Validation study of small-angle X-ray scattering tensor tomography

Small-angle scattering tensor tomography (SASTT) is a recently developed technique able to tomographically reconstruct the 3D reciprocal space from voxels within a bulk volume. SASTT extends the concept of X-ray computed tomography, which typically reconstructs scalar values, by reconstructing a tensor per voxel, which represents the local nanostructure 3D organization. In this study, the nanostructure orientation in a human trabecular-bone sample obtained by SASTT was validated by sectioning the sample and using 3D scanning small-angle X-ray scattering (3D sSAXS) to measure and analyze the orientation from single voxels within each thin section. Besides the presence of cutting artefacts from the slicing process, the nanostructure orientations obtained with the two independent methods were in good agreement, as quantified with the absolute value of the dot product calculated between the nanostructure main orientations obtained in each voxel. The average dot product per voxel over the full sample containing over 10 000 voxels was 0.84, and in six slices, in which fewer cutting artefacts were observed, the dot product increased to 0.91. In addition, SAXS tensor tomography not only yields orientation information but can also reconstruct the full 3D reciprocal-space map. It is shown that the measured anisotropic scattering for individual voxels was reproduced from the SASTT reconstruction in each voxel of the 3D sample. The scattering curves along different 3D directions are validated with data from single voxels, demonstrating SASTT\u27s potential for a separate analysis of nanostructure orientation and structural information from the angle-dependent intensity distribution

Propagation of generalized vector Helmholtz-Gauss beams through paraxial optical systems

We introduce the generalized vector Helmholtz-Gauss (gVHzG) beams that constitute a general family of localized beam solutions of the Maxwell equations in the paraxial domain. The propagation of the electromagnetic components through axisymmetric ABCD optical systems is expressed elegantly in a coordinate-free and closed-form expression that is fully characterized by the transformation of two independent complex beam parameters. The transverse mathematical structure of the gVHzG beams is form-invariant under paraxial transformations. Any paraxial beam with the same waist size and transverse spatial frequency can be expressed as a superposition of gVHzG beams with the appropriate weight factors. This formalism can be straightforwardly applied to propagate vector Bessel-Gauss, Mathieu-Gauss, and Parabolic-Gauss beams, among others

Tomographic reconstruction of a three-dimensional magnetization vector field

Using x-ray magnetic nanotomography the internal magnetization structure within extended samples can be determined with high spatial resolution and element specificity, without the need for assumptions or prior knowledge of the magnetic properties of a sample. Here we present the details of a new algorithm for the reconstruction of a three-dimensional magnetization vector field, discussing both the mathematical description of the problem, and details of the gradient-based iterative reconstruction routine. To test the accuracy of the algorithm the method is demonstrated for a complex simulated magnetization configuration obtained from micromagnetic simulations. The reconstruction of the complex three-dimensional magnetic nanostructure, including the surroundings of magnetic singularities (or Bloch points), exhibits an excellent qualitative and quantitative agreement with the simulated magnetic structure. This method provides a robust route for the reconstruction of internal three-dimensional magnetization structures obtained from x-ray magnetic tomographic datasets, which can be acquired with either hard or soft x-rays, and can be applied to a wide variety of three-dimensional magnetic systems

Three-dimensional magnetization structures revealed with X-ray vector nanotomography

In soft ferromagnetic materials, the smoothly varying magnetization leads to the formation of fundamental patterns such as domains, vortices and domain walls<sup>1</sup>. These have been studied extensively in thin films of thicknesses up to around 200 nanometres, in which the magnetization is accessible with current transmission imaging methods that make use of electrons or soft X-rays. In thicker samples, however, in which the magnetization structure varies throughout the thickness and is intrinsically three dimensional, determining the complex magnetic structure directly still represents a challenge<sup>1, 3</sup>. We have developed hard-X-ray vector nanotomography with which to determine the three-dimensional magnetic configuration at the nanoscale within micrometre-sized samples. We imaged the structure of the magnetization within a soft magnetic pillar of diameter 5 micrometres with a spatial resolution of 100 nanometres and, within the bulk, observed a complex magnetic configuration that consists of vortices and antivortices that form cross-tie walls and vortex walls along intersecting planes. At the intersections of these structures, magnetic singularities—Bloch points—occur. These were predicted more than fifty years ago<sup>4</sup> but have so far not been directly observed. Here we image the three-dimensional magnetic structure in the vicinity of the Bloch points, which until now has been accessible only through micromagnetic simulations, and identify two possible magnetization configurations: a circulating magnetization structure<sup>5</sup> and a twisted state that appears to correspond to an ‘anti-Bloch point’. Our imaging method enables the nanoscale study of topological magnetic structures<sup>6</sup> in systems with sizes of the order of tens of micrometres. Knowledge of internal nanomagnetic textures is critical for understanding macroscopic magnetic properties and for designing bulk magnets for technological applications<sup>7</sup>

