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. Types of Imperfections
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
Line defects
Area/Planar defects
• Dislocations
Edges, Screws, Mixed
• Grain Boundaries
Stacking Faults
Anti-Phase and Twin Boundaries

3. Point Defects Self-interstitial: atom crowded in ‘holes’
Vacancy: a vacant lattice site
It is not possible to create a crystal free of vacancies.
About 1 out of 10,000 sites are vacant near melting.
Self-interstitials are much less likely in metals, e.g.,
as it is hard to get big atom into small hole - there is
large distortions in lattice required that costs energy.
Thermodynamics (temperature and counting) provides an expression for
Vacancy Concentration:
Qv=vacancy formation energy
kB= 1.38 x 10–23 J/atom-K = 8.62 x 10–5 eV/atom-K
Defects ALWAYS cost energy!

4. 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

5. Imperfections in Solids
Application of Hume–Rothery rules – Solid Solutions
Would you predict more Al or Ag to dissolve in Zn?
More Al because size is closer
7.43 % vs 8.48 % – FCC in HCP
and val. Is higher – but not too
Much!
2. More Zn or Al
in Cu?
Surely Zn since size is closer
thus causing lower distortion
(4% vs 12%)
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.

6. Imperfections in Solids
Linear Defects (Dislocations)
Are one-dimensional defects around which atoms are misaligned
Edge dislocation:
extra half-plane of atoms inserted in a crystal structure
b  to dislocation line
Screw dislocation:
spiral planar ramp resulting from shear deformation
b  to dislocation line
Burger’s vector, b: measure of lattice distortion and is measured as a distance along the close packed directions in the lattice

7. Imperfections in Solids
Edge Dislocation

8. Burgers Vector.
The magnitude and the direction of the displacement are defined by a vector, called the Burgers Vector.
(a), starting from the point P, we go up by 4 steps, then move towards right by 5 steps, move down by 4 steps and finally move towards left by 5 steps to reach the starting point P. Now the Burgers circuit gets closed.
When the same operation is performed on the defect crystal (b) we end up at Q instead of the starting point!
So, we have to move an extra step to return to P, in order to close the Burgers circuit.
BV= = b

9. Motion of Edge Dislocation
• Dislocation motion requires the successive bumping
of a half plane of atoms (from left to right here).
• Bonds across the slipping planes are broken and
remade in succession.
Atomic view of edge
dislocation motion from
left to right as a crystal
is sheared.
(Courtesy P.M. Anderson)

10. Edge Dislocations Exiting Crystal Form Steps
Burger’s
Vector = b
Shear stress
The caterpillar or rug-moving analogy

11. Screw dislocation. The spiral stacking of crystal planes leads to the Burgers vector being parallel to the dislocation line.

12. Imperfections in Solids
Screw Dislocation
Screw Dislocation b
Dislocation
line
Burgers vector b
(b)
(a)
Adapted from Fig. 4.4, Callister 7e.

13. Edge, Screw, and Mixed Dislocations
Adapted from Fig. 4.5, Callister 7e.

14. Motion – Edge dislocation
Applied shear

15. Motion Screw Dislocation
Apply shear

16. Imperfections in Solids
Dislocations are visible in electron micrographs
Adapted from Fig. 4.6, Callister 7e.

17. Dislocations & Crystal Structures
• Structure: close-packed
planes & directions
are preferred.
view onto two
close-packed
planes.
close-packed directions
close-packed plane (bottom)
close-packed plane (top)
• Comparison among crystal structures:
FCC: many close-packed planes/directions;
HCP: only one plane, 3 directions;
BCC: none
• Specimens that
were tensile
tested.
Mg (HCP)
tensile direction
Al (FCC)

18. Planar Defects: Surfaces
All defects cost energy (energy is higher than perfect crystal)
Surfaces, grain, interphase and twin boundaries, stacking faults
Planar Defect Energy is Energy per Unit Area (J/m2 or erg/cm2)
Surfaces: missing or fewer number of optimal or preferred bonds.
surface

19. SURFACE IMPERFECTIONS
Surface imperfections arise from a change in the stacking of atomic planes on or across a boundary.
The change may be one of the orientations or of the stacking sequence of atomic planes.
In geometric concept, surface imperfections are two- dimensional. They are of two types external and internal surface imperfections.

20. EXTERNAL SURFACE IMPERFECTIONS
They are the imperfections represented by a boundary. At the boundary the atomic bonds are terminated.
The atoms on the surface cannot be compared with the atoms within the crystal. The reason is that the surface atoms have neighbors on one side only. Where as the atoms inside the crystal have neighbors on either sides. This is shown in figure. Since these surface atoms are not surrounded by others, they possess higher energy than that of internal atoms.
For most metals, the energy of the surface atoms is of the order of 1J/m2.

