Chapter 3-

CRYSTAL SYSTEMS

General lattice that is in the shape of a parallelepiped or prism. a, b, and c are called lattice parameters.

x, y, and z here are called material axes, and the coordinates described by them are termed “material coordinates”

Chapter 3-

CRYSTAL SYSTEMS

There are only seven possible types of crystal structures or unit cells which can fill 3D space if stacked together.

For example, the cubic unit to the upper left can become a BCC or an FCC lattice.

Chapter 3-

CRYSTALLOGRAPHIC POINTS, DIRECTIONS, and PLANES

Example: determine point ¼ 1 ½ for the following lattice:

Chapter 3-

CRYSTALLOGRAPHIC DIRECTIONS

x y z

Projections a/2 b 0c

Projections (in terms of a, b, and c)

1/2 1 0

Reduction to nearest integer

1 2 0

Final result [120]

Directions are free vectors, i.e. they can be moved around as long as parallelism is maintained.

Chapter 3-

CRYSTALLOGRAPHIC DIRECTIONS

A “family” of equivalent directions are enclosed in angle brackets. For example, <100> stands for the [100], [-100],[010],[0-10],[001], and [00-1] directions. This is just a convenience.

Chapter 3-

CRYSTALLOGRAPHIC DIRECTIONS for Hexagonal Crystals

The directions utilize a four-axis, or Miller-Bravais, coordinate system, and not three axes like in cubic crystals

cBasal plane

Chapter 3-

CRYSTALLOGRAPHIC DIRECTIONS

Exercise (in groups of two): Draw a sketch of the crystallographic directions [2 2 1], and [1 -1 1] for a cubic crystal,And [1 0 0 1] for a hexagonal crystal

Chapter 3-

CRYSTALLOGRAPHIC PLANESPlanes are described by Miller Indices, e.g. (hkl)

Chapter 3-

CRYSTALLOGRAPHIC PLANESPlanes are described by Miller Indices, e.g. (hkl)

Example: If plane intersects the origin, then shift plane to another neighboring cell before determining its indices

x y z

Intercepts a -b c/2

Intercepts (in terms of a, b, and c) -1 1/2

Reciprocals 0 -1 2

Reductions (unnecessary)

Final result (0-12)

Chapter 3-

CRYSTALLOGRAPHIC PLANESExample: construct a (0-11) plane within a cubic unit cell

Note that (0-11) plane is equivalent to (01-1) plane

For convenience, a family of equivalent planes has indices enclosed in curly braces. For example, {111} stands for (111), (-1-1-1), (-111), (1-1-1), (1-11),(-11-1),(11-1), and (-1-11)

Chapter 3-

CRYSTALLOGRAPHIC PLANES

Exercise (in groups of two): Draw a sketch of the crystallographic planes (1 1 -1) and (120)

Chapter 3-

CLOSE-PACKED CRYSTALLOGRAPHIC DIRECTIONS

AND PLANES

In FCC, the close-packed planes are the {111} planes, and the close-packed directions are the <110> directions.

In BCC, the close-packed planes are the {110} planes, and the close-packed directions are the <111>

In HCP, the close-packed planes are the (0001) basal plane, and the close-packed direction are the [1000], [0100], and [0010] directions and their negatives.

Chapter 3-17

• Some engineering applications require single crystals:

• Crystal properties reveal features of atomic structure.

(Courtesy P.M. Anderson)

--Ex: Certain crystal planes in quartz fracture more easily than others.

--diamond single crystals for abrasives

--turbine bladesFig. 8.30(c), Callister 6e.(Fig. 8.30(c) courtesyof Pratt and Whitney).(Courtesy Martin

Deakins,GE Superabrasives, Worthington, OH. Used with permission.)

CRYSTALS AS BUILDING BLOCKS

Chapter 3-18

• Most engineering materials are polycrystals and NOT a single crystal.POLYCRYSTALS

• Each "grain" is a single crystal.• If crystals are randomly oriented,overall component properties are not directional.• Crystal sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Chapter 3-18• Nb-Hf-W plate with an electron beam weld.

Adapted from Fig. K, color inset pages of Callister 6e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

POLYCRYSTALS

Chapter 3-19

• Single Crystals-Properties vary with direction: anisotropic.

-Example: the modulus of elasticity (E) in BCC iron:

• Polycrystals

-Properties may/may not vary with direction.-If grains are randomly oriented: isotropic. (Epoly iron = 210 GPa)-If grains are textured, anisotropic.

E (diagonal) = 273 GPa

E (edge) = 125 GPa

200 m

Data from Table 3.3, Callister 6e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Adapted from Fig. 4.12(b), Callister 6e.(Fig. 4.12(b) is courtesy of L.C. Smith and C. Brady, the National Bureau of Standards, Washington, DC [now the National Institute of Standards and Technology, Gaithersburg, MD].)

SINGLE VS POLYCRYSTALS

Chapter 3-

d=n/2sinc

x-ray intensity (from detector)

c20

• Incoming X-rays diffract from crystal planes.

• Measurement of: Critical angles, c, for X-rays provide atomic spacing, d.

Adapted from Fig. 3.2W, Callister 6e.

X-RAYS TO CONFIRM CRYSTAL STRUCTURE

reflections must be in phase to detect signal

spacing between planes

d

incoming

X-rays

outg

oing

X-ra

ys

detector

extra distance travelled by wave “2”

“1”

“2”

“1”

“2”

Chapter 3- 21

• Atoms can be arranged and imaged!

Carbon monoxide molecules

arranged on a platinum (111)

surface.

Photos produced from the work of C.P. Lutz, Zeppenfeld, and D.M. Eigler. Reprinted with permission from International Business Machines Corporation, copyright 1995.

Iron atoms arranged on a copper (111)

surface. These Kanji characters

represent the word “atom”.

SCANNING TUNNELING MICROSCOPY

Chapter 3-22

• Demonstrates "polymorphism" The same atoms in a matter can have more than one crystal structure.

DEMO: HEATING ANDCOOLING OF AN IRON WIRE

Temperature, C

BCC Stable

FCC Stable

914

1391

1536

shorter

longer!shorter!

longer

Tc 768 magnet falls off

BCC Stable

Liquid

heat up

cool down

Chapter 3-

• 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).

• Material properties generally vary with single crystal orientation (i.e., they are anisotropic), but properties are generally non-directional (i.e., they are isotropic) in polycrystals with randomly oriented grains.

23

SUMMARY