X-RAY DIFFRACTION

X- Ray Sources
Diffraction: Braggs Law
Crystal Structure Determination

Elements of X-Ray Diffraction

B.D. Cullity & S.R. Stock
Prentice Hall, Upper Saddle River (2001)
X-Ray Diffraction: A Practical Approach
C. 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 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

Mo Target impacted by electrons accelerated by a 35 kV potential

Intensity

K
White
radiation

0.2

0.6

Characteristic radiation
due to energy transitions
in the atom

1.0

Wavelength ()

1.4

Target Metal

Of K radiation ()

Mo

0.71

Cu

1.54

Co

1.79

Fe

1.94

Cr

2.29

Incident X-rays

SPECIMEN

Fluorescent X-rays

Heat

Electrons

Scattered X-rays

Compton recoil

Photoelectrons

Coherent
Incoherent (Compton modified)
From bound charges
From loosely bound charges

Transmitted beam
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

E1 h 1

(1 , 1 )

Electron knocked aside

E2 h 2

(2 , 2 )

2 1 0.0243(1 Cos 2 )
No fixed phase relation between the incident and scattered waves
Incoherent does not contribute to diffraction
(Darkens the background of the diffraction patterns)

Fluorescent X-rays

Knocked out electron

from inner shell

Vacuum

Energy
E
levels L

L3

E L2

L2

E L1

L1

EK

Nucleus

Characteristic x-rays
(Fluorescent X-rays)
(1016s later seems like scattering!)

 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-rays

Secondary
emission

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

BRAGGs EQUATION

Deviation = 2
Ray 1

Ray 2

dS
in

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

For constructive interference: n = 2d Sin

In plane scattering is in phase

Incident and scattered
waves are in phase if
Scattering from across planes 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 scattering

and NOT reflection

Note that in the Braggs 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
Braggs condition !!
Note: shift (systematic) is
actually not a problem!
d

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

n AY YB

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

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

n

d
[2 Sin Sin ] 2 d Sin
Sin

n 2 d Sin

d
Sin( )
XY

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)
AY
Cos 1
a

XC
Cos 2
a

a Cos 1 a Cos 2 h
a Cos 1 Cos 2 h

Generalizing into 3D

a Cos 1 Cos 2 h
b Cos 3 Cos 4 k
c Cos 5 Cos 6 l

Laues equations
S0 incoming X-ray beam
S Scattered X-ray beam

a (S S0 ) h

b (S S0 ) k

c (S S0 ) l

This is looking at diffraction from atomic arrays and not planes

 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?

 Braggs equation is a negative law

If Braggs eq. is NOT satisfied NO reflection can occur
If Braggs 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 ) and d110= 2.22
n

Sin

0.34

20.7

First order reflection from (110)

43.92

Second order reflection from (110)

Also written as (220)

d hkl

0.69

a
h2 k 2 l 2

a
d 220
8
a
d110
2

d 220 1

d110 2

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

from (nh nk nl)

n 2d hkl sin
d hkl
2
sin
n

2d nh nk nl sin

Intensity of the Scattered electrons

Scattering by a crystal

A
Electron Polarization factor

B
Atom Atomic scattering factor (f)

C
Unit cell (uc) Structure factor (F)

A Scattering by an Electron

(0 , 0 )

Em
issi

on
in

Sets electron into oscillation

Coherent
(definite phase relationship)

all
di
rec
tion
s

(0 , 0 )

Scattered beams

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

For an polarized wave

For a wave oscillating in z direction

P
r
x

Intensity of the scattered beam due to an electron (I) at a point P

such that r >>

e 4 Sin 2

I I 0 2 4
2
mc r

The reason we are able to

neglect scattering from the
protons in the nucleus
The scattered rays are also plane polarized

e 4 Sin 2

I I 0 2 4
2
mc r

For an unpolarized wave

E is the measure of the amplitude of the wave

E2 = Intensity

E E E
2

2
y

2
z

I0
I0 y I0z
2

IPy = Intensity at point P due to Ey

I Py I 0 y

2
e 4 Sin y

2 4
2

mc
r

IPz = Intensity at point P due to Ez

I Pz

2
e 4 Sin z
I0z 2 4

mc
r

Total Intensity at point P due to Ey & Ez

2
2
e 4 Sin y Sin z

I P I0 2 4
2

mc
r

2
2
e 4 Sin y Sin z

I P I0 2 4
2

mc
r

Sin 2 y Sin 2 z
1 Cos 2 y 1 Cos 2 z 2 Cos 2 y Cos 2 z

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

Hence

1 Cos 2 (x )
2 Cos 2 y Cos 2 z 2 1 Cos 2 (x )

Cos
(x )

