Everything SAXS: small-angle scattering pattern collection and correction

The first of these three, pinhole-collimated instruments (schematically shown in figure 3) have become very popular due to their flexibility in terms of samples and easy availability of data reduction and analysis procedures. While initially eschewed for slit-collimated instruments due to the drastically higher primary beam intensity of the latter, improvements in point-source x-ray generators as well as 2D focusing optics have reduced the weight of this argument somewhat. This class of instruments also dominates the small-angle scattering field at synchrotrons as well as neutron sources due to their aforementioned flexibility.

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These instruments typically consist of a point-based x-ray source followed by x-ray optics. These optics are either used to parallelize the photons emanating from the source, or focus the x-rays to a spot on the detector or sample. After the x-ray optics, the beam is then further collimated using either three or more collimators made from round pinholes or sets of slit blades, separated by tens of centimetres to several metres (a particular effect of the collimation on beam properties is given in section 2.2.4). While the third collimator was required to remove slit or pinhole scattering from the second collimator [15, 196, 138], the recent development of single-crystalline 'scatterless' slits remove the need for the third collimator [58, 109].

There are two main instrument variants in circulation as to what happens after the collimation section. One type of instrument ends the in-vacuum collimation section with an x-ray transparent window, allowing for an in-air sample placement and environment before entering another x-ray transparent window delimiting the vacuum section to the detector (this sample-to-detector vacuum section is also known as the 'flight tube'). As this introduction of two x-ray transparent windows and an air path generates a non-negligible amount of small-angle scattering background, it does not lend itself well to samples with low scattering power [43]. The second instrument variant, therefore, consists of a vacuum sample chamber (and often a vacuum valve which can be closed to maintain the vacuum in the flight tube during the sample change procedure), and thus allows an uninterrupted flightpath from collimation through the sample into the flight tube. While this generates the least unwanted scattering, it does add restrictions to the sample and sample environments that can be put in place [138].

At the end of the flight tube sits the in-vacuum beamstop, whose purpose is to prevent the transmitted beam from damaging the detector or causing unwanted parasitic scattering, and can be one of three types. This beamstop can be a normal beamstop, which blocks all of the transmitted beam. It is useful in many cases, however, to have an estimate for the amount of radiation flux present in the transmitted beam. To this end, the beamstop can be replaced or augmented with a small PIN diode, which measures the flux directly (albeit on arbitrary scale), or the beamstop can be made 'semitransparent', meaning that the beamstop is adapted to pass through a heavily attenuated amount of radiation which subsequently falls onto the detector. The presence of either of the two latter options can be used to benefit the accuracy of the data reduction step, leading to more accurate data and therefore more accurate results.

Finally, the flight tube exits in a window followed (almost) immediately by the detector. For detectors with a large detecting area, this exit window (and the flight tube exit section) must be engineered to be strong and large, sometimes leading to visible parasitic scattering from the window material. It is therefore recommended to keep the detector small, allowing for a small and modular flight tube with very little exit window issues. Alternatively, for very modern systems, some detectors can work in-vacuum as well which removes this last (small) source of parasitic scattering and allows for step-less translation of the detector and beamstop within this vacuum, drastically increasing the flexibility in angular measurement range.

One alternative to this type of instrument was the 'Huxley–Holmes' camera which contained two separate optical components for monochromatization and focusing, to achieve a very low background [202]. While this instrument is performing well, the authors currently recommend going for a more common configuration instead consisting of focusing optics followed by scatterless slits [201].

A second type of instrument exists which is much more compact than the pinhole-collimated systems, is less expensive and illuminates a larger amount of sample to collect more scattering. This type of instrument is often referred to as a 'Kratky' or 'block-collimated' camera, perhaps best explained by Kratky [103] and Glatter and Kratky [63]. This camera is commonly built on an x-ray source emitting a beam with a line-shaped cross-section, and collimates the x-ray beam using rectangular blocks of metal⁴. While this instrument is sometimes referred to as an ultra-small-angle scattering instrument, it is typically used as a normal small-angle scattering instrument.

The line-shaped cross-section of the x-ray beam does bring with it a major drawback, in that the collected scattering pattern is substantially different from the pattern one would obtain from a pinhole-collimated instrument, and therefore needs a modified data correction procedure. Effectively, the scattering pattern is distorted or blurred due to a superposition of intensity contributions from various scattering points along the line-shaped beam. While the collected 'slit-smeared' scattering patterns can be subjected to a numerical correction to compensate for this smearing effect, such desmearing processes in the best case merely amplify the noise in the system and in the worst case introduces artefacts which could be mistaken for real features [188]. This desmearing procedure will be discussed in more detail in section 3.4.9. Furthermore, analysis of samples containing an anisotropic structure becomes more tedious, leaving the instrument most suited to isotropically scattering samples.

There are a number of instruments preceding the block-collimated camera, which employed a line-shaped x-ray beam collimated with a series of slits [149, 198, 72]. While these formed the basis of the first SAXS instruments, and are by definition slit-collimated instruments, they are no longer in widespread use.

A third type of instrument is one particularly suitable for ultra-small-angle scattering purposes (for the analysis of larger structures typically from several nanometres to several microns), and is known as the 'Bonse–Hart' camera [17]. These instruments utilize the high angular selectivity of crystalline reflections to single out a very narrow band of scattering angles for observation, i.e. using the crystals as angle selectors both for collimation- as well as analysis purposes. While the idea of using crystalline reflections was not new [49, 149], the advantage of the implementation by Bonse and Hart [17] was the use of multiple reflections in channel-cut crystals to improve the off-angle signal rejection in a straightforward manner [26].

The incident beam is collimated to a highly parallel beam through multiple crystalline reflections rejecting all but the angles in reflection condition. The sample is placed into this parallel beam effecting small-angle scattering as the beam passes through the sample. A second crystal (a.k.a. 'analyser crystal') is then used to pick out a single angular band of the scattered radiation. Through rotation of the analyser crystal, the scattered intensity at various angles can be evaluated with an extremely high angular precision or resolution. A few standalone instruments have been constructed on synchrotrons [26, 40, 84, 81], and several more have been built as complementary instruments around laboratory x-ray sources (tube—as well as rotating anode sources) [17, 65, 106, 26].

The instrument angular resolution is defined mostly by the rocking curve of the crystalline reflection, also known as the 'Darwin width', which is the angular bandpass window of the crystalline reflection [5, 82]. While a large variety of crystalline materials can be used in the instrument, the channel-cut crystals are usually made from either silicon or germanium due to their high degree of crystalline perfection over large sizes [5]. These crystals have Darwin widths (FWHM) for the common (111) and (220) x-ray reflections of about 0.0002°, thus defining the q-resolution for most such instruments to be about 0.001 nm⁻¹ (where the multiple reflections do not change the Darwin width, but improve the off-reflection rejection) [26, 178, 40, 65, 106]. When a higher resolution is required, for example to measure larger structures, the combination of high-energy and higher-order crystalline reflections can lead to a ten-fold increase in resolution [82]. Neutrons can also be used instead of x-rays, as the Darwin width for a neutron reflection is about 0.105 times that of the x-ray counterpart, which can therefore lead to a similar increase in resolution [5].

