Main

At extreme compressions, found in the cores of large rocky exoplanets, it was traditionally believed that solid matter would assemble in close-packed, metallic crystal structures with such stability that structural rearrangements would be highly unlikely9,10. However, the advent of modern computational tools for predicting structures and electronic properties at any arbitrary volume has suggested a new paradigm for a wide range of highly compressed materials1,2,3,4,5,6. Calculations for matter at terapascal pressures indicate that solid materials may adopt non-close-packed lattice arrangements to accommodate clouds of localized valence electrons forced away from the core electrons by Coulombic interactions and Pauli exclusion11. Theory predicts that the structure of elemental Mg will transform multiple times at pressures approaching 1 TPa as the influence of these quantum effects grows. Close-packed structures such as hexagonal close-packed (hcp) and face-centred cubic (fcc), which are stable in Mg at ambient and moderate pressures, are predicted to be replaced with hexagonal and cubic structures containing more interstitial space within the lattice1,2,4,5. Similar behaviour has been observed in the light alkali metals at more modest pressures due to their high compressibility and low electronegativity7,8. It remains unknown whether such behaviour is ubiquitous in high-pressure solids, as terapascal pressures are inaccessible using standard experimental techniques: such pressures are incredibly challenging to achieve at present using static compression methods12 and sample heating associated with shock compression precludes the study of solid matter at 1 TPa.

To reach the terapascal regime to study solid Mg, we used the National Ignition Facility (NIF) and directly probed the crystal structure at peak pressure with nanosecond-duration X-ray diffraction. Our experiments reveal four new, distinct, non-close-packed structures of Mg at 0.31 TPa, 0.56 TPa, 0.96 TPa and 1.32 TPa. Diffraction at the highest pressures is consistent with the formation of simple hexagonal and simple cubic phases in agreement with decade-old computational predictions4,5. These results represent the observation of solid–solid phase transformations occurring at terapascal pressures and offer experimental insights into how pressure-induced quantum effects can drive structural stability at extreme conditions.

We used 16 beams of the NIF to launch an ablatively driven uniaxial-compression wave into a Mg sample (see Methods). The ramp-shaped laser pulse allowed the sample to achieve high compression at cool enough temperatures to remain in the solid state. Pressure in the sample gradually increased over the 25 ns duration of the laser pulse, before pressure uniformity was achieved for several nanoseconds at peak compression. The target package consisted of a 50-μm-thick polycrystalline Mg foil sandwiched between a Be anvil, coated with a thin Au preheat shield, and a diamond window (Supplementary Fig. 1 and Extended Data Table 1). This target assembly was placed at the front of the target diffraction in situ (TARDIS) diagnostic (Fig. 1)13. A Ge backlighter foil, positioned near the front surface of the TARDIS, was irradiated by 24 beams of the NIF for 2 ns to produce quasi-monochromatic He α radiation (characteristic X-ray wavelength of 1.209 Å)14. The X-ray pulse was timed to probe the sample once it had achieved a uniform pressure state. Diffraction patterns were collected in a transmission geometry on X-ray sensitive image plates placed on the inside top, bottom and rear surfaces of the TARDIS box. Figure 1 shows a representative diffraction pattern in stereographic projection such that the Debye–Scherrer diffraction rings appear as concentric circles. The rear surface velocity of the diamond tamper was measured in each experiment using the velocity interferometry system for any reflector diagnostic15, which determines the sample pressure history13 using the known equation of state of diamond16,17. Sample temperatures were not measured in these experiments but were estimated to range from 2,500 K at 0.31 TPa to 5,000 K at 1.32 TPa using radiation-coupled hydrodynamic simulations (Supplementary Information).

Fig. 1: Experimental set-up of the terapascal NIF diffraction experiments.
figure 1

a, The Mg sample is situated at the front of the TARDIS box, where it is compressed by a temporally shaped laser pulse. A typical target assembly is shown. Laser irradiation of a Ge foil generates an ~10.2 keV X-ray source that probes the sample at peak compression. The diffraction pattern is collected on X-ray sensitive image plates situated within the TARDIS diagnostic. b, Representative free-surface velocity history (yellow line) as measured by the velocity interferometry system for any reflector (VISAR) in each experiment. c, A stereographic projection of a representative diffraction data collected at 0.41 TPa.

