Calculations of electron inelastic mean free paths. X. Data for 41 elemental solids over the 50 eV to 200 keV range with the relativistic full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 41 elemental solids (Li, Be, graphite, diamond, glassy C, Na, Mg, Al, Si, K, Sc, Ti, V, Cr, Fe, Co, Ni, Cu, Ge, Y, Nb, Mo, Ru, Rh, Pd, Ag, In, Sn, Cs, Gd, Tb, Dy, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Bi) for electron energies from 50 eV to 200 keV. The IMFPs were calculated from measured energy loss functions for each solid with a relativistic version of the full Penn algorithm. The calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root‐mean‐square (RMS) deviation in these fits was 0.68%. The IMFPs were also compared with IMFPs from a relativistic version of our predictive Tanuma, and Powell and Penn (TPP‐2M) equation that was developed from a modified form of the relativistic Bethe equation. In these comparisons, the average RMS deviation was 11.9% for energies between 50 eV and 200 keV. This RMS deviation is almost the same as that found previously in a similar comparison for the 50 eV to 30 keV range (12.3%). Relatively large RMS deviations were found for diamond, graphite, and cesium as in our previous comparisons. If these three elements were excluded in the comparisons, the average RMS deviation was 8.9% between 50 eV and 200 keV. The relativistic TPP‐2M equation can thus be used to estimate IMFPs in solid materials for energies between 50 eV and 200 keV. We found satisfactory agreement between our calculated IMFPs and those from recent calculations and from measurements at energies of 100 and 200 keV. Copyright © 2015 John Wiley & Sons, Ltd.     

Calculations of electron inelastic mean free paths. XIII. Data for 14 organic compounds and water over the 50 eV to 200 keV range with the relativistic full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 14 organic compounds (26-n-paraffin, adenine, β-carotene, diphenyl-hexatriene, guanine, Kapton, polyacetylene, poly (butene-1-sulfone), polyethylene, polymethylmethacrylate, polystyrene, poly(2-vinylpyridine), thymine, and uracil) and liquid water for electron energies from 50 eV to 200 keV with the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants and from calculated atomic scattering factors for X-ray energies. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.17%. The IMFPs were also compared with a relativistic version of our predictive Tanuma–Powell–Penn (TPP-2M) equation. The average RMS deviation in these comparisons was 7.2% for energies between 50 eV and 200 keV. This average RMS deviation is smaller than that found in a similar comparison for our group of 41 elemental solids (11.9%) and for our group of 42 inorganic compounds (10.7%) for the same energy range. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations for energies between 200 eV and 10 keV. We also found reasonable agreement between our IMFPs for organic compounds and measured IMFPs for energies between 50 eV and 200 keV. Substantial progress for IMFP measurements for liquid water has been made in recent years through the invention of liquid water microjet photoelectron spectroscopy and droplet photoelectron imaging. We found that the IMFPs from these experiments and the associated analyses were larger than our IMFPs by factors between two and four for energies between about 30 eV and 1000 eV. The energy dependences of the measured IMFPs are, however, similar to that of our IMFPs in the same energy range. Since IMFPs calculated from the same algorithm for a number of inorganic compounds agree reasonably well with measured IMFPs for energies between 100 eV and 200 keV, the large differences between IMFPs for water from recent experiments and our results are surprising and need to be resolved with additional experiments.     

Calculations of electron inelastic mean free paths. XII. Data for 42 inorganic compounds over the 50 eV to 200 keV range with the full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 42 inorganic compounds (AgBr, AgCl, AgI, Al₂O₃, AlAs, AlN, AlSb, cubic BN, hexagonal BN, CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF₂, MgO, NaCl, NbC_0.712, NbC_0.844, NbC_0.93, PbS, PbSe, PbTe, SiC, SiO₂, SnTe, TiC_0.7, TiC_0.95, VC_0.76, VC_0.86, Y₃Al₅O₁₂, ZnS, ZnSe, and ZnTe) for electron energies from 50 eV to 200 keV. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants for 15 compounds while calculated ELFs were utilized for the other 27 compounds. Checks based on ELF sum rules showed that the calculated ELFs were superior to the measured ELFs that we had used previously. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.60%. The IMFPs were also compared with a relativistic version of our predictive Tanuma-Powell-Penn (TPP-2M) equation. The average RMS deviation in these comparisons was 10.7% for energies between 50 eV and 200 keV. This average RMS deviation is almost the same as that found in a similar comparison for a group of 41 elemental solids (11.9%) although relatively large deviations were found for cubic BN (65.6%) and hexagonal BN (34.3%). If these two compounds are excluded in the comparisons, the average RMS deviation becomes 8.7%. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations and from experiments.     

