1. A (0001) plane for an HCP unit cell is show below.
Each of the 6 perimeter atoms in this plane is shared with three other unit cells, whereas the center atom is shared with no other unit cells; this gives rise to three equivalent atoms belonging to this plane.
In terms of the atomic radius R, the area of each of the 6 equilateral triangles that have been drawn is , or the total area of the plane shown is .
. And the planar density for this (0001) plane is equal to
or the total area of the plane shown is
(b) the atomic radius for titanium is nm. Therefore, the planar density for the (0001) plane is

2. CHAPTER 4: IMPERFECTIONS IN SOLIDS
ISSUES TO ADDRESS...
• What are the solidification mechanisms?
• What types of defects arise in solids?
• Can the number and type of defects be varied
and controlled?
• How do defects affect material properties?
• Are defects undesirable?

3. • How do defects affect material properties?
sterling silver (7.5% copper) much harder and
stronger than silver
silicon transistors rely on controlled “doping”
• Are defects undesirable? NO 1

5. Imperfections in Solids
Solidification- result of casting of molten material
2 steps
Nuclei form
Nuclei grow to form crystals – grain structure
Start with a molten material – all liquid
nuclei
crystals growing
grain structure
liquid
Adapted from Fig.4.14 (b), Callister 7e.
Crystals grow until they meet each other

6. Polycrystalline Materials
Grain Boundaries
regions between crystals
transition from lattice of one region to that of the other
slightly disordered
low density in grain boundaries
high mobility
high diffusivity
high chemical reactivity
Adapted from Fig. 4.7, Callister 7e.

7. Solidification
Grains can be - equiaxed (roughly same size in all directions)
- columnar (elongated grains)
~ 8 cm
heat
flow
Shell of equiaxed grains due to rapid cooling (greater T) near wall
Columnar in area with less undercooling
Adapted from Fig. 4.12, Callister 7e.
Grain Refiner - added to make smaller, more uniform, equiaxed grains.

8. Imperfections in Solids
There is no such thing as a perfect crystal.
What are these imperfections?
Why are they important?
Many of the important properties of materials are due to the presence of imperfections.

9. Types of Imperfections
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
• Dislocations
Line defects
• Grain Boundaries
Area defects

10. 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

12. 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)
Each lattice site
is a potential
vacancy site

13. 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

14. 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

15. 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. vancancy conc. in the bulk.

16. Completely pure metal (or other substance) – Impossible!!
Pay – really pay – for % - NIST standards
still 1022 to 1023 impurities per m3
Alloy – deliberately add “impurity” to engineer properties. Copper (7.5%) in silver

17. Point Defects in Alloys
Two outcomes if impurity (B) added to host (A):
• 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.

18. Terminology - Alloys
Solvent – component in highest concentration – also called – host
Solute – component present in minor concentration

19. Remember metals- often crystallize in
close packed structures – high packing density.
More interstitial or substitutional ??
Usually < 10%

20. Imperfections in Solids
Conditions for substitutional solid solution (S.S.)
W. Hume – Rothery rule
1. r (atomic radius) < 15%
2. Proximity in periodic table
i.e., similar electronegativities
3. Same crystal structure for pure metals
4. Valency
All else being equal, a metal will have a greater tendency to dissolve a metal of higher valency than one of lower valency

21. Carbon in Iron Element C Fe Radii 0.017 nm 0.124 Unit cell Hex BCC
Electronegativity
2.5
1.6
Valence +4
+2 (+3)
Solid solution ?
Interstitial or substitional?
C - 2s2 2p2
Fe – 3d6 4s2

22. Copper and Nickel Element Cu Ni Radii 0.128 nm 0.125 Unit cell FCC
Electronegativity
1.9
1.8
Oxidation (Valence)
+1 (+2) +2
Expect to form solid solution?
Ni – 3d8 4s2 (28)
Cu - 3d10 4s1 (29)

23. Imperfections in Solids
Application of Hume–Rothery rules – Solid Solutions
1. Would you predict more Al or Ag to dissolve in Zn?
2. More Zn or Al
in Cu?
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.

24. IMPURITIES • Impurities must also satisfy charge balance • Ex: NaCl
• Substitutional cation impurity
• Substitutional anion impurity 11

25. Imperfections in Solids
Specification of composition
weight percent
m1 = mass of component 1
nm1 = number of moles of component 1
atom percent