Table of contents

Evaluating a fractal curve's "approximate length" by walking a compass defines a "compass exponent." Long ago, I showed that for a self-similar curve (e.g., a model of coastline), the compass exponent coincides with all the other forms of the fractal dimension, e.g., the similarity, box or mass dimensions. Now walk a compass along a self-affine curve, such as a scalar Brownian record B(t). It will be shown that a full description in terms of fractal dimension is complex. Each version of dimension has a local and a global value, separated by a crossover. First finding: the basic methods of evaluating the global fractal dimension yield 1: globally, a self-affine fractal behaves as it if were not fractal. Second finding: the box and mass dimensions are 1.5, but the compass dimension is D = 2. More generally, for a fractional Bownian record B_H(t), (e.g., a model of vertical cuts of relief), the global fractal dimensions are 1, several local fractal dimensions are 2-H, and the compass dimension is 1/H. This 1/H is the fractal dimension of a self-similar fractal trail, whose definition was already implicit in the definition of the record of B_H(t).

In the resolvent methods for solving the Schrödinger equation, the eigenvalues are represented by simple poles of the so-called Weinstein function, whereas the eigenfunctions are represented by expressions of the form ∞/∞. It is shown that there exists a simple identity which permits the evaluation of the eigenfunction, and which shows the connection between the resolvent methods and partitioning technique, rational approximations, and perturbation theory. It is also shown that this identity may be extended to the solution of the Liouville equation.

The self-similar structure of mode lockings for circle maps is studied by means of the associated Farey trees. We investigate numerically several classes of scaling relations implicit in the Farey organization of mode lockings and discuss the extent to which they lead to universal scaling laws.

We experimentally show that nonlinear parallel arrays can be made to compute with attractors. This leads to fast adaptive behavior in which dynamical associations can be made between different inputs which initially produce sharply distinct outputs. We first define a set of simple local procedures which allow a general computing structure to change its state in time so as to produce classical Pavlovian conditioning. We then examine the dynamics of coalescence and dissociation of attractors with a number of quantitative experiments. We also show how such arrays exhibit generalization and differentiation of inputs in their behavior.

We show how the general approaches based on the order parameter concept and the slaving principle, so far developed for classical systems, can be extended to quantum systems. Possible applications are the formation of spatial or temporal structures formed by electrons or holes in semiconductors (Gunn effect and filamentation).

For a microscopic quantum-mechanical system we consider the measurement of an observable A which is assumed to characterize the system uniquely. The measurement process takes place through interaction between the microscopic system and the measurement apparatus, which in turn is influenced by external fields. The detailed influence from the initial state of the measuring apparatus and from the external fields on the density matrix of the microscopic system is stochastic and may be taken into account as a random walk in the simplex characterizing its diagonal elements (in the representation where A is diagonal). The diffusion equation corresponding to this random walk is derived. It is shown that the corners are attractors of this process, i.e. each single measurement ends up in an eigenstate of A. The relative frequencies for the different results are given by the squared moduli of the corresponding amplitudes. The reduction of the wave packet is thus the result of a statistical process.

We give criteria for anomaly cancellations in chiral Yang-Mills supergravities (including dual formulations of the theories) in 6, 8 and 10 dimensions.

The total photoionization cross section for uranium has been calculated in the energy range 60–540eV using local-density based random phase approximation (LDRPA). The result is compared to compilations of experimental photoabsorption data and is found to clear up a number of discrepancies between experiment and other calculations the region between onset of 5d- and 4f-absorption.

The coupled-cluster or exp S formalism, where the wave operator is expressed in exponential form, is treated for a general, multi-Configurational model space. It is shown that the cluster operator, S, is rigorously connected when the model space is complete or "quasi-complete" in the sense that it contains all configurations that can be formed by distributing the valence electrons within certain groups of valence orbitals with given occupation number in each group. For this class of model spaces also the linked-diagram theorem is valid in the sense that the diagrams of the wave operator and the effective Hamiltonian do not contain any separate, closed part. The diagrams of the effective Hamiltonian are connected, while those of the wave operator may contain disconnected, open parts.

For a more general, multi-configurational (incomplete) model space a formal expansion of coupled-cluster type is still possible, but it is found that the cluster operator is no longer necessarily connected, which leads to unlinked diagrams in the expansion of the wave operator and the effective Hamiltonian. A general procedure for generating the cluster operator in this case is described and applied particularly in the pair approximation.