As the channel-cut crystals only collimate in one direction, these instruments suffer from a similar slit-smearing effect as the Kratky-type instruments discussed in section 2.2.2. Desmearing of the data is therefore required, unless effort and intensity is expended to collimate the beam in the perpendicular direction as well [105, 65]. An additional drawback to these instruments is the requirement for a step-scanning evaluation of the scattering curve⁵, which increases measurement times considerably. Due to the fast intensity falloff at higher angles, and the extremely narrow angular acceptance window of the analyser crystal, this instrument performs best at ultra-small angles but has much reduced efficiency at larger angles. These properties render this type of instrument a useful addition to existing SAXS instrumentation, but is less frequently encountered as a standalone instrument. Given the particulars of the data, Bonse–Hart data correction may require additional consideration (e.g. for the determination of the sample transmission factor and the effects of the rocking curve on the data) [26, 178, 203].

While the difference between a Kratky camera and a Bonse–Hart camera initially seemed to be in favour of the Kratky camera [104], it gradually became clear that both instruments have their place in the lab. For small-angle scattering measurements on weakly scattering systems at common small angles (i.e. 0.1 ≤ q (nm⁻¹) ≤ 3), a Kratky camera performs very well, while for measurements to very small angles (i.e. below q (nm⁻¹) ≈ 0.1) the Bonse–Hart approach would be the preferred instrument [37, 26].

In typical scattering measurements, only a fraction of the volume is irradiated with coherent radiation (i.e. with in-phase electromagnetic fields), therefore only that fraction of the irradiated sample volume contributes to the scattered intensity [189]. In other words, the irradiated sample volume typically contains a multitude of so-called 'coherence volumes', each of which contributes to the scattering pattern. As there is no inter-volume coherence, it is the sum of the scattering intensities (as opposed to the sum of the amplitudes) from each of these volumes that is detected [110].

These coherence volumes are defined by two components, the longitudinal component (parallel to the beam direction) and the transversal component (perpendicular to the beam direction, see figure 4). The longitudinal component is dependent on the degree of monochromaticity of the radiation, and is large for monochromatic radiation and quite small for polychromatic beams [110]. The transversal dimension ζ_t of the coherence volume is defined through the collimation, in particular through the dimensions of the beam-defining collimator and its distance to the sample, and can be estimated as [189]:

$$\begin{eqnarray} &&{\zeta }_{\mathrm{t}}=\frac{\lambda l}{w} \end{eqnarray} \tag{ 1 }$$

where λ is the wavelength of the radiation, l the distance between the beam-defining collimator and the sample, and w the size of the collimator opening (ζ_t can be calculated for each direction for collimators with nonuniform openings).

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The estimation of the transversal coherence length is an important check for experiments. Scattering objects with dimensions close to or larger than the transversal coherence length may not contribute significantly to the small-angle scattering as the coherence volume will be within a uniform region of material (an effect seen amongst others by Rosalie and Pauw [150]). This effect can be exploited to investigate the actual transversal coherence length in an instrument as shown by Gibaud et al [61]. For a more detailed treatment of coherence (i.e. when it is approaching significance or what happens when a single coherence volume encompasses the sample), the reader is referred to the aforementioned literature.

The first step for any data correction is to read in the information from detectors. While for point- and linear position sensitive detectors (PSDs), the choice has almost universally been made for the convenience of ASCII data, for image detectors this has not been so straightforward.

Therefore, whenever a detector system is bought, particular attention needs to be paid to the data format of the images one obtains. For some reason, quite a few detector manufacturers worldwide prefer their own image data formats over more standard image formats (a list of some of these formats can be found in the documentation accompanying the NIKA package [80]). This tendency hinders data preservation efforts (though one should preserve corrected and reduced data rather than the original data, a point discussed in section 3.6) and sometimes causes read-in issues of the data in data reduction packages. Two cases in particular have come to the attention of the author, the Rigaku data format and the Bruker data format, which will be used to illustrate the issue.

The Rigaku data format has all the characteristics of a 16-bit TIFF image, and will actually load as such. Without going into details, 16 bits (per image value) would get you a maximum per-pixel value of 2¹⁶: 65 536. This value would be insufficient for storing the number of photons obtained for example from the aforementioned PILATUS hybrid pixel detector, which therefore uses a 32-bit image format. The Rigaku format treats such count numbers slightly differently in order to store them in 16 bits: 15 bits behave like normal bits up to a value of 2¹5 (32 768), the 16th bit acts not as a standard bit but as a 'multiply-by-32'-flag⁶. While this is documented [148], the danger lies in the compatibility of their data format with standard binary data: the intensities will be wrong, but the scientist ignorant of this issue will not immediately notice something is awry.

The Bruker data format, on the other hand, is unlikely to be compatible with any standard image reading routines, and authoritative information on the image format is not very easy to obtain. Some of their image formats appears to use an 8-bit image format (i.e. with per-pixel maximum values of 256), with a subsequent 'overflow' list detailing pixels that have exceeded this 8-bit limit. Implementation and read-in of these data is therefore cumbersome, perhaps even unnecessarily complicated given the alternatives.

In the best case, detector systems adhere to known and common image formats [44]. Active development is ongoing for supporting detector data of these and more complicated multi-chip detectors and instruments in the NeXus format [94, 99]. The NeXus format itself is based on the very versatile, portable, well-documented and open HDF5 data storage format [53]. Such standards will hopefully resolve some of the challenges related to data ingestion into data reduction procedures.

Spurious signals can be detected for a range of reasons: from external sources such as cosmic rays, nearby x-ray sources or atmospheric radioactive decay, or from internal sources such as the employed electronics. These often appear as spikes or streaks in the detected signal, varying in location and amount from image to image. Integrating (e.g. CCD) detectors without energy discrimination are most heavily affected by these phenomena, whilst photon counting, energy discriminating detectors often only show a single extra count (or streak of 1 extra count) upon event occurrence.

Given their potentially high values, zingers can significantly affect the recorded signal, and should be removed in CCD-based detectors. The trick for their detection and subsequent removal is to record multiple images per measurement and mask all statistically significant differences. A suitable computational procedure is described by Barna et al [7] and Nielsen et al [128].

No two detection surfaces (pixels) are exactly the same due to manufacturing tolerances, slight damage or differences in the underlying electronics, to name but a few. Therefore, every detector apart from point detectors (i.e. every spatially resolved detector) has to be corrected for interpixel sensitivity, with the notable exception of image plates⁷. These interpixel sensitivity variations can easily be on the order of 15% for some detectors [182]. The correction is straightforward in theory: collect a uniform, high amount of scattering on the detector, assume the per-pixel detector response should be identical for this scattering, and use the relative difference in detected signal between the pixels as a normalization matrix for future measurements. In practice, though, uniformly distributing a large number of photons (of the right energy) on the detector surface can be a challenge.

