MATERIALS SCIENCEMATERIALS SCIENCE&&

ENGINEERING ENGINEERING

Anandh Subramaniam & Kantesh BalaniMaterials Science and Engineering (MSE)

Indian Institute of Technology, Kanpur- 208016Email: 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.htm

A Learner’s GuideA Learner’s Guide

Close Packed CrystalsClose Packed Crystals

Close Packed CrystalsClose Packed Crystals

Initially we consider ‘usual’ type of close packed crystals, which are made of single kind of sphere.

In other types of close packed crystals (e.g. tetrahedrally close packed crystals, also called topologically close packed crystals), more than one size of sphere may be involved.

One may even conceive of close packing of ellipsoids and other non-spherical objects.

Cubic Close Packed (CCP- commonly called FCC crystal also) and Hexagonal Close Packed (HCP) are two common examples of close packed crystals.

The term close packed crystal implies closest packed crystal (having a packing fraction of 0.74).

The proof that this is the densest crystallographic packing of spheres possible is a difficult one (and will not be considered here).

CCP and HCP are just two examples among a series of close packed structures which can be envisaged (shown in coming slides).

Every atom in these structures has a coordination number of 12 forming a Cubeoctahedron or a Twinned Cubeoctahedron (around the central atom).

The common starting point is a close packed layer of atoms with 6-fold symmetry. Identical layers are stacked one on another with a shift. The shift is such that the atoms in the above (and below) layers sit in the ‘valleys’ formed

by a layer. All such possibilities (see coming slides) lead to Close Packed Crystals. The original 6-fold symmetry present in a single layer is lost on this kind of packing

(you must be aware of the 3-fold present in CCP and HCP crystals!). Yes! HCP crystal has NO true 6-fold axis!

CCP Coordination Polyhedron Cubeoctahedron

HCP Coordination Polyhedron Twinned Cubeoctahedron

Starting Point Hexagonal layer Three positions A (the first layer atomic positions), B & C (Valleys) are shown The second layer (of hexagonal packing of atoms) can be positioned in valley B (or

equivalently in valley C)

Step-1

Step-2A

AB

Part of the hexagonal layer shown

The third layer can be positioned with atoms directly above the A layer (Option-1) or with atoms above the C layer (Option-2)

Step-3

Layer-3

(Option-1)

(Option-2)

C-site vacant

ABC

ABA

Continuing this ABAB sequence we get the HCP structure

Continuing this ABCABC sequence we get the CCP structure(Though not obvious!)

ABCABC… & ABAB… are just but two amongst the infinite possibilities At each stage of construction we have a choice of putting an atomic layer at A, B or C

position Possibilites include:

ABCAB.ABCAB.ABCAB… ABCABCAB.ABCABCAB.ABCABCAB…

Hence we can construct crystals with larger and larger unit cells. If we randomly put the layers we will not get a crystal in the ‘true sense’.

(We can think of these as 2D crystals, which are not periodic in 3rd dimension). Few stages in the infinite choice tree is shown below.

A

B

C

A

C

A

B

B

C

A

B

B

C

B

C

Track a branch to infinity or truncate at some stage and repeat to get a structure

In the ABCABC… packing we start with a layer having 6-fold symmetry. Interestingly, this packing leads to a 4-fold axis at an angle of 54.74 to the original 6-fold axis and to the familiar Cubic Close Packed crystal (‘FCC’ unit cell)

Actually a –3 (3 bar) roto-inversion pseudo-axis

Rigid sphere-like atoms without long range interactions can arrange in any of the infinite possibilities shown before.

Not only can we have ordered sequences, but also disordered close packed sequences (the diorder is in the way ‘A’, ‘B’ & ‘C’ appear and not within a given plane (say ‘A’)

If Cobalt is annealed above and below 450C a disordered sequence of ABC packing is obtained (T >450C Co → ABCABC… packing, T <450C Co → ABAB… packing)

Layers Stacking Example Stacking symbol2 AB Mg (hP2, P63/mmc) 2H3 ABC Cu (cF4, Fm–3m) 3C4 ABAC La (hp4, P63/mmc) 4H9 ABABCBCAC Sm (hR3, R–3m) 9R

Some examples of various stacking sequences

Lipson & Stokes (Proc. Roy. Soc. A, 181, 101. 1943) showed the formation of trigonal graphite with stacking sequence ABCA instead of ABAB.Note: Graphite is not a close packed structure.

SiC (not close packed structure) shows many polytypes. Common ones are: 3C-SiC (cubic unit cell, zincblende); 2H-SiC; 4H-SiC; 6H-SiC (hexagonal unit cell, wurtzile ); 15R-SiC (rhombohedral unit cell).

Among the polytypes of diamond the following is the decreasing order of stability: 3C > 6H > 9R > 4H > 2H.

Lattice parameter(s) a = 3.77Å, c = 12.159Å

Space Group P63/mmc (194)

Strukturbericht notation

Pearson symbol hp4

Other examples with this structure

Wyckoff position

SiteSymmetry x y z Occupancy

La1 2a -3m 0 0 0 1

La2 2c -6m2 0.33 0.67 0.25 1

La

A layer

A layer

B layer

Closed packed crystal

[0001]

[1120]

Note: All atoms are La

C layer

More views

A layer

A layer

B layer

C layer

A

B

C