Ultrafast X-ray diffraction measurements of shock-compressed iron and iron alloys


 The extreme pressures achievable with dynamic compression holds great promise for studying planetary interiors. Phase stability of Fe-Si alloys, which are complex to address, is particularly relevant to understanding telluric planetary cores due to the widely varying properties produced by small changes in Si concentration. Here we report the study of phase stability of pure iron and Fe-Si alloys by x-ray diffraction measurements carried out on shocked samples using an x-ray free electron laser (XFEL). Our setup combined with the brilliance of the XFEL allows us to observe the rapid onset of high-pressure solid-solid phase transformation in Fe and Fe-Si8.5wt%; we observe no such evidence in Fe-Si16wt% up to 110 GPa on the nanosecond timescale. Density Functional Theory calculations provide the conceptual framework to rationalize these observations. Taken together our experiments and calculations support recent dynamic compression measurements and shed light on conflicting static compression results. Our work highlights the need to properly consider the differing intrinsic timescales of the static and dynamic experiments when comparing results, and the complementarity of the techniques in assessing phase diagram and transition mechanisms.

becomes approximately hydrostatic. This approach ensures precise knowledge of the in-situ sample composition at the nanosecond timescale and gives access to phenomena like high-pressure crystalline phase transformation kinetics 20;21;41;42 . For example, Smith et al. 20 and Amadou et al. 21 have inferred that the bcc-hcp transformation occurs in pure Fe on a timescale of ∼1 ns, consistent with a nucleation and growth process. They also find a higher transformation onset stress than reported using static compression. On the other hand, kinetics in non-metals can occur on much longer timescales 43;44 . Recent measurements have also observed phase transformation boundary lowering in dynamic compression 45;46 . The application of dynamic compression experiments to the determination of stable phase relations for systems of geophysical interest holds great potential but is not as straight-forward as it might appear.
Here, we explore the phase stability of iron, Fe-Si 8.5wt% and Fe-Si 16wt% in conditions near those existing at the Earth's core-mantle boundary using laser-driven shock wave compression. We investigate the cubic phase stabilization of iron alloys for increasing Si content and compare approaches in a pressuretemperature regime previously explored with LH-DAC and at higher temperature than studied with dynamic compression. Our ultrafast structural measurements of shocked samples, which for the same pressure generate higher entropy and therefore temperature than in ramp compression, help further understanding about the origins of this disagreement.
X-ray free electron laser (XFEL) sources now provide bright enough x-ray beams to record highquality XRD measurements in a single, femtosecond-scale pulse making a wide range of new studies possible. We have used the extremely bright, ∼10 fs x-ray pulse generated at the Spring-8 Angstrom Compact Free Electron Laser [47][48][49] (SACLA) facility in Japan to determine the crystal structure during the initial shock in Fe, Fe-Si 8.5wt% , and Fe-Si 16wt% as well as on release into sapphire in Fe-Si 16wt% . The short duration and high brightness of the x-ray pulse combined with the short penetration depth of the x-rays in the reflection probe geometry allows us to effectively take snapshots of the crystallographic phase. We have observed the rapid onset of the hcp phase but not completion of the transformation in Fe and Fe-Si 8.5wt% during the initial shock. On the other hand, we observe no evidence of a structural phase transformation in Fe-Si 16wt% either during the initial shock or on release, which is in disagreement with the prevailing view coming from early static measurements 6;23 . Our new measurements were only possible with the combination of XFEL pulse, reflection geometry as shown in figure 1, and exploiting textured samples.

