1. Introduction

The electron and positron material interaction has a great importance in several domains of the analytical techniques of the material such as electron probe microanalysis, electron energy-loss spectroscopy, Auger electron spectroscopy, positron annihilation spectroscopy, etc. Electron-transport calculations are usually performed by means of either analytical theory or Monte Carlo simulation. It is important to know that the latter (Monte Carlo method) has become a powerful tool in the calculation and prediction of radiation effects in the solids. Both methods require an accurate knowledge of the cross-section for elastic scattering of electrons as functions of the projectile kinetic energy E. For that reason, Rouabah et al[1-3] have been interested to propose a simplified expression of the transport cross-section of electron and positron by basing on analytical expression reported by Jablonski[4]. The direct calculation of the deviation between their interpolated results and the accurate values shows that there are drastic deviations. In addition, Rouabah et al have combined between the Vicanek and Urbassek theory[5] (deterministic model) and the Monte Carlo simulation (probabilistic model) to investigate the transport of 0.5–4keV electrons in solid targets[1, 6]. However, when we applied a combination between two different methods we should be very careful especially if the first model is stochastic (probabilistic) and the second is analytic (deterministic). After a careful analysis of Refs.[1-3, 6-8] we notice that there is an abnormal problem in number of questions: the expression of the electron TCS[1-3], the combination between Monte Carlo and analytic model[1, 6], the backscattering coefficient results[1,6-8], the range of penetration[1, 6-8] and the large deviation of their results[1-3, 6-8]. Moreover, some errors are repeated in all Rouabah’s papers[1-3, 6-8] (see below). We note that we have focused our attention on Rouabah’s paper “[1]” which was the base of number of their other works.

So, the present work shows necessary considerations to study the transport of the slow electrons and positrons in solid targets.

This paper is organized as follows. In Section 2 we describe our comments on the methods used in the calculations: the transport cross-sections, the normalized combination between the Monte-Carlo simulation and the Vicanek and Urbassek theory, the electron range, the mean number of wide angle collisions and the analytical backscattering coefficient. Finally, Section 3 contains the conclusion.

2. Methods and Comments

Our comment on the Rouabah et al works could be summarized on three subjects: their transport cross-section (TCS), their combination between analytic and stochastic models and their calculated range using Monte Carlo method.

2.1. Comment on Rouabah et al TCSs

Rouabah et al suggested a simplified expression of the electron transport cross section using a simple fit. However, actually, they have attributed in their work[1-3, 6-8], an old screened TCS (denoted ) to Jablonski[4]. Indeed, firstly, we will present the demonstration that is more than 60 years old; then we present the true Jablonski TCS.

2.1.1. The Screened Cross Sections and The Wentzel Model (1927)

According to Fernandez-Varea et al[9]: “The Wentzel[10] approach for describing elastic scattering of particles with charge Z'e (Z ' =- 1 and + 1 for electrons and positrons, respectively) by atoms of atomic number Z is based on the simplified scattering potential

(1)

where the exponential factor tries to reproduce the effect of screening. The screening radius R may be estimated from the Thomas-Fermi model of the atom, which yields”

(2)

where is the Bohr radius () and .

“However, it is more expedient to determine R so as to obtain agreement with more accurate elastic scattering cross sections”[9].

Generally, R has been taken using the next form (see Nigam[11] (1959)):

(3)

where and µ is a constant (generally µ>1 which signifies that ).

It could be noted that, at high incident electron energies we take µ = µ^∞[12]. Moreover, the first order Born approximation is valid only at that case (see below). Thus:

(4)

The agreeable value of depends on the used screened potential and the energy[12]. For example, some authors take as value corresponding to the Thomas Fermi potential (i.e. see[11-14]).

The transport cross section corresponds to the screened Differential Cross Section (DCS) (which is obtained from the first order Born approximation), which gives[9]

(5)

The screening parameter A is given by[7]:

(6)

where p is the momentum and β is the velocity of the scattered particle in units of the speed of light c.

