Introduction

A most fundamental property of the elemental metals is their crystallographic arrangements of the atoms1. This has been studied extensively experimentally under ambient conditions. More recently a lot of interest has been directed towards the crystal-structure behavior when the metals are compressed1,2,3,4,5,6. Perhaps the most anomalous behavior has been discovered for cerium where two different phases α and γ, where the former appears at slight pressure at room temperature, have been found to have the same crystal structure (face-centered cubic, fcc). In the present work we present compelling evidence that cerium under very high pressure adopts still another fcc atomic arrangement we call ω. Thus, cerium displays a phase diagram with three different phases having the same crystal structure, a most remarkable behavior that even more clearly than before singles out cerium as a very special metal.

Cerium is the second metal in the lanthanide series, but the first with an appreciable occupation of 4f-electron states. The 4f electron likely plays a decisive role in the phenomenon that cerium may be most known for, namely, its isostructural transition between the α and γ phases, see the phase diagram in Fig. 1. A recent summary of the various explanations and discussions around the α- γ cerium puzzle has been given with plenty of references7 and there are ongoing debates in the literature. However, we will here model pressurized cerium and its behavior under extreme compression and the approach is directly relevant for the α phase but less so for the γ phase, as it disappears already at very moderate pressure, see Fig. 1.

Fig. 1
figure 1

Experimentally determined phase diagram of cerium14. The red phase line is from Ref.12. The α and γ phases are fcc, β double hexagonal, α′ orthorhombic, α′′ monoclinic, δ bcc, and ε tetragonal. Extrapolations of the phase line to zero temperature suggest the bct phase to stabilize between 15 and16 GPa.

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Experimentally, there have been quite a few studies of compressed cerium through the years8,9,10,11,12,13,14,15. These investigations all suggest the occurrence of a tetragonal phase at pressures beyond at least about 10 GPa, but there are also reports of orthorhombic and monoclinic phases and mixtures thereof prior to the tetragonal phase as shown in Fig. 1.

In this paper, we do not consider the mechanism that stabilizes the γ phase over the α phase and vice versa, but rather emphasize cerium’s response to high pressure. Generally, for f-electron materials, compression tends to broaden the f-bands as orbital overlap increases and consequently the f electrons become more band like or delocalized (itinerant). Johansson’s16 idea is that the 4f electron behaves in this manner already in the α phase and our study is consistent with this view, while there are other explanations given in the literature7. Regardless, as cerium is being pressurized, there can be no doubt that the 4f electrons delocalize and we will provide ample arguments below that this is happening. Indeed, the pressure-induced phase transitions are directly linked to 4f-electron bonding, not unlike the 5f electrons in the early actinide metals, where the 5f electrons distort the crystal into low-symmetry structures due to a Peierls-like symmetry breaking mechanism17. This effect in the actinides is driven exclusively by the 5f electrons17 and an analogous scenario is found here for the 4f electrons in cerium under compression. This interrelation was already pointed out in Refs.16,18,19 and it means that the earlier actinide metals (Th-Pu) now have a direct link to the Periodic Table. Namely, cerium is the bridgehead element connecting the actinide metals with the lanthanide metals completing the consistency of the periodic arrangement of the elemental metals.

At the highest compressions, with volumes approaching 10% of the ambient volume, the Peierls distortion is no longer a sufficiently strong mechanism available for stabilization while core-states and valence-band hybridizations20, and electrostatic core repulsion become more dominant. These interactions dictate high-symmetry crystals with efficient packing of the atoms.

The high compressions considered in this investigation may seem to be of only academic interest, but extreme conditions in densities (and temperatures) need to be known to constrain equation-of-state models that in turn are required to close the equations of hydrodynamics21. On the experimental side there are progress as well to explore extremes and recent laser-driven ramp compression measurements have accomplished pressures in the tenths of TPa22, reaching conditions relevant for high energy density physics. Magnesium was explored up to 1.3 TPa (1300 GPa)23 and copper up to 2.3 TPa (2300 GPa)22 in the National Ignition Facility closing in on the pressures required to observe the ω phase in cerium (~ 5 TPa).

