ABOUT SOME EXTREMAL PRINCIPLES IN NON-EQUILIBRIUM THERMODYNAMICS

Laboratoire de Génie Chimique, Université de Toulouse, CNRS, INP, UPS, Toulouse, France sergio.dacunha@toulouse-inp.fr , vincent.gerbaud@toulouse-inp.fr

Extremum principles in non-equilibrium thermodynamics

• 1931 : Onsager's early variation principle in NET [1] is another formulation of its famous reciprocal laws for coupled phenomena.

-Limitations: Onsager's law is rigorously derived for continuous thermodynamic systems close to equilibrium, based on the principle of microscopic reversibility, and is valid for linear flux-forces relationships 𝐽 = 𝐿. 𝑋 or 𝑋 = 𝑅. 𝐽 and when local equilibrium holds [1]

• Among popular principles, Prigogine's minEP in the 40's can be deduced from Ziegler's maxEP in the 60's [2] : The system maximizes entropy production at every instant, but its evolution in time decreases entropy production. This is illustrated in Figure 1.

-Prigogine used Onsager's law to derive minEP. It states that, given fixed 𝑋 1 , 𝑋 2 , … , 𝑋 𝑘 forces and forces 𝑋 𝑘+1 , 𝑋 𝑘+2 , … , 𝑋 𝑛 free to vary, the system reaches a minimum of entropy production at the steady state.

-Martyushev et al. discussed strict restrictions on Ziegler's principle [3] : (i) 𝜎 is a function of fluxes, 𝜎 𝑎 ≡ 𝜎 𝐽 ; (ii) 𝜎 equals the sum of fluxes times forces in the system, 𝜎 𝑏 ≡ 𝜎 = 𝐽. 𝑋.

• With 2 forces 𝑋 1 , 𝑋 2 and 2 fluxes 𝐽 1 , 𝐽 2 only in the system, Ziegler's principle is easily represented with the 3-D coordinates 𝐽 1 , 𝐽 2 , 𝜎. Assumption (i) is displayed by a surface (orange) 𝜎 𝑎 = 𝐽. 𝑋 = 𝑅. 𝐽. 𝐽, and assumption (ii) is a plane (blue). Generally their intersection is a curve, and maxEP postulates that the system state is at every instant the maximum of this curve w.r.t coordinate 𝜎 (black dot curve). • For minEP, when 𝑋 1 is fixed, the free force 𝑋 2 change corresponds to the blue plane rotation, 𝜎 𝑏 = 𝐽. 𝑋. The stationary state 𝐽 2 = 0 corresponds to a minimum of entropy production (red dot in figure 1 far right)

• Later, Ziegler proposed his "Orthogonality Principle" as a generalization of Onsager's law for nonlinear flux-force relationships [1,2] and attempted to demonstrate it in the particular case of homogeneous rheological bodies. Ziegler also proposed an equivalent formulation for the orthogonality principle called "Principle of Maximum Dissipation Rate" (maxEP) [1,2] and postulated its validity for continuous system under certain conditions. • So, is Ziegler's principle the superior one? answer is NO.

• Despite strong arguments that have ruled out several critics, restrictions apply [see 2, 3],

• [2,3] has also overlooked that maxEP is inaccurate for a system with coupled reactions far from equilibrium [see for ex. 4] • Many of the arguments in favour of Ziegler's principle are based on the fact that it yields Onsager's reciprocal relations in the linear domain. However, Ziegler designed his maxEP principle so that it would coincide with Onsager's law near equilibrium. Hence, this argument is not valid. • Beretta's Steepest Entropy Ascent (SEA) law, would be a stronger candidate [5] : "every non-equilibrium state of a system or local subsystem for which entropy is well defined must be equipped with a metric in state space with respect to which the irreversible component of its time evolution is in the direction of steepest entropy ascent compatible with the conservation constraints."

• It was shown to be coherent with many advanced nonequilibrium frameworks [6]

• It was shown to give an extension of Onsager's reciprocal laws as a non-linear relation valid far from equilibrium within the framework of quasiequilibrium approximation. • A key point is that the metric operator defines the direction of SEA and the characteristic time driving the dynamic evolution of the system defines the evolution rate in such direction. Future work: investigation is in progress, e.g. to decipher whether the black dot trajectory has special properties (geodesic? shortest time?) in the metric proposed by Beretta ; e.g. apply Beretta's proposal to chemical kinetics as in [4] Sergio Da Cunha, Nataliya Shcherbakova, Vincent Gerbaud

Context and motivations

• A very large literature has been devoted to extremum principles in non-equilibrium thermodynamics (NET), like Prigogine's minEP or Ziegler's maxEP (EP = entropy production) • An extremum principle that remains valid outside of linear NET would be a powerful tool in the study of coupled phenomena and of dynamics.

• But does such a principle exist? Any restrictions and limitations? This is what we are examining. Honestly, our results are timid but in progress.

The system evolves with respect to time, but it respects maxEP principle at any instant.

At stationary state (J 2 = 0), Prigogine's minEP holds, a result of maxEP through the system evolution.

Ziegler's maxEP principle defines the state of a system for given forces X 1 and X 2 .

