[a]Lawrence Livermore National Laboratory, Livermore, CA 94550, USA [b]Gordon
and Betty Moore Foundation, Palo Alto, CA 94304, USA [c]Department of Earth
and Planetary Sciences, Johns Hopkins University, Baltimore, Maryland 21218,
USA

# A structural study of hcp and liquid iron under shock compression up to 275
GPa

Saransh Singh<EMAIL_ADDRESS>Richard Briggs  Martin G. Gorman  Lorin X.
Benedict  Christine J. Wu  Amy L. Coleman  Federica Coppari  Amalia Fernandez-
Pañella  Christopher McGuire  Melissa Sims  June K. Wicks  Jon H. Eggert
Dayne E. Fratanduono  Raymond F. Smith

###### Abstract

We combine nanosecond laser shock compression with _in-situ_ picosecond X-ray
diffraction to provide structural data on iron up to 275 GPa. We constrain the
extent of hcp-liquid coexistence, the onset of total melt, and the structure
within the liquid phase. Our results indicate that iron, under shock
compression, melts completely by 258(8) GPa. A coordination number analysis
indicates that iron is a simple liquid at these pressure-temperature
conditions. We also perform texture analysis between the ambient body-
centered-cubic (bcc) $\alpha$, and the hexagonal-closed-packed (hcp) high-
pressure $\epsilon-$phase. We rule out the Rong-Dunlop orientation
relationship (OR) between the $\alpha$ and $\epsilon-$phases. However, we
cannot distinguish between three other closely related ORs: Burger’s, Mao-
Bassett-Takahashi, and Potter’s OR. The solid-liquid coexistence region is
constrained from a melt onset pressure of 225(3) GPa from previously published
sound speed measurements and full melt (246.5(1.8)-258(8) GPa) from X-ray
diffraction measurements, with an associated latent heat of melting of 362(56)
- 542(83) J/g. This value is lower than recently reported theoretical
estimates and suggests that the contribution to the earth’s geodynamo energy
budget from heat release due to freezing of the inner core is smaller than
previously thought. Melt pressures for these nanosecond shock experiments are
consistent with gas gun shock experiments that last for microseconds,
indicating that the melt transition occurs rapidly.

††journal: Physical Review B

##  1 Introduction

Iron is a cosmochemically abundant element that plays a significant role in
terrestrial planetary interiors as the dominant core constituent. The ultra-
high pressure properties of iron are essential for interpreting the dynamics
and interior structure of the earth and rocky exoplanets (anzellini2013,
Smith2018, Kraus2022). Within the earth, it is estimated that the solid inner
core is comprised of Fe alloyed with $\sim$5-10% of impurities (e.g., Si, S,
O, C, H, and Ni) by weight (Hirose2021). Surrounding the solid inner core is
the Fe-rich outer liquid core, which is estimated to have $\sim$8-16$\%$
impurity content by weight (Hirose2021). According to the standard model,
convection in the outer core is driven by processes associated with
solidification and growth of the inner core. One source is the buoyancy
generated by the exclusion of incompatible light elements from the solid.
Another is latent heat release from re-crystallization. Planetary magnetic
fields arising from convection within the outer liquid Fe-rich cores play an
important role in the atmospheric evolution and surface environment of planets
(Foley2016). There is a strong need to constrain material properties of Fe
close to melting at the extreme pressures found within planetary interiors to
understand these processes better.

High-pressure static compression studies on Fe have revealed a phase
transformation from the ambient body-centered-cubic (bcc) $\alpha$-phase to a
hexagonal-closed packed (hcp) $\epsilon$-phase at 15.3 GPa with an associated
5% volume collapse (barge1990, boehler1990, taylor1991). A large (shear-stress
dependent) hysteresis of the transition is observed with a midpoint pressure
of 12.9 GPa (barge1990, boehler1990, taylor1991). The melting of Fe has also
been reported as a function of pressure by static compression techniques
(anzellini2013, Sinmyo2019, Kuwayama2020, boehler1993, boehler1990), and
theoretical calculations (belonoshko2000, alfe2009, swift2020). Under near
instantaneous uniaxial shock compression, the $\alpha$$\rightarrow$$\epsilon$
phase transformation in polycrystalline Fe samples has been observed, after a
period of stress relaxation, to initiate at 12.9 GPa (bancroft1956,
Jensen2009b, barker1974, barker1975). At higher pressure, changes in sound
speed within mm-thick samples shock compressed over microseconds were used to
infer the onset of melt at 225(3) GPa, and its completion by 260(3) GPa
(nguyen2004). Under nanosecond laser shock-compression of 15-$\mu$m thick
samples, melt onset was reported to be between 241.5(3)-242.4(2.3) GPa through
_in situ_ X-ray diffraction (XRD) measurements (Turneaure2020). A further
constraint on the high-pressure Fe melt line was reported in a recent laser
driven shock-ramp XRD experiments (Kraus2022).

In the experiments reported here, we employ nanosecond laser shock compression
combined with _in situ_ picosecond XRD and velocimetry techniques to show that
Fe is fully melted along the Hugoniot at 258(8) GPa. We report on the first
liquid structural and density measurements of Fe under shock compression. This
data, combined with previous sound speed data (nguyen2004), provides a direct
determination of the extent of solid-liquid coexistence and a constraint on
the latent heat of fusion.

Figure 1:  $|$ Experimental setup. The target design consists of a polyimide
ablator, a 21-$\mu$m-thick Fe foil, and a LiF window for velocimetry
measurements (dolan2006). Raw diffraction data from laser shock compressed Fe
at an estimated pressure of 275(9) GPa.

##  2 Experimental Method

Laser shock compression experiments were conducted at the Dynamic Compression
Sector (DCS) beamline of the Advanced Photon Source (APS), located at the
Argonne National Laboratory (wang2019). Targets consisted of a polyimide
ablator ($\sim$35-50 $\mu$m thickness) glued to high purity 21 $\mu$m-thick Fe
foil (see Fig. 1). An [100] orientated LiF window was glued to the Fe sample
to facilitate measurements of the Fe/LiF particle velocity. A 1-3 $\mu$m-thick
epoxy held these three layers together. The LiF window is coated with 0.1
$\mu$m-thick Ti to enhance reflectivity for velocimetry measurements (VISAR)
(dolan2006). The full density 99.99% pure Fe foils were supplied by
Goodfellow, USA (initial density = 7.874 g/cm3).

A 5 or 10 ns approximately flat-top laser pulse at 351 nm, with energies
between 15 and 80 J, was focused within a 580 $\mu$m diameter focal spot on
the front surface of the polyimide layer. This setup uniaxially shock-
compressed the target assembly and the generated longitudinal stress states in
Fe between 25 and 275 GPa. A point-VISAR Doppler velocity interferometer was
used to measure the time history of the Fe/LiF interface, $u_{p}$(t)
(dolan2006), and through standard impedance-matching techniques, this allowed
a determination of shock pressure in the iron sample. Simultaneously, an X-ray
pulse ($\sim$80 ps) was timed to probe the compressed sample during shock
transit within the Fe, which produced an X-ray diffraction pattern recorded in
a transmission geometry (Fig. 1), with contributions from the compressed Fe
(behind the shock front) and uncompressed Fe (ahead of the shock front). The
X-ray energy had a peak of $\sim$23.56 keV with a pink beam profile. The
experimental geometry is also described in Refs. (wang2019, briggs2019,
Coleman2022, Sims2022).

