research papers
Synchrotron totalscattering data applicable to dualspace structural analysis
^{a}Department of Chemistry, Aarhus University, Langelandsgade 140, Aarhus C, 8000, Denmark, ^{b}RIKEN SPring8 Center, 111 Kouto, Sayocho, Sayogun, Hyogo, 6795148, Japan, and ^{c}JST, PRESTO, 418 Honcho, Kawaguchi, Saitama 3320012, Japan
^{*}Correspondence email: katok@spring8.or.jp, bo@chem.au.dk
Synchrotron powder Xray diffraction (PXRD) is a well established technique for investigating the atomic arrangement of crystalline materials. At modern beamlines, Xray scattering data can be collected in a totalscattering setting, which additionally opens up the opportunity for directspace structural analysis through the atomic pair distribution function (PDF). Modelling of PXRD and PDF data is typically carried out separately, but employing a concurrent structural model to both direct and reciprocalspace data has the possibility to enhance totalscattering data analysis. However, totalscattering measurements applicable to such dualspace analyses are technically demanding. Recently, the technical demands have been fulfilled by a MYTHEN microstrip detector system (OHGI), which meets the stringent requirements for both techniques with respect to Q range, Q resolution and In the present study, we evaluate the quality of totalscattering data obtained with OHGI by separate direct and reciprocalspace analysis of Si. Excellent agreement between structural parameters in both spaces is found, demonstrating that the totalscattering data from OHGI can be utilized in dualspace structural analysis e.g. for in situ and operando measurements.
Keywords: dualspace structural analysis; totalscattering data; synchrotrons; pair distribution functions; powder Xray diffraction.
1. Introduction
The study of the solid phase of matter has improved significantly in recent decades causing immense progress in fields such as life science and materials science. The progress can be mainly attributed to two aspects. One is the increasing number of large science facilities such as spallation neutron sources and synchrotrons, which offer users access to increasingly bright and brilliant neutrons and Xrays. The other is a steady improvement in detector technology leading to short acquisition times and high data quality. The enhanced quality of data is a call for increasingly sophisticated analysis techniques, which can give detailed structural descriptions of the solid phase in terms of chemical bonding, microstructure and lattice defects.
Powder Xray diffraction (PXRD) is a well established technique in the materials science community for determining phase purity, composition, microstructural features and the crystallographic structure of a powdered sample. Previous studies have shown that the accuracy of PXRD data collected is comparable with that of singlecrystal Xray diffraction (SCXRD) data in the case of crystalline materials with high symmetry (Tolborg et al., 2017; Svane et al., 2019).
For a perfect crystal, only the Bragg scattering from PXRD or SCXRD is needed to describe the crystalline structure. However, almost all crystalline materials exhibit disorder to some extent such as vacancies, dislocations, stacking faults or more complicated deviations from the average structure, such as incommensurable substructures or correlated thermal motion. Disorder and deviations from the average structure lead to diffuse scattering.
In a totalscattering experiment, Bragg and diffuse scattering are measured impartially. Analysis of totalscattering data can be carried out in e.g. has been used to model planar defects, morphology and correlated thermal motion of nanocrystals with poorly defined Bragg scattering (Scardi & Gelisio, 2016; Moscheni et al., 2018; Bertolotti et al., 2020). Totalscattering data can also be treated in via a Fourier transformation to obtain the pair distribution function (PDF) (Egami & Billinge, 2012). For crystalline structures with well defined Bragg scattering and minor degrees of diffuse scattering from shortrange disorder, modelling of the PDF is the method of choice (Scardi & Gelisio, 2016). Both techniques have also been successful for modelling of liquid and amorphous phases.
using the Debye scattering equation. This techniqueConsidering the obvious scientific motivation, only a few totalscattering beamlines have been developed at synchrotrons for simultaneous, i.e. single shot, measurement of highquality Bragg and diffuse scattering, referred to herein as dualspace quality. The primary reason is the inherent tradeoff between range and resolution of the scattering vector, Q. Typical PXRDdedicated beamlines have focused on high angularresolution measurement of Bragg scattering with point detectors and relatively low energy Xrays (Fitch, 2004), while the focus of PDF dedicated beamlines has been on wide Qrange measurements with large area detectors and highenergy Xrays (Chupas et al., 2007). Consequently, the average structure analysis, e.g. using the (Rietveld, 1969; Young, 2002), has been performed separately from local structural analysis using PDF methods. This situation makes it complicated to directly compare the local and average structure in crystalline solids with disorder using Xrays.
