Optical Theorem and the Inversion of Cross Section Data for Atom Scattering from Defects on Surfaces D

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Optical theorem and the inversion of cross section data for atom scattering from defects on surfaces D. A. Hamburger Department of Physics and The Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel R. B. Gerber Department of Physical Chemistry and The Fritz Haber Center for Molecular Dynamics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel and Department of Chemistry, University of California, Irvine, California 92717 ͑Received 3 November 1994; accepted 24 January 1995͒ The information content and properties of the cross section for atom scattering from a defect on a flat surface are investigated. Using the Sudden approximation, a simple expression is obtained that relates the cross section to the underlying atom/defect interaction potential. An approximate inversion formula is given, that determines the shape function of the defect from the scattering data. Another inversion formula approximately determines the potential due to a weak corrugation in the case of substitutional disorder. An optical theorem, derived in the framework of the Sudden approximation, plays a central role in deriving the equations that conveniently relate the interaction potential to the cross section. Also essential for the result is the equivalence of the operational definition for the cross section for scattering by a defect, given by Poelsema and Comsa, and the formal definition from quantum scattering theory. This equivalence is established here. The inversion result is applied to determine the shape function of an Ag atom on Pt͑111͒ from scattering data.© 1995 American Institute of Physics.

I. INTRODUCTION latter definition to the standard formal one from scattering theory, which we shall employ. Here we will establish this The pioneering work by Poelsema, Comsa, and equivalence. An analytical study of the cross section is most co-workers1–5 on atom-surface scattering in the last decade, easily performed by relating the cross section through an has led to the introduction and application of the concept of optical theorem to the scattering amplitude, as has been done the cross section of a defect on a surface in studies of ques- by several authors.6,8,17 We derive an optical theorem for tions related to surface disorder. Extensive experimental atom-surface scattering by use of the Sudden approximation work in the field has focused primarily on the measurement ͑SA͒.23 This will lead to a simple explicit form for the cross of the dependence of the cross section on the surface param- section, and is expected to be fairly accurate, as the SA per- eters, usually the coverage dependence. Some efforts have forms best for the calculation of specular amplitudes. By also been directed to the dependence on the scattering pa- employing our optical theorem result, we will answer a num- rameters, such as incidence energy and angle. These cross ber of specific questions related to the information that can section studies have attracted significant theoretical be extracted from the single defect cross section. interest,6–13 concentrating almost exclusively on the depen- First we shall examine the relative role of the long range dence on the scattering parameters. Relatively little attention dispersion forces vs the short range repulsive part of the has been paid to the inverse question of the information con- potential in determining the interaction of a He atom with an tent of the cross section. Most work in this direction has been adsorbed defect. It is a well-recognized fact in surface scat- experimental, again focusing on the use of cross section data tering theory,6,10,22,24 that the magnitude of the total cross to study coverage-dependent questions, such as the onset of section of a defect is dominated by small angle deflections Ostwald ripening,14 and whether growth proceeds in a layer- caused by the long range forces. We will show that they are by-layer fashion.15–20 With very few exceptions,13,21 most mostly effective in setting the magnitude of the cross section, studies to date do not deal with the issue of the structural whereas at least for the high-energy regime the short range information contained in the cross section. For instance, can forces tend to determine its qualitative shape. the cross section be used to extract the shape of an object on A second issue that will be pursued is whether the cross the surface, or its electron density profile? Can the interac- section can be used to perform an inversion of the He-defect tion potential between an atom and a surface defect be ob- interaction potential. An affirmative answer will be given in tained from the cross section data? In the present paper, we two important cases: the cross section can be used to invert address these questions theoretically, by focusing on what the semiclassical shape function of a defect on a surface, and can be learned from studying the cross section of an indi- to approximately invert the interaction potential of a He atom vidual defect on a surface. with a weak corrugation, e.g., due to dilute substitutional To make contact with experiment, through the opera- disorder. The results are applied to obtain the shape function tional definition of the cross section proposed by Poelsema of an Ag atom on Pt͑111͒, using cross section data from and Comsa,22 it is required to show the equivalence of the Zeppenfeld.25

J.Downloaded¬19¬Sep¬2003¬to¬142.150.226.143.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jsp Chem. Phys. 102 (17), 1 May 1995 0021-9606/95/102(17)/6919/12/$6.00 © 1995 American Institute of Physics 6919 6920 D. A. Hamburger and R. B. Gerber: Inversion of scattering data from defects

The structure of the paper is as follows: we discuss the tional definition proposed by Poelsema and Comsa22 for standard and operational definitions of the cross section in atom-surface scattering, and applied extensively for cross Sec. II and prove their equivalence. In Sec. III, we briefly section calculations from diffraction data. Their formula review the SA, and derive the optical theorem for atom- reads: surface scattering within the SA. We then proceed to discuss a number of applications of the optical theorem, in Sec. IV. dI ϭϪ 1 s ⌺op nlim0 , ͑3͒ The first is a discussion of the influence of the short range → I0 dn repulsive part of the atom-surface interaction potential. The second is perhaps the central result of this paper: an inver- where ⌺op is the total scattering cross section, Is , I0 repre- sion scheme for the shape function of an adsorbate on a sent the specular scattering intensity from the corrugated and surface from cross section measurements. We apply this re- flat ͑smooth, defect-free͒ surfaces respectively, and n is the 9 sult to data for Ag/Pt͑111͒. The third application is the ap- number of defects per total surface area. Xu et al. employed proximate inversion of the potential from the cross section this formula to derive an expression for the cross section data. Concluding remarks are presented in Sec. V. based on Gaussian wave packets, and showed that their expression can be interpreted as an optical theorem. We will II. DEFINITION OF THE CROSS SECTION show that the operational definition ͓Eq. ͑3͔͒, is in fact identical to the standard definition of quantum scattering theory Consider a scattering experiment of an atom beam with ͓Eq. ͑2͔͒, without recourse to a specific method of calculat- wave-vector kϭ(k ,k ,k ), impinging on a flat surface ͑the x y z ing the quantities that appear in the formulas. This is true, zϭ0 plane͒. Let Rϭ(x,y) denote a vector in the plane. A provided the proper conditions are established to relate two- typical experimental setup will involve a highly monochro- body and atom-surface collisions. The conditions under matic beam with a lateral spread in the surface plane. This which the QDCS is defined, are as follows. beam is suitably described by a set of states ͉⌿in͘ϭ͉⌽R͘. After impinging upon the surface, the beam is scattered, in ͑1͒ The incident particles have a sharply defined momentum part by the defect whose cross section we are interested in. p0 and are randomly displaced in a plane perpendicular To derive the cross section, we start from the definition of the to p0 . quantum differential scattering cross section ͑QDCS͒ for ͑2͒ The distribution of scatterers must be such as to avoid two-body collisions.26 This definition generalizes quite natu- coherent scattering off two or more centers. rally for atom-surface scattering, as we will demonstrate. Let When interpreted in terms of atom-surface scattering, we w(d⍀ ⌽ ) denote the probability that a particle incident R see that the first condition may be traded by random lateral upon a← target as a wave packet ⌽, displaced laterally by R displacements parallel to the surface plane ͑maintaining a with respect to the origin, emerges after an elastic collision sharply peaked momentum͒. The second requires that the in the solid angle d⍀. Then the average number of observed defects are distributed very dilutely. This is, however, exactly scatterings into d⍀ is: the assumption underlying the operational definition, and of course holds for scattering off a single defect. Thus the con- N ͑d⍀͒ϭ w͑d⍀ ⌽ ͒Ϸ d2Rn w͑d⍀ ⌽ ͒, sc ͚ ← Ri ͵ inc ← R ditions assumed for the definition of the two-body cross sec- i tion are naturally extendible to the atom-surface case, and where ninc is the incident density, which we assume to be suit the operational definition. In order to demonstrate the uniform over an area A, i.e., nincϭNinc /A, so that: equivalence of the definitions, we first replace Nsc(⍀)/Ninc by (I(d⍀)/I )d⍀ in Eq. ͑1͒, where I is the off-specular N d 0 inc ␴ intensity. I is the incident intensity, which is clearly equal to Nsc͑d⍀͒ϭ d⍀, ͑1͒ 0 A d⍀ the total scattered intensity in the case of a flat surface, and is where d␴/d⍀ is the differential cross section, defined as: hence indeed the same quantity as defined in the discussion of the operational definition above. With these replacements, d ␴ 2 the total cross section is: d⍀ϵ␴͑d⍀͒ϭ d Rw͑d⍀ ⌽R͒. ͑2͒ d⍀ ͵A ← d␴ A Thus, as is well known, the QDCS is seen to be a mea- ⌺ϵ d⍀ϭ I͑⍀͒d⍀. ͑4͒ sure of the target area effective in scattering the incident ͵ d⍀ I0 ͵ beam. It must be emphasized that the definition of the QDCS excludes the forward direction. Classically speaking, the rea- Next consider taking the limit in the operational defini- son is that one cannot distinguish a forwardly scattered par- tion. This can be done by assuming that only one defect is ticle from one that did not interact with the target at all. The present on the surface, so that for all practical purposes analogue in surface scattering from a defect, is that one can- dnϭ1/A, A being the total surface area, and dIsϭIsϪI0 . not distinguish a specularly scattered particle from one that Then from Eq. ͑3͒: was scattered by the flat part of the surface. More formally, the reason is that inclusion of the forward direction would A ⌺opϭ ͑I0ϪIs͒. ͑5͒ lead to a divergence of the cross section. We would now like I0 to discuss the connection between the latter definition, which is widely accepted for two-body processes, and the opera- But clearly, I0ϪIsϭ͐I(⍀)d⍀, so that ⌺opϭ⌺, as claimed.

