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Expanding Lorentz and spectrum corrections to large volumes of
for singlecrystal timeofflight neutron diffraction^{a}Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA, and ^{b}Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA
^{*}Correspondence email: choffmann@ornl.gov
Evidence is mounting that potentially exploitable properties of technologically and chemically interesting crystalline materials are often attributable to local structure effects, which can be observed as modulated diffuse scattering (mDS) next to Bragg diffraction (BD). BD forms a regular sparse grid of intense discrete points in
Traditionally, the intensity of each Bragg peak is extracted by integration of each individual reflection first, followed by application of the required corrections. In contrast, mDS is weak and covers expansive volumes of close to, or between, Bragg reflections. For a representative measurement of the diffuse scattering, multiple sample orientations are generally required, where many points in are measured multiple times and the resulting data are combined. The common postintegration data reduction method is not optimal with regard to counting statistics. A general and inclusive data processing method is needed. In this contribution, a comprehensive data analysis approach is introduced to correct and merge the full volume of scattering data in a single step, while correctly accounting for the of the individual measurements. Development of this new approach required the exploration of a data treatment and correction protocol that includes the entire collected volume, using neutron timeofflight or wavelengthresolved data collected at TOPAZ at the Spallation Neutron Source at Oak Ridge National Laboratory.Keywords: modulated diffuse scattering; local structure modeling; Lorentz and spectrum corrections; singlecrystal timeofflight neutron diffraction.
1. Introduction
Crystalline materials are primarily characterized by longrange order, which is reflected in the diffraction pattern as a series of delta functions, i.e. Bragg diffraction (BD). Local variations of structural subunits yield a modulated diffuse scattering (mDS) pattern. Recent examples show that a description of the local structure proved essential to interpret the structure–property relationships that underpin the material's functionality (Aebischer et al., 2006; Tu et al., 2013; Welberry & Goossens, 2008; Rao et al., 2013). The Bragg structure is represented by a smallest overall with translational symmetry defining the longrange order in Bragg data processing and structure are essentially routine and largely automated for monochromatic Xray diffraction. However, interpretation of the local structure from diffuse diffraction data (including measurement and appropriate application of corrections in order to prepare the data for integration) and computational modeling are much more complex (Weber & Bürgi, 2002; Bürgi et al., 2005; MichelsClark et al., 2013; Welberry & Goossens, 2008). The size of the distortion in real space, D, affects the size of the scattering distribution in , following (Nield & Keen, 2001). The inverse relationship implies that the scattered intensities resulting from small local disturbances in are distributed throughout and are generally orders of magnitude weaker in intensity than Bragg scattering. The diffuse scattering intensity distribution depends on the nature of the local structure and may be spread in one, two or three dimensions of (Welberry, 2004). Separating the diffuse intensities from the contributions of various independent sources of background is not trivial and needs to be addressed on a case by case basis (Bürgi et al., 2005).
The majority of diffuse scattering exploration has been done for data collected on monochromatic synchrotron or homelaboratory Xray sources. Common limitations, due to the accumulating nature of most twodimensional area Xray detectors, are data overflow, detector
and oversaturation of Bragg reflections. This necessitates separate individual data collection and data treatment of BD and mDS, and impairs consistent and simultaneous data processing and analysis. Preparation of mDS data for analysis in modeling software means that the collected raw data require complete reciprocal volume reconstructions from a series of singlewavelength wedges (= frames), which entails careful scaling schemes, removal of overexposed Bragg peaks and subtraction of background to extract representative diffuse intensities in one, two and threedimensional space as lines, layers and volumes, respectively. The dimensionality of diffuse data is specific to the experiment and sample. Generally, a similar treatment is applied to BD and mDS in neutron diffraction.Here we introduce a comprehensive protocol of combined BD and mDS data processing, equally treating the full scattering volume of
Data treatment and analysis are explored for timeofflight (TOF) Laue neutron scattering patterns collected on the SNS TOPAZ singlecrystal diffractometer. The instrument is equipped with stateoftheart area detectors, with continuous readout of individual neutron events. This circumvents limitations of data overflow and oversaturation of Bragg reflections, eliminating the necessity of a separate weighting scheme for the diffuse intensities. The total scattering (Bragg + Diffuse) data of singlecrystal samples are collected in volumes of Every neutron is counted as an event and saved. Since event data are not accumulated into predefined bins, histograms can be defined dynamically as part of the data processing protocol.The underlying premise of conventional data processing is based on the assumption that each Bragg peak is approximated as diffraction at one wavelength, resulting in discrete intensity points in
