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Jiamin Gong, Lihao Yang, [Hideki Yoshikawa](https://orcid.org/0000-0002-7389-8865), [Shigeo Tanuma](https://orcid.org/0000-0003-2628-9941), [Bo Da](https://orcid.org/0000-0002-0785-8662), Chuanhong Jin, Zejun Ding

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[Determining electron inelastic mean free paths by iterative Monte Carlo analysis of the backscattered electron spectrum](https://mdr.nims.go.jp/datasets/c4668c13-ed1c-477d-b24a-984fa40ccde3)

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Determining electron inelastic mean free paths by iterative Monte Carlo analysis of the backscattered electron spectrumViewOnlineExportCitationRESEARCH ARTICLE |  NOVEMBER 17 2025Determining electron inelastic mean free paths by iterativeMonte Carlo analysis of the backscattered electron spectrumJiamin Gong  ; Lihao Yang  ; Hideki Yoshikawa; Shigeo Tanuma  ; Bo Da   ; Chuanhong Jin  ;Zejun Ding  J. Appl. Phys. 138, 195301 (2025)https://doi.org/10.1063/5.0302645Articles You May Be Interested InEvaluation of dielectric function models for calculation of electron inelastic mean free pathJ. Appl. Phys. (May 2022)Electron inelastic scattering and secondary electron emission calculated without the single poleapproximationJ. Appl. Phys. (December 2008)Optical Constants and Inelastic Electron-Scattering Data for 17 Elemental MetalsJ. Phys. Chem. Ref. Data (December 2009) 12 December 2025 05:02:49https://pubs.aip.org/aip/jap/article/138/19/195301/3372726/Determining-electron-inelastic-mean-free-paths-byhttps://pubs.aip.org/aip/jap/article/138/19/195301/3372726/Determining-electron-inelastic-mean-free-paths-by?pdfCoverIconEvent=citejavascript:;https://orcid.org/0009-0002-4394-8822javascript:;https://orcid.org/0000-0002-5948-7736javascript:;javascript:;https://orcid.org/0000-0003-2628-9941javascript:;https://orcid.org/0000-0002-0785-8662javascript:;https://orcid.org/0000-0001-8845-5664javascript:;https://orcid.org/0000-0001-5767-1145https://crossmark.crossref.org/dialog/?doi=10.1063/5.0302645&domain=pdf&date_stamp=2025-11-17https://doi.org/10.1063/5.0302645https://pubs.aip.org/aip/jap/article/131/17/175301/2836996/Evaluation-of-dielectric-function-models-forhttps://pubs.aip.org/aip/jap/article/104/11/114907/146249/Electron-inelastic-scattering-and-secondaryhttps://pubs.aip.org/aip/jpr/article/38/4/1013/383748/Optical-Constants-and-Inelastic-Electronhttps://servedbyadbutler.com/redirect.spark?MID=188841&plid=3384786&setID=1044475&channelID=0&CID=1606475&banID=524135939&PID=0&textadID=0&tc=1&rnd=2309570019&scheduleID=3549501&adSize=1640x440&data_keys=%7B%22%22%3A%22%22%7D&metadata=%5B%5D&mt=1765515769438215&spr=1&referrer=http%3A%2F%2Fpubs.aip.org%2Faip%2Fjap%2Farticle-pdf%2Fdoi%2F10.1063%2F5.0302645%2F20806495%2F195301_1_5.0302645.pdf&request_uuid=d634ab2b-8b4c-4a03-8214-ac19eea8f7ff&hc=fc76e7ad71ac3f22ae37d0edba49ef5a911a1040&location=Determining electron inelastic mean free paths byiterative Monte Carlo analysis of the backscatteredelectron spectrumCite as: J. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645View Online Export Citation CrossMarkSubmitted: 16 September 2025 · Accepted: 27 October 2025 ·Published Online: 17 November 2025Jiamin Gong,1 Lihao Yang,2,3 Hideki Yoshikawa,4 Shigeo Tanuma,5 Bo Da,4,a) Chuanhong Jin,1and Zejun Ding2,3,a)AFFILIATIONS1State Key Laboratory of Silicon and Advanced Semiconductor Materials, School of Materials Science and Engineering,Zhejiang University, Hangzhou, Zhejiang 310027, People’s Republic of China2Hefei National Research Center for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei,Anhui 230026, People’s Republic of China3Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China4Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japan5Materials Data Platform Center, National Institute for Materials Science, Tsukuba, Ibaraki 305-0044, Japana)Authors to whom correspondence should be addressed: DA.Bo@nims.go.jp and zjding@ustc.edu.cnABSTRACTThis study presents a novel method for extracting inelastic mean free paths (IMFPs) of electrons in solids from the backscattered electronspectrum (BES) using iterative Monte Carlo simulations. In our approach, the IMFP is parameterized using the Tanuma–Powell–Penn(TPP-2M) formula and a classical