Use of the radial distribution function with the Chebyshev distance to characterize orientation of two-dimensional square crystals

In the present work, the orientation of two-dimensional square crystals is studied using the radial distribution function (RDF) defined with the Chebyshev distance. Traditional RDF methods, based on the Euclidean metric, fail to detect changes in crystal orientation due to their isotropic nature. To overcome this limitation, we introduce a modification of the RDF employing the Chebyshev metric, which adds sensitivity to directional changes within square lattices. This approach expands the information conventionally obtained from the RDF by incorporating not only translational properties but also orientational information. The study provides a mathematical analysis of the peak distribution in this enhanced RDF approach when applied to square crystals rotated through various orientations. We found that the orientation of square crystals can be efficiently determined by calculating the abscissa of the first peak in the Chebyshev RDF, which reduces computational times, even for fine numerical resolutions of differential annular widths as small as 10^{-3} or less. This approach can be applied to any two-dimensional (2D) Bravais lattice structure, with potential modifications based on lattice parameters and is robust for analyzing imperfect lattices with noise and vacancies. In this study, we demonstrate its application on rotated 2D Bravais square lattices, laying the groundwork for future studies on other 2D lattice structures. Additionally, we explore the use of this method to characterize jamming in 2D granular cubic systems, focusing on the orientation of oblique square grains in tilted vibrated layers under cyclic shear.