Phase of reflection of very low energy electrons at crystal surfaces

The amplitude coefficients of electron reflection at crystal surfaces are complex numbers, each of which may be characterized by a reflection intensity (the squared modulus of the coefficient) and a phase. The phase of reflection of very low energy (? 10 eV) electron reflection is described on the basis of existing theory, and experimental approaches to phase determination are reviewed. Theoretical properties of the phase are described on the basis of the two-beam dynamical theory of diffraction. The model considered is an idealized substrate crystal with an attached selvedge (surface region). The indirect effect of inelastic scattering (absorption) is included by going to complex values of the electron energy or of the surface-normal component K of the propagation vector. In the absence of a selvedge the phase is determined solely by the band structure of the substrate crystal. If a selvedge is present there are large additional effects on the phase associated with zeros of the amplitude coefficient of reflection on the complex K plane. The experimental approaches considered are: (1) measurement of the kinetic energy distributions of ions produced in the field ion microscope, and (2) measurement of the periodic deviations from the Schottky line in field-assisted thermionic emission and photo-emission. Recent results of phase determination for W (011) surface by method (1) are summarized and compared with theoretical expectations.     

Surface-state parameters from very low energy electron reflection data

A parametric form of the amplitude of elastic reflection of very low energy electrons is derived. The amplitude expression conforms to the results of an earlier analysis of a simple case of electron reflection called the quasi two-beam case. The parameters in the amplitude expression refer to: (1) the surface states of the crystal; (2) the band structure of the substrate crystal; and (3) absorption (inelastic scattering) in the energy range of the experiment The amplitude expression also includes parameters relating to (4) the behavior of the amplitude at infinity and at negative energy.The amplitude expression is used to parameterize existing experimental results for nickel (001) and for the surface formed by adsorption of sodium on nickel (001) to form the centered (2 × 2) structure. The parameterization employs previously-computed values of parameters relating to the nickel band structure [category (2) above], and parameters in categories (1), (3) and (4) are adjusted to fit the electron-reflection data. In the case of the sodium-covered surface it is shown that the shape of the intensity-energy curve and the general level of intensity relative to that for clean nickel depends critically on the surface-state parameters. Two surface states are needed to fit the intensity data The values of the surface-state parameters are: location relative to vacuum level: 2.5 ± 0.1, 6.9 ± 0.2eV; width: 4.2 ± 0.4, 7.5 ± 1.0eV. The classification and significance of surface-state resonances is discussed briefly.     

Vibrational properties of thin films adsorbed on crystals with strong spatial dispersion

The effect of strong spatial dispersion of the substrate crystal onto the dynamics of thin epitaxially adsorbed films is described in a simple model through the following quantities: i) phonon reflection coefficients ii) depth-dependent local densities of states (LDOS) and iii) finite-lifetime surface states called leaky waves. The correspondence of these three types of characteristics accessible in different experimental methods is established. The bulk band of the substrate crystal consists of two distinct frequency ranges separated by an edge singularity: above the singularity the substrate supports two different waves for a given frequency, whereas only one wave can exist for each frequency in the low-frequency range. The resonances in the low-frequency range are found to correspond to maxima in the LDOS, to maxima in the amplitude of a near field arising in phonon reflection and to leaky waves involving a single leakage wave packet. The antiresonances in the same frequency range are characterised by minima in LDOS and in the near field, whereas the corresponding leaky waves involve two leakage wave packets. The only leaky waves found in the high-frequency range involve two leakage wave packets and are related to resonances. The antiresonances then are characterised by an anomalous increase in the extraordinary reflected wave. The edge singularity manifests itself in an additional quasi resonance, whose features depend on the coupling between the substrate and the thin film.     