  1. R.C. Dewar, C.H. Lineweaver, R.K. Niven and K. Regenauer-Lieb, Beyond the Second Law. An overview. in R.C. Dewar et al. (eds.), Beyond the Second Law -Entropy Production and Non-Equilibrium Systems, chap. 1, Springer: Berlin/Heidelberg, Germany, 2014. 2. L.M. Martyushev and V.D. Seleznev, Maximum Entropy Production Principle in Physics, Chemistry and Biology, Physics Reports, 426, 1-45, 2006. 3. L.M. Martyushev and V.D. Seleznev. The restrictions of the maximum entropy production principle. Physica A 410, 17-21, 2014. 4. J. Bataille, D.G.B. Edelen and J. Kestin, Nonequilibrium Thermodynamics of the Nonlinear Equations of Chemical Kinetics, J. Non-Equilib. Thermodyn., vol. 3, 153-168, 1978. 5. G.P. Beretta The fourth law of thermodynamics: steepest entropy ascent Philosophical Transactions of the Royal Society A, 378, 20190168, 2020. 6. G.P. Beretta GP. Steepest entropy ascent model for far-non-equilibrium thermodynamics. Unified implementation of the maximum entropy production principle. Phys. Rev. E 90, 042113, 2014.


ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Maxime Debiossac

Asier Zugarramurdi

Zhao Mu

Petru Lunca-Popa

Andrew J Mayne

Philippe Roncin

Helium diffraction on SiC grown graphene: Qualitative and quantitative descriptions with the hard-corrugated-wall model

Monolayer epitaxial graphene grown on 6H-SiC(0001) was recently investigated by grazing-incidence fastatom diffraction and analyzed with an ab initio electronic density calculation and with exact atomic diffraction methods. With these results as a reference, the hard-corrugated-wall model (HCW) is used as a complementary analytic approach to link binary potentials to the observed atomic corrugation. The main result is that the HCW model reproduces the macroscopic corrugation of the moiré pattern on a quantitative level, suggesting that soft-wall corrections may be neglected for macroscopic superstructures, allowing straightforward analysis in terms of a one-dimensional corrugation function.

I. INTRODUCTION

Graphene is a promising two-dimensional material whose electronic properties depend strongly on its binding to the substrate. This anchoring is visible both in the detailed topography and in the local conductivity, measured in scanning tunneling microscopy (STM) experiments where individual contributions are difficult to disentangle. In contrast, atomic force microscope (AFM) and helium atom scattering (HAS ) are techniques that probe only the topography of the electronic density of the topmost layer [START_REF] Borca | Electronic and geometric corrugation of periodically rippled, self nanostructured graphene epitaxially grown on Ru[END_REF][START_REF] Gibson | Helium atom scattering from graphene grown on Rh(111)[END_REF][START_REF] Anemone | Quality of graphene on sapphire: Long-range order from helium diffraction versus lattice defects from raman spectroscopy[END_REF][START_REF] Maccariello | Observation of localized vibrational modes of graphene nanodomes by inelastic atom scattering[END_REF][START_REF] Al Taleb | Helium diffraction and acoustic phonons of graphene grown on copper foil[END_REF][START_REF] Tamtögl | Graphene on Ni(111): Electronic corrugation and dynamics from helium atom scattering[END_REF][START_REF] Taleb | Phonon dynamics of graphene on metals[END_REF][START_REF] Shichibe | Probing interlayer interactions between graphene and metal substrates by supersonic rare-gas atom scattering[END_REF] and are only perturbed by weak polarization effects. This is also the case of grazingincidence fast-atom diffraction (GIFAD of FAD), which uses helium atoms with an energy E 0 in the keV range but at such a low incidence angle θ that the energy E ⊥ = E 0 sin 2 θ of the movement normal to the surface is in the sub-eV range. This technique was used recently by Zugarramurdi et al. [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF] together with ab initio calculations [START_REF] Varchon | Electronic structure of epitaxial graphene layers on SiC: Effect of the substrate[END_REF] to investigate the structure of an epitaxial graphene monolayer grown on 6H-SiC(0001). Here, we use the much simpler hard-corrugated-wall model (HCW) to fit the experimental data without a priori knowledge of the surface topography or of the interaction potential. This qualitative description is then evaluated qualitatively by using the HCW with the same interaction potential used in the exact diffraction calculation. The excellent agreement supports the use of the HCW model to investigate the various forms of graphene with fast helium atoms.

II. GIFAD

Thermal helium diffraction at crystal surfaces was a seminal experiment of quantum mechanics almost a century ago [START_REF] Estermann | Beugung von molekularstrahlen[END_REF]. The use of fast atoms in the keV range at grazing incidence (see Fig. 1) is much more recent simply because it was not anticipated that atoms with a subpicometer wavelength λ could be coherently scattered by surface atoms with a thermal fluctuation larger than λ. Surprisingly, GIFAD was discovered independently in France [START_REF] Khemliche | Dispositif et procédé de caractérisation de surfaces[END_REF][START_REF] Rousseau | Quantum Scattering of Fast Atoms and Molecules on Surfaces[END_REF] and in Germany [START_REF] Schüller | Diffraction of Fast Atomic Projectiles during Grazing Scattering from a LiF(001) Surface[END_REF].

