Friedel-pair based indexing method for

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M. Moscickia, P. Keneseib, J. Wrightc, H. Pintoa, T. Lippmannd, A. Borbélya,∗, A.R. Pyzallae ... Article history: ... may be less than 0.1◦, and the position of the center of mass of the grains can be ... As a result of in situ measurement of grain rotations in alu- ... tor coordinates of Friedel reflection-pairs ((h, k, l) and (−h, −k, −l)).
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Friedel-pair based indexing method for characterization of single grains with hard X-rays M. Moscicki a , P. Kenesei b , J. Wright c , H. Pinto a , T. Lippmann d , A. Borbély a,∗ , A.R. Pyzalla e a

Max-Planck-Institut für Eisenforschung GmbH, Max-Planck Strasse 1, 40237 Düsseldorf, Germany Eötvös Loránd University, Institute of Physics, Department of Materials Physics, 1518 Budapest, POB 32, Hungary European Synchrotron Radiation Facility, Grenoble, France d GKSS-Research Center, Max-Planck-Str., D-21502 Geesthacht, Germany e Helmholtz Zentrum für Energie und Materialien GmbH, Glienicker Strasse 100, 14109 Berlin, Germany b c

a r t i c l e

i n f o

Article history: Received 25 February 2009 Received in revised form 29 April 2009 Accepted 2 May 2009 Available online xxx Keywords: X-ray diffraction Friedel-pair In situ testing Crystallographic orientation change

a b s t r a c t A new evaluation procedure is presented to characterize the orientation and position of single grains within the bulk of a polycrystalline sample. Considering the symmetry properties of Friedel-pairs the contributions to reflection spot positions arising from grain orientation and position could be clearly separated. The proposed method avoids simultaneous fitting of all grain parameters with the goal of a higher accuracy. Depending on the number of reflections considered the accuracy of grain orientation may be less than 0.1◦ , and the position of the center of mass of the grains can be accurate within one-third of the pixel size. © 2009 Elsevier B.V. All rights reserved.

1. Introduction In situ investigation of materials under loading conditions enables new insights into the kinetics of governing physical processes. Nondestructive investigation techniques using hard synchrotron radiation enable for example to study the deformation behavior of single grains embedded in the bulk of the sample. The three-dimensional X-ray diffraction (3DXRD) method developed at the European Synchrotron Radiation Facility (ESRF) [1] was successfully applied to study texture evolution during deformation [2], the response of single grains to external stresses [3], 3D grain growth during recrystallization [4] and phase transformations [5]. From the point of view of structural materials consisting mainly of polycrystals the micro- and meso-scales are of significant importance, since only investigations at this level give the unbiased information necessary to understand grain interactions. Determination of local strains and stresses in single grains during loading represents one of the major challenges that is expected to be solved by the 3DXRD technique [6]. A method for measuring local strains and crystallographic orientation in polycrystalline samples has been also developed based on X-ray microbeams with broad-bandpass energy [7].

As a result of in situ measurement of grain rotations in aluminium [8], Winther et al. [9] concluded that the simple Taylor model [10] describes reasonably well experimental data of many grains, however, discrepancies have also been stated for grain orientations in the center of the stereographic triangle and near the 100 pole. In order to help the evaluation of such in situ investigations we propose here a new indexing method, which beside the evaluation of the crystallographic orientation determines the grain position in the sample as well. The evaluation algorithm uses detector coordinates of Friedel reflection-pairs ((h, k, l) and (−h, −k, −l)) and requires the measurement of diffraction spots in a relatively large rotation interval of about 180◦ . Such a setup is usually applied for in situ strain measurements. The advantage of using Friedelpairs resides in their symmetry properties enabling the separation of grain position effects from grain orientation. Based on their symmetry the basic diffractometer equations can be simplified and the number of unknown parameters in the fit can be reduced. The method assumes a perfectly aligned detector lying perpendicular to the incoming beam. With the help of a reference powder sample the calibration of the setup can be done for detector tilt, sampledetector distance and position of the beam center [11]. 2. Description of the scattering geometry

∗ Corresponding author. Tel.: +49 211 6792 970; fax: +49 211 6792 390. E-mail address: [email protected] (A. Borbély).

