Chapter 4

Crystal Defects and

Noncrystalline Structure
Imperfection
ME 2105 Dr. Lindeke

In our pervious Lecture when

discussing Crystals we
ASSUMED PERFECT ORDER

In real materials we find:

Crystalline Defects or lattice irregularity
Most real materials have one or more errors in perfection
with dimensions on the order of an atomic diameter to many
lattice sites
Defects can be classification:
1. according to geometry
(point, line or plane)
2. dimensions of the defect

Forming a liquid solution of water and

alcohol. Mixing occurs on the
molecular scale.
We can define this
mixture/solution on
a weight or atomic
basis
A similar discussion
can apply to
mixtures of metals
called alloys

Point Defects in the solid

state are more predictable

Vacancies:

-vacant atomic sites in a structure.

Vacancy
distortion
of planes

Self-Interstitials:

-"extra" atoms positioned between atomic sites.

distortion
of planes

selfinterstitial

POINT DEFECTS
The simplest of the point defect is a vacancy, or vacant lattice site.
All crystalline solids contain vacancies.
Principles of thermodynamics is used explain the necessity of the
existence of vacancies in crystalline solids.
The presence of vacancies increases the entropy (randomness) of
the crystal.
The equilibrium number of vacancies for a given quantity of
material depends on and increases with temperature as follows: (an
Arrhenius model)
Total no. of atomic sites

Energy required to form vacancy

Equilibrium no. of vacancies

Nv= N exp(-Qv/kT)

T = absolute temperature in Kelvin

k = gas or Boltzmanns constant

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.

Solid solution of nickel in copper shown

along a (100) plane. This is a
substitutional solid solution with nickel
atoms substituting for copper atoms
on fcc atom sites.

Imperfections in Solids
Conditions for substitutional solid solution
(S.S.)
Hume Rothery rules
1. r (atomic radius) < 15%
2. Proximity in periodic table
i.e., similar electronegativities

3. Same crystal structure for pure metals

4. Valency equality
All else being equal, a metal will have a greater
tendency to dissolve a metal of higher valency than
one of lower valency (it provides more electrons to the
cloud)

Imperfections in Solids
Application of HumeRothery rules Solid
Solutions
Element Atomic Crystal Electro- Valence
1. Would you predict
more Al or Ag
More Al
size is closer
and
tobecause
dissolve
in Zn?
val. Is higher but not too much
because of structural differences
FCC in HCP

2. More Zn or Al
Surelyin
Zn Cu?
since size is closer thus
causing lower distortion (4% vs
12%)

Radius Structure nega(nm)

tivity
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

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

Table on p. 106, Callister 7e.

Imperfections in Solids
Specification of composition
weight percent

m1
C1
x 100
m1 m2

m1 = mass of component 1

atom percent

n m1
C
x 100
n m1 n m 2
'
1

nm1 = number of moles of component 1

Wt. % and At. % -- An example

Typically we work with a basis weight (100g or 1 kg) or moles
given: alloy by weight -- 60% Cu, 40% Ni
600 g
nCu
9.44m
63.55 g / m
400 g
nNi
6.82m
58.69 g / m
9.44
'
CCu

.581 or 58.1%
9.44 6.82
6.82
'
CNi
.419 or 41.9%
9.44 6.82

Converting Between: (Wt% and At

%)
C1 A2
C
100
C1 A2 C2 A1
'
1

C2 A1
C
100
C1 A2 C2 A1
'
2

C A1
C1 '
100
'
C1 A1 C2 A2

Converts from
wt% to At% (Ai
is atomic
weight)

'
1

C2' A2
C2 '
100
'
C1 A1 C2 A2

Converts from
at% to wt% (Ai
is atomic
weight)

Interstitial solid solution applies to carbon in iron. The carbon atom is small enough to fit
with some strain in the interstice (or opening)
among adjacent Fe atoms in this important
steel structure. [This unit-cell structure can be
compared with that shown in Figure 3.4b.]

