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P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Atomic packing in solids

http://folk.uio.no/ravi/CMP2015

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Atomic packing in solids

1


Periodic table of

elements

2


Periodic table

of elements

3


4

Electronegativity Review

Electronegativity: A measure of the attractive force that one atom in a covalent bond has for the electrons of the bond


(Primitive cubic)

Crystal Structure of Metals5


Calculation of an octahedral interstitial site

6


Closest Packing

Holes

7

P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Atomic packing in solids 8

Packing Of Spherical Vegetables


Atomic packing in Metals

Closest Packing:

Assumes that metal atoms are uniform, hard

spheres.

Spheres are packed in layers. Like oranges in

grocery store display

abab packing - 3rd directly over 1st layer - called

hexagonal closest pack (hcp)

9


Atomic packing in Metals (con’t)

abca packing - 3rd layer not directly over 1st, 4th layer is

over 1st - cubic closest pack (ccp) or face centered cubic

(fcc) see next slide

10


Face Centered Cubic (FCC)

11


Metallic Bonding Nearest Neighbors

The Indicated Sphere Has 12 Nearest Neighbors

Each sphere in closest packed

(both fcp and hcp) has 12

equivalent nearest neighbors.

12


Unit cell atoms

fcc and hcp

8 x 1/8 spheres and 6 x 1/2

spheres = 4 total atoms in

unit cell

13


Close-Packing of Spheres

Many solids have close-packed structures in which spherical

particles are arranged in order to leave the minimum amount of

space.

There are two forms of close-packing; cubic close-packing and

hexagonal close-packing.

In both cases, each sphere has a coordination number of 12,

which means the each sphere has 12 equidistant neighbors.

There are 6 neighbors in one plane, 3 below and 3 above.

14


Hexagonal

close- packing

Cubic close-

packing

Hexagonal and cubic close packing15


Hexagonal Closest Packing

16


Cubic Closest Packing

17


The abab staching in

hcp lattice.

The Indicated Sphere Has 12

Nearest Neighbors

18


19

Packing in Metals

Model: Packing uniform, hard spheres to best use available

space. This is called closest packing. Each atom has 12

nearest neighbors.

hexagonal closest packed (“hcp”)

cubic closest packed (“ccp”)


Co-ordination number

•Number of spheres which are

touching a particle sphere

•In ionic crystals Number of oppositely charged

ions surrounding a particular ion

e.g., Co-ordination number of

Cl– and Na+ in NaCl molecule is 6

each.

20


Simple cubic

How do we count nearest neighbors?

Coordination Number

Draw a few more unit cells…...

21


Simple cubic

Highlight the nearest neighbors….

Coordination Number


coordination number of 6

What about body centered cubic?????

Simple Cubic

COORDINATION NUMBER

How many nearest neighbors???

23


24

In the three types of cubic unit cells:

Simple cubic

Coordination Number

CN = 6

Body Centered cubic CN = ?

Lets look at this…….


25

In bcc lattices, each sphere has a coordination number of 8

Body-centered cubic packing (bcc)

COORDINATION NUMBER?????


26

The Position of Tetrahedral Holes in a Face-

Centered Cubic Unit Cell


27

Cubic Closest

Packing in NaCl


28

The Net Number of Spheres in a Face-

Centered Cubic Unit Cell


Alloys

An alloy is a blend of a host metal and one or more

other elements which are added to change the

properties of the host metal.

Ores are naturally occurring compounds or

mixtures of compounds from which elements can be

extracted.

Bronze, first used about 5500 years ago, is an

example of a substitutional alloy, where tin atoms

replace some of the copper atoms in the cubic array.

