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

2. 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

3. + = 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

4. 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)