{'fraction_non_alphanumeric': 0.0356713680349124, 'fraction_numerical': 0.019986085636582128, 'mean_word_length': 4.381892443839346, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Probing the gamma-alpha Transition in Bulk Ce under Pressure: A Direct Investigation by Resonant Inelastic X-ray Scattering

We report on the most complete investigation to date of the 4f-electron properties at the gamma-alpha transition in elemental Ce by resonant inelastic x-ray scattering (RIXS). The Ce 2p3d-RIXS spectra were measured directly in the bulk material as a function of pressure through the transition. The spectra were simulated within the Anderson impurity model. The occupation number nf was derived from the calculations in both gamma- and alpha-phases in the ground state along with the f doubleoccupancy. We find that the electronic structure changes result mainly from band formation of 4f electrons which concurs with reduced electron correlation and increased Kondo screening at high pressure.

The γ-α transition in Ce is archetypical of localizationdelocalization phenomenon encountered in f -electron systems. This isostructural phase transition accompanied by a large volume contraction that ends in a tricritical point is a manifestation of subtle interactions between f -levels self-consistently embedded in a sea of conduction electrons. The Ce anomalous behavior at the transition is commonly described by one or the other of two possible scenarios involving either a Mott transition [1] or Kondo hybridization [2,3]. Recent theoretical approaches seem to point to a somewhat intermediate behavior at finite temperature in either phase, with electron correlations and screening as mandatory ingredients [4,5]. More precisely, both phases can be described by strongly correlated 4f electrons, but differ in their degree of localization: the occupation number n f is almost equal to unity in the γ-phase and is reduced in the mixed-valent regime in the α-phase. As stated in Ref. 5, important information about the γ-α transition and more particularly the effects of electron correlation in Ce, is contained not only in n f , but also in the probability of double occupancy of the f sites. 4f electron states are clearly identifiable in spectroscopic data obtained by x-ray photoelectron (XPS) or x-ray absorption (XAS) spectroscopies. They are thus the two probes that have contributed most to unraveling the electronic properties of Ce in the past, and proof of a mixed-valent behavior has been accumulated unambiguously from such experiments over the years. Well separated features, each assigned to a different f valency indicate that there is more than just a single component. The f configurations are mixed in the ground state, but the degeneracy is lifted in the XPS or XAS final state, as the core-hole is screened differently by the f 0 , f 1 , or f 2 states. This should normally allow a direct estimation of the various f -electron weights, and therefore help to characterize the degree of hybridization of the f -electron, at the origin of the heavy-fermion like behavior of Ce. However, the electron-count derived in the final state is not representative of the Ce ground state and the Ce electronic properties at the γ-α transition have, in fact, never been observed directly in the bulk material.
Recently, resonant inelastic x-ray scattering (RIXS) has emerged as a means of probing the mixed-valent behavior in rare-earths in considerable detail [6]. The experiment consists in measuring the 3d → 2p decay following a resonant excitation close to the rare-earth L 2,3 edge (2p → 5d transition). The RIXS process, subsequently denoted 2p3d-RIXS, benefits from the selective resonant enhancement of the different valent states, through a proper choice of the incident energies. In a previous series of experiments, we have studied the γ-α transition with temperature in Ce-Sc and Ce-Th alloys by 2p3d-RIXS [7]. The chemical pressure induced by alloying normally prevents contamination by the intermediate β-phase. The sudden changes in the electronic properties at the transition temperature and the accompanying hysteresis both pointed to a first-order transition, consistent with a pure γ-α isostructural change. However, electron interactions with the dopant element (Sc or Th) necessarily intervenes in the Ce-4f electronic properties.
Here we investigate elemental Ce by 2p3d-RIXS directly subjected to high pressure to induce the γ-α transition. Without the alloying effects inherent in previous experiments [6,7] it has now become possible to implement the Anderson impurity model (AIM) to analyze the spectra and, from there, estimate the ground state f -counts in both phases and in particular the variation in n f and double occupancy across the transition. This combination of RIXS spectroscopic measurements with advanced simulations provides the most complete picture of the bulk electronic properties of elemental Ce at the γ-α transition to date.
