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Two versions of the sudden approximation are introduced to decouple and solve the equations that describe atom- surface scattering with many open diffraction channels. Both approximations require a high incident beam wave number compared with the magnitude of the reciprocal space vector of the lattice. In this framework, simple explicit expressions are obtained for the observable diffraction intensifies, making calculations feasible even for systems with hundreds of open diffraction channels. Further considerable simplifications ensue when the approximations are specialized to the case of a Lennard-Jones-Devonshire potential, or to that of a weakly corrugated surface. The approximations were applied to the systems He/LiF(001); Ne/LiF(001) and Ne/W(110) and the results are compared with other calculations or with experiment. The sudden approximation is found to be of good accuracy in these cases.

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... L'intensité diffractée est ensuite calculée analytiquement en résolvant l'équation de Schrödinger en imposant la nullité de la fonction d'onde sur le mur dur. D'autres modèles [39,40,41] se sont développés par la suite afin de relaxer la contrainte de mur dur en considérant une interaction douce avec la surface. Plus récemment et dans un autre contexte, Henkel et al. [33] ont obtenu une formule analytique pour la diffraction en incidence normale et oblique. ...

This work was devoted to the study of fast atom diffraction (energies in the keV range)at grazing incidence angles (~ 1°) along or close to a low indexed direction of crystalline surfaces.This specific scattering geometry bears two advantages : (i) the diffraction pattern as a wholeis collected within seconds on a position sensitive detector ; (ii) the low energy associated to themotion normal to the surface quenches decoherence due to electronic excitations and stronglyreduces decoherence due to thermal vibrations. The high sensitivity of probe atoms to thesurface electron density (repulsive part) and to the Van der Waals attractive well reveals Fanoresonances where the trapped atoms preserve their coherence over distances as long as 0.2μm.As a complement to these fundamental studies, fast atom diffraction has been proved to be arobust mean to probe the dynamics of epitaxial growth of semiconductors (GaAs). Finally, workperformed on monolayer graphene grown on 6H-SiC(0001) suggest the possibility to use fastatoms to monitor graphene growth in real time, a key process to measure the level of alterationof the intrinsic graphene electronic structure.

... This is the well-known expression for the SA scattering amplitude [21]. The SA has been tested extensively by comparison to exact coupled-channel calculations on Ne/W(110) and He/LiF(001) [22], as well as by comparison to exact time-dependent propagation methods in the case of scattering from defects [23]. ...

The sudden approximation is applied to invert structural data on randomly corrugated surfaces from inert atom scattering intensities. Several expressions relating experimental observables to surface statistical features are derived. The results suggest that atom (and in particular He) scattering can be used profitably to study hitherto unexplored forms of complex surface disorder.

This is the story of a career in theoretical chemistry during a time of dramatic changes in the field due to phenomenal growth in the availability of computational power. It is likewise the story of the highly gifted graduate students and postdoctoral fellows that I was fortunate to mentor throughout my career. It includes reminiscences of the great mentors that I had and of the exciting collaborations with both experimentalists and theorists on which I built much of my research.
This is an account of the developments of exciting scientific disciplines in which I was involved: vibrational spectroscopy, molecular reaction mechanisms and dynamics, e.g., in atmospheric chemistry, and the prediction of new, exotic molecules, in particular noble gas molecules.
From my very first project to my current work, my career in science has brought me the excitement and fascination of research. What a wonderful pursuit!
Expected final online publication date for the Annual Review of Physical Chemistry, Volume 72 is April 20, 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

The chapter starts with an overview of the history of the theory of surface phonons. Next the history of helium atom and electron inelastic scattering experiments are briefly reviewed. Present day helium atom scattering experiments are described next and some important related concepts are introduced. The chapter closes with a short history of the classical and quantum theory of inelastic surface scattering.

We present elastic He-beam scattering data of the Pd(111)/H system. Diffraction intensities were measured as a function of surface temperature in the range 140°K–320°K. Two remarkable features are observed : the first is the presence of C3v symmetry at (1 × 1) saturation coverage (140°K) and its transformation to C6v symmetry at lower coverages (270°K). The second feature is the anomalous attenuation of the specular He beam accompanying this transformation. Taken together these features provide strong evidence of a fundamental change in the surface charge density corrugation. A classical interpretation of the motion of hydrogen either fails to reproduce the measured attenuation or leads to contradictory and unphysical conclusions regarding the H-metal bond length or surface equilibrium. An alternative quantum mechanical interpretation is developed and is shown to provide consistent and satisfactory explanation of the measurements.

In this paper we present formal and computational studies of Δmj — transitions occuring in heteronuclear and homonuclear molecule — corrugated surface collisions. The model potential is a pairwise additive one which correctly incorporates the fact that Δmj — transitions occur only for corrugated surfaces (provided the quantization axis is chosen to be the average surface normal). The principal results are; a) Δmj — transitions are sensitive to lattice symmetry b) strong selection rules obtain for specular scattering c) the magnitude of Δmj — transition probabilities are strongly sensitive to surface corrugation d) the ratio of molecular length to lattice dimension (r/a) has a strong influence on the magnitude of Δmj — transition probabilities (with the probabilities increasing as (r/a) increases) e) Δmj — rainbows are predicted to occur. Computations are reported illustrating the above for both heteronuclear and homonuclear molecule — corrugated surface collisions.

A very exhaustive analysis of selective adsorption resonances, for the 4He—Cu(11α) systems, is presented. Emphasis on results, supplied by the Close-Coupling and Golden-Rule formulations, is made with different types of elastic scattering: non-resonant, resonant, critical, at threshold and temperature dependent conditions. Connected with the critical kinematic effect, new results are predicted.

