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 Close Packed Crystals

 Initially we consider ‘usual’ type of close packed crystals, which are made of single kind of sphere.  In other types of close packed crystals (e.g. tetrahedrally close packed crystals, also called topologically close packed crystals), more than one size of sphere may be involved.  One may even conceive of close packing of ellipsoids and other non-spherical objects.  Cubic Close Packed (CCP- commonly called FCC crystal also) and Hexagonal Close Packed (HCP) are two common examples of close packed crystals.  The term close packed crystal implies closest packed crystal (having a packing fraction of 0.74).  The proof that this is the densest crystallographic packing of spheres possible is a difficult one (and will not be considered here).  CCP and HCP are just two examples among a series of close packed structures which can be envisaged (shown in coming slides).  Every atom in these structures has a coordination number of 12  forming a Cubeoctahedron or a Twinned Cubeoctahedron (around the central atom).

 The common starting point is a close packed layer of atoms with 6-fold symmetry.  Identical layers are stacked one on another with a shift.  The shift is such that the atoms in the above (and below) layers sit in the ‘valleys’ formed by a layer.  All such possibilities (see coming slides) lead to Close Packed Crystals.  The original 6-fold symmetry present in a single layer is lost on this kind of packing (you must be aware of the 3-fold present in CCP and HCP crystals!). Yes! HCP crystal has NO true 6-fold axis!CCPHCP

CCP  Coordination Polyhedron  Cubeoctahedron HCP  Coordination Polyhedron  Twinned Cubeoctahedron

 Starting Point  Hexagonal layer  Three positions A (the first layer atomic positions), B & C (Valleys) are shown  The second layer (of hexagonal packing of atoms) can be positioned in valley B (or equivalently in valley C) Step-1 Step-2 A AB Part of the hexagonal layer shown

 The third layer can be positioned with atoms directly above the A layer (Option-1) or with atoms above the C layer (Option-2) Step-3 Layer-3 (Option-1) (Option-2) C-site vacant ABC ABA Continuing this ABAB sequence we get the HCP structure Continuing this ABCABC sequence we get the CCP structure (Though not obvious!)

 ABCABC… & ABAB… are just but two amongst the infinite possibilities  At each stage of construction we have a choice of putting an atomic layer at A, B or C position  Possibilites include:  ABCAB.ABCAB.ABCAB…  ABCABCAB.ABCABCAB.ABCABCAB…  Hence we can construct crystals with larger and larger unit cells.  If we randomly put the layers we will not get a crystal in the ‘true sense’. (We can think of these as 2D crystals, which are not periodic in 3 rd dimension).  Few stages in the infinite choice tree is shown below. A B C A C A B B C A B B C B C Track a branch to infinity or truncate at some stage and repeat to get a structure

 In the ABCABC… packing we start with a layer having 6-fold symmetry. Interestingly, this packing leads to a 4-fold axis at an angle of  to the original 6-fold axis and to the familiar Cubic Close Packed crystal (‘FCC’ unit cell) Actually a –3 (3 bar) roto-inversion pseudo-axis

 Rigid sphere-like atoms without long range interactions can arrange in any of the infinite possibilities shown before.  Not only can we have ordered sequences, but also disordered close packed sequences (the diorder is in the way ‘A’, ‘B’ & ‘C’ appear and not within a given plane (say ‘A’)  If Cobalt is annealed above and below 450  C a disordered sequence of ABC packing is obtained (T >450  C Co → ABCABC… packing, T <450  C Co → ABAB… packing) LayersStackingExampleStacking symbol 2ABMg (hP2, P6 3 /mmc)2H 3ABCCu (cF4, Fm–3m)3C 4ABACLa (hp4, P6 3 /mmc)4H 9ABABCBCACSm (hR3, R–3m)9R Some examples of various stacking sequences  Lipson & Stokes (Proc. Roy. Soc. A, 181, ) showed the formation of trigonal graphite with stacking sequence ABCA instead of ABAB. Note: Graphite is not a close packed structure.  SiC (not close packed structure) shows many polytypes. Common ones are: 3C-SiC (cubic unit cell, zincblende); 2H-SiC; 4H-SiC; 6H-SiC (hexagonal unit cell, wurtzile ); 15R-SiC (rhombohedral unit cell).  Among the polytypes of diamond the following is the decreasing order of stability: 3C > 6H > 9R > 4H > 2H.

Lattice parameter(s)a = 3.77Å, c = Å Space GroupP6 3 /mmc (194) Strukturbericht notation Pearson symbolhp4 Other examples with this structure Wyckoff position Site Symmetry xyzOccupancy La12a-3m0001 La22c-6m La A layer B layer Closed packed crystal [0001] Note: All atoms are La C layer

More views A layer B layer C layer A B C