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Abrar Hussain, Lihao Yang, Shifeng Mao, [Bo Da](https://orcid.org/0000-0002-0785-8662), Károly Tőkési, Z.J. Ding

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[Determination of electron backscattering coefficient of beryllium by a high-precision Monte Carlo simulation](https://mdr.nims.go.jp/datasets/b263c318-8edf-4d4f-b269-a21702406828)

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Determination of electron backscattering coefficient of beryllium by a high-precision Monte Carlo simulationNuclear Materials and Energy 26 (2021) 100862Available online 8 December 20202352-1791/© 2020 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license(http://creativecommons.org/licenses/by-nc-nd/4.0/).Determination of electron backscattering coefficient of beryllium by a high-precision Monte Carlo simulation Abrar Hussain a,1, Lihao Yang a,1, Shifeng Mao b,*, Bo Da c, Károly Tőkési d,*, Z.J. Ding a,* a Hefei National Laboratory for Physical Sciences at Microscale and Department of Physics, University of Science and Technology of China, Hefei, Anhui 230026, PR China b Department of Engineering and Applied Physics, University of Science and Technology of China, Hefei, Anhui 230026, People’s Republic of China c Research and Services Division of Materials Data and Integrated System (MaDIS), National Institute for Materials Science, 1-1 Namiki Tsukuba, Ibaraki 305-0044, Japan d Institute for Nuclear Research, (ATOMKI), Debrecen, Hungary   A R T I C L E  I N F O   Keywords: Monte Carlo simulation Backscattering Beryllium Boron Carbon Contamination A B S T R A C T   We present an up-to-date Monte Carlo simulation of electron backscattering coefficient of beryllium, which is an important material in fusion reactor, at an impact energy range of electrons between 100 eV and 100 keV. The Mott’s cross section calculated with more accurate scattering potential and a relativistic dielectric functional formalism with full Penn algorithm and experimental optical data are used in the modelling of electron elastic and inelastic scatterings, respectively. This Monte Carlo simulation modelling enabled us to derive comprehensive theoretical values of backscattering coefficient, which are found significantly smaller than the previous published experimental data particularly at energies below 10 keV. Simulation of electron backscattering spectra was then carried out with and without Auger electron emission, which confirmed that Auger electrons have negligible contribution to backscattering coefficient. To validate our simulation results, we have also performed calculation for amorphous boron and carbon, while the comparison with the available experimental data shows reasonable agreement. Further simulation for carbon and water covered Be sample has revealed that the surface contamination in low vacuum conditions with several atomic/molecular layers can drastically alter the measurement values for beryllium at low energies. The low backscattering coefficient values of beryllium are partly attributed to the extremely strong forward elastic scattering for low atomic number element. Therefore, we recommend to use the here reported backscattering coefficient data of beryllium for applications.   1. Introduction Beryllium is selected as the first wall material in ITER [1] due to the following advantages. As a low Z material, it can be compatible with high performance core plasma without significant fuel dilution and radiation. It is non-reactive with hydrogen isotopes to avoid the tritium retention issue. Furthermore, it has good thermal conductivity and oxygen gettering ability [2]. Although beryllium has important application in the fusion reactor, there are very limited and old data available for studies of its interactions with charged particles, especially electrons, in the incident energy range of 0.1–100 keV. Only a few investigations exist for the determination of the electron backscattering coefficient [3]. The backscattering coefficient is a quantity to characterize the amount of backscattered electrons by bombarding a solid surface with an incident electron beam. When primary electrons penetrate a material they undergo elastic and inelastic collisions during their motion in matter before either losing all energies in the bulk or emitting from sample surface. During the inelastic collision processes secondary electrons can be generated. Empirically one separates the emitted electrons according to their kinetic energies and defines them as secondary electrons if their energies are below 50 eV and as backscattered electrons otherwise. Denoting the backscattering coefficient by η (the number of backscattered electrons of energies greater than 50 eV per an incident electron) and the secondary electrons yield by δ, the total electron yield is hence the sum of them. During the last decades many experiments have been performed to explore the general tendency of dependence of backscattering coefficient on atomic number of the target and incident electron energy. It was observed that the backscattering coefficient increases with atomic * Corresponding authors. E-mail addresses: sfmao@ustc.edu.cn (S. Mao), tokesi@atomki.hu (K. Tőkési), zjding@ustc.edu.cn (Z.J. Ding).   1 A. Hussain and L.H. Yang contributed equally to the work. Contents lists available at ScienceDirect Nuclear Materials and Energy journal homepage: www.elsevier.com/locate/nme https://doi.org/10.1016/j.nme.2020.100862 Received 3 June 2020; Received in revised form 21 November 2020; Accepted 25 November 2020   mailto:sfmao@ustc.edu.cnmailto:tokesi@atomki.humailto:zjding@ustc.edu.cnwww.sciencedirect.com/science/journal/23521791https://www.elsevier.com/locate/nmehttps://doi.org/10.1016/j.nme.2020.100862https://doi.org/10.1016/j.nme.2020.100862https://doi.org/10.1016/j.nme.2020.100862http://crossmark.crossref.org/dialog/?doi=10.1016/j.nme.2020.100862&domain=pdfhttp://creativecommons.org/licenses/by-nc-nd/4.0/Nuclear Materials and Energy 26 (2021) 1008622number at high energies above 10 keV [4–6]. The general behaviour of the primary energy dependence of backscattering coefficient was also found as [7–9]: a) for low atomic number solids (Z < 20) it decreases with increasing of incident energy; b) for high-Z elements (Z > 26) it increases with incident energy; c) it is almost constant for intermediate value Z-elements [10]. It was also observed that, at low primary energies η is surface sensitive [11]. Joy has compiled these historical measurement data into a database for many elemental solids and compounds without including uncertainty and judging the reliability [12]. For Be sample, Martin et al. had performed experimental investigations of the energy and angular distribution of backscattered electrons at high energies, in the incident energy range between 43.5 keV and 124 keV [13]. Reimer and Tollkamp had measured η as a function of primary energy for a normal incident beam in a scanning electron microscope. They have found that η increases with decreasing primary energy particularly below 5 keV [14]. Bronshtein and Fraiman had performed backscattering coefficient measurements up to 4 keV primary energy [15]. Shimizu had measured the backscattering coefficients with a spherical symmetric electron energy analyser in an ultrahigh vacuum (UHV) system, and compared experimental data with his Monte Carlo calculations [16]. These measurement data are scattered in a rather large range, from 57% difference at 1 keV down to 40% at ~ 10 keV. The only available theoretical data were provided by Shimizu who used a Monte Carlo simulation technique to calculate backscattering coefficient as a function of the incident electron energy above 5 keV [16]. He used the stopping power formula of Spencer and Fano [17] in the continuous slowing down approximation for the calculation of electron energy loss and Rutherford scattering formula for describing electron elastic cross section [16], but the approximations of this theoretical framework are now known to be poor, particularly at low electron energies [6,18]. Monte Carlo method is a powerful theoretical tool widely used for study of particle interaction with matter [18–20]. In recent decades we have continuously improved Monte Carlo modeling of electron beam interaction with solids and surfaces by employing modern knowledge and theories of electron