X-ray Diffraction: Principles and Practice

1
X-ray Diffraction Principles and Practice

• Ashish Garg and Nilesh Gurao
• Department of Materials Science and Engineering
• Indian Institute of Technology Kanpur

2
Layout of the Lecture

• Materials Characterization
• Importance of X-ray Diffraction
• Basics
• Diffraction
• X-ray Diffraction
• Crystal Structure and X-ray Diffraction
• Different Methods
• Phase Analysis
• Texture Analysis
• Stress Analysis
• Particles Size Analysis
• ..
• Summary

3
Materials Characterization

• Essentially to evaluate the structure and
  properties
• Structural Characterization
• Diffraction
• X-ray and Electron Diffraction
• Microscopy
• Spectroscopy
• Property Evaluation
• Mechanical
• Electrical
• Anything else

4
Time Line

• 1665 Diffraction effects observed by Italian
  mathematician Francesco Maria Grimaldi
• 1868 X-rays Discovered by German Scientist
  Röntgen
• 1912 Discovery of X-ray Diffraction by Crystals
  von Laue
• 1912 Braggs Discovery

5
Electromagnetic Spectrum
6
Generation of X-rays
7
Commercial X-ray Tube
8
X-ray Spectrum from an Iron target

• Short Wavelength Limit
• Continuous spectrum
• Characteristic X-ray Moseleys Law

9
Use of Filter

• Ni filter for Cu Target

10
Crystal Structures

• Lattice
• Point lattice
• Motif
• Lattice Parameters

11
Primitive vs Non-primitive

• Primitive Unit-cell has only one lattice point
  per unit cell.

12
Crystal Systems and Bravais Lattices
13
Crystal Planes
14
Crystal Directions
How to locate a direction Example 231
direction would be 1/3 intercept on cell
a-length 1/2 intercept on cell b-length and 1/6
intercept on cell c-length Directions are always
denoted with uvw with square brackets and
family of directios in the form ltuvwgt
15
Stereographic Projection
2-D projection of poles of planes in a crystal
16
Structure of Common Materials

• Metals
• Copper FCC
• ?-Iron BCC
• Zinc HCP
• Silver FCC
• Aluminium FCC
• Ceramics
• SiC Diamond Cubic
• Al2O3 Hexagonal
• MgO NaCl type

17
Diffraction

• A diffracted beam may be defined as a beam
  composed of a large number of scattered rays
  mutually reinforcing each other

18
Scattering Modes

• Random arrangement of atoms in space gives rise
  to scattering in all directions weak effect and
  intensities add
• By atoms arranged periodically in space
• In a few specific directions satisfying Braggs
  law strong intensities of the scattered beam
  Diffraction
• No scattering along directions not satisfying
  Braggs law

19
Types of Diffraction

• Fresnel near field
• Fraunhofer far field

Spherical wavefront
Parabolic wavefront
Planar wavefront
Rayleigh- Sommerfield
Fresenel
Fraunhofer
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Diffraction of light through an
aperture
d
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Minima
Maxima
n 1, 2,..
n 0, 1,..
23
Youngs Double slit experiment
Constructive Interference
d sin? m?, m 1,2,3..
d sin? (m½)?, m 1,2,3..
Destructive Interference
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Interference
25
Interference and Diffraction
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Braggs Law

• n?2d.sin?
• n Order of reflection
• d Plane spacing
• ? Bragg Angle

Path difference must be integral multiples of the
wavelength ?in?out
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Braggs Law
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Geometry of Braggs law

• The incident beam, the normal to the reflection
  plane, and the diffracted beam are always
  co-planar.
• The angle between the diffracted beam and the
  transmitted beam is always 2? (usually measured).
• Sin ? cannot be more than unity this requires
  n? lt 2d, for n1, ? lt 2d
• ? should be less than twice the d spacing we
  want to study

