Cubic crystal field

M = DlS -iS S+iy] + E S -S )+g)iBS-H as a consequence of axially symmetric and rhombic contributions to the crystal field cubic ligand field parameter electronic charge... 

Due to the intermediate coupling the sign of the crystal field matrix element 6 is reversed compared to the pure Russell-Saunders state. Thus for 8-fold cubic coordination a F7 ground state was found. From EPR measurements on Pu3"1" diluted in fluorite host lattices, a magnetic moment at T=0 K can be calculated, ranging from li ff = 1.333 (in Ce02) to y ff = 0.942 (in SrCl2) (24,... 

S is the ionic spin and D, ID, and a are crystal field parameters describing the strength of the axial, the rhombohedral, and the cubic crystal field terms, respectively. The coordinate system of the cubic crystal field (ij, rj, g) may differ from that used to describe the axial and the rhombohedral crystal field interactions (x,y,z). 

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) ... 

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. 

These situations are known as the weak, strong, and medium field cases respectively. As an example of case (c) consider the Cr + ion (3d ). The ground term for the free atom is F and some 14,000 cm. above this state is a state. Under the action of a crystal field of cubic symmetry, such as that... 

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... 

Mallinson et al. (1988) have performed an analysis of a set of static theoretical structure factors based on a wave function of the octahedral, high-spin hexa-aquairon(II) ion by Newton and coworkers (Jafri et al. 1980, Logan et al. 1984). To simulate the crystal field, the occupancy of the orbitals was modified to represent a low-spin complex with preferential occupancy of the t2g orbitals, rather than the more even distribution found in the high-spin complex. The complex ion (Fig. 10.14) was centered at the corners of a cubic unit cell with a = 10.000 A and space group Pm3. Refinement of the 1375 static structure factors (sin 8/X < 1.2 A 1) gave an agreement factor of R = 4.35% for the spherical-atom model with variable positional parameters (Table 10.12). Addition of three anharmonic thermal... 

The effective and ordered moments in a cubic crystal field were also calculated using the obtained gj values and the wave functions given by Lea, Leask and WolP In the case of the Tg quadruplet the magnetic moment is anisotropic and depends upon the crystal field parameter x. The effective moment is space averaged and given as a function x ° The value of the ordered moment along (001) is given as a function of x also. 

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.

Cubic crystal field (eightfold coordination) in cubic crystal... 

In tetrahedral and cubic symmetry, the crystal-field levels are inverted, giving a single orbital state lowest. For dl in an octahedral field, the 4F state also has the single orbital state lowest, so we would expect the d1 configuration in a tetrahedral or cubic field to behave in a similar fashion and fit the spin Hamiltonian given in Eq. (158) solved for S=f. For most of the examples listed in Table XV, the tetrahedral symmetry is not distorted so that D = E=0 and no fine structure is reported. The 5=f character of the spin state is revealed in these cases by the fact that Eq. (80) must be added to the spin Hamiltonian to explain the ESR results on d1 in tetrahedral and cubic fields (131). For Co2+ in Cs3CoCl5, D= —4.5 cm-1 (222) and in CdS, D > 2 cm-1 (223). 

The crystal field does, however, have a dramatic effect on the magnetic anisotropy of lanthanide complexes. For complexes of less than cubic symmetry the three principal values of the susceptibility tensor are unequal. For uniaxial symmetry, Xx — Xy Xz and for biaxial symmetry Xx Xy Xz- Very extensive studies632-640 have been carried out on the single crystal susceptibilities of the D3d lanthanide hexakis(antipyrene) triiodides over the temperature range 80-300 K, and crystal field parameters were obtained. This crystal field-induced anisotropy is responsible for the effectiveness of lanthanide complexes as NMR shift reagents, and single crystal anisotropies of lanthanide complexes have been determined in this connection also.563... 

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