First laboratory Xray diffraction contrast tomography ...

research papers Journal of

Applied Crystallography

First laboratory X-ray diffraction contrast tomography for grain mapping of polycrystals

ISSN 0021-8898

A. King,a* P. Reischig,b,c J. Adrienb and W. Ludwigb,d Received 10 April 2013 Accepted 11 August 2013

# 2013 International Union of Crystallography Printed in Singapore – all rights reserved

a

Synchrotron Soleil, 91192 St Aubin, France, bUniversite´ de Lyon, INSA-Lyon, MATEIS CNRS UMR 5510, 69621 Villeurbanne, France, cDelft University of Technology, 2600 AA Delft, The Netherlands, and dEuropean Synchrotron Radiation Facility, 38043 Grenoble, France. Correspondence e-mail: [email protected]

The first results of three-dimensional grain mapping using a laboratory tomograph equipped with a microfocus W target X-ray tube source, operated at 90 kV and 350 mA, are presented. Adapted algorithms exploit the polychromatic radiation spectrum and the projection magnification arising from the cone-beam geometry. The first map of grain shapes and crystallographic orientations from a titanium sample containing 42 grains is presented and its validity confirmed by a phase contrast reconstruction of the grain boundaries. Perspectives are given for the further development of the technique to accommodate samples with more grains or with greater intragranular orientation spread.

1. Introduction The vast majority of metallic and ceramic materials are polycrystalline, and the interaction between the crystalline grain structure and various physical processes is increasingly the subject of study (Bantounas et al., 2007; Herbig et al., 2011; King et al., 2008; Krupp, 2007; Rolland du Roscoat et al., 2011; Schaef et al., 2011). The nondestructive three-dimensional mapping of crystallographic orientation in polycrystals is thus increasingly important for materials science. In recent years, several such techniques have emerged. These include three-dimensional X-ray diffraction microscopy (3DXRD) (Poulsen, 2004; Schmidt et al., 2008), high-energy X-ray microscopy (Suter et al., 2006) and diffraction contrast tomography (DCT) (Johnson et al., 2008; Ludwig et al., 2009; Reischig et al., 2013). These techniques are characterized by the use of extended, parallel and monochromatic synchrotron X-ray beams. Grain shapes, positions and crystallographic orientations are determined from the shapes, intensity distributions and the positions of diffraction spots recorded on a high-resolution two-dimensional detector placed close to the sample. A related family of techniques uses focused polychromatic synchrotron radiation (Larson et al., 2002). In these methods, Laue diffraction patterns recorded on a low-resolution detector give grain orientations, and spatial resolution is obtained by scanning the sample and an analyser wire with respect to the focused beam (‘differential aperture microscopy’). Initially, these techniques used X-ray energies on the order of 15 keV in reflection geometry to investigate small sample volumes. Recently, the concept has been adapted to higher X-ray energies, allowing the investigation of millimetre-sized samples in transmission geometry (Hofmann et al., 2012). All of these X-ray techniques have in common the

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use of synchrotron radiation, which imposes strong restrictions on access to instruments, meaning that only limited experimental time is available to scientists. Electron backscatter diffraction in the scanning electron microscope allows two-dimensional crystallographic orientation mapping on surfaces, but can only be extended to three dimensions using destructive serial sectioning techniques (Matteson et al., 2002; Rowenhorst et al., 2006; Venables & Harland, 1973). Recent developments have applied 3DXRD to electron diffraction in the transmission electron microscope, but this is applicable only to samples of less than one micrometre in thickness (Lui et al., 2011). Laboratory X-ray microtomography has become an essential tool in many materials science institutes (e.g. Limodin et al., 2009; McDonald et al., 2009). A microfocus X-ray tube provides a divergent polychromatic X-ray beam, contrasting with the parallel beam typically used in synchrotron microtomography. The sample is placed close to the source and a divergent cone-beam geometry is used to project an enlarged radiograph onto a detector placed some distance away. The spatial resolution in the image is determined by the source size, detector resolution, and the ratio between the source– sample and source–detector distances. The spatial resolution and field of view are easily adjusted by changing the sample position. Features generating absorption contrast can be reconstructed in three dimensions using adapted backprojection algorithms (Feldkamp et al., 1984). However, only in some specific cases of multiphase materials can absorption contrast reveal grain shapes, owing to differences in chemical composition or density, and even then the grain orientations are not accessible. There is thus a clear motivation for a technique that can exploit diffraction contrast imaging using standard laboratory J. Appl. Cryst. (2013). 46, 1734–1740

