chapter 3: the structure of crystalline solids chapter 3_0.pdfchapter 3 - 7 • tend to be densely...

Chapter 3 - 1

ISSUES TO ADDRESS...

• How do atoms assemble into solid structures?

(for now, focus on metals)

• How does the density of a material depend on

its structure?

• When do material properties vary with the

sample (i.e., part) orientation?

Chapter 3: The Structure of Crystalline Solids

Instructor: Eng. Tamer Haddad

Chapter 3 - 2

• Non dense, random packing

• Dense, ordered packing

Dense, ordered packed structures tend to have lower

energies.

Energy and Packing

Energy

r

typical neighborbond length

typical neighborbond energy

Energy

r

typical neighborbond length

typical neighborbond energy


Chapter 3 - 3

• atoms pack in periodic, 3D arraysCrystalline materials...

-metals

-many ceramics

-some polymers

• atoms have no periodic packing

Noncrystalline materials...

-complex structures

-rapid cooling

crystalline SiO2

noncrystalline SiO2"Amorphous" = NoncrystallineAdapted from Fig. 3.22(b),

Callister 7e.

Adapted from Fig. 3.22(a),

Callister 7e.

Materials and Packing

Si Oxygen

• typical of:

• occurs for:


Chapter 3 - 4

Crystal Structure:-

• The manner in which the atoms, ions or molecules are

spatially arranged.

• There is a large number of different crystal structures all

having long range atomic order.

Atomic Hard Sphere Model

When describing crystalline

structures, atoms of solid

spheres and well defined

diameters

Lattice : 3D array of spheres centersInstructor: Eng. Tamer Haddad

Chapter 3 - 5

Section 3.3 – Unit CellsUnit cell:

It is often convenient to subdivide the structure

into small repeat entities called unit cell.

smallest repetitive volume which contains the

complete lattice pattern of a crystal.

• The unit cell is the basic structural unit or building block of

the crystal structure and defines the crystal structure by virtue

of its geometry and the atom positions within.Instructor: Eng. Tamer Haddad

Chapter 3 - 6

Section 3.4 – Metallic Crystal Structures

• How can we stack metal atoms to minimize

empty space?

2-dimensions

vs.


Chapter 3 - 7

• 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.

We will examine three such structures...

Metallic Crystal Structures


Chapter 3 - 8

Adapted from Fig. 3.1, Callister 7e.

• Atoms touch each other along face diagonals.--Note: All atoms are identical; the face-centered atoms are shaded

differently only for ease of viewing.

Face Centered Cubic Structure (FCC)

ex: Aluminum, Copper, Silver, and Gold.


Chapter 3 - 9

Face Centered Cubic Structure (FCC)

a

2 a

Adapted from

Fig. 3.1(a),

Callister 7e.

a: cube edge length

R: atomic radius

Unit cell edge length for face-

centered Cubic

a = 2R√2

• Each corner atom is shared among eight unit cells,

whereas a face-centered atom belongs to only two..

4 atoms/unit cell may be assigned to a given unit cell: 6 face x 1/2 + 8

corners x 1/8 Instructor: Eng. Tamer Haddad

Chapter 3 - 10

Two Important Characteristics of a crystal structure:

(I) Coordination Number

Coordination number:

For metals, each atom has the same number of nearest-

neighbor or touching atoms, which is the coordination

number.

• For FCC, Coordination # = 12


Chapter 3 -

a

2 a

11

(II) Atomic Packing Factor (APF)

APF = Volume of atoms in unit cell*

Volume of unit cell

*assume hard spheres

• APF for a face-centered cubic structure = 0.74

maximum achievable APF

APF =

4

3p ( 2a/4)34

atoms

unit cell atom

volume

a3

unit cell

volume


Chapter 3 - 12

• Coordination # = 8

Adapted from Fig. 3.2,

Callister 7e.

• Atoms touch each other along cube diagonals.--Note: All atoms are identical; the center atom is shaded

differently only for ease of viewing.

Body Centered Cubic Structure (BCC)

ex: Chromium, iron and tungsten.

