research papers
Refining shortrange order parameters from the threedimensional diffuse scattering in singlecrystal electron diffraction data
^{a}University of Antwerp, Department of Physics, Groenenborgerlaan 171, B2020 Antwerp, Belgium, ^{b}University of Oxford, Inorganic Chemistry Laboratory, South Parks Road, Oxford OX1 3QR, United Kingdom, ^{c}FriedrichAlexanderUniversität ErlangenNürnberg, Kirstallographie und Strukturphysik, Staudtstraße 3, 91058 Erlangen, Germany, ^{d}Czech Academy of Sciences, Department of Structure Analysis, Na Slovance 2, 182 21 Prague, Czechia, and ^{e}Aarhus University, Department of Chemistry and iNANO, Langelandsgade 140, 8000 Aarhus, Denmark
^{*}Correspondence email: romy.poppe@uantwerpen.be
Our study compares shortrange order parameters refined from the diffuse scattering in singlecrystal Xray and singlecrystal electron diffraction data. Nb_{0.84}CoSb was chosen as a reference material. The correlations between neighbouring vacancies and the displacements of Sb and Co atoms were refined from the diffuse scattering using a Monte Carlo in DISCUS. The difference between the Sb and Co displacements refined from the diffuse scattering and the Sb and Co displacements refined from the Bragg reflections in singlecrystal Xray diffraction data is 0.012 (7) Å for the on diffuse scattering in singlecrystal Xray diffraction data and 0.03 (2) Å for the on the diffuse scattering in singlecrystal electron diffraction data. As electron diffraction requires much smaller crystals than Xray diffraction, this opens up the possibility of refining shortrange order parameters in many technologically relevant materials for which no crystals large enough for singlecrystal Xray diffraction are available.
Keywords: 3D electron diffraction; 3DED; singlecrystal diffuse scattering; 3D difference pair distribution functions; 3DΔPDF.
1. Introduction
Structure solution and
are used to solve and refine the average structure of crystalline materials from the intensities of the Bragg reflections. Parameters that are most commonly refined are the average atomic positions, site occupancies and atomic displacement parameters. However, for many materials, the local structure differs significantly from the average structure. When the deviations from the average structure are ordered on a local scale, they are referred to as local order or shortrange order. Materials with shortrange order have diffraction patterns that contain both Bragg reflections and structured diffuse scattering. Because the properties of many materials depend on the shortrange order, refining shortrange order parameters is essential for understanding and optimizing material properties.The development of threedimensional electron diffraction (3DED) in 2007 (Kolb et al., 2007, 2008) allowed the acquisition of electron diffraction data with less multiple scattering compared with inzone electron diffraction patterns. The main advantage of 3DED is that it allows us to determine the of materials for which no crystals large enough for singlecrystal Xray diffraction are available (Gemmi et al., 2019). When refining the average structure from singlecrystal Xray diffraction data using a standard leastsquares it is assumed that the intensities of the Bragg reflections are approximately proportional to the square of the absolute value of the Because electrons are scattered multiple times while going through the crystal, this assumption is not valid for 3DED data. A leastsquares in which the intensities are calculated using the of diffraction was developed by Palatinus and coworkers (Palatinus et al., 2013; Palatinus, Corrêa et al., 2015; Palatinus, Petříček et al., 2015) and is implemented in Jana2020 (Petříček et al., 2023).
In most studies on the diffuse scattering in singlecrystal electron diffraction data, the diffuse scattering in experimental data is qualitatively compared with the diffuse scattering in calculated data (Withers et al., 2003, 2004; Fujii et al., 2007; Goodwin et al., 2007; Brázda et al., 2016; Zhao et al., 2017; Neagu & Tai, 2017; Gorelik et al., 2023). A quantitative analysis of the diffuse scattering in 3DED data has only been reported in the case of 1D diffuse scattering (Krysiak et al., 2018, 2020; Poppe et al., 2022). In this article, we refine shortrange order parameters from the 3D diffuse scattering in 3DED data. The defective halfHeusler system Nb_{0.84}CoSb (Fig. 1), previously studied using singlecrystal Xray diffraction (Roth et al., 2021), was chosen as a reference material. The correlations between neighbouring vacancies and the displacements of the Sb and Co atoms were refined from the diffuse scattering using a Monte Carlo in DISCUS. The Nb occupancy and the displacements of Sb and Co atoms were also refined from the Bragg reflections in 3DED data.
