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University of Ljubljana

Faculty of Mathematics and Physics

Department of physics

Seminar – 4th year

X-ray diffraction contrast tomography

Author: Martin Knapič

Adviser: Janez Dolinšek

Ljubljana, 28. May 2011

Abstract

X-ray diffraction contrast tomography is a technique which allows simultaneous determination of crystal grain shapes, grain orientations and absorption microstructure of polycrystals. The technique is based on attenuation of X-rays in polycrystals and shares a common experimental setup with conventional X-ray absorption contrast tomography. Unlike absorption contrast tomography, diffraction contrast tomography also uses information about diffraction contribution to the attenuation coefficient. This allows it to reveal grain microstructure, which can not be determined from maps of attenuation coefficient.

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Table of contents 1. Introduction.........................................................................................................................................................2 2. Basics..................................................................................................................................................................2 2.1 Polycrystal...........................................................................................................................................2 2.2 X-ray absorption and attenuation........................................................................................................3 2.3 Theory of diffraction...........................................................................................................................3 2.3.1 Bragg law...........................................................................................................................3 2.3.2 Structure factor and diffracted intensity............................................................................4 2.3.3 Ewald sphere......................................................................................................................5 2.4 Synchrotron radiation..........................................................................................................................6 3. Grain mapping procedure....................................................................................................................................6 3.1 Basic assumptions................................................................................................................................7 3.2 Removal of absorption background.....................................................................................................7

3.2.1 Filtered backprojection reconstruction algorithm...............................................................8 3.3 Summation of spots belonging to the same reflection.........................................................................9 3.4 Spot sorting..........................................................................................................................................9 3.5 Reconstruction......................................................................................................................................9 3.5.1 Algebraic reconstruction technique...................................................................................10 3.6 Orientation determination...................................................................................................................11 3.7 Results................................................................................................................................................12 3.8 Limitations..........................................................................................................................................13

3.9 Comparison with other three-dimensional X-ray imaging methods...................................................13 4. Conclusion...........................................................................................................................................................14 5. References...........................................................................................................................................................14

1. Introduction: X-rays were discovered in 1895 by Wilhelm Conrad Röntgen (1845-1923). Their property of penetrating opaque bodies was immediately found usefull. Today X-ray absorption is used in a wide range of applications, such as medical imaging, detection of failures in metals, analysis of paintings... Another important property of X-rays is diffration on crystals, which can be used to determine crystal structure. X-ray diffraction contrast tomography (DCT)[1] uses both properties of X-rays: it measures attenuation in material, which is a consequence of both absorption and diffraction. Measured transmitted intensities during 180° sample rotation are used to calculate attenuation coefficient and absorption coefficient. The angular dependence of the diffraction contribution to attenuation coefficent can be used to determine the shapes of crystal grains in the undeformed polycrystalline sample and their orientation.

2. Basics

2.1 Polycrystal Polycrystal is a material that consists of crystal grains. Almost all common metals and many ceramics are polycrystalline. Each grain inside the polycrystal is a crystal with translational symmetry. The crystal grains can be identical in chemical composition and lattice structure, but differ in lattice orientation. Orientation is defined as the rotation needed to rotate basis vectors of monocrystal grain’s Bravais lattice into the basis vectors of laboratory coordinate system. The rotation can be represented in several different ways, usually with rotation matrix, Euler angles or Rodrigues vectors. Symmetries of the Bravais lattice imply that several different rotations can represent the same orientation. A single crystal grain is usually not a perfect crystal, since it contains defects such as point defects and dislocations. As a consequence the basis vectors of Bravais lattice in different regions of the same grain will not be exactly parallel, so orientation is not exactly defined. The typical angle between orientations of different parts of the same grain is called orientation spread.

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Texture [2] is a polycrystal parameter that describes “randomness” of grain orientations. A polycrystal with completely random orientation is said to have no texture, while a polycrystal with strong texture will have prefered orientations that are more common. The full 3D representation of crystallographic texture is given by

the orientation distribution function )(gODF which is defined as a volume fraction of grain with orientation

g :

gdgdV

VgODF

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)(1)( = (1)

Different crystal grains are separated by grain boundaries [3], which represent weak points of material due to obvious deviation from perfect single crystal. When material is exposed to mechanical stresses or corrosion, cracks in material usualy occur along grain boundaries. Grain size, orientation spread and texture are the most important polycrystal parameters which determine applicability of diffraction contrast tomography.

