X-RAY DIFFRACTION

1
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)
• X-Ray Diffraction A Practical Approach
• C. Suryanarayana M. Grant Norton
• Plenum Press, New York (1998)

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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
An accelerating (/decelerating) 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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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)
• Refraction of X-rays is neglected for now.

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Incoherent Scattering (Compton modified) From
loosely bound charges

• Here the particle picture of the electron
  photon comes in 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 with waves scattered
  from other atoms

Secondary emission
Incoming X-rays
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Sets Electron cloud into oscillation
Sets nucleus (with protons) 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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• Note that in the Braggs equation
• The interatomic spacing (a) along the plane does
  not appear
• Only the interplanar spacing (d) appears
• ? Change in position or spacing of atoms along
  the plane should not affect Braggs condition !!

Note shift (systematic) is actually not a
problem!
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Note shift is actually not a problem! ? Why is
systematic shift not a problem?
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Consider the case for which ?1 ? ?2
Constructive interference can still occur if the
difference in the path length traversed by R1 and
R2 before and after scattering are an integral
multiple of the wavelength ? (AY - XC) h ?
(h is an integer)
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Generalizing into 3D
Laues equations
?S0 ? incoming X-ray beam ?S ? Scattered X-ray
beam
This is looking at diffraction from atomic arrays
and not planes
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• A physical picture of scattering leading to
  diffraction is embodied in Laues equations
• Braggs method of visualizing diffraction as
  reflection from a set of planes is a different
  way of understanding the phenomenon of
  diffraction from crystals
• The plane picture (Braggs equations) are
  simpler and we usually stick to them
• Hence, we should think twice before asking the
  question if there are no atoms in the
  scattering planes, how are they scattering waves?

22

• 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
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

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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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Intensity of the Scattered electrons
Scattering by a crystal
A
Electron
Polarization factor
B
Atom
Atomic scattering factor (f)
C
Unit cell (uc)
Structure factor (F)
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A
Scattering by an Electron
Emission in all directions
Sets electron into oscillation
Coherent(definite phase relationship)
Scattered beams

• The electric field (E) is the main cause for the
  acceleration of the electron
• The moving particle radiates most strongly in a
  direction perpendicular to its motion
• The radiation will be polarized along the
  direction of its motion

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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) at a point P such that r gtgt ?
The reason we are able to neglect scattering from
the protons in the nucleus
The scattered rays are also plane polarized
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E is the measure of the amplitude of the wave E2
Intensity
For an unpolarized wave
IPy Intensity at point P due to Ey
Total Intensity at point P due to Ey Ez
IPz Intensity at point P due to Ez
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Sum of the squares of the direction cosines 1
Hence
?
In terms of 2?
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• In general P could lie anywhere in 3D space
• For the specific case of Bragg scatteringThe
  incident direction ? IOThe diffracted beam
  direction ? OPThe trace of the scattering plane
  ? BBAre all coplanar
• ? OP is constrained to be on the xz plane

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E is the measure of the amplitude of the wave E2
Intensity
For an unpolarized wave
IPy Intensity at point P due to Ey
The zx plane is ? to the y direction hence, ?
90?
IPz Intensity at point P due to Ez
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? Scattered beam is not unpolarized
Very small number

• Forward and backward scattered intensity higher
  than at 90?
• Scattered intensity minute fraction of the
  incident intensity

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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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B
Scattering by an Atom

• BRUSH-UP
• The conventional UC has lattice points as the
  vertices
• There may or may not be atoms located at the
  lattice points
• The shape of the UC is a parallelepiped (Greek
  parallelepipedon) in 3D
• There may be additional atoms in the UC due to
  two reasons? The chosen UC is non-primitive?
  The additional atoms may be part of the motif

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C
Scattering by the Unit cell (uc)

• Coherent Scattering
• Unit Cell (UC) is 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
41
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
42
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)
For n atoms in the UC
Structure factor is independent of the shape and
size of the unit cell
If the UC distorts so do the planes in it!!
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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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• If the blue planes are scattering in phase then
  on C- centering the red planes will scatter out
  of phase (with the blue planes- as they bisect
  them) and hence the (210) reflection will become
  extinct
• This analysis is consistent with the extinction
  rules (h k) odd is absent

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• In case of the (310) planes no new
  translationally equivalent planes are added on
  lattice centering ? this reflection cannot go
  missing.
• This analysis is consistent with the extinction
  rules (h k) even is present

