Diffraction Lecture 25: Rietveld Refinements

Scientific concepts, discoveries, and nature/physics phenomena

Inverse crystallography problem (powder vs single crystal)

• Goal of crystallography: work backward from a diffraction pattern to determine the crystal structure.
• Single crystals:
  • Peak positions and intensities can be separated more cleanly (via:
    1. indexing
    2. structure factors
    3. phase determination
    4. structural refinement).
• Powder diffraction:
  • The analysis steps are not cleanly separated because many peaks overlap.
  • Refinement must fit the entire pattern at once.

Diffraction physics in a nutshell

• Unit cell dimensions determine Bragg peak angles (constructive interference).
• Structure factors connect peak intensities to the atomic arrangement within the unit cell.
• Background scattering can include:
  • inelastic scattering (Compton)
  • and other instrumental contributions.

Rietveld method (key discovery)

• Hugo Rietveld (late 1960s) developed a method for analyzing powder diffraction by fitting the full diffraction pattern simultaneously using a structural model.
• Motivation: rather than matching individual reflection intensities (difficult due to overlap), fit the pattern globally.

Modeling a powder diffraction pattern (4-part decomposition)

A typical Rietveld-style pattern model can be decomposed into:

1. Peak positions
   • Determined from lattice parameters (where peaks occur).
2. Peak intensities / peak areas
   • Related to structure factors (crystallographic information).
3. Peak profile / peak shape
   • Peaks are not delta functions; width/lineshape matter.
4. Background
   • Smooth scattering + instrumental and inelastic effects.

Space group assignment and structure solution strategy

• From powder diffraction, determine:
  • Index peaks → lattice and peak positions
  • Systematic absences → space group
• Often, instead of a detailed structure factor-based phase solution, use:
  • analogies to similar compounds’ structures
  • estimates based on the number of atoms per unit cell
  • multiplicities of Wyckoff positions
• Then refine atomic positions using Rietveld refinement.

Peak shape models

• Common baseline peak shapes:
  • Gaussian
  • Lorentzian (with long tails)
• Real instruments often produce a mixture:
  • Pseudo-Voigt function with mixing parameter η
    • η = 1 → pure Lorentzian
    • η = 0 → pure Gaussian
• Peak width variation with diffraction angle:
  • Thompson–Cox–Hastings (TCH) function
  • Uses refinable parameters u, v, w, x, y (as described) to model FWHM vs 2θ
• Instrumental vs sample broadening
  • Sample effects discussed include:
    • small crystallite size
    • microstrain

Instrument-specific effects on lineshape

Different diffractometers/detectors alter the Gaussian/Lorentzian balance and the peak-width evolution:

• Point detector vs position-sensitive detector
• Benchtop (e.g., Cu source) vs synchrotron

Examples described:

• Benchtop (Rigaku) data: low-angle peaks more Lorentzian; Gaussian component grows with angle.
• Position-sensitive detector: adds extra Gaussian contribution even at low angles.
• Synchrotron: extremely sharp peaks at low angles; moderate broadening at higher angles.

Guidelines for collecting powder diffraction data for successful Rietveld refinement

• Choose geometry/mode
  • Bragg–Brentano (flat plate) vs Debye–Scherrer (capillary transmission).
• Absorption constraints
  • For capillaries, ensure μR < 1, ideally < 0.1 (otherwise results become unreliable).
  • If absorption is too high:
    • use higher-energy X-rays
    • or dilute the sample.
• Flat plate considerations
  • Avoid beam spillover (from a divergent slit choice), as it distorts intensities and peak shapes.
  • Preferred orientation (texture) is a known issue for flat plate measurements, breaking the random-orientation assumption.
• Capillary advantages
  • Reduces preferred orientation issues (when absorption allows).

• Data coverage and resolution requirements

  • Collect a wide d-spacing range (include high-angle peaks) to reduce overlap and improve weak peak discrimination.
  • Choose a step size small relative to the sharpest peak:
    • typically 5–10× smaller than FWHM (e.g., ~0.01–0.02 for Cu lab data; smaller for synchrotrons).

• Avoid preprocessing that damages refinements

  • Do not smooth
  • Do not subtract background beforehand
  • Do not remove Kα2 before refinement (these should be modeled during Rietveld refinement).
• Fluorescence and absorption edges
  • With Cu radiation, elements like Fe and Co can cause strong fluorescence, increasing background and affecting scattering.
  • Certain lanthanides (Cs–Dy) have problematic L-edges with Cu radiation.
  • Some transition elements may also be affected (e.g., 5d-related effects were mentioned).

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Methodology / workflow outlined in the subtitles

Powder diffraction + Rietveld refinement workflow (high-level)

1. Collect powder diffraction pattern
2. Index peaks to determine lattice/peak positions
3. Analyze systematic absences to assign a space group
4. Generate an initial structural model
   • by analogy to related structures, or
   • using number of atoms/unit cell and Wyckoff multiplicities
5. Refine the structure by fitting the entire pattern
   • refine simultaneously:
     • peak positions
     • peak intensities
     • peak profile/shape
     • background

Pattern decomposition into four modeled components

• (1) Peak positions
  • lattice-parameter dependent
• (2) Peak intensities
  • peak area/intensity linked to structure factors
• (3) Peak profile/shape
  • Gaussian/Lorentzian mixture (Pseudo-Voigt with η)
  • width varying with 2θ (TCH model)
• (4) Background
  • modeled with few terms (e.g., Chebyshev polynomials)
  • or via interpolation between selected background points

Background modeling guideline

• Start with few polynomial terms (e.g., ~3–4 rather than many).
• Add terms only if necessary to avoid:
  • background “wiggling”
  • strong correlation between background parameters and peak intensities.

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Researchers / sources featured

• Dr. Curtis Moore (instructor; previously discussed single crystal methods)
• Hugo Rietveld (developer of the Rietveld method; described as Dutch crystallographer, late 1960s)
• Art Slate (PhD supervisor mentioned by the lecturer)
• Noah Hozapfel (name associated with data refined in the lecturer’s group)
• Lynn McCusker (author of refinement guidelines paper)
• Bob von Dreele (author of refinement guidelines paper)
• Dave Cox (author of refinement guidelines paper)
• Daniel Luer (author of refinement guidelines paper)
• Paulo Scardi (author of refinement guidelines paper)
• Argonne National Laboratory
  • referenced as a source/location mentioned for absorption edge lookup tooling (no individual named)