2.2 X-ray absorption and attenuation The decrease of X-ray intensity in the material is proportional to intensity and the infinitesimal traveled path ds:

dsrIdI

EFF )(µ−= , (2)

where EFFµ is effective attenuation coefficient [4]. If the X-ray intensity decreases only due to absorption, the effective attenuation coefficient is equal to absorption coefficient. However X-ray intensity can also decrease due to diffraction, in which case the effective attenuation coefficient will be a sum of absorption coefficient and diffraction coefficient [1]:

( )dsrrdsrIdI

DIFFRABSEFF )()()( µµµ +−=−= (3)

Unlike the absorption coefficient, diffraction coefficient is not independent of the X-ray beam direction, since it only appears when Bragg condition is fulfiled. The solution of the above differential equation is:

( )∫−⋅= dsrII EFF )(exp0 µ (4)

2.3 Theory of diffraction

2.3.1 Bragg’s law

Essential requirement for diffraction [5], [6] is constructive interference. The amplitude of electromagnetic radiation scattered from different atoms will be maximized only when the contributions from different atoms all have the same phase (considering π2 shift symmetry). For atoms lying in the same plane, this condition is fulfiled when the incident and diffracted rays have the same inclination ϑ agains the scattering plane. Radiation scattered from other parallel planes will have the same phase only when the difference of paths is equal to an integer multiple of the wavelength, resulting in Bragg’s law [5]:

λϑ ⋅= nd )sin(2 , (5) whered is interplanar distance, λ is the wavelength and n is an integer number. The number of atoms in the crystal grain is assumed to be large, which is true for typical grain sizes in micrometer range. Elastic scattering is assumed, so the wavelength of radiation does not change. The geometry is presented on the figure bellow.

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Figure 1: Diffraction occurs when the difference of paths for scattering on two atoms from neighnbouring lattice planes is equal to an integer multiple of wavelength. The wavelength should be of the same order of magnitude as the distances between neighbouring atoms, typically a fraction of nanometre, which corresponds to photon energies in keV range (X-rays). The concept of Bragg diffraction applies equally to neutron diffraction and electron diffraction.

2.3.2 Structure factor and diffracted intensity

Let jr be the position of j-th atom, k the incident beam’s wave vector and 'k the diffracted beam’s wave

vector. Then the phase change from the point 0r to the point r is:

00 ')'()(')( rkrkrkkrrkrrk jjj ⋅−⋅+⋅−=−⋅+−⋅=ϕ (6) The phase difference against a reference atom at the coordinate system’s origin is:

jrk ⋅∆=δ , (7)

where 'kkk −=∆ is the scattering vector. Electric field scattered from j-th atom is [5]:

( ))'(exp),( trkrkirf

trE jj ω

φ−⋅∆+⋅= , (8)

where φ is the amplitude of the incident radiation, ω is the angular frequency of the X-rays and jf is the atomic form factor. For Bravais lattice with a basis, the total electric field is calculated by summing over all

atoms in unit cell (index j) and all unit cells (index n), where jρ describes the atom’s position inside the unit cell:

( ) ( )

( ) ( ) ( ) ( ) )(exp)'(exp)(exp)'(exp

))('(exp)'(exp),( ,

kFrkir

trkikiftrkrkir

trkrkirf

trkrkirf

trE

nnj

jjnn

n jjn

j

n jjn

j

∆⋅∆−⋅=⋅∆−⋅∆+⋅

=−+⋅∆+⋅=−⋅∆+⋅=

∑∑∑

∑∑∑∑φωρωφ

ωρφ

ωφ

(9) Here we defined structure factor:

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( )∑ ⋅∆=∆j

jj kifkF )(exp)( ρ (10)

From equation (9) we can see that constructive interference will occur when:

( ) 1exp =⋅∆ nrki (11)

Which means that k∆ must be a vector of the reciprocal lattice. Calculated amplitude of electric field 0E obtained from equation (9) can be used to calculate the intensity of

diffracted radiation I :

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200EcI ε

= , (12)

where c is the speed of light and 0ε is vacuum permittivity.

