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)

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

A accelerating charge radiates electromagnetic radiation

Inte

nsit

y

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)

Incoherent Scattering (Compton modified) From loosely bound charges

Here the particle picture of the electron & photon comes 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

Incoming X-raysSecondaryemission

Sets Electron cloud into oscillation Sets nucleus 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

A B

X Y

Atomic Planes

Extra path traveled by scattered waves XB

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

A B

X Y

But this is still reinforced scatteringand NOT reflection

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

8

220

ad

2110

ad

2

1

110

220 d

d

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

Crystal structure determination

Monochromatic X-rays

Panchromatic X-rays

Monochromatic X-rays

Many s (orientations)Powder specimen

POWDER METHOD

Single LAUETECHNIQUE

Varied by rotation

ROTATINGCRYSTALMETHOD

THE POWDER METHOD

2222

22

2222

222

222

222

sin)(

sin4

)(

sin4

2

lkh

alkh

lkh

a

lkhad

dSin

Intensity of the Scattered electrons

Electron Atom Unit cell (uc)Scattering by a crystal

A B C

Scattering by an Electron

),( 00 Sets electron into oscillation

Emission in all directions

Scattered beams),( 00 Coherent

(definite phase relationship)

A

x

z

r

P

Intensity of the scattered beam due to an electron (I)

2

2

42

4

0 r

Sin

cm

eII

For a wave oscillating in z direction

For an polarized wave

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

222zy EEE

zy III

000

2

242

4

02

2

42

4

0

12rcm

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

222r

Cos

cm

eI

r

Sin

cm

eII zzPz

2

200

42

4 2

r

CosII

cm

eIII zy

PzPyP

2

2

42

40 21

2 r

Cos

cm

eIIP

Scattered beam is not unpolarized

Polarization factorComes into being as we used unpolarized beam

2

21 2

42

4

20 Cos

cm

e

r

IIP

Rotational symmetry about x axis + mirror symmetry about yz plane Forward and backward scattered intensity higher than at 90 Scattered intensity minute fraction of the incident intensity

Very small number

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 Amplitude

atoman 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 scatteringIncoherent (Compton)

scattering

Z Sin() /

C Scattering by the Unit cell (uc)

Coherent Scattering Unit Cell (uc) 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

h

adAC h 00

:::: ACMCN

xABRBS ::::

haxx

AC

AB

)(2 0021SindMCN hRR

h

ax

AC

ABRBSRR

31

2

a

xh

hax

RR 2

231

xcoordinatefractionala

x xhRR 231

Extending to 3D )(2 zhykxh Independent of the shape of uc

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

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 Amplitude

ucin atoms allby scattered waveof AmplitudeFactor StructureF

)](2[ zhykxhii feAeE )(2 zhykxh In complex notation

2FI

)](2[

11

jjjj zhykxhin

jj

n

j

ij

hkln efefF

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

nnie )1(

)(2

Cosee ii

Structure factor calculations

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

)](2[ jjjj zhykxhij

ij efefF

fefefF hkhi 0)]000(2[

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[ jjjj zhykxhij

ij efefF

]1[ )()]2

(2[0

)]02

1

2

1(2[)]000(2[

khikh

i

hkhihkhi

efefef

efefF

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

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

)](2[ jjjj zhykxhij

ij efefF

]1[ )()]2

(2[0

)]2

1

2

1

2

1(2[)]000(2[

lkhilkh

i

hkhihkhi

efefef

efefF

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[ jjjj zhykxhij

ij efefF

]1[ )()()(

)]2

(2[)]2

(2[)]2

(2[)]0(2[

hlilkikhi

hli

lki

khii

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)