With the advent of increasingly better methods for the production of UHV, atomic and molecular beams and crystal surfaces including their characterization, studies of gas—solid interactions developed as an important branch of surface physics. Measured in the very first atomic scattering and diffraction experiments and correctly recognized and interpreted from the very beginning, selective adsorption resonances now represent a special highlight in atom-surface scattering investigations allowing for a precise determination of bound state energies in the physical atom—surface interaction potential and thus providing a unique test for any method of calculating such interaction potentials. The theory of the scattering and diffraction of atoms (molecules) from solid (crystalline) surfaces also greatly benefitted from the stringent task of correctly describing the many detailed and characteristic structures caused by selective adsorption resonances. The following sections will give a review of the historical development of selective adsorption studies, the discovery of diffraction-, rotation- and phonon-mediated resonances together with some of their theoretical implications (interaction potentials are discussed in Chap. 3).

Atom scattering at thermal energies has proven to be one of the most sensitive experimental methods for obtaining detailed microscopic information on surfaces [8.1]. In some cases the necessary theory is very simple, as for example in identifying surface structures from the positions of diffraction peaks, or obtaining surface phonon dispersion relations from the positions of the phonon peaks in an inelastic experiment. In most other cases, however, sophisticated theory which often involves intensive numerical calculations is necessary in order to fully exploit the extreme sensitivity of the method to surface structure, disorder and surface vibrations. As with thermal neutrons, the wavelengths of small mass and low energy atoms such as He are comparable to interparticle spacings in solids, and the energies are comparable to maximum crystal phonon energies. Thus such particles are ideally suited for studies of both surface structure and surface vibrations. The theory of scattering of atoms from an extended target such as a surface has similarities with many of the highly developed techniques used to interpret scattering from bulk solids or liquids, as for example neutron, X-ray or electron scattering. The major difference from bulk scattering is that the presence of the surface breaks the translational symmetry normal to the surface, hence momentum is no longer conserved in that direction. One immediate consequence of this is that diffraction peaks from ordered surfaces are two-dimensional in character and can be observed for all incident beam conditions.

We propose a novel method to measure the fractal dimension of a submonolayer metal adatom system grown under conditions of limited diffusivity on a surface. The method is based on measuring the specular peak attenuation of He atoms scattered from the surface, as a function of incidence energy. The (Minkowski) fractal dimension thus obtained is that of contours of constant electron density of the adatom system. Simulation results are presented, based on experimental data. A coverage dependent fractal dimension is found from a two-decade wide scaling regime.

Quantum‐mechanical approximation methods and calculations for rotational transitions in molecule‐surface collisions are reviewed. The methods are analyzed with regard to predictions of several observable effects: (1) Large ΔJ (rotational quantum number) transitions, and rotational rainbow scattering. (2) The relation between rotational and diffractive transitions. (3) Scaling properties of the rotationally‐inelastic scattering amplitudes. (4) Trapping and resonances induced by rotational‐translational energy transfer. The methods examined with regard to some of these effects include: coupled‐channel calculations; the Sudden approximation with regard to both the rotational and the diffractive transitions, and the hard corrugated wall model. It is concluded that whilst available methods provide a qualitative description of the effects mentioned, quantitative treatment of real systems remains an open problem. The main difficulties in formulating a satisfactory quantitative model are examined. Finally, the article presents new results on molecular reorientation processes (ΔM J transitions) in molecule‐surface collisions. It is shown, using the Sudden approximation, that molecular reorientation probabilities should reflect sensitively on surface structural corrugation.

The angular intensity distribution of He beams scattered from compact clusters and from diffusion limited aggregates, epitaxially grown on metal surfaces, is investigated theoretically. The purpose is twofold: to distinguish compact cluster structures from diffusion limited aggregates, and to find observable {\em signatures} that can characterize the compact clusters at the atomic level of detail. To simplify the collision dynamics, the study is carried out in the framework of the sudden approximation, which assumes that momentum changes perpendicular to the surface are large compared with momentum transfer due to surface corrugation. The diffusion limited aggregates on which the scattering calculations were done, were generated by kinetic Monte Carlo simulations. It is demonstrated, by focusing on the example of compact Pt Heptamers, that signatures of structure of compact clusters may indeed be extracted from the scattering distribution. These signatures enable both an experimental distinction between diffusion limited aggregates and compact clusters, and a determination of the cluster structure. The characteristics comprising the signatures are, to varying degrees, the Rainbow, Fraunhofer, specular and constructive interference peaks, all seen in the intensity distribution. It is also shown, how the distribution of adsorbate heights above the metal surface can be obtained by an analysis of the specular peak attenuation. The results contribute to establishing He scattering as a powerful tool in the investigation of surface disorder and epitaxial growth on surfaces, alongside with STM. Comment: 41 pages, 16 postscript figures. For more details see http://www.fh.huji.ac.il/~dani

The diffraction of thermal He atoms from mixed Xe+Kr monolayers on Pt(111) was measured, and the results were compared with theoretical studies of these systems. The results shed light on the structural properties of these disordered systems, and on their relation to the He diffraction intensities. Experimentally, the specular (0,0), the (1,0), and the (2,0) Bragg peak intensities were measured for monolayers of different Kr:Xe concentration ratios. The theoretical calculations included Monte Carlo simulations of the mixed disordered monolayers, and quantum calculations in the Sudden approximation of the scattering intensities from the simulated disordered structures. The following main results were obtained: (1) Both experiment and the Monte Carlo simulations suggest that the mixed Xe+Kr monolayers are periodic for all Xe:Kr concentration ratios, the lattice constant varies linearly with the Xe:Kr ratio. The domain size of the 2D crystals, from experiment and theory, is found to be larger than 100 Å. (2) The Monte Carlo simulations suggest that the Xe+Kr monolayers form an almost ideal substitutionally disordered lattice. (3) Using a semiempirical Debye–Waller factor, reasonable agreement is found between the theoretical and the measured diffraction intensities, thus supporting the calculated structural model for the disordered surface. (4) The theoretical scattering calculations show that in addition to the diffraction peaks, there are also intensity maxima at non‐Bragg positions. These are entirely due to the lattice disorder, and are identified as a recently found new type of Rainbow effect that can furnish important information on disordered surfaces. The results demonstrate the power of He scattering as a tool for exploring substitutionally disordered surfaces.

