Dec 7, 2012 - projectile in the surface plane and the slow motion in the direction perpendicular to the surface appear nearly decoupled. Fast-atom diffraction ...
PHYSICAL REVIEW A 86, 062903 (2012)
Transition from fast to slow atom diffraction Asier Zugarramurdi and Andrei G. Borisov Institut des Sciences Mol´eculaires d’Orsay (ISMO), UMR 8214 CNRS–Universit´e Paris-Sud, Bˆatiment 351, Universit´e Paris-Sud, 91405 Orsay CEDEX, France (Received 22 September 2012; published 7 December 2012) For energetic atomic beams grazingly incident at a surface along low-index directions, the fast motion of the projectile in the surface plane and the slow motion in the direction perpendicular to the surface appear nearly decoupled. Fast-atom diffraction (FAD) experiments reveal two-dimensional (2D) diffraction patterns associated with exchange of the reciprocal vector perpendicular to the low-index direction of fast motion. These results are usually interpreted within the axial-channeling approximation, where the effective 2D potential experienced by the projectile is set as an average of the 3D surface potential along the atomic strings forming the channel. In this work, using the example of grazing scattering of He atoms at a LiF(001) surface, we address theoretically the range of validity of the axial-channeling approximation. Full quantum wave-packet-propagation calculations are used to study the transition from the 2D (fast atom) to the 3D diffraction pattern characteristic for low-energy atomic and molecular projectiles scattered from surfaces. Along with exact calculations, a semianalytical perturbative treatment based on the Lippmann-Schwinger equation allows an explanation of why the diffraction processes involving the exchange of reciprocal-lattice vectors along the fast-motion direction are exponentially small in typical FAD conditions. DOI: 10.1103/PhysRevA.86.062903
PACS number(s): 34.35.+a, 68.49.Bc, 34.50.Cx
I. INTRODUCTION
For a long time it has been assumed that fast projectiles scattered from surfaces follow classical trajectories. This is primarily because of the numerous sources of energy losses leading to the decoherence of the beam and disappearance of quantum wave effects such as diffraction. Many surface scattering experiments have been successfully explained within this classical framework [1–4]. However, recent studies have demonstrated that quantum coherence can be preserved for atomic projectiles such as He atoms of energies ∼0.2– 2 keV grazingly scattered along low-index directions from monocrystal surfaces. Well-resolved diffraction patterns in the scattered beam have been observed with dielectric, semiconductor, and even metal surfaces [5–10]. The diffraction is possible because, under grazing angles, the de Broglie wavelength associated with the slow projectile motion normal to the surface is comparable to interatomic spacings at the surface. Furthermore, for the glancing-incidence geometry the turning point of the projectile trajectory is far enough from the surface that the incoherent energy losses due to electron and phonon excitations are reduced [11,12]. Since the first works reporting on grazing-incidence fastatom diffraction (FAD or GIFAD) at surfaces, the technique has proven its power as a surface analysis tool. One of the advantages of FAD over other surface analysis techniques is the extreme sensitivity of the diffraction pattern to the details of the interaction potentials between the projectile and the topmost layer of the atoms at the target. Thus, valuable information on the surface structure and reconstruction could be extracted [10,12,13]. From the point of view of the interpretation of the experimental data, most of the fully quantum treatments and the semiclassical approaches reported so far rely on a simplifying assumption on the projectile-surface interaction. It is argued that for the incidence along the low-index directions the projectile “feels” only the potential averaged along the atomic 1050-2947/2012/86(6)/062903(9)
