Anandh Subramaniam & Kantesh Balani 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
The rotations compatible with translational symmetry are (1, 2, 3, 4, 6) In general any n-fold rotational axis is possible, where ‘n’ divides 360 without leaving a reminder. E.g. 1, 2, 3, 4, 5, 8, 12- fold axes are possible. Apart from 1 (trivial), 2, 3, 4, 6- fold axes, other rotations are not compatible with translational symmetry. We will see next “why” this is so. First, let us observe two rows of lattice points in a hexagonal lattice. Note that after the action of the 6-fold axes (twice over) the spacing between the lattice points in the next row is 2t. (AA’, BB’). Some of the disallowed crystallographic symmetries like 5, 8, 12- fold are seen in QUASICRYSTALS. In quasicrystals translational periodicity is absent, but they have inflationary symmetry.
Consider two rows of lattice points: R1 and R2 A is taken to A’ by the lattice translation vector t (modulus is t) Point A is taken to B’ by a rotation axis and similarly A’ is taken to B by a rotation axis If B and B’ have to qualify as lattice points then BB’ must be an integral multiple of AA’, i.e. b = mt (where m is an integer) m & (1 m) are integers Say (1 m) is M Thus Cos can take only half-integral values. The permitted values are in the table in the next slide.
M Cos n = 2/ 3 3/2 - 2 1 2 1/2 2/3 3 /2 4 1 ½ /3 6 /2 4 1 ½ /3 6 1 3/2 When we talk about n-fold, we also include the roto-reflection and roto-inversion axes.
Funda Check Is 5-fold symmetry not allowed in crystals because of the fact that pentagons cannot tile the plane. (Noting that pentagons have 5-fold symmetry). Yes, the following two statements are true: ‘A’ regular pentagon cannot tile the plane ‘monohedrally’ 5-fold symmetry is not allowed in crystals. This does not imply that this is the reason (for the fact) that crystals cannot have 5-fold symmetry. E.g. A regular tetrahedron cannot tile space monohedrally; however, tetrahedral symmetry (23) is observed in crystals! A interesting aside: though regular pentagons cannot tile the plane, a non-regular version of the same can tile the plane (found in Egypt- as created by ancient artisans). The number of such non-regular pentagons which can tile the plane is ‘not known’ (i.e. the number of ways in which a non-regular pentagon can tile the plane is not known).