Fig. 1 shows a schematic drawing of the experimental setup. During measurement the sample is rotated in steps of ω around

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Introducing Eqs. (2), (3) and the normalized Eq. (4) into Eq. (1) a system of three equations is obtained, which represents the basis of the so-called “forward simulation”. Based on it the spot locations (u, v) with regard to the beam center B can be predicted as a function of position and orientation of the grains. In case of a perfectly aligned detector, having its plane perpendicular to the direct beam and pixel columns lying parallel to the rotation axis of the sample, Eq. (1) becomes:

  D u



=

v

x cos ω − y sin ω x sin ω + y cos ω z

   +

1 0 0



+ 2 sin  R

h k l



t, (6)

where ( h ,  k ,  l )T = gc /|g| is the unit vector along gc . The detector spot coordinates are given by:

Fig. 1. Schematic geometry of the experimental setup.

u = x sin ω + y cos ω + 2 sin (R21 h + R22 k + R23 l )t,

(7)

v = z + 2 sin (R31 h + R32 k + R33 l )t,

(8)

where the vertical axis Oz of the laboratory coordinate system, which coincides with Zs in Fig. 1. At given ω positions (or small intervals in case when orientation gradients are present) the (h, k, l) lattice planes of given grains fulfill the Bragg condition and give rise to diffraction spots recorded by the 2D detector placed at distance D behind the sample. The scattered intensity is integrated during each ω step after which a detector image is saved. Based on simple geometric considerations sketched in Fig. 1 the vector rd describing the position of a diffraction spot with regard to the beam center on the detector is given by: rd = rω + eK t − De1 ,

(1)

where rω = (xω , yω , z)T points to the center of mass (CM) position of the grain in the laboratory frame when the Bragg condition is fulfilled; e1 = (1, 0, 0)T and eK are unit vectors in the direction of the incoming and diffracted beams, respectively. The parameter t is determined from the intersection condition between the scattered wave vector K and the detector plane. Usually the grain position r = (x, y, z)T in the sample coordinate system (at ω = 0) is of practical interest, which means that rω should be related to r through the matrix describing the rotation of the grain around the vertical axis Oz:



xω yω z



 

= ˝z (ω)

x y z



=

cos ω sin ω 0

−sin ω cos ω 0

0 0 1

x y z

.

(2)

(3)

where eω is the unit vector along gω ,the diffraction vector in the laboratory frame. K1 denotes the incoming wave vector and  the Bragg angle. gω is linked to the reciprocal lattice vector gh k l = (h, k, l)T through the formula [1,12]: gω = ˝z SU−1 Bg h k l = R(Bg h k l ) = Rg c ,

(4)

where gc is the diffraction vector in the Cartesian grain system and is obtained from gh k l through the transformation matrix B [13]. U is the orientation matrix of the grain (defined according the passive convention) and S relates the sample coordinate system to the laboratory system. For simplicity we introduce the matrix R denoting the product of the three orthogonal matrices z , S and U−1 : R = ˝z SU

−1

,

D − x cos ω + y sin ω . 1 + 2 sin (R11 h + R12 k + R13 l )

(9)

The rotation angle ω in Eqs. (7)–(9) is determined by the Bragg’s law, which imposes the geometrical relationship between eω and e1 : e1 eω = −sin .

