Cubic-field splitting

The condition for localized electrons with spontaneous atomic moments is U > w that for itinerant electrons with no spontaneous moment is Un < w. The intermediate case Un = w is of considerable theoretical interest. In the case of iron oxides, a U5 = 3 eV ensures a localized 3 d majority-spin configuration at both Fe " and Fe " ions since the cubic-field splitting Ag < Ag, is small enough to leave the ions in the high-spin state. However, localization of the minority-spin electron, particularly in the mixed Fe / Fe " state, does not necessarily follow. Similarly, a much smaller U4 will be seen to make U4 = w for Fe " in the perovskites A Fe03. 

Note that, of course, the cubic crystal field splitting dzi, dx2 y2 (eg) at 6Dq and dxz, dyz, dxy at -4Dq is reproduced if Ds and Dt are both zero. Note also that the centre of gravity of neither of the eg nor of the t2g sets is maintained as the symmetry departs from cubic. This means that, in low symmetry, the concept of the cubic field splitting is not clearly defined. For small departures from cubic symmetry the lack of definition is not serious in practice, but to maintain the concept in, say, a square-planar complex, as is sometimes done, requires care. 

Finally, a a band on the itinerant-electron side of the Mott-Hubbard transition in the presence of localized t spins S = 3/2 signals covalent-mixing parameters X and therefore a cubic-field splitting (Eq. 5 of Goodenough, this volume) that approaches Aex- To test this deduction, hydrostatic pressure was used to induce a transition from the high-spin to the low-spin states in CaFeOs the transition was observed to occur at a pressure Pc 30 GPa [50]. 

In a cubic field three spin-allowed transitions are expected because of the splitting of the free-ion, ground term and the presence of the term. In an octahedral field the splitting is the same as for the octahedral d ion and the same energy level diagram (p. 1029) can be used to interpret the spectra as was used for octahedral Cr Spectra of octahedral Ni usually do consist of three bands which are accordingly assigned as ... 

Magnetic exchange, 1,257, 267 polymetallic complexes, 1,138 Magnetic moment, 1, 259 Magnetic properties atomic systems, 1,260 cubic field systems, 1,263 free-atom states and terms, I, 260 lower symmetry, 1, 264 zero-field splitting, 1,262 Magnetic susceptibility, 1,256, 259 Magnetism... 

The spectrum of Mn2+ in zeolites has been used to study the bonding and cation sites in these crystalline materials. This is a 3d5 ion hence, one would expect a zero-field splitting effect. A detailed analysis of this system was carried out by Nicula et al. (170). When the symmetry of the environment is less than cubic, the resonance field for transitions other than those between the + and — electron spin states varies rapidly with orientation, and that portion of the spectrum is spread over several hundred gauss. The energies of the levels are given by the equation... 

Sharma, R.R., Das, T.P., and Orbach, R. 1968. Zero-field splitting of S-state ions. III. Corrections to part I and II and application to distorted cubic crystals. Physical Review 171 378-388. 

Make a sketch of a cubic complex (start with a tetrahedral complex and add four ligands). Analyze the repulsion of each of the d orbitals and sketch the splitting pattern that would exist in a cubic field. 

F ure 5.6 The energy-level scheme for the crystal field splitting of a d electron in different symmetries (a) octahedral, (b) tetrahedral, and (c) cubic. 

If, in addition to the cubic crystal field, a component of lower symmetry is present, such as one having tetragonal or trigonal symmetry (as for the Al sites in a-AUOa), further splitting will occur as shown in Fig. 24. Crystal field splittings for other configurations in both the weak and strong field cases are summarized in a review article by Moffitt and Ballhausen... 

Table 5. The levels and cubic crystal-field splitting (zJ) for Ce3+ in several host lattices. All values in kK (23)...

Composition Coordination of Ce3+ 5d levels (observed) ) Deduced cubic levels Cubic crystal-field splitting (A) Spherical 5d... 

Fig. 1.5 Plot of energies of eg and t2g d-functions against k in a cubic lattice, for the case where the crystal-field splitting is large compared with the overlap energy integral.

In any cubic field, or in the case of octahedral coordination by eight anions, the 3d-state of a transitional-metal cation splits into six states of t2g symmetry with orbital wave functions of the form... 

A prediction of crystal field theory as outlined in the preceding subsections is that the crystal field splitting parameter, A, should be rather critically dependent upon the details of the crystal lattice in which the transition metal ion is found, and that the splittings of the /-orbital energies should become larger and quite complicated in lattices of symmetry lower than cubic. The theory could not be expected to apply, for example, to the spectra of transition metal ions in solution. 

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