Calculations of electron inelastic mean free paths

We report calculations of electron inelastic mean free paths (IMFPs) for 50–2000 eV electrons in 14 elemental solids (Li, Be, diamond, graphite, Na, K, Sc, Ge, In, Sn, Cs, Gd, Tb, and Dy) and for one solid (Al) using better optical data than in our previous work. The new IMFPs have also been used to test our TPP‐2M equation for estimating IMFPs in these materials. We found surprisingly large root‐mean‐square (RMS) deviations (39.3–71.8%) between IMFPs calculated from TPP‐2M and those calculated here from optical data for diamond, graphite and cesium; previously we had found an average RMS deviation of 10.2% for a group of 27 elemental solids. An analysis showed that the large deviations occurred for relatively small computed values of the parameter β in the TPP‐2M equation (β ～ 0.01 for diamond and graphite) and also for relatively large values of β (β ～ 0.25 for Cs). Although such extreme values of β are unlikely to be encountered for many other materials, the present results indicate an additional limitation in the reliability of the TPP‐2M equation. We also show that the parameter N_v in the TPP‐2M equation should be computed for the rare‐earth elements from the number of valence electrons and the six 5p electrons. Copyright © 2004 John Wiley & Sons, Ltd.     

Calculations of electron inelastic mean free paths. XI. Data for liquid water for energies from 50 eV to 30 keV

We calculated electron inelastic mean free paths (IMFPs) for liquid water from its optical energy‐loss function (ELF) for electron energies from 50 eV to 30 keV. These calculations were made with the relativistic full Penn algorithm that has been used for previous IMFP and electron stopping‐power calculations for many elemental solids. We also calculated IMFPs of water with three additional algorithms: the relativistic single‐pole approximation, the relativistic simplified single‐pole approximation, and the relativistic extended Mermin method. These calculations were made by using the same optical ELF in order to assess any differences of the IMFPs arising from choice of the algorithm. We found good agreement among the IMFPs from the four algorithms for energies over 300 eV. For energies less than 100 eV, however, large differences became apparent. IMFPs from the relativistic TPP‐2M equation for predicting IMFPs were in good agreement with IMFPs from the four algorithms for energies between 300 eV and 30 keV, but there was poorer agreement for lower energies. We calculated values of the static structure factor as a function of momentum transfer from the full Penn algorithm. The resulting values were in good agreement with results from first‐principle calculations and with inelastic X‐ray scattering spectroscopy experiments. We made comparisons of our IMFPs with earlier calculations from authors who had used different algorithms and different ELF data sets. IMFP differences could then be analyzed in terms of the algorithms and the data sets. Finally, we compared our IMFPs with measurements of IMFPs and of a related quantity, the effective attenuation length. There were large variations in the measured IMFPs and effective attenuation lengths (as well as their dependence on electron energy). Further measurements are therefore required to establish consistent data sets and for more detailed comparisons with calculated IMFPs. Copyright © 2016 John Wiley & Sons, Ltd.     

