Theory of inelastic atom-surface scattering: A three-dimensional treatment

Inelastic scattering of atoms by solid surfaces is treated within the first order distorted wave Born approximation. The solid is treated as a semi-infinite elastic continuum and the phonon excitations of this system are properly taken into account, including the Rayleigh waves. No restriction concerning the scattering angles (in and out) are made. In particular momentum transfers parallel to the surface are taken into account. The contribution of all these effects to the sticking coefficient are considered in detail. It turns out, that the Rayleigh phonons contribute about 25% to the local density of phonon states at the surface and roughly 50% to the sticking coefficient. The three dimensional motion of the scattered particle (in particular the transfer of parallel momentum) increases the sticking coefficient by a factor of about three as compared to one dimensional treatments.     

On the Debye-Waller factor for atom-surface scattering

The Debye-Waller factor in atom-surface scattering is considered. It is found that the equation suggested by Beeby is valid for soft potentials provided that (1) the attractive portion of the potential is stationary, (2) the repulsive portion of the potentials moves but does not distort, and (3) the effect of trajectories that reflect off the attractive well and multiply scatter from the repulsive wall is negligible.     

Theory of inelastic atom-surface scattering: Examples of energy distributions

The theory of the energy distribution of atoms scattered inelastically by solid surfaces which was developed previously is applied to various examples. The dependence of the results on a number of parameters is studied in detail. The importance of many phonon contributions as compared to the validity of first order distorted wave Born approximation is considered in particular. It turns out that low energy He atoms scattered by heavy transition metals provide a good example for which one phonon emission (or absorption) dominates. All other noble gases show appreciable many phonon contributions increasing, of course, with increasing mass of the noble gas and temperature of the solid. For heavy noble gases such as Kr and Xe the energy distribution approaches a gaussian, the width of which is due to the thermal and zero-point motion of the lattice. This width is quite large and thus probably masks most of the fine structure of the energy distribution occuring in classical trajectory calculations. We have also tried to apply the theory to light diatomic molecules. Although the results are less certain, partly because of the neglect of the internal motion of the molecules and partly because of uncertainties in the interaction parameters, one probably can expect appreciable many phonon effects already for H₂ and, of course, more so for N₂ and O₂. Recent experimental results on the Debye-Waller factor of Ne/Cu can be reproduced with reasonable potential parameters.     

Model calculations of atom-surface scattering

The elastic scattering of light mass, thermal-energy atoms from simple surfaces is investigated. The surface is represented by the model of a single planar square array of hard spheres. The effect of the surface potential well is treated semiclassically by simply shifting the energy of the incident atom ; furthermore a constant imaginary term is added to the energy to account for inelastic scattering and adsorption. As in the multiple scattering formalism of LEED the total scattering matrix of the lattice is expanded in terms of the individual gas atom-surface atom t-matrices. Propagation of the incident atom on the surface is described in terms of a one particle Green's function propagator with complex energy. The terms in the multiple scattering series are summed to all orders, by using standard matrix inversion techniques. The size of the matrix to be inverted limits to ten the total number of phase shifts that are included in the calculation. Thermal effects are included through angle dependent Debye-Waller factors.Model calculations have been performed to study the intensity of the specular and the diffracted beams as a function of the angles of incidence. The importance of surface temperature (introduced by the Debye-Waller factors), the incident energy and the depth of the potential well of the gas-surface interaction are discussed. The main feature of the results is the decrease of the intensity of the specular beam in going from glancing incidence to normal incidence and the presence of structure due to the appearance and disappearance of diffracted beams across the surface. The azimuthal behavior of the specular beam is in agreement with experimental observations.     

Resonance theory in atom-surface scattering

We study the problem of analytic extension of the resolvent for Hamiltonians arising in scattering of atoms by a quantum surface. We prove that the resolvent extends holomorphically to some regions of the lower half plane with isolated singularities called Landau resonances which are branch points of the resolvent. We study also the effect of impurities on the singularities of the resolvent and show that the presence of impurities adds poles to the Landau resonances.     

Application of a one-dimensional classical model of atom-oscillator scattering to atom-surface accommodation coefficient theory

A simplified model is presented for classical-mechanical calculation of the energy accommodation coefficient in gas-surface scattering: conservation of the tangential component of momentum of a gas atom is assumed, allowing restriction to a consideration of the changes in only the normal component of momentum, and the solid is represented by a single atom joined by a harmonic spring to a rigid wall; the gas atom-solid atom interaction is modelled by a (different) harmonic spring, the force-constant of which depends on the gas temperature. The model was devised to help point a way to solving some of the outstanding problems in more realistic (lattice) models: one major problem is the existence or non-existence of thermal equilibrium of a gas and a surface if their temperatures are equal, and another is the handling of the phenomenon of trapping, or sticking, of a gas atom at a surface. Some light is shed on both of these (and on some other) problems. Wherever possible, comparison is made with the best available experimental data.     

Theory of inelastic atom-surface scattering: Average energy loss and energy distribution

Theoretical results for the energy distributionP() of atoms scattered by solid surfaces are presented. An exact expression for the first moment (the average energy loss) ofP() is derived involving the force-force correlation of the scattering particle and the susceptibility of the lattice. An approximate result forP() is derived in which the same two quantities occur. The forces are treated statically (i.e. neglecting the response of the lattice) and semiclassically (i.e. neglecting their noncommutativity). Quantum effects of the lattice are taken into account properly. No approximations are made concerning the number of excited phonons. An expression for the reduction of elastic scattering due to inelastic processes (generalized Debye-Waller factor) is found containing the dynamic and finite size effects discussed in the literature.     