Accordingly, there is an increasing need for dualspacequality total Xray scattering data that permit average and localstructure analysis on an equal basis, similar to the progress achieved for neutron timeofflight diffractometers (Bowron et al., 2010; Neuefeind et al., 2012; Smith et al., 2019). Dualspacequality totalscattering data have been utilized to study e.g. oxygen disorder in δBi_{2}O_{3} (Hull et al., 2009), where `big box' modelling was carried out with the RMCProfile software (Tucker et al., 2007).
Dualspace analysis requires the measurement of totalscattering data with a high Q range and Q resolution need to be high in order to obtain well resolved diffraction peaks to a high order. The microstrip detector module MYTHEN (DECTRIS) (Schmitt et al., 2003) has the potential for dualspacequality totalscattering measurements because of a flexible arrangement to cover a wide Q range, high spatial resolution given by a sharp linespread function and a photoncounting architecture with high signaltonoise ratio. Even so, data obtained through MYTHEN modules have not been successfully applied to dualspace analysis. The unsuccessful attempts can be ascribed to the difference in Xray response between microstrip channels, which is referred to as Xray response nonuniformity (XRNU) (Kato et al., 2019). XRNU is a major contributing factor in the of a detector system. Although MYTHEN has a counter of 24 bits, which is equivalent to a of 10^{7}, a noise level of 1% caused by XRNU reduces the effective to 10^{4}. All types of Xray detectors have been reported to suffer from XRNU (Amemiya, 1995; Williams & Shaddix, 2007; Bergamaschi et al., 2010; Skinner et al., 2012; Wernecke et al., 2014). The conventional approach to the problem, the socalled flatfield calibration (Hammersley et al., 1995; Moy et al., 1996), succeeded in reducing the XRNU noise level from several percent down to 1%. However, the flatfield calibration, which needs a uniform reference intensity, has failed to reduce the level further because it is impossible to produce a completely uniform intensity. Recently, Kato et al. (Kato et al., 2019; Kato & Shigeta, 2020) have developed a datadriven approach to the problem without using a uniform intensity, which is referred to as ReLiEf (responsetolight effector). The ReLiEf approach has succeeded in reducing the XRNU noise level in MYTHEN modules down to 0.1%, which is equivalent to a of 10^{6}. The ReLiEf algorithm has been integrated into a totalscattering measurement system called OHGI (overlapped highgrade intelligencer) (Kato et al., 2019), which consists of fifteen overlapping MYTHEN modules installed at the RIKEN Materials Science beamline BL44B2 (Kato et al., 2010; Kato & Tanaka, 2016) at SPring8. A unique combination of OHGI, ReLiEf and intermediate energy (30 keV) Xrays facilitate the collection of singleshot dualspacequality total Xray scattering data with a wide Q range (Q_{max} > 25 Å^{−1}), high Q resolution (Q step < 10^{−3} Å^{−1}) and high signaltonoise ratio (dynamic range > 10^{5}). The precision in the intensity of the instrument has been shown to closely follow the even for weak signals, and the pristine data quality has been demonstrated for amorphous SiO_{2} and TiO_{2} nanoparticles (Kato et al., 2019).
and high signaltonoise ratio in order to be sensitive to Bragg and diffuse scattering. Furthermore, both theIn the present study, we evaluate totalscattering data obtained through OHGI by analysis of a NIST standard reference material Si powder. Si is a highly crystalline material with minimal disorder and therefore serves as an appropriate reference material for a benchmark test. The structural features of Si have been fully understood from the viewpoint of both theory and experiment. One feature of the OHGI totalscattering data is outstanding reciprocalspace range and resolution, which allows computation of longrange PDFs (r > 500 Å) from a singleshot measurement. The primary purpose is to confirm consistency between structural parameters, such as lattice parameters and atomic displacement parameters (ADPs) obtained separately by Rietveld and PDF analysis. Moreover, the instrumental effects on the extremely well resolved PDFs are assessed by a boxcarrefinement scheme.
