X-ray Diffraction
Layout of the Lecture

Materials Characterization


Basics




Diffraction
X-ray Diffraction
Crystal Structure and X-ray Diffraction
Different Methods






Importance of X-ray Diffraction
Phase Analysis
Texture Analysis
Stress Analysis
Particles Size Analysis
………..
Summary
Materials Characterization


Essentially to evaluate the structure and
properties
Structural Characterization

Diffraction




X-ray and Electron Diffraction
Microscopy
Spectroscopy
Property Evaluation



Mechanical
Electrical
Anything else
Time Line




1665: Diffraction effects observed by Italian
mathematician Francesco Maria Grimaldi
1868: X-rays Discovered by German Scientist
Röntgen
1912: Discovery of X-ray Diffraction by
Crystals: von Laue
1912: Bragg’s Discovery
Electromagnetic Spectrum
Generation of X-rays
Commercial X-ray Tube
X-ray Spectrum from an Iron target
12400
 Short Wavelength Limit SWL (nm) 
V
 Continuous spectrum
I CS  AiZV m
 Characteristic X-ray Moseley’s Law
  C(Z   )
I K  Bi (V V k)n
λSWL
Use of Filter

Ni filter for Cu Target
Crystal Systems and Bravais Lattices
Structure of Common Materials

Metals






Copper: FCC
-Iron: BCC
Zinc: HCP
Silver: FCC
Aluminium: FCC
Ceramics



SiC: Diamond Cubic
Al2O3: Hexagonal
MgO: NaCl type
Diffraction

A diffracted beam may be defined as a beam
composed of a large number of scattered rays
mutually reinforcing each other
Scattering
Interaction with a single particle
Diffraction
Interaction with a crystal
Scattering Modes


Random arrangement of atoms in space gives rise
to scattering in all directions: weak effect and
intensities add
By atoms arranged periodically in space


In a few specific directions satisfying Bragg’s law: strong
intensities of the scattered beam :Diffraction
No scattering along directions not satisfying Bragg’s law
Diffraction of light through an aperture
d
1
0.9
0.8
0.7
0.6
Intensity
0.5
0.4
0.3
0.2
0.1
0
-15
-10
-5
0
5
10
15
1
0.9
0.8
0.7
0.6
Intensity
0.5
0.4
0.3
0.2
0.1
0
-15
-10
-5
0
Minima
sinθ n

λ
a
5
10
15
Maxima
sin 
n = 0, 1,..

2n  1

2a
n = 1, 2,..
Young’s Double slit experiment
d sinθ = mλ, m = 1,2,3…..
Constructive Interference
d sinθ = (m+½)λ, m = 1,2,3…..
Destructive Interference
Interference
Phase Difference = 0˚
Phase Difference = 180˚
Phase Difference = 90˚
Interference and Diffraction
Bragg’s Law
n=2d.sin
n: Order of reflection
d: Plane spacing
a
= 2 2 2
h k l
: Bragg Angle
in
out
2
Path difference must be integral multiples of the wavelength
in=out
Braggs Law
n
d
2 sin 
1
sin   d
2

Geometry of Bragg’s law



The incident beam, the normal to the reflection plane,
and the diffracted beam are always co-planar.
The angle between the diffracted beam and the
transmitted beam is always 2 (usually measured).
Sin  cannot be more than unity; this requires
nλ < 2d, for n=1, λ < 2d
λ should be less than twice the d spacing we want to study
Order of reflection



Rewrite Bragg’s law λ=2 sin d/n
A reflection of any order as a first order reflection from planes,
real or fictitious, spaced at a distance 1/n of the previous spacing
Set d’ = d/n
λ=2d’ sin