One solution is to irradiate directly with a low-power x-ray source placed some distance from the detector, as discussed in detail by Barna et al [7]. This solution needs small corrections for area dilation and air absorption, in addition to a few more detector-specific ones, and needs a separate check of the uniformity of the source. The advantage is that it can be tuned to the energy of interest, and that a sufficient number of photons is easily acquired [146, 41, 59].

Alternatively, doped glasses can be used to obtain a flat-field image, as suggested by Moy et al [121]. This has the advantage of reduced complexity in setting up the flat-field measurement, but may suffer from nonuniform images [7] and has a reduced photon flux. Some use the uniform scattering of water as flat-field measurement data, despite water not scattering uniformly at very small angles (though this can be corrected for), and the scattering intensity at larger angles being quite low for obtaining good per-pixel statistics [125]. Similarly, samples with known scattering behaviour can be used for such purposes [59].

Another solution common in laboratory settings is the use of radioactive sources (emitters) which can be easily accommodated in most instruments [126]. The major drawback of that solution is the differences between the emitter energy and the energy used during normal measurements, and a very low detected count rate necessitating impractically long collection times for decent flat-field images. The alternative suggested by Né et al [126] is the image collection during slow and well-controlled scanning of an emitter over the detector surface, with the challenge of achieving a homogeneous exposure.

The alternatives which place the radiation source at the sample location share one further advantage in case of detectors using phosphorescent screens. The advantage is that through placement of the radiating source at the sample location, one simultaneously corrects for the dependency of the response of phosphorescent screens to the direction of incident photons. If this is not done, one might consider correcting for this effect separately [7, 199].

Given these challenges, it is therefore recommended for (time-stable) detectors to obtain flat-field images from the manufacturer who should be equipped to record these. The corrected intensity I_j,cor for datapoint j can be retrieved from the input intensity I_j using a flat-field image F_j (which can be normalized to 1 to avoid large numbers):

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}=\frac{{I}_{j}}{{F}_{j}}. \end{eqnarray} \tag{ 2 }$$

If there are uncertainties available when performing this step, they will propagate as:

$$\begin{eqnarray} &&{\sigma }_{\mathrm{r},j,\mathrm{cor}}={I}_{j,\mathrm{cor}}\sqrt{\left [\frac{{\sigma }_{\mathrm{r},j}}{{I}_{j}}\right ]^{2}+\left [\frac{\sigma ({F}_{j})}{{F}_{j}}\right ]^{2}}. \end{eqnarray} \tag{ 3 }$$

After arrival of a photon on a detection surface or in a detection volume, a certain amount of time is needed for the detector to recover from this event before a second photon can be detected. This time is called the 'dead time': a photon arriving in this timespan will not be detected. More precisely, the electronic pulses generated by the arrival of two near-simultaneous photons will start to overlap, causing either rejection of both photons due to the compound pulse being too high (energy rejection), or the two pulses being counted as one. This is discussed clearly by Laundy and Collins [107].

This correction can be unnecessary for some of the modern hybrid detectors at the count-rates they are commonly subjected to. The PILATUS detector, for example, only shows a >2% deviation from a linear response at an incident photon rate of more than 450 000 photons/pixel s⁻¹ [101]. Gas-based detectors, especially 1D and 2D wire detectors very much need this correction.

One aspect of this correction that is of high importance is that when the data uncertainty is calculated based on counting statistics (i.e. Poisson statistics), these uncertainties should be calculated from the detected photons, not from the deadtime-corrected photons. This implies that there is a count rate characteristic for each detector beyond which the data accuracy decreases! This phenomenon is evident from Laundy and Collins [107].

The number of deadtime-corrected counts I_jcor can be obtained from the detected number of counts I_j collected in time t by numerically finding a solution for [107]:

$$\begin{eqnarray} &&{I}_{j}={I}_{j,\mathrm{cor}}\exp (-{I}_{j,\mathrm{cor}}T) \end{eqnarray} \tag{ 4 }$$

with

$$\begin{eqnarray} &&T=\frac{{\tau }_{1}+{\tau }_{2}}{t} \end{eqnarray} \tag{ 5 }$$

where τ₁ is the minimum time difference required between a prior pulse and the current pulse for the current pulse to be recorded correctly. Similarly, τ₂ is the minimum arrival time difference required between the current pulse and a subsequent pulse for the current pulse to be recorded correctly. As pulses follow an asymmetric profile like a log-normal function, these two times can be different (for a 1 μs pulse shaping time this can be τ₁ = 3.0 μs and τ₂ = 2.0 μs [107]).

At this point we can also estimate the uncertainty (standard deviation) σ_r,j for the corrected counts through [107]:

$$\begin{eqnarray} \fl {\sigma }_{\mathrm{r},j}&=&\left [\{ (1-{I}_{j,\mathrm{cor}}T)^{2}{I}_{j}\} \left \{ 1+2\exp \left (-{I}_{j,\mathrm{cor}}\frac{\max ({\tau }_{1},{\tau }_{2})}{t}\right )\right .\vphantom {\left \{\exp \left (- I_{j,\mathrm {cor}} T \right ) \right \}^{-1^{x^{x^{x^x}}}}} \right .\nonumber\\ &&-\left .\left .~\vphantom {\frac {\max (\tau _1,\tau _2)}{t}} 2\left (1+{I}_{j,\mathrm{cor}}T\right )\exp \left (-{I}_{j,\mathrm{cor}}T\right )\right \} ^{-1}\right ]^{\frac{1}{2}}. \end{eqnarray} \tag{ 6 }$$

Interestingly, if τ₁ and τ₂ are known, the true uncertainty σ_r,j can be retrieved from the deadtime-corrected values through insertion of equation (4) into (6), which may be of use in detector systems where the deadtime correction is performed by the detector system itself.

Most non-photon counting detectors do not necessarily give an output linearly proportional to the incident amount of radiation. This used to be especially severe for films, which required accurate corrections for each film type [23]. For more modern detection systems the effect appears small (i.e. on the order of 1%), but may nevertheless be considered especially when approaching the limits of the dynamic range [123, 124, 67]. It is relevant for image plates [118, 29, 126, 8] and may be considered for some CCD detectors as well [67]. It may even be relevant for some gas-based photon counting detectors insofar it is not already accounted for with the deadtime correction [8].

This correction can be applied by characterizing the detector response for various fluxes of incident radiation, for example through attenuating monochromatic radiation using a series of calibrated foils to reduce the incident flux [67]. Simply collecting radiation for a longer time may obfuscate the detector response to incident flux with other time-dependent effects especially for image plates [29], unless this is explicitly taken into account [67]. Furthermore, the energy of the incident radiation has to be identical to the energy used for normal measurements, as the gamma correction can be energy dependent [85].

One alternative solution to circumvent the need for this correction is to determine the range of incident radiation amounts where the detector response is linear, and to stay within that range. For samples which exhibit scattering covering a wider dynamic range than thus supported, attenuators can be devised in the beam path to locally attenuate the signal [125]. Introducing additional elements into the beam path may, however, cause scattering or act as a high-pass energy filter leading to 'radiation hardening', and such modifications should therefore not be applied without thorough considerations of the consequences.

