Received 18 June 2012
An introduction to three-dimensional X-ray diffraction microscopy1
Three-dimensional X-ray diffraction microscopy is a fast and nondestructive structural characterization technique aimed at studies of the individual crystalline elements (grains or subgrains) within millimetre-sized polycrystalline specimens. It is based on two principles: the use of highly penetrating hard X-rays from a synchrotron source and the application of `tomographic' reconstruction algorithms for the analysis of the diffraction data. In favourable cases, the position, morphology, phase and crystallographic orientation can be derived for up to 1000 elements simultaneously. For each grain its average strain tensor may also be derived, from which the type II stresses can be inferred. Furthermore, the dynamics of the individual elements can be monitored during typical processes such as deformation or annealing. A review of the field is provided, with a viewpoint from materials science.
Following the first demonstration experiments more than a decade ago (Poulsen et al., 1997, 2001; Lauridsen et al., 2001; Margulies et al., 2002; Fu et al., 2003), three-dimensional X-ray diffraction (3DXRD) has proliferated in a number of directions. Similar to the development of transmission electron microscopy (TEM), the development of 3DXRD reflects the great diversity in needs for structural characterization. More specifically, it is a consequence of the fact that in a 3DXRD experiment one must prioritize between spatial, angular and time resolution.
In the following a snapshot is provided of the `classical' 3DXRD methodology, instrumentation and scientific use. To ease the presentation, four standard modes of operation are defined. Mode I relates to fast statistical descriptions of grain properties (such as volume, strain tensor, orientation and phase) without the acquisition of information on grain position; in Mode II, in addition, the three-dimensional centre-of-mass positions of the grains are known. In Modes III and IV a full volumetric mapping is provided of grains and orientations for undeformed and deformed specimens, respectively.
The aim of this review article is to provide an overview of the field from the perspective of materials science. For illustration a number of examples of work will be summarized but it is not the intention to be exhaustive. The focus is on presenting methods that have matured sufficiently to be offered to users, while the most recent exploratory work is only briefly outlined in an outlook section. For a more physics-based introduction to the 3DXRD methodology we refer to Poulsen (2004), Banhart (2008) and Bernier et al. (2011), while a mathematical treatment is given by Alpers et al. (2007). A review of applications of 3DXRD for multigrain crystallography - primarily of interest for chemistry and structural biology - is given by Sørensen, Schmidt et al. (2012).
The review will not deal with the variant of 3DXRD known as diffraction contrast tomography (DCT; Ludwig et al., 2008, 2009; Johnson et al., 2008) as DCT is discussed in another article (Reischig et al., 2013) in the virtual special issue on diffraction imaging in which the present article features. Likewise, for reasons of space, work on high angular resolution 3DXRD (reciprocal space mapping of individual grains) will only be mentioned briefly - here we refer the reader to the recent summary of Pantleon et al. (2010).
Alternative approaches to provision of three-dimensional grain and orientation maps of bulk materials based on X-ray diffraction exist. These are based on inserting wires (Larson et al., 2002), slits (Bunge et al., 2003) or collimators (Wroblewski et al., 1999) between the sample and the detector and scanning the sample with respect to these elements. Not surprisingly such methods will be slower than the tomographic approach of 3DXRD, but they can be associated with other advantages, such as improved spatial resolution or improved options for measuring the local elastic strain. In particular we mention the technique of `differential-aperture X-ray microscopy' (DAXM; Larson et al., 2002; Ice et al., 2005, 2011; Levine et al., 2006; Barabash et al., 2009). Using a polychromatic microbeam with energies of 8-20 keV and scanning a wire, a resolution of 200 nm in three dimensions has been demonstrated. Another related effort is the work on probing heterogeneous diluted materials by diffraction tomography (Bleuet et al., 2008). Using a pencil beam and scanning the sample, this is a powerful technique, for instance, to map the spatial distribution of crystallographic phases (but not for individual grains) in small-grained multiphase materials.
The classical 3DXRD setup - sketched in Fig. 1 - is quite similar to conventional tomography settings at synchrotrons. A (nearly) parallel monochromatic X-ray beam impinges on the sample as a uniform field. The sample is mounted on an rotation stage, where is the rotation around an axis perpendicular to the incoming beam. As an option, x, y and z translations may be added as well as additional rotations.
| || Figure 1 |
Sketch of the classical 3DXRD setup. Two types of detectors are used: high-spatial-resolution detectors close to the sample and a low-resolution detector far away from the sample. The Bragg angle, 2, the rotation angle, , and the azimuthal angle, , are indicated for the diffracted beam arising from one grain of a coarse-grained specimen, and for a combination of two high-resolution detectors. The sketch is outdated in the sense that the beamstop now typically is removed, and the direct beam is transmitted through central holes in the two screens for possible tomography characterization by a fourth detector.
Any part of the illuminated structure that fulfils the Bragg condition will generate a diffracted beam. This beam is transmitted through the sample and probed by two-dimensional detectors. To probe the complete structure, and not just the part that happens to fulfil the Bragg condition, the sample is rotated. Hence, exposures are made for equi-angular settings of with a step of . To provide a uniform sampling the sample is rotated by during each exposure. Key to 3DXRD is the idea to mimic a three-dimensional detector by positioning several two-dimensional detectors at different distances from the centre of rotation, L, and exposing these either simultaneously (many detectors are semi-transparent to hard X-rays) or subsequently.
Typically two types of detectors are used: near-field detectors with a spatial resolution of 1-5 µm in close proximity to the sample (L in the range 2-20 mm) and far-field detectors with a pixel size of 50-200 µm (L in the range 10-50 cm). The former provide information on position and orientation degrees of freedom, while the latter probe crystallographic properties in general, including strain and orientation. With the near-field detector, images may be acquired at several - typically two or three - distances, as illustrated in Fig. 1. This enables ray tracing of the diffracted beam and adds robustness to the algorithms. The drawback of this setup is that all detectors need very careful alignment and the near-field detector restricts space for the sample environment.
The incoming beam may illuminate the full sample or be focused in one direction to probe only a layer within the material (see Fig. 1). In the following we shall refer to these as the three-dimensional and two-dimensional cases, respectively. Notably, the two-dimensional case is restricted to use at third-generation synchrotron sources, while the three-dimensional case also applies to second-generation sources. In the two-dimensional case, three-dimensional information is generated simply by repeating data acquisition for a set of layers and by stacking the resulting reconstructions.
An example of a set of exposures obtained in the two-dimensional case is provided in Fig. 2. On the innermost detector, the diffraction spots are distributed almost randomly, and the position and shape reflect the position and morphology of the associated grain. On the far-field detector, the diffraction spots are positioned on the Debye-Scherrer ring familiar from powder diffraction. In this case there is little spatial information in the pattern, but a high angular resolution, ideal for stress characterization and crystallography, is achieved.
| || Figure 2 |
Typical 3DXRD data from a coarse-grained undeformed polycrystal. The sample is an annealed aluminium polycrystal with grains of average size 70 µm. The X-ray energy was 55 keV. The left and middle images are `near-field' exposures with two screens (see Fig. 1) at distances from the sample of L = 7 mm and L = 17 mm, respectively. The elongated central holes in the two screens are clearly visible. On the right is a corresponding `far-field' image made with a low-resolution detector at L = 190 mm. In the latter case the diffraction spots are positioned on Debye-Scherrer rings.
At the point of writing, the classical setup has been implemented into dedicated 3DXRD microscopy stations at beamline ID11 at the European Synchrotron Radiation Facility, ESRF, at sector 1-ID at the Advanced Photon Source, APS [under the name High Energy Diffraction Microscopy, HEDM - see Lienert et al. (2011) for details], and at the HEMS beamline at PETRA-III.
Associated with the classical setup at ESRF is a comprehensive software package, FABLE, complete with graphical user interfaces (GUIs) and options for use of parallel computing (http://sourceforge.net/apps/trac/fable/ ).