An nth order reflection from (hkl) planes of spacing d may be
considered as a first order reflection from the (nh nk nl) plane of
spacing d’ = d/n
*The term reflection is only notional due to symmetry between incoming and
outgoing beam w.r.t. plane normal, otherwise we are only talking of diffraction.
Reciprocal lattice vectors
Used to describe Fourier analysis of electron concentration
of the diffracted pattern.
Every crystal has associated with it a crystal lattice and a
reciprocal lattice.
A diffraction pattern of a crystal is the map of reciprocal
lattice of the crystal.
Real space
Reciprocal space
Crystal Lattice
Reciprocal Lattice
Crystal structure
Diffraction pattern
Unit cell content
y
x
y’
x’
Structure factor
y’
x’
Reciprocal space
bc
a* 
a  (b  c )
ca
b* 
a  (b  c )
a b
c* 
a  (b  c )
Reciprocal lattice of FCC is BCC
and vice versa
001
a
010
b
c
100
NaCl crystals in a tube facing X-ray beam
Powder Diffractometer
(400)
(410)
(220)
(330)(221)
(310)
(311)
(222)
(320)
(321)
(211)
(210)
(200)
(111)
(110)
(100)
Calculated Patterns for a Cubic Crystal
Structure Factor
N
Fhkl   f n e
2 i ( hun  kvn lwn )
Intensity of the diffracted beam  |F|2
1
−
−
−
−
h,k,l : indices of the diffraction plane under consideration
u,v,w : co-ordinates of the atoms in the lattice
N : number of atoms
fn : scattering factor of a particular type of atom
Bravais Lattice
Reflections possibly present Reflections necessarily absent
Simple
All
None
Body Centered (h+k+l): Even
(h+k+l): Odd
Face Centered
h, k, and l: mixed
h, k, and l unmixed i.e. all
odd or all even
Systematic Absences
Permitted Reflections
Simple Cubic
BCC
(100), (110), (111), (200), (210), (211),
(220), (300), (221) ………
(110), (200), (211), (220), (310), (222)….
FCC
(111), (200), (220), (311)…..
Diffraction Methods
Method
Wavelength
Angle
Laue
Variable
Fixed
Rotating
Crystal
Powder
Fixed
Fixed
Specimen
Single
Crystal
Variable (in Single
part)
Crystal
Variable
Powder
Powder Method
Sample
Incident Beam
Film
• Useful for determining lattice parameters with high precision and for
identification of phases
Indexing a powder pattern
Bragg’s Law
n = 2d sin
For cubic crystals
a
d hkl  2
h  k 2  l2
which gives rise to
θ
 S1
(for front reflections) or
2W
  S2 
θ  1   (for back reflections)
2 W
sin 2 θ
2
 2
2
2
2
h k l
4a
which is a constant
Indexing
SS1 (mm) () () sin2
1
(mm)
38
19.0
38
19.0
0.11
45
22.5
45
22.5
0.15
66
33.0
66
33.0
0.30
78
39.0
78
39.0
0.40
83
41.5
83
41.5
0.45
97
49.5
97
49.5
0.58
113
56.5
113
56.5
0.70
118
59.0
118
59.0
0.73
139
69.5
139
69.5
0.88
168
84.9
168
84.9
0.99
FCC; wavelength=1.54056Å
Simple
BCC
Cubic
2 2
h22+k
sin
 +l
2/
+l2 22/ Lattice
22+k
22+lh
22 2+k2sin
hsin
h22+k22+l22Parameter, a (Å)
0.11
3
0.15
4
0.30
8
0.40
11
0.45
12
0.58
16
0.70
19
0.73
20
0.88
24
0.99
27
21
0.037
42
0.038
63
0.038
84
0.036
5
10
0.038
6
12
0.036
8
14
0.037
9
16
0.037
10
18
0.037
11
20
0.037
0.055
0.11
0.038
0.75
0.050
0.10
0.050
0.10
0.045
0.09
0.048
0.097
4.023
3.978
3.978
4.039
3.978
Not
NotSimple
BCC
Cubic
4.046
But
what is the lattice
4.023
0.046
0.081
parameter?
4.023
0.049
0.088
4.023
0.050
0.09
4.023
Not Constant
0.050
0.0925
Constant; so it is FCC
Diffraction from a variety of materials
(From “Elements of X-ray
Diffraction”, B.D. Cullity,
Addison Wesley)
Reality
Crystallite size can be
calculated using
Scherrer Formula
0.9
t
B cos  B
Instrumental broadening must be subtracted
(From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley)
Intensity of diffracted beam
•
•
•
•
•
•
polarization factor
structure factor (F2)
multiplicity factor
Lorentz factor
absorption factor
temperature factor
For most materials the peaks and their intensity are
documented
 JCPDS
 ICDD
Name and formula
Reference code:
00-001-1260
PDF index name:
Nickel
Empirical formula:
Ni
Chemical formula:
Ni
Crystallographic parameters
Crystal system:
Cubic
Space group:
Fm-3m
Space group number:
225
a (Å):
3.5175
b (Å):
3.5175
c (Å):
3.5175
Alpha (°):
90.0000
Beta (°):
90.0000
Gamma (°):
90.0000
Measured density (g/cm^3):
8.90
Volume of cell (10^6 pm^3):
43.52
Z:
4.00
RIR:
Status, subfiles and quality
Status:
Marked as deleted by ICDD
Subfiles:
Inorganic
Quality:
Blank (B)
References
Primary reference:
Hanawalt et al., Anal. Chem., 10, 475, (1938)
Optical data:
Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107
Unit cell:
The Structure of Crystals, 1st Ed.
Stick pattern from JCPDS
http://ww1.iucr.org/cww-top/crystal.index.html
Actual Pattern
Lattice parameter, phase diagrams
Texture, Strain (micro and residual)
Size, microstructure (twins and
dislocations)
Bulk electrodeposited nanocrystalline nickel
Powder X-ray diffraction
is essentially a misnomer and should be replaced by
Polycrystalline X-ray diffraction
Information in a Diffraction Pattern





