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Journal of Applied Crystallography logoLink to Journal of Applied Crystallography
. 2020 May 5;53(Pt 3):671–678. doi: 10.1107/S1600576720002940

Calculation of total scattering from a crystalline structural model based on experimental optics parameters

Satoshi Hiroi a,b,c,*, Koji Ohara c, Satoru Ohuchi d, Yukihiro Umetani d, Takashi Kozaki d, Emiko Igaki d, Osami Sakata a,b
PMCID: PMC7312155  PMID: 32684882

A calculation procedure for X-ray total scattering and the pair distribution function from a crystalline structural model is presented. It allows one to easily and precisely deal with diffraction-angle-dependent parameters such as the atomic form factor and the resolution of the optics.

Keywords: total scattering measurement, atomic pair distribution function, local structure, resolution function

Abstract

Total scattering measurements enable understanding of the structural disorder in crystalline materials by Fourier transformation of the total structure factor, S(Q), where Q is the magnitude of the scattering vector. In this work, the direct calculation of total scattering from a crystalline structural model is proposed. To calculate the total scattering intensity, a suitable Q-broadening function for the diffraction profile is needed because the intensity and the width depend on the optical parameters of the diffraction apparatus, such as the X-ray energy resolution and divergence, and the intrinsic parameters. X-ray total scattering measurements for CeO2 powder were performed at beamline BL04B2 of the SPring-8 synchrotron radiation facility in Japan for comparison with the calculated S(Q) under various optical conditions. The evaluated Q-broadening function was comparable to the full width at half-maximum of the Bragg peaks in the experimental total scattering pattern. The proposed calculation method correctly accounts for parameters with Q dependence such as the atomic form factor and resolution function, enables estimation of the total scattering factor, and facilitates determination of the reduced pair distribution function for both crystalline and amorphous materials.

1. Introduction  

Many total scattering measurements using high-energy synchrotron X-rays or spallation pulse neutrons have been performed for structural analysis of disordered systems such as amorphous and liquid materials (Amann-Winkel et al., 2016 ▸; Maruyama et al., 2010 ▸; Bennett et al., 2015 ▸). An X-ray diffraction measurement generally provides the average structure of crystalline powder materials, whereas an X-ray total scattering measurement with high-energy X-rays provides details about the local structure such as the interatomic distance and coordination number of atoms. The use of software such as PDFgui (Proffen & Billinge, 1999 ▸; Farrow et al., 2007 ▸) to support the local structural refinement of crystalline structural models has become more popular, thereby accelerating the analysis of total scattering data for powder materials.

The experimental pair distribution function (PDF), G(r), obtained by X-ray total scattering measurement, is defined by the Fourier transform of the scattering intensity per electron from the position of the atomic centre, S(Q), that is,

1.

where Q is the magnitude of the scattering vector [Q = (4π/λ) sin θ, Å−1], and λ and θ are the wavelength (Å) of the incident X-rays and scattering angle, respectively. The range of the integration in equation (1) should be from Q min to Q max of the experimental S(Q) due to the finite condition of the measurement. In the case of PDF calculation from a crystal structure, the following equation is generally used:

1.

where N and bi are the number of atoms in the system and the scattering power, respectively (Proffen & Billinge, 1999 ▸). The Dirac delta function, δ(r − rij), corresponds to the atomic position of atom j of atomic species β with respect to the central atom i of α. In local structural analysis by the PDF such as in the PDFgui software, the delta function is substituted into another distribution function to take into account the mean-square displacement and the correlated motion (Jeong et al., 1999 ▸). If the PDF obtained by a neutron scattering measurement is used for local structural analysis, the PDF given by equation (2) corresponds to the experimental one because the scattering power of atoms is independent of Q. In the case of X-ray total scattering, however, bi = f i(0) should be assumed to obtain G calc(r) because the atomic form factor depends on Q. This results in a different interatomic correlation from the experimental PDF in the short-range region for a multi-component system. In order to avoid this situation, a calculation process for the PDF taking into account the Q dependence of the atomic form factor has been developed (Masson & Thomas, 2013 ▸). This technique allows one to directly calculate the exact PDF considering the atomic form factor from the crystal structure.