scattering in matter. The systematically developed Monte Carlo simulation models and methods by Ding’s group (DingMC) in different approaches for applications to electron spectroscopies and electron microscopies are comprised of two categories: 1. Classical trajectory Monte Carlo (CTMC) simulations, where the simulation of electron elastic and inelastic scattering in amorphous-like surfaces/solids/ thin-films are treated by conventional Monte Carlo sampling techniques from respective scattering cross sections, including (a) CTMC-SEM, simulation of secondary electrons and backscattered electrons emitted from bulk solids as signals in scanning electron microscopy and background in Auger electron spectroscopy [21–38]; (b) CTMC-3DSEM, simulation for complex 3D sample geometries particularly for critical dimension scanning electron microscopic imaging [39–49]; [c] CTMC- EPMA, simulation of continuous and characteristic X-ray signals in electron probe microanalysis [50–52]; (d) CTMC-SES, simulation of Auger electron and/or X-ray photoelectron signals in surface electron spectroscopies [53–56]; (e) CTMC-REELS, simulation of electron elastic peak spectroscopy and reflection electron energy loss spectroscopic spectrum from surfaces [57–66]; (f) CTMC-RMC, a reverse Monte Carlo method for deriving optical constants of solids from reflection electron energy loss spectroscopy spectra [67–73]; (g) CTMC-CHARG, simulation of specimen charging phenomena in insulators and semiconductors [74–78]; (h) CTMC-ATOMIC, simulation for atomic thin layers with substrate [63,66] or without substrate, like graphene, particularly for deriving electron inelastic scattering mean free path [79–81]; 2. Quantum trajectory Monte Carlo (QTMC) simulation for crystalline materials, which combines the simulation of Bohmian quantum trajectories for electron elastic scattering/diffraction [82–84] with the Monte Carlo sampling of electron inelastic scattering: QTMC-ARSEI, for atomic resolution secondary electron imaging in scanning transmission electron microscopy [85–87]. In this work we employ an up-to-date simulation model of CTMC-SEM to study electron backscattering coefficient. This model configuration uses the best available scattering potential for the calculation of relativistic elastic scattering cross section; in addition, the relativistic form of dielectric functional formulation of electron inelastic scattering cross section in full Penn algorithm is adopted. The backscattering energy spectrum and backscattering coefficient from beryllium sample are investigated theoretically at electron impact energies between 0.1 and 100 keV with and without the Auger electron emission taken into account. Our simulations consider both the individual elastic and inelastic scattering of primary electrons and include the high energy cascade secondary electron production during electron transport inside the bulk beryllium. The primary purpose of this work was to try to reduce the uncertainty range of the experimental data distribution by a high accuracy Monte Carlo simulation, and especially predict the reasonable values at low primary energies below 1 keV where the experimental data are rare. However, the simulation has led to a new finding of extremely low backscattering coefficient values, particularly below 10 keV, as compared with the previous experimental data and also with that of the closest elemental solids, boron and carbon. While the comparisons with experiments in cases of boron and carbon are very reasonable, there is a quite large difference between the present calculation and experimental data. The present calculation predicts backscattering coefficient values by the lower limit of experimental data range at very high primary energies, and even becomes much smaller than experimental data with decreasing primary energy. By considering the surface contamination very likely to be presented in early experiments the difference between experimental observation and theoretical calculation can be explained. The reason for such very low backscattering coefficient values for beryllium has been analysed to be attributed to the especially strong forward elastic scattering of beryllium atoms, and electron scattering mean free paths. 2. Monte Carlo model In our simulation both the elastic and inelastic scatterings of primary electrons and also the secondary electron production during transport of the electrons inside beryllium are taken into account. Details of the calculation procedure are described elsewhere [27,32], here the outline of our calculation procedure is briefly introduced while we mainly emphasis on the new feature of this up-to-date CTMC-SEM model. Elastic cross section For the elastic scattering of electrons, we used the Mott’s differential cross section [88], dσedΩ= |f (θ) |2 + |g(θ) |2 (1)  where f(θ) and g(θ) are the scattering amplitudes and can be calculated with the partial wave expansion method by solving Dirac equation, f (θ) =12ik∑∞l=0{(l + 1)(e2iδ+ℓ − 1)+ l(e2iδ−ℓ − 1) }Pℓ(cosθ) (2)  g(θ) =12ik∑∞ℓ=1{− e2iδ+ℓ + e2iδ−ℓ}P1ℓ(cosθ) (3)  where δ+l and δ−l are spin up and spin down phase shifts of the ℓth partial wave, respectively; Pℓ(cosθ) and P1ℓ(cosθ) are the Legendre and the first order associated Legendre functions, respectively. The scattering potential used in the present model is expressed as, V(r) = Vst(r)+Vex(r)+Vcp(r) (4)  where Vst is the electrostatic potential, Vex is the electron exchange potential and Vcp is the correlation-polarization potential. The electrostatic potential for the interaction between an electron and the target atom is Vst(r) = − e[φn(r) + φe(r) ] (5)  where φn and φe are respectively the components of nucleus and the A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086230.1 1 10 100012345elastic cross section (10-15  cm2 /sr)scattering angle (deg.) 0.1 keV 0.5 keV    1 keV    5 keV  10 keVBe(a)0.1 1 10 1000123elastic cross section (10-15  cm2 /sr)scattering angle (deg.) 0.1 keV 0.5 keV    1 keV    5 keV  10 keVB(b)0.1 1 10 1000.00.40.81.21.6elastic cross section (10-15 cm2 /sr)scattering angle (deg.) 0.1 keV 0.5 keV    1 keV    5 keV  10 keV(c)C0.1 1 10 100012345100 eV Be B Celastic cross section (10-15 cm2 /sr)scattering angle (deg.)(d)0.1 1 10 10001234500 eV Be B Celastic cross section (10-15 cm2 /sr)scattering angle (deg.)(e)0.1 1 10 1000.00.51.01.52.02.55 keV Be B Celastic cross section (10- 15 cm2 /sr)scattering angle (deg.)(f)Fig. 1. Electron differential elastic scattering cross section as a function of scattering angle at different energies for: (a) beryllium, (b) boron and (c) carbon. Comparison for all the three elements at the energies of: (d) 0.1 keV, (e) at 0.5 keV and (f) 5 keV. A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 1008624electron cloud of electrostatic potential, φn(r) = e(1r∫ r0ρn(r′)4πr′2dr′+∫ ∞rρn(r′)4πr′ dr′)(6)  φe(r) = − e(1r∫ r0ρe(r′)4πr′2dr′+∫ ∞rρe(r′)4πr′dr′)(7)  where ρn is the nuclear charge distribution and ρe is the electron charge distribution of the target atom. A Fermi distribution is used for the determination of the nuclear charge distribution [89], ρn,F(r) =ρ0exp[(r − Rn)/z ] + 1(8)  where Rn = 1.07 × 10− 13A1/3 (cm) and z = 5.46× 10− 14 (cm), A is the atomic mass (amu) of the element. The constant ρ0, which is twice the proton density at r = Rn, is to be determined by normalization. The most accurate electron densities available for free atoms are obtained from self-consistent relativistic Dirac-Fock (DF) calculations. The numerical DF densities calculated by the multiconfiguration DF program of Desclaux [90] are used here for determination of the electron charge distribution. The Furness-McCarthy exchange potential [91] is used for the electron exchange potential: Vex,FM(r) =12[E − Vst(r) ] −12{[E − Vst(r) ]2 + 4πa0e4ρe(r)}1/2(9) When the projectile is far from the atom, the polarization potential energy can be approximated by means of the Buckingham potential, Vcp,B(r) = −αde22(r2 + d2)2 (10)  where αd is dipole polarizability of the target atom and d is a phenomenological cut-off parameter that serves to prevent the polarization potential from diverging at r = 0. The experimental values of the atomic dipole polarizabilities from Ref. [92] are usually used in Eq. (10). Mittleman and Watson [93] proposed an expression for the cut-off parameter, d4 = 12αda0Z− 1/3b2pol, where Z is the atomic number and bpol is an adjustable energy-dependent parameter, b2pol = max{(E − 50 eV)/(16 eV),1 }. Perdew and Zunger [94] proposed a parameterization of the correlation potential: Vcp(r) =⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−e2a0(0.0311lnrs − 0.0584 + 0.00133rslnrs − 0.0084rs), rs < 1;−e2a00.1423 + 0.1748r1/2s + 0.0633rs(1 + 1.0529r1/2s + 0.3334rs)2 , rs⩾1.