29


Metal Alloys

Substitutional Alloy:

some metal atoms

replaced by others of

similar size.

brass = Cu/Zn

30


Metal Alloys(continued)

Interstitial Alloy:

Interstices (holes) in closest

packed metal structure are

occupied by small atoms.

steel = iron + carbon

31


Bronze


Substitutional Alloy

Examples

Where a lattice atom is replaced by an atom of

similar size

• Brass, one third of copper atoms are replaced by

zinc atoms

• Sterling silver (93% Silver and 7%Cu)

• Pewter (85% Sn, 7% Cu, 6% Bi, and 2% Sb)

• Plumber’s solder (67% Pb and 33% Sn)

33


Alloys

– Interstitial Alloy

When lattice holes (interstices) are filled with smaller atoms

Steel best known interstitial alloy, contains carbon atoms in the holes of an iron crystal

– Carbon atoms change properties

Carbon a very good covalent bonding atom changes the non-directional bonding of the iron, to have some direction

Results in increased strength, harder, and less ductile

The larger the percent of carbon the harder and stronger the steel

Other metals can be used in addition to carbon, thus forming alloy steels


Carbon Steel

Unlike bronze the carbon atoms fit into the holes formed by

the stacking of the iron atoms. Alloys formed by using the

holes are called interstital alloys.

P.Ravindran, PHY074- Condensed Matter Physics, Spring 2015 : Atomic packing in solids 10–36

Two Types of

Alloys

Substitutional

Interstitial


The fraction of the volume that is actually occupied by spheres…..

cellunit the of volume

cellunit the in spheres the by occupied volumevf

Efficiency Of Packing


38

Fraction(f) Of The Volume Occupied By The Spheres In The Unit

Cell.

unitcell

spheresv

V

Vf

Vspheres= number of spheres x volume single sphere

Vunit cell = a3 cubic unit cell of edge length a

Lets get NUMBER OF SPHERES

cellunit the of volume

cellunit the in spheres the by occupied volumevf

Packing Efficiency


Packing Efficiency

•Counting atoms in a unit cell!

•Atoms can be wholly in a unit

cell or atoms shared

between adjacent unit cells

In the lattice Counts 1 for atom in cell

Counts for 1/2 atom on a face.

Counts for 1/4 atom on a face.

Counts as 1/8 for atom on a corner.


40

What is the number of spheres in the fcc unit cell?

Total spheres = 8 (1/8) + 6 (1/2)

= 1 + 3 = 4

Face-centred Cubic Unit Cell

Note: 1/8 of a sphere on 8 corners and ½ of aSphere on 6 faces of the cube


Packing Spheres into Lattices

The most efficient way to pack hard spheres is

Spheres are packed in layers in which each sphere is surrounded by

six others.

CLOSEST PACKING

First Layer

41


Packing Spheres into Lattices:

Next Layer The next spheres fit into

a “dimple” formed by three spheres in the first layer.

There are two sets of dimples…...

42


43


Next Layer The next spheres fit into

The two types of “dimples” formed by three spheres in the

first layer.The second layer…..

NOTE: the

inverted

triangleTriangle

not

inverted


Tetrahedral site Octahedral site

Closed packed structure44



Second Layer

Once one is put on the others are forced into half of the dimples of the same type….



Once one is put on the others are forced into half of the dimples of the same type….

Second Layer



Second Layer

Note that the second layer only occupies half the dimples in the first layer.

Inverted triangle dimples are

not filled.


Packing Spheres into LatticesSecond Layer

Note that the second layer only occupies half the dimples in the

first layer.

Occupied

dimple

Unoccupied

DIMPLE

THE THIRD LAYER…...

48


Packing Spheres Into Lattices

The aba arrangement of layers, option 1.

A

B

A

HEXAGONAL CLOSEST PACKING

A HEXAGONAL UNIT CELL.


HCP

Hexagonal Unit Cell

ABA ARRANGEMENT HAS A HEXAGONAL UNIT CELL.


SUMMARY

NOW OPTION TWO…..

EXPANDED VIEW

HEXAGONAL CLOSED PACKED STRUCTURE


1

11


(1) a dimple directly above sphere in the first layer

THIRD LAYER, Choose a dimple

NOTE: the inverted triangle


OPTION 2!

THIS DIMPLE DOES NOT

2 2

2

LIE DIRECTLY OVER THE SPHERES OF THE FIRST LAYER.


THIRD LAYER

53

../../../VideoDemos/AN11_028.MOV


GREEN SPHERES DO NOT LIE DIRECTLY OVER THE SPHERES OF THE FIRST LAYER.

THE THIRD LAYER IS DIFFERENT FROM THE FIRST…….


THIRD LAYER

OPTION 2


A

B

C

NOT THE SAME AS OPTION ONE!

