# Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View

## Abstract

**:**

## 1. Introduction

- The
**Fermatian level**, which refers to the analysis of the problem assuming a bare hard-wall-like (fully repulsive) model to describe the He-CO/Pt(111) interaction. Because the trajectories here are of the type of sudden impact (free propagation except at the impact point on the substrate wall, where the trajectory is deflected according to the usual law of reflection), they are going to be straight-like rays, as in optics (this is why it is referred to as Fermatian). - The
**Newtonian level**, where the He-CO/Pt(111) interaction is modeled in terms of a potential energy surface that smoothly changes from point to point. This model has a repulsive wall that avoids He atoms to approach the substrate beyond a certain distance (for a given incidence energy), and an attractive tail that accounts for van der Waals long-range attraction. The existence of these two regions, repulsive and attractive, gives rise to an attractive channel around the CO adsorbate and that continuous along the flat Pt surface, inducing the possibility of temporary trapping for the He atoms. - The
**Bohmian level**is the upper one and, to some extent, makes an important difference with the previous models because here the trajectories are not only dependent on the interaction potential model, but also on the particular shape displayed by the wave function at each point of the configuration space at a given time (the “guiding” or “pilot” wave).

## 2. Potential Model and Computational Details

## 3. Wave-Function Approach

#### 3.1. Diffraction from a Repulsive Hard-Wall Potential

#### 3.2. Diffraction from the Potential Model (2)

## 4. Trajectory-Based Description

#### 4.1. Fermatian Level

- Trajectories to the left of ${\mathcal{F}}_{1}$ or to the right of ${\mathcal{F}}_{7}$ only interact with the clean Pt surface and hence their deflection and incidence angles are equal. These trajectories, plus ${\mathcal{F}}_{5}$ only contribute to mirror reflection from the flat surface, only contributing the intensity for $\Delta K=0$, since ${\theta}_{d}={\theta}_{i}$—hence, this contribution will be more prominent as the range of impact parameters increases.
- Any trajectory between ${\mathcal{F}}_{1}$ and ${\mathcal{F}}_{\alpha}$ is deflected in an angle that goes from ${\theta}_{i}$ to ${0}^{\circ}$ as the impact parameter increases. The same deflection angles are found for trajectories between ${\mathcal{F}}_{\beta}$ and ${\mathcal{F}}_{5}$, although here the trend is that the angle increases from ${0}^{\circ}$ to ${\theta}_{i}$ as b increases. Here, we have two sets of pairs of homologous trajectories: trajectories from the first set undergo double collisions (first with the flat surface and then with the adsorbate) and trajectories from the latter only have a single collision (with the flat surface). For any of these pairs, the angular distance between their impact points on the adsorbate surface is $\pi /2-{\theta}_{i}$, as can be seen in Figure 6a.
- There are also pairs of homologous trajectories with deflection angles between ${\theta}_{i}$ and $\pi /2$. These are the trajectories confined between ${\mathcal{F}}_{5}$ and ${\mathcal{F}}_{6}$, with single collisions (with the adsorbate), and between ${\mathcal{F}}_{6}$ and ${\mathcal{F}}_{7}$, with double collisions (first with the adsorbate and then with the flat surface). This second set corresponds to trajectories impinging on the adsorbate within the sector B. In this case, the angular distance between impact points is not a constant, but depends on the deflection angle as $\pi /2-{\theta}_{d}$. This distance gradually vanishes as both trajectories approach ${\mathcal{R}}_{6}$ and is maximum when the trajectories coincide with the separatrices ${\mathcal{R}}_{5}$ and ${\mathcal{R}}_{7}$. A representative set is depicted in Figure 6b.
- Trajectories ${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{4}$ are both deflected backwards along the incidence direction, i.e., ${\theta}_{d}=-{\theta}_{i}$. Accordingly, trajectories between ${\mathcal{F}}_{\alpha}$ and ${\mathcal{F}}_{2}$ are deflected between ${0}^{\circ}$ and $-{\theta}_{i}$ after undergoing double collisions (first with the flat surface and then with the adsorbate), while trajectories between ${\mathcal{F}}_{4}$ and ${\mathcal{F}}_{\beta}$ (to the right of ${\mathcal{F}}_{4}$) undergo single collisions. The angular distance between impact points of homologous pairs of trajectories is now $\pi /2-{\theta}_{d}$, although not all trajectories between ${\mathcal{F}}_{4}$ and ${\mathcal{F}}_{\beta}$ have a correspondent between ${\mathcal{F}}_{\alpha}$ and ${\mathcal{F}}_{2}$. This is because the flat surface intersects the adsorbate surface at a distance $z={z}_{r}$ above its center of mass instead of at $z=0$. Thus, instead of reaching a maximum deflection of $-{\theta}_{i}$, we have ${\theta}_{d}^{\mathrm{max}}=-{\theta}_{i}+{(\mathrm{sin})}^{-1}({z}_{r}/a)$, which is the deflection for the trajectory ${\mathcal{F}}_{2}^{\prime}$. An illustrative pair of homologous trajectories of this kind is displayed in Figure 6c.
- The trajectory ${\mathcal{F}}_{3}$ separates the sets of homologous trajectories that are backward deflected, with the second collision taking place from the flat surface. One set is confined within trajectories ${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$, with double collisions (first with the adsorbate and then with the flat surface), and the other set, with single collisions, is delimited by ${\mathcal{F}}_{3}$ and ${\mathcal{F}}_{4}$ (trajectories to the left of ${\mathcal{F}}_{4}$). Unlike the previous set of backward-scattered homologous pairs, here all trajectories are paired, with the angular distance between their impact points being $\pi /2-{\theta}_{d}$, as before. A representative pair is displayed in Figure 6d.