research papers X-ray sources and equipment, in order that three-dimensional polycrystalline microstructures can be studied nondestructively without a synchrotron source. Enabling three-dimensional mapping of grain shapes and orientations with widely available laboratory equipment will allow more systematic use of these techniques, important for developing true scientific understanding of how polycrystals respond to loading, corrosion or thermomechanical processing. In this article, we describe laboratory DCT (Lab-DCT), a new technique that is performed using a standard laboratory tomograph, and present the first experimental results obtained.

2. Technique Lab-DCT is performed using standard laboratory X-ray tomography instrumentation. A single detector is used to collect both absorption contrast radiographs in the direct beam and diffraction spot images. The addition of slits to define the incident beam is the only hardware modification necessary to a standard instrument. The projection magnification is adjusted so that the image of the sample fills only the central part of the detector. Fig. 1 shows a schematic diagram of the experimental setup. As the sample is rotated through 360 around a fixed axis, grains diffract the beam on to the outer part of the detector. The recorded diffraction spot images are used to calculate the orientations and positions of grains, and to reconstruct their three-dimensional shapes. The direct-beam radiographs are used to produce an absorption contrast reconstruction of the sample as in regular tomography. Figs. 2(a) and 2(b) show examples of the diffraction spots from a sample and the corresponding direct-beam image.

rotated, diffraction spots precess across the detector in curved paths. Fig. 3(a) shows the sum of diffraction spot images recorded during a 15 sample rotation, showing the diffraction spot paths. The Lab-DCT analysis method described here is based on identifying these paths of diffraction spots. A diffraction spot arising from the (hkl) planes of a grain will appear on one side of the detector and move towards the sample, to lower diffraction angles and correspondingly higher beam energies. As the maximum energy in the beam spectrum is reached, the spot gradually disappears, reappearing on the other side of the sample after further rotation [scattering from the ðhklÞ planes]. The spot then moves away from the sample to higher diffraction angles and lower energies, until it disappears from the detector. The only exceptions are those scattering vectors that are close to parallel with the rotation axis, which may never reach high enough diffraction angles to exceed the edges of the detector. The spot intensity changes as a function of angle, according to the incident-beam spectrum, scattering efficiency, self-absorption by the sample and

2.1. Data acquisition

A polychromatic X-ray spectrum from a tube source means that diffraction events are not observed at discrete angles (2). Instead, different beam energies are diffracted at varying diffraction angles as the lattice planes rotate. As the sample is Figure 2 (a) Diffraction spot contrasts after processing to remove other contributions. (b) The simultaneously acquired direct-beam image. The sample and the outline of the beam are marked in both (a) and (b). The outline of the grain, which gives rise to the circled diffraction spot, is marked in (b).

Figure 3 Figure 1 A schematic view of the Lab-DCT setup. The technique can be performed in using standard laboratory tomography instrumentation with minimal modifications. Dimensions shown refer to measurements for which results are shown. J. Appl. Cryst. (2013). 46, 1734–1740

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(a) Composite of images for 15 sample rotation, showing lines of spots. No filtering of the direct beam is used. (b) An enlarged region of (a), with guide lines marking the spot path and characteristic beam energies, with (c) an Hf filter and (d) an Er filter applied. Note the increased background noise due to an incorrect dark current correction. (e) Narrow bandwidth image produced by dividing image (b) by (c).

