Hierarchic Computations in Geometric Space—Showing Sharp Diffraction in Quasicrystals

A quasi-structure factor method is used to show how sharp diffraction patterns are produced by aperiodic quasicrystals. Icosahedral symmetry is described for the tensor rank 3 solids with edge-sharing unit cells that are pentagonally close packed in hierarchic structures having a geometric reciprocal lattice. The hierarchic symmetry replaces translational symmetry in crystal diffraction. Details in the calculation show how the symmetry is simulated in diffraction.

Diffraction Line Width in Quasicrystals—Sharper than Crystals

A quasicrystal has a structure intermediate between crystals and compound glasses. The disorder in glass makes its diffraction diffuse, so it is surprising that quasicrystals diffract more sharply than crystals. The greater sharpness is computed to be due to the hierarchic structure with unit cell alignment in 3-dimensional space. Electron microscope phase contrast images map the comparatively heavy Mn atoms in icosahedral Al6Mn, where the transition metal locates the centers of unit cells inside clusters and superclusters. Because the solid is aperiodic, each diffracted beam is a product of multiple interplanar spacings combined, and this contrasts with the unique relationship between spacing and incident angle in Bragg diffraction from crystals. Simulated quasi-structure factors add the relative phase shifts that are in geometric series from cell to cluster to superclusters of increasing order. The scattering becomes coherent in best fit, angular configuration between the aperiodic solid and a longitudinally periodic X-ray or electron probe. The quasi-structure factors express angular divergence in each diffracted beam from its corresponding Bragg condition, and the divergence provides a special metric, essential for atomic measurement in the geometric solids. The fit is reinforced at all levels from the unit cell to cluster to high order superclusters. The optics operates under a new quasi-Bragg law in a new geometric space. In this paper, we proceed to examine the effect of specimen size on line resolution in diffraction, first analytically and secondly in simulation. The line resolution follows a power law on the supercluster order, matching its atomic population.

Strong Analog Classical Simulation of Coherent Quantum Dynamics

A strong analog classical simulation of general quantum evolution is proposed, which serves as a novel scheme in quantum computation and simulation. The scheme employs the approach of geometric quantum mechanics and quantum informational technique of quantum tomography, which applies broadly to cases of mixed states, nonunitary evolution, and infinite dimensional systems. The simulation provides an intriguing classical picture to probe quantum phenomena, namely, a coherent quantum dynamics can be viewed as a globally constrained classical Hamiltonian dynamics of a collection of coupled particles or strings. Efficiency analysis reveals a fundamental difference between the locality in real space and locality in Hilbert space, the latter enables efficient strong analog classical simulations. Examples are also studied to highlight the differences and gaps among various simulation methods.