Saturation and self-absorption effects in the angle-dependent 2p3d resonant inelastic X-ray scattering spectra of Co3+

It is shown that the 2p3d resonant inelastic X-ray scattering intensity is distorted by saturation and self-absorption effects, i.e. by incident-energy-dependent saturation and by emission-energy-dependent self-absorption.


Introduction
Resonant inelastic X-ray scattering (RIXS) is a developing technique that is used to investigate the ground and excited states of transition metals, in particular when combined with the angular degrees of freedom (i.e. angle-dependent measurements). The angular dependence of RIXS has been examined by Michel van Veenendaal using crystal field calculations for L-and M-edges in the case of a single valence hole (or equally a single electron) ( van Veenendaal, 2006). Based on these calculations, one is able to discriminate between different transitions of the d-states via their spin characteristics using certain photon polarizations or experimental geometries for RIXS ( van Veenendaal, 2006;Moretti Sala et al., 2011). By including the spin-orbit coupling, exchange field (2J ex ) or external magnetic field, the system symmetry is further reduced (van Schooneveld et al., 2012). This implies that RIXS can even have the sensitivity to investigate the fine structure of spin-orbit coupling in multielectron systems. However, this sensitivity is commonly hampered by geometrical aspects of the experiment.
The RIXS cross-section can be explained as a combination of an absorption (photon-in) process and an emission (photonout) process. The absorption process excites the electrons to higher energy levels and the emission process emits photons when the electrons decay to the lower energy levels. Two geometrical effects have to be taken into account here: (i) the probing depth is dependent on the X-ray absorption spectroscopy (XAS) cross-section (saturation) and (ii) the emitted ISSN 1600-5775 photons can be re-absorbed (self-absorption). Consequently, the emission intensity might be distorted according to the photon energy and the experimental geometry, which is well known in fluorescence yield XAS (FY-XAS) (Jaklevic et al., 1977;Trö ger et al., 1992;Eisebitt et al., 1993;Chakarian et al., 1998;Nakajima et al., 1999;Achkar et al., 2011). As such, it also distorts the RIXS spectra (Chabot-Couture et al., 2010;Dallera et al., 1997).
In this article, we raise awareness that the 2p3d RIXS intensity is distorted not only by the incident-energy-dependent saturation but also by emission-energy-dependent selfabsorption. A model is proposed to describe the angledependent coefficients of the RIXS intensity for the saturation and self-absorption effects, which has been used to correct the 2p3d RIXS intensity of magnons in cuprate (Minola et al., 2015), where we applied the self-absorption-added coefficients to generate a theoretical prediction. The angle-dependent 2p3d RIXS of a LaCoO 3 single crystal and a 55 nm LaCoO 3 film on an SrTiO 3 substrate (LaCoO 3 /SrTiO 3 film) were compared to illustrate these effects. Furthermore, we show that the fine structure of the spin-orbit coupling and exchange interaction within the 3d 6 manifold of a Co 3+ ion can potentially be revealed using RIXS angle-dependent measurements.

Kramers-Heisenberg formula
The RIXS cross-section is determined by the Kramers-Heisenberg equation (Kramers & Heisenberg, 1925), where A f (! in ) is the scattering amplitude which is defined as Here h -! in and h -! out are the energies of the incident and the emitted photons; E g , E f and E m are the eigenvalues of the ground state |gi, the final state |fi and the intermediate state |mi, respectively; À gives the lifetime broadening of the intermediate state. V I (V E ) is the transition operator induced by the electro-magnetic field of the incident (emitted) photon and can be expanded as " " " I ÁP P I expðik I Á r I Þ (" " " E ÁP P E expðik E Á r E Þ). " " " I (" " " E ) andP P I (P P E ) describe the electric field polarization vector and the momentum operator of the incident(emitted) photon. k I (k E ) and r I (r E ) give the wavevector and position of the electro-magnetic wave. For 2p3d RIXS, the 2p3d (3d2p) transition operator is approximated as an electric dipole operator " " " I ÁP P I (" " " E ÁP P E ) for the absorbed (emitted) channel, i.e. expðik I Á r I Þ½expðik E Á r E Þ ' 1. The scattering amplitude can also be described by Green's func- where the groundstate energy (E g ) is taken as zero. Note that we can ignore the scattering coefficient r 2 e ð! out =! in Þ because ð! out =! in Þ ' 1 (! in ' ! out ) and the classical electron radius r e is constant.
