Ptychography is a lensless imaging technique that aims to reconstruct an object from a set of coherent diffraction patterns originating from different and partially overlapping sample illumination areas. For a successful convergence of the iterative algorithms used, the sample scan positions have to be known with very high accuracy. Here, we present a method that allows to directly encode this information in the diffraction patterns without the need of accurate position encoders. Our approach relies on combining ptychography with another coherent imaging method, namely Fourier-transform holography. We have imaged two different objects using coherent soft-X-ray illumination and investigate the influence of experimental and numerical position refinement on the reconstruction result. We demonstrate that holographically encoded positions significantly reduce the experimental and numerical requirements. Our ptychographic reconstructions cover a large field of view with diffraction-limited resolution and high sensitivity in the reconstructed phase shift and absorption of the objects.
© 2016 Optical Society of America
Through the combination of modern coherent X-ray sources and today’s computational power, X-ray nanoscale imaging based on solving the phase problem by numerical inversion of the object’s diffraction pattern is made possible with reasonable expense . As these techniques are independent of any image-forming optical element, the achievable spatial resolution is limited, in principle, only by the wavelength of the incident light and, in practice, by the available photon flux and radiation damage of the specimen. Furthermore, coherent X-ray imaging methods give access to the full complex transmission function of the object, i.e., they map both absorption and phase shift.
The general idea of coherent diffractive imaging (CDI) is to iteratively search for a solution to the phase problem that is in agreement with both the measured diffraction pattern and a priori real-space information about the object such as its physical extent . In ptychography, additional redundancy is introduced into the phase retrieval problem by recording scattering patterns of several overlapping object areas, i.e., by scanning a constant illumination across the sample . As a result, the convergence of the iterative algorithms is faster and more robust to, e.g., noise and missing data compared to classical CDI from a single diffraction pattern [4,5].
The redundancy in the data has been exploited to recover the illumination function [6–8], missing data in the diffraction patterns , modes in incoherent illumination [9,10], and refined positions of the illuminated areas on the sample [8,11–15]. Consequentially, these recovery possibilities relax the requirement of accurate knowledge of some of the a priori information, allowing for images to be reconstructed from undersampled diffraction data  and—to a certain extent—from inaccurate scan positions.
In order to successfully reconstruct a ptychographic data set, the precise knowledge (or retrieval) of the scan positions has been found to be one of the crucial conditions [11,13–15]. The development of position correction algorithms—even down to the correction of sub-pixel errors—significantly improved the image resolution and quality. Nevertheless, these correction algorithms still rely on input positions that have to be determined during the imaging experiment with sufficient accuracy. If the position error becomes comparable to the scan step size [11,14], the phase retrieval algorithms are likely to fail in recovering true positions and, ultimately, the image with high resolution. In addition, the numerical recovery of scan positions—in particular for large deviations—significantly increases computational time.
In this paper, we present a simple and efficient method to determine scan positions without the need for extended computational power. The method is based on holographically encoding the positions directly into the recorded scattering patterns. Our approach, thus, works without any a priori knowledge of the illumination position and dispenses the need of accurate encoders for the sample motion, e.g., via stepping motors. With our position correction method, robust positions are provided within the certainty allowed by the maximal numerical aperture (pixel certainty). In combination with conventional numerical position correction methods, sub-pixel refinement is also possible.
The holographic information is encoded by the principle of Fourier-transform holography (FTH) [17,18]. Its experimental implementation requires only minor changes in the ptychography setup if a pinhole aperture upstream of the sample restricts the illumination on the sample, which is the common ptychography setup when using soft X-rays [19,20]. Besides this aperture defining the probe, we introduce an additional small aperture, which acts as a source for a coherent reference beam [Fig. 1]. Due to the interference between the reference beam and X-rays scattered from the object, each diffraction pattern can now be perceived as a hologram, which can be directly inverted to a real-space image by a single Fourier transformation. We use these images of the object area illuminated in each exposure to directly determine the position shift between adjacent exposures.
Our approach is similar to previous soft-X-ray FTH experiments using an optics mask which is separated from the sample substrate and, thus, movable in respect to the sample in order to extend the limited field of view (FOV) [21,22]. However, through the subsequent ptychographic reconstruction, we are additionally able to (i) overcome limitations in the spatial resolution given by the size of the reference pinhole, (ii) independently recover the object’s transmission function and the illumination function, (iii) increase the photon efficiency, i.e., signal-to-noise ratio (SNR) in the images, (iv) recover missing data at low spatial frequencies, and (v) align the images in respect to each other with sub-pixel accuracy.
