I. INTRODUCTION
Synchrotron powder diffraction is a well-understood technique to probe the structure of materials for purposes like structure determination and/or refinement, in situ characterization, or trace phase quantification. Experiments should be designed to be as sensitive as possible to the details of interest for a specific problem. Because of this, versatility is a key feature for a modern diffraction instrument. While flexibility is appealing, practicality and quality of the results also need to be preserved. The focus of the present study is synchrotron X-ray diffraction with two-dimensional (2D) detectors where there are many parameters which can be varied. An incident photon energy should be chosen to suit both the chemical composition of the sample and the Q-range to be measured. Increasing sample to detector distance improves the reciprocal space resolution, which can be good for monitoring phase transitions which involve a lattice distortion or for following an evolving quantitative phase analysis, but with the drawback that it reduces the Q-range and count rate per pixel (Norby, Reference Norby1997). Placing the detector as close as possible increases the Q-range which improves real-space resolution for pair distribution function, or PDF, methods (Egami and Billinge, Reference Egami and Billinge2003).
Recent developments in X-ray focusing optics mean it is now possible to create X-ray beams with sizes ranging from nanometers to millimeters (Kirkpatrick and Baez, Reference Kirkpatrick and Baez1948; Snigirev et al., Reference Snigirev, Snigireva, Grigoriev, Yunkin, Di Michiel, Vaughan, Kohn and Kuznetsov2009; Vaughan et al., Reference Vaughan, Wright, Bythkov, Rossat, Gleyzolle, Snigireva and Snigirev2011; Cesar Da Silva et al., Reference Cesar da Silva, Pacureanu, Yang, Bohic, Morawe, Barrett and Cloetens2017). The bandpass (∆E/E) of the X-ray beam can even be chosen by selecting different optics or changing the bending of a curved crystal monochromator. Features of the detector, like point spread function (PSF), can also be controlled (e.g. by binning or changing a phosphor screen). Because of this versatility, experimental conditions are multiple and the measured data depend on a convolution of all of these effects. Although the qualitative effect of many parameters on the resultant diffraction pattern is well known, in the first part of this paper, we have computed their effects by geometrical derivation, before checking the results on a reference material.
The second part of this paper shows two examples of applications where the choice of experimental parameters was not obvious: how best to detect small distortions in a material with a small particle size and how to deal with mixtures of crystalline and amorphous samples? The first example is the case of a barium-perovskite (BaTiO3) with a particle size of about 100 nm, a material which has been intensively studied for its piezoelectric and ferroelectric applications (Chandler et al., Reference Chandler, Roger and Hampden-Smith1993; Bhalla et al., Reference Bhalla, Gou and Roy2000). Several temperature-dependent crystallographic transitions are well established from the literature (Kwei et al., Reference Kwei, Lawson, Billinge and Cheong1993). BaTiO3 is cubic at high temperatures, exhibiting a paraelectric behavior down to 393 K. At lower temperature it converts to tetragonal (until 278 K), then orthorhombic (until 183 K), and then rhombohedral structures at lower temperatures. All of these three modifications are characterized by the off-centering of titanium species from the octahedral units, which causes ferroelectric behavior.
The second example is the case of a SiO2.Al2O3.Na2O.CaO glass, a material whose network is made by corner-sharing structural units (i.e. silicon and aluminum centered tetrahedra) which is interrupted by network modifier cations, like calcium and sodium. The modifiers have strong implications on the technological properties of the material because the structure changes its degree of polymerization (Shelby, Reference Shelby2005). High-energy X-ray diffraction capabilities to probe non-crystalline materials have already been well established (Yavari et al., Reference Yavari, Le Moulec, Inoue, Nishiyama, Lupu, Matsubara, Botta, Vaughan, Di Michiel and Kvick2005; Bernasconi et al., Reference Bernasconi, Dapiaggi, Bowron, Ceola and Maurina2016); therefore, our goal is removing the effect of progressive additions of crystalline SiO2 quartz to the original glass sample. This approach can be of interest in the field of ceramic materials when dealing with glazes whose microstructure is dominated (more than 90 wt. %) by a glassy matrix (Bernasconi et al., Reference Bernasconi, Diella, Marinoni, Pavese and Francescon2012), or to follow glass structure evolution in the presence of recrystallization.