###  2.1 X-ray diffraction data processing

CeO2, and powdered Si calibrants were used to determine the sample to detector
distance, beam center, tilt, and rotation. XRD images were azimuthally
averaged and analyzed using HEXRD python package (HEXRDgithub). First, powder
diffraction rings from known standards (CeO2 and Si) were used for detector
calibration. The pink beam profile function detailed in Ref. (vondreele2021)
was used for this step. Next, several intensity corrections were performed to
enable accurate density determinations from the diffraction profiles. The
intensity corrections included: (i) subtracting the dark counts, (ii)
subtracting the ambient Fe pattern for the case of liquid XRD analysis and
subtracting LiF single-crystal Laue peaks for all experiments, (iii)
correcting for the solid angle subtended by each pixel on the detector (iv)
accounting for the polarization factor (for liquid diffraction profiles only)
and (v) correcting for the attenuation due to varying path length of the
diffracted X-rays. Figure 2 shows representative intensity corrected
integrated diffraction profiles for shock-compressed Fe (see also Figs.
S12-S14).

Figure 2: X-ray diffraction Azimuthally averaged diffraction data as a
function of increasing shock pressure (right axis). The dashed vertical lines
are diffraction angles for the ambient bcc Fe. The traces are colored coded to
depict crystal structure: bcc+hcp (green), hcp (black), and liquid (blue) (see
also Figs. S12-S14)

By increasing the incident drive laser energy to achieve stresses on the
Hugoniot between 25 and 275 GPa, a range of XRD profile shapes were observed
indicating different Fe structures. Over the stress range studied, Fe exhibits
an hcp structure from 25-210 GPa. For the two highest pressure shots (258 and
275 GPa), diffuse scattering around Q = 3.8 $\AA^{-1}$ is present in these
profiles with no evidence of compressed hcp; this is characteristic of the
complete melting [Q = 4$\pi$sin($\theta$)/$\lambda$, where $\lambda$ is the
X-ray wavelength, and $\theta$ is the Bragg scattering angle].

###  2.2 Stress determination

Figure 3: Pressure determination using Fe-LiF impedance matching. Using Sesame
EOS table 2140 for Fe for a range of assumed LiF up values, the main plot
shows the calculated Fe pressure calculated through impedance-matching
assuming (i) the reflected shock approximation (RSA) and (ii) the release
adiabat (RA) approach. In these calculations, the RA path does not include a
strength model. While the RA approach is physically correct, the RSA is often
used in impedance matching because it simplifies the analysis, as one needs
only the Hugoniot data to perform the analysis. In contrast, the RA approach
relies on a theoretical model (celliers2005, kerley2013). The upper plot
represents underestimating of the calculated Fe pressure from the RSA approach
as a function of increasing pressure. This offset has been accounted for in
our pressure estimate.

Standard impedance matching techniques were employed to determine the stress
in the Fe layer from the measurement of the Fe/LiF particle velocity, and
using $U_{S}$-$u_{p}$ fits (shock velocity-particle velocity, both in $km/s$)
to measure shock compression data for LiF (Rigg2014) (Eqn. 1) and Fe (Eqn. 2).
The Fe/LiF particle velocity was calculated after accounting for the
refractive index of LiF under shock compression (Rigg2014).

$U_{S,LiF}=5.215(\pm 0.048)+1.351(\pm 0.025)\times u_{p,LiF}$ (1)
$U_{S,Fe}=a+b\times u_{p,Fe}-c\times u_{p,Fe}^{2}+d\times u_{p,Fe}^{3},$ (2)

where a = 3.4188 ($\pm$0.0539), b = 2.1663 ($\pm$0.0823), c = 0.1992
($\pm$0.0369), and d = 0.021219 ($\pm$0.00497). The uncertainties from
impedance matching have contributions from the scatter of EOS data for both Fe
(Walsh1957, Altshuler1958, Altshuler1962, Altshuler1968, Altshuler1974,
Altshuler1977, Altshuler1981, Altshuler1996, Altshuler1960, Mcqueen1960,
Mcqueen1970, Skidmore1962, Krupnikov1963, Balchan1966, Ragan1984, Hixson1991,
Brown2000, Trunin1972, Trunin1992, Trunin1993, Trunin1994, Trunin2001) and LiF
(Davis2016, Rigg2014, hawreliak2023) and the steadiness of the shock wave as
determined in the velocity histories (S5a,c and e). Reported uncertainty in
the Hugoniots for Fe and LiF place a _minimum_ uncertainty of $\pm$0.4 GPa for
Fe shock stress determination through impedance matching. This value was
calculated for $P_{Fe}$=250 GPa, based on fits to Hugoniot data with
1-$\sigma$ confidence bands. Confidence bands rely only on the fit
coefficients and the estimated uncertainties in the coefficients. This value
represents a systematic uncertainty and is combined with estimated random
uncertainties (described below) to give the total pressure uncertainty values
reported in Table LABEL:table:Summ_table. While the use of reflected Hugoniots
is a standard approximation for the pressure determination through impedance
matching (kerley2013, celliers2005), we note that at high pressure, the use of
a release adiabat is needed (Fig. 3). In our analysis, we initially conducted
impedance matching using the reflected shock approximations and then made an
additional correction based on the trend in Fig. 3, upper.

The measured Fe/LiF particle velocity profiles, shown in Figure S5, are
characterized by an initial shock followed by a time-dependent distribution of
velocity states. The stress in the sample (as reported in Table
LABEL:table:Summ_table) is calculated through impedance matching while
considering the velocity distribution after the initial shock (see Fig. 4b).
The calculated  random uncertainty in stress is a contribution of the
following: (i) the standard distribution of velocity states above the initial
shock, and (ii) the accuracy with which fringe shifts can be measured in the
point-VISAR system, taken here as 0.024 km/s (2% of a fringe shift
(dolan2006)). Other contributors to stress uncertainty that is not explicitly
treated relate to uncertainties in the refractive index of LiF (Rigg2014),
uncertainties in the timing of the X-ray probe with respect to the VISAR, and
uncertainties in the measurements of sample thickness. In addition, the point
VISAR system integrates spatially and therefore does not provide any
information on the distribution of stress states which may arise due to non-
planarities within the drive (as in Ref. (Coleman2022). Moreover, in the case
of a detected Fe/LiF velocity profile which is characterized by an initial
shock followed by an increase in velocity, there is a progressive increase in
shock strength throughout the sample as late-time characteristics catch up and
reinforce, to some extent, the shock front during transit. To account for
these additional uncertainties, we increase our total pressure uncertainties
by an additional $\pm$ 5 GPa. We note that our pressure uncertainties are
similar to Ref. (Coleman2022) but are significantly higher than reported in
Refs. (Turneaure2020, Sharma2019). The high-pressure constraint for solid-
liquid coexistence in Fe is shot 19-C-068, which is the lowest pressure
observation of liquid only. The calculated pressure for this shot based on
impedance matching as described above is 258(8) GPa.

As an additional check on the pressure and pressure distribution within the
sample during the X-ray probe time, the pressure history in Fe is also
determined by simulating the experimental conditions with a 1D hydrocode,
HYADES (Larsen1994), which calculates the hydrodynamic flow of pressure waves
through the target assembly in time ($t$) and Lagrangian space ($h$) (Fig.
4a). The inputs to the hydrocode are the thicknesses of each of the
constituent layers of the target, including the measured epoxy layer
thicknesses ($\sim$1-3-$\mu$m), an equation-of-state (EOS) description of each
of the materials within the target, and laser intensity as a function of time,
$I_{Laser}$($t$). Based on experience over hundreds of shots, we find that
pressure (GPa) in the polyimide ablator scales approximately as
4.65$\times$$I_{Laser}$0.8, with $I_{Laser}$($t$) (PW/m2) calculated from
measurements of laser power divided by an estimate of the laser spot size
(Fig. S4). A series of forward calculations were run with iterative
adjustments of $I_{Laser}$($t$) (few % level) until convergence was reached
between the calculated $u_{p,LiF}$($t$) and the measured $u_{p,LiF}$($t$)
curves (Fig. 4b). Once achieved, the calculated $P$($h$,$t$), at the X-ray
probe time, is used to estimate the volume-integrated pressure and pressure
distribution in the Fe sample. This method of pressure determination
explicitly accounts for any temporal non-steadiness in the compression wave.
The results of the hydrocode simulations in Fig. 4 are for s19-C-068, which in
our experiments represents the first liquid-only shot (the upper-pressure
constraint of the solid-liquid coexistence). The determined pressure in the Fe
sample from the hydrocode simulations is 255.0(2.2) GPa, which is in agreement
with the impedance matching approach (258(8) GPa) described above.