III. OPTICAL THEOREM FOR SURFACE SCATTERING B. Derivation of the optical theorem IN THE SUDDEN APPROXIMATION The ensuing discussion closely follows that of Taylor26 We very briefly review the SA. For more details see, for two-body scattering. We proceed to derive the optical e.g., the review by Gerber.23 theorem for surface scattering within the SA. The probabilities w(d⍀ ␾ ) in Eq. ͑2͒ are: A. The Sudden approximation ← R ϱ 2 2 Basically, the SA requires that the momentum transfer in w͑d⍀ ⌿in͒ϭd⍀ ͵ p dp ͉⌿out͑p͉͒ ͑pϭបk͒ ͑9͒ parallel to the surface be small compared with the momen- ← 0 27,28 tum transfer normal to the surface, i.e., since w(d3p ⌿ )ϭd3p͉⌿ (p)͉2 is the probability that ← in out long after the collision the particle incident as ⌿in , emerging ͉qЈϪq͉Ӷ2k , ͑6͒ z as ⌿out , has momentum in (p,pϩdp). Only the direction of p is of interest, as we are assuming elastic scattering. The where kz is the incident wave number in the z direction, q is ‘‘in’’ and ‘‘out’’ states are related by the S-matrix: the incident wave vector in parallel to the surface plane, and ͉⌿ ͘ϭS͉⌿ ͘. In the momentum representation: qЈ is any intermediate or final wave vector in parallel to the out in surface plane which plays a significant role in the scattering ⌿ ͑p͒ϭ d3pЈ ͗p͉S͉pЈ͘⌿ ͑pЈ͒. ͑10͒ process. The S-matrix element for scattering from q to qЈ out ͵ in may then be shown to be given by the function: The combination of Eqs. ͑2͒, ͑9͒, and ͑10͒ yields the differential cross section. Following the discussion in the 1 ͗q͉S͉qЈ͘ϭ dR e2i␩͑R͒ei⌬q•R, ͑7͒ preceding section, it can be seen that there is nothing which A ͵surf essentially restricts the use of Eqs. ͑2͒, ͑9͒, and ͑10͒ to two- body scattering, and with the restriction of scattering angles where Rϭ(x,y) and ␩(R) is the scattering phase shift com- to the half-space above the surface (zϾ0), they may as well puted for fixed R, given in the WKB approximation by: be applied to atom-surface scattering. We now proceed to calculate the differential cross section within the Sudden ap- ϱ 2mV͑R,z͒ 1/2 2 proximation. Let us denote pϭប(q,kz), where q is the ␩͑R͒ϭ dz kzϪ 2 Ϫkz Ϫkzzt͑R͒. ͵z ͑R͒ ͫͩ ប ͪ ͬ wave-vector component in parallel to the surface, and t ͑8͒ ⌬qϵqЈϪq. We require the familiar ‘‘scattering-amplitude’’ f , in terms of which the optical theorem will be expressed. Here zt(R) is the classical turning point for the integrand To find its form within the SA, we first write down a general in Eq. ͑8͒. The normalization is chosen such that the unit-cell expression for the S-matrix element, in the case of scattering A is the average area available for one adsorbate, i.e., is from a surface with a defect. This expression must be com- determined by the coverage. This choice is suitable for low posed of two terms, reflecting specular ͑the analogue of for- coverages, when the simultaneous interaction of a He atom ward scattering in the gas phase͒ and off-specular scattering. with two neighboring adsorbates is small. Of course, an ad- The first arises in the case of scattering from the flat part of sorbate may ‘‘spill over’’ beyond its unit-cell due to the long the surface, whereas the second is due to scattering from the range of the potential, and hence the integration in Eq. ͑7͒ is defect. Energy is assumed to be conserved, so the general over the entire surface. Condition ͑6͒ for the validity of the expression for the S-matrix element ͗p͉S͉pЈ͘ contains an on- Sudden approximation is expected to break down for sys- 2 shell delta-function of the energy E pϵp /2m: tems of high corrugation. For instance, certain isolated adsorbates on an otherwise flat surface can represent, for real- i ͗p͉S͉pЈ͘ϭ␦͑p ϩpЈ͒␦͑ប⌬q͒ϩ ␦͑E ϪE ͒ istic parameters, a very substantial local corrugation. z z 2␲mប p pЈ Nevertheless, previous calculations have shown that the Sud- ϫf͑p pЈ͒. ͑11͒ den approximation reproduces rather well many features of ← the scattering from isolated adsorbates.29 Features for which The identification of the scattering-amplitude by com- it breaks down are, e.g., intensity peaks due to double colli- paring Eqs. ͑7͒, ͑11͒ is now made as follows. First, the sion events, in which the incoming atom first hits the surface phase-shift ␩ is constant for scattering from a flat surface, and then the adsorbate ͑or vice versa͒, which are a particu- and may well be chosen zero. In this case the SA yields larly sensitive manifestation of a strong corrugation ͑we note ͗q͉S͉qЈ͘ϭ␦(⌬q) ͓Eq. ͑7͒ with ␩(R)ϭ0], in agreement that a double-collision version of the Sudden approximation with the general form for specular scattering. Second, we has recently been developed30͒. However, due to the assump- consider the remainder after subtraction of the specular part tion ͑6͒, the Sudden approximation is particularly useful for from the RHS of Eq. ͑7͒: the evaluation of specular intensities at incidence angles 2␲ close to the specular direction, as confirmed by comparison g͑q qЈ͒ϭ dR ͑e2i␩͑R͒Ϫ1͒ei⌬q•R ͑12͒ with numerically exact wave packet calculations.10 In the ← iA ͵surf context of the present work, only the specular intensities are so that: required ͓which are obtained by setting qЈϭq in Eq. ͑7͔͒, which constitutes the most favorable condition for the Sud- 1 i ͗q͉S͉qЈ͘ϭ ␦͑⌬q͒ϩ g͑q qЈ͒. ͑13͒ den. A 2␲ ←

