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X-ray Diffraction and Its
Applications in Science &
Engineering
CONVENTIONS OF LATTICE DESCRIPTION
 Unit cell is the smallest unit of a
crystal, which, if repeated, could
generate the whole crystal.
 A crystal’s unit cell dimensions are
defined by six numbers, the lengths
of the 3 axes, a, b, and c, and the
three interaxial angles, ,  and .
CRYSTAL STRUCTURE
 A crystal lattice is a 3-D
arrangement of unit cells.
 Space lattice is an imaginative
grid system in three dimensions
in which every point (or node)
has an environment that is
identical to that of any other
point or node.
Space Lattice + Basis = Crystal structure
CRSYSTALLOGRAPHIC DIRECTIONS
A crystallographic direction is defined as the line between two vectors. The
following steps are utilized in the determination of the three dimensional
indices:
1. A vector of conventional length is positioned such that it passes through
the origin of the coordinate system.
2. The length of the vector projection on each of the three axes is
determined; these are measured in terms of the unit cell dimensions a,b
and c.
3. These three numbers are multiplied or divided by a common factor to
reduce them to the smallest integer values.
4. The three indices are enclosed in square brackets :
[uvw]. The u, v and w integers
correspond to the reduced projections along
the x, y and z axes, respectively
CRYSTALLOGRAPHIC PLANES
The crystallographic planes are specified by the three miller indices (hkl). Any
two planes parallel to each other are equivalent and have identical indices.
The following procedure is used to determine h, k and l index numbers :
1. The length of the intercept for each axis is determined in terms of the
lattice parameters a, b and c.
2. The reciprocals of these numbers are taken.
3. These numbers are changed to the set of smallest integers by multiplying or
division by a common factor.
4. Finally, the integer indices, not separated by commas, are enclosed within
parentheses, thus: (hkl).
INDEXING OF PLANES AND DIRECTIONS IN CUBIC
SYSTEMS
z
(100)
c
(110)
y
a
x
z
c
(111)
b
b
a
X
y
CRYSTAL SYSTEMS
Seven Types of crystal systems are :
Crystal system
Cell Length
Cell Angles
Cubic
a=b=c
α = β = γ =90º
Tetragonal
a=b#c
α = β = γ =90º
Orthorhombic
a#b#c
α = β = γ =90º
Hexagonal
a=b#c
α = β =90º, γ = 120º
Rhombohedral
a=b=c
α = β = γ # 90º
Monoclinic
a#b#c
β = γ =90º # α
Triclinic
a#b#c
α#β#γ
BASICS OF X-RAY DIFFRACTION
WHAT ARE X-RAYS?
 X-rays are electromagnetic waves having wavelength in the
range of 0.1-100 Aº and energies in the range of 120 eV to 120
keV.
 X-rays up to about 10 keV (1-100 Aº wavelength) are classified
as "soft" X-rays, and from about 10 to 120 keV ( 0.1-1 Aº ) as
"hard" X-rays, due to their penetrating abilities.
ELECTROMAGNETIC SPECTRUM
PRODUCTION OF X-RAYS
A beam of electrons is generated from the hot ungsten filament and these
electrons are accelerated towards the anode with a high potential difference
between the cathode and anode (Target). Anode is mainly Cu, Mo, Al and Mg.
After striking the anode the electrons generate the X-rays.
While monochromatic source is preffered, the X-ray beam actually consists of
several characteristic X-ray lines.
ORIGIN OF X-RAY
ee-
e-
Continuous X-ray
e-
Characteristics X-ray
SPECTRAL CONTAMINATION IN DIFFRACTION PATTERNS
K1
K2
K1
K2
 Kβ will give extra peak in the XRD pattern which can be eliminated
by adding filters.
BRAGG’s LAW
Bragg’s Law is used to expalin the
intereference pattern of the X-rays
scattered by the crystals
n  2d hkl sin 
Where,
n
an integer
λ
wavelength of the
incident X-ray
dhkl
interplanar spacing
WHAT IS X-RAY DIFFRACTION ?
 The periodic lattice found in crystalline structure may act as diffraction grating
for wave particles of electromagnetic radiation with wavelength of a similar order
of magnitude (1Aº).
 The atomic planes of a crystal causes an incident beam of X-rays to interfere with
one another as they come out from the crystal. This phenomenon is called X-ray
diffraction.
ESSENTIAL PARTS OF THE DIFFRACTOMETER
 X-ray Tube: the source of X rays
 Incident-beam optics: condition the X-ray beam before it hits
the sample
 The goniometer: the platform that holds and moves the
sample, and detector.
 The sample & sample holder
 Receiving-side optics: condition the X-ray beam after it has
encountered the sample
 Detector: count the number of X rays scattered by the sample
APPLICATIONS OF XRD
 XRD is a nondestructive technique
 To identify crystalline phases and orientation
 To determine structural properties: strain, grain size, epitaxy, phase
composition, preferred orientation, order-disorder transformation,
thermal expansion
 To measure thickness of thin films and multilayers
 To determine atomic arrangement
 Detection limits: ~ 3% in a two phase mixture; can be ~ 0.1 % with
synchrotron radiation
SAMPLE PREPARATION FOR XRD
 An ideal powder sample should have many crystallites in
random orientations
 If the crystallites in a sample are very large, there will not be a
smooth distribution of crystal orientations. You will not get a
powder average diffraction pattern.
