Crystalline field Octahedral

To illustrate how the perturbation problem must be solved, we now describe one of the simplest cases, corresponding to an octahedral crystalline field acting on a single d valence electron. 

In Appendix A2, we have formally applied the perturbation method to find the energy levels of a d ion in an octahedral environment, considering the ligand ions as point charges. However, in order to understand the effect of the crystalline field over d ions, it is very illustrative to consider another set of basis functions, the d orbitals displayed in Figure 5.2. These orbitals are real functions that are derived from the following linear combinations of the spherical harmonics ... 

So far, we have discussed the crystalline field acting on the ion A due to an octahedral environment of six B ligand ions. In many optically ion activated crystals, such as Ti +rAlaOj, the local symmetry of the active ion A is slightly distorted from the perfect octahedral symmetry Oh symmetry). This distortion can be considered as a perturbation of the main octahedral field. In general, this perturbation lifts the orbital degeneracy of the tag and eg levels and then produces additional structure in the tag eg absorption/emission bands. 

On the other hand, the crystalline field due to main symmetries other than Oh symmetry can be also related to this same case. For this purpose, it is useful to represent the octahedral structure of our reference ABe center as in Figure 5.5(a). In this representation, the B ions lie in the center of the six faces of a regular cube of side 2a and the ion A (not displayed in the figure) is in the cube center the distance A-B is equal to a. 

The regular cube used in Figure 5.5 to represent different symmetry centers suggests that these symmetries can be easily interrelated. In particular, following the same steps as in Appendix A2, it can be shown that the crystal field strengths, lODq, of the tetrahedral and cubic symmetries are related to that of the octahedral symmetry. Assuming the same distance A-B for all three symmetries, the relationships between the crystalline field strengths are as follows (Henderson and Imbusch, 1989) ... 

This means that the fifthly degenerate d energy level splits into two levels in an octahedral crystalline field one triply degenerate and the other doubly degenerate. 

Figure A2.2 The effect of an octahedral crystalline field on a d energy level.

Fig. 5. Octahedral-site splitting of (a) the or-bitally fivefold-degenerate d1 manifold by a cubic crystalline field and (b) the high-spin Mn(III) configuration.

DBTDL is described as an octahedral complex (31, 32) four of its coordination vacancies are occupied by the ligands whereas the alcohol and the isocyanate can occupy the two last positions. The methoxy group of the activated alcohols is unable to participate to the crystalline field and has only a steric and therefore, negative effect on the rate. 

A variety of structural information can be inferred from magnetic susceptibility data. " The site symmetry of a paramagnetic ion can be determined from the localized atomic moment. The orbital contribution to atomic moment is strongly influenced by the crystalline fields, and therefore magnetic susceptibility measurements provide information about the local symmetry of the ions. For example, Co ions have a nearly spin-only atomic moment of in a tetrahedral interstice, whereas octahedrally coordinated Co has a moment of about 3.7 Ub. 

The above simple picture of solids is not universally true because we have a class of crystalline solids, known as Mott insulators, whose electronic properties radically contradict the elementary band theory. Typical examples of Mott insulators are MnO, CoO and NiO, possessing the rocksalt structure. Here the only states in the vicinity of the Fermi level would be the 3d states. The cation d orbitals in the rocksalt structure would be split into t g and eg sets by the octahedral crystal field of the anions. In the transition-metal monoxides, TiO-NiO (3d -3d% the d levels would be partly filled and hence the simple band theory predicts them to be metallic. The prediction is true in TiO... 

In the same samples, a second absorption feature was detected that is associated with the dopant ions themselves. These ligand-field transitions allow distinction among various octahedral and tetrahedral Co2+ species and are discussed in more detail in Section III.C. The three distinct spectra observed in Fig. 4(b) correspond to octahedral precursor (initial spectrum), tetrahedral surface-bound Co2+ (broad intermediate spectrum), and tetrahedral substitutional Co2+ in ZnO (intense structured spectrum). Plotting the tetrahedral substitutional Co2+ absorption intensity as a function of added base yields the data shown as triangles in Fig. 4(b). Again, no change in Co2+ absorption is observed until sufficient base is added to reach critical supersaturation of the precursors, after which base addition causes the conversion of solvated octahedral Co2+ into tetrahedral Co2+ substitutionally doped into ZnO. Importantly, a plot of the substitutional Co2+ absorption intensity versus added base shows the same nucleation point but does not show any jump in intensity that would correspond with the jump in ZnO intensity. Instead, extrapolation of the tetrahedral Co2+ intensities to zero shows intersection at the base concentration where ZnO first nucleates, demonstrating the need for crystalline ZnO to be... 

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