At no relativistic domain (), the equation (6) can be rewritten as follows:

(7)

Or

(8)

Let’s put,

(9)

Consequently, the equation (5) can be rewritten as follows:

(10)

At no relativistic domain, can be rewritten as follows:

(11)

From equation (9) the energy E is given by:

(12)

Consequently, the equation (11) can be rewritten as follows:

(13)

We have shown that the first order Born approximation applied on the screened potential gives the same transport cross section attributed by Rouabah et al to Jablonski[4]. Consequently the age of is more than 60 years old (since the works of Wentzel (1927) and Nigam 1959). We note that Rouabah et al have repeated this error (the attribution of to Jablonski[4]) in all their works[1-3, 6-8 ]. Consequently, all these works must be reviewed.

2.1.2. The Demonstration that the TCS Derived by Jablonski[4] is not

We recall that Jablonski said in his abstract[4]: “Analytical description of photoelectron and Auger-electron transport in solids requires values of the transport cross sections for electron energies between 50 and 2000 eV. An analytical formula is proposed to provide needed values of for energies in this range and for all elements.”

To proof that derived by Jablonski[4] is not we can base on Jablonski “himself”[4] as follows:

1. The great proof is the above calculations (see above “section I”).

2. The symbolization differences between and : Jablonski[4] used the symbolization in the Abstract, equations: (2[definition], 21,24a, 25, 27, 32 ..) however he used the separate symbolization (i.e. without multiplying it by ) only in equations (8 and 15) and he don’t use it in his results section.

3. The index B in : Jablonski (himself) said[4]:“the index B denotes the first-order Born approximation.”

4. The index “∞” in which signifies “At sufficiently high incident electron energies[12]”, which is not the case of Jablonski study[4] {50-2000 eV}. Knowing that Jablonski himself said[4]: “ The constant µ in the screening parameter depends on energy, especially at energies below 1 keV. The use of the asymptotic value µ^∞ is justified only at energies exceeding 5 keV”.

5. The calculated deviation mentioned in Jablonski abstract[4] correspond to (see equation (27) in Jablonski paper[4])

6. We note that the index B of denotes first order Born approximation. However, the first order Born approximation is not valid at lower energies (see[4]; E must verify ; thus, the electron energy should exceed 229.2 eV for carbon or 1277.7 eV for gold) which is not the case, too, of 50-2000 eV especially for heavy atoms. In addition Jablonski himself said[4]:“Thus, the accuracy of the first-order Born approximation for is generally rather poor. To obtain a more accurate analytical expression for , we need an additional analytical function G(ε₀) correcting the cross section ”.

7. We recall also that Jablonski (himself) said[4]:”An improved analytical expression is derived here to provide reasonably accurate values of in the energy range of interest for surface-sensitive electron spectroscopies, i.e.,50–2000 eV.” We note that the deviation of by comparison to that obtained by quantum methods reached drastic values. In Table 1 we presented , (tabulated by Mayol and Salvat[15] who used the Relativitic Partial Waves Expansion Method (RPWEM)) and the percentage deviation between them. For example, we remark that at E=250 eV the percentage deviation of reaches ≈1500 % for Au (Z=79). Has Jablonski proposed such approximation and said “…reasonably accurate values of ..”?

8. In the abstract of[4], Jablonski said: “For atomic numbers up to 30, the mean deviation between accurate values of and values from the analytical formula reaches 0.5%. For several elements with larger atomic numbers, this deviation increases to 5%, although it is much lower for the majority of cases” . However, from Table 1, we can remark that the deviation of is always greater than 2%. Moreover, we found that the deviations reached drastic values: 30%, 50%, 200 %, 1500%,.. Consequently, and certainly, is not the TCS mentioned in the abstract of Jablonski paper[4].

