Exploring high-symmetry structures in non-Cartesian coordinates: preparation and characteristics of cylindrically symmetric-rotating crystals

Since their discovery in 1981, quasicrystals have challenged traditional crystallography, showcasing the diversity of material structures. Quasicrystals are ordered structures in higher-dimensional Cartesian coordinate systems projected into lower dimensions. Exploring whether materials can grow into high-symmetry ordered structures in different mathematical coordinate systems is an intriguing topic. This paper reports a new crystallization growth mode in InSiO thin films, exhibiting high symmetry in spherical and cylindrical coordinate systems, named cylindrical symmetric rotating crystals (CSRC). These crystals show rotational symmetry in their lattice structure, consistent with cylindrical symmetry. They are also part of spherical crystals, which theoretically have one of the highest symmetries. Theoretical predictions suggest that a two-dimensional slice of a spherical crystal in conventional grating mode SEM images will display relatively complete Kikuchi patterns due to the wave properties of incident electrons, unlike crystals or quasicrystals, which show surface morphology from the particle properties of incident electrons. We formed InSiO CSRC by heating InSiO amorphous films and observed relatively complete Kikuchi patterns that quantitatively relate to the incident electron beam direction and energy. These findings align with theoretical predictions, indicating that the Kikuchi patterns in SEM images result from Bragg diffraction through coherent electron scattering within the crystals. Quantitative analysis of these patterns provides information on the lattice structure, constants, crystal orientation, stress information, defect concentration, and Brillouin zone of the InSiO material. This method of obtaining microstructural information directly from SEM images applies only to CSRC, not to conventional or quasicrystal samples.