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research papers detector efficiency as a function of energy. The spot behaviour is illustrated schematically in Fig. 4. Because the spots move as the sample is rotated, images are acquired at discrete angular steps in order that the spot images are not blurred. This is in contrast with the monochromatic synchrotron technique, in which continuous rotation is necessary to ensure that all parts of a diffracting grain are observed. It is important that the beam spectrum contains a sufficiently wide range of energies that diffraction spots arising from several families of crystal lattice planes can be observed within the range of solid angle covered by the detector. To best exploit the projection magnification in the conical beam, the diffraction angles (Bragg angles, 2) should be similar to the cone opening angle covered by the detector, favouring the use of relatively high beam energies (Fig. 1). Because of the projection magnification, the physical pixel size of the detector does not limit the spatial resolution as it does in the parallelbeam case. Using a low-spatial-resolution detector greatly improves the detection efficiency, particularly for high X-ray energies, and to some extent compensates for the muchreduced flux when compared with a synchrotron source. The technique is therefore particularly well suited to highly attenuating materials requiring high beam energies for sufficient transmission. 2.2. Data processing

The principles of the data processing are similar to those used for standard synchrotron DCT, and the initial treatment here of the images was performed using tools from the standard DCT processing software (Johnson et al., 2008; Ludwig et al., 2009; Reischig et al., 2013). A constant dark current is offset and a moving median background correction applied in order to remove slowly changing features from the images, to leave only diffraction spot contrasts (Fig. 2a). The diffraction spots are segmented from the images and metadata describing

Figure 4 Determination of diffraction geometry from observed data. The spotintensity profile is used to determine the centre of the line. The construction used reveals the scattering vector and a path through the sample on which the grain must be found.

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the spots stored in a database. The direct-beam image record is simultaneously treated as a conventional tomographic data acquisition. The data analysis is based on using an automatic search algorithm to identify the paths of spots as they precess across the detector as the sample rotates. The identification of such lines containing several spots appears to be robust, provided that the majority of spots are not affected by overlaps. The geometry of the diffraction events is determined by analysing the paths of the spots, as shown in Fig. 4. Two versions of this analysis have been implemented using MATLAB (version 7.14.0; MathWorks Inc., Natick, MA, USA): a computationally simple version, in which the spot paths are approximated to straight lines, and a more advanced analysis in which the true curved paths are fitted. (a) Straight lines. For most lattice planes (as an indication, those giving rise to diffracted beams with angles 2 < 15 , 20 < < 160 ), the lines of spots can be treated as straight lines to a good approximation. Each line of spots, extending from one side of the detector to the other, is fitted with a straight line in three dimensions (x and y position in the detector plane as a function of the sample rotation). The centre point of each line is then identified by considering the diffraction spot intensities. The centre point can be considered as the sample rotation angle and detector position at which diffraction would occur with an angle of 0 . A line drawn from this point on the detector to the X-ray source must therefore pass through the diffracting grain, as shown in Fig. 2. The diffracting plane normal must be perpendicular to this line and coplanar with the line of diffraction spots. The error implied by the assumption of straight lines can be estimated by simulating the paths of a diffraction spot across the detector for a given experimental configuration. The difference between the true plane normal and the plane normal calculated from a straight line fit through the simulated points gives a measure of the error in each observation. Grain orientations are found by a best fit to multiple observations of plane normals, so the final error in orientation should be less than the error in a single plane normal. (b) Curved lines. For higher diffraction angles and for scattering vectors that are nearly parallel to the rotation axis, the diffraction spot paths become noticeably curved. The path can be described by considering the direction of the scattering vector and the position of the grain within the sample. At each sample rotation angle, the position of the grain in the incident cone beam defines the incident-beam direction. The directions of the incident beam and the scattering vector determine the direction of the diffracted beam, and the hence the diffraction angle (2) and the wavelength diffracted from the polychromatic beam. The intersection of the diffracted beam with the detector plane defines the spot position. Given a line of diffraction spot positions, the grain position and the scattering vector can be fitted using a suitable least-squares algorithm. In either case, the ðhklÞ family for each reflection is identified using the profile of diffracted intensity as a function of 2 in each line. By considering the beam spectrum (in the data presented here, this exhibited a clear maximum around the


