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Grazing incidence fast atom diffraction for He atoms impinging on a Ag(110) surface

Experimental diffraction patterns produced by grazing scattering of fast helium atoms from a Ag(110) surface are used as a sensitive tool to test both the scattering and the potential models. To describe the elastic collision process we employ the surface eikonal (SE) approximation, which is a semi-classical method that includes the quantum interference between contributions coming from different projectile paths. The projectile-surface potential is derived from an accurate density-functional theory (DFT) calculation that takes into account the three degrees of freedom of the incident projectile. A fairly good agreement between theoretical and experimental momentum distributions is found for incidence along different low-indexed crystallographic directions.


I. INTRODUCTION
Since the unexpected observation of grazing incidence diffraction of fast atoms (GIFAD) on crystal surfaces [1,2], extensive research, both experimental and theoretical, has been devoted to the subject [3][4][5][6][7][8][9][10]. The first experimental evidences of this phenomenon were reported at insulator materials [1,2], where the presence of a wide band-gap helps to suppress inelastic processes, thus preventing quantum decoherence [11]. Soon afterwards the effect was observed at semi-conductor [12] and metallic surfaces [7,13] even though, in the case of metals, energy loss values were found to be significant [7,14,15]. In addition, GIFAD patterns have displayed an exceptional sensitivity to the projectile-surface interaction, making it possible to study very subtle contributions, like the ones produced by surface rumpling [5,16,17] or by adsorbed structures [18,19]. Nowadays GIFAD is becoming a promissory tool for examining the electronic and morphological characteristics of solid-vacuum interfaces [12,[20][21][22].
The aim of this work is to investigate the diffraction patterns produced by fast He atoms grazingly impinging on a Ag(110) surface. Since this collision system corresponds to the first and simplest metallic case for which GIFAD effects were experimentally observed [7], it provides a useful prototype to test both the theoretical method and the surface potential model.
To describe the scattering process we employ a distorted wave theory -the surface eikonal (SE) approximation -that makes use of the eikonal wave function to represent the elastic collision with the surface, while the projectile motion is classically described using different initial conditions [6]. The SE approach has been used to evaluate GIFAD distributions from insulator surfaces, providing results in good agreement with the experimental data [23][24][25].
It has also been applied to the elastic scattering of fast N atoms from a (111) silver surface, for which asymmetries in the diffraction patterns might be originated by second atomic layer effects in the surface potential [26].
Due to the strong dependence of the interference patterns on the atom-surface interaction, a crucial issue of the theoretical description is the detailed representation of the projectile-surface potential. In Refs. [7,14] the He-Ag(110) potential was simulated as a two-dimensional sinusoidal function, whose corrugation amplitude was derived from experimental data by means of the Hard-Wall approximation. Here we use a potential energy surface (PES) that was built from a large set of ab initio data obtained with the DFT-based "Quantum Espresso" code [27], combined with a sophisticated interpolation technique [28]. From such ab initio values we derived a three-dimensional (3D) PES, taking into account the projectile's three degrees of freedom. No average of the surface potential along the incidence direction was considered in the calculation.
In this article, eikonal projectile distributions derived by using the DFT potential are compared with experimental data for three different incidence directions: [110], [001], and [112]. The paper is organized as follows. The theoretical method, including details of the potential calculation, is summarized in Sec. II, results are presented and discussed in Sec.
III, and in Sec. IV we outline our conclusions. Atomic units (a.u.) are used unless otherwise stated.

A. The transition amplitude
Within the SE approximation, the scattering state of the projectile is represented with the eikonal wave function [6], where R P is the position vector of the incident atom, φ i ( R P ) = (2π) −3/2 exp(i K i · R P ) is the initial unperturbed wave function, with K i the initial projectile momentum, and the sign + indicates the outgoing asymptotic conditions. In Eq. (1) the function η(t) denotes the eikonal-Maslov phase, which depends on the classical position of the projectile at the time t, R P (t), as [24]: where V SP is the projectile-surface interaction and φ M = νπ/2 is the Maslov correction that takes into account the phase change of the scattering wave function as it passes through a focus, with the Maslov index ν defined as in Ref. [29].
By introducing Eq. (1) in the usual definition of the T-matrix element [30], the SE transition matrix per unit area A reads [24].
where R os is the initial position of the projectile on the surface plane and is the transition amplitude associated with the classical path R P = R P ( R os , t). The vector represents the component of the projectile velocity that is perpendicular to the surface plane, with z along the surface normal, aiming towards the vacuum region.
From Eq. (3), the differential probability, per unit of surface area, for elastic scattering , where m P is the projectile mass, and θ f and ϕ f are the final polar and azimuthal angles, respectively, with ϕ f measured with respect to the incidence direction in the surface plane. Details are given in Refs. [6,24].

