Journal of the ICRU Vol 7 No 1 (2007) Report 77 Oxford University Press
doi:10.1093/jicru/ndm005
EXECUTIVE SUMMARY to about 100 eV, and are used to evaluate the limitations of the static-field approximation. The non-relativistic (Schro¨dinger) and relativistic (Dirac) theories of scattering by a central potential are described, together with calculational methods and approximations relevant to elastic scattering of electrons and positrons by neutral atoms and positive ions. Information is also provided on various atomic-charge distribution models, approximate exchange potentials, and optical-model potentials that have been used in elastic-scattering calculations. The theoretical differential cross-sections for atoms presented in the Report were calculated by using the Dirac partial-wave expansion method, which consistently accounts for spin and relativistic effects. Accurate and robust numerical methods for the solution of the radial Dirac equations and for the summation of the partial-wave series are described. These methods were implemented in the computer code system ELSEPA (Salvat et al., 2005), which calculates scattering amplitudes, differential cross sections, spin-polarization functions, and total cross sections for elastic scattering by neutral atoms from the static-field approximation and optical-model potentials. At high energies (higher than about 20Z keV, where Z is the atomic number), where the partial-wave series converge too slowly, the atomic differential cross-section is calculated as the product of the partial-wave crosssection for scattering by the bare nucleus and a pre-calculated screening function. This type of numerical calculation is feasible for energies up to about 500 MeV. For higher energies, the differential cross-section of the bare nucleus is obtained from the first Born approximation. The program ELSEPA can also calculate elastic scattering by molecules (within the independent-atom approximation) and by positive ions for electrons and positrons with kinetic energies up to about 1 MeV. Experimental techniques for measuring elasticscattering differential cross-sections and related quantities are surveyed. Available experimental cross-section data for atoms, and for water molecules, are compared with calculated theoretical
# International Commission on Radiation Units and Measurements 2007
Downloaded from http://jicru.oxfordjournals.org/ at University of California, San Francisco on September 3, 2014
Elastic scattering has a direct influence on the spatial distribution of energy deposited by electrons and positrons. Because of elastic scattering, the trajectories of these particles are tortuous and, consequently, they penetrate distances substantially shorter than their path lengths. Thus, although energy transfers in elastic collisions are negligible, elastic-scattering properties of the medium have a strong effect on the spatial distribution of dose from electrons and positrons. Knowledge of differential cross-sections for elastic scattering of these particles is needed for modeling the energy deposition from any form of ionizing radiation, because of the production of energetic electrons, and occasionally positrons, in the interactions of the primary radiation. Specific applications that require accurate information on elastic scattering extend from radiation dosimetry, radiation therapy, radiation processing, radiation sensors, and radiation protection to atmospheric studies, plasma physics, and material analysis (through electronprobe microanalysis, analytical electron microscopy, Auger-electron spectroscopy, and x-ray photoelectron spectroscopy). The present Report contains a review of theoretical methods and experimental techniques for studying elastic scattering of electrons and positrons by single atoms, positive ions, and molecules, as well as in condensed materials. The emphasis of the theoretical sections is in the scattering by neutral atoms and in the range of intermediate and high energies, from a few keV upwards. At these energies, the target atom behaves nearly as a frozen distribution of electric charge, and the process can be described as scattering of the projectile by the electrostatic field of the atom (the so-called static-field approximation). In the case of electron scattering, exchange effects can be accounted for by means of an approximate local potential. More elaborate optical-model potentials, which account for the effects of atomic charge polarizability and inelastic absorption, are also described. Calculations with semi-empirical opticalmodel potentials provide results consistent with experimental information for lower energies, down
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
6
Downloaded from http://jicru.oxfordjournals.org/ at University of California, San Francisco on September 3, 2014
database and a graphical interface are also provided. The structure of the database has been designed to meet the needs of Monte Carlo simulation of electron and positron transport. High-energy simulation schemes employ condensed-history methods in which the global effect of the collisions of an electron or positron along a given path length is described by means of a multiple-scattering theory. The Report contains a brief description of the multiple-scattering theories of Goudsmit and Saunderson, and of Lewis, with corrections to account for the scattering effect of inelastic collisions. Programs for calculating multiplescattering distributions from these two theories, using the differential cross-sections of the database, are included in the compact disc. The transport of electrons with energies lower than about 100 keV can be simulated by sampling individual elastic interactions, a process that requires only the differential cross-sections provided in the database. Simulations at energies lower than about 10 keV may require cross-sections more accurate than those obtained from the static-field approximation. The code ELSEPA can be used to generate these cross-sections from the semi-empirical opticalmodel potentials described in the Report.
