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.
Highlights
The central concept for describing the structure of crystalline solids is that of the unit cell, the periodic repetition of which results in a spatially extended crystal exhibiting particular symmetries.For both three- and two-dimensional (3D and 2D) crystals, the enumeration of possible unit cell symmetries was first carried out by E.S
Apart from the explicit expressions for the symmetry-adapted Fourier series and the symmetry relations between Fourier coefficients, we provide visualisations showing how direct-space symmetry is transformed into coefficient-space symmetry
̸ c0,0, it follows from Equation (8) and the p2 symmetry property (Table 5) that for wallpaper groups having a 2-fold rotation axis at the origin, all coefficients ck1,k2 must be real for f (x, y) to be real
Summary
The central concept for describing the structure of crystalline solids is that of the unit cell, the periodic repetition of which results in a spatially extended crystal exhibiting particular symmetries. The information compiled in the “International Tables” [4,5,6,7,8] is useful for interpreting diffraction experiments: X-ray, neutron, and electron diffraction are the de facto methods for crystal structure determination and provide access to a crystal’s space group and to the contents of its unit cell. The bulk of the paper is formed by the 17 tables with detailed information on the Fourier series and Fourier coefficients for each of the wallpaper groups (Section 5). Its equilibrium properties (e.g., electronic density) are expected to display the 2D symmetry of its underlying atomic structure (wallpaper group p6mm, see below)
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