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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
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Abstract
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 rootmeansquare (RMS) deviation in these fits was 0.68 %. The IMFPs were also compared with IMFPs from a relativistic version of our predictive TPP2M 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 TPP2M 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 keV and 200 keV.
Summary
We report new calculations of 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 experimental optical data using the probability for energy loss per unit distance traveled by an electron with relativistic kinetic energy T with the relativistic full Penn algorithm for energies up to 200 keV.
The calculated IMFPs could be fitted with a modification of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies between 50 eV and 200 keV. The average RMS deviation in these fits was 0.68 %. We also developed a relativistic version of our TPP2M equation [Eqns (26) and (29)] that could be used to estimate IMFPs for electron energies between 50 eV and 200 keV. This predictive IMFP equation is based on the modified relativistic Bethe equation. The four parameters in the relativistic TPP2M equation are calculated using the same equations that were developed for our original TPP2M equation. The latter equation was based on our earlier IMFP calculations for a group of 27 elemental solids and a group of 14 organic compounds with electron energies between 50 eV and 2 keV [6].
We compared our calculated IMFPs with values from the relativistic TPP2M equation and found an average RMS deviation of 11.9 % for the 41 solids; this average RMS deviation was almost the same as that found in a previous comparison for the 50 eV to 30 keV range (12.3 %). Large RMS deviations were found for diamond, graphite, and cesium (70.7 %, 46.6 %, and 34.7 %, respectively) as shown in Table 4; possible reasons for these large deviations were discussed in a previous paper [8]. If the RMS deviations for diamond, graphite, and cesium are excluded, the average RMS deviation for the remaining 38 elements was 8.9 %. This value is slightly superior to the corresponding average RMS deviation of 9.2 % found with IMFPs for the 50 eV to 30 keV range for the same elements [8] and 10.2 % for the 50 eV to 2 keV range for our original group of 27 elemental solids [6]. We therefore believe that the relativistic TPP2M equation should be useful for estimating IMFPs in most materials for electron energies between 50 eV and 200 keV with an average RMS uncertainty of about 10 %. Nevertheless, we point out that possible allotropic effects remain to be examined.
We compared our calculated IMFPs with those from recent calculations and experiments. Our calculated IMFPs for Al and Si and energies between 10 eV and 200 keV agree well with those of FernandezVarea et al. [12] that were calculated from a relativistic opticaldata model. There is similar good agreement between our IMFPs for Cu and Au and those of FernandezVarea et al. for energies between 500 eV and 200 keV. There are small but systematic differences at lower energies that must be due to differences of the optical energyloss functions or to the different dispersion relations that were used in each IMFP algorithm.
We also compared our calculated IMFPs with measured IMFPs from TEM experiments at 100 keV for 11 elemental solids and at 200 keV for 32 elemental solids. We found satisfactory agreement in these comparisons with an overall average RMS difference between them of 13.6 % (or 12.3 % with the exclusion of Be in the comparisons). These average RMS differences are similar to the estimated uncertainty of about 10 % for the IMFP measurements. We also compared IMFPs from the relativistic TPP2M equation proposed in the present work with the IMFPs determined from TEM experiments. We again found good agreement in these comparisons except for Hf at 100 keV and diamond, Y, and In at 200 keV. The average RMS difference between IMFPs from the TPP2M equation and the measured IMFPs is 17.4 %.
Notice
This is the peer reviewed version of the following article: Shinotsuka, H., Tanuma, S., Powell, C. J., and Penn, D. R. (2015) 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. Surf. Interface Anal., 47: 871– 888, which has been published in final form at https://doi.org/10.1002/sia.5789. This article may be used for noncommercial purposes in accordance with Wiley Terms and Conditions for Use of SelfArchived Versions.
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 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)
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 23/07/2015
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 01/04/2021
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 19/07/2021
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