x-ray diffraction

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X-RAY
CRYSTALLOGRAPHY
Presented by
Mahfooz Alam
M.Tech. ( Materials Engineering)
Reg No:- 17ETMM10

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WALK OF THE TALK
E.M Spectrum
Production of X-rays?
Crystallography
Bragg’s Equation
Experimental Methods
Conclusion

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ELECTROMAGNETIC
SPECTRUM
Discovery of X-rays: Roentgen, 1895 (Nobel Prize 1901)

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Motivation
 For electromagnetic radiation to be diffracted the spacing in the
grating ( grating refers to a series of obstacles or a series of scatters)
should be of the same order as the wavelength.
 In crystals the typical interatomic spacing ~ 2-3 Å**  so the
suitable radiation for the diffraction study of crystals is X-rays.
 Hence, X-rays are used for the investigation of crystal structure.
** If the wavelength is of the order of the lattice spacing, then diffraction effects will be
prominent.
 Three possibilities (regimes) exist based on the wavelength () and the spacing
between the scatters .
  < a  transmission dominated.
  ~ a  diffraction dominated.
  > a  reflection dominated.

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Generation of X-rays
X-rays can be generated by decelerating electrons.
Hence, X-rays are generated by bombarding a target (say Cu) with an electron
beam.
The resultant spectrum of X-rays generated (i.e. X-rays versus Intensity plot) is
shown in the next slide. The pattern shows intense peaks on a ‘broad’ background.
The intense peaks can be ‘thought of’ as monochromatic radiation and be used
for X-ray diffraction studies.
Target
Metal
 Of K
radiation (Å)
Mo 0.71
Cu 1.54
Co 1.79
Fe 1.94
Cr 2.29

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Cont.…
Characteristic X-ray
Continuous Spectrum

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X-ray Spectrum

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Crystal Systems
7 crystal systems of varying
symmetry are known
These systems are built by
changing the lattice parameters:
a, b, and c are the edge lengths
, , and  are interaxial angles
Fig. 3.4, Callister 7e.
Unit cell: Smallest repetitive volume which contains
the complete lattice pattern of a crystal.

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Crystal Systems
Crystal structures are divided
into groups according to unit
cell geometry (symmetry).
Cont....

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BRAVAIS LATTICE

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Specimen
Incident X-rays
Transmitted beam
Fluorescent X-rays Electrons
Compton recoil Photoelectrons
Scattered X-rays
Coherent
From bound charges
• When X-rays hit a specimen, the interaction can
result in various signals/emissions/effects.
• The coherently scattered X-rays are the ones
important from a XRD perspective.
Incoherent
From loosely bound charges
Interaction of X-ray with Specimen
11

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 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 can undergo constructive or destructive interference.
Incoming X-rays
Secondary
emission
Cont.…

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 Ray-2 travels an extra path as compared to Ray-1 (= ABC). The path difference between
Ray-1 and Ray-2 = ABC = (d Sin + d Sin) = (2d.Sin).
 For constructive interference, this path difference should be an integral multiple of :
n = 2d Sin  the Bragg’s equation..
Bragg’s Equation
Braggs Law
The diffracted beam appears to be reflected from a set of crystal lattice planes.
Angle of incidence = Angle of reflection.
Diffraction laws: Bragg & Bragg, 1912-1913 (Nobel Prize 1915)

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2 sinhkl hkln d 
λ= 2
dhkl
n
sin θ
n n n n n n2 sinh k l h k ld 
1nhnk nl
hkl
d
d n

300
100
1
3
d
d

200
100
1
2
d
d

Hence, (100) planes are a subset of (200) planes
Order of Reflection
 is the angle between the incident x-rays and the set of parallel atomic planes (which have
a spacing dhkl). Which is 10
It is NOT the angle between the x-rays and the sample surface (note: specimens could be
spherical or could have a rough surface).

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Bravais Lattice
Determination
Lattice Parameter
Determination
Strain
Applications of XRD
Crystallite Size
Crystalline, Amorphous
Nature of Materials

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Crystal Structure Determination
Monochromatic X-rays
Panchromatic X-rays
Monochromatic X-rays
Many s (orientations)
Powder specimen
Powder
Method
Single 
Laue
Technique
 Varied by rotation
Rotating Crystal
Method
λ fixed
θ variable


λ fixed
θ rotated


λ variable
θ fixed


 As diffraction occurs only at specific Bragg angles, the chance that a reflection is observed
when a crystal is irradiated with monochromatic X-rays at a particular angle is small (added to this
the diffracted intensity is a small fraction of the beam used for irradiation).
 The probability to get a diffracted beam (with sufficient intensity) is increased by either varying
the wavelength () or having many orientations (rotating the crystal or having multiple
crystallites in many orientations).
 The three methods used to achieve high probability of diffraction are shown below.
Only the powder method is commonly used in materials science.

