Sodium’s elusive tI50 phase predicted to be a body-centered tetragonal electride

Abstract

Sodium exhibits an anomalous melting curve with a minimum at high pressure, where several complex crystalline phases coinciding with this minimum have been observed but remain unresolved. Solving these structures would help better understand sodium’s unusual melting behavior. In this study, we determine the crystal structure of one such phase, the tI50 phase. Using a data-derived potential-assisted structure search, we identify the tI50 phase as a body-centered tetragonal structure with 50 atoms per unit cell, crystallizing in the I4/m space group. The predicted lattice parameters deviate by less than 0.38% from experimental values, and enthalpy calculation confirms this structure as the ground state for sodium across a ~20 GPa pressure range. The structure features a cage-like polyhedral network with interstitial electride states and remains weakly metallic, unlike the insulating hP4 phase. Resolving this unique structure advances understanding of sodium’s diverse structural behavior under high pressure.

Sodium exhibits an anomalous melting curve with a minimum at high pressure, but the crystalline phases coinciding with this minimum remain unresolved. Here, using a data-derived potential-assisted structure search, the authors predict the crystal structure of one of those phases, tI50, and show that it exhibits weak metallic character.

Introduction

Recent high-pressure experiments have fundamentally challenged the longstanding assumption that metals, when subject to high pressure, would favor densely packed structures with nearly free electrons (NFE). At extreme pressures, compression becomes so intense that it can induce core-valence or even core-core overlaps, which lead to interesting and unexpected changes in the electronic characteristics of metals^1,2. This phenomenon is particularly notable in alkali metals such as Li and Na, where such overlaps are expected to result in an insulating state with a narrow band gap^1,2, a prediction that has been confirmed experimentally^3,4. Contrary to the intuitive expectation that compression would enhance interatomic interactions and broaden the electronic bands^5,6 -- thereby making alkali metals even more free-electron-like, research has revealed that these elements adopt remarkably complex structures under high pressure. Sodium, in particular, exhibits a series of pressure-induced phase transitions at room temperature: bcc → fcc at 65 GPa⁷ → cI16 at 103 GPa^8,9 → oP8 119 GPa⁸ → tI19 at 125 GPa⁸ → hP4 at 200 GPa⁶. Notably, the hP4 phase is optically transparent and insulating. Its unusual electronic and optical properties are the result of p-d hybridizations of valence electrons being repelled into the lattice interstices, where they become localized due to repulsion from the core electrons⁶. These interstitial electrons in this phase behave like anions without nuclei, forming what is known as the ‘electride state’, a concept that originated in low-pressure molecular systems¹⁰. Subsequent theoretical work extended these ideas to molecular orbital analyses of a variety of lithium-ammonia solutions, which provides insights into electron localization in extended systems¹¹. The interstitial quasi-atom (IQA) model was later proposed to provide a theoretical framework for predicting electride formation in compressed solids^12,13. The hP4 phase of sodium has been observed to persist up to nearly 500 GPa under ramp compression¹⁴, with predictions of further structural transformations at higher pressures or under varying temperature conditions^15,16.

An especially intriguing feature of the Na phase diagram is its anomalous melting curve^17,18. Diffraction experiments have revealed a peak melting temperature of about 1000 K at 30 GPa, followed by a sustained decline as pressure increases, bringing the melting point to near room temperature at 120 GPa¹⁷. This makes Na one of the few known elements with an experimentally accessible liquid state at such extreme densities^17,19. It is hypothesized that the forces responsible for sodium’s transition into an electride state also play a fundamental role in driving this unusual melting behavior¹⁴. In 2008, Gregoryanz et al. made a critical discovery: the minimum of the melting curve coincides with the presence of seven distinct crystalline phases identified through single-crystal diffraction experiments⁸. To date, three of these phases (cI16, oP8, and tI19) have been fully characterized, one of which is also observed in Li under high pressure²⁰. However, the remaining four phases (tI50, aP90, oC120, and mP512) remain unresolved due to their structural complexity with large numbers of atoms in the unit cells (as indicated by their Pearson notations). Resolving these complex structures will help advance the understanding of Na’s unusual melting behavior¹⁷.