(10)

Independent if the rotation axis of the sample is aligned along Oz or not, Eq. (10) can be reduced to the following form: a sin ω + b cos ω + c = 0,

(11)

where a and b are constants depending on  h ,  k ,  l and tilt components of the sample rotation axis and c = sin . If the angle between a diffraction vector and sample rotation axis is larger than the Bragg angle , Eq. (11) has two solutions in the [0, 2] interval and are given by:



ω1,2 = arctan





ca2 ± |a|b

a2 + b2 − c 2

a(a2 + b2 )



,−

cb ± |a|

a2 + b2 − c 2

(5)



a2 + b2 (12)

 

The direction of the scattered intensity eK is calculated from the Bragg’s law K = K1 + gω , which can be written in terms of the unit vectors: eK = e1 + 2 sin eω ,

t=

3. Indexing diffraction patterns 3.1. Evaluation of grain orientation Depending on the parameters of interest such as grain orientation or both orientation and position of the grains, different indexing methods can be developed. If only the orientation of the grains has to be evaluated usually a large sample-detector distance D is chosen, so that coordinates x, y and z in Eqs. (7) and (8) can be neglected beside D. This leads to the simplified diffractometer equation [1], when each spot on the detector is associated with a diffraction vector placed at the origin of the sample coordinate system. The diffraction vector can be simply given in terms of the Bragg angle  and the azimuthal angle  of the diffraction spot ( is measured clockwise from the projection of the rotation axis on the detector, when looking from the sample towards the detector downstream the beam [1]):



gω =

2 

cos(2) − 1



⎣ −sin(2)sin() ⎦ . sin(2)cos()

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(13)

.

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The indexing method introduced by Ref. [12] considers this simplified diffractometer equation and it is based on a comparison −1 between experimental diffraction vectors g S exp = S −1 ˝z g ω and candidate vectors gS sim = U−1 gc corresponding to candidate orientation matrices U. To speed up computations the comparison is performed in the sample coordinate system (CS) (indicated by the subscript S) for all independent orientations U enabled by the crystal symmetry. Candidate U matrices belong usually to the nodal points of a regular grid defined in the fundamental zone of the orientation space. Those orientations are selected, which fulfill the completeness criteria for the diffraction vectors [1]. The method is robust and has the advantage that it can be applied to scans performed over smaller ω intervals, adequate to study the kinetics of faster processes like second-phase nucleation [5] or recrystallization [4]. The method proposed here differs from that of Lauridsen et al. [12] and it is based on a direct approach considering only the experimentally measured diffractions vectors. It avoids scanning of the orientation space, however, requires larger ω intervals to be measured. If the crystallographic axes of the unit cell are mutually perpendicular to each-other then two h 0 0 type reflections per grain are enough for setting up the orientation matrix (the normalized h 0 0 diffraction vectors are the rows of U). The minimum ω interval necessary to apply the method is 90◦ –2 2 0 0 and corresponds to unfavorably oriented grains with 0 0 l axis lying within a cone of vertex angle  around the rotation axis. Once a candidate orientation matrix Uh 0 0 was found it is transformed to the fundamental zone of the orientation space and candidate diffraction vectors according to the measured diffraction orders h k l are generated. The candidate orientation is accepted if the completeness criterion between the measured and generated diffraction vectors is fulfilled. A generated diffraction vector is considered to match an experimental vector if the angle between them is less than a given threshold, which depends on the integration step ω. Checking for completeness is mandatory since due to applied finite thresholds non-interrelated h 0 0 type reflection pairs can be selected to define the candidate orientation. This may happen predominantly in case of textured polycrystals. To obtain a more accurate orientation than Uh 0 0 the system of equations (g S

exp )i

= (U ∗ )

−1 c

g ,

(14)

can also be solved in a least-square sense for the elements of the orientation matrix. The subscript i indicates the number of diffraction vectors (equations) taken into account and is typically about 20 if the first four reflection orders of an fcc crystal (1 1 1, 2 0 0, 2 2 0 and 3 1 1) and a scan interval ω of 90◦ are considered. In some cases the solution of Eq. (14) must be refined by removing outlier reflections, which incidentally fulfill the imposed angular condition. If the disorientation angle between the new orientation U* and the candidate Uh 0 0 is larger than a few degrees, the outliers are excluded. Due to their nearly diverging Lorentz factor spots associated with detector polar angles  ∈ [−20◦ , 20◦ ] and  ∈ [−160◦ , 200◦ ] (see Fig. 1 for definition of ) are also excluded from the fit. Such reflections are extended over many ω integration steps, so the ω-coordinate of their center of mass can have a larger error. 3.2. Refinement of grain orientation based on Friedel-pairs A more accurate evaluation of the grain orientation becomes possible if adequate experimental conditions related to symmetry properties of Eqs. (7)–(9) are selected. An experimental possibility is offered by the use of Friedel-pairs, which are used for example for accurate centering of crystals at the mechanical center of the goniostat [14]. Considering the geometry of the 3DXRD method the

