J R Manson

Hocine Khemliche

Philippe Roncin

Theory of grazing incidence diffraction of fast atoms and molecules from surfaces

Keywords: number͑s͒: 79.20.Rf, 34.35.ϩa, 68.49.Bc II. THEORY

Prompted by recent experimental developments, a theory of surface scattering of fast atoms at grazing incidence is developed. The theory gives rise to a quantum-mechanical limit for ordered surfaces that describes coherent diffraction peaks whose thermal attenuation is governed by a Debye-Waller factor, however, this Debye-Waller factor has values much larger than would be calculated using simple models. A classical limit for incoherent scattering is obtained for high energies and temperatures. Between these limiting classical and quantum cases is another regime in which diffraction features appear that are broadened by the motion in the fast direction of the scattered beam but whose intensity is not governed by a Debye-Waller factor. All of these limits appear to be accessible within the range of currently available experimental conditions.

I. INTRODUCTION

Recently, two laboratories have demonstrated that it is possible to diffract beams of fast-moving atoms and molecules from ordered surfaces under conditions of grazing incident angles, a method that has been given the name grazing incidence fast atom diffraction ͑GIFAD͒. [1][2][3] The range of energies goes from several hundred eV up to tens of keV for the projectiles used which include He, H atoms, and H 2 molecules. The target surfaces were single-crystal alkali halides. These energies correspond to de Broglie wavelengths several orders of magnitude smaller than the lattice spacing of the target crystal. Because of the long travel path near the surface one might expect sufficient dephasing to occur which would destroy all quantum coherence, and furthermore a simple application of the Debye-Waller attenuation due to thermal roughness of the surface would imply totally negligible diffraction intensities except possibly for the most grazing incidence conditions. However, when the incident beam was aligned along a high-symmetry direction intense and clearly measurable diffraction peak signals were observed in the direction normal to the scattering plane.

Two different theoretical groups have carried out numerical simulations of the diffraction process using propagation of wave-packet techniques and have shown that one can obtain a reasonable description of the scattering intensities as functions of the initial beam parameters such as energy and incident angle. 1,[START_REF] Simonović | presented at the ICPEAC Conference[END_REF] These descriptions indicate that along the fast velocity component of the incident beam the interaction potential of the surface is averaged so that the total potential appears as if it were relatively smooth with one-dimensional corrugations in the high-symmetry direction. Also, it has been shown that simple calculations of diffraction intensities based on the eikonal approximation for a one-dimensional repulsive hard wall potential can predict the diffraction intensities reasonably well. 1 In this paper a theory is developed starting from semiclassical quantum mechanics that produces analytical expressions which describe the scattering process at levels of coherence ranging from purely quantum-mechanical diffraction to the fully incoherent classical limit. The basic starting approximations are the eikonal approximation and assumption of a target crystal whose thermal vibrations are in the harmonic limit. The scattering process is described as a series of sequential collisions along the path of the projectile as it moves near the crystal. Under these conditions very little energy is lost due to motion in the fast direction parallel to the surface and it is shown that the Debye-Waller factor becomes sufficiently large to permit measurable diffraction intensities. Under conditions where quantum peaks are observed the very small energy loss associated with the fast direction ensures that the dephasing is sufficiently weak to permit quantum scattering in the other two mutually perpendicular directions. In addition to the purely quantum diffraction limit the theory can be taken to the classical limit through use of the Bohr correspondence principle. In between the clearly distinct quantum and classical limits there is a regime where the dephasing allows for distinct quantum peak features that are broadened by a limited coherence length and these quantum diffraction features do not obey Debye-Waller attenuation. Each of these regimes has its own signature characteristics that should be measurable in the energy, angular, and temperature dependence of the measurements.

In the remainder of this paper the theory is developed together with a description of the various possible quantummechanical and classical limits in Sec. II. In Sec. III some numerically calculated results are shown and discussed. Discussion of the results is presented in Sec. IV and some concluding remarks are made in Sec. V. Some of the results used in developing the various models of Sec. II are developed in the Appendixes A-C.

w͑k f ,k i ͒ = 2 ប ͳ ͚ ͕n f ͖ ͉T fi ͉ 2 ␦͑E f -E i ͒ ʹ , ͑1͒
where T fi is the transition matrix element taken with respect to the final and initial states of the system of projectile plus target, and E i and E f are the initial and final global energies of the entire system. The angular brackets indicate an average over all initial states of the surface target and the ͚ ͕n f ͖ indicates a sum over all final states of the target.

The usual procedure is to write the energy delta function as its time-integral Fourier representation and to shift to the interaction picture in which the time dependence is governed by the target Hamiltonian, [START_REF] Weinstock | [END_REF] a process that is sometimes called the Glauber-Van Hove transformation: [START_REF] Glauber | ͑Minutes of the[END_REF]7 w͑k f ,k i ͒ = 1

ប 2 ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ͗T if ͑t͒T fi ͑0͒͘, ͑2͒
where E f and E i are the final and initial translational energies of the projectile atom. The fast atom diffraction experiment is one in which one component of the incident momentum, chosen as បk ix , is much larger than the component in the direction normal to the surface which is chosen to be the z direction. Thus, it will be a reasonable approximation to assume that the transition operator separates into the product of an operator in the normal direction and an operator in the directions parallel to the surface T ˆ= T ˆzT ˆxy . ͑3͒

The matrix element of the transition operator, taken with respect to final and initial states of the projectile, then separates into the product T fi = ͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͓͒⌿ f ͑x,y͉͒T ˆxy ͉⌿ i ͑x,y͔͒. ͑4͒

The squared normal transition operator ͉͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͉͒ 2 is proportional to the probability of making a transition from k iz to k fz . There are two useful limits in which this can be evaluated exactly. The first is the nearly trivial case of a surface whose potential is a rigid hard repulsive wall in which case it is a constant. This is the evaluation used in the standard application of the eikonal approximation for elastic scattering. 8 The other useful limit is given by the classical motion of a grazing-angle projectile moving in a onedimensional exponentially repulsive potential such as

V͑z͒ = V 0 e -⌫z .
͑5͒

If the atoms of the surface are subject to small random displacements about their equilibrium positions the scattered distribution becomes a lognormal distribution in the final angle 9,10 as shown below in Appendix A 2:

͉͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͉͒ 2 ϰ P͑ f ͒ = ͱ 2 1 ⌫ f exp ͭ -2 ⌫ 2 2 ͫ ln ͩ f f ͪͬ 2 ͮ , ͑6͒
where 2 is the mean-square displacement. The origin of the surface displacement is thermal motion, although the dis-placement appears static to the fast-moving projectiles because the collision times are much shorter than typical phonon periods. Thus 2 is given by the mean-square thermal displacement normal to the surface, which in the hightemperature limit for a Debye model of the phonon density of states is

2 = ͗u z 2 ͘ = 3ប 2 T S M C k B ⌰ D 2 , ͑7͒
where k B is Boltzmann's constant, T S is the surface temperature, M C is the surface mass, and ⌰ D is the surface Debye temperature.

The matrix elements of the operator T ˆxy will be evaluated semiclassically within the eikonal approximation following the work of Bortolani and Levi. 11 For purely elastic scattering from a rigid potential the wave function in the asymptotic region far from the surface is of the form

i ͑r͒ = e ik i •r -͚ K A͑K͒e i͑K i +K͒•R e ik fz z , ͑8͒
where K i is the surface-parallel component of the incident wave vector k i and

K = K f -K i . The perpendicular compo- nent k fz of k f is determined by energy conservation E f = E i which implies k fz 2 = k iz 2 -K 2 -2K • K i . ͑9͒
The coefficient A͑K͒ of the outgoing asymptotic wave is related to the transition matrix element through [START_REF] Manson | Helium Atom Scattering[END_REF] ͓⌿ f ͑x,y͉͒T ˆxy

͉⌿ i ͑x,y͔͒ = ie i␦ f ប 2 k fz mL A͑K͒, ͑10͒
where m is the particle mass, L is the quantization length, and ␦ f is an unimportant phase.

The eikonal approximation consists in first applying the Rayleigh ansatz, i.e., assuming that the wave function of Eq. ͑8͒ is valid near the surface and then applying the correct hard wall boundary condition

͓R,z = ͑R͔͒ = 0, ͑11͒
where ͑R͒ is the corrugation function of the repulsive wall. The eikonal approximation is obtained by further assuming that k fz is weakly dependent on K which allows for a simple evaluation of the coefficient of the outgoing wave

A͑K͒ = 1 L 2 ͵ dRe -iK•R e -i⌬k z ͑R͒ ,

͑12͒

where

⌬k z = k iz + k fz .
Equation ͑12͒ is the eikonal solution to the purely elastic problem. The more general problem of a potential with thermal vibrations can be treated by introducing the displacement function u͑R , t͒ of the surface into the coefficient A͑K͒ using the transformations R → Ru ʈ ͑R , t͒ and ͑R͒ → ͑R͒ u z ͑R , t͒. Applying this transformation to the transition matrix through Eq. ͑12͒ casts the transition rate of Eq. ͑2͒ into the form

w͑k f ,k͒ i = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 ͉͑⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͉͒͒ 2 ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ϫ 1 L 4 ͵ dR ͵ dRЈe -iK•͑R-RЈ͒ e -i⌬k z ͓͑R͒-͑RЈ͔͒ ϫ͗e i⌬k•u͑R,t͒ e -i⌬k•u͑RЈ,0͒ ͘, ͑13͒
where ⌬k = k f -k i = ͑K , ⌬k z ͒. Equation ͑13͒ is equivalent to the similar expression for surface scattering in the eikonal approximation developed in Ref. 11. The thermal average in Eq. ͑13͒ can be readily carried out in the harmonic approximation, and after taking proper account of the commutation relations of the displacement operators at different times and positions the result is [START_REF] Maradudin | Theory of Lattice Dynamics in the Harmonic Approximation[END_REF] w͑k f ,k i ͒ = 1

ប 2 ͩ ប 2 k fz mL ͪ 2 ͉͑⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͉͒͒ 2 ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ϫ 1 L 4 ͵ dR ͵ dRЈe -iK•͑R-RЈ͒ e -i⌬k z ͓͑R͒-͑RЈ͔͒
ϫe -2W͑⌬k͒ e 2W͑⌬k;R,RЈ,t͒ . ͑14͒

The argument of the Debye-Waller factor exp͕-2W͖ is given by

2W͑⌬k͒ = ͗͑⌬k • u͒ 2 ͘, ͑15͒
and a standard approximation is to assume that all cross terms in Eq. ͑15͒ average to zero and that the mean-square displacements in all three mutually perpendicular cartesian directions are the same. This leads to

2W͑⌬k͒ → ⌬k 2 ͗u z 2 ͘ = 3ប 2 ⌬k 2 T S M C k B ⌰ D 2 .