This study investigates the long-elusive structures of Na observed by Gregoryanz et al. using state-of-the-art computational techniques. A comprehensive search for Na structures was conducted at 120 GPa, near where the complex phases were experimentally observed. The search is carried out across a polymorphic space of 250,000 randomly generated structures, incorporating unit cells with up to 120 atoms in all possible space groups using data-derived potential²¹ within a sensible random structure search approach²². The data-derived potential is constructed through iterative exploration of configuration space without prior knowledge of any local minima. With this search, we report with confidence the determination of the tI50 structure, a body-centered tetragonal structure with the I4/m space group. Enthalpy calculation at 0 K establishes the tI50 phase as the thermodynamic ground state for Na at pressures between 143 and 166 GPa, while phonon calculations confirm its dynamic stability. The tI50 phase, like other Na phases at similar pressures, exhibits electride states due to the reordering of energies for orbitals centered on interstitial voids and those centered on ionic cores. The identification of the tI50 structure resolves a critical puzzle in the complex structural landscape of Na under high pressure.

Results and discussion

In the initial test of data-derived potential, the structure search successfully yielded all the known ground-state structures of Na, including bcc, fcc, cI16, oP8, and hP4, at pressures closely matching their respective stable pressure ranges. This is a noteworthy result, especially considering that the training set contained only randomly generated structures with eight atoms, with none of these ground-state structures included. Despite the relative simplicity of these structures, finding nontrivial physics, such as electride states, supports the validity of the potential model. The production search using unit cells containing 50 atoms at 120 GPa, followed by screening and density functional theory (DFT)²³-based optimization, revealed that energetically competitive structures in tetragonal unit cells corresponding to several space groups, which are, in increasing order of enthalpy, I4/m, I-4, P4₂2₁2, I-42m, I4/mmm, P-4n2, P4/mmm, P-4, and P-42c. A local search within these space groups confirmed that the structure with the lowest enthalpy at this pressure is the I4/m structure. Figure 1a shows the calculated enthalpies of Na structures as functions of pressure. It shows that the I4/m structure becomes more stable than the cI16 structure at around 146 GPa and remains the thermodynamic ground state for Na until approximately 167 GPa, where the oP8 phase becomes more stable.

Fig. 1. Equation of states and vibrational energies of the tI50 phase.

a Calculated enthalpies of dense sodium phases as functions of pressure. b, c Calculated vibrational free energies per atom for four dense sodium phases at 150 GPa and 170 GPa. In both (b) and (c), the vibrational free energy of the fcc structure at the denoted pressure is used as the zero-energy reference level.

To investigate the temperature-dependent stability of the I4/m structure, we calculated its vibrational contribution to free energy at finite temperature within the quasi-harmonic approximation,

$$F\left(V,T\right)=E_0\left(V\right)+k_{B}T{\int }_0^{\infty }g\left(\omega \right)\ln \left[2\sinh \left(\frac{\hslash \omega }{2k_{B}T}\right)\right]d\omega ,$$ 1

where E₀(V) is the static crystal energy, and the second term is the vibrational free energy F_phonon, with ω as the phonon frequency and g(ω) the normalized phonon density of states (PHDOS). Figure 1b, c compares the calculated vibrational free energies of fcc, cI16, oP8 and I4/m structures at 150 and 170 GPa, respectively. At 150 GPa and 300 K, the vibrational free energy of I4/m structure is ~ 1.22 meV per atom lower than that of the cI16 structure, suggesting that the former is thermodynamically favored at room temperature (Fig. 1b). At 170 GPa and 300 K, the I4/m structure is slightly higher in vibrational free energy than the oP8 structure by ~1 meV per atom (Fig. 1c). A correction of the stable range of the I4/m structure can be made accounting for these free energy corrections: it would refine its stable pressure range to 143–166 GPa at room temperature, representing a modest shift from the athermal enthalpy range of 146–167 GPa. This calculated pressure range is reasonably consistent with the experimental observation of the tI50 phase near 118 GPa⁸. It is interesting to note that a similar discrepancy of ~60 GPa was also reported between predicted and observed stability pressures for the hP4 phase in earlier studies⁴ – such discrepancies are common in high-pressure experiments and not solely due to the athermal nature of DFT. Despite minor temperature effects, our results strongly support I4/m as the observed tI50 phase.

