introduced, but their higher-order effect can be neglected for the present discussion.

want to acknowledge everyone that contributed to this process.

Firstly, I would like to express my sincere gratitude to my thesis director Nicholas Brookes, who put his trust in me since the beginning of the project, and my thesis supervisor Kurt Kummer, which has always been understanding and patient with me. They have believed in my skills and ideas more than I did and have been forgiving of my initial inexperience, allowing me to work serenely towards my goals. By doing so, they have been able to take out the best of me.

Together with my advisors, I want to acknowledge the daily dedication and hard work that the ID32 beamline staff puts into the maintenance of the instrumentation. I learned a lot from the strengths of each member of this united team and they all deserve to be listed by name. I admired the determination and genuineness with whom Flora Yakhou-Harris, the ID32 beamline operation manager, can face every situation. Andrea Fondacaro, the beamline technician, besides knowing everything about the instrumentation, is the most sympathetic, generous and helpful person I ever met, always ready to give wise and disinterested advice. The two Post-Docs have been good companions for relaxed chats and laughs during the breaks. Emilio Velez-Fort, with his easygoing and positive approach to problems, and Davide Betto, with his unshakable enthusiasm and willingness, make a perfect working pair. I would also like to mention the newcomers, Emilie Lefrançoise and Ivan Spinelli, which already integrated perfectly in the group and I wish them all the best. So, I thank the whole ID32 team for the enjoyable, relaxing and pleasant time spent together.

I am also very thankful to Pr. Giacomo Ghiringhelli and Pr. Lucio Braicovich of Politecnico di Milano, with whom I moved the first steps in this fascinating world during my master thesis and which taught me all the secrets of RIXS spectroscopy. Beside designing and commissioning the ERIXS spectrometer and polarimeter, they kept enlightening me with fruitful discussions, which have been very useful for this PhD project. I also would like to acknowledge the rest of the Politecnico di Milano team, Yingying Peng and Greta Dellea, which have been a nice company during the long, stressful and sometimes disorienting nights (and weeks) of commissioning.

Thanks to Maurits Haverkort for providing the SolidStatePhysics package for Mathematica and the Quanty code, with whom all the calculations in this thesis have been performed. The samples have been prepared at the Max-Planck-Institute for Chemical Physics of Solids in Dresden, in the group of Christoph Geibel. I thank him and his team members, Silvia Seiro and Nubia Caroca-Canales, for their excellent work.

I want to express my gratitude to Pr. Olivier Isnard, Pr. Gerrit van der Laan, Pr. Liu Hao Tjeng and Dr. Andrea Severing for agreeing to serve on my thesis committee. It is an honor for me that such experienced and prominent scientists considered my work worth of their precious time.

I wish to thank all my friends, new and old. And finally, special thanks to my family and my girlfriend, without whom none of my achievements would have been possible.

Properties of cerium intermetallics

Cerium intermetallic compounds are solid state materials containing, in addition to Cerium, metallic and optionally non-metallic elements, arranged in a crystal structure different from that of the constituents. They belong to the group of strongly correlated materials, meaning that they are characterized by a strong interplay among the electrons, which cannot be considered as a uniform sea of non-interacting particles. Typically, correlated materials have ions with incompletely filled d-or f -electron shells for which the charge distribution is highly anisotropic. The behavior of the electrons cannot be described as non interacting charged particles in a mean field and many-body approaches need to be applied to account for their correlated motion. Many peculiar physical properties can arise in correlated materials, some of which are still matter of debate among researchers.

In particular, the physics of lanthanide-and actinide-based intermetallic compounds derives from the electrons in the partially filled f orbitals, which behave like localized magnetic moments in a sea of itinerant conduction electrons. The interaction between f and conduction electrons leads to two competing effects: On the one hand the conduction electrons can mediate a coupling among f electrons at different lattice sites via the Ruderman-Kittel-Kasuya-Yoshida (RKKY) interaction, favouring a magnetically ordered ground state. On the other hand, the local moments on each site tend to be screened by the conduction electrons, producing a non-magnetic state due to the so-called Kondo screening. Both the RKKY and Kondo effects arise at low temperatures and their competition leads to remarkable magnetic, thermodynamic and electronic behaviours [START_REF] Stewart | Heavy-fermion systems[END_REF][START_REF] Misra | Chapter 1: Overview of heavy fermion systems[END_REF][START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF]. These compounds are often referred to as heavy fermions, since the interaction of conduction electrons with localized f states may result in the observation of an enhanced electron effective mass in measurements as specific heat or resistivity. Furthermore, these systems range from magnetically ordered (ferromagnetic or, more often, antiferromagnetic) to Kondo-screened or intermediate valent, and from metallic or superconducting to Kondo-insulating. Quantum critical phenomena and non-Fermi liquid behavior arise when passing from one regime to another.

Because of this exceptional diversification of uncommon properties, the scientific interest in Kondo and heavy-fermion materials continuously increased in the last 80 years [START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF].

Many aspects of heavy-fermion physics are still not understood and the presence of so many large anomalies made these systems "a paradise for experimentalists" but "a nightmare for theoreticians" [START_REF] Khomskii | Basic aspects of the quantum theory of solids: order and elementary excitations[END_REF]. The most studied heavy fermion materials are Ce and Yb based systems, for their large variety and relative simplicity due to the presence of a single 4f electron or hole. However, many other compounds (as those of Pr, Sm, U and Pu) exhibit the same fascinating phenomena.

This thesis focuses in particular on the study of the characteristics of f levels in Ce compounds that grow in the ThCr 2 Si 2 crystal structure, also called the 122 family. In the following, a short review of heavy fermions' history and properties is presented, following references [START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF][START_REF] Khomskii | Basic aspects of the quantum theory of solids: order and elementary excitations[END_REF][START_REF] Stigari | Hybridization and crystal-field effects in Kondo insulators studied by means of core-level spectroscopy[END_REF][START_REF] Riseborough | Heavy fermion semiconductors[END_REF] and the references therein. The many possible effects arising from the interaction between the f states and conduction electrons are reviewed, before going into the details of how to study the energy and symmetry of the 4f levels.

Historical outline

The study of the Kondo effect dates back to 1934, when de Haas et al. [START_REF] De Haas | The electrical resistance of gold, copper and lead at low temperatures[END_REF] measured the electrical resistivity of a gold wire with a small amount of impurities and found an unexpected increase of the resistivity at low temperatures, with a minimum around 4 K. The same behavior was later found in dilute magnetic alloys [START_REF] Van Den | Anomalies in dilute metallic solutions of transition metals[END_REF], suggesting a connection to the presence of magnetic impurities. A theoretical explanation for this anomaly was missing until 1964, when Kondo [START_REF] Kondo | Resistance minimum in dilute magnetic alloys[END_REF] analyzed the scattering of electrons with magnetic impurities on the basis of a model developed by Anderson [START_REF] Anderson | Localized magnetic states in metals[END_REF] three years earlier. The so-called Anderson impurity model (AIM) predicted an antiferromagnetic exchange interaction between the localized magnetic moment of the impurity and the free electrons in a metal. Kondo applied the thirdorder perturbation theory to study the scattering rate caused by such screened magnetic impurities and found a term proportional to lnpT ´1q, which caused the resistivity increase at low temperatures. This term, combined with the the electron-phonon and electronelectron scattering contributions, which on the contrary increase with temperature, causes the observed minimum in the resistivity. The minimum occurs around a characteristic temperature T K (Kondo temperature) at which the antiferromagnetic coupling between the magnetic impurities and the conduction electrons becomes relevant. As a result of the coupling, the local moment is screened and a non-magnetic Kondo-singlet state forms at each site, with an energy scale k B T K .

In 1975 another unexpected experimental result, by Andres et al. [START_REF] Andres | 4 f-virtual-bound-state formation in CeAl 3 at low temperatures[END_REF], set the discovery of a new class of materials: heavy fermions (HF). They measured a value of the specific heat coefficient in CeAl 3 about a thousand times larger than the one of typical metals, meaning that the effective electron mass in this compound was significantly enhanced compared to the free electron. Soon after, in 1977, Doniach [START_REF] Doniach | The Kondo lattice and weak antiferromagnetism[END_REF] suggested the periodical arrangement of Kondo impurities in the crystal structure (Kondo lattice) as a cause of the heavy-fermion behavior.

The already large interest in heavy fermions gained an additional boost with the discovery of unconventional superconductivity. Curiously, the very first observation of superconductivity in heavy-fermion materials was almost simultaneous with the discovery of the materials themselves. Bucher et al. [START_REF] Bucher | Electronic properties of beryllides of the rare earth and some actinides[END_REF] measured superconductivity in UBe 13 already in 1975, but dismissed it as an artifact because the presence of magnetic moments was supposed to forbid superconductivity, according to the predictions of the BCS theory (Bardeen-Cooper-Schrieffer [START_REF] Cooper | Bound electron pairs in a degenerate Fermi gas[END_REF]). More convincing results were obtained in 1979 by Steglich et al. on CeCu 2 Si 2 [START_REF] Steglich | Superconductivity in the presence of strong Pauli paramagnetism: CeCu 2 Si 2[END_REF]. Their discovery opened up the field of unconventional superconductivity, proving that magnetism and superconductivity are not disconnected phenomena. This boosted an intensive work on these compounds, with quick progresses both in the theoretical and experimental fields. Many heavy fermion superconducting systems were discovered and new models (as the Anderson's large-degeneracy expansion) were developed for their interpretation. The phase diagram of heavy fermion materials was further enriched in 1994, with the observation of a zero-temperature quantum phase transition, i.e. a quantum critical point (QCP), in CeCu 6 [START_REF] Löhneysen | Non-Fermi-liquid behavior in a heavy-fermion alloy at a magnetic instability[END_REF].

The Doniach phase diagram

The phase diagram of HF (Figure 1.1) is governed by the competition between the Kondo screening and the RKKY interaction. Both effects depend on the exchange coupling J between f and conduction electrons and the density of states n around the Fermi level. An antiferromagnetic (AFM) ordering develops in the region where T RKKY ą T K , while for T RKKY ă T K a paramagnetic Fermi liquid (FL) phase is formed. In between the two phases, a quantum critical point (QCP) can arise, in the vicinity of which exotic properties as non-Fermi liquid behaviour (NFL) and unconventional superconductivity may be found. Figure redrawn after Ref. [START_REF] Aoki | Heavy fermions in a high magnetic field[END_REF]. The Kondo temperature, below which the f -electron magnetic moments are screened by the conduction electrons, has a strong exponential dependence T K 9 expp´1{ p2Jnqq .

(1.1)

The Kondo screening of the magnetic moments would lead to a non-magnetic ground state, with localized singlet states formed by the antiferromagnetic coupling of f and conduction electrons. The RKKY interaction, on the other hand, favours a magnetic ordering of the localized moments mediated by oscillations of the conduction electrons' spin density. In absence of Kondo screening, it sets in below the temperature

T RKKY 9 J 2 n . (1.2)
The RKKY ordering is the dominating effect at low values of Jn while when Jn is large the Kondo effect becomes relevant and reduces the actual ordering temperature. The magnetic ordering gradually disappears, leaving a paramagnetic ground state with strongly interacting electrons that can be described in terms of a so-called Fermi liquid (FL). This is the proper heavy fermion state, which is discussed in the next section. Between the two regimes, one often reaches a quantum critical point (QCP), which is the indication of a phase transition taking place at the absolute zero. In the vicinity of the QCP, large quantum magnetic fluctuation arise, leading to non-Fermi liquid behaviour [START_REF] Stewart | Non-Fermi-liquid behavior in d-and f -electron metals[END_REF], i.e. deviations from the typical FL properties, and possibly to unconventional superconductivity [START_REF] Pfleiderer | Superconducting phases of f -electron compounds[END_REF].

The Doniach phase diagram in Figure 1.1 qualitatively illustrates the various possible behaviours of a heavy fermion system as a function of T and a parameter δ " Jn. δ is the parameter usually varied in experiments, by applying hydrostatic pressure or chemical doping.

The heavy fermion state

The heavy fermion (HF) behaviour arises from the strong coupling of the f electrons to the conduction band, which causes a partial delocalisation of the f electrons and the formation of heavily-dressed quasiparticles. The coupling becomes effective below a characteristic temperature T ˚, while at high temperatures the thermal fluctuations overcome the Kondo effect and the material acts like a normal metal with a paramagnetic behavior of the noninteracting localized moments. Lowering T , the localized moments start to be screened by the spins of the conduction electrons and a Kondo singlet forms at each site. This causes an increase of the electrical resistivity with decreasing T , until at T ˚the resistivity suddenly drops, due to the development of phase coherence in the scattering channel [START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF].

This is due to the interaction, mediated by the conduction electrons, of the dense order of localized magnetic moment, which form a so-called Kondo lattice. The strongly coupled state between the f electrons and the conduction band causes an effective hybridization between the localized f and the itinerant states. As a result, the f electrons contribute to the Fermi surface in the HF state, as can be observed in Shubnikov-de Haas or de Haas-van

Alphen experiments [START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF]. The characteristic temperature for the onset of the Kondo lattice state T ˚is related to the single-ion Kondo temperature T K . In the following, for simplicity, both temperatures will be referred to as T K , to indicate in general the temperature range at which the Kondo effects start to be relevant.

Many properties of the HF state are explained by the Landau's Fermi liquid (FL) theory [START_REF] Landau | The theory of a Fermi liquid[END_REF], which considers the electron correlations by starting from a non-interacting electron fluid and adiabatically turning on the interactions. This transforms the ground state of the Fermi gas into the ground state of the interacting system. The observed heavy quasiparticles can be described in terms of Landau quasiparticles, which differ from free electrons for their dynamics (i.e. they show an enhanced mass and a reduced magnetic moment).

At low temperature FLs show a linear dependence of the specific heat C9γT with enhanced values of γ, a quadratic dependence of the resistivity ρpT q9AT 2 and a characteristic (and small) temperature dependence of the magnetic susceptibility [START_REF] Stigari | Hybridization and crystal-field effects in Kondo insulators studied by means of core-level spectroscopy[END_REF][START_REF] Panwar | Magnetic susceptibility of heavy fermions and mixed-valence systems[END_REF][START_REF] Carneiro | Finite-temperature contributions to the magnetic susceptibility of a normal Fermi liquid[END_REF], often saturating at high values (enhanced Pauli paramagnetism). Moreover, two scaling laws χ{γ " const [START_REF] Wilson | The renormalization group: Critical phenomena and the Kondo problem[END_REF] and A{γ 2 " const [START_REF] Kadowaki | Universal relationship of the resistivity and specific heat in heavy-fermion compounds[END_REF] are also typical of the FL character of HF compounds.

In contrast, signatures of the non-Fermi liquid phase are a logarithmic dependence of the specific heat, a resistivity ρpT q9AT n with 1 ă n ă 1.5 and often a temperature dependence of the magnetic susceptibility χpT q9χ 0 p1 ´cpT {T 0 q 1{2 q, instead of the χpT q9χ 0 p1 `aT 2 log T q of the FL state [START_REF] Aoki | Heavy fermions in a high magnetic field[END_REF].

Heavy fermion superconductivity and quantum criticality

If the antiferromagnetic ordering temperature is lowered to the absolute zero by the competition with the Kondo effect, a quantum critical point (QCP) is achieved. The QCP corresponds to a quantum phase transition from the antiferromagnetic to the paramagnetic state, that takes place at 0 K and can be induced, for example, by application of pressure.

In the vicinity of the quantum phase transition, the amplitude and the coherence length of quantum magnetic fluctuations grows rapidly. Contrary to a classical finite-temperature phase transition, where the fluctuations are restricted to a small fraction of a reduced temperature scale, the critical behaviour is extended to a wide portion of the phase diagram and influence the material properties [START_REF] Knebel | Antiferromagnetism and superconductivity in cerium based heavy-fermion compounds[END_REF]. As a result, exotic phenomena as non Fermi liquid behavior [START_REF] Stewart | Non-Fermi-liquid behavior in d-and f -electron metals[END_REF] and unconventional superconductivity [START_REF] Pfleiderer | Superconducting phases of f -electron compounds[END_REF] may arise.

Two different magnetic QCP are possible [START_REF] Coleman | Quantum criticality[END_REF][START_REF] Si | Locally critical quantum phase transitions in strongly correlated metals[END_REF]. In the spin density wave model the antiferromagnetic order arises from a spontaneous spatial modulation of the spins of the heavy charge carriers, which exist on both side of the QCP. On the other hand, in the Kondo breakdown scenario the transition from the paramagnetic to the antiferromagnetic state is considered in terms of a local criticality, due to the dissolution of the Kondo singlets and a localization of the f electrons, with a consequent change of the Fermi surface. In this case, the heavy quasiparticles are therefore only present in the paramagnetic state.

Other quantum criticalities, of non-magnetic nature, are also possible and have been considered responsible of specific features of the heavy fermion superconductivity (SC). The superconducting phase of CeCu 2 Si 2 exhibits a double-dome structure in the phase diagram, with maximal critical temperatures of T c1 " 600 mK at P c1 " 0.45 GPa and T c2 " 2 K at P c2 « 4 GPa [START_REF] Holmes | Valence instability and superconductivity in heavy fermion systems[END_REF][START_REF] Thomas | Precise determination of the pressure dependence of T c in the heavy-fermion superconductor CeCu 2 Si 2[END_REF]. The two SC phases can be separated by Ge substitution of the Si atoms, indicating that they could be related to different causes [START_REF] Yuan | Observation of two distinct superconducting phases in CeCu 2 Si 2[END_REF]. Studies on CeCu 2 Ge 2 , which at ambient pressure is an antiferromagnet, linked the SC1 phase to a magnetic QCP.

On the other hand, the origin of SC2 is still a matter of debate. At first, critical valence fluctuations were proposed to provide the glue for superconductivity [START_REF] Holmes | Valence instability and superconductivity in heavy fermion systems[END_REF][START_REF] Miyake | New trend of superconductivity in strongly correlated electron systems[END_REF]. In this model, the f electrons are supposed to enter in a weakly correlated phase, accompanied by a change in the Cerium f levels occupation, energy and degeneracy. This abrupt valence change has not been observed in experiment [START_REF] Rueff | Pressure-induced valence crossover in superconducting CeCu 2 Si 2[END_REF], but a variation of the crystal field splitting of the f levels has been found around P « 4 GPa [START_REF] Fujiwara | High pressure NQR measurement in CeCu 2 Si 2 up to sudden disappearance of superconductivity[END_REF]. Alternatively, LDA+DMFT calculations suggested an orbital transition between two crystal field levels with different hybridizations [START_REF] Hattori | Meta-orbital transition in heavy-fermion systems: Analysis by dynamical mean field theory and self-consistent renormalization theory of orbital fluctuations[END_REF][START_REF] Pourovskii | Theoretical prediction and spectroscopic fingerprints of an orbital transition in CeCu 2 Si 2[END_REF]. The corresponding critical orbital fluctuations would be coupled with the itinerant charges, providing the glue for the Cooper pairs.

The discovery of HFSC, in the vicinity of an antiferromagnetic phase and in certain cases triggered by pressure, opened up the variegated field of unconventional superconductivity. In contrast to the classical BCS superconductors, electronic interactions are considered to drive the formation of the Cooper pairs. The involvement of the heavy f electrons in the pairing mechanism was confirmed by measurements of the specific heat, the London prenetration depth and the superconducting coherence length, all showing the large mass renormalization of the charge carriers [START_REF] Coleman | Heavy Fermions: electrons at the edge of magnetism, volume 1 of Handbook of Magnetism and Advanced Magnetic Materials[END_REF].

Intermediate valence

In the previous treatment the f occupation has always been considered as (almost) integral (for Cerium ions n f « 1), with the f electrons acting as localized moments. Depending on the respective energy of the f (E f ) and Fermi (E F ) levels, upon hybridization the f occupation may undergo a considerable change. Valence changes can be accompanied by changes in the system's properties, an example being the aforementioned SC phase related to critical valence fluctuations in CeCu 2 Si 2 .

As a matter of fact, both the Ce ion valence and the exchange integral J depend on the hybridization strength V and the distance between E f and E F . Following Ref. [START_REF] Khomskii | Basic aspects of the quantum theory of solids: order and elementary excitations[END_REF], one can relate the sample properties to n f and J by redrawing the Doniach phase diagram in function of E f (Figure 1.2). Considering that (in the limit of infinite on-site repulsion of the f electrons)

J9 2V 2 E F ´Ef , ( 1.3)
J increases when the f level approaches E F . Of course this expression is not valid for E F ´Ef Ñ 0, but its qualitative tendency is still correct. Starting with E f ! E F (regime I in Figure 1.2), the deep f level is occupied by one electron (n f " 1) and the corresponding low value of J will let the RKKY interaction win over the Kondo effect. As discussed above, an ordered magnetic phase with localized magnetic moments will form at low temperatures.

By moving the f level closer to the Fermi level, the Kondo energy scale becomes relevant and the f magnetic moments are screened. This causes a decrease of the magnetic ordering temperature, which is eventually driven to zero. The Kondo interaction gives rise to a resonance peak in the density of states close to E F , of width " T K . The Kondo behaviour may initially coexist with the remaining magnetic ordering (region IIa in Figure 1.2) until the non magnetic HF liquid behaviour completely sets in. Since the f level is still relatively deep, its occupation is again close to one (n f « 1). The intermediate valence should be interpreted as a superposition of states with dif-ferent distributions of the electrons between the f states and the conduction band, i.e.

|Ψy " a|f 1 c n y `b|f 0 c n`1 y, where c represents the conduction band. In presence of such a superposition of states, the system experiences quantum fluctuations among the different valence configurations and the average occupation n f is given by the weights |a| 2 of each contribution. Referring to this picture, one also usually speaks of mixed valence.

To conclude the description of Figure 1.2, when E f ą E F the f states are eventually depopulated. The f electrons are completely transferred to the conduction band, all the interactions with the localized states disappear and the system enters a non magnetic metallic phase.

Significance of the 4f states

In this introductory section, the properties of heavy fermion systems, which include Cerium intermetallics, were presented with a general overview, without considering the detailed characteristics of the f levels involved. Instead, as one can imagine, the properties of f states are of fundamental importance for the determination of the interaction with the conduction band.

There are 7 f orbitals, i.e. solutions of the Schrödinger equation for the hydrogen atom with quantum number l " 3, each with two possibilities for the electron spin. Due to the intra-atomic (spin-orbit) and inter-atomic (crystal field) interactions the 14 one-electron f states are split in energy and, in presence of more than one f electron, the e ´´e ínteractions further complicate the f energy scheme.

First of all, the splitting lifts the ground state degeneracy, which, for a Ce 3`i on in a (nonmagnetic) crystal environment, can be 2, 4 or 6 depending on the specific crystal electric field. This simple consideration has already important consequences, since the results of many theoretical works, as the Anderson impurity large N expansion, depend on the f states' degeneracy. The size of the splitting should be introduced in the discussion too, because, when compared to the other typical energy scales of the system, it determines which f levels may actually take part in the interactions. Moreover, each f level is characterized by a different projection of the total angular momentum J z , which gives the magnetic moment of the Ce ion. Since all the exotic phenomena presented above are driven by magnetism, the knowledge of the magnetic moment of the unhybridized ion provides much information about the magnetic interactions ruling the observed phenomena. Lastly, the unhybridized f states are highly anisotropic and their specific symmetry and charge distribution are very important for modeling the properties of heavy-fermion systems [START_REF] Zwicknagl | Quasi-particles in heavy fermion systems[END_REF][START_REF] Gunnarsson | Crystal-field and configuration dependence of hopping-matrix elements for CeCu 2 Si 2[END_REF]. Indeed, strongly anisotropic Fermi liquid behavior has been observed [START_REF] Matsumoto | Anisotropic heavy-Fermi-liquid formation in valence-fluctuating α-YbAlB 4[END_REF] and the crystal-field anisotropy is supposed to influence the band structure, the Fermi surface [START_REF] Zwicknagl | CeCu 2 Si 2 : Renormalized band structure, quasiparticles and co-operative phenomena[END_REF] and even the formation of the SC state [START_REF] Eremin | Feedback spin resonance in superconducting CeCu 2 Si 2 and CeCoIn 5[END_REF][START_REF] Nevidomskyy | Layered Kondo lattice model for quantum critical β-YbAlB 4[END_REF][START_REF] Flint | Tandem pairing in heavy-fermion superconductors[END_REF][START_REF] Willers | Determining the in-plane orientation of the ground-state orbital of CeCu 2 Si 2[END_REF].

Despite the efforts of the last 40 years, the experimental determination of the energy, symmetry and charge distribution of the f states is still a challenge and different techniques can lead to different conclusions. In this work, the new capabilities of Resonant Inelastic X-ray Scattering (RIXS) spectroscopy at the rare earth M 4,5 edges are shown to be ideal probes of the 4f levels and used for solving this old puzzle in prototypical cases.

Cerium 4f states in the free ion and in the crystal environment of solids

As just discussed, many fascinating and interesting phenomena arise in Cerium intermetallic compounds from the presence of 4f states and their strong interaction with the surroundings.

The property of 4f levels (degeneracy, energy, symmetry...) are of great importance for the full understanding of the underlying physics. In this section, the Hamiltonian describing the electronic states in a Cerium ion is discussed, starting from the free ion and more and more adapting it to the specifics of Cerium intermetallic compounds of the 122 family, as those studied in this thesis. The arguments in this section apply as well to the description of the excited states that come into play in the spectroscopic techniques used to study the electronic levels of the Cerium ion and that will be discussed in the next section and in chapter 2.

Energy levels of the free ion

The central field approximation. The calculation of the energy levels of an atomic system is based on the Schrödinger equation

HΨ " EΨ. (1.4)
Considering the atom as made up of a point nucleus of infinite mass and charge Ze, surrounded by N electrons, of mass m and charge ´e, the systems' Hamiltonian is given by [START_REF] Condon | The Theory of Atomic Spectra[END_REF][START_REF] Goldschmidt | Chapter 1: Atomic properties (free atom)[END_REF]:

H " N ÿ i"1 pp 2 i {2m ´Ze 2 {r i `ξpr i ql i ¨si q `N ÿ iąj"1 e 2 {r ij . (1.5)
The first summation in the Hamiltonian considers only single-electron terms: (a) the kinetic energy of each (i-th) electron; (b) its potential energy in the field produced by the nucleus and (c) the spin-orbit interaction. The second summation, on the other hand, takes into account the interactions acting between the electrons in the atom, modeled as a sum of electrostatic repulsions between two charges ´e at a time separated by a distance r ij . This last term prevents the separation of the variables describing each electron, therefore an approximation is needed in order to solve the Schrödinger equation for many-electrons atoms. The repulsion cannot be neglected as small in the first place and subsequently treated as a perturbation, because, due to the large amount of electrons, its total effect can be comparable to the interaction between the electrons and the nucleus.

The central field approximation [START_REF] Slater | The theory of complex spectra[END_REF][START_REF] Condon | The Theory of Atomic Spectra[END_REF] considers each electron as moving independently in the field of the nucleus plus a central field due to the charge distribution of the other electrons. In this way the most of the electronic repulsion is taken into account as an average potential screening the nucleus and the remaining part can be eventually treated as a perturbation. The Hamiltonian for the central field approximation which is the starting point of the calculation is given by:

H 0 " N ÿ i"1 pp 2 i {2m `U pr i qq (1.6)
where ´U prq{e is the spherically symmetric effective potential. The remaining terms, i.e.

the spin-orbit interaction and the deviations from the average potential can be treated as a perturbation

H 1 " H ´H0 . (1.7)
The solution of the approximated multi-electron Schrödinger equation

H 0 Ψ " E 0 Ψ (1.8)
can now be expressed using single-electron wave functions ψ ki , that describe the independent motion of an electron in a central field U prq, with k representing the set of quantum numbers (nlm l ). It is possible to separate the radial and angular part ψ k " R nl Y lm l pθ, φq, where Y lm l are spherical harmonics [START_REF] Condon | The Theory of Atomic Spectra[END_REF] and R nl a radial function depending on U prq. The quantum numbers follow the usual notation, meaning that n is the principal quantum number of the electron shell, l the orbital angular momentum quantum number (subshell) and m " m l the magnetic quantum number. In the general potential U prq, the energy values are degenerate only with respect to m l , but there is no degeneracy with respect to l, the latter being a peculiarity of the Coulomb potential e 2 {r and therefore present in the hydrogen atom (or any one-electron ion) only [START_REF] Goldschmidt | Chapter 1: Atomic properties (free atom)[END_REF]. The spin of the electron is introduced by multiplying ψ by one of the two spin wave functions, depending on the two possible spin projections m s " ˘1 2

along the z axis.

