1. 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

2. Size Factor compounds: (i) Laves phases (ii) Frank-Kasper Phases D  These phases have a formula: AB 2  Laves phases can be regarded as Tetrahedrally Close Packed (TCP)* structures with an ideal ratio of the radii (r A /r B ) = (3/2) 1/2 ~1.225 [or usually r A /r B  (1.1, 1.6)]  If r A /r B = 1.225 then a high packing density is achieved with the chemical formula AB 2 with a average coordination number of 13.3  Crystal structures:  Hexagonal → MgZn 2 (C15), MgNi 2 (C36)  FCC → MgCu 2 (C14)  There are more than 1400 members belonging to the ‘Laves family’  Many ternary and multinary representatives of the Laves phases have been reported with excess of A or B elements. Some ternary Laves phases are known in systems with no corresponding binary Laves phases.  The range of existence of the three phases (C15, C36, C14) in ternary Laves phases is influenced by the e/a ratio D(i) Laves Phases * Also called Topologically Close Packed structures?

3.  Laves phases containing transition metals as components have interesting Physical and mechanical properties. Engineering materials based on Laves phases are being developed for:  High temperature applications (for use in turbine blade fine precipitates of Laves phases is shown to improve fatigue strength)  Hydrogen storage applications (in nickel-metal hydride batteries)

4. MgZn 2 (Laves) Lattice parameter(s)a = 5.18 Å, c = 8.52 Å Space GroupP 6 3 /mmc (194) Strukturbericht notationC15 Pearson symbolhP12 Other examples with this structure NbCr 2 Wyckoff position Site Symmetry xyzOccupancy Mg4f3m0.330.670.0621 Zn12a-3m0001 Zn26hmm20.830.660.251 MgZn 2 Laves Phase Mg Zn2 Zn1 [0001] Hexagonal C14 Zn: Vertex-1, Edge-1, Inside cell-6 → 8 Mg: Inside cell-4 → 4 Unit cell formula: Mg 4 Zn 8

5. MgZn 2 Laves Phase More views Constructing the hexagonal laves phase Start with a layer of Zn atoms Put Mg atoms in the depressions formed in the layer (above and below) Add a hexagonal array of Zn atoms in the depressions formed by the Mg atoms (above and below) This gives us half the unit cell in ‘c’ direction

6. Mg (8a) Cu (16d) MgCu 2 (Laves) Lattice parameter(s)a = 7.048 Å Space GroupFd-3m (227) Strukturbericht notationC14 Pearson symbolcF24 Other examples with this structure Au 2 Pb MgCu 2 Laves Phase Cubic [001] Wyckoff position Site Symmetry xyzOccupancy Cu16d-3m0.625 1 Mg8a-43m0001 C15 Very frequent structural type Unit cell formula: Mg 8 Cu 16 Mg: Vertex-1, FC-3, Inside cell-4 → 8 Cu: Inside cell-16 → 16

7. More views MgCu 2 Laves Phase Successive layers are build on the depressions on the previous layer

8. More views Tetrahedra of Cu Note: the solid lines in the figures are for visualization of atomic positions etc. (they are not meant to show bonds) MgCu 2 Laves Phase Not to scale

9. D(ii) Frank-Kasper  Have coordination numbers (CN): CN =12, CN = 14, CN = 15, CN = 16

10. Al 12 W (Frank-Kasper) Lattice parameter(s)a = 7.58 Å Space GroupIm-3 (204) Strukturbericht notation Pearson symbolcI26 Other examples with this structure Al 12 Mn, Al 12 Mo Wyckoff position Site Symmetry xyzOccupancy Al24gm00.1840.3091 W2am-30001 Al 12 W Frank-Kasper Phase Al W [001] Cubic CN =12 Unit cell formula: Al 24 W 2 W: Vertex-1, BC-1 → 2 Cu: FC-12, Inside cell-12 → 24 Motif: 12Al +W (consistent with stoichiometry) Lattice: Body Centred Cubic

11. More views Al 12 W Frank-Kasper Phase  Icosahedral coordination around W atoms  Local icosahedral symmetry is destroyed in the long range packing  Note that icosahedral symmetry is not found in crystals  This phase is closely related to quasicrystals

12. More views Al 12 W Frank-Kasper Phase [100] [110] [111]