WE CALL THE THIRD LAYER C THIS TIME!


THIRD LAYER OPTION 2


A

B

C

The ABC Arrangement of layers.

We call the third layer C this time!

NOW THE FOURTH LAYER…….

OPTION 2


Third Layer

56


57

A

B

C

Fourth Layer The Same As First.


FOURTH LAYER

PUT SPHERE IN SO THAT


58

A

B

C

The ABCA Arrangement………..

Fourth Layer The Same As First.


A

This Is Called Cubic Closed Packed

….


Unit Cell Of CCP

Face- Centred Cubic Unit Cell (FCC)

This ABCA arrangement has a

A Comparison…..

CUBIC UNIT CELLL


COMPARISON

HCP CCPNOTICE the flip…...

NEAREST NEIGHBORS…..

60


Packing in Ionic Crystals

Ions pack themselves to maximize the attractions and

minimize repulsions between the ions.

AAA pack = primitive unit, coordination# 8

ABA packing = body-centered unit, coordination# 12

ABCA packing = face-centered unit, coordination# 12


Structures of Metals

Closest Packing

1. Hexagonal

2. Cubic

A-B A-B-C

62


COORDINATION NUMBER

The number of nearest neighbors that a lattice point has in a

crystalline solid

Lets look at hcp and ccp…...


HCP CCP

Coordination Number =12

Coordination Number


Spaces In Ionic Solids Are Filled With

Counter Ions

In NaCl, the Cl- ions form a

unit cell that is face centered

cubic

Na+ ions, being smaller, fill

the spaces between the Cl-

ions

If we count the atoms in the

unit cell we have 6 of each,

thus a 1:1 Na+:Cl- ratio

65


The three cubic unit cells.

Simple Cubic

coordination number = 6

Atoms/unit cell = 1/8 * 8 = 1

1/8 atom

at 8

corners

66



Body-centered

Cubic


1/8 atom

at 8

corners1 atom at

center

Atoms/unit cell = (1/8*8) + 1 = 2

67



Face-centered

Cubic


Atoms/unit cell =

(1/8*8)+(1/2*6) = 4

1/8 atom

at 8

corners

1/2 atom

at 6 faces

68


End centered cubic unit cell

Total no. of constituents per unit cell.

= Total contribution by constituents at corners

+ Total contribution by particles on the faces.

=8 x 1/8 + 1/2 x 2

=1 + 1

=2

69


Hexagonal close packed

structure

Consists of atoms with ABABAB stacking.

Each atom is surrounded by 12 closest neighbors.

Packing efficiency is 74%.

70


Cubic close packed structure

Consists of atoms with ABCABCABC stacking.

Each atom is surrounded by 12 closest neighbours.

Packing efficiency is 74%.

71


Packing Structures

• Both structures have coordination number of 12

o 74% of total volume of structure is occupied by spheres

o 26% empty space between spheres

• When unequal-sized spheres are packed into lattice

o Large particles usually assume one of the close packing

arrangements

o Small particles fill the holes between larger particles

72


Fraction of an Atom that Occupies a Unit Cell

for Various Positions in the Unit Cell

Positions in Unit Cell Fraction in Unit Cell

Center 1

Face ½

Edge ¼

Corner 1/8

73


Calculating Mass of Atoms

Make sure you know the volume or solve for it

Know the atoms per unit cell (given)

Know the density of atom (given)

Solve for mass using d =m / V

This gives you mass per unit cell and multiple by atoms per

unit cell

74


Cubic unit

cells

The unit cell forms the basic repeating unit of a crystal lattice.

The cubic unit cell is the simplest type.

75


Coordination Numbers

Coordination # Percent of molecule occupied

Single Cubic 6 52%

Face 12 74%

Body 8 68%

Hexagonal 12 74%

76


Geometry of a Cube

2

Diagonal Face

222

ef

eef

3

3

DiagonalBody

2222

eb

efeb

77


Atomic Radius and Cell Dimensions

Simple Cubic

r = e/2

1 atom/unit cell (in metals)

Body-Centered Cubic

b = 4r = e(3)1/2

r = e(3)1/2/4

2 atoms/unit cell (in metals)

Face-Centered Cubic

f = 4r = e(2)1/2

r = e(2)1/2/4

4 atoms/unit cell (in metals)

78


• Tend to be densely packed:

Metallic Crystal Structures

vs.