The tetragonal I4/m structure optimized at 117 GPa has lattice parameters of a = 7.245 Å and c = 9.228 Å (Supplementary Table 1). These theoretical lattice parameters closely resemble the experimentally determined values for the tI50 phase, which are a = 7.234(1) Å and c = 9.193(3) Å at 117(2) GPa⁸. The discrepancy between theoretical and experimental lattice parameters is less than 0.38%, establishing the accuracy and reliability of our prediction. This close agreement, combined with its low enthalpy, identifies the I4/m structure as a compelling candidate for the tI50 phase. Notably, the I4/m structure represents a unique crystal structure that has not been previously observed in any element at either ambient or high pressures. In the I4/m structure, the optimized Wyckoff positions are as follows: 16e (0.5806, 0.6972, 0.1184); 16i (0.7167, 0.5905, 0.6814); 8g (0.0000, 0.5000, 0.6321); 8h (0.3637, 0.2092, 0.5000); 2a (0.0000, 0.0000, 0.0000). The crystallographic information file (.cif) for the optimized I4/m structure is provided as Supplementary Data 1. The body-centered tetragonal unit cell is characterized by a dense, cage-like polyhedral network of Na atoms (Fig. 2a, b). The shortest Na-Na contacts within the I4/m structure (at 117 GPa) range from 2.202 Å to 2.462 Å. The central atom at the 2a site is surrounded by atoms from the 16e and 8 g sites, forming a 12-fold coordination hub. The atoms at the 8g sites are 8-coordinated, while those at the 8h sites are 9-coordinated, with additional coordination connecting to the central atom. All 16e atoms exhibit 8-fold coordination. Different coordination patterns give rise to a cage-like structure, where enhanced interactions under high pressure increase the packing efficiency. Notably, the I4/m structure exhibits a higher density than its precursor, the cI16 structure, with a calculated density increase of 0.47% at 117 GPa. This enhanced density reflects the efficiency of the polyhedral network in accommodating the high-pressure environment. Interestingly, a previous prediction has suggested that Na can adopt a cage-like structure at terapascal pressures¹⁵; however, the current finding indicates that a similar atomic arrangement can form at significantly lower pressures.

Fig. 2. Crystal structure and vibrational properties of the tI50 phase.

a, b Optimized I4/m structure of sodium viewed along different crystallographic axes. c Phonon dispersion curves for the I4/m structure of sodium calculated at 150 GPa.

The density functional perturbation theory (DFPT)²⁴ is employed to evaluate the dynamical stability of the I4/m structure at 150 GPa and 0 K. The absence of imaginary phonon modes across the entire Brillouin zone confirms the dynamical stability of this structure (Fig. 2c). An interesting feature of the phonon dispersion curves is the appearance of an envelope-like parabolic shape that emerges from combining segments from different phonon branches as the k-vector traverses from Γ to M, and then to X, and back to Γ. A similar profile is observed when the k-vector moves from Γ to P, and then to N. These parabolic envelopes, peaking slightly above 12 THz, represent composite dispersions arising from the interaction of multiple phonon modes. It indicates transitions from acoustic to optical phonon modes that lead to phonon mode hybridization or band crossings along high-symmetry paths in the Brillouin zone. The shape and peak frequency of these envelopes reflect the high stiffness and strong interatomic forces in the I4/m structure under high pressure.