The diatomic series: (a) XH where X ranges from Be to F and (b) the homonuclear set N₂, O₂ and F₂, are first considered. From the work of Allan et al. on the kinetic energy T₂ = (ℏ²/72m) ∫(∇ρ)²/ρ dτ associated with the inhomogeneity of the electron density ρ(r), it is demonstrated that T₂/N is approximately linear in the total number of electrons N for both the series (a) and (b). The bare nuclear-nuclear potential energy U_nn of series (a) is thereby directly related to T₂/N.

Some polyatomic molecules are then embraced by considering the `neon-like' molecules HF, H<sub>2</sub>O, NH<sub>3</sub> and CH<sub>4</sub> and also, more briefly, the `argon-like' series HCl to SiH₄. In both series, T₂ can be related approximately to ∫_o^∞ρ(r)dr where ρ(r) is the spherical average of the molecular electron density about the heavier atom as origin. In turn, this integral is shown to be (4π)^-1 ∫_o^∞kf(k)dk where f(k) is the X-ray scattering factor. Chemical interest in T₂ in the `neon-like' series is again plain from the fact that T₂ closely correlates with the quantity Δ = E - (3/2)E_s, where E is the total molecular energy and E_s the sum of orbital energies.

Finally, a brief discussion is given of the relation of molecular dissociation energy to the density gradient kinetic energy T₂.

In "real space" methods basis functions of atomic orbital type play a central role. They can be more or less localized and are in general non-orthogonal. Some aspects of the concepts of localization and ortho-normalization, and in particular their interdependence are discussed.

Solutions are obtained in one, two and three dimensions for reaction diffusion equations including two- and three-body recombination as well as ionization of neutrals successively produced by recombinations in dense plasmas.

A recently presented model for liquid-glass transition by Leutheusser and by Bengtzelius, Sjögren, and Sjölander is briefly summarized and discussed. Some conceptual points, which evidently were not spelled out clearly enough in the original articles, are clarified here.

We have studied arrays of superconducting tunnel junctions made of aluminium, with and without an applied high frequency field. The radiation frequency was around 73 GHz, which corresponded to 85% of the energy gap frequency. For sufficiently high critical current densities (j_c ⩾ 20 A/cm²) these A1 junctions exhibited non-equilibrium effects. This was manifested in a heavy back-bending of the current-voltage characteristic in the gap region. The gap suppression could be up to 16% of the energy gap. A back-bending of the curve could also be observed at the onset of the photon induced step. From the gap reduction we could estimate an effective quasi-particle relaxation time to about 650 ns at an assessed film temperature of 0.48K. This can be compared to a theoretical value of 1400 ns at the same temperature. A kink in the current-voltage characteristic was seen at an injection rate of the order of the value predicted from the T* model of a non-equilibrium superconductor. A strong r.f. field caused pair breaking and gap reduction. Smaller r.f. powers, however, gave a gap enhancement and we observed up to about 5% increases of the gap as compared to the suppressed value caused by the tunnel current injection without mm waves.

The self-consistent method, based on Bogoliubov's inequality, is a scheme which provides optimised vibrational frequencies Ω. We have derived an explicit equation for Ω when a single particle is bound to a potential which is specified by its series expansion in the particle displacement from the potential minimum. This equation is applied to the potentials V tan² (πx/a) and V[(x/a) - (a/x)]². It is discussed to what extent the true vibrational entropy can be approximated by a proper choice of a vibrational frequency in the harmonic entropy formula. We find that Ω obtained by the self-consistent scheme may be less adequate than the result of low order perturbation theory.

The XXZ spin Hamitonian H = - (S_i^xS_j^x + S_i^yS_j^y + λS_i^zS_j^z) is simulated for quantum spins. The XY model (λ = 0) has a Kosterlitz- Thouless Phase transition T_KT = 0.45 ± 0.05, slightly above which the specific heat has a finite peak. The vortex density has a non-zero value in the low-temperature limit, probably due to quantum fluctuations. The critical temperature holds up at least to λ = -0.8. The Ising transition temperature T_Ising is measured for Ising-like antiferromagnets λ < -1 and is found to drop off to zero in a manner consistent with the reciprocal-logarithmic behavior T_c α-1/log|-λ-1|, as proposed for the classical model.