FCC and HCP close-packed BCC not close-packed

- Minimize empty space

79



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1H He

1 2

2Li Be B C N O F Ne

2,1 2,2 2,3 2,4 2,5 2,6 2,7 2,8

3Na Mg Al Si P S Cl Ar

2,8,1 2,8,2 2,8,3 2,8,4 2,8,5 2,8,6 2,8,7 2,8,8

4K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr

2,8,8,1 2,8,8,2 2,8,9,22,8,10,

22,8,11,

22,8,13,

12,8,13,

22,8,14,

22,8,15,

22,8,16,

22,8,18,

12,8,18,

22,8,18,

32,8,18,

42,8,18,

52,8,18,

62,8,18,

72,8,18,

8

5Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe

2,8,188,1

2,8,188,2

2,8,189,2

2,8,1810,2

2,8,1812,1

2,8,1813,1

2,8,1814,1

2,8,1815,1

2,8,1816,1

2,8,1818,0

2,8,1818,1

2,8,1818,2

2,8,1818,3

2,8,1818,4

2,8,1818,5

2,8,1818,6

2,8,1818,7

2,8,1818,8

6

Cs Ba * Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn

2,8,1818,8,1

2,8,1818,8,2

2,8,1832,10,2

2,8,1832,11,2

2,8,1832,12,2

2,8,1832,13,2

2,8,1832,14,2

2,8,1832,15,2

2,8,1832,17,1

2,8,1832,18,1

2,8,1832,18,2

2,8,1832,18,3

2,8,1832,18,4

2,8,1832,18,5

2,8,1832,18,6

2,8,1832,18,7

2,8,1832,18,8

Fr Ra ** Rf Db Sg Bh Hs Mt Uun Uuu Uub

2,8,18,3218,8,1

2,8,18,32

18,8,2

2,8,18,32

32,10,2

2,8,18,32

32,11,2

2,8,18,32

32,12,2

2,8,18,32

32,13,2

2,8,18,32

32,14,2

2,8,18,32

32,15,2

2,8,18,32

32,17,1

2,8,18,32

32,18,1

2,8,18,32

32,18,2

Element Groups (Families)

