Keywords: Bragg coherent X-ray diffraction imaging, electron backscatter diffraction, strain calculation, phase interpolation, crystal orientation

2.1. Microcrystal fabrication

Samples were produced by sputter deposition of a thin film onto a single-crystal sapphire wafer (C-plane orientation). One substrate with a film thickness of 375 nm was produced for the Fe–Ni microcrystals. It was dewetted in a vacuum furnace purged with a gas mixture of 5% hydrogen, balance argon, at 1523 K for 24 h. The resulting crystals exhibit a face-centred cubic (f.c.c.) structure, range from 0.5 to 1.5 µm in size [Fig. 1 ▸(a)] and adhere to the substrate surface. The substrate was cleaved to make substrates 1 and 2, both containing Fe–Ni microcrystals. Substrate 3 contained Co–Fe microcrystals that were produced in a similar way. The procedure and details for substrate 3 can be found elsewhere (Yanget al., 2022 ▸).

Each substrate was coated with 10 nm of amorphous carbon via thermal evaporation using a Leica ACE600 coater to assist with scanning electron microscopy (SEM) imaging. To facilitate reliable measurement of multiple reflections from a specific microcrystal, a ZEISS NVision 40 Ga focused ion beam (FIB) instrument was used to remove the surrounding crystals within a 40 µm radius using currents from 6 nA to 150 pA and an acceleration voltage of 30 kV. Only SEM imaging was used to position the FIB milling scans to prevent large lattice strains caused by FIB imaging (Hofmannet al., 2017b ▸). The isolated crystals on each substrate are shown in Fig. 2 ▸. Crystal 1B [Fig. 1 ▸(b)] was used for the computation of the strain and rotation tensors (Fig. 7).

Energy-dispersive X-ray spectroscopy (EDX) was used to determine the elemental composition of each crystal [Fig. 1 ▸(c)]. EDX showed a homogeneous distribution of all elements throughout the dewetted crystals (Fig. 3 ▸). EDX was performed on a ZEISS Merlin instrument using an Xmax 150 detector (Oxford Instruments) with an elliptical region encapsulating crystal 2B on the substrate for 16 s with an accelerating voltage of 10 kV.

2.2. Electron backscatter diffraction

Crystal orientation was determined by EBSD using a ZEISS Merlin instrument equipped with a Bruker Quantax EBSD system and a Bruker eFlash detector tilted at 4°. Electron backscatter patterns (EBSPs) were recorded with the sample tilted at 70° (Fig. 4 ▸) using an accelerating voltage of 30 kV and a current of 15 nA. The EBSPs were 800 × 600 pixels and a step size of 19.8 nm was used between consecutive points on the sample. The diffraction patterns were indexed and the Euler angles extracted for each pattern using the BrukerESPRIT 2.1 EBSD software. The Euler angles were exported and analysed usingMTEX, a MATLAB toolbox for texture analysis (Bachmannet al., 2011 ▸), to produce inverse pole figure (IPF) maps for all crystals (Fig. 2 ▸).

Here we use the Bunge convention (Bunge, 1982 ▸) to describe each crystal orientation (crystal frame) relative to the substrate (sample frame). The crystal orientation matrixUB is composed ofU, which describes the rotation of the crystal reference frame, andB [equation (2)], which characterizes the unit-cell parameters.

Using the same convention as Brittonet al. (2016 ▸), the unit cell has vectorsa,b andc with lengthsa,b andc, respectively. Angle α describes the angle betweenb andc, β the angle betweenc anda, and γ the angle betweena andb.B is used to transform the base vectors to Cartesian base vectors:

where

Since all crystals in this study have an f.c.c. structure,B is the 3 × 3 identity matrix multiplied by the lattice constant.

UB provides the direction and radial position of specifichkl reflections,H_hkl, in laboratory coordinates (Busing & Levy, 1967 ▸),

Several different coordinate frames are used in EBSD that refer to different aspects of the measurement (Fig. 4 ▸). In this paper, all coordinate systems and rotation matrices will be right handed and we will use the same notation as Brittonet al. (2016 ▸). For EBSD, the following subscripts describe specific coordinate systems:

(i) ‘d’ is the detector frame that describes the EBSPs.

