Consequences of local gauge symmetry in empirical tight-binding theory

A method for incorporating electromagnetic fields into empirical tight-binding theory is derived from the principle of local gauge symmetry. Gauge invariance is shown to be incompatible with empirical tight-binding theory unless a representation exists in which the coordinate operator is diagonal. The present approach takes this basis as fundamental and uses group theory to construct symmetrized linear combinations of discrete coordinate eigenkets. This produces orthogonal atomic-like "orbitals" that may be used as a tight-binding basis. The coordinate matrix in the latter basis includes intra-atomic matrix elements between different orbitals on the same atom. Lattice gauge theory is then used to define discrete electromagnetic fields and their interaction with electrons. Local gauge symmetry is shown to impose strong restrictions limiting the range of the Hamiltonian in the coordinate basis. The theory is applied to the semiconductors Ge and Si, for which it is shown that a basis of 15 orbitals per atom provides a satisfactory description of the valence bands and the lowest conduction bands. Calculations of the dielectric function demonstrate that this model yields an accurate joint density of states, but underestimates the oscillator strength by about 20% in comparison to a nonlocal empirical pseudopotential calculation.Comment: 23 pages, 7 figures, RevTeX4; submitted to Phys. Rev.

Particle hydrodynamics with tessellation techniques

Lagrangian smoothed particle hydrodynamics (SPH) is a well-established approach to model fluids in astrophysical problems, thanks to its geometric flexibility and ability to automatically adjust the spatial resolution to the clumping of matter. However, a number of recent studies have emphasized inaccuracies of SPH in the treatment of fluid instabilities. The origin of these numerical problems can be traced back to spurious surface effects across contact discontinuities, and to SPH's inherent prevention of mixing at the particle level. We here investigate a new fluid particle model where the density estimate is carried out with the help of an auxiliary mesh constructed as the Voronoi tessellation of the simulation particles instead of an adaptive smoothing kernel. This Voronoi-based approach improves the ability of the scheme to represent sharp contact discontinuities. We show that this eliminates spurious surface tension effects present in SPH and that play a role in suppressing certain fluid instabilities. We find that the new `Voronoi Particle Hydrodynamics' described here produces comparable results than SPH in shocks, and better ones in turbulent regimes of pure hydrodynamical simulations. We also discuss formulations of the artificial viscosity needed in this scheme and how judiciously chosen correction forces can be derived in order to maintain a high degree of particle order and hence a regular Voronoi mesh. This is especially helpful in simulating self-gravitating fluids with existing gravity solvers used for N-body simulations.Comment: 26 pages, 24 figures, currentversion is accepted by MNRA

A multi-scale homogenization scheme for modeling anisotropic material’s elastic and failure response

The effect of small-scale random defects such as microcracks or inclusions are critical to the prediction of material failure, yet including these in a fracture simulation can be difficult to perform efficiently. Typically, work has focused on implementing these through a statistical characterization of the micro- or meso-scales. This characterization has traditionally focused on the spatial distribution of faults, assuming the material is purely isotropic. At the macro-scale, many materials can be assumed to be fully isotropic and homogeneous, but at the small scale may show significant anisotropy or heterogeneity. Other materials may be effectively anisotropic in bulk, such as rock bedding planes. Statistical volume elements (SVE) are one homogenization methodology used to retain this heterogeneity or anisotropy when characterizing a material. Unlike a Representative Volume Element (RVE), the choice of SVE including size, boundary conditions applied, shape, and type, may affect the given material properties. In addition, the size which an RVE exists is well-studied for homogeneity, but there is less study of the isotropic limit. This work introduces a multi-scale methodology using SVEs to study material heterogeneity and anisotropy. Results are given for macroscopic fracture simulations using this SVE-based homogenization scheme. In addition, the rate of convergence to the RVE limit for both the homogeneous and isotropic limit of two types of SVE, Regular Square and Voronoi Square, are shown. This methodology shows promise for characterization of both isotropic and anisotropic materials