I P I0 2 4
2

mc
r

In terms of 2
2
e 4 1 Cos (2 )

I P I0 2 4
2

mc
r

e 4 Cos 2 2
I I 0 2 4
mc
r2

P
r
2

In general P could lie anywhere in 3D space

For the specific case of Bragg scattering:
The incident direction IO
The diffracted beam direction OP
The trace of the scattering plane BB
Are all coplanar
OP is constrained to be on the xz plane

E is the measure of the amplitude of the wave

E2 = Intensity

For an unpolarized wave

E 2 E y2 E z2
I0
I0 y I0z
2
IPy = Intensity at point P due to Ey
I Py I 0 y

Sin 2

e
m2c 4

I0 y

e4 1
2 4
2
mc r

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

IPz = Intensity at point P due to Ez

I Pz I 0 z

Sin
2
e
2
m 2c 4
r2

I0z

e 4 Cos 2 2

2 4
2
mc
r

I P I Py I Pz

I 0 y I 0 z Cos 2 2

e
2 4
m c

I 0 e 4 1 Cos 2 2

IP
2 4
2
2 mc
r

r2

Scattered beam is not unpolarized

Very small number

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

I 0 e 4 1 Cos 2 2

I P 2 2 4
r mc
2

Polarization factor
Comes into being as we used
unpolarized beam

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

Sin( )

f Atomic Scattering Factor

Amplitude of wave scattered by an atom

Amplitude of wave scattered by an electron

30

20

Schematic

10

0.2

0.4

0.6

0.8

Sin( )
(1)

1.0

Z
Sin() /

Coherent scattering

Incoherent (Compton)
scattering

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 paralllepipedon) 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

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

R1'

Ray 1 = R1
Ray 3 = R3

B
M

(h00) plane

R
Ray 2 = R2

R3'

R2'
N

C
Unit Cell

d(h00)

AC d h 00

a
h

MCN :: AC ::
RBS :: AB :: x
AB x
x

AC a
h

R1R2 MCN 2d h 00 Sin( )

R1R3 RBS
R1R3

AB
x

a
AC
h

2 x
x
2 h
a
a
h

Extending to 3D

x
fractional coordinate x
a

2 ( h x k y l z )

R1R3 2 h x

Independent of the shape of UC

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

 2 ( h x k y l z )

In complex notation

E Aei fei[2 ( h x k y l z )]

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)
F Structure Factor

hkl
n

fj e
j 1

Amplitude of wave scattered by all atoms in uc

Amplitude of wave scattered by an electron
i j

fj e

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

I F2

For n atoms in the UC

j 1

Structure factor is independent of the shape and size of the unit cell
If the UC distorts so do the planes in it!!

Structure factor calculations

A

Simple Cubic

ni

(1)

e ( odd n ) i 1

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

e ( even n ) i 1

e ni e ni
e i e i
Cos ( )
2

F fj e
F f
F2 f 2

i j

fj e

ei[2 ( h 0 k 0

l 0)]

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

f e0 f

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

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

F fj e

i j

fj e

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

F f ei[2 ( h 0 k 0

l 0)]

i[ 2 (

h k
)]
2

f e0 f e

C- centred Orthorhombic

f e

1
1
i [2 ( h k l 0)]

2
2

f [1 ei ( h k ) ]
Real

v en
(h + k) e
oth odd
b
r
o
n
e
Both ev

F f [1 e i ( h k ) ] Mixture

of odd a
nd e ve n
(h + k)
odd

F 2f

F2 4 f 2

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

F 0

F2 0

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

F is independent of the l index

 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

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

F fj e

i j

fj e

F f ei[2 ( h 0 k 0
f e0 f e

i[ 2 (

Body centred
Orthorhombic

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

l 0)]

f e

h k l
)]
2

1
1
1
i[2 ( h k l )]
2
2
2

f [1 ei ( h k l ) ]
Real

F f [1 e i ( h k l ) ]

l)
(h + k +
(h + k +

e ve n

l ) odd

F 2f

F2 4 f 2

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

F 0

F2 0

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

Face Centred Cubic

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

F fj e

i j

fj e

i [ 2 ( 0 )]

i [ 2 (

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

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

hk
)]
2

i [ 2 (

k l
)]
2

i [ 2 (

l h
)]
2

F fe
e
e
e

f [1 e i ( h k ) e i ( k l ) e i ( l h ) ]
Real
F f [1 e i ( h k ) e i ( k l ) e i ( l h ) ]
(h, k, l) unmixed

