X-ray orientation microscopy using topo-tomography and multi-mode diffraction contrast tomography

Highlights

• Generalization of 6D-DCT reconstruction framework.

• Improved spatial resolution by combination of acquisition modes.

• Multi-mode X-ray orientation microscopy with flexible geometry.

• Orientation imaging with Topotomography.

Abstract

Polycrystal orientation mapping techniques based on full-field acquisition schemes like X-ray Diffraction Contrast Tomography and certain other variants of 3D X-ray Diffraction or near-field High Energy Diffraction Microscopy enable time efficient mapping of 3D grain microstructures. The spatial resolution obtained with this class of monochromatic beam X-ray diffraction imaging approaches remains typically below the ultimate spatial resolution achievable with X-ray imaging detectors. Introducing a generalised reconstruction framework enabling the combination of acquisitions with different detector pixel size and sample tilt settings provide a pathway towards 3D orientation mapping with a spatial resolution approaching the one of state of the art X-ray imaging detector systems.

Introduction

Experimental capabilities to map crystal orientation and elastic strain fields in the bulk of polycrystalline materials by means of X-ray diffraction have seen tremendous progress over the past years. A whole portfolio of different X-ray diffraction based techniques have reached maturity and are now routinely applied to a broad variety of topics in materials science covering fields like grain coarsening [1], [2], plastic deformation [3], [4], various modes of materials failure [5], [6], [7] and phase transformations [8], [9].

Very much like modern electron microscopes offer a variety of imaging and diffraction modes in the same instrument, state of the art synchrotron beamlines offer multi-modal X-ray characterization. In the case of the materials science beamline at the European Synchrotron Radiation Facility this portfolio includes phase contrast tomography (PCT) [10], [11] as a Fresnel diffraction based imaging mode, diffraction contrast tomography (DCT) [12] as a Bragg diffraction based imaging mode for mapping the grain structure in polycrystalline sample volumes, and Topo-tomography (TT) [13] as a Bragg diffraction based imaging mode for mapping individual grains by rotation around one of the scattering vectors. These techniques typically employ high resolution imaging detectors (0.5–5 $\mathrm{\mu }$m pixel size), whereas so-called far-field techniques like three-dimensional X-ray diffraction (3DXRD), as well as (nano) scanning X-ray diffraction computed tomography (nXRD-CT) [14] employ diffraction detectors with larger pixels (50–200 $\mathrm{\mu }$m). These latter techniques typically yield sufficient angular resolution to reveal the small elastic distortions of the crystal unit cell and are therefore often used to obtain complementary information in strained materials [15], [16] (see also contribution by J. Wright [36] for more details on these last two techniques and the Materials Science endstation ID11 at ESRF).

The data generated by imaging or diffraction modalities are usually reconstructed independently and results are combined in a post-processing step, as illustrated in previous studies of stress corrosion cracking [5] and fatigue cracking [6], [17], [18] which captured crack propagation by repeated PCT observations on grain microstructures which were previously characterized by 3D grain mapping techniques on the same instrument and during the same experimental session. There are, however, also first examples of combined analysis schemes for data acquired in different diffraction modalities. For instance, grain shape reconstructions by near-field High Energy Diffraction Microscopy (NF-HEDM) [19] are commonly seeded by indexing information obtained from far-field (FF-HEDM) [20][44] and instrument alignment for topo-tomographic observations of individual grains is inferred from concomitant DCT observations [21], [22].

The ultimate spatial resolution of near-field polycrystal grain mapping techniques is inherently limited by the need to capture diffraction signals from a number of different hkl reflections. For instance, for metals with highly symmetric crystal structures, the X-ray imaging detector is typically positioned at a distance such that the innermost 3 to 5 hkl families are intercepted by the screen, giving rise to several tens up to hundred observable diffraction blobs per grain. In order to avoid overlaps between the transmitted and the diffracted beams, the footprint of the illuminated sample volume has to be kept small and it typically does not exceed one quarter of the lateral dimensions of the detector. Consequently, in the limiting case of a single crystal, the ultimate spatial resolution of the resulting grainmap is already compromised by a factor of four with respect to the full resolution of the detector system. For polycrystalline samples containing up to ten and more grains through-thickness the spatial sampling (number of voxels per grain) degrades accordingly and the physical voxel size in the resulting grain map is often well below the ultimate spatial resolution achievable with state of the art X-ray imaging detectors (see Fig. 1).

In order to overcome the limits in resolution dictated by the detector system, two options exist: one can either focus the beam and switch to a 3D point scanning approach like nXRD-CT [14], [23] or one can ”zoom-in” on individual grains inside the sample volume using Dark Field X-ray Microscopy (DFXM) [24], [25]. Both methods can provide access to sub-micrometer spatial resolution which, neglecting instrument error motion and sample drifts, is ultimately limited by the performance of the X-ray optical elements. However, in both cases this gain in spatial resolution comes at the expense of reduced temporal resolution, since these methods imply multi-dimensional scanning procedures (see contributions by H. Simons et al. [24] and J. Wright et al. [36] for more detail on these techniques).

In this article, we propose a different strategy to improve the spatial resolution of full-field grain mapping techniques. As will be shown, the combination of limited projection data acquired at high spatial resolution (e.g. TT scans of individual grains or partial near-field diffraction data acquired on a high resolution detector covering only the innermost hkl families) with data acquired in the conventional setting at lower spatial resolution can result in significant improvements in the overall reconstruction quality.

In order to enable joint reconstruction of the 3D orientation field from disparate projection data (i.e. different detector positions, rotation axis, pixel resolution and sample tilt settings) we introduce a generalization of the six-dimensional reconstruction framework proposed by Poulsen [26] and Viganò [27], [28]. This model builds on kinematical diffraction and we further assume that the position, average orientation and the orientation space sub-volume occupied by the grain are known from previous polycrystal indexing and analysis steps, not further detailed here. In a nutshell, in addition to the regular sampling of real space, a regular sampling of 3D orientation space is introduced (see Fig. 2 for an illustration of this concept). Each real space volume element (voxel) is assigned a finite set of discrete orientations which are used to model (”probe”) the local orientation distribution of the grain. Using three position and three orientation space coordinates we operate in a six-dimensional position-orientation space: each of its elements holds a scalar quantity describing the volume fraction of material occupied by one of the sampled orientations at one of the sampled positions. Using such a description, the diffracted intensities $\mathbf{b}$ observed on the detector can be expressed by the action of a linear forward projection operator $\mathbf{A}$ on the set of unknown position-orientation space elements $\mathbf{x}$ as: $\mathbf{A}\mathbf{x}=\mathbf{b}$. As detailed in Section 2 this equation represents a large-scale system of linear equations. Approximate solutions can be found using iterative tomographic optimization schemes based on iterative forward and back-projection operations and exploiting prior knowledge (e.g. smoothness, non-negativity) about the solution. A final processing step consists in converting the scalar 6D position-orientation output of the optimization algorithm back into a 3D vector field representation (e.g. 3 Euler angles) by calculating for each voxel the average of the 3D orientation distribution associated to it.

We now outline the structure of this article. In Section 2 we present the generalized six-dimensional mathematical framework. In Section 3 we present and compare the results obtained on a synthetic test case for which we have simulated selected combinations of low resolution and high resolution DCT and TT acquisitions. Some practical experimental aspects and limitations are discussed in Section 4 before we conclude the article in Section 5.