Instrumental broadening and the radial pair distribution function with 2D detectors

The instrumental resolution function for pair distribution functions (PDFs) with 2D detectors is expressed and used to simulate the impact of various parameters on experimental PDF data sets.


Introduction
The instrument resolution function -a fundamental characteristic of any diffractometer -describes the peak profile measured and its angular dependence, due to the specific geometry and conditions of that particular diffractometer.It is routinely characterized with standard, highly crystalline samples and, in many cases, it can also be calculated theoretically (Caglioti et al., 1958;Gozzo et al., 2006) or modelled with ray-tracing simulations (Rebuffi et al., 2017).Any diffraction pattern is then a convolution of the scattering function from the sample and the instrumental resolution function for the diffractometer on which it was measured.The accurate decoupling of these contributions is important for structural and microstructural refinement from Rietveld and pair distribution function (PDF) analyses of powder diffraction data.
With the benefits of low noise, high dynamic range, high efficiency of detection and fast read-out, experimental setups based on large-area photon-counting detectors with thick Si or CdTe sensors are extensively used for total scattering measurements at synchrotron sources and are becoming more common in the home laboratories.The separation of sample and instrument contributions to the resulting data is straightforward in reciprocal space and the known resolution function can be used (Neder & Proffen, 2020).Analysis of interatomic distances in direct space, on the other hand, requires a Fourier transform of the diffraction intensity, resulting in the PDF.The Fourier transform of the diffraction pattern gives a product of the Fourier transforms of the resolution and scattering functions.The simplest possible instrumental resolution function is a single Gaussian broad-ening, which is trivial to model as its Fourier transform results in an envelope function that exponentially decays with interatomic distance.However, the situation becomes more complicated if we have to account for the Q dependence of the parameters of the resolution function, in particular for the change of full width at half-maximum (FWHM) as a function of the scattering vector.
A simple Gaussian Q-dependent resolution function has been proposed for which an analytical solution to the realspace data has been found.The FWHM of the diffraction peaks, where � 2 0 is the variance of the Q-independent broadening component summing up instrumental and sample-related contributions and � is a constant encompassing the instrumental parameters for a certain diffractometer (Qiu et al., 2004).This solution was first presented by Thorpe et al. (2002) and was recently revisited and derived for Lorentzian and pseudo-Voigt profiles (Beyer et al., 2022), but crucially the resolution function for a measurement with large-area detec-tors was recently reported (Chernyshov et al., 2021), and has a very different Q dependence to that in equation ( 1).
Here we consider the effect of the resolution function for such experimental setups on the PDF data.First, we reexpress the resolution function as a function of scattering vector and compare it with a standard diffraction pattern collected on a large-area detector.Further analysis is done within the theoretical scheme proposed by Thorpe et al. (2002).Second, we compare diffraction patterns measured with a powdered LaB 6 standard using different sets of instrument parameters.Finally, we evaluate corresponding PDF data and illustrate the effect of the resolution function numerically.

Theory
The instrumental broadening of diffraction lines for a large 2D detector oriented normal to the beam is derived by Chernyshov et al. (2021).The square of the instrumental FWHM as a function of diffraction angle, 2�, is given by Hereinafter we assume the angular uncertainty is the variance of a normal distribution.The coefficients in equation ( 2) are defined by the experimental parameters as follows: where D is the sample-to-detector distance, p is the pixel size (or point spread function, if any) of the detector, t is the thickness of the detector's sensitive layer, c is the size of the sample (e.g. the diameter of the capillary with a powder sample) and � denotes the angular divergence of the scattered beam.