trajectory Monte Carlo simulation is used to model electron transport processes and generate a theoreticalBES. Through an iterative process, we optimize the TPP parameters to achieve the best fit between the simulated and experimental spectraover a wide energy range. This method was applied to determine the IMFPs of 25 targets, such as C (graphite), Mg, Al (100), Al (111), Si,Ti, V, Cr, Fe, Co, Ni, Cu (100), Cu (110), Cu (111), Mo, Ru, Rh, Ag, Sn, W, Re, Ir, Pt, Au, and Bi, in the energy range from 200 to 4900 eV,using a comprehensive experimental BES data set. The results demonstrate that the IMFPs obtained with our BES method show an averageroot-mean-square deviation of approximately 15% from values calculated using the full-Penn algorithm and the standard TPP-2M formula.This approach not only provides a new technique for the IMFP measurement but also offers a novel perspective for utilizing the continuousbackground signal in surface electron spectroscopy, which is typically overlooked.© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/). https://doi.org/10.1063/5.0302645INTRODUCTIONThe electron inelastic mean free path (IMFP) is a fundamentalparameter that describes electron transport in solids and governs thesurface sensitivity of core electron spectroscopy techniques.1–3Accurate IMFP values are, therefore, critical for quantitative analysisin x-ray photoelectron spectroscopy (XPS) and Auger electron spec-troscopy (AES).4–8 Over the years, several experimental and theoreti-cal approaches have been developed to determine the IMFPs.A widely used experimental technique is elastic peak electronspectroscopy (EPES),4–11 which determines the IMFP by analyzingthe intensity of elastically backscattered electrons. The method relieson comparing the elastic peak intensities of a sample to those of a ref-erence material with a known IMFP value. Consequently, the accu-racy of the resulting IMFP is directly dependent on the reliability ofthe reference data. While it requires such careful calibration and refer-ence data, EPES provides a direct and robust experimental approachfor IMFP determination.Another common approach is the overlayer-film method,12,13where the attenuation of a substrate’s photoemission or Augersignal is measured as a function of the thickness of a depositedJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-1© Author(s) 2025 12 December 2025 05:02:49https://doi.org/10.1063/5.0302645https://doi.org/10.1063/5.0302645https://pubs.aip.org/action/showCitFormats?type=show&doi=10.1063/5.0302645http://crossmark.crossref.org/dialog/?doi=10.1063/5.0302645&domain=pdf&date_stamp=2025-11-17https://orcid.org/0009-0002-4394-8822https://orcid.org/0000-0002-5948-7736https://orcid.org/0000-0003-2628-9941https://orcid.org/0000-0002-0785-8662https://orcid.org/0000-0001-8845-5664https://orcid.org/0000-0001-5767-1145mailto:DA.Bo@nims.go.jpmailto:zjding@ustc.edu.cnhttps://creativecommons.org/licenses/by-nc-nd/4.0/https://creativecommons.org/licenses/by-nc-nd/4.0/https://doi.org/10.1063/5.0302645https://pubs.aip.org/aip/japthin film. While straightforward, this technique typically yields aneffective attenuation length (EAL) rather than the absolute IMFP,14which includes contributions from both inelastic and elastic scatter-ing; however, to extract the true value of the IMFP, careful correc-tion procedures are required.Furthermore, reflection electron energy loss spectroscopy(REELS)15–20 provides a powerful pathway to obtain the IMFP bydetermining the material’s energy loss function (ELF).21 The ELF,given by Im{�1/ε(ω)}, where ε(ω) is the optical dielectric function,governs the probability of inelastic scattering22–26 and, thus, deter-mines the IMFP.27,28 The recently developed reverse Monte Carlo(RMC) method allows a highly accurate evaluation of the ELF fromthe experimentally measured REELS spectrum.29–36 The REELS-RMCmethod, thus, provides an accurate approach to the IMFP determina-tion by connecting experimental energy loss data to the material’sdielectric response at the cost of extensive computation.IMFP determination based on secondary electrons has alsobeen proposed.37 Secondary electrons are low-energy electrons(typically < 50 eV) excited by the incident electrons from thevalence or conduction band in the material. Due to the low energynature