Crystalline field perturbation

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). 

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. 

The planar XY3-ions like N03 (nitrate) and COj (carbonate) are perennial favourites because the D3h-structure is especially sensitive to the perturbing crystalline field. The following normal modes are generated by the Z)3A-struc-... 

It is well-known that the electron repulsion perturbation gives rise to LS terms or multiplets (also known as Russell-Saunders terms) which in turn are split into LSJ spin-orbital levels by spin-orbit interaction. These spin-orbital levels are further split into what are known as Stark levels by the crystalline field. The energies of the terms, the spin-orbital levels and the crystalline field levels can be calculated by one of two methods, (1) the Slater determinantal method [310-313], (2) the Racah tensor operator method [314-316]. 

The HCp operator represents the nonspherically symmetric components of the one-electron CF interactions, i.e. the perturbation of the Ln3+ 4fN electron system by all the other ions. The states arising from the 4fN configuration are well-shielded from the oscillating crystalline field (so that spectral lines are sharp) but a static field penetrates the ion and produces a Stark splitting of energy levels. The general form of the CF Hamiltonian Hcf is given by... 

The detailed Hamiltonians appropriate to the electronic and nuclear properties of the rare earths in general have been excellently summarised elsewhere [1]. They are not given here explicitly because of the more limited depth of treatment. Note, however, that the vectorial addition of the total orbital, L, and spin angular momentum, S, denoted by J is the most useful quantum number for describing electronic states. Any crystalline field potential then acts as a perturbation to the appropriate / state. This is opposite to the situation found in Fe, where the crystalline field is the dominant term. 

The characterization of electronic excited states has attracted much attention in connection with photochemistry. For example, transition metal complexes are characterized by a variety of absorption spectra in the visible and ultraviolet (UV) regions. The absorption spectra essentially give us information about the electronic excited states corresponding to dipole-allowed transitions due to their high symmetries, while some of the data in crystalline fields indicate the existence of several excited states to which dipole transitions are forbidden in the absence of perturbation. Most photochemical reactions of metal complexes, which are occasionally important as homogeneous photocatalytic reactions, involve both allowed and forbidden excited states. Thus, the systematic understanding of the nature of these excited states is essential in designing photochemical reactions. 

Data on the absorption of permanganate ion in different crystalline fields indicate the existence of several excited states to which dipole transitions are forbidden in the absence of perturbing effects. The correlation of these data with the results of our calculations gives important evidence as to their correctness. 

Fig. 1. Error bounds for the nuclear resonance line shape of crystalline CaF2, broadened by a Lorentzian slit function (i.e., the energy absorption by the coupled nuclear spins, due to an exponentially damped harmonic perturbation by a radiofrequency magnetic field).

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