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[

lkh

il

ik

ih

i

Cl

hli

lki

khii

Na

eeeef

eeeefF

][

]1[)()()()(

)()()(

lkhilikihi

Cl

hlilkikhi

Na

eeeef

eeefF

]1[

]1[)()()()(

)()()(

khihlilkilkhi

Cl

hlilkikhi

Na

eeeef

eeefF

]1][[ )()()()( hlilkikhilkhi

ClNaeeeeffF

NaCl: Face Centred Cubic

(h, k, l) unmixed

]1][[ )()()()( hlilkikhilkhi

ClNaeeeeffF

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

Zero for mixed indices

][4 )( lkhi

ClNaeffF

][4 ClNa

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

ClNaffF

][4 ClNa

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

ClNaffF

Presence of additional atoms/ions/molecules in the uc (as a part of the motif ) can alter the intensities of some of the reflections

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

2

1

2

1

SinCos

SinfactorLorentz

21 2CosIP

Multiplicity factor

Lattice Index Multiplicity Planes

Cubic (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) 21

(321) 48

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

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

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

(210) 6

(211) 21

(321) 48

0

5

10

15

20

25

30

0 20 40 60 80

Bragg Angle (, degrees)

Lor

entz

-Pol

ariz

atio

n f

acto

r

Polarization factor Lorentz factor

2

1

2

1

SinCos

SinfactorLorentz 21 2CosIP

CosSin

CosfactoronPolarizatiLorentz

2

2 21

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

CosSin

CospFI

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

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

Reciprocal LatticeProperties are reciprocal to the crystal lattice

32*

1

1aa

Vb

13

*2

1aa

Vb

21

*3

1aa

Vb

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 OXMBArea

OXMBArea

aaV

bb

1

)(

)(

121

*3

*3

001

*3

1

db

The reciprocal lattice is created by interplanar spacings

** as written usuall ii ab

A reciprocal lattice vector is to the corresponding real lattice plane

*3

*2

*1

* blbkbhghkl

hklhklhkl d

gg1**

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

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 Lattice = SC

No missing reflections

Figures NOT to Scale

000

200

222

002

101

022

020110

BCC

Lattice = BCC

Reciprocal Lattice = FCC

220

011

202

100 missing reflection (F = 0)

22 4 fF

Weighing factor for each point “motif”

Figures NOT to Scale

000200

222

002022

020

FCC

Lattice = FCC

Reciprocal Lattice = BCC

220

111

202

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

22 16 fF

Weighing factor for each point “motif”

The Ewald* Sphere

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

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

01

10

02

00 20

2

(41)

Ki

KD

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

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 diffraction pattern from Al

420

111

200 22

0

311

222

400 33

1

422

1 & 2 peaks resolved

Radiation: Cu K, = 1.54 Å

Note: Peaks or not idealized peaks broadend Increasing splitting of peaks with g Peaks are all not of same intensity

n 2 Sin Sin2 ratio Index

1 38.52 19.26 0.33 0.11 3 111

2 44.76 22.38 0.38 0.14 4 200

3 65.14 32.57 0.54 0.29 8 220

4 78.26 39.13 0.63 0.40 11 311

5 82.47 41.235 0.66 0.43 12 222

6 99.11 49.555 0.76 0.58 16 400

7 112.03 56.015 0.83 0.69 19 331

8 116.60 58.3 0.85 0.72 20 420

9 137.47 68.735 0.93 0.87 24 422

Determination of Crystal Structure from 2 versus Intensity Data

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

Bravais LatticeReflections which

may be presentReflections

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 centredh and k unmixed

C centredh 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

Extinction Rules

n 2→ Intensity Sin Sin2 ratio

Determination of Crystal Structure from 2 versus Intensity Data

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 …

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

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.

Bravais lattice determination

Lattice parameter determination

Determination of solvus line in phase diagrams

Long range order

Applications of XRD

Crystallite size and Strain

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

SinCos

SinfactorLorentz

21 2CosIP

Diffraction angle (2) →

Inte

nsit

y →

90 1800

Crystal

90 1800

Diffraction angle (2) →

Inte

nsit

y → Liquid / Amorphous solid

90 1800

Diffraction angle (2) →

Inte

nsit

y →

Monoatomic gas

300 310

Schematic of difference between the diffraction patterns of various phases