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Powder diffraction : Developed independently in two countries:
Debye and Scherer in Germany, 1916
Hull in the United States, 1917
Definition: Powder diffraction is a scientific technique using X-ray,
neutron, or electron diffraction on powder or microcrystalline samples for
structural characterization of materials.
 Every possible crystalline orientation is represented equally in a
powdered sample. The resulting orientational averaging causes the three
dimensional reciprocal space that is studied in single crystal diffraction to
be projected onto a single dimension.
Powder Diffraction

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Powder Diffraction is more aptly named polycrystalline diffraction
Samples can be powder, sintered pellets, coatings on substrates,
engine blocks, …
If the crystallites are randomly oriented, and there are enough of
them, then they will produce a continuous Debye cone.
In a linear diffraction pattern, the detector scans through an arc that
intersects each Debye cone at a single point; thus giving the
appearance of a discrete diffraction peak.
Cont..

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 In the power diffraction method a 2 versus intensity (I) plot is obtained from the
diffractometer (and associated instrumentation).
 The ‘intensity’ is the area under the peak in such a plot (NOT the height of the peak).
Powder diffraction pattern from Al
Radiation: Cu K,  = 1.54 Å
Increasing 
Increasing d
Usually in degrees ()
Intensity versus 2 Data in
Powder Method

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Structure Allowed Reflections
SC All
BCC (h + k + l) even
FCC h, k and l unmixed
DC
Either,  h, k and l are all odd or
 all are even & (h + k + l) divisible by 4
Selection / Extinction Rules

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2→  Sin Sin2 
Ratios
of Sin2
Dividing Sin2 by
0.134/3 = 0.044667
Whole
number
ratios
Index
1 21.5 0.366 0.134 1 3 111
2 25 0.422 0.178 1.33 3.99 4 200
3 37 0.60 0.362 2.70 8.10 8 220
4 45 0.707 0.500 3.73 11.19 11 311
5 47 0.731 0.535 4 11.98 12 222
6 58 0.848 0.719 5.37 16.10 16 400
7 68 0.927 0.859 6.41 19.23 19 331
 FCC lattice
E.g.:-
Given the positions of the Bragg peaks we find the lattice type

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How are real diffraction patterns
different from the ideal one's?
 We have seen real and ideal diffraction patterns. In ideal patterns the peaks are ‘’ functions.
• Real diffraction patterns are different from ideal ones in the following ways:
 Peaks are broadened
Could be due to instrumental, residual ‘non-uniform’ strain (microstrain), grain size etc. broadening.
 Peaks could be shifted from their ideal positions
Could be due to uniform strain→ macrostrain.
 Relative intensities of the peaks could be altered
Instrumental broadening
Crystal defects (‘bent’ planes)
Peak Broadening
Small crystallite size

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Structure Factor (F)
Multiplicity factor (p)
Polarization factor
Lorentz factor
Relative Intensity of diffraction lines in a powder pattern
Absorption factor
Temperature factor
The resultant wave scattered by all atoms of the unit cell
Number of equivalent scattering planes
Effect of wave polarization
Combination of 3 geometric factors
Specimen absorption
Thermal diffuse scattering
Lorentzfactor=( 1
Sin2θ)(Cosθ)( 1
Sin2θ)
I P= (1+Cos2
(2θ))

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Crystallite Size Determination
 Scherrer use X-rays to estimate the crystallite size.
B(2θ)=
Kλ
Lcosθ
size (L) Peak width (B) is inversely proportional to crystallite
 as the crystallite size gets smaller, the peak gets broader
 The constant of proportionality, K (the Scherrer constant) depends on
the how the width is determined, the shape of the crystal, and the size
distribution
 Most common values for K are:
0.94 for FWHM of spherical crystals with cubic symmetry
0.89 for integral breadth of spherical crystals
K actually varies from 0.62 to 2.08
Factors
•how the peak width is defined
•how crystallite size is defined
•the shape of the crystal
•the size distribution

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Broadeing 2 2 tan 

   
d
b
d
Non-uniform Strain
Uniform Strain
No Strain
do
2
2
2
  d  strain
Lattice Strain

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Diffraction angle (2) →
Intensity→
90 1800
Crystal
90 1800
Diffraction angle (2) →
Intensity→
Liquid / Amorphous solid
90
1800
Diffraction angle (2) →
Intensity→
Monoatomic gas
Diffractogram of various
phases
Sharp peaks
Diffuse Peak
No peak

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Conclusion
• Two types of X-ray are produced using Cathode Ray Tube
1) Continuous X-ray
2) Characteristics X-ray
• X-ray is be used in different characterization technique
technique to characterize the materials.
• X-ray Crystallography has a wide range of application in
chemical,physical,biological, material and biochemical sciences.
• The real diffractogram obtained is different from ideal ones.

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References
1) Elements of X-Ray Diffraction
B.D. Cullity & S.R. Stock
Prentice Hall, Upper Saddle River (2001)
2) An Introduction to Material Science and Engineering
by :- William D Callister
3) Nptel Lecture by Bala Subramaniam
4)Website
 www.matter.org.uk/diffraction/
www.ngsr.netfirms.com/englishhtm/Diffraction

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I would like to thank Dr.-Ing Vadali V.S.S Srikanth and Dr.Jai
Prakash Gautam for motivating and guiding us throughout the
course.
I would like to thank Dr. Swati Ghosh Acharya for teaching X-ray
diffraction in material characterization course.
I would like to thank all my friends.
I am highly obliged to all non-teaching staff of SEST.
I would like to thank AICTE for providing me the scholarship to
pursue my M.Tech.
Acknowledgment

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Thank You