For convenience, we provide more detailed information on the 3DXRD microscope setup at beamline ID11 at ESRF. Close to the sample, a so-called three-dimensional detector unit has been installed, comprising two semi-transparent two-dimensional detectors. Both of these contain a scintillator screen, which converts X-rays to visual photons. Via a mirror and conventional microscope optics the visual light is acquired by a CCD. The three-dimensional detector is designed such that the first screen is at a distance of 3-10 mm from the sample, while the distance between the first and second screens is fixed at 10 mm. The pixel sizes of the two screens are 1.5 and 4.5 µm, and the corresponding active areas 3.072 × 3.072 mm and 9.216 × 9.216 mm, respectively. At a much larger distance of 20-50 cm, a far-field detector is placed, typically a FReLoN camera with a pixel size of 50 µm and an active area of 102.4 × 102.4 mm. These three detectors all probe the dark-field diffracted beam. The two screens and mirrors in the three-dimensional detector have a small hole in the centre that allows the direct beam to pass through to be detected by a high-resolution camera, which probes the attenuation of the direct beam. This can be used for simultaneous absorption and phase contrast tomography.
For completeness, below we briefly outline four alternative 3DXRD-type settings and compare these with the classical 3DXRD setup.
In this case only one (near-field) detector is used. The direct beam is made to have dimensions that imply that it illuminates a square in the centre of the detector, constituting approximately 10% of the total detector area. In this way, absorption contrast tomography and 3DXRD-type analysis is directly integrated. A standard tomography detector and parallel-beam setup can be used, implying that DCT with little effort can be implemented on many existing beamlines. Furthermore, in the language of electron microscopy, DCT makes it possible to combine tomography, bright-field imaging (reconstruction based on the extinction spots) and dark-field imaging (reconstruction based on the diffraction spots). The main disadvantage of DCT is that the setup is less versatile than the classical 3DXRD setup; in particular the lack of a far-field detector implies that there are fewer options for crystallography.
Topotomography is an alternative tomographic procedure, which allows the reconstruction of a single grain using a setup with additional tilt stages (Ludwig et al., 2001, 2007). For a selected grain, the scattering vector of a suitable reflection is aligned parallel to the rotation axis of the tomographic setup. This particular setting assures that the diffraction condition is maintained while the sample is turned 360° around the rotation axis. Grain mapping can be performed from data acquired in a diffracted beam by means of existing reconstruction algorithms such as the (cone beam) filtered backprojection algorithm well known from tomography. The disadvantage is the restriction to one grain only.
At the expense of spatial resolution and time resolution, in this setup the angular resolution is improved by two orders of magnitude. As a result very detailed reciprocal space maps can be made of individual embedded grains (Jakobsen et al., 2006, 2007).
(a) The characterization of large sample volumes is often impossible as the number of grains illuminated becomes so large that the diffraction spots are likely to overlap. For such cases it is relevant to develop approaches where the volume of interest is divided into a set of subvolumes, to be characterized successively.
(b) The use of a near-field detector is prohibitive for certain in situ studies, which involve large samples or spacious sample surroundings. Hence, it is relevant to consider grain-mapping methods based on the use of a far-field detector only.
The box-scan algorithm is based on scanning a `box-shaped' beam with respect to the detector. The size of the box is optimized to characterize as many grains as possible simultaneously within the constraint of avoiding excessive spot overlap. The scanning procedure is optimized to minimize the number of settings required. The result is a procedure that, in terms of data acquisition speed, is slower than classical 3DXRD or DCT but on the other hand faster than procedures based on scanning the sample with respect to a point beam.
A vital difference between the two types of detectors shown in Fig. 1 relates to data acquisition time: with the current technology the far-field detectors provide a time resolution that is several orders of magnitude better than the near-field ones. This makes it necessary to carefully consider the priority of time and space resolution in a 3DXRD experiment. In practice we may distinguish between four modes of operation, as sketched in Fig. 3.
| || Figure 3 |
Sketch of four typical 3DXRD modes of operation, listed in direction of increasing spatial information. Mode I: statistical mode, with no spatial information. Mode II: information on centre-of-mass position, volume, average orientation and average strain tensor for each grain. Mode III: three-dimensional grain map for undeformed specimens. Mode IV: three-dimensional orientation map for deformed specimens. In all cases, orientations are represented by a grey scale.
Mode I is based only on far-field information. Mode I is an option for very fast measurements, where only a small interval is probed by a conventional far-field detector. The intensity, shape and position of individual diffraction spots are followed as a function of time. As a consequence only a subset of the illuminated grains are characterized and no spatial information is available. Also, the orientation and strain characterization will not be complete. Examples of work will be given below.
Mode I is often also relevant for studies where the emphasis is on acquiring detailed crystallographic information for each grain (see §4.1 below), and it is the mode used for high-angular-resolution 3DXRD.
In Mode II the aim is to acquire information on the phase, centre-of-mass position and volume, as well as the average orientation and average strain tensor of each grain. For a given detector, often the accuracy on the centre-of-mass positions can be substantially better than the spatial resolution of a boundary position of the grains, and hence Mode II mapping may be performed with lower-resolution - and more efficient - detectors. For coarse-grained specimens a far-field detector may be sufficient. The result of the box-scan procedure is typically also a grain-centre map. It should be emphasized that, with knowledge of the grain centres and volumes, a primitive three-dimensional grain map can be made by Laguerre tessellation. The quality of such a map is sufficient to identify neighbours with a reasonable fidelity (Lyckegaard et al., 2011).
Modes III and IV aim at the generation of complete three-dimensional maps: that is, volumetric mapping. In these cases one or more high-spatial-resolution detectors are needed. In Mode III the material is assumed to be undeformed, implying that the orientation is constant within each grain, and the aim is to provide a grain map. DCT and topotomography work typically relate to this mode of operation. In Mode IV, the aim is to study deformed material, where the orientation varies locally within each grain. This makes it relevant to measure orientation maps, where each voxel in the sample is associated with its own orientation. The two modes will be detailed below.
In many experiments combinations of these modes are relevant. As an example one may wish to start by mapping extended parts of a sample in order to identify subvolumes of particular interest. Then one focuses on such parts and performs a fast centre-of-mass study on these parts during in situ processing. This procedure is complemented by mapping extended parts again in Mode III or IV after the processing is completed.
It should be noted that none of the modes above enables three-dimensional mapping of variations of the elastic strain within grains and the corresponding type III stresses. Generalizations of the software are being pursued at the moment, but this work is still at best exploratory. In contrast, mapping the type III stresses is an integral part of the polychromatic DAXM method.
An important simplification arises in the case where the diffraction pattern is composed of a set of primarily non-overlapping diffraction spots. Three examples of such data sets are shown in Fig. 2. In this case, polycrystalline indexing schemes can be applied readily (Lauridsen et al., 2001). A large number of problems within polycrystal and powder research can be tailored to apply to this situation.
The starting point for such schemes is to apply image analysis routines, which identify the spots and determine their properties (position, integrated intensity, shape...). This typically involves the steps of preprocessing, geometry calibration and peak searching. Within FABLE, a full suite of GUIs exist for these procedures, the main one being ImageD11. Peak searching is performed in two steps: first diffraction spots are identified by means of segmentation as connected components in (ydet, zdet, ) space, where (ydet, zdet) are detector coordinates. For data from deformed materials, exhibiting broad diffraction spots, segmentation methods using several thresholds are superior (see Kenesei, 2010).
Next, the directions of the corresponding diffracted X-rays are determined. In the setup illustrated in Fig. 1, with several near-field detectors, ray tracing may be used. Alternatively, in order to acquire data for a full 360° rotation in , Friedel pairs can used (Ludwig et al., 2009; Moscicki et al., 2009). In the case of far-field data only, one may simply assume that all grains are positioned at the centre of rotation.
Indexing involves finding the orientation matrices of the grains in the sample and sorting the G vectors (scattering vectors) according to grain of origin. Following the presentation of the original polycrystalline indexing program GRAINDEX by Lauridsen et al. (2001), several alternative approaches have been proposed (Ludwig et al., 2009; Moscicki et al., 2009; Bernier et al., 2011; Schmidt, 2012). Generally speaking, the sorting can be based on three principles: orientation, position or grain volume (the integrated intensity of any diffraction spot is proportional to the volume of the associated grain of origin if fully and homogeneously illuminated).
