# Measurement of competing pathways in a shock-induced
phase transition in zirconium by femtosecond diffraction

Saransh Singh111Lawrence Livermore National Laboratory, Livermore, CA, USA
<EMAIL_ADDRESS>Martin G. Gorman22footnotemark: 2 Patrick G.
Heighway333Department of Physics, Clarendon Laboratory, University of Oxford,
Oxford, UK Joel V. Bernier44footnotemark: 4
David McGonegle 55footnotemark: 5666Oxford Centre for High Energy Density
Science, Department of Physics, Clarendon Laboratory, University of Oxford,
Oxford, UK Hae Ja Lee777Linac Coherent Light Source, SLAC National Accelerator
Laboratory, Menlo Park, CA, USA Bob Nagler88footnotemark: 8 Jon H.
Eggert99footnotemark: 9 Raymond F. Smith1010footnotemark: 10

††journal: Nature Materials

The traditional picture of solid-solid phase transformations assumes an
ordered parent phase transforms into an ordered daughter phase via a single
unique pathway. Zirconium and its prototypical phase transition from hexagonal
close-packed (hcp) to simple hexagonal (hex-3) structure has generated
considerable controversy over several decades regarding which mechanism
mediates the transformation. However, a lack of _in situ_ measurements over
the relevant atomistic timescales has hindered our ability to identify the
true pathway. In this study, we exploit femtosecond X-ray diffraction coupled
with nanosecond laser compression to give unprecedented insights into the
complexities of how materials transform at the lattice level. We observe
single-crystal zirconium changing from hcp to a hex-3 structure via not one
but three competing pathways simultaneously. Concurrently, we also observe a
broad diffuse background underlying the sharp Bragg diffraction during the
transition. We corroborate our observation of the diffuse signal with
multimillion-atom molecular dynamics simulations using a machine-learned
interatomic potential. Our study demonstrates that the traditional mechanistic
view of transitions may fail for even an elemental metal and that the
mechanisms by which materials transform are far more intricate than generally
thought.

Solid-solid phase transitions have been an active area of scientific inquiry
for the last hundred years (Bridgman1912, Bain1924, Kurdjumow1930, Shoji1931,
Wassermann1933, Burgers1934, Nishiyama1934). The lattice-level restructuring
brought on by applying sufficient heat or pressure is conventionally measured
using X-ray diffraction techniques, yielding direct interatomic separation and
coordination measurements. Ultrabright sources such as X-ray free-electron
lasers (XFELs) make it possible to realize such measurements in situ in solids
dynamically compressed to extreme, planetary-scale pressures over mere
nanoseconds. Recent research efforts have focused increasingly on using
diffraction to pin down not only the structure of exotic high-pressure phases
but the atomistic pathway by which that phase is reached (Yagi1992,
Hawreliak2006, Pandolfi2022).

The conventional picture of the atomistic pathways is based upon atomic
mapping from an ordered parent phase to an ordered daughter phase via lattice
strain and atomic shuffles. Each transition pathway is differentiated by the
orientation with which it leaves the daughter phase relative to its parent.
Thus, by measuring the orientation relationship (OR) between specific
crystallographic planes and directions within the parent and daughter phases
using diffraction, one can experimentally constrain the atomistic transition
mechanism.

Due to the commercial and technological importance of zirconium (Zr), titanium
(Ti), and their alloys, the transformation mechanism between the ductile
$\alpha-$phase (hcp) and the brittle high-pressure $\omega-$phase (simple
hexagonal) has been the subject of intense theoretical and experimental
scrutiny (SILCOCK1958481, SARGENT1971, RABINKIN1981, UZ1973, Kutsar1990,
Swinburne2016, Song1995, Jyoti1997, jyoti2008, osu1070481734, Trinkle2003,
Wenk2013, ADACHI20151, wang2019, ZONG2014a, ZONG2014b, Guan2016, Rawat2017,
ZONG2019). Existing literature suggests that two distinct mechanisms can
mediate the $\alpha\to\omega$ phase transition. The majority of shock-
compression studies on recovered samples have observed what we will call the
variant I OR (Song1995, Jyoti1997, jyoti2008). In contrast, the majority of
static-compression and high-pressure torsion studies report variant II
(SILCOCK1958481, SARGENT1971, RABINKIN1981, Wenk2013, ADACHI20151, wang2019)
(see supplementary materials Table LABEL:Table:OR). However, several studies
deviate from this trend (UZ1973, Kutsar1990, Swinburne2016) for reasons that
remain unclear. A complete understanding of this model phase transition thus
remains elusive.

In this study, we approach the $\alpha\to\omega$ phase transition from a new
direction by combining uniaxial laser-shock compression of high-purity, well-
oriented single-crystal Zr with _in-situ_ diffraction measurements using a
high-brightness femtosecond X-ray free electron laser (XFEL) source. We
produce near-instantaneous snapshots of atoms in the high-pressure shocked
state as the material is transforming using the femtosecond probe duration,
revealing lattice-level details of the phase transition mechanism.

Fig. 1:  $|$ Experimental setup for X-ray diffraction measurements of shock-
compressed single-crystal Zr samples. a. The 10 keV, 50 fs output of the LCLS-
XFEL probed the shock-compressed Zr and produced an X-ray diffraction signal
as recorded on CSPad detectors (Hermann2013). Here, the diffraction angle
($2\theta$) increases radially, with azimuthal angle ($\phi$) defined around
the Debye-Scherrer cones. This data provides information on crystal structure,
sample density, and microstructural texture, while velocimetry measurements
provided information on shock timing and pressure uniformity during the X-ray
probe period (Celliers2004). The diffraction data shown [10.6 GPa] is
consistent with multiple reflections from the compressed $\alpha$ phase (red
arrows) and a single intense peak from the $\omega-$phase (cyan arrows). b. We
present the same diffraction data in polar coordinates. The Debye-Scherrer
rings appear as straight lines in this view. We show the $\phi$-averaged
lineout in the lower panel, where the green curve represents the detector
background. The peaks from the $\alpha$ and $\omega-$phase are shaded with red
and cyan colors, respectively. We reproduce an enlarged view of the lower
diffraction-angle region in the inset. The lineout to the right of the inset
shows the azimuthal variation of the diffuse signal between the Bragg peaks
noted by the dashed vertical lines.

We performed laser-driven shock-compression experiments at the Matter in
Extreme Conditions (MEC) endstation of the Linac Coherent Light Source (LCLS)
(Nagler2015, brown2017). We compressed eight targets with peak pressures
ranging from 5.3 to 21.3 GPa. We show a schematic of the experimental setup
and target design used at MEC in Fig. 1a. The target consisted of a plastic
ablator and a 40-$\mu$m-thick Zr single crystal with the $[0001]$ direction
parallel to the sample normal (shock direction). We used an ultrabright XFEL
pulse incident normal to the target to perform volume-integrated X-ray
diffraction. We used Cornell-Stanford Pixel Array Detectors (CSPADs) to record
the scattered X-rays from the shocked Zr sample in transmission geometry
(Hermann2013).

In Fig. 1a, we show the 2D orthographic view of diffraction data for a shock
pressure of 10.6 GPa (marginally above the phase transition pressure). While
the Zr sample loses its simple single crystalline nature upon shock-
compression, it retains a highly oriented microstructure, as seen in the
localized intensity around the Debye-Scherrer diffraction rings. At this
pressure, we observe diffraction signals from both the compressed $\alpha$ and
$\omega-$phases. We do not measure any diffraction from the uncompressed
material ahead of the shock front due to the highly oriented nature of the
zirconium foil (supplementary materials Fig. S1). We also observe a diffuse
background signal superimposed on top of sharp Bragg peaks. This diffuse
signal (marked by dashed lines in the Fig. 1b inset) in the intervening
regions between the Bragg peaks is observed to be non-uniform azimuthally and
has an angular period of $60^{\circ}$ indicating a sixfold symmetry. The
azimuthally localized diffuse signal reveals itself only in our single-crystal
scattering configuration; we do not observe it in our polycrystalline Zr
diffraction data (see supplementary materials Fig. S2 and accompanying text).

We show the azimuthally averaged lineouts at five pressures above the
$\alpha-\omega$ phase boundary in Fig. 2a. We have shaded the peaks
originating from the $\alpha-$phase in red and the $\omega-$phase in cyan. The
$\alpha-$phase, which dominates the diffraction signal at 10.6 GPa, gradually
recedes in intensity with increasing pressure. The diffraction signal at 18.1
GPa is almost exclusively from the high-pressure $\omega-$phase. We also
observe a reduction in the integrated diffuse signal (hatched area) with an
increase in pressure.

To accurately account for the observed intensity distribution in each Debye-
Scherrer ring, we constructed a forward model to calculate the expected
diffraction signal for our experimental geometry. This model accounts for
X-ray beam energy and direction, detector positions, and crystallographic
information about the sample, such as crystal structure, unit cell dimensions
on compression, phase volume fraction, and crystallographic texture of each
phase (see Methods section for details).

Fig. 2:  $|$ Evolution of diffraction signal as a function of shock pressure.
a. Waterfall plot of azimuthally averaged lineouts from single crystal Zr
targets. The diffuse scattering signal decreases with increasing pressure,
with a corresponding increase in the number of peaks observed for the
$\omega-$phase. b. Experimental and modeled diffraction patterns from shock-
compressed single-crystal zirconium at 15.2 GPa. We modeled the
crystallographic texture of the $\omega-$phase as the sum of unimodal
distributions centered around the $12$ orientations of the variant I OR, $3$
orientations of variant II OR, and $6$ orientations of the variant III OR. We
show the region with diffraction from the $\\{10\bar{1}0\\}_{\alpha}$,
$\\{10\bar{1}1\\}_{\omega}$ and $\\{11\bar{2}1\\}_{\omega}$ planes.
Contributions to the $\\{10\bar{1}1\\}_{\omega}$ diffraction from the ORs of
variants I and III are azimuthally separated and indicated by the cyan and
blue arrows, respectively. Diffraction from the $\\{11\bar{2}1\\}_{\omega}$
plane originates entirely from variant II OR. c. Volume fraction of each OR
formed within the $\omega-$phase shows a strong pressure dependence. We
observe the dominance of variant III OR at low pressures, which variant II OR
eventually supplants at higher pressures.

