Detector tilt considerations in high-energy Bragg coherent diffraction imaging: a simulation study
Siddharth Maddali, Marc Allain, Peng Li, Virginie Chamard, Stephan O. Hruszkewycz
DDetector tilt considerations in Bragg coherent diffraction imaging:A simulation study
S. Maddali a , M. Allain b , P. Li b , V. Chamard b , S. O. Hruszkewycz a a Argonne National Laboratory, 9700 S. Cass Ave, Lemont, IL 60439 (USA) b Aix-Marseille University, CNRS, Centrale Marseille, Institute Fresnel, Marseille, France
Abstract
This paper addresses three-dimensional signal distortion and image reconstruction issues in x-ray Bragg coherent diffractionimaging (BCDI) in the event of a non-trivial, non-orthogonal orientation of the area detector with respect to the diffractedbeam. Growing interest in novel BCDI adaptations at fourth-generation synchrotron light sources has necessitated impro-visations in the experimental configuration and the subsequent data analysis. One such possibly unavoidable improvisationthat is envisioned in this paper is a photon-counting area detector whose face is tilted away from the perpendicular to theBragg-diffracted beam during acquisition of the coherent diffraction signal. We describe a likely circumstance in which onewould require such a detector configuration. Using physically accurate diffraction simulations from synthetic scatterersin the presence of such tilted detectors, we analyze the general nature of the observed signal distortion qualitatively andquantitatively, and provide a prescription to correct for it during image reconstruction. Our simulations and reconstructionsare based on an adaptation of the known theory of BCDI sampling geometry as well as recently developed geometry-awareprojection-based methods of wavefield propagation. Such configurational modifications and their numerical remedies arepotentially valuable in realizing unconventional coherent diffraction measurement configurations and eventually paving theway for novel materials characterization experiments in the future.
Keywords:
Bragg coherent diffraction imaging, scattering geometry, detector orientation, coordinate transformation, shearcorrection, Bragg ptychography
1. Introduction
Bragg coherent diffractionimaging (BCDI) is a synchrotron-based lensless imaging technique for spatial resolution of lat-tice structure and distortions on the scale of a few tens of nanometers [1, 2, 3, 4]. As a means of obtaining three-dimensional(3D) real-space images via phase retrieval inversion algorithms [5, 6, 7, 8], BCDI is a valuable means of materials character-ization owing to its ability to spatially resolve specific components of the 3D lattice strain tensor in deformed crystals, ina nondestructive manner. BCDI and the related imaging technique of Bragg ptychography [9, 10, 11, 4] together constitutean important set of nano-scale imaging modalities for compact as well as extended single crystal materials.With the several orders of magnitude increase in coherent flux at fourth-generation synchrotron light sources ( e.g. ,ESRF-EBL and the upcoming APS-U), coherent diffraction methods will play an increasingly important role in the 3D char-acterization of materials structure at the nanoscale. The need to incorporate these techniques into existing measurementpipelines will create a requirement for flexible and unconventional diffractometer geometries for smooth functioning. Sucha situation could arise in reciprocal space locations where it is difficult to rotate a detector arm into the required position,but BCDI capabilities are required nevertheless.As an example, an experiment might require enhanced strain resolution corresponding to a higher-order Bragg reflec-tion that may be outside the accessible rotation range of a conventional detector arm. In such a case, one may favor analternate configuration such as a wall-mounted BCDI detector configuration in which the physical detector chip will not beperpendicular to the exit beam. Another example from a recent work describes the unprecedented nanoscale strain mappingon individual crystalline domains in a poly-grain material by exploiting the partial coherence of a high-energy x-ray beam(52 keV) typically used for meso-scale orientation and strain characterization [12]. The high beam energy necessitated asample-detector distance of ∼ m in order to resolve diffraction fringes, which was achieved with a wall-mounted BCDIdetector whose face was perpendicular to the downsteram direction, instead of the diffracted beam. These examples indicatethe possible increasing need for BCDI with unconventional detector configurations at future beamlines. Email address: [email protected] (S. Maddali) a r X i v : . [ phy s i c s . i n s - d e t ] A ug igure 1: BCDI schematic with all the essential degrees of freedom and