Actually the TCS derived by Jablonski lies in his paper[4]: “Thus, the accuracy of the first-order Born approximation for is generally rather poor. To obtain a more accurate analytical expression for, we need an additional analytical function which correcting

(14)

With

(15)

where A₀ , A₁ , A₂ , A₃ , and A₄ are fitted constants for each element.”

2.1.3. Rouabah et al TCS[1]

Before quoting our point of view of this section, we recall that Rouabah et al used the same transport cross section expressed by equation (13) where the only difference is that has been taken as a free parameter. Thus, to determine, they have adjusted to Mayol et al TCS[15]. After a fitting process, they suggested the next interpolation form of given by:

(16)

Since Rouabah et al[1] adjusted to [15]; by basing on the equation (13), it is easily to conclude that "μ^∞" is given by:

(17)

The equation (17) clearly shows that μ^∞ (as adjusted parameter) is a function of Z and E, but Rouabah et al[1] expressed it using equation (16) which depends only on Z. May be the authors of[1] or any one said that “Rouabah et al. expression depends implicitly on E. In fact what was done μ^∞ is calculating for each element as a function of energy. Then finding an optimal value for μ^∞ for each element (i.e. for each z). Finally, one fitted μ^∞ as a function of z for all the elements of interest. So, μ^∞ depends explicitly on z but implicitly on E”. Our response will be presented as follows:

● To said that an arbitrary function depends implicitly on the energy (as example), we should find a relation between at least one parameter (x or y ..) depends explicitly on the energy otherwise is independent explicitly and implicitly on the energy.

For example, if and we can said that f(x,y,..) depends explicitly on x and implicitly on E. Indeed, the equation (16) does not any relation with the energy; neither explicit nor implicit.

● Rouabah et al[1] have adjusted to [15]. So, the large deviation between them (see below) is the proof of the weak point of this choice.

● Elsewhere, we note that the TCS results of[1] (denoted by in their tables (1-4)[1]) don’t correspond to their fit given by equation (16). After a number of tests we think that they have used the next expression:

2.1.4. The invalidity of Rouabah et al TCS[1]

To show that Rouabah et al TCS fit[1] is not accurate, we can based on their data results themselves. In fact, Table (2) represents their TCS[1], Mayol and Salvat TCS[15] and the percentage deviation between them. We think that, these deviations are clearly invalidates their proposal fit (for Al: 21% and 12 % at E=250 and 500 eV… respectively, for Cu: 41%, 20% and 10.2 % at E=250, 500 and 1000 eV respectively, for Au: 96% and 30.3 % at E=250 and 500 eV respectively ….). Unfortunately, this Rouabah et al TCS[1] has been used in other Rouabahah’s papers[1, 6-8]. Consequently all their works[1, 6-8] must be reviewed and revised.

2.1.5. The invalidity of Rouabah et al TCS[2]

Rouabah et al[1], in order to show that their model[1] is valid, have presented three tables carrying out a comparative study: their TCS ()[1], TCS of their previous work ()[2], given by equation (13) and the TCS tabulated by Mayol and Salvat ()[15] where it is clear that is in better agreement with than both and. By contrast, this latter is not the proof of the validity of their fit[1], but it is the proof of the invalidity of [2] and. Concerning the invalidity of at lower energies; it has been discussed above. However, the invalidity of their “[2]” can be proved as follows:

Rouabah et al carried out an erratum[16] on their results of[2] in which they precise[16]:

“(i) in all the text, the statement percentage deviation has to be replaced by relative deviation.

(ii) In table (4) and fig.1, the deviations D and D’ are expressed in absolute units and not as percentages.”.

We think that it will be better that if the table (4) and Fig. 1 of[2], expressed as percentage commonly used in literature (i.e. more appropriate than absolute units).

