research papers X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. I.

Journal of Applied Crystallography ISSN 0021-8898 X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. I. Direct beam case Received 28 June 2007 Accepted 16 January 2008 Wolfgang Ludwig, a,b * Søeren Schmidt, c Erik Mejdal Lauridsen c and Henning Friis Poulsen c a MATEIS, INSA-Lyon, Villeurbanne, France, b European Synchrotron Radiation Facility, France, and c Centre for Fundamental Research: Metal Structures in Four Dimensions, Risø National Laboratory, Roskilde, Denmark. Correspondence e-mail: wolfgang.ludwig@insa-lyon.fr # 2008 International Union of Crystallography Printed in Singapore all rights reserved The principles of a novel technique for nondestructive and simultaneous mapping of the three-dimensional grain and the absorption microstructure of a material are explained. The technique is termed X-ray diffraction contrast tomography, underlining its similarity to conventional X-ray absorption contrast tomography with which it shares a common experimental setup. The grains are imaged using the occasionally occurring diffraction contribution to the X-ray attenuation coefficient each time a grain fulfils the diffraction condition. The three-dimensional grain shapes are reconstructed from a limited number of projections using an algebraic reconstruction technique. An algorithm based on scanning orientation space and aiming at determining the corresponding crystallographic grain orientations is proposed. The potential and limitations of a first approach, based on the acquisition of the direct beam projection images only, are discussed in this first part of the paper. An extension is presented in the second part of the paper [Johnson, King, Honnicke, Marrow & Ludwig (2008). J. Appl. Cryst. 41, 310 318], addressing the case of combined direct and diffracted beam acquisition. 1. Introduction Over the past ten years considerable effort has been put into the development of novel three-dimensional grain mapping techniques for polycrystalline materials. In contrast to threedimensional reciprocal space mapping techniques (Fewster, 1997; Jakobsen et al., 2006) where the focus is given to orientation and strain characterization of individual crystals, these new approaches aim at real space description of bulk polycrystalline materials in terms of three-dimensional shapes and orientations of all grains present in the illuminated sample volume. These new real space mapping techniques, based on the diffraction of synchrotron beams, can be divided into two classes. The first set of techniques, known under the name of three-dimensional X-ray diffraction microscopy (3DXRD), employ reconstruction algorithms of the kind known in tomography [see e.g. Poulsen (2004) for a recent review]. The second class comprise techniques such as differential aperture X-ray microscopy (DAXM; Larson et al., 2002), where threedimensional information is obtained by scanning the sample in three directions. Recent investigations where such nondestructive bulk mapping techniques were used to analyse microstructural changes related to deformation and annealing processes in metallic samples clearly underline their potential for improving our understanding of complex processes in classical fields of physical metallurgy (Offerman et al., 2002; Schmidt et al., 2004; Iqbal et al., 2006; Levine et al., 2006; Jakobsen et al., 2006). In the current paper, a radically different data acquisition strategy, aiming at simultaneous reconstruction of the absorption and grain microstructure of a material, is proposed. The procedure is termed X-ray diffraction contrast tomography (DCT), reflecting its similarities to conventional X-ray absorption contrast tomography. During acquisition of an optimized tomographic scan, undeformed grains embedded in the bulk of a polycrystalline sample give rise to distinct diffraction contrasts which can be observed in the transmitted beam each time a grain fulfils the Bragg diffraction condition. By extracting and sorting these contrasts into groups belonging to individual grains, one is able to reconstruct the three-dimensional grain shapes by means of parallel beam, algebraic reconstruction techniques (ARTs; Gordon, 1970). The method applies to undeformed polycrystalline mono- or multiphase materials containing a limited number of grains per sample cross section. The similarity to absorption tomography implies that the method can be easily implemented at any modern synchrotron facility providing a microtomographic imaging setup. In the next section the methodology of diffraction contrast tomography is explained in more detail. The analysis procedure is illustrated with the example of a recrystallized Al 1050 alloy multicrystal in x3 of the paper. The potential and the 302 doi:10.1107/s0021889808001684 J. Appl. Cryst. (2008). 41, 302 309