Figure 4: $|$ Hyades simulation of shot 19-C-068. a. Calculated map of
pressure distribution throughout the target package, where the horizontal
dashed lines indicate material boundaries in Lagrangian coordinates. For this
shot, the 80 ps X-ray pulse probed the sample 0.7 ns before shock breakout
into the LiF window. The resultant X-ray diffraction pattern is volume
integrated and therefore includes a scattering contribution from the ambient
material ahead of the shock front as well as the shocked material (behind the
shock front). In this simulation, Fe EOS table #2150 from Ref. (kerley1993)
(see also Fig. S15), and #7271 for LiF (Davis2016) were used. b. Agreement
between measured (solid) with calculated $u_{p,LiF}$($t$) (dashed). Using the
impedance matching approach for determining pressure, we obtain a value of
258(8) GPa. For this estimate, we consider the velocity distribution over the
period between the two blue arrows. These peak velocity values are most
closely associated with the location of the peak of the diffraction data,
which is used to calculate density. During the X-ray probe period, the
hydrocode simulations calculate a value of 255.5(2.2) GPa, consistent with the
impedance matching estimates.

##  3 Results

###  3.1 Liquid structure measurements

The high flux of the X-ray source at DCS coupled with angular coverage up to
almost 8 $\AA^{-1}$ permits quantitative analysis of liquid density recorded
using ps X-ray diffraction. Due to the non-monochromatic “pink” X-ray beam,
the raw liquid diffraction intensities are artificially shifted to higher Q
(Bratos2014). The corrections outlined in Ref. (singh2022) were applied to
account for this artificial shift and derive quantitative densities. This
method has already been applied to other simple metallic systems such as Ag,
Sn, and Cu and shows good agreement with the Hugoniot pressure-density
relationship (Coleman2022, singh2022, Sims2022).

Figure 5:  $|$ Liquid structure measurements. a. Liquid structure factor
(solid lines) and b. corresponding radial distribution function (solid lines)
for dynamically compressed Fe at 258(8) GPa (s19-C-068) and 275(9) GPa
(s19-C-117). Also shown is liquid XRD data taken at lower pressure under
laser-heated static compression (markers and dashed lines) (Kuwayama2020).

Figures 5a,b presents the structure factor, $S(Q)$, and the corresponding
radial distribution functions, $g(r)$ of the two shots with complete melting
(solid black lines). The lowest and highest pressure liquid structure data
presented in Ref. (Kuwayama2020) have also been shown (markers and dashed
lines). That data was collected using a laser-heated diamond anvil cell
apparatus up to a maximum pressure of 116 GPa, providing the density of liquid
iron along several isotherms that lie close to the melting temperature
(Kuwayama2020). The $S(Q)$ and $g(r)$ follow the expected behavior of shifting
to higher $Q$ and lower $r$. The coordination number (CN) was determined by
integrating the area under the first peak in the radial distribution function.
Two different integration schemes were followed for determining the CN: (i)
twice the area up to the peak and (ii) the area up to the first minimum when
plotting 4$\pi$nr2g(r) against r (n is the number density). We find that the
coordination number at 258(8) and 275(9) GPa are $\sim$11.7 and 12.2,
respectively, using the first method, and 12.7 and 12.9 using the second
method. Either method shows simple liquid behavior.

Figure 6: Pressure versus density. $P$-$\rho$ data from this study (bcc - blue
squares, hcp - red squares, and liquid - cyan squares), and a previous XRD
study (Turneaure2020) (hcp - red open squares, hcp+liquid - red crossed
circles). Previously published Hugoniot (Smith2018) (gray triangles), and 300
K static data (Dewaele2006, dubrovinsky2000b, Kuwayama2020) (blue and white
filled circles) are also shown, as well as an isentrope determined from laser
ramp-compression experiments (blue curve) (Smith2018). High-temperature liquid
static data is shown by the open cyan circle along with calculated 4000 K and
5000 K isotherms (Kuwayama2020). Three Hugoniot models are also shown (green
curves) (Wu2023, benedict2022, sjostrom2018). (Inset) A magnified view shows
agreement for melt completion pressures along the Hugoniot between our XRD
data and previous sound speed measurements (nguyen2004). However, previous XRD
data (Turneaure2020) and sound speed measurements (nguyen2004) disagree on the
melt onset pressures along the Hugoniot. These combined XRD studies constrain
the hcp+liquid coexistence along the Hugoniot between 241.5(3)-258(8) GPa,
whereas it is 225(3)-258(8) GPa using the combined sound speed measurements
and XRD data from this study. Figure 7: Fe pressure-temperature phase map. a.
High $P$-$T$ region of the Fe phase map, which shows the stability regions for
the $\epsilon$-hcp, and the liquid phases. The melt line is constrained by
static (anzellini2013), and nanosecond timescale shock+ramp compression
experimental data (Kraus2022). Three theoretical bounds on the Hugoniot are
represented by the solid (benedict2022) and dashed (sjostrom2018), and dashed-
dotted green curves (Wu2023). The extent of solid-liquid coexistence in the
Benedict (benedict2022) and Wu (Wu2023) models are set by the measured changes
in sound speed under shock compression of Nguyen and Holmes (nguyen2004) (see
c. and Fig. S1), which are inferred in that study to reflect the onset and
completion of melt. Our XRD data point for melt completion at 258(8) GPa is
consistent with previous sound speed determinations of solid-liquid
coexistence (green band). However, the XRD data point for melt onset reported
in (Turneaure2020) at 241.5(3) GPa is inconsistent with previous measurements.
This may be related to the insensitivity of X-ray diffraction to measure small
volumes of liquid (see Supplementary Materials). Also shown is the range of
possible temperatures for the earth’s outer core (Dziewonski1981). A more
inclusive representation of experimental and theoretical studies of Fe in
$P$-$T$ space is shown in Fig. S2. b. Energy vs. pressure plot, which shows
our shock-compression data, previous Hugoniot data ((Smith2018), grey
triangles), theoretical predictions for the Hugoniot (Wu2023, sjostrom2018,
benedict2022), and solidus and liquidus curves (derived from EOS in Ref.
(Wu2023)). c. Sound speed data from Ref. (nguyen2004).

