Phonons of the anomalous element cerium

Abstract

Many physical and chemical properties of the light rare-earths and actinides are governed by the active role of f electrons, and despite intensive efforts the details of the mechanisms of phase stability and transformation are not fully understood. A prominent example which has attracted a lot of interest, both experimentally and theoretically over the years is the isostructural γ - α transition in cerium. We have determined by inelastic X-ray scattering, the complete phonon dispersion scheme of elemental cerium across the γ → α transition, and compared it with theoretical results using ab initio lattice dynamics. Several phonon branches show strong changes in the dispersion shape, indicating large modifications in the interactions between phonons and conduction electrons. This is reflected as well by the lattice Grüneisen parameters, particularly around the X point. We derive a vibrational entropy change , illustrating the importance of the lattice contribution to the transition. Additionally, we compare first principles calculations with the experiments to shed light on the mechanism underlying the isostructural volume collapse in cerium under pressure.

The rich phase diagram of elemental cerium (see Fig. 1) reflects how even modest changes in pressure and/or temperature can effect the correlation between electrons. The famous isostructural transition between the low density fcc phase (γ) and the high density fcc phase (α) was discovered more than seven decades ago 1, 2) and remains the only solid–solid transition in an element that ends at a critical point. At ambient temperature, the first order transition occurs at 0.75 GPa, and leads to a volume decrease of the fcc lattice by 15%. It is generally believed that the transition arises from changes in the degree of localization and correlation of the 4f electron. Currently, there are two competing models for the transition. Within the Mott transition picture (3, 4) the 4f electron is localized and nonbinding in the γ-phase and becomes itinerant (metallic) and binding in the α-phase. Accordingly, the phase transition is dependent on the kinetic energy of the 4f electron. Within the Kondo volume collapse model (5, 6) the 4f electrons, hybridized with the spd conduction electrons, are in a local moment regime and do not participate in bonding, whereas they form coherent quasi-particle bands in the α-phase and thus participate in bonding. Recent theoretical work actually suggests that the two scenarios are quite similar at finite temperatures (7–9). In a wider context it is now understood that many physical and chemical properties of the light rare-earths and actinides are governed by the active role of f electrons. However, despite intensive efforts, the detailed mechanisms of phase stability and transformation remain poorly understood.

Fig. 1.

Cerium metal: (A) Schematic phase diagram in the low-pressure region. The blue circles indicate the thermodynamic points at which the IXS data were recorded; β: dhcp phase, δ: bcc phase, CP denotes the critical point. (B) Three-dimensional view of the face-centered real space structure. (C) Representation of the first Brillouin zone for the face-centered cubic lattice.

Aside from the underlying physical model, the role played by the crystalline lattice in the transition is currently an area of debate. In particular, the role of entropy in the transition is an area of controversy. Indeed, while an analysis (4) of experimental data, later complemented by total energy calculations within the LDA + DMFT scheme, shows that entropic effects play the dominant role in the transition (10), the relative importance of the spin and lattice contributions is still unclear. The phonon density of states (PDOS) of a Ce_0.9Th_0.1 alloy at 150 K (γ-phase) and 140 K (α-phase) determined by inelastic neutron scattering (INS) (11) display very small differences across the transition, so that changes in the vibrational entropy, , have been suggested to be negligible. In contrast, high-pressure X-ray and neutron diffraction studies on polycrystalline Ce concluded that accounts for roughly half of the total entropy change (12). Further evidence for a large role of the vibrational energetics is provided by recent ultrasonic investigations (13, 14) and a combined high-pressure and high-temperature X-ray diffraction study (15). In fact, a fit to the pressure-volume diffraction data by the Kondo volume collapse model plus a quasiharmonic representation of the phonons indicates dramatic changes in the lattice Grüneisen parameter across the transition. However, the experimental results can be equally well represented by calculations based on the Mott model of the phase transition (16).

In the present study we directly assess the role of the lattice in the γ - α transition of elemental cerium using high energy resolution inelastic X-ray scattering (IXS). We have determined the complete phonon dispersions for several pressures across the transition and have performed a Born–von Kármán analysis to derive elastic and thermodynamic parameters. Furthermore, we complement these results by comparing the experimental data with ab initio lattice dynamics calculations, utilizing both trivalent (4f electron localized) and tetravalent (4f electron part of the conduction band) potentials.

