Two-dimensional Kβ-Kα fluorescence spectrum by nonlinear resonant inelastic X-ray scattering

High sensitivity of the Kβ fluorescence spectrum to electronic state is widely used to investigate spin and oxidation state of first-row transition-metal compounds. However, the complex electronic structure results in overlapping spectral features, and the interpretation may be hampered by ambiguity in resolving the spectrum into components representing different electronic states. Here, we tackle this difficulty with a nonlinear resonant inelastic X-ray scattering (RIXS) scheme, where we leverage sequential two-photon absorption to realize an inverse process of the Kβ emission, and measure the successive Kα emission. The nonlinear RIXS reveals two-dimensional (2D) Kβ-Kα fluorescence spectrum of copper metal, leading to better understanding of the spectral feature. We isolate 3d-related satellite peaks in the 2D spectrum, and find good agreement with our multiplet ligand field calculation. Our work not only advances the fluorescence spectroscopy, but opens the door to extend RIXS into the nonlinear regime.

X-ray fluorescence spectroscopy has been a standard technique for non-destructive element analysis due to the one-to-one relationship between elements and emission photon energies.Recent investigation [1][2][3] on the spectral shape has revealed a close correlation of the Kβ fluorescence spectrum to the electronic states, such as the spin and oxidation state of first-row transition-metal compounds.Now, X-ray fluorescence spectroscopy becomes a powerful tool for investigating even biological samples, e.g., the oxidation state of Mn in the oxygen-evolving complex of photosystem II [4][5][6][7] , in addition to the various field of physics and chemistry [1][2][3] .The Kβ fluorescence is a radiative transition from a 1s core-hole state (1s 1 ) to a 3p core-hole state (3p 5 ) and does not explicitly involve a 3d state.It is an exchange interaction between 3d and 3p electrons that reflects the 3d electronic state in the Kβ fluorescence 8 .At the same time, the exchange and other interactions split the electronic states and result in heavily overlapping spectral features as we will see later.This could make it difficult to extract the key spectral components related to the electronic states of interest and hamper the in-depth discussion.
A good playground for the fluorescence spectrum is copper metal, which is a simple system but has been investigated for decades both experimentally and theoretically [9][10][11][12][13][14][15] .There are two approaches to analyzing the fluorescence spectrum.As for the Kβ spectrum, it can be analytically decomposed into five Lorentzians 10 .This phenomenological analysis may have a problem that each component is not necessarily connected to a particular electronic state.The Kβ spectrum can also be well reproduced as a sum of theoretical spectra calculated for possible electronic states 11,12 .However, different theoretical models can fit the spectrum, and thus the theoretical fitting may not discriminate subtle changes or differences in the electronic state.In order to perform more deterministic analysis, we need to get more spectroscopic information from the fluorescence process.
In this article, we discuss the expansion of the one-dimensional fluorescence spectrum to two dimensions.We propose a fusion of nonlinear absorption and resonant inelastic X-ray scattering (RIXS) and demonstrate our idea on copper metal.Our analysis of the 2D fluorescence spectrum resolves 3d-related satellite peaks, which are hidden by a strong main peak in the conventional Kβ fluorescence spectrum.We also present multiplet ligand calculations and interpret the experimentally resolved satellites based on the theoretical prediction.

Nonlinear RIXS scheme
Our idea to extract more information from the Kβ fluorescence process is to expand the one-dimensional spectrum to two dimensional by adding a second axis.The spectral components, which overlap in the Kβ fluorescence spectrum, may separate along the second axis direction.Suppose we could use an absorption process from 3p 5 to 1s 1 .This is an inverse process of the Kβ fluorescence and should be governed by the same transition matrix elements.After this absorption process, the resultant 1s 1 state might decay by emitting the Kα fluorescence, i.e., a 1s 1 → 2p 5 radiative transition.Consequently, we could have two tuneable parameters, the photon energies for the absorption and emission processes.This absorption-emission process is to be treated in a similar manner to RIXS [16][17][18] because the absorption is resonant.If we had measured the Kβ emission, the process should be described as resonance fluorescence 19 .