2 atoms/unit cell: 1 center + 8 corners x 1/8Instructor: Eng. Tamer Haddad

Chapter 3 - 13

Atomic Packing Factor: BCC

a

APF =

4

3p ( 3a/4)32

atoms

unit cell atom

volume

a3

unit cell

volume

length = 4R =

Close-packed directions:

3 a

• APF for a body-centered cubic structure = 0.68

aR

Adapted from

Fig. 3.2(a), Callister 7e.

a2

a3


Chapter 3 -Instructor: Eng. Tamer Haddad

14

Hexagonal Close-Packed Structure

(HCP)• Not all metals have unit cells with cubic symmetry; hexagonal

unit cell are another common metallic crystal structure.

ex: Cadmium, magnesium, titanium, and zinc.

Chapter 3 - 15

• Coordination # = 12

• APF = 0.74

Hexagonal Close-Packed Structure

(HCP)

6 atoms/unit cell =

one-sixth of each of the 12 top and bottom face corner

atoms, one-half of each of the 2 center face atoms, and all 3

mid plane interior atoms.

• c /a = 1.633 with deviations for some HCP metals.


Chapter 3 - 16

Theoretical Density Computations, r

where n = number of atoms/unit cell

A = atomic weight

VC = Volume of unit cell = a3 for cubic

NA = Avogadro’s number

= 6.023 x 1023 atoms/mol

Density = r =

VCNA

n Ar =

CellUnitofVolumeTotal

CellUnitinAtomsofMass


Chapter 3 - 17

• Ex: Cr (BCC)

A = 52.00 g/mol

R = 0.125 nm

n = 2

rtheoretical

a = 4R/ 3 = 0.2887 nm

ractual

aR

r = a3

52.002

atoms

unit cellmol

g

unit cell

volume atoms

mol

6.023x 1023

Theoretical Density, r

= 7.18 g/cm3

= 7.19 g/cm3


Chapter 3 - 18

Densities of Material Classes

rmetals > rceramics > rpolymers

Why?

Data from Table B1, Callister 7e.

r(g

/cm

)

3

Graphite/ Ceramics/ Semicond

Metals/ Alloys

Composites/ fibers

Polymers

1

2

20

30Based on data in Table B1, Callister

*GFRE, CFRE, & AFRE are Glass,Carbon, & Aramid Fiber-ReinforcedEpoxy composites (values based on60% volume fraction of aligned fibers

in an epoxy matrix).10

3

4

5

0.3

0.4

0.5

Magnesium

Aluminum

Steels

Titanium

Cu,Ni

Tin, Zinc

Silver, Mo

TantalumGold, WPlatinum

Graphite

Silicon

Glass -sodaConcrete

Si nitrideDiamondAl oxide

Zirconia

HDPE, PSPP, LDPE

PC

PTFE

PETPVCSilicone

Wood

AFRE*

CFRE*

GFRE*

Glass fibers

Carbon fibers

Aramid fibers

Metals have...• close-packing

(metallic bonding)

• often large atomic masses

Ceramics have...• less dense packing

• often lighter elements

Polymers have...• low packing density

(often amorphous)

• lighter elements (C,H,O)

Composites have...• intermediate values

In general


Chapter 3 - 19

Section 3.6 – Polymorphism/Allotropy • For metals and non-metals, two or more distinct crystal

structures for the same material (polymorphism/allotropy)

due to temperature and pressure)

carbon

diamond, graphite BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system


Ambient

Conditions

High

Pressures

Chapter 3 -

Crystalline and Non-crystalline Materials

Instructor: Eng. Tamer Haddad20

Single Crystals For a crystalline solid, when the periodic and repeated

arrangement of atoms is perfect or extends throughout the

entirety of the specimen without interruption,

Chapter 3 -Instructor: Eng. Tamer Haddad

21

Polycrystalline Crystals

Most crystalline solids are composed of a collection of many

small crystals or grains.

Ex. Solidification of polycrystalline specimen.

Grain boundary

Chapter 3 - 22

• Atoms may assemble into crystalline or

amorphous 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).

SUMMARY

• 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.


• Materials can be single crystals or polycrystalline.

Chapter 3 -

Question 3-10

Instructor: Eng. Tamer Haddad23

Some hypothetical metal has the simple cubic crystal structure

shown in Figure 3.23. If its atomic weight is 74.5 g/mol and the

atomic radius is 0.145 nm, compute its density.

chapter 3: the structure of crystalline solids chapter 3_0.pdfchapter 3 - 7 • tend to be densely...

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