The resolution function in Q space
Consider a Gaussian centred at Q 0 in Q space: where (�Q) is the variance.The angular uncertainty of the corresponding Bragg angle reads (Chernyshov et al., 2021) Given that Þ the variance of the angular resolution function becomes Alternatively, the variance as a function of Q can be written as the tenth-order even polynomial function: with the coefficients expressed via components of the resolution associated with sample size, beam divergence, sample-todetector distance and detector parameters -pixel size and thickness of the sensitive layer (see Appendix A).
The effect of some instrumental parameters on (�Q) 2 is shown in Fig. 1.It is clear that the default assumption from equation ( 1) does not hold for measurements with a large-area detector set normal to the incoming beam.
There is also a high-angle/high-Q limit of (�Q) 2 = 0 when the scattered X-ray is normal to the incoming beam and parallel to the detector surface.Therefore, Q max < ½2ð2Þ 1=2 ��=� and 2� < ð�=2Þ.

The resolution function in PDF r space
Following the considerations suggested by Billinge & Thorpe (2002), we can derive the effect of instrumental broadening on the PDF.Assuming the Gaussian approximation holds, the observed radial PDF, G c (r), can be expressed using the following convolution: where G(r 0 ) is the actual radial PDF and C(r, r 0 ) is a transmission function, i.e. the real-space instrumental contribution.C(r, r 0 ) can be expressed as the Fourier transform of the resolution function in Q space [equation (4)]: While there is no analytical solution, this integral can be evaluated numerically with the polynomial expression for variance [equation ( 10)] and the coefficients listed in Appendix A. Equation ( 12) can be written in the following form: The exponent in front of the integral is a damping function that suppresses the PDF at high r.Note that the damping parameter is a function of various experimental parameters: The integral expresses the position-dependent broadening of PDF features, while the damping parameter, � 2 0 ¼ a 0 , sets a limit on the size of the structural correlations or structural coherence, i.e. particle size, that can be probed for certain experimental conditions.Using an intensity threshold of I=I max > 0:018 [expðÀ r 2 max a 0 Þ ¼ expðÀ 4Þ] to denote the decay of signal to only noise defines the maximum interatomic distance that is measurable: For a detector with size, V, in any direction perpendicular to the incoming beam (the aperture of the detector), the detectable scattering vectors are �Q max , where Therefore, for a detector with fixed position and aperture, we can define the relation For a typical detector with vertical size 300 mm and pixel size 0.172 mm, positioned 150 mm from the sample, a wavelength of 0.3 A ˚would correspond to a Q max of 22 A ˚À 1 .
Given a capillary diameter and beam size of 0.3 mm and neglecting beam divergence, one gets r max = 83 A ˚; therefore, for the given experimental conditions, it is not possible to reliably determine particle sizes larger than 8 nm.Increasing the capillary and beam size to 0.7 mm decreases r max to 4 nm.Obviously, a non-zero beam divergence will further reduce r max , and an increase in accuracy (signal-noise ratio) will lead to higher r max .
The shape and width of PDF features are given by the integral in equation ( 13).It is easy to see that, in contrast to the resolution expected by default [equation ( 1)], the decrease in width of diffraction lines with Q does not result in a Gaussian as derived by Thorpe et al. (2002).The function 13) is now limited by Q max , and the corresponding Fourier transform may be approximated by a sinc(rQ max ) function, at least for small r 0 .We investigate the broadening by numerical calculation of the resolution function as a function of r for a set of r 0 values and find that it is rather insensitive to the actual shape of the resolution function.This conjecture is illustrated in Appendix B; it implies that there is effectively no instrumental position-dependent broadening in the PDF pattern.
We have therefore split instrumental broadening of Bragg lines into two components: a term that leads to a decay of the intensity of the PDF pattern with interatomic distance [equation ( 14)], and a Q-dependent part that results in a broadening of PDF peaks.The first component sets a limitation on the maximal size of coherent regions [equation ( 14)].The second component defines instrumental contribution in the width of PDF peaks, which is predominantly defined by 1=Q max .For the majority of synchrotron experiments, the detector and beam size are fixed while the sample-to-detector distance and X-ray wavelength may be varied; characteristic dependencies of r max and Q max are shown in Appendix C.
The above derivations assume a high degree of monochromatization of the incoming beam.However, it becomes common practice to increase the bandwidth �� to have higher  intensity, which adds the following term to equation ( 5) (Chernyshov et al., 2021): The corresponding contribution in the line broadening in Q space, as follows from equation ( 8), is written Therefore, the effect of increased bandwidth is an increase of the a 2 coefficient [see equation ( 10) and Appendix A], that affects the width of PDF features but does not reduce the maximal size of a structurally coherent region.Since the bandwidth contribution is always positive, in contrast to the other coefficients in the series equation ( 10), we can consider the final broadening as a convolution of a Gaussian [�ð��=�Þ] with a sinc function (Appendix B).Therefore, the effect of a broad bandwidth will dominate for measurements done up to very high Q.
We conclude the theoretical part with a comment on the geometry where the detector is tilted by an angle � to cover a larger angular range.The corresponding expression for the resolution as a function of diffraction angle is derived by Chernyshov et al. (2021): where x = cosð2�Þ or x = cosð2� À �Þ and M = ½4ð2 ln 2Þ 1=2 ð��=�Þ� 2 , and the last term takes energy bandwidth into account [equation ( 18 )].For such a geometry the resolution function in Q space can be evaluated numerically.