2.3.3 Ewald sphere

We have shown that the diffraction occurs when k∆ is a vector of the reciprocal lattice. This can be graphicaly presented with the concept of Ewald sphere [7] . While Bragg’s law is usefull for representation of diffraction condition in real space, Ewald sphere represents the diffraction condition in reciprocal space. Elastic scattering implies the incident and diffracted wave vectors are equal in length and can be drawn inside a sphere as on figure below:

Figure 2: Ewald sphere [7], representing incident and scattered wave vectors ),( fi KK , their difference, the

scattering vector K∆ and reciprocal lattice points.

Since diffraction arises when k∆ is equal to one of the reciprocal lattice points, we can draw the reciprocal lattice with one point at incident wave vector’s tip and predict diffraction events whenever the sphere intersects one of the reciprocal lattice points. It is easy to imagine how rotation of the crystal will bring successive reciprocal lattice points to the sphere’s surface and cause diffraction.

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In any real experiment the incident X-rays will have a finite bandwidth, so Ewald sphere will be broadened to a finite thickness. Individual diffraction events are therefore dispersed over an angular interval, so integration over the angle is required to capture the whole diffraction. Lorentz factor L is introduced to compensate for the fact that reciprocal lattice points cross the sphere at different angles and consequently spend different amounts of time in the (broadened) sphere.

2.4 Synchrotron radiation A synchrotron is a particular type of cyclic particle accelerator in which the magnetic field (to turn the particles so that they circulate) and the electric field (to accelerate the particles) are carefully synchronised with the travelling particle beam. An accelerated charged particle will emit electromagnetic radiation, so particles in the synchrotron will emit radiation as their direction of movement is changed when passing through the magnetic field.

Figure 3: A charged particle in a synchrotron is radially accelerated by the magnetic field and emits synchrotron radiation [8], colimated by relativistic effects. Until the 1980s electron synchrotrons were considered to be primarily the issue of particle physicists. The electromagnetic radiation they created was an inconvenient by-product (the radiation represented a source of energy loss and a safety hazard). However, as the remarkable properties of synchrotron radiation were increasingly understood by physicists and chemists in other areas of research, the emphasis began to shift towards the design and construction of synchrotrons as dedicated radiation sources [8].

The irradiated power of synchrotron radiation increases with the particle energy and corresponding Lorentz term γ . Relativistic effects also help to colimate the radiation in the direction of the particle speed, leading to very high radiance (power per unit solid angle per unit projected source area), many orders of magnitude higher than that of X-rays produced in conventional X-ray tubes. The radiation also has some other remarkable properties: it is horizontally polarised in the plane of the electron orbit and circularly polarised above and below the orbit. This has advantages which can be put to use in both synchrotron diffraction and spectroscopy. The synchrotron radiation power spectrum extends from radio frequency to X-rays with a maximum at wavelength of about 1 nm.

3. Grain mapping procedure The picture bellow presents the experimental setup of X-ray diffraction contrast tomography:

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Figure 4: X-ray diffraction contrast tomography [1] requires measurement of the transmitted X-ray intensity during continuous rotation of the sample over an angle of 180° . The experiment includes the monochromated synchrotron beam, investigated sample fixed to the rotating axis and electronic high resolution X-ray detector system. The synchrotron beam needs to be partially coherent: the coherence length has to be greater than the part of the grain that diffracts simultaneously (the whole grain does not diffract simultaneously because of orientation spread). During a continous 180° rotation 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 local reduction of the transmitted intensity and can be detected as an “extinction spot” on the detector screen behind the sample. Orientation spread of the grains, finite X-ray beam bandwidth and beam divergence will expand diffraction events and corresponding extinction spots over an interval of the sample rotation angle ω . For this reason the grain projection images must be integrated over an angular range ω∆ in order to illuminate the whole sample. The integration interval must be large enough to compensate the above mentioned broadening effects but small enough to preserve the maximum available diffraction contrast in the projection images. The refraction (and multiple diffractions) are neglected, so we assume the X-rays travel along straight lines.

3.1 Basic assumptions According to the equation (4), the transmitted intensity at each point of the screen vu, is:

( )∫−⋅= dxrvuIvuI EFF )(exp),(),( 0 µ , (13)

where r defines the position inside the sample and the integral is calculated along the beam direction x. The coherent scattering contribution to the attenuation coefficient arising from subvolumes of individual grain

),,,,( LFr hklDIFFR ϕλµ depends on the wavelength λ , the local effective misorientation )(rϕ with respect

to the maximum of a given hkl reflection, the structure factor hklF and the Lorentz factor L associated with the reflection. We assume that after integration over the local misorientation ranges ϕ the diffraction contributions arising

from different reflections of the same grain can be separated into a spatially varying part )(~ rDIFFRµ and a multiplicative scaling function ),,( gm ωλ , where g represents the orientation matrix of a grain.