We present the results of the forward model calculation for the 15.2 GPa shot
in Fig. 2b. We determined the phase fractions of the $\omega-$phases to be
close to 100% for this shot. The azimuthal intensity distribution is
consistent with the presence of $\omega-$phase orientations from three
distinct phase transition pathways: variants I, II, and III. For clarity, we
only show the variant contributions to two reflections. While variants I and
II have been reported in numerous studies, variant III has never been observed
experimentally. We present the pressure dependence of the phase-fraction
evolution, the relative abundance of the different phase transition ORs within
the $\omega-$phase, and the integrated diffuse background signal in Fig. 2c.
The phase fraction of the $\alpha-$phase decreases with increasing pressure as
expected. In the $\omega-$phase volume, we observe three competing phase
transition pathways vying for dominance. variant III dominates at lower
pressures and is eventually superseded by variant II as the pressure
increases. We refer the readers to supplementary materials Fig. S3 for our
semi-quantitative method of determining the phase fractions of different
phases and the relative abundance of the various phase transition ORs and Fig.
S4 for a comprehensive collection of the forward model results with increasing
pressure.

Fig. 3:  $|$ Molecular dynamic simulations with machine learning potential. a.
Visualization of a Zr single crystal with initial dimensions of $40\times
40\times 100$ nm3 shock-compressed along [0001] to 14 GPa, with atoms colored
according to their local phase (above) and shear stress (below) after 18 ps,
performed using ovito (OVITO). Below is the time evolution of the $\alpha$,
$\beta$, $\omega$, and non-crystalline phase fractions (the last group
includes both partially disordered nano-clusters and atoms on grain
boundaries). The circular inset shows the local arrangement of atoms in a
region of phase coexistence, viewed in a plane normal to the shock. b.
Synthetic diffraction generated 158 ps into the simulation using 10 keV
X-rays. The top panel shows the full diffraction pattern in polar coordinates,
with a semitransparent mask added to simulate gaps in the CSPad detectors. The
lower panel compares azimuthally averaged lineouts from the simulation (14
GPa) and experiment (15.2 GPa). The right panel shows the azimuthal variation
of the simulated diffuse signal centered at $2\theta\sim 30^{\circ}$, with
averages taken over the windowed region (red-boxed region). We overlay the
azimuthal distribution of the azimuthally localized diffuse signal from the
experimental data (shot 93) in blue.

The azimuthally localized yet diffuse diffraction alludes to further
complexity during the phase transition. To investigate the origin of this
diffuse signal, we performed large-scale classical molecular dynamics (MD)
simulations of [0001]-oriented Zr single crystals with initial dimensions of
$40\times 40\times 100$ nm3 shocked to 14 GPa using LAMMPS (LAMMPS). We
modeled the interatomic interactions using the machine-learned potential of
Zong et al. (ZONG2019), which was tailored to model allotropy in Zr at
pressures of up to 30 GPa. These simulations allow us to see in microscopic
detail the dynamics unfolding in the wake of the shock and to analyze the
structure factor of the complex atomistic configuration that results.

We show in Fig. 3a a visualization of the computational cell 18 ps into the
simulation, with atoms colored according to their local phase (above) and
shear stress (below). The shock is led by an elastic precursor, behind which
the pressure and shear stress are elevated to 10.0 GPa and 3.5 GPa,
respectively, but the crystal retains a strained $\alpha-$phase structure.
Trailing this precursor is a phase transition front that raises the pressure
to 14 GPa while relaxing the shear stress to 0.9 GPa, heralding the onset of
the $\alpha\to\omega$ transition. The parent $\alpha$ phase first transforms
to a short-lived ($\sim$10 ps) intermediate $\beta$ (bcc) phase before taking
one of two pathways: the $\beta$ nano-grains either promptly transform to the
$\omega$ structure via a displacive mechanism (UZ1973), or they rapidly
amorphize and subsequently reconstitute into an anisotropic yet non-
crystalline structure via slower diffusive motion. The resulting clusters of
partially disordered atoms form 10% of the cell by mass and occupy the
interstices between the larger $\omega-$grains (which constitute 60%). Though
non-crystalline, the atomistic structure of these nano-clusters shows
orientational order and a structure factor qualitatively similar to that of
the $\omega-$phase (see supplementary materials Fig. S5 for the radial and
angular distribution functions and Fig. S6 for the structure factors of the
$\omega$ and non-crystalline structures).

In Fig. 3b, we show synthetic diffraction generated by the phase-transformed
region of the simulation cell with 10 keV X-rays. Sharp diffraction peaks
attributable to $\omega$ grains with the variant II OR are visible at Bragg
angles similar to those measured in the experiment and at identical azimuthal
angles. The difference in peak intensities is due to the strong sensitivity of
the Bragg condition to the $\omega$ phase’s $c/a$ ratio, which at 14 GPa takes
values of 0.605 and 0.621 in the experiment and simulation, respectively. In
addition to these Bragg peaks, we observe a diffuse diffraction signal
centered at $29^{\circ}$ akin to that seen in the data. Like the experimental
signal, this diffuse diffraction is anisotropic and exhibits the same sixfold
modulations around the Debye-Scherrer ring. While it is tempting to attribute
the diffuse scattering to the partially disordered nano-clusters, the truth is
subtler: both the crystalline $\omega-$phase and disordered material
contribute to the diffuse signal. Moreover, their contributions do not
decompose additively, suggesting they diffract coherently. The signal in
question is diffuse because the active scattering vectors are situated far
from the sharp maxima in the aggregate’s structure factor. Our simulations
suggest that we are measuring tenuous interpeak scattering intensity that is
dwarfed by intense Bragg diffraction in a conventional polycrystal experiment.

The new variant III OR we observe in our experiments has never been observed
in quenched $\omega-$Zr captured from recovery experiments. This observation
is in contrast to numerous experimental static and dynamic compression studies
reporting variant I and variant II ORs (SILCOCK1958481, SARGENT1971, UZ1973,
RABINKIN1981, Kutsar1990, Song1995, Jyoti1997, jyoti2008, Wenk2013,
ADACHI20151, wang2019). In a recovery study of shock and pressure released Zr,
samples that completely transformed to the $\omega-$phase during shock loading
retained between $0$ to $48\%$ of the high-pressure phase in recovered samples
(gorman2020). The lack of any experimental observation of the variant III OR
in recovered samples could be related to an OR dependence of the
$\omega\rightarrow\alpha$ reverse phase transition. Upon pressure release, we
hypothesize a complete reversion of the variant III $\omega-$phase
orientations to the $\alpha-$phase, but hysteresis for the other variants.

In our MD simulations, the vast majority (99%) of the $\omega-$phase assumes
the variant II OR, with only trace amounts of variant I present. This
prediction contrasts with the multi-mechanism picture painted by the
experiment. However, the simulation properties can rationalize this behavior.
Zong _et al._ (ZONG2019), in their MD study using the identical interatomic
potential as our study, have reported a switch in the $\omega-$phase OR from
variant II to variant I upon artificially reducing shear stress in their
simulations. We posit that variant I is made rare by the artificially high
shear-stress environment created in these idealistic simulations in perfect,
defect-free crystals. In the presence of defects, some plasticity before the
phase transition will bring about the low shear-stress environment that favors
variant I OR over variant II. Recent static high-pressure diffraction
measurement by Singh _et al._ (Singh2023) on highly oriented Zr also supports
this physical model. In contrast to other static-compression experiments
(SILCOCK1958481, SARGENT1971, RABINKIN1981, Wenk2013, ADACHI20151, wang2019),
the authors in that study observed variant I OR under static-compression when
neon was used as a pressure medium to ensure purely hydrostatic conditions. We
believe the absence of variant III in our simulations results from the
interatomic potential’s inability to accurately model the energy landscape of
Zr along the variant III pathway. As this transformation pathway was
undiscovered until now, its configuration space did not form part of the
database on which the potential was trained. Exploration of this new pathway
via first-principles atomistic simulations is warranted.

Our MD simulations predict a small amount of the intermediate bcc
$\beta-$phase in the tens of ps timescale. Our forward diffraction model
predicts that the $\\{220\\}_{\beta}$ reflection should be observable in our
experiments for the expected Burger’s OR (Burgers1934) between the hcp
$\alpha-$phase and the bcc $\beta-$phase. Although not conclusive, our
diffraction data hints at the presence of this intermediate bcc phase. We
refer the readers to the supplementary materials Fig. S7, S8 and accompanying
text for more details.

In summary, the dynamic behavior of the elemental metal Zr under shock
compression is far more intricate than previously thought, involving multiple
competing phase transition pathways and a partially disordered state in a
single ns shock volume. While we initially attempted to use the more
conventional EAM potential for Zr, only the machine-learned interatomic
potential partially reproduced our experimental observations. Our experiments
and analysis highlight the importance of in situ measurements on highly
oriented crystals in validating our best theoretical models at extreme
conditions. While our results focus on zirconium, we expect our results to
apply to the transformation dynamics of other material systems at high
pressure and temperature.

Methods

Target. A schematic of the experimental setup and target design used at MEC is
shown in Fig. 1. The target consisted of a 50-$\mu$m thick polyimide ablator,
a 40-$\mu$m-thick Zr [0001] single crystal. The single crystal Zr samples were
supplied by Princeton Scientific Corp. The samples were orientated to with $<$
1∘ of the [0001] axis. An impurity analysis confirmed $>$99.95% purity and an
ambient-pressure density of 6.52 g/cm3.