strings in the direction of fast motion [10,12,14–16]. This axial-surface-channeling (ASC) approximation allows general three-dimensional (3D) scattering problem to be reduced to a 2D scattering in the plane perpendicular to the surface and defined by the normal vector along the corresponding lowindex direction. The diffraction patterns can be thus simulated at a relatively low computational cost, as compared to the 3D studies of FAD [15,16] which are much more computationally demanding. Our contribution provides a quantitative assessment of the validity of the ACS approximation. We answer the question: What are the scattering conditions when the 3D structure of projectile-surface interaction potentials becomes important? To this end we study theoretically the scattering of He atoms from the perfectly periodic and rigid ionic crystal LiF(001) surface. The decoherence of the beam due to the thermal vibrations of the lattice and electronic excitations is left for future work. We consider the incident beam along the 110 and 100 directions. A large range of scattering conditions (energies and incident angles) is encompassed with fully quantum calculations based on the wave-packet-propagation approach. We compare the results obtained with full account of the three-dimensional (3D) projectile-surface interaction potential with those obtained within the ASC approximation. Results of our study show that the 2D ASC approximation holds in typical FAD conditions so that it can be confidently used for the efficient modeling of experimental data at low computational cost. At the same time, with decreasing projectile velocity component parallel to the surface the 2D approximation progressively breaks down. The diffraction acquires the 3D character where the reciprocal-lattice vector can be exchanged along the beam direction. The paper is organized as follows. Section II describes the wave-packet-propagation technique and the construction of the projectile-surface interaction potential. Section III presents results and their discussion. A summary and conclusions are
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projectile. The energy of the incident particles perpendicular to the surface, E⊥ = kz2 /2M, is small. The scattering matrix describing the diffraction of the incident beam at the surface is calculated numerically with the wave-packet-propagation (WPP) method as described in Ref. [6]. It is worth noting that the first applications of the WPP technique to atom-surface scattering problems were reported by Yinnon and Kosloff [17]. Here we give a brief outline of the specific application of the WPP to FAD studies. The method is based on the direct solution of the time-dependent Schr¨odinger equation for the wave function of the atomic projectile: 1 + Vs (r) + Vabs (r) (r,t), (1) i∂t (r,t) = − 2M
FIG. 1. (Color online) Schematic representation of the geometry of the LiF(001) surface for the incident beam along the 110 (left side) and 100 (right side) directions. In the top panels the geometry in real space is shown together with the direct lattice vectors as used in the calculation. Orange (big) spheres, F− ; blue (small) spheres, Li+ ions of the lattice. The bottom panels show the corresponding reciprocal lattice together with the primitive vectors. Black arrows show the primitive reciprocal-lattice vectors for the (super)cell used in the calculation for the 110 (100) configurations. For the 100 configuration the red arrows show the primitive reciprocal lattice vectors of the actual unit cell. Relevant values are given in Table I.
given in Sec. IV. We use atomic units throughout this work unless otherwise stated. II. METHODS A. Wave-packet-propagation method
First we establish the notation conventions to be used through the paper. We study the elastic scattering of 3 He and 4 He atom beams from the rigid LiF(001) surface. The atomic projectile impacts at the surface along a low-index symmetry direction which is assumed to be the x direction as sketched in Fig. 1. The surface lies in the (x,y) plane and the z axis is perpendicular to the surface and pointing into the vacuum. For the typical FAD conditions, the perfectly aligned (ky = 0) incident beam is characterized by fast motion in the x direction with corresponding momentum kx and energy parallel to the surface of E = kx2 /2M, where M stands for the mass of the TABLE I. Parameters of the ionic LiF crystal. The lattice constant is given by a = 7.592a0 . The absolute values of the primitive vectors of the direct lattice at the surface are a1 = a2 = α. G1 = G2 = G are the absolute values of the primitive reciprocal-lattice vectors used in the (super)cell. Direction 110 100
α (units of a0 ) √ a/ 2 a
G (units of a0−1 ) 1.170 0.828
where r = (x,y,z), is the Laplace operator, and M is the projectile mass. The atom-surface interaction potential Vs (r) and the complex absorbing potential Vabs (r) are defined below. We stress here that Vs is the full 3D potential so that we describe the complete 3D diffraction process. At this point we do not rely on the ASC approximation and the motions perpendicular and parallel to the surface are coupled a priori. Owing to the periodicity of the problem in the surface (x,y) plane the sought wave function is represented as a Bloch wave 2
