Anandh Subramaniam & Kantesh Balani 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: 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

Superlattices and Ordered Structures 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. Click here to revise concepts about Sublattices & Subcrystals Click here to see XRD patterns from ordered structures * 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 revise concepts about Sublattices & Subcrystals Order-Disorder Transformations 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. Click here to revise concepts about Sublattices & Subcrystals

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

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, Cu3Au..] 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

NiAl Lattice: Simple Cubic NiAl Let us consider some more ordered structures NiAl This is similar to CuZn Lattice: Simple Cubic SC Unit cell formula: NiAl Motif: 1Ni + 1Al NiAl Lattice parameter(s) a = 2.88 Å, Space Group P 4/m 3 2/m (221) Strukturbericht notation B2 Pearson symbol cP2 Other examples with this structure CsCl, CuZn

Lattice: Simple Tetragonal CuAu (I) Cu Au Cu Lattice: Simple Tetragonal Au Motif: 2Au +2Cu (cosistent with stoichiometry) Unit cell formula: Cu2Au2 CuAu Lattice parameter(s) a = 3.96Å, c = 3.67Å Space Group P4/mmm (123) Strukturbericht notation L10 Pearson symbol tP4 Other examples with this structure TiAl Wyckoff position x y z Au1 1a Au2 1c 0.5 Cu 2e

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 Cu subcrystal-2 Origin for the Cu subcrystal-1 Origin for the Au subcrystal-2 Origin for the Au subcrystal-1 All ‘subcrystals’ are tetragonal (primitive)

Motif: 3Cu +1Au (consistent with stoichiometry) Cu3Au Lattice: Simple Cubic Motif: 3Cu +1Au (consistent with stoichiometry) Cu3Au Lattice parameter(s) a = 3.75 Å Space Group Pm-3m (221) Strukturbericht notation L12 Pearson symbol cP4 Other examples with this structure Ni3Al, TiPt3

Ni3Al Ni3Al This is similar to Cu3Au Lattice parameter(s) a = 3.56 Å Space Group Pm-3m (221) Strukturbericht notation L12 Pearson symbol cP4 Other examples with this structure Cu3Au, TiPt3

Strukturbericht notation Al3Ni [100] [010] [001] Formula for Unit cell: Al12Ni4 Al3Ni Lattice parameter(s) a = 6.62 Å, b = 7.47 Å, c = 4.68 Å Space Group P 21/n 21/m 21/a (Pnma) (62) Strukturbericht notation DO20 Pearson symbol oP16 Other examples Fe3C Wyckoff position Site Symmetry x y z Occupancy Ni 4c .m. 0.3764 0.25 0.4426 1 Al1 0.0388 0.6578 Al2 8d 0.1834 0.0689 0.1656

Fe3Al Fe Al Fe3Al Unit cell formula: Fe12Al4 Fe2 (¼,¼,¼) Fe1 (½,½,0) 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: Fe12Al4 Fe3Al Lattice parameter(s) a = 5.792 Å Space Group Fm-3m (225) Strukturbericht notation DO3 Pearson symbol cF16 Other examples with this structure Fe3Bi Wyckoff position Fe1 4a Fe2 8c 0.25 Al 4b 0.5

Motif: 3Fe +1Al (consistent with stoichiometry) Lattice: Face Centred Cubic 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

Fe3Al More views Fe Al [100]

Fe1 and Fe2 have different environments Fe3Al Fe3Al More views Fe1 and Fe2 have different environments Fe2 (¼,¼,¼) Fe1 (½,½,0) Fe1 (0,0,0) Fe1 (0,0,0) Cube of Fe Tetrahedron of Fe Tetrahedron of Al Fe2 (¼,¼,¼) Fe1 (½,½,0)

Spin Ordering 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

Other examples with this structure Antiferromagnetic ordering MnF2 is antiferromagnetic below 67K (TN) (b) (a) MnF2 Lattice parameter(s) a = 4.87, b = 4.87, c = 3.31 (Å) Space Group P42/mnm (136) Pearson symbol tP6 Other examples with this structure TiO2 Wyckoff position Site Symmetry x y z Occupancy Mn 2a m.2m 1 F 4f m.mm 0.305

Other examples with this structure MnO Disordered Lattice parameter(s) a = 4.4 (Å) Space Group Fm-3m (225) Pearson symbol cF8 Other examples with this structure NaCl Wyckoff position Site Symmetry x y z Occupancy Mn 4a m-3m 1 O 4b 0.5

Anti ferromagnetic MnO, TN =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

Other examples with this structure MnAu2 Nice example of antiferromagnetic ordering where the spins are not anti-parallel. TN = 363K Helical spin structure Metamagnetic behaviour- field induced transition to ferromagnetism MnAu2 Lattice parameter(s) a =3.37, b = 3.37, c = 8.79 Å Space Group I4/mmm (139) Pearson symbol tI6 Other examples with this structure MoSi2 Wyckoff position Site Symmetry x y z Occupancy Mn 2a 4/mmm 1 Au 4e 4mm 0.335

Cu3Au Disordered Ordered Disordered Ordered - Ni3Al, FCC L12 (AuCu3-I type) Ordered

Superlattice lines