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

Mystery of the missing entries in the Bravais List!  Few illustrations  Another point of view: There are 7 crystal systems Each of them “can have” P, I, F, C type lattice (4 types) Hence the “potential” number is (7  4) = 28 But 14 of these are only distinct Points to note:  Every lattice that you can construct is present somewhere in the list  the issue is where to put them  Concept of choice of unit cell is also invoked along with the classification of Bravais lattices  The factors which are taken to into account are (like for the unit cell):  Symmetry  Size

Mystery of the missing entries in the Bravais List! 1CubicCube   Hence even though this lattice remains as it is  it is called Simple Tetragonal (which is smaller in size) Hence Cannot be called Cubic But then Cubic crystals need not have any 4-fold axes!! (cubic lattices do need to have!) What we choose PIFC

FCT = BCT 2TetragonalSquare Prism (general height)   Mystery of the missing entries in the Bravais List! What we choose PIFC Smaller sized Body Centred Cell is chosen Face Centred Tetragonal = Body Centred Tetragonal

CCT = ST 2TetragonalSquare Prism (general height)   Mystery of the missing entries in the Bravais List! What we choose PIFC Smaller sized Simple Cell is chosen C Centred Tetragonal = Simple Tetragonal

4Hexagonal 120  Rhombic Prism   Mystery of the missing entries in the Bravais List! PIFC Putting a lattice point at body centre destroys the 6-fold axis Hence body centred hexagonal lattice NOT possible

4Hexagonal 120  Rhombic Prism   Mystery of the missing entries in the Bravais List! PIFC Putting a lattice point at body centre destroys the 6-fold axis

4Hexagonal 120  Rhombic Prism   Mystery of the missing entries in the Bravais List! PIFC Putting a lattice point at body centre destroys the 6-fold axis

& what we don’t Mystery of the missing entries in the Bravais List! As the FCC lattice has higher symmetry than the BCT cell (the one with higher symmetry is chosen) Cubic F  Tetragonal I (not chosen)

Some example of the allowed and disallowed transfers Crystal systemBravais lattices 1.CubicPIFC 2.TetragonalPIF 3.OrthorhombicPIFC 4.HexagonalP 5.TrigonalP 6.MonoclinicPC 7.TriclinicP Slide Courtesy Dr. Rajesh Prasad x