Also compared to the standard ptychography method (for implementations in the soft-X-ray regime see [19,20]), our extension has several advantages. The high certainty in the scan position improves the convergence of the phase retrieval algorithms and, therefore, allows to, e.g., use less overlap between adjacent scan positions or to recover missing information in the diffraction patterns due to a beamstop in the center of the detector. Moreover, the direct image retrieved from the holographic reconstruction can be used as a real-time feedback during the experiment to identify an area of interest and, additionally, as a starting guess for ptychography. As we will show, this combination of two different imaging methods also helps to improve the consistency of the reconstructed images and to identify imaging artifacts of either method.
2. Experimental details
We have applied our holography-guided ptychography method to image two different objects both prepared on the same 100 nm thick Si3N4 substrate [Fig. 2(a)]. First, a Siemens sector star served as a test structure for an assessment of the technique and for characterization of the illumination. Subsequently, a cluster of differently shaped fossil diatom shells was imaged. Diatoms are hard-shelled algae with silica cell walls forming complex structures in the nanometer regime. The diatoms were first deposited on the Si3N4 membrane from aqueous suspension and then air-dried. Subsequently, the Siemens star pattern was produced by focused electron-beam induced deposition (FEBID) of platinum from a methylcyclopentadienyl-platinumtrimethyl (MeCpPtMe3) precursor gas at an accelerating voltage of 15 kV and 340 pA beam current .
The beam-defining asymmetric aperture with 10 µm in diameter was fabricated into another 400 nm thick Si3N4 membrane, which was coated with gold of 1100 nm thickness, using focused-ion-beam (FIB) milling. The holographic reference pinhole of 300 nm in diameter was placed in a distance of 20 µm from the aperture center [Fig. 2(b)].
The measurements were carried out at the undulator beamline UE52-SGM at the BESSY II synchrotron source with a photon energy of 200 eV, corresponding to a wavelength of λ = 6.2 nm. The pinhole mask was brought into close proximity to the sample plane, while avoiding any contact between both elements and, therefore, sample damage. The distance between sample and mask was found later to be 45 µm by numerical focusing, i.e., free-space propagation of the FTH images. Although it is still possible to numerically focus the FTH reconstruction (down to the resolution defined by the reference hole) for much larger distances on the order of several 100 µm, the holographic reconstruction will suffer from an uneven illumination due to Fresnel diffraction from the aperture illuminating the object .
The sample was mounted on positioning stages driven by piezoelectric stepper actuators in closed-loop operation using optoelectronic encoders for all three dimensions parallel (x) and transversal (y, z) to the beam axis (attocube ANP series). The sample positions of interest were identified during the experiment by real-time FTH reconstructions. The diffraction patterns were recorded in the far-field in a distance of 0.325 m from the sample plane with a charge-coupled device (CCD) detector (Princeton Instruments) with a chip containing 2048 × 2048 pixels with 13.5 µm in size. The maximal scattering angle fully covered by the detector was 2.44° corresponding to a real-space pixel size in the reconstructions of 73 nm. The intense direct beam was blocked by a beamstop in order to exploit the limited dynamic range of the CCD camera efficiently for detecting low intensities at large scattering angles. The diameter of the area shadowed by the beamstop was 1.03 mm, which is approximately twice as large as the central Airy disk resulting from the beam-defining aperture.
For the ptychography scan of the Siemens star, 19 overlapping regions were recorded on a quadratic raster grid with a target step size of 2.5 µm in both directions, corresponding to an overlapping ratio of 0.75. At each scan position, 20 images with 4 s acquisition time and 4 s detector read-out time were recorded. The positioning stages reached a new target position within a time below 1 s. In total, the time for the full scan adds up to about 1 h also including 4 dark images for background subtraction. The diatoms were imaged in 59 overlapping regions on the same grid as the Siemens star. For each diffraction pattern, we recorded 5 images with 0.5 s acquisition time and 4 s read-out time, resulting in a total scan time (again including dark images) of about 25 min.
