1. X-ray diffraction

2. Equipment
Bruker D8 Analytical X-ray Systems

5. X-ray beam source
Bruker D8 ADVANCE uses an x-ray tube with a Cu anode as the primary x-ray beam source. In this component x-rays are generated when a focused electron beam accelerated across a high voltage field bombards a stationary solid Cu target. As electrons collide with atoms in the target and slow down, a continuous spectrum of x-rays is emitted, which is termed Bremsstrahlung radiation.
The high energy electrons also eject inner shell electrons in atoms through the ionization process. When a free electron fills the shell, an x-ray photon with energy characteristic of the target material is emitted.
Common targets used in x-ray tubes include Cu and Mo, that emit 8 keV and 14 keV x-rays with corresponding wavelengths of 1.54 Å and 0.8 Å, respectively.

6. Wavelengths for X-Ray source
Copper
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cobalt
Cu Ka1 Å Å
Co Ka1 Å Å
Cu Ka2 Å Å
Co Ka2 Å Å
Cu Kb Å Å
Co Kb Å Å
Molybdenum
Chromium
Mo Ka1 Å Å
Cr Ka1 Å Å
Mo Ka2 Å Å
Cr Ka2 Å Å
Mo Kb Å Å
Cr Kb Å Å
Often quoted values from Cullity (1956) and Bearden, Rev. Mod. Phys. 39 (1967) are incorrect.
Values from Bearden (1967) are reprinted in international Tables for X-Ray Crystallography and most XRD textbooks.
Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)

7. 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

8. θ - 2θ Scan
The θ - 2θ scan maintains these angles with the sample, detector and X-ray source
Normal to surface
Only planes of atoms that share this normal will be seen in the θ - 2θ Scan
NanoLab/NSF NUE/Bumm

9. Occurs throughout the bulk Takes place at any angle
Powder diffraction data can be collected using either transmission or reflection geometry, as shown below. Because the particles in the powder sample are randomly oriented, these two methods will yield the same data
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

10. 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.

11. How does it work?
In powder XRD method, a sample is ground to a powder (±10µm) in order to expose all possible orientations to the X-ray beam of the crystal values of , d and  for diffraction are achieved as follows:
1.  is kept constant by using filtered X- radiation that is
approximately monochromatic.
2. d may have value consistent with the crystal structure
3.  is the variable parameters, in terms of which the
diffraction peaks are measured.

12. How does XRD Works???
Every crystalline substance produce its own XRD pattern, which because it is dependent on the internal structure, is characteristic of that substance.
The XRD pattern is often spoken as the “FINGERPRINT” of a mineral or a crystalline substance, because it differs from pattern of every other mineral or crystalline substances.

13. Basic Component Of XRD Machine
Therefore any XRD machine will consist of three basic component.
Monochromatic X-ray source ()
Sample-holder (goniometer).
Data collector- such as film, strip chart or magnetic medium/storage.
By varying the angle , the Bragg’s Law conditions are satisfied by different d-spacing in polycrystalline materials. Plotting the angular positions and intensities of the resultant diffraction peaks produces a pattern which is characterised of the sample

14. X-ray Components
A typical X-ray instrument is built by combining high performance components such as X-ray tubes, X-ray optics, X-ray detectors, sample handling device etc. to meet the analytical requirements. A consequent modular design is the key to configure the best instrumentation. .

15. Diffraction Pattern Collected Where A Ni Filter Is Used To Remove Kβ
K alpha 1 and K alpha 2 overlap heavily at low angles and are easier to discriminate at high angles. Kb

16. Typical experimental data from Bruker XRD
TiO2
2-theta
intensitas 20
405
357
381
371
376
356
370
395
373
335
397 I
101 Anatase
110 Rutile 2

17. 101 Anatase
110 Rutile

18. 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

19. 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

20. 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

23. Sample preparation

24. Make a mine powder

25. Sample holder

26. Side Drift Mount
Designed to reduce preferred orientation – great for clay samples, (and others with peaks at low 2-theta angles)

27. Film, pellets, crystals mineral specimens

28. Sample holder

29. Specimen Holders for X-ray Diffraction

31. Match The Sample/Measurement Conditions With The Diffraction Pattern1 2 3

35. Misinterpreting X-Ray Diffraction Results

36. Rock Salt Why are peaks missing? The sample is made from Morton’s Salt
JCPDF#
111
200
220
311
222
The sample is made from Morton’s Salt
JCPDF# is supposed to fit it (Sodium Chloride Halite)

37. It’s a single crystal
111
200
220
311
222 2q
The (200) planes would diffract at °2q; however, they are not properly aligned to produce a diffraction peak
The (222) planes are parallel to the (111) planes.
At °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

38. A random polycrystalline sample that contains thousands of crystallites should exhibit all possible diffraction peaks
200
220
111
222
311 2q 2q 2q
For every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract (the plane perpendicular bisects the incident and diffracted beams).
Basic assumptions of powder diffraction are that for every set of planes there is an equal number of crystallites that will diffract and that there is a statistically relevant number of crystallites, not just one or two.