Alkali Earth Alkaline Earth Transition Metals

Rare Earth Other Metals Metalloids

Non-Metals Halogens Noble Gases

http://www.chemicalelements.com/show/electronconfig.html

http://www.chemicalelements.com/elements/h.html

http://www.chemicalelements.com/elements/he.html

http://www.chemicalelements.com/elements/li.html

http://www.chemicalelements.com/elements/be.html

http://www.chemicalelements.com/elements/b.html

http://www.chemicalelements.com/elements/c.html

http://www.chemicalelements.com/elements/n.html

http://www.chemicalelements.com/elements/o.html

http://www.chemicalelements.com/elements/f.html

http://www.chemicalelements.com/elements/ne.html

http://www.chemicalelements.com/elements/na.html

http://www.chemicalelements.com/elements/mg.html

http://www.chemicalelements.com/elements/al.html

http://www.chemicalelements.com/elements/si.html

http://www.chemicalelements.com/elements/p.html

http://www.chemicalelements.com/elements/s.html

http://www.chemicalelements.com/elements/cl.html

http://www.chemicalelements.com/elements/ar.html

http://www.chemicalelements.com/elements/k.html

http://www.chemicalelements.com/elements/ca.html

http://www.chemicalelements.com/elements/ga.html

http://www.chemicalelements.com/elements/ge.html

http://www.chemicalelements.com/elements/as.html

http://www.chemicalelements.com/elements/se.html

http://www.chemicalelements.com/elements/br.html

http://www.chemicalelements.com/elements/kr.html

http://www.chemicalelements.com/elements/rb.html

http://www.chemicalelements.com/elements/sr.html

http://www.chemicalelements.com/elements/in.html

http://www.chemicalelements.com/elements/sn.html

http://www.chemicalelements.com/elements/sb.html

http://www.chemicalelements.com/elements/te.html

http://www.chemicalelements.com/elements/i.html

http://www.chemicalelements.com/elements/xe.html

http://www.chemicalelements.com/elements/cs.html

http://www.chemicalelements.com/elements/ba.html

http://www.chemicalelements.com/elements/tl.html

http://www.chemicalelements.com/elements/pb.html

http://www.chemicalelements.com/elements/bi.html

http://www.chemicalelements.com/elements/po.html

http://www.chemicalelements.com/elements/at.html

http://www.chemicalelements.com/elements/rn.html

http://www.chemicalelements.com/elements/fr.html

http://www.chemicalelements.com/elements/ra.html

http://www.chemicalelements.com/elements/rf.html

http://www.chemicalelements.com/elements/db.html

http://www.chemicalelements.com/elements/sg.html

http://www.chemicalelements.com/elements/bh.html

http://www.chemicalelements.com/elements/hs.html

http://www.chemicalelements.com/elements/mt.html

http://www.chemicalelements.com/elements/uun.html

http://www.chemicalelements.com/elements/uuu.html

http://www.chemicalelements.com/elements/uub.html


Packing fraction

• Packing fraction is the fraction of total volume of a

cube occupied by constituent particles.

Packing fraction(PF) =

Volume occupied by effective number of particles

Volume of the unit cell

82


Packing of spheres in the unitcell.

simple cubic

(52% packing

efficiency)

body-centered

cubic

(68% packing

efficiency)

83


hexagonal

unit cell

closest packing of

first and second

layers

layer a

layer a

layer b

layer c

hexagonal

closest

packing cubic closest

packing

abab… (74%)

abcabc… (74%)

face-centered

unit cell

84

Packing of spheres in the unitcell.


Packing fraction of simple cubic crystal

For simple cubic crystal(scc)

Packing fraction =

3

3

41 r

3

a

3

3

4r

3 a 2r 0.5232r

For bcc, body diagonal, 4r 3a

Packing fraction =

3

3

42 r

3 0.68 i.e., 68%

4r

3

aA B

85


Packing fraction of face cubic crystal

For fcc, face diagonal, 4r = 2a

Packing fraction =

3

3

44 r

3 0.74 i.e., 74%

4r

2

A B

C

a

86


Calculating the Packing Factor

74.018)2/4(

)3

4(4)(

Factor Packing

24r/ cells,unit FCCfor Since,

)3

4)(atoms/cell (4

Factor Packing

3

3

0

3

0

3

r

r

r

a

a

Calculate the packing factor for the FCC cell.

SOLUTION

In a FCC cell, there are four lattice points per cell; if there is one atom

per lattice point, there are also four atoms per cell. The volume of one

atom is 4πr3/3 and the volume of the unit cell is . 3

0a


Density

The number of particles present per unit cell.

Suppose the edge length of the unit cell = a

Number of atoms present in one unit cell = Z

Atomic mass of the element = M

88


Density

Density of unit cell (r) =Mass contained in one unit cell

Volume of the unit cell

Mass contained in one unit cell =

Number of particles in one unit cell × Mass of one particle

Since mass of one particle = Atomic mass

Avogadro ' s number

89


• Coordination # increases with

Coordination # and Ionic Radii

ZnS

(zincblende)

NaCl(sodium

chloride)

CsCl(cesium chloride)

rcation

ranion

Issue: How many anions can you

arrange around a cation?

2

rcationranion

Coordination

Number

< 0.155

0.155 - 0.225

0.225 - 0.414

0.414 - 0.732

0.732 - 1.0

3

4

6

8

linear

triangular

Tetrahedron

Octahedron

cubic

Coordination

Geometry

(Cation-anion radius ratio)



Cation Site Size

Determine minimum rcation/ranion for OH site (C.N. = 6)

a 2ranion

2ranion 2rcation 2 2ranion

ranion rcation 2ranion

rcation ( 2 1)ranion

2ranion 2rcation 2a

4140anion

cation .r

r


This is the NaCl structure.

Two interpenetrating fcc arrays, one of Na+ ions and one

of Cl- ions.

The Na+ sit in the holes of the black (Cl-) lattice

SO HOW WE DESCRIBE IONIC SOLIDS???

Cl-

Na+

NaCl structure93


The anion is usually larger than the cation.

