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symmetry-adapted fourier series for the wallpaper groups壁纸symmetry-adapted傅里叶级数组.pdf

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symmetry-adapted fourier series for the wallpaper groups壁纸symmetry-adapted傅里叶级数组

Symmetry 2012, 4, 379-426; doi:10.3390/sym4030379 OPEN ACCESS symmetry ISSN 2073-8994 /journal/symmetry Article Symmetry-Adapted Fourier Series for the Wallpaper Groups Bart Verberck , Departement Fysica, Universiteit Antwerpen, Groenenborgerlaan 171, 2020 Antwerpen, Belgium; E-Mail: bart.verberck@ua.ac.be ¨ ¨ ¨ Institut fur Theoretische Physik und Astrophysik, Universitat Wurzburg, Am Hubland, D-97074 ¨ Wurzburg, Germany Received: 11 June 2012; in revised form: 27 June 2012 / Accepted: 5 July 2012 / Published: 17 July 2012 Abstract: Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefﬁcients. Visualising the equivalencies of Fourier coefﬁcients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefﬁcient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefﬁcient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride. Keywords: wallpaper groups; Fourier series; symmetry-adapted functions 1. Introduction The central concept for describing the