Advantages of a curved image plate for rapid laboratory-based x-ray total scattering measurements: Application to pair distribution function analysis

Interpretation of the pair distribution function or PDF is an approach for interrogating the structure of crystalline and amorphous materials including, but not limited to, Li-ion battery electrodes,¹ hydrogen storage materials,² and catalysts.³ It utilizes the Fourier transform of the total scattering structure factor, S(Q), that is accessible through a simple scattering experiment and, in its most powerful form, compares this to the same function calculated from a refined model. Unlike traditional crystallography, there is no requirement for long-range order and this opens up myriad applications, introduction to which can be found in the very readable review articles of Billinge and Kanatzidis⁴ and Mancini and Malavasi.⁵ 

Over the last 10–15 years, renewed interest⁶ has been given to collecting total scattering data for PDF analysis using laboratory-based instruments, and most manufacturers now openly market instrumentation configured for such measurements. In fact, it is reasonably simple to adapt any diffractometer for collecting PDF quality data as long as the requirements for accessing high Q data can be achieved through a combination of wavelength, instrument geometry, and appropriate counting times. Instruments originally designed for either powder or single crystal diffraction can be adapted in this way, and examples in the literature include a PANalytical X’Pert Pro^7,8 and PANalytical Empyrean,⁹ Bruker D8 Advance,¹⁰ and, most recently, a STOE STADI P diffractometer.¹¹ The quality of the data is most noticeable in the signal/noise ratio (counting statistics), and this is most strongly affected by the type of detector and required scans times. Most laboratory-based equipment based on a powder diffractometer will have some form of 1D detector that speeds up the collection time compared to point counters, but collecting high quality data is still a time-consuming business. A recent paper by Thomae et al. describes an instrument that combines 4 such 1D detectors and a novel scanning mode that reduces the time to around 6 h.¹¹ 

Herein, we describe the first use of a single, large area, curved image plate system based on a Rigaku Spider (currently marketed as RAPID II) single crystal diffractometer fitted with an Ag tube and a graphite monochromator. Eliminating the need to scan the detector reduces the typical counting time to about 1–2 h. This gain in efficiency throughout the data collection process (sample, empty holder, and empty instrument) enables routine data collection of several samples per day while giving data quality congruent with full model fitting and refinement.

Although extended treatments of total scattering and PDF theory are available in the literature, for example, Underneath the Bragg Peaks by Egami and Billinge,¹² it is pertinent here to include a brief overview so the reader can understand the importance of rigorous data collection and processing strategies. The x-ray PDF is obtained by the Fourier transform of the total scattering x-ray structure factor, S(Q), where Q is as large as possible. We use the historical definition of structure factors for S(Q) and F(Q) for consistency with historical work in the field of amorphous scattering.¹³ The exact value is dependent on the instrument geometry and, most significantly, the wavelength but typically ranges from Q_min = 0.2 Å⁻¹ to Q_max = 20–40 Å⁻¹ (including laboratory and synchrotron sources). For a high-quality PDF, the maximum range should be used, and one form of the function is given as follows:

where S(Q) is the total scattering structure factor obtained from the total scattering I(Q) after normalization for the scattering cross section of each of the contributing atoms and corrections including background scattering, Compton and multiple scattering, absorption, and geometrical considerations. It should be noted that the literature contains various definitions and symbols to describe alternative correlation functions, and these are well explained by Keen.¹³ 

To obtain the most accurate PDF, careful correction of data must be implemented. This requires collection of the sample itself, the empty container (capillary), and the empty diffractometer to enable subtraction of all scattering not coming from the sample. This subtraction and corrections for other factors, such as Compton scattering mentioned above, can be implemented in a number of freely available software packages such as GudrunX.¹⁴ 

As previously discussed, PDF experiments are subject to specific data requirements that are rarely met by laboratory-based diffractometers in standard configurations. The main limitation is the restricted Q_max value that can be accessed, which needs to be as large as possible to enable a usable Fourier transform of the total scattering. An instrument fitted with a molybdenum tube will have a theoretical Q_max value of 17.68 Å⁻¹ (for sin θ = 1), which enables the generation of qualitative PDFs for some systems, but is still suboptimal. The most suitable tube commonly available is silver, which gives a potential Q_max value of 22.4 Å⁻¹ (for sin θ = 1). The actual value obtainable depends on physical limitations of the diffractometer geometry and scattering power of the sample.

Another major consideration is the detector; the accessible 2θ range will, in combination with the wavelength, dictate the Q_max, and considerations regarding the intrinsic noise and sensitivity will impact the acquisition time. The reduced flux of the shorter wavelength (silver radiation typically being approximately a third of the intensity of copper) is also a major factor to contend with. Therefore, a detector that is sensitive to lower intensity has a high efficiency at higher energy and exhibits a low noise level during measurements is crucial to a laboratory-based total scattering instrument.

