This document provides an overview of x-ray diffraction principles and practices. It begins with an introduction to materials characterization and the importance of x-ray diffraction. It then covers the basics of diffraction and Bragg's law. The document discusses different x-ray diffraction methods and techniques for analyzing crystal structure, phase, texture, stress, and particle size. It provides examples of analyzing diffraction patterns from single and multiple phases. Finally, it touches on concepts like broadening, texture, and pole figures.
2. Layout of the Lecture
Materials Characterization
Importance of X-ray Diffraction
Basics
Diffraction
X-ray Diffraction
Crystal Structure and X-ray Diffraction
Different Methods
Phase Analysis
Texture Analysis
Stress Analysis
Particles Size Analysis
………..
Summary
3. 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
4. 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
8. 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
9. 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
10. d
Diffraction of light through an aperture
-15 -10 -5 0 5 10 15
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Intensity
12. Bragg’s Law
n=2d.sin
n: Order of reflection
d: Plane spacing
=
: Bragg Angle
Path difference must be integral multiples of the wavelength
in=out
in out
2
2 2 2
a
h k l
14. 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
15. 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
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
λ=2d’ sin
*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.
16. 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.
17. Real space Reciprocal space
Crystal Lattice Reciprocal Lattice
Crystal structure Diffraction pattern
Unit cell content Structure factor
x
y
y’
x’
y’
x’
18. Reciprocal space
Reciprocal lattice of FCC is BCC
and vice versa
)
(
*
)
(
*
)
(
*
c
b
a
b
a
c
c
b
a
a
c
b
c
b
a
c
b
a
a
b
c
100
001
010
21. 2 Circle diffratometer 2 and
3 and 4 circle diffractometer 2θ, ω, φ, χ
6 circle diffractometer θ, φ, χ and δ, γ, µ
www.serc.carleton.edu/
Hong et al., Nuclear Instruments and Methods in Physics Research A 572 (2007) 942
25. Structure Factor
2 ( )
1
n n n
N
i hu kv lw
hkl n
F f e
− 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 unmixed i.e. all
odd or all even
h, k, and l: mixed
Intensity of the diffracted beam |F|2
27. Diffraction Methods
Method Wavelength Angle Specimen
Laue Variable Fixed Single
Crystal
Rotating
Crystal
Fixed Variable (in
part)
Single
Crystal
Powder Fixed Variable Powder
28. Laue Method
• Uses Single crystal
• Uses White Radiation
• Used for determining crystal orientation and quality
Transmission Zone axis
crystal
Incident beam
Film
Reflection
Zone axis
crystal
Incident beam Film
30. Diffraction from a variety of materials
(From “Elements of X-ray
Diffraction”, B.D. Cullity,
Addison Wesley)
31. Reality
0.9
cos
B
t
B
Crystallite size can be
calculated using
Scherrer Formula
Instrumental broadening must be subtracted
(From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley)
32. • 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
Intensity of diffracted beam
33. 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.
35. Bulk electrodeposited nanocrystalline nickel
Lattice parameter, phase diagrams
Texture, Strain (micro and residual)
Size, microstructure (twins and
dislocations)
Actual Pattern
36. Powder X-ray diffraction
is essentially a misnomer and should be replaced by
Polycrystalline X-ray diffraction
37. Information in a Diffraction Pattern
Phase Identification
Crystal Size
Crystal Quality
Texture (to some extent)
Crystal Structure
38. Analysis of Single Phase
Intensity
(a.u.)
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
39. 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
40. Presence of Multiple phases
More Complex
Several permutations combinations possible
e.g. d1; d2; and d3, the first three strongest lines
show several alternatives
Then take any of the two lines together and match
It turns out that 1st and 3rd strongest lies belong to
Cu and then all other peaks for Cu can be
separated out
Now separate the remaining lines and normalize
the intensities
Look for first three lines and it turns out that the
phase is Cu2O
If more phases, more pain to solve
d (Å) I/I1
3.01 5
2.47 72
2.13 28
2.09 100
1.80 52
1.50 20
1.29 9
1.28 18
1.22 4
1.08 20
1.04 3
0.98 5
0.91 4
0.83 8
0.81 10
*
*
*
*
*
*
*
Pattern for Cu
d (Å) I/I1
2.088 100
1.808 46
1.278 20
1.09 17
1.0436 5
0.9038 3
0.8293 9
0.8083 8
Remaining Lines
d (Å) I/I1
Observed Normalized
3.01 5 7
2.47 72 100
2.13 28 39
1.50 20 28
1.29 9 13
1.22 4 6
0.98 5 7
Pattern of Cu2O
d (Å) I/I1
3.020 9
2.465 100
2.135 37
1.743 1
1.510 27
1.287 17
1.233 4
1.0674 2
0.9795 4
41. Broadeing 2 2 tan
d
b
d
Lattice Strain
Non-uniform Strain
Uniform Strain
No Strain
do
2
2
2
d strain
42. 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
43. Texture
Fiber Texture
A particular direction [uvw] for all grains is more or less parallel to
the wire or fiber axis
e.g. [111] fiber texture in Al cold drawn wire
Double axis is also possible
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]
44. 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
45. Degree of orientation
Substrate
Film
Side view
[uvw] corresponding
to planes parallel to
the surface
But what if the planes when looked from top have random orientation?
