X-RAY DIFFRACTIONX-RAY DIFFRACTION

X- Ray Sources Diffraction: Bragg’s Law Crystal Structure Determination

Elements of X-Ray DiffractionB.D. Cullity & S.R. Stock

Prentice Hall, Upper Saddle River (2001)

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

Plenum Press, New York (1998)

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

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

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

Beam of electrons Target X-rays

An accelerating (/decelerating) charge radiates electromagnetic radiation

Inte

nsity

Wavelength ()

Mo Target impacted by electrons accelerated by a 35 kV potential

0.2 0.6 1.0 1.4

White radiation

Characteristic radiation → due to energy transitions in the atom

K

K

Target Metal Of K radiation (Å)

Mo 0.71

Cu 1.54

Co 1.79

Fe 1.94

Cr 2.29

Heat

Incident X-rays

SPECIMEN

Transmitted beam

Fluorescent X-raysElectrons

Compton recoil PhotoelectronsScattered X-rays

CoherentFrom bound charges

Incoherent (Compton modified)From loosely bound charges

X-rays can also be refracted (refractive index slightly less than 1) and reflected (at very small angles) Refraction of X-rays is neglected for now.

Incoherent Scattering (Compton modified) From loosely bound charges

Here the particle picture of the electron & photon comes in handy

),( 11 Electron knocked aside

),( 22

11 hE

22 hE

)21(0243.012 Cos

2

No fixed phase relation between the incident and scattered wavesIncoherent does not contribute to diffraction

(Darkens the background of the diffraction patterns)

Vacuum

Energylevels

KE

1LE

2LE

3LE

Nucleus

K

1L

2L

3L

Characteristic x-rays(Fluorescent X-rays)

(10−16s later seems like scattering!)

Fluorescent X-raysKnocked out electron

from inner shell

A beam of X-rays directed at a crystal interacts with the electrons 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 coherent

The emission will undergo constructive or destructive interference with waves scattered from other atoms

Incoming X-raysSecondaryemission

Sets Electron cloud into oscillation

Sets nucleus (with protons) into oscillation

Small effect neglected

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

BRAGG’s EQUATION

d

dSin

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

For constructive interference: n = 2d Sin

Ray 1

Ray 2

Deviation = 2

Incident and scattered waves are in phase if

Scattering from across planes is in phase

In plane scattering is in phase

Extra path traveled by incoming waves AY

Extra path traveled by scattered waves XB

These can be in phase if and only if incident = scattered

But this is still reinforced scatteringand NOT reflection

Note that in the Bragg’s equation: The interatomic spacing (a) along the plane does not appear Only the interplanar spacing (d) appears Change in position or spacing of atoms along the plane should not affect

Bragg’s condition !!

d

Note: shift (systematic) is actually not a problem!

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 ] 2dn Sin Sin d SinSin

2n d Sin

Consider the case for which 1 2

Constructive interference can still occur if the difference in the path length traversed by R1 and R2 before and after scattering are an integral multiple of the wavelength (AY − XC) = h (h is an integer)

1Cosa

AY 2Cos

aXC

hCosaCosa 21

hCosCosa 21

Laue’s equations

S0 incoming 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

A physical picture of scattering leading to diffraction is embodied in Laue’s equations

Bragg’s 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’ (Bragg’s 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?”

Bragg’s equation is a negative law If Bragg’s eq. is NOT satisfied NO reflection can occur If Bragg’s eq. is satisfied reflection MAY 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

n = 2d Sin

n is an integer and is the order of the reflection

For Cu K radiation ( = 1.54 Å) and d110= 2.22 Å

n Sin

1 0.34 20.7º First order reflection from (110)

2 0.69 43.92ºSecond order reflection from (110)

Also written as (220)

222 lkh

adhkl

8220ad

2110ad

21

110

220 dd

sin2 hkldn

In XRD nth order reflection from (h k l) is considered as 1st order reflectionfrom (nh nk nl)

sin2n

dhkl

sin2 n n n lkhd

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)

