To obtain the lattice parameters accurately by the Rietveld method, the relationship between the lattice parameters and the peak-shift, which is the deviation in diffraction angle from the theoretical Bragg position, was studied. We show that the fitting accuracy of lattice parameters is related directly to the well reproducibility of the peak-shift. This study unveils that the peak-shift consists of the experimental and the analytical ones. The analytical peak-shift erroneously lowers a reliability factor Rwp, which has, so far, been the conventional criterion of fit. The conventional Rietveld method obtains a unit-cell which is a homothetic (proportional) unit-cell of the true one. We propose an additional criterion based on the peak-shift to obtain the true lattice parameters accurately. Our criterion can achieve reproducibility reasonably well for the experimental peak-shift, leading to highly improved accuracy of the lattice parameters.
Structural study for powder materials relies on the Rietveld method, which is capable of refining the structural and magnetic parameters from diffraction data1,2,3. However, it is fundamentally difficult to determine accurately the refinement parameters4,5,6,7. In the Rietveld method, the weighted sum of squares residual, Sr, between the observed and the calculated intensities of powder diffraction data is minimized in a nonlinear least-squares method. The calculated intensity includes the peak-shift that is absolutely inevitable in the experiment. To evaluate quantitatively the best fit of the data, several reliability-factors such as Rwp, Rp, Re, R F , S and χ2 are proposed3,4. The most accepted factor is the weighted-profile R, termed as Rwp, where the numerator includes Sr that is minimized during the refinements. The goodness-of-fit, S or χ2 ≡ S2, is used as another useful numerical criterion4. The S-value of 1.3 or less is empirically considered to be satisfactory. However, a poor counting statistics or a high background also makes S smaller; the S-value sometimes turns out to be less than 1.0. On the other hand, S may possibly be larger than 1.3 even for the best fitting with an appropriate model. Young and co-workers have suggested that these values to be given in publication8.
Strictly speaking, there is no general agreement on these criteria in the Rietveld method. Other studies have concluded that viewing the profile-plots is more effective than R-values to determine the quality of a refinement5,6,7. As such, the refined structural parameters have been found to differ from researcher to researcher. For instance, Hill summarized the results of Rietveld refinements on the project undertaken by the Commission on Powder Diffraction of the International Union of Crystallography7. Several specialists analysed the standard PbSO4 powder diffraction pattern, measured by a conventional Bragg–Brentano diffractometer using Cu Kα radiation. The lattice parameters a, b and c are in the range of 8.4764–8.4859 Å, 5.3962–5.4024 Å and 6.9568–6.9650 Å, respectively. The accuracy of the lattice parameters is of an order of 0.01 Å (=10 × 10−3 Å), which is incomparably large considering that the linear thermal expansion coefficient is of an order of 10−5 K−1 to 10−6 K−1 for general solid materials9. Furthermore, the weighted mean parameters for a-, b- and c-axes are 8.4804(4) Å, 5.3989(3) Å and 6.9605(2) Å, respectively. They are in good agreement with those determined from single-crystal X-ray diffraction data7,10 which is generally accepted to be high-accuracy. These facts mean that either smaller or larger lattice parameter compared to the true one is possibly obtained depending on a researcher by the Rietveld method. This is a critical disadvantage to study the dependences of lattice parameters on temperature, composition, pressure and so on. A technique to determine refinement parameters accurately is needed for the Rietveld method.
We shed light on the peak-shift that tends to be overlooked. This paper proposes an additional criterion, focusing on a fitting accuracy along the horizontal-axis of powder diffraction data, to determine the lattice parameters accurately. In the following, we demonstrate that our criterion enables the well reproducibility of the peak-shift, leading to highly improved accuracy of the lattice parameter by two or more digits lower compared to that obtained by the conventional Rietveld method.
The X-ray diffraction pattern of standard reference material (SRM) 660a (lanthanum hexaboride)11 from the National Institute of Standards and Technology (NIST) collected with Cu Kα1 radiation was used in this study. We focused on the maximum diffraction angle (2θmax) of the data used in the analysis. We conducted several conventional Rietveld refinements in the 2θ-range from 18° to 2θmax, where 2θmax was in between 52° and 152°. There were five Bragg-peaks for 2θmax = 52° and twenty-four Bragg-peaks for 2θmax = 152°. The representative results for 2θmax = 152° and 92° are demonstrated.