Results
Direct laser ablation forms an expanding surface plasma that drives our samples to high pressure. The expansion causes the target to recoil, which generates a pressure wave that travels into the sample away from the ablating surface, forming a shock. On some shots, the VISAR diagnostic monitored Fe/sapphire interface motion -the inset in Fig. 1 shows example data. Figure 2 shows integrated XRD patterns of ambient (dashed green line) and in situ high-pressure (solid purple line) of Fe (a), Fe-Si 8.5wt% (b), and Fe-Si 16wt% (c) recorded during the initial shock. Labels give the phase and reflection plane of each observed peak; asterisks denote the high-pressure phase peaks. We describe the identification process for each of the reflections in detail below. Fe and Fe-Si 8.5wt% patterns include reflections from a ∼100 nm deposition of Au used to aid sample alignment (see Methods), which prevented VISAR collection when fielded.
On shots where VISAR was available, we determine pressure using impedance matching between the iron and sapphire window 50 . Comparing the observed peak particle velocity and breakout time to simulations using the 1D radiation hydrodynamics code MULTI 51 helps to confirm the pressure regime. The presented data are all measured for the same laser intensity and sample design. In general, the 3/20 10  pressures inferred from VISAR measurements and with MULTI agree with the pressure inferred using an equation of state 6;33;52 for the volume measured by the x-ray diffraction. We do not see the typical multi-wave behavior associated with the bcc-hcp transformation due to the higher velocity of the hcp phase. This is due to the strength of the shock and prevents use of typical wave-profile analysis to determine phase transformation kinetics, instead requiring a direct structural probe like XRD. Across our data, the rise time of the shock is observed to be 0.5±0.1 ns, which combined with the directly measured strain gives an average strain rate ofε = 1.
where d 0 and d are the ambient and compressed d-spacings, respectively). Table 1 summarizes the reflections observed in the data shown in Fig. 2. The volume calculation assumes hydrostatic compression (for both cubic and hcp phases) and a c/a ratio of 1.61 for the hcp phase, as found by other laser-shock experiments 53 and close to the value reported by static studies 54 . The data in Fig. 2 were collected during the initial shock, i.e., along the principal Hugoniot. For Fe-Si 16wt% we have also performed additional shots at various pressures ranging from ∼10 -110 GPa both on the initial shock and after breakout into sapphire and find no evidence of phase transformation.

Phase Identification
As seen in Fig. 3 (a), the preferred orientation of crystallites (texture, due to the deposition process) in our Fe samples produces azimuthally narrow XRD spots. This is in contrast to the typical (azimuthally broad) Debye-Scherrer rings associated with XRD from powder samples with no texture like the Au reflections seen in Fig. 3 (b) and (c). While the geometry, sample texture, and detector size prevent observation of some reflections, the azimuthally narrow XRD spots arising from our textured samples make it possible to    The textural similarity between the ambient and high-pressure diffraction is the key to our analysis. The high-pressure bcc {110} and {211} reflections in Fig. 3 (b) have the same pattern as their ambient counterparts in (a) and yield the same bcc lattice parameter to within ∼1%, smaller than the uncertainty of the measurement. We identify the hcp {002} reflection (red in Fig. 3 (b)) by combining knowledge of the initial bcc texture and a priori knowledge of the parent-child orientation relationship, addressed by both static and dynamic compression XRD experiments. In these descriptions the bcc {110} becomes the hcp {002} plane while maintaining its initial orientation. This relationship combined with their similar d-spacings for our pressure produces the similar azimuthal bcc {110} and hcp {002} diffraction patterns we observe. Furthermore, as already mentioned, identifying this reflection as the hcp {002} and assuming a c/a ratio of 1.61, in line with other measurements in shocked samples, gives a volume consistent with our MULTI simulations and VISAR data from other shots. Sample texture and/or being below the detection threshold explains the absence of hcp {100} and {101} reflections. The Fe-Si 8.5wt% samples (shown in Fig. 3 (c)) do not have the same texture as the Fe samples, but they have a similar level of agreement between the bcc {110} and {211} reflections as seen in Fig. 2 (b), and so we interpret the three observed lines in the same way.

Fe-Si 16wt%
The Fe-Si 16wt% samples form the B2 structure at ambient conditions. Fischer et al. 6 report the D0 3 structure at ambient conditions in samples with similar Si content. However, the D0 3 and B2 phases are structurally alike and produce similar diffraction patterns with D0 3 producing a few additional very weak reflections that may have been below our detection threshold. Furthermore, physical vapor deposition synthesis at relatively low temperature does not favor Si ordering 25;59 . The inset in Fig. 1 shows the Fe-Si 16wt% phase diagram, reproduced from Fischer et al. with the shock Hugoniot and example release adiabats (calculated using QEOS 60 ) for Fe-Si 16wt% . Based on the results from Fischer, we expected to see a mixture of B2 + hcp in the conditions of our experiment.
We find no evidence of a structural phase transformation during the initial shock over a range of pressures (∼10-110 GPa) or upon release from high pressure. The lack of some of the weaker reflections at  The Debye-Scherrer rings form ellipses due to the geometry (see Fig. 1); the diffuse background is likely due to X-rays generated in the front-surface laser ablation and fluorescence from the XFEL interacting with the Au layer.

