Atomic Scattering Cross Sections Research Articles

The use of the R-matrix method in atomic calculations

A brief account is given of the applicability of the R-matrix method to the calculation of e --atom scattering cross sections, atomic polarizabilities and van der Waals coefficients, photoionization cross sections, nonlinear, optical harmonic coefficients, Born ionization cross sections and free-free transition processes. The current state of the computer codes is discussed.

Low-energy electron diffraction intensities from the clean copper (001) surface

Experimental absolute intensity curves of the 00, 11, 20 and 22 diffraction beams from a clean copper (001) surface obtained for incident electron energies 1–600 eV are presented. A complete band gap extending to about 2 eV above the vacuum level in the electron energy band structure for copper accounts for most intense peak in the curves: the first Bragg peak around 1 eV in the 00 beam intensity. With increasing electron energy the peak intensities decrease and the peak widths increase. In the electron energy region 30–140 eV strong multiple scattering gives rise to intense secondary Bragg peaks and unsymmetrical peaks in the curves. These effects are due to large atomic scattering cross section of the copper atoms probably related to the 3p and 3s core states.

Multiple-Scattering Treatment of Low-Energy Electron-Diffraction Intensities

A theory of intensities in the diffraction of low-energy (≲100-eV) electrons by crystals is formulated on the basis of Lax's multiple-scattering equations. The formulation is self-consistent and is thus applicable regardless of the magnitude of the atomic-scattering factor. The qualitative conclusions of the theory are discussed in detail with reference to simplified model crystals. The theory predicts that the reflectivity curves (reflectivity associated with a given beam versus electron energy) should show two types of peaks in addition to the ordinary Bragg peaks: (a) a resonance peak in the specular reflectivity curve; (b) secondary Bragg peaks in all reflectivity curves. The resonance peak is predicted at an energy just below that of the first appearance of a nonspecular beam. It derives from multiple scattering in a single atom layer and is associated with a resonance (zero and phase discontinuity) in the effective field. The secondary Bragg peaks are predicted at energies given by conditions similar in form to the ordinary Bragg condition. They derive from multiple scattering between atom layers and are associated with the interference between four or more plane wave components of the effective field. Both types of peaks are prominent only when the atomic-scattering cross section is large. All qualitative results are illustrated by an exact numerical application to a model crystal of isotropic (s-wave) scatterers. The relationship with existing theories is discussed. It is shown that the conventional modified kinematical approach to low-energy electron diffraction is a limiting case that is approached when inelastic scattering is dominant. It is also shown that when certain assumptions are introduced, the most important being that of neglecting intralayer multiple scattering and the ``two-beam'' assumption, the theory reduces to the Darwin dynamical theory.

A new calculation of electron scattering cross sections and a theoretical discussion of image contrast in the electron microscope

The current basis for the theoretical discussion of the contrast in an image in the electron microscope is the 1954 Lenz theory. The basis of this theory is reviewed and the reasons are given for the two interpretations of Lenz's theory in its application to all the elements which have arisen in the literature; sample calculations based on these interpretations and theoretical reasons for favouring one of them are given. Since 1954, theoretical descriptions of the atom have improved and the Hartree-Fock wave functions have replaced the Hartree wave functions and the Thomas-Fermi-Dirac potentials have superseded the Thomas-Fermi potentials. In this article the results from the newer atomic models are incorporated into the theory of single-electron scattering as applied to electron microscopy. The new method of calculation avoids the rigid expressions used by Lenz for the potential distribution within an atom and the atomic radius by using directly the values of the atomic scattering factors for X-rays and for electrons evaluated from the latest atomic models. A feature of the new expressions for the elastic and inelastic atomic scattering cross sections is their flexibility; the results of future developments of the atomic models can be readily incorporated into the calculations. The variations of the elastic, inelastic and total mass-scattering cross sections and differential atomic scattering cross sections with the accelerating voltage of the electron beam, the angular aperture of the objective lens and the atomic number of the amorphous specimen giving rise to the scattering, are discussed in detail for both the newly developed theory and the Lenz theory in its two interpretations. Most of the results are given for accelerating voltages in the range 20-100 kV but a few results for the new theory are given for voltages of 500 kV and 1000 kV. The differences and similarities between the results of the two theories are discussed and a section is included on the agreement between theoretical and experimental parameters describing the image contrast. The theoretical values of the total mass-scattering cross sections are appreciably higher than the experimental values, but the theoretical predictions about the variations of the cross sections with the atomic number of the scattering element and the instrumental parameters are in general agreement with the experimental results. Methods are suggested by which the basis of the new calculations of the cross sections may be improved.