To further investigate the thermal stability of the tI50 phase, ab initio molecular dynamics (AIMD) simulations were performed on a supercell of the I4/m structure at high pressure (150 GPa) and room temperature (300 K). After equilibration, a 20-ps thermal trajectory was collected in an isothermal-isobaric (NPT) ensemble to monitor structural evolution under conditions of constant pressure and finite temperature. To further verify this and check any potential temperature drift, an additional 12-ps trajectory was obtained in a canonical (NVT) ensemble. The time evolution of total energy, pressure, and temperature as functions of time in both NPT and NVT ensembles are presented in Supplementary Figs. 3 and 4. Within the NPT ensemble, the total energy remains well equilibrated with only small fluctuations. The pressure stays centered around 150 GPa with small fluctuations due to atomic motions. The system maintains an average temperature near 300 K as well. In the NVT simulation, the system is heated up from 0 K and equilibrated quickly to 300 K and remains well equilibrated at this temperature. After equilibration, the total energy stabilizes around a constant mean value, and the pressure, which is not controlled in this ensemble, remains centered around 145.5 GPa. The slight pressure drop is due to the expansion of the system at finite temperature. The absence of any abrupt changes in energy, volume, or pressure throughout both trajectories indicates that the I4/m structure is thermodynamically stable under these conditions, providing further support for its identification as the experimentally observed tI50 phase.

The band structure and density of states (DOS) of the I4/m structure calculated at 150 GPa are shown in Fig. 3. The absence of a band gap near the Fermi level confirms the metallic behavior of this phase. However, the band structure exhibits notable pressure-induced features that significantly deviate from the NFE behavior of Na observed under ambient conditions. At 150 GPa, the NFE characteristics are only observed at the bottom of the valence bands, where the 2s-electron dominates the occupied states. The electronic DOS indicates an increasing contribution of 2p-electrons as the energy rises within the valence bands, which is unusual for NFE metals, where s-electrons are expected to dominate. This observation suggests a pressure-induced s-p hybridization and a reconfiguration of orbital occupancy. These behaviors have been observed previously in other high-pressure phases of Na^25–27. The occupancy of d-electrons emerges in the valence bands and grows with increasing energy and eventually reaches a level slightly higher than that of the s-electrons at the Fermi level. This indicates an occurrence of pressure-induced s-d transition, commonly seen in alkali metals. Notably, the calculated electronic DOS at the Fermi level is 0.163 states eV^–1 atom^–1, substantially lower than the 0.489 states eV^–1 atom^–1 of the bcc phase at ambient pressure⁷. This indicates that the I4/m phase is a relatively weak metal. Furthermore, the presence of flat bands near −2 eV, which are associated with a high DOS, implies potential van Hove singularities that could enhance electron correlations, possibly leading to interesting properties such as superconductivity. While Na is not a superconductor at ambient pressure²⁸, superconducting behavior has been reported in various high-pressure phases²⁹. Within this pressure range, the cI16 phase remains an NFE metal characterized by a spherical Fermi surface²⁷, whereas the oP8 phase behaves as a semimetal with a pseudogap near the Fermi level³⁰. The identification of the I4/m structure as an intermediate between the cI16 and oP8 phases is compelling, as it exhibits features characteristic of both phases. The occurrence of the I4/m structure also coincides with the minimum of the melting curve, suggesting that this phase is structurally unique. The high symmetry of the I4/m structure, combined with its large unit cell, may lead to zone folding effects, potentially resulting in unconventional electronic states and topological features.

Fig. 3. Electronic structure of the tI50 Phase.

Electronic band structure and projected density of states for the I4/m structure of sodium calculated at 150 GPa.