Forward model. Given a material system with a known crystal structure, lattice
parameters, temperature, and crystallographic orientation distribution
function, the forward model computes the expected diffraction signal in
$2\theta-\phi$ space. In any diffraction experiment in the transmission
geometry and a single sample view, only a single ring of the complete pole
figure is observable. Therefore, given an orientation distribution function,
the forward model modulates the powder diffraction intensity in the azimuthal
direction by the observable pole density for each one of the reflections. Fig.
S9 shows the complete pole figures for a unimodal orientation distribution
centered around the starting orientation of the $\alpha-$phase. We show
similar pole figures for the $\omega-$phase for the three observed ORs in
Figures S10, S11 and S12. The dotted magenta circle shows the observable ring
of the complete pole figure for each of the reflections. The computation
proceeds by the following steps:

  1. 1.

For a given material and reciprocal lattice reflection, compute the powder
diffraction intensity. The factors include the volume fraction of the phase,
unit cell volume, structure factor, Lorentz-polarization factor, the
multiplicity of the reflection, and the Debye-Waller factor.

  2. 2.

Modulate the powder diffraction intensity in the azimuthal direction by
multiplying the magnitude of the pole density around the observable ring.

  3. 3.

Plot the intensity from the previous step as a pseudo-voigt function in
$2\theta$ dimension. The full-width half maxima is a function of $2\theta$ via
the Cagliotti parameters (CAGLIOTI1958) and particle size, micro-strain
parameters. The Thomas-Cox-Hastings profile function (Thompson1987) gives the
mixing factor.

  4. 4.

Apply the mask in $2\theta-\phi$ derived from the composite CSPAD detector.

We performed all computations relating to the orientation distribution
function and pole density using the finite element representation of the
Rodrigues space fundamental zone (KUMAR1998, Barton2002).

Laser compression. The output of the 527-nm drive laser at MEC, temporally
shaped into a 15-ns flattop pulse, was focused into a 250-$\mu$m spot on the
front of the polyimide. This generated an ablatively-driven shock into the
target assembly, which an applied uniaxially load along the c-axis of the Zr
crystal. Shock pressure in the sample is directly proportional to the laser
intensity, and, therefore precise control of the on-target energy allowed for
precise tuning of the pressure within the Zr sample (brown2017). We set the
X-ray probe time in our experiments to ensure the same shock-compressed Zr
volume from shot-to-shot, as determined by the velocimetry measurements (Fig.
S13a).

Molecular dynamics simulations. Large-scale classical molecular dynamics (MD)
simulations of dynamically compressed zirconium were performed using the open-
source code LAMMPS (LAMMPS). Interatomic interactions were modeled using the
machine-learned potential developed by Zong et al. (ZONG2019). This short-
range classical potential is trained on a reference energy database generated
using ab initio MD simulations and nudged elastic band calculations. The
database includes atomistic configurations representing the $\alpha$, $\beta$,
and $\omega$ phases of Zr – as well as those found along the high-symmetry
transition pathways from $\beta\to\alpha$, $\beta\to\omega$, and
$\alpha\to\omega$ – at pressures of up to 30 GPa. In our estimation, this
potential is best suited to treating allotropy in Zr at high pressures.

The crystals we simulate are initially defect-free blocks of $\alpha$-Zr with
their [0001] direction aligned with the compression axis, $z$. The crystals
are subjected to periodic boundary conditions in the transverse directions,
$x$ and $y$, to simulate laterally confining material present behind a
uniform, uniaxial shock front. All crystals span at least 40 x 40 nm2 in the
$x$ and $y$ directions to mitigate finite-size effects that might otherwise
influence grain growth. Shock waves were driven into the crystal using a
momentum mirror ramped to the final particle velocity $U_{P}$ over 250 ps. A
1-fs timestep is used throughout.

To relax computational costs, we use an iterative scheme whereby the length of
the crystal is grown incrementally at a rate that keeps pace with the shock
front. At every timestep, there exists a tracer layer at a distance
$\delta=U_{S}^{e}t_{\text{eq}}$ from the crystal’s rear surface, where
$U_{S}^{e}$ is the speed of the elastic precursor wave, and $t_{\text{eq}}$ is
the time required for a pristine block of crystal to thermalize ($\sim 1$ ps).
When the tracer velocity exceeds $0.5U_{P}^{e}$, a new block of Zr of
thickness $\delta$ is seamlessly appended to the rear surface; the new
material thermalizes for at least time $t_{\text{eq}}$ before the shock front
sweeps through it. By comparing simulations carried out under an EAM potential
with and without this incremental scheme, we have verified that the stress
waves generated by the addition of new material (which cause oscillations in
the pressure and shear stress of $\pm 1\%$ and $\pm 5\%$ of their peak values,
respectively) do not substantively change the final thermodynamic state
reached by the system. By building the ‘runway’ required by the shockwave
iteratively – rather than constructing the crystal with its ultimate
dimensions at the outset – we avoid needless simulation of ambient material
waiting to be compressed and thus double our computational efficiency.

We also set a ‘cutoff time’ after which we isolate a relatively small portion
of the crystal far behind the shock front for study, and delete the remainder.
Once a sufficiently large volume of phase-transformed Zr has accumulated near
the front surface, we find the greatest $z$-coordinate at which the average
particle velocity is no less than 95% of the Hugoniot particle velocity
$U_{P}$, insert at this limit a second momentum mirror co-moving with the
first at fixed velocity $U_{P}$, and remove all atoms beyond the second
mirror. The remaining block of Zr, which comprises at least 2 million atoms,
evolves under constant-volume boundary conditions until the global phase
fractions stabilize.

To partition the crystal into its constituent $\alpha$, $\beta$, $\omega$, and
non-crystalline phase fractions, we use the following pipeline [executed in
OVITO (OVITO)]:

  1. 1.

Following Zong et al. (ZONG2019, Zong2020), we first classify the atoms
according to their Ackland-Jones parameter (AJP) (AcklandJones2006). Atoms in
the $\omega$ phase take AJP values of 2 or 3, depending on whether they are
situated on $\omega$’s graphene-like planes or the basal planes that sandwich
them, respectively. For brevity, we refer to atoms classified thus as
‘graphitic’ and ‘basal,’ respectively.

  2. 2.

Atoms in the $\omega$ phase are identified by analyzing their nearest and
next-nearest neighbors’ AJP values. We count the number of graphitic and basal
atoms ($n_{g}$ and $n_{b}$, respectively) with which each central atom is
coordinated within a cutoff radius of 3.5 Å. Within this radius, basal atoms
should have two basal neighbors and 12 graphitic neighbors; graphitic atoms
should have five graphitic neighbors and six basal neighbors. Basal and
graphitic atoms whose coordination numbers $(n_{g},n_{b})$ take the values
$(2,12)$ or $(6,5)$, respectively, are assigned to the $\omega$ phase. We give
a tolerance of $\pm 1$ when comparing coordination numbers to combat thermal
fluctuations.

  3. 3.

Atoms not assigned to the $\omega$ phase are subsequently classified by
adaptive common-neighbor analysis (aCNA). Atoms are assigned to $\alpha$
(hcp), $\beta$ (bcc), or ‘other’ structures. The third classification includes
atoms in the bulk partially disordered phase, those found on grain boundaries,
and ‘false negatives’ whose local crystalline environment was so strongly
perturbed by thermal fluctuations as to be unrecognizable to the AJP or aCNA.

  4. 4.

Atoms whose assigned phase differs from that of all but no more than one of
its nearest neighbors are reassigned to the structure of those neighbors. This
corrective step further mitigates misidentifications from thermal noise.

Whenever identifying phases, we use atomic coordinates that have been time-
averaged over 200 fs (approximately one oscillation period) to further reduce
the number of false negatives incorrectly assigned a non-crystalline
structure.

To synthesize diffraction patterns from a set of (identical) atoms, we first
calculate the ionic structure factor on the locus of $q$-vectors forming the
Ewald sphere using the standard expression

$S_{i}(\textbf{q})=\sum_{\alpha=1}^{N}e^{-i\mathbf{q}\cdot\mathbf{r}_{\alpha}}\
,$ (1)

where the $\\{\mathbf{r}_{\alpha}\\}$ are the (instantaneous) coordinates of
the $N$ atoms of interest. The raw ionic scattering intensity $\lVert
S_{i}(\mathbf{q})\rVert^{2}$ is calculated and post-multiplied by atomic-form,
Lorentz-polarization, and self-attenuation factors.

Data availability

The data supporting the findings of this study are available from the
corresponding authors upon reasonable request.

Acknowledgements

We thank the Linac Coherent Light Source (LCLS) operations staff and the
Target Engineering Team at Lawrence Livermore National Laboratory (LLNL) for
assistance in these experiments. The authors would also like to acknowledge
Hongxiang Zong and Graeme J. Ackland for generously sharing their machine
learning-derived interatomic potential for Zr and Justin Wark for his helpful
suggestions during the writing of this manuscript. The research was supported
by the Laboratory Directed Research and Development Program at LLNL (project
nos. 17-ERD-014 and 21-ERD-032). This work was performed under the auspices of
the US Department of Energy by Lawrence Livermore National Laboratory under
Contract No. DE-AC52-07NA27344 (LLNL-JRNL-XXXXXX). P. G. H. gratefully
acknowledges the support of AWE via the Oxford Centre for High Energy Density
Science (OxCHEDS).

Author Contributions

S.S., M.G., P.G.H., D.M., J.B., R.B., J.E., and R.S. were involved in all
aspects of the experimental design and analysis. B.N. and H-J.L. were involved
in all aspects of the experimental acquisition of data. The manuscript was
written by S.S., M.G., P.G.H., J.E., and R.S.and reviewed by all authors.