͑16͒

The position and time dependent correlation function appearing in the exponential of Eq. ͑14͒ is given by 2W͑⌬k;R,RЈ,t͒ = ͗⌬k • u͑R,t͒⌬k • u͑RЈ,0͒͘. ͑17͒ If the surface is periodic the correlation function also exhibits this periodicity and can be expanded in terms of the normal modes of vibration and appears as [START_REF] Maradudin | Theory of Lattice Dynamics in the Harmonic Approximation[END_REF]14 2W͑⌬k;R,RЈ,t͒ = ͚ ␣,␣Ј=1

3 ⌬k ␣ ⌬k ␣͚ Q, ប 2N C M C ͑Q͒ e ␣ ͑Q,͒ ϫe ␣ Ј ͑Q,͒e iQ•͑R-RЈ͒ ͕͓n ͑Q͒ + 1͔ ϫe -i ͑Q͒t + n ͑Q͒e i ͑Q͒t ͖, ͑18͒
where e ␣ ͑Q , ͒ is the ␣ Cartesian component of the polarization vector of the mode with parallel wave vector Q and perpendicular quantum number ͑ is discrete for surface modes and is a continuous variable for the bulk modes corresponding to the perpendicular wave vector͒ and N C is the number of modes. The mode frequency is ͑Q͒ and n ͑Q͒ is the Bose-Einstein function

n ͑Q͒ = 1 e ប ͑Q͒ k B T S -1
.

͑19͒

Comparison of the correlation function of Eqs. ͑17͒ or ͑18͒ and the Debye-Waller exponent of Eq. ͑15͒ shows that

W͑⌬k͒ = W͑⌬k;R = RЈ,t = 0͒, ͑20͒
a fact that becomes important later when discussing the classical multiphonon limit of the transition rate. Equation ͑14͒ is the starting point for developing and discussing the various regimes of scattering that can occur in fast atom diffraction from surfaces. These regimes range from the purely quantum-mechanical case to completely incoherent classical scattering and in Secs. II A and II D below we discuss several of these possibilities.

A. Classical scattering

The classical regime occurs when the scattering is completely incoherent as for example when a large number of phonons are transferred in the collision and the quantum coherence of the incident beam is completely destroyed. One case in which this occurs is when the Debye-Waller factor becomes negligibly small implying that its argument 2W is large. 14,15 The value of the Debye-Waller argument is essentially a measure of the number of phonons transferred in the collision, and it will be large when either ⌬k or the meansquare displacement ͑or effectively the temperature T S ͒ becomes large. When the Debye-Waller argument 2W becomes large the correlation function 2W of Eq. ͑18͒ also becomes large and furthermore its most important contributions come from the region of small t and small R -RЈ. In this case the leading terms in the expansion of 2W are

2W͑⌬k;R,RЈ,t͒ Ϸ 2W͑⌬k͒ -it ͚ ␣,␣Ј=1 3 ⌬k ␣ ⌬k ␣ Ј ͚ Q, ប 2N C M C e ␣ ͑Q,͒e ␣ Ј ͑Q,͒ -t 2 ͚ ␣,␣Ј=1 3 ⌬k ␣ ⌬k ␣ Ј ͚ Q, ប ͑Q͒ 4N C M C e ␣ ͑Q,͒e ␣ Ј ͑Q,͓͒2n ͑Q͒ + 1͔ -͚ ␣,␣Ј=1 3 ⌬k ␣ ⌬k ␣ Ј ͚ Q, ប͓Q • ͑R -RЈ͔͒ 2 4N C M C ͑Q͒ e ␣ ͑Q,͒e ␣ Ј ͑Q,͓͒2n ͑Q͒ + 1͔, ͑21͒
where the Debye-Waller argument 2W͑⌬k͒ is the same as that of Eq. ͑15͒

The simplest approximation for evaluating the expansion of Eq. ͑21͒ is to assume a Debye-Model with bulklike symmetry in which all cross terms vanish and for this case it simplifies to 2W͑⌬k;R,RЈ,t͒ Ϸ 2W͑⌬k͒ -

i⌬E 0 t -⌬E 0 k B T S t 2 - ⌬E 0 k B T S ͑R -RЈ͒ 2 2ប 2 v R 2 , ͑22͒
where the classical recoil energy is given by ⌬E 0 = ប 2 ⌬k 2 / 2M C , and the velocity parameter v R is given by the relation 16 1

v R 2 = 1 k B T S ͚ ␣,␣Ј=1 3 ⌬k ˆ␣⌬k ˆ␣ Ј ͚ Q, ប͑Q • R ˆ͒2 2N C ͑Q͒ ϫe ␣ ͑Q,͒e ␣ Ј ͑Q,͓͒2n ͑Q͒ + 1͔, ͑23͒
where R ˆis a unit vector in the direction of R -RЈ. The parameter v R of Eq. ͑23͒ is a weighted average of phonon speeds parallel to the surface and it can be evaluated for several simple models of the surface phonon density. 16 These models produce values that are of order of the bulk acoustic phonon velocities or the Rayleigh wave velocity.

With the simple evaluation of the correlation function of Eq. ͑22͒ the transition rate of Eq. ͑13͒ can be evaluated, a process equivalent to a steepest descents approximation. Ignoring for the moment the final term involving the spatial dependence in Eq. ͑22͒ the result is

w͑k f ,k i ͒ = 2 ប ͩ ប 2 k fz mL ͪ 2 ͉͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͉͒ 2 ͉A͑K͉͒ 2 ϫ ͱ 1 4k B T S ⌬E 0 exp ͭ -͑E f -E i + ⌬E 0 ͒ 2 4k B T S ⌬E 0 ͮ .

͑24͒

Equation ͑24͒ is an expression that describes a single classical collision. For example, if the prefactor transition matrix elements are chosen to be a constant, i.e., if

ͩ ប 2 k fz mL ͪ 2 ͉͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͉͒ 2 ͉A͑K͉͒ 2 = C, ͑25͒
the choice appropriate for hard-sphere scattering where the classical cross section is uniform in all angular directions, then it becomes the classical transition rate well known in neutron scattering. 17 It is also an expression that has been demonstrated to explain single collision events in low energy alkali ion scattering from metal surfaces 18,19 as well as energy-resolved measurements of rare-gas scattering from molten metal surfaces. 20 The spatial dependence appearing in Eq. ͑22͒ that was ignored in deriving the transition rate of Eq. ͑24͒ will become of importance to the discussion below in Sec. II D. However, it is of interest to note here that another classical expression for the transition rate was obtained by Brako and Newns 16 under the assumption of a smooth surface with ther-mal vibrations. If all terms of Eq. ͑22͒, including the term in the spatial dependence, are used to evaluate Eq. ͑13͒, but at the same time assuming that the surface is flat, i.e., setting the corrugation function ͑R͒ equal to a constant, the result is

w͑k f ,k i ͒ ϰ ͱ 1 4k B T S ⌬E 0 1 4k B T S ⌬E 0 ϫ exp ͭ -͑E f -E i + ⌬E 0 ͒ 2 + 2ប 2 v R 2 K 2 4k B T S ⌬E 0 ͮ ,

͑26͒

where the transition matrix prefactors have again been set equal to a constant. This expression has also been shown to describe rare-gas scattering from metal surfaces. 21,22 For the conditions considered here for fast atom scattering at grazing incidence conditions Eq. ͑26͒ will not be a useful approximation. For the expected values of v R , which is in the neighborhood of phonon speeds, the Gaussian-like function in parallel momentum will be very broad and will not have an appreciable effect on the overall transition rate.

Thus, the basis for describing classical scattering will be Eq. ͑24͒, but as mentioned above it describes a singlescattering event in which the projectile particle is assumed to make a transition from its initial state of momentum បk i to the final state of momentum បk f . However, the discussion in Appendix B shows that this is not at all the case. Instead a grazing-angle projectile will follow a trajectory along the surface and will encounter successive, incoherent collisions with approximately N atoms before being deflected into the final state of momentum បk f . Under grazing-angle conditions, which imply small total scattering angles , the transition rate of Eq. ͑24͒ becomes a very sharp Gaussian as a function of final energy E f with an average energy loss given by

E i -E f Ϸ ⌬E 0 Ϸ E i 2 , ͑27͒
where = m / M C is the mass ratio and the final term on the right of Eq. ͑27͒ arises because in a small-angle collision with little energy loss ͉⌬k͉Ϸk i . This same result can be obtained from the Baule conditions for scattering through small angles as shown in Appendix B.

As shown in more detail in Appendix B the multiplescattering events along the trajectory can be viewed as a series of N convolutions of the Gaussian-like differential reflection function. The energy loss during these N collisions is N times that of Eq. ͑27͒ but with the total scattering angle replaced by the fractional angle suffered at each of the N collisions which is approximately / N. Given that the square of the full width at half maximum ͑FWHM͒ of the Gaussian approximation to Eq. ͑24͒ is proportional to 4k B T S ⌬E 0 and that for the convolution of Gaussians the squared width of the convolution is the sum of the squares of the individual widths, the final result is a transition rate that looks exactly like Eq. ͑24͒ except that the recoil energy is replaced by

⌬E = ⌬E 0 N = E i 2 N . ͑28͒
As shown in Appendix B this energy loss is very small and, moreover, is nearly entirely associated with the motion in the slow directions. The energy loss in the fast direction is negligible in comparison because it can be readily shown that ⌬k x 2 is smaller than ⌬k z 2 + ⌬k y 2 by a factor of i 2 which for grazing angles gives a reduction of several orders of magnitude. This small energy loss in the fast direction implies that the scattered projectiles will lie close to the energy conservation circle defined by

k fz 2 + k fy 2 = k iz 2 + k iy 2 , ͑29͒
which also implies that the total scattering angle is given in all cases to a good approximation by 2 i .

Combining the above simple approximation of the total scattering angle with the evaluation of N =6/ ⌫a i for the exponentially repulsive potential of Eq. ͑5͒ as obtained in Appendix B leads to

⌬E = 4E i i 2 N = 2 3 E i ⌫a i 3 , ͑30͒
where a is the distance between surface atoms in the fast direction. This then leads to a final expression for the classical transition rate which looks similar to Eq. ͑24͒ but with important modifications: ͑1͒ ⌬E 0 is replaced by the much smaller ⌬E; ͑2͒ the scattering amplitude ͉A͑K͉͒ 2 separates into the product of ͉A͑⌬k x ͉͒ 2 ͉A͑⌬k y ͉͒ 2 . However, in the fast direction the surface is smoothly averaged and ͉A͑⌬k x ͉͒ 2 is a constant. For the slow direction parallel to the surface ͉A͑⌬k y ͉͒ 2 is the probability of being deflected out of the scattering plane. In the classical limit it will contain the rainbow features of the surface profile, which for example, can be calculated from the classical limit of the eikonal approximation of Eq. ͑12͒. ͑3͒ Finally, the transition matrix element in the normal direction is given by the lognormal distribution of Eq. ͑A22͒ with f given by the relation for the energy conservation circle of Eq. ͑29͒

w͑k f ,k i ͒ = 2 ប ͩ ប 2 k fz mL ͪ 2 P͑ f ͉͒A͑⌬k y ͉͒ 2 ͱ 1 k B T S ⌬E ϫexp ͭ -͑E f -E i + ⌬E͒ 2 4k B T S ⌬E ͮ .

