Theory of resonant X-ray emission spectroscopy

A short review is presented of recent topics in resonant X-ray emission spectroscopy (RXES), especially of the polarization dependence of RXES in d and f electron systems. Polarization-dependent RXES provides important information on the symmetry of electronic states. A scattering-angle dependence in the fluorescence yield of rare-earth metals, RXES in polarized and depolarized configurations of TiO₂, and the magnetic circular dichroism in RXES of ferromagnetic systems are discussed together with typical experimental data.

1. Introduction

Resonant X-ray emission spectroscopy (RXES) is one of the most powerful tools in solid-state spectroscopy with synchrotron radiation (Kotani & Shin, 2000). In this paper we report theoretical calculations on the polarization dependence of RXES in d and f electron systems, such as rare-earth metals and transition-metal compounds, together with their experimental results. Using synchrotron radiation with an undulator beamline, the incident photon is polarized linearly or circularly in RXES experiments. However, it is very difficult to detect the polarization of the emitted photon directly.

The polarization-dependent RXES is described theoretically by the coherent second-order optical formula in the form

where Ω and ω are, respectively, the energies of the incident and emitted photons with polarization directions β and α; |g〉, |i〉 and |j〉 are initial, intermediate and final states of the material system, respectively, and E_g, E_i and E_j are their energies. The operator T represents the optical dipole transition, and Γ_i is the spectral broadening due to the core-hole lifetime in the intermediate state. Tanaka & Kotani (1993) performed, before experimental observations, theoretical calculations of F_xx and F_zx in the Cu 2p → 3d → 2p RXES of La₂CuO₄, where the crystalline c axis is taken in the z direction. They showed that the polarization dependence of RXES provides important information on the symmetry of electronic states.

A few years after the theoretical calculation, F_α,β of La₂CuO₄ was observed experimentally by using a sophisticated geometrical arrangement where a spectrometer can be rotated around the incident beam axis and a sample can also be rotated around its surface normal (Duda, 1996; Duda et al., 1998). Unfortunately, the resolution is not very good, but the theoretical and experimental results are in fair agreement within the experimental resolution.

More recently, polarization-dependent RXES has been studied more precisely, without detecting the polarization of the emitted photon, by switching the experimental geometry (polarized and depolarized configurations defined later) or by changing the scattering angle.

In this paper three typical examples are presented: scattering-angle dependence (§2), polarized and depolarized configurations (§3), and magnetic circular dichroism (§4) in RXES.

2. Scattering-angle dependence in fluorescence yield of rare-earth metals

We take the incident photon polarization in the z direction and denote a scattering angle by θ, as shown in Fig. 1. The polarization of the emitted photon is not detected. Then the RXES spectrum is expressed as (Nakazawa et al., 1998)

where C_q^(k) is defined by

with spherical harmonics Y_kq, and the angle-dependent factor A_q(θ) is given by

Figure 1 Geometrical arrangement and coordinate axes in the calculation of RXES and FY.

Experimental observations of the θ dependence of RXES were made by Pompa et al. (1997) for the fluorescence yield (FY) of 3d → 4f → 3d RXES in La and Sm metals, instead of the RXES itself. FY is the spectroscopy where the intensity of RXES integrated over the emitted photon energy is measured as a function of the incident photon energy. The results are shown as dots in Figs. 2(a) and 2(b). It is found that in La the intensity of FY changes drastically with θ, but the spectral shape of FY is unchanged. In Sm, on the other hand, the change in the intensity is less dramatic but the spectral shape changes with θ.

Figure 2 Calculated (line) and experimental (dots) results of the scattering-angle-dependent FY for (a) La^3+ and (b) Sm^3+.

Theoretical calculations of these FY spectra were performed by Nakazawa et al. (1998). From (2), the spectrum of FY is given by

where

Since the spectral structure of RXES in rare-earth metals is mainly determined by the intra-atomic multiplet structure, the calculation was made with a free-ion model. The calculated results are shown as solid curves in Figs. 2(a) and 2(b). The theoretical and experimental results are in good agreement with each other. Nakazawa et al. (1998) discussed the reason for the angle dependence of FY for La and Sm. For La, FY intensity decreases rapidly as θ increases. Since the initial and final electronic configuration of La³⁺ is 4f⁰ with J = 0, only the component q = 0 contributes to the FY intensity (note that F_zz ≠ 0 and F_xz = F_yz = 0 in this case), which is proportional to cos²θ as seen from (4).