II. EXPERIMENTAL SECTION
Experiments have been performed at the material science beamline ID11 (ESRF, Grenoble). The diffraction data have been acquired in transmission geometry, with a fast readout fiber optic coupled CCD detector (Frelon camera, 2048 × 2048 pixels, 16 bits, Labiche et al., Reference Labiche, Mathon, Pascarelli, Newton, Ferre, Curfs, Vaughan, Homs and Carreira2007) with a pixel size of 50 µm. All the investigated samples were rotating during the data acquisition, with the exception of the glass with crystalline phase impurity. We have divided the rest of this section and the next section into three different parts: the instrument resolution, the nanocrystalline BaTiO3 powder, and the mixed crystalline/amorphous sample.
A. Instrumental resolution
We have verified the Q (reciprocal space) and r (real space) resolution by measuring silicon NIST SRM640c powder using different capillary diameters and sample to detector distances. Two common diameters of Kapton capillaries have been selected (0.5 and 1.5 mm) and three sample to detector distances have been chosen (9, 12, and 22 cm). The X-ray beamsize was around 75 µm square and defined by slits. For each configuration, ten images with an exposure time of 3 s have been collected and then averaged to improve the counting statistics. Ten empty capillary images were collected as well, to be able to subtract the background and eventually generate a PDF (described below). For configurations with the shorter detector distances of 9 and 12 cm, the detector was offset to maximize the Q-range by putting the beam center in the corner of the image, and keeping the edge of the camera vertical. This has the drawback that only quarter of rings have been measured instead of full rings, but it gives a factor of almost 2 increase of the Q-range and the azimuth range still averages over horizontal and vertical beam polarizations. This experiment was performed at the Hf K-edge energy (65.352 keV).
B. One hundred nanometer grain-sized BaTiO3
A nanocrystalline material from Aldrich (CAS 12047-27-7) whose particle size is around 100 nm from Brunauer-Emmett-Teller (BET) analysis was loaded into a borosilicate glass capillary (diameter of 0.5 mm). Data were collected with wavelength λ = 0.12465 Å (E~100 keV) with a sample detector distance of around 15 cm for reciprocal space analysis to give Q MAX around 8 Å−1. Further data with a sample detector distance of 8.7 cm were collected for real space analysis (Q MAX around 31 Å−1) at the Pt K-edge energy (78.395 keV). Five hundred images with an exposure time of 5 s/image have been collected to provide extremely good counting statistics for the PDF analysis. The same data collection strategy was adopted to measure the empty capillary.
C. Glass with crystalline phase impurity
Small quartz additions have been performed on a complex aluminosilicate-based glass (composition in weight % is 69.69 SiO2, 13.54 Al2O3, 7.06 CaO, and 11.71 Na2O) obtained by quenching from 1200 °C. Full compositions are given in Table I. After a preliminary homogenization of the amorphous and crystalline phases by gently grinding in an agate mortar, each composition was loaded into a Kapton capillary (diameter of 1.5 mm) and then measured at the beamline at an energy of 60 keV (λ equals to 0.20535 Å). The larger capillary diameter of 1.5 mm (c.f. 0.5 mm for BaTiO3) was selected to enhance the weak signal at the expense of some Q resolution; a high Q resolution is not as important when studying glassy samples. A sample detector distance of 8.5 cm has been chosen to correspond to a Q MAX of about 23.5 Å−1. For each composition, five different positions along the capillary have been measured to increase sampling statistics (for the glass) and to improve the effectiveness of our further data processing by median filtering of the five different regions per sample. Then 500 images of 0.5 s were collected for each region. The short exposure time was selected to avoid saturation of the detector because of strong Bragg's single-crystal diffraction of quartz, while the large number of images overcomes statistical noise. For the empty capillary, 500 images of 0.5 s of exposure time were collected as well.
Table I. Quartz wt. % addition to the glass phase.
Experimental packing fractions are also provided as well as sample density, considering a density of 2.469 and 2.65 g cm−3 for glass and quartz, respectively.
