1. X-RAY DIFFRACTION X- Ray Sources Diffraction: Bragg’s 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)

2. Hence, X-rays can be used for the study of crystal structures
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

3. K K Mo Target impacted by electrons accelerated by a 35 kV potential
Characteristic radiation → due to energy transitions in the atom K
White radiation
Intensity
0.2
0.6
1.0
1.4
Wavelength ()

4. Target Metal
 Of K radiation (Å) Mo
0.71 Cu
1.54 Co
1.79 Fe
1.94 Cr
2.29

5. Incident X-rays Fluorescent X-rays Electrons Scattered X-rays
SPECIMEN
Heat
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.

6. (Darkens the background of the diffraction patterns)
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 waves Incoherent  does not contribute to diffraction
(Darkens the background of the diffraction patterns)

7. Fluorescent X-rays Energy levels Characteristic x-rays
Knocked out electron from inner shell
Vacuum
Energy levels
Characteristic x-rays
(Fluorescent X-rays)
(10−16s later  seems like scattering!)
Nucleus

8. The secondary radiation is in all directions
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

9. Sets Electron cloud into oscillation
Sets nucleus (with protons) into oscillation
Small effect  neglected

10. Oscillating charge re-radiates  In phase with the incoming x-rays

11. d dSin The path difference between ray 1 and ray 2 = 2d Sin
BRAGG’s 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

15. Incident and scattered waves are in phase if
In plane scattering is in phase
Incident and scattered waves are in phase if
Scattering from across planes is in phase

16. But this is still reinforced scattering and NOT reflection
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 scattering and NOT reflection

17. Note that in the Bragg’s 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 Bragg’s condition !! d
Note: shift (systematic) is actually not a problem!

18. Note: shift is actually not a problem
Note: shift is actually not a problem!  Why is ‘systematic’ shift not a problem?

19. 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)

20. This is looking at diffraction from atomic arrays and not planes
Generalizing into 3D
Laue’s equations
S0  incoming X-ray beam
S  Scattered X-ray beam
This is looking at diffraction from atomic arrays and not planes

21. A physical picture of scattering leading to diffraction is embodied in Laue’s equations
Bragg’s 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’ (Bragg’s 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. Diffraction = Reinforced Coherent Scattering
Bragg’s equation is a negative law  If Bragg’s eq. is NOT satisfied  NO reflection can occur  If Bragg’s 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

23. n is an integer and is the order of the reflection
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)

24. In XRD nth order reflection from (h k l) is considered as 1st order reflection from (nh nk nl)

26. A B C Intensity of the Scattered electrons Scattering by a crystal
Polarization factor B
Atom
Atomic scattering factor (f) C
Unit cell (uc)
Structure factor (F)

27. A Scattering by an Electron Coherent (definite phase relationship)
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

28. Intensity of the scattered beam due to an electron (I) at a point P
For an polarized wave z P
For a wave oscillating in z direction r  x
Intensity of the scattered beam due to an electron (I) at a point P
such that r >> 
The reason we are able to neglect scattering from the protons in the nucleus
The scattered rays are also plane polarized

30. For an unpolarized wave
E is the measure of the amplitude of the wave
E2 = Intensity
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

31. Sum of the squares of the direction cosines =1
Hence 
In terms of 2

32. In general P could lie anywhere in 3D space x z r P  2
In general P could lie anywhere in 3D space
For the specific case of Bragg scattering: The incident direction  IO The diffracted beam direction  OP The trace of the scattering plane  BB’ Are all coplanar
 OP is constrained to be on the xz plane

33. For an unpolarized wave E is the measure of the amplitude of the wave
E2 = Intensity
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

34.  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

35. Polarization factor Comes into being as we used unpolarized beam

36. B Scattering by an Atom f → (Å−1) →
Scattering by an atom  [Atomic number, (path difference suffered by scattering from each e−, )]
Scattering by an atom  [Z, (, )]
Angle of scattering leads to path differences
In the forward direction all scattered waves are in phase
f →
(Å−1) →
0.2
0.4
0.6
0.8
1.0 10 20 30
Schematic

37. Incoherent (Compton) scattering
Coherent scattering
Incoherent (Compton) scattering
Z   
Sin() /  

38. 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 parallēlepipedon) 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

39. 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

40. x d(h00) a B  A R S B M N C Ray 1 = R1 Ray 3 = R3 Ray 2 = R2
Unit Cell x R S
Ray 2 = R2 B
d(h00) a M N
(h00) plane C

41. Independent of the shape of UC
Extending to 3D
Independent of the shape of UC
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!!

