In this paper we show that the diffraction condition for the scattering of atoms from surfaces leads to the appearance of a distinct type of classical singularity. Moreover, it is also shown that the onset of classical trapping or classical chaos is closely related to the bifurcation set of the diffraction-order function around the surface points presenting the rainbow effect. As an illustration of this dynamic, application to the scattering of He atoms by the stepped Cu(115) surface is presented using both a hard corrugated one-dimensional wall and a soft corrugated Morse potential

Borondo, F.

Guantes, R.

Jaffé, Charles

Margalef-Roig, J.

Miret-Artés, S.

English

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Faculty Scholarship1996Classical Singularities In Chaotic Atom-SurfaceScatteringS. Miret-ArtésJ. Margalef-RoigR. GuantesF. BorondoCharles JafféFollow this and additional works at: https://researchrepository.wvu.edu/faculty_publicationsThis Article is brought to you for free and open access by The Research Repository @ WVU. It has been accepted for inclusion in Faculty Scholarshipby an authorized administrator of The Research Repository @ WVU. For more information, please contact ian.harmon@mail.wvu.edu.Digital Commons CitationMiret-Artés, S.; Margalef-Roig, J.; Guantes, R.; Borondo, F.; and Jaffé, Charles, "Classical Singularities In Chaotic Atom-SurfaceScattering" (1996). Faculty Scholarship. 485.https://researchrepository.wvu.edu/faculty_publications/485Classical singularities in chaotic atom-surface scatteringS. Miret-Artés and J. Margalef-RoigInstituto de Matema´ticas y Fı́sica Fundamental, Consejo Superior de Investigaciones Cientı´ficas, Serrano 123, 28006 Madrid, SpainR. Guantes and F. BorondoDepartamento de Quı´mica, C-IX, Universidad Auto´noma de Madrid, Cantoblanco-28049 Madrid, SpainCharles Jaffe´Deparment of Chemistry, West Virginia University, Morgantown, West Virginia 26506~Received 4 June 1996!In this paper we show that the diffraction condition for the scattering of atoms from surfaces leads to theappearance of a distinct type of classical singularity. Moreover, it is also shown that the onset of classicaltrapping or classical chaos is closely related to the bifurcation set of the diffraction-order function around thesurface points presenting the rainbow effect. As an illustration of this dynamic, application to the scattering ofHe atoms by the stepped Cu~115! surface is presented using both a hard corrugated one-dimensional wall anda soft corrugated Morse potential.@S0163-1829~96!05639-1#Surface rainbows correspond to extrema, either maximaor minima, of the classical deflection~CD! function, which isdefined as the variation of the final or outgoing angle of thescattering particles with respect to the impact points on thesurface~expressed in terms of the impact parameter!. Withinthe classical framework, and under certain conditions, theseextrema lead to singularities in diffraction patterns usuallyknown ascaustics. In quantum mechanics, caustics are re-placed by finite scattering intensities in all Bragg directions.Garibaldi et al.1 and Berry2 showed that classical rainbowpatterns provide the envelope of the quantum-mechanicaldiffraction peak intensities. Care must be exercised in theassignment of experimentally observed features asrainbows.3 This is due to the fact that, in addition to therainbow singularities, other classical and quantum-mechanical effects occur that lead to features that are easilymistaken for rainbows. These effects often take place whenthe classical dynamics associated with the scattering processis chaotic. In these circumstances, the scattering particle istrapped on the surface for a finite time. This temporary~vi-brational! trapping has been conjectured to be associatedwith selective adsorption resonances. However, as has beenrecently stated,4 the relation between trapping and these reso-nances needs further investigation to clearly understand bothphenomena. In addition to the occurrence of such quantum-mechanical