29
Order of reflection

• Rewrite Braggs law ?2 sin? d/n
• A reflection of any order as a first order
  reflection from planes, real or fictitious,
  spaced at a distance 1/n of the previous spacing
• Set d d/n
• An nth order reflection from (hkl) planes of
  spacing d may be considered as a first order
  reflection from the (nh nk nl) plane of spacing
  d d/n

?2d sin?
The term reflection is only notional due to
symmetry between incoming and outgoing beam
w.r.t. plane normal, otherwise we are only
talking of diffraction.
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Reciprocal lattice vectors

• Used to describe Fourier analysis of electron
  concentration of the diffracted pattern.
• Every crystal has associated with it a crystal
  lattice and a reciprocal lattice.
• A diffraction pattern of a crystal is the map of
  reciprocal lattice of the crystal.

31
Real space Reciprocal space Crystal
Lattice
Reciprocal Lattice Crystal
structure Diffraction
pattern Unit cell content
Structure factor
x
y
y
x
y
x
32
Reciprocal space
Reciprocal lattice of FCC is BCC and vice versa
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Ewald sphere
k'
k
Ewald sphere
Limiting sphere
34
Ewald sphere
J. Krawit, Introduction to Diffraction in
Materials Science and Engineering, Wiley New York
2001
35
Braggs Law
Real space
Reciprocal space
S
?
?
?
?
S0
S0
36
Orienting crystal
Intensity vs 2?
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Diffraction vs Reflection

• Diffraction occurs from subsurface atoms where as
  reflection is a surface phenomenon
• Diffraction occurs at only specific angles where
  as reflection can occur at any angle
• Despite this we call diffracting planes as
  reflecting planes

????
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Two Circle Diffractometer

• For polycrystalline Materials

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Four Circle Diffractometer

• For single crystals

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• 2 Circle diffratometer ? 2? and ?
• 3 and 4 circle diffractometer ? 2?, ?, f, ?
• 6 circle diffractometer ? ?, f, ? and d, ?, µ

www.serc.carleton.edu/
Hong et al., Nuclear Instruments and Methods in
Physics Research A 572 (2007) 942
41
NaCl crystals in a tube facing X-ray beam
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Powder Diffractometer
43
Calculated Patterns for a Cubic Crystal
44
Structure Factor
Intensity of the diffracted beam ? F2

• h,k,l indices of the diffraction plane under
  consideration
• u,v,w co-ordinates of the atoms in the lattice
• N number of atoms
• fn scattering factor of a particular type of
  atom

Bravais Lattice Reflections possibly present Reflections necessarily absent
Simple All None
Body Centered (hkl) Even (hkl) Odd
Face Centered h, k, and l unmixed i.e. all odd or all even h, k, and l mixed
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Systematic Absences
Permitted Reflections
Simple Cubic (100), (110), (111), (200), (210), (211), (220), (300), (221)
BCC (110), (200), (211), (220), (310), (222).
FCC (111), (200), (220), (311)..
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Diffraction Methods
Method Wavelength Angle Specimen
Laue Variable Fixed Single Crystal
Rotating Crystal Fixed Variable (in part) Single Crystal
Powder Fixed Variable Powder
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Laue Method

• Uses Single crystal
• Uses White Radiation
• Used for determining crystal orientation and
  quality

48
Rotating Crystal Method
49
Powder Method

• Useful for determining lattice parameters with
  high precision and for identification of phases