According to Bragg's law, this produces a sparse grid leaving most of to background. As this only yields a minute volume of interest for analysis, and computing resources traditionally have been limited, it is generally sufficient to integrate the raw Bragg intensities first, and then apply scattering symmetry, instrument and wavelength dependent corrections only on the integrated Bragg intensities. The Bragg peaks are saved individually. However, in the case of diffuse data, which are distributed over considerably larger volumes of a singlepoint correction is not applicable. Moreover, the sparse grid of BD needs to be transformed into a fine grid that covers the mDS volume, with adequate resolution to account for variations in scattering geometry and in wavelength, for the case of timeofflight Laue data. Both requirements (dense grid and corrections) can be met simultaneously by processing the volume of as a whole in momentum space. Because the Bragg and modulated diffuse scattering are system dependent, the appropriate grid size needs to be adjustable for every case. Therefore, event based collection, without predefined histograms, is the ideal data acquisition mode.In this work, data processing is examined theoretically, expanding the correction protocol and simultaneously taking the wavelength variations of TOF into account, to normalize a complete data set. Original and new data analysis procedures are compared for both Bragg and diffuse scattering for a single crystal of Er^{3+} doped βNaLaF_{4} (hexagonal phase), from a family of light emitting sodium lanthanide tetrafluorides. Information about the sample is presented in §2.1. The current data processing approach to Bragg peaks is shown in §2.2. The proposed revised protocol is described theoretically in §2.3, while an example of implementation is shown in §2.4. The results are discussed in §3.
2. Materials and methods
2.1. Example material description: technically exploitable βNaLaF_{4}
Several of the rare earth doped sodium lanthanide tetrafluoride compounds are efficient light upconversion phosphors. They have been studied for their ability to act as luminescence host matrices (Haase & Schafer, 2011), a function based on efficient luminescent sites, which are generally triggered by locally asymmetric environments. The average (Bragg) determined by Xray diffraction (Burns, 1965) showed high symmetry and revealed no obvious link to the material properties. However, spectroscopic investigations indicated that the average structure displays an idealized high symmetry for the La atoms located at the suspect center positions (Aebischer et al., 2006; Tu et al., 2013). Subsequent investigation of the local structure arrangement through analysis of the associated diffuse diffraction data supported the spectroscopic findings. This shows that complementary information provided by mDS is needed to allow insight on the true structure–property relationship (Auzel, 2004; Krämer et al., 2004; Aebischer et al., 2005, 2006; Tu et al., 2013), and understanding of the local structure is critical for targeted development and improvement of materials, in this case upconversion properties.
Neutron diffraction provides complementary data to Xray diffraction, owing to differences in scattering strength for the elements present. It is a mechanism to verify and enhance the description of the local structure derived from the diffuse Xray data. Aebischer et al. (2006) reported a qualitative estimate of a diffuse Xray pattern, which is dominated by La^{3+} (57 electrons) relative to F and Na (with only nine and 11 electrons, respectively). The combination of heavy and light elements present in βNaLaF_{4} makes neutron diffraction a particularly useful technique as Na and F have more favorable coherent scattering lengths of 3.63 fm (Na) and 5.654 fm (F) compared with 8.24 fm for La (Sears, 1992).
A highflux neutron beam is required for quantitative diffuse scattering studies. The TOPAZ singlecrystal diffractometer at the SNS was used for data collection on βNaLaF_{4}. Subsequently, the new processing protocol for neutron data was applied and a quantitative local structure model was simulated using ZODS, computational modeling software in development (Chodkiewicz et al., 2010). The methodology described may be applied in general to neutron diffraction data measured in TOF event mode.
2.2. Standard neutron scattering experiment
Niobium doped vanadium is a purely incoherent scatterer and therefore isotropic by definition, providing a mechanism to correct for incident r = 0.1375 cm) niobium doped vanadium sample. A background measured with an empty instrument was collected for a similar time, scaled to the same incident as the vanadium data and then subtracted. A spherical absorption correction was applied to the niobium doped vanadium data after background removal. The resulting data set was used for detector efficiency calculation and for calculating the wavelength dependent incident flux.
in momentum space. Diffraction data for the were collected for 48 h from a spherical (A single crystal of βNaLaF_{4} (approximately one cubic millimetre) was selected from a batch of crystals grown using the Bridgeman technique by collaborators at the University of Bern (Bürgi, 2011). The crystal was mounted on an aluminium pin using Super Glue (cyanoacrylate) and placed on the goniometer equipped with a 100 K nitrogen cryostream (Cryostream 700 Plus). Neutron event data were collected using a pulsed neutron beam with a wavelength band of 0.5–3.5 Å, at 60 Hz. The first Bragg reflections were integrated live using the EventViewer program in ISAW (Worlton et al., 2004) during data collection. The quality of the singlecrystal sample was determined by examining Bragg peaks in the live data, in a matter of minutes. A single crystal without splitting or multiple Bragg peaks was selected for the study.