In the conventional Rietveld refinement, the lattice parameters are acnv,(152) = 4.15655(1) Å with Rwpcnv,(152) = 8.203% and acnv,(92) = 4.15811(22) Å with Rwpcnv,(92) = 8.610%, where the superscripts ‘cnv’, (152) and (92) refer to the “conventional”, 2θmax = 152° and 92°, respectively. Here, acnv,(152) and acnv,(92) are 0.37 × 10−3 Å (or 0.0089%) smaller and 1.19 × 10−3 Å (or 0.0286%) larger than aSRM ≃ 4.15692(1) Å, respectively11. The Rietveld refinements with a fixed value of aSRM were conducted. The reliability factors Rwpfix,(152) and Rwpfix,(92) are 8.355% and 8.623%, respectively, where the superscript ‘fix’ refers to the “fixed”. Significantly, Rwpfix is larger than Rwpcnv, implying that Rwp is an incomplete criterion of fit. Note that a difference between Rwpfix and Rwpcnv is not caused by the difference of the number of parameters in each refinement because Re, which corresponds to mathematically expected Rwp, is Refix,(152) = Recnv,(152) = 8.203% and Refix,(92) = Recnv,(92) = 4.090%, and are the same with each other independent on the number of parameters.
where Z is the zero-point shift (also known as the zero error), Ds the specimen-displacement parameter and Ts the specimen-transparency parameter. Manually estimated peak-shift, Δ2θm, is also plotted. Note that the 2θ-regions with grey background in Fig. 1 are not used in the Rietveld refinement. Clearly, Δ2θRfix and Δ2θm correspond well with each other within an error bar in the analysis 2θ-region (white area). In contrast, Δ2θRcnv differs from Δ2θRfix and Δ2θm especially in the large 2θ-region.
Figure 1c and d show the 2θ-dependence of the difference, Δ2θdif ≡ Δ2θRcnv − Δ2θRfix, which could be zero when a = aSRM. Otherwise, the absolute value of Δ2θdif increases with 2θ. Moreover, Δ2θdif is not negligible with respect to the magnitude compared to Δ2θRfix and Δ2θm (Fig. 1a and b). Note that Δ2θdif can be expressed by Eq. (1) with a different set of values of (Zcnv, Dscnv, Tscnv) and (Zfix, Dsfix, Tsfix). Most importantly, in the analysis 2θ-range, the 2θ-dependence of Δ2θdif corresponds well with that of Δ2θana, which is expressed as:
where A is the proportional coefficient. Here, Eq. (2) is not obtained by fitting the experimental data but is formulated by rearranging the following two Bragg’s equations and, therefore holds for any crystal system:
The coefficients Acnv,(152) and Acnv,(92) are acnv,(152)/aSRM = 0.999911 and acnv,(92)/aSRM = 1.000286, respectively. Equally important is that Δ2θm + Δ2θana as well as Δ2θRfix + Δ2θana are in good agreement with Δ2θRcnv in the analysis 2θ-range and enhance against Δ2θRcnv beyond 2θmax (Fig. 1e and f).
Criteria of fit
To investigate a criterion of fit in detail and study how the peak-shift affects the result, we have conducted several Rietveld refinements with a fixed value of Z. Figure 2 shows the Z- and a-dependences in the conventional criterion as well as by the criteria set in this study. The sums are carried out over all the Bragg-peaks in the analysis 2θ-range for Σ|Δ2θR| and the whole 2θ-range for Σall|Δ2θR|. Note that Σ|Δ2θR| and Σall|Δ2θR| are calculated from the result after the refinement. The convergence in the refinement is judged by using Rwp. For Σ|Δ2θR|, the number of Bragg-peaks in the sum depends on 2θmax, and is 24 for 2θmax = 152° and 13 for 2θmax = 92°. In contrast, the number of Bragg-peaks is always 25 for Σall|Δ2θR|, including a reflection with the Miller indices of 432 and 520 (lattice spacing d ≃ 0.772 Å) at 2θ ≃ 172° that is measureable in principle but is not observed in the data.
The conventional criterion Rwp shows a parabolic curve with the minimum values of 8.203% at acnv,(152) = 4.15655(1) Å and 8.610% at acnv,(92) = 4.15811(22) Å as shown in Fig. 2a and b. Importantly, the minimum of Rwp is not at aSRM, which is a strong evident that Rwp itself is an insufficient criterion to obtain the true lattice parameter. Further, the range of Rwp for 2θmax = 92° is much smaller than that for 2θmax = 152°. It suggests that for the smaller 2θmax, it is more difficult to distinguish the minimum Rwp correctly.