the relevant coordinate frames of reference. In anticipation of the increased demand for BCDI capabilities with non-traditional diffractometer configurations, thispaper addresses the issue of real-space image reconstruction from a BCDI signal distorted by an arbitrary detector tilt.Specifically, we describe (i) a Fourier transform -based forward model of far-field propagation that correctly computes thediffraction pattern from a synthetic scatterer (collected on an area detector that is tilted away from the plane perpendicularto the exit beam) and (ii) the custom adaptation of a known BCDI post- phase retrieval method of shear-correction [13] tocorrectly obtain the original distortion-free scatterer from this diffraction pattern.It is worth noting that a limited form of this geometric quantification is performed by the xrayutilities software pack-age [14]. Specifically, it addresses the case in which the detector tilt can be decomposed as two independent tilts along themutually perpendicular sampling directions dictated by the pixels. Our manuscript generalizes this treatment to an arbi-trary detector tilt relative to the perpendicular to the exit beam that need not correspond to this decomposition criterion, e.g. , when the detector face is rotated about the exit beam. Such a measurement is ddescribed in Ref. [12].As a recap of the representatice experimental schmematic [13], we refer to Fig. 1 that depicts the following features:• The scattering geometry with the incident and diffracted wave vectors k i and k f respectively, along with the scatteringangle θ B .• The orthonormal laboratory frame B lab = [ˆ s ˆ s ˆ s ] .• The orthonormal detector frame B det = [ˆ k ˆ k ˆ k ] , where ˆ k and ˆ k span the plane normal to the exit beam. Wehereafter referred to this as the ‘imaging plane’ as opposed to the ‘detector plane’, by which we mean the plane definedby the detector pixels. In conventional BCDI, the detector plane is held parallel to the imaging plane.• The rocking angle θ , in this case about ˆ s , but in general about any direction permissible by the experimental ar-rangement.• Most importantly, the discrete sampling steps in Fourier space corresponding to the detector pixels, represented asthe columns of a × matrix: B recip = [ q i q j q k ] . q i and q j are the reciprocal-space steps defined by the pixeldimensions along ˆ k and ˆ k . q k denotes the incremental migration of the 3D diffraction signal through the imagingplane by virtue of the θ rotation and is independent of the detector pixel dimensions.The azimuth and elevation angles δ and γ in Fig. 1 are specific to the configuration at the 34-ID-C end station of theAdvanced Photon Source and denote the typical parameterizations of the detector position. In BCDI the column vectors of B recip are not mutually orthogonal owing to the geometry of q k (Ref. [13] contains a derivation of this fact). However thedetector is typically arranged with its face is normal to the exit beam such that q i (cid:107) ˆ k and q j (cid:107) ˆ k , as seen in Fig. 1.Let the dimensions of the acquired data set be N × N × N , where N and N denote the pixel span of the detector and N the number of discrete steps in the rocking direction ( θ in Fig. 1). The thesis of Ref. [13] is: given that B recip = [ q i q j q k ] can be computed from the experimental geometry, the real-space sampling steps associated with the three axes of theii igure 2: Schematic of a BCDI diffraction geometry with different detector configurations: (i) the traditional mounting normal to the exit beam which isarranged by diffractometer rotation stages and a radial arm, and (ii) an unconventional wall-aligned mounting ( i.e. normal to the incident beam). The latteris easily achieved with inexpensive translation stages that move the detector parallel to the far wall of the experimental enclosure. phase-retrieved object array may be analogously expressed as the columns of another matrix B real . Generally, the mutualnon-orthogonality of the columns of B real implies a non-orthogonality associated with the axes of the 3D array obtainedfrom phase retrieval. It has been shown [13] that B real is given by: B real = B − T recip D (1)where D = N − N − N − and − T equivalently denotes the inverse of the transpose or the transpose of the inverse.The phase-retrieved array containing the real-space scatterer, combined with knowledge of the shear encoded in B real ,is sufficient for accurate, un-distorted rendering of the scatterer with one of several available software packages for 3Dvisualization. For the interested reader, the method to directly compute gradients on a grid of such sheared sample points(required to convert the scatterer’s complex phase to a spatially resolved lattice strain field) is provided in the appendixof [13].