͑31͒

The relationship between the transition rate and the differential reflection coefficient actually measured in the experiments is made explicit in Appendix C. The differential reflection coefficient described by Eq. ͑31͒ has a width in the final polar angle ͑or equivalently in the direction of k fz ͒ dictated by the width of the lognormal distribution P͑ f ͒. Its spread in the azimuthal direction ͑or equivalently the k fy direction͒ is determined by the form factor ͉A͑⌬k y ͉͒ 2 which depends on the shape of the surface potential along the y direction. The intensity will exhibit peaks at the rainbow angles which correspond to the inflection points in the surface profile and the spread in the azimuthal direction will be limited to the classically allowed region, which is typically within the limits of the rainbow features allowed by the largest tilt angle of the surface profile.

B. Purely quantum-mechanical scattering

The case of purely quantum-mechanical scattering in all three dimensions is formally no different from the case of ordinary atom-surface diffraction at low energies. This is most easily discussed in terms of the eikonal approximation 11 or the kinematical approximation 14 in which case the transition rate becomes the same as Eq. ͑14͒ except with ͉͑⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͉͒͒ 2 taken to be constant

w͑k f ,k i ͒ = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ϫ 1 L 4 ͵ dR ͵ dRЈ ϫe -iK•͑R-RЈ͒ e -i⌬k z ͓͑R͒-͑RЈ͔͒ e -2W͑⌬k͒ e 2W͑⌬k;R,RЈ,t͒ .

͑32͒

This is then usually developed into a series ordered in numbers of phonons transferred by expanding the exponential of the correlation function. The elastic scattering is the zeroth order term in this series given by

w͑k f ,k i ͒ = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ϫ ͯ 1 L 2 ͵ dRe -iK•R e -i⌬k z ͑R͒ ͯ 2 e -2W͑⌬k͒ . ͑33͒
The integral over time becomes a simple ␦͑E f -E i ͒ stating the conservation of energy. If the corrugation function ͑R͒ is periodic in both directions parallel to the surface the eikonal approximation matrix element reduces to an integral over a single unit cell multiplied by a Kronecker ␦ function restricting the parallel momentum transfer to reciprocal-lattice vectors G of the surface

w͑k f ,k i ͒ = 2 ប ͩ ប 2 k fz mL ͪ 2 ͚ G ͉A͑G͉͒ 2 e -2W͑⌬k͒ ␦ K,G ␦͑E f -E i ͒, ͑34͒
where the scattering amplitude is

A͑G͒ = 1 S uc ͵ uc dRe -iG•R e -i⌬k z ͑R͒ . ͑35͒
As stated above, Eq. ͑34͒ is the eikonal approximation result for diffraction from a two-dimensional periodic and strongly repulsive surface. However, in general if ប 2 k fz A͑G͒ / mL is replaced by the transition matrix element of the full elastic interaction potential Eq. ͑34͒ becomes exact and is not an approximation.

There are several reasons why it is unlikely that the purely quantum-mechanical diffraction described by the case of Eq. ͑34͒, other than possibly a peak at the specular position, will be observed in the k fx direction in fast atom diffraction: ͑1͒ for energies of hundreds of eV the Debye-Waller factor would be extremely small except for the most grazing inci-dent conditions, while experimentally it is known that diffraction features in the transverse k fy direction are observed for energies into the keV range at incident angles of the order of 1°above the surface plane. 1,2 ͑2͒ The experimental evidence indicates that in the fast direction the atomic projectile experiences an averaged potential that is not corrugated, implying that the matrix elements will be small for nonzero order diffraction peaks in the k fx direction. 23 ͑3͒ The conditions of energy and parallel momentum conservation determine the value of the perpendicular momentum and similar to Eq. ͑9͒ above it is restricted to the Laue circle given by

k fz 2 = k iz 2 -G 2 -2G • K i . ͑36͒
With k ix ӷ k iz the cross term on the right-hand side of Eq. ͑36͒ will dominate and for positive G x values k fz 2 Ͻ 0 and those beams will be evanescent. 1 For negative G x values k fz will become substantially larger than k iz implying that the final diffraction angle will be larger than i and will not satisfy grazing conditions for the exiting beam. This again leads to the conclusion that such diffraction features with G x 0 would have negligible intensity.

C. Quantum behavior in only two dimensions

A more likely scenario for producing diffraction is that the problem reduces to a mixed quantum-classical case, with purely quantum-mechanical behavior in the two slow directions ͑the z and y directions͒ and classical behavior in the fast direction. In this case the motion of the atomic projectile in the fast direction is regarded as a series of N incoherent and successive collisions with the surface atoms just as in Sec. II A above meaning that the lognormal distribution in final angles is retained. However, the quantum treatment of Sec. II B above is carried out for the y direction. The result is

w͑k f ,k i ͒ = 2 ប ͩ ប 2 k fz mL ͪ 2 P͑ f ͒ ͚ G y ͉A͑G y ͉͒ 2 ϫe -2W͑⌬k͒/N ␦ ⌬k y ,G y ͑E f ͒, ͑37͒
where ͑E f ͒ is the density of final states in translational energy identified in Sec. II D below. The scattering amplitude in the y direction calculated with the eikonal approximation is now

A͑G y ͒ = 1 a y ͵ 0 a y dye -iG y y e -i⌬k z ͑y͒ , ͑38͒
which for a simple one-dimensional sinusoidal corrugation function ͑y͒ = ha y sin͑2y / a y ͒, where h is the corrugation strength, leads to the well-known Bessel function solution 8 A͑G y ͒ = J ͉m͉ ͑ha y ⌬k z ͒, ͑39͒

where J n ͑q͒ is the ordinary Bessel function of order n and argument q.

An important modification has been applied to the Debye-Waller factor; it appears divided by a factor of N. If the interaction with the surface is considered to be a series of incoherent collisions with N surface sites, a Debye-Waller factor should be associated with each of these collisions. Such a multiplicative cumulation of Debye-Waller factors is used in low-energy electron diffraction ͑LEED͒ for describing the multiple collisions that the electrons incur as they penetrate the surface. 24 However, the Debye-Waller argument 2W for each of these small collisions is smaller than that of Eq. ͑16͒ by a factor of 1 / N 2 for reasons similar to those that were discussed in connection with Eq. ͑28͒ above, i.e., because ͉⌬k͉Ϸk i and for each small collision Ϸ 2 i / N. Thus, this argument implies that the Debye-Waller factor of Eq. ͑37͒ is transformed according to

e -2W → ͑e -2W/N 2 ͒ N = e -2W/N . ͑40͒
This same result can be arrived at by a somewhat different argument that is more rigorous in the sense that it clearly exhibits all necessary approximations. 25 The Debye-Waller factor is the average of the phase induced by the thermal displacement

e -W = ͗e i⌬k•u ͘ = e -1 2 ͗͑⌬k • u͒ 2 ͘ . ͑41͒
If the total phase is considered to be the sum of the individual phases picked up in each of the small collisions

⌬k • u Ϸ ͚ n=1 N ⌬k n • u n , ͑42͒
and if the individual scattering events are considered uncorrelated the sum of all cross terms in the mean-square thermal average ͗͑⌬k • u͒ 2 ͘ will go to zero leading to a sum containing N mean-square phases. Assuming a similar phase for each small collision again results in the same expression as in Eq. ͑40͒.

There is a seemingly apparent conflict inherent in the result of Eq. ͑37͒ because the density of states in final energy ͑E f ͒, if evaluated strictly, is a ␦ function in the difference between final and initial translational energies. However, if the motion in the fast direction is classical then the translational energy of the projectile cannot be conserved because some energy will be exchanged with the surface through the classical collisions. The conservation of energy is very nearly preserved because the exchange of energy associated with the fast direction is negligibly small for the same reasons as explained above in Sec. II A, i.e., as long as all angles are grazing ⌬k x 2 is smaller than the momentum transfer in the other two directions by a factor of order i 2 . Thus the condition of energy conservation is weakly relaxed, and this allows for the range of k fz values implied by final polar angle width of the lognormal distribution. The expression for ͑E f ͒ that allows for a small exchange of energy is determined in Eq. ͑52͒ of Sec. II D below.

The scattered intensity spectrum described by Eq. ͑37͒ exhibits a number of interesting features. In the k fy direction there are narrow diffraction peaks whose intensity are given by the transition matrix element of the potential which in the eikonal approximation is Eq. ͑38͒. In the actual experiment, the widths in this direction would be limited by the transverse coherence length of the experiment. In the k fz direction these diffraction features are broad with a width determined by the angular spread of the lognormal distribution. All of these diffraction features lie roughly along the Laue circle given by

k fz 2 = k iz 2 -G y 2 -2G y k iy , ͑43͒
or equivalently

cos f = cos i cos i cos f . ͑44͒
In each of the diffraction features k fz increases from the bottom to the top of the broad streak which means that the intensity can vary considerably as is seen from the phase variation in the scattering amplitude of Eq. ͑38͒. This quantum interference effect can be either destructive or constructive, and when it is destructive it can even make portions of the streak disappear. This quantum interference effect can also cause a given streak to appear as if it were shifted above or below the Laue circle. Finally, the Debye-Waller factor is considerably larger than that implied by the simple expression of Eq. ͑16͒ making it possible to observe diffraction peaks with measurable intensities. The Debye-Waller factor essentially depends on momentum transfer only in the quantum-mechanical directions, i.e., ⌬k 2 Ϸ ⌬k y 2 + ⌬k z 2 because at such small scattering angles ⌬k x 2 is negligible in comparison. All of these features are observed in the experimental measurements. 1,2

D. Near-classical scattering with spatial correlations

The two cases above in Secs. II B and II C discuss conditions in which quantum-mechanical diffraction peaks can arise in the scattered intensity. There is yet a third case in which spatial correlations will give rise to diffraction features in an otherwise nearly classical scattered spectrum. This case is similar to the case of diffraction features observed in the multiphonon background in electron-diffraction experiments such as reflection high-energy electron diffraction ͑RHEED͒ where they are known as Kikuchi lines. [START_REF] Ichimiya | Reflection High-Energy Electron Diffraction ͑Cambridge[END_REF] The starting point for this treatment is Eq. ͑14͒ assuming an averaged flat potential in the fast direction and a corrugation function ͑y͒ of periodic length a y in the y direction

w͑k f ,k i ͒ = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 P͑ f ͒ ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប ϫ 1 L 2 ͵ -ϱ +ϱ dy ͵ -ϱ +ϱ dyЈe -i⌬k y ͑y-yЈ͒
ϫe -i⌬k z ͓͑y͒-͑yЈ͔͒ e -2W͑⌬k͒ e 2W͑⌬k;y,yЈ,t͒ . ͑45͒

The periodicity of the corrugation function is expressed as ͑y + na y ͒ = ͑y͒ with n an integer and the fact that the correlation function W depends only on the difference in displacement y -yЈ allows one to replace the spatial integrals by a sum of integrals over a single unit cell. Since the correlation function has very weak spatial dependence over the distance of a single period ͑or equivalently, under the assumption that each unit cell vibrates rigidly͒ this leads to

w͑k f ,k i ͒ = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 P͑ f ͒ ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប 1 L 2 ͵ 0 a y dy ϫ ͵ 0 a y dyЈe -i⌬k y ͑y-yЈ͒ e -i⌬k z ͓͑y͒-͑yЈ͔͒ e -2W͑⌬k͒ ϫ ͚ n=-ϱ +ϱ ͚ nЈ=-ϱ +ϱ
e -i⌬k y a y ͑n-nЈ͒ e 2W͑⌬k;na y -nЈa y ,t͒ .

