7c x raydiffraction

8/12/2019 7c X RayDiffraction

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X-RAY DIFFRACTION

X- Ray Sources

Diffraction: Braggs Law

Crystal Structure Determination

Elements of X-Ray DiffractionB.D. Cullity & S.R. StockPrentice Hall, Upper Saddle River (2001)

X-Ray Diffraction: A Practical ApproachC. Suryanarayana & M. Grant Norton

Plenum Press, New York (1998)


2/74

For electromagnetic radiation to be diffracted the spacingin the grating should be of the same order as the wavelength

In crystals the typical interatomic spacing ~ 2-3 so thesuitable radiation is X-rays

Hence, X-rays can be used for the study of crystal structures

Beam of electrons TargetX-rays

An accelerating (/decelerating) charge radiates electromagnetic radiation


3/74

Intensity

Wavelength ()

Mo Target impacted by electrons accelerated by a 35 kV potential

0.2 0.6 1.0 1.4

Whiteradiation

Characteristic radiation due to energy transitionsin the atom

K

K


4/74

Target Metal Of K radiation ()

Mo 0.71

Cu 1.54

Co 1.79

Fe 1.94

Cr 2.29


5/74

A beam of X-rays directed at a crystal interacts with theelectrons of the atoms in the crystal

The electrons oscillate under the influence of the incoming

X-Rays and become secondary sources of EM radiation

The secondary radiation is in all directions

The waves emitted by the electrons have the same frequency

as the incoming X-rays coherentThe emission will undergo constructive or destructive

interference with waves scattered from other atoms

Incoming X-rays

Secondary

emission


6/74

Sets Electron cloud into oscillation

Sets nucleus (with protons) into oscillation

Small effect neglected


7/74

Oscillating charge re-radiates In phase with the incoming x-rays


8/74

BRAGGs EQUATION

d

The path difference between ray 1 and ray 2 = 2d Sin

For constructive interference: n= 2d Sin

Ray 1

Ray 2

Deviation = 2


9/74


10/74


11/74


12/74

Incident and scatteredwaves are in phase if

Scattering from across planes is in phase

In plane scattering is in phase


13/74

Extra path traveled by incoming wavesAY

Extra path traveled by scattered wavesXB

These can be in phase if and only ifincident= scattered

But this is still reinforced scatteringand NOT reflection


14/74

Note that in the Braggs equation:

The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appearsChange in position or spacing of atoms along the plane should not affect

Braggs condition !!

d

Note: shift (systematic) isactually not a problem!


15/74

Note: shift is actually not a problem! Why is systematic shift not a problem?

n AY YB [180 ( )] ( )AY XY Cos XY Cos

( )YB XY Cos [ ( ) ( )] [2 ]n AY YB XY Cos Cos XY Sin Sin

( )d

SinXY

[2 ] 2d

n Sin Sin d SinSin

2n d Sin


16/74

Consider the case for which 12

Constructive interference can still occur if the difference in the path lengthtraversed by R1and R2before and after scattering are an integral multiple of the

wavelength(AY XC) = h (h is an integer)

1Cos

a

AY

2Cosa

XC hCosaCosa 21

hCosCosa 21


17/74

Laues equations

S0incoming X-ray beam

S Scattered X-ray beam

hSSa )(0

kSSb )( 0

lSSc )(0

hCosCosa 21Generalizing into 3D

kCosCosb 43

lCosCosc 65

This is looking at diffraction from atomic arrays and not planes


18/74

A physical picture of scattering leading to diffraction is embodied in Laues equations

Braggs method of visualizing diffraction as reflection from a set of planes is a different

way of understanding the phenomenon of diffraction from crystals

The plane picture (Braggs equations) are simpler and we usually stick to them

Hence, we should think twice before asking the question: if there are no atoms in the

scattering planes, how are they scattering waves?


