spain. 2 Servéis Científico-Técnics. Universitat de Barcelona. Martí i Franqués s/n. 08028 Barcelona. Spain. 3 Facultat de Física (ECM), Universitat de Barcelona. Diagonal 647. 08028 Barcelona, Spain. Abstract The FORTRAN 77 code PHOTAC to compute photon attenuation coefficients of elements and compounds is described. The code is based on the semi- analytical approximate atomic cross sections proposed by Baró et al. (1994). Photoelectric cross sections are calculated directly from a simple analytical expression. Atomic cross sections for coherent and incoherent scattering and for pair production are obtained as integráis of the corresponding diiferential cross sections. These integráis are evaluated, to a pre-selected accuracy, by using a 20-point Gauss adaptive integration algorithm. Calculated attenuation coefficients agree with recently compiled databases to within ~ 1%, in the energy range from 1 keV to 1 GeV. The complete source listing of the program PHOTAC is included. 1 Introduction In this report we describe the numerical calculation of photon attenuation coefncients of elements and compounds from the set of analytical approximate cross sections proposed in a recent paper by Baró et al. (1994) (hereafter referred to as Pl). These analytical cross sections were primarily developed to facilitate the Monte Cario simulation of photon transport (Jenkins et ai, 1988; Ljungberg and Strand, 1989: Zerby, 1963). Each element is characterized by a small number of parameters. which were obtained by fitting the partial attenuation coefficients from recent compilations (Cullen et al., 1989: Cullen et al, 1990; Berger and Hubbell, 1987). for the atomic numbers Z = 1 to 92. The adopted analytical forms are flexible enough to reproduce the tabulated attenuation coefficients to within ~ 0.5%, which is less than uncertainties in the compiled data. The FORTRAN 77 code PHOTAC described here gives calculated attenuation coeffi- cients for arbitran7 substances for photon energies in the range from 1 keV to ~ 1 GeV, with an accuracy that is only attainable by interpolación in very large databases. Of course, PHOTAC also involves parametric tables for the different elements and interaction mechanisms, but the required numerical information is much less than the large amount of data contained in current photon-interaction databases. On the other hand, the analytical formulae adopted in PHOTAC, although approximate, provide more complete information than a mere table of attenuation coefficients. In particular, these formulae allow the easy calculation of the differential cross sections for the relevant interaction processes. Actually, PHOTAC is intended not only to compute attenuation coefficients, but also to present the parametric tables given in Pl in a form readily usable for other computa- tional purposes, e.g. Monte Cario simulation. To this end, these parametric tables have been included in the block-data subprogram PHOTON, which serves as a complete datábase for photon-interaction calculations in the aforesaid energy range. This block-data loads the parametric tables in sepárate common blocks, so that the cross section parameters for different elements and interactions are directly transferable to other subprograms. In section 2. the basic formulae derived in Pl are given in a form readily suited for the numerical evaluation of the attenuation coefficients. The calculation of integráis, which is the most delicate part in terms of calculation speed. is considered in section 3. In section 4, the code PHOTAC is briefly described: attenuation coefficients obtained from PHOTAC and from the XCOM program of Berger and Hubbell (1987) are compared. Details on the practica] use of PHOTAC are given in appendix A. Appendix B contains the source listing of the program. including the block-data PHOTON. 