Slide 1

Chapter 3: The Structure of Crystalline Solids

Solid materials may be classified according to the regularity with which atoms or ions are arranged with respect to one another. A crystalline materials: is one in which atoms are situated in a repeating or periodic array over large atomic distances (long-range order ). All metals, many ceramics materials, and certain polymers form crystalline structure under normal solidification conditions

Non crystalline or amorphous materials : No long range order only short range order For example: Amorphous Si, Glasses

Some of theproperties of crystalline solids depend on the crystal structure of the materials

Chapter 3: The Structure of Crystalline Solids Some of the properties of crystalline solids depend on the crystal structure of the materials

Unit cell The basic structure unit or building block of the crystal structure

Introduction to Materials Science for Engineers, James F. Shacklford, Prentice Hall, NJ, 1996

Chapter 3: The Structure of Crystalline Solids All possible structures reduce to a small number of basic unit cell geometries : -There are only seven, unique unit cell shapes that can be stacked together to fill three-dimensional space ( seven crystal systems)

.

Introduction to Materials Science for Engineers, James F. Shacklford, Prentice Hall, NJ, 1996

Chapter 3: The Structure of Crystalline Solids The possibility of how the atoms stacked together within a given unit cell : 14 Bravais lattices

Lattice points : theoretical points arranged periodical in threedimensional space

Introduction to Materials Science for Engineers, James F. Shacklford, Prentice Hall, NJ, 1996

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. - Electron cloud shields cores from each other • Have the simplest crystal structures. Three crystal structures are found for most of the common Metals Face-centered cubic Body-centered cubic

Hexagonal close-packed

Simple Cubic Structure (SC) • Rare due to low packing density (only Po • Close-packed directions are cube edges.

has this structure)

• Coordination # = 6 (# nearest neighbors)

(Courtesy P.M. Anderson)

Atomic Packing Factor (APF) Volume of atoms in unit cell* APF =

Vs Vc

=

Volume of unit cell *assume hard spheres

• APF for a simple cubic structure = 0.52

a Vc= a3

= 2R = (2R )3

a volume

R=0.5a

atoms unit cell

close-packed directions contains 8 x 1/8 = 1 atom/unit cell

4 1

atom π (0.5a)

3

3 APF =

a3

volume unit cell

Body Centered Cubic Structure (BCC) • Atoms touch each other along cube diagonals. --Note: All atoms are identical; the center atom is shaded differently only for ease of viewing.

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

• Coordination # = 8

Adapted from Fig. 3.2,

Callister 7e.

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

Atomic Packing Factor: BCC • APF for a body-centered cubic structure = 0.68

3a a

2a Close-packed directions: length = 4R

R

a =

a

Vc =

a3

= 3 a

4R 3 =

64 R3 3

atoms unit cell

4 2

π ( 3 a/4 ) 3

volume atom

3

APF = Adapted from Fig. 3.2(a), Callister 7e.

a3

volume unit cell

3

Face Centered Cubic Structure (FCC) • Atoms touch each other along face diagonals.

ex: Al, Cu, Au, Pb, Ni, Pt, Ag •

Coordination # = 12

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

Adapted from

Callister 7e.

Atomic Packing Factor: FCC maximum achievable APF

• APF for a face-centered cubic structure = 0.74

Close-packed directions: length = 4R = a= 2R

2a

2 a

2

Vc = a3 = 16 R3

2

Unit cell contains:

a

6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell atoms unit cell

4 4

π ( 2 a/4 ) 3

3

volume atom

APF =

a3 Adapted from

Callister 7e.

volume unit cell

Chapter 3: The Structure of Crystalline Solids

Adapted from Callister 7e.

Chapter 3: The Structure of Crystalline Solids

Adapted from Callister 7e.

Hexagonal Close-Packed Structure (HCP)

• Coordination # = 12 • APF = 0.74 • c/a = 1.633

6 atoms/unit cell ex: Cd, Mg, Ti, Zn

Close-packed stacking sequence for FCC • ABCABC... Stacking Sequence • 2D Projection B A A sites B sites C sites • FCC Unit Cell A B C

B

C B

B C B

C B

B

Close-packed stacking sequence for HCP

• ABAB... Stacking Sequence • 3D Projection

c

a

• 2D Projection A sites

Top

layer

B sites

Middle layer

A sites

Bottom layer

stacking sequence

HCP

FCC

Polymorphism • Two or more distinct crystal structures for the same material (allotropy/polymorphism) titanium α, β-Ti

iron system liquid 1538ºC δ-Fe BCC 1394ºC γ-Fe FCC 912ºC BCC α-Fe

carbon diamond, graphite

Diamond

graphite

The percent change in volume = (final volume-initial volume)/(initial volume)x 100 ∆V % = ((Vf-Vi)/Vi) X 100

Single vs polycrystalline • Single Crystals For a crystalline solid, when the periodic and repeated arrangement of atoms is perfect or extend throughout the entirety of the specimen without interruption, the result is a single crystal.