Phase Identification
Crystal Size
Crystal Quality
Texture (to some extent)
Crystal Structure
Intensity (a.u.)
Analysis of Single Phase
2(˚)
d (Å)
(I/I1)*100
27.42
3.25
10
31.70
2.82
100
45.54
1.99
60
53.55
1.71
5
56.40
1.63
30
65.70
1.42
20
76.08
1.25
30
84.11
1.15
30
89.94
1.09
5
I1: Intensity of the strongest peak
Procedure






Note first three strongest peaks at d1, d2, and d3
In the present case: d1: 2.82; d2: 1.99 and d3: 1.63 Å
Search JCPDS manual to find the d group belonging to the
strongest line: between 2.84-2.80 Å
There are 17 substances with approximately similar d2 but only 4
have d1: 2.82 Å
Out of these, only NaCl has d3: 1.63 Å
It is NaCl……………Hurrah
Specimen and Intensities
Substance
File Number
2.829 1.999 2.26x 1.619 1.519 1.499 3.578 2.668
(ErSe)2Q
19-443
2.82x 1.996 1.632 3.261 1.261 1.151 1.411 0.891
NaCl
5-628
2.824 1.994 1.54x 1.204 1.194 2.443 5.622 4.892
(NH4)2WO2Cl4
22-65
2.82x 1.998 1.263 1.632 1.152 0.941 0.891 1.411
(BePd)2C
18-225
Caution: It could be much more tricky if the sample is oriented or textured or your goniometer is not
calibrated
Presence of Multiple phases






d (Å)
I/I1
Pattern of C
3.01
5
d (Å)
Pattern for Cu 2.47
More Complex
d (Å)
I/I1
2.13
Several permutations combinations possible
2.088
100
2.09
1.808lines
46
e.g. d1; d2; and d3, the first three strongest
1.80
1.278
20
show several alternatives
1.50
17
Then take any of the two lines together1.09
and match
1.29
1.0436
5
It turns out that 1st and 3rd strongest lies belong to1.28
0.9038
3
1.22
Cu and then all other peaks for Cu can be
0.8293
9
1.08
separated out
0.8083
8
1.04Lines
Remaining
Now separate the remaining lines and normalize
0.98I/I
the intensities
d
1