Total cross sections for He scattering from isolated imperfections on surfaces are calculated using the Sudden approximation, and in some cases also by a numerically exact, time‐dependent quantum‐mechanical wave packet method. Systems studied include: CO adsorbates on Pt(111); mono‐, di‐, and trivacancies on Pt(111). The main results are: (1) the incidence angle and energy dependence of the cross section for He/[Pt(111)+CO] are very sensitive to the CO distance from the Pt plane. Interactions with the adsorbate image have little effect on the cross section. (2) The cross sections for clusters of vacancies are given within 10% or better, by the geometric sum of the monovacancy cross sections, the latter being treated as circles centered at each monovacancy. (3) The dependence of the cross section on the energy is sufficiently sensitive to distinguish between the ‘‘electron density hole’’ and ‘‘electron density hump’’ models for vacancies and vacancy clusters. (4) The Sudden approximation compares well with the exact quantum‐mechanical results at typical experimental energies, when the incidence angle is not too far from the normal. These results indicate that experimental measurements of He scattering cross sections as a function of energy and incidence angle, combined with Sudden or wave packet scattering calculations, can provide detailed information on surface defects and their interactions with gas‐phase atoms.

A recent multiple collision model for molecular dissociation on flat, rigid surfaces is modified to include the influence of surface corrugation and vi

Exact quantum-mechanical calculations are present for He scattering from one-dimensional models of disordered, mixed Xe + Ar overlayers. A time-dependent wavepacket approach is used with a recent technique for solving the Schrödinger equation. Results are given for several overlayers of different Xe : Ar concentration ratios. The dependence of scattering intensities on the disordered structures is discussed. The results provide a reference for testing approximations for scattering from disordered surfaces.

He scattering from substitutionally disordered Xe+Kr monolayers is studied theoretically. The angular distribution of the scattered He, and lifetimes of scattering resonances at the surface, are calculated for different Xe:kr mixing ratios. In addition to Bragg diffraction, new intensity maxima are found. These are interpreted as rainbow effects (disorder rainbows), and may be very useful in characterizing disordered surfaces. Also, trapping resonances are found, which are sharply sensitive to the percolation thresholds for the mixtures. The results show He scattering can be a powerful tool in studying 2D percolation behavior.

It is shown that within the coordinate representation sudden approximation,the general amplitude Sm′,n′, mn is expressed in terms of system dependent dynamical factors, which are the amplitude Sm̄n̄,oo, and system independent factors which contain all momentum transfer information.

We examine the conditions for the validity of the sudden approximation and investigate the utility and validity of the weak-coupling limit of the sudden approximation. We also discuss purely classical calculations in the sudden approximation, showing that these contain the inter-relations recently established in the quantum coordinate representation sudden approximation.

A general iterative inversion procedure based on functional sensitivity analysis is presented for determining the gas–surface interaction potential from low energy elastic scattering data. Formally, Tikhonov regularization, singular function analysis, and a recently developed exact transformation technique are implemented to render the inversion stable and efficient. Specifically, the simulation of helium scattering from a rigid periodic xenon monolayer on the graphite (0001) face is considered. It is found that the functional sensitivity densities of the diffraction intensities with respect to the He–Xe/C(0001) potential contain profound information, thus are invaluable in guiding the inversion of scattering data to yield the potential. Although, unequivocal determination of the full three-dimensional potential from the inevitably incomplete experimental data may be difficult, we demonstrate that simulated input data consisting of a finite number of polar scan specular intensities can be used to accurately recover the underlying He–Xe/C(0001) potential. The recovered potential has been obtained without imposing any explicit functional form on the potential per se. The resulting procedure is quite promising for treating real laboratory data.

Exact coupled-channel calculations are presented for the scattering of Ne from W(110) and He from LiF(001), using symmetry to partly decouple the scattering equations. The results are used to test the recently proposed sudden approximation. For Ne/W(110), typical of all metals, the sudden approximation gives excellent quantitative accuracy. For the very unfavorable system He/LiF(001) good semiquantitative agreement is found with the exact results. It is concluded that the sudden approximately provides an efficient and accurate tool for atom—surface scattering calculations.

In this work a quantum mechanical approach to the study of reactive diatom (stationary) surfacae collisions is presented. Following the general theory the infinite order sudden approximation (IOSA) is introduced and the corresponding S matrix elements derived. The case of noncorrugated surfaces is treated in detail and a new quasiselection rule is found which governs the reactive process.

The well-known classical path approximation is applied to a calculation of diffraction intensities in the scattering of atoms from a rigid crystal with a soft interaction potential. A general expression is derived for the diffraction intensities which can be applied to potentials with several higher-order terms in the Fourier series. For an uncorrugated Morse potential with a first-order exponential corrugation term an analytic solution is obtained which is compared with the infinite order suddent (IOS) approximation calculations for Ne/W(110) and He/LiF(100). Both approximations are very accurate for the weakly corrugated Ne/W system. For He/LiF the present approximation is more accurate than the sudden (IOS) approximation and has the added advantage of providing an analytic solution. Several improvements are suggested.

Simulated specular scattering intensity of He particles from Xe and CO adsorbed on Pt surfaces shows distinctive features of the surface order parameters which are, however, not observed in experiments. Inferences may be drawn about the influence of long-range order on the scattering properties.

