A Dimensionally Consistent Size-Strain Plot Method for Crystallite Size and Microstrain Estimation

TL;DR

The paper identifies a long-standing dimensional inconsistency in the SSP equation used to extract crystallite size $D$ and microstrain $\sigma$ from XRD peak broadening. It traces the origin through the reciprocal-unit framework of Langford and Halder–Wagner, showing that a missing factor led to a dimensionless and physically meaningless form that proliferated in the literature. By deriving a dimensionally consistent SSP equation from the HW formulation, it demonstrates unit-invariant, physically meaningful estimations and establishes the analytical equivalence between HW and SSP representations. The work emphasizes rigorous re-evaluation of established analytical tools and has broad practical implications for the interpretation of vast XRD-based microstructural data.

Abstract

X-ray diffraction (XRD) peak broadening analysis remains a cornerstone for quantifying crystallite size and lattice microstrain in materials. Among various approaches, the Size Strain Plot (SSP) method is widely employed for its conceptual simplicity and ease of use. However, this study reveals that the equation most commonly applied in SSP analysis is dimensionally inconsistent, a critical flaw that has gone largely unnoticed and replicated across decades of materials research. This pervasive error raises concerns about the validity of a significant body of published microstructural data. By tracing the historical origin of the misformulated equation, we demonstrate how a seemingly minor oversight evolved into a widely accepted standard practice within the field. We then present a dimensionally consistent formulation that restores physical meaning and analytical reliability to the SSP method. The corrected framework re-establishes the SSP approach as a robust and physically valid tool for XRD-based microstructural characterization. \en

Figures (4)

• Figure 1: Peak-shape fitting of a representative XRD reflection using four analytical functions: Voigt, pseudo-Voigt, pseudo-Voigt (Thompson--Cox--Hastings, TCH), and Pearson VII. The upper panel shows the experimental data (black dots) and the fitted profiles (solid lines), normalised to the same peak height for clarity. The lower panel presents the corresponding residuals ($y_{\rm obs} -y_{cal}$), highlighting that the Voigt, pseudo-Voigt, and pseudo-Voigt(TCH) functions provide the closest agreement with the experimental peak shape, whereas the Pearson VII profile exhibits noticeable deviations in the tail region.
• Figure 2: Representative Size--Strain Plot (SSP) constructed using the dimensionally inconsistent formulation [Eq. (\ref{['eq:zakssp']})]. The resulting intercept, slope, and the corresponding size and strain parameters are summarised in Table \ref{['tab:comparison_results']}.
• Figure 3: Size--Strain Plot (SSP) constructed using the dimensionally consistent formulation [Eq. (\ref{['eq:correctssp_eqn']})]. The fitted intercept and slope, together with the derived domain-size and microstrain parameters, are summarised in Table \ref{['tab:comparison_results']}.
• Figure 4: Halder--Wagner plot constructed in reciprocal units according to Eq. (\ref{['eq:HalderW']}). The $x$–$y$ values generated by this relation correspond precisely to those of the dimensionally consistent Size--Strain Plot (SSP) in Figure \ref{['fig:ssp_eq8']}. Consequently, the fitted trend and all derived parameters are identical, confirming the analytical equivalence of the two formulations. A numerical comparison of the yielded parameters is provided in Table \ref{['tab:comparison_results']}.