Transcript Slide 1

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