Results

As functions of atomic volume, we calculate the total energy for the α (fcc), β (dhcp), α′ (orthorhombic), α′′ (monoclinic), δ (bcc), ε (bct), and hcp phases. The results of these calculations are shown in Fig. 2, where we present the energy differences of the structures relative to the α phase. Because the monoclinic angle of the α′′ phase relaxes to 90° in our calculations for all studied volumes it is mechanically unstable and not shown. If the angle is kept fixed at 92°, the value suggested by some experiments10, the monoclinic phase has an energy that is only slightly larger than that of the fcc or bct phases because the phase is the result of a small distortion of fcc or bct. Furthermore, the calculated energy of the δ phase is always more than 19 mRy/atom greater than the fcc phase and is therefore excluded from Fig. 2. We notice also that the observed orthorhombic α’ phase is never energetically stable in our model, but close enough to the fcc phase that it is a probable phase in a real sample of cerium at finite temperatures.

Fig. 2
figure 2

DFT total energy differences between various phases relative to fcc. Phase transitions to bct (19 GPa), hcp (1700 GPa), and fcc (5100 GPa) are indicated.

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We can identify three phase transitions in Fig. 2: fcc → bct → hcp → fcc. The enthalpies for the phases in the first transition (fcc → bct) coincide at 19 GPa with a small volume collapse of about 0.2%. The small volume collapse is no surprise because at the transition, the structural difference between the fcc and bct phases is only a small change of the axial c/a ratio. Consequently, the enthalpies for these structures are close to parallel as functions of pressure which makes the transition point delicate to calculate.

Experimentally, at finite temperatures, there are reports of transitions from the α phase to either the α′ or a mixture between α′ and α′′ phases, see Fig. 1. Apparently, the phase stabilities of α′, α′′ and the mixture depend on how the cerium sample is prepared with heat treatments for the experiment24. Our calculations do not account for sample preparation or thermal effects and should therefore be compared to measurements at very low (or zero) temperature. Extrapolating the phase lines to zero temperature in Fig. 1 (dashed lines) suggests a bct (ε) phase somewhere between 15 and 16 GPa. In a more recent measurement, a complete transition to the ε phase was observed at 17.6 GPa13 at room temperature. These experimental transition pressures are thus in good agreement with our prediction of 19 GPa but somewhat higher than results from previous calculations 11.5 and 14.5 GPa25, 10 GPa26, and 12 GPa27. Although the DFT methodologies used in25,26, and to a degree in27, are close to present approach, the differences in the predicted transition pressure reflect the sensitivity in determining a transition from coinciding nearly parallel enthalpies. Minor differences in various numerical approximations including the electron exchange and correlation functional, spin–orbit coupling and orbital polarization, basis set size, k-point integrations, and robustness in relaxing the ε phase, all play roles in determining the theoretical transition pressure. The present calculations are, however, considered considerably more accurate than the others relative to these approximations.

In our model the bct phase is found to be stable in cerium over a very wide pressure range but at a pressure of 1700 GPa cerium undergoes a transition to the hcp phase, close to the previous theoretical assessment (1600 GPa)26. The β (dhcp) phase is also rather competitive in cerium but never the lowest-enthalpy phase in our compression calculations. Upon even higher compressions, cerium once more returns to the fcc phase at 5100 GPa. We will refer to this new fcc phase as ω cerium. Hence, cerium has interestingly and without equal, three distinct fcc phases. One has a large volume (γ), another a small volume (α), and finally a phase at very high compression, ω, that remains stable up to the highest compression considered. In Table 1, we list our calculated pressure up to about 100 Mbar (10,000 GPa or 10 TPa) for cerium as a function of atomic volume for your reference.

