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Ordered Structures Please revise the concept of Sublattices and ‘Subcrystals’ by clicking here before proceeding with this topic MATERIALS SCIENCE &ENGINEERING.

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Presentation on theme: "Ordered Structures Please revise the concept of Sublattices and ‘Subcrystals’ by clicking here before proceeding with this topic MATERIALS SCIENCE &ENGINEERING."— Presentation transcript:

1 Ordered Structures Please revise the concept of Sublattices and ‘Subcrystals’ by clicking here before proceeding with this topic MATERIALS SCIENCE &ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email:, URL: AN INTRODUCTORY E-BOOK Part of A Learner’s Guide

2  Often the term superlattice # and ordered structure is used interchangeably.  The term superlattice implies the superlattice is made up of sublattices.  An ordered structure (e.g. CuZn, B2 structure*) is a superlattice. An ordered structure is a product of an ordering transformation of an disordered structure (e.g. CuZn BCC structure*)  But, not all superlattices are ordered structures. E.g. NaCl crystal consists of two subcrystals (one FCC sublattice occupied by Na + ions and other FCC sublattice by Cl  ions). So technically NaCl is a superlattice (should have been called a supercrystal!) but not an ordered structure. Superlattices and Ordered Structures Click here to revise concepts about Sublattices & Subcrystals * Explained in an upcoming slide. # Sometimes the term superlattice is used wrongly: e.g. in the case of Ag nanocrystals arranged in a FCC lattice the resulting Nano-crystalline solid is sometimes wrongly referred to as a ‘superlattice’. Click here to see XRD patterns from ordered structures

3  On interesting class of alloys are those which show order-disorder transformations  Typically the high temperature phase is dis-ordered while the low temperature phase is ordered (e.g. CuZn system  next slide)  The order can be positional or orientational  In case of positionally ordered structures:  The ordered structure can be considered as a superlattice  The ‘superlattice’ consists of two or more interpenetrating ‘sub-lattices’  with each sublattice being occupied by a specific elements (further complications include: SL-1 being occupied by A-atoms and SL-2 being occupied by B & C atoms- with probabilistic occupation of B & C atoms in SL-2, which is disordered)  Order and disorder can be with respect to a physical property like magnetization. E.g. in the Ferromagnetic phase of Fe, the magnetic moments (spins) are aligned within a domain. On heating Fe above the Curie temperature the magnetic moments become randomly oriented, giving rise to the paramagnetic phase.  Even vacancies get ordered in a sublattice. Order-Disorder Transformations Click here to revise concepts about Sublattices & Subcrystals

4 Positional Order G = H  TS High T disordered Low T ordered 470 ºC Sublattice-1 (SL-1) Sublattice-2 (SL-2) BCC SC SL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (½, ½, ½) In a strict sense this is not a crystal !! Probabilistic occupation of each BCC lattice site: 50% by Cu, 50% by Zn Diagrams not to scale

5 ORDERING  A-B bonds are preferred to AA or BB bonds e.g. Cu-Zn bonds are preferred compared to Cu-Cu or Zn-Zn bonds  The ordered alloy in the Cu-Zn alloys is an example of an INTERMEDIATE STRUCTURE that forms in the system with limited solid solubility  The structure of the ordered alloy is different from that of both the component elements (Cu-FCC, Zn-HCP)  The formation of the ordered structure is accompanied by change in properties. E.g. in Permalloy ordering leads to → reduction in magnetic permeability, increase in hardness etc. [~Compound]  Complete solid solutions are formed when the ratios of the components of the alloy (atomic) are whole no.s → 1:1, 1:2, 1:3 etc. [CuAu, Cu 3 Au..]  Ordered solid solutions are (in some sense) in-between solid solutions and chemical compounds  Degree of order decreases on heating and vanishes on reaching disordering temperature [  compound]  Off stoichiometry in the ordered structure is accommodated by: ◘ Vacancies in one of the sublattices (structural vacancies) NiAl with B2 structure Al rich compositions result from vacant Ni sites ◘ Replacement of atom in one sublattice with atoms from other sublattice NiAl with B2 structure Ni rich compositions result from antisite defects

6 Let us consider some more ordered structures NiAl Lattice parameter(s)a = 2.88 Å, Space Group P 4/m  3 2/m (221) Strukturbericht notationB2 Pearson symbolcP2 Other examples with this structureCsCl, CuZn SC Motif: 1Ni + 1Al Lattice: Simple Cubic Unit cell formula: NiAl This is similar to CuZn

7 CuAu Lattice parameter(s)a = 3.96Å, c = 3.67Å Space GroupP4/mmm (123) Strukturbericht notationL1 0 Pearson symboltP4 Other examples with this structureTiAl CuAu (I) Cu Au Cu Au Wyckoff position xyz Au11a000 Au21c0.5 0 Cu2e00.5 Motif: 2Au +2Cu (cosistent with stoichiometry) Lattice: Simple Tetragonal Unit cell formula: Cu 2 Au 2