The Q-broadening effect of scattering intensity is the other factor that should be considered in structural analysis by total scattering measurements. Peak broadening observed in total scattering intensities is a result of smearing by the experimental resolution. The Q-broadening effect is considered by multiplying G calc(r) by a damping function when the PDF is calculated directly from the crystal structure by equation (2). A Gaussian-type damping function is generally used in PDF analysis software due to the assumption of a Gaussian function with a constant full width at half-maximum (FWHM) against Q for the Q-broadening effect (Proffen & Billinge, 1999 ▸; Farrow et al., 2007 ▸). However, the Q-broadening function generally depends on the Q value due to the resolution function of the apparatus and intrinsic effects such as crystallographic domain size and lattice distortion. Novel total scattering instruments in synchrotron facilities, multi-detector systems or flat-panel detector systems are widely installed in order to reduce the time for accumulation of counts. The FWHMs of the Q-broadening function in such cases show complicated Q dependence because the optical conditions on each detector are different in terms of the width of the receiving slit and detector position.

In general, a Q-broadening function such as the resolution function from the diffraction apparatus does not cause extensive modulation of the PDF in the short-range region, whereas it affects the damping behaviour of the amplitude in the long-range region over 20 Å (Proffen & Billinge, 1999 ▸; Farrow et al., 2007 ▸). In the case of a total scattering measurement for metal nanoparticles, the PDF indicates rapidly damped oscillation at long range, reflecting the average nanoparticle size and shape (Masadeh et al., 2007 ▸). The contributions from the resolution function of the apparatus and the intrinsic effects of the material can be separated by precise estimation of the Q-broadening function. Furthermore, from a structural analysis point of view, the local structural information obtained by a total scattering measurement is often combined with information from an X-ray diffraction measurement to determine the exact crystal structure (Tucker et al., 2007 ▸). The PDF obtained by a total scattering measurement includes structural information of not only short-range correlation but also long-range order because the measurement system is exactly the same as that for a diffraction measurement except for the optics. Therefore, total scattering has the potential for full-range structural analysis by precise estimation of the Q-broadening function without the need to combine it with the other diffraction profile.

Our aim in this work is to develop a calculation method for the total scattering and the PDF from a crystalline structural model based on the RMCPOW software developed by Mellergård, using a correctly predicted resolution function of the optics of the diffraction apparatus (Mellergård & McGreevy, 1999 ▸, 2000 ▸, 2001 ▸). Furthermore, we show that this method can provide a wide-range PDF over 100 Å from a crystal structure with ease. In Section 2, the procedure of this method is introduced in detail. The Q-broadening function related to the resolution function of the experimental conditions and the intrinsic effects of the target material and the thermal vibration in the crystal are also discussed in this section. In Section 3, the experimental conditions of an X-ray total scattering measurement for a CeO2 powder at BL04B2 at the synchrotron facility SPring-8, Japan, are precisely described in order to determine the resolution function of the optics. In Section 4, we show results of the structural analysis by the PDF of the CeO2 powder using the developed method. In Section 5, we provide a summary and conclusions.

2. Calculation of the structure factor and reduced pair distribution function  

2.1. Basics  

Mellergård et al. developed the reverse Monte Carlo method for powder diffraction (RMCPOW) and reported the diffraction intensity from a crystalline structural model with a volume of V consisting of N atoms to be

2.1.

where Q is the magnitude of the scattering vector, τ is the vector of the reciprocal-lattice point, F(τ) is the crystal structure factor on τ and Inline graphic is called the Q-broadening function, accounting for the optics of the measurement and intrinsic effects of a powder sample. A pseudo-Voigt function was used as the basic shape of the Q-broadening function, the same as for general Rietveld analysis, for instance with RIETAN-FP (Izumi & Momma, 2007 ▸). Equation (3) corresponds to the one-dimensional differential scattering cross section from a perfect crystal. The FWHM of the Q-broadening function, ΔQ, is determined by the contributions from the resolution function of the apparatus and the material intrinsic effects (see the next subsection). If inter-phase correlation in the sample is ignored, the total scattering intensity of a material containing multiple phases can be obtained by a mole-fraction-weighted linear combination of the partial scattering intensities for each phase (Sánchez-Gil et al., 2015 ▸). We obtain S(Q) by normalizing the diffraction intensity I(Q) with respect to the atomic form factor, f(Q):

2.1.

where 〈f(Q)2〉 and 〈f(Q)〉2 are defined as Inline graphic and Inline graphic, respectively, and ci is the mole fraction of the ith element. After calculating S(Q) using equations (3) and (4), we obtained the calculated G(r) by Fourier transform of the S(Q) described by equation (1) in the same manner as calculation of the experimental G(r).

