X-Ray Diffraction & Crystal Defects(21!10!2011)

8/3/2019 X-Ray Diffraction & Crystal Defects(21!10!2011)

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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.

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X-Ray Diffraction & Crystal Defects(21!10!2011)