X-RAY DIFFRACTION

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X-RAY DIFFRACTION

• X- Ray Sources
• Diffraction Braggs Law
• Crystal Structure Determination

Elements of X-Ray Diffraction B.D. Cullity
S.R. Stock Prentice Hall, Upper Saddle River
(2001)
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• For electromagnetic radiation to be diffracted
  the spacing in the grating should be of the
  same order as the wavelength
• In crystals the typical interatomic spacing
  2-3 Å so the suitable radiation is X-rays
• Hence, X-rays can be used for the study of
  crystal structures

Target
X-rays
Beam of electrons
A accelerating charge radiates electromagnetic
radiation
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Mo Target impacted by electrons accelerated by a
35 kV potential
K?
Characteristic radiation ? due to energy
transitions in the atom
K?
White radiation
Intensity
1.4
0.6
0.2
1.0
Wavelength (?)
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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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Incident X-rays
Heat
SPECIMEN
Fluorescent X-rays
Electrons
Scattered X-rays
Compton recoil
Photoelectrons
Coherent From bound charges
Incoherent (Compton modified) From loosely bound
charges
Transmitted beam
X-rays can also be refracted (refractive index
slightly less than 1) and reflected (at very
small angles)
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Incoherent Scattering (Compton modified) From
loosely bound charges

• Here the particle picture of the electron
  photon comes handy

Electron knocked aside
2?
No fixed phase relation between the incident and
scattered wavesIncoherent ? does not contribute
to diffraction (Darkens the background of the
diffraction patterns)
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Fluorescent X-rays
Knocked out electronfrom inner shell
Vacuum
Energylevels


Characteristic x-rays (Fluorescent
X-rays) (10-16s later ? seems like scattering!)
Nucleus
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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 will undergo constructive or
  destructive interference

Secondary emission
Incoming X-rays
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Sets Electron cloud into oscillation
Sets nucleus into oscillation Small effect ?
neglected
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Oscillating charge re-radiates ? In phase with
the incoming x-rays
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BRAGGs EQUATION
Deviation 2?
Ray 1
Ray 2
?
?
?
d
?
?
dSin?

• The path difference between ray 1 and ray 2 2d
  Sin?
• For constructive interference n? 2d Sin?

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In plane scattering is in phase
Incident and scattered waves are in phase if
Scattering from across planes is in phase
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Extra path traveled by incoming waves ? AY
These can be in phase if and only if ?
?incident ?scattered
Extra path traveled by scattered waves ? XB
But this is still reinforced scatteringand NOT
reflection
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• Braggs equation is a negative law? If Braggs
  eq. is NOT satisfied ? NO reflection can occur?
  If Braggs eq. is satisfied ? reflection MAY
  occur
• Diffraction Reinforced Coherent Scattering

Reflection versus Scattering
Reflection Diffraction
Occurs from surface Occurs throughout the bulk
Takes place at any angle Takes place only at Bragg angles
100 of the intensity may be reflected Small fraction of intensity is diffracted
X-rays can be reflected at very small angles of
incidence
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• n? 2d Sin?
• n is an integer and is the order of the
  reflection
• For Cu K? radiation (? 1.54 Å) and d110 2.22
  Å

n Sin? ?
1 0.34 20.7º First order reflection from (110)
2 0.69 43.92º Second order reflection from (110) Also written as (220)
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In XRD nth order reflection from (h k l) is
considered as 1st order reflectionfrom (nh nk nl)
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Crystal structure determination
Many ?s (orientations) Powder specimen
POWDER METHOD
Monochromatic X-rays
Single ?
LAUETECHNIQUE
Panchromatic X-rays
ROTATINGCRYSTALMETHOD
? Varied by rotation
Monochromatic X-rays
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THE POWDER METHOD
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Intensity of the Scattered electrons
C
A
B
Electron
Atom
Unit cell (uc)
Scattering by a crystal
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A
Scattering by an Electron
Emission in all directions
Sets electron into oscillation
Coherent(definite phase relationship)
Scattered beams
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For an polarized wave
z
P
r
For a wave oscillating in z direction
?
x
Intensity of the scattered beam due to an
electron (I)
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E is the measure of the amplitude of the wave E2
Intensityc
For an unpolarized wave
IPy Intensity at point P due to Ey
IPz Intensity at point P due to Ez
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? Scattered beam is not unpolarized
Very small number