The biggest hurdle to realize our idea is the fact that all core levels are occupied in the ground state, and consequently, the 3p 5 → 1s 1 absorption process is strictly forbidden by the Pauli principle.The key is sequential two-photon absorption (Fig. 1a).In this nonlinear process, the first photon creates a 3p core hole, and opens a new resonant absorption channel from 3p 5 to 1s 1 within a very short period of the 3p core-hole lifetime.The second photon can excite a 1s core electron to the 3p core hole if the photon energy is carefully tuned to the resonance.We note that the sequential absorption on the core-to-core resonance was already reported in the soft-X-ray region 20,21 , however, has not been combined with the X-ray emission spectroscopy.
As we use the nonlinear process and RIXS, we refer to the above process as 3p 5 → 1s 1 → 2p 5 nonlinear RIXS.An inverse process, i.e., 2p 5 → 1s 1 → 3p 5 nonlinear RIXS, is also possible but may be less efficient due to the shorter lifetime for the 2p 5 state than 3p 5 , and an order of magnitude lower Kβ yield than Kα.We label the excited state after the first absorption as i, the intermediate state after the second absorption as n, the final state after the emission as f, and the ground state as g (Fig. 1a).The excited state i may include multiple hole state due to a shake-off process and cascade of the Auger decay.The shake-off process can create two-hole states having one 3p hole and an additional hole in 3s, 3p, or 3d subshell 22 .The pump photon energy higher than the L 1 absorption edge may create a 2s, 2p, or 3s hole, and initiate the Auger cascade, during which multi-hole state with at least one 3p hole may appear.Among various multi-hole states possible, however, a 3p 5 3d 9 state is found important to interpret the experimental result, as we will see later.
First, we formulate the nonlinear RIXS process.We exclude the Auger cascade from the formulation for simplicity.The absorption of the first photon can be treated as an independent process from the subsequence because a promoted high-energy photo-electron hardly interacts with the rest system.Therefore, this part can be described by rate equations 23 .Below, we will omit the electrons in the high-energy continuum from the configuration expression, e.g., |i〉 = |3p 5 〉 instead of |i〉 = |3p 5 ε〉, unless we need to indicate it explicitly.On the other hand, the absorption of the second photon is resonant, and thus the i → n absorption and the n → f emission should be described as a coherent second-order process like the conventional RIXS.Combining all processes, the differential cross-section for the nonlinear RIXS may be expressed as: where ω p is the pump angular frequency, ω e is the angular frequency at which the emission is measured, dΩ is an element of solid angle into which the photon specified by the subscript is emitted, σ gi is the g → i absorption cross-section, and τ i is the lifetime of i.We introduced a normalized dimensionless instrumental function, Φ, which represents the finite photon-energy resolution of the detecting system.The dimension of σ 2 ð Þ RIXS is (length) 4 (time), which is the same as the crosssection for two-photon absorption.The double differential scattering cross-section of the RIXS part may be written as 24 : by keeping only a resonant term in the Kramers-Heisenberg formula and applying the dipole approximation.Here, ω 1 and ω 2 are the angular frequencies of the incident and emitted photons, respectively, m is the electron mass, r 0 is the classical radius of electron, Γ n is the lifetime broadening of n, ω ab =(E a −E b )/ħ with E x being the energy of the state x, and r is the position operator of the electron.Summation over all electrons is omitted for simplicity.We point out here that the initial state of the RIXS part includes many excited states with different energies, whereas the conventional RIXS starts from the ground state.