(r,t) = eikx x+iky y−ik t/2M ψ(r,t),
(2)
where ψ(r,t) √ is a periodic function of the x and y coordinates and k = kx2 + ky2 is the momentum parallel to the surface. Note that the rapidly oscillating time-evolution exponent associated with the fast motion is introduced in Eq. (2). The actually calculated ψ(r,t) is then a slow function of time so that numerical convergence is improved. Consistent with the choice of the periodicity of the problem in Cartesian coordinates we use (i) a square surface unit cell with dimensions √a2 × √a2 for the case of incidence along the 110 direction; (ii) the supercell geometry with an a × a square supercell for the case ˚ is of incidence along the 100 direction. Here, a = 4.02 A the LiF lattice constant. A schematic representation of the geometry is given in Fig. 1. From Eqs. (2) and (1) one obtains the time-dependent Schr¨odinger equation describing the evolution of ψ(r,t) : i∂t ψ(r,t) = [Tˆ + Vs (r) + Vabs (r)]ψ(r,t),
(3)
U
where the kinetic energy operator is given by 2 k2 ∂ 1 + ikx − Tˆ = − 2M ∂x 2M 2 1 ∂ 1 ∂2 − . + iky − 2M ∂y 2M ∂z2
(4)
For the time evolution of the wave function we use the shorttime propagation with the split-operator technique [18,19]: ψ(r,t + δt) = e−iU δt/2 e−i T δt e−iU δt/2 ψ(r,t). ˆ
(5)
With the wave function represented on an equidistant mesh in the x, y, and z coordinates, the action of the exponential of the kinetic energy operator is calculated via the pseudospectral Fourier-grid Hamiltonian approach [20,21]. We use typically
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Nx × Ny × Nz = 32 × 32 × 1536 points covering the unit (super)cell in x and y, and a z range starting at 0.5a0 above the surface atomic layer and extending up to 77a0 into the vacuum. The propagation time step is typically δt = 0.15 a.u. The initial wave packet ψ(r,t = 0) ≡ ψ0 (r) is set in the vacuum side above the surface. It has the form of a Gaussian wave packet impinging at the surface along the z direction: ψ0 (r) = e−iυz Mz e−(z−z0 )
2
/γ
2
.
(6)
The mean velocity υz and the width of the wave packet γ are taken such that the plane-wave spectrum of ψ0 (r) covers the energy range of interest. Typically we use Mυz = 17.6 a.u., and γ = 0.46 a.u. With initial conditions given by Eq. (6) and the wave-function representation given by Eq. (2) one fixes in the calculation the parallel energy. On the other hand, the range of normal energies included in the initial wave packet allows one to obtain with a single time propagation the energy-resolved scattering matrix (diffraction probabilities) for different incident angles given by in = arctan(kz /k ). (For the scattering matrix calculations with single time propagation, see, e.g., the book by Tannor [22]). Once the wave packet is propagated, the energy-resolved scattering matrix is extracted with the virtual-detector method [23]. In the asymptotic region at large enough z above the surface the time-dependent wave function is projected on the reciprocal-lattice-vector basis: 2
g (r,t) = eikx x+iky y−ik t/2M 1 × e−imG1 x−inG2 y ψ(r,t)dxdy. A C
(7)
The integral in Eq. (7) is evaluated in the 2D unit cell C of area A. The reciprocal-lattice vector is given by g = mex G1 + ney G2 , where ex(y) are the unit vectors in the x (y) directions, and G1 (G2 ) are the absolute values of the corresponding reciprocal primitive vectors. In the present case G1 = G2 = G. The vectors g define the diffraction orders which will also be denoted by the pair (m,n). The energy-resolved results are obtained with the time-toenergy transform: ∞ 1 eiωt g (r,t)dt, (8) g (r,ω) = 2π 0 where ω corresponds to the total energy. The energy of the incident beam perpendicular to the surface is obviously given by E⊥ = ω − k2 /2M. In practice the upper limit of the integration corresponds to large but finite propagation time T . In the present system convergence is typically achieved for T ∼ 2 × 106 a.u. Finally, the normalized intensities of open diffraction orders (or diffraction probabilities) are calculated as the ratio between the flux in the given diffracted beam and the incident flux through the plane parallel to the surface:
∂ Im g∗ (r,ω) ∂z g (r,ω) 3D , (9) R(m,n) (ω) = Jin (ω) where Jin (ω) stands for the incident flux. The expression given by Eq. (9) is evaluated in the asymptotic region at some z = zd with zd being the position of the virtual detector. The (m,n) diffraction order is open when motion along the coordinate z
normal to the surface is possible, i.e., when 2Mω (kx + mG)2 + (ky + nG)2 ,
(10)
or, equivalently, 2ME⊥ 2G(mkx + nky ) + (m2 + n2 )G2 .