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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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Mixed indices
Two odd and one even (e.g. 112) two even and one
odd (e.g. 122)
(h, k, l) mixed
e.g. (100), (211) (210), (032), (033)
All odd (e.g. 111) all even (e.g. 222)
Unmixed indices
(h, k, l) unmixed
e.g. (111), (200), (220), (333), (420)
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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
Mixed indices
(h, k, l) mixed
e.g. (100), (211) (210), (032), (033)
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Unmixed indices
(h, k, l) unmixed
e.g. (111), (222) (133), (244)
If (h k l) is even
e.g. (222),(244)
If (h k l) is odd
e.g. (111), (133)
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? Presence of additional atoms/ions/molecules in
the UC can alter the intensities of some of the
reflections
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Selection / Extinction Rules
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Reciprocal Lattice
Properties are reciprocal to the crystal lattice
BASIS VECTORS
B
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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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

• There are two ways of constructing the Reciprocal
  Crystal
• 1) Construct the lattice and decorate each
  lattice point with appropriate intensity
• 2) Use the concept as that for the real crystal

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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 Crystal 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 Crystal 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 Crystal BCC
Figures NOT to Scale
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In a strict sense this is not a crystal !!
Ordered Solid solution
High T disordered
BCC
470ºC
G H ? TS
Sublattice-1
Sublattice-2
SC
Low T ordered
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Ordered
BCC
FCC
Ordered
FCC
BCC
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• There are two ways of constructing the Reciprocal
  Crystal
• 1) Construct the lattice and decorate each
  lattice point with appropriate intensity
• 2) Use the concept as that for the real crystal

1) SC two kinds of Intensities decorating the
lattice 2) (FCC) (Motif 1FR 1SLR)
?? FR ? Fundamental Reflection?? SLR ?
Superlattice Reflection
1) SC two kinds of Intensities decorating the
lattice 2) (BCC) (Motif 1FR 3SLR)
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The Ewald Sphere
Paul Peter Ewald (German physicist and
crystallographer 1888-1985)
70
7. Paul-Peter-Ewald-Kolloquium Freitag, 17.
Juli 2008
organisiert von Max-Planck-Institut für
MetallforschungInstitut für Theoretische und
Angewandte Physik,Institut für
Metallkunde,Institut für Nichtmetallische
Anorganische Materialiender Universität
Stuttgart Programm
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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

See Cullitys book A15-4
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Braggs equation revisited

• Draw a circle with diameter 2/?
• Construct a triangle with the diameter as the
  hypotenuse and 1/dhkl as a side (any triangle
  inscribed in a circle with the diameter as the
  hypotenuse is a right angle triangle) AOP
• The angle opposite the 1/d side is ?hkl (from the
  rewritten Braggs equation)

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The Ewald Sphere construction
Crystal related information is present in the
reciprocal crystal
The Ewald sphere construction generates the
diffraction pattern
Radiation related information is present in the
Ewald Sphere
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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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Ewald sphere ? X-rays
?(Cu K?) 1.54 Å, 1/? 0.65 Å-1 (2/? 1.3
Å-1), aAl 4.05 Å, d111 2.34 Å, 1/d111 0.43
Å-1
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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
Cone of diffracted rays
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POWDER METHOD
Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections
Film may be replaced with detector
http//www.matter.org.uk/diffraction/x-ray/powder_
method.htm
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The 440 reflection is not observed
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The 331 reflection is not observed
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THE POWDER METHOD
Cubic crystal
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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
Altered in crystals with lower symmetry
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Multiplicity factor
Altered in crystals with lower symmetry (of the
same crystal class)
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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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THE POWDER METHOD
Cubic crystal
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Determination of Crystal Structure from 2? versus
Intensity Data
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FCC
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The ratio of (h2 K2 l2) derived from
extinction rules
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Powder diffraction pattern from Al
Radiation Cu K?, ? 1.54056 Å
111

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

311
220
200
420
331
422
222
400
?1 ?2 peaks resolved
X-Ray Diffraction A Practical Approach, C.
Suryanarayana M. Grant Norton, Plenum Press,
New York (1998)
93
Actually, the variation in 2? is to be seen
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Determination of Crystal Structure from 2? versus
Intensity Data
? ?1 , ?2 peaks are resolved (?1 peaks are
listed)
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Error in d spacing
For the same ?? the error in Sin? ? with ??
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Error in d spacing
Error in d spacing decreases with ?
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Applications of XRD
Bravais lattice determination
Lattice parameter determination
Determination of solvus line in phase diagrams
Long range order
Crystallite size and Strain
More
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Schematic of difference between the diffraction
patterns of various phases
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Crystallite size and Strain

