research papers
Dynamic crystal rotation resolved by highspeed synchrotron Xray Laue diffraction
^{a}The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610031, People's Republic of China, ^{b}Key Laboratory of Advanced Technologies of Materials, Ministry of Education, Southwest Jiaotong University, Chengdu, Sichuan 610031, People's Republic of China, and ^{c}Advanced Photon Source, Argonne National Laboratory, Argonne, IL 60439, USA
^{*}Correspondence email: sluo@pims.ac.cn
Dynamic compression experiments are performed on singlecrystal Si under split Hopkinson pressure bar loading, together with simultaneous highspeed (250–350 ns resolution) synchrotron Xray Laue diffraction and phasecontrast imaging. A methodology is presented which determines crystal rotation parameters, i.e. instantaneous rotation axes and angles, from two unindexed Laue diffraction spots. Twodimensional translation is obtained from dynamic imaging by a single camera. Highspeed motion of crystals, including translation and rotation, can be tracked in real time via simultaneous imaging and diffraction.
Keywords: synchrotron Xray Laue diffraction; crystal rotation.
1. Introduction
Grain or crystal rotation is a key mechanism of accommodating elastic or plastic deformation in crystalline solids (Chen et al., 2013). In quasistatic experiments, crystal rotation is normally resolved with electron backscattering diffraction (Zaafarani et al., 2008) or Xray Laue diffraction (Maaß et al., 2007, 2008). Singleshot multiframe in situ realtime measurements on crystal rotation during dynamic loading have been an experimental challenge, especially for high strain rate loading in the nano to microsecond regimes with a split Hopkinson pressure bar (SHPB) (Kolsky, 1949) or a gas gun (Fowles et al., 1970). In powder technology and fluid mechanics, rotation of particles or granules plays a significant role in the gas–solid interactions, including twophase flows, heat transfer and coal combustion (Kajishima, 2004; Goldschmidt et al., 2004; Sun & Battaglia, 2006). Detection of highspeed particle rotation is in urgent need. For example, Wu et al. (2008) used optical imaging to measure particle rotation at about 500 revolutions per second (r s^{−1}) in a circulating Xray diffraction is a useful complement for rotation measurements given its penetration capability and higher resolution. Xray diffraction was used to study the quasistatic packing and deformation characteristics of sands (Hall et al., 2011). Other applications may involve measuring remotely translation and rotation of a fastmoving object.
Synchrotron Xray Laue diffraction possesses unique advantages for latticelevel measurements under dynamic loading and has been under intensive development along with synchrotron Xray imaging (Luo et al., 2012; Hudspeth et al., 2013; Lambert et al., 2014; Turneaure et al., 2009; Wei et al., 2004) for studying highstrainrate deformation, fracture and phase change of a variety of materials. In particular, a realtime in situ simultaneous Xray imaging and diffraction technique has been demonstrated recently at the Advanced Photon Source (Fan et al., 2014; Hudspeth et al., 2015). In this work, we choose singlecrystal Si, brittle and of high modulus, for dynamic loading with a SHPB, and perform simultaneous Xray imaging and Laue diffraction, in order to establish an illustrative case for quantitative rotation and translation measurements. A theoretical procedure is introduced to resolve threedimensional highspeed rotation of a single crystal with two unindexed diffraction spots. Threedimensional crystal rotation at approximately 300 r s^{−1} is resolved. Highspeed motion of crystals, including twodimensional translation and threedimensional rotation, can be tracked in real time from simultaneous imaging and diffraction.
2. Methodology
The experimental methodology for simultaneous synchrotron Xray imaging and diffraction has been established at beamline 32IDB of the Advanced Photon Source (Fan et al., 2014; Hudspeth et al., 2015). The schematic setup for simultaneous diffraction and phase contrast imaging is shown in Fig. 1, along with the coordinate system. We perform highstrainrate loading on a brittle Si singlecrystal with SHPB. Prior to loading, the crystallographic directions [01], [2] and [111] are oriented approximately along the x, y and z directions, respectively. Dimensions of the sample perpendicular to the Xray direction () are 3 mm × 3 mm and the thickness along the Xray direction is 1 mm. After the gas gun of the SHPB is fired, a compression pulse is imposed on the sample. A tensile pulse ensues upon reflection between the bar–sample interface, and the incident bar is then separated from the sample. The sample retains its overall integrity and is subjected to rigidbody motion, including translation and rotation, when a stress is below the fracture strength of Si (∼1 GPa).