Lastly, while it cannot be considered a true nonlinearity correction, for image plates the measured intensity is also a function of measurement time (i.e. the delay after exposure before measuring) [126]. Internal decay causes a reduction of the measurable signal over time, with a fast decay component (with a half-time on the order of minutes) and a slow decay component (on the order of hours). Effectively, this can even cause intensity variations on the order of several per cent during the read-out of the image plate. A decay time correction should therefore be considered for accurate reproduction of intensity, and such a correction is described amongst others by Hammersley et al [67]. It should be noted that this time decay is likely also dependent on the energy of the used x-rays as it is for protons [16].

This correction is applied if the nonlinear behaviour of the intensity can be expressed as a function of the incident radiation γ(I):

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}=\gamma ({I}_{j}). \end{eqnarray} \tag{ 7 }$$

The relative datapoint uncertainty scales similarly:

$$\begin{eqnarray} &&{\sigma }_{\mathrm{r},j,\mathrm{cor}}={\sigma }_{\mathrm{r},j}\frac{\gamma ({I}_{j})}{{I}_{j}}. \end{eqnarray} \tag{ 8 }$$

There are two factors adding to the detected signal even without the presence of an x-ray beam, these are the detector 'dark current' and the omnipresent natural radiation. While these are two separate effects, their correction is identical and can be simultaneously considered. The cause of the dark current signal depends on the detector. Some detector electronics add their own 'pedestal' bias to prevent negative voltages entering the analogue-to-digital converter (ADC) [7, 125], which can be considered a form of dark current. CCD chips may also exhibit a baseline noise, also known as 'read noise', photomultiplier tubes (PMTs) in image plate systems detect a small leak current without any incident photons and ion chambers also detect a small current without radiation. Natural background radiation furthermore adds a time-proportional level of noise in any detector [126].

The dark current components are homogeneously distributed over the entire detector, and can thus (for statistical purposes) be corrected for by subtraction of a single value from each detected pixel value. This single value is a summation of all three dark current components: a time-independent component, a time-dependent component and a flux-dependent component. To elaborate, the time-independent component would be the base amount ('pedestal')-level, applicable to detectors based on PMTs and CCDs [39]. Naturally occurring background radiation can be considered part of the time-dependent component, visible in every detector. One important note here is that the image plates start collecting natural radiation from the time of their last erasure rather than from the start of the measurement [126]. Some detectors may also show a time-dependent dark current in addition to the natural background [124]. These two components can be easily determined through evaluation of the total detected signal as a function of exposure time without an applied x-ray beam. The last component, the flux-dependent dark current level is a specific complication encountered in some image-intensifier-based CCD detectors, and requires the simultaneous determination of the dark current signal alongside the measured signal through partial masking of the detection surface with x-ray absorbent material [143, 125].

This can be expressed mathematically as:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}={I}_{j}-\left ({D}_{\mathrm{a}}+{D}_{\mathrm{b}}t+{D}_{\mathrm{c}}\left (\int _{j}{I}_{j}\right )\right ) \end{eqnarray} \tag{ 9 }$$

where D_a is the time-independent component, D_b the time-dependent factor times the measurement time t, and D_c the flux-dependent component for those detectors suffering from that particular complication (determined simultaneously with the measurement). Image plates furthermore have a natural decay which means that the time-dependent component may not be truly linear over large timescales. It is therefore best practice to determine the dark current contribution using exposure times similar to the measurement times. For accurate determination of the dark current contribution when measurement times are small, the averaging of multiple exposures on the timescale of the measurement can improve statistics [125].

As the dark current is ideally pixel-independent, D_a,D_b and D_c can be determined to high precision when averaged over the entire detector. This should render their relative uncertainties σ(D)/D rather small thus only having a minor effect on the intensity uncertainty. The uncertainty should propagate (assuming Poisson statistics) approximately as:

$$\begin{eqnarray} &&\fl {\sigma }_{\mathrm{r},j,\mathrm{cor}}\nonumber\\ &&=\sqrt{{\sigma }_{\mathrm{r},j}^{2}+\sigma ({D}_{\mathrm{a}})^{2}+(t \sigma ({D}_{\mathrm{b}}))^{2}+\sigma \left ({D}_{\mathrm{c}}\left (\int _{j}{I}_{j}\right )\right )^{2}}. \end{eqnarray} \tag{ 10 }$$

Among the more complicated detector corrections is that of the geometric distortion, which can be severe for some detectors (in particular for wire detectors and image-intensifier-based CCD detectors), small for others (i.e. <1% for fibre-optically coupled CCD detectors) [124], to non-existent for direct-detection systems. The electronics and design of image intensifiers in CCD cameras and electronics of wire detectors can give rise to pixels being assigned incorrect geometric positions, leading to geometric distortion [7]. Even image plate readers can show this effect due to the read-out mechanics [108], and it therefore seems a necessary correction for all detectors save those based on direct-detection (e.g. the PILATUS detector). In order to put the detected pixels back in their right 'place', i.e. in a location corresponding to the arrival location of the detected photon on the detector surface, a geometric distortion correction must take place.

The most common method for this is to place a mask with regularly spaced holes in front of the detector, which is subsequently irradiated with more-or-less uniform photons originating from the sample position. This then allows for the evaluation of where on the detector the photons are observed versus where the photons actually arrived through the holes in the mask [108, 185].

These corrections only really can take care of smoothly varying distortions, and are ill-suited for corrections of abrupt distortions as those found upon occurrence of discontinuous shears in fibre-optically coupled detectors [7]. Corrections for these distortions must be considered separately [35]. Rather than trying to correct the actual image by, for example, inserting or interpolating pixel values (e.g. [185]), one good way of dealing with these corrections is to determine a coordinate look-up table ('displacement maps') for each pixel. These maps can subsequently be used during the data averaging procedure (see section 3.4.10) to put the detected datapoints in the right bins [92, 93].

Image plates, besides the small geometric distortion mentioned above also require a specific correction: one that corrects for variance in subsequent placements of image plates. Since it is mechanically challenging to reproducibly place an image plate to within 50 μm (or approximate grain size), every image plate may be slightly offset. The variance in placement for a given image plate placement and read-out procedure (ideally designed to minimize placement variance) can be quantified and evaluated for significance of severity. If necessary, symmetry in the scattering patterns can be exploited to determine the beam centre of every image. A procedure for achieving this is described by Le Flanchec et al [108].

Due to the detector specificity of the required correction and the relatively complex procedure, the methods for correcting image distortions are not reproduced here. Geometric distortions should not affect the datapoint uncertainties.

In virtually any detection system there will be 'broken' pixels, either pinned to the maximum or minimum value, or simply giving incorrect response to the incident radiation. Additionally, pixels masked by the beamstop or the beamstop holder should be ignored as well. For masking these, an oft used technique is to record a scattering pattern of a strong scatterer, after which a Boolean array can be manually generated, indicating for each pixel whether it should be masked or not. For space-saving purposes, this Boolean array can be reduced to a list of pixel indices to be masked.

This array (or list of pixel indices) can subsequently be used in the averaging procedure to not consider invalid pixels in the procedure. Such masking does not affect the uncertainties.