Table 1 DFT pressure as a function of atomic volume.
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In Fig. 3 we show calculated normalized c/a axial ratios for the bct, hcp, and dhcp phases as functions of volume (solid symbols) together with the same measured quantity (open symbols) for the bct phase. The onset of the bct phase in the experiments are indicated by the dashed vertical lines. Here, we normalize the axial ratios relative to their ideal value, which for bct is \(\sqrt{2}\), for hcp \(\sqrt{8/3}\), and for dhcp \(2\sqrt{8/3}\). In other words, for bct, a normalized value of unity corresponds exactly to the fcc structure. The ideal value also corresponds to the most efficient packing of hard spheres given the symmetry of the crystal structure. We notice the smoothness of the calculated axial ratios and this behavior is a testament to the robustness of the structural-relaxation procedure we will describe in the Computational methods section. For the bct phase, the c/a axial ratio compares rather well between theory and measurements considering that the model deals with pristine (no defects or phase mixing) cerium at zero temperature, while the experimental sample is an imperfect material at room temperature. The latest experimental data13 represented by open yellow diamonds, compare most favorably with the calculations and show an onset of the bct phase just above 20 Å3 in close agreement with our prediction. Notice that the c/a ratio smoothly increases with compression and approaches a maximum normalized value close to 1.17 (c/a = 1.65). But at sufficiently high compression, the normalized c/a value converges towards unity for all phases as the ideal packing of the atoms becomes more important due to electrostatic repulsion of the ions.

Fig. 3
figure 3

DFT and experimental normalized c/a axial ratios as functions of atomic volume. The hcp, dhcp, and bct ratios are normalized with \(\sqrt{8/3}\), \(2\sqrt{8/3}\), and \(\sqrt{2}\), respectively. For bct, when the normalized c/a ratio equals unity, the fcc structure is recovered. Filled symbols are results from calculations and open symbols refer to measured axial ratios in the bct phase9,11,13. Dashed vertical lines indicate the experimental transition volumes to the bct phase.

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In Fig. 4 we compare our calculated equation of state (pressure as a function of volume) with that of Olsen et al.9, Vohra et al.11, and Ma et al.13 and a newly constructed EOS model28. The EOS model is made to best represent low-pressure experimental data and smoothly integrate the higher-pressure DFT data for a thermodynamically consistent description. A detailed presentation of this cerium EOS model will be presented in the future. Close to zero pressure, the DFT model underestimates the atomic volume of the α phase (26.5 versus 28.0 Å3), but already at minor compressions the calculations agree very well with the measurements9,11,13 up to the highest measured pressure just above 200 GPa. It is noteworthy that our model predicts an equilibrium volume about 5% smaller than reported from experiments for the α phase. Although not catastrophic for a parameter-free first-principles theory that do not account for thermal expansion and possible defects in the sample, this discrepancy is larger than one would first expect.

Fig. 4
figure 4

DFT and EOS model28 (solid lines) with measured (open symbols)9,11,13 pressures as functions of atomic volume.

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In the analysis by Smith and Kmetko29, cerium is on the border of a localized-delocalized f-electron behavior, while the 5f actinide Th, positioned in the same column in the Periodic Table, is safely in the delocalized-electron regime. Thorium could thus be considered a 5f analog to α cerium, assuming the 4f electron is delocalized. But, for thorium our theory predicts an atomic volume of 32.30 Å3, while the zero-temperature experimental volume is deduced to be 32.55 Å330 i.e. a small 0.8% difference. This is notably much less than the corresponding discrepancy for its 4f analog α cerium. Of course, thorium does not have another competing phase with larger volume as is the case for α cerium with its neighbor γ cerium, that could influence the atomic volume. Such a competing γ-type phase in thorium could only possibly appear at very expanded volumes. Therefore, a precursor effect like that from the γ phase in cerium is not available in thorium. Directly related to this is the fact that the experimentally observed volume for γ cerium is notably anomalous, being smaller than expected, as was already pointed out by Pauling31 and Zachariasen and Ellinger32,33. In this case the neighborhood of the α phase acts as a precursor and accordingly reduces the γ volume.