Fig. 1 ▸(a) shows a diagram of the concept, properly considering the resolution function, intrinsic structural information and atomic form factor. The Q-broadening function in equation (3) affects the calculated G(r), especially damping the amplitude in the long-range region. If the Q-broadening function accounting for the resolution function of the diffraction apparatus and intrinsic sample effects such as crystallographic domain size and lattice distortion is correctly estimated, the calculated G(r) will be consistent with the experimental G(r) without any approximations. As a result, the calculated G(r) will enable us to discuss the atomic disorder in the crystalline material precisely.

Figure 1.

Figure 1

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(a) Schematic diagram of the procedure to obtain the reduced pair distribution function from experimental and calculated total scattering data. Unit cell (b) and 3 × 3 × 3 supercell (c) of the CeO2 crystal.

It is possible to get not only G(r) but also I(Q) and S(Q) simultaneously by the above procedure. In fact, structural analysis by I(Q) or S(Q) can be carried out in the same manner as Rietveld analysis. However, structural analysis using the PDF has advantages in the detection of local disorder from the averaged structure because G(r) provides the interatomic correlation in the short-range region. From another point of view, the number of measuring points in the total scattering measurement has to be reduced in order to accumulate counts of scattered photons effectively unless an area-type detector system is installed. Many measuring points are required in order to precisely determine the position and shape of the Bragg peaks of crystalline materials. On the other hand, an appropriate PDF for structural analysis is easily obtained with no dependence on experimental data because the bin size of G(r) can be arbitrarily chosen by dr in equation (1). Therefore, G(r) should be preferred for structural analysis of crystalline materials by total scattering measurement.

2.2. Q-dependent FWHM of the Q-broadening function, ΔQ  

The Q-broadening function, R(Q − |τ|) in equation (3), affects the estimation of S(Q), especially the height and width of the Bragg reflections. The Q-broadening function includes contributions from the resolution function of the optics of the diffraction apparatus and imperfections in the crystal structure such as finite crystallographic domain size and lattice distortion. Therefore, in general, the FWHM of the Q-broadening function, ΔQ, depends on the diffraction angle or Q value. ΔQ can be deduced by theoretical calculation using optical information on the beamline and optimization of the intrinsic effect of the sample without any measurement of standard samples.

The resolution function of the apparatus is determined by the energy resolution of the incident X-rays, ΔE, of energy E, the acceptance angle of the diffracted beam position, Δθ, and the divergence of the incident X-rays, Δα. The partial contributions to the Q-broadening function from these components correspond to ΔQE = QΔE/E, ΔQ θ = Q/tan θΔθ and ΔQ α = 4π/λsin(Δα/2), respectively. The intrinsic effect of the material can also be coarsely estimated by the Scherrer equation or the Williamson–Hall relation (Guinier, 1994 ▸; Langford & Wilson, 1978 ▸; Yogamalar et al., 2009 ▸). Using the Williamson–Hall relation, the contribution of the intrinsic effect, ω (Å−1), to the Q-broadening function is written as

2.2.

where K is the Scherrer constant, D is the crystallographic domain size (Å) and σ is relative distortion of lattice spacing in the crystal. D and σ in equation (5) can be used as the fitting variables in the procedure. We can obtain ΔQ from the root sum square of the resolution function of the apparatus and the intrinsic effect in the same manner as Warren (1941 ▸), that is,

2.2.

If equation (2) is used to calculate G(r) from the crystal structure, a damping function depending on r is multiplied by G(r) in order to estimate the smearing effect. In general, the damping function frequently used for the calculation is a Gaussian-type function. This function is defined if a Gaussian function with constant FWHM for the Q value is assumed for the Q-broadening function. However, ΔQ generally has a Q dependency as mentioned above; furthermore, ΔQ often possesses a complicated dependency on Q if multiple point detectors or a flat-panel detector are installed at the beamline. In the case of a multiple point detector system for the total scattering measurement, for example, the receiving slit width in front of each detector and the distance between the sample and each detector are often different in order to increase the detected counts for the wide-angle region. In this situation, it is difficult to deal exactly with the smearing effect for the scattering intensity by the damping function because ΔQ is a discontinuous and complicated function.

