Polymorphism or Allotropy

Many elements or compounds exist in more than one crystalline

form under different conditions of temperature and pressure.

This phenomenon is termed polymorphism and if the material is

an elemental solid is called allotropy.

Example: Iron (Fe – Z = 26)

liquid above 1539 C.

δ-iron (BCC) between 1394 and 1539 C.γ-iron (FCC) between 912 and 1394 C.α-iron (BCC) between -273 and 912 C.

α iron γ iron δ iron912oC 1400oC 1539oC

liquid iron

BCC FCC BCC

Another example of allotropy is carbon.

Pure, solid carbon occurs in three crystalline forms – diamond,

graphite; and large, hollow fullerenes. Two kinds of fullerenes are

shown here: buckminsterfullerene (buckyball) and carbon nanotube.

Crystallographic Planes and DirectionsAtom Positions in Cubic Unit Cells

A cube of lattice parameter a is considered to have a side equal to

unity. Only the atoms with coordinates x, y and z greater than or

equal to zero and less than unity belong to that specific cell.

y

z

x

1,0,1

1,0,0

1,1,1

½, ½, ½

0,0,0

0,0,1 0,1,1

1,1,0

0,1,0

1,,0 <≤ zyx

Directions in The Unit Cell

For cubic crystals the crystallographic directions indices are the

vector components of the direction resolved along each of the

coordinate axes and reduced to the smallest integer.

y

z

x

1,0,1

1,0,0

1,1,1

½, ½, ½

0,0,0

0,0,1 0,1,1

1,1,0

0,1,0A

Example direction A

a) Two points origin coordinates

0,0,0 and final position

coordinates 1,1,0

b) 1,1,0 - 0,0,0 = 1,1,0

c) No fractions to clear

d) Direction [110]

y

z

x

1,1,1

0,0,0

0,0,1

B

C

½, 1, 0

Example direction B

a) Two points origin coordinates

1,1,1 and final position

coordinates 0,0,0

b) 0,0,0 - 1,1,1 = -1,-1,-1

c) No fractions to clear

d) Direction ]111[___

Example direction C

a) Two points origin coordinates

½,1,0 and final position

coordinates 0,0,1

b) 0,0,1 - ½,1,0 = -½,-1,1

c) There are fractions to clear.

Multiply times 2. 2( -½,-1,1) =

-1,-2,2

d) Direction ]221[__

Notes About the Use of Miller Indices for Directions

� A direction and its negative are not identical; [100] is not equal to

[bar100]. They represent the same line but opposite directions. .

� direction and its multiple are identical: [100] is the same direction

as [200]. We just forgot to reduce to lowest integers.

� Certain groups of directions are equivalent; they have their

particular indices primarily because of the way we construct the co-

ordinates. For example, a [100] direction is equivalent to the [010]

direction if we re-define the co-ordinates system. We may refer to

groups of equivalent directions as directions of the same family. The

special brackets < > are used to indicate this collection of directions.

Example:

The family of directions <100> consists of six equivalent directions

00]1[],1[000],1[0[001],[010],[100], >100< ≡

Miller Indices for Crystallographic planes in Cubic Cells

� Planes in unit cells are also defined by three integer numbers,

called the Miller indices and written (hkl).

� Miller’s indices can be used as a shorthand notation to identify

crystallographic directions (earlier) AND planes.

Procedure for determining Miller Indices

� locate the origin

� identify the points at which the plane intercepts the x, y and z

coordinates as fractions of unit cell length. If the plane passes

through the origin, the origin of the co-ordinate system must be

moved!

� take reciprocals of these intercepts

� clear fractions but do not reduce to lowest integers

� enclose the resulting numbers in parentheses (h k l). Again, the

negative numbers should be written with a bar over the number.

y

z

x

A

Example: Miller indices for plane A

a) Locate the origin of coordinate.

b) Find the intercepts x = 1, y = 1, z = 1

c) Find the inverse 1/x=1, 1/y=1, 1/z=1

d) No fractions to clear

e) (1 1 1)

More Miller Indices - Examples

a

b

c

0.50.5

a

b

c

2/3

1/5

a

b

c

a

b

c

a

b

c

ab

c

Notes About the Use of Miller Indices for Planes

� A plane and its negative are parallel and identical.