It was rapidly noticed that the fast motion along x and the slow motion in the (y,z) plane appear decoupled. In the slow-motion plane, the situation is similar to that of a hyperthermal helium atom, with energy E ⊥ , interacting with a one-dimensional (1D) array of quasiatoms [START_REF] Danailov | Computer simulation of the reflection of energetic ions from crystal surfaces at glancing incidence[END_REF]. The fast projectiles are sensitive to the potential averaged along the axial channel, so that only the surface corrugation across the channel is resolved. This forms the basis of the twodimensional (2D) axial-surface-channeling approximation (ASCA) [START_REF] Rousseau | Quantum Scattering of Fast Atoms and Molecules on Surfaces[END_REF][START_REF] Schüller | Diffraction of Fast Atomic Projectiles during Grazing Scattering from a LiF(001) Surface[END_REF][START_REF] Danailov | Computer simulation of the reflection of energetic ions from crystal surfaces at glancing incidence[END_REF][START_REF] Zugarramurdi | Transition from fast to slow atom diffraction[END_REF][START_REF] Farías | Pronounced out of plane diffraction of H 2 molecules from a Pd(1 1 1) surface[END_REF]. The practical interest of GIFAD is that keV atoms are efficiently detected and the full diffraction pattern is confined to a narrow cone which can be recorded in a few seconds using a microchannel plate [START_REF] Atkinson | Dynamic grazing incidence fast atom diffraction during molecular beam epitaxial growth of GaAs[END_REF].

III. THEORETICAL DESCRIPTIONS

From a theoretical point of view, the scattering of a helium atom with a surface is quite a complex quantum system. First, a potential-energy landscape describing the interaction energy V (x,y,z) between the surface and the projectile is required. The transition matrix formalism has been used to identify the different regimes [START_REF] Manson | Theory of grazing incidence diffraction of fast atoms and molecules from surfaces[END_REF], taking into account phonon excitations. In addition, a purely elastic diffraction regime where the surface atoms are frozen at their equilibrium position providing a periodic three-dimensional (3D) potential-energy function V (x,y,z) has been investigated using the wave-packet technique [START_REF] Rousseau | Quantum Scattering of Fast Atoms and Molecules on Surfaces[END_REF][START_REF] Debiossac | Combined experimental and theoretical study of fast atom diffraction on the β 2 (2 × 4) reconstructed GaAs(001) surface[END_REF][START_REF] Aigner | Suppression of Decoherence in Fast Atom Diffraction at Surfaces[END_REF] as well as close-coupling calculations [START_REF] Debiossac | Transient Quantum Trapping of Fast Atoms at Surfaces[END_REF] and semiclassical calculations [START_REF] Gravielle | Fast atom diffraction for grazing scattering of Ne atoms from a LiF(0 0 1) surface[END_REF][START_REF] Winter | Fast atom diffraction during grazing scattering from surfaces[END_REF]. The ASCA simplifies the problem, and its range of validity has been investigated in detail [START_REF] Zugarramurdi | Transition from fast to slow atom diffraction[END_REF]. In ASCA, the actual 3D potential is averaged along the low-index direction, taken here along x to generate an effective 2D potential Ṽ (y,z) = V (x,y,z) x . Schematically, the "egg-carton" view of the surface atoms with 2D symmetry is replaced by a "corrugated iron sheet" or washboard description FIG. 1. Schematic view of a GIFAD setup. A few hundred eV beam of neutral helium atoms impinges at a grazing incidence angle onto the surface. When the SiC grown graphene is aligned along a low index direction, diffraction features can be observed on a positionsensitive detector located about 1 m downstream.

with only 1D symmetry along y and translational invariance along x. A tiny misalignment of the beam with respect to the low-index-surface crystallographic axis does not alter the 1D symmetry but simply changes both the effective energy and the initial direction of the effective particle [START_REF] Seifert | Transition from axial to planar surface channeling for fast atom diffraction[END_REF][START_REF] Zugarramurdi | Theoretical study of the effect of beam misalignment in fast atom diffraction at surfaces[END_REF][START_REF] Debiossac | Atomic diffraction under oblique incidence: An analytical expression[END_REF][START_REF] Pollak | Second-order semiclassical perturbation theory for diffractive scattering from a surface[END_REF]. With these simplifications, calculating the diffraction of helium atoms onto any periodic topology becomes computationally more affordable.

