The sketch of the Ar ^q+ ion grazing scattering from the tungsten (W) surface is shown in Figure 1. Based on the COBM [16]; [18], resonant capture (RC), resonant loss (RL), autoionization (AI), peeling off (PO), and side feeding (SF) are adopted to describe the neutralization of Ar ^q+ ions, resulting in the corresponding charge-exchange energy loss. As the Ar ^q+ ion approaches the tungsten surface, it induces the dielectric response phenomenon, creating an “image charge” that accelerates the HCI. When the Ar ^q+ ion reaches a critical distance R _c , the electrons from the solid surface can be captured into the high-lying shells of the Ar ^q+ ion in a very short time, leaving the low shells empty. As a result, the hollow atom (HA) is formed. Then, the HA decays to its ground state via Auger transition and X-ray emission [19]; [20]; [21]; [22]. R _c is given by the following equation: R c ≈ 1 2 W 8 q 1 + 2 , (1)

where W is the work function of the metallic surface and q ₁ is the initial charged state of projectiles. This critical distance marks the beginning of neutralization and the production of the HA. The work function of the metallic surface plays a vital role in the ion–target interaction process, as indicated by Eq. 1. In this study, the energy losses of ions interacting with three tungsten surfaces are compared. The work functions of W (111), W (100), and W (110) are 4.47 eV, 4.63 eV, and 5.25 eV, respectively [23].

When the interaction potential between an ion and target changes dynamically, the well-established Thomas–Fermi (TF) statistical atom model can be used, so the Coulomb potential between a projectile and a target atom can be approximated as [9]; [24] W TFM = Z 1 Z 2 r ϕ r / a F , (2) where Z ₁ and Z ₂ are the nuclear charges of the projectile and target atom, respectively, r is the inter-atomic distance, and ϕ r / a F represents one particular interatomic “screening function (Eqs 3, 4).” In the simulation, the Thomas–Fermi–Moliere screening function was adopted [9]; [25]: ϕ r / a F = ∑ i a i e x p − b i r / a F , (3) in which a _i = {0.35, 0.55, 0.1} and b _i = {0.3, 1.2, 6.0}. a _F is the screening length [25]; [26]; [27]. a F = 0.8854 Z 1 − q 1 2 / 3 + Z 2 − q 2 2 / 3 , (4)

where q ₁ and q ₂ are the charge states of the projectile and target atom, respectively.

As the HCI approaches the surface, the outer-orbital electrons from ions may be peeled off by the electrons on the target atoms when the distance between the HCI and free electron gas (FEG) edge of the surface is less than a certain threshold value [28]; [29]. SF may occur when the projectile approaches the surface closely enough. Furthermore, more details about the inelastic collision process (electron energy loss contributions) are described in Section 3.

Tungsten targets with 25 atoms per layer were constructed using Vesta software, exhibiting distinct atom arrangements. For bcc W, the lattice constant a is 3.16 Å. The face spans of the targets are 3 6 a and 1 2 a , 2 2 a (equal to 0.88 Å, 1.58 Å, and 2.24 Å) corresponding to W (111), W (100), and W (110), respectively. Notably, the W (110) surface represents the closest packed surface of bcc W, while W (111) exhibits the most open structure. The surface atom densities of the outermost layer are 34.09%, 58.87%, and 83.26% for W (111), W (100), and W (110), respectively [30].

To accurately estimate the energy loss of ions, it is essential to describe correctly the crystal structure and the inhomogeneous electron density distribution of the surface. In this case, the electron densities of W surfaces were calculated using the VASP [15]. The choice of interatomic potentials is crucial for the calculation of surface electron density. F. J. Domínguez-Gutiérrez et al. discussed various interatomic potentials for tungsten, considering different potential models like EAM, EAM/ZBL, MEAM, ABOP, and tabGAP for nanoindentation simulations [31]. However, only the ground state of tungsten is needed in this work. Therefore, based on the first principle of density functional theory (DFT), VASP utilizes standard pseudopotentials from the potential library and a plane-wave basis set to perform ab initio total-energy calculations for metallic systems. This package is widely used for electronic structure calculations across various materials, including metallic and semiconducting surfaces, as well as liquid and amorphous semiconductors, even for systems of considerable size.

