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[TANUMA, Shigeo](https://orcid.org/0000-0003-2628-9941), C.J. Powell, D.R. Penn

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This is the peer reviewed version of the following article: Calculations of electron inelastic mean free paths.VIII. Data for 15 elemental solids over the 50-2000 eV range, which has been published in final form at  https://doi.org/10.1002/sia.1997. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions. This article may not be enhanced, enriched or otherwise transformed into a derivative work, without express permission from Wiley or by statutory rights under applicable legislation. Copyright notices must not be removed, obscured or modified. The article must be linked to Wiley’s version of record on Wiley Online Library and any embedding, framing or otherwise making available the article or pages thereof by third parties from platforms, services and websites other than Wiley Online Library must be prohibited.[Creative Commons BY-NC Attribution-NonCommercial 4.0 International](https://creativecommons.org/licenses/by-nc/4.0/)

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[Calculations of electron inelastic mean free paths.VIII. Data for 15 elemental solids over the 50-2000 eV range](https://mdr.nims.go.jp/datasets/e235ed80-9897-4fef-b0c0-33a87f8ca20f)

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IMFP VIII_Author Manuscript2Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   1 Calculations of Electron Inelastic Mean Free Paths. VIII. Data for 15 Elemental Solids over the 50 eV to 2000 eV Range S. Tanuma,1 C. J. Powell,2* and D. R. Penn2 1Materials Analysis Station, National Institute for Materials Science, 1-2-1 Sengen,  Tsukuba 305-0047, Japan 2National Institute for Standards and Technology, Gaithersburg, MD 20899, USA    We report calculations of electron inelastic mean free paths (IMFPs) for 50 eV to 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 deviations (from 39.3 % to 71.8 %) between IMFPs calculated from TPP-2M and those calculated here from optical data for diamond, graphite, and cesium. An analysis showed that these deviations occurred for relatively small computed values of the parameter b in the TPP-2M equation (b ~ 0.01 for diamond and graphite) and for a relatively large value of b (b ~ 0.25 for Cs). While such extreme values of b are unlikely to be encountered for many other materials, the present results indicate an aditional limitation in the reliability of the TPP-2M equation. We also show that the parameter Nv in the TPP-2M equation should be computed for the rare-earth elements from the number of valence electrons and the six 5p electrons.   *  Correspondence to: C. J. Powell, National Institute of Standards and Technology, Gaithersburg, MD 20899-8370, USA. E-mail: Cedric.Powell@nist.gov.  Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   2 INTRODUCTION  We have previously reported calculations of inelastic mean free paths (IMFPs) of 50 eV to 2000 eV electrons from experimental optical data for 27 elemental solids, 15 inorganic compounds, and 14 organic compounds, as well as analyses of these results.1-7 The optical data were checked for internal consistency using two sum rules;8 these checks indicated that the optical data for the group of elemental solids and the group of organic compounds were more reliable than those for the group of inorganic compounds. We therefore analyzed the IMFPs for the groups of elemental solids and organic compounds to derive an equation, designated TPP-2M, based on the Bethe equation9 for inelastic scattering of electrons in matter.5 We found that the four parameters in TPP-2M could be empirically related to several material parameters (atomic or molecular weight, density, number of valence electrons per atom or molecule, and the bandgap energy for nonconductors). The TPP-2M equation could then be used to estimate IMFPs for other materials over the 50 eV to 2000 eV energy range.  We report here IMFP calculations for 50 eV to 2000 eV electrons in an additional 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 earlier work.1,2 The new IMFPs are useful because they provide tests of TPP-2M for three additional classes of materials: alkali metals, allotropes of carbon, and rare-earth elements. The rare-earth elements have unusual electronic properties, and we discuss how TPP-2M should be evaluated for these elements.  IMFP CALCULATIONS  Our method for the IMFP calculation has been described previously.1,2 We make use of the Penn10 algorithm that is expected to give useful results for electron energies above 50 eV.4 We also calculated IMFPs for energies between 10 eV and 40 eV, and display these results in the Figures below to show the IMFP trends vs. electron energy more completely; these results should be considered only as rough estimates. The IMFP values were calculated for most materials using Eq. (14) of Ref. 1 for electron energies between 10 eV and 200 eV and the simpler Eq. (16) of Ref. 1 was used for energies between 300 eV and 2000 eV; for graphite and Al, Eq. (14) was used for energies between 10 eV and 300 eV while Eq. (16) was used for energies of 400 eV and above. We note that the difference in IMFPs obtained using these two equations at 300 eV is very small (for example, less than 0.2 % for graphite). All energies are expressed with respect to the Fermi level.  Table 1 contains material-property data used in the IMFP calculations or in the later analysis of the IMFP results. We show there values of the atomic weight M, the bulk density r, the number of valence electrons per atom (Nv),7 the free-electron plasmon energy ( ), the bandgap energy (Eg), and the Fermi energy (EF). Data for glassy carbon are included in Table 1 and later Tables since IMFPs for this material1,2 will also be analyzed.  The IMFPs were calculated using values of the energy loss function , where e(w) is the complex dielectric constant for the material of interest as a function of photon energy . Values of the energy loss function were calculated from the sources of optical data and energy-loss-function measurements listed in Table 2.11-21 Since no direct 2/1)/(8.28 MNE vp r=)](/1Im[ we-wAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   3 measurements were available for photon energies larger than about 40 eV, it was necessary to make use of atomic photoabsorption data for extrapolations to photon energies beyond the particular measurement range.18 These extrapolations are not considered a serious source of error because e(w) for photon energies larger than about 40 eV is mainly determined by atomic properties except in the vicinity of core-electron excitation thresholds. The latter differences are unimportant in the IMFP calculation because an integration is made of the energy loss function over .1   We evaluated the accuracy of our sets of  data using two useful sum rules.3,8 These sum rules are the oscillator-strength sum rule (or the f-sum rule) and a limiting form of the Kramers-Kronig integral (or KK-sum rule). The f-sum can be evaluated as the total effective number of electrons per atom Zeff contributing to the inelastic scattering:3,8              (1)  where , ,  is the density of atoms, and Na is Avogadro's number. When the upper limit  in Eq. (1) is equal to infinity, Zeff should be equal to Z, the total number of electrons per atom. The KK-sum can be expressed as:3,8             (2)  where n(0) is the limiting value of the refractive index at low photon energies (below those where absorption maxima are observed). In the limit , .  