Figure 6 shows our data in pressure-density space, color-coded with measured
phase (blue: bcc, red: hcp, cyan: liquid), and plotted against other shock,
ramp, and static compression datasets. Also plotted is the $P$-$\rho$ curve
estimated from the Preliminary Reference Earth Model (PREM) (Dziewonski1981)
for the outer liquid and solid inner core (IC). At 180 GPa, where the $P$-$T$
conditions along the Fe Hugoniot (benedict2022) are comparable to those found
within the outer liquid core ($\sim$180 GPa, $\sim$4080 K, see Fig. 7), there
is a 10.1% difference in the density. The lower density of the Fe-rich outer
core is expected due to the inclusion of 2.7-11.5% by weight of low-Z
impurities (Hirose2021). The inset to Fig. 6 shows the magnified view of the
structural measurements from this study (solid horizontal line). The green
hatched region shows the coexistence region determined using sound speed
measurements from a previous study (nguyen2004). A dashed horizontal line also
shows the melt onset pressure from a recent XRD study in Ref. (Turneaure2020)
at $241.5(3)$ GPa.  Based on the XRD study presented in this work (melt
completion) and in the XRD study of Ref. (Turneaure2020) (melt onset), the
liquid-solid coexistence lies between 241.5(3)-258(8) GPa. This coexistence is
smaller than the 225(3)-260(3) GPa range inferred by Nguyen and Holmes
(nguyen2004) from experiments that measure changes in sound speed as a
function of pressure for microsecond shock compression of mm-thick Fe samples
(see Supplementary Materials Fig. S1) – although a reanalysis of this data is
planned based on a reevaluation of uncertainty analysis (akin2015).

One contributing factor for the discrepancy in melt onset pressures between
the two experimental approaches (XRD and sound speed) _may_ lies in a lower
sensitivity of XRD in detecting the weak diffuse signal of the incipient
liquid phase. The sound speed technique is expected to be sensitive to the
onset of melting.Both the shear and bulk moduli contribute to the sound
velocity in a solid. In contrast, the shear modulus $\rightarrow$ 0 in a
liquid, resulting in a sharp decrease in sound velocity on crossing the melt
curve (nguyen2004) (see Fig. S1). This is expected to be more sensitive than
XRD measurements to the onset of small volumes of liquid (see Supplementary
Methods). However, because of the sparsity of data points in the measurements
presented in Ref. (nguyen2004) (no data between 247 and 260 GPa, see Fig. 7c),
the sound speed measurements are less constraining for the completion of the
melt.

The completion of melt, however, is well constrained through XRD as small
volumes of a textured solid are easily observed. We note that while the lowest
pressure liquid-only data presented here is at 258(8) GPa, the pressure for
melt completion is constrained by this value and the observation of high-
pressure hcp from a previous XRD study (Turneaure2020), to be in the range
246.5(1.8)-258(8) GPa.

The consistency in the pressures for melt completion between experiments with
shock compression durations which differ by over three orders of magnitude is
consistent with a lack of time-dependence in the melt of Fe. This is in
contrast to the reported strong time-dependence reported for the low-pressure
$\alpha$$\rightarrow$$\epsilon$ phase transformation (Smith2013).

Figure 7a shows the pressure-temperature ($P$-$T$) phase map of Fe in the
vicinity of the melt line with three recent Hugoniot models (sjostrom2018,
benedict2022, Wu2023). These hugoniot models were constructed to cross the
melt line in agreement with the extrapolated melt of the static compression
study of Anzellini _et al._ (anzellini2013) and the dynamic ns-compression
work of Kraus _et al._ (Kraus2022). The pressure extent of the solid-liquid
coexistence for the Wu _et al._ , and Benedict _et al._ Hugoniots were set to
agree with the 225(3)-260(3) GPa range inferred from sound speed measurements
from Nguyen and Holmes (nguyen2004) (see also Supplementary Materials Fig.
S1).

The internal energy of the solidus and liquidus curves from the Wu EOS model
is plotted in Fig. 7b (Wu2023). By using the Rankine-Hugoniot equations, based
on conservation of energy, E = $\frac{1}{2}$
P(($\rho$-$\rho_{0}$)/$\rho$$\rho_{0}$) (zel2002), where P is the
experimentally-determined shock pressure, $\rho$ is the density from X-ray
diffraction measurements, and $\rho_{0}$ is the initial density (7.874 g/cm3),
we can compare our measured Hugoniot data with the calculated Hugoniot and
melting curve, and provide a valuable description of the Fe phase diagram in
pressure-energy space (Fig. 7b).

###  3.2 $\alpha\rightarrow\epsilon$ solid-solid phase transition

Iron undergoes bcc $\alpha$ to hcp $\epsilon$ phase transformation around $13$
GPa (Jensen2009). The geometrical mapping from the bcc to the hcp phase has
been an active area of study for decades with several phase transition
mechanisms proposed in the literature (burgers1934, Mao1978, rong1984,
potter1973). In our study, XRD data provides information on microstructural
changes in Fe across the $\alpha$$\rightarrow$$\epsilon$ phase boundary. A
forward diffraction and texture analysis approach is employed to relate
measured azimuthal ($\phi$) diffraction intensities with those predicted from
mechanistic transformation pathway models.

Texture measurement using X-ray diffraction usually involves measuring pole
figures and inverting the pole figures to obtain the full orientation
distribution function. These measurements are routine for static compression
experiments to measure the orientation relationship between ambient and high-
pressure phases. However, due to the ns timescales of shock compression
experiments, the measurement of the complete pole figure is not feasible in
our measurements. Only a single ring of the complete pole figure is measured,
which makes crystallographic texture determination using pole figure inversion
ill-conditioned. Therefore, we employ a forward diffraction model to measure
the orientation relationship (OR). Given all experiment parameters, the model
computes the intensity distribution in $2\theta-\phi$ space. These include:

  1. 1.

The crystal structure, lattice parameters, and phase fraction of the ambient
and high-pressure phase, including the Debye-Waller factors. This information
is used to compute the powder diffraction intensity in $2\theta$.

  2. 2.

Crystallographic texture of the ambient and high-pressure phase. This
information modulates the powder intensity in the $\phi$ direction.

  3. 3.

X-ray source specification such as polarization and meaningful peak shapes for
the X-ray source. In our case, we use the “pink” beam function, which is a
convolution of back-to-back exponential functions with a Gaussian and
Lorentzian function described in Ref. (vondreele2021).

The crystallographic texture is represented using a finite element
representation of the Rodrigues space fundamental zone (Barton2002,
Bernier2006, Singh2020).

The samples used in lower pressure shots (P $<60$ GPa, see Table
LABEL:table:Summ_table) were highly textured and had significant intensity
variation in the azimuthal direction of the Debye-Scherrer rings (see Supp.
Mat. Fig. S11a). Upon compression, the high-pressure hcp $\epsilon-$phase had
a repeatable intensity distribution with the ambient pressure texture. The
texture is used to compute the complete pole figures using the methodology
described in Ref. (Barton2002). The pole figures computed for the ambient bcc
phase using the forward model are shown in Fig. S6. Only a single ring of the
entire pole figure is measured in the experiment, shown by the red dotted
circle. The intensity variation along this ring modulates the powder
diffraction intensity for each diffraction line. The resulting diffraction
pattern after applying the azimuthal modulation is shown in Figs. 9c, S11a.

Figure 8:  Pole figures for different mechanisms Pole figures for some low
index planes ambient cubic BCC phase in the cube orientation and high pressure
hexagonal closed packed phase as a result of the different phase transition
mechanisms: (a) Mao-Bassett-Takahashi OR given by the
$(110)_{\text{bcc}}~{}||~{}(0001)_{\text{hcp}}$ and
$[001]_{\text{bcc}}~{}||~{}[2\bar{1}\bar{1}0]_{\text{hcp}}$ (6 variants), (b)
Burger’s OR given by $(110)_{\text{bcc}}~{}||~{}(0001)_{\text{hcp}}$ and
$[1\bar{1}1]_{\text{bcc}}~{}||~{}[11\bar{2}0]_{\text{hcp}}$ (12 variants), (c)
Potter OR given by $(110)_{\text{bcc}}~{}||~{}(1\bar{1}01)_{\text{hcp}}$ and
$[1\bar{1}1]_{\text{bcc}}~{}||~{}[11\bar{2}0]_{\text{hcp}}$ (24 variants) and
(d) Rong-Dunlop OR given by $(021)_{\text{bcc}}~{}||~{}(0001)_{\text{hcp}}$
and $[100]_{\text{bcc}}~{}||~{}[11\bar{2}0]_{\text{hcp}}$ (12 variants).

