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

Hexagonal Close Packed Crystal While SC, FCC, BCC are Lattices; HCP, DC (Diamond Cubic) are Crystals HCP Crystal = Hexagonal Lattice + two identical spheres as motif {at (0,0,0) and (2/3, 1/3, 1/2)}  Note: there is only one type of Hexagonal Lattice  (Simple Hexagonal Lattice) (If the spheres are of different size then the crystal is not HCP) Like CCP Crystal (CCP Crystal = FCC lattice + single sphere motif) HCP is not just close packed but is closest packed! (Packing fraction of 0.74  74% of space is filled with spheres) Points to be noted: HCP has a packing fraction of 74% assuming a hard sphere packing. This packing also implies an ideal c/a ratio (see below). As real crystals do not have an ideal c/a ratio (table below), this also implies that they are not closest packed in the ‘strictest sense’. Metal c/a Cd 1.886 Zn 1.856 Mg 1.624 Zr 1.590 Ti 1.588 Be 1.586 IDEAL c/a

= + Hexagonal Lattice HCP crystal Motif Two atom Motif The atoms at c = ½ [e.g. (2/3, 1/3, 1/2)] are part of the motif and is not located at a lattice point + Motif As we have seen that HCP crystal is constructed starting with a hexgonal close packed layer, with ABABAB.. packing. The 6-fold axis (present in a single layer) is lost on ABABAB.. packing and only a 3-fold remains. Though 3 unit cells are often shown in diagrams- the rhombic prism (blue outline) is the unit cell. Two atom Motif

Spheres at: (0,0,0), (⅔, ⅓,½) Note: Atoms are coloured differently but are the same

Q: If the HCP crystal has only a 3-fold axis why does it come under the ‘Hexagonal class*’ of crystals? A: The HCP crystal has ABAB.. packing. The C site is vacant! A 63 Screw axis passes through the C-sites (along [0001] direction). This six (6) is responsible for the crystal coming under the hexagonal crystal system (even though there is no usual 6-fold axis) * As hexagonal word would imply 6-fold