The electron-plasmon Fröhlich-type Hamiltonian introduced by Lundqvist, is formulated in terms of the Feynman path integral. It is shown that the average propagator for the plasmaron, the dressing of the electron by plasmons, can be evaluated using the path integral method developed by one of us (V.S.) for treating the polarons and the disordered systems. The average propagator is then used to obtain the ground state energy of the plasmaron. It is shown that for strong coupling, r_s → ∞, the leading term of the ground state energy is found to be proportional to 1/r_s in agreement with the Wigner calculation while for weak coupling, r_s → 0, the ground state energy contains a constant term plus a divergent term similar to that of the Gell-Mann and Brueckner result. Extensions of this method to the calculation of other physical parameters are discussed.

This paper is concerned with the a priori theory of the photoemission process in solids. Scattering-theoretical and response-theoretical approaches are discussed and a new demonstration of their equivalence is given. The elastic photoemission spectrum is analyzed in detail and an expression exact to all orders in interparticle interaction is obtained. This expression is similar to the renormalized three-current correlation formula used by Pendry and by others but includes also a screening and exchange-correlation correction to the optical field. The loss part of the spectrum is also discussed and a new quantum-mechanical approximation for the energy losses is obtained. For long mean free paths this correctly reduces to a three-step model in which the energy dissipation is given by a semi-classical transport equation.

In order to assess the accuracy of the approximation to exchange and correlation recently presented by Langreth, Predew, and Mehl (LPM) we have used it to calculate the cohesive energy and the charge density of metallic Be. In accordance with previous results on bulk Si we find the LPM approximation to be superior to the LD approximation for both the energy and the density. The LD energy error is reduced by approximately a factor of three in the LPM approximation. We also show that a comparison between theoretical and experimental form factors cannot be used to distiguish between different approximations used in computing the density of the valence electrons. Differences in valence densities are overshadowed by errors in the core density. Finally we have discovered an error in previous pseudopotential calculations of the electronic density of solids. We also present a procedure for correcting the error, which enables us to obtain accurate valence densities within the ab initio pseudopotential approach.

We compare the Hartee-Fock (HF), Talman-Shadwick (TS), exact Slater (S), Slater Xα, and local-density (LD) schemes, and investigate how well the associated one-electron potentials describe electronic excitations. We discuss in some detail the role of self-interaction in the different schemes, and demonstrate that the Xα and LD schemes effectively represent ionic (V^N-1) potentials in the core region. In fact, for excited f electrons, the LDA is very similar to a relaxed HF average V^N-1 potential with a 4d core hole. Furthermore, we demonstrate how the self-interaction shifts of core levels arise because of incomplete local cancellation between the orbital-specific Hartree self-interaction and the average, orbital-independent exchange hole.

The atomic dipole polarizability is introduced via the LS-dependent HF scheme and, in numerical applications to 4d ionization of atomic Ba, we compare the effective HF ¹P V^N-1 one-electron potential with the ionic HF_avV^N-1 and the neutral-atom HF V^N potentials. We also compare it with the RPAE, with and without inclusion of relaxation, and demonstrate the effects of external screening by adding and removing electrons (Ba^1- - Ba²⁺).

Finally we motivate in some detail our opinion that the self-interaction in the LDA may be considered as simulating relaxation owing to a systematic partial cancellation between self-interaction and relaxation effects for outer and intermediate shells.

The effects of lattice relaxation on the midgap adsorption due to a soliton in a ring of trans-polyacetylene is studied within the adiabatic approximation. The Franck-Condon factors are calculated for processes involving the emission of up to three phonons. The absorption edge is found to be broadened by processes involving an even number of phonons. On this background appear extremely sharp peaks, due to processes involving an odd number of phonons. In the latter case, the shape oscillation mode of the soliton is found to be the dominant mode. Our results also indicate a third localized phonon mode about the soliton, in agreement with the results of Ito et al. [24].

