Validating Phase Quantification of Crystalline Materials using X-Ray Diffraction

X-ray diffraction (XRD) is a common method to perform phase identification on crystalline materials. This technique, when combined with elemental analysis, can be a very powerful tool capable of distinguishing similar materials with different properties, such as polymorphs of the same compound. This is often an important factor in materials development. In this article, we are going to show you how XRD can be used to both identify and quantify the crystalline components in a mixture.

In this example article we have created three mixtures that contain different amounts of three white powders: calcite (CaCO₃), anatase (TiO₂), and rutile (TiO₂). Notice that anatase and rutile are polymorphs, which have the same stoichiometry. This means that these two components will be indistinguishable to methods of identification that rely on elemental composition. Since x-ray diffraction measures the crystal structure of a material, it is a perfect fit to determine the compositions of these three mixtures, even though two of the components are polymorphs.

There are two common methods used for quantification of phase fraction from x-ray diffraction patterns, and we will evaluate the accuracy of both methods in this article. The methods are known as reference intensity ratios (RIR) and whole pattern fitting (WPF). A prerequisite for performing quantification using either of these methods is that the primary phases in the diffraction patterns are indexed to high quality reference diffraction patterns. Once this is done, phase quantification analysis can be performed.

To begin the evaluation, the three samples were weighed using an analytical balance. Table 1 lists the known compositions in weight percentage (wt%) of Samples 1, 2, and 3.

RIR Method

Figure 1 displays an example of the quantification process using the RIR method. The experimental diffraction patterns were each indexed using quality reference patterns obtained from the ICDD database. These reference patterns are listed just above the x-axis near the bottom of the diffraction pattern.

In this example, quantification was performed iteratively on seven groups of peaks, and the quality of the fitted results for each set of peaks is displayed in the difference plot near the top of Figure 1. The strongest peak for each of the phases present are labelled with the computed weight percent for that phase, and color coded accordingly.

WPF Method

This method employs Rietveld refinement techniques to completely fit a simulated diffraction pattern to an experimental pattern. The first parameter optimized by this method (if it is a mixture) is composition, and then more granular diffraction-related parameters such as lattice constants and site occupancy are refined.

Figure 2 displays an example of the quantification process using the WPF method. The results are displayed in a very similar format to the RIR method.

Results and Discussion

How do these methods stack up against the true compositions? To evaluate this, three replicate diffraction patterns were collected from each sample. The replicates were analyzed and quantified separately using both the WPF and RIR methods. The mean values were plotted for each sample in a bar chart, and the black line at the top of the blue bars shows the standard deviation of the replicate measurements (Figure 3).

Finally, the accuracy was assessed in three different composition ranges. If you look at Table 1 you will notice that the ratios used in each sample are all close to 60%, 30%, and 10%. This was done so that the results could be grouped by composition instead of sample, so we can see how precision and accuracy vary with respect to composition.

The relative standard deviation (RSD) and percent error (%Error) were used to measure the precision and accuracy, respectively, of the mean compositions and standard deviations that were obtained from the replicates. Table 2 summarizes theses values.

From Table 2 we can see that there is an inverse correlation between concentration and RSD for both the RIR and WPF methods (as concentration increases, RSD decreases). This means that precision is improving as concentrations increase. This trend also holds true for accuracy: the %Error improves at higher concentrations.

Additionally, both methods are reasonably accurate at 60 and 30 wt%, but at 10 wt% they deviate from the actual concentration by more than 10% of the value, which is on the high side. This is probably because the concentration is nearing the XRD detection limit, which is around 3-5 wt%. Therefore, neither of these methods should be applied to concentrations much lower than 10 wt% in a mixture.

Summary

Hopefully, this analysis highlighted some of the benefits and considerations of using diffraction to quantify concentration in crystalline samples. One of the main advantages is that the component list can include specific phases that may even be polymorphs with the same composition. This was demonstrated here with the anatase and rutile polymorphs. The precision and accuracy were good enough to distinguish between these two compounds down to concentrations of at least 10 wt% in a mixture of phases.

Two methods of calculating the concentration were also evaluated and compared to one another. They were RIR and WPF; both methods performed well on these mixtures of calcite, anatase, and rutile. The choice of method for a given project may depend on experimental details like how the peaks of each phase are distributed throughout the diffraction pattern. It is nice to have two options to work with when thinking about how to approach client samples. Please contact McCrone Associates if phase quantification is required for your materials characterization needs.