The laboratory-based diffractometer described herein is a modified Rigaku Spider/Rapid II utilizing a point focus sealed silver tube (40 kV, 35 mA) in transmission geometry (Debye–Scherrer geometry) (Fig. 1). The beam is monochromated using a flat graphite (200) crystal, allowing Kα radiation to pass but suppressing K_β giving a wavelength of λ = 0.560 886 Å. The sample position is fixed at 127.4 mm from the detector, with the beam stop placed 13 mm behind the sample.

The beam width is set by the double pin-hole collimator and, where possible, matched to the inner capillary diameter; the distance from the end of the collimator to the sample is kept short (10 mm) to reduce air scatter between the collimator and the sample. The detector is a unique Fujifilm image plate with a 2θ range of −60° to 144°, totaling in a full usable area of 2θ = 204° (asymmetrically spanning 2θ = 0°). The sample is mounted on an Eulerian three-axis goniometer, with the sample set to oscillate in Φ 0°–85° (continuous rotation is not supported by the control software). Exposure times are sample specific but range from 30 min to 6 h. Samples are mounted in borosilicate glass capillaries ranging from 0.5 to 2.0 mm in diameter depending on the scattering power of the sample.

The 2D images are integrated to a 1D profile using Rigaku’s 2DP software.¹⁵ The maximum available Q_max for sin θ = 1 is 22.4 Å⁻¹, but a value of 20.68 Å⁻¹ is routinely used to allow full capture of the available range in χ (also denoted β) and provides a compromise between absolute Q and counting statistics. Thus, the area integrated typically spans 3.0°–134.8° in 2θ and 135.0°–225.0° in χ, as shown in Fig. 2. Note that the size of the detector allows for capture of higher 2θ angles (204.0°), but the direct beam is centered to allow ∼75° 2θ to be captured on the negative side—as required by the original single crystal use. For this application, it is unfortunate that the detector is fixed and cannot be moved to make more efficient use of its large area. The integration of the 2D area described above into a 1D profile is somewhat simplified by the curved nature (in one direction) of the image plate and the removal of the need to stitch overlapping images together. To avoid any scaling issues, only the positive 2θ side of the detector is used in conjunction with a constant χ range, and this has the disadvantage of wasting a significant amount of detector real estate. Standard Lorentz and polarization corrections are applied, the latter required due to the use of a monochromator.

Further data reduction and Fourier transform of the 1D profile are performed using GudrunX¹⁴ with appropriate instrument specific parameters such as beam profile and goniometer geometry (see Table S1 in the supplementary material). The use of a monochromator greatly reduces, and thus simplifies, the corrections for bremsstrahlung and K_β contamination. The instrument beam profile is such that K_α1 and K_α2 splitting can be observed only for cases of highly crystalline materials. GudrunX takes the intensity weighted average K_α wavelength, which results in an accurate Fourier transform to the PDF and subsequent model refinement. During this step, the data are corrected for absorption, multiple scattering, and non-sample scattering using measurements of the empty instrument (background) and empty capillary (container) (see Fig. S3 for the individual contributions). The calculated Compton scattering is also subtracted, and the data are scaled to optimally follow the total scattering cross section of the sample. During the Fourier transform, the Lorch function (which helps smooth termination ripples¹⁶) is set at 0.1 for crystalline materials and between 0.15 and 0.2 for amorphous and liquid samples. The data are output in a number of forms including the normalized total scattering structure factor S(Q), the reduced total scattering factor F(Q), and the pair distribution function D(r).

The experimental pair distribution functions for all standard samples are refined against a model using a small-box approach in TOPAS v6 academic.¹⁷ The instrument specific parameters used in TOPAS refinements are dQ (peak width), Lor (the Lorentzian contribution to the pseudo-Voigt peak shape), and α (the convolution term). Ideal values for these were obtained by a TOPAS Rietveld refinement of the silicon diffraction data with these parameters being extracted from the fitted peak shapes. For the PDF fits presented here, the following values were used: dQ = 0.072 19, Lor = 0.409 58, and α = 0.011 75. The full Rietveld fit for silicon can be found in Fig. S1 of the supplementary material.

To test the ability of the modified instrument setup and evaluate the implementation of data processing procedures, a number of standard materials were chosen for measurement. These were lanthanum hexaboride (Sigma-Aldrich), silicon 604f (NIST), 5 nm anatase from Get-Nano-Materials, and amorphous silica (Alfa Aesar). In addition, water was chosen as a more challenging sample with direct relevance to ongoing studies of aqueous metal salts.

First, we discuss LaB₆ that has an extremely strong x-ray scattering profile and is often used as an x-ray line profile standard. In the context of total scattering, it was desirable to have a material that scattered strongly to high angles to test the full angular range of the image plate and to evaluate the integration parameters used for extracting the required line profile from the 2D image to high Q values. The resulting PDF refinement is presented in Fig. 3 and clearly shows a well-fitting model, providing confidence in the parameters used to extract the PDF from the total scattering data.