So, to prove that the TCS of[2] is inaccurate, we can combine between their results published in[2] and their erratum[16]. Take for example their table (4) of their work[2], the colon D (which corresponds to the deviation of their results) of Au (Z=79) we find the next results: 4.62, 1.61, 0.49, 0.29, 0.16, .. According to their erratum[16] “the deviations D are expressed in absolute units and not as percentages” so, these values (4.62, 1.61,…) are in absolute units; by consequence these deviations become in percentage as follows: 462%, 161%, 49%, 29%, 16% .. These deviations are totally unacceptable for the fitting process. Indeed, we should multiply their published deviations by 100 (on other word the error factor is 10⁴%). Consequently, we note and we reconfirm that their erratum[16] also invalidates their work[2].

2.1.6. The Invalidity of Rouabah et al TCS[3]

Rouabah et al[3] repeated approximately the same above mistake of[1] in the case of positron. After an adjustment process to Dapor results[17], they are suggested two interpolation forms of given by:

(18)

(19)

To show that Rouabah et al interpolated TCS[3] is not accurate; we can base only on their data results. In fact, Table (3) represents their TCSs[3], Dapor TCS[17] and the percentage deviation between them. These deviations are clearly invalidates their proposal.

2.2. The Combination between Monte Carlo Simulation and Vicanek And Urbassek Formula

Before giving our point of view; we recall that Rouabah et al[1] have used Monte Carlo simulation (MC) to calculate the range and Vicanek and Urbassek formula to calculate the backscattering coefficient (BSC).

In their MC simulation they used the screened Rutherford cross section, given by:[18]

(20)

where z, θ,e and E are the atomic number, the scattering angle, the electron charge, and the electron energy in eV, respectively. The parameter β represents an atomic screening parameter which is given by, β_N =µZ^2/3/E, where:

(21)

(22)

The Transport cross section (denoted in the following) obtained by integrating is given by:

(23)

In Vicanek and Urbassek formula, the BSC () is expressed as[5],

(24)

with : and.

In relation (24), is the mean number of wide angle collisions defined as,

(25)

where is the transport cross-section, is the range of penetration and is the number of atoms per unit of volume in the solid target given by:

(26)

where N_Av , and A are the Avogadro number, the density and the atomic mass of the target respectively.

So, to calculate , Rouabah et al[1] used MC simulation to calculate the range and for this raison, they think that they have carried out a combination between MC simulation and Vicanek and Urbassek formula. We think that the word “combination” is not appropriate because, we think that to do a combination between MC and the analytic model; the authors of“[1]” should be used the same elastic and inelastic models (i.e. the same input data). To show that the authors of[1] have not carried out a normalized combination, we have presented in Table (4) and where we remarked a big difference between them (i.e. the TCS used by[1] in Vicanek and Urbassek expression do not correspond to that obtained by integrating used by them in their MC code).

Besides, is this combination between Monte Carlo method and Vicanek and Urbassek theory evident and realistic? Most part of the response is mentioned in our paper[19]. Indeed, we have showed that: firstly, the Monte Carlo method is more recommended, generally, to be used in the calculation of the backscattering coefficient than Vicanek and Urbassek theory (with a condition to use the same input data)[19].

Consequently, we did not need to use this combination. Secondly, the use of this combination should be done by taking into account the normalization condition[19].

To make clear the normalization problem, we can give the following example:

Let’s note by N₀, N_a and N_b: the incident particles number, the absorbed particles number and the backscattered particles number. We suppose that N₀=10 (to facilitate calculations)

We suppose also that Monte Carlo method gives backscattering coefficient BSC1=0.3 (so we conclude that in this case N_b=3 and Na=7), and their model (combination between Monte Carlo and Vicanek and Urbassek model) gives BSC2=0.2 (so we conclude that in this case N_b=2 and N_a=8). Now, if we take BSC2 as a reference for calculations (accurate results), the absorbed particles number is 8, but when we use Monte Carlo, N_a is 7. Consequently, this is a contradiction: which is the correct 7 or 8? The problem of the normalization becomes very difficult when the target is a thin film[19] which is the case of their paper[6].

Consequently, both their works[1, 6] must be reviewed.