Top view
46. Pole Figure
4 Peaks at ~50
Excellent in-plane
orientation
1
1
1
1
2
2
2
2
1
1
1
2
2
2
3
3
3
1
2
3
2 sets of peaks at ~ 5, 65
and 85°
Indicating a doublet or
opposite twin growth
3 sets of peaks
at ~ 35 and 85°
indicating a
triplet or triple
twin growth
(117) Pole Figures for Bismuth Titanate Films
SrTiO3 (100) SrTiO3 (110) SrTiO3 (111)
47. Rocking Curve
An useful method for evaluating the quality of oriented samples such as
epitaxial films
is changed by rocking the sample but B is held constant
Width of Rocking curve is a direct measure of the range of orientation present in
the irradiated area of the crystal
17.4 17.6 17.8 18.0 18.2
Intensity
(a.u.)
()
17.5 17.6 17.7 17.8
FWHM = 0.07°
(0010) Rocking curve of (001)-
oriented SrBi2Ta2O9 thin film
32.4 32.6 32.8 33.0 33.2
()
32.4 32.6 32.8 33.0 33.2
(2212) Rocking curve of (116)-
oriented SrBi2Ta2O9 thin film
FWHM = 0.171°
B
Normal
48. Order Disorder Transformation
Structure factor is dependent on the presence
of order or disorder within a material
Present in systems such as Cu-Au, Ti-Al, Ni-
Fe
Order-disorder transformation at specific
compositions upon heating/cooling across a
critical temperature
Examples: Cu3Au, Ni3Fe
49. Order Disorder Transformation
Structure factor is dependent on the presence
of order or disorder within a material.
Complete Disorder
Example: AB with A and B atoms
randomly distributed in the lattice
Lattice positions: (000) and (½ ½ ½)
Atomic scattering factor
favj= ½ (fA+fB)
Structure Factor, F, is given by
F = Σf exp[2i (hu+kv+lw)]
= favj [1+e( i (h+k+l))]
= 2. favj when h+k+l is even
= 0 when h+k+l is odd
The expected pattern is like a BCC crystal
A
B
50. Order Disorder Transformation
Complete Order
Example: AB with A at (000) and B at (½ ½ ½)
Structure Factor, F, is given by
F = fA e[2i (h.0+k.0+l.0)]+ fA e[2i (h. ½+k. ½+l. ½)]
= fA+fB when h+k+l is even
= fA-fB when h+k+l is odd
The expected pattern is not like a BCC crystal,
rather like a simple cubic crystal where all the
reflections are present.
Extra reflections present are called as
superlattice reflections
A
B
54. Geometry and Configuration
Source Incident Beam
Optics
Sample Diffracted Beam
Optics
Detector
Incident Beam Part Diffracted Beam Part
Theta-Theta Source and detector move θ, sample fixed
Theta-2Theta Sample moves θ and detector 2θ , source fixed
Vertical configuration Horizontal sample
Horizontal configuration Vertical sample
55. 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
Sample translation
56. 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)
58. Small angle anode Large angle anode
Small focal spot Large focal spot
59. Rotating anode of W or Mo for high flux
Microfocus rotating anode 10 times brighter
Liquid anode for high flux 100 times brighter
and small beam size
Gallium and Gallium, indium, tin alloys
Synchrotron provides intense beam but access is limited
Brighter than a thousand suns
60. High brilliance and coherence
X-ray bulb emitting all radiations from IR to X-rays
http://www.coe.berkeley.edu/AST/srms
Synchrotron
61. Moving charge emits radiation
Electrons at v~c
Bending magnet, wiggler and undulator
Straight section wiggler and undulator
Curved sections Bending magnet
62. Filter to remove Kβ For eg. Ni filter for Cu Kβ
Reduction in intensity of Kα
Choice of proper thickness
63. Slits To limit the size of beam (Divergence slits)
To alter beam profile
(Soller slit angular divergence )
Narrow slits Lower intensity
+
Narrow peak
67. 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
68. 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
69.
70. 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
71. 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
72. 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