2 Basic formulae We consider the interaction of unpolarized photons of energy E~f (assumed to be given in eV) with atoms of atomic number Z. We limit our considerations to the dominant inter- action processes in the energy range from 1 keV up to ~ 1 GeV, i.e. coherent (Rayleigh) scattering, incoherent (Compton) scattering, photoelectric effect and electrón-positrón pair production. In the following. K stands for the photon energy in units of the electrón rest energy me2 — 0.511 x 106 eV, i.e. -A. a) me1 Analytical form factors and incoherent scattering functions are usually expressed in terms of the dimensionless variable = 20.6074 —, (2) 4TTTZ me where q is the'momentum transfer. 2.1 Coherent Scattering The atomic differential cross section (DCS) for coherent scattering is given by dcr l + cos2<9 co 2 [F(q,Z)f.2 (3) r[F(qZ)} 1133 The quantity re = 2.8179 x 10~~ era is the classical radius of the electrón and F(q. Z) is the atomic form factor, which is evaluated from the analytical expression 3 4 + Ü2X + 2 4 F{q,Z)={ (l + a4x + a5x )- ' (4) max {f{x, Z), FK{q, Z)} if Z > 10 and f{x, Z) < 2, where Fn(q,Z) is the contribution to the form factor of the K-shell electrons, which is calculated analytically according to the relativistic hydrogenic approximation (see Pl). The parameters Oi,....a5, which are given in the block-data PHOTON, were obtained by fitting the non-relativistic form factors given by Hubbell et al. (1975) (see Pl). The magnitude of the momentum transfer q is related to the polar scattering angle 6 through q = {mc)K [2(1 -eos 6))1'2 . (5) The total atomic cross section is obtained as 2 2 aco = -r C (l + eos 6) [F(q. Z)f d(cos 9). (6) J-\ v ' For energies larger than ~ Z/2 MeV, the total cross section becomes proportional to E~2. 3 2.2 Incoherent Scattering The atomic DCS for incoherent scattering is considered as a function of the fractional energy of the secondary photon, r = El ¡'E~r The DCS is obtained as the product of the Klein-Nishina DCS and the incoherent scattering function S(q, Z). It can be written in the form d(7in irrz i 2 ^ f + + 1 -f K T S?1 Z). 7 dr The magnitude of the momentum transfer is given by q2 = (mc)2/í [2 + K - 2r(l + n) + T2K] . (8) The incoherent scattering function is evaliiated by means of the analytical approxima- tion S(Z) Z11 j (9) with the parameters given in the block-data PHOTON, which were determined by fitting the S(x,Z) functions tabulated by Hubbell et al. (1975) (see Pl). The total incoherent scattering cross section per atom is obtained as íl áo"m i nn\ <rin = / —— dr. (10) Jrmm dr where (11) is the mínimum fractional energy of the secondary photon. For energies larger than ~ 10 MeV, binding effects are negligible (S(x. Z) ~ Z) and a-m does not differ significantly from the Klein-Nishina total cross section. which is given bv 2.3 Photoelectric effect The photoelectric cross section, <rph, is evaluated by using the analytical expressions = FKa*h(E-nZ) if £c<£~, l 2 = exp (AK - BKy + CKy~ + DKy~ ) if UK < E-, < Ec 1 2 = exp (AL - BLy + CLy- + DLy- ) if ULl < £7 < UK = exp(AL2 - BL2y) if UL2 < £-, < UL1 = exp (AL3 - BL3y) if UL3 < E-, < UL2 1 2 = exp {AM - Buy + C^y' + Duy~ ) if ^u < £>• < UL3 = exp(.4M2 - BU2y) if UM2 < E-, < UMI = exp(A\[3 - B\\3y) if UM3 < E,,. < Í7M2 r = exp(,4M4 - J5.M4j/) if LM4 < E.,. < L''M3 = exp(AM5 - BMSV) if UMÓ < En < l\\i 1 2 r = exp (AN - BNy + CN?/- + DNy~ ) if t- Nl < E, < t/M5 1 2 = 0.9800 exp (AN - 5Ny + CNJ/" + DNy- ) if [/*N2 < £~r < t/Ni J 2 r = ' 0.9665 exp (AN - 5Ny + CNy- + DNy~ ) if L- N3 < £7 < L'N2 1 2 = 0.9094 exp (AN - 5Ny + Cuy- + DNy- ) if £/N4 < E7 < Í7N3, where Ec = 5(Z + 15) keV and y = ln(^/eV), These expressions give the atomic cross 24 2 section in barns (= 10~ cm ). The subshell binding energies UK, • • •, U^¡4 included in the block-data PHOTON have been taken from the compilation by Lederer and Shirley (1978). For energies E~t > Ec.
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