1. Applications of X-Ray Diffraction
Dr. T. Ramlochan
March 2010

2. Applications of powder diffraction
Diffraction pattern gives information about peak positions, intensity, and shape
Identification (qualitative) – most common use of powder diffraction is identification of crystalline phases (search/match); peak positions and intensity related to unique crystal structure
Quantification – determination of phase amounts in a polycrystalline material; peak intensity and shape related to concentration
RIR – reference intensity ratio
Whole-pattern fitting (Rietveld analysis) – computationally intensive; can only be applied with powerful analytical software.
Determination of crystallographic structure (i.e., unit cell) – much of X-ray diffraction is concerned with discovering and describing crystal structure

3. Phase identification
Diffraction patterns are unique fingerprints (distinct and reproducible) of the crystal structure of materials that can be used to determine phase composition of a polycrystalline material
“…every crystalline substance gives a pattern; the same substance always gives the same pattern; and in a mixture of substances each produces its pattern independently of the others.”
-- A.W. Hull (1919), “A New Method of Chemical Analysis”
Gypsum (CaSO4·H2O) layered structure and diffraction pattern

4. Phase identification
Phase identification is essentially an exercise of pattern comparison between the unknown and a database of single-phase reference patterns
Diffractogram is reduced to a table of ‘d-I’ pairs (d-spacing and relative intensity); some important information is lost

5. Search/Match
Search/Match – legacy (manual) method used database of index cards or books with d-I for reference materials
Tedious, time intensive, human error
OK for single phase, but multi-phase very difficult
(1) file number, (2) three strongest lines, (3) lowest-angle line, (4) chemical formula and name of the substance, (5) data on diffraction method used, (6) crystallographic data, (7) optical and other data, (8) data on specimen, (9) diffraction pattern.

6. Search/Match
International Center for Diffraction Data (ICDD) current Powder Diffraction File (PDF ) database contains 186,107 entries of almost every known inorganic (159,809) and many organic (28,610) crystalline substances
Includes cell parameters, d-spacing, chemical formula, relative intensity, RIR
Software used to determine peak positions and intensities and used to search/match or compare with all (or subset/restrictions) of the ICDD PDF-2 database
Some knowledge about the material can be used to limit search
Reference patterns with high ‘score’ are visually compared with the sample data and best match(es) are selected by user
Works well for abundant phases, not so well for minor phases
Human error

9. Phase identification problems
Accurate line position is very important
Specimen displacement
geometry of diffraction requires that specimen lie on the focusing circle and be at the center of the diffractometer circle or will cause angular errors (e.g., if sample is “high” the detected ∆2θ will be positive)
Strain in crystal lattice
Macrostrain causes uniform strain in unit cell; unit cell dimensions and distances between planes altered; shifts the location of the diffraction peak in the pattern

10. Quantitative phase analysis: Intensity ratios
Concentration (wt%) (and density) of a particular phase is proportional to its intensity (peak area minus background, height?)
More crystallites, more reflections, greater intensity
Klug and Alexander (1954) were first to describe a technique for quantification using intensities of the crystalline phases in a mixture
Ratio of peak intensity from unknown phase ‘A’ (I?) to a standard ‘B’ (IS) is a linear function of the mass fraction of ‘A’ in the original sample
Use known amount of internal standard mineral (e.g., rutile) to calibrate the intensities of the unknown phase
Calibration (regression) curve

11. Quantitative phase analysis: Intensity ratios
Use single peak only, generally most intense peak (I100%)
Requires fully resolved peak
Chose internal standard – must not overlap any other peaks; simple well-defined pattern (e.g., face centered cubic); same crystallite size as sample
Spiking (standard addition) – add extra wt% of desired phase to mixture and acquire at least 2 scans
External standard
Calibration (regression) curve

12. Reference intensity ratio (RIR)
General formula for relating intensity ratio to mass fraction
ICDD PDF-2 uses corundum (Al2O3) as reference B and gives k for 50:50 mixture of phase A and corundum
RIR is I/Icor using intensity of the strongest peak (100%)
If I1/Icor is k1 and I2/Icor is k2, then I1/I2 is k1/k2
If we know RIRs for every phase in mixture, we can determine the relative amounts of each phase (do not need corundum) because sum of all mass fractions equal 1
Quick way to get ‘semi-quantitative’ information, but often inaccurate due irregularities in sample

13. Problems with intensity ratios
Uses single peak (I100%)
Ideal sample is homogeneous and the crystallites have a random distribution of all possible planes
Each possible reflection from a given set of h, k, l planes will have an equal number of crystallites contributing to it
Can only occur if particles are spherical
Crystal fragment shape is influenced by cleavage
Platy or acicular crystals will have dominant direction when compacted
Cubic (110) (d2) family will have twice the intensity of the (100) (d1) family due to multiplicity