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high-pressure is due to the background and noise generated by the ablation plasma, but does not invalidate our conclusions: we cannot distinguish bcc from B2 or D0 3 (going forward all references to the B2 phase have this caveat) but we can clearly discard the presence of the hcp phase.  . Pressure-volume relations for Fe, Fe-Si 8.5wt% , and Fe-Si 16wt% plotted with similar alloys. The black lines/gray shading show volume and uncertainty of our measurements. Comparative Hugoniot/static data: in (a), the Sesame 2150 calculated shock Hugoniot is plotted with shock Hugoniot data from Brown et al. 52 ; in (b), static compression phase relation data for Fe-Si 8.7wt% from Hirao et al. 33 taken at 300 K and shock Hugoniot data for Fe-Si 6.9wt% from Marsh et al. 61 are plotted; in (c), static compression data for Fe-Si 16wt% from Fischer et al. 23 heated between 300-2200 K. Panels (b) and (c) also show the Sesame 2150 shock Hugoniot for comparison.

Discussion
The advent of laser-driven dynamic compression has vastly expanded the range of accessible pressure and temperature conditions. However, the intrinsically high strain-rate and uniaxial strain can activate exotic material responses like strain-rate dependence of phase boundaries and large flow stress. While wave profile analysis can investigate dynamic effects like these 20;21 , the approach lacks direct, simultaneous phase determination. Recent dynamic compression studies using XRD have found different phase boundaries than in static compression 45;46 , which ultimately raises questions about the applicability of dynamic compression to the study of equilibrium phase diagrams of interest in Earth and planetary science. Identifying and understanding these kinds of disagreements in conditions already studied with static compression is therefore imperative. This careful approach is fundamental to developing any new technique and obviously needed for dynamic compression.
The ultrafast x-ray pulse combined with the reflection geometry allows extremely high temporal resolution. In our setup, the rearmost 1 µm of the sample (x-ray side) scatters ∼64% of the detected x-rays. This depth corresponds to ∼130 ps in shock transit time of flight in our conditions. The x-ray pulse probes the sample before the initial shock transit is complete -any compressed signal is therefore probing the leading edge of the shock. This effectively sets an upper bound of ∼130 ps on the timescale for the onset of the bcc-hcp phase transformation in iron and Fe-Si 8.5wt% under these conditions. This time resolution allows us to confirm the differences observed between static and dynamic compression from Fischer et al. 23 and Wicks et al. 24 as well as suggesting preliminary insight into the origin of these differences.
We have observed both the transient bcc and high-pressure hcp phase in pure iron during the initial shock, a result of the transient compression dynamics. However, the nature of strain relaxation in dynamic compression is largely undetermined. A series of large computer simulations of the behavior of polycrystalline and single crystal iron by Kadau et al. 62;63 give insight into these dynamics. They propose that the compression process starts with initial elastic compression, followed by plastic deformation in the bcc phase, and finally transformation into the high-pressure hcp phase. Our data simultaneously visualizes the transiently compressed bcc phase and the transformed high-pressure hcp phase. This highlights the importance of investigating phase transformation dynamics in compression experiments before applying these results to constrain phase diagrams. As well, our work clearly shows the advantage of using textured samples for this purpose. This approach could be especially useful for identifying phase transformation orientation relationships, or for resolving uncertainties in ambiguous diffraction data. Performing this experiment with a powder sample could have hypothetically led one to mistakenly interpret the driven bcc {110} reflection as the hcp {100} plane. This faulty assumption yields a c/a ratio of 1.64 for the hcp phase, significantly higher than observed in either dynamic 53 or static 64 experiments. As a non-ideal c/a ratio in the hcp structure could be related to elastic anisotropy, an incorrect assessment would lead, for instance, to erroneous conclusions concerning the seismological anisotropy exhibited by Earth's inner core 27;28 .