The full anti-symmetrized many-electron wave function ΨpK 1 K 2 ...., K N q is then given by the Slater determinant of the single-electron wave functions. The antisymmetry ensures that the Pauli exclusion principle is not violated.

ΨpK 1 , ..., K N q " 1 ? N ! ψ 1 pK 1 q ψ 1 pK 2 q ¨¨¨ψ 1 pK N q ψ 2 pK 1 q ψ 2 pK 2 q ¨¨¨ψ 2 pK N q . . . . . . . . . . . .

ψ N pK 1 q ψ N pK 2 q ¨¨¨ψ N pK N q (1.9)
where K " pnlm l m s q. The resulting functions are known as zero-order wave functions.

Since the eigenvalues E 0 are degenerate with respect to m l and m s , the energy levels in the central field approximation are characterized by pn 1 l 1 qpn 2 l 2 q...pn N l N q. The sequence of quantum numbers specifies the electronic configuration of the system.

Electrostatic and spin-orbit interactions. The zero-order wave functions make up the basis for calculating the first-order corrections due to the perturbation potential that represents the remaining electrostatic and spin-orbit interactions [START_REF] Haverkort | Spin and orbital degrees of freedom in transition metal oxides and oxide thin films studied by soft X-ray absorption spectroscopy[END_REF][START_REF] De Groot | Multiplet effects in X-ray spectroscopy[END_REF][START_REF] Goldschmidt | Chapter 1: Atomic properties (free atom)[END_REF]. These terms are different for different states of the same configuration, which split into levels under their influence. In the first order perturbation theory each configuration is considered separately, assuming that the distances between the configurations are much larger than the splitting caused by the perturbation. [START_REF] Goldschmidt | Chapter 1: Atomic properties (free atom)[END_REF] The energy corrections are obtained by calculating matrix elements of the kind:

@ Ψ|H 1 |Ψ 1 D (1.10)
where H 1 represents the perturbation potential. The calculation details are described by Condon et al. [START_REF] Condon | The Theory of Atomic Spectra[END_REF] for both the spin-obit (single-electron operator) and the electrostatic (two-electrons operator) perturbations.

The separation of the wave function in an angular and a radial part is essential in the evaluation of the integrals. In particular, the radial part of the spin orbit interaction can be easily reduced to a parameter ζ nl , that stands for the integral

ζ nl " ż 8 0 R 2 nl prqξprqr 2 dr. (1.11)
The spin-orbit interaction of each electron will then be written in the form:

ζ nl l ¨s.

(1.12)

The electrostatic part can be expressed in a series of Legendre polynomials. The angular dependence can be solved analytically and consists of integrals over spherical harmonics.

Several selection rules can be applied for its simplification. The radial dependence is given by the so-called Slater integrals:

F k pn a l a , n b l b q "e 2 ż 8 0 ż 8 0 r k ă r k`1 ą R 2 nala pr 1 qR 2 n b l b pr 2 qr 1 2 r 2 2 dr 1 dr 2 G k pn a l a , n b l b q "e 2 ż 8 0 ż 8 0 r k ă r k`1 ą R nala pr 1 qR n b l b pr 2 qR n b l b pr 1 qR nala pr 2 qr 1 2 r 2 2 dr 1 dr 2 (1.13)
which cannot be solved analytically. k denotes the order of the multipole components in the interaction, r ă " minpr i , r j q and r ą " maxpr i , r j q. The general formulation of the matrix element of the two-electron interaction is thus

ÿ i f i F i `ÿ i g i G i (1.14)
where F i (f i ) and G i (g i ) are respectively the Slater-Condon parameters for the radial (angular) part of the direct and exchange parts of the electrostatic interaction [START_REF] De Groot | Multiplet effects in X-ray spectroscopy[END_REF]. The values for the Slater integrals and the spin-orbit parameters for a free ion can be determined with the code by R. D. Cowan [START_REF] Robert D Cowan | The theory of atomic structure and spectra[END_REF] based on Hartree-Fock approximation. The effect of the electrostatic interaction is to split the electronic configurations into levels, the so-called Russel-Saunders (or LS) terms. The corresponding wave function is characterized by L, S, M L and M S , which are all good quantum numbers describing the atomic levels (respectively, total orbital and spin angular momentum of the atom, and their components along a quantization axis) as far as the spin-orbit interaction is not taken into account. The usual notation of terms symbols is 2S`1 X, where X corresponds to a letter according to the value of L (X " S, P, D, F... for L " 0, 1, 2, 3...). Since the terms with the same L and S have the same energy, the energy levels have a p2L `1qp2S `1q degeneracy, that comes from all the possible combinations of M L and M S quantum numbers.

The spin-orbit interaction couples the spin and orbital angular momenta, that in principle are no more conserved independently. The total angular momentum J " L `S is still conserved and therefore J becomes a more convenient quantum number than L and S for the description of the energy levels.

In the Russel-Saunders (or LS) coupling picture, the spin orbit is considered as a perturbation on the energy levels determined by the electrostatic interaction. This picture holds always for light rare-earth atoms, while for transition metals it works only in absence of a crystalline field (whose effects are stronger than the spin-orbit). The energy corrections are given by the diagonal matrix elements of the spin orbit Hamiltonian calculated on the LS terms, each of whom is split in a close group of levels, still characterized, to a good approximation, by S and L, and distinguished by different values of J [START_REF] Goldschmidt | Chapter 1: Atomic properties (free atom)[END_REF]. The group as whole, indicated by 2S`1 X J , is called a multiplet and each level is p2J `1q-fold degenerate.

In the simple case of the ground state of a trivalent Cerium ion, only one electron is present in the 4f subshell, all the other subshells being filled and contributing only with a spherical potential in the central-field approximation Hamiltonian. The electron has S " 1{2 and L " 3, which can combine giving two different values of J, |L ´S| " 5{2 and L `S " 7{2. Following the Hunds' rule, the level with the lowest J will be the ground state, described by the term 2 F 5{2 . The only parameter needed for the description of the electronic levels is here the 4f spin-orbit parameter. On the other hand, after the excitation of a core electron in the 4f subshell, as in X-ray Absorption Spectroscopy (XAS) or Resonant Inelastic X-ray Scattering (RIXS), the interactions that have to be considered are many more. The electrostatic repulsion between the 4f electrons, the electrostatic interaction between the core hole and the 4f electrons and the spin-orbit coupling for the core hole increase dramatically the number of the possible atomic levels. In the case of an excited 3d 9 4f 2 configuration, the number of possible states is 10 ¨14¨13

2

" 910, grouped in multiplets with the same L, S, and J but different M J . Selection rules for dipole transition will restrict the number of possible excited states which nonetheless remains large.

The correct and full implementation of the electron-electron repulsion, with the full multiplet atomic theory, is of fundamental importance for the calculation of the spectra and interpretation of the experimental data. A discussion about the consequences of simplifications in the electron-electron Hamiltonian can be found in Ref. [START_REF] Haverkort | Spin and orbital degrees of freedom in transition metal oxides and oxide thin films studied by soft X-ray absorption spectroscopy[END_REF]. The Slater integrals to be considered in all the 3d 10 4f n Ñ 3d 9 4f n`1 transitions are given and discussed in Ref.

[50].

Atoms in a crystal: the crystal electric field

Reduction of Slater integrals. In a solid, the presence of both "free" electrons and "fixed" ions introduce further modifications in the atomic Hamiltonian. First of all, the presence of the other charges partially screens the electron-electron repulsion. This affects in particular the monopole part of the repulsion [START_REF] Haverkort | Spin and orbital degrees of freedom in transition metal oxides and oxide thin films studied by soft X-ray absorption spectroscopy[END_REF], which anyway is usually not considered in multiplet calculations because it only affects the average energy of a configuration. On the other hand, the multiplet splitting within a configuration is experimentally found to be much less reduced from the free ion values. Experimentally, it has been noticed that a reduction of the Slater-Condon parameters to 80% of their Hartree-Fock values can take into account the screening effects. In the case of the presence of a 3d hole in a rare earth ion, Figure 1.3: Square of the atomic radial wave functions for 4f , 5s, 5d and 6s states, as calculated by the Hartree-Fock method for Ce 3`. On the x-axis the distance r from the nucleus is given in units of the Bohr radius a 0 . For comparison, in Cerium intermetallics of the 122 family, the a crystallographic axis is " 4 Å « 8a 0 . Figure reproduced from Ref [START_REF] Stigari | Hybridization and crystal-field effects in Kondo insulators studied by means of core-level spectroscopy[END_REF].

the reduction of the repulsion between the 3d hole and the 4f electrons is usually reduced to 60%, in order to reproduce the 3d Ñ 4f X-ray absorption spectra [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF].

Crystal-field model. In addition to the average screening effect, the presence of the ions surrounding the atom leads to a further splitting of its p2J `1q-degenerate multiplet levels. This effect is small for electronic states of the 4f subshell because they are in general well localized in the ion and shielded by other electrons (see Figure 1.3). Therefore it can be treated as a perturbation. The overlap with the wave functions of the surrounding ions is negligible for 4f orbitals, while the shells with n " 5, 6 are much more affected by surrounding ions, with 5d and 6s electrons being involved in the bondings. This is the reason why the lanthanides lose first their 5d and 6s electrons upon ionization, so that Ce 3ì s found with a 4f 1 configuration in its ground state.

Since 4f wave functions do not participate significantly in the bondings, their interaction with the surrounding ions of a crystal can be modeled, to a good approximation, as an interaction with an electrostatic field, called the crystal electric field (CEF) [START_REF] Haverkort | Spin and orbital degrees of freedom in transition metal oxides and oxide thin films studied by soft X-ray absorption spectroscopy[END_REF][START_REF] Fulde | Chapter 17: Crystal fields[END_REF][START_REF] Rotter | McPhase User Manual[END_REF][START_REF] Liu | Electronic Energy Level Structure[END_REF].

The CEF Hamiltonian can be written as:

H CEF " ´e ¨VCEF pr, θ, φq, (1.15)
where V CEF is the CEF potential. It is important to stress the effective nature of the CEF potential: the CEF-split states are given mostly by an anisotropic hybridization of the 4f states with the ligands' bands, therefore deviations from the CEF model, which depicts a single ion in a static electric field, are possible. [START_REF] Stigari | Hybridization and crystal-field effects in Kondo insulators studied by means of core-level spectroscopy[END_REF] There are of course some physical requirements for the CEF potential: first of all it must reflect the symmetry of the crystal. It is indeed the lowering of the symmetry, from the spherical environment of the free atom to the local point symmetry in the crystal, that lifts the p2J `1q degeneracy of the multiplets. The lower the symmetry, the larger the number of CEF-split levels. To express considerations on the symmetry of the CEF potential, and to perform the calculations, it is particularly convenient to expand it in spherical harmonics:

V CEF pr, θ, φq " 8 ÿ k"0 k ÿ m"´k A m k r k C m k pθ, φq, (1.16)
where

C m k pθ, φq " b 4π 2k`1 ¨Y m k pθ, φq " b pk´mq!
pk`mq! ¨Pk pcosθq ¨eimφ are the renormalized spherical harmonics, and P k pcosθq the Legendre polynomials. A m k are the coefficients of the expansion. This expansion is possible as long as the overlap with ligands is neglected for 4f wave functions, so that V CEF satisfies Laplace's equation. [START_REF] Taylor | Physics of rare earth solids[END_REF] To calculate the CEF states' energies as a first order perturbation, one has to evaluate the matrix elements on the basis of ψ k " R l n Y m l pθ, φq:

H CEF i,j " ´e 8 ÿ k"0 k ÿ m"´k A m k A R li ni prq|r k |R lj nj prq E A Y mi li pθ, φq|C m k pθ, φq|Y mj lj pθ, φq E .
(1.17)

The integrals over the radial part

A R li ni prq|r k |R lj nj prq
E can, for a free ion, be calculated with the Hartree-Fock approximation using Cowan's code [START_REF] Robert D Cowan | The theory of atomic structure and spectra[END_REF], while the integrals over spherical harmonics can be solved analytically; they are usually re-written using 3j symbols [START_REF] Robert D Cowan | The theory of atomic structure and spectra[END_REF]:

A Y mi li pθ, φq|C m k pθ, φq|Y mj lj pθ, φq E " p´1q mi b p2l i `1qp2l j `1q ˜li k l j 0 0 0 ¸˜l i k l j ´mi m m j ¸.
(1.18)

where the 3j are defined as:

˜j1 j 2 j 3 m 1 m 2 m 3 ¸" p´1q j1´j2´m3 ? 2j 3 `1 xj 1 m 1 j 2 m 2 |j 3 ´m3 y. (1.19)
It is now possible to use some properties in order to reduce the number of CEF parameters needed to fully describe the Hamiltonian. The triangular inequality of 3J symbols imposes that only matrix elements with k ď l i `lj can be non-vanishing, therefore only the CEF parameters A m k that respect this restriction will affect the CEF levels. Furthermore, k`l i `lj has to be even to have a non-zero expectation value, otherwise the angular part would consist of an integral over an odd function, which vanishes. Symmetry conditions on the V CEF , which has to reflect the site symmetry of the rare earth ion, further reduce the number of useful A m k parameters. This restriction is introduced because H CEF must be invariant under the point groups' symmetry operations. For example, if the high symmetry axis has q-fold rotational symmetry, V CEF has to fulfill the condition V CEF pr, θ, φq " V CEF pr, θ, φ`2 π q q. This condition on the angular part of V CEF translates into e imφ " e impφ`2 π q q , that is true if m " N ¨q with N P Z. This condition on m in the CEF Hamiltonian directly implies that the CEF can only mix states with ∆J z " q [START_REF] Sastri | Modern aspects of rare earths and their complexes[END_REF].

Further conditions on the A m k allow to simplify the parametrization [START_REF] Newman | Crystal field handbook[END_REF][START_REF] Görller-Walrand | Chapter 155: Rationalization of crystal-field parametrization[END_REF]. For example, the requirement of an hermitian Hamiltonian imposes A m k " p´1q m pA ´m k q ˚, so that it is sufficient to determine the parameters A m k with m ě 0.

Crystal-field parameters. In most cases, the A m k are considered as free parameters and are chosen in order to fit experimental data [START_REF] Haverkort | Spin and orbital degrees of freedom in transition metal oxides and oxide thin films studied by soft X-ray absorption spectroscopy[END_REF][START_REF] Rotter | McPhase User Manual[END_REF][START_REF] Taylor | Physics of rare earth solids[END_REF]. No assumptions are made on the potential, other than the ones dictated by symmetry. Moreover, the parameters A m k are always multiplied by the radial integral

A R li ni prq|r k |R lj nj prq E .
The calculation of the integral with an atomic Hartree-Fock based method may lead to wrong expectation value, because the radial wave functions may be different for a free atom and an atom in a solid. To avoid this problem, the radial expectation values is included in the free phenomenological parameter, defining:

Ǎm k " A m k A R li ni prq|r k |R lj nj prq E . (1.20)
This is the notation that will be used in this thesis. It must be said that in literature many other conventions can be found to express the CEF parameters, a full list and explanation of which is given in Ref [START_REF] Rotter | McPhase User Manual[END_REF]. The Stevens parameters B m k are by far the most common in literature. These parameters result directly from the Stevens approximation [START_REF] Stevens | Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions[END_REF], which deserves some explanation.

Since the splitting of the LSJ multiplets produced by the spin-orbit coupling is much larger than the effect of the CEF potential, one may assume that the CEF will only be able to mix states within the same LSJ multiplet (but with different values of J z ). This approximation considerably reduces the size of the basis on which the matrix elements of H CEF need to be evaluated, simplifying the calculations. The study of the CEF is usually performed in order to know the properties of the ground state and of the first low-energy excitations, therefore one can restrict the calculation to the lowest LSJ multiplet. J can be then considered as a good quantum number [START_REF] Stigari | Hybridization and crystal-field effects in Kondo insulators studied by means of core-level spectroscopy[END_REF]. Stevens, after expressing V CEF as sums of (non-normalized) tesseral harmonics in the Cartesian coordinates, showed that the sum of polynomials in x i , y i , z i can be replaced by a sum of polynomials in J x , J y , J z operators, which transform likewise under symmetry operations [START_REF] Fulde | Chapter 17: Crystal fields[END_REF]. The new operators are much more convenient because they act on the basis as a whole, while the x i , y i , z i polynomials acted on a particular one-electron wave function [START_REF] Fulde | Chapter 17: Crystal fields[END_REF]. After the conversion, Stevens Hamiltonian is expressed as:

H Stevens " ÿ k l ÿ m"0 B m k O m k pJq. (1.21)
O m k are the Stevens operators and B m k the Stevens parameters, to be determined experimentally. To understand the relations between B m k and Ǎm k parameters, one must follow backwards the Stevens method. The Stevens parameters are defined as [START_REF] Stevens | Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions[END_REF][START_REF] Rotter | McPhase User Manual[END_REF] :

B m k " Ãm k @ r k D xΘ k y . (1.22)
Where @ r k D is a short notation for the usual expectation value of the radial part of the matrix element and xΘ k y " xJ|Θ k |Jy is a prefactor introduced during the x i , y i , z i Ñ J x , J y , J z substitution. These prefactors were tabulated by Stevens [START_REF] Stevens | Matrix elements and operator equivalents connected with the magnetic properties of rare earth ions[END_REF] and depend only on k and J, which in this approximation is a good quantum number, therefore constant for a given multiplet. Ãm k are the coefficients of the expansion of the CEF potential:

V Stevens CEF " ÿ l l ÿ m"0 Ãm k r k f m k px, y, zq. (1.23)
f m k px, y, zq are tesseral Harmonics in their Cartesian expression, without any numerical normalization factor. Re-introducing this prefactor as p k,m (tabulated in Ref. [START_REF] Rotter | McPhase User Manual[END_REF]) and expressing the tesseral harmonics as sum of spherical harmonics, it is easily found that [START_REF] Görller-Walrand | Chapter 155: Rationalization of crystal-field parametrization[END_REF]:

B m k " Ǎm k p k,m c 4π 2k `1 xΘ k y if m " 0 B m k " Ǎm k p k,m c 8π 2k `1 xΘ k y if m ‰ 0. (1.24)
The calculations in this thesis were performed using the SolidStatePhysics package for Mathematica by Maurits Haverkort. There the prefactors λ k,m " p k,m b 4π 2k`1 are defined in the documentation as Stevens prefactors.

The Stevens notation is extensively used in all the techniques that limit their analysis to the lowest J multiplet, as Inelastic Neutron Scattering (INS) and many indirect techniques as magnetic susceptibility. In a RIXS spectrum, on the other hand, features coming States of Ce 3`i n tetragonal crystal field. The specific case of the ground state of a Ce 3`i on can now be discussed. As stated above, we can focus on one configuration (4f 1 ) and split its LS terms by perturbating them with spin-orbit coupling first and then with the CEF. Since there is one electron in the 4f subshell, L " 3, S " 1{2 and the only possible term symbol is 2 F . Spin-orbit coupling splits levels with different J values ( J " |L ˘S| " 5{2, 7{2 ) leading to 2 F 5{2 and 2 F 7{2 . The Hund's rules set that for a less-than-half-filled subshell, the ground state is the one with the lowest J values, i.e. 2 F 5{2 .

The p2J `1q degeneracy of each multiplet is lifted by the CEF. The Kramers' theorem states that in the absence of a magnetic field the energy levels of a system with an odd number of fermions (here, electrons) can at most be split into double-degenerate levels, the so-called Kramers' doublets. The Cerium intermetallic compounds discussed in this thesis are of the 122 family, which exhibits tetragonal D 4h symmetry that completely splits the 2 F 5{2 and 2 F 7{2 multiplets in 3 and 4 Kramers' doublets respectively.

The CEF parameters determine the energy of the CEF levels and the J z mixing of the CEF wavefunctions. Which of the Ǎm k have to be used is set by the constraints outlined before. Since L " 3, we have that k ď l i `lj " 3 `3 " 6 and the requirement on k`l i `lj " even implies that k " even. Therefore k " 0, 2, 4, 6 are the possible values of k for a 4f 1 configuration. The harmonic functions with k " 0 only have a spherical contribution to the potential and don't affect the splitting, thus their effect is usually neglected and is included in the effective spherically symmetric potential. The presence of the four-fold rotational axis imposes m " 0, 4. This implies that the CEF Hamiltonian will mix in the lowest 2 F 5{2 multiplet, J z " ˘5{2 with J z " ¯3{2. J z " ˘1{2, on the other hand, stays a pure J z states as long as the inter-multiplet mixing is neglected, otherwise a small mixing with J z " ¯7{2 is in principle possible.

Limiting the analysis to the 2 F 5{2 multiplet and neglecting the inter-multiplet mixing, another condition imposes that k ď 2J " 5 [START_REF] Sievers | Asphericity of 4f -shells in their Hund's rule ground states[END_REF], letting us discard the terms with k " 6.

We are therefore left with the three parameters Ǎ0

2 , Ǎ0 4 and Ǎ4 4 to model the effects of CEF on the 2 F 5{2 multiplet 1 .

The CEF wave functions result from an admixture of different J z states within the same multiplet and can therefore be expressed as linear combination of the |J " 5{2, J z y " |J z y wave functions. After applying the CEF Hamiltonian on the |J " 5{2, J z y basis, the resulting states are given by: Γ 1 7 " 1 pa 2 `b2 q pa|˘5{2y `b|¯3{2yq

Γ 2 7 " 1 pa 2 `b2 q pb|˘5{2y ´a|¯3{2yq Γ 6 " |˘1{2y, (1.25)
where the double group Bethe's notation Γ k is used to label the states according to their symmetry. The relative energy of the CEF levels and the mixing of the J z wave functions can be expressed as a function of Ǎ0 (1.27)

The relative energies of the states are:

E Γ 1 7 " 1 105 " p´12 Ǎ0 2 ´5 Ǎ0 4 q ´2b 4p9 Ǎ0 2 ´5 Ǎ0 4 q 2 `350 Ǎ4 4 2  E Γ 2 7 " 1 105 " p´12 Ǎ0 2 ´5 Ǎ0 4 q `2b 4p9 Ǎ0 2 ´5 Ǎ0 4 q 2 `350 Ǎ4 4 2  E Γ6 " 8 35 Ǎ0 2 `2 21 Ǎ0 4 .
(1.28)

1 For a complete description of the splitting of the 2 F 7{2 multiplet, we would need to add Ǎ0 6 and Ǎ4 6 to the set.

It can be easily noticed that

E Γ 1 7 ď E Γ 2 7 .
Apart from this, these expressions do not provide any further information about the order of the states in energy. In our strictly phenomenological approach, this needs to be determined with experiments.

The Ǎ4

4 parameter is always present elevated to the power of two in (1.28), therefore its sign does not affect the energy splitting. However, it affects the sign of α and this has consequences on the orientation of the states with respect to the crystallographic axes.

This will be discussed in more detail in section 4.3.

Experimental determination of the crystal field acting on the 4f states

Several techniques have been used over the years to investigate the CEF effects in rare earth ions, depending on the studied material class and the specific physical property of interest.

Each of these methods comes with some limitations and the gained information on the CEF is usually only partial and does not necessarily agree with the results obtained by another method. In this section, the established techniques are briefly reviewed, in order to be able to compare their peculiarities to those of Resonant Inelastic X-ray Scattering, discussed in the next chapters.

Classic approaches

The CEF splitting affects various physical properties of the rare earth compounds, therefore different physical quantities can be measured and related to the CEF scheme.

The most straightforward methods to probe the energy of the CEF-split levels involve optical (absorption and fluorescence) measurements [START_REF] Dieke | Spectra and energy levels of rare earth ions in crystals[END_REF][START_REF] Hufner | Optical spectra of transparent rare earth compounds[END_REF][START_REF] Ofelt | Intensities of crystal spectra of rare earth ions[END_REF], but these are feasible only on transparent samples, while in metallic systems the signal coming from 4f ´4f transitions would be in general overwhelmed by that of other excitations occurring in the same energy range. Since a simple and direct optical measurement of the CEF-levels' splitting is not doable, many indirect measurements arose to study the CEF scheme in rare earth compounds.

The majority of the methods exploit the magnetic properties of the rare earth ion. Electron Paramagnetic Resonance (EPR) or Electron Spin Resonance (ESR) [START_REF] Baberschke | Crystal electric field effects in the ESR of dilute alloys[END_REF][START_REF] Barnes | NMR, EPR and Mössbauer effect: metals, alloys and compounds[END_REF], measuring the Zeeman splitting of a given CEF-level, can study the properties of the ground state wave function and its magnetic field-induced mixing with other states, but it is suitable only for very small CEF splitting, mostly on S-states ions [START_REF] Fulde | Magnetic excitations in crystal-field split 4f systems[END_REF][START_REF] Fulde | Chapter 17: Crystal fields[END_REF]. The so-called hyperfine methods, as Nuclear Magnetic Resonance (NMR), Mössbauer spectroscopy and Muon Spin Resonance (µSR) study the (magnetic) effect of the CEF-ground state on the nucleus [START_REF] Barnes | NMR, EPR and Mössbauer effect: metals, alloys and compounds[END_REF], but, being rather indirect techniques, may often not give a unique set of CEF parameters [START_REF] Fulde | Magnetic excitations in crystal-field split 4f systems[END_REF].

Magnetic susceptibility and specific heat

Magnetic susceptibility (χ) measurements have a more direct insight into the magnetic properties of the ground state and are more commonly used for Cerium intermetallic compounds.

The electron in the 4f sub-shell is responsible for the atomic magnetic moment µ, that can be related to the J z mixing of the occupied states. The projection of µ along a direction i is given by µ i " ´µB g J J i and, in presence of an applied magnetic field B ext " B ext i , an energy term B ext ¨pL `2Sq " g J B ext ¨J " g J B ext i J i is added in the atomic Hamiltonian. The magnetization of a (paramagnetic) sample at finite temperature T can then be calculated via a Boltzmann statistic, by averaging the magnetic moment of each level, weighted with the respective Boltzmann factor. Since in Cerium intermetallics of the 122 family the tetragonal symmetry sets a natural quantization axis, measurements with B r100s and B r001s show a magnetic anisotropy, which is directly related to the ground state's J z value. Moreover, the temperature dependence of χ i pT q " BMipT q BHi provides information about the thermally occupied excited states. This method has again the drawback of being indirect: in the data analysis, indeed, the CEF-parameters are varied until the calculated curve agrees with experiments, but different sets of parameters, yielding different energy splittings of the 4f levels may fit the same curve. Furthermore, at low temperatures, the Kondo effect and the antiferromagnetic RKKY interaction strongly affect both the magnetic anisotropy and the temperature dependence of χpT q. Their effect can be considered in simulations by introducing some further parameters, but this implies further uncertainty in the data interpretation. [START_REF] Borchi | Kondo effect in cerium intermetallics: Magnetic susceptibility[END_REF][START_REF] Liu | The susceptibilities and molecular-field constants of rare-earth compounds from mean-field theory[END_REF][START_REF] Sales | Susceptibility of interconfiguration-fluctuation compounds[END_REF] Another indirect way to determine the CEF-levels is via specific heat measurements.

Starting with all the ions in their ground state at the absolute zero, upon increasing T higher energy levels are thermally populated and therefore more energy must be provided to the sample. At higher temperatures, one can imagine that all the states are equally populated and no further energy is required for the CEF excitations. The magnetic specific heat is therefore expected to show a maximum at a temperature such that the thermal energy becomes comparable to the separation of the levels (usually at T À 1{2∆E CEF ). This phenomenon is referred to as Schottky anomaly [START_REF] Taylor | Physics of rare earth solids[END_REF][START_REF] Fulde | Chapter 17: Crystal fields[END_REF][START_REF] Meyer | The heat capacities of seven rare-earth ethylsulphates at low temperature[END_REF]. The contribution of phonons to the total specific heat is usually considered as independent, and either neglected or estimated from non-magnetic parent compounds. The accuracy of the measurements is therefore limited by the presence of spin-phonon interactions. In addition to this, Kondo [START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF]. The fit of the data lead to a CEF scheme Γ 7 -Γ 6 -Γ 7 , with 31 and 80 meV splittings. The CEFrelated part is the one at high temperatures (ą 100 K), while the decreasing of χ at low temperatures has to be ascribed to magnetism and other phenomena. Note that another magnetic susceptibility study, in Ref. [START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF], gave 27 and 59 meV splittings and a Γ 7 -Γ 7 -Γ 6 scheme. and exchange effects among rare earth ions further distort the specific heat curves and their contribution has to be considered. Figure 1.6 shows the 4f part of the specific heat in YbIr 2 Si 2 after subtraction of the specific heat measured from the non-magnetic reference LuIr 2 Si 2 . The data is compared to calculations of the magnetic part of the specific heat including the Kondo effect for two different CEF schemes obtained from inelastic neutron scattering and photoemission spectroscopy [START_REF] Patil | Crystalline electric field splitting of 4f states in YbIr 2 Si 2 : An ARPES view[END_REF].