#### 4.2. Newtonian Level

#### 4.3. Bohmian Level

## 5. Conclusions

**Hard-wall model.**This model is in the form of an impenetrable (fully repulsive) wall, where the interaction is reduced to a sudden impact on the He atoms on the such a wall. The first model allows an exact asymptotic analytical treatment, convenient to elucidate the main mechanism observed in the diffraction pattern produced by single adsorbed particles on nearly flat surfaces, namely reflection symmetry interference.**Potential energy function.**This interaction model is determined from fitting to the experimental data and constitutes a refinement of the previous one in the sense that there is detailed information on the intensity of the interaction between the incoming atom and the substrate at each point (in this regard, the hard-wall model is just a crude approximation). Thus, in spite of its lack of analyticity, unlike the hard-wall model, it provides us with a more realistic description of the diffraction process in real time, rendering information on additional physics, such as rainbow features or surface trapping.

**Fermatian level.**This first level is the simplest one, based on computing what has been here denoted as Fermatian trajectories, which are just the direct analog to optical rays reflected on a hard wall in a medium with constant refractive index. According to this trajectory model:- -
- These trajectories have revealed that there are pairs of homologous trajectories, such that one of the peers undergoes single scattering off the interaction potential, while the other undergoes double scattering. The fact that a trajectory collides with the CO/Pt system at one point (single collision) or at two different points (double collision) is a function of the impact parameter. Accordingly, a simple mapping can be establish, which helps to easily localize regions of impact parameters that are going to produce homologous pairs of trajectories.
- -
- The mechanism of reflection symmetry interference is associated with these paired trajectories, which is explained in the same way that we explain interference from two coherent sources: interference maxima and minima arise depending on whether the path difference between the two paths (or virtual rays) joining each source with a given observation point on a distant screen is equal to an integer number of wavelengths or to half an integer, respectively. Although these paths are nonphysical (they are just a mathematical construct), they allow us to understand in simple terms the appearance of the alternating structure of bright and dark interference fringes. In the present case, the path length arises from the extra path length of the trajectory affected by the double collision with respect to the homologous pair with single collision.
- -
- In addition, it has also been seen that two specific trajectories are deflected parallel to the surface, which can be interpreted as a mechanism precursor of the surface trapping mechanism that appears in more refined models, such as the Newtonian and the Bohmian ones.