2. Methods
2.1. Totalscattering measurements
Totalscattering data of an Si powder (NIST SRM640d) were collected at 100 and 300 K using OHGI (Kato et al., 2019) at the RIKEN Materials Science beamline BL44B2 (Kato et al., 2010; Kato & Tanaka, 2016) at SPring8. The incident Xrays had an energy of 27.546 (6) keV (λ = 0.4501 (1) Å), which was calibrated through Le Bail (Le Bail, 2005) of LaB_{6} powder (NIST SRM660b) data. This corresponds to a Q_{max} of ∼27 Å^{−1}. The detectable of OHGI was set to 13.8 keV, which is equivalent to half of the incident energy, to minimize the effect of double A single data set from OHGI has a 2θ step of 0.01°. In the present experiments, two data sets were collected by shifting OHGI 0.005° along 2θ. These were then integrated into a single data set with an effective resolution of 0.005°. The Si sample was packed into a glass capillary with an inner diameter of 0.3 mm. In addition, data were measured on an empty capillary at 100 and 300 K to subtract the background and instrument scattering before data normalization. By using an incident energy of 27.546 (6) keV and a capillary diameter of 0.3 mm, the absorption effect on the scattering intensity was negligible. Data processing was based on the assumption that the incident beam from the bending magnet Xray source was completely polarized in the horizontal plane. The dimensions of the incident beam were fixed by a collimator 3 mm in the horizontal direction and 0.5 mm in the vertical direction. The total datacollection time at each temperature was ∼1 h.
To compare instrumental profile resolutions with beamlines that have in situ or operando capabilities, LaB_{6} powder data collected at two other synchrotron beamlines, P02.1 (Dippel et al., 2015) and P21.1, at PETRA III (DESY in Hamburg, Germany) were analyzed. LaB_{6} powder data collected using an imaging plate (IP) detector at BL44B2 at SPring8 have also been used to compare the instrumental profile resolutions.
2.2. Reciprocalspace refinements
Rietveld and Le Bail refinements of the totalscattering data were carried out using the TOPASAcademic Version 6 software (Coelho, 2018). An angular range from 6 to 110° 2θ was selected for data analysis. The background scattering was fitted using a sevendegree Chebyshev polynomial. Bragg peak profiles were modelled using the Thompson–Cox–Hastings (TCH) pseudoVoigt peakprofile function (Thompson et al., 1987). The number of peakprofile parameters was minimized by iteratively assessing the correlation matrix and R factors. Peak shifts and peak asymmetry, caused by misaligned sample capillaries and axial divergence, respectively, were insignificant. Two R factors, R_{wp} and R_{Bragg}, were used to assess the reliability of fit; the former for evaluating the statistical significance of each data point, the latter for evaluating the difference between model and data at the calculated peak positions.
For the Le Bail refinements of LaB_{6}, the refined parameters were a scale factor, background parameters, incidentbeam wavelength and peakprofile parameters. The lattice parameter was fixed to the certified value of SRM660b (Black et al., 2011). The range was limited to 80° in 2θ owing to restrictions on the number of parameters in TOPAS. For the Rietveld refinements of Si (space group #227 , origin choice 2), the following parameters were employed: a scale factor, background parameters, the lattice parameter, TCH peakprofile parameters (U, W and Y) and the isotropic ADP. To avoid local minima, the ADPs were optimized by refining the model with 10 000 iterations, where a random number between −50 and 50% of the ADP after a convergent iteration was added to the value for the next iteration. The ADP with the lowest R_{wp} among all convergent iterations was selected as the final value.
2.3. Directspace refinements
PDFs of Si were computed using the PDFgetX3 algorithm (Juhás et al., 2013). The PDFgetX3 data normalization is semiquantative and not strictly correct for an arbitrary system. However, in the case of a homoatomic sample, such as Si, the normalization will be correct. The Q range was 1.0–27.0 Å^{−1} and the ad hoc correction parameter r_{poly} was set to 1.05. Leastsquares refinements of the PDF model were also carried out using TOPASAcademic Version 6, which allowed for refinements of longrange PDFs within reasonable time frames (Coelho et al., 2015; Coelho, 2018). Refinements were carried out in a range of 1.0–500.0 Å with a step of 0.01 Å, with all points included. A convoluted sinc function was implemented to account for Fourier ripples (Chung & Thorpe, 1997). parameters for Si were as follows: a scale factor, the lattice parameter, the isotropic ADP, and instrumental parameters Q_{damp} and Q_{broad}. The R_{wp} value was used to assess the fit. To avoid local minima, the ADPs were optimized by refining the model with 1000 iterations, where a random number between −25 and 25% of the ADP after a convergent iteration was added to the value for the next iteration. The ADP with the lowest R_{wp} among all convergent iterations was selected as the final value.
