Differential Inverse Mean Free Paths Research Articles

How the choice of model dielectric function affects the calculated observables

It is investigated how the model used to describe a dielectric function (i.e. a Mermin, Drude, Drude–Lindhard, Levine–Louie with relaxation time dielectric function) affects the interpretation of a REELS experiment, the calculation of the electron inelastic mean free path as well proton stopping and straggling. Three dielectric functions are constructed that are based on different models describing a metal, but have identical loss functions in the optical limit. A loss function with the same shape, but half the amplitude, is used to derive four different model dielectric functions for an insulator. From these dielectric functions we calculate the differential inverse mean free path, the mean free path itself, as well as the stopping force and straggling for protons. The similarity of the underlying physics between proton stopping, straggling and the electron inelastic mean free path is stressed by describing all three in terms of the differential inverse inelastic mean free path. To further highlight the reason why observed quantities depend on the model dielectric function used we study partial differential inverse inelastic mean free paths, i.e. those obtained by integrating over only a limited range of momentum transfers. In this way it becomes quite transparent why the observable quantities depend on the choice of model dielectric function.

Inelastic interactions of low-energy electrons with biological media

A study of inelastic interactions of low-energy electrons with biological media is of fundamental importance in the understanding of radiation-induced actions leading to the biological damage. In the present work, electron energy loss properties in liquid water and solid DNA were calculated based on the dielectric response theory for valence band electrons and atomic models for inner-shell electrons. For electron interactions with the valence band, an extended Drude dielectric function was fitted to experimental optical data and then checked by sum rules. The valance band was considered as composed of many subbands with different oscillator strength, binding energy and damping coefficient. The real and imaginary parts of the dielectric function and the energy loss function were all checked. For electron interactions with inner shells, the generalized oscillator strength was calculated using the sum-rule-constrained binary-encounter model and the local-plasma approximation. The differential inverse mean free paths for the valence band and inner shells were calculated using the Born approximation. Exchange effect was estimated using an approach based on the Moller cross-section. Results of the present work were compared with similar calculations by other authors.

Influence of the direction of motion on the inelastic interaction between electrons and solid surfaces

Theoretical derivations of the inelastic differential inverse mean free path (DIMFP) and inelastic mean free path (IMFP) for electrons crossing solid surfaces were made for different crossing angles and electron distances relative to the crossing point at the surface. Individual contributions from volume and surface excitations were separated and analyzed for electrons traveling inside and outside the solid. Extended Drude dielectric functions were employed to calculate the DIMFP and inverse IMFP for electrons incident into and escaping from Cu. It was found that the DIMFP and inverse IMFP for electrons moving inside the solid were approximately independent of crossing angle and position of electrons due to the compensation of volume and surface excitations. For electrons moving deep inside the solid, the DIMFP and inverse IMFP reduced to the values for electrons moving in an infinite solid. As electrons traveling in the vacuum, the DIMFP and inverse IMFP became greater for glancing incident and escaping angles since surface excitations were more probable. The surface excitation parameter (SEP) for electrons traveling in vacuum showed an angular dependence. The SEP of escaping electrons was found larger than that of incident electrons due to the attractive force exerted by the induced surface charges. The calculated SEP was found to follow a simple expression.

Auger-electron cascades in diamond and amorphous carbon

We have analyzed cascades of secondary electrons in diamond and amorphous carbon generated by the thermalization of a single Auger electron. The elastic electron mean free path was calculated as a function of impact energy in the muffin-tin potential approximation. The inelastic scattering cross section and the energy loss of electrons ~expressed in terms of the differential inverse mean free path! were estimated from two ‘‘optical’’ models that utilize the measured dielectric constants of the materials. Using these data, a Monte Carlo model describing the time evolution of the cascade was constructed. The results show that at most around 20‐ 40 secondary cascade electrons are released by a single Auger electron in a macroscopic sample of diamond or amorphous carbon. Consideration of the real band structure of diamond reduces this number further. The release of the cascade electrons happens within the first 100 fs after the emission of the primary Auger electron. The results have implications to planned experiments with femtosecond x-ray sources.

Electron inelastic interactions with overlayer systems

An overlayer system composed of a thin film on the top of a semi-infinite substrate was studied in this work for electron inelastic interactions. Analytical expressions for the depth-dependent inelastic differential and integral inverse mean free paths were derived for both incident and escaping electrons. The interface (film-substrate) effect and the surface (vacuum-film) effect were analyzed by comparing the results of an overlayer system and a semi-infinite system. It was found that the interface effect extended to several angstroms on both sides of the interface for a 500 eV electron incident into or escaping from the vacuum–SiO2–Si and the vacuum–Au–Ni systems. An application of the spatial-varying inelastic differential inverse mean free paths was made by Monte Carlo simulations of the electron elastic backscattering from an overlayer system. Good agreement was found between results calculated presently and data measured experimentally on the elastic reflection coefficient.

Surface excitation parameters of low-energy electrons crossing solid surfaces

The surface excitation parameter, which describes the influence of surface excitations by electrons for the vacuum side in electron spectroscopies, has been calculated for electrons of 200–2000 eV energies crossing surfaces of Cu, Ag, Au, Fe, Pd, Ni, MgO and SiO2. These calculations were performed for both incident and escaping electrons by the use of dielectric response theory. Spatially varying differential inverse mean free paths for surface excitations as a function of electron distance from the surface were found. The results showed that small differences existed in the surface excitation parameter among different metals but large differences occurred between metals and semiconductors or insulators. Calculated surface excitation parameters were fitted very well to a simple formula, i.e. Ps=aE−b, where Ps is the surface excitation parameter and E is the electron energy. © 1998 John Wiley & Sons, Ltd.