A dynamical Lie algebraic approach to statistical dynamics of the rotationally inelastic gas-surface scattering is described. This method is applied to the study of the scattering of NO from Ag(111) surface. Statistical average values of some physical observables, such as the translational-to-rotational (T→R) energy transfer and the interaction potential, and their dependence on various dynamic variables of the system are given analytically. The calculations predict a strong dependence of the average energy transfer and average interaction potential on temperature and the incident translational energy. The results imply that the dynamical Lie algebraic method appears to have a wide range of validity for describing the statistical dynamics of gas-surface scattering.

This paper shows that the He diffraction specular intensity does not depend in a simple, predictable way on single statistical parameters describing the surface defect lateral distribution. In particular it shows the lack of correlation between specular intensity and the geometrical overlap of single defect cross sections, a parameter widely used in the literature for a qualitative interpretation of He atom diffraction data. An approach to the problem of surface structure determination is attempted on the basis of a cluster expansion of the scattering matrix at reflection angle.

Calculations are reported on He scattering from clusters of different numbers of CO molecules adsorbed in an ordered way on flat Pt(111) surfaces. The sudden approximation is used to obtain both the integral cross sections for scattering by adsorbates and the angular intensity distribution of the scattered atoms. The cross section values have been discussed on the basis of the Comsa and Poelsema overlap approach that, already successfully tested for clusters of vacancies, has been found to hold good even in the case of molecular adsorbates as surface defects. The angular intensity distribution curves show Fraunhofer interferences and rainbow maxima, as in the case of isolated adsorbates, and periodic diffraction peaks caused by the ordered structure of CO molecules on the Pt surface. Basically it has been demonstrated that from these peaks it is possible to obtain information on the geometric structure of the adlayer if the adsorbates form a two-dimensional crystal on the surface.

On the basis of the sudden approximation, a theoretical analysis has been performed on He scattering from two CO molecules adsorbed on Pt(111) at different mutual distances dCO (10 au≲dco≲ 50 au) and for the incident collision wavenumber kz from 1.0 to 4.0 bohr−1. The values obtained for the cross section Σ show that Comsa and Poelsema's hypothesis of cross-section geometric overlap (GO) breaks down. In fact, in a dCO range of nearly 22 au, and at all collision energies, the Σ values are higher than twice the cross section of the isolated adsorbed CO. Obviously this finding is not consistent with the GO hypothesis: the corresponding overlap between the two CO should be negative and therefore meaningless.

The scattering of a diatomic molecule from a solid surface is analyzed within an impulsive collision approximation. It is shown that for a particular simple molecule-surface potential function not only does such an assumption permit the separation of the rotationally inelastic contribution from the total scattered intensity, but also it allows one to estimate the importance of translation-rotation energy transfer on the basis of just a few parameters which appear in the chosen diatom-surface potential model. The implications and limitations of the use of such a model in the analysis of scattering data are discussed.

A formal theory of rotationally inelastic scattering of 2Π diatomic molecules from surfaces is developed. The Λ-doublet degeneracy of the molecule gives rise to an effective molecule-surface potential which has an additional dependence, not present in Σ-state molecule-surface systems, on the angle of rotation about the diatomic axis. Close-coupled scattering equations are derived for pure Hund's case (a) and pure Hund's case (b) representations of the molecular electronic-rotational states. The coordinate-representation sudden approximation is then applied, yielding simplifying expressions for various intensities of scattering from the surface. Propensity rules for rotational transitions between the Λ-doublets and between the spin-orbit manifolds for both the low and high final total molecular angular momentum regimes are derived. Trends in recent experimental observations on these transitions for the NO/Ag(111) system are explained within the present theoretical framework.

Since the specular intensity and scattering matrix are related by a quadratical relation, a term is present in the intensity which scales as the square of the defect surface coverage. One can directly account for the term in every calculation which provides the scattering matrix. The explicit formulas under the sudden approximation are reported and their implications discussed. Any relation between cross-section and specular intensity which does not take it into account is demonstrated to be inadequate even at very low coverage values. Consequently cross-sections calculated on phenomenological grounds cannot give correct information on surface structure.

This article reviews recent progress in understanding the dynamics of molecular dissociation in impact on crystalline surfaces, a topic pursued by a combination of theoretical methods (trajectory calculations and impulsive collision models) with molecular beam scattering experiments. The studies reviewed deal with molecules such as I2 and ICl in collision with single-crystal surfaces of chemically inert insulators, e.g., MgO(100), sapphire, and diamond. Dissociation in such systems is an elementary, single-collision process, advantageous for pursuing understanding on a first-principles basis. The main findings include the following: (1) Dissociation occurs by a centrifugal mechanism involving high rotational excitation upon impact. (2) Large energy transfer to the solid takes place in cases such as I2/MgO(100) and I2/sapphire, the mechanism of which is shown by theoretical simulations to involve a shock-wave excitation of the solid upon the molecular impact. The effect of energy transfer to the solid on the dissociation probability depends strongly on the system, and the theoretical model provides an interpretation for this. (3) There are important qualitative differences between the results for a homonuclear collider and those for a related mass asymmetric molecule (e.g., I2 vs. ICl) regarding the energy dependence of the dissociation probability and the energy distribution of the products. It is argued that for the simplest systems studied a coherent picture seems to emerge for the various aspects of the dissociation process with good consistency between theoretical and experimental results. Major open problems and future directions in the field are discussed.

A simple model for gas-surface scattering is presented which permits treatment of inelastic effects in diffractive systems. This model, founded on an impulsive collision assumption, leads to an intensity distribution which is just a sum of contributions from n-phonon scattering events. Furthermore, by using a convenient form'for the repulsive interaction potential, ana- lytic expressions are obtained for the elastic and one-phonon intensities that are in qualitative agreement with experimental results.