Inelastic neutron scattering

Inelastic Neutron Scattering (INS) [START_REF] Fulde | Chapter 17: Crystal fields[END_REF][START_REF] Fulde | Magnetic excitations in crystal-field split 4f systems[END_REF][START_REF] Goremychkin | Crystal field effects in RCu 2 2Si 2 compounds (R= Ce, Pr, Nd, Ho, Er): Inelastic neutron scattering investigation[END_REF][START_REF] Cribier | Difusion des neutrons et effet Stark cristallin dans les oxydes de terre rare[END_REF] can measure the splitting between the 4f levels directly and has become the dominant method for studying crystal fields in rare earth intermetallic compounds. In such an experiment monochromatic neutrons are scattered from the sample and are analysed according to their energy and momentum, determining the energy-and momentum Q transfer. The differential cross section for a transition between two CEF-split states is (in dipole approximation) proportional to |xj|J K |iy| 2 , where J K is the J component perpendicular to the scattering vector Q, the transition being due to a magnetic interaction. Transitions between two CEF-levels are visible in the INS spectrum as peaks in correspondence to the energy transfer required for the excitation, that is the energy of the CEF-splitting. Moreover information about the ground state's spatial distribution is contained in the intensity ratios of the quasielastic and inelastic excitation lines.

The magnetic scattering is often superimposed by phonon contributions which are complicated to correct for. In principle one can separate the magnetic and nuclear scattering by using polarised neutrons and analysing their polarisation after the scattering [START_REF] Moon | Polarization analysis of thermal-neutron scattering[END_REF]. However, this is usually not performed because of the limited intensities. The most used method to separate the two contribution relies on the different behavior of the two scattering processes when the momentum transfer and temperature are varied. One can also obtain information on the phonon spectra by measuring a nonmagnetic reference compound with a similar crystal structure. This procedure, even if well established, is very difficult to apply on strongly correlated electron systems. The electron-electron interactions cause the CEF-transition peaks to be very broad and they can be lost in the complex phonon background.

Another drawback of INS is the limited energy range that can be probed. For energy losses larger than " 60meV, the contribution of multiple (phonon) scattering processes becomes very large and covers the CEF peaks. Therefore splittings larger than " 60meV cannot be observed.

Finally, the count rate is always an issue in INS. In order to increase the signal, large samples are required, but it is not always possible to grow single crystals of centimeter-size.

Powder are therefore used, but this means a partial loss of the information coming from the peaks' intensity. Furthemore, many elements are strong absorber of neutrons, therefore INS experiments in some compounds are practically impossible.

Recently established methods

X-ray absorption spectroscopy

In X-ray Absorption Spectroscopy (XAS) X-ray photons excite an electron from a filled core level to an empty valence one, leaving a core hole. The amount of photons that interacted with the sample is measured by comparing the flux transmitted by the sample with the incident one or by measuring the electrons (Total Electron Yield) or photons (Fluorescence Yield) emitted during the recombination of the core hole. XAS spectra are labelled depending on the core level excited: its principal quantum number (n " 1, 2, 3, 4...) is replaced by a letter (K, L, M , N ...) and a number is added to represent, in order of increasing energy, the spin-orbit-split subshells. To study the properties of lanthanide ions 3d 3{2,5{2 Ñ 4f and 4d 3{2,5{2 Ñ 4f transitions are used, corresponding to M 4,5 and N 4,5 edges, which are both in the soft-X-ray range.

The transition from an initial state i to a final state f is governed by the Fermi's golden rule: |xf |T |iy| 2 , where T is the transition operator. In the case of λ-polarised soft Xrays interacting with a core electron, the operator can be well approximated as a dipolar interaction T 9e kλ ¨p. Fermi's golden rule corresponds to an integral over the whole space, which is not zero only for an even integrand. Since the dipole operator T is odd, the initial and final state must have different parity to have a non-vanishing integral, i.e. a dipole-allowed transition. By applying these symmetry considerations, together with the conservation of angular momentum and spin (photons carry an angular momentum 1 and no spin) the dipole selection rules are found: ∆l " ˘1; ∆j " 0, ˘1; ∆m l " 0, ˘1; ∆m s " 0;

∆L " 0, ˘1; ∆S " 0; ∆J " 0, ˘1 but no J " 0 Ñ J " 0 transitions. The notation for the quantum numbers is the usual.

The integral is also very dependent on the light polarisation, i.e. the orientation of the dipolar electric field with respect to the initial and final wavefunctions. The angular and polarisation dependencies give rise to dichroism in X-ray absorption spectra, meaning that spectra acquired with different light polarisations will be different. This effect is used to study the magnetic properties of the ions (mostly with circular light polarisation), or the orbital anisotropy in the ground state (linear polarisation). The first known study addressed to the investigation of 4f states in rare earth intermetallic compounds via X-ray Absorption Spectroscopy was performed on CeRh 3 B 2 in the nineties. Crystal-field-related dichroism was observed in the calculated [START_REF] Jo | Calculation of linear dichroism in the Ce 4d-Core XAS and 4f electronic state of CeRh 3 B 2[END_REF] and measured [START_REF] Yamaguchi | Soft-X-ray linear-dichroism and magnetic-circular-dichroism studies of CeRh 3 B 2 : Large crystalfield splitting and anomalous ferromagnetism[END_REF] spectra. Unfortunately, the model employed for the data interpretation was applicable only for the rather large CEF splitting of CeRh 3 B 2 and lead to a much too high Kondo temperature [START_REF] Hansmann | Determining the crystal-field ground state in rare earth heavy fermion materials using soft-X-ray absorption spectroscopy[END_REF]. Moreover, the crystal field symmetry was not fully taken into account [START_REF] Givord | The cerium magnetic form factor and diffuse polarization in CeRh 3 B 2 as functions of temperature[END_REF].

Recently, Hansmann et al. [START_REF] Hansmann | Determining the crystal-field ground state in rare earth heavy fermion materials using soft-X-ray absorption spectroscopy[END_REF] conceived a simple and powerful method to study the 4f ground wavefunction that stimulated a comeback for this technique to probe the CEF in rare earth intermetallics. With a single-ion approach, justified by the fact that usually E Kondo ! E CEF , they observed that the M 4,5 spectra calculated supposing pure J z states for the 4f electrons show defined polarisation dependences. This polarisation dependence, or X-ray Linear Dichroism (XLD), strongly depends on the ground state wavefunction's J z

(Figure 1.8).
The idea is to fit the experimental spectrum with an incoherent sum of the spectra simulated for pure J z states. The resulting coefficients of the sum give the J z mixing of the 4f electron ground state. Considering a Γ 7 ground wavefunction, given by an admixture α|˘5{2y `β|¯3{2y, Figure 1.8 illustrates the spectra obtained with this method for various This method provides a very good description of the ground state wave function, but information about the symmetry and energy of the higher energetic levels is missing. In principle, a temperature dependence of the XAS due to thermal population of excited states may provide an estimate of the energy of higher levels if the splittings are small enough. Due to the only limited information provided by XAS there is usually an uncertainty on the CEF scheme left and other measurements, such as INS, are needed for a complete description. Some difficulties may also arise from the fact that the mixed valence of the Ce ion leads to an extra spectral component originating from the Ce 4`c ontribution, partly superimposed to the Ce 3`, that needs to be considered separately to the dichroic effects [START_REF] Sundermann | CeRu 4 Sn 6 : a strongly correlated material with nontrivial topology[END_REF][START_REF] Thole | Strong magnetic dichroism predicted in the M 4,5 X-ray absorption spectra of magnetic rare-earth materials[END_REF]. Moreover, to study the ground state, experiments must be performed at low temperatures k B T ! ∆E CEF to avoid the thermal population of higher levels, but at these low temperatures the sample may be in an antiferromagnetic phase. The magnetic ordering is another cause of dichroism in the absorption spectra [START_REF] Chen | Exchange field effect in the crystal-field ground state of CeM Al 4 Si 2[END_REF][START_REF] Thole | Strong magnetic dichroism predicted in the M 4,5 X-ray absorption spectra of magnetic rare-earth materials[END_REF] and it needs to be identified apart from the ground state mixing effects.

Lastly, if the XAS signal is detected as TEY, due to the short mean free path of electrons in condensed matter, the signal is given by a portion of the sample close to the surface, which must be accurately prepared in order to avoid artefacts. Figure 1.10: Photoemission spectroscopy from the 4f subshell of lanthanides. If the 4f levels are occupied, as in the case of Yb 3`, the CEF splittings are observable with direct PES, each peak corresponding to a level. This is not possible on Ce 3`, because only the ground state is populated. In Resonant PES, a core electron is resonantly excited into the 4f subshell and the super Coster-Kronig decay from the 4f levels is observed. The emitted electrons will have different energies depending on the 4f levels occupied in the intermediate state, which allows to determine the splitting between the levels.

A final remark shall be made on the techniques presented up to now, in particular the most commonly used: magnetic susceptibility, INS and XAS. In their usual formulation, they all rely on dipole excitations and are therefore insensitive to symmetries higher than twofold. There are, in tetragonal systems, two possibilities for the orientation of the CEF wavefunctions, distinguished by a 45 0 rotation about the crystal c axis. The wavefunction orientation is of relevant importance for the study of the hybridization effects with the neighboring atoms. It does not affect the splitting, the symmetry or the mixing α 2 , therefore it is not detectable with any of the techniques only sensitive to these quantities. Recently, core-level photoemission [START_REF] Mori | Probing strongly correlated 4f -orbital symmetry of the ground state in Yb compounds by linear dichroism in core-level photoemission[END_REF] and Non Resonant X-ray Scattering [START_REF] Willers | Determining the in-plane orientation of the ground-state orbital of CeCu 2 Si 2[END_REF] demonstrated their sensitivity to the in-plane orientation of the 4f orbitals. We will show that the same information can be obtained easily with Resonant Inelastic X-ray Scattering that, as a second order process, is not limited to dipole excitations only.

Photoemission spectrocopy

One of the most used experimental techniques to gain information about the electronic states in matter is photoemission spectroscopy (PES), which analyses the energy (and momentum) of the electrons emitted by a sample exposed to monochromatic UV or X-ray radiation. In PES is successfully used to study the splitting and hybridisation-induced dispersion of 4f CEF-levels, as long as they are populated so that electrons can be photoemitted. This is in general the case of heavier lanthanides as europium or ytterbium [START_REF] Höppner | Interplay of Dirac fermions and heavy quasiparticles in solids[END_REF][START_REF] Patil | Crystalline electric field splitting of 4f states in YbIr 2 Si 2 : An ARPES view[END_REF] (Figure 1.10, left panel). Cerium, on the other hand, has at most one electron in the 4f shell and photoemission from the unoccupied CEF-levels is not possible. It is in principle possible to thermally populate the higher energy CEF-levels and to observe other CEF-related features [START_REF] Ehm | High-resolution photoemission study on low-T K Ce systems: Kondo resonance, crystal field structures, and their temperature dependence[END_REF], but this does not provide the full CEF scheme and comes with several technical difficulties. The study of empty levels is usually carried out via Inverse Photoemission Spectroscopy (IPES) [START_REF] Weibel | Resonant inverse photoemission: A new probe of correlated systems[END_REF], but the resolution (" 0.1 eV) is not good enough to study the CEF splitting.

Recently, Resonant Photoemission Spectrocopy was used to probe the splitting and dispersion of CEF levels in cerium compounds [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF]. By tuning the photon energy to the M 4,5

or N 4,5 edges, 3d or 4d core level electrons are excited into the valence subshell (4f ), generating a core hole. If at least another electron is present in the valence subshell, a super Coster-Kronig Auger decay is possible. The core hole is filled by a 4f electron and the released energy is transferred to another 4f electron, which is emitted (Figure 1.10, right).

A typical drawback of PES-RPES is the surface sensitivity: it is discussed in Ref. [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF] that not only the surface quality, but also which atom terminates the surface after the cleaving (Ce or Si in the case of CeRh 2 Si 2 ) influences the features observable in the spectrum.

Moreover, the enhanced surface sensitivity, caused by the short inelastic mean free path of electrons, may be a limitation to draw conclusions about the CEF effects in the bulk. Lastly, it should be stressed that in the photoemission process, one electron is removed from the ion and its energy is measured only afterwards. During the photoemission, the other electrons may reorganize to screen the generated hole, thus emitting the photoelectron with a slightly different energy. The measured value is therefore the energy of the electronic level without the electron, that can be different from the actual energy of an electron in that level.

Scope of this thesis

The strongly interacting 4f CEF-split levels are the origin of the manifold of exotic phenomena arising in rare earth intermetallic compounds, as discussed in section 1.1. This makes the study of the crystalline field effects in these compounds of primary importance, so that many techniques have been developed during the years to investigate the rare earths' 4f levels, the most relevant of which were discussed in section 1.3. Drawbacks and limitations are present in every technique and only a partial information about the CEF can in general be obtained.

In this thesis work, Resonant Inelastic soft X-ray Scattering (RIXS) is presented for the first time as a technique able to probe directly 4f CEF levels' splittings and symmetries, with bulk sensitivity and free of a complex background. A RIXS investigation of 4f levels became possible only in the last two years, with the inauguration of the ID32 beamline at ESRF and its high resolution RIXS spectrometer, first of a new generation of instruments that are being built in many facilities around the world. In chapter 2 the possible RIXS processes in 4f systems are discussed and data obtained with 250 meV resolution using the predecessor of the current spectrometer is presented. This data already shows all the possible features visible in a rare-earth RIXS spectrum, including first hints on CEF effects.

Chapter 3 describes the instrumentation for high resolution soft-RIXS experiments, focusing on the details and innovations of ID32 and its ERIXS spectrometer. In chapter 4 CeRh 2 Si 2 is used as an example to demonstrate how information about the splitting, symmetry and orientation of the states can be obtained from the spectra. RIXS is also sensitive to the dispersion of 4f levels and this will be shown in chapter 5, where a simple model for the interpretation of the data will be also provided. Finally, chapter 6 deals with the limits of the presented method. CeCu 2 Si 2 spectra will be used to show an analysis of the CEF in a system with splittings smaller than the instrumentation's resolution, while CeCo 2 Ge 2 will represent a case with very large splittings, where the usual hypothesis and approximations made in CEF analysis are not valid. Finally, in the Summary and outlook the most important results will be summarized and the possible future of the use of RIXS on rare earth compounds is outlined.

Chapter 2

Resonant inelastic soft X-ray scattering from 4f states Resonant Inelastic X-ray scattering (RIXS) is a synchrotron-based spectrocopic technique able to directly probe elementary excitations in a large variety of systems [START_REF] Ament | Theoretical demonstration of how the dispersion of magnetic excitations in cuprate compounds can be determined using resonant inelastic X-ray scattering[END_REF][START_REF] Glatzel | Resonant X-ray spectroscopy to study K absorption pre-edges in 3d transition metal compounds[END_REF][START_REF] Matsubara | Polarization dependence of resonant X-ray emission spectra in early transition metal compounds[END_REF][START_REF] Ament | Resonant inelastic X-ray scattering studies of elementary excitations[END_REF][START_REF] Singh | Advanced X-ray absorption and emission spectroscopy: in situ catalytic studies[END_REF][START_REF] Kotani | Resonant inelastic X-ray scattering spectra for electrons in solids[END_REF][START_REF] Chiuzbȃian | Localized electronic excitations in NiO studied with resonant inelastic X-ray scattering at the Ni M threshold: Evidence of spin flip[END_REF][START_REF] Ghiringhelli | Low energy electronic excitations in the layered cuprates studied by copper L 3 Resonant Inelastic X-Ray Scattering[END_REF][START_REF] Ament | Resonant inelastic X-ray scattering on spin-orbit coupled insulating iridates[END_REF]. In this chapter the basic concepts for the understanding of RIXS spectrocopy are presented, paying particular attention to its applications for the study of rare earth intermetallic compounds at the M 4,5 edge. The scattering process is described, starting from a basic description and then deriving and discussing in detail the formula for the RIXS cross section. The peculiar properties that make RIXS so powerful and suitable for the study of strongly correlated systems are then listed and commented on, together with the possible excitations accessible by this technique.

Results of a first, exploratory RIXS study of cerium intermetallics will be presented, showing the typical features observable. The data, acquired on ID08, the former soft-RIXS beamline of ESRF, also serve to depict the state of the art of M 4,5 RIXS before the instrumentation upgrades of the last years, setting the starting point for this thesis work.

Fundamentals of resonant inelastic X-ray scattering

RIXS is a photon in -photon out process in which a photon is scattered off a sample leaving behind an excitation. The energy of the incident photon is set to an X-ray absorption resonance of the sample, largely increasing the scattering cross section and ensuring chemical sensitivity. Depending on the details of the light-matter interaction, the sample may be left in different excited states which can be characterised by looking at the changes in the photon's energy, momentum and/or polarisation. By comparing the photon's properties Figure 2.1: Simplified scheme of an M 5 RIXS process on Ce 3`, leading to a f f excitation between CEF-split levels which can be detected as an energy loss of the scattered photon. The peaks in the photon energy loss spectrum (right) directly correspond to the different CEF-split 4f levels that can be occupied by the 4f electron in the final state. before and after the interaction, information can be retrieved about the sample's ground and excited states.

The whole interaction can be, in a simplified approach, interpreted as a two-step process. First, the photon is resonantly absorbed by a core electron, which is promoted to an empty electronic level (A), leaving a deep core hole. The sample is thus in a highly energetic intermediate state which decays within a few femtoseconds via either an Auger process (as in RPES) or a radiative transition. In the latter case, an electron from a valence level (B ) recombines with the core hole, emitting a second X-ray photon. If the electronic levels A and B differ, a net electronic transition from A to B took place in the sample, as depicted in Figure 2.1 in the case of a 4f Ñ 4f (or f f ) transitions in Ce 3`. Other kinds of excitations, like magnons or phonons, and charge transfer may occur, depending on many possible phenomena taking place in the highly excited intermediate state. It must be stressed that the given scheme, although it provides an easy interpretation of the peaks of a RIXS spectrum, is missing an important detail. The highly excited intermediate state has a linewidth broadening of " 1 eV, large if compared with the splittings among the different possible intermediate states. Many intermediate states can therefore contribute to the scattering process and must be considered at the same time.

In order to avoid the thermal population of the excited states, RIXS experiments are usually performed at low temperatures. In this way, before the interaction, the sample lies in its ground state, which constitutes the reference energy for all the RIXS transitions.

Information about the excitations occurring in the sample can be obtained by imposing the energy and momentum conservation of the whole system, made of the photons and the scatterer:

ω i " ω o `Eexc , k i " k o ` q exc , (2.1)
where ω i ( ω o ) is the energy of the incident (outgoing) photon and k i ( k o ) is the corresponding momentum. Moreover a third conservation law applies to the angular momentum, regarding the polarisation of the ingoing ( i ) and outgoing ( o ) photons, as well as the total angular momentum of the sample. If the energy, momentum and polarisation of both the ingoing and outgoing photons are known, the corresponding properties of the excitation can be retrieved.

In a RIXS experiment the energy ω i and polarisation i of the incident photons are controlled to a high degree, thanks to the use of synchrotron beamlines, equipped with undulators and monochromators. So are the absolute values of the momenta, since |k i | "

ω i {c and |k o | " ω o {c « ω i {c,
where it is used the fact that E exc ! ω i and therefore

ω i « ω o .
The directions of the momenta are imposed by the geometry of the experiment, i.e. by the angles defined by the beamline, the sample and the spectrometer, which selects the photons scattered in a certain solid angle. The parameters to be determined in order to have a full description of the process, and of the excitations, are therefore the energy and polarisation of the scattered photons. These can be measured with a spectrometer and a polarimeter respectively.

A last observation can be made before facing the details of the calculation of the RIXS cross section. RIXS was described as consisting of two radiative transitions, absorption and emission, each one of which is regulated by the dipole selection rules. The total RIXS transition, however, does not follow these rules. f f excitations provide a typical example: the transition 4f Ñ 4f would be forbidden by dipole, since ∆l " 0, but the two steps 3d Ñ 4f (∆l " `1) and 4f Ñ 3d (∆l " ´1) are allowed, making the whole transition possible. RIXS can therefore access transitions forbidden by dipole selection rules.

The Kramers-Heisenberg cross section

The RIXS cross section can be derived by considering the Hamiltonian of the system and calculating the transition probabilities using perturbation theory [START_REF] Ament | Resonant inelastic X-ray scattering studies of elementary excitations[END_REF][START_REF] Sala | Magnetic and orbital resonant inelastic X-ray scattering[END_REF]. Quantum electrodynamics is used to describe the interaction between the electrons and the photons' electromagnetic field, the latter being represented by a vector potential Apr, tq (so that

Eprq " ´∇φ ´BA{Bt and B " ∇ ˆA, where φ is a continuously differentiable scalar function). In the non-relativitic case and in the limit that the potentials related to both the electrons and the photons in the system are small compared to twice the mass of the electron (eφ{2mc 2 , e|A|{2mc ! 1), the Hamiltonian for a system with N electrons interacting with an electromagnetic field is written (in SI units) as:

H " N ÿ j ˆrp j `eApr j qs 2 2m
`e 2m σ j ¨Bpr j q `e 2p2mcq 2 σ j ¨tEpr j q ˆrp j `eApr j qs ´rp j `eAs ˆEpr j qu

ḢCoulomb `ż dk ÿ ω k ˆc: ,k c ,k `1 2 ˙, (2.2)
where p j , r j and σ j are the momentum and position operators and the Pauli matrices acting on the electron j, respectively. c p:q ,k annihilates (creates) a photon with wave-vector k and polarisation . The first term of H is the kinetic energy of the electrons in presence of an electromagnetic field, the second yields the Zeeman splitting and the third takes into account the spin-orbit coupling. H Coulomb contains the remaining electrostatic interactions of electrons with the charges in the sample (among which, the potential of the nucleus, the electron-electron interaction and crystal field). The last term is the energy of the radiation summed on all the modes, each mode having energy ω k .

To derive the photon scattering cross section, it is convenient to split the Hamiltonian H in an electron-photon interaction part H 1 and the remaining terms H 0 " H el `Hem , which describe the electrons and and photons in absence of the interaction. H 1 can then be treated as a perturbation of H 0 . Since H 0 does not mix the electron and photon terms, its eigenfunctions are written as product of those of H el and H em .

Let's consider a single photon in the initial state, with momentum k i , energy ω i and polarisation i , that is scattered to k o , ω o , o . In the process, the sample changes its state from the ground |gy to a final |f y one. The total wavefunctions of the system can thus be written as |gy " |g; 1 i,ki y and |fy " |f ; 1 o,ko y, where |1 o ,ko y is the one-photon wavefunction, result of the application of the creation operator c ,k on the vacuum state |0y. The energy corresponding to the |gy and |fy states are, respectively, E g " E g ` ω i and

E f " E f ` ω o .
The transition rate w for the scattering process can be written, in the second order perturbation theory, as

w " 2π ÿ f ˇˇˇx f|H 1 |gy `ÿ n |xf|H 1 |ny|xn|H 1 |gy E g ´En ˇˇˇ2 ¨δpE f ´Eg q , ( 2.3)
where n runs over the intermediate states |ny, eigenstates of the unperturbed Hamiltonian H 0 with energy E n . In a general (non resonant) scattering event, the first order amplitude dominates the second order, which is usually neglected. In the case of RIXS, instead, the incoming X-ray energy matches an absorption edge, therefore, for some |ny, E g « E n , making the second term very large.

Within the limits considered here (non-relativistic electrons in small potentials), the resonant part of the second order amplitude can be written as

w " e 2 2m 2 V 0 ? ω ki ω ko ÿ n N ÿ j,j 1 xf | o ¨pj 1 e ´iko¨r j 1 |nyxn| i ¨pj e ´iki¨rj |gy E g ` ω ki ´En `iΓ n . (2.4)
V is the volume of the system and a broadening Γ n is introduced in the denominator to account for the short lifetime (" 1 fs) of the intermediate states. Considering (2.4) as the second order term in (2.3), and neglecting the first order term, one obtains the Kramers-Heisenberg expression for the RIXS cross section.

The formula can be further simplified by means of the dipole approximation, that holds true for soft X-ray photons interacting with atomic electrons. e ik¨r can indeed be considered to be approximately constant at the length scale of the atomic wavefunctions, therefore e ik¨r " 1 `ik ¨r `... « 1. The scattering amplitude becomes

w " e 2 2m 2 V 0 ? ω ki ω ko ÿ n N ÿ j,j 1 xf | o ¨pj 1 |nyxn| i ¨pj |gy E g ` ω ki ´En `iΓ n « « e 2 2m 2 V 0 ? ω ki ω ko ÿ n N ÿ j,j 1 xf | o ¨rj 1 |nyxn| i ¨rj |gy E g ` ω ki ´En `iΓ n , (2.5)
where the operator p was substituted with the commutator imrH 0 , rs{ and the equality xn| i ¨rH 0 rs|gy " xn|H 0 i ¨r|gy ´xn| i ¨H0 r|gy " pE n ´Eg qxn| i ¨r|gy « ω ki xn| i ¨r|gy was used. Finally, one can obtain the double differential cross section by multiplying w by the density of (outgoing) photons' states in the solid angle dΩ (that is

V k 2 o d|k o |dΩ{p2πq 3 " V ω 2
ko d ω ko dΩ{ c 3 p2πq 3 ) and dividing by the photon flux c{V . The RIXS cross section results:

d 2 σ d ωdΩ " r 2 e m 2 ω k i ω 3 ko ÿ f ˇˇˇÿ n N ÿ j,j 1 xf | o ¨rj 1 |nyxn| i ¨rj |gy E g ` ω ki ´En `iΓ n ˇˇˇ2 δpE g ´Ef ` ωq , (2.6)
where r e " e 2 {p4π 0 mc 2 q is the classical electron radius and ω " ω ki ´ ω ko .

The δ function assures the energy conservation, so that the difference of the photon energies equals the excitation energy of the system. The RIXS excitations will be therefore observed at constant energy losses. This distinguishes proper RIXS features from fluorescence that occurs at fixed ω ko . Therefore the energy loss ω " ω ki ´ ω ko increases linearly with the energy of the incident photons. These features are in general given by the radiative decay of an electron from shells different to those reached by the excited electron in the intermediate state. The information about ω ki is lost and ω ko only depends on the ∆E between the atomic levels involved in the emission of the photon.

It shall be noted that the cross section formula contains two dipole operators and both transitions must be allowed in order to go from the state |gy to |f y. The cross section formula correctly reproduces also the interference among the possible excitations paths discussed in the previous section, because the sum over the intermediate states n is carried out before the absolute square. The sum over all the possible intermediate states might lead to very demanding computations in complex systems, when one wants to simulate a RIXS spectrum. Software programs such as Quanty or SolidStatePhysics overcome this problem by introducing Green's functions for the description of the scattering process, that make the calculations much faster.

Lastly, it must be stressed that the resonance acts as an amplifier increasing remarkably the cross section of the process and enhancing the sensitivity of the measurements, which would be otherwise impossible, due to the very small numerators in the formula. Despite this, RIXS is still a very photon hungry technique, and sources with very high brilliance, as well as sensitive instrumentation and detectors are necessary to perform experiments.