(11)  where rs =1a0[34πρe(r)]1/3(12)  is the radius of the sphere that contains (on average) one electron of the gas, in units of the Bohr radius a0. A more accurate correlation- polarization potential combines the long-range polarization potential (Eq. (10)) with the correlation potential Vcp(r) obtained from the local- density approximation (LDA) which is expressed as [95], Vcp,LDA(r) ≡{max{Vco(r),Vcp,B(r)}, r < rcp;Vcp,B(r), r⩾rcp.(13)  where rcp is the outer radius at which Vcp(r) and Vcp,B(r) cross. All the calculation of elastic cross section are performed by the Fortran 77 code, ELSEPA [96]. Fig. 1 shows the primary electron energy and scattering angle dependent elastic cross sections of Be. Generally, with increasing incident energy the elastic cross sections are decreasing. The highest elastic cross sections are in forwards angles. Then the elastic cross sections decrease rapidly with the increasing scattering angle. We note that at low primary energies, the elastic cross sections increase above 900, showing the importance of the electron backscattering at low impact energies. Fig. 1(d) also shows the variation of electron elastic cross section with scattering angles and material nature. It is observed that elastic cross section for lower atomic number element Be is higher than the higher atomic number elements, B and C. These variations strongly affect the results of backscattering coefficients. The integration of the differential cross section over total solid angles gives us the total elastic cross sections as, σe =∫dσedΩdΩ = 2π∫ π0sinθ{|f (θ) |2 + |g(θ) |2}dθ (14) Fig. 2(a) shows the total elastic cross sections as a function of energy for the simulated materials i.e. Be, B and C. The total elastic cross section 101 102 103 104 1050123101 102 103 104 10510-310-210-1100(a)total elastic cross section (10-15 cm2 )energy (eV) Be B C101 102 103 104 10510-210-1100101102(b)EMFP (nm)energy (eV) Be B C (glassy) C (graphite) C (amorphous)Fig. 2. (a) Total elastic scattering cross section and (b) elastic mean free path as functions of energy for five materials, beryllium, boron and carbon allotropes (glassy, amorphous and graphite). A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 1008625for the energy lower than almost 250 eV is found lower for the heavier element C, however, at the higher energy, this variation becomes reverse. In order to compare with electron inelastic mean free path (IMFP) λin, we also plot in Fig. 2(b) the electron elastic mean free path, λe = 1/naσe, where na is atom number density given in Table. 1, for five materials, i.e. beryllium, boron and carbon allotropes (glassy, amorphous and graphite), considered in simulation. Inelastic cross section For the description of the electron inelastic scattering processes, we used the dielectric function formalism with which the specific electronic excitations a particular sample can be considered. In this model, the differential inverse inelastic mean free path (DIIMFP) for moving electrons in a material is given as: d2λ− 1ind(ħω)dq=2γ21 + γ1πa0EIm{− 1ε(q,ω)}1q(15)  where γ = 1+E/(m0c2) is the relativistic correction factor, m0 the mass of electron, m0c2 the static energy of an electron. The coefficient, 2γ2/(1 + γ), in Eq. (15) is due to the relativistic effect; ε(q,ω) is the complex dielectric function of medium as a function of energy loss ħω and momentum transfer ħq for characterizing the electronic excitation; a0 is Bohr’s radius and λin is the electron inelastic mean free path (IMFP). The term Im{ − 1/ε(q,ω) } is known as the energy loss function (ELF) which determines the probability of inelastic scattering events. Penn had suggested an algorithm, the full Penn algorithm (FPA), for the extension of the electron energy loss function from the optical limit of q → 0, Im{ − 1/ε(q = 0,ω) }, into the (q,ω)-plane, Im{ − 1/ε(q,ω) } [97]. Using the Lindhard dielectric function εL(q,ω;ωp), the ELF can be written as: Im{− 1ε(q,ω)}=∫ ∞0g(ωp)Im{− 1εL(q,ω;ωp)}dωp (16)  where g(ω) is the expansion coefficient, and it is related to the optical ELF, Im{ − 1/ε(0,ω) }, according to the following equation: g(ω) = 2πω Im{− 1ε(0,ω)}(17) It was shown that this approach yields closer simulation results to the experimental data, either backscattering coefficient or the energy distribution of electrons, than the single-pole approximation (SPA) [32] by which Im{− 1/εL(q,ω;ωp) }is simplified as a Dirac δ-function along the plasmon dispersion line [23]. From Eq. (16) the energy loss distribution and IMFP can be obtained respectively by performing the integrations: dλ− 1ind(ħω) =∫ q+q−dqd2λ− 1ind(ħω)dq(18)  λ− 1in =∫ E− EF0dλ− 1indω dω (19)  where the integration limits, ħq± =̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅m0E(2 + E/(m0c2) )√±̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅m0(E − ħω)(2 + (E − ħω)/(m0c2) )√, are obtained from the energy and momentum conservations. Tanuma et al. have performed extensive calculation of IMFP for nonrelativistic electrons [98–101] and relativistic electrons [102–104]. Recently a machine learning technique is employed for predicting the IMFP and improving empirical formulation [105]. According to the ELF as shown by Fig. 3, the dominant contribution of the energy loss originates from the electronic excitation around several tens eV, and particularly all these elemental materials (Be, B and C) demonstrates a strong plasmon peak around 18–28 eV. For carbon the peak position and intensity varies with the crystalline form: the peak position of (π + σ) plasmon is 20 eV for glassy carbon, 24.8 eV for amorphous carbon and 28.3 eV for graphite. In addition, there is a weak π plasmon around 5.4–6.9 eV for carbon. Fig. 3 also shows the sharp K- shell ionization edges around hundred eV, which will play a significant role when the Auger process is taken into account during the simulation. To check the accuracy of the ELFs used, we applied two well-known sum rules, the perfect screening sum rule (ps-sum rule) and the oscillator strength sum rule (f-sum rule) [113]. The f-sum rule Zeff is given by, Zeff =2πΩ2P∫ ωmax0ωIm{− 1ε(ω)}dω (20)  where ħΩP =̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅4πnae2/m√. For a very large value of ωmax, expectation value of Zeff must be atomic number Z, the total number of electrons per atom or molecule. The ps-sum rule Peff can be obtained from the Kramers-Kronig relation as [114,115]: Peff =2π∫ ωmax01ωIm{− 1ε(ω)}dω+Re{1ε(0)}(21)  where Re{1/ε(0) } = 0 for conductors. The expectation value of Peff is unity in the limit of ωmax→∞. Fig. 4 shows the calculated results of f-sum rule and ps-sum rule for beryllium (Z = 4), boron (Z = 5) and carbon allotropes (Z = 6). For Be the corresponding limit values are Zeff = 3.96 and Peff = 1.05, 10-1 100 101 102 103 104 10510-1310-1210-1110-1010-910-810-710-610-510-410-310-210-11001010 10 20 30 40 5001234K-shellenergy loss functionenergy loss (eV) Be B C (glassy) C (graphite) C (amorphous)Fig. 3. Energy loss functions, Im{ − 1/ε(0,ω) }, of beryllium, boron and carbon allotropes (glassy, amorphous and graphite) used in our simulation. The experimental data of dielectric function, ε(0,ω), were taken from Palik [106] in the range of 0.02–300 eV and from Henke et al. [107] in the range of 0.3–30 keV for Be, from Prieto et al. [108] in the range of 0–80 eV and from Henke et al. [107] in the range of 0.08–30 keV for B, from Abril et al. [109] in the range of 0–200 eV and from Henke et al. [107] in the range of 0.2–30 keV for amorphous C, from Venghaus [110] in the range of 0.02–40 eV and from Henke et al. [107] in the range of 0.05–30 keV for graphite C, from Hagemann [111] in the range of 0.0001–30 keV for glassy C and atomic scattering factor was used in 30 keV-100 keV [112] for all the simulated materials. Table 1 Mass density ρ and atom number density na for five elemental materials, i.e. beryllium, boron and carbon allotropes (glassy, amorphous and graphite), where NA is Avogadro constant and M is atomic weight.  