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Theory and practice of X-ray diffraction experiment

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Title: Theory and practice of X-ray diffraction experiment


1
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
2
Grain size and types of diffraction experiment
  • Single crystal
  • Bulk powder
  • Nanopowder
  • Amorphous/glass

3
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
4
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
5
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.
6
Explanation of Bragg derivation
nl2dsinq
7
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
8
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
9
Goniometry in matrix representation
10
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
11
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.
12
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
13
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).

14
Structure factor
15
Lorenz correction
Accounts for the different speed with which the
reciprocal vectors move through the Ewald sphere
16
Polarization correction
17
Other intensity corrections
  • Absorption correction
  • Extinction correction
  • Preferred orientation
  • Illuminated volume
  • Incident intensity

18
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.
19
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
20
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

21
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

22
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
23
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
24
Figures of merit
25
Demonstration of single-crystal data processing
with GSE_ADA
26
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.

27
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

28
High resolution powder diffraction with analyzer
crystal
29
Bragg-Brentano powder diffractometer
30
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

31
Energy dispersive powder diffraction
Ge solid state semiconductor detector
Multiple wavelengths
32
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.

33
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.
34
Peak width analysis
13nm nano powder
FW(S) cos(q) K l / size 4 strain sin(q)
Bulk fine powder
35
SM and Quantitative analysis demonstration
36
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

37
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

38
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
39
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.

40
(No Transcript)
41
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

42
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.
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