2. Experimental
2.1. Synthesis
The samples used in this study were previously used by Roth et al. (2021) and are referred to as `SC0.81' and `Q0.84 #2'. Two different synthesis methods were used to prepare these samples. The SC0.81 sample has nominal stoichiometry Nb_{0.81}CoSb and was slowly cooled (SC) using an induction furnace. The Q0.84 #2 sample has nominal stoichiometry Nb_{0.84}CoSb and was thermally quenched (Q) from the melt. Details of the synthesis can be found in Roth et al. (2021) for the slowly cooled sample and in Yu et al. (2018) for the thermally quenched sample. The thermally quenched sample Nb_{0.84}CoSb (Q0.84 #2) has only shortrange Nbvacancy order, whereas the slowly cooled sample Nb_{0.81}CoSb (SC0.81) also has longrange Nbvacancy order.
2.2. 3DED data collection
Samples for 3DED data collection were prepared by dispersing the powder in ethanol. A few droplets of the suspension were deposited on a copper grid covered with an
film. Ultrathin grids were used to reduce the experimental background.3DED data were acquired with an aberrationcorrected cubed FEI Titan 80–300 electron microscope operated at 300 kV using a GATAN US1000XP CCD camera (4096 × 4096 pixels with 16bit dynamic range). The crystal was illuminated in selected area electron diffraction (SAED) mode with an exposure time of 1 s per frame. Electron diffraction patterns were acquired with a Fischione tomography holder (tilt range ±80°) in a stepwise manner using an inhouse developed script.
The 3DED data used for the dynamical
were collected with a step size of 0.1° on crystals 80–200 nm in size. The crystals were entirely illuminated during the whole data collection. The 3DED data used for the diffuse scattering analysis were collected with a step size of 0.1 or 0.2° on crystals 200–3000 nm in size. For larger crystals, only a thin part of the crystal was illuminated, which was recentred inside the aperture every few degrees.Because the Bragg reflections are three orders of magnitude stronger than the diffuse scattering, the acquisition of highquality diffuse scattering data requires careful subtraction of the experimental background. PETS2 (Palatinus et al., 2019) was used to process the 3DED data including background subtraction of the individual frames, integration of the Bragg reflection intensities for the dynamical and applying symmetry with m3m in the reconstruction of the 3D Pixels with negative intensities were used as negative values. The 3D of all 3DED data was indexed with a cubic with the cell parameter a = 5.89864 (3) Å and F43m (Zeier et al., 2017).
Details on the acquisition of the singlecrystal Xray diffraction data can be found in Roth et al. (2021).
2.3. Dynamical of the average crystal structure
Dynamical refinements of the average Jana2020. Because the F43m is noncentrosymmetric, a inversion matrix was defined. Refined parameters include the Nb occupancy; the twin fraction or a thickness parameter; harmonic displacement parameters of Sb, Nb and Co; and one scale factor per virtual frame. The Nb occupancy was allowed to refine freely, whereas the occupancies of Sb and Co were fixed to 1. The intensities in the dynamical were calculated for a wedgeshaped crystal (Palatinus, Petříček et al., 2015). The dynamical parameters were set to g_{max} = 2.4 Å^{−1}, S_{g}^{max}(matrix) = 0.025 Å^{−1}, S_{g}^{max}(refine) = 0.1, = 0.66 and N_{int} = 100. The meaning of the dynamical parameters can be found in the supporting information.
were performed in2.4. Monte Carlo refinement
DISCUS (Proffen & Neder, 1997) was used to build a model of the shortrange Nbvacancy order in the thermally quenched sample Nb_{0.84}CoSb (Q0.84 #2) and of the longrange Nbvacancy order in the slowly cooled sample Nb_{0.81}CoSb (SC0.81). Both models can be found in Poppe (2023a).
The intensities in ). The F(Q) was calculated using the discrete Fourier transform (DFT):
were calculated according to the standard formula for kinematic scattering (Neder & Proffen, 2008where N is the number of atoms in the crystal, f_{j}(Q) is the atomic form factor of atom j, is a vector and is the vector of atom j.