Scattering by an Electron

),( 00 Sets electron into oscillation

Emission in ‘all’ directions

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

x

z

r

P

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

2

2

42

4

0 rSin

cmeII

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 the protons in the nucleus

The scattered rays are also plane polarized

2

2

42

4

0 rSin

cmeII

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

222zy EEE zy III

000

2

24

0 2 4 2y

Py y

SineI Im c r

IPy = Intensity at point P due to Ey

IPz = Intensity at point P due to Ez

24

0 2 4 2z

Pz z

SineI Im c r

Total Intensity at point P due to Ey & Ez

2 24

0 2 4 2y z

P

Sin SineI Im c r

2 24

0 2 4 2y z

P

Sin SineI Im 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 Im c r

24

0 2 4 2

1 (2 )P

CoseI Im c r

In terms of 2

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 BB’Are all coplanar

OP is constrained to be on the xz plane

x

z

r

P

2

2

2

42

4

02

rCos

cmeII

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

222zy EEE

zy III00

0

2

242

4

02

2

42

4

012rcm

eIr

Sin

cmeII 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

0222

rCos

cmeI

r

Sin

cmeII zzPz

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

-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

200

42

4 2rCosII

cmeIII zy

PzPyP

2

2

42

40 21

2 rCos

cmeIIP

Scattered beam is not unpolarized

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

Very small number

Polarization factorComes into being as we used unpolarized beam

221 2

42

4

20 Cos

cme

rIIP

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+C

os(2

t)^2)

/2

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

electronan by scattered waveof Amplitudeatoman by scattered waveof Amplitude

Factor Scattering Atomicf

f →

)(Sin

(Å−1) →0.2 0.4 0.6 0.8 1.0

10

20

30

Schematic

)(Sin

Coherent scattering Incoherent (Compton) scattering

Z Sin() /

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 (Greek parallēlepipedon) 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 Scattering by the Unit cell (uc) Coherent Scattering Unit Cell (UC) is representative of the crystal structure Scattered 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 ones scattered from top and bottom planes

d(h00)

B

Ray 1 = R1

Ray 2 = R2

Ray 3 = R3

Unit Cell

x

M

C

N

RB

S

A

'1R

'2R

'3R

(h00) planea

hadAC h 00

:::: ACMCNxABRBS ::::

haxx

ACAB

)(2 0021SindMCN hRR

h

ax

ACABRBSRR

31

2

axh

hax

RR 22

31 xcoordinatefractional

ax xhRR 2

31

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

Note: R1 is from corner atoms and R3 is from atoms in additional positions in UC

2

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

The resultant amplitude of all the waves scattered by all the atoms in the UC gives the scattering factor for the unit cell

The unit cell scattering factor is called the Structure Factor (F)

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

electronan by scattered waveof Amplitudeucin atoms allby scattered waveof AmplitudeFactor StructureF

[2 ( )]i i h x k y l zE Ae fe 2 ( )h x k y l z In complex notation

2FI

[2 ( )]

1 1

j j j jn n

i i h x k y l zhkln j j

j j

F f e f e

Structure factor is independent of the shape and size of the unit cell

For n atoms in the UC

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

nnie )1(

)(2

Cosee ii

Structure factor calculations

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

[2 ( )]j j j ji i h x k y l zj jF f e f e

[2 ( 0 0 0)] 0i h k lF 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

B Atom at (0,0,0) & (½, ½, 0) and equivalent positions

[2 ( )]j j j ji i h x k y l zj jF f e f e

1 1[2 ( 0)][2 ( 0 0 0)] 2 2

[ 2 ( )]0 ( )2 [1 ]