Thus, close to the phase transition between the two fcc phases, both sides of the transition are affecting each other, reducing their difference in volume. When exposed to temperature, the admixture of the two phases will increase. Eventually, they will coalesce, and the phase separation line will end in a critical point. These precursor phenomena can also easily be identified in the compressibility data taken by Lipp et al.34 in the pressure–temperature region close to the α-γ phase transition in cerium. In any event, we conclude that the larger discrepancy for α cerium relative to thorium in our theory emanates from the existence of the larger volume γ phase and its action as a precursor.

The largest deviation between theory and measurements for cerium is at zero pressure, but our DFT model is quickly improving with compression. This agrees with that the interfering γ phase is further away with pressure. In addition, this trend is consistent with a study of 64 solids in the Periodic Table showing that density-functional theory is improving with pressures up to at least 100 GPa30 compared to experimental data.

Discussion

Cerium shows a most interesting behavior upon compression where the fcc crystal structure plays a central role. Few metals have multiple distinct fcc phases in their phase diagram, but cerium shows three, the γ, α, and the predicted high-pressure ω phase, in order of decreasing atomic volume. The subject of the γ and α phase transition is enormous and still debated in the literature, see discussions and references in7, but our present focus is on the compressibility of cerium and in that context, γ is not a key player. Moving on to the α phase, the reason it forms in the fcc phase was explained straightforwardly by Johansson et al.35 who argue that the 4f-electron contribution to the chemical bonding is essential and without it, its ground-state structure would rather have been analogous to Group IVB elements (Ti, Zr, Hf; bcc, hcp, or hexagonal-ω structures depending on volume). In this context, it is interesting to also compare cerium with its nearest neighbors lanthanum and praseodymium. These metals respond differently to compression, no formation of any low-symmetry structure in La, but stabilization of an orthorhombic phase in Pr at about 20 GPa36. The reason is simply that La has practically no 4f-band occupation even under compression, while Pr has a substantial number of 4f-electrons, driving the transition.

As has already pointed out, the calculated atomic volume for α cerium is somewhat smaller than expected and the discrepancy with room-temperature experimental data is about 5%. We argue that the presence of the larger volume γ phase plays a precursor role for the difference between zero temperature theory and room temperature experiments. Contrarily, the γ phase volume appears smaller than expected due to the presence of the α phase, as mentioned in the previous section. As the temperature increases, the state (α or γ) which is metastable may be thermally populated with an increasing probability of finding γ cerium37. This scenario is supported by low-temperature neutron diffraction measurements that show phase mixing and its dependence on the heat and mechanical treatment of the sample38. In addition, there are nonlinear changes in the sound speed across the transition, also suggesting a sluggishness of the transition34.

So, what is driving the destabilization of fcc into the tetragonal bct phase? Generally, distortion of the crystal structures in itinerant (or delocalized) f-electron metals is due to a Peierls-like symmetry-breaking mechanism17, but to understand more specifically which structure is plausible one must consider the amount of f electrons involved in the bonding (f-band occupation). In Fig. 5 we plot the 4f-band occupation that shows a noticeable increase with compression. The increase is mostly due to promotion of 6s and 6p electrons into the 4f band. There are fundamentally two reasons for this behavior, and we discuss it in deeper detail in Ref.39. First, the kinetic contribution is larger for orbitals with more radial nodes, causing the energy for the 6s and 6p bands to rise faster than the 4f band upon compression. Second, for a less than half-filled band, as is the case for the 4f band here, bonding states are still available to be populated, and these states are energetically favorable over anti-bonding states, particularly for contracted volumes. The combined effect leads to a 4f-band occupation increase with compression and analogous trends are predicted for the 5f electrons in the early actinides (Th-Pu)39.

Fig. 5
figure 5

DFT 4f.-band occupation as a function of atomic volume.

