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

Size Factor compounds: (i) Laves phases (ii) Frank-Kasper PhasesD

These phases have a formula: AB2

Laves phases can be regarded as Tetrahedrally Close Packed (TCP)* structures with an ideal ratio of the radii (rA/rB) = (3/2)1/2 ~1.225 [or usually rA/rB (1.1, 1.6)]

If rA/rB = 1.225 then a high packing density is achieved with the chemical formula AB2 with a average coordination number of 13.3

Crystal structures: Hexagonal → MgZn2 (C15), MgNi2 (C36) FCC → MgCu2 (C14)

There are more than 1400 members belonging to the ‘Laves family’ Many ternary and multinary representatives of the Laves phases have been reported with

excess of A or B elements. Some ternary Laves phases are known in systems with no corresponding binary Laves phases.

The range of existence of the three phases (C15, C36, C14) in ternary Laves phases is influenced by the e/a ratio

D(i) Laves Phases

* Also called Topologically Close Packed structures?

Laves phases containing transition metals as components have interesting Physical and mechanical properties. Engineering materials based on Laves phases are being developed for: High temperature applications

(for use in turbine blade fine precipitates of Laves phases is shown to improve fatigue strength)

Hydrogen storage applications (in nickel-metal hydride batteries)

MgZn2 (Laves)Lattice parameter(s) a = 5.18 Å, c = 8.52 Å

Space Group P 63/mmc (194)

Strukturbericht notation C15

Pearson symbol hP12

Other examples with this structure

NbCr2

Wyckoff position

SiteSymmetry

x y z Occupancy

Mg 4f 3m 0.33 0.67 0.062 1

Zn1 2a -3m 0 0 0 1

Zn2 6h mm2 0.83 0.66 0.25 1

MgZn2 Laves Phase

Mg

Zn2

Zn1

[0001]

HexagonalC14

Zn: Vertex-1, Edge-1, Inside cell-6 → 8Mg: Inside cell-4 → 4

Unit cell formula: Mg4Zn8

MgZn2 Laves PhaseMore views Constructing the hexagonal laves phase

Start with a layer of Zn atoms Put Mg atoms in the depressions formed in the layer (above and below)

Add a hexagonal array of Zn atoms in the depressions formed by the Mg atoms (above and below)

This gives us half the unit cell in ‘c’ direction

Mg (8a) Cu (16d)

MgCu2 (Laves)Lattice parameter(s) a = 7.048 Å

Space Group Fd-3m (227)

Strukturbericht notation C14

Pearson symbol cF24

Other examples with this structure

Au2Pb

MgCu2 Laves Phase Cubic[001]

Wyckoff position

SiteSymmetry

x y z Occupancy

Cu 16d -3m 0.625 0.625 0.625 1

Mg 8a -43m 0 0 0 1

C15

Very frequent structural

type

Unit cell formula: Mg8Cu16

Mg: Vertex-1, FC-3, Inside cell-4 → 8Cu: Inside cell-16 → 16

More views MgCu2 Laves Phase

Successive layers are build on the depressions on the previous layer

More views

Tetrahedra of Cu

Note: the solid lines in the figures are for visualization of atomic positions etc. (they are not meant to show bonds)

MgCu2 Laves Phase

Not to scale

D(ii) Frank-Kasper

Have coordination numbers (CN): CN =12, CN = 14, CN = 15, CN = 16

Al12W (Frank-Kasper)Lattice parameter(s) a = 7.58 Å

Space Group Im-3 (204)

Strukturbericht notation

Pearson symbol cI26

Other examples with this structure

Al12Mn, Al12Mo

Wyckoff position

SiteSymmetry

x y z Occupancy

Al 24g m 0 0.184 0.309 1

W 2a m-3 0 0 0 1

Al12W Frank-Kasper Phase

Al

W

[001]

Cubic CN =12

Unit cell formula: Al24W2

W: Vertex-1, BC-1 → 2Cu: FC-12, Inside cell-12 → 24

Motif: 12Al +W (consistent with stoichiometry)

Lattice: Body Centred Cubic

More views Al12W Frank-Kasper Phase

Icosahedral coordination around W atoms Local icosahedral symmetry is destroyed in the long range packing Note that icosahedral symmetry is not found in crystals This phase is closely related to quasicrystals

More views Al12W Frank-Kasper Phase

[100]

[110]

[111]