x-ray diffraction. introduction:introduction: x-ray diffraction techniques are very useful for...

X-Ray Diffraction

• Introduction:Introduction:

• X-ray diffraction techniques are very useful for crystal X-ray diffraction techniques are very useful for crystal structure analysis and identification of different types of structure analysis and identification of different types of crystals.crystals.

• Experimental study of crystalline materials became Experimental study of crystalline materials became

possible only after the discovery of possible only after the discovery of x-raysx-rays. .

• Diffraction occurs when waves traveling through an Diffraction occurs when waves traveling through an object whose dimensions are order of object whose dimensions are order of wavelengthwavelength..

• Typical inter atomic spacing in crystals is Typical inter atomic spacing in crystals is 2-3°A2-3°A..

• The x-rays have wavelengths (The x-rays have wavelengths (0.02°A to 100°A0.02°A to 100°A) in this ) in this range . Hence x-ray diffraction is utilized to study the range . Hence x-ray diffraction is utilized to study the crystal structures.crystal structures.

Bragg’s lawBragg’s law

• Bragg’s law states that, the Bragg’s law states that, the path path differencedifference between the two reflected between the two reflected rays by the crystal planes should be rays by the crystal planes should be an an integral multiple of wavelengthintegral multiple of wavelength of of incident x-rays for producing incident x-rays for producing maxima maxima or or constructive interferenceconstructive interference..

Plane 1

Plane 2

Plane 3

A

B

C D

P

Q

R

S

d90° ө 90°ө

The path difference between these two rays is CB+BDThe path difference between these two rays is CB+BD

n sin 2d

sin dBDCB

nBDCB

Where n = 1,2,3,…..first , second …order etc.Where n = 1,2,3,…..first , second …order etc.

• For 1For 1stst order sin order sinθθ11 = = λλ/2d./2d.

• For 2For 2ndnd order sin order sinθθ22 = 2 = 2λλ / 2d. / 2d.

• For 3For 3rdrd order sin order sinθθ3 3 = 3= 3λλ / 2d. / 2d.

• where where θθ11, , θθ22 and and θθ33 are the glancing angles for n=1,2 and 3 are the glancing angles for n=1,2 and 3

respectively.respectively.

X-ray Diffraction Techniques

• There are two main experimental X-ray diffraction methods by There are two main experimental X-ray diffraction methods by which the crystal structure can be analyzed.which the crystal structure can be analyzed.

1.Laue method.1.Laue method.

2.Powder method.2.Powder method.

Laue Method:

Consider heterogeneous beam of x-rays in the wavelength of Consider heterogeneous beam of x-rays in the wavelength of 0.2°A to 2°A0.2°A to 2°A originating from a suitable sou originating from a suitable source.rce.

In this technique, the crystal stationary in a In this technique, the crystal stationary in a heterogeneousheterogeneous beam of x-rays.beam of x-rays.

The diffraction pattern consists of a The diffraction pattern consists of a bright central spotbright central spot and a set and a set of spots arranged in a definite pattern about the central spot.of spots arranged in a definite pattern about the central spot.

The The symmetrical patternsymmetrical pattern caused by diffraction of x-rays by caused by diffraction of x-rays by crystal planes is called the Laue patterncrystal planes is called the Laue pattern..

crystal

Photographic plate

Laue pattern

X-rays

slits

Photographic film

P

oθ

White x-radiation

crystal

2θ

D

• The crystal is fixed, for an incident angle ‘θ’, the diffracted angle becomes ‘2θ’.

• Consider, ‘D’ is the distance between the crystal and photographic film and OP = R.

Tan2θ = OP/OC OP = OC tan2θ R = D tan2θ

• This method is used to study the This method is used to study the orientationorientation of the of the crystal and verify crystal and verify crystal symmetrycrystal symmetry..

Powder method (Debye – Scherer method)

• The powder method is applicable to finely divided crystalline The powder method is applicable to finely divided crystalline powder.powder.

• It is used for accurate determination of lattice parameters in It is used for accurate determination of lattice parameters in crystals.crystals.

