Dobson Chapter5 Imperfections 2011

Dobson Chapter5 Imperfections 2011 - X‐ay

Diffrac+on

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Diffrac+on X‐ray
Diffrac+on X-ray diffraction (XRD) is commonly used to characterize crystal structure. The technique is based on Bragg’s law, which relates angles corresponding to constructive interference to the inter-plane spacing (d). • Diffrac+on
gra+ngs
must
have
spacings
comparable
to
the wavelength
of
diffracted
radia+on. • Can’t
resolve
spacings
<
λ • Spacing
is
the
distance
between
parallel
planes
of
atoms. 1 X‐ray
Diffrac+on:
Interference
/
Phase X‐ray
Diffrac+on ” θ ” “2 “1 ” “2 θ i go t ou or “1 g in m co s in ray X‐ extra
 distance
 travelled
 by
wave
“2” s ay ‐r X g

 n ct te de ” •

Incoming
X‐rays
diffract
from
crystal
planes. λ d Measurement
of
cri+cal angle,
θc,
allows computa+on
of
planar spacing,
d. Bragg’s Law: nλ = 2d sin θ reflec+ons
must
 be
in
phase
for
 a
detectable
signal Adapted
from
Fig.
3.20, Callister
&
Rethwisch
8e. spacing
 between
 planes X‐ray
 intensity
 (from
 detector) d
 = nλ 2
sin θc 
 θ θc X‐ray
Diffrac+on The
magnitude
of
the
distance
(d)
between
two
parallel
planes
in
a
cubic
crystal
is
a func+on
of
the
Miller
indices
(h,k,l)
as
well
as
the
laTce
parameters. Interplanar separation for a plane having indices h, k, and l dhkl = a h2 + k2 + l2 X‐ray
Diffrac+on
PaUern z z Intensity (relative) c a x z c b y (110) a x c b y a x (211) b (200) Diffraction angle 2θ Diffraction pattern for polycrystalline α-iron (BCC) Adapted
from
Fig.
3.22,
Callister
8e. hUp://www.youtube.com/watch?v=lwV5WCBh9a0 y X‐ray
Diffrac+on
PaUern Summary •

Atoms
may
assemble
into
crystalline
or
amorphous
structures. •

Common
metallic
crystal
structures
are
FCC,
BCC,
and
HCP. 



Coordina+on
number
and
atomic
packing
factor
are
the
same 



for
both
FCC
and
HCP
crystal
structures. •

We
can
predict
the
density
of
a
material,
provided
we
know 



the
atomic
weight,
atomic
radius,
and
crystal
geometry
(e.g., 



FCC,
BCC,
HCP). •

Crystallographic
points,
direc+ons
and
planes
are
specified
in 




terms
of
indexing
schemes.
Crystallographic
direc+ons
and 




planes
are
related
to
atomic
linear
densi+es
and
planar 




densi+es. Summary •

Materials
can
be
single
crystals
or
polycrystalline. 




Material
proper6es
generally
vary
with
single
crystal 




orienta+on
(i.e.,
they
are
anisotropic),
but
are
generally 




non‐direc+onal
(i.e.,
they
are
isotropic)
in
polycrystals 




with
randomly
oriented
grains. •

Some
materials
can
have
more
than
one
crystal
structure.
This 



is
referred
to
as
polymorphism
(or
allotropy). •

X‐ray
diffrac+on
is
used
for
crystal
structure
and 




interplanar
spacing
determina+ons. IMPERFECTIONS
IN
SOLIDS Chapter 5 • Classify, describe, and illustrate imperfections in crystal structures • Classify methods of examining defects IMPERFECTIONS
IN
SOLIDS • Solidifica+on‐
result
of
cas+ng
of
molten
material – 2
steps • Nuclei
form • Nuclei
grow
to
form
crystals
–
grain
structure • Start
with
a
molten
material
–
all
liquid nuclei liquid crystals growing grain structure Adapted from Fig. 5.19 (b), Callister & Rethwisch 3e. • Crystals grow until they meet each other Polycrystalline
Materials Grain
Boundaries • • • • regions
between
crystals transi+on
from
laTce
of
one region
to
that
of
the
other slightly
disordered low
density
in
grain
boundaries – high
mobility – high
diffusivity – high
chemical
reac+vity Adapted from Fig. 5.12, Callister & Rethwisch 3e. Solidifica+on Grains
can
be ‐
equiaxed


