Experiments using broadband Laue x-ray diffraction (XRD) were used to examine the lattice structure of dynamically compressed [100]-oriented single crystal iron samples at the Dynamic Compression Sector at the Advanced Photon Source. These experiments used 1 $\mu $m thick iron single crystal samples sandwiched between a polyimide ablator and a polycarbonate window. A 100 J, 10 ns duration laser pulse incident on the polyimide ablator was used to shock compress the iron samples to initial stresses greater than 25 GPa, exceeding the $\u223c13$ GPa alpha (body-centered-cubic or bcc structure) to epsilon (hexagonal-close-packed or hcp structure) phase transition stress. XRD measurements were performed at various times relative to the shock wave entering the iron sample: early times, $<\u223c150$ ps while the initial shock waves propagated through the iron; intermediate times, after the iron equilibrated with the ablator and window reaching a plateau stress state (12 or 17 GPa) lasting several nanoseconds; and late times, during uniaxial strain release. The early time measurements show that in $<\u223c150$ ps, the high-pressure hcp phase is relaxed with a $c/a$ ratio of 1.61, contrary to previous laser shock experiments where a $c/a$ ratio of 1.7 was inferred. In the plateau stress state and partially released states, XRD measurements showed that the hcp structure retained a $c/a$ ratio of 1.61 with no observable changes in the microstructure. Upon stress release at $\u223c1$ GPa/ns release rate, the reverse phase transition (hcp to bcc) to the original single crystal orientation (implying a transformation memory effect) was observed to reach completion somewhere between 13 and 11 GPa, indicating little stress hysteresis under rapid uniaxial strain release. A similar memory effect for the reverse hcp to bcc transformation has been previously observed under hydrostatic compression. However, the bcc/hcp orientation relationships differ somewhat between dynamic and static compression experiments, implying that the transformation pathway under uniaxial dynamic strain differs from the Burgers mechanism.

## I. INTRODUCTION

The alpha (body-centered-cubic or bcc) to epsilon (hexagonal-closed-paked or hcp) phase transition in iron has been extensively studied for more than half a century.^{1–10} The first evidence for a high-pressure polymorphic transformation in iron was obtained by Bancroft *et al.*, who observed three transmitted shock waves through ARMCO iron samples and interpreted the third wave as being due to a phase transformation.^{1} Bridgman, the leading high-pressure scientist at the time, questioned how a structural phase transition could occur on a timescale associated with shock loading experiments (nanoseconds to a few microseconds).^{11} Subsequent high-pressure static compression x-ray diffraction (XRD) measurements demonstrated that bcc iron transforms to the hcp structure at $\u223c13$ GPa, the same stress observed in the shock wave experiments.^{1,2} Computational investigations using non-equilibrium molecular dynamics (NEMD) simulations showed the bcc to hcp transformation in iron to be martensitic with compression and shuffle of the lattice that occurred on the picosecond timescale of the simulations.^{12} The NEMD simulations agreed with laser driven shockwave compression experiments using *in situ* XRD measurements to probe single crystal iron samples.^{13,14} These simulations and experiments concluded that uniaxial strain dynamic compression of bcc iron along a [100]-lattice direction resulted in a highly strained high-pressure hcp lattice structure different from results of static compression experiments performed in diamond anvil cells.^{8,10,15} In the dynamic experiments and simulations on single crystal iron, a $c/a$ ratio of $\u223c1.7$ was observed,^{12–14} while static compression experiments resulted in a $c/a$ ratio of $\u223c1.61$.^{2,8} This difference was attributed to the different timescales and loading histories between the dynamic and static experiments. However, subsequent dynamic compression experiments on polycrystalline iron foils showed a rapid phase transition to the hcp structure with a $c/a$ ratio of 1.6.^{16} The previous single crystal iron laser shock experiments were limited to the XRD measurements being no later than about a nanosecond after loading due to sample thickness and x-ray penetration depth constraints.^{13,14} Therefore, an outstanding question is whether the large shear strain (and shear stress) associated with $c/a=1.7$ in hcp iron formed by shock compression of single crystal Fe along a [100] direction would relax at later times with the $c/a$ ratio approaching the equilibrium value of $\u223c1.61$.^{8}

The primary focus of this work is to investigate the longer timescale response of single crystal iron shock loaded along the [100] axis via laser drive to determine whether lattice relaxation occurs in the high-pressure hcp iron phase during the several nanosecond duration high pressure state. A secondary objective is to examine the reverse phase transformation back to the bcc structure during rapid uniaxial strain release. To achieve these objectives, thin (1 $\mu $m thick) iron single crystal samples were sandwiched between a polyimide ablator and a polycarbonate window allowing the high-pressure state to be supported for an extended period of time. The short stress wave transit time in the thin iron samples results in nearly spatially uniform stress states during the high-pressure plateau state and during stress release. A similar approach is employed in dynamic compression ramp wave experiments where thin samples are sandwiched between two diamonds to avoid large stress gradients in the sample due to the ramp wave.^{17}

## II. EXPERIMENTAL METHODS

Figure 1 shows a schematic diagram of the experimental setup for single pulse x-ray diffraction measurements in iron laser shocked along the [100] direction. The experiments were performed in the laser shock station (Hutch-C) of the Dynamic Compression Sector (DCS) at the Advanced Photon Source.^{18} The iron samples were oriented such that the angle between the incident x-ray beam and the sample normal ([100] crystallographic direction) was 52$\xb0$. The Rayonix SX165 x-ray detector plane was nominally perpendicular to the incident x-ray beam.

Two experimental campaigns used the 100 J DCS laser system with either a 500 or 1000 $\mu $m diameter focal spot generated by a phase plate to shock load the single crystal iron samples.^{19} The laser drive duration was 10 ns, and the laser energy on the target was reduced below 100 J using beam splitters to control the shock stress. Experiments using the 1000 $\mu $m diameter focal spot with 70% laser energy resulted in a plateau stress of $12.0\xb11.2$ GPa. Experiments using the 500 $\mu $m diameter focal spot with 40% laser energy, and experiments using the 1000 $\mu $m diameter focal spot with 100% laser energy both resulted in a plateau stress of $17\xb11.7$ GPa. Experiments labeled 20-C-0XY were performed with a 50 $\mu $m thick polyimide ablator and the 500 $\mu $m laser drive spot configuration. Experiments labeled 20-C-4XY were performed with a 75 $\mu $m thick polyimide ablator and the 1000 $\mu $m laser drive spot configuration.

The iron single crystal samples were grown on an oriented NaCl substrate at the Department of Physics and Astronomy, University of Aarhus, Denmark. The iron samples were epoxy bonded to the 250 $\mu $m thick polycarbonate window before the NaCl was dissolved in water. A polyimide ablator was then epoxy bonded to the other side of the iron sample following the guidelines for laser shock targets at the DCS.^{18} The polyimide was coated with aluminum on the free surface side to absorb the drive laser energy and to prevent laser light from preheating the sample. The 250 $\mu $m thick polycarbonate window was coated with aluminum on the sample side to provide a reflecting surface for velocity interferometry measurements to determine the particle velocity history at the iron/polycarbonate interface; the stress history in the sample is determined from these measurements.

