Introduction

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, S r, 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 R wp, R p, R e, R F , S and χ 2 are proposed3,4. The most accepted factor is the weighted-profile R, termed as R wp, where the numerator includes S r that is minimized during the refinements. The goodness-of-fit, S or χ 2 ≡ S 2, 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 (2θ max) of the data used in the analysis. We conducted several conventional Rietveld refinements in the 2θ-range from 18° to 2θ max, where 2θ max was in between 52° and 152°. There were five Bragg-peaks for 2θ max = 52° and twenty-four Bragg-peaks for 2θ max = 152°. The representative results for 2θ max = 152° and 92° are demonstrated.

Results

Rietveld refinements

In the conventional Rietveld refinement, the lattice parameters are a cnv,(152) = 4.15655(1) Å with R wp cnv,(152) = 8.203% and a cnv,(92) = 4.15811(22) Å with R wp cnv,(92) = 8.610%, where the superscripts ‘cnv’, (152) and (92) refer to the “conventional”, 2θ max = 152° and 92°, respectively. Here, a cnv,(152) and a cnv,(92) are 0.37 × 10−3 Å (or 0.0089%) smaller and 1.19 × 10−3 Å (or 0.0286%) larger than a SRM ≃ 4.15692(1) Å, respectively11. The Rietveld refinements with a fixed value of a SRM were conducted. The reliability factors R wp fix,(152) and R wp fix,(92) are 8.355% and 8.623%, respectively, where the superscript ‘fix’ refers to the “fixed”. Significantly, R wp fix is larger than R wp cnv, implying that R wp is an incomplete criterion of fit. Note that a difference between R wp fix and R wp cnv is not caused by the difference of the number of parameters in each refinement because R e, which corresponds to mathematically expected R wp, is R e fix,(152) = R e cnv,(152) = 8.203% and R e fix,(92) = R e cnv,(92) = 4.090%, and are the same with each other independent on the number of parameters.

Figure 1a and b show the 2θ-dependence of the peak-shift Δ2θ R computed with the following equation12:

$${\rm{\Delta }}2{\theta }_{{\rm{R}}}=Z+{D}_{{\rm{s}}}\,\cos \,\theta +{T}_{{\rm{s}}}\,\sin \,2\theta ,$$
(1)

where Z is the zero-point shift (also known as the zero error), D s the specimen-displacement parameter and T s the specimen-transparency parameter. Manually estimated peak-shift, Δ2θ m, is also plotted. Note that the 2θ-regions with grey background in Fig. 1 are not used in the Rietveld refinement. Clearly, Δ2θ R fix and Δ2θ m correspond well with each other within an error bar in the analysis 2θ-region (white area). In contrast, Δ2θ R cnv differs from Δ2θ R fix and Δ2θ m especially in the large 2θ-region.

Figure 1
figure 1

Diffraction angle dependence of the peak-shifts. (a,b) Δ2θ R cnv, Δ2θ R fix and Δ2θ m. Δ2θ R cnv is obtained by conventional Rietveld refinement. Lattice parameter is fixed at a SRM in Rietveld refinement for Δ2θ R fix. Δ2θ m is manually calculated by comparing 2θ-angle values of Bragg-peaks in the raw data and certification by NIST. (c,d) Difference Δ2θ dif ≡ Δ2θ R cnv − Δ2θ R fix and analytical peak-shift Δ2θ ana. Horizontal dash line with zero intensity is drawn as visual guide. (e,f) Sums Δ2θ m + Δ2θ ana and Δ2θ R fix + Δ2θ ana. Δ2θ R cnv is again plotted for comparison. For all the panels, open squares at approximately 2θ = 172° are the calculated data for a reflection with the Miller indices of 432 and 520 that is not observed in the data. Note that the 2θ-regions with grey background are not used in the Rietveld refinement. Left and right panels are for 2θ max = 152° and 92°, respectively. The error bars indicate the measurement error of the diffraction data in (a,b,e) and (f).

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Figure 1c and d show the 2θ-dependence of the difference, Δ2θ dif ≡ Δ2θ R cnv − Δ2θ R fix, which could be zero when a = a SRM. Otherwise, the absolute value of Δ2θ dif increases with 2θ. Moreover, Δ2θ dif is not negligible with respect to the magnitude compared to Δ2θR fix and Δ2θ m (Fig. 1a and b). Note that Δ2θ dif can be expressed by Eq. (1) with a different set of values of (Z cnv, D s cnv, T s cnv) and (Z fix, D s fix, T s fix). Most importantly, in the analysis 2θ-range, the 2θ-dependence of Δ2θ dif corresponds well with that of Δ2θ ana, which is expressed as:

$${\rm{\Delta }}2{\theta }_{{\rm{ana}}}=2\{{\sin }^{-1}(\frac{1}{A}\,\sin \,\theta )-\theta \},$$
(2)