D. Data processing
The momentum transfer Q can be computed from Bragg's law as:
$${Q = \displaystyle{{{\rm 4}\pi} \over \lambda} {\rm sin}\left( {\displaystyle{{{\rm arctan}(R/D)} \over {\rm 2}}} \right)}$$
where λ is the incident wavelength, R is the active radius of the 2D detector, and D is the sample to detector distance. For these measurements, the incident beam was focused by Al and Be lenses in an in-vacuum transfocator to increase the flux density at the sample position (Vaughan et al., Reference Vaughan, Wright, Bythkov, Rossat, Gleyzolle, Snigireva and Snigirev2011). The images have been processed using the pyFAI software (Kieffer and Wright, Reference Kieffer and Wright2013) to get the 2θ-intensity profile. An incident angle correction (Wu et al., Reference Wu, Rodrigues and Coppens2002; Bernasconi et al., Reference Bernasconi, Wright and Harker2015) has been applied using previously measured X-ray phosphor transmission values (in the range 0.6–0.8 at these energies). Before transformation to real space, the data also need to be corrected for contributions from the empty capillary, Compton scattering, and self-absorption, in order to obtain the coherent intensity I C. This quantity is normalized to the average X-ray atomic form factor of the sample in order to get the total scattering function S(Q) using the equation:
$${S(Q) = \displaystyle{{I_{\rm C} -\langle\, f(Q^{\rm 2} )\rangle + \langle\, f(Q)\rangle ^{\rm 2}} \over {\langle\, f(Q)\rangle ^{\rm 2}}}} $$
This function is Fourier transformed to give the reduced pair distribution function G(r) or the pair distribution function g(r) by the equation:
$${G(r) = {\rm 4}\pi \rho _0 r(g(r) - {\rm 1}) = \displaystyle{{\rm 2} \over \pi} \int\limits_{\rm 0}^\infty Q \left[ {S(Q) - {\rm 1}} \right]{\rm sine}(Qr)dQ}$$
where ρ 0 is the average number density of the material. PDFgetX3 (Juhas et al., Reference Juhas, Davis, Farrow and Billinge2013) and GudrunX (Soper, Reference Soper2010a,Reference Soperb) are software packages which can be used to carry out these steps.
In the cases of silicon NIST SRM640c and 100 nm grain-sized BaTiO3, the PDFgetX3 was used. This software works well for crystalline and even nanostructured materials. In the case of glass plus crystalline phase, GudrunX was used instead as it allowed us better control of the various parameters and what is signal vs. background.
III. RESULTS AND DISCUSSION
A. Q-space resolution effects
We look at the effect of some parameters on the Q-space resolution of a 2D detector. Consider a sample with thickness t, irradiated by a parallel beam, whose size is b, located at a distance D from the area detector, whose PSF is ps. The wavelength λ is not unique because of the bandpass bp of the monochromator. Figure 1 represents the geometry of the diffraction experiment in the absence of detector tilts, where also the footprint F of the beam diffracted by the sample, at a given 2θ angle, is shown.
Figure 1. Geometry of the diffraction experiment in transmission with a 2D detector.
To compute the Q-space resolution, the quantity between the Q values at the end of the footprint, for a given 2θ angle, needs to be considered. This quantity, which we will call as deltaQ, corresponds to the difference between the two lengths R1 and R2. R1 is the distance between the beam center on the detector and the inner point of the diffracted beam on the detector, which is related to the inner front of the sample, corresponding to:
$${R{\rm 1} = {\rm tan(2}\theta {\rm )}\left( {D - \displaystyle{t \over {\rm 2}}} \right) - \displaystyle{b \over {\rm 2}} - \displaystyle{{\,ps} \over {\rm 2}}}$$
R2 is the distance between the beam center on the detector and the outer point of the diffracted beam on the detector, which is related to the outer back of the sample, corresponding to:
$${R{\rm 2} = {\rm tan(2}\theta {\rm )}\left( {D + \displaystyle{t \over {\rm 2}}} \right) + \displaystyle{b \over {\rm 2}} + \displaystyle{{\,ps} \over {\rm 2}}}$$
When moving these quantities in Q-space from 2θ space by substitution with Eq.(1), one should consider the bandpass effect which implies a reduction of R1 because of the smaller wavelength tail of the beam (i.e. λ LOW) and a raise of R2 because of the higher wavelength tail of the beam (i.e λ HIGH). So deltaQ corresponds to:
$$\eqalign{&{\rm delta}Q = \cr & \quad \displaystyle{{4\pi} \over {\lambda_{\rm low}}} {\rm sin}\left({\displaystyle{{{\rm arctan}\left({(R2/D)}\right)} \over {2}}}\right) - \displaystyle{{4\pi} \over {\lambda_{\rm high}}} {\rm sin}\left({\displaystyle{{{\rm arctan}\left({(R1/D)}\right)} \over {2}}} \right)}$$
The quantities λ LOW and λ HIGH correspond to (λ–bp λ) and (λ + bp λ), respectively. Figure 2 displays the behavior of deltaQ by varying its controlling parameters t, b, ps, bp, λ, and D, starting from an initial configuration where these values correspond to 0.5 mm, 75 µm, four pixels (assuming a pixel size of 50 µm), 2 × 10−3, 0.1899 Å and 9 cm, respectively.