Polycarbonate was selected as the window material due to its high x-ray transmission and low x-ray scattering. The scattering efficiency from the 1 $\mu $m thick iron samples is low and scattering and/or absorption from higher-Z windows was found to significantly degrade the signal-to-noise ratio of the Fe sample x-ray scattering measurement. However, polycarbonate is not an ideal window material for examining the shock response of Fe due to the shock impedance mismatch between polycarbonate and Fe. The loading history consists of initial shockwaves propagating through the sample followed by a series of reverberating release and compression waves that ring back and forth in the sample until the iron reaches stress equilibrium with the window and ablator. The wave transit times are short ($\u223c150$ ps) because the Fe samples are very thin, and the plateau stress is quickly established.

A broadband x-ray probe beam was generated using a 2.7 cm period undulator with an average 13.5 mm magnet gap and a 5 mm gap taper across the undulator. The x-rays were focused using Kirkpatrick–Baez mirrors with platinum coatings to a spot size less than 100 $\mu $m at the target. The x-ray focal spot is centered relative to the laser drive spot and is also much smaller than the laser drive spot. The x-ray mirrors also acted as a high energy x-ray filter. Taking x-ray absorption of the target and beamline windows into account as well as the high energy reflectivity cutoff of the x-ray mirrors, the useful x-ray bandwidth extends from $\u223c9$ to 40 keV, which is sufficiently broad for Laue x-ray diffraction measurements. Shutters and high-speed choppers were used to isolate a single x-ray pulse ($\u223c100$ ps duration), which was used as the probe pulse for x-ray diffraction patterns on both ambient and laser driven samples.

Figure 2 shows a representative XRD pattern for the ambient iron sample (bcc structure) used in experiment 20-C-066. Prior to each experiment, an x-ray diffraction image of the ambient sample was used to determine the precise orientation of the iron sample relative to the detector plane and incident x-ray beam.

## III. EXPERIMENTAL RESULTS AND DISCUSSION

### A. Continuum results and x-ray measurement timing

For each of the three laser drive conditions, a series of nominally identical laser shock experiments were performed with the delay between the x-ray probe and the laser drive beam varied to examine the iron crystal structure at different times relative to the shock wave entering the Fe sample from the polyimide ablator. Point-VISAR^{20} and line VISAR^{21} measurements were made in each experiment and the variations in recorded iron/polycarbonate interface particle velocity histories were within 5% between experiments with nominally the same drive conditions.

The iron stress history is calculated from the measured interface particle velocity history and known properties of the polycarbonate windows. Polycarbonate was characterized using symmetric impact experiments at the Institute for Shock Physics to determine the stress–particle velocity relationship and the index window correction factor (0.9) used to calculate the actual particle velocity from the apparent particle velocity. More details on the velocity measurements and stress determination are given in Appendix A.

Representative stress histories at the sample/window interface are shown in Fig. 3 for each of the three drive conditions. The difference in the duration of the plateau stress in the $\u223c17$ GPa experiments between the different campaigns is due to the different ablator thicknesses and the associated time for the release wave to travel from the polyimide ablation surface to the iron sample. For all experiments, the stress release comes from the ablator drive surface after the laser drive pulse turns off rather than from the window free surface.

Due to the different mechanical impedances of the iron sample and the polycarbonate window/polyimide ablator, the stress histories in the samples differ significantly from the interface stress histories shown in Fig. 3 for $\u223c500$ ps after initial shock loading. The initial shock waves in the iron samples bring the iron to over 25 GPa stress before undergoing partial stress release due to wave reflection at the polycarbonate window. Subsequent reverberations between the window and the ablator establish near stress equilibrium within $\u223c500$ ps. For the slow time variations in stress after 500 ps, the iron sample window/ablator impedance mismatches become unimportant and, to a good approximation, the stress throughout the driven iron sample equals the measured sample/window interface stress.

The times at which x-ray measurements were recorded relative to shock breakout at the iron/polycarbonate interface are indicated in Fig. 3 for each experiment. These times were determined by measuring shock breakout in the VISAR signal, which had a calibrated timing with respect to the x-ray pulse time. The uncertainty in the relative timing of the x-ray measurement and the shock breakout is estimated to be $\u223c400$ ps, which is larger than the $\u223c150$ ps transit time of the stress waves through iron.

For the discussion of the XRD results in Secs. III B–III D, experiments are divided into three regions: early times ($<500$ ps after shock loading) where either the initial shock waves are propagating through the sample or significant stress reverberations are still present in the sample, intermediate times at which the iron is approximately at the plateau stress (several nanoseconds in duration); and late times up to 20 ns after the shock breakout where the iron stress is decreasing due to a release wave from the ablator free surface.

### B. X-ray diffraction results: Initial shock and early time measurements

Determination of whether the XRD measurement was made before the initial shock breakout at the polycarbonate window is based upon the observed x-ray diffraction features because the timing uncertainty between the x-ray measurement and shock breakout at the polycarbonate window is greater than the $\u223c150$ ps wave transit time through the 1 $\mu $m thick iron sample. For experiments 20-C-065 and 20-C-066, the x rays probed the sample while the initial shock waves were still propagating through the sample. This is known because three sets of diffraction spots are observed in these two experiments. An XRD pattern from experiment 20-C-066 is shown in Fig. 4. The three features are (1) sharp ambient bcc iron spots surrounded by blue diamonds in Figs. 4(b)–4(g) indicating that a portion of the sample is still unshocked, (2) sharp shifted bcc iron spots surrounded by red squares in Figs. 4(b)–4(g) corresponding to elastically compressed bcc iron, and (3) new significantly elongated spots corresponding to hcp iron indicated by yellow and orange triangles in Fig. 4(a). Analysis of both experiments 20-C-065 and 20-C-066 results in the same findings and here we present detailed results and analysis for experiment 20-C-066.

#### 1. Elastic bcc

From the experimental geometry and a Laue XRD spot position on the detector, the direction of the associated iron reciprocal lattice vector (normal to the diffracting planes) can be calculated. The Laue XRD technique uses a broad spectrum of x rays to record a large collection of diffraction spots, but since the wavelength of the diffracted x rays is unknown the length of the reciprocal lattice vector (inversely related to the lattice plane spacing) cannot be calculated. Therefore, a shift of a single bcc Laue spot demonstrates a change in the associated lattice plane orientation but does not give enough information to determine whether the change is due to strain (the symmetric part of the displacement gradient matrix $Ju$) or rotation (the anti-symmetric part of $Ju$). Measurements of multiple diffraction spots from multiple diffraction planes are needed to determine the complete displacement gradient matrix. The approach used here is to map the XRD data for a given Laue spot onto a 2D plane representing reciprocal lattice vector rotations (or equivalently reciprocal lattice vector displacements). The reciprocal lattice vector, $G(hkl)$, corresponding to the $(hkl)$ plane of the ambient bcc Fe defines the origin of the 2D map to measure the rotation (or displacement) associated with the lattice changes. See Appendix B for more details on how the XRD data are mapped into the reciprocal space.