(ii) ‘s’ is the EBSD sample frame that is related to the detector frame by sample (θ_sample) and detector (θ_detector) tilts about thex axis. Thex_s andy_s axes correspond to the directions of EBSD scan points.

(iii) ‘m’ is the SEM map frame. This corresponds to how the EBSPs overlay on the SEM maps and thus how the EBSD orientation is referenced.

The EBSD software returns the orientation matrix from EBSD measurements in the SEM map frame (Fig. 4 ▸). The corresponding orientation matrix is referred to asU_EBSD, m. It can be constructed from a series of rotations using Euler angles, where each angle describes a rotation about a coordinate axis. Here we use right-handed rotation matrices to describe vector rotations about thex axis,

and thez axis,

and Bunge-convention Euler angles, ϕ₁, Φ and ϕ₂ (Brittonet al., 2016 ▸). Equivalently, equations (5) and (6) correspond to left-handed rotations through angle θ for the coordinate system. To transform vectors from the crystal coordinate frame to the laboratory frame, we use (Brittonet al., 2016 ▸)

We note that the Euler angles in equation (7) are negative because the original angles as defined are for left-handed rotation matrices. First a rotation of −ϕ₁ is applied about the originalz axis, followed by a rotation of −Φ about the newx axis and finally a rotation of −ϕ₂ about the newz axis. For consistency with the convention used here (Busing & Levy, 1967 ▸), we express equation (7) as

Here, the EBSD software already accounts for the instrument tilts θ_sample and θ_detector and returns Euler angles in the SEM map frame (subscript m) (Fig. 4 ▸).

To input the orientation matrix into thespec orientation calculator on beamline 34-ID-C, we must define twohkl reflections corresponding to out-of-plane [equation (13)] and in-plane [equation (14)] reflections (Hofmannet al., 2017a ▸). One concern is the consistency in the indexing of crystals on the 34-ID-E Laue instrument and in EBSD measurements. Due to the cubic structure of the present crystals, there are equivalent orientation matrices that differ by 90° rotations about the crystal axes. These rotations are accounted for using anR_crystal rotation matrix that captures rotations by 90(n_x, y, z)°, wheren_x, y, z ∈ {−1, 0, 1, 2}, about thex,y andz axes,

To align the EBSD map frame to the BCDI laboratory frame, a −90° rotation about thex axis is required [Fig. 5 ▸(a)]. Combining this with equations (8) and (9) leads to the formation of theUB_34C, EBSD matrix,

These matrix-based orientation operators provide a generalized approach to the transformation of orientation, irrespective of the software implementation.

2.3. Microbeam Laue X-ray diffraction

Microbeam Laue diffraction was used to verify the lattice orientation of each crystal independently. This was performed on beamline 34-ID-E at the APS. Further details about the instrument can be found elsewhere (Liuet al., 2004 ▸; Hofmannet al., 2017a ▸).

The sample was positioned with its surface inclined at a 45° angle to the incident beam [see Fig. 5 ▸(b)] and diffraction patterns were recorded using a Perkin–Elmer flat-panel detector above the sample. 2D fluorescence measurements of the FeKα₁ peak (6.40 keV) were used to identify the spatial position of the crystals, using a monochromatic 17 keV (Δλ/λ ≃ 10⁻⁴) X-ray beam focused to 0.25 × 0.25 µm (horizontal × vertical) using Kirkpatrick–Baez (KB) mirrors.

Next, a polychromatic X-ray beam was used to collect a Laue diffraction pattern of each crystal (Fig. 6 ▸). The pattern shows weak Bragg reflections from the microcrystals and strong Bragg peaks from the single-crystal sapphire substrate. The two sets of peaks were indexed and fitted using theLaueGo software (https://www.aps.anl.gov/Science/Scientific-Software/LaueGo). From the indexing, we could determine theUB matrix [equation (4)]. TheUB matrix determined by Laue diffraction on the 34-ID-E instrument is referred to asUB_Laue,

wherea*,b* andc* are the column (represented by vertical lines) reciprocal-space vectors returned byLaueGo in units of nm⁻¹ in the 34-ID-E laboratory frame.