An accurate and simple universal curve for the energy‐dependent electron inelastic mean free path

The values of inelastic mean free paths (IMFPs) calculated from optical data for the three material categories of elements, inorganic compounds and organic compounds are re‐assessed to provide a simple equation giving an estimate of the IMFP, knowing only the identities of the elements in an analysed layer and the atomic density of that layer. This simple equation is required for quantification of the thicknesses for layers of mixed elements in which the required parameters for use of the popular equation, TPP‐2M, are insufficiently known. It describes the published values, calculated from optical data for energies above 100 eV, to a similar root mean square (RMS) deviation as that for TPP‐2M in the three material categories. The RMS deviation for all three categories averages 8.4%, provided the inorganic data are ‘corrected’ for the published sum rule errors. If, in an analysed layer, only elements are identified and the atomic density is unknown, i.e. only the average Z value of the layer is known, a simpler relation is provided for the IMFP in monolayers with only one unknown parameter Z that exhibits an RMS deviation from the IMFPs calculated from optical data of 11.5%. Copyright © 2011 Crown copyright.     

Evaluation of elastic‐scattering cross sections for electrons and positrons over a wide energy range

Quantification of surface‐ and bulk‐analytical methods, e.g. Auger‐electron spectroscopy (AES), X‐ray photoelectron spectroscopy (XPS), electron‐probe microanalysis (EPMA), and analytical electron microscopy (AEM), requires knowledge of reliable elastic‐scattering cross sections for describing electron transport in solids. Cross sections for elastic scattering of electrons and positrons by atoms, ions, and molecules can be calculated with the recently developed code ELSEPA (Elastic Scattering of Electrons and Positrons by Atoms) for kinetic energies of the projectile from 10 eV to 50 eV. These calculations can be made after appropriate selection of the basic input parameters: electron‐density distribution, a model for the nuclear‐charge distribution, and a model for the electron‐exchange potential (the latter option applies only to scattering of electrons). Additionally, the correlation‐polarization potential and an imaginary absorption potential can be considered in the calculations. We report comparisons of calculated differential elastic‐scattering cross sections (DCSs) for silicon and gold at selected energies (500 eV, 5 keV, 30 keV) relevant to AES, XPS, EPMA, and AEM, and at 100 MeV as a limiting case. The DCSs for electrons and positrons differ considerably, particularly for medium‐ and high‐atomic‐number elements and for kinetic energies below about 5 keV. The DCSs for positrons are always monotonically decreasing functions of the scattering angle, while the DCSs for electrons have a diffraction‐like structure with several minima and maxima. A significant influence of the electron‐exchange correction is observed at 500 eV. The correlation‐polarization correction is significant for small scattering angles at 500 eV, while the absorption correction is important at energies below about 10 keV. Copyright © 2005 John Wiley & Sons, Ltd.     

Variation of the electron inelastic mean free path during depth profiling of the Fe/Si interface as determined by quantitative REELS

The aim of this work is to determine the dependence of the electron inelastic mean free path (IMFP) at the Fe/Si interface during depth profiling by sputtering with 3 keV Ar⁺ ions. In order to estimate the variation of the IMFP at the interface, reflection electron energy‐loss spectroscopy (REELS) measurements were performed after different sputtering times at the Fe/Si interface with three different primary electron energies (i.e. 0.5, 1 and 1.5 keV). Even though it is highly likely that a compound (i.e. Fe_xSi) is formed at the interface, all the experimental REELS spectra could be analysed as a linear combination of those corresponding to pure Si and Fe. Using the model developed by Yubero and Tougaard for quantitative analysis of these REELS spectra we could estimate the IMFP values along the depth profile at the interface. The resulting IMFPs are observed to vary linearly with the average composition (as determined by REELS) at the Fe/Si interface as it is sputter depth profiled. The energy dependence of the IMFP for different compositions is presented and discussed. For completeness, we have determined the energy‐loss functions as well as the IMFPs of the pure elements (i.e. Fe and Si). Copyright © 2004 John Wiley & Sons, Ltd.     

Effect of Different Electron Elastic-Scattering Cross Sections on Inelastic Mean Free Paths Obtained from Elastic-Backscattering Experiments

Inelastic mean free paths (IMFPs) of electrons with energies between 100eV and 5,000eV have been frequently obtained from measurements of elastic-backscattering probabilities for different specimen materials. A calculation of these probabilities is also required to determine IMFPs. We report calculations of elastic-backscattering probabilities for gold at energies of 100eV and 500eV with differential elastic-scattering cross sections obtained from the Thomas-Fermi-Dirac potential and the more reliable Dirac-Hartree-Fock potential. For two representative experimental configurations, the average deviation between IMFPs obtained with cross sections from the two potentials was 11.4%.     