Debye-Waller effects in atom-surface scattering

The temperature sensitivity of helium atom reflection from NaF ranges from slight in the quantum regime (long wavelength, near-glancing incidence) to considerable in the classical regime (short wavelength, near-normal incidence). The latter is describable by the Debye-Waller theory only over a limited range of angles. In this range, no Beeby well-depth correction is required. The surface Debye temperature of the NaF is found to be 425 ± 20 K.     

Elementary resonance processes in atom-surface scattering

In atom-surface scattering, the resonance phenomenon is being very actively investigated. In this work, and within the closecoupling formalism, a systematic classification of the elementary resonance processes is proposed according to the different selective adsorption/desorption mechanisms which are thought to be the most important. In doing so, new mechanisms can be theoretically predicted and experimentally observed. As an extension, the processes of capture, trapping and sticking are also considered, and some operational definitions to calculate their respective probabilities are proposed.     

Multipole contributions to atom-surface scattering potential

We calculate multipole corrections to the standard van der Waals (dipole) potential in atom-surface scattering. The quadru- and octupoles give at most a 15% deeper potential at relevant physisorption distances, and the degree of metal screening is shown to have very little effect on this conclusion.     

The extinction theorem in atom-surface scattering

We present a generalization of the extinction theorem of optics to treat atom-surface scattering. The method, which is valid for a wide class of atom-surface potentials, is not limited to the case of small roughness. Its connections with Rayleigh approximation are discussed.     

Phonon coupling in inelastic atom-surface scattering

Inelastic scattering of a thermal He beam from a LiF surface has been observed in terms of velocity and intensity distributions as a function of scattering angle. The results allow a correlation with theoretical predictions assuming single Rayleigh phonon coupling for both phonon creation and annihilation processes. A marked decrease of inelastic intensities with momentum or energy transfer is demonstrated, which depends strongly on the temperature of the scattering surface.     

Interpretation of selective adsorption in atom-surface scattering

The elastic theory for resonant atom-surface scattering is shown to explain simply the occurrence of both minima and maxima at resonance and the strength and shape of the absorption lines, on the assumption that the repulsive part of the potential is of the form V(z?ζ), where ζ is the surface corrugation and V can be regarded as a hard wall at the locus of turning points. Explicit formulae are obtained in the semiclassical limit. For a simple sinusoidal corrugatrion with rectangular symmetry, the signature of an isolated resonance is determined unambiguously by the parity of ¦m¦+¦m' ? m¦?¦m'¦+¦n¦+¦n' ?n¦$\text{}\text{?}$s|n'¦, where (m,n) is the resonant vector and (m',n') the scattering vector, in reciprocal lattice units. The predicted signatures of isolated resonances agree with experiments and more elaborate calculations for the systems He/LiF and He/Graphite. Calculated splittings in the surface band structure also show reasonably good agreement. An alternative formulation of resonant scattering, as suggested by Wolfe and Weare, can be based in an appropriate sorting of successive orders of distorted wave perturbation theory. For small corrugations there is good correspondence between the two approaches.     

An exact iterative solution of the atom-surface scattering problem for realistic potentials

A general method, based on high order distorted wave perturbation theory, is developed to obtain exact reflection intensities for low energy atoms scattered by solid surfaces. Resonance shapes are calculated using projection techniques to obtain uniform convergence. Important differences with the hard corrugated wall model arise when the calculations for a corrugated Morse potential are compared with experiment for helium scattered by a Cu(110) surface.     

Quantum theory of atom-surface scattering: Debye-Waller factor

The Debye-Waller factor, introduced historically for X-rays, was used later for electrons, neutrons, and atoms as well. In this process of extension, however, the assumptions on which the Debye-Waller theory rested became more and more questionable until in the case of atoms (whose scattering from surfaces is both strong and slow) serious modifications are necessary. In the present article four models are discussed in order. In Model 1 a fast atom impinges on a surface whose atoms all vibrate deviating from their equilibrium positions by the same vector displacement ?. In Model 2 again the impinging atom is fast, but the atoms in the surface vibrate incoherently rather than coherently. It is shown that both Models 1 and 2 yield the conventional Debye-Waller result in the infinite crystal atom mass limit (for Model 2 Einstein oscillators have also to be assumed) and it is also shown how corrections to this result can be built. Turning then to slow impinging atoms, in Model 3 a slow atom impinges on a hard crystal surface, interacting with the rapidly varying potential of the vibrating solid. Model 3 is discussed in detail and it is shown that the Debye-Waller exponent can be written in terms of a time integral of the product of two correlations: the force correlation and the displacement correlation. The result is a dramatic increase of diffraction of relatively heavy atoms (with respect to the conventional theory). Finally, in Model 4 the impinging atom is again slow but the crystal is soft rather than hard. This case is more difficult to treat but a preliminary analysis again indicates a dramatic increase of diffraction since the soft solid adjusts itself to the instantaneous atom position leading to elastic scattering. The experimental implications of the present theory, especially for neon scattering from surfaces, are discussed.     

Multiphonon atom-surface scattering from corrugated surfaces: derivation of the inelastic scattering spectrum for diffraction states

A strictly quantum mechanical derivation of the energy and parallel momentum resolved scattering spectrum formula that combines the effects of the diffraction of atoms from corrugated surfaces and multiple inelastic scattering by dispersive phonons is presented. The final result is expressed in the compact and numerically tractable form of a Fourier transform of a cumulant expansion in which each term embodies an interplay between the processes of projectile diffraction and multiphonon scattering to all orders in the respective interaction potentials. The Debye-Waller reduction of the intensities of diffraction peaks is explicitly formulated.