Special analytic properties of the amplitude for scattering on the Woods-Saxon potential

It is shown that the s-wave partial amplitude f(k) for scattering on the real-valued Woods-Saxon potential V(r)=?V ₀/[1+exp((r?R)/d)] has very special analytic properties: the trajectories of the poles of the function k cotδ [of the zeros of the amplitude f(k)] coincide with the lines of the dynamical singularities [spurious poles of f(k)], so that the zeros and the poles compensate each other. In contrast to what is obtained for Yukawa-like potentials, the scattering length does not vanish here at zero energy. The results reported in this article were obtained analytically under the assumption that exp(-R/d)?1. The problem of revealing the poles of the function k cotδ in a partial-wave analysis of neutron scattering on nuclei is discussed.     

Resonance properties from a coupled channel meson exchange model

In this talk, I present the results on the pole structure of pion-nucleon scattering in an analytic model based on meson exchange. The analytic properties of scattering amplitudes provide important informa-tion. Besides the cuts, the poles and zeros on the different Riemann sheets determine the global behavior of the amplitude on the physical axis. Pole positions and residues allow for a parameterization of resonances in a well-defined way, free of assumptions for the background and energy dependence of the resonance part. This is a necessary condition to relate resonance contributions in different reactions.     

Diffraction effects in low-energy electron emission

The origin of diffraction peaks in the energy distribution of intensity of low-energy (< 1000 eV) electron emission from crystals is discussed from the standpoint of the dynamical theory of diffraction. The emitted electrons are considered to originate at relatively incoherent point sources in the crystal. The two-beam approximation of dynamical theory is used. The theory accounts for the chief regularities of diffraction peaks: temperature-dependence of peak intensities like that for low-energy electron diffraction (LEED) peaks, correlation of peak energies with X-ray absorption fine structure, and correlation of peak energies with the energies of normal-incidence LEED peaks in specular reflection. It is shown that the conditions for diffraction peaks coincide with the conditions for emergence of Kikuchi lines. It is predicted that for energies just above those of diffraction peaks, such emergences should be observable in the angular distribution of emission as intensity minima for emission along low-index crystal axes. Theory of Kikuchi band profiles is developed in an Appendix.     

Electronic surface resonances and crystal surface structure

A new method of determination of the lateral structure of crystal surfaces is presented. The method is based on earlier work showing the existence of resonances in the elastic scattering of low-energy electrons at crystal surfaces. The method consists of: (a) Measurement of the surface resonance band structure Ek_∥E, k_∥ respectively denote the electron energy and surface-parallel momentum for which resonances occur) and (b) Interpretation of E(k_∥) to determine the lateral variation of the effective potential acting on electrons at the surface.The surface resonance band structure is measured by a net-current electron reflection method. The measurement method is basically the same as used previously but here its precision is greatly enhanced by the use of digital methods of data handling including a digital filter to remove background due to inelastic and non-resonance elastic scattering. The surface resonance band structure is interpreted by a two-dimensional nearly-free electron scheme. In this scheme the interaction elements are Fourier coefficients of an effective potential which is an average of a pseudopotential with respect to the depth distribution of electron density in a surface resonance — the surface-weighted pseudopotential. Experimental surface resonance band structure for Ni(001), Ni(001) p(2 × 2)O and two different Ni(001) c(2 × 2)O surfaces (one of them with an oxygen-saturated Ni substrate) are presented for E = 1–30 eV and k_∥ running halfway from $\text{}\text{?}$gG towards H? in the surface Brillouin zone for Ni(001). The experimental results are fitted, using the nearly-free electron scheme, to determine the Fourier coefficients of the surface-weighted pseudopotential. Surface potential variations synthesized from the above data are discussed in comparison with the atomic arrangements known from LEED. It is demonstrated that the new method can give a correct picture of the lateral structure of surfaces. It is emphasized that these results are obtained without costly equipment or computations called for by other methods.     

Analytic properties of the scattering amplitude and resonances parameters in a meson exchange model

The analytic properties of scattering amplitudes provide important information. Besides the cuts, the poles and zeros on the different Riemann sheets determine the global behavior of the amplitude on the physical axis. Pole positions and residues allow for a parameterization of resonances in a well-defined way, free of assumptions for the background and energy dependence of the resonance part. This is a necessary condition to relate resonance contributions in different reactions. In the present study, we determine the pole structure of pion–nucleon scattering in an analytic model based on meson exchange. For this, the sheet structure of the amplitude is determined. To show the precision of the resonance extraction and discuss phenomena such as resonance interference, we discuss the S₁₁ amplitude in greater detail.     