The holograms were reconstructed by a 2-dimensional Fourier transformation with subsequent free-space propagation of 45 µm . The beamstop shadow in the holograms was smoothed with a Gaussian convolution before reconstruction. All reconstructions shown were zero-padded by a factor of 2. The position shift between adjacent FTH reconstructions was determined by using selected sharp features and, additionally, by spatial cross-correlation of the images. The scan position obtained from the FTH images were then used as input for the following ptychographic reconstruction.
The ptychograms were reconstructed using the extended ptychographic iterative engine (ePIE)  with optional position correction (pcPIE) . If not otherwise specified, the algorithms were applied in the following order: 300 iterations of ePIE, 250 iterations of pcPIE with sub-pixel correction and 50 ePIE iterations. The convergence was monitored using the standard error metric as defined in . The Fourier-space constraint was not applied in the area of the beamstop shadow where diffraction data is missing. We achieved the best results in recovering this data when using a non-zero object starting guess. We have started with a transversaly constant amplitude and zero phase. The position correction was performed as described in  by calculating three additional guesses per iteration and scan position with randomly shifted positions in the interval (–6 pixels, 6 pixels) independently for the y and z direction corresponding to a maximal shift of 438 nm in each direction. The object guess resulting in a diffraction pattern with the smallest deviation from the measured data is then further used for the next iteration.
The holographically reconstructed images can also be employed to create a first guess for the probe function. We have generated a binary mask representing the aperture from a non-propagated FTH reconstruction. The guess for the probe is retrieved by free-space propagation of this mask by the distance between aperture and sample plane. For the first 10 iterations of ePIE, the probe function was not updated.
3. Experimental results and discussion
In Fig. 3, typical diffraction patterns from the Siemens star and the diatom data set are shown. In both scattering patterns, we observe characteristic streaks from the asymmetric object aperture that can be used to determine the orientation of the beam defining aperture. Even though the accumulation time for each image in the data set of the Siemens star (80 s) was much higher than for the diatoms (2.5 s), the scattering intensity at high momentum transfer in the former is still lower. This finding is related to the low thickness and high carbon content of the deposited structure as will be discussed later.
First object: Siemens star
Figures 4(a) and 4(b) display two adjacent holographic reconstructions of the Siemens star object with post-processing treatment as described. Due to the fixed position of the object aperture in respect to the reference hole on the optics mask, the imaged FOV always appears at the same position in the numerical reconstruction matrix . The fringes around the object aperture appear due to Fresnel diffraction over the distance between aperture and sample. The blue arrow marks the shift of the same selected feature between two adjacent scan positions. Using the intensity maximum of the spatial cross-correlation of both images (excluding the diffraction from the aperture), we are able to determine the shift of the sample between both images with one-pixel accuracy [Fig. 4(c)]. Using more sophisticated image registration algorithms , the shift can possibly be determined even with sub-pixel accuracy. Such correlation and registration methods are also able to deal with noisy images in the case of weakly scattering objects that hardly produce a distinguishable FTH image from a single exposure.
The spatial resolution of the holographic reconstructions was assessed by the fading Siemens star spokes toward its center to about 190 nm.
In Fig. 4(d), the positions obtained from the FTH images are compared to the encoder readout from the positioners. According to the readout, the chosen step size of 2.5 µm was realized within a small error (< 20 nm) (except for two positions). To correct for backlash of the whole positioning system, the scan positions were always approached from the same direction. When moving into a new line an additional backlash correction was applied. The comparison shows a significant deviation of the reported positions from the actual positions obtained from the holographic data. A considerable backlash is present at the beginning of each line and the resulting mean step size of (2.1 ± 0.2) µm is smaller than intended. As these errors systematically add up (after 18 movements, we observe a deviation of 4.7 µm), it is expected that the pcPIE position correction assuming individual small random deviations at each scan position is likely to fail in recovering the true positions. On the other hand, the positioning errors also lead to deviations from the regular scan pattern, which is known to produce artifacts in the ptychographic reconstruction.
In order to combine the separate FTH reconstructions into a single image with extended FOV, we have performed ptychographic reconstructions based on the ePIE algorithm using different methods for position correction as presented in Figs. 5(a)–5(c). The reconstruction in Fig. 5(a) was obtained by using the standard reconstruction scheme with numerical position correction as outlined in the experimental details (300 iterations ePIE, 250 pcPIE, 50 ePIE) starting with the positions reported from the encoders. The next reconstruction in Fig. 5(b) is based on the positions from the FTH images, but has been performed without numerical correction (i.e., 600 iterations ePIE). And finally, for the reconstruction in Fig. 5(c), the positions from FTH and numerical correction down to sub-pixels were implemented.
