Coexistence of bound and virtual-bound states in shallow-core to valence x-ray spectroscopies

Subhra Sen Gupta, J. A. Bradley, M. W. Haverkort, G. T. Seidler, A. Tanaka, and G. A. Sawatzky

Phys. Rev. B 84, 075134 – Published 10 August 2011

Abstract

With the example of the non-resonant inelastic x-ray scattering (NIXS) at the O $_{45}$ edges ( $5 d \to 5 f$ ) of the actinides, we develop the theory for shallow-core to valence excitations, where the multiplet spread is larger than the core-hole attraction, such as if the core and valence orbitals have the same principal quantum number. This involves very strong final state configuration interaction (CI), which manifests itself as huge reductions in the Slater-Condon integrals, needed to explain the spectral shapes within a simple renormalized atomic multiplet theory. But more importantly, this results in a cross-over from bound (excitonic) to virtual-bound excited states with increasing energy, within the same core-valance multiplet structure, and in large differences between the dipole and high-order multipole transitions, as observed in NIXS. While the bound states (often higher multipole allowed) can still be modeled using local cluster-like models, the virtual-bound resonances (often dipole-allowed) cannot be interpreted within such local approaches. This is in stark contrast to the more familiar core-valence transitions between different principal quantum number shells, where all the final excited states almost invariably form bound core-hole excitons and can be modeled using local approaches. The possibility of observing giant multipole resonances for systems with high angular momentum ground states is also predicted. The theory is important to obtain ground state information from core-level x-ray spectroscopies of strongly correlated transition metal, rare-earth, and actinide systems.

Received 9 May 2011

DOI:https://doi.org/10.1103/PhysRevB.84.075134

Authors & Affiliations

Subhra Sen Gupta^1,*, J. A. Bradley², M. W. Haverkort³, G. T. Seidler², A. Tanaka⁴, and G. A. Sawatzky¹

¹Department of Physics and Astronomy, University of British Columbia, Vancouver, BC V6T 1Z1, Canada
²Department of Physics, University of Washington, Seattle, Washington 98105, USA
³Max Planck Institute for Solid State Research, Heisenbergstraße 1, D-70569 Stuttgart, Germany
⁴Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan

^*subhra@phas.ubc.ca

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Vol. 84, Iss. 7 — 15 August 2011

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Figure 1
Calculated $q$ -dependent NIXS spectra [ $S (\vec{q}, ω)$ ] within a simple renormalized atomic multiplet model for the $5 f^{0}$ system Th $^{4 +}$ (a), compared with the corresponding experimental NIXS spectra[20] for ThO $_{2}$ (c); and for the $5 f^{2}$ system U $^{4 +}$ (b), compared with the corresponding experimental NIXS spectra[20] for UO $_{2}$ (d). The respective component spectra (calculated) for the $l = 1, 3, 5$ channels are shown at the bottom of panels (a) and (b). Also, in each case, the calculated spectra with high resolution (dashed lines) is shown below the calculated spectra with experimental resolution (thick solid lines), to bring out the finer details of the multiplet structure. The inset to panel (c) shows the relative variation of the cross-sections (transition probabilities) for the different multipole channels as a function of $q$ .Reuse & Permissions
Figure 2
(a) Plots of the $5 d$ , $5 f$ , and $6 f$ atomic H-F radial wavefunctions for Th $^{4 +}$ ( $5 f^{0}$ system) (inset at the top); and calculated NIXS [ $S (\vec{q}, ω)$ ] for the same, including final-state CI with the $6 f$ level, for 60% of atomic $C I$ ( $5 f$ - $6 f$ ) values (main). Both the $5 f$ -like and $6 f$ -like regions are shown. The bare component ( $l$ $=$ $1, 3, 5$ ) spectra (not to scale) are shown at the bottom ( $6 f$ region magnified in the bottom inset for clarity). (b) Schematic illustrating the high sensitivity of the dipole term vis-a-vis the high multipole terms, due to the strongly term-dependent CI with the $6 f$ .Reuse & Permissions
Figure 3
(a) Experimental $q$ -averaged NIXS for ThO $_{2}$ , compared with the component ( $l$ $=$ $1, 3, 5$ ) summed calculated spectra for atomic values of ( $F_{d f}^{k}$ , $G_{d f}^{k}$ ), and the $C I$ ( $5 f$ - $6 f$ ) varied from a scaling of 20% to their full atomic values. Only peak-positions are relevant. The best agreement is obtained at 60% scaling of $C I$ ( $5 f$ - $6 f$ ). Variation of the high-energy dipole peak position with varying scaling of: (b) $C I$ ( $5 f$ - $6 f$ ), and (c) ( $F_{d f}^{k}$ , $G_{d f}^{k}$ ). While the latter shows a simple linear trend, the former shows a parabolic behavior with signs of saturation.Reuse & Permissions
Figure 4
Schematics illustrating the differences between transitions occurring between different $n$ -shells and within the same $n$ -shell (see text): (a) The final valence state of XAS or NIXS in the absence of the core-hole; (b) the effect of a scaler core-hole potential ( $Q$ ) on this final state; (c) its fanning out due to core-valence multiplet effects, in the case $n > n^{'}$ , when all final state terms are excitonic bound states; and (d) formation of the core-valence multiplet in the case when $n = n^{'}$ , leading to a gradual crossover from bound-states (lower terms) to virtual-bound resonances (higher terms) as a function of energy.Reuse & Permissions
Figure 5
(a) Schematic showing the model used to simulate the O $_{45}$ NIXS of ThO $_{2}$ (Th $^{4 +}$ ), which captures both the excitonic high multipole features as well as the GDR. It includes a Fano-effect of the mixing of the atomic $5 d$ - $5 f$ transition, with that to a fictitious $7 p$ -like rectangular broad band, via the $C I$ ( $5 f$ - $7 p$ ) matrix elements. (b) The calculated $S (\vec{q}, ω)$ from the model (lines) resulting in a broad, asymmetric lineshape for the high-energy dipole term, that agrees well with the experimental $q$ -dependent NIXS for ThO $_{2}$ (symbols).[20] A dipole term at lower energy ( $\sim$ 89 eV) is not broadened into a resonance, but appears as an excitonic feature just like the high multipole terms.Reuse & Permissions
Figure 6
Calculated N $_{45}$ NIXS for a model more-than-half-filled rare-earth system Tm $^{3 +}$ , (e.g., in Tm $_{2}$ O $_{3}$ ). The transition involved is $4 d^{10} 4 f^{12} \to 4 d^{9} 4 f^{13}$ . (a) The bare component spectra for the $l = 1, 3, 5$ channels. (b)-(g) Calculated NIXS spectra [ $S (\vec{q}, ω)$ ] plotted as a function of increasing $q$ , for $q = 4, 8, 12, 16, 20$ , and 24 Å $^{- 1}$ . The vertical scales in the different panels are not to be compared.Reuse & Permissions

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