To convertUB_Laue into aUB matrix for use on the BCDI instrument 34-ID-C,UB_34C, Laue, the 45° rotation of the sample in the Laue laboratory frame must be accounted for [Fig. 5 ▸(b)]. To align the microbeam Laue and BCDI laboratory frames, a rotation of the sample by 45° about thex axis is required (Hofmannet al., 2017a ▸), leading to

2.4. Bragg coherent X-ray diffraction imaging

BCDI was performed on beamline 34-ID-C at the APS. Anin situ confocal microscope was used to position the microcrystal within the X-ray beam (Beitraet al., 2010 ▸). The sample was illuminated using a 9 keV (λ = 0.138 nm) coherent X-ray beam, with a bandwidth of Δλ/λ ≃ 10⁻⁴, from an Si(111) monochromator. The X-ray beam was focused to a size of 1.1 × 1.1 µm (horizontal × vertical, FWHM) using KB mirrors. Beam-defining slits were used to select the coherent portion of the beam at the entrance to the KB mirrors. For beamline 34-ID-C, the transverse coherence length is ξ_h > 10 µm and the longitudinal coherence length is ξ_w ≃ 0.7 µm at a photon energy of 9 keV (Leakeet al., 2009 ▸).

The sample needs to be positioned such that a specifichkl Bragg diffraction condition is met to produce a diffraction pattern in the far-field Fraunhofer regime. The orientation matrix determined by EBSD or Laue diffraction is communicated to thespec software used on 34-ID-C by defining two reflections that correspond tohkl values associated with laboratoryx (in-plane) andy (out-of-plane) directions [note the angles referred to here are set elsewhere (Hofmannet al., 2017a ▸)]:

(i) The primary reflection (out-of-plane,y direction),H_⊥, where the instrument angles are set to δ_spec = 0°, γ_spec = 20°, θ_spec = 0°, χ_spec = 90° and ϕ_spec = −10°. The fractional is then

(ii) The secondary reflection (in-plane,x direction),H_∥, where the instrument angles were set to δ_spec = 20°, γ_spec = 0°, θ_spec = 10°, χ_spec = 90° and ϕ_spec = 0°. The fractional is then

HereUB refers toUB_34C, EBSD orUB_34C, Laue. These two fractionalhkl vectors are then entered intospec as known reflections. On this basis, the expected angular positions of the {111} and {200} reflections from the sample were calculated. Not all {111} and {200} reflections could be measured as some may exceed the angular range of the sample and detector motors. Occasionally, manual motor adjustments of a few degrees were required to locate the Bragg peak. Once a Bragg peak was found, the sample tilt and positioning were refined such that the centre of mass was on the centre of the detector positioned 0.5 m away from the sample. At this aligned position, the angular and translational positions were saved for each Bragg peak and used to determine the true beamline orientation matrixUB_34C by minimizing the least-squares error associated with the measured reflections,

where are the Miller indices in crystal coordinates.

Differences between the true and predicted positions of the reflections usingUB_34C arise from a number of different sources. The largest error is the repeatability of the sample position in different coordinate systems. The use of Thorlabs 1X1 kinematic mounts, which have angular errors of less than a millidegree, helps with the precise re-mounting of samples. There is also uncertainty in the goniometer precision and alignment with the diffractometer, which can influence the angle readout. Furthermore, the centre of the detector may not be perfectly aligned to the calculated position for a given detector distance or angle. The position of the measured Bragg peak is limited by the energy resolution of the incident X-ray as it affects the Bragg angle.

CXDPs were collected on a 256 × 256 pixel module of a 512 × 512 pixel Timepix area detector (Amsterdam Scientific Instruments) with a GaAs sensor and a pixel size of 55 × 55 µm positioned 1.0 m from the sample to ensure oversampling. CXDPs were recorded by rotating the crystal through an angular range of 0.6° and recording an image every 0.005° with 0.1 s exposure time and 50 accumulations at each angle.

To optimize the signal-to-noise ratio and increase the dynamic range of the CXDPs, three repeated scans for each of the 111,, 200, and 002 reflections were performed and aligned to maximize their cross correlation. Once aligned, the minimum acceptable Pearson cross correlation for summation of CXDPs from a specific Bragg reflection was chosen to be 0.976, similarly to previous BCDI studies (Hofmannet al., 2018 ▸, 2020 ▸). CXDPs were corrected for dead time, dark field and white field prior to cross-correlation alignment. Details regarding the recovery of the real-space images using phase retrieval algorithms are given in AppendixA and the computation of the strain is given in AppendixB.