Correlation coefficients c_{mn}^{uvw} between neighbouring vacancies in DISCUS are defined as (Welberry, 1985; Neder & Proffen, 2008)
where P_{mn}^{uvw} is the probability that sites m and n are occupied by the same atom type, uvw the interatomic vector, and is the Nb occupancy. Negative values of c_{mn}^{uvw} indicate that sites m and n tend to be occupied by different atom types whereas positive values of c_{mn}^{uvw} indicate that sites m and n tend to be occupied by the same atom type. A correlation value of zero describes a random distribution of the two atom types. The maximum negative value of c_{mn}^{uvw} for a given occupancy is /(1 − ) ( P_{mn}^{uvw} = 0), the maximum positive value is +1( P_{mn}^{uvw} = ).
The Monte Carlo h0l plane from both singlecrystal Xray diffraction data and 3DED data acquired on the thermally quenched sample (Q0.84 #2). The 3D Xray and electron diffuse scattering data are available in Poppe (2023d). The Bragg reflections were subtracted using MANTID (Arnold et al., 2014), and the 3D diffuse scattering data were cropped on a grid with 215 × 215 pixels for −6 ≤ h,l ≤ 6. The intensities in the h0l plane were converted to input for DISCUS using a custom Python script (Poppe, 2023c).
was applied to the diffuse scattering in theThe model of the shortrange Nbvacancy order was implemented in a differential evolutionary algorithm. Three parameters were refined: the correlation between nextnearest neighbour vacancies and the displacements of Sb and Co atoms. A more detailed explanation of the DISCUS models and the Monte Carlo is given in the supporting information. The Monte Carlo (Poppe, 2023b) took about seven days for 19 cycles on a desktop computer using eight cores in parallel. The 24 children were calculated in parallel, whereas the individual crystals and the individual lots were calculated in series.
2.5. The 3DΔPDF
The 3D difference pair distribution function (3DΔPDF) was calculated in MANTID. MANTID was installed on a highperformance computing (HPC) cluster, and the 3DΔPDF was calculated in parallel on a node with 128 GB RAM. The Bragg reflections were subtracted, and the 3DΔPDF was obtained by Fourier transform of the 3D diffuse scattering/satellite reflections.
For the simulated 3DΔPDF maps, Scatty (Paddison, 2019) was used to calculate the 3D diffuse scattering from the DISCUS models. Scatty uses a fast Fourier transform (FFT) algorithm to calculate the in equation (1), which accelerates the calculation of the diffuse scattering by a factor 10^{2}–10^{3} compared with the discrete Fourier transform (DFT). Lanczos resampling was used to reduce the highfrequency noise. For the shortrange Nbvacancy order model, the calculation of the 3D diffuse scattering took about eight days. The diffuse scattering was calculated for expansion order 10, expansion maximum error 0.05 and window 2. To obtain an error smaller than 5%, the outer part of the 3D (about 18%) was calculated using DFT. For the longrange Nbvacancy order model, calculation of the 3D diffuse scattering took only 35 min. The diffuse scattering was calculated for expansion order 1 and window 2.
For the experimental 3D diffuse scattering data, the 3D h,k,l ≤ 25.2. For the 3D diffuse scattering data calculated in Scatty, the 3D was reconstructed on a grid with 401 × 401 × 401 voxels for −20 ≤ h,k,l ≤ 20.
was reconstructed on a grid with 901 × 901 × 901 voxels for −25.2 ≤3. Results and discussion
3.1. Dynamical of the average crystal structure
The percentage of vacancies on the Nb sites and the average displacements of Sb and Co atoms in the thermally quenched sample (Q0.84 #2) were refined from the Bragg reflections in the 3DED data using a dynamical et al., 2021).
and were compared with reference values refined from the Bragg reflections in singlecrystal Xray diffraction data (Roth3DED data were acquired on three different crystals, and each dataset was processed separately using PETS2 (Palatinus et al., 2019). Analogous to the of the average structure from the Bragg reflections in singlecrystal Xray diffraction data (Roth et al., 2021), the dynamical of the average structure was done in two stages. In the first stage (centre model), the displacements of Sb and Co atoms were fixed to zero. In the second stage (split model), Sb atoms were offcentred at (1/2 + Δ, 1/2, 1/2), Co atoms were offcentred at (1/4 − δ, 1/4 − δ, 3/4 − δ), and the displacements of Sb and Co atoms (Δ and δ) were allowed to refine freely.