Competing Interests

The authors declare no competing interests.

Measurement of competing pathways in a shock-induced
phase transition in zirconium by femtosecond diffraction

Saransh Singh, Martin G. Gorman, Patrick G. Heighway, Joel V. Bernier,
David McGonegle, Hae-Ja Lee, Bob Nagler, Jon H. Eggert, Raymond F. Smith

Supplementary Materials

This Supplementary Material contains additional information relevant to the
paper. We present a more complete set of experimental data collected and
analysis conducted as part of this experiment. The document also contains all
the relevant pole figure information to reproduce Fig. 2b, as well as the
forward model results for the other shots. We also present our data hinting at
the presence of a ns persisting body-centered cubic (bcc) intermediate phase.

Supplementary Material Tables:

Table LABEL:Table:OR: Summary of ORs reported for $\alpha\rightarrow\omega$
for Zr, Ti and their alloys.

Table LABEL:Table:OR_bcc: Summary of ORs reported for hcp $\rightarrow$ bcc.

Table LABEL:Table:lp: Lattice parameters for all data points collected in this
study.

Table LABEL:Table:numvariants1: Number of orientation variants for
$\alpha\rightarrow\omega$ phase transition ORs.

Table LABEL:Table:numvariants2: Number of orientation variants for
$\alpha\rightarrow\beta$ phase transition ORs.

Supplementary Material Figures:

Figure S1: 2-D diffraction signal and azimuthally averaged lineouts for the
ambient single crystal zirconium.

Figure S2: Diffraction lineouts for shock compression of polycrystalline-Zr
samples showing no diffuse background.

Figure S3: Contribution of diffraction signal form $\alpha$ and different
orientations of $\omega-$phase observed in our experiments.

Figure S4: Simulated diffraction pattern using the forward model for different
pressures.

Figure S5: Pair-correlation functions of simulated $\beta-$phase,
$\omega-$phase, and partially disordered nanoclusters at 14 GPa.

Figure S6: Comparison of the simulated structure factors in a high-symmetry
plane for the $\omega$ nanocrystal and partially disordered nanoclusters.

Figure S7: A repeat shot for the the data presented in Figs. 1, S4a.

Figure S8: Simulated pole figure and diffraction signal if an intermediate bcc
$\beta-$phase was observed.

Figure S9: Simulated pole figure for single crystal $\alpha-$Zr shock
compressed to 10.6 GPa.

Figure S10: Simulated pole figures for high pressure $\omega-$Zr formed at
15.2 GPa for the variant I ORs observed in this study.

Figure S11: Simulated pole figures for high pressure $\omega-$Zr formed at
15.2 GPa for the variant II ORs observed in this study.

Figure S12: Simulated pole figures for high pressure $\omega-$Zr formed at
15.2 GPa for the variant III ORs observed in this study.

Figure S13: Representative VISAR trace and hugoniot points measured in our
experiments.

Figure S14: Distribution of a few low index crystallographic plane normals for
different proposed ORs for the $\alpha\rightarrow\omega$ phase transition in
the literature.

Figure S15: Distribution of a few low index crystallographic plane normals for
different proposed ORs for the $\alpha\rightarrow\beta$ phase transition in
the literature.

Figure S16: Simulated pole figures for high pressure $\omega-$Zr for MD
simulations in Fig. 2.

Figure S17: Supercell mapping using atomic shuffle and strain for the TAO-I
$\alpha\rightarrow\omega$ phase transition mechanism giving rise to variant I
OR.

Figure S18: Supercell mapping using atomic shuffle and strain for the Silcock
$\alpha\rightarrow\omega$ phase transition mechanism giving rise to variant II
OR.

## Orientation Relationships

Table S1: $|$ Overview of studies determining the OR during $\alpha\rightarrow\omega$ transformation in Ti, Zr and their alloys Authors |  Experimental or Simulation details |  $\boldsymbol{(\textrm{hkl})_{\alpha}}~{}~{}~{}~{}~{}||~{}~{}~{}~{}\boldsymbol{(\textrm{hkl})_{\omega}}\newline \boldsymbol{[\textrm{uvw}]_{\ \alpha}}~{}~{}||~{}~{}~{}~{}\boldsymbol{[\textrm{uvw}]_{\omega}}$
---|---|---
Silcock (SILCOCK1958481)
(1958) |  Experimental x-ray diffraction study of the Ti alloys. OR between $\alpha-$ and $\omega-$phase was theoretically determined by studying projections of different planes. |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Sargent and Conrad (SARGENT1971)
(1971) |  Experimental study on pressure soaked at 2.8 GPa using Zone Axis diffraction patterns (ZADP) in a Transmission Electron Microscopy (TEM) |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
(variant II)
Usikov and Zilbershtei (UZ1973)
(1973) |  Experimental study on pressure soaked Ti and Zr using ZADPs in a TEM |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$
(variant I)
Rabinkin, Talianker and Botstein (RABINKIN1981)
(1981) |  Static compression of Zr foils in a Diamond Anvil Cell (DAC). Orientation relation derived from selected area diffraction patterns of recovered samples |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Kutsar et al. (Kutsar1990)
(1990) |  Experimental shock compression using gas gun apparatus of Zr foils. Orientation relationship derived from ZADP in a TEM of recovered sample |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Song and Gray (Song1995)
(1995) |  Experimental shock compression using gas gun apparatus of Zr foils. Orientation relationship derived from ZADP in a TEM of recovered sample |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[1\bar{1}00]_{\alpha}||[11\bar{2}\bar{3}]_{\omega}$
(variant I)
Jyoti et al. (Jyoti1997)
(1997) |  Experimental shock compression using gas gun apparatus of Zr foils. Orientation relationship derived from ZADP in a TEM of recovered samples |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$
(variant I)
Trinkle, Hennig and Wilkins (Trinkle2003)
(2003) |  Geometric mapping of $\alpha$ and $\omega-$supercells together with ab initio simulations |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$
(variant I for lowest energy TAO-I pathway)
Jyoti et al. (jyoti2008)
(2008) |  Experimental shock compression using gas gun apparatus of Zr foils. Orientation relationship derived from ZADP in a TEM of recovered samples |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$
(variant I)
Wenk et al. (Wenk2013)
(2013) |  Static compression of Zr wires using Deformation-DIA (D-DIA) and Diamond Anvil Cell (DAC) |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Zong et. al. (ZONG2014a)
(2014) |  Molecular dynamics of different single crystal Ti using a modified embedded atom method (MEAM) potential |  $(0001)_{\alpha}||(1\bar{1}00)_{\omega}$
$[10\bar{1}0]_{\alpha}||[11\bar{2}3]_{\omega}$
(for c-oriented Ti; closely related to variant III)
$(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(for $a$ and $a+b$ oriented Ti; variant II)
Zong et al. (ZONG2014b)
(2014) |  Combination of experimental and simulations. Experimental shock compressed Ti samples using gas gun apparatus with orientation relationship for the reverse $\omega\rightarrow\alpha$ measured using synchrotron x-ray diffraction of recovered samples. Simulations performed using molecular dynamics simulations and a spline based MEAM potential for Ti |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Adachi et al. (ADACHI20151)
(2015) |  Experimental study of $\alpha\rightarrow\omega$ phase transition during high-pressure torsion using Neutron diffraction |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
Guan and Liu (Guan2016)
(2016) |  First-principles density functional theory calculation, including the heterojunction energy of the $\alpha$ and $\omega-$phases |  $(11\bar{2}2)_{\alpha}||(1\bar{1}00)_{\omega}$
$[1\bar{1}00]_{\alpha}||[11\bar{2}0]_{\omega}$
(at higher pressures; closely related to variant I)
$(10\bar{1}0)_{\alpha}||(1\bar{1}00)_{\omega}$
$[0001]_{\alpha}||[11\bar{2}0]_{\omega}$
(at lower pressures; variant II)
$(0001)_{\alpha}||(02\bar{2}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[2\bar{1}\bar{1}0]_{\omega}$
(via an intermediate cubic FCC phase)
Swinburne et al. (Swinburne2016)
(2016) |  Experimental laser shock compression of rolled Zr foils coupled with in-situ x-ray diffraction |  $(10\bar{1}0)_{\alpha}~{}||~{}(10\bar{1}1)_{\omega}\newline [0001]_{\alpha}~{}||~{}[1\bar{2}10]_{\omega}$
Rawat and Mitra (Rawat2017)
(2017) |  Molecular dynamics simulation of $c-$oriented Ti using $4$ different potentials |  $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$
(variant I)
Zong et al. (ZONG2019)
(2019) |  Molecular dynamics simulation of shock compression in differently oriented single crystal Zr using machine learned ab initio potential |  $(0001)_{\alpha}~{}||~{}(11\bar{2}0)_{\omega}$
(variant II)
Wang et al. (wang2019)
(2019) |  Experimental study of $\alpha\rightarrow\omega$ phase transition during high-pressure torsion using high energy x-ray diffraction |  $(0001)_{\alpha}||(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$
(variant II)
This study
(2023) |  Experimental study of $\alpha\rightarrow\omega$ phase transition using laser compression and in-situ X-ray diffraction on [0001] oriented Zr single crystals |  $(0001)_{\alpha}||(1\bar{1}00)_{\omega}$
$[11\bar{2}0]_{\alpha}||[11\bar{2}3]_{\omega}$
(variant III)
Table S2: $|$ Overview of studies determining the OR during hcp $\rightarrow$ bcc transformation Authors |  Experimental details |  $\boldsymbol{(\textrm{hkl})_{\text{\small bcc}}}~{}~{}~{}~{}||~{}~{}~{}~{}\boldsymbol{(\textrm{hkl})_{\text{\small hcp}}}\newline \boldsymbol{[\textrm{uvw}]_{\text{\small bcc}}}~{}~{}||~{}~{}~{}~{}\boldsymbol{[\textrm{uvw}]_{\text{\small hcp}}}$
---|---|---
Burger’s (Burgers1934)
(1934) |  Experimental X-ray diffraction study on the orientation relationship between the hcp $\alpha-$ and bcc $\beta-$phase in Zr. |  $(110)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[1\bar{1}1]_{\text{\small bcc}}||[11\bar{2}0]_{\text{\small hcp}}$
Pitsch-Schrader (Pitsch1958)
(1958) |  Experimental $\epsilon-$Fe |  $(110)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[1\bar{1}0]_{\text{\small bcc}}||[10\bar{1}0]_{\text{\small hcp}}$
Mao-Bassett-Takahashi (Mao1967)
(1967) |  Experimental X-ray diffraction study on the orientation relationship between the ambient $\alpha-$phase and high-pressure $\epsilon-$phase in Fe |  $(110)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[001]_{\text{\small bcc}}||[2\bar{1}\bar{1}0]_{\text{\small hcp}}$
Gjönnes-Östmer (Gjonnes1970)
(1970) |  Experiment study of precipitates in Mg alloy |  $(110)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[1\bar{1}2]_{\text{\small bcc}}||[1\bar{2}10]_{\text{\small hcp}}$
Potter (potter1973)
(1973) |  Experimental study on orientation relationship between bcc $\alpha-$phase in Vanadium-Nitrogen system and hcp V3N precipitate using transmission electron microscopy |  $(110)_{\text{\small bcc}}||(1\bar{1}01)_{\text{\small hcp}}$
$[1\bar{1}1]_{\text{\small bcc}}||~{}~{}[11\bar{2}0]_{\text{\small hcp}}$
Crawley-Milliken (Crawley1974)
(1974) |  Experimental study on Mg-alloy using selected area diffraction in the transmission eletron microscope |  $(111)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[11\bar{2}]_{\text{\small bcc}}||~{}~{}[2\bar{1}\bar{1}0]_{\text{\small
hcp}}$
Rong-Dunlop (rong1984)
(1984) |  Experimental study on orientation relationship between bcc ferrite and hcp M2C (M = Cr, Mo, Fe) precipitates in ASP23 high strength steel using transmission electron microscopy |  $(021)_{\text{\small bcc}}||(0001)_{\text{\small hcp}}$
$[100]_{\text{\small bcc}}||[11\bar{2}0]_{\text{\small hcp}}$
Song-Du-Sun (Song2002)
(2002) |  Experimental study on orientation relationship between hcp M2C (M = Mo, V) carbide precipitates and bcc ferrite matrix in M50NiL low carbon bearing steel using transmission electron microscopy |  $(010)_{\text{\small bcc}}||(01\bar{1}1)_{\text{\small hcp}}$
$[100]_{\text{\small bcc}}||[2\bar{1}\bar{1}0]_{\text{\small hcp}}$