8 Q & A  How to understand the CuAu ordered structure in terms of the language of superlattices? The crystal is simple (primitive) tetragonal The formula for the UC is 2Cu + 2Au  we need two sublattices for Au and two sublattices for Cu Cu Au Origin for the Au subcrystal-1 Origin for the Au subcrystal-2 Origin for the Cu subcrystal-1 Origin for the Cu subcrystal-2 All ‘subcrystals’ are tetragonal (primitive)

9 Cu 3 Au Lattice parameter(s)a = 3.75 Å Space GroupPm-3m (221) Strukturbericht notationL1 2 Pearson symbolcP4 Other examples with this structureNi 3 Al, TiPt 3 Cu 3 Au Cu Au Motif: 3Cu +1Au (consistent with stoichiometry) Lattice: Simple Cubic

10 This is similar to Cu 3 Au Ni 3 Al Lattice parameter(s)a = 3.56 Å Space GroupPm-3m (221) Strukturbericht notationL1 2 Pearson symbolcP4 Other examples with this structureCu 3 Au, TiPt 3

11 Al 3 Ni Lattice parameter(s)a = 6.62 Å, b = 7.47 Å, c = 4.68 Å Space GroupP 2 1 /n 2 1 /m 2 1 /a (Pnma) (62) Strukturbericht notationDO 20 Pearson symboloP16 Other examplesFe 3 C Al 3 Ni [001] [010] [100] Wyckoff position Site Symmetry xyzOccupancy Ni4c.m.0.37640.250.44261 Al14c.m.0.03880.250.65781 Al28d10.18340.06890.16561 Formula for Unit cell: Al 12 Ni 4


13 Fe 3 Al Lattice parameter(s)a = 5.792 Å Space GroupFm-3m (225) Strukturbericht notationDO 3 Pearson symbolcF16 Other examples with this structureFe 3 Bi Fe 3 Al Al Fe Fe2 (¼,¼,¼) Fe1 (½,½,0) Fe1 (0,0,0) Wyckoff position Fe14a000 Fe28c0.25 Al4b0.500 Dark blue: Fe at corners Lighter blue: Fe at face centres V. Light Blue: Fe at ( ¼,¼,¼) Fe: Vertex-1, FC-3, (¼,¼,¼)-8 → 12 Al: Edge-3, BC-1 → 4 Unit cell formula: Fe 12 Al 4

14 Al Fe Motif: 3Fe +1Al (consistent with stoichiometry) Lattice: Face Centred Cubic Fe Assignment: (i) try to put the motif at each lattice point and obtain the entire crystal (ii) Chose alternate motifs to accomplish the same task

15 More views [100] Al Fe Fe 3 Al

16 More views Fe 3 Al Fe2 (¼,¼,¼) Fe1 (½,½,0) Fe1 and Fe2 have different environments Tetrahedron of Fe Tetrahedron of Al Fe2 (¼,¼,¼) Fe1 (0,0,0) Fe1 (½,½,0) Cube of Fe

17  In Ferromagnets, Ferrimagnets and Antiferromagnets, spin (magnetization vector) is ordered.  A schematic of the possible orderings is shown in the figure below (more complicated orderings are also possible!).  We shall consider Antiferromagnetism as an example to show the formation of superlattices (ordered structures).  Above the Curie or Néel temperature the spin structure will become disordered and state would be paramagnetic Spin Ordering

18  MnF 2 is antiferromagnetic below 67K (T N ) Antiferromagnetic ordering MnF 2 Lattice parameter(s)a = 4.87, b = 4.87, c = 3.31 (Å) Space GroupP4 2 /mnm (136) Pearson symboltP6 Other examples with this structureTiO 2 Wyckoff position Site Symmetry xyzOccupancy Mn2am.2m0001 F4fm.mm0.305 01 (a) (b)

19 Wyckoff position Site Symmetry xyzOccupancy Mn4am-3m0001 O4bm-3m0.5 1 MnODisordered Lattice parameter(s)a = 4.4 (Å) Space GroupFm-3m (225) Pearson symbolcF8 Other examples with this structureNaCl

20  Anti ferromagnetic MnO, T N =122K  Perfect order not obtained even at low temperatures  Rhombohedral angle changes with lowering of temperature  Rhombohedral, a = 8.873,  = 90  26’ at 4.2K  even above Néel temperature order persists in domains about 5nm in size

21  Nice example of antiferromagnetic ordering where the spins are not anti-parallel.  T N = 363K  Helical spin structure  Metamagnetic behaviour- field induced transition to ferromagnetism MnAu 2 Lattice parameter(s)a =3.37, b = 3.37, c = 8.79 Å Space GroupI4/mmm (139) Pearson symboltI6 Other examples with this structureMoSi 2 Wyckoff position Site Symmetry xyzOccupancy Mn2a4/mmm0001 Au4e4mm000.3351


23 Ordered Disordered Cu 3 Au DisorderedOrdered  - Ni 3 Al, FCC L1 2 (AuCu 3 -I type)

24 Superlattice lines

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