I. INTRODUCTION
Scientific interest in performing total scattering experiments (Bragg and diffuse scattering) is increasing, owing to the ability of this technique to unravel the disordered structure in crystalline phases as well as in glasses and liquids. For this reason, scientists are continuously looking toward improvements in data quality, both at large-scale synchrotron facilities and from conventional laboratory instruments, and also on the data reduction, which is quite delicate for these kinds of experiments.
Regarding data quality, high scattering angles and/or high-energy radiation are required to explore a wide momentum transfer (Q) range (Q = 4πsinθ/λ). This needs to be combined with good statistics, especially at high Q where the diffuse scattering is dominant and the data are most sensitive to small atomic shifts. For proper background subtraction and normalization, the data need to be acquired with a stable incident beam (Billinge, Reference Billinge, Dinnebier and Billinge2009).
At synchrotron radiation facilities, one common way to satisfy these requirements is to perform the experiment in transmission geometry, with the data collected using a two-dimensional (2D) detector; this setup allows also fast data acquisition that is particularly suitable when time resolution is required (Ponchut, Reference Ponchut2006).
As for crystallographic experiments, pair distribution function (PDF) analysis requires the sample–detector parameter calibration, azimuthal integration, and incident angle corrections (Wu et al., Reference Wu, Rodrigues and Coppens2002), in order to extract the 1D scattering intensity I(Q) from the 2D image. Afterwards, to extract information from deviations in the periodic atomic arrangement, coherent intensity I C is extracted from the measured intensity I M, following equation (Juhas et al., Reference Juhas, Davis, Farrow and Billinge2013):
$${I_{\rm M}}\left(Q \right)=a\left(Q \right){I_{\rm C}}\left(Q \right)+b\lpar Q\rpar \comma \;$$
where a(Q) and b(Q) are multiplicative and additive corrections like self-absorption and X-ray polarization, and Compton and container scattering, respectively. The I C function, which includes both Bragg and diffuse scattering, is then normalized for the scattering factors (atomic form factors for X-rays which are Q-dependent; and scattering length b for neutrons which are independent of Q). This gives the total scattering function S(Q), as shown in the equation:
$$S\left(Q \right)=\displaystyle{{\left({{I_{\rm C}}\left(Q \right)- f\,{{\lpar Q\rpar }^2}+f\,{{\lpar Q\rpar }^2}} \right)} \over {\,f\,{{\lpar Q\rpar }^2}}}.$$
This S(Q) function is sine Fourier transformed to obtain G(r), defined as the pair distribution function, following the equation:
$$G\left(r \right)=\; \left({\displaystyle{2 \over \pi }} \right)\vint_{{Q_{{\rm min}}}}^{{Q_{{\rm max}}}} {Q\left[{S\left(Q \right)- 1} \right]\sin Qr\, dQ} .$$
This contains the real-space information which reveals the local structure of the investigated sample and also information about grain size from G(r) from amplitudes at high r, that are particularly important for nanomaterials (Masadeh et al., Reference Masadeh, Bozin, Farrow, Paglia, Juhas, Billinge, Karkamkar and Kanatzidis2007).
The present paper is aimed at evaluating some effects of the processed data range (e.g. Q MAX) and of the acquisition statistics (e.g. azimuthal integration range) on the G(r) of some glassy and crystalline materials.
The experiments have been performed at the Material Science ID11 beamline [European Synchrotron Radiation Facility (ESRF)] that is particularly suitable for total scattering experiments because of its high-energy source (18–140 keV), combined with a transmission geometry and a fast 2D charge coupled device (CCD) area detector (Frelon camera, Labiche et al., Reference Labiche, Mathon, Pascarelli, Newton, Ferre, Curfs, Vaughan, Homs and Carreira2007).