Measurement of σ 2 ð Þ RIXS
Since the lifetime of the 3p core-hole state, τ i , is on a sub-femtosecond timescale, nonlinear RIXS requires intense femtosecond X-rays.We used an X-ray free-electron laser, SACLA 25 , which delivered an X-ray beam with the central photon energy at 8905 eV, corresponding to the Kβ photon energy of copper, a full-width-at-half-maximum (FWHM) bandwidth of 43 eV, and an FWHM pulse duration as short as 8 fs (ref.26).We used a Si (111) double-crystal monochromator to select a 1.0 eV monochromatic beam, and to scan the pump-photon energy over the Kβ range, and then, focused the beam down to 0.72(H) × 0.92(V) μm 2 (FWHM) by a Kirkpatrick-Baez (KB) mirror 27 (see Methods and Supplementary Fig. 2 for details).The average pulse energy in front of the KB mirror was 8.0 μJ at 8905 eV, from which we estimate the peak intensity and the peak photon-flux density on the focus to be 1.5 × 10 17 W/cm 2 and 11 photons/Å 2 /fs, respectively.We measured the Kα emission spectrum from a 10-μm-thick copper foil set on the focus using a single-shot polychromator.
Figure 1b shows a typical pump pulse-energy dependence of the emission count rate measured at ω e = 8048 eV, i.e., the peak of the Kα 1 fluorescence spectrum.The pump photon energy was tuned to the Kβ fluorescence peak at ω p = 8905 eV (Supplementary Fig. 3).The pulseenergy dependence is well reproduced by a quadratic term only, as expected for sequential two-photon absorption.This indicates that non-resonant fluorescence by higher harmonic radiation and multiphoton processes of more than two are negligible under the present condition.The coefficient of the quadratic term is proportional to Analyzing the pulse-energy dependence of the Kα emission counts at each pump and emission photon energies, we map out σ 2 ð Þ RIXS on the ω p −ω e plane as shown in Fig. 1c.

2D fluorescence spectrum
The σ 2 ð Þ RIXS plane shows two prominent peaks at (ω p , ω e )=(8905 eV, 8027 eV) and (8905 eV, 8048 eV), which correspond to the peak photon energies for the Kβ and Kα 1,2 fluorescence spectra.These two peaks are not simple Lorentzian but are accompanied by additional weak peaks.To make clear the weak structures, three characteristic sections along constant ω p lines are shown in Fig. 2. We also plot the Kα fluorescence spectrum measured at an excitation photon energy of 9100 eV well above the K edge at 8980 eV for comparison.Among the three emission spectra, only that at ω p = 8905 eV, i.e., the Kβ peak, is found similar to the Kα fluorescence spectrum.
Here, we employ the phenomenological approach to analyze the emission spectra.The Kα emission spectrum is much simpler than Kβ because the exchange interaction between the 3d and 2p electrons is weaker than that between 3d and 3p.We may associate analytically decomposed Lorentzians with particular electronic transitions.The Kα 1 emission peak around ω e = 8048 eV can be decomposed into two Lorentzians labeled F and G.The spectral shape of the main peak G coincides with the Kα 1 fluorescence spectrum.Component G is considered to originate predominantly from the 1s 1 → 2p 5 diagram transition.Comparing with the previous analysis on the Kα fluorescence spectrum 10,13 , we assign the weaker peak F to a satellite due to the 1s 1 3d 9 → 2p 5 3d 9 transition.However, we consider that component F may contain just a part of the 1s 1 3d 9 → 2p 5 3d 9 transition, and the rest may be included in the main component G.The Kα 2 peak also consists of two Lorentzians D and E, which are considered to have the same origins as F and G, respectively.The branching ratio (Kα 2 /Kα 1 ) of the nonlinear RIXS is found slightly higher than the Kα fluorescence, which may be due to the resonant excitation at a single photon energy of 8905 eV.