39. Hint: Why are the intensities different?
Which of these diffraction patterns comes from a nanocrystalline material? 66 67 68 69 70 71 72 73 74 2 q
(deg.)
Intensity (a.u.)
Hint: Why are the intensities different? 1o
0.0015o
These two diffraction patterns come from the exact same sample (silicon).
The apparent difference in peak broadening is due to the instrument optics, not due to specimen broadening
These diffraction patterns were produced from the exact same sample
The apparent peak broadening is due solely to the instrumentation
0.0015° slits vs. 1° slits optical cofigurations
Scan speed ( stepsize)

40. Crystallite Size Broadening
Scherrer’s Formula
Peak Width B(2q) varies inversely with crystallite size
The constant of proportionality, K (the Scherrer constant) depends on the how the width is determined, the shape of the crystal, and the size distribution
the most common values for K are 0.94 (for FWHM of spherical crystals with cubic symmetry), 0.89 (for integral breadth of spherical crystals with cubic symmetry, and 1 (because 0.94 and 0.89 both round up to 1).
K actually varies from 0.62 to 2.08
For an excellent discussion of K, refer to JI Langford and AJC Wilson, “Scherrer after sixty years: A survey and some new results in the determination of crystallite size,” J. Appl. Cryst. 11 (1978) p
Remember:
Instrument contributions must be subtracted

41. Scherrer’s Formula t = thickness of crystallite / crystallite size
K = constant dependent on crystallite shape (0.89)
l = x-ray wavelength
B = FWHM (full width at half max) or integral breadth
qB = Bragg Angle

42. Scherrer’s Formula What is B? B = (2θ High) – (2θ Low)
B is the difference in angles at half max
Peak
2θ low
2θ high
Noise

43. When to Use Scherrer’s Formula
Crystallite size <1000 Å
Peak broadening by other factors
Causes of broadening
Size
Strain
Instrument
If breadth consistent for each peak then assured broadening due to crystallite size
K depends on definition of t and B
Within 20%-30% accuracy at best
Sherrer’s Formula References
Corman, D. Scherrer’s Formula: Using XRD to Determine Average Diameter of Nanocrystals.

44. Scherrer’s Example

45. Scherrer’s Example = 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) )
t = 0.89*λ / (B Cos θB) λ = 1.54 Ǻ
= 0.89*1.54 Ǻ / ( * Cos (98.25/ 2 ) )
= 1200 Ǻ
B = ( )*π/180 =
Simple Right!
Target Metal
 Of K radiation (Å) Mo
0.71 Cu
1.54 Co
1.79 Fe
1.94 Cr
2.29

46. Methods used to Define Peak Width
46.7
46.8
46.9
47.0
47.1
47.2
47.3
47.4
47.5
47.6
47.7
47.8
47.9 2 q
(deg.)
Intensity (a.u.)
Full Width at Half Maximum (FWHM)
the width of the diffraction peak, in radians, at a height half-way between background and the peak maximum
Integral Breadth
the total area under the peak divided by the peak height
the width of a rectangle having the same area and the same height as the peak
requires very careful evaluation of the tails of the peak and the background
FWHM
46.7
46.8
46.9
47.0
47.1
47.2
47.3
47.4
47.5
47.6
47.7
47.8
47.9 2 q
(deg.)
Intensity (a.u.)

47. Remember, Crystallite Size is Different than Particle Size
A particle may be made up of several different crystallites
Crystallite size often matches grain size, but there are exceptions
TEM images collected by Jane Howe at Oak Ridge National Laboratory.

48. Anistropic Size Broadening
The broadening of a single diffraction peak is the product of the crystallite dimensions in the direction perpendicular to the planes that produced the diffraction peak.

49. Instrumental Peak Profile
A large crystallite size, defect-free powder specimen will still produce diffraction peaks with a finite width
The peak widths from the instrument peak profile are a convolution of:
X-ray Source Profile
Wavelength widths of Ka1 and Ka2 lines
Size of the X-ray source
Superposition of Ka1 and Ka2 peaks
Goniometer Optics
Divergence and Receiving Slit widths
Imperfect focusing
Beam size
Penetration into the sample
47.0
47.2
47.4
47.6
47.8 2 q
(deg.)
Intensity (a.u.)
Patterns collected from the same sample with different instruments and configurations at MIT

51. What Instrument to Use?
The instrumental profile determines the upper limit of crystallite size that can be evaluated
if the Instrumental peak width is much larger than the broadening due to crystallite size, then we cannot accurately determine crystallite size
For analyzing larger nanocrystallites, it is important to use the instrument with the smallest instrumental peak width
Very small nanocrystallites produce weak signals
the specimen broadening will be significantly larger than the instrumental broadening
the signal:noise ratio is more important than the instrumental profile

52. Smaller Crystals Produce Broader XRD Peaks

53. Comparison of Peak Widths at Crystallite Sizes
FWHM (deg)
100 nm
0.099
50 nm
0.182
10 nm
0.871
5 nm
1.745
Rigaku XRPD is better for very small nanocrystallites, <80 nm (upper limit 100 nm)
PANalytical X’Pert Pro is better for larger nanocrystallites, <150 nm

54. Decrease
crystallite size
A = anatase, R = rutile, B = brokite, (B)=TiO2(B)
Wahyuningsih, S., 2009

55. Polycrystalline films on Silicon
Why do the peaks broaden toward each other?
Solid Solution Inhomogeneity
Variation in the composition of a solid solution can create a distribution of d-spacing for a crystallographic plane
CeO2
19 nm 45 46 47 48 49 50 51 52 2 q
(deg.)
Intensity (a.u.)
ZrO2
46nm
CexZr1-xO2
0<x<1

56. Many factors may contribute to the observed peak profile
Instrumental Peak Profile
Crystallite Size
Microstrain
Non-uniform Lattice Distortions
Faulting
Dislocations
Solid Solution Inhomogeneity
The peak profile is a convolution of the profiles from all of these contributions

57. Thank you for your attending!
Workshop & Analysis Informations:
Dr. Sayekti Wahyuningsih, M.Si
Dr. Yoventina Iriani, M.Si
Laboratorium MIPA Terpadu
FMIPA Universitas Sebelas Maret
Phone / fax : (0271)