# Overview

This section is adapted from Kutsal, M., Bernard, P., Berruyer, G., Cook, P. K., Hino, R., Jakobsen, A. C., … Detlefs, C. (2019). The ESRF dark-field x-ray microscope at ID06. IOP Conference Series: Materials Science and Engineering, 580, 012007. https://doi.org/10.1088/1757-899X/580/1/012007. More details and references can be found in this publication.

Many, if not most, technological materials are composed of crystalline elements that are hierarchically organized over length scales ranging from millimetres to nanometres, spanning up to 6 orders of magnitude. The same is true for bio-minerals, ice, sand and geological materials in general.  Crystalline elements and substructures such as grains, domains and atomic-scale defect networks determine many of the macroscopic physical and mechanical properties of these materials. Understanding the interplay of physical phenomena and structural dynamics at, and between, these different lengths scales is therefore a critical and persistent issue across materials and geological sciences. Our understanding, however, is still limited by the lack of a non-destructive three-dimensional (3D) probe of the local crystal lattice (structure, strain, and orientation) that can be rapidly switched between different length scales and that enables acquisition of movies during processing.

The need to probe the local crystallography favours a diffraction-based approach. Existing 3D techniques, however, have shortcomings with respect to multi-scale characterization. Electron-based methods can provide very high spatial resolution, but are either limited to thin foils or involve serial sectioning. Scanning and coherent x-ray methods are limited to small sampling volumes. Furthermore, all methods face the challenge that the illuminated part of a bulk sample may comprise millions or even billions of structural elements whose diffracted signals overlap, rendering data analysis and interpretation complicated and in many cases impossible.

Aiming to overcome these limitations, dark-field x-ray microscopy (DFXM) is a full-field imaging technique for non-destructive mapping of the structure, orientation, and strain of deeply embedded crystalline elements in 3D. By placing an x-ray objective lens in the diffracted beam, direct space resolutions of 30—100 nm can be reached while maintaining a comfortable working distance of 10 cm or more between the sample and x-ray lens.  The spatial resolution and field of view can be adapted by changing the focal length of the lens and thus zooming'' in or out. Furthermore, through its narrow angular and real space field of view, the objective also filters stray diffraction signals, suppressing unwanted overlap and isolating the individual structural element of interest.

The instrument at ID06-HXRM is designed to combine DFXM with coarse scale 3D grain mapping techniques such as 3D X-Ray Diffraction (3DXRD) and diffraction contrast tomography (DCT), as well as classical tomography and diffraction topography. This combination enables the user to rapidly progress from fast overviews of the entire specimen to detailed studies of local phenomena in a single experimental setting, without the need to dismount the sample.

## The geometry and principle of DFXM

 Principle of dark-field x-ray microscopy. Monochromatic x-rays illuminate a crystalline element of interest, and the diffracted radiation is imaged by means of an x-ray objective and a 2D detector. The objective is here a compound refractive lens, comprising N lenslets each with a thickness of T. d1, d2, and fN are the sample plane-entry of CRL distance, exit of CRL - image plane distance and the focal distance, respectively. Scanning the sample tilt (χ, φ), and scattering (2θ) angles facilitate mapping of orientation and axial strain respectively, while different projections can be obtained by rotating the sample about its scattering vector, Q.

The geometry and operational principle of dark-field x-ray microscopy (see Figure 1) is conceptually similar to dark-field transmission electron microscopy (TEM): the diffracted beam passes through an x-ray objective lens, creating a magnified image of a specific region of interest with contrast from local variations in lattice symmetry, orientation and strain. The sample-to-detector distance d1+NT+d2 is  2-6 m, enabling magnification ratios of up to 50 while still maintaining sufficient space around the sample for complex sample environments.

A defining feature of the dark-field x-ray microscope is the x-ray objective. Like a visible light microscope, the x-ray objective can be reconfigured to adjust the magnification, field-of-view, and numerical aperture, hence allowing adjustment of the spatial resolution according to specific experimental requirements. So far primarily compound refractive x-ray lenses made of Beryllium or SU-8 polymer have been used. Irrespective of the choice, the imaging system is associated with a Fourier plane, the back focal plane downstream of the objective, see Figure 1.  Similar to dark field microscopy operation in a TEM, the direct space (imaging) information in the image plane and the Fourier space (diffraction) information in the back focal plane may be combined in a variety of ways.

Like the TEM, the dark-field x-ray microscope can be operated in a variety of modes. Most experiments so far have used a one-dimensionally-focusing condenser to create a narrow line-beam that illuminates a `layer’ of the material (typically 200 μm × 200 μm × 500 nm), which is then imaged at an oblique angle. Experiments typically involve the use of a succession of modalities including

• Rocking curve imaging in section topography:  The dependence of the intensity on the Bragg angle (rocking scan) is analyzed pixel by pixel. It is possible to combine rocking curve imaging in magnified and non-magnified (using a near-field camera, see below) modes without unmounting the sample.
• Mosaicity scans: By systematically varying the sample tilts, (χ, φ), a spatially resolved local pole figure can be acquired.
• Strain scans: By scanning longitudinally (θ-2θ-scan) the strain component along the scattering vector is imaged. Typically, this is combined with a rocking scan or with integration over the rocking profile at each 2θ) setting. As an alternative, strain mapping may be performed in the back focal plane.
• Reciprocal space maps: A high resolution reciprocal space map of the illuminated volume  is available either in the back focal plane.

3D mapping can be performed in two ways. Firstly, by stacking layers of the kind described above. This is performed by translating the sample through the planar beam in small increments. A second, faster method is magnified topo-tomography. Here, projections of the sample are acquired while the sample is rotated about the scattering vector, and a 3D representation is reconstructed using tomographic principles. Again, data can be taken in magnified and non-magnified mode. Experimental protocols and reconstruction codes for magnified topo-tomography are currently under development.

By setting 2θ  to zero, magnified bright field imaging is obtainable. Translating the sample along the optical axis makes it possible to acquire pure absorption contrast images or phase contrast images at any Fresnel number. Moreover, by placing a phase plate in the back focal plane, Zernike type phase contrast images are created.

## Complementary imaging on longer length scales

It is necessary to combine DFXM with more traditional imaging in order to identify local regions of interest and to provide overviews during processing studies. The microscope at ID06-HXRM enables this is several ways:

• A near-field camera with high resolution placed close to the sample can provide classical absorption or phase contrast tomography, non-magnified diffraction topographies, and DCT measurements.
• A diffraction camera with large pixels can provide 3DXRD type mapping as well as classical (sample averaged) diffraction information.
• Far-field imaging can provide high resolution reciprocal space maps. Conveniently this modality can be obtained with the same detector as shown in Figure 1 by simply translating the objective out of the diffracted beam.

The coordinate system used at ID06-HXRM has x along the beam axis, y horizontal and away from the storage ring (to port, or outboard), and z upwards.