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research papers tungsten K lines at 58.0 and 59.3 keV), the ðhklÞ family of each line was identified. Each line of spots therefore reveals the orientation and the ðhklÞ type of a set of lattice planes, and a path through the sample on which the grain must lie. This information is equivalent to that obtained from a Friedel pair of spots in the monochromatic and parallel-beam synchrotron technique, and so the standard DCT grain-indexing algorithm (Indexter) can be used to determine grain positions and crystallographic orientations (Ludwig et al., 2009; Reischig et al., 2013). 2.2.1. Image formation and reconstruction. The threedimensional shapes of the individual grains are reconstructed from the indexed diffraction spot images. Both the directbeam radiograph and the diffraction spot images are enlarged by the conical projection. The interaction of the polychromatic cone beam with the parallel lattice planes results in diffraction spot images which are mirrored along the scattering direction, and magnified astigmatically, as illustrated schematically in Fig. 5. Similarly, the outline of the grain that gives rise to the diffraction spot circled in Fig. 2(a) is marked on Fig. 2(b). The shapes of the grain and the spot illustrate the mirroring along the scattering direction. The inverse transform, correcting the inversion and astigmatism, is applied before using a conebeam geometry simultaneous iterative reconstruction technique (SIRT) algorithm to reconstruct the three-dimensional grain shapes (Palenstijn et al., 2013). The individual grain reconstructions are assembled to produce a full three-dimensional map. Postprocessing morphological dilation is used to fill any unassigned voxels in the grain map (Johnson et al., 2008). A similar cone-beam (SIRT) algorithm is used to reconstruct the absorption contrast volume of the sample from the direct-beam radiographs.

3. Results 3.1. Sample

The sample used for this first proof of concept measurement was a ‘dog bone’-shaped tensile test sample with a 0.4 mm gauge diameter. The material used was the metastable -titanium alloy Ti21S (Dey et al., 2007). Previous DCT investigations using the same material have shown that it has low mosaicity, with an intragranular orientation spread of 0.05–0.15 per grain. A heat treatment of 24 h at 1063 K was used to precipitate a layer of -titanium on the grain boundaries. The and phases have sufficiently different densities that they can be distinguished using phase contrast tomography (PCT), allowing the grain shapes (although not their crystallographic orientations) to be reconstructed using established techniques (Cloetens et al., 1997). In this way, the accuracy of the Lab-DCT grain reconstruction can be validated, using the method described by Ludwig et al. (2009). This approach has the advantage of being nondestructive and of providing the complete three-dimensional grain shapes for verification. 3.2. Method

Measurements were performed using a Phoenix X-ray v|tome|x laboratory tomograph at INSA de Lyon, using a J. Appl. Cryst. (2013). 46, 1734–1740

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microfocus X-ray generator with a tungsten transmission anode and an accelerating voltage of 90 kV and a current of 350 mA. The effective source size under these conditions has been measured as 3–4 mm. The detector is a PaxScan 2520 V from Varian Medical Systems, which uses a CsI scintillator and amorphous silicon architecture. It has a physical pixel size of 0.127 mm and a field of view of 1920 1536 pixels and was located 577 mm from the source. The sample was placed 23 mm from the source, giving an effective pixel size of 5 mm. Images were binned 2 2 to increase sensitivity, doubling the effective pixel size to 10 mm. A set of manual slits was placed between the source and sample to define the size of the direct beam. The conical beam geometry implies that the slit opening was about 350 mm in order to obtain a beam size of about 700 mm at the sample position. These dimensions used for this particular sample are indicated in Fig. 1. In all, 720 radiographs were recorded covering 360 in steps of 0.5 . The exposure time for each image was 10 s, giving a total acquisition time of around 2 h. Images were processed as described in x2, and 17 919 diffraction spots were segmented from the images. Lines of spots were identified automatically and fitted using straight lines. Taking the example of a plane normal inclined at 45 to the rotation axis, the error involved in assuming straight lines for this experimental configuration was estimated to be less than 0.5 . A total of 538 lines of diffraction spots were used to produce geometrical information for the grain-indexing step. Spot images were treated as described in x2 and used as projections to reconstruct the grain shapes. 3.3. Reconstruction