2.3. Thermal vibration effect  

All of the atoms in a crystal vibrate around their average position. In the case of Rietveld analysis using the diffraction profile, the thermal vibration is taken into account by employing the Debye–Waller factor. On the other hand, the thermal vibration can be simply evaluated by using a distribution function such as a Gaussian instead of a delta function as in equation (2) in the case of direct calculation of G(r) from the crystal structure. Unfortunately, neither the Debye–Waller factor nor the distribution function procedure can be employed for evaluation of the thermal vibration if G(r) is calculated from the crystal structure using the procedure described in Section 2.1. If the Debye–Waller factor is applied for evaluation of the thermal vibration in the process, for example, calculation of S(Q) by equation (4) fails in the high-Q region because the factor makes the denominator rapidly small. Here, we employed a method generating the atomic distribution around the average position by a normally distributed random number in order to evaluate the thermal vibration. First, a supercell containing nx × ny × nz unit cells of the crystal structure is generated as shown in Figs. 1 ▸(b) and 1 ▸(c). Next, all of the atoms in the supercell are moved from their average position by a random number following a Gaussian distribution with the mean-square displacement as the variance. Note that the generated relative positions of the equivalent atoms in the supercell are not the same. Using this atomic configuration for the procedure, the calculated G(r) considering thermal vibration can be obtained. If a larger supercell size is applied for the calculation, that is, the number of equivalent atoms is increased, the procedure provides a G(r) having a higher reliability because the atomic distribution from the average position asymptotically goes to the Gaussian distribution. However, the user needs to determine a suitable supercell size for the calculation because a larger supercell size produces many reciprocal-lattice points. In this study, we employed a supercell containing 3 × 3 × 3 cubic cells of CeO2 crystal structure as shown in Fig. 1 ▸(c) for calculation of G(r). In this case, the number of equivalent Ce and O atoms included in the supercell was 108 and 216, respectively.

3. Total scattering measurement at BL04B2  

X-ray total scattering measurements for the CeO2 powder were performed to determine S(Q) and evaluate the broadening function. The CeO2 powder (674a, NIST) was loaded in a 1 mm inner-diameter quartz capillary. The incident X-ray beam, having an energy of 61.377 keV, was monochromated by the Si 220 reflection on a bent monochromator. The beam size and divergence were limited by a pair of slits between the monochromator and CeO2 sample. The diffracted X-ray photons from the sample were counted by four CdTe detectors (X-123CdTe, Amptek) for the low-angle region and two highly pure Ge solid-state detectors (Ge-SSDs) (GL0515, CAN­BERRA Industries) for the high-angle region (Fig. S1 in the supporting information). The six detectors were installed every 8°. The measurement angular range was from 0.3 to 49°, and the maximum magnitude of the scattering vector, Q max, was 25.8 Å−1. The width of the receiving slits in front of all of the detectors, S RS, was 0.3 mm.

For BL04B2, the incident X-rays are supplied from a bending magnet as a light source and the beam divergence is limited to 0.72 mrad by the front-end shield (Isshiki et al., 2001 ▸). The X-rays are irradiated onto the bent Si monochromator, tilted 3° in the horizontal direction, in the optical hatch. The focal position of the reflected X-rays on the CdTe detectors is located 11 000 mm from the monochromator. A pair of incident slits are located 3140 and 8640 mm from the monochromator, and the sample capillary is placed 9670 mm from the monochromator (Fig. S1). The distances between the sample and detectors, LSD, are 1330 and 950 mm for the CdTe detectors and Ge-SSDs, respectively. The X-ray energy resolution, ΔE/E, at beamline BL04B2 depends on the beam divergence, in other words, the width of the incident slits. In the optics, the energy resolution corresponds to ΔE/E = Δαcot θB, where θB is the Bragg angle of the monochromator crystal, that is, θB = 3°. Here, the divergence was evaluated from the width, S IS, of the incident slit, located 3140 mm from the bent monochromator, and the distance L SF between the slit and focal point on the CdTe detector (Fig. S1). Therefore, we obtained the following relation for the energy resolution: ΔE/E = (S IS/L SF)cot θB. The optical parameters of beamline BL04B2 used for the calculation are summarized in Table 1 ▸.

