Semiempirical determination of the atom-surface interaction

We discuss three methods of determining V(z), the lateral average (G = 0 Fourier component) of the atom-surface interaction, from the bound state spectrum found in beam scattering. One method uses the Rydberg-Klein-Rees technique, which yields the width of the potential (i.e., separation of classical turning points) as a function of energy. This method incorporates also the known asymptotic form V ～ ?C₃z^?3, whereC₃ is derived from the polarizability and dielectric function of atom and solid, respectively. The second method uses a hybrid potential, constructed from a Morse potential with shifted zero of energy connected to the asymptotic form,?C₃z^?3, requiring continuity of V and $\text{d}\text{V}\text{d}\text{z}$. The third potential is a Lennard-Jones 3–9 interaction. Results are presented for H and He scattering from LiF and NaF.     

Physical adsorption for non-planar geometries

We propose an approximation V_a(r) for the Van der Waals interaction V(r) between an atom and a non-planar solid. V_a(r) is both simpler to compute than V(r) and of considerably more convenient form. The approximation is found to be quite good except for very large z (the atom-surface separation). In the latter case, our comparison of V_a and V permits one to estimate corrections to V_a.     

Determining potential energy constants for atom- and molecule-surface interactions

Methods developed for diatomic molecule spectroscopy are adapted for use in analysing the energies of atom-surface bound states in order to determine certain features of the surface-averaged potential energy function. In cases for which appropriate data are available, these simple graphical methods can yield a model-independent estimate of the potential well depth e, a value for the coefficient (C₃) of the long-range z^?3 term in the atom-surface potential, and estimates of both the total number of bound states and the energies of any unobserved levels lying near the dissociation limit. Application of these techniques to the data for atomic hydrogen on (001)LiF and NaF and for atomic helium on (001)LiF yielded: ? = 18.6(±1.0), 18.2 (±3.6) and 8.6(±0.8) meV, and C₃ = 250(±90), 180(±110) and 95(±40) meV ų, respectively. Application of this approach to the data for molecular hydrogen on (001)LiF led to a new set of vibrational assignments and showed that ? = 37(±4) meV, and that the H₂ and D₂ data of O'Keefe et al. and the H₂ binding energies which Tsuchida obtained from the data of Frisch and Stern are all internally consistent.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. III. Localization Properties

In this paper, we study the asymptotic localization properties with high probability of the Kth eigenfunction (associated with the Kth largest eigenvalue, K?1 fixed) of the multidimensional Anderson Hamiltonian in torus V increasing to the whole of lattice. Denote by z _K,V ∈V the site at which the Kth largest value of potential is attained. It is well-known that if the tails of potential distribution are heavier than the double exponential function and satisfies additional regularity and continuity conditions at infinity, then the Kth eigenfunction is asymptotically delta-function at the site z _τ(K),V (localization centre) for some random τ(K)=τ _V (K)?1. We study the asymptotic behavior of the index τ _V (K) by distinguishing between three cases of the tails of potential distribution: (i) for the “heavy tails” (including Gaussian), τ _V (K) is asymptotically bounded; (ii) for the light tails, but heavier than the double exponential, the index τ _V (K) unboundedly increases like |V|^o(1); (iii) finally, for the double exponential tails with high disorder, the index τ _V (K) behaves like a power of |V|. For Weibull’s and fractional-double exponential types distributions associated with the case (ii), we obtain the first order expansion formulas for logτ _V (K).     

The dispersion force of physical adsorption: II. Nonlocal theory

We present a nonlocal theory of the dispersion force of physisorption for non-polar atoms or molecules. The calculation uses linear response theory and the surface impedance method. The results generalize those presented previously, which relied on a local theory of the dielectric response. Also included in the treatment is the role of diffuseness of the interface between substrate and vacuum. Adatom dipole and quadrupole fluctuations are treated. Several diverse contributions to the potential are thus calculated, giving rise to a substantial modification of the conventional expression V ? -C₃(z-z₀)^?3, especially at small separation z.     