Results of the dynamical . The refined atomic displacement parameters for Sb, Co and Nb are listed in Table S1 of the supporting information. The standard uncertainties on the Nb occupancy and the average displacements of Sb and Co atoms only consider random errors in the intensities of the Bragg reflections and are thus underestimated. Differences between the refined Nb occupancy and the refined Sb and Co displacements for the three crystals can be due to real differences in the Nb occupancy and the Sb and Co displacements between the three crystals or can be due to systematic errors in the calculation of the intensities in the dynamical Systematic errors can be attributed to (i) strong multiple scattering caused by the high atomic numbers of Nb, Sb and Co; (ii) the relatively low datatoparameter ratio (i.e. the number of observed reflections per refined parameter); (iii) a high crystal mosaicity; and (vi) no optimization of the frame orientation angles. The orientations of the frames could not be optimized due to the limited number of reflections on each frame. The limited accuracy of the goniometer of the TEM stage or small unpredictable movements of the crystal during the acquisition of the data may cause the orientation of a frame, as calculated from the orientation matrix, to be inaccurate (Palatinus, Corrêa et al., 2015). Note that the R values and the refined parameters have a slight dependence on the dynamical parameters. For example, changing the value for from 0.66 to 0.8 changed the refined Nb occupancy by 1.9% for crystal 1, by 0.3% for crystal 2 and by 0.1% for crystal 3.
for the centre model and for the split model are shown in Table 1

The average Nb occupancy of the three crystals (x) was calculated using equation (3):
where x_{i} is the Nb occupancy of each crystal and s_{i} is the on the Nb occupancy. For the split model, the average Nb occupancy refined from the Bragg reflections in 3DED data [0.813 (11)] differs by only 0.014 (11) from the Nb occupancy refined from the Bragg reflections in singlecrystal Xray diffraction data [0.827 (2)]. Refined occupancies are more accurate for highsymmetry unit cells, such as for Nb_{0.84}CoSb, than for lowsymmetric unit cells. Palatinus and coworkers previously compared the Fe/Mg occupancy in (Mg_{x}Fe_{1−x})_{2}Si_{2}O_{6} refined from the Bragg reflections in 3DED data with the Fe/Mg occupancy refined from the Bragg reflections in singlecrystal Xray diffraction data, and found a difference in Fe/Mg occupancy of 0.028 (8) (Palatinus, Corrêa et al., 2015).
For the singlecrystal Xray diffraction data, the R value improved significantly after of the displacements of Sb and Co atoms (Roth et al., 2021), whereas for the 3DED data, the R value stayed approximately the same (Table 1). The average displacements refined from the Bragg reflections in 3DED data [0.175 (6) Å for Sb and 0.180 (8) Å for Co] were calculated using equation (3) and differ by 0.040 (5) Å from the average displacements refined from the Bragg reflections in singlecrystal Xray diffraction data [0.141 (1) Å for Sb and 0.130 (1) Å for Co]. Palatinus and coworkers previously compared the atomic positions refined from the Bragg reflections in 3DED data with the displacements refined from the Bragg reflections in singlecrystal Xray diffraction data for three different samples and found a difference in bond length between 0.01 and 0.02 Å (Palatinus, Corrêa et al., 2015). The difference in bond length is thus larger for Nb_{0.84}CoSb than for the samples reported by Palatinus, Corrêa et al. (2015), which is probably due to differences in the crystal mosaicity. The atomic numbers of the elements of the samples in Palatinus, Corrêa et al. (2015) are lower than the atomic numbers of Nb, Sb and Co. Because the average between two scattering events is smaller in samples with elements with higher atomic numbers, the intensities in the 3DED data acquired on Nb_{0.84}CoSb are more influenced by multiple scattering than the intensities in the 3DED data acquired on the samples in Palatinus, Corrêa et al. (2015).
Fig. 2 shows the difference Fourier maps [] in the x0.5z plane after of the average structure of the thermally quenched sample (Q0.84 #2) with the centre model, for both the singlecrystal Xray diffraction data and the 3DED data acquired on the three different crystals. The average positions of Sb and Nb atoms in the x0.5z plane are shown on the left. The red/blue features in the difference Fourier map represent the residual electron density for Xray diffraction and the residual electrostatic potential for electron diffraction. In the difference Fourier map of the singlecrystal Xray diffraction data, four maxima in the residual electron density can be observed around the average position of the Sb atom, which indicate splitting of the Sb position. Splitting of the Sb position is unclear from the difference Fourier maps of the 3DED data acquired on crystal 1 and crystal 2, but is clear from the difference Fourier map of the 3DED data acquired on crystal 3, even though the maxima in the residual electrostatic potential are broader than for Xray diffraction. Differences between the difference Fourier maps of the three crystals are likely due to systematic errors in the calculation of the intensities in the dynamical (due to e.g. strong multiple scattering, low datatoparameter ratio, a high crystal mosaicity, no optimization of the frame orientation angles). This also explains the difference between the average Sb displacement refined from the Bragg reflections in 3DED and singlecrystal Xray diffraction data.