Using the strong Bragg peaks, the ^{−1} deviation from individual units. As the is of hexagonal primitive () symmetry, no further cell transformation was necessary. The UB matrix was used to plan the experiment using the Crystal Plan software (Zikovsky et al., 2011), which maximizes the coverage of by optimizing the crystal orientations with a The data were collected using 11 goniometer settings for approximately 11 h each. A threedimensional volume of was collected within the wavelength band at each orientation. All orientations combined cover the volume of to be analyzed.
(Niggli cell) and orientation in the instrument, which together define the UB matrix, were determined using a fast Fourier transform (FFT) algorithm and leastsquares refined. This UB matrix was then used to index the found Bragg peaks within the specified tolerance of ±0.1 ÅNeutron events, which were detected in detector space (x, y and TOF), were mapped to threedimensional The integration algorithm first found strong Bragg peaks and indexed them using the UB matrix. The singlecrystal Bragg peaks were integrated by defining the shape of the integration domain in as a sphere around the center of the Bragg peak (Schultz et al., 2014). Integration was performed by summing all neutron events inside a chosen radius (in Å^{−1}) around each peak center point in Since the error associated with each event is assumed to be random and independent (Poisson distribution), the errors were summed in quadrature. If the chosen radius resulted in an integration volume that was either partially or entirely off the detector edge, the peak was discarded. The background was estimated by defining a second integration volume, a shell of a specified thickness around the peak, and subtracted.
Using the statistics versus integration radius utility written by Schultz et al. (2014) and the βNaLaF_{4} data, the best radius for integration in was between 0.13 and 0.16 Å^{−1}, based on the number of peaks integrated and their ratios. The integrated intensities were corrected for the wavelength dependence of the incident spectrum, detector efficiency, sample absorption and the Lorentz factor, as described by Schultz et al. (2014). The processed intensities (structure factors) of the indexed peaks were used for structure In §3, the values were used for comparison with the new general correction protocol, described in §2.3.
The diffuse scattering data were then treated using the standard procedure. The raw event data of each orientation were processed using the MANTID (Manipulation and Analysis Toolkit for Instrument Data) (Arnold et al., 2014) software. A geometric detector calibration was applied to correct detector positions relative to the sample and incident beam. The detector spatial distortion was taken into account through calibration at the detector firmware level (Riedel et al., 2015). The number of events collected per orientation varies according to the neutron and the exposure time. In order to combine the data from multiple orientations, the integrated raw events were normalized by the associated beam monitor counts after absorption, spectrum/detector dependent and Lorentz corrections had been applied. The result was a series of individual corrected and normalized data sets representing different volumetric regions of reciprocal space.
A (left). Since combining orientations of inherently low statistics (diffuse data) relative to Bragg diffraction is the goal, the nonadditive nature of the neutron differential must explicitly be taken into account. This is done by averaging the at every point. Adding the processed data sets and applying the arithmetic mean, or an unweighted mathematical average, by dividing the sum of data per bin (= grid point) by the number of data sets contributing per bin, results in more coverage but does not improve the data (see Fig. 1, right). Visually, the graininess of the figure does not decrease. Data sets obtained by applying symmetry operations can be treated the same way. Arithmetic averaging is sufficient when all regions of are measured with the same statistical weight.
reconstruction was attempted by combining the processed data sets. Simply adding three data sets increases the intensity in the overlapping regions, as can be seen in Fig. 1Combining 11 orientations of diffuse TOF diffraction data by arithmetic averaging is depicted in Fig. 1 (right). Welberry et al. (2005) show that the approach works well for good quality data. However, simple averaging of good quality and noisy measurements yields noisy data. The expectation from a physics point of view is that combining a statistically improved with a poor measurement should improve the overall data quality. If the data quality does not improve, regions where quality is poor are usually discarded. Multiple measurements with lower statistics can be useful for data analysis, if the data are treated properly (compare Fig. 1, right, and Fig. 2, bottom left). Here we present an approach to take into account the of the measured data at varying neutron flux.
