Anandh Subramaniam & Kantesh Balani We had already seen that the following guidelines are used w.r.t to choice of unit cell: When possible we chose a primitive unit cell The factors governing the choice of unit cell are:  Symmetry of the Unit Cell  should be maximum (corresponding to lattice)  Size of the Unit Cell  should be minimum  Convention  if above fails to resolve the issue we use some convention (We will see later - using an example- that convention is not without common sense!) MATERIALS SCIENCE & ENGINEERING 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 AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s Guide

Alternate choice of unit cells for Orthorhombic lattices In the case of the orthorhombic lattices the importance of convention comes into play E.g. for the case of the C centred OR lattice two possible choices of UC exist: the primitive rhombic prism and the rectangular prism A casual glance may make it seem that the rectangular prism has a higher symmetry than the rhombic prism → actually this is not true and both cells have the same symmetry (2/m 2/m 2/m) Alternate choice of unit cell for “C”(C-centred orthorhombic) case. The new (orange) unit cell is a rhombic prism with (a = b  c,  =  = 90o,   90o,   120o) Both the cells have the same symmetry  (2/m 2/m 2/m) In some sense this is the true Ortho- “rhombic” cell

In fact a consistent alternate set of axis can be chosen for Orthorhombic lattices and there is not clear winner based on size either! z = 0 & z = 1 Conventional Alternate choice (“ortho-rhombic”) P C (2ce the size) I F (2ce the size) F I (½ the size) C P ( ½ the size) z = ½ Note: All spheres represent lattice points. They are coloured differently but are the same

If based on symmetry or size there is no reason to chose the rectangular prism over the rhombic prism, then why do we chose the square prism UC? The ‘simpler’ answer is convention- but, is there a reason behind this convention? The next slide shows that there is actually a ‘sub-conscious’ intuitive reason for the same (though which may not be conceived as ‘that scientific’.  We artificially introduce some two fold axis like 2x, 2y etc. – along the axes of the unit cells in ‘c’ or ‘z’ projection. For the rectangular prism this is consistent with the symmetry in projection, while for the rhombic prism this leads to new points on the plane (not consistent with the symmetry in projection). Hence, at the intuitive level (‘perhaps in the functioning of our brains’) the rectangular prism is seen as having higher symmetry!

2d produces this additional point not part of the original lattice Intuitively one might feel that the orthogonal cell has a higher symmetry  is there some reason for this? 2d produces this additional point not part of the original lattice This is in addition to our liking for 90! The 2x and 2y axes move lattice points out the plane of the sheet in a semi-circle to other points of the lattice (without introducing any new points). The 2d axis introduces new points which are not lattice points of the original lattice. The motion of the lattice points under the effect of the artificially introduced 2-folds is shown as dashed lines (---).