(11)
As stated in Sec. I, the purpose of the present work is to assess the validity of the ASC approximation for FAD, where the effect of the surface potential corrugation in the beam direction is neglected. Indeed, it has been argued that because of the large energy difference between the fast and the slow motions, in the FAD technique the reciprocal-lattice-vector exchange leading to the diffraction happens perpendicular to the axial channel defined by the low-index direction at the surface [6, 10,12,14–16]. This motivates the use of the averaged potential: 1 L V2D (y,z) = Vs (x,y,z)dx, (12) L 0 where L is the period of the lattice along the fast-motion direction x. Under this approximation only the corrugation along y is retained. It is noteworthy that similar conclusions on the dominance of the diffraction out of the scattering plane defined by the specular trajectory have been drawn in analysis of grazing scattering of H2 molecular beams at surfaces at 100 meV energies [24–27]. In order to directly compare results of the full 3D diffraction calculations with those obtained under the ASC approximation we have performed a WPP study of 2D diffraction with the potential given by Eq. (12). In this case the fast motion associated with the x coordinate factorizes out. The numerical problem reduces to solving the 2D Schr¨odinger equation describing the quantum motion in the (y,z) plane, i∂t ψ2D (y,z,t) = Hψ2D (y,z,t),
(13)
with the Hamiltonian H given by 2 1 ∂ H=− + iky 2M ∂y −
ky2 2M
−
1 ∂2 + V2D (y,z) + Vabs (z). 2M ∂z2
(14)
The methodology for obtaining the diffraction probabilities in this case is similar to that described above for the 3D case except that only the y and z variables are retained and the vector g is substituted by the scalar nG. Since there is no coupling between fast motion along the x axis and slow y and z motion, the diffraction takes place in the (y,z) plane. For that reason we will refer to this process as 2D diffraction and denote the corresponding diffraction probability as R2D n . In this case the condition for the diffraction order to be open is given by the 2D counterpart of Eq. (11): 2ME⊥ 2Gnky + (nG)2 .
(15)
In what follows below, for both the 2D and 3D cases we will consider the experimental geometry with perfect alignment of the beam along the low-index direction, i.e., ky = 0 in Eq. (2). Results obtained with ky = 0 appear completely sufficient for the purpose of the present study. We have checked that the conclusions of our work are not altered for grazing beams
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with small misalignement ky G. It is worth noting that, provided the Schr¨odinger equation is linear, the scattering matrix calculated at different values of ky allows one to obtain the diffraction profile corresponding to the incident beam with finite extension in the y direction. This latter case has been used in earlier theoretical treatments to model the finite spatial coherence of the beam [12]. B. Potentials
Following an earlier publication [6] the potential for the He-LiF(001) interaction Vs (r) has been constructed as a sum of binary interactions VF (VLi ) with F− (Li+ ) sites of the ionic crystal lattice: VF (r − RF ) + VLi (r − RLi ). (16) Vs (r) = RF ,RLi
The summation runs over F− and Li+ sites located at RF and RLi , respectively. The binary interaction potentials have been obtained from quantum chemistry calculations performed with the HONDO7 program for a He atom interacting with F5 Li5 and F4 Li5 clusters embedded in the point-charge lattice [28]. Perfect LiF(001) surface geometry is assumed. We have not accounted for the surface rumpling demonstrated in Ref. [14]. Indeed, it affects the precise dependence of the diffraction pattern on the incidence conditions. However, the present work is aimed at a discussion of the gross features of the diffraction process. In particular, we seek for the conditions at which the ASC approximation breaks down and diffraction along the fast direction sets in. The finite size of the computation mesh is an inevitable problem of WPP calculations. The complex absorbing potential [29] serves then to impose outgoing-wave boundary conditions at large z above the surface so that the situation of semi-infinite free space above the crystal is reproduced. We use Vabs (z) of the form: −iC1 (z − C2 )2 , z C2 , Vabs (z) = (17) 0 otherwise, with the length parameter C2 = 20a0 and strength C1 = 10−4 a.u. III. RESULTS AND DISCUSSION A. Calculated diffraction probabilities
We start the discussion of the results by analyzing the specular reflection probability of the projectile beam, which is equivalent to R3D (0,0) , the probability of diffraction into the (m = 0,n = 0) order in the 3D case. In the upper panels of Figs. 2 and 3 we show the results of the full 3D WPP study for 3 He incidence along the 110 and 100 directions, respectively. Qualitatively similar results have been obtained in calculations with 4 He projectiles (not shown). The specular reflection probability is presented as a function of the energy E⊥ of the incident beam perpendicular to the surface. The encompassed E⊥ range is typical for FAD experiments on LiF(001) [5,6,10]. Different curves correspond to different parallel energies of the incident beam E = kx2 /2M in the range of 1–50 eV. Note, however, that for the smallest parallel energies considered here the incidence angle in becomes large so that grazing
FIG. 2. (Color online) 3 He incidence along the 110 direction. Upper panel: Specular reflection probability [intensity of the (m = 0,n = 0) diffraction order] as a function of the energy of the incident beam perpendicular to the surface. The 3D WPP results are shown for different energies parallel to the surface. Note that for E 50 eV the 3D results are almost indistinguishable from that obtained with the averaged 2D potential. Lower panel: Differences between the full 3D and 2D specular reflection probabilities R given by Eq. (18).