A static compression study of single crystal Fe unambiguously demonstrated
that the OR between the single crystal $\alpha$ and high-pressure
$\epsilon-$phase was the Burger’s OR [$(110)_{\alpha}||(0001)_{\epsilon}$ and
$[1\bar{1}1]_{\alpha}||[11\bar{2}0]_{\epsilon}$] (Dewaele2015). All $12$
orientation variants predicted by Burger’s mechanism were observed in the
study. However, the mechanism during dynamic compression experiments is still
unknown. An alternative mechanism has been proposed in Ref. (Mao1978)
resulting in the OR $(110)_{\alpha}||(0001)_{\epsilon}$ and
$[001]_{\alpha}||[2\bar{1}\bar{1}0]_{\epsilon}$. We refer to this as the Mao-
Bassett-Takahashi (MBT) OR. The MBT mechanism results in $6$ orientation
variants that can form from a single $\alpha-$phase grain, half the number of
variants formed in the Burger’s OR (burgers1934). We also considered
experimentally observed bcc-hcp ORs for other material systems (potter1973,
rong1984). The distribution of some low-index $\epsilon-$phase planes formed
as a result of these ORs are shown in Fig. 8a-d. Three of the four ORs produce
very similar distributions of the crystallographic planes. The
$\\{0001\\}_{\epsilon}$ planes formed by Burger’s and MBT OR are identically
distributed, while the angular separation of the $\\{11\bar{2}0\\}_{\epsilon}$
plane is only $5.26^{\circ}$. The OR observed by Potter (potter1973) produces
twice the number of variants as the Burger’s OR, with each pair of the
variants separated by $\sim$$2^{\circ}$ to the Burger’s variant. The full ORs
are listed in Table 1.

Figure 9: $|$ Texture as a result of Burger’s OR. a. Forward model simulation
of XRD pattern and b. experimentally measured XRD pattern for Fe shock
compressed to 25(7) GPa (shot 19-C-122). The diffraction data is a combination
of the ambient $\alpha$-phase (ahead of the shock wave) and the compressed
$\epsilon$-phase (e.g., see top panels). The orientation distribution of the
high-pressure phase was determined by applying Burger’s OR to the ambient
$\alpha-$phase orientation distribution. Averaged profiles in 2-$\theta$ are
shown in lower panels, and profiles averaged in $\phi$ over the regions of
interest defined by the solid (model), and dashed boxes (data) are shown in c.

The pole figure of some low-index $\epsilon-$phase planes formed due to these
ORs is shown in Fig. 8. Note that these pole figures are for the cube
orientation of the ambient bcc phase such that
$(001)_{\alpha}||\textbf{z}_{\textrm{lab}}$ and
$[100]_{\epsilon}||\textbf{x}_{\textrm{lab}}$. Since the ambient
$\alpha-$phase has a pole figure different from the cube orientation as shown
in Fig. S6, the pole figures for the high-pressure $\epsilon-$phase formed as
a result of the different ORs listed in Table. 1 is calculated by applying the
starting orientation of the ambient bcc phase to the pole figures shown in
Fig. 8. The calculated pole figures for the high-pressure hcp phase following
the Burger’s, Mao-Bassett-Takahashi, Potter, and Rong-Dunlop orientation
relationships are shown in Figs. S7, S8, S9, S10 respectively, and the
resulting diffraction pattern are shown in Figs. 9d, S11b-d respectively.

Fig. 9a shows the results of the forward calculation for shot 19-C-122 [25(7)
GPa], with experimental data shown in Fig. 9b. The lattice parameters were
determined by performing a LeBail fit to the 1-D lineouts (see Figs. S12, S13,
S14). Here, Burger’s OR was applied to the ambient phase texture to obtain the
$\epsilon-$phase texture, with good agreement to the experimentally measured
XRD pattern. The diffraction signal consists of ambient $\alpha-$ ($20\%$) and
high-pressure $\epsilon-$phase ($80\%$), with a complete transformation to all
variants of the $\epsilon-$phase.

The results of the forward model are compared with the experimental
measurements by comparing the azimuthal variation of some lower-angle
diffraction lines. While we rule out the Rong-Dunlop OR (rong1984) based on
this calculation, our data are unable to distinguish between the ORs from
Burger (burgers1934), Mao-Bassett-Takahashi (Mao1967), or the Potter
mechanisms (potter1973). High-quality single crystal samples having a small
orientation spread, together with highly monochromatic X-rays need to be
employed to unambiguously identify the phase transition OR for iron during
dynamic compression.

In a shock compression study of single crystal iron along the $[100]$
direction (hawreliak2008), the authors reported formation of polycrystalline
hcp $\epsilon-$phase with the grain size between $2$ and $15$ nm, with the
conclusion that ”single-crystal iron becomes nanocrystalline in shock
transforming from $\alpha$ to $\epsilon$ phase.” While our in-situ diffraction
results do not disagree with these results, it presents a more nuanced picture
of the grain refinement. The overall texture of the ambient $\alpha$ and the
high-pressure $\epsilon-$phase is explained well by a unimodal orientation
distribution. This is akin to a Gaussian distribution around specific
orientations. The full width at half-max (fwhm) for the ambient and high-
pressure phase that gives the best agreement with experimental data is $\sim
2^{\circ}$ and $\sim 4^{\circ}$, respectively. While each grain retains the
mean orientation predicted by the phase transformation mechanism, the
orientation distribution within each grain points to microstructure refinement
with the possibility that each “sub-grain” with that grain is nm-sized. There
is no large-scale grain refinement to result in an almost powder-like
diffraction pattern from the high-pressure $\epsilon-$phase.

Table 1: $|$ Commonly observed orientation relationships between bcc and hcp phases Authors |  Details |  $\boldsymbol{(\textrm{hkl})_{\alpha}}~{}~{}~{}~{}||~{}~{}~{}~{}\boldsymbol{(\textrm{hkl})_{\epsilon}}\newline \boldsymbol{[\textrm{uvw}]_{\ \alpha}}~{}~{}||~{}~{}~{}~{}\boldsymbol{[\textrm{uvw}]_{\epsilon}}$
---|---|---
Burger’s (burgers1934)
(1934) |  Experimental X-ray diffraction study on the orientation relationship between the hcp $\alpha-$ and bcc $\beta-$phase in Zr. |  $(110)_{\alpha}~{}~{}||~{}~{}(0001)_{\epsilon}$
$[1\bar{1}1]_{\alpha}~{}~{}~{}||~{}~{}[11\bar{2}0]_{\epsilon}$
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 |  $(110)_{\alpha}~{}~{}||~{}~{}(0001)_{\epsilon}$
$[001]_{\alpha}~{}~{}~{}||~{}~{}[2\bar{1}\bar{1}0]_{\epsilon}$
Potter (potter1973)
(1973) |  Experimental study on orientation relationship between bcc $\alpha-$phase in Vanadium-Nitrogen system and hcp V3N precipitate using transmission electron microscopy |  $(110)_{\alpha}~{}~{}||~{}~{}(1\bar{1}01)_{\epsilon}$
$[1\bar{1}1]_{\alpha}~{}~{}~{}||~{}~{}[11\bar{2}0]_{\epsilon}$
Rong-Dunlop (rong1984)
(1973) |  Experimental study on orientation relationship between bcc $\alpha-$ferrite and hcp M2C (M = Cr, Mo, Fe) precipitates in ASP23 high strength steel using a transmission electron microscope |  $(021)_{\alpha}~{}~{}||~{}~{}(0001)_{\epsilon}$
$[100]_{\alpha}~{}~{}~{}||~{}~{}[11\bar{2}0]_{\epsilon}$

##  4 Discussion

Latent heat release ($\Delta$Hm) associated with freezing at the Earth’s inner
core boundary (ICB) is regarded as one of the primary drivers ($\sim$20% of
total energy contribution (LABROSSE2015)) of convection currents in the outer
core. This value has not been directly constrained experimentally at ICB
conditions, and therefore models describing the core dynamics rely on
theoretical calculations of $\Delta$Hm (sun2018, cuong2021, sun2022). The
entropy of fusion ($\Delta S_{m}=\Delta$H${}_{m}/T_{m}$) for simple systems is
expected to approach the gas constant, R asymptotically at high pressure
(Stishov1975), commonly referred to as Richard’s rule. Since our liquid
structure measurement of Fe on the Hugoniot demonstrate that Fe remains a
simple liquid at these conditions, such behavior is expected.

