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MATERIALS SCIENCE & ENGINEERING. Part of. A Learner’s Guide. AN INTRODUCTORY E-BOOK. 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. - PowerPoint PPT Presentation

Ordered StructuresPlease revise the concept of Sublattices and Subcrystals by clicking herebefore proceeding with this topic

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 StructuresClick 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

One 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. A vacancy in a vacancy sublattice is an atom!!The disordered phase is truly an amorphous structure and is considered a crystal in probabilistic occupational sense. [Click here to know more]Order-Disorder TransformationsClick here to revise concepts about Sublattices & Subcrystals

Positional OrderG = H TSHigh T disorderedLow T ordered470CSublattice-1 (SL-1)Sublattice-2 (SL-2)BCCSCSL-1 occupied by Cu and SL-2 occupied by Zn. Origin of SL-2 at (, , )In a strict sense this is not a crystal !!Probabilistic occupation of each BCC lattice site: 50% by Cu, 50% by ZnDiagrams not to scaleIn the order-disorder transformation shown in the figures belowthe high temperature phase is disordered and has a BCC lattice, while the low temperature structure is simple cubic (B2 structure). B2 structure

ORDERINGA-B bonds are preferred to AA or BB bonds e.g. Cu-Zn bonds are preferred compared to Cu-Cu or Zn-Zn bondsThe ordered alloy in the Cu-Zn alloys is an example of an INTERMEDIATE STRUCTURE that forms in the system with limited solid solubilityThe 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 compoundsDegree 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

Let us consider some more ordered structuresNiAlMotif: 1Ni + 1AlLattice: Simple CubicUnit cell formula: NiAlThis is similar to CuZnTwo interpenetrating Simple Cubic crystals (origin of crystal-1 at (0,0,0) and origin of crystal-2 at (,,))

NiAlLattice parameter(s)a = 2.88 Space GroupP 4/m 3 2/m (221)Strukturbericht notationB2Pearson symbolcP2Other examples with this structureCsCl, CuZn

CuAu (I)CuAuCuAuMotif: 2Au +2Cu (consistent with stoichiometry)Lattice: Simple Tetragonal

CuAuLattice parameter(s)a = 3.96, c = 3.67 Space GroupP 4/m 2/m 2/m (123)Strukturbericht notationL10Pearson symboltP4Other examples with this structureTiAl

Wyckoff positionxyzAu11a000Au21c0.50.50Cu2e00.50.5

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

Cu3AuCuAuMotif: 3Cu +1Au (consistent with stoichiometry)Lattice: Simple Cubic

Cu3AuLattice parameter(s)a = 3.75 Space GroupP 4/m 3 2/m (221)Strukturbericht notationL12Pearson symbolcP4Other examples with this structureNi3Al, TiPt3

This is similar to Cu3AuNi3Al

Ni3AlLattice parameter(s)a = 3.56 Space GroupP 4/m 3 2/m (221)Strukturbericht notationL12Pearson symbolcP4Other examples with this structureCu3Au, TiPt3

Al3Ni[001][010][100]Formula for Unit cell: Al12Ni4

Al3NiLattice parameter(s)a = 6.62 , b = 7.47 , c = 4.68 Space GroupP 21/n 21/m 21/a (Pnma) (62)Strukturbericht notationDO20Pearson symboloP16Other examplesFe3C

Wyckoff positionSite SymmetryxyzOccupancyNi4c.m.0.37640.250.44261Al14c.m.0.03880.250.65781Al28d10.18340.06890.16561

Fe3AlAlFeFe2 (,,)Fe1 (,,0)Fe1 (0,0,0)Dark blue: Fe at corners Lighter blue: Fe at face centresV. Light Blue: Fe at (,,)Fe: Vertex-1, FC-3, (,,)-8 12Al: Edge-3, BC-1 4

Fe3AlLattice parameter(s)a = 5.792 Space GroupF 4/m 3 2/m (225)Strukturbericht notationDO3Pearson symbolcF16Other examples with this structureFe3Bi

Wyckoff positionFe14a000Fe28c0.250.250.25Al4b0.500

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

More views[100]AlFeFe3Al

Fe3AlMore viewsFe3AlFe2 (,,)Fe1 (,,0)Fe1 and Fe2 have different environmentsTetrahedron of FeTetrahedron of AlFe2 (,,)Fe1 (0,0,0)Fe1 (0,0,0)Fe1 (,,0)Cube of Fe

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 Nel temperature the spin structure will become disordered and state would be paramagneticSpin Ordering

(a) Ferromagnetic

(b) Antiferromagnetic

(e) Helical magnetic (spin)

(c) Ferrimagnetic

(d) Canted Antiferromagnetic

MnF2 is antiferromagnetic below 67K (TN)Antiferromagnetic ordering(a) (b)

MnF2Lattice parameter(s)a = 4.87, b = 4.87, c = 3.31 ()Space GroupP42/mnm (136)Pearson symboltP6Other examples with this structureTiO2

Wyckoff positionSite SymmetryxyzOccupancyMn2am.2m0001F4fm.mm0.3050.30501

(a) Neutron diffraction patterns taken from MnF2 above and below the Nel temperature (TN=67K). Note the strong 100 superlattice peak in the antiferromagnetic state. (b) Antiferromagnetic state showing antiparallel spin arrangement in Mn ions. There is a contraction along the spin orientation axis on ordering. [after R.A. Erickson, Phys. Rev. 90 (1953) 779].

Wyckoff positionSite SymmetryxyzOccupancyMn4am-3m0001O4bm-3m0.50.50.51

MnODisorderedLattice parameter(s)a = 4.4 ()Space GroupFm-3m (225)Pearson symbolcF8Other examples with this structureNaCl

Anti ferromagnetic MnO, TN =122KPerfect order not obtained even at low temperaturesRhombohedral angle changes with lowering of temperatureRhombohedral, a = 8.873, = 9026 at 4.2Keven above Nel temperature order persists in domains about 5nm in size

Nice example of antiferromagnetic ordering where the spins are not anti-parallel.TN = 363KHelical spin structureMetamagnetic behaviour- field induced transition to ferromagnetismMnAu2

MnAu2Lattice parameter(s)a =3.37, b = 3.37, c = 8.79 Space GroupI4/mmm (139)Pearson symboltI6Other examples with this structureMoSi2

Wyckoff positionSite SymmetryxyzOccupancyMn2a4/mmm0001Au4e4mm000.3351

OrderedDisorderedCu3Au

DisorderedOrdered- Ni3Al, FCCL12 (AuCu3-I type)

Superlattice lines

How to understand the statement that: on ordering the symmetry decreases?Funda CheckThe ordered structure has lower symmetry. This can be: reduction in point group symmetry or increase in the length of the shortest lattice translation vector. The reduction in point group symmetry could be due to ordering of a physical property (e.g. magnetic moment from electron spins). Shortest vector: [111] length = 3/2Shortest vector: [100] length = 1Reduction in the length of the shortest repeat periodicity (vector) on orderingCubic (P4/m 3 2/m) becomes Tetragonal (P 4/m 2/m 2/m) 50% chance of occupation by A