Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections
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Atomic form factors, incoherent scattering functions, and photon scattering cross sections Cite as: Journal of Physical and Chemical Reference Data 4, 471 (1975); https://doi.org/10.1063/1.555523 Published Online: 15 October 2009 J. H. Hubbell, Wm. J. Veigele, E. A. Briggs, R. T. Brown, D. T. Cromer, and R. J. Howerton ARTICLES YOU MAY BE INTERESTED IN Erratum: Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections Journal of Physical and Chemical Reference Data 6, 615 (1977); https:// doi.org/10.1063/1.555554 Relativistic atomic form factors and photon coherent scattering cross sections Journal of Physical and Chemical Reference Data 8, 69 (1979); https:// doi.org/10.1063/1.555593 Theoretical Form Factor, Attenuation, and Scattering Tabulation for Z=1–92 from E=1–10 eV to E=0.4–1.0 MeV Journal of Physical and Chemical Reference Data 24, 71 (1995); https:// doi.org/10.1063/1.555974 Journal of Physical and Chemical Reference Data 4, 471 (1975); https://doi.org/10.1063/1.555523 4, 471 © 1975 American Institute of Physics for the National Institute of Standards and Technology. Atomic Form Factors, Incoherent Scattering Functions, and Photon Scattering Cross Sections J. H. Hubbell * Institute for Basic Standards, National Bureau of Standards, Washington, D.C. 20234 Wm. J. Veigele t and E. A. Briggs t Kaman Sciences Corp., Colorado Springs, Colorado 80907 R. T. Brown * and D. T. Cromer * University of California, Los Alamos Scientific Lab., Los Alamos, New Mexico 87544 and R. J. Howenon:j: University of California, Lawrence Livermore Lab., Livermore, California 94551 Tabulations are presented of the atomic form factor, F(x,Z), and the incoherent scattering function, S(x,Z), for values of x (=sin (IJ/2)/X,) from 0.005 A-I to 109 A-1, for all elements Z=l to 100. These tables are constructed from available state-of-the-art theoretical data, including the Pirenne formulas for Z = 1, configuration-interaction results by Brown using Brown-Fontana and Weiss correlated wavefunctions for Z=2 to 6, non-relativistic Hartree-Fock results by Cromer for Z=7 to 100, and a relativistic K-shell analytic expression for F(x,Z) by Bethe and Levinger for x> 10 A-I for all elements Z =2 to 100. These tabulated values are graphically compared with available photon scattering angular distribution measurements. Tables of coherent (Rayleigh) and incoherent (Compton) total scattering cross sections, obtained by numerical integration over combinations of F2(X.z) with the Thomson for mula and S (x,Z) with the Klein-Nishina formula, respectively, are presented for all elements Z= I to 100, for photon energies 100 eV (1..=124 A)lto 100 MeV (0.000124 A). The incoherent scattering cross sections also include the radiative and double-Compton corrections as given by Mork. Similar tables are presented for the special cases of terminally-bonded hydrogen and for the H2 molecule, interpolated and extrapolated from values calculated by Stewart et al. and by Bentley and Stewart using Kolos Roothaan wavefunctions. Key words: Atomic form factor; Compton scattering; cross sections; gamma rays; incoherent scatter ing function; photons; Rayleigh scattering; tabulations; x-rays. Contents Page 1. Introduction and Notation .... ........... ........... 472 2.4. Hartree-Fock Independent-Particle 1.1. Introduction... ... ................................ 472 Model .......................................... 478 1.2. Physical Constants; Units; Notation; 2.5. Cuufi.!!;ul'aLiuu-IuLt::racLiou Cakulatiums Basic Photon Scattering Formulas ....... 472 for Z = 2 to 6 ................................. 479 1.3. Generalized Expressions for the Atomic 2.6. Bethe-Levinger Relativistic K-Shell Form Factor F (x.Z) and the In- Formula for F (x.Z) ........................ 479 coherent Scattering Function S (x,Z) ... 474 3. Composition of the Present F (x,Z) and 2. Theoretical Models and Approximations for S (x,Z) Tables .............................. .. 480 F (x,Z) and S (x,Z) ......... ...... ........ ........ 475 4. Comparison of Theoretical F (x,Z) and 2.1. Atomic, Terminally-Honded, and Mo- S(x,Z) Values with Available Ex- lecular Hydrogen... .......................... 475 periments ....... ~ .............. ; .............. 481 2.2. Pauling and Sherman Model.................. 477, 5. Total Cross Sections for Coherent (Rayleigh) 2.3. Thomas-Fermi Statistical Model............. 477 and Incoherent (Compton) Scattering of Photons by Atomic Electrons ................... 490 5.1. Integration Procedures ....................... 490 * Work supported by the NBS Office of Standard Reference Data, Some support was also received from the Defense Nuclear Agency, 5.2. Radiative and Double-Compton Correc t Now at Resource Science. Inc .• Colorado Springs, Colorado 80903. tions to the Incoherent Scattering :t Work performed u~der the auspices of the USERDA. Cross Section ................................ 491 6. Discussion ............................................. 