• Rotational symmetry about x axis mirror
  symmetry about yz plane
• Forward and backward scattered intensity higher
  than at 90?
• Scattered intensity minute fraction of the
  incident intensity

Polarization factorComes into being as we used
unpolarized beam
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B
Scattering by an Atom
Scattering by an atom ? Atomic number, (path
difference suffered by scattering from each e-,
?)

• Angle of scattering leads to path differences
• In the forward direction all scattered waves are
  in phase

Scattering by an atom ? Z, (?, ?)
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Coherent scattering Incoherent (Compton) scattering
Z ? ? ?
Sin(?) / ? ? ? ?
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C
Scattering by the Unit cell (uc)

• Coherent Scattering
• Unit Cell (uc) representative of the crystal
  structure
• Scattered waves from various atoms in the uc
  interfere to create the diffraction pattern

The wave scattered from the middle plane is out
of phase with the ones scattered from top and
bottom planes
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Ray 1 R1
Ray 3 R3
?B
A
?
Unit Cell
x
S
R
Ray 2 R2
B
d(h00)
a
M
N
(h00) plane
C
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Independent of the shape of uc
Extending to 3D
Note R1 is from corner atoms and R3 is from
atoms in additional positions in uc
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In complex notation

• If atom B is different from atom A ? the
  amplitudes must be weighed by the respective
  atomic scattering factors (f)
• The resultant amplitude of all the waves
  scattered by all the atoms in the uc gives the
  scattering factor for the unit cell
• The unit cell scattering factor is called the
  Structure Factor (F)

Scattering by an unit cell f(position of the
atoms, atomic scattering factors)
Structure factor is independent of the shape and
size of the unit cell
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Structure factor calculations
Simple Cubic
A
Atom at (0,0,0) and equivalent positions
? F is independent of the scattering plane (h k l)
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B
C- centred Orthorhombic
Atom at (0,0,0) (½, ½, 0) and equivalent
positions
Real
(h k) even
Both even or both odd
e.g. (001), (110), (112) (021), (022), (023)
Mixture of odd and even
(h k) odd
e.g. (100), (101), (102) (031), (032), (033)
? F is independent of the l index
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Body centred Orthorhombic
C
Atom at (0,0,0) (½, ½, ½) and equivalent
positions
Real
(h k l) even
e.g. (110), (200), (211) (220), (022), (310)
(h k l) odd
e.g. (100), (001), (111) (210), (032), (133)
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D
Face Centred Cubic
Atom at (0,0,0) (½, ½, 0) and equivalent
positions
(½, ½, 0), (½, 0, ½), (0, ½, ½)
Real
(h, k, l) unmixed
e.g. (111), (200), (220), (333), (420)
(h, k, l) mixed
e.g. (100), (211) (210), (032), (033)
Two odd and one even (e.g. 112) two even and one
odd (e.g. 122)
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E
Na at (0,0,0) Face Centering Translations ?
(½, ½, 0), (½, 0, ½), (0, ½, ½) Cl- at (½, 0, 0)
FCT ? (0, ½, 0), (0, 0, ½), (½, ½, ½)
NaCl Face Centred Cubic
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Zero for mixed indices
(h, k, l) mixed
e.g. (100), (211) (210), (032), (033)
(h, k, l) unmixed
If (h k l) is even
If (h k l) is odd
? Presence of additional atoms/ions/molecules in
the uc (as a part of the motif ) can alter the
intensities of some of the reflections
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Relative Intensity of diffraction lines in a
powder pattern
Structure Factor (F)
Scattering from uc
Multiplicity factor (p)
Number of equivalent scattering planes
Polarization factor
Effect of wave polarization
Lorentz factor
Combination of 3 geometric factors
Absorption factor
Specimen absorption
Temperature factor
Thermal diffuse scattering
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Multiplicity factor
Lattice Index Multiplicity Planes
Cubic (100) 6 (100) (010) (001) (? 2 for negatives)
(110) 12 (110) (101) (011), (?110) (?101) (0?11) (? 2 for negatives)
(111) 8 (111) (11?1) (1?11) (?111) (? 2 for negatives)
(210) 24 (210) 3! Ways, (?210) 3! Ways, (2?10) 3! Ways, (?2?10) 3! Ways,
(211) 21
(321) 48
Tetragonal (100) 4 (100) (010)
(110) 4 (110) (?110)
(111) 8 (111) (11?1) (1?11) (?111) (? 2 for negatives)
(210) 6
(211) 21
(321) 48
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Lorentz factor
Polarization factor
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Intensity of powder pattern lines (ignoring
Temperature Absorption factors)