of the secondary electrons, their escape depth is stronglydependent on the IMFP of the material. By analyzing the energydistribution and angle distribution of secondary electrons, theIMFP may be derived.While these methods utilize specific features of the electronenergy spectrum—such as the elastic peak, characteristic photo-emission lines, or discrete energy loss features—the broad, continu-ous background of backscattered electrons has remained largelyunexploited for IMFP determination. This background signal con-tains rich information about the cumulative effects of inelastic scat-tering events that an electron undergoes before escaping a solid.In this work, we present a novel approach that leverages thisoften-discarded BES to determine the IMFP. Our method combinesexperimental BES data (∼200/300 eV to E0−100 eV, where E0denotes the primary electron energy) with an iterative Monte Carlosimulation. We parameterize the IMFP using the Tanuma–Powell–Penn (TPP-2M) formula38 and iteratively adjust its parametersuntil the simulated BES achieves an optimal fit with the experimen-tal spectrum. A key innovation of our approach is the developmentof “artificial” ELFs. These artificial ELFs, each constructed to satisfythe fundamental physical sum rules, can be used for materialswhose optical data are not available for calculating their true ELFs.As we will demonstrate, the resulting IMFP is robust and showslimited dependence on the precise shape of the ELF. This robust-ness enables reliable IMFP measurements for a wide range of mate-rials, significantly expanding the applicability of experimentalIMFP determination.EXPERIMENTGoto has performed an extensive experimental measurementof the BES data, which are employed in this study, for a variety ofelemental solids with a cylindrical mirror analyzer (CMA) devel-oped in the 1990s.39,40 The electron gun, aligned coaxially with theCMA, ensures that electrons are incident perpendicular to thesample surface. The emission angle of the analyzer ranged between36:3� and 48:3�, with respect to the surface normal. The incidentelectron energy ranged from 400 to 5000 eV, with a beam currentof ∼1 μA. The spectra were collected in the EN(E) mode, with anenergy resolution of fe ¼ ΔEs/E ¼ 0:25% and an energy step of0.05 eV. Figure 1 shows the measured EN(E) spectra of cobalt atincident electron energies of 0.4, 1, 3, and 5 keV.THEORETICAL METHODSArtificial optical ELFIn this study, we have employed the up-to-date classical trajec-tory Monte Carlo (CTMC) code41 to simulate backscattered elec-tron spectra. Specifically, the electron trajectories are tracked bysampling their step lengths, energy losses, and angular deflectionsaccording to the respective probability distributions. Theelectron-atom elastic scattering was described by Mott’s crosssection,42 which determines the rate of elastic scattering andangular distribution of electrons in the scattering events. A dielec-tric functional approach is used to model electron inelastic scatter-ing,43 where the differential inverse inelastic mean free path(DIIMFP) governs the distribution of energy loss and momentumtransfer during the inelastic scattering events. The key to a dielec-tric functional model lies in obtaining the momentum-dependentELF, given by Im[�1/ε(q, ω)]. Here, the full-Penn algorithm (FPA)is used to extrapolate the optical ELF of the material into the q-ωplane.28 For materials whose optical ELFs are unknown, wepropose a method for constructing an artificial optical ELF as anapproximation: a Drude-type formula is used to describe the ELFof a material in the energy loss range of 0–100 eV,Im�1ε(ω; ωp, γ ¼ ωp)� �¼ Aωγω2pω2 � ω2p� �2þ ω2ω2p, (1)where A, ωp, and γ represent the amplitude, energy, and half-widthof a Drude oscillator, respectively, and we have taken γ ¼ ωp. ForFIG. 1. The energy spectra of backscattered electrons measured at differentprimary energies for Co.