As expected, the pcPIE reconstruction based on the encoder positions fails in retrieving the actual positions and, consequentially, in reconstructing a satisfactory image. The Siemens star is only visible as a blurry disk. This finding demonstrates that the numerical position correction has to rely on reasonably precise input parameters. In contrast, the Siemens star object is distinguishably reconstructed when using the position correction from the FTH images. When comparing amplitude and phase reconstruction, the phase image appears to have higher contrast, less noise and less artifacts (in particular toward the center of the star). The reconstruction applying pcPIE based on the FTH positions as input further improves the image quality compared to the ePIE run without numerical correction. The background (i.e., the transparent Si3N4 membrane) appears much more uniform and the spatial resolution improves from 230 nm (ePIE) to 150 nm (pcPIE) with the latter clearly surpassing the resolution of the FTH images. In particular, the image quality in the reconstructed amplitude benefits from the position correction as the amplitude now explicitly reproduces the phase image. After 250 iterations of pcPIE with sub-pixel corrections, the average correction of the positions was (0.8 ± 0.7) pixels, corresponding to (58 ± 51) nm. We thus conclude that, firstly, the small correction demonstrates that indeed the positions of the illumination could be retrieved within the resolution of one pixel (73 nm) using our holographic implementation. Secondly, the pcPIE reconstructions again show the high sensitivity of the phase retrieval process to the accuracy of the scan positions.
While the spatial resolution in the ptychographic reconstruction is better than in the FTH reconstructions, we also observe small differences in the structure of the Siemens star between both imaging methods. In particular toward the center of the Siemens star, the spokes appear as being bent in the ptychographic reconstructions in Figs. 5(b) and 5(c) (see also magnification in Fig. 5(d)). This finding is not supported by the FTH images [Fig. 4(a)]. We have, thus, made an attempt to employ the FTH reconstruction as starting guess for ptychography in order to trigger the iterative phase retrieval. At the position of the center of the Siemens star, we have replaced the real part of the otherwise uniform starting guess with an FTH reconstruction of the Siemens star. Using this starting guess and our standard reconstruction scheme (with numerical position correction), we obtain the reconstruction as presented in Fig. 5(e) and Fig. 6. The described artifacts are clearly reduced in this image. The remaining slight bending only appears at spatial frequencies much smaller than resolved by the FTH images. This example shows how a combination of the two different imaging methods helps to improve the consistency of the reconstructed object.
In Fig. 6, we show the full final reconstruction (using FTH positions, FTH starting guess, and numerical position correction) of the Siemens star together with the reconstructed illumination function. Below the actual FOV, also sample regions illuminated by X-rays passing through the reference hole at different scan positions become visible in the reconstruction. In the illumination function, the light field passing through both the object aperture and the reference hole of the optics mask is convincingly reconstructed. Due to the distance between mask and sample, Fresnel fringes appear at the rim of the object aperture and in the diverging reference wave. By numerical free-space propagation of the light field into the mask plane, we obtain sharp edges at the object aperture and an almost point-like source for the reference beam, both in agreement with the SEM image as shown in Fig. 2.
