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X-Ray Diffraction
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IntroductionIntroduction
XX--ray diffraction techniques are very useful for crystalray diffraction techniques are very useful for crystalstructure analysis and identification of different types ofstructure analysis and identification of different types ofcrystals.crystals.
Experimental study of crystalline materials became possibleExperimental study of crystalline materials became possible
only after the discovery ofonly after the discovery ofXX--raysrays..
Diffraction occurs when waves traveling through an objectDiffraction occurs when waves traveling through an objectwhose dimensions are order ofwhose dimensions are order ofwavelengthwavelength..
Typical inter atomic spacing in crystals isTypical inter atomic spacing in crystals is 22--33AA..
The xThe x--rays have wavelengthsrays have wavelengths (0.02(0.02A to 100A to 100A)A) in this range .in this range .Hence xHence x--ray diffraction is utilized to study the crystalray diffraction is utilized to study the crystalstructures.structures.
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Braggs law
Braggs law states that, the path difference
between the two reflected rays by the crystalplanes should be an integral multiple of
wavelength of incident x-rays for producing
maxima or constructive interference.
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Plane 1
Plane 2
Plane 3
A
B
C D
P
Q
R
S
d90 90
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The path difference between these two rays is CB+BDThe path difference between these two rays is CB+BD
PU
U
P
nsin2d
sindBDCB
nBDCB
!
!!
!
Where n = 1,2,3,..first , second order etc.Where n = 1,2,3,..first , second order etc.
For 1For 1stst order sinorder sin11 == /2d./2d.
For 2For 2ndnd order sinorder sin22 = 2= 2 / 2d./ 2d. For 3For 3rdrd order sinorder sin33 = 3= 3 / 2d./ 2d.
wherewhere 11,, 22 andand 33 are the glancing angles for n=1,2are the glancing angles for n=1,2
and 3 respectively.and 3 respectively.
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X-Ray Diffraction Techniques
There are two main experimental XThere are two main experimental X--Ray diffractionRay diffractionmethods by which the crystal structure can bemethods by which the crystal structure can be
analyzed.analyzed.
1.Laue Method.1.Laue Method.
2.Powder Method.2.Powder Method.
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Laue Method
Consider heterogeneous beam of XConsider heterogeneous beam of X--Rays in the wavelength ofRays in the wavelength of0.20.2A to 2A to 2AA originating from a suitable source.originating from a suitable source.
In this technique, the crystal is stationary in aIn this technique, the crystal is stationary in a heterogeneousheterogeneousbeam of Xbeam of X--Rays.Rays.
The diffraction pattern consists ofThe diffraction pattern consists ofaa bright central spotbright central spot and aand aset of spots arranged in a definite pattern about the central spot.set of spots arranged in a definite pattern about the central spot.
TheThe symmetrical patternsymmetrical pattern caused by diffraction of Xcaused by diffraction of X--Rays byRays bycrystal planes is called the Laue patterncrystal planes is called the Laue pattern..
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Crystal
Photographic Plate
Laue
Pattern
X-Rays
Slits
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Photographic film
P
o
X-Rays
Crystal
2D
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If The Crystal is fixed, for an incident angle , thediffracted angle becomes 2.
Consider, D is the distance between the crystal andphotographic film and OP = R.
Tan2 = OP/OC
OP = OC tan2
R = D tan2
This method is used to study theThis method is used to study the orientationorientation of theof thecrystal and verify thecrystal and verify the Crystal symmetryCrystal symmetry..
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Powder ( Debye Scherer) Method
The Powder Method is applicable to finely divided CrystallineThe Powder Method is applicable to finely divided Crystallinepowder.powder.
It is used for accurate determination of lattice parameters inIt is used for accurate determination of lattice parameters incrystals.crystals.
The powdered specimen is kept inside a small capillary tube.The powdered specimen is kept inside a small capillary tube.
A narrow pencil of monochromatic XA narrow pencil of monochromatic X--Ray is diffracted from theRay is diffracted from thepowder and recorded by the Photographic film as a series of linespowder and recorded by the Photographic film as a series of lines
of varying curvature.of varying curvature.