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-2© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japthe energy loss beyond 100 eV, the ELF is calculated using the atomicscattering factors from the Henke database44 (100 eV–30 keV) andthe EPDL97 database45 (30 keV–10MeV). Based on the two con-straints of the f-sum rule and the ps-sum rule, one can derive theunique value for the parameters A and ωp in Eq. (1) and thereby con-struct a Drude-type ELF. The f-sum rule and the ps-sum rule23,46 aregiven, respectively, asZeff jELF ¼2πΩ2pð10ωIm[�1/ε(ω)]dω ¼ Z, (2)Peff jELF ¼2πð101ωIm[�1/ε(ω)]dωþ Re[�1/ε(0)] ¼ 1, (3)where ħΩp ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4πna/mpand na is the atomic density and Z is theatomic number of the material.Figure 2 presents a comparison between the artificial ELFsand the optical ELFs obtained from the literature47 for the materialsAu, Bi, Al, and Si.In the Monte Carlo simulation, the IMFP of electrons, λin, isparameterized in the form of the modified Bethe equation (theTPP-2M formula) as follows:λin ¼ EE2p β ln(γE)� CEþ DE2� � , (4)where E is the kinetic energy of an electron (with respect to theFermi energy). Ep ¼ 28:8ffiffiffiffiffiffiffiffiffiffiffiffiffiffiNvρ/Mpis the plasmon energy of a free-electron metal, where Nv is the number of valence electrons peratom or molecule, ρ is the mass density of the bulk material, andM is the atomic mass. β, γ, C, and D are the parameters in theequation. Here, we only consider the BES for the electron energylarger than 200 eV so that the parameters C and D are inconse-quential as previously reported.48,49 Thus, we adopt only twoparameters β and γ to describe the IMFP as follows:λin ¼ EE2pβ ln(γE): (5)FIG. 2. The comparison of the optical ELFs (the black line) with the artificial ELFs (the red line) for (a) Au, (b) Bi, (c) Al, and (d) Si.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-3© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japExtracting IMFP from BESBased on the above analysis, the simulated current intensityIsim(E) can be obtained through a Monte Carlo simulation. Tocompare with the experimental BES data, it is necessary to convo-lute the simulated current intensity for the finite energy resolutionwith a Gaussian distribution function whose full width at halfmaximum (FWHM), ΔEe, is expressed as follows:6ΔEe ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiΔE2s þ ΔE2g þ ΔE2Rq, (6)where ΔEs is the energy resolution of the CMA, ΔEg represents theinherent broadening of the electrons emitted from the electrongun, and ΔER is the intrinsic recoil broadening, which is the broad-ening of the elastic peak resulting from the energy loss due to theelastic scattering of electrons with surface atoms. Since we employ arelatively high incident electron energy (400–5000 eV) and onlyconsider the spectrum in the energy range far away from the elasticpeak, the primary factor influencing ΔEe is ΔEs, whereas the broad-ening effects caused by the electron gun and intrinsic recoil areminimal. Therefore, we take ΔEe ¼ ΔEs ¼ feE, which leads to thefollowing equation:σ ¼ ΔEe2� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� ln 2p ¼ feE2� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2� ln 2p , (7)where σ is the standard deviation of the Gaussian function. Thesimulated spectrum Isim(E) is then convoluted with the Gaussianfunction.The procedure for determining the IMFP is described asfollows:(1) Calculate the elastic scattering cross section and the DIIMFP ofthe material.(2) Initialize arbitrary numerical values for the parameters βand γ.(3) Calculate the backscattered electron energy distribution Isim(Ei)by a Monte Carlo simulation.(4) Compute the potential energy U, defined as a measure of thedifference between the simulated spectrum Isim(Ei) and theexperimental spectrum Iexp(Ei),U ¼Xi�Isim(Ei)� Iexp(Ei)�2, (8)where the summation spans all measured data points in theenergy range, typically from 200 or 300 eV to E0 � 100 eV,with E0 being the incident electron energy.(5) Modify β and γ within a constrained range to generate anupdated simulated spectrum I0sim(Ei) and the correspondingpotential energy U 0, which is given byU 0 ¼XiI0sim(Ei)� Iexp(Ei)� �2: (9)(6) Evaluate the change in potential energy usingΔU ¼ U 0 � Umin, where Umin is the historical minimum. IfΔU , 0, update the optimal parameters to the modifiedparameters β and γ and set Umin ¼ U 0.