Theoretical description of the electronic vibrational spectroscopy of atoms adsorbed on metal surfaces is developed which takes into account both open and closed (i.e. resonant) scattering channels of incoming electrons. Inelastic scattering cross section for the lowest order vibrational transition is formulated in the two-beam approximation, and the matrix elements are explicitly calculated in the point dipole model. The proper inclusion of the resonant channel scattering leads to the enhancement of the inelastic peaks near resonances, and, assuming the laterally screened induced electron-dipole interaction, explains the partial breakdown of the dipole selection rules and the relative strengths of parallel and normal vibrational excitations, as observed in HREELS of hydrogen adsorbed on transition metal surfaces. The angular dependences of the loss peaks on and off resonances are also discussed.

We have used measurements of Raman scattering-excitation profile spectra for totally symmetric and non-totally symmetric vibration modes of free-base phthalocyanine molecules adsorbed on Ag, as a modulation spectroscopic probe, to obtain information about the energies, widths and vibronic character of the singlet (π to π*) valence electron excitations and about the character of the bonding of the adsorbate by the substrate. The data show that the Q_x and Q_y excitations are red-shifted relative to the corresponding excitations of isolated free-base phthalocyanine molecules, and broadened, and that the Q_x - Q_y energy separation is reduced by about a factor of two. On the other hand the frequencies of the various vibration modes are unshifted. We find that the resonant Raman scattering at the Q_x and Q_y excitations is due primarily to a Herzberg-Teller mechanism that involves vibronic coupling of the nearly degenerate Q_x and Q_y states with one another and with higher singlet states. These results together with the absence of any sizable xz, yz and zz components of the contribution to the Raman tensor from the Q_x and Q_y excitations suggest that the π electrons are not primarily involved in the bonding of the phthalocyanine to the Ag substrate. The decrease in the Q_x - Q_y separation is tentatively attributed to a weakening of the hydrogen bonds at the central nitrogen atoms that may result from the bonding of the molecules by the substrate via the lone pair electrons of the central nitrogens atoms.

The exchange-correlation functional of Langreth and Mehl [Phys. Rev. B28, 1809 (1983)] is generalized to include spin dependence. It is then applied to the calculation of ground-state and electron removal energies in atoms. A functional for the correlation energy alone is also given, for use in cases where exchange can be calculated exactly.

A simple expression for the effective two-body interaction in the electron liquid is derived using a diagrammatic analysis. The interaction is expressed in terms of the local field factors and the Lindhard polarizability. It includes the contribution of both density and spin fluctuations.

This paper gives a brief survey over some recent work on surface coatings for energy efficiency and solar applications. Specifically, it covers coatings for

- selective absorption of solar energy (for solar collectors),

- selective emission of infrared radiation in the 8-13 μm range (for passive cooling devices),

- visible transmission combined with near-infrared reflectance (for windows with decreased inlet of solar energy),

- solar transmission combined with thermal-infrared reflectance (for windows with decreased outlet of thermal radiation),

- variable transmission of solar or visible radiation (for "smart windows" with dynamic control of radiant energy).

One general conclusion from the survey is that the research field embodying these types of coatings remains, with a few exceptions, in a state of rapid progress.

The classical McIntyre formula for p-polarized light incident on a thin film on a substrate is extended in general terms to include a realistic description of the interfaces and the possible excitation of plasma waves in the film. An earlier extension is critized and criteria are given for when the classical result is applicable.

Vibrational excitation of diatomic molecules scattered from solid surfaces is considered. Emphasis is placed on a mechanism in which charge transfer between the molecule and surface creates a temporary molecular ion. A classical mechanics analysis of this effect has been presented previously. Here a mixed picture is offered in which the center-of-mass translational motion is treated classically with a trajectory approximation (TA) and the intramolecular vibrational motion quantum mechanically. A procedure for insuring energy conservation and microscopic reversibility in the TA is given. Both state-to-state T → V probabilities and mean energy transfer to the vibrational system are calculated for parameters which model N₂ and I₂ molecules and the results are considered in the light of the "exact" classical mechanics results.

The correlation energy of the two dimensional electron gas is calculated using the method of Singwi-Sjölander. Our results for the unpolarized case are in good agreement with prior results of Jonson and Ceperley. We also report results for the spin polarized case where prior good results were lacking.

