1. No category

Theory and practice of X-ray
diffraction experiment
Power Point presentation from lecture # 2 is available at:
http://dl.dropbox.com/u/23622306/UIUC/Lecture%202.ppt
Power Point presentation from lecture # 4 is available at:
http://dl.dropbox.com/u/23622306/UIUC/lecture%204.ppt
Grain size and types of diffraction experiment
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Single crystal
Bulk powder
Nanopowder
Amorphous/glass
Reciprocal space
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z
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hkl
X-ray
x
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y
Direct space relates to atoms in the unit cell, while reciprocal
space relates to peaks in diffraction experiment.
In diffraction experiment planes of atoms in the crystal act as
a “selective” mirror and reflect (in the same way as in optics)
incoming beam of x-rays if specific geometric conditions are
satisfied.
Vectors in reciprocal space correspond to families of planes
in the crystal (the direction of the rec. vector is normal to the
family of planes).
In a conventional crystal (as opposed to incommensurately
modulated crystal or quasi-crystal) vectors in reciprocal
space can occupy only points on a 3-dimensional grid.
Grid coordinates of reciprocal vectors are known as Miller
indices hkl
Geometry of the reciprocal space grid is determined by the
unit cell of the crystal. And can be directly measured.
D-spacing is equal to inverse of the reciprocal vector length.
Coordinates of vectors in reciprocal space are described in
laboratory (instrument-related) reference coordinate system.
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UB matrix relates the Miller indices of reciprocal vector (hkl) with
its Cartesian coordinates in lab reference system (xyz) at zero
goniometer position.
Orientation matrix
xyz=UB hkl
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Columns in the orientation matrix are coordinates of the principal
vectors in reciprocal space 1 0 0 , 0 1 0 and 0 0 1.
UB matrix is composed of two sub-matrices, U and B
U describes the orientation of the crystal axes with respect to
the laboratory reference system, while B stores information
about the unit cell parameters.
By inverting the above equation one can calculate what are the
Miller indices of a measured reciprocal vector xyz
hkl=UB-1 xyz
z
hkl
X-ray
x
y
Diffraction condition and Bragg equation for a single crystal
hkl
1q
diffracted X-ray
nl=2dsinq
2q
incident X-ray
1q
2q
Diffraction geometry is analogous to reflection in a selective
mirror:
hkl
Diffraction is effective only at selected incidence angles with
respect to the “reflecting” plane, determined by the Bragg
equation.
When diffraction is effective “reflection” angle equals to the
incidence angle, with the reciprocal vector bisecting the two.
Intensity of the “reflected” beam is not equal to the intensity of
the incident beam, it is related to the mean electron density
within the direct lattice plane.
Explanation of Bragg derivation
nl=2dsinq
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In diffraction experiment we measure the direction and intensity of the diffracted
beams as well as the orientation of the crystal corresponding to each of the
diffraction peaks.
Diffraction does not occur at any arbitrary crystal orientation. In general to bring a
given reciprocal vector to a diffraction position one may need to rotate the sample
using goniometer or change the incident wavelength (an exception is
polychromatic experiment in which a range of incident wavelengths is available).
When the diffraction event occurs, the following relation between the incident and
diffracted beam vectors, incident wavelength and the reciprocal vector is satisfied:
Ewald construction
1/l(S0+Sd)=R xyz
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R is the goniometer rotation matrix bringing the crystal from the zero-goniometer
position to the position at which maximum peak intensity is observed.
In the Ewald construction the center of the crystal and the center of reciprocal
space do not coincide.
S0
Sd
Rxyz
Radius of the Evald sphere is 1/l
Experimental approach to
single crystal diffraction
Effective diffraction vs. observed diffraction
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Cause effective diffraction and find (observe) diffraction signal
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Measure Sd and R for a number of diffraction peaks
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Calculate xyz using the Ewald equation
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Find orientation matrix/index the peaks (assign Miller indices)
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Measure/calculate peak intensities I(hkl)
Wide oscillation ± 13°
Monochromatic SXD experiment
Goniometry in matrix representation
Atomic scattering factor
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The incoming X-rays are scattered by the electrons of the atoms. As the wavelength of the X-rays (1.5
to 0.5 A) is of the order of the atom diameter, most of the scattering is in the forward direction. For
neutrons of the same wavelength the scattering factor is not angle dependent due to the fact that the
atomic nucleus is magnitudes smaller than the electron cloud. It is also obvious that the X-ray
scattering power will depend on the number of electrons in the particular atom. The X-ray scattering
power of an atom decreases with increasing scattering angle and is higher for heavier atoms. A plot of
scattering factor f in units of electrons vs. sin(theta)/lambda shows this behavior. Note that for zero
scattering angle the value of f equals the number of electrons.