• Braggs equation assumes? Crystal is perfect
  and infinite? Incident beam is perfectly
  parallel and monochromatic
• Actual experimental conditions are different from
  these leading various kinds of deviations from
  Braggs condition? Peaks are not ? curves ?
  Peaks are broadened
• There are also deviations from the assumptions
  involved in the generating powder patterns?
  Crystals may not be randomly oriented (textured
  sample) ? Peak intensities are altered

• In a powder sample if the crystallite size lt 0.5
  ?m? there are insufficient number of planes to
  build up a sharp diffraction pattern? peaks are
  broadened

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XRD Line Broadening
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XRD Line Broadening

• Unresolved ?1 , ?2 peaks ? Non-monochromaticity
  of the source (finite width of ? peak)
• Imperfect focusing

Instrumental
Bi
Crystallite size

• In the vicinity of ?B the -ve of Braggs
  equation not being satisfied

Bc
Strain

• Residual Strain arising from dislocations,
  coherent precipitates etc. leading to broadening

Bs
Stacking fault
In principle every defect contributes to some
broadening
Other defects
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Crystallite size

• Size gt 10 ?m ? Spotty ring (no. of grains in
  the irradiated portion insufficient to produce a
  ring)
• Size ? (10, 0.5) ? ? Smooth continuous ring
  pattern
• Size ? (0.5, 0.1) ? ? Rings are broadened
• Size lt 0.1 ? ? No ring pattern (irradiated
  volume too small to produce a diffraction ring
  pattern diffraction occurs only at low angles)

Spotty ring
Rings
Diffuse
Broadened Rings
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Effect of crystallite size on SAD patterns
Single crystal
Spotty pattern
Few crystals in the selected region
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Effect of crystallite size on SAD patterns
Ring pattern
Broadened Rings
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Subtracting Instrumental Broadening

• Instrumental broadening has to be subtracted to
  get the broadening effects due to the sample

• 1
• Mix specimen with known coarse-grained ( 10?m),
  well annealed (strain free) ? does not give any
  broadening due to strain or crystallite size (the
  only broadening is instrumental). A brittle
  material which can be ground into powder form
  without leading to much stored strain is good.
• If the pattern of the test sample (standard) is
  recorded separately then the experimental
  conditions should be identical (it is preferable
  that one or more peaks of the standard lies close
  to the specimens peaks)

• 2
• Use the same material as the standard as the
  specimen to be X-rayed but with large grain size
  and well annealed

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For a peak with a Lorentzian profile
Hendrik Antoon Lorentz
Longer tail
On the theory of reflection and refraction of
light
For a peak with a Gaussian profile
A geometric mean can also used
Johann Carl Friedrich Gauss (1777-1855), painted
by Christian Albrecht Jensen
University of Göttingen
http//en.wikipedia.org/wiki/Carl_Friedrich_Gauss
109
Scherrers formula
For Gaussian line profiles and cubic crystals

• ? ? Wavelength
• L ? Average crystallite size (? to surface of
  specimen)
• k ? 0.94 k ? (0.89, 1.39) 1 (the accuracy
  of the method is only 10)

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Strain broadening

• ? ? Strain in the material

Smaller angle peaksshould be used to separate
Bs and Bc
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Separating crystallite size broadening and strain
broadening
Plot of Br Cos? vs Sin?
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Example of a calculation
Sample Annealed AlRadiation Cu k? (? 1.54 Å)
Intensity ?
Sample Cold-worked AlRadiation Cu k? (? 1.54
Å)
2? ?
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Intensity ?
2? ?
40
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X-Ray Diffraction A Practical Approach, C.
Suryanarayana M. Grant Norton, Plenum Press,
New York (1998)
113
Annealed Al
Cold-worked Al
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end
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Iso-intensity circle
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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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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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421 missing
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Ewald sphere ? X-rays
?(Cu K?) 1.54 Å, 1/? 0.65 Å-1, aCu 3.61 Å,
1/aCu 0.28 Å-1
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Multiplicity factor