Figure 2. (Colour online) Computed effects of different parameters on deltaQ as defined from Eq. (6). Initial configuration (i.e. blue curve) corresponds to a sample thickness of 0.5 mm, a beamsize of 75 μm, a point spread of 4 pixels (assuming a pixel size of 50 μm, a bandpass of 2*10−3, a wavelength of 0.1899 Å, and a sample to detector distance of 9 cm. Variations from this initial configuration are described in the different legends.
The Q-resolution is clearly improved by:
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(i) a thinner sample (blue vs. green curves);
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(ii) a smaller beamsize (blue vs. gray curves);
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(iii) a narrower PSF (blue vs. red curves);
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(iv) a lower bandpass (blue vs. black curves);
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(v) a larger sample to detector distance giving a smaller Q MAX (blue vs. orange curves);
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(vi) a longer wavelength giving a smaller Q MAX (blue vs. cyan curves).
Points i–v all reduce the signal. The effect of wavelength is less obvious as it depends on the incident spectrum, sample absorption and required Q MAX and the intrinsic sample cross-sections for Compton vs. Thompson scattering.
B. Experimental resolution
A total of six configurations have been explored by changing capillary size and sample to detector distance. To compare their different Q-resolution, the full width at half maximum (FWHM) of each peak has been extracted by using a Pearson VII function (Hall et al., Reference Hall, Veeraraghavan, Rubin and Winchell1977). The true peak shape depends on the detector PSF (Holton et al., Reference Holton, Nielsen and Frankel2012) and beam shape and this goes beyond the scope of the current work where we focus on the FWHM resolution.
By considering these experimental results, it is possible to check the previous calculations from Eq. (6). The measured FWHM was only one choice of a peak width parameter, other measures like integral breadth (etc.) are also possible and a more detailed ray-tracing computation might reveal the precise peak shape. For simplicity, we compared the FWHM from the data as half of deltaQ (e.g. we approximate the peak as a triangle where deltaQ is the length of the base). Figure 3 shows the FWHMs of the six investigated configurations vs. Q, and deltaQ/2 from Eq. (6). A good general agreement was found. In the calculations, beamsize, PSF, bandpass, and wavelength were 75 µm, 4 pixels (assuming a pixel size of 50 µm), 2 × 10−3, and 0.1899 Å.
Figure 3. (Colour online) Measured FWHMs of the six investigated configurations versus Q (symbols) and deltaQ/2 from Eq. (6) (lines).
It is not intuitive to evaluate these effects in real space. We have excluded the largest detector distance (i.e. 22 cm) because Q MAX would be too low for a meaningful real space analysis (i.e. 7.5 Å−1). For the other data, we have compared the G(r) real space resolution and the damping at high r values, which depends on the exponential of δQ 2 r 2/2 (Proffen et al., Reference Proffen, Page, McLain, Clausen, Darling, TenCate, Lee and Ustundag2005). In this case of silicon NIST SRM 640c, the best real space resolution is provided by the data which have the largest Q-range (about 24 Å−1) as seen in Figure 4 (left side).
Figure 4. (Colour online) Silicon NIST SRM 640c G(r)s for different Qrange and capillary size. On the left the real space resolution is displayed. On the right side the real space dampening is displayed. For blue curve the QMAX is about 24 Å−1 while for green curves QMAX is about 20 Å−1.
