Chapter 3 -

Remember Miller Indices?

• For directions:– Determine coordinates

for “head” and “tail” of the direction

– “head”-”tail”– Clear fraction/reduce

results to lowest integers.

– Enclose numbers in [] and a bar over negative integers.

• For planes:– Identify points at which

the plane intercepts the x, y, z axis.

– Take reciprocals of these intercepts.

– Clear fractions and do NOT reduce to the lowest integers.

– Enclose the numbers in parentheses () and a bar over negative integers.

Chapter 3 -

Special note for directions…

• For Miller Indices of directions:– Since directions are vectors, a direction and its

negative are not identical!• [100] ≠ [100] Same line, opposite directions!

– A direction and its multiple are identical!• [100] is the same direction as [200] (need to reduce!)• [111] is the same direction as [222], [333]!

– Certain groups of directions are equivalent; they have their particular indices because of the way we construct the coordinates.

• Family of directions: <111>=[111], [111],[111],[111],…

Chapter 3 -

Special note for planes…

• For Miller Indices of planes:– Planes and their negatives are identical (not the case

for directions!)• E.g. (020) = (020)

– Planes and their multiples are not identical (Again, different from directions!) We can show this by defining planar densities and planar packing fractions.

• E.g. (010) ≠ (020) See example!– Each unit cell, equivalent planes have their particular

indices because of the orientation of the coordinates.• Family of planes: {110} = (110),(110),(110),(101), (101),…

– In cubic systems, a direction that has the same indices as a plane is perpendicular to that plane.

Chapter 3 -

Calculate the planar density for the (010) and (020) planes in simple cubic polonium, which has a lattice parameter of 0.334 nm.

Example: Calculating the Planar Density

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

a0a0

Chapter 3 -

2142

2

atoms/cm 1096.8atoms/nm 96.8

)334.0(

faceper atom 1

face of area

faceper atom (010)density Planar

SOLUTIONThe total atoms on each face is one. The planar density is:

However, no atoms are centered on the (020) planes. Therefore, the planar density is zero. The (010) and (020) planes are not equivalent!

(a0)2

Chapter 3 - 6

Planar Density of (100) IronSolution:  At T < 912C iron has the BCC structure.

(100)

Radius of iron R = 0.1241 nm

R3

34a

2D repeat unit

= Planar Density = a2

1

atoms

2D repeat unit

= nm2

atoms12.1

m2

atoms= 1.2 x 1019

12

R3

34area

2D repeat unit

Chapter 3 - 7

Planar Density of (111) Iron

333 2

2

R3

16R

34

2a3ah22area

ah2

3

0.5

= = nm2

atoms7.0m2

atoms0.70 x 1019

3 2R6

16Planar Density =

atoms

2D repeat unit

area

2D repeat unit

a2

a2

h

There are only (3)(1/6)=1/2 atoms in the plane.

Chapter 3 -

In-Class Exercise 1: Determine planar density

Determine the planar density for BCC lithium in the (100), (110), and the (111) planes.

atomic radius for Li = 0.152 nm

Chapter 3 -

Solution for plane (100)

510.33510.03

152.04

3

4

152.0

0 nmnmr

BCCa

nmrLi

For (100):

214

28/10115.8

10510.3

1_ cmatoms

cm

atomdensityplanar

Chapter 3 -

Solution for plane (110)

For (110):

It is important to visualize how the plane is cutting across the unit cell – as shown in the diagram!

215

28/10148.1

10510.32

2_ cmatoms

cm

atomsdensityplanar

Chapter 3 -

Solution for plane (111)

02a

For (111):Note: Since the (111) does NOT pass through the center of the atom in the middle of the BCC unit cell, we do not count it!

2000 866.0

2

33

2

1

2

1_ aaabhareaplane

214

28/10686.4

10510.3866.0

2/1_ cmatoms

cm

atomdensityplanar

02a

Chapter 3 -

In-Class Exercise 2: Determine planar density

Determine the planar density for FCC nickel in the (100), (110), and (111) planes.

atomic radius for Nickel= 0.125 nm

Remember when visualizing the plane, only count the atoms that the plane passes through the center of the atom. If the plane does NOT pass through the center of that atom, we do not count it!

Chapter 3 -

Solution for plane (100)

536.33536.02

125.04

2

4

125.0

0 nmnmr

FCCa

nmrNi

a0

For (100):

215

28/10600.1

10536.3

2_ cmatoms

cm

atomsdensityplanar

Chapter 3 -

Solution for plane (110)

215

28/10131.1

10536.32

2_ cmatoms

cm

atomsdensityplanar

For (110):

a

0

02a

It is important to visualize how the plane is cutting across the unit cell – as shown in the diagram!