Experimentally, this value can be constrained at the high $P$-$T$ conditions
accessed along the Hugoniot by a measure of the extent of solid-liquid
coexistence and knowledge of the equation of state of Fe (see Supplementary
Materials). The measurement of melt completion in this study together with
previous measurements of melt onset (nguyen2004), can be used to constrain the
latent heat of fusion at 258 GPa to 542(83) J/g, equivalent to the entropy of
fusion, $\Delta S_{m}=0.66\textrm{R}$ (see Supp. Mat. Sec. S1). This value is
significantly lower than the expected value of R, and approximately half of
three recent theory predictions at this pressure (cuong2021), and suggests
that the energy contribution to the geodynamo budget due to the freezing of Fe
is smaller than previously thought. We also note that the latent heat at the
same pressure derived from the Fe EOS model of Wu _et al._ (Wu2023) is 666.9
J/g, corresponding to $\Delta S_{m}=0.82\textrm{R}$. If the previous XRD
measurement of $241.5(3)$ GPa (Turneaure2020) is used to constrain the melt
onset, then a latent heat of fusion of $243.4(77)$ J/g is obtained. This
corresponds to $\Delta S_{m}=0.3\textrm{R}$, a value much lower than expected
from Richard’s rule (Stishov1975), especially for a simple liquid system such
as Fe (see Fig. 5a,b). If this value is true, it likely points to new physics
in liquid Fe at planetary interior conditions.

Assuming that the latent heat of fusion at the inner core boundary pressure
(330 GPa) changes negligibly due to higher pressure and a small amount of
impurities, the total power output due to the solidification of the inner core
can be computed. This quantity is the ratio of the heat generated due to the
solidification and the time over which the solidification occurred. The heat
generated due to solidification is given by the product of
$\Delta\textrm{H}_{m}$ of pure iron and the mass of the inner core ($1.1\times
10^{23}$ Kg (SOROKHTIN2011)). The latest estimates for the age of Earth’s
inner core range between $0.565-4$ billion years (LABROSSE1997, stacey2008,
Bono2019, Zhang2020). This results in power output, when considering our
estimates of latent heat, of $0.47(7)-3.35(5)$ TW, which is lower than
previous estimates (Anderson1997). This discrepancy suggests the importance of
compositional convection and/or radioactive heating (Kanani2003, Gando2011) in
maintaining the Earth’s geodynamo.

The depression in the melting point over pure material can be estimated,
assuming an ideal mixing model and thermodynamic equilibrium between pure
solid and liquid with impurities. It is given by (Anderson1997)

$\log(1-\chi_{s})=\frac{-\Delta\textrm{H}_{m}}{R}\left(\frac{1}{T}-\frac{1}{T^{*}}\right).$
(3)

Here, $\chi_{s}$ is the mole fraction of impurities, $R$ is the gas constant,
$T^{*}$ is the melt temperature of the pure material, and $T$ is the melt
temperature in the presence of impurities. For the same level of impurity, a
smaller latent heat will result in a larger drop in the melting temperature.
It is estimated that the liquid outer core contains approximately $2.7-11.5\%$
light impurities by weight (Hirose2021). The melting temperature at the inner
core boundary (ICB; P = 330 GPa) using the melt curve in Ref. (Kraus2022) is
approximately $6220$ K. Using the estimate of the latent heat in this study,
the temperature of outer core side of ICB is constrained to a range of
$3600-5650$ K. We note that the upper bound of this range is more accurate
than the lower bound as Eqn. 3 holds better for dilute solutions, i.e., small
$\chi_{s}$.

##  5 Conclusions

In conclusion, we present a detailed structural study of Fe under shock
compression from 25 to 275 GPa. Liquid scattering data were analyzed to obtain
the scattering factors, the radial distribution function and density on the
shock Hugoniot. The upper bound shock pressure for complete melt was found to
be 258(8) GPa. A forward diffraction model is used to perform texture analysis
of the ambient bcc $\alpha-$phase and high-pressure hcp $\epsilon-$phase. Some
commonly observed orientation relationships between bcc and hcp phases are
tested for consistency with our diffraction data. We are able to rule out the
Rong-Dunlop OR. However, our data can not distinguish between Burger’s, MBT,
and Potter’s orientation relationship. Our XRD data, along with previous melt
onset determination using sound speed measurements (nguyen2004), suggest an
hcp-liquid coexistence along the Hugoniot (225(3)-258(8) GPa). Our data is
consistent with an internal energy change due to melting of 542(83) J/g at
258(8) GPa, which is approximately half of recent theoretical studies, and a
value that suggests that the energy contribution to the geodynamo budget due
to freezing of Fe at the inner core boundary is smaller than previously
thought. The small latent heat also suggests a larger decrease in the melting
temperature in the presence of impurities than previously estimated.

## Acknowledgements

We would like to thank Carol Ann Davis for her help in preparing the Fe
targets. Paulo Rigg, Pinaki Das, Ray Gunawidjaja, Yuelin Li, Adam Schuman,
Nicholas Sinclair, Xiaoming Wang, and Jun Zhang at the Dynamic Compression
Sector are gratefully acknowledged for their expert assistance with the laser
experiments. We thank Yoshi Toyoda for his assistance with VISAR measurements
during the laser experiments, and Richard Kraus for helpful discussions on
pressure uncertainies. This work was performed under the auspices of the US
Department of Energy by Lawrence Livermore National Laboratory under contract
number DE-AC52-07NA27344. This publication is based upon work performed at the
Dynamic Compression Sector, which is operated by Washington State University
under the US Department of Energy (DOE)/National Nuclear Security
Administration under Award No. DE-NA0003957. This research used resources from
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.
DE-AC02-06CH11357.

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A structural study of hcp and liquid iron
under shock compression up to 275 GPa

Saransh Singh, Richard Briggs, Martin G. Gorman, Lorin X. Benedict, Christine
J. Wu, Amy L. Coleman, Federica Coppari, Amalia Fernandez-Pa$\tilde{n}$ella,
Christopher McGuire, Melissa Sims, June K. Wicks, Jon H. Eggert, Dayne E.
Fratanduono, Raymond F. Smith

Supplementary Materials

Supplementary Material Tables:

Table LABEL:table:Summ_table: Summary of data.

Table LABEL:Table:Hm_summ: Table of values used for estimating the latent heat
of fusion.

Supplementary Material Figures:

Figure S1: hcp+liquid coexistence.

Figure S2: Temperature versus Pressure map for Fe.

Figure S3: Fit to Hugoniot data.

Figure S4: Laser focal spot spatial profile.

Figure S5: VISAR traces and Laser power profiles.