Figure 5 shows an example reciprocal space mapping for the (1$3\xaf$0) Laue diffraction spot where $hx$=(3,1,0) and $hy$=(0,0,1) are the orthogonal axes, which the rotations $\beta $ and $\gamma $ are measured about, respectively. Using this mapping, it is possible to measure the effective change in direction $\Delta g$ of the (1,$3\xaf$,0) reciprocal lattice vector between the ambient and “elastic” spot, $P$, as

which is orthogonal to the (1$3\xaf$0) reciprocal lattice vector, $G(13\xaf0)$. Thus, the ambient $G(13\xaf0)$=(1,$3\xaf$,0) reciprocal lattice vector changes to $P=\Delta g+(1+\alpha )G(13\xaf0)=(1+\alpha )(1,\u22123,0)+0.022(3,1,0)$ in the shocked state. The $\alpha $ term accounts for isotropic volume changes that may occur in the lattice that the Laue XRD measurement is not sensitive to. By applying this mapping to all of the shifted bcc spots shown in Figs. 4(b)–4(g) and determining the mapped spot centroids, a displacement gradient matrix defining the change in the shocked bcc Fe lattice shape and orientation relative to the ambient bcc Fe lattice can be calculated using standard matrix algebra as described in Appendix C,

Here, $\alpha $ times the identity matrix is an isotropic strain component which broadband Laue XRD is not sensitive to. The coordinate system used to express the bcc iron displacement gradient matrix is the x–y–z coordinate system shown in Fig. 6. The small off diagonal terms suggest no lattice rotations have occurred and that the lattice has a 7.3% compressive strain along the [100] iron shock loading direction.

Figures 4(b)–4(g) show that forward calculations of simulated Laue XRD spots for bcc Fe compressed 7.3% along the [100] direction indeed match the observed bcc Fe Laue spot shifts, confirming the accuracy of the displacement gradient matrix. The shifted bcc peaks have breadths similar to the ambient bcc peaks suggesting that plastic deformation has not occurred in the bcc phase and the shifted peaks are therefore labeled “elastic.” The 7.3% elastic compression of the bcc iron is similar to the elastic compressions observed previously in laser based single crystal Fe XRD experiments^{13,14} and NEMD simulations.^{12} We note that this large elastic strain is consistent with the observed increase in initial elastic strain with decreasing sample thickness observed in gas gun and laser experiments.^{22,23} However, in contrast to previous experiments on gas guns with 1–2 mm thick samples, there is no observable plastic relaxation observed in the bcc material before the phase transition.^{6,24}

#### 2. Phase transition to hcp

The observed hcp Laue diffraction spots correspond to two hcp variants with orthogonal $c$-axes. Yellow and orange symbols in Fig. 4 are predicted Laue spot positions based on the ambient iron crystal orientation and the assumed twofold degenerate transformation pathway with an hcp $c/a$ ratio of 1.61 and no anisotropic lattice compression (the hcp basal plane is an undistorted hexagon). Figure 6 shows the assumed orientation relationship between the bcc and hcp lattice based on the previous shock compression work.^{13,14} An $x$–$y$–$z$ Cartesian coordinate system is defined based on the cubic axes of the ambient bcc crystal as

where $aibcc$ are the real space bcc lattice vectors with the lattice constant $abcc=287$ pm and $bibcc$ are the associated reciprocal lattice vectors. The real space and reciprocal space lattice vectors for hcp Variant #1 are

Similarly, for hcp Variant #2, the real space and reciprocal space lattice vectors are

with $c/a=1.633$ the above structure has an ideal hcp structure. Unless otherwise noted, we use $c/a=1.61$ for our reference hcp lattice when performing reciprocal space maps for the observed hcp Laue spots since $c/a=1.61$ has been observed as the equilibrium value for hydrostatically compressed Fe.^{8,10} The reference hcp lattice assumes a perfect hexagonal basal plane with $|a1hcp1|=|a2hcp1|$ and $|a1hcp2|=|a2hcp2|$.

We compare the choices of $c/a=1.61$ and $c/a=1.7$ for the hcp reference configuration by mapping multiple diffraction spots into reciprocal lattice space using both choices for $c/a$. Deviations in Laue spot positions from the origin in the reciprocal space map mean that the lattice plane orientation deviates from that of the reference configuration. Figure 7 compares the mapping in reciprocal space of several intense hcp peaks for hcp Variant #1 from experiment 20-C-066 using reference configurations with $c/a=1.61$ and 1.70. The reciprocal space mapping for an hcp reference configuration with $c/a=1.61$ clearly has mapped spots closer to the origin than the mapped spots using a reference configuration with $c/a=1.70$ indicating that the hcp lattice has a $c/a$ ratio closer to 1.61 than 1.7. Figure 8 shows similar mapping for the two observed diffraction spots from lattice planes parallel to the hcp *c*-axis; these spot locations do not depend on the $c/a$ ratio.

Figures 7 and 8 show red rings around the spots which correspond to the FWHM contours for two-dimensional Gaussian fits to the reciprocal space mapped diffraction spots. Using the reciprocal lattice vectors corresponding to the centroids of the collection of observed hcp diffraction spots, a displacement gradient matrix, $J$, can be calculated (see Appendix C). The displacement gradient matrix is composed of a symmetric part, $\epsilon ij$, representing the strains and an anti-symmetric part, $\omega ij$, representing lattice rotations,

A nonzero $\omega ij$ signifies a deviation from the bcc/hcp orientation relationships shown in Fig. 6. For hcp Variant #1 from experiment 20-C-066, using a reference hcp configuration with $c/a=1.61$ the strain and rotation matrices are

and

The strain and rotation matrices for the hcp iron in Eqs. (7) and (8) are expressed in a different coordinate system than the one used for the bcc displacement gradient matrix given in Eq. (2). For Eqs. (7) and (8), a Cartesian coordinate system is used where the $c$-direction is parallel to the $z\u2032$-axis and the $a1$-direction is parallel to the $x=x\u2032$-axis (same as the loading direction). Thus, the strain matrix, $\epsilon ij$ can be expressed as

Since the Laue diffraction technique is insensitive to isotropic strains, the diagonal elements of the strain matrix are all shifted by the same value such that $\epsilon aa=0$, as in Eq. (7); this is equivalent to setting the hcp reference unit cell *a*-axis lengths to $|a1|$. The reason for expressing the strain and rotation matrices in the $x\u2032\u2212y\u2032\u2212z\u2032$ coordinate system is to give the diagonal strain components specific meanings. Under uniaxial strain dynamic loading, the stress along the shock loading direction can, in general, differ from the stress transverse to the shock loading direction due to strength; such stress differences would distort the basal plane of the hcp lattice. The hcp $a1$-direction is parallel to the shock loading direction for both hcp variants so $\epsilon cc$ can be used to determine the strain along the hcp *c*-axis relative to the reference hcp configuration allowing $c/a$ to be calculated where $a$ is the magnitude of $a1$. Similarly, $\epsilon y\u2032y\u2032$ will give the difference in strain between the loading direction and $y\u2032=z\u2032\xd7x\u2032$, a direction orthogonal to both the loading direction and the hcp *c*-axis. From this strain component, the magnitude of the hcp $a2$ lattice vector can be calculated providing a measure of $a2/a1$ which quantifies the distortion of the hcp basal plane from hexagonal.