Table 3 lists the errors in the f-sum and KK-sum for the present group of materials from evaluations of Eqs. (1) and (2) with DEmax = 30 keV. For Z > 50, we expect Zeff to be less than Z since K-shell excitations cannot contribute to the integral of Eq. (1); for In (Z = 49) and Sn (Z = 50), the K-shell contributions are small (~ 0.1). Although there are two electrons in the K shell, the contributions of these electrons to the f-sum will be less than two.9 Examination of Zeff as a function of DEmax for Na, Al, K, and Sc (elements that have K-shell binding energies appreciably less than 30 keV), for example, shows that the K-shell contributions to the f-sum integral range from 1.54 to 1.71 so that the average K-shell contribution for these elements is 1.65. For the elements In through Dy, Zeff values from Eq. (1) were therefore compared with  rather than with Z. The average root-mean-square (RMS) error for the sets of  data based on the f-sum rule is 4.0 % while that for the KK-sum is 8.4 %. The larger RMS error for the KK-sum is mainly due to the relatively large KK-sum errors for K and Sn. The KK-sum errors for these two materials must be associated with errors in the optical data for relatively low photon energies since the corresponding errors in the f-sum are much smaller. The RMS error for the f-sum errors in Table 3 is less than a factor of two smaller than that for our original group of 27 elemental solids (9.9 %) while the RMS error for the KK-w)](/1Im[ we-òDDD-DW= max022 )()](/1Im[)/2(Epeff EdEEZ epw=DE 2/12 )/4( menap p=W MNn aa /r=maxED)0()()](/1Im[)/2( 201max -D - +DD-D= ò nEdEEPEeff ep¥®D maxE 1®effP)65.1( -Z)](/1Im[ ED- eAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   4 sum is a little smaller than that for the original group (9.5 %).1 We also note that the sum-rule errors for the new set of  data for Al in Table 3 (1 % and 1.8 %) are appreciably smaller than the corresponding errors (12 % and 11 %) for the original set of Al data.1   As examples of our analysis and as material for later discussion, we show the energy-loss functions, , of diamond, graphite, and glassy C1,2 in Fig. 1 as a function of energy-loss, DE. These functions are also interesting because there is a common set of optical data for diamond and graphite from Ref. 18 for  eV (Table 2). The small differences in the magnitudes of  for  eV are due to the different densities of diamond, graphite, and glassy C (Table 1). We also note here that values of  for diamond and graphite were obtained from Refs. 19 and 20, respectively, rather than from Refs. 11 and 12 because the former data satisfied the sum rules [Eqs. (1) and (2)] much better than the latter data. Figure 2 shows plots of  for the three forms of carbon as a function of , where              (3)  and R is the Rydberg energy (13.606 eV). The IMFP is inversely proportional to .1 It is apparent from Fig. 2 that the larger contributions to  for carbon come from valence-electron excitations with DE values between 5 eV and 200 eV and much smaller contributions from K-shell excitations.   Figures 3 and 4 show plots of Zeff and Peff, respectively, for diamond, graphite, and glassy C as a function of DEmax. We see from Fig. 3 that Zeff saturates for each material close to the expected value of Z = 6. It is clear from Fig. 4 that the main contributions to Peff come from excitations with  eV.   IMFP RESULTS  Table 4 shows IMFPs calculated from the optical data for 50 eV to 2000 eV electrons in the 15 elemental solids. Plots of IMFP vs. electron energy for Li, Be, diamond, graphite, glassy C, Na, Al, Ge, In, Sn, and Cs are shown in Figs. 5-15 as examples of our results; IMFP results for glassy C1,2 have been included here for comparison with those of diamond and graphite. Similar plots for K, Gd, Tb, and Dy have been published previously.7 Information on uncertainties of the calculated IMFPs is given in our paper IV4 and Ref. 22. The uncertainties of IMFPs for energies between 50 eV and 200 eV will be larger than those for higher energies.   As in our previous work,2,3,5 we analyzed the IMFP dependence on energy for each material with a modified form of the Bethe equation9 for inelastic electron scattering in matter:               (4) )](/1Im[ we-)](/1Im[ ED- e50³w)](/1Im[ ED- e 40³DE)](/1Im[ ED- e2totM maxEDòDDD-W= max0222 )()](/1Im[)/2(Eptot EdERM ep2totM2totM50£DE)]}/()/()ln([/{ 22 EDECEEE p +-= gblAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   5  where l is the IMFP (in Å), E is the electron energy (in eV), and b, g, C, and D are parameters. The second and third terms in the denominator of Eq. (4) were added in order to describe the IMFP dependence on energy for energies less than 200 eV.23,24 The solid lines in Figs. 5-15 show fits of Eq. (4) to the calculated IMFPs over the 50 eV to 2000 eV range. Values of b, g, C, and D from the fits for each material are shown in Table 5.  To check our analysis, we have also calculated values of b valid for "high" electron energies. These values of b, denoted bopt, represent the slopes of Fano plots (in which E/l is plotted versus lnE) in the asymptotic Bethe region.25 Values of bopt can be obtained from the following relation:3     (eV-1 Å-1)            (5)  We have evaluated  from Eq. (3) (with DEmax = 30 keV) and determined bopt from Eq. (5); these values are shown in Table 5. We find that bopt is typically less than b by amounts varying from 1 % (K) to 16 % (Ge) although bopt exceeds b for graphite and Sc by 1 % and 6 %, respectively. It is expected25 that the slopes of the Fano plots for energies between 50 eV and 2000 eV (given by b) should exceed the asymptotic slopes (given by bopt).  We have previously analyzed calculated IMFPs for a group of 27 elemental solids2 and a group of 14 organic compounds5 and found that the four parameters in Eq. (4) could be related empirically to other material-property data:5   ,          (6a)  ,             (6b)  ,             (6c)  ,             (6d)  .            (6e)  Equations (4) and (6) represent our TPP-2M formula for estimating IMFPs in materials.  The dashed lines in Figs. 5-15 indicate IMFPs calculated from TPP-2M and the data in Table 1. Table 6 shows values of the root-mean-square (RMS) deviations between IMFPs from TPP-2M and IMFPs calculated from optical data for the indicated materials. The unexpectedly large values of the RMS deviations for diamond  (71.8 %), graphite (49.5 %), and Cs (39.3 %) will be discussed further below. If these three elements are disregarded, the average RMS deviation for the remaining elements is 11.8 %, only slightly larger than the average RMS deviation of 10.2 % for our original group of 27 elements.5 We have recently discussed the appropriate value for the parameter Nv (the number of valence electrons per atom) for certain elements where there has been ambiguity in the choice of this parameter.7 Seah et al.26 had earlier reported an analysis of Auger-electron intensities for 61 elemental solids and of photoelectron intensities for 58 elemental solids in which they vtotopt NM 8.28/2=b2totM1.05.022 069.0)/(944.010.0 rb +++-= gp EE5.0191.0 -= rgUC 91.097.1 -=UD 8.204.53 -=4.829// 2pv EMNU == rAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   6 recommended that Nv be calculated from the number of electrons with binding energies less than or equal to 28 eV. The elements K, Gd, Tb, and Dy were among those for which there were differing recommendations for Nv that were discussed previously.7 There were also different recommendations for the elements In, Sn, and Cs, and so we show IMFPs from TPP-2M in Figs. 13-15 for these elements that were obtained from values of Nv proposed by Seah et al. and ourselves.7 These results will be discussed further below.  