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We observed four distinct phases of Mg up to 1.32 TPa. Figure 2a displays four representative diffraction images collected at 0.31 TPa, 0.56 TPa, 0.96 TPa and 1.32 TPa, projected such that the Debye–Scherrer diffraction rings appear as vertical lines of constant scattering angle. The azimuthally averaged one-dimensional (1D) profiles for all experiments are shown in Fig. 2b, with the diffraction peaks from compressed Mg shaded. In some experiments, non-sample diffraction peaks were observed either from the Be pusher, Au heat shield or single-crystal diamond window (Extended Data Figs. 1–6). Six diffraction reflections from the compressed Mg sample were visible at 0.31 TPa (Fig. 2b). From an analysis of the location of these reflections, it is clear that the body-centred cubic (bcc) phase that is predicted to be stable at these conditions cannot explain all observed reflections. However, the diffraction data are consistent with a bcc-like structure, given that the location of the strongest reflections are similar to those predicted for bcc phases, in addition to the observation of weaker diffraction peaks (Fig. 3a). We propose a structural candidate, a bc6 structure, that explains all reflections and gives a reasonable density (Supplementary Fig. 2), but this structure was calculated to have considerably higher enthalpy than the bcc phase. We are therefore unable to unambiguously determine the structure of this phase. A similar diffraction pattern, suggestive of a similar structure, was also collected at 0.41 TPa (Fig. 2b and Extended Data Fig. 6).

Fig. 2: X-ray diffraction from extremely compressed magnesium.
figure 2

a, Four representative 2D diffraction images collected at 0.31 TPa, 0.56 TPa, 0.96 TPa and 1.32 TPa (top to bottom) and projected into ϕ (azimuthal angle)–2θ (diffraction angle) space. b, Corresponding azimuthally averaged diffraction data. Diffraction from the Mg sample is highlighted by shading and the Miller indices of the determined phases are labelled where possible. Non-sample, or non-diffraction features in the 2D image have been masked out for clarity (red dashed outlines in a). Weak reflections inconsistent with the bcc phase are highlighted in the 0.31 TPa and 0.41 TPa profiles with black asterisks. Diffraction peaks originating from the Be pusher are observed in some experiments and are also highlighted.

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Fig. 3: Comparison with theoretical predictions.
figure 3

a, Comparison of measured d spacings (circles) with density function theory (DFT) simulations conducted at zero temperature (0 K) for the bcc, Fmmm, simple hexagonal and simple cubic phases of Mg and the bcc phase of Be (coloured lines). The simulated bcc lines are shown to highlight the similarities with some of the measured d spacings at 0.31 TPa and 0.41 TPa. One marker at 0.56 TPa is shaded grey/purple to indicate that the observed diffraction peak intensity had contributions from both the Mg sample and Be pusher. b, Calculated pressure–density relations for various phases of Mg with experimentally-determined values overlaid. An extrapolation of the PV relation of bcc Mg from 300 K static compression23 (solid black line) and the PV relation for shock-compressed Mg from a tabular EOS (SESAME EoS 2860)24,25 (dashed black line) are also shown for comparison. Error bars represent 1σ uncertainties in pressure.

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At 0.56 TPa, five diffraction reflections from the Mg sample were observed (Fig. 2b). The fcc phase that is predicted to be stable at these conditions could not explain all observed reflections5. However, as before, diffraction peaks in similar locations to those predicted for fcc, as well as the observation of several weak reflections, pointed towards the actual structure being a distortion of fcc. We found that a structure with space group symmetry Fmmm and four atoms in the conventional unit cell (Z = 4) explained all observed reflections (Fig. 3a) and was energetically competitive with fcc (Supplementary Fig. 7). This new phase of Mg is an orthorhombic distortion of fcc with the main structural changes involving an ~11% contraction of the a lattice parameter and an ~15% expansion of the c lattice parameter with little relative change to the b axis. The measured lattice parameters are a = 2.58(1) Å, b = 2.91(1) Å and c = 3.35(1) Å where the numbers in parenthesis represent 1σ uncertainties.