We perform simultaneous Xray imaging and diffraction measurements on the sample. The probe Xrays are from an APS `undulator A' light source and the undulator gap is 25 mm. The spectral flux–photon energy curves of the undulator source are presented elsewhere (Luo et al., 2012). The probe Xrays () illuminate the sample perpendicular to the loading direction. The transmitted Xrays are collected by an imaging scintillator, Cedoped Lu_{3}Al_{5}O_{12} (LuAG:Ce), and recorded with a Photron FastCam SAZ (). The exposure time is 0.35 µs and the frame interval is 10 µs. At the same time, the scattered light is collected by a diffraction scintillator, Cedoped Lu_{2–2x}Y_{2x}SiO_{5} (LYSO:Ce), and is recorded with another SAZ after passing a microchannel plate (MCP, Quantum Leap E, Stanford Computer Optics, Inc.), synchronized with the imaging camera. The exposure time is 0.25 µs and the frame interval is 10 µs. The number of activated pixels is 384×408. The diffraction camera is mounted on a rotation stage for continuous rotation in the circumferential direction. A Huber 410 goniometer is used to control the rotation stage with a resolution of 0.001°. The scintillator, MCP and SAZ together are equivalent to an area detector (). The pixel size is calibrated to be 60 µm × 60 µm for the current geometry. The sampletodiffraction detector distance (d) is 230 mm and the angle between the incident beam and the normal of the diffraction detector plane is 24.5°.
Each Laue diffraction spot corresponds to et al., 2015; Liu et al., 2014; Wang et al., 2011; Barabash et al., 2010) and grain rotation (Maaß et al., 2007, 2008); there are many dedicated software packages available for this analysis (Tamura, 2014; Micha & Robach, 2014; Huang, 2010). A method based on digital image correlation using Laue spots as speckles has been developed recently for strain analysis (Borbély, 2015). However, the effective area of detectors available for dynamic events is very limited and the sample–detector distance has to be sufficiently large in order to accommodate the imaging camera as well as protective components for impact loading. As a result, the number of recorded diffraction spots is small (e.g. 1 or 2). Meanwhile, a conventional Hopkinson bar setup is not intended for highaccuracy alignment as required for Laue diffraction indexing. So indexing the spots and orienting a crystal are currently impractical, and the aforementioned Laue analysis tools are not applicable. In our experiments, the sample moves stressfree, so the remaining strain after lowspeed impact is small and its contribution to diffraction spot movement is negligible compared with rotation. Therefore, we establish a quantitative relationship between threedimensional crystal rotation and twodimensional movement of two or more diffraction spots, which is elaborated in Appendix A. In this method, neither crystal orientation nor indices of diffraction spots are needed to obtain rotation parameters (rotation axes and rotation angles).
planes with certain spatial orientation. deformation and crystal rotation can both contribute to its shift. Laue diffraction pattern analysis has been widely applied to obtain strain tensor (Li3. Results and discussion
The dynamic loading experiment was conducted at a projectile velocity of 20 m s^{−1}. Results from simultaneous imaging and diffraction are presented in Figs. 2 and 3.
During recoil, the incident bar separates from the Si sample at 1 m s^{−1}. The sample remains essentially intact except that minor fragments form from stress concentrations during impact and flies in free rigidbody motion along the loading direction (Fig. 2). The twodimensional phasecontrast images supply information on planar translation and rotation of the sample, but not on outofplane motion. The translation velocities are approximately 5 m s^{−1} and 1 m s^{−1} in the x and y directions, respectively, and the inplane rotation speed is small.
Eight representative frames of diffraction patterns containing two diffraction spots are chosen for analysis (Fig. 3, columns 1 and 3). Two diffraction spots shift simultaneously from the lower right to upper left of the screen, without change in spot shape. This shift is attributed to sample or crystal rotation, rather than compression or tension, because the sample has already been subjected to unloading and detached from the incident bar during the time window of interest here. The minute residual deformation (Noyan & Cohen, 2013) cannot induce such a pronounced shift. Note that there is only one diffraction spot recorded prior to frame 1 (Fig. 3). As the sample undergoes rotation during impact (before frame 1), indexing the spots before and after frame 1 is not possible even though its initial orientation is known. However, a quantitative analysis of crystal rotation can be conducted from two Laue spots with the scheme detailed in Appendix A.
As each harmonic of the undulator radiation has a finite bandwidth (∼2% for the APS undulator A light source) (Fan et al., 2014), some diffraction spots do not vanish from the field of view during rotation, within the time window of observation. As discussed above, relative translation and rotation of the sample between adjacent frames on the Oxy plane is tiny due to short frame intervals (Fig. 3), so the smallangle assumption in equation (8) of Appendix A is reasonable.
Each pixel on the diffraction detector is mapped onto the –γ plane. Given the finite size of a diffraction spot, we use Gaussian fitting to obtain the coordinates of its center (, γ). The diffraction spots are plotted on the –γ plane in columns 2 and 4 of Fig. 3. Then, the relative rotation axes (expressed in three projection quantities) and the rotation angles between adjacent frames are calculated (Table 1). The rotation directions during the free motion of the sample appear to be random. The rotation angles are extremely small between two neighboring frames (0.05–0.13°), consistent with the smallangle assumption. The sample mainly undergoes translation along the loading (x) direction with concurrent, small and irregular rotation. The high angular resolution of dynamic Laue diffraction is advantageous for such applications as determining grain rotation in a polycrystalline solid.