2. A tilted detector
The shear-correcting coordinate inversion method summarized in Section 1 generalizes to any BCDI configuration pro-vided the sampling basis matrix B recip is properly parameterized according to the experimental degrees of freedom. Thedetector plane was assumed perpendicular to the exit beam, an arrangement typically ensured in BCDI by fixing the detec-tor on a radial arm, facing inwards and pointed at the mounted scatterer. In this section, we demonstrate the flexibility ofthe sampling basis formalism of Eq. (1) in addressing the general case when q i and q j are not aligned parallel to ˆ k and ˆ k respectively. As demonstrated in Ref. [12], such a situation may arise in the design of future BCDI facilities in whichdetector motion is dictated not by diffractometer rotations such as γ and δ , but relatively inexpensive translation stages,an example of which is shown in Fig. 2. Such a design would greatly simplify the design of a BCDI measurement, with theburden of correcting for the tilt-induced signal distortion being placed on numerical methods.We first consider a simplifying assumption. We assume an ideal detector with perfectly fabricated pixels which faithfullyregister an incident photon in its entirety. The pixels are not susceptible to energy redistribution due to the passage of theincident radiation through multiple adjacent pixels owing to the slanted propagation path. This undesirable feature of real-world detectors would result in a blurring effect of the acquired signal whose correction involves knowledge of the intricaciesof detector chip design. Further corrections to address this issue will be detector-specific and therefore outside the scope ofthis work.We now refer to the schematic in Fig. 3. In 3(a), we see a simplified one-dimensional detector arranged to capturethe peak of a Bragg reflection at its center, but tilted away from the imaging plane by an angle ξ . Here we denote thewavelength of the radiation as λ , the distance between the object and the center of the detector as D and the pixel pitch as p .This arrangement renders the extent of angular space queried by the detector smaller by a factor of cos ξ (note the region ofangular information ‘lost’ to the BCDI measurement). The Fourier space norm of the pixel step is no longer p/λD as shownin [13], but ( p/λD ) cos ξ , as indicated by the dashed arrow. The scattered intensity in the region of lost information does notcontribute to the acquired signal. In the case that the diffraction fringes were to have reached the detector edges, this wouldiii igure 3: (a) Difference in sampling geometry when the imaging plane of the physical area detector is placed perpendicular to the exit beam, compared towhen it is tilted with respect to this orientation. The tilt of the detector effectively results in a smaller fraction of angle space being queried. The effectivesampling step | q | of the pixel is smaller than if the detector were perpendicular to k f by a factor of cos ξ , as seen by the dashed arrow. (b) A view of asimilar scenario in two dimensions looking in the direction of the exit beam, with the ⊗ symbol denoting the direction perpendicular to the figure plane,inwards. For a general detector tilt away from the perpendicular position, the effective sampling steps of the pixels are the projections of | q i | and (cid:12)(cid:12) q j (cid:12)(cid:12) inthe plane perpendicular to k f . result in a measurement with missing higher-order Fourier components and therefore a reconstruction with necessarilylower spatial resolution. A more general treatment of this one-dimensional detector case is found in Ref. [14].We wish to generalize this idea to the case of a two-dimensional area detector arbitrarily tilted with respect to the exitbeam ( i.e. when the pixel steps q i and q j no longer lie in the imaging plane). To do this, it suffices to compute the projectionsof q i and q j in the imaging plane. Fig. 3(b) is viewed with the imaging plane (red) coincident with the plane of the figure.The exit beam k f (denoted by ⊗ ) enters the imaging plane perpendicularly. Also shown is the projection of the tilteddetector face (blue), effectively a sheared window when seen from the viewpoint of the incoming far-field diffraction. Theregion of lost information is now the shaded area in between the blue and red quadrilaterals. Because of this, the respectiveprojections { q (cid:48) i , q (cid:48) j } of the pixel steps { q i , q j } are not orthogonal in general, even though they still span the imaging plane.The projection operator P into the imaging plane is: P = I − ˆ k ˆ k T (2)where I is the × identity matrix and ˆ k is the third axis of the detector frame (along the direction of the exit beam),treated as a × column vector. With this, we can compute the effective sampling steps in the imaging plane: q (cid:48) i = P q i (3) q (cid:48) j = P q j (4)The basis vectors q i and q j in the matrix expression for B recip are respectively replaced by q (cid:48) i and q (cid:48) j from Eqs. (3) and (4).We finally note that the third sampling vector q k is not modified by the tilt of the detector, since it depends only on theBragg reflection of interest and the direction of scatterer rocking. The computation of B real then follows in the usual mannerdescribed in Eq. (1).