As for the case of Fe, we have observed multiple phases during the initial shock in Fe-Si 8.5wt% . The similarity of the data argues for the same interpretation and the same underlying phenomena. The measured d-spacing of the transient bcc {110} and {211} reflections corresponds to a hydrostatic pressure of 38 GPa; the high-pressure hcp {002} reflection corresponds to a pressure of 132±26 GPa using the 300 K static compression equation of state from Hirao et al. 33 . We make no thermal corrections since the Hirao data agrees well with the shock Hugoniot data reported in Marsh et al. 61 , as seen in Fig.4, for Fe-Si 6.9wt% ; measurements of Fe-Si 5wt% at slightly lower pressure by Edmund et al. 65 suggest the thermal pressure is ∼15 GPa for our conditions, similar to that found by Fischer et al. in Fe-Si 16wt% and less than our error bars. The inferred bcc pressure is consistent with our calculations for the transformation pressure described below. Given the rapid onset, the bcc-hcp phase transformation process in Fe-Si 8.5wt% is likely a non-diffusive process like the Martensitic bcc-hcp phase transformation in iron. Assuming the Clapeyron slope is positive, these results would seem to support our findings.
On the other hand, we observe no evidence for a phase transformation in Fe-Si 16wt% . These results are in disagreement with Fischer et al. 6 Edmund et al. 25 have studied the bcc-hcp transformation onset 9/20 pressure over a range of Fe-Si alloy concentrations at ambient temperature. They find a slightly increasing hcp onset pressure up to 8.7wt% Si; the phase boundary then increases by ∼11.5 GPa/wt% up to 11.8wt%, and the hcp phase in Fe-Si 17wt% is not observed up to 65 GPa. The reported absence of transition in Fe-Si 17wt% at 300 K up 65 GPa, together with the positive Clapeyron slope of the transition, appear in contrast to the presence of hcp phase at 50 GPa and 1000 K argued by Fischer, and rather in agreement with our observations.
To supplement our measurements and provide further insight, we have performed structural prediction calculations using Density Functional Theory (DFT) at 0 K. The phase diagrams for Fe-Si 9.1wt% (Fe 10 Si 2 cell) and Fe-Si 16.7wt% (Fe 10 Si 4 or Fe 5 Si 2 cells) were calculated for the pressure range between 0 and 120 GPa, by estimating the relative enthalpy of the main phases at the GGA-PBE level of approximation 66 for the exchange-correlation energy functional. These concentrations approximate the experimental conditions while managing the calculation size. Non-magnetic B2 and hcp phases use the larger Fe 10 Si 4 cell, while ferromagnetic D0 3 uses the smaller Fe 5 Si 2 cell due to the heavier computational cost of magnetic DFT calculations. The calculations aiming to predict the ferromagnetic phases include magnetic effects through the implementation of a standard basic collinear spin-polarization model. The Ab Initio Random Structure Search 67 (AIRSS) code was employed for performing crystal structure predictions, coupled with efficiency enhancements described in the Methods section and detailed in a separate publication 68 Figure 5. Phase stability calculations using density functional theory at 0 K for Fe-Si 9.1wt% (a) and Fe-Si 16.7wt% (b). Figure 5 shows predictions of enthalpy vs. pressure from these calculations. The thermodynamically stable phase is the one with the structure having the lowest enthalpy. For Fe-Si 9.1wt% , the ferromagnetic bcc phase dominates at low pressures; we find hcp is most stable above ∼40 GPa. In the case of Fe-Si 16.7wt% , the ferromagnetic D0 3 phase dominates at low pressures. Starting at ∼60 GPa, the B2 phase stabilizes, with the hcp phase being slightly less stable than B2 by around 20-50 meV/atom. The fcc phase remains metastable at all pressures in both alloys. The increasing stabilization of the cubic phases for increasing Si concentration as seen in Fig. 5 agrees with previous results 6;23-25;69 and our measurements that small differences in Si concentration impact phase stability. This also highlights a possible advantage of using laser-generated compression, which is largely immune to diffusive concentration changes due to the short timescale.
The different timescales of the experiments could also explain these apparent inconsistencies. Our findings are in agreement with Wicks et al. 24 , suggesting that the process required to form the hcp phase observed by Fischer in Fe-Si 16wt% may not be active on the 1-10 nanosecond timescale. Static compression experiments typically take place on timescales of at least seconds, which in principle eliminates transient effects, but also enables diffusion. Atomic rearrangement is common in shock waves as a result of, for example, plasticity (due to the large applied stress) and Martensitic phase transformations. However, dynamic compression occurs on a timescale that largely prevents diffusive processes that change bulk alloy concentrations like those used to form steels. Recent studies in other materials have revealed diffusive phase transformations active on timescales as fast as tens of nanoseconds using dynamic compression 43 and can evolve with characteristic times on the order of hundreds of seconds or more using static compression 44 , highlighting a broad range of possible behaviors. In our case, the rapid stabilization of the hcp structure for iron and for Fe-Si 8.5wt% , but not for Fe-Si 16wt% , may imply that in this latter case the formation of hcp phase may require non-Martensitic action like diffusive de-mixing.