**Newtonian level.**On the next level, the Newtonian one, classical trajectories are obtained for the realistic potential energy surface describing the interaction between the He atoms and the substrate. In this case, it is not so simple to distinguish between single and double collisions, because the deflection of the trajectories near the surface, where the interaction between the He atoms and the CO/Pt surface is stronger, changes gradually very smoothly. However, we have been able to extract a series of interesting conclusions:- -
- By means of an energy diagram (asymptotic energy along the z direction as a function of the impact parameter), we been able to devise a method that allows to determine in a simple fashion pairs of homologous (Newtonian) trajectories. This diagram is thus a suitable method to determine a behavioral mapping of initial conditions (impact parameters) for a given incidence direction (incident energy).
- -
- Accordingly, also at this level, it is possible to find an underlying mechanism responsible for the reflection symmetry interference found in the corresponding quantum intensity patterns. Actually, interference patterns could be reconstructed in the same way as with the Fermatian model, although in this case we would be dealing with a space-dependent refractive index (the potential function) and the Newtonian trajectories would play the role of Feynman’s paths. Nonetheless, although such a reconstruction is possible and the techniques are well known, this does not mean that trajectories, Fermatian or Newtonian, contain any information on the interference process; in both cases, they are only a tool to determine the interference pattern.
- -
- Regarding the trapping phenomenon, it has been found to be more prominent, with an important amount of trajectories remaining trapped permanently along the surface. This is, however, only a temporary feature, since it may disappear as son as the trapped atoms find another adsorbate. In such a case, the collisions with this adsorbate may provoke an effective transfer of energy from the parallel to the normal direction, such that the will be able to eventually leave the surface.
- -
- Finally, due to the attractive well surrounding the adsorbate, we have also observed the appearance of rainbow features, i.e., high accumulations of trajectories along particular deflection directions. However, rather than contributing with a specific, localized feature in the corresponding quantum intensity pattern, rainbows seem to manifest affecting them globally, i.e., giving rise to features that appear at different places. This has been noticed by computing exactly the same with an alternative repulsive adsorbate model, which lacks the surrounding attractive well and therefore does not give rise to the formation of rainbows.

**Bohmian level.**The upper level here considered is the Bohmian one, where things change substantially if we note that the transition from the Fermatian level to the Newtonian one can be seen as a refinement associated with having a more accurate description of the interaction potential model, changing a hard wall by a “soft” wall. These are the main findings at this level:- -
- First of all, since Bohmian trajectories are associated with a particular wave function, there is no freedom to choose a given set of initial conditions because depending on the positions selected relative to the region covered by the initial wave function, the trajectories are going to exhibit a different behavior. Thus, we have seen that while some of them are deflected quite far from the physical surface (more intense interaction region), other trajectories move just on top of it, displaying signatures of vorticality.
- -
- To better understand that point, notice that Fermatian trajectories are only ruled by the law of reflection, while Newtonian trajectories are ruled by correlations between the two degrees of freedom, x and y, that can be locally established within the interaction region (i.e., the region where the interaction potential is stronger, near the substrate). In the case of Bohmian trajectories, the dynamics is not directly ruled by the interaction potential, but by a wave field that is able to (non-classically) convey information from everywhere in the configuration space (through its phase). This makes a substantial difference between classical (Fermatian or Newtonian) and Bohmian trajectories, which may lead us to think that direct comparisons or analogies must be taken with care. That is, nothing of what has been seen at the previous levels remains at the upper one, since it is not possible to form pairs of homologous trajectories.
- -
- In this case, and contrary to the two previous models, the trajectories contain information about the interference process and, therefore, can be used to determine the fringe structure of the pattern by simply making statistics over them. If they are properly distributed across the region of the configuration space covered by the initial probability density, they will eventually distribute according to the final probability distribution by virtue of the continuity equation that they satisfy.
- -
- Regarding rainbow features, present in the Newtonian model and also, with a weak precursor, in the Fermatian one, the only a similar behavior is observed, although it is difficult to establish a unique correspondence with the phenomenon of the two previous models. In the Bohmian case, taken the trajectories that start with the same value ${z}_{0}$, it is seen that their final positions show, for some range of ${x}_{0}$ values, a certain “precession” as ${x}_{0}$ increases. However, it has not been possible to uniquely identify this phenomenon with the classical rainbow. In the case of surface trapping, on the contrary, there same effect has been observed in the three models (again, in the Fermatian model it is only a weak precursor).
- -
- Finally, it has also been observed that, depending on how close or far a Bohmian trajectory is started from the physical substrate surface, it will be able to reach this surface or just bounce backwards quite far from it (from what we could call an effective nonphysical surface). Actually, if the trajectories start close to the surface, they are influenced by the web of maxima developed (and sustained for some time) around the adsorbate, displaying a rich vortical dynamics.