i h k li 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[ )( khiefF

fF 2

0F

22 4 fF

02 F

Both even or both odd

Mixture of odd and even

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

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

(h + k) even

(h + k) odd

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

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

C Atom at (0,0,0) & (½, ½, ½) and equivalent positions

[2 ( )]j j j ji i h x k y l zj jF f e f e

1 1 1[2 ( )][2 ( 0 0 0)] 2 2 2

[ 2 ( )]0 ( )2 [1 ]

i h k li h k l

h k li i h k l

F f e f e

f e f e f e

Body centred Orthorhombic

Real

]1[ )( lkhiefF

fF 2

0F

22 4 fF

02 F

(h + k + l) even

(h + k + l) odd

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

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

D Atom at (0,0,0) & (½, ½, 0) and equivalent positions

[2 ( )]j j j ji i h x k y l zj jF f e f e

]1[ )()()(

)]2

(2[)]2

(2[)]2

(2[)]0(2[

hlilkikhi

hlilkikhii

eeef

eeeefF

Face Centred Cubic

Real

fF 4

0F

22 16 fF

02 F

(h, k, l) unmixed

(h, k, l) mixed

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

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

(½, ½, 0), (½, 0, ½), (0, ½, ½)

]1[ )()()( hlilkikhi eeefF

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

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 16 fF (h, k, l) unmixed

e.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

E Na+ at (0,0,0) + Face Centering Translations (½, ½, 0), (½, 0, ½), (0, ½, ½) Cl− at (½, 0, 0) + FCT (0, ½, 0), (0, 0, ½), (½, ½, ½)

)]2

(2[)]2

(2[)]2

(2[)]2

(2[

)]2

(2[)]2

(2[)]2

(2[)]0(2[

lkhilikihi

Cl

hlilkikhiiNa

eeeef

eeeefF

][

]1[)()()()(

)()()(

lkhilikihiCl

hlilkikhiNa

eeeef

eeefF

]1[

]1[)()()()(

)()()(

khihlilkilkhiCl

hlilkikhiNa

eeeef

eeefF

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

eeeeffF

NaCl: Face Centred Cubic

]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

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

effF

][4 ClNa

ffF If (h + k + l) is even 22 ][16 ClNa

ffF

][4 ClNa

ffF If (h + k + l) is odd 22 ][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

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

Bravais Lattice Reflections which may be present

Reflections necessarily absent

Simple all None

Body centred (h + k + l) even (h + k + l) odd

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

End centred h and k unmixed C centred

h and k mixedC centred

Bravais Lattice Allowed ReflectionsSC All

BCC (h + k + l) evenFCC h, k and l unmixed

DC

h, k and l are all oddOr

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

Selection / Extinction Rules

h2 + k2 + l2 SC FCC BCC DC

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, 221

10 310 310

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

Reciprocal LatticeProperties are reciprocal to the crystal lattice

32*

11 aaV

b

13*2

1 aaV

b

21*

31 aaV

b

B

O

P

M

A

C

B

O

P

M

A

C

O

P

M

A

C

O

P

M

A

C

O

P

M

A

C

*b3

2a

1a

3a

OPCellHeight of OAMBAreaOAMBArea

aaV

bb

1)(

)(

121

*3

*3

001

*3

1d

b

The reciprocal lattice is created by interplanar spacings

** as written usuall ii ab

B

BASIS VECTORS

21*

3 to is aandab

A reciprocal lattice vector is to the corresponding real lattice plane

*3

*2

*1

* blbkbhghkl

hklhklhkl d

gg 1**

The length of a reciprocal lattice vector is the reciprocal of the spacing of the corresponding real lattice plane

Planes in the crystal become lattice points in the reciprocal lattice ALTERNATE CONSTRUCTION OF THE REAL LATTICE

Reciprocal lattice point represents the orientation and spacing of a set of planes

Reciprocal Lattice

(01)

(10)(11)

(21)

10 20

11

221202

01 21

00

The reciprocal lattice has an origin!