Energy conservation is implicitly assumed in the SA ex- 2 i/បR•͑pЉϪpЈ͒ pression Eq. ͑7͒. Still, comparing Eq. ͑13͒ and Eq. ͑11͒, g is ͵ d Re ␦͑EpϪEpЈ͒␦͑EpϪEpЉ͒ not quite the scattering amplitude yet, since it does not in- m clude the action of the S-matrix on the wave-function part in 2 ប 27 ϭ͑2␲͒ ␦͑pЉϪpЈ͒␦͑EpϪEpЈ͒. ͑19͒ perpendicular to the surface. The SA derivation yields only kz the action on the part parallel to the surface. However, we Ј can obtain the corresponding factor by comparison to the Inserting this into Eq. ͑18͒ and applying the momentum well known partial-wave expansion result. If we assume a delta-function, we obtain: plane wave He-beam at normal incidence to a perfectly flat d⍀ ϱ 1 surface with a spherically symmetric He-adsorbate interac- 2 3 ␴͑d⍀͒ϭ ͵ p dp ͵d pЈ ␦͑EpϪEpЈ͒ tion, the partial-wave expansion conditions apply. The gas- m 0 kzЈ phase expression for the scattering amplitude is then:26 1 ϫ͉g͑q qЈ͉͒2͉⌽͑pЈ͉͒2 . ͑20͒ 2␲ ← ͑kЈ͒2 f ͑k kЈ͒ϭ Ym͑kˆ͓͒e2i␦l͑Ep͒Ϫ1͔Ym͑kˆЈ͒*, ͑14͒ ik ͚ l l ← l,m Next we use ␦(E pϪE pЈ)ϭm/p ␦(pϪpЈ) to do the ra- m ˆ ˆ dial integration over p: where Y l (k) is the spherical harmonic whose argument k denotes the polar angle (␪,␾)ofkˆ, and ␦ (E ) is the phase- l p 1 shift in the angular momentum basis ͕͉E,l,m͖͘. Comparing ␴͑d⍀͒ϭd⍀ d3pЈ ͉g͑q qЈ͒⌽͑pЈ͉͒2, ͑21͒ Eqs. ͑12͒, ͑14͒, we see that the contribution of the compo- ͵ kzЈkЈ ← nent perpendicular to the surface is a factor of 1/k, so that where p ϭ p . To simplify the last integral and obtain the the SA scattering amplitude is: ͉ ͉ ͉ Ј͉ standard form, we recall that ⌽ represents the incident wave- 1 function ͑at zero lateral displacement͒. The simplification is f ͑p pЈ͒ϭ g͑q qЈ͒. ͑15͒ ← k ← obtained by the physically plausible assumption that ⌽ is sharply peaked about the incidence momentum p0 . More We emphasize at this point our strategy for deriving the quantitatively, we assume that ͓1/(kzЈkЈ)͔g(q qЈ) does not optical theorem: the analogue to the gas-phase forward vs vary appreciably in the region of ⌽’s peak, so← that it can be non-forward scattering, is the flat-surface vs defect scatter- taken outside of the integral sign, and be replaced by its ing. Within the SA, this distinction is easily made by sepa- values at k0 . Using further the normalization condition rating the vanishing part from the non-zero part of the phase- ͐d3pЈ ͉⌽(pЈ)͉2ϭ1, we finally obtain the familiar looking shift. result for the differential cross section: We are now ready to calculate the differential cross section within the SA. Substituting Eq. 11 into Eq. 10 yields: 1 ͑ ͒ ͑ ͒ 2 ␴͑d⍀ k0͒ϭd⍀ ͉g͑q q0͉͒ , ͑22͒ i ← k0zk0 ← 3 ⌿out͑p͒ϭ⌿in͑q,Ϫkz͒ϩ d pЈ ␦͑E pϪE p ͒ 2␲mប ͵ Ј where from Eq. ͑12͒ ϫ f ͑p pЈ͒⌿ ͑pЈ͒. ͑16͒ ← in 4␲ g͑q q ͒ϭ dR sin͑␩͑R͒͒ei␩͑R͒ei͑q0Ϫq͒•R. ͑23͒ The first term is the wave scattered from the flat part of ← 0 A ͵ the surface, the second by the defect. Next we restrict obser- vation to defect scattering only. Using the lateral displace- What we are after is the total cross section for scattering ment assumption from Sec. II we then obtain by a defect. We next employ the differential cross section result to express this in terms of an optical theorem for sur- i face scattering. To do so, we use the manifest unitarity of the ⌿ ͑p͒ϭ d3pЈ ␦͑E ϪE ͒f ͑p pЈ͒ out 2␲mប ͵ p pЈ ← SA expression, Eq. ͑7͒. Separating once again the specular and non-specular parts of the S-matrix, now in operator Ϫi/បR•pЈ ϫe ⌽͑pЈ͒. ͑17͒ form: Sϭ1ϩR, and applying the unitarity condition, we † † Substituting Eq. ͑17͒ into Eq. ͑9͒ and then into Eq. ͑2͒: have: RϩR ϭϪRR . Inserting a complete set and taking matrix elements: d ϱ ⍀ 2 2 3 ␴͑d⍀͒ϭ d R p dp d pЈ ␦͑EpϪEp ͒ 2 2 ͵ ͵ ͵ Ј 3 2␲m ប A 0 ͗pЈ͉R͉p͘ϩ͗p͉R͉pЈ͘*ϭϪ͵ d pЉ͗pЉ͉R͉pЈ͘*͗pЉ͉R͉p͘. ͑24͒ Ϫi/បR•pЈ 3 ϫf͑p pЈ͒e ⌽͑pЈ͒͵d pЉ ␦͑EpϪEpЉ͒ ← But it is clear from Eq. ͑11͒ that: ϫf*͑p pЉ͒ei/បR•pЉ⌽͑pЉ͒. ͑18͒ ← i 2 ͗pЈ͉R͉p͘ϭ g͑q qЈ͒␦͑E ϪE ͒. The R integration yields (2␲) ␦(qЉϪqЈ). It is easily 2␲mp ← p pЈ checked, using some well known properties of the delta- function, that consequently: Inserting this into Eq. ͑24͒ and factoring out a common ␦:

FIG. 1. Cross section in the cone case (nϭ1), for two values of the corru- FIG. 2. Cross section in the convex case (nϭ2), for two values of the gation parameter h. The cross sections are almost identical, except for the corrugation parameter h. The stronger the corrugation, the faster and more pronounced are the oscillations, and the larger the cross section for small behavior at small kz . kz .