We combine Gaussian wave packets and the coupled channel method to develop a theory of H2 diffraction and rotational excitation by collision with surfaces. This improves our previous work on H2 diffraction since it eliminates the mean trajectory approximation; it also extends Heller’s work to problems in which the dynamics require the creation of new packets which must be coupled to each other as they are propagated through the interaction region. The approximations involved in the above Gaussian wave packet can be removed by extending a method proposed by Fleck, Morris, and Feit, which propagates the Gaussian wave function exactly and efficiently.

A simple, approximate treatment of atom–surface elastic scattering is given. The treatment requires the incident beam energy to be high, such that all the atoms leading to different channels which contribute significantly to diffraction may be described by the same normal motion. This requirement can be realized in experiments with supersonic beams for a (near-)normal incidence. Diffraction intensities are given in an analytic form that requires little computational effort. Applications to the systems He/LiF and Ne/W are shown, and the results are compared with other calculations and experimental data.

The theoretical interpretation of experimental results on He scattering from metal adatoms epitaxially grown on metal substrates makes use, in general, of an interaction potential that, as far as the He-adatom interaction is concerned, is a pairwise sum of Lennard-Jones (VLJ) terms. This approach results in more corrugated metal surfaces than expected on the basis of both experimental data and qualitative considerations on the contribution of the electron densities of the metal adatoms to the repulsive part of the interaction potential. The present work suggests a very simple way to significantly reduce or eliminate the surface corrugation by averaging the He-adsorbate potential over a unit cell of the substrate lattice. The so obtained potential (VAV) was compared with VLJ for the HeAg/Pt(111) colliding system on the basis of turning point surfaces, specular and diffractive intensities vs. surface coverage and angular intensity distribution of scattered atoms, calculated under the sudden approximation. These data show that VAV, although too simple to correctly reproduce all the features of the He-adatom interaction, could represent a useful tool in the study of the growth of metal adlayers.

An effective mirror model for the scattering of atomic beams from stiff, slightly corrugated solid surfaces is examined. A simple formula for the corrugation function in terms of only two parameters, characterizing some general features of a gas-solid pairwise interaction, is derived. Applications to diffraction of He and Ne from LiF indicate that deviations from simple harmonic corrugation functions may be more important than previously assumed.

A modified sudden approximation is introduced for the calculation of elastic atom scattering intensities from surfaces. On the basis of a comparison with the limit of hard wall interaction potentials, this new sudden approximation will have an extended range of validity for scattering into final states with larger parallel momentum exchange. As for any coordinate representation sudden approximation, the method is equally applicable to both periodic and non-periodic surfaces. This paper reports the preliminary results of an investigation of the atom-surface interaction potentials at step edge defects using the new approximation. Comparisons with experimental data of diffuse scattering of He atoms from monatomic steps, on an otherwise perfect Pt(111) crystalline surface, indicate the presence of an attractive well in the vicinity of the step.

Theoretical techniques for describing laser-stimulated surface processes in a vacuum and at a gas-surface interface are presented. For adspecies-surface systems, the laser excitation of vibrational degrees of freedom is considered, and quantum-mechanical and classical models and also an “almost first-principles” treatment of the competition between multiphoton absorption and multiphonon relaxation are discussed. The laser excitation of electronic degrees of freedom is considered with respect to surface states of semiconductors and metals, for the predissociation of diatomic adspecies on metal substrates, for ionization, and for resonance fluorescence of a gaseous atom near a metal. In connection with gas-surface interactions, the influence of laser radiation on diffraction patterns and energy transfer in atom-surface scattering is explored. Collisional ionization and ion neutralization in the presence of laser radiation are discussed. The roles of partial pressure and surface coverage in laser-stimulated surface processes are analyzed. Finally, some ideas on surface waves and annealing are presented.

In this review we cover recent advances in the theory of the selective adsorption phenomenon that appears in light atom/molecule scattering off solid surfaces. Due to the universal van der Waals attractive interaction incoming gas particles can get trapped by the surface, this giving rise to the formation of quasi-bound states or resonances. The knowledge of the position and width of these resonances provides relevant direct information about the nature of the gas–surface interaction as well as about the evaporation and desorption mechanisms. This information can be obtained by means of a plethora of theoretical methods developed in both the energy and time domains, which we analyze and discuss here in detail. In particular, special emphasis is given to close-coupling, wave-packet, and trajectory-based formalisms. Furthermore, a novel description of selective adsorption resonances from a stochastic quantum perspective within the density matrix and Langevin formalisms, when correlations and fluctuations of the surface (considered as a thermal bath) are taken into account, is also proposed and discussed.

A model is proposed for vibrational deexcitation of diatomic molecules by collisions with a solid surface. The expressions obtained are analyzed to yield insight into the collision dynamics and used to predict the rotational and translational energy distributions, and other properties of interest. The method is developed in the approximation of a stationary surface, and is closely related to a recent model for vibrational relaxation in atom–molecule collisions. From considerations based on the scales of the relevant energy spacings and coupling strengths applied to the vibrational, rotational, and diffraction states involved, the scattering equations are greatly simplified by several approximations. For a simple but realistic class of potentials, analytical expressions are obtained for the deactivation probabilities pertaining to all final translational–rotational channels. Using the expressions of the model, a detailed study is made of: (i) The rotational–translational energy distribution produced by the vibrational energy release, and its dependence on system parameters; (ii) isotope and collision‐energy dependence of the deactivation probabilities; (iii) scaling properties of the transition probabilities with regard to ΔJ = J′−J, the change in rotational quantum number. The model is applied numerically to collisions of vibrationally excited H2, D2, T2, HD with a noncorrugated surface over a wide range of energies. The most striking feature of the model results is that a highly dominant fraction of the vibrational energy goes into molecular rotation, the main channel being an almost resonant V–R process in all cases.