Electromagnetic scattering as a radiation reaction problem

Electromagnetic scattering is treated as a dynamical problem, by equating the external torque exerted by the incident wave on the sphere to the self torque due to the radiated (scattered) wave. ForI _mech=0,our scattering amplitude is equal to the usualP-wave amplitude of the electromagnetic scattering on an infinitely conducting sphere. The poles of the scattering amplitude, in particular their dependence onI _mech, are studied. For example, a pole on the positive imaginary axis, which usually corresponds to a bound state, corresponds to a runaway solution in our case. Non-decaying resonances are also discussed.     

Probing interface electronic structure with overlayer quantum-well resonances: Al/Si(111)

The dispersion of quantum-well resonances in ultrathin epitaxial Al films on Si(111) reveals energy- and wave vector-dependent reflection properties at the Al/Si interface. The substrate electronic structure strongly influences the phase shift of the electron waves upon reflection at the interface. Thus the details of the substrate electronic structure need to be taken into account for a complete analysis of metallic quantum-well resonances. Furthermore, the assumption of loss of parallel wave vector information upon reflection or transmission through a lattice-mismatched interface is challenged. The changes induced in the electronic structure of the overlayer can be used to probe the ground-state substrate band edges.     

Temporary trapping of a light atom in between two heavy atoms

The temporary trapping of a light atom in between two heavy atoms is associated with short lived overlapping resonances. It is shown for the electronically excited ArHCl that these quantum diffraction resonances can be associated, within the framework of the adiabatic approximation, with the non-physical poles of the scattering matrix when the light atom is scattered through one-dimensional dynamical barriers.     

Kondo state of Co impurities at noble metal surfaces

We use scanning tunneling microscopy and spectroscopy to study the properties of magnetic Co adatoms on noble metal surfaces at 6 K. Due to spin-flip scattering of the substrate electrons at the impurity the many-body Kondo state forms. This state is characterized by an energy, the Kondo temperature T_K. We measure T_K of adatom systems and a resonant scattering phase shift locally and are thus able to discuss the coupling of the Co adatom to the metal electronic system. From the resonant scattering phase shift of the surface-state electrons scattering off a Co adatom on Ag(111), we find that the coupling to the surface state is rather weak. On the other hand, increasing the number of nearest neighbor substrate atoms increases the coupling of a Co adatom to the host metal and increases T_K. This shows the dominant character of the coupling of the Co atom to the bulk states of the substrate crystal. PACS 72.10.Fk; 68.37,Ef; 72.15.Qm     

On the location of poles of Ruelle's zeta function

We discuss examples of one-dimensional lattice spin systems of classical statistical mechanics whose generalized zeta function has all its poles and zeros on the real axis. The close relation between certain hyperbolic dynamical systems and these spin systems lets one expect that this is also true for some of the dynamical systems. In fact, we have found several one-dimensional expansive systems, among them the Gauss map whose zeta functions have their zeros, respectively their poles, on the real axis. Such a behaviour is closely related to the spectral properties of the sytems transfer operator which in the cases considered is a positive nuclear operator in a Banach space of holomorphic functions. We formulate a general conjecture concerning the spectrum of this class of operators.     

Instabilities and quasi-localized states in nonlinear Fano-like systems

We study the dynamical scattering in one-dimensional systems with a nonlinear side-coupled defect. Such structures exhibit the nonlinear Fano resonances, where nothing can propagate through. We developed a numerical model to study dynamical scattering. According to our analysis the scattering waves become dynamically unstable in the vicinity of the nonlinear Fano resonances, due to modulational instability caused by the presence of nonlinearity. It results in a time-growing amplitude of the nonlinear defect. We also demonstrate the existence of the nonlinear quasi-localized state, supported by such structures.     