A separable approximation to the two-body t-matrix for short-range central local potentials

The S-wave potential in momentum space, V₀(p, p′), corresponding to a local two-body central short-range potential may be expanded in separable form with the help of suitable quadrature formulae. With a two-term separable expansion for a variety of nuclear potentials, the resulting t-matrix is found to be in close agreement with the corresponding result with a six-term expansion for small values of p and p′ which are important in the calculation of the trinucleon binding energy.     

Interpretation of auger energy shifts of silicon in solid silicon compounds

Chemical shifts of Auger transitions and photoelectron binding energies of silicon have been measured and interpreted using the quasi-atomic approach. The Si KL_2,3L_2,3 and L_2,3V₁V₁ Auger transitions and the binding energies of Si 2p and of the valence electrons at the maximum of the density of states V₁ have been investigated in solid silicon and in the compounds SiC, Si₃N₄, SiO₂, Na₂SiF₆ and T₃Si (T = V, Cr, Mn, Fe, Co, Ni). The relaxation-energy shift ΔR^ea_S(2p, 2p) describing the polarization effect (final-state effect) has been evaluated by AES and XPS measurements. Furthermore, the extra-atomic relaxation energy R^ea_D(2p) of the 2p electrons has been determined experimentally for silicon atoms in differing environments. This allows estimation of the potential parameter V(2p) describing the potential effect (initial-state effect). In general R^ea_D(2p) was found to be more sensitive to changes in chemical bonding than V2p). The behaviour of the quasi-atomic Si V₁ electrons seems to be the converse.     

Surface states in a one-dimensional crystal

The present short paper considers the role of the shape of the surface potential, particularly its long range character, on the existence and energies of the electronic surface states. For that purpose, a one-dimensional crystal is being considered represented by a Kronig-Penney potential

$\text{V}{}_{\mathrm{I}}\text{(z)=?U}{}_0\text{+}\text{∑}\text{n}\text{h}{}^2\text{m}\text{P}\text{a}\text{(s+na)}$

, (P < 0) for z < ? terminated by an image potential of the form V_II (z) = ?Ce²/z(z >?). The physical values of U₀ and a given only two gaps between energies ?U₀ and O. It is found that for a step barrier surface potential at z = ? there is only one surface state in each gap. On the contrary, for an image type potential, the number of surface states depends on the value assumed for ? or C(U₀ = Ce²/?). Varying ? or C, one can modify the shape of the potential from almost a step barrier to an image potential (C = 1) and study the existence of surface states in every case. In particular for C ? 1 (? ? 1 Å) more than one surface state in the higher gap are obtained.     

Electro-optical properties of strontium-barium niobate crystals and their relation to the domain structure of the crystals

The electro-optical coefficients r _ij and half-wave voltage V_λ/2 of strontium-barium niobate crystals poled in the ferroelectric phase are shown to vary along the polar axis. The r _ij (z) and V_λ/2(z) dependences indicate the presence of a residual domain density D(z) and clearly depend on the sign of the polarizing field, with r _ij being minimum (D being maximum) near the negative electrode. This character of the D(z) distribution and, hence, the r _ij (z) and V_λ/2(z) coordinate dependences can be explained by predominant domain nucleation near the negative electrode, which is revealed when the switching processes are studied using 90° (Rayleigh) light scattering from domain walls.     

Evaluation of dipole and quadrupolar contributions to the asymptotic atom-surface interaction

The position of the reference plane z₀ as well as the dipole and quadrupolar contributions to the long-range atom-surface interaction are evaluated within a simple model. The model treats the substrate as a bulk material and a selvedge, each of which has constant electronic density. The required polarizabilities of the atom are treated as one-term Padé approximants. It is then straightforward to calculate the electronic susceptibility and evaluate expressions for C₃, C′₅, and z₀ that have been recently derived.     