resonances, we show in this work that trapping isrelated to a type of classical singularity. This appears in thetransformation from the CD function to the diffraction order~DO! function, which was originally introduced by Milleret al.5 in his theory of the classical S-matrix for elastic atom-surface scattering. This function gives the parallel momen-tum transfer in the scattering process for each value of theimpact parameter.For any incident energy, the surface rainbow angles aredependent on the incident scattering angle and occur at cer-tain impact parameters, which correspond to the inflectionpoints of the surface. As the incident scattering angle is in-creased, these angles also increase. The onset of classicaltrapping, that is, classical chaos, takes place when a surfacerainbow angle reaches6p/2, according to Ref. 4. A rainbowangle of6p/2 implies that the scattering particle is travelingparallel to the surface and likely will encounter the surfacemore than once. As will be shown later, if we analyze plotsof the incident angle versus the impact parameter, above agiven threshold of the incident angle, not only do the inflec-tion points of the surface lead to this classical trapping, but,also, more impact parameters contribute to it. In these plots,two well-separated regions delimited by the two branches ofa parabola are observed in such a way that the inner regioncorresponds to multiple scattering events and the outer re-gion to single scattering events. The points of this paraboladefine the singularity that we are discussing in this paper.In order to demonstrate the origin of this singularity, weconsider the classical in-plane scattering intensity as formu-lated in Ref. 5:I J~Ei ,u i !5(jUdJ~bj ;Ei ,u i !db U21, ~1!whereEi is the incident energy,u i is the incident scatteringangle, b is the normalized impact parameter~defined byb5x/a), and a is the surface unit-cell length consideredone-dimensional along thex direction; the sum is over allimpact parametersbj that contribute to a given integer value,J, of the DO function,J(b;Ei ,u i). This function arises as analternative way to express the diffraction condition and canbe written asJ~b;Ei ,u i !5aA2mEi2p\@sinu f~b!2sinu i #, ~2!u f(b) being the CD function. Equation~2! is thus moreadapted to a classical analysis and along with Eq.~1! givesthe envelope of the diffraction pattern. In order to obtain theintensity in the different Bragg directions, one has to evalu-ate this function for integer values of the diffraction order,PHYSICAL REVIEW B 15 OCTOBER 1996-IVOLUME 54, NUMBER 15540163-1829/96/54~15!/10397~4!/$10.00 10 397 © 1996 The American Physical SocietyJ. The scattering intensity will have singularities wheneverthe derivative of the DO function with respect tob is zero or,using the chain rule, whenS dJ~u f ;Ei ,u i !du f D S du f~b!db D50. ~3!Clearly, if either of these two factors or both are equal tozero, they give rise to different kinds of singularities in thescattering intensity. Zeros of the first factor are responsiblefor the new singularity that we are describing in this paper,and the zeros of the second factor correspond to the well-known surface rainbows. We consider these two factorsseparately.The second factor in Eq.~3! is the derivative of the CDfunction with respect tob. If we want to perform this deriva-tive analytically, it is necessary to specify a simple interac-tion model for this scattering. For this purpose, we describethe surface by a hard corrugated one-dimensional wall. Thediffraction amplitudes in the eikonal approximation are ex-pressed as1Ag51aE dx exp$2 i @gx1qgzj~x!#%, ~4!whereg5(2pn/a,0) is the reciprocal lattice vector withnan integer,qgz is the momentum transfer of the particles inthe z direction, andj(x) is the corrugation function. In theframework of this approximation, multiple scattering is ne-glected. The CD function is then obtained by evaluating theintegral of Eq.~4! in the stationary phase approximation to-gether with the diffraction condition, Eq.