DISCUSSION  We will discuss the reliability of the TPP-2M equation for predicting IMFPs, the IMFP results for the low-density solids, and the choice of Nv in the TPP-2M equation for In, Sn, and Cs and for the rare-earth elements Gd, Tb, and Dy.  Reliability of the TPP-2M Equation  We first consider the IMFPs for the three allotropes of carbon in Figs. 7-9. At an energy of 2,000 eV, for example, the IMFPs calculated from optical data vary from 28.1 Å (diamond) to 31.6 Å (graphite) and 46.8 Å (glassy C) whereas the corresponding values from TPP-2M are almost identical (48.5 Å, 46.2 Å, and 46.3 Å). According to the Bethe theory for inelastic scattering of electrons in matter,9 as further reviewed and discussed by Inokuti,23 the total cross section, , for inelastic scattering by an atom at a sufficiently high energy is proportional to . Since the IMFP is inversely proportional to the product of  and the density of atoms, we thus expect the IMFP for the three allotropes of carbon to be inversely proportional to  (where we have assumed, for simplicity, that there is no significant difference in the dependences of the energy-loss functions on momentum transfer for the three carbon allotropes).6 Table 7 shows the asymptotic values of  for the three forms of carbon (for  keV) and the values of . The values of  for diamond and graphite are 56 % and 66 % of those for glassy C; these percentages correspond closely to those found of IMFPs at 2,000 eV calculated from optical data (60 % and 67 %). Since the IMFPs from TPP-2M for diamond and graphite do not vary as expected from this simple analysis, we deduce that these values must have a much larger standard uncertainty than previously estimated for elemental solids.5 The large RMS deviations in Table 6 between IMFPs from the TPP-2M equation and IMFPs calculated from optical data for diamond, graphite, and Cs are not believed to be associated with inadequacies of the optical data for these solids (Table 2) because the errors in the f-sum and KK-sum rules are less than or comparable to those found for other elemental solids (Table 3, and Table 1 of paper I). To determine whether the large deviations in Table 6 might be associated in some way with extreme values of one or more of the parameters in Eqs. (4) and (6), we repeated key steps in the development of our TPP-2M equation.5 This analysis was performed with IMFPs calculated from optical data for the previous groups of elemental solids,1,2,7 inorganic compounds,3,7 and organic compounds5 with some exclusions and additions. We excluded IMFPs for materials with average errors in their f-sum and KK-sum evaluations exceeding 15 %  (Nb, Al2O3, GaAs, InAs, InSb, LiF, and NaCl) and we tots2totM tots2totMr2totM30max =DE 12 )( -totMr 12 )( -totMrAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   7 added IMFPs for some additional inorganic compounds (MgO, and a series of non-stoichiometric carbides of Ti, V, and Nb).27   Figure 16 shows a plot of "observed" values of b, bobs, found from fits of Eq. (4) to the calculated IMFPs for each material versus  (as shown in Fig. 10 of paper V). The long-dashed and short-dashed lines in Fig. 16 are plots of Eq. 6(a) with  g/cm3 and  g/cm3, respectively. There is a general linear trend of the plotted points but, as found previously, there are deviations that depend on density. The plotted point in Fig. 16 for Cs, however, lies appreciably above the lines. Since IMFPs computed from Eq. (4) are inversely proportional to b, we can conclude either that bobs from the fit to IMFPs calculated from the Cs optical data is too large and thus the Cs IMFPs in Table 4 are smaller than they should be, or that b from Eq. 6(a) is too small and thus that IMFPs for Cs from TPP-2M (with Nv = 1, as shown in Table 1) are larger than they should be. We will comment further on the appropriate choice of Nv for Cs later.  Figure 17 shows a plot of  as a function of density (as suggested by Eq. 6(a) and as illustrated previously in Fig. 11 of paper V). The solid line is a plot of the variation expected from Eq. 6(a), namely . It can be seen that most points scatter about the solid line although the point for Cs is again a significant outlier. A plot of  versus density on logarithmic scales (not shown) indicates that the points for Li and K in Fig. 17 deviate significantly from the linear dependence of the other points. It is not clear at present whether the deviations for Li and K in Fig. 17 are due to inadequacies of the IMFPs for these solids (e.g., as a result of unknown limitations of the optical data) or to inadequacies of Eq. 6(a). While it is possible that Eq. 6(a) could be modified to account for the low-density behavior exemplified by Li and K, we are reluctant to do this based on data for only two materials. In addition, it seems unlikely that TPP-2M would be used in practice for material densities appreciably less than those of the polymers examined in paper V.  Figure 18 shows plots, on two different scales, of b from Eq. 6(a) versus bobs. As found previously for Fig. 12 of paper V, most points in Fig. 18 scatter about the dashed line . There are, however, deviations at both low and high values of bobs. As expected from Figs. 16 and 17, the point for Cs in Fig. 18(b) (with the largest values of b and bobs and the smallest value of Ep for the materials considered here) deviates appreciably from the dashed line. Despite the deviations of the points for Li and K from the lines in Figs. 16 and 17, the points for these solids lie reasonably close to the line in Fig. 18. As a result, the RMS deviations between IMFPs from TPP-2M and the IMFPs calculated from optical data for Li and K are reasonably small (13.8 % and 2.3 %, respectively, from Table 6). It is therefore unnecessary to modify TPP-2M further to account for the deviations of the points for these solids in Figs. 16 and 17.  We identify three points in Fig. 18(a) for low values of bobs. While the absolute deviations of b for these points (diamond, graphite, and Pd) from the dashed line in Fig. 18(a) are less than the maximum deviation for other points, the relative deviation for each of these 5.022 )( -+ gp EE1=r10=r5.022 )(944.0 -+-= gpobsr EEbb1.0069.010.0 rb +-=r1.0+rbobsbb =Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   8 three points is much larger. These large relative deviations lead to the substantial RMS deviations between IMFPs from TPP-2M and those from optical data (71.8 %, 49.5 %, and 20.1 % for diamond, graphite, and Pd,5 respectively). The RMS deviation for Pd is appreciably less than those for diamond and graphite because the material parameters of Pd lead to lesser sensitivity to variations of b than for the other two materials.   Elastic-peak electron spectroscopy (EPES) has been identified as the preferred technique for measurements of IMFPs.22 Two groups have made IMFP measurements by EPES for graphite,28,29 and their results are plotted in Fig. 8; Lesiak et al.29 have also reported similar measurements for glassy carbon, and these results are shown in Fig. 9. Most of the IMFPs reported by Tanuma et al.29 for graphite agree better with IMFPs from TPP-2M rather than the IMFPs calculated from optical data; one value (at 250 eV) is much larger than expected from the energy dependence of nearby points, probably due to a diffraction effect. The graphite IMFPs of Lesiak et al.28 show moderate scatter; some values in Fig. 8 are close to the TPP-2M curve while others are close to the solid line derived from the IMFPs computed from optical data. Overall, the general agreement of IMFPs from EPES with IMFPs from TPP-2M suggests that the TPP-2M equation is robust and that there must be some unexpected uncertainty in the graphite IMFPs obtained from optical data beyond that expected from the uncertainty in the corresponding optical data, as shown in Figs. 2 and 3 and as listed in Table 3, and in the model used for the IMFP calculation.4,22 For glassy carbon, the measured IMFPs of Lesiak et al. in Fig. 9 are larger than both values from TPP-2M and from the optical data (which agree well with each other).   