IV. GRAPHENE PREPARATION

Graphene was prepared by epitaxial growth on a highly nitrogen doped SiC wafer (resistivity: 0.04 cm, dopant density: 3 × 10 18 atoms/cm 3 ). We proceed by starting systematically with the formation of the SiC(0001) 3 × 3 reconstruction of the Si face [START_REF] Baffou | State selective electron transport through electronic surface states of 6HSiC(0001)33[END_REF]. Graphene was produced by annealing the Si-terminated surface at 1325 • C for 25 min [START_REF] Yang | Topology and electron scattering properties of the electronic interfaces in epitaxial graphene probed by resonant tunneling spectroscopy[END_REF][START_REF] Yang | Manipulation at a distance: Atomic scale observation of ballistic electron transport in single layer graphene[END_REF]. The sample was then characterized by STM. Previous observations of layer-by-layer graphene growth on the Si face of SiC have shown characteristic topographical features due to electronic and geometric contributions that enable the buffer layer, singlelayer Gr, and bilayer Gr to be unequivocally distinguished in STM [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF][START_REF] Yang | Topology and electron scattering properties of the electronic interfaces in epitaxial graphene probed by resonant tunneling spectroscopy[END_REF][START_REF] Yang | Manipulation at a distance: Atomic scale observation of ballistic electron transport in single layer graphene[END_REF][START_REF] Yang | Quantum interference channeling at graphene edges[END_REF]. From the STM images, it is estimated that 90% of the substrate is covered by a single layer. After growth and characterization, the graphene sample was transferred in ambient atmosphere to the GIFAD setup. A mild annealing at 600 • C in UHV was sufficient to obtain clear diffraction patterns with a 300 eV He atom beam.

V. PREVIOUS RESULTS

In Ref. [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF] the reference surface topology was taken from an extensive density-functional-theory calculation of the 6H-SiC(0001) surface which includes the intermediate buffer layer and the terminal graphene layer [START_REF] Varchon | Electronic structure of epitaxial graphene layers on SiC: Effect of the substrate[END_REF]. The interaction potential V (x,y,z) with the helium atom was calculated by attaching an effective Lennard-Jones (LJ) C-He binary potential [START_REF] Carlos | Interaction between a He atom and a graphite surface[END_REF] optimized from HAS experiments on graphite considered to be a good model for graphene in terms of the binary potential [START_REF] Bartolomei | Global potentials for the interaction between rare gases and graphene based surfaces: An atom-bond pairwise additive representation[END_REF].

The main results can be summarized as follows. Diffraction was observed along the [100] and [110] directions of graphene corresponding to the zigzag and armchair directions, respectively. The 13 × 13R30 • moiré structure of the terminal graphene layer was observed; a precise quantitative measure of the size of the unit cells and relative orientation were obtained and are illustrated in Fig. 2. Quantitatively, the LJ potential used to describe the effective C-He interaction was found to be ideally suited in the 10-300 meV energy range where the diffraction on the carbon honeycomb backbone is reproduced without any adjustment. However, the amplitude of the moiré structure was overestimated in the calculation [START_REF] Varchon | Electronic structure of epitaxial graphene layers on SiC: Effect of the substrate[END_REF], and the best fit to the data required a scaling factor of 66 %, i.e., a reduction by 1/3 [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF].

VI. SIMPLIFIED DESCRIPTION WITHIN THE HARD-CORRUGATED-WALL MODEL

We investigate here the performance of a simpler model, the hard-corrugated-wall model. The diffraction problem is considered to be a simple reflection of the helium atom at the potential-energy line Z(y) corresponding to Ṽ 2D (y, Z(y)) = E ⊥ . This neglects the progressive deceleration/acceleration before and after reflection. The scattering problem is now equivalent to the diffraction of optical rays at a 1D mirror grating described by an arbitrary corrugation function Z(y). In analogy with optics, the diffracted intensities I m are given by the Fourier transform of the corrugation function Z(y) [START_REF] Garibaldi | Quantum theory of atom surface scattering: Diffraction and rainbow[END_REF]:

I m = k f z k iz 1 a y a y 0 e -imG y -2i k⊥ Z(y) dy 2 . ( 1
)
Integration is over one projected lattice unit a y = 2π/G y and k⊥

= (k iz + k f z )/2,
where k iz and k f z are the initial and final components of the wave vector in the z direction; 2 k⊥ is therefore the momentum transfer in the z direction, which, for quasispecular scattering mG y k iz , is close to k ⊥ ≡ k iz . At first glance, the approximation looks severe, but knowing that most helium-surface interaction potentials have a pronounced exponential decay towards the vacuum, most of the momentum transfer indeed takes place close to the surface of the classical turning point. When the range of the potential and the corrugation amplitude are limited, i.e., much smaller than the lattice parameter, this surface is close to the corrugation function Z(y), and the HCW is known to be qualitatively correct [START_REF] Garibaldi | Quantum theory of atom surface scattering: Diffraction and rainbow[END_REF][START_REF] Armand | Scattering by a hard corrugated wall: An exact solution[END_REF][START_REF] Hofmann | The transition from single phonon to multiphonon energy transfer in atom surface collisions[END_REF]. However, the accuracy of the HCW model can be limited, and corrugation amplitudes with departures of up to 30% from exact theoretical treatments have been reported [START_REF] Winter | Fast atom diffraction during grazing scattering from surfaces[END_REF]. The main interest of this method is to reduce the complex scattering problem to that of calculating of a 1D integral. To limit the number of free parameters in the fitting procedures, the corrugation function is expanded, for instance, in a Fourier series Z(y) = n α n cos(nG [

I m = J 2 m (k ⊥ z c ).