Summary
We have studied phase relations in shocked Fe, Fe-Si 8.5wt% , and Fe-Si 16wt% . Using femtosecond XRD measurements, we have directly identified a transient mixture of bcc+hcp phases in the initial shock in Fe and Fe-Si 8.5wt% , which is a result of the kinetics of the bcc-hcp phase transformation. Differently from early static experiments 6;23 , we see no evidence for a high-pressure phase transformation in Fe-Si 16wt% during either the initial shock or release. The different timescales of the experiments could explain this observation and may imply a different cubic-hcp phase transition mechanism for high Si concentrations. Finally, these results suggest that we could leverage the temporal and out-of-equilibrium aspects of laser shock compression to study phase stabilization processes and mechanisms in both Fe-Si alloys and other materials.

Acknowledgments
The XFEL experiments were performed at the BL3 of SACLA with the approval of the Japan Synchrotron

Experimental Details
We performed powder X-ray diffraction measurements of Fe, Fe-Si 8.5wt% , and Fe-Si 16wt% sandwich layered samples. A 10 µm Parylene-N layer is used as an ablator. A second 200 nm thick Ta layer, prevents pre-heating from the laser-ablator interaction. The ∼ 3 µm Fe(-Si x ) sample layer is deposited using physical vapor deposition directly onto the 25 µm thick c-cut single crystal sapphire layer, which is used as a tamper and VISAR window. In some samples, the VISAR-facing (rear) side of the sapphire window was coated with a ∼ 100 nm thick Au layer, which was used as an alignment fiducial. The Au layer melts under direct XFEL exposure allowing precise coalignment of the optical drive laser and XFEL to the sample. The samples have been characterized by ambient diffraction performed using a Rigaku diffractometer and coupled ion polishing and SEM microscopy -a summary of these ambient properties is given in Table 2 Table 2. Summary of sample initial conditions. The density is calculated from the weighted average of observed diffraction lines corresponding to the cubic phase. We observe no sign of C contamination.
High pressure was achieved dynamically via laser-generated shock wave using the EH5 end station at SACLA. Our samples were irradiated with ∼ 10 J of 532 nm radiation in a 3.4 ns pulse focused with a lens to a ∼ 250 x 500 µm 2 spot projected onto the angled target plane. The laser was incident at an angle of ∼ 57 • from normal giving a target-plane irradiance of ∼ 1 × 10 12 W/cm 2 . The laser interacts with the ablator layer, launching a shock wave into the sample; the difference in shock impedance at the CH and Ta interfaces causes a significant pressure increase in the Ta that propagates to the probe layer. Sapphire has similar shock impedance as Fe and is used to minimize wave reflections at the interface.
We probed the crystal lattice structure with X-ray diffraction by using the ∼ 10 fs SACLA XFEL pulse, with 10 11 photons/pulse tuned to 10.1 keV. Scattered X-rays were collected using a 3.2 megapixel CCD detector placed 76.6 mm from the interaction point with a vertical orientation as shown in figure 1. The sample was probed in a reflection geometry with the X-ray pulse incident on the rear surface at 20 • from the sample surface. With respect to the geometry shown in figure 1, the X-ray pulse is slitted to 200 µm in the out of the page direction and focused with Kirkpatrick-Baez mirrors to 13 µm in the vertical direction, projecting to ∼38 x 200 µm 2 on the sample surface. The detector position is determined with CeO 2 powder diffraction; the detector position calibration and integrations were performed using Dioptas 70 . showing the pressure as a function of time and Lagrangian position. The peak irradiance is 8×10 11 W/cm 2 , which is lower than our experimental irradiance as typical of 1D radiation hydrodynamics simulations, produces the best fit to our data. The sample layers are indicated on the bottom (with the Ta heat shield included, but not labeled).
The reported uncertainty in reflection positions is determined by the width of the best-fit Gaussian to the deconvolution of the ambient and driven reflections. Finally, the Fe/sapphire interface velocity was monitored using a line-imaging VISAR with a field of view of 400 µm. The synchronization of the main optical beam with the XFEL beam was performed in the same way as described by Albertazzi, et al. 71 and whose dynamics are discussed by Hartley, et al. 72 The coarse timing is tuned with a photodiode that measures emission from a polyimide sample. Fine tuning is accomplished with a 500 nm thick polycrystalline Au sample. The timing baseline, i.e., zero probe delay, is defined as the time when the diffraction from the Au sample begins to broaden due to sample heating.