## Lattice Parameters

Table S3: Table of Lattice Parameters |  |  |
---|---|---|---
Shot $\\#$ |  Pressure (GPa) |  $\boldsymbol{\alpha}-$phase lattice constants (Å), density (g cm-3), c/a ratio |  $\boldsymbol{\omega}-$phase lattice constants (Å), density (g cm-3), c/a ratio
Shot 244 |  5.3 |  $a=3.17,c=5.075$, $\rho$ = 6.86, c/a = 1.60 |  —
Shot 96 |  10.4 |  $a=3.17,c=4.8$, $\rho$ = 7.25, c/a = 1.51 |  $a=4.9,c=2.96$, $\rho$ = 7.48, c/a = 0.604
Shot 93 |  10.6 |  $a=3.168,c=4.8$, $\rho$ = 7.26, c/a = 1.51 |  $a=4.9,c=2.98$, $\rho$ = 7.34, c/a = 0.608
Shot 97 |  12.3 |  $a=3.11,c=4.8$, $\rho$ = 7.53, c/a = 1.54 |  $a=4.87,c=2.95$, $\rho$ = 7.5, c/a = 0.606
Shot 92 |  14.1 |  $a=3.15,c=4.8$, $\rho$ = 7.34, c/a = 1.52 |  $a=4.86,c=2.94$, $\rho$ = 7.56, c/a = 0.605
Shot 94 |  15.2 |  — |  $a=4.8,c=2.96$, $\rho$ = 7.69, c/a = 0.617
Shot 91 |  18.1 |  — |  $a=4.77,c=2.96$, $\rho$ = 7.79, c/a = 0.621
Shot 99 |  21.3 |  $a=3.15,c=4.52$, $\rho$ = 7.8, c/a = 1.5 |  $a=4.78,c=2.948$, $\rho$ = 7.79, c/a = 0.617

Fig. S1: $|$ Establishing detector background. The $2$-D (top) and azimuthally
averaged (bottom) diffraction signal from an uncompressed Zr sample (0 GPa).
The lack of any Bragg diffraction is a result of the highly-oriented nature of
the starting sample. For this shot the XFEL X-rays were timed to probe the
sample before the shock arrived at the Zr, therefore any small contribution
from laser plasma X-rays is recorded on the detectors as a background level
for the high pressure shots (e.g., Fig. 2).

##  S1 Polycrystalline Zr data

Shock compression experiments along with in-situ x-ray diffraction were
conducted for polycrystalline Zr samples in the same pressure ranges as
discussed in the main text. A waterfall plot of the azimuthally averaged
intensities for the ambient sample and shock compressed Zr in intermediate
pressure regime [regime (ii)] of 10 – 15 GPa is shown in Fig. S2(a). X-ray
diffraction was recorded while the shock was transiting the Zr sample, which
resulted in a diffraction contribution from both the compressed Zr, and the
uncompressed volume ahead of the shock front. The location of the ambient
pressure peaks have been marked with dashed vertical lines. The compressed
$\alpha-$peaks have been labelled with an asterix (∗) and the $\omega-$phase
has been labelled with a blue arrow. Compared to polycrystalline samples, in-
situ x-ray diffraction during shock compression of single crystal Zr exhibits
a diffuse background superimposed on top of the crystalline diffraction, which
reduces in magnitude as the shock pressure is increased (Fig. 2a). While there
have been no systematic experimental studies relating the shock response and
phase transition pathways along various high symmetry directions of hcp
metals, orientational anisotropy in the transition pathway and orientation
relationships has been reported in large scale molecular dynamics simulations
for titanium shocked along the $[0001],[10\bar{1}0])$ and $[11\bar{2}0]$
directions(ZONG2014a). We hypothesize a similar orientational dependence on
the phase transition OR for Zr. This dependence could be related to the
different response of the single crystal and polycrystalline response of Zr
under shock compression. This question will be answered in future planned
experiments.

Fig. S2: $|$ Diffraction data from shock-compressed polycrystalline Zr
samples. a. Waterfall plot of azimuthally-averaged diffraction intensities as
a function of diffraction angle for poly-Zr shocked to $\sim$10 $\rightarrow$
15 GPa [regime(ii)] – comparable to the pressure range in Fig. 2. All volume-
integrated XRD data was taken during shock transit within the Zr layer, and
therefore each trace contains a contribution from the ambient pressure
$\alpha-$Zr volume ahead of the shock front (vertical dashed lines).
Compressed $\alpha-$Zr peaks are labelled with an asterix. In contrast to the
single crystal data in Fig. 2, the polycrystalline samples exhibit no evidence
of a disordered background (see dotted horizontal lines). b. Magnified view
shows the emergence of the $\\{11\bar{2}1\\}_{\omega}$ peak.  Fig. S3:
Breakdown of the simulated diffraction pattern presented in Fig. 2a-d as a
weighted sum of diffraction from the $\alpha-$phase, and variant I, variant II
and variant III orientations of the $\omega-$phase. Left most column presents
the same experimental data as Fig. 2a-d respectively. The % weight indicated
in the figure correspond to the phase fractions. Red box indicates no
contribution from the particular phase or orientation variants.  Fig. S4:  $|$
Evolution of diffraction signal as a function of shock pressure. The
experimentally measured 1D azimuthally averaged (top) and full 2D diffraction
signal (middle). The forward model predictions are shown in the bottom panel.
a. At 10.6 GPa, the 2D and integrated diffraction signals show a combination
of both $\alpha-$ (red filled) and $\omega-$phases (cyan filled). A large
diffuse scattering background also accompanies the Bragg diffraction. b. At
pressures above the onset of the phase transition [12.3 GPa], the
$\\{0001\\}_{\omega}$ and $\\{11\bar{2}1\\}_{\omega}$ peaks start to emerge. A
reduction in the diffuse background accompanies the emergence of these new
peaks. c. An increase in shock pressure to 15.2 GPa leads to a further
reduction in the phase fraction of the $\alpha-$phase and the diffuse
scattering intensity. The new $\omega-$phase peaks observed in b. increase in
intensity. d. At much higher pressures [18.1 GPa], the diffraction signal is
composed almost entirely of the $\omega-$phase, with a small amount of
diffraction from the compressed $\alpha-$phase, pointing to near complete
conversion of the $\alpha\rightarrow\omega$ phase transition. There is a
substantial reduction in the diffuse background signal and further evolution
in the azimuthal intensity distribution.

##  S2  Real-space structure of the partially disordered material in molecular
dynamics simulations

The nanoclusters of noncrystalline material that form between the $\omega$
grains have an atomistic structure differing from that of the prevailing
$\omega-$phase and residual $\beta-$phase. While the structure is not
perfectly crystalline, it does exhibit a degree of translational and
orientational order (hence our referring to this structure as ‘partially’
disordered). Figs. S5(a-c) show radial distribution functions (RDFs) for
$\beta$, $\omega$, and partially disordered nanoclusters. The disordered
structure exhibits translational order, though the correlation between its
atomic separations is weaker than in the $\beta$ or $\omega$ crystalline
phases. Figs. S5(d-f) also shows the angular distribution functions (ADFs),
defined as the probability of finding two atoms at a given fixed distance and
separated by a displacement vector with direction
$(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$ [with (0,0,1) being
the shock direction]. We evaluate each structure’s ADF at a separation
corresponding to the first maximum in its respective RDF. We observe that the
partially disordered structure shows strong orientational order despite it
being noncrystalline. The strongly directional bonding within the partially
disordered structure also manifests in its highly anisotropic structure
factor, shown in Fig. S6.

