Full Penn Research Articles

Determination of electron inelastic mean free path and stopping power of hafnium dioxide

We present inelastic mean free path (IMFP) and stopping power data of the hafnium dioxide applying the relativistic dielectric response theory. The energy loss function (ELF) derived from reflection electron energy loss spectroscopy spectrum with the reverse Monte Carlo method was used for the first time to obtain the IMFP and stopping power of HfO2. The probability of the energy loss is determined by the dielectric response function εq,ω as a function of the frequency ω and the wavenumber q of the electromagnetic disturbance. Two algorithms, namely the full Penn algorithm (FPA) and the super-extended Mermin algorithm (SMA), were employed to expand the optical energy loss function, Im-1/ε0,ω, into the q,ω-plane. The results indicate that the IMFP and the stopping power obtained by using both algorithms are consistent at high electron energies, but show differences at low electron energies (less than ∼ 70 eV). This discrepancy arises from the consideration of the finite plasmon lifetimes effect in the SMA model, while it is neglected in the FPA model. Additionally, we observed that the band gap has a significant influence on the IMFP and the stopping power at low electron energies. Typically, the inclusion of the band gap leads to an increase in the IMFP, because transition channels with energies larger than E-Eg-Ev are prohibited.

Calculations of electron inelastic mean free paths (IMFPs). XIV. Calculated IMFPs for LiF and Si3N4 and development of an improved predictive IMFP formula

We report inelastic mean free paths (IMFPs) of Si3N4 and LiF for electron energies from 50 eV to 200 keV that were calculated from their optical energy‐loss functions using the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. Our calculated IMFPs, designated as optical 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 root mean square (RMS) deviations in these fits were less than 1% for Si3N4 and LiF. The IMFPs were also compared with the relativistic version of our predictive Tanuma–Powell–Penn (TPP‐2M) equation. We found that IMFPs calculated from the TPP‐2M equation are systematically larger than the optical IMFPs for both LiF and Si3N4. The RMS differences between IMFPs from the TPP‐2M equation and the optical IMFPs were 49.3% for LiF and 17.3% for Si3N4 for energies between 50 eV and 200 keV. These RMS differences are much larger than those for most of the inorganic compounds in our previous IMFP calculations where the average RMS difference was 10.7% for 42 inorganic compounds. We also report the development of an improved predictive IMFP formula, which we designate as the JTP equation. This formula is a refinement of the TPP‐2M equation and is based on the recent IMFP calculations for 100 materials including the present IMFPs for Si3N4 and LiF (41 elemental solids, 45 inorganic compounds, and 14 organic compounds) for 83 electron energies between 50 eV and 200 keV. Our predictive JTP equation gave satisfactory results in comparisons of optical IMFPs and IMFPs calculated from the JTP equation. The RMS difference between the 8300 optical IMFPs used for optimization and the IMFPs calculated from the JTP equation was 10.2%. This value is appreciably less than the RMS difference of 16.0% found in a similar comparison of the optical IMFPs and IMFPs from the TPP‐2M equation. Furthermore, IMFPs from the JTP equation were compared with measured IMFPs for energies between 50 eV and 200 keV for 16 elemental solids and 37 inorganic compounds. We found that the JTP equation gave satisfactory results that were comparable with previous comparisons of the optical IMFPs and measured IMFPs. We believe that the JTP equation will be applicable to a wider range of materials than the TPP‐2M equation.

Influence of energy loss function to the Monte Carlo simulated electron backscattering coefficient

We report an improved calculation of the electron backscattering coefficients (BSCs) for beryllium, molybdenum and tungsten at electron energies of 0.1–100 keV based on an up-to-date Monte Carlo simulation method with different input of energy loss function (ELF) data. The electron inelastic cross-section is derived from the relativistic dielectric functional formalism, where the full Penn’s algorithm is applied for the extension of the ELF from the optical limit of q to 0 into the left( {q,omega } right)-plane. We have found that the accuracy of energy loss function may affect largely the calculated BSC. We also show that this has close relationship with the f- and ps-sum rules.