In the last step of our analysis, we have employed the high quantitative accuracy of the ptychographic reconstruction of the object’s transmission function to assess the composition and thickness d of the deposited structure. Typically, Pt structures prepared with FEBID have a high content of carbon originating from the precursor gas [23,25]. Neglecting other impurities, we denote the Pt content of the deposited material with c and the C content with 1 – c. In contrast to previous assessments of that kind as, e.g., in [19,20,26], we have used the phase shift Δφ and the absorption Δμ of the structure in order to determine both c and d. The refractive index of a compound material is given by [27,28]:
Here, r0 denotes the classical electron radius and na the overall density of atoms in the compound. Each element q of the compound is characterized by its atomic content cq, and its atomic scattering factors and . Essentially, the atomic constants and can directly be calculated from the tabulated atomic scattering factors , or from the classical optical constants (δ and β) of each element normalized to its atomic density. The phase shift the X-rays acquire when traveling through our two-component material is, thus, calculated as :
The phase φ of the Siemens star and the phase φ0 of the substrate are taken from the reconstruction. Similarly, the absorption Δµ of the material determined from the amplitude of the Siemens star (A) and of the membrane (A0) is given by:
These two equations lead to an expression for the Pt content:
In our ptychographic reconstruction [Fig. 6], we determined Δφ = 0.15 ± 0.03 and Δμ = 0.077 ± 0.017 for the Siemens star in respect to the surrounding substrate. As a result, we derive a Pt content c = 0.19 ± 0.09. This result is in line with energy dispersive X-ray spectroscopy (EDX) measurements at similar nanostructures showing a Pt content of 16–17% . By estimating the compound density—that is otherwise unknown—from the densities of the pure elements (nPt = 66 nm−3 and nC = 113 nm−3) as , we derive a layer thickness of d = (16 ± 3) nm. The absorption contrast is mainly (to 86%) generated by the Pt atoms that would add up to a thickness of only 3 nm of the pure element. On the other hand, the high C content is responsible for most (71%) of the phase contrast.
Second object: diatoms
Based on our experience treating the Siemens star data set, we have reconstructed the much larger data set for the diatoms with a combination of ePIE and pcPIE as described in the experimental details. From the holographic images, we again retrieved the corrected position for each exposure. As the mean step size of (2.1 ± 0.4) µm for the scan was smaller than intended (2.5 µm), the average overlapping ratio between adjacent exposures also increases to 0.89. The ptychographic reconstruction was initialized using the positions from the holograms, the probe function from the Siemens star reconstruction and a constant amplitude with zero phase as initial object guess. The probe function was continuously updated after the first 10 iterations; the position correction was started after 300 ePIE iterations. The numerical position correction (250 iterations of pcPIE) shifted the images by (2.4 ± 2.1) pixels ((175 ± 153) nm) on average.
We present the final result of the reconstruction in Fig. 7. The investigated FOV is approximately 34 µm × 22 µm containing four distinct, differently shaped diatom shells, some small diatom fragments and part of the Si frame supporting the membrane at the top of the FOV. Due to the low energy of the X-rays used, some objects have a high absorption up to almost opacity (in particular, the Si frame). For the same reason, we observe unphysical phase discontinuities, which were “unwrapped” in the image shown , as well as regions with undefined, noisy phase in the phase contrast image.
The high absorption of some objects and resulting high contrast with respect to the nearly transparent substrate also adversely affects the ptychographic reconstruction process. In our first reconstructions (not shown), we have observed that the reconstructed absorption of those objects was too low—as again witnessed by the holographic reconstructions. Correspondingly, the missing diffraction intensities due to the beamstop shadow, which were retrieved by the ePIE algorithm, were too high in the final Fourier-space guess. We successfully applied the following solution to this problem: first, we conducted the reconstruction scheme as described in the experimental methods, i.e., 300 iterations of ePIE and 250 iterations of pcPIE. During another three iterations of ePIE, we dynamically scaled the recovered intensities by matching the intensities in the outer rim (of 6 pixels width) of the beamstop area with the known intensities in a rim (also of 6 pixels width) adjacent to the beamstop area. Subsequently, the reconstruction was refined using 300 standard iterations of ePIE resulting in the images as presented in Fig. 7.
Both the absorption contrast and the phase contrast image resemble the SEM image of the diatoms [Fig. 2(a)] to a very high degree when considering the different contrast mechanisms. Many details visible in the surface-sensitive SEM image can also be found in our ptychographic projection image except for regions where the total absorption is too high. The phase contrast image can be directly scaled to an integrated height image using the relation [(similar to Eq. (2)] d = Δφλ/(2πδ) where Δφ denotes the phase difference to the surrounding substrate, and 1–δ the real part of the refractive index of silica. However, due to the projection nature of the X-ray images, the actual 3D shape of the diatoms is not directly accessible. A similar diatom as the spherical sieve-like diatom in the top-right corner has already been imaged using tomographic X-ray holography showing a certain curvature of the overall shape . The long ladder-like diatom is actually shaped as a bisected hollow cylinder as the SEM image suggests. Due to this shape the diatom appears to be much thicker at the long edges. A slice through one of these edges [Fig. 8], thus, represents a slice through the wall of the diatom. The slice as shown in Fig. 8(d) also reveals the very faint bridges in the openings of the diatom shell with a phase shift of 0.14 rad corresponding to a SiO2 thickness of 17 nm. For comparison, the standard deviation of the phase across the uniform substrate is only 17 mrad.