Strengths of the technique

Many characteristics of the scattering process described above make RIXS a powerful and unique technique for studying elementary excitations in condensed matter. The most important aspects are shortly reviewed here, paying particular attention to the features that allow to provide complementary information to, and overcome the drawbacks of, the established techniques used to investigate 4f states in rare earths. First of all it is clear from the aforementioned conservation laws that RIXS can simultaneously probe the energy and but this is usually not exploited for f f excitations. The strong q dependence of the cross sections for phonon and magnetic scattering, indeed, causes the magnetic (f f ) contribution to be lost in the phonon background at high scattering angles. Furthermore, INS experiments are often performed on powdered samples, therefore averaging q on all the possible orientations of the crystal.

The use of powders is dictated in INS by the need of large samples (« grams, i.e. cm 3 sizes), difficult to grow as single crystals, in order to compensate for the small scattering cross sections. This is obviated in RIXS by the use of resonance, that amplifies the signal and allows to measure sub-millimeter size single crystals or thin films.

The resonant nature of RIXS also provides element sensitivity. This becomes very important when one wants to focus on excitations related to a specific subshell of a sample's constituent, as is the case of f f excitations. This holds true also for phonons and magnons, that are visible in a RIXS spectrum only if they are somehow coupled with the electronic levels involved in the RIXS process. The large and complex phonon background present in INS is therefore greatly, if not completely, suppressed.

Chemical sensitivity is also guaranteed in RIXS spectroscopy. Different oxidation states (or valences) of the same element have slightly shifted absorption edges and one can tune the incident photon energy to the desired one. This becomes helpful in the study of mixed valence compounds, where only a specific valence can be selected, neglecting the mixing (Figure 2.2). In contrast, the valence mixing is directly visible in a XAS spectrum, and it must be considered to correctly analyse the data.

The RIXS process is also charge neutral, meaning that no electron is added to or removed from the sample during the process. In the case of f f excitations, one also does not change the occupancy of the 4f shell. This guarantees that the energy splittings measured in a RIXS spectrum are free of any screening effect, in contrast to e.g. photoemission. Another important feature that distinguishes RIXS from PES is the bulk sensitivity: soft X-rays can travel tens of nanometers inside the sample, making the surface contribution negligible compared to the rest of the volume probed. If desired, the surface contribution can be enhanced by selecting a scattering geometry in grazing incidence. The quality of the surface is however important for the study of features very close (or superimposed) to the peak at zero energy loss, because the roughness of the surface contributes to the diffuse (elastic) scattering. Lastly, the possibility of investigating the polarisation dependence of each feature provides further information on the nature of each excitation.

Excitations accessible to RIXS

The RIXS process may leave several types of excitations in the sample, depending on the selected absorption edge and the properties of the material. The most important features are outlined here, while a more complete review can be found in Refs. [START_REF] Ament | Resonant inelastic X-ray scattering studies of elementary excitations[END_REF][START_REF] Kotani | Resonant inelastic X-ray scattering spectra for electrons in solids[END_REF].

Orbital excitations. If the valence subshell involved in the RIXS process is partially filled (i.e. other electrons than the photoexcited one can decay to fill the core hole) an orbital excitation can occur; this is the case of f f excitations depicted in Figure 2.1. 3d orbital (dd) excitations have been widely studied, because of the very large CEF splittings (ą 1 eV) that made them detectable with the much lower resolution of the first RIXS spectrometers.

These excitations usually provide local information about the ionic orbitals. In some special cases, degenerate orbitals can be present and these may couple with the neighboring sites by superexchange, allowing collective orbital excitations, called orbitons. It must be stressed that in a RIXS spectrum the peaks due to orbital excitations do not necessarily correspond to the one-electron levels as depicted in Figure 2.1. This happens only when one single electron (or hole) occupies the valence subshell as in Ce 3`( 4f 1 ) and Cu 2`( 3d 9 ). In other ions, as Eu 3`( Figure 2.3), the electronic interactions within the electrons in the valence subshell give rise to a complex multiplet structure, resulting in a manifold of peaks.

RIXS is also sensitive to many other elementary excitations, due to various phenomena that might take place in the intermediate excited state.

Collective magnetic excitations. In materials with magnetic correlations, collective magnetic excitations are possible. In general, photons carry an angular momentum L ph " 1 that can be exchanged during the interaction, meaning that a RIXS process can transfer ∆L ph z " 0, 1, 2 to the sample. In the example of 3d materials, where the magnetism is related to the spin of valence electrons, the transferred angular momentum may rotate one or two spins (∆S z " 1 or ∆S z " 2), giving rise to magnons or bi-magnons. The rotation of the spin occurs in the intermediate state, in which the spin of the 2p core hole, due to the strong spin-orbit coupling, is not a good quantum number and it is allowed to change.

Since magnetic excitations imply a change of the angular momentum (i.e. the polarisation) of the photon, they are usually well recognisable via a polarisation analysis.

Phonons. Due to the extra valence electron, the intermediate state presents a local valence charge disequilibrium that might affect the bondings with the neighbouring atoms, eventually generating lattice vibrations, i.e. phonons. These have been observed at the L and K edges [START_REF] Braicovich | Magnetic excitations and phase separation in the underdoped La 2´x Sr x CuO 4 superconductor measured by resonant inelastic X-ray scattering[END_REF][START_REF] Yavaş | Observation of phonons with resonant inelastic X-ray scattering[END_REF]. The electron-phonon coupling is anyway small for 4f orbitals and a phonon contribution in M edge RIXS spectra has neither been predicted nor observed by experiments.

Charge transfer. The presence of the core hole in the intermediate states can also cause rearrangement of the ligands' charge in order to screen it, yielding a charge transfer from the ligand sites. The process is usually interpreted as a transfer of a ligand (or valence band) electron to the excited ion due to a stronger hybridisation taking place in the intermediate state. Charge transfer excitations (E Loss " 1 to 10 eV) can provide much information about the hopping processes and the on-site repulsion.

Fluorescence. Coster-Kronig conversion and other fluorescence features may also be visible in RIXS spectra, but they lie at much higher energies (" 20 eV) and are not usually studied with RIXS. They will be anyway shown in the next section in the specific case of cerium compounds.

Elastic scattering. Lastly, the peak at zero energy loss, even if it does not represent any excitation, must be mentioned in the list of RIXS spectral features. It is due to the photons that don't undergo any inelastic process during the scattering, therefore it is referred to as elastic peak. Most of the elastic intensity comes from diffuse scattering from the sample's surface and is particularly intense in specular geometry. It can be reduced by going offspecular and/or choosing opportune experimental conditions so that i ¨ o " 0. Elastic scattering can bring information about orderings present in the sample. Charge density waves and stripes have been found on layered cuprates [START_REF] Ghiringhelli | Long-range incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x[END_REF] thanks to the study of the elastic peak in RIXS spectra. These features might be difficult to observe in conventional diffraction experiments without energy discrimination because they produce a small modulation of the elastic intensity, that is lost in the background of the inelastic features. RIXS spectroscopy can separate the inelastic and the elastic contributions, i.e. it can be used as a very sensitive energy resolved diffraction. 2.2 M 4,5 resonant inelastic X-ray scattering on cerium: a first, exploratory study Before the advent of ERIXS, the high resolution spectrometer installed on ID32 at ESRF, the energy resolution of soft-RIXS instrumentation (" 125 meV for SAXES, at Swiss Light Source), was too low for resolving the f f excitations (" 30 ´60 meV) in cerium. In the following, data acquired in 2013 with the spectrometer AXES on ID08 beamline at ESRF (" 250 meV resolution) are presented, in order to review, with the help of an example, all the possible features that were already visible in RIXS spectra of cerium compounds before the latest upgrades. These data have been published in Ref. [START_REF] Amorese | 4f excitations in Ce Kondo lattices studied by resonant inelastic X-ray scattering[END_REF] as part of this thesis work. A uses CEF parameters obtained from INS [START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF], with a Kramer's doublet ground state and a quartet at 29 meV (0-29-29 meV splitting scheme), while B1 and B2 are obtained by sligtly modifying the CEF (respectively giving splitting of 0-29-34 meV and 0-33-41 meV). The three schemes provide a similar fit of the magnetic susceptibility curves (see Ref. [START_REF] Amorese | 4f excitations in Ce Kondo lattices studied by resonant inelastic X-ray scattering[END_REF] for details). Even with 250 meV resolution, small variations of the CEF have visible effects on the spectra. decay of the excited electron back into the core hole. The emitted photon has the same energy of the absorbed one, giving only an elastic peak at zero energy loss. On the other hand, CeCu 2 Si 2 and CeRu 2 P 2 spectra have a two-peak structure, due to the presence of a 4f electron that makes f f excitations possible. The first peak, around zero energy loss, contains all transitions within the 2 F 5{2 multiplet, included the elastic peak, while the other one, at about 300 meV, consists in all transitions to a 2 F 7{2 final state. As expected, the inelastic intensity is higher on resonance. Figure 2.5 compares the spectra of the three compounds. CeCu 2 Si 2 and CeRu 2 P 2 spectra cannot be fit by calculations without the CEF effects, implying that RIXS is sensitive to them. Moreover, the two spectra present clear differences, ascribable to different CEF schemes. The right panel of Figure 2.5 show attempts of fitting the CeCu 2 Si 2 spectrum with calculated spectra, using CEF parameters from Ref.

First evidence of 4f excitations

[74] (A) or alternative parameters obtained by the fit of magnetic susceptibility data (B1, B2). It is clear that, due mostly to changes in the peaks' intensities in the three cases, the three slightly different CEF schemes provide distinct spectra. In comparison, the blue curve (B2) best fits the energy position and the intensity ratio of the experimental data.

The corresponding CEF parameters yield an energy of 33 and 41 meV for the lowest two excited CEF levels, in contrast to the 29 meV proposed in Ref [START_REF] Goremychkin | Crystal field effects in RCu 2 2Si 2 compounds (R= Ce, Pr, Nd, Ho, Er): Inelastic neutron scattering investigation[END_REF], and represented by the red curve A1.

Even though a complete determination of the CEF scheme was impossible at the time, due to the lack of resolution, this was the first proof of the ability of RIXS to probe excitations within the 4f subshell.

Charge transfer excitations

Charge transfer (CT) processes are a direct consequence of the hybridisation of 4f states with the ligand orbitals and involve the hopping of a ligand electron into the 4f subshell.

Usually CT excitations are in the energy range from 1 to 10 eV and they could therefore be studied even with the low resolution of the early days of RIXS.

The basic process was theorized for the (mostly 4f 0 ) mixed valence compound CeO 2 by Nakazawa et al. in Refs. [START_REF] Nakazawa | Theory of resonant X-ray emission spectra in Ce compounds[END_REF][START_REF] Nakazawa | Theory of polarization dependence in resonant X-ray emission spectroscopy of Ce compounds[END_REF] and later confirmed by experiments of Watanabe et al. [START_REF] Watanabe | Polarization dependence of resonant soft X-ray emission spectra in Ce compounds[END_REF]. The CT RIXS process is depicted in Figure 2.6. The initial state consists in an admixture of 4f 0 and 4f 1 v configurations, where v represents a valence hole. The admixture generates a bonding ground state, a nonbonding state located 4.5 eV higher in energy and an antibonding state at 7 eV (4 eV and 6 eV respectively according to measurements in Ref. [START_REF] Watanabe | Polarization dependence of resonant soft X-ray emission spectra in Ce compounds[END_REF]). Similarly, bonding, nonbonding and antibonding states are present in the intermediate state after the 3d Ñ 4f excitation, given by the admixture of 3d 9 4f 1 and 3d 9 4f 2 v. The presence of the core hole brings the two configurations closer in energy in the intermediate state, resulting in a stronger hybridisation and thus allowing the charge transfer during the RIXS process. When the photon energy is tuned to the main peaks of the M 4,5 XAS, the bonding intermediate state is selected and a transition to the nonbonding final state is possible, in addition to bonding final state (which includes the elastic peak and f f excitations, if possible). A transition to the antibonding state is also possible, but 4 eV loss, together with a weaker signal around 6 eV, while only the 6 eV feature is visible when the incident photon energy is tuned to the XAS satellite. Both excitations seem to move towards higher energy losses with increasing photon energy. This happens because, by changing the photon energy, it is possible to span the finite bandwidths of the intermediate and final states, always finding a path that allows the CT transition. The transition to the nonbonding final state at 4 eV is independent on the incident photons' polarisation, while the transition to the antibonding state at 6 eV seems enhanced with vertical polarised photons (Figure 2.7, top right). This further confirms the calculations [START_REF] Nakazawa | Theory of polarization dependence in resonant X-ray emission spectroscopy of Ce compounds[END_REF], which predicts a weaker signal on the antibonding peak with horizontal polarised light. In order to interpret the RIXS map of CeCu 2 Si 2 and CeRu 2 P 2 an extension to the scheme of Figure 2.6 is needed. A suitable model was developed by Nakazawa et al. [START_REF] Nakazawa | Theoretical study on the effect of electron-hole pair excitations in resonant X-ray emission spectroscopy for Ce intermetallic compounds[END_REF] for the description of charge transfers in CeRh 3 . Both the metallic nature of the compounds and the higher 4f population (n f " 0.8 in their calculations) are taken into account, providing a better description also for cerium compounds of the 122 family. The basis set includes 4f 0 , 4f 1 v, 4f 2 v 2 , 4f 0 cv, 4f 1 cv 2 and 4f 2 cv 3 configurations and more excitations become therefore possible. By using this model, it is still possible to assign in Figure 2.7 a peak to the transition to the nonbonding state (at 1 eV for CeCu 2 Si 2 and 1.5 eV for CeRu 2 P 2 ), which overlaps with the low energy features, and the intense peak on the XAS satellite to the antibonding state (2.2 and 2.8 eV respectively). Other weak features are attributed to e-h pairs and f 2 excitations. It must be noticed that the energy required for a CT excitation diminishes from NCCO to CeRu 2 P 2 and again to CeCu 2 Si 2 . Following the argument of Dallera et al. [START_REF] Dallera | Charge-transfer excitations in lanthanum compounds measured by resonant inelastic X-ray scattering at the M 5 edge[END_REF], this effect can be related to the electronegativity of the atoms surounding the rare earth ion. This could also explain the results of Watanabe et al., that found lower energies for CT excitations for CeRh 3 compared to CeO 2 and to calculations in Ref. [START_REF] Nakazawa | Theoretical study on the effect of electron-hole pair excitations in resonant X-ray emission spectroscopy for Ce intermetallic compounds[END_REF]. The presence of the CT excitations at low energies, partly overlapping with the f f signal, implies that the valence band and the 4f levels are energetically very close, giving rise to the possibility of hybridisation.

To complete the description of the CT excitations in cerium intermetallic compounds,

Coster-Kronig conversion and fluorescence

Other features at higher energy losses are visible in RIXS spectra of cerium compounds.

Strictly speaking, these features are not related to a RIXS process, because different electronic levels are involved in the absorption and emission steps. The information on the intermediate state, and on the energy of the absorbed photon, can be in general lost in this kind of processes. If this is the case, the energy of the emitted photon is constant and only dependent on the energy of the levels involved in the emission process. These excitations are visible as lines with slope 1 in the RIXS map E in vs. E Loss " E in ´Eout " E in ´constant and are generally referred to as fluorescence. In both the M 4 and M 5 RIXS maps of CeCu 2 Si 2 (Figure 2.9) a peak is present around 22 eV energy loss. In agreement with the experimental observations of Dallera et al [START_REF] Dallera | Charge-transfer excitations in lanthanum compounds measured by resonant inelastic X-ray scattering at the M 5 edge[END_REF] it can be assigned to a 5p Ñ 3d radiative deexcitation. As measured by Lenth et al. [START_REF] Lenth | Giant resonance enhancement of the 4f , 5p, and 5s states in the photoemission spectra of rare-earth compounds[END_REF], the 5p subshell lies 22 eV lower in energy than the 4f , therefore a 5p Ñ 3d deexcitation generates a photon 22 eV less energetic than the one absorbed in the 3d Ñ 4f transition (Figure 2.10, left). The energy loss is 2 eV higher in the M 4 RIXS maps, if compared to the M 5 RIXS map with horizontal polarised photons. The data at the M 5 edge with vertical polarised light, on the other hand, show a broad peak from 21 to 23 eV energy loss. Multiple losses are indeed possible, due to the combination of 5p Ñ 3d emission and CT excitations, that can leave the system in the final state in the 5p4f n`2 v configuration, different from the pure 5p emission final configuration 5p4f n`1 [START_REF] Tanaka | Theory of the resonant 5p Ñ 3d X-ray emission spectra in CeO 2[END_REF][START_REF] Dallera | Charge-transfer excitations in lanthanum compounds measured by resonant inelastic X-ray scattering at the M 5 edge[END_REF]. The various possible CT processes are not well resolved in the presented maps because of the poor statistics and the strong smoothing applied in order to make these weak features appear out of the noise. In comparison, the 5p Ñ 3d emission peaks are strong around the main XAS peaks, but disappear quickly with higher photon energies, probably . because other deexcitation processes become more probable. This does not allow to probe the fluorescent behaviour of the process. In the small range of E in for which it is visible, the peak seems indeed to appear at constant energy loss, instead of constant E out .

Another feature is visible at similar energy losses, but only in the M 4 RIXS maps, implying that the corresponding deexcitation path is possible only in presence of a 3d 3{2 hole. This is the case of the MMN Coster-Kronig process (Figure 2.10, right). In the intermediate state, an electron deexcites from the 3d 5{2 to the 3d 3{2 subshell in order to fill the hole, and its energy is transferred to a valence electron, which is photoemitted. A radiative deexcitation then occurs from the valence to the 3d 5{2 hole left by the Coster-Kronig conversion. The emitted photon will differ from the absorbed one in the 3d 3{2 Ñ 4f transition by the distance in energy between the 3d 3{2 and 3d 5{2 subshells. This can be measured as the distance between the M 4 and M 5 edges in a XAS spectrum, about 17 eV. Interestingly, the energy loss of this feature increases with increasing incident photon energy, i.e. is a fluorescence process. This means that during the Coster-Kronig process the information on the absorption and on the intermediate state is lost and the deexcitation takes place from a relaxed state. The presence of the extra Coster-Kronig deexcitation path causes 3d 3{2 holes to have a shorter lifetime and this results in a larger lifetime broadening of M 4 XAS spectra compared to the M 5 .

Chapter 3

Advances in soft X-ray RIXS instrumentation at the ESRF During the last 20 years, the increasing interests in RIXS, driven by its versatility and ability to probe elementary excitations, has boosted a continuous technical improvement of the instrumentation. In this chapter, the ID32 beamline of European Synchrotron Radiation Facility (ESRF) and its spectrometer, which represent the state-of-the-art in instrumentation for soft-RIXS spectroscopy, are reviewed.

The basic requirements to perform RIXS spectroscopy are a monochromatic incident Xray beam, an analyser of the scattered photons' energy and a detector. The X-ray photons are provided by synchrotron radiation and are monochromatized along the beamline, while a dispersive spectrometer is usually employed for the energy analysis. On ID32, innovations and extensions to this basic scheme have been introduced. The continuous rotation of the spectrometer and a 6-degrees-of-freedom control of the sample position give a unique and unprecedented flexibility for the choice of the scattering geometry. Moreover, a polarimeter allows for the first time the analysis of the polarisation of the scattered beam. Lastly, the use of a centroid reconstruction algorithm allowed to overcome some limitations of the commercially available CCD detectors.

The ID32 beamline at the ESRF

The high flux of monochromatic X-ray photons is produced in the ESRF storage ring.

Bunches of electrons are first accelerated up to 6 GeV in the linear accelerator and booster ring of the ESRF and then kept in the 844 m circular trajectory, made up of 32 straight sections, by 64 bending magnets. Quadrupoles and sextupoles guarantee the collimation of the electron beam and klystrons-powered RF cavities maintain the energy of the electrons constant. Various operation modes are possible depending on the needs of the 40 beamlines served. The most suitable modes for RIXS spectroscopy, due to its small cross section, are those providing the maximum electron current (200 mA) in the ring (i.e. the uniform multibunch, 7/8+1 and hybrid modes). In these operation modes, electrons are reinjected every 12 hours in the storage ring to compensate for the losses due to the finite lifetime (69 hours) of the electron beam. The ring is going to be upgraded in the program ESRF Upgrade Phase II, starting in 2019, to offer higher brilliance and coherence and to keep constant the current in the ring, by implementing the top-up mode.

In the 32 nd straight section three APPLE II [START_REF] Sasaki | Design of a new type of planar undulator for generating variably polarized radiation[END_REF] undulators constitute the photon source of the ID32 beamline. The central undulator is 2.5 m long and the other two 1.6 m, all having a period of 88 mm. The APPLE II structure consists in four arrays of permanent magnets, oriented following a specific pattern, as shown in Figure 3.1. The shape of the magnetic field inside the undulator can be controlled by acting on the vertical gap and the relative longitudinal position (phase) of the arrays. The magnetic field on the undulator axis imposes a wiggling trajectory to the electrons, which are forced to emit photons. ID32 undulators can produce photon beams with peak intensity in the 0.4 -1.7 keV energy range, with " 100% circular or linear (vertical, horizontal or inclined) polarisation.

The beam needs to be monochromatized to a high degree in order to be used for spectroscopy. A first filtering takes place in the optics hutch, where the combination of a watercooled double mirror and a Bremsstrahlung stop removes the photons of higher energy. The beam is also collimated by the (toroidal) mirror in the horizontal direction and focused in the vertical, at the entrance slit of the beamline monochromator. Focusing the beam at the beginning (and end) of each section allows to consider the different parts of the beamline as independent. The vertical focus is critical all along the beamline because, due to the use of a dispersive monochromator and spectrometer, it is directly related to the energy resolution.

On the other hand, the horizontal direction is less important, and the horizontal beam size is kept larger in order to reduce the heat load on the optics.

The monochromator set-up used for RIXS spectrocopy consists in a variable line spacing plane grating mirror (VLS -PGM) [START_REF] Fujisawa | Varied line spacing plane grating monochromator for undulator beamline[END_REF][START_REF] Petersen | Review of plane grating focusing for soft X-ray monochromators[END_REF] of the SX700 Peterson type. This configuration allows to keep the outgoing monochromatic beam parallel to the incoming one, always at the same height, no matter the energy selected. The grating consists in a plane (coated) Silicon mirror with a certain groove density, which, by diffraction, spatially disperses photons in the vertical direction according to their energy. The variable line spacing gives the grating focussing properties. The exit slit selects part of the first diffraction order of the outgoing beam, i.e. a certain range of energies, and is then refocussed onto the sample. Refocussing is done by a cylindrical mirror (horizontal direction) and a elliptical mirror equipped with mechanical benders (vertical focus). The beam size at the sample is " 4 µm in the vertical direction and " 60 µm in the horizontal. Therefore sub-millimeter size samples can be measured.

Two VLS gratings are available for the ID32 RIXS branch, one with an average of 800 lines/mm (for high flux and intermediate resolution) and the second with 1600 lines/mm (for very high resolution). Another VLS grating with 300 lines/mm provides a moderate energy resolution for the other ID32 branch, which hosts an X-ray magnetic circular dichroism (XMCD) endstation [START_REF] Kummer | The high-field magnet endstation for X-ray magnetic dichroism experiments at ESRF soft X-ray beamline ID32[END_REF] and an open room for non-routine experiments. In this case, a The temperature control of the monochromator is of high relevance, because the beam heat load can locally modify the shape of the elements, degrading their performance. This is particularly true for the first element, the plane mirror, which takes most of the thermal load. Its temperature is therefore kept constant, by liquid Nitrogen cooling, at 125 K, at which the linear thermal expansion coefficient of Silicon is zero. For the gratings, instead, water cooling is sufficient.

The whole beamline is kept in ultra high vacuum (UHV) conditions (" 10 ´10 mbar), mostly to avoid the damage of the optics by the interaction with high energy photons and carbon atoms present in air. Moreover, soft X-rays are highly interacting with matter and, in addition to the requirement of vacuum, no window can be put on the photons' path. ID32 is indeed a windowless beamline, whose vacuum is directly connected to the ESRF storage ring.

Sample stage and scattering geometry

The sample chamber is always kept in vacuum " 10 ´9 mbar and the samples are mounted on a shuttle (Figure 3.3) which is inserted in the chamber and positioned on the sample stage via a load-lock system. The sample is electrically insulated in order to allow total electron yield (TEY) measurements with a wire connected to an ammeter that records the drain current from the sample. Low temperatures down to 20 K can be reached on the sample, thanks to a liquid Helium cooling system.

The sample stage consists in a 4-circle UHV diffractometer that allows the motorized translation (sub-µm resolution) and rotation (" 0.01 ˝resolution) in vacuum of the sample (Figure 3.4).

The flexibility of the experimental geometry, given by the goniometer and the continuous rotation of the spectrometer, is an innovation of ID32 compared to the other existing instrumentations. It allows for the first time in soft-RIXS spectroscopy to have a complete 3D control of the momentum transferred to the sample. This is, for instance, crucial to study the dispersion relations in 3D materials or for being able to exploit geometrical dependencies in the scattering cross sections as shown in the next chapters.

To understand the scattering geometry, a careful definition of the angles is necessary.

The scattering angle is the angle between the incident and the outgoing beams, i.e. between the beamline and the spectrometer axes. Following the diffraction notation, this angle is indicated as 2θ. 2θ " 0 ˝defines the forward scattering and 2θ " 180 ˝is the backscattering geometry. On ID32, 2θ is allowed to vary continuously from 50 ˝to 150 ˝, without breaking the vacuum. The plane containing the scattering angle is the scattering plane.

The angle θ is the rotation of the sample around the vertical axes and it is defined as 0 when the sample surface is parallel to the incident beam and 90 ˝when the sample faces the beam. Note that θ " 2θ 2 only in specular geometry, but in general it can deviate by an angle δ from this value. χ is the rotation of the sample around the axes defined by the intersection between its surface and the scattering plane, while ϕ is the rotation angle around the normal to the sample surface.

The RIXS spectrometer

The purpose of the spectrometer is to collect the photons scattered by the sample in a certain solid angle and analyze their energy. In the soft X-ray range, this is typically performed by a dispersive grating. On ID32 the single optical element scheme [START_REF] Ghiringhelli | SAXES, a high resolution spectrometer for resonant X-ray emission in the 400-1600 eV energy range[END_REF] was adopted, meaning that both focusing and dispersing actions are provided by a concave spherical VLS grating.

In order to minimize the aberrations for each energy, the grating can be moved along the spectrometer axis, changing the sample-grating distance r 1 , and can be rotated to change the angle of incidence. Photons of different energies are focused in different positions by the grating, defining a focal plane. Vertical and horizontal motions of the detector allow its positioning on the focal plane, in correspondence of the energies of interests. The detector is a 2D position-sensitive CCD camera. In case of the ID32 spectrometer, the energy dispersion only occurs in the vertical direction and a monochromatic beam produces a horizontal line on the acquired images. Different energies in the beam are found as lines at different height.

The energy resolution can be improved by going far from the grating: ERIXS is indeed an extremely long spectrometer, with a grating-detector distance r 2 « 8.5 m and a sample-grating distance r 1 « 2.5 m (twice the sizes of SAXES spectrometer at Swiss Light Source [START_REF] Ghiringhelli | SAXES, a high resolution spectrometer for resonant X-ray emission in the 400-1600 eV energy range[END_REF]). At such distances, the photons hitting the detector are those contained in an extremely small solid angle, causing an enormous intensity loss. In order to increase the acceptance of the spectrometer a parabolic mirror can be inserted in the beam path before the grating (Figure 3.6) (at about 1 m distance from the sample) to collimate the photons in a horizontal fan of 20 mrad onto the detector. The vertical angular acceptance of the spectrometer, on the other hand, is about 2 mrad and is defined by the grating pitch angle (" 2 ˝), length (" 20 cm) and distance from the sample (r 1 ). Two gratings are available on ERIXS: one with a groove density of 1400 lines/mm (for higher photon throughput and medium resolution) and one with 2500 lines/mm (for higher resolution).

The total resolution for spectrocopy is given by combining the resolution of the beamline and spectrometer. The first is determined by the diffraction properties of the gratings, the aperture of the monochromator's exit slit and by the aberrations and slope errors that affect the focusing properties. The spectrometer resolution is given by the vertical beam size on the sample (which acts as the spectrometer's source), the diffraction and focusing properties of the grating and the detector resolution. To reduce the vertical size of the spectrometer's source, slits were used in the past between the sample and the grating. On ID32 the use of slits is unnecessary, thanks to the small focus provided by the beamline on the sample. On the other hand, the detector contribution to the resolution has become particularly important and a lot of work has been done to improve its performances, as it will be explained in the following.