The full opening angle of the diffraction cone 4The full opening angle of the diffraction cone 4 is determined is determinedby measuring the distance s between two corresponding arcs.by measuring the distance s between two corresponding arcs.
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r2
2
Incident X-Raybeam
S
Crystal
Powder 4r
s
rs4
!
!
Where r is theWhere r is thespecimen to filmspecimen to filmdistance.distance.
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Applications of Powder Method
Study of d-spacing.
Study of mixtures.
Study of alloys.
Stress determination in metals.
Determination of particle size.
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Defects in Crystals
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IntroductionIntroduction
InIn anan idealideal crystal,crystal, thethe atomicatomic arrangementarrangement iisperfectlyperfectly regularregular andand continuouscontinuous butbut realreal crystalscrystalsnevernever perfectperfect..
TheyThey alwaysalways containcontain aa considerableconsiderable densitydensity defectsdefectsandand imperfectionsimperfections thatthat affectaffect thetheirr physical,physical,chemicalchemical ,mechanical,mechanical andand electronicelectronic propertiesproperties..
CrystallineCrystalline imperfectionsimperfections cancan bebe classifiedclassified onon thethebasisbasis ofof theirtheir geometrygeometry underunder fourfour mainmain divisionsdivisionsnamelynamely
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1.Vacancies or Schottky
2.Interstitialcies or Frenkel
3.Compositional defects.
a. Substitutional
b. interstitial
4.Electronic defects
Defects
Point defects
(0-dimensional)
Line defects
(1-dimensional)
Surface defects
(2-dimensional)
Volume defects
(3-dimensional)
1.Edge dislocation
2.Screw dislocation
1.Grain boundaries
2.Tilt boundaries
3.Twin boundaries
4.Stacking faults
1.Cracks
2.Voids or air bubbles
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PointPoint Defectsefects
Point imperfections are alsoPoint imperfections are also
called zero dimensionalcalled zero dimensional
imperfections.imperfections.
One or two atomic diametersOne or two atomic diameters
is the typical size of a pointis the typical size of a pointimperfection.imperfection. Perfect Crystal
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Vacancy:A Vacancy refers to an atomic site from where the
atom is missing.
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Compositional defects
A Substitutional impurity is apoint imperfection and itrefers to a foreign atom that
substitutes for or replaces aparent atom in the crystal.
A small sized atom occupying
the void space in the parentcrystal disturbing the parentatoms from their regular sitesis an interstitial impurity.
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Frenkel Defect:An atom leaves the regular site
and occupies interstitial position. Such defects are
called Frenkel defects.
The point imperfections in silver halides and CaF2are of the Frenkel type.
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Schottky defect:
A pair of one cat-aion and one anion can be
missing from an ionic crystal as shown in a figure.such a pair of vacant ion sites is called Schottky
defect.
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Electronic defects
Errors in charge distribution in solids arecalled electronic defects.
These defects are produced, when thecomposition of an ionic crystal does not
correspond to the exact Stoichiometricformula.
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Calculation of number of vacancies ata giventemperature.
All most in all crystals vacancies are presentand the main causefor these defects is thermalagitation.
Let us consider Ev is the energy required to move an atom fromlattice site inside the crystal to lattice site on the surface.
Therefore the amount of energy required to produce n number ofisolated vacancies can be written as
vnEU !
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The total number of ways to move n number ofatoms out of N number ofatoms in a crystalon
to its surfa
ce will
be
!)!(
!
nnN
NP
!
The increase in entropy due to formation of nvacancies can be written as
}log{
log
!)!(!nnN
N
B
B
K
PKS
!
!
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But the free energy TSUF !
}logn!n)!log(NT{logN!KnEF
)n!n)!(N
N!Tlog(KnEF
Bv
Bv
!
!
Using Sterlings approximation, log x! = x log x - x
nlogn}n)n)log(N(NT{NlogNKnEF Bv !
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At thermalequilibrium, free energy is constantand minimum with respect to n, hence
}TK
ENexp{n
Nnif
}TK
Eexp{
n
nN
}n
nNTlog{KE
logn}1n)log(NT{1KE
0nlogn})n)n)log(N(NT{NlogNK(nEdn
d
odndF
B
v
B
v
Bv
Bv
Bv
$
!