For a better insight into the angular dependence of RIXS, many authors have already reformulated the scattering amplitude by using the spherical tensor expansion (Luo et al., 1993;van Veenendaal, 2006;Juhin et al., 2014;Moretti Sala et al., 2011). In the case of the 2p3d RIXS within the fast-collision approximation, the interference of the intermediate states can be ignored and Green's function can be simplified by where " E E j AE is the mean energy for the spin-orbit coupled 2p 3/2 and 2p 1/2 edges (Luo et al., 1993;van Veenendaal, 2006). Then the 2p3d RIXS crosssection formula can be expanded as Van Veenendaal pointed out that the cross-section is now described by the effective dd transition operators W Qq from a 3d n ground state to another 3d n final state with a polarization weighting factor T Qq (" E ," I ) ( van Veenendaal, 2006). W Qq is a one-particle operator which is constructed by the orbital-and spin-dependent tensors. The system geometry [ Fig. 1(a)] is reflected by the polarization tensors T Qq (" E ," I ). P dipole is the reduced matrix element between the 2p (core) and 3d (valence) levels. With this simplification, the beam polarization and the electronic configuration can be decoupled from each other and they therefore can be linked directly to the experimental geometry. In this work, the ground and excited states' energies and the 2p3d RIXS spectra were computed using the program Quanty, where both the tetragonal distortion and ligand-to-metal charge transfer could be considered (Haverkort et al., 2012;de Groot, 2005   the coordinates of the crystal field operator, where the axes are given by a, b and c as indicated in Fig. 1(a). Because no polarization analyzer was used, unpolarized emitted photons have been assumed and were composed by the summation of two orthogonal linear polarized beams (Juhin et al., 2014). The energy diagrams of the final (excited) states |fi were obtained from the cluster calculation using the program Quanty (Haverkort, 2010). The exchange interaction is estimated to be 6 meV (T c ' 70 K) which is neglected in the current simulation and discussion since it shows only tiny effects on the spectra. The electronic configurations of Co 3+ ions in the LaCoO 3 crystal and the LaCoO 3 /SrTiO 3 film were suggested to be the pure 1 A 1g (O h ) configuration and a mixture of the 1 A 1g (O h ) and the 5 B 2g (D 4h ) configurations, respectively (Tomiyasu et al., 2017;Wang et al., 2019). The model parameters of the 1 A 1g (O h ) ground state configuration are: F 2 dd = 9.371 eV, F 4 dd = 5.859 eV, d = 0.055 eV, 10Dq = 0.595 eV, U = 6.5 eV, Q = 7.5 eV, Á = 1.5 eV, T t 2g = 1.8 eV and T e g = 3.118 eV. For the 5 B 2g (D 4h ) ground state configuration, Ds = À0.12 eV and Dt = À0.018 eV are applied in addition to the 1 A 1g (O h ) case. The calculated excited states are indicated by the term symbol notations in Fig. 1(b). We note that the 5 B 2g (D 4h ) state is the subgroup of the 5 T 2g (O h ) state in the tetragonal distortion [ Fig. 1(c)].