Analysis of the formation of ion beams in varying electric fields of a stationary plasma thruster

We propose a technique for analyzing the distribution function of the velocity components (radial V _r and azimuthal V _φ) of ions in a beam. This technique is used for studying the ion beam emerging from a stationary plasma thruster (SPT). It is shown that the beam contains ions with a radial velocity component in the range V _r /V _z = ? 1.2 to +0.74, as well as ions with the azimuthal velocity component in the range V _?/V _z = ±0.9. Numerical calculations lead to the conclusion that ions acquire the azimuthal velocity component in the field of the azimuthal wave of the plasma potential evolving in the SPT channel.     

Relativistic effects in one-dimensional hydrogen atom

The dependence of stationary levels of a Dirac particle in the regularized Coulomb potential V _??(z) = ?q/(|z| + ??) on the cutoff parameter ?? is studied. It is shown that, in 1 + 1 D, the energy spectrum of a Dirac particle in such a potential reveals some specific features which nonanalytically depend on the coupling constant q and are essentially relativistic in nature. These properties turn out to be most important for ?? ? 1, explicitly demonstrating the existence of a physically reasonable energy spectrum for an arbitrarily small ?? > 0 and, at the same time, the absence of regular limit ?? ?? 0 (hence, the absence of a well-defined spectral problem for the Dirac equation without regularization for arbitrary q in 1 + 1 D).     

Description of ultracold atoms in a one-dimensional geometry of a harmonic trap with a realistic interaction

We compute the ground-state energy of two atoms in a one-dimensional geometry of a harmonic optical trap. We obtain a dependence of the energy on a one-dimensional scattering length, which corre-sponds to various strengths of the interaction potential V _int (x) = V ₀ exp {?2cx ²}. The calculation is performed by numerical and analytical methods. For the analytical method we choose the oscillator representation method (OR), which has been successfully applied to computations of bound states of various few-body systems. The main results of this paper are (1) a numerical investigation of the validity range of the previously used pseudopotential method and (2) an investigation of the validity range of the OR for the potential V(x) = V _conf (x) + V _int (x) = x ²/2 + V ₀ exp {?2cx ²}.     

Dynamical dark energy models – dynamical system approach

We study the Friedmann-Robertson-Walker model with dynamical dark energy modelled in terms of the equation of state p _X = w _X (a(z)) ρ _X in which the coefficient w _X is parameterized by the scale factor a or redshift z. We use methods of qualitative analysis of differential equations to investigate the space of all admissible solutions for all initial conditions on the two-dimensional phase plane. We show advantages of representing this dynamics as a motion of a particle in the one-dimensional potential V(a). One of the features of this reduction is the possibility of investigating how typical big rip singularities are in the future evolution of the model. The properties of potential function V can serve as a tool for qualitative classification of all evolution paths. Some important features like resolution of the acceleration problem can be simply visualized as domains on the phase plane. Then one is able to see how large is the class of solutions (labelled by the inset of the initial conditions) leading to the desired property.     

From nuclear stopping to interatomic potential via the power law formalism

A general method for inverting the energy-dependent nuclear stopping cross sections S_n(?) to derive the interatomic potential V(x) is described. A correspondence between ? and x is derived for mapping S_n(?) to V(x) and vice versa. The method is illustrated by using the recent range-energy data of keV indium and xenon ions in amorphous silicon to self-consistently deduce the In-Si and Xe-Si potentials.     

Semiclassical description of inelastic atom scattering by surfaces

Inelastic scattering of atoms of moderate energies (say<5 eV) by solid surfaces is almost entirely due to energy exchange with lattice vibrations. It can give valuable information about the atom-surface interaction potential and the vibrational dynamics at surfaces. Theoretically this process represents a challenging many-body problem, calling for suitable approximation methods. Work in progress (K. Burke, L. D. Chang, and W. Kohn) is described. (1) We have solved a simple model problem in which the normal modes of the lattice are schematized by a single one-dimensional harmonic oscillator, initially in its ground state (T=0). The classical solution gives a unique energy loss. We have calculated the leading quantum correction and find a Gaussian final energy distribution whose width is proportional toh ^1/2. Our exact results are in general different from the so-called trajectory approximation. (2) We are about to propose a new type of atom-surface scattering experiment, which will provide a direct measure of the quantum corrections to classical scattering.     

