Topic

# Symmetry (geometry)

About: Symmetry (geometry) is a research topic. Over the lifetime, 7710 publications have been published within this topic receiving 113087 citations.

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TL;DR: In this paper, a dichotomy method for indexing powder diffraction patterns for low-symmetry lattices is studied in terms of an optimization of bound relations used in the comparison of observed data with the calculated patterns generated at each level of the analysis.

Abstract: The dichotomy method for indexing powder diffraction patterns for low-symmetry lattices is studied in terms of an optimization of bound relations used in the comparison of observed data with the calculated patterns generated at each level of the analysis. A rigorous mathematical treatment is presented for monoclinic and triclinic cases. A new program, DICVOL91, has been written, working from the cubic end of the symmetry sequence to triclinic lattices. The search of unit cells is exhaustive within input parameter limits, although a few restrictions for the hkl indices of the first two diffraction lines have been introduced in the study of triclinic symmetry. The efficiency of the method has been checked by means of a large number of accurate powder data, with a very high success rate. Calculation times appeared to be quite reasonable for the majority of examples, down to monoclinic symmetry, but were less predictable for triclinic cases. Applications to all symmetries, including cases with a dominant zone, are discussed.

1,459 citations

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TL;DR: The left-right symmetry of the starting gauge interactions is a "natural" symmetry if it is broken in no way except possibly by mass terms in the Lagrangian as mentioned in this paper.

Abstract: It is remarked that left-right symmetry of the starting gauge interactions is a "natural" symmetry if it is broken in no way except possibly by mass terms in the Lagrangian. The implications of this result for the unification of coupling constants and for parity nonconservation at low and high energies are stressed.

1,117 citations

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TL;DR: In this article, Rossmann and Blow describe how they have detected the existence of partial, approximate symmetry from a knowledge of the intensities alone, and the effect of noncrystallographic symmetry, whether partial or total, results in decreasing the size of the structure to be determined, while the number of observable intensities remains the same.

Abstract: In this paper, Rossmann & Blow describe how they have detected the existence of partial, approximate symmetry from a knowledge of the intensities alone. The effect of noncrystallographic symmetry, whether partial or total, results in decreasing the size of the structure to be determined, while the number of observable intensities remains the same.

863 citations

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TL;DR: In this article, it was shown that reflection symmetry can be employed to generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken.

Abstract: Second-order topological insulators are crystalline insulators with a gapped bulk and gapped crystalline boundaries, but with topologically protected gapless states at the intersection of two boundaries. Without further spatial symmetries, five of the ten Altland-Zirnbauer symmetry classes allow for the existence of such second-order topological insulators in two and three dimensions. We show that reflection symmetry can be employed to systematically generate examples of second-order topological insulators and superconductors, although the topologically protected states at corners (in two dimensions) or at crystal edges (in three dimensions) continue to exist if reflection symmetry is broken. A three-dimensional second-order topological insulator with broken time-reversal symmetry shows a Hall conductance quantized in units of e^{2}/h.

807 citations