Classical scattering of Atoms by a model surface

A classical model of atom/surface scattering is presented in which the gas-surface interaction consists of a corrugated hard wall plus attractive square well. New analytic solutions for the scattering intensity for monoenergetic beams are given for both one- and two-dimensional surfaces. Explicit formulae for trapping probabilities and rainbow angles are obtained. An oven beam velocity distribution is incorporated into the model, and calculated results are presented both for in-plane and out-of-plane scattering. Comparisons are made with the Ne/LiF data of Smith, O'Keefe, and Palmer and with the classical calculations of McClure. Objections to the classical rainbow scattering interpretation are discussed.     

Cutoffs and shadows in classical scattering of atoms from surfaces

A commonly used model for the scattering of atoms from crystalline surfaces is that of a classical hard sphere interacting with a hard, corrugated surface. This model is simple enough to permit the study of the effect on the scattering of variation of parameters of the model. We examine here how changes in the amplitude and symmetry of the corrugations, the size of the sphere, and the depth of an attractive well near the surface modify the scattering. We show, in particular, how edge effects due to shadowing of parts of the surface and second hits by the sphere distort the classical rainbow structure. We show the results as topological and intensity plots for special cases. The edge effects give new information on the structure of the surface. For suitable real systems, experimental measurement of the angular and energy dependence of the fraction of the scattering that is elastic may show some of the features described.     

Anomalous scattering from dielectric bispheres in the specular direction

Resonant scattering from dielectric bispheres in the specular direction (the so-called specular resonance), previously known only in the microwave range, has been observed at the optical wavelength. Systematic experiments with micrometer-sized dielectric bispheres assembled by micromanipulation, together with rigorous numerical calculations, reveal that this scattering is a precursor of the classical rainbow and is a general phenomenon observed in the wide range of size parameters (>5 for n=1.59) for various refractive indices.     

Simple approach to refraction effects in atom-surface scattering

A simple method of calculating low energy atom-surface diffraction intensities is presented for the case in which the scattering is dominated by a single rainbow pattern. Good agreement is obtained when these calculations are compared with recent results for the scattering of He by a Pt(997) surface.     

Work function changes produced by chemisorption due to site heterogeneity

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.     

Temperature dependence in atom–surface scattering

It is shown that a straightforward measure of the temperature dependence of energy resolved atom–surface scattering spectra measured under classical conditions can be related to the strength of the surface corrugation. Using classical perturbation theory combined with a Langevin bath formalism for describing energy transfer, explicit expressions for the scattering probabilities are obtained for both two-dimensional, in-plane scattering and full three-dimensional scattering. For strong surface corrugations results expressed as analytic closed-form equations for the scattering probability are derived which demonstrate that the temperature dependence of the scattering probability weakens with increasing corrugation strength. The relationship to the inelastic rainbow is briefly discussed.     

Classical theory of atom–surface scattering: The rainbow effect

The scattering of heavy atoms and molecules from surfaces is oftentimes dominated by classical mechanics. A large body of experiments have gathered data on the angular distributions of the scattered species, their energy loss distribution, sticking probability, dependence on surface temperature and more. For many years these phenomena have been considered theoretically in the framework of the “washboard model” in which the interaction of the incident particle with the surface is described in terms of hard wall potentials. Although this class of models has helped in elucidating some of the features it left open many questions such as: true potentials are clearly not hard wall potentials, it does not provide a realistic framework for phonon scattering, and it cannot explain the incident angle and incident energy dependence of rainbow scattering, nor can it provide a consistent theory for sticking. In recent years we have been developing a classical perturbation theory approach which has provided new insight into the dynamics of atom–surface scattering. The theory includes both surface corrugation as well as interaction with surface phonons in terms of harmonic baths which are linearly coupled to the system coordinates. This model has been successful in elucidating many new features of rainbow scattering in terms of frictions and bath fluctuations or noise. It has also given new insight into the origins of asymmetry in atomic scattering from surfaces. New phenomena deduced from the theory include friction induced rainbows, energy loss rainbows, a theory of super-rainbows, and more. In this review we present the classical theory of atom–surface scattering as well as extensions and implications for semiclassical scattering and the further development of a quantum theory of surface scattering. Special emphasis is given to the inversion of scattering data into information on the particle–surface interactions.     

Heat transfer in a spherical turbid medium with conduction and radiation

The heat transfer through a spherical media with conduction and radiation is considered. The medium is considered to be turbid and anisotropically scattering with diffusely reflecting boundaries of constant temperatures. The radiative transfer problem is solved using the Galerkin method. An iterative method is used to solve the nonlinear relation between the radiative transfer equation and the conductive energy equation. Calculations are carried out and compared for a homogeneous, isotropically scattering medium with isothermal, transparent boundaries. The results show good agreement with previous work. Calculations are carried out for inhomogeneous media with isotropic, and forward and backward anisotropic scattering. The boundaries of the media are considered to be isothermal and may be transparent or diffusely reflecting boundaries. The calculations are used to study the effects of the single scattering albedo, the anisotropic scattering parameter, the conduction-radiation parameter and the heat source.     