The combined (beamline + spectrometer) response function is a Gaussian of " 30 meV FWHM at the Ce M 5 edge (" 882 eV) if both the beamline and the spectrometer are set in the high resolution mode, " 60 meV FWHM in the high efficiency mode and " 42 meV in the medium resolution mode that offers a good compromise between throughput and resolving power (Figure 3.7).

As mentioned above, the whole spectrometer arm can rotate around the sample chamber over 100 ˝without breaking the vacuum. This is possible thanks to a double-differential pumped steel ribbon with a nozzle, designed by the Cinel Company. The rotation of the whole spectrometer takes place on air pads, floating on low roughness marble and granite surfaces. Two motors allow the synchronized movement of the spectrometer and the sample chamber's ribbon, to whom the scattering arm is joined with bellows.

Polarisation analysis in the scattered beam

In addition to the spectral analysis of the scattered photons, a simultaneous measurement of their polarisation is also possible on ERIXS, by inserting a polarisation-selective optical element (polarimeter) in the beam path. Polarimetry is not trivial for soft-RIXS and the only former study of a general scheme1 to analyse photons' (linear) polarisation is the one of Braicovich et al. [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft X-ray scattering[END_REF], performed on AXES spectrometer at ID08. That study set the basis for the ERIXS polarimeter, designed by G. Ghiringhelli and L. Braicovich in collaboration with ESRF. X-ray polarimetry is performed with reflective optics. In the hard X-rays range Bragg optics are both efficient and intrinsically polarisation selective, allowing since many years hard-RIXS experiments with polarisation analysis [START_REF] Ishii | Polarization-analyzed resonant inelastic x-ray scattering of the orbital excitations in KCuF 3[END_REF]. For soft X-rays, on the other hand, decent reflectivity is achievable only with multilayer mirrors, out of the non-polarisationselective grazing-incidence condition. Theoretically, the optimal angle of incidence for polarisation analysis is θ pol " 45 ˝(the Brewster's angle), with detection at 2θ pol " 90 ˝. In such a geometry the reflectivity is zero for light polarised with the E vector perpendicular to the multilayer surface, giving 100% polarisation selectivity, but the overall efficiency is extremely reduced (r « 10 ´3) and the acquisition of RIXS spectra would be impossible. A The set-up is sketched in Figure 3.8. When used, the polarimeter is introduced in the beam path just before the detector and deflects the X-rays to another, identical detector. The spectrum acquired by this second detector is

I pol " r π ¨Iπ `rσ ¨Iσ , (3.1)
where π (σ) indicates the polarisation perpendicular (parallel) to the multilayer's surface, and r πpσq is the polarisation dependent reflectivity of the multilayer. On the other hand, the spectrum acquired without using the polarimeter (direct beam) is:

I dir " I π `Iσ . (3.2)
By combining Equations 3.1 and 3.2 and knowning r π and r σ it is possible to find I π and I σ , the components of the scattered beam parallel and perpendicular to the scattering plane:

I π "
´Ipol ´rσ ¨Idir r σ ´rπ , I σ " I pol ´rπ ¨Idir r σ ´rπ .

(3.3) Figure 3.9: Pictures of the multilayers and drawing of the detector chamber with the polarimeter.

The values of r π and r σ need to be carefully estimated at the beginning of each experiment.

In a real measurement, they not only account for the polarisation-dependent reflectivity of the multilayer, but include all the possible contributions that modify the intensity of the deflected beam with respect to the direct beam, like for instance a different efficiency of the two detectors. r π and r σ are calibrated by using a pure elastic line, as the Bragg peak of a for the two channels. This is achieved by putting the two detectors at the same distance from the multilayer, in order to preserve the total distance from the grating, which is the focusing element. A careful calibration of the pixel Ñ eV conversion (see next section) for both detectors is also crucial.

To better understand the design of the ERIXS' polarimeter it is worth discussing its differences with the prototype tested on AXES at ID08. The horizontal beam divergence was non-negligible in AXES, due to the absence of the parabolic collimating mirror (as in Fig- Figure 3.12: A typical image acquired in a RIXS experiment and its integration along the isoenergetic lines to get the spectrum. The zoom of the image in a low-intensity region reveals the spots produced by the X-ray photons. Each spot extends to more than one single pixel, limiting the resolution.

multilayer and find the optimal position (that is to say, the optimal multilayer's spacing)

for the given energy. Two multilayers are mounted in ERIXS as shown in Figure 3.9, the larger one covering the energy range between 700 and 1000 eV and the smaller one for the energies between 450 and 650 eV.

Detection and image processing

As aforementioned, the detectors used in ERIXS are 2D position sensitive CCD cameras and the acquired images consist in horizontal lines at different heights, each corresponding to a photon energy. This allows the parallel acquisition of the whole RIXS spectrum.

For the detection of soft X-rays, that have very short penetration length in matter, thinned back-illuminated CCD cameras are needed, working in vacuum and without windows. Moreover, the low intensity of the RIXS signal requires a very uniform background, with low darkcurrent and readout noise, obtained by cooling of a high quality Silicon chip.

Currently, ERIXS uses two Andor iKon-L detectors, with 2048 ˆ2048 pixels of 13.5 13.5

µm 2 size, one for the direct beam and the second to detect the beam deflected by the polarimeter. The chip is kept at ´80 ˝C by water-cooled Peltier cells, providing negligible darkcurrent and an extremely low readout noise of 3 electrons rms at 50 kHz pixel reading frequency or 6.5 electrons rms at 1 MHz.

The usual data reduction process consists in the integration of the acquired images along the isoenergetic lines, as shown in Figure 3.12. The lines don't appear perfectly horizontal in the image because the detector is mounted slightly tilted, in order to spread the signal over multiple pixel lines and be less affected by the chip's inhomogeneities. The slope also allows to perform an oversampling of the image in smaller virtual pixels while integrating, giving a better definition of the spectrum.

After the integration, the vertical axis must be converted in energy. The conversion factor pixel Ñ eV is obtained by measuring the changes in the position of the elastic line while gradually changing the incident photon energy.

It is clear at this point that the spatial resolution of the detector directly affects the energy resolution of the spectra. The resolution of CCD cameras for X-rays is limited by the size of the electron cloud excited by each photon absorbed in the detector. The charge cloud is generated by diffusion and electron-electron scattering processes, and has approximatively a gaussian shape of " 25 µm FWHM, larger than the pixel size. When the charge is collected in the pixels, each cloud produces a spot in the image spread over more than one pixel [START_REF] Ghiringhelli | SAXES, a high resolution spectrometer for resonant X-ray emission in the 400-1600 eV energy range[END_REF][START_REF] Amorese | Enhancing spatial resolution of soft X-ray CCD detectors by single-photon centroid determination[END_REF][START_REF] Amorese | Single photon acquisition in CCD detectors for high resolution resonant inelastic soft x-ray scattering[END_REF], as visible in the inset of Figure 3.12. If the detector is used without any further arrangement, its resolution alone would add a contribution of about 75 meV at the Ce M 5 edge while it has to be significantly smaller to achieve the target value of 30 meV for the total combined resolution. An incline γ " 25 ˝allows to reduce the detector contribution down to 30 meV (Figure 3.13), but this value is still too large and would prevent from reaching the target performances of ERIXS.

Because of its importance, a lot of effort has been put into improving the detector's spatial resolution. On ERIXS, a simple but powerful solution has been implemented, without the need of buying specific (and expensive) detectors. The idea is to use the intensity distribution in the pixels of a spot to retrieve the information about photon's impact position. This procedure must be used on each single photon's spot and it is thus named single photon counting. I have spent a significant amount of time implementing a respective algorithm and characterising the improvements in resolution with commercially available CCD detectors.

The basic assumption is that the impact position of the photon is in the charge cloud's center, which must be found by using the intensity distribution among the pixels of a spot.

This approach is also called centroiding. Centroiding is used since a long time for photons of higher energies, but its applicability in the soft X-rays range was limited by a low signal to noise ratio. The amount of electrons excited by a photon in Silicon is proportional to its energy. The electron-hole pair generation energy in Si is 3.65 e ´{eV, meaning that 880 eV photons generate clouds of about 244 electrons, which spread among " 9 ´16 pixels. The outer pixels can collect less than 10-15 electrons which is comparable to the readout noise of the electronics. Thanks to the fast advances in the detector industry, commercial CCD cameras can now reach very low noise values (3-6 electrons rms), allowing the application of centroiding techniques in the soft X-rays range [START_REF] Soman | Improving the spatial resolution of a soft X-ray Charge Coupled Device used for resonant inelastic X-ray scattering[END_REF], without the need of a detector specifically designed for single photon counting [START_REF] Hall | Improving the resolution in soft X-ray emission spectrometers through photon-counting using an electron multiplying CCD[END_REF].

The solution implemented for the ERIXS spectrometer consists in a post-processing algorithm acting on images obtained with the normal acquisition mode. The spots are found on the image by comparing the pixels' intensities with empirical thresholds and the centroid position is assigned by a simple (but robust) center of mass calculation, i.e. weighting the pixels' positions with their intensities [START_REF] Amorese | Enhancing spatial resolution of soft X-ray CCD detectors by single-photon centroid determination[END_REF][START_REF] Amorese | Single photon acquisition in CCD detectors for high resolution resonant inelastic soft x-ray scattering[END_REF]. The spots' centers are then assigned to channels of arbitrary size following the isoenergetic lines' slope. Each spot center counts for one photon and the number of photons in each channel gives the intensity of the corresponding point of the spectrum. The choice of the channels' width has to be made by a trade off between the definition-resolution of the spectrum (narrower channels) and the intensity in each channel (larger channels). Usual channel widths used to analyse ERIXS data are between 0.2 and 0.4 pixels.

Preliminary off-line tests [START_REF] Amorese | Single photon acquisition in CCD detectors for high resolution resonant inelastic soft x-ray scattering[END_REF][START_REF] Amorese | Enhancing spatial resolution of soft X-ray CCD detectors by single-photon centroid determination[END_REF] proved that a spatial resolution of the images better than 6 µm can be obtained, which translates in about 7 meV energy resolution (in the ERIXS' high resolution mode). This value is negligible if compared to the beamline and spectrometer contributions, of about 20 meV each, and allowed to reach the target combined resolution of 30 meV at the Ce M 5 edge. The tests also revealed the presence of both intrinsic and random errors in the centroid reconstruction, some of which could be in principle corrected [START_REF] Langini | Advances in the detection at the new XMCD and RIXS beamline of the ESRF[END_REF]. For the time being, the correction is not implemented in the analysis of ERIXS' data because the effect of these errors (less than 2 µm) is negligible if compared to the usual integration channel size (5 µm) or to the other contributions to the combined resolution.

The left panel of Figure 3.14 shows the resolution gain obtained for high resolution RIXS spectrocopy, estimate from the broadening of a pure elastic reference. Considering a pixel Ñ eV conversion factor of 16.8 meV/pixel, the total combined resolution is improved from 43 to 30 meV when using single photon centroiding. The right panel shows the same gain in the case of the medium resolution set-up (from 68 to 55 meV, with a conversion factor of Another advantage of using single photon counting is a complete suppression of the detector's background, because all the base modulations do not contribute to the final spectrum, which is made by counting the number of centroids in each integration channel. This background elimination is very useful for the detection of low intensity signals or when two spectra need to be subtracted, as it happens for the polarisation analysis.

For the ID32 spectrometer, the single photon counting algorithm is now the routine way of reducing the 2D CCD images to a 1D spectrum.

Chapter 4

Full characterisation of the 4f levels in CeRh 2 Si 2

The recent improvements in Resonant Inelastic soft X-ray Scattering (soft-RIXS) instrumentation open up multiple possibilities for studying the properties of condensed matter.

The experimental resolution is now comparable with many low-energy excitations, which became visible in soft-RIXS spectra. Among these, the excitations between two 4f levels can provide much information about the relative energy, symmetry and charge distribution of the 4f states.

This chapter investigates the capabilities of high resolution soft-RIXS to study the crystal-field split 4f states of cerium ions in cerium intermetallics of the 122 family. Since this work represents the very first observation of 4f crystal-field features in soft-RIXS, stress will be put on the methods employed to get the information from the spectra. The example of a typical heavy fermion of the 122 family, CeRh 2 Si 2 , will be used to show and comment the steps of the data analysis, but the main outcome of this chapter are in general applicable to any cerium-based compound.

To complete the picture, the findings will be compared with results obtained with other techniques. The choice of the most important experimental settings will also be discussed, providing a further insight in the many opportunities offered by soft-RIXS spectroscopy. CeRh 2 Si 2 was not the first heavy fermion discovered to superconduct under pressure, but this behavior was unexpected on the basis of thermopower measurements, unlike its predecessors CeCu 2 Ge 2 and CePd 2 Si 2 [START_REF] Jaccard | Pressure induced heavy fermion superconductivity of CeCu 2 Ge 2[END_REF][START_REF] Grosche | Magnetic and superconducting phases of CePd 2 Si 2[END_REF][START_REF] Thompson | Effect of pressure on the Néel temperature of Kondo-lattice systems[END_REF]. Since then, many studies enriched the list of its properties.

Properties of CeRh

Of particular interest is the ambiguity in the nature of the 4f electrons [START_REF] Boursier | Comparison between Ce and Yb heavy fermion compounds: CeRh 2 Si 2 versus YbRh 2 Si 2[END_REF]. On the one hand, the high magnetic ordering temperature, the large magnetic moment per Ce ion (m AF1 « 1.28´1.5 µ B [START_REF] Grier | Magnetic ordering in CeM 2 Si 2 (M= Ag, Au, Pd, Rh) compounds as studied by neutron diffraction[END_REF][START_REF] Kawarazaki | Ground-state magnetic structure of CeRh 2 Si 2 and the response to hydrostatic pressure as studied by neutron diffraction[END_REF][START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF] and m AF2 « 2.4 µ B [130]1 ), the large entropy [START_REF] Graf | Magnetic phase transitions in CeRh 2 Si 2 : specific heat, susceptibility and resistance studies[END_REF] and the small Fermi surface [START_REF] Araki | Fermi surface instability in CeRh 2 Si 2 under pressure[END_REF] suggest an high localization of the 4f electrons. On the other hand, the quasi-elastic linewidth observed in inelastic neutron scattering spectra indicates a large Kondo scale (T K « 30 K) and the small value of p c implies the vicinity to magnetic instabilities; both of these observations entail a significant hybridization of the 4f electrons.

At ambient pressure, the magnetic phase wins and masks the hybridization, whose effects are anyway present and observable in photoemission spectrocopy [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF].

Despite the knowledge of the many peculiar physical properties of CeRh 2 Si 2 , the energy splittings, symmetries and charge distributions of the Ce 4f states are still not determined.

Ab initio calculations [START_REF] Vildosola | Spectral properties and crystal-field splittings in CeM 2 Si 2 (M= Ru, Rh, or Pd) compounds[END_REF] seem to give splittings that are too small and need comparison with experiments to restrict the value of the free parameters. Yet, due to the complex physics, the experimental techniques traditionally used to determine the crystal electric field (CEF) in cerium intermetallics have given contradicting results and are far from being conclusive. For example, different measurements of the dependence of the magnetic susceptibility on temperature set the CEF splittings of the 2 F 5{2 multiplet to 27 and 59 meV with states Γ 7 -Γ 7 -Γ 6 in order of increasing energy [START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF] or 31 and 78 [START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF] with a Γ 7 -Γ 6 -Γ 7 order. These experiments, determining the CEF splittings only indirectly and being affected by the magnetic properties of the sample, do not always guarantee a unique result.

Considering more direct techniques, an INS-XAS study [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF] measured energy splittings of 32 and 52 meV and Γ 7 -Γ 7 -Γ 6 order of the states, but the hybridized 4f states give broad peaks in the spectrum, with a low contrast over the phonon background. Moreover, the energy of the observable excitations is limited to " 60meV in INS, preventing from seeing states at higher energies. Recently a resonant photoemission [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF] study measured 48 and 62 meV splittings with an order Γ 7 -Γ 6 -Γ 7 of the states. The electron removal in the final state of photoemission does not guarantee that the splittings observed are the same of the ground state and moreover the results can easily be affected by surface effects.

Therefore a technique that probes directly the CEF states' symmetry and splitting with bulk sensitivity and with a clear interpretation of the results was still needed. Soft-RIXS spectroscopy with " 30 meV resolution, comparable to the CEF splittings, can overcome the limitation of the traditional techniques. CeRh 2 Si 2 was therefore considered an optimal compound to perform the first soft-RIXS test, whose outcome, due to the absence of an established interpretation of its CEF, is both unbiased by previous knowledge and of high scientific interest.

From RIXS spectra to the crystal electric field

The high-resolution M 5 RIXS spectra of CeRh 2 Si 2 are shown in Figure 4.3. Two different energies of the incoming photons were chosen, 0.3 eV before (881.7 eV) and after (882.3 eV) the first peak of the XAS (indicated in Figure 4.2 with A and C). As visible by the RIXS spectra and discussed in section 2.2 and in section 4.5 the two incident energies were chosen to enhance the signal in different spectral regions, around 0 and 300 meV energy loss, respectively. The scattering angle was fixed to 2θ " 150 ˝, with the crystallographic axis a and c lying in the scattering plane (φ " 0 ˝, χ " 0 ˝), c making the same angle with the incoming and detected beams (specular geometry, i.e. δ " 0 ˝and θ " 75 ˝). Spectra were measured with polarisations of the incident beams both parallel (π) and perpendicular (σ) to the scattering plane. The spectrum at 881.7 eV and π polarisation was acquired for 8 hours, as calculated by simulations. The intensity of the elastic peak needed to be corrected in order to better fit the spectrum, because the signal at zero energy loss is influenced by contributions not considered in simulations as low-energy phonons, surface quality. It can be noticed that the peaks, representing the 4f levels, are split in two groups, corresponding to the 2 F 5{2 (three peaks, including the elastic one, in the range E loss " 0 ´100 meV) and 2 F 7{2 (4 peaks at E loss " 250 ´400meV ) multiplets. The 2 F 5{2 peaks have been broadened by a resolution-limited Gaussian of " 35 meV FWHM, while a larger Gaussian broadening of about 60 meV (not resolution-limited) was needed for the excitations in the 2 F 7{2 multiplet.

A low and flat background (grey shaded area) was added to take into account the signal from excitations in the conduction band.

In the simulations, the Slater integrals were reduced to about 55% of their atomic

Hartree-Fock values for the 4f -4f electronic interactions and to 88% for the 3d ´4f interactions. These values were found by comparing experimental XAS spectra of powdered Ce 3`s amples to simulations (without CEF), and show a negligible deviation [START_REF] Thole | 3d X-ray-absorption lines and the 3d 9 4f n`1 multiplets of the lanthanides[END_REF] from the " 60% and " 80% reductions suggested in literature [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF]. Similarly, the 4f and 3d spin 4 " ´76 meV and Ǎ4 4 " 194 meV, resulting in a 46-73 meV CEF splitting of the 2 F 5{2 multiplet and an order Γ 1 7 ´Γ6 ´Γ2 7 of the three levels. It is important to stress once again, before facing the details of the data analysis, that the full determination of the 4f CEF levels can in principle be obtained with a single RIXS spectrum. This is a rather unique feature of RIXS spectrocopy, because other techniques are only sensitive to certain properties of the 4f levels, therefore the information gained about the CEF is never complete and may lead to wrong interpretations. A RIXS spectrum simultaneously provides not only information about the CEF levels' splittings, given by the position of the peaks, but also about their mixing, symmetry and charge-density spatial distribution. This information is held in the relative intensities of the peaks, i.e. by the cross sections of the excitations, which are highly influenced by the states' characteristics.

Moreover, the dependency of the peaks' intensity on controlled experimental parameters as the incident photon energy, scattering geometry and polarisation can be also exploited to remove any residual ambiguity.

In order to relate the states' properties to the peak intensities, the comparison of the acquired data with calculated RIXS spectra, each representative of different CEF schemes, is essential. This is a very powerful tool of RIXS spectrocopy, for which the spectrum can be calculated with dedicated codes and the absence of a complex background allows for a precise evaluation of the intensities.

The methods to retrieve the basic information from the RIXS spectra are explained in this section. The various steps of the data analysis, each revealing a specific characteristic of the CEF scheme, are discussed, eventually providing a recipe to be followed for future experiments.

Splitting and symmetry

The most important properties of the 4f levels to be determined are the energy (CEF splitting) and the J z mixing of the states. In order to simplify the analysis one can focus on the lowest three levels ( 2 F 5{2 multiplet). The detailed characterization of the 2 F 7{2 levels is only of secondary importance because they lie at too high energies (" 280 meV ą 3000 K) to influence the properties of the material. Moreover, this choice allows to reduce the set of CEF parameters needed to Ǎ0

2 , Ǎ0 4 and Ǎ4 4 , since the higher orders of the CEF potential don't affect the 2 F 5{2 levels (see section 1.2.2).

An estimate of the size of the 4f CEF splitting, corresponding to the distance between the peaks, can be obtained just by observing the RIXS spectra. A first intense inelastic peak is visible in Figure 4.3 at 46 meV and another less intense peak, almost covered by the tail of the first, can be located at slightly higher energies via a Gaussian fit. Since the features are overlapping, unknown parameters as the peaks' relative intensity affect the fit, which can provide only a rough estimate of the splittings (error " ˘10 meV). This first estimate can then be optimized with the help of simulations. In fact, the calculation of the cross sections assesses the relative peaks' intensities, reducing the number of unknown variables. This leads to an iterative process between the fit and simulations, to optimize the guess on the peaks' positions and to re-calculate the cross sections; eventually providing results with an accuracy better than 2 meV. Knowledge of the size of the splittings, although of first importance, does not provide a full description of the CEF levels. It defines the distances between the ground and the first and second excited states, but does not tell which level corresponds to each of the three possible wavefunctions with symmetry Γ 1 7 , Γ 2 7 and Γ 6 (see Equation 1.25). This information can, however, be obtained by comparison with calculations, since the RIXS cross sections are highly affected by the symmetries of the states. The various arrangements of the three wavefunctions on the three CEF split levels give indeed different shapes of the calculated spectra, with different relative intensities of the inelastic peaks, as shown in Figure 4.4 (right). As a general trend, it shall be noticed that, when Γ 1 7 is the ground state (red and blue lines) the RIXS transitions to the excited Γ 6 level is stronger than the excitation in the Γ 2 7 . On the other hand, the excitations between a Γ 6 ground state and the Γ 1,2 7 levels (green line) have almost equal cross sections. By comparing these observations to the experimental data, one can easily deduce that the scheme best representing the CEF splittings in CeRh 2 Si 2 is: Γ 1 7 (ground state, 0 mev) -Γ 6 (46 meV) -Γ 2 7 (73 meV).

J z mixing

The information about the energy and symmetry of the states still does not provide a full characterization of the CEF levels. In fact, even if the Γ wavefunctions are assigned to the levels, the amount of their J z mixing is unknown. To better understand this statement, one shall refer to Equations 1.27 and 1.28, that show the relations holding, in Stevens' approximation, between the energy and mixing of the 2 F 5{2 levels and the CEF parameters Ǎ0 2 , Ǎ0 4 and Ǎ4 4 . As discussed there, the full characterization of the (unhybridized) 4f states is equivalent to the definition of the set of CEF parameters describing them. Trying to revert 2 of the ground state (Γ 1 7 " α|˘5{2y `β|¯3{2yq) mixing coefficients and of its J z expectation value. Right: Full CEF scheme (including the 2 F 7{2 multiplet) in function of Ǎ0

2 .

Equation 1.28 to find the CEF parameters as a function of the splittings, one ends up with an underdetermined system, made of two equations (to impose ∆E 1 " E Γ6 ´EΓ 1 7 " 46 meV and ∆E 2 " E Γ 2 7 ´EΓ 1 7 " 73 meV) and three unknowns ( Ǎ0 2 , Ǎ0 4 and Ǎ4 4 ). It is anyway possible to express two parameters as a function of the third, obtaining:

A 0 4 " 7∆E 1 ´7 2 ∆E 2 ´12 5 A 0 2 A 4 4 " ˘1 10
c 7 2 b ´p20∆E 1 ´25∆E 2 ´12A 0 2 qp20∆E 1 `25∆E 2 ´12A 0 2 q . ( 4.1)
These relations define, as a function of Ǎ0 2 , sets of parameters generating the same splittings ∆E 1 and ∆E 2 , but different J z mixing of the wavefunctions. The dependence of the ground-state mixing on Ǎ0

2 is shown in Figure 4.5 (left). The ground state mixing can vary continuously from a pure |5{2y (α 2 " 0, β 2 " 1) to a pure |3{2y (α 2 " 1, β 2 " 0) state. The one-to-one dependence that holds between Ǎ0

2 and the mixing parameters makes sufficient to determine α 2 or the J z expectation value to find Ǎ0

2 and completely define the CEF. These physical quantities are measurable, for example, with XAS spectroscopy or magnetic measurements (as magnetic susceptibility).

RIXS is not directly sensitive to α or J z , but these are not the only measurable properties affected by Ǎ0

2 . Figure 4.5 (right panel) shows the energy of all the 7 CEF-split 4f levels as a function of Ǎ0

2 . As expected, the energy splitting of the 2 F 5{2 levels is constant, apart from a small deviation due to the limits of Stevens' approximation, that can be taken into account and corrected. Yet there are large modifications of the 2 F 7{2 levels' splitting scheme. These, together with a cross section change, give sizeable effects on the RIXS spectra. Strictly speaking, for a precise description of the 2 F 7{2 levels, the Ǎ0

6 and Ǎ4 6 parameters should be 84 Looking at the excitations in the 2 F 7{2 multiplet (200-400 meV), the simulations with Ǎ0 2 " 60 meV best reproduce the data (left column). In the low energy region (right column), instead, the sensitivity on Ǎ0

2 is much lower. Looking at the excitations in the 2 F 7{2 multiplet (200-400 meV), the simulations with Ǎ0

2 " 60 meV best reproduce the data (left column). In the low energy region (right column), instead, the sensitivity on Ǎ0

2 is much lower. 2 and the best agreement with all the experimental spectra is obtained for Ǎ0 2 « 60 meV (and thus, by Equation 4.1 Ǎ0 4 « ´77 meV and Ǎ4 4 « 194 meV, the value of A 4 4 being increased of 8% compared to the formula in order to correct for the errors of Stevens' approximation). On the other hand, the low energy region (right columns) presents a much lower dependence on Ǎ0

2 , only due to a small variation of the cross sections (the small errors on the peaks' positions, due to Stevens' approximation, were corrected here). The subtle differences among the calculated spectra are of the size of the estimation errors on the peaks' position and broadening and shall not be considered crucial. However, the calculation with Ǎ0

2 « 60 meV always provides a good fit of the data. The extra intensity visible at 100 meV is not due to CEF excitations and will be discussed later.

The low dependency of the 2 F 5{2 signal on Ǎ0

2 is actually an advantage more than a limitation. It indeed allows to study the energy splitting and symmetry of the 2 F 5{2 levels using calculated spectra, as done in Figure 4.4, without the need of knowing the precise value of Ǎ0

2 , which can be determined afterwards.

Comparison with other techniques

It is interesting at this point to check the found value Ǎ0 2 « 60 meV against data of other techniques usually employed to probe α and J z .

The temperature dependence of magnetic susceptibility probes the magnetic moment per Ce ion, related with J z . Two curves of the magnetic susceptibility can be found in literature, reported in Figure 4.8 with empty diamonds (from Ref. [START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF]) and circles (Ref. [START_REF] Mori | Effect of pressure on the magnetization of single crystal CeRh 2 Si 2[END_REF]).

The paramagnetic region (T " T N1 " 37 K) is well reproduced by the dotted curves, both calculated with the same CEF splitting and symmetry schemes determined with RIXS (46-73 meV and Γ 1 7 ´Γ6 ´Γ2 7 ). The dotted grey lines in Figure 4.8 are obtained with Ǎ0 2 " 60 meV, which gives a set of parameters identical to the one derived above from the RIXS data.

