Francesc Salvat

elsepa—Dirac partial-wave calculation of elastic scattering of electrons and positrons by atoms, positive ions and molecules ☆

The fortran 77 code system elsepa for the calculation of elastic scattering of electrons and positrons by atoms, positive ions and molecules is presented. These codes perform relativistic (Dirac) partial-wave calculations for scattering by a local central interaction potential V(r)V(r). For atoms and ions, the static-field approximation is adopted, with the potential set equal to the electrostatic interaction energy between the projectile and the target, plus an approximate local exchange interaction when the projectile is an electron. For projectiles with kinetic energies up to 10 keV, the potential may optionally include a semiempirical correlation–polarization potential to describe the effect of the target charge polarizability. Also, for projectiles with energies less than 1 MeV, an imaginary absorptive potential can be introduced to account for the depletion of the projectile wave function caused by open inelastic channels. Molecular cross sections are calculated by means of a single-scattering independent-atom approximation in which the electron density of a bound atom is approximated by that of the free neutral atom. Elastic scattering by individual atoms in solids is described by means of a muffin-tin model potential. Partial-wave calculations are feasible on modest personal computers for energies up to about 5 MeV. The elsepa code also implements approximate factorization methods that allow the fast calculation of elastic cross sections for much higher energies. The interaction model adopted in the calculations is defined by the user by combining the different options offered by the code. The nuclear charge distribution can be selected among four analytical models (point nucleus, uniformly charged sphere, Fermis distribution and Helms uniform–uniform distribution). The atomic electron density is handled in numerical form. The distribution package includes data files with electronic densities of neutral atoms of the elements hydrogen to lawrencium (Z=1Z=1–103) obtained from multiconfiguration Dirac–Fock self-consistent calculations. For comparison purposes, three simple analytical approximations to the electron density of neutral atoms (corresponding to the Thomas–Fermi, the Thomas–Fermi–Dirac and the Dirac–Hartree–Fock–Slater models) are also included. For calculations of elastic scattering by ions, the electron density should be provided by the user. The exchange potential for electron scattering can be selected among three different analytical approximations (Thomas–Fermi, Furness–McCarthy, Riley–Truhlar). The offered options for the correlation–polarization potential are based on the empirical Buckingham potential. The imaginary absorption potential is calculated from the local-density approximation proposed by Salvat [Phys. Rev. A 68 (2003) 012708]. Program summary Title of program:ELSEPA Catalogue identifier: ADUS Program summary URL:http://cpc.cs.qub.ac.uk/cpc/summaries/ADUS Program obtainable from: CPC Program Library, Queens University of Belfast, N. Ireland License provisions: none Computer for which the program is designed and others in which it is operable: Any computer with a FORTRAN 77 compiler Operating systems under which the program has been tested: Windows XP, Windows 2000, Debian GNU/Linux 3.0r0 (sarge) Compilers: 1Compaq Visual Fortran v6.5 (Windows); GNU FORTRAN, g77 (Windows and Linux) Programming language used: FORTRAN 77 No. of bits in a word: 32 Memory required to execute with typical data: 0.6 Mb No. of lines in distributed program, including test data, etc.:135 489 No. of bytes in distributed program, including test data, etc.: 1 280 006 Distribution format: tar.gz Keywords: Dirac partial-wave analysis, electron elastic scattering, positron elastic scattering, differential cross sections, momentum transfer cross sections, transport cross sections, scattering amplitudes, spin polarization, scattering by complex potentials, high-energy atomic screening functions Nature of the physical problem: The code calculates differential cross sections, total cross sections and transport cross sections for single elastic scattering of electrons and positrons by neutral atoms, positive ions and randomly oriented molecules. For projectiles with kinetic energies less than about 5 MeV, the programs can also compute scattering amplitudes and spin polarization functions. Method of solution: The effective interaction between the projectile and a target atom is represented by a local central potential that can optionally include an imaginary (absorptive) part to account approximately for the coupling with inelastic channels. For projectiles with kinetic energy less that about 5 MeV, the code performs a conventional relativistic Dirac partial-wave analysis. For higher kinetic energies, where the convergence of the partial-wave series is too slow, approximate factorization methods are used. Restrictions on the complexity of the program: The calculations are based on the static-field approximation. The optional correlation–polarization and inelastic absorption corrections are obtained from approximate, semiempirical models. Calculations for molecules are based on a single-scattering independent-atom approximation. To ensure accuracy of the results for scattering by ions, the electron density of the ion must be supplied by the user. Typical running time: on a 2.8 GHz Pentium 4, 1 the calculation of elastic scattering by atoms and ions takes between a few seconds and about two minutes, depending on the atomic number of the target, the adopted potential model and the kinetic energy of the projectile. Unusual features of the program: The program calculates elastic cross sections for electrons and positrons with kinetic energies in a wide range, from a few tens of eV up to about 1 GeV. Calculations can be performed for neutral atoms of all elements, from hydrogen to lawrencium (Z=1Z=1–103), ions and simple molecules.