3.2. Singlecrystal electron diffraction versus singlecrystal Xray diffraction
Inzone selected area electron diffraction (SAED) patterns were previously acquired on Nb_{0.8}CoSb (Xia et al., 2019). The main advantage of 3DED is that it allows the acquisition of 3D electron diffuse scattering data with less multiple scattering compared with inzone SAED patterns.
3DED data were acquired on the thermally quenched sample Nb_{0.84}CoSb (Q0.84 #2) and the slowly cooled sample Nb_{0.81}CoSb (SC0.81). The crystals used for the diffuse scattering analysis were larger than the crystals used for the dynamical in the previous section. Fig. 3 shows the h0l plane reconstructed from singlecrystal Xray diffraction and singlecrystal electron diffraction data acquired on both samples. The h0l plane for −20 ≤ h,l ≤ 20 is shown in Fig. S3. The h0.5l and hhl planes are shown in Figs. S4 and S5.
The diffuse scattering intensity at lower scattering angles is higher for electron diffraction than for Xray diffraction, which can be explained by differences in the atomic form factors for electrons and Xrays. Fig. S6 shows the Xray and electron atomic form factors of Co, Nb and Sb as a function of the scattering angle. The Xray atomic form factor is the Fourier transform of the electron density, whereas the electron atomic form factor is the Fourier transform of the electrostatic potential. Because the electrostatic potential is broader than the electron density, the electron atomic form factors decrease faster to zero than the Xray atomic form factors, which explains the difference in the intensity distribution of the diffuse scattering.
The bottom row in Fig. 3 shows the h0l plane of the 3D diffuse scattering calculated in Scatty from the structure models calculated in DISCUS. For the thermally quenched sample (Q0.84 #2), the diffuse scattering was calculated from the shortrange order model with a for nearestneighbour vacancies of c_{(1/2,1/2,0)} = −0.19 and a for nextnearest neighbour vacancies of c_{(1,0,0)} = −0.012. The values for c_{(1/2,1/2,0)} and c_{(1,0,0)} were determined based on a series of Monte Carlo simulations and gave the best visual agreement between the observed and calculated diffuse scattering. Note that the values for c_{(1/2,1/2,0)} and c_{(1,0,0)} are different from those used for the calculation of the diffuse scattering by Roth et al. (2021). The diffuse scattering was calculated for an Sb displacement of 0.141 Å and a Co displacement of 0.130 Å (displacements refined from the Bragg reflections in singlecrystal Xray diffraction data in Table 1). For the slowly cooled sample (SC0.81), the diffuse scattering was calculated from the longrange order model. Symmetry with the m3m was applied to the 3D diffuse scattering calculated in Scatty (Fig. S7).
3.3. Monte Carlo refinement
The Monte Carlo ). Increasing the crystal size, the number of crystals and the number of lots reduces the noise but increases the time. Figs. S10 and S11 show that increasing the crystal size from 25 × 25 × 25 unit cells to 30 × 30 × 30 unit cells, increasing the number of crystals from 10 to 20, and increasing the number of lots from 50 to 500 improves the quality of the calculated diffuse scattering only marginally.
was performed on a model crystal with 25 × 25 × 25 unit cells. The limited crystal size introduces highfrequency noise in the calculated diffuse scattering. The diffuse scattering was averaged over ten crystals to reduce the highfrequency noise. For each crystal, the diffuse scattering was also averaged over 50 lots with a size of 12 × 12 × 12 unit cells, randomly distributed within the model crystal. The lot size should be larger than the correlation length of the longest correlations but smaller than or equal to the crystal size divided by two (Paddison, 2019The Monte Carlo c_{(1/2,1/2,0)} and nextnearest neighbour vacancies c_{(1,0,0)}. Because nearest and nextnearest neighbour vacancies avoid each other, both c_{(1/2,1/2,0)} and c_{(1,0,0)} should be negative. According to equation (2), the maximum negative value is c_{(1/2,1/2,0)} ≃ −4.78 , where P_{mn}^{(1/2,1/2,0)} is the probability that two nearest neighbouring Nb sites m and n are occupied by Nb atoms. However, for Nb_{0.84}CoSb a value of P_{mn}^{(1/2,1/2,0)} = 0 cannot be achieved and the maximum achievable negative value is c_{(1/2,1/2,0)} ≃ −0.20.