limitations of the direct beam approach are discussed in x4. The procedure is summarized in Appendix A. In part II of the paper (Johnson et al., 2008), an approach combining features of DCT and 3DXRD is presented, which increases the applicability of DCT. 2. Principle Let us assume a plane monochromatic X-ray wave impinging on a polycrystalline material consisting of plastically undeformed grains with negligible internal orientation spread. During a continuous 180 movement around the tomographic rotation axis, each of the Ewald spheres associated with the individual grains will, from time to time, pass through reciprocal space lattice points, giving rise to diffracted beams. Each of these diffraction events is associated with a local reduction of the transmitted intensity recorded on the highresolution imaging detector behind the sample (Fig. 1). The strength and visibility of the above-mentioned diffraction contrast will depend on details of the acquisition conditions and on the quality of the sample in terms of texture, grain size and the mosaic spread of the individual grains. The latter three parameters determine the probability of overlapping diffraction contrasts in the projection images. Since these overlaps render the extraction of contributions from individual grains increasingly difficult, one has to tailor the transverse sample dimensions in order to limit the probability of spot overlap. From an experimental point of view, the divergence and the energy bandwidth of the incoming radiation have to be chosen such that the corresponding broadening of the reflection curves is small compared with the intrinsic scattering width of the grains under investigation. The dispersive, oblique angle diffraction geometry implies that the individual grain projection images have to be integrated over an angular range! (rotation around the tomographic rotation axis) in order to Figure 1 The acquisition geometry used for diffraction contrast tomography is identical to that used in synchrotron microtomography. During continuous rotation of the sample around the vertical rotation axis, a large number of projection images are recorded on an electronic highresolution detector system. During sample rotation, each of the individual grains will fulfil the Bragg diffraction condition many times. For low-index reflections generally associated with high structure factors, the additional contribution to the X-ray attenuation coefficient is strong enough to be visible as extinction spots in the transmitted beam. illuminate the whole grain volume. Again, the integration interval! has to be chosen to be smaller than or equal to the sum of the above-mentioned crystal reflection broadening effects, in order to preserve the maximum available diffraction contrast in the projection images. In the following subsections the basic assumptions enabling direct interpretation of the diffraction contrasts in terms of grain projections are outlined and the principal steps of the data analysis procedure are explained in more detail. 2.1. Basic assumptions Neglecting effects from dynamical theory of X-ray diffraction 1 one can assume that the intensity decay along any ray passing through a diffracting grain can be described by an exponential function with an effective attenuation coefficient eff (r) = abs (r) + diffr (r), Iðu; vþ ¼I 0 ðu; vþ exp R eff ðrþ dx ; ð1þ where r =(x, y, z) defines the position inside the sample and the integral is calculated along the beam direction x. The integration along straight lines further implies that we neglect any refraction effects arising from local variations of the X-ray refractive index decrement (n =1 +i, 10 6 ) inside the sample. 2 In this simple model abs comprises all relevant contributions to the X-ray attenuation coefficient other than coherent scattering from crystals aligned for Bragg diffraction. The coherent scattering contribution to the attenuation coefficient arising from sub-volumes of an individual grain, diffr ðr;; ;F hkl ; LÞ, depends on the wavelength, the local effective misorientation (r) with respect to the maximum of a given hkl reflection, the structure factor F hkl and the Lorentz factor L associated with the reflection. We further assume that after integration over the local misorientation ranges the diffraction contributions arising from different reflections of the same grain can be separated into a spatially varying part diffr (r) and a multiplicative scaling function mð;!; gþ, where g represents the orientation matrix of the grain. The spatially varying part diffr (r) expresses the local diffraction power of the crystal and accounts for local variations of the scattered intensity, which may arise from differences in defect density and/or the presence of second phase inclusions inside the grains. This contribution is supposed to be independent of the actual orientation of the crystal lattice with respect to the beam. The scaling function mð;!; gþ expresses the fact that these diffraction contributions can only be observed for particular rotation angles, each time a reflection fulfils the Bragg condition. In practice the scaling function mð;!; gþ is equal to zero for most of the rotation angles! and scales 1 Equivalent to the assumption of ideal imperfect crystals, composed of very small mosaic blocks. 2 The refraction of hard X-rays as a result of phase gradients can in practice only be observed when placing an analyser crystal between the sample and the detector an imaging technique known as diffraction enhanced imaging (Chapman et al., 1997). The deviation from straight ray propagation and the resulting spatial distortion of the projection images in the current study is small compared with the spatial resolution employed and can therefore, as is common practice in X-ray absorption imaging, be neglected. J. Appl. Cryst. (2008). 41, 302 309 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I 303