On the other hand the calculation of black curves needed, in order to reproduce the larger magnetic anysotropy, a slightly higher value of the parameter Ǎ0

2 " 76 meV (in this second case Ǎ0

4 " ´116 meV and Ǎ4 4 " 166 meV). The corresponding values of J z , 1.41 and 1.79, are in good agreement with the magnetic moment per cerium ion measured by other techniques (m AF " g J J z µ B " 6{7J z µ B " 1.2 ´1.52µ B ) [START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF][START_REF] Kawarazaki | Ground-state magnetic structure of CeRh 2 Si 2 and the response to hydrostatic pressure as studied by neutron diffraction[END_REF][START_REF] Grier | Magnetic ordering in CeM 2 Si 2 (M= Ag, Au, Pd, Rh) compounds as studied by neutron diffraction[END_REF].

The measurements of the magnetic properties therefore confirm, within the experimental uncertainties, the results found by the analysis of above. It is worth mentioning that in Ref. 2 " 76 meV and 60 meV. An exchange field along c was introduced to better reproduce the low-T region, with a value of H exc " ´9 meV for the χ c simulation (violet line) and " ´37 meV for χ a (teal).

[70] a scheme Γ 1 7 ´Γ6 ´Γ2 7 with 31-78 meV splittings (instead of 46-73 meV) were supposed by the fit of the same magnetic susceptibility curve of figure Figure 4.8. The errors on the splitting could be due to the lower sensitivity to the excited levels, whose thermal population is still low in the measured temperature range.

To complete the discussion about the magnetic susceptibility data, it must be noticed that the fit can be improved in the low-T region by introducing a mean magnetic field along the c axis to reproduce the exchange interactions leading to the AFM magnetic order [START_REF] Quezel | Magnetic ordering in TbRh 2 Si 2 and CeRh 2 Si 2[END_REF].

The exchange field was set to " ´9 meV to calculate χ c (violet line) and " ´37 meV for χ a (teal). The factor « 4 between the two values is similar to the ratio between the two molecular field parameters λ a and λ c used in Ref. [START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF]. Another technique recently employed to determine the (ground state) J z mixing is X-ray absorption spectroscopy. Data from Ref.

[51] are shown in Figure 4.9, together with simulations with varying Ǎ0

2 values (and therefore varying J z mixing α 2 of the ground state). The calculation with Ǎ0 2 « 50 meV provide a good fit of the data, thus not disproving the value Ǎ0 2 « 60 meV found above with RIXS and magnetic susceptibility. But the quality of the fit improves going towards lower values of Ǎ0

2 . In particular, the spectrum with polarisation c is better fit, in the M 5 (880-885 eV) region, by the calculation with Ǎ0 2 " 20 meV. This value would be in agreement with the results of Ref. [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF], that lead to a mixing α 2 " 0.53. However it should be remembered that Figure 4.9: XAS experimental spectra from Ref. [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF] and calculations with the CEF levels' splitting and symmetry found in this work and varying Ǎ0

2 and the mixing α.

the XAS spectra are affected by both the magnetic order (the experiment was performed at 18 K ă T N2 " 26 K ă T N1 " 37 K) and the mixed valence of CeRh 2 Si 2 . Therefore, after considering the XAS experimental uncertainties and the small change of the calculated spectra in the range 20 meV ă Ǎ0 2 ă 80 meV, it's safe to say that XAS spectroscopy's results are compatible enough with those of RIXS spectroscopy.

To conclude the comparison, the XAS-INS study of Ref. [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF] gave splittings 30 -52 meV and a scheme Γ 1 7

´Γ2

7 ´Γ6 of the CEF levels, in clear contrast with the RIXS data presented in the previous section which in turn agrees well with the findings based on magnetic susceptibility and photoemission measurements. A possible reason for this disagreement could lie in the limitations of XAS and INS. XAS probes the ground state's mixing, while the excited levels can be accessed only by thermal population, thus indirectly. Unfortunately, the magnetic ordering, the splittings and the excited levels' symmetry all give different temperature dependencies of the spectra, difficult to disentangle. On the other hand, the INS spectra can probe only a limited range of excitation energies (in that paper, E Loss ă 60 meV) and the determination of the peak's position can be affected by the phonon background subtraction. for all possible polarisation channels they show indeed a strong dependence on the states orientation. This is shown in Figure 4.11. For ϕ " 0 ˝, i.e. the (100) axis in the scattering plane (upper row of spectra), the strongest feature, due to the transition Γ 1 7 ´Γ6 at 46 meV, is found in the crossed-polarisation channels (πσ and σπ) for Ǎ4 4 ą 0 (left) and in the non-crossed channel for Ǎ4 4 ă 0 (left). On the other hand, all the other cross sections are approximately constant. Note that the Γ 6 state is almost insensitive to the sign of Ǎ4 CeCu 2 Si 2 [START_REF] Willers | Determining the in-plane orientation of the ground-state orbital of CeCu 2 Si 2[END_REF]. Furthermore, thanks to their different polarisation dependence, the 2 F 5{2 peaks can be easily distinguished in the experimental spectra despite the lower resolution (60 meV) and their positions further confirm the splittings determined above.

Interestingly, the simulations correctly reproduce also the intensity at zero energy loss for the crossed-polarisation channels (red lines). This channel is indeed free of the contributions of phonon excitations and surface reflections, since these two processes don't rotate the polarisation. On the contrary, the elastic intensity of the non-crossed channel needed as usual to be corrected to better fit the data. The non-crossed elastic contribution was anyway reduced by cleaving the sample in vacuum and slightly misaligning the sample relative to specular geometry (δ " 0 ˝, χ « 0 ˝).

Other spectral features

The non-crossed spectra in Figure 4.12 show some extra intensity around 100 meV, which is not reproduced by the multiplet calculations. The residuals, after smoothing, are shown in Since this extra feature appears in Figure 4.12 in the non-crossed polarisation channels (ππ and σσ), while it is completely absent in the crossed channels (πσ and σπ), it cannot be correlated to any of the f f transitions. Moreover, it cannot arise from magnetic excitations, since these rotate the photons' polarisation to fulfill the angular momentum conservation, giving a signal in the crossed channel [START_REF] Braicovich | The simultaneous measurement of energy and linear polarization of the scattered radiation in resonant inelastic soft X-ray scattering[END_REF].

Due to its polarisation dependence, this extra feature could be assigned to phonon or charge excitations. Electronic conduction bands originated from Rh 4d states and strongly hybridizing with the Ce 4f states have been observed by photoemission [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF] to start at about 100 meV and disperse towards higher energies (see Figure 1.10). The 100 meV peak can therefore arise from electronic excitations, possibly of charge transfer (CT) nature, into the 4d states. This provides also an explanation for the slowly varying background introduced to better fit the data (e.g. in Figure 4.3), which, starting at " 100 meV, could be due to higher energy excitations in the dispersing 4d band.

Temperature dependence and magnetism

To avoid the thermal population of the excited levels, which would give additional peaks in the spectra, RIXS data are usually acquired at cryogenic sample temperatures. This was also found to prevent radiation damage effects. Obviously, the properties of the sample at the measuring temperature must be considered, because different sample phases (e.g. heavy fermion, Kondo region, magnetic order...) may give rise to different spectral features.

In the specific case of CeRh 2 Si 2 , data were acquired at T sample " 25 K, below T N1 " 37 K, at (disordered phase) are presented in Figure 4.14. They show a perfect overlap in the inelastic region (E loss ą 25meV ), meaning that, within the present experimental resolution (here " 50 meV), no magnetism-induced modification of the CEF levels can be observed. Therefore, the results of this chapter are not affected by the antiferromagnetic ordering at 25 K. On the other hand, there is a relevant increase of the (quasi) elastic intensity with temperature.

Magnetic excitations in CeRh 2 Si 2 can be compatible with this energy range [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF], but their study is impossible because other effects may dominate the temperature dependence of the quasi-elastic peak. Kondo-related effects might be present, as well as increased atomic vibrations, which give additional low energy features and a stronger diffuse scattering. Spectra calculated using the usual CEF parameters (solid lines in Figure 4.14) fit the experimental data with the same accuracy at the three temperatures, further confirming the absence of inelastic magnetic features. As usual, the calculated elastic intensity needed to be corrected. However, the goodness of the resolution-limited Gaussian fit between 0 and 50 meV doesn't suggest the presence of additional features in this energy range. At 100 meV the quality of the fit is again strongly affected by the broad CT peak, whose relative intensity is higher in the present scattering geometry. This could be mostly due to a relative reduction of the cross section for CEF excitations when going off-specular (δ " 15 ˝).

Although no magnetism-related effect is observed in RIXS spectra with the present resolution, it is anyway interesting to understand what could be expected, with the help of calculations. In the simplest approach, an interaction with a magnetic field along c , representing the internal exchange field aligning the moments [START_REF] Grier | Magnetic ordering in CeM 2 Si 2 (M= Ag, Au, Pd, Rh) compounds as studied by neutron diffraction[END_REF], is added to the singleion Hamiltonian. The field induces a Zeeman splitting of the 4f levels, shown in the top panel of Figure 4.15. The results are plotted as a function of the ground state splitting ∆ gs produced by the magnetic interaction. Based on INS data [START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF], on our RIXS measurements and on the fact that T N1 " 37 K 3 meV, the ground state splitting should be relatively small, let's say ∆ gs ă 10meV . The splitting of the higher CEF states is negligible in this 7 peaks (calculated as Γ 6 {pΓ 6 `Γ2 7 q and Γ 2 7 {pΓ 6 `Γ2 7 q) resulting from simulations at various scattering angles (in specular geometry, δ " 0 ˝). The two possible signs of Ǎ4

4 give different dependencies on 2θ, but converge at 150 ˝, allowing, at this angle, to neglect the Ǎ4 4 sign and simplify the analysis.

range, while there is a slight increase of their average energy, due to lowering of the ground state's energy. Anyway, the maximum combined shift induced by the magnetism is small and not easily detectable by our measurements. Therefore the only effect observable in RIXS spectra would be the excitation in the ∆ gs split state. Yet its detection is impossible for the moment, being limited by the experimental resolution and by the uncertainties affecting the quasi-elastic region. Moreover, the calculated intensity of this magnetic feature is rapidly decreasing if compared to that of the adjacent elastic peak (bottom left panel of Figure 4.15).

Therefore, if present, the ∆ gs excitation would be hidden in the tails of the elastic peak, and would be very difficult to detect. Regarding the transitions in the higher CEF levels, only the lowest of the two Γ 6 split states and the highest of the Γ 2 7 give non negligible intensities in the considered scattering conditions (2θ " 150 ˝, δ " 15 ˝and σ polarisation, as those of Figure 4.14). As shown in the bottom right panel of Figure 4.15, both peaks should decrease with increasing magnetic interaction, but the ratio between the two is constant with ∆ gs . This further confirms what is observed in the experimental data: the shape of the inelastic part of the spectrum is not modified by the arising of the magnetic ordering and no magnetism-related effects are detectable in spectra.

Choice of scattering geometry and incident photon energy

To determine the CEF scheme from the spectra, the choice of the experimental parameters is particularly important. The relative intensity of the CEF peaks depends indeed on the 4 . Starting from the top panels (ϕ " 0 ˝), if Ǎ4 4 ą 0 the Γ 6 relative intensity is higher in the crossed channels (red solid lines) and lower in the non-crossed ones (blue solid lines), while the intensity ratio is inverted if Ǎ4 4 ă 0 (dotted lines). The difference between the two situations reaches its maximum at 150 ˝, making this angle the most suitable also for the determination of the Ǎ4 4 sign, by using the ERIXS polarimeter. The bottom panel shows the same plots for ϕ " 45 ˝, which presents an exactly opposite description, with the crossed-and non-crossed channel swapped. It is also worth to notice that towards 2θ " 150 ˝, the intensity ratios for π an σ incident photons' polarisation become similar, consistently with the proximity of this angle to the backscattering condition (180 ˝) at which the two polarisations would be equivalent for our sample orientation. The incident photon energy has almost no effect on the intensity ratio between the Γ 6 and Γ 2 7 transitions. On the other hand, the relative intensity of the excitations in the 2 F 5{2 and 2 F 7{2 multiplets is strongly dependent on E in . The existence of a trend was already suggested by the low-resolution data, as visible in Figure 2.4, and then confirmed by the high-resolution spectra in Figure 4.3. To further investigate the observed effects, RIXS maps of CeCu 2 Si 2 are shown in Figure 4.18. The spectra were acquired at 2θ " 90 ˝, angle at which the elastic intensity is strongly reduced in the π polarisation channel. This allows to easily identify on the corresponding map (left panel) the 2 F 5{2 resonance energy, located approximately in correspondence of the first M 5 XAS peak. The RIXS map built from σ polarised spectra (right) is in fact dominated by the elastic intensity, which follows the shape of the XAS spectrum and, being more intense, covers the 2 F 5{2 resonance. On the other hand, the excitation in the 2 F 7{2 multiplet (at around 300 meV E loss ) is enhanced with incident energies halfway between the two main M 5 XAS peaks. Interestingly, the difference between the 2 F 5{2 and 2 F 7{2 resonance energies is approximately the energy distance separating the two multiplets (" 300meV ). As a consequence, each CEF level is no longer characterised by one single energy E CEF across the entire Brillouin zone, but forms a heavy band in a narrow energy range around E CEF in part of the BZ. This deviation from the CEF model, and its effects on the spectra, should be considered for a correct description of the splittings. In RIXS, as well as INS, the 4f ´dispersion causes a broadening or, more generally, a shape modification of the peaks as well as an apparent shift of the RIXS (and INS) peaks' position in the spectra. The main difference between RIXS-INS and PES lies in the method employed for the determination of the CEF splitting. PES extracts electrons from the 4f states and measures their energy and momentum: by applying the conservation laws, it can retrieve information on the energy E and momentum k of each electron before the interaction and hence determine the dispersion relation Epkq of the CEF levels. This also holds for resonant photoemission Figure 5.1: High-resolution (" 35 meV) RIXS spectra at 2θ " 90 ˝, specular geometry. The 46-73 meV splitting scheme (CFA), obtained by analyzing 2θ " 150 ˝data, doesn't fit the experimental data, while there they are accurately reproduced by using new values for the splittings, 43-63 meV (CFB), in simulations. [START_REF] Olson | Valence-band dispersion in angle-resolved resonant photoemission from LaSb[END_REF][START_REF] Olson | Crystal momentum conservation in resonant photoelectron emission[END_REF]. RIXS and INS, instead, measure the energy and momentum transfer required for an electronic transition between two 4f levels. The shape and energy of the spectral features are therefore affected by the dispersion Epkq of both levels, whose contributions are complexly entangled. Moreover, only the transferred momentum ∆k, fixed by the scattering geometry, is known, while the electrons' momenta in the initial and final states are undefined.

Note that this is only when studying electronic transitions. The energy and momentum of magnons and phonons is well defined by the energy and momentum transferred to the sample in the excitation process that creates them. The present chapter investigates the effects of the CEF levels' dispersion in RIXS spectroscopy and the possible errors induced in the determination of the CEF. The first section introduces the problem by showing a dependence of the observed CEF peaks' positions on the scattering angle. In fact, the effects of the dispersion affecting the RIXS features vary with the transferred momentum ∆k, i.e. with the scattering geometry. This will be explained in the second section, where a simple model for the interpretation of the peaks' shifts will be introduced. The comparison between data and models with different states' dispersions can even, for simple situations, provide a qualitative description of the levels' k´dependence, thus giving a more accurate description of the splittings. In the last section the CEF splittings obtained from RIXS spectra with various momentum transfers will be presented, confirming the qualitative agreement with the presented model.

Angular dependence of the CEF excitation energies

In the previous chapter, the analysis of CeRh 2 Si 2 RIXS spectra gave an order Γ 1 7 ´Γ6 Γ2

7 of the 2 F 5{2 CEF levels, with 46-73 meV splittings. The set of CEF parameters best reproducing the experimental data was Ǎ0

2 " 60 meV, Ǎ0 4 " ´76 meV and Ǎ4 4 " 194 meV, hereinafter referred to as CFA. The scattering geometry employed, (2θ " 150 ˝, δ " 0 ˝and χ " 0 ˝) corresponds to a momentum transfer of about 1.4 rlu, parallel to k z . The transition therefore occurs between states of neighboring Brillouin zones, which, once folded in the same zone, lie 0.4 rlu apart along k z , i.e. almost half the Brillouin zone size.

To investigate the effects of the CEF levels dispersion, the same set of data (π and σ polarisation, E in " 881.7, 882.3 eV) was acquired at 2θ " 90 ˝, corresponding to ∆k « 1 r.l.u. k z . Once the initial and final states are folded in the same Brillouin zone, they lie approximately in the same position, opposite to the case above, where the initial and final states were separated by approximately the maximum distance along k z . On the other hand, the comparison of 2θ " 150 ˝data with these two CEF schemes confirms that CFA better reproduces the experimental spectra at this angle. Therefore, the scattering geometry affects the estimate of the CEF splittings from RIXS data. This is due to the angular dependence of ∆k, which determines how the initial and final states' dispersions combine to shape (and shift) the peaks in the RIXS spectrum. The influence of the levels' dispersion on the CEF splitting determination will be discussed in the next section, together with a simple model to interpret the observed shifts of the peaks.

It is interesting to notice that the magnetic susceptibility curves are equally fitted by simulations with both CFA and CFB, provided that a suitable value of Ǎ0

2 , which gives the right J z (or α 2 ), is chosen. Figure 5.3 presents calculated curves for CFB, with Ǎ0

2 " 56 meV (J z " 1.41 µ B ) and Ǎ0

2 " 71 meV (J z " 1.76 µ B ), respectively fitting the experimental data from Refs. [START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF] and [START_REF] Mori | Effect of pressure on the magnetization of single crystal CeRh 2 Si 2[END_REF]. An identical fit was obtained using CFA in Figure 4.8, despite the relevant differences in the values of the splittings between the two CEF schemes.

Interpretation of the observed ∆k dependence

High resolution RIXS is normally employed to map the dispersion of collective excitations as phonons or magnons and Epkq is readily obtained by the angular dependence of the peak positions. In fact, these quasiparticles, not present before the interaction, are created with a defined energy E " E loss and momentum k " ∆k by the RIXS process.

On the other hand, f f excitations occur between two dispersing states, E i pk i q and E f pk f q. The condition on the momentum conservation defines a value for ∆k " k f ´ki , but not k i or k f individually. As a consequence, transitions in any point of the 4f levels' dispersions simultaneously contribute to the same RIXS feature. A peak in the spectrum corresponds to the energy required for a given excitation E loss " E excit. " E f ´Ei and, in case of dispersing levels, transitions occurring in different points of the reciprocal space will be characterized by different energies E f pk f q´E i pk i q " E f pk i `∆kq´E i pk i q. Therefore, an f f excitation will not give a single peak at ∆E CEF , but an intensity distribution proportional to:

I ∆k pE loss q " ż σpE f pk `∆kq ´Ei pkqqdk , (5.1)
where the cross section σ can depend on k. Under the assumption of constant cross section, the shape of a CEF feature is given by the energy difference, integrated over the reciprocal space, between the dispersions of the two levels "staggered" by ∆k. For strongly asymmetric shapes of I ∆k pE loss q, the measured peak's position can deviate substantially from the splitting ∆E CEF , that would be observable in absence of hybridization and dispersion.

By changing ∆k, the staggering between the two dispersions changes, modifying the feature's intensity distribution and, potentially, the peak's position in the spectrum. This is the origin of the shifts observed between experimental data at 2θ " 150 ˝and 90 ˝. The dependence of the observed peak's energy on ∆k is strictly related to the shape of the dispersions, therefore, by comparing the experimental shifts with E i pkq and E f pkq from ARPES measurements and band structure calculations, it is in principle possible to retrieve a qualitative description of the level's k´dependence.

An intuitive representation of the discussed effects is depicted in Figures 5.4 ARPES data in Figure 1.11 both Γ 1 7 and Γ 6 display a hole-like dispersion while Γ 2 7 shows opposite, electron-like behaviour [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF] .

As discussed above, at 2θ " 90 ˝the transition occurs between two equivalent points in the k´space (∆k " p0 0 1q rlu). At 2θ " 150 ˝, instead, the initial and final states are 0.4 rlu apart along k z . Following Equation 5.1, the (k´dependent) excitation energy is calculated as the distance between the ground level and the excited one shifted by ∆k. This relative ∆k z band shift is depicted in the middle rows of Figures 5.4 and 5.5, to visualize the excitation energies in different points of the reciprocal space for the two scattering geometries. The shape of a RIXS feature in the spectrum will be given by the distribution in energy of all the possible transitions with fixed ∆k between the two levels (blue bars in the bottom rows of Figures 5.4 and 5.5). At 90 ˝the excitation in the Γ 2 7 level (Figure 5.4) occurs mostly between flat portions of both the initial and final levels, giving a strong peak at the CEF splitting energy ∆E CEF . The transitions between the dispersing regions give only a minor contribution and produce a small tail towards higher energies. After the convolution with the experimental resolution (green line), the position of the peak is only slightly shifted (" 1 ´2 meV) towards higher energies compared to its theoretical value ∆E CEF in absence of dispersion (black line).

The effect of the levels' dispersion is much more relevant at 150 ˝, because, due to the relative shift of 0.4 rlu of the two bands, larger portions of the Brillouin zone contribute with transitions of energy E ą ∆E CEF . As a result, the observed peak's energy is about 11 meV larger than the value of the CEF splitting. Compared to the presented experimental splitting estimates (63 meV at 90 ˝and 73 meV at 150 ˝), this model reproduces the measured " 10 meV variation between the two scattering angles and suggests a value of about 62 meV for the Γ 1 7 -Γ 2 7 splitting for those regions of the BZ where the CEF levels do not show dispersion.

The Γ 6 peak undergoes much smaller shifts, because both Γ 1 7 and Γ 6 show a very similar dispersion at the Z point, keeping the distribution of the transition energies always close to ∆E CEF . If the shapes of the two dispersions were exactly the same, no shift of the peaks would be observed, and the measured splitting would coincide with ∆E CEF . To reproduce the observed shifts (43 meV at 90 ˝and 46 meV at 150 ˝) a stronger dispersion must be assigned to the Γ 6 state. This moves the RIXS peak towards lower energies in the 90 spectra, due to the arising of transitions with E ă ∆E CEF , while at 150 ˝the shift gets lost in the broader linewidth, caused by transitions energies both larger and smaller than ∆E CEF . A splitting ∆E CEF " 47 meV was imposed between the flat regions of the two dispersions, in order to get peaks' positions consistent with the experiments. Surprisingly, the estimated 47-62 meV scheme perfectly agrees with the splittings measured between the non-dispersing portions of RPES data [START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF], suggesting that the dispersion effects should always be considered in order to correctly compare the results of different techniques. The simple model presented already provides an intuitive interpretation of the observed angular dependence of the peaks and a tool for a qualitative analysis.

Mapping the reciprocal space

Folded back into the first Brillouin zone, the momentum transferred in the two scattering geometries correspond to positions close to the Γ point (∆k 2θ"90 ˝,δ"0 ˝) and the Z point (∆k 2θ"150 ˝,δ"0 ˝), i.e. the zone center and the zone boundary along k z . To further investigate the ∆k´dependence of the RIXS peaks, we acquired high-resolution spectra at various angles (Figure 5.6) and compared them with spectra calculated for the two CEF schemes CFA (46-73 meV) and CFB (43-63 meV) obtained at 2θ " 150 ˝(« Z point) and 2θ " 90 ˝(Γ point), respectively. Contrary to those acquired in specular geometry, the new spectra at 2θ " 150 ˝, with δ " 20 ˝and 60 ˝off-specular, are in better agreement with the CFB splitting scheme. Their transferred momenta, both with ∆k z 0.3 rlu and, respectively, ∆k x « 0.2 rlu and ∆k x « 0.5 rlu (∆k y " 0) move away from the Z point. On the other hand, the spectrum acquired in specular geometry at 2θ " 134 ˝, with a transferred momentum closer to the Z point, is reproduced by calculations with CFA splittings. It seems therefore that the CFA scheme only suits spectra whose momentum transfers lie within a region around the Z point, while other spectra are better reproduced by CFB. To map the reciprocal space in more points, we acquired 60 meV-resolution spectra (Figure 5.7) varying δ in steps of 5 ˝, for both 2θ " 150 ˝and 90 ˝, and φ " 0 ˝(∆k xy " ∆k x , ∆k y " 0) and 45 ˝(∆k x " ∆k y " ∆k xy { ?

2). The CFA scheme correctly reproduces the spectra at 150 ˝scattering angle for small values of δ, while at about δ " ˘20 ˝the calculations start to deviate from the experimental data. At larger δ, instead, the the spectra are better reproduced using the CFB scheme. On the other hand, 90 ˝data are all in agreement, within the experimental uncertainties, with the CFB scheme. No difference is observed between the two orientations of the sample ϕ " 0 ˝and 45 ˝. All these findings enforce the conclusions above that the CFA scheme describes the experimental observations in a restricted region around Z (approximately 150 ˝, δ " 0 ˝), while CFB fits the data everywhere else. Due to the noise and the lowered energy resolution, it was not possible to investigate the transition between the CFA and CFB schemes, or to try fitting the spectra at new angles using other splittings. By plotting in the reciprocal space the transferred momenta (∆k z , ∆k xy ) of all the CeRh 2 Si 2 spectra presented, one obtains the maps in Figure 5.8, where the colors of the dots indicate which CEF scheme better fits the data.

After folding the points in their equivalent positions within the first Brillouin zone, the special character of spectra with ∆k near the Z becomes evident. A more complete model than the one presented above might be needed to fully understand the details of this ∆k dependence of the CEF peaks in RIXS. Nonetheless, this first observation of an angular dependence of f f excitations is crucial to better interpret the results of RIXS and INS. With a detailed theoretical analysis of these effects, it could be possible to completely determine the three-dimensional dispersion of 4f levels from the angular dependence of the 4f features. Thanks to the reduction of the elastic intensity with π polarisation, a first inelastic peak is visible at 30 meV (dashed line). A shoulder is visible at higher energies (dotted line), in addition to a broad tail up to 170 meV. Spectra have been renormalized to easy the comparison between σ and π polarizations.

other (T c2 " 2 K and P c2 « 4 GPa) to orbital or valence fluctuations [START_REF] Holmes | Valence instability and superconductivity in heavy fermion systems[END_REF][START_REF] Thomas | Precise determination of the pressure dependence of T c in the heavy-fermion superconductor CeCu 2 Si 2[END_REF][START_REF] Miyake | New trend of superconductivity in strongly correlated electron systems[END_REF][START_REF] Hattori | Meta-orbital transition in heavy-fermion systems: Analysis by dynamical mean field theory and self-consistent renormalization theory of orbital fluctuations[END_REF][START_REF] Pourovskii | Theoretical prediction and spectroscopic fingerprints of an orbital transition in CeCu 2 Si 2[END_REF].

Ce is close to 3`valence and the sample's magnetic properties at T ą T K « 10 K describe a system of non-interacting local 4f moments. Below this temperature a nonmagnetic state develops and the electrons at the Fermi level acquire huge effective masses. The specific heat discontinuity at T c is enhanced by a factor of 10 3 if compared to normal superconductors, similar to the normal-state electronic specific-heat coefficient, confirming the participation of the heavy fermions in the superconducting behavior [START_REF] Stewart | Heavy-fermion systems[END_REF][START_REF] Grewe | Handbook on the physics and chemistry of rare earths[END_REF][START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF].

Boosted by these discoveries, several studies have been addressed to the determination of the CEF splitting scheme in CeCu 2 Si 2 . The first INS investigation [START_REF] Horn | Magnetic neutron scattering and crystal-field states in CeCu 2 Si 2[END_REF] assigned two peaks to the lowest CEF levels, with splittings of 12 meV and 30 meV and this result was roughly confirmed by another work on non-stoichiometric CeCu 2`x Si 2 [START_REF] Holland-Moritz | Crystal-field splitting and magnetic relaxation in CeCu 2.02 Si 2[END_REF]. Inelastic scattering of polarised neutrons [START_REF] Stassis | Polarized-neutron study of the paramagnetic scattering from CeCu 2 Si 2[END_REF][START_REF] Uemura | Polarized neutron scattering from CeCu 2 Si 2 in applied magnetic fields[END_REF] revealed the presence of the peak at 30 meV, but nothing was detected at lower energies. The discrepancy was attributed to to the lower resolution and the higher noise of the polarised spectra. Later, the presence of a crystal-field level at 12 meV was further disproved by specific-heat and INS measurements [START_REF] Bredl | Specific heat and thermal expansion of CeCu 2 Si 2 at low temperature[END_REF] [103] and the most accepted CEF scheme is now the one proposed in Ref. [START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF], with two closely spaced excited levels at about 30 meV. Anyway, all these studies could only investigate excitation energies up to approximately 50 meV. Therefore CeCu 2 Si 2 constitutes an optimal test for the ability of RIXS of studying small CEF splittings and, at the same time, investigate higher energy excitations.