(roughly
same
size
in
all
direc+ons) ‐
columnar


(elongated
grains) ~ 8 cm heat flow Columnar in area with less undercooling Adapted from Fig. 5.17, Callister & Rethwisch 3e. Shell of equiaxed grains due to rapid cooling (greater ΔT) near wall Grain Refiner - added to make smaller, more uniform, equiaxed grains. IMPERFECTIONS
IN
SOLIDS There
is
no
such
thing
as
a
perfect
crystal. • What
are
these
imperfec+ons? • Why
are
they
important? Many
of
the
important
proper+es
of
materials are
due
to
the
presence
of
imperfec+ons. Types
of
Imperfec+ons •

Vacancy
atoms •

Inters++al
atoms •

Subs+tu+onal
atoms Point
defects •

Disloca+ons Line
defects •

Grain
Boundaries Area
defects Point
Defects •

Vacancies:
Very
Common ‐vacant
atomic
sites
in
a
structure. Temperature
dependent!! Vacancy distor+on
 of
planes Equation for the equilibrium concnetration of point defects at a given temperature Number of total atomic sites Vacancy formation energy " !Qv # N v = N exp $ % & kT ' Equilibrium number of vacancies Boltzmann’s Constant Temperature Es+ma+ng
Vacancy
Concentra+on Find
the
equilibrium
#
of
vacancies
in
1
m3
of
Cu
at
1000°C. •

Given: ρ 
=
8.4g
/cm 3 A Cu=
63.5
g/mol 
 Q v 
=
0.9
eV/atom N A 
=
6.02
x
1023 atoms/mol For
1
m3 
 ,
N
= ρx Avagadro’s Number NA A Cu 
 *Don’t forget to use the correct units =


8.0
x
1028
sites Es+ma+ng
Vacancy
Concentra+on Find
the
equilibrium
#
of
vacancies
in
1
m3
of
Cu
at
1000°C. •

Given: k 
=
8.62
x
10‐5
eV/atom‐K Q v 
=
0.9
eV/atom T =
1273K N =


8.0
x
1028
sites " !Qv # N v = N exp $ % & kT ' •

Answer: N v 
=
 (2.7
x
10‐4)(8.0
x
1028)
sites
=
2.2
x
1025
vacancies/m3 
 Point
Defects •

Self‐Inters++als:
Rare ‐"extra"
atoms
posi+oned
between
atomic
sites
in
a
monatomic
crystal. distor+on
 of
planes self‐ inters++al *The reason these are rare is that they introduce significant distortions into the crystal lattice. Point
Defects
in
Ceramics • Vacancies -- vacancies exist in ceramics for both cations and anions • Interstitials -- interstitials exist for cations -- interstitials are not normally observed for anions because anions are large relative to the interstitial sites Cation Interstitial Cation Vacancy Anion Vacancy Adapted from Fig. 5.2, Callister & Rethwisch 3e. (Fig. 5.2 is 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.) Point
Defects
in
Ceramics • Frenkel Defect -- a cation vacancy-cation interstitial pair. • Shottky Defect -- a paired set of cation and anion vacancies. Shottky Defect: Frenkel Defect Adapted from Fig. 5.3, Callister & Rethwisch 3e. (Fig. 5.3 is 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.) Stoichiometry: Exact ratio of cations to anions as predicted by the chemical formula. *Some materials can exist in non-stoichiometric states when one of the ions is present in two different valence states. Imperfec+ons
in
Metals 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. Imperfec+ons
in
Metals Condi+ons
for
subs+tu+onal
solid
solu+on: • W.
Hume
–
Rothery
rule – 1.