But the interstitial solubility is quite low since the size mismatch of the site to
the radius of a carbon atom is only about 1/4

Random, substitution solid solution

can occur in Ionic Crystalline
materials as well. Here of NiO in
MgO. The O2 arrangement is
unaffected. The substitution occurs
among Ni2+ and Mg2+ ions.

A substitution solid solution of Al2O3 in MgO

is not as simple as the case of NiO in MgO.
The requirement of charge neutrality in the
overall compound permits only two Al3+ ions
to fill every threeMg2+ vacant sites, leaving
oneMg2+ vacancy.

Iron oxide, Fe1xO with x 0.05, is an

example of a nonstoichiometric compound.
Similar to the case of Figure 4.6, both Fe2+
and Fe3+ ions occupy the cation sites, with
one Fe2+ vacancy occurring for every two
Fe3+ ions present.

Defects in Ceramic
Frenkel Defect Structures
--a cation is out of place.

Shottky Defect
--a paired set of cation and anion vacancies.
Shottky
Defect:

from W.G. Moffatt, G.W. Pearsall,

and J. Wulff, The Structure and
Properties of Materials, Vol. 1,
Structure, John Wiley and Sons,
Inc., p. 78.

Frenkel
Defect

Equilibrium concentration of defects

~e

QD / kT

Line Defects
Are called Dislocations:

And:
slip between crystal planes result when dislocations
move,
this motion produces permanent (plastic)
deformation.
Schematic
of Zinc (HCP):
before deformation

after tensile elongation

Adapted from Fig. 7.8, Callister 7e.

slip steps which

are the physical
evidence of large
numbers of
dislocations
slipping along
the close packed
plane {0001}

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 (the bergers vector) is (perpendicular) to
dislocation line

Screw dislocation:
spiral planar ramp resulting from shear deformation
b is (parallel) to dislocation line
Burgers vector, b: is a measure of lattice
distortion and is measured as a distance along the
close packed directions in the lattice

Edge Dislocation
Edge Dislocation

Fig. 4.3, Callister 7e.

Definition of the Burgers vector, b,

relative to an edge dislocation. (a) In
the perfect crystal, an m n atomic
step loop closes at the starting point.
(b) In the region of a dislocation, the
same loop does not close, and the
closure vector (b) represents the
magnitude of the structural defect.
For the edge dislocation, the Burgers
vector is perpendicular to the
dislocation line.

Screw dislocation. The spiral stacking of crystal

planes leads to the Burgers vector being parallel to
the dislocation line.

Mixed dislocation. This dislocation has both edge

and screw character with a single Burgers vector
consistent with the pure edge and pure screw
regions.

Burgers vector for the

aluminum oxide structure.
The large repeat distance
in this relatively complex
structure causes the
Burgers vector to be
broken up into two (for
O2) or four (for Al3+)
partial dislocations, each
representing a smaller slip

Imperfections in Solids
Dislocations are visible in (T) electron micrographs

Adapted from Fig. 4.6, Callister 7e.

Dislocations & Crystal Structures

Structure: closepacked
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 super-close many near close

Specimens that
were tensile
tested.

Mg (HCP)
tensile direction

Al (FCC)

Planar Defects in Solids

One case is a twin boundary (plane)
Essentially a reflection of atom positions across the
twinning plane.

Adapted from Fig. 4.9, Callister 7e.

Stacking faults
For FCC metals an error in ABCABC packing sequence
Ex: ABCABABC

Simple view of the surface of a crystalline

material.

A more detailed model of the elaborate ledgelike

structure of the surface of a crystalline material.
Each cube represents a single atom. [From J. P.
Hirth and G. M. Pound, J. Chem. Phys. 26, 1216
(1957).]