We need to point out that IMFPs derived from EPES measurements are based on a model in which it is assumed that elastic scattering of the backscattered electrons can be described by differential cross sections for elastic scattering by atoms in an amorphous or disordered solid.22,30 The sample of interest is typically bombarded by rare-gas ions both to clean the surface and to amorphize the solid at least to the information depth for the detected EPES signal.31 For crystalline samples such as diamond and graphite, an amorphized surface region might be sufficiently different in local atomic structure and bonding from the starting material to cause appreciable differences between the IMFPs derived from EPES experiments and those calculated from the optical data for the corresponding crystalline materials. In the experiments of Tanuma et al.,29 however, the EPES measurements were performed without any ion sputtering. Further EPES experiments are needed to verify and extend the work of Lesiak et al. and Tanuma et al. (e.g., to perform the experiments at different scattering angles and/or with large analyzer acceptance angles for which diffraction effects might be shown to be minimal on samples that had had little or no ion bombardment). Additional tests should also be performed to characterize the surfaces before and after any ion bombardment; reflection electron energy-loss experiments could indicate, by comparison with energy-loss functions obtained from optical data (Fig. 1), whether a carbon sample was predominantly diamond, graphite, or glassy carbon. Additionally, Auger-electron spectroscopy can be utilized to distinguish the different forms of carbon.32-34  In principle, the large RMS deviations between IMFPs from TPP-2M and IMFPs calculated from optical data in Table 6 for diamond, graphite, and Cs could arise from Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   9 limitations of the TPP-2M equation, some unrecognized source of uncertainty in the optical data for these materials, and/or a deficiency of the algorithm used for the IMFP calculations. The optical data for these materials appear to be satisfactory, as revealed by our sum-rule checks (Figs. 2 and 3, and Table 3). In addition, there is a single prominent peak in the energy-loss function for each of these materials that can be identified as being due to bulk-plasmon excitation; these energy losses agrees within 13 % with the values of Ep in Table 1. The free-electron-gas expression of Lindhard and coworkers35 for the dielectric function on which the Penn IMFP algorithm10 is based would therefore appear to be valid.4 By elimination, we believe that there is a limitation of the TPP-2M equation at extreme values of the parameter b. The large RMS deviations for diamond, graphite, and Cs in Table 6 occur for values of b that are smaller (diamond and graphite, where b ~ 0.01) than those for most other materials we have investigated to date and much larger (Cs, where b ~ 0.25) than those of other materials. We note that the first and third terms on the right-hand side of Eq. 6(a) are based on an empirical fit to deviations resulting from limitations of the second term in providing a value for b (as plotted in Fig. 17). As noted above, the fractional uncertainty in the derived b can be large when the values of b are small (as found for diamond, graphite, and Pd). A large uncertainty in the value of b for Cs (as suggested by Fig. 18(b)) could result simply from the fact that Eq. 6(a) is not valid for large values of b. Further IMFP calculations for materials with values of Ep and bobs comparable to those of diamond, graphite and Cs would be required to determine whether a more complex expression than Eq. 6(a) would provide more reliable values of b for such materials.  Li, Be, Na, Al, K, and Cs  We were initially interested in IMFPs for Li, Be, Na, Al, K, and Cs for two main reasons. First, these solids have low densities, and we wished to check whether our TPP-2M equation, developed mainly from computed IMFPs for high-density metals and low-density polymers,5 would be valid for relatively light metals. Second, four of the solids are alkali metals, a class of solids that we had not previously investigated. Although there was no reason to expect that IMFPs for alkali metals would be qualitatively different from those for other solids (other than being generally larger, as illustrated by the results in Table 4, because of their lower densities), we wished to compute IMFPs for as many different types of solids as possible. The RMS deviations in Table 6 between IMFPs from the TPP-2M equation [Eqs. (4) and (6)] for Li, Be, Na, Al, and K vary between 2.3 % (for K) and 25.3 % (for Be); the RMS deviation for Cs was discussed in the previous section. The RMS deviations for Li, Be, Al, and K are comparable to those found for our original groups of 27 elemental solids and 14 organic compounds for which the average RMS deviations were 10.2 % and 8.5 %, respectively.5 We also note that the new RMS deviation for Al (12.7 %) is much smaller than that we found previously (22.7 %)5 when we computed IMFPs from what we now know to be an inferior set of optical data.1,2  Choice of Nv in the TPP-2M Equation Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   10  We have recently discussed the appropriate choice of the parameter Nv in the TPP-2M equation and have pointed out that there can be ambiguity in choosing a value of this parameter for some elements.7 This ambiguity arises for elements with shallow core levels (generally with binding energies less than about 30 eV) that may contribute significant intensity to the energy-loss function. As a result, the IMFP estimated from TPP-2M with Nv computed only from the number of valence electrons may be an overestimate of the IMFP.7  Figures 13-15 show IMFPs computed from TPP-2M for In, Sn, and Cs where Nv has been calculated from the number of valence electrons for each element and from the values recommended by Seah et al.26 (M. P. Seah and I. S. Gilmore, personal communication). From an analysis of measured Auger-electron and photoelectron intensities for some 60 elemental solids, Seah et al. recommended that Nv be computed from the number of electrons with binding energies less than or equal to 28 eV. Figure 14 for Sn shows that IMFPs obtained from TPP-2M with  agree much better with the IMFPs from optical data than values obtained with . For In and Cs, however, the IMFPs calculated from optical data are roughly midway between those obtained from TPP-2M with the alternative values of Nv. The RMS deviations between IMFPs from TPP-2M with the alternative values of Nv and IMFPs from the optical data are 15.1 % and 39.5 % for In and Cs, respectively; these RMS deviations are comparable to those listed in Table 6 (19.5 % and 39.3 %, respectively) for the Nv values listed in Table 1.  