(7) Repeat steps (5) and (6) until Umin converges below a prede-fined threshold or the maximum iteration count is reached.The optimal parameters β and γ, which minimize the differ-ence between the simulated and experimental spectra, are thensubstituted into Eq. (5) to determine the IMFP of the material.RESULTS AND DISCUSSIONWe have determined the IMFPs from BES for 25 targets[C (graphite), Mg, Al (100), Al (111), Si, Ti, V, Cr, Fe, Co, Ni, Cu(100), Cu (110), Cu (111), Mo, Ru, Rh, Ag, Sn, W, Re, Ir, Pt, Au,and Bi] using the DIIMFPs calculated from both optical ELFs fromthe literature47 and artificial ELFs. As illustrated in Fig. 3 for cobalt,the simulated energy spectrum obtained using the DIIMFP derivedfrom the optical ELF shows excellent agreement with the experi-mental spectrum. Similarly, an agreement is also achieved with theartificial ELF for other materials such as Fe, Mo, Ir, Pt, and so on,as shown in Fig. 4. Figure 5 presents the corresponding IMFPcurves extracted from energy spectra at the ten incident electronenergies ranging from 400 to 5000 eV, along with the averagedvalues obtained through curve fitting.The observed differences at Auger peaks [Figs. 3(c) and 3(d)]are expected, as our model does not account for Auger electrongeneration. This simplification is justified because Auger processesprimarily contribute to peak features rather than the continuousbackground spectrum of backscattered electrons and consequentlyhave negligible influence on the IMFP determination from spec-trum shape analysis.Our analysis focuses on the backscattered electron spectrumin the energy range of Emin to Emax, where Emin is 200 or 300 eVand Emax is E0 � 100 eV. All reported IMFPs then correspond tothis defined energy range. For the cobalt sample, ten independentelectron energy spectra were acquired at different incident energies,each producing a distinct IMFP curve (scattered data points inFig. 5). The final IMFP parameters were determined through a sys-tematic two-step analytical process: (1) averaging individual IMFPcurves across the 200–5000 eV range to obtain a composite curveand (2) a linear regression analysis based on Eq. (5). This fittingprocedure yielded the characteristic IMFP parameters through theslope β and intercept β ln γ of the Fano plot, where E[E2pλ]�1 isplotted against ln E. The experimental conditions and the derivedIMFP parameters are summarized in Table I.Similarly, the IMFPs for other materials are determined.Table II lists the IMFP parameters β and γ derived from the opticalELF data for the 25 elemental materials. The sources of theseoptical ELF data are shown in Table 2 of Ref. 47. The same analyti-cal procedure was applied to artificial ELFs, and the resulting IMFPparameters are given in Table III.Figure 6 presents a comparison of the IMFPs for Co, Ni, Cu(110), and W, calculated using different methods, along with theexperimental and theoretical IMFP data from the literature. Forclarity, the following nomenclature is adopted: Optical_IMFP:results derived from BES with the optical ELF; Artificial_IMFP:results derived from BES with the artificial ELF; FPA_IMFP:Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-4© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japIMFPs calculated using the FPA method;27 and TPP_IMFP: IMFPscalculated using the TPP-2M formula.47For cobalt [Fig. 6(a)], the Optical_IMFP and Artificial_IMFPexhibit excellent agreement with each other and closely match bothFPA_IMFP and TPP_IMFP. In the case of copper [Fig. 6(c)] andtungsten [Fig. 6(d)], Optical_IMFP and Artificial_IMFP are consis-tent but diverge from FPA_IMFP and TPP_IMFP. For nickel, ascan be observed from both Fig. 6(b) and Table IV, the resultsobtained by the BES method not only differ significantly fromFPA_IMFP and TPP_IMFP but also reveal considerable discrepan-cies between Optical_IMFP and Artificial_IMFP.Further comparison with IMFP data from theliterature5,7,8,50–63 indicates that notable variations exist amongmeasurements obtained by different researchers and/or methods.We found that in the EPES method, the choice of reference materi-als significantly influences the results. For instance, in the IMFPmeasurements of Co and Ni, it can be observed that the IMFPvalues obtained by Krawczyk et al.50 and Beilschmidt et al.54 usingdifferent reference materials present considerable differences. In thecase of Co, the Krawczyk_Pd (using Pd as the reference sample)IMFP values are higher than those obtained by the BES method,whereas the Krawczyk_Al (using Al as the reference sample) IMFPFIG. 