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We know from the behavior of the d-transition metals40,41, as well as the early actinides42, that the gradual population of the dominating bands contributing to the chemical bonding (3d, 4d, 5d, or 5f) dictate the ground-state crystal structures. To illustrate this, we remind ourselves the concept of canonical band theory40 that conveniently isolates the band energies to specific orbitals such as d or f, for example. In this simple approach one can calculate the energy difference between different crystal structures as functions of occupation of a specific band.

In Fig. 6 we present the canonical-band energy differences relative to the fcc phase (c/a = \(\sqrt{2}\),) for bct crystals with different c/a axial ratios, focusing on the f orbitals. This is done for four different occupations of the f band (0.5, 1.0, 1.5, and 2.0). Notice that for small occupations (0.5 and 1.0) the fcc crystal is stable over bct regardless of c/a axial ratio, in agreement with the assessment by Johansson et al.37. But, for higher f occupations (1.5 and 2.0) the canonical band energy favors the bct structure with a c/a ratio larger than the ideal fcc value. This result, and the conclusion from Fig. 5, nicely explain first the stability of the bct phase in cerium under compression and second the axial-ratio increase during the compression. Another aspect of the canonical-band energies shown in Fig. 6 is that the bcc (δ) phase (c/a = 1) is much higher in energy than the fcc phase for all occupations, effectively ruling out this phase for cerium at low temperatures. Of course, the energy of the bcc phase in the full-fledged electronic-structure calculations is also much higher than the fcc phase for all studied atomic volumes as mentioned in the previous section. Because the canonical band picture does not distinguish between 4 and 5f electrons, the same behavior under compression, i.e., an fcc → bct transition with an increasing c/a in the bct phase, is predicted to occur for thorium (cerium’s 5f counterpart or analog), and indeed that is happening26,27,43,44.

Fig. 6
figure 6

Canonical-band theory energies (solid circles) for the bct phase relative to the fcc phase as functions of c/a axial ratio for 0.5, 1.0, 1.5, and 2.0 f-band occupations. The lines connecting the symbols are guides to the eye.

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The general argument of a Peierls-distortion stabilized lowering of the crystal symmetry (bct over fcc) is illustrated in Fig. 7. Here we plot the electronic density of states (DOS) for fcc and bct (c/a = 1.65) at a compressed volume, 9.5 Å3 (pressure = 564 GPa). At this compression, cerium is comfortably stable in the bct phase, see Fig. 2. Notice in Fig. 7 that the fcc DOS has a dominant high-intensity peak close to the Fermi level (zero energy) that is absent from the bct DOS. This peak is energetically unfavorable for the fcc phase and consequently bct has lower energy. The narrow bands that give rise to a high DOS at the Fermi level in cerium (and the early actinides) is the crux to the symmetry-breaking argument we have referred to as a Peierls distortion. Lowering the crystal symmetry, thus removing symmetry constraints of some energy eigenvalues to be degenerate (resulting in high DOS), lowers the DOS at the Fermi level and the total energy17. This mechanism is of course only relevant in a situation with narrow bands, but as cerium experiences extreme compressions, the 4f band broaden considerably and the Peierls distortion disappears. In Fig. 8 we show DOS for fcc and bct, now at a very compressed volume, 3.6 Å3 (pressure = 8688 GPa). We notice immediately that the DOS is much wider but also that there are no peaks for either phase at the Fermi level. This rules out a Peierls distortion and rather the band energy combined with electrostatic ion repulsion and core states and valence-band hybridization20 determine the crystal structure that has once more returned to fcc, ω cerium.

Fig. 7
figure 7

DFT total electronic density of states for fcc (black) and bct (c/a = 1.65, red) at about 9.5 Å3. Fcc has a pronounced peak near the Fermi level (zero energy) that is absent from the bct phase.

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Fig. 8
figure 8

DFT total electronic density of states for fcc (black) and bct (c/a = 1.65, red) at about 3.6 Å3. No destabilizing major peak in the density of states are found for either phase at the Fermi level (zero energy).