Table 1. Optical parameters in high-energy X-ray total scattering measurements.

(S IS, S RS) (mm) ΔE/E Δα (mrad) L SD CdTe (mm) L SD Ge (mm)
(2.0, 0.3) 0.004856 0.2545 1330 950
(1.0, 0.3) 0.002429 0.1273 1330 950
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4. Results and discussion  

We performed X-ray total scattering measurements of the CeO2 powder at BL04B2 under two sets of optical conditions. Fig. 2 ▸ shows the experimental structure factor of CeO2 determined from observed X-ray total scattering patterns with two incident slit widths S IS. An obvious difference was observed in the angular resolution upon changing S IS because of the limited divergence of the incident X-rays. As previously explained, the energy resolution of the incident X-rays simultaneously changed because of the divergence.

Figure 2.

Figure 2

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Experimental (points) and calculated (lines) structure factors for CeO2 powder with different incident slit widths, S IS = 2.0 mm (a) and S IS = 1.0 mm (b).

We optimized the crystallographic domain size, lattice distortion and mean-square displacement parameters, U Ce and U O, of the CeO2 powder to obtain the smallest residual error between the experimental and calculated G(r) in the r range less than 200 Å. Here, we employed the Scherrer constant K = 1.0747, assuming a spherical crystallographic domain in equation (5) (Guinier, 1994 ▸; Langford & Wilson, 1978 ▸; Yogamalar et al., 2009 ▸). The calculated structure factors which give the optimized parameters are also shown in Fig. 2 ▸ as red and blue curves for comparison with the experimental data. It is noted that no interatomic correlated motions of Ce and O atoms were considered in the optimizing process. The calculated S(Q) values for the two types of optical conditions were consistent with those for each experimental S(Q) in the high-Q range. Table 2 ▸ summarizes the optimized parameters obtained from the S(Q) calculation for each set of optical conditions. The crystallographic domain size and lattice distortion differed slightly between the optical conditions because the intrinsic component was hidden by the component of the poor instrumental resolution in the Q-broadening function. In contrast, almost the same mean-square displacement of the Ce atom between the optical conditions was obtained because the short-range correlation is not affected so much by the Q-broadening function. However, the difference in mean-square displacement of the O atom between the optical conditions was slightly larger due to the small scattering power. The isotropic displacement parameters obtained by the proposed method were B iso = 0.214 and 0.413 Å2 (S IS = 1.0 mm) for the Ce and O atoms, respectively. Both displacement parameters are smaller than B iso = 0.335 and 0.514 Å2 obtained by an X-ray diffraction measurement at BL15XU, SPring-8 (Tanaka et al., 2013 ▸). The difference is consistent with the difference in the real-space resolution, in other words Q max.

Table 2. Optimized structural parameters obtained by the proposed method.

Optical condition (S IS, S RS) (mm) Domain size (nm) Distortion parameter (×10−5) U O (×10−3 Å2) U Ce (×10−3 Å2)
(2.0, 0.3) 33.0 9.23 7.22 2.60
(1.0, 0.3) 38.0 4.49 5.23 2.71
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To evaluate the Q-broadening function obtained by equation (6), we confirmed the FWHM values of the Bragg peaks using experimental X-ray total scattering profiles. Fig. 3 ▸ shows the Q dependence of the FWHM values of the Bragg peaks in the total scattering and calculated ΔQ which were determined from the optical parameters and optimized structural parameters. Although a small discontinuity is observed at Q = 17.66 Å−1 because of the positions of the CdTe and the high-purity Ge solid-state detectors, good agreement was observed between the experimental FWHM values of the Bragg peaks and the calculated Q-broadening functions, especially at lower angles, regardless of the difference in optical conditions. It was difficult to determine the correct FWHM in the high-Q region because of the poorer resolution and overlapping peaks.

Figure 3.

Figure 3

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FWHM values of Bragg peaks for experimental total scattering patterns (points) and estimated ΔQ (coloured solid curves). The colours correspond to different optical conditions. Coloured dashed curves indicate the intrinsic components of crystallographic domain size and lattice distortion of the target material. Black curves correspond to the estimated resolution function derived from the diffraction apparatus.

