1. Crystallographic Planes
Miller Indices:
Reciprocals of the (three) axial intercepts for a plane, cleared of fractions & common multiples. All parallel planes have same Miller indices.
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)

2. Crystallographic Planesz x y a b c
Example 1
a b c
Intercepts 
Reciprocals
1/ / /
Reduction
Miller Indices (110)
Example 2
a b c z x y a b c
Intercepts
½  
Reciprocals
1/½ 1/ 1/
Reduction
Miller Indices (200)

3. Crystallographic Planesz x y a b c
Example 3
a b c
Intercepts
½ ¾
Reciprocals
1/½ 1/ /¾ /3
Reduction
Miller Indices (634)

4. Single Crystals • Single Crystals -Properties vary with
E (diagonal) = 273 GPa
E (edge) = 125 GPa
• Single Crystals
-Properties vary with
direction: anisotropic.
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.)
-Example: the modulus
of elasticity (E) in BCC iron:

5. Fracture surface of failed stainless steel bolt

6. 200X – As cast aluminum from exercise equipment

7. Polycrystalline Materials
Properties may/may not vary with direction.
If grains are randomly oriented: isotropic
(Epoly iron = 210 GPa)
If grains are textured: anisotropic
200 mm

8. Crystal Defects Types of Defects • Vacancies • Interstitial atoms
There is no such thing as a perfect crystal
We use/engineer the imperfections to control properties
Types of Defects
• Vacancies
• Interstitial atoms
• Substitutional atoms
Point defects
• Dislocations
Linear defects
• Grain Boundaries
Interfacial defects

9. Point Defects Vacancy self- interstitial • Vacancies:
-vacant atomic sites in a structure.
Vacancy
distortion
of planes
• Self-Interstitials:
-"extra" atoms positioned between atomic sites.
self-
interstitial
distortion
of planes

10. Equilibrium Concentration: Point Defects
• Equilibrium concentration varies with temperature!
No. of defects
Activation energy æ - ö N Q v ç v ÷ =
exp ç ÷
No. of potential è ø N k T
defect sites.
Temperature
Boltzmann's constant
-23
(1.38 x 10
J/atom-K) -5
(8.62 x 10
eV/atom-K)
k = R/Na
Each lattice site
Notes:
Form is of an Arrehnius Equation
PV=nRT; n is the number of moles
PV=NkT; N is the number of molecules
is a potential
vacancy site

11. Measuring Activation Energyç ÷ N v =
exp - Q k T æ è ö ø
• We can get Qv from
an experiment.
• Measure this... N v T
exponential
dependence!
defect concentration
• Replot it... 1/ T N v ln - Q /k
slope

12. Estimating Vacancy Concentration
• Find the equil. # of vacancies in 1 m3 of Cu at 1000C.
• Given: 3 r
= 8.4 g / cm A
= 63.5 g/mol Cu Q
= 0.9 eV/atom N
= 6.02 x 1023
atoms/mol A v
8.62 x 10-5
eV/atom-K
0.9 eV/atom
1273K ç ÷ N v =
exp - Q k T æ è ö ø
= 2.7 x 10-4
For 1 m3
, N = N A Cu r x
1 m3
= 8.0 x 1028 sites
• Answer: N v =
(2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025 vacancies

13. Observing Equilibrium Vacancy Conc.
• Low energy electron
microscope view of
a (110) surface of NiAl.
• Increasing T causes
surface island of
atoms to grow.
• Why? The equil. vacancy
conc. increases via atom
motion from the crystal
to the surface, where
they join the island.
Reprinted with permission from Nature (K.F. McCarty, J.A. Nobel, and N.C. Bartelt, "Vacancies in
Solids and the Stability of Surface Morphology",
Nature, Vol. 412, pp (2001). Image is
5.75 mm by 5.75 mm.) Copyright (2001) Macmillan Publishers, Ltd. I
sland grows/shrinks to maintain
equil. vacancy conc. in the bulk.

14. Alloys
Book makes the point that cannot truly refine to a purity level greater than % (“4 9s”)
We just calculated that there are 8 x 1028 atoms in one m3 of Cu
Multiply % by 8 x 1028 = 8 x 1022 Impurity Atoms
In real world we don’t work with pure metals but rather we work with alloys
Alloys are “metal soup” in which impurities have been added intentionally (or unintentionally through processing) to produce specific properties

15. Solid Solutions Two outcomes if impurity (B) added to host (A): OR
• Solid solution of B in A (i.e., random dist. of point defects) OR
Substitutional solid soln.
(e.g., Cu in Ni)
Interstitial solid soln.
(e.g., C in Fe)
• Solid solution of B in A plus particles of a new
phase (usually for a larger amount of B)
Second phase particle
--different composition
--often different structure.

16. BCC Interstitial Sites
Source: C. Barrett and T.B. Massalski, Structure of Metals, 3rd Revised Edition, Pergamon, 1980.

17. FCC Interstitial Sites
Source: C. Barrett and T.B. Massalski, Structure of Metals, 3rd Revised Edition, Pergamon, 1980.

18. Conditions for Substitutional Solid Solutions
Hume – Rothery Rules
1. r (atomic radius) < 15%
2. Proximity in periodic table
i.e., similar electronegativities. If the difference in electronegativity is too great will tend to form intermetallic compounds instead of solid solutions
3. Same crystal structure for pure metals
4. Similar Valency
Metals have a greater tendency to dissolve metals of higher valency than lower valency
William Hume-Rothery was a British Metallurgist who founded the
Metallurgy Department at Oxford in the 1950s

19. Conditions for Interstitial Solid Solutions
Hume – Rothery Rules
Solute atoms must be similar in size to the interstitial locations in lattice structure
2. Proximity in periodic table
i.e., similar electronegativities

20. Application of Hume–Rothery Rules
1. Would you predict more Al or Ag to dissolve in Zn?
2. More Zn or Al
in Cu?
3. Will C form substitutional or interstitial solid solution with iron?
Element Atomic Crystal Electro- Valence Radius Structure nega- (nm) tivity
Cu FCC C H O Ag FCC Al FCC Co HCP Cr BCC Fe BCC Ni FCC Pd FCC Zn HCP
Table on p. 106, Callister 7e.