Crystalline field Splitting

Finally, we must say that the calculation of the crystalline field splitting for multielectron d" states is much more complicated than for d states. For d" states (n > 1), electrostatic interactions among the d electrons must be taken into account, together with the interactions of these valence electrons with the crystalline field. 

Fig. 4. Crystalline field splitting observed for polyethylene (a) film quenched from melt (b) film annealed at 80° C.

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... 

Weak crystalline field //cf //so, Hq. In this case, the energy levels of the free ion A are only slightly perturbed (shifted and split) by the crystalline field. The free ion wavefunctions are then used as basis functions to apply perturbation theory, //cf being the perturbation Hamiltonian over the / states (where S and L are the spin and orbital angular momenta and. 1 = L + S). This approach is generally applied to describe the energy levels of trivalent rare earth ions, since for these ions the 4f valence electrons are screened by the outer 5s 5p electrons. These electrons partially shield the crystalline field created by the B ions (see Section 6.2). 

The Rn,mi functions are related to the average probability of finding an electron in an specific orbital at a distance r from the nucleus of the central ion. We do not consider this part of the function in our calculation, because it is unaffected by the crystalline field (it does not lead to energy splitting). 

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. 

One example of a concrete system where one observes optical spectrum caused by the Ai-E electronic transition is the N-V center in diamond. This center consists of a substitutional N atom and three nearest C atoms (one of the nearest C atoms is replaced by the vacancy) and it has a trigonal symmetry. The ZPL line at 637 nm of this center corresponds to the electronic transition between the triplet 3A and the 3E electronic states. In the standard model of this center the electronic states of the center come from the occupation and the splitting of the aj and t2 levels arising from three C radicals. The crystalline field of a trigonal symmetry splits the t2 level into a number of states including the ground (Aj) state and the first excited E-state (see, e.g. Refs. [17-25]). Our experimental study of the optical transition between the E and the Aj electronic states indeed showed the 7 3 dependence of the ZPL width at low temperatures. 

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.

If the orbital angular momentum is quenched by the crystalline fields, so that e is small, then it is possible to show (347) that for a resonance experiment the spectroscopic splitting factor g is given by... 

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