X-ray phase contrast tomography. For the morphological study of the micro-scale structures of the Li battery cathode, synchrotron hard x-ray tomography based on phase contrast with high spatial resolution becomes our method of choice. Holotomography measurements of the samples were conducted at the ID16A-NI nano-imaging beamline\cite{Cesar_da_Silva_2017} of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. This beamline offers a unique combination of nanofocus (~20nm) and a very high photon flux (up to 1012 photons/s at ΔE/E~1%). Two pairs of multilayer coated Kirkpatrick-Baez (KB) optics are used to focus the x-rays at 17 keV and 33.6 keV respectively. This experiment were performed at 17 keV. Besides the benefits of the ability to perform nano-tomography with a high energy, the magnifying geometry of the cone beam also allows large field of view (FOV) with 100 nm and 70 nm voxel size. Due to free space propagation of the x-ray beam, the contrast in the images is dominated by phase contrast, related to the real part of the complex refractive index, which is determined by the electron density of the material. By measuring the Fresnel diffraction patterns at different effective propagation distances, the phase maps of the sample can be retrieved via holographic reconstruction, the so called phase retrieval procedure\cite{Cloetens_1999} implemented using GNU Octave software. In our measurements, the sample was placed downstream of the KB focus and magnified radiographs were recorded onto an x-ray detector using a FReLoN charged-coupled device (CCD) with a 2048x2048 binned pixels array. For every tomography scan, 1500 projections were acquired with 0.2s exposure time at 100 nm, or 70nm pixel size. Tomographies at four different focus to sample distances were acquired to complete one holotomography scan, which were subsequently used for phase retrieval. The 2D phase maps retrieved from the angular projections were then used as input for a tomographic reconstruction based on the filtered back projection (FBP) algorithm method (ESRF PyHST software package)\cite{Mirone_2014}. The reconstructed 3D volumes are proportional to the changes in electron density of the sample.
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