scattering conditions are not fulfilled. In order to assess the validity of the axial-channeling approximation we also show the specular reflection probabilities calculated with the 2D averaged potential. This is equivalent to R2D 0 , the probability of diffraction into the (n = 0) diffraction order. Obviously, in the 2D case the dependence on the parallel energy is absent. We obtain that for both scattering directions the full 3D and the 2D results nicely agree at high energies of the scattered beam. Basically, the ASC approximation holds down to E 50 eV, i.e., well below the low-beam-energy limit in FAD experiments. Only for energies E 20 eV do some differences start to be perceivable. The higher is the perpendicular energy of the incident beam, the larger is the error introduced by the ASC approximation. This is because the projectile gets closer to the surface and then the amplitute of the oscillations of the 3D surface potential probed by the projectile becomes larger. For E decreasing below 20 eV the differences between the 3D and 2D results progressively increase until the 2D approximation breaks down at E ∼ 5 eV. This is evident from the reflection curves as a loss of the typical FAD structure approximately given in this case by the J0 Bessel function [10,30–33]. As we will show below, the breakdown of the ASC approximation is linked with
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FIG. 4. (Color online) 3 He incidence along the 110 direction. Interpolated image of the (m = 0,n) diffraction probabilities with n = 0,1,2. The results are shown as functions of the energies of the incident beam parallel and perpendicular to the surface. FIG. 3. (Color online) As Fig. 2 but for 3 He incidence along the 100 direction. Here the encompased perpendicular energy region is larger in order to include a comparable number of oscillations.
the onset of diffraction where the reciprocal-lattice vectors are exchanged along the fast-motion direction x. The breakdown of the ASC approximation can be also seen from the results presented in the lower panels of Figs. 2 and 3. The differences R(E⊥ ,E ) between the 2D and 3D specular reflection probabilities are shown as functions of the energies of the incident beam perpendicular and parallel to the surface: 3D R(E⊥ ,E ) = R2D 0 (E⊥ ) − R(0,0) (E⊥ ,E ).
(18)
The plotted R(E⊥ ,E ) appear as oscillating curves which level off at zero for large E where the difference between 2D and 3D calculations vanish. For the smallest E addressed here, R(E⊥ ,E ) becomes comparable to the absolute value of the specular reflection probability over the entire perpendicular energy range. This implies that the diffraction is 3D and not 2D. At somewhat higher E the error introduced by the 2D approach increases progressively toward higher perpendicular energies E⊥ . As discussed above, this reflects the role of the surface corrugation in the possibility of diffraction along the beam. The calculated 3D diffraction into the (0,n) diffraction orders with reciprocal-lattice-vector exchange perpendicular to the axial-channeling direction shows the same trends as discussed for the specular reflection probability (0,0). Some of the results are shown in Fig. 4 where the contour maps of the diffraction probabilities are plotted as functions of the energies of the incident beam parallel and perpendicular to the surface. At large parallel energies, the probability of (0,n) diffraction becomes independent of E (the bright stripes tend
to be parallel to the ordinate axis) and depends only on the energy of the incident beam perpendicular to the surface. 2D For E above 50 eV we find that R3D (0,n) (E⊥ ,E ) = Rn (E⊥ ), i.e., the ASC approximation holds. With decreasing E the results start to display E dependence which a priori cannot be captured by the 2D model. Observe that for smaller E⊥ the 2D ASC approximation holds up to smaller E since the classical turning point is moved away from the surface. The breakdown of the 2D description of the diffraction process is associated with loss of intensity of the (0,n) diffraction peaks. This is directly connected with the onset of the diffraction into the (m = 0,n) orders, corresponding to the exchange of reciprocal-lattice vectors with nonzero projection on the fast-motion direction. These diffracted beams catch part of the flux, decreasing overall the diffraction probability per channel, and modifying the energy dependence of the (0,n) diffraction peaks. In Figs. 5 and 6 we show the calculated probabilities for diffraction with exchange of the reciprocal-lattice vector g along the fast-motion direction such that g · ex = 0. This corresponds to (m = 0,n) diffraction, which is possible only in the 3D case and usually assumed to be negligible for FAD scattering conditions. Note that a positive (negative) value of m corresponds to an increase (decrease) of the momentum parallel to the surface in the diffracted beam. It follows from the present results that the smaller is the energy of the incident beam parallel to the surface, the higher is the probability to exchange a reciprocal-lattice vector along the incidence direction x. In the displayed energy range, appreciable diffraction exists only for energies below E ∼ 10 eV, i.e., for nongrazing incidence angles. Diffraction probabilities of some percent are reached in this case. It is worth noting that for incidence along the 100 direction the calculations are performed in the supercell geometry.