)(~ rDIFFRµ represents the local “diffraction power” of the crystall, which can vary due to different defect density or second phase inclusions inside the grain. In practice the scaling function ),,( gm ωλ is equal to zero for most of the rotation angles ω and scales proportional to the structure and Lorentz factors for rotation angles corresponding to the different reflections of the grain:

LFm hkl2∝ (14)

From equation (13) we can express line integrals of the effective attenuation coefficient:

∫∫∫ +==

− dxrmdxrdxr

vuIvuI

DIFFRABSEFF )(~)()(),(),(ln

0

µµµ (15)

3.2 Removal of absorption background

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To obtain information about the orientation of the crystals, we must separate the absorption contribution to line integrals of the attenuation coefficient (15) from diffraction contribution. The calculation of absorption coefficient can be done in a similar way as in X-ray computed tomography (CT), where diffraction is not considered, therefore absorption and attenuation coefficients are identical. A standard method for calculation of the attenuation coefficient from the measured projections in non-diffracting case is filtered backprojection reconstruction algorithm. This method can be used for calculation of absorption coefficient in the diffracting case because the angular intervals for which diffraction causes a significant contribution to the effective attenuation coefficient are negligible compared to the 180° rotation range of a tomographic scan (the angular average of diffraction coefficient is negligible compared to the average of absorption coefficient). It is also possible to reject diffraction events before aplication of the algorithm by calculating the average intensity values over an angular range wider than the typical angular intervals over which a diffraction spot extends. Then deviations from the average intensity can be identified as diffraction spots and replaced by the average intensity.

3.2.1 Filtered backprojection reconstruction algorithm

The X-rays and the movement of the points in the sample is perpendicular to the rotation axes, which means that attenuation on different planes perpendicular to the rotation axes can be reconstructed independently (slice-by-slice approach). The basic problem of tomography is the following: we have a set of 1-D projections taken at different angles. How do we reconstruct the 2-D sample from which these projections were taken? If the projections were taken at continuously changing angle, then we can solve the problem with filtered backprojection algorithm. The algorithm is based on Fourier slice theorem [4], which says that the 1D Fourier transform of the projection of the sample to a line is the same as the Fourier transform on that line.

Figure 5: Fourier slice theorem [9]: projection followed by one dimensional Fourier transform produces a slice of two dimensional Fourier transform.

Fourier slice theorem can be expressed with operator equation:

2111 FSPF = (16)

Where 1F is Fourier transform in one dimension, 2F is Fourier transform in two dimensions, 1P is the operator

of projection to a certain line and 1S is the slice operator, which extracts 1-D slice from a function.

Therefore it is possible to obtain 2D Fourier transform of the sample by transforming a large number of projections in different directions. 2D Fourier transform can then be used to obtain the attenuation coefficient in

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space domain using inverse Fourier transform. Since the projections will usually contain noise, filtering is used. It is performed with convolution of a projection with a filtering function, which is more easily implemented by multiplication in frequency domain. Having reconstructed the absorption coefficient, one can subtract the line integrals of absorption coefficient from the line integrals of attenuation coefficient, obtaining the line integrals of diffraction coefficient.

∫∫∫∫ −

−=−= dxr

vuIvuIdxrdxrdxrm ABSABSEFFDIFFR )(

),(),(ln)()()(~

0

µµµµ (17)

3.3 Summation of spots belonging to the same reflection: In general, difraction spots can spread over a range of succesive imaging and only part of the diffracting grain may be visible in each of the individual images due to orientation spread of the grain. One therefore first has to sum contributions belonging to the same reflection. This can be accomplished by applying a three-dimensional segmentation algorithm based on morphological image reconstruction. While working through the stack of successive projection images, each individual region of interest will be summed with contributions from neighbouring images as long as the regions are connected in the third dimension. 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 ovelap from spots originating from other diffracting grains in front of or behind the grain of consideration. Although 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.