2.4. Boxcar refinements in direct space
To examine the behaviour of the PDF as a function of correlation length r, a boxcarrefinement scheme was employed (Proffen & Kim, 2009; Usher et al., 2016). A narrow section of the directspace range, i.e. a box, was defined and subsequently moved in fixed steps through the entire range. The box width was set to 10 Å and the centroid was moved in 10 Å steps in the range 1–500 Å. Note that the first box had a width of 9 Å. At first, three parameters were included in the model; scale factor, the isotropic ADP and the lattice parameter. The lattice parameter was found to be consistent between all ranges and was therefore fixed at the value found from Rietveld refinements. A convoluted sinc function was included to account for the Fourier ripples. The final parameters were a scale factor and the isotropic ADP. Parameters in each box were refined with 100 iterations. Similar to the description in Section 2.3, the final ADP was optimized by using a random number between −50 and 50% after each convergent iteration.
3. Quality of totalscattering data
Fig. 1 shows the results of Rietveld analysis of the totalscattering data of Si collected at 100 K in the Q range from 1.9 to 23 Å^{−1}. The R_{wp} and R_{Bragg} values are 6.05% and 2.10%, respectively. The low R factors demonstrate that the simple Rietveld model employed sufficiently describes the data, although further improvement possibly could be achieved by accounting for chemical bonding effects via multipole modelling (Svane et al., 2021).
To evaluate the instrumental resolution of OHGI, the LaB_{6} data were compared with other synchrotron beamlines with different detectors dedicated to in situ or operando measurements. The resolution has also been compared with the IP detector at beamline BL44B2 at SPring8 and the Aarhus IP detector (AVID) (Wahlberg et al., 2016; Tolborg et al., 2017). Fig. 2(a) shows the peak shapes of the most intense reflection and Fig. 2(b) shows the square root of the full widths at half maxima (FWHM) as a function of Q, which was calculated from the TCH parameters refined by the Le Bail method. For comparison, the FWHM that were originally computed in 2θ were transformed into the corresponding values in Q by using the approximation for sufficiently small values given by ΔQ = 4π cos θ/λΔθ. The vertical and horizontal grey lines on Fig. 2(b) show levels of Q = 25 Å^{−1} and (FWHM)^{1/2} = 0.27 Å^{−1/2}, respectively. Given a constant Gaussian peakprofile function, this value of (FWHM)^{1/2} is where the corresponding PDF is damped down to one percent at r = 100 Å. It is thus necessary to collect data with an instrumental resolution below the horizontal line to produce longrange PDFs. The vertical line denotes the minimum Q range for producing PDFs with a directspace resolution of ∼0.125 Å (calculated from Δr ≃ π/Q_{max}). This resolution is still too low for peak separation in some structures (Qiu et al., 2004) but serves as a minimum requirement for highquality PDFs.
The results in Fig. 2(b) clearly indicate that OHGI satisfies the two criteria for longrange highresolution totalscattering measurements, enabling dualspace analysis.. The Perkin–Elmer (PE) area detectors employed at P02.1 or P21.1 with different sampletodetector distances do not satisfy the criteria owing to the tradeoff relationship between Q range and Q resolution. Both AVID and the IP detector at BL44B2 have comparable resolutions with OHGI but their Q ranges do not satisfy the criteria.
The instrumental Q resolution of totalscattering data depends on the divergence and energy resolution of the primary beam, as well as the point (or line) spread function of the detector. The instrumental Q range is determined by the energy of the primary beam and the architecture of the detector system. As shown in Fig. 2(b), both have been optimized for OHGI to yield dualspacequality totalscattering data. In Fig. 3, the experimental FWHM from singlepeak fitting and FWHM as calculated from the refined TCH parameters from the Rietveld analysis of Si at 100 K are shown. A peakprofile function of has been fitted to the TCH FWHM, where Γ_{Q0} is a constant contribution to the peak width and κ is a linear broadening coefficient. This peakprofile function follows refined TCH FWHM very closely but a deviation from the experimental FWHM is seen at high Q. The discrepancy in FWHM is also noticeable in the difference curve at high Q in Fig. 1, where every peak has a higher maximum intensity than the model owing to overestimation of the FWHM. Inspection of the individual fits (see Fig. S1 in the Supporting information) shows that the shape of the generic pseudoVoigt function used for singlepeak fitting gives an adequate description, which means that the discrepancy is solely in the width of the peaks. Many combinations of TCH parameters were tested to improve the Rietveld model at high Q but those reported in Fig. 3 gave the lowest agreement factors owing to the high intensity, and consequently high weight, of the low Q diffraction peaks. The refined TCH shape is primarily Gaussian with a Lorentzian mixing parameter between 18.4% and 7.86% for peaks in the lowest and highest reciprocalspace regions, respectively.