Comparison of electron elastic-scattering cross sections calculated from two commonly used atomic potentials

We have analyzed differential cross sections (DCSs) for the elastic scattering of electrons by neutral atoms that have been derived from two commonly used atomic potentials: the Thomas–Fermi–Dirac (TFD) potential and the Dirac–Hartree–Fock (DHF) potential. DCSs from the latter potential are believed to be more accurate. We compared DCSs for six atoms (H, Al, Ni, Ag, Au, and Cm) at four energies (100, 500, 1000, and 10 000 eV) from two databases issued by the National Institute of Standards and Technology in which DCSs had been obtained from the TFD and DHF potentials. While the DCSs from the two potentials had similar shapes and magnitudes, there can be pronounced deviations (up to 70%) for small scattering angles for Al, Ag, Au, and Cm. In addition, there were differences of up to 400% at scattering angles for which there were deep minima in the DCSs; at other angles, the differences were typically less than 20%. The DCS differences decreased with increasing electron energy. DCSs calculated from the two ...

Elastic scattering of electrons and positrons by atoms. Schrödinger and Dirac partial wave analysis

Abstract Two FORTRAN 77 codes are described which provide a complete description of elastic scattering of electrons and positrons by atoms using the static field approximation with non-relativistic (Schrodinger) and relativistic (Dirac) partial wave analysis. The delivered information includes phase shifts, differential cross-sections, scattering amplitudes and percentage points of the single scattering angular distribution. The scattering field may be internally generated by the codes (which incorporate an accurate analytical approximation to the Dirac-Hartree-Fock-Slater field of free atoms) or read from the input file. Solid state effects for scattering in solids are described by means of a simple muffin-tin model. For electron scattering, exchange corrections are also taken into account. Phase shifts are obtained by using the RADWEQ subroutine package [Comput. Phys. Commun. 62 (1991) 65] to solve the radial equations. The relativistic code provides reliable cross-section data for kinetic energies between ≈ 1 keV and ≈ 1 MeV.

On the theory and simulation of multiple elastic scattering of electrons

Abstract Multiple elastic scattering of electrons in matter is analyzed on the basis of accurate single scattering differential cross sections obtained from partial wave calculations. We give a brief derivation of Molieres multiple scattering theory that clarifies its physical content and points out its limitations. In particular, it is shown that transport mean free paths calculated from the Moliere single scattering cross section differ significantly from the values obtained from partial wave calculations. We present a mixed simulation algorithm that overcomes most of the limitations of the currently available condensed Monte Carlo codes. This algorithm takes advantage of the fact that most of the collisions experienced by a high-energy electron along a given path length are soft, i.e. the scattering angle is less than a selected small value χs. The global effect of these soft collisions is described by using a multiple scattering approximation. Hard collisions, with scattering angle larger than χs, occur in a moderately small number and are described as in detailed simulations. This mixed algorithm can be applied to any single scattering differential cross section, it leads to the correct spatial distributions and it completely avoids problems related to boundary crossing. Moreover, when the single scattering law underlying Molieres theory is adopted, the algorithm can be formulated in a completely analytical way.