proportional to the structure and Lorentz factors for rotation angles corresponding to the different reflections of the grain: m / F hkl 2 L: After logarithmic subtraction of the intensity distribution of the incoming beam (flat-field correction) we obtain access to the line integrals of the effective attenuation coefficient eff along the beam direction dx: ln Iðu; vþ I 0 ðu; vþ ¼ R eff ðrþ dx ¼ R abs ðrþ dx þ m R diffr ðrþ dx: 2.2. Removal of the absorption background In general the angular intervals for which diffraction (from different reflections) gives rise to a significant contribution to the effective attenuation coefficient are negligible compared with the 180 rotation range of a tomographic scan. One may therefore obtain a good estimate of the local absorption coefficient abs by neglecting the diffraction contributions and by applying a conventional filtered backprojection reconstruction algorithm (see e.g. Kak & Slaney, 1988) to the DCT projection data. Having reconstructed the absorption coefficient distribution, one can calculate the local absorption background R abs ðrþ dx by forward projection and subtract this contribution from the raw projection images. An alternative way to estimate the absorption background consists in application of a one-dimensional median filter to a threedimensional stack of projection images, built from equally spaced images centred around the current projection and covering an adequate angular range (typically several times the maximum width of the reflections observed). By calculating pixel by pixel the one-dimensional median value along the stack dimension (see Fig. 3 in part II), one can efficiently reject diffraction events and obtain a good estimate of the slowly varying absorption background. After logarithmic Figure 2 (a) Two-dimensional backprojection of the (thresholded) central lines of the extinction spots [dashed line in Fig. 3(c)] selected by the first filtering step. Erroneous projections not belonging to the grain of interest could be easily filtered out at this step, since they typically would not superpose in the grain position. (b) Two-dimensional ART reconstruction of the corresponding grain cross section, using all 27 projections available for this particular grain. (c) Overlay of the ART reconstruction (Fig. 2b) with the corresponding absorption contrast reconstruction obtained by conventional filtered backprojection reconstruction. ð2þ subtraction of the absorption background the resulting images only contain contrasts arising from diffraction events and the grey values correspond to projections of the local diffraction contribution m R diffr ðrþ dx along the beam direction. 2.3. Spot summation and segmentation In general, diffraction spots can spread over a range of successive images and only part of the diffracting grain may be visible in each of the individual images. One therefore first has to sum the contributions belonging to the same reflection. This can be accomplished by applying a three-dimensional segmentation algorithm based on morphological image reconstruction (see e.g. Vincent, 1993). While working through the stack of successive projection images, each individual region of interest (i.e. an extinction contrast above a certain threshold) will be summed with contributions from neighbouring images as long as the regions are connected in the third dimension. The summed extinction spot as well as some region properties (centroid, bounding box, area, intensity,! range) are stored in a database. The main challenge of spot summation is related to the fact that one has to deal with faint contrasts, which in addition may be affected by overlap from spots originating from other diffracting grains in front of or behind the grain of consideration. Although the separation of the different contributions seems feasible by eye, the above-mentioned spot segmentation algorithm based on morphological image reconstruction may fail to separate the overlapping parts. 2.4. Spot sorting Next, the segmented extinction spots are sorted into sets belonging to the same grain. In the case of a limited number of grains as is considered in this first part of the paper, the spot sorting can be achieved by means of the following two spatial filtering steps. First, the top and bottom vertical limits of the extinction spots belonging to the same grain have to be identical and independent of the current! rotation angle. Depending on the accuracy of segmentation and the total number of grains per cross section, this preliminary list of spots may in certain cases still contain spots belonging to different grains, having similar vertical extent and position. These outliners can in general be eliminated by a second filtering step where the central lines of the thresholded spots are backprojected into the sample plane, taking into account the respective rotation angles. Only spots superimposing on a common location in the sample plane will be retained as valid projections of the same grain (Fig. 2a). By applying an intensity threshold to the backprojected image based on these accepted spots, one obtains a first crude estimate of the twodimensional grain shape and its position within the sample plane. 