3. Comparison between the simulated (red) and experimental (black) backscattered electron spectra for cobalt (Co) at primary electron energies of (a) 400 eV, (b)1000 eV, (c) 3000 eV, and (d) 5000 eV. The incident electron beam current was maintained at 1 μA for all measurements. The DIIMFP is derived from the optical ELF.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-5© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japvalues are lower than the BES results. The results of Werneret al.8,51 tend to converge with the present values at high energies.For Ni, the results of Nagatomi and Goto53 and Beilschmidt et al.54using Pt (Beilschmidt_Pt) and Au (Beilschmidt_Au) as referencesamples align closely with Optical_IMFP, whereas Beilschmidt_AlIMFP data exhibit close agreement with the Artificial_IMFPresults. In contrast, the data of Gergely et al.55 and Werner56 showbetter consistency with the TPP-2M47 and FPA27 predictions. ForCu, the IMFP measurements by Doliński et al.57,58 are in closeagreement with the BES results. However, another set of data pub-lished by Doliński et al.59 aligns more closely with Werner’s fittingcurve.51 Similarly, the measurements from Tanuma et al.60 showexcellent consistency with Werner’s fitting curve.51 The IMFP datapoints reported by Lesiak et al.61 nearly coincide with theFPA_IMFP27 values, while the results by Gergely et al.62 exhibitbetter agreement with the TPP_IMFP values.47 The measurementsby Tanuma et al.7 fall between the results obtained by the BESmethod and the FPA method.27 Notably, Jablonski’s measurementresult5 lies precisely at the intersection of the TPP_IMFP47 andWerner’s curve.51 For W, the results of Gergely et al.62 show betteragreement with FPA_IMFP and TPP_IMFP, while those of Werneret al.8,51 and Tanuma et al.7 exhibit closer consistency with the BESFIG. 4. Comparison between the simulated (red) and experimental (black) backscattered electron spectra for (a) Fe, (b) Mo, (c) Ir, and (d) Pt at the primary electronenergy of 1000 eV. The incident electron beam current was maintained at 1 μA for all measurements. The DIIMFPs are derived from the artificial ELFs.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-6© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japresults. Notably, Lesiak’s measured IMFP values63 are lower thanthe other data sets.Based on the above analysis, it can be observed that theIMFPs of materials obtained by different theoretical and experi-mental methods exhibit certain differences. Even with the sameEPES method, the selection of different reference samples leads tovariations in results. The diversity of measurement uncertaintysources—including measurement methods, instruments, andapproximations in theoretical models—makes it very difficult toassess the accuracy of IMFP results.Given that the most comprehensive IMFP data sets currentlyavailable are derived from FPA_IMFP and TPP_IMFP, we haveconducted a focused comparison between these reference data andthe IMFP values extracted using the BES method.A data deviation analysis of the IMFPs extracted for these 25targets was conducted by comparing the IMFPs obtained using ourmethod with those derived using the FPA method27 and theTPP-2M formula.47 The root-mean-square (RMS) deviationbetween the two sets of IMFP data, A and B, is defined as follows:RMSA,B ¼ 100�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1nXni¼1λin,A(Ei)� λin,B(Ei)λin,B(Ei)  2s, (10)where Ei represents the electron energy. In our comparison, theIMFP results obtained using the BES method are designated as setA and the IMFP data for comparison (either the FPA or theFIG. 5. IMFPs of Co derived using the BES method at ten primary electronenergies (scattered data points). The red solid line represents the averagedIMFP by fitting the ten individual results.TABLE I. Summary of the experimental conditions and the derived IMFP parametervalues for Co using the BES method.No. E0 (eV) Emin (eV) Emax (eV)β(eV−1 nm−1) γ (eV−1)1 400.25 200.00 300.25 0.112 09 0.062 102 600.30 200.00 500.30 0.118 72 0.062 103 800.10 200.00 700.10 0.127 39 0.062 104 1000.00 300.00 900.00 0.126 06 0.065 575 1199.80 300.00 1099.80 0.135 46 0.065 146 1499.60 300.00 1399.60 0.138 66 0.068 057 1998.80 300.00 1898.80 0.139 81 0.074 148 2996.50 300.00 2896.50 0.141 98 0.073 619 3993.50 300.00 3893.50 0.146 49 