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An Introduction to X-ray Powder Diffraction Analysis
When a beam of X-rays illuminates a single crystal, many “spots” are generated. The positions of the
spots are determined by the size and shape of the unit cell and the symmetry. The intensities of the
spots are determined by the arrangement of the atoms within the crystal. After measuring the intensities
of all of the diffraction spots (reflections), it is generally possible to determine the positions of the
atoms in the unit cell (the structure) in a straightforward manner. Sometimes, however, the sample is
more complex (twinning, aperiodic structure, diffuse scattering), and the structural analysis becomes a
challenge for even the most skilled crystallographers.
Most real solids are not single
crystals (that is why we prize
gem single crystals so much!),
but are composed of large
numbers of tiny crystals, so are
described as polycrystalline. In
the diffraction pattern, the effect
is that each of the spots is
spread out into a ring. If the
crystallites
are
oriented
randomly, the rings are uniform,
and no information is lost by
measuring along a radius vector
Image 1
of the complete 3-dimensional
diffraction pattern. The result is a plot of diffracted intensity vs. angle - a powder pattern:
Image 2
Qualitative Phase Analysis
Because the positions of the peaks in a powder pattern are determined by the size, shape, and
symmetry of the unit cell and the peak intensities are determined by the arrangement of atoms within the
cell, the powder pattern is a characteristic “fingerprint” of a phase. In a mixture of phases, the diffraction
patterns overlap (but do not otherwise interfere), so it is possible to identify the components of a
mixture. In practice, the experimental powder pattern is searched against the Powder Diffraction File, a
database containing the patterns of > 700,000 pure compounds, produced by the International Centre
for Diffraction Data (www.icdd.com). Using such search/match techniques, it is easy to identify the
compound giving rise to the previous powder pattern as vanadium(III) tris (dihydrogen phosphate),
V(H2PO4) 3.
Image 3
Using contemporary algorithms and the PDF database, the major components of mixtures can generally
be identified. New compounds not present in the database, are (of course) tougher to identify, but such
compounds present interesting research problems! Identifying minor or trace phases can require
special techniques. To learn more about how to identify trace phases, come to the workshop on Trace
Phase Identification Using Chemical Information at the 2010 Denver X-ray Conference
(www.dxcicdd.com), taught by Tim Fawcett and Jim Kaduk.
Quantitative Phase Analysis
As the concentration of a phase in a mixture varies, the intensities of all of the peaks from this phase
vary in concert (ideally). Thus, the concentrations of phases in a mixture can be determined by
measuring the intensities of peaks in the powder pattern. The traditional techniques for quantitative
phase analysis are described in standard textbooks, such as:
Ron Jenkins and Robert L. Snyder, Introduction to X-ray Powder Diffractometry. Wiley (1996).
Vitalij K. Pecharsky and Peter Y. Zavalij, Fundamentals of Powder Diffraction and S tructural
Characterization of Materials, S econd Edition. Springer (2009).
Abraham Clearfield, Joseph Reibenspies, and Nattamai Bhuvanesh, Principles and Applications of
Powder Diffraction. Wiley (2008).
Robert E. Dinnebier and Simon J. L. Billinge, Powder Diffraction: Theory and Practice. RSCX
Publishing (2008).
While these traditional single-peak or peak-cluster techniques can be used if necessary, Poly
Crystallography generally determines quantitative phase analyses by the Rietveld method. In a Rietveld
refinement, we use the crystal structures of all of the phases and diffraction physics to do a leastsquares modeling of the full diffraction pattern. Among the refined parameters are scale factors for
each phase, from which the quantitative analysis is derived. The Rietveld method corrects for and/or
models many of the systematic errors that can plague a powder pattern, and thus can yield more
accurate and more precise results than traditional methods. With normal care, accuracy and precision of
about ±2 wt% can be achieved, and with additional effort, accuracy and precision as good as ±0.1 wt%
can be attained. When an internal standard is added to the sample, the concentration of amorphous
material can also be quantified.