Δr
(atomic
radius)
<
15% – 2.

Proximity
in
periodic
table • i.e.,
similar
electronega6vi6es – 3.

Same
crystal
structure
for
pure
metals – 4.

Valence • All
else
being
equal,
a
metal
will
have
a
greater
tendency
to
dissolve a
metal
of
higher
valence
than
one
of
lower
valence Imperfec+ons
in
Ceramics Electroneutrality (charge balance) must be maintained when impurities are present Na + Cl • Substitutional cation impurity Ex: NaCl cation vacancy Ca 2+ Na + Na + without impurity Ca 2+ impurity • Substitutional anion impurity O 2- without impurity Cl Cl O2- impurity Ca 2+ with impurity an ion vacancy with impurity Impurities in Solids Subs/tu/onal
defects
occur as
one
atom
or
ion
is subs+tuted
for
an

atom
or ion
within
the
laTce. The
rela6ve
size
of
the
atoms involved
influence
the distor6on
of
the
crystal. In
metals,
subs+tu+onal
defects lead
to
alloy
forma+on.

Alloys
can be
described
as
solid
solu/ons. Impurities in Metals In
solu6on
terms,
alloy
forma+on
is
influenced
by: Atomic
size–
similarity
favors
solubility
(±15%). Crystal
structure
–
similarity
favors
solubility. Electronega6vity
–
similarity
favors
solubility;
large differences
lead
to
intermetallic
compound forma+on. Valences
–
a
metal
will
have
greater
tendency
to dissolve
another
of
higher
valence
rather
than
lower. Eg:
Copper
/
Nickel
subs6tu6on R=0.128
and
0.125,
both
are
FCC,
electro
1.9
and
1.8
valences +1
and
+2 Imperfec+ons
in
Metals Applica+on
of
rules
–
Solid
Solu+ons 1.
Would
you
predict more
Al
or
Ag to
dissolve
in
Zn? 2.
More
Zn
or
Al in
Cu? Element Cu C H O Ag Al Co Cr Fe Ni Pd Zn Atomic Crystal Radius Structure (nm) 0.1278 0.071 0.046 0.060 0.1445 0.1431 0.1253 0.1249 0.1241 0.1246 0.1376 0.1332 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 Table
on
p.
118,
Callister
&
Rethwisch
8e. Impuri+es
in
Solids Composi+on
(or
concentra+on)
of
an
alloy
is
frequently
quoted
in
either
weight percent
or
atom
percent.

Note
that
one
oUen
finds
reports
of
“percent composi6on”
with
no
further
specfica6on‐
weight
and
atom
percent
are
NOT
the same. Specifica+on
of
composi+on/concentra+on: – 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 Specifica+on
of
Composi+on This
idea
of
composi+onal
variance
gives
rise
to
the
concepts
of average
density
and
average
atomic
weight. ! ave 100 = C1 C2 + !1 ! 2 Aave 100 = C1 C2 + A1 A2 *The
book
covers
how
you
can
convert
weight
%
to
atom
%
and atom%
to
weight
%
as
well
as
other
conversions…
STUDY
THESE! E L 1 Imperfec+ons
in
solids Linear
Defects
(Disloca+ons) – Are
one‐dimensional
defects
around
which
atoms
are
misaligned • Edge
disloca+on: – extra
half‐plane
of
atoms
inserted
in
a
crystal
structure – b
⊥
to
disloca+on
line Burger’s
vector,
b:
measure
of
laTce
distor+on • Screw
disloca+on: – spiral
planar
ramp
resul+ng
from
shear
deforma+on – b
||
to
disloca+on
line Imperfec+ons
in
solids Edge
Disloca/on Fig.
4.3,
Callister
7e. Screw
Disloca+on Front View Top View hUp://www.youtube.com/watch?v=z3MzDiyLtWc&feature=related Imperfec+ons
in
Solids Screw
Disloca+on Screw
Disloca+on b Disloca+on line Burgersvector
b (b) (a) Adapted
from
Fig.
4.4,
Callister
7e. Imperfec+ons
in
Solids Disloca+ons
are
visible
in
electron
micrographs Titanium
alloy Adapted
from
Fig.
4.6,
Callister
7e. Disloca+ons Disloca+ons
in
materials
provide
pathway/
mechanism for
plas/c
deforma/on‐
an
irreversible
change
in shape
arising
from
applied
stress
or
force. (*Elas5c
deforma5ons
are
those
that
“recover”
when
the
force
is removed). The
presence
of
slip
(disloca+ons)
accounts
for
the difference
in
theore+cal/measured
strength
of
metals. Disloca6ons
result
in
metal
duc6lity
and
also
influence electronic
and
op6cal
proper6es. Thus,
material
proper+es
can
be
influenced
by controlling/altering
the
disloca+ons
within
a
material. Disloca+ons Virtually
all
crystalline
materials
contains
some
disloca+ons
that were
introduced
either
during
solidifica+on,
during
plas+c deforma+on,
or
as
a
result
of
thermal
stress. Interfacial
Defects A
grain
can
be
defined
as
a
region
of
space
possessing
the
same crystallographic
orienta+on.