Figure 6.2: Experimental data in a scattering geometry that enhances the cross section for Γ 1 7 Ñ Γ 2 7 transition. The shoulder is here more visible, suggesting the presence of a peak at 47 meV. Simulations with a 30-47 meV splitting scheme prodivide a good fit of the data ( Ǎ0

2 " 40 meV, Ǎ0 4 " ´50 meV and Ǎ4 4 " 117 meV). The quality of the fit further improves by adding a Gaussian at 95 meV. On the other hand, the CEF proposed in Ref. [START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF] give a very different spectra.

Determination of the crystal electric field splittings

Splittings smaller than the experimental resolution can be detected by exploiting the angular and polarisation dependence of the cross sections. With the present experimental resolution of " 35 meV, the CEF peak expected at 30 meV might be hidden by the neighboring (and usually much stronger) elastic peak, which must be strongly reduced to allow for the detection of the f f transitions. A consistent portion of the elastic intensity is due to diffuse scattering and has a in ¨ out polarisation dependence [START_REF] Ament | Resonant inelastic X-ray scattering studies of elementary excitations[END_REF]. This contribution can be minimized by using π polarised photons and detecting the signal at 2θ " 90 ˝: in these conditions the polarisation vectors of the outgoing photons are always perpendicular to in , giving a zero scalar product. The effect is shown in Figure 6.1, where spectra acquired with both σ and π incident polarisation are compared, the latter showing a reduced elastic peak intensity. In these spectra, an inelastic peak arises at 30 meV, (indicated by the dashed line) in agreement with the previous studies.

In addition to this first peak, the spectra show some intensity up to 170 meV. In part, it can be assigned to excitations in the conduction band, similarly to what was discussed before for CeRh 2 Si 2 . Yet, a small shoulder is visible in all spectra on the high-energy side of the first f f excitation (dotted line). This feature may represent another f f excitation at higher energy, not detected by the previous studies due to the limitations on the maximum Figure 6.3: Fit of the inverse magnetic susceptibility data from Ref. [START_REF] Batlogg | Magnetism and superconductivity in CeCu 2 Si 2 single crystals[END_REF] using the proposed set of CEF parameters. excitation energy. To verify this hypothesis, we selected experimental conditions that enhance the excitation in the Γ 2 7 state, which was found to be the highest of the three 2 F 5 {2 levels in CeRh 2 Si 2 (Figure 6.2). The presence of a second feature over 30 meV is clear from the experimental spectrum. By assigning the first peak to the Γ 6 state and the second to the Γ 2 7 , the best fit is obtained with 30 -47 meV splittings and CEF parameters Ǎ0 2 " 40 meV, Ǎ0

4 " ´50 meV and Ǎ4 4 " 117 meV. The quality of the fit improves by adding a broad gaussian centered at 95 meV, to account for the extra intensity probably due to CT-like features.

On the other hand, the CEF scheme proposed in Ref. [START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF] can't reproduce correctly the experimental data.

The new set of CEF parameters can also fit the inverse magnetic susceptibility curves χ ´1pT q presented in Ref. [START_REF] Batlogg | Magnetism and superconductivity in CeCu 2 Si 2 single crystals[END_REF], which were used in Ref. [START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF] to test the soundness of the 30 -30 meV scheme (Figure 6.3). Interestingly, a weak feature at 50 meV has been observed by Raman scattering in Ref. [START_REF] Cooper | Raman scattering study of the electronic and vibrational excitations in CeCu 2 Si 2[END_REF] and it was there assigned to a phonon mode of the Silicon ions. Yet, its symmetry and temperature dependencies, as well as its absence at low temperatures on the parent compound LaCu 2 Si 2 , are compatible with a CEF transition.

All the data acquired can be fitted using the proposed scheme, with two separated f f excitations and a broad feature at higher energies (Figure 6.4). Unfortunately the spectra are not well defined, mostly due to a weak second peak, and it is difficult to estimate the accuracy of the simulations. Therefore, despite the intriguing results, the energy resolution The simulations best reproducing the data have a slightly modified scheme, with the second peak positioned at 55 meV instead of 47 meV. The shoulder is anyway weak and the broad feature at 95 meV strongly modifies the shape of the spectra, not allowing for a clear identification of the features. is still insufficient to investigate more in detail the CeCu 2 Si 2 CEF splittings and the nature of the observed spectral features. Moreover, the broad and relatively intense peak at 95 meV introduces uncertainty in the data analysis and it should be better identified to provide a more reliable interpretation of the data.

Determination of the wavefunctions' orientation without polarisation analysis

The orientation of the CEF wavefunctions in CeCu 2 Si 2 was determined by Willers et al.

with non-resonant X-ray scattering [START_REF] Willers | Determining the in-plane orientation of the ground-state orbital of CeCu 2 Si 2[END_REF], reaching the same conclusions found in section 4.3 for CeRh 2 Si 2 . The investigation of such effects on this sample can serve as an example of the RIXS ability of studying the CEF properties in presence of small splittings.

With such close and overlapping features an analysis of the polarisation of the scattered beam, as the one carried out for CeRh 2 Si 2 , is not possible. The intensity loss (" 90%) on the multilayer requires indeed to lower the resolution to " 60 meV, in order to recover a reasonable photon flux on the detector. With this resolution, the features would be indistinguishable on the polarised spectra. Anyway, it is possible to find an experimental condition that maximized the dependence of the (non-polarised) cross sections on the property of interest. As shown in Figure 4.16, the intensity ratio between the Γ 6 and Γ 2 7 peaks depends on the scattering angles and on the sign of the Ǎ4 4 parameter, which determines the wavefunctions' orientation. At about 90 ˝, depending on the Ǎ4 4 sign, the two transitions can have The only estimate of the CEF scheme given in literature results from a point-charge model calculation and predicts splittings of 14 and 22 meV for the lowest CEF levels [START_REF] Fujii | Magnetic properties of CeCo 2 Ge 2 and NdCo 2 Ge 2 single crystals[END_REF].

Direct observation of the entire 4f splitting scheme

The determination of the 4f splittings is straightforward from CeCo 2 Ge 2 RIXS data. The 7 CEF split states produce well separated peaks, which can be easily identified on the spectrum shown in Figure 6.6. The level's energies determined by fitting with Gaussians are 0 -62 -120 meV and 260 -315 -365 -460 meV for the three 2 F 5{2 and the four 2 F 7{2 -split states respectively. Moreover the features show different widths, especially those belonging to the 2 F 7{2 multiplet.

The splittings are extremely large if compared to the other 122 cerium compounds, as a possible consequence of the strong 4f hybridization. The same applies for the peak widths, which may reflect the dispersive character of the 4f states observed in Ref. [START_REF] Venturini | The Ce 4f electronic structure in CeCo 2 Ge 2 : a soft x-ray resonant photoemission investigation[END_REF]. The influence of effects other than purely electrostatic might also explain the strong disagreement with the point-charge model predictions.

The possibility of a detailed study of such large splittings is a specific characteristic of RIXS spectroscopy. The large energy scales can indeed be the reason why 4f splittings in Figure 6.7: Calculated levels' energies varying Ǎ0 2 , after imposing the 62-120 meV splittings of the 2 F 5{2 multiplet ( Ǎ0 6 , Ǎ4 6 " 0). Without the higher order parameters, the 2 F 7{2 splittings (dotted lines) are not correctly reproduced for any Ǎ0

2 . The calculated peak intensities vary considerably with Ǎ0

2 .

CeCo 2 Ge 2 have not been observed with techniques such as INS and specific heat, limited in maximum excitation energy.

Crystal electric field analysis

The observation of the 2 F 7{2 levels' splittings on the spectra allows, in principle, for a more complete determination of the CEF potential. In fact, so far the analysis have been limited

to the 2 F 5{2 splittings, which, within Stevens' approximation, are determined by Ǎ0 2 Ǎ0

4 and Ǎ4

4 . The 2 F 7{2 splittings are affected also by parameters of higher order, namely Ǎ0 6 and Ǎ4 6 . The effect of these parameters is usually neglected when studying the CEF on cerium but must now be considered for a better interpretation of the CeCo 2 Ge 2 RIXS spectra.

As a starting point, the method presented in section 4.2.3 can be employed, neglecting, for the moment, Ǎ0

6 and Ǎ4 6 . The reduced set of parameters is determined as a function of Ǎ0 2 by imposing the measured 2 F 5{2 splittings in Equation 1.28. Figure 6.7 reports the calculated levels' energies and RIXS peaks' intensities varying Ǎ0 2 (for σ polarisation, 2θ " 150 ˝and specular geometry). The calculated 2 F 7{2 splittings don't match the measured values (dotted lines) for any Ǎ0

2 , confirming the need of higher order parameters to correctly describe these levels. 2 " 17 meV, Ǎ0

4 " ´73 meV, Ǎ4 4 " 377 meV, Ǎ0 6 " 220 meV and Ǎ4 6 " 200 meV.

Moreover, the intensity of the first two peaks at 62 and 120 meV varies with Ǎ0 2 . This dependence was already observed in the analysis of CeRh 2 Si 2 spectra (Figures 4.6 and 4.7), but it is here enhanced by the stronger CEF effects. Such a strong intensity variation can be used to determine the value of Ǎ0 2 , by comparison between calculations and measurements. In fact, the other methods presented in chapter 3 to find the Ǎ0 2 value cannot be used for CeCo 2 Ge 2 . The magnetic susceptibility and XAS data are strongly affected by the 4f mixed valence, while the RIXS signal of the 2 F 7{2 multiplet is influenced also by the Ǎ0 6 and Ǎ4 6 parameters.

After defining the possible (reduced) sets of parameters that reproduce the 2 F 5{2 splittings, Ǎ0 6 and Ǎ4 6 shall be determined. This can be done by applying the same analysis of above, but now focusing on the 2 F 7{2 states. The measured splittings are imposed in calculations and the best CEF scheme is chosen by comparing simulations to experimental data. In principle, the introduction of parameters of higher order should keep the 2 F 5{2 splittings unvaried, as long as the relations among Ǎ0

2 , Ǎ0 4 and Ǎ4 4 are not changed. Yet, for such large splittings the Stevens' condition ∆E CEF ! ∆E spin-orbit is not satisfied and there is a non-negligible mixing between the two spin-orbit multiplets. As a consequence, the splittings of the 2 F 5{2 -derived states will be slightly modified by the introduction Ǎ0 6 and Ǎ4

6 and small corrections of the parameters are needed. The final set of parameters is: Ǎ0

2 " 17 meV, Ǎ0 4 " ´73 meV, Ǎ4 4 " 377 meV, Ǎ0 6 " 220 meV and Ǎ4

6 " 200 meV. It allows to perfectly reproduce the experimental spectra at 2θ " 150 ˝, as shown in Figure 6.8. Yet, simulations with these parameters are not able to fit the spectra acquired at 2θ " 90 ˝scattering angle (Figure 6.9). In particular, there is a strong disagreement between the calculated and observed intensities, while the shift of the Therefore the approach presented in this thesis, based on the CEF theory, fails to interpret the CeCo 2 Ge 2 spectra. It is possible that the peak intensities are affected by interactions that cannot be explained with a simple CEF model. Further work is needed to understand the causes of the large splittings observed in CeCo 2 Ge 2 and their effects on the RIXS cross sections. This first detection of unexpectedly large splittings is very interesting for the physics of cerium intermetallics, and would not have been possible with other techniques. Moreover, RIXS spectroscopy, by studying the polarisation and angular dependence of each feature, allows for a more precise analysis and interpretation of the data, thus providing a privileged tool for future studies on large 4f splittings. energies higher than expected, which would disprove the currently accepted crystal field scheme. Yet the results are uncertain due to both the insufficient resolution and a broad feature in the same energy range probably of charge transfer nature. Large splittings allow instead to clearly resolve and study all the f f transitions, including the excitations into the 2 F 7{2 states. This was shown with the example of CeCo 2 Ge 2 . However, in this case the presented crystal-field approach failed in the interpretation of the data. The simulations could not in fact reproduce the cross-sections variation with the scattering geometry, suggesting the influence of phenomena not considered within the crystal-field theory.

The demonstrated ability of RIXS to probe the f f transitions opens up many possibilities for future studies. In particular, the angular dependence of the measured splittings needs to be addressed both with systematic experiments and theoretical work. A detailed mapping of these effects will provide deeper insight in the 4f states' energies, hybridization and dispersion. Yet, this will be possible only if a refined model for the interpretation of the measured effects will be available. The broad peak detected around 100 meV in both CeRh 2 Si 2 and CeCu 2 Si 2 and the anomalous spectra of CeCo 2 Ge 2 also deserve further investigation. The interactions of the 4f levels with the surroundings cause indeed modifications of the spectra that need to be understood for a more accurate comparison between simulations and measurements.

In the future, the RIXS investigation of 4f levels will benefit from further technological advances. New soft-RIXS spectrometers are in construction in many facilities, as NSLS-II in Brookhaven (NY, USA), MAX-IV in Lund (Sweden), Diamond Light Source (UK) and NSRRC (Taiwan). With the expected improvements in the experimental resolution smaller splittings will become accessible and an even more detailed analysis will be possible. This thesis explored the potential of RIXS and laid the groundwork for new studies on the physics of 4f electrons. Soft-RIXS demonstrated to be complementary to traditional techniques and free of many of their disadvantages. Its greatest limitation, the experimental resolution, is being overcome thanks to the rapid improvements of the instrumentation. With the present work, a new field of research opens up and brings a wealth of new possibilities to probe the fascinating properties of rare-earth compounds.

Cerium intermetallic compounds exhibit fascinating magnetic, electronic and thermodynamic properties at low temperatures. Their physics is governed by two competing interactions between the conduction band and localised 4f electrons: the Kondo effect tends to screen the 4f magnetic moments while RKKY exchange favours ordered local moments. From this competition, a complex phase diagram arises and its understanding requires a full characterisation the 4f levels. The crystal electric field (CEF) effects lift the degeneracy of the spin-orbit multiplets and the resulting energy splittings (tens of meV) as well as the symmetry of the ground and excited states must be determined experimentally. CEF effects in the lanthanides can be studied with a number of experimental probes including magnetic susceptibility, inelastic neutron scattering, x-ray absorption spectroscopy or photoemission spectroscopy. However, each of these techniques has limitations which can prevent an accurate characterisation. This thesis discusses the potential of Resonant Inelastic soft X-ray Scattering (soft-RIXS) to become a new probe of the CEF in cerium compounds. The latest advances in instrumentation have pushed the energy resolution in soft RIXS down to 30 meV at the Ce M5 edge, good enough to resolve the electronic transitions between the lowest lying 4f levels. This is demonstrated using the example of CeRh2Si2 spectra. The observed energies of the spin-orbit and CEF excitations provide a direct measurement of the 4f splittings and the comparison of the experimental spectra with full-multiplet single-ion calculations unambiguously identifies the symmetry of the lowest CEF levels. The orientation of the wave-functions in the crystal ab plane can also be determined in a straightforward manner by measuring the polarisation of the inelastically scattered photons. The interaction with itinerant states can lead to a momentum dependence of the 4f levels' energy and this leaves signatures in the excitation spectra that can be observed in RIXS. A simple model is proposed to explain the observed q dependence but further theoretical work is needed to properly account for the momentum dependence of the 4f excitations. Using the examples of CeCu2Si2 and CeCo2Ge2 it is demonstrated that RIXS can probe CEF splittings that are significantly smaller or much larger than those in CeRh2Si2. With better energy resolution and more advanced modelling RIXS studies of the CEF in Cerium and other lanthanide ions will become even more accurate in the future.

Résumé

Les composés intermétalliques à base de cérium présentent à basse température des propriétés magnétiques, électroniques et thermodynamiques tout à fait fascinantes. Deux interactions majeures entre la bande de conduction et les électrons localisés de la couche 4f entrent en compétition et régissent la physique de ces composés: l'effet Kondo tend à écranter les moments magnétiques 4f alors que les interactions RKKY favorisent plutôt des moments localisés et ordonnés. Un diagramme de phase complexe naît de cette compétition et le comprendre exige une caractérisation complète des niveaux 4f. La dégenerescence des multiplets liés au couplage spin-orbit est levée par les effets de champ cristallin, et les écarts entre les niveaux d'énergie qui en résultent ainsi que les symétries des états fondamentaux et excités doivent être déterminés expérimentalement. Les effets de champ cristallin dans les lanthanides peuvent être sondés par diverses techniques expérimentales dont la susceptibilité magnetique, la diffusion inélastique de neutrons, la spectroscopie par absorption de rayons X ou de photoémission. Cependant, chacune de ces techniques présente des limitations propres qui empêchent une caractérisation précise de ces effets. Nous présentons dans cette thèse les capacités de la technique de diffusion résonante inélastique des rayons X mous (soft-RIXS) à sonder les effets de champ cristallin dans les composés à base de cérium. Les dernieres avancées en instrumentation ont permis de repousser les limites de la résolution en énergie jusqu'à 30 meV au seuil M5 du Cérium, suffisant pour séparer les transitions électroniques des plus bas des niveaux 4f. Le composé CeRh2Si2 a été étudié à titre d'exemple. Les énergies des excitations de couplage spin-orbit et de champ cristallin observées expérimentalement fournissent une mesure directe des écarts des niveaux 4f, tandis que la comparaison entre les spectres expérimentaux et des calculs basés sur le multiplet complet dans l'approximation à un ion permet d'identifier sans ambigüité la symétrie des états de plus basse énergie du champ cristallin. L'orientation des fonctions d'ondes dans les plans ab du cristal peut aussi être déterminée de manière très simple en mesurant la polarisation des photons diffusés de manière inélastique. L'interaction avec les états itinérants peut introduire une variation de l'énergie des niveaux 4f en fonction du vecteur de diffusion q, qui se traduit par des signatures observables dans les spectres de diffusion inélastique résonante de rayons X. Un modèle simple est proposé pour expliquer la variation observée en fonction de q, mais des calculs théoriques plus poussés apparaissent nécessaires pour rendre compte de manière plus exacte de ces variations. L'étude des composés CeCu2Si2 et CeCo2Ge2 montre que la diffusion inélastique des rayons X peut sonder des effets de champ cristallin bien plus faibles ou bien plus importants que ceux observés dans CeRh2Si2. Dans le futur, une meilleure résolution en énergie et des modélisations plus poussées devraient encore améliorer la précision des études des effets de champ cristallin dans les composés de Cérium et autres lanthanides par diffusion inélastique de rayons X.

6

  Crystal field excitations in other Ce 122 compounds 6.1 CeCu 2 Si 2 : crystal field splittings below the experimental resolution . . . . . . 6.2 CeCo 2 Ge 2 : large splittings and limitations of the crystal field model . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Qualitative illustration of the Doniach phase diagram. The different dependencies of T K and T RKKY on an external control parameter δ produce diverse ground states. An antiferromagnetic (AFM) ordering develops in the region where T RKKY ą T K , while for T RKKY ă T K a paramagnetic Fermi liquid (FL) phase is formed. In between the two phases, a quantum critical point (QCP) can arise, in the vicinity of which exotic properties as non-Fermi liquid behaviour (NFL) and unconventional superconductivity may be found.Figure redrawn after Ref. [17].

Figure 1 . 2 :

 12 Figure 1.2: Figures reproduced from Refs. [4, 5] representing, as a function of E f , (a) the extended Doniach phase diagram and (b) the energy scheme and density of state of the system.

Figure 1 . 4 :

 14 Figure 1.4: Energy splitting of the 4f 1 configuration in tetragonal symmetry and the charge distribution of each CEF state arising from the 2 F 5{2 multiplet.

Figure 1 . 5 :

 15 Figure 1.5: Magnetic susceptibility data for CeRh 2 Si 2 , reproduced from Ref.[START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF]. The fit of the data lead to a CEF scheme Γ 7 -Γ 6 -Γ 7 , with 31 and 80 meV splittings. The CEFrelated part is the one at high temperatures (ą 100 K), while the decreasing of χ at low temperatures has to be ascribed to magnetism and other phenomena. Note that another magnetic susceptibility study, in Ref.[START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF], gave 27 and 59 meV splittings and a Γ 7 -Γ 7 -Γ 6 scheme.

Figure 1 . 6 :

 16 Figure 1.6: Magnetic part of the specific heat of YbIr 2 Si 2 as function of temperature. The data is compared to calculations for two CEF schemes obtained with photoemission spectroscopy (solid line) and inelastic neutron scattering (dashed line). Figure reproduced from [73].

Figure 1 . 7 :

 17 Figure 1.7: INS data for CeRh 2 Si 2 , reproduced from Ref.[START_REF] Willers | Spectroscopic determination of crystal-field levels in CeRh 2 Si 2 and CeRu 2 Si 2 and of the 4f 0 contributions in CeM 2 Si 2 (M =Cu, Ru, Rh, Pd, and Au)[END_REF]. The open dots represent the phonon scattering, scaled from the high to low scattering angles. The red lines represent the signal attributed to CEF peaks. Combining these data with XAS measurements (Figure1.9), a CEF scheme Γ1 7 -Γ 2 7 -Γ 6 , with 30 and 52 meV splittings, was obtained.

Figure 1 . 8 :

 18 Figure 1.8: Left: Figure taken from Ref[START_REF] Höppner | Interplay of Dirac fermions and heavy quasiparticles in solids[END_REF], showing the dichroism of XAS spectra in the case of pure J z states (depicted in the insets) as ground wavefunctions for the 4f electron. Right: calculated spectra for a Γ 7 ground state, for various values of the α 2 mixing parameter (see section 4.2.2 for more information). To observe the XLD, XAS spectra with light polarisation c and with a K c are compared, a and c representing the crystallographic axes of the tetragonal unit cell.

Figure 1 . 9 :

 19 Figure 1.9: XAS data and simulations for CeRh 2 Si 2 , reproduced from Ref. [51]. Calculations with a Γ 7 ground state |α| " 0.73 reproduce the experimental spectra. The order of the levels obtained from these data, combined with those of Figure 1.7 was Γ 1 7 -Γ 2 7 -Γ 6 , with 30 and 52 meV splittings.

Figure 1 . 11 :

 111 Figure 1.11: Resonant Photoemission data of CeRh 2 Si 2 from Ref [88]. The two features labelled A and B correspond to the signal due to CEF levels. The splittings estimated for the 2 F 5{2 multiplet are 48 (A) and 62 meV (B), with a Γ 1 7 -Γ 6 ´Γ2 7 scheme. A dispersion of the levels due to hybridisation with the approaching valence band is clearly visible.

Figure 2 . 2 :

 22 Figure 2.2: RIXS spectra of the mixed valence compound EuNi 2 P 2 at different incident energies accross the M 5 edge (left) and corresponding XAS spectrum (right). While in the XAS spectrum features coming from Eu 2`a nd Eu 3`a re superimposed, the RIXS spectra clearly changes depending on the resonance selected.

Figure 2 . 3 :

 23 Figure 2.3: M 5 RIXS spectra of EuCo 2 Si 2 at different incident energies accross the edge (left) and simulations for a Eu 3`( right) ion, showing the manifold of multiplet excitations. Crystal field was not considered in this simulation.

Figure 2 . 4 :

 24 Figure 2.4: (Left) M 4,5 XAS spectra, with π polarisation and 40 ˝incidence, of three different cerium compounds, together with calculated spectra for pure Ce 4`a nd Ce 3`i n absence of crystal field. Both total electron yield (blue) and fluorescence yield (magenta) are shown, the latter enhancing features with low intensity. The coloured dots represent the energy at which RIXS spectra were acquired. (Right) Corresponding M 5 RIXS spectra, with σ (red) and π (blue) polarisation, 2θ " 130 ˝, θ " 40 ˝(normal emission).

XAS spectra of Nd 1 . 84 Figure 2 . 4 .

 18424 Figure 2.4. By comparing the experimental spectra with the calculated ones for isolated cerium ions, NCCO and CeRu 2 P 2 can be considered respectively as (almost) pure Ce 4`a nd Ce 3`, while CeCu 2 Si 2 shows spectral signatures of both configurations. M 5 RIXS spectra of NCCO (Figure 2.4, right) confirm its 4f 0 nature. Since no electron is present in the 4f subshell before the 3d Ñ 4f excitation, the only possible 4f Ñ 3d emission process is the

Figure 2 . 5 :

 25 Figure 2.5: (Left) RIXS spectra CeCu 2 Si 2 and CeRu 2 P 2 (Ce3`) with incoming photon energy between the two M 5 XAS peaks with π polarisation. The spectra are not fit by calculations without CEF, proving that RIXS is sensitive to its effects. The difference between the two spectra in the low energy range (ă 0.5 eV) are due to different CEF schemes. NCCO (Ce 4`) does not show any low energy inelastic features. (Right) RIXS spectra for CeCu 2 Si 2 and corresponding spectra calculated using different sets of CEF parameters. The calculations without CEF (0) don't fit the data, while introducing the CEF the fit improves. A uses CEF parameters obtained from INS[START_REF] Goremychkin | Crystal-field excitations in CeCu 2 Si 2[END_REF], with a Kramer's doublet ground state and a quartet at 29 meV (0-29-29 meV splitting scheme), while B1 and B2 are obtained by sligtly modifying the CEF (respectively giving splitting of 0-29-34 meV and 0-33-41 meV). The three schemes provide a similar fit of the magnetic susceptibility curves (see Ref.[START_REF] Amorese | 4f excitations in Ce Kondo lattices studied by resonant inelastic X-ray scattering[END_REF] for details). Even with 250 meV resolution, small variations of the CEF have visible effects on the spectra.

Figure 2 . 6 :

 26 Figure 2.6: Charge transfer process as described in Refs[START_REF] Nakazawa | Theory of resonant X-ray emission spectra in Ce compounds[END_REF][START_REF] Nakazawa | Theory of polarization dependence in resonant X-ray emission spectroscopy of Ce compounds[END_REF]

Figure 2 . 7 :

 27 Figure 2.7: M 4 RIXS maps with vertical polarisation of the incident light and corresponding XAS spectra. Vertical lines indicate the spectra used to build the map. For NCCO, the comparison between the two polarisations is also shown.

Figure 2 . 8 :

 28 Figure 2.8: M 4,5 RIXS maps for CeCu 2 Si 2 with horizontal (H) and vertical (V) polarisation. Zoom on CT region.

Figure 2 .

 2 Figure 2.8 shows the M 4 and M 5 RIXS maps of CeCu 2 Si 2 with both vertical and horizontal polarised light. The excitations and their energy positions are the same at the two edges, but the intensity ratio between the CT and the f f signal is higher at the M 4 edge. The polarisation dependence of CT transitions observed for NCCO is confirmed in CeCu 2 Si 2 spectra, where the intensity at 2 eV is always stronger with vertical polarisation of the incident light.

Figure 2 . 9 :

 29 Figure 2.9: M 4,5 RIXS maps for CeCu 2 Si 2 with horizontal (H) and vertical (V) polarisation. Zoom on high energy losses.

Figure 2 . 10 :

 210 Figure 2.10: Scheme of the 5p Ñ 3d emission and the MMN Coster-Kronig transition that may occur during a RIXS process.

  The whole ID32 beamlime was built in the framework of a major ESRF upgrade program (ESRF Upgrade Phase I) started in 2009 and ended in 2015. The design and commissioning of the ERIXS (European-RIXS) spectrometer and its polarimeter are the result of a long-standing and close collaboration between the group of Professors Giacomo Ghiringhelli and Lucio Braicovich of Politecnico di Milano (Dipartimento di Fisica) and the ESRF (in particular, the ID32 team led by Nicholas Brookes). I had the privilege of following the development of the beamline and spectrometer and dedicated a considerable portion of the

Figure 3 . 1 :

 31 Figure 3.1: Sketch [111] and photo [112] of an APPLE II undulator.

Figure 3 . 2 :

 32 Figure 3.2: Scheme of the optical layout and floor plan of ID32 beamline.

Figure 3 . 3 :

 33 Figure 3.3: Sample mounted on a shuttle (seen through a loadlock window) and the sample stage. The sample is usually glued in the middle of the sample holder. A piece of black carbon tape is also placed on the holder to provide a purely elastic reference for the spectra.