Fig. S5: $|$ Pair-correlation functions for equilibrated $\beta$, $\omega$,
and partially disordered material in Zr simulated under the Zong potential
(ZONG2019) compressed to 14 GPa. (a-c) Standard one-dimensional radial
distribution functions (RDFs) calculated out to 8 Å separation. (d-f) Angular
distribution functions (ADFs) for select $\beta$, $\omega$, and partially
disordered nanoclusters ($N_{\text{atom}}\sim 150)$, calculated at the
separation indicated by the shaded bars in (a-c), time-averaged over at least
1 ps. Direction $\theta=0^{\circ}$ corresponds to the shock direction;
$\theta=90^{\circ}$ is the ‘equatorial plane’ containing directions normal to
the shock. (g) Lineouts around the equatorial plane for the $\beta$, $\omega$,
and partially disordered nanoclusters, with signal levels scaled arbitrarily
for ease of comparison.

##  S3  Reciprocal-space structure of the partially disordered material in
molecular dynamics simulations

In Fig. S6, we show the squared modulus of the structure factor of the
$\omega$ nanocrystal and the partially disordered nanoclusters, calculated in
the high-symmetry $q_{y}q_{z}$ plane. The structure factor of the $\omega$
grains exhibit maxima at positions expected from their known variant II
orientations. The structure factor of the partially disordered material also
shows clear maxima at locations that coincide with those of the $\omega-$phase
structure factor. We note that the greater width of the disordered material’s
maxima is partly attributable to it forming in relatively small nanoclusters.
The similar forms of the two structure factors suggest that the $\omega$
nanocrystal and the partially disordered nanoclusters contribute to the
diffraction pattern in qualitatively similar ways. In Fig. S6, we also
indicate the region of reciprocal space from which the diffuse diffraction
signal arises. This signal comes from the tenuous cloud of intensity
surrounding the $\\{10\bar{1}1\\}_{\omega}$ and $\\{11\bar{2}0\\}_{\omega}$
reciprocal lattice vectors (and their equivalent maxima in the partially
disordered material) that marginally miss the Bragg condition.

Fig. S6: $|$ Comparison of the structure factors in a high-symmetry plane for
the $\omega$ nanocrystal and the partially disordered nanoclusters (left and
right, respectively) that form during shock-compression of Zr simulated under
the Zong potential (ZONG2019). The cross-section of the viewing plane with the
Ewald sphere (corresponding to a photon energy of 10 keV) and with the ideal
Polanyi spheres for the $\omega$ structure are shown by dashed green and black
lines, respectively. The black circle marks the region of reciprocal space
from which the diffuse scattering signal (centered on $2\theta\sim
30^{\circ}$) originates.

##  S4 Intermediate bcc $\beta-$phase

Fig. S7: $|$ X-ray diffraction data taken at peak shock pressure of 10.4 GPa
(shot 96). a. 2D diffraction signal in the $2\theta-\phi$ space. b.
Azimuthally averaged lineout showing the presence of sharp Bragg diffraction
together with the diffuse scaatering signal. This was a repeat shot for the
one shown in Fig. 1 and Fig. S4a. The demonstrates excellent reproducibility
in the observations of texture and enhanced disordered background. c. Inset
shows the high $2\theta$ region. A lineout for one of the
$\\{02\bar{2}2\\}_{\alpha}$ peaks is also shown. This lineout shows two peaks
separated in $2\theta$. The azimuthal location of the first peak at $\sim
60^{\circ}$ is consistent with the presence of an intermediate bcc
$\beta-$phase. We discuss this further in section S4.

An intermediate bcc $\beta-$phase was initially hypothesized in Ref. (UZ1973)
to explain the experimentally observed OR within recovered samples. Zong et
al. (ZONG2019, Zong2020) also reported a metastable $\beta-$phase in molecular
dynamics simulations of shock-compressed single crystal Zr along the $c-$axis
using interatomic potentials derived from a machine learning method. Finally,
Armstrong et al. (Armstrong2021) experimentally reported this phase in dynamic
compression experiments on pure polycrystalline Zr using in-situ x-ray
diffraction.

Our MD simulations also show the transition of the hcp $\alpha-$phase to the
bcc $\beta-$phase via rapid atomic shuffling in the $(0001)_{\alpha}$ plane
[the Burgers mechanism (Burgers1934)] immediately behind the phase-transition
front. After that, most of the transient $\beta$ phase decays into the stable
$\omega$ phase via further in-plane shuffling (to produce variant II) or out-
of-plane shuffling (variant I). However, the $\beta-$phase dominates for only
a few picoseconds, and rapidly decays to a prohibitively small mass fraction
$(\sim 5\%)$.

We checked the consistency of our data with the presence of an intermediate
bcc $\beta-$phase persisting in the ns timescale. The orientation relationship
between the hexagonal close-packed $\alpha$ and bcc $\beta-$phase in Zr has
been experimentally determined to be the Burger’s OR
($(0001)_{\alpha}~{}||~{}(110)_{\beta}$ and
$[2\bar{1}\bar{1}0]~{}||~{}[\bar{1}11]_{\beta}$) (Burgers1934) with a total of
$6$ orientation variants. The stereographic projection of some important low
index planes formed due to the Burger’s OR for the starting orientation of
$\alpha-$Zr in our experiment is shown in Fig. S8a. To search for evidence of
a bcc $\beta-$phase in our data we considered the peak labelled as
$\\{10\bar{1}1\\}_{\omega}$ in Figs. 2a, S7 as originating from the
$\\{110\\}$ plane of the intermediate $\beta-$phase. This fixes the lattice
parameter and density of the $\beta-$phase to a value of $7.31$ g cm-3. The
calculated lattice parameter and the known ORs let us compute the diffraction
signal originating from the $\beta-$phase.

Fig. S8: Pole figures for a. a few important planes resulting due to the
Burger’s orientation relationship between the hexagonal closed pack $\alpha$
and bcc $\beta-$phase b. a unimodal orientation distribution of the
$\beta-$phase resulting from the Burger’s OR c. resulting diffraction signal
due to the Burger’s OR on the CSPAD detectors d. a few important planes
resulting due to the Mao-Bassett-Takahashi (MBT) orientation relationship
between the hcp $\alpha-$ and bcc $\beta-$phase e. a unimodal orientation
distribution of the $\beta-$phase resulting from the MBT OR and f. resulting
diffraction signal due to the MBT OR on the CSPAD detectors. The green circles
represent the orientations which meet the Bragg’s conditions and observable in
our experiments. If an intermediate $\beta-$phase was formed following the
Burger’s OR, then the $(220)$ planes should show a strong diffraction signal.
Since this is not observed in our experiments, we conclude the absence of an
intermediate bcc $\beta-$phase in the $\alpha\rightarrow\omega$ transition.

We show the pole figure for unimodal orientation distribution centered around
the $6$ orientations generated by the Burger’s OR in Fig. S8b. We represent
the region of the pole figure observable in our experiments by the green
overlays. Finally, we show the expected diffraction signal on the CSPAD
detectors in Fig. S8c. Our model predicts no diffraction from the
$\\{110\\}_{\beta}$ planes but observable diffraction signal from the
$(220)_{\beta}$ reflection. We predict the $(220)_{\beta}$ diffraction signal
at the same azimuthal angle as the $(02\bar{2}2)_{\alpha}$ reflection. In
addition, the expected $2\theta$ location for the two reflections is close. We
look closer at this region of the CSPAD detectors in Fig. S7inset. The
$2\theta$ lineout for one of the $(02\bar{2}2)_{\alpha}$ shows evidence for
two peaks. This observation is consistent with a tiny volume fraction of the
intermediate $\beta-$phase. We note that this feature is repeatable (see Fig.
1b). While the above observation suggests that we are measuring diffraction
from an intermediate $\beta-$phase as seen in our MD simulations, we do not
believe that the signal level or detector coverage is high enough to be
conclusive. We plan to repeat these measurements with better detector coverage
in future experiments.

We also checked the consistency of other hcp-bcc ORs with our data. We list
the different ORs we considered when interpreting our data in Table
LABEL:Table:OR_bcc. For example, we show the pole figures for unimodal
orientation distribution centered around the $3$ orientations generated by the
MBT OR in Fig. S8e, with the green overlays showing the region of the pole
figure observable in our experiments. We present the expected diffraction
signal on the CSPAD detectors in Fig. S8f. The diffraction produced by this OR
is very similar to Burger’s OR. Therefore, we are unable to distinguish
between the two.

We also note that while the Rong-Dunlop (rong1984) and the Song Du-Sun
(Song2002) ORs, for hcp metal-carbon precipitates in bcc iron, produce an
azimuthal intensity distribution matching our experiments for the currently
labeled $(10\bar{1}1)_{\omega}$ reflection, and indistinguishable from our
variant III, these ORs have never been experimentally observed or
theoretically predicted in Zr. Since these ORs have only been observed in high
strength steel for hcp carbide precipitates within the bcc ferrite matrix, we
deem it highly unlikely that these ORs are present in the $\beta-$phase during
the $\alpha\rightarrow\omega$ phase transition in Zr.