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Indexing a powder pattern
Braggs Law n? 2d sin? For cubic crystals
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Indexing
FCC wavelength1.54056Å FCC wavelength1.54056Å FCC wavelength1.54056Å FCC wavelength1.54056Å FCC wavelength1.54056Å FCC wavelength1.54056Å
S1 (mm) ?(?) sin2? h2k2l2 sin2?/ h2k2l2 Lattice Parameter, a (Å)
38 19.0 0.11 3 0.037 4.023
45 22.5 0.15 4 0.038 3.978
66 33.0 0.30 8 0.038 3.978
78 39.0 0.40 11 0.036 4.039
83 41.5 0.45 12 0.038 3.978
97 49.5 0.58 16 0.036 4.046
113 56.5 0.70 19 0.037 4.023
118 59.0 0.73 20 0.037 4.023
139 69.5 0.88 24 0.037 4.023
168 84.9 0.99 27 0.037 4.023
Constant so it is FCC Constant so it is FCC
BCC BCC BCC BCC BCC BCC
S1 (mm) ?(?) sin2? h2k2l2 sin2?/ h2k2l2 Not BCC
38 19.0 0.11 2 0.055 Not BCC
45 22.5 0.15 4 0.038 Not BCC
66 33.0 0.30 6 0.050 Not BCC
78 39.0 0.40 8 0.050 Not BCC
83 41.5 0.45 10 0.045 Not BCC
97 49.5 0.58 12 0.048 Not BCC
113 56.5 0.70 14 0.050 Not BCC
118 59.0 0.73 16 0.046 Not BCC
139 69.5 0.88 18 0.049 Not BCC
168 84.9 0.99 20 0.050 Not BCC
Not Constant Not BCC
Simple Cubic Simple Cubic Simple Cubic Simple Cubic Simple Cubic Simple Cubic
S1 (mm) ?(?) sin2? h2k2l2 sin2?/ h2k2l2 Not Simple Cubic
38 19.0 0.11 1 0.11 Not Simple Cubic
45 22.5 0.15 2 0.75 Not Simple Cubic
66 33.0 0.30 3 0.10 Not Simple Cubic
78 39.0 0.40 4 0.10 Not Simple Cubic
83 41.5 0.45 5 0.09 Not Simple Cubic
97 49.5 0.58 6 0.097 Not Simple Cubic
113 56.5 0.70 8 0.0925 Not Simple Cubic
118 59.0 0.73 9 0.081 Not Simple Cubic
139 69.5 0.88 10 0.088 Not Simple Cubic
168 84.9 0.99 11 0.09 Not Simple Cubic
Not Constant Not Simple Cubic
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Current Setup
User can choose to move the sample and detector
(fixed source) or detector and source (sample
stage fixed) at the same time
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Diffraction from a variety of materials

• (From Elements of X-ray Diffraction, B.D.
  Cullity, Addison Wesley)

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Reality
Instrumental broadening must be subtracted

• (From Elements of X-ray Diffraction, B.D.
  Cullity, Addison Wesley)

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Intensity of diffracted beam

• polarization factor
• structure factor (F2)
• multiplicity factor
• Lorentz factor
• absorption factor
• temperature factor

• For most materials the peaks and their
  intensity are documented
• JCPDS
• ICDD

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Name and formula Reference code 00-001-1260 PDF
index name Nickel Empirical formula Ni Chemical
formula Ni Crystallographic parameters Crystal
system Cubic Space group Fm-3m Space group
number 225 a (Å) 3.5175 b (Å) 3.5175 c
(Å) 3.5175 Alpha () 90.0000 Beta ()
90.0000 Gamma () 90.0000 Measured density
(g/cm3) 8.90 Volume of cell (106 pm3)
43.52 Z 4.00 RIR - Status, subfiles and
quality Status Marked as deleted by
ICDD Subfiles Inorganic Quality Blank
(B) References Primary reference Hanawalt et
al., Anal. Chem., 10, 475, (1938) Optical
data Data on Chem. for Cer. Use, Natl. Res.
Council Bull. 107 Unit cell The Structure of
Crystals, 1st Ed.
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Stick pattern from JCPDS
http//ww1.iucr.org/cww-top/crystal.index.html
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Actual Pattern
Lattice parameter, phase diagrams Texture, Strain
(micro and residual) Size, microstructure (twins
and dislocations)
Bulk electrodeposited nanocrystalline nickel
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Powder X-ray diffraction is essentially a
misnomer and should be replaced by
Polycrystalline X-ray diffraction
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Information in a Diffraction Pattern