Saturation and self-absorption effects
To simulate the saturation and the self-absorption effects, the transmitted photons (I T ) within a small distance d' can be described by the differential Lambert-Beer relation with the absorption factor of the incident beam (! in ), The integration of equation (4) from zero to the path length ' becomes: Here the constant I 0 gives the finite incoming photon flux. We note that this relation also depends on the surface profile of the sample. For a flat sample, the absorption probing depth d depends on the angle between the incident photon and the sample surface [ Fig. 2(a)], which gives the relation ' = d=sin . Thus we can rewrite equation (5) as a function of d, For the emission process, the total emitted flux is the integration of the beam of emitted photons from the surface to the probing depth d. This implies the path length ' 0 = d=sin , where is the angle between the scattered beam and sample surface [ Fig. 2(a)]. Then the total emitted flux becomes Here, dI S (z) stands for the scattering possibility per absorbed photon within a finite distance dz at a certain depth z, from which we can expand dI S (z) into equation (4) with a scattering possibility S(! in ,! out ) as The scattering possibility S(! in ,! out ) is determined by the state transition possibility. To a good approximation, the multiplication S(! in ,! out ) (! in ) represents the scattering intensity of RIXS. We note that our simulation calculated the scattering intensity of RIXS including the interference effects. In addition, the Auger decay channels are also state dependent. This will induce additional scaling factors on the emission intensity.
In the case of the 2p3d channels, the energy dependence in Auger decay is less than 10% (de Groot et al., 1994) and we approximate them as constant. By integrating equation (7), the emission intensity including both the saturation and the self-absorption effects is written as (as a function of probing depth d) This equation has already been discussed for the cases of FY-XAS spectra (Jaklevic et al., 1977;Trö ger et al., 1992;Eisebitt et al., 1993;Chakarian et al., 1998;Nakajima et al., 1999;Achkar et al., 2011;Chabot-Couture et al., 2010) by assuming that (! out ) is approximately constant. The spectral saturation is described by the term ð! in Þ=sin and is sensitive to the incidence angle ; the self-absorption effect is determined by the emission angle . For RIXS, however, (! out ) cannot be treated as a constant because the emitted photon energy is analyzed.  2.2.1. Thin samples. If the sample thickness (D) is much smaller than the attenuation length (), i.e. ) D ' d ! 0, we can use the approximation of the exponential function, exp(Àx) ' 1 À x. Then we obtain Equation (10) shows that the emission intensity is proportional to the incoming photon flux I 0 , the absorption factor of the incident beam (! in ), the scattering factor S(! in ,! out ), the sample thickness D and the angle . Thus, the total emission intensity only depends on the rotation angle , which changes the overall intensity as a function of angle but shows no energy-dependent spectral distortion caused by the selfabsorption effect. This self-absorption-free approximation is also applied to dilute specimens. (! in ) and (! out ) are the total absorption factors at certain energies ! in and ! out , which can also be described by the summation over all individual elements 'X', i.e. ð!Þ = P X X ð!Þ and Sð! in ; ! out Þ = P X S X ð! in ; ! out Þ. The FY-XAS or RIXS intensity is proportional to the multiplication S(! in ,! out ) (! in ). By ignoring the inter-atom interaction, this multiplication of the two summation factors (!)S(! in ,! out ) is replaced by the summation of the multiplications P X X ð!Þ S X ð! in ; ! out Þ. Then equation (10) is expressed as In the case of the Co L 3 edge, the absorption factor and scattering possibility of the other elements are constant so that equation (11) becomes The state-dependent scattering possibility S Co (! in ,! out ) implies that the partial fluorescence yield X-ray absorption is not identical to the normal absorption. Liu et al. show good agreement of such state-dependent behavior between experimental and calculated results in dilute specimens (Liu et al., 2018). If S Co (! in ,! out ) is approximated as a constant, the scattered spectrum is equivalent to the normal absorption Co (! in ).

Thick samples.