!
!
!
!
Hence equilibrium concentration of vacanciesincreases with increase of temperature.
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Calculation of number Schottky defects ata
given tempera
ture: In ionic crystals, the number of schottky defects ata given
temperature, can be calculatedassumingan equal number ofpositive and negative ion vacancies are present.
Let us consider Ep is the energy required to move an ion Pairfrom lattice site inside the crystal to alattice site on the surface.
Therefore the amount of energy required to produce n number
of isolated ion pair vacancies will be
pnEU !
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The total number of ways to move n numbers ofion pairs out of N number of ionic molecules in acrystalon to the surface will be
2
2
]!)!(
!log[
log
]!)!(
![
nnN
NKS
PKS
nnN
NP
B
B
!
!
!
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The free energy
2
Bp ]
n!n)!(N
N!Tlog[KnEF
TSUF
!
!
Using stirling approximation xxxx ! log!log
nlogn]n)n)log(N(NT[NlogN2KnEF
n]nlognn)n)log(N(N2[NlogN]n!n)!(N
N!log[
n]nlognn)(Nn)n)log(N(NN2[NlogN]n!n)!(N
N!log[
Bv
2
2
!
!
!
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At thermalequilibrium, free energy is constantand
minimum with respect to n, hence
}T2K
ENexp{n
Nnif
}
T2K
En)exp{(Nn
]n
nNlog[
T2K
E
]
n
nNTlog[2KE
0]dn
dF[
B
p
B
p
B
p
BP
T
$
!
!
!
!
H
ence it is concl
uded tha
tthe number of Schottkydefects increases withincrease of temperature.
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Calculation of number of FrenkelDefects at giventemperature:
In ionic crystalan ion may be displaced from the regularlattice into an interstitialsite or void space.
If it is so, thena
va
ca
ncya
nda
n interstitial
defect will
beformed.
A Frenkel imperfection in silver halides and calcium
fl
uoridea
re of the Frenkel
type.Frenkeland Schottky defects togetherare calledIntrinsic defects.
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Let us consider Ei is the energy required tomove an atom from lattice site inside the crystalto alattice site on the surface.
The amount of energy required to produce n
number of isolated vacancies
inEU !
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The total number of ways to move n numbers ofions out of N number ionic molecules in a crystalon to the surface will be,
]}n!n)!(N
!N][
n!n)!(N
N!Tlog{[KnEF
TSUfreeenergy
]}n!n)!(N
!N][
n!n)!(N
N!log{[KS
logpKentropy
]n!n)!(N
!N][
n!n)!(N
N![p
i
iBi
i
iB
B
i
i
!
!
!
!
!
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}log2)log()()log()(loglog{
log2)log()()log()(loglog
]}!)!(
!][!)!(
!log{[
nnnNnNnNnNNNNNTKnEF
nnnNnNnNnNNNNN
nnN
N
nnN
N
iiiiBi
iiii
i
i
!
!
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At equilibrium, the freeenergy is constantand
minimum with respect ton, hence
TK
ENNn
TK
ENNn
nNNTKE
n
NN
TK
nNnN
nnNnNTKE
dn
dF
B
ii
B
i
iBi
i
B
i
i
Bi
T
2exp)(
2
}log{
2
1log
]log2}[log{
}log{
,
}))((log{
0][
2
1
2
2
$
$
$
$
""""
!
!
H
ence it is concl
uded tha
tnumber of Frenkeldefects, is proportional(NNi)1/2
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Line defects
Line defects are one dimensionalimperfections in the geometrical sense.
There are in general two types ofdislocations:1. Edge dislocation2. Screw dislocation
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Edge dislocation
Ina
perfect crystal
,a
tomsa
rea
rra
nged in both vertical a
ndhorizontalplanes parallel to the side faces.
If one of these vertical planes does not extended to fulllength but ends in between, within the crystal as shown in
figure, it is called edge dislocation.
Edge dislocations are symbolically represented by or ordepending on whether the incomplete plane st arts from thetop or from the bottom of the crystal.
These two configurations are referred to as positive andnegative edge dislocations.
