X-ray diffraction
Equipment
Bruker D8 Analytical X-ray Systems
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.
Wavelengths for X-Ray source
Copper
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cobalt
Anodes
Bearden
(1967)
Holzer et al.
(1997)
Cu Ka1
1.54056Å
1.540598 Å
Co Ka1
1.788965Å
1.789010 Å
Cu Ka2
1.54439Å
1.544426 Å
Co Ka2
1.792850Å
1.792900 Å
Cu Kb
1.39220Å
1.392250 Å
Co Kb
1.62079Å
1.620830 Å
Cr Ka1
2.28970Å
2.289760 Å
Molybdenum
Anodes
Chromium
Anodes
Mo Ka1
0.709300Å
Mo Ka2
0.713590Å
0.713609 Å
Cr Ka2
2.293606Å
2.293663 Å
Mo Kb
0.632288Å
0.632305 Å
Cr Kb
2.08487Å
2.084920 Å

Often quoted values from Cullity (1956) and Bearden, Rev. Mod.
Phys. 39 (1967) are incorrect.


0.709319 Å
Values from Bearden (1967) are reprinted in international Tables for XRay Crystallography and most XRD textbooks.
Most recent values are from Hölzer et al. Phys. Rev. A 56 (1997)
BRAGG’s EQUATION
Deviation = 2
Ray 1
Ray 2




d
 The path difference between ray 1 and ray 2 = 2d Sin
 For constructive interference: n = 2d Sin
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

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
Occurs from surface
Takes place at any angle
~100 % of the intensity may be
reflected
Diffraction
Occurs throughout the bulk
Takes place only at Bragg angles
Small fraction of intensity is
diffracted
Incident X-rays
SPECIMEN
Fluorescent X-rays
Electrons
Scattered X-rays
Coherent
From bound charges
Heat
Compton recoil
Photoelectrons
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.
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.
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.
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
X-ray Components
A typical X-ray instrument is built by combining
high performance components such as Xray 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.
.
Diffraction Pattern Collected
Where A Ni Filter Is Used
To Remove Kβ
Kb
E keV   h 
hc


6.02
  
Typical experimental data from Bruker XRD
TiO2
2-theta
I
14000
101 Anatase
12000
10000
8000
6000
4000
110 Rutile
2000
0
0
10
20
30
2
40
50
60
70
intensitas
20
405
20.05001
357
20.10002
381
20.15002
371
20.20003
376
20.25004
356
20.30005
370
20.35006
395
20.40006
373
20.45007
335
20.50008
397
101 Anatase
110 Rutile
Examples of 3D Reciprocal Lattices weighed in with scattering power (|F|2)
SC
001
011
101
Lattice = SC
111
000
010
100
No missing reflections
110
Reciprocal Crystal = SC
Figures NOT to Scale
002
022
BCC
202
222
011
101
000
Lattice = BCC
020
110
200
100 missing reflection (F = 0)
Weighing factor for each point “motif”
220
Reciprocal Crystal = FCC
F2  4 f 2
Figures NOT to Scale
002
022
FCC
202
222
111
Lattice = FCC
020
000
200
100 missing reflection (F = 0)
220
110 missing reflection (F = 0)
Weighing factor for each point “motif”
Reciprocal Crystal = BCC
F 2  16 f 2
Figures NOT to Scale
Sample preparation
Make a mine powder
•
Sample holder
Side Drift Mount
Designed to reduce preferred orientation – great for clay samples, (and
others with peaks at low 2-theta angles)
Film, pellets, crystals
mineral specimens
Sample holder
Specimen Holders for X-ray Diffraction
Match The Sample/Measurement
Conditions With The Diffraction Pattern
1
2
3
Misinterpreting X-Ray
Diffraction Results
Rock Salt
JCPDF# 01-0994
Why are peaks missing?
200
220
111
222
311
• The sample is made from Morton’s Salt
• JCPDF# 01-0994 is supposed to fit it (Sodium Chloride Halite)
It’s a single crystal
200
220
111
222
311
2
At 27.42 °2, Bragg’s law
fulfilled for the (111) planes,
producing a diffraction peak.
The (200) planes would diffract at 31.82
°2; however, they are not properly
aligned to produce a diffraction peak
The (222) planes are parallel to the (111)
planes.
A random polycrystalline sample that contains thousands of
crystallites should exhibit all possible diffraction peaks
200
220
111
222
311
2
2
2
• 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.
Intensity (a.u.)
Which of these diffraction patterns comes
from a nanocrystalline material?
Hint: Why are the
intensities different?
1o
0.0015o
66
67
68
69
70
71
72
73
74
2 (deg.)


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)
http://prism.mit.edu/xray
Crystallite Size Broadening
B2  


Scherrer’s Formula
Peak Width B(2) 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




0.94
t cos
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) p102-113.
Remember:

Instrument contributions must be subtracted
Scherrer’s Formula
K 
t
B  cos B
t = thickness of crystallite / crystallite size
K = constant dependent on crystallite shape (0.89)
 = x-ray wavelength
B = FWHM (full width at half max) or integral breadth
B = Bragg Angle
Scherrer’s Formula
What is B?
B = (2θ High) – (2θ Low)
Peak
2θ high
2θ low
B is the difference in
angles at half max
Noise
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.
Scherrer’s Example
Au Foil
10000
9000
8000
98.25 (400)
7000
Counts
6000
5000
4000
3000
2000
1000
0
95
95.5
96
96.5
97
97.5
98
98.5
2 Theta
99
99.5
100
100.5
101
101.5
102
Scherrer’s Example
0.89  
t
B  cos B
t
= 0.89*λ / (B Cos θB)
λ = 1.54 Ǻ
= 0.89*1.54 Ǻ / ( 0.00174 * Cos (98.25/ 2 ) )
= 1200 Ǻ
B = (98.3 - 98.2)*π/180 = 0.00174
Target Metal
 Of Ka radiation (Å)
Mo
0.71
Cu
1.54
Co
1.79
Fe
1.94
Cr
2.29
Simple Right!
Methods used to Define Peak Width
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
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 (deg.)
Intensity (a.u.)

FWHM
Intensity (a.u.)

46.7
46.8
46.9
47.0
47.1
47.2
47.3
47.4
2 (deg.)
47.5
47.6
47.7
47.8
47.9
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
http://prism.mit.edu/xray
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.
http://prism.mit.edu/xray


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
http://prism.mit.edu/xray
Intensity (a.u.)
Instrumental Peak Profile
47.0
47.2
47.4
47.6
47.8
2 (deg.)
Patterns collected from the same
sample with different instruments
and configurations at MIT
B ( FWHM )  Bi  Bc  Bs  BSF  ...
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
http://prism.mit.edu/xray
Smaller Crystals Produce Broader XRD Peaks
Comparison of Peak Widths at
Crystallite Sizes


Crystallite Size
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
http://prism.mit.edu/xray
Decrease
crystallite size
A = anatase, R = rutile, B = brokite, (B)=TiO2(B)
Wahyuningsih, S., 2009
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
ZrO2
46nm
CeO2
19 nm
Intensity (a.u.)

45
46
CexZr1-xO2
0<x<1
47
48
49
2(deg.)
50
51
52
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
http://prism.mit.edu/xray
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) 663375