A model for the description of thermal attenuation in atom, molecule/surface scattering is presented. It is based on the energy sudden approximation for all degrees of freedom, i.e., phonons, diffraction, and rotation, and leads to a generalized Debye–Waller factor that depends on the rotational transition and is valid for arbitrary interaction potentials. The traditional Debye–Waller factor is recovered for a hard potential. Assuming a Debye frequency spectrum for the phonons we present two model calculations for molecule/surface scattering. In the first case we assume a pairwise interaction between the atoms of the molecule and the surface atoms and observe a temperature dependence of the rotational transition probabilities, which is due to both the rotational energy transfer and the rotational dependence of the Debye–Waller factor. In the second case we model NO/Ag(111) scattering and conclude that a variation of the surface temperature has only a slight influence on the final rotational state distribution which is in accordance with the experimental findings of Auerbach et al. The mean rotational energy transfer shows a slight linear increase with the temperature as recently observed by Kubiak et al.

The rotationally inelastic diffraction of H2 from a corrugated surface is investigated using a mean trajectory model. The center of mass motion is treated using Gaussian wave packets, which propagate on a rotationally averaged potential. This trajectory in turn drives the rotational transitions. The method is nonperturbative and allows for changes in mj, the rotational orientation, and agrees well with recent close coupling calculations. A connection is also made with the recent semiclassical trajectory work of DePristo. The effects of the attractive well depth on rotational excitation and diffraction are considered.

Performing the classical limit of the coordinate‐representation‐sudden approximation of Gerber et al. [J. Chem. Phys. 73, 4397 (1980)], we discuss rainbow effects in diatom‐surface scattering. Under special conditions, which are stated in this article, rainbows can be classified into surface rainbows and rotational rainbows. The latter are expected to be common features of diatom‐surface scattering provided: (i) the collision is impulsive and (ii) many rotational states are energetically open. Simple analytic expressions for the rainbow states are derived using a repulsive model potential and the dependence on collision and potential parameters is discussed. The predictions are all substantiated by calculations performed within the sudden approximation and using this model potential.

The scattering of He atoms from a CO molecule adsorbed on a Pt surface is studied theoretically by methods that include: (1) Numerically exact solutions of the time‐dependent Schrödinger equation for the scattered wavepacket; (2) The sudden approximation; (3) Classical trajectories. The methods are used to obtain detailed insight into the collision dynamics, and to predict and understand interesting features in the angular intensity distribution of the scattered atoms. The analysis and interpretation of the exact quantum results is facilitated by calculations of the probability current density of the scattered particles. Some of the main results are: (i) The angular intensity distribution exhibits nonspecular maxima of two types: Several of the peaks are rainbow effects induced by the adsorbate, while others (at angles nearer to the specular) are Fraunhofer diffraction interferences. Both types of peaks contain useful, largely complementary, information on adsorbate geometry and on the He/adsorbate interaction. (ii) The angular intensity distribution is quantitatively sensitive to the adsorbate distance from the surface, suggesting possible determination of that distance from experimental data. (iii) The corrugation due to the adsorbate leads to scattering resonances associated with temporary trapping of the scattered atom at the defect site. This is a new effect of potential importance for experimental studies of atom/defect interactions. The results obtained here suggest that He scattering from isolated adsorbates exhibits distinct, substantial effects, measurement of which should yield very useful data on the adsorbates and on their interactions with gas‐phase atoms.

The dynamical Lie algebraic method is used for the description of statistical mechanics of rotationally inelastic molecule–surface scattering. A main advantage of this method is that it can not only give the expression for evolution operator in terms of the group parameters, but also provide the expression for the density operator for a given system. The group parameters may then be determined by solving a set of coupled nonlinear differential equations. Thus, the expressions of the statistical average values of the translational-to-rotational energy transfer, the interaction potential, and their dependence on the main dynamic variables for the system are derived in terms of the density operator formalism in statistical mechanics. The method is applied to the scattering of NO molecules from a static, flat Ag(111) surface to illustrate its general procedure. The results demonstrate that the dynamical Lie algebraic method can be useful for describing statistical dynamics of gas–surface scattering. © 2000 American Institute of Physics.

It was recently proposed that the sudden approximation should be a powerful tool for the calculation of the angular intensity distribution in high‐energy atom scattering from disordered surfaces. In the present study the sudden approximation is applied to scattering from one‐ and two‐dimensional models of: (1) Isolated adsorbed impurities on crystalline surfaces (Ar on Cu); (2) Mixed overlayers on an underlying surface (Xe+Ar mixtures on a smooth surface). The results are tested against numerically exact quantum‐mechanical wave packet calculations. Except for very low collision energies, the sudden approximation gives results of excellent quantitative accuracy for both types of noncrystalline surfaces. At low energies, several features of the intensity distribution are not produced correctly by the sudden: These are found to be due mainly to double collision effects. The accuracy and validity range of the method are discussed in the light of the results obtained in the test calculations.

Rotational and reorientational transitions in molecular collisions with solid surfaces are investigated by a model based on a sudden approximation with respect to both the rotational and the diffraction states that play a role in the scattering. The approximation developed leads to computationally simple expressions and provides detailed insight into the physical properties of the processes involved. A detailed quantitative study is made of the rotational state distribution produced by the collision, the variation of rotational excitation probabilities with the scattering angle, and related questions. A number of factorizations, sum‐rule, and scaling properties are predicted for ‖ Sjmj,00;j′m′j′,mn @qL ‖2, the transition probability between the initial (jmj) and the final (j′m′j′) rotational states for scattering into the (mn) diffraction channel. The strongest sum rules and scaling laws are obtained using additional approximations beyond the sudden decoupling. Among the latter results: (1) The j,j′ dependence of ‖ Sj0,00;j’m’0,mn ‖2 is determined entirely by the difference variable Δj=j′−j. (2) The diffractive intensity distribution summed over all final rotational states is the same as that obtained for a mass‐equivalent atom (with an interaction that is the orientation‐averaged molecule–surface potential). (3) The rotational state distribution, summed over all diffraction states, equals that calculated from a corresponding flat surface. (4) All rotational transition probabilities for the (m,n) diffraction spot can be obtained from the diffraction–rotational transition probabilities in the (m,0) and (n,0) diffraction spots. The above and other properties are tested numerically in the framework of the full sudden approximation for a model of H2/LiF(001) in the energy range 0.5–0.9 eV. They are found to hold to excellent accuracy. Systematics of the results with regard to variation of the surface corrugation parameter are noted.