Inelastic scattering of electrons in solids from a generalized oscillator strength model using optical and photoelectric data

Inelastic scattering of electrons in solids is computed from a generalized oscillator strength model based on optical and photoelectric data. The optical oscillator strength is extended into the non-zero momentum transfer region by using free-electron gas dispersion for the weakly bound electrons. The applicability of this method to non-conduction valence electrons and to inner shells is discussed. A different extension method, which reproduces ionization thresholds, is used for inner-shell ionization. The calculations are simplified by using a two-modes model for the Lindhard theory of the free-electron gas. Exchange effects are accounted for by means of a modified Ochkur approximation. Inelastic mean free paths and stopping powers obtained from this optical-data model for four materials (Al, Si, Cu and Au) and for electrons with energies from 10 eV to 10 keV are presented.

Overview of physical interaction models for photon and electron transport used in Monte Carlo codes

The physical principles and approximations employed in Monte Carlo simulations of coupled electron–photon transport are reviewed. After a brief analysis of the assumptions underlying the trajectory picture used to generate random particle histories, we concentrate on the physics of the various interaction processes of photons and electrons. For each of these processes we describe the theoretical models and approximations that lead to the differential cross sections employed in general-purpose Monte Carlo codes. References to relevant publications and data resources are also provided.

Cross Sections for Inner-Shell Ionization by Electron Impact

An analysis is presented of measured and calculated cross sections for inner-shell ionization by electron impact. We describe the essentials of classical and semiclassical models and of quantum approximations for computing ionization cross sections. The emphasis is on the recent formulation of the distorted-wave Born approximation by Bote and Salvat [Phys. Rev. A 77, 042701 (2008)] that has been used to generate an extensive database of cross sections for the ionization of the K shell and the L and M subshells of all elements from hydrogen to einsteinium (Z = 1 to Z = 99) by electrons and positrons with kinetic energies up to 1 GeV. We describe a systematic method for evaluating cross sections for emission of x rays and Auger electrons based on atomic transition probabilities from the Evaluated Atomic Data Library of Perkins et al. [Lawrence Livermore National Laboratory, UCRL-ID-50400, 1991]. We made an extensive comparison of measured K-shell, L-subshell, and M-subshell ionization cross sections and of ...

Semiempirical cross sections for the simulation of the energy loss of electrons and positrons in matter

Abstract Differential cross sections (DCS) suitable for Monte Carlo simulation of the energy loss of fast electrons and positrons in matter are proposed. These DCSs are based on high-energy approximations, complemented with semiempirical ingredients which make them accurate for a wide energy range. The DCS for collision losses, including Fermis density effect correction, is obtained from the Born approximation and a simple generalized oscillator strength model. Collision stopping powers, and also mean free paths and energy straggling parameters, can be evaluated in a purely analytical way. The DCS for bremsstrahlung emission is given in the form of the Bethe-Heitler DCS with exponential screening including a Coulomb correction. The described DCSs have moderately simple analytical expressions which allow random sampling of the energy loss in detailed Monte Carlo simulations by using purely analytical methods. Suitable sampling algorithms are also described.

Analytical cross sections for Monte Carlo simulation of photon transport

Abstract Simple analytical approximations to the photon-interaction cross sections in the energy range from 1 keV up to ∼ 1 GeV are proposed. The analytical formulae are obtained as combinations of familiar approximations and numerical fits. Differential cross sections for coherent and incoherent scattering are given in terms of new analytical approximations to the atomic form factor and the incoherent scattering function. Pair production is described by means of the Bethe-Heitler differential cross section for exponential screening, together with both a Coulomb correction and an empirical low-energy correction. The calculation of partial attenuation coefficients for these interactions requires only a single numerical quadrature. Simple analytical algorithms for random sampling from the proposed differential cross sections are described. A new parametrization of the photoelectric cross section is also given. The resulting total attenuation coefficients agree with recent compilations to within ∼0.5%. The parametric tables included in this paper contain all the information required for Monte Carlo simulation, as well as for the evaluation of attenuation coefficients, for elements with Z = 1 to 92 and photon energies down to either the L1 edge of 1 keV, whichever is the largest.

Theory of conversion electron Mössbauer spectroscopy (CEMS)

Abstract A theory of conversion electron Mossbauer spectroscopy (CEMS) including second order effects, i.e. secondary electron emission, and detection coincidence corrections has been derived. The theory is applicable to surface films containing any number of distinct layers. The partial spectra are given in terms of analytical functions of the Mossbauer parameters and the physical characteristics of each layer in the sample. Numerical results are compared to available experimental data.