• The powdered specimen is kept inside a small capillary The powdered specimen is kept inside a small capillary tube.tube.

• A narrow pencil of monochromatic x-ray is diffracted from A narrow pencil of monochromatic x-ray is diffracted from the powder and recorded by the photographic film as a the powder and recorded by the photographic film as a series of lines of varying curvature.series of lines of varying curvature.

• The full opening angle of the diffraction cone ‘4The full opening angle of the diffraction cone ‘4θθ’ is ’ is determined by measuring the distance ‘s’ between two determined by measuring the distance ‘s’ between two corresponding arc.corresponding arc.

w1

w2

r

2θ2θ

Incident x-ray beam

S

Crystal powder

• Where r is the specimen to film distance.Where r is the specimen to film distance.

X-ray powder diffraction has large number of applications.

• Study of d-spacing.• Study of mixtures.• Study of alloys.• Stress determination in metals. • Determination of particle size.

rs

rs

4

4

Defects in Crystals

IntroductionIntroduction

• In an ideal crystal, the atomic arrangement iIn an ideal crystal, the atomic arrangement is perfectly regular and continuous but real perfectly regular and continuous but real crystals never perfect.crystals never perfect.

• They always contain a considerable density They always contain a considerable density defects and imperfections that affect thedefects and imperfections that affect their r physical, chemical ,mechanical and electronic physical, chemical ,mechanical and electronic properties.properties.

• Crystalline imperfections can be classified on Crystalline imperfections can be classified on they basis of their geometry under four main they basis of their geometry under four main divisions namelydivisions namely

1.Vacancies or Schottky2.Interstitialcies or Frenkel3.Compositional defects.

a. Substitutionalb. interstitial

4.Electronic defects

Defects

Point defects (0-dimensional)

Line defects (1-dimensional)

Surface defects (2-dimensional)

Volume defects (3-dimensional)

1.Edge dislocation2.Screw dislocation

1.Grain boundaries2.Tilt boundaries3.Twin boundaries4.Stacking faults

1.Cracks2.Voids or air bubbles

Point Point DefectsefectsPoint imperfections are also called zero dimensional Point imperfections are also called zero dimensional imperfections. One or two atomic diameters is the imperfections. One or two atomic diameters is the typical size of a point imperfection.typical size of a point imperfection.

Perfect Crystal

Vacancy:

A vacancy refers to an atomic site from where the atom is missing.

Compositional defects.

• A substitutianal impurity is a point imperfection and it refers to a foreign atom that substitutes for or replaces a parent atom in the crystal.

• A small sized atom occupying the void space in the parent crystal disturbing the parent atoms from their regular sites is a interstitial impurity.

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 CaF2 are of the Frenkel type.

Schottky defect: A pair of one cation 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.

Electronic defects:

• Errors in charge distribution in solids are called electronic defects.

• These defects are produced, when the composition of an ionic crystal does not correspond to the exact Stoichiometric formula.

• Calculation of number of vacancies at a given temperature.

• All most in all crystals vacancies are present and the main cause for these defects is thermal agitation.

• Let us consider Ev is the energy required to move an atom from lattice site inside the crystal to lattice site on the surface.

• Therefore the amount of energy required to produce n number of isolated vacancies can be written as

vnEU

• The total number of ways to move n number of atoms out of N number of atoms in a crystal on to its surface will be

!)!(

!

nnN

NP

The increase in entropy due to formation of n vacancies can be written as

}log{

log

!)!(!

nnNN

B

B

K

PKS

But the free energy

TSUF

}!log)!log(!{log

)!)!(

!log(

nnNNTKnEF

nnN

NTKnEF

Bv

Bv

Using Stirling is a approximation, log x! = x log x - x

}log)log()(log{ nnnNnNNNTKnEF Bv

At thermal equilibrium, free energy is constant and minimum with respect to n, hence

}exp{

}log{

}log1)log(1{

0})log)log()(log{(

TK

E

n

nNn

nNTKE

nnNTKE

nnnNnNNNTKnEdn

d

odn

dF

B

v

Bv

Bv

Bv

}exp{TK

ENn

Nifn

B

v

Hence equilibrium concentration of vacancies decreases with temperature.