Experimental illustration of resolution effects
Powdered LaB 6 samples were measured at the BM31 beamline (Swiss-Norwegian Beamlines at the European Synchrotron Radiation Facility, Grenoble, France) using a MAR345 2D detector with a wavelength of 0.270793 A ˚, at a distance of 181 mm and a maximum Q of 23 A ˚À 1 .The sample was loaded in various sizes of glass capillary (0.1, 0.2, 0.3, 0.4, 0.5 and 0.7 mm in diameter).Corresponding backgrounds were measured using the same size capillaries and were subtracted from the monitor-normalized and integrated patterns.Calibration of the instrument and integration of images were done using pyFAI (Ashiotis et al., 2015).The FWHMs of individual Bragg lines as a function of Q are given in Appendix E and are rather scattered at high Q due to strong overlap of Bragg lines; however, the trend corresponds well to the expected instrumental resolution [equations ( 2), (10), see also Fig. 1].
PDFgetX3 was used to convert the 1D total scattering patterns to total scattering structure functions [S(Q) and reduced form, F(Q)] and PDF [G(r)] (Juha ´s et al., 2013).The PDFs are shown in Fig. 2 with different r ranges (normalized to the maximum intensity).With 0.7 mm capillaries the signal decays to 2% of the maximum at 120 A ˚{converting the pattern to an approximate g(r) function, G(r) = 4�� 0 r[g(r) À 1], the signal decays to 2% at 75 A ˚}.The lattice parameter, appropriate atomic positions, damping and atomic displacement parameters were refined against these PDFs using Diffpy-CMI (Juha ´s et al., 2015).The damping was refined using a combined Gaussian-Lorentzian profile as described by Beyer et al. (2022).The Lorentzian component for the 0.1 mm capillary refined to 0.007, and this fixed value was used for all the other refinements, while the Gaussian component was freely refined.The results of the refinements are shown in Appendix D, Figs.7-9, with the R p values shown in Table 1.
The refined Q damp and U atomic displacement parameters are shown in Fig. 3.The largest impact of capillary size is, predictably, on Q damp , while peak broadening is much less affected, as shown by the smaller effect on the atomic displacement parameters.Although the refined value for U 33 for boron appears to reduce with capillary size (0.025 to 0.022 going from the 0.1 to 0.7 mm capillaries), this trend is within the error of the refinement.Q damp as a function of capillary size was fitted with a quadratic function, the result of which was Q damp = 0.0133 + 0.0125c 2 , in agreement with equation ( 14).The small impact of capillary size on the broadening in total scattering patterns can also be seen visually in Fig. 2(a) by observing that in the range 0-20 A ˚in r, the PDF profiles appear to match almost exactly.