• Phase Identification
• Crystal Size
• Crystal Quality
• Texture (to some extent)
• Crystal Structure

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Analysis of Single Phase
2?() d (Å) (I/I1)100
27.42 3.25 10
31.70 2.82 100
45.54 1.99 60
53.55 1.71 5
56.40 1.63 30
65.70 1.42 20
76.08 1.25 30
84.11 1.15 30
89.94 1.09 5
I1 Intensity of the strongest peak
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Procedure

• Note first three strongest peaks at d1, d2, and
  d3
• In the present case d1 2.82 d2 1.99 and d3
  1.63 Å
• Search JCPDS manual to find the d group belonging
  to the strongest line between 2.84-2.80 Å
• There are 17 substances with approximately
  similar d2 but only 4 have d1 2.82 Å
• Out of these, only NaCl has d3 1.63 Å
• It is NaClHurrah

Specimen and Intensities Substance File Number
2.829 1.999 2.26x 1.619 1.519 1.499 3.578 2.668 (ErSe)2Q 19-443
2.82x 1.996 1.632 3.261 1.261 1.151 1.411 0.891 NaCl 5-628
2.824 1.994 1.54x 1.204 1.194 2.443 5.622 4.892 (NH4)2WO2Cl4 22-65
2.82x 1.998 1.263 1.632 1.152 0.941 0.891 1.411 (BePd)2C 18-225
Caution It could be much more tricky if the
sample is oriented or textured or your goniometer
is not calibrated
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Presence of Multiple phases
d (Å) I/I1
3.01 5
2.47 72
2.13 28
2.09 100
1.80 52
1.50 20
1.29 9
1.28 18
1.22 4
1.08 20
1.04 3
0.98 5
0.91 4
0.83 8
0.81 10

• More Complex
• Several permutations combinations possible
• e.g. d1 d2 and d3, the first three strongest
  lines show several alternatives
• Then take any of the two lines together and match
• It turns out that 1st and 3rd strongest lies
  belong to Cu and then all other peaks for Cu can
  be separated out
• Now separate the remaining lines and normalize
  the intensities
• Look for first three lines and it turns out that
  the phase is Cu2O
• If more phases, more pain to solve ?

Remaining Lines Remaining Lines Remaining Lines
d (Å) I/I1 I/I1
d (Å) Observed Normalized
3.01 5 7
2.47 72 100
2.13 28 39
1.50 20 28
1.29 9 13
1.22 4 6
0.98 5 7

Pattern of Cu2O Pattern of Cu2O
d (Å) I/I1
3.020 9
2.465 100
2.135 37
1.743 1
1.510 27
1.287 17
1.233 4
1.0674 2
0.9795 4
Pattern for Cu Pattern for Cu
d (Å) I/I1
2.088 100
1.808 46
1.278 20
1.09 17
1.0436 5
0.9038 3
0.8293 9
0.8083 8
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Lattice Strain
?? ? ?d ? strain
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Texture in Materials

• Grains with in a polycrystalline are not
  completely randomly distributed
• Clustering of grains about some particular
  orientation(s) to a certain degree
• Examples
• Present in cold-rolled brass or steel sheets
• Cold worked materials tend to exhibit some
  texture after recrystallization
• Affects the properties due to anisotropic nature

66
Texture

• Fiber Texture
• A particular direction uvw for all grains is
  more or less parallel to the wire or fiber axis
• e.g. 111 fiber texture in Al cold drawn wire
• Double axis is also possible
• Example 111 and 100 fiber textures in Cu
  wire
• Sheet Texture
• Most of the grains are oriented with a certain
  crystallographic plane (hkl) roughly parallel to
  the sheet surface and certain direction uvw
  parallel to the rolling direction
• Notation (hkl)uvw

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Texture in materials
? uvw i.e. perpendicular to the surface of all
grains is parallel to a direction uvw
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Pole Figures

• (100) pole figures for a sheet material
• (a) Random orientation (b) Preferred orientation