Molecular beam techniques were employed to investigate the scattering of nearly monoenergetic He and Ne atoms of thermal energies from the clean (001) face of LiF at low temperature. By using good angular resolution and sensitivity, diffraction peaks were resolved over a wide range of incident and final angles. Taking into account the velocity distribution within the supersonic beam, the diffraction probabilities of the elastic peaks are extracted from the measured scattered intensity. A comparison with elastic diffraction theories is given, particularly with the quantum theory of surface rainbow developed by Levi and coworkers. This theory is found to be in satisfactory agreement with the experimental results. Information on the surface roughness is obtained: to the exploring incident atoms the surface appears as a simple sinusoidal hard wall in two dimensions, with a corrugation amplitude of about 0.3 Å. Qualitative information on the attractive part of the atom-surface potential is also obtained. Owing to the low surface temperature, the inelastic scattering is relatively weak. From the measurement (for helium), or from the evaluation (for neon) of the total elastic scattered intensity observed at normal incidence, a Debye-Waller factor of 0.25 for He, and of 0.037 for Ne at 80 K is derived. The surprisingly large value found for NeLiF cannot be explained by the conventional theory, and is suggested to be connected with the long collision time. The inelastic structures observed in the scattering of neon, particularly at large outgoing angles, are discussed.

The elastic scattering of an atom from a solid surface is considered in the close-coupling formulation. The resulting sets of coupled differential equations are solved by a technique developed by Gordon and previously applied to atom-molecule inelastic scattering. For the example of He+LiF, utilizing the Lennard-Jones, Devonshire model potential, it is shown that basis sets consisting of only low order diffraction channels can be seriously in error. Results are presented using typically 29 coupled channels but as many as 41 channels are included for individual cases. Several calculations including only five coupled channels are also presented. These results are shown to be very useful in interpreting the causes of various effects seen in the 29 channel computations even though providing a poor over-all approximation to the 29 channel results. Various diffraction processes (with emphasis on the specular intensity) are studied as functions of the coupling between the specular and diffracted beams, the incident direction, and for the two isotopes He3, He4. The direct numerical approach used here automatically includes the effect of bound state resonances. These are seen to have a significant effect on the specular intensity. This is found to be the usual behavior at the threshold for the first nonspecular channel, but can also occur when there are already many channels open. Many of the trends observed in experimental data are also observed in the present results.

Methods for the integration of initial value problems for the ordinary differential equation dy/dx = f(x,y) which are a combination of one step procedures (e.g. Runge-Kutta) and multistep procedures (e.g., Adams' method) are discussed. A generalization of a theorem from Henrici [3] proves that these methods converge under suitable conditions of stability and consistency. This, incidentally, is also a proof that predictor-corrector methods using a finite number of iterations converge. Four specific methods of order 4, 6, 8 and 10 have been found. Numerical comparisons of the first three of these have been made with Adams', Nordsieck's and the Runge-Kutta methods.

Bound state resonances related to the band structure of adsorbed atoms and their usefulness for determining the periodic components of atom-solid interaction potential are theoretically investigated. A variety of specular intensity patterns associated with bound state resonances near the Brillouin zone boundaries are exhibited. The (10) and (11̄) bound state resonances give rise to two split specular minima with the splitting depending essentially on v10 for a fixed beam energy; however, the detailed features are dependent on other periodic components. For incidence along a crystal symmetry direction, symmetrization of basis states not only makes numerical computation very efficient, but also implies that there is only one specular minimum for a pair of bound states which are equivalent by symmetry. The (01) and (10) resonances along and near the x = y direction are presented to illustrate the symmetrization principle. The depth of one of the specular minima decreases and finally vanishes as the symmetry direction is approached. The single specular minimum corresponds to a resonance with the bound state which is a symmetric linear combination of (01) and (10) states in a potential well of v0 + V11. As expected, the shift in positions of specular minima caused by the periodic surface potential increases with decreasing beam energy.

The recent quantum mechanical theory of scattering of atoms by solid surfaces, proposed by Cabrera, Celli, Goodman and Manson (CCGM), is applied to calculations of the elastic scattering of 3He and 4He by LiF. The model used is a very special case of the CCGM theory, and is chosen because it allows straightforward calculations and because of its ease of interpretation. Because of uncertainties in the relevant physical parameters in the model and the effects of the simplifications and assumptions made therein, the calculations have only qualitative significance. Such calculations are useful, however, as they provide a basis for (a) discussion of the physical processes involved in elastic scattering, (b) preliminary qualitative correlation of existing experimental data and (c) specific suggestions for future experiments. Existing experimental data support the qualitative predictions of the model.

A quantum mechanical theory of the scattering of atoms by solid surfaces is presented. The theory is applied to a detailed discussion of elastic scattering (diffraction) processes, and the extension to inelastic scattering (phonon exchange) processes is discussed briefly. A great advantage of the theory is that scattering intensities of any size are easily handled; the moduli of the scattering matrix elements are not restricted to be small. If the results are expanded to lowest order in these moduli, then the “first order distorted wave Born approximation” is recovered. An example of the results obtained is that the intensity of the specularly scattered beam is by no means always larger than other diffracted intensities; this result is in agreement with experiments, and is a decided improvement over the usual first order treatments.