If the sample is thick (large D), d becomes infinity; the incident photons are fully absorbed. Thus the exponential term in equation (9) becomes zero. Then we obtain the scattering intensity as a function of the photon energy and sample geometry, This relation has been applied to a method of inverse partial fluorescence yield to obtain bulk-sensitive absorption spectra (Achkar et al., 2011), where the key aspect of inverse partial fluorescence yield is that one applies the partial fluorescence yield away from the target edges, so that the multiplication of X (! in ) and S X (! in ,! out ) is approximately constant. For example, in CoO, the element cobalt and the element oxygen are considered. Then the emission intensity becomes When accumulating the oxygen K edge fluorescence signal, the emission intensity of cobalt is negligible [ Co (! in )S Co (! in ,! out ) ' 0]. The emission spectrum becomes the summation of the emitted photon energy (! out ) at the oxygen K edge, which leads to O (! in )S O (! in ,! out ) that is approximated as constant with respect to the incident energy of the cobalt L 2,3 edge. It also implies that the absorption factors of the emitted photons [ Co (! out ) and O (! out )] are approximately constant. Then the emission intensity is simplified as The inverse intensity becomes I À1 where C 1 , C 2 and O (! in ) are constant with respect to the incident energy of the cobalt L 2,3 edge. Thus the emission intensity is mainly dominated by the absorption factor Co (! in ) which provides a bulk-sensitive absorption spectrum free from self-absorption and saturation effects.
For RIXS, equation (13) shows that the scattering intensity of RIXS should be multiplied by the self-absorption-added coefficients 1=½ð! in Þ þ ð! out Þðsin = sin Þ when saturation and self-absorption effects are included (Chabot-Couture et al., 2010). Fig. 2(b) shows this saturation relation as a function of . Here, (! in ) is fixed at one and (! out ) is scaled from zero to a hundred (infinity) to simulate the trends. The scattering angle is 140 . Depending on the competition between (! in ) and (! out ), three cases can occur: (i) (! out ) ) (! in ) (I ems ' 0). In this condition, the absorption factor of the emitted photons is much bigger than the incident beam absorption. The scattered photons can more easily be reabsorbed in the emission process, which implies that the emission intensity tends to vanish [the line (! out ) ' 100 in Fig. 2 When the absorption factor of the emission beam is much smaller than the incident beam, the angle-dependent term is negligible together with the small (! out ) value. In other words, the spectrum distortion is angle-independent [the line (! out ) ' 0 in Fig. 2(b)].
(iii) (! out ) ' (! in ). In this case, none of the factors can be omitted, as shown by the lines (! out ) ' 1. The selfabsorption effect is dependent on the geometry and the photon energy of both the incident and emitted beams. From the equation, we specify the two extreme conditions. The first case is that the incident beam is propagating along the sample surface plane (grazing incidence), where ' 0. Here, the research papers 982 Ru-Pan Wang et al. 2p3d RIXS spectra of Co 3+ emission intensity I ems ' I 0 f½S X ð! in ; ! out Þ X ð! in Þ=ð! in Þg provides a maximum value. But the absorption cross-section is saturated due to the competition between the absorption factor on certain element X (! in ) and the total absorption factor (! in ). The second case describes the propagation of the emitted beam being along the sample surface plane (grazing exit). It implies that the scattered photons are totally reabsorbed and no scattering intensity should be expected (I ems ' 0).
We present in Fig. 3 the influence of the self-absorption and saturation effects on the low-spin Co 3+ 1 A 1g (O h ) ground state as an example for bulk LaCoO 3 . The theoretical Co (! out ) and Co (! in ) can be obtained by calculating the XAS spectra. Co (! out ) is approximately equal to the total Co (!). For a fixed incident photon energy, the incident absorption factor Co (! in ) is always the same. Note that the total (!) includes not only the contribution of the target element but also the contributions from other elements. We assume that the other absorption channels result in a constant background B of about 5% of the total (!) (the value is estimated in Appendix A). Fig. 3 shows the self-absorption-added coefficients {1=½ð! in Þ þ ð! out Þðsin = sin Þ} as a function of rotation angle and energy loss (energy transfer). The emitted signal can be classified into two parts: the dd excitations and the fluorescence. The emitted photons' energies of dd excitations are close to the L 3 edge absorption, which implies that ð! dd out Þ ' ð! in Þ, so case (iii) is applied. In contrast, the emission energy of the fluorescence feature is $2.5 eV lower than the elastic peak for the main line of L 3 RIXS, so it experiences considerably less absorption. According to the calculation, the absorption factor 2.5 eV before the edge is considerably smaller. We can assume that ð! F out Þ ( ð! in Þ which leads to case (ii) and shows no angular dependence. This shows that the overall intensity of the angle-dependent spectra can be normalized to the fluorescence feature.