When ω p was detuned from the Kβ peak, the Kα emission spectrum changed shape as expected for RIXS.We need at least three Lorentzians, G´, B, and C, to reproduce the Kα 1 emission spectrum at ω p = 8897 eV.The broad peak labeled G´lies near the constant energy transfer line, ω p −ω e = 857 eV, which passes through the peak G on the two-dimensional spectrum.Therefore, we consider G´is the tail of G.This is a characteristic of the coherent process, where the conservation of energy must be satisfied between the states i and f, but not for the intermediate state n in Eq. ( 2).On the other hand, the Kα 1 emission spectrum for ω p = 8912 eV has no sign of the tail of G, and can be reproduced by two Lorentzians, I and J.The three peaks, C, G, and J, have similar central photon energies, indicating that they have the same origin, while larger energy difference among the satellite peaks, B, F, and I, implies that they might come from different electronic states.Unlike Kα 1 , the Kα 2 emission spectra can be reproduced with one or two components.

Analysis of σ 2
ð Þ RIXS Now, we apply the same procedure as Fig. 2 to decompose all emission spectra into Lorentzian components in order to understand the overall structure of σ 2 ð Þ RIXS .We independently fitted the emission spectrum at each ω p with a minimum number of Lorentzians and no constant background (Supplementary Fig. 4).One 2D spectral feature may be sliced into several one-dimensional Lorentzians.We did not impose any constraint on the fitting parameters, such as the peak photon energy, to connect a Lorentzian to its neighbors along the ω p direction.Figure 3a and b show the decomposition result in the Kα 1 and Kα 2 regions, respectively.The components in the Kα 1 region can be classified into several clusters, while those for Kα 2 are not dissolved well as is pointed out above.
To decipher the spectral feature, we compare theoretical calculation based on Eq. ( 1).We used the standard multiplet ligand field RIXS due to the fitting with a quadratic function (Fig. 1b) is smaller than the marker size.Solid lines in the middle panel represent the Kα fluorescence spectrum measured by one-photon excitation at 9100 eV well above the K edge.Dashed lines represent the fitting with Lorentzian(s), and each component is plotted by a dotted line.theory 16,17,28 with octahedral (O h ) crystal fields of a face-centered cubic structure of copper metal (see the "Methods" section for details).In the rest of this article, we focus on the 3p 5 3d 9 satellites, i.e., transitions involving |i〉 = |3p 5 3d 9 〉.We ignored the i-dependence of τ i in Eq. ( 1), and calculated the scattering cross-section of the RIXS part as 3p 5 3d 9 ε → 1s 1 3d 9 ε → 2p 5 3d 9 ε using Eq. ( 2), where we considered a Cu 2+ ion instead of explicitly taking into account the shake-off process 10 .
Figure 3c and d RIXS calculated for the 3p 5 3d 9 satellites.Our simulation predicted three peaks for both the Kα 1 and Kα 2 emission.Since the 3p 5 3d 9 configuration has two holes, the possible LS terms are 1,3 P, 1,3 D, and 1,3 F, corresponding to the orbital angular momentum of L = 1, 2, and 3, respectively 17 .The superscript indicates the spin multiplicity, e.g., 2 S + 1 = 1 for singlet and 3 for triplet, where S is the spin angular momentum.These L terms split due to the direct Coulomb interaction (Fig. 3e).Each L term further splits by the 3p−3d exchange interaction.The 3p and weaker 3d spin-orbit interactions remove the degeneracy of the triplet, and mixes up all the LS terms.The spin-orbit interaction is weaker than the exchange interaction, and the effect would be experimentally indistinguishable.We retain the LS notation for the sake of clarity.In addition to the above interactions within an atom, the crystal field also splits the 3d level into e g and t 2g .