For brevity in the following we shall restrict ourselves to indexing based on orientation relationships. One route to indexing in this case is to utilize the fact that for each scattering vector we can determine a line in three-dimensional orientation space: the projection line. By definition, the orientation of the associated grain is on this line. Furthermore, the projection lines for all scattering vectors belonging to a given grain will intersect at one point. Hence, determining orientations becomes a question of finding intersections/vertices. In that connection the choice of representation of orientation space has turned out to be of major importance for the speed of algorithms. For high-symmetry-space-group materials such as cubic materials, a representation by Rodrigues vectors (Frank, 1988) seems ideal. For general use, a new representation - by frustrums - has been proposed (Kazantsev et al., 2009).
Alternatively, several indexing programs, such as GRAINDEX and GrainSpotter (Schmidt, 2012) and the software by Bernier et al. (2011), act as forward-simulation programs, which for a given orientation calculate theoretical G vectors. These algorithms loop over orientation space and attempt at any given orientation to match the theoretical G vectors with experimental G vectors in the vicinity. Grains are defined mainly by a completeness criterion on the ratio between observed and expected G vectors.
The core of GrainSpotter (Schmidt, 2012) is an indexing engine with theoretical O(N) complexity, where N is the number of measured G vectors. Uniquely the program uses a combination of two representations of orientation space: Rodrigues space for local operations and quaternion space for global operations. Grains are indexed by segmenting the orientation space (either systematically or randomly) into subvolumes, called local Rodrigues spaces, and for each subvolume identifying possible orientations without searching the full three-dimensional subvolume and by using each candidate G vector for the subvolume in question only once. Grains are defined mainly by a completeness criterion on the ratio between observed and expected G vectors. In connection with the use of look-up tables and code optimization, the indexing of 500 grains can be as fast as a couple of minutes, on one computer core, sufficient for online operation. Another unique feature of GrainSpotter is a pseudo-twin filter. When the crystallographic symmetry is high, a subset of G vectors associated with a given orientation can form a pattern consistent with another not crystallographically equivalent orientation at a completeness of up to 1/3. Adding reflections from other grains, which are positioned arbitrarily in the vicinity of the theoretical G vectors, false grains with completeness above 50% may appear if the filter is not applied.
Bernier et al. (2011) have presented an extensive formulation for orientation indexing and cell refinement. This was demonstrated on simulated diffraction data from an aggregate of 819 individual grains. The paper also contains a study of the effect of varying the sample rotation range and the influence of various experimental uncertainties. This software is optimized for strain and stress analysis.
Once, the indexing program has identified grains it is at times relevant to run optimization programs that filter out remaining `bad reflections' and redistribute some of the reflections among grains based on figure-of-merit functions, which may also involve new criteria such as position or intensity (Oddershede, Schmidt et al., 2010; Bernier et al., 2011). This optimization step is often combined with determining the strain and centre of mass of the grains (see §6).
There are two main limitations to the indexing:
(a) Spot overlap. The probability of spot overlap on the detector is determined by the number of grains illuminated, the texture of the sample, the size of the unit cell and - most critically - the orientation spread of each grain. Experience shows that an overlap fraction of the order of 10% is acceptable for the indexing, in the sense that overlapping spots are weeded out by the algorithms owing to their erroneous position and/or intensity. Extensive simulations of the amount of overlap as function of unit-cell size and resolution are provided by Sørensen, Schmidt et al. (2012). It follows from these and the 10% rule of thumb that for typical samples of relevance to hard-materials science one can aim at indexing up to a few thousand grains simultaneously. On the other hand, plastic deformation introduces orientation spread within the grains, which often prohibits indexing of embedded grains in materials that are deformed to more than 10%.
(b) Small grains. Often grain sizes are lognormally distributed. This in connection with the rapid decline in integrated intensity with increasing Bragg angle, 2, implies that the indexing algorithm may be faced with three types of grains: (1) large grains where essentially all spots are correctly identified by peak searching, (2) very small grains where essentially no spots pass the peak-searching threshold and the indexing program consequently does not assign a grain, and (3) intermediate size grains, where a fraction of peaks - typically at low 2 - are found. To deal in an optimal way with the latter type of grains it is relevant to use different thresholds at different 2 values, and often several iterations are required.
3DXRD has the potential to become an important tool for crystallography as well. Conventionally, X-ray crystallography is based on either single-crystal or powder diffraction measurements. Such methods apply to two extremes of sample morphology: a homogeneous monodomain specimen or a monodispersed powder comprising a huge number of crystals. Unfortunately, the morphology of a sample is often a heterogeneous assembly of a finite number of poor crystals, including several phases. In contrast, with 3DXRD such samples are accessible, and furthermore once a grain has been indexed all the tools of conventional single-crystal diffraction analysis are available. In 2004 it was shown that structural solution and refinement on multigrain data can be on a par with single-crystal work and superior to results from state-of-the-art powder diffraction (Vaughan et al., 2004). Within the project TotalCryst (http://www.totalcryst.dk ) the possibilities for multigrain crystallography have been explored with success within the disciplines of pharmacy, chemistry and structural biology (Sørensen, Schmidt et al., 2012). Other applications are within reciprocal-space mapping (Jakobsen et al., 2006), and characterization of sizes, strains and defect populations of individual grains (Ungar et al., 2010).
Mode I is characterized by the use of only a far-field detector and by the fact that no grain map is obtained. Nevertheless, this mode of operation is of great practical interest. The facts that far-field detectors tend to be much more efficient than near-field detectors and that there is no need to place optical elements close to the sample imply that this mode is ideal for fast in situ measurements within complicated sample environments. For very fast data acquisitions one simply repeats acquisitions, oscillating within a small range around a given setting. One can then monitor any potential change in integrated intensity, spot position and maximum position of the diffraction spots appearing in this interval as a function of time. From this information one can immediately infer changes in volume and radial strain of the associated grain, and detect rotations of the axis of the reflection. The time resolution may be a fraction of a second.
By increasing the rotation range one can acquire sufficient reflections to perform indexing of the grains, providing robustness and an elimination of a possible bias to results caused by probing only certain texture components. Furthermore, if the range approaches 90° one can determine the evolution in the full-strain tensor and the full orientation of each grain. Still, the time resolution can be as good as 10 s.
As a result, unique statistical information is acquired about the dynamics of the individual grains. Distributions of grain properties can be used both as input to and validation of three-dimensional simulations of the grain scale dynamics.
Traditionally, nucleation and growth phenomena have been analysed using ensemble average properties, such as the volume fraction of transformed material. However, the predictive power of average properties is limited by the neglect of heterogeneities. For example, nucleation may take place preferentially at specific sites, and the growth rate of nuclei may depend strongly on orientation, size, stoichiometry or relationships with neighbouring volumes. 3DXRD is an ideal tool to study the effect of heterogeneities, and as such it has been used for a series of studies related to recrystallization (Lauridsen et al., 2003), solidification (Iqbal et al., 2005), plastic deformation (Martins et al., 2004), and phase transformations in steel (Offerman et al., 2002; Jimenez-Melero et al., 2007), superconductors (Liu et al., 1999), ceramics and ferroelectrics. In all cases it was demonstrated that the ensemble average `Avrami-type' models are at best gross simplifications.
More specifically, Lauridsen, Juul Jensen and co-workers studied recrystallization kinetics during isothermal annealing in Al and Cu deformed to medium, high and very high strains (e.g. Lauridsen et al., 2003, 2006; Poulsen et al., 2011). The uniqueness of these measurements relates to the fact that it is the kinetics of individual recrystallizing grains that are followed with a time resolution down to seconds and not an overall average kinetics curve, such as is obtained with standard methods. The 3DXRD Mode I measurements have for all the investigated samples shown that each individual grain has its own kinetics curve and no two grains are alike. The data are important because they reveal that the general assumption in recrystallization modelling of one or a few typical growth rates is not sound. Instead a distribution of growth rates has to be incorporated into the recrystallization modelling, and it has been shown that this can significantly change the expected overall recrystallization kinetics and thus the interpretation of such curves (Godiksen et al., 2007).
Notably, 3DXRD studies are also relevant for crystalline structures with a grain size smaller than the spatial resolution of any existing near-field detector. Evidently, in such cases the generation of a grain map is not possible. However, by focusing the beam to say 0.2 × 0.2 µm, diffraction spots from individual grains as small as 30 nm can be detected on a far-field detector. By tracing the integrated intensities and positions of such spots as a function of time, one can infer changes in volume, orientation and strain of the grains of origin.