2.3. New data processing approach for diffuse scattering
To correctly account for the ):
of the data, we begin with the definition of the differential scattering (Lovesey, 1984where N is the number of scattered neutrons per unit time in an infinitesimal volume () of around a momentum transfer , divided by the incident (Φ) and the solid angle of the detector (). When using multiple detectors, multiple experimental configurations (sample orientations measured for different times or different wavelengths) or polychromatic incident beams, the previous equation needs to be rewritten as
where the summation occurs over all detectors and configurations that contribute to the scattering in the i, with solid angle , counting N_{i} neutrons in such a region, the is only that part of the incident beam that can contribute to the scattering around . This is especially important for polychromatic beams.
volume element . For a given detectorWhile counting all neutrons that scatter in a certain region of the is applied to an incoherent scatterer, such as niobium doped vanadium. The differential scattering for this material is constant, since the scattering is isotropic:
is straightforward, the calculation of the that contributes to that particular scattering is not so obvious. However, as will be shown, this quantity can easily be measured. Equation (2)where is the total
If the measurement for the vanadium sample occurs in the same conditions (sample orientation and incident flux) as that for the sample of interest, thenwhere V_{i} are the neutron counts from vanadium. The sum yields
The N_{i} and V_{i} values should be corrected to account for sample and vanadium absorption.
For a single contribution to a particular point in et al. (1989) and used in the software for data processing at the SXD instrument at ISIS (Keen & Nield, 1996). In these previous papers, the authors divide the scattering intensity from the sample by the scattering intensity from the vanadium (N_{i}/V_{i} in our notation), then apply some corrections to account for inelasticity, multiple scattering etc. In the case of multiple contributions, or multiple symmetry operations (Welberry et al., 2005), their protocol uses a simple arithmetic mean of various N_{i}/V_{i} contributions. However, a weighted average should be used instead.
this is equivalent to the protocol described by HoweWe can rewrite equation (5) as
The weight in this case is the number of counts from the incoherent scatterer, in exactly the same region of the w_{i} = V_{i}. The simple arithmetic mean is a particular case where w_{i} = 1 or all w_{i} contributions are equal.
As stated earlier, the measurement of vanadium has to occur in the same conditions (sample orientation and incident flux) as the measurement of the sample of interest. This does not mean that the vanadium is not isotropic; it is just the simplest way to correctly keep track of the weights in equation (6). If we assign the same and orientation to the vanadium as for the sample, the same detectors will contribute with the same incident fluxes for sample and vanadium to each individual region in the reciprocal space.
The same reasoning can be used for symmetrization. Symmetry operations can be applied to the data, as long as the same symmetry operations are applied to the [00L] direction) are shown in Fig. 2. The symmetrized data are considered new data sets, so normalization is calculated independently for each rotation, and symmetrized data and normalizations are added to the original ones, before performing the division. Contrary to the procedure used for Fig. 1, the overlap region has now been correctly taken into account. This general protocol works for all regions of including both Bragg and diffuse scattering.
data, to correctly count the weight at each point in the The results (11 different orientations and a 180° rotation around theMoreover, by correctly accounting for the statistical weights in the overlap region (Fig. 2) we intrinsically apply both Lorentz and spectrum corrections, as noted by Mayers (1984). To prove that this is indeed the case, the way these corrections are calculated must be revisited. The coherent scattering differential is related to the unitcell by (Lovesey, 1984)
where N_{c} is the number of coherent scatterers, v_{c} is the unitcell volume of the scattering crystal, is the momentum transfer to the sample and is a vector. To obtain the number of scattered neutrons from a particular Bragg peak, we integrate over the solid angle of the detector and multiply by the incident integrated as follows:
In structure refinements, I_{c} is measured by counting all scattered neutrons in a particular region of in our case a sphere (but not necessarily).
Given the definition of wavelength, λ, in terms of momentum, k, it follows that
Following the convention that the incident beam is along the direction and is in the horizontal plane, perpendicular to the incident beam,
where θ is the conventional polar angle of a spherical coordinate system (not the crystallographic angle) and φ is the azimuthal angle.
The Jacobian for the transformation from to spherical coordinates of is given by
Simplifying of the Jacobian yields for the integration volume element
The last three terms on the right hand side represent the scattering solid angle .
Using the transformation to spherical coordinates, the intensity for a Bragg peak is given by
Applying equation (16), and integrating the δ function, yields
where the sample volume is V = N v_{c}.
The
is then related to the integrated intensity byThe first fraction represents the spectrum correction; the last is the Lorentz correction and takes into account the amount of time a given reflection remains in the diffraction condition (Buras & Gerward, 1975). Note that in common crystallographic convention the scattering angle is called , so the Lorentz correction appears as . The Lorentz correction has a different form if the experiment is performed on a monochromatic incident beam and the integrated intensity is measured by rocking the crystal.