As a selfconsistency check, we conduct forward calculations using the rotation parameters in Table 1 to compute the diffraction spot positions on the diffraction detector. Given the rotation axes and angles as well as the initial positions of diffraction spots, their positions at any instant can be obtained as elaborated in Appendix A. The simulated diffraction spot positions are listed in Table 2 and plotted as empty circles in columns 1 and 3 of Fig. 3. The comparison shows a complete coincidence between the simulations and the measurement.

The twodimensional translation and full rotation parameters as a function of time (Table 1) are illustrated in Fig. 4, using a schematic object. Therefore, rigidbody motion, including translation and threedimensional rotation, of the sample can be strictly mapped through simultaneous twodimensional Xray imaging and diffraction. Translation mapping can be extended into three dimensions if an extra camera is used at a different view angle. Simultaneous highspeed tracking of translation and rotation in three dimensions offers great potential in a multitude of applications.
We present the simplest case in the above discussion. For strong shocks, both
strain and grain rotation contribute to the movement of a diffraction spot. Strain can be evaluated from conventional methods such as known equation of state, and it is then decoupled from rotation to a certain extent. However, more diffraction spots are certainly desirable.4. Conclusion
We have performed highspeed simultaneous synchrotron Xray Laue diffraction and phasecontrast imaging measurements on singlecrystal Si under SHPB loading, and developed a methodology for quantifying rotation parameters, i.e. rotation axis and rotation angles. The exposure time is 250–350 ns and the frame interval is 10 µs. Highspeed crystal rotation at about 300 r s^{−1} is resolved. Highspeed motion of crystals, including twodimensional translation and threedimensional rotation, can be tracked in real time from simultaneous imaging and diffraction.
APPENDIX A
Determining rotation parameters from two Laue diffraction spots
The geometry of diffraction is defined in Figs. 5(a) and 5(b). Incident Xrays () are scattered by planes () and the scattered Xrays () yield a diffraction spot () on a diffraction detector () located at an arbitrary position. The origin, O, is the transmission spot on the detector plane of the incident Xray beam propagating along the −z direction. For convenience, we introduce the virtual detector plane containing O, i.e. the Oxy plane, perpendicular to the incident Xray beam. A diffraction spot () on the detector is mapped onto the virtual detector plane or vice versa. Discussions refer to the virtual detector plane () unless stated otherwise.
The diffraction angle and . The azimuthal angle of a diffraction spot, γ, is the angle formed between and the x axis. The distance between the sample and the origin is . In terms of and the mapped coordinates (x,y) of a diffraction spot:
The unit vector n is the normal of certain parallel planes. The angle between n and is defined as η, and . Therefore,
n is expressed in terms of γ and θ obtained from twodimensional diffraction patterns, so the instantaneous orientation of the planes is determined.
The rotation axis and rotation angle are chosen to describe sample rotation, and are denoted by a unit vector and α, respectively. Here, superscript T denotes the transpose. The corresponding rotation matrix can be expressed in terms of u and α as
After sample rotation, instantaneous normal vector at time t_{1} changes to = (n_{x2},n_{y2},n_{z2})^{T} at t_{2}. The rotation matrix, , relates and via
The objective is to solve and α from equation (7), which fully characterizes the rigidbody rotation of the sample. This equation can be solved numerically. For small rotation angles, it can be simplified with and . Equation (7) is then reduced to
Subtracting on both sides of equation (8) yields
where , and .
For convenience, the Euler vector E is defined as
It follows that
There are three unknowns, E_{x}, E_{y}, E_{z}. But these three equations are linearly dependent or contradictory. Thus, in order to determine , another diffraction spot is needed. Given two diffraction spots from different sets of planes, this problem becomes overdetermined (six equations versus three unknowns) and the rotation parameters can be fully solved as
Note that even more diffraction spots can be used and the uncertainties are reduced.
For each diffraction spot, a set of and thus a set of (n_{x},n_{y},n_{z}) are obtained. Two independent spots yield two sets of (n_{x},n_{y},n_{z}). The rotation parameters, α and u, are solved from four diffraction spots for frames 1 and 2. Repeating such a process between two adjacent frames yields the rotation history of the sample. A MATLAB code is developed to perform such analysis in a batch mode.
To validate the above derivations, we perform forward analysis, i.e. calculate the diffraction spots on the detector from rotation parameters α and u obtained from the diffraction spots of frames 1 and 2 as described above. is calculated from the initial positions of a diffraction spot at frame 1. After rotation, changes to , which is calculated with the Rodrigues' rotation formula (Rodrigues, 1840)
Then,
On the virtual detector plane,
The simulated diffraction spot (x_{2},y_{2}) is then mapped onto the detector and should coincide with the directly measured spot at frame 2.
Acknowledgements
The MATLAB program developed in this work has benefited from the HisPoD diffraction simulation code. We are grateful to J. Wang at APS, B. X. Bie, D. Fan and L. Lu at PIMS for their help with the experiments. This work was partially supported by the 973 project (2014CB845904) and NSF (11472253) of China. Use of the Advanced Photon Source, an Office of Science User Facility operated for the US Department of Energy (DOE) Office of Science by Argonne National Laboratory, was supported by the US DOE under contract No. DEAC0206CH11357.
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