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Thin Film Specimen
Film or Coating
Substrate

• Smaller volume i.e. less intensity of the
  scattered beam from the film
• Grazing angle
• Useful only for polycrystalline specimens

70
Thin Film XRD

• Precise lattice constants measurements derived
  from 2?-? scans, which provide information about
  lattice mismatch between the film and the
  substrate and therefore is indicative of strain
  stress
• Rocking curve measurements made by doing a q scan
  at a fixed 2 ? angle, the width of which is
  inversely proportionally to the dislocation
  density in the film and is therefore used as a
  gauge of the quality of the film.
• Superlattice measurements in multilayered
  heteroepitaxial structures, which manifest as
  satellite peaks surrounding the main diffraction
  peak from the film. Film thickness and quality
  can be deduced from the data.
• Glancing incidence x-ray reflectivity
  measurements, which can determine the thickness,
  roughness, and density of the film. This
  technique does not require crystalline film and
  works even with amorphous materials.

71
Thin Films Specimens
?B1??B2 i.e. No Diffraction from hkl plane
Diffraction from hkl plane occurs
Single Crystal Substrate
Single Crystal Substrate

• If the sample and substrate is polycrystalline,
  then problems are less
• But if even if one of them is oriented, problems
  arise
• In such situations substrate alignment is
  necessary

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Oriented thin films

• Bismuth Titanate thin films on oriented SrTiO3
  substrates
• Only one type of peaks
• It apparent that films are highly oriented

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Degree of orientation
Side view
But what if the planes when looked from top have
random orientation?
Top view
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Pole Figure
SrTiO3 (110)
SrTiO3 (111)
SrTiO3 (100)

• 4 Peaks at 50?
• Excellent in-plane orientation

• 2 sets of peaks at 5, 65 and 85
• Indicating a doublet or opposite twin growth

• 3 sets of peaks at 35 and 85
• indicating a triplet or triple twin growth

(117) Pole Figures for Bismuth Titanate Films
75
Texture Evolution
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Rocking Curve

• An useful method for evaluating the quality of
  oriented samples such as epitaxial films
• ? is changed by rocking the sample but ?B is held
  constant
• Width of Rocking curve is a direct measure of the
  range of orientation present in the irradiated
  area of the crystal

77
Order Disorder Transformation

• Structure factor is dependent on the presence of
  order or disorder within a material
• Present in systems such as Cu-Au, Ti-Al, Ni-Fe
• Order-disorder transformation at specific
  compositions upon heating/cooling across a
  critical temperature
• Examples Cu3Au, Ni3Fe

78
Order Disorder Transformation

• Structure factor is dependent on the presence of
  order or disorder within a material.

• Complete Disorder
• Example AB with A and B atoms randomly
  distributed in the lattice
• Lattice positions (000) and (½ ½ ½)
• Atomic scattering factor
• favj ½ (fAfB)
• Structure Factor, F, is given by
• F Sf exp2??i (hukvlw)
• favj 1e(? i (hkl))
• 2. favj when hkl is even
• 0 when hkl is odd
• The expected pattern is like a BCC crystal

79
Order Disorder Transformation

• Complete Order
• Example AB with A at (000) and B at (½ ½ ½)
• Structure Factor, F, is given by
• F fA e2??i (h.0k.0l.0) fA e2??i (h. ½k.
  ½l. ½)
• fAfB when hkl is even
• fA-fB when hkl is odd
• The expected pattern is not like a BCC crystal,
  rather like a simple cubic crystal where all the
  reflections are present.
• Extra reflections present are called as
  superlattice reflections

80
Order-Disorder Transformation
Disordered Cu3Au
Ordered Cu3Au
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Instrumentation
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Diffractometer

• Source
• Optics
• Detector

Incident Beam Part
Diffracted Beam Part
Sample
Diffracted Beam Optics
Source
Detector
Incident Beam Optics
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Geometry and Configuration

