Multiplet Effects in X-ray Absorption - Inorganic Chemistry and ...

F. de Groot / Coordination Chemistry Reviews 249 (2005) 31–63 37J ′ = J = 0. Within LS coupling also S = 0 and L = 1.The dipole selection rule reduces the number of final statesthat can be reached from the ground state. The J-value in theground state is zero, proclaiming that the J-value in the finalstate must be one, thus only the three term symbols 1 P 1 , 3 P 1and 3 D 1 obtain finite intensity. The problem of calculatingthe 2p absorption spectrum is hereby reduced to solving the3 × 3 energy matrix of the final states with J = 1.To indicate the application of this simple calculation, wecompare a series of X-ray absorption spectra of tetravalenttitanium 2p and 3p edges and the trivalent lanthanum 3dand 4d edges. The ground states of Ti IV and La III are, respectively,3d 0 and 4f 0 and they share a 1 S ground state.The transitions at the four edges are, respectively, 3d 0 →2p 5 3d 1 ,3d 0 → 3p 5 3d 1 ,4f 0 → 3d 9 4f 1 and 4f 0 → 4d 9 4f 1 .These four calculations are equivalent and all spectra consistof three peaks. What changes are the values of theatomic Slater–Condon parameters and core hole spin–orbitcouplings, as given in table. The important factor for thespectral shape is the ratio of the core spin–orbit couplingand the F 2 value. Finite values of both the core spin–orbitand the Slater–Condon parameters cause the presence of thepre-peak. It can be seen in Table 3 that the 3p and 4d spectrahave small core spin–orbit couplings, implying small p 3/2(d 5/2 ) edges and extremely small pre-peak intensities. Thedeeper 2p and 3d core levels have larger core spin–orbit splittingwith the result of a p 3/2 (d 5/2 ) edge of almost the sameintensity as the p 1/2 (d 3/2 ) edge and a larger pre-peak. Notethat none of these systems comes close to the single-particleresult of a 2:1 ratio of the p edges or the 3:2 ratio of thed edges. Fig. 4 shows the X-ray absorption spectral shapes.They are given on a logarithmic scale to make the pre-edgesvisible.Table 3The relative intensities, energy, core hole spin–orbit coupling and F 2Slater–Condon parameters are compared for four different 1 S 0 systemsEdge Ti 2p Ti 3p La 3d La 4dAverage energy (eV) 464.00 37.00 841.00 103.00Core spin–orbit (eV) 3.78 0.43 6.80 1.12F 2 Slater–Condon (eV) 5.04 8.91 5.65 10.45IntensitiesPre-peak 0.01 10 −4 0.01 10 −3p 3/2 or d 5/2 0.72 10 −3 0.80 0.01p 1/2 or d 3/2 1.26 1.99 1.19 1.99The G 1 and G 3 Slater–Condon parameters have an approximately constantratio with respect to the F 2 value.In Table 4 the ground state term symbols of all 3d N systemsare given. Together with the dipole selection rules thisstrongly limits the number of final states that can be reached.Consider, for example, the 3d 3 → 2p 5 3d 4 transition: The3d 3 ground state has J = 3/2 and there are, respectively, 21,35 and 39 terms of the 2p 5 3d 4 configuration with J ′ = 1/2,3/2 and 5/2. This implies a total of 95 allowed peaks out ofthe 180 final state term symbols. From Table 4 some specialcases can be found, for example, a 3d 9 system makes a transitionto a 2p 5 3d 10 configuration, which has only two termsymbols, out of which only the term symbol with J ′ = 3/2is allowed. In other words, the L 2 edge has zero intensity.The 3d 0 and 3d 8 systems have only three, respectively, fourpeaks, because of the limited amount of states for the 2p 5 3d 1and 2p 5 3d 9 configurations.Atomic multiplet theory is able to accurately describe the3d and 4d X-ray absorption spectra of the rare earths. Incase of the 3d metal ions, atomic multiplet theory can notsimulate the X-ray absorption spectra accurately because theFig. 4. The La III 4d and 3d plus T IIV 3p and 2p X-ray absorption spectra as calculated for isolated ions. The intensity is given on a logarithmic scale tomake the pre-edge peaks visible. The intensities of titanium have been multiplied by 1000.