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Chapter 3: Structures of Metals & Ceramics Structures The properties of some materials are directly related to their crystal structures. Significant property differences exist between crystalline and noncrystalline materials having the same composition. Energy and Packing Energy

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structures
Structures
  • The properties of some materials are directly related to their crystal structures.
  • Significant property differences exist between crystalline and noncrystalline materials having the same composition.
energy and packing
Energy and Packing

Energy

typical neighbor

bond length

typical neighbor

r

bond energy

• Dense, ordered packing

Energy

typical neighbor

bond length

r

typical neighbor

bond energy

• Non dense, random packing

Dense, ordered packed structures tend to have lower energies.

materials and packing
Materials and Packing

Crystalline materials...

• atoms pack in periodic, 3D arrays

• typical of:

-metals

-many ceramics

-some polymers

crystalline SiO2

Si

Oxygen

Noncrystalline materials...

• atoms have no periodic packing

• occurs for:

-complex structures

-rapid cooling

"Amorphous" = Noncrystalline

noncrystalline SiO2

metallic crystal structures
 Metallic Crystal Structures
  • How can we stack metal atoms to minimize empty space?

2-dimensions

vs.

metallic crystal structures6
Metallic Crystal Structures

• Tend to be densely packed.

• Reasons for dense packing:

  • Typically, only one element is present, so all atomic radii are the same.
  • - Metallic bonding is not directional.
  • - Nearest neighbor distances tend to be small in
  • order to lower bond energy.
  • - The “electron cloud” shields cores from each other

• They have the simplest crystal structures.

simple cubic structure sc
Simple Cubic Structure (SC)

• Rare due to low packing density (only Po has this structure)

• Close-packed directions are cube edges.

• Coordination # = 6

(# nearest neighbors)

atomic packing factor apf
Atomic Packing Factor (APF)

volume

atoms

atom

4

a

3

unit cell

p

(0.5a)

3

R=0.5a

volume

close-packed directions

unit cell

contains 8 x 1/8 =

1

atom/unit cell

Volume of atoms in unit cell*

APF =

Volume of unit cell

*assume hard spheres

• APF for a simple cubic structure = 0.52

1

APF =

3

a

body centered cubic structure bcc
Body Centered Cubic Structure (BCC)

• Atoms touch each other along cube diagonals.

All atoms are identical.

ex: Cr, W, Fe (), Tantalum, Molybdenum

• Coordination # = 8

2 atoms/unit cell: 1 center + 8 corners x 1/8

atomic packing factor bcc
Atomic Packing Factor: BCC

a

3

a

2

Close-packed directions:

R

3 a

length = 4R =

a

atoms

volume

4

3

p

(

3

a/4

)

2

unit cell

atom

3

APF =

volume

3

a

unit cell

• APF for a body-centered cubic structure = 0.68

a

face centered cubic structure fcc
Face Centered Cubic Structure (FCC)

• Atoms touch each other along face diagonals.

--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag

• Coordination # = 12

4 atoms/unit cell: 6 face x 1/2 + 8 corners x 1/8

atomic packing factor fcc
Atomic Packing Factor: FCC

2 a

Unit cell contains:

6 x1/2 + 8 x1/8

=

4 atoms/unit cell

a

atoms

volume

4

3

p

(

2

a/4

)

4

unit cell

atom

3

APF =

volume

3

a

unit cell

• APF for a face-centered cubic structure = 0.74

maximum achievable APF

Close-packed directions:

2 a

length = 4R =

hexagonal close packed structure hcp
Hexagonal Close-Packed Structure (HCP)

A sites

Top

layer

c

Middle layer

B

sites

A sites

Bottom layer

a

• ABAB... Stacking Sequence

• 3D Projection

• 2D Projection

6 atoms/unit cell

• Coordination # = 12

ex: Cd, Mg, Ti, Zn

• APF = 0.74

• c/a = 1.633

slide18

Detector

Xray-Tube

Xray-Tube

Sample

Metal Target

(Cu or Co)

Detector

X-Ray Diffractometer

sample

d – distance between the

same atomic planes

λ – monochromatic

wavelength

θ – angle of diffracto-

meter

Bragg´s Equation

d = λ/2 sinθ

theoretical density r

nA

VCNA

Mass

of

Atoms

in

Unit

Cell

Total

Volume

of

Unit

Cell

 =

Theoretical Density, r

Density =  =

where n = number of atoms/unit cell

A =atomic weight

VC = Volume of unit cell = a3 for cubic

NA = Avogadro’s number

= 6.022 x 1023 atoms/mol

theoretical density r21

R

a

2

52.00

atoms

g

 =

unit cell

mol

atoms

3

a

6.022x1023

mol

volume

unit cell

Theoretical Density, r
  • Ex: Cr (BCC)