Fig. 6 shows the grain map reconstructed using Lab-DCT. Grains have been coloured according to their orientation, expressed as Rodrigues’ vectors and mapped into a red– green–blue colour space. The map consists of 42 grains with an average diameter of 170 mm. The same volume of the sample

Figure 5 Diffraction spot image formation in polychromatic, cone-beam geometry creates projections that are astigmatically magnified and inverted along the scattering direction. The letter ‘G’ is chosen as a suitably asymmetrical shape for illustration.


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research papers was imaged using synchrotron PCT measurements at beamlines ID11 and ID19 of the ESRF. These volumes were rescaled and aligned to the Lab-DCT grain map. Sections through the grain map and the PCT reconstruction are shown. It can be seen that all large grains are reconstructed and that the reconstructed grain shapes are realistic. Marked in the images is a grain boundary that was not decorated by the -titanium phase. Inspection of the Lab-DCT grain map shows that this is a low-angle grain boundary (6.8 intergranular misorientation; note the similar colours of the two grains). The main sources of error in the reconstruction appear to arise from small partial grains at the surface of the sample that have been subdivided when the sample was cut from a larger piece of material.

4. Discussion The method presented here has been demonstrated using a sample with low mosaicity and relatively large grains. It is useful to discuss several factors that relate to the application of the technique to more general materials for scientific applications and to the use of polychromatic laboratory sources for this type of measurement.

reasonable to split the acquisition over several scans. Under these conditions, the scattering power of a grain is proportional to the grain volume. Therefore, smaller-grained samples will require longer exposure times or more intense sources. If the sample size can be reduced as well as the grain size, the sample can be positioned closer to the X-ray source, providing greater flux density and stronger magnification. In this case, exposure time would be expected to scale as the inverse of the grain diameter. 4.2. New laboratory sources

Currently, the acquisition time for a data set is limited by the integration time required to form usable diffraction spot images given the available X-ray flux. In future, developments of liquid jet anode laboratory sources may provide this increase (Hemberg et al., 2003). Current designs offer an order of magnitude increase in flux compared with equivalent solid anode sources, but have not yet been optimized for the very small source sizes required for cone-beam projection microscopy [Excillum JXS-D1 Metal Jet Source Data Sheet 2011-1010 (downloaded 23 July 2013); http://www.excillum.com/ images/pdf/JXS-D1_2011-10-10.pdf]. 4.3. Effects of polychromaticity

4.1. Application

To be a useful scientific tool, Lab-DCT should be capable of mapping of the order of one thousand grains in a sample. This should be easily achieved, as the 42-grain sample mapped here was far from the condition where diffraction spot overlap would become limiting. Thus, several hundred grains in a scan should be possible, and one thousand grains can be mapped by scanning several adjacent volumes. It should be noted that relatively few images are required compared with the synchrotron technique (5–10 times fewer) and hence it is

The polychromaticity of the beam limits the described approach to samples exhibiting small lattice deformations, as these lead to an irregular distortion of the diffraction spots. Nevertheless, if the lattice deformations are small enough, the advantages of using a polychromatic instead of a monochromatic beam are twofold. Firstly, spots from a given lattice plane and grain precess across the detector, diffracting different X-ray energies at each rotational angle of the sample. This increases the number of available projection images per grain, which improves the orientation and shape reconstruc-

Figure 6 The reconstructed grain map, compared with the PCT reconstruction showing the grain boundaries that was used for validation purposes. In the DCT reconstruction colour represents crystallographic orientation.