The charge-exchange energy loss is one of the focuses of this study. As previously mentioned, the charge transfer occurs mainly through resonant and Auger processes. In the simulation, the coupled rate equation (Eq. 7) were used to judge whether the charge-exchange processes occurred [33]. The population P _n of the n _th shell can be described by the rate equation [9]; [16]: d P n d t = θ S n − P n I n R C − I n R L P n + θ S n − P n ω n fin ∑ n ′ > n A n ′ , n ω n ′ ini − 2 ω n ini ∑ n ′ < n A n , n ′ ω n ′ fin − I n P O P n + θ S n − P n I n S F + θ S n − P n ∑ n ′ > n Γ n ′ , n − ∑ n ′ < n Γ n , n ′ , (7)

where I n R C and I n R L are the currents of the resonant captured and lost electrons, respectively, I n P O is the decay rate of the peeling-off process, I n S F is the electron capture rate of the SF process, ω n ini and ω n fin are the (empirical) statistical factors used to correct the Auger rate, A n ′ , n = 5.06 × 1 0 − 3 | △ n | 3.46 , Γ n , n ′ is the radiative decay rate, S _n is the number of electrons fully filled in the n _th shell, and θ is the unit step function.

Accompanying the charge exchange of the HCI with the metallic surface, the charge-exchange energy loss takes place. This charge-exchange energy loss can be approximated as a soft collision [34]. In the simulation, P, T, and m denote the projectiles, target atoms, and electrons, respectively. When the high-lying levels of the HCI capture surface electrons, giving rise to the process P + (T + m) → (P + m) + T, the charge-exchange energy loss can be calculated using the following expression[35]: Δ E cap = B T − B P + E P m P 4 m T m P + m m P B T − B P E P − m m P 2 , (8) where m _P , m _T , and m represent the masses of the projectiles, lattice atoms, and electrons, while B _P and B _T signify the binding energy of the exchanged electrons, and E _P corresponds to the energy of the projectiles (Eq. 8). However, in the (P + m) + T → P + m + T process, when the electron resonates loss with unoccupied states on the crystal surface, the contribution of B _T to the energy loss can be neglected. As a result, the expression for charge-exchange energy loss can be written as [35] Δ E loss = 1 2 m P v 2 + m P m P + m B + E P m P 4 m T ⋅ B E P 2 , (9)

where v represents the laboratory-frame velocity of the projectiles and B = B _P + E _c , where E _c stands for the mean kinetic energy of electrons separated from the projectiles (Eq. 9), a concept thoroughly discussed in [36].

The interaction of Ar ¹⁷⁺ ions with the W target was simulated using a Monte Carlo program developed by our group. The theoretical basis of the simulation is described as follows.

The force experienced by particles at a certain position above the surface can be expressed as F ⃗ q , R = F i m e ̂ x + ∑ F ⃗ TFM R , (10)

where q represents the charge state of projectiles, R indicates the position relative to the topmost layer, and the first term on the right side of the equation represents the attractive force of the induced image charges. The second term represents the sum of the repulsive forces exerted by all atoms on the particles. The surface potential Eq. 2 is employed in Eq. 10.