Before commenting further on the Nv results for In, Sn, and Cs, we turn to similar results for Gd, Tb, and Dy in Figs. 4-6 of paper VII.7 We recommended there that, for the rare-earth elements, Nv should be computed from the sum of the number of valence electrons for the solid state, either two or three as discussed by Netzer and Matthew,36 and the six 5p electrons that contribute strongly to the energy-loss function. Although the 5p electrons for the lanthanides have binding energies between 18 eV and 27 eV,37 they have been included in the calculation of Nv because of the strong 5p-6d excitations that occur close to the threshold energy for excitation (i.e., the binding energy).38 In contrast, the number of 4f electrons has not been included in the Nv calculation, even though they have binding energies of less than 9 eV.39 The 4f orbitals are highly localized, and they contribute weakly to the energy-loss function because of the substantial "delayed onsets" in plots of photoabsorption cross section versus photon energy.  We now consider the plot in Fig. 19(a) of the energy-loss function for Gd as representative of those for other lanthanide elements.38 Here and in following plots, we show the energy-loss function for energy losses up to 200 eV to show the relative contributions of valence-electron and shallow-core-electron excitations to the IMFP. The strongest feature in Fig. 19(a), an energy-loss at about 13 eV, is due to bulk-plasmon excitation.38 This energy-loss is close to the calculated free-electron bulk-plasmon energy (based on three valence electrons/atom) of 11.4 eV. Although the 5p electrons in Gd are core levels (with a 5p3/2 binding energy of 21.4 eV),37 they contribute strongly to the energy-loss function due to a strong 5p → 5d resonance excitation.38 In contrast, excitations of 4f electrons (with a binding energy of about 8 eV)37 are relatively weak with no obvious structure. Excitations of 4d electrons begin at the expected 4d3/2 threshold (about 141 eV).37  4=vN14=vNAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   11  Figures 19(b) and 19(c) show plots of Zeff and , respectively, for Gd as a function of  from evaluations of Eqs. (1) and (3). It is clear from Fig. 19(b) that  (the value of Nv given in Table 1 for Gd, Tb, and Dy) when reaches 50 eV; the seven 4f electrons36 in Gd contribute to the monotonic increase of Zeff in Fig. 19(b) with increasing  until the beginning of the 4d excitations at  eV. It is clear from Fig. 19(c), however, that the valence and 5p excitations in Gd contribute strongly to the value of  while other excitations are responsible for the gradual increase of  for  eV. The asymptotic value of  is 4.71 while the value of  for  eV is about 3.13. Unlike the situation for C in Fig. 4 where core-electron excitations contribute minimally to the value of , such excitations with  eV contribute about 34 % of the value of  and thus to the IMFP.1,2 Nevertheless, it is apparent from Fig. 4 of paper VII that variation of Nv in TPP-2M for Gd from 9 to 16 decreases the derived IMFP by less than 12 %. The choice of  is thus justified by the strong contributions of the three valence and six 5p electrons to the loss spectrum (for energy losses less than 50 eV) and the dominant contribution of these excitations to .  We now discuss further the choice of Nv in TPP-2M for In, Sn, and Cs. Figures 20(a) and 21(a) shows the energy-loss functions of Cs and Sn, respectively. The energy-loss function for Cs is similar to that of Gd in Fig. 19(a), with a strong bulk-plasmon excitation of the valence electrons at an energy loss of 3.1 eV (close to the free-electron value of 3.43 eV) and excitations of 5p and 4d electrons (with binding energies of about 12.2 and 75 eV, respectively).37 Plots of Zeff and  for Cs in Figs. 20(b) and 20(c) show clear plateaus in the vicinity of  eV. The value of Zeff increases from 5.9 at  eV to 6.2 at  eV, while  is about 6.6 at  eV compared to the asymptotic value of 8.52 at  keV. As for Gd and the other rare-earth elements, it now appears that we should include the six 5p electrons in the count of Nv for Cs; that is, Nv for Cs should be 7 instead of 1. The dot-dashed line in Fig. 15 shows IMFPs computed from TPP-2M with this value of Nv. The IMFP values are only slightly larger than those for Nv = 9, and the RMS deviation between IMFPs obtained with Nv = 7 and the IMFPs calculated from optical data is 34.9 %, only a slight reduction from the value of 39.3 % found with Nv = 1. We conclude that, while Nv = 7 for Cs appears to be a better choice than Nv = 1, there is still substantial disagreement with IMFPs obtained from optical data. The value of b from TPP-2M is now 0.0777 (instead of 0.249 with Nv = 1), and the point for Cs in Fig. 18(b) would lie much further from the dashed line than the plotted Cs point. Further tests of TPP-2M are therefore required to assess its validity for relatively large values of b and bobs.  We now consider the choice of Nv for In and Sn which have similar energy-loss functions. The energy-loss function for Sn in Fig. 21(a) shows a strong volume-plasmon with an energy loss of 14 eV, close to the free-electron value of 14.3 eV with Nv = 4. Other features in Fig. 21(a) can be identified as being due to excitations of 4d and 4p electrons 2totMmaxED 9»effZmaxEDmaxED 141max =DE2totM2totM 50max >DE2totM 2totM 50max =DE2totM 50>DE2totM9=vN2totM2totM50max =DE 50max =DE74max =DE 2totM 50max =DE30max =DEAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   12 (with binding energies of about 24 and 85 eV, respectively).37 The plot of Zeff for Sn in Fig. 21(b) shows a step at about  eV, where Zeff = 1.82, and a monotonic increase for larger values of with a plateau in the vicinity of  eV where Zeff = 18.0; this plot is qualitatively different from the corresponding plot for Cs in Fig. 20(b). The plot of  for Sn in Fig. 21(c) shows a step at about  eV (where ) and an increase to a plateau at about 150 eV (where ). The asymptotic value of  is 5.58 (for  keV). These results suggest that the ten 4d electrons should be included in the count of Nv although Fig. 14 indicates that this would lead to larger deviations of the TPP-2M IMFPs from the values computed from optical data. While the arguments based on Fig. 21 indicate that Nv =14 would be a better choice than Nv = 4, the opposite conclusion is reached from Fig. 14. For In, plots of Zeff and  versus are similar to those for Sn in Figs. 21(b) and (c). The IMFPs computed from optical data for In in Fig. 13, however, are roughly midway between those obtained from TPP-2M with Nv = 3 and Nv = 13. Experimental IMFP measurements are therefore required to test whether the IMFPs for In and Sn derived from optical data are correct and/or whether the Nv values for these solids in Table 1 should be revised to include ten 4d electrons.  SUMMARY  We report results of IMFP calculations for 50 eV to 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 earlier work.1,2 The new IMFPs have also been used to test our TPP-2M equation5 for estimating IMFPs in additional classes of materials (alkali metals, allotropes of carbon, and rare-earth elements).  We found unexpectedly large RMS deviations between IMFPs calculated from TPP-2M and IMFPs derived from optical data for three solids: diamond (71.8 %), graphite (49.5 %), and Cs (39.3 %). We analyzed possible origins of these large RMS deviations, and concluded that they occurred for extreme values of the parameter b in TPP-2M, as calculated from Eq. (6a). When the calculated value of b is small (~ 0.01), as it is for diamond and graphite, the relative uncertainty of b can be large. When the calculated value of b is large (~ 0.25), as it is for Cs, TPP-2M does not provide reliable IMFPs. Fortunately, for current applications, there seem to be few materials for which such extreme values of b will be encountered. Unfortunately, the lack of materials with these extreme values of b makes it difficult for us to derive empirical modifications of Eq. (6a) for such situations.  