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Once shown to be thermodynamically favored (lowest enthalpy) relative to the other studied phase, it is preferable to also confirm the ω phase dynamical stability. In Fig. 9 we show the fcc (ω) phonon spectra at 3.4 Å3. Clearly, all phonon branches are positive and there is no indication of crystal instabilities.

Fig. 9
figure 9

DFT phonons for the fcc ω phase at the atomic volume of 3.4 Å3.

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We have thus concluded that cerium shows three isostructural phases, γ, α, and ω in its phase diagram in addition to the other phases. The reason cerium can display this remarkable behavior is due to the action of the 4f electron. We argue in γ that the 4f electron is absent from the bonding being localized and other electrons from the valence band drive the crystal structure. Here, the inert localized and non-bonding 4f electron screens the nuclear charge in such a way that the remaining 3 valence electrons behave like 3 standard lanthanide conduction electrons. For α cerium this is dramatically changed because now the 4f electron character has modified to become delocalized or bonding and thus being directly responsible for the crystal structure. During compression the tetragonal phase, also dictated by an increasing presence of 4f bonding, is dominant but finally morphs back into the fcc ω phase. In the ω phase the 4f electron has totally lost its normal f character and has become dissolved into the surrounding electron bath environment because of the band-broadening that follows the extreme compression.

Although we are confident that the predicted phases are correct, there has been development of sophisticated algorithms for structural searches under compression, but f-electron materials such as cerium pose significant challenges45 to these approaches.

Computational methods

We model cerium with ab initio theory as a metal with delocalized 4f-band electrons. This approach may be less relevant for the γ phase, but our focus is on the α phase and cerium’s response to high compressions. Cerium does not spontaneously spin polarize in this model and it remains nonmagnetic in our study. Nonetheless, spin–orbit coupling, and orbital polarization play some roles because the 4f band is very narrow and these interactions, at least in the low-pressure regime, influence the phase stability and compressibility. At expanded volumes (approaching the γ volume) cerium indeed develops magnetic moments in the model consistent with γ cerium although in a higher energy state. The spin–orbit coupling and orbital polarization46 mechanisms increase the computational burden only marginally and are implemented in the electronic-structure code without the need for adjustable parameters. We note that with a scalar-relativistic basis set, as used here, one needs to avoid spin–orbit interaction on the p states for accurate results as has been discussed in detail47. The general effect of these interactions on magnetism and chemical bonding for the rare-earth metals was described in Ref.48. However, as the 4f bands broaden during compression, they become less important.

The fundamental and necessary assumption in DFT relates to the treatment of electron exchange and correlation, and the generalized gradient approximation is preferred for f-electron materials and the PBE form is adopted49. The DFT implementation is a robust all-electron full-potential linear muffin-tin orbitals (FPLMTO) method50. The essential inputs for the FPLMTO calculations, once an electron exchange and correlation functional has been decided, are the atomic number (58 for cerium) and the crystal structure.

The FPLMTO method does require a choice of basis set and because we are applying enormous compressions, we define semi-core 5s, 5p, and 4d states that are allowed to fully hybridize with valence 6s, 6p, 5d, and 4f states for a total of 22 band electrons per cerium atom. In FPLMTO the computational region is divided into non-overlapping muffin-tin spheres and a remaining interstitial part. Outside the muffin-tin spheres the wave functions are Hankel or Neumann functions which are represented by a Fourier series using reciprocal lattice vectors. The size of the muffin-tin spheres is for most calculations chosen so that SMT/SWS = 0.86, where SMT is the muffin-tin sphere radius and SWS is the Wigner–Seitz radius (that is the radius of a sphere with a volume that equals the atomic volume). One exception is our calculation of the 4f-band occupation that are performed for the α phase, where we use SMT/SWS = 0.90. The reason for the choice of a larger muffin-tin sphere in this case is because the character of the electrons is not determined in the interstitial region but only in the muffin-tin sphere that is therefore maximized (close to touching spheres) for best accuracy. Test calculations with a smaller SMT/SWS ratio (0.80) at the various phase transitions indicate differences in the energy separation between the involved phases varying from 0.5 (20 GPa) to 1.0 (5100 GPa) mRy/atom (6.8–13.6 meV/atom). These are very small energies, and our results are thus only very weakly dependent on the necessary choice of this ratio.