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FIG. 5. (Color online) Calculated diffraction probabilities for a He beam incident at the LiF(001) surface along the 110 direction. Selected (m,n) diffraction orders with m = 0 are shown in different panels. Results are shown as a function of the energy of the incident beam perpendicular to the surface. Different lines are used for the data obtained with different energies of the incident beam parallel to the surface, as explained in the legend on the top. Because of the symmetry of the surface potential the ±n diffraction orders are degenerate.
3
The actual primitive vectors of the surface reciprocal lattice are given by ξ 1/2 = ey G ± ex G as shown in Fig. 1. The (m,n) diffraction order within the supercell representation corresponds then to the ( m+n , m−n ) diffraction order of the 2 2 actual reciprocal lattice. The perpendicular-energy thresholds for the opening of the diffraction orders as given by Eq. (11) are low or even nonexistent for diffraction with negative m, while they are clearly observed for m > 0 diffraction orders. With increasing E the thresholds for m > 0 diffraction are shifted to larger E⊥ . In typical FAD conditions diffraction with m > 0 is impossible. In contrast, for sufficiently high E , diffraction with m < 0 is always possible energetically. However, the diffracted beams in this case are leaving the surface at large exit angles with respect to the surface plane:
tan( ) =
FIG. 6. (Color online) As in Fig. 5 but for the 100 direction of the incident beam.
Another interesting feature worth commenting on is the similarity of the perpendicular-energy dependence of the results at fixed parallel energy as appears in the left and right panels of Figs. 5 and 6. However, the results in the right panels are shifted by a few hundred meV toward higher perpendicular energies as compared to those in the left panels. This observation can be understood from the detailed balance principle (DBP) [34,35]. Applied to the present case the DBP implies the following: At fixed total energy the probability of scattering i → f from initial state (i) defined by the perpendicular energy E⊥ and momentum parallel to the surface k into the final state (f ) defined by E⊥ and k is equal to the probability of inverse scattering f → i. Consider now the diffraction process. Let the initial state be defined by E⊥ and the momentum parallel to the surface k = ex kx ; then obviously E = kx2 /2M. The final state corresponding to diffraction with exchanged reciprocal-lattice vector g = mex G + ney G is then defined by the perpendicular energy E⊥ = E⊥ − [(m2 + n2 )G2 + 2mGkx ]/2M, and the parallel momentum k = ex kx + g. Here, E = |ex kx + g|2 /2M.
2ME⊥ + 2|m|Gkx − (m2 + n2 )G2 . (kx − |m|G)2
(20)
(21)
From the DBP the probability of the direct process (19) 062903-6
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(E⊥ ,k ) −→ (E⊥ ,k )
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is equal to the probability of the inverse process −g
(E⊥ ,k ) −→ (E⊥ ,k ).