Chapter 3 -

Solution for plane (111)

02a

02a

For (111):Again try to visualize the plane, count the number of atoms in the plane:

02a

2000 866.0

2

32

2

1

2

1_ aaabhareaplane

215

28/10847.1

10536.3866.0

2_ cmatoms

cm

atomsdensityplanar

Chapter 3 -16

Home Exercise: Determine planar densityDetermine the planar density for (0001) plane for an HCP unit cell Titaniumatomic radius for titanium is 0.145 nm

Chapter 3 - 17

• Some engineering applications require single crystals:

• Properties of crystalline materials often related to crystal structure.

--Ex: Quartz fractures more easily along some crystal planes than others.

--diamond single crystals for abrasives

--turbine blades(Co and Ni superalloys)

Fig. 8.33(c), Callister 7e.(Fig. 8.33(c) courtesyof Pratt and Whitney).(Courtesy Martin Deakins,

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

Crystals as Building Blocks

Chapter 3 -

Poly crystal Material

Grains

Single crystal

Chapter 3 - 19

• Most engineering materials are polycrystals.

• Nb-Hf-W plate with an electron beam weld.• Each "grain" is a single crystal.• If grains are randomly oriented, overall component properties are not directional.• Grain sizes typ. range from 1 nm to 2 cm (i.e., from a few to millions of atomic layers).

Adapted from Fig. K, color inset pages of Callister 5e.(Fig. K is courtesy of Paul E. Danielson, Teledyne Wah Chang Albany)

1 mm

Polycrystals

Isotropic

Anisotropic

Chapter 3 - 20

• 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.

200 m

Data from Table 3.3, Callister 7e.(Source of data is R.W. Hertzberg, Deformation and Fracture Mechanics of Engineering Materials, 3rd ed., John Wiley and Sons, 1989.)

Adapted from Fig. 4.14(b), Callister 7e.(Fig. 4.14(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 PolycrystalsE (diagonal) = 273 GPa

E (edge) = 125 GPa

Chapter 3 - 21

Section 3.6  – Polymorphism

• Two or more distinct crystal structures for the same material (allotropy/polymorphism)   titanium

  , -Ti

carbon

diamond, graphite

BCC

FCC

BCC

1538ºC

1394ºC

912ºC

-Fe

-Fe

-Fe

liquid

iron system

Chapter 3 - 22

Section 3.16 - X-Ray Diffraction

• Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.

• Can’t resolve spacings • Spacing is the distance between parallel planes of

atoms.  

Chapter 3 -

(c) 2003 Brooks/C

ole Publishing / Thom

son Learning

(a)Destructive (out of phase) x-ray beam gives a weak signal.

(b)Reinforcing (in phase) interactions between x-rays and the crystalline material. Reinforcement occurs at angles that satisfy Bragg’s law.

Chapter 3 - 24

X-Rays to Determine Crystal Structure

X-ray intensity (from detector)

c

d n

2 sinc

Measurement of critical angle, c, allows computation of planar spacing, d.

• Incoming X-rays diffract from crystal planes.

Adapted from Fig. 3.19, Callister 7e.

reflections must be in phase for a detectable 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 -

(c) 2003 Brooks/C

ole Publishing / Thom

son Learning

(a) Diagram of a diffractometer, showing powder sample, incident and diffracted beams.

(b) (b) The diffraction pattern obtained from a sample of gold powder.

Chapter 3 - 26

X-Ray Diffraction Pattern

Adapted from Fig. 3.20, Callister 5e.

(110)

(200)

(211)

z

x

ya b

c

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC)

Inte

nsity

(re

lativ

e)

z

x

ya b

cz

x

ya b

c

Chapter 3 -

Bragg’s Law:

sin2 hkldn Bragg’s Law:

222

0

lkh

adhkl

Interplanar

spacing:

dMiller Indices

Where is half the angle between the diffracted beam and the original beam direction

is the wavelength of X-ray

d is the interplanar spacing

Chapter 3 - 28

• 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.

• Crystallographic points, directions and planes are specified in terms of indexing schemes. Crystallographic directions and planes are related to atomic linear densities and planar densities.

Chapter 3 - 29

• Some materials can have more than one crystal structure. This is referred to as polymorphism (or allotropy).

SUMMARY

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

• X-ray diffraction is used for crystal structure and interplanar spacing determinations.