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MME 2001 MATERIALS SCIENCE 1 27.10.2015
outline Planar density
Interplanar spacing
Structure-property relations
X-ray diffraction
Imperfections in solids
point defects
vacancies
impurities
solid solutions
line defects
dislocations
Quiz
at 14:50
planar densities The parameter corresponding to the linear density
for crystallographic planes is planar density, and
planes having the same planar density values are
also equivalent.
(001) / {001}
planar density Planar Density of Atoms
PD =
a
[110]
area of the plane
number of atoms inside the plane
(110) Although six atoms have
centers that lie on this plane,
only one-quarter of each of
atoms A, C, D, and F, and one-
half of atoms B and E, for a
total equivalence of just 2
atoms, are on that plane.
A
B
C
D
1/4
1/2
x
planar density
a
[110] (110) area of (110) plane is
equal to the product of
its length and width.
length = 4R
width (vertical
dimension)= 2R2
Planar density of (110) planes of FCC crystal
area of this plane =
(4R)( 2R2) = 8R2 2
area of (110) plane = 8R2 2
PD110 = 2 atoms / 8R2 2
= 1 / 4R22
planar density Planar Density of Atoms PD =
a
[110] Area of the plane
Number of atoms
(110)
Corner atoms (A, C, D and F) only quarter
of each is in the plane: 4 x ¼ = 1
Face atoms (B and E): only half of each is
in the plane: 2x1/2: 1
Total number of atoms inside the plane = 2
planar density
a
[110] (110)
Calculate the planar atomic density for the (110)
plane in FCC copper (a = 0.3615 nm) in
atoms/cm2.
A, C, D, F: corner atoms;
shared by 4 unit cells
B and E: edge atoms; shared
by 2 unit cells
planar density
a
[110] (110)
Number of atoms on the (110) plane in FCC:
4 atoms at the corners, each shared by 4 unit cells,
2 atoms at the edges, each shared by 2 unit cells
4x1/4 + 2x1/2 = 2 atoms
area of (110) plane = a x a2
a = 0.3615 nm
PD110 = 2 atoms /a22
= 2/a2
= 1.082 x 1015 atoms/cm2
Planar Density of (100) Iron Solution: At T < 912C iron has the BCC structure
(100)
Radius of iron R = 0.1241 nm
R 3
4 a =
(100) plane
= PD= a 2
1 atoms
= nm2
atoms 12.1
m2
atoms = 1.2 x 1019
1 2
R 3
4 area
Planar Density of (111) Iron (111) plane 1 atom in plane/unit surface cell
3 3 3 2
2
R 3
16 R
3
4 2 a 3 ah 2 area =
= = =
atoms in plane
atoms above plane
atoms below plane
a h 2
3 =
a 2
1 = =
nm2
atoms 7.0
m2
atoms 0.70 x 1019
3 2 R 3
16 Planar Density =
atoms
area Radius of iron R = 0.1241 nm
planar density (111) Plane in a FCC crystal:
(3 corner atoms x 1/6) + (3 side atoms x ½) = 2 atoms
Planar density of (111) plane in the FCC:
PD(111) = 2
2.a x 2.a
4
3.a2 =
3 2
1 2
PD(111) > PD (110)
(111) planes are more
densely packed
PD110 = 2/a2
Linear and planar densities
Linear and planar densities are important
considerations relative to the process of
slip—that is, the mechanism by which metals
plastically deform.
Slip occurs on the most densely packed
crystallographic planes and, in those planes,
along directions having the greatest atomic
packing.
Spacing between crystal planes
Relation between planes and directions In the cubic system planes and directions having same
indices are perpendicular to each other,i.e. İf [uvw] direction
is perpendicular to (hkl) plane, then h=u, k=v and l=w
Ex: {100} planes and <100> directions are perpendicular to
each other.
If [uvw] direction is parallel to (hkl), that is if [uvw] lies in
the plane (hkl) then hu + kv + lw = 0.