More information about quantitative phase analysis, using both traditional and full-pattern techniques, is
available at the ICDD Clinic Session II - Advanced Methods in X-ray Powder Diffraction, given each year in
June at ICDD Headquarters.
ab initio Crystal Structure Determination
Most crystal structures are determined using single crystal methods. Many real materials are not
obtainable as single crystals, however. Improvements in data quality, algorithms, and computer powder
make it increasingly possible to solve crystal structures using powder diffraction data. Techniques used
include Patterson and direct methods, real space geometry optimization, Monte Carlo simulated
annealing, charge flipping, stealth and guile, and working by analogy. Determination of a crystal
structure from scratch can be straightforward, but more often is a true research project. Poly
Crystallography Principal Scientist Jim Kaduk is well-known for his success in solv ing structures using
powder diffraction data.
Lattice Parameter Determination
Many materials, particularly such inorganic examples as minerals and alloys, are solid solutions different elements may occupy the same site in the crystal structure. Because different elements have
different sizes, changes in composition result in changes in the lattice parameters (unit cell
dimensions). Accurate and precise determination of the lattice parameters of a solid solution can thus
be used to carry out quantitative analysis at the microscopic scale. An example is the variation in the
hexagonal a lattice parameter of dolomite with Mg content:
Image 4
So, from the experimental lattice parameters of dolomite from a Rietveld refinement, we can determine
the composition of this phase - even in the presence of other minerals.
Microstructure Analysis - Crystallite Size and Strain
The widths and shapes of the peaks in a powder pattern are determined by many factors, including
contributions both from the diffractometer and the specimen. Once the instrumental factors are
understood by measuring a sample having no size or strain broadening, the crystallite size and
microstrain can be determined from the experimental peak widths.
Size broadening is easier to understand. As the crystallites (coherently scattering domains, not
necessarily the same size as the particles) get smaller, the diffraction peaks get wider. Using the know
diffraction physics, we can work backward, and compute the average crystallite size from the observed
peak widths. An “infinite” crystallite is ~2000 Å in diameter, and by the time the crystallites get as small
as 30-50 Å the peaks are hard to see. So, for nanocrystalline and microcrystalline materials, accurate
crystallite sizes can be obtained using X-ray diffraction. A common application is studying the sintering
of metals in supported catalysts. For simple materials, even crystallite size distributions can be derived
by analyzing the details of the peak shapes.
Microstrain broadening is the result of small changes in local lattice parameters resulting from defects,
imperfections, and variations in the crystal structure. Solid solutions often exhibit microstrain
broadening. The absolute value of mictostrain broadening may be hard to interpret, but changes in such
broadening are often important practically. The dependence of mictrostrain broadening with diffraction
angle is different than that of size broadening, so the two effects can be separated, if a large enough
range of diffraction is observed.
Texture Analysis
Nonrandom distributions of crystallites, such as those resulting from platy or needle-like crystals or
extruded and rolled specimens, can change the relative intensities of the diffraction peaks from a
phase. These deviations from expected random intensities can be analyzed to derive the nature and
extent of preferred orientation (texture) in a material. The physical properties of a material, such as
tensile strength or barrier properties, can be a sensitive function of the texture. Texture is often
visualized by a pole figure - a map (stereographic projection) of the distribution of one particular vector
in a crystal structure in space.
Image 5
Pair Distribution Function Analysis - Amorphous Materials
The X-ray scattering from amorphous materials leads to broad features in the powder pattern. These
features can be analyzed using old mathematics (the Debye function), or more rigorously using new
approaches and high-energy synchrotron or neutron diffraction. The results of either analysis is a radial
distribution function (pair correlation function), which gives the electron density weighted distribution of
interatomic vectors in the material. Such functions can be hard to interpret on an absolute basis, but
changes in catalyst supports during use can be detected and quantified.
Image 6
Image 7
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