The starting material was an ingot of high-purity polycrystalline Ce kept in silicon oil. Maximum care was taken to avoid oxidation of the sample during the loading and measurement phases. A small chip of Ce was cut off directly in silicon oil and loaded in a membrane diamond anvil cell with rubies for pressure calibration. The oil served both as pressure transmitting medium and to prevent oxidation of the sample once in the pressure cell. At the energies of the Ce L 3 edge (5.73 keV) and of the 3d → 2p 3/2 (Lα 1 ) emission (4.84 keV), absorption of x-ray by diamonds is critical. The pressure cell was equipped with a pair of perforated diamonds each caped with a 0.5 mm thick diamond to maximize throughput. We used Rh as gasketting material (for details, see Ref. [8]). The RIXS experiments were performed at ID-12 at ESRF. The incident energy was selected by a fixed-exit Si(111) monochromator and the beam focused at the sample position in a 40×300 (V×H) µm 2 spot. We privileged the transmission geometry with ≈ 10 • scattering angle. This configuration optimizes the scattering volume and hence the emitted signal strength. It also ensures that the sample probed is homogeneous with respect to pressure gradient. The RIXS spectra were acquired using a bent crystal spectrometer [9]. The pressure cell was mounted outside the spectrometer, with the emitted x-rays entering the spectrometer chamber through a thin Be window. Count rates at the maximum of the Ce Lα 1 line were of the order of 10 Hz because of the low transmission by the pressure cell. Figure 1a shows the experimental L 3 XAS spectra as a function of pressure. The white line shows a marked decrease in intensity as Ce is driven through the γ-α transition, while the feature denoted 4f 0 progressively builds up at higher energy. The overall spectral shape and the spectral changes at the transition are consistent with early results by Lengeler et al. [10]. In the latter, however, the α-phase was obtained from γ-Ce by low temperature quenching and the presence of parasitic βphase cannot be ruled out. Following the interpretation from Gunnarsson and Schönhammer [11], the XAS final states split into multiple components because of the 2p core-hole Coulomb potential acting on the mixed-valent ground state c 0 |4f 0 + c 1 |4f 1 + c 2 |4f 2 , where |c n | 2 represent the weight of the individual components and the superscripts indicate the dominant configuration in each case. As an example, the white line is mostly attributed to the 4f 1 final-state configuration, while the high-energy hump is mainly 4f 0 -like. We use this standard shorthand notation for the subsequent referencing of the spectroscopic features. Note that the 4f 2 component, expected to show up in the pre-edge region, is masked by the 2p 3/2 core-hole lifetime. A series of 2p3d-RIXS spectra were measured at 0 kbar in the pressure cell at incident energies E 0 in the Ce L 3 edge region. The general behavior of the RIXS spectra as a function of the incident energy follows closely the results obtained in Ce-Sc and Ce-Th alloys in the γ-phase [7,12]. The series is not reproduced here, but we briefly summarize the main findings: as the incident energy is tuned to the pre-edge region, a shoulder shows up in the RIXS spectra on the low energy-transfer side of the main emission (see Fig. 2); this feature resonates at E 0 = 5718.3 eV (denoted 4f 2 in Fig. 1); at higher incident energies, the main line continues growing in intensity until it reaches a maximum at the white line energy (4f 1 label in Fig. 1), where the fluorescence regime sets in. Accordingly the low-energy shoulder observed in the RIXS spectra at 5718.3 eV can be traced back to a 4f 2 -dominated final state (see Fig. 2), while the main emission line comes from transitions to mainly 4f 1 final states. In Fig. 2, we can follow the evolution of the 2p3d-RIXS spectra measured at E 0 = 5718.3 eV as the pressure is increased. The spectrum at 1.5 kbar barely shows a difference with the standard pressure data. However, a striking increase (≈ 40%) in the 4f 2 /4f 1 intensity ratio is observed as the systems passes the γ-α transition pressure.
In a previous experiment, we had extracted the corrected 4f 2 /4f 1 ratio after deconvolution from lifetime broadening using a phenomenological approach. Here, we take a major step forward by carrying out full multiplet calculations within the Anderson impurity model. Details of the model calculations and Hamiltonian have been described in previous work [16,17]. We use a basis set consisting of 4f 0 , 4f 1 v, and 4f 2 v2 configurations in the ground state, where v denotes a hole in I: Parameters used in the calculations as a function of pressure and volume (after Ref. [13]): ǫ 0 f is the binding energy of the bare f -level and V the hybridization energy. Weights of the 4f -components and the resulting occupation number n f , and variation ∆n f are also indicated. TK is the Kondo temperature. the valence band below the Fermi energy (ε F ). The intermediate states are thus described by linear combinations of 2p 5 4f 0 5d, 2p 5 4f 1 v5d, and 2p 5 4f 2 v 2 5d. The final state contains 3d 9 4f 0 5d, 3d 9 4f 1 v5d, and 3d 9 4f 2 v 2 5d. The atomic Slater integrals and spin-orbit interaction parameters are obtained using Cowan's Hartree-Fock program with relativistic corrections [18] and scaled to 80% to account for intra-atomic configuration interaction effects. The configuration-dependent hybridization strength with (R c ,R v ) = (0.6, 0.8) are also used in the present analysis, following Ref. [19]. The parameters of U f f (on-site Coulomb repulsion), U f c (2p) ((U f c (2p) = U f c (3d) + 0.5 eV) is the attractive core-hole potential), and W (conduction-band width) are taken as 6.0, 10.5 eV, and 2.0 eV respectively. The hybridization strength V (ε) between the 4f and the conduction band (CB) states depends on the CB energy ε. The CB states are supposed symmetric around ε F = 0 and we use a semielliptical form of V (ε) 2 discretized on a logarithmic εscale [20]. The RIXS cross-section is derived from the Kramers-Heisenberg formula (see Eq. 5 in Ref. [16]). The simulated XAS and RIXS spectra are represented in Fig. 1b and Fig. 2 respectively for the three measured pressures. The XAS spectra are well reproduced throughout the transition. The overall agreement for RIXS is equally good, except on the high energy-transfer side. The discrepancy likely results from a fluorescence-like contribution to the spectra, which is not taken into account in the calculations. The parameters used in the calculations are reported in Table I for pressures of 0 kbar, 10 kbar, and 20 kbar, along with the estimated weights for the different 4f n -components. The parameter selfconsistency was checked against XAS, RIXS spectra, and valence-band photoemission data from Ref. [21]. These were shown to be already well described by a simplified version of IAM, which nevertheless takes into account the actual density of states [22].