Results and Discussion

The experimental phonon dispersions are reported in Fig. 2, together with a Born–von Kármán force-constant model fit, including interactions up to the sixth nearest neighbor, and the results of the ab initio calculations. For comparison the data at ambient pressure, obtained by inelastic neutron scattering (17), are shown as well. While no strong pressure evolution within the γ-phase is observed, significant changes occur across the γ - α transition. This proves without any quantitative analysis that lattice degrees of freedom have to be considered in explaining the phase transition. The most striking differences occur along the [110] and the [111] directions. While the T[111] branch shows no strong pressure evolution, there is a considerable stiffening of the L[111] branch, leading to an increasing energy difference of the phonons at the L point. Furthermore, the weak overbending of the T[111] branch at ξ = 0.3 gradually disappears in the α-phase. This overbending has previously been related to the fcc → dhcp (γ → β) transformation, which occurs at 260 K and ambient pressure (17). We note, however, that the phonon energy of the T[111] branch at the L point remains low, suggesting that along the [111] direction each atomic plane can easily slide relative to its immediate neighboring atomic plane to form a new stacking arrangement. A similar soft-mode behavior has been observed in Pu (18) and La (19). The second marked change is related to the L[110] branch. Within the γ-phase this branch is rather flat between the K and the X point. In contrast to this, in the α-phase the maximum phonon energy is attained at ξ ≈ 0.6, and the branch disperses downwards for higher ξ values toward the X point. This effect is even more pronounced at the highest pressure of 2.5 GPa. Further changes are visible for the T₁[110] and the T[001] branches, which show changes in their dispersion, with an increasingly flat region around the X point. Finally, the L[001] and T[001] phonon branch have very similar slopes in the γ-phase, indicating that the respective elastic moduli C₁₁ and C₄₄ are very similar—a very unusual behavior, to this extent not encountered in any other element. In fact, close inspection of the phonon dispersion fit at 0.8 GPa suggests that C₁₁ < C₄₄.

Fig. 2.

Phonon dispersion of cerium metal at ambient conditions, 0.4 and 0.6 GPa in the γ-phase, and at 0.8 and 2.5 GPa in the α-phase. LA branches (circles), T_1〈110〉 branch (triangles), and T_2〈001〉 branches (diamonds). The black solid lines are the results of a Born–von Kármán fit. The blue (green) lines represent calculations with the trivalent potential (scaled by 0.9) and the red lines show dispersion relations derived from the tetravalent potential. The corresponding phonon density of states (PDOS) is reported in the right panels.

In Fig. 2 B–E, the blue and red lines represent calculated phonon dispersion relations. For γ-Ce (at 0.4 GPa), the phonon dispersion curves derived from the trivalent potential (one localized f electron) systematically overestimate the phonon energies. An overall scaling, however, of the phonon energies by 0.9 (see Fig. 2C), green lines) provides a very good agreement; in particular the overbending of the T[111] branch and the kink of the T₁[110] branch are well reproduced. Such a simple scaling cannot be employed for the calculations using the tetravalent potential (for which the 4f states are treated as part of the valence band), and furthermore, the shape of the T[111] and T₁[110] branches is not reproduced. It is worth noting that similar agreements between experiment and calculation in the γ phase were reported (20, 21). The trivalent potential therefore provides a better description of the lattice dynamics, in agreement with the generally accepted picture of a localized 4f electron in the γ-phase. We note that the kink observed in the T₁[110] branch suggests a Kohn anomaly similar to that observed in Th (22) and δ-Pu (18). Because this anomaly is not present in the calculations using the tetravalent potential, we argue it originates from electronic effects. Additionally, it is interesting that this kink accompanies the softening near the L point, which implies that the lower dimensionality of atomic layers in the vicinity of the fcc → dhcp (γ → β) transition enhances the Kohn anomaly. For α-Ce (0.8 and 2.5 GPa) the phonon dispersion calculations with the tetravalent potential follow the experimental data very well for most of the phonon branches, whereas results for the trivalent potential yield unstable modes. The observed softening of the TA mode near the X point at 0.8 GPa is well described by theory as well as the weaker but still visible overbending of the T[111] branch. Strong deviation only occurs for the T₁[110] and T[111] branches.

A quantitative representation of the phonon energy changes across the transition is provided by the mode Grüneisen parameters , displayed in Fig. 3. While positive values of γ_q (increase of phonon energy with increasing pressure/decreasing volume) are common behavior, negative values can occur as precursor of phase transitions or as signatures of electron–phonon coupling. Besides the low-q portion of the L[001] branch and a small portion of the T[111] branch around q = 0.3, the most pronounced regions of negative γ_q’s occur around the X point (T[001] for q > 0.65, T₂[110] for q > 0.58, and L[110] above q = 0.85). This points to significant changes in the electron–phonon interactions in these regions of reciprocal space.

Fig. 3.

Evolution of the mode Grüneisen parameters across the γ → α transition for ρ = 7.00 and 8.42 g/cm³, respectively, in cerium as derived from the Born–von Kármán fit to the experimental dispersion. L modes: circles, T₁ modes: triangles, T₂ mode: diamonds.

