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)

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
A accelerating 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)

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 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
Secondary
emission
Incoming X-rays

9. Sets Electron cloud into oscillation Sets nucleus 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

12. 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

13. 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

14. 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

15. 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)

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

17. 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

18. THE POWDER METHOD

19. A B C Intensity of the Scattered electrons Scattering by a crystal
Atom
Unit cell (uc)

20. A Scattering by an Electron Coherent (definite phase relationship)
Emission in all directions
Sets electron into oscillation
Coherent (definite phase relationship)
Scattered beams

21. Intensity of the scattered beam due to an electron (I)
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)

23. For an unpolarized wave
E is the measure of the amplitude of the wave
E2 = Intensityc
IPy = Intensity at point P due to Ey
IPz = Intensity at point P due to Ez

24.  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 factor Comes into being as we used unpolarized beam

25. 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

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

28. 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

29. 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

30. 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

31. 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

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

33. 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

34. 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)

35. 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)

36. 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

37. Zero for mixed indices If (h + k + l) is even If (h + k + l) is odd
(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

38. 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

39. Multiplicity factor Lattice Index 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 4
[(100) (010)]
[(110) (110)]

40. Polarization factor
Lorentz factor

41. 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

44. 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

45. Reciprocal Lattice
Properties are reciprocal to the crystal lattice
The reciprocal lattice is created by interplanar spacings

46. 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

47. Reciprocal Lattice
The reciprocal lattice has an origin!

48. Note perpendicularity of various vectors

49. 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

50. SC Lattice = SC Reciprocal Lattice = 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 Lattice = SC
Figures NOT to Scale

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

52. FCC Lattice = FCC Reciprocal Lattice = 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 Lattice = BCC
Figures NOT to Scale

53. 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
* Paul Peter Ewald (German physicist and crystallographer; )
See Cullity’s book: A15-4

54. 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

55. Diffraction cones and the Debye-Scherrer geometry
Film may be replaced with detector

56. Peaks or not idealized  peaks  broadend
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
220
311
200
331
420
422
222
400
1 & 2 peaks resolved

57. Determination of Crystal Structure from 2 versus Intensity Data
ratio
Index 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
116.60
58.3
0.85
0.72 20
420 9
137.47
68.735
0.93
0.87 24
422

62. 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

64. Reflections which may be present Reflections necessarily absent
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

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

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

67. 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

68. 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

69. 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)

70. 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

72. 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
300
310