� Planes and its multiple are parallel planes: (100) is parallel to the

plane (200) and the distance between (200) planes is half of the

distance between (100) planes.

Certain groups of planes are equivalent (same atom distribution);

they have their particular indices primarily because of the way we

construct the co-ordinates. For example, a (100) planes is

equivalent to the (010) planes. We may refer to groups of

equivalent planes as planes of the same family. The special

brackets { } are used to indicate this collection of planes.

� In cubic systems the direction of miller indices [h k l] is normal

o perpendicular to the (h k l) plane.

� in cubic systems, the distance d between planes (h k l ) is given

by the formula where a is the lattice

constant.

Example:

The family of planes {100} consists of three equivalent planes (100)

, (010) and (001)

222 lkh

ad

++=

A “family” of crystal planes contains all those planes are crystallo-

graphically equivalent.

• Planes have the same atomic packing density

• a family is designated by indices that are enclosed by braces.

- {111}:

)111(),111(),111(),111(),111(),111(),111(),111(

• Single Crystal

• Polycrystalline materials

• Anisotropy and isotropy

c

(110) = (1100)- -

(100) = (1010)-

a1

a2

a3

(001) = (0001)

(110) = (1100)- -

Two Types of Indices in the Hexagonal System

a1

a2

a3

c

a1

a2

a3

c

Miller: (hkl) (same as before)

Miller-Bravais: (hkil) → i = - (h+k)

a3 = - (a1 + a2)

a1 ,a2 ,and c are independent, a3 is not!

Structure of Ceramics

Ceramicskeramikos - burnt stuff in Greek - desirable properties of

ceramics are normally achieved through a high temperature heat

treatment process (firing).

Usually a compound between metallic and nonmetallic elements

Always composed of more than one element (e.g., Al2O3, NaCl,

SiC, SiO2)

Bonds are partially or totally ionic, can have combination of ionic

and covalent bonding (electronegativity)

Generally hard, brittle and electrical and thermal insulators

Can be optically opaque, semi-transparent, or transparent

Traditional ceramics – based on clay (china, bricks, tiles,

porcelain), glasses.

“New ceramics” for electronic, computer, aerospace industries.

Crystal Structures in Ceramics with

predominantly ionic bonding

Crystal structure is defined by

The electric charge: The crystal must remain electrically

neutral. Charge balance dictates chemical formula (Ca2+ and F-

form CaF2).

Relative size of the cation and anion. The ratio of the atomic

radii (rcation/ranion) dictates the atomic arrangement. Stable

structures have cation/anion contact.

Coordination Number: the number of anions nearest neighbors for a

cation.

As the ratio gets larger (that is as rcation/ranion ~ 1) the coordination

number gets larger and larger.

Holes in sphere packing

Triangular Tetrahedral Octahedral

Calculating minimum radius ratio

for a triangle:

1550

2

330

21

2

1

.

cos

)30=( cos

=

==+

°=

+=

=

a

c

ca

a

ca

a

r

r

rr

r

AO

AB

rrAO

rAB

αα

A C

BO

AC

B

O

4140

2

245

21

2

1

.

cos

)45=( cos

=

==+

°=

+=

=

a

c

o

ca

a

ca

a

r

r

rr

r

AO

AB

rrAO

rAB

αα

for an octahedral hole

C.N. = 2

rC/rA < 0.155

C.N. = 3

0.155 < rC/rA < 0.225

C.N. = 4

0.225 < rC/rA < 0.414

C.N. = 6

0.414 < rC/rA < 0.732

C.N. = 8

0.732 < rC/rA < 1.0

Ionic (and other) structures may be derived from the occupation of

interstitial sites in close-packed arrangements.