At 0.96 TPa, two diffraction peaks from the Mg sample were observed. Although we are unable to unambiguously determine the structure from these data, the diffraction is consistent with the simple hexagonal structure (space group symmetry P6/mmm (Z = 1)), predicted to be the stable structure at these conditions5. Assuming a simple hexagonal structure, the measured c/a ratio (c/a = 0.87(1)) agrees well with the prediction from previous density functional theory (DFT) calculations (c/a = 0.90)2,5 supporting our assignment of this phase. The measured lattice parameters are a = 1.87(1) Å and c = 1.63(1) Å. Furthermore, the d spacings of the two observed reflections were compared with DFT predictions for energetically favourable high-pressure phases (Fig. 3a and Supplementary Fig. 2) and shown to be consistent with only the simple hexagonal phase, within experimental uncertainty. Given the low signal-to-noise ratio of the diffraction data at 0.96 TPa, there may be other weak diffraction lines present that we were unable to resolve. However, as we saw no evidence of other diffraction lines, we chose to make the simplest interpretation of the data that was also supported by theoretical predictions5. One diffraction peak observed at 0.41 TPa, 0.56 TPa and 0.96 TPa is consistent with a bcc phase of Be, the material used as the pusher in these experiments (Fig. 3a, Supplementary Figs. 5 and 6 and Supplementary Table 1). This phase has been predicted to become stable above 0.3 TPa (refs. 18,19,20,21).

At 1.32 TPa, the observed diffraction pattern also showed two peaks. Again, while we were unable to definitively determine the structure of the Mg sample at these conditions, the two diffraction peals were consistent with the simple cubic structure (space group symmetry Pm3m (Z = 1)) predicted to be stable at these conditions5. Observed d spacings were again compared with DFT predictions for the candidate phases at 1.32 TPa (Fig. 3a and Supplementary Fig. 10), and only the simple cubic phase assignment is consistent within experimental uncertainty. As with the diffraction patterns collected at 0.96 TPa, we cannot rule out the presence of additional weak diffraction peaks in the data but chose the simplest interpretation of the data, which is that the sample forms the simple cubic structure, in agreement with theoretical predictions5. The simple cubic lattice parameter is a = 1.60(1) Å. To our knowledge this is the highest-pressure structural phase transition observed and corresponds to an ~5.5-fold compression of Mg.

The measured pressures and densities of the assigned Fmmm, simple hexagonal and simple cubic phases are compared with DFT predictions in Fig. 3b. The density for the phases observed at 0.31 TPa and 0.41 TPa was calculated assuming a bcc phase, using the reflections consistent with bcc, as a likely phase assignment was not found for the apparently distorted structure (see Extended Data Table 2 for more information on measured diffraction locations, densities and uncertainties). We investigated this unexpected observation of non-close-packed structures at 0.31 TPa, 0.41 TPa and 0.56 TPa (fcc was predicted to be stable to 0.76 TPa; ref. 5) by performing structure searches at 0 K. We found several lower-symmetry phases (Supplementary Fig. 7) that are energetically competitive at these conditions. Among them is the orthorhombic Fmmm phase observed experimentally, which lies within 20 meV of the stable fcc phase. This suggests that elevated temperatures present in the ramp-compressed Mg (~4,200 K predicted, see the Supplementary Information) may stabilize the Fmmm phase—a surprising result, as symmetry-breaking transitions are commonly considered to be low-temperature phenomena22. However, phonon free-energy calculations within the quasi-harmonic approximation show that the Fmmm phase of Mg has larger entropy (S) than fcc such that at elevated temperatures, the Gibbs free energy (G = H − TS), where T is temperature and H is enthalpy, of Fmmm becomes lower than fcc (Supplementary Fig. 7). The Fmmm phase was also found to be dynamically stable with no negative phonon frequencies (Supplementary Fig. 8).