Similarly, the intensity for an
process is calculated aswhere N_{i} is the number of incoherent scatterers, with a total .
The corresponding incoherent integrated intensity is given by
If the data are integrated over a small volume, it can be assumed that λ and θ are approximately constant and can be taken outside of the integral together with . The integral of remains, which is the integration volume in and may be chosen as a userdefined constant .
Following the considerations for the coherent intensity, the
intensity is given similarly byGiven that the integration is defined to be over the same small volume for the Bragg peak and the ], the term and Lorentz factor are identical, yielding
[equation (25)where c is a constant that is wavelength and detector independent.
The ratios of the integral form of the Lorentz correction, equation (24), and the analytical form, equation (25), for selected incident neutron wavelengths and two integration box sizes (0.10 and 0.05 Å^{−1}) were calculated and compared in order to estimate the quality of a constant λ and θ approximation for a defined integration region, , as shown in Fig. 3. For this comparison we assume that the neutron is independent of wavelength for the integration volume. It is apparent that the deviation from the ideal case (ratio = 1) increases with increasing d spacing for all wavelengths. The increase is faster for shorter wavelengths. The deviation is more pronounced for the larger, 0.10 Å^{−1}, integration box than the smaller, 0.05 Å^{−1}, box. This is an expected effect, since an integration region in represents a larger region in detector space at low scattering angles.
We should note that the arithmetic averaging of N_{ i}/V_{ i} contributions is noisier and has larger error bars than the new A comparison of intensities from Figs. 1 and 2 is shown in Fig. 4.
An even more convincing proof is obtained by performing a onedimensional cut, as shown in Fig. 5. The cuts are along the [0K0] direction, at H = 0, L = −4.5. The arithmetic mean approach produces data that are less regular and have larger error bars than the proposed protocol.
The smaller error resulting from the new protocol can be shown analytically. If we assume the vanadium is measured for a much longer time than the sample (as is most often the case), this implies that the resulting N_{i}/V_{i} values approach a single constant I. The ratio of scattering cross sections from the two methods is
measurement error is negligible in comparison to that of the coherent scattering. Additionally, in the limit of very good statistics for coherent scattering measurements, allUsing Poisson statistics, the error bars for N_{i} are equal to N_{i}^{1/2}. The ratio of the variances for the scattering differential cross sections, Err_{old} and Err_{new}, is then given by adding in quadrature the errors from each contribution:
Since , we can use the Cauchy–Schwartz inequality (Cauchy, 2015) on the previous equation to yield
Equality is achieved only when the weights V_{i} are equal. The error bars for the new method are therefore smaller.
2.4. Experimental testing of the new data processing protocol
A direct comparison between the integration and correction methods described in §§2.2 and 2.3 is impossible for diffuse scattering data, since the integration area is continuous, although it may be done for Bragg scattering. Two integrated data sets of βNaLaF_{4} were used for benchmarking the original and new data processing methods through structure the first from the original postintegration Braggonly correction method and the second from the new comprehensive correction preintegration method. For post integration, sample and Lorentz corrections were calculated at peak centers and applied to the integrated intensities. Bragg peaks measured at different sample orientations are considered separately, even if they correspond to the same position in the reciprocal space.
In considering the normalization of diffuse scattering data measured on TOPAZ, other corrections including multiple scattering, Debye–Waller factor and any inelasticity effects are negligible and can be ignored. However, it is worth noting that the inelasticity effect is dependent on the ratio of scattered to incident flight paths (Mayers, 1984), which is insignificant for the TOPAZ instrument, but might be important for other instruments.
Using the same processing protocol for the Bragg data as for diffuse scattering data, the raw data were binned on a regular grid in the HKL space. The spacing of the grid was chosen in such a way that the expected Bragg peaks were in the center of each HKL box and were completely contained inside individual bins. After the corrections had been applied to all the intensity contained in such a box was considered the integrated intensity of a Bragg peak. The average intensities of the surrounding boxes were calculated as background and subtracted from the signal.