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) contour plot of the He-CO/Pt(111) interaction potential model (2) (see text for details). The energy difference between consecutive repulsive/attractive contour levels (red dashed lines/blue solid lines) is 10 meV/1 meV. The thick black solid line denotes the repulsive boundary for an approximate hard-wall model set for an incidence energy of 10 meV (see text for details). In the right-hand side panels, energy profiles along the z direction for: (

**b**) $x=0$ Å; (

**c**) $x=3.31$ Å and (

**d**) $x=6.35$ Å. In these panels, the total interaction potential is denoted with black solid line, while red dashed and blue dash-dotted lines refer to the Morse and Lennard–Jones contributions, respectively.

**Figure 2.**(

**a**) Relative diffraction intensity (black solid line) produced by a radial hard-wall model for incidence conditions ${\theta}_{i}={0}^{\circ}$ and ${E}_{i}=10$ meV. To compare with, the Fraunhofer and illuminated-face intensities are also shown, which are denoted with red dashed line and blue dash-dotted line, respectively; (

**b**) As in panel (

**a**), but for ${E}_{i}=40$ meV.

**Figure 3.**On the left-hand side, contour plots illustrating three different instants of the evolution of the probability density near the surface for incidence conditions ${\theta}_{i}={0}^{\circ}$ and ${E}_{i}=10$ meV: (

**a**) when the density starts being influenced by the adsorbate; (

**b**) when the density is totally interacting with the substrate (i.e., with both the adsorbate and the flat Pt surface) and (

**c**) when the density starts leaving the substrate. In panel (

**d**), on the right-hand side, plot of the probability density far from the influence of the adsorbate ($t=11$ ps). Arrows and capital letters denote different diffraction directions to be identified in the intensity plot displayed in Figure 4a: ${A}_{i}$: directions identifying interference features associated with the superposition of the circular and planar wavefronts, contributing to the central maxima of the intensity pattern; ${B}_{i}$: associated with features arising from the reflection symmetry interference phenomenon; C: surface trapping.

**Figure 4.**(

**a**) Relative diffraction intensity (black solid line) produced by an axial-symmetric potential model based on (2) for incidence conditions ${\theta}_{i}={0}^{\circ}$ and ${E}_{i}=10$ meV. For comparison, the intensities before (red dashed line) and after (black solid line) removing the plane-wave contribution (see text for details) are both plotted; (

**b**) the same as in panel (

**a**), but for a fully repulsive model of the CO adsorbate, obtained after removal of the attractive part of the Lennard–Jones function (4). To compare, in panels (

**c**,

**d**), the same as in panels (

**a**,

**b**), respectively, but for ${E}_{i}=40$ meV.

**Figure 5.**Separatrix Fermatian trajectories for the repulsive hard-wall model set for an incidence energy ${E}_{i}=10$ meV. For a better illustration of all possible cases, an incidence angle ${\theta}_{i}={20}^{\circ}$ has been chosen. Separatrices displaying forward/backward deflection are denoted with solid/dashed lines. Separatrices delimiting regions leading to double scattering in the rear/front part of the adsorbate (regions C-A/B) are denoted with red/blue color. Separatrices deflected perpendicularly to the flat surface are denoted with green color. The shadow region S, which depends on the incidence angle, cannot be reached by any trajectory (except for ${\theta}_{i}={0}^{\circ}$, where this area goes to zero).

**Figure 6.**In panels (

**a**–

**d**), pairs of homologous Fermatian trajectories with different deflection angle ${\theta}_{d}$: (

**a**) forward deflection, with ${\theta}_{i}\ge {\theta}_{d}\ge 0$; (

**b**) forward deflection, with $\pi /2\ge {\theta}_{d}\ge {\theta}_{i}$; (

**c**) backward deflection, with $-{\theta}_{i}+\delta \le {\theta}_{d}\le 0$, where $\delta ={(\mathrm{sin})}^{-1}({z}_{r}/a)$ and (

**d**) backward deflection, $-\pi /2\le {\theta}_{d}\le -{\theta}_{i}$. In panels (

**e**,

**f**), boundaries of the regions where any trajectory impinging on them will display double collisions (see text for further details).