A generalization of Takayanagi's fully quantum mechanical strong coupling approximation for rotationally inelastic collisions, valid for small energy exchange, is discussed and tested numerically on a model problem. The method is computationally very simple, and the results are most encouraging.

Extensive computations on the scattering of atoms from a hard corrugated surface model (HCS) under the Rayleigh hypothesis by using the GR numerical method are presented. The computational applicability, the limits of convergence, as well as the sensitivity of the method are studied as a function of the scattering and corrugation parameters. We also calculate a set of curves, called topographic curves, of constant diffraction probabilities versus the incident angle and the corrugation parameters for a square unit cell. The effect of the strength and number of the Fourier coefficients of the corrugation function are analyzed obtaining rainbow disappearance when these coefficients increase in strength and number. A tentative analysis of the crystallography of the LiF(001) surface by using monoenergetic thermal helium beams is given in terms of the hard sphere and ’’contact’’ hard sphere model proposed here. The agreement with the experimental data on He/LiF(001) is excellent. We also discuss an empirical method for determining Debye–Waller factors by fitting the calculated peaks to the observed ones. We calculate the mean square displacement of the atoms vibrating at the surface for different directions on the He/LiF (001) system. These displacements and consequently the Debye–Waller factors are different than if the incident particles were electrons. Finally, several proposals for future experimental work are made.

A new method, of very general applicability and very easily programmed for an electronic computer, is proposed for the numerical integration of functions of many independent variables. This new method renders obsolete, in most applications, the commonly used Monte Carlo procedure and the more recent, original method of Haselgrove. In the new scheme the sample points are distributed systematically rather than at random and the ensemble of points forms a unique, closed, symmetrical pattern, which effectively fills the space of the multidimensional integration. The paper contains an extensive statistical‐analytic treatment of the error characteristics of the new method, one that enables advance quantitative estimation of upper limits of error in the integration of various types of functions. For continuous functions with bounded first derivatives, the error is shown ultimately to disappear at least as rapidly as the inverse square of the number of sample points; moreover, for runs of practical length the error limits with the new scheme are smaller—by a factor ranging from 2 to perhaps 104 or more—than those of any previous general procedure. The method employs certain rational constants which govern the arrangement of sample points. Tables of such constants, suitably optimized, which will permit the integration of functions with up to 12 independent variables, are provided along with a discussion of a method by which such constants may be obtained.

A systematic method is discussed for decoupling the internal angular momentum of molecules involved in a collision from their relative angular momentum. This leads to a large class of rotational approximations of varying degrees of complexity and accuracy. These approximations may be used directly for computing rotational transitions or they may be used for reducing the rotational complexity involved in accurate vibrational calculations. It is shown how this approach may be used to study the infinite−order sudden approximation and how that approximation may be extended to more complex potentials. It is shown also how one may use results of the jz−conserving approximation to obtain more complete information on the scattering matrix. The present approach may be used to deduce new angular momentum decoupling approximations and analyze such approximations arrived at through other considerations.

A semiclassical formalism is presented which predicts specular atomic scattering from close‐packed surfaces and either ’’rainbow’’ or diffractive scattering from strongly periodic surfaces. The intensities are a series of δ functions at the Laue conditions modulated by the product of the classical intensities and a structure factor accounting for interference among multiple trajectories within a unit cell. Narrow energy distributions in the incident beam and/or long wavelengths result in well‐resolved diffracted beams, while broad distributions and/or small wavelengths result in coalesced beams which display only the ’’rainbowlike’’ intensity envelope. A simple elastic model using a 6–9 potential has been used to predict He and Ne scattering from W(112). For helium the peak positions and relative peak heights agree with the measured scattering essentially within experimental reproducibility, but the peak widths calculated with this elastic model are too narrow and the intensity maxima too large. Neon calculations reproduce qualitatively the rainbow patterns observed in the experiments but neither the intensities nor the positions of the rainbow features are correct, indicating the existence of large inelastic effects.

The semiclassical theory of gas‐solid‐surface collisions is applied to the diffractive He☒LiF system. It is shown that uniform semiclassical results are in good agreement with high quality quantum‐mechanical results. The present study suggests that classically forbidden contributions should not always be ignored when both classically allowed and forbidden contributions are present. The experimental results of O'Keefe et al. are examined.

A quantum theory of elastic scattering of atoms from crystal surfaces is presented, based on a hard corrugated surface model. It is shown in detail how the rainbow effect arises and determines the diffraction probabilities, such a rainbow effect being the quantum analogon of McClure's classical rainbow. Further topics considered are the influence of a potential well and the reasons why diffraction hardly occurs from metal surfaces. The basis for a possible extension to inelastic scattering is sketched.

The elastic scattering of atoms from solid surfaces is examined within the semiclassical framework. Explicit expression for diffraction intensities are obtained which utilize classical trajectory information as the computational device. Effect of lattice disorder are examined.The results of Beeby and Weinberg are considered.An alternative method for extracting the surface atom displacements and the attractive well depth of the atom-surface interaction is discussed.

The elastic scattering of low-energy light atoms from a perfect crystalline surface is studied by an iterative integration scheme using the Green function. The atom-solid interaction is represented by the often used Morse type surface potential. A varying number of closed and open channels is included in the calculation, according to necessicity. For a beam incident along the cyrstal symmetry directions, a scheme to utilize the symmetry condition for efficient computation is proposed. The diffraction intensities at a bound state resonance (selective adsorption) are calculated by properly selecting a reference potential for the calculation of the Green function. The calculations yield the resonant diffraction intensity patterns in agreement with previous calculations using a different numerical technique and with the experimental observations for the HeLiF and HeNaF systems. A calculation including 69 allowed diffracted beams (open channels) for the HeLiF system at normal incidence is also presented and comparison with experimental results is made to estimate the periodic potential parameter.