In order to test the proposed procedure with a conventional structure βNaLaF_{4} measurements. This ensures correction with at the same position in of the sample with the same statistical weight.
the Bragg peaks of each orientation were corrected and integrated individually and not combined as has been done to visualize the diffuse scattering. The same point of the sample can be measured in different sample orientations on a different detector and/or at a different neutron wavelength. For every sample orientation we scaled the vanadium data to the same timeintegrated incident We associated the same lattice parameters and sample orientation with vanadium as for the correspondingThe scattering from vanadium has the same energy dependent spectrum in each detector pixel, up to a pixel dependent multiplicative constant (the efficiency of the detector). This constant varies within 6% of the average for each detector module. However, detector pixels are located at different distances from the sample, so neutrons with the same energy arrive at different detectors at different times; for simplicity all neutron flight times were converted to momentum. Any quantity that depends on neutron velocity could have been selected, such as wavelength, but a constant grid in wavelength results in a logarithmic grid when transformed to the
of the sample. A linear grid in momentum space yields a linear grid in momentum transfer.From equation (26), a low neutron count for vanadium will produce large errors. To decrease background, and avoid regions with a low number of counts, both the coherent and incoherent data were cropped to lie in the [1.85, 9.5] Å^{−1} interval. This also eliminates detected events due to the very high energy neutrons associated with the prompt pulse (Carpenter & Yelon, 1986). To increase statistics, the momentum dependence of vanadium scattering in detector pixels was averaged with similar energy response. The data were binned using a 0.01 Å^{−1} interval, as the averaging cannot be done in event mode. The bin width was chosen so that it is small enough to be representative of a continuous spectrum but within computational limits for processing the large number of events (20 billion). The binned data were smoothed over each of the 256 by 256 pixels per detector and this average value was expanded over all pixels in each detector. The average values of each detector pixel were multiplied by the integrated value of the detector counts over the momentum range.
3. Results and discussion
The integrated intensities from the standard workflow and new protocol were compared to determine the linearity of their correlation. The intensity statistics as a function of spherical integration radius indicated an optimal integration radius of approximately 0.14 Å^{−1} in space (§2.2). Integration radii from 0.13 to 0.16 Å^{−1} in 0.005 Å^{−1} increments were tested.
A grid commensurate with the lattice was chosen so that all the Bragg peaks were in the same position inside corresponding HKL boxes and to reduce the number of computations. To integrate approximately the same volume in HKL space as is used for spherical integration, the box size was defined as a cube of 0.1 units in each direction.
The same peaks as integrated with the spherical integration method were used in the new integration procedure. Peaks with were used for structure r = 0.937 (R Development Core Team, 2008), and the residuals are randomly distributed about y = 0, indicating a high positive linear correlation between the two sets of intensities. The error bars in Fig. 6 show the standard deviations in both methods, which are comparable.
Outlier reflections, where , were excluded from the final The integrated intensities using the original workflow and new protocol were compared. The correlation between the intensities of the two methods calculated as the sample covariance divided by the product of each data set's standard deviation is high,Full anisotropic refinements, including a GSAS (Larson & Von Dreele, 2000) program, using the EXPGUI (Toby, 2001) graphical interface, on structure factors of βNaLaF_{4} obtained from both methods. A summary of the is given in Table 1.
correction, were performed with the

The
results show that the Bragg structure statistics are similar for both integration methods, with 13 reflections rejected in the final round of for the box method. The different shape of the integration and background regions could explain more reflections being accepted for the sphere integration.In the case of major structural differences or are highly correlated (R^{2} = 0.995). The F statistic is 3465 with a corresponding p value of 2.2 ×10^{16}, indicating that there is excellent agreement between structural results from sphere and grid integration and the null hypothesis that there is no linear correlation between the data sets should be rejected (R Development Core Team, 2008).
issues, the anisotropic displacement parameters (ADPs) would be uncorrelated. However, the results shown in Fig. 7An Xray data set was collected as a basis for comparison with the neutron data using a crystal from the same batch [see supporting information; performed using SHELX97 (Sheldrick, 2008)]. The ADPs of both neutron integration methods were highly correlated to the Xray ADPs; R^{2} was approximately 0.88 for spherical integration and 0.85 for the new grid integration (R Development Core Team, 2008). The slightly lower correlation of ADPs between the Xray and neutron experiments can be explained by the difference in scattering cross sections measured by different methods. These Bragg results demonstrate the validity of the proposed method for the reduction of diffuse intensity data for local structural analysis.
As a proof of principle the new normalization protocol was then used to integrate the diffuse intensities of four layers (L = −2.5, −3.5, −4.5 and −5.5). A was defined by doubling the c axis, resulting in integer values for L for computational modeling (Chodkiewicz et al., 2010). We masked the aluminium powder rings from the sample pin corresponding to the 111, 200, 220 and 311 reflections using the MANTID (Arnold et al., 2014) software. The data sets were binned with a step size = 0.1 in units. In principle, a much smaller step size may be needed to extract the diffuse intensities and their associated standard uncertainties for quantitative computational modeling (MichelsClark et al., 2013) as the grid definition is not restricted by the necessity of integer HKL values. Intensities were calculated on the same grid as the experimental data using a Monte Carlo model of the local structure, as described by Aebischer et al. (2006), in a diffuse scattering modeling software currently under development, ZODS (Zurich Oak Ridge Disorder Simulations; Chodkiewicz et al., 2010). The four simulated diffuse layers are shown in Fig. 8. The measured data in Fig. 2 are in good agreement with the simulation. Comparing with Xray data from Aebischer et al. (2006) the complementarity of scattering length differences between Xrays and neutrons is evident and can be used as a seed for comprehensive analysis development.