We describe an ionic solid as a lattice of the larger ions with the

smaller ions occupying holes in the lattice.

Consist of two interpenetrating lattices of the

two ions (cations and anions) in the solid.

NOTE:

Ionic Solids

HOLES????

94


The yellow dots form a FCC lattice!

Holes In A FCC Lattice

95


The yellow dots form a fcc lattice!

HOW MANY HOLES??????

Holes In A FCC Lattice96


Holes In a FCC Lattice

The holes:

THIRTEEN: ONE IN THE CENTRE

How many??

12 on the edges.

What shape is the hole ?

97


98

OCTAHEDRAL HOLES:

There is one octahedral hole in the centre of the unit cell.

Central Hole

If each one is occupied by an atom?


There are 4 complete octahedral holes per fcc unit cell.

The Octahedral Holes

If each one is occupied by an atom?

How many atoms per unit cell?

Number of atoms = 1 + 12 x (1/4) = 4

1/4 atom1 atom

Notice that the number of octahedral holes is the same as the number of atoms forming the unit cell!!

99


The Octahedral Holes

Other holes…..

Between two layers…..

10

0


There are other holes!

Where are the other holes in the FCC unit cell?

Can you spot them??????

Look at one of the small cubes

Other Holes10

1


Small Cube

Take a point at the centre of this cube

There are eight of these….

10

2


There is one tetrahedral hole in each of the eight smaller cubes in

the unit cell.

All the holes are completely within the cell, so there are 8 tetrahedral

holes per fcc unit cell

Notice that there are twice as many tetrahedral holes as atoms forming the lattice! That would be 8 holes.

Tetrahedral Holes10

3


Tetrahedral Holes

Formed by three spheres in one layer and

There is one more hole……….

one sphere in another layer sitting in the dimple they form.

104


Formed from the space between three ions in a plane.

Trigonal Holes

Formed by three spheres in one layer.

The smallest hole!

105


Atomic Size Ratios and the Location of Atoms in Unit Cells

Packing Type of Hole Radius Ratio

hcp or ccp Tetrahedral 0.22 - 0.41

hcp or ccp Octahedral 0.41 - 0.73

Simple Cubic Cubic 0.73 - 1.00

About Holes in Cubic Arrays



M+ or M2+ cations always occupy the holes

Consequently the radius of the cation must be

This causes the X– anions to be pushed apart,

greater than the size of the hole!

which reduces the X– – X– repulsion.

with the largest coordination number without rattling around!

Which hole will a cation occupy??????

Tight Fit

So we will investigate the size of these holes!

They occupy the holes that result in maximum attraction and minimum repulsion.

108


Investigate the size of these holes!

Which hole will a cation occupy??????

The size of the hole depends upon the

size of the ion (usually anion) that forms the lattice into which

the cations are to go……...

OCTAHEDRAL HOLE IN FCC….

109


Look at plane

Draw a square.

Put in spheres.

Fit a small sphere in

This will be the cation

These are the anions

Draw diagonal

Put in distances……..

Octahedral Holes In FCC 110


R

R

R

R

2r

Radius of ion = R

Look at plane

Radius of hole = r+(2R)2(2R)2 = (2R + 2r)2

8R2 = (2R + 2r)2

rRR 2222

rR 2)222(

rR 12

0.414R = r

Octahedral Holes In FCC111


R

R

R

R

2r

Radius of anion = R

Look at plane

Radius of hole = r

0.414R = r

The size of cation that just fits has a

radius that is

0.414 x radius of anion(R)

roctahedral hole = 0.414 R

What about the tetrahedral hole?

Octahedral Holes In FCC112


Using similar calculations, we can find the radius of other types

of holes as well:

rtetrahedral = 0.225 R

r = radius of ion fitting into hole (usually the cation)

The ratio between the radius of a hole in a cubic lattice

R is the radius of the ion forming the lattice (usually the anion).

fcc

RADIUS RATIO:

and the radius of the ions forming the hole

roctahedral = 0.414 R

What about other cubic cell systems??

DO IT!!!!!!!

113


rcubic = 0.732 Ranion

If the M+ cations (e.g. Cs+) are sufficiently large,

The next best closest packed X– array adopted by the anions is

a simple cubic structure, giving cubic holes which are large

enough to hold the cations.