Roughly speaking, the 3p 5 3d 9 configuration can be categorized energetically into three groups of two terms each, i.e., 1 F and 1 P, 3  , because the separation (crystal-field splitting), which was set to 1 eV in the calculation, is larger than a calculated energy splitting of 0.21 eV between 1s 1 3d 9 1 D and 1s 1 3d 9 3 D by the exchange interaction.The energy split in |n〉 = |1s 1 3d 9 〉 is smaller than those in |i〉 = |3p 5 3d 9 〉.Thus, three satellite peaks in Fig. 3c and 3d correspond to the three groups in |i〉 = |3p 5 3d 9 〉.This also applies to the Kβ fluorescence and is the origin of a three-peak structure, which has been theoretically predicted but experimentally unresolved [9][10][11][12] .The final state, f, is dominated by the spin-orbit interaction for the 2p orbital.We note that the 1s 1 t 2g 6 e g 3 and 1s 1 t 2g 5 e g 4 levels form bands in the real system, and they may be experimentally inseparable in the emission spectrum.
Comparing Fig. 3a with Fig. 3c, we interpret the experimentally resolved clusters as follows.A cluster of strong components (open circles) around (8905 eV, 8048 eV) is the main peak by the 3p 5 → 1s 1 → 2p 5 RIXS process.The main peak may be doublet, because the 3p 5 level splits into 3p 1/2 and 3p 3/2 by the spin-orbit interaction.The separation is reported at about 2.5 eV (ref.10), and hence, the 3p 5 (3p 1/2 ) related peak may appear around ω p = 8903 eV, where we found another cluster (open diamonds).The diagonal streak (open triangles and a part of open diamonds) extending from the main peak towards the lower left have similar energy transfers, ω p −ω e = 857 eV.We assign the streak to the tail of the main peak.However, no counterpart was found in the higher photon-energy region.We consider that the components lying along ω e = 8048 eV apart from the main peak are unseparated satellites and the Kα 1 fluorescence from relaxed states.Three clusters labeled B, F, and I (see also Supplementary Fig. 5) are considered to be the satellites involving the 3p 5 3d 9 1 F/ 1 P, 3 D/ 3 P, and 1 D/ 3 F terms, respectively.It is such separation of the 3p 5 3d 9 satellites from the main peak that we expected for the 2D fluorescence spectrum, and has not been experimentally resolved for the onedimensional Kβ fluorescence spectrum [9][10][11][12] .The theoretical calculation predicted almost the same emission photon energy for these three components, while we find the energies are slightly different.The gravities of the clusters B, F, and I are ω e = 8045.4± 0.06, 8045.6 ± 0.11, and 8046.8 ± 0.04 eV, respectively.This implies that transition probabilities to each state in |n〉 = |1s 1 3d 9 〉 may differ among the 3p 5 3d 9 1 F/ 1 P, 3 D/ 3 P, and 1 D/ 3 F terms in the real system.
The 3p 5 3d 9 1 F/ 1 P, 3 D/ 3 P, and 1 D/ 3 F components less disperse along ω e , therefore, we discuss the ω p dependence by integrating σ 2 ð Þ RIXS over ω e within the Kα 1 region (Fig. 4).The theoretical calculation well reproduces the experimentally deduced spectrum.The agreement is satisfactory, considering that we did not make any empirical adjustments during the calculation.Especially, the energy separations among the three peaks determined mainly by the direct and exchange interactions are in good agreement.However, the relative peak height among the three peaks differs between theory and experiment.This discrepancy may be due to partial extraction of the satellite contribution in our phenomenological analysis of the Kα 1 emission spectra, and ignored contributions from multi-hole states other than 3p 5 3d 9 .

Discussion
We successfully realized our nonlinear RIXS scheme on copper metal and measured the 2D fluorescence spectrum as σ 2 ð Þ RIXS .Although the main part of the spectrum is simple due to the complete 3d subshell in copper metal, we showed that we could isolate the weaker satellite peaks and identify them arising from the two-hole 3p 5 3d 9 configuration with the help of multiplet ligand field calculation.Because of the Auger cascade, the initial state of the RIXS part, i, is not exactly the same as the final state of the Kβ fluorescence.Nevertheless, σ 2 ð Þ RIXS retained the three-peak satellite structure expected for the Kβ fluorescence spectrum.A possible reason is that solid-state effects could multi-hole states other than the 3p 5 3d 9 configuration (see Supplementary Note 1).