The
intersec6on
of
two
regions
of
different orienta6on
is
thus
known
as
a
grain
boundary.

These
regions
are
classified as
defects,
with
respect
to
perfect
crystallographic
order. Interfacial
Defects Tilt
boundaries
(lev)
and
twinning
(mirror
plane,
right) represent
specific
forms
of
grain
boundaries.

Due
to
their ordered
nature,
these
defects
are
lower
energy
than
typical (random)
grain
boundaries. Twinned quartz crystal Microscopic
Examina+on • Crystallites
(grains)
and
grain
boundaries
vary considerably
in
size.
Can
be
quite
large. • Crystallites
(grains)
can
be
quite
small
(mm
or
less)
– necessary
to
observe
with
a
microscope. Op+cal
Microscopy • Up to 2000X magnification. • Polishing removes surface features (e.g., scratches) • Etching changes reflectance, depending on crystal orientation. crystallographic planes Adapted from Fig. 5.18(b) and (c), Callister & Rethwisch 3e. (Fig. 5.18(c) is courtesy of J.E. Burke, General Electric Co.) Micrograph of brass (a Cu-Zn alloy) 0.75mm Op+cal
Microscopy Grain boundaries... • are imperfections, • are more susceptible to etching, • may be revealed as dark lines, • change in crystal orientation across boundary. polished surface (a) surface groove grain boundary Fe-Cr alloy (b) Adapted from Fig. 5.19(a) and (b), Callister & Rethwisch 3e. (Fig. 5.19(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].) Microscopy Op+cal
resolu+on
ca.
10‐7
m
=
0.1
µm
=
100
nm For
higher
resolu+on
we
need
higher
frequency
/
smaller
λ – X‐Rays?

Difficult
to
focus. – Electrons • wavelengths
ca.
3
pm
(1000
pm
=
1
nm) – (Magnifica+on
‐
1,000,000
x) • Atomic
resolu+on
possible • Electron
beam
focused
by
magne+c
lenses. SEM TEM Scanning
Tunneling
Microscopy • 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”. Microscopy 43 Summary ‐>
Point,
Line,
and
Area
defects
exist
in
solids. ‐>
Burger’s
vector
(b)
is
a
measure
of
the
magnitude
and
direc+on
of deforma+on. ‐>
The
number
and
type
of
defects
can
be
varied
and
controlled (e.g.,
T
controls
vacancy
concentra+on) ‐>
Defects
affect
material
proper+es
(e.g.,
grain
boundaries
control crystal
slip) ‐>
Defects
may
be
desirable
or
undesirable
(e.g.,
disloca+ons
may
be good
or
bad,
depending
on
whether
plas+c
deforma+on
is
desirable or
not.) ...
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This note was uploaded on 02/09/2012 for the course EMA 3010 taught by Professor Unknown during the Spring '08 term at University of Florida.

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