3

Fig. 2. Schematic drawing of the position of Friedel-pairs on the detector, during a complete ω-turn of the sample. Spots 1–2 (as well as 3–4) are denoted as ω-pair, while spots 1–3 (2–4) as -pairs.

following remark can be made. During a complete ω turn of a grain each h k l reflection as well as its Friedel-pair is measured twice. The positions of the corresponding spots on the detector are shown schematically in Fig. 2. Selecting a h k l reflection located in the first detector quadrant (point 1) the related Friedel-pairs (h¯ k¯ ¯l reflections) will be positioned at points 2 and 3. The second emergence of the h k l reflection has the number 4. Based on the relative position of diffracting lattice planes h k l with respect to the incoming beam it can be rationalized that one Friedel reflection appears after an additional half turn of the sample (with regard to ω1 ), when the back side of the h k l planes (i.e. the h¯ k¯ ¯l planes) are closing the Bragg angle with the incoming beam. The corresponding spot has the number 2. A similar remark is valid for spots 3 and 4, too. Such reflection pairs are denoted in the following as ω-pairs and satisfy the relationship ω2,3 = ω1,4 ± 180◦ . If the position of the reflecting grain is not very far from the rotation axis, the second Friedel-pair (spot 3) can be considered in a first approximation as just the inversion of spot 1 with regard to the beam center. Reflection pairs 1–3 (and 2–4) are denoted as -pairs and are interrelated through the approximate relationship 3,4 ∼ = 1,2 ± 180◦ . Considering the simplified Eq. (13) for the ω-pair 1–2 the relationship between their azimuthal coordinates  can be worked out: −1

−1

˝z (ω1 )g ω1 = −˝z (ω2 )g ω2 ,

(15)

which simplifies to: 2 ∼ = 540◦ − 1 .

(16)

Due to symmetry considerations the azimuthal angle of spot 4 is given by: 4 ∼ = 360◦ − 1 .

(17)

Based on Eq. (15) by considering ω3 instead of ω2 and 3 = 1 ± 180◦ one can also predict the relationship between the ω angles of the -pairs. For pair 1–3 we have:



ω3 = ω1 + 2 arctan

tan  sin 1



.

(18)

A similar equation is obtained for -pair 2–4 if subscript 1 is exchanged in Eq. (18) by 4 and subscript 3 by 2. Relations (16)–(18) are very useful in practice offering a fast method to group reflections into Friedel-pairs or pair-sets. The symmetry properties of Eqs. (7)–(9) can be exploited by selecting the ω-reflection pairs separated by 180◦ in rotation angle ω. Considering spot 1 as reference (with coordinates (u, v) in Fig. 2) the location of its ω-pair (spot 2) is given by the coordinates (uF , vF ),

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which are obtained by substituting ( h ,  k ,  l ) = (− h , − k , − l ) and ω = ω + 180◦ into Eqs. (7)–(9):

where the functions f(x, y) and f(x, y, z) represent the right hand side of the Eqs. (23) and (24).

uF = −x sin ω − y cos ω + (D + x cos ω − y sin ω)

4. Results for simulation and experimental data

2 sin (R21 h + R22 k + R23 l ) × , (19) 1 + 2 sin (R11 h + R12 k + R13 l ) 2 sin (R31 h + R32 k + R33 l ) vF = z − (D + x cos ω − y sin ω) . 1 + 2 sin (R11 h + R12 k + R13 l ) (20) Eqs. (7)–(9) and (19) and (20) can be used for example to eliminate the grain center coordinates (x, y, z):

v − vF R31 h + R32 k + R33 l = , u + uF R21 h + R22 k + R23 l

(21)

which quantity depends only on the grain orientation and the diffraction vector. Based on it a least-square functional including all experimentally found ω-pairs can be set up and minimized with regard to the Euler angles (ϕ1 , ˚, ϕ2 ): 2