Topological analysis of the Electron Localization Function (ELF)³¹ reveals the bonding and electron localization patterns in the I4/m phase (Fig. 4a, b). While ELF mapping highlights regions of interstitial electron localization, it does not on its own confirm electride character, as similar features may also arise in metallic systems without non-nuclear charge maxima^32,33. A recent study on the hP4 phase of sodium has established a benchmark approach that combines ELF mapping with orbital analysis and topology features of electron density, namely, the presence of non-nuclear attractors (NNAs), to identify electride states in sodium under pressure³⁴. In the I4/m structure, large lobes of ELF are developed within octahedral interstitial voids, suggesting evidence for electron localization (Fig. 4b). A total of 30 such ELF lobes are found around four distinct Wyckoff sites in the unit cell: 2b (0.0000, 0.0000, 0.5000); 4e (0.0000, 0.0000, 0.2500); 8h (0.1052, 0.2861, 0.0000); and 16i (0.3085, 0.1568, 0.2143). However, the presence of ELF maximum alone is insufficient; an electride must also exhibit NNA, i.e., a local maximum in the electron density with a negative Laplacian, and such NNA should coincide with ELF basins. To this end, we performed a complete topological analysis of the electron density utilizing the Quantum Theory of Atoms in Molecules (QTAIM) framework³⁵ (Supplementary Method 2). The results reveal well-defined NNAs inside the ELF basins at all four distinct Wyckoff positions mentioned above (Supplementary Table 2). These NNAs exhibit local maxima of electron density with all three principal curvatures being negative, which corresponds to (3, −3) critical points. At 150 GPa, the electron density ${\rho }_{\mathrm{NNA}}$ at four unique NNAs ranged from 0.1970 to 0.2204 e⁻ Å^–3, and their Laplacian ${\nabla }^2{\rho }_{\mathrm{NNA}}$ range from −0.5576 to −0.4237 e⁻ Å^–5. These values are comparable, though slightly lower, than those reported for Na-hP4 at higher pressures³⁴, and fall within the accepted range for electride classification. As a comparison, conventional hydrogen bonds exhibit electron density ${\rho }_{\mathrm{BCP}}$ ranging 0.04–0.24 e⁻ Å^–3 and Laplacian ${\nabla }^2{\rho }_{\mathrm{BCP}}$ between 0.578 and 0.873 e⁻ Å^–5 at their bond critical points (BCP)³⁶.

Fig. 4. The electron localization function of the tI50 phase at 150 GPa.

a ELF contour map showing the (0 0 1/2) plane. b ELF isosurface with the value of 0.9, shown within the I4/m unit cell. Na atoms are represented by white spheres. The Wyckoff sites corresponding to octahedral interstitial voids are marked by different colors: 2b (red), 4e (black), 8h (green) and 16i (cyan). These sites are in close proximity to the centers of ELF lobes.

This analysis suggests that the I4/m structure of sodium is metallic with electride character, with interstitial electrons acting as anionic species. If these interstitial electron blocks are viewed as nuclei-free pseudo atoms, the resulting stochiometry would be A₅B₃, where A corresponds to the sodium atoms and B to the electride pseudo atoms. The closest analogy to this structure is Ti₅Ga₃, which crystallizes in a tetragonal cell with the I4/mcm space group³⁷. The I4/m structure can be viewed as a symmetry-lowered variant of I4/mcm, where mirror planes through the unit cell corners are lost. Interestingly, although the formation of electride states has been associated with insulating behavior, as observed in the hP4 phase, the electride state in the I4/m structure only induces weak metallicity. Electronic structure calculations at various pressures provide no evidence of a transition to an insulating phase within the I4/m structure. This is likely due to the lack of strong p-d hybridization responsible for insulating behavior in the hP4 structure^2,4,34. In a recent study, the formation of the electrides leading to an insulating state is attributed to constructive overlap of large lobes of p-d hybridized orbitals within a penta-capped trigonal prism geometry³⁴. The I4/m structure lacks such geometry (its electrons are squeezed to octahedral voids) and orbital arrangement, which leads to a weaker electron confinement in the interstitial regions and thus maintains a metallic character despite the presence of NNAs.

In summary, this study uncovers the crystal structure and properties of the elusive tI50 phase of sodium, first observed experimentally fifteen years ago. Using a data-driven-potential-assisted structure search, we determined that the tI50 phase has a body-centered tetragonal structure with 50 atoms in the unit cell and the I4/m space group. The predicted lattice parameters, a = 7.245 Å and c = 9.228 Å, closely match experimental values, with less than 0.38% deviation. This phase is calculated to be the thermodynamic ground state for sodium between 143 and 166 GPa, with dynamical stability established by phonon calculations. A notable feature of this structure is its weak metallicity associated with the electride states at high symmetry sites, which, if treated as nuclei-free atoms, the tI50 phase would resemble an A₅B₅ structural type. These findings advance our understanding of sodium’s complex structural landscape under extreme pressure.