B. Projectile-surface interaction
The interaction energy of the He atom with the Ag(110) surface is described with a full adiabatic 3D PES that depends on the atomic position R P = (X, Y, Z). The PES is constructed from a grid of ab initio energies for 42 Z values and 6 (XY ) sites, chosen as indicated in Fig. 1, over which an interpolation is performed [28].
All ab initio data are obtained from the DFT-based "Quantum Espresso" code [27].
The values of relevant input parameters are chosen so that ab initio energies are calculated to a prescribed accuracy (differences < 5 meV with respect to the converged result).
The exchange-correlation energy is calculated within the generalized gradient approximation (GGA), using the Perdew-Burke-Ernzerhof energy functional (PBE) [31]. The electron-core interaction is described with ultra-soft pseudopotentials [32]. to relaxation amount to -9.14% and +4.11% for the first and second interlayer distances respectively, in accord with experimental results [34]. Once relaxed, the slab is kept frozen for the calculations that follow.
Given the closed-shell electronic structure of the He atom (1s 2 ), we perform a non spinpolarized calculation of the ground state. A quality check of the interpolation shows that the error introduced is < 1 meV, well below the prescribed accuracy for the ab initio data.
In Figs. 2 a) and b) we show the equipotential curves along the directions [001] and [110], respectively, both starting from a site corresponding to an atom of the first layer.

III. RESULTS
In this work we use the SE approximation to study momentum distributions of 3 He atoms elastically scattered from a Ag(110) surface under grazing incidence conditions. Within the SE approach the perpendicular momentum distribution is derived from the double differential probability dP (SE) /dΩ f as where K f s = K f cos θ f is the final momentum parallel to the surface and Q tr = K f cos θ f sin ϕ f is the component of the momentum transfer in the transverse direction, which is perpendicular to the incidence direction on the surface plane.
Like in Refs. [7,14], projectile impact along three different channels of the silver surface -[110], [001], and [112] -is considered (see Fig. 1). Note that in contrast with the notation employed in Refs. [7,14], here the figures will be labeled with the direction of the incident beam, rather than with the probed direction, which is perpendicular to K i . For each of these directions, we evaluated T (SE) if from Eq. (3) using 4 × 10 5 classical trajectories with random initial positions R os that vary within an area A. In this work, in order to resemble the incident wave packet we use Gaussian distributions along the incidence and transverse directions, with full widths at half maximum determined by the size of A, to evaluate the random initial positions of projectile trajectories [30,35]. For an almost perfect plane-wave beam, the area A is large and includes several reduced unit cells.
In Fig. 3 we compare theoretical and experimental perpendicular momentum spectra for He atoms impinging along the [110] direction. The incidence energy is E i = K 2 i /(2m P ) = 500 eV and the polar incidence angle, measured with respect to the surface plane, is The two different mechanisms -Bragg diffraction and supernumerary rainbows -that are present in GIFAD patterns [3] can be analyzed separately with the SE model. The SE transition matrix can be factorized as [25] T where T In order to compare with the experimental data, in Fig. 3 we also plot SE differential probabilities convoluted with a Lorentz function (red solid line) to simulate the experimental conditions. The parameters of the line broadening are taken from the observed linewidths, as stated in Ref. [7]. Such a convolution takes into account not only the experimental divergence of the incident beam but also the broadening introduced by both thermal vibrations of lattice atoms and inelastic processes, which contribute to spoil the coherence [36]. We found a good agreement between the convoluted SE results and the experiment in the whole range of perpendicular momenta, with the exception of the maxima of order m = ±3 whose intensity is overestimated by the SE curve. This fact is related to the sharpness of the rainbow peaks of the envelope function T , being sensitive to the potential contour across the incidence direction. Then, as a consequence of the much lower corrugation of the channel, observed in Fig. 2, the projectile momentum spectrum is narrower than the one of Fig. 3. [35] When few reduced unit cells are considered in the transverse direction, such a soft random function avoids spurious interferences originated by the uniform random distribution, which are associated with the one-slit problem.
[36] In order to account for the incoherent background reported in Ref. [7], a constant term was also added to the convoluted SE probabilities.