Abstract
To obtain the lattice parameters accurately by the Rietveld method, the relationship between the lattice parameters and the peakshift, 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 peakshift. This study unveils that the peakshift consists of the experimental and the analytical ones. The analytical peakshift erroneously lowers a reliability factor R _{wp}, which has, so far, been the conventional criterion of fit. The conventional Rietveld method obtains a unitcell which is a homothetic (proportional) unitcell of the true one. We propose an additional criterion based on the peakshift to obtain the true lattice parameters accurately. Our criterion can achieve reproducibility reasonably well for the experimental peakshift, leading to highly improved accuracy of the lattice parameters.
Introduction
Structural study for powder materials relies on the Rietveld method, which is capable of refining the structural and magnetic parameters from diffraction data^{1,2,3}. However, it is fundamentally difficult to determine accurately the refinement parameters^{4,5,6,7}. 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 leastsquares method. The calculated intensity includes the peakshift that is absolutely inevitable in the experiment. To evaluate quantitatively the best fit of the data, several reliabilityfactors such as R _{wp}, R _{p}, R _{e}, R _{ F }, S and χ ^{2} are proposed^{3,4}. The most accepted factor is the weightedprofile R, termed as R _{wp}, where the numerator includes S _{r} that is minimized during the refinements. The goodnessoffit, S or χ ^{2} ≡ S ^{2}, is used as another useful numerical criterion^{4}. The Svalue 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 Svalue 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 coworkers have suggested that these values to be given in publication^{8}.
Strictly speaking, there is no general agreement on these criteria in the Rietveld method. Other studies have concluded that viewing the profileplots is more effective than Rvalues to determine the quality of a refinement^{5,6,7}. 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 Crystallography^{7}. Several specialists analysed the standard PbSO_{4} 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 materials^{9}. Furthermore, the weighted mean parameters for a, b and caxes are 8.4804(4) Å, 5.3989(3) Å and 6.9605(2) Å, respectively. They are in good agreement with those determined from singlecrystal Xray diffraction data^{7,10} which is generally accepted to be highaccuracy. 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 peakshift that tends to be overlooked. This paper proposes an additional criterion, focusing on a fitting accuracy along the horizontalaxis of powder diffraction data, to determine the lattice parameters accurately. In the following, we demonstrate that our criterion enables the well reproducibility of the peakshift, 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 Xray diffraction pattern of standard reference material (SRM) 660a (lanthanum hexaboride)^{11} from the National Institute of Standards and Technology (NIST) collected with Cu Kα_{1} 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 2θrange from 18° to 2θ _{max}, where 2θ _{max} was in between 52° and 152°. There were five Braggpeaks for 2θ _{max} = 52° and twentyfour Braggpeaks 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, respectively^{11}. 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 2θdependence of the peakshift Δ2θ _{R} computed with the following equation^{12}:
where Z is the zeropoint shift (also known as the zero error), D _{s} the specimendisplacement parameter and T _{s} the specimentransparency parameter. Manually estimated peakshift, Δ2θ _{m}, is also plotted. Note that the 2θregions 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 2θregion (white area). In contrast, Δ2θ _{R} ^{cnv} differs from Δ2θ _{R} ^{fix} and Δ2θ _{m} especially in the large 2θregion.
Figure 1c and d show the 2θdependence 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 Δ2θ_{R} ^{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 2θrange, the 2θdependence of Δ2θ _{dif} corresponds well with that of Δ2θ _{ana}, 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:
$$\begin{array}{c}2d\,\sin \,\frac{2\theta }{2}=\lambda ,\\ 2(Ad)\,\sin \,\frac{2\theta +{\rm{\Delta }}2{\theta }_{{\rm{ana}}}}{2}=\lambda .\end{array}$$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 2θrange 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 peakshift affects the result, we have conducted several Rietveld refinements with a fixed value of Z. Figure 2 shows the Z and adependences in the conventional criterion as well as by the criteria set in this study. The sums are carried out over all the Braggpeaks in the analysis 2θrange for ΣΔ2θ _{R} and the whole 2θrange for Σ^{all}Δ2θ _{R}. Note that ΣΔ2θ _{R} and Σ^{all}Δ2θ _{R} are calculated from the result after the refinement. The convergence in the refinement is judged by using R _{wp}. For ΣΔ2θ _{R}, the number of Braggpeaks 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 Braggpeaks is always 25 for Σ^{all}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.
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 ΣΔ2θ _{R} shows a Vshaped 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 peakshift (Fig. 2c and d). The lattice parameter a ^{sum} is closer to a _{SRM} compared with a ^{cnv}. The magnitude of ΣΔ2θ _{R} for 2θ _{max} = 92° is smaller than that for 2θ _{max} = 152°, which is reasonable considering the number of Braggpeaks in the sum. Our proposed criterion Σ^{all}Δ2θ _{R} shows a sharper Vshaped curve than ΣΔ2θ _{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 Σ^{all}Δ2θ _{R} increases and the Vshape 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 report^{7}. Next, a ^{sum}, which is determined with the minimum of ΣΔ2θ _{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 Σ^{all}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 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 peakshifts determined with the minima of R _{wp}, ΣΔ2θ _{R} and Σ^{all}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 peakshift.