We have calculated the cohesive energy and the energy difference between the fcc and bcc phases of the 5d elements Lu-Au using the recently developed Linear Augmented Slater Type Orbital method. We find the crystal structure to be correctly predicted and the structural energy differences to be significantly larger than the most commonly quoted values. We also find that the cohesive energies of these elements can be understood within the local spin density functional theory provided the experimental promotion energy from the local density dⁿs configuration to the preferred d^n-1s² configuration is included.

The time required for decay of a metastable state, through tunneling, is a well settled subject. During this lifetime of the initial state, the particle oscillates within its initial well. By contrast, questions about the duration of the actual tunneling process, when it finally occurs, have led to widely divergent answers. In recent years, Jonson, Stevens, and we, have, through independent approaches, pointed out that there is an effective barrier traversal velocity obtained by dividing the magnitude of the imaginary momentum, under the barrier, by the particle mass. Here we present a fourth approach to this earlier answer, by considering a time modulated stream incident on the barrier. At low modulation frequencies the transmitted beam reproduces the incident modulation without lag, and without change in modulation depth. As the modulation frequency is increased we eventually depart from this simple behavior, and that is taken as a measure of the transversal time. We also provide some of the details omitted in our earlier analysis.

The integer quantum Hall effect is derived for a finite rectangular sample and rather general boundary conditions using a Kubo formula approach. The precise quantization of the Hall resistance in units of h/e² is subject to constraints on the size of the sample, the strength of the magnetic field and the shape of the confining potential at the sample boundaries. The derivation assumed a (bulk) mobility gap but not necessarily a gap in the density of states in between Landau levels.

By neglecting certain terms in the Hamiltonian (of importance only in the immediate neighbourhood of the threshold energy) we are able to write an exact expression for the photoemission intensity which involves the standard time inverted LEED state and broadened hole-states. If inelastic extrinsic losses are neglected our expression is similar to the one used by e.g. Feibelman and Eastman [1], Pendry [2] and Spanjaard et al. [3].

Our theory also allows inelastic losses (including interference between intrinsic and extrinsic losses) to be evaluated. The expressions are similar to those obtained by Inglesfield [4], but the damping effect no longer has to be put in by hand. Our theory further can handle not only core electron but also valence electron photoemission and it is not restricted to metals but applies to all types of solids.

We have also traced why and when Inglesfield's one step theory agrees with a semiclassical calculation.

The fraction of atoms which appear in a charge state different from the dominant adiabatic (usually neutral) one after scattering or sputtering from a metal surface is calculated. We use a time-dependent Anderson Hamiltonian with an intraatomic Coulomb repulsion term. An approximate solution is found by using only a small subset of electronic states. The intraatomic correlation is taken into account by not allowing double occupation of the atomic orbital. The dependence of the non-adiabatic fraction upon the atom velocity v is similar to the one found by using a non-interacting Hamiltonian, and is proportional to exp (–c/v). Some experimental results are discussed, but true quantitative comparison is at present not possible.

We have used the real-space renormalization group method to investigate the Aubry model of incommensurate potential. The exact numerically solved density of states exhibits the self-similarity property and is nowhere dense. From the behaviour of single-particle Green's function, we have shown that the localized-to-extended transition is energy dependent. The Avron-Simon singularity is interpreted physically as the extremely large fluctuation of the local density of states from one site to the other.

Concentrations of particulate bromine and lead have been measured during one summer and one winter period. The measurements were made simultaneously in five sites in a city on the Swedish west coast. A rural site about 60 km from the city was used to measure the background aerosol.

Aerosol sampling was made with six dichotomous virtual impactors, which fractionate the aerosol into two modes, one fine particle mode (aerodynamic diameter, a.d. < 3.5 μm) and one coarse particle mode (3.5 μm < a.d. < 18 μm). The aerosol was collected onto thin teflon filters.

Element concentrations were obtained by Energy Dispersive X-Ray Fluorescence Analysis. The element concentrations were related to air mass trajectories.

The Br/Pb ratio proved to be the same on a given date for the city sites and the background site. A dependence on the air mass history was found, suggesting that it is the quality of the air basin in the region that influences the Br/Pb ratio even for fresh car exhaust. The Br/Pb ratio was the same for fine and coarse particles, indicating that the ratio is determined before coagulation with larger particles occur.

The ratios between coarse and fine particles containing lead and bromine respectively were also studied. The results suggest that lead and bromine are actually attached to the same particles.