3.4 Spot sorting Now that the spots belonging to the same reflection are summed, the reflections should be sorted into sets belonging to the same grain. Since the direction of the incoming X-rays is perpendicular to the rotation axes, the top and bottom vertical limits of all reflections belonging to the same grain must be identical. Because of limited accuracy of vertical limits, different grains with similar vertical limits can not always be distinguished, so a second test criterion is applied: when the spots belonging to the same grain are backprojected into the sample plane (taking into account the rotation angles), they must intersect at the same position in the sample. This also gives a crude estimate of the two-dimensional grain shape and its position in the sample plane.

Figure 6: Backprojections of extinction spots arising from the same grain intersect at the same point and give an aproximate shape of the grain [1]:

3.5 Reconstruction: In section 3.2 we have learned how to remove absorption background and obtain information about the

coefficient of diffraction in the form of line integrals ∫ dxrm DIFFRi )(~µ . The dependency on the scaling

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function belonging to i-th diffraction of a certain grain ),,( gmm iii ωλ= can be eliminated using the fact that the area integral of a parallel projection image is equal to the volume integral of diffraction coefficient and hence is independent of the projection direction:

( ) CdVrdudvdxr DIFFRDIFFR == ∫∫∫∫∫ ∫ )(~)(~ µµ , (18)

where the constant C is the same for all projections of the same grain. Dividing the known values of

∫ dxrm DIFFRi )(~µ by their area integral gives line integrals of diffraction coefficient, multiplied by a constant

value C1

:

( ) ∫∫∫ ∫

∫ = dxrCdudvdxrm

dxrmDIFFR

DIFFRi

DIFFRi )(~1)(~

)(~µ

µ

µ (19)

Now the problem is very similar to the calculation of absorption coefficient, which we already solved in section 3.2 by filtered backprojection algorithm. The difference is that instead of a continuous array of projection images through the whole 180° angular range, we only have a few tens of parallel projection images for each grain, corresponding to different low-index (hkl) families, so filtered backprojection algorithm can not be applied. The problem can be solved using a standard algebraic reconstruction technique (ART) algorithm [10], which can reconstruct attenuation in completely asymmetric object from projection images taken from only 5-10 different angles.

3.5.1 Algebraic reconstruction technique

We use slice-by-slice approach again. The diffraction contribution to attenuation coefficient )(~ rDIFFRµ is to be

calculated from its line integrals ),( tp iθ :

∫= dsstrtp iDIFFRi )),,((~),( θµθ , (20)

where iθ is the direction of the X-ray beam (i indexes different diffractions), t is the coordinate perpendicular to

the beam and s is the coordinate parallel to the beam. The problem is translated into discrete form with ip

representing the integrals of ),( tp iθ over a band of width τ and jf representing the average diffraction

contribution to attenuation coefficient ( DIFFRµ~ ) on different squares with side length δ . Then ip are the area

integrals of DIFFRµ~ and can be approximated by summation of contributions from different squares (on which

diffraction contribution to attenuation coefficient is approximated by the average value jf ). Thus we get a sistem of linear equations:

ij

jji pfw =∑ , , (21)

Where jiw , is the area of intersection between band i and square j .

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Figure 7: Discretization of the problem of reconstruction of diffraction contribution to attenuation coefficient from the known projections at several different angles [9]. When resolution is high and the number of different projection angles is small, the system (21) will be underdetermined. The underdetermined system is solved by an iterative algorithm which returns a minimum norm solution. The pictures bellow show the two dimensional ART reconstruction of the diffraction contribution to attenuation coefficient in the grain cross section obtained from 27 projections.

Figure 8: a) two dimensional ART reconstruction of grain cross section. b) Overlay of the ART reconstruction with corresponding absorption contrast reconstruction obtained by conventional filtered backprojection algorithm [1]

3.6 Orientation determination: An optional step of the analysis procedure is the determination of grain orientations. Since our experiment includes rotation of the sample, the angle ω must be specified to describe Bravais lattice orientation and the orientation can be transformed to different angles ω by mapping it with an additional rotation matrix. Compared with conventional far field acquisition geometry where the scattering vector and intensity are measured for each diffraction spot, diffraction contrast tomography lacks much information: wave vectors of diffraction beams and consequently scattering vectors are unknown and only decrease of the transmitted intensity can be used to obtain indirect information about diffracted intensity.