Electron density around an isolated atom (r) is spherically
symmetric, with max at nucleus position, and falls off smoothly
with distance.
f s  

  r e
2i r s 
dr  f s 

Cromer and Mann 9-parameter equation
Anomalous dispersion
The scattering factor contains additional (complex) contributions from anomalous dispersion effects
(essentially resonance absorption) which become substantial in the vicinity of the X-ray absorption
edge of the scattering atom. These anomalous contributions can be calculated as well and their
presence can be exploited in the MAD phasing technique.
Excitation Scans
We can observe the Δf” by measuring
the absorption of the x-rays by the
atom. Often we us the fluorescence of
the absorbing atom as a measure of
absorptivity. That is, we measure an
“excitation” spectrum.
How to get Δf ’?
The real, “dispersive” component is
calculated from Δf” by the KramersKronig relationship. Very roughly, Δf’ is
the negative first derivative of Δf”.
From Ramakrishnan’s study of GH5
Thermal vibrations and Debye-Waller factor
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There is an additional weakening of the scattering power of the atoms by the so
called Temperature-, B- , or Debye-Waller factor. This exponential factor is also
angle dependent and effects the high angle reflections substantially (one of the
reasons for cryo-cooling crystals is to reduce the attenuation of the high angle
reflections due to this B-factor).
Structure factor
1
 r  
Vc
F e
 2i h r 
h
h
Fh    r e
V
N
Fh   f i sh e
i 1
I h  Fh Fh
*
2i hx i 
2i hr 
dr
Lorenz correction
Accounts for the different speed with which the reciprocal vectors move
through the Ewald sphere
L
cos q
sin 2 2q
Polarization correction
105
100
95
I
90
85
80
75
70
-200
-150
-100
-50
0
azimuth (deg)
50
100
150
200
Other intensity corrections
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Absorption correction
Extinction correction
Preferred orientation
Illuminated volume
Incident intensity
Symmetry of peak intensities. Friedel pairs and Laue classes
Friedel's law, named after Georges Friedel, is a property of Fourier
transforms of real functions.1
Given a real function , its Fourier transform
Laue
classes
Non-centrosymmetric
groups
having the same
Laue class
1
has the following properties.
2/m
2, m
mmm
222, 2mm
-3
-3m
where is the complex conjugate of .
Centrosymmetric points are called Friedel's pairs.
The Friedel pair symmetry is broken by anomalous dispersion. The
effect is strongly pronounced for incident energies close to the
absorption edge.
3
32, 3m
4/m
4,
4/mmm
422,
6/m
6,
6/mmm
622,
m-3
m3m
, 42m
, 62m
23
432,
32
Systematic absences and space group determination
A centered hkl
B centered
C centered
F centered
I centered
R (obverse)
R (reverse)
k + l = 2n
h + l = 2n
h + k = 2n
k + l = 2n, h + l = 2n, h + k = 2n
h + k + l = 2n
-h + k + l = 3n
h - k + l = 3n
Screw || [100]
21, 42
41, 43
Screw || [010]
21, 42
41, 43
Screw || [001]
21, 42, 63
31, 32, 62, 64
41, 43
61, 65
h00
h = 2n
h = 4n
0k0
k = 2n
k = 4n
00l
l = 2n
l = 3n
l = 4n
l = 6n
Glide reflecting in a
b glide
c glide
n glide
d glide
Glide reflecting in b
a glide
c glide
n glide
d glide
Glide reflecting in c
b glide
a glide
n glide
d glide
0kl
k = 2n
l = 2n
k + l = 2n
k + l = 4n
h0l
h = 2n
l = 2n
h + l = 2n
h + l = 4n
hk0
k = 2n
h = 2n
k + h = 2n
k + h = 4n
Point detector experiment
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Scintillator detector converts x-ray to electric signal
Center the sample on rotation axis and with the beam
Search for peaks (usually 10-20) at different 2theta and sample orientations
Center the peaks that were found
Determine orientation matrix
Find more peaks to constrain and refine the orientation matrix better
Calculate position for a list of peaks that need to be measured based on orientation
matrix
Position each peak individually, record a rocking curve/peak profile
Integrate, scale and correct peak intensities
Solve/refine the structure
Area detector experiment
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CCD detector detects visible light. Phosphor screen in front of the detector converts
x-ray to visible light.