2.5. Reconstruction Before the projections of a given grain set can be input to a tomographic reconstruction algorithm, one has to eliminate the dependence on the scaling function m (different families 304 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I J. Appl. Cryst. (2008). 41, 302 309

of hkl reflections give rise to different diffracted intensities). If the structure and Lorentz factors are known (i.e. after indexing and orientation determination), one may normalize the projections images by dividing them by these multiplicative factors. Otherwise one may eliminate the dependency on m by dividing each of the projection images by its own integral. 3 Having performed a full 180 scan and given the fact that several low-index (hkl) families give rise to sufficiently strong diffraction contrasts, one typically obtains a few tens of parallel beam projection images for each of the grains. The parallel beam projection geometry allows decomposition of the three-dimensional reconstruction problem into a series of independent two-dimensional reconstruction tasks (slice-byslice approach), which can be solved using a standard ART algorithm (Gordon et al., 1970). By stacking the reconstructed two-dimensional slices (Fig. 2b) the corresponding threedimensional grain volume can be assembled. The outlined procedure is repeated for each of the grains in the sample volume. 2.6. Orientation determination So far we have not used any crystallographic information and for none of the observed extinction spots do we know which (hkl) family the spot is associated with. This lack of information complicates the orientation determination compared with the conventional far-field acquisition geometry where usually four parameters are known for each measured diffraction spot (three scattering vector components and intensity). For the latter type of data only the set of scattering vectors is needed to determine the crystallographic orientation. This can be achieved by using a Rodrigues space approach (Frank, 1988) as presented in the second part of the paper or, alternatively, by using a forward approach, i.e. scanning through the full orientation space in small steps. For each orientation the deviation between the expected and measured scattering vectors is calculated and the correct orientation is characterized by having the smallest deviation. Since in the current situation only one of the three parameters of the scattering vector is known, namely the rotation angle!, one has to include knowledge about the intensity of the diffraction spots in order to determine orientations. In a first step the integrated extinction spot intensities are corrected for X-ray absorption in the sample matrix. Next, the above-mentioned forward simulation approach is applied and potential orientations are characterized by having a good match between the expected and measured! rotation angles. It turns out that several solutions may exist, especially for highly symmetrical space groups. To select the right solution one may use the scattering vectors assigned to the extinction spots and convert the integrated intensities into crystal volumes by normalization with the corresponding structure 3 This makes use of the fact that the area integral (two-dimensional) of a parallel projection image is equal to the volume integral (three-dimensional) of the underlying object function and hence is independent of the projection direction. and Lorentz factors. The correct solution is then characterized by having the smallest internal spread in the crystal volumes. Note that the three-dimensional grain shape reconstruction procedure is based on spatial filtering criteria only and can therefore be performed without analysis of the grain orientations. In this respect the orientation determination can be considered as an optional step of the analysis procedure. 3. Results The experiments presented in this paper have been performed at the high-resolution imaging beamline ID19 at the European Synchrotron Radiation Facility (ESRF). The polychromatic synchrotron beam was monochromated to 20 kev using an Si 111 double-crystal monochromator, and two-dimensional projection images were recorded on a high-resolution detector system based on a transparent luminescent screen (Martin & Koch, 2006), light optics and a CCD camera (Labiche et al., 2007). The sample-to-detector distance was 20 mm and an effective pixel size of 2.8 mm was chosen for this experiment. The exposure time for each projection image was 2 s and a total of 9000 projections were recorded during a continuous motion! scan over 180. The resulting angular integration range