Methodology
The single crystal of LaCoO 3 was prepared from the polycrystalline sample which was a stoichiometric mixture of highpurity powders of La 2 O 3 and Co 3 O 4 using the floating-zone method. The LaCoO 3 /SrTiO 3 thin film was fabricated using pulsed laser deposition in combination with in situ reflection high-energy electron diffraction (RHEED). It was grown under a 0.2 mbar O 2 background pressure and at a deposition temperature of 750 C. The laser flux was adjusted to 1.9 J cm À2 . The sample thickness was 55 nm which is determined by X-ray reflectivity (XRR) measurement. More sample details are given by Tomiyasu et al. (2017) and Wang et al. (2019).
The Co 2p3d RIXS measurements were performed at beamline BL05A1 at the Taiwan Light Source (Lai et al., 2014). The scattering angle ( + ) was set at 140 for the LaCoO 3 single crystal and at 90 for the LaCoO 3 /SrTiO 3 film. Both samples were measured at the Co L 3 edge ($780 eV) at 20 K. The experimental resolution of RIXS was calibrated as $90 meV and the recorded incident energy broadening was $1000 meV (FWHM). The calculations of 2p3d RIXS spectra were performed using the program Quanty with intrinsic lifetime broadenings of $300 meV and 20 meV (FWHM) for the intermediate and final states, respectively (Haverkort et al., 2012). In addition to the intrinsic lifetime broadening, we applied 1000 meV and 60 meV Gaussian broadening to simulate the experiment incident energy window and the energy loss resolution. However, the self-absorption effect depends on the intrinsic absorption factor (), which is independent of the instrumental broadening. Thus, the selfabsorption-added coefficients in Fig. 3 include only intrinsic lifetime broadening. The experimental spectra were normalized to the fluorescence feature, where the fluorescence feature was subtracted from the normalized spectra (see also Appendix B). The simulations were normalized to the charge transfer features. Fig. 4(a) shows the angle-dependent RIXS of the LaCoO 3 single crystal at the maximum of the Co L 3 edge. The elastic peak shows a maximum intensity at the specular angle ( = ) due to strong reflection. In contrast, the intensity of the dd excitations decreases when increases. In comparison, we show in Fig. 4(b) the -dependent calculation of the 1 A 1g (O h ) ground state, where the angular dependence can be observed. The 3 T 1g (O h ) state slightly changes its intensity and shifts its position due to the competition of spin-orbit fine structures, which will be discussed later. However, the trends are not in good agreement with the experimental results because the The self-absorption-added coefficients {1=½ð! in Þ þ ð! out Þðsin = sin Þ} as a function of rotation angle and emitted photon energy loss for a lowspin Co 3+ 1 A 1g (O h ) ground state, where + = 140 . The absorption factor is plotted in the inset. It indicates also the conditions of case (ii) (! in ) ) (! out ) and case (iii) (! in ) ' (! out ). We note that the absorption factor has been normalized to the maximum and the selfabsorption-added coefficients have been normalized to the maximum of the region of interest. saturation and the self-absorption effects should be considered in a concentrated sample. By including the saturation and the self-absorption effects, the multiplied results show good agreement for the trends of the angular dependence except that the elastic peak is overestimated [ Fig. 4(c)]. Although the spectra were distorted, the small energy shifts of the 3 T 1g excited states are still preserved theoretically. Unfortunately, better statistics and energy resolution are required to observe this tiny shift in the experiment.