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For conventional far field acquisition methods only the set of scattering vectors is needed to determine the crystallgraphic orientation. One of the ways to do this is 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 diffraction contrast tomography only provides rotation angles iω at which diffractions occur (without knowledge of the scattering vectors), one has to include knowledge about intensity of diffraction spots in order to determine orientations. In a first step the integrated extinction spots (integration over angle ω ) 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 expected and measured iω rotation angles. It turns out that several solutions may exist, especially for highly symmetrical space groups. The integrated intensities are used to distinguish those solutions. With the knowledge of scattering vectors, the structure and Lorentz factors it is possible to calculate how much power will be diffracted per unit of crystal volume. The diffracted power is equal to the decrease of transmitted power. By inverting this correlation the integrated transmitted intensities can be used to calculate the crystal volume where diffraction occurs. This volume should be equal for several different diffractions belonging to the same grain. So the correct orientation of the grain can be identified by having the smallest differences between calculated crystal volumes.

3.7 Results The authors of X-ray diffraction contrast tomography [1] have demonstrated the applicability of the method by an experiment at the high-resolution imaging beamline ID19 at the European Synchrotron Radiation Facility (ESRF). The polychromatic synchrotron beam was monocromated to 20 keV by diffraction on Si 111 double-crystal monochromator. The coherence length was 250 mµ . Two-dimensional projection images were recorded on a high-resolution detector system based on transparent luminescent screen, light optics and CCD camera. The sample-to-detector distance was 20 mm and an effective pixel size 2.8 mµ . The exposure time for each projection was 2 seconds and a total of 9000 projections were recorded during a continuous motion rotation of the sample over an angle of 180°. The resulting angular integration range of 0.02° per projection allowed maximizing the available extinction contrast. 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. The sample was a cylinder with 550 mµ diameter and a length of 2 mm, prepared from aluminium alloy Al 1050. The sample was given mechanical and heat treatment in order to increase grain size and decrease orientation spread. After treatment the average grain size was 200 mµ and orientation spreads were below 0.1°. However part of the sample exhibited a second family of smaller grains with considerably higher orientation spreads. The following results relate to the first family of the grains.

Figure 9 [1]: a) the projection image at an angle when one of the grains fulfils the Bragg diffraction condition. b) the absorption background c) diffraction images obtained by logarithmic subtraction of absorption background Figure 9 a) shows one of the projection images for which one of the large grains fulfils the Bragg condition, figure 9 b) shows the absorption background obtained from the same image and figure 9 c) shows summed partial diffraction images with absorption background removed by logarithmic subtraction. In the current example about 300 such grain projection images were extracted from the raw projection images and then sorted into groups belonging to the same grains. Then the grain projections were normalized (divided by their area integral) and algebraic reconstruction technique was used to determine grain shapes. For visual representation, different grains are assigned different colours

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Figure 10 [1]: a) Sample cross section showing different grains in different colors. b) 3D representation of grain shapes. Since diffraction contrast for different grains is calculated independently, the borders of adjacent grains don’t match perfectly. This can be used to estimate grain shape reconstruction accuracy, which was better than 10 mµ in the presented example,while the absorption image reconstruction was of the order of 3 mµ and orientation space resolution was better than 0.1 °. In the central part of figure 10 b) no grain could be identified with the current approach.

3.8 Limitations: In principle, X-ray diffraction contrast tomography can be performed on any polycrystal (metals, ceramics). However there are limitations regarding orientation spread and grain size. We have seen that diffraction contrast tomography failed in the central region of the sample on figure 10 b). Independend measurement with 3DXRD (three dimensional X-ray diffraction microscopy) [11] revealed the presence of two grain populations: a family of large grains (100-500 mµ ) with orientation spread below 0.05° (region labelled “A” in figure 9 a)) and a second family of smaller grains (20-100 mµ ) with considerably higher orientation spreads of the order of 0.2°-1°. For the second family the contrasts associated with one reflection can spread over up to several tens of consecutive images due to high orientation spread. This combined with smaller grain size (and therefore higher number of diffraction spots) greatly increases the probability of spot overlap. Smaller grain size also reduces the contrast due to shorter X-ray path through individual grain. The cross section size of the sample and consequent number of grains per cross section also affects the employability of the method since greater number of grains per cross section will increase the probability of spot overlap. In order for the method to work, the maximum number of grains per cross section has to be selected as a function of the macroscopic sample texture and orientation spread of the individual grains. The authors of X-ray diffraction contrast tomography [1] have later extended the method so that it includes acquisition of diffracted beams in addition to transmitted beam [12]. This way the method can function on polycristal samples containing more than 100 grains per cross section.

