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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.