The main effect, according to the calculation, is the sharp decrease in the 4f 1 component to the advantage of the 4f 0 -related feature, which gains intensity as Ce becomes more α-like. Such a trend is consistent with the spectral changes in the XAS spectra. Formally, the transfer of spectral weight from the 4f 1 (5d 1 ) configuration toward a more 4f 0 (5d 2 ) configuration in the α-phase can be understood as a partial delocalization of the 4f electrons. Interestingly enough, the highly hybridized 4f 2 state also shows a sizable (∼40%) increase with pressure. The increased contribution from the 4f 2 component at high pressure stresses the growing interaction between the 4f and the conduction electrons, a characteristic feature of Kondo-like behavior. This is supported by the increase of the hybridization parameter V as reported in Table I in the high pressure regime. The growth of double occupancy at low volume has another important conse-quence: it points to less correlation in the α-phase as electron hopping is favored. Therefore, the new picture that arises from the RIXS analysis at the γ-α transition is that of the coexistence of competing effects: partial delocalization of the 4f electrons through band formation with the conduction states on the one hand, and reduced electron-electron correlations on the other hand that allows the system to accommodate stronger on-site repulsion.
Finally, the change in n f is obtained from the calculated weights of the 4f -components, according to n f = 0|c 0 | 2 + 1|c 1 | 2 + 2|c 2 | 2 . We obtain n f = 0.97 in the γphase (P = 1.5 kbar), and n f = 0.81 in the α-phase (P = 20 kbar). The results are clearly consistent with earlier estimations [12,23,24] for the γ-phase, but not for the α-phase where values differ substantially. Our RIXS-derived n f -value is 10-15% lower. These new values of the f -occupation for Ce are particularly opportune as they can be compared to recent ab initio calculations. Table I compares the change in n f at the transition derived from the RIXS data to values calculated within dynamical mean-field theory (DMFT) as a function of volume change. (We use the equation of state derived by Jeong et al. [13] in Ce for pressure to volume conversion.) The discontinuous dependence of n f at the transition is well accounted for by DMFT [5,14] in the low temperature limit. On the other hand, the drop in n f at the transition is largely underestimated (4-10% in the DMFT calculations against ≈ 20% according to the RIXS results). The discrepancy with our experimental finding calls for an improved basis set and functional.
n f deviates from unity in Ce as a direct consequence of non-zero hybridization. A remarkable manifestation of this Kondo behavior is the occurrence of a quasiparticle resonance at E F in the single-particle spectral function ρ f (ω). According to the Friedel sum rule, ρ f (E F ) depends only on the ground state value of n f , the degeneracy N f of the local orbital, and a coupling parameter ∆ ∝ V 2 /W , and varies as n f (1−n f ). The sharp decrease of n f in the α-phase indicates a strong enhancement of the quasiparticle peak and that of the renormalization of the bare particle corresponding to (1 − n f ). But the former effect is partly smeared out at temperatures comparable to the Kondo temperature T K [5]. T K is here the key quantity to characterize the 4f -electron coupling with the Fermi sea. It can be evaluated thanks to the Friedel sum rule and given the approximate relationship (1 − n f )/n f ∼ (πk B T K )/(N f ∆) [25] in the limit of large N f . We obtain 70 K in the γ-phase and 1700 K in the αphase assuming ∆ ∼ 110 meV. The temperatures show a fair agreement with neutron scattering data [26] obtained in Ce-Sc alloys but differ very significantly from the generally accepted XPS-derived values [24]. They are consistently smaller by a factor ∼ 2 in the α-phase.
We thank B. Amadon for fruitful discussions on DMFT calculations.