4. Summary
In this work, we present a comprehensive correction and normalization protocol for total scattering data. The ratio of integrated intensities, over the same small volume in the GSAS Bragg refinements of structure factors from the new and old methods produce highly comparable structural results, with low R values and highly correlated ADPs. This enables the use of corrected diffuse data as input to local structure modeling software. It is imperative that all coherent data and all incoherent data are processed comprehensively, each of them summed separately for different orientations, and normalized only at the end. The same principle applies to symmetrization: symmetrize data, add it to the original data, symmetrize normalization and add it to the original normalization, then divide the sums. This way the statistical weights of the measurements are preserved. The methods presented here work equally well on event or histogrammed data, and provide a first step for automated processing of total scattering neutron TOF singlecrystal diffraction data.
for the coherent and intensity is shown to implicitly account for the term and Lorentz contributions.Supporting information
Comparisons of Xray data sets and neutron data. DOI: https://doi.org//10.1107/S1600576716001369/fs5119sup1.pdf
Acknowledgements
We acknowledge Dr Nicholas Sauter for providing support for TMC as a postdoctoral fellow at Lawrence Berkeley National Laboratory. Professor HansBeat Bürgi, University of Bern/University of Zürich, has been mentoring structure modeling of NaLaF_{4} with ZODS as part of TMC's PhD thesis. Dr Michal Chodkiewicz has been the force behind the ZODS program development. The single crystals for the experiment were generously provided by Dr Karl Krämer at the University of Bern, Switzerland, through HansBeat Bürgi. A special thank you is extended to Arthur Schultz for thoughtful discussions. Xray data were collected by Dr R. Custelcean at Chemical Sciences Division, Oak Ridge National Laboratory. TMC was supported through a Sinergia grant (CRSIKO_122706) from the Swiss National Science Foundation (SNF) for part of the PhD research, as well as by the University of Tennessee and Oak Ridge National Laboratory. Research at ORNL's Spallation Neutron Source was sponsored by the Scientific User Facilities Division, Office of Basic Energy Sciences, US Department of Energy. Oak Ridge National Laboratory is managed by UTBattelle LLC under contract No. DEAC0500OR22725 with the US Department of Energy. Computing time to simulate the diffuse scattering model used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the US Department of Energy under contract No. DEAC0205CH11231.
References
Aebischer, A., Heer, S., Biner, D., Krämer, K. W., Haase, M. & Güdel, H. U. (2005). Chem. Phys. Lett. 407, 124–128. Web of Science CrossRef CAS Google Scholar
Aebischer, A., Hostettler, M., Hauser, J., Krämer, K. W., Weber, T., Güdel, H. U. & Bürgi, H. B. (2006). Angew. Chem. Int. Ed. 45, 2802–2806. Web of Science CrossRef CAS Google Scholar
Arnold, O. et al. (2014). Nucl. Instrum. Methods Phys. Res. Sect. A, 764, 156–166. Web of Science CrossRef CAS Google Scholar
Auzel, F. (2004). Chem. Rev. 104, 139–174. Web of Science CrossRef PubMed CAS Google Scholar
Buras, B. & Gerward, L. (1975). Acta Cryst. A31, 372–374. CrossRef IUCr Journals Web of Science Google Scholar
Bürgi, H. B. (2011). Personal communication. Google Scholar
Bürgi, H. B., Hostettler, M., Birkedal, H. & Schwarzenbach, D. (2005). Z. Kristallogr. 220, 1066–1075. Google Scholar
Burns, J. H. (1965). Inorg. Chem. 4, 881–886. CrossRef CAS Web of Science Google Scholar
Carpenter, J. M. & Yelon, W. B. (1986). Methods in Experimental Physics, edited by K. Sköld & D. L. Price, Vol. 23, Part A, pp. 99–196. New York: Academic Press. Google Scholar
