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Powder X-ray Diffraction Basic Course - Seventh Installment: Lattice Constants

Keigo Nagao

Summer 2024 Volume 40, No. 2 , 20-26

In the seventh lecture of the Powder X-ray Diffraction Basic Course, we will describe lattice constants. Lattice constants are a set of fundamental parameters used in various analyses by the X-ray diffraction method. There are three methods for lattice constant calculation; 1. a simple method using the Miller indices of one observed diffraction peak and the d value converted from the peak position using Bragg’s law; 2. the least-squares method using multiple diffraction peaks and the Miller indices of each peak; 3. the WPPF (Whole Powder Pattern Fitting) method using a wide range of diffraction patterns. Analysis methods 2 and 3 allow angle calibration using an angle standard reference material. There are external and internal standard methods, each with their own advantages and disadvantages. The WPPF method also allows angle calibration using a peak shift model function without measurement of angle standards. In-situ measurements are also possible using a temperature control attachment, where the temperature of the sample can be changed and the diffraction pattern is acquired in situ. High-temperature measurements of LSMO (lanthanum strontium manganese oxide), a solid oxide fuel cell, revealed that the behavior of the lattice expansion was different for the a- and c-axes.

Highlights

  • Lattice constants are the unit cell dimensions and angles, and they are highly sensitive to changes in temperature, pressure, composition, and material state.
  • Three main ways to calculate them are described: a quick single-peak approach, a multi-peak least-squares approach, and WPPF, which fits a broad portion of the entire diffraction pattern and is now the most commonly used method.
  • High-angle diffraction peaks give more accurate d-spacings, so they are especially valuable when precise lattice constants are needed.
  • Angle calibration strongly affects accuracy. External and internal standards each have tradeoffs, and WPPF can also correct peak shifts with a model function without measuring a standard every time.
  • High-temperature XRD showed that LSMO expands differently along the a- and c-axes, and composition-dependent measurements of Y(InxMn1−x)O₃ followed a near-linear trend consistent with Vegard’s law.

Summary

Lattice constants are the numbers that describe the size and shape of a crystal’s repeating unit cell. In XRD, they are obtained from diffraction peak positions because those peak positions are related to the spacing between crystal planes through Bragg’s law. Different crystal systems place different constraints on how many independent lattice parameters are needed.

A simple way to estimate lattice constants is to assign Miller indices to a peak, convert the peak position to a d-spacing, and calculate the corresponding unit-cell dimension. That approach is fast, but it is limited because it depends heavily on one or a few peaks and can be affected by peak overlap and peak-shape asymmetry.

A more reliable approach uses several diffraction peaks and refines the lattice constants by least squares. An even more powerful approach is WPPF, which fits a wide angular range of the measured pattern using a physically meaningful peak-shape model. Because it uses much more of the data, WPPF is better suited to modern measurements and to patterns where peaks are close together.

Accurate lattice constants also depend on angle calibration. That can be done with a standard material measured separately, mixed into the sample, or by using a peak-shift correction model in WPPF. Practical examples show why this matters: barium titanate allows very precise c/a evaluation, LSMO shows anisotropic thermal expansion at high temperature, and Y(InxMn1−x)O₃  shows lattice changes that track composition.

Frequently asked questions

They are the parameters that define the size and shape of the crystal unit cell. In the most general case, there are six: three lengths, a, b, and c, and three angles, α, β, and γ. Higher-symmetry crystal systems need fewer independent parameters, while low-symmetry systems need more.

They are fundamental structural values and are often used to track how a material changes with temperature, pressure, composition, or degradation. Because diffraction peak positions respond to changes in interplanar spacing, XRD can detect small shifts in lattice dimensions that reflect real changes in the material.

A small angular error changes the calculated d-spacing less favorably at low angles than at high angles. The discussion based on the differentiated Bragg relation shows that measurements become more sensitive to true d changes as θ approaches 90°, so higher-angle peaks generally improve accuracy.

The simple method uses one indexed peak and its d-spacing to calculate a parameter quickly. The least-squares method uses multiple indexed peaks and refines lattice constants from all of them together, which improves reliability. WPPF goes further by fitting a broad portion of the whole diffraction pattern, making it especially useful for modern 1D detector data and for patterns with overlapping peaks.

Because real diffraction peaks are not perfectly symmetric. Effects such as umbrella and horizontal divergence can shift the apparent peak position, often toward lower angle, which makes the calculated d-spacing and lattice constant come out slightly too large. Peak overlap can make this worse unless profile fitting is used carefully.

Angle calibration corrects systematic peak-position errors so that refined lattice constants are closer to their true values. It can be done with a certified standard measured separately from the sample, with a standard mixed into the sample, or in WPPF through a peak-shift model. Without good calibration, even high-quality diffraction data can produce inaccurate lattice parameters.

The external standard method measures a certified reference separately, which is convenient because the sample does not need to be mixed and the standard peaks do not interfere with the sample peaks. Its weakness is that differences in sample height or X-ray absorption between the standard and the test specimen can introduce calibration errors. The internal standard method mixes the standard into the sample, which helps cancel absorption and height-related effects, but it requires mixing and consumes the reference material.

They show that lattice constants are useful for tracking real material behavior. Barium titanate gave a highly precise c/a ratio using WPPF with a peak-shift model. LSMO showed that the a- and c-axes both expand with temperature but not by the same amount, so the thermal expansion is anisotropic. Y(InxMn1−x)O₃ showed nearly linear changes in lattice parameters with composition, which is consistent with Vegard’s law and can help estimate solid-solution ratio.

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