Anandh Subramaniam & Kantesh Balani MATERIALS SCIENCE & ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

Connection between Geometry & Symmetry From reading some of the topics so far one might get an impression that the: “Geometry of the unit cell has no connection with the symmetry” !! One may even be tempted to conclude (e.g.) that choice of: “Square Prism Unit Cell (Tetragonal UC) for the Tetragonal Crystal is purely coincidental Let us consider an example to clarify the matter The crux of the issue lies in the fact that many of the examples considered are “ideal geometrical” examples  which have been used to illustrate basic concepts In real crystals with atomic entities  nature decides the final outcome and we are left to analyze the outcome in the language of crystallography

+ = Let us consider a cubic unit cell decorated with a two atom motif UC of cubic lattice This is an ideal geometrical example If the entities are atomic a better picture would be We will use two views

But then these atoms will not be hanging loosely in space But then these atoms will not be hanging loosely in space!  The atoms would be touching each other  an even better picture would be Side View Front View New Unit Cell As you can see this automatically leads to an unit cell in which a = b  c i.e. a Square Prism unit cell of Tetragonal crystal system Hence a lower symmetry motif can lead to an altered geometry of the unit cell (of lower symmetry)