2.5. Sample mounting

For the SEM, EDX and EBSD analyses, samples were mounted on 12.5 mm diameter SEM specimen pin stubs using silver paint. For microbeam Laue X-ray diffraction, they were attached to a Thorlabs 1X1 kinematic mount. From here, a Thorlabs kinematic mount adapter between 34-ID-E and 34-ID-C was used to mount the samples for BCDI. This adapter consists of two 1X1 mounts sandwiched together to enable sample orientation to be well preserved between the beamlines. There is no kinematic mount adapter between the SEM and BCDI instruments. Moreover, the use of magnets in the kinematic mounts inhibits their use for electron microscopy. Across the different instruments, the sample orientation was maintained throughout as shown in Fig. 5 ▸, which has an arbitrary sample feature to illustrate the respective orientations.

3.1. Orientation matrix comparison

The angular mismatch between twoUB matrices,UB₁ andUB₂, can be determined by convertingUB₁(UB₂)⁻¹ into a rotation vector. A rotation matrixR can be converted using Rodrigues’ rotation formula in matrix exponential form,

wherew_m is an antisymmetric matrix,

which contains the elements of the rotation vector. The rotation vector is defined by a rotation axis multiplied by a rotation θ. If the two orientation matrices are different,UB₁(UB₂)⁻¹ can be converted to a rotation vector where the angular mismatch is θ. IfUB₁ =UB₂, thenUB₁(UB₂)⁻¹ =I₃ and therefore θ = 0.

Here we setUB₁ andUB₂ asUB_34C, EBSD,UB_34C, Laue orUB_34C. First we rearrange equation (16) to calculatew_m:

where here refers to the matrix natural logarithm. Next we reconstruct the rotation vector using equation (17) and calculate its magnitude to obtain the mismatch,

To calculate the angular mismatch betweenUB_34C, EBSD and other orientation matrices, the permutation ofn_x, y, z that produced the smallest error was chosen. Tables 1 ▸–3 ▸ ▸ show the angular mismatch ofUB_34C, Laue andUB_34C, EBSD compared withUB_34C for substrates 1–3. The average angular mismatch for all crystals between orientation matricesUB_34C, Laue andUB_34C is 4.48°, that betweenUB_34C, EBSD andUB_34C is 6.09°, and that betweenUB_34C, Laue andUB_34C, EBSD is 1.95°. Generally,UB_34C, Laue is most similar toUB_34C. This is expected, as there is a Thorlabs kinematic mount adapter between 34-ID-E and 34-ID-C for the precise angular alignment of samples. A larger difference is observed when comparingUB_34C, EBSD andUB_34C. This is due to the manual removal of the SEM pin stub from the electron microscope, which then needs to be re-secured to the kinematic mount. The cylindrical pin permits a greater degree of rotational freedom, thereby increasing the angular mismatch whenUB_34C, EBSD is considered. Despite this increased angular freedom in the pin, a 2° increase in the angular uncertainty when usingUB_34C, EBSD instead ofUB_34C, Laue is still a very accurate result. This means only a slightly larger angular range needs to be explored in alignment.

The alignment of crystals for BCDI using EBSD remains much more time efficient because microbeam Laue diffraction pre-alignment on 34-ID-E is no longer required. The use of EBSD for pre-alignment of BCDI samples also affords greater flexibility in experiment type. The ability to pre-characterize samples offsite should substantially increase throughput and make MBCDI a much more widely accessible technique, especially on beamlines without pink-beam capability or access to a nearby Laue instrument.

3.2. Determination of strain

MBCDI allows the strain and rotation tensors to be computed if at least three reflections are measured, thus providing more information about the crystal defects present (Hofmannet al., 2017b ▸, 2018 ▸, 2020 ▸; Phillipset al., 2020 ▸). Fig. 7 ▸ shows the strain and rotation tensors reconstructed from five measured Bragg reflections of crystal 1B. The ε_xx, ε_yy and ε_zz slices show defects close to the edge that may not be resolved if analysing a single reflection alone (Yanget al., 2021 ▸), as some crystal defects, such as dislocations, are visible only whenQ_hkl · b ≠ 0, whereb is the Burgers vector (Williams & Carter, 2009 ▸).