As soon as the ion–surface distance R < 5.0 a.u. is reached, the short-distance EEL is calculated immediately. The energy loss for a step length is given by d E = S e R · d l , (11) where S _e is the position-dependent electron stopping power. When considering the stopping power of incident ions, the dielectric effect of the target material needs to be considered. The induced potential generated by ions grazing on a solid surface is V ind r ⃗ , t = V r ⃗ , t − V coul r ⃗ , t = Q 2 π ∫ d 2 k / / k / / e x p i k / / ⃗ r ⃗ − v / / ⃗ t − k / / z 0 ′ + | z ′ | 1 − ε 1 + ε . (12) The local dielectric function for metals is written as [37] ε ω = 1 − ω p 2 ω ω + i γ , (13) where ω p = 4 π n 0 and n ₀ is the electron density. Eqs 12, 13 indicate that the induced potential on the solid surface is related to the electron density n ₀, which is associated with the polarization capability (dielectric constant) of the target. From a qualitative perspective, when ions graze scattering from a surface with a high charge density, the uniformity of electron distribution on the surface is disrupted. However, the electrons tend to restore and involve certain relaxation time so that ion motion is hindered. Therefore, the energy loss of ions per unit distance is inversely proportional to ion velocity. Then, the expression for S _e in Eq. 11 is as follows [37]; [38]: S e R = 2 k F 2 v v F ∫ 0 1 k d k σ n k 2 H k , z 0 ′ , (14) where k _F = v _F is the Fermi wave number, z 0 ′ is the position from the FEG edge of the solid surface, and σ _n (k) is the space Fourier transform of the total charge distribution localized at the incident ion based on the Brandt–Kitagawa (B-K) model [39]. The factors in Eq. 14 can be seen in Eqs 15–20.

When the projectile is located outside the FEG ( z 0 ′ > 0 ) , the position-dependent ionization degree can be expressed as a double exponent [37]: q z 0 = q 0 ⁡ exp − exp − z 0 − z s L , (15) where q ₀ is the initial ionization degree, L is a characteristic length, and z s = L ⁡ ln Γ 0 L / v n , with v _n being the perpendicular velocity and Γ ₀ being the typical resonant ionization rate, which is taken to be of the order of 10¹⁵ s ⁻¹. The maximum value of q(z ₀) is reached at z ₀ = z _s [37].

When the projectile enters the FEG ( z 0 ′ < 0 ) , an empirical model is used [40]. It should be noted that there are no known theories within the current research domain that can accurately describe this situation. Therefore, the empirical model was utilized to offer an initial explanation. However, further research is encouraged to explore more accurate models in depth. The ionization degree is given by q b = 1 − exp 0.803 y r 0.3 − 1.3167 y r 0.6 − 0.38157 y r − 0.008983 y r 2 , (16) where y r = v r / Z 1 2 / 3 , with v _r being the ion velocity relative to the target–electron velocity, defined as follows: v r = v 1 + v F 2 5 v 2 , v ⩾ v F , v r = 3 v F 4 1 + 2 v 2 3 v F 2 − 1 15 v v F 4 , v < v F . . (17) Then, in Eq. 14, H k , z 0 ′ = 2 f i k 1 + f r k 2 exp − 4 k F z 0 ′ k , z 0 ′ > 0 (18) or H k , z 0 ′ = f i k + f i k , 2 z 0 ′ + 2 f r 2 k , z 0 ′ f i k 1 + f r k 2 − 4 f r k , z 0 ′ f i k , z 0 ′ 1 + f r k , z 0 ′ < 0 , (19) where f _r (k) = f _r (k, 0) and f _i (k) = f _i (k, 0); the detailed expression of f _r (k, z) and f _i (k, z) is given in [41], in which we learned that a parameter γ ₀ links to all of factors of Eq. 14, and γ 0 = 1 − k / k 0 4 α r s / π , where α ≡ (4/9π)^1/3 and r s = ( 3 / 4 π n e ) 1 / 3 is an important non-dimensional parameter to describe the properties of electron gas in the expression of γ ₀ k k 0 = 1 − α r s π + α 2 6 r s 6 d d r s r s − 2 d d r s E c r s , (20) where E _c (r _s ) is the correlation energy of electron gas, which is given in [42]. Further details are given in [9]; [24,27].