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Perfect Crystal
An incomplete plane in aCrystal results in an
edge dislocation
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Perfect crystal
Edge dislocated crystal
Extra half plane
Slip plane
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The edge dislocation containing an extr a plane of atoms lying above the positive slip plane (or)
Burgers plane are conventionally called the positiveedge dislocation.
If the extra half plane of atoms containing belowthe slip plane called the negative edge dislocation.
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Positive and negative dislocations
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Burgers vector
The magnitude and the direction of thedisplacement are defined by a vector called
the Burgers vector.
Consider two cryst als one perfect andanother with edge dislocation.
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Perfect crystal
P
An incomplete plane in aCrystal results in an edge
dislocation
Fig 1. Fig 2.
PQb
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From fig. 1.
Starting from the point P, we go up by 6 steps, then movetowards right by 5 steps, and move down by 6 steps and
finally move towards left by 5 steps to reach the startingpoint P, the burgers circuit gets closed in a perfect crystal.
Where as in fig 2.
We end up at Q instead of the starting point P. In order toreach the point P, we have to move an extra step QP. Sothat the burgers circuit is closed.
The magnitude and the direction of the step QP defines theBurgers vector (BV)
BV = QP = b
The Burgers vector is perpendicular to the edgedislocation line and it is parallel in Screw
dislocation.
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Screw dislocation
The second basic type of dislocation is the Screw orBurgers dislocation.
In this, the atoms are displaced in two separate
planes perpendicular to each other. In a figure the plane ABCD is the slippedarea. The upper portion of the crystalhas been sheared by
an atomic distance to the right relative to the lower
portion. No slip has taken place to the right of AD and AD is
a dislocation line.
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A
B
CD
Shear vector
Here, the dislocation is parallelto its Burgers vector
or shear vector. The designation screw for this lattice defect is
derived from the fact that the lattice planes of thecrystalspiral the dislocation line AD.
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Surface defects
Surface defects arise from a change in the st ackingofatomic planes (or) across a boundary.
The change may be one of the orient ations (or) ofthe stacking sequence of the planes.
Surfa
ce defectsa
re two types.1. Externalsurface imperfections2. Internalsurface imperfections
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Externalsurface defects
Since these surface atoms are not entirel y surrounded byothers, they posses higher energy than that of internalatoms.
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Internal
surfa
ce imperfections Internalsurface defects are four types.
1. Grain boundaries2. Tilt boundaries3. Twin boundaries
4. Stacking faults
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1. Grain boundary
The boundary between two miss oriented crystallinematerialis calledgrain boundary.
During nucleation or cryst allization this mayhappen.
Grain bound aries also known as high angleboundaries.
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Grain boundaries
Poly crystal
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2. Tilt (angle) boundaries
This is calledlow-angle boundary as the orientationdifference between two neighboring crystals is lessthan 10.
Low angle boundaries can be described by suitablearray of edge dislocations.
A low angle tilt bound ary is composed of edgedislocation lying one above the other in boundary.
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Tilt boundary
b
D
D
bThe low angle (or) tilt will be =
b Magnitude of burgers vector
D Average vertical distance between
dislocations
=b/
b
D
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A
B
C
A
Stacking sequence of ABCABC..in FCC
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STACKING SEQUENCE IN HCP AB AB.
A
B
A
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While growing if the plane A indicated by arrowa
bove is missing then we get the sequence.ABCABCBCABC.
Thus we find that the st acking in the missing regionbecomes HCP.
This thin region is a surface imperfection and is
calledastacking fault.
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4.Twin boundary
The atomic arrangement on one side of a twinboundary is a mirror reflection of the arrangementon the other side, such a boundary and the region
between the pair of bound aries is called twinnedregion.
Twin boundaries are easily identified under anoptionalmicroscope.
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Volume defects
Volume defects such as cracks may arise when there is only smallelectrostatic dissimilarity between the stacking sequences of closepacked planes in metals.
When clusters of atoms are missing ,a large
Vacancy or void is got which is also a volume imperfection.
Foreign particle inclusions, large voids or non crystalline regionswhich have the dimensions of the order of 0.20nm are also called
volume imperfection.
We can study the volume defects by using interferometrictechniques and optical microscopes.