(ϕ1 , ˚, ϕ2 ) =

N 

v − vF i=1

R31 h + R32 k + R33 l − u + uF R21 h + R22 k + R23 l

2 ,

(22)

i

where N represents the number of Friedel-pairs. As starting value for the minimization the solution U* of Eq. (14) can be taken. Since the matrix elements of R contain the rotation angle ω it should be emphasized that the experimental values of ωi are inadequate for minimization, they should be recalculated for each candidate Euler angles. A slightly different arithmetic applied to Eqs. (7)–(9) and (19) and (20) allows the evaluation of grain center coordinates:



u − uF = x 2 sin ω −







sin ω.

(24)

u + uF u + uF cos ω + y 2 cos ω + sin ω , D D (23)

and

v + vF = 2z − x

v − vF D

cos ω + y

v − vF D

These equations contain the rotation angle ω, which should be taken from the orientation refinement. They do not contain the Bragg angle , which is falsified in the experiment due to the displacement of the grain from the rotation axis. These two equations allow for setting up a second functional, which can be minimized to obtain the grain center coordinates:

2 (x, y, z) =

N

i=1

2

(u − uF − f (x, y))i +

N

i=1

2

(v + vF − f (x, y, z))i , (25)

To test the indexing method outlined above we have considered copper as model material and spot locations corresponding to the first four diffraction orders (1 1 1, 2 0 0, 2 2 0 and 3 1 1) were generated for grains of known orientation and position. The integration range ω was subdivided in two subintervals (0–120◦ ) and (180–300◦ ), which corresponds to the experimental conditions imposed by the tensile testing device used for real in situ measurements at beamline HARWI II of the DESY synchrotron in Hamburg. All parameters used for simulations were according to the experiment: beam energy of 70 keV, detector pixel size (100 ␮m for MAR345 image plate scanner) and a detector-sample distance of 887 mm. Indexing computer simulated patterns arising from 100 grains with random orientation shows that the original orientation of the grains was found with an accuracy better than 0.1◦ and this does not deteriorate much if a random error of about 1 pixel is considered for azimuthal angle  (Fig. 3a). Such errors may be caused for example by residual strains present in single grains. The -error is given in pixels rather than in degrees to account for the higher accuracy of high-order reflections. An -error of 1 pixel corresponds to angular errors of about 0.08◦ and 0.04◦ for 1 1 1 and 3 1 1 reflections, respectively. Fig. 3b shows the influence of the -error on the recovered grain positions, which initially were randomly chosen within a circle of 1 mm in diameter. Recovered grain positions are much more sensitive than the crystallographic orientation. It can be stated, however, that on the average the recovered grain positions are accurate within one-third of the pixel size (for -error of 1 pixel). This means that the center of mass position of grains recovered from far-field measurements (sample-detector distance of the order of 1 m) cannot be used for grain structure reconstructions unless the grain size is much larger than the pixel size of the detector. The accuracy of crystallographic orientation and position of the indexed grains depends on the integration step ω, too. Since this parameter acts mainly as bounds for selecting the diffraction vectors associated to a candidate orientation it has a smaller effect on the final results; the value is recalculated during minimization of in functionals (22) and (25) having finally an error smaller than the disorientation ı. Fig. 4 shows the disorientation of the indexed grains as a function of integration step ω. For ω steps smaller than 2◦ the recovered orientation differs by less than 0.5◦ from the original. The -error in this case was considered diffraction order independent and had a quite large value of 0.25◦ . For the analyses of experimental diffraction patterns the position of diffraction spots were corrected according to Ref. [15] for the tilt of the detector plane and rotation of tilt axis with regard

Fig. 3. Distribution of the absolute error of recovered grain orientations (a) (disorientation with regard to the initial orientation) and center of mass positions (b). The results were obtained by applying the developed evaluation software to index diffraction patterns of 100 grains with random position and orientation. Integration step ω = 0.5◦ . The recovered grain orientations show less dependence on the additional random errors , while the grain positions are more sensitive to it.