We recently discussed the appropriate choice of the parameter Nv in TPP-2M.7 For most elements, we recommended that Nv be calculated in most cases from the number of valence electrons per atom in a given solid. For the rare-earth elements, however, we recommended that Nv be computed from the sum of the number of valence electrons in the solid state (either two or three) and the six 5p electrons that have binding energies between 18 eV and 27 eV. We provide a rationale here for this choice using Gd as an example. Due to a resonance excitation,38 the Gd 5p electrons contribute strongly to the energy-loss function 20max =DEmaxED 150max =DE2totM 20max =DE 80.12 =totM09.52 =totM 2totM30max =DE2totM maxEDAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   13 and to the IMFP. In contrast, the 4f electrons contribute weakly to the energy-loss function and IMFP. The same rationale indicates that the six 5p electrons of Cs should be included in the count for Nv; this choice, however, does not significantly reduce the RMS deviation between IMFPs from TPP-2M and those from optical data. We also consider inclusion of the ten 4d electrons in the count of Nv for In and Sn since these electrons also contribute strongly to the energy-loss function. Experimental IMFP measurements will be needed to test whether the Nv values for these elements should be revised to include ten 4d electrons.  Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   14 REFERENCES 1. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1988; 11: 577, Paper I in this series. 2. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1991; 17: 911, Paper II in this series. 3. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1991; 17: 929, Paper III in this series. 4. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1993; 20: 77, Paper IV in this series. 5. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1994; 21: 165, Paper V in this series. 6. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 1997; 25: 25, Paper VI in this series. 7. Tanuma S, Powell CJ, Penn DR. Surf. Interface Anal. 2003; 35: 268, Paper VII in this series. 8. Tanuma S, Powell CJ, Penn DR. J. Electron Spectrosc. Relat. Phenom.. 1993; 62: 95. 9. Bethe H. Ann. der Physik 1930; 5: 325. 10. Penn DR. Phys. Rev. B 1987; 35: 482. 11. Handbook of Optical Constants of Solids, ed. by E. D. Palik, Academic Press, New York (1985). 12. Handbook of Optical Constants of Solids II, ed. by E. D. Palik, Academic Press, New York (1991). 13. Handbook of Optical Constants of Solids III, ed. by E. D. Palik, Academic Press, New York (1998). 14. Brousseau-Lahaye B, Colliex C, Frandon J, Gasgnier M, Trebbia P. Phys. Stat. Sol. (b) 1975; 69: 257. 15. Marton L, Toots J. Phys. Rev. 1967; 160: 602. 16. Daniels, J. Opt. Commun. 1971; 3: 13. 17. Colliex C, Gasgnier M, Trebbia P. J. de Phys. 1976; 37: 397. 18. Henke BL, Davis JC, Gullikson EM, Perera RCC. Lawrence Berkeley Laboratory Report 26259 (1988); Henke BL, Gullikson EM, Davis JC. At. Data Nucl. Data Tables 1993; 54: 181. 19. Daniels J, von Festernberg C, Raether H, Zeppenfeld K. Springer Tracts in Modern Physics 1970; 54: 77. 20. Venghaus H. Phys. Stat. Sol. (b) 1975; 71: 609. 21. Hagemann H-J, Gudat W, Kunz C. Deutsches Elektronen-Synchrotron Report SR-74/7 (1974); Hagemann H-J, Gudat W, Kunz C. J. Opt. Soc. Am. 1975; 65: 742. 22. Powell CJ, Jablonski, A. J. Phys. Chem. Ref. Data 1999; 28: 19. 23. Inokuti M. Rev. Mod. Phys. 1971; 43: 297. 24. Ashley JC. J. Electron Spectrosc. Relat. Phenom. 1982; 28: 177. 25. Powell CJ. Surf. Interface Anal. 1987; 10: 349; Ultramicroscopy 1989; 28: 24. 26. Seah, MP, Gilmore IS, Spencer SJ. Surf. Interface Anal. 2001; 31: 778. 27. Tanuma S, Powell CJ, Penn DR (to be published). 28. Lesiak B, Jablsonki A, Prussak Z, Mrozek P. Surf. Science 1989; 223: 213. Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   15 29. Tanuma S, Goto K, Ichimura S (to be published). 30. Jablonski A, Salvat F, Powell CJ. J. Phys. Chem. Ref. Data 2004; 33: 409. 31. Jablonski A, Powell CJ. Surf. Science 2004; 551: 106. 32. Haas TW, Grant JT, Dooley, GJ. J. Appl. Phys. 1972; 43: 1853. 33. Steffen HJ, Roux CD, Marton D, Rabalais JW. Phys. Rev. B 1991; 44: 3981. 34. Ramaker DE, in Surface Analysis by Auger and X-Ray Photoelectron Spectroscopy, D. Briggs and J. T. Grants, eds. IM Publications: Chichester, 2003, p. 465. 35. Lindhard J, Scharff. K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 1953; 27: No. 15; Lindhard J, Scharff M, Schiott HE. K. Dan. Vidensk. Selsk. Mat.-Fys. Medd. 1963; 33: No. 14. 36. Netzer F, Matthew JAD. Rep. Prog. Phys. 1986; 49: 621. 37. Wagner CD, Naumkin AV, Kraut-Vass A, Allison JW, Powell CJ, Rumble JR. NIST X-ray Photoelectron Spectroscopy Database, SRD 20, Version 3.4; http://srdata.nist.gov/xps. 38. Netzer FP, Matthew JAD, in Handbook on the Physics and Chemistry of Rare Earths, K. A. Gschneidner, L. Eyring, and S. Hufner, eds. North-Holland: Amsterdam, 1987, Vol. 10, p.547. 39. Cardona M, Ley L (eds). Photoemission in Solids I. Springer-Verlag: Berlin, 1978. Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   16 Table 1. Values of the material parameters used in the IMFP calculations and the analysis of the IMFP results for the indicated materials. ___________________________________________________________________________ Material          M         r                Nv                  Ep         Eg               EF                (g cm-3)                (eV)                (eV)         (eV) ___________________________________________________________________________ Li     6.941 0.534 1   7.99 0   4.74 Be     9.0122 1.848 2 18.44 0 14.3 Diamond   12.011 3.515 4 31.16 5.5 20.4 Graphite   12.011 2.25 4 24.93 0 20.4 Glassy C   12.011 1.8 4 22.30 0 20.4 Na   22.9898 0.971 1   5.92 0   3.24 Al   26.9815 2.699 3 15.78 0 11.2 K   39.0983 0.862 1   4.28 0   2.12 Sc   44.9559 2.989 3 12.86 0   5.8 Ge   72.59 5.32 4 15.59 0.67 12.6 In 114.818 7.31 3 12.59 0   4.82 Sn 118.71 7.31 4 14.29 0   5.51 Cs 132.905 1.88 1   3.43 0   1.73 Gd 157.25 8.23 9 19.77 0   3.5 Tb 158.9253 8.25 9 19.69 0   4.0 Dy 162.5 8.78 9 20.08 0   3.5 _____________________________________________________________________________Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   17 Table 2. Sources of optical data and energy-loss-function measurements used in the IMFP calculations. ___________________________________________________________________________ Material Photon energy range (eV) Source of data ___________________________________________________________________________ Li   0.14 – 10.6  Ref. 12    12 – 28  Interpolation of energy-loss-function data  with aDEn equation    30 – 30,000  Ref. 18 Be   0.02 – 300  Ref. 12    300 – 30,000  Ref. 18    Diamond  5.5 – 45  Ref. 19    45 – 30,000  Ref. 18 Graphite  0.02 – 40  Ref. 20     50 – 30,000  Ref. 18 Glassy C  0.1 – 30,000  Ref. 21 Na   0.55 – 4  Ref. 12    4 – 9   Interpolation of extinction-coefficient data  with a linear equation    9 – 34   Ref. 12 (extinction-coefficient data)    4 – 34   Interpolation of refractive-index data using  cubic-spline function    35 – 30,000  Ref. 18 Table 2 (continued) Al   0.04 – 10,000  Ref. 11 Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   18    10,044 – 30,000 Ref. 18 K   0.55 – 10.33  Ref. 12    11 – 32  Interpolation of refractive-index data with a       linear equation    11 – 32  Ref. 12 (extinction-coefficient data)    35 – 30,000  Ref. 18 Sc   1.4 – 48.2  Ref. 14    49 – 30,000  Ref. 18 Ge   0.7 – 7   Ref. 11    7.66 – 25.2  Ref. 15    26 – 30  Interpolation with a quintic equation    30.7 – 30,000  Ref. 18  In   0.12 – 16.75  Ref. 13    17 – 28.5  Interpolation of refractive-index data with       a linear equation    17 – 28.5  Ref. 13 (extinction-coefficient data)    30 – 30,000  Ref. 18 Sn   0.1 – 21  Ref. 13     23 – 50  Interpolation of refractive-index data with       a quartic equation    23 – 50  Ref. 13 (extinction-coefficient data)   Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   19 Table 2 (continued)    50.9 – 30,000  Ref. 18  Cs   0.51 – 30  Ref. 13    31.5 – 30,000  Ref. 18 Gd   0.9 – 39.7  Ref. 16    50 – 30,000  Ref. 18 Tb   0.5 – 46.6  Ref. 17    50.9 – 30,000  Ref. 18 Dy   1.3 – 49.6  Ref. 16    52.3 – 30,000  Ref. 18 ___________________________________________________________________________  Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   20 Table 3. Errors in the f-sum [Eq. (1)] and the KK-sum [Eq. (2)] rules for the indicated materials with DEmax = 30 keV. For In, Sn, Cs, Gd, Tb, and Dy, the f-sum error was evaluated by comparing Zeff from Eq. (1) with Z – 1.65 instead of Z as described in the text. ___________________________________________________________________________ Material   Error in f-sum rule (%)  Error in KK-sum rule (%) ___________________________________________________________________________ Li 2.1 1.8 Be 2.4 6.6 Diamond -0.5 -0.2 Graphite 4.8 7.7 Glassy C -3.9 -0.2 Na 1.2 3.5 Al 1.0 1.8 K -6.8 -17.0 Sc 9.0 -7.5 Ge 2.0 4.2 In -0.2 -8.3 Sn 0.6 -23.1 Cs -7.5 0.4 Gd -0.1 2.6 Tb 2.4 6.6 Dy 3.2 -1.2 ______________________________________________________________________________ Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   21 Table 4. Inelastic mean free paths as a function of energy for 15 elemental solids. ____________________________________________________________________________________________________________     Electron       Inelastic mean free path (Å)      energy         (eV)       Li          Be  Diamond Graphite Na        Al         K          Sc         Ge        In         Sn        Cs        Gd        Tb        Dy ____________________________________________________________________________________________________________ 50 4.6 3.5 6.9 4.8 5.0 3.5 7.1 4.6 4.0 4.7 5.8 6.3 4.1 4.1 4.5 100 7.0 4.3 4.7 4.4 7.6 4.6 10.0 4.9 5.0 5.4 6.4 9.3 4.6 4.2 4.6 150 9.2 5.2 5.1 5.2 9.8 5.7 12.8 5.6 6.1 6.3 7.3 12.1 5.5 4.9 5.2 200 11.3 6.2 5.7 6.0 11.8 6.8 15.6 6.5 7.2 7.1 8.2 14.8 6.5 5.7 6.0 300 15.3 8.2 7.1 7.6 15.8 8.9 21.2 8.4 9.3 8.8 10.0 20.0 8.4 7.3 7.7 400 19.0 10.0 8.5 9.3 19.5 10.9 26.4 10.2 11.1 10.6 11.9 24.7 10.1 8.8 9.2 500 22.7 11.7 9.9 10.8 23.1 12.7 31.4 11.9 12.9 12.3 13.8 29.3 11.8 10.2 10.6 600 26.2 13.4 11.3 12.4 26.6 14.5 36.3 13.7 14.6 14.0 15.6 33.8 13.4 11.6 12.1 700 29.7 15.1 12.6 13.9 30.1 16.3 41.1 15.4 16.3 15.7 17.5 38.1 15.0 13.0 13.5 800 33.1 16.7 13.9 15.4 33.4 18.0 45.8 17.0 18.0 17.4 19.3 42.4 16.5 14.3 14.8 900 36.4 18.3 15.1 16.8 36.8 19.7 50.3 18.6 19.6 19.0 21.0 46.7 18.0 15.6 16.2 1000 39.7 19.9 16.4 18.2 40.0 21.3 54.8 20.2 21.2 20.6 22.8 50.8 19.5 16.9 17.5 1100 43.0 21.5 17.6 19.6 43.3 23.0 59.2 21.8 22.8 22.1 24.5 54.9 21.0 18.2 18.8 1200 46.2 23.0 18.8 21.0 46.5 24.6 63.6 23.3 24.4 23.7 26.2 59.0 22.5 19.5 20.1 1300 49.4 24.5 20.0 22.3 49.6 26.2 67.9 24.9 26.0 25.2 27.8 63.0 23.9 20.7 21.4 1400 52.6 26.1 21.2 23.7 52.7 27.8 72.2 26.4 27.5 26.7 29.5 67.0 25.4 22.0 22.7 Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   22 1500 55.7 27.6 22.4 25.0 55.8 29.4 76.4 27.9 29.0 28.1 31.1 71.0 26.8 23.2 23.9 1600 58.8 29.0 23.5 26.4 58.9 31.0 80.6 29.3 30.5 29.6 32.7 74.9 28.2 24.4 25.1 1700 61.9 30.5 24.7 27.7 62.0 32.6 84.8 30.8 32.0 31.0 34.3 78.7 29.6 25.6 26.4 1800 65.0 32.0 25.8 29.0 65.0 34.1 88.9 32.3 33.5 32.5 35.8 82.6 31.0 26.8 27.6 1900 68.0 33.4 27.0 30.3 68.0 35.6 93.0 33.7 35.0 33.9 37.4 86.4 32.3 28.0 28.8 2000 71.0 34.9 28.1 31.6 71.0 37.2 97.1 35.1 36.5 35.3 38.9 90.2 33.7 29.2 30.0 _________________________________________________________________________________________________________ Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   23 Table 5. Values of the parameters b, g, C, and D found in the fits of Eq. (4) to the IMFP data for each material together with values of bopt calculated from Eq. (5). ______________________________________________________________________________ Material          bopt           b  g   C  D     (eV-1 Å-1)     (eV-1 Å-1)          (eV-1)            (Å-1)         (eV Å-1) ______________________________________________________________________________ Li 0.0646 0.0670 0.370 2.43 58.8 Be 0.0279 0.0305 0.130 1.07 14.8 Diamond 0.0123 0.0138 0.104 1.30 26.4 Graphite 0.0190 0.0188 0.117 1.19 18.2 Glassy C 0.0147 0.0159 0.117 0.91 18.4 Na 0.1236 0.1306 0.243 5.95 202.6 Al 0.0367 0.0422 0.086 0.46 13.9 K 0.1726 0.1746 0.316 5.31 9.1 Sc 0.0604 0.0569 0.217 6.55 152.1 Ge 0.0409 0.0484 0.054 0.17 17.8 In 0.0672 0.0725 0.073 4.12 140.1 Sn 0.0485 0.0544 0.054 2.43 93.8 Cs 0.2960 0.3177 0.196 3.33 43.0 Gd 0.0279 0.0303 0.078 0.85 16.9 Tb 0.0327 0.0345 0.087 1.85 43.0 Dy 0.0311 0.0330 0.077 1.79 47.2 ______________________________________________________________________________Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   24 Table 6. Root-mean-square (RMS) deviations between IMFPs from TPP-2M [Eqs. (4) and (6) and the values of Nv in Table 1] and IMFPs calculated from optical data for the indicated materials and energies between 50 eV and 2,000 eV. ______________________________________________________________________________ Material       RMS deviation (%) ______________________________________________________________________________ Li 13.8 Be 25.3 Diamond 71.8 Graphite 49.5 Glassy C 1.7 Na 4.9 Al 12.7 K 2.3 Sc 23.6 Ge 6.3 In 19.5 Sn 6.5 Cs 39.3 Gd 6.5 Tb 13.5 Dy 6.5 ______________________________________________________________________________Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   25 Table 7. Values of  (for  keV) and  for the three allotropes of carbon.  ______________________________________________________________________________ Material                  (cm3 g-1) ______________________________________________________________________________ diamond    1.552    0.183 graphite    2.038    0.218 glassy carbon   1.690    0.329 ______________________________________________________________________________ 2totM 30max =DE 12 )( -totMr2totM 12 )( -totMrAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   26 Figure Captions Fig. 1 Plots of the energy loss function, , as a function of energy loss, DE, for diamond and graphite (from the optical data shown in Table 2). Fig. 2. Plots of  [from Eq. (3)] for diamond and graphite as a function of . Fig. 3. Plots of Zeff [from Eq. (1)] for diamond and graphite as a function of . Fig. 4. Plots of Peff [from Eq. (2)] for diamond and graphite as a function of . Fig. 5. Inelastic mean free paths (solid circles) calculated for lithium as a function of electron energy. IMFP values are shown for the 10 eV to 40 eV range to illustrate trends but these results are not considered to be reliable (see text). The solid line is a fit to the modified Bethe equation [Eq. (4)]; values of the parameters found in the fit are shown in Table 5. The dashed line shows IMFPs calculated from the TPP-2M predictive formula [Eqs. (4) and (6)] using the parameters listed in Table 1. Fig. 6. IMFP results for beryllium as a function of electron energy; see caption to Fig. 5. Fig. 7. IMFP results for diamond as a function of electron energy; see caption to Fig. 5. Fig. 8. IMFP results for graphite as a function of electron energy; see caption