of 0.02 per projection allowed maximizing the available extinction contrast. The sample, a small cylinder of 550 mm diameter and 2 mm in length, was prepared from aluminium alloy Al 1050 (99%Al) by means of electrical discharge machining. In order to produce a coarse-grained microstructure with a low degree of mosaic spread, the material was first cold rolled to 90% reduction, machined to the cylindrical shape and then given a two-step heat treatment. After a first 6 h recrystallization annealing at 543 K the sample was kept for 4 h at 903 K and then air cooled. These processing steps resulted in a grain Figure 3 (a) Integrated, monochromatic beam projection image of the cylindrical AA1050 sample, with one grain fulfilling the Bragg condition. The intensity diffracted out of the direct beam gives rise to locally enhanced X-ray attenuation in the transmitted beam, visible as extinction spots. (b) Corresponding absorption background, calculated by median filtering of adjacent projection images. (c) Extinction spot, corresponding to a projection of the grain volume and obtained by logarithmic subtraction of the images to the left. J. Appl. Cryst. (2008). 41, 302 309 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I 305

population with average size of the order of 200 mm and orientation spreads below 0.1. However, part of the sample exhibited a second family of smaller grains with considerably higher orientation spreads. The results presented in the following paragraphs relate to the first family of grains. Fig. 3(a) shows one of the monochromatic beam projection image for which one of the large grains happens to fulfil the Bragg condition. The projection was integrated over an! range of 0.02, which in this case is close to the total orientation spread of the grain. The integration range has been chosen such that in about half of the cases the observed extinction spots extended over more than a single image. Fig. 3(b) shows the absorption background obtained by the previously mentioned median filter applied to a stack of adjacent projection images. Logarithmic subtraction of this absorption background and summation of the partial diffraction images finally yields the grain projection images required for tomographic reconstruction (Fig. 3c). In the current example about 300 such grain projections were automatically extracted from the raw projection images (see movie M1, supporting material 4 ). As explained in x2.4, the grain projections were then sorted into groups belonging to the same grain. Movie M2 shows such a subset of projections, which were filtered out by application of the two spatial selection criteria. After normalization of the projection images belonging to a given set, the three-dimensional grain shape of the corresponding grain was reconstructed in a slice-by-slice approach using a two-dimensional ART reconstruction algorithm. The reconstruction in Fig. 2(b) was obtained using the 27 projections available for this grain and shows a fairly homogenous distribution of the diffraction contribution inside the grain. Fig. 2(c) shows an overlay of the ART reconstruction (Fig. 2b) with the corresponding (filtered backprojection) absorption contrast reconstruction of the corresponding sample cross section. Since in the current example only a few hundred out of the 9000 projection images show strong extinction contrasts, one can reconstruct this absorption image directly from the raw projection images (Fig. 3a) without introducing notable artefacts arising from the diffraction contribution. Alternatively, one could use the median filtered projection images (Fig. 3b), resulting in a slight loss of spatial resolution. As expected, the maps of the attenuation coefficient do not reveal the grain microstructure of the material. One can, however, detect the presence of small iron-rich intermetallic inclusions, commonly encountered in this type of alloy (see arrow in Fig. 2c) In order to enable simultaneous visualization of the grain and the absorption microstructure of the entire sample, the individual grain sub-volumes have to be segmented, labelled (colour coded) and assembled into a single three-dimensional data set (Fig. 4b). The result of an image overlay, produced from a slice through the three-dimensional grain volume and 4 Supplementary data for this paper are available from the IUCr electronic archives (Reference: HX5063). Services for accessing these data are described at the back of the journal. the corresponding slice through the three-dimensional absorption contrast reconstruction, is shown in Fig. 4(a). From inspection of this compound map it can be concluded that the technique can provide space-filling reconstruction of grain structures with an accuracy better than 10 mm. The absorption image on the other hand provides the resolution determined by the detector system and the mechanical precision of the instrument (of the order of 3 mm in the configuration used here). 