The scattering effect of photons depends on the polarization of the incident radiation and the direction of the scattered radiation [168]. This phenomenon causes a slight reduction in intensity. While this effect is commonly corrected for in wide angle diffraction studies, it is often considered negligible in small-angle scattering data correction [141, 14, 145, 128]. When quantified, the correction amounts to nearly 1% for scattering angles 2θ of about 5°. This correction applies both to unpolarized radiation as well as polarized radiation, in the former the correction is isotropic, and in the latter anisotropic (as shown in figure 5). Depending on the angular range collected, the polarization correction may be considered for a slight increase in accuracy.

The correction factor for 2D detector images is given by Hura et al [78] as:

$$\begin{eqnarray} {I}_{j,\mathrm{cor}}&=&{I}_{j}[{P}_{\mathrm{i}}(1-(\sin (\psi )\sin (2 \theta ))^{2})\nonumber\\ &&+~(1-{P}_{\mathrm{ i}})(1-(\cos (\psi )\sin (2 \theta ))^{2})] \end{eqnarray} \tag{ 11 }$$

where ψ is the azimuthal angle on the detector surface (defined here clockwise, 0 at 12 o'clock) 2θ the scattering angle, and P_i the fraction of incident radiation polarized in the horizontal plane (azimuthal angle of 90°)⁸. The correction for unpolarized radiation is achieved when P_i = 0.5, most synchrotron beam lines have a P_i ≈ 0.95.⁹ As this is a correction between datapoint values, only the relative uncertainty σ_r,j is affected similarly to the effect of polarization on the intensity:

$$\begin{eqnarray} {\sigma }_{\mathrm{r},j,\mathrm{cor}}&=&{\sigma }_{\mathrm{r},j}[{P}_{\mathrm{i}}(1-(\sin (\psi )\sin (2 \theta ))^{2})\nonumber\\ &&+~(1-{P}_{\mathrm{ i}})(1-(\cos (\psi )\sin (2 \theta ))^{2})]. \end{eqnarray} \tag{ 12 }$$

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Any material inserted into the beam absorbs a certain amount of radiation [78, 3]. This affects the amount of background scattering impinging on the detector as well as the amount of scattering of the remaining radiation by the sample (differing slightly depending on the path through the sample as well as shown in section 3.4.7). As the amount of radiation scattered by the sample is typically small, the absorption or transmission factor can be determined by measuring the flux directly before and after the sample.

There are three commonly applied methods for measuring the sample absorption, one 'in situ' method commonly found at synchrotrons and two offline methods. At synchrotrons, so-called ionization chamber detectors can be installed (usually in air) directly before and after the sample position. These detectors are very straightforward in their construction, typically consisting of two electrodes suspended in air [161, 179, 117]. They are particularly suitable as they exhibit no parasitic scattering and can be used to monitor the incident flux as well as the absorption over the duration of the measurement. Assuming a non-identical but linear response for both upstream (u) and downstream (d) detectors, the readings without sample (indicated with subscript ₀) and after sample insertion (subscripted ₁) can be used to calculate the transmission factor T_r through:

$$\begin{eqnarray} &&{T}_{\mathrm{r}}=\frac{\frac{{I}_{\mathrm{d},1}}{{I}_{\mathrm{u},1}}}{\frac{{I}_{\mathrm{d},0}}{{I}_{\mathrm{u},0}}}=\frac{{I}_{\mathrm{d},1}{I}_{\mathrm{u},0}}{{I}_{\mathrm{u},1}{I}_{\mathrm{d},0}}. \end{eqnarray} \tag{ 13 }$$

It is not always possible to insert ionization chambers, for example when working with a completely evacuated instrument. In that case, two alternative solutions can be applied to measure the beam flux sequentially before and after insertion of the sample. In one solution, the beamstop is modified to either: (1) allow for a small fraction of the direct beam to pass through and be detected by the main detector (i.e. a 'semitransparent beamstop'), or (2) where the beamstop is augmented with a small¹⁰ pin-diode measuring the direct beam flux [120, 45]. The second option is to place a strong scatterer in the beam downstream from the sample position, and measure the integrated scattering signal from the strong scatterer [138]. For the beamstop modification case, the ratio of the two fluxes (before and after insertion of the sample) is the transmission factor. In the last case, the ratio of the two integrated intensities on the detector is the transmission factor:

$$\begin{eqnarray} &&{T}_{\mathrm{r}}=\frac{{I}_{1}}{{I}_{0}} \end{eqnarray} \tag{ 14 }$$

where I₀ is the intensity of the primary beam without sample, and I₁ the intensity of the beam after insertion (and downstream) of the sample. A transmission factor correction for highly absorbing samples scattering to wide angles is discussed in section 3.4.7. The transmission factor is dependent on the linear absorption coefficient μ and thickness d of a material through:

$$\begin{eqnarray} &&{T}_{\mathrm{r}}=\exp \left (-\mu d\right ). \end{eqnarray} \tag{ 15 }$$

The transmission correction can be applied by dividing the detected intensity with the transmission factor. Furthermore, the detected intensity is proportional to the incident flux on the material f_s, which can be similarly corrected for:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}=\frac{{I}_{j}}{{T}_{\mathrm{r}}{f}_{\mathrm{s}}}. \end{eqnarray} \tag{ 16 }$$

The relative intensity uncertainty remains largely unaffected by this correction (if the background is small and/or shows little localization), but the absolute uncertainty is directly related to the uncertainties of the measured transmission and flux:

$$\begin{eqnarray} &&{\sigma }_{\mathrm{a},\mathrm{cor}}=\mathop{\sum }\limits _{j}{I}_{j,\mathrm{cor}}\sqrt{\left [\frac{{\sigma }_{\mathrm{a}}}{\mathop{\sum }_{j}{I}_{j}}\right ]^{2}+\left [\frac{\sigma ({T}_{\mathrm{r}})}{{T}_{\mathrm{r}}}\right ]^{2}+\left [\frac{\sigma ({f}_{\mathrm{s}})}{{f}_{\mathrm{s}}}\right ]^{2}}.\nonumber\\ \end{eqnarray} \tag{ 17 }$$

The time and thickness corrections are nearly identical to the transmission and flux corrections (section 3.4.2) and equally straightforward: the detected intensity is proportional to the measurement time¹¹, and the amount of scattered radiation is proportional to the amount of material in the beam. The thickness correction is applied to correct for differences in the amount of material in the beam.