Characteristics of secondary electron emission from few layer graphene on silicon (111) surface

As a typical two-dimensional (2D) coating material, graphene has been utilized to effectively reduce secondary electron emission from the surface. Nevertheless, the microscopic mechanism and the dominant factor of secondary electron emission suppression remain controversial. Since traditional models rely on the data of experimental bulk properties which are scarcely appropriate to the 2D coating situation, this paper presents the first-principles-based numerical calculations of the electron interaction and emission process for monolayer and multilayer graphene on silicon (111) substrate. By using the anisotropic energy loss for the coating graphene, the electron transport process can be described more realistically. The real physical electron interactions, including the elastic scattering of electron–nucleus, inelastic scattering of the electron–extranuclear electron, and electron–phonon effect, are considered and calculated by using the Monte Carlo method. The energy level transition theory-based first-principles method and the full Penn algorithm are used to calculate the energy loss function during the inelastic scattering. Variations of the energy loss function and interface electron density differences for 1 to 4 layer graphene coating GoSi are calculated, and their inner electron distributions and secondary electron emissions are analyzed. Simulation results demonstrate that the dominant factor of the inhibiting of secondary electron yield (SEY) of GoSi is to induce the deeper electrons in the internal scattering process. In contrast, a low surface potential barrier due to the positive deviation of electron density difference at monolayer GoSi interface in turn weakens the suppression of secondary electron emission of the graphene layer. Only when the graphene layer number is 3, does the contribution of surface work function to the secondary electron emission suppression appear to be slightly positive.

Evaluation of dielectric function models for calculation of electron inelastic mean free path

This work investigates the detailed difference between dielectric function models, the Mermin model and the full Penn algorithm (FPA) model, for the determination of an electron inelastic mean free path (IMFP) with optical energy loss function (ELF), as an extension of our previous study [Da et al., Surf. Interface Anal. 51, 627 (2019)] by using the simple Drude-type ELF. In the conventional normal Mermin (NM) model, the approximations of ELF by the Drude equation will introduce inevitable fitting error. In order to enhance the accuracy of the NM model, our previous proposed extended Mermin model [Da et al., Phys. Rev. Lett. 113, 063201 (2014)], which is renamed as a super-extended Mermin algorithm (SE-MA) now, is employed to eliminate the error by expanding the definition of Drude oscillators used in the NM. In the SE-MA, the Drude-like oscillators allow the existence of negative strengths to express the fine structures of phonon–electron scattering and the plasmon lifetime broadening effect. Because in our previous study, the simple Drude-type ELF cannot include these complex structures, in this work, the electron IMFPs are calculated for five realistic materials, Al, Si, Cu, Au, and MgO. The difference between IMFPs calculated by the SE-MA model and the FPA model is material dependent and is significant in the low energy region, which is analyzed by using the Fano plot. This is due to the more important role played by the plasmon lifetime broadening effect.

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.

Stopping power and CSDA range of electrons in liquid water, LiF, CaF2, and Al2O3 from the energy gap up to 433 keV

Stopping powers (SP) for electrons with energies greater than 10 keV are well described through the Bethe approximation. However, for energies below, such an approach is inadequate. SP in LiF, CaF2 and Al2O3 as the luminescent materials commonly used for dosimetry study and liquid-water are scarce at energies below 10 keV. This work investigated SP and CSDA-range of electrons in in these compounds from the energy gap up to 433 keV using the full Penn algorithm. In particular, the influence of the exciton states was evaluated. The experimental optical data were selected after a rigorous evaluation process. The results indicate that the exciton states affect the SP mainly at energies below 100 eV by up to 48%, 39%, 32% and 15% for liquid-water, LiF, CaF2 and Al2O3, respectively. Furthermore, we observed a certain relationship between the minimum excitation-energy (exciton state) and the energy at which the SP is maximum.

Electron backscattering coefficients of molybdenum and tungsten based on the Monte Carlo simulations

Monte Carlo simulation is employed for the calculation of electron backscattering coefficients of molybdenum (Mo) and tungsten (W) at normal incidence angle and at energies between 100 eV to 100 keV. For the modelling of the electron elastic scattering we have applied the partial-wave expansion method using the Mott's cross-section, where the scattering potential includes the electron electrostatic potential, exchange potential and correlation-polarization potential. A relativistic dielectric functional approach with full Penn's algorithm is adopted for the calculation of the electron inelastic cross-section. The effect of phonon excitation, the interband transition of the loosely bound valance electrons, and the inner-shells electrons excitations are considered in the energy loss functions of the materials via the optical constants. We found that our present simulated data agree well with the experimental data at energies above 20 keV for both elements while they are well above the experimental data at lower incident energies. The disagreement can be explained by the possible surface contamination of carbonaceous atomic layers exist in the measurements by additional simulation for the contaminated surfaces. Moreover, for the better and detailed understanding of the energy dependence of the backscattering coefficients, we also present the backscattered electron energy spectra, angular distributions, and depth distributions for both samples.