~2!, to beu f~b!5u i62utan21@j8~b!/a#u, ~5!where the1 sign is used when the tan21 function givesnegative angles and the2 sign otherwise. Thus, the secondfactor in Eq.~3! isS du f~b!db D5 2uj9~b!/au11@j8~b!/a#2 . ~6!From this equation, it is seen that a surface rainbow occursfor each inflection point of the surface corrugation function,j9(b* )50, whereb* designates the inflection points. As itis well known, this singularity is related to the topology ofthe surface.The first factor in Eq.~3! is easily obtained by differenti-ating Eq.~2! and this yieldsS dJ~u f ;Ei ,u i !du f D5aA2mEi2p\ cosu f . ~7!The singularity occurs when the deflection angle is againequal to6p/2 ~onset of classical trapping! but for impactparameters, in general, different fromb* . Prior to the onsetof classical trapping this singularity does not occur. It arisesas aconsequence of the diffraction conditiona d a classicalimage of this singular behavior could be given by the skip-ping stones on a river. Thus in order to skip a stone on a riverit is necessary not only to get the incident angle correct butalso to reach certain impact points on the water surface.The singularity condition can be also expressed in a veryuseful way in terms ofu i andb by substituting Eq.~5! in Eq.~7! and equating the result to zero. Then thelocus of thissingularity is given by the following planar curvetanu i~b!5612@j8~b!/a#22uj8~b!/au. ~8!Notice that this function has a minimum atb5b* withu i5u i* Moreover, at this point, in which both factors of Eq.~3! vanish simultaneously, the second derivative of the DOfunction with respect tob is also equal to zero. Therefore theset of conditions~i! j9(b* )50 butj-8(b* )Þ0, and~ii ! Eq.~8! with b5b* , define thebifurcation setof the DO function@J8(b* )5J9(b* )50#. This point also coincides with the on-set of classical trapping and chaos.4 For incident anglesabove the critical valueu i* the rainbow singularitybifurcatesinto two branches, the new singularities, for which only Eq.~8!, and not condition~i!, is fulfilled.In order to clearly illustrate this dynamics, we consider adetailed analysis of the scattering of He atoms from thestepped Cu~115! surface for an incident energy of 63 meV.Rainbow diffraction patterns have been observed experimen-tally for this surface,6,7 and theoretical calculations havebeen also reported within the close-coupling3,8 and semiclas-icalS-matrix3,9 formalisms. The interaction potential modelfor this surface and at this energy has been taken from theliterature6 and corresponds to a fitting to experimental re-sults. The corresponding parameters are given in Table I.In Fig. 1 the bifurcation diagram@Eq. ~8!# correspondingto the one-dimensional corrugation function,j(x)5Vx(x),given in Table I is shown. Hereb*50.22 and the onset ofclassical trapping is predicted atu i*568.73°. The regionabove the curve corresponds to conditions of multiple scat-tering events or trapping, and the region below to singlescattering events. Also, in the inset of this figure we show theclassical diffraction pattern corresponding to the minimum ofthe curve. As can be seen the asymptotic behavior is verydifferent for both singularities. In the rainbow singularity~left! the tendency is controlled by surface structural proper-ties @see Eq.~6!#, while in the new singularity~right! theasymptotic behavior is of cosine type@see Eq.~7!#.In Fig. 2 we show the DO functions obtained from Eq.~2!~dashed line! and by numerical integration of the Hamiltonequations~solid line! at u i528°. The differences in bothTABLE I. Potential energy surface for the scattering of He fromthe Cu~115! surface at 63 meV~Ref. 6!.Potential energy:V(x,z)5VM(z)1VC(x,z)Morse potential:VM(z)5D(12e2az)2D56.35 meV,a51.05 Å21Coupling potential:VC(x,z)5Vz(z)Vx(x)Vz(z)5D exp(22az)Vx~x!5( n512 Frncos2npxa 1snsin2npxa GFourier coefficients at 63 meV:r 150.2147 ,s1520.0252,r 250.0131 ,s2520.0044Unit cell length for Cu~115!: a56.625 Å10 398 54BRIEF REPORTScurves are only due to the presence of the attractive part ofthe potential. Although the numerical agreement is not good,the main qualitative features of the dynamics are rather wellreproduced in the analytical solution given by the eikonalapproximation. At the same incident energy, we display inFig. 