(23)
Since, for E 1 eV the wave vector kx G, the G2 terms can be neglected in Eq. (20). As well, from Eq. (21) it follows that E ≈ E . Thus, at approximately the same parallel energy the probability of diffraction with exchanged reciprocal-lattice vector g equals the probability of diffraction with −g taken at E⊥ − mGkx /M. This fully explains the calculated results. It is worth mentioning that for the (m,n = 0) diffraction orders the detailed balance principle implies guiding. That is, for incidence at some azimuthal angle (tan = nG/kx ) with respect to the axial-channeling direction, one of the diffracted beams will exit perfectly aligned along the axial channel. The efficiency of guiding is given by the probability of the diffraction of the perfectly aligned incident beam into the (m,n = 0) diffraction order. B. Transition matrix elements in FAD
We have seen from the WPP results that in FAD conditions transitions involving momentum exchange along the fast-motion direction are unlikely. The diffraction involves reciprocal-lattice-vector exchange in the direction perpendicular to the axial channel and can be well modeled using the 2D potential averaged along the axial-channeling direction. In the following we develop some arguments within a simple perturbative approach which help in understanding the above exact results. We start with the expansion of the periodic potential Vs of the projectile-surface interaction in a Fourier series in the plane parallel to the surface for the geometry where the beam is aligned along the 110 direction: Vs (x,y,z) = Vmn (z)eimGx einGy , (24) m,n
where m and n are integers. Assuming that the Vmn (z) terms are small, and using the scattering-theory approach based on the Lippmann-Schwinger equation, the probability of (m,0) diffraction with reciprocal-lattice-vector exchange along the fast-motion direction is given by [36] 2 M 2 dzVm0 (z)ϕm0 (z)ϕ00 (z) . R= (25) km0 k00 Here, k00 and km0 are the initial- and final-state momenta perpendicular to the surface, where the corresponding energies 2 2 E⊥ = k00 /2M and E⊥ = km0 /2M are linked by Eq. (20) with n = 0. The wave functions ϕ00 (z) and ϕm0 (z) describe the motion perpendicular to the surface along z in the initial and final states of the diffraction. They satisfy the 1D stationary Schr¨odinger equation with in-plane averaged potential V00 (z) and energies E⊥ and E⊥ , respectively. The above derivation of the diffraction probability R is based on first-order perturbation theory, where the specular reflection is assumed to dominate the scattering. In Fig. 7 we show the probability of (−1,0) diffraction calculated within the exact 3D WPP approach and compare it with the perturbative prediction given by Eq. (25). The dotted line in the same figure shows the increase of the energy perpen-
FIG. 7. (Color online) Left y axis: Calculated (−1,0) diffraction probabilities for a 3 He beam incident at the LiF(001) surface along the 110 direction. Results are presented as functions of the energy parallel to the surface for two energies of the incident particles perpendicular to the surface. Continuous lines: Full 3D WPP. Dashed lines: Perturbative approach. The normalized factor 1/18 is applied to the perturbative results. Right y axis: Perpendicular energy difference between initial and final states of the diffraction E⊥ = E⊥ − E⊥ . The dotted line shows E⊥ as a function of the parallel energy.
dicular to the surface as it results from diffraction. It is given by the difference in energies perpendicular to the surface of the diffracted and incident beams: E⊥ = E⊥ − E⊥ , and can be directly obtained from Eq. (20). The results are presented as functions of the energy of the incident beam parallel to the surface for two fixed values of E⊥ . For large parallel energies the diffraction probabilities are extremely small so that the system does not “probe” the surface potential corrugation along the beam. This fully supports the use of the 2D averaged potential in simulations of FAD. With decreasing E the diffraction probabilities grow exponentially. For E 10 eV (5 eV) 3D diffraction with reciprocal-lattice vector exchange along the axial channel reaches the level of several percent for E⊥ = 1 eV (0.5) eV so that the 2D approximation cannot be used. In this parallel-energy range the exact and perturbative results show distinctly different behaviors with oscillating structure of the former and exponential growth of the latter. The fact that the perturbative description overestimates R overall and fails at low E points to the importance of the coupling to other diffraction orders. In particular, only some percent of the population remains in the scattered specular beam (see Figs. 2 and 3). Thus the perturbative estimate can be used here only as a hint toward understanding the exponential behavior of the diffraction probability with E . The decrease of the (−1,0) diffraction probability with increasing E is directly linked to the amount of the perpendicular energy to be transferred to the projectile during the surface scattering event. The larger is the energy change the less probable is the corresponding transition. This can be understood with the help of Fig. 8 where we have plotted the densities of the wave functions ϕ00 (z) and ϕ10 (z) corresponding to the (−1,0) diffraction with incident energies E⊥ = 0.2 eV and E = 500 eV so that E⊥ = 2.8 eV. The
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PHYSICAL REVIEW A 86, 062903 (2012) IV. SUMMARY AND CONCLUSIONS
FIG. 8. (Color online) In-plane averaged potential V00 (z) (dashed line) and V10 (z) harmonic of the potential (dotted line) together with probability densities (wave functions squared) describing the particle motion perpendicular to the surface before (black, lower line) and after (blue, upper line) diffraction. The probability densities are shifted according to the perpendicular energy of the corresponding state E⊥ . The nonshaded area corresponds to the vacuum region above the classical turning point. For further details see the main text of the paper.
