GESUS, an Interactive Computer Application for Teaching and Learning the Space Groups of Symmetry

Adolfo Miras,Domingo Martín,Agustín Cota

doi:10.3390/educsci12020085

Abstract

This paper describes a digital application designed for learning Space Groups of Symmetry (SGS). It teaches how to recognize the operations performed by the symmetry elements, both point (or 2D) operators (proper and improper rotations, including mirroring, inversion, and other rotoinversions), and space (or 3D) operators (screw axes and glide planes), as well as their combinations with the translations of the lattice. The software applies a 3D space vision to identify the elements of symmetry compatible with the proposed structural models. The symbols of internationally accepted representation are used. The system, class, and space group of the crystal that agree with the proposed model are solved. Two settings are taken into account in the Monoclinic system. The application self-evaluates and assesses the knowledge acquired, allowing each exercise to be re-done until it is correctly completed with the appropriate recommendations. This application constitutes a useful and easy-to-use tool for SGS learning. It is aimed at beginner students of crystallography, with elementary knowledge about elements of symmetry, Bravais lattices, crystal classes, and wallpaper groups.

Highlights

Knowledge of crystalline structures and their symmetry properties is essential for understanding the implications derived from the physical and chemical properties of most solids
Symmetry is used as a fundamental property of the orderly arrangements of atoms found in crystals, as it characterizes the external well-formed crystals, identifies repeating patterns of molecules, and simplifies both data collection and most calculations
To the best of our knowledge, there are no applications that interactively teach how to deduce the position of each symmetry element using the “international standard” space group diagram, either by taking into account the arrangement of motifs or combinations of symmetry operators

Summary

Introduction

Knowledge of crystalline structures and their symmetry properties is essential for understanding the implications derived from the physical and chemical properties of most solids. When the structural data of a compound are described in the literature, the space group symbol, unit cell dimensions, and coordinates “x, y, z” for the atoms are provided For this reason, curricula with topics related to crystalline matter (Chemistry, Materials Engineering, Geology, Condensed Matter Physics, etc.) usually include subjects such as Crystallography, in which space symmetry is taught. To the best of our knowledge, there are no applications that interactively teach how to deduce the position of each symmetry element using the “international standard” space group diagram (the main difference from the above-mentioned software or applications), either by taking into account the arrangement of motifs or combinations of symmetry operators It is to be understood as an international standardized space group diagram, as the symmetry element diagram displays the location and orientation of the symmetry elements of the space group. Those diagrams are orthogonal projections of the space-group unit cell onto the plane of projection along a basis vector of the conventional crystallographic coordinate system

A Crystallographic Overview and Its Implications for GESUS

Target Users

Conclusions