Considering the measured intensity of OHGI, the precision and accuracy are significantly influenced by XRNU, as described in Section 1. To correct the data for XRNU, correction factors were obtained with ReLiEf (Kato et al., 2019; Kato & Shigeta, 2020) at the incident wavelength and that were identical to those used for sample measurement. To investigate for any systematic intensity errors, totalscattering data obtained using the appropriate XRNU correction factors have been examined in terms of ADPs of Si at 100 and 300 K. The ADPs of Si at these temperatures are established from theory and previously reported experiments.
Table 1 shows the extracted ADPs of Si with reference values (Wahlberg et al., 2016; Tolborg et al., 2017; Flensburg & Stewart, 1999; Sang et al., 2010). The ADP values at both temperatures from OHGI agree well with those from other experiments even though they are somewhat smaller in all cases, except for AVID #1. This inconsistency can be explained by the coexistence of Bragg scattering and thermal diffuse scattering (TDS). Since the integrated intensity of each Bragg peak at high 2θ angles tends to be overestimated owing to TDS (Willis & Pryor, 1975), the refined ADP values become an underestimation of the true values if TDS is not accounted for. The fact that this effect is noticeable at 300 K is a testimony to the high precision of weak scattering on the OHGI instrument. The precision is on par with AVID, which has previously served as a benchmark for stateoftheart PXRD data quality.

4. Pair distribution functions
Fig. 4(a) shows the PDF of Si at 300 K obtained from the totalscattering data collected with OHGI. The longrange PDF clearly demonstrates that interatomic correlations can be observed at least up to r = 500 Å thanks to the high Q resolution. To examine the characteristics of the longrange PDF, boxcar refinements were carried out. The results are shown in Figs. 4(b)–4(d). When the box was shifted to higher correlations lengths, two conspicuous effects were confirmed. One, shown in Fig. 4(b), is a decrease in PDF peak intensity as a function of r, and the other, shown in Fig. 4(c), is a gradual increase in PDF peak width with increasing r. In PDF refinements with PDFgui (Farrow et al., 2009), these two effects, i.e. the rdependent damping and peak broadening, can be described by the correction parameters Q_{damp} and Q_{broad}, respectively.
The Q_{damp} parameter models the width of a Gaussian envelope function. This description is formulated by assuming constant Gaussian peaks for the reciprocalspace peakprofile function. According to the Fourier convolution theorem, the PDF should consequently be multiplied by the Fourier transformation of a constant Gaussian peak profile, which is also a Gaussian. For convenience, the envelope function is typically expressed such that Q_{damp} = Γ_{Q0}, where Γ_{Q0} is the FWHM of the peak profiles. As seen in Fig. 4(b), the refined scale factors were successfully fitted by a Gaussian envelope with the exception of those at low r. The misfit may be attributed to the Lorentzian component of the peak profiles, which is not taken into account in the Q_{damp} description.
The Q_{broad} parameter models the PDF peak broadening caused by the broadening of the reciprocal peak profiles (Thorpe et al., 2002). This parameter becomes especially significant for refinements with a wide range in r. In the derivation of Q_{broad}, the Qdependent broadening is assumed to be in accordance with the form , which reproduced the FHWM function in the on the OHGI data (see Fig. 3). Consequently, the PDF peak broadening can be expressed by the following function, as implemented in PDFgui (Farrow et al., 2009),
Here, Γ_{r} is the total PDF peak width and Γ_{r0} is the constant contribution. The two terms δ_{1} and δ_{2} are parameters for correlated atomic motion at higher and lower temperatures than the Debye temperature, respectively (Jeong et al., 2003). Although the Debye temperature of Si is much higher than room temperature, δ_{1} rather than δ_{2} was employed as a parameter for robustness. In Fig. 4(c), it is shown that this description gives an adequate fit with the refined FWHM from the boxcar Fig. 5 shows the wholerange of the 100 K Si PDF up to r = 500 Å. The refined model yielded a low R_{wp} factor, especially considering the high number of data points. The refined ADP value is also reasonable and close to that found in the PXRD analysis, see Table 2. In addition, the Q_{damp} parameter was one order of magnitude smaller than that at typical PDF beamlines and was comparable with that at highresolution powderdiffraction beamlines (Saleta et al., 2017). Once again, these results clearly demonstrate that OHGI can yield highly reliable and well resolved totalscattering data.

