José M. Fernández-Varea

PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter

Abstract A mixed algorithm for Monte Carlo simulation of relativistic electron and positron transport in matter is described. Cross sections for the different interaction mechanisms are approximated by expressions that permit the generation of random tracks by using purely analytical methods. Hard elastic collisions, with scattering angle greater than a preselected cutoff value, and hard inelastic collisions and radiative events, with energy loss larger than given cutoff values, are simulated in detail. Soft interactions, with scattering angle or energy loss less than the corresponding cutoffs, are simulated by means of multiple scattering approaches. This algorithm handles lateral displacements correctly and completely avoids difficulties related with interface crossing. The simulation is shown to be stable under variations of the adopted cutoffs; these can be made quite large, thus speeding up the simulation considerably, without altering the results. The reliability of the algorithm is demonstrated through a comparison of simulation results with experimental data. Good agreement is found for electrons and positrons with kinetic energies down to a few keV.

An algorithm for Monte Carlo simulation of coupled electron-photon transport

Abstract An algorithm for Monte Carlo simulation of coupled electron-photon transport is described. Electron and positron tracks are generated by means of PENELOPE, a mixed procedure developed by Baro et al. [Nucl. Instr. and Meth. B 100 (1995) 31]. The simulation of photon transport follows the conventional, detailed method. Photons are assumed to interact via coherent and incoherent scattering, photoelectric absorption and electron-positron pair production. Photon interactions are simulated through analytical differential cross sections, derived from simple physical models and renormalized to reproduce accurate attenuation coefficients available from the literature. The combined algorithm has been implemented in a FORTRAN 77 computer code that generates electron-photon showers in arbitrary materials for the energy range from ∼1 GeV down to 1 keV or the binding energy of the L-shell of the heaviest element in the medium, whichever is the largest. The code is capable of following secondary particles that are generated within this energy range. The reliability of the algorithm and computer code is demonstrated by comparing simulation results with experimental data and with results from other Monte Carlo codes.

Experimental benchmarks of the Monte Carlo code penelope

Abstract The physical algorithms implemented in the latest release of the general-purpose Monte Carlo code penelope for the simulation of coupled electron–photon transport are briefly described. We discuss the mixed (class II) scheme used to transport intermediate- and high-energy electrons and positrons and, in particular, the approximations adopted to account for the energy dependence of the interaction cross-sections. The reliability of the simulation code, i.e. of the adopted interaction models and tracking algorithms, is analyzed by means of a comprehensive comparison of simulation results with experimental data available from the literature. The present analysis demonstrates that penelope yields a consistent description of electron transport processes in the energy range from a few keV up to about 1 GeV.

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.

Monte Carlo simulation of 0.1–100 keV electron and positron transport in solids using optical data and partial wave methods

Abstract A new Monte Carlo code for the detailed simulation of the transport of low-energy electrons and positrons in solids is presented, including a critical discussion of concepts and approximations in the scattering model. Inelastic scattering is calculated using a Bethe surface model based on optical and photoelectric data for the solid, making possible a good accuracy at low energies, and a high resolution (∼1 eV) in simulated energy loss spectra. Exchange corrections for electrons and relativistic corrections for energies up to ∼100 keV are included. Elastic scattering is calculated by means of a differential cross section obtained by relativistic partial wave analysis for an exchange corrected muffin-tin Dirac-Hartree-Slater atomic potential. In the simulation, no adjustments of parameters to empirical scattering data are made. For comparison, measurements have been made of the characteristic low energy loss spectrum of 100 keV electrons through a thin silicon film. Simulated results for electrons and positrons are also compared with other available experimental data, in particular at low (a few keV) energies. In general, very good agreement is obtained.

Accurate numerical solution of the radial Schrödinger and Dirac wave equations

Abstract A FORTRAN 77 subroutine package for the numerical solution of the Schrodinger and Dirac wave equations for central fields is presented. The considered fields are such that the function ν ( r ) ≡ rV ( r ) is finite for all r and tends to constant values for r → 0 and r → ∞. This includes finite-range fields as well as combinations of Coulomb and short-range fields. The potential energy function V ( r ) used in the calculation is the natural cubic spline that interpolates a table of values provided by the user. The radial wave equations are solved by using piecewise exact power series expansions of the radial functions, which are summed up to the prescribed accuracy so that truncation errors can be completely avoided. Normalized radial wave functions, eigenvalues for bound states and phase shifts for free states are evaluated.

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.

Limitations (and merits) of PENELOPE as a track-structure code

Abstract Purpose: To outline the limitations of PENELOPE (acronym of PENetration and Energy LOss of Positrons and Electrons) as a track-structure code, and to comment on modifications that enable its fruitful use in certain microdosimetry and nanodosimetry applications. Methods: Attention is paid to the way in which inelastic collisions of electrons are modelled and to the ensuing implications for microdosimetry analysis. Results: Inelastic mean free paths and collision stopping powers calculated with PENELOPE and two well-known optical-data models are compared. An ad hoc modification of PENELOPE is summarized where ionization and excitation of liquid water by electron impact is simulated using tables of realistic differential and total cross sections. Conclusions: PENELOPE can be employed advantageously in some track-structure applications provided that the default model for inelastic interactions of electrons is replaced by suitable tables of differential and total cross sections.