19/74

Braggs equation is a negative law

If Braggs eq. is NOT satisfied NO reflection can occur

If Braggs eq. is satisfied reflectionMAY occur

Diffraction = Reinforced Coherent Scattering

Reflection versus Scattering

Reflection Diffraction

Occurs from surface Occurs throughout the bulk

Takes place at any angle Takes place only at Bragg angles

~100 % of the intensity may be reflected Small fraction of intensity is diffracted

X-rays can be reflected at very small angles of incidence


20/74

n= 2d Sin

n is an integer and is the order of the reflection

For Cu Kradiation (= 1.54 ) and d110= 2.22

n Sin

1 0.34 20.7 First order reflection from (110)

2 0.69 43.92Second order reflection from (110)

Also written as (220)

222 lkh

adhkl

8220

ad

2110

ad

2

1

110

220

d

d


21/74

sin2 hkldn

In XRD nthorder reflection from (h k l) is considered as 1storder reflectionfrom (nh nk nl)

sin2 n

dhkl

sin2 nnn lkhd


22/74


23/74

Intensity of the Scattered electrons

Electron

Atom

Unit cell (uc)

Scattering by a crystal

A

B

C

Polarization factor

Atomic scattering factor (f)

Structure factor (F)


24/74

Scattering by an Electron

),( 00

Sets electron into oscillation

Scattered beams),( 00

Coherent(definite phase relationship)

A

The electric field (E)is the main cause for the acceleration of the electron The moving particle radiates most strongly in a direction perpendicular to its

motion The radiation will be polarized along the direction of its motion


25/74

x

z

r

P

Intensity of the scattered beam due to an electron (I) at a point Psuch that r >>

2

2

42

4

0r

Sin

cm

eII

For a wave oscillating in z direction

For an polarized wave

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180 210 240 270 300 330 360

t

Cos(t)

The reason we are able to

neglect scattering from theprotons in the nucleus

The scattered rays are also plane polarized


26/74

2

2

42

4

0r

Sin

cm

eII

F l i d E is the measure of the amplitude of the wave


27/74

For an unpolarized wave E is the measure of the amplitude of the waveE2= Intensity

222

zy EEE zy III

000

2

240 2 4 2

y

Py y

SineI I

m c r

IPy= Intensity at point P due to Ey

IPz= Intensity at point P due to Ez

240 2 4 2

z

Pz z

SineI I

m c r

Total Intensity at point P due to Ey& Ez

2 240 2 4 2

y z

P

Sin SineI I

m c r


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2 240 2 4 2

y z

P

Sin SineI I

m c r

2 2 2 2 2 21 1 2y z y z y zSin Sin Cos Cos Cos Cos

2 2 2 1x y zCos Cos Cos Sum of the squares of the direction cosines =1

2 2 2 22 2 1 ( ) 1 ( )y z x xCos Cos Cos Cos Hence

24

0 2 4 2

1 ( )xP

CoseI I

m c r

2

4

0 2 4 21 (2 )

PCoseI I

m c r

In terms of 2


29/74

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180

2t

[Cos(2t)]^2

In general P could lie anywhere in 3D space For the specific case of Bragg scattering:

The incident direction IOThe diffracted beam direction OPThe trace of the scattering plane BBAre all coplanar

OP is constrained to be on the xz plane

x

z

r

P

2

2

2

42

4

0

2

r

Cos

cm

eII

F l i d E is the measure of the amplitude of the wave


30/74

For an unpolarized wave E is the measure of the amplitude of the waveE2= Intensity

222

zy EEE

zy III

000

2

242

4

02

2

42

4

0

12

rcm

eI

r

Sin

cm

eII yyPy

IPy= Intensity at point P due to Ey

IPz= Intensity at point P due to Ez

2

2

42

4

02

2

42

4

0

222

r

Cos

cm

eI

r

Sin

cm

eII zzPz

The zx plane is to the y direction: hence, = 90

1


31/74

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 30 60 90 120 150 180 210 240 270 300 330 360

t

Cos(t)

2

2

00

42

4 2

r

CosII

cm

eIII

zy

PzPyP

2

2

42

4

0 21

2 r

Cos

cm

eIIP

Scattered beam is not unpolarized

Forward and backward scattered intensity higher than at 90Scattered intensity minute fraction of the incident intensity

Very small number


32/74

Polarization factorComes into being as we usedunpolarized beam

2

212

42

4

2

0 Cos

cm

e

r

IIP

0

0.2

0.4

0.6

0.8

1

1.2

0 30 60 90 120 150 180 210 240 270 300 330 360

2t

(1+Cos(2t)^2)/2

B


33/74

B Scattering by an Atom

Scattering by an atom [Atomic number, (path difference suffered by scattering from each e, )]