In the case of compounds transition metals with incomplete 3d subshell, the exchange interaction splits the main peak into the Kβ´and Kβ 1,3 peaks, which are known as a useful indicators of the spin and oxidation state [1][2][3] .For example, the Kβ´peak of Mn (II) compounds is a transition to a 5 P final state with the 3p spin antiparallel to the five 3d spins, while the Kβ 1,3 peak is to a parallel-spin 7 P final state.However, the Kβ 1,3 peak is also shaped by the 5 P term with a different 3d spin configuration 29 .The spectrum becomes more complicated for Mn (III) and Mn (IV) , which may relate to the oxygen-evolving complex [4][5][6][7] .The nonlinear RIXS could serve for a detailed understanding of the spin and oxidation state by resolving such overlapping terms.Our analysis still relies on the phenomenological decomposition of σ 2 ð Þ RIXS , which worked relatively well due to the simpler Kα 1 spectrum for copper metal.However, we anticipate that further theoretical and experimental investigation would lead to sophisticated analysis free from phenomenological modeling.
In the present study, we incorporate the sequential two-photon absorption into RIXS to realize core-to-core absorption.It would be possible to use absorption channels from core to valence 30 .Apart from realizing absorption to the occupied state, other nonlinear processes are also interesting candidates for nonlinear RIXS.Inclusion of direct two-photon absorption is a straightforward application of the present study.Combination of direct two-photon absorption [31][32][33] and RIXS might be useful to investigate directly the 3d state because the 1s → 3d transition is allowed and the 1s → 4p is forbidden by the selection rule.Stimulated emission [34][35][36] , which can enhance the fluorescence yield, and select a particular radiative transition of interest, is another interesting nonlinear process to be combined.

Experimental setup
SACLA was operated in self-amplified spontaneous emission (SASE) mode and at a repetition rate of 60 Hz.The half of X-ray pulses were delivered to experimental station 4 of BL3, where we conducted our experiment.The broad-band SASE was tuned to cover the whole Kβ spectral range so that the nonlinear RIXS plane could be measured simply by scanning the monochromator without changing the accelerator and the undulator parameters.The focus size was measured by knife-edge scans with 200-μmϕ gold wires.The pulse energy at the focal position was determined shot by shot using a calibrated beam monitor 37 .The shot-by-shot fluctuation was used to reveal the pulseenergy dependence of the emission intensity (Supplementary Fig. 6).The copper foil was continuously shifted to expose the undamaged fresh surface.The foil was tilted by 30°from the normal incidence to measure the X-ray emission at right angles to the incoming beam in the polarization plane (Supplementary Fig. 2).This geometry suppresses stray X-rays due to elastic and Compton scattering.
The single-shot polychromator consisted of four cylindrically bent Si (444) crystals set in the von Hamos geometry 2 and a multi-port charge-coupled device 38 (MPCCD).The crystal dimension was 100 mm along the circumferential direction and 25 mm along the dispersion direction, and the curvature radius was 500 mm.The MPCCD was able to detect a single photon and was suitable to measure the weak nonlinear signal.The nominal photon-energy resolution of the polychromator is calculated to be 0.075 eV/pixel.The pixel number of MPCCD was 1024 along the dispersed direction.The corresponding photon-energy range of the polychromator was about 77 eV, which was enough to cover both the Kα 1 and Kα 2 spectral ranges.The absolute photon energy of the polychromator was calibrated to the Kα 1 peak in literature 10 .The photon energy of the monochromator was calibrated to the absorption edge of a copper foil beforehand.