While a strong scatterer highlights the performance of the instrument, it is not suitable for rigorously evaluating the data reduction strategy as the coherent scattering dominates the incoherent scattering and the quality of the extracted PDF is less sensitive to the adequacy of the various corrections. To address this, silicon was chosen as a sample with a lower scattering power and being more representative of real materials. It is also the most commonly supplied standard with new diffractometers and has potential as a useful comparison between different labs. The data and refinement results are presented in Fig. 4 and show excellent data quality and goodness of fit (Rwp = 0.12) out to values of r = 80 Å, where, as seen in the inset, there is no longer information observable in the experimental PDF.

This drop-off of the signal above 80 Å is indicative of some fundamental characteristics of the instrument as it is the Q-resolution (broadness of the Bragg peaks) and not the r-space resolution (size of Q_max) that causes this broadening. This is because broad Bragg peaks are linked to a drop-off in intensity at higher r as can be seen in the data for LaB₆ (Fig. 3) and contrasted with that of Thomae et al. (Fig. 2 of their work) where the higher resolution instrument [curved Ge(111) monochromator focusing the beam on the detector] produces less of a Q-dependent intensity drop-off.¹¹ To aid comparison of the Q resolution of our instrument to others in the literature, a PDFgui refinement of LaB₆ was performed, providing values for q_damp of 0.036 and q_broad of 0.011 (see Fig. S2 in the supplementary material for full refinement details) compared to 0.011 and 0.010 reported by Thomae et al.¹¹ These numbers are as expected and reflect the differences in Q resolution of these two instruments.

Nanomaterials are currently of great interest in numerous fields^18–20 and are ideally suited for analysis via the PDF approach.^21,22 As a further standard material, TiO₂ (anatase) nanoparticles were chosen as this would allow data quality comparison with other lab-based systems.¹¹ The measurement and refined fit are presented in Fig. 5 and show excellent data quality (Rwp = 0.11). The loss of long-range order is clearly visible in the experimental PDF (inset of Fig. 5) and can be interpreted with some assumptions, such as a monodispersive sample, as the size of the nanoparticles. Refined in the TOPAS as the spherical damp value, this gives a size of ∼46 Å (4.6 nm), which is in good agreement with that quoted by the supplier (5.0 nm) and also with that obtained from interpretation of the Bragg peak width using the Scherrer equation of 56(3) Å [LaB₆ and Si gave near identical values of 72.1(5) and 72.6(16) Å, respectively, and were used to correct the TiO₂ value for instrument broadening].²³ Although the TiO₂ nanoparticle data presented by Thomae et al. are not comparable, they are different samples and the refinement and data reduction strategies are different, which, nonetheless, provides a significantly lower Rwp.¹¹ It should be noted that the resolution of the instrument needs to be ruled out as the cause of the observed signal drop-off.

The application of total scattering to the study of dissolved species in liquid media is of particular interest to us, and herein, we present the data for water—a common solvent and a useful standard material for these more challenging systems. There is much written on the structure of water and much associated debate²⁴—we do not intend to add to this but merely show that data of refinable quality can be collected in the laboratory. Figure 6 shows the reduced total scattering factor for water collected for 180 min and the corresponding EPSR refinement.²⁵ Considering the short exposure time, these data compare favorably with other studies in the literature, for example, those contained in the review by Soper [Fig. 10(b)].²⁶ 

While we do not intend to provide a rigorous statistical analysis, the following basic observations can be made. Once the capillary size has been chosen, the main data collection variable is the scan or image exposure time. To evaluate the effect this has on the processed data, amorphous silica was chosen as a weakly scattering material in which noise was likely to dominate the scattering at high Q. This is most clearly seen in the total scattering structure factor S(Q) plots, and a series of measurements with increasing exposure times are shown in Fig. 7. The propagation of noise into the final PDF can be seen in Fig. 8 as high frequency ripples at longer r for the shorter data collection times. A perceptible improvement in the form of the very low r region can also be seen although this is somewhat dependent on various parameters adjusted during the data reduction as well as the value of Q_max chosen.

In this work, we have reported the successful modification of an image plate-based single crystal diffractometer to collect high quality total scattering data. Through the full PDF refinement of a number of standard materials, we have demonstrated the high quality of these data and the suitability of the data reduction protocols.

Although the required switch to shorter wavelength radiation reduces the intensity of the incident beam significantly (scales approximately as λ³), the large image plate detector (high quantum efficiency at hard x-ray energies²⁷) is the ideal solution for providing an excellent signal/noise ratio in comparatively short exposure times.

While the instrument and procedures described in this article produce excellent quality total scattering data, an enhancement can be imagined employing an Ag rotating anode generator and a narrow band monochromator such as Ge(111) with the same image-plate detector and shifted beam center, which would give a high Q resolution instrument with excellent counting statistics and large Q_max.

In conclusion, we have demonstrated that a large curved image plate, with virtually zero background response, is ideal for collecting total scattering data from samples including all but the most crystalline powders, amorphous materials, and liquids and whose scattering is dominated by the weak diffuse component.

See the supplementary material for additional material including figures and tables.

D.J.M.I. acknowledges the EPSRC for DTP funding (Grant No. EP/R513325/1).

The data that support the findings of this study are openly available in University of Southampton repository at https://doi.org/10.5258/SOTON/D1676.