Figure 3 . 4 :

 34 Figure 3.4: Definition of the angles and the 4-circle goniometer.

Figure 3 . 5 :

 35 Figure 3.5: Picture of the ID32 spectrometer.

Figure 3 . 6 :

 36 Figure 3.6: Scheme of soft-RIXS grating spectrometer based on a single optical element. Panel (a): Perspective view. Panels (b) and (c): Top view, in the absence of horizontal focusing (b) and in the case of horizontal collimation by a parabolic mirror (c). Figure from [118].

Figure 3 .

 3 Figure 3.7: A M 5 RIXS spectrum of CeCo 2 Ge 2 is used to compare the resolution performances of AXES spectrometer (ID08) and ERIXS (ID32). For the latter, spectra acquired in both the high efficiency and high resolution modes are presented.

Figure 3 . 8 :

 38 Figure 3.8: Scheme of ID32 spectrometer with the polarimeter.

  multilayer (paying attention not to saturate and damage the detector) or the diffused scattering by a polycrystalline non-resonant material, usually carbon tape, placed in the sample position. The intensity ratios I pol {I dir measured with incident π and σ polarisation, give the two estimates for r π and r σ , respectively. Usual values for the measurements presented in this thesis are r π « 9.1% and r π « 14.2%, in good agreement with measurements of the multilayer's reflectivity [121].To avoid artifacts in the subtraction of the two spectra I pol and I dir in Equation 3.3, one must put particular attention to having the same response function (the same resolution)

ure 3 . 6 ,Figure 3 . 10 :Figure 3 . 11 :

 36310311 Figure 3.10: Top view of a RIXS spectrometer and polarimeter in absence of a collimating mirror. Since different portions of the beam hit the multilayer with slightly different angles θ, the spacing of the multilayer needs to be graded in the horizontal direction in order to fulfil the Bragg conditions for the whole beam.

Figure 3 . 13 :

 313 Figure 3.13: Sketch of the improvement in the effective spatial resolution for an inclined CCD. The gain in spatial resolution is sin γ, being γ the angle between the detector's surface and the optical axis.

Figure 3 . 14 :

 314 Figure 3.14: Left: Spatial resolution gain obtained with single photon counting (SPC) instead of the usual integration of raw images (Raw). The latter is rescaled to the SPC maximum in order to highlight the different broadenings. Right: Comparison of two detectors, the Princeton PI-SX 2048 and the Andor iKon-L (currently mounted on ERIXS). The resolution improvement, here shown for the medium resolution set-up, is the same for the two devices, but the latter has an higher readout frequency, allowing the reduction of the duty cycle.

2 2 CeRh 2 Figure 4 . 1 .

 22241 Figure 4.1. Despite being already known as a heavy fermion system with two distinct

Figure 4 . 1 :

 41 Figure 4.1: Crystal structure of cerium intermetallic compounds of the 122 family. For CeRh 2 Si 2 a " b " 4.09 Å and c " 10.18 Å [127]. The samples grow as flat platelets with the crystallographic c axis pointing out of the surface.

Figure 4 . 2 :

 42 Figure 4.2: M 5 XAS spectra of CeRh 2 Si 2 with photon polarisations parallel to the a and c axes. The photon energies corresponding to the RIXS spectra presented in the following are indicated as A (881.7 eV), B (882 eV) and C (882.3 eV).

Figure 4 .

 4 Figure 4.3 also shows the calculated spectra that best fit the data, with very good agreement. The sticks indicate the energy and intensity of the inelastic 4f ´4f transitions,

Figure 4 . 3 : 2 "

 432 Figure 4.3: M 5 RIXS spectra of CeRh 2 Si 2 at 2Θ " 150 ˝and specular geometry. The experimental spectra are perfectly fit by simulations with CEF parameters Ǎ02 " 60 meV, Ǎ04 " ´76 meV and Ǎ4 4 " 194 meV. i.e. with a splitting of 46-73 meV of the 2 F 5{2 multiplet, with an order Γ 1 7 ´Γ6 ´Γ2 7 of the three states.

Figure 4 . 4 (

 44 left) compares spectra calculated with different splitting schemes, among which the 46 ´73 meV option shows undoubtedly the best agreement with experimental data.

Figure 4 . 4 :

 44 Figure 4.4: Left: Calculated spectra with different values of the CEF splittings (and same mixing and symmetries of the states); the best fit of the experimental data is obtained when the second inelasic peak is located at 73 meV. The three Gaussians represent the peaks constituting the simulated spectrum. Right: The cross sections of the transitions is affected by the states' symmetry. The best fit is obtained with a Γ 1 7 ´Γ6 ´Γ2 7 order of the states in energy.

Figure 4 . 5 :

 45 Figure 4.5: Left: Dependence on Ǎ02 of the ground state (Γ 1 7 " α|˘5{2y `β|¯3{2yq) mixing coefficients and of its J z expectation value. Right: Full CEF scheme (including the 2 F 7{2 multiplet) in function of Ǎ02 .

Figure 4 . 6 :

 46 Figure 4.6: Comparison between experimental data (π polarisation) and simulations with the determined splitting scheme and various Ǎ0 2 values.

Figure 4 . 7 :

 47 Figure 4.7: Comparison between experimental data (σ polarisation) and simulations with the determined splitting scheme and various Ǎ0 2 values.

Figures 4 .

 4 Figures 4.6 and 4.7 present the calculated RIXS spectra for various values of Ǎ02 and their comparison with experimental data. The signal from the 2 F 7{2 levels (left columns) is, as expected, highly dependent on Ǎ02 and the best agreement with all the experimental spectra is obtained for Ǎ0 2 « 60 meV (and thus, by Equation4.1 Ǎ0 4 « ´77 meV and Ǎ4 4 « 194 meV, the value of A4 4 being increased of 8% compared to the formula in order to correct for the errors of Stevens' approximation). On the other hand, the low energy region (right columns)

Figure 4 . 8 :

 48 Figure 4.8: Temperature dependence of the magnetic susceptibility from Refs. [139] (Data 1) and [70] (Data 2) and corresponding fit performed with the CEF scheme found by the RIXS data and, respectively, Ǎ02 " 76 meV and 60 meV. An exchange field along c was introduced to better reproduce the low-T region, with a value of H exc " ´9 meV for the χ c simulation (violet line) and " ´37 meV for χ a (teal).

Figure 4 . 11 :

 411 Figure 4.11: Spatial distribution of 2 F 5{2 wavefunctions for positive and negative Ǎ44 and corresponding simulated spectra with polarisation analysis, for both π and σ incident polarisations and two orientations of the sample relative to the scattering plane (ϕ " 0 ˝and ϕ " 45 ˝). The energy of the incident photons was set to the first peak of the XAS spectrum, i.e. 882 eV (labelled as B in Figure4.2). 2θ " 150 ˝, χ " 0 ˝.

4 .

 4 It has an almost purely J z " |˘1{2y character, which makes it invariant under rotations around the c axis. The two Γ 7 states instead are characterized by a C 4 rotational axis and are both strongly affected by the 45 ˝rotation due to the sign of Ǎ4 4 . This dependence on the states' orientation is further confirmed by the spectra in the lower row of the Figure, which shows that a rotation of the crystal of 45 ˝degrees around the c axis (ϕ " 45 ˝) is equivalent to changing the sign of Ǎ4 4 . Comparing these simulations to the RIXS data (Figure 4.12) they are in perfect agreement for Ǎ4 4 ą 0, for both sample orientations and polarisations. This result is in line with the orientation of the ground state found with Non-resonant Inelastic X-ray Scattering (NIXS) by Willers et al. on the ambient-pressure superconducting parent compound

Figure 4 .

 4 Figure 4.12: M 5 RIXS data and simulations ( Ǎ4 4 ą 0) with polarisation analysis of the scattered beam. 2θ " 150 ˝, χ « 0 ˝, δ « 0 ˝, E in " 882 eV. Spectra with incident π polarisation have been acquired for 12 hours each, while spectra with σ polarisation for 3 (ϕ " 0 ˝) and 6 hours (ϕ " 45 ˝). The experimental resolution was lowered to 60 meV to increase the photon flux. The storage ring was operating at 90 mA in 16 bunch top-up filling mode.

Figure 4 .

 4 Figure 4.13: Left: Residuals (smoothed) of the fit in Figure 4.12 showing the presence of an extra feature at 100 meV, in the non-crossed channel. Right: The energy position of this feature agrees well with the energy of the hybridized Rh4d-Ce4f states seen in ARPES [88].

Figure 4 .

 4 Figure 4.13. A similar behavior was visible in the zoom of the unpolarised high-resolution data of Figures 4.6and 4.7, left column.

  and 4.7, left column.

Figure 4 .

 4 Figure 4.14: RIXS spectra of CeRh 2 Si 2 at three different temperatures across the AFM transition, acquired with 2θ " 150 ˝, δ " 15 ˝and σ polarisation.No temperature dependence is visible in the inelastic part, while the elastic peak grows with temperature, possibly due to an increased diffuse scattering. Simulations without antiferromagnetic ordering fit equally well the three spectra, confirming that our analysis is not affected by the magnetism. Calculations were performed considering the Boltzmann population of excited levels which was always lower than 2.8%.

Figure 4 . 15 :

 415 Figure 4.15: Effect of AFM order on the energy of the 4f levels and the RIXS cross sections. The plots are shown as a function of the ground state splitting ∆ gs induced by the magnetic exchange field along c. Top panel: Splittings of the lowest 4f CEF levels. The levels giving a non-negligible contribution to the RIXS spectra at 2θ " 150 ˝, δ " 15 ˝and σ polarisation (experimental conditions of Figure 4.14) are indicated by thicker lines and dots. Bottom panels: calculated cross sections for the possible tansitions.

Figure 4 . 16 :

 416 Figure 4.16: Relative intensities of the Γ 6 and Γ 27 peaks (calculated as Γ 6 {pΓ 6 `Γ2 7 q and Γ 2 7 {pΓ 6 `Γ2 7 q) resulting from simulations at various scattering angles (in specular geometry, δ " 0 ˝). The two possible signs of Ǎ44 give different dependencies on 2θ, but converge at 150 ˝, allowing, at this angle, to neglect the Ǎ4 4 sign and simplify the analysis.

Figure 4 .

 4 Figure 4.18: M 5 RIXS maps for CeCu 2 Si 2 with π and σ polarisations of the incident photons and corresponding XAS spectra. The resonant energies for the excitations in the 2 F 5{2 and 2 F 7{2 multiplets, about 882 and 882.5 eV respectively, are visible in the map acquired with π polarisation, while the σ polarisation spectra are more affected by the strong elastic peak, which follows the XAS signal.

2 2 In

 22 photoemission spectrocopy (ARPES) in several heavy fermion compounds, based both on cerium, such as CeIrIn 5[START_REF] Fujimori | Direct observation of a quasiparticle band in CeIrIn 5 : An angle-resolved photoemission spectroscopy study[END_REF], and other rare-earths, as YbRh 2 Si 2 , YbIr 2 Si 2 and EuRh 2 Si 2[START_REF] Vyalikh | Photoemission insight into heavy-fermion behavior in YbRh 2 Si 2[END_REF] [144] [145][START_REF] Höppner | Interplay of Dirac fermions and heavy quasiparticles in solids[END_REF]. In CeRh 2 Si 2 , angle resolved resonant PES revealed a curvature of the CEF levels (Figure1.11) near k x,y " 0 rlu , induced by the approaching hole-like Rh 4d´derived band[START_REF] Patil | ARPES view on surface and bulk hybridization phenomena in the antiferromagnetic Kondo lattice CeRh 2 Si 2[END_REF].

Figure 5 . 2 :

 52 Figure 5.2: Comparison of the CFA and CFB splitting schemes with high-resolution data at 150 ˝scattering angle, δ " 0 ˝. The fits confirm that the CFA scheme is the most accurate for data acquired in this geometry.

Figure 5 . 3 :

 53 Figure 5.3: Fit of the temperature dependence of the magnetic susceptibility and simulations with the new CEF splittings (43-63 meV). Data 1 are from Refs. [139] and Data 2 from[70].

and 5. 5 , 2 spectra. The 2 FFigure 5 . 4 :

 52254 Figure 5.4: Simple model for the intepretation of the angular dependence observed for the Γ 1 7 Ñ Γ 2 7 excitation. Top row: A guess for the levels' dispersions along k z and the ∆k z for the RIXS processes at the two angles. Middle row: The top band is shifted according to ∆k z , in order to allow an easy representation of the various possible transition energies. Bottom row: The histogram (blue) of the transition energies and its convolution with the experimental resolution (green and red lines), which gives the shape of the RIXS feature on the spectra. The RIXS peaks are considerably shifted in energy compared to the ∆E CEF position observable in absence of dispersion.

Figure 5 . 5 :

 55 Figure 5.5: Simplified model for the intepretation of the angular dependence of the Γ 1 7 Ñ Γ 6 excitation. Top row: A guess for the levels' dispersions along k z and the ∆k z of the two angles. Middle row: The top band is shifted according to ∆k z , to easily represent the transition energies. Bottom row: The histogram (blue) of the possible energies and its convolution with the experimental resolution (green and red lines), which gives the shape of the RIXS feature on the spectra. Due to the similar dispersions of the two levels, the energy distribution is less asymmetric and the peaks are closer to ∆E CEF . A " ´3 meV shift is anyway visible in the 90 ˝case, where the deviation from the ∆E CEF transition energy occurs only towards lower energies.

Figure 5 . 6 :

 56 Figure 5.6: High resolution RIXS spectra acquired in various angles and fits using the CFA and CFB schemes. The CEF best reproducing the data varies with the scattering geometry. It is anyway possible, with the present resolution, to fit the data with one of the two schemes found above.

Figure 5 . 7 :

 57 Figure 5.7: 60-meV resolution RIXS spectra at various δ for both 2θ " 150 ˝and 90 ˝, and ϕ " 0 ˝and 45 ˝.The CFA scheme fits the data at 150 ˝around δ " 0 ˝, while at large positive and negative values of δ the experimental spectra are reproduced more accurately by calculations with CFB. At 90 ˝, instead, the CFB scheme provides a good agreement for every δ.

Figure 5 . 8 :

 58 Figure 5.8: Left: Map on the reciprocal space of the transferred momenta of all the scattering geometries tested, indicating the CEF scheme better fitting each spectrum. The larger dots represent the high resolution data (shown in Figures 5.1, 5.2, 5.6 and 4.14). Right: Folding the points in the same half of the Brillouin zone, it can be noticed that the CFA scheme fits only spectra with ∆k around the Z point.

Figure 6 . 1 :

 61 Figure 6.1: RIXs spectra of CeCu 2 Si 2 .Thanks to the reduction of the elastic intensity with π polarisation, a first inelastic peak is visible at 30 meV (dashed line). A shoulder is visible at higher energies (dotted line), in addition to a broad tail up to 170 meV. Spectra have been renormalized to easy the comparison between σ and π polarizations.

Figure 6 . 4 :

 64 Figure 6.4: CeCu 2 Si 2 experimental data at 2θ " 120 ˝and δ " 0 ˝, and simulated spectra.The simulations best reproducing the data have a slightly modified scheme, with the second peak positioned at 55 meV instead of 47 meV. The shoulder is anyway weak and the broad feature at 95 meV strongly modifies the shape of the spectra, not allowing for a clear identification of the features.

Figure 6 . 6 :

 66 Figure 6.6: CeCo 2 Ge 2 RIXS spectrum and a fit with 7 Gaussians representing the 7 CEF levels. The peak widths vary between 35 and 65 meV.

Figure 6 . 8 :

 68 Figure 6.8: CeCo 2 Ge 2 RIXS spectra at 2θ " 150 ˝and simulations with Ǎ02 " 17 meV, Ǎ04 " ´73 meV, Ǎ4 4 " 377 meV, Ǎ0 6 " 220 meV and Ǎ4 6 " 200 meV.

Figure 6 . 9 :

 69 Figure 6.9: CeCo 2 Ge 2 RIXS spectra at 2θ " 90 ˝and calculated cross sections. The simulations give intensity of the peaks that are very different from the measured ones.

Previously, a system sensitive to circular dichroism was implemented by Duda et al.[START_REF] Duda | A helicity resolving soft X-ray emission spectrometer for studying magnetic circular dichroism with laboratory excitation sources[END_REF], but its applicability was limited to the Fe L 3 edge.

Many values of the magnetic moment can be found in literature[START_REF] Kawasaki | Si-NMR study of antiferromagnetic heavy-fermion compounds CePd 2 Si 2 and CeRh 2 Si 2[END_REF][START_REF] Grier | Magnetic ordering in CeM 2 Si 2 (M= Ag, Au, Pd, Rh) compounds as studied by neutron diffraction[END_REF][START_REF] Kawarazaki | Ground-state magnetic structure of CeRh 2 Si 2 and the response to hydrostatic pressure as studied by neutron diffraction[END_REF][START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF], due to: (a) different microscopic magnetic structures used to model the antiferromagnetic ordering and (b) the possible presence of magnetic fluctuations, which are integrated on a different timescale by each technique[START_REF] Grier | Magnetic ordering in CeM 2 Si 2 (M= Ag, Au, Pd, Rh) compounds as studied by neutron diffraction[END_REF][START_REF] Kawarazaki | Ground-state magnetic structure of CeRh 2 Si 2 and the response to hydrostatic pressure as studied by neutron diffraction[END_REF][START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF][START_REF] Kawasaki | Si-NMR study of antiferromagnetic heavy-fermion compounds CePd 2 Si 2 and CeRh 2 Si 2[END_REF]. The values presented here are considered the most reliable by the author. In particular, it must be noticed that values between 1.28´1.6 µ B moment per cerium ion agree with magnetic susceptibility data[START_REF] Mori | Effect of pressure on the magnetization of single crystal CeRh 2 Si 2[END_REF][START_REF] Hideki | Magnetic properties of CeRh 2 Si 2 and CePd 2 Si 2 single crystals[END_REF][START_REF] Settai | Single crystal growth and magnetic properties of CeRh 2 Si 2[END_REF] for T ą T N 1 (paramagnetic region); this result is independent on the specifics of the antiferromagnetic structure and is a property of each individual site.

Acknowledgements

In these three years of PhD, I grew both as a scientist and as a person and I

Procedure for the determination of the crystal electric field parameters

It is useful at this point to summarize the procedure employed to get the CEF parameters from the RIXS spectra (Figure 4.10). One should at first focus on the three 2 F 5{2 levels, in order to reduce the number of variables. A first guess of the splittings ∆E 1 and ∆E 2 can be found by fitting the experimental data with three resolution-limited gaussians and thus estimating the position of the two inelastic peaks. The values of the energy splitting allow to reverse the expressions of E Γ 1 7 , E Γ 2 7 and E Γ6 in Equation 1.28 and get Ǎ0 4 and Ǎ4 4 as a function of Ǎ0

2 . Since the symmetries are not assigned yet to the levels, three expressions are obtained for each of the two parameters, corresponding to the three possible schemes

. The determination of the symmetry can be performed by comparing the relative intensity of the inelastic peaks with cross section calculations (Figure 4.4, right panel). To do so, one needs to make a first guess of Ǎ0

RIXS with polarisation analysis

In the previous section, all the main properties of 4f levels (namely the splitting, symmetry and J z mixing) have been determined from the study of (unpolarised) RIXS spectra. More information can be obtained from RIXS spectra by analysing the polarisation of the scattered beam.

Determination of wavefunction orientation in the ab plane

When substituting the value of Ǎ0 2 in Equation 4.1, two solutions with opposite signs are possible for Ǎ4

4 . This occurs because the states energies in Equation 1.28 only depend on p Ǎ4

4 q 2 and are therefore insensitive to its sign. The same holds true for the J z mixing, since in this case the measured quantity is α 2 , which only depends on the square of Ǎ4 4 (see Equation 1.27). On the other hand, Ǎ4

4 determines the sign of α and therefore the sign of the linear combination of the Γ 7 wavefunctions. This affects the states' angular distribution, which rotates 45 ˝degrees in the crystal ab plane (as shown at the top of Figure 4.11).

The orientation of the states in a symmetry higher than twofold cannot be investigated by the standard dipole-based techniques (as INS, XAS, χpT q, ...). Resonant Inelastic Xray Scattering, being a second order process, is not constrained by dipole symmetry and can therefore access the information on the states' orientation. This is particularly clear with polarisation analysis of the scattered photons. If one calculates the RIXS spectra Figure 4.17: Calculated intensity ratios Γ 6 {pΓ 6 `Γ2

7 q for the crossed and non-crossed polarisation channels at various scattering angles (and specular geometry). The two possible signs of Ǎ4

4 give different Γ 6 ratios for the crossed and non-crossed spectra (which translate in different spectral shapes), allowing to determine the Ǎ4 4 sign with a polarisation analysis. The difference is maximum at 2θ " 150

scattering angle and on energy and polarisation of the incident beam. Figure 4.16 shows the intensity ratios for the transitions from the Γ 1 7 into the Γ 6 and Γ 2 7 levels calculated for the found CEF parameters as a function of the scattering angle 2θ in the full range of the ERIXS spectrometer (50 ˝-150 ˝) and keeping the condition of specular geometry δ " 0 ˝. As a general trend, the Γ 2 7 relative intensity decreases, and the Γ 6 contribution increases, by increasing the scattering angle. The scattering geometry can therefore be used to enhance the signal of a specific excitation. For example, at 150 ˝the higher intensity of the Γ 6 peak allowed to easily locate it on the tail of the elastic peak. Most importantly, at 150 the peaks' intensity ratio is the same for Ǎ4 4 ą 0 and Ǎ4 4 ă 0, if the two crossed and noncrossed polarisation channels are not separated. This allows to neglect the Ǎ4 4 sign in the determination of the CEF levels' symmetry and energy, therefore restricting the possible variables and simplifying the analysis.

On the other hand, the two possible signs of Ǎ4

4 give clearly different spectra if the channels for crossed-and non-crossed polarisations are studied individually. This is shown in Figure 4.17, which reports the calculated ratio Γ 6 {pΓ 6 `Γ2 7 q for the two polarisation Chapter 6

Crystal field excitations in other Ce 122 compounds

In the previous chapters, the ability of RIXS of probing the 4f levels in cerium intermetallics was discussed using the example of CeRh 2 Si 2 . In that case, the splittings were comparable to the experimental resolution. It is useful at this point to investigate the limits of applicability of RIXS spectroscopy to study the 4f states' properties within the CEF model. To this end, two opposite cases will be considered in this chapter. In CeCu 2 Si 2 the CEF splittings are smaller than our energy resolution and it will be shown that it is nonetheless possible to obtain information on the CEF by exploiting the polarisation and angular dependence of the cross sections. At the other end, CeCo 2 Ge 2 presents extremely large splittings and all the f f excitations are clearly resolved in the spectra, allowing even a detailed analysis of the 2 F 7{2 levels. Yet, for such large effects on the 4f levels, the usual CEF approach loses its validity. The Stevens' approximation doesn't hold true anymore and the mixing between the two spin-orbit multiplets can't be neglected. Moreover, and most importantly, the strong interaction of the 4f electrons with the surroundings cause a significant broadening of the spectral features and modify the peaks' intensities compared to single-ion calculations which introduces uncertainties in the analysis.

CeCu 2 Si 2 : crystal field splittings below the experimental resolution

CeCu 2 Si 2 is the first compound in which heavy fermion superconductivity was discovered [START_REF] Steglich | Superconductivity in the presence of strong Pauli paramagnetism: CeCu 2 Si 2[END_REF]. Further studies found two distinct superconducting domes in the phase diagram, one (T c1 " 600 mK, P c1 " 0.45 GPa) related to a magnetic quantum critical point and the Figure 6.5: The cross section dependence on the Ǎ4 4 sign can be exploited to determine the wavefunciton orientation, without the need of a polarisation analysis. Supposing that the CEF scheme proposed for CeCu 2 Si 2 is correct, simulations with Ǎ4 4 ą 0 are in better agreement with the experimental data, confirming the results of Ref. [START_REF] Willers | Determining the in-plane orientation of the ground-state orbital of CeCu 2 Si 2[END_REF]. similar intensities or Γ 6 is two times more intense than Γ 2 7 . Figure 6.5 shows the comparison of experimental data with the two possibilities, confirming the findings that Ǎ4 4 ą 0. Note that in these data ϕ " 45 ˝and so the cross section dependence on the Ǎ4 4 sign is inverted if compared to what shown in Figure 4.16.

6.2 CeCo 2 Ge 2 : large splittings and limitations of the crystal field model CeCo 2 Ge 2 shows intermediate 4f occupation at low temperatures, while at room temperature Ce ions are close to a trivalent state [START_REF] Kong | Thermodynamic and transport properties of single crystalline RCo 2 Ge 2 (R= Y, La-Nd, Sm-Tm)[END_REF][START_REF] Fujii | Magnetic properties of CeCo 2 Ge 2 and NdCo 2 Ge 2 single crystals[END_REF][START_REF] Ansari | The near-edge xray absorption spectroscopy of RT 2 Si 2 and RT 2 Ge 2 compounds[END_REF]. Anomalies in the magnetic susceptibility and in the resistivity indicate the arising of a strong Kondo state at T K « 120 K [START_REF] Fujii | Magnetic properties of CeCo 2 Ge 2 and NdCo 2 Ge 2 single crystals[END_REF], in agreement with specific-heat measurements [START_REF] Kong | Thermodynamic and transport properties of single crystalline RCo 2 Ge 2 (R= Y, La-Nd, Sm-Tm)[END_REF]. Around this temperature, it has also been observed a large thermal expansion in both the a and c lattice constants. Below 20 K, the quadratic dependence of the resistivity suggests a Fermi-liquid behavior [START_REF] Fujii | Magnetic properties of CeCo 2 Ge 2 and NdCo 2 Ge 2 single crystals[END_REF].

No experimental study of the 4f CEF splittings has been reported. The intermediate valence effects [START_REF] Fujii | Magnetic properties of CeCo 2 Ge 2 and NdCo 2 Ge 2 single crystals[END_REF] and other non-local contributions to the magnetism [START_REF] Kong | Thermodynamic and transport properties of single crystalline RCo 2 Ge 2 (R= Y, La-Nd, Sm-Tm)[END_REF] make it impossible to deduce the CEF parameters from susceptibility curves and no INS study has Chapter 7

Summary and outlook

Resonant Inelastic soft X-ray Scattering (soft-RIXS) has been presented as a new tool to probe the properties of 4f states in cerium intermetallic compounds. The physics of 4f electrons, typically characterized by low energy scales, became accessible to soft-RIXS in the last years, with the construction and commissioning of the ID32 beamline and the ERIXS high-resolution spectrometer at ESRF. The design and functioning of this world-leading instrument have been presented in detail, paying particular attention to the most important innovations. The example of CeRh 2 Si 2 has been used to discuss the information that can be retrieved from high-resolution RIXS spectra at the M 5 edge. In particular, it was possible to determine the 4f states' splittings and symmetry within the crystal electric field model, by comparing the experimental spectra to full-multiplet single-ion simulations. A general procedure for data analysis was also outlined, providing a guideline for the future works.

Further insight was obtained by analyzing the polarisation and momentum dependencies of the f f transitions. The polarisation analysis of the scattered beam revealed the orientation of the wavefunctions in the crystal ab plane and confirmed the crystal-field splitting scheme.

A variation of the measured splittings with the scattering geometry was detected, unveiling effects of the 4f states' dispersion in RIXS spectroscopy. A first interpretation of the observed peaks' shifts was illustrated with a simplified model. The final result of the presented analysis on CeRh 2 Si 2 gave a Γ 1 7 -Γ 6 -Γ 2 7 splitting scheme for the lowest 4f levels ( 2 F 5{2 multiplet), with energies and dispersions compatible with resonant photoemission data. No temperature dependence of the f f transitions across T Nèel has been detected. The example of CeRh 2 Si 2 shows that a complete crystal-field analysis can be performed with RIXS on 4f splittings comparable to the experimental resolution (35 meV). An investigation of smaller splittings is possible, by exploiting the angular and polarisation dependence of the f f cross-sections. The first spectra of CeCu 2 Si 2 revealed the presence of a peak at