492 Copyright © 1975 by the U.S. Secretary of Commerce on behalf of the United States. This copyright will be assigned to the American Institute of Physics and the American Chemical 7. References ............................................. 492 Society, to whom all requests regarding reproduction should be addressed. 8. Measurement References (Whole-Atoms) ...... 493 471 J. Phys. Chern. Ref. Data, Vol. 4",No. 3, 1975 472 HUBBELL ET At. 1. Introduction and Notation Although the form-factor and scattering-function approach is an approximation valid primarily for small 1.1. Introduction angles and neglects electron pre-collision motion A number of systematic calculations and tabulations effects, the more rigorous theoretical treatments (see, of atomic form factor (see, e.g., [1]-[14)1 and incoherent e.g., refs. [30]-[38]) are not sufficiently tra~table for scattering function (see, e.g., [9], [12], and [14]-[19]) extensive systematic calculation and tabulation. Within values have ~een presented in the crystallographic and these limitations, the tables presented in this report chemical physics literature. These tabulations, in some· fill what the authors consider a present need for an cases presented in the form of parameters for simple extended-range, state-of-the-art set of evaluated the analytical formulas (see, e.g., [10], [13], and [19]) to oretical scattering cross section differential and in facilitate their use in machine computations, are gen tegral data for use in ca1culating radiation attenuation, erally limited to momentum transfer arguments 2 transport and energy deposition in medical physics, 0/"-) sin (4)/2) (or (1/"-) sin 4>, where 4>= 8/2 = the reactor shielding, industrial radiography, weapons angle of incidence and reflection from a crystal lattice effects, and in a variety of other applications in addition plane) below - 2.0 A-I. A few higher-x exceptions to x-ray crystallography. include tabulations by Hanson et al. [7] extending. to We caution, however, that in some regions of the 6.0 A-1, by Cromer and Mann [17], Cromer [18] and tables there are large uncertainties, particularly in Tavard et al. [9] extending to 8.0 A-I, and results by F(x, Z) for x > 10 A-I where use of the Bethe-Levinger Cromer [20] published by Veigele et al. [12] 'extending to formula forces a smooth Z-dependence from which the 80.0 A-I. available measurements suggest departures of as much Integral coherent and incoherent scattering cross sec as a factor of two or more for some elements. For tions for photon 3 energies up to 24.8, 99.2, and 992.0 x> 100 A-I, where no measurements are available ke V can be calculated using the above atomic form for comparison, the combination of the above and other factor and incoherent scattering function sets extend uncertainties may result in departures of F (x, Z) ing up to momentum transfer arguments 2, 8, and 80 values (in table I) from physical reality by one or more A-1, respectively, without extrapolation. For calculating orders of magnitude. these scattering cross section~ for higher photon ener We caution also that for energies' above a few MeV gies, various ad hoc extrapolation procedures have been and extending up into the asymptotic region, additional employed (see, e.g., refs. [21], [22], [23], and [24]). scattering processes such as Delbriick, nuclear reson The incoherent scattering integrated cross section ance and proton Compton, not considered in this work, is relatively insensitive to variations in the high-x may become considerably more important than those extrapolated incoherent scattering function values, derived from the F(x, Z) and S(x, Z) values described all of which approach the fl-ee-electron limiting ca5e ann tahnlatp.nin thi~ work_ as x increases. The coherent scattering cross section, The specific sources of theoretical data (previous on the other hand, increases its sensitivity to the atomic calculations, formulas, normalizations), used in construct form factor with increasing x. Although the asymptotic ing the present F(x,Z) and S(x,Z) tabulations (table I), high-x dependence4 has recently been shown by Goscin· and the x- and Z -region in which each is applied, are 4 listed in section 3 of this work. ski and Lindner [25] and by Smith [26] to be F(x,Z) -x- , the differing extrapolation procedures [21]-[24] have 1.2. Physical Constants 5: Units: Notation: led to substantially differing integrated cross section Basic Photon Scattering Formulas values. Although the integrated coherent scattering cross c velocity oflight= 2.99792458' 108 m s -1, section is a small fraction of the total attenuation coeffi e elementary charge = 1.6021892 . 10-19 cient in the extrapolation region, it can still result in Coulomb total attenuation coefficient variations of the order of 1 %, or in one extreme case [21] 5-10%. For the present =4.803242 ·lO-lOcm3/2g1/2s-1(e.s.u.) tabulations the atomic form factor was extrapolated to = 1.5189186 . 10-14 m3/2 kgl/2 s-1, high-x by means of a relativistic theoretical expression given by Bethe and Levinger [29]. electron rest-mass 9.109534· 10-31 kg, 1 Figures in brackets indicate literature references at the end of this paper.