Moreover, we note that Rouabah et al[6] have used the Vicanek and Urbassek formula to calculate the backscattering coefficient in function of the film thickness. Let’s ask a question: who showed that the mathematical expression of the backscattering coefficient developed by Vicanek and Urbassek (equation (24)) is applicable for thin films? Knowing that this formula is valid only for semi-infinite solid case or for thin film with a thickness for which we can consider it as a semi-infinite solid. We note that Vicanek and Urbassek said -as example- the next clear expression “The present scheme has to be completed by semi-infinite medium boundary condition[5]”. This latter is the second proof of the no evidence of their BSC results in function of the film thickness, in one hand, and the equation (24) is not applicable for thin films, in another hand.

2.3. The Electron Range, the Mean Number of Wide Angle Collisions and the Backscattering Coefficient Results

Rouabah et al[1] used Monte Carlo simulating individual electron scattering events where the elastic model is that given by equations (20-22) and the inelastic processes are handled in terms of Gryzinski`s excitation function[20]. The Gryzinski`s differential cross section is given by,

(27)

where , , and E are the energy loss, the mean electron binding energy, and the primary projectile energy, respectively.

The electron range (R) calculated by Rouabah et al[1] can be deduced by using their data tables[1]. So, from the equations (25-26) we conclude that R can be written as follows:

(28)

Consequently:

(29)

(30)

So their electron range R_Al and R_Cu can be calculated by combining between the equations (29, 30) and their data (Tables 1, 2 and 4 of[1]).

We note that we have used the same code used by the authors of[1] but, unfortunately, we have not found the same results. So, in Table (5) we have presented their R_Al and R_Cu by substituting their data in the equations (29, 30)[1] and that obtained by using the same Monte Carlo code (denoted R^MC in the following).

The problem is the following: how to show that their results (denoted R^R) do not correspond to the true results? We assume that there are three methods (or we propose three methods) as follows:

The first method: is to use the Monte Carlo simulation using the same scheme explained by[18] and calculating the range (this way is the typical method for verification).

The second method: is to find a published work calculated the range for Al and Cu using the same Monte Carlo scheme. For this point, we note that we have not found any previous works calculating this one using the same procedures except the work of Bentabet[19] but the elastics models are not the same.

The third method: this latter is possible and practice. So, this method has been raised by Jablonski et al[21-22] «who used two types of algorithms one simulating individual electron scattering events and the other implementing CSDA. They found that both algorithms were in satisfactory agreement for primary energies exceeding 1 keV. At lower energies there were deviations up to 10% occurred due to numerical approximations[21]». We note that our work[19] confirm the later “notification of Jablonski et al”. On other words, since the authors of[1] have used the Monte Carlo simulating individual electron scattering events to calculate the range then their range should be near to that obtained by using CSDA (the deviation must be less than 10% particularly for E>1keV) otherwise there is a problem in their accuracy results.

The range calculated using the CSDA is given by where E₀ and S(E) are the primary energy and the stopping power of primary particle respectively. The integration was performed, generally, from the primary energy E₀ to the cutoff energy instead of 0 eV.

Indeed, we note that in our work[19] we have calculated the range using CSDA for Al. So, Table (6) represents their range (R^R), the range by using the same code (R^MC), the range using CSDA (R^CSDA) and the deviation between them. We note that Rouabah et al[1] have taken 20 eV as a cutoff energy however the R^CSDA calculated by[19] has been done for the cutoff energy equal to 100 eV. Consequently, when we added the range between 20 eV to 100 eV the deviation becomes greater (reaches ≈13 % for E=2keV). Therefore, on the basis of Jablonski et al notification, while the deviation is more than 10% (particularly for E>1 keV) then the results of[1] are incorrect. Consequently, all their results concerning the mean number of wide angle collisions and the backscattering coefficients are incorrect. We note that this error has been repeated by Rouabah et al in their work[6]. By consequence, both their works[1, 6] must be reviewed.