The main limitation for such studies is spot overlap. To overcome this problem two approaches have been pursued. The first approach is based on reducing the number of illuminated grains by investigating foils (Gundlach et al., 2004). Provided the foil thickness is at least ten times the grain size, for many annealing processes the grains at the centre of the foil may be considered bulk grains. This methodology has been applied to a series of coarsening studies of subgrains in aluminium deformed to both medium and very high strains.
The second approach is valid for a range of materials where the smallest structural units are nearly perfect crystals. An example is plastically deformed metals, where the subgrains are almost free of dislocations. For such materials, the spot overlap probability is defined by the angular resolution of the instrument. Hence, by improving the resolution of the instrument, spot overlap is reduced. These considerations have led to the installation of a high-angular-resolution 3DXRD setup at the APS, which provides reciprocal-space maps of individual grains with an angular resolution - in 2, and - that is two orders of magnitude better than the specifications for standard 3DXRD. This instrument has produced a wealth of information on the dynamics of subgrains (Jakobsen et al., 2006, 2007; Pantleon et al., 2010).
In this mode, the objective is, for each grain in the illuminated volume, to determine its centre-of-mass position, grain volume, phase and average orientation as well as the elastic strain tensor components averaged over the grain (from the latter one can infer the type II stress). The setup is typically similar to that of Mode I: that is, comprising only a far-field detector. The difference is the angular-rotation range, which generally speaking should be as close to 180° as possible. In comparison to the more comprehensive mapping methods (Modes III and IV), data acquisition is faster and the setup leaves ample space for auxiliary equipment.
Centre-of-mass grain mapping involves a combination of indexing and optimization/refinement steps. In the HEDM software the indexing and refinement are carried out interchangeably allowing new reflections to be assigned to a grain after refinement (Edmiston et al., 2011; Bernier et al., 2011). Using FABLE the indexing is carried out by GrainSpotter (Schmidt, 2012), and the refinement is subsequently done in the FitAllB module (Oddershede, Schmidt et al., 2010): a 12-parameter-per-grain fitting algorithm based on a least-squares minimization of grain positions, orientations and strain tensor elements. For both software packages - and from both simulated and real data - it has been shown that for an undeformed material the grain positions can be determined with an accuracy of 10 µm, the volumes with a relative error of 20%, the orientations to 0.05° and the components of the elastic strain tensors to 1 × 10-4, provided data are acquired within a large range. If the experimental setup leaves space for a near-field detector the spatial resolution can be substantially improved: three-dimensional maps with a 2 µm resolution have been reported (Oddershede, Schmidt et al., 2010).
All of the indexing and refinement algorithms build on the assumption that it is possible to unambiguously separate the diffraction spots and determine the centre-of-mass position of each in terms of 2, and . As already mentioned, this assumption tends to break down when the material deforms plastically. This segmentation problem is one of the main limitations of the grain-centre mapping technique. For slightly deformed materials simple thresholding is sufficient to segment the diffraction peaks, but for larger deformations a more sophisticated connectivity search such as the DIGIgrain algorithm (Kenesei, 2010) is required.
So far the grain-centre mapping technique has been used to study, for example, twinning in hexagonal-close-packed materials (Aydiner et al., 2009), stress-induced phase transformations in shape-memory alloys (Berveiller et al., 2011), lattice rotations and elastic strain evolutions during plastic deformation (Oddershede, Wright et al., 2010; Oddershede et al., 2011), intergranular force transmission in sand (Hall et al., 2011), and the stress-field evolution around a growing crack (Oddershede et al., 2012). Below we present two applications in more detail.
Three-dimensional stress mapping based on scanning neutrons or hard X-ray beams across the sample and analysing the resulting powder patterns has become a routine tool (Reimers et al., 2008). However, for studying local phenomena such as the stress field and plastic deformation zone at an edge or around a crack tip, the number of grains in the volume of interest may not be sufficient for powder diffraction methods to apply. By using each grain as an independent probe a much improved map - in terms of spatial resolution - can be obtained.
As an example of this type of application, the stress field and associated plastic deformation zone around a notch in a coarse-grained Mg AZ31 sample were measured under tensile load (Oddershede et al., 2012). Selected contours of the stress field and plastic deformation zone at 170 MPa, i.e. just before yielding, are shown in Fig. 4, which also shows the initial outline of the notch relative to the mapped volume. At higher applied loads - 205 MPa - a stress relaxation was observed in front of the notch, which was attributed to the initiation and propagation of a crack. The white contour levels shown in Fig. 4(b) originate from a three-dimensional finite-element-based continuum model. Evidently a good overall correspondence between the measured and simulated stress fields and plastic deformation zones exists.
| || Figure 4 |
Tensile deformation of an Mg sample with a notch. (a) Contour plot of the experimental and simulated stress along the tensile z axis at 170 MPa, i.e. just before yielding. The concentration of tensile stress in front of the notch is evident. (b) The extent of the plastic deformation zone as estimated from the experimental contour plots of the lattice rotation overlaid by the simulated contour levels of the effective plastic strain (p in %). From Oddershede et al. (2012).
The first studies of lattice rotations of individual grains were performed on around 100 randomly selected grains (Margulies et al., 2001; Poulsen et al., 2003; Winther et al., 2004). The obtained data were used to determine the orientation dependence of the lattice rotations and pinpoint active slip systems.
As a recent example, the positions and orientations of almost 2000 grains in a volume of 0.7 × 0.7 × 1 mm within a recrystallized interstitial free steel sample were measured (Oddershede, Wright et al., 2010). The sample was tensile deformed in ex situ steps to a maximum deformation of 9% elongation. The grain-centre mapping was performed layer by layer perpendicular to the tensile z axis using a focused planar beam with a height of 10 µm and translating the sample the same distance along z between each data collection. Because the 10 µm beam height was substantially less than the 70 µm average grain size along the translational direction, the grains could be observed in several subsequent layers as illustrated in Fig. 5(a) for the undeformed sample. Of the 1843 grains identified in the undeformed sample, 1343 could be tracked to 3% deformation (Fig. 5b, open symbols). At 9% deformation the sample was thinned down to reduce overlap of diffraction spots: as a result 113 grains in one subvolume were tracked all the way to 9% deformation (Fig. 5b, filled symbols). The use of a complementary near-field detector enabled the grain positions to be refined with an accuracy as good as 2 µm in the undeformed state and 8 µm at 9% deformation.
| || Figure 5 |
Reconstructing a three-dimensional volume containing 1850 grains of IF steel at multiple loads by scanning with a planar beam. The units on the axes are micrometres. (a) The centre-of-mass position of the grain fraction illuminated in each layer of the undeformed sample shown as a sphere color coded according to orientation. From Oddershede, Schmidt et al. (2010). (b) The centre-of-mass positions of the 1343 grains that could be followed to 3% (open symbols) and the 113 grains that could be tracked all the way to 9% (filled symbols) tensile deformation. From Oddershede, Wright et al. (2010).
In Mode III we assume that the crystalline orientation within each grain is constant. For this case, three-dimensional maps comprising several thousand grains are now routinely made with the classical setup as well as with DCT, with a spatial resolution in the range 1-5 µm and with a data acquisition time of a few hours.2
The grain orientations can be found by indexing as outlined above. Hence, mathematically speaking the task at hand is to determine the three-dimensional boundary network with as high a precision as possible. Inspired by tomography, one solution is the use of reconstruction algorithms. The development of such algorithms is nontrivial, as the complexity in terms of the dimensionality and sheer size of the reconstruction space is much larger than for classical tomography. Another difference is that in 3DXRD the number of useful projections is given by the number of observable reflections and as such is intrinsically limited (Poulsen, 2004).
For historical reasons, the work with the classical 3DXRD setup has focused on layer-by-layer solutions, where the layers are reconstructed independently. Several approaches have been developed for this two-dimensional case:
(i) Forward projection. For each voxel in the layer of interest in the sample, a forward-projection algorithm scans orientation space to test which orientations match the diffraction patterns. Cross-talk between the voxels in the layer is neglected. Hence, one can only infer a set of possible orientations. However, in practice, the constraint that all voxels belonging to a given grain must have the same orientation often implies that there is one and only one (possible) orientation for each voxel.