elements ρ(g/cm3)  na = NAρ/M(1023 atoms/cm3)  Be  1.85  1.236 B  2.34  1.3034 C (glassy)  1.80  0.9024 C (amorphous)  2.10  1.0528 C (graphite)  2.25  1.1280  A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 1008626respectively, which are quite close to the nominal values of 4 and 1. Accordingly, the relative errors of sum rules for Be are 1.3% and 5%, respectively. In the previous work of Shinotsuka and Tanuma et. al [102] these errors for Zeff and Peff are 2.4% and 6.6% for Be, respectively. We can conclude that our ELF has enough accuracy for accurate Monte Carlo simulation for Be. While for boron and carbon, there are larger deviations from expectation values, which are listed in Table 2. Fig. 5(a) shows the comparison on the calculated IMPFs between Be, B and C. For Be, our calculated IMPF agrees very well with Shinotsuka et al. [102] in the energy range of 1–104 eV. The graphite demonstrates smaller IMFP values, i.e. stronger electron inelastic scattering probability, above 100 eV as compared with glassy and amorphous carbon. This is due to the higher intensity and larger energy values of plasmon peaks of graphite as shown by Fig. 3, while the sum rules indicate that the intensity of ELF is overestimated for graphite. The electron stopping power is the energy loss travelled per unit distance due to inelastic scattering, S = − dE/dx, which was widely used in the early Monte Carlo simulations in the continuous slowing-down approximation. The universal Bethe stopping power equation is written as [116,117], S = 785ZρAEln(1.166EJ)(in eV/Å) (22)  where J is the mean ionization energy of the target atoms (in eV) and is given by [118,119], J ={11.5Z, Z⩽12;9.76Z + 58.5Z − 0.19, Z⩾13. (23) However, this stopping power presents nonphysical negative values at low energies; several extending formulations have been proposed [120–122]. On the other hand, the electron stopping power can be derived from the dielectric theory [23,123]: S =∫ E− EF0ω dλ− 1indω dω (24) Fig. 6 shows the comparison of stopping powers for Be and C allotropes between the present data and the results from other sources [12,116,117,120–123]. In this comparison, the mean excitation energy of Be and graphite is set respectively as 60 and 80 eV [119]. For both Be and carbon cases our results agree well with Shinotsuka’s results [123]. Particularly, the present stopping powers of graphite agree well with the experimental data [12], while other theoretical or empirical expressions 100 101 102 103 104 105012345676.305.05(a)3.964.62f-sum ruleenergy loss (eV) Be B C (glassy) C (graphite) C (amorphous)5.6810-1 100 101 102 103 1040.00.20.40.60.81.01.20.9030.9461.0041.050ps-sum ruleenergy loss (eV) Be B C (glassy) C (graphite) C (amorphous)(b) 1.106Fig. 4. Validation of sum rules for beryllium, boron and carbon allotropes (glassy, graphite and amorphous) as functions of energy loss: (a) f-sum rule; (b) ps- sum rule. 100 101 102 103 104100101(a)IMFP (nm)energy (eV) Be B C (glassy) C (graphite) C (amorphous)100 101 102 103 104100101102 (b)in/eenergy (eV) Be B C (glassy) C (graphite) C (amorphous)Fig. 5. (a) Electron inelastic mean free path and (b) the ratio of electron inelastic mean free to elastic mean free path, as functions of electron energy for all the simulated materials. Table 2 Relative errors of sum rules for five elemental materials, i.e. beryllium, boron and carbon allotropes (glassy, amorphous and graphite).  elements Zeff (%)  Peff (%)  Be  1.3  5.0 B  − 8.0  − 10.0 C (glassy)  − 5.3  0.4 C (amorphous)  − 16.0  − 5.4 C (graphite)  5.0  11.0  A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 1008627show certain differences. The simulations of the backscattered electron spectra were carried out with and without the Auger electron emission [55,56] taken into account. If the energy loss is less than the binding energy of the corresponding inner-shell (i.e. K-shell for Be), ħω < EB (where EB is the binding energy of the inner-shell presented in the optical ELF), then a secondary electron is assumed to be excited from the Fermi sea by transferring ħω energy from the primary electron to a valence electron of energy with the excitation probability being proportional to a joint density of states of free electrons ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅E0(E0 + EF)√. If ħω > EB, the secondary is excited from the inner-shell and has kinetic energy of ħω − EB. Detailed description of the calculations can be found in Refs. [28,32]. In addition, the relaxation of the excited atom may proceed via two paths: the emission of an Auger electron or the emission of a photon. The sum of the corresponding Auger transition probability, PA, and the corresponding fluorescence transition probability, PX, must be 1. The Auger transition probability, i.e. Auger channel parameter PXA, for any Auger channel from level X, is given by [124], PXA = 1 − Z4/(Z4 + Z40)(25)  where Z is the atomic number of an element, and Z0 = 32.4, 89.4, 155.9 and 300 for X = K, L, M and N, respectively. If ħω > EB, Auger electron with the energy of 98 eV for Be will be excited with a certain probability ( PX=KA ). After undergoing multiple elastic and inelastic scattering collisions inside the sample some electrons will reach back to the surface and only part of them can escape if the following condition is satisfied: Ecos2β > U0 (26)  where β is the angle between the electron moving direction and the surface normal, the inner-potential, U0, is the sum of work function and Fermi energy. This condition arises from the fact that the target-vacuum interface has an energy barrier to be overcome by the escaping electron. The electron emission probability from the surface of the material is given by the quantum mechanical transmission function as [18], T(E, β) =⎧⎪⎪⎪⎨⎪⎪⎪⎩4̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅1 − U0/Ecos2β√[1 +̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅1 − U0/Ecos2β√ ]2, if Ecos2β > U0;0, otherwise.(27)  3. Results and discussion We have performed first the Monte Carlo simulation of the energy spectra of backscattered electrons for mono-energetic primary beam incident on an ideally flat Be surface. For each primary energy we used 1x108 primary 102 103 104101102103104105106intensity (a.u.)energy (eV)(a)10 keV8 keV5 keV3 keV2 keV1 keV0.5 keV0.2 keV0.1 keV102 103 104101102103104105106(b)10 keV8 keV5 keV3 keV2 keV1 keV0.5 keV0.2 keV0.1 keVintensity (a.u.)energy (eV)Fig. 7. Backscattered electron energy spectra of beryllium for different primary energies and at normal incidence condition: a) without Auger process; b) with Auger process taken into account. 100 101 102 103 104051015 present Bethe Joy(a)Stopping power (eV/Å)energy (eV) Jablonski Eq. (8) Jablonski Eq. (10) Nguyen-Truong ShinotsukaBe100 101 102 103 104051015Graphite present Bethe Joy Exp. Exp.(b)Stopping power (eV/Å)energy (eV)Graphite Jablonski Eq. (8) Jablonski Eq. (10) Nguyen-Truong ShinotsukaC present (amorphous) present (glassy) Shinotsuka (glassy) Fig. 6. Comparison of stopping powers between the present data and (a) other theoretical results [116,117,120–123] for beryllium, (b) other theoretical/experimental results for carbon [12,116,117,120–123]. A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086280.1 1 10 1000.050.100.15Be(a)Monte Carlo with Auger without AugerExperimentReimer 1980  Bronshtein 1969Shimizu 1974 Philibert 1963Neubert 1980 Drescher 1970backscattering coefficientprimary energy (keV)0.1 1 10 1000.000.050.100.15 B (b) Monte Carlo         Experimentpresent         Hunger 1979Tanuma 2009backscattering coefficientprimary energy (keV)0.1 1 10 1000.000.050.100.15primary energy (keV)(c) C Monte Carlo C (glassy) C (graphite) C (amorphous)Experiment Hunger 1979  Bronshtein 1969 Philibert 1963  Heinrich 1966 Neubert 1980  Sternglass 1954 Palluel 1947  Bishop 1966 El Gomati as-inserted  2011 El Gomati cleaned  2011backscattering coefficient0.1 1 10 1000.00.10.20.3 Monte Carlo C (glassy)  Salvat 2006 C (graphite)  Demers 2002 C (amorphous)  Ritchie 2005 Dapor 2017 Lin 2005 Tanuma 2009 Kim 2015 C (d)backscattering coefficientprimary energy (keV)0.1 1 10 1000.050.100.15(e)backscattering coefficientprimary energy (keV)MC   Reuter  Be  B  C (glassy)            C (graphite)            C (amorphous)0.1 1 10 1000.000.050.100.15C/Be(f) ExperimentReimer 1980    Bronshtein 1969Shimizu 1974   Philibert 1963Neubert 1980   Drescher 1970Monte Carlo0-layer