43. Structure factor calculations
Simple Cubic A
Atom at (0,0,0) and equivalent positions
 F is independent of the scattering plane (h k l)

44. Real C- centred Orthorhombic
Atom at (0,0,0) & (½, ½, 0) and equivalent positions
C- centred Orthorhombic
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

45. 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

46. 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

47. Real Body centred Orthorhombic
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)

48. Real Face Centred Cubic
Atom at (0,0,0) & (½, ½, 0) and equivalent positions
Face Centred Cubic
(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)

49. Mixed indices Mixed indices CASE h k l A o e B (h, k, l) mixed
Two odd and one even (e.g. 112); two even and one odd (e.g. 122)
Mixed indices
CASE h k l A o e B
(h, k, l) mixed
e.g. (100), (211); (210), (032), (033)
Unmixed indices
All odd (e.g. 111); all even (e.g. 222)
Unmixed indices
CASE h k l A o B e
(h, k, l) unmixed
e.g. (111), (200), (220), (333), (420)

50. 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

51. Zero for mixed indices Mixed indices CASE h k l A o e B Mixed indices
(h, k, l) mixed
e.g. (100), (211); (210), (032), (033)

52. If (h + k + l) is even If (h + k + l) is odd Unmixed indices CASE h kB e
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)

53.  Presence of additional atoms/ions/molecules in the UC can alter the intensities of some of the reflections

54. Reflections which may be present Reflections necessarily absent
Selection / 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 mixed C 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 odd Or all are even & (h + k + l) divisible by 4

55. h2 + k2 + l2 SC
FCC
BCC DC 1
100 2
110 3
111 4
200 5
210 6
211 7 8
220 9
300, 221 10
310 11
311 12
222 13
320 14
321 15 16
400 17
410, 322 18
411, 330 19
331

57. Reciprocal Lattice BASIS VECTORS B
Properties are reciprocal to the crystal lattice
BASIS
VECTORS B
The reciprocal lattice is created by interplanar spacings

58. 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

59. Reciprocal Lattice
The reciprocal lattice has an origin!

60. Note perpendicularity of various vectors

61. Physics comes in from the following:
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 (|Fhkl|2)  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

62. 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

63. SC Lattice = SC Reciprocal Crystal = SC
Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2) SC
001
011
101
111
Lattice = SC
000
010
100
110
No missing reflections
Reciprocal Crystal = SC
Figures NOT to Scale

64. BCC Lattice = BCC Reciprocal Crystal = FCC 002 022 202 222 011 101 020
000
Lattice = BCC
110
200
100 missing reflection (F = 0)
220
Reciprocal Crystal = FCC
Weighing factor for each point “motif”
Figures NOT to Scale

65. FCC Lattice = FCC Reciprocal Crystal = BCC 002 022 202 222 111 020 000
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

66. Ordered Solid solution
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

67. SC
Ordered
BCC
Disordered
Ordered
- NiAl, BCC
B2 (CsCl type)
- Ni3Al, FCC
L12 (AuCu3-I type)
FCC SC
Ordered
FCC
BCC

68. 1) SC + two kinds of Intensities decorating the lattice
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)