3.9 Comparison with other three-dimensional X-ray imaging methods: Other methods for analysis of materials with X-rays include microtomography [13], “three-dimensional X-ray diffraction microscopy” (3DXRD)[11] and three dimensional crystal microscope ([14], [15]). Microtomography (commonly known as industrial CT scanning) can determine absorption in material with 1 mµ spatial resolution, but can not in general reveal the grain structure of crystalline materials. On the other hand 3DXRD or three-dimensional crystal microscope allow characterization of 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 (DCT) may be regarded as a combination of conventional absorption contrast tomography and 3DXRD that partly overcomes these limitations. DCT allows the possibility of adjusting the detector system’s field of view to the sample dimensions and therefore potentially provides higher spatial resolution than any other grain mapping approach based on recording of diffracted

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beams. Direct beam acquisition geometry of DCT also doesn’t constrain the range of acceptable sample-to-detector distances, which is an essential prerequisite for performing in-situ imaging experiments, generally involving the use of bulky sample environments (furnace, tensile rig. etc.).

4. Conclusions: We have learned the basic theory and the principle of operation of X-ray diffraction contrast tomography. The technique is applicable to undeformed polycrystalline samples containing a limited number of grains per cross section and exhibiting typical orientation spreads of the order of one tenth of a degree. It has a variety of possible applications: diffraction contrast tomography can determine the position of grain boundaries and help to estimate their strength based on orientations of the neighbouring grains, so it could be usefull for research of stress corrosion cracking or fatique crack propagation (either experimental or finite element simulations). Grain shapes and sizes can also be used to predict material mechanical properties. By increasing monochromatic flux one may also think about in-situ observation of grain-coarsening processes such as recrystallization and grain growth. Compared with alternative 3DXRD grain mapping approaches, diffraction contrast tomography has the advantage of providing simultaneously access to a sample’s three dimensional grain and absorption microstructure. Because of the close similarities to conventional absorption contrast tomography, diffraction contrast tomography can be performed on mechanically simple, high resolution imaging setups available nowadays at any modern synchrotron source.

5. References 1. W. Ludwig, S. Schmidt, E. M. Lauridsen, H. F. Poulsen: X-ray diffraction contrast tomography: a novel technique for three dimensional grain mapping of polycrystals (2008) 2. http://en.wikipedia.org/wiki/Texture_(crystalline) (28.5.2011) 3. http://en.wikipedia.org/wiki/Grain_boundary (28.5.2011 4. http://en.wikipedia.org/wiki/Attenuation_coefficient (28.5.2011) 5. H. P. Myers, Introduction to solid state physics 6. http://www.xtal.iqfr.csic.es/Cristalografia/index-en.html (28.5.2011) 7. http://en.wikipedia.org/wiki/Ewald_sphere (28.5.2011) 8. http://pd.chem.ucl.ac.uk/pdnn/pdindex.htm#diff2 (28.5.2011) 9. A. C. Kak, M. Slaney, Principles of computerized images (1988) 10. R. Gordon, R. Bender, G. T. Herman, algebraic reconstruction techniques (ART) for three dimensional electron microscopy and X-ray photography (1970) 11. H. F. Poulsen, Three-Dimensional X-ray Diffraction Microscopy. Mapping Polycristals and their dynamics. (2004) 12. G. Johnson, A. King, M. G. Honnicke, J. Marrow, W. Ludwig: X-ray diffraction contrast tomography: a novel technique for three-dimensional grain mapping of polycrystals. II. The combined case (2008) 13. http://en.wikipedia.org/wiki/Microtomography(28.5.2011) 14. B. C. Larson, W. Yang,, G. E. Ice, J. D. Budai, J. Z. Tischler, Three dimensional X-ray structural microscopy with submicrometre resolution (2002) 15. G. E. Ice, B. C. Larson, W. Yang, J. D. Budai, J. Z. Tischler, J. W. L. Pang, R.I. Barabash, W. Liu, Polychromatic X-ray microdiffraction studies of mesoscale structure and dynamics