Transfer Operators and Dynamical Zeta Functions for a Class of Lattice Spin Models

 We investigate the location of zeros and poles of a dynamical zeta function for a family of subshifts of finite type with an interaction function depending on the parameters . The system corresponds to the well known Kac-Baker lattice spin model in statistical mechanics. Its dynamical zeta function can be expressed in terms of the Fredholm determinants of two transfer operators and with the Ruelle operator acting in a Banach space of holomorphic functions, and an integral operator introduced originally by Kac, which acts in the space with a kernel which is symmetric and positive definite for positive β. By relating via the Segal-Bargmann transform to an operator closely related to the Kac operator we can prove equality of their spectra and hence reality, respectively positivity, for the eigenvalues of the operator for real, respectively positive, β. For a restricted range of parameters we can determine the asymptotic behavior of the eigenvalues of for large positive and negative values of β and deduce from this the existence of infinitely many non-trivial zeros and poles of the dynamical zeta functions on the real β line at least for generic . For the special choice , we find a family of eigenfunctions and eigenvalues of leading to an infinite sequence of equally spaced ``trivial' zeros and poles of the zeta function on a line parallel to the imaginary β-axis. Hence there seems to hold some generalized Riemann hypothesis also for this kind of dynamical zeta functions. Received: 14 March 2002 / Accepted: 24 June 2002 Published online: 14 November 2002     

The phase of the Riemann zeta function

We, offer an alternative interpretation of the Riemann zeta functionζ(s) as a scattering amplitude and its nontrivial zeros as the resonances in the scattering amplitude. We also look at several different facets of the phase of theζ function. For example, we show that the smooth part of theζ function along the line of the zeros is related to the quantum density of states of an inverted oscillator. On the other hand, for ℜs>1/2, we show that the memory of the zeros fades only gradually through a Lorentzian smoothing of the delta functions. The corresponding trace formula for ℜs≫1 is shown to be of the same form as generated by a one-dimensional harmonic oscillator in one direction along with an inverted oscillator in the transverse direction. Quite remarkably for this simple model, the Gutzwiller trace formula can be obtained analytically and is found to agree with the quantum result.     

Mode oscillation and harmonic distortions associated with sinusoidal modulation of semiconductor lasers

Techniques to deal with Feshbach resonances are applied to describe resonant light scattering off one dimensional photonic crystal slabs. Accurate expressions for scattering amplitudes, free of any fitting parameter, are obtained for isolated as well as overlapping resonances. They relate the resonance properties to the properties of the optical structure and of the incident light. For the most common case of a piecewise constant dielectric function, the calculations can be carried out essentially analytically. After establishing the accuracy of this approach we demonstrate its potential in the analysis of the reflection coefficients for the diverse shapes of overlapping, interacting resonances.     

INVESTIGATIONS ABOUT THE DYNAMICAL ELECTRON SCATTERING BY A HEXAGON-PRISM β-Si3N4 WHISKER

The dynamical electron scattering effects by a hexagon-prism β-Si₃N₄ whisker were investigated. The 002 dark field images of the wedges of the whisker along [110] direction showed sinusoidal fringes, in which there existed great differences in fringe period under two- and N-beam conditions. Fine structures of the corresponding diffraction spots under two-beam condition were also very different from those under many-beam condition. Such dynamical scattering effects were analyzed quantitatively.     

Resonance theory of elastic wave scattering from a cylindrical cavity

Resonant excitation of a fluid-filled cylindrical cavity in an elastic medium by an incident compressional wave is investigated on the basis of the resonance theory of nuclear scattering. It is shown that the scattering amplitude consists of a series of narrow resonances super-imposed upon, and interfering with, a broad background that corresponds to the scattering from an empty cavity. The resonances may be analyzed in a most enlightening fashion by studying them separately in each partial wave of the normal-mode series. They are seen to correspond to excitations of the eigenvibrations of the cavity fluid caused by a phase-match of “creeping waves”, similar to the “Regge poles” of nuclear physics.