The number of k points used for integrations in the reciprocal space are 5000–7500 depending on crystal structure and to each energy eigenvalue we apply a Gaussian broadening with a 20 mRy width. Choosing a smaller 10 mRy value results in insignificant changes of the results. Other computational parameters (basis set, Fourier mesh, k points) are carefully converged as well.

We limit our study to the following structures: fcc (α), body-centered cubic (bcc, δ), body-centered tetragonal (bct, ε), hexagonal close-packed (hcp), double hexagonal close-packed (dhcp, β), body-centered orthorhombic (α′), and body-centered monoclinic (α′′). These structures are the same, with the same notations, as has been presented in the literature14. Except for the cubic structures, these all needs to be “structurally relaxed”, i.e., a search for the lowest energy is done with respect to non-symmetry constrained structural quantities: axial ratios b/a, c/a, and monoclinic angle for α′′, b/a and c/a for α′, and c/a for bct, hcp, and dhcp. In practice, the calculations are performed on a volume grid of about 100 volumes and the axial ratios that minimizes the total energy is determined from fitting a 6-order polynomial to the total energies from calculations of 10 values for each grid point. The minimum total energy and corresponding crystal structure are then extracted from these fits for each volume which ensures a very smooth behavior of both the total energy and axial ratio. All attempts to relax the monoclinic structure (α′′) result in either the bct (ε) or the fcc (α) structure (monoclinic angle relaxes to 90°), depending on volume, in agreement with the DFT modeling by Ravindran et al.25.

We calculate the pressure from fitting a Murnaghan form of the equation of state utilizing only 5 volume and energy pairs at a time, determining the middle-point’s pressure. These fits then overlap so that about 100 Murnaghan fits were conducted to determine the pressure in the studied volume range. In our studied wide pressure regime, it is not possible to accurately represent the pressures with only one analytical form.

To isolate the dependence of the f electrons on the phase transitions, we employ so-called canonical band theory. This is a concept that Skriver40 introduced that in a simple model focuses on a particular orbital (d or f, for example) so that its influence on the phase stability can be analyzed. This type of simple modeling has previously been applied to help understand the ground-state crystallographic phases of the d-transition metals41 and the early actinides (Th-Pu)42 where the 3d, 4d, 5d, or 5f orbitals are dominating the bonding.

To confirm the mechanical stability of the predicted fcc ω phase we calculate its phonons applying the small displacement method for an fcc 3 × 3 × 3 (27 atoms) supercell51. Here we utilized 128 k points while other details of the calculations are as described above. The forces required for this method can be evaluated using a linear-response anzats (Hellmann–Feynman forces), but those are numerically difficult to calculate and inaccurate in this case. Rather, we apply a scheme that is based on electronic energy shifts due to small atomic displacements to obtain the forces, see further discussions in52.

In addition to the first-principles modeling, we have adopted an equation-of-state (EOS) model for cerium that accounts for experimental data where available, up to about 200 GPa. This free-energy model smoothly adjusts to the DFT results at higher compressions28.

Significance statement

Crystal structure is a most fundamental property of a metal. Cerium is a remarkably unusual element known to possess two separate phases with different bonding characteristics, yet the same crystal structure (α and γ phases). This unique situation has greatly impacted science with research exploring this peculiarity for half a century. Through our present theoretical investigations, we have made the remarkable discovery that there is even one more isostructural phase that we name ω-cerium, again with a different stabilization mechanism. With three distinct phases (α, γ, and ω), yet with the same crystal structure, cerium is without equal among elements in Nature. Here we not only predict this most unique situation but also explain the fundamental reasons for it.