The terms in the $\omega $ rotation matrix [Eq. (8)] are all consistent with zero within uncertainty. Results from the other experiments for both hcp variants are the same indicating that Fig. 6 accurately represents the bcc/hcp iron orientation relationship.

For hcp Variant #1 from experiment 20-C-066, the strain matrix in Eq. (7) indicates that $c/a1=1.63$ and $a2/a1=1.00$. Similar analyses were performed for the other early time experiments for hcp variants having enough observable high intensity Laue spots to determine the displacement gradient matrix. Table I provides a summary of all the hcp lattice parameter ratios (determined from the strain matrices) for the early time experiments. The average $c/a$-ratio is 1.61 and all hcp variants for all experiments are consistent with this value within experimental uncertainty. Furthermore, there is no observable distortion of the hcp basal plane (or no extra strain along the loading direction) as the average ratio of $a2/a1$ is 1.00. These findings demonstrate that for single crystal Fe shocked along [100], the hcp Fe forms in a nearly fully stress relaxed state in sub-nanosecond timescales.

. | hcp . | $ca1$ . | $a2a1$ . |
---|---|---|---|

Experiment . | variant . | . | . |

20-C-064 | 1 | 1.61 ± 0.04 | 0.988 ± 0.006 |

20-C-064 | 2 | 1.61 ± 0.03 | 1.000 ± 0.006 |

20-C-065 | 1 | 1.63 ± 0.03 | 1.000 ± 0.007 |

20-C-065 | 2 | 1.63 ± 0.07 | 1.017 ± 0.011 |

20-C-066 | 1 | 1.63 ± 0.03 | 1.000 ± 0.007 |

20-C-066 | 2 | 1.60 ± 0.04 | 1.017 ± 0.011 |

20-C-401 | 1 | 1.62 ± 0.05 | 0.986 ± 0.010 |

20-C-424 | 1 | 1.59 ± 0.04 | 0.985 ± 0.010 |

20-C-424 | 2 | 1.61 ± 0.03 | 1.021 ± 0.010 |

. | hcp . | $ca1$ . | $a2a1$ . |
---|---|---|---|

Experiment . | variant . | . | . |

20-C-064 | 1 | 1.61 ± 0.04 | 0.988 ± 0.006 |

20-C-064 | 2 | 1.61 ± 0.03 | 1.000 ± 0.006 |

20-C-065 | 1 | 1.63 ± 0.03 | 1.000 ± 0.007 |

20-C-065 | 2 | 1.63 ± 0.07 | 1.017 ± 0.011 |

20-C-066 | 1 | 1.63 ± 0.03 | 1.000 ± 0.007 |

20-C-066 | 2 | 1.60 ± 0.04 | 1.017 ± 0.011 |

20-C-401 | 1 | 1.62 ± 0.05 | 0.986 ± 0.010 |

20-C-424 | 1 | 1.59 ± 0.04 | 0.985 ± 0.010 |

20-C-424 | 2 | 1.61 ± 0.03 | 1.021 ± 0.010 |

The present results are in agreement with previous findings for laser shocked polycrystalline iron^{16,25} but are inconsistent with previous NEMD simulations of Fe shocked along a [100] direction which showed no plastic relaxation of the hcp phase and a $c/a$ ratio of $\u223c1.7$,^{12} which is well outside the error bars of the current experiments. In principle, it could be argued that these differences could be due to different timescales; the present experiments are sub-nanosecond and include x-ray probe times $<150$ ps after initial shock loading, but the NEMD simulations^{12} are still an order of magnitude shorter timescale. The present results also disagree with the interpretation of previous laser shock experiments on single crystal iron shocked along the [100] direction^{13,14} where it was concluded that the hcp iron $c/a$ ratio was 1.7, consistent with the NEMD simulation results.^{12}

The discrepancy in $c/a$ values for shocked hcp iron from the present work and previous work^{13,14} is likely due to limitations in the earlier work. The current diffraction data have multiple diffraction spots, which are exclusively due to a particular hcp variant. Thus, the lattice structure of each variant can be interpreted without having the other variant pollute the interpretation. In Fig. 4, the two data points where upright and inverted triangles are in close proximity represent an example of diffraction planes where this complication can occur. Analysis of the previously published single crystal iron laser shock XRD data^{13,14} primarily used hcp peaks where a parent bcc diffraction peak yielded diffraction peaks with contributions from both hcp variants. Figure 9 shows the shifted bcc ($1\xaf$10) diffraction from elastically compressed bcc Fe for experiment 20-C-066, which has hcp ($1\xaf1$01) diffraction spots of both variants in close proximity. The $c/a$ value of 1.61 causes these two hcp diffraction spots to split. If only these or similar diffraction peaks were used to interpret the data, then the peaks could be interpreted as a single broader peak that would suggest a $c/a$ ratio near 1.7 rather than the $c/a$ value of 1.61 determined from the present experiments. We emphasize that in the present experiments, we uniquely identify several spots for each hcp variant, which allows us to determine the hcp lattice orientation, $c/a$ ratio, and any distortions of the hcp basal plane.

### C. Plateau stress and initial stress release

Several experiments with x-ray timing either in the plateau region or after the initial onset of stress release resulted in the observation of Laue spots from both hcp variants. The hcp lattice parameter ratios were determined using the same analysis methods used in the early time experiments and results are shown in Fig. 10. The $c/a$ ratio remains near 1.61 and $a2/a1$ remains near unity indicating that as the stress partially releases and recompresses during reverberations and during initial stress release after the plateau the hcp lattice does not develop significant shear strains.

Additionally, an examination of the shapes of hcp diffraction spots mapped into reciprocal space shows that the hcp microstructure is the same during initial shock compression (e.g., experiment 20-C-065), in the plateau region, and during initial stress release. Figure 11 shows the ($1\xaf$100) spot mapped into reciprocal space for several experiments with x-ray times ranging from the first 150 ps after shock loading to $\u223c8$ ns after shock loading. The breadth and shape of the diffraction peaks are the result of the sample microstructure. If we attribute the entire spot breadths to small hcp Fe grain size, we can estimate the number of unit cells along the two directions orthogonal to the diffraction plane normal. The standard estimate for crystallite broadening in polycrystal samples uses the Debye–Sherrer model, which measures the broadening of diffraction along the reciprocal lattice vector. As stated previously, the Laue diffraction technique is insensitive to the reciprocal lattice vector length so our measurements are not sensitive to broadening in that direction. Here, we model the diffraction spot broadening assuming the crystallite acts like an aperture that limits the number of cells that will diffract. In this case, the diffracted intensity is given by

where $\theta =\theta B+\Delta \theta $ is half the scattering angle, $N$ is the number of unit cells, $\theta B$ is the Bragg angle, and $\Delta \theta $ is a small angular displacement from the diffraction peak. For diffraction $2\pi d\lambda sin\theta B=\pi $, and we can expand the intensity assuming $\Delta \theta $ is small and solve for $I/I0=1/2$ to get

Using $\theta B=7.2\xb0$ for ($1\xaf$100) hcp peaks together with the widths measured in Fig. 11, we estimate a lower bound for the grain size along the hcp Fe *c*-axis of $N[0001]=6$ unit cells and a lower bound for the grain size along an hcp Fe [11$2\xaf$0] direction of $N[112\xaf0]=12$ unit cells. Other broadening mechanisms such as strains and mosaic spread may also contribute to the spot breadths, so the grain size estimates given above are only lower bounds. Independent of the various contributions to the diffraction spot sizes, it is important to note that for both low and high stresses and at all measurement times, the spot shapes are very similar, suggesting little to no evolution in the microstructure. Similar analysis of other diffraction spots shows that hcp spot shapes mapped into reciprocal space also do not change as a function of stress and time.