3. Simulation results
We now demonstrate the reconstruction of a synthetic digital object from simulated BCDI scans acquired at variousarbitrary detector tilts. For all the following simulations, we adopt the self-conjugate detector frame B det = [ˆ k ˆ k ˆ k ] instead of the laboratory frame B lab in which to render the original and reconstructed objects, as well as the Fourier-spacesignal. For simplicity, the synthetic object in question is a phase-less pyramid with a square base, with well-defined facetsand edges. The forward model devised in order to simulate the signal acquired using tilted detectors are described in detailin Appendix A, along with a summary of the various detector tilts used in the simulations. These manipulations in fact formthe theoretical basis of the back-projection method, whose detailed derivation is the subject of Ref. [15]. Here, in the interestsof highlighting the tilt-induced shear and its correction, we bypass the phase retrieval process altogether and merely obtainthe scatterer ‘reconstructions’ from the inverse FFT of the simulated wave fields, and compare their morphologies beforeiv igure 4: Demonstrations of the simulated signal distortion for different detector tilts, along with the subsequent real-space distortions in the pyramid. Ineach sub-figure, the inverse FFT of the simulated wave field is used as a substitute for the phase-retrieved object itself. (a) (i) central cross-section of thediffraction pattern, seen here in the (ˆ k , ˆ k ) plane, (ii) central cross-section of the ‘reconstructed’ pyramid obtained by IFFT, (iii) The axes of the orthonor-mal coordinate frame for each of the isosurface plots in this figure, (iv) isosurface plot of the synthetic, phase-less pyramid; (b) (i) central slice of the signalwhen the detector is tilted by ◦ about ˆ k , (ii) central slice of the ‘reconstructed’ object subsequently obtained by IFFT, (iii) isosurface plot of the pyramidwithout accounting for tilt-induced distortion, (iv) isosurface plot when tilt-induced distortion has been accounted for; (c) the corresponding images whenthe detector is tilted by ◦ about ˆ k , (d) the corresponding images when the detector is tilted by ◦ about the in-plane vector [1 / √ / T ; (e) thecorresponding images when the detector is tilted by ◦ about [1 / √ / T , followed by a ◦ rotation about ˆ k . In each 2D image, the axis unitsare in pixels while in each isosurface plot, the axis units are in nanometers. Note that the last image in each of the panels (b), (c), (d) and (e), which depictsthe shear-corrected pyramid morphology, is in agreement with the last image in panel (a), which is the ground truth. vnd after the distortions that arise from Eqs. (3) and (4) have been corrected. Under these circumstances, the inverse FFT issimply a proxy for the phase retrieval solution in the limit of infinite signal to noise ratio.Fig. 4 shows the simulated diffraction signals of the phase-less pyramid than has been arbitrarily oriented in the detectorframe, when the detector is tilted in different ways with respect to the exit beam. We note the ‘stretched’ nature of thediffraction patterns in Figs. 4(b), 4(c), 4(d) and 4(e) along various directions owing to the tilted detector, when compared tothe diffraction pattern in 4(a). For example, 4(b) shows the diffraction when the detector is tilted by ◦ about ˆ k , whichappears like the pattern in 4(a), but stretched along ˆ k . This signal is missing higher-order fringe information along thisdirection, which translates to deficiency of spatial resolution that manifests in the blurred edges of the object cross-sectionobtained by IFFT. Similarly, Fig. 4(c) shows the signal when the detector is rotated by ◦ about ˆ k , resulting in a stretchalong ˆ k and a corresponding blurring along the edges of the recovered object. Figs. 4 show the distorted diffraction signalin the case of more complicated detector tilts (see Appendix A for a full summary).In each case, the distortions in the pyramidal shape in the absence of the shear correction are shown along with theshear-corrected 3D isosurface rendering (third and fourth images in each panel respectively). The latter shows much betteragreement with the original pyramid in Fig. 4(a) in terms of object size, shape and orientation. Further, we note that theloss of higher-order Fourier-space information due to the tilt results in the ‘frayed’ edges and non-uniform amplitude crosssection in the real-space object (the second panel in each Fig. 4 sub-figure), a sign of reduced spatial information. Thestriations on the pyramid faces in panels (b), (c), (d) and (e) can also be explained by these less-than-perfect edges in theamplitude cross-sections.