• Theta-Theta Source and detector move
  ?, sample fixed
• Theta-2Theta Sample moves ? and
  detector 2? , source fixed

• Vertical configuration Horizontal
  sample
• Horizontal configuration Vertical sample

Incident Beam Part
Diffracted Beam Part
Sample
Diffracted Beam Optics
Source
Detector
Incident Beam Optics
84
Sample translation

• XYZ translation
• Z translation sample alignment
• Sample exactly on the diffractometer circle
• Knife edge or laser
• Video microscope with laser
• XY movement to choose
• area of interest

85
X-ray generation

• X-ray tube (? 0.8-2.3 ?)
• Rotating anode (? 0.8-2.3 ?)
• Liquid metal
• Synchrotron (? ranging from infrared to X-ray)

86
X-ray tube
X-rays
Be window
Metal anode
W cathode
Electrons
Rotating anode
87
Large angle anode
Small angle anode
Small focal spot
Large focal spot
88

• Rotating anode of W or Mo for high flux
• Microfocus rotating anode 10 times
  brighter
• Liquid anode for high flux 100
  times brighter
• and small beam size
• Gallium and Gallium, indium, tin alloys
• Synchrotron provides intense beam but access is
  limited
• Brighter than a thousand suns

89
Synchrotron

• High brilliance and coherence
• X-ray bulb emitting all radiations from IR to
  X-rays

http//www.coe.berkeley.edu/AST/srms
90

• Moving charge emits radiation
• Electrons at vc
• Bending magnet, wiggler and undulator
• Straight section wiggler and
  undulator
• Curved sections Bending magnet

91

• Filter to remove Kß For eg. Ni
  filter for Cu Kß
• Reduction in intensity of Ka
• Choice of proper thickness

92

• Slits To limit the size of beam
  (Divergence slits)
• To alter beam
  profile
• (Soller slit angular
  divergence )
• Narrow slits Lower intensity
• Narrow peak

93

• Mirror focusing and remove Ka2
• Mono-chromator remove Ka2

Si
Graphite
94
Beam Profile
Parallel beam
Mirror
Source
Soller slit
Detector
Mirror
Sample
Para-focusing
Source
Detector
Sample
95
Point focus
Source
Detector
Sample
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Comparison
Parallel beam Para-focusing
X-rays are aligned X-rays are diverging
Lower intensity for bulk samples Higher intensity
Higher intensity for small samples Lower intensity
Instrumental broadening independent of orientation of diffraction vector with specimen normal Instrumental broadening dependent of orientation of diffraction vector with specimen normal
Suitable for GI-XRD Suitable for Bragg-Brentano
Texture, stress Powder diffraction
97
Detectors

• Single photon detector (Point or 0D)
• scintillation detector NaI
• proportional counter, Xenon gas
• semiconductor
• Position sensitive detector (Linear or 1D)
• gas filled wire detectors, Xenon gas
• charge coupled devices (CCD)
• Area detectors (2D)
• wire
• CCD
• 3D detector

Photoelectron or Electron-hole pair
Photomultiplier tube or amplifier
Electrical signal
X-ray photon
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99

• Resolution ability to distinguish between
  energies
• Energy proportionality ability to produce
  signal proportioanl to energy of x-ray photon
  detected
• Sensitivity ability to detect low intensity
  levels
• Speed to capture dynamic phenomenon
• Range better view of the reciprocal space

100
Data collection and analysis

• Choose 2? range
• Step size and time per step
• Hardware slit size, filter, sample alignment
• Fast scan followed with a slower scan
• Look for fluorescence
• Collected data Background subtraction, Ka2
  stripping
• Normalize data for comparison I/Imax

101
Summary

• X-ray Diffraction is a very useful to
  characterize materials for following information
• Phase analysis
• Lattice parameter determination
• Strain determination
• Texture and orientation analysis
• Order-disorder transformation
• and many more things
• Choice of correct type of method is critical for
  the kind of work one intends to do.
• Powerful technique for thin film characterization