Simple Cubic

they can no longer fit into octahedral holes of a fcc lattice.

YOU can show that...

114


The cubic hole

The coordination number in the cubic hole is ?

The coordination number in the

fcc tetrahedral hole is ?

4!

The coordination number in the

fcc octahedral hole is ?

6!

8

In contrast for a fcc lattice…...

rcubic = 0.732 Ranion

115


Summary:

Face centred cubic:

Trigonal hole Too small to be occupied

Tetrahedral hole CN = 4 rcation = 0.225Ranion

Octahedral hole CN = 6

8 of these

rcation = 0.414Ranion 4 of these

Simple cubic:

Cubic hole CN = 8 rcation = 0.732Ranion 1 of these

For a given anion

rtrigonal < rtetrahedral < roctahedral < rcubic

116


Into Which Hole Will The Ion Go??

TETRAHEDRAL

The hole filled is tetrahedral if:

0.225Ranion < rcation < 0.414Ranion

rtetrahedral < rcation < roctahedral

117



OCTAHEDRAL

The hole filled is octahedral if:

0.414Ranion < rcation < 0.732Ranion

roctahedral < rcation < rcubic

118



CUBIC

The hole filled is cubic if:

0.732Ranion < rcation

Lets look at these ideas in action…….

rcubic < rcation

119


Na+

has a radius of 98pm.Cl

-has a radius of 181pm.

Consider a fcc array of Cl- then:

Radius of the tetrahedral hole is 0.225 x 181=41pm

Radius of the octahedral hole is 0.414 x 181=75pm

Consider a sc array of Cl- then:

Radius of the cubic hole is 0.732 x 181=132pm

So the best fit is the octahedral hole in the fcc array!

The 98pm is bigger than 75pm but less than 132!

OR USING RATIOS…….

NaCl120


Na+

has a radius of 98pm.Cl

-has a radius of 181pm.

54.0181

98

pm

pm

r

r

r

r

Cl

Na

anion

cation

225.0tet

anion

tet

cation

r

r414.0

oct

anion

oct

cation

r

r732.0

cubic

anion

cubic

cation

r

r

0.54 lies between 0.414 and 0.732

so the sodium cations will occupy octahedral holes

in a fcc (ccp) lattice

NaCl121


1:1 stoichiometry is required

How many complete octahedral holes in face centred cubic array

of Cl- ?????

So stoichiometry is ok!!

4

How many Cl- needed to form the fcc array??? 4

Therefore 4 Cl- and 4 Na+

NaClIs the stoichiometry ok??? 122


Example: Predict the structure of Li2S

Li+

is 68 pm S2-

is 190pm

36.0190

68

2

pm

pm

r

r

r

r

S

Li

anion

cationCalculate ratio..

Examine the cation-anion radius ratios to find which type of

holes the smaller ions fill

STEP ONE:

225.0tet

anion

tet

cation

r

r414.0

oct

anion

oct

cation

r

r

COMPARE with ratios….

Which is the best hole???? TETRAHEDRAL!!!!

123


This requires tetrahedral holes.

Example: Predict the structure of Li2S

Li+ is 68 pm S2- is 190pm

36.0190

68

2

pm

pm

r

r

r

r

S

Li

anion

cation

face- centred cubic array

Lets look at the structure…...

Calculate ratio..

Which lattice has tetrahedral holes???

Thus the S2- will form a fcc lattice ...

124


FCC unit cell with tetrahedral holes

ANION

CATION

There are 8 tetrahedral

holes.

How many are occupied?

Four anions in the unit cell.

STEP TWO: Determine what fraction of those holes must be

filled to give the correct chemical formula

125


FCC unit cell with tetrahedral holes

S2-

Li+

How many are occupied?

Li2S needs two Li+ for each S2-

Four anions in the

unit cell.

There are 8 tetrahedral

holes.

Therefore all the tetrahedral holes are occupied!

126

Li2S is a face centered lattice of S2- with

all of the tetrahedral holes filled by Li+

ions.