cross-sections to reveal both the capabilities and the limitations of the static-field approximation and the optical-model potential. Experimental techniques and theoretical methods for elastic scattering of low-energy electrons in single crystals and disordered solids are discussed. Results from other experimental techniques, mostly surfaceelectron spectroscopies, that are sensitive to elastic scattering are also considered. These techniques provide indirect evidence, usually through Monte Carlo simulations, for the reliability of the theoretical atomic cross-sections and their applicability for describing elastic scattering in condensed matter. The Report is accompanied by a compact disc with an extensive numerical database of differential cross-sections, total cross-sections, and transport cross-sections for elastic scattering of electrons and positrons by neutral atoms. This database covers all elements from hydrogen to lawrencium (Z ¼ 1 to 103) and the energy range from 50 eV to 100 MeV. It was generated using the static-field approximation for neutral atoms with finite nuclei and self-consistent Dirac – Fock electron densities. The calculations were performed by using the code system ELSEPA which is included in the compact disc. Programs for extracting information from the
doi:10.1093/jicru/ndm005
EXECUTIVE SUMMARY to about 100 eV, and are used to evaluate the limitations of the static-field approximation. The non-relativistic (Schro¨dinger) and relativistic (Dirac) theories of scattering by a central potential are described, together with calculational methods and approximations relevant to elastic scattering of electrons and positrons by neutral atoms and positive ions. Information is also provided on various atomic-charge distribution models, approximate exchange potentials, and optical-model potentials that have been used in elastic-scattering calculations. The theoretical differential cross-sections for atoms presented in the Report were calculated by using the Dirac partial-wave expansion method, which consistently accounts for spin and relativistic effects. Accurate and robust numerical methods for the solution of the radial Dirac equations and for the summation of the partial-wave series are described. These methods were implemented in the computer code system ELSEPA (Salvat et al., 2005), which calculates scattering amplitudes, differential cross sections, spin-polarization functions, and total cross sections for elastic scattering by neutral atoms from the static-field approximation and optical-model potentials. At high energies (higher than about 20Z keV, where Z is the atomic number), where the partial-wave series converge too slowly, the atomic differential cross-section is calculated as the product of the partial-wave crosssection for scattering by the bare nucleus and a pre-calculated screening function. This type of numerical calculation is feasible for energies up to about 500 MeV. For higher energies, the differential cross-section of the bare nucleus is obtained from the first Born approximation. The program ELSEPA can also calculate elastic scattering by molecules (within the independent-atom approximation) and by positive ions for electrons and positrons with kinetic energies up to about 1 MeV. Experimental techniques for measuring elasticscattering differential cross-sections and related quantities are surveyed. Available experimental cross-section data for atoms, and for water molecules, are compared with calculated theoretical
# International Commission on Radiation Units and Measurements 2007
Downloaded from http://jicru.oxfordjournals.org/ at University of California, San Francisco on September 3, 2014
Elastic scattering has a direct influence on the spatial distribution of energy deposited by electrons and positrons. Because of elastic scattering, the trajectories of these particles are tortuous and, consequently, they penetrate distances substantially shorter than their path lengths. Thus, although energy transfers in elastic collisions are negligible, elastic-scattering properties of the medium have a strong effect on the spatial distribution of dose from electrons and positrons. Knowledge of differential cross-sections for elastic scattering of these particles is needed for modeling the energy deposition from any form of ionizing radiation, because of the production of energetic electrons, and occasionally positrons, in the interactions of the primary radiation. Specific applications that require accurate information on elastic scattering extend from radiation dosimetry, radiation therapy, radiation processing, radiation sensors, and radiation protection to atmospheric studies, plasma physics, and material analysis (through electronprobe microanalysis, analytical electron microscopy, Auger-electron spectroscopy, and x-ray photoelectron spectroscopy). The present Report contains a review of theoretical methods and experimental techniques for studying elastic scattering of electrons and positrons by single atoms, positive ions, and molecules, as well as in condensed materials. The emphasis of the theoretical sections is in the scattering by neutral atoms and in the range of intermediate and high energies, from a few keV upwards. At these energies, the target atom behaves nearly as a frozen distribution of electric charge, and the process can be described as scattering of the projectile by the electrostatic field of the atom (the so-called static-field approximation). In the case of electron scattering, exchange effects can be accounted for by means of an approximate local potential. More elaborate optical-model potentials, which account for the effects of atomic charge polarizability and inelastic absorption, are also described. Calculations with semi-empirical opticalmodel potentials provide results consistent with experimental information for lower energies, down
ELASTIC SCATTERING OF ELECTRONS AND POSITRONS
6
Downloaded from http://jicru.oxfordjournals.org/ at University of California, San Francisco on September 3, 2014
database and a graphical interface are also provided. The structure of the database has been designed to meet the needs of Monte Carlo simulation of electron and positron transport. High-energy simulation schemes employ condensed-history methods in which the global effect of the collisions of an electron or positron along a given path length is described by means of a multiple-scattering theory. The Report contains a brief description of the multiple-scattering theories of Goudsmit and Saunderson, and of Lewis, with corrections to account for the scattering effect of inelastic collisions. Programs for calculating multiplescattering distributions from these two theories, using the differential cross-sections of the database, are included in the compact disc. The transport of electrons with energies lower than about 100 keV can be simulated by sampling individual elastic interactions, a process that requires only the differential cross-sections provided in the database. Simulations at energies lower than about 10 keV may require cross-sections more accurate than those obtained from the static-field approximation. The code ELSEPA can be used to generate these cross-sections from the semi-empirical opticalmodel potentials described in the Report.
cross-sections to reveal both the capabilities and the limitations of the static-field approximation and the optical-model potential. Experimental techniques and theoretical methods for elastic scattering of low-energy electrons in single crystals and disordered solids are discussed. Results from other experimental techniques, mostly surfaceelectron spectroscopies, that are sensitive to elastic scattering are also considered. These techniques provide indirect evidence, usually through Monte Carlo simulations, for the reliability of the theoretical atomic cross-sections and their applicability for describing elastic scattering in condensed matter. The Report is accompanied by a compact disc with an extensive numerical database of differential cross-sections, total cross-sections, and transport cross-sections for elastic scattering of electrons and positrons by neutral atoms. This database covers all elements from hydrogen to lawrencium (Z ¼ 1 to 103) and the energy range from 50 eV to 100 MeV. It was generated using the static-field approximation for neutral atoms with finite nuclei and self-consistent Dirac – Fock electron densities. The calculations were performed by using the code system ELSEPA which is included in the compact disc. Programs for extracting information from the