VII. THE ZIGZAG DIRECTION

Along the zigzag direction, Fig. 3 displays a typical diffraction pattern obtained with 300 eV helium atoms at an angle of incidence of 0.89 • corresponding to an energy E ⊥ = 72 meV. The simplest parameter that can be extracted is the peak separation, measured at 0.223

• ± 0.002 • , i.e., a momentum G [100] = 2.95 ± 0.03 Å-1 .
This corresponds to a projected periodicity of 2.13 Å, matching very well the value of 3b/2, where b = 1.42 Å is the C-C bond length and a = √ 3b is the 2D lattice parameter of graphene at room temperature (see Fig. 2). It is worth noticing that the diffraction pattern exhibits elongated streaks in the vertical direction, rather than small spots of the size of the primary beam. This indicates that the diffraction is not perfectly elastic as observed, for instance, on surfaces grown by molecular beam epitaxy [START_REF] Debiossac | Combined experimental and theoretical study of fast atom diffraction on the β 2 (2 × 4) reconstructed GaAs(001) surface[END_REF] or on a freshly cleaved crystal [START_REF] Debiossac | Transient Quantum Trapping of Fast Atoms at Surfaces[END_REF][START_REF] Busch | Evidence for longitudinal coherence in fast atom diffraction[END_REF]. This is probably due to the presence of defects in the periodic arrangement which limit the coherence length of the surface and reduce the accuracy of the measurements.

The intensity distribution along the Laue circle is reported in Fig. 4. With a perpendicular energy close to 10 meV, mainly specular reflection is observed, and the line profiles are quite narrow [Fig. 4(a)]. At larger perpendicular energies, Fig. 4(b) shows that the line profile has a more complex structure with shoulders, indicating the possible presence of sidebands. Despite investigating different regions of the graphene layer, it was not possible to record a diffraction pattern where these satellite peaks could be resolved. We therefore label the diffraction peaks with respect to G [100] associated with the C-C backbone periodicity, and the intensity is considered as that of the whole structure, including the contributions of the comparatively large base. Figure 5 shows the smooth evolution of the intensities of the diffraction orders as a function of the angle of incidence, expressed here as a perpendicular energy. The dashed lines drawn between the experimental values are the fit of the HCW model for a sinusoidal corrugation function: approaches the surface. A closer look reveals that the dashed lines tend to overestimate the specular intensity at low energy. This could easily be modeled by an increasing corrugation amplitude at low energy. Such behavior is not uncommon at higher E ⊥ , where it indicates that the interaction potential is softer on top than on the bottom part of the corrugation function [START_REF] Momeni | Grazing incidence fast atom diffraction (GIFAD): Doing RHEED with atoms, e[END_REF]. However, such an effect does not seem realistic at all below 30 meV. This behavior is well reproduced here by adding a small van der Waals contribution E V dW to the effective energy

I m = J 2 m ( k⊥ z c )
k ⊥ = √ 2M(E ⊥ + E V dW
). This is equivalent to the Beeby correction energy [START_REF] Sanz | A trajectory description of quantum processes[END_REF] used in the Debye-Waller evaluation of thermal decoherence. The value of E V dW = 10 meV for the depth of the van der Waals potential energy is only an estimation and was not optimized. More accurate values can be derived from bound-state resonances [START_REF] Debiossac | Transient Quantum Trapping of Fast Atoms at Surfaces[END_REF][START_REF] Jardine | Ultrahigh resolution spin-echo measurement of surface potential energy landscapes[END_REF] when the is large enough. Another option to retrieve the corrugation amplitude is to fit an individual diffraction image with z c as the only free parameter. Figure 6 shows that z c is constant above 30 meV, giving a value of z c = 0.074 ± 0.003 Å. Below 30 meV, the sudden increase in the corrugation amplitude is probably due to the van der Waals attraction, which becomes comparatively important.

VIII. THE ARMCHAIR DIRECTION

Along the [110] armchair direction, the diffraction pattern is completely different. Figure 7 shows a single group of closely packed lines separated by G [110] = 0.39 ± 0.02 Å-1 . This indicates a projected lattice vector L T = 16.1 ± 0.5 Å consistent with the moiré structure [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF]. The counterpart of the C-C backbone structure expected at 5.11

Å-1 = 2π/1.23 Å is conspicuous by its absence (arrows in Fig. 8). Figure 8 displays the diffracted intensity on the Laue circle fitted using Voigt profiles. These are a convolution of a Gaussian and a Lorentzian, where the Gaussian width is that of the primary beam profile W G = 0.17 Å-1 , while the Lorentzian component W L was left free but is common to all peaks in a given diffraction pattern. Figure 9 shows that this width W L increases gradually with E ⊥ and reaches a value larger than the peak separation (the reciprocal lattice vector) when E ⊥ > 90 meV. Beyond this value the diffraction features are blurred. Defining the visibility V is as the remaining modulation of an infinite array of Lorentzians, V is decays exponentially with the ratio of W L /G, but more work is needed to link this visibility to the coherence ratio of the diffracted signal and to the Debye-Waller factor specific to GIFAD [START_REF] Manson | Theory of grazing incidence diffraction of fast atoms and molecules from surfaces[END_REF][START_REF] Rousseau | Surface analysis with grazing incidence fast atom diffraction (GIFAD)[END_REF]. Further studies are also needed to investigate the possible connection between the minimum width W 0 and the structural disorder.