was used to refine the correlations between nearest neighbour vacanciesThe effect of the ratio c_{(1/2,1/2,0)}/c_{(1,0,0)} and the displacements of Sb and Co atoms on the intensity distribution of the diffuse scattering is shown in Fig. S12. Because the intensity distribution of the diffuse scattering depends on the ratio c_{(1/2,1/2,0)}/c_{(1,0,0)}, c_{(1/2,1/2,0)} was fixed to −0.20 (a value close to the actual determined based on a series of Monte Carlo simulations) and c_{(1,0,0)} was refined. In total, three parameters were refined: the correlation between nextnearest neighbour vacancies [c_{(1,0,0)}], the distance between a vacancy i and a neighbouring Sb atom k(τ_{ik}), and the distance between a vacancy i and a neighbouring Co atom k′(). Because nextnearest neighbour vacancies avoid each other, starting values for c_{(1,0,0)} were chosen in the range [−0.60, −0.01]. The average distance between a vacancy i and a neighbouring Sb atom k is τ_{ik} = 2.801. Because Sb atoms move towards neighbouring vacancies, starting values for τ_{ik} were chosen in the range [2.545, 2.945]. The average distance between a vacancy i and a neighbouring Co atom k′ is = 2.680. Because Co atoms move away from neighbouring vacancies, starting values for were chosen in the range [2.550, 2.950].
The Monte Carlo h0l plane from the singlecrystal Xray diffraction data and the 3DED data of the thermally quenched sample (Q0.84 #2) in Fig. 3. The spatial resolution of the observed diffuse scattering is determined by various effects including the convergence of the beam, the detector point spread function and the crystal mosaicity. To account for resolution effects, the intensity of each pixel in the calculated h0l plane was convoluted with a Gaussian function (Fig. S13). The standard deviation of the Gaussian function (σ = 0.006 Å^{−1} for Xray diffraction and σ = 0.008 Å^{−1} for electron diffraction) was estimated from the intensity profile of unsaturated Bragg reflections.
was applied to the diffuse scattering in theThe refinements were interrupted manually after 19 . The standard uncertainties of the shortrange order parameters are underestimated. Systematic errors could be due to (i) the limited number of correlations included in the model, (ii) nonperfect background subtraction, (iii) inaccurate resolution function correction, (iv) distortions in the reconstructed 3D diffuse scattering (e.g. due to small crystal movements during data acquisition or the instability of the goniometer of the TEM sample stage) and (v) no correction for multiple scattering.
cycles. The refined shortrange order parameters are listed in Table 2

In DISCUS, there is a difference between target correlations and displacements (the refined correlations and displacements) and achieved correlations and displacements (the actual correlations and displacements) (Neder & Proffen, 2008). The achieved correlations and displacements were calculated from the final structure model after the The achieved Co displacements are identical to the target Co displacements, whereas the achieved Sb displacements are lower than the target Sb displacements. The reason is that an Sb atom with two neighbouring vacancies on opposite sides of the Sb atom will not move away from its average position. A Co atom may also have two neighbouring vacancies, but these two neighbouring vacancies cannot be on opposite sides of the Co atom [Fig. 1(b)]. Each Co atom with at least one neighbouring vacancy will thus move away from its average position.
The R value in Table 2 is much higher for the shortrange order parameters refined from the diffuse scattering in the 3DED data than for the shortrange order parameters refined from the diffuse scattering in the singlecrystal Xray diffraction data. The evolution of the R values and the shortrange order parameters during the applied to the diffuse scattering in singlecrystal Xray diffraction and 3DED data are shown in Figs. S14 and S15, respectively.
The diffuse scattering calculated for the refined shortrange order parameters is shown in Fig. 4. At lower scattering angles, the observed diffuse scattering intensities are higher than the calculated diffuse scattering intensities. The satellite reflections are also sharper in the observed diffuse scattering than in the calculated diffuse scattering, especially for the diffuse scattering in the 3DED data.