72
100
2.465
52
2.135
20
9
If more phases, more pain to solve

1.743
1.510
4
1.287
20
*
1.233
3
5
0.91
4
that
the
Observed
Normalized
0.83
8
3.01
5
2.47
72
100
2.13
28
39
0.81
*
*
18
(Å)
Look for first three lines and it turns out
phase is Cu2O
3.020
28
*
*
710
*
*
1.0674
0.9795
1
Lattice Strain
do
No Strain
2
Uniform Strain
  d  strain
2
Non-uniform Strain
Broadeing b   2  2
d
tan 
d
2
Texture in Materials



Grains with in a polycrystalline are not completely
randomly distributed
Clustering of grains about some particular
orientation(s) to a certain degree
Examples:



Present in cold-rolled brass or steel sheets
Cold worked materials tend to exhibit some texture after
recrystallization
Affects the properties due to anisotropic nature
Texture

Fiber Texture

A particular direction [uvw] for all grains is more or less parallel to
the wire or fiber axis


Double axis is also possible


e.g. [111] fiber texture in Al cold drawn wire
Example: [111] and [100] fiber textures in Cu wire
Sheet Texture


Most of the grains are oriented with a certain crystallographic plane
(hkl) roughly parallel to the sheet surface and certain direction [uvw]
parallel to the rolling direction
Notation: (hkl)[uvw]
Texture in materials










 [uvw] i.e. perpendicular to
the surface of all grains is
parallel to a direction [uvw]
Also, if the direction [u1v1w1]
is parallel for all regions, the
structure is like a single
crystal
However, the direction
[u1v1w1] is not aligned for all
regions, the structure is like a
mosaic structure, also called
as Mosaic Texture
Thin Film Specimen
Grazing angle (very small, ~1-5)
B
B
Film or Coating
Substrate


Smaller volume i.e. less intensity of the scattered
beam from the film
Grazing angle

Useful only for polycrystalline specimens
Thin Film XRD




Precise lattice constants measurements derived from 2- scans, which
provide information about lattice mismatch between the film and the
substrate and therefore is indicative of strain & stress
Rocking curve measurements made by doing a q scan at a fixed 2 
angle, the width of which is inversely proportionally to the dislocation
density in the film and is therefore used as a gauge of the quality of the
film.
Superlattice measurements in multilayered heteroepitaxial structures,
which manifest as satellite peaks surrounding the main diffraction peak
from the film. Film thickness and quality can be deduced from the data.
Glancing incidence x-ray reflectivity measurements, which can
determine the thickness, roughness, and density of the film. This
technique does not require crystalline film and works even with
amorphous materials.
Thin Films Specimens
B1B
(hkl) plane of the
substrate



B2
B
B1B2
Diffraction
from hkl
plane
i.e.
No Diffraction
from
hkl
occurs
plane
Single Crystal Substrate
If the sample and substrate is polycrystalline, then
problems are less
But if even if one of them is oriented, problems arise
In such situations substrate alignment is necessary
30
40
0028
0024

0026
0022
0020
0018

50
60
70
80
90
2216

20

10
△
△
SrTiO3 (110)
40
50
60
70
10
20
80
30
40
90

30
4016/
0416
20

10
014

△
△
Log Intensity (a.u.)