Further insight into the nature of the transition is gained by the determination of the vibrational entropy S_vib, the mean square atomic displacements , and the Debye temperature Θ_D from the phonon density-of-states (PDOS), using well-established integral equations (23). These are reported in Table 1. We obtain a value for the vibrational entropy change per atom across the transition of 0.33 k_B. This is consistent with estimations using our derived values of the Debye temperature Θ_D and for the two phases just below and above the transition: and . These values are in very good agreement with ultrasound measurements (24) on a polycrystalline sample, which yield 0.32 k_B, and in reasonable agreement with the estimation of 0.22 k_B from ultrasonic investigations (13), but are about a factor two lower than the ones reported by Jeong et al., derived from their neutron and X-ray diffraction work (12). Despite this discrepancy our results definitely confirm that the lattice contribution to the phase transition cannot be neglected, considering that the total entropy change, derived from the Clausius–Clapeyron relation .

Table 1.

Vibrational entropy S_V, Debye temperature Θ_D, and the mean square atomic displacements , derived from the cerium phonon-density-of-states

P (GPa)	a (Å)	SV (kB)	ΘD (K)	(Å2)
0.0001 (γ)	5.160	6.88	117	0.023
0.4 (γ)	5.120	6.75	122	0.020
0.6 (γ)	5.105	6.76	123	0.021
0.8 (α)	4.799	6.44	138	0.017
2.5 (α)	4.687	6.14	157	0.015

P and a denote the pressure and the lattice parameter, respectively.

Conclusions

In conclusion, our work provides extensive, previously unreported results of the phonon dispersion in elemental cerium as a function of pressure across the γ - α transition. We observe phonon softening in distinct regions of reciprocal space that reflect significant changes in the electron–phonon interaction and emphasize the importance of the lattice in this intriguing phase transition. The lattice contribution to the phase transition is about 0.33 k_B, to be compared to the total entropy change of 1.5 k_B. The present study confirms predictions of a recent P-T X-ray diffraction data analysis of large changes in the lattice Grüneisen parameters across the transition. The absence of significant pressure evolution in the phonon dispersion in the γ-phase, once compared with the large softening of the bulk modulus observed in X-ray and neutron diffraction reinforces the notion of a long range elastic instability not detectable at the relatively large momentum transfer (ξ≥0.2) investigated here. Ab initio calculations of the γ-phase, treating the 4f electron as localized (trivalent potential), yield the best agreement with experiment, while in the α-phase best agreement is achieved when the 4f electron is treated as part of the valence band (tetravalent potential). Our experimental results provide a unique benchmark to test theoretical models and will stimulate further experimental efforts to improve our understanding of this unique isostructural phase transformation in cerium metal.

Methods

The single crystal samples were prepared from the same ingot grown at the Ames Laboratory, which was used in the historical first INS experiments of γ-Ce at ambient conditions (17, 25). Crystals of 60–80 μm diameter and 20 μm thickness were prepared by laser cutting and mechano-chemical polishing techniques (26). Samples with a [100] or [110] surface normal were loaded in diamond anvil cells together with a small ruby chip, using either helium or neon as pressure transmitting medium. The crystalline quality and purity was checked by rocking curve scans, and typical values ranged between 1–2 °. The crystal quality degraded only slightly in the α-phase. The pressure was determined online by the ruby fluorescence technique.

The IXS experiment was performed on beamline ID28 at the European Synchrotron Radiation Facility in Grenoble. We utilized the silicon (9 9 9) configuration at 17794 eV, yielding an overall energy resolution of 3 meV full-width-half-maximum (FWHM), and the focusing optics in Kirkpatrick–Baez configuration with a focal spot size at the sample of 30 × 60 μm² (horizontal × vertical, FWHM). The momentum resolution was set by slits in front of the crystal analyzers to 0.16° × 0.48°, corresponding to q_y = 0.009 and q_z = 0.027 in reciprocal lattice units.

The ab initio calculations are based on the generalized gradient approximation (GGA) (27) with the Perdew–Burke–Ernzerhof parameterization (28) for the exchange-correlation functional to density functional theory (29, 30). We employed the projected augmented wave (PAW) approach, as implemented in the VASP software (31). The PAW potential with the valence state [5s²5p⁶]6s²5d¹4f¹ (four valence electrons) and [5s²5p⁶]6s²5d¹4f⁰ (three valence electrons) has been used (semicore states are bracketed). Convergence was achieved with a cutoff energy of 500 eV and a k-point mesh of 24 × 24 × 24 generated with the Monkhorst–Pack (MP) method (32). For the calculation of the phonon dispersion curves, the PHONON 4.22 code with the ab initio force-constant method was employed (33). We used a 3 × 3 × 3 supercell containing 108 atoms, which was sampled by using a 6 × 6 × 6 k-points mesh generated by the MP scheme. We utilized the experimentally determined lattice constants in the calculations.