F 4f

F 2 16 f 2
e.g. (111), (200), (220), (333), (420)

(h, k, l) mixed

F 0

F2 0
e.g. (100), (211); (210), (032), (033)

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)

Mixed indices

CASE

CASE A : [1 ei ( e ) ei ( o ) ei ( o ) ] [1 1 1 1] 0
CASE B : [1 ei ( o ) ei (e ) ei ( o ) ] [1 1 1 1] 0
(h, k, l) mixed
Unmixed indices

F 0

F 0
2

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

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

Unmixed indices CASE

CASE A : [1 ei ( e ) ei ( e ) ei ( e ) ] [1 1 1 1] 4
CASE B : [1 ei ( e ) ei ( e ) ei ( e ) ] [1 1 1 1] 4
(h, k, l) unmixed

F 4f

F 2 16 f 2
e.g. (111), (200), (220), (333), (420)

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

Cl at (, 0, 0) + FCT (0, , 0), (0, 0, ), (, , )
NaCl:

i [ 2 (

hk
)]
2

i [ 2 (

k l
)]
2

i [ 2 (

l h
)]
2

Face Centred Cubic

F f Na e i[ 2 ( 0)] e
e
e

i[ 2 ( h2 )] i[ 2 ( k2 )] i[ 2 ( 2l )] i[ 2 ( h 2k l )]
f Cl e
e
e
e

F f Na [1 e i ( h k ) e i ( k l ) e i ( l h ) ]
f Cl [e i ( h ) e i ( k ) e i ( l ) e i ( h k l ) ]
F f Na [1 e i ( h k ) e i ( k l ) e i ( l h ) ]
f Cl e i ( h k l ) [e i ( k l ) e i ( l h ) e i ( h k ) 1]
F [ f Na f Cl e i ( h k l ) ][1 e i ( h k ) e i ( k l ) e i ( l h ) ]

F [ f Na f Cl e i ( h k l ) ][1 e i ( h k ) e i ( k l ) e i ( l h ) ]
F [Term 1][Term 2]
Mixed indices

Mixed indices

Zero for mixed indices

CASE

CASE A : Term 2 [1 e i ( e ) e i ( o ) e i ( o ) ] [1 1 1 1] 0
CASE B : Term 2 [1 e i ( o ) e i ( e ) e i ( o ) ] [1 1 1 1] 0
(h, k, l) mixed

F 0

F2 0

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

Unmixed indices

Unmixed indices CASE

CASE A : Term 2 [1 e i ( e ) e i ( e ) e i ( e ) ] [1 1 1 1] 4
CASE B : Term 2 [1 e i ( e ) e i ( e ) e i ( e ) ] [1 1 1 1] 4

(h, k, l) unmixed

F 4[ f Na f Cl e i ( h k l ) ]
e.g. (111), (222); (133), (244)

F 4[ f Na f Cl ]

If (h + k + l) is even

F 2 16[ f Na f Cl ]2
e.g. (222),(244)

F 4[ f Na f Cl ]

If (h + k + l) is odd

F 2 16[ f Na f Cl ]2
e.g. (111), (133)

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

the intensities of some of the reflections

Selection / Extinction Rules

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 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 odd

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

h2 + k2 + l2

SC

FCC

100

110

111

111

200

200

210

211

BCC

DC

110
111
200
211

7
8

220

220

300, 221

10

310

11

311

311

12

222

222

13

320

14

321

220

220

310
311
222
321

15
16

400

17

410, 322

18

411, 330

19

331

400

400

400

411, 330
331

331

Reciprocal Lattice
BASIS
VECTORS

Properties are reciprocal to the crystal lattice

* 1
b1 a2 a3
V

*
*
bi usuall written as ai

* 1
b2 a3 a1
V
*
b3

* 1
b b3 a1 a2
V
Area (OAMB)
1

Area (OAMB) Height of Cell OP

*
3

1
b
d 001
*
3

* 1
b3 a1 a2
V

a3

a2

B
B

b3 is to a1 and a2
O

The reciprocal lattice is created by interplanar spacings

M
A

a1

 A reciprocal lattice vector is to the corresponding real lattice plane

*
*
*
*
g hkl h b1 k b2 l b3
The length of a reciprocal lattice vector is the reciprocal of the spacing of the
corresponding real lattice plane

*
hkl

*
1
g hkl
d hkl

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
02

(01)

a2

(11)
(21)
a1

a1

(10)

12

22

11

01

*
b2

00

*
b1

*
11

21
*
g 21

10

20
a1

The reciprocal lattice has an origin!