Cauchy, A. (2015). Cauchy's Inequality, https://proofwiki.org/wiki/Cauchy's_Inequality. Google Scholar
Chodkiewicz, M., Bürgi, H.B. & Weber, T. (2010). Acta Cryst. A66, s215–s216. CrossRef IUCr Journals Google Scholar
Haase, M. & Schafer, H. (2011). Angew. Chem. Int. Ed. 50, 5808–5829. Web of Science CrossRef CAS Google Scholar
Howe, M. A., McGreevy, R. L. & Howells, W. S. (1989). J. Phys. Condens. Matter, 1, 3433. CrossRef Web of Science Google Scholar
Keen, D. A. & Nield, V. M. (1996). SXD Diffuse Scattering Data Analysis Manual. Technical Report RALTR96095. Council for the Central Laboratory of the Research Councils, Didcot, Oxford, UK. Google Scholar
Krämer, K. W., Biner, D., Frei, G., Güdel, H. U., Hehlen, M. P. & Lüthi, S. R. (2004). Chem. Mater. 16, 1244–1251. Google Scholar
Larson, A. C. & Von Dreele, R. B. (2000). General Structure Analysis System (GSAS). Technical Report LAUR 86–748. Los Alamos National Laboratory, New Mexico, USA. Google Scholar
Lovesey, S. W. (1984). Theory of Neutron Scattering from Condensed Matter, Vol. 1, International Series of Monographs on Physics, No. 72. Oxford: Clarendon Press. Google Scholar
Mayers, J. (1984). Nucl. Instrum. Methods Phys. Res. 221, 609–618. CrossRef CAS Web of Science Google Scholar
MichelsClark, T. M., Lynch, V. E., Hoffmann, C. M., Hauser, J., Weber, T., Harrison, R. & Bürgi, H. B. (2013). J. Appl. Cryst. 46, 1616–1625. Web of Science CrossRef CAS IUCr Journals Google Scholar
Nield, V. M. & Keen, D. A. (2001). Diffuse Neutron Scattering from Crystalline Materials, Oxford Series on Neutron Scattering in Condensed Matter, Vol. 14. New York: Oxford University Press. Google Scholar
R Development Core Team (2008). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. http://www.Rproject.org. Google Scholar
Rao, B. N., Datta, R., Chandrashekaran, S. S., Mishra, D. K., Sathe, V., Senyshyn, A. & Ranjan, R. (2013). Phys. Rev. B, 88, 224103. Web of Science CrossRef Google Scholar
Riedel, R., Donahue, C., Visscher, T. & Montcalm, C. (2015). Nucl. Instrum. Methods Phys. Res. Sect. A, 794, 224–233. Web of Science CrossRef CAS Google Scholar
Schultz, A. J., Jørgensen, M. R. V., Wang, X., Mikkelson, R. L., Mikkelson, D. J., Lynch, V. E., Peterson, P. F., Green, M. L. & Hoffmann, C. M. (2014). J. Appl. Cryst. 47, 915–921. Web of Science CrossRef CAS IUCr Journals Google Scholar
Sears, V. F. (1992). Neutron News, 3(3), 29–37. Google Scholar
Sheldrick, G. M. (2008). Acta Cryst. A64, 112–122. Web of Science CrossRef CAS IUCr Journals Google Scholar
Toby, B. H. (2001). J. Appl. Cryst. 34, 210–213. Web of Science CrossRef CAS IUCr Journals Google Scholar
Tu, D., Lui, Y., Zhu, H., Li, R., Lui, L. & Chen, X. (2013). Angew. Chem. Int. Ed. 52, 1128–1133. Web of Science CrossRef CAS Google Scholar
Weber, T. & Bürgi, H.B. (2002). Acta Cryst. A58, 526–540. Web of Science CrossRef CAS IUCr Journals Google Scholar
Welberry, T. R. (2004). Diffuse Xray Scattering and Models of Disorder. New York: Oxford University Press. Google Scholar
Welberry, T. R. & Goossens, D. J. (2008). Acta Cryst. A64, 23–32. Web of Science CrossRef CAS IUCr Journals Google Scholar
Welberry, T. R., Gutmann, M. J., Woo, H., Goossens, D. J., Xu, G., Stock, C., Chen, W. & Ye, Z.G. (2005). J. Appl. Cryst. 38, 639–647. Web of Science CrossRef CAS IUCr Journals Google Scholar
Worlton, T., Chatterjee, A., Hammonds, J., Bouzek, C., Mikkelson, D., Mikkelson, R., Miller, M., Serum, B. & Peterson, P. (2004). Neutron News, 15(3), 14–15. CrossRef Google Scholar
Zikovsky, J., Peterson, P. F., Wang, X. P., Frost, M. & Hoffmann, C. (2011). J. Appl. Cryst. 44, 418–423. Web of Science CrossRef CAS IUCr Journals Google Scholar
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