 Crystallites should be <10 mm in size to get good powder
statistics
 Large crystallite sizes and non-random crystallite orientations
both lead to peak intensity variation.
Intensity
X-RAY DIFFRACTION PATTERN OF AIR
20
40
2θ (degrees)
60
X-RAY DIFFRACTION PATTERN OF AMORPHOUS SOLIDS
XRD PATTERNS OF NANO-PARTICLES
Ceria
Dried
ZrO2
CERIA
ZrO2
INTENSITY
DIFFRACTION PATTERN OF A SINGLE CRYSTAL
A single crystal will produce only one family of peaks
in the diffraction pattern
INTENSITY
DIFFRACTION PATTERN OF A POLYCRYSTALLINE SAMPLE
A polycrystalline samples contain thousands of crystallites, therefore
all possible diffraction peaks should be observed.
EXTINCTION RULES FOR CUBIC CRYSTALS
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
1
100
2
110
3
111
4
200
5
210
6
211
211
8
220
220
9
300, 221
10
310
11
311
12
222
13
320
14
321
BCC
FCC
DC
111
111
110
200
200
7
220
220
311
311
310
222
321
222
INFORMATION PROCURRED FROM X-RAY DATA
1. Phase identification
2. Volume fraction of the phases
3. Crystallite size
4. Strain
PHASE IDENTIFICATON
 The diffraction pattern for every phase is as unique as your
fingerprint
 Phases with the same chemical composition can have drastically
different diffraction patterns.
 Obtain XRD pattern
 Measure d-spacings
 Obtain integrated intensities
 Compare data with known standards in the JCPDS file, which
are for random orientations (there are more than 50,000 JCPDS
cards of inorganic materials).
JCPDS CARD
1.file number 2.three strongest lines 3.lowest-angle line 4.chemical formula
and name 5.data on diffraction method used 6.crystallographic data 7.optical
and other data 8.data on specimen 9.data on diffraction pattern.
Joint Committee on Powder Diffraction Standards, JCPDS (1969)
Replaced by International Centre for Diffraction Data, ICDF (1978)
QUANTITATIVE PHASE ANALYSIS
 The four main methods of quantitative phase analysis:
(1) External standard method
(2) direct comparison method
(3) internal standard method
(4) Reference intensity ratio method (RIR)
 With high quality data, you can determine how much of each phase is
present.
 The ratio of peak intensities varies linearly as a function of weight
fractions for any two phases in a mixture.
 RIR method is fast and gives semi-quantitative results.
 Whole pattern fitting/Rietveld refinement is a more accurate but more
complicated analysis.
CRYSTALLITE SIZE
 Crystallites smaller than ~120nm create broadening of diffraction peaks.
This peak broadening can be used to quantify the average crystallite size
of nano particles using the Scherrer ‘s equation
00-043-1002> Cerianite- - CeO
2
Intensity (a.u.)
B 2  
K
 cos 
  wavelength of X - ray
  Full width at half maxima
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41
2 (deg.)
EFFECT OF LATTICE STRAIN IN DIFFRACTION PEAK AND
POSITION
NO STRAIN
Uniform Strain
(d1-d0)/d0
Peak moves, no shape changes
Non-Uniform Strain
d1# constant
Peak broadens
Continued……………..
 Uniform strain causes the unit cell to expand/contract in an
isotropic way. This simply leads to a change in the unit cell
parameters and shift of the peaks. There is no broadening
associated with this type of strain.
 Non-uniform strain leads to systematic shifts of atoms from their
ideal positions and to peak broadening. This type of strain arises
from the following sources:
. Point defects (vacancies, site-disorder)
. Plastic deformation (cold worked metals, thin films)
. Poor crystallinity
STRUCTURAL DETERMINATION
To determine the structure of monoatomic cubic crystals, the
following equation is used:
sin 2  
2
4a 2
(h 2  k 2  l 2 )
n is assumed to be 1
Θ values are determined from the diffraction pattern
Λ is wavelength of X-ray
UNIT CELL LATTICE PARAMETER REFINEMENT
 By
accurately
measuring
peak
positions over a long
range of 2theta and d
spacings,
we
can
determine
the
unit
cell lattice parameters
of the phases in our
sample by using the
following
formulas
for
different
the
crystal system.
INSTRUMENTAL SOURCES OF ERROR
Specimen displacement
 Instrument misalignment
 Error in zero 2θ position
 Peak distortion due to Kα2 and Kβ wavelengths
CONCLUSIONS
Non-destructive, fast, easy sample prep
High-accuracy for d-spacing calculations
Can be done in-situ
Single crystal, poly, and amorphous materials
Standards are available for thousands of material systems
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