Crystalline field levels

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]. 

Entropy evaluations from published cryothermal data on the lanthanide (III) oxides are summarized in Table II with an indication of the lowest temperature of the measurements and the estimated magnetic entropy increments below this temperature. Their original assignment of crystalline field levels from thermal data still appears to be in good accord with recent findings e.g., 17). Unfortunately, measurements on these substances were made only down to about 8°K. because the finely divided oxide samples tend to absorb the helium gas utilized to enhance thermal equilibration between sample and calorimeter. 

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). 

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. 

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.

Most metal oxides are ionic crystals and belong to either the class of semiconductors or insulators, in which the valence band mainly comprises the frontier orbitals of oxide ions and the conduction band contains the frontier orbitals of metal ions. In forming an ionic metal oxide ciTstal from metal ions and oxide ions, as shown in Fig. 2-21, the crystalline field shifts the frontier electron level of metal ions to higher energies to form an antibonding band (the conduction... 

Fig. 23. Energy level scheme of a single 3d electron showing the effect of crystalline fields (CF) of various symmetry. Electron occupation of levels is indicated by a circle in (d) and by arrows in (e) to denote spin polarization.

Fig. 1.—Energy levels in crystalline fields of Oh and coin-pressed Civ symmetry, with (—3Ds + 5Dt) > 0.

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. 

Our aim is to derive theoretical crystalline field energy levels of f2 and fit the experimental levels of PrCl3. According to the crystal structure proposed by Zachariasen [319], the nearest neighbors of the Pr3+ ion are nine Cl- ions arranged in a tricapped trigonal... 

The results of the crystalline field calculations are given in Table 8.44. At the time Margolis did the fitting, 40 out of 61 Stark levels of PrCb were known. Since then, 6 more levels have been identified and a few of the previous levels have been slightly modified to within 1 cm-1. We have listed these experimental values [322,330-332] along with our calculated values in this table. The crystal field parameters used were the same values derived by Margolis, i.e.,... 

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... 

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