For the lower stress experiments where the plateau stress was $12.0\xb11.2$ GPa, in addition to the observation of hcp Laue spots, bcc spots were also observed, indicating a mixed hcp/bcc phase plateau stress region for $12.0\xb11.2$ GPa stress. This is likely due to the final stress after reverberations above and below the transformation stress being very close to the 13 GPa transformation stress. The bcc peaks observed in the mixed phase are consistent with an isotropic bcc lattice with the ambient bcc lattice orientation. Figure 12 shows the bcc ($1\xaf$2$3\xaf$) peak in the reciprocal space for the ambient sample and for the shocked sample in experiment 20-C-401 using an isotropic bcc lattice as the reference configuration. The x-ray measurement was made 510 ps after shock break out at the polycarbonate window, indicating Fe is in the beginning of the $12.0\xb11.2$ GPa stress region. The bcc peaks observed in the mixed phase at the $12.0\xb11.2$ GPa plateau state are very weak so it was not possible to quantitatively determine the displacement gradient matrix for the bcc phase in the mixed phase. However, the spots mapped into reciprocal space are consistent with an isotropically compressed bcc lattice because they are not significantly shifted relative to the ambient bcc spot as shown in Fig. 12. The observation of an isotropically compressed bcc lattice in the mixed phase region is consistent with the near-equilibrium state of the hcp lattice in the mixed phase further supporting the absence of significant stress deviators in the dynamically compressed iron.

### D. Stress release

Five experiments were performed with the x-ray diffraction measurement occurring during the stress release portion of the loading history (see Fig. 3). For experiment 20-C-070, the stress was released to $13.0\xb11.2$ GPa, and only the hcp structure with $c/a=1.64\xb10.04$ was observed. For experiment 20-C-069, the stress was released to $10.9\xb11.5$ GPa and only bcc diffraction spots corresponding to the initial bcc crystal orientation were observed; the same result was observed for the other three release experiments with lower stress. Figure 13 shows reciprocal space mapped images of the bcc (1$3\xaf$0) diffraction spot from experiment 20-C-068 where the lattice has returned to its ambient single crystal orientation, but with some additional spot broadening relative to the ambient bcc spot.

The reverse hcp to bcc phase transition occurs rapidly during release (within 2 ns) and does not show significant hysteresis with complete reverse transformation occurring between 13 and 11 GPa. In contrast, in static compression experiments significant hysteresis was observed in the reverse hcp to bcc transformation during pressure release.^{10,15} In recent static compression work on single crystal Fe, the bcc phase only appeared at 12 GPa upon release (while the forward transition occurs above 15 GPa), indicating significant stress hysteresis.^{8} Also notable is that the reverse transformation appears to follow the reverse mechanistic pathway during rapid uniaxial strain loading with the final bcc diffraction pattern consistent with the original single crystal orientation. A similar memory effect was observed for the reverse transformation in the hydrostatic experiments.^{8}

## IV. CONCLUSIONS

The interpretation of previous *in situ* XRD data for Fe single crystals laser shocked along a [100] direction was that the Fe bcc to hcp transition occurred rapidly by elastic compression of the bcc lattice along the [100] shock loading direction, driving an instability that caused alternate lattice planes to shuffle to break the cubic symmetry.^{13,14} This transition pathway caused the formation of two observable variants of the hcp structure in the shock compressed state.^{13,14} It was argued that this process could be rapid because it did not require any plastic deformation; only local shuffling of atoms was required, which was consistent with NEMD simulations for Fe shocked along [100].^{12} However, this mechanism results in a highly strained hcp lattice. In particular, a perfect hcp structure requires a 18.4% volume compression accomplished by uniaxial lattice compression along the [100] Fe shock loading direction, resulting in a $c/a$-ratio of $3$ or $\u22481.73$ far from the equilibrium value of $\u223c1.61$.^{8,10} Additionally, for arbitrary shock stresses, the volume compression in the peak state will differ from 18.4%, resulting in a distorted basal plane with different magnitudes for the hcp lattice vectors $a1$ and $a2$ in the hcp basal plane. Analysis of the early *in situ*, nanosecond-timescale XRD experiments in Fe shocked along a [100] direction supported this picture as the XRD data indicated a large $c/a$ ratio of $\u223c1.7$ for the hcp Fe.^{13,14}

The primary objective of the present work was to examine dynamically compressed Fe shocked along a [100] direction over longer timescales to determine if stress relaxation of the hcp lattice from the highly strained state reported earlier^{13,14} occurs. The present experiments utilized a Laue diffraction approach and multiple observed diffraction spots could be uniquely identified as belonging to one of two hcp variants. By examining the locations of several hcp diffraction spots mapped into reciprocal space, a displacement gradient matrix was obtained, which quantifies any strains or rotations of the hcp variants relative to a reference hcp lattice with a prescribed reference $c/a$ ratio and orientation. The same bcc/hcp orientation relationships reported previously^{13,14} were found in the present work. However, an important finding in our work is that the hcp structure is in a near-equilibrium stress state (hydrostatic stress) within 150 ps of shock compression with $c/a=1.61$ and an undistorted hcp basal plane; the same equilibrium lattice state was retained during the plateau stress state and during stress release prior to the reverse hcp to bcc transformation. Thus, stress relaxation to a near-equilibrium state occurs both in the initial shock wave (in $<$150 ps) and as the longitudinal stress is varied at later times. A possible reason for the apparent misinterpretation of the hcp $c/a$ ratio determined from earlier *in situ* laser shock Fe single crystal XRD experiments^{13,14} was discussed. The hcp lattice strain state observed in our shocked single crystal Fe samples are consistent with previous *in situ* XRD experiments on shocked polycrystalline Fe.^{16}

Although the present results for Fe shock compressed along the [100] direction result in the same $c/a$ ratio for hcp Fe observed in hydrostatically compressed single crystal Fe^{8} and polycrystalline Fe,^{10} there are some differences in the resulting bcc/hcp orientation relationships. The diffraction patterns for hydrostatically compressed single crystal Fe and polycrystal Fe were consistent with Burgers path for the bcc to hcp transition.^{8,10,26} Along the Burgers path, an atomic shuffle reorders the stacking of the lattice planes in the $[011]bcc$ direction to break the cubic symmetry and a shear of the ($1\xaf$12)_{bcc} planes in the $[1\xaf11\xaf]bcc$ direction occurs to achieve $c/a=1.61$ and $a1=a2$. Whether shearing occurs during the phase transformation or during stress relaxation of the hcp lattice after the shuffle is not clear, but shearing causes a $\theta \u223c\xb15\xb0$ rotation about the hcp $c$-axis for each of the six degenerate compression and shuffle processes resulting in 12 unique hcp variants.