Methods

Crystal structure predictions

The search for Na structures was carried out using the ab initio random structure search (AIRSS) method²² and an ephemeral data-derived potential (EDDP), which utilizes a set of shallow neural networks²¹. EDDP training was performed in seven iterative cycles. Initially, 2000 randomly generated structures were used to train the first EDDP model. In each following iteration, around 1100 new structures, generated via AIRSS using the updated EDDP, were added to the dataset for retraining. After rejecting the most unreasonable structures, this iterative process produced a final dataset of 7763 structures on which the final EDDP was trained. Out of 512 individual neural network fits, 20 were selected by non-negative least squares (NNLS) to construct the EDDP. For the EDDP training, unit cells with 8 Na atoms were employed, while the AIRSS search for Na structures used unit cells containing 50 atoms. The parameters for random structure generation included a volume range of 5.6–12.6 ų per atom, a cut-off radius of 4.5 Å, function exponents ranging from 2 to 10, and features accounting for up to three-body interactions. During the AIRSS search, the minimum interatomic separation was set between 1.0 and 3.0 Å. Both the EDDP generation and structure search were conducted at 120 GPa. In total, 250,000 random structures were generated for each unit cell size using AIRSS and optimized to local minima using the EDDP. Of these structures, 112 structures were selected for further analysis. Details of the structure search are presented in Supplementary Method 1, including Supplementary Figs. 1 and 2.

Electronic structure, vibrational properties, and topological calculations

DFT calculations were performed using the all-electron projector augmented wave (PAW) method, as implemented in the Vienna Ab initio Simulation Package (VASP)³⁸. The Perdew-Burk-Ernzerhof (PBE) functional was applied at the generalized gradient approximation (GGA) level³⁹. To account for increased core overlap at high densities, the 2s, 2p, and 3s electrons were treated as valence states in the PAW potential⁴⁰. A plane wave basis set with an energy cutoff of 645 eV and k-point meshes with a spacing of 2π × 0.032 Å⁻¹ were used for single-point energy calculations (see convergence test in Supplementary Fig. 5). Self-consistent calculations were considered converged when the energy change fell below 10⁻⁶ eV per atom, and all structures were optimized until the Hellmann–Feynman forces on atoms were reduced to below 1 meV per atom. For the projected-orbital analysis, the Wigner–Seitz radius of Na atoms was rescaled to 1.289 Å at 150 GPa, to achieve the percentage of the total unit cell volume covered by the spheres around each atom used by the projection to be approximately 100%. Phonon dispersion calculations were performed using density DFPT with VASP and post-processed using the Phonopy package⁴¹. The AIMD simulations were performed on a 2 × 2 × 2 supercell comprising 200 atoms using the VASP³⁸ package with a time step of 1 fs to accurately capture all atomic motions. The simulation cell adopted a supercell of the I4/m structure optimized at 150 GPa and 0 K. Structural stability of the supercell was examined over 20 ps in an NPT ensemble and a 12-ps temperature equilibration in a NVT ensemble. The ELF calculation was performed using VASP and examined using VESTA-v3 software⁴². The topological analysis of the electron density, based on the Quantum Theory of Atoms in Molecules (QTAIM)³⁵, was performed using the Critic2 code⁴³.

Author contributions

Y.Y. designed the project. A.A., H.W., and Y.Y. performed the calculations. H.W. and Y.Y. analyzed the data. M.B. contributed to the Methodology. A.A. and Y.Y. wrote the manuscript with input from all authors.

Peer review

Peer review information

Communications Chemistry thanks Zenner Pereira and the other, anonymous, reviewer for their contribution to the peer review of this work.

Data availability

The data that support the findings of this study are readily available and can be requested from Yansun Yao.

Code availability

AIRSS and EDDP are free and open-source codes available at AIRSS | Materials Theory Group and https://www.mtg.msm.cam.ac.uk/Codes/EDDP, respectively. At the same time, VASP is a commercial code available from https://www.vasp.at. Instructions on how to obtain and use these codes are available on their respective websites.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s42004-025-01566-3.