II. EXPERIMENTAL METHODS
A. Sample
Si NIST SRM640c was used to perform the initial calibration before a range of crystalline and glass materials were investigated. The crystalline samples were nanocrystalline CeO2 and an industrial grade ZnO, whose mineralogical purity have been checked by matching with CeO2 and ZnO d-spacings from Wyckoff (Reference Wyckoff1963) and Xu and Ching (Reference Xu and Ching1993), respectively.
The glassy materials were two aluminosilicate glasses with different amount of zinc which had been prepared by quenching from 1200 °C and ground in a zirconia mortar; their molar compositions are summarized in Table I.
Molar composition (expressed as oxide fractions) of the two investigated glass samples.
The powders were loaded in kapton capillaries with internal diameter of 1.5 mm.
B. Data collection
Measurements were performed in transmission geometry at the ID11 experimental hutch 1 with a 2D Frelon camera. Energy calibration was performed on Si NIST SRM640c material, following the method of Hong et al. (Reference Hong, Chen and Duffy2012) at seven different sample–detector distances.
All data were collected with the detector as close as possible to the sample (distance ~10 cm) and then moving it laterally and vertically with respect to the beam in order to have circular and quarter of circle sections of diffraction rings, as shown in Figure 1. Consequently, the Q MAX at the edge of the detector was 16 and 28 Å−1, respectively. The maximum achievable theta angle equals [arctan(D/R)]/2, where D is the sample–detector distance and R is the radius of the inscribed circle in the 2D-detector. However, if the radius of the circumscribed circle is considered, then a higher Q MAX can be obtained in both configurations (e.g. 22 and 36 Å−1, respectively), with the drawback that the azimuthal integration is limited only to the image diagonal. Thus, a larger range Q MAX is available from the image corners, but these data are not available at all azimuthal angles, which may eventually cause problems to properly account for factors such as polarization, etc.
(Colour online) Examples of 2D images collected with the beam in the CCD corner on sample A (left side) and with the beam in the CCD center on ZnO sample (right side).
To increase the statistics, 20 images per samples were collected, with an exposure time that was 5 and 20 s for crystalline and glass samples, respectively.
C. Data reduction
The 2D images need to be integrated to obtain intensity as function of 2θ, or Q, or d-spacing, which was done using the FIT2D program (Hammersley et al., Reference Hammersley, Svensson, Hanfland, Fitch and Hausermann1996). This reduction method needed (i) dark current images with the same exposure time to subtract from the raw image to remove the electronic signal not generated by X-ray radiation; (ii) flood image from the detector that describes the different response of each detector pixel to X-ray radiation; and (iii) the spatial distortion of the detector.
Afterwards, because of the high energy of our measurements, the absorption of the X-ray beam in the phosphor may be less than 100% (“thin phosphor regime”), with a consequent detector response that depends on the incident angle α and the effective thickness of the phosphor. For this reason, following Wu et al. (Reference Wu, Rodrigues and Coppens2002), observed intensity I OBS have been corrected using a phosphor contribution, expressed as (1 − T ⊥ phosphor)/{1-exp[ln(T ⊥ phosphor)/cos(α)]}, that considers α and phosphor transmission at a perpendicular incidence condition (T phosphor). For completeness, we also considered the absorption contribution that comes from the protective faceplate in front of the phosphor that plays an opposite role, attenuating more of the signal at high α than at low α. The contribution from this faceplate, can be expressed as T faceplate/{exp[ln(T faceplate/cos(α)]}, where T faceplate is the faceplate transmission at the perpendicular incidence condition. Therefore, the adopted equation to obtain the equivalent perpendicular-incidence intensity I from I OBS using the Frelon camera is:
$$\eqalign{I_{\bot}&= \displaystyle{I_{\rm OBS}\left(1 - T_{\bot}^{\rm phosphor} \right) \over \left(1 - e^{\left(\ln \left(T_{\bot}^{\rm phosphor} \right) {\big /}\cos \alpha \right)} \right)} \cr &\quad \times\displaystyle{T_{\bot}^{\rm faceplate} \over \left(e^{\left(\ln \left(T_{\bot}^{\rm faceplate} \right) {\big /} \cos \alpha \right)} \right)}.}$$
T ⊥ faceplateand T ⊥ phosphor were directly measured at the ID11 beamline at the energy of 78 keV. The T ⊥ faceplate term changes when an absorbing screen is deliberately placed in front of the detector in order to reduce the background because of X-ray florescence (lower-energy X-rays are preferentially absorbed).