4. Conclusion
We have derived a geometric correction for the morphology of a reconstructed scatterer in a BCDI measurement witha detector tilted with respect to the diffracted beam. The correction method demonstrated is seen to be a straightforwardgeneralization of the mathematial theory developed in Ref. [13]. We have successfully validated our theory by developinga customized forward model of the distorted diffraction signal acquired by a tilted detector and applying the coordinatetransform theory from Ref. [13] to recover the original object as it would have been obtained by phase retrieval.
5. Acknowledgements
The conceptualization of the tilted-detector BCDI measurement and development of the required wave propagators weresupported by the US Department of Energy (DOE), Office of Science, Basic Energy Sciences, Materials Science and Engineer-ing Division. The generalized, geometry-aware propagator theory underlying this effort was developed with the support ofthe European Research Council (European Unionfis Horizon H2020 research and innovation program grant agreement No.724881).
Appendix A. Simulating diffraction with a tilted detector
Consider a compact crystalline scatterer denoted by the complex scalar field ψ ( x ) ≡ ψ ( x, y, z ) whose coordinatesare defined in the orthonormal detector frame B det = [ˆ k ˆ k ˆ k ] . In a BCDI measurement, the squared modulus of itsFourier transform is measured slice by slice using an area detector whose Fourier-space imaging plane is displaced byinteger multiples of q k , defined by a single step along the rocking curve. If q k ≡ [ q (1) k q (2) k q (3) k ] T in the detector frame,then the n th slice of the 3D Fourier transform is given by the projection-slice theorem [16, 17]: Ψ n ( k x , k y ) (cid:124) (cid:123)(cid:122) (cid:125) slice of 3D FFT = (cid:90) R dx (cid:90) R dy e − ι π (cid:104) x (cid:16) k x + nq (1) k (cid:17) + y (cid:16) k y + nq (2) k (cid:17)(cid:105) (cid:124) (cid:123)(cid:122) (cid:125)
2D FFT (cid:90) R dz e − ι πznq (3) k (cid:124) (cid:123)(cid:122) (cid:125) projection ψ ( x, y, z ) (A.1) i.e. the n th slice of the scattered 3D wave field whose intensity is accessed by the area detector is equal to the 2D Fouriertransform of the modulated projection of the scatterer, evaluated at the 2D points ( k x + nq (1) k , k y + nq (2) k ) . The modulationin question is the phase factor e − i πznq (3) k . The expression (A.1) is evaluated numerically by means of the two-dimensionalDFT operator F and the projection operator R along the ˆ k -direction by: Ψ n ( k x , k y ) = F R e − ι π x T q k ψ ( x ) (A.2)vine may rewrite Eq. (A.1) more explicitly in terms of the two-dimensional quantities x ≡ [ x y ] T , k ≡ [ k x k y ] T and q k, ≡ [ q (1) k q (2) k ] T : Ψ( k ) = (cid:90) R dx (cid:90) R dy e − ι π x T ( k + n q k, ) (cid:90) R dz e − ι πznq (3) k ψ ( x , z ) (A.3)In order to model the tilt of the detector face, we employ the axis-angle parameterization of a rotation matrix R ( α, ˆ n ) (described in [13] Eq. (19)), acting on the columns of the × matrix [ q i q j ] . We recall that these columns represent thepixel steps in perpendicular directions along the face of the detector. The tilt is modeled by the following two rotationsapplied in order:1. A rotation R = R ( ξ, ˆ n ( φ n )) by an angle ξ about an axis ˆ n ( φ n ) ≡ [cos φ n sin φ n T in the (ˆ k , ˆ k ) imaging planefollowed by. . .2. A rotation R = R ( φ, ˆ k ) by an angle φ about the exit beam direction.In short, the pixel sampling steps (originally aligned along ˆ k and ˆ k ) are transformed due to a tilted detector in the followingmanner: (cid:2) q i q j (cid:3) detector −−−−→ is tilted R ( φ, ˆ k ) R ( ξ, ˆ n ( φ n )) (cid:2) q i q j (cid:3) = R R (cid:2) q i q j (cid:3) (A.4)The in-plane sampling vectors described in Eq. (3) and (4) are obtained by: (cid:2) q (cid:48) i q (cid:48) j (cid:3) = P R R (cid:2) q i q j (cid:3) (A.5)where P is the projection