Ordered StructuresOrdered Structures

Please revise the concept of Sublattices and ‘Subcrystals’ by clicking here

before proceeding with this topic

Please revise the concept of Sublattices and ‘Subcrystals’ by clicking here

before proceeding with this topic

MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING 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-BOOKAN INTRODUCTORY E-BOOK

Part of

http://home.iitk.ac.in/~anandh/E-book.htmhttp://home.iitk.ac.in/~anandh/E-book.htm

A Learner’s GuideA Learner’s GuideA Learner’s GuideA Learner’s Guide

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 structuresClick 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 Transformations

Click here to revise concepts about Sublattices & Subcrystals

Positional Order

G = H TS

High T disordered

Low T ordered

470ºC

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

In the order-disorder transformation shown in the figures below

the high temperature phase is disordered and has a BCC lattice, while the low temperature structure is simple cubic (B2 structure).

B2 structure

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

Let us consider some more ordered structures

NiAl

NiAlLattice 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

SC

Motif: 1Ni + 1Al

Lattice: Simple Cubic

Unit cell formula: NiAl

This is similar to CuZn

Two interpenetrating Simple Cubic crystals (origin of crystal-1 at (0,0,0) and origin of crystal-2 at (½,½,½))

CuAuLattice parameter(s) a = 3.96Å, c = 3.67Å

Space Group P 4/m 2/m 2/m (123)

Strukturbericht notation L10

Pearson symbol tP4

Other examples with this structure TiAl

CuAu (I)

Cu

Au

Cu

Au

Wyckoff position

x y z

Au1 1a 0 0 0

Au2 1c 0.5 0.5 0

Cu 2e 0 0.5 0.5

Motif: 2Au +2Cu (consistent with stoichiometry)

Lattice: Simple Tetragonal

Unit cell formula: Cu2Au2

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)

Cu3AuLattice parameter(s) a = 3.75 Å

Space Group P 4/m 3 2/m (221)

Strukturbericht notation L12

Pearson symbol cP4

Other examples with this structure Ni3Al, TiPt3

Cu3Au

Cu

Au

Motif: 3Cu +1Au (consistent with stoichiometry)

Lattice: Simple Cubic

This is similar to Cu3AuNi3Al

Ni3AlLattice parameter(s) a = 3.56 Å

Space Group P 4/m 3 2/m (221)

Strukturbericht notation L12

Pearson symbol cP4

Other examples with this structure Cu3Au, TiPt3

Al3NiLattice 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

Al3Ni

[001][010]

[100]

Wyckoff position

SiteSymmetry

x y z Occupancy

Ni 4c .m. 0.3764 0.25 0.4426 1

Al1 4c .m. 0.0388 0.25 0.6578 1

Al2 8d 1 0.1834 0.0689 0.1656 1

Formula for Unit cell: Al12Ni4

Fe3AlLattice parameter(s) a = 5.792 Å

Space Group F 4/m 3 2/m (225)

Strukturbericht notation DO3

Pearson symbol cF16

Other examples with this structure Fe3Bi

Fe3Al

Al

Fe

Fe2 (¼,¼,¼)

Fe1 (½,½,0)Fe1 (0,0,0)

Wyckoff position

Fe1 4a 0 0 0

Fe2 8c 0.25 0.25 0.25

Al 4b 0.5 0 0

Dark blue: Fe at cornersLighter blue: Fe at face centresV. Light Blue: Fe at (¼,¼,¼)

Fe: Vertex-1, FC-3, (¼,¼,¼)-8 → 12Al: Edge-3, BC-1 → 4 Unit cell formula: Fe12Al4

AlFe

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

More views

[100]

Al

Fe

Fe3Al

Fe3AlMore views Fe3Al

Fe2 (¼,¼,¼)

Fe1 (½,½,0)

Fe1 and Fe2 have different environments

Tetrahedron of FeTetrahedron of Al

Fe2 (¼,¼,¼)

Fe1 (0,0,0)

Fe1 (0,0,0)

Fe1 (½,½,0)

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 Néel temperature the spin structure will become disordered and state would

be paramagnetic

Spin Ordering

(a) Ferromagnetic

(b) Antiferromagnetic

(c) Ferrimagnetic

(d) Canted Antiferromagnetic

(e) Helical magnetic

(spin)

MnF2 is antiferromagnetic below 67K (TN)

Antiferromagnetic ordering

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

SiteSymmetry

x y z Occupancy

Mn 2a m.2m 0 0 0 1

F 4f m.mm 0.305 0.305 0 1

(a) Neutron diffraction patterns taken from MnF2 above and below the Néel 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].

(a) (b)

Wyckoff position

SiteSymmetry

x y z Occupancy

Mn 4a m-3m 0 0 0 1

O 4b m-3m 0.5 0.5 0.5 1

MnO DisorderedLattice parameter(s) a = 4.4 (Å)

Space Group Fm-3m (225)

Pearson symbol cF8

Other examples with this structure NaCl

Anti ferromagnetic MnO, TN =122K

Perfect order not obtained even at low temperatures Rhombohedral angle changes with lowering of temperature Rhombohedral, a = 8.873, = 9026’ at 4.2K even above Néel temperature order persists in domains about 5nm in size

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

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

SiteSymmetry

x y z Occupancy

Mn 2a 4/mmm 0 0 0 1

Au 4e 4mm 0 0 0.335 1

Ordered

Disordered Cu3Au

Disordered Ordered

- Ni3Al, FCC L12 (AuCu3-I type)

Superlattice lines

How to understand the statement that: “on ordering the symmetry decreases”?Funda Check

The 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: ½[111] length = 3/2

Shortest vector: [100] length = 1

Reduction in the length of the shortest repeat periodicity

(vector) on ordering

Cubic (P4/m 3 2/m) becomes Tetragonal

(P 4/m 2/m 2/m)

50% chance of occupation by A or B

50% chance of occupation by A or B

Cubic (of spins are

time averaged)

becomes Tetragonal