͑46͒

Making the classical limit expansion of the correlation function as in Eqs. ͑21͒ and ͑22͒ leads to

w͑k f ,k i ͒ = 1 ប 2 ͩ ប 2 k fz mL ͪ 2 P͑ f ͒ ͵ -ϱ +ϱ dte i͑E i -E f ͒t/ប e -i⌬Et-⌬Ek b T S t 2 ϫ ͉A͑⌬k y ͉͒ 2 ͚ n=-ϱ +ϱ ͚ nЈ=-ϱ +ϱ e -i⌬k y a y ͑n-nЈ͒ ϫe -⌬Ek b T S ͑n -nЈ͒ 2 a y 2 /2ប 2 v R 2 , ͑47͒
where ⌬E 0 has been replaced by ⌬E of Eq. ͑30͒ to account for the multiple collisions with N surface unit cells along the fast direction, and as in Eq. ͑38͒

A͑⌬k y ͒ = 1 a y ͵ 0 a y dye -i⌬k y y e -i⌬k z ͑y͒ . ͑48͒
The discrete sums reduce to a single summation given by L a y ͚ n=-ϱ +ϱ e -i⌬k y a y n e -⌬Ek b T S n 2 a y

2 /͑2ប 2 v R 2 ͒ Ϸ L a y បv R ͱ 2⌬Ek B T S ͚ G y e 2ប 2 v R 2 ͑⌬k y -G y ͒ 2 /͑4k B T S ⌬E͒ , ͑49͒
where L is a quantization length parallel to the surface. Clearly, this summation is a periodic function in ⌬k y → ⌬k y + G y where G y =2 / a y with an integer, and the right-hand side of Eq. ͑49͒ becomes a good approximation if

⌬Ek B T S a y 2 / ͑2ប 2 v R 2 ͒ Ͻ 1.
The time integral in Eq. ͑45͒ is the same Gaussian integral encountered in Sec. II A above so the final result for the transition rate is

w͑k f ,k i ͒ = 2 ប ͩ ប 2 k fz mL ͪ 2 P͑ f ͒ 1 ͱ 4⌬Ek B T S ϫexp ͭ -͑E f -E i + ⌬E͒ 2 4⌬Ek B T S ͮ ͉A͑⌬k y ͉͒ 2 បv R ͱ 2⌬Ek B T S ϫ ͚ G y e 2បv R 2 ͑⌬k y -G y ͒ 2 /͑4k B T S ⌬E͒ . ͑50͒
Equation ͑50͒ describes periodic diffraction features lying roughly centered on the Laue circle of Eq. ͑43͒. In the k fy direction these features are Gaussian-like with a width parameter k 0 given by

k 0 = ͱ ⌬Ek B T S a y 2 2ប 2 v R 2 .

͑51͒

In the k fz direction they appear as long streaks whose width is governed by the lognormal distribution in final polar angles. The strength of these peaks at any point ͑k fz , k fy ͒ in the wave vector space perpendicular to the fast direction is given by the form factor ͉A͑⌬k y ͉͒ 2 which in this treatment is approximated by the eikonal expression of Eq. ͑48͒. This form factor can vary substantially over the k fz extent of the diffraction feature, just as discussed above in connection with Sec. II C even to the point of making parts of a diffraction feature disappear because of destructive quantum interference arising from the corrugation within the unit cell of the potential. There is no Debye-Waller factor associated with these diffraction features. Instead the temperature dependence varies inversely with a power of the temperature, in this case T S -3/2 , typical of classical scattering. Also typical of classical scattering, the widths of all peaks increases as ͱ T S for temperatures large enough where zero-point motion is not important. The energy spread within these peaks is quite narrow and given by

͑E f ͒ = 1 ͱ 4⌬Ek B T S exp ͭ -͑E f -E i + ⌬E͒ 2 4⌬Ek B T S ͮ , ͑52͒
which approaches the energy ␦ function ␦͑E f -E i + ⌬E͒ for small ⌬E. Equation ͑52͒ also identifies the density of final energy states ͑E f ͒ introduced above in Eq. ͑37͒.

III. CALCULATIONS

In the following several calculations are shown that illustrate some of the different cases presented in the sections above. The system considered is 3 He scattering from LiF͑001͒ incident along the ͗100͘ direction with an incident energy of 200 eV. The averaged surface corrugation appears as a one-dimensional periodic washboard with a spacing b / 2, where b = 4.03 Å is the LiF lattice vector dimension. The effective one-dimensional reciprocal-lattice vector is G y =4 / b = 3.12 Å -1 .

Figure 1 shows a set of experimental data taken from Ref. 1 compared with two different calculations. The first calculation shown as solid curves, which was also presented in Ref. 1, is a quantum wave-packet propagation using a realistic, static ground-state potential. The second set of calculations shown as dashed curves is carried out using the simple eikonal approximation for a sinusoidal hard corrugated wall in which the diffracted peak intensities are given by the square of the Bessel function of Eq. ͑39͒ multiplied by the ratio of incident to outgoing flux. The corrugation strength parameter is h = 0.024. The measured intensities of the specular and three higher order diffraction peaks are shown as points plotted as functions of the final energy associated with motion normal to the surface ranging from 0 to 0.6 eV, and were obtained for the fixed incident energy beam by increasing the incident angle from near grazing to about 3.5°. For each of the higher diffraction orders, the intensity reported is the sum of the intensities of the two peaks symmetrically positioned on either side of the specular. Both theoretical calculations provide a reasonable description of the data. The quality of the agreement given by the eikonal approximation allows interpretations of the observed diffraction spectra in terms of straightforward quantum interference within a simple corrugated hard surface model, as was noted in Ref.

1. Furthermore, the good agreement with this simple scattering model lends validity to our use of the eikonal approximation here as the starting point for developing a more complete dynamical scattering theory.

It is clearly evident from Fig. 1 that the range of normal energies over which diffraction features are observed is large, with the ratio of the smallest to the largest measured energy being nearly 100. This means that the potentialenergy landscape of the surface can be sampled over a range of normal energies of at least two orders of magnitude, which is considerably larger than can be done with ordinary He atom scattering at thermal energies. In terms of total incident energy, experiments have exhibited diffraction features at energies from 100 eV up to well into the keV range. 1,2 Figure 2 shows a calculation combining both the twodimensional ͑2D͒ quantum behavior of Sec. II C and the near-classical scattering of Sec. II D for the conditions of Fig. 1 at an incident angle i = 2.68°which corresponds to the normal energy of 0.44 eV. The differential reflection coefficient dR / d f d f is plotted as a function of f ͑measured in degrees͒ with the polar angle chosen to lie along the Laue circle of Eq. ͑43͒ or ͑44͒. The intensity plotted on the vertical axis is measured in arbitrary units and the measured data points are normalized to the calculations at the position of the second-order diffraction peak.

For these calculations the surface Debye temperature is chosen to be 530 K ͑a value corresponding to a bulk Debye temperature of 750 K divided by ͱ 2͒, the mass M C = 19 amu corresponding to that of a fluorine atom, the exponential potential range parameter is ⌫ = 2.0 Å -1 , the surface temperature is 300 K, and the velocity parameter is chosen to be v R = 4000 m / s which is the Rayleigh wave speed of LiF͑001͒. [START_REF] Kaye | Tables of Physical and Chemical Constants and Some Mathematical Functions ͑Longman[END_REF] The scattering amplitude A͑⌬k y ͒ that determines the relative intensities of the diffraction features is the same eikonal Bessel function approximation used in Fig. 1 with h = 0.024. The evaluation of the number N =6/ ⌫a i of LiF lattice sites encountered along the path of the He atom during the interaction is approximately N ϳ 21 for these conditions.

The broad diffraction features centered about the diffraction points are calculations of the nearly classical transition rate of Eq. ͑50͒. The sharp vertical lines located precisely at the diffraction positions ⌬k y = G y are the quantum diffraction peaks calculated from Eq. ͑37͒. The relative intensity of the two sets of calculations with respect to each other is arbitrarily chosen. In this particular example the zeroth order or specular diffraction peak at f = 0 has nearly zero intensity and peaks through third order are visible on either side of the specular position.

Figure 3 shows, for the same conditions as Fig. 2, the diffraction feature calculated with the nearly classical transition rate of Eq. ͑50͒ but this time plotted as a function of f measured in degrees for values of f fixed at the positions of the first and third order diffraction peaks in the upper and lower panels, respectively. This exhibits a rather broad feature with an FWHM of about half a degree which is essentially the width of the lognormal distribution in final polar angle. The polar angle distribution for the diffraction peaks calculated with the quantum model of Eq. ͑37͒ would be identical. For the third order diffraction peak a small shift of the distribution along the Laue circle to polar angles smaller than the specular angle f = 2.68°is evident.

Figures 2 and3 show the intensity integrated over all final energies because this is what is experimentally measured by the channel-plate detector. In order to illustrate the energy dependence the full three-dimensional differential reflection coefficient dR͑k f , k i ͒ / dE f d⍀ f is plotted as a function of final energy E f in Fig. 4 for final angles at the specular position and at the second-order diffraction peak in the upper and lower panels, respectively. The calculations are carried out with the nearly classical transition rate of Eq. ͑50͒ for the same conditions as in Fig. 2. The two peaks are quite similar, and in fact the energy spectrum in the vicinity of all diffraction peaks looks very much the same. This figure demon- strates the remarkably small energy losses experienced by these projectiles in spite of the high incident energy and relatively large incident angle. The full width at half maximum is less than 50 meV and the most probable final energy is approximately 10 meV smaller than the 200 eV incident energy, or a fractional energy loss of about 5 ϫ 10 -5 . This energy loss is in agreement with the value of about 7.2 meV given by the expression of Eq. ͑30͒.