69. The Ewald Sphere
* Paul Peter Ewald (German physicist and crystallographer; )

70. 7. Paul-Peter-Ewald-Kolloquium
Freitag, 17. Juli 2008
organisiert von:
Max-Planck-Institut für Metallforschung Institut für Theoretische und Angewandte Physik, Institut für Metallkunde, Institut für Nichtmetallische Anorganische Materialien der Universität Stuttgart
Programm
13:30
Joachim Spatz (Max-Planck-Institut für Metallforschung) Begrüßung
13:45
Heribert Knorr (Ministerium für Wissenschaft, Forschung und Kunst Baden-Württemberg Begrüßung
14:00
Stefan Hell (Max-Planck-Institut für Biophysikalische Chemie) Nano-Auflösung mit fokussiertem Licht
14:30
Antoni Tomsia (Lawrence Berkeley National Laboratory) Using Ice to Mimic Nacre: From Structural Materials to Artificial Bone
15:00
Pause Kaffee und Getränke
15:30
Frank Gießelmann(Universität Stuttgart) Von ferroelektrischen Fluiden zu geordneten Dispersionen von Nanoröhren: Aktuelle Themen der Flüssigkristallforschung
16:00
Verleihung des Günter-Petzow-Preises 2008
16:15
Udo Welzel (Max-Planck-Institut für Metallforschung) Materialien unter Spannung: Ursachen, Messung und Auswirkungen- Freund und Feind
ab 17:00
Sommerfest des Max-Planck-Instituts für Metallforschung

71. The Ewald Sphere
The reciprocal lattice points are the values of momentum transfer for which the Bragg’s 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 Cullity’s book: A15-4

72. Bragg’s 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 Bragg’s equation)

73. 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

74. K = K =g = Diffraction Vector
Ewald Sphere
The Ewald Sphere touches the reciprocal lattice (for point 41)
 Bragg’s equation is satisfied for 41
K = K =g = Diffraction Vector

75. 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

76. Crystal structure determination
Many s (orientations)
Powder specimen
POWDER METHOD
Monochromatic X-rays
Single 
LAUE TECHNIQUE
Panchromatic X-rays
ROTATING CRYSTAL METHOD
 Varied by rotation
Monochromatic X-rays

77. THE POWDER METHOD
Cone of diffracted rays

78. POWDER METHOD Diffraction cones and the Debye-Scherrer geometry
Different cones for different reflections
Film may be replaced with detector

79. The 440 reflection is not observed

80. The 331 reflection is not observed

81. THE POWDER METHOD
Cubic crystal

82. 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

83. Multiplicity factor * Altered in crystals with lower symmetry Lattice
Index
Multiplicity
Planes
Cubic
(with highest symmetry)
(100) 6
[(100) (010) (001)] ( 2 for negatives)
(110) 12
[(110) (101) (011), (110) (101) (011)] ( 2 for negatives)
(111)
[(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) 24
(211)  3 ways, (211)  3! ways, (211)  3 ways
(321)
48*
Tetragonal 4
[(100) (010)] ( 2 for negatives)
[(110) (110)] ( 2 for negatives) 8 8*
(210) = 2 Ways, (210) = 2 Ways, (210) = 2 Ways, (210) = 2 Ways 16
[Same as for (210) = 8]  2 (as l can be +1 or 1)
16*
Same as above (as last digit is anyhow not permuted)
* Altered in crystals with lower symmetry

84. Multiplicity factor Cubic hkl hhl hk0 hh0 hhh h00 48* 24 24* 12 8 6
Hexagonal
hk.l
hh.l
h0.l
hk.0
hh.0
h0.0
00.l
12* 2
Tetragonal
h0l
00l
16* 8* 4
Orthorhombic
0kl
0k0
Monoclinic
Triclinic
* Altered in crystals with lower symmetry (of the same crystal class)

85. Polarization factor
Lorentz factor

86. Valid for Debye-Scherrer geometry I → Relative Integrated “Intensity”
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

87. THE POWDER METHOD
Cubic crystal

88. Determination of Crystal Structure from 2 versus Intensity Data
2→ 
Intensity
Sin
Sin2 
ratio

89. FCC 2→  Intensity Sin Sin2  ratio 1 21.5 0.366 0.134 3 2 25 0.422
0.178 4 37
0.60
0.362 8 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

90. h2 + k2 + l2 SC
FCC
BCC DC 1
100 2
110 3
111 4
200 5
210 6
211 7 8
220 9
300, 221 10
310 11
311 12
222 13
320 14
321 15 16
400 17
410, 322 18
411, 330 19
331

91. The ratio of (h2 + K2 + l2) derived from extinction rulesSC 1 2 3 4 5 6 8 …
BCC 7
FCC 11 12 DC 16

92. Peaks or not idealized  peaks  broadened
Powder diffraction pattern from Al
Radiation: Cu K,  = Å
111
Note:
Peaks or not idealized  peaks  broadened
Increasing splitting of peaks with g 
Peaks are all not of same intensity
220
311
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