• Valid for Debye-Scherrer geometry
• I ? Relative Integrated Intensity
• F ? Structure factor
• p ? Multiplicity factor

• POINTS
• As one is interested in relative (integrated)
  intensities of the lines constant factors are
  omitted ? Volume of specimen ? me , e ?
  (1/dectector radius)
• Random orientation of crystals ? in a with
  Texture intensities are modified
• I is really diffracted energy (as Intensity is
  Energy/area/time)
• Ignoring Temperature Absorption factors ? valid
  for lines close-by in pattern

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Crystal Lattice Motif

• In crystals based on a particular lattice the
  intensities of particular reflections are
  modified ? they may even go missing

Diffraction Pattern
Position of the Lattice points ? LATTICE
Intensity of the diffraction spots ? MOTIF
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Reciprocal Lattice
Properties are reciprocal to the crystal lattice
The reciprocal lattice is created by interplanar
spacings
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• A reciprocal lattice vector is ? to the
  corresponding real lattice plane

• The length of a reciprocal lattice vector is the
  reciprocal of the spacing of the corresponding
  real lattice plane

• Planes in the crystal become lattice points in
  the reciprocal lattice ? ALTERNATE CONSTRUCTION
  OF THE REAL LATTICE
• Reciprocal lattice point represents the
  orientation and spacing of a set of planes

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Reciprocal Lattice
The reciprocal lattice has an origin!
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Note perpendicularity of various vectors
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• Reciprocal lattice is the reciprocal of a
  primitive lattice and is purely geometrical ?
  does not deal with the intensities of the points
• Physics comes in from the following
• For non-primitive cells (? lattices with
  additional points) and for crystals decorated
  with motifs (? crystal lattice motif) the
  Reciprocal lattice points have to be weighed in
  with the corresponding scattering power (Fhkl2)
  ? Some of the Reciprocal lattice points go
  missing (or may be scaled up or down in
  intensity)? Making of Reciprocal Crystal
  (Reciprocal lattice decorated with a motif of
  scattering power)
• The Ewald sphere construction further can select
  those points which are actually observed in a
  diffraction experiment

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Examples of 3D Reciprocal Lattices weighed in
with scattering power (F2)
SC
001
011
111
101
Lattice SC
000
010
100
110
No missing reflections
Reciprocal Lattice SC
Figures NOT to Scale
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002
022
BCC
202
222
011
101
020
000
Lattice BCC
110
200
100 missing reflection (F 0)
220
Reciprocal Lattice FCC
Weighing factor for each point motif
Figures NOT to Scale
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002
022
FCC
202
222
111
020
000
Lattice FCC
200
220
100 missing reflection (F 0)
110 missing reflection (F 0)
Weighing factor for each point motif
Reciprocal Lattice BCC
Figures NOT to Scale
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The Ewald Sphere

• The reciprocal lattice points are the values of
  momentum transfer for which the Braggs equation
  is satisfied
• For diffraction to occur the scattering vector
  must be equal to a reciprocal lattice vector
• Geometrically ? if the origin of reciprocal space
  is placed at the tip of ki then diffraction will
  occur only for those reciprocal lattice points
  that lie on the surface of the Ewald sphere