A potential criteria Σ|Δ2θR| shows a V-shaped curve with the minimum values at asum,(152) = 4.15684(0) Å and asum,(92) = 4.15625(2) Å, where the superscript ‘sum’ refers to the “sum” of the peak-shift (Fig. 2c and d). The lattice parameter asum is closer to aSRM compared with acnv. The magnitude of Σ|Δ2θR| for 2θmax = 92° is smaller than that for 2θmax = 152°, which is reasonable considering the number of Bragg-peaks in the sum. Our proposed criterion Σall|Δ2θR| shows a sharper V-shaped curve than Σ|Δ2θR| with the minimum values at aall,(152) = 4.15686(0) Å and aall,(92) = 4.15696(2) Å, where the superscript ‘all’ refers to the sum of “all” values of Δ2θR (Fig. 2e and f). The lattice parameter aall is much closer to aSRM compared with acnv and asum. With decreasing 2θmax, the magnitude of Σall|Δ2θR| increases and the V-shape becomes sharper.
Figure 3a demonstrates the 2θmax-dependence of the lattice parameters obtained by several criteria. First, acnv, which is obtained by the conventional Rietveld method, shows a large deviation from aSRM and strong dependence on 2θmax. The maximum deviation from aSRM is >10 × 10−3 Å, which is in the same order as that in Hill’s report7. Next, asum, which is determined with the minimum of Σ|Δ2θR|, approaches toward aSRM with increasing 2θmax. The smallest deviation from aSRM is 0.08 × 10−3 Å at 2θmax = 152°. Subsequently, aall, which is determined by using Σall|Δ2θR|, corresponds well with aSRM even for the smaller 2θmax. The deviation from aSRM is 0.60 × 10−3 Å at the most and within 0.06 × 10−3 Å above 2θmax = 74°. The accuracy is improved by two or more orders of magnitude compared with that of the conventional Rietveld method.
Figure 3b shows the 2θmax-dependence of Rwp’s. It is clear that Rwp increases with decreasing 2θmax. For all 2θmax, the values of Rwpcnv are smaller than those of Rwpall despite the fact that acnv does not correspond to aSRM. The difference, Rwpall − Rwpcnv, becomes smaller with decreasing 2θmax and is 0.02% or less below 2θmax = 120° as shown in Fig. 3c. It becomes zero at some 2θmax’s, implying the impossibility in distinguishing the true solution exclusively by the Rwp-value.
Figure 3d–f show the peak-shifts determined with the minima of Rwp, Σ|Δ2θR| and Σall|Δ2θR|. Clearly, Δ2θRcnv does not reproduce Δ2θm, reflecting a mismatch of the lattice parameter between acnv and aSRM. Although Δ2θRsum is closer to Δ2θm than Δ2θRcnv, it deviates from Δ2θm above 2θmax as shown in the inset of Fig. 3e as an example. Additionally, Δ2θRall well reproduces Δ2θm for the all 2θmax (Fig. 3f). These facts indicate that the fitting accuracy relates directly to the well reproducibility of the peak-shift.
The present study reveals several critical findings. Firstly, the 2θ-dependence of peak-shift does not obey Eq. (1) in the calculation; instead follows the equation:
where ζ is the zero-point shift, δs the specimen-displacement parameter, τs the specimen-transparency parameter and A the proportional coefficient to lattice spacing (Fig. 1e and f). Equation (3) holds for any crystal system and can be simply rewritten as:
where Δ2θexp is the experimental peak-shift by the geometry (includes design-geometry of instrument as well as specimen-geometry) and Δ2θana is the analytical peak-shift caused by the mismatch of the lattice parameters. Notably, Δ2θana exists in the calculation only when A ≠ 1. Considering Eqs (3) and/or (4), one cannot obtain the true peak-shift when Δ2θana ≠ 0 (A ≠ 1). Secondly, Δ2θana can be fitted very well by Eq. (1) in the analysis 2θ-range (Fig. 1c and d). The finite value of Δ2θana, therefore, induces a false peak-shift with irrelevant lower-Rwp (Fig. 2a and b). As a result, a homothetic unit-cell, which is proportional to the true one, is obtained in the conventional Rietveld method. To obtain the correct unit-cell, Δ2θana = 0 should be imposed. Finally, we have proposed an additional criterion, Σall|Δ2θR|, which measures the fitting accuracy along the horizontal-axis of the diffraction data and is capable of preventing Δ2θana from enhancing. By combining our criterion with Rwp, we can well reproduce the peak-shift (Fig. 3f). Consequently, we can determine the lattice parameter within the accuracy of 0.06 × 10−3 Å (Fig. 3a). Incidentally, we deduce that there was no need to consider too much detail about the peak-shift in the early stage of developing the method because the angle-dispersive neutron data was used3,13. Neutron has high transparency against the materials. Enough high-angle data, e.g. 2θmax = 144° (ref.3), with quite broad Bragg-peaks were generally obtained using an old-fashioned reactor source. As a result, the 2θ-dependence of peak-shift was approximately constant and could easily be reproduced. In fact, Rietveld applied a zero-shift parameter as the peak-shift function which is independent on 2θ (ref.3).