We employed the Williamson–Hall equation, equation (5), in order to determine the intrinsic component of the Q-broadening function, ω, derived from the crystallographic domain size and lattice distortion. The intrinsic components are also shown in Fig. 3 ▸ as dashed curves. In these experimental conditions, although the intrinsic components were quite small because of the poor resolution function of the apparatus, the Q-broadening functions were fitted to the FWHM of the experimental Bragg peaks by optimization of the crystallographic domain size and lattice distortion. We confirmed that the intrinsic effect on the Q-broadening function of the total scattering measurement could be estimated by the same procedure as Rietveld refinement of a powder diffraction pattern. While crystallographic domain size and lattice distortion are not local structural information, the optimized intrinsic effect should help us to get the correct local atomic structure.

A calculated G(r) that properly considers the resolution function of the apparatus and intrinsic structure effect can be obtained by equation (1) because G(r) was already used for optimization of the structural parameters. Figs. 4 ▸(a) and 4 ▸(b) show the comparison of the experimental G(r) with the calculated one for the CeO2 powder. We obtained good agreement of G(r)s not only at short range but also at long range up to 200 Å, regardless of the different optical conditions. The agreement at long range corresponds to good estimation of the Q-broadening function because the oscillation in this region reflects the Bragg peaks caused by translation symmetry. From an experimental point of view, the resolution function arising from the apparatus should be considered to obtain the calculated G(r) of the crystalline material. In total scattering measurements, multiple point detectors having different optical conditions, such as receiving slit size and distance between the sample and the detectors, are generally used in order to reduce the measurement time. Therefore, the proposed method for dealing with any Q-broadening function related to resolution function should be useful to obtain the local structural information from experimental total scattering. Fig. 4 ▸(c) shows a comparison of experimental and calculated G(r)s in the short-range region corresponding to nearest-neighbour distance. A wide correlation at 1.5–3.5 Å in the experimental G(r) is confirmed. Our procedure and Masson’s method successfully reproduced the correlation in the calculated G(r)s, except for termination ripples, whereas the G(r) calculated by equation (2) showed a different width of the first peak. A possible explanation for this difference is that these contributions arise from ion-core vibration and the electron distribution between the Ce and O ions. In this method, the electron distribution around a specific element was already taken into account as the Q-dependent atomic form factor in equation (3). In this method, we failed to find the valid structural parameters to completely reproduce the second peak corresponding to Ce–Ce correlation because correlated motion of atoms was not considered. The correlated motion related to phonons should be dealt with by the RMCPOW method using the structural parameters of this method. Otherwise, other lattice vibrational models such as those of piezoelectric materials have to be assumed for structural optimization.

Figure 4.

Figure 4

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Experimental G(r)s (points) and calculated G(r)s (curves) of CeO2 powder using the proposed method for S IS = 2.0 mm (a) and S IS = 1.0 mm (b). (c) Experimental and calculated G(r)s in the short-range region. Calculated G(r)s obtained in a different way [by equation (2) and Masson’s method (Masson & Thomas, 2012 ▸)] are also shown in this figure for comparison.

In the conventional PDF analysis method for crystalline material, the Q-broadening effect of scattering intensity is considered by multiplying the ideal G(r) by a damping function. In general, a Gaussian-type damping function, B(r) = exp[−(rQ damp)2/2], is employed in the analysis (Proffen & Billinge, 1999 ▸). The damping function works well if ΔQ has no dependence on Q because B(r) is obtained by Fourier transform of the Gaussian-type Q-broadening function with constant FWHM. However, ΔQ generally depends on Q due to the optical conditions and the intrinsic effects of the material as mentioned in Section 3. Calculated G(r)s employing the optical conditions of BL04B2 are shown in Fig. 5 ▸. In this figure, Gaussian-type damping functions (or envelopes of the PDF) having Q damp = 0.0085, 0.0170 and 0.0255 are also shown as black curves. The damping function decreases rapidly in the long-range region over 100 Å while the amplitude of the calculated G(r) is preserved. It may be difficult to determine a well fitted G(r) if a Gaussian is used as the damping function because of the difference in the damping speed at short and long range in real space. Particularly in the case of incident X-rays with poor energy resolution, assumption of the constant FWHM of the Q-broadening function is not appropriate due to a significant increase in ΔQ in the high-Q region. A precise estimation of ΔQ enables one to analyse the whole real-space region of the experimental G(r) with a single structural model.

Figure 5.