21. AREA DEFECTS: GRAIN BOUNDARIES
• are boundaries between crystals.
• are produced by the solidification process, for example.
• have a change in crystal orientation across them.
• impede dislocation motion.
Metal Ingot
~ 8cm
grain
boundaries
Adapted from Fig. 4.10, Callister 6e. (Fig is from Metals Handbook, Vol. 9, 9th edition, Metallography and Microstructures, Am. Society for Metals, Metals Park, OH, 1985.) 15
Adapted from Fig. 4.7, Callister 6e.

22. Planar Defects in Solids
One case is a twin boundary (plane)
Essentially a reflection of atom positions across the twin plane.
Stacking faults
For FCC metals an error in ABCABC packing sequence
Ex: ABCABABC
Adapted from Fig. 4.9, Callister 7e.

23. STACKING FAULTS
Whenever the stacking of atomic planes is not in a proper sequence throughout the crystal, the fault caused is known as stacking fault.
For example, the stacking sequence in an ideal FCC crystal may be described as A-B-C-A-B-C- A-B-C-……. But the stacking fault may change the sequence to A-B-C-A-B-A-B-A-B-C. The region in which the stacking fault occurs (A-B-A-B) forms a thin region and it becomes HCP.
This thin region is a surface imperfection and is called a stacking fault.

24. Microscopic Examination
Crystallites (grains) and grain boundaries. Vary considerably in size. Can be quite large
ex: Large single crystal of quartz or diamond or Si
ex: Aluminum light post or garbage can - see the individual grains
Crystallites (grains) can be quite small (mm or less) – necessary to observe with a microscope.

25. MACROSCOPIC EXAMINATION: The shape and average size or
diameter of the grains for a polycrystalline specimen are large
enough to observe with the unaided eye.

26. MICROSCOPIC EXAMINATION
Applications
To Examine the structural elements and defects that influence the properties of materials.
Ensure that the associations between the properties and structure (and defects) are properly understood.
Predict the properties of materials once these relationships have been established.
Structural elements exist in ‘macroscopic’ and ‘microscopic’ dimensions

27. Optical Microscopy Polarized light
metallographic scopes often use polarized light to increase contrast
Also used for transparent samples such as polymers

28. Optical Microscopy • Useful up to 2000X magnification.
• Polishing removes surface features (e.g., scratches)
• Etching changes reflectance, depending on crystal orientation.
0.75mm
crystallographic planes
Adapted from Fig. 4.13(b) and (c), Callister 7e. (Fig. 4.13(c) is courtesy
of J.E. Burke, General Electric Co.
Micrograph of
brass (a Cu-Zn alloy)

29. Optical Microscopy Grain boundaries... Fe-Cr alloy
grain boundary
surface groove
polished surface
(a)
• are imperfections,
• are more susceptible
to etching,
• may be revealed as
dark lines,
• change in crystal
orientation across
boundary.
Adapted from Fig. 4.14(a) and (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].)

30. GRAIN SIZE DETERMINATION
The grain size is often determined when the properties of a polycrystalline material are under consideration. The grain size has a significant impact of strength and response to further processing
Linear Intercept method
Straight lines are drawn through several photomicrographs that show the grain structure.
The grains intersected by each line segment are counted
The line length is then divided by an average number of grains intersected.
The average grain diameter is found by dividing this result by the linear magnification of the photomicrographs.

31. N = 2 n-1 ASTM (American Society for testing and Materials)
ASTM has prepared several standard comparison charts, all having different average grain sizes. To each is assigned a number from 1 to 10, which is termed the grain size number; the larger this number, the smaller the grains.
VISUAL CHARTS each with a number
Quick and easy – used for steel
N = 2 n-1
No. of grains/square inch
Grain size no.

32. High resolutions and magnification (50,000x SEM); (TEM 1,000,000x)

34. Scanning Tunneling Microscopy (STM)
• Atoms can be arranged and imaged!
Photos produced from the work of C.P. Lutz, Zeppenfeld, and D.M. Eigler. Reprinted with permission from International Business Machines Corporation, copyright 1995.
Carbon monoxide molecules arranged on a platinum (111) surface.
Iron atoms arranged on a copper (111) surface. These Kanji characters represent the word “atom”.

35. Summary • Point, Line, and Area defects exist in solids.
• The number and type of defects can be varied
and controlled (e.g., T controls vacancy conc.)
• Defects affect material properties (e.g., grain
boundaries control crystal slip).
• Defects may be desirable or undesirable
(e.g., dislocations may be good or bad, depending
on whether plastic deformation is desirable or not.)