Figure 2 is an example of sample A raw data without (red curve) and with (black curve) the application of Eq. (4). For this sample, at a Q-value of 25 Å−1, I OBS would be overestimated by up to 8.3% without the application of Eq. (4). Moreover, the effect of the two distinct correction terms is also plotted, showing that at the energy of 78 keV the contribution of the phosphor is dominant with respect to the contribution from the faceplate.
Finally, the data (with correction for incident angle) were processed with PDFgetX3 (Juhas et al., Reference Juhas, Davis, Farrow and Billinge2013), one of the most common total scattering pair distribution programs to obtain S(Q) and G(r), in agreement with Eqs (1)–(3).
III. RESULTS AND DISCUSSION
Energy calibration using Si NIST SRM640c as reference material, provided a wavelength coming from the double bent laue ID11 monochromator of 0.158 636(50) Å and a sample detector distance of 97.018 mm.
The resolution of the present experimental setup was estimated by looking at the evolution of Si peaks full-width at half-maximum (FWHM) as function of Q, as displayed in Figure 3.
(Colour online) Instrumental resolution obtained by Pseudo-Voigt fit (used in 0–8 Å−1 range) and Gaussian fit (used in 8–13 Å−1 range) of Si NIST SRM640c peaks at different Q-values (beam was in the CCD corner). For each selected peak, the R 2 of the Pseudo-Voigt/Gaussian fit to determine FWHM was always above the values of 0.99. Afterwards, 20 different points have been fitted with a 2nd degree polynomial function (red line).
In order to evaluate the role of Q range and statistics on the resultant G(r), different integration ranges were selected when using the FIT2D program, summarized in Table II.
Integration strategy adopted for evaluate Q MAX role (upper part) and statistic role (lower part) during processing with FIT2D.
When reducing the azimuthal range from 90° to 20° to increase the Q MAX, the polarization effect may be an issue that needs to be taken into account and corrected (Kahn and Fourme, Reference Kahn and Fourme1982); however, in the present work, if differently oriented slices of 20° are selected when doing the azimuthal integration, the resultant crystalline and glass G(r)s did not exhibit significant differences after processing with FIT2D and PDFgetX3.
Applying Eq. (4) influences the peak amplitudes in G(r), increasing peak heights as displayed in Figure 4.
(Colour online) Zincite G(r)s without (red line) and with (black line) the application of incident angle correction using Eq. (4). Curves have been obtained with a Q MAX of 28 Å−1 and an azimuthal range of 90°.
A. Crystalline samples
For the crystalline samples, both nanocrystalline CeO2 and industrial grade ZnO were used to evaluate the effect of Q range and the role of counting statistics. These data were processed with PDFgetX3, using an r poly values of 1.3 Å for CeO2 and 1.2 Å for ZnO. These values were selected from a graphical inspection of G(r).
In Figure 5, the S(Q) for CeO2 is shown (data collected with the beam center in the corner of the detector), along with a plot of the resultant G(r)s as function of different Q MAX values. The different real-space resolution (mainly function of π/Q MAX) is displayed with the higher Q MAX (i.e. 28 Å−1, black line) by sharper peaks and truncations that are more frequent but lower in amplitude. This effect has been determined for both CeO2 and ZnO, in terms of FWHM, by fitting with a Gaussian to the first metal–metal peaks in G(r), which are the peaks with the higher amplitude. The results are summarized in Table III.