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:

$$\begin{array}{c}2d\,\sin \,\frac{2\theta }{2}=\lambda ,\\ 2(Ad)\,\sin \,\frac{2\theta +{\rm{\Delta }}2{\theta }_{{\rm{ana}}}}{2}=\lambda .\end{array}$$

The coefficients A cnv,(152) and A cnv,(92) are a cnv,(152)/a SRM = 0.999911 and a cnv,(92)/a SRM = 1.000286, respectively. Equally important is that Δ2θ m + Δ2θ ana as well as Δ2θ R fix + Δ2θ ana are in good agreement with Δ2θ R cnv in the analysis 2θ-range and enhance against Δ2θ R cnv beyond 2θ max (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|Δ2θ R| and the whole 2θ-range for Σall|Δ2θ R|. Note that Σ|Δ2θ R| and Σall|Δ2θ R| are calculated from the result after the refinement. The convergence in the refinement is judged by using R wp. For Σ|Δ2θ R|, the number of Bragg-peaks in the sum depends on 2θ max, and is 24 for 2θ max = 152° and 13 for 2θ max = 92°. In contrast, the number of Bragg-peaks is always 25 for Σall|Δ2θ R|, 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.

Figure 2
figure 2

Conventional and candidate criteria of fit. Z- and a-dependences of (a), R wp fix for 2θ max = 152°, (b), R wp fix for 2θ max = 92°, (c), Σ|Δ2θ R| for 2θ max = 152°, (d), Σ|Δ2θ R| for 2θ max = 92°, (e), Σall|Δ2θ R| for 2θ max = 152°, (f), Σall|Δ2θ R| for 2θ max = 92°. Vertical dot line indicates a = a SRM. The minimum for each criterion is shown by the arrow. (a), R wp fix for 2θ max = 92° is also plotted for comparison (small dots). Note that relationship between Z and a for 2θ max = 152° (left panels) and that for 2θ max = 92° (right panels) are not the same.

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The conventional criterion R wp shows a parabolic curve with the minimum values of 8.203% at a cnv,(152) = 4.15655(1) Å and 8.610% at a cnv,(92) = 4.15811(22) Å as shown in Fig. 2a and b. Importantly, the minimum of R wp is not at a SRM, which is a strong evident that R wp itself is an insufficient criterion to obtain the true lattice parameter. Further, the range of R wp for 2θ max = 92° is much smaller than that for 2θ max = 152°. It suggests that for the smaller 2θ max, it is more difficult to distinguish the minimum R wp correctly.

A potential criteria Σ|Δ2θ R| shows a V-shaped curve with the minimum values at a sum,(152) = 4.15684(0) Å and a sum,(92) = 4.15625(2) Å, where the superscript ‘sum’ refers to the “sum” of the peak-shift (Fig. 2c and d). The lattice parameter a sum is closer to a SRM compared with a cnv. The magnitude of Σ|Δ2θ R| for 2θ max = 92° is smaller than that for 2θ max = 152°, which is reasonable considering the number of Bragg-peaks in the sum. Our proposed criterion Σall|Δ2θ R| shows a sharper V-shaped curve than Σ|Δ2θ R| with the minimum values at a all,(152) = 4.15686(0) Å and a all,(92) = 4.15696(2) Å, where the superscript ‘all’ refers to the sum of “all” values of Δ2θ R (Fig. 2e and f). The lattice parameter a all is much closer to a SRM compared with a cnv and a sum. With decreasing 2θ max, the magnitude of Σall|Δ2θ R| increases and the V-shape becomes sharper.

Figure 3a demonstrates the 2θ max-dependence of the lattice parameters obtained by several criteria. First, a cnv, which is obtained by the conventional Rietveld method, shows a large deviation from a SRM and strong dependence on 2θ max. The maximum deviation from a SRM is >10 × 10−3 Å, which is in the same order as that in Hill’s report7. Next, a sum, which is determined with the minimum of Σ|Δ2θ R|, approaches toward a SRM with increasing 2θ max. The smallest deviation from a SRM is 0.08 × 10−3 Å at 2θ max = 152°. Subsequently, a all, which is determined by using Σall|Δ2θ R|, corresponds well with a SRM even for the smaller 2θ max. The deviation from a SRM is 0.60 × 10−3 Å at the most and within 0.06 × 10−3 Å above 2θ max = 74°. The accuracy is improved by two or more orders of magnitude compared with that of the conventional Rietveld method.

