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Journal of
Applied
Crystallography
ISSN 0021-8898
Received 28 June 2007
Accepted 16 January 2008
X-ray diffraction contrast tomography: a novel
technique for three-dimensional grain mapping
of polycrystals. I. Direct beam case
Wolfgang Ludwig,a,b* Søeren Schmidt,c Erik Mejdal Lauridsenc and Henning Friis
Poulsenc
a
MATEIS, INSA-Lyon, Villeurbanne, France, bEuropean Synchrotron Radiation Facility, France, and
Centre for Fundamental Research: Metal Structures in Four Dimensions, Risø National Laboratory,
Roskilde, Denmark. Correspondence e-mail: [email protected]
c
# 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;
302
doi:10.1107/S0021889808001684
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
J. Appl. Cryst. (2008). 41, 302–309
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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.
J. Appl. Cryst. (2008). 41, 302–309
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 diffraction1 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),
R
Iðu; vÞ ¼ I0 ðu; vÞ exp 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, ’ 106) 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; ; ’; Fhkl ; LÞ, depends on the wavelength , the local
effective misorientation ’(r) with respect to the maximum of a
given hkl reflection, the structure factor Fhkl 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.
Wolfgang Ludwig et al.
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proportional to the structure and Lorentz factors for rotation
angles corresponding to the different reflections of the grain:
2
m / Fhkl L:
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
R
contribution m diffr ðrÞ dx along the beam direction.
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:
R
Iðu; vÞ
ln
¼ eff ðrÞ dx
I0 ðu; vÞ
R
R
ð2Þ
¼ abs ðrÞ dx þ m diffr ðrÞ dx:
2.3. Spot summation and segmentation
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
R
background 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.
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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
J. Appl. Cryst. (2008). 41, 302–309
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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.
J. Appl. Cryst. (2008). 41, 302–309
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.
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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 material4). 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.
306
Wolfgang Ludwig et al.
X-ray diffraction contrast tomography I
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
J. Appl. Cryst. (2008). 41, 302–309
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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.
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
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)
Figure 4
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.
J. Appl. Cryst. (2008). 41, 302–309
(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.
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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.
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
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.
308
Wolfgang Ludwig et al.
X-ray diffraction contrast tomography I
(1) Flat-field correction of raw projection images containing
absorption and diffraction contrasts (Fig. 3a):
!
Itot
¼
image! dark
;
flat dark
R
!
Itot
¼ exp abs ðrÞ þ diffr ðrÞ dx :
ð3Þ
!
defines the local transmission of the X-ray beam (ranging
Itot
between 0 and 1).
(2) Reconstruction of the three-dimensional absorption
microstructure from projection images ln(Itot) 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):
!n !nþ1
!
!
!þn
’ Median! Itot
; Itot ; . . . ; Itot
; . . . ; Itot
Iabs
;
R
!
Iabs ffi exp abs ðrÞ dx ;
R
!
!
P!diffr ¼ lnðItot
=Iabs
Þ ¼ diffr ðrÞ dx:
ð4Þ
!
Iabs
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).
J. Appl. Cryst. (2008). 41, 302–309
research papers
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).
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).
A3. Spot sorting
References
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(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.).
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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
J. Appl. Cryst. (2008). 41, 302–309
Wolfgang Ludwig et al.
X-ray diffraction contrast tomography I
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