Atom scattering as a probe of the surface electron-phonon interaction at conducting surfaces

An atomic projectile colliding with a surface at kinetic energies in the thermal or hyperthermal range interacts with and is reflected by the electronic density well in front of the first layer of target atoms, and it is generally accepted that the repulsive interaction potential is proportional to the density of electrons extending outside the surface. This review develops a complete treatment of the elastic and inelastic scattering of atoms from a conducting surface in which the interaction with the electron density and its vibrations is treated using electron-phonon coupling theory. Starting from the basic principles of formal scattering theory, the elastic and inelastic scattering intensities are developed in a manner that identifies the small overlap region in the surface electron density where the projectile atom is repelled. The effective vibrational displacements of the electron gas, which lead to energy transfer through excitation of phonons, are directly related to the vibrational displacements of the atomic cores in the target crystal via electron-phonon coupling. The effective Debye-Waller factor for atom-surface scattering is developed and related to the mean square displacements of the atomic cores. The complex dependence of the Debye-Waller factor on momentum and energy of the projectile, including the effects of the attractive adsorption well in the interaction potential, are clearly defined. Applying the standard approximations of electron-phonon coupling theory for metals to the distorted wave Born approximation leads to expressions which relate the elastic and inelastic scattering intensities, as well as the Debye-Waller factor, to the well known electron-phonon coupling constant λ. This treatment reproduces the previously obtained result that the intensities for single phonon inelastic peaks in the scattered spectra are proportional to the mode specific mass correction components λ_Q,ν defined by the relationship λ = 〈λ_Q,ν〉. The intensities of elastic diffraction peaks are shown to be a weighted sum over the λ_Q,ν, and the Debye-Waller factor can also be expressed in terms of a similar weighted summation. In the simplest case the Debye-Waller exponent is shown to be proportional to λ and for simple metals, metal overlayers, and other kinds of conducting surfaces values of λ are extracted from available experimental data. This dependence of the elastic and inelastic scattering, and that of the Debye-Waller factor, on the electron-phonon coupling constant λ shows that measurements of elastic and inelastic spectra of atomic scattering are capable of revealing detailed information about the electron-phonon coupling mechanism in the surface electron density.     

Elastic and inelastic diffraction of Mössbauer γ-rays from KCN

Mössbauer γ-ray diffraction was used to discriminate between the elastic and inelastic scattering intensities from the (1 1 1) to (5 5 5) Bragg reflections of a single crystal of KCN. The energy resolution of our experiment was 28 neV. We observe pronounced inelastic peaks at each Bragg point, while the elastic scattering dies out rapidly due to a large Debye-Waller factor. Thus in case of (4 4 4) and (5 5 5) the inelastic scattering is larger in magnitude than the elastic one.     

Attenuation of phonon scattering intensities by point defects

The cross section of the coherent, inelastic neutron scattering is characteristically decreased in the presence of defect-induced lattice distortions. This effect is in analogy to the attenuation of Bragg intensities due to a static Debye-Waller factor. The integrated scattering intensities from transversal acoustic (TA-) phonons of the system NbN_0.014 have been measured and are shown to be attenuated with respect to the scattering intensities from the phonons of a pure Nb crystal. We discuss the obtained results through comparison with various model calculations.     

Path integral methods for inelastic scattering

Corrections to the primitive semi-classical amplitude for multiple inelastic scattering are obtained from a path integral formulation of scattering theory. The path integrals are calculated by making an expansion about a classical orbit describing elastic scattering. Terms are collected to give a series in inverse powers of the reduced mass m of relative motion of the target and projectile. The leading term is the primitive semi-classical amplitude for multiple excitation and explicit formulae are given for the corrections of order $\text{1}\text{m}$. These are calculated in detail for a one-dimensional model. It is shown that some, but not all, of the corrections can be included by evaluating the primitive amplitude with a symmetrized orbit.     

On the neutron diffraction in crystals in the field of a sound wave

Diffraction of neutrons in crystals under influence of a sound wave is considered. The probability of scattering of neutrons at the elastic interaction with the crystal is calculated. On the contrary, scattering of neutrons by an acoustical phonon has inelastic character. The possibility to control the Debye-Waller factor is shown.     

Mössbauer gamma-ray diffraction from the molecular crystal KCN

Mössbauer gamma-ray diffraction was applied to separate the elastic and inelastic scattering intensities from the (200), (400) and (600) Bragg reflections of KCN. The energy resolution of our experiment was 60 neV. The Debye-Waller factor extracted from the elastic data and the thermal diffuse inelastic data both increase towards phase transition, theoretically a logarithmic singularity was predicted.     

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.     

Eikonal description of quantal scattering

Elastic and inelastic quantal scattering is described by a theory in which the contribution of a range of impact parameters to the scattering amplitude is determined by a phase integral (“eikonal”) which is integrated along a real curved “quantal” trajectory. This amplitude reduces to the Glauber expression in the high-energy, forward-angle limit, and to the usual semiclassical amplitude in the classical limit. The formulation can be applied to the study of heavy-ion scattering. The quantal trajectories are investigated analytically for the case of Coulomb scattering. A numerical analysis of elastic ¹⁶O¹⁶O scattering is carried out. The results show appreciable improvement as compared with the Glauber approximation.     

Higgs boson production by W and Z collisions

A general formulation for the WW → H and ZZ → H mechanisms of heavy Higgs boson production in hadron colliders is presented. The known results in the effective W approximation are obtained from the quark-parton model limit of the double inelastic scattering configuration. In addition contributions from the elastic scattering configurations and from α_s-processes of QCD are estimated.     