Variation in S _e is evident in the profile of the relationship between S _e and ion–surface distance z on the W (111) surface, as shown in Figure 2. As the ion–surface distance z decreases, S _e increases exponentially with z, coinciding with a rapid and double-exponential increase in the ionization degree of incident ions. However, when z < 2.87 a.u., S _e decreases with decreasing z until approximately z = 0.56 a.u. and then shows an increasing trend. In the range 0.56 a.u. < z < 2.87 a.u., ions enter the FEG on the surface, and the growing screening effect of the target electrons on ions reduces the polarization capability of the ions on the surface. When z < 0.56 a.u., the increase in S _e can be attributed to the contribution of inner-shell electron ionization of the target atoms.

Figure 3 presents two-dimensional contour-color plots depicting the electron density n _e and the corresponding S _e on a plane (z = 1.49 a.u.) for W (111), W (100), and W (110) surfaces, respectively. These results reveal the varying corrugation in S _e across different crystal surfaces, with reduced S _e in areas of higher electron density. This observation corroborates the finding in the vertical direction and suggests that S _e also depends on the inhomogeneous electron density distribution on the x–y plane.

The three-dimensional spatial distribution of S _e on the W (111), W (100), and W (110) surfaces is shown in Figures 4A–C. It is noteworthy that the corrugation of S _e is most pronounced on the W (111) surface, whereas the corrugation of S _e is flat on the W (110) surface. These features are attributed to W (110) and W (111), corresponding to the closest packed and most open atom arrangements, respectively. It is also observed that the S _e distribution under the axial channeling (red arrow, grazing along low-index directions) is relatively more uniform compared to the planar channeling (grazing along random directions).

Furthermore, simulations were performed involving the grazing scattering of 180-keV Ar ¹⁷⁺ ions from tungsten surfaces across a full azimuthal range (0°–360°) at an incident angle of 0.8°. The surface condition was set as either a uniform or an inhomogeneous electron gas. The energy loss was enhanced when ions grazed under axial channeling for surfaces with uniform or inhomogeneous electron gas (Figure 9). For the W (111) and W (110) surfaces, the energy loss varies with a period of 180° (Figures 9A, C), while the energy loss changing period is 90° for the W (100) surface (Figure 9B). The feature is in good agreement with the periodic arrangement of atoms on each crystal surface. The results show that information about the structure of the crystal surface can be obtained by studying the relationship between the energy loss of projectiles and the azimuthal orientation of the surface.

For the W (111) surface (Figure 9A), the influence of the inhomogeneous distribution of surface electron density on the energy loss becomes evident only under axial channeling, because the closest distance is larger for ions grazing on the W (111) surface under planar channeling compared to that under axial channeling (Figure 10B). Therefore, ions can reach a deeper position, in which the change in electron density corrugation will be more intense for ions grazing under axial channeling. Meanwhile, the trajectory length of ions grazing on the W (111) surface is longer, resulting in greater energy loss than that on the W (100) and W (110) surfaces (Figure 10A).

For the W (100) surface (Figure 9B), the energy loss significantly depends on the corrugation of the electron distribution. The difference is approximately 300 eV when ions graze scattering under planar channeling, and the deviations can reach 400 eV when ions move under axial channeling. First, ions can reach deeper positions from the surface than from the W (111) surface (Figure 10B). Second, the inhomogeneous distribution of the electron density on the W (100) surface is more pronounced than on the W (110) surface (Figure 4). The results show that the dependence of the electron density distribution on the energy loss of ions grazing on the W (100) surface is stronger than that on the W (111) and W (110) surfaces.

In Figure 9C, the ordered arrangement of atoms on the W (110) surface allows a smoother electron density distribution, leading to minimal impact of both uniform and inhomogeneous electron gas conditions on ion energy loss (less than 100 eV under planar channeling).

These observations highlight the necessity for a detailed characterization of the crystal surface structure and the inhomogeneous distribution of surface electron density for ions grazing on surfaces.