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The average distance of the equal hard spheres is introduced to evaluate the density of a given arrangement. The absolute smallest value is two radii because the spheres can not be closer to each other than their diameter. The absolute densest arrangement of two, three and four spheres is defined, which gives the absolute highest density in one, two and three dimensions. The absolute highest density of equal spheres in three dimensions is the tetrahedron formed by the centers of four spheres touching each other with density of 0.7796. The density of this tetrahedron unit can be maintained only locally because the tetrahedron units can not be expanded to form a tightly packed arrangement in three dimensions. The maximum number of tetrahedron units that one sphere is able to accommodate is twenty which corresponds to the density of 0.684. The only compatible formation of equal spheres which can be mixed with tetrahedron is octahedron. In order to mix the tetrahedron and octahedron units certain geometrical constrains must be satisfied. It is shown that the only possible mixture of tetrahedrons and octahedrons units is the one which accommodates eight tetrahedron and six octahedron vertexes which is identical to FCC and an alternative proof for the Kepler conjecture. It is suggested that there is a density gap between the FCC density and the highest density of disordered arrangements and that the icosahedrons configuration with its 0.684 density represents the upper bound on the disordered arrangements.
We present a short review of our studies of disorder influence upon Ginzburg - Landau expansion coefficients in Anderson - Hubbard model with attraction in the framework of the generalized DMFT+$Sigma$ approximation. A wide range of attractive potent
Using computed x-ray tomography we determine the three dimensional (3d) structure of binary hard sphere mixtures as a function of composition and size ratio of the particles, q. Using a recently introduced four-point correlation function we reveal th
We study disorder effects upon the temperature behavior of the upper critical magnetic field in attractive Hubbard model within the generalized $DMFT+Sigma$ approach. We consider the wide range of attraction potentials $U$ - from the weak coupling li
Amorphous solids or glasses are known to exhibit stretched-exponential decay over broad time intervals in several of their macroscopic observables: intermediate scattering function, dielectric relaxation modulus, time-elastic modulus etc. This behavi
We report B_c2 data for LaO_{0.9}F_{0.1}FeAs_{1-delta} in a wide T and field range up to 60 Tesla. The large slope of B_c2 approx ~ -6 Tesla/K near an improved T_c = 28.5 K of the in-plane B_c2(T) contrasts with a flattening starting at 23 K above 30