1a

2a

1a1

1a

*11g *

21g*b2

*b1

1020

11

2212

02

01

21

00

(01)

(10)(11)

(21)

1a

2a

*b2

*b1

1a

(01)

(10)(11)

(21) Note perpendicularity of various vectors

Reciprocal lattice is the reciprocal of a primitive lattice and is purely geometrical does not deal with the intensities of the points

Physics comes in from the following: For non-primitive cells ( lattices with additional points) and for crystals

decorated with motifs ( crystal = lattice + motif) the Reciprocal lattice points have to be weighed in with the corresponding scattering power (|Fhkl|2) Some of the Reciprocal lattice points go missing (or may be scaled up or down in intensity) Making of Reciprocal Crystal (Reciprocal lattice decorated with a motif of scattering power)

The Ewald sphere construction further can select those points which are actually observed in a diffraction experiment

In crystals based on a particular lattice the intensities of particular reflections are modified they may even go missing

Crystal = Lattice + Motif

Diffraction Pattern

Position of the Lattice points LATTICE

Intensity of the diffraction spots ‘MOTIF’

There are two ways of constructing the Reciprocal Crystal:

1) Construct the lattice and decorate each lattice point with appropriate intensity

2) Use the concept as that for the real crystal

Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)

Figures NOT to Scale

000

100

111

001

101

011

010

110

SC

Lattice = SC

Reciprocal Crystal = SCNo missing reflections

Figures NOT to Scale

000

200

222

002

101

022

020110

BCC

Lattice = BCC

Reciprocal Crystal = FCC

220

011

202

100 missing reflection (F = 0)

22 4 fF

Weighing factor for each point “motif”

Figures NOT to Scale

000200

222

002 022

020

FCC

Lattice = FCC

Reciprocal Crystal = BCC

220

111

202

100 missing reflection (F = 0)110 missing reflection (F = 0)

22 16 fF

Weighing factor for each point “motif”

Ordered Solid solution

G = H TS

High T disordered

Low T ordered

470ºC

Sublattice-1

Sublattice-2

BCC

SC

In a strict sense this is not a crystal !!

Disordered Ordered- NiAl, BCC B2 (CsCl type)- Ni3Al, FCC L12 (AuCu3-I type)

BCC SC

BCCFCC

FCC

SC

Ordered

Ordered

There are two ways of constructing the Reciprocal Crystal:

1) Construct the lattice and decorate each lattice point with appropriate intensity

2) Use the concept as that for the real crystal

1) SC + two kinds of Intensities decorating the lattice

2) (FCC) + (Motif = 1FR + 1SLR)

1) SC + two kinds of Intensities decorating the lattice

2) (BCC) + (Motif = 1FR + 3SLR)

FR Fundamental Reflection SLR Superlattice Reflection

The Ewald Sphere* Paul Peter Ewald (German physicist and crystallographer; 1888-1985)

organisiert von:Max-Planck-Institut für MetallforschungInstitut für Theoretische und Angewandte Physik,Institut für Metallkunde,Institut für Nichtmetallische Anorganische Materialiender Universität Stuttgart Programm

13:30 Joachim Spatz (Max-Planck-Institut für Metallforschung) Begrüßung

13:45 Heribert Knorr (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg Begrüßung

14:00 Stefan Hell (Max-Planck-Institut für Biophysikalische Chemie)Nano-Auflösung mit fokussiertem Licht

14:30 Antoni Tomsia (Lawrence Berkeley National Laboratory)Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone

15:00 Pause Kaffee und Getränke

15:30 Frank Gießelmann(Universität Stuttgart) Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanoröhren: Aktuelle Themen der Flüssigkristallforschung

16:00 Verleihung des Günter-Petzow-Preises 2008

16:15 Udo Welzel (Max-Planck-Institut für Metallforschung) Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind

ab 17:00 Sommerfest des Max-Planck-Instituts für Metallforschung

7. Paul-Peter-Ewald-Kolloquium

Freitag, 17. Juli 2008

The Ewald Sphere

The reciprocal lattice points are the values of momentum transfer for which the Bragg’s equation is satisfied