Comparing the RHS of Eq. ͑26͒ with that of Eq. ͑22͒,we 1 ͓g͑qЈ q͒Ϫg͑q qЈ͒*͔ observe that it is just the total cross section for scattering at p ← ← incidence momentum p, i.e., i 1 3 2 ϭ d pЉ ␦͑E pϪE p ͒g͑qЉ qЈ͒*g͑qЉ q͒ ϭ 2␲m ͵ Љ ← ← ⌺͑p͒ϵ ͵ d⍀pЈ ␴͑d⍀pЈ p͒ ͵ d⍀pЈ ͉g͑qЈ q͉͒ ← kkz ← 1 whence finally, the Optical Theorem: ϫ 2 . ͑25͒ ͑pЉ͒ 8␲ ⌺͑k͒ϭ Im͓g͑q q͔͒. ͑27͒ The optical theorem results by considering the specular kkz ← scattering (qϭqЈ) and integrating over pЉ. In contrast to From Eq. ͑23͒ we obtain an explicit form for the total two-body scattering, in the atom-surface scattering case only cross section: the space above the surface is available, which can be taken 2 1 32␲ care of by a factor of 2. Then, using ⌺͑k͒ϭ dR sin2␩͑R͒. ͑28͒ ϱ Akk ͵ ͐0 dpЈ h(pЈ)␦(EpϪEpЈ)ϭ(m/p) h(p) and performing the z surf radial integration: Equation ͑28͒ is the optical theorem for surface scattering within the SA. As a final remark, we did not consider the 1 ϱ 3 2 effect of Bragg scattering from a periodical underlying sur- d pЉ•••ϭ d⍀pЉ pЉ dpЉ ␦͑EpϪEpЉ͒ ͵ 2͵ ͵0 face. As shown by Xu et al.,9 this contribution should be 1 counted along with the specular amplitude, and its incorpo- ϫg*͑qЉ qЈ͒g͑qЉ q͒ 2 ration in the present framework should pose no particular ← ← pЉ difficulty. m ϭ d⍀ g*͑qЉ qЈ͒g͑qЉ q͒. IV. APPLICATIONS 2p͵ pЉ ← ← We now illustrate a number of applications of our result. Using this and the specularity condition in Eq. ͑25͒,we obtain: A. Solvable model of a single adsorbate It has been known for many years that the long range 2i i 1 attractive interaction is dominant in determining the He- Im͓g͑q q͔͒ϭ d⍀ ͉g͑qЈ q͉͒2. ͑26͒ p ← 2␲ 2p ͵ pЈ ← adsorbate cross section.8,10,22,24 We wish to identify the con-

reasonable sharp edge for ␣ϭ1, yet the cases nϭ2,4 should be a crude, but reasonable model of the main features of the adsorbate shape function. In order to check the inﬂuence of the sharp edge, we may consider instead of ␨ a concave shape function, which trades the edge with a sharp tip at ␣ϭ0:

0, ␣Ͼ1 ␰͑R͒ϭh␬͉͑R͉/l͒ ␬͑␣͒ϭ .͑31͒ ͭ͑1Ϫ␣͒n,0р␣р1 This model is solvable for nϭ1,2. Using the symmetry, we have from the expression for the cross section within the SA ͓Eq. ͑28͔͒, for the convex case:

64 2 2 ⑀ ␲ ␥ 2 ⌺͑k͒ϭ ͵ d␣␣sin ͓hkz␨͑␣͔͒, ͑32͒ kkz 0

␨ ␬ where ⑀ϵRs /l. Deﬁning ␤ϵhkz , and letting In(␤), In(␤) stand for the integrals ͑without the prefactors͒ in the convex and concave cases respectively, they evaluate to:

⑀2 cos͑2␤͒Ϫcos͑2␤͑1Ϫ⑀͒͒ I␨͑␤͒ϭI␬͑␤͒ϭ ϩ 1 1 4 8␤2 FIG. 3. Cross section in the convex case, for three values of the convexity ⑀sin͑2␤͑1Ϫ⑀͒͒ parameter n, at a constant corrugation hϭ1 Å. The higher the convexity, the ϩ , faster and more pronounced are the oscillations, and the larger the cross 4␤ section for small kz . ⑀2 sin͑2␤͒Ϫsin͑2␤͑1Ϫ⑀2͒͒ I␨͑␤͒ϭ Ϫ , 2 4 8␤ tribution of the short range repulsive forces to the cross sec- ⑀2 1 ␲ tion. For this purpose, we will completely ignore the long I␬͑␤͒ϭ ϩ ͓F ͑Ϫ2ͱ␤/␲͒ϪF ͑2ͱ␤/␲͑⑀Ϫ1͔͒͒ range attractive forces and consider a hard wall He-adsorbate 2 4 4ͱ␤ c c interaction, where the adsorbate is described by a shape func- sin͑2␤͒Ϫsin͑2␤͑⑀Ϫ1͒2͒ tion ␰(R). Then the potential takes the form: ϩ , 8␤ 0, zу␰͑R͒ V͑r͒ϭ . ⑀2 1 ␲ ͭ ϱ, zϽ␰͑R͒ ␨ 2 I ͑␤͒ϭ Ϫ ͑cos͑2␤͒F ͑2⑀ ͱ␤/␲͒ 4 4 8ͱ␤ c From the expression for the phase-shift Eq. ͑8͒, it is clear that in this case: 2 ϩsin͑2␤͒Fs͑2⑀ ͱ␤/␲͒͒, ͑33͒

␩͑R͒ϭϪkz␰͑R͒. ͑29͒ where Fc , Fs are the Fresnel integrals: We now consider a fully solvable model of a single ad- 2 2 sorbate, which although highly artiﬁcial, will provide us with x ␲y x ␲y Fc͑x͒ϭ cos dy, Fs͑x͒ϭ sin dy. valuable insight regarding the role of the repulsive interac- ͵0 ͩ 2 ͪ ͵0 ͩ 2 ͪ tion in determining the general nature of the cross section. Let us assume a cylindrically symmetric shape function of An analysis of the models in the simple case of ⑀ϭ1 the form: yields the following results ͑see Fig. 1—cone, Figs. 2, 3— convex cases, Fig. 4—concave cases͒: 0, ␣Ͼ1 ␰͑R͒ϭh␨͉͑R͉/l͒ ␨͑␣͒ϭ , ͑30͒ ͑1͒ In both the concave and convex cases the cross section 1Ϫ␣n,0р␣р1 ͭ generally decreases with increasing incidence wave- where h is the height of the adsorbate above the surface number kz . ͑effectively the strength of the coupling between the He and ͑2͒ The convex model exhibits more noticeable oscillations. the adsorbate͒. lϭ␥L (0р␥р1) is a characteristic range of These are due to interference between He particles strik- the He-adsorbate interaction. The unit-cell area ͑see Sec. ing the top of the adsorbate and the surface. Since the III A͒ is taken as Aϭ␲L2. The linear extent of the entire top has a larger surface area in the convex case, the surface is denoted Rs . We will demonstrate this choice of increased oscillations are expected. ␨(␣) to be analytically solvable in terms of known functions ͑3͒ The cross section is generally larger in the convex case. for nϭ1,2,4. For nϭ1 one obtains a cone, for nϭ2,4 the This is also expected due to the larger surface area of the shape is a convexly deformed cone. It has a physically un- top.