o/t fcc(ccp) hcp

all oct. NaCl NiAs

all tetr. CaF2 (ReB2)

o/t (all) (Li3Bi) (Na3As)

½ t sphalerite (ZnS) wurtzite (ZnS)

(½ o CdCl2 CdI2)

Comparison between structures with filled octahedral and tetrahedral holes

Location and number of tetrahedral

holes in a fcc (ccp) unit cell

- Z = 4 (number of atoms in the unit cell)

- N = 8 (number of tetrahedral holes in the

unit cell)

Crystals having filled Interstitial Sites

NaCl structure has Na+ ions

at all 4 octahedral sites

Octahedral, Oh, Sites

Na+ ions

Cl- ions

Ionic Crystals prefer the NaCl

Structure:

• Large interatomic distance

• LiH, MgO, MnO, AgBr, PbS,

KCl, KBr

Crystals having filled Interstitial Sites

Both the diamond cubic structure

And the Zinc sulfide structures

have 4 tetrahedral sites occupied

and 4 tetrahedral sited empty.

Tetrahedral, Th, Sites

Zn atoms

S atoms

Covalently Bonded Crystals Prefer

this Structure

• Shorter Interatomic Distances

than ionic

• Group IV Crystals (C, Si, Ge, Sn)

• Group III--Group V Crystals

(AlP, GaP, GaAs, AlAs, InSb)

• Zn, Cd – Group VI Crystals

(ZnS, ZnSe, CdS)

• Cu, Ag – Group VII Crystals

(AgI, CuCl, CuF)

Rock Salt Structure (NaCl)

Cl NaCoordination = 6

NaCl, MgO, LiF, FeO, CoO

NaCl structure: rC = rNa = 0.102 nm,

rA = rCl = 0.181 nm rC/rA = 0.56

AX Type Crystal Structures

Cesium Chloride Structure (CsCl)

CsClCoordination = 8

Is this a body centered cubic structure?

CsCl Structure: rC = rCs = 0.170 nm,

rA = rCl = 0.181 nm → rC/rA = 0.94

Zinc Blende Structure (ZnS)

ZnSCoordination = 4

radius ratio = 0.402

ZnS, ZnTe, SiC have this crystal structure

AmXp-Type Crystal StructuresIf the charges on the cations and anions are not the same, a compound

can exist with the chemical formula AmXp , where m and/or p ≠ 1. An

example would be AX2 , for which a common crystal structure is

found in fluorite (CaF2).

Fluorite CaF2

Fluorite (CaF2): rC = rCa = 0.100 nm, rA= rF = 0.133 nm ⇒ rC/rA = 0.75

From the table for stable geometries we

see that C.N. = 8

Other compounds that have this crystal

structure include UO2 , PuO2 , and ThO2.

AmBnXp-Type Crystal Structures

It is also possible for ceramic compounds to have more than one

type of cation; for two types of cations (represented by A and B),

their chemical formula may be designated as AmBnXp . Barium

titanate (BaTiO3), having both Ba2++++ and Ti4+ cations, falls into this

classification. This material has a perovskite crystal structure and

rather interesting electromechanical properties

Perovskite -

an Inorganic Chameleon

ABX3 - three compositional variables, A,

B and X

• CaTiO3 - dielectric

• BaTiO3 - ferroelectric

• Pb(Mg1/3Nb2/3)O3 - relaxor

ferroelectric

• Pb(Zr1-xTix)O3 - piezoelectric

• (Ba1-xLax)TiO3 – semiconductor

• (Y1/3Ba2/3)CuO3-x -

superconductor

• NaxWO3 - mixed conductor;

electrochromic

• SrCeO3 - H - protonic conductor

• RECoO3-x - mixed conductor

• (Li0.5-3xLa0.5+x)TiO3 - lithium ion

conductor

• LaMnO3-x - Giant magneto-

resistance

The perovskite structure CaTiO3

- TiO6 – octahedra

- CaO12 – cuboctahedra

(Ca2+ and O2- form a cubic close

packing)

→ preferred basis structure of

piezoelectric, ferroelectric and

superconducting materials