For example, _
[110] lies in the plane (111)
Since 1x(-1) + 1x1 + 1x0 = 0
Structure property correlation
Al: ductile Fe: not ductile Mg: not ductile
FCC BCC HCP
Plastic deformation via SLIP
Sliding of crystal planes over one another!
SLIP occurs on most densely packed planes in the
most closely packed planes!
slip plane + slip direction: SLIP SYSTEM
Structure property correlation In FCC,
{111} planes are closely packed and there are
4 unique {111} planes.
Each of these planes contains
3 closely packed <111> directions.
In HCP,
The basal plane (0001) is the close packed.
It contains 3 close packed <1120> directions.
4x3=12
slip
systems
1x3=3
slip
systems
Higher number of slip systems allows greater plastic
deformation before fracture imparting ductility to FCC
metals!
Structure property correlation
4x3=12 slip systems
1x3=3 slip systems
Structure property correlation Close packed planes are planes with greatest
interplanar spacing. This allows slip to takeplace
easily on these planes.
BCC structure has 48 possible slip systems.
However, there is no close packed plane. Hence
plastic deformation before fracture is not
substantial.
Slip might occur in {110}, {112} and {123} planes in
the <111> directions.
Learning check
Sketch within a cubic unit cell the plane (312):
—
Which one
shows the
plane (221)
correctly
drawn?
learning check
Which one
shows the
(212) plane
correctly
drawn?
learning check
learning check
Which one
shows the
(111) plane
correctly
Drawn?
_
learning check
122
131
213
123
331
121
learning check
Sketch within a cubic unit cell the following
planes:
Determine the Miller indices for the planes in the
following unit cell:
A: (403) B: (112)
learning check Determine the Miller indices for the planes in the
following unit cell:
A: (322)
B: (202)
B: (221) A: (324)
X-ray diffraction waves 1 and 2 have the same wavelength and
remain in phase after a scattering event.
The amplitudes of the scattered waves add together
in the resultant wave.
X-ray diffraction Waves 3 and 4 have the same wavelength and
become out of phase after a scattering event.
The amplitudes of the two scattered waves
cancel one another.
+
_
X-ray diffraction n = SQ + QT
n = dhkl sin + dhkl sin
n = 2dhkl sin
X-Rays to Determine Crystal Structure
• Incoming X-rays diffract from crystal planes.
reflections must be in phase for a detectable signal
spacing between planes
d
extra distance travelled by wave “2”
X-ray diffraction The magnitude of the distance between two
adjacent and parallel planes of atoms (i.e., the
interplanar spacing d) is a function of the Miller
indices (h, k, and l) as well as the lattice
parameter(s).
For cubic symmetry,
in which a is the lattice parameter (unit cell edge
length).
X-ray diffraction Schematic diagram of an
x-ray diffractometer;
T x-ray source,
S specimen,
C detector, and
O the axis around which
the specimen and
detector rotate.
interplanar spacing computations
diffraction angle computations For BCC iron, compute the diffraction angle for the (220) set
of planes. The lattice parameter for Fe is 0.2866 nm. Also,
assume that monochromatic radiation having wavelength of
0.1790 nm is used and the order of reflection is 1.
X-Ray Diffraction
Diffraction gratings must have spacings comparable to the wavelength of diffracted radiation.
Can’t resolve spacings
Spacing is the distance between parallel planes of atoms.
X-Ray Diffraction Pattern
(110)
(200)
(211)
z
x
y a b
c
Diffraction pattern for polycrystalline a-iron (BCC)
Inte
nsi
ty (
rela
tive)
z
x
y a b
c
z
x
y a b
c
Diffraction angle 2
imperfections in solids
imperfections in solids
imperfections in solids
homogenization/annealing
rolling/extrusion Atomic
composition
bonding
X’tal structure
Addition and manipulaton
of defects
Microstructure:
material
properties Therm
o-
mechanic
al
pro
cess
ing
imperfections in solids
● perfect order is assumed to exist throughout
crystalline materials on an atomic scale.
● However, such an idealized solid does not
exist; all contain large numbers of various
defects.
● Crystalline defect refers to a lattice
irregularity having one or more of its
dimensions on the order of an atomic
diameter.
imperfections in solids
● Many of the properties of materials are
profoundly sensitive to deviations from
crystalline perfection!