0.071 7710 4989.50 300.00 4889.50 0.145 74 0.072 16Mean … 200.00 4889.50 0.172 38 0.031 42TABLE II. List of the parameters β and γ in the TPP-2M formula for 25 elementalsolids derived using the BES method and optical ELFs.Elementβ(eV−1 nm−1)γ(eV−1) Elementβ(eV−1 nm−1)γ(eV−1)Ag 0.229 36 0.026 99 Mg 0.545 20 0.153 20Al (100) 0.412 57 0.067 20 Mo 0.369 51 0.072 11Al (111) 0.335 52 0.296 33 Ni 0.164 91 0.039 79Au 0.275 38 0.042 01 Pt 0.276 19 0.052 00Bi 0.711 88 0.033 39 Re 0.362 08 0.021 31C (graphite) 0.186 50 0.059 46 Rh 0.249 86 0.061 25Co 0.172 38 0.031 42 Ru 0.428 75 0.024 59Cr 0.264 14 0.065 68 Si 0.274 29 0.197 51Cu (100) 0.172 10 0.013 07 Sn 0.580 59 0.030 26Cu (110) 0.124 88 0.187 98 Ti 0.486 02 0.042 78Cu (111) 0.128 74 0.080 24 V 0.332 21 0.043 60Fe 0.241 91 0.023 59 W 0.478 12 0.025 67Ir 0.284 90 0.027 66TABLE III. List of the parameters β and γ in the TPP-2M formula for 25 elementalsolids derived using the BES method and artificial ELFs.Elementβ(eV−1 nm−1)γ(eV−1) Elementβ(eV−1 nm−1)γ(eV−1)Ag 0.273 09 0.025 74 Mg 0.591 53 0.129 70Al (100) 0.363 32 0.069 89 Mo 0.366 03 0.065 56Al (111) 0.289 24 0.360 82 Ni 0.204 98 0.065 61Au 0.264 64 0.042 76 Pt 0.277 49 0.043 06Bi 0.873 69 0.027 48 Re 0.344 31 0.020 61C (graphite) 0.187 66 0.063 50 Rh 0.240 64 0.057 39Co 0.176 02 0.028 87 Ru 0.408 30 0.022 33Cr 0.261 48 0.053 34 Si 0.255 77 0.232 33Cu (100) 0.183 82 0.012 00 Sn 0.757 16 0.024 06Cu (110) 0.138 54 0.129 39 Ti 0.476 30 0.041 74Cu (111) 0.145 05 0.054 82 V 0.357 45 0.036 51Fe 0.250 22 0.018 19 W 0.473 25 0.024 72Ir 0.275 19 0.025 10Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-7© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japTPP-2M data) are designated as set B. Table IV lists the RMSbetween the IMFPs obtained in this study and the correspondingvalues derived using the FPA method and the TPP-2M formulareported in the literature.27,47The results demonstrate that for most materials except Ni, theRMS differences between the obtained two sets of IMFPs and thosecalculated using either the FPA method or the TPP-2M formula arerelatively small. This indicates that the approach of extracting IMFPsfrom BES is basically practicable. The proposed scheme exhibitsstrong extensibility—by employing a universal approach based onthe artificial ELF, only the material’s chemical formula and densityare required for modeling electron inelastic scattering to derive theIMFP from BES. The artificial ELF has two key characteristics: (1) itsatisfies the sum rules and (2) its ELF shape may be quite differentfrom the realistic ELF. The first feature offers an advantage by ensur-ing that the modeling of energy loss in electron inelastic scattering ina Monte Carlo simulation does not lose too much accuracy due tothe second feature, and these inaccuracies would be smeared out inthe multiple inelastic scattering events of backscattered electrons. Forcertain materials (e.g., Au, Cr, Fe, Mg, Mo, Pt, Rh, Ru, and W), theartificial ELFs yield IMFPs that are in better agreement withFPA_IMFP and TPP_IMFP results, as they satisfy the sum rules.FIG. 6. Comparison of IMFPs for: (a) Co, (b) Ni, (c) Cu (110), and (d) W, calculated using different methods: BES with the optical ELF (red), BES with the artificial ELF(green), the FPA model (black),27 and the TPP-2 M formula (blue).47 The experimental and theoretical IMFP data are taken from the literature.5,7,8,27,47,50–63Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 138, 195301 (2025); doi: 10.1063/5.0302645 138, 195301-8© Author(s) 2025 12 December 2025 05:02:49https://pubs.aip.org/aip/japFrom the rightmost column of Table IV, it is evident that dif-ferent ELFs, or different energy loss distributions, do not signifi-cantly alter the final IMFP results. The bottom row of Table IVindicates that the average RMS value between Optical_IMFP andArtificial_IMFP is as low as 7.32%. It should be noted that thisdoes not imply the difference introduced by the artificial ELF is7.32%. For example, Sn shows a high RMS value of 17.30%between Optical_IMFP and Artificial_IMFP. However, their devia-tions from the reference standards are much lower: 8.65% and11.38% from FPA_IMFP and 10.01% and 12.25% from TPP_IMFP,respectively.It can be observed that the RMS differences between thecurrent study results and those using the