Scattering by an atom [Z, (, )] Angle of scattering leads to path differences

In the forward direction all scattered waves are in phase

electronanbyscatteredwaveofAmplitudeatomanbyscatteredwaveofAmplitude

FactorScatteringAtomicf

f

)(Sin(1)

0.2 0.4 0.6 0.8 1.0

10

20

30

Schematic

)(Sin


34/74

Coherent scattering Incoherent (Compton) scattering

Z Sin() /

B S i b A


35/74

B Scattering by an Atom

BRUSH-UP

The conventional UC has lattice points as the vertices

There may or may not be atoms located at the lattice points

The shape of the UC is a parallelepiped (Greekparalllepipedon)in 3D

There may be additional atoms in the UC due to two reasons:

The chosen UC is non-primitive

The additional atoms may be part of the motif

C S tt i b th U it ll ( )


36/74

C Scattering by the Unit cell (uc)

Coherent ScatteringUnit Cell (UC) is representative of the crystal structureScattered waves from various atoms in the UC interfere to create the diffraction pattern

The wave scattered from the middle plane is out of phase with the onesscattered from top and bottom planes


37/74

d(h00)

B

Ray 1 = R1

Ray 2 = R2

Ray 3 = R3

Unit Cell

x

M

C

N

R

B

S

A

'

1R

'

2

R

'

3R

(h00) planea

a


38/74

h

adAC h 00

::::ACMCN

xABRBS ::::

ha

xx

AC

AB

)(2 0021 SindMCN hRR

ha

x

AC

ABRBSRR 31

2

axh

hax

RR 22

31 xcoordinatefractional

a

x xhRR 231

Extending to 3D 2 ( )h x k y l z Independent of the shape of UC

Note: R1is from corner atoms and R3is from atoms in additional positions in UC

2

[2 ( )]i i h x k y l zE A f

2 ( )h k l In complex notation


39/74

If atom B is different from atom A the amplitudes must be weighed by the respectiveatomic scattering factors (f)

The resultant amplitude of all the waves scattered by all the atoms in the UC gives thescattering factor for the unit cell The unit cell scattering factor is called theStructure Factor (F)

Scattering by an unit cell = f(position of the atoms, atomic scattering factors)

electronanbyscatteredwaveofAmplitudeucinatomsallbyscatteredwaveofAmplitudeFactorStructureF

[2 ( )]i i h x k y l z E Ae fe

2 ( )h x k y l z p

2FI

[2 ( )]

1 1

j j j j

n ni i h x k y l z hkl

n j j

j j

F f e f e

Structure factor is independent of theshapeandsize of the unit cell

For natoms in the UC

If the UC distorts so do the planes in it!!

nni)(S f l l i


40/74

nnie )1(

)(2

Cos

ee ii

Structure factor calculations

A Atom at (0,0,0) and equivalent positions

[2 ( )]j j j ji i h x k y l z

j jF f e f e

[2 ( 0 0 0)] 0i h k l F f e f e f

22 fF F is independent of the scattering plane (h k l)

nini ee

Simple Cubic

1)( inodde

1)( inevene


41/74

B Atom at (0,0,0) & (, , 0) and equivalent positions


j jF f e f e

1 1[2 ( 0)]

[2 ( 0 0 0)] 2 2

[ 2 ( )]0 ( )2

[1 ]

i h k l i h k l

h ki

i h k

F f e f e

f e f e f e

F is independent of the l index

C- centred Orthorhombic

Real

]1[ )( khi

efF

fF 2

0F

224fF

02 F

e.g. (001), (110), (112); (021), (022), (023)

e.g. (100), (101), (102); (031), (032), (033)


42/74

If the blue planes are scattering in phase then on C- centering the red planes will scatter out

of phase (with the blue planes- as they bisect them) and hence the (210) reflection will

become extinct

This analysis is consistent with the extinction rules: (h + k) odd is absent


43/74

In case of the (310) planes no new translationally equivalent planes are added on lattice

centering this reflection cannot go missing. This analysis is consistent with the extinction rules: (h + k) even is present