transition amplitude is given by the integral in Eq. (25). It thus reflects the overlap between the slowly oscillating initial state weighted by the exponentially decreasing V10 (z) harmonic of the potential and the final state, which is an extremely fast-oscillating function because of its high energy (large energy transfer from the parallel motion). The integral in Eq. (25) is mainly accumulated at the classical turning point of the final state where the initial state is exponentially small. The demonstration performed here for the (−1,0) diffraction order can be generalized, leading to the conclusion that this is the perpendicular-energy mismatch between the initial and final states that suppresses the diffraction with reciprocallattice-vector exchange along the fast-motion direction. A similar discussion based on classical arguments can be found in Ref. [24]. By choosing systems with small reciprocal-lattice vector along the beam direction, the perpendicular-energy mismatch between the incident and diffracted beams can be sufficiently reduced to enable 3D diffraction. This is particularly evident for the (−1,0) diffraction order, where in typical FAD conditions in Eq. (20), the normal-energy exchange is given by E⊥ ≈
Gkx . M
In summary, we have addressed in a systematic way the validity of the 2D axial-surface-channeling approximation in description of the FAD at surfaces when the projectiles are incident along low-index directions. For the test case of He-atom-beam grazing scattering at the LiF(001) surface we have performed a wave-packet-propagation study of both (i) the full 3D diffraction with the possibility of reciprocal-latticevector exchange along the fast-motion direction; and (ii) the 2D diffraction from the surface potential averaged along the fast-motion direction where only the potential modulation perpendicular to the axial channel is retained. The calculations have been performed for incidence along the 110 and 100 directions, and a wide range of scattering conditions has been encompassed (perpendicular and parallel to the surface energies of the beam). We have found that in the typical experimental conditions of FAD reciprocal-latticevector exchange along the beam is highly unlikely. The ASC approximation then perfectly describes the diffraction pattern. For parallel energies below some tens of eV, 3D diffraction progressively sets in until the 2D ASC approximation completely breaks down at energies of some eV. Basically, this can be interpreted as a transition from fast to slow atom diffraction. Indeed, the diffraction in this case is 3D and should be described in the same manner as for thermal helium beam analysis [37–41]. Analysis of the data based on perturbation theory points at perpendicular-energy mismatch between the incident and diffracted beams as the reason for the vanishing diffraction along the beam in typical FAD conditions for typical surfaces. It also allows us to conclude that, provided the corresponding reciprocal-lattice vector is small, diffraction along the beam can be observed in FAD. According to our results, complex reconstructions or vicinal surfaces possess sufficiently large periods of surface corrugation and might show the 3D effects even in typical FAD conditions. In this respect, recently Winter and co-workers [42] have demonstrated experimentally the existence of several Laue circles on Al2 O3 (1120) for H atom beams of 300 eV, under an azimuthal rotation with respect to the 0001 direction. The study presented here has been performed for a perfectly periodic rigid surface. The decoherence of the scattered beam by thermal disorder [11,14,43] and electronic excitations [44, 45] and its possible impact on the transition from 2D to 3D diffraction are left for future work. ACKNOWLEDGMENTS
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Thus one would infer that for targets with large surface period, as can be the case for reconstructed or vicinal surfaces, 3D diffraction along the beam direction should be possible.
We acknowledge funding from the French Agence Nationale de la Recherche (Grant No. ANR-2011-EMMA003-01). A.Z. acknowledges helpful discussions concerning experimental issues in FAD with colleagues at ISMO, Orsay.
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