● the impact is not always adverse!
● often specific characteristics are
deliberately fashioned by the introduction
of controlled quantity of defects.
• vacancy atoms
• interstitial atoms
• substitutional atoms
Point defects
0-dimensional
Types of Imperfections
• dislocations Line defects
1-dimensional
• grain boundaries Area defects
2-dimensional
• cavities/porosity volume defects
3-dimensional
vacant atomic site/missing atom
Point Defects / vacancies
Vacancy
distortion of planes
All crystalline solids contain vacancies.
vacancies increase the entropy (the randomness) of the
crystal: thermodynamically favorable!
the number of vacancies increases exponentially with T!
For most metals, Nv/N at Tm is on the order of 104; one
lattice site out of 10,000 will be empty.
Point Defects / self-interstitial
self- interstitial
distortion of planes
● own atom crowded into an interstitial site!
● introduces relatively large distortions in the
surrounding lattice as the atom is much larger than
the interstitial position.
● is not highly probable!
● exists in very small concentrations
● significantly lower than for vacancies.
Most common defects in crystalline solids are point
defects.
At high temperatures,
atoms frequently and
randomly change their
positions leaving behind
empty lattice sites.
In general, diffusion
(mass transport by atomic
motion) can only occur
because of vacancies.
Point Defects
distance
vacancy atom
energy
Em
Point Defects in ionic crystals Charge neutrality must be
maintained!
Schottky imperfection
Formation of equivalent
(not necessarily equal)
numbers of cationic and
anionic vacancies
Frenkel imperfection
Formation of an ion
vacancy and an ion
interstitial
Schottky defect
Frenkel defect
Boltzmann's constant
(1.38 x 10 -23 J/atom-K)
(8.62 x 10 -5 eV/atom-K)
N v
N = exp
Q v
k T
No. of defects
No. of potential
defect sites
Activation energy
Temperature
Each lattice site is a potential vacancy site
• Equilibrium concentration varies with temperature!
Equilibrium Concentration: Point Defects
• We can get Qv from
an experiment. N v
N = exp
Q v
k T
Measuring Activation Energy
• Measure this...
N v
N
T
exponential dependence!
defect concentration
• Replot it...
1/ T
N
N v ln
- Q v /k
slope
N v
N = Q v
k T
ln =
Q v
k T
1
Equilibrium # of vacancies in 1 m3 of Cu at 1000C?
A Cu = 63.5 g/mol = 8.4 g / cm 3
Q v = 0.9 eV/atom N A = 6.02 x 1023 atoms/mol
Estimating Vacancy Concentration
For 1 m3 , N = N A
A Cu x x 106 cm3 = 8.0 x 1028 sites/
8.62 x 10-5 eV/atom-K
0.9 eV/atom
1273K
N v
N = exp
Q v
k T
= 2.7 x 10-4
N v = (2.7 x 10-4)(8.0 x 1028) sites = 2.2 x 1025 vacancies
1 m3 m3
Equilibrium # of vacancies in 1 m3 of Fe at 850C?
A Fe = 55.85 g/mol = 7.65 g / cm 3
Q v = 1.08 eV/atom N A = 6.02 x 1023 atoms/mol
Estimating Vacancy Concentration
8.62 x 10-5 eV/atom-K
1.08 eV/atom
1123K
N v N = exp Q v
k T
N v = 1.18 x 1024 vacancies
N A
A Fe = exp
Q v
k T
N = (6.022x1023 atoms/mol) (7.65g/cm3)
55.85 g/mol
1.08 eV/atom
(8.62x10-5 eV/atom-K)(1123K) exp -
v
fraction of atom sites that are vacant for lead at its
melting temperature of 327°C?
Q v = 0.55 eV/atom N A = 6.02 x 1023 atoms/mol
Estimating Vacancy Concentration
8.62 x 10-5 eV/atom-K
0.55 eV/atom
600K
N v
N = exp
Q v
k T
= 2.41 x 10-5
see you next week!