Body centred


44/74

C Atom at (0,0,0) & (, , ) and equivalent positions


j jF f e f e

1 1 1[2 ( )]

[2 ( 0 0 0)] 2 2 2

[ 2 ( )]0 ( )2

[1 ]

i h k l i h k l

h k li

i h k l

F f e f e

f e f e f e

Orthorhombic

Real

]1[ )( lkhi

efF

fF 2

0F

224fF

02 F

e.g. (110), (200), (211); (220), (022), (310)

e.g. (100), (001), (111); (210), (032), (133)

F C d C bi


45/74

D Atom at (0,0,0) & (, , 0) and equivalent positions


j jF f e f e

]1[)()()(

)]2

(2[)]2

(2[)]2

(2[)]0(2[

hlilkikhi

hli

lki

khi

i

eeef

eeeefF

Face Centred Cubic

Real

fF 4

0F

22 16fF

02 F

(h, k, l) unmixed

(h, k, l) mixed

e.g. (111), (200), (220), (333), (420)

e.g. (100), (211); (210), (032), (033)

(, , 0), (, 0, ), (0, , )

]1[ )()()( hlilkikhi eeefF

Two odd and one even (e.g. 112); two even and one odd (e.g. 122)

Mixed indices Two odd and one even (e g 112); two even and one odd (e g 122)


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Mixed indices CASE h k l

A o o e

B o e e

( ) ( ) ( )CASE A: [1 ] [1 1 1 1] 0i e i o i oe e e

( ) ( ) ( )CASE B: [1 ] [1 1 1 1] 0i o i e i oe e e

0F 02 F(h, k, l) mixed e.g. (100), (211); (210), (032), (033)

Mixed indices Two odd and one even (e.g. 112); two even and one odd (e.g. 122)

Unmixed indices CASE h k l

A o o o

B e e e

Unmixed indices

fF 4 22 16fF (h, k, l) unmixede.g. (111), (200), (220), (333), (420)

All odd (e.g. 111); all even (e.g. 222)

( ) ( ) ( )CASE A : [1 ] [1 1 1 1] 4i e i e i ee e e

( ) ( ) ( )CASE B: [1 ] [1 1 1 1] 4i e i e i ee e e

Na+ at (0 0 0) + Face Centering Translations ( 0) ( 0 ) (0 )


47/74

E Na at (0,0,0) + Face Centering Translations (, , 0), (, 0, ), (0, , )Clat (, 0, 0) + FCT (0, , 0), (0, 0, ), (, , )

)]2

(2[)]2(2[)]

2(2[)]

2(2[

)]2

(2[)]2

(2[)]2

(2[)]0(2[

lkhi

li

ki

hi

Cl

hli

lki

khi

i

Na

eeeef

eeeefF

][

]1[

)()()()(

)()()(

lkhilikihi

Cl

hlilkikhi

Na

eeeef

eeefF

]1[

]1[

)()()()(

)()()(

khihlilkilkhi

Cl

hlilkikhi

Na

eeeef

eeefF

]1][[

)()()()( hlilkikhilkhi

ClNa eeeeffF

NaCl:Face Centred Cubic

)()()()( hlilkikhilkhi


48/74

]1][[ )()()()( hlilkikhilkhiClNa

eeeeffF

Zero for mixed indices

Mixed indices CASE h k l

A o o e

B o e e

]2][1[ TermTermF

0]1111[]1[2:ACASE )()()( oioiei eeeTerm

0]1111[]1[2:BCASE )()()( oieioi eeeTerm

0F 02 F(h, k, l) mixed e.g. (100), (211); (210), (032), (033)

Mixed indices

Unmixed indices CASE h k li d i di


49/74

(h, k, l) unmixed ][4)( lkhi

ClNa

effF

][4 ClNa ffF If (h + k + l) is even22 ][16 ClNa ffF

][4 ClNa

ffF If (h + k + l) is odd22 ][16 ClNa ffF

e.g. (111), (222); (133), (244)

e.g. (222),(244)

e.g. (111), (133)

Unmixed indices CASE h k l

A o o o

B e e e

4]1111[]1[2:ACASE)()()(

eieiei

eeeTerm

4]1111[]1[2:BCASE )()()( eieiei eeeTerm

Unmixed indices


50/74

Presence of additional atoms/ions/molecules in the UC can alterthe intensities of some of the reflections