Important remarks:

■ We are recognized that the notification “to say that the algorithm range is true it must be no deviations between the ranges obtained analytically and that obtained by using CSDA in Monte Carlo scheme otherwise the used Monte Carlo code is wrong (except a statistical fluctuation, due to the use of the Monte Carlo simulation)” was the notice of the reviewer chosen by the Journal Nucl. Instrum. Methods Phys. Res. B to review the paper[19]. We think that this notification is the key to confirm the validity of the Monte Carlo Algorithm range.

■ In the case of the Monte Carlo method the statistical fluctuation error is calculated as , where N is the number of initial particles[22]. Since 104 particle histories were used by the authors of[1], this statistical error is found to be about 1%. On other words, the deviation of their ranges[1, 6] is not due to the statistical fluctuation but there is an error calculation.

3. Conclusions

In summary, in this comment we have showed that Rouabah et al transport cross sections are inaccurate and in reality were not based on Jablonski’s[4], as well. Moreover, their combination between Monte Carlo and Vicanek and Urbassek theory is not normalized and all their results concerning the mean number of wide angle collisions and the backscattering coefficients must be reviewed and revised. In other words, our work about Rouabah et al papers[1-3, 6-8, 16] can be summarized by point as follows:

1. (Transport cross section) attributed in[1-3, 6-8] to Jablonski[4] is not true, but it is an old cross-section.

2. Actually the transport cross section of Jablonski[4] is that given by

3. The tabulated results of (their tables 1-3 of[1]) do not correspond to their fit by using:

4. Depends on Z and E but not only on Z.

5. The passage from depends on Z and E to depends only on Z is not justifiable, if is not impossible.

6. The deviation of (15 %.., 20%,.., 25%, ..30%,...40%, …) shows clearly the invalidity of their fit[1-3].

7. The combination between Monte Carlo and Vicanek and Urbassek theory has been used without a normalized manner[1, 6].

8. The ranges calculated by Rouabah et al[1, 6] are not correct.

9. The mean number of wide angle collisions of Rouabah et al are not correct[1].

10. Some (if it is not all) their backscattering coefficients [1, 6-8] are not correct.

11. We recall that Rouabah et al have calculated the range by using Monte Carlo simulation where they used the screened Rutherford cross section and the Gryzinski model to describe the elastic and inelastic collisions respectively. We confirm that either by using CSDA or scattering by event of Monte Carlo schemes, the range calculated by Rouabah et al[1, 6, 8] do not correspond to the true values (of Gryzinski range).

12. Their erratum[16] itself invalidates their work[2].

Table (1). Transport Cross Section (in A°2) and the deviation of       to      . D: is the deviation.      .      : The electron transport cross section tabulated by Mayol and Salvat[15].      : TCS of Born approximation given by (13) with µ∞=1.22

Table (2). Energy dependence of the transport cross-sections and the deviation of       to       in function of Z expressed in A˚ 2.      : The electron transport cross section tabulated by Mayol and Salvat[15].      : The electron transport cross section tabulated by[1].

Table (3). Transport cross –section (in A°2).       : Rouabah et al TCS given by (13, 18)[3].       : Rouabah et al TCS given by (13, 19)[3].       Dapor TSC[17]. D1: percentage deviation between       and      . D2: percentage deviation between       and

Table (4). Energy dependence of the transport cross-sections in Al and Cu expressed in A˚ 2.      : Present work by using the elastic model of[18].

Table (5). the electron penetration range. RR : the electron range of Rouabah et al[1] by using equations (29-30). RMC the electron range by using the same code used by Rouabah et al[1]

Table (6). The electron penetration range in Al. RR: the electron range of Rouabah et al by using equations (29-30). RMC the electron range by using the same Monte Carlo code which has been used by Rouabah et al[1]. RMC2 the electron range by using Monte Carlo simulation calculated by[19]. RCSDA the electron range obtained by using CSDA scheme calculated by[19].