Der Regenbogeneffekt und Interferenzerscheinungen bei molekularen Stößen

Crossed beam measurements of differential elastic scattering cross sections for collisions of alkali atoms with mercury and xenon atoms as well as with different molecules are presented. Special attention is given to the rainbow effect including supernumary rainbows and to interference effects connected with rainbow scattering. The observation of well resolved primary and supernumary rainbows together with interference patterns make the determination of a three parameter potential model possible. Partial wave numerical calculations of differential cross sections are used to evaluate potential parameters and to discuss the limits of the semiclassical rainbow theory. Elastic scattering between reactive collision partners is investigated too and interpreted in terms of a simple optical model.     

Refractive scattering and the nuclear rainbow in the interaction of12, 13C with12C at 20MeV/N

The elastic and inelastic scattering and the neutron transfer have been measured for the systems¹²C +¹²C and¹³C +¹²C at 20MeV/N up to θ_cm= 60° with theQ3D -spectrometer. The angular distributions of the elastic scattering show an enhanced cross section at angles larger than 40°. It can be identified as refractive scattering with the clear signature of a nuclear rainbow.L-cut-off calculations show that these contributions come fromL-values which are significantly lower than the grazingL-value. The deflection function has a broad minimum in thisL-range which is typical for rainbow scattering. TheS-matrix is decomposed by a phenomenological parametrization into a refractive and a diffractive part. The interference of these amplitudes plays an important role in the rainbow enhancement. The spatial localization of the refractive scattering is deduced from the turning points of the corresponding trajectories; a localization between 2.5 fm and 4 fm is found. Semi-classical calculations with complex trajectories in the single-turning-point approximation show good agreement with the quantummechanical calculations. Refractive contributions are not observed in the inelastic scattering. This can be explained by reducing the strength of the conventional collective form factor in the internal region. In contrast to this the enhancement at large angles is seen in the one-neutron transfer channels where the refractive scattering is dominant. This is the first observation of such contributions to heavy-ion transfer reactions.     

Sound scattering at fluid-fluid rough surface

Extinction theorem was used to deduce the first order scattering cross-section including the double scattering effects for the fluid-fluid rough surface. If the double scattering effects are neglected in the present method, the scattering cross-section agrees with the result obtained by the perturbation method based on Rayleigh hypothesis. Calculations of scattering strength were carried out, and comparisons with the first-order perturbation method based on Rayleigh hypothesis were also done. The results show that double scattering effects are obvious with the increase of the root mean square of surface height and the grazing angle when the valid condition k ₁ h < 1 is satisfied. Supported by the National Basic Research Program of China (Grant No. 2005CB422307)     

Scattering of CO and N(2) molecules by a graphite surface

Measurements of angular distributions for the scattering of well-defined incident beams of CO and N(2) molecules from a graphite surface are presented. The measurements were carried out over a range of graphite surface temperatures from 150 to 400?K and a range of incident translational energies from 275 to over 600?meV. The behavior of the widths, positions and relative intensities of the angular distributions for both CO and N(2) were found to be quite similar. The experimental measurements are discussed in comparison with calculations using a classical mechanical model that describes single collisions with a surface. Based on the behavior of the angular distributions as functions of temperature and incident translational energy, and the agreement between measured data and calculations of the single-collision model, it is concluded that the scattering process is predominantly a single collision with a collective surface for which the effective mass is significantly larger than that of a single carbon atom. This conclusion is consistent with that of earlier experiments for molecular beams of O(2) molecules and Xe atoms scattering from graphite. Further calculations are carried out with the theoretical molecular scattering model in order to predict translational and rotational energy transfers to and from the molecule during scattering events under similar initial conditions.     

Capture by stabilized continuum: Classical and quantum aspects

We review here the results of our investigations concerning chaotic atomic scattering in the presence of a laser field. Particular emphasis is put on the existence of classical stable resonance structures, induced by the intense laser field, which are embedded in the field-free continuum. We show that phase space structures in the vicinity of a resonance island play an important role in the chaotic scattering behavior and form the basis for a mechanism to enhance the lifetimes of the collisional partners. Quantum calculations, based on a wave packet propagation method, show that quantum solutions are strongly influenced by the classical phase space structures. More specifically, a wave packet is found to spread differently in the regular and chaotic regions; in the latter case it spreads exponentially with time until saturation occurs, defining the saturation time. We also investigate the dependence of the spreading rates in both the regular and chaotic regimes. Calculations with an ensemble of classical trajectories are also presented to further illustrate the smoothing effects of varying.     