Paul Peter Ewald (German physicist and
crystallographer 1888-1985)
See Cullitys book A15-4
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Ewald Sphere
The Ewald Sphere touches the reciprocal lattice
(for point 41) ? Braggs equation is satisfied
for 41
?K K ?g Diffraction Vector
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Diffraction cones and the Debye-Scherrer geometry
Film may be replaced with detector
http//www.matter.org.uk/diffraction/x-ray/powder_
method.htm
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Powder diffraction pattern from Al
Radiation Cu K?, ? 1.54 Å
111

• Note
• Peaks or not idealized ? peaks ? broadend
• Increasing splitting of peaks with ?g ?
• Peaks are all not of same intensity

311
220
200
331
422
420
222
400
?1 ?2 peaks resolved
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Determination of Crystal Structure from 2? versus
Intensity Data
n 2? ? Sin? Sin2 ? ratio Index
1 38.52 19.26 0.33 0.11 3 111
2 44.76 22.38 0.38 0.14 4 200
3 65.14 32.57 0.54 0.29 8 220
4 78.26 39.13 0.63 0.40 11 311
5 82.47 41.235 0.66 0.43 12 222
6 99.11 49.555 0.76 0.58 16 400
7 112.03 56.015 0.83 0.69 19 331
8 116.60 58.3 0.85 0.72 20 420
9 137.47 68.735 0.93 0.87 24 422
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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
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Extinction Rules
Bravais Lattice Reflections which may be present 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
End centred h and k unmixed C centred h and k mixedC centred
Bravais Lattice Allowed Reflections
SC All
BCC (h k l) even
FCC h, k and l unmixed
DC h, k and l are all oddOrall are even(h k l) divisible by 4
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Determination of Crystal Structure from 2? versus
Intensity Data
n 2?? ? Intensity Sin? Sin2 ? ratio





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The ratio of (h2 K2 l2) derived from
extinction rules
SC 1 2 3 4 5 6 8
BCC 1 2 3 4 5 6 7
FCC 3 4 8 11 12
DC 3 8 11 16
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2?? ? Intensity Sin? Sin2 ? ratio
1 21.5 0.366 0.134 3
2 25 0.422 0.178 4
3 37 0.60 0.362 8
4 45 0.707 0.500 11
5 47 0.731 0.535 12
6 58 0.848 0.719 16
7 68 0.927 0.859 19
FCC
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h2 k2 l2 SC FCC BCC DC
1 100
2 110 110
3 111 111 111
4 200 200 200
5 210
6 211 211
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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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16 400 400 400 400
17 410, 322
18 411, 330 411, 330
19 331 331 331
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Consider the compound ZnS (sphalerite). Sulphur
atoms occupy fcc sites with zinc atoms displaced
by ¼ ¼ ¼ from these sites. Click on the animation
opposite to show this structure. The unit cell
can be reduced to four atoms of sulphur and 4
atoms of zinc. Many important compounds adopt
this structure. Examples include ZnS, GaAs, InSb,
InP and (AlGa)As. Diamond also has this
structure, with C atoms replacing all the Zn and
S atoms. Important semiconductor materials
silicon and germanium have the same structure as
diamond.
Structure factor calculation Consider a general
unit cell for this type of structure. It can be
reduced to 4 atoms of type A at 000, 0 ½ ½, ½ 0
½, ½ ½ 0 i.e. in the fcc position and 4 atoms of
type B at the sites ¼ ¼ ¼ from the A sites. This
can be expressed as The structure factors for
this structure are F 0 if h, k, l mixed (just
like fcc) F 4(fA ifB) if h, k, l all odd F
4(fA - fB) if h, k, l all even and h k l 2n
where nodd (e.g. 200) F 4(fA fB) if h, k, l
all even and h k l 2n where neven (e.g. 400)
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Applications of XRD
Bravais lattice determination
Scattering from uc
Lattice parameter determination
Number of equivalent scattering planes
Determination of solvus line in phase diagrams
Effect of wave polarization
Long range order
Combination of 3 geometric factors
Crystallite size and Strain
Specimen absorption
Temperature factor
Thermal diffuse scattering
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Schematic of difference between the diffraction
patterns of various phases
300
310
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