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.     

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.     

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.     

Debye-Waller factor in atom-surface scattering: Elastic and inelastic channels

Explicit expressions for the Debye-Waller factor for the elastic and one-phonon channels are presented to lowest order in the phonon displacement, using a hard wall model to represent the atom-surface interaction. The periodicity of the crystal is accounted for; thus we explicitly generalize to all elastic channels the reflectance result found by Garcia et al. within the plane-surface model, and we include the contribution of the umklapp processes to the inelastic channels. We show how for high incident energy of the atom all Debye-Waller factors reduce to the standard result.     

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.     

Theory of the sticking coefficient for atom-surface scattering

A calculation of the sticking coefficient is presented for a one-dimensional system with harmonic interactions. The physical basis for the results is discussed and it is shown how general conclusions may be drawn for three-dimensional systems.     

A semiclassical treatment of atom-surface scattering: He + LiF(001)

The study of the diffraction of atoms by periodic surfaces is considered within a semiclassical framework. The semiclassical theory for atom-surface scattering is briefly reviewed, and the results are applied to HeLiF(001) scattering. Several model potentials for the atom-surface interaction are discussed, and some of the effects of these potentials on the classical trajectories are noted. The model potential which best describes the experimental results is found to approach a corrugated hard wall; the long range attractive part of the potential is found to be of secondary importance for incident angles θ_i < 50°.     

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.     

Theory of inelastic scattering from magnetic impurities

We use the numerical renormalization group method to calculate the single-particle matrix elements T of the many-body T matrix of the conduction electrons scattered by a magnetic impurity at T=0 temperature. Since T determines both the total and the elastic, spin-diagonal scattering cross sections, we are able to compute the full energy, spin, and magnetic field dependence of the inelastic scattering cross section sigma(inel)(omega). We find an almost linear frequency dependence of sigma(inel)(omega) below the Kondo temperature T(K), which crosses over to a omega(2) behavior only at extremely low energies. Our method can be generalized to other quantum impurity models.     

Theory of indirect resonant inelastic X-ray scattering

We express the cross section for indirect resonant inelastic X-ray scattering in terms of an intrinsic dynamic correlation function of the system that is studied with this technique. The cross section is a linear combination of the charge response function and the dynamic longitudinal spin density correlation function. This result is asymptotically exact for both strong and weak local core-hole potentials. We show that one can change the relative charge and spin contribution to the inelastic spectral weight by varying the incident photon energy.     

Inelastic atom-surface scattering: A comparison of classical and quantum treatments

The validity of the classical approximation is investigated for the calculation of the energy transfer and sticking coefficient of particles scattered by harmonic solids. If in zeroth order the static approximation for the force between particle and solid is valid, the energy transfer can be calculated classically, if the WKB approximation for the matrix element of the force holds. This is the case except at low impact energies. The characteristic energy below which quantum effects become important depends strongly on the long range behavior of the force.For the sticking coefficient one has to distinguish quantum effects of particle and solid. Quantum effects for the particle can again be neglected if the WKB approximation for the matrix elements is valid. The solid, however, can be treated classically only, if the number of excited phonons is large compared to unity. This requires energy transfers large compared to the Debye energy of the solid.Extract from doctoral thesis of R.S. submitted to Fakultät für Physik, Techn. Univ. München, 1979. Work supported in part by DFG, Sonderforschungsbereich 128     

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.     

Adiabatic treatment of inelastic deuteron-nucleus scattering

Inelastic scattering of 80 MeV deuterons from ⁵⁸Ni and ¹²⁰Sn is analyzed in terms of an adiabatic coupled-channels Born approximation (ACCBA) method. The ACCBA method is a practical means to treat deuteron plus nucleus as a three-body system. Inclusion of the breakup modes significantly reduces the inelastic cross section. The significance of considering both the s-wave and d-wave breakup modes is pointed out.     

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.     

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.     

Theory of inelastic neutron scattering from molecular fluids

The time-dependent angular pair correlation function is discussed and its use in the analysis of inelastic neutron scattering experiments from polyatomic fluids is described, including both the coherent and incoherent spectra. The set of formal results given here permits a systematic interpretation of neutron inelastic scattering spectra on simple molecular liquids. Neutron spectra second moments are reviewed, and a new result for the fourth moment is given for the incoherent spectrum. Numerical results for the moments are obtained. The fourth moment depends on the mean squared torque and the mean squared force acting on a molecule in the fluid, and may provide a means for studying intermolecular forces in dense fluids. In addition, a method of calculating the correlation function for weak anisotropic forces is outlined.     

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.