The resolution improvement (blue curves) allows weaker peaks at 10.42, 11.76, and 12.15 Å to be distinguished from the strong neighbors at 10.16, 11.52, and 12.37 Å, respectively. These features are not clear in the data collected with Q MAX of 20 Å−1 in the green curves. The damping of G(r) is reduced in the data with lower Q-range and smaller capillary size (see Figure 4, right side, green curve with triangles), because of the improved Q-resolution. The data with the largest Q-range have the most damped signal (blue curve with circles). We have evaluated G(r) features using the PDFGUI program (Farrow et al., Reference Farrow, Juhas, Liu, Bryndin, Bozin, Bloch, Proffen and Billinge2007) in terms of Q DAMP and Q BROAD parameters in the 1–40 Å real space range. This range has been selected as region where G(r) clearly shows features for all four configurations. In the refinement strategy, the silicon isotropic thermal displacement parameter (i.e. U ISO in PDFGUI) has been kept fixed at 0.0056 Å2 (from a previous Rietveld determination using data collected at the high-resolution beamline, ID22, ESRF). Correlated atomic motion effects have been also considered in the modeling (Jeong et al., Reference Jeong, Heffner, Graf and Billinge2003). The results of these fits are summarized in Figure 5, where a very good fit is displayed for all four configurations.
Figure 5. (Colour online) PDFGUI silicon NIST SRM 640c G(r) graphic fit for all the different investigated sample to detector distances and capillary size conditions. QDAMP and QBROAD are also displayed.
Although the damping is well modeled by the fitting routine, we have quantified how it affects the accuracy of the results. To compare the four configurations, silicon coordination numbers have been extracted by integrating the radial distribution function R(r) in a range of interest:
$${R(r) = G(r)r + {\rm 4}\pi r^2 \rho _{\rm 0}} $$
Coordination numbers have been calculated by normalizing the integral of the second peak in G(r) to be 12 for all four experimental conditions. The value 12 is because of the silicon structure (space group Fd-3m), and all the other integrals have been scaled in the same way. The second peak is preferred to the first because there are fewer spurious ripples which may bias the values. The results are displayed in Figure 6.
Figure 6. (Colour online) Experimental coordination numbers of silicon NIST SRM 640c obtained by fitting the radial distribution function R(r).
A general good agreement has been found, with the better accuracy provided by the data collected with the larger capillary size and with the higher Q-range (blue circles in Figure 6) in the 1–7 Å range (average error is 7.49%). At short radius values, the real space resolution (larger Q-range) and higher counting statistics (larger sample) are more important. However, if the 7–11 Å range is considered, better accuracy (average error is 9.85%) is provided by the data with a smaller capillary size and a higher Q-range (blue triangles in Figure 6). Increasing the Q-range has helped in both cases, but a larger sample degrades the results at higher r values because of increased damping. Full results are summarized in Table II. Our coordination analysis does not go beyond the values of 11 Å because of peak of overlaps. We conclude that the results are useful provided that (i) the statistics are good enough to avoid spurious ripples for first coordination shells and (ii) the signal is not damped for further shells. Moreover, accuracy is improved by increasing the Q-range.
Table II. Average percentage error in silicon coordinations as determined by radial distribution function analysis.
C. One hundred nanometer grain-sized BaTiO3
We started our analysis with Rietveld refinement by using the Topas program (Coelho, Reference Coelho2005). Tetragonal (space group P4mm) and cubic (space group Pm-3m) crystallographic modifications have been used to model the data. Because oxygen has only eight electrons compared with 56 for Barium, the X-ray data are not sensitive to oxygen; therefore, we have kept the oxygen coordinates fixed and we have refined only cell parameters, Ti-off centering (when present) and isotropic thermal displacement parameters. The instrumental contribution was previously evaluated from Silicon NIST SRM 640c. A much better fit was found for the non-centro symmetric (P4mm) model compared with the centro-symmetric (Pm-3m), which confirms that the average structure maintains the polarization (as expected). Moreover, a small amount of a BaCO3 phase impurity was observed and quantified (around 3 wt. %). The graphical fit is displayed in Figure 7, while full results are displayed in Table III.
Figure 7. (Colour online) Graphical fit of BaTiO3 Rietveld refinement. HKLs are provided for BaTiO3 (in green) and for BaCO3 (in grey).
Table III. BaTiO3 results from Rietveld refinement of the tetragonal and cubic crystallographic models at room temperature.