Figure 10 displays the evolution of the five central diffraction orders (m = -2,-1,0,1,2) derived from the fit using Voigt profiles. As in Fig. 5, the positive and negative diffraction orders have been added together. Note that, due to the deconvolution, the statistical dispersion is significantly larger than along the [100] direction. where the valleys and bumps are identical. Figure 10 indicates that this sinusoidal shape tends to underestimate the population of large diffraction orders, i.e., larger deflection angles, suggesting that steeper slopes are present [larger derivative of Z(y)]. This can be provided by adding higher-order terms in the harmonic expansion of Z(y), but we have chosen to use a Gaussian shape Z(y) = z c e -(y/2L T σ ) 2 where only one additional parameter is involved, namely, the width σ/L T relative to the lattice parameter L T = 16 Å. The result reported in Fig. 10 is for a relative FWHM of 1/3. Note that, for symmetric profiles, i.e., if a value y s such that Z(yy s ) = Z(y sy) exists, the HCW model is not sensitive to the sign of the corrugation function Z(y) because it displays the same corrugation amplitude and the same slope distribution. This means that bumps with a FWHM of 2/3 and narrower FIG. 10. Diffracted intensities along the [110] direction as a function of E ⊥ . The lines correspond to the HCW model with two different functions Z(y) modeling the shape of the moiré bumps. Solid lines are for Gaussian shapes, while the dashed ones are for a sinusoidal shape (see text). valleys of 1/3 would produce the same results. These simple models are only indications; it is not completely meaningful to compare the present data where only a few diffraction orders are observed with a complex corrugation function Z(y). As a rule of thumb, the number of significant points in the unit cell should compare to the number of observed diffraction orders. For larger corrugation amplitudes as measured along the [1 10] direction of the β 2 (2 × 4) reconstruction of GaAs(001), up to a hundred diffraction orders have been recorded, and much more complex corrugation profiles have been derived [START_REF] Debiossac | Combined experimental and theoretical study of fast atom diffraction on the β 2 (2 × 4) reconstructed GaAs(001) surface[END_REF].

IX. DISCUSSION

The experiment provides model-independent information such as the peak positions, peak profile, width, and intensities. Conversely, the corrugation amplitudes reported here are derived through the HCW model. On the one hand, the HCW model can be seen as a powerful tool to describe, with only one or two parameters, the rapidly varying set of data presented in Figs. 5 and8. On the other hand, the question remains regarding how accurate this description is. It is important to note that the observed intensities result from interferences between trajectories bouncing along the top or bottom of Z(y). Assuming a corrugation amplitude varying slowly with the energy E ⊥ , the intensity of the specular peak undergoes a full oscillation from dark to bright and dark again in a wavenumber interval δk ⊥ given by δk ⊥ z c = 1, i.e., δk ⊥ = z -1 c . This offers a robust and redundant evaluation of the corrugation amplitude z c . However, the HCW model is known to be only qualitative when soft-wall corrections are needed, i.e., when the sudden momentum transfer approximation is not valid. In this case the classical turning points are no longer close to the equipotential line. The other well-known situation where the HCW fails to be quantitative occurs when the corrugation amplitude becomes large compared to the size of the lattice unit cell, giving rise to possible multiple-scattering effects. Since an exact diffraction calculation has been performed, it is easy to check if the corrugation amplitude extracted from the data is quantitative or not by applying the HCW model to the same potential-energy landscape. It is straightforward to reconstruct the surface potential from the effective LJ potentials [START_REF] Carlos | Interaction between a He atom and a graphite surface[END_REF] used in [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF]. However, as shown in Fig. 3 and4 and in Ref. [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF], the diffraction along the [100] direction is not directly affected by the moiré superstructure; thus, along this direction the graphene can be modeled as a flat layer. The pairwise potential used in the present study is given by

V LJ (R) = 4 σ 12 R 12 - σ 6 R 6 ,
with = 1.4 meV and σ = 2.74 Å [START_REF] Carlos | Interaction between a He atom and a graphite surface[END_REF]. We now use the string approach where averaging is calculated separately for each row of perfectly aligned atoms, with the associated cylindrical coordinate ρ = y 2 + z 2 and n = 1/d x being the surface atom density along x. The cylindrical potential for each string is then given by For both directions investigated here, only two strings of carbon atoms having equal linear densities are present inside the projected lattice unit. Along the [100] zigzag direction, the two rows group together with a separation of 0.71 Å, while the next-nearest row, in the next lattice unit, is twice as far away at 1.42 Å. This produces the observed corrugation where the highest points correspond to the middle of these double rows and the lowest points sit in between them. Along the [110] armchair direction, the two rows are evenly spaced, minimizing the corrugation amplitude, which is indeed almost ten times lower than along the [100] direction. The highest points of the corrugation function correspond here to the top of the row, while the lowest points are in between.

V LJ (ρ) = nπ 63σ 12 64ρ 11 -
The projected potential-energy landscape is then easily calculated by adding the contributions of a few lattice units. The equipotential energy lines are calculated numerically and are displayed in Fig. 11. The corrugation amplitude calculated at E ⊥ = 100 meV is z c = 0.074 Å, exactly the one derived from the experimental data. This 0.001 Å accuracy is accidental since this LJ binary potential predicts that z c should vary from 0.07 to 0.08 Å between E ⊥ = 50 meV and E ⊥ = 300 meV, which is not observed in the data. This indicates that the stiffness of the LJ potential may not be entirely correct. This is not too surprising since the LJ form of the repulsive part is purely empirical, while the prefactor has been adjusted to fit the bound states inside the potential well (see Fig. 12).