These two corrections are applied through normalization of the measured intensity with the sample measurement time t_s and thickness d_s:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}=\frac{{I}_{j}}{{t}_{\mathrm{s}}{d}_{\mathrm{s}}}, \end{eqnarray} \tag{ 18 }$$

with uncertainty

$$\begin{eqnarray} &&{\sigma }_{\mathrm{a},\mathrm{cor}}=\mathop{\sum }\limits _{j}{I}_{j,\mathrm{cor}}\sqrt{\left [\frac{{\sigma }_{\mathrm{a}}}{\mathop{\sum }_{j}{I}_{j}}\right ]^{2}+\left [\frac{\sigma ({t}_{\mathrm{s}})}{{t}_{\mathrm{s}}}\right ]^{2}+\left [\frac{\sigma ({d}_{\mathrm{s}})}{{d}_{\mathrm{s}}}\right ]^{2}}.\nonumber\\ \end{eqnarray} \tag{ 19 }$$

Scaling the data to reflect the materials' differential scattering cross-section can be a great boon to the value of the data. This scaling allows for the evaluation of the scattering power of the sample in material specific absolute terms which can lead to e.g. the determination of the volume fraction of scatterers or their specific surface area and to check the validity of assumptions made. Furthermore, it allows for proper scaling between techniques, and can help distinguish multiple scattering effects. This scaling gives the scattering profile the units of scattering probability per unit time, per sample volume, per incident flux and per solid angle, which if worked out comes to m⁻¹ sr⁻¹ (though centimetres are sometimes used instead of metres) [195, 14, 42, 204].

This scaling can be achieved in two ways; either through direct calibration with samples whose scattering power can be calculated, or through the use of secondary standards. A discussion of the benefits and drawbacks of either have been well explained by Dreiss et al [42]. The most straightforward method is using a secondary standard such as Lupolen or calibrated glassy carbon samples [152, 204], as they do not require detailed knowledge on detector behaviour and beam profiles, and scatter significantly allowing for rapid collection of sufficient intensity to perform the calibration. The scattering of these samples in absolute intensity units is determined separately, and its calibrated datafile should come with the sample [195, 204]. By comparing the intensity in the calibrated datafile with the locally collected intensity, a calibration factor C can be determined:

$$\begin{eqnarray} &&C=\frac{\left (\frac{\delta \Sigma }{\delta \Omega }\right )_{\mathrm{st}}}{{I}_{\mathrm{st},\mathrm{cor}}} \end{eqnarray} \tag{ 20 }$$

where the subscript _st denotes the calibration standard, and I_st,cor is the measured and corrected intensity and $\left (\frac{\delta \Sigma }{\delta \Omega }\right )_{\mathrm{st}}$ is the known scattering pattern (calibrated datafile) from the calibration sample. The calibration factor can be determined by a least-squares fit or linear regression (with least squares allowing for inclusion of counting statistics and optional flat background contribution).

The accuracy of this determination depends on a large amount of uncertainties. Practically, though, an accuracy of about 10% can be achieved [122]. It may be approximated as:

$$\begin{eqnarray} &&\left [\frac{\sigma (C)}{C}\right ]^{2}=\left [\frac{\mathop{\sum }_{j}\sigma \left (\left (\frac{\delta \Sigma }{\delta \Omega }\right )_{\mathrm{st},j}\right )}{\mathop{\sum }_{j}\left (\frac{\delta \Sigma }{\delta \Omega }\right )_{\mathrm{st},j}}\right ]^{2}+\left [\frac{\mathop{\sum }_{j}{\sigma }_{\mathrm{a}}}{\mathop{\sum }_{j}{I}_{j}}\right ]^{2}. \end{eqnarray} \tag{ 21 }$$

The approximation is finally applied to the measured data as:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}={I}_{j}C. \end{eqnarray} \tag{ 22 }$$

Whose absolute uncertainty follows:

$$\begin{eqnarray} &&{\sigma }_{\mathrm{a},\mathrm{cor}}=\mathop{\sum }\limits _{j}{I}_{j,\mathrm{cor}}\sqrt{\left [\frac{{\sigma }_{\mathrm{a}}}{\mathop{\sum }_{j}{I}_{j}}\right ]^{2}+\left [\frac{\sigma (C)}{C}\right ]^{2}}. \end{eqnarray} \tag{ 23 }$$

In any scattering measurement, it is of great importance to isolate the (coherent) scattering of the objects under investigation from all other parasitic scattering contributions such as windows, solvents, gases, collimators, sample holders and any incoherent scattering components. For example, this would be the removal of the scattering of capillaries and solvents from the scattering pattern of a suspension or solution, or the removal of instrumental background (scattering from windows, air spaces, etc) from the scattering pattern collected for a polymer film. A background measurement thus contains as many as possible of the components present in the sample measurement, minus the actual sample. A detailed discussion of suitable background samples can be found in Brlet et al [18].

In most cases, the background measurement should be performed with as many variables identical to the sample measurement, and subjected to the same data corrections¹². This means that the background measurement should be measured for the same amount of time as the sample measurement. However, in case the detector behaviour is well characterized and the signal-to-noise ratio (i.e. the sample-to-background scattering signal) is large, the sample-to-background measurement time ratio may be skewed (to favour sample measurement time) in order to improve the statistics after background subtraction [169, 134]. The correction is applied as:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}={I}_{j}-{I}_{j,\mathrm{b}} \end{eqnarray} \tag{ 24 }$$

with I_j,b the background measurement intensity for datapoint j. The uncertainties, both absolute and relative, propagate as follows:

$$\begin{eqnarray} &&{\sigma }_{\mathrm{r},j,\mathrm{cor}}=\sqrt{{\sigma }_{\mathrm{r},j}^{2}+{\sigma }_{\mathrm{r},j,\mathrm{b}}^{2}} \end{eqnarray} \tag{ 25 }$$

$$\begin{eqnarray} &&{\sigma }_{\mathrm{a},\mathrm{cor}}=\sqrt{{\sigma }_{\mathrm{a}}^{2}+{\sigma }_{\mathrm{a},\mathrm{b}}^{2}}. \end{eqnarray} \tag{ 26 }$$

Most detectors are flat with uniform, square pixels, but we wish to collect the intensity over a solid angle of a (virtual) sphere. The projection of the detector pixels on the sphere results in a difference in solid angle covered by each pixel (illustrated in figure 6) [14, 7, 108]. Therefore, we need to correct the intensity for the difference between these areas¹³.

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The correction for this effect achieved by means of a few geometrical parameters. This correction is given by [14] as:

$$\begin{eqnarray} &&\frac{{L}_{\mathrm{P}}^{2}}{{p}_{x}{p}_{y}}\frac{{L}_{\mathrm{P}}}{{L}_{0}} \end{eqnarray} \tag{ 27 }$$

where L_P is the distance from the sample to the pixel, L₀ the distance from the sample to the point of normal incidence (usually identical to the direct beam position except in case of tilted detectors), and p_x and p_y are the sizes of the pixels in the horizontal and vertical direction, respectively. As it is unnormalized, this correction factor typically assumes very large values. When normalized to assume a value of 1 at the point of normal incidence, the correction becomes:

$$\begin{eqnarray} &&{I}_{j,\mathrm{cor}}={I}_{j}\frac{{L}_{\mathrm{P}}^{3}}{{L}_{0}^{3}}. \end{eqnarray} \tag{ 28 }$$

Its magnitude is shown in figure 7, and is generally less than 1% for scattering angles lower than 5°. It very quickly becomes more severe beyond those angles.

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When scattering occurs in a sample, the scattered radiation has to travel some distance through the sample. Depending on the sample geometry and the scattering angle, this scattered radiation has to travel through more or less material. The direction-dependent absorption thus occurring can induce an angle-dependent scattered intensity reduction which is most severe for scattering to wider angles and for samples with a high attenuation coefficient [18, 175, 200, 11]. This is essentially a correction of the transmission factor correction described in section 3.4.2.

