Theory and practice of X-ray diffraction experiment

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Theory and practice of X-ray diffraction
experiment
Power Point presentation from lecture 2 is
available at http//dl.dropbox.com/u/23622306/UIU
C/Lecture202.ppt
Power Point presentation from lecture 4 is
available at http//dl.dropbox.com/u/23622306/UIU
C/lecture204.ppt
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Grain size and types of diffraction experiment

• Single crystal
• Bulk powder
• Nanopowder
• Amorphous/glass

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Reciprocal space

• 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.

y
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Orientation matrix

• UB matrix relates the Miller indices of
  reciprocal vector (hkl) with its Cartesian
  coordinates in lab reference system (xyz) at zero
  goniometer position.
• xyzUB hkl
• 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
• hklUB-1 xyz

z
hkl
X-ray
x
y
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Diffraction condition and Bragg equation for a
single crystal
1q
hkl
diffracted X-ray
nl2dsinq
2q
incident X-ray
1q
2q
Diffraction geometry is analogous to reflection
in a selective mirror 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.
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Explanation of Bragg derivation
nl2dsinq
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Ewald construction

• 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
• 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.

1/l(S0Sd)R xyz
Rxyz
Sd
S0
Radius of the Evald sphere is 1/l
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Experimental approach to single crystal
diffraction

• Effective diffraction vs. observed diffraction
• Cause effective diffraction and find (observe)
  diffraction signal
• Measure Sd and R for a number of diffraction
  peaks
• Calculate xyz using the Ewald equation
• Find orientation matrix/index the peaks (assign
  Miller indices)
• Measure/calculate peak intensities I(hkl)

Wide oscillation 13
Monochromatic SXD experiment
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Goniometry in matrix representation
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Atomic scattering factor

• 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.
Cromer and Mann 9-parameter equation
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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.
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Excitation Scans We can observe the ?f 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 ?f ? The real, dispersive component
is calculated from ?f by the Kramers-Kronig
relationship. Very roughly, ?f is the negative
first derivative of ?f.
From Ramakrishnans study of GH5
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Thermal vibrations and Debye-Waller factor

• 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).

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Structure factor
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Lorenz correction
Accounts for the different speed with which the
reciprocal vectors move through the Ewald sphere
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Polarization correction
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Other intensity corrections

• Absorption correction
• Extinction correction
• Preferred orientation
• Illuminated volume
• Incident intensity

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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                                  has
the following properties. 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.
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Systematic absences and space group determination
A centered hkl k l 2n B centered h l
2n C centered h k 2n F centered k l 2n,
h l 2n, h k 2n I centered h k l
2n R (obverse) -h k l 3n R (reverse) h -
k l 3n
Glide reflecting in a 0kl       b glide k
2n     c glide l 2n     n glide k l
2n     d glide k l 4n Glide reflecting in
b h0l       a glide h 2n     c glide l
2n     n glide h l 2n     d glide h l
4n Glide reflecting in c hk0       b glide k
2n     a glide h 2n     n glide k h
2n     d glide k h 4n
Screw 100 h00       21, 42 h 2n     41,
43 h 4n Screw 010 0k0       21, 42 k
2n     41, 43 k 4n Screw 001 00l       21,
42, 63 l 2n     31, 32, 62, 64 l
3n     41, 43 l 4n     61, 65 l 6n
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Point detector experiment

• 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

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Area detector experiment

• 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

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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
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General algorithm for analysis of single-crystal
XRD data
Image Initial peak list (Sd) Step scan analysis
(R) Orientation matrix (UB) Predicted peak
list Integrated intensities I(hkl) Scaled and
corrected intensities
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Figures of merit
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Demonstration of single-crystal data processing
with GSE_ADA
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Powder diffraction

• Sample is composed of a very large (preferably
  gt106) 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.

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Different experimental approaches to powder
diffraction

• Polychromatic EDX approach
• 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)
• Monochromatic approach with point detector
• 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
• Mono beam
• Area detector at fixed or variable angle/position
• Non-scanning signal accumulation
• Does not suffer from preferred orientation
  problems

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High resolution powder diffraction with analyzer
crystal
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Bragg-Brentano powder diffractometer
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Integration of 2-dimensional diffraction pattern

• Calibration of detector geometry
• 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 geometry calibration
• Goniometer zeros
• Goniometer axis alignment with the detector
  orientation
• Instrumental function
• Calibrating with a diffraction standard
• Cake transform

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Energy dispersive powder diffraction
Ge solid state semiconductor detector
Multiple wavelengths
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Powder pattern fitting techniques

• 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
• 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.
• 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.

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Peak width analysis

• Instrumental function
• Focused beam divergence away from the sample
• Detector pixel size
• Phosphor point spread function
• Sample-related factors
• 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.
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Peak width analysis
13nm nano powder
FW(S) cos(q) K l / size 4 strain sin(q)
Bulk fine powder
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SM and Quantitative analysis demonstration
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Tools of the tradePowder diffraction

• Image integration
• Fit2d
• Powder3d
• Powder pattern analysis
• Jade
• GSE_Shell
• Indexing
• MDI Jade
• Treor (index permutation method)
• Ito (zone indexing method)
• Dicvol (dichotomy algorithm)
• Database and SM
• American Mineralogist Crystal Structure Database
• Project RRUFF
• Crystallography Open Database
• ICSD
• PDF
• Peak and pattern fitting, refinement of structure
• GSAS

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Tools of the tradeSingle-crystal diffraction

• Commercial
• Bruker Saint/SMART, APEX
• HKL Research Denzo/HKL2000
• Oxford (Agilent) Crysalis
• Academic
• XDS (Wofgang Kabsch) Linux
• GSE_ADA (Przemek Dera) Windows

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Structure determination
Structure identification Logical
choices Search and match Indexing-based Structur
e refinement (requires the structure model to be
approximately known) Rietveld method Single-crys
tal refinement Structure solution (ab
initio) direct methods charge
flipping simulated annealing
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High pressure apparatus

• 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.

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(No Transcript)
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High-pressure crystallography challenges

• Experimental challenges
• Very small sample (lt0.01mm)
• Angular access restricted (low completeness)
• Absorption limits the incident energy to gt15 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

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Homework (detailed instructions will be sent by
Friday 9/16), due on Tuesday 9/27

• 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.