FIG. 5. Cross section for k 0, in the concave and convex cases, as a FIG. 4. Cross section in the concave case for two values of the concavity z→ parameter n, at a constant corrugation hϭ1 Å. The larger the concavity, the function of the concavity/convexity parameter. The more concave ͑i.e., smaller the cross section. Oscillations are virtually unnoticeable. peaked͒, the smaller the cross section. The opposite happens the more convex ͑i.e., rectangular͒ the shape function.

͑4͒ The cross section has a finite limit for kz 0, which can easily be found by a first-order Taylor expansion→ of the section is entirely ͑apart from oscillations͒ due to the 2 Ϫ2 sin in Eq. ͑32͒. This limit is as a matter of fact beyond kz factor. the region of validity of the SA. However, relying on the In conclusion, it appears that the cross sections we ob- impressive success of the SA under unfavorable condi- tained exhibit all the general features of the experimentally tions of significant corrugation,29 we consider the limit observed ones: the oscillations, the decrease with increasing anyway: its dependence on the corrugation parameter h incidence energy and the finite limit for vanishing incidence and the concavity/convexity parameter n is of high inter- energy. We conclude that while the long range attractive part est. We find: is important in the setting the magnitude of the cross section 32␲2h2n2 values ͑the stronger the interaction, the larger the cross sec- ␨ ⌺ ͑k 0͒ϭ ͑convex͒, tion͒, at least for the high-energy regime the energy depen- → ͑1ϩn͒͑2ϩn͒ dent features are already determined by the short-range re- 32␲2h2 pulsive interaction. ⌺␬͑k 0͒ϭ ͑concave͒. → ͑1ϩn͒͑1ϩ2n͒ In the next section we will significantly simplify the re- lation Eq. ͑32͒ between the cross section and the shape func- The corrugation parameter h is thus seen to be of major tion, without any further approximations, in order to be able importance in determining the cross section, which in the to eventually invert the shape function from cross section limit of low incidence energy scales as its square. The de- measurements. pendence on the exponent n is, as expected, opposite in the concave vs convex case ͑see Fig. 5͒: in the former, the cross section monotonically decreases with n, whereas it increases B. Inversion of shape function from measurement of in the latter. This is once again due to size of the surface area cross section of the adsorbate top in each case. Two further remarks are in In this section we will demonstrate how the optical theo- order: rem result may be used to approximately solve the inversion ͑1͒ Due to the absence of the attractive part in our model, problem for the He-adsorbate potential. The SA was already the resulting cross section values are too small. Typical employed by Gerber, Yinnon, and co-workers31,32 for an ap- values for, e.g., CO/Pt͑111͒2,3 are in the range 100–300 proximate inversion of the potential, by using the full angu- Å2. This deficiency of the hard wall model was already lar intensity distribution. Here we will demonstrate a less observed by Jonsson et al.6,24 general result for the inversion of the adsorbate shape func- ␨ ␬ ͑2͒ The integrals In(␤), In(␤) tend asymptotically to a con- tion, which will, however, have the advantage of relying on a stant value with kz . Therefore the decrease in the cross simpler measurement, of the specular intensities alone.

We consider the case of a cylindrically symmetric adsorbate. If our unit-cell and the surface are circles of radii L and Rs respectively, then from Eq. ͑28͒ we have for the adsorbate cross section: 2 32␲ Rs k , ϭ 2 sin2 R RdR ⌺͑ z ␪in͒ 2 2 ␲ ͵ ␩͑ ͒ ␲L kz cos͑␪in͒ 0

R 1 1 2 s ϭ 2 RsϪ͵ cos͑2␩͑R͒͒RdR , ͑34͒ ␣kz ͫ2 0 ͬ where 2 L cos͑␪in͒ ␣ϵ . 32␲2 6 ͓We note in passing that the often discussed cos(␪in) factor, ␪in being the angle between the incident direction and the surface normal, arises naturally in our formalism.͔ Our demonstration in Sec. II that this cross section is equivalent to that measurable by a specular attenuation experiment, en- sures that the last equation can be regarded as a simple integral equation for the phase-shift ␩ in terms of measurable quantities Rs , L, and ⌺(kz ,␪in). The measurable part is:

M͑k ͒ϵ 1 R2Ϫ␣k2⌺͑k ,␪ ͒. ͑35͒ FIG. 6. General form of the shape function ⌰(R) and its inverse R(⌰). The z 2 s z z in shape function should have a smooth top and monotonously decrease over a Let us now make the hard wall assumption again, which substantial range. We ignore the possibility of a subsequent increase. leads to the form of Eq. ͑29͒ for the phase-shift. By doing so, we will be able to obtain the form of the shape function ␰(R) of the adsorbate. Of course the concept of a shape which allows us to write I as a Fourier transform of a function is rather artificial, yet it is of great interest in char- ⌰ product: acterizing surface roughness, and is not unrelated to what is Ϫ1 obtained in STM measurements. Defining for convenience: I⌰ϭF ͓F͑⌰͒b͑⌰͒; kz͔ ϩ F ͓F͑⌰͒b͑⌰͒; kz͔, ͑42͒ ⌰͑R͒ϭ2␰͑R͒ ͑36͒ where we introduced a notation for the Fourier transform, we can express the integral part of Eq. ͑34͒ as: 1 ϱ ixy Rs F ͓ f ͑x͒; y͔ϭ ͵ dx f͑x͒e . I⌰ϵ RdRcos͑kz⌰͑R͒͒, ͑37͒ ͱ2␲ Ϫϱ ͵0 To isolate F(⌰) we apply an inverse Fourier transform so that Eq. ͑34͒ becomes: to Eq. ͑42͒, yielding: I ϭM͑k ͒. ͑38͒ Ϫ1 ⌰ z F͑⌰͒b͑⌰͒ ϩ F͑Ϫ⌰͒b͑Ϫ⌰͒ϭF ͓I⌰͑kz͒; ⌰͔. We next assume ⌰(R) to be a monotone, single-valued ͑43͒ function, so that we can consider its inverse R(⌰) ͑see Fig. But ⌰Ͼ0 so that by definition of the box function the 6͒. Thus: second term in Eq. ͑43͒ vanishes, and we obtain from it,

⌰͑Rs͒ dR͑⌰͒ upon inserting the definitions of M ͓Eq. ͑35͔͒ and F ͓Eq. I⌰ϭ ͵ d⌰ R͑⌰͒cos͑kz⌰͒ ͑40͔͒: ⌰͑0͒ d⌰ 2 d R2 ͱ ␲ ͑ ͑⌰͒͒ 2 Ϫ1 2 1 ⌰͑Rs͒ ϭ␲R ␦͑⌰͒ Ϫ ␣F ͓k ⌺͑k ,␪ ͒; ⌰͔ ϭ d⌰ F͑⌰͒͑eikz⌰ϩeϪikz⌰͒ ͑39͒ 4 d⌰ s z z in ͱ2␲ ͵⌰͑0͒ for ⌰͑Rs͒р⌰р⌰͑0͒. ͑44͒ where we defined: If we assume that the shape function never fully van- ͱ2␲ d͑R2͑⌰͒͒ ishes (⌰(R )Ͼ0) then ␦(⌰)ϭ0, so that finally: F͑⌰͒ϭ . ͑40͒ s 4 d⌰ d R2 L2 cos ͑ ͑⌰͒͒ ͑␪in͒ Ϫ1 2 R(⌰) is confined, so it is convenient to define a ‘‘box’’ ϭϪ F ͓kz⌺␪ ͑kz͒; ⌰͔. ͑45͒ d⌰ 8␲2ͱ2␲ in function: The last equation is the sought after inversion. Its RHS 1, ⌰͑Rs͒р⌰р⌰͑0͒ should be considered as calculated from the experimental b͑⌰͒ϭ ͑41͒ ͭ 0, else data on the cross section as a function of the incidence wave

