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 R2 3
, or the
2
total area of the plane shown is .6R 3
And the planar density for this (0001) plane is equal to
PD0001 =
or the total area of the plane shown is

3 atoms
6R2
3

number of atoms cent ered on (0001) plane
area of (0001) plane
1
2R2
3
(b) the atomic radius for titanium is 0.145 nm. Therefore, the planar density for the (0001) plane is
PD0001 (Ti)
1
2R 2
1
2
19
2

13
.
73
nm

1
.
373
x
10
m
3 2 3(0.145nm) 2

Chapter 4 - 1
Types of Imperfections
• Vacancy atoms
• Interstitial atoms
• Substitutional atoms
Point defects
• Dislocations
Line defects
Edges, Screws, Mixed
• Grain Boundaries
Area/Planar defects
• Stacking Faults
• Anti-Phase and Twin Boundaries
Chapter 4 - 2
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 
Nv
 exp  
N
 kBT 
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!
Chapter 4 - 3
Imperfections in Solids
Conditions for substitutional solid solution (S.S.)
• W. Hume – Rothery rule
rsolute  rsolvent
x100%
rsolvent
– 1. r (atomic radius)
– 2. Proximity in periodic table
< 15%
• 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
Chapter 4 - 4
Imperfections in Solids
Application of Hume–Rothery rules – Solid Solutions
1. Would you predict
more Al or Ag
to dissolve in Zn?
Element
Atomic Crystal
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%)
Electronegativity
Valence
FCC
1.9
+2
FCC
FCC
HCP
BCC
BCC
FCC
FCC
HCP
1.9
1.5
1.8
1.6
1.8
1.8
2.2
1.6
+1
+3
+2
+3
+2
+2
+2
+2
Radius Structure
(nm)
Cu
C
H
O
Ag
Al
Co
Cr
Fe
Ni
Pd
Zn
0.1278
0.071
0.046
0.060
0.1445
0.1431
0.1253
0.1249
0.1241
0.1246
0.1376
0.1332
Table on p. 106, Callister 7e.
Chapter 4 - 5
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
Chapter 4 - 6
Imperfections in Solids
Edge Dislocation
Chapter 4 - 7
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= QP = b
Chapter 4 - 8
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)
Chapter 4 - 9
Edge Dislocations Exiting Crystal Form Steps
Burger’s
Vector = b
Shear stress
The caterpillar or rug-moving analogy
Chapter 4 - 10
Screw dislocation. The spiral stacking of crystal planes
leads to the Burgers vector being parallel to the dislocation
line.
Chapter 4 - 11
Imperfections in Solids
Screw Dislocation
Screw Dislocation
b
Dislocation
line
Burgers vector b
(b)
(a)
Adapted from Fig. 4.4, Callister 7e.
Chapter 4 - 12
Edge, Screw, and Mixed Dislocations
Mixed
Edge
Adapted from Fig. 4.5, Callister 7e.
Screw
Chapter 4 - 13
Motion – Edge dislocation
Applied shear
Chapter 4 - 14
Motion Screw Dislocation
Apply shear
Chapter 4 - 15
Imperfections in Solids
Dislocations are visible in electron micrographs
Adapted from Fig. 4.6, Callister 7e.
Chapter 4 - 16
Dislocations & Crystal Structures
• Structure: close-packed
planes & directions
are preferred.
view onto two
close-packed
planes.
close-packed plane (bottom)
close-packed directions
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)
Chapter 4 - 17
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
Chapter 4 - 18
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 twodimensional. They are of two types external and internal
surface imperfections.
Chapter 4 - 19
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.
Chapter 4 - 20
AREA DEFECTS: GRAIN BOUNDARIES
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
heat
flow
Adapted from Fig. 4.7, Callister 6e.
Adapted from Fig. 4.10, Callister 6e. (Fig.
4.10 is from Metals Handbook, Vol. 9, 9th edition,
Metallography and Microstructures, Am. Society for
Metals, Metals Park, OH, 1985.)
Chapter 4 - 15
21
Planar Defects in Solids
• One case is a twin boundary (plane)
– Essentially a reflection of atom positions across the twin
plane.
Adapted from Fig. 4.9, Callister 7e.
• Stacking faults
– For FCC metals an error in ABCABC packing sequence
– Ex: ABCABABC
Chapter 4 - 22
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-BA-B-C. The region in which the stacking fault occurs (A-BA-B) forms a thin region and it becomes HCP.
 This thin region is a surface imperfection and is called a
stacking fault.
Chapter 4 - 23
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.
Chapter 4 - 24
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.
Chapter 4 - 25
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
Chapter 4 - 26
Optical Microscopy
• Polarized light
– metallographic scopes often use polarized
light to increase contrast
– Also used for transparent samples such as
polymers
Chapter 4 - 27
Optical Microscopy
• Useful up to 2000X magnification.
• Polishing removes surface features (e.g., scratches)
• Etching changes reflectance, depending on crystal orientation.
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)
0.75mm
Chapter 4 - 28
Optical Microscopy
Grain boundaries...
• are imperfections,
• are more susceptible
to etching,
• may be revealed as
dark lines,
• change in crystal
orientation across
boundary.
polished surface
surface groove
grain boundary
(a)
Fe-Cr alloy
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].)
(b)
Chapter 4 - 29
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.
Chapter 4 - 30
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 (@100x) each with a number
Quick and easy – used for steel
Grain size no.
No. of grains/square inch
N = 2 n-1
Chapter 4 - 31
High resolutions and magnification (50,000x SEM); (TEM 1,000,000x)
Chapter 4 - 32
Chapter 4 - 33
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”.
Chapter 4 - 34
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.)
Chapter 4 - 35
Chapter 4 - 36