• polycrystalline Most crystalline solids are composed of a collection of many small crystals or grains; such materials are termed polycrystalline

grains grain boundary

Non crystalline Solids

Crystalline silicon dioxide

Non crystalline silicon dioxide

Non crystalline solids (amorphous) : lack a systematic and regular arrangement of atoms (disordered)

Section 3.8 Point Coordinates

Crystallographic Directions Algorithm

z

y x

ex: 1, 0, ½

1. Vector repositioned (if necessary) to pass through origin. 2. Read off projections in terms of unit cell dimensions a, b, and c 3. Adjust to smallest integer values 4. Enclose in square brackets, no commas [uvw]

=> 2, 0, 1 => [ 201 ]

Miller indices of directions

Miller indices of directions

Miller indices of directions

½ ,1, 0

=> 1,2,0 =>

[120 ]

1 ,-1, 0

=> 1,-1,0 =>

[110 ]

Miller indices of directions

z

y ½,1,0

x -½ ,-1, 1 => -1,-2,2 =>

[122]

Miller indices of directions

A set of directions, which are structurally equivalent families of directions



= [111], [111], [111], [111], [111], [111], [111], [111]

Crystallographic Planes Algorithm 1. Read off intercepts of plane with axes in terms of a, b, c 2. Take reciprocals of intercepts 3. Reduce to smallest integer values 4. Enclose in parentheses, no commas i.e., (hkl) All parallel planes have same Miller indices.

Crystallographic Planes z

example 1. Intercepts 2. Reciprocals

a

b 1 1/1 1 1

3.

Reduction

1 1/1 1 1

4.

Miller Indices

(110)

c

c

∞ 1/∞ 0 0

y b

a x z

example 1. Intercepts 2. Reciprocals

a

3.

Reduction

1/2 1/½ 2 2

4.

Miller Indices

(200)

b

∞ 1/∞ 0 0

c

∞ 1/∞ 0 0

c

y a x

b

Crystallographic Planes z example

a

b

c

1.

Intercepts

1/2

1

3/4

2.

Reciprocals

1/½

1/1

1/¾

3.

Reduction

4.

Miller Indices

c •

2

1

4/3

6

3

4

(634)

a

•

x

Family of Planes {hkl} Ex: {100} = (100), (010), (001), (100), (010), (001)

•

b

y

Crystallographic Planes

All parallel planes have same Miller indices.

Crystallographic Planes

Anisotropy - Directionality of properties -Substances in which measured properties are independent of the direction of measurement are isotropic • Single Crystals -Properties vary with direction: anisotropic. -Example: the modulus of elasticity (E) in BCC iron:

E (diagonal) = 273 GPa

• Polycrystals -Properties may/may not vary with direction. -If grains are randomly oriented: isotropic. -If grains are textured, anisotropic.

E (edge) = 125 GPa

Densities of Material Classes In general ρ ρ ρ metals > ceramics > polymers

Metals have...

• low packing density (often amorphous) • lighter elements (C,H,O)

Composites have... • intermediate values

10

Silver, Mo Cu,Ni Steels Tin, Zinc

(g/cm 3)

Polymers have...

20

Platinum Gold, W Tantalum

5 4 3

ρ

• close-packing (metallic bonding) • often large atomic masses • less dense packing • often lighter elements

Graphite/ Ceramics/ Semicond

Polymers

Composites/ fibers

30

Why?

Ceramics have...

Metals/ Alloys

2

Titanium Aluminum Magnesium

B ased on data in Table B1, Callister *GFRE, CFRE, & AFRE are Glass, Carbon, & Aramid Fiber-Reinforced Epoxy composites (values based on 60% volume fraction of aligned fibers in an epoxy matrix). Zirconia Al oxide Diamond Si nitride Glass -soda Concrete Silicon G raphite

1

0.5 0.4 0.3

Glass fibers PTFE Silicone PVC PET PC HDPE, PS PP, LDPE

GFRE* Carbon fibers CFRE * Aramid fibers AFRE *

Wood

Data from Table B1, Callister 7e.

DENSITY COMPUTATIONS A knowledge of the crystal structure of a metallic solid permits computation of its theoretical density through the relationship