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MATERIALS SCIENCE & ENGINEERING

MATERIALS SCIENCE & ENGINEERING. Part of. A Learner’s Guide. AN INTRODUCTORY E-BOOK. 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.

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MATERIALS SCIENCE & ENGINEERING

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  1. MATERIALS SCIENCE & ENGINEERING Part of A Learner’s Guide AN INTRODUCTORY E-BOOK 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 http://home.iitk.ac.in/~anandh/E-book.htm

  2. How do these symmetries create this lattice? (in combination with translation ‘ofcourse’! t mh 2-fold2 i1 2-fold1 mv1 mv2 i2 Subscript 1  At lattice points Subscript 2  Between lattice points Click to proceed

  3. One of the 2-folds (2-fold2) and one of the inversion centres (i2) have been chosen for illustration t mv1* mv2 2-fold2 i2 Note: mh cannot create the lattice starting from a point m1*  this is actually (mv1 + t) ! t will be applied to all these operators  else we will get no lattice!

  4. Only points being added to the right are shown t mv1* mv2 2-fold2 i2

  5. t mv1* mv2 2-fold2 i2

  6. t mv1* mv2 2-fold2 i2

  7. t • Only points being added to the right are shown • Note that only a partial lattice is created • Similarly 2-fold1 and i1 will create partial lattices mv1* mv2 2-fold2 i2 and so forth..

  8. Q & A

  9. 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 them Origin of the Point Groups Symmetry operators (without translational component) acting at a point will leave a finite set of points around the point

  10. 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 them

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