C 1-layer C2-layer C 3-layer Cbackscattering coefficientprimary energy (keV)0.1 1 10 1000.000.050.100.150.20H2O/Be(g) ExperimentReimer 1980    Bronshtein 1969Shimizu 1974   Philibert 1963Neubert 1980   Drescher 1970Monte Carlo0-layer H2O 1-layer H2O2-layer H2O 3-layer H2Obackscattering coefficientprimary energy (keV)0.1 1 10 1000.000.050.100.15Monte Carlo 0-layer Be 1-layer H2O 2-layer H2O 3-layer H2O 1-layer C 2-layer C 3-layer C(h)backscattering coefficientprimary energy (keV)Fig. 8. Comparison on the electron backscattering coefficients as functions of primary energy for (a) beryllium, (b) boron and (c) carbon (in different forms), between the present Monte Carlo simulation and experimental data. (d) A comparison on the Monte Carlo simulated backscattering coefficients for carbon between the present simulation and other calculation results. (e) A comparison on the present Monte Carlo simulation for all of the simulated materials, and with Reuter’s formula for 20 keV. (f) A comparison on electron backscattering coefficients between the experimental data of beryllium and Monte Carlo simulation for carbon contaminated beryllium surface, which is covered by amorphous carbon at different numbers of atomic layers. (g) A comparison on electron backscattering coefficients between the experimental data of beryllium and Monte Carlo simulation for water adsorbed on beryllium surface, which is covered by amorphous water at different numbers of atomic layers. (h) A comparison on the Monte Carlo simulated backscattering coefficients between the absorbed amorphous carbon and water layers on Be surface.   A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 1008629electron trajectories during a simulation. Fig. 7 shows the simulated backscattered electron energy spectra of beryllium at the incident angle of 00 with respect to the surface normal for several incident electron energies. The simulations were carried out without and with the Auger electron emission taken into account in Fig. 7(a) and 7(b), respectively. In order to smooth the simulated spectra and present finite width of the elastic peak, the convolution has been performed by using a Gaussian function with the standard deviation of 1 eV. We can see clearly the enhanced contributions in the spectra due to the Auger peak around 100 eV compared to the peakless structure in the spectra when the Auger process was neglected. Below the elastic peak there are multiple plasmon peaks corresponding to the plasmon peak of 18 eV observed in ELF of Be (Fig. 3). The log–log plot indicates the tendency of intensity variation with the primary energy. The highest elastically backscattered electron intensity is observed for the lowest incident electron energy. The backscattering coefficient is the area of the spectra; with increasing primary energy the spectra intensity reduces while the energy region expands, so backscattering coefficient may not monotonically change with primary energy. Counting number of electrons in Fig. 7 according to their emission energy and dividing by the number of incident electrons, we can obtain the backscattering coefficient η. Fig. 8(a)-8(c) shows the comparison on the primary energy dependence of backscattering coefficients, η(Ep), of elemental solids, Be, B and C, with the available experimental data. It can be seen that η first increases relatively rapidly with primary energy and then decreases slowly. The maximum η is found around 330 eV. The present results in Fig. 7 are displayed in an extended energy range for the linear relationship of η(Ep)̃lnEp as given in Ref. [125]. In theoretical point of view, the present simulation, combined the best available Monte Carlo technique with the best available ELF, is expected to result in the much more accurate backscattering coefficients than previous theoretical data. From central limit theorem it was expected that the Monte Carlo simulation would produce the theoretical data falling within the range of experimental data distribution, as in the case of previous study for Cu [30]. But quite surprisingly we have found that, in Fig. 8(a), the simulation yields the backscattering coefficient values for Be much smaller than the previously reported experimental data, particularly than the Bronshtein’s data [15] at low primary energies below 1 keV, and reach the lower limit of experimental data distribution range above 10 keV [14–16,126–128]. To explore the reason for this difference between the simulation and experiment, we first take the Auger process into consideration in the simulation. However, as indicated by Fig. 8(a) that the contribution of Auger process to backscattering coefficient is negligible. Then we have performed calculations also for the elemental materials having the nearest atomic numbers, i.e. boron and carbon, to validate our Monte Carlo simulation. For the close atomic number elements, boron and carbon are believed to have quite the similar η(Ep)curve shape as well as the close values as beryllium. Fig. 8(b) compare the present calculated results with the available experimental [8] and simulated [129] results for B. Our simulation agrees very well with those data above 5 keV. Fig. 8(c) shows the comparison between present calculated results with the available experimental data [8,15,127,128,130–134] for C. But for carbon the situation is rather complex because there are different structural forms of a carbon solid having varied density values. Fig. 8(c) shows that the experimental data measured by different researchers on carbon are very different. The main reason for this is considered to be due to the sample preparation, where the information about the sample, especially the form and the density of the sample, was not given in the literature in most cases. Below 2 keV Sternglass’s data values [131] are very low while Bronshtein’s data values [15] are rather high, and El- Gomati’s data values [133] are moderate and are almost constant between 300 eV and 5 keV. El-Gomati et al. [133] further demonstrated that as expected the surface cleaning by ion bombardment alters the η values very slightly since the contamination is mostly hydrocarbons. Therefore, we have performed calculations for three typical carbon allotropes with the known density values given in Table 1. Fig. 8(c) shows that graphite has the lowest backscattering coefficients and amorphous carbon has the largest values, while the glassy carbon presents the intermediate backscattering coefficient values which agree excellently well with Bronshtein’s experimental data [15] at low primary energies below 1 keV and are quite close to other experimental data at higher energies. It is also interesting to note from Table 1 and Fig. 8(c) that the graphite has higher density but lower η values than glassy carbon, which should not be realistic. This is because, on the one hand, the crystalline form affects the electronic excitation as displayed by ELF in Fig. 3. On the other hand, the sum rules are strongly underestimated for boron and amorphous carbon indicating the underestimation of inelastic cross section, and are strongly overestimated for graphite indicating the overestimation of inelastic cross section. Fig. 5 shows that graphite has the smaller IMFP, and also the ratio λin/λe, as compared with glassy and amorphous carbon. Therefore, in Monte Carlo simulation electrons suffer much more inelastic events and less elastic scattering events in graphite than in glassy carbon. This explains why the simulated η values are lower for heavier carbon (graphite) than the lighter glassy carbon and amorphous carbon. Because the ELF used for glassy carbon has quite satisfactory sum rule values, the simulated backscattering coefficients for glassy carbon should be much more reliable than amorphous carbon and graphite. Then, the simulation in agreement with Bronshtein’s data [15] in the lower energy region below 1 keV and other experimental data in the higher energy region above 5 keV is quite reasonable. Fig. 8(d) also illustrates the comparison with other reported Monte Carlo simulations on carbon [129,135–139] by using different simulation models and codes, mostly employed the stopping power equation in the continuous slowing down approximation for electron inelastic scattering. But this approximation is believed to be much poorer than the present individual inelastic scattering simulation based on