4. Discussion With the advent of high-energy third-generation synchrotron sources X-ray microtomography has evolved in recent years into a routine three-dimensional characterization technique with 1 mm spatial resolution available as a matter of routine. One of the shortcomings of this nondestructive imaging technique is obviously related to its insensitivity with respect to the crystalline microstructure of the material; apart from some special cases where phase transformations, segregation or wetting processes lead to significant changes in material composition at the level of the grain boundaries, absorption and phase contrast imaging do not in general reveal the grain structure of crystalline materials. On the other hand, recently established three-dimensional grain mapping techniques such as 3DXRD (Poulsen, 2004) or the three-dimensional crystal microscope (Larson et al., 2002; Ice et al., 2005) allow characterization of the three-dimensional grain shape, orientation and in some cases the strain state of individual grains, but do not provide access to the absorption microstructure of the material. Diffraction contrast tomography may be regarded as a combination of conventional absorption contrast tomography and 3DXRD that partly overcomes these limitations. 4.1. Comparison with existing three-dimensional X-ray imaging techniques In this first part of the paper the feasibility of simultaneous grain and absorption microstructure characterization has been demonstrated for the case of a polycrystalline sample, fulfilling certain conditions on sample versus grain size, grain orientation spread and texture. Compared with 3DXRD and the three-dimensional crystal microscope, the sample requirements for the direct beam variant of DCT are more restrictive in the sense that one cannot handle the case of plastically deformed materials. On the other hand, DCT equally applies to multiphase materials. Provided the crystalline phases can be distinguished by differences in absorption and/or phase contrast, one can actually relax the above-mentioned restrictions, since the additional spatial information contained in the absorption image will help in solving some of the segmentation and overlap problems giving rise to the restrictions in the case of monophase materials. The possibility of adjusting the field of view of the highresolution detector system to the sample dimensions implies that the direct beam approach potentially provides higher 306 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I J. Appl. Cryst. (2008). 41, 302 309

spatial resolution than any other grain mapping approach based on recording the diffracted beams. In the latter case one always has to find a compromise between the concurring requirements for ultimate spatial resolution and a large field of view for capturing the diffracted beams. In practice, the experimental setup used in this paper implied a trade-off between time and spatial resolution. 5 The narrow bandwidth combined with the effective pixel size of 2.8 mm resulted in a total scanning time of 6 h at beamline ID19. Collimation of the X-ray beam by means of compound refractive lenses and/or installation of the experiment on a high section or an in-vacuum undulator beamline can be expected to result in considerable enhancement of the time and/or spatial resolution performance of the technique. The large number of available projections per grain and the well defined parallel beam acquisition geometry are additional factors contributing to the high spatial resolution provided by this tomographic imaging technique. Another interesting aspect of the direct beam acquisition geometry is related to the fact that it does not constrain the range of acceptable sample-to-detector distances. This is an essential prerequisite for performing in-situ imaging experiments, generally involving the use of bulky sample environment (furnace, tensile rig etc.). Provided the incoming beam has a sufficient degree of coherence, this flexibility also allows exploitation of Fresnel diffraction (in-line holography) as an additional contrast mechanism, adding extra information to the simultaneously acquired tomographic image of the sample microstructure (Cloetens et al., 1997, 1999). The small angular increment (0.02 ) used during the DCT scanning procedure can be expected to provide an orientation space resolution better than 0.1. This orientation information together with the precise knowledge of the local grain boundary normals will allow reliable identification of special grain boundary configurations. Last but not least it shall be noted that, in situations where a full rotation of the sample is difficult because of geometrical constraints, a potential variant to the technique could consist in performing energy scans for a small set of accessible sample rotations. By scanning a large enough energy range one can ensure that for each sample orientation at least one reflection will be acquired for each of the grains. 