Inelastic electron interactions in overlayers on substrates

The inelastic interaction of electrons with overlayers on substrates is investigated within the framework of dielectric response theory. A general formula for the position-dependent inelastic differential inverse mean free path (DIIMFP), for electron transport in a layered system has been derived. The position dependence reveals specific details of the changes between bulk, interface and surface excitation modes. Numerical calculations of the DIIMFP and the IIMFP have been made for the case of a Au film deposited on a Si substrate, for which experimental optical constants and an extended Drude dielectric function have been used.

Elastic reflection of low-energy electrons from polycrystalline gold targets

Monte Carlo calculations to study the influence of solid-state, polarization, exchange and relativistic effects on the angular distribution and the reflection coefficient for electrons elastically reflected from gold surfaces have been performed. Elastic differential cross sections were determined using relativistic Dirac equations with a solid-atom potential including the polarization and exchange effects. Inelastic differential inverse mean free paths were computed using the extended Drude dielectric function for volume and surface excitations. These showed that the polarization effect made a contribution to elastic differential cross sections at small scattering angles, whereas, the exchange and relativistic effects contributed to these cross sections at large scattering angles. These effects are important to the angular distribution and the elastic reflection coefficient for electron energies below 400 eV. Monte Carlo results calculated in this work are in very good agreement with experimental data.

Electron differential inverse mean free path for surface electron spectroscopy

A new, general expression for the position-dependent differential inverse mean free path (DIMFP) of an electron penetrating into vacuum from a solid is derived. This DIMFP can be divided up into a bulk and a surface term. It is found that the surface effect is restricted to a surface layer extending on both sides of the vacuum-solid interface. An extended Drude dielectric function, which allows the characteristic oscillator strength, damping constant, and critical-point energy for each subband of valence electrons, is employed to estimate electron DIMFPs near Al and Au surfaces. Our results are relevant to the understanding of inelastic electron scattering near a solid surface.

The surface effect on Au 4f X-ray photoelectron spectra

An investigation of the surface effect on Au 4f X-ray photoelectron spectra was made for different emission angles. It was found that surface plasmon excitations made significant contributions to the spectra in single and plural energy-loss peaks and in the multiple loss background. Our approach involved Monte Carlo simulations of photoelectrons in the interior region of the solid and Poisson stochastic processes in the surface region. The partial-wave expansion method with the Hartree-Fock-Wigner-Seitz potential for the solid was used to calculate differential electron elastic scattering cross sections. An extended Drude dielectric function model was employed to compute electron differential inverse mean free paths for volume excitations. The inelastic mean free paths in the surface region became spatially-varying and were calculated. Differential probabilities for surface excitations were determined using the same model dielectric function. The computed X-ray photoelectron spectra including surface excitations agreed quite well with experimental data.

Quantitative analysis in X-ray photoelectron spectroscopy: influence of surface excitations

A take-off-angle-dependent convolution formula has been derived to obtain angle-resolved XPS spectra by including the effect of surface excitations into the Landau formula. An extended Drude dielectric function is employed to estimate the electron differential inverse mean free paths for bulk excitations. The same dielectric function is applied to evaluate the probability of surface excitations for emitted electrons by including the recoil effect and without the small-scattering-angle approximation. Through numerical calculations for Au 4f XPS spectra the effects of surface excitations have been shown to be significant in XPS spectra. Moreover, the influence of surface excitations on the energy-loss structure within 30 eV is enhanced for large take-off angles. Calculated spectra that include the surface excitations are in better agreement with experimental data.

Differential cross sections for plasmon excitations and reflected electron-energy-loss spectra.

Electron differential inverse mean free paths for volume-plasmon excitations and differential probabilities for surface-plasmon excitations have been calculated using dielectric response theory. A model dielectric function which satisfied sum rules and agreed with optical data was established for these calculations. In surface-plasmon calculations, we considered electron-impact emissions of the reflected electron-energy-loss spectroscopy. Formulations were made for obliquely incident electrons with the recoil effect and without the small-scattering-angle assumption. For volume-plasmon excitations, we evaluated corrections due to the exchange and ${\mathit{Z}}_{1}^{3}$ effects. Comparison between calculated results and experimental data extracted from reflected electron-energy-loss spectra showed good agreement. Calculated differential cross sections have been used to solve the transport equation for the angular and energy flux spectra of reflected electrons. Contributions to the spectra from single and plural plasmon excitations were analyzed. It was found that calculated spectra were in good agreement with measured data.

Straggling and plasmon excitation in the energy loss spectra of electrons transmitted through carbon

A model is described for the calculation of differential inverse mean free paths (DIMFPs) for the inelastic interactions of electrons with solids. The energy loss function Im{-1/ ε( q, ω)} is represented as a sum of Drude-type functions. The adjustable parameters in the energy loss function are fixed by a fit to the energy loss function in the optical limit ( q → 0) obtained from data on the optical constants of glassy carbon. The extension of the optical energy loss function to arbitrary values of q in this model leads to an analytical expression for the DIMFP. Energy loss spectra for low energy electrons (less than 2000 eV) transmitted through carbon foils of various thicknesses are determined by solving the transport equation using a convolution method and the DIMFPs determined as described above. Comparisons of these calculated spectra with those measured by Jacobi show that the model provides a good description of both the broad straggling distribution and the fine structure due to plasmon excitation.