Figure S6: Calculated pole figures for the ambient bcc-phase

Figure S7: Calculated pole figures for the high pressure hcp-phase following
the Burger’s orientation relationship.

Figure S8: Calculated pole figures for the high pressure hcp-phase following
the Mao-Bassett-Takahashi orientation relationship.

Figure S9: Calculated pole figures for the high pressure hcp-phase following
the Potter orientation relationship.

Figure S10: Calculated pole figures for the high pressure hcp-phase following
the Rong-Dunlop orientation relationship.

Figure S11: Simulated diffraction pattern for (a) ambient $\alpha-$Fe and
high-pressure $\epsilon-$phase following the (b) Mao-Bassett-Takahashi, (c)
Potter and (d) Rong-Dunlop orientation relationships. The equivalent results
for the Burger’s orientation relationship reported in Fig. 9.

Figure S12: X-ray Diffraction Lineouts (s19-C-122, s19-C-109, s19-C-129,
s19-C-128).

Figure S13: X-ray Diffraction Lineouts (s19-C-121, s19-C-123, s19-C-124,
s19-C-126).

Figure S14: X-ray Diffraction Lineouts (s19-C-125, s19-C-127, s19-C-088).

Figure S15: Sesame EOS #2150 for Fe.

Table S1:  Summary table of all shots taken in the experiment.  Shot $\\#$ |  X-ray $\\#$ |  Pressure (GPa) |  Phase |  density (g/cm3)
---|---|---|---|---
19-C-122 |  438 |  25(7) |  hcp |  9.208(5)
19-C-109 |  333 |  38(7) |  bcc + hcp |  8.036(3)
9.482(7)
19-C-129 |  496 |  33(6) |  bcc + hcp |  8.038(3)
9.242(3)
19-C-128 |  489 |  35(6) |  hcp |  9.301(4)
19-C-121 |  432 |  32(8) |  bcc + hcp |  8.023(3)
9.455(4)
19-C-123 |  448 |  57(7) |  hcp |  9.942(7)
19-C-124 |  454 |  92(7) |  hcp |  10.633(5)
19-C-126 |  468 |  93(8) |  hcp |  10.72(2)
19-C-125 |  462 |  96(7) |  hcp |  10.721(3)
19-C-127 |  474 |  103(7) |  hcp |  10.867(4)
19-C-088 |  182 |  210(13) |  hcp |  12.081(6)
19-C-068 |  037 |  258(8) |  liquid |  12.51 $\pm$ (0.3, 0.06)
19-C-117 |  393 |  275(9) |  liquid |  12.7 $\pm$ (0.35, 0.1)
Figure S1: Pressure versus sound speed. Sound speed data from Nguyen and
Holmes (nguyen2004cp) (green symbols) as a function of shock pressure. While
these data provide no direct structural information, changes in sound speed
are interpreted as the onset and completion of melt, as indicated. In those
experiments, the shock duration applied to mm-thick Fe samples was $\sim\mu$s.
In contrast the combined XRD and laser shock compression experiments reported
here use nanosecond shock compression of 21 $\mu$m thick samples. Our lowest
pressure liquid only point is at 258(8) GPa (solid vertical line). This data
combined with observations from a previous XRD study (Turneaure2020cp) places
the completion of shock melt in the range 246.5(1.8)-258(8) GPa. The
hcp+liquid coexistence region from our data and an earlier XRD study
(Turneaure2020cp) are consistent with with onset (vertical dashed line) and
completion of melt of 241.5(3.0) GPa and 258(8) GPa, respectively.  Figure S2:
Temperature versus Pressure. Pressure-temperature phase map of Fe with three
solid phases and a liquid phase. The melt line as constrained by static
compression studies is bound by yellow shaded region (anzellini2013cp,
boehler1990cp, boehler1993cp, Sinmyo2019cp, konopkova2021, morard2018,
jackson2013). Bounds provided by theoretical calculations are shown by the
blue shaded region (alfe2009cp, swift2020cp, belonoshko2000cp, wu2011). Three
theoretical bounds on the Hugoniot are represented by the green curves
(Wu2023cp, sjostrom2018cp, benedict2022cp). Determined pressure and estimated
temperatures for previous sound speed measurements under shock compression (no
direct lattice level observations) for the onset of shock melt (colored
squares (nguyen2004cp, Brown1986cp, yoo1993)) and full melt (white square
(nguyen2004cp)) are also shown, as well as the calculated melt onset
temperature from a recent laser shock + XRD study (Turneaure2020cp).

##  S1 Latent Heat of Fusion

Following Ref. (Anderson1997cp), the latent heat of fusion, $\Delta H_{m}$ is
given by the expression

$\Delta H_{m}=\frac{T_{m}\Delta V_{m}}{\textrm{d}T_{m}/\textrm{d}P}.$ (4)

Here, $T_{m}$ is the melt curve of iron, $\Delta V_{m}$ is the volume change
due to melting and $\textrm{d}T_{m}/\textrm{d}P$ is the Clapeyron slope.
$T_{m}$ and $\textrm{d}T_{m}/\textrm{d}P$ were determined using the Simon fit
melt curve in Ref. (Kraus2022cp) given by

$T_{m}=5530\left(\frac{P-260}{293}+1\right)^{0.552}.$ (5)

Here, $T_{m}$ is the melt temperature in K and $P$ is the pressure in GPa. The
$\Delta V_{m}$ term is determined by the change in density of the melt onset
point of 225 GPa in Ref. (nguyen2004cp),
$\rho^{\textrm{sol}}_{\textrm{start}}$, and the density of the first purely
liquid point at 258 GPa (shot 19-C-068) in our study,
$\rho^{\textrm{liq}}_{\textrm{end}}$. As the pressure increases between these
two data points, the $\Delta V$ from the volume difference between these
points has contribution both from volume change due to melting, as well as
volume change due to compression. The density change due to compression was
estimated by using the theoretical value for the bulk modulus in the
coexistence (Wu2023cp) using the equation

$\Delta\rho_{\textrm{comp}}=\int_{P_{\textrm{start}}}^{P_{\textrm{end}}}\frac{\rho}{K}\textrm{d}P.$
(6)

$\Delta V_{m}$ can be estimated using

$\Delta
V_{m}=(\frac{1}{\rho^{\textrm{liq}}_{\textrm{end}}}-\frac{1}{\rho^{\textrm{sol}}_{\textrm{start}}+\Delta\rho_{\textrm{comp}}})$
(7)

The table below shows the values and associated uncertainties used for
estimating the latent heat. Assuming uncorrelated parameters, a final value of
$\Delta H_{m}={\color[rgb]{0,0,0}542(83)~{}J/g}$ is obtained.