1 ⌰͑0͒ 2 2 I⌰ϭ Rs Ϫkz d⌰ R ͑⌰͒sin͑kz⌰͒ . ͑46͒ 2 ͫ ͵0 ͬ

Equating this according to Eq. ͑38͒ to M(kz), and assuming that we have guessed a functional form for ⌰(R) with a set of free parameters ͕␣i͖, the inversion result may now be stated in the form: cos͑␪in͒ ⌺ ͑k ͒ϭg͑k ;͕␣ ͖͒ ͑4␲͒2 ␪in z z i FIG. 7. Cross section data for single Ag atom on Pt͑111͒͑circles͒, along with fits for Lorentzian ͑solid line͒ and Gaussian ͑dashed line͒ shape func- 1 ⌰͑0͒ 2 tion models. The agreement with the Gaussian model is somewhat better. g͑kz ;͕␣i͖͒ϵ 2 ͵ d⌰ R ͑⌰;͕␣i͖͒sin͑kz⌰͒. L kz 0 ͑47͒ 2 l ͑cos͑kz⌰0͒Ϫ1͒ ⌰0 si͑kz⌰0͒ The remaining task is to find the set ␣ which gives the gL͑kz ;͕l,⌰0͖͒ϭ ϩ , ͕ i͖ L2 k2 k best fit for the chosen form of g to the cross section data. In ͩ z z ͪ ͑49͒ the next section we shall implement this approach on experimental data. Alternatively, the last result may be viewed as a l2 1 theoretical expression for the cross section, within the SA gG͑kz ;͕l,⌰0͖͒ϭ 2 2 ͑⌫ϩln͑kz⌰0͒Ϫci͑kz⌰0͒͒, ͑50͒ L kz and a hard wall model. where si(x),ci(x) are the sine and cosine integrals respectively, and ⌫ is Euler’s constant ͑0.5772...͒. Both gL and gG contain an oscillatory part, expressing the interference C. Fit of shape function for Ag on Pt(111) between parts of the He beam striking the top of the adsor- Ϫ2 We now employ our inversion result to obtain a fit for a bate and the flat surface, and both decay as kz for large shape function of an Ag atom on a Pt͑111͒ surface, based on kz . The parameters l ͑the He-adsorbate interaction range͒ He scattering measurements by Zeppenfeld.25 The shape and L ͑the unit-cell extent͒ appear only in the combination 2 2 function is a legitimate and highly relevant object of study in l /L , and may hence be treated as a single dimensionless the field of surface roughness. It basically corresponds to an parameter ␥ϵl/L. This result is easily seen to hold for any equipotential surface, and therefore within a certain shape function of the form ⌰(R/l), as was already demon- approximation,33,34 also to the core electron densities. As a strated in Sec. IV A. The dependence of the cross section on result, it is a complementary quantity to what is measured in the He-adsorbate interaction range thus contains no sur- STM experiments, where one probes the electron densities at prises: the cross section scales as the square of this range. It the Fermi level. Assuming cylindrical symmetry, the shape is the adsorbate height ⌰0 , i.e., the coupling coefficient be- function should have the general form depicted in Fig. 6: it tween the He and the adsorbate, which plays the interesting should be flat and smooth at its top, and decay to zero height role. It determines the frequency of oscillations of the cross far away from the nucleus. This leads us to consider two section, and affects its magnitude. Using the results for the simple forms for the shape function: a Lorentzian and a Lorentzian and Gaussian models, Eqs. ͑49͒ and ͑50͒, respectively, in the expression for the cross section, Eq. ͑47͒ we Gaussian, with two free parameters l, ⌰0 each. Thus: calculated best-fits for the parameters ⌰0 and ␥. The fits are 25 ⌰0 2 shown along with the empirical cross section values in Fig. ⌰ ͑R͒ϭ or ⌰ ͑R͒ϭ⌰ eϪ͑R/l͒ . ͑48͒ L 1ϩ͑R/l͒2 G 0 7. The deviations are 1.47% for the Lorentzian, 1.14% for the Gaussian. The fits do not follow the oscillations in the 2 Inverting these for R (⌰) leads to integrals g(kz ;͕␣i͖) data closely, but these are uncertain anyway due to the large ͓Eq. ͑47͔͒ expressible in terms of known functions: experimental error (ϳ20%). The values found are:

FIG. 8. Behavior of the cross sections of the Lorentzian and Gaussian mod- FIG. 9. Shape functions of the Ag adatom for the Lorentzian and Gaussian els over a large range of incidence wave numbers. The two models predict a models, using the parameters obtained by ﬁtting the predicted cross sections similar behavior, differing essentially by a shift along the y-axis. to the measured one. The Gaussian model has a smoother top and is less narrowly peaked, in agreement with intuitive expectation for the ‘‘correct’’ shape. Note the different scales along the x- and y-axes.

Lorentzian: ⌰0ϭ1.51Å, ␥ϭ0.93,

Gaussian: ⌰0ϭ1.05 Å, ␥ϭ1.83. weak corrugation due to substitutional disorder. Let the sur- The values for the adsorbate ‘‘height’’ ⌰0 are in reasonable 25 face be flat and located at zϭ0. The interaction potential agreement with those obtained from STM measurements. with an incident He atom may be written approximately as: That ␥Ͼ1 for the Gaussian model expresses the fact that the 2 2 He-adsorbate interaction range extends beyond the adsorbate U͑R,z͒ϭUs͑z͒ϩUd͑R ϩz ͒, ͉Us͉ӷ͉Ud͉, ͑51͒ unit-cell. In any inversion problem, the question of stability 2 is of major importance. In the present case, convergence to where Rϭ(x,y), Vsϭ (ប /2m)Us is the contribution of the 2 the fitted values is overall better for the Gaussian model, surface, and Vdϭ (ប /2m)Ud is the interaction with the de- which also has a smaller deviation: the Gaussian model con- fect, assumed spherically symmetric. We next assume verges to the same values for initial guesses of Vs(z) is known and wish to determine Vd from the scattering data. We note that a similar inversion has already been con- 0Ͻ⌰0Ͻ5 Å, whereas convergence is obtained for the sidered by Gerber, Yinnon, and co-workers,31,32 but for de- Lorentzian model only for the smaller range of 0Ͻ⌰0Ͻ3Å. The initial value of the parameter ␥ has almost no influence fectless, crystalline surfaces. When the condition on the convergence. When compared over a large range of ͉Us͉ӷ͉Ud͉ is used in Eq. ͑8͒, a first order expansion yields: kz values ͑see Fig. 8͒, the two models are seen to essentially 1 differ in their prediction of the cross section merely by a ␩͑R͒Ϸ␩0͑R͒ϩ 2 ␩1͑R͒, shift. The situation is different, however, when we compare ϱ the sought-after quantity: the shape function. The corre- 2 1/2 ␩0͑R͒ϭ ͵ dz ͓͑kzϪUs͑z͒͒ Ϫkz͔Ϫkzzt͑R͒, sponding shape functions are shown in Fig. 9. The Lorentz- zt͑R͒ ian model is much more peaked and narrow. Intuitively, it ϱ 2 2 Ud͑R ϩz ͒ seems that the Gaussian shape is better suited to describe the R ϭϪ dz . 52 ␩1͑ ͒ ͵ 2 1/2 ͑ ͒ adatom shape function, in agreement with the relative devia- zt͑R͒ ͑kzϪUs͑z͒͒ tions and convergences. The turning-point function zt(R) may either be measured by the inversion procedure described in Sec. IV B for D. Inversion of potential for weakly corrugated the shape function ␰, or approximated as the solution to surfaces Vs(z)ϭE, i.e., neglecting Vd altogether. In either case Consider a surface of substrate type A, in which one of ␩0(R) is a known quantity. Let us next use our approxima- the atoms is replaced by an atom of type B, such that the tion for ␩ in Eq. ͑34͒. Then after a first order expansion radii differ only slightly. This is an example of a surface with about 2␩0:

R R 2 1 2 s dz s ␣kz ⌺͑kz ,␪in͒Ϫ Rsϩ dR Rcos͑2␩0͒ f ͑␨͒ϭ dR Rsin͑2␩0͑R͒͒Ud͑R,␨͒. 2 ͵0 d␨͵0

Rs The above expression has the form of the familiar Abel Ϸ dR Rsin͑2␩0͑R͒͒␩1͑R͒. ͑53͒ transform, for which an explicit inversion exists: ͵0 1 ␨ dI k The LHS consists of measurable/known quantities and 2 ͑ z͒ 2 Ϫ 1/2 f ͑␨͒ϭϪ dkz 2 ͑␨Ϫkz ͒ . ͑56͒ we denote it M 1(kz). The RHS contains the unknown func- ͵ ␲ 0 dkz tion Ud which we are after; we denote it I(kz). Using the We rewrite this, using IϭM 1 ͓Eq. ͑53͔͒ and changing deﬁnition of ␩1 ͓Eq. ͑52͔͒: variables from R to ␳, as:

R ϱ 2 2 s Ud͑R ϩz ͒ ␳͑Rs͒ dR I͑kz͒ϭϪ dR Rsin͑2␩0͑R͒͒ dz 2 1/2 . JϵϪ d␳ Ud͑␳,␨͒ R͑␳͒sin͑2␳͒ϭG͑␨,kz͒, ͵0 ͵z R ͑k ϪU ͑z͒͒ ͵ t͑ ͒ z s ␳͑0͒ d␳ ͑54͒ ͑57͒ A practical approach at this point would be to expand where: 2 2 2 2 ␨ Ud(R ϩz ) in some convenient basis ͕f n(R ϩz )͖: 1 d␨ dM1͑kz͒ 2 2 2 2 G͑␨,k ͒ϭ dk2 ͑␨Ϫk2͒Ϫ 1/2. ͑58͒ Ud(R ϩz )ϭ͚cn f n(R ϩz ), so that: z ͵ z 2 z ␲ dz 0 dkz R ϱ 2 2 s fn͑R ϩz ͒ The function G(␨,k ) consists entirely of known quanti- c dR Rsin͑2␩ ͑R͒͒ dz z ͚ n ͵ 0 ͵ 2 1/2 ties. We are still left with the task of isolating the potential 0 zt͑R͒ ͑kzϪUs͑z͒͒ Ud . But it is clear that J in Eq. ͑57͒ is almost identical to ϭϪM ͑k ͒. ͑55͒ 1 z I⌰ in Eq. ͑39͒. In order to be able to use the Fourier transform technique applied to I , we must now introduce one This linear system can then be solved for as many c as ⌰ n additional assumption: there are data points ͑in k ). A convenient choice of ͑non- z ␩ (R) may reasonably well be approximated orthogonal͒ basis functions are Gaussians, since then the 0 in the ‘‘hard-wall’’ form: ␩ (R)ϭϪk ␰(R). double integral decouples into a product ͑neglecting the 0 z Here ␰(R) is to be considered known from the R-dependence of z ). This approach should give a reasonable t inversion of cross section data as described in approximation to the defect potential. Powerful methods to Sec. IV B. ͑Hamburger et al.21 have shown that deal with related numerical inversion problems, employing this form holds quite well also for realistic functional sensitivity analysis, have been developed by Ho potentials.͒ and Rabitz.35,36 However, with some further assumptions it is also pos- Changing variables to ⌰ϭ2␰(R): sible to perform an actual inversion for U , by combining d ⌰͑Rs͒ dR the technique of Sec. IV B and an Abel transform. We pro- Jϭ d⌰ U ͑⌰,␨͒ R͑⌰͒ sin͑k ⌰͒ ͵ d d⌰ z ceed to show this. ⌰͑0͒ ͫ ͬ Our ﬁrst two additional assumptions are as follows: 1 ⌰͑Rs͒ ikz⌰ Ϫikz⌰ ϭ d⌰ F͑⌰͒͑e Ϫe ͒ϵI⌰ , ͑59͒ ͑1͒ The interaction with the surface is purely repulsive, so ͱ2␲ ͵⌰͑0͒ that ␨ϭU (z) is a monotonously decreasing function. s where now: ͑2͒ Also ␳ϭ␩0(R) is a monotone function. That this is reasonable can be seen by considering a hard-wall system; ͱ2␲ d͑R2͑⌰͒͒ F͑⌰͒ϭ U ͑⌰,␨͒. ͑60͒ then a monotonously decreasing shape function ͑see Sec. 4i d⌰ d IV B͒ induces a monotonously increasing phase shift, by Eq. ͑29͒. Repeating the arguments leading to Eq. ͑45͒ and remem- bering that now I ϭG(␨,kz), we ﬁnally obtain: Under these assumptions we may consider the inverse ␪ functions z(␨) and R(␳). The transformation to the variable d͑R2͑⌰͒͒ Ϫ1 4i U ͑⌰,␨͒ϭ F Ϫ1͓G͑␨,k ͒;⌰͔. ␨ decouples the integration limits if we make the fair ap- d ͫ d⌰ ͬ z 2 ͱ2␲ proximation that kz ϭUs(zt(R)) ͑i.e., neglecting Ud), for ͑61͒ then: Formally, this completes the inversion. ϱ 2 2 0 Ud͑R ϩz ͒ dz Ud͑R,␨͒ dz ϭ d␨ ͵ 2 1/2 ͵2 d␨ 2 1/2 zt͑R͒ ͑kzϪUs͑z͒͒ kz ͑kzϪ␨͒ V. CONCLUDING REMARKS and hence: In this paper we investigated theoretically the cross section of a defect on a surface. We found that this quantity, 0 when available from experiments, contains important infor- I͑k ͒ϭ d␨ ͑k2Ϫ␨͒Ϫ 1/2f ͑␨͒, z ͵ 2 z mation on the defect and its interaction with an in-coming kz atom. We demonstrated that the cross section can be used for where such purposes as an inversion to yield the shape function of