4.2. Limitations One of the main limitations of the current approach is related to the stringent requirements concerning the acceptable grain orientation spread. As can be seen from the reconstruction of the full sample volume (Fig. 4b), the central part of the thin cylindrical sample shows unfilled gaps where no grain could be identified with the current approach. The independent measurement of the grain size and orientation spread by means of 3DXRD (far-field acquisition geometry) 5 Note that conventional absorption microtomography scans can be routinely performed with higher spatial resolution by employing large bandwidth multilayer monochromators, providing two orders of magnitude increase in flux compared with the Si 111 double-crystal monochromator used in this study. reveals the presence of two grain populations with clearly distinct size and orientation spread distributions: a family of large grains (100 500 mm) with orientation spread below 0.05 [region labelled A in Fig. 3(a)] and a second family of smaller grains (20 100 mm) with considerably higher orientation spreads of order 0.2 1 [labelled B in Fig. 3(a)]. The latter family gives rise to the irregular distribution of locally enhanced intensity, discernible in the regions not occupied by the first family of grains. For these grains, the simultaneously diffracting grain volume for a given! position is small, and the contrast associated with one reflection may spread over up to several tens of consecutive images. The combined effects of the reduced contrast and the increased probability of spot overlap lead to the breakdown of the direct beam approach in these cases. In order to avoid such a situation, the total number of grains per sample cross section has to be selected (by appropriate sample dimensioning) as a function of macroscopic sample Figure 4 (a) Two-dimensional sample cross section showing overlay of the absorption contrast microstructure with the segmented and colour-coded grain microstructure, assembled from the individual grain reconstructions. (b) Rendition of the segmented and assembled three-dimensional grain volume data set. J. Appl. Cryst. (2008). 41, 302 309 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I 307

texture and the orientation spread of the individual grains; the stronger the texture and orientation spread, the higher the probability of spot overlap. Given a random sample texture and orientation spreads of the order of 0.1, 6 one can expect the method to work with samples containing up to a few tens of grains per cross section. Note that no size restriction applies to the sample dimension parallel to the rotation axis direction. Concerning the assumptions made on the image formation process in x2.1, one may argue that the quality of the reconstructed grain maps justifies the approximation of kinematical scattering made there. However, closer inspection of the individual projection images (Fig. 3c) reveals a spotty grain substructure with local intensity enhancements, which cannot be explained in the framework of kinematic scattering theory. It has been shown in an accompanying study on the same grain (Ludwig et al., 2007) that these features can be attributed to the direct image (Tanner, 1976) contrasts known from the dynamical theory of X-ray diffraction. Long-range strain fields, associated with micrometre-sized intermetallic inclusions, present in this type of aluminium alloy lead to local enhancement of the diffracted intensity around these inclusions. Owing to the limited number of projections and the varying diffraction conditions (each grain projection is associated with a different reflection) the DCT grain reconstructions (Fig. 2b) show a higher level of background nonuniformity and the direct image contrasts stemming from the intermetallic inclusions can no longer be resolved. 4.3. Potential applications Taking into account the above-mentioned restrictions one may still think of a variety of applications where the simultaneous access to the materials absorption and/or phase contrast microstructure and the three-dimensional grain microstructure can be expected to provide unprecedented insight. One may, for instance, consider the characterization of undeformed, polycrystalline samples before exposing the material to chemical and/or mechanical degradation processes such as stress corrosion cracking or fatigue crack propagation, to mention just two of them. The grain mapping, as well as the characterization of the subsequent crack propagation, can be performed on the same instrument, taking advantage of the insitu imaging capability of state of the art microtomographic imaging instruments. Experimental data of this type are currently scarce and would provide invaluable input for various types of models and numerical simulations. By increasing the monochromatic flux, one may also think of insitu observation of grain-coarsening processes such as recrystallization and grain growth. 