Density functional calculation of core‐electron binding energies of isomers of C3H6O2 and C3H5NO

The core‐electron binding energies of six isomers of C₃H₆O₂ and four isomers of C₃H₅NO were calculated by a DFT/uGTS/scaled‐pVTZ approach. An average absolute deviation from experiment of 0.15 eV was found for 14 C, N, and O 1s energies. The results confirm the distinctive nature of the X‐ray photoelectron spectra (XPS) of isomers and support the use of electron spectroscopy complemented by accurate theoretical predictions as a tool for chemical analysis. © 1999 John Wiley & Sons, Inc. Int J Quant Chem 76: 44–50, 2000     

Attenuation lengths in organic materials

Measurements are reported for attenuation lengths in overlayers of guanine, poly(styrene), poly(methyl methacrylate) and poly(2‐vinylpyridine) in the energy range 700–1400 eV to evaluate the accuracy of theoretical computations and generic equations. The layers are deposited on gold, either by evaporation and condensation or by spin casting. Measurements are made by X‐ray photoelectron spectroscopy (XPS) for the substrate Au 4f, 4d and 4p peaks as well as the overlayer N 1s peak in guanine and poly(2‐vinylpyridine). These measurements all correlate with the theory of Tanuma, Powell and Penn, to an RMS deviation of 11%. Correlations with the predictions using the generic equation known as TPP‐2M exhibit a poorer RMS deviation of 20%. Use of an ‘average’ organic material for the inelastic mean free paths for the four materials also leads to a 20% RMS deviation. Additional data are also presented for Irganox 1010 that is used, increasingly, as a reference sample for secondary ion mass spectrometry. Comparisons for 5 materials with the Gries G1 formula gives an RMS deviation of 13%, but a small change to interpolate between Gries' classes of organic materials with H/C being around either 1 or 2, leads to a reduced RMS deviation of 10%. This final version, G1‐SS, requiring only the material density and chemical formula, is very simple to use for all organic materials where these data are known or can be estimated. © Crown copyright 2010. Reproduced with the permission of Her Majesty's Stationery Office. Published by John Wiley & Sons, Ltd.     

Simple universal curve for the energy‐dependent electron attenuation length for all materials

An analysis is presented for a simple, universal equation for the computation of attenuation lengths (L) for any material, necessary for quantifying layer thicknesses in Auger electron spectroscopy (AES) and X‐ray photoelectron spectroscopy (XPS). Attenuation lengths for selected materials may be computed from the inelastic mean free path (λ_Opt) computed, in turn, from optical data. The computation of L involves the transport mean free path and gives good L values where values of λ_Opt are available. However, λ_Opt values are not available for all materials. Instead, λ may be calculated from the TPP‐2M relation, but this requires the accurate estimation of a number of materials parameters that vary over a wide range. Although these procedures are all soundly based, they are impractical in many analytical situations. L is therefore simply reexpressed, here, in terms of the average Z of the layer which may be deduced from the AES or XPS analysis, the average atomic size a (varies in a small range) and the kinetic energy E of the emitted electron. For strongly bonded materials, such as oxides and alkali halides, a small extra term is included for the heat of formation. A new equation, S3, is established with a root mean square (RMS) deviation of 8% compared with the values of attenuation length calculated from λ_Opt available for elements, inorganic compounds, and organic compounds. This excellent result is suitable for practical analysis. In many films, an average value of a of 0.25 nm is appropriate, and then L may be expressed only in terms of the average Z and E. Then, L expressed in monolayers, equation S4, exhibits an RMS deviation of 9% for many elements. These results are valid for the energy range 100 to 30 000 eV and for angles of emission up to 65°. Copyright © 2012 Crown copyright.     