Conclusions
We considered the instrumental line broadening for the case when powder diffraction of monochromatic radiation detected by a large-area detector is used for analysis of the PDF.We have shown that for a detector set normal to the incident beam, the broadening of PDF peaks predominantly depends on the maximal Q value reached in the experiment and to a much lesser extent on the instrumental broadening of the diffraction lines.
We have also derived the instrumental damping of the PDF signal as a Q-independent contribution to instrumental broadening defined by sample/beam size, pixel size of the detector and beam divergence.The overall instrumental contribution to the width of Bragg lines leads to a decay of PDF intensity with interatomic radius: thus the broader the lines in the diffraction pattern, the smaller the particle sizes that can be analysed by the PDF method.The corresponding analytical expression is derived and given by equation ( 14), with a graphical illustration shown in Appendix C. The practical importance of the derived results may be better illustrated by the following set of experimental parameters: � = 0.2 A ˚, sample-to-detector distance is 300 mm, Pilatus 2M detector with pixel size 0.172 mm and beam/capillary size 0.3 mm.The setup offers Q max = 33 A ˚, which seems to be good for a generic PDF experiment; however the PDF intensity will be damped down at 5 nm by the instrumental resolution irrespective of the actual particle sizes.The diffraction experiment may therefore be planned, from the resolution point of view, as an optimizing trade-off between the damping of PDF intensity and the width of PDF peaks.The analytical expressions derived here are based on a series of approximations, such as a symmetric Gaussian line shape, for the sake of simplicity.Experimental data collected with a set of geometrical conditions confirm the expected effects of the instrumental broadening.Nevertheless, accurate numerical simulations with more realistic line shapes are necessary to improve the estimates presented here.

APPENDIX A
Figure 4 Comparison between the different resolution functions described in equations ( 1) and ( 10).Left-hand column: the instrumental resolution function (FWHM broadening as a function of q).Middle column: exponential function in equation ( 12).Right-hand column: Fourier transform of the exponential function.Calculations were performed using the following parameters: Q max = 20 A ˚À 1 , D = 100 mm, � = 0.25 A ˚, p = 0.172 mm, c = 0.3 mm and t = 0.1 mm.

Figure 5
Experimental instrumental resolution at D = 890 mm, Pilatus2M 1 mm CdTe detector.Circle markers show the FWHM of diffraction lines of LaB 6 , and the grey line shows the model calculation.Observed (black), calculated (red) and difference (green) plots for the fitting of the X-ray diffraction data from the 0.1 mm capillary.
was subtracted from the data prior to peak fitting.The FWHMs plotted against Q in Fig. 8 clearly resemble the calculated functions in Fig. 1 (capillary size).The data shown are filtered to show the points where the relative peak intensity is 5% or more above the background, as the low signal levels at high Q values led to a large number of outliers which obscured the overall trends.The final fits for the 0.1 mm capillary are shown in Fig. 9.The fitting method assumes the background between peaks to be linear, and despite the initial background correction this leads to some artefacts in the difference plot at low Q.The effect of these on the FWHM values from the Gaussian fits is negligible (see Figs. 10,11).

Figure 1
Figure 1 Calculated resolution functions for changing experimental parameters of a simulated diffraction experiment.(a) shows the effects on change in detector sensitive layer thickness (t) with a capillary size (c) of 0.1 mm and a sample-to-detector distance (D) of 100 mm.(b) varies c with t = 0.1 mm and D = 100 mm.(c) showcases D with c = 0.1 mm and t = 0.1 mm.Subplot (d) presents a calculated resolution function given a typical experimental setup with a 1.0 mm capillary situated 200 mm from a detector with a sensitive layer thickness of 1.0 mm.

Figure 3 (
Figure 3 (a) Q damp with a quadratic fit and (b) U atomic displacement parameters refined for the PDF patterns for each capillary size.

Figure 6
Figure 6Calculated instrumental resolution at D = 200 mm, Pilatus2M 1 mm CdTe detector for small and large capillary sizes.Inset: r max as a function of capillary size.

Figure 10 FWHM
Figure 10FWHM of LaB 6 peaks for capillary sizes 0.1 to 0.7 mm from Gaussian peak fitting, filtering out peaks with low intensity.Each data set was filtered to a linear regression to clarify the trends at high Q.

Table 1 R
p values for the PDF refinements for each capillary size.