CsCl: Cs+ is 167 pm Cl- is 181pm Calculate ratio

0.92 is greater than 0.732

92.0181

167

pm

pm

r

r

r

r

Cl

Cs

anion

cation

the cesium cations will occupy cubic holes of a simple cubic lattice.

732.0cubic

anion

cubic

cation

r

rCompare…...

127


There are the same number of cubic holes and lattice points in the

cubic lattice.

Hence stoichiometry OK!

CsCl is composed of a simple cubic lattice of chloride anions with

cesium cations in all the cubic holes.

128


Cesium Chloride129


ZnS: Zn2+ is 64 pm S2- is 190 pm Calculate ratio

35.0190

64

2

2

pm

pm

r

r

r

r

S

Zn

anion

cation

225.0tet

anion

tet

cation

r

r414.0

oct

anion

oct

cation

r

rCOMPARE

This requires

tetrahedral holes.

The sulfide ions will form a face-centered cubic array

because….

that is the only type to possess tetrahedral holes.

130


We need an equal number of zinc and sulfide ions.

There are the twice as many tetrahedral holes(8) as S2-(4) that

form the fcc lattice.

Therefore, half the

tetrahedral holes will be

filled.

131

ZnS is composed of a fcc lattice of sulfide anions with zinc cations in half the

tetrahedral holes.


There are two forms of ZnS

This is an example

of polymorphism.

One is the zinc blende that we have talked about!

The other is wurtzite based on hcp lattice.

132


AX- Type Crystal Structure

Ceramic materials which have equal number of cations

and anoins.

AX compounds

– A = cation

– X = anion

Consists of :

– Rock Salt/ Sodium Chloride (NaCl ) structure

– Cesium Chloride Structure

– Zinc Blende Structure


• On the basis of ionic radii, what crystal structure

would you predict for FeO?

• Answer:

5500

1400

0770

anion

cation

.

.

.

r

r

based on this ratio,

--coord # = 6

--structure = NaCl

Example 1: Predicting Structure of FeO

Ionic radius (nm)

0.053

0.077

0.069

0.100

0.140

0.181

0.133

Cation

Anion

Al3+

Fe2+

Fe3+

Ca2+

O2-

Cl-

F-


AX- Crystal Structure: Rock Salt Structure

Same concepts can be applied to ionic solids in general.

Example: Sodium Chloride (NaCl ) / rock salt structure

rNa = 0.102 nm

rNa/rCl = 0.564

cations prefer OH sites

rCl = 0.181 nm

The coordination number is 6


AX-Crystal Structure :Rock Salt Structure

MgO and FeO also have the NaCl structure

O2- rO = 0.140 nm

Mg2+ rMg = 0.072 nm

rMg/rO = 0.514

cations prefer OH sites

So each oxygen has 6 neighboring Mg2+


AX-Crystal Structures: Cesium Chloride

Structure

939.0181.0

170.0

Cl

Cs

r

r

Cesium Chloride structure:

cubic sites preferred

So each Cs+ has 8 neighboring Cl-


AX-Crystal Structures: Zinc Blende

So each Zn2+ has 4 neighboring O2-

Zinc Blende structure?? 529.0

140.0

074.0

2

2

O

ZnHO

r

r

• Size arguments predict Zn2+

in OH sites,

• In observed structure Zn2+

in TD sites

Why is Zn2+ in TD sites?

– bonding hybridization of

zinc favors TD sites

Ex: ZnO, ZnS, SiC


AX2 Type Crystal Structures

Fluorite structure

• Calcium Fluorite (CaF2)

• cations in cubic sites

• UO2, ThO2, ZrO2, CeO2

• Antifluorite structure –

cations and anions

reversed.

•Charges of cation and anions are

not the same


ABX3 Crystal Structures

Ceramic compound have more than one type of cation

Perovskite

Ex: complex oxide

BaTiO3

(Barium Titanate)


(c) 2003 Brooks/Cole Publishing / Thomson Learning™

Figure 3.31 Connection between anion polyhedra. Different possible connections include sharing of corners, edges,

or faces. In this figure, examples of connections between tetrahedra are shown.


The perovskite unit cell showing the A and B site cations and oxygen ions

occupying the face-center positions of the unit cell. Note: Ions are not show to

scale.


Crystal Structure of Ceramic

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