The other indication provided by the LJ potential is the depth of the Van der Waals potential-energy well. This is easily visualized in the planar form obtained by integration along both the x and y coordinates

V LJ (z) = 2a s π 2σ 12 5z 10 - σ 6 z 4 ,
where a s = 2/5.24 Å2 is the surface density, i.e., the number of atoms in the unit cell divided by its area. The attractive parts of neighboring atoms add up significantly because the location of the well is at a distance larger than the lattice parameter. In contrast Fig. 11 shows that the repulsive part of the planar, axial, and radial potentials is not too different in the energy range of E ⊥ ∼ 10-100 meV probed here. The resulting atomic radius, string radius, and minimum distance of approach FIG. 12. The Lennard-Jones effective C-He potential [START_REF] Carlos | Interaction between a He atom and a graphite surface[END_REF] V (R) is reported together with its value integrated axially V (ρ) = x V (R) and its planar average V (z) = x y V (R).

to the surface, defined as the distance where the potential energy equals the initial kinetic energy, are close to each other.

Along the armchair [110] direction, the graphene carbon backbone is not observed. This is only due to the averaging, which reduces the 2D egg-carton corrugation from 0.2 Å, corresponding to center of the hexagon and top of a C atom, down to an effective projected corrugation of only 0.007 Å. Consequently, the graphene backbone is very flat when observed along the [110] direction. The same averaging effect also affects the moiré structure but along the zigzag direction because the moiré domes are aligned along the [110] direction, as indicated by the R30 • index. The corrugation amplitude of 0.14 Å derived from the data in Fig. 10 is in accordance with the value of 0.21 Å plotted in Fig. 4(c) of Ref. [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF] before multiplying by the scaling factor of 0.656, resulting in a value of 0.14 Å. The main result here is that fast-atom diffraction on graphene is almost a perfect system to apply the HCW approach. This conclusion might be less valid for HAS because a 3D potential V (x,y,z) must be considered, so that the minimum of the corrugation function is in the center of the hexagon and is not weakened by any averaging. At E ⊥ = 100 meV, the 3D corrugation amplitude is almost three times larger than the 2D average along the [100] direction Ṽ (y,z). This comparatively strong attenuation is specific to the hexagonal symmetry and is even more pronounced along the [110] direction. For rectangular lattices there is often a direction where the measured corrugation amplitude compares with its 3D value derived from HAS.

X. HCW IN GIFAD AND HAS

The corrugation amplitude of the moiré pattern of graphene deposited on various substrates has been investigated by thermal energy helium diffraction (HAS). Corrugations of 0.06 Å on Ni(111) [START_REF] Tamtögl | Graphene on Ni(111): Electronic corrugation and dynamics from helium atom scattering[END_REF], 0.15 Å on Ru(0001), and up to 0.9 Å on Rh(111) [START_REF] Gibson | Helium atom scattering from graphene grown on Rh(111)[END_REF] have been observed, indicating variations of the graphene interaction with the underlying metal substrate.

The value measured here by GIFAD on 6H-SiC(0001) is 0.14 Å, corresponding to a 2D corrugation of 0.27 Å [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF], where the difference between these two values originates from the averaging of the surface potential (ASCA) due to the grazing-incidence geometry. The value appears rather large for a system assumed to be weakly bound to its substrate, but here, the observed corrugation is expected to reflect mainly the corrugation of the underlying buffer carbon layer, which is indeed strongly bound to the SiC substrate.

It should be noted that Eq. ( 1) is valid only for normal incidence of the projectile. In GIFAD this corresponds to a primary beam well aligned with the probed direction [START_REF] Winter | Fast atom diffraction during grazing scattering from surfaces[END_REF][START_REF] Zugarramurdi | Theoretical study of the effect of beam misalignment in fast atom diffraction at surfaces[END_REF], whereas in a typical HAS setup this orthogonality of both the beam and detector is difficult to achieve. As a result the intensities observed in HAS for opposite diffraction orders, i.e., +m and -m, differ, limiting the interest of the HCW model in data analysis (see, e.g., Fig. 3 of [START_REF] Gibson | Helium atom scattering from graphene grown on Rh(111)[END_REF]). Recently, two independent papers [START_REF] Debiossac | Atomic diffraction under oblique incidence: An analytical expression[END_REF][START_REF] Pollak | Second-order semiclassical perturbation theory for diffractive scattering from a surface[END_REF] have proposed analytical formulas taking into account the oblique incidence. These predictions have not yet been compared, but both should significantly improve the comparison of HAS data with simple descriptions of the surface corrugation function.