To understand the angle dependence of FY for Sm, the calculated results of f_q(Ω) [given by (6)] for q = 0 (caused by the diagonal component F_zz of RXES) and q = ±1 (caused by off-diagonal components F_xz and F_yz) are shown in Fig. 3 for Sm. It is seen that f₀ and f_±1 for Sm are much different both in their intensity and in their spectral shape. With the change of the angle θ, the weight between f₀ and f_±1 is changed, so that the angle dependence of FY is understood.

Figure 3 Calculated results of FY for Sm^3+, which is decomposed into the polarization component of the emitted X-ray.

3. RXES in polarized and depolarized configurations of TiO₂

Let us consider the geometrical configuration shown in Fig. 1. The angle θ between the incident and emitted photon directions is fixed at 90°, and the incident photon polarization is taken in either the y or the z direction, which is denoted by the `polarized' or `depolarized' configuration, respectively. The polarization of the emitted photon is not detected. In these two configurations the spectra of RXES are expressed, from (2), as

The calculation of the Ti 2p → 3d → 2p RXES of TiO₂ was made by Matsubara et al. (2000, 2001) with a TiO₆ cluster model with O_h symmetry. The results are shown in Fig. 4 (lower panel), where the incident photon energy is taken at the positions a–h in the XAS spectrum (upper panel), and the RXES spectra are plotted as a function of the emitted photon energy with the elastic (Rayleigh) scattering line as the origin. The spectral structure is divided into three categories: (1) elastic line at 0 eV, (2) inelastic spectra at 7 and 9 eV, and (3) inelastic line at 14 eV. The mechanism of these spectra can be explained by the energy-level scheme shown in Fig. 5. TiO₂ is nominally in the d⁰ state, but actually the d⁰ configuration is strongly mixed with a charge-transferred configuration by the covalency hybridization, where denotes a hole in the ligand state (O 2p molecular orbit). The ground state is the bonding state between the d⁰ and d¹ configurations, and the anti-bonding state is located about 14 eV above the ground state. Both bonding and anti-bonding states are specified by an irreducible representation A_1g of the O_h symmetry group, because their symmetry is the same as the d⁰ state. In addition to these states, there are non-bonding states with A_2g, T_1g, T_2g, E_g, symmetries about 7–9 eV above the ground state. When a Ti 2p electron is excited to the 3d state by the incident photon, we have 2p⁵3d¹ and 2p⁵3d² configurations, which are mixed strongly by the covalency hybridization. The main peak of the Ti 2p XAS corresponds to the bonding state between the 2p⁵3d¹ and 2p⁵3d² configurations, while the satellite corresponds to the anti-bonding state between them. X-ray absorption is almost forbidden in the non-bonding states. In Fig. 5 we disregard the effects of the spin–orbit splitting of the 2p states and the crystal field splitting of the 3d states, for simplicity. If we take into account these effects, the main peak (and also the satellite) splits into four peaks, as seen in Fig. 4.

Figure 4 Calculated results of Ti 2p XAS and 2p → 3d → 2p RXES of TiO2 using the TiO6 cluster model.

Figure 5 Schematic representation of the Ti 2p → 3d → 2p RXES of TiO2 using the TiO6 cluster model.

The resonantly excited intermediate states, which correspond to the main line and the satellite of XAS, decay radiatively to each of the final states of RXES, i.e. the bonding, non-bonding and anti-bonding states. The categories (1), (2) and (3) of RXES spectra correspond to the bonding, non-bonding and anti-bonding final states, respectively. The spectrum (3) occurs for the incident photon energy tuned to the satellite of the XAS spectrum. This is because the XAS satellite corresponds to the anti-bonding intermediate state of RXES, so that it enhances dramatically the intensity of the anti-bonding final state (3). The spectra (1) and (3) are allowed only for the polarized configuration, while the spectrum (2) is allowed for both polarized and depolarized configurations.

Let us consider the mechanism of the polarization dependence of the RXES spectra. It is easily shown from (7) and group theoretical consideration that the final states with A_1g, T_1g, T_2g and E_g irreducible representations are allowed for the polarized configuration, whereas those with only T_1g and T_2g irreducible representations are allowed for the depolarized configuration. Therefore, the elastic peak (bonding state) and the 14 eV inelastic peak (anti-bonding state) are allowed for the polarized configuration, but they are forbidden for the depolarized configuration. The non-bonding states are allowed both for the polarized and depolarized configurations.