• Calculation of number Schottky defects at a given temperature.

• In ionic crystals, the number of schottky defects at a given temperature, can be calculated assuming an equal number of positive and negative ion vacancies are present.

• Let us consider Ep is the energy required to move an ion Pair from lattice site inside the crystal to a lattice site on the surface.

• Therefore the amount of energy required to produce n number of isolated ion pair vacancies will be

pnEU

• The total number of ways that to move n numbers of ion pairs out of N number of ionic molecules in a crystal on to the surface will be

2

2

]!)!(

!log[

log

]!)!(

![

nnN

NKS

PKS

nnN

NP

B

B

• The free energy….

2]!)!(

!log[

nnN

NTKnEF

TSUF

Bp

Using stirling approximation xxxx log!log

]log)log()(log[2

]log)log()(log[2]!)!(

!log[

]log)()log()(log[2]!)!(

!log[

2

2

nnnNnNNNTKnEF

nnnnNnNNNnnN

N

nnnnNnNnNNNNnnN

N

Bv

At thermal equilibrium, free energy is constant and minimum withrespect to n,hence

}2

exp{

}2

exp{)(

]log[2

]log[2

0][

TK

ENn

Nifn

TK

EnNn

n

nN

TK

En

nNTKE

dn

dF

B

p

B

p

B

p

BP

T

Calculation of number of Frenkel Defects at given temperature: In ionic crystal an ion may be displaced from the regular lattice into an interstitial site or void space. If it is so, then a vacancy and an interstitial defect will be formed.

A Frenkel imperfections in silver halides and calcium fluoride are of the Frenkel type.

Frenkel and Schottky defects together are called Intrinsic defects.

• Let us consider Ei is the energy required to move an atom from lattice site inside the crystal to a lattice site on the surface.

• The amount of energy required to produce n number of isolated vacancies…

inEU

• The total number of ways to move n numbers of ions out of N number ionic molecules in a crystal on to the surface will be,

]}!)!(

!][

!)!(

!log{[

]}!)!(

!][

!)!(

!log{[

log

]!)!(

!][

!)!(

![

nnN

N

nnN

NTKnEF

TSUfreeenergy

nnN

N

nnN

NKS

pKentropy

nnN

N

nnN

Np

i

iBi

i

iB

B

i

i

}log2)log()()log()(loglog{

log2)log()()log()(loglog

]}!)!(

!][

!)!(

!log{[

nnnNnNnNnNNNNNTKnEF

nnnNnNnNnNNNNN

nnN

N

nnN

N

iiiiBi

iiii

i

i

At equilibrium, the free energyis constant and minimum withrespect to n, hence

TK

ENNn

TK

ENNn

nNNTKEn

NNTK

nNnNn

nNnNTKE

dn

dF

B

ii

Bi

iBi

iB

i

iBi

T

2exp)(

2}log{

2

1log

]log2}[log{

}log{

,

}))((

log{

0][

2

1

2

2

Hence it is concluded that number of Frenkel defects,is proportional (NNi)1/2

Line defects

• Line defects are one dimensional imperfections in the geometrical sense.

• There are in general two types of dislocations:

1. Edge dislocation

2. Screw dislocation

Edge dislocation

• In a perfect crystal, atoms are arranged in both vertical and horizontal planes parallel to the side faces.

• If one of these vertical planes does not extended to full length but ends in between, within the crystal as shown in figure, it is called edge dislocation.

• Edge dislocations are symbolically represented by ┴ or ┬ or depending on whether the incomplete plane starts from the top or from the bottom of the crystal.

• These two configurations are referred to as positive and negative edge dislocations.

Perfect crystal

An incomplete plane in aCrystal results in an edgedislocation

Perfect crystal

Edge dislocated crystal

Extra half plane

Slip plane

• The edge dislocation containing an extra plane of atoms lying above the positive slip plane (or) Burgers plane are conventionally called the positive edge dislocation.