Results and discussion
For an isotropic 1 A 1g ground state, in general, no angular dependence in a dipole 2p3d transition (2p XAS) is observed (Merz et al., 2010). However, the effective dd transition operator W Qq determines the 2p3d RIXS cross-section, where the intermediate state plays an important role. This transition operator can be quadruple-like (rank 2 transition) because of the combination of two dipole transitions (rank 1 transition). On the one hand, the incident absorption channel of RIXS excites the ground state to intermediate states which preserve a mixture of different symmetries. On the other hand, due to the spin-orbit coupling, the final excited state splits into different branches with a small energy difference. In combination, the effective transition preserves spin and orbit characteristics and the polarization weighting factor T Qq (" E , " I ) provides the intensity of states as a function of geometry (van Veenendaal, 2006;Haverkort et al., 2010). In order to confirm this idea, 20 meV resolved 2p3d RIXS simulations were performed. Figs. 5(a) and 5(b) show the 2p3d RIXS spectra of three geometries with the scattering angle ( + ) set to 90 . With a finer broadening, the energy splittings caused by spinorbit coupling are better resolved [Figs. 5(a) and 5(b)]. The 3 T 1g states split into the E g , T 2g , T 1g and A 1g states, where we have indicated the double-group labels in italic. In the case of vertically-polarized incident photons (V-polarization), the elements in the polarization tensor equally contribute in all geometries. Thus, no angular dependence is expected [cf. Fig. 5(b)]. In contrast, the horizontally-polarized incident beam (H-polarization) induces the polarization tensor to change as a function of rotation angle. Thus, the angular dependence is pronounced and depends on the spin and orbit characteristics of the initial and the final state. Figs. 5(c) and 5(d) present the angle-dependent behavior of different polarizations by plots of the RIXS cross-section probed at the maximum of the L 3 edge as a function of rotation angle. Here the E g and T 2g groups of the 3 T 1g state are selected for comparison. We note that the broadening of the spectra in Fig. 4 has been increased due to the limiting experimental resolution. Hence the angular dependence reflects the intensity competition and energy shift. One way to avoid the self-absorption effect is by reducing the sample thickness, so that the emission intensity is an energy-independent function of the rotation angle [cf. equa-   tion (10)]. Once we normalize the spectra to the fluorescence feature, the intensity of the dd excitations can be represented properly. We examine the angle-dependent 2p3d RIXS of the 55 nm LaCoO 3 film grown on the SrTiO 3 substrate. This thickness is on the limit of the attenuation length of the Co L 3 edge (25-125 nm) but can still be considered to be free from self-absorption effects (see further arguments in Appendix A). The dd excitations are now less influenced by the selfabsorption effect and are mainly determined by the polarization selectivity -see Fig. 6. Most of the angle-dependent features agree with the calculation of the 5 B 2g (D 4h ) ground state (gray arrows). As has been discussed, the ground state of the Co 3+ ions in the LaCoO 3 /SrTiO 3 film shows a mixture of the low-spin 1 A 1g (O h ) state and high-spin 5 B 2g (D 4h ) state (Wang et al., 2019). Compared with the 5 B 2g (D 4h ) state, the angular dependence of RIXS is negligible for the 1 A 1g (O h ) state (Fig. 6). The angle-independent features of the 1 A 1g (O h ) state are indicated by black arrows, but are almost invisible compared with the 5 B 2g (D 4h ) ground state. Some discrepancies remain, for example the 1.3 eV feature indicated with the red arrow in the 5 B 2g (D 4h ) calculation is not evident in the experimental data. This could be because: (i) the mixture of the spin states suppresses the intensity of the 1.3 eV feature (Wang et al., 2019); (ii) we assumed that the CoO 6 clusters are well aligned to the pseudo-cubic orientation (002) -experimentally, not all the CoO 6 clusters should be aligned exactly along the pseudo-cubic orientation; (iii) the discussions above do not take the trigonal symmetry reduction into account. The CoO 6 clusters are naturally aligned in a trigonal field for a relaxed LaCoO 3 crystal (the space group of the unit cell is R " 3 3c). By including the trigonal distortion, the polarization selectivity might be different from the tetragonal distortion.