These results were produced using a refined method for the computation of the strain and rotation tensor. A general approach for the computation of both these tensors is described in AppendixB. This relies on the recovery of the phase of the CXDP. The intensity of the CXDP is the squared magnitude of the Fourier transform of the complex crystal electron densityf. The solution for the recovered phase is non-unique, as global phase offsetsC can produce the same CXDP,i.e.. After phase retrieval, the phase values are bound between [−π, π], which describes the periodic nature of the crystal structure but not necessarily the true complex crystal electron density. For instance, if the projected displacement in the direction ofQ_hkl is greater than π/|Q_hkl|, then a phase jump, where the phase difference is 2π between two pixels, will occur. These phase jumps cause discontinuities in the derivatives of the phase ∂ψ_hkl(r)/∂j, wherej corresponds to the spatialx,y orz coordinate, leading to spurious large strains. Typically, phase unwrapping algorithms can be used to remove phase jumps, but dislocations have characteristic phase vortices (Clarket al., 2015 ▸) that end at dislocation lines, meaning that phase jumps associated with dislocations cannot be unwrapped.

To account for this, Hofmannet al. (2020 ▸) demonstrated an approach that involves producing two additional copies of the phase with phase offsets of and, respectively. This shifts the phase jumps to different locations and, by choosing the phase gradient with the smallest magnitude for each voxel, the correct phase derivatives can be found. Here, we employ a more efficient method used in coherent X-ray diffraction tomography following Guizar-Sicairoset al. (2011 ▸), which has also been applied in the Bragg geometry using ptychography (Liet al., 2021 ▸). Rather than making multiple copies of the phase, we take the derivative of the complex exponential of the phase and determine the phase gradient using the chain rule:

Here is a circle expressed using Euler’s formula, where the phase jumps disappear. Fig. 8 ▸ shows the difference between the two methods for computing the lattice rotation and strain tensors. We can see that the procedure based on phase offsets (Hofmannet al., 2020 ▸) fails to resolve the details fully in the regions with high strain,i.e. around the edges and central region of missing intensity. The new approach of computing phase gradients successfully deals with these complex regions.

Furthermore, we apply this to the interpolation of the phase from detector conjugated space to sample space, by interpolating the complex quantity instead of ψ_hkl. This avoids the blurring of phase jumps that occurs during direct interpolation of ψ_hkl, shown in Fig. 9 ▸.

This refinement of strain and rotation tensor computation allows for a more accurate reconstruction of crystal defects and their associated nanoscale lattice strains. It also reduces the time required to compute the tensors, since phase offsets need to be applied before and after mapping the crystal from detector conjugated space to orthogonal sample space. This will play an important role in the analysis of large MBCDI data sets, such as those obtained fromin situ oroperando experiments that reveal crystal defects interacting with their environment. Reconstruction accuracy in multi-Bragg-peak phase retrieval algorithms that involve a shared displacement field constraint applied to all reflections would also be improved by implementing this more accurate interpolation of phase values (Newton, 2020 ▸; Gaoet al., 2021 ▸; Wilkinet al., 2021 ▸).

Supplementary video SV1. DOI:10.1107/S1600576722007646/te5099sup1.mp4

Supplementary video SV2. DOI:10.1107/S1600576722007646/te5099sup2.mp4

Supplementary video SV3. DOI:10.1107/S1600576722007646/te5099sup3.mp4

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Supplementary Materials

Data Availability Statement

The processed diffraction patterns, final reconstructions and data analysis scripts, including a script to compute the orientation matrix using EBSD, are publicly available athttps://doi.org/10.5281/zenodo.6383408.

Also available are three supplementary videos, as follows:

(i) SV1 showsxy plane slices through the lattice strain and rotation tensors in Fig. 7 ▸.

(ii) SV2 showsyz plane slices through the lattice strain and rotation tensors in Fig. 7 ▸.

(iii) SV3 showszx plane slices through the lattice strain and rotation tensors in Fig. 7 ▸.