5. Conclusions The feasibility of a novel, nondestructive synchrotron imaging technique capable of reconstructing the three-dimensional 6 For the case of metallic polycrystals, such low levels of orientation spread are commonly encountered in recrystallization or solidification microstructures. absorption microstructure, grain shapes and orientations in undeformed polycrystalline samples has been demonstrated. Given the close similarities to conventional absorption or phase contrast tomography one can take advantage of the mechanically simple, high-resolution imaging setups available nowadays at any modern synchrotron source. The applicability of the technique presented in this first part of the paper is limited to polycrystalline samples containing a limited number of grains per cross section and exhibiting typical orientation spreads within individual grains of the order of one tenth of a degree. Compared with alternative 3DXRD grain mapping approaches, diffraction contrast tomography has the advantage of providing simultaneously access to a sample s threedimensional grain and absorption (phase contrast) microstructure. Since the detector field of view and hence the spatial resolution can be adapted to the sample size, the ultimate resolution is superior to grain mapping techniques based on the acquisition of diffracted beams. Owing to the fact that the detector can be placed far behind the sample, diffraction contrast tomography is the only three-dimensional grain mapping technique enabling the use of complicated spacious sample environments. APPENDIX A Summary of the data analysis procedure A1. Preprocessing (1) Flat-field correction of raw projection images containing absorption and diffraction contrasts (Fig. 3a): I tot! ¼ image! dark flat dark ; I! tot ¼ exp R abs ðrþþ diffr ðrþ dx : I tot! defines the local transmission of the X-ray beam (ranging between 0 and 1). (2) Reconstruction of the three-dimensional absorption microstructure from projection images ln(i tot ) by means of a conventional filtered backprojection tomographic reconstruction algorithm. (3) Calculation of the absorption background (Fig. 3b) from a three-dimensional projection image stack (pixel-by-pixel one-dimensional median filter, operating along the new stack dimension): I abs! Median! I tot! n ; I tot! nþ1 ;...; I tot;!...; I tot!þn ; I abs! ffi exp R abs ðrþ dx ; P! diffr ¼ lnði tot=i! absþ! ¼ R ð4þ diffr ðrþ dx: I abs! describes the local transmission of the X-ray beam and P! diffr gives a mathematical projection of the (diffraction contribution to the) attenuation coefficient. (4) Removal of absorption background and calculation of diffraction contrast projections (Fig. 3c). ð3þ 308 Wolfgang Ludwig et al. X-ray diffraction contrast tomography I J. Appl. Cryst. (2008). 41, 302 309

A2. Spot segmentation (5) Segment individual diffraction contrasts in projection images P! diffr (morphological image reconstruction algorithm). (6) Sum contributions belonging to the same diffraction spot from adjacent images in!. (7) Calculate region properties of summed diffraction spots (centre of mass, intensity, area, bounding box,! range) and save summed spot image (Fig. 3c). A3. Spot sorting (8) Select an arbitrary spot. (9) Select subset S1 of spots with identical vertical bounds (vertical position of grains does not change during rotation around!). (10) Backproject central line of this subset of spots onto sample plane (Fig. 2a). (11) Select subset S2 of spots (out of set S1) which backproject onto a common position in the (xy) sample plane: subset S2 is saved as a grain data set. (12) Repeat until all spots have been attributed to grains (or identified as outliners, overlaps etc.). A4. Grain reconstruction (applied to each grain data set independently) (13) Determine grain orientation from the known set of! rotation angles and integrated spot intensities (optional). (14) Normalize the diffraction spot images to a common integrated intensity. (15) Reconstruct the three-dimensional grain shape by means of algebraic reconstruction techniques (ARTs). (16) Segment and label the reconstructed grain volume data set (assign a unique gray value to all voxels belonging to a given grain). A5. Visualization (17) Assign a colour representing crystallographic orientation to each reconstructed grain volume (optional). (18) Assemble the three-dimensional sample volume by merging the individual grain volumes into a single volume (Fig. 4b). (19) Produce transparent image overlay of the threedimensional absorption image with the corresponding threedimensional grain map in order to visualize absorption and grain information simultaneously (Fig. 4a). We thank P. Cloetens, G. Berruyer and A. Homs for their assistance in setting up the continuous motion scanning procedure. R. Godiksen is acknowledged for providing the two-dimensional ART reconstruction algorithm used in this paper. WL thanks J. Y. Buffiere, J. Baruchel and D. J. Jensen for fruitful discussions and their support, without which the work presented in this paper would not have been possible. SS, EML and HFP acknowledge support by the Danish National Research Foundation, by the EU program TotalCryst and by the Danish National Science Research Council (via Dansync). References Chapman, D., Thomlinson, W., Johnston, R. E., Washburn, D., Pisano, E., Gmur, N., Zhong, Z., Menk, R., Arfelli, F. & Sayers, D. (1997). Phys. Med. Biol. 42, 2015 2025. Cloetens, P., Ludwig, W., Baruchel, J., van Dyke, D., van Landuyt, J., Guigay, J. P. & Schlenker, M. (1999). Appl. Phys. Lett. 75, 2912 2914. Cloetens, P., Pateyron-Salomé, M., Buffiere, J. Y., Peix, G., Baruchel, J., Peyrin, F. & Schlenker, M. (1997). J. 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