(01)

a2

(11)

(10)

(21)
a1

a1

*
b2

*
b1

(01)

(11)
(21)

(10)

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

Crystal = Lattice + Motif

In crystals based on a particular lattice the intensities of particular

reflections are modified they may even go missing

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)

SC

001

011

101

Lattice = SC

111

000
010
100
110

No missing reflections

Reciprocal Crystal = SC

Figures NOT to Scale

002

022

BCC
202

222
011
101

000

Lattice = BCC

020
110

200
100 missing reflection (F = 0)
Weighing factor for each point motif

220

Reciprocal Crystal = FCC

F2 4 f 2
Figures NOT to Scale

002

022

FCC
202

222
111

Lattice = FCC

020

000
200

100 missing reflection (F = 0)

220
110 missing reflection (F = 0)

Weighing factor for each point motif

Reciprocal Crystal = BCC

F 2 16 f 2
Figures NOT to Scale

Ordered Solid solution

In a strict sense this is not a crystal !!

High T disordered

BCC

G = H TS

470C
Sublattice-1

Sublattice-2

Low T ordered

SC

Ordered
BCC

SC

Disordered

Ordered

- NiAl, BCC

B2 (CsCl type)

- Ni3Al, FCC

L12 (AuCu3-I type)

FCC

Ordered

FCC

SC

BCC

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)
FR Fundamental Reflection
SLR Superlattice Reflection

1) SC + two kinds of Intensities decorating the lattice

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

The Ewald Sphere

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

7. Paul-Peter-Ewald-Kolloquium
Freitag, 17. Juli 2008 organisiert von:

Max-Planck-Institut fr Metallforschung
Institut fr Theoretische und Angewandte Physik,
Institut fr Metallkunde,
Institut fr Nichtmetallische Anorganische
Materialien
der Universitt Stuttgart
Programm

13:30

Joachim Spatz (Max-Planck-Institut fr Metallforschung)

Begrung

13:45

Heribert Knorr (Ministerium fr Wissenschaft, Forschung und Kunst Baden-Wrttemberg

Begrung

14:00

Stefan Hell (Max-Planck-Institut fr Biophysikalische Chemie)

Nano-Auflsung 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 Getrnke

15:30

Frank Gieelmann(Universitt Stuttgart)

Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanorhren: Aktuelle Themen der
Flssigkristallforschung

16:00

Verleihung des Gnter-Petzow-Preises 2008

16:15

Udo Welzel (Max-Planck-Institut fr Metallforschung)

Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind

ab 17:00

Sommerfest des Max-Planck-Instituts fr Metallforschung

The Ewald Sphere

The reciprocal lattice points are the values of momentum transfer for
which the Braggs 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 Cullitys book: A15-4

Braggs equation revisited

n 2 d hkl Sin hkl Sin hkl

2 1 d hkl

d hkl
2

Draw a circle with diameter 2/

Construct a triangle with the diameter as the hypotenuse and 1/d hkl 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 Braggs equation)

*
1
*
g hkl
g hkl

d hkl

The Ewald Sphere construction

Crystal related information is present in the reciprocal crystal

Sin hkl

2 1 d hkl

The Ewald sphere construction generates the diffraction pattern

d hkl
2

Radiation related information is present in the Ewald Sphere

Ewald Sphere

Reciprocal Space

Ki

KD

The Ewald Sphere touches the

reciprocal lattice (for point 41)

02
01

00

K
10

20

(41)

Braggs equation is satisfied

for 41

K = K = g = Diffraction Vector

Ewald sphere X-rays

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

Crystal structure determination

Monochromatic X-rays

Many s (orientations)
Powder specimen

POWDER
METHOD

Panchromatic X-rays

Single

LAUE
TECHNIQUE

Varied by rotation

ROTATING
CRYSTAL
METHOD

Monochromatic X-rays

THE POWDER METHOD

Cone of diffracted rays

POWDER METHOD
Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections

Film may be replaced with detector

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

The 440 reflection is not observed

The 331 reflection is not observed

THE POWDER METHOD

2dSin

Cubic crystal

d Cubic

a
h2 k 2 l 2

2
2
4
a
sin

2
2
h k2 l2

2
4
a
(h 2 k 2 l 2 ) 2 sin 2

2
2
2
a2
(
h

l
)
2
4 sin

(h 2 k 2 l 2 ) sin 2

Relative Intensity of diffraction lines in a powder pattern

Structure Factor (F)