##  S5  Orientation Variants

Fig. S9: Pole figures (log scale) for a unimodal orientation distribution of
the $\alpha-$phase centered around the orientation of the ambient crystal,
i.e. $c-$axis pointing in the direction of the x-rays and $a-$axis normal to
it, pointing east. The observable rings in our experimental geometry for each
reflection is shown by the dotted green circle. When the circle overlaps with
non-zero intensity in the pole figure, then diffraction is observed for that
reflection. The variation of intensity around the circle is reflected in the
observed diffraction signal.  Fig. S10: Pole figures (log scale) for a
unimodal orientation distribution of the $\omega-$phase centered around the
orientations from the variant I (TAO-I) OR. The observable rings in our
experimental geometry for each reflection is shown by the dotted green circle.
When the circle overlaps with non-zero intensity in the pole figure, then
diffraction is observed for that reflection. The variation of intensity around
the circle is reflected in the observed diffraction signal.  Fig. S11: Pole
figures (log scale) for a unimodal orientation distribution of the
$\omega-$phase centered around the orientations from variant II (Silcock) OR.
The observable rings in our experimental geometry for each reflection is shown
by the dotted green circle. When the circle overlaps with non-zero intensity
in the pole figure, then diffraction is observed for that reflection. The
variation of intensity around the circle is reflected in the observed
diffraction signal.  Fig. S12: Pole figures (log scale) for a unimodal
orientation distribution of the $\omega-$phase centered around the
orientations from variant III OR. The observable rings in our experimental
geometry for each reflection is shown by the dotted green circle. When the
circle overlaps with non-zero intensity in the pole figure, then diffraction
is observed for that reflection. The variation of intensity around the circle
is reflected in the observed diffraction signal.

Each of the orientation relationships listed in Table LABEL:Table:OR describes
one particular change of basis between the parent $\alpha$ and the daughter
$\omega-$phase. However, due to crystallographic symmetry, it is possible to
form other $\omega-$phase with a different orientation. For example, the
variant I OR described by $(0001)_{\alpha}||(01\bar{1}1)_{\omega}$ and
$[11\bar{2}0]_{\alpha}||[10\bar{1}1]_{\omega}$ can also form such that
$(0001)_{\alpha}||(1\bar{1}01)_{\omega}$ and
$[11\bar{2}0]_{\alpha}||[\bar{1}011]_{\omega}$ etc. Therefore, when computing
the diffraction signal resulting from the resultant $\omega-$phase, it is
important to consider all the orientation variants due to a particular OR. In
this section, we list the number of variants for each of the OR described
above and show distribution of different crystallographically important planes
and directions for each of the ORs. Given the transformation matrix,
$\mathbf{T_{\alpha\rightarrow\omega}}$ between the parent and daughter basis,
and the point group symmetry, $\mathbf{G}^{\alpha}$ and $\mathbf{G}^{\omega}$
of the two phases, the common symmetries between the two phases,
$\mathbf{H}^{\alpha,\omega}$ is given by

$\mathbf{H}^{\alpha,\omega}=\mathbf{G}^{\alpha}~{}\cap~{}\mathbf{T_{\alpha\rightarrow\omega}}\mathbf{G}^{\omega}\mathbf{T_{\alpha\rightarrow\omega}}^{-1}.$

The number of orientation variants, $N_{\alpha\rightarrow\omega}$ is then
given by the following expression

$N_{\alpha\rightarrow\omega}=\frac{|\mathbf{G}^{\alpha}|}{|\mathbf{H}^{\alpha,\omega}|}.$

Here, $|\mathbf{A}|$ represents the number of elements in group $\mathbf{A}$.
Each orientation variant, $\alpha_{i}$ can be calculated from symmetry
elements, $g_{i}^{\alpha}$ the parent phase point group,
$\mathbf{G}^{\alpha}$. The orientation variants for each OR can be computed
using the symmetry group of the parent phase. If
$\mathbf{T_{\alpha\rightarrow\omega}}$ denotes the transformation matrix
between the $\alpha$ and the $\omega-$phase given by the OR, then the
orientation variants are the unique elements in the set,
$\mathbf{T^{\prime}_{\alpha\rightarrow\omega}}$, given by

$\mathbf{T^{\prime}_{\alpha\rightarrow\omega}}=\mathbf{G}^{\alpha}\mathbf{T_{\alpha\rightarrow\omega}}.$

The reader is referred to (Cayron2007) for a more comprehensive description of
the mathematical structure of orientation variants. All orientation variants
of the different ORs in the literature and their respective pole figures were
computed to compare with our experimental measurements. Fig. S14 presents the
stereographic projections of a few crystallographically important planes of
the $\omega-$phase for the different ORs.

Table S4: $|$ Number of orientation variants for different $\alpha\rightarrow\omega$ orientation relationships Orientation relationship |  # variants
---|---
$(0001)_{\alpha}~{}||~{}(01\bar{1}1)_{\omega}$
$[11\bar{2}0]_{\alpha}~{}||~{}[10\bar{1}1]_{\omega}$
(variant I) |  12
$(0001)_{\alpha}~{}||~{}(11\bar{2}0)_{\omega}$
$[11\bar{2}0]_{\alpha}~{}||~{}[0001]_{\omega}$
(variant II) |  3
$(0001)_{\alpha}~{}||~{}(1\bar{1}00)_{\omega}$
$[11\bar{2}0]_{\alpha}~{}||~{}[11\bar{2}3]_{\omega}$
(variant III) |  6
$(0001)_{\alpha}~{}||~{}(1\bar{1}00)_{\omega}$
$[10\bar{1}0]_{\alpha}~{}||~{}[11\bar{2}3]_{\omega}$
closely related to variant III (ZONG2014a)) |  6
$(10\bar{1}0)_{\alpha}~{}||~{}(10\bar{1}1)_{\omega}$
$[0001]_{\alpha}~{}||~{}[1\bar{2}10]_{\omega}$
(Swinburne et al. (Swinburne2016)) |  6
$(11\bar{2}2)_{\alpha}~{}||~{}(1\bar{1}00)_{\omega}$
$[1\bar{1}00]_{\alpha}~{}||~{}[11\bar{2}0]_{\omega}$
(closely related to variant I (TAO-I) (Guan2016)) |  6
$(10\bar{1}0)_{\alpha}~{}||~{}(1\bar{1}00)_{\omega}$
$[0001]_{\alpha}~{}||~{}[11\bar{2}0]_{\omega}$
(variant II (Silcock/Heterojunction II) (Guan2016)) |  3
$(0001)_{\alpha}~{}||~{}(02\bar{2}1)_{\omega}$
$[11\bar{2}0]_{\alpha}~{}||~{}[2\bar{1}\bar{1}0]_{\omega}$
(Guan and Liu (intermediate FCC) (Guan2016)) |  6
Table S5: $|$ Number of orientation variants for different $\alpha\rightarrow\beta$ orientation relationships Orientation relationship |  # variants
---|---
$(110)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[1\bar{1}1]_{\text{\small bcc}}~{}||~{}[11\bar{2}0]_{\text{\small hcp}}$
(Burger’s (Burgers1934)) |  6
$(110)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[1\bar{1}0]_{\text{\small bcc}}~{}||~{}[10\bar{1}0]_{\text{\small hcp}}$
(Pitsch-Schrader (Pitsch1958)) |  3
$(110)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[001]_{\text{\small bcc}}~{}||~{}[2\bar{1}\bar{1}0]_{\text{\small hcp}}$
(Mao-Bassett-Takahashi (Mao1967)) |  3
$(110)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[1\bar{1}2]_{\text{\small bcc}}~{}||~{}[1\bar{2}10]_{\text{\small hcp}}$
(Gjönnes-Östmer (Gjonnes1970)) |  12
$(110)_{\text{\small bcc}}~{}||~{}(1\bar{1}01)_{\text{\small hcp}}$
$[1\bar{1}1]_{\text{\small bcc}}~{}||~{}[11\bar{2}0]_{\text{\small hcp}}$
(Potter (potter1973)) |  12
$(111)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[11\bar{2}]_{\text{\small bcc}}~{}||~{}[2\bar{1}\bar{1}0]_{\text{\small
hcp}}$
(Crawley-Milliken (Crawley1974)) |  2
$(021)_{\text{\small bcc}}~{}||~{}(0001)_{\text{\small hcp}}$
$[100]_{\text{\small bcc}}~{}||~{}[11\bar{2}0]_{\text{\small hcp}}$
(Rong-Dunlop (rong1984)) |  6
$(010)_{\text{\small bcc}}~{}||~{}(01\bar{1}1)_{\text{\small hcp}}$
$[100]_{\text{\small bcc}}~{}||~{}[2\bar{1}\bar{1}0]_{\text{\small hcp}}$
(Song-Du-Sun (Song2002)) |  6
Fig. S13: $|$ Velocimetry measurements and pressure determination. a. A
representative line-visar image taken over a 300 $\mu$m field of view
(Celliers2004). The determined Zr-free surface velocity is shown as the yellow
trace. This data was used to verify: (i) the relative timing of the x-ray
probe and the compression wave, (ii) the temporal steadiness of the laser
shock. b. Pressure-density Hugoniot data established from previous work (open
black circles) (al1981, walsh1957, mcqueen1970, rigg2014). Density determined
from x-ray diffraction data ($\alpha-$phase: red open symbols; $\omega-$phase:
blue filled symbols). In total eight single crystal shots where taken. The
diffraction patterns from shots 93, 94 and 97 are shown in Fig. 2. The
diffraction pattern for shot 96 is shown in Fig. S7. To determine pressure in
our experiments, we fix the measured density of the $\alpha$ and
$\omega-$phases to the fit through previously determined Hugoniot data (grey
curve). Below shock pressures of 10.2 GPa only compressed $\alpha-$peaks are
observed in the X-ray diffraction data. The highest pressure we observe
compressed $\alpha-$Zr is 12.3 GPa. For shock pressures below this value, the
$\omega-$Zr pressure and density within the mixed phase assembly is higher
than the $\alpha-$phase density.  Fig. S14: Distribution of the
$\\{0001\\},\\{10\bar{1}1\\}$ and $\\{11\bar{2}0\\}$ planes of the
$\omega-$phase for different ORs reported in experimental and theoretical
studies. References for each of these studies can be found in Table
LABEL:Table:OR. The heterojunction I OR reported in Guan and Liu (Guan2016) is
reported to be equivalent to variant I OR. While the ORs are similar, they are
not equivalent. Heterojunction II OR is identical to the variant II or Silcock
OR. Similar distribution for the other crystallographic planes can be computed
and used to calculated the expected diffraction signal on the CSPAD detectors.
Direct comparison with observed diffraction can be used to identify the
resultant orientation relationship between $\alpha$ and $\omega-$phases.  Fig.
S15: Distribution of the $\\{001\\},\\{110\\}$ and $\\{111\\}$ planes of the
bcc $\beta-$phase for different ORs reported in experimental studies.
References for each of these studies can be found in Table LABEL:Table:OR_bcc.
The Pitsch-Schrader (Pitsch1958) is equivalent to Mao-Bassett-Takahashi OR
(Mao1967). Direct comparison with observed diffraction can be used to identify
the presence of an intermediate $\beta-$phase lasting in the ns timescale.
Fig. S16: Pole figures (linear scale) calculated from the structure factor of
the 50 largest $\omega$ grains in the molecular dynamics simulation described
in Fig. 3. The observable rings in our experimental geometry for each
reflection is shown by the dotted green circle. When the circle overlaps with
non-zero intensity in the pole figure, then diffraction is observed for that
reflection. The variation of intensity around the circle is reflected in the
observed diffraction signal. Red circles mark the locations where the
structure factor is expected to be strongest for variant II OR. Note that the
$c/a$ ratio in the simulations is such that
$\lvert\\{11\bar{2}0\\}\rvert\approx\lvert\\{10\bar{1}1\\}\rvert$ and
$\lvert\\{30\bar{3}0\\}\rvert\approx\lvert\\{21\bar{3}1\\}\rvert$, making
their plot plots practically inseparable.