The workhorse for the studies at beamline 1-ID at the APS is the forward-projection software by Suter et al. (2006). Their algorithm uses a grid search of SO(3) followed by Monte Carlo optimization around candidate orientations. The best orientation based on a confidence or completeness criterion is then chosen. They address the speed issue by the use of a massive parallelization of the code. This software has been tested in detail using both simulations and experimental data. As an example we mention work on high-purity well annealed nickel (Hefferan et al., 2009). The reconstruction of one of the layers is reproduced here as Fig. 6. In this case statistics on grain orientations, intragranular misorientations and nearest-neighbour grain misorientations were compared with statistics from electron backscatter diffraction (EBSD). The correspondence was satisfactory, and it was furthermore demonstrated that the orientation resolution with the setup at 1-ID is better than 0.1°. Extensive three-dimensional maps of this kind have been generated with a spatial resolution of close to 1 µm. At the time of writing these maps are the most comprehensive obtained by any 3DXRD technique.
| || Figure 6 |
Map of one layer from a cylinder of high-purity nickel. Each element in the two-dimensional triangular mesh was fitted independently. The colour scale represents orientation space. Black lines in the maps show mesh edges separating triangles with more than a 2° misorientation. From the work by Hefferan et al. (2009) performed at the HEDM beamline at the APS.
(ii) Grain-by-grain reconstructions using algebraic reconstruction algorithms. Using hard X-rays, generally speaking for polycrystalline materials kinematical scattering is a very good approximation (Poulsen, 2004). The fact that there is a linear relationship between scattering volume and integrated intensity implies that a wealth of mathematical reconstruction principles can be applied to the problem of grain mapping. The first work of this kind was based on using a variant of the algebraic reconstruction technique (ART; Poulsen & Fu, 2003). The grain-mapping algorithm assumes that the orientations and centre-of-mass positions of all grain sections are known a priori, determined by an indexing program. The method attempts to reconstruct the boundary of each grain separately. For a specific grain, this is done by associating a `grain density' with each voxel in the layer. Once the solution has converged, the grain boundary is defined by setting a threshold. A complete grain map is obtained by superposing the solutions - the boundaries - of the individual grains.
A great advantage of using ART (or variants such as the simultaneous iterative reconstruction and simultaneous algebraic reconstruction techniques) in this scheme is that projections from as few as ten reflections are sufficient to generate a quality map. Nevertheless, for certain studies it is of interest to reduce this number even more, e.g. in order to enable very fast data acquisition procedures. In that connection the emergence within the past decade of the new mathematical discipline of discrete tomography (Herman & Kuba, 2007) has been excellent timing. The two-dimensional grain-mapping problem has in fact become a test case for the development of the algorithms within this discipline. As an example we mention that the discrete algebraic reconstruction technique, DART, proposed by J. Batenburg, has been shown to provide outstanding maps even in the case of using only three projections (Batenburg et al., 2010).
This whole class of algorithms, however, suffer from the problem that each grain is mapped individually. At the end these maps are morphed together in various ways, e.g. by heuristic smoothing operations (Alpers et al., 2006; Batenburg et al., 2010) or by the standard image-analysis technique of using dilations and erosions to generate space-filling maps. However, these procedures of `stitching together' the grain map are prone to experimental error as the centre-of-mass positions of the grains need to be known with high accuracy. In practice, this problem has imposed a limitation on the use of the grain-by-grain reconstruction algorithms.
(iii) Monte Carlo-based reconstruction. Stochastic approaches are attractive as they easily enable genuine simultaneous reconstructions of all grains rather than the grain-by-grain reconstructions discussed above. On the other hand, they tend to be slow and to `get stuck' in local minima if the configuration space is too large. Two approaches have been demonstrated for reducing the size of the orientation space. The first is a restoration approach, where a coarse or ambiguous grain map first is generated by one of the methods presented above (Alpers et al., 2006). A Monte Carlo-based routine is then used to `restore' the correct voxel affiliations in the regions close to grain boundaries. The second is an indexing approach. In this approach there is no need for an initial grain map. Instead the grain map is reconstructed on the basis of only the output from the indexing program - if there are N grains this implies that each voxel is free to take any of the associated N orientations (Alpers et al., 2006). Thanks to these limitations in solution space both approaches have been demonstrated to provide excellent reconstructions on standard PCs of relatively large two-dimensional grain maps within less than a minute.
(iv) GrainSweeper (S. Schmidt, personal communication). This algorithm is a two-step procedure for identifying the crystallographic orientation in each voxel. First, a rough orientation distribution function is constructed by adding up all geodesics in Rodrigues-Frank (RF) space related to pixels on the CCD(s) that geometrically may have been emitted from the voxel in question. Only pixels with intensities above a certain user-defined threshold are used. Afterwards, the set of local maxima in RF space (orientations) are verified against the CCD data by means of forward projections of the orientations. Normally, only the orientation with the highest completeness (ratio of measured signals to expected signals) is assigned to the voxel. To speed up the process the user can choose to apply a connectivity search in neighboring voxels, i.e. to test, recursively, if the same or similar orientation is valid for voxels that touch the current voxel.
The prime limitation for Mode III mapping in general is that the geometry of the setup has to be known with a high accuracy; the tolerances on parameters such as the tilts of the detector, the sample-detector distance etc. are very tight. Today such calibration is performed prior to measurements, implying that the setup must be sufficiently stable that the geometry is fixed during the entire experiment.
The first applications were in the field of recrystallization, as here it can be sufficient to study one grain only. From a series of studies on several pure metals and involving also topotomography, the group around Juul Jensen (Schmidt et al., 2004; van Boxel et al., 2010) has drawn the following conclusions:
(1) Occasionally, facets may form and migrate at almost constant rates for extended periods of time.
(2) The nonfaceted segments of boundaries do not migrate at a constant rate, even though the deformed matrix of the deformed single crystal, and thus the driving force, is relatively homogeneous. On the contrary, boundary segments move forward for a while, then stop, move again etc. (stop-go motion).
(3) Protrusions and retrusions typically form locally on the migrating boundaries.
In particular, the latter two phenomena are surprising. From this work, it has been concluded that protrusions/retrusions may contribute an additional driving force as a result of boundary curvature, which locally may be of similar magnitude to that provided by the energy stored in the deformed matrix. On an overall scale it is further shown that the formation of protrusions/retrusions affects the stop-go motion of the boundary, and if the protrusions/retrusions are large, they contribute a net increase in the overall migration rate of the boundary (van Boxel et al., 2010). In other words, the work has shown that new theory is needed to describe the local and overall motion of boundaries during recrystallization.
In this study the evolution of the morphology of several hundred grains was monitored simultaneously for the first time (Schmidt et al., 2008). The Al-0.1%Mn sample material had grain sizes between 30 and 100 µm. The sample was annealed in an external furnace. Before the first annealing and following each annealing step a cylindrical volume with a diameter of 700 µm and height of 350 µm was fully characterized by mapping out the sample in layers using a planar beam shape at 50 keV. The grain volumes were reconstructed using GrainSweeper. In total, five annealing steps were made at annealing temperatures between 673 and 723 K. Initially, 491 grains were present, leaving only 49 grains after the final annealing step. Fig. 7 shows four layers of the fully reconstructed initial and final grain volumes.
| || Figure 7 |
Grain growth in Al-0.1%Mn. Shown are the maps for the first four layers in the initial state (above) and the corresponding maps for the same layers after heating (below). The fully reconstructed grain volumes - covering 40 layers - comprised 491 and 49 grains, respectively. The colour scale is related to the crystallographic orientation of the individual grain. From Schmidt et al. (2008).
A basic limitation for Mode III operation is the assumption that there are no variations in orientation within a given grain. This is evidently not the case for deformed specimens, and annealed samples may exhibit growth fronts within grains. Mode IV operation is relevant for such cases.
Orientation imaging microscopy (OIM) is well known from electron microscopy, where a scanning method - EBSD - is used to create maps of the orientations in a layer of the sample point by point. Three-dimensional mapping is obtained by means of serial sectioning techniques. The 3DXRD equivalent is a three-dimensional map where the orientation in each voxel of the sample is reconstructed independently. Grains and grain boundaries are then defined in a post-processing step, similar to that used by EBSD software. In contrast to the grain-by-grain reconstruction case adopted in Mode III, a space-filling map is automatically generated.