The displacements refined from the diffuse scattering in singlecrystal Xray diffraction data are 0.142 (11) Å for Sb and 0.112 (8) Å for Co. The displacements refined from the diffuse scattering in 3DED data are 0.142 (23) Å for Sb and 0.071 (21) Å for Co. The standard uncertainties only consider random errors in the intensities of the Bragg reflections and are thus underestimated. The difference between the Sb and Co displacements refined from the diffuse scattering and the Sb and Co displacements refined from the Bragg reflections in singlecrystal Xray diffraction data [0.141 (1) Å for Sb and 0.130 (1) Å for Co] is 0.012 (7) Å for the R value is higher for the applied to the diffuse scattering in 3DED data (63.5%) than for the applied to the diffuse scattering in singlecrystal Xray diffraction data (37.2%), which indicates that the correlations refined from the diffuse scattering in singlecrystal Xray diffraction data are likely to be more accurate. The higher Rvalue for electron diffraction could be due to artefacts in the subtraction of the experimental background or could be due to residual multiple scattering.
on the diffuse scattering in singlecrystal Xray diffraction data, and 0.03 (2) Å for the on the diffuse scattering in 3DED data. The local Sb and Co displacements refined from the diffuse scattering are thus close to the average Sb and Co displacements refined from the Bragg reflections in singlecrystal Xray diffraction data, for both Xray and electron diffraction. As 3DED requires much smaller crystal sizes than singlecrystal Xray diffraction, this opens up the possibility to refine shortrange order parameters in materials for which no crystals large enough for singlecrystal Xray diffraction are available. TheThe Monte Carlo
of the shortrange order parameters took about seven days for 19 cycles. The time is proportional to the number of pixels/voxels in and the number of refined shortrange order parameters. Therefore, the was carried out against the diffuse scattering in one 2D plane, and only three parameters were refined. Refining shortrange order parameters against the 3D diffuse scattering and refining the correlations between further nearest neighbour vacancies could additionally improve the match between observed and calculated intensities.3.4. The 3DΔPDF
The 3D difference pair distribution function (3DΔPDF) is the Fourier transform of the 3D diffuse scattering in singlecrystal diffraction data (Schaub et al., 2007) and is often used to determine the origin of the observed diffuse scattering. Recently, Schmidt, Klar et al. (2023) showed that the 3DΔPDF can also be reconstructed from the diffuse scattering in 3DED data. The 3DΔPDF was obtained by removing the Bragg reflections and Fourier transforming the 3D diffuse scattering/satellite reflections (see Fig. S16). 3DΔPDF provides information about correlations between neighbouring atoms that are not represented by the average structure. Positive/negative 3DΔPDF values mean that the probability of finding scattering densities separated by the corresponding interatomic vector is higher/lower in the real structure than in the average structure (Weber & Simonov, 2012). Because Xrays are scattered by the electron cloud, the scattering densities in the Xray 3DΔPDF are electron densities. Electrons are scattered by the atomic nucleus and the electron cloud, and the scattering densities in the electron 3DΔPDF are thus atomic charge densities.
Fig. 5 shows the x0z plane of the Xray and electron 3DΔPDF, for both the thermally quenched sample (Q0.84 #2) and the slowly cooled sample (SC0.81). The 3DΔPDF was reconstructed from the 3D diffuse scattering data of which the h0l plane is shown in Fig. 3. The calculated 3DΔPDF maps are in good agreement with the experimental ones. The displacement of Co atoms can be identified from the x0.27z plane of the 3DΔPDF, for which a similar comparison is shown in Fig. S17. Because the Bragg reflections close to the central beam were overexposed due to the limited dynamical range of the CCD, the Bragg reflections in the electron diffraction data could not entirely be subtracted (Fig. S18). The experimental electron 3DΔPDF is thus affected by blooming artefacts due to saturated Bragg reflections (deformation of the features in the 3DΔPDF and weak additional features).
The electron 3DΔPDF and the Xray 3DΔPDF of Nb_{0.84}CoSb contain the same information about correlations between neighbouring atoms, and they can thus both be used to determine the origin of the diffuse scattering. The maximum observable correlation length in the 3DΔPDF (8.93 r.l.u.) is determined by the voxel size in (Δh, Δk, Δl ≃ 0.056). The intensity of the diffuse scattering goes faster to zero for electrons than for Xrays (Fig. 3), which results in broader features in the electron 3DΔPDF maps than in the Xray 3DΔPDF maps (Fig. 5), but which does not hinder the qualitative interpretation of the 3DΔPDF. Besides, the diffuse scattering is broader for electron diffraction than for Xray diffraction, which results in a faster decay of the features in the electron 3DΔPDF maps than in the Xray 3DΔPDF maps, as can be seen in Fig. S19.