Determination of electron backscattering coefficient of beryllium by a high-precision Monte Carlo simulation

We present an up-to-date Monte Carlo simulation of electron backscattering coefficient of beryllium, which is an important material in fusion reactor, at an impact energy range of electrons between 100 eV and 100 keV. The Mott’s cross section calculated with more accurate scattering potential and a relativistic dielectric functional formalism with full Penn algorithm and experimental optical data are used in the modelling of electron elastic and inelastic scatterings, respectively. This Monte Carlo simulation modelling enabled us to derive comprehensive theoretical values of backscattering coefficient, which are found significantly smaller than the previous published experimental data particularly at energies below 10 keV. Simulation of electron backscattering spectra was then carried out with and without Auger electron emission, which confirmed that Auger electrons have negligible contribution to backscattering coefficient. To validate our simulation results, we have also performed calculation for amorphous boron and carbon, while the comparison with the available experimental data shows reasonable agreement. Further simulation for carbon and water covered Be sample has revealed that the surface contamination in low vacuum conditions with several atomic/molecular layers can drastically alter the measurement values for beryllium at low energies. The low backscattering coefficient values of beryllium are partly attributed to the extremely strong forward elastic scattering for low atomic number element. Therefore, we recommend to use the here reported backscattering coefficient data of beryllium for applications.

Monte Carlo simulation study of electron yields from compound semiconductor materials

A systematic study has been performed based on a Monte Carlo simulation for the investigation of secondary electron yields, backscattering coefficients, and total electron yields for eight compound semiconductor materials, i.e., AlN, TiN, VN, VC, GaAs, InAs, InSb, and PbS, at different incident electron energies in the range 0.1–10 keV. Our Monte Carlo simulation model is based on the Mott cross section for electron elastic scattering as calculated by a partial wave method and a dielectric functional approach to electron inelastic scattering with the full Penn algorithm. We used Palik's optical data for lower photon energies below 100 eVs and Henke's data for higher photon energies. The cascade production of secondary electrons in electron inelastic scattering and low energy is included in the simulation. The simulated results of electron backscattering coefficients are compared with the experimental data available in the literature. Considering the fact that the experimental data for these compound materials are not available, we have compared them with experimental data for elements having the nearest mean atomic numbers. The simulation predicted much larger backscattering coefficient values than the empirical Staub formula.

Electron Inelastic Mean Free Paths for LiF, CaF2, Al2O3, and Liquid Water from 433 keV down to the Energy Gap.

We report new calculations, which include the influence of the band gap and exciton states, of the electron inelastic mean free path (IMFP) for liquid water, LiF, CaF2, and Al2O3 from the band gap to 433 keV. Among compounds, liquid water is the most studied due to its role in radiobiological research, whereas LiF and CaF2 are the most widely used thermoluminescent dosimeters in environmental monitoring and medical and space dosimetry. Due to its sensitivity, the optically stimulated luminescent dosimeter, Al2O3, has recently begun to be used for personnel monitoring. Previous treatments have modified the integration domain to consider the indistinguishability between the incident electron and the ejected one or the bandgap energy for nonconductors but not to accommodate exciton states within the band gap, and no published IMFP data are available for CaF2. Our calculation was carried out using an electron-beam–solid-state interaction model through the relativistic full Penn algorithm. Integration limits that consider the band gap, the valence band width, and exciton interactions have been used. The results suggest that, at electron energies below 100 eV, the different choices of models for integration limits and the exciton interaction can affect the IMFP by 9–29%. At higher energies, the differences associated with the choice of energy-loss function and other input parameters are around 2.5–7.5%.

Conventional vs. model-based measurement of patterned line widths from scanning electron microscopy profiles

Scanning electron microscopy (SEM) is a practical tool to determine the dimensions of nanometer-scale features. Conventional width measurements use arbitrary criteria, e.g., a 50 % threshold crossing, to assign feature boundaries in the measured SEM intensity profile. To estimate the errors associated with such a procedure, we have simulated secondary electron signals from a suite of line shapes consisting of 30 nm tall silicon lines with varying width, sidewall angle, and corner rounding. Four different inelastic scattering models were employed in Monte Carlo simulations of electron transport to compute secondary electron image intensity profiles for each of the shapes. The 4 models were combinations of dielectric function theory with either the single-pole approximation (SPA) or the full Penn algorithm (FPA), and either with or without Auger electron emission. Feature widths were determined either by the conventional threshold method or by the model-based library (MBL) method, which is a fit of the simulated profiles to the reference model (FPA + Auger). On the basis of these comparisons we estimate the error in the measured width of such features by the conventional procedure to be as much as several nanometers. A 1 nm difference in the size of, e.g., a nominally 10 nm transistor gate would substantially alter its electronic properties. Thus, the conventional measurements do not meet the contemporary requirements of the semiconductor industry. In contrast, MBL measurements employing models with varying accuracy differed one from another by less than 1 nm. Thus, a MBL measurement is preferable in the nanoscale domain.