(Colour online) Total scattering F(Q) function of nanocrystalline CeO2, using an azimuthal integration range of 90° and Q MAX of 28 Å−1, on the left side. Comparison of G(r) resulting from Fourier transform of the same F(Q) but changing the Q MAX from 28 Å−1 (black line), 24 Å−1 (red line), and 20 Å−1 (blue line).
Full-width half-maximum (FWHM) of the first metal–metal peak in G(r) and refined metal and oxygen U ISO parameter in nanocrystalline CeO2 and industrial grade ZnO, as function of Q MAX and azimuthal range. Moreover, the R W values from the PDFGUI fit are summarized. Fits have been performed in the 0.5–30 Å range.
When refining these different patterns with popular software such as PDFGUI (Farrow et al., Reference Farrow, Juhas, Liu, Bryndin, Bozin, Bloch, Proffen and Billinge2009), after previous tuning of resolution parameters (i.e. Q DAMP and Q BROAD) a good agreement between the refined metal and oxygen U ISO parameters has been observed, as a function of the different Qranges. However, when Q MAX of 16 Å−1 is used, oxygen U ISO was systematically and clearly bigger (as displayed in Table III).
The explanation for the UISO problem is the limited real-space resolution of the configuration with Q MAX of 16 Å−1, which drives the fit to a smearing of some peaks. As example, we refer to ceria G(r) where, in the 9–10 Å range, one can observe a Ce–Ce peak at about 9.35 Å and a Ce–O peak at about 9.65 Å with a Q MAX of at least 20 Å−1, whereas only the Ce–Ce peak is present if Q MAX of 16 Å−1 is used (see Figure 6).
G(r) magnification if Q MAX of 20 Å−1 (black line) and 16 Å−1 (dashed line) are used. Owing to the different real-space resolution, the former is able to show both Ce–Ce and Ce–O peaks, located at about 9.35 and 9.65 Å, respectively, whereas the latter is able to show only the Ce–Ce peak.
As far as the effect of statistics on the resultant G(r), FWHM of the peaks does not vary significantly if 90° or 20° of azimuthal integration range are used (Q MAX was fixed at 28 Å−1) and also the refined values of metal and oxygen U ISO parameters are almost unchanged, as showed in Table III.
Because of that, the idea of increased Q range at the expense statistics looks reasonable: in fact, this may allow a better real-space resolution (0.098 Å instead of 0.112 Å), but it may induce less well-resolved signal and noise in the S(Q). Therefore, PDFGUI Q DAMP parameter, which describes PDF Gaussian envelope because of the limited Q-resolution, has been refined on Si NIST SRM640c, giving values of 0.0593 (33) for a Q MAX of 32 Å−1 and azimuthal range of 20°, and 0.0599 (28) for a Q MAX of 28 Å−1 and azimuthal range of 90°. As for the non-systematic peak broadening induced by noise at high Q, the PDFGUI Q BROAD parameter resulted 0.0602 (59) in the case of Q MAX of 32 Å−1 and azimuthal range of 20°, and 0.0607 (51) for a Q MAX of 28 Å−1 and azimuthal range of 90°. Agreement is also present in terms of refined metal and oxygen U ISO parameter, for CeO2 and for ZnO, as summarized in Table III.
These considerations are important when dealing with some common problems in PDF data collection. For example, sample environments (e.g. systems for temperature control such as ovens, cryostream, heat blowers, etc.) can cause shadowing on regions of the detector for large scattering angles. Evidence here suggests that a reduced azimuthal range does not affect the quality of the G(r).
From these samples and this experimental setup it is shown that a better real-space resolution can be achieved by reducing azimuthal range to increase Q MAX, without affecting the statistics. Moreover, the configuration with the beam in the center of CCD is not recommended, because of its limited Q MAX.
B. Glass samples
The diffraction pattern of amorphous materials is completely dominated by diffuse scattering, making the role of counting statistics crucial to minimize truncations during Fourier transform. On the other hand, Q MAX is relevant when looking to resolve distances in complex compositions, as larger Q MAX improves the real-space resolution.
