arXiv:cond-mat/0602169v1 [cond-mat.str-el] 7 Feb 2006
Probing the γ-α Transition in Bulk Ce under Pressure:
A Direct Investigation by Resonant Inelastic X-ray Scattering
J.-P. Rueff,1,2J.-P. Iti´ e,1M. Taguchi,3C. F. Hague,2J.-M. Mariot,2R. Delaunay,2J.-P. Kappler,4and N. Jaouen5,1
1Synchrotron SOLEIL, L’Orme des Merisiers, Saint-Aubin, BP 48, 91192 Gif-sur-Yvette Cedex, France
2Laboratoire de Chimie Physique–Mati` ere et Rayonnement (UMR 7614),
Universit´ e Pierre et Marie Curie, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France
3Soft X-ray spectroscopy Lab, RIKEN/SPring-8, 1-1-1, Sayo, Sayo, Hyogo 679-5148, Japan
4IPCMS (UMR 7504), 23 rue du Lœss, BP 43, 67034 Strasbourg Cedex, France
5ESRF, 6 rue Jules Horowitz, BP 220, 38043 Grenoble Cedex, France
(Dated: February 4, 2008)
We report on the most complete investigation to date of the 4f-electron properties at the γ-
α 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 γ- and α-phases in the ground state along with the f double-
occupancy. 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
PACS numbers: 78.70.En, 71.27.+a, 71.20.Eh
The γ-α transition in Ce is archetypical of localization-
delocalization phenomenon encountered in f-electron
systems. This isostructural phase transition accompa-
nied by a large volume contraction that ends in a tri-
critical point is a manifestation of subtle interactions be-
tween f-levels self-consistently embedded in a sea of con-
duction electrons. The Ce anomalous behavior at the
transition is commonly described by one or the other
of two possible scenarios involving either a Mott tran-
sition  or Kondo hybridization [2, 3]. Recent theo-
retical approaches seem to point to a somewhat inter-
mediate behavior at finite temperature in either phase,
with electron correlations and screening as mandatory
ingredients [4, 5]. More precisely, both phases can be de-
scribed by strongly correlated 4f electrons, but differ in
their degree of localization: the occupation number nf
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 nf, but also in the prob-
ability of double occupancy of the f sites. 4f electron
states are clearly identifiable in spectroscopic data ob-
tained 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 prop-
erties of Ce in the past, and proof of a mixed-valent be-
havior 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 configura-
tions 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
This should normally allow a direct estimation of the
various f-electron weights, and therefore help to charac-
terize 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 repre-
sentative 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 be-
havior in rare-earths in considerable detail . The ex-
periment consists in measuring the 3d → 2p decay follow-
ing a resonant excitation close to the rare-earth L2,3edge
(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 .
The chemical pressure induced by alloying normally pre-
vents contamination by the intermediate β-phase. The
sudden changes in the electronic properties at the transi-
tion 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 di-
rectly subjected to high pressure to induce the γ-α tran-
sition. Without the alloying effects inherent in previous
experiments [6, 7] it has now become possible to imple-
ment 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 nf 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 poly-
crystalline Ce kept in silicon oil.
taken to avoid oxidation of the sample during the load-
ing 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 L3 edge (5.73 keV) and
of the 3d → 2p3/2(Lα1) 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 through-
put.We used Rh as gasketting material (for details,
see Ref. ). 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) µm2spot. We
privileged the transmission geometry with ≈ 10◦scatter-
ing angle. This configuration optimizes the scattering
volume and hence the emitted signal strength. It also
ensures that the sample probed is homogeneous with re-
spect to pressure gradient. The RIXS spectra were ac-
quired using a bent crystal spectrometer . The pres-
sure 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 Lα1line were of the order of 10 Hz because of
the low transmission by the pressure cell.
Figure 1a shows the experimental L3XAS 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 4f0progressively
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. . In the latter,
however, the α-phase was obtained from γ-Ce by low
temperature quenching and the presence of parasitic β-
phase cannot be ruled out. Following the interpretation
from Gunnarsson and Sch¨ onhammer , the XAS final
states split into multiple components because of the 2p
core-hole Coulomb potential acting on the mixed-valent
ground state c0|4f0?+c1|4f1?+c2|4f2?, where |cn|2rep-
resent 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 4f1final-state configuration, while the high-energy
hump is mainly 4f0-like. We use this standard shorthand
notation for the subsequent referencing of the spectro-
scopic features. Note that the 4f2component, expected
to show up in the pre-edge region, is masked by the 2p3/2
Maximum care was
5715 572557355745 57155725 57355745
FIG. 1: (color online). Experimental (a) and calculated (b)
L3 XAS spectra in elemental Ce as a function of pressure.
The spectra are normalized to a unity jump at the absorption
edge. Solid arrows point to the 4f0and 4f1components.
The dashed arrow indicates the incident energy where the
4f2component resonates in the 2p3d-RIXS spectra. Ticks in
panel b) are the multiplet states (shown at 0 kbar).
A series of 2p3d-RIXS spectra were measured at 0 kbar
in the pressure cell at incident energies E0in the Ce L3
edge region. The general behavior of the RIXS spec-
tra 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 E0 = 5718.3 eV (denoted 4f2in Fig. 1); at higher
incident energies, the main line continues growing in in-
tensity until it reaches a maximum at the white line en-
ergy (4f1label 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 4f2-dominated final state (see Fig. 2), while the main
emission line comes from transitions to mainly 4f1fi-
nal states. In Fig. 2, we can follow the evolution of the
2p3d-RIXS spectra measured at E0= 5718.3 eV as the
pressure is increased. The spectrum at 1.5 kbar barely
shows a difference with the standard pressure data. How-
ever, a striking increase (≈ 40%) in the 4f2/4f1intensity
ratio is observed as the systems passes the γ-α transition
In a previous experiment, we had extracted the cor-
rected 4f2/4f1ratio after deconvolution from lifetime
broadening using a phenomenological approach. Here,
we take a major step forward by carrying out full mul-
tiplet calculations within the Anderson impurity model.
