Kramers-Kronig relations beyond the optical approximation

We extend Kramers-Kronig relations beyond the optical approximation, that is to dielectric functions $\varepsilon(q,\omega)$ that depend not only on the frequency but on the wave number as well. This implies extending the notion of causality commonly used in the theory of Kramers-Kronig relations to include the fact that signals cannot propagate faster than light in vacuo. The results we derive do not apply exclusively to electrodynamics but also to other theories of isotropic linear response in which the response function depends on both wave number and frequency.Comment: 1 figur

Bethe stopping-power formula and its corrections

The classical and quantum theories leading to the asymptotic Bethe formula of the stopping power of matter for charged particles heavier than the electron are briefly reviewed. Models and approximations for the practical calculation of various corrections that extend the validity range of the formula are described. The asymptotic formula and the associated shell correction were determined previously from an extensive database of atomic generalized oscillator strengths, calculated for an independent-electron model with the Dirac-Hartree-Fock-Slater (DHFS) self-consistent potential, with due account for relativistic departures from the Bethe sum rule. The nonrelativistic Bloch correction is extended to the relativistic domain by means of the Lindhard-Sørensen formulation, and an accurate parametrization for point projectiles with small charges is proposed. The density-effect correction and the Barkas correction are obtained from a semiempirical model of the optical oscillator strength (OOS), built from the calculated DHFS contributions of inner electron subshells plus the OOS of outer-shell electrons represented by an analytical expression, which is determined by the composition, mass density, and empirical mean excitation energy, or I value of the material. Inclusion of the shell, density-effect, Lindhard-Sørensen, and Barkas corrections into the asymptotic formula leads to the corrected Bethe formula. A general strategy is proposed to determine the stopping power in terms of only the I value of the material. It is shown that, with the empirical I values recommended in Report 37 of the International Commission on Radiation Units and Measurements, the stopping powers calculated numerically from the corrected formula are in close agreement with available measurements of the stopping power of elemental materials for protons and alpha particles with energies higher than 0.75 and 5 MeV, respectively

Surface excitations in the modelling of electron transport for electron- beam-induced deposition experiments

The aim of the present overview article is to raise awareness of an essential aspect that is usually not accounted for in the modelling of electron transport for focused-electron-beam-induced deposition (FEBID) of nanostructures: surface excitations are on the one hand responsible for a sizeable fraction of the intensity in reflection-electron-energy-loss spectra for primary electron energies of up to a few keV and, on the other hand, they play a key role in the emission of secondary electrons from solids, regardless of the primary energy. In this overview work we present a general perspective of recent works on the subject of surface excitations and on low-energy electron transport, highlighting the most relevant aspects for the modelling of electron transport in FEBID simulations.Comment: 17 pages, 5 figure

Simulation of electron transport in electron beam induced deposition of nanostructures

We present a numerical investigation of energy and charge distributions during electron-beam-induced growth of W nanostructures on SiO2 substrates using Monte Carlo simulation of electron transport. This study gives a quantitative insight into the deposition of energy and charge in the substrate and in already existing metallic nanostructures in the presence of the electron beam. We analyze electron trajectories, inelastic mean free paths, and distribution of backscattered electrons in different deposit compositions and depths. We find that, while in the early stages of the nanostructure growth a significant fraction of electron trajectories still interact with the substrate, as the nanostructure becomes thicker the transport takes place almost exclusively in the nanostructure. In particular, a larger deposit density leads to enhanced electron backscattering. This work shows how mesoscopic radiation-transport techniques can contribute to a model which addresses the multi-scale nature of the electron-beam-induced deposition (EBID) process. Furthermore, similar simulations can aid in understanding the role played by backscattered electrons and emitted secondary electrons in the change of structural properties of nanostructured materials during post-growth electron-beam treatments.Comment: 22 pages, 14 figures, 1 tabl

Electromagnetic interaction models for Monte Carlo simulation of protons and alpha particles

Electromagnetic interactions of protons and alpha particles are modeled in a form that is suitable for Monte Carlo simulation of the transport of charged particles. The differential cross section (DCS) for elastic collisions with neutral atoms is expressed as the product of the DCS for collisions with the bare nucleus and a correction factor that accounts for the screening of the nuclear charge by the atomic electrons. The screening factor is obtained as the ratio of the DCS for scattering of the projectile by an atom with a point nucleus and the parameterized Dirac-Hartree-Fock-Slater (DHFS) electron density, calculated from the eikonal approximation, and the Rutherford DCS for collisions with the bare point nucleus. Inelastic collisions, which cause electronic excitations of the material, are described by means of the plane-wave Born approximation, with an empirical simple model of the generalized oscillator strength (GOS) that combines several extended oscillators with resonance energies and strengths determined from the atomic configurations and from the empirical mean excitation energy of the material. The contributions from inner subshells are renormalized to agree with realistic ionization cross sections calculated numerically from the DHFS self-consistent model of atoms by means of the plane-wave Born approximation. The resulting DCS allows analytical random sampling of individual hard inelastic interactions.Comment: 48 pages, 12 figure

Theory and calculation of the atomic photoeffect

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SBETHE: Stopping powers of materials for swift charged particles from the corrected Bethe formula

The Fortran program sbethe calculates the stopping power of materials for swift charged particles with small charges (electrons, muons, protons, their antiparticles, and alphas). The electronic stopping power is computed from the corrected Bethe formula, with the shell correction derived from numerical calculations with the plane-wave Born approximation (PWBA) for atoms, which were based on an independent-electron model with the Dirac–Hartree–Fock–Slater self-consistent potential for the ground-state configuration of the target atom. The density effect correction is evaluated from an empirical optical oscillator strength (OOS) model based on atomic subshell contributions obtained from PWBA calculations. For projectiles heavier than the electron, the Barkas correction is evaluated from the OOS model, and the Lindhard–Sørensen correction is estimated from an accurate parameterization of its numerical values. The calculated electronic stopping power is completely determined by a single empirical parameter, the mean excitation energy or I value of the material. The radiative stopping power for electrons, and positrons, is evaluated by means of Seltzer and Berger's cross section tables for bremsstrahlung emission. The program yields reliable stopping powers and particle ranges for arbitrary materials and projectiles with kinetic energy larger than a certain cutoff value , which is specific of each projectile kind. The program is accompanied by an extensive database that contains tables of relevant energy-dependent atomic quantities for all the elements from hydrogen to einsteinium. sbethe may be used to generate basic information for dosimetry calculations and Monte Carlo simulations of radiation transport, and as a pedagogical tool