Table 2 lists the structural parameters and R_{wp} factors obtained from reciprocal and directspace refinements of the Si data at 100 and 300 K. The lattice parameters at both temperatures from agreed with those from on a scale of 10^{−4} Å. The ADP at 100 K from was consistent with that from within the estimated standard deviation. In contrast, the ADP at 300 K from was significantly smaller than that from As discussed in Section 3, correlated atomic motion results in TDS in and around the Bragg peaks, which causes an artificial decrease in the ADP in when not accounted for. In the addition of the δ parameters to the model makes it possible to separate the effects of TDS from the ADP.
The overall agreement between the structural parameters for Si in reciprocal and ), (iii) how are various structural effects causing diffuse scattering (such as TDS) handled in both spaces simultaneously and (iv) how do the pseudoVoigt peak profiles with nonnegligible Lorentzian components affect the PDF.
demonstrates that OHGI can provide a measurement basis for singleshot dualspace structural analysis. In the present study, dualspace analysis using a single data set was performed separately. A concurrent dualspace analysis would need to overcome the following challenges for treatment of reciprocal and directspace data on an equal basis: (i) how is the agreement of the model in individual spaces weighted, (ii) how is the structural PDF model of polyatomic specimens calculated (Neder & Proffen, 20205. Conclusions
In conclusion, we found that synchotron totalscattering data obtained through OHGI at BL44B2 at SPring8 were of unprecedented quality for both accurate PXRD and PDF analysis. Both the lattice parameters and ADPs of Si at 100 and 300 K in Q_{damp} and Q_{broad} were found to adequately describe the effects of the primarily Gaussian reciprocalspace peak profiles on the longrange PDFs (r = 500 Å). These results clearly demonstrate that the data quality of singleshot measurements from OHGI is applicable to dualspace analysis and can bridge the gap between the analysis of the average and local structures of crystalline materials.
were found to be consistent with those in The correction parametersSupporting information
Supporting information. DOI: https://doi.org//10.1107/S2052252521001664/ro5024sup1.pdf
Acknowledgements
Synchrotronradiation experiments were performed at the RIKEN Materials Science beamline BL44B2 at SPring8 with the approval of the RIKEN SPring8 Center (Proposal Nos. 20160037 and 20180024). The authors thank Mr. Kazuya Shigeta (Nippon Gijutsu Center Co. Ltd) for technical contribution to BL44B2. The authors also thank the staff at PETRA III (DESY) beamlines P02.1 (Powder Diffraction and TotalScattering Beamline) and P21.1 (Swedish Materials Science Beamline), specifically Dr AnnChristin Dippel for ongoing collaboration. Dr Philip Chater is thanked for insightful correspondence and the implementation of several functions in the TOPASAcademic Version 6 software. Nikolaj Roth and Lasse R. Jørgensen are thanked for fruitful discussions.
Funding information
The following funding is acknowledged: Villum Foundation and the Danish Ministry for Higher Education and Science (SMART Lighthouse) and JST, PRESTO (grant No. JPMJPR1872 to Kenichi Kato).
