Argon scattering on clean and smooth W(110) surface

This study was performed for a wide range of initial parameters of the impinging beam and metal surface as well. Total of 1920 cases were computed. The current investigations were conducted for various angles of incidents (from βⁱ⁼⁰° to 70° with 5°

steps), a series of surface temperatures (Tsurf=350K, 400K, 450K, 500K), and varied velocities of impinging Ar atoms (from Vⁱ=100m/s to 1600m/s with 50m/s steps).

Any analytical or numerical study based on mathematical descriptions has assumptions and simplifications. Consequently, results of such works must be compared with related experimental results in order to prove the reasonability of assumptions.

Figure 4.1-2 shows the correlation of the mean kinetic energies of the impinging atoms

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(initial energy of the beam) with the mean kinetic energy of the beam scattered (average energy of scattered atoms) by the tungsten surface. Solid line in Figure 4.1-2 is the result from Janda experiments [102] and black cross symbols represent MD results obtained in the current study. Both results show the linear relationship between the mean kinetic energies of the impinging atoms and mean kinetic energy of the scattered atoms when the angle of the incidents was βⁱ⁼⁴⁵°. The linear relationship from MD results can be fitted by the following function:

(² )

s i

B S B surf

E b E b k T

< >= < > + , (4.1-1)

where kB is the Boltzhmann’s constant, and bS=0.18 and bB=0.77 are the proportionality factors. Analysis of Equation (4.1-1) allowed us to conclude that if the energy of the incident beam was less than bS(2kBTsurf)/(1-bB), then the surface transferred energy to the gas (E^S>Eⁱ), otherwise the gas gave up energy to the surface (Eⁱ>E^S).

The difference between the Janda experimental data and the MD simulation results does not exceed 9%. This difference can be explained by the fact that even the best MD models are a rough approximation of real processes. For example, the simulated substrate was ideal, i.e. it had no irregularities or scratches, while real substrate may have some faults. Based on the above, we conclude that achieving 9%

difference is a reasonable result.

Another important parameter of the gas-surface interaction is the momentum change of the atoms, which can be characterized by an angular distribution of scattered atoms. Figure 4.1-3 illustrates the normalized distributions of probability density of the polar angle of the scattered atoms (normalization was carried out by the peak value in order to improve visibility). Circles and crosses correspond to two cases of surface temperature, Tsurf=350K and Tsurf=500K, respectively. Furthermore, the temperature and

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incident angle of the argon beam were equal to TG=295K and βⁱ⁼⁴⁵°, respectively. As can be seen, the argon scattering distribution peaks increased in intensity with the increase of surface temperature. This finding is in line with the results of Weinberg [78].

Figure 4.1-4 shows the average energy of the scattered beam as a function of incident beam energy <Eⁱ> for different values of the incident angle βⁱ of the beam. It is clear that all plots presented in Figure 4.1-4 have two regions: (1) a linear region, where the mean energy of the scattered atoms linearly depended on the incident energy; and (2) a nonlinear region, corresponding to relatively low energies of impinging Ar atoms , where the mean energy of the scattered atoms were independent of the incidents’

energy. It should be noted that the slope of the linear regions on the plots shown in Figure 4.1-4 increases with increasing incident angles of Ar atom beams. The same effect was also noted by Agrawal and Raff [103]. The linearity of dependence between

<E^s> and <Eⁱ> can be clarified by using correlation coefficient presented in Table 4.1-1.

If the distribution of random numbers corresponded to a Maxwell-Boltzmann distribution, the relationship between root-mean-square deviation (RMSD) σ ^{and the} mean value μ would be described by the following relation (see e.g. [104]):

1 3 8 surface as a function of both incident beam energy and incident polar angle is presented in Figure 4.1-5a and Figure 4.1-5b. One can see that the value of σ^V/μ^V approaches 0.42 (shown in Equation (4.1-2)) with decreasing incident energy regardless of the incident

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polar angle βⁱ. Figure 4.1-6 shows that the velocities of scattered argon atoms having low (high) incident energy were distributed according to a Maxwell distribution (shown by a solid (dashed) line and crosses (diamonds)). Taking into account the ratio Eⁱmin≤ε^WAr≈0.95 (where E^imin was the minimal value of the kinetic energy of the impinging beam considered in this study), one can conclude that Ar atoms with kinetic energy Eⁱ≤ε^WAr sank into the potential well, induced by the surface atoms, and were thus scattered by the surface, regardless of the initial conditions. A similar effect was mentioned by Fan and Manson [79] and observed experimentally in Sec. 2 of Ref. [105]

and in Sec. 3.D of Ref [81].