In the hydrostatic experiments on single crystals, this extra rotation is directly observed in the diffraction pattern where each of the six degenerate transformation mechanisms resulted in two hcp orientations with relative rotations about the *c*-axis of $\u223c10\xb0$.^{8} In contrast, in the present dynamic compression experiments, only two of the six compression and shuffle transformation pathways are observed due to the shock loading direction breaking the degeneracy. Furthermore, the additional splitting of those two pathways into four variants with the $\u223c\xb15\xb0$ rotations about the $c$-axis associated with the ($1\xaf$12)_{bcc} shear was not observed. Previous examinations of the bcc to hcp transition mechanism for dynamically compressed single crystal iron along the [100] direction shows that uniaxial compression of the bcc lattice before (or during) the transition would reduce the effective rotation caused by the ($1\xaf$12)_{bcc} shear.^{14} Allowing up to 12% uniaxial compression of the bcc lattice before the phase transition, (this would create a $c/a$ = 1.61) results in a $\xb12\xb0$ rotation about the hcp $c$-axis to achieve $a2/a1=1$. This rotation is significantly larger than the components of the rotation matrix $\omega $ determined from the present experiment. A $\xb12\xb0$ rotation is also larger than the angular broadening of the diffraction spots for rotations about $hy=(0,0,1)$ in Fig. 11, which is only $\xb11\xb0$.

Thus, it is clear that the mechanism proposed for the hydrostatic experiments cannot fully explain the bcc/hcp orientation relationships observed in the dynamic compression single crystal experiments. This may be due to the difference in the initial state of the bcc lattice before the phase transition in the dynamic and quasi-static experiments. The quasi-static experiments have an ideal bcc structure due to hydrostatic compression, while in the dynamic experiments, the structure is tetragonal due to the elastic compression of the bcc lattice along the [100]-direction. The shifting of the $[011]bcc$ planes to break the cubic symmetry in the Burgers mechanism creates a distorted basal plane that would have a large shear strain with $a2/a1=0.86$. The shear along the ($1\xaf$12)_{bcc} planes in the $[1\xaf11\xaf]bcc$ direction is responsible for increasing $a2$ and decreasing $a1$, relieving the shear stress in the lattice. Without the lattice shear, the mechanism through which the hcp lattice forms in a fully relaxed state in under 150 ps during shock compression remains an open question.

The reverse transformation from hcp to bcc was also examined using *in situ* XRD during uniaxial strain release at $\u223c1$ GPa/ns. The reverse transformation from hcp Fe to bcc Fe was bracketed to begin and to reach completion between 13 and 11 GPa, indicating significantly less hysteresis than that observed in hydrostatic compression experiments.^{8,10} Additionally, under stress release, only the ambient bcc crystal orientation was observed (without shear strains), indicating under rapid uniaxial strain release, the reverse transformation has a memory effect. A similar memory effect was observed in hydrostatic compression experiments and attributed to a reverse Burgers pathway to go from hcp back to bcc.^{8}

## ACKNOWLEDGMENTS

The authors are grateful to Pinaki Das, Yuelin Li, Kory Green, Drew Rickerson, Korey Mercer, Paulo Rigg, Adam Schuman, Nick Sinclair, Xiaoming Wang, Austin Spencer, Ray Gunawidjaja, and Jun Zhang at the Dynamic Compression Sector (Advanced Photon Source, Argonne National Laboratory) for their assistance with the experiments, particularly with the required remote operations. This publication is based upon work supported by the U.S. Department of Energy (DOE)/National Nuclear Security Administration (NNSA) under Grant No. DE-NA0003900. This publication is also based upon the work performed at the Dynamic Compression Sector, which is operated by Washington State University under DOE/NNSA Award No. DE-NA0003957. This research used resources of the Advanced Photon Source, a DOE Office of Science User Facility operated for the DOE Office of Science by Argonne National Laboratory under Contract No. DEAC02-06CH11357.

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.

### APPENDIX A: STRESS DETERMINATION

The stress in the iron sample is determined by using the Hugoniot stress in polycarbonate and velocity measurements at the interface between the iron sample and the polycarbonate window. Figure 14 shows a representative line-VISAR image from experiment 20-C-064 where the axes show the spatial and temporal calibration of the diagnostic. For the 20-C-0XY experiments, the laser spot size was 500 $\mu $m in diameter, which is close to the field of view of the line-VISAR streak camera. The green dashed lines correspond to the edges of the laser spot. There are observable release effects at the edge of the laser spot where the breakout of the shock is at a later time than the center. The red dashed box in Fig. 14 denotes the XRD probe region. Assuming the lateral release velocity is on the order of 10 km/s, edge release waves do not affect the x-ray probe region until 20 ns after shock breakout which is the latest time an XRD measurement is made.

The variation in the break out time of the shockwave in the x-ray probe region is less than the 100 ps resolution of the line-VISAR. The shock transit time through the polyimide ablator is $\u223c10$ ns,^{18} so the variation in the breakout time suggests the shock speed variations across the probe region are $<1$%, and the shock pressure variations across the probe region are $<2$%. Note this is different from the uncertainty in the pressure, which is determined by the velocity resolution of the VISAR systems.

The breadth of the diffraction spot can be used to constrain the sample flatness as distortions (bends) in the sample would broaden the diffraction spot. Assuming the full 0.5$\xb0$ angular width of the diffraction spot is due to foil curvature, over the 100 $\mu $m probe region there is up to 250 nm variation. This suggests that over the timescale of the experiment, the compression is planar and uniaxial.

Figure 15(a) shows the velocity extracted for experiment 20-C-064 from the line-VISAR image shown in Fig. 14. The solid red curve is the average velocity over the planar image ($\u2212$200 to 200 $\mu $m in the raw data), the light red curve is the uncertainty. Figure 15(b) is a collection of six experiments showing about a 5% variation in velocity across different experiments up to 12 ns after breakout. There is more uncertainty and variation in the late time release velocity.

To calibrate the stress response of the polycarbonate, symmetric impact experiments were performed at the Institute for Shock Physics (ISP). A schematic diagram of the experiment is shown in Fig. 16(a). An aluminum coating on the impactor gives a reflective surface, which is used to record the projectile velocity before impact and the impactor/sample interface apparent velocity after impact. The velocity measurements of the impactor and impact interface are shown in Fig. 16(b) in red. The impactor/sample interface particle velocity will be half the projectile velocity after impact as the impact is symmetric. In this manner, using the measured velocity of the interface after impact, the optical correction factor $up\u2217=0.9up$ is calculated, where $up\u2217$ is the apparent material velocity and $up$ is the actual known material velocity. The shock velocity is inferred by the transit time through the 3 mm sample to the sample/window interface. The sample/window interface measured apparent velocity is shown in green in Fig. 16(b).