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research papers tion accuracy. Secondly, diffraction occurs simultaneously from the entire grain volume, even though the incident angle varies across the grain in the conical beam. In the case of a monochromatic conical beam, only partial grain volumes would be visible at any given rotation angle. As mentioned previously, to obtain sharp diffraction spot images stepwise rotation is used as opposed to continuous rotation with a monochromatic beam. The current grain-reconstruction algorithms are based on backprojection and assume no mosaic spread within grains. In the example given, a grain has an angular size seen from the source of about 0.5 (170 mm grains, 23 mm from the source). For the diffraction spots to be reasonable projections, the intragranular orientation spread should be much less than this, i.e. 0.1 per grain. This limits the technique to materials with very low mosaicity. This argument does imply that the problems of mosaicity can be mitigated by moving the sample closer to the source, increasing the angular extent per grain, until the overlap between spot images on the detector becomes a limitation. To achieve better reconstructions, it will be necessary to incorporate subgrain misorientation into the reconstruction algorithms as discussed by Schmidt et al. (2008) and Suter et al. (2006). Grain volumes and centres of mass can, nevertheless, still be obtained from less perfect samples. The data processing route presented is also applicable to data acquired using other polychromatic radiation sources, including cold or thermal neutrons and polychromatic or broadband monochromatic synchrotron beams.

of grains, it might be more effective to reduce the vertical beam height and hence the volume of material sampled. However, a sample with a large number of grains in a cross section might require this approach.

5. Conclusions Lab-DCT is a new technique enabling the reconstruction of the grain structure of polycrystals using standard laboratory tomography instrumentation with minimal modifications. The technique and the data processing route are described. The algorithms described are also applicable to other experiments using polychromatic radiation, including neutron and synchrotron X-ray sources, and future laboratory X-ray sources. The first grain map derived from experimental data is presented. The validity of the reconstructed grain shapes has been confirmed by imaging the same region of the sample using synchrotron PCT to image a second crystallographic phase precipitated on the grain boundaries. It is hoped that by allowing grain mapping experiments to be performed using laboratory equipment, the new technique will enable researchers to make more compete studies of the influence of crystallographic microstructure on material behaviour and properties. James Marrow, Oxford University, is acknowledged for suggesting a laboratory-based DCT technique in 2007, prior to the developments shown here. The majority of this work was carried out while AK was employed at the ESRF, Grenoble.

4.4. Filtering to reduce bandwidth

An alternative route for the control of excessive mosaicity, or to allow greater numbers of grains to be treated in a single scan by reducing spot overlap, is to use filtering to create a pseudo-monochromatic data acquisition at the energy of the tungsten K lines. This technique was described by Webster et al. (1986) for coherent and Compton scattering experiments. Filters are chosen with absorption edges just above and below the K line energy. Ideally, erbium and ytterbium would be used. For trial purposes erbium (Er K edge 57.5 keV) and hafnium (Hf K edge 65.4 keV), both 0.25 mm thick, were used. The filters have essentially the same attenuation effect, except in the immediate vicinity of the absorption edge. Therefore, when an image acquired using one filter is divided by an image acquired using the other filter, all contributions are removed with the exception of those arising from energies between the two absorption edges, a window of about 4 keV if Yb and Er are used (i.e. E/E = 7%). Thus most spot overlaps can be removed. Fig. 3(b)–3(e) show trial data illustrating this process. In the images shown, an error in the calibration of the dark current has produced increased background noise in one of the acquisitions, reducing the final image quality. Note this approach has the drawback that two consecutive scans must be performed and that the filters reduce the incident flux, and therefore must be expected to cost a factor of four in acquisition time. In the case of spot overlap due to a large number J. Appl. Cryst. (2013). 46, 1734–1740

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