Mathematically speaking, the task in Mode IV is therefore to reconstruct a vector field r(x), where r symbolizes orientation and x position in direct space. For such fields no transform or algebraic reconstruction algorithm is readily available. Stochastic approaches are possible. The two most popular programs today for OIM are the forward-simulation program by Suter et al. (2006) and GrainSweeper by Schmidt, both presented above. Examples of GrainSweeper applications are given below.
The computing complexity is evidently a concern. To reduce this complexity, and to increase the likelihood of the problem being well posed, both programs operate with several detector distances and with two-dimensional data: that is, layer-by-layer reconstructions.
If the degree of deformation is sufficiently small that the grains can still be identified in far-field data, one may contemplate generating a three-dimensional centre-of-mass map prior to OIM reconstruction. This can then be used to constrain solution space and effectively reduce the computing time required by orders of magnitude. For such a case, Rodek et al. (2007) presented a forward-projection Monte Carlo algorithm, where `the grains are grown' from the seeds provided by the three-dimensional grain-centre map. This approach was tested on simulated microstructures typical of deformation up to 30%. At low noise levels, reconstructions were nearly perfect.
The main problem of all of these formalisms is, however, the assumption that each voxel is represented by one and only one orientation. With the present generation of detectors, the spatial resolution of 3DXRD-based OIM is 2-5 µm. For deformed metals, this should be contrasted with the typical size of subgrains, which is of the order of 0.2-2 µm. Hence, it is evident that at best OIM can represent a kind of local average. Note that this is more of an issue for 3DXRD than for scanning methods like EBSD and DAXM, because the diffraction signal from various voxels may overlap. It is not known when a 3DXRD vector-field formalism may break down and provide unphysical maps. As we shall see below, early work has provided encouraging evidence of the suitability for use at moderate degrees of deformation, but more work is needed in terms of validation and defining the limitations.
Recently, for small degrees of deformation, an alternative strategy for use of OIM based on insertion of an absorbing grid between the sample and the near-field detector has been proposed. Initial results are presented by King et al. (2010).
For nondeformed samples and using a connectivity search, a typical OIM reconstruction (resolution equivalent to smallest detector pixel size and 2002-4002 voxels) by GrainSweeper takes less than an hour. For heavily deformed samples the reconstruction takes of the order of 24 h on four computer cores. If a centre-of-mass map is available each layer can be reconstructed in less than a minute.
Two examples of reconstructed OIMs are provided in Fig. 8. To the left is a layer within an NaCl single crystal, which had been compressed by 16.5%. This was part of a geosciences model system study of the development of grain boundaries and substructure dynamics (Borthwick et al., 2012). The formation of subgrains is clearly visible.
| || Figure 8 |
Examples of OIM work by the use of GrainSweeper. In both cases, orientations are mapped on a colour scale. Black voxels indicate voxels with no orientation assignment. (Left) Reconstruction based on three-dimensional detector data of a layer within an NaCl single crystal with a prehistory of 16.5% compression following by annealing. Subgrains are visible. Courtesy of S. Schmidt. (Right) Reconstruction of a layer within a 30% cold-rolled Al1050 sample using only a far-field detector. The white and black lines denote misorientations of 2 and 15°, respectively. Three grains are identified and marked as A, B and C. Superposed on the OIM map are four red dots (numbered 2, 4, 5 and 6) marking the positions of nuclei appearing during the subsequent annealing. The longer dimension of the cross section is 1 mm. Adapted from West et al. (2009).
To the right is reproduced the very first 3DXRD OIM image from the work of West et al. (2009). At that point in time the three-dimensional detector at beamline ID11 at ESRF was not yet operational, and as such this map is generated solely on the minute amount of spatial information that can be extracted from a far-field detector. The spatial resolution is correspondingly poor: of the order of 30 µm, and with extended areas of voxels with no assignment of an orientation (shown in black). The statistical noise on orientations is of the order of 2°.
The motivation for the latter work was to study bulk nucleation, appearing during recrystallization of metals. In situ studies of such phenomena are challenging as the nuclei are small (the critical nucleation radius is typically around 1 µm) and the exact positions of the nucleation sites are not known, implying that one has to monitor large volumes of the sample in order to capture these rare events. Furthermore, for the events to be characteristic of bulk properties, the nuclei should form within grains that are deeply embedded in the material.
Notably, despite the mapping resolution of 30 µm, nuclei as small as 1 µm could be observed in the far-field data provided these appear with orientations that are different from those present in the illuminated part of the deformed microstructure. By comparing the raw data before and after annealing a number of such spots corresponding to new texture components were identified. It was possible to index most of these and to determine the centre-of-mass position of these nuclei. Within the vicinity of the layer shown in Fig. 8 four nuclei were found - these are marked in the figure as red dots with numbers 2, 4, 5 and 6 attached to them. The unique feature of this kind of mapping is that the properties of the new nuclei can be correlated directly with the deformed microstructure at the same position before annealing. The main discovery as such is the proof that nuclei of orientations not present within the microstructure can form - this calls for new nucleation mechanisms to be formulated.
The basic 3DXRD algorithms were all developed for monophase materials. However several programs like GrainSpotter and FitAllB can now be applied to multiphase materials by first running them once for each phase and then performing a global optimization step.
A powerful way to generalize the mapping procedure is to combine 3DXRD with classical absorption or phase contrast tomography. Such a combination is inherent to DCT, but has also become popular with the classical 3DXRD setup. This combination immediately suggests studies of, for example, the correlation between crack propagation (in situ tomography), microstructure (3DXRD in Mode III) and local stress development (Mode II). Examples of such work are the studies on intergranular stress corrosion cracking (King et al., 2008) and the work on stress-field evolution around a growing crack (Oddershede et al., 2012).
Tomography data are also very useful in order to define the external shape of the sample with high fidelity and in connection with the use of markers. More ambitiously, one may contemplate using tomography data to subdivide a sample representation into volumes representing various phases and subsequently to perform grain mapping one phase at a time for each phase using the tomography map to define the phase boundaries.
The current status of such hybrid methods is illustrated by the recent work on natural chalk by the nanogeoscience group in Copenhagen (Sørensen, Hakim et al., 2012). Chalk primarily consists of calcite, CaCO3, but can contain a number of other minerals, e.g. clays and quartz. In this study, first tomography was used to provide a three-dimensional representation of the porous network. From the tomography it was found that many high-density inclusions are buried in the bulk of the nanocrystalline chalk. To identify the mineral type of these inclusions a complementary 3DXRD experiment was performed on the same sample. Based on existing knowledge, a catalogue of 11 possible phases were made to guide which minority phases to search for in the 3DXRD data. By trial and error, from this list successful indexing of four mineral types were achieved and their position determined by a centre-of-mass grain map. Next, on the basis of the position, these inclusions were easily identified in the tomogram. In this way a three-dimensional map of both the porous structure and the various second phase elements could be derived.
It is of obvious interest to improve the spatial resolution for three-dimensional mapping. Generally speaking, the current limitation for 3DXRD is the near-field detectors. The coupling via visual light becomes inefficient and geometrically challenging near the wavelength of the optical light used. A radically new detector technology has been suggested by the group at the Technical University of Denmark (DTU) (http://www.faqs.org/patents/app/20100276605 ) and simulations predict that this approach could provide 100 nm resolution in the energy range of 10 keV and above. However, a fully operational detector of this kind is at least five years down the line.
The joint focus of the ESRF and DTU groups is currently on hard X-ray microscopy, where an objective is used, enabling a magnification of the beam and hence a remedy to the detector resolution limitation. An account of the status and future plans for the upgrade of beamline ID11, ESRF, as of mid 2010 is provided by Vaughan et al. (2010).