Simulation of carrier capture in quantum well lasers due to strong inelastic scattering

The effects of strong inelastic scattering on carrier transport over and capture into the quantum wells of quantum well lasers are simulated. In contrast to most semiconductor devices, strong scattering is beneficial to the operation of quantum well lasers. However, such strong inelastic scattering in nanostructures can be expected to produce intermediate degrees of phase coherence, limiting the applicability of both classical models, such as Bethe thermionic emission theory, and commonly used quantum mechanical treatments, such as Fermi's Golden Rule. Two computational approaches are demonstrated for simulating such transport with intermediate degrees of phase coherence. First, absorbing potentials are used within Schrödinger's equation to represent inelastic scattering. This simple approach both exhibits much of the essential physics of such transport and is computationally efficient. Then a more rigorous approach, Schrödinger equation (based) Monte Carlo (SEMC), is demonstrated. While SEMC is rigorously quantum mechanical, the numerical algorithm has more in common with semiclassical Monte Carlo methods than path integral-based quantum Monte Carlo methods. Both of these methods demonstrate nonlinear variations in carrier capture with variations in scattering, and the destruction of quantum resonances for transmission over the quantum well.     

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.     

A connection between Raman intensities and EXAFS Debye-Waller factors in amorphous solids

For tetrahedrally-bonded amorphous systems a connection is derived between the Raman scattering intensity and the Debye-Waller factor in the Extended X-ray Absorption Fine Structure (EXAFS). Specifically it is argued that mechanisms contributing to the reduced Raman intensity would yield spectra proportional to the projected density of modes contributing to the EXAFS Debye-Waller factor, provided correlations in bond-bond motions are negligible. Model calculations support this conjecture. An expression for the coupling constant appearing in the Raman intensity is also obtained.     

Dissipative tunneling at finite temperature

A simple theory is presented for the influence of a weakly coupled interaction system on the tunneling of a particle out of a metastable well. It is based on the standard model of momentum and energy transfer to an infinite set of oscillators and is applied to the case of phase tunneling in a Josephson contact. The distribution of the energy transfer and in particular the Debye-Waller factor for elastic processes is determined by the imaginary part of the dielectric function. For small damping γ the main influence of dissipation on the total tunneling probability is contained in a factor exp —AMγ(Δq)². The numerical coefficientA and the distance Δq under the barrier depend on the considered tunneling state andA(T) vanishes at a temperatureT ^* above which classical activation prevails. The tunneling probability of any level is therefore predicted to increase with temperature. In additional general expressions are derived for the correlation functions of a damped quantum oscillator in terms of the classical response of the interaction system.     

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 resonances in the Debye-Waller factor of atom surface scattering

He atom surface scattering by dispersionless phonons is treated employing coupled channel (CC) calculations. At low energies, they predict a behavior opposite to perturbative Born or "exponentiated" Born approximation: strong resonant phonon stimulated elastic and inhibited inelastic scattering. The corresponding resonances have not been observed in earlier CC results since these have considered only the temperature dependence of the Debye-Waller factor at higher energy or omitted the attractive well. The resonances can be interpreted in terms of bound states in the attractive well with several excited vibrational quanta. They may be observable for, e.g., He scattering by a cold Xe/Cu surface.     

Quantum theory of atom-surface scattering: Incommensurate and fluid adsorbates

The problem of atomic scattering from adsorbate-covered surfaces, treated earlier for the case of commensurate overlayers, is considered again in the eikonal approximation for incommensurate lattice phases and for fluid phases. Stochastic methods are employed and for a specific model (hard bosses on a plane) it is shown how the statistical and geometric problems can be separately solved. In order to explain the meaning of the coherent and incoherent scattering contributions a time-dependent theory is introduced and it is shown that the incoherent “elastic” scattering is in fact weakly inelastic and (for classical diffusion with diffusion coefficient D) has an energy width of the order ?DQ², where Q is the parallel momentum transfer. The problem of the decay of substrate diffraction intensities when the coverage of random impurities is increased is also 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.     

Molekularstrahlmessungen von Stoßquerschnitten für Übergänge zwischen definierten Rotationszuständen zwei-atomiger Moleküle

Previously reported experimental results for inelastic cross sections for rotational excitation of TlF molecules in low-lying, well defined rotational states are interpreted in terms of a time dependent perturbation theory formulation of the high energy approximation. In order to calculate inelastic cross sections for the observed small angle scattering the Born approximation and the classical deflection function are shown to be applicable. In this approximation the ΔJ selection rules are characteristic of the individual terms in the expansion of the potential, whereas the ΔM selection rules depend on the orientation of the molecule with respect to the scattering trajectory. An approximation for dealing with a scattering gas consisting of molecules is introduced and the appropriate orientation averaging is carried out for the case of a generalized electrostatic potential. The measured results for the transition TlF(2.0)→ (3.0) in collisions with the rare gases and CH₄ and SF₆ are more than a factor three larger than calculated results for the induced dipole-quadrupole (α, μ, Q) interaction. Rough argeement is found between calculated results for a dipole-quadrupole interaction and the experimental results for the above mentioned transition produced by the scattering gases O₂, N₂ (air), N₂O, and H₂O. Finally, the dipole-dipole potential appears to provide an explanation for the large inelastic cross sections observed with NH₃ and CF₂Cl₂. Calculated inelastic and total cross sections are however considerably larger (about a factor 2) than the measured results with NH₃. Possible explanations are discussed.