Different from previous BaTiO3 room temperature investigations (Petkov et al., Reference Petkov, Gateshki, Niederberger and Ren2006; Rabuffetti and Brutchey, Reference Rabuffetti and Brutchey2012), our analysis has not provided negative values for the mean-square atomic displacement of oxygen and titanium: this can be attributed to the particle size of our sample (around 100 nm), that does not excessively suppress the Bragg peaks, in contrast with Petkov et al., and with Rabuffetti and Brutchey, where the particle size was at least one order of magnitude smaller. However, the nanometric size motivated us to look for possible local variations in the sample and we therefore moved to pair distribution analysis.
Figure 8 displays the room temperature data of the BaTiO3. In Q space, the importance of high statistics (500 images, red curve) compared with single image (blue curve) is shown on the top right side. In real space, spurious ripples are greatly reduced and in spite of the instrumental damping of the signal, the G(r) has signal up to around 55–60 Å and clear features in the low r region as displayed on the bottom left side.
Figure 8. (Colour online) PDF data of the investigated BaTiO3. Top left: raw data (sample in blue, empty capillary in green). Top right: reduced total scattering function Q.F(Q) for 1 single vs 500 images with a magnified inset for the high Q region. Bottom left: G(r) in the 0–80 Å range. Bottom right: G(r) is plotted in the local 1–7.2 Å range for 1 single vs 500 images.
An initial fit of the G(r) has been performed in the 1.5–45 Å range using the PDFgui program with the tetragonal and the cubic models. We have not considered larger r values because of the damping of the signal. Scale factor, cell parameters, Ti-off centering (when present), and isotropic thermal displacement parameters have been refined. Q DAMP and Q BROAD parameters have been fixed from a silicon NIST SMR 640c collected at the same conditions. As expected, the tetragonal model better describes the structure and gives an R WP values of 17.22% compared with 20.53% for the cubic model. However, the fit is not satisfactory, and this suggests that a unique structure is not able to describe our sample in the 1.5–45 Å range. Down to around 15 Å, the tetragonal model works very well and yields good fit (R WP was 13.19%). Full results of PDFGUI fits in the 15–45 Å range are summarized in Table IV.
Table IV. BaTiO3 pair distribution function results in the average region (15–45 Å range).
At lower r values, the fit becomes worse, which is an evidence of local structural variations. We focus on the 1.5–15 Å range to highlight differences between the local and average structure. If the tetragonal crystallographic modification is refined in the 1.5–15 Å range, the local model has larger tetragonality (c/a axis ratio) and smaller Ti displacements. This is counterintuitive because it is hard to understand how a lattice is more metrically elongated with a reduced local displacement of the titanium atom. Similar findings were shown by previous authors (Smith et al., Reference Smith, Page, Siegrist, Redmond, Walter, Seshadri, Brus and Steigenwald2008; Rabuffetti and Brutchey, Reference Rabuffetti and Brutchey2012) which concluded that the local crystal structure is less sensitive to disorder than the average structure, leading to abnormally high values of c/a in the low range. Unless there is some difference in the samples, these controversial results indicate that the tetragonal structure is not right for the local structure here. The Ti thermal isotropic displacement parameter is 65% larger at low r than in the 15–45 Å range, which suggests Ti-off centering is not only in the c direction but actually off-axis. If the symmetry is reduced to an orthorhombic modification, we have observed that (i) the Ti thermal isotropic displacement parameter is halved and (ii) the fit is improved. Fit results for 1.5–15 Å range are summarized in Table V. This improvement no longer manifests in the average range (i.e. 15–45 Å), as a confirmation that the average structure is well described by the tetragonal modification.
Table V. Pair distribution function results in the local region (1.5–15 Å range).
Considering these results, our BaTiO3 sample displays an average tetragonal structure, which has a local orthorhombic-like distortion on length scales of about four unit cells. Further lowering of the local symmetry might improve the fit in the 1.5–15 Å range. Moreover, the use of G(r) obtained by averaging 500 images instead of using one single image improved the PDFGUI fit, lowering the R W from 22 to 18 wt. %. Further information, from neutron diffraction (for oxygen), or electron microscopy or from modeling approaches, might all be included in a future study.