a dielectric functional formalism, which describes the specific electronic excitation channel in a material rather than the simple Z-dependence in stopping power equation, particularly considering the fact that the Bethe stopping power equation is invalid at low energies and one has to introduce the modified mean ionization energy value. Depending on different approaches to stopping power equation, some of these codes yielded extreme low η values. e.g. by NISTMonte [137] and Geant4 [139] codes, while others presented extreme high values at low energies [20]. Furthermore, Fig. 8(e) compares our simulated values with those calculated from Reuter’s empirical fitting formula for high energy backscattering coefficients [140],η = − 0.0254 + 0.016Z − 1.86× 10− 4Z2 +8.3× 10− 7Z3, is given at 20 keV for Be, B and C. Our present simulation also agrees well with the formula for B and C, and slightly smaller for Be. Looking at experimental data solely, an obvious character of Bronshtein’s data [15] for Be and C in comparing with others is that the η(Ep)curve has a maximum around several hundred eV but not monotonously increasing when primary energy Ep is reduced. The tendency agrees well with the present calculation. The clue from this observation is that the reason for the uncertainty involved in experimental data can be partly accounted for. Experimentally the effects of surface roughness and cleanness existed in all experiments can affect the obtained results more or less. Like aluminum, beryllium is easy to be oxidized. Moreover, the presence of a surface carbonaceous layer on the specimen surface may also have a significant influence. The measurement performed without ultrahigh vacuum condition and ion beam sputtering would inevitably involve surface contamination due to residual gases containing hydrocarbon, carbon dioxide, oxygen and water molecules on a surface [141]. By vacuum baking and electron beam irradiation these adsorbed molecule species may change their structure or even dissociate into carbon, hydrogen, oxygen and even silicon atom species. Carbon, oxygen and silicon atoms have a large effect for increasing the η value of bulk beryllium. It would be quite reasonable to consider that the higher η values of experimental data than our calculation come mainly from surface contamination. The Drescher’s [126] and Reimer’s [14] data were measured in a scanning electron microscope without in situ surface cleaning procedure, and the samples A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086210-6 -4 -2 0 2 4 6024681012(d)z (nm)x (nm)  Be  B  C (glassy)e-1e-2e-3Fig. 10. Monte Carlo simulated relative intensity map of the electron trajectory spatial distribution for a normal incident electron beam of 0.5 keV: (a) Be, (b) B, (c) C (glassy). (d) A comparison of contour map for the three materials. 0 2 4 6 8 10012340.5 keV(a)intensity (arb. units)depth (nm) Be B C (glassy)0 2 4 6 8 10 12 140.00.51.01.52.02.51keV(b)intensity (arb. units)depth (nm) Be B C (glassy)Fig. 9. Backscattered electron intensity as a function of maximum penetration depth for a normal incident electron beam of (a) 0.5 keV and (b) 1 keV. The distribution is normalized by the area under the curve. A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086211include electropolished metals, evaporated films and mechanically polished specimens. It is then not surprising that their data defines the upper limit of experimental data distribution range of beryllium. Shimizu had performed the measurement in an ultrahigh vacuum with an argon ion bombardment of the sample [16], and his data are lower than that of Reimer at high energies and approaches the data of Reimer when primary energy is lowered down to 1 keV. The sample property was not described in the Neubert’s experiment [127]. Bronshtein et al. had prepared pristine beryllium surface by vacuum deposition of thin beryllium layers of various thicknesses onto a platinum substrate, and the target was heated to a high temperature for degassing [15]. The sample cleanness in this experiment explains why the Bronshtein’s data are much lower than others. Therefore, the surface cleanness can be one of the most responsible reasons of the disagreement between our calculation and experimental results. Because the physical properties, e.g. optical constants, of these complex adsorbates is generally unknown, we will consider in this work the contaminates made of amorphous carbon and water. Amorphous carbon is the dominate contamination ingredient as experimentally found, whose electrical properties has been measured by X-ray photoelectron spectroscopy [142] and whose optical data are available. For water its optical data are also available in the photon energy range of 1.24 × 10− 7 − 3 × 104 eV [36], while its optical data in the energy range of 30 keV-100 keV can be determined from the atomic scattering factor of H and O [112]. Therefore, we have performed another simulation for a bulk Be solid whose surface is either covered with an amorphous carbon or water layer with varying thickness. The simulation method for this contamination layer covered sample uses CTMC-ATOMIC, which is 0 10 20 30 40 50 60 70 800.00.20.40.6backscattering coefficientincident angle (deg.) 0.2 keV 0.5 keV    1 keV    2 keV  10 keV(a)Be0 10 20 30 40 50 60 70 800.00.20.40.6(b) MC Neubert Exp. Drescher Exp. 20 keV        20 keV        25 keV 25 keV  40 keV 40 keV        60 keV 60 keVbackscattering coefficientincident angle (deg.)Fig. 11. Backscattering coefficients of Be as a function of incident angle with respect to the surface normal at different incident energies: (a) 0.2–10 keV; (b) 20–60 keV for comparison with experiments. 18016515013512010590756045301500.1 keV (a)1801651501351201059075604530150(b)0.5 keV18016515013512010590756045301501 keV (c)18016515013512010590756045301505 keV (d)Fig. 12. Angular distributions of backscattered electrons of Be at several different incident angles, i.e. 30◦(green), 60◦(blue) and 90◦(red), and primary energies: (a) 0.1 keV, (b) 1 keV, (c) 3 keV and (d) 5 keV. A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086212basically the same as our previous Monte Carlo simulation of reflection electron energy loss spectrum for carbon contaminated sample by a surface excitation model [63], the difference is only that the present Monte Carlo model as described above is just a bulk excitation model because here we are not interested in weak surface excitation features near the elastic peak. The simulation results shown in Fig. 8(f) & 8(g) indeed indicate the trend of increasing the value at low primary energies below 1 keV by absorption of carbon layer and water layer, respectively; two carbon layers or one water layer are most likely existed in Bronshtein’s sample [15]. This simulation explains very well part of the experimental data. Furthermore, additional existence of residual oxygen and silicon atoms on beryllium surface from decomposition of trisiloxanes actually will further enhance the contamination effect and even at high primary energies because their atomic numbers are greater than carbon. Hydrogen atoms will slightly weaken the commination effect because of the smaller atomic number and elastic scattering cross section compared with that of carbon. It has been also suggested by El-Gomati et al. that great care is needed in surface preparation at low primary beam energies for accurate η determination [11]. To understand the mechanism leading to the Z-dependence of backscattering coefficients in more detail, we consider that the electron penetration spatial region, i.e. the interaction volume, may present some useful information. Fig. 9 shows the distributions of backscattered electron intensity on the maximum penetration depth at primary energies of 0.5 keV and 1 keV for three elemental solids. In the deep interior of the solid electrons have no enough energy to escape from the surface to be backscattered electrons. Therefore, the larger distribution range and smaller mean depth value estimated from the distribution for the lighter elements and at the higher primary energies indicate the lower η values. Fig. 10 demonstrates the electron trajectory spatial distributions for three elemental solids. The colour scale from red to blue shows the intensity of the electron trajectory spatial distribution from high value to low value. The intensity is normalized