In parallel with these efforts, recently the 3DXRD methodology has been transferred to TEM. Using conical scanning dark-field microscopy in connection with a variant of the GrainSweeper reconstruction algorithm, orientation mapping was demonstrated with a three-dimensional spatial resolution of 1 nm (Liu et al., 2011). The first work relates to the three-dimensional mapping of a 150 nm-thick foil of nanocrystalline aluminium (see Fig. 9). The volume in the figure corresponds to a small fraction of the entire illuminated volume, which comprised of the order of 1000 grains. The acquisition procedure used, however, is slow in the sense that about 100 000 dark-field images were required. It is too early to comment on the general applicability of this technique, but it is evident that a main limitation - similar to TEM tomography - is the restriction to electron-transparent foils.
| || Figure 9 |
A TEM equivalent of 3DXRD. The three-dimensional grain-orientation map comprises a subvolume of the fully reconstructed volume within the 150 nm-thick aluminium film specimen. The colour scale represents different crystal orientations with a tolerance of 2°. The spatial resolution is estimated to be 1 nm. From Liu et al. (2011).
Among other directions of exploration at the moment we mention the following:
(a) The generalization of 3DXRD/DCT towards extraction of subgrain information. King et al. (2010) have demonstrated the application of reticulography in connection with polycrystals. In this setup an absorbing grid is inserted between the sample and the detector, close to the sample. One diffraction spot from one grain is investigated (at a time). Distortions in the crystal lattice result in variations in the direction of the diffracted beam, which will give rise to a distorted image of the grid on the detector. By analysing these distortions, a two-dimensional map of components of the grain misorientation can be determined. Three-dimensional information is provided by the combined analysis of several diffraction spots.
(b) The ambition to determine local crystallographic information (such as a three-dimensional grain map) from a deeply embedded volume in thick specimens. Classical 3DXRD as presented above is not suitable for such work, because of the massive spot overlap. One solution is to place sophisticated collimators in the diffracted beam, such as conical slits (Nielsen et al., 2000) or spiral slits (Martins & Honkimäki, 2005). Some software-based solutions for extending the range of application of 3DXRD towards diffraction patterns with more overlap are outlined by Sørensen, Schmidt et al. (2012).
The author gratefully acknowledges the European Research Council for a grant. The Danish Research Council is acknowledged for funding synchrotron experiments (via Danscatt). All figures showing synchrotron work - except Fig. 6 - are based on data acquired at the ESRF.
Alpers, A., Poulsen, H. F., Knudsen, E. & Herman, G. T. (2006). J. Appl. Cryst. 39, 582-588.
Alpers, A., Rodek, L., Poulsen, H. F., Knudsen, E. & Herman, G. T. (2007). Advances in Discrete Tomography and its Applications, edited by G. T. Herman & A. Kuba, pp. 271-301. Berlin: Birkhäuser.
Aydiner, C. C., Bernier, J. V., Clausen, B., Lienert, U., Tome, C. & Brown, D. W. (2009). Phys. Rev. B, 80, 024113.
Banhart, J. (2008). Editor. Advanced Tomographic Methods in Materials Research and Engineering, pp. 249-277. Oxford University Press.
Barabash, R. I., Ice, G. E., Liu, W. & Barabash, O. M. (2009). Micron, 40, 28-36.
Batenburg, K. J., Sijbers, J., Poulsen, H. F. & Knudsen, E. (2010). J. Appl. Cryst. 43, 1464-1473.
Bernier, J. V., Barton, N. R., Lienert, U. & Miller, M. P. (2011). J. Strain Analysis, 46, 527-554.
Berveiller, S., Malard, B., Wright, J., Patoor, E. & Geandier, G. (2011). Acta Mater. 59, 3636-3645.
Bleuet, P., Welcomme, E., Dooryhée, E., Susini, J., Hodeau, J. L. & Walter, P. (2008). Nat. Mater. 7, 468-472.
Borthwick, V. E., Schmidt, S., Piazolo, S. & Gundlach, C. (2012). Geochem. Geophys. Geosyst. 13, Q05005.
Boxel, S. van, Schmidt, S., Ludwig, W., Zhang, Y. B., Sørensen, H. O., Pantleon, W. & Juul Jensen, D. (2010). Proceedings of the 31st Risø International Sympsium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 449-456. Roskilde: DTU Press.
Bunge, H. J., Wcislak, L., Klein, H., Garbe, U. & Schneider, J. R. (2003). J. Appl. Cryst. 36, 1240-1255.
Edmiston, J. K., Barton, N. R., Bernier, J. V., Johnson, G. C. & Steigmann, D. J. (2011). J. Appl. Cryst. 44, 299-312.
Frank, F. C. (1988). Metall. Trans. A, 19, 403.
Fu, X., Poulsen, H. F., Schmidt, S., Nielsen, S. F., Lauridsen, E. M. & Jensen, D. J. (2003). Scr. Mater. 49, 1093-1096.
Godiksen, R. B., Schmidt, S. & Juul Jensen, D. (2007). Scr. Mater. 57, 345-348.
Gundlach, C., Pantleon, W., Lauridsen, E. M., Margulies, L., Doherty, R. & Poulsen, H. F. (2004). Scr. Mater. 50, 477-481.
Hall, S. A., Wright, J., Pirling, T., Ando, E., Hughes, D. J. & Viggiani, G. (2011). Granul. Matter, 13, 251-254.
Hefferan, M., Li, S. F., Lind, J., Lienert, U., Rollett, A. D., Wynblatt, P. & Suter, R. M. (2009). Comput. Mater. Continua, 14, 209-219.
Herman, G. T. & Kuba, A. (2007). Editors. Advances in Discrete Tomography and its Applications. Boston: Birkhäuser.
Ice, G. E., Budai, J. D. & Pang, J. W. (2011). Science, 334, 1234-1239.
Ice, G. E., Larson, B. C., Yang, W., Budai, J. D., Tischler, J. Z., Pang, J. W. L., Barabash, R. I. & Liu, W. (2005). J. Synchrotron Rad. 12, 155-162.
Iqbal, N., van Dijk, N. H., Offerman, S. E., Moret, M. P., Katgerman, L. & Kearley, G. J. (2005). Acta Mater. 53, 2875-2880.
Jakobsen, B., Poulsen, H. F., Lienert, U., Almer, J., Shastri, S. D., Sørensen, H. O., Gundlach, C. & Pantleon, W. (2006). Science, 312, 889-892.
Jakobsen, B., Poulsen, H. F., Lienert, U. & Pantleon, W. (2007). Acta Mater. 55, 3421-3430.
Jimenez-Melero, E., van Dijk, N. H., Zhao, L., Sietsma, J., Offerman, S. E., Wright, J. P. & van der Zwaag, S. (2007). Acta Mater. 55, 6713-6723.
Johnson, G., King, A., Honnicke, M. G., Marrow, J. & Ludwig, W. (2008). J. Appl. Cryst. 41, 310-318.
Kazantsev, I. G., Schmidt, S. & Poulsen, H. F. (2009). Inverse Probl. 25, 105009.
Kenesei, P. (2010). PhD thesis, Eötvös Loránd University, Hungary.
King, A., Johnson, G., Engelberg, D., Ludwig, W. & Marrow, J. (2008). Science, 321, 382-385.
King, A., Reischig, P., Martin, S., Fonseca, J., Preuss, M. & Ludwig, W. (2010). Proceedings of the 31st Risø International Symposium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 43-57. Roskilde: DTU Press.
Larson, B. C., Yang, W., Ice, G. E., Budai, J. D. & Tischler, J. Z. (2002). Nature (London), 415, 887-890.
Lauridsen, E. M., Poulsen, H. F., Nielsen, S. F. & Juul Jensen, D. (2003). Acta Mater. 51, 4423-4435.
Lauridsen, E. M., Schmidt, S., Nielsen, S. F., Margulies, L., Poulsen, H. F. & Juul Jensen, D. (2006). Scr. Mater. 55, 51-56.
Lauridsen, E. M., Schmidt, S., Suter, R. M. & Poulsen, H. F. (2001). J. Appl. Cryst. 34, 744-750.
Levine, L. E., Larson, B. C., Yang, W., Kassner, M. E., Tischler, J. Z., Delos-Reyes, M. A., Fields, R. J. & Liu, W. (2006). Nat. Mater. 5, 619-622.
Lienert, U., Brandes, M. C., Bernier, J. V., Mills, M. J., Miller, M. P., Li, S. F., Hefferan, C. M., Lind, J. & Suter, R. M. (2010). Proceedings of the 31st Risø International Symposium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 69-77. Roskilde: DTU Press.