The role of the neutron electric form factor ind(e,e′ N)N including polarization observables

The influence of the neutron electric form factor on various observables in two-body break-up of deuterons by electrons such as differential cross section, beam, target and beam-target asymmetries and outgoing nucleon polarization as well is investigated for different kinematic regions. The electron-deuteron vector asymmetryA _ed ^V and the outgoing nucleon polarization component P′_x(n) are the most promising observables in and off the quasi-free region for a determination ofG _En. Also the single polarization observablesA _d ^T and P_y ⁰(p) and the double polarization observable P′_z(n) show significant influences fromG _En.     

Measuring entanglement-induced quantum correlations in the oscillatory motion of a trapped ion

The preparation of a wide class of entangled vibrational states involving two or three phononic modes of a three-dimensional trapped ion has been reported in the literature from both theoretical and experimental points of view. Here, the time evolution of such a system from an initial condition wherein two oscillatory modes (M _x, M _y) are reciprocally entangled and both are disentangled to the third mode (M _z) is studied. By coupling one of the entangled oscillators (M _x) with the third oscillator (M _z), a correlation between the two uncoupled modes (M _y, M _z) is induced, well visible when the mean value of a suitable operator is considered. A method of measuring the expectation value of a vibrational observable is briefly sketched and then exploited in order to reveal such nonclassical behavior.     

Green's functions of three-particle subresolvents containing separable potentials

Analytic expressions for Green's functions of three-particle subresolvents (including one spectator particle) are given, i.e. the configuration space representation of the operator-valued functionG _α (z)=(z?K?V _α )^?1 whereK is the total kinetic energy of the three-particle system andV _α is the interaction of the pairΒγ. A sum of separable potentials is well suited to exhibit the structure of the corresponding Riemann surface such that from these results also some conclusions may be drawn about the full Green's operator of the three-particle system.     

Extremal Theory for Spectrum of Random Discrete Schrödinger Operator. II. Distributions with Heavy Tails

We study the asymptotic structure of the first K largest eigenvalues λ _k,V and the corresponding eigenfunctions ψ(?;λ _k,V ) of a finite-volume Anderson model (discrete Schrödinger operator) \(\mathcal{H}_{V}= \kappa \Delta_{V}+\xi(\cdot)\) on the multidimensional lattice torus V increasing to the whole of lattice ? ^ν , provided the distribution function F(?) of i.i.d. potential ξ(?) satisfies condition ?log(1?F(t))=o(t ³) and some additional regularity conditions as t→∞. For z∈V, denote by λ ⁰(z) the principal eigenvalue of the “single-peak” Hamiltonian κΔ _V +ξ(z)δ _z in l ²(V), and let \(\lambda^{0}_{k,V}\) be the kth largest value of the sample λ ⁰(?) in V. We first show that the eigenvalues λ _k,V are asymptotically close to \(\lambda^{0}_{k,V}\). We then prove extremal type limit theorems (i.e., Poisson statistics) for the normalized eigenvalues (λ _k,V ?B _V )a _V , where the normalizing constants a _V >0 and B _V are chosen the same as in the corresponding limit theorems for \(\lambda^{0}_{k,V}\). The eigenfunction ψ(?;λ _k,V ) is shown to be asymptotically completely localized (as V↑?) at the sites z _k,V ∈V defined by \(\lambda^{0}(z_{k,V})=\lambda^{0}_{k,V}\). Proofs are based on the finite-rank (in particular, rank one) perturbation arguments for discrete Schrödinger operator when potential peaks are sparse.