Motoo Hori

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. I. Perturbation expansions for the effective permittivity of cell materials

A perturbation formulation is developed for the effective permittivity of random heterogeneous materials that are statistically homogeneous but not necessarily statistically isotropic. The formal perturbation solution is expressed explicitly by means of the many‐point correlation functions of the permittivity field. For a broad class of random multiphase materials called cell materials, the second‐ and third‐order perturbation effects are determined as functions of the depolarizing factor tensors of cells. In the special case where the medium is statistically isotropic, the formulas giving the effective permittivity to third order are shown to be in substantial accord with the results obtained by previous investigators.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. II. Bounds for the effective permittivity of statistically anisotropic materials

Upper and lower bounds are derived for the effective permittivity of random heterogeneous materials that are statistically homogeneous but not necessarily statistically isotropic. As trial functions in standard variational principles, we use modified perturbation expansions for the electric field and the electric displacement. The bounds are expressed in terms of the many‐point correlation functions of the spatial variation of permittivity. For a wide class of random multiphase materials called cell materials, explicit calculation is performed taking account of the three‐point correlation effects.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. IV. Effective−medium theory and cumulant expansion method

Several perturbation solutions for the effective permittivity in a completely random medium are evaluated and the validity of the approximations is discussed. It is shown through a diagram technique that the effective−medium theory in a macroscopically inhomogeneous material is equivalent to the coherent−potential approximation in a discordered binary alloy not only in its physical concept but also in its mathematical structure. A cumulant expansion method which substantially takes into account the clustering effects is proposed while the effective−medium theory or the coherent−potential approximation is, by its nature, a single−site approximation and neglects the clustering effects. The numerical results obtained by the effective−medium theory and by the cumulant method for a binary mixture of a conducting material and an insulating material are compared with the computer simulation data on the effective conductivity of a three−dimensional random network. The solution of the cumulant method gives a remar...

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. VII. Comparison of different approaches

A general analysis is given for the theoretical evaluation of the effective permittivity of random heterogeneous materials that are statistical homogeneous. The perturbative and variational formulations presented in Papers I–VI in this series of work are reconstructed from a slightly different point of view and compared with recent approaches developed by other authors. Perturbation expansions for the effective permittivity are derived through electric field, electric displacement, Lorentz field, and T matrix. The validity of various approximate solutions involving the effective‐medium approximation and the cumulant expansion method is discussed with the aid of a diagrammatic representation of the perturbation series. It is confirmed that, at the present stage, the cumulant theory is the best approximation for a three‐dimensional system, while the effective‐medium theory is the best for a two‐dimensional system. The meaning and applicability of variational approaches are also reviewed.

Statistical theory of effective electrical, thermal, and magnetic properties of random heterogeneous materials. V. One− and two−dimensional systems

A perturbation theory is developed for the effective permittivity of one− and two−dimensional random heterogeneous materials that are statistically homogeneous. Closely following the formulations for three−dimensional systems presented in Papers I−IV in this series of work, we derive the formal perturbation solutions for two−dimensional systems, evaluate the second−order and third−order terms for cell materials, and determine upper and lower bounds of the effective permittivity. Here statistical isotropy is not necessarily required. Several approximate perturbation solutions for completely random systems (which are statistically isotropic) are obtained by summing some selected partial series of the perturbation expansion up to an infinite order and numerical results are illustrated. We analyze validity of approximations by means of diagram representation of the perturbation series. The effective−medium theory serves as a good approximation for two−dimensional systems and gives the critical percolation con...

Percolation in Penrose tiling and its dual: in Comparison with Analysis for Kagomé, Dice and Square lattices

Abstract We combine the idea of finite-size scaling and Monte Carlo simulations to study percolation in Penrose tiling (nonperiodic with mixed-valued coordination), its dual (non periodic with coordination number z = 4), dice (periodic with mixed-valued coordination), kagome (periodic with z = 4), and square (periodic with z = 4). We estimate percolation thresholds and some critical exponents of these lattices both for bond and site percolation 1 . Our analysis suggests that ‘ universality ’ concerning critical exponents holds even in lattices without periodicity and/or without single-valued coordination.

Molecular orbital study for electronic structures of metallic microclusters

Abstract In order to investigate the electronic structure of potassium microclusters, we execute molecular-orbital calculations up to the number of atoms N = 48. In particular, we evaluate the total energy, the ionization potentials ( IP ), and level spacing Δϵ near the highest-occupied molecular orbital as functions of N . The atomic configurations composed of pairs of K atoms tend to be stable, and the size dependences of IP and Δϵ indicate that K N is metallic even when N is as small as 3. On the basis of our results, we also discuss the data for K in comparison with the data for Hg microclusters.