IV. DISCUSSION

In this section we discuss the nature of various scattering regimes that can occur in fast atom diffraction and describe the responses to controllable initial experimental conditions, thus suggesting possibilities for making experimental measurements that could explore and distinguish each of these cases. Such a discussion should begin with mention of purely quantum diffraction in all three spatial dimensions. This would be essentially no different than a description of the well-known technique of ordinary He atom diffraction at low ͑thermal͒ energies, and is unlikely to apply to the present experiments because of the high energies and correspondingly extremely large velocities in the fast direction.

However, here we review the salient features of three regimes of diffraction and scattering that appear to be possible. These are ͑A͒ quantum behavior in two dimensions, ͑B͒ near-classical scattering with spatial correlations, and in addition, ͑C͒ possible contributions to the incoherent background are briefly reviewed. Each of these regimes has very characteristic features in its dependence on the experimentally controllable variables.

A. Quantum behavior in two dimensions

In this case, discussed in detail above in Sec. II C, the motion in the fast direction, parallel to the surface, is classical while the motion normal to the surface and also that parallel to the surface but transverse to the fast direction is quantum mechanical. In spite of the classical motion with multiquantum energy transfer in one of the three dimensions, quantum features can be observed in the other dimensions because the energy losses in the fast direction are exceedingly small, too small to cause a complete loss of quantum coherence.

This regime is characterized by diffraction features that appear as streaks in the k fz direction, with the length of the streaks governed by the lognormal distribution P͑ f ͒ in final polar angles f . In the k fy direction these peaks are narrow, their widths being limited by the instrument resolution. These peaks are observed to be distributed roughly around the Laue circle whose wave vector components are given by Eq. ͑36͒.

The intensity dependence of these diffraction features obeys a variant of the classic Debye-Waller behavior. The temperature dependence of the peak intensity is an exponentially decreasing function of temperature divided by a power of T S , and in the treatment shown in Eq. ͑37͒ it is seen that power is one. The argument of the Debye-Waller factor is considerably reduced over the standard value that would be calculated according to Eq. ͑16͒ using the initial and final momenta of the diffraction features. The 2W argument of the Debye-Waller factor is effectively divided by N, the number of collision sites along the path of motion. It is this effect that allows the Debye-Waller factor to be large enough that the diffraction features are measurable. Also, the motion in the fast direction does not contribute appreciably to the momentum transfer in the Debye-Waller argument of Eq. ͑16͒ and this is the cause of the extremely small energy losses in that dimension. For beams incident along high-symmetry directions of the surface, where the Laue circle of Eq. ͑36͒ appears on the channel-plate detector as symmetric about the specular spot, it is of interest to note that the Debye-Waller exponent 2W gets smaller, and hence the Debye-Waller factor becomes larger, with increasing G y reciprocal-lattice vectors, and this is because of the associated decrease of the final perpendicular momentum transfer. If the incident beam is slightly misaligned with respect to a high-symmetry direction, which would imply a small nonvanishing K i in Eq. ͑36͒, the Laue circle becomes skewed with respect to the specular point and the preceeding statement becomes true only for those reciprocal-lattice vectors G y whose corresponding perpendicular momentum transfers decrease, since some perpendicular momentum transfers may increase. The temperature dependence of the width of the diffraction features differs in one respect from the standard Debye-Waller behavior. In the k fy direction, the widths are independent of temperature, which is the normal Debye-Waller behavior. However, the widths of the streaks in the k fz direction increases in proportion to ͱ T S . This increase in width comes from the lognormal distribution in Eq. ͑37͒ which defines the overall length of the streaks. It is also of interest to note that the lognormal distribution shows that the width of the streaks in the k fz direction will be proportional to ⌫, the range parameter of the exponentially decreasing interaction potential in the surface-normal direction.

The relative intensities of the diffraction features associated with different G y reciprocal-lattice vectors, as well as the relative intensity within the streak of the diffraction feature due to a single given G y , are governed by the corrugation landscape of the potential as expressed in the transition matrix amplitudes A͑G y ͒ of Eq. ͑37͒. For example, the corrugation of the surface can produce rainbow effects and supernumerary rainbow effects in the diffraction peak distributions, as has been recently reported. 3 Intensity modulation due to the shape of the potential can also be observed within the streak of a single diffraction feature and such effects can cause the streak to appear "shifted" above or below its expected position on the Laue circle, as is apparent in the observations. 1 In addition to the above responses of the diffraction features to the controllable experimental variables, it is of interest to discuss several possibilities that might be open for future experimental investigation. The first of these is the unlikely possibility of observing quantum diffraction effects in the high velocity direction due to reciprocal-lattice vectors in the G x direction. It is of interest to note the reasons why this is unlikely. For the case of positive G x ͑meaning the direction of G x is parallel to the fast velocity component͒ at the energies used in the present experiments the correspond-ing k fz components given by the Laue equation ͑36͒ become rapidly evanescent and hence unobservable. On the other hand, for negative G x the value of k fz becomes sufficiently large that the diffraction features are projected several degrees in polar angle above the specular position. This means that these peaks would also be unobservable because a flat ordered surface cannot scatter intensity into such large scattering angles.

A possible effect closely related to supernumerary rainbows is the observation of quantum interference due to scattering of terraces at different atomic height intervals on the surface as seen in thermal energy He atom diffraction [START_REF] Lapujoulade | [END_REF] and in reflection high energy electron diffraction. [START_REF] Ichimiya | Reflection High-Energy Electron Diffraction ͑Cambridge[END_REF] This phenomenon should also manifest itself in the present fast atom diffraction-type experiments, although the effect may be limited by the relatively short coherence length of the experiment.

An effect observed very early in the history of He atom diffraction at thermal energies was the observation of selective adsorption or resonances with the bound states of the potential. 29,30 In principle these resonances, which are characterized by sharp increases or decreases in diffraction peak intensities as an experimentally controllable parameter is varied through resonance conditions, should be present in the current experiments. The simplest statement of the condition for selective adsorption resonance is that the value of ប 2 k fz 2 / 2m must be negative ͑corresponding to an evanescent beam͒ and nearly equal to the energy of a bound state of the potential. In the present experiments, the simplest way to look for selection adsorption events would be by slightly varying the incident azimuthal angle i away from a highsymmetry direction. When the incident beam is aligned along a high-symmetry direction with i = 0°the diffraction features observed on the channel-plate detector lie roughly along the Laue circle of Eq. ͑36͒ which appears symmetric with respect to the specular spot. However, if i deviates slightly from zero, for example upon a small azimuthal rotation of the target crystal, the observed Laue circle becomes skewed and asymmetric. Along with this asymmetric shifting of the Laue circle, many of the low-index diffraction features will become evanescent and have k fz values that pass through conditions for resonance.

One of the most important uses of thermal energy He atom diffraction has been the measurement of single phonon transfers and surface phonon-dispersion relations. Because of the large disparities of energy between that of typical phonons and the kinetic energies used in these experiments it is unlikely that most single surface phonon inelastic events will be measurable. However, it is worth mentioning that there is one type of inelastic process that could be measured by these experiments, and this would be energy exchange with dispersionless excitations, e.g., surface phonons, surface plasmons, surface polaritons, etc. whose energies do not depend on momentum transfer parallel to the surface. For example, diffraction should still be observable even in the event that the atomic projectiles exchange one or more dispersionless inelastic quanta because such processes would not destroy the quantum coherence of the system.

B. Near-classical scattering with spatial correlations

This case is described in detail in Sec. II D and is analogous to Kikuchi lines observed in RHEED due to simultaneous multiple phonon transfer and diffraction. The diffraction features appear similar to those due to two-dimensional diffraction described in the above section, except that there is no Debye-Waller factor and there is temperature dependent broadening in both the k fz and k fy directions. This type of diffraction could appear under conditions where the 2D quantum behavior of Sec. IV A has negligible intensity, or the two can appear simultaneously as shown in the calculations presented in Fig. 2. The expression for the transition rate is given in Eq. ͑50͒ and the major difference with the 2D diffraction case of Eq. ͑37͒ is that the product of the Debye-Waller factor and the discrete delta-function dependence in parallel momentum transfer have been replaced by a periodic density of states consisting of broader peaks. The physical reason for these differences is the fact that multiple phonon quanta are transferred in this case, and the spatial region of coherence from which an interference diffraction pattern is built is no longer infinitely large, but becomes finite in extent.

The diffraction features still manifest as relatively long streaks in the k fz direction whose length is governed by the lognormal distribution and these streaks appear aligned roughly along the Laue circle. In the k fy direction the peaks are now broadened beyond the instrumental limit, and the temperature dependence of the FWHM of this broadening is the same in both directions, i.e., proportional to ͱ T S . The temperature dependence of the maximum intensity of the diffraction features, as shown in Eq. ͑50͒, is inversely proportional to T 3/2 .

C. Classical incoherent limit

The classical incoherent limit is essentially a part of the background intensity that should be observed in the experiments. In this paper two different mechanisms from which it can arise are discussed. The first of these, presented in Sec. II A, arises from scattering from regions of the surface whose heights are randomly distributed but whose average interaction potential is a repulsive exponential with range parameter ⌫. The appearance of this background intensity is a feature broadened in the k fz direction by the lognormal distribution and broad in the k fy direction according to the extent permitted by the associated classical scattering form factor ͉A͑⌬k y ͉͒ 2 . The temperature dependence of the maximum of this background intensity varies approximately inversely with the surface temperature, while the FWHM is proportional to ͱ T S . Because the peak width in the polar angle f is governed mainly by the lognormal function, this width is only weakly dependent on incident energy. The peak intensity also is inversely proportional to ⌫ ͱ , while the FWHM is directly proportional to this same quantity.

The total-energy loss is evaluated in Appendix A in the classical limit of large incident energy and high temperatures and is given by ⌬E =2E i ⌫a i 3 / 3. This energy loss for the present grazing-angle geometry with small i is negligible and roughly five orders of magnitude smaller than the inci-dent energy. Most of this energy loss can be ascribed to motion in the direction perpendicular to the surface, and it can be readily shown that the fraction of energy that can be ascribed to diminishing the fast បk fx component of momentum is smaller than the above ⌬E by a factor proportional to i 2 . It is of interest to note that ⌬E is proportional to the range parameter of the repulsive exponential potential, implying that measurement of the energy losses over a range of incident energies, or when combined with measurements of intensity and FWHM dependence, may provide a means to measure this important physical parameter.