Selection / Extinction Rules


51/74

Bravais LatticeReflections whichmay bepresent

Reflectionsnecessarilyabsent

Simple all NoneBody centred (h + k + l) even (h + k + l) odd

Face centred h, k and l unmixed h, k and l mixed

End centredh and k unmixed

C centred

h and k mixed

C centred

Bravais Lattice Allowed Reflections

SC All

BCC (h + k + l) even

FCC h, k and l unmixed

DC

h, k and l are all oddOr

all are even& (h + k + l) divisible by 4

h2+ k2+ l2 SC FCC BCC DC


52/74

1 100

2 110 110

3 111 111 111

4 200 200 2005 210

6 211 211

7

8 220 220 220 220

9 300, 22110 310 310

11 311 311 311

12 222 222 222

13 320

14 321 321

15

16 400 400 400 400

17 410, 322

18 411, 330 411, 330

19 331 331 331

Crystal structure determination


53/74

y

Monochromatic X-rays

Panchromatic X-rays

Monochromatic X-rays

Many s (orientations)Powder specimen

POWDERMETHOD

Single LAUETECHNIQUE

Varied by rotation

ROTATINGCRYSTALMETHOD


54/74

POWDER METHOD


55/74

http://www.matter.org.uk/diffraction/x-ray/powder_method.htm

Diffraction cones and the Debye-Scherrer geometry

Film may be replaced with detector

POWDER METHOD

Different cones for different reflections


56/74

The 440 reflection is not observed


57/74

The 331 reflection is not observed

THE POWDER METHOD


58/74

THE POWDER METHOD

2222 sin)( lkh

2

2

2222

sin4

)( a

lkh

)(sin4

222

2

22 lkha

222

lkh

adCubic

dSin2

222

222 sin4

lkh

a

Cubic crystal

Relative Intensity of diffraction lines in a powder pattern


59/74

Structure Factor (F)

Multiplicity factor (p)

Polarization factor

Lorentz factor

RelativeIntensity of diffraction lines in a powder pattern

Absorption factor

Temperature factor

Scattering from UC

Number of equivalent scattering planes

Effect of wave polarization

Combination of 3 geometric factors

Specimen absorption

Thermal diffuse scattering

2

1

2

1

Sin

Cos

Sin

factorLorentz

21 2CosIP

Multiplicity factor


60/74

Lattice Index Multiplicity Planes

Cubic

(with highest

symmetry)

(100) 6 [(100) (010) (001)] (2 for negatives)

(110) 12 [(110) (101) (011), (110) (101) (011)] (2 for negatives)

(111) 12 [(111) (111) (111) (111)] (2 for negatives)

(210) 24*(210) 3! Ways, (210) 3! Ways,

(210) 3! Ways, (210) 3! Ways

(211) 24(211) 3 ways, (211) 3! ways,

(211) 3 ways

(321) 48*

Tetragonal

(with highestsymmetry)

(100) 4 [(100) (010)] (2 for negatives)

(110) 4 [(110) (110)] (2 for negatives)

(111) 8 [(111) (111) (111) (111)] (2 for negatives)

(210) 8*(210) = 2 Ways, (210) = 2 Ways,

(210) = 2 Ways, (210) = 2 Ways

(211) 16 [Same as for (210) = 8] 2 (as l can be +1 or 1)

(321) 16* Same as above (as last digit is anyhow not permuted)

* Altered in crystals with lower symmetry

Multiplicity factor


61/74

Cubichkl hhl hk0 hh0 hhh h00

48* 24 24* 12 8 6

Hexagonal hk.l hh.l h0.l hk.0 hh.0 h0.0 00.l24* 12* 12* 12* 6 6 2

Tetragonalhkl hhl h0l hk0 hh0 h00 00l

16* 8 8 8* 4 4 2

Orthorhombichkl hk0 h0l 0kl h00 0k0 00l

8 4 4 4 2 2 2

Monoclinichkl h0l 0k0

4 2 2

Triclinichkl

2

* Altered in crystals with lower symmetry (of the same crystal class)

