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X-Ray Form Factor, Attenuation, and Scattering Tables

Theoretical Form Factor, Attenuation and Scattering Tabulation
for Z=1-92 from E=1-10 eV to E=0.4-1.0 MeV

C. T. Chantler

School of Physics, University of Melbourne
Parksville, Victoria, 3052 Australia

Tables for form factors and anomalous dispersion are widely used in the UV, x-ray and γ-ray communities, and have existed for a considerable period of time. Much of the recent theoretical basis for these was contributed by Cromer, Mann and Liberman while much of the experimental data was synthesized by Henke et al. More recent developments in both areas have led to new and revised tables. These works have employed numerous simplifications compared to detailed relativistic S-matrix calculations; the latter do not lend themselves to convenient tabular application for the range of Z and energy of general interest. Conversely, the former tables appear to have large regions of limited validity throughout the range of Z and energies, and in particular have important limitations with regard to extrapolation to energies outside tabulated ranges.

In the present study, the primary interactions of x-rays with isolated atoms from Z = 1 (hydrogen) to Z = 92 (uranium) are described and computed within a self-consistent Dirac-Hartree-Fock framework. This has general application across the range of energy from 1-10 eV to 400-1000 keV, with limitations (described below) as the low- and high-energy extremes are approached. Tabulations are provided for the f1 and f2 components of the form factors, together with the photoelectric attenuation coefficient for the atom, µ, and the value for the K-shell, µK, as functions of energy and wavelength. Also provided are estimated correction factors as described in the text, conversion factors, and a simple estimate for the sum of the scattering contributions (from an isolated atom).

The method used herein is primarily theoretical and considers intermediate assumptions which limit the precision and applicability of previous theoretical tabulations. Particular concern involves the application of the dispersion relation to derive Re(f) from photoelectric absorption cross-sections. The revised formulation presented here explicitly avoids most of the limitations of previous works. Revised formulae can lead to significant qualitative and quantitative improvement, particularly above 30-60 keV energies, near absorption edges, and at 0.03 keV to 3 keV energies. Recent experimental syntheses are often complementary to this approach. Examples are given where the revised theoretical tables are in better agreement with experiment than are those based on experimental syntheses.

Table of Contents

          J. Phys. Chem. Ref. Data 24, 71-643 (1995)
  1. Introduction and Importance of Form Factors
  2. Form Factors and Standard Definitions
  3. Available Tabulations and Basis of Current Formalism
  4. Basic Transform Equations
  5. Avoidance of Singularities in f′(E) Above Edges
  6. Avoidance of Imprecision in f′(E) Below and Near Edges
  7. Integration Precision Requirements
  8. Choice of Interpolation Formalism
  9. Results for Hydrogen and Helium
  10. How to Use These Tables
  11. Limitations of the Current Tabulation
  12. Other Current Concerns
  13. Summary and Typical Uncertainties
  14. Acknowledgments
  15. References
  16. Explanation of Tables and Tabulated Figures


See also J. Phys. Chem. Ref. Data 2000

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Last update: July 1999