A(atomic weight) =52.00 g/mol

n = 2 atoms/unit cell

R = 0.125 nm

a = 4R/ 3 = 0.2887 nm

theoretical

= 7.18 g/cm3

ractual

= 7.19 g/cm3

factors that determine crystal structure

-

-

-

-

-

-

+

+

+

-

-

-

-

-

-

unstable

-

F

2+

Ca

+

CaF

:

2

anions

cation

-

F

A

X

m

p

m, p values to achieve charge neutrality

Factors that Determine Crystal Structure

1.Relative sizes of ions – Formation of stable structures:

--maximize the # of oppositely charged ion neighbors.

stable

stable

2.Maintenance of

Charge Neutrality:

--Net charge in ceramic

should be zero.

--Reflected in chemical

formula:

atomic bonding in ceramics
Atomic Bonding in Ceramics

CaF2: large

SiC: small

• Bonding:

-- Can be ionic and/or covalent in character.

-- % ionic character increases with difference in

electronegativity of atoms.

• Degree of ionic character may be large or small:

coordination and ionic radii
Coordination # and Ionic Radii

r

cation

r

anion

r

ZnS

cation

(zinc blende)

r

anion

NaCl

(sodium

chloride)

CsCl

(cesium

chloride)

• Coordination # increases with

To form a stable structure, how many anions can

surround a cation?

Coord

#

linear

< 0.155

2

triangular

0.155 - 0.225

3

tetrahedral

0.225 - 0.414

4

octahedral

0.414 - 0.732

6

cubic

0.732 - 1.0

8

computation of minimum cation anion radius ratio
Computation of Minimum Cation-Anion Radius Ratio

a

  • Determine minimum rcation/ranionfor an octahedral site (C.N. = 6)

Measure the radii (blue and yellow spheres)

a= 2ranion

Substitute for “a” in the above equation

example problem predicting the crystal structure of feo
Example Problem:Predicting the Crystal Structure of FeO

• Answer:

Cation

Ionic radius (nm)

3+

Al

0.053

2

+

Fe

0.077

3+

Fe

0.069

2+

Ca

0.100

based on this ratio,

-- coord # = 6 because

0.414 < 0.550 < 0.732

-- crystal structure is similar to NaCl

Anion

2-

O

0.140

-

Cl

0.181

-

F

0.133

• On the basis of ionic radii, what crystal structure would you predict for FeO?

rock salt structure
Rock Salt Structure

Same concepts can be applied to ionic solids in general.

Example: NaCl (rock salt) structure

rNa = 0.102 nm

rCl = 0.181 nm

  • rNa/rCl = 0.564
  • cations (Na+) prefer octahedralsites
mgo and feo
MgO and FeO

MgO and FeO also have the NaCl structure

O2- rO = 0.140 nm

Mg2+ rMg = 0.072 nm

  • rMg/rO = 0.514
  • cations prefer octahedral sites

So each Mg2+ (or Fe2+) has 6 neighbor oxygen atoms

ax crystal structures
AX Crystal Structures

AX–Type Crystal Structures include NaCl, CsCl, and zinc blende

Cesium Chloride structure:

 Since 0.732 < 0.939 < 1.0, cubicsites preferred

So each Cs+ has 8 neighbor Cl-

ax 2 crystal structures
AX2 Crystal Structures

Fluorite structure

  • Calcium Fluorite (CaF2)
  • Cations in cubic sites
  • UO2, ThO2, ZrO2, CeO2
  • Antifluorite structure –
  • positions of cations and anions reversed
abx 3 crystal structures
ABX3 Crystal Structures
  • Perovskite structure
  • Ex: complex oxide
  • BaTiO3
summary
SUMMARY

• Atoms may assemble into crystalline or amorphous structures.

• Common metallic crystal structures are FCC, BCC and HCP. Coordination number and atomic packing factor are the same for both FCC and HCP crystal structures.

• We can predict the density of a material, provided we know the atomic weight, atomic radius, and crystal geometry (e.g., FCC, BCC, HCP).

• Interatomic bonding in ceramics is ionic and/or covalent.

• Ceramic crystal structures are based on:

-- maintaining charge neutrality

-- cation-anion radii ratios.