References
Amemiya, Y. (1995). J. Synchrotron Rad. 2, 13–21. CrossRef CAS Web of Science IUCr Journals Google Scholar
Bergamaschi, A., Cervellino, A., Dinapoli, R., Gozzo, F., Henrich, B., Johnson, I., Kraft, P., Mozzanica, A., Schmitt, B. & Shi, X. (2010). J. Synchrotron Rad. 17, 653–668. Web of Science CrossRef CAS IUCr Journals Google Scholar
Bertolotti, F., Vivani, A., Moscheni, D., Ferri, F., Cervellino, A., Masciocchi, N. & Guagliardi, A. (2020). Nanomaterials, 10, 743. Web of Science CrossRef Google Scholar
Black, D. R., Windover, D., Henins, A., Filliben, J. & Cline, J. P. (2011). Powder Diffr. 26, 155–158. Web of Science CrossRef CAS Google Scholar
Bowron, D. T., Soper, A. K., Jones, K., Ansell, S., Birch, S., Norris, J., Perrott, L., Riedel, D., Rhodes, N. J., Wakefield, S. R., Botti, A., Ricci, M. A., Grazzi, F. & Zoppi, M. (2010). Rev. Sci. Instrum. 81, 033905. Web of Science CrossRef PubMed Google Scholar
Chung, J. S. & Thorpe, M. (1997). Phys. Rev. B, 55, 1545–1553. CrossRef CAS Web of Science Google Scholar
Chupas, P. J., Chapman, K. W. & Lee, P. L. (2007). J. Appl. Cryst. 40, 463–470. Web of Science CrossRef CAS IUCr Journals Google Scholar
Coelho, A. A. (2018). J. Appl. Cryst. 51, 210–218. Web of Science CrossRef CAS IUCr Journals Google Scholar
Coelho, A. A., Chater, P. A. & Kern, A. (2015). J. Appl. Cryst. 48, 869–875. Web of Science CrossRef CAS IUCr Journals Google Scholar
Dippel, A.C., Liermann, H.P., Delitz, J. T., Walter, P., SchulteSchrepping, H., Seeck, O. H. & Franz, H. (2015). J. Synchrotron Rad. 22, 675–687. Web of Science CrossRef CAS IUCr Journals Google Scholar
Egami, T. & Billinge, S. J. (2012). Underneath the Bragg Peaks. Amsterdam: Elsevier. Google Scholar
Farrow, C. L., Juhas, P., Liu, J. W., Bryndin, D., Bozin, E. S., Bloch, J., Proffen, T. & Billinge, S. J. L. (2009). PDFgui user guide. Google Scholar
Fitch, A. N. (2004). J. Res. Natl Inst. Stand. Technol. 109, 133–142. Web of Science CrossRef CAS PubMed Google Scholar
Flensburg, C. & Stewart, R. F. (1999). Phys. Rev. B, 60, 284–291. Web of Science CrossRef CAS Google Scholar
Hammersley, A. P., Svensson, S. O., Thompson, A., Graafsma, H., Kvick, Å. & Moy, J. P. (1995). Rev. Sci. Instrum. 66, 2729–2733. CrossRef CAS Web of Science Google Scholar
Hull, S., Norberg, S. T., Tucker, M. G., Eriksson, S. G., Mohn, C. E. & Stølen, S. (2009). Dalton Trans. pp. 8737. Google Scholar
Jeong, I. K., Heffner, R. H., Graf, M. J. & Billinge, S. J. L. (2003). Phys. Rev. B, 67, 9. Web of Science CrossRef Google Scholar
Juhás, P., Davis, T., Farrow, C. L. & Billinge, S. J. L. (2013). J. Appl. Cryst. 46, 560–566. Web of Science CrossRef IUCr Journals Google Scholar
Kato, K., Hirose, R., Takemoto, M., Ha, S., Kim, J., Higuchi, M., Matsuda, R., Kitagawa, S. & Takata, M. (2010). AIP Conf. Proc. 1234, 875–878. CrossRef CAS Google Scholar
Kato, K. & Shigeta, K. (2020). J. Synchrotron Rad. 27, 1172–1179. Web of Science CrossRef IUCr Journals Google Scholar
Kato, K. & Tanaka, H. (2016). Adv. Phys. X, 1, 55–80. Google Scholar
Kato, K., Tanaka, Y., Yamauchi, M., Ohara, K. & Hatsui, T. (2019). J. Synchrotron Rad. 26, 762–773. Web of Science CrossRef CAS IUCr Journals Google Scholar
Le Bail, A. (2005). Powder Diffr. 20, 316–326. Web of Science CrossRef CAS Google Scholar
Moscheni, D., Bertolotti, F., Piveteau, L., Protesescu, L., Dirin, D. N., Kovalenko, M. V., Cervellino, A., Pedersen, J. S., Masciocchi, N. & Guagliardi, A. (2018). ACS Nano, 12, 12558–12570. Web of Science CrossRef CAS PubMed Google Scholar
Moy, J. P., Hammersley, A. P., Svensson, S. O., Thompson, A., Brown, K., Claustre, L., Gonzalez, A. & McSweeney, S. (1996). J. Synchrotron Rad. 3, 1–5. CrossRef CAS Web of Science IUCr Journals Google Scholar