Details of the model calculations and Hamiltonian have
been described in previous work [16, 17].
basis set consisting of 4f0, 4f1v, and 4f2v2 configu-
rations in the ground state, where v denotes a hole in
We use a
TABLE I: Parameters used in the calculations as a function of pressure and volume (after Ref. ): ǫ0
of the bare f-level and V the hybridization energy. Weights of the 4f-components and the resulting occupation number nf,
and variation ∆nf are also indicated. TK is the Kondo temperature.
fis the binding energy
v (˚ A3)
bDMFT at 632 K, from Ref. 5
cDMFT at 400 K, from Ref. 14
dDMFT at 400 K, from Ref. 15
885890 895900905 910 915
E0 − E1 (eV)
Intensity (arb. u.)
P = 1.5 kbar
P = 10 kbar
P = 20 kbar
FIG. 2: (color online). 2p3d-RIXS spectra in elemental Ce as
a function of pressure (circles). All the spectra were measured
at a fixed incident energy E0 set to 5718.3 eV. To emphasize
changes with pressure, the spectrum measured at ambient
conditions (dashed line) is repeated for each pressure curve.
Superimposed are the calculated spectra (thick lines). The
spectra are normalized to the maximum intensity and offset
for clarity. Ticks indicate the multiplet states.
the valence band below the Fermi energy (εF). The in-
termediate states are thus described by linear combina-
tions of 2p54f05d, 2p54f1v5d, and 2p54f2v25d. The fi-
nal state contains 3d94f05d, 3d94f1v5d, and 3d94f2v25d.
The atomic Slater integrals and spin-orbit interaction
parameters are obtained using Cowan’s Hartree–Fock
program with relativistic corrections  and scaled to
80% to account for intra-atomic configuration interac-
tion effects. The configuration-dependent hybridization
strength with (Rc,Rv) = (0.6,0.8) are also used in the
present analysis, following Ref. . The parameters of
Uff (on-site Coulomb repulsion), Ufc(2p) ((Ufc(2p) =
Ufc(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 εF = 0 and we use a semi-
elliptical form of V (ε)2discretized on a logarithmic ε-
scale . The RIXS cross-section is derived from the
Kramers-Heisenberg formula (see Eq. 5 in Ref. ).
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 through-
out 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 ac-
count in the calculations. The parameters used in the cal-
culations are reported in Table I for pressures of 0 kbar,
10 kbar, and 20 kbar, along with the estimated weights
for the different 4fn-components. The parameter self-
consistency was checked against XAS, RIXS spectra, and
valence-band photoemission data from Ref. . These
were shown to be already well described by a simplified
version of IAM, which nevertheless takes into account the
actual density of states .
The main effect, according to the calculation, is the
sharp decrease in the 4f1component to the advantage
of the 4f0-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 trans-
fer of spectral weight from the 4f1(5d1) configuration to-
ward a more 4f0(5d2) configuration in the α-phase can
be understood as a partial delocalization of the 4f elec-
trons. Interestingly enough, the highly hybridized 4f2
state also shows a sizable (∼40%) increase with pressure.
The increased contribution from the 4f2component at
high pressure stresses the growing interaction between
the 4f and the conduction electrons, a characteristic fea-
ture 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 forma-
tion 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
Finally, the change in nf is obtained from the calcu-
lated weights of the 4f-components, according to nf =
0|c0|2+ 1|c1|2+ 2|c2|2. We obtain nf = 0.97 in the γ-
phase (P = 1.5 kbar), and nf = 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 nf-value is 10–15% lower. These new val-
ues 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 nf at the transition de-
rived 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.  in Ce for pressure to volume conver-
sion.) The discontinuous dependence of nf at the tran-
sition is well accounted for by DMFT [5, 14] in the low
temperature limit. On the other hand, the drop in nf
at the transition is largely underestimated (4–10% in the
DMFT calculations against ≈ 20% accordingto the RIXS
results). The discrepancy with our experimental finding
calls for an improved basis set and functional.
nf 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 EF in the single-particle spectral function
ρf(ω). According to the Friedel sum rule, ρf(EF) de-
pends only on the ground state value of nf, the degen-
eracy Nf of the local orbital, and a coupling parameter
∆ ∝ V2/W, and varies as nf(1−nf). The sharp decrease
of nf in the α-phase indicates a strong enhancement of
the quasiparticle peak and that of the renormalization
of the bare particle corresponding to (1 − nf). But the
former effect is partly smeared out at temperatures com-
parable to the Kondo temperature TK . TK 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 −nf)/nf∼ (πkBTK)/(Nf∆)  in the limit of large
Nf. We obtain 70 K in the γ-phase and 1700 K in the α-
phase assuming ∆ ∼ 110 meV. The temperatures show
a fair agreement with neutron scattering data  ob-
tained in Ce-Sc alloys but differ very significantly from
the generally accepted XPS-derived values . They are
consistently smaller by a factor ∼ 2 in the α-phase.
We thank B. Amadon for fruitful discussions on DMFT
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