The other, and radically different, type of background expected to be measured in high energy atomic scattering experiments is that due to binary collisions with topological defects such as adatoms or step edges on the surface. This contribution is described by Eq. ͑24͒ taken together with the constant form factor of Eq. ͑25͒. This is discussed further starting with Eq. ͑B2͒ of Appendix B. This type of binary scattering is spread over a broad range of energies and angles, with a FWHM proportional to ͱ E i T S and an intensity at its peak maximum inversely proportional to the same quantity, and its general features under the assumption of hard-sphere scattering are outlined in Eqs. ͑B3͒-͑B7͒. In addition to its larger energy and angular widths, this type of background can be distinguished from that discussed in the two paragraphs above by its large energy losses, the fractional energy loss under the grazing-angle conditions of the present experiments would be approximately E i i 2 as opposed to the cubic behavior in i discussed above for ⌬E. This type of classical incoherent scattering is well documented and has been extensively measured in experiments in low-energy ion scattering over energy ranges similar to those of interest here. 19

V. CONCLUSIONS

The objective of this paper is to discuss the variety of scattering regimes that can occur in the diffraction of fast atomic and molecular projectiles at grazing angles from surfaces, and in addition there is a brief description of the background intensity that may be produced. The conclusion is that there are two rather distinct types of quantum effects that may be present in the diffraction features that have been experimentally observed, namely, quantum diffraction in the two slow directions and near-classical scattering with spatial correlations. The latter is similar to Kikuchi features observed in electron diffraction. At the present state of the experimental measurements, it is not possible to clearly distinguish between these two different regimes of quantum diffraction. However, the two regimes present clear and measurable differences in their response to experimentally controllable parameters such as incident energy, incident angle, and surface temperature. In particular, only the case of quantum diffraction in the two slow directions has a thermal attenuation governed by Debye-Waller type behavior. The thermal attenuation of the near-classical diffraction varies inversely with a power of the temperature. Thus, more precise measurements covering a greater range of experimental parameters should make it possible to distinguish between the possible diffraction mechanisms.

ACKNOWLEDGMENTS

One of us ͑J.R.M.͒ would like to express appreciation to the Centre National de la Recherche Scientifique ͑CNRS͒, the Université de Paris-Sud, and the Laboratoire des Collisions Atomiques et Moléculaires ͑LCAM͒ for support and hospitality during a portion of this work.

APPENDIX A: CLASSICAL MOTION IN AN EXPONENTIAL POTENTIAL

For a point particle moving in a one-dimensional exponentially repulsive potential

V͑z͒ = V 0 e -⌫z , ͑A1͒
the expression of conservation of energy is

E z = 1 2 mv z 2 + V 0 e -⌫z , ͑A2͒
where E z is the energy associated with the normal motion and the normal velocity is

v z ͑t͒ = dz dt = ͱ 2E z m ͑1 -e -⌫͑z-z 0 ͒ ͒, ͑A3͒
where z 0 =ln͑V 0 / E z ͒ / ⌫ is the classical turning point. Equation ͑A3͒ admits an analytic solution that can be parametri-

cally expressed in terms of v iz = ͱ 2E z / m as z͑t͒ = z 0 + v iz t + 2 ⌫ lnͩ 1 + e -⌫v iz t 2 ͪ, ͑A4͒
as can be readily verified by direct differentiation.

Energy loss in a classical grazing incidence collision

Beginning from the known expression for the energy transfer in a two-body collision with a small total scattering angle

⌬E 0 = E i -E f = 2 E i , ͑A5͒
where is the mass ratio; the small fractional energy loss occurring over a short distance parallel to the surface during which the particle is deflected by the small angle ␦ is

␦E = ͑␦͒ 2 E i = E i ͩ ␦ ␦t ͪ 2 ͑␦t͒ 2 . ͑A6͒
The assumption of grazing-angle scattering also implies that the parallel velocity v

x ӷ v z implying that ␦ ␦t Ϸ ␦v z ͑t͒ ␦t 1 v x Ϸ ␦ 2 z͑t͒ ͑␦t͒ 2 1 v x , ͑A7͒
and for a surface of small periodic length a one can approximate

v x = dx dt Ϸ a ␦t . ͑A8͒
Taking the infinitesimal limit, Eq. ͑A6͒ becomes

␦E = E i aͩ d 2 z͑t͒ dt 2 ͪ 2 1 v x 3 dt. ͑A9͒
Equation ͑A9͒ identifies the power loss spectrum, which can be evaluated from Eq. ͑A4͒ as 9,10

dE͑t͒ dt = E i a 1 v x 3 ⌫ 2 v 0 4 4 cosh 4 ͑⌫v 0 t/2͒ . ͑A10͒
Integrating Eq. ͑A10͒ over all times along the path of motion and recognizing that the total scattering angle for the grazing-angle collision is Ϸ 2 i leads to a result that is cubic in the incident angle, rather than the quadratic form suggested by Eq. ͑A5͒: 9,10

⌬E = 2 3 E i ⌫a i 3 . ͑A11͒
The result of Eq. ͑A11͒ can be understood in physical terms through the following simple argument: in a grazingangle collision the interaction can be assumed to be spread roughly equally over N sequential and incoherent collisions with N lattice sites in a line along the surface, and clearly for such grazing-angle conditions N ϰ 1 / i . If each of these individual collisions has a fractional energy loss equivalent to Eq. ͑A5͒ and if the fractional angular deflection is the uniform value ␦ Ϸ͑ i + f ͒ / N Ϸ 2 i / N then the sum of all energy losses over the N small collisions along the trajectory is

⌬E = 4E i i 2 N . ͑A12͒
Since N is known to be inversely proportional to i , Eq. ͑A12͒ gives the same cubic dependence on incident angle as the more exact result of Eq. ͑A11͒. This argument also identifies an approximation for estimating the number of surface cells lying along the path of the collision 9,10 N Ϸ 6 ⌫a i . ͑A13͒

The lognormal angular distribution

If the surface potential is exponentially repulsive as in Eq. ͑12͒ and additionally if each element of the surface is subject to a random displacement in the vertical direction the distribution of final scattering angles is lognormal. 9,10 This is perhaps most easily understood through noting that in a grazingangle scattering event with v iz Ӷ v x the final angle is approximately given by

f Ϸ i Ϸ v iz v x Ϸ ͱ 2E z /m v x , ͑A14͒
and this can be related to the classical turning point through

f Ϸ ͱ 2V 0 /m v x e -⌫z 0 /2 = f e -⌫z 0 /2 . ͑A15͒
Now suppose that the surface is subject to random displacements u z in the vertical direction whose probability distribution is Gaussian and given by P͑u z ͒ = 1 ͱ 2 2 exp ͭ -u z 2 2 2 ͮ . ͑A16͒

Even though ultimately the displacements u z may be due to thermal motion of the surface atoms the motion of the projectile is so fast that they appear static, i.e., the collision time is much shorter than a typical phonon period. One can then obtain the mean value of f through

͗ f ͘ f = ͵ -ϱ +ϱ
du z e -⌫͑z 0 -u z ͒/2 1 ͱ 2 2 exp ͭ -u z 2 2 2 ͮ , ͑A17͒

which gives the result

͗ f ͘ f = e ⌫ 2 2 /8
. ͑A18͒

If the integration variable in Eq. ͑A17͒ is changed to f according to

ln ͩ f f ͪ = ⌫ 2 ͑z 0 -u z ͒, ͑A19͒
the result is

͗ f ͘ f = 1 ⌫ f ͱ 2 2 ͵ d f expͭ-2 ͓ ln ͑ f f ͔͒ 2 ⌫ 2 2 ͮ.

͑A20͒

However, relating Eq. ͑A20͒ to the standard form in terms of the distribution function P͑ f ͒ given in the small-angle approximation by

f = ͵ d f f P͑ f ͒,

͑A21͒

identifies P͑ f ͒ as the lognormal distribution

P͑ f ͒ = ͱ 2 2 1 ⌫ f expͭ- 2 ͓ ln ͑ f f ͔͒ 2 ⌫ 2 2 ͮ. ͑A22͒
For an ordered surface, the roughness will be caused by thermal motion. So even though the surface appears static to the rapidly moving projectile, the width of the Gaussian distribution can be associated with the mean-square thermal displacement 2 = ͗u z 2 ͘. ͑A23͒

APPENDIX B: CONVOLUTION OF MULTIPLE COLLISIONS WITH THE SURFACE

The differential reflection coefficient describing classical scattering is Eq. ͑24͒

w͑k f ,k i ͒ = ͩ ប 2 k fz mL ͪ 2 ͉͓⌽ f ͑z͉͒T ˆz͉⌽ i ͑z͔͉͒ 2 ͉A͑K͉͒ 2 ϫ ͱ ប 4k B T S ⌬E 0 exp ͭ -͑E f -E i + ⌬E 0 ͒ 2 4k B T S ⌬E 0 ͮ .

͑B1͒

under many conditions, including the case of grazing-angle scattering at high energies considered here, this becomes very nearly Gaussian in form leading to a differential reflection coefficient Eq. ͑C2͒ of the form

d 3 R͑k f ,k i ͒ dE f d 2 ⍀ f Ϸ ͱ 1 4k B T S ⌬E 0 exp ͭ -͑E f -E i + ⌬E 0 ͒ 2 4k B T S ⌬E 0 ͮ .

͑B2͒

When this happens the most probable energy is given by the Baule relation for an elastic two-body collision E f = f͑ , ͒E i where is the mass ratio, is the scattering angle, and f͑,͒ = ͩ ͱ 1 -2 sin 2 + cos 1 + ͪ 2

.

͑B3͒

The recoil energy evaluated at the most probable final energy is ⌬E 0 = g TA ͑ , ͒E i with g TA ͑,͒ = ͓1 + f͑,͒ -2 ͱ f͑,͒cos ͔. ͑B4͒

The mean-square energy deviation is ͗E f 2 ͘ =2g͑ , ͒E i k B T S where g͑,͒ = g TA ͑,͒ ͑ 1 + -cos ͱ f͑,͒ ͒ 2 .