Neder, R. B. & Proffen, Th. (2020). J. Appl. Cryst. 53, 710–721. Web of Science CrossRef CAS IUCr Journals Google Scholar
Neuefeind, J., Feygenson, M., Carruth, J., Hoffmann, R. & Chipley, K. K. (2012). Nucl. Instrum. Methods Phys. Res. B, 287, 68–75. Web of Science CrossRef CAS Google Scholar
Proffen, T. & Kim, H. (2009). J. Mater. Chem. 19, 5078–5088. Web of Science CrossRef CAS Google Scholar
Qiu, X., Božin, E. S., Juhas, P., Proffen, T. & Billinge, S. J. L. (2004). J. Appl. Cryst. 37, 110–116. Web of Science CrossRef CAS IUCr Journals Google Scholar
Rietveld, H. M. (1969). J. Appl. Cryst. 2, 65–71. CrossRef CAS IUCr Journals Web of Science Google Scholar
Saleta, M. E., Eleotério, M., Mesquita, A., Mastelaro, V. R. & Granado, E. (2017). J. Synchrotron Rad. 24, 1098–1104. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sang, X. H., Kulovits, A. & Wiezorek, J. M. K. (2010). Acta Cryst. A66, 685–693. Web of Science CrossRef IUCr Journals Google Scholar
Scardi, P. & Gelisio, L. (2016). Sci. Rep. 6, 1–7. Web of Science CrossRef PubMed Google Scholar
Schmitt, B., Brönnimann, C., Eikenberry, E. F., Gozzo, F., Hörmann, C., Horisberger, R. & Patterson, B. (2003). Nucl. Instrum. Methods Phys. Res. A, 501, 267–272. Web of Science CrossRef CAS Google Scholar
Skinner, L. B., Benmore, C. J. & Parise, J. B. (2012). Nucl. Instrum. Methods Phys. Res. A, 662, 61–70. Web of Science CrossRef CAS Google Scholar
Smith, R. I., Hull, S., Tucker, M. G., Playford, H. Y., McPhail, D. J., Waller, S. P. & Norberg, S. T. (2019). Rev. Sci. Instrum. 90, 115101. Web of Science CrossRef PubMed Google Scholar
Svane, B., Tolborg, K., Jørgensen, L. R., Roelsgaard, M., Jørgensen, M. R. V. & Brummerstedt Iversen, B. (2019). Acta Cryst. A75, 600–609. Web of Science CrossRef IUCr Journals Google Scholar
Svane, B., Tolborg, K., Kato, K. & Iversen, B. B. (2021). Acta Cryst. A77, https://doi.org/10.1107/S2053273320016605. Web of Science CrossRef IUCr Journals Google Scholar
Thompson, P., Cox, D. E. & Hastings, J. B. (1987). J. Appl. Cryst. 20, 79–83. CrossRef ICSD CAS Web of Science IUCr Journals Google Scholar
Thorpe, M. F., Levashov, V. A., Lei, M. & Billinge, S. J. L. (2002). In From Semiconductors to Proteins: Beyond the Average Structure, edited by S. J. L. Billinge & M. F. Thorpe. Boston, Massachusetts: Springer Google Scholar
Tolborg, K., Jørgensen, M. R. V., Christensen, S., Kasai, H., Becker, J., Walter, P., Dippel, A.C., AlsNielsen, J. & Iversen, B. B. (2017). Acta Cryst. B73, 521–530. Web of Science CrossRef IUCr Journals Google Scholar
Tucker, M. G., Keen, D. A., Dove, M. T., Goodwin, A. L. & Hui, Q. (2007). J. Phys. Condens. Matter, 19, 335218. Web of Science CrossRef PubMed Google Scholar
Usher, T. M., Iamsasri, T., Forrester, J. S., Raengthon, N., Triamnak, N., Cann, D. P. & Jones, J. L. (2016). J. Appl. Phys. 120, 184102. Web of Science CrossRef ICSD Google Scholar
Wahlberg, N., Bindzus, N., Bjerg, L., Becker, J., Dippel, A.C. & Iversen, B. B. (2016). Acta Cryst. A72, 28–35. Web of Science CrossRef IUCr Journals Google Scholar
Wernecke, J., Gollwitzer, C., Müller, P. & Krumrey, M. (2014). J. Synchrotron Rad. 21, 529–536. Web of Science CrossRef CAS IUCr Journals Google Scholar
Williams, T. C. & Shaddix, C. R. (2007). Rev. Sci. Instrum. 78, 123702. Web of Science CrossRef PubMed Google Scholar
Willis, B. T. M. & Pryor, A. W. (1975). Thermal Vibrations in Crystallography. London: Cambridge University Press. Google Scholar
Young, R. A. (2002). The Rietveld Method. Oxford University Press. Google Scholar
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