This is the case for the two investigated aluminosilicate glasses: they differ in zinc content (see Table I) and have been compared following the same criteria as for the crystalline samples, as expressed in Table II, using PDFgetX3 program to process the data.
For these kinds of glasses, the Zn–O distance should be located at about 1.95 Å (Cassingham et al., Reference Cassingham, Stennet, Bingham, Hyatt and Aquilanti2011). This was confirmed by some preliminary EXAFS measurements performed at BM23 beamline (ESRF).
As shown in Figure 7, Q range strongly affects Zn–O peak positions: there is a large error if Q MAX of 16 Å−1 is used (black line), while it is close to the expected values (i.e. 1.95 Å) in the other cases. This is because of the better real-space resolution provided by a higher Q MAX, especially when it is 28 Å−1 (blue line). A similar effect is also present for Si–O distance, which is commonly located at 1.6 Å for these glasses (Bowron et al., Reference Bowron, Soper, Jones, Ansell, Birch, Norris, Perrott, Riedel, Rhodes, Wakefield, Botti, Ricci, Grazzi and Zoppi2010) but is shifted to 1.65 if Q MAX of 16 Å−1 is used.
(Colour online) Samples A and B G(r)s as function of different adopted Q MAX, obtained with PDFGetX3 program. Black curve is obtained with Q MAX of 16 Å−1, putting the beam in the CCD center, while blue and red curves have been obtained by putting the beam in the CCD corner, using an azimuthal range of 90° and a Q MAX of 28 (blue line) and 20 Å−1 (red line).
Unlike for crystalline samples, the effect of statistics becomes more relevant in the case of glass samples. If Q MAX is fixed at 28 Å−1, S(Q) is more noisy at high Q if a reduced azimuthal range is used or if only one single image is considered (see Figure 8, left side). This noise can affect the resultant G(r)s in the long range, generating amplified oscillations (red and blue curves), whereas in the short range the three curves are superimposable.
(Colour online) On the left side, the different noise in high Q in S(Q) is displayed, whereas on the right, differences in the resultant G(r)s are shown in the long range. The black line refers to an azimuthal range of 90°, averaging 20 images the red line refers to a reduced azimuthal range of 20°, averaging 20 images; whereas the blue line refers to an azimuthal range of 90°, considering one single image.
In consideration of that, the idea to increase Q MAX (from 28 to 32 Å−1) by reducing the azimuthal range (from 90° to 20°) must be evaluated as a function of the investigated sample and eventual presence of sample environment. Reducing the azimuthal range allows Q MAX to be increased but also introduces noise and truncation at large distances in real space (r, as shown in Figure 8). In the present case, an increase of Q MAX from 28 to 32 Å−1 does not significantly improve the resolution of atomic distances at low r.
A careful S(Q) re-binning of the data might help to reduce some noise effects.
IV. CONCLUDING REMARKS
The effects of different Q ranges and statistics on the pair distribution function of different crystalline and glass samples have been investigated.
First of all, in order to obtain a PDF that is as accurate as possible, a careful preliminary data reduction is recommended, especially with the application of incident angle correction which may affect the raw intensity at high Q, where the diffuse scattering has a strong influence, and consequently the G(r).
At these experimental conditions, putting the beam in the corner of the CCD was crucial, leading to cut-off Q MAX value of at least 20 Å−1, making in turn the G(r) more accurate, both for crystalline and glass samples.
Counting statistics effect depend on the nature of the sample: for crystalline samples it does not lead to G(r) peak broadening, whereas in the case of glass samples noise is more evident on the long range, enhancing the amplitude of oscillations.
ACKNOWLEDGEMENTS
Authors thank Gavin Vaughan (European Synchrotron Radiation Facility) and Monica Dapiaggi (University of Milan) for helpful discussions, and Giovanni Agostini (European Synchrotron Radiation Facility) for Zn-Edge EXAFS measurements at the BM23 beamline, whose results will be more in details described in a future work.
