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The use of a mathematical model is proposed in order to denoise X-ray two-dimensional patterns. The method relies on a generalized diffusion equation whose diffusion constant depends on the image gradients. The numerical solution of the diffusion equation provides an efficient reduction of pattern noise as witnessed by the computed peak of signal-to-noise ratio. The use of experimental data with different inherent levels of noise allows us to show the success of the method even in the case, experimentally relevant, when patterns are blurred by Poissonian noise. The corresponding MatLab code for the numerical method is made available.

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A mm thick free-standing gel containing lipid vesicles made of 2-oleoyl-1-palmitoyl-sn-glycero-3-phosphocholine (POPC) was studied by scanning Small Angle X-ray Scattering (SAXS) and X-ray Transmission (XT) microscopies. Raster scanning relatively large volumes, besides reducing the risk of radiation damage, allows signal integration, improving the signal-to-noise ratio (SNR), as well as high statistical significance of the dataset. The persistence of lipid vesicles in gel was demonstrated, while mapping their spatial distribution and concentration gradients. Information about lipid aggregation and packing, as well as about gel density gradients, was obtained. A posteriori confirmation of lipid presence in well-defined sample areas was obtained by studying the dried sample, featuring clear Bragg peaks from stacked bilayers. The comparison between wet and dry samples allowed it to be proved that lipids do not significantly migrate within the gel even upon drying, whereas bilayer curvature is lost by removing water, resulting in lipids packed in ordered lamellae. Suitable algorithms were successfully employed for enhancing transmission microscopy sensitivity to low absorbing objects, and allowing full SAXS intensity normalization as a general approach. In particular, data reduction includes normalization of the SAXS intensity against the local sample thickness derived from absorption contrast maps. The proposed study was demonstrated by a room-sized instrumentation, although equipped with a high brilliance X-ray micro-source, and is expected to be applicable to a wide variety of organic, inorganic, and multicomponent systems, including biomaterials. The employed routines for data reduction and microscopy, including Gaussian filter for contrast enhancement of low absorbing objects and a region growing segmentation algorithm to exclude no-sample regions, have been implemented and made freely available through the updated in-house developed software SUNBIM.

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The Debye Function Analysis of diffraction patterns from nanosized mineral crystals showing different average degrees of maturity was carried out on engineered bone samples. The analysis relied on a bivariate family of atomistic hydroxyapatite nanocrystal models and provided information about crystal structure, size and shape distributions of the mineral component of the newly formed bone. An average rod-like shape of nanocrystals was found in all samples, with average sizes well matching the collagen I gap region. The diffraction patterns investigated through the Debye Function Analysis were used as signal models to perform the Canonical Correlation Analysis of high resolution X-ray micro-diffraction patterns collected on porous and resorbable hydroxyapatite/silicon-stabilized tricalcium phosphate (Si-TCP) implants. The nosologic maps clearly showed a size gradient in the new formed bone that validates the mechanism (mimicking the bone remodelling in orthotopic bones) of a continuous deposition of bone by osteoblasts, an increasing mineralization of the newly deposited bone, a growth of the new crystals, at the same time that osteoclasts adhere to the scaffold surface and resorb the bioceramic. The comparison of samples at different implantation times proved that the selective resorption of Si-TCP component from the scaffold was already evident after two and almost complete after six months.

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The hierarchical structure of bone makes the X-ray microdiffraction scanning techniques one of the most effective tool to investigate the structural features of this tissue at different length scales: the atomic/nanometer scale of the X-ray scattering signals and the macroscopic scale of the scanned sample area. The potentiality of the microdiffraction approach has been verified also by investigations on tissue-engineered bone substitutes used to repair large hard bone defects. The aim of this review is to present the most representative and recent results obtained through high-resolution scanning microdiffraction techniques studying both natural and tissue-engineered bone. The rapid evolution of the instrumental set-ups and the advanced methods of data analysis are described. Recent examples in which X-ray microbeams were used for imaging quantitative features of natural bone tissue and engineered bone substitutes are presented along with the qualitative and quantitative information extracted from the two-dimensional patterns collected on bone samples and on ex vivo cell seeded bioceramic implants. Thanks to the microdiffraction approach, several aspects of the mechanisms leading to the generation of the new bone, coupled to the scaffold resorption in the tissue-engineered constructs, have been tentatively interpreted. The potential of X-ray microdiffraction as an imaging tool in the field of bone tissue engineering is discussed and the key role of high-spatial resolution, availability of automatic tools (for dealing with the huge amount of experimental data) and advanced analysis techniques is elucidated. Finally, future perspectives in the field are presented.

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We present an ab initio numerical many-body GW calculation of the band plot in freestanding graphene. We consider the full ionic and electronic structure introducing e-e interaction and correlation effects via a self-energy containing non-Hermitian and dynamical terms. With respect to the density-functional theory local-density approximation, the Fermi velocity is renormalized with an increase of 17%, in better agreement with the experiment. Close to the Dirac point the linear dispersion is modified by the presence of a kink, as observed by angle-resolved photoemission spectroscopy. We demonstrate that the kink is due to low-energy pi-->pi* single-particle excitations and to the pi plasmon. The GW self-energy does not open the band gap.

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The probabilistic formula provided by Hauptman and Giacovazzo for estimating three-phase invariants when anomalous scatterers are present is revisited. Its main defects are: (a) it is absolutely resistant to any attempt at interpreting it in terms of parameters accessible via the experiment; (b) its calculation is time consuming and requires computing resources. A distribution based on interpretable estimates of the parameters is proposed. The role of the old and the new expressions in the single-wavelength anomalous diffraction (SAD) techniques is discussed, and compared with the role of analogous formulas estimating triplet invariants from isomorphous diffraction data.

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The probabilistic formula derived via the rigorous method of the joint probability distribution function to estimate protein phases in the single-wavelength anomalous diffraction (SAD) case [Giacovazzo & Siliqi (2001). Acta Cryst. A57, 40-46] has been revised. A simple but equally effective formula is provided, allowing an easy interpretation of the role of the structural parameters accessible via a diffraction experiment. In particular, the formula is able to simultaneously combine the contribution arising from the anomalous differences with a Sim-like contribution, and also to take the errors into account.

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In the previous paper [Giacovazzo & Siliqi (2002). Acta Cryst. A58, 590-597], a probabilistic approach for the SIR-MIR and the SIRAS-MIRAS cases has been described. The mathematical technique is able to take into account the errors arising from measurements, lack of isomorphism and heavy-atom model substructure. An automatic procedure is here described in which the conclusive formulas of that probabilistic approach have been implemented. The procedure has been successfully applied to several test structures: it can automatically provide, starting from the experimental data, high-quality electron-density maps.