operator from Eq. (2). As mentioned in Section 2, q (cid:48) i and q (cid:48) j are no longer mutually perpendicular,even though they span the imaging plane. Of course, we ignore the extreme tilt of ξ = 90 ◦ , in which case q (cid:48) i (cid:107) q (cid:48) j and theyno longer span the imaging plane.We note in passing that we have expressed the tilt of the detector as a general rotation matrix, a quantity known torequire 3 parameters to be unambiguously specified. In our case, these parameters are: (1) the in-plane orientation φ n of the first rotation axis ˆ n ( φ n ) , (2) the angle of rotation ξ about this axis, and (3) the angle of rotation φ about the ˆ k direction. The xrayutilities library [14], on the other hand, restricts itself to a two-parameter tilt of the detector aboutmutually perpendicular directions and explicitly stops short of a full parameterization. In this sense, the formalism beingdeveloped here is more general and capable of addressing the experimental configurations beyond the scope of xrayutilities .We reiterate that there is no effect of the tilted detector on the third sampling vector q k , which is determined solelyby the manner of rotation of the scatterer (‘rocking’) during the measurement. We next define the projection operator K ≡ [ˆ k ˆ k ] T that extracts the first two components of its 3D vector operand i.e. for any detector-frame 3D vector [ x y z ] T ,we have K [ x y z ] T = [ x y ] T .We now seek the two-dimensional shearing operation that distorts the wave field in the imaging plane due to the detectortilt, in the manner described in Section 2. Put differently, we seek the × shear matrix S that satisfies the followingcondition: KP R R [ q i q j ] = SKP [ q i q j ] (A.6)which gives us the formal expression for the two-dimensional in-plane distortion operator: S = KP R R [ q i q j ] (cid:0) KP [ q i q j ] (cid:1) − (A.7)Thus, from Eq. (A.7) we are now able to determine the 2D sample points k (t)2D accessed by the tilted detector (superscript (t) stands for tilt ), in terms of the sample points k if the detector were not tilted: k (t)2D = Sk (A.8)Using Eq. (A.8) to substitute for k in Eq. (A.3), we get: Ψ n (cid:16) S − k (t)2D (cid:17) = (cid:90) R dx (cid:90) R dy e − ι π ( S − T x ) T ( k (t)2D + n Sq k, ) (cid:90) R dz e − ι πznq (3) k ψ ( x , z )= det( S ) (cid:90) R d ˜ x (cid:90) R d ˜ ye − ι π ˜ x T ( k (t)2D + n ˜ q k, ) (cid:90) R dz e − ι πznq (3) k ψ (cid:16) S T ˜ x , z (cid:17) (A.9)where the final expression above is obtained with the change of integration variables S − T x ≡ ˜ x = [˜ x ˜ y ] T and inaddition, the substitution Sq k, ≡ ˜ q k, .We note that up to the multiplicative term det( S ) , the expression (A.9) is completely analogous to Eq. (A.3), whoseoperator version is Eq. (A.2). Eq. (A.9) tells us that the far-field coherent diffraction can in fact be simulated on an arbitrarilytilted detector in a computationally efficient manner using Eq. (A.2), provided the following conditions are satisfied:vii. The signal sampling shear S in the imaging plane is computed using Eq. (A.7).2. The real-space orthogonal grid x of the synthetic scatterer is sheared in its first two dimensions by: x −→ S − T x .3. The Fourier-space incremental step q k due to sample rocking is sheared in its first two dimensions by: q k, −→ Sq k, .4. The scatterer ψ is re-sampled in its first two dimensions by: ψ ( x , z ) −→ ψ (cid:16) S − T x , z (cid:17) This is the method adopted to obtain the 3D wave fields and subsequently the diffraction patterns incident upon a tilteddetector. The parameters θ , φ and φ n used to generate the rotation matrices R ( ξ, ˆ n ( φ n )) and R ( φ, ˆ k ) for the simulated ξ ( ◦ ) φ n ( ◦ ) φ ( ◦ ) Sub-figure in Fig. 460 0 0 (b)
60 90 0 (c)
60 60 0 (d)
60 60 73 (e)
Table A.1: Various tilt parameters for the results shown in Fig. 4. detector tilts in Fig. 4 are listed in Table A.1
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