94. Determination of Crystal Structure from 2 versus Intensity Data
ratio
Index
a (nm) 1
38.52
19.26
0.33
0.11 3
111 2
44.76
22.38
0.38
0.14 4
200
65.14
32.57
0.54
0.29 8
220
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
*  1 , 2 peaks are resolved (1 peaks are listed)

95. Error in d spacing
For the same  the error in Sin  with 

96. Error in d spacing
Error in d spacing decreases with 

97. Applications of XRD Bravais lattice determination
Lattice parameter determination
Determination of solvus line in phase diagrams
Long range order
Crystallite size and Strain
More

98. Crystal Intensity → Diffraction angle (2) → Intensity →90
180
Crystal
Schematic of difference between the diffraction patterns of various phases 90
180
Diffraction angle (2) →
Intensity →
Monoatomic gas 90
180
Diffraction angle (2) →
Intensity →
Liquid / Amorphous solid

99. Crystallite size and Strain
Bragg’s 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 Bragg’s 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 < 0.5 m  there are insufficient number of planes to build up a sharp diffraction pattern  peaks are broadened

101. XRD Line Broadening

102. XRD Line Broadening Instrumental Crystallite size Strain
Unresolved 1 , 2 peaks  Non-monochromaticity of the source (finite width of  peak)
Imperfect focusing Bi
Crystallite size
In the vicinity of B the −ve of Bragg’s 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

104. Size  (10, 0.5)   Smooth continuous ring pattern
Crystallite size
Size > 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 < 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

105. Effect of crystallite size on SAD patterns
Single crystal
“Spotty” pattern
Few crystals in the selected region

106. Effect of crystallite size on SAD patterns
Ring pattern
Broadened Rings

107. 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 specimen’s peaks) 2
Use the same material as the standard as the specimen to be X-rayed but with large grain size and well annealed

108. 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 ( ), painted by Christian Albrecht Jensen
University of Göttingen

109. Scherrer’s formula  → Wavelength
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%)

110. Smaller angle peaks should be used to separate Bs and Bc
Strain broadening
 → Strain in the material
Smaller angle peaks should be used to separate Bs and Bc

111. Separating crystallite size broadening and strain broadening
Plot of [Br Cos] vs [Sin]

112. Example of a calculation
Sample: Annealed Al Radiation: Cu k ( = 1.54 Å)
Intensity →
Sample: Cold-worked Al Radiation: Cu k ( = 1.54 Å)
2 → 40 60
Intensity →
2 → 40 60
X-Ray Diffraction: A Practical Approach, C. Suryanarayana & M. Grant Norton, Plenum Press, New York (1998)

113. Annealed Al
Peak No.
2 ()
hkl
Bi = FWHM ()
Bi = FWHM (rad) 1
38.52
111
0.103
1.8  10−3 2
44.76
200
0.066
1.2  10−3 3
65.13
220
0.089
1.6  10−3
Cold-worked Al
2 ()
Sin()
hkl
B ()
B (rad)
Br Cos (rad) 1
38.51
0.3298
111
0.187
3.3  10−3
2.8  10−3
2.6  10−3 2
44.77
0.3808
200
0.206
3.6  10−3
3.4  10−3
3.1  10−3 3
65.15
0.5384
220
0.271
4.7  10−3
4.4  10−3
3.7  10−3

115. end

116. Iso-intensity circle

117. 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

118. Consider the compound ZnS (sphalerite)
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 n=odd (e.g. 200)
F = 4(fA + fB) if h, k, l all even and h+ k+ l = 2n where n=even (e.g. 400)

120. 421 missing

122. (Cu K) = 1.54 Å, 1/ = 0.65 Å−1, aCu = 3.61 Å, 1/aCu = 0.28 Å−1
Ewald sphere  X-rays
(Cu K) = 1.54 Å, 1/ = 0.65 Å−1, aCu = 3.61 Å, 1/aCu = 0.28 Å−1

123. Multiplicity factor Lattice Index Planes Cubic with highest symmetry
(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)
(321) 48
Tetragonal 4
[(100) (010)]
[(110) (110)]