Table S2: Table of values used for estimating the latent heat of fusion Parameter |  Value (units) |  Source
---|---|---
$T_{m}$
(melting temperature) |  5461$\pm$530 K |  Ref. (Kraus2022cp)
$\textrm{d}T_{m}/\textrm{d}P$
(Clapeyron slope of melt curve) |  10.52$\pm$1.21 K/GPa |  Ref. (Kraus2022cp)
$\rho^{\textrm{sol}}_{\textrm{start}}$
(density at melt start) |  12.318$\pm$0.078 g/cm3 |  Hugoniot Fit in Fig. S3
$\rho^{\textrm{liq}}_{\textrm{end}}$
(density at melt end) |  12.413$\pm$0.077 g/cm3 |  Hugoniot Fit in Fig. S3
$P_{\textrm{start}}$
(Pressure at last purely hcp point) |  225$\pm$3.0 GPa |  Ref. (Turneaure2020cp)
$P_{\textrm{end}}$
(Pressure at first purely liquid point) |  258$\pm$8 GPa |  This study
$K_{T_{m}}$
(Isothermal bulk modulus in the solid-liquid coexistence) |  linear between 936-978 GPa |  Ref. (Wu2023cp)

##  S2 Sensitivity of XRD techniques to measuring small volumes of
liquid/solid

Using XRD techniques our ability to constrain the extent of solid-liquid
coexistence is limited by the ability to determine small volumes of liquid
(melt onset) and solid (melt completion). For comparable volumes, scattering
off liquid is more difficult to detect than a solid. Liquid diffraction is
texture free (more powder like) and is broader in 2$\theta$ and lower in
amplitude than scattering off a comparable volume of solid. In a mixed phase,
if the signal from the liquid is below the threshold of detection then the
analysis will show only compressed hcp as being present. In our transmission
geometry shock compression in the mixed phase will result in a volume-
integrated XRD pattern comprising of ambient-pressure bcc (ahead of the shock
front), and compressed liquid+hcp behind (behind the shock front). Azimthual
averaging of the XRD pattern (see Figs. S12, S13, and S14) results in an
overlap of the three contributions. Determining small volumes of liquid is
further complicated by the pink beam at DCS which results in overlapping of
signal from closely spaced lines.

A recent study on Ge on the same experimental platform, using the same X-ray
spectral distribution at the Dynamic Compression Sector has reported a
detectable liquid volume thickness as low as $\sim$2 $\mu$m (Exp. 3 in Ref.
(Renganathan2023)). Since Fe is close in atomic number to Ge, we expect the
detectability limit in Fe to be similar to the value in Ge. The thickness of
the Fe samples used in Ref. (Turneaure2020) were 15µm, with a reported 89% (or
13.35 $\mu$m) of compressed material. Based on the Ge study in Ref.
(Renganathan2023) it would be expected that at least 2 $\mu$m of Fe liquid
thickness would be detectable (or 15% of the compressed volume).

Figure S3: Hugoniot Fit. The red curve is based on a fit to previous Hugoniot
data (grey triangles), and the dashed black lines are $1\sigma$ prediction
bands which represents the bounds within which there is an expectation 68.3%
of shock data points should fall within. The upper plots shows residuals to
the fit.  Figure S4: DCS laser spot. (left) Equivalent target plane image of
DCS 580 µm diameter laser spot with a normalized intensity scale. The 0.5
intensity contour is highlighted. (right) Normalized intensity as a function
of distance for 50 $\mu$m wide vertical (red) and horizontal (blue) lineouts
from the image on the left (vertical lineout averaged between blue line in
left image). The intensity lineouts are described by a super-Gaussian spatial
profile (dashed blue curve), as indicated. Within this distribution 40% of the
total energy is contained within the central 400-$\mu$m diameter spot (blue
shaded region).  Figure S5: Summary of Fe/LiF particle velocity traces (a),
(c), (e), and the associated laser power versus time profiles (b), (d), (f).
The velocity traces were offset in time such that time=0 equates to the shock
arrival time at the Fe/LiF interface. For some shots the point VISAR systems
did not record usable data and instead a PDV VISAR record was used
(dolan2006cp). The shot number, laser energy for each shot, as well as the
measured phases are listed in the legend. For pressure determination through
impedance matching (as described in the main text) we considered the velocity
distribution between the arrows (as indicated). For traces with only one arrow
we accounted for velocities from t=0. When multiple velocity traces were
obtained for a given shot we performed a weighted average and velocity
distribution analysis over all the velocity information available (from
multiple velocity records).  Figure S6:  Calculated pole figure for ambient
${\alpha-}$Fe Calculated pole figures for the initial texture of the ambient
cubic Fe sample. The red circle shows the region of the $2\theta-\phi$ space
which are measured with our diffraction geometry. Color bar units are m.r.d.
(multiples of random distribution), where 1 is random texture and $>$1 is non
random.  Figure S7:  Calculated pole figure for Burger’s OR for high-pressure
${\epsilon-}$Fe Pole figures from applying the Burger’s orientation
relationship to the ambient sample texture shown in Fig. S6. The red circle
shows the part of the $2\theta-\phi$ space probed in our diffraction
measurements. Color bar units are m.r.d. (multiples of random distribution),
where 1 is random texture and $>$1 is non random. Figure S8:  Calculated pole
figure for MBT OR for high-pressure ${\epsilon-}$Fe Pole figures from applying
the Mao-Bassett-Takahashi orientation relationship to the ambient sample
texture shown in Fig. S6. The red circle shows the part of the $2\theta-\phi$
space probed in our diffraction measurements. Color bar units are m.r.d.
(multiples of random distribution), where 1 is random texture and $>$1 is non
random. Figure S9:  Calculated pole figure for Potter’s OR for high-pressure
${\epsilon-}$Fe Pole figures from applying the Potter orientation relationship
to the ambient sample texture shown in Fig. S6. The red circle shows the part
of the $2\theta-\phi$ space probed in our diffraction measurements. Color bar
units are m.r.d. (multiples of random distribution), where 1 is random texture
and $>$1 is non random. Figure S10:  Calculated pole figure for Rong-Dunlop OR
for high-pressure ${\epsilon-}$Fe Pole figures from applying the Rong-Dunlop
orientation relationship to ambient the sample texture shown in Fig. S6. The
red circle shows the part of the $2\theta-\phi$ space probed in our
diffraction measurements. Color bar units are m.r.d. (multiples of random
distribution), where 1 is random texture and $>$1 is non random. Figure S11:
Simulated diffraction patterns Comparison of simulated diffraction signal with
the experimental diffraction signal for (a) ambient $\alpha-$Fe and high-
pressure hcp phase following (b) Mao-Bassett-Takahashi (MBT) (c) Potter and
(d) Rong-Dunlop ORs. The azimuthal lineout for the band between
$14^{\circ}-18^{\circ}$ diffraction angle is shown on the right for the
simulated (solid) and experimental (dashed) signal. The azimuthally averaged
lineout is shown at the bottom. The inset show presents a high-resolution view
of a small section of the simulated and experimental data. We are not able to
distinguish between the Burger’s, MBT and Potter’s OR based on our data.
However, we are able to rule out the Rong-Dunlop OR.  Figure S12:
Azimuthally-integrated profiles for experiments s19C122, s19C109, s19C129, and
s19c128 (blue) and fit using the HEXRD software (red dotted) (HEXRDgithubcp).
All profiles include a contribution from the unshocked-volume (ambient fcc-
phase) ahead of the shock front (black ticks), and the shock-compressed fcc-
phase (red ticks) and/or bcc-phase (green ticks). The % difference between the
experimental profile and the fit is shown on the right hand axis. Figure S13:
Azimuthally-integrated profiles for experiments s19C121, s19C123, s19C124, and
s19C126 (blue) and fit using the HEXRD software (red dotted) (HEXRDgithubcp).
All profiles include a contribution from the unshocked-volume (ambient fcc-
phase) ahead of the shock front (black ticks), and the shock-compressed fcc-
phase (red ticks) and/or bcc-phase (green ticks). The % difference between the
experimental profile and the fit is shown on the right hand axis. Figure S14:
Azimuthally-integrated profiles for experiments s19C125, s19C127, and s19C088
(blue) and fit using the HEXRD software (red dotted) (HEXRDgithubcp). All
profiles include a contribution from the unshocked-volume (ambient fcc-phase)
ahead of the shock front (black ticks), and the shock-compressed fcc-phase
(red ticks) and/or bcc-phase (green ticks). The % difference between the
experimental profile and the fit is shown on the right hand axis. Figure S15:
Sesame EOS #2150 for Fe as used in simulation reported in Fig. 4, compared to
previously published Hugoniot data.

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