Polarization factor Lorentz factor


62/74

0

5

10

15

20

25

30

0 20 40 60 80

Bragg Angle ( , degrees)

Lorentz-Polarization

factor

2

1

2

1

SinCos

SinfactorLorentz 21 2CosIP

CosSin

CosfactoronPolarizatiLorentz

2

2 21

Intensity of powder pattern lines (ignoring Temperature & Absorption factors)


63/74

Intensity of powder pattern lines (ignoring Temperature&Absorptionfactors)

CosSin

CospFI

2

22 21

Valid for Debye-Scherrer geometry

I Relative IntegratedIntensityF Structure factorp Multiplicity factor

POINTS

As one is interested in relative (integrated) intensities of the lines constant factorsare omittedVolume of specimen me, e (1/dectector radius)

Random orientation of crystals in a with Textureintensities are modified

Iis really diffracted energy(as Intensity is Energy/area/time)

Ignoring Temperature & Absorption factors valid for lines close-by in pattern

THE POWDER METHOD


64/74

THE POWDER METHOD

2222 sin)( lkh

2

2

2222

sin4

)( a

lkh

)(sin4

222

2

22 lkha

222

lkh

adCubic

dSin2

222

222 sin4

lkh

a

Cubic crystal


65/74

n 2 Intensity Sin Sin2 ratio

Determination of Crystal Structure from 2versus Intensity Data


66/74

2 Intensity Sin Sin2 ratio

1 21.5 0.366 0.134 3

2 25 0.422 0.178 4

3 37 0.60 0.362 8

4 45 0.707 0.500 115 47 0.731 0.535 12

6 58 0.848 0.719 16

7 68 0.927 0.859 19

FCC

h2+ k2+ l2 SC FCC BCC DC

1 100


67/74

1 100

2 110 110

3 111 111 111

4 200 200 200

5 210

6 211 211

7

8 220 220 220 220

9 300, 22110 310 310

11 311 311 311

12 222 222 222

13 320

14 321 32115

16 400 400 400 400

17 410, 322

18 411, 330 411, 330

19 331 331 331


68/74

The ratio of (h2+ K2+ l2) derived from extinction rules

SC 1 2 3 4 5 6 8

BCC 1 2 3 4 5 6 7

FCC 3 4 8 11 12

DC 3 8 11 16

Powder diffraction pattern from Al Radiation: Cu K, = 1.54056


69/74

420

111

200

220

311

222

4

00 3

31

422

1& 2peaks resolved

Note: Peaks or not idealized peaks broadened

Increasing splitting of peaks with g

Peaks are all not of same intensity

X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)

Determination of Crystal Structure from 2versus Intensity Data


70/74

n 2 Sin Sin2 ratio Index a (nm)1 38.52 19.26 0.33 0.11 3 111 0.404482 44.76 22.38 0.38 0.14 4 200 0.40457

3 65.14 32.57 0.54 0.29 8 220 0.40471

4 78.26 39.13 0.63 0.40 11 311 0.40480

5* 82.47 41.235 0.66 0.43 12 222 0.40480

6* 99.11 49.555 0.76 0.58 16 400 0.40485

7* 112.03 56.015 0.83 0.69 19 331 0.40491

8* 116.60 58.3 0.85 0.72 20 420 0.40491

9* 137.47 68.735 0.93 0.87 24 422 0.40494

* 1, 2peaks are resolved (1peaks are listed)

1

d


71/74

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 30 60 90

t

Sin(t)

Sind

2

22

)(

Sin

Cos

d

dd

Tan

dd

Sin

Cos

d

dd

)(

Error in d spacing

14


72/74

0

2

4

6

8

10

12

14

0 20 40 60 80 100

t

Cot(t)

Tan

dd

Sin

Cos

d

dd

)(

Error in d spacing

Error in d spacing decreases with

Applications of XRD


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Bravais lattice determination

Lattice parameter determination

Determination of solvus line in phase diagrams

Long range order

Crystallite size and Strain

More

y

CrystalS h ti f diff b t


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Diffraction angle (2)

Intensity

90 1800


Intensit

y

Liquid / Amorphous solid

90 180


Intensity

Monoatomic gas

Schematic of difference betweenthe diffraction patterns of various phases

7c x raydiffraction

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