͑B5͒

The trajectory approximation to g͑ , ͒ is g TA ͑ , ͒ and clearly f͑,͒ = 1g TA ͑,͒, ͑B6͒

and for small scattering angles g͑,͒ Ϸ g TA ͑,͒ Ϸ 2 . ͑B7͒

The interaction with the surface is described in terms of successive collisions with N surface atoms and each of these collisions has a scattering angle which if averaged over the whole path is approximately n =2 i / N since the final scattering intensity is predominantly along the circle of energy conservation. The differential reflection for the nth collision is given by

d 3 R͑k n ,k n-1 ͒ dE n d 2 ⍀ n Ϸ 1 ͱ n exp ͭ -͓E n -f n ͑, n ͒E n-1 ͔ 2 n 2 ͮ ,

͑B8͒

where the definition of n is obvious from Eq. ͑B2͒. The differential reflection coefficient resulting from this series of collisions is the convolution of N different successive collisions

d 3 R͑k f ,k i ͒ dE f d 2 ⍀ f = ͵ dE N d⍀ N d 3 R͑k f ,k N ͒ dE N d 2 ⍀ N ¯͵ dE 1 d⍀ 1 ϫ d 3 R͑k 2 ,k 1 ͒ dE 2 d 2 ⍀ 2 dR 3 ͑k 1 ,k i ͒ dE 1 d 2 ⍀ 1 . ͑B9͒
The usual rule that the convolution of Gaussians is a Gaussian whose squared width is the sum of the squared widths of each element of the convolution becomes slightly more com-plicated in this case because of the factor f n E n-1 in the argument. In this case the final result is

d 3 R͑k f ,k i ͒ dE f d 2 ⍀ f = 1 ͱ T exp ͭ -͑E n -f T E i ͒ 2 T 2 ͮ , ͑B10͒
where

f T = ͟ n=1 N f n → 1 -Nͩ 2 i N ͪ 2

͑B11͒

and

T 2 = ͚ n=1 N ͟ j=1 N-j f j 2 n 2 → N n 2 = Nͩ 2 i N ͪ 2 E i k B T S , ͑B12͒
where the expressions on the right-hand side of Eqs. ͑B11͒ and ͑B12͒ are obtained after making the small-angle approximation that all f n are nearly equal to unity and n are small, and both are independent of n. Recalling the definition of ⌬E from Eq. ͑30͒ the differential reflection coefficient now becomes that of Eq. ͑31͒:

w͑k f ,k i ͒ = ͩ ប 2 k fz mL ͪ 2 P͑ f ͉͒A͑⌬k y ͉͒ 2 ͱ ប k B T S ⌬E ϫexp ͭ -͑E f -E i + ⌬E͒ 2 4k B T S ⌬E ͮ . ͑B13͒

APPENDIX C: EXPERIMENTALLY MEASURED QUANTITIES

The physical quantity measured by these experiments is the differential reflection coefficient per unit final energy and solid angle denoted as dR͑k f , k i ͒ / dE f d⍀ f . This is obtained from the transition rate by first forming the differential reflection coefficient per unit volume of wave vector space

d 3 R͑k f ,k i ͒ dk f = w͑k f ,k i ͒ ⌬k 3 j i , ͑C1͒
where in terms of the quantization length L in each Cartesian direction the element of volume in phase space is ⌬k 3 = ͑2 / L͒ 3 and the incident flux normal to the surface is j i = បk iz / mL. Changing to spherical coordinates according to dk f = k f 2 dk f d⍀ f and recalling that E f = ប 2 k f 2 / 2m leads to

dR 3 ͑k f ,k i ͒ dE f d 2 ⍀ f = L 4 m 2 ͑2ប͒ 3 k f k iz w͑k f ,k i ͒. ͑C2͒
The channel-plate detector used in the experiments measures the integral over all final particle energies and thus the measured quantity is the differential reflection coefficient per unit final polar and azimuthal angles which is expressed as

d 2 R d f d f = cos͑ f ͒ ͵ 0 ϱ dE f d 3 R͑k f ,k i ͒ dE f d 2 ⍀ f , ͑C3͒
where the polar angle used here is measured with respect to the surface plane and not the usual one measured with respect to the surface normal.

0 1 FIG. 1 .

 11 FIG. 1. ͑Color online͒ Comparison of the simple eikonal approximation with experiment and wave-packet calculations from Ref. 1, for 200 eV 3 He atoms scattered by a LiF͑001͒ surface aligned along the ͗100͘ direction. The evolution of the relative intensities of the observed diffraction orders is shown as a function of normal energy. The solid curves are the wave-packet calculations, the dashed curves are the eikonal approximation, and measurements are exhibited as points.

5 FIG. 2 .FIG. 3 .FIG. 4 .

 5234 FIG.2. ͑Color online͒ The scattered intensity as a function of f with f lying on the Laue circle for a 200 eV beam of3 He incident on LiF͑001͒ ͗100͘ at an incident angle i = 2.68°with respect to the surface, corresponding to Fig.1above for the normal energy 0.44 eV. The experimental points are normalized to the calculations for the second-order diffraction peak. All other parameters are specified in the text. The broad features are the calculations for the nearly classical case of Eq. ͑50͒ and the sharp vertical lines at the diffraction peak positions represent the intensities of the quantum diffraction peaks calculated from Eq. ͑37͒.


Maxime Debiossac

Philippe Roncin

Atomic diffraction under oblique incidence: An analytical expression

Keywords: number(s): 34.35.+a, 68.49.Bc, 34.50.Cx

] to model cold-atom diffraction by optical standing waves, is applied to the diffraction of fast atoms on crystal surfaces at grazing incidence (GIFAD or FAD). We first show that the interaction time and interaction length embedded in the obliquity factor is well suited to explain the transition from three-dimensional to two-dimensional (2D) diffraction. The situation of a slightly misaligned primary beam, corresponding to oblique incidence in the effective 2D system, is addressed pointing out discrepancies such as the absence of net deflection of the atomic beam. Guided by time-reversal considerations, we propose an arbitrarily symmetrized form significantly improving the agreement with experimental data recorded in oblique incidence.

Since the development of quantum atom optics, in particular interferometry with cold atoms and molecules [START_REF] Berman | Atom Interferometry[END_REF][START_REF] Juffmann | [END_REF], wave matter diffraction phenomena have been observed on light gratings, in transmission [3], and reflection [4]. In this context, the development of a simple analytical formula to model both situations as a thin phase grating by Henkel et al. [5] has received considerable attention. In a different context, diffraction of thermal atoms and molecules at crystal surfaces, first observed by Estermann and Stern in 1930 [6], has developed as a specific surface science technique able to reveal the corrugation amplitude of the surface electronic density in the sub-Å range. More recently, the diffraction of fast, keV, atoms off crystalline insulator, metal, or semiconductor surfaces at grazing incidence (GIFAD) has been demonstrated both under static [7][8][9][10] and conventional thin-film growth conditions [11]. Theoretically, extensive semiclassical techniques based on the eikonal approximation with quantum treatment of the rainbow [12,13], sophisticated wave-packet approaches [9,14,15], and close coupling calculations [16] have been developed reaching excellent accuracy but are computationally too heavy to be used in real time during experiments. At normal incidence, analytical formula already exist that provided few picometer sensitivity on the corrugation amplitude of simple lattice cell [15] or even super structure such as the moire pattern of graphene [17], but no equation is available that can reproduce even qualitatively diffraction intensity when the beam is misaligned by a fraction of a degree from the crystal axis.

I. ATOMIC DIFFRACTION BY A THIN PHASE GRATING

In Ref. [5] Henkel et al. study the reflection of atoms on the evanescent field created above a dielectric by internal reflection of a laser standing wave. We briefly recall their results while renaming variables according to the context of atom surface scattering. Only in plane diffraction, i.e., the 2D problem is treated explicitly and the approximation starts from the unperturbed atom trajectory corresponding to specular reflection on an planar exponential atom-surface potential V 0 (z) = V p exp(-z/R c ). In this simple expression V p is a pre-exponential factor, which has no importance here, while R c is the range of the potential defining the characteristic interaction time with the surface τ int = R c /v iz , with v iz the velocity component toward the surface. The classical trajectory is analytic [18] allowing semiclassical wave functions to be derived. The action integral S(x) describing the momentum transfer is expanded as a power series S(x) = S 0 + εS 1 (x) + ε 2 S 2 (x) + O(ε 3 ) of the corrugated part of the potential taken here as a perturbation; V (x,z) = V p (z) * [1 + εcos(2πx/a x )] a x being the lattice parameter and G x = 2π/a x the reciprocal lattice vector. As a result, they derive the diffracted intensities I n for any oblique incidence:

I n = k f z k iz J 2 n 2k iz εR c χ 2R c G x k ix k iz , (1)
where J n is the Bessel function of the first kind of order n, k f z /k iz a geometrical factor due to normalization of the wave function, and χ (2R c G x k ix k iz ) with χ (u) = u/sinh(u) is the obliquity factor associated with the incidence angle θ in defined by tan(θ in ) = k ix /k iz .

At normal incidence (θ in = 0), the obliquity factor is equal to 1 and can be forgotten. For weak contrast ε 1 the classical turning point Z t is shown to have a cosine dependence Z t (x) = cst + εR c cos(G x x). Introducing the full (top to bottom) corrugation amplitude h c = 2εR c the diffracted intensity becomes I n = k ix k iz J 2 n (2k iz h c ) similar to [19] and to the corrugated hardwall model where a Fresnel-Huygens ray tracing model retrieves the intensities diffracted from a corrugated mirror. At this normal incidence, the interest of Eq. ( 1) appears therefore limited because the newly introduced R c parameter is entangled with the corrugation amplitude h c without adding new prediction.

For incidence angle departing from normal, the obliquity factor is less than one and corresponds a reduced coupling leading to a reduced population of high diffraction orders. The argument u describes the number of lattice sites span within the range R c . More precisely the interaction length L x simply defined as L x = v ix τ int allows the argument u to be be rewritten as u = 4πL x /a x = 2G x L x proportional to the number L x /a x of lattice sites within the range during the interaction length. Since the interaction oscillates with a spatial period a x , the obliquity factor describes the progressive cancellation of all the contributions of successive lattice sites as the angle of incidence becomes more and more grazing. This was also discovered numerically by Farias et al. [20,21] when reducing down to 10 deg the angle of incidence of thermal hydrogen with respect to the surface plane (θ in = 80 deg to the normal). Note also that the above defined trajectory length was shown to be a very useful concept to derive a fundamental parameter such as the energy and momentum exchange with the lattice [12,22,23] or Auger neutralization rates at insulator surface [24] or metals [25] in the context of grazing incidence.

II. GRAZING INCIDENCE FAST ATOM DIFFRACTION (GIFAD)

GIFAD setups have been detailed in numerous papers [7][8][9]. It consists of a primary He + ion beam extracted at the desired final energy from a commercial ion source. The He + ions are converted to a fast neutral He 0 atom via resonant charge exchange inside an effusive gas cell. The He 0 beam is then collimated before entering the chamber hosting the target surface. Figure 1 defines the two relative angles of the primary neutral beam with respect to a selected low index direction of the crystal surface. The diffracted beams are collected ≈1 m downstream onto a position sensitive detector. In practice, both and φ angles hardly exceed 1 deg. It should be noted that the geometry is exactly the one used in RHEED setup widely used in the molecular beam epitaxy chamber to monitor thin film growth where GIFAD can therefore be an alternative technique [11].