It is important to note that both the impact surface and the velocity profile after propagating through a 3 mm polycarbonate sample are a sharp jump to a steady velocity, suggesting at these stresses, the viscoelastic properties observed at lower stress are over driven and do not need to be included in determining the stress state. From multiple symmetric impact polycarbonate experiments, the polycabonate shock velocity, $us$, as a function particle velocity for stresses from 7 to 22 GPa is $us(up)=(2.372\xb10.046)(km/s)+(1.54\xb10.03)up$. This is consistent with previously published shock results for polycarbonate.^{27,28} The standard Hugoniot relationship, $P=\rho 0usup$ is used to determine the stress, $P$, where $\rho 0=1.197$ g/cc is the ambient density of polycarboante. The inferred stresses for each experiment are given in Table II. For the experiments where the initial shockwave is still in the sample, the stress is determined using impedance matching with the polycarbonate and iron Hugoniots.^{6} Since the timescale of reverberations rivals that of the VISAR diagnostics the sample/window interface velocity profile is fit assuming the reverberations transit the sample in 150 ps. For the early time experiments, when the initial shockwave has transited the sample, the iron stress is not in equilibrium with the polycarbonate and there will be large stress gradients due to the reverberations in the iron sample. For these experiments, the stress range given is the maximum and minimum reverberation stresses in the sample. In early time experiments when the elastic wave is not observed, the stress range includes the full range of stress the sample reverberates through.

Experiment . | Interface velocity . | Stress . |
---|---|---|

. | (km/s) . | (GPa) . |

20-C-057 | 2.33 ± 0.12 | 16.7 ± 1.7 |

20-C-059 | 2.33 ± 0.12 | 16.7 ± 1.7 |

20-C-060 | 2.39 ± 0.12 | 17.3 ± 1.7 |

20-C-061 | 2.28 ± 0.11 | 16.0 ± 1.6 |

20-C-064 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-065 | 2.00 ± 0.11 | 29 ± 5 ^{a} |

20-C-066 | 2.00 ± 0.11 | 29 ± 5 ^{a} |

20-C-067 | 2.44 ± 0.12 | 18.0 ± 1.8 |

20-C-068 | 1.11 ± 0.2 | 5.4 ± 1.5 |

20-C-069 | 1.77 ± 0.14 | 10.9 ± 1.5 |

20-C-070 | 2.01 ± 0.12 | 13.0 ± 1.2 |

20-C-401 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-407 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-410 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-416 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-418 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-421 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-423 | 2.33 ± 0.12 | 16.7 ± 1.6 |

20-C-424 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-425 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-428 | 1.11 ± 0.12 | 5.4 ± 1.0 |

Experiment . | Interface velocity . | Stress . |
---|---|---|

. | (km/s) . | (GPa) . |

20-C-057 | 2.33 ± 0.12 | 16.7 ± 1.7 |

20-C-059 | 2.33 ± 0.12 | 16.7 ± 1.7 |

20-C-060 | 2.39 ± 0.12 | 17.3 ± 1.7 |

20-C-061 | 2.28 ± 0.11 | 16.0 ± 1.6 |

20-C-064 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-065 | 2.00 ± 0.11 | 29 ± 5 ^{a} |

20-C-066 | 2.00 ± 0.11 | 29 ± 5 ^{a} |

20-C-067 | 2.44 ± 0.12 | 18.0 ± 1.8 |

20-C-068 | 1.11 ± 0.2 | 5.4 ± 1.5 |

20-C-069 | 1.77 ± 0.14 | 10.9 ± 1.5 |

20-C-070 | 2.01 ± 0.12 | 13.0 ± 1.2 |

20-C-401 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-407 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-410 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-416 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-418 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-421 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-423 | 2.33 ± 0.12 | 16.7 ± 1.6 |

20-C-424 | 2.00 ± 0.11 | 7 → 30 ^{b} |

20-C-425 | 1.90 ± 0.1 | 12.0 ± 1.2 |

20-C-428 | 1.11 ± 0.12 | 5.4 ± 1.0 |

^{a}

Stress is determined by shock impedance matching with iron.

^{b}

Stress range of the all the reverberating waves.

### APPENDIX B: CRYSTALLOGRAPHY NOTATIONS AND MAPPING XRD LAUE SPOTS TO RECIPROCAL SPACE

There are many different formulations that can be used to describe the scattering of radiation from matter. For analysis of the present experiments, we use the wave vector-scattering vector relationship where the difference between the wave vector describing the incident x rays, $ki$, and the wave vector describing the scattered x rays, $ks$, is equal to the scattering vector $G=ks\u2212ki$.

For crystalline materials that have atoms located on a periodic lattice with basis vectors $a1$, $a2$, and $a3$ defining the unit cell, there is a corresponding set of three basis vectors $b1$, $b2$, and $b3$, which define a reciprocal unit cell in Fourier space,

where $i$, $j$, and $k$ cycle through the corresponding indices (1, 2, and 3) is in either the real or reciprocal space.

For x-ray diffraction, a peak in scattered intensity occurs when the scattering vector $G$ is equal to a reciprocal lattice vector, $G(hkl)=hb1+kb2+lb3=(h,k,l)$, where $h$, $k$, and $l$ are integers. We use the notation $(h,k,l)$ to indicate the reciprocal lattice vector $G(hkl)$. The intensity of the diffraction peak corresponding to $G(hkl)$ depends on the types and locations of the atoms in the unit cell through the structure factor. Reciprocal lattice points located at $G(hkl)$ with structure factors equal to zero will not diffract.

The reciprocal lattice vector $G(hkl)$ is normal to a set of real space lattice planes and $1/|G(hkl)|$ is equal to the real space lattice plane spacing $dhkl$. For bcc Fe, we refer to real space crystallographic planes by the unitless Miller indices ($hkl$). For hcp Fe, we refer to real space crystallographic planes by the unitless Bravais–Miller indices ($hkil$), where $i=\u2212(h+k)$.

The bcc real space lattice directions are denoted by the unitless indices [$uvw$]. The hcp real space lattice directions are denoted by the unitless indices [$uvtw$], where $t=\u2212(u+v)$. For both crystal structures, the real space lattice direction is parallel to the vector $ua1+va2+wa3$.

For elastic scattering, $|ks|=|ki|=1/\lambda $, where $\lambda $ is the wavelength of the diffracting x rays. This restriction forces $|G(hkl)|=2|ki|sin(\theta G)$, where 2$\theta G$ is the angle between the incoming and outgoing x rays scattered from $G(hkl)$ as shown in Fig. 17(a). This is equivalent to Bragg’s law

where $\theta G$ is the x-ray angle of incidence relative to the diffracting planes $(hkl)$.