Our present findings possibly accelerate designing novel materials since a comparative study between the experiment and theory14 may be achieved with high-accuracy. The criterion we set in this report would be applicable for structure determination from powder diffraction15,16,17 including indexing the diffraction peaks18,19,20 and the profile decomposition21,22,23 as well as a quality management of mass production of materials in industry. Further study related to the structural parameters in the unit-cell is desirable.
In summary, an additional criterion, Σall|Δ2θR|, to determine accurately the lattice parameter by the Rietveld method from powder diffraction data has been proposed. The refinements of the same data with different fixed-values of peak-shift parameter lead to different values of reliability factor, Rwp. The refined lattice parameter at the minimum Rwp-value is different from the correct one. The peak-shift includes the analytical Δ2θana as well as the experimental Δ2θexp in the calculation. Δ2θana must be neutralized for the analysis because it results in a false unit-cell that is proportional to the true one with the incorrect lower Rwp-value when Δ2θana ≠ 0. Our criterion allows well reproducibility of the peak-shift through the highly accurate determination of lattice parameter by two or more digits lower than that compared with the conventional Rietveld method.
Powder diffraction data
Diffraction pattern used in this study was measured by Le Bail and distributed on the website24. The data file with a name of “660a-2.dat” in a compressed file, x-celerator.zip, was used. The X-ray diffraction pattern for SRM 660a11 was carefully collected in the range of 2θ = 18.003°−151.995° with a step of 0.008° by using a conventional diffractometer (Philips X’Pert, equipped with an X’Celerator detector) with Cu Kα1 radiation. Twenty-four diffraction peaks were observed. Total measurement time was more than 17 h and the largest intensity was more than 100000 counts, realizing very good statistics and a high signal-to-background ratio. The diffraction peaks were very sharp as a full width at half-maximum (FWHM) of a diffraction peak were approximately 0.03° at the lowest angle and 0.17° at the highest angle. The peaks were fairly symmetric for the in-house data. The lattice parameter aSRM = 4.1569162(97) Å ≃ 4.15692(1) Å at 22.5 °C has been clarified by NIST11.
Data analysis approach
The Rietveld program RIETAN-FP25 was selected to analyse the data in this study. Taking the rounding error of the program into consideration, a value of the wavelength λ (1.540593 Å for Cu Kα1 radiation) in RIETAN-FP is the same as that used in the computation for SRM 660a11 by NIST (λ = 1.5405929(5) Å)26. Note that the other major Rietveld programs use a slightly different value for Cu Kα radiation as default. For example, in GSAS27, GSAS-II28, FullProf29, Z-Rietveld30 and TOPAS31, the wavelengths of Cu Kα1 are 1.5405 Å, 1.54051 Å, 1.54056 Å, 1.54056 Å and 1.540596 Å, respectively. The profile function of a Thompson-Cox-Hastings pseudo-Voigt function32 was used. Howard’s method33, which is based on the multi-term Simpson’s rule integration, was employed for the profile asymmetry. Profile cut-off was 0.001%. The background function was the sixth order of Legendre polynomials.
In addition to the conventional Rietveld refinement, several sets of Rietveld refinement, with different fixed-values for the first term Z of the peak-shift, were performed. Here, the other refinement parameters were refined. This is because we have assumed that a parameter Z, which is different from the true value Ztrue, gives Rwp larger than that for the true value Rwptrue. The range of Z between −0.2° and 0.2° was chosen considering a FWHM of a diffraction peak. For each set, a total of 157 calculation-steps were conducted to confirm that our procedure was enough to converge. Thus, our calculation is the fixed routine one that is applied to the same data set starting from different fixed Z-values for Rietveld refinement.
The peak-shift Δ2θm ≡ 2θSRM − 2θobs was calculated by using the raw data. The list of ideal Bragg-peak angle, 2θSRM, was provided in the certificate of SRM 660a11. The observed diffraction angle 2θobs for each reflection was chosen at the strongest intensity in the diffraction data near 2θSRM. The measurement error of 2θobs was assumed to be the same as the step of 0.008° in the data.
The data that support the findings of this study are distributed by Prof. Armel Le Bail and available in the website, http://www.cristal.org/powdif/low_fwhm_and_rp.html.
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The authors thank Armel Le Bail for kindly allowing us to use his high-quality data. We are grateful to Biswajit Paik for critically reading an early version of the manuscript.
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