For diffraction to occur the scattering vector must be equal to a reciprocal lattice vector

Geometrically if the origin of reciprocal space is placed at the tip of ki then diffraction will occur only for those reciprocal lattice points that lie on the surface of the Ewald sphere

See Cullity’s book: A15-4

hklhkl Sindn 2

2

12 hkl

hklhkl

dd

Sin

Draw a circle with diameter 2/ Construct a triangle with the diameter as the hypotenuse and 1/dhkl as a side (any

triangle inscribed in a circle with the diameter as the hypotenuse is a right angle triangle): AOP

The angle opposite the 1/d side is hkl (from the rewritten Bragg’s equation)

Bragg’s equation revisited

hklhklhkl d

gg 1**

212 hkl

hklhkl

dd

Sin

Radiation related information is present in the Ewald Sphere

Crystal related information is present in the reciprocal crystal

The Ewald sphere construction generates the diffraction pattern

The Ewald Sphere construction

01

10

02

00 20

2

(41)

KiKD

K

Reciprocal Space

K = K =g = Diffraction Vector

Ewald Sphere

The Ewald Sphere touches the reciprocal lattice (for point 41)

Bragg’s equation is satisfied for 41

(Cu K) = 1.54 Å, 1/ = 0.65 Å−1 (2/ = 1.3 Å−1), aAl = 4.05 Å, d111 = 2.34 Å, 1/d111 = 0.43 Å−1

Ewald sphere X-rays

Crystal structure determination

Monochromatic X-rays

Panchromatic X-rays

Monochromatic X-rays

Many s (orientations)Powder specimen

POWDER METHOD

Single LAUETECHNIQUE

Varied by rotationROTATINGCRYSTALMETHOD

THE POWDER METHOD

Cone of diffracted rays

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

The 440 reflection is not observed

The 331 reflection is not observed

THE POWDER METHOD

2222 sin)( lkh

22

2222 sin4)( alkh

)(sin4

2222

22 lkha

222 lkhadCubic

dSin2

222

222 sin4

lkha

Cubic crystal

Structure Factor (F)

Multiplicity factor (p)

Polarization factor

Lorentz factor

Relative Intensity 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

21

21

SinCos

SinfactorLorentz

21 2CosIP

Multiplicity factor

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 highest

symmetry)

(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

Cubichkl hhl hk0 hh0 hhh h0048* 24 24* 12 8 6

Hexagonalhk.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 00l16* 8 8 8* 4 4 2

Orthorhombichkl hk0 h0l 0kl h00 0k0 00l8 4 4 4 2 2 2

Monoclinichkl h0l 0k04 2 2

Triclinichkl2

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

Multiplicity factor

0

5

10

15

20

25

30

0 20 40 60 80

Bragg Angle (, degrees)

Lor

entz

-Pol

ariz

atio

n fa

ctor

Polarization factor Lorentz factor

21

21

SinCos

SinfactorLorentz 21 2CosIP

CosSinCosfactoronPolarizatiLorentz 2

2 21

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

CosSinCospFI 2

22 21

Valid for Debye-Scherrer geometry I → Relative Integrated “Intensity” F → Structure factor p → Multiplicity factor

POINTS As one is interested in relative (integrated) intensities of the lines constant factors

are omitted Volume of specimen me , e (1/dectector radius)

Random orientation of crystals in a with Texture intensities are modified I is really diffracted energy (as Intensity is Energy/area/time) Ignoring Temperature & Absorption factors valid for lines close-by in pattern

THE POWDER METHOD

2222 sin)( lkh

22

2222 sin4)( alkh

)(sin4

2222

22 lkha

222 lkhadCubic

dSin2

222

222 sin4

lkha

Cubic crystal

n 2→ Intensity Sin Sin2 ratio

Determination of Crystal Structure from 2 versus Intensity Data

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 11

5 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

2 110 110

3 111 111 111

4 200 200 200

5 210

6 211 211

7

8 220 220 220 220

9 300, 221

10 310 310

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

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

420

111

200 22

0

311

222

400 33

1

422

1 & 2 peaks resolved

Radiation: Cu K, = 1.54056 Å

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)