For conventional powder diffraction measurements, $|ki|$ is set to a single value $1/\lambda $ and for a random distribution of grain orientations, grains which satisfy the relationship $G(hkl)=ks\u2212ki$ will scatter x rays forming a diffraction cone. In contrast, for single crystal Laue XRD, a broadband x-ray source is used and a particular reciprocal lattice vector direction $G(hkl)$ is fixed by the crystal orientation and the x-ray wavelength satisfying $G(hkl)=ks\u2212ki$ will result in diffracted x rays provided those x-ray wavelengths are contained in the incident x-ray beam. If the crystal lattice parameter magnitudes are known (such as for the ambient bcc Fe), the x-ray wavelengths contributing to each ($hkl$) diffraction spot can be calculated. However, once the crystal has been shocked the lattice parameters are generally not precisely known and, hence, the x-ray wavelength contributing to each ($hkl$) diffraction spot cannot be calculated. As discussed below, this latter fact means that the Laue diffraction technique is not sensitive to isotropic compression of the lattice; the Laue diffraction technique provides information regarding lattice shear strains and lattice rotations.

Figure 17 shows an example of how the 2D reciprocal lattice space maps are formed from the measured diffraction pattern. As an example, we consider the mapping of the area near the bcc Fe ($13\xaf0$) diffraction spot shown on the detector in Fig. 4(a) to the 2D reciprocal lattice map shown in Fig. 5. In general, for either ambient or shocked state diffraction data, the approach requires using a reference configuration for the crystal lattice defined by the crystal reciprocal lattice basis vectors and their orientation in real space. A particular reciprocal lattice vector $G0(hkl)$ from the reference configuration is then chosen and the 2D reciprocal space map is in a plane normal to $G0(hkl)$, which passes through the origin of the reciprocal space map. In our example, the ambient bcc Fe lattice with the measured ambient orientation is chosen as the crystal lattice reference configuration and the $G0(hkl)=G(13\xaf0)$ ambient reciprocal lattice vector is used to define the reciprocal space map plane and origin as shown in Fig. 17(a).

Figure 17(a) shows the x-ray scattering configuration. From the known experimental geometry, each detector pixel corresponds to a unique direction for the outgoing wavevector $ks(Q)$ and a unique direction for the outgoing scattering vector $Q$. However, the magnitudes of $ks(Q)$ and $Q$ are unknown except for an ideal single crystal with known lattice parameters, conditions not met for the shocked single crystal Fe. Thus, diffracted intensity at a single point on the detector can correspond to any location along the line $O\u2212Q\u2032$ in the reciprocal space [see Fig. 17(a)], provided the spectrum of x rays incident on the sample contains the necessary wavelength corresponding to a particular scattering vector $Q$ that contains finite scattering intensity. Thus, the measured diffracted intensity in a detector pixel corresponding to the direction of $Q$ is effectively an integration of the intensity in the reciprocal space along the $Q$ vector direction. Therefore, mapping the measured detector x-ray signal to reciprocal space results in two-dimensional maps; the intensity distribution in the reciprocal space along the $Q$ vector direction cannot be determined.

In constructing the reciprocal space maps, two orthogonal reciprocal lattice vectors $hx$ and $hy$ are selected each of which is orthogonal to the reciprocal lattice vector $G0(hkl)$ defining the origin of the reciprocal space map. In the present example, $G0(hkl)=G(13\xaf0)$ and the two orthogonal reciprocal lattice vector directions used for the 2D reciprocal space map are $hx=(3,1,0)$ and $hy=(0,0,1)$. The reciprocal space maps are plotted in terms of small rotation angles $\gamma $ about the $hy$ axis and $\beta $ about the $hx$ axis, which together rotate $G0(hkl)$ to be parallel to $Q$.

The reciprocal lattice displacement vector $\Delta g$ is defined as the vector, which added to $G0(hkl)$ results in a vector parallel to $Q$ that ends on the 2D reciprocal space map plane as shown in Fig. 17(a); for this example, $\Delta g$ is parrallel to $hx$. Thus, any point $P$ in the 2D map corresponds to a reciprocal space scattering vector $P=(1+\alpha )[G0(hkl)+\Delta g]$. The $\alpha $ term corresponds to isotropic lattice strains and cannot be determined from the Laue XRD experiments. However, $\Delta g$ is related to lattice shear strains and lattice rotations relative to the reference configuration. For small reciprocal lattice vector displacements, the components of $\Delta g$ along the $hx$ and $hy$ direction can, to first order, be linearly mapped to the rotation angles $\gamma $ and $\beta $,

where $\Delta gx\u2225hx$ and $\Delta gy\u2225hy$ are the displacements parallel to $hx$ and $hy$, respectively.

The diffraction spots mapped into reciprocal space have finite breadth and their centroids are determined by fitting a two-dimensional Gaussian to the mapped spots resulting in $\Delta gx$ and $\Delta gy$ values for each mapped spot. As described in Appendix C, the displacement gradient matrix for the shocked iron can be calculated relative to the reference lattice configuration by calculating $\Delta gx$ and $\Delta gy$ for a number of reciprocal space mapped diffraction spots.

### APPENDIX C: SOLVING FOR THE DISPLACEMENT GRADIENT MATRIX

This section describes how the displacement gradient matrix $Jij=\u2202ui/\u2202xj$ relating the actual Fe crystal lattice to a reference Fe crystal lattice configuration is calculated. A particular point $P$ in the reciprocal space map corresponds to the reciprocal lattice vector $P$,

where $\alpha $ is a scale factor associated with the unknown length of the reciprocal lattice vector $P$ and the latter approximation assumes that both $\alpha $ and $\Delta g$ are small. A reciprocal lattice displacement gradient matrix $jij$ relates each reference reciprocal lattice vector to the corresponding measured reciprocal lattice vector $P$,

For small reciprocal lattice displacement gradients,

keeping only first order terms.

The reciprocal lattice displacement gradient matrix can be decomposed into two parts,

where $I$ is the identity matrix, the $\alpha I$ term corresponds to isotropic reciprocal lattice strains, and $j\u2032$ represents reciprocal lattice shear strains and reciprocal lattice rotations. The $\alpha $ value cannot be solved from the Laue diffraction data so $\alpha $ remains indeterminate in our solution for $j$. A nonzero $\alpha I$ value in $j$ simply translates to an isotropic strain term $\u2212\alpha I$ in the real space displacement gradient, $J$. In Eq. (C2), both $G0(hkl)$ and $Pj$ are known and $jij$ is unknown. By applying Eq. (C2) to a number of linearly independent diffraction spots from the reference configuration $G0(hkl)n$ to their corresponding shifted reciprocal lattice vectors $Pn$, we obtain the matrix equation,

From this point, conventional matrix methods can be used to solve for the reciprocal lattice displacement gradient matrix $j$,

where $\Gamma $ is the matrix of $G0(hkl)n$ vectors, $\Omega $ is the matrix of measured reciprocal lattice vectors $Pn$ , $\Gamma T$ denotes the transpose of matrix $\Gamma $, and $()\u22121$ denotes the matrix inverse.

Once $j$ has been determined the real space displacement gradient matrix is given by

where $\omega $ is the antisymmetric portion of $Jij$ and represents displacements due to rotations and $\epsilon $ is the strain matrix, which is known to within a multiple of the identity matrix $\alpha I$, which represents isotropic lattice strains. As discussed in the main text, $J$ and $\epsilon $ are expressed in different coordinate systems for bcc and hcp Fe with the coordinate system choices judicially chosen to highlight the desired lattice parameter ratios.

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