Typical optical
micrograph of a grain
structure, 100. The
material is a low-carbon
steel. The grain
boundaries have been
lightly etched with a
chemical solution so that
they reflect light
differently from the
polished grains, thereby
giving a distinctive
contrast. (From Metals
Handbook, 8th ed., Vol. 7:
Atlas of Microstructures of
Industrial Alloys,

Simple grain-boundary
structure. This is termed a
tilt boundary because it is
formed when two adjacent
crystalline grains are tilted
relative to each other by a
few degrees (). The
resulting structure is
equivalent to isolated edge
dislocations separated by
the distance b/, where b is
the length of the Burgers
vector, b. (From W. T. Read,
Dislocations in Crystals,
McGraw-Hill Book Company,
New York, 1953. Reprinted
with permission of the

The ledge Growth leads to structures with Grain

Boundries The shape and average size or
diameter of the grains for some polycrystalline
specimens are large enough to observe with the
unaided eye. (Macrosocipic examination)

Specimen for the calculation of the grain-size

number, G is defined at a magnification of 100.
This material is a low-carbon steel similar to that
shown in Figure 4.18. (From Metals Handbook, 8th
ed., Vol. 7: Atlas of Microstructures of Industrial
Alloys, American Society for Metals, Metals Park,
OH, 1972.)

Optical Microscopy
Useful up to 2000X magnification (?).
Polishing removes surface features (e.g.,
scratches)
Etching changes reflectance, depending on
crystal
orientation since different Xtal planes
have different
reactivity.

crystallographic plane
Courtesy of J.E. Burke, General
Electric Co.

Micrograph of
brass (a Cu-Zn alloy)
0.75mm

Optical Microscopy
Since Grain
boundaries...
are planer imperfections,
are more susceptible
to etching,
may be revealed as
dark lines,
relate change in crystal
orientation across
boundary.

polished surface

(a)

surface groove
grain boundary
(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].)

ASTM grain
size number
G-1

N=2

number of grains/in2
at 100x
magnification

Fe-Cr alloy
(b)

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 G-1

NOTE: The ASTM grain size is related (or

relates) a grain area AT 100x MAGNIFICATION

Determining Grain Size, using a

micrograph taken at 300x
We count 14 grains
in a 1 in2 area on
the (300x) image
To report ASTM
grain size we
needed a measure
of N at 100x not
300x
We need a
conversion method!

M
G 1
NM
2
100
M is mag. of image
N M is measured grain count at M
now solve for G:
log( N M ) 2 log M log 100 G 1 log 2

G
G

log N m 2 log M 4
log 2

log 14 2 log 300 4

0.301

1
1 7.98 8

For this same material, how many

Grains would I expect /in2 at 100x? At
50x?

N 2

G 1

81

128 grains/in (at 100x)

2

Now, how many grain would I expect at 50x?

2

100
100

NM 2
128*

50
M

2
2
N M 128* 2 512 grains/in
8 1

At 100x

Two-dimensional schematics give a comparison of

(a) a crystalline oxide and (b) a non-crystalline
oxide. The non-crystalline material retains shortrange order (the triangularly coordinated building
block), but loses long-range order (crystallinity).
This illustration was also used to define glass in
Chapter 1 (Figure 1.8).

Bernal model of an amorphous metal structure. The

irregular stacking of atoms is represented as a
connected set of polyhedra. Each polyhedron is
produced by drawing lines between the centers of
adjacent atoms. Such polyhedra are irregular in
shape and the stacking is not repetitive.

A chemical impurity such

as Na+ is a glass
modifier, breaking up the
random network and
leaving nonbridging
oxygen ions. [From B. E.
Warren, J. Am. Ceram.
Soc. 24, 256 (1941).]

Schematic illustration of
medium-range ordering in a
CaOSiO2 glass. Edge-sharing
CaO6 octahedra have been
identified by neutron-diffraction
experiments. [From P. H.
Gaskell et al., Nature 350, 675
(1991).]

Summary
Point, Line, Surface and Volumetric defects exist in
solids.
The number and type of defects can be varied and
controlled
T controls vacancy conc.
amount of plastic deformation controls # of dislocations
Weight of charge materials determine concentration of
substitutional or interstitial point defects

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.
Inclusions can be intention for alloy development