Very recently, experimental measurements of RXES for TiO₂ have been made by Harada et al. (2000) for the polarized and depolarized configurations. The result is shown in Fig. 6. The three categories of RXES spectra (1)–(3) are clearly seen, in addition to the spectra indicated by vertical bars, which are absent in our TiO₆ cluster-model calculation (Fig. 4) but are explained as originating from the effect of the periodic arrangement of the Ti ions (Idé & Kotani, 1998). The elastic scattering peak at 0 eV (category 1) and the inelastic one at 14 eV (category 3) are allowed only for the polarized configuration, and the intensity of the 14 eV peak is dramatically enhanced when the incident photon energy is tuned to the satellite of the XAS spectrum. Near the middle of the elastic (0 eV) and inelastic (14 eV) scattering peaks, there are inelastic scattering spectra (category 2) which are allowed both for the polarized and depolarized configurations. These results are in good agreement with the calculated ones.

Figure 6 Experimental results of Ti 2p XAS and 2p → 3d → 2p RXES of TiO2.

It is to be remarked that the reason that the elastic and the 14 eV inelastic lines are forbidden for the depolarized configuration is that the off-diagonal components, F_xz and F_yz, vanish if both initial and final states of RXES are the non-degenerate A_1g state. In other words, if the initial and final states of the material system are the same, the polarizations of the incident and emitted photons should also be the same, because the angular momentum of the total system (including the photon and the material) is conserved. The reason that the FY of La vanishes for θ = 90° is also essentially the same.

4. MCD in RXES of ferromagnetic systems

Another interesting phenomenon on the polarization dependence of RXES is the magnetic circular dichroism (MCD) in ferromagnetic systems. We consider the situation where a photon with circular polarization (+ or − helicity) is incident on the sample with an angle φ from the sample surface normal, and the emitted photon is detected with an angle φ′ from the surface normal without analyzing the polarization, where the surface normal is perpendicular to the magnetization of the sample, and all the directions of incident and emitted photons, surface normal and the magnetization are in the same plane. Then the MCD of RXES, i.e. the difference of RXES for the incident photons with + and − helicities, is given by

where F_q′q is defined by

with the polar axis [for C_q⁽¹⁾] taken in the direction of the magnetization.

An example is shown in Fig. 7. Here, the experimental data (dots) (Krisch et al., 1996; de Groot et al., 1997) are shown for the X-ray emission and its MCD for ferromagnetic Gd metal, where a 2p_3/2 core electron is excited to a high-energy continuum (instead of the excitation near to the threshold) and a 3d_5/2 electron makes a radiative transition to the 2p_3/2 state. The angles φ and φ′ are taken as 60° and 30°, respectively. The calculated results (de Groot et al., 1997) with a Gd³⁺ ion model are also shown by the solid curves. The agreement with the experimental results is good. Experiments and calculations similar to this but for the 4d core level (instead of the 3d level) were also made by Krisch et al. (1996) and by de Groot et al. (1997). Also, similar calculations of normal X-ray emission spectroscopy (NXES) and MCD were made by Jo & Tanaka (1998) for both 3d and 4d core levels of Nd³⁺, Gd³⁺ and Dy³⁺ ions.

Figure 7 Gd 3d–2p3/2 X-ray emission spectrum (upper) and its MCD (lower) under the excitation of a 2p3/2 core electron to a high-energy continuum by circular polarized incident X-rays. Solid curves and dots are the calculated and experimental results, respectively.

Finally, it is to be mentioned that the MCD of RXES, where the incident photon energy is tuned to the absorption threshold, is a more interesting subject. Krisch et al. (1996) and Iwazumi et al. (1997) observed the MCD of Gd 3d_5/2 → 2p_3/2 RXES in the 2p_3/2 excitation threshold. The data should include important information on the magnetic polarization of both ground and excited states. Since the final state of XAS is the same as the intermediate state of RXES, a theoretical analysis of these data of MCD in RXES should be a critical test of the validity of possible mechanisms proposed for the MCD in XAS, and some study along this line is now in progress (Fukui et al., 2001).