The key finding of this study is the experimental observation that non-close-packed structures do become stable at terapascal pressures and over fivefold compression, confirming the predictions of theoretical calculations for Mg and other elemental systems over the past decade1,2,4,5. The drop in coordination number from 12 in ambient hcp Mg to 8 in the simple hexagonal phase at 0.96 TPa and 6 in the simple cubic phase at 1.32 TPa contradicts our traditional expectation that crystals should pack more efficiently with increasing compression9,10. However, similar predictions at terapascal pressures for several other elemental systems such as C (ref. 3), Al (ref. 4) and Si (ref. 6) suggest that this behaviour could be relatively general. At extreme compression, as valence electron density is forced away from atomic sites due to Coulomb effects and Pauli exclusion, the system can be stabilized by forming electride structures in which charge is localized into interstitial sites in the lattice. Structures such as simple hexagonal and simple cubic, which are energetically unfavourable configurations at ambient conditions due to their low atomic packing fractions (60% for simple hexagonal and 52% for simple cubic), emerge as the most stable configurations at terapascal pressures as their larger interstitial regions mean that they can accommodate localized charge density more easily than close-packed structures (74% atomic packing fraction) (Supplementary Fig. 7). In Fig. 4, we show calculations of the electronic localization function for the observed and predicted structures of Mg (yellow regions in each unit cell) (see Methods). Our calculations, as well as previous studies4,5, predicted that valence charge accumulates in the interstitial regions of simple hexagonal Mg, centred at the 2d (\(\frac{1}{3}\), \(\frac{2}{3}\), \(\frac{1}{2}\)) Wyckoff site of the simple hexagonal lattice (Fig. 4) and the 1b (\(\frac{1}{2}\), \(\frac{1}{2}\),\(\frac{1}{2}\)) Wyckoff site of the simple cubic lattice (Fig. 4) forming pseudo-ionic MgE2 and MgE structures (where E denotes localized-electron pseudo-anions), analogous to the AlB2 and CsCl structures. We suggest that the direct observation of the simple hexagonal and simple cubic phases of Mg is experimental evidence of how core–valence and core–core electron interactions can influence material structure at terapascal pressures. Figure 4 shows our revised temperature–pressure phase diagram of Mg to 7,000 K and 1.4 TPa. Our experiments complement previous 0 K theoretical investigations and have mapped out high-temperature regions of the Mg phase space, discovering new distorted cubic structures between 0.31 TPa and 0.56 TPa, as well as the simple hexagonal and simple cubic structures at 0.96 TPa and 1.32 TPa.

Fig. 4: Pressure–temperature phase diagram of Mg.
figure 4

The phase diagram was constructed using static compression (solid black line)23 and previous DFT calculations (dashed black lines)5. Black circles with upward arrows represent estimates of Mg temperatures for the NIF experiments based on hydrocode simulations that incorporate radiation coupling and thermal transport models (Supplementary Information). Error bars represent 1σ uncertainties in pressure. The calculated Hugoniot is shown (red line). The primitive unit cells of the stable phases and the localized valence charge are shown (yellow regions) (conventional unit cell is shown for the bcc structure). High-temperature regions representing the bcc-like and Fmmm regions are indicated.

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Using X-ray diffraction, we report four new phases of Mg up to 1.32 TPa giving experimental insight into phase transformations in extremely compressed solids above 1 TPa. Our diffraction measurements show clear evidence of non-close-packed, open structures emerging as the most stable configurations at terapascal pressures, providing experimental evidence of the formation of novel electride states and confirming decade-old predictions.

Methods

Experimental methods

Experiments were performed at the NIF at the Lawrence Livermore National Laboratory. Targets consisted of a 50 μm polycrystalline Mg foil, sandwiched between a 200 μm diamond window and a 20 μm Be pusher that was bonded to a 40 μm Be ablator (Fig. 1). We used a double-sided non-contact ZYGO white-light interferometer to measure a 3D surface optical profile (x–y spatial dimensions and z height relative to a reference measurement) of all parts before assembly and again during assembly after each subsequent epoxy layer was added, resulting in a thickness uncertainty of less than 0.2 μm for each component. More details can be found in ref. 13. Thin coatings of Re and Au were deposited between the pusher and ablator to protect the sample from X-ray heating from the drive plasma and the X-ray source. The target package was mounted onto a 400-μm-diameter pinhole made of U + 6%Nb, which was positioned on the front of the TARDIS diagnostic (Fig. 1)13. After background subtraction13, the 2D diffraction images were azimuthally averaged to give 1D diffraction profiles. To perform the calibration of the system, geometry diffraction lines from a compressed sample were used as input features for an optimization routine as described in Rygg et al.13. Each diffraction line was constrained to have the same 2θ along a given Debye–Scherrer arc; this assumes that the effects of pressure non-uniformity, sample strength or finite grain size do not cause 2θ distortions of the diffraction lines. Samples were compressed using 16 beams of NIF with temporally shaped laser pulses of 25 ns duration and peak intensities up to 1 × 1014 W cm−2. The drive beams were incident 40–46° from normal and 1 mm continuous phase plates were used to smooth the beam profile. The drive beams were pointed into four quadrants to achieve an ~1.8 mm drive spot. This configuration has been demonstrated to achieve transverse uniformity in the drive sample13. A typical pulse shape used in these experiments is shown in Supplementary Fig. 11. Quasi-monochromatic X-rays were generated by irradiating a Ge foil with 24 beams of NIF with a 2 ns laser pulse with a peak intensity of 2 × 1015 W cm−2. The X-rays were emitted isotropically from the Ge foil. Diffracted X-rays from the Mg sample were collimated by the pinhole and collected on a set of X-ray sensitive image plates in the TARDIS diagnostic. Plastic and Ge filters were placed in front of the image plates to absorb unwanted X-rays. While we use the term ‘pressure’ throughout the manuscript, our samples were uniaxially compressed. The presence of deviatoric stress states (not measured) due to high-pressure strength could result in states that deviate from ideal hydrostaticity.