Quantitative interior x-ray nanotomography by a hybrid imaging technique

Hierarchical structures appear often in life and materials sciences, and their characterization profits greatly from imaging methods that allow seamless probing of various length scales without sacrificing image quality. X-ray tomography is particularly adept at probing 3D structures; however, zooming in on a region of interest results in a loss of quantitativeness of image contrast and suffers from artifacts unless a priori knowledge or assumptions about the sample are used. Here, we demonstrate a hybrid technique that exploits a micrometer-resolution overview to realize ab initio nanoscale interior tomography with quantitative contrast. In a study of avian eggshell, a model for bionanoporous materials, our approach reveals a complex arrangement of vesicles with sizes ranging from hundred nanometers to a few micrometers. We anticipate that such an approach can be widely adopted and benefited from at synchrotron and laboratory sources, for instance, where such zooming capabilities are already present or can be readily realized

Röntgenblick für Magnete

Wirbel und Wände – das Innere eines Magneten ist komplexer als es den Anschein hat. Mit Hilfe von Röntgenstrahlen ist es unserer Gruppe aus Forschern vom Paul Scherrer Institut, der ETH Zürich und der Universität Glasgow gelungen, die magnetische Struktur mikrometergroßer Proben mit bisher unerreichter Auflösung abzubilden

Ultrastructure organization of human trabeculae assessed by 3D sSAXS and relation to bone microarchitecture

Although the organization of bone ultrastructure, i.e. the orientation and arrangement of the mineralized collagen fibrils, has been in the focus of research for many years for cortical bone, and many models on the osteonal arrangement have been proposed, limited attention has been paid to trabecular bone ultrastructure. This is surprising because trabeculae play a crucial role for the mechanical strength of several bone sites, including the vertebrae and the femoral head. On this account, we first validated a recently developed method (3D sSAXS or 3D scanning small-angle X-ray scattering) for investigating bone ultrastructure in a quantitative and spatially resolved way, using conventional linearly polarized light microscopy as a gold standard. While both methods are used to analyze thin tissue sections, in contrast to polarized light microscopy, 3D sSAXS has the important advantage that it provides 3D information on the orientation and arrangement of bone ultrastructure. In this first study of its kind, we used 3D sSAXS to investigate the ultrastructural organization of 22 vertebral trabeculae of different alignment, types and sizes, obtained from 4 subjects of different ages. Maps of ultrastructure orientation and arrangement of the trabeculae were retrieved by stacking information from consecutive 20-?m-thick bone sections. The organization of the ultrastructure was analyzed in relation to trabecular microarchitecture obtained from computed tomography and to relevant parameters such as distance to trabecular surface, local curvature or local bone mineralization. We found that (i) ultrastructure organization is similar for all investigated trabeculae independent of their particular characteristics, (ii) bone ultrastructure exhibiting a high degree of orientation was arranged in domains, (iii) highly oriented ultrastructural areas were located closer to the bone surface, (iv) the ultrastructure of the human trabecular bone specimens followed the microarchitecture, being oriented mostly parallel to bone surface, and (v) local surface curvature seems to have an effect on the ultrastructure organization. Further studies that investigate bone ultrastructure orientation and arrangement are needed in order to understand its organization and consequently its relation to bone biology and mechanics

High-speed tensor tomography: iterative reconstruction tensor tomography (IRTT) algorithm

The recent advent of tensor tomography techniques has enabled tomographic investigations of the 3D nanostructure organization of biological and material science samples. These techniques extended the concept of conventional X-ray tomography by reconstructing not only a scalar value such as the attenuation coefficient per voxel, but also a set of parameters that capture the local anisotropy of nanostructures within every voxel of the sample. Tensor tomography data sets are intrinsically large as each pixel of a conventional X-ray projection is substituted by a scattering pattern, and projections have to be recorded at different sample angular orientations with several tilts of the rotation axis with respect to the X-ray propagation direction. Currently available reconstruction approaches for such large data sets are computationally expensive. Here, a novel, fast reconstruction algorithm, named iterative reconstruction tensor tomography (IRTT), is presented to simplify and accelerate tensor tomography reconstructions. IRTT is based on a second-rank tensor model to describe the anisotropy of the nanostructure in every voxel and on an iterative error backpropagation reconstruction algorithm to achieve high convergence speed. The feasibility and accuracy of IRTT are demonstrated by reconstructing the nanostructure anisotropy of three samples: a carbon fiber knot, a human bone trabecula specimen and a fixed mouse brain. Results and reconstruction speed were compared with those obtained by the small-angle scattering tensor tomography (SASTT) reconstruction method introduced by Liebi et al. [Nature (2015), 527, 349–352]. The principal orientation of the nanostructure within each voxel revealed a high level of agreement between the two methods. Yet, for identical data sets and computer hardware used, IRTT was shown to be more than an order of magnitude faster. IRTT was found to yield robust results, it does not require prior knowledge of the sample for initializing parameters, and can be used in cases where simple anisotropy metrics are sufficient, i.e. the tensor approximation adequately captures the level of anisotropy and the dominant orientation within a voxel. In addition, by greatly accelerating the reconstruction, IRTT is particularly suitable for handling large tomographic data sets of samples with internal structure or as a real-time analysis tool during the experiment for online feedback during data acquisition. Alternatively, the IRTT results might be used as an initial guess for models capturing a higher complexity of structural anisotropy such as spherical harmonics based SASTT in Liebi et al. (2015), improving both overall convergence speed and robustness of the reconstruction