0

2

4

6

8

10

12

14

0 30 60 90t

1/Cos

(t)

2n d Sin

2d d Cosd

12

dd d Cos

Actually, the variation in 2 is to be seen

n 2 Sin Sin2 ratio Index a (nm)

1 38.52 19.26 0.33 0.11 3 111 0.40448

2 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

Determination of Crystal Structure from 2 versus Intensity Data

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

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 30 60 90

t

Sin(t)

For the same the error in Sin with

Sind

2

22)(

SinCos

ddd

Tandd

SinCos

ddd

)(

Error in d spacing

0

2

4

6

8

10

12

14

0 20 40 60 80 100

t

Cot

(t)

Tandd

SinCos

ddd

)(

Error in d spacing

Error in d spacing decreases with

Bravais lattice determination

Lattice parameter determination

Determination of solvus line in phase diagrams

Long range order

Applications of XRD

Crystallite size and Strain

More

Diffraction angle (2) →

Inte

nsity

→

90 1800

Crystal

90 1800

Diffraction angle (2) →

Inte

nsity

→

Liquid / Amorphous solid

90 1800

Diffraction angle (2) →

Inte

nsity

→

Monoatomic gas

Schematic of difference between the diffraction patterns of various phases

Crystallite size and Strain

Bragg’s equation assumes: Crystal is perfect and infinite Incident beam is perfectly parallel and monochromatic

Actual experimental conditions are different from these leading various kinds of deviations from Bragg’s condition Peaks are not ‘’ curves Peaks are broadened

There are also deviations from the assumptions involved in the generating powder patterns Crystals may not be randomly oriented (textured sample) Peak intensities are altered

In a powder sample if the crystallite size < 0.5 m there are insufficient number of planes to build up a sharp diffraction pattern peaks are broadened

XRD Line Broadening

Instrumental

Crystallite size

Strain

Stacking fault

XRD Line Broadening

Other defects

Unresolved 1 , 2 peaks Non-monochromaticity of the source (finite width of peak) Imperfect focusing

In the vicinity of B the −ve of Bragg’s equation not being satisfied

‘Residual Strain’ arising from dislocations, coherent precipitates etc. leading to broadening

In principle every defect contributes to some broadening

Bi

Bc

Bs

...)( SFsci BBBBFWHMB

...)( SFsci BBBBFWHMB

Crystallite size

Size > 10 m Spotty ring

(no. of grains in the irradiated portion insufficient to produce a ring)

Size (10, 0.5) Smooth continuous ring pattern

Size (0.5, 0.1) Rings are broadened

Size < 0.1 No ring pattern

(irradiated volume too small to produce a diffraction ring pattern &

diffraction occurs only at low angles)Spotty ring

Rings

Broadened RingsDiffuse

Effect of crystallite size on SAD patterns

Single crystal

“Spotty” pattern

Few crystals in the selected region

Effect of crystallite size on SAD patterns

Ring patternBroadened Rings

Subtracting Instrumental Broadening

Instrumental broadening has to be subtracted to get the broadening effects due to the sample

1 Mix specimen with known coarse-grained (~ 10m), well annealed (strain free)

does not give any broadening due to strain or crystallite size (the only broadening is instrumental). A brittle material which can be

ground into powder form without leading to much stored strain is good. If the pattern of the test sample (standard) is recorded separately then the

experimental conditions should be identical (it is preferable that one or more peaks of the standard lies close to the specimen’s peaks)

2 Use the same material as the standard as the specimen to be X-rayed but with

large grain size and well annealed

rsci BBBBB

...)( SFsci BBBBFWHMB

For a peak with a Lorentzian profile

222ir BBB For a peak with a Gaussian profile

222 )( iir BBBBB A geometric mean can also used

Longer tail

Johann Carl Friedrich Gauss (1777-1855), painted by Christian Albrecht Jensen

http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Hendrik Antoon LorentzOn the theory of reflection and refraction of light

