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.     

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. 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. IX. Data for 41 elemental solids over the 50 eV to 30 keV range

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 30 keV. The IMFPs were calculated from experimental optical data using the full Penn algorithm for energies up to 300 eV and the simpler single‐pole approximation for higher energies. The calculated IMFPs could be fitted to a modified form of the Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 30 keV. The average root‐mean‐square (RMS) deviation in these fits was 0.48%. The new IMFPs were also compared with IMFPs from the predictive TPP‐2M equation; in these comparisons, the average RMS deviation was 12.3% for energies between 50 eV and 30 keV. This RMS deviation is almost the same as that found previously in a similar comparison for the 50 eV–2 keV range. Relatively large RMS deviations were found for diamond, graphite and cesium. If these three elements were excluded in the comparison, the average RMS deviation was 9.2% between 50 eV and 30 keV. We found satisfactory agreement of our calculated IMFPs with IMFPs from recent calculations and from elastic‐peak electron‐spectroscopy experiments. Copyright © 2010 John Wiley & Sons, Ltd.     

Stopping powers and inelastic mean free path from quantitative analysis of experimental REELS spectra for electrons in Ti,Fe, Ni,and Pd

Stopping power (SP) and inelastic mean free path (IMFP) of electrons in Ti, Fe, Ni, and Pd have been determined by using dielectric models. We have used energy loss function (ELF) determined from quantitative analysis of experimental reflection electron energy loss spectroscopy (REELS) spectra as the input parameter for this model. ELF in this study was determined from the previously published quantitative analysis of REELS spectra. The SP of Fe, Ni, Pd, and Ti was compared with several calculation methods for energies from 100 eV to 10 keV and shows SP in this study, which are in best agreement for medium to high energies (greater than or equal to 300 eV). The IMFP obtained in this study shows the best agreement with online database TPP2M and NIST and also calculation by Tanuma with a root mean square (rms ) less than 12%. The present approach shows ELF from quantitative analysis of REELS spectra has a high potential for the experimental determination of SP and IMFP of metals.     

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.     

Review of absolute doubly differential cross-section experiments and cross-section ratios for electron bremsstrahlung from rare gas atom and thin-film targets

We discuss doubly differential cross-section experiments for electron bremsstrahlung from free gas atom and thin-film targets for electron energies of 100 keV or less. We compare cross-section ratios for different target atoms with two theoretical models: ordinary bremsstrahlung and total bremsstrahlung calculated in the stripping approximation. Ratios of cross sections have been used to improve the comparison between experiment and theory when only relative cross sections are available or when the error in the absolute cross section is large. We also discuss additional background processes that may be more important in gas target experiments.     

Evaluation of robustness to surface conditions of the target factor analysis method for determining the dielectric function from reflection electron energy loss spectra: Application to GaAs

Target factor analysis (TFA) of a series of angle‐resolved reflection electron energy loss spectra (REELS) was recently demonstrated to be a useful method to determine bulk energy loss functions (ELFs), which by the TFA are separated from the surface‐loss structures of REELS. The dielectric function is then readily derived by Kramers–Kronig analysis of the ELF. The advantage of the method compared with other methods, which are also based on the analysis of REELS, is that the condition of the outermost surface region is unimportant because the excitations that occur there are removed by the TFA and ideally a pure bulk component is determined. Our method is thus particularly useful for determining the ELF from compound materials that are hard to clean without modifying the outermost atomic layers. In this paper, the robustness of the method was studied by applying it to three GaAs samples with different surface compositions caused by different surface cleaning methods. The results showed that when electrons of energy 3000–4500 eV were used, the resulting bulk ELFs were essentially identical except for small differences for the sample that had the largest thickness of the modified surface layer. It is concluded that this is a useful method, provided that the thickness of the modified layer is kept to a minimum by using shallow angle sputtering and by using REELS electrons at a sufficiently high energy that a major part of the electron trajectories are at a depth larger than the thickness of the modified surface layer. Copyright © 2013 John Wiley & Sons, Ltd.     

Virtual point detector for HPGe detectors for 26.6–1332 keV photon energies by experiment and Monte Carlo simulation

Variation of virtual point detector (VPD) position inside HPGe detector as a function of source photon energy for the energy range from 26.6 to 1,332 keV was investigated. Although VPD concept was well established for HPGe detectors from 59.5 to 10 MeV, a new attempt was made to obtain VPD positions for photon energies below 59.5 keV. It was found that VPD position shows different functional behavior for the energy ranges 26.6–59.5 keV and 59.5–1,332 keV. The VPD position decreases with increasing energy for 26.6, 31.7, 36.4, and 37.3 keV and increases with the energy until it reaches a plateau. The functional behavior of VPD position for the energy range 26.6–59.5 keV was attributed to the dead layer thickness of the Ge crystal. Monte Carlo simulations were performed to investigate the behavior of VPD position with various dead layer thickness ranging from 100 to 800 μm. It was seen that VPD position increases with increasing energy for 31.7, 59.5, and 80.1 keV more significant at relatively lower energies, but constant for the energies 661–1,332 keV.     

Challenges and techniques to effectively characterize the efficiency of broad-energy germanium detectors at energies less than 45 keV

With the now common availability of large-volume thin-window germanium detectors, it is possible to routinely measure very low energy (<45 keV) gamma and X-rays while maintaining good sensitivity for high-energy gamma rays. The effective calibration of such detectors down to these low energies is often problematic or not possible because of the lack of calibrated sources or knowledge of the source geometry. New methods have been recently developed that extend Canberra’s ISOCS/LabSOCS mathematical efficiency computation methods down to energies as low as 10 keV. Key to these developments is the capability to characterize the efficiency versus spatial location of a detector at the factory and thus eliminate the requirement to have “in the field” low-energy source standards. In this paper, the challenges for performing reliable efficiency characterizations below 45 keV and techniques developed to overcome these challenges are discussed. Response characterization results are presented for various types of low-energy and broad-energy detectors manufactured by Canberra.     

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.     

Electron and positron elastic scattering in gaseous and liquid water: A comparative study

In the present work, we present both theoretical differential and total cross sections for the elastic scattering process of positrons and electrons in liquid and vapour water for energies ranging from 10 eV to 10 keV. The calculations are performed in the partial-wave formalism by means of a complex interaction potential taking into account static potential as well as fine effects like exchange and polarization contributions. The theoretical results obtained in this free-parameter quantum-mechanical treatment are compared to available experimental data and good agreement is generally observed. Moreover, quantitative differences are reported between the positron and electron scattering, in vapour as well as in liquid water.     

Kβ/Kα X-ray intensity ratios for elements in the range 16⩽Z⩽92 excited by 5.9, 59.5 and 123.6 keV photons

The K shell intensity ratios K_β/K_α for 59 elements in the atomic region 16⩽Z⩽92 have been measured at excitation energies of 5.9, 59.5 and 123.6 keV. K X-rays emitted by samples have been counted by a Si(Li) detector with resolution 160 eV at 5.9 keV. The measured values were compared with the theoretical values calculated using Scofield's tables based on the Hartree–Slater and Hartree–Fock theories and available experimental values. Reasonable agreement is typically obtained between present and theoretical values.