Correlation between average energy of scattered atoms and parameters of incidence was approximated using the least square method (this method was used to obtain all approximation functions discussed in current work). Eq. (4.1-3) represents the dependence between <E^s>, Eⁱ and βⁱ. It should be noted, that the units of Eⁱ and β^{i are}

Coefficients A0 and B0 in Eq. (4.1-3) depend on polar angle βⁱ and their values are shown in Table 4.1-2.

One can easily check that if βⁱ⁼⁴⁵° then coefficients b^S and bB in equation (4.1-3) have the values of 0.155 and 0.837, respectively, which are less than 9% difference from the values shown in Ref. [102]. It should be noted that the factor bS of equation

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(4.1-3) increases with increasing incident angles of Ar atom beams. The same effect was also noted by Agrawal and Raff [103].

Correlation of the normalized root mean square deviation of scattered atom’s velocities with parameters of incidence shown in Figure 4.1-5 can be approximated by Eq. (4.1-4) and the coefficients A1n,k are listed in Table 4.1-3.

( )

{

( ) } ^{( )}

Both polar and azimuthal angles of the scattered atoms were varied within ranges of [-π/2;π/2], thus the normalized RMSD corresponding to a uniform distribution for random numbers within this range was [106]:

180 30 3

12 12

uni

σ =b a⁻ = = (4.1-5a)

Thus, the normalized RMSDs of scattering of the scattering angles α^{s and} β^s, respectively, are defined as follows:

uni 30 3 4.1-8, respectively, were normalized by the σ^uni from Equation (4.1-5a). Based on the results shown in Figure 4.1-7, one could conclude that the mean azimuthal angle of the scattered beam α^s was independent of incident kinetic energy and incident angle βⁱ when an atom’s beam impinged normally on the surface (data points represented by

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cross symbols). Figure 4.1-8 presents correlation similar to those shown in Figure 4.1-7, but for the polar angle β^s of the scattered beam.

Normalized RMSDs of the azimuthal angle α^s and polar angle β^s of scattered atoms versus kinetic energy of impinging atoms and incident polar angle shown in Figure 4.1-7 and Figure 4.1-8 are approximated by the polynomials (4.1-6) and (4.1-7), respectively:

It is obviously from Figure 4.1-8 that the RMSD of polar angle of scattered atoms is inversely proportional to the incident energy, i.e. the angular distribution becomes narrower while the incident energy increases. Figure 4.1-9 shows that fraction of atoms scattered in backward direction is sufficient in case of low incident energy Ar atoms and it decreases when the energy of falling beam becomes higher. These conclusions are in line with experimental results shown in Ref. [81, 105].

Coefficients A2 n,k and A3 n,k for Eq. (4.1-6) and (4.1-7) are listed in Table 4.1-4 and Table 4.1-5, respectively.

Figure 4.1-10a and Figure 4.1-10b illustrate the correlation of the average polar angle β^s of the scattered atoms with incident kinetic energy and polar angle β^{i. This is} approximated by the polynomial Eq. (4.1-8) having coefficients A4n,k which are listed in Table 4.1-6.

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The results presented in Figure 4.1-10a show that an increase in incident energy caused linearization of dependence between the incident and the scattered polar angle, while the average polar angle of the scattered argon atoms that had low incident energy tended toward zero. Correlation coefficient between average value of polar angle β^{s of} scattered atoms and incident angle βⁱ of the beam is shown in Figure 11. Figure 4.1-11 proves that correlation between angles mentioned above becomes linear when the incident energy is high enough.

It is clear that the distribution of the angle β^s did not become fully uniform (the angle did not become independent of other parameters) in the range of studied incident parameters of Ar beam. On the other hand, the distribution of the angle α^{s became} uniform (as shown by the cross symbols in Figure 4.1-12) when the incident beam had low energy (<Eⁱ>/(2kBTsurf)≈0) or was a normal falling beam (βⁱ≈0), and became Gaussian when the incidents had high energy (shown as crosses and a dashed line in Figure 4.1-13). Comparing angle β^s in Figures 4.1-12 and 4.1-13, one can see that the scattering of Ar atoms was more specular in the case of the high energy falling beam (Figure 4.1-13) than in the case of the low energy one (Figure 4.1-12). It should be noted that scattering does not become fully specular in the range of studied parameters, because Ar atoms change their energy and momentum upon collision. This issue can also be explained in terms of relation between Eⁱmin and ε^WAr.

Probability distribution functions (PDF) presented in Figures 4.1-12 and 4.1-13 have 30 bins and the critical value of Chi-square corresponding to 0.05 level of significance is 42.56. Values of χ² for the curves fitting the data points presented in Figure 4.1-12 are 9.18 and 13.01, corresponding to uniform PDF of α^s and normal distribution of _β^s, respectively. The values of χ² for functions fitting the distributions of

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angles _α^s and _β^s of scattered atoms are 11.13 and 12.94, respectively. All these values do not exceed the critical value of the Chi-square. It means that these distribution functions can fit data points well.