University of Göttingen

Scherrer’s formula

( )cB

kBLCos

→ Wavelength L → Average crystallite size ( to surface of specimen) k → 0.94 [k (0.89, 1.39)]

~ 1 (the accuracy of the method is only 10%)

For Gaussian line profiles and cubic crystals

0

2

4

6

8

10

12

14

0 30 60 90t

1/Cos

(t)

Strain broadening

( )s BB Tan

→ Strain in the material

Smaller angle peaksshould be used to separate Bs and Bc

0

2

4

6

8

10

12

14

0 20 40 60 80 100

t

Tan(

t)

Separating crystallite size broadening and strain broadening

scr BBB )(

CosLkBc )( TanBs

)()(

TanCosLkBr

)()( SinL

kCosBr

Plot of [Br Cos] vs [Sin]

Example of a calculation

Sample: Annealed AlRadiation: Cu k ( = 1.54 Å)

Sample: Cold-worked AlRadiation: Cu k ( = 1.54 Å)

2 →

Inte

nsity

→

2 →

Inte

nsity

→40 60

40 60X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)

Annealed Al

Peak No. 2 () hkl Bi = FWHM () Bi = FWHM (rad)

1 38.52 111 0.103 1.8 10−3

2 44.76 200 0.066 1.2 10−3

3 65.13 220 0.089 1.6 10−3

Cold-worked Al

2 () Sin() hkl B () B (rad) Br Cos (rad)

1 38.51 0.3298 111 0.187 3.3 10−3 2.8 10−3 2.6 10−3

2 44.77 0.3808 200 0.206 3.6 10−3 3.4 10−3 3.1 10−3

3 65.15 0.5384 220 0.271 4.7 10−3 4.4 10−3 3.7 10−3

222ir BBB

3107.1 L

k nmLSizeGrain 90)(

end

Iso-intensity circle

Extinction Rules

Structure Factor (F): The resultant wave scattered by all atoms of the unit cell

The Structure Factor is independent of the shape and size of the unit cell; but is dependent on the position of the atoms within the cell

Structure factor calculation

Consider a general unit cell for this type of structure. It can be reduced to 4 atoms of type A at 000, 0 ½ ½, ½ 0 ½, ½ ½ 0 i.e. in the fcc position and 4 atoms of type B at the sites ¼ ¼ ¼ from the A sites. This can be expressed as:

The structure factors for this structure are:

F = 0 if h, k, l mixed (just like fcc)

F = 4(fA ± ifB) if h, k, l all odd

F = 4(fA - fB) if h, k, l all even and h+ k+ l = 2n where n=odd (e.g. 200)

F = 4(fA + fB) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)

Consider the compound ZnS (sphalerite). Sulphur atoms occupy fcc sites with zinc atoms displaced by ¼ ¼ ¼ from these sites. Click on the animation opposite to show this structure. The unit cell can be reduced to four atoms of sulphur and 4 atoms of zinc.

Many important compounds adopt this structure. Examples include ZnS, GaAs, InSb, InP and (AlGa)As. Diamond also has this structure, with C atoms replacing all the Zn and S atoms. Important semiconductor materials silicon and germanium have the same structure as diamond.

421 missing

Ewald sphere X-rays

(Cu K) = 1.54 Å, 1/ = 0.65 Å−1, aCu = 3.61 Å, 1/aCu = 0.28 Å−1

0.28 Å 1

0.65 Å 1

Multiplicity factor

Lattice Index Multiplicity Planes

Cubicwith highest

symmetry

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

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

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

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

(211) 24

(321) 48

Tetragonal (100) 4 [(100) (010)]

(110) 4 [(110) (110)]

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

(210) 6

(211) 24

(321) 48