Scattering from UC

Multiplicity factor (p)

Number of equivalent scattering planes

Polarization factor

Effect of wave polarization

I P 1 Cos 2 2
Lorentz factor

Combination of 3 geometric factors

1
1

Lorentz factor
Cos

Sin 2
Sin 2
Absorption factor

Specimen absorption

Temperature factor

Thermal diffuse scattering

Multiplicity factor
Lattice

Index

Multiplicity

Planes

Cubic
(with highest
symmetry)

(100)

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

(110)

12

[(110) (101) (011), ( 110) ( 101) (0 11)] ( 2 for negatives)

(111)

12

[(111) (11 1) (1 11) ( 111)] ( 2 for negatives)

(210)

24*

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

(2 10) 3! Ways, ( 2 10) 3! Ways

(211)

24

(211) 3 ways, (21 1) 3! ways,

( 211) 3 ways

(321)

48*

(100)

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

(110)

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

(111)

[(111) (11 1) (1 11) ( 111)] ( 2 for negatives)

(210)

8*

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

(2 10) = 2 Ways, ( 2 10) = 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)

Tetragonal
(with highest
symmetry)

* Altered in crystals with lower symmetry

Multiplicity factor

Cubic
Hexagonal
Tetragonal
Orthorhombic
Monoclinic
Triclinic

hkl
48*
hk.l
24*
hkl
16*
hkl
8
hkl
4
hkl
2

hhl
24
hh.l
12*
hhl
8
hk0
4
h0l
2

hk0
24*
h0.l
12*
h0l
8
h0l
4
0k0
2

hh0
12
hk.0
12*
hk0
8*
0kl
4

hhh
8
hh.0
6
hh0
4
h00
2

h00
6
h0.0
6
h00
4
0k0
2

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

00.l
2
00l
2
00l
2

Lorentz factor

Polarization factor

I P 1 Cos 2 2

1
1

Cos

Sin 2
Sin 2

Lorentz factor

1 Cos 2 2

Lorentz Polarization factor

2
Sin Cos

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

2
2 1 Cos 2

I F p
2
Sin Cos

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

2dSin

Cubic crystal

d Cubic

a
h2 k 2 l 2

2
2
4
a
sin

2
2
h k2 l2

2
4
a
(h 2 k 2 l 2 ) 2 sin 2

2
2
2
a2
(
h

l
)
2
4 sin

(h 2 k 2 l 2 ) sin 2

Determination of Crystal Structure from 2 versus Intensity Data

n

Intensity

Sin

Sin2

ratio

Intensity

Sin

Sin2

ratio

21.5

0.366

0.134

25

0.422

0.178

37

0.60

0.362

45

0.707

0.500

11

47

0.731

0.535

12

58

0.848

0.719

16

68

0.927

0.859

19

FCC

h2 + k2 + l2

SC

FCC

100

110

111

111

200

200

210

211

BCC

DC

110
111
200
211

7
8

220

220

300, 221

10

310

11

311

311

12

222

222

13

320

14

321

220

220

310
311
222
321

15
16

400

17

410, 322

18

411, 330

19

331

400

400

400

411, 330
331

331

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

SC

BCC

FCC

11

12

DC

11

16

Radiation: Cu K, = 1.54056

111

Powder diffraction pattern from Al

422

420

331

400

222

311

220

200

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

1 & 2 peaks resolved

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

n 2 d Sin
d
2 d Cos
d
d
1

d 2 d Cos

Actually, the variation in 2 is to be seen

Determination of Crystal Structure from 2 versus Intensity Data

n

Sin

Sin2

ratio

Index

a (nm)

38.52

19.26

0.33

0.11

111

0.40448

44.76

22.38

0.38

0.14

200

0.40457

65.14

32.57

0.54

0.29

220

0.40471

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 , 2 peaks are resolved (1 peaks are listed)