Hydrodynamic Simulations
Simulations are performed with the incident laser pulse profile and measured target layers. Figure 6 shows the simulated pressure as a function of time and Lagrangian space for peak irradiance of 8×10 11 W/cm 2 . We find a good match for a lower irradiance than used in the experiment, which is typical for 1D simulations. The location of the CH, Fe, and sapphire layers are indicated on the bottom. We used the SESAME 2150 equation of state tables 73 for pure Fe and a QEOS 60 equation of state for Fe-Si 16wt% . Here we observe a complicated wave interaction due to surface reflections in the multi-layer sample. However, the short attenuation length of 10.1 keV X-rays in Fe (7.46 µm) and the oblique probe angle heavily bias our recorded signal to a small volume near the Fe/sapphire interface with ∼64% of the signal coming from a depth of 1 µm (normal to the Fe/sapphire interface surface). This is consistent with the lack of gradients or asymmetry seen in the diffraction.

Density Functional Theory
For each of the Fe 10 Si 2 , Fe 5 Si 2 and Fe 10 Si 4 crystalline cells, 10,000 random structures have been generated. These random structures were generated with the AIRSS code 67 by imposing a minimum interatomic distance of 1.5Å and were comprised in the volume range between 7 -14Å 3 /atom. At each desired pressure, the enthalpy H=U+P ext × V where U is the internal energy, P ext is the externally-applied pressure, and V is the volume, of the initial random structures was calculated with the Quantum ESPRESSO 13/20 ab-initio code 74;75 , as a preliminary indication of their likely thermodynamic favorability. The 1% of the structures with the lowest initial enthalpy were selected for further processing. These favorable structures were subsequently clustered using Daura's algorithm 76 (cutoff value of 2.5) based on the PIV metric 77 with a switching function decaying from 0.9 to 0.1 between 1 and 6Å. This clustering resulted in a few tens of cluster centers: the geometry of such atomic configurations was optimized with a precision better than 1 meV to obtain the final structures and enthalpies. We adopted the following parameters for DFT calculations: PBE exchange-correlation functional, ultra-soft Vanderbilt pseudopotentials, kinetic-energy cutoff = 60 Ry, charge density cutoff = 600 Ry,8×8×8 k-point grid, and Methfessel-Paxton smearing with width = 0.01 Ry.
The identity of the final relaxed phases was confirmed with three different methods: space group and symmetry assignment using the spglib library 78 , enthalpy equivalence between structures, and structure factor comparison calculated using the VESTA crystal visualization software package 79 . The appearance of the crystal structures was also visualized and inspected with VESTA.

Data Availability
Data that support the findings are available from the corresponding author upon request.