Lienert, U., Li, S. F., Hefferan, C. M., Lind, J., Suter, R. M., Bernier, J. V., Barton, N. R., Brandes, M. C., Mills, M. J., Miller, M. P., Jakobsen, B. & Pantleon, W. (2011). J. Microsc. 63, 70-77.
Liu, H. H., Schmidt, S., Poulsen, H. F., Godfrey, A., Liu, Z. Q., Sharon, J. A. & Huang, X. (2011). Science, 332, 833-834.
Liu, Y. L., Wang, W. G., Poulsen, H. F. & Vase, P. (1999). Supercond. Sci. Technol. 12, 376-381.
Ludwig, W., Cloetens, P., Härtwig, J., Baruchel, J., Hamelin, B. & Bastie, P. (2001). J. Appl. Cryst. 34, 602-607.
Ludwig, W., Lauridsen, E. M., Schmidt, S., Poulsen, H. F. & Baruchel, J. (2007). J. Appl. Cryst. 40, 905-911.
Ludwig, W., Reischig, P., King, A., Herbig, M., Lauridsen, E. M., Johnson, G., Marrow, T. J. & Buffiere, J. Y. (2009). Rev. Sci. Instrum. 80, 33905-33909.
Ludwig, W., Schmidt, S., Lauridsen, E. M. & Poulsen, H. F. (2008). J. Appl. Cryst. 41, 302-309.
Lyckegaard, A., Alpers, A., Ludwig, W., Fonda, R. W., Margulies, L., Götz, A., Sørensen, H. O., Dey, S. R., Poulsen, H. F. & Lauridsen, E. M. (2010). Proceedings of the 31st Risø International Symposium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 329-366. Roskilde: DTU Press.
Lyckegaard, A., Lauridsen, E. M., Ludwig, W., Fonda, R. & Poulsen, H. F. (2011). Adv. Eng. Mater. 13, 165-170.
Margulies, L., Lorentzen, T., Poulsen, H. F. & Leffers, T. (2002). Acta Mater. 50, 1771-1779.
Margulies, L., Winther, G. & Poulsen, H. F. (2001). Science, 291, 2392-2394.
Martins, R. V. & Honkimäki, V. (2005). Textures Microstruct. 35, 145-152.
Martins, R. V., Margulies, L., Schmidt, S., Poulsen, H. F. & Leffers, T. (2004). Mater. Sci. Eng. A, 387, 84-88.
Moscicki, M., Kenesei, P., Wright, J., Pinto, H., Lippmann, T., Borbély, A. & Pyzalla, A. R. (2009). Mater. Sci. Eng. A, 524, 64-68.
Nielsen, S. F., Wolf, A., Poulsen, H. F., Ohler, M., Lienert, U. & Owen, R. A. (2000). J. Synchrotron Rad. 7, 103-109.
Oddershede, J., Camin, B., Schmidt, S., Mikkelsen, L. P., Sørensen, H. O., Lienert, U., Poulsen, H. F. & Reimers, W. (2012). Acta Mater. 60, 3570-3580.
Oddershede, J., Schmidt, S., Poulsen, H. F., Margulies, L., Wright, J., Moscicki, M., Reimers, W. & Winther, G. (2011). Mater. Charact. 62, 651-660.
Oddershede, J., Schmidt, S., Poulsen, H. F., Sørensen, H. O., Wright, J. & Reimers, W. (2010). J. Appl. Cryst. 43, 539-549.
Oddershede, J., Wright, J., Margulies, L., Huang, X., Poulsen, H. F., Schmidt, S. & Winther, G. (2010). Proceedigns of the 31st Risø International Symposium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 369-374. Roskilde: DTU Press.
Offerman, S. E., van Dijk, N. H., Sietsma, J., Grigull, S., Lauridsen, E. M., Margulies, L., Poulsen, H. F., Rekveldt, M. T. & van der Zwaag, S. (2002). Science, 298, 1003-1005.
Pantleon, W., Wejdemann, C., Jakobsen, B., Lienert, U. & Poulsen, H. F. (2010). Proceedings of the 31st Risø International Symposium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 79-100. Roskilde: DTU Press.
Poulsen, H. F. (2004). Three-Dimensional X-ray Diffraction Microscopy. Berlin: Springer.
Poulsen, H. F. & Fu, X. (2003). J. Appl. Cryst. 36, 1062-1068.
Poulsen, H. F., Garbe, S., Lorentzen, T., Juul Jensen, D., Poulsen, F. W., Andersen, N. H., Frello, T., Feidenhans'l, R. & Graafsma, H. (1997). J. Synchrotron Rad. 4, 147-154.
Poulsen, H. F., Margulies, L., Schmidt, S. & Winther, G. (2003). Acta Mater. 51, 3821-3830.
Poulsen, H. F., Nielsen, S. F., Lauridsen, E. M., Schmidt, S., Suter, R. M., Lienert, U., Margulies, L., Lorentzen, T. & Juul Jensen, D. (2001). J. Appl. Cryst. 34, 751-756.
Poulsen, S. O., Lauridsen, E. M., Lyckegaard, A., Oddershede, J., Gundlach, C., Curts, C. & Juul Jensen, D. (2011). Scr. Mater. 64, 1003-1006.
Reimers, W., Pyzalla, A. R., Schreyer, A. & Clemens, H. (2008). Editors. Neutron and Synchrotron Radiation in Engineering Material Science. Weinheim: Wiley-VCH.
Reischig, P., King, A., Nervo, L., Viganó, N., Guilhem, Y., Palenstijn, W. J., Batenburg, K. J., Preuss, M. & Ludwig, W. (2013). J. Appl. Cryst. 46. Submitted.
Rodek, L., Poulsen, H. F., Knudsen, E. & Herman, G. T. (2007). J. Appl. Cryst. 40, 313-321.
Schmidt, S. (2012). GrainSpotter, http://sourceforge.net/apps/trac/fable/wiki/grainspotter .
Schmidt, S., Nielsen, S. F., Gundlach, C., Margulies, L., Huang, X. & Jensen, D. J. (2004). Science, 305, 229-232.
Schmidt, S., Olsen, U. L., Poulsen, H. F., Sørensen, H. O., Lauridsen, E. M., Margulies, L., Maurice, C. & Juul Jensen, D. (2008). Scr. Mater. 59, 491-494.
Sørensen, H. O., Hakim, S., Pedersen, S., Schmidt, S., Hem, C. P., Pasarin, I. S., Frandsen, C., Balogh, Z. I., Olsen, U. L., Oddershede, J., Christiansen, B. C., Feidenhans'l, R. & Stipp, S. L. S. (2012). Can. Mineral. 50, 501-509.
Sørensen, H. O., Schmidt, S., Wright, J. P., Vaughan, G. B. M., Techert, S., Garman, E. F., Oddershede, J., Davaasambu, J., Paithankar, K. S., Gundlach, C. & Poulsen, H. F. (2012). Z. Kristallogr. 227, 63-78.
Suter, R. M., Hennessy, D., Xiao, C. & Lienert, U. (2006). Rev. Sci. Instrum. 77, 123905.
Ungar, T., Ribarik, G., Balogh, L., Salem, A. A., Semiatin, S. L. & Vaughan, G. B. M. (2010). Scr. Mater. 63, 69-72.
Vaughan, G. B. M., Schmidt, S. & Poulsen, H. F. (2004). Z. Kristallogr. 219, 813-825.
Vaughan, G. B. M., Wright, J. P. et al. (2010). Proceedings of the 31st Risø International Sympsium on Materials Science, edited by N. Hansen, D. Juul Jensen, S. F. Nielsen, H. F. Poulsen & B. Ralph, pp. 457-476. Roskilde: DTU Press.
West, S., Schmidt, S., Sørensen, H. O., Winther, G., Poulsen, H. F., Margulies, L., Gundlach, C. & Juul Jensen, D. (2009). Scr. Mater. 61, 875-878.
Winther, G., Margulies, L., Schmidt, S. & Poulsen, H. F. (2004). Acta Mater. 52, 2863-2872.
Wroblewski, T., Clauss, O., Crostack, H.-A., Ertel, A., Fandrich, F., Genzel, Ch., Hradil, K., Ternes, W. & Woldt, E. (1999). Nucl. Instrum. Methods Phys. Res. Sect. A, 428, 570-582.