1. Today’s objective
We have learnt
•How do crystallites arrange in a polycrystalline
material
•How to represent polycrystal information in
stereographic projection
• To get an overview of diffraction phenomenon,
in general, and X-ray diffraction, in particular
2. X-Ray Diffraction (XRD): Suitable for the study
of the structure of crystalline materials
Why
The typical interatomic spacing in a crystal is of the
order of Å , the wavelength of X-ray is of the same order
This makes crystals to act as diffraction grating for X-
radiation
X-rays can be conveniently produced in Laboratory
3. • Interplanar spacing, hence lattice parameter
• Orientation of a single crystal or grain
• Measure the size, shape and internal strain of small
crystalline regions
• Crystal structure of an unknown material
Based on the diffraction principles, the following
can be measured in a crystalline materials:
4. Diffraction
• Diffraction is essentially a scattering phenomena where
at some particular angle the scattered radiation forms
constructive interface (arises when an electromagnetic
waves interact with the periodic structure )
• Diffraction is basically Reinforced Coherent Scattering
5. Understanding constructive interference
If two waves A and
B are propagating
in same phase, the
resulting wave C
will have magnitude
of addition of both
A and B
Bragg law is satisfied when the wavelength satisfies; nλ ≤ 2d
A
A
+
B
B
C =
λ
λ
λ
6. Constructive
interference will
occur when:
λ = AB + BC
AB=BC
n λ = 2AB
sin ɵ =AB/d
AB=d sin ɵ
n λ =2d sin ɵ
λ= 2dhklsin ɵhkl
A
B
C
ɵ
z
d
90-
ɵ
90-ɵ
ɵ
λ= 2dhklsinɵhkl
If 2ɵ: Bragg angle, and
λ: X-ray wavelength
Essentially, it gives relationship between the angle of
incidence ,wavelength of the incident radiation and the
spacing between parallel lattice plane of a crystal.
7. • There are three variables: λ,
ɵ, and d
λ is known (X-ray Source)
ɵ is measured in the
experiment (2ɵ)
• ‘d’ can be calculated using
the Bragg’s relation
• For the planes (hkl), the cell
parameter a can be
calculated
When the diffraction condition is met there will be a
diffracted X-ray beam
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
sin
)
(
sin
4
)
(
sin
4
2
l
k
h
a
l
k
h
l
k
h
a
l
k
h
a
d
dSin
8. θ - 2θ Scan
The θ - 2θ scan maintains these angles with the sample,
detector and X-ray source
Normal to surface
Only (hkl) planes of atoms that share the surface normal
will be seen in the θ - 2θ Scan
2θ
θ
surface
9. Crystal = Lattice + Motif
Diffraction from a crystal
• The structure of a crystal can be defined as:
• A beam of X-rays directed at a crystal interacts with
the electrons of the atoms in the crystal , undergoes
diffraction and gives rise to intensity distribution in the
diffracted output, which is characteristic of the crystal
structure. The output is known as diffraction pattern.
• Diffraction pattern consists of a set of peaks with
certain height (intensity) and spaced at certain intervals
(not the same interval between each of the peaks)
10. • Therefore, based on arrangement of atoms in a
crystal, intensities of particular diffraction peak is
modified sometimes the pattern go missing
• Lattice decides the position of the peaks (spacing
between the peaks), while the motif decides the
height of the peaks.
11. Examples of diffraction from crystals
Lattice = SC
No missing reflections 100 missing reflection (F = 0)
Lattice = BCC
Lattice = FCC
100 missing reflection (F = 0)
110 missing reflection (F = 0)
Extinction Rules
• Structure Factor (F):
The resultant wave
scattered by all atoms
of the unit cell
• The Structure Factor is
independent of the
shape and size of the
unit cell; but is
dependent on the
position of the atoms
within the cell
12. Bravais Lattice
Diffraction
Condition
Reflections
necessarily absent
Simple all None
Body centred (h + k + l) even (h + k + l) odd
Face centred h, k and l unmixed h, k and l mixed
h2 + k2 + l2 Simple Cubic Face Centred Cubic Body Centred Cubic
1 100
2 110 110
3 111 111
4 200 200 200
5 210
6 211 211
7
8 220 220 220
9 300, 221
Diffraction Extinction Criteria for different
materials with different crystals structures
14. 2θ1 2θ2
2θB
2θB
Why are the peaks broad??
θB: Bragg Angle
If uniform crystallites (no
misorientation within
them, imaginary)
But there is always some misorientation
within the grains (each crystallites always
misorientated with each other , reality)
Full width
at half
maximum
(FWHM)
Uniform
Crystallites
Misorientatio
n of
Crystallites
15. t = thickness of crystallite
K = constant dependent on crystallite shape (~ 0.89)
l = x-ray wavelength
B = FWHM (full width at half maximum)
ɵB = Bragg Angle
• Scherrer Formula: Relation between the crystallites size and the
FWHM
B
B
K
t
cos
• Crystallites Size: Essentially uniform agglomeration of crystals
posses same reflection patterns
Crystallite size <1000 Å
Error >20%
• Limitations of Scherrer equation
Peak broadening can be contributed by other factors, like, size,
strain and instrument
16. Questions
1. Out of the following, which can be measured using X-ray diffraction:
(a) Interplanar spacing, hence lattice parameter
(b) Orientation of a single crystal or grain
(c) Grain boundary character
(d) Size, shape and internal strain of small crystalline regions
2. Which of the following grain sizes can not be measured using X-ray
diffraction:
(a) 10 m
(b) 0.1 m
(c) 0.01 m
(d) 0.001 m
4. Determine the values of 2ɵ and (hkl) for the first three lines on the powder
patterns of substances with the following structures (Cu Kα=3.14Å) in FCC
unit cell (a = 3.00Å).
5. Calculate the crystallite size for FWHM B for = 10, 45, and 80°.