Starting from the first GIFAD observations [7][8][9] it was realized that diffracted intensities corresponding to the exchange of reciprocal lattice vector G x (Fig. 1) are not observed. In terms of energy, and assuming G x ∼ G y , such an exchange along x costs k i /G times more than when the exchanged vector is perpendicular to the initial momentum (i.e., along y). Interestingly the other argument raised [9,20,21], in addition to A fast He atom is incident with a total wave vector k i close to the 110 direction of LiF(100) surface with polar incidence angle φ lab and misalignment angle lab . This 3D scattering problem can be reduced to a 2D problem of the He atom probing, with a wave vector k i,2D = (k iy ,k iz ), the corrugation function ξ (y) under oblique incidence θ 2D = arctan(k iy /k iz ). The diffracted beams are depicted with dashed vectors with the specular one in red.

the energy increase, is that the associated coupling decreases at the grazing angle for geometric reasons and this is exactly what is described in the obliquity factor. For the geometry used in Sec. III, the numerical value is vanishingly small forcing specular reflection in the scattering plane. If any, diffraction has be out of plane, as actually observed (from there on G y will be used instead of G x ). As a result the movement along the x direction can be reduced to a pure translation and eliminated. Of course, doing so one has to consider the effective interaction potential defined as the average of the 3D V (x,y,z) potential along the x direction [9]. The 3D egg boxlike turns into a corrugated iron plate, both having the same exponential range R c . If we define the effective interaction parameters as those in the (y,z) plane (the detector plane) one gets k 2D = k i ( sin(φ lab ), sin( lab )) and tan(θ 2D ) = tan( lab )/tan(φ lab ). The most important result being that the effective interaction energy can be continuously adjusted between a few meV to few eV allowing the exploration of the electron density on a range of distances to the atomic plane comparatively wider than HAS. The progressive transition from 3D to 2D diffraction have been studied in detail with wave packet technique [26,27] and partially observed [28] using a surface with comparatively large super lattice (large a involving many atoms per unit cell) and rather oblique incidence.

For grazing incidence and small misalignment angles, the effective 2D oblique incidence angle can be written as tan(θ 2D ) ≈ lab /φ lab so that for φ lab = 1 deg, a 1 deg misalignment angle yields a 45 deg effective oblique incidence. This outlines that quasinormal effective incidence requires that lab is orders of magnitude smaller than φ lab which is extremely demanding in terms of mechanical accuracy, particularly when thermal treatment is required. This explains that many GIFAD images are recorded with a slightly oblique incidence.

The systematic experimental investigation of the diffracted intensity as a function of the misalignment angle was first reported on LiF surface [29] followed by a detail analysis and associated calculation with close coupling equations [14]. Note that during such a scan the modulus of the effective wave vector k 2D is not constant (only k iz is constant). Both studies indicate that the diffracted profile becomes strongly asymmetric (I +n = I -n ) resulting in a net deflection of the diffracted beam taken as a whole, i.e., the average deflection angle outin ∝ n nI n is not zero and shows a characteristic behavior with a linear dependence around = 0, allowing automatic alignment of the crystal [14]. The prediction of Eq. ( 1) would here be a flat horizontal line because it predicts a symmetric population of the diffraction orders (I +n = I -n ).

In their work on beam deflection [14], the authors use time-reversal symmetry [START_REF] Bohm | Quantum Mechanics: Foundations and Applications[END_REF] to link the data corresponding to oblique incidence to those recorded under normal incidence taking into account the modified energy in the effective (y,z) plane. This symmetry is not present in Eq. (1), it does not predict the same scattering probability for 0 → n and n → 0. This shortcoming probably derives from the fact that the obliquity factor is evaluated only along the specular trajectory. One could try to imagine half trajectories where the action integral would be evaluated separately on the way in and out. This being beyond our forces we have tried to force the time symmetry by modifying arbitrarily formula (1) so that it gives the same result if one exchanges the labels "i" and "f " [START_REF] Hinch | [END_REF]. In practice the initial momenta k iy and k iz are replaced by their average between initial and final values 1) becomes

k iz → k z = (k iz + k f z )/2 and k iy → k y = (k iy + k fy )/2, with k fy = k iy + nG y and k f z = √ k 2 iz -2nG y k iy -n 2 G 2 y , Eq. (
I n = k f z k iz J 2 n 2k z h c χ 2R c G y k y k z . ( 2
)
It should be noted that the argument inside the Bessel function now depends on the diffraction order.

III. RESULTS AND DISCUSSION

We now compare the prediction of Eq. ( 2) in the case of fast atom diffraction on a LiF(100) surface when the incident beam is aligned close to the 110 direction. This surface and direction has been extensively studied and quasisinusoidal equipotential curves have been reported in the sub eV effective energy range with a corrugation amplitude small enough to prevent multiple scattering. The parameters R c = 1.18 Å and h c = 0.325 Å have been fitted to reproduce the DFT potential of LiF surface given in [32].

In Fig. 2 we show the diffraction chart for semiclassical calculations in oblique incidence obtained from Eq. ( 1) [Fig. 2(d)] and from Eq. ( 2) [Fig. 2(b)] together with the experimental data and a quantum calculation based on the close coupling formalism [14] for 4 He at E i = 460 eV incident at φ in = 0.93 • aligned close to the 110 direction of LiF(100). For the specular order, both semiclassical models give the same diffracted intensity but for higher diffraction orders (i.e., for larger oblique incidence angle θ i ) the diffracted intensity is no longer symmetric with respect to k iy = 0 in the case of Eq. ( 2). In order to emphasize the link with classical mechanics we have calculated the extrema of the momentum transfer from the classical equation of motion. The result is shown as dashed lines in Fig. 2(b) and the position of the rainbow angle where the momentum transfer is maximum is highlighted (k iy,rainbow ≈ 4G y ). At this value of misalignment, Fig. 3 compares, in a more quantitative way, diffraction intensities [experiment, quantum, and semiclassical from Eq. ( 2)].

We see from Figs. 2 and 3 that the semiclassical approach proposed here reproduces surprisingly well the experimental data which is very encouraging for practical applications. Of course a deeper analysis of this ad hoc modification is certainly needed on the theoretical point of view. For instance, the geometrical prefactor k f z /k iz is not symmetric either, other symmetrized forms are also possible and may be easier to justify, etc. More pragmatic, possible extensions beyond pure sinusoidal corrugation function can also be investigated. Intuitively, there is a reasonable hope that the actual constraint is that the corrugation function integrates to zero over a lattice unit, i.e., for symmetric cells. Another very useful extension would be to link directly this model to the 2D row model developed for grazing incidences [33], where the corrugation amplitude may depend on the effective energy. Concerning the limitations, these should the same as exposed in [5] knowing that the intensities are supposed to be better described than the associated phases. The limit for the semiclassical approach Experimental data [14]. (b) Semiclassical calculations using Eq. ( 2). (c) "Exact" quantum calculation [14]. (d) Semiclassical calculations using Eq. ( 1). Note that both horizontal and vertical scale have the same units (G y ) relevant for diffraction. In addition, the misalignment angle lab is reported also in degrees in the laboratory frame (red) as well as the corresponding obliquity angle θ 2D = tan( lab )/ tan(φ lab ) (blue). (a) and (c) have been adapted from [14]. The dashed line in (b) correspond to the maximum transfer of momentum obtained from classical calculations. The rainbow angle of k iy ≈ 4G y used for quantitative comparison in Fig. 3 is marked by the arrows labeled "rb." λ 2D R c is easily fulfilled for helium particles with E 2D above 1 meV because the range R c is always on the order of magnitude of few bohr radius (0.523 Å) [34]. This is because R c derives from fundamental atom-atom interaction potential which, in the asymptotic region probed here, is governed by the simple overlap of the ground state wave function scaling as I 1/2 p with I p the ionization potential and I p does not vary significantly within the periodic table. As far as simple lattice cells are concerned, the lattice parameter a is on the order of 10 atomic units so that the condition of weak corrugation h c a y is not too restrictive if no surface reconstruction involving several atomic layers is present.

IV. SUMMARY AND CONCLUSION

We have shown that the perturbation method developed for the diffraction on light grating [5] is well suited to account for reflection on surfaces at normal incidence. We have suggested that the obliquity factor has a strong physical support explaining the collapse of in plane diffraction as the angle of incidence becomes grazing. However, as is, this obliquity factor seems to violate the time-reversal symmetry and fails to account for the scattering asymmetry. By imposing a more symmetric form we have shown that the agreement with experiment is greatly improved. Our modified Eq. ( 2) should have the same physical limitations as the original formula, i.e., it only applies to diffraction in the semiclassical regime and from weakly corrugated potential where multiple scattering events do not occur. Optimistically, the good agreement presented here suggest that scans exploring large misalignment could allow experiment measurement of the effective range R c of the interaction potential. If so, the technique would be very useful because, once again, R c is a very important physical parameter governing the interaction length, the number of active sites, the momentum, and energy transfer to the surface. This would provide a direct (self-consistent) link with the angular dependent effective number of sites N (φ) included in the Debye-Waller factor modified for grazing incidence [12,23,35]. This drastic reduction of the original Debye-Waller factor by e -N(φ) is at the heart of the ability of GIFAD to explore the surface at high temperature [11,16] as required for operation inside a molecular beam epitaxy chamber. This work was motivated by the urgent need for a simple approach withstanding a slight misalignment in the GIFAD diffraction data. Since the result is analytic we believe that, in complement with more rigorous methods, it will prove useful as a fast method to qualitatively benefit from the picometer sensitivity of GIFAD.

1 FIG. 1 .

 11 FIG.1. (Color online) Schematic of the scattering geometry. A fast He atom is incident with a total wave vector k i close to the 110 direction of LiF(100) surface with polar incidence angle φ lab and misalignment angle lab . This 3D scattering problem can be reduced to a 2D problem of the He atom probing, with a wave vector k i,2D = (k iy ,k iz ), the corrugation function ξ (y) under oblique incidence θ 2D = arctan(k iy /k iz ). The diffracted beams are depicted with dashed vectors with the specular one in red.

  FIG. 2. (Color online) Diffraction charts for 4 He with E i = 460 eV and φ lab = 0.93 • incident on a LiF surface close to the 110 direction, i.e., k i = 942 Å-1 and k iz = 15.3 Å-1 , G y is 2.2 Å-1 corresponding to a deflection angle of 0.134 deg = sin -1 (G y /k y ). The interaction parameters are R c = 1.18 Å and h c = 0.325 Å. (a)Experimental data[14]. (b) Semiclassical calculations using Eq. (2). (c) "Exact" quantum calculation[14]. (d) Semiclassical calculations using Eq. (1). Note that both horizontal and vertical scale have the same units (G y ) relevant for diffraction. In addition, the misalignment angle lab is reported also in degrees in the laboratory frame (red) as well as the corresponding obliquity angle θ 2D = tan( lab )/ tan(φ lab ) (blue). (a) and (c) have been adapted from[14]. The dashed line in (b) correspond to the maximum transfer of momentum obtained from classical calculations. The rainbow angle of k iy ≈ 4G y used for quantitative comparison in Fig.3is marked by the arrows labeled "rb."

20 FIG. 3 .

 203 FIG. 3. (Color online) Diffraction probabilities extracted fromFig.2for k iy = 4G y (rainbow position) for experiment (black pattern), quantum theory (dashed pattern), and semiclassical theory from Eq. (2) (blue pattern).

ACKNOWLEDGMENTS

We acknowledge C. Westbrook, J. R. Manson, and P. Atkinson for helpful discussions and corrections in the writing of the manuscript. This work has been funded by the Agence Nationale de la Recherche (Contract ANR-2011-EMMA-003-01).