Comparison of the Mermin and Penn models for inelastic mean‐free path calculations for electrons based on a model using optical energy‐loss functions

We present detailed comparisons of calculated inelastic mean‐free paths (IMFPs) of electrons with energies up to 10 keV from the full Penn algorithm (FPA) model and the Mermin model using prototypical Drude‐type energy‐loss functions (ELFs). From systematic comparisons of IMFPs obtained from the prototypical ELFs using these two models, we obtain evidence that supports the Mermin model as the more reasonable model for electron IMFP calculations than the FPA model. The deviations between these two models are mainly because of finite plasmon lifetimes and show a strong dependence on the width of the ELFs and electron energy. For instance, the theoretical description of the ELF based on the Mermin model predicts lower values for the IMFPs than those obtained by the FPA approach in most cases. As for those prototypical ELFs with neither very sharp nor very broad structures that describe the ELF of actual materials, the Mermin IMFPs are smaller than the FPA IMFPs by about 5% for energies between 1 and 10 keV. We also found that over the 130 to 10 keV range, the modified Bethe equation fits IMFPs calculated with both algorithms within a 1% relative difference.

Determination of electron inelastic mean free path of three transition metals from reflection electron energy loss spectroscopy spectrum measurement data

The energy loss functions (ELFs) of three transition metals (Cr, Co and Pd) have been derived from reflection electron energy loss spectroscopy spectrum with a theoretical analysis of the measured data [Xu et al., J. Appl. Phys. 123, 043306 (2018)]. In this work, we update our previous ELFs in a wider photon energy region (0–200 eV) with a better accuracy, which is verified by sum rules and a root-mean-square deviation. The electron inelastic mean free paths (IMFPs) of Cr, Co and Pd have been calculated with the obtained ELFs by adopting a dielectric response theory. We employ both the single-pole approximation and full Penn algorithm for the calculation of IMFPs, and the calculated results are compared with other references.

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, NbC0.712, NbC0.844, NbC0.93, PbS, PbSe, PbTe, SiC, SiO2, SnTe, TiC0.7, TiC0.95, VC0.76, VC0.86, 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.

Monte Carlo simulation of incident electrons passing through thin metal layer

A Monte Carlo simulation using two schemes, the discrete energy loss approach and the continuous slowing down approximation, was implemented in C++ to calculate the energy transmission coefficient and average energy loss for low-energy (1–10 keV) incident electrons passing through a thin metal layer. The simulation model uses the Ashley model for electron inelastic scattering, the electron elastic scattering cross section taken from the NIST database, and the stopping power derived from the full Penn algorithm. The results of the two schemes agree well with each other and can be used to quantitatively evaluate the shielding effect of a thin coated metal layer on incident electrons for a diamond amplified photocathode.

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 (FPA) 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 (SPA), the relativistic simplified SPA, and the relativistic extended Mermin method. These calculations were made 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 FPA. The resulting values were in good agreement with results from first-principles 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 (EAL). There were large variations in the measured IMFPs and EALs (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.

Selective association of electrocardiographic abnormalities with insulin sensitivity and beta-cell function in type 2 diabetes mellitus: a cross-sectional analysis.

We investigated the association of electrocardiographic (ECG) abnormalities with markers of insulin resistance and pancreatic beta-cell dysfunction in a cross-sectional study of type 2 diabetes patients. Electrocardiographic criteria were evaluated in the Penn Diabetes Heart Study participants (n = 1671; 64% male; 61% Caucasian), including a sub-sample (n = 710) that underwent oral glucose tolerance testing. The Matsuda Insulin Sensitivity Index and homeostasis model assessment of insulin resistance (HOMA-IR) estimated insulin sensitivity; Insulinogenic Index and homeostasis model assessment of beta-cell function assessed beta-cell function. Multivariable regression modelling was used to analyse associations of ECG changes with these indices. In unadjusted analyses, subjects in the highest quartile of Matsuda index had the lowest prevalence of Q-waves (6.3% versus 15.3%, p = 0.005). In adjusted models, an inverse association was seen between Q-waves and log Matsuda index [one standard deviation increase; OR = 0.59 (95% CI 0.43-0.87 p = 0.001)]. In the full Penn Diabetes Heart Study, there was a direct association between Q-waves and HOMA-IR [one standard deviation increase; OR = 1.43 (95% CI 1.13-1.81, p = 0.003)]. In adjusted models, left ventricular hypertrophy also was inversely associated with Matsuda index and directly with HOMA-IR. Higher Insulinogenic Index scores were associated with a lower prevalence of nonspecific ST changes [OR = 0.78 (95% CI 0.62-0.98, p = 0.032)]. In type 2 diabetic patients, both oral glucose tolerance testing-derived and HOMA-derived measures of insulin resistance were associated with pathologic Q-waves and left ventricular hypertrophy on ECGs. Copyright © 2016 John Wiley & Sons, Ltd.

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

Our inelastic mean free paths were calculated correctly.

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.