Theoretical calculation
Our aim in this study is to calculate the 3d 9 components of σ 2 ð Þ RIXS , i.e., the 3d 9 → 3p 5 3d 9 ε and 3p 5 3d 9 ε → 1s 1 3d 9 ε→2p 5 3d 9 ε RIXS.We used the standard multiplet ligand field theory 16,17,28 with octahedral (O h ) crystal field and did not make any empirical adjustments to the term averaged values of the Slater integrals for the 3d 9 , 3p 5 3d 9 , 2p 5 3d 9 , and 1s 1 3d 9 configurations.In the spirit of the early phase of the theoretical study, this is a sensible approach and has a more general interest beyond the specific experiment discussed here.The intra-atomic correlation not included in this scheme was treated approximately by reducing the values of the Slater integrals.The Slater integrals and the spin-orbit coupling constants were calculated by the Hartree-Fock code 39 , and then the Slater integrals were scaled down to 80% as usual 17 .The crystal field describes the breaking of the 3d orbital degeneracy in copper metal due to the effect of the electrical field of neighboring copper atoms.The crystal-field splitting parameter was set to 10Dq = 1.0 eV.We assumed that the shake-up process is negligible 12 , and the shake-off probability does not strongly depend on the details of the multiplet.We calculated σ gi in Eq. ( 1) as a 3d 9 → 3p 5 3d 9 ε transition process instead of 3d 10 → 3p 5 3d 9 ε 2 (ref.10).Because of a similar reason, we set τ i = 0.15 eV, however, it may strongly depend on the spin state in other 3d configurations 28 .

Fig. 1 |
Fig. 1 | Nonlinear RIXS of copper metal.a Schematic nonlinear RIXS process.The system is excited by sequential two-photon absorption including the inverse process of the Kβ fluorescence, and then decays by the Kα emission.Uninvolved orbitals and electrons are omitted for simplicity.b The pulse-energy dependence of the 8048 eV emission count rate at a pump photon energy of 8905 eV (circles).Vertical bars indicate the standard error of the mean.Solid line represents the best fit with a quadratic function without a linear or constant term.The coefficient of the quadratic term gives σ 2 ð Þ RIXS .c Image plot of σ 2 ð Þ RIXS .The pump photon energy covers the whole Kβ spectral range, while the emission photon energy does the main parts around the Kα 1 and Kα 2 peaks.

1s 1 t2g 6 eg 3 Fig. 3 |
Fig. 3 | Decomposition of Kα emission spectra, and theoretical calculation for 3p 5 3d 9 satellite peaks.a Decomposition of the Kα 1 emission spectra.Each symbol and its size represent the peak center and area of the Lorentzian, respectively.Components denoted by the same symbol are considered to have the same origin.The cluster ID for closed diamonds (B), squares (F), and hexagons (I) is taken from the corresponding peak in Fig. 2. Open circles and open diamonds are considered to be the main peak and the Kα 1 fluorescence from relaxed state (see text for details).Triangles are assigned to the tail of main peak.Dashed line indicates the constant energy transfer line, ω p −ω e = 857 eV, which pass through the main peak.b Decomposition of the Kα 2 emission spectra.Circles indicate the center of the Lorentzian with the peak area represented by the size.c, d Theoretical σ 2 ð Þ RIXS calculated for the 3p 5 3d 9 satellite peaks.The calculated spectrum is shifted to align with the experimental data.e Schematic energy-level diagram of copper metal for | i〉 = |3p 5 3d 9 〉, |n〉 = |1s 1 3d 9 〉, and |f〉 = |2p 5 3d 9 〉.The energy difference is taken from the theoretical calculation.

Fig. 4 |
Fig. 4 | Experimental and theoretical ω p dependence of ω e -integrated σ 2 ð Þ RIXS for 3p 5 3d 9 satellite.Symbols represent the peak area of the fitted Lorentzian.Vertical bar indicates the error of the Lorentzian fitting.Diamonds, squares, and circles represent the same components as shown in Fig. 3a.Solid line represents the ω eintegrated value over the Kα 1 region of the calculated σ 2 ð Þ or 1 D and 3 F, which are separated by several electron volts.The two terms in each group happen to have close energies.The intermediate state, |n〉 = |1s 1 3d 9 〉, is characterized by 1s 1 t 2g D and 3 P,