3~a! the analytical CD functions@Eq. ~5!# and in Fig.3~b! the analytical DO functions@Eq. ~2!# at three differentincident angles: 62°~dotted line!, 68.73° ~solid line! and75° ~dashed line!, respectively. Whereas the CD functions donot present any qualitative change~number of extrema!, theDO functions present around the maximum drastic changes.From a largeplateauwe pass to the occurrence of threeextrema. The minimum value ofJ corresponds to the oldmaximum atb* . Numerically the onset of chaos appears at54° and not at 68.73°; this difference is again due to thepresence of the soft interaction potential.To our knowledge, the effect described in this paper hasnot been observed experimentally in any diffraction experi-ment. To a large degree this could be due to technical diffi-culties but they are easily surmounted. In order to observe it,one needs to place the detector parallel to the surface andthen to vary the incident scattering angle in order to find itscritical valueu i* ; the outgoing beam being a Bragg peak.This would require us to change the typical experimentalsetup, where the angle between the source and the detector(uSD) is usually aroundp/2, to a more planar configuration~in general greater than 100°).10 For example, in our casewhere the critical incident angle value is 54°, with an inci-dent wave vector of 5 Å21 the (10) diffraction peak shouldpresent such an effect by using an experimental configura-tion with uSD5144°. Similar features to those discussed forthe rainbow effect are expected here in the diffraction inten-sities. Finally, due to the fact that this singularity has itsorigin in the diffraction condition we think that it could alsobe observed in any scattering of low energy particles~such asneutrons, electrons, etc.! from surfaces.ACKNOWLEDGMENTSThis work has been supported in part by DGICYT~Spain!under Grants Nos. PB89-122, PB92-53, and PB92-181, andby the NSF ~USA! under Grant No. RII-8922106. R.G.gratefully acknowledges a grant from the Ministerio de Edu-cación y Ciencia~Spain!.FIG. 1. Bifurcation diagram for the singularities in the He-Cu~115! scattering. The location of the rainbow singularity,b*50.22,u i*568.73°, is marked with a full circle. Above thispoint the rainbow singularity bifurcates, giving rise to the two sin-gularity branches described in the text. In the inset the classicaldiffraction pattern corresponding to the bifurcation point is shown.FIG. 2. Analytical~solid line! and numerical~dashed line! DOfunctions for the scattering of He from Cu~115! at an incident en-ergy of 63 meV and incident angle of 28°.FIG. 3. Analytical CD function~a! and DO function~b! at threeincident angles: 62°~dotted line!, 68.73° ~solid line! and 75°~dashed line!. The incident energy for the scattering of He fromCu~115! is 63 meV.54 10 399BRIEF REPORTS1U. Garibaldi, A. G. Levi, R. Spadacini, and G. E. Tommei, Surf.Sci. 48, 649 ~1975!.2M. V. Berry, J. Phys. A8, 566 ~1975!.3R. Guantes, F. Borondo, C. Jaffe´, and S. Miret-Arte´s, Surf. Sci.338, L863 ~1995!.4F. Borondo, C. Jaffe´, and S. Miret-Arte´s, Surf. Sci.317, 211~1994!.5R. I. Massel, R. P. Merril, and W. H. Miller, J. Chem. Phys.64,45 ~1976!.6G. Gorse, B. Salanon, F. Fabre, A. Kara, J. Perreau, G. Armand,and J. Lapujoulade, Surf. Sci.147, 611 ~1984!.7S. Miret-Artés, J. P. Toennies, and G. Witte, Phys. Rev. B~to bepublished!.8M. Hernandez, S. Miret-Arte´s, P. Villarreal, and G. Delgado-Barrio, Surf. Sci.290, L693 ~1993!.9R. Guantes, F. Borondo, C. Jaffe´, and S. Miret-Arte´s, Phys. Rev.B 53, 14 117~1996!.10J. P. Toennies~private communication!.10 400 54BRIEF REPORTS

Classical Singularities In Chaotic Atom-Surface Scattering

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Classical Singularities In Chaotic Atom-Surface Scattering

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