##  S6 TAO-I mechanism (variant I) during $\alpha\rightarrow\omega$
transformation

Fig. S17: The TAO-I mechanism (variant I) is a mapping of two triclinic
supercells of $\alpha$ and $\omega-$phases. This pathway is the lowest energy
$\alpha\rightarrow\omega$ pathway during homogeneous nucleation. A combination
of lattice strain and atomic shuffle transforms the alpha supercell to the
omega supercell. The atomic shuffle are shown by the red arrows, with the tail
of the arrow showing the original location in the $\alpha-$phase, and the head
the final location in the $\omega-$phase. The primitive unit cell (dashed
line) along with the alpha supercell is shown in the left panel. The
$(0001)_{\alpha}$ plane, shown in red, is fully converted to the
$(\bar{1}011)_{\omega}$ plane shown in blue.

The TAO-I mechanism is a mapping of two triclinic supercells of the $\alpha$
and $\omega-$phases using a combination of lattice strain and atomic shuffle.
This is the lowest energy pathway under hydrostatic pressure conditions if
only homogeneous nucleation is considered (Guan2016). The TAO-I mechanism
produces variant I OR between the $\alpha$ and $\omega$ unit cells given by
$(0001)_{\alpha}||(01\bar{1}1)_{\omega}$,
$[11\bar{2}0]_{\alpha}||[\bar{1}011]_{\omega}$. The $\alpha-$supercell
(primed) is constructed from the hexagonal basis by the following change of
basis (osu1070481734)

$\displaystyle\begin{pmatrix}\mathbf{a}^{\prime}_{\alpha}\\\
\mathbf{b}^{\prime}_{\alpha}\\\
\mathbf{c}^{\prime}_{\alpha}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{\alpha}\quad\mathbf{b}_{\alpha}\quad\mathbf{c}_{\alpha}\end{pmatrix}\begin{pmatrix}2&1&1\\\
1&2&1\\\ 0&0&1\end{pmatrix}.$

Similarly, the $\omega-$supercell (primed) is constructed from the hexagonal
basis (un-primed) by the following change of basis

$\displaystyle\begin{pmatrix}\mathbf{a}^{\prime}_{\omega}\\\
\mathbf{b}^{\prime}_{\omega}\\\
\mathbf{c}^{\prime}_{\omega}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{\omega}\quad\mathbf{b}_{\omega}\quad\mathbf{c}_{\omega}\end{pmatrix}\begin{pmatrix}1&1&0\\\
0&1&0\\\ 1&1&2\end{pmatrix}.$

These supercells and location of atoms are shown in Fig. S17. The magnitude of
strains and atomic shuffle for the TAO-I mechanism can be calculated based on
the experimentally measured lattice parameters of the two phases. More details
about calculating these strains and shuffles can be found in (osu1070481734).
For shot $93$ (Figs. 1, 2a), the principle strain were calculated to be

$\epsilon_{x}=-0.09989\quad\epsilon_{y}=0.0189\quad\epsilon_{z}=0.0960.$

The strain magnitude for the other shots are of comparable magnitude. The
maximum magnitude of the shuffle for an atom in the unstrained $\alpha-$basis
is given by (osu1070481734)

$\delta_{\textrm{max}}=\frac{\sqrt{a_{\alpha}^{\prime 2}+c_{\alpha}^{\prime
2}}}{12},$

while the root mean squared shuffle is given by

$\delta_{\textrm{RMS}}=\frac{\sqrt{3a_{\alpha}^{\prime 2}+c_{\alpha}^{\prime
2}}}{6}.$

For the lattice parameters in shot 93 (Figs. 1, 2a) of our experiment,
$\delta_{\textrm{max}}=0.6634$ Å and $\delta_{\textrm{RMS}}=0.6026$ Å. The
shuffle magnitudes for the other shots are of comparable magnitude.

##  S7  Silcock mechanism (variant II) during $\alpha\rightarrow\omega$
transformation

Fig. S18: The Silcock mechanism (variant II) is a mapping of the a different
pair of $\alpha$ and $\omega-$phase supercells. The atomic shuffle is shown by
the red arrows, with the tail of the arrow showing the original location in
the $\alpha-$phase, and the head the final location in the $\omega-$phase.
This is followed by lattice strains mapping the $(0001)_{\alpha}$ plane to the
$(11\bar{2}0)_{\omega}$ plane. The $\alpha$ and $\omega-$phase supercells are
shown with the solid red and blue line respectively.

The Silcock mechanism is a mapping of two orthorhombic supercells of the
$\alpha$ and $\omega-$phase susing a combination of lattice strains and
shuffles. This mechanism needs large atomic shuffles but modest lattice
strain. The Silcock mechanism produces the variant II OR given by
$(0001)_{\alpha}||(11\bar{2}0)_{\omega}$,
$[11\bar{2}0]_{\alpha}||[0001]_{\omega}$. For the Silcock mechanism, the
orthorhombic $\alpha-$supercell is constructed from the hexagonal basis by the
following change of basis (osu1070481734)

$\displaystyle\begin{pmatrix}\mathbf{a}^{\prime}_{\alpha}\\\
\mathbf{b}^{\prime}_{\alpha}\\\
\mathbf{c}^{\prime}_{\alpha}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{\alpha}\quad\mathbf{b}_{\alpha}\quad\mathbf{c}_{\alpha}\end{pmatrix}\begin{pmatrix}1&0&-3\\\
0&0&-6\\\ 0&1&0\end{pmatrix}.$

Similarly, the orthorhombic $\omega-$supercell is constructed from the
hexagonal basis by the following change of basis

$\displaystyle\begin{pmatrix}\mathbf{a}^{\prime}_{\omega}\\\
\mathbf{b}^{\prime}_{\omega}\\\
\mathbf{c}^{\prime}_{\omega}\end{pmatrix}=\begin{pmatrix}\mathbf{a}_{\omega}\quad\mathbf{b}_{\omega}\quad\mathbf{c}_{\omega}\end{pmatrix}\begin{pmatrix}0&-1&2\\\
0&0&4\\\ -1&0&0\end{pmatrix}.$

These supercells and location of atoms are shown in Fig. S18.The lattice
strains and magnitude of atomic shuffles for the Silcock mechanism can be
calculated using the measured lattice parameters of the $\alpha$ and
$\omega-$phase. For shot 93 (Figs. 1, 2a), the principle strains were
calculated to be

$\epsilon_{x}=-0.0669\quad\epsilon_{y}=-0.0052\quad\epsilon_{z}=0.009.$

The magnitude of shuffle for all atom in the unstrained $\alpha-$basis is
equal and is given by (osu1070481734)

$\delta_{\textrm{max}}=\delta_{\textrm{RMS}}=\frac{\sqrt{a^{\prime
2}_{\alpha}+c^{\prime 2}_{\alpha}/324}}{4}.$

For the lattice parameters in shot 93 (Figs. 1, 2left) of our experiment,
$\delta_{\textrm{max}}=\delta_{\textrm{RMS}}=0.8243$ Å.

##  S8  variant III orientation relationship during $\alpha\rightarrow\omega$
transformation

In addition to variant I and II ORs, we observe the presence of a third,
previously experimentally unreported orientation relationship in our
measurements. This OR is very closely related to the one reported by Zong et
al. (ZONG2014a) in a MD simulation study of shock compression along the c-axis
of Ti using a modified embedded atom model (MEAM) potential. Our proposed OR,
which we refer to as variant III, is $30^{\circ}$ rotated from the one
reported in that study and is given by
$(0001)_{\alpha}||(1\bar{1}00)_{\omega}$,
$[11\bar{2}0]_{\alpha}||[11\bar{2}3]_{\omega}$. We note that the lattice
strains and shuffle leading to this orientation relationship is unknown.