**Figure 7.**(

**a**) Set of classical Newtonian trajectories for the interaction potential model (2) for an incidence energy ${E}_{i}=10$ meV. As in Figure 5, for a better illustration of all possible cases, also an incidence angle ${\theta}_{i}={20}^{\circ}$ has been chosen. Moreover, the incident part of the trajectories has been represented with dashed line; (

**b**) classical deflection function. While surface trapping gives rise to two kind of discontinuous regions, rainbow features manifest as local maxima (1) and minima (2); (

**c**) Asymptotic-energy diagram. Here, trapping is detected through the two regions of the curve below the threshold ${E}_{z}=0$ meV, while rainbows manifest with two local minima (green squares; for rainbow 2, see enlargement of region A in the inset). Orange circles denote conditions leading to perpendicular deflection with respect to the flat surface.

**Figure 8.**Pairs of homologous Newtonian trajectories with the same asymptotic value for their energy along the z, obtained from the energy diagram of Figure 7c: (

**a**) ${E}_{1}>{E}_{z}>0$; (

**b**) $0>{E}_{z}$; (

**c**) ${E}_{i}{cos}^{2}{\theta}_{i}>{E}_{z}>{E}_{1}$ and (

**d**) ${E}_{2}>{E}_{z}>{E}_{i}{cos}^{2}{\theta}_{i}$, where ${E}_{1}$ and ${E}_{2}$ are the energies (along the z direction) corresponding to rainbows 1 and 2 (see Figure 7b,c), respectively, and ${E}_{i}{cos}^{2}{\theta}_{i}$ is the incidence energy (also along the z direction), with ${E}_{i}=10$ meV. Forward/backward deflected trajectories are denoted with solid/dashed line. Trajectories undergoing single/double collision/s are denoted with black/blue colors. All trajectories are started from a distance ${z}_{i}=10.27$ Å above the flat Pt surface (beyond $z\approx 6.35$ Å, the interaction potential model (2) is negligible; see Figure 1a).

**Figure 9.**Deflection function for a purely repulsive model of the CO adsorbate (black solid line), obtained after removal of the attractive part of the Lennard–Jones function (4). Here, for simplicity, the incidence conditions are ${\theta}_{i}={0}^{\circ}$ and ${E}_{i}=10$ meV. To compare with, the deflection function corresponding to the full interaction potential model (2) is also represented (blue solid line), obtained for the same incidence conditions.

**Figure 10.**Set of Bohmian trajectories with initial positions uniformly distributed along the x direction and fixed value along the z direction: ${z}_{i}={\langle z\rangle}_{0}=10.27$ Å, which corresponds to the center of the incident wave packet ($t=0$) above the clean Pt surface (beyond $z\approx 6.35$ Å, the interaction potential model (2) is negligible; see Figure 1a). On the right-hand side, enlargement of the left plot near the adsorbate to illustrate the dynamical behavior displayed by the trajectories close to the substrate surface.

**Figure 11.**Set of Bohmian trajectories with initial positions uniformly distributed along the x direction and fixed value along the z direction: (

**a**) ${z}_{0}={\langle z\rangle}_{0}-3.18$ Å; (

**b**) ${z}_{0}={\langle z\rangle}_{0}-2.12$ Å and (

**c**) ${z}_{0}={\langle z\rangle}_{0}-1.06$ Å, where ${\langle z\rangle}_{0}=10.27$ Å, which corresponds to the center of the incident wave packet ($t=0$) above the clean Pt surface (beyond $z\approx 6.35$ Å, the interaction potential model (2) is negligible; see Figure 1a). In the corresponding lower panels, enlargement of the upper plots near the adsorbate to illustrate the dynamical behavior displayed by the trajectories close to the substrate surface.

**Figure 12.**As in Figure 11, but considering: (

**a**) ${z}_{0}={\langle z\rangle}_{0}+1.06$ Å; (

**b**) ${z}_{0}={\langle z\rangle}_{0}+2.12$ Å and (

**c**) ${z}_{0}={\langle z\rangle}_{0}+3.18$ Å.

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Sanz, Á.S. Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View. *Entropy* **2018**, *20*, 451.
https://doi.org/10.3390/e20060451

**AMA Style**

Sanz ÁS. Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View. *Entropy*. 2018; 20(6):451.
https://doi.org/10.3390/e20060451

**Chicago/Turabian Style**

Sanz, Ángel S. 2018. "Atom-Diffraction from Surfaces with Defects: A Fermatian, Newtonian and Bohmian Joint View" *Entropy* 20, no. 6: 451.
https://doi.org/10.3390/e20060451