Theoretical methods

First-principles structure searches, enthalpy calculations and molecular dynamics simulations were performed with the Vienna ab initio simulation package (VASP)26 within the Perdew–Burke–Ernzerhof generalized gradient approximation27 of DFT. We used a ten electron projector augmented wave pseudopotential from the VASP library with an 800 eV plane wave cutoff and 1.7 Bohr outmost core radius. We performed first-principles molecular dynamics simulations of liquid Mg along the 10,000 K isotherm over the pressure range of interest. These were carried out with 128-atom supercells in the canonical (constant volume, temperature and particle number) ensemble, with the (\(\frac{1}{4}\), \(\frac{1}{4}\),\(\frac{1}{4}\)) k-point sampling of the Brillouin zone, Fermi–Dirac smearing and a Nosse–Hoover thermostat with a 1 fs ionic time step. The liquid was quenched at pressures around 350 and 650 GPa until signs of solidification appeared. From the lowest temperature liquid (T ≈ 5,000 K) obtained at each pressure random configurations were taken with cluster sizes between 2 and 10 Å, for which full structural optimizations were carried out. These optimizations were carried out at (target) pressures of 0.32, 0.6 and 0.96 TPa. Overall, about 300, 400 and 100 configurations were generated and optimized at each of the three pressures, respectively, with the number of atoms ranging from 1 to 30. To confirm that this search procedure was exhaustive, we carried out additional genetic structure search and adaptive metadynamics calculations using the Universal Structure Predictor Evolutionary Xtallography (USPEX) package28,29,30,31,32. These were carried out with 20–30 hereditary generations, starting with 60 random structures with unit cells of 1 to 28 atoms, 50% of each generation were used to create the next, and k-point sampling with a reciprocal space resolution of 0.039. However, no additional competitive structures were found. The enthalpies of the most competitive structures within a 50 meV per atom window found by the liquid sampling method method at 0.32, 0.6 and 0.96 TPa were then computed in the range from 0 to 1.3 TPa. For the final structural calculations and enthalpy computations, the Brillouin zone was sampled with dense k-point meshes sufficient to ensure enthalpy convergence better than 1 meV per atom (22 × 22 × 22 for fcc and simple cubic, 28 × 28 × 28 for bcc and simple hexagonal, 18x18x18 for hcp, 24 × 24 × 24 for Fmmm (Supplementary Table 2), 36 × 36 × 36 for Immm and R-3m, and 6 × 6 × 6 for Pnma. The accuracy of the VASP pseudopotential in describing Mg at the highest pressure considered here was verified by carrying out additional calculations with the ABINIT code33 and using a norm-conserving pseudopotential with a 1.56 Bohr outmost core radius and 1,360 eV cutoff. Finally, phonon and Gibbs free-energy calculations were performed within the DFT perturbation theory34 as implemented in ABINIT. The dynamical matrices were computed on uniform 5 × 5 × 5 q-point meshes in the Brillouin zone, from which interatomic force constants were obtained and used to interpolate the phonon dispersions on 38 × 38 × 38 q-point meshes. As discussed in the main text, we found from first-principles theory that Fmmm trends towards stability over fcc Mg at very high temperature. However, we note that the energy differences between fcc and Fmmm are relatively small and anharmonic effects are neglected in the quasi-harmonic Gibbs free-energy calculations, and therefore this method cannot be used to accurately predict the transition temperature, but instead only indicate the trend with increasing temperature. To quantify the degree of interstitial electron localization (yellow regions of unit cells in Fig. 4), we computed the electron localization function (ELF) from 0 K unit cells at selected pressures. ELF = 0.5 corresponds to the electron gas (that is, delocalized electrons) and ELF = 1.0 corresponds to perfect localization. We deemed ELF > 0.75 to correspond to localized electrons. We then partitioned the charge density in regions where ELF > 0.75 and ELF < 0.75. Integrating the interstitial region where ELF > 0.75 gave the number of localized electrons. It is worth noting that absolute magnitudes depend on the choice of ELF cutoff, but the general trend is not very sensitive to this choice.