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

How do these symmetries create this lattice? (in combination with translation ‘ofcourse’! 2-fold 1 i2i2 i1i1 2-fold 2 m v2 m v1 mhmh Subscript 1  At lattice pointsSubscript 2  Between lattice points t Click to proceed

t m v1 * Note: m h cannot create the lattice starting from a point m v2 2-fold 2 i2i2 m 1 *  this is actually (m v1 + t) ! t will be applied to all these operators  else we will get no lattice! One of the 2-folds (2-fold 2 ) and one of the inversion centres (i 2 ) have been chosen for illustration

t m v1 * m v2 2-fold 2 i2i2  Only points being added to the right are shown

t m v1 * m v2 2-fold 2 i2i2

t m v1 * m v2 2-fold 2 i2i2

t m v1 * m v2 2-fold 2 i2i2  Only points being added to the right are shown  Note that only a partial lattice is created  Similarly 2-fold 1 and i 1 will create partial lattices and so forth..

Q & A

Time for some Q & A  Why do we have to invoke translation (‘ofcourse’!) to construct the lattice?  Without the translation the point will not move!  There are some symmetry operators like Glide Reflection which can create a lattice by themselves as they have translation built into themGlide Reflection Origin of the Point Groups Point Groups Symmetry operators (without translational component) acting at a point will leave a finite set of points around the point

 Many of the symmetry operators seem to produce the same effect. Then why use them?  There will always be some redundancy with respect to the effect of symmetry operators (or their combinations)  This problem is pronounced in lower dimension where many of them produce identical effects. There are no left or right handed objects in 1D hence a 2-fold, an inversion centre and a mirror all may produce the same effect. Analogy: This is like a tensor looking like a ‘vector’ in 1-D, looking like a ‘scalar’ in 0D!  Hence, when we go to higher dimensions some of the differences will become clear  If translation is doing all the job of creating a lattice, then why the symmetry operators?  As we know lattices are being used to make crystals  crystals are based on symmetry  One should note that as translation can create a lattice an array of symmetry operators can also create a lattice (this array itself can be considered a lattice or even a crystal!)  Symmetry operators are present in the lattice even if one decides to ignore themmake crystals