2. X – Ray Diffraction
STUDY OF CRYSTAL STRUCTURE.
2
BY,
ASSISTANT PROFESSOR RINKESH KURKURE
HEAD DEPARTMENT OF PHYSICS,
BHARAT COLLEGE, BADLAPUR(W)
3. OVERVIEW
Crystal structures and atoms position in cubic cell with plane.
Electromagnetic Spectrum and X- ray
Production of X-rays.
Interaction of X-rays with matter.
What can we analyzed by X- ray
Bragg’s Law
Scherrer’s Formula
Data analysis of XRD.
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4. Crystal structures and
atoms position in cubic
cell with planes
A CRYSTAL STRUCTURE IS A
UNIQUE ARRANGEMENT OF
ATOMS, IONS OR
MOLECULES
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5. SPACE LATTICE AND UNIT CELL.
Atoms, arranged in repetitive 3-Dimensional pattern,
in long range order (LRO) give rise to crystal
structure.
Properties of solids depends upon crystal structure
and bonding force.
An imaginary network of lines, with atoms at
intersection of lines, representing the arrangement of
atoms is called space lattice.
Unit cell is that block of atoms which repeats itself to
form space lattice.
Materials arranged in short range order are called
amorphous materials.
Unit Cell
Space Lattice
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6. BRAVIAS LATTICE
Cubic Unit Cell
a = b = c
α = β = γ = 900
Simple Cubic Body Centered Face centered Simple Body Centered
Tetragonal
a =b ≠ c
α = β = γ = 900
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7. BRAVIAS LATTICE.
Orthorhombic
a ≠ b ≠ c
α = β = γ = 900
Simple
Base Centered
Face Centered Body Centered Simple
Rhombohedral
a = b = c
α = β = γ ≠ 900
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8. BRAVIAS LATTICE
Hexagonal
a = b ≠ c
α = β =900 γ = 1200
Hexagonal Monoclinic TriclinicBase Centered
Monoclinic
Monoclinic
a ≠ b ≠ c
α = γ = 900 β ≠ 900
Triclinic
a ≠ b ≠ c
α ≠ β ≠ γ ≠ 900
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9. Atom Positions
in Cubic Unit
Cells
Atom positions are located using unit
distances along the axes.
In cubic crystals, Direction Indices are vector
components of directions resolved along
each axes, resolved to smallest integers.
Direction indices are position coordinates of
unit cell where the direction vector emerges
from cell surface, converted to integers.
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10. Atom Positions in Cubic Unit Cells
Cartesian coordinate system is use to locate atoms.
In a cubic unit cell
Y axis is the direction to the right.
X axis is the direction coming out of the paper.
Z axis is the direction towards top.
Negative directions are to the opposite of positive
directions.
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11. ELECTROMAGNETIC
SPECTRA
The electromagnetic spectrum is the
range of all possible frequencies of
electromagnetic radiation.
The "electromagnetic spectrum" of an
object has a different meaning, and is
instead the characteristic distribution of
electromagnetic radiation emitted or
absorbed by that particular object.
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14. X-RAYS – PRODUCTION
Information about crystal structure are obtained using X-Rays.
The X-rays used are about the same wavelength (0.05-0.25 nm) as distance between
crystal lattice planes.
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15. CHARACTERISTIC X-RAYS
X-Ray spectrum of Molybdenum is obtained
when Molybdenum is used as target metal.
Kα and Kβ are characteristic of an element.
For Molybdenum Kα occurs at wave length of
about 0.07nm.
Electrons of n=1 shell of target metal are
knocked out by bombarding electrons.
Electrons of higher level drop down by
releasing energy to replace lost electrons
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16. INTERACTION OF X-RAYS WITH
MATTER
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19. What can we analyzed by X- rayABSORPTION
Photoelectron
effect
Photoelectron XPS
Florescence
X-ray
XRF
Auger electron AES
Scattering
Thomson scattering XRD
CRYSTAL
STRUCTURE
ANALYSIS
Compton Scattering
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21. BRAGG’S LAW
For rays reflected from different planes to be in
phase, the extra distance traveled by a ray should
be a integral multiple of wave length λ .
nλ = MP + PN (n = 1,2…)
n is order of diffraction
If dhkl is interplanar distance,
Then MP = PN = dhkl.Sinθ
Therefore,
λ = 2 dhkl.Sinθ
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23. Now for two planes A and B we have
2 2 2 2
2
2
2 2 2 2
2
2
( )
4
( )
4
A A A
A
B B B
B
h k l
Sin
a
h k l
Sin
a
Dividing each other, we get
)(
)(
222
222
2
2
BBB
AAA
B
A
lkh
lkh
Sin
Sin
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25. Smaller Crystals Produce Broader XRD Peaks
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26. t = thickness of crystallite
K = constant dependent on crystallite shape (0.89)
= x-ray wavelength
B = FWHM (full width at half max) or integral breadth
B = Bragg Angle
Scherrer’s Formula
BcosB
K
t
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27. What is B?
B = (2θ High) – (2θ Low)
B is the difference in angles at half
max
2θ high
Noise
2θ low
Peak
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Scherrer’s Formula
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28. When to Use Scherrer’s Formula
Crystallite size <1000 Å
Peak broadening by other factors
Causes of broadening
Size
Strain
Instrument
If breadth consistent for each peak then assured
broadening due to crystallite size
K depends on definition of t and B
Within 20%-30% accuracy at best 5/27/2016
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29. Bragg Example
d = λ / (2 Sin θB) λ = 1.54 Ǻ
= 1.54 Ǻ / ( 2 * Sin ( 38.3 / 2 ) )
= 2.35 Ǻ
INTERPRETATION OF EXPRIMENTAL
DATA
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30. Au Foil
98.25 (400)
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
95 95.5 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 101 101.5 102
2 Theta
Counts
Scherrer’s Example
BB
t
cos
89.0
t = 0.89*λ / (B Cos θB) λ = 1.54 Ǻ
= 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) )
= 1200 Ǻ
B = (98.3 - 98.2)*π/180 = 0.00174
INTERPRETATION OF EXPRIMENTAL
DATA
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31. INTERPRETATION OF EXPRIMENTAL
DATA
For BCC crystals, the first two sets of diffracting planes are {110} and {200}
planes.
Therefore
For FCC crystals the first two sets of diffracting planes are {111} and {200}
planes
Therefore
75.0
)002(
)111(
222
222
2
2
B
A
Sin
Sin
5.0
)002(
)011(
222
222
2
2
B
A
Sin
Sin
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Identification of Crystal structure
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35. Short cut
For BCC structure, diffraction occurs only on planes
whose miller indices when added together total to an
even number.
I.e. (h+k+l) = even Reflections present
(h+k+l) = odd Reflections absent
For FCC structure, diffraction occurs only on planes
whose miller indices are either all even or all odd.
I.e. (h,k,l) all even Reflections present
(h,k,l) all odd Reflections present
(h,k,l) not all even or all odd Reflections
absent.
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