D. Glass with crystalline phase impurity
For each of the nine samples including quartz, five different volumes in the sample have been irradiated by the beam. For each volume of each sample, 500 images were averaged to improve the statistics. At this level, for each sample, the median between the five different images were generated, in order to remove/reduce the spots generated by single-crystal diffraction of quartz grains. Finally, for each sample, the median image was azimuthally integrated using pyFAI. A threshold value to get a successful removal of quartz spots was found: when the quartz amount is lower/equals to about 2.5 wt. %, the median filtering works well, providing a 2θ vs. intensity pattern that is substantially identical to the pattern of pure glass. On contrary, this was not the case when quartz amounts were larger than 2.5 wt. %. This effect is displayed in Figure 9.
Figure 9. 2θ vs intensity pattern of glass samples with different quartz amounts. On the left side, as collected pattern are displayed while on the right side the median filtering has been applied to reduce Bragg signal from quartz.
For data with quartz amounts <2.5 wt. %, the data were processed with GudrunX, to get the F(Q) and the g(r) of the samples, and afterwards an empirical potential structure refinement (EPSR, Soper, Reference Soper2010a,Reference Soperb) was applied to evaluate whether the median filtering was successful even when extracting structural information from the diffraction pattern. The same potentials of Bernasconi et al. (Reference Bernasconi, Dapiaggi, Bowron, Ceola and Maurina2016) were used. Non-bridging oxygen (NBO) fractions, which represent the fraction of oxygen shared by one single structural unit [i.e. silicon and aluminum centered tetrahedra (Shelby, Reference Shelby2005)], have been used as a criterion to compare the samples. NBO fractions of 0.136, 0.133, and 0.126 have been found for samples GL0, GL1, and GL2, respectively. This trend towards lower NBO fractions can be attributed to some parasitic signals from the crystalline phase: the larger the signal, the more ordered and polymerized is the structure. These results underline that, for the present case of interest, the median filtering is not completely successful, but at the same time, this underlines the sensitivity of the data and the modeling approach to weak Bragg/diffuse scattering contributions. The effect of these Bragg/diffuse scattering contributions can be minimized by reducing the capillary size and or the beam size, with the drawback that statistics will be worse and, in turn, the EPSR reproducibility will be even worse, in a similar manner remarked by Bernasconi et al., Reference Bernasconi, Dapiaggi, Bowron, Ceola and Maurina2016. Making a step-wise rotation of the sample to change the pattern of single-crystal diffraction spots might also help, but nevertheless, the diffuse background from the crystalline material can never be removed. Increasing the quartz content of the sample does indeed reduce the overall NBO of the sample, in line with our observation.
IV. CONCLUSIONS
The effect on the reciprocal space resolution of different parameters like sample to detector distance, sample thickness, beam size, detector PSF, energy and bandpass (∆E/E) have been evaluated and computed after geometrical derivation by Eq. (6). The effect of two of these parameters (sample to detector distance and sample thickness) have been then experimentally measured at a high-energy beamline on a silicon NIST SRM 640c and compared with calculations by Eq. (6), showing satisfactory results. The analysis has been extended to real space, where an increase of Q-range is the most desirable requirement after a previous evaluation of PDF damping, an effect that can be reduced by a narrower capillary. This is most important for crystalline samples which have structured G(r) at high r values. For amorphous glasses, the G(r) is anyway rather smooth at high r and so a larger diameter capillary helps to improve counting statistics.
Applications in materials science are illustrated by nanosized BaTiO3 (particle size around 100 nm) and an aluminosilicate-based glass sample with different crystalline phase additions.
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(i) For the BaTiO3, the combination of reciprocal and real space analyses assigned an average tetragonal structure up to around 15 Å, with a lower local symmetry (orthorhombic at least) in the local r-range region. This kind of analysis of local vs. long range structures helps us to understand material properties that conventional structural approaches do not reveal.
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(ii) For the glass, it has been possible to remove the contribution of a crystalline phase Bragg contribute, with an underestimation in the NBO fractions of about 0.003, when the addition is around 1.2 wt. %, and of about 0.01, when the addition is around 2.5 wt. %. With this case, besides demonstrating how high-energy diffraction measures the local structure of a glass, we have shown a method to reduce the effects of an undesired crystalline phase.
Acknowledgements
The authors thank the ESRF for the provision of synchrotron beamtime. The authors also thank the anonymous referees for their constructive suggestions.