Center the sample on rotation axis and with the beam
Collect diffraction images while rotating the sample
Determine detector coordinates and sample orientations for each diffraction peak
Reconstruct the reciprocal space in 3-d and determine the orientation matrix (index)
Predict peak positions in recorded diffraction images and retrieve peak intensities
(structure factor amplitudes)
Solve/refine the structure
Area detector and step
scan approach
With wide rotation image we determine the direction of the
diffracted beam Sd but the rotation angles (necessary to calculate
R) at which each peak occurs are unknown.
Step scan allows to determine R for each peak by finding a step
image at which the peak has maximum intensity.
With R and Sd we can calculate xyz.
Step scan images
General algorithm for
analysis of single-crystal
XRD data
zoom
fitting
Residual
Image
Initial peak list (Sd)
Step scan analysis (R)
Orientation matrix (UB)
Predicted peak list
Integrated intensities I(hkl)
Scaled and corrected intensities
Figures of merit
Rint
F  F


F
2
o
2
o
2
o
Rsigma
 F 


F
2
o
2
o
Demonstration of single-crystal data processing with
GSE_ADA
Powder diffraction
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Sample is composed of a very large (preferably >106)
number of single crystal specimens with random
distribution of crystal orientations.
Diffraction for all reciprocal vectors occurs (is effective) at
any arbitrary orientation of the sample. Sample rotation is
not necessary.
Single crystal diffraction peaks (directional beams) become
cones of radiation.
Peaks corresponding to different reciprocal vectors
(different hkls) that have similar length (d-spacing) overlap.
Diffraction signal carries only information about the length
of the reciprocal vectors, but not their orientation.
Different experimental approaches to powder diffraction
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Polychromatic EDX approach
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Monochromatic approach with point detector
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White beam
Point energy-dispersive detector at fixed angle
Non-scanning signal accumulation
Can provide access to large d-spacing range w/o much of angular access
Suffers from preferred orientation problems
Peak usually quite broad
Peak intensity interpretation difficult (correction)
Mono beam
Point detector at variable angle
Scanning signal accumulation
Suffers from preferred orientation problems
Can provide patterns with very narrow peak profiles
Monochromatic approach with area detector
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Mono beam
Area detector at fixed or variable angle/position
Non-scanning signal accumulation
Does not suffer from preferred orientation problems
High resolution powder diffraction with analyzer
crystal
Bragg-Brentano powder diffractometer
Integration of 2-dimensional diffraction pattern
Calibration of detector geometry
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Goniometer geometry calibration
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Detector coordinates of the point of intersection of the
incident beam and the detector surface
Sample-to-detector distance
Detector non-orthogonality with respect to the beam
Pixel size
Goniometer zeros
Goniometer axis alignment with the detector orientation
Instrumental function
Calibrating with a diffraction standard
Cake transform
0.07
0.065
0.06
peak width [deg]
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0.055
0.05
0.045
0.04
190
210
230
250
270
detector distance [mm]
290
310
Energy dispersive powder diffraction
Ge solid state semiconductor detector
Multiple wavelengths
Powder pattern fitting techniques
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Individual peak fitting
– Peak positions are not constrained – they are individually refined
– Peak profiles are not related with each other
– Closely overlapping peaks are very hard to deal with
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LeBail (unit-cell constrained) refinement
– Individual peak positions are not refined – they are calculated from the unit cell
parameters which are refined
– Individual peak profiles are not refined – there is a global function (e.g. Cagliotti
function) that connects all peak profiles.
