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X-ray Diffraction and Its Applications in Science &
Engineering
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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 .
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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
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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
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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).
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INDEXING OF PLANES AND DIRECTIONS IN CUBIC SYSTEMS
z
x
ya
b
c
X
y
z
a
b
c
(100) (110)
(111)
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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
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BASICS OF X-RAY DIFFRACTION
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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.
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ELECTROMAGNETIC SPECTRUM
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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.
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e-
e-
e-e-
ORIGIN OF X-RAY
Continuous X-ray Characteristics X-ray
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Kβ will give extra peak in the XRD pattern which can be eliminated
by adding filters.
K1
K1K2
K2
SPECTRAL CONTAMINATION IN DIFFRACTION PATTERNS
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Bragg’s Law is used to expalin the
intereference pattern of the X-rays
scattered by the crystals
sin2 hkldn Where,n an integerλ wavelength of the incident X-raydhkl interplanar spacing
BRAGG’s LAW
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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.
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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
ESSENTIAL PARTS OF THE DIFFRACTOMETER
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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
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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.
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20 6040
2θ (degrees)
Inte
nsity
X-RAY DIFFRACTION PATTERN OF AIR
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X-RAY DIFFRACTION PATTERN OF AMORPHOUS SOLIDS
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Dried ZrO2
Ceria
CERIA ZrO2
XRD PATTERNS OF NANO-PARTICLES
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DIFFRACTION PATTERN OF A SINGLE CRYSTAL
A single crystal will produce only one family of peaks in the diffraction pattern
INTE
NSI
TY
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DIFFRACTION PATTERN OF A POLYCRYSTALLINE SAMPLEIN
TEN
SITY
A polycrystalline samples contain thousands of crystallites, therefore all possible diffraction peaks should be observed.
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EXTINCTION RULES FOR CUBIC CRYSTALS
Bravais Lattice Allowed Reflections
SC All
BCC (h + k + l) even
FCC h, k and l unmixed
DCh, k and l are all odd
Orall are even
& (h + k + l) divisible by 4
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h2 + k2 + l2 SC BCC FCC DC
1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220 220
9 300, 221
10 310 310
11 311 311 311
12 222 222 222
13 320
14 321 321
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1. Phase identification
2. Volume fraction of the phases
3. Crystallite size
4. Strain
INFORMATION PROCURRED FROM X-RAY DATA
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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).
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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)
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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.
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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
23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 412 (deg.)
Inte
nsity
(a.u
.)
00-043-1002> Cerianite- - CeO2
maxima halfat width Full
ray-X of wavelength
cos2
KB
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EFFECT OF LATTICE STRAIN IN DIFFRACTION PEAK AND POSITION
NO STRAIN
Uniform Strain(d1-d0)/d0
Peak moves, no shape changes
Non-Uniform Straind1# constantPeak broadens
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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
Continued……………..
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STRUCTURAL DETERMINATION
To determine the structure of monoatomic cubic crystals, the following equation is used:
)(4
sin 2222
22 lkh
a
n is assumed to be 1
Θ values are determined from the diffraction pattern
Λ is wavelength of X-ray
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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 the different
crystal system.
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INSTRUMENTAL SOURCES OF ERROR
Specimen displacement
Instrument misalignment
Error in zero 2θ position
Peak distortion due to Kα2 and Kβ wavelengths
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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