Conclusions
We have presented the angular dependence of 2p3d RIXS analyzed with the scattering cross-section including saturation and self-absorption effects. The angle-dependent spectra are a better probe of the symmetry type than using only the difference between two orthogonal polarizations, due to the angular relation between the ground/excited states symmetry and the polarization tensor. A remarkable consequence is that even the isotropic low-spin 1 A 1g (O h ) ground state shows angle-dependent 2p3d RIXS spectra, despite the absence of angular dependence in the 2p XAS spectra (Merz et al., 2010;Wang et al., 2019). Unfortunately, the high Co concentration in the LaCoO 3 single crystal distorts the RIXS spectra due to the saturation and self-absorption effects which limits the distinguishability. By applying a model to consider these effects, the trends are explained as a function of both the energy and the rotation angle. In contrast, for the LaCoO 3 /SrTiO 3 film, the self-absorption effect is small. Thus the LaCoO 3 /SrTiO 3 film shows better agreement with the angular trend although some features are not reproduced well. The possible reason for this mismatch could be a mixture of ground states, non-perfect empirical parameters and/or reduced symmetry.
APPENDIX A Estimation of the attenuation length of the elements at the Co L 3 edge ($780 eV) In the main text, we use the relative relation of the absorption factor () to describe the trends of the saturation and selfabsorption effects. Here we provide the experimental quantities of different elements in the LaCoO 3 to estimate the following questions: (a) What is the maximum attenuation length of the Co L 3 edge? (b) What is the critical film thickness in order to claim the absence of the self-absorption effect? (c) What is the reasonable weighting for B ?
According to the tabulated data (Henke et al., 1993), the attenuation length at 780 eV for the La, O and Co elements in the LaCoO 3 ( = 7.25 g cm À3 ) are $950, $810 and $380 nm, respectively. The values show that the attenuation length of Co is much longer than our film thickness (55 nm), which leads to the consequence that our film can be self-absorption free.
However, the attenuation length values at the absorption edge are likely overestimated using Henke's table. For cobalt metal ( = 8.9 g cm À3 ), the estimated attenuation length is $75 nm but the experimental results indicate that the attenuation length was $25 nm at the peak maximum (Chen et al., 1995). By considering that the mass weight of Co is about 25% in the LaCoO 3 ( Co = 1.8 g cm À3 ), the attenuation length of the Co L 3 edge peak maximum is $125 nm (five times larger). This value is still larger than our film thickness of $55 nm.
On the other hand, the self-absorption effect on the RIXS profile will be dependent on the intrinsic lifetime broadening. If we set a lifetime broadening of 300 meV, the selected energy at a few hundred meV away from maximum of the L 3 resonance peak has an attenuation length a few times longer. This The 2p3d RIXS results of the 55 nm LaCoO 3 /SrTiO 3 film compared with the 1 A 1g (O h ) and 5 B 2g (D 4h ) ground states. Both the (a) V-and (b) Hpolarized are calculated and measured at a geometry for a scattering angle of 90 .
implies that we can treat the 55 nm LaCoO 3 film as being free from self-absorption effect at the orbital excitations (>200 meV). However, we remark that minor self-absorption effects could be visible for the elastic and low-loss features (<150 meV).
Here we estimate the background absorption. The weighting of is proportional to the inverse of the attenuation length. It gives values of 2-10%, 3-12% and 95-78% with respect to elements La, O and Co (attenuation lengths are 950, 810 and 25-125 nm), respectively. The background absorption factor ( B ) is the off-resonant contribution from the other elements (except Co) and is estimated to be about 5-22%. We set B as $5% of max in our model. In order to compare the calculation with the experimental results, we normalized the spectra to the summed area of the charge transfer/fluorescence features. These features (>2 eV) are expected to be self-absorption free according to our discussion. We use five Gaussian functions to simulate the experimental fluorescence profile (gray area in Fig. 7). This fluorescence profile was used for the normalization procedure. After the spectra were normalized, the fluorescence profile was subtracted from the normalized spectra.