• If the extra half plane of atoms containing

below the slip plane called the negative edge dislocation.

Positive and negative dislocations

Burgers vector

• The magnitude and the direction of the displacement are defined by a vector called the Burgers vector.

• Consider two crystals one perfect and another with edge dislocation.

Perfect crystal

P

An incomplete plane in aCrystal results in an edgedislocation

Fig 1. Fig 2.

PQb

• From fig. 1. • Starting from the point P, we go up by 6 steps, then

move towards right by 5 steps, and move down by 6 steps and finally move towards left by 5 steps to reach the starting point P, the burgers circuit gets closed.

• From fig 2. • We end up at Q instead of the starting point P.

• Now we have to move an extra step QP to return to ‘P’ in order to close the burgers circuit.

• The magnitude and the direction of the step defines the Burgers vector (BV)

BV = QP = b

• The Burgers vector is perpendicular to the edge dislocation line.

Screw dislocation

• The second basic type of dislocation is the Screw or Burgers dislocation.

• In this, the atoms are displaced in two separate planes perpendicular to each other.

• In a figure the plane ABCD is the slipped area.

• The upper portion of the crystal has 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.

A

B

CD

Shear vector

• Here, the dislocation is parallel to its Burgers vector or shear vector.

• The designation ‘screw’ for this lattice defect is derived from the fact that the lattice planes of the crystal spiral the dislocation line AD.

Surface defects

• Surface defects arise from a change in the stacking of atomic planes (or) across a boundary.

• The change may be one of the orientations (or) of the stacking sequence of the planes.

• Surface defects are two types.

1. External surface imperfections

2. Internal surface imperfections

External surface defects

• Since these surface atoms are not entirely surrounded by others, they posses higher energy

than that of internal atoms.

Internal surface imperfections

• Internal surface defects are four types.

1. Grain boundaries

2. Tilt boundaries

3. Twin boundaries

4. Stacking faults

1. Grain boundary

• The boundary between two miss oriented crystalline material is called grain boundary.

• During nucleation or crystallization this may happen.

• Grain boundaries also known as high angle boundaries.

Grain boundaries

Poly crystal

2. Tilt (angle) boundaries

• This is called low-angle boundary as the orientation difference between two neighboring crystals is less than 10º.

• Low angle boundaries can be described by suitable array of edge dislocations.

• A low angle tilt boundary is composed of edge dislocation lying one above the other in boundary.

Tilt boundaryθ

b

D

D

bThe low angle (or) tilt will be θ=

b Magnitude of burger’s vector

D Average vertical distance between dislocations

=b/θ

b

D

θ

3. Stacking faults

• Stacking faults are planar surface imperfections caused by faults in the stacking sequence of atomic planes (or) layers in crystals.

• In FCC crystal we have three different stacking layers ABC while in HCP stacking we have only two different layers BC.

• Hence when FCC crystal grows we have the stacking as

…….ABCABCABC……..

A

B

C

A

Stacking sequence of ABC ABC…..in FCC

STACKING SEQUENCE IN HCP AB AB….

A

B

A

• While growing if the plane A indicated by arrow above is missing then we get the sequence

………….ABCABCBCABC…………….

• Thus we find that the stacking in the missing region becomes HCP.

• This thin region is a surface imperfection and is called a stacking fault.

4.Twin boundary

• The atomic arrangement on one side of a twin boundary is a mirror reflection of the arrangement on the other side, such a boundary and the region between the pair of boundaries is called twinned region.

• Twin boundaries are easily identified under an optional microscope.

Twin boundary

Fig: The atomic arrangement in a twin boundary

Volume defects

• Volume defects such as cracks may arise when there is only small electrostatic dissimilarity between the stacking sequences of close packed planes in metals.

• When clusters of atoms are missing ,a largeVacancy or void is got which is also a volume imperfection.

• Foreign particle inclusions, large voids or non crystalline regions which have the dimensions of the order of 0.20nm are also called volume imperfection.

• We can study the volume defects by using interferometric techniques and optical microscopes.