The 3DΔPDF of Nb_{0.84}CoSb is similar to the 3DΔPDF of Zr_{0.82}Y_{0.18}O_{1.91} (Schmidt, Neder et al., 2023; Schmidt, Klar et al., 2023) and the origin of the diffuse scattering (correlations between neighbouring vacancies and the relaxation of the Zr, Y and O atoms around these vacancies) is also similar. However, the 3DΔPDF of Zr_{0.82}Y_{0.18}O_{1.91} was only interpreted in a qualitative way and no of the shortrange order parameters was applied to the diffuse scattering. The features in the 3DΔPDF maps of Nb_{0.84}CoSb in Fig. 5 are much sharper than those in the 3DΔPDF maps of Zr_{0.82}Y_{0.18}O_{1.91} (for both electrons and Xrays) (Schmidt, Neder et al., 2023; Schmidt, Klar et al., 2023), which can be explained by the different Qrange. The 3D diffuse scattering data of Zr_{0.82}Y_{0.18}O_{1.91} were acquired for −10 ≤ h,k,l ≤ 10, whereas the 3D diffuse scattering data of Nb_{0.84}CoSb were acquired for −20 ≤ h,k,l ≤ 20. The Qrange is thus higher for Nb_{0.84}CoSb (∼21.3 Å^{−1}) than for Zr_{0.82}Y_{0.18}O_{1.91} (∼12.2 Å^{−1}). Fig. S20 illustrates that the width of the features in the 3DΔPDF is inversely proportional to the Qrange.
Shortrange order parameters can also be refined from the 3D diffuse scattering using a 3DΔPDF in Yell (Simonov et al., 2014). However, at the moment, the 3DΔPDF in Yell can only be applied to the diffuse scattering in singlecrystal Xray diffraction data. Refining correlations from the diffuse scattering in 3DED data is thus only possible using a Monte Carlo refinement.
4. Conclusions
In this article, we have demonstrated the possibility to refine shortrange order parameters from the 3D diffuse scattering in 3DED data using a Monte Carlo DISCUS. As 3DED requires much smaller crystal sizes than singlecrystal Xray diffraction, this opens up the possibility to refine shortrange order parameters in materials for which no crystals large enough for singlecrystal Xray diffraction are available.
inThe correlations between neighbouring vacancies and the displacements of Sb and Co atoms were refined from the diffuse scattering in both singlecrystal Xray diffraction and 3DED data acquired on Nb_{0.84}CoSb. The local Sb and Co displacements refined from the diffuse scattering in singlecrystal Xray diffraction data [0.142 (11) and 0.112 (8) Å] and 3DED data [0.142 (23) and 0.071 (21) Å] are close to the average Sb and Co displacements refined from the Bragg reflections in singlecrystal Xray diffraction data [0.141 (1) Å and 0.130 (1) Å]. The R value was higher for the applied to the diffuse scattering in 3DED data (63.5%) than for the applied to the diffuse scattering in singlecrystal Xray diffraction data (37.2%), which indicates that the correlations refined from the diffuse scattering in singlecrystal Xray diffraction data are likely to be more accurate. The higher R value for electron diffraction could be due to artefacts in the subtraction of the experimental background or residual multiple scattering. Since it is not possible to include multiple scattering in the calculation of diffuse scattering intensities, and since the probability for multiple scattering to occur increases with increasing sample thickness, it is important to acquire 3DED data on small crystals.
5. Related literature
The following references are cited in the supporting information: Klar et al. (2023); Price et al. (2005); Proffen & Welberry (1998); Roth et al. (2020); Warren et al. (1951); Welberry & Weber (2016).
Supporting information
Supporting figures. DOI: https://doi.org/10.1107/S2052252523010254/vq5004sup1.pdf
Acknowledgements
We would like to thank Dr Joe Paddison for includng the atomic scattering factors of electrons in Scatty.
Funding information
The computational resources and services used in this work were provided by the HPC core facility CalcUA of the Universiteit Antwerpen and VSC (Flemish Supercomputer Center), funded by the Research Foundation: Flanders (FWO) (grant Nos. G035619N; G040116N) and the Flemish Government.
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