14. Problems with intensity ratios
Non-random distribution of the crystallites is referred to as preferred orientation (texture)
Most common cause of deviation of experimental data from the ideal intensity pattern
Intensity ratios are greatly distorted by preferred orientation
Minimise by back-loading sample, slurry with acetone

15. Problems with intensity ratios
Compositional variations (e.g., site occupancy)
Substitution of one atom for another in unit cell can alter intensities (impure phases or solid solution)
Brownmillerite Ca2 Alx Fe2-x O5

16. Problems with intensity ratios
Peak overlap – can exaggerate intensity or obscure peak (not fully resolved)
Particular problem with clinker/cement

17. Problems with intensity ratios
Crystallite size (not necessarily same as particle size)
Large crystallites (i.e., thousands of unit cells) will produce sharp, very intense diffraction peaks only at the precise location of the Bragg angle (due to cancelling of diffractions by incoherent scattering).
Small crystallites (~1 µm) will produce broad peaks due to incoherent scattering at angles close to the Bragg angle
Large crystals increase microabsorption
<45 µm recommended, 2-10 µm optimal size, uniform size*, over grinding can cause amorphous layer

18. Problems with intensity ratios
Microabsorption
Strongly absorbing minerals (e.g., C4AF) will have reduced intensities
Weakly absorbing minerals (e.g., periclase) will have greater than average intensities.
Reduced by fine grinding
Not really a problem with internal standard approach

19. Problems with intensity ratios
Strain in crystal lattice
Non-uniform microstrain (due to dislocations, vacancies, impurities, etc.) results in peak broadening and possibly asymmetry

20. Other errors: Flat specimen error
For correct diffraction geometry, sample should be curved and lie on focusing circle; flat sample causes an asymmetric peak broadening at towards lower 2θ angles
Fixed slit (vs. variable slit)
Larger area of sample irradiated at low 2θ values have less depth of penetration; at higher 2θ, irradiated area is smaller, but depth of penetration greater. These tend to offset, so get constant volume being irradiated.
Axial divergence
Occurs if X-ray beam diverges out of the plane of the focusing circle; causes peak asymmetry at low angles; soller slits and curved crystal monochromator limit divergence

21. Whole-pattern fitting (i.e., Rietveld analysis)
Rietveld (1969) developed a method to refine crystal structure information using neutron powder diffraction
Uses ‘initial’ crystal structure as a starting point to calculate the expected diffraction profile based on physics
Differences between the calculated profile and the measured profile are minimised by a least squares iterative approach
Uses all peaks and the complete profile (i.e., peak position, peak intensity, and peak shape) in the analysis
Many variables (errors) can be accounted for (e.g., preferred orientation, crystallite size, strain, peak overlap, peak asymmetry, site occupancy, absorption, diffraction geometry, etc.)
pattern is sum total of all of the effects
Can be used for standardless quantification of polycrystalline materials
Capable of much greater accuracy than intensity ratio methods

22. Rietveld analysis How does it work?
Need to already know what is in the mixture (phase identification)
Need solved unit cell structural information for each compound in mixture
Unit cell parameters (dimensions, angles, crystal system, space group)
Atom type and positions (and site occupancy)
Inorganic Crystal Structure Database (ICSD)
Maintained by FIZ Karlsruhe and NIST
Contains 89,064 entries of inorganic crystal structure data (does not contain everything you may want)

23. ICSD database
Calcite 80869

25. ICSD database
Super cell for alite (C3S) contains 226 atoms

27. Very computer intensive
4320 calculated peaks contribute to final profile

28. Rietveld analysis benefits
Rietveld analysis has advantages over conventional Bogue calculations based on XRF data
Bogue assumes pure compounds (know this is not true)
Gives more accurate determination of major clinker phases
underestimates alite (up to 20%), overestimates belite (up to 10%) (Stutzman, 2004; Glasser, 2004), underestimates ferrite by ~2-3% (for high Fe2O3 clinker) (Feldman et al., 2005)
Polymorphism – can detect and quantify different polymorphs of compounds (e.g., cubic C3A, orthorhombic C3A)
Can detect and quantify other phases (e.g., free lime, periclase, calcium sulphates (different forms), alkali sulphates, etc.)

29. What about amorphous materials?
Rietveld gives a ‘normalised’ fit – assumes that everything is crystalline and sum of mass fractions add up to 1 (or 100%)
Can use internal standard (all peaks) to calibrate scale factors
Difference between calculated and actual is due to amorphous content
e.g., Determination of glass content of blast-furnace slag