Bismuth Titanate thin
films on oriented
SrTiO3 substrates
Only one type of peaks
It apparent that films
are highly oriented
SrTiO3 (100)
*
△

0016
0012
0014
008

0010
006
004
Oriented thin films
50
o
60
2 ( )
△
SrTiO3 (111)
70
80
90
Degree of orientation
[uvw] corresponding
to planes parallel to
the surface
Film
Substrate
Side view
But what if the planes when looked from top have random orientation?
Top view
Instrumentation
Diffractometer
 Source
 Optics
 Detector
Incident Beam Part
Diffracted Beam Part
Sample
Source
Incident Beam
Optics
Diffracted Beam
Optics
Detector
Geometry and Configuration
 Theta-Theta
Source and detector move θ, sample fixed
 Theta-2Theta
Sample moves θ and detector 2θ , source fixed
 Vertical configuration
 Horizontal configuration
Incident Beam Part
Source Incident Beam
Optics
Horizontal sample
Vertical sample
Diffracted Beam Part
Sample Diffracted Beam
Optics
Detector
Sample translation
 XYZ translation
 Z translation
sample alignment
 Sample exactly on the diffractometer circle
 Knife edge or laser
 Video microscope with laser
 XY movement to choose
area of interest
X-ray generation
 X-ray tube (λ = 0.8-2.3 Ǻ)
 Rotating anode (λ = 0.8-2.3 Ǻ)
 Liquid metal
 Synchrotron (λ ranging from infrared to X-ray)
X-ray tube
X-rays
Be window
W cathode
Rotating anode
Electrons
Metal anode
Small angle anode
Small focal spot
Large angle anode
Large focal spot
 Rotating anode of W or Mo for high flux
 Microfocus rotating anode
 Liquid anode for high flux
and small beam size
10 times brighter
100 times brighter
 Gallium and Gallium, indium, tin alloys
 Synchrotron provides intense beam but access is limited
 Brighter than a thousand suns
 Filter
to remove Kβ
For eg. Ni filter for Cu Kβ
 Reduction in intensity of Kα
 Choice of proper thickness
 Slits
To limit the size of beam (Divergence slits)
To alter beam profile
(Soller slit angular divergence )
 Narrow slits
Lower intensity
+
Narrow peak
Mirror
focusing and remove Kα2
Mono-chromator
Si
remove Kα2
Graphite
Beam Profile
Mirror
Parallel beam
Source
Soller slit
Detector
Mirror
Sample
Para-focusing
Detector
Sample
Source
Point focus
Detector
Sample
Source
Comparison
Parallel beam
Para-focusing
X-rays are aligned
X-rays are diverging
Lower intensity for bulk
samples
Higher intensity
Higher intensity for small
samples
Lower intensity
Instrumental broadening
independent of orientation
of diffraction vector with
specimen normal
Instrumental broadening
dependent of orientation of
diffraction vector with
specimen normal
Suitable for GI-XRD
Suitable for Bragg-Brentano
Texture, stress
Powder diffraction
Detectors
 Single photon detector (Point or 0D)
 scintillation detector NaI
 proportional counter, Xenon gas
 semiconductor
 Position sensitive detector (Linear or 1D)
 gas filled wire detectors, Xenon gas
 charge coupled devices (CCD)
 Area detectors (2D)
 wire
 CCD
 3D detector
X-ray photon
Photoelectron or
Electron-hole pair
Photomultiplier tube or
amplifier
Electrical signal
 Resolution: ability to distinguish between energies
 Energy proportionality: ability to produce signal proportioanl to
energy of x-ray photon detected
 Sensitivity: ability to detect low intensity levels
 Speed: to capture dynamic phenomenon
 Range: better view of the reciprocal space
Data collection and analysis
 Choose 2θ range
 Step size and time per step
 Hardware: slit size, filter, sample alignment
 Fast scan followed with a slower scan
 Look for fluorescence
 Collected data: Background subtraction, Kα2 stripping
 Normalize data for comparison I/Imax
Summary

X-ray Diffraction is a very useful to characterize materials
for following information








Phase analysis
Lattice parameter determination
Strain determination
Texture and orientation analysis
Order-disorder transformation
and many more things
Choice of correct type of method is critical for the kind of
work one intends to do.
Powerful technique for thin film characterization