2 Sin

d (d ) Cos

d
2Sin 2
d ( d ) Cos
d

d
d
Sin
Tan

Error in d spacing

Fo
r

the

sa

me

t
he

err

or

in

Si
n

wi
th

d ( d ) Cos
d

d
d
Sin
Tan

Error in d spacing

Error in d spacing decreases with

Applications of XRD
Bravais lattice determination
Lattice parameter determination
Determination of solvus line in phase diagrams
Long range order
Crystallite size and Strain
More

Intensity

Crystal

90

180

Diffraction angle (2)

Monoatomic gas

Intensity

Schematic of difference between

the diffraction patterns of various phases

Intensity

Diffraction angle (2)

Liquid / Amorphous solid

0

90

180

Diffraction angle (2)

90

180

Crystallite size and Strain

Braggs 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 Braggs 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

XRD Line Broadening

Instrumental
Bi

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

Crystallite size

In the vicinity of B the ve of Braggs equation not being satisfied

Strain

Residual Strain arising from dislocations, coherent precipitates

etc. leading to broadening

Bc

Bs

Stacking fault
In principle every defect contributes to some broadening

Other defects

B ( FWHM ) Bi Bc Bs BSF ...

B ( FWHM ) Bi Bc Bs BSF ...

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

Diffuse

Broadened Rings

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 pattern

Broadened 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 specimens peaks)
2
Use the same material as the standard as the specimen to be X-rayed but with large
grain size and well annealed

B ( FWHM ) Bi Bc Bs BSF ...

For a peak with a Lorentzian profile

B Bi Bc Bs Br

Hendrik Antoon Lorentz

Longer tail

For a peak with a Gaussian profile

A geometric mean can also used

On the theory of reflection and refraction of light

Br2 B 2 Bi2
Br2 ( B Bi ) B 2 Bi2
http://en.wikipedia.org/wiki/Carl_Friedrich_Gauss

Johann Carl Friedrich Gauss (1777-1855),

painted by Christian Albrecht Jensen
University of Gttingen

Scherrers formula

For Gaussian line profiles and cubic crystals

Wavelength
k
Bc
L Average crystallite size ( to surface of specimen)
L Cos ( B )

k 0.94 [k (0.89, 1.39)]

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

Strain broadening

Bs Tan( B )
Strain in the material

Smaller angle peaks

should be used to
separate Bs and Bc

Separating crystallite size broadening and strain broadening

Br Bc Bs

Br

k
Bs Tan ( )
Bc
L Cos ( )

k
Tan ( )
L Cos ( )

Br Cos ( )

k
Sin( )
L

Plot of [Br Cos] vs [Sin]

Example of a calculation

Intensity

Sample: Annealed Al
Radiation: Cu k ( = 1.54 )

Sample: Cold-worked Al
Radiation: Cu k ( = 1.54 )

40

60

Intensity

2
X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)

40

60

Annealed Al
Peak No.

2 ()

hkl

Bi = FWHM ()

Bi = FWHM (rad)

38.52

111

0.103

1.8 103

44.76

200

0.066

1.2 103

65.13

220

0.089

1.6 103

Cold-worked Al
2 ()

Sin()

hkl

B ()

B (rad)

Br2 B 2 Bi2

Br Cos (rad)

1 38.51

0.3298

111

0.187

3.3 103

2.8 103

2.6 103

2 44.77

0.3808

200

0.206

3.6 103

3.4 103

3.1 103

3 65.15

0.5384

220

0.271

4.7 103

4.4 103

3.7 103

k
1.7 10 3
L

Grain Size ( L) 90 nm

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

Consider the compound ZnS (sphalerite). Sulphur atoms occupy fcc sites with zinc atoms
displaced by 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.
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 , 0 , 0 i.e. in the fcc position and 4 atoms of type B at the sites
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)

421 missing

Ewald sphere X-rays

0.65

0.28

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

Multiplicity factor
Lattice

Index

Multiplicity

Planes

Cubic
with highest
symmetry

(100)

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

(110)

12

[(110) (101) (011), ( 110) ( 101) (0 11)] ( 2 for

negatives)

(111)

[(111) (11 1) (1 11) ( 111)] ( 2 for negatives)

(210)

24

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

(2 10) = 3! Ways, ( 2 10) = 3! Ways,

(211)

24

(321)

48

(100)

[(100) (010)]

(110)

[(110) ( 110)]

(111)

[(111) (11 1) (1 11) ( 111)] ( 2 for negatives)

(210)

(211)

24

(321)

48

Tetragonal