by the number of the incident electrons. The contour lines in Fig. 10(d) represent the normalized intensities having values of e− 1, e− 2 and e− 3, for different degrees of attenuation according to natural logarithm. The diffusion rate can be seen from these contour lines. The broad distribution in horizontal direction represents the larger scattering angles in electron elastic scattering events and hence the larger η values. This spatial distribution is determined by the combination effect of differential elastic scattering cross section (Fig. 1), elastic mean free path (Fig. 2(b)) and inelastic mean free path (Fig. 5(a)) while the latter two are related to material density. Cross section for electron elastic scattering with atoms depends on Z, but not monotonically. Fig. 2(a) shows that at higher energies, cross section is larger for the heavier element (C) than the lighter element (Be), and at lower energies it is reversed; the transition happens around 300–400 eV, which coincides with the energy Ep for the maximum η. But after considering the density factor, the behaviour of elastic mean free path λe is not a simple relation of Z, e.g. solid carbon has different forms and densities. The density also affects the value of inelastic mean free path λin simultaneously. Therefore, instead of considering the Z-dependence only from λe one has to consider the ratio λin/λe (Fig. 5(b)). The greater λin and smaller λe, the more elastic collisions than inelastic collisions and, hence, the larger the value of η. But, as it can be seen in Fig. 5(b) that this ratio is bigger for Be than C below 300–400 eV. Therefore, we must consider electron differential scattering cross section. For the lighter element Be, its forward elastic scattering is much stronger than the heavier element C; then electrons undergo more frequent small angle scattering in Be than in C. In fact, all these factors, i. e. total and differential scattering cross sections (both elastic and inelastic), together contribute to the smaller η values of Be. We have also calculated backscattering coefficients as functions of incident angle and emission angle. Fig. 11 shows the incident angle dependence for Be at several primary energies, and the comparison with experiment. The η value increases with the incident angle, defined as the angle of electron beam with respect to the surface normal. This is because at lower incident angles, most of electrons are going deeper inside the materials and are more difficult for them to escape back from the material surfaces. The comparison of the simulation with the Neubert’s experimental data [127] at incident energies of 20, 40 and 60 keV all fit very well in Fig. 11(b), where the Drescher’s experimental data [126] for 25 keV incident energy are also displayed. As results of our Monte Carlo simulations we can also evaluate the emission angular distributions of the backscattered electrons. Fig. 12 shows the simulated angular distributions of back scattered electrons at different incident angles and primary energies. Other than the total yield of backscattered electrons in Fig. 11, this figure displays how the emission angular distribution changes with the incident angle. Because beryllium is a light element and forward scattering is strong, then the emission maximum appears in the forward angular region, but not symmetrical about surface normal. The angular position for the maximum shifts with incident energy. 4. Conclusions In this work, we have presented a Monte Carlo simulation of the electron backscattering coefficient of Be, B and C (glassy, graphite and amorphous) at impact energy range between 0.1 keV and 100 keV. Both the high energy cascade secondary electrons and Auger electrons produced during the transport of the incident electrons inside the bulk beryllium were taken into account. This up-to-date CTMC-SEM modelling uses the Mott’s cross section calculated with more accurate scattering potential and a relativistic dielectric functional formalism with full Penn algorithm and experimental optical data. Sum rule checks indicate that, beryllium and glassy carbon have good sum rule values and the simulation of backscattering coefficients for them should be reasonable; while boron and amorphous possess an underestimated sum rule and graphite has overestimated ones, hence, the calculation of backscattering coefficients for them would tend to be overestimated and underestimated, respectively. The simulation results show that the Auger electrons contribute negligibly to the backscattering coefficient. For beryllium the simulation results are found well below experimental data at low energies and approaches the lower limit of the experimental data distribution range at high energies above 10 keV. Further simulations for carbon or water absorbed on beryllium surfaces indicate that a contamination with a thickness of one or two atomic/molecular layers can largely increase the backscattering coefficient of beryllium below 1 keV. Therefore, the early experimental data measured either in bad vacuum conditions and/or without previous ion-sputter cleaning of the surface are likely to be overestimated. The very low values of the backscattering coefficients for extreme low-Z elements (Be and B) are partly attributed to the strong elastic forward scattering, while previously it was assumed that the backscattering coefficient is dominated by the total cross sections of elastic and inelastic scattering. We believe we have derived the presently most accurate backscattering coefficients for beryllium and glassy carbon among all available theoretical and experimental data. We highly recommend to use these data in applications at least until further accurate experimental data for clean Be surface become available. Both reliable experimental and theoretical benchmark results need to be established in future. CRediT authorship contribution statement Abrar Hussain: Investigation, Visualization, Writing - original draft. Lihao Yang: Methodology, Software, Validation, Writing - original draft. Shifeng Mao: Methodology, Software, Supervision. Bo Da: Methodology, Supervision. Károly Tőkési: Conceptualization, Writing - review & editing, Supervision. Z.J. Ding: Conceptualization, Software, Validation, Resources, Writing - review & editing, Supervision, Project administration, Funding acquisition. Declaration of Competing Interest The authors declare that they have no known competing financial A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 10086213interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The work was supported by the National Key Research and Development Project (2019YFF0216404) and Education Ministry through “111 Project 2.0” (BP0719016), the National Research, Development and Innovation Office (NKFIH) under Grant KH126886 and the European Cost Actions CA15107 (MultiComp). This work at National Institute for Materials Science was supported by Grant for Basic Science Research Projects from the Sumitomo Foundation. We thank also Dr. H.M. Li and the supercomputing center of USTC for the support of parallel computing.  6. Appendix A: uncertainty analysis Quantification of the computed data quality has been assessed by uncertainty. Recently the estimation of the uncertainty of the computer simulated results is becoming an important issue [143,144]. The sources of uncertainty for electron scattering calculations have been considered in Ref. [144]; following the general consideration on the uncertainty classification, the expression for the combined standard uncertainty [145] of the simulated backscattering coefficient can be expressed here as: u2simulation = u2modelling + u2numerical + u2specimen (A1)  where the three terms are due to physical modelling, numerical calculation and specimen parameters. The uncertainty due to numerical calculation mainly originates from two parts, i.e. statistical fluctuation and the general computation uncertainty: u2numerical = u2statistical + u2computation (A2) The statistical fluctuation term is given by ustatistical∝1/̅̅̅̅N√, where N is the number of samples, i.e. the number of incident electrons here, which can be omitted in this work because 1×108 primary electron trajectories have been used in a simulation. The general computation uncertainty term, ucomputation, is usually caused by the truncation error and the finite precision of computations