We have assessed the spatial resolution at several object edges (10%–90% criterion) in the reconstructed images with one example shown in Fig. 8. The resolution of the ptychographic reconstruction (120 nm) is significantly better than the resolution from the holographic reconstructions (220 nm) and is predominately limited by the maximum recorded scattering angle as we observe strong scattering up to the rim of the CCD. Our findings again demonstrate that the ptychographic phase retrieval algorithm can retrieve more spatial information from the scattering data than encoded in the holographic interference pattern. However, without the information encoded in the holograms, the error in the position readout would have been too large to reliably employ the numerical position corrections only.
We have applied the calculations of the material composition as presented for the Siemens star also for the diatoms. As the diatom material is known to be SiO2, the calculation serves as a control experiment. When we assume that the material is solely made of Si and O, we derive an atomic Si content of 0.35 ± 0.10 which is in agreement with expected content in silica.
As a final remark, we would like to point out that the position errors reported here are orders of magnitude larger than typically present in state-of-the-art ptychographic imaging setups . Piezoelectric positioning stages, potentially equipped with interferometric position control, with 10 nm accuracy and better are nowadays commercially available. We, thus, refrain from proposing to apply our method combination at highly advanced and automated ptychographic imaging setups. However, we see potential application for setups that combine very different X-ray methods or for setups that realize elaborate sample environments. For example, one could think about spectroscopic measurements of nano-structured samples where an in situ real-space image would help to understand the structure–function relation. Due to space constraints—particularly for soft-X-ray experiments in high vacuum—it might be difficult to implement ultra-high-resolution stages. Sample environments such as magnetic fields, extreme temperatures, or flowing gases or liquids may impair the position stability and control. Our method can be implemented with fairly every positioning system with micrometer resolution. The intermediate FTH imaging step not only provides the scan positions, it can also be used as a real-time monitor of the sample area scanned, as a reliable cross-check for the ptychographic reconstruction and as a fallback if ptychography fails to retrieve a solution at certain scan positions or even completely.
In addition, ptychography experiments at X-ray sources based on non-linear processes such as free-electron lasers (FELs) or high-harmonic generation (HHG) sources could benefit from a holographic monitor of the beam position. At these sources, the beam position and pointing direction considerably fluctuates from pulse to pulse. Ptychographic imaging has been used to characterize the illumination function of the FEL beam on a test sample, where position refinement has been mandatory in order to retrieve an image of the sample and of the average illumination .
We have presented an advanced coherent X-ray imaging method combining Fourier-transform holography with ptychography. We particularly use the position information retrieved from the holograms as input for the ptychographic reconstruction. In addition, we have employed the holographic reconstruction as immediate feedback at the experiment, to identify artifacts in the ptychographic reconstruction, and as a starting object and probe guess for iterative image retrieval. The final image is retrieved from the ptychographic reconstruction as it provides a larger field of view and higher spatial resolution. The ptychographic reconstruction also disentangles the object’s transmission function from the illumination.
We have shown that the ptychographic reconstruction fails when relying on scan positions from the positioner encoders with errors on the order of the step size. Our reconstructions demonstrate that the scan positions have to be known or recovered with a precision smaller than the intended spatial resolution of the reconstructed images.
For both investigated objects, the Siemens star and the diatoms, the quality of both the absorption and the phase contrast images is very high allowing us to extract additional information such as material thickness and composition of the objects. We speculate that the high image quality and reliable convergence of the iterative algorithms could also be related to the broad spatial-frequency spectrum of the probe due to the superimposed divergent reference beam . We reached a phase sensitivity (as limited by the noise in the actual reconstructions) of 17 mrad. The achieved spatial resolution of 120 nm to 150 nm is already diffraction limited and can readily be improved using a setup that allows to probe higher momentum transfer, e.g., via a smaller sample–CCD distance. Our holography-guided ptychography approach, thus, provides a route for robust lensless imaging with reduced experimental and numerical effort in the soft-X-ray regime with high relevance to applications in the water window or applications under resonant X-ray scattering conditions.
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