Its correction for plate-like samples to a scattering pattern takes the form of:

$$\begin{eqnarray} {I}_{j,\mathrm{cor}}=\left \{ \begin{array}{@{}ll@{}} \displaystyle {I}_{j}\frac{1-{T}^{\left [\frac{1}{\cos (2 \theta )}-1\right ]}}{\ln (T)-\frac{1}{\cos (2 \theta )}\ln (T)},&\displaystyle \tqs \text{for }2 \theta \not = 0\\ \displaystyle {I}_{j},&\displaystyle \tqs \text{for }2 \theta =0 \end{array}\right . \end{eqnarray} \tag{ 29 }$$

which can be expressed in terms of linear absorption coefficient μ and thickness d as:

$$\begin{eqnarray} &&\fl {I}_{j,\mathrm{cor}}=\left \{ \begin{array}{@{}l@{}} \displaystyle s{I}_{j}\exp (\mu d)\frac{-\exp (-\mu d)+\exp \left (-\mu d/\cos (2 \theta )\right )}{\mu d-\mu d/\cos (2 \theta )},\tqs \hspace{-3pc}\\\tqs \quad \displaystyle \text{for }2 \theta \not = 0\\ \displaystyle {I}_{j},\tqs \text{for }2 \theta =0 \end{array}\right . \end{eqnarray} \tag{ 30 }$$

where 2θ denotes the scattering angle. As the numerator and denominator of the fraction tend to zero for 2θ = 0, at that point I_j,cor = I_j must be substituted. This correction is only valid for plate-like samples, for which it is still straightforward to derive. For spherical samples and cylindrical samples, the direction-dependent attenuation becomes much more complicated [175, 200], and an extra level of difficulty is added for off-centre beams [11].

Figure 8 shows the magnitude of the correction depending on the transmission factor and scattering angle. As previously remarked, the effect is most severe for highly absorbing samples and wide angles.

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As the correction is rather minimal for small-angle scattering, its effects on the uncertainties are expected equally negligible. Given the estimated complexity of the uncertainty propagation in this case, its derivation is here omitted.

Multiple scattering occurs when a scattered photon still travelling through the material undergoes a subsequent scattering event. As the probability for any photon to scatter (irrespective of whether it has scattered or not) is proportional to the scattering cross-section of the material and the amount of sample in the beam, multiple scattering becomes more dominant for strongly scattering, thick samples [154, 30, 115, 114]. It effects a 'smearing' of the true scattering profile, which can significantly affect analyses [24, 30].

When the possibility of multiple scattering exists for a particular sample measurement (i.e. with a transmission factor below approximately 1/e and strongly scattering samples) it is prudent to test whether it is a significant contribution. This can be performed experimentally by measuring samples with different thicknesses or by changing the incident wavelength. If the scattering profile after corrections significantly differ, chances are that multiple scattering may need to be accounted for [114]. Alternatively, the multiple scattering effect can be estimated analytically [154] or using Monte Carlo based procedures [30, 160].

Like any smearing effect, correcting data (also known as 'desmearing') for multiple scattering effects is much more involved than smearing the model fitting function. When given the choice, implementing a smearing procedure in the fitting model rather than desmearing the data is preferred [60]. Correcting for multiple scattering is generally a complex, iterative procedure where the multiple scattering smearing profile is estimated and removed from the data [114]. It becomes even more complicated for samples with direction-dependent sample thicknesses and hence different multiple scattering probabilities [12, 174]. One avenue for simplifying the correction and estimation is by approximation of the multiple scattering effect as mainly consisting of double scattering [24, 60, 11].

Besides the smearing effect of the multiple scattering phenomenon, there are more aspects which contribute to a reduction in definition, or sharpness of the scattering pattern (graphically explained in figure 9). These are: (a) the wavelength spread, (b) the beam divergence and (c) the beam profile at the sample, (d) finite sample thickness effects, and (e) the finite resolution of the detector due to detection position sensing limitations [140, 6, 70]. All these aspects effect a smearing or 'blurring' of the observed scattering pattern that can reduce the definition of sharp features, which in turn may lead to (for example) an overestimation of polydispersity. This paragraph will introduce the aforementioned five aspects.

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The wavelength spread $\frac{\Delta \lambda }{\lambda }$ directly affects the smearing in q-space as it introduces a spectrum rather than a single value for the wavelength, leading to a distribution R_w(q,〈q〉) of q around the mean 〈q〉. This smearing effect by R_w(q,〈q〉) is proportional to [140]:

$$\begin{eqnarray} &&{I}_{\mathrm{sm}}(\langle q\rangle )=\int \nolimits _{0}^{\infty }R(q,\langle q\rangle )I(q)\hspace{0.167em} \mathrm{d}q. \end{eqnarray} \tag{ 31 }$$

For most pinhole-collimated, crystal-monochromated x-ray instruments, the wavelength spread, and therefore R_w(q,〈q〉) is very small compared to the other resolution-limiting factors such as collimation effects and detector spread. The wavelength spread in these systems typically assumes values smaller than 10⁻³ and is therefore often not considered [140]. For Bonse–Hart instruments, however, the monochromaticity is tied to the instrument resolution and its consideration is therefore a necessity [26, 178, 203].

The smearing effect due to the sample thickness is similarly very small. This is the smearing effect introduced by a variation in the sample-to-detector distance due to the sample radius (as the exact origin of a scattering event may lie anywhere within the sample). The maximum relative deviation in q: $\frac{\Delta q}{q}$ can be easily derived from geometrical considerations, resulting in a function of the scattering angle 2θ, the mean distance from the sample to the detector L₀ and the sample radius r:

$$\begin{eqnarray} &&\frac{\Delta q}{q}=2-\frac{2}{\sin \theta \sqrt{\left (\frac{{L}_{0}+{r}_{\mathrm{s}}}{{L}_{0}\tan \theta }\right )^{2}+1}}. \end{eqnarray} \tag{ 32 }$$

Evaluation of the deviation given by equation (32) for typical values of 2θ,L₀ and r (5°, 1 m and 1 mm, respectively) shows that the resulting uncertainty in q due to the sample thickness is only a fraction of a per cent (0.2% for the example values given). Incidentally, the same equation can be used to determine the uncertainty in q arising from uncertainties in the sample-to-detector distance measurement.

Besides the previous two, the incident beam characteristics (in particular its profile and divergence defined by the collimation) and detector position sensing inaccuracies also cause a smearing of the detected scattering pattern [140, 6, 70]. The finite beam size, defined by the collimation, has a smearing effect similar to the previously discussed sample thickness effect, where the scattering events are distributed in space over the entire irradiated sample cross-section. This leads to a distribution of scattering angles impinging on each detector position rather than a single value. This finite beam width effect, with the beam radius denoted as r_b, can be derived in the same way as the sample thickness effect, leading to:

$$\begin{eqnarray} &&\frac{\Delta q}{q}=2-\frac{2}{\sin \theta \sqrt{\left (\frac{{L}_{0}}{{L}_{0}\tan \theta -{r}_{\mathrm{b}}}\right )^{2}+1}}. \end{eqnarray} \tag{ 33 }$$

Evaluating this for a variety of configurations shows that this effect may assume significant values for smaller angles and large beams. This is one of the aspects limiting the final resolution of an instrument, along with the beam divergence and detector position resolution.