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to detector pixel columns. Fig. 5a shows the orientation change of 9 selected grains after a global strain of 4%. Crystallographic grain rotations in copper are not in conflict with an interpretation that the rotation behavior is similar to aluminium [8]. Applicability of present indexing algorithm in practice is confirmed by the analysis of grain positions recovered for the undeformed and deformed states and which are shown in Fig. 5b. It is visible that the relative position of the grains with respect to each-other has little changed after deformation, however, all grains moved laterally with a significant amount. 5. Conclusion

Fig. 4. Influence of the integration step ω on the average orientation error of indexed grains (the data contain a random azimuthal error of  = 1 pixel). Orientation refinement based on Friedel-pairs reduces the error by a factor of about 3 compared to the orientation U200 based on 200 reflections.

A new indexing method for obtaining orientation of single grains within the framework of 3DRXD framework was proposed. The algorithm uses coordinates of Friedel reflection pairs and allows a clear separation of grain orientation and grain position contributions. Conditions have been worked out for grouping reflections in pairs leading to a fast indexing algorithm. According to test simulations small errors in peak position do not influence much the recovered grain orientations. Grain center positions can be, however, significantly influenced. The applicability of the method in practice was exemplified by analyzing the in situ diffractograms of a tensile deformed copper sample. Acknowledgement P.K. and A.B. acknowledge the support of the EU-NSF FP6 project DIGIMAT. References [1] H.F. Poulsen, in: G. Hohler (Ed.), Springer Tracts in Modern Physics, vol. 205, Springer, New York, 2004. [2] L. Margulies, G. Winther, H.F. Poulsen, Science 291 (2001) 2392. [3] B. Jakobsen, H.F. Poulsen, U. Lienert, J. Almer, S.D. Shastri, H.O. Sørensen, C. Gundlach, W. Pantleon, Science 312 (2006) 889. [4] S. Schmidt, S.F. Nielsen, C. Gundlach, L. Margulies, X. Huang, D. Juul Jensen, Science 305 (2004) 229. [5] S.E. Offerman, N.H. van Dijk, J. Sietsma, S. Grigull, E.M. Lauridsen, L. Margulies, H.F. Poulsen, M.Th. Rekveldt, S. van der Zwaag, Science 298 (2002) 1003. [6] L. Margulies, T. Lorentzen, H.F. Poulsen, T. Leffers, Acta Mater. 50 (2002) 1771. [7] J.-S. Chung, G. Ice, J. Appl. Phys. 86 (1999) 5249. [8] H.F. Poulsen, L. Margulies, S. Schmidt, G. Winther, Acta Mater. 51 (2003) 3821. [9] G. Winther, L. Margulies, S. Schmidt, H.F. Poulsen, Acta Mater. 52 (2004) 2863. [10] G.J. Taylor, J. Inst. Met. 62 (1938) 307. [11] A.P. Hammersley, S.O. Svensson, M. Hanfland, A.N. Fitch, D. Häusermann, High Pressure Res. 14 (1996) 235–248. [12] E.M. Lauridsen, S. Schmidt, R.M. Suter, H.F. Poulsen, J. Appl. Cryst. 34 (2001) 744. [13] W.R. Busing, H.A. Levy, Acta Cryst. 22 (1967) 457. [14] H.E. King Jr., L.W. Finger, J. Appl. Cryst. 12 (1979) 374–378. [15] P. Kenesei, M. Moscicki, J. Wright, G. Vaughan, A. Borbély, A.R. Pyzalla, in preparation.

Fig. 5. (a) Orientation change of 9 grains after 4% of uniaxial global strain. Most of the grains follow the predictions of the Taylor theory. (b) Recovered center of mass position of the grains. Full and open symbols mark the undeformed and deformed states, respectively.

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