Surface excitation parameter for 12 semiconductors and determination of a general predictive formula

The surface excitation parameter (SEP) is theoretically calculated for 12 semiconductors (GaN, GaP, GaSb, GaAs, InSb, InAs, InP, SiC, ZnSe, ZnS, Si and Ge) and for Ni (which is usually used as a reference in experiments) for electron energies between 300 eV and 3400 eV, and for angles between 0° and 70° to the surface normal. We use our previous definition of SEP, as the change in excitation probability, for an electron, caused by the presence of the surface in comparison with an electron moving the same distance in an infinite medium. The calculations are performed within the dielectric response theory by means of the QUEELS‐ε(k, ω)‐REELS software determining the energy‐differential inelastic electron scattering cross‐sections for reflection‐electron‐energy‐loss spectroscopy (REELS), and for which the only input is the dielectric function of the medium. By fitting to these SEP values as well as our previous ones, i.e. from 27 materials, including metals, oxides, polymers and semiconductors, we also establish a simple equation depending on the generalized plasmon energy and the energy band gap of the material which allows to estimate the SEP when the dielectric function is not available. Copyright © 2009 John Wiley & Sons, Ltd.     

Exploration of density functional methods for one‐electron reduction potential of nitrobenzenes

Performance of the set of density functional approaches for calculation of one‐electron reduction potentials of nitroaromatic compounds was investigated. To select the most precise and affordable method, we selected a set of model molecules and investigated effects of basis set, density functional, and solvation model on the calculation of reduction potentials. It was found that the mPWB1K/TZVP method provides the most accurate gas phase electron affinity values (RMS error is 0.1 eV). This method in conjunction with the PCM (Bondi) method yields also the most accurate difference in solvation energies of neutral oxidized form and anion‐radical reduced form. The final E⁰ values were calculated with RMS error of 0.10 V, compared with experimental values. © 2009 Wiley Periodicals, Inc. J Comput Chem 2010     

Do solid surface potential barriers retard convoy peak electrons?

Using a coaxial cylindric electron spectrometer and an electrostatic ion energy analyzer in tandem, a direct measurement of the difference of the energy of convoy peak electron and the electron equivalent ion energy of protons emerging from the downstream surface of C, Au and Al foils is performed in the proton energy range from 60 to 250 keV. This measurement is made possible using the accepted evidence that for a gas target these energies are equal. It is found that also for the beam foil convoy peak electrons, within an experimental average uncertainty of about ±0.1 eV, there is no difference between these energies. If one accepts that the origin of convoy electrons is from inside the solid, the conclusion is that no retardation by the solid surface potential barrier, which is of the order of a few eV, is observed. This is attributed to the strong electron-ion Coulomb interaction which almost completely overshadows the force exerted on the electron by the field of the surface barrier.     

Memory effect on the inelastic interaction of electrons moving parallel to a solid surface

It is generally assumed that two successive inelastic interactions between an electron and a solid are independent of each other. In other words, the electron has no memory of its previous interaction. However, the previous interaction of the electron generates a potential that should influence its succeeding inelastic interaction. The aim of this work is to establish a model to account for the memory effect of an electron between two successive inelastic interactions. On the basis of the dielectric response theory, formulae for differential inverse inelastic mean free paths (DIIMFPs) and inelastic mean free paths (IMFPs) considering the memory effect were derived for electrons moving parallel to a solid surface by solving the Poisson equation and applying suitable boundary conditions. These mean free paths were then calculated with the extended Drude dielectric function for a Cu surface. It was found that the DIIMFP and the IMFP with the memory effect for electron energy E lay between the corresponding values without the memory effect for electron energy E and previous energy E₀. The memory effect increased with increasing electron energy loss, E₀ ? E, in the previous inelastic interaction. Copyright © 2005 John Wiley & Sons, Ltd.     

Elastic and inelastic cross sections for low-energy electron collisions with pyrimidine

We present theoretical elastic and electronic excitation cross sections and experimental electronic excitation cross sections for electron collisions with pyrimidine. We use the R-matrix method to determine elastic integral and differential cross sections and integral inelastic cross sections for energies up to 15 eV. The experimental inelastic cross sections have been determined in the 15-50 eV impact energy range. Typically, there is quite reasonable agreement between the theoretical and experimental integral inelastic cross sections. Calculated elastic cross sections agree very well with prior results.