We classify the point symmetries at the different points in the Brillouin zone for the 17 two-dimensional space groups and the symmetries of the Bloch waves for the 10 two-dimensional crystallographic point groups. Simple examples involving breakdown of Friedels law, Gjonnes-Moodie lines, and reflection and refraction at interfaces are presented.
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http://dx.doi.org/10.1016/j.ultramic.2010.08.003 | DOI Listing |
Acta Crystallogr A Found Adv
July 2022
Université de Lorraine, CNRS, CRM2, Nancy, France.
Diffraction enhancement of symmetry (DES) is the phenomenon according to which the symmetry of the diffraction pattern of a crystal can be higher than the point symmetry of the structure that has produced it. The most well known example is that of Friedel's law, which is however violated in the presence of resonant scattering. This phenomenon is addressed in monoarchetypal modular structures and it is shown that a sufficient condition for DES is that both the module and the family of stacking vectors are invariant under an isometry that is not a symmetry operation for the structure.
View Article and Find Full Text PDFACS Nano
July 2020
Department of Electrical Engineering, The Pennsylvania State University, University Park, Pennsylvania 16802, United States.
The inspection of Friedel's law in ultrafast electron diffraction (UED) is important to gain a comprehensive understanding of material atomic structure and its dynamic response. Here, monoclinic gallium telluride (GaTe), as a low-symmetry, layered crystal in contrast to many other 2D materials, is investigated by mega-electronvolt UED. Strong out-of-phase oscillations of Bragg peak intensities are observed for Friedel pairs, which does not obey Friedel's law.
View Article and Find Full Text PDFUltramicroscopy
August 2020
School of Applied and Engineering Physics, Cornell University, Ithaca, USA.; Kavli Institute at Cornell for Nanoscale Science, Ithaca, USA. Electronic address:
Friedel's law guarantees an inversion-symmetric diffraction pattern for thin, light materials where a kinematic approximation or a single-scattering model holds. Typically, breaking Friedel symmetry is ascribed to multiple scattering events within thick, non-centrosymmetric crystals. However, two-dimensional (2D) materials such as a single monolayer of MoS can also violate Friedel's law, with unexpected contrast between conjugate Bragg peaks.
View Article and Find Full Text PDFPhys Chem Chem Phys
March 2017
EaStCHEM, School of Chemistry, University of Edinburgh, David Brewster Road, EH9 3FJ Edinburgh, UK.
An approach for calculating elastic X-ray scattering from polyatomic molecules in specific electronic, vibrational, and rotational states is presented, and is used to consider the characterization of specific states in polyatomic molecules using elastic X-ray scattering. Instead of the standard independent atom model (IAM) method, the X-ray scattering is calculated directly from ab initio wavefunctions. The role of molecular symmetry and Friedel's law is examined, with the molecules BF, CH, NF, and 1,3-cyclohexadiene used as specific examples.
View Article and Find Full Text PDFNat Commun
July 2016
School of Electrical Engineering and Computer Science, University of Ottawa, 800 King Edward Avenue, Ottawa, Ontario, Canada K1N 6N5.
Recently, synthetic optical materials represented via non-Hermitian Hamiltonians have attracted significant attention because of their nonorthogonal eigensystems, enabling unidirectionality, nonreciprocity and unconventional beam dynamics. Such systems demand carefully configured complex optical potentials to create skewed vector spaces with a desired metric distortion. In this paper, we report optically generated non-Hermitian photonic lattices with versatile control of real and imaginary sub-lattices.
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