Discussion
The present study reveals several critical findings. Firstly, the 2θdependence of peakshift does not obey Eq. (1) in the calculation; instead follows the equation:
where ζ is the zeropoint shift, δ _{s} the specimendisplacement parameter, τ _{s} the specimentransparency 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 Δ2θ _{exp} is the experimental peakshift by the geometry (includes designgeometry of instrument as well as specimengeometry) and Δ2θ _{ana} is the analytical peakshift 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 peakshift when Δ2θ _{ana} ≠ 0 (A ≠ 1). Secondly, Δ2θ _{ana} can be fitted very well by Eq. (1) in the analysis 2θrange (Fig. 1c and d). The finite value of Δ2θ _{ana}, therefore, induces a false peakshift with irrelevant lowerR _{wp} (Fig. 2a and b). As a result, a homothetic unitcell, which is proportional to the true one, is obtained in the conventional Rietveld method. To obtain the correct unitcell, Δ2θ _{ana} = 0 should be imposed. Finally, we have proposed an additional criterion, Σ^{all}Δ2θ _{R}, which measures the fitting accuracy along the horizontalaxis 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 peakshift (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 peakshift in the early stage of developing the method because the angledispersive neutron data was used^{3,13}. Neutron has high transparency against the materials. Enough highangle data, e.g. 2θ _{max} = 144° (ref.^{3}), with quite broad Braggpeaks were generally obtained using an oldfashioned reactor source. As a result, the 2θdependence of peakshift was approximately constant and could easily be reproduced. In fact, Rietveld applied a zeroshift parameter as the peakshift function which is independent on 2θ (ref.^{3}).
Our present findings possibly accelerate designing novel materials since a comparative study between the experiment and theory^{14} may be achieved with highaccuracy. The criterion we set in this report would be applicable for structure determination from powder diffraction^{15,16,17} including indexing the diffraction peaks^{18,19,20} and the profile decomposition^{21,22,23} as well as a quality management of mass production of materials in industry. Further study related to the structural parameters in the unitcell is desirable.
In summary, an additional criterion, Σ^{all}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 fixedvalues of peakshift 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 peakshift 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 unitcell 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 peakshift 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 website^{24}. The data file with a name of “660a2.dat” in a compressed file, xcelerator.zip, was used. The Xray diffraction pattern for SRM 660a^{11} 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. Twentyfour 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 signaltobackground ratio. The diffraction peaks were very sharp as a full width at halfmaximum (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 inhouse data. The lattice parameter a _{SRM} = 4.1569162(97) Å ≃ 4.15692(1) Å at 22.5 °C has been clarified by NIST^{11}.
Data analysis approach
Rietveld refinements
The Rietveld program RIETANFP^{25} 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 RIETANFP is the same as that used in the computation for SRM 660a^{11} 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 GSAS^{27}, GSASII^{28}, FullProf^{29}, ZRietveld^{30} and TOPAS^{31}, 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 ThompsonCoxHastings pseudoVoigt function^{32} was used. Howard’s method^{33}, which is based on the multiterm Simpson’s rule integration, was employed for the profile asymmetry. Profile cutoff 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 fixedvalues for the first term Z of the peakshift, 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 calculationsteps 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 Zvalues for Rietveld refinement.
Peakshift estimation
The peakshift Δ2θ _{m} ≡ 2θ _{SRM} − 2θ _{obs} was calculated by using the raw data. The list of ideal Braggpeak angle, 2θ _{SRM}, was provided in the certificate of SRM 660a^{11}. 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.
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Acknowledgements
The authors thank Armel Le Bail for kindly allowing us to use his highquality data. We are grateful to Biswajit Paik for critically reading an early version of the manuscript.
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Affiliations
Physonit Inc., 610 Minamihorikawa, Kaita, Aki, Hiroshima, 7360044, Japan
 Masami Tsubota
Department of Electrical Engineering, Faculty of Engineering, Fukuoka Institute of Technology, 3301 Wajirohigashi, Higashiku, Fukuoka, 8110295, Japan
 Jiro Kitagawa
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Contributions
M.T. planned and overseen this research. Data analyses were performed by M.T. and crosschecked the results by J.K. All authors contributed to the manuscript preparation.
Competing Interests
M.T.’s institution is currently applying for patents in Japan (JP appl. No. 2017110500 filed on 2 June 2017) related to this work. J.K. declares no competing financial interests.
Corresponding author
Correspondence to Masami Tsubota.
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