Space-filling Potentials Research Articles

Correlation of surface fractal dimension with frictional angle at critical state of sands

INTRODUCTION The fractal approach is potentially a useful tool for studying the surface formation of sand particles (Anvir et al., 1985). A simple descriptor of surface texture is provided by the surface fractal dimension, which is assumed as an indicator to describe the surface roughness. The surface fractal dimensions of Ottawa sand and Madagascar quartz sand are 2.02 0.01 and 2.14 0.06 respectively, measured by sorption of gases and liquids (Anvir et al., 1985). Many researchers have estimated the surface fractal dimension of sand particles and have given it different significances (Moore & Donaldson, 1995; Vallejo, 1995; Vallejo & Zhou, 1995; Barak et al., 1996; Pardini & Gallart, 1998; Gori & Mari, 2001; Itabashi et al., 2001). Moore & Donaldson (1995) quantified soil microstructure as reflected by interconnectivity among the soil grains using fractal principles. A high fractal dimension implied a high space-filling potential. From the point of view of soil strength and compressibility, one would expect a soil with a high fractal dimension to be stronger and less compressible than one with a low fractal dimension (Moore & Donaldson, 1995). Vallejo (1995) and Vallejo & Zhou (1995) described the shape of granular materials using the surface fractal dimension. Pardini & Gallart (1998) applied laser technology to measure the effect of freezing and thawing on the surface roughness and surface fractal dimension of regolith. Gori & Mari (2001) measured the surface fractal dimension of sand particles using the imagery method, and tried to correlate surface fractal dimension with the internal friction angle of sand. It was thought that the surface fractal dimension of sand particles was able to describe all shape characteristics, such as particle shape, roundness, sphericity, and texture. Itabashi et al. (2001) evaluated the surface fractal dimension of sand particles through measuring the grain perimeter using a different ruler size on an enlarged photomicrograph. Itabashi et al. (2001) stated that the surface fractal dimension of sand particles represented the particle shape of microparticles and the surface roughness of microparticles. The values of the surface fractal dimension of sand particles were found in the range 2.01–2.14 from published data (Vallejo, 1995; Vallejo & Zhou, 1995; Barak et al., 1996; Pardini & Gallart, 1998; Gori & Mari, 2001; Itabashi et al., 2001). Quartz sand particles exhibit smoother surfaces, which result in surface fractal dimensions close to 2.0 (Barak et al., 1996). As regards the surface roughness and particle shape of sand particles, the fractal dimension has proved to be a good parameter for describing the particle microstructure, and it has some advantages over other methods used previously. In the work described in this paper, the surface fractal dimension was used to measure the surface roughness of sand particles, and the surface fractal dimension of 10 sand samples taken from different locations in Japan was measured using the imagery method. The relationship between the surface fractal dimension and the critical state angle of shearing resistance for sands was also studied.

Full potential photoemission theory: application to MgO and CaO

The full potential photoemission theory is applied for the first time to the alkaline-earth oxides MgO and CaO. This theory is a straightforward generalization of the one-step model of photoemission for the case of anisotropic, space-filling potentials and several atoms per unit cell. It allows the calculation of angle-resolved ultraviolet photoemission (ARUPS) and bremsstrahlung isochromat spectra (ARBIS) from compounds. The calculations are based on space-filling crystal potentials calculated self-consistently with the full potential Korringa-Kohn-Rostoker (KKR) band structure method. As a first application of this generalized one-step model we present calculated angle-resolved photoemission spectra from the (100) surfaces of MgO and CaO and compare the results with corresponding experimental data. The results include the self-consistent band structures and a comparison of muffin-tin and full potential results. The overall agreement between experiment and theory is excellent.

Multiple-scattering theory for space-filling cell potentials.

The multiple-scattering theory (MST) method of Korringa, and of Kohn and Rostoker for determining the electronic structure of solids, originally developed in connection with potentials bounded by non- overlapping spheres (muffin-tin potentials), is generalized to the case of space-filling potential cells of arbitrary shape. Both variational and nonvariational formalisms are used in effecting this generalization. In contrast to the case of muffin-tin potentials, different forms of MST exhibit different convergence rates for the energy and the wave function. Numerical results are presented that illustrate the differing convergence rates of the variational and nonvariational forms of MST for space-filling potentials. The generalized MST described here should be useful quite generally for constructing global solutions to linear partial differential equations from sets of locally exact solutions.

Multiple Scattering Theory with Space Filling Potentials

ABSTRACTA brief account of the current state or our understanding of full cell multiple scattering theory is given. The development of this understanding is also briefly outlined.

Korringa-Kohn-Rostoker Electronic Structure Method for Space-Filling Cell Potentials

ABSTRACTThe multiple scattering theory (MST) method of Korringa, and of Kohn and Rostoker for determining the electronic structure of solids, originally developed in connection with potentials bounded by noa-overlapping spheres (Muffin-tin (MT) potentials), is generalized to the case of space-filling potential cells of arbitrary shape through the use of a variational formalism. This generalized version of MST retains the separability of structure and potential characteristic of the application of MST to MT potentials. However, in contrast to the MT case, different forms of MST exhibit different convergence rates for the energy and the wave function. Numerical results are presented which illustrate the differing convergence rates of the variational and nonvariatonal forms of MST for space-filling potentials.

Full-potential Korringa-Kohn-Rostoker band theory applied to the Mathieu potential.

The band theory of Korringa, Kohn, and Rostoker (KKR) based on the Green-function method is extended to space-filling potentials. A numerical test using the Mathieu potential shows good convergence for the bands up to 1.5 Ry with {ital l}{le}4 included in the angular-momentum expansion for the wave functions. Our results strongly support the applicability of the full-potential KKR to bulk electronic-structure problems.

Validity, accuracy, and efficiency of multiple-scattering theory for space-filling scatterers.

The equations of multiple-scattering theory (MST) are solved for the case of a lattice of square-well potentials in the shape of squares that fill the plane. We find that multiple-scattering theory is exact within reasonably achievable numerical accuracy. There is a problem with the convergence of MST in its traditional formulation that has obscured the fact that it is an exact theory even for non-muffin-tin, space-filling potentials. We show that this problem can be easily circumvented and describe a rapidly convergent, exact formulation of MST.

Multiple-scattering Green-function method for space-filling cell potentials.

It is shown that the equations of multiple-scattering theory (MST) originally derived for scattering off collections of muffin-tin (MT) potentials, i.e., potential cells bounded by non overlapping spheres, remain valid in the case of arbitrarily shaped, non overlapping, and particularly space-filling potential cells. Specifically, it is shown that in the angular momentum representation the total scattering (transition) matrix, the Greenfunction, the Bloch function for a translationally invariant material, and the Lloyd formula for the change in the integrated density of states have forms that are invariant with respect to the partition of a given potential into non overlapping cells, and with respect to the choice of the cell centers. Ananalytic proof is provided for the vanishing of near-field corrections (NFC's)long conjectured to arise when the spheres bounding individual cells over lap one another or adjacent potentials. Thus, the well-known MST expressions, originally derived for the case of MT potentials, for obtaining the solution of the Schroedinger equation and hence determining the band structure and the charge density of materials, ordered or disordered, are rigorously valid in the completely general case of arbitrarily shaped cells. The differences between this work and previous attempts to generalize MST to non-MT space-filling potentials aremore » discussed. It is pointed out that in calculations involving non-MT potential cells, particular attention must be paid to the question of convergence of expansions in angular momentum eigenstates. This convergence is tested numerically in terms of cluster calculations and through the calculation of the electronic structure of elemental bcc Nb and fcc Zr and Rh.« less