1. 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
URL: home.iitk.ac.in/~anandh
AN INTRODUCTORY E-BOOK
Part of
A Learner’s Guide

2. 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’.

3. 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

4. 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)

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, 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

6. 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

7. 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

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 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)

9. 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

10. 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

11. Strukturbericht notation
Al3Ni
[100]
[010]
[001]
Formula for Unit cell: Al12Ni4
Al3Ni
Lattice parameter(s)
a = Å, b = Å, c = 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

13. 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 = 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

14. 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

15. Fe3Al
More views Fe Al
[100]

16. 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)

17. 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

18. 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

19. 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

20. 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

21. 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

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

24. Superlattice lines