– Peak intensities are refined free (not constrained by the structure model)
– Gives much more reliable way of refining positions and intensities of closely
overlapping peaks.
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Rietveld (model-biased) refinement
– Individual peak positions are not refined – they are calculated from the unit cell
parameters which are refined
– Individual peak profiles are not refined – there is a global function (e.g. Cagliotti
function) that connects all peak profiles.
– Individual peak intensities are not refined free, they are calculated from the
structure model which is refined.
Peak width analysis
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Instrumental function
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Sample-related factors
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Focused beam divergence away from the sample
Detector pixel size
Phosphor point spread function
Scherrer formula
Hall-Williamson plot
Cagliotti function
Deconvolution
size = K l / FW(S) cos(q)
FW(S) cos(q) = K l / size + 4 strain sin(q)
FW(S)D = FWHMD - FW(I)D
The dimensionless shape factor K has a typical value of about 0.9, but varies with the actual
shape of the crystallite.
Scherrer formula is not applicable to grains larger than about 0.1 μm, which precludes those
observed in most metallographic and ceramographic microstructures.
Intensity(Counts)
Peak width analysis
13nm nano powder
FW(S) cos(q) = K l / size + 4 strain sin(q)
[CNMDAC3_023.chi]
5000
4500
4000
Bulk fine powder
3500
3000
2500
2000
1500
1000
500
0
4
5
6
7
8
9
10
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12
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T wo-T heta (deg)
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16
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21
S&M and Quantitative analysis
demonstration
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Image integration
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Powder pattern analysis
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American Mineralogist Crystal Structure Database
Project RRUFF
Crystallography Open Database
ICSD
PDF
Peak and pattern fitting, refinement of structure
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MDI Jade
Treor (index permutation method)
Ito (zone indexing method)
Dicvol (dichotomy algorithm)
Database and S&M
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Jade
GSE_Shell
Indexing
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Fit2d
Powder3d
GSAS
Fullprof
PowderCell
Endeavour
Jade
Structure determination ab initio
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Endeavour
Fox
Expo
Tools of the trade
Powder diffraction
Tools of the trade
Single-crystal diffraction
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Commercial
– Bruker Saint/SMART, APEX
– HKL Research Denzo/HKL2000
– Oxford (Agilent) Crysalis
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Academic
– XDS (Wofgang Kabsch) Linux
– GSE_ADA (Przemek Dera) Windows
Structure determination
Structure identification
“Logical” choices
Search and match
Indexing-based
Structure refinement (requires the structure model to be approximately known)
Rietveld method
Single-crystal refinement
Structure solution (ab initio)
direct methods
charge flipping
simulated annealing
High pressure apparatus
Incident x-ray beam
Incident x-ray beam
Beryllium seats
Diamond anvil
Diamond anvil
Metal
Metal
gasket
gasket
Steel frame
Steel frame
Diffracted x-ray beam
Diffracted x-ray beam
• Sample is immersed in hydrostatic
liquid, which freezes at some point
during compression (usually below 10
GPa).
• Diffraction pressure calibrant is
placed in the sample chamber along
with the sample.
• Both incident and diffracted beam
travel though diamonds, Be disks,
pressure medium, and sample.
High-pressure crystallography challenges
Experimental challenges:
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Very small sample (<0.01mm)
Angular access restricted (low completeness)
Absorption limits the incident energy to >15 keV
Absorption and extinction affect intensity measurement
High background (scattering, Compton, etc.)
Multiple SXD diffraction signal
Contamination by PXD signal
Poor sample quality (strain, multi-grain assemblages)
More challenging sample centering
Beam size vs. sample size
Homework (detailed instructions will be sent by
Friday 9/16), due on Tuesday 9/27
1.
2.
3.
4.
5.
Download and install Rosetta. Feel free to use any other program
of your preference to complete the tasks described below (in
which case you can skip 1).
Download the example unknown powder patterns xx1.chi, xx2.chi,
xx3.chi
Identify minerals present in each sample
Download the example QA powder patterns of quartz-albite
mixture QA25.chi, QA50.chi. The number in file name stands for
the albite content (as prepared).
Verify how accurate is the quantitative analysis done with
Rosetta.