Ordered Structures
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Transcript of Ordered Structures
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: [email protected], 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