Figure 5

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Difference in damping behaviour of the calculated G(r). In this figure, we assumed U Ce = U O = 0 in order to precisely determine the difference in the Q-broadening function. Black curves are envelopes of G(r) in the constant ΔQ model corresponding to Q damp = 0.0085, 0.0170 and 0.0255.

Finally, we describe the calculation cost of the process to calculate G(r) from a crystal model. The calculation cost should be small for optimization of structural parameters of crystalline materials. The total calculation cost of the procedure described in Section 2.1 to obtain G(r) from a crystal structure is mainly decided by the number of reciprocal-lattice points. The reciprocal-lattice points used for the calculation depend on Q max and the volume of the simulation cell (Mellergård & McGreevy, 1999 ▸). The number of reciprocal-lattice points in Q max = 25.8 Å−1 is approximately 1.24 × 106 for a supercell containing a 3 × 3 × 3 cubic unit cell of CeO2. The number of reciprocal-lattice points is not prohibitive if a modern computer can be used for the calculation. In fact, the calculated G(r) in the range of r < 200 Å for the CeO2 crystal was obtained in 10 s or less by the procedure using a modern CPU (Intel core i7-7Y75, 1.3 GHz/3.6 GHz). In contrast, if equation (2) is employed to calculate G(r) from a crystal structure in such an r range, the interatomic distances in all of the atomic pairs in r < r max have to be calculated in principle, that is, the calculation time strongly depends on r max. The number of atomic pairs in r < 200 Å for the CeO2 crystal is approximately 3.6 × 1012. (It is noted that this is the worst case; in other words, it is possible to reduce the number of pairs by translational symmetry.) Therefore, the procedure described in Section 2.1 allows us to calculate G(r) in quite a wide range of more than several tens of nanometres in a short time.

5. Conclusion  

A calculation method to obtain total scattering data, S(Q), from a crystalline structural model was proposed. This calculation method enables PDF information, including the ripples, to be obtained from Fourier transform of S(Q), and the information is consistent with that of the experimental reduced PDF, G(r). The basic equation for the calculation, which was previously reported by Mellergård et al., requires a Q-broadening function derived from the diffraction apparatus and intrinsic effects. We presented the procedures used to estimate the resolution function from optical information of the diffraction apparatus and attempted to calculate S(Q) for CeO2 under the optical conditions at beamline BL04B2 at the SPring-8 synchrotron radiation facility in Japan. The estimated Q-broadening function from the resolution function and Williamson–Hall equation was consistent with the FWHM of the Bragg peaks in the experimental total scattering profiles. Almost the same intensity and width of the Bragg peaks as those in the experimental total scattering patterns of a CeO2 crystal were obtained over the entire Q range using the Q-broadening function and appropriate mean-square displacement parameters. The calculated G(r) obtained by Fourier transform of the S(Q) corresponded to the experimental G(r) in not only the short-range region less than 50 Å but also the long-range region up to 200 Å. The amplitude of the calculated G(r) was slowly damped in the long-range region above 100 Å compared with the Gaussian-type damping function. A precise estimation of the Q dependence of the Q-broadening function enables analysis of the whole real-space region of the experimental G(r) with a single structural model. This method also gives us a calculated G(r) that considers the Q-dependent parameters including the atomic form factor in the case of an X-ray total scattering measurement. It enables us to discuss the ionic core disorder in the crystalline material because the electron distribution in each atom is naturally included by this method. Furthermore, our calculation method enables structural analysis of multiple phases, including amorphous phases, in the material. In the near future, we plan to develop software for local structural refinement by pattern fitting of experimental total scattering data using our calculation method.

Supplementary Material

Optical details of BL04B2 beamline. DOI: 10.1107/S1600576720002940/to5208sup1.pdf

j-53-00671-sup1.pdf (541.8KB, pdf)

Acknowledgments

The authors would like to thank Dr Y. Yoneda for in-depth discussions on the PDFgui software. The authors would also like to thank Tiffany Jain, MS, from the Edanz Group (https://www.edanzediting.com/ac), for editing a draft of this manuscript.

Funding Statement

This work was funded by Panasonic NIMS Center of Exellence for Advanced Functional Materials grant . Japan Society for the Promotion of Science grant 19H0581.

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Supplementary Materials

Optical details of BL04B2 beamline. DOI: 10.1107/S1600576720002940/to5208sup1.pdf

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