A more significant difference between GIFAD and HAS lies in the ability of HAS to measure phonon modes of the graphene surface by inelastic scattering [START_REF] Maccariello | Observation of localized vibrational modes of graphene nanodomes by inelastic atom scattering[END_REF][START_REF] Al Taleb | Helium diffraction and acoustic phonons of graphene grown on copper foil[END_REF][START_REF] Tamtögl | Graphene on Ni(111): Electronic corrugation and dynamics from helium atom scattering[END_REF][START_REF] Taleb | Phonon dynamics of graphene on metals[END_REF][START_REF] Shichibe | Probing interlayer interactions between graphene and metal substrates by supersonic rare-gas atom scattering[END_REF]. In the present case some modes specific to quasifree standing graphene were identified [START_REF] Maccariello | Observation of localized vibrational modes of graphene nanodomes by inelastic atom scattering[END_REF], providing a very interesting link with dynamical behavior of the graphene layer as probed by atomic force microscopy [START_REF] Koch | Elastic response of graphene nanodomes[END_REF]. It would certainly be worth trying to obtain comparable information from scattering of fast atoms. On the one hand, a few meV resolution on top of few hundred eV projectiles does not seem realistic. On the other hand, Shichibe et al. [START_REF] Shichibe | Probing interlayer interactions between graphene and metal substrates by supersonic rare-gas atom scattering[END_REF] have shown that this inelastic behavior also has specific signatures in the scattering profile perpendicular to the surface plane. Provided that topological defects are under control, these aspects could probably be investigated at grazing incidences as well.

XI. CONCLUSION

The hard-corrugated-wall model was found to be fully quantitative with predictions almost indistinguishable from that of an exact close-coupling calculation. The reason for the very good agreement lies in the compact interaction potential between the helium atom and the carbon atoms of the surface. In other words, the so-called soft potential effects, which tend to separate the turning-point surface from the isoenergy surface, are probably very small. The other favorable condition is that the specific averaging associated with the hexagonal symmetry generates a 1D apparent corrugation that is significantly weaker than the 2D corrugation. As a result, for E ⊥ below 100 meV, the HCW model can safely be used to analyze the numerous moiré structures of graphene, and as demonstrated in [START_REF] Zugarramurdi | Determination of the geometric corrugation of graphene on SiC(0001) by grazing incidence fast atom diffraction[END_REF], the LJ effective binary potential [START_REF] Carlos | Interaction between a He atom and a graphite surface[END_REF] can also be useful to generate a potential-energy landscape from the atomic positions. For flat graphene, this landscape is derived analytically from the LJ effective binary potential. Furthermore, along the armchair direction, only the overall periodic shape of the moiré structure is needed since the individual carbon atoms of the graphene backbone do not contribute to the diffraction pattern.

FIG. 2 .

 2 FIG. 2. He-Gr/SiC(001) interaction potential [9] at height z = 2.3 Å (left). Dashed colored lines indicate the three equivalent zigzag ([100]) and armchair ([110]) directions named according to the graphene honeycomb structure (right). The lattice parameter (a) and the C-C bond length (b) are indicated. The observed moiré corrugation is high along the armchair direction connecting neighboring moiré bumps and reduced along the zigzag direction connecting a bump to a neighboring valley. The situation is reversed on the graphene backbone, where the zigzag direction connects holes via the center of C-C bond and has the largest corrugation.

  FIG.3. Diffraction pattern recorded with 300 eV helium atoms aligned along the graphene [100] (zigzag) direction. The radius of the Laue circle intercepting the specular spot as well as that of the primary beam, before target insertion, indicates the angle of incidence.

  with a top to bottom amplitude z c = 0.074 Å.

FIG. 4 .

 4 FIG.[START_REF] Maccariello | Observation of localized vibrational modes of graphene nanodomes by inelastic atom scattering[END_REF]. For 300 eV helium at (a) 0.33 • and (b) 0.89 • along the [100] direction, the diffracted intensity on the Laue circle (see Fig.3) is reported as a function of the momentum transfer in the perpendicular direction (here[110]).Good agreement with a constant value of the corrugation indicates that the shape of the corrugation function does not change significantly in this energy range as the perpendicular energy increases, i.e., as the turning point of the trajectory

FIG. 6 .

 6 FIG. 6. Corrugation amplitude z c derived from each diffraction image along the [100] direction from a fit of the diffracted intensities I m by Bessel function I m = J 2 m ( k⊥ z c ) (see text).

FIG. 7 .

 7 FIG. 7. Diffraction pattern recorded along the graphene [110] (armchair) direction with 300 eV helium at 0.88 • incidence corresponding to E ⊥ = 70 meV.

FIG. 9 .

 9 FIG. 8. The intensity of the Laue circle in Fig. 7 is fitted by identical Voigt or Lorentzian profiles with a width W L sitting at Bragg position y n = nG [110] . Note the absence of intensity at 5.11 Å-1 indicated by the vertical arrows.

FIG. 11 .

 11 FIG. 11. (a) Equipotential energy lines of a flat graphene. Note that the coordinates are perpendicular to the beam direction. At E ⊥ = 100 meV a full corrugation amplitude of 0.074 Å is measured. (b) The lowest point in energy is sitting in the middle of the C-C bond.

ACKNOWLEDGMENTS

We are most grateful to A. G. Borisov for continuous discussions and fruitful advice. This research was supported by Triangle de la physique (Grant No. 2012-040T-GIFAD) and by the Agence Nationale de la Recherche (Grant No. ANR-11-EMMA-0003).