Figure 3
figure 3

Comparison of obtained lattice parameters and peak-shifts. (a), 2θ max-dependence of lattice parameters a cnv, a sum and a all. The error bars represent the standard error σ in the Rietveld refinement. Horizontal dot line indicates a SRM. Inset: Enlarged view of a in the range of 75° ≤ 2θ max ≤ 165°. (b), 2θ max-dependence of reliability factors R wp cnv, R wp sum and R wp all. c, 2θ max-dependence of difference R wp all − R wp cnv. (d,e,f), 2θ-dependences of peak-shifts Δ2θ R cnv, Δ2θ R sum and Δ2θ R all. Bold dot lines and dash lines are for 2θ max = 152° and 92°, respectively. (e), Inset: Enlarged view of Δ2θ R sum for 2θ max = 92°. Δ2θ R sum starts to deviate at approximately 2θ max as shown by the arrow with increasing 2θ.

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Figure 3b shows the 2θ max-dependence of R wp’s. It is clear that R wp increases with decreasing 2θ max. For all 2θ max, the values of R wp cnv are smaller than those of R wp all despite the fact that a cnv does not correspond to a SRM. The difference, R wp all − R wp cnv, becomes smaller with decreasing 2θ max and is 0.02% or less below 2θ max = 120° as shown in Fig. 3c. It becomes zero at some 2θ max’s, implying the impossibility in distinguishing the true solution exclusively by the R wp-value.

Figure 3d–f show the peak-shifts determined with the minima of R wp, Σ|Δ2θ R| and Σall|Δ2θ R|. Clearly, Δ2θ R cnv does not reproduce Δ2θ m, reflecting a mismatch of the lattice parameter between a cnv and a SRM. Although Δ2θ R sum is closer to Δ2θ m than Δ2θ R cnv, it deviates from Δ2θ m above 2θ max as shown in the inset of Fig. 3e as an example. Additionally, Δ2θ R all well reproduces Δ2θ m for the all 2θ max (Fig. 3f). These facts indicate that the fitting accuracy relates directly to the well reproducibility of the peak-shift.

Discussion

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:

$${\rm{\Delta }}2{\theta }_{{\rm{R}}}=\zeta +{\delta }_{{\rm{s}}}\,\cos \,\theta +{\tau }_{{\rm{s}}}\,\sin \,2\theta +2\{{\sin }^{-1}(\frac{1}{A}\,\sin \,\theta )-\theta \},$$
(3)

where ζ is the zero-point shift, δ s the specimen-displacement parameter, τ s 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:

$${\rm{\Delta }}2{\theta }_{{\rm{R}}}={\rm{\Delta }}2{\theta }_{\exp }+{\rm{\Delta }}2{\theta }_{{\rm{ana}}},$$
(4)

where Δ2θ exp is the experimental peak-shift by the geometry (includes design-geometry of instrument as well as specimen-geometry) and Δ2θ ana is the analytical peak-shift caused by the mismatch of the lattice parameters. Notably, Δ2θ ana exists in the calculation only when A ≠ 1. Considering Eqs (3) and/or (4), one cannot obtain the true peak-shift when Δ2θ ana ≠ 0 (A ≠ 1). Secondly, Δ2θ ana can be fitted very well by Eq. (1) in the analysis 2θ-range (Fig. 1c and d). The finite value of Δ2θ ana, therefore, induces a false peak-shift with irrelevant lower-R wp (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, Δ2θ ana = 0 should be imposed. Finally, we have proposed an additional criterion, Σall|Δ2θ R|, which measures the fitting accuracy along the horizontal-axis of the diffraction data and is capable of preventing Δ2θ ana from enhancing. By combining our criterion with R wp, 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. 2θ max = 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|Δ2θ R|, 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, R wp. The refined lattice parameter at the minimum R wp-value is different from the correct one. The peak-shift includes the analytical Δ2θ ana as well as the experimental Δ2θ exp in the calculation. Δ2θ ana 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 R wp-value when Δ2θ ana ≠ 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.

Methods

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 Kα 1 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 a SRM = 4.1569162(97) Å ≃ 4.15692(1) Å at 22.5 °C has been clarified by NIST11.

Data analysis approach

Rietveld refinements

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 Kα 1 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 Kα 1 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 Z true, gives R wp larger than that for the true value R wp true. 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.

Peak-shift estimation

The peak-shift Δ2θ m ≡ 2θ SRM − 2θ obs was calculated by using the raw data. The list of ideal Bragg-peak angle, 2θ SRM, was provided in the certificate of SRM 660a11. The observed diffraction angle 2θ obs for each reflection was chosen at the strongest intensity in the diffraction data near 2θ SRM. The measurement error of 2θ obs was assumed to be the same as the step of 0.008° in the data.

Data Availability

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.