The collimation defines both the beam size at the sample position as well as the divergence of the beam. The divergence of the beam at the sample position similarly effects an angular spread $\frac{\Delta \theta }{\theta }$ that smears the definition of q. A good derivation of this is available in the literature [140, 138]. Lastly, the finite position sensitivity of the detector introduces yet another smearing contribution, at minimum defined by the size of the individual detection elements, but commonly defined by the point-spread function of the detector [138].

While these beam- and detector-contributions can be considered separately, the compound smearing contributions can be directly evaluated as the image of the direct beam on the detector (with which the 'true' scattering convolves) [141]. It has recently been demonstrated that the compound method produces more accurate results than the separate approach in practice [38].

These smearing effects are often not considered in the evaluation or correction of pinhole-collimated x-ray scattering instruments (if they are considered, they are usually incorporated as a model smearing rather than a data desmearing). There are some notable exceptions by Le Flanchec et al [108] and Stribeck and Nochel [172], in the latter example it is applied to allow for improved intercomparability of 2D scattering patterns collected with differing collimation.

Beam desmearing corrections are more commonly applied for instruments with line-collimated beams (e.g. instruments discussed in sections 2.2.2 and 2.2.3), though model smearing rather than data desmearing is still recommended [65, 151]. As slit-smeared instruments have existed as long as small-angle scattering itself, desmearing procedures are available of many types and vintages. Some notable ones include Lake [105]; Strobl [173]; Vonk [191]; Glatter [62] and Singh et al [166]. One commonly implemented iterative desmearing procedure is described by Lake [105, 188]. The disadvantages of any desmearing procedure are their tendency to amplify small differences leading to increased noise levels, and their arbitrary cut-off criterion [173]. The former disadvantage is partially offset by the improved data accuracy of the initial data (due to the increased flux of slit-collimated instruments), and the second can be overcome through introduction of cut-off criteria [188].

At some point in the data correction procedure for isotropically scattering samples, a data reduction step is performed, known as 'integration', 'averaging' or 'binning'. For isotropically scattering samples, a reduction in dimensionality of the data usually accompanies this procedure (e.g. from 2D images to 1D plots), performed by grouping and averaging pixels with similar scattering angle q irrespective of their azimuthal angle on the detector (denoted ψ). For anisotropically scattering samples, pixels with similar q and ψ can be combined to form a new 2D dataset but with a reduced number or different distribution of datapoints [135, 132], though some dispense with binning altogether [131].

The advantages of this step are threefold. Firstly, the data become more manageable, allowing for faster fitting and improved data visualization. Secondly, the relative data uncertainties become smaller for the averaged data. Lastly, the standard deviation between similar pixels in a group (or bin) can provide a good estimate for the actual uncertainty on the average value if this standard deviation exceeds the photon counting statistics-based estimate propagated until this step¹⁴.

More specifically [133]: for radial averaging the many datapoints I_j collected from each pixel on the detector are reduced into a small number of q-bins I_qbin before the data analysis procedures. In this reduction step, each measured datapoint collected between the bin edges (class limits) q_n and q_n+1 is averaged and assumed valid for the mean $\bar {q}=\langle q\in [{q}_{n},{q}_{n+1}]\rangle$, i.e.:

$$\begin{eqnarray} &&{I}_{q\mathrm{bin}}(\bar {q})=\langle {I}_{j}(q\in [{q}_{n},{q}_{n+1}])\rangle , \end{eqnarray} \tag{ 34 }$$

with the uncertainty defined as

$$\begin{eqnarray} &&\fl {\sigma }_{\mathrm{r},q\mathrm{bin}}=\max \left \{ \begin{array}{@{}l@{}} \displaystyle \frac{1}{{N}_{q\mathrm{bin}}}\sqrt{\mathop{\sum }\limits _{q\in [{q}_{n},{q}_{n+1}]}\sigma _{\mathrm{r},j}^{2}} \\ \displaystyle \sqrt{\frac{1}{{N}_{q\mathrm{bin}}-1}\mathop{\sum }\limits _{q\in [{q}_{n},{q}_{n+1}]}\left ({I}_{j}-{I}_{q\mathrm{bin}}\right )^{2}} \end{array}\right . \end{eqnarray} \tag{ 35 }$$

where the summation is over all datapoints j falling within the bin edges and N_qbin is the total number of datapoints in the bin. As previously mentioned and evident from equation (35), the maximum value is chosen between the propagated uncertainty and the sample standard deviation in the bin. In other words, if the sample standard deviation of the pixels in the bin exceeds the estimate based on the previously propagated uncertainty, the sample standard deviation is considered a more accurate estimate. This can be further augmented to never have a relative uncertainty estimate less than 1% of the intensity, as it is (even with the most stringent corrections) challenging to get more accurate than this [78].

There is still a choice to be made in this procedure, that is the spacing between the bin edges. Normally, this is chosen either uniform or logarithmically spaced (with more data points at low values) [80]. However, for data with sharp features, a more involved choice might be preferred [153, 22].

For each measurement (indeed, each datafile), the average transmission factor for the duration of the measurement has to be calculated (e.g. for correct background subtraction). The methods for this have been given before in section 3.4.2. The on-line measurement techniques allow for constant collection of the incident beam flux as well as the transmission factor during the measurement (often with a frequency of several Hertz), which should also be stored. Deviations in the transmission factor during the measurement are a good indication of sample instability or motion.

The background corrections have been discussed in detail in section 3.4.5. Repeat the background measurement for each change in geometrical configuration or change of solvent or capillary type and size.

The actual sample measurement should be measured long enough for collection of sufficiently accurate intensities, and should be measured multiple times in sequence to check for sample instability (and possible dezingering). Multiple samples of the same material should be measured to determine the statistical uncertainty of the physical parameters resulting from the eventual pattern analysis. For dynamic systems this is not always possible, but repeated runs of the same dynamic system should provide some insights on the final uncertainties.

If the details of the dark current components are not known (see section 3.3.6), measure a new darkcurrent image (a measurement with the beam shutter closed) in case a CCD or CMOS detector is used, for each measurement duration in your measurement repertoire. Since the darkcurrent images often have a time-dependent and a time-independent component, it is necessary to measure the darkcurrent images for the same time as the actual sample measurement.

Thus, the user should determine for each sample:

• The sample name (for logging).
• The sample filename.
• The sample measurement duration.
• The sample transmission factor.
• The sample thickness (thickness in the direction and location of the direct beam).
• The incident flux onto the sample.
• Remarkable aspects regarding the sample.
• The relevant background name (identifier for logging).
• The relevant background filename (all information collected for the sample (flux, transmission, etc) should also be collected for the background measurement).
• When using image plates, the times of: the last erasure, the start of exposure, the end of exposure, the start of read-out and the end of the read-out procedure.