Double-scatter Mueller matrices for vector electromagnetic scattering from single grooves and lines on a silicon surface

A recently developed numerical method is used to calculate the single- and double-scattered Mueller matrices for scattering of vector-electromagnetic waves from rough surfaces. Calculations are performed for the case of a single groove and a single rib on a silicon surface and results are compared to the published experimental results. The calculated results show good agreement with the experimental results for the groove case and poorer agreement with the rib case, probably due to errors in the experimental construction of the rib. It is found that, for the cases studied here, variations of the sign with scatter angle in individual Mueller matrix elements are associated with the presence of double- (or multiple-)scattered light, as has been found previously for scalar diffraction calculations in Gaussian randomly rough surfaces.     

Selective adsorption resonances at rainbow conditions in the scattering of atoms by stepped surfaces: application to the He/Cu(117) system

For stepped surfaces, dynamics is not parity reversal invariant with respect to the step configuration and the way the atomic beam approaches the surface, downstairs or upstairs scattering, should dramatically change rainbow patterns. We show that only one close-coupling calculation provides us information of the two possible scatterings at once. Furthermore, it is possible to find some incident conditions where surface rainbow is displayed for both configurations. At the same time, if resonances are analysed at rainbow conditions, we could enhance resonant features in the diffraction channel displaying rainbow. An application to the study of diffraction of ⁴He atoms by the Cu(117) surface is presented.     

Regenbogenstreuung an Kalium-Quecksilber

Differential cross section measurements for the scattering of K by Hg at thermal energies are reported. Rainbow angles are compared with literature data and agreement within 2% is found. The data indicate a departure of the rainbow cross section from the Airy integral approximation. The measured variation of the rainbow angle with collision energy cannot safely be explained by those intermolecular potentials for which numerical calculations are available.     

Theory of molecule-surface scattering at thermal and hyperthermal energies

A theoretical model of molecule-surface scattering is developed which includes energy and momentum transfers between the surface and projectile for both translational and rotational motions and internal mode excitation for the projectile molecule. The translation and rotation motions are treated in the classical limit, while a quantum treatment for internal vibrational mode excitation is used. The results of calculations are compared with recent high-precision measurements of the scattering of a beam of C(2)H(2) molecules from a clean, ordered LiF(001) surface at energies of up to nearly 1 eV. The calculated results for angular distributions and rotational excitations are in good agreement with experiment.     

Interband transitions and electronic Raman continuum in systems with strong inelastic scattering: Applications to YBa2Cu3O7-δ

We consider a model for the electronic Raman continuum which takes into account strong inelastic scattering and interband transitions. Calculations are based on four-vertex Raman scattering diagrams (Kawabata formalism) within the RPA for Coulomb interaction and the ladder diagram Bethe-Salpeter equation for the vertex. We apply this method to an analysis of the nature of the electronic Raman continuum in the normal state of the high-T _c superconductor YBa₂Cu₃O₇. In numerical calculations we take into account all the self-energy effects and make simulations for vertex corrections assuming that inelastic scattering is due to electron-phonon interaction. Theab-plane polarized continuum contains a large contribution from interband processes and does not depend strongly on temperature and inelastic scattering strength. The in-plane anisotropy is determined by interband transitions rather than by anisotropy of the Fermi surface. The ZZ continuum contains only weak contribution from interband transitions. It can be crudely described within a single band model with inelastic scattering and is strongly dependent on the relaxation rates of inelastic scattering. The nature of the oxygen-deficiency dependence of the Raman spectra is also commented upon.     

Spectral power determinations of compressibility and density variations in model media and calf liver using ultrasound

A model of scattering is used to relate average differential scattering cross section and power spectra of scattering medium variations. The model expresses the average differential scattering cross section as a sum of the power spectrum of medium compressibility variations, the power spectrum of density variations weighted by the square of the cosine of the scattering angle, and the cross-power spectrum of compressibility and density variations weighted by twice the cosine of the scattering angle. Known values of the average differential scattering cross section at a minimum of three different scattering angles and temporal frequencies corresponding to the same spatial frequency are used to calculate each of the three power spectra. Since noise and statistical fluctuations are present in actual measurements of average differential scattering cross section, the calculations of power spectra are obtained from an overdetermined set of equations to which a solution is found by using a singular value decomposition. Data derived from a model for scattering from a cloud of correlated particles are employed to show the influence of additive noise. Calculations are also made from measurements of scattering from three suspensions of particles that have a different average radius in each suspension but are similarly modeled by scattering from a cloud. Additionally, the calculations are applied to measurements of average differential scattering cross section of calf liver. The results show that determination of the power spectra of scattering medium variations can be made under practical conditions, and also imply that density variations contribute significantly to scattering by calf liver.