with floating-point numerals. It has been shown that the incorrect interpolation process between the tabulated elastic cross section values can cause even larger effect, an obvious deviation on the simulation results [11]. All these factors have been carefully considered in our CTMC-SEM code and this uncertainty term is also negligible. The specimen term, uspecimen, is caused by the uncertainties of the parameters used for describing the physical properties of the specimen, such as, mass density, work function, Fermi energy etc. These parameters are usually well defined for most of materials, and particularly, electron backscattering is not sensitive to work function. While some exceptions do exist, for example, the density of amorphous carbon has a rather large value range in between 1.4-3.0 g/cm3 [146,147], which varies with the ways of production. The carbon layer covered on the Be specimen in the experiment is amorphous, whose density should be related to the particular experimental condition rather than a fixed value. As ELF is also related to the mass density, while it is known for amorphous carbon only for the density of 2.1 g/cm3, which is the value we used for calculations of bulk amorphous carbon and carbon contaminated beryllium surface. Of course, the denser the amorphous carbon is the stronger the contamination effect will be. As for clean Be, the specimen term of uncertainty, uspecimen, is negligible. It can be understood from our discussions given above, the change of the η value by contamination in a certain experiment should be attributed to the experimental uncertainty but not to the effect of theoretical simulation. The uncertainty term due to physical modelling, umodelling, is considered to be the most important term. It is composed of two parts here as, u2modelling = u2elastic + u2inelastic (A3)  where uelastic and uinelastic represent the uncertainties involved in a particular modelling of elastic and inelastic scattering, respectively. The Mott’s cross section and dielectric functional theory (FPA) are regarded as the most accurate models for the description of electron elastic and inelastic scattering, respectively, up to now (see Appendix-B for a comparison with several other other approaches). However, the inputs to the calculations of Mott’s cross section and IMFP, i.e. the scattering potential and the ELF, respectively, may still contain some uncertainties. The deviation of ELF can be estimated by the errors of sum rules (see Table 2), which confirm that the uinelastic term is negligible for beryllium and glassy carbon. On the other hand, the scattering potential may have an impact on the calculated Mott’s cross section and, hence, the simulation results of backscattering coefficient. In ELSEPA code the details for the calculation of elastic cross section are specified by the input arguments which are listed in part in Table A1 for the arguments, MNUCL, MELEC, MEXCH and MCPOL with their detailed description about the options [96]. A label “abcd” is used to denote the scattering potential, i.e. “MNUCL=a”, “MELEC=b”, “MEXCH=c”, and “MCPOL=d”. Note that the nuclear charge distribution will affect the calculated elastic cross section only for projectiles with kinetic energies higher than 50 MeV [96]. In this work, “MNUCL=3” is used and we will only focus on the influence of options for electron distribution model (MELEC), electron exchange potential (MEXCH) and electron correlation- polarization potential (MCPOL). Then totally 48 (4×4×3) different combinations of scattering potentials have been employed to calculate the uncertainty range of the elastic cross section. Fig. A1(a) shows the range of total elastic scattering cross section as a function of electron energy as determined from 48 different scattering potentials. The “3412” curve is the total elastic scattering cross section used in present work, while the grey area represents the maximum error range calculated based on all other scattering potentials for free Be atom by using ELSEPA code [96]. Potential “3200” will lead to the smallest total elastic scattering cross section, and the potentials “3111”, “3121” and “3131” will lead to the maximum ones. Fig. A1(b) shows the corresponding backscattering coefficient by using different Mott’s cross sections shown in Fig. A1(a). As the full range from minimum to maximum values are considered the error is within 100% confidence limits. It can be seen that even with the largest elastic cross section, the calculated backscattering coefficients for pure Be are still much lower than the experimental data. This fact indicates that the uncertainty involved in the present Monte Carlo modelling has less effect to the conclusions made. A. Hussain et al.                                                                                                                                                                                                                                Nuclear Materials and Energy 26 (2021) 100862147. Appendix B: comparison with other models In addition to the models adopted in this work, i.e. the Mott’s cross section and dielectric functional approach (FPA), there are some other theoretical models or approaches used for the description of electron elastic and inelastic scattering in a Monte Carlo simulation. Here we select several typical combinations of modelling [50] to investigate the sensitivity of backscattering coefficient to models [45]. Due to its simple analytical form, the screened Rutherford cross section had been widely used to describe elastic electron-atom collisions in the early years. The screened Rutherford differential scattering cross section is expressed by dσdΩ=Z2e44E2(1 − cosθ + 2β)2 (A4)  where β is the atomic screening parameter, and is given by β = 2.61Z2/3/E where E is in eV [148,149]. On the other hand, the continuous slowing down approximation (CSDA) with the use of stopping power formula, either Bethe’s equation, Eq. (22), or the empirically modified expression [120], has been also widely used in a Monte Carlo modelling of electron inelastic scattering for various calculations including backscattering coefficient [150]. Fig. B1 illustrates the comparison on the electron backscattering coefficients of beryllium with different models for electron elastic and inelastic scattering. Here the stopping power in CSDA approach is obtained from our present FPA modelling as shown in Fig. 6. It is very clear that the use of screened Rutherford formula overestimates the backscattering coefficient, mostly due to the incorrect description of differential cross section, and, the CSDA approach underestimates the backscattering coefficient. 0.01 0.1 1 10 10010-410-310-210-1100101Be(a) Error 3412 (present) 3111 3200total elastic cross section (×10-15 cm2 )energy (keV)0.1 1 10 1000.050.100.15Be(b) ExperimentReimer 1980   Bronshtein 1969Shimizu 1974  Philibert 1963Neubert 1980  Drescher 1970Monte Carlo Error 3412 (present) 3111 3200backscattering coefficientprimary energy (keV)Fig. A1. Ranges of (a) total elastic cross section and (b) backscattering coefficient for beryllium for 100% confidence limits as determined from different scattering potentials. Table A1 part of the input arguments in the calculation of elastic cross section by ELSEPA code. For more details, please refer to Ref. [96].  Argument Values Description MNUCL  Nuclear charge distribution model  1 Point nucleus (P).  2 Uniformly charged sphere (U).  3 Fermi distribution (F).  4 Helm’s uniform-uniform distribution (Uu). MELEC  Electron distribution model  1 Thomas-Fermi-Molière distribution (TFM).  2 Thomas-Fermi-Dirac distribution (TFD).  3 Dirac-Hartree-Fock-Slater distribution (DHFS).  4 Numerical Dirac-Fock distribution (DF), read from the database files. MEXCH  Electron exchange potential  0 No exchange potential.  1 Furness-McCarthy potential (FM).  2 Thomas-Fermi potential (TF).  3 Riley-Truhlar potential (RT). MCPOL  Correlation-polarization potential  0 No correlation-polarization potential.  1 Buckingham potential (B).  2 Local-density approximation (LDA).  A. Hussain et al.                                                                                                                                                                                                                                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