David A. Liberman

Anomalous scattering factors Δf′ and Δf″ have been calculated relativistically for Cr, Fe, Cu, Mo, and Ag Kα radiations for the atoms Li through Cf. An interpolation scheme for other wavelengths is included in a separate report. Relativistic calculations of the photoelectric cross section have been made and the integral for the principal contribution to Δf′ has been evaluated numerically without approximation to the form of the cross section-vs-energy curve, as has been done in previous calculations. Many of the results are significantly different from previous calculations. Where experimental values exist, agreement for Δf″ is improved. For the rare gases, except for xenon, agreement between Δf′ and experiment is improved. Because of the more rigorous evaluation of Δf′ from cross-section information, it is presumed that the present Δf′ values are more accurate than previous calculated values. Calculated mass absorption coefficients for the elements are included as incidental information.

Self-consistent-field solutions have been obtained of the two first-order linear differential equations that result from the Dirac equation. These solutions for the major and minor components of the ($\mathrm{nlj}$) eigenfunctions were obtained by the relativistic equivalent of the method Herman and Skillman used for atomic structure calculations in which exchange was handled by Slater's ${\ensuremath{\rho}}^{\frac{1}{3}}$ method. Detailed comparisons between the energy eigenvalues and electronic energy levels determined from x-ray spectroscopic data have been made for ${\mathrm{Cu}}^{+1}$, Fe, W, and Pt, and particularly for Hg and U. The agreement was very good---better than that obtained in any previous self-consistent-field calculations available to the authors.

The Fortran program eccpa calculates differential and integrated cross sections for elastic collisions of charged particles with atoms by using the classical-trajectory method and several quantum methods and approximations. The collisions are described within the framework of the static-field approximation, with the interaction between the projectile and the target atom represented by the Coulomb potential of the atomic nucleus screened by the atomic electrons. To allow the use of fast and robust calculation methods, the interaction is assumed to be the same in the center-of-mass frame and in the laboratory frame. Although this assumption neglects the effect of relativity on the interaction, it allows using strict relativistic kinematics. The equation of the relative motion in the center-of-mass frame is shown to have the same form as in the non-relativistic theory, with a relativistic reduced mass and an effective potential. The wave equation for the relative motion, as obtained from the correspondence principle, is formally identical to the non-relativistic Schrödinger equation with the reduced mass and the effective potential, and it reduces to the familiar Klein-Gordon equation when the mass of the target atom is much larger than that of the projectile. Collisions of spin 1/2 projectiles are also described by solving the Dirac wave equation. Various approximate solution methods are described and applied to a generic potential represented as a sum of Yukawa terms, which allows a good part of the calculations to be performed analytically. The program eccpa is useful for assessing the validity and the relative accuracy of the various approximations, and as a pedagogical tool.Program Title: eccpaCPC Library link to program files: https://doi.org/10.17632/c3tn9hyfvb.1Licensing provisions: CC by NC 3.0Programming language: Fortran 90/95Nature of problem: The program computes differential cross sections (DCSs) for elastic collisions of charged particles (electrons, positrons, muons, antimuons, protons, antiprotons, and alphas) with neutral atoms. Calculations are performed within the static-field approximation with screened Coulomb potentials expressed as a sum of Yukawa terms with their parameters fitted to approximate the atomic electrostatic potentials resulting from the Thomas–Fermi model and from self-consistent Dirac–Hartree–Fock–Slater calculations. The program eccpa provides DCSs computed with four different approaches: the classical trajectory method, the Born approximation, the partial-wave expansion method with approximate phase shifts, and the eikonal approximation. The user is allowed to select the atomic number of the target atom, the potential model, the kind of projectile and its kinetic energy. Calculation results are written in a number of output files with formats suited for visualization with a plotting program. A Java graphical user interface allows running the program and visualizing the results interactively.Solution method: A relativistic extension of the classical trajectory method is formulated on the assumption that the interaction in the center-of-mass frame is central, which is a fundamental requirement of the adopted calculation schemes; the DCS in the laboratory frame is then obtained from the relativistic (Lorentz) transform of the DCS calculated in the center-of-mass frame. This scheme qualifies as semi-relativistic, because it accounts for relativistic kinematics in a rigorous way, but disregards the differences between the interactions observed from the laboratory and the center-of-mass frames. We consider the elementary quantum formulation based on the relativistic Schrödinger (or Klein–Gordon) wave equation obtained from the correspondence principle. Accurate DCSs for potential scattering can be computed by using the partial-wave expansion method, at the expense of considerable numerical work. To avoid the difficult calculation of phase shifts from the numerical solution of the radial wave equation, we adopt a simplified strategy that combines the (first) Born approximation, for both the scattering amplitude and the phase shifts, and the Wentzel–Kramers–Brillouin (WKB) approximation for the phase shifts. We also describe the semi-classical eikonal approximation, which is known to yield reliable DCSs for collisions with small scattering angles. The case of collisions of electrons and positrons is considered on similar grounds, with the scattering amplitudes obtained from the Dirac equation.The numerical work is simplified by approximating the interaction potential as a sum of Yukawa terms, which allows performing a good part of the calculations analytically. Integrals of functions given by analytical formulas are calculated by means of an adaptive algorithm that combines the 20-point Gauss-Legendre quadrature formula with a bisection scheme; this algorithm allows strict control of numerical errors and gives results with a relative accuracy better than about 10−10 for well-behaved integrands. The whole calculation for a given energy of the projectile takes no longer than a few seconds on a modern personal computer, quite irrespectively of the energy and of the atomic number of the target atom.Additional comments including restrictions and unusual features: The adopted interaction potentials correspond to atoms with point nuclei. The use of a parameterization instead of numerical tables of the potential (obtained, e.g., from atomic structure calculations) has a minor effect on the calculated DCSs. This effect is limited to large scattering angles, where the actual DCS does differ from calculations with screened Coulomb potentials due to the effect of the finite size and structure of the atomic nucleus, which is disregarded here.DCSs obtained from the partial-wave expansion method and with the eikonal approximation provide a fairly accurate description of collisions with small and moderate deflection angles. They can be used, e.g., in Monte Carlo simulations of the transport of fast charged particles in matter. The information generated by the program allows assessing the accuracy of calculations with the various approaches, and permits identifying the ranges of validity of the classical trajectory method and the Born approximation.

A model for condensed matter is described in which the ions surrounding a particular atom are replaced by a positive charge distribution which is constant outside of a sphere containing the atom and zero inside. The orbital functions, both bound and free, are obtained as solutions of the Dirac equation and are used to self-consistently determine the potential function. In order to obtain the desired equation-of-state data from the calculations, three different and somewhat arbitrary prescriptions are used to separate quantities pertaining to the atom from those of the electron gas in which it is imbedded. Results are shown for 14 elements, including the $5d$ transition metals, in the neighborhood of normal solid density. Equation-of-state data for nickel, copper, and zinc are also given and are compared with experiment.

Recent experimental measurements have revealed systematic differences between measured values of f' and those calculated by the method of Cromer & Liberman [J. Chem. Phys. (1970), 53, 1891-1898] when the incident X-ray is near to and on the long-wavelength side of an edge. The source of the discrepancy has been identified and some previously published values of f' are corrected. These corrections also apply to Table 2.3.1 in International Tables for X-ray Crystallography, Vol. IV [(1974), Birmingham: Kynoch Press].