FPA method and theTPP-2M formula are, on average, around 15%. Although the differ-ences reach approximately 30% for some materials [e.g., Ru, Cu(110), Cu (100), Mo, and Ni], the results obtained using the BESmethod are quite satisfactory overall, considering that each methodmay have inherent uncertainty and result in deviation from thetrue values.CONCLUSIONIn this work, we have demonstrated a novel method for deter-mining the electron IMFP by analyzing the backscattered electronspectrum (BES). This approach offers distinct advantages over con-ventional techniques. Specifically, it is less sensitive to surface con-ditions than methods based on secondary electrons and avoids thecomplex surface excitation corrections required in EPES analysis.While the resulting IMFP values do not necessarily represent animprovement in accuracy over established theoretical models, theyshow good agreement, with average differences of ∼15% from FPAand TPP-2M predictions for 25 different elemental solids. Thisconsistency validates the BES method as a powerful addition to theexisting analytical tools for IMFP determination. Crucially, ourwork transforms the continuous background in electron spectra, asignal that is typically discarded, into a valuable source for quanti-tative material analysis. The method is applicable for electron ener-gies above ∼200 eV, where the influence of secondary electrons isnegligible.ACKNOWLEDGMENTSB. Da acknowledges JSPS KAKENHI (No. JP21K14656) andthe Kurata Grants from the Hitachi Global Foundation and fromthe Iketani Science & Technology Foundation. The authors in ZJUacknowledge financial support by the National Key R&D Programof China under Grant No. 2022YFB4401602 and the 111 Projectunder Grant No. B16042. They thank Professor K. Goto for hisexperimental support and assistance. They also extend their grati-tude to Dr. H. M. Li and the supercomputing center of USTC forparallel computing support.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.Author ContributionsJiamin Gong: Data curation (equal); Formal analysis (equal);Investigation (equal); Methodology (equal); Validation (equal);Visualization (equal); Writing – original draft (equal). Lihao Yang:Data curation (equal); Formal analysis (equal); Investigation (equal);Methodology (equal); Software (equal). Hideki Yoshikawa:Resources (equal); Supervision (equal). Shigeo Tanuma:Conceptualization (equal); Resources (equal); Supervision (equal).Bo Da: Conceptualization (equal); Funding acquisition (equal);Project administration (equal); Resources (equal); Supervision(equal). Chuanhong Jin: Funding acquisition (equal); Projectadministration (equal); Resources (equal); Supervision (equal);Writing – review & editing (equal). Zejun Ding: Conceptualization(equal); Funding acquisition (equal); Methodology (equal); Projectadministration (equal); Resources (equal); Software (equal);Supervision (equal); Writing – review & editing (equal).DATA AVAILABILITYThe data that support the findings of this study are availablefrom the corresponding authors upon reasonable request.TABLE IV. RMS between the IMFPs derived in this work and those obtained usingthe FPA method27 and the TPP-2M formula.47ElementRMSa RMSbRMScFPA TPP-2M FPA TPP-2MAg 4.69 4.60 11.44 13.79 14.67Al (100) 7.56 2.69 20.91 9.82 12.45Al (111) 6.56 13.12 10.79 5.95 12.03Au 20.89 12.67 18.13 9.66 3.53Bi 2.82 11.02 12.22 22.79 13.36C (graphite) 15.59 21.52 12.92 23.32 2.33Co 1.51 4.38 2.25 4.67 1.06Cr 20.33 24.30 15.00 19.24 6.82Cu (100) 16.39 29.02 17.11 28.92 3.11Cu (110) 32.19 24.96 33.84 26.78 3.03Cu (111) 20.87 12.25 22.37 13.98 3.22Fe 16.70 14.62 13.17 12.09 7.56Ir 6.43 1.69 1.85 8.32 7.08Mg 17.40 7.06 11.99 2.15 4.59Mo 24.71 29.49 22.15 27.09 3.44Ni 14.58 15.64 39.98 40.74 29.84Pt 25.00 17.01 21.28 12.86 5.12Re 8.85 5.41 15.60 10.68 6.48Rh 18.35 23.42 13.79 19.14 5.63Ru 33.99 36.80 28.43 31.43 8.74Si 4.57 5.44 7.00 9.19 3.98Sn 8.65 10.01 11.38 12.25 17.30Ti 3.12 17.96 4.47 15.74 2.78V 7.26 14.58 9.34 16.48 2.55W 11.76 17.90 9.77 16.05 2.39Mean 14.03 15.10 15.49 16.53 7.32aset A = Optical_IMFP; set B = FPA_IMFP or TPP_IMFP.bset A = Artificial_IMFP; set B = FPA_IMFP or TPP_IMFP.cset A = Optical_IMFP; set B = Artificial_IMFP.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. 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