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Coupled-channel calculations and the accuracy of the sudden approximation for atom---surface scattering
Abstract
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.
... 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]. It is particularly noteworthy that the SA yielding an inverted potential of remarkable accuracy [4] for simulated Ne/W(110) data. ...
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.
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.
These lectures describe the application of the quantum theory of scattering to atom-surface interaction processes. The treatment will be introductory, self-contained and fairly inclusive. A complete list of references is not given, however; these can be found in recent reviews [1–6].
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.
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.
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.
We present a detailed investigation of a new effect in molecule/surface scattering, rotationally mediated selective adsorption, which has been first observed by Cowin et al. (J. Chem. Phys. 75 (1981) 1033) in rotationally inelastic HD/Pt (111) scattering. It is due to resonance between an asymptotically closed rotational channel and a vibrational bound state of the molecule/surface system. Exact close-coupling and diffractionally sudden calculations are performed for HD scattered from a rigid, flat and weakly corrugated surface. Both variation of the collision energy and variation of the incident angle are considered. In the latter case we include the averaging over the collision energy of the initial beam and compare qualitatively our results with the experiment. Taking into account the resonances below they j = 0 → 1 threshold we are able to explain all of the experimentally observed resonances and to assign them with the appropriate rotational-vibrational quantum numbers. The largest deviation between experimental and theoretical resonance angles is ~ 1.6° at a collision energy of E = 109 meV. An interesting interference effect is found in the case of weak surface corrugation. Although the diffractionally sudden approximation generally gives only poor results, it also reproduces this effect and provides a simple explanation of it in terms of the Breit-Wigner representation of the S-matrix.
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.
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.
Classical theory of collisions is formulated which also includes the uncertainty principle. The theory is used for calculating transition probabilities in the collinear He-H2 collision, and the results are compared with quantum calculations. Very good agreement is found for all the parameters investigated.
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.
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.
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.
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.
The angular intensity distribution of atomic beams scattered from partially disordered solid surfaces is investigated. Quantitative relations are established between correlation functions that describe the structural disorder, and several features of the scattered intensity distribution. 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. Simple relations are obtained between the structural correlation functions and the scattering at near‐specular angles, at extreme off‐specular angles, and for directions close to that at which the surface scattering rainbow appears for the corresponding ordered lattice system. Detailed expressions for the case of step disorder are presented. The results suggest that molecular beam scattering should be a powerful probe of structural properties of disordered surfaces.
The interaction potential for the elastic diffractive scattering of low‐energy He atoms from the highly corrugated LiF(001) crystal surface is derived from semi ab initio pair potentials in the framework of the recently developed Tang–Toennies potential model [J. Chem. Phys. 80, 3726 (1984)]. In addition to the sum of all He atom‐crystal ion two‐body potentials the induced dipole potential caused by the electric field of the ion lattice is taken into account, leaving only one free parameter, the C6 dispersion constant of the He–F− interaction. By simple adjustment of this parameter, it is possible to fit all of the experimental bound states of the atom‐surface potential well, to within experimental error. Diffraction probabilities calculated by the close coupling method with this potential are shown to be in good agreement with the available experimental results. Two different empirical potential models based on the Morse potential are also investigated, but do not provide as good a description of the bound states and diffraction intensities.
It is shown that atom–surface interaction potentials can be recovered from the diffraction peak intensities measured in beam scattering experiments by a direct, simple inversion method. The potential is determined uniquely, apart from a term depending only on the coordinate perpendicular to the surface. The method does not assume any parametrized functional form for the potential. The results are obtained for a rigid surface, using the sudden approximation for the scattering dynamics. Application to a test case using simulated data for Ne/W (110) yielded an inverted potential of remarkable accuracy.
A recently developed time-dependent quantum-mechanical Fourier method is employed for the simulation of scattering atoms from surfaces of solids. The method allows exact calculation of scattering intensities, resonance strengths and lifetimes, in systems with a very large number of populated diffraction peaks. The study applies the method to the He/stepped copper surface. The accuracy of the method is checked by comparing to close coupling results for the He/W system.
Classical theory of collisions is cast in a form which also includes the uncertainty principle. This theory is used for analyzing the vibration energy transfer in the collinear collision which approximates the He-H2 system. The results are compared with the quantum calculations and several classical and semiclassical approaches. Very good agreement with quantum theory is found, for all the parameters investigated.
A semiclassical formulation of inelastic atom-surface scattering is presented. This formulation is a mixture of classical S-matrix theory and a classical path model. A Debye-Waller factor enters this theory very naturally as the probability of elastic reflection in the presence of inelastic channels. Because of its importance the Debye-Waller factor is discussed in some detail. Finally, assuming a simplified model of the gas-surface system, the whole scattering problem is solved analytically.
The theoretical formalism of the distributed approximating functions (DAF) is applied to solve accurately 3D-atom-surface scattering problems. Formulated in coordinate space, the DAF approach starts from an entirely new idea: providing a “uniform” approximation everywhere to a wavepacket, and results naturally in a near-local or banded free propagator. The banded Toeplitz structure of the DAF free propagator matrix on a uniform grid makes possible the application of the most efficient codes in the matrix-vector multiplication in evolving the wavefunction of a quantum system in time, and with extremely small memory requirements. The numerical study conducted in this paper demonstrates that the DAF method outperforms the most powerful available FFT method both in CPU time and storage requirements. The DAF approach gives the same accurate results as the FFT does, and, in some cases, yields more accurate results.
A new propagation scheme for the time dependent Schrödinger equation is based on a Chebychev polynomial expansion of the evolution operator Û=exp(−i Ĥt). Combined with the Fourier method for calculating the Hamiltonian operation the scheme is not only extremely accurate but is up to six times more efficient than the presently used second order differencing propagation scheme.
We have developed a new method for solving quantum dynamical scattering problems, using the time-independent Schrödinger equation (TISE), based on a novel method to generalize a "one-way" quantum mechanical wave equation, impose correct boundary conditions, and eliminate exponentially growing closed channel solutions. The approach is readily parallelized to achieve approximate N(2) scaling, where N is the number of coupled equations. The full two-way nature of the TISE is included while propagating the wave function in the scattering variable and the full S-matrix is obtained. The new algorithm is based on a "Modified Cayley" operator splitting approach, generalizing earlier work where the method was applied to the time-dependent Schrödinger equation. All scattering variable propagation approaches to solving the TISE involve solving a Helmholtz-type equation, and for more than one degree of freedom, these are notoriously ill-behaved, due to the unavoidable presence of exponentially growing contributions to the numerical solution. Traditionally, the method used to eliminate exponential growth has posed a major obstacle to the full parallelization of such propagation algorithms. We stabilize by using the Feshbach projection operator technique to remove all the nonphysical exponentially growing closed channels, while retaining all of the propagating open channel components, as well as exponentially decaying closed channel components.
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.
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.
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.
It has been shown that a partial decoupling of the coupled channel equations for atom-surface scattering can be obtained by the use of symmetry and this greatly reduces the numerical èffort required to solve the equations. The method is illustrated for the systems Ne/W(110), Ne/LiF(001) in the simplified framework of a recently-proposed sudden approximation.
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 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.
Computer solution of ordinary differential equations
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