to Fig. 5. Measurements of IMFPs by elastic-peak electron spectroscopy from Lesiak et al. (ref. 28) and Tanuma et al. (ref. 29) are shown as open circles and solid diamonds, respectively. Fig. 9. IMFP results for glassy carbon (from Ref. 2) as a function of electron energy; see caption to Fig. 5. Measurements of IMFPs by elastic-peak electron spectroscopy from Lesiak et al. (ref. 28) are shown as open circles. Fig. 10. IMFP results for sodium as a function of electron energy; see caption to Fig. 5. )](/1Im[ ED- e2totM maxEDmaxEDmaxEDAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   27 Fig. 11. IMFP results for aluminum as a function of electron energy; see caption to Fig. 5. Fig. 12. IMFP results for germanium as a function of electron energy; see caption to Fig. 5. Fig. 13. IMFP results for indium as a function of electron energy. The long-dashed and short-dashed lines show IMFPs calculated from the TPP-2M equation [Eqs. (4) and (6)] with Nv = 3 and Nv = 13, respectively. See also caption to Fig. 5. Fig. 14. IMFP results for tin as a function of electron energy. The long-dashed and short-dashed lines show IMFPs calculated from the TPP-2M equation [Eqs. (4) and (6)] with Nv = 4 and Nv = 14, respectively. See also caption to Fig. 5. Fig. 15. IMFP results for cesium as a function of electron energy. The long-dashed, dot-dashed, and short-dashed lines show IMFPs calculated from the TPP-2M equation [Eqs. (4) and (6)] with Nv = 1, Nv = 7, and Nv = 9, respectively. See also caption to Fig. 5. Fig. 16. The solid circles show values of b, bobs, found from fits of Eq. (4) to the calculated IMFPs for groups of elements, inorganic compounds, and organic compounds as a function of . The long-dashed and short-dashed lines are plots of Eq. 6(a) with  g/cm3 and  g/cm3, respectively. Fig. 17. Values of  (solid circles) as a function of density. The solid line is a plot of , the variation expected from Eq. 6(a). Fig. 18. Values of  b from Eq. 6(a) (solid circles) versus bobs. The dashed line is a plot of . Fig. 19. Plot of (a) the energy loss spectrum of Gd versus energy loss DE for energy losses up to 200 eV, (b) Zeff versus  for Gd from Eq. (1), and (c)  versus  for Gd from Eq. (3). 5.022 )( -+ gp EE1=r 10=r5.022 )(944.0 -+-= gpobsr EEbb1.0069.010.0 rb +-=robsbb =maxED 2totM maxEDAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   28 Fig. 20. Plot of (a) the energy loss spectrum of Cs versus energy loss DE for energy losses up to 200 eV, (b) Zeff versus  for Cs from Eq. (1), and (c)  versus  for Cs from Eq. (3). Fig. 21. Plot of (a) the energy loss spectrum of Sn versus energy loss DE for energy losses up to 200 eV, (b) Zeff versus  for Sn from Eq. (1), and (c)  versus  for Sn from Eq. (3). maxED 2totM maxEDmaxED 2totM maxEDAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   29                             Fig. 1.   10-1110-910-710-510-310-110110-1 100 101 102 103 104 105DiamondGraphiteGlassy CEnergy Loss FunctionDE (eV)Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   30                 Fig. 2.   -10123456710-2 10-1 100 101 102 103 104 105DiamondGraphiteGlassy CZ effDEmax (eV)Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   31                     Fig. 3.  -0.200.20.40.60.811.210-1 100 101 102 103 104 105DiamondGraphiteGlassy CP effDEmax (eV)Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   32                  Fig. 4.    00.511.522.510-1 100 101 102 103 104 105DiamondGraphiteGlassy CMtotDEmax (eV)2Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   33           Fig. 5.   010203040506070800 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oLiAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   34           Fig. 6.  010203040500 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oBeAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   35           Fig. 7.   010203040500 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oDiamondAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   36                 Fig. 8.  010203040500 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MLesiak et al.Tanuma et al.Inelastic Mean Free Path (A)Electron Energy (eV)GraphiteoAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   37            Fig. 9. 01020304050600 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MLesiak et al.Inelastic Mean Free Path (A)Electron Energy (eV)oGlassy CAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   38            Fig. 10.  010203040506070800 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oNaAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   39            Fig. 11.  010203040500 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oAlAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   40            Fig. 12.  05101520253035400 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2MInelastic Mean Free Path (A)Electron Energy (eV)oGeAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   41            Fig. 13.   010203040500 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2M (N  = 3)TPP-2M (N  = 13)Inelastic Mean Free Path (A)Electron Energy (eV)oInvvAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   42            Fig. 14.   05101520253035400 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2M (N  = 4)TPP-2M (N  = 14)Inelastic Mean Free Path (A)Electron Energy (eV)oSnvvAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   43            Fig. 15. 0204060801001201400 500 1000 1500 2000Calc. IMFPsMod. Bethe Eq. FitTPP-2M (N  = 1)TPP-2M (N  = 7)TPP-2M (N  = 9)Inelastic Mean Free Path (A)Electron Energy (eV)oCsvvvAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   44                  Fig. 16.  00.050.10.150.20.250.30.350 0.05 0.1 0.15 0.2 0.25 0.3b obs(Ep + Eg)-0.52 2LiNaKCsAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   45                   Fig. 17.   -0.06-0.04-0.0200.020.040.060 5 10 15 20 25b rDensity (g cm-3)CsKLiAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   46                  Fig. 18(a).  00.010.020.030.040.050.060.070.080 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08b  (eV-1 A-1)bobs (eV-1 A-1)oodiamondgraphitePdAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   47                 Fig. 18(b).   00.050.10.150.20.250 0.05 0.1 0.15 0.2 0.25 0.3 0.35b  (eV-1 A-1)bobs (eV-1 A-1)ooCsKNaLiAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   48                 Fig. 19(a).  00.20.40.60.811.21.40 50 100 150 200Energy Loss FunctionDE (eV)Gd4d4fV, 5pAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   49             Fig. 19(b).  051015200 50 100 150 200Z effDEmax (eV)GdAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   50              Fig. 19(c).   0123450 50 100 150 200MtotDEmax (eV)Gd2Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   51              Fig. 20(a). 00.511.522.530 50 100 150 200Energy Loss FunctionEnergy Loss (eV)Cs4dV5pAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   52              Fig. 20(b). 051015200 50 100 150 200Z effDEmax (eV)CsAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   53               Fig. 20(c).  024680 50 100 150 200MtotDEmax (eV)Cs2Author Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   54              Fig. 21(a).  00.511.522.530 50 100 150 200Energy Loss FunctionEnergy Loss (eV)Sn4dV4pAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   55               Fig. 21(b).  051015200 50 100 150 200Z effDEmax (eV)SnAuthor Manuscript: Published in final edited form as: Surf. Interface Anal. Volume 37, Issue 1, Pages 1 - 14 (2005). https://doi.org/10.1002/sia.1997   56               Fig. 21(c).   01234560 50 100 150 200MtotDEmax (eV)Sn2