Downloaded¬19¬Sep¬2003¬to¬142.150.226.143.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jspJ. Chem. Phys., Vol. 102, No. 17, 1 May 1995 6930 D. A. Hamburger and R. B. Gerber: Inversion of scattering data from defects a defect, an approximate inversion to give the interaction sions and allowing us to use their Ag/Pt͑111͒ data prior to potential with a substituted atom, and evaluation of the role publication. We would also like to thank Dr. A. T. Yinnon for of the long-range repulsive forces in the interaction with an very useful discussions. incident atom. The single most important advantage of the 1 B. Poelsema and G. Comsa, Faraday Discuss. Chem. Soc. 80,16͑1985͒. cross section is probably that it is so easily measurable, by 2 B. Poelsema, S.T. deZwart, and G. Comsa, Phys. Rev. Lett. 49, 578 employing the definition of the cross section proposed by ͑1982͒. Poelsema and Comsa.1 This operational definition was 3 B. Poelsema, S.T. deZwart, and G. Comsa, Phys. Rev. Lett. 51, 522 shown here to be equivalent to the standard formal definition ͑1983͒. 4 B. Poelsema, L.K. Verheij, and G. Comsa, Phys. Rev. Lett. 49, 1731 of quantum scattering theory, and hence either can be used ͑1982͒. interchangeably. Using the formal definition and the Sudden 5 B. Poelsema, R.L. Palmer, and G. Comsa, Surf. Sci. 136,1͑1984͒. approximation, we derived a form of the Optical theorem 6 H. Jonsson, J. Weare, and A.C. Levi, Surf. Sci. 148, 126 ͑1984͒. 7 which is quite amenable to analytical study. Future work may W.-K. Liu, Faraday Discuss. Chem. Soc. 80,17͑1985͒. 8 B. Gumhalter and D. Lovric´, Surf. Sci. 178, 743 ͑1986͒. benefit from this expression for the cross section. For ex- 9 H. Xu, D. Huber, and E.J. Heller, J. Chem. Phys. 89, 2550 ͑1988͒. ample, it seems that extracting various probability distribu- 10 A.T. Yinnon, R. Kosloff, R.B. Gerber, B. Poelsema, and G. Comsa, J. tions characterizing dilute surface disorder such as a distri- Chem. Phys. 88, 3722 ͑1988͒. ͑ 11 bution of radii for hemispherical defects͒ should prove S.D. Bosanac and M. Sunic, Chem. Phys. Lett. 115,75͑1985͒. 12 B. Gumhalter and W.-K. Liu, Surf. Sci. 148, 371 ͑1984͒. possible within this framework. Traditional approaches 13 G. Petrella, A.T. Yinnon, and R.B. Gerber, Chem. Phys. Lett. 158, 250 tended to rely on differential cross section measurements for ͑1989͒. inversion applications. Our general message for future work 14 B. Poelsema, A.F. Becker, G. Rosenfeld, R. Kunkel, N. Nagel, L.K. Ver- heij, and G. Comsa, Surf. Sci. 272, 269 ͑1992͒. is that the total cross section, which is much easier to obtain 15 R. Kunkel, B. Poelsema, L.K. Verheij, and G. Comsa, Phys. Rev. Lett. 65, experimentally, will be a highly fruitful subject of study for 733 ͑1990͒. the purpose of both structural and dynamical characterization 16 Y.-W. Yang, H. Xu, and T. Engel, J. Phys. Chem. 97, 1749 ͑1993͒. 17 of isolated defects. Indeed, a very promising possibility for H. Xu, Y.-W. Yang, and T. Engel, Surf. Sci. 255,73͑1991͒. 18 P. Dastoor, M. Arnott, E.M. McCash, and W. Allison, Surf. Sci. 272, 154 future work should be one of measuring cross sections for ͑1992͒. surface defects of interest, and using this data in a corre- 19 B.J. Hinch, C. Koziol, J.P. Toennies, and G. Zhang, Europhys. Lett. 10, sponding theoretical effort to extract information on the po- 341 ͑1989͒. 20 sition of the defect, its geometric shape, and its interaction S.A. Safron, J. Duan, G.G. Bishop, E.S. Gillman, and J.G. Skofronick, J. Phys. Chem. 97, 1749 ͑1993͒. potential with the incident atom. The latter is especially im- 21 D.A. Hamburger, A.T. Yinnon, I. Farbman, A. Ben-Shaul, and R.B. Ger- portant, since the interaction potential contains information ber, Surf. Sci. 327, 165 ͑1995͒. on the electronic density structure of the defect. We empha- 22 B. Poelsema and G. Comsa, in Scattering of Thermal Energy Atoms from size that cross sections measurements, when performed over Disordered Surfaces, Springer Tracts in Modern Physics, Vol. 115 ͑Springer, Berlin, 1989͒. a wide range of energies and incidence angles, may well 23 R.B. Gerber, In Delgado-Bario, editor, Dynamics of Molecular Processes become a new type of surface microscopy. We are currently ͑IOP, Bristol, 1993͒. pursuing efforts along this line, in cooperation with the ex- 24 H. Jonsson, J. Weare, and A.C. Levi, Phys. Rev. B 30, 2241 ͑1984͒. 25 periments of P. Zeppenfeld. P. Zeppenfeld ͑private communication͒. 26 J.R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions ͑Krieger, New York, Florida, 1987͒. ACKNOWLEDGMENTS 27 R.B Gerber, A.T. Yinnon, and J.N. Murrel, Chem. Phys. 31,1͑1978͒. 28 R.B. Gerber, Chem. Rev. 87,29͑1987͒. This research was supported by the German-Israeli 29 A.T. Yinnon, R.B. Gerber, D.K Dacol, and H. Rabitz, J. Chem. Phys. 84, Foundation for Scientific Research ͑G.I.F.͒, under Grant No. 5955 ͑1986͒. 30 D.A. Hamburger and R.B. Gerber ͑to be published͒. I-215-006.5/91 ͑to R.B.G.͒. Part of this work was carried out 31 R.B. Gerber and A.T. Yinnon, J. Chem. Phys. 73, 3232 ͑1980͒. with support from the Institute of Surface and Interface Sci- 32 A.T. Yinnon, E. Kolodney, A. Amirav, and R.B. Gerber, Chem. Phys. Lett. ence ͑ISIS͒ at UC Irvine. The Fritz Haber Center at the He- 123, 268 ͑1986͒. 33 brew University is supported by the Minerva Gesellschaft fu¨r N. Esbjerg and J.K. Norskov, Phys. Rev. Lett. 45, 807 ͑1980͒. 34 N. D. Lang and J.K. Norskov, Phys. Rev. B 27, 4612 ͑1983͒. die Forschung, Munich, Germany. We are very grateful to 35 T.-S. Ho and H. Rabitz, J. Chem. Phys. 94, 2305 ͑1991͒. Dr. P. Zeppenfeld and M. A. Krzyzowski for helpful discus- 36 T.-S. Ho and H. Rabitz, J. Chem. Phys. 96, 7092 ͑1992͒.

Downloaded¬19¬Sep¬2003¬to¬142.150.226.143.¬Redistribution¬subject¬to¬AIP¬license¬or¬copyright,¬see¬http://ojps.aip.org/jcpo/jcpcr.jspJ. Chem. Phys., Vol. 102, No. 17, 1 May 1995