Crystalline field

The data on the absorption spectra of permanganate ion in different crystalline fields is interpreted in terms of the symmetries of the excited states predicted by our calculations. 

Crystal-field theory (CFT) was constructed as the first theoretical model to account for these spectral differences. Its central idea is simple in the extreme. In free atoms and ions, all electrons, but for our interests particularly the outer or non-core electrons, are subject to three main energetic constraints a) they possess kinetic energy, b) they are attracted to the nucleus and c) they repel one another. (We shall put that a little more exactly, and symbolically, later). Within the environment of other ions, as for example within the lattice of a crystal, those electrons are expected to be subject also to one further constraint. Namely, they will be affected by the non-spherical electric field established by the surrounding ions. That electric field was called the crystalline field , but we now simply call it the crystal field . Since we are almost exclusively concerned with the spectral and other properties of positively charged transition-metal ions surrounded by anions of the lattice, the effect of the crystal field is to repel the electrons. 

A number of theoretical works have been devoted to the study of the hydrogen-deuterium exchange reaction. Hauffe (25) examined this reaction from the standpoint of the boundary layer theory of chemisorption. Dowden and co-workers (26) undertook a theoretical investigation of the hydrogen-deuterium exchange reaction from the viewpoint of the theory of crystalline fields. 

In crystalline field theory, the valence electrons belong to ion A and the effect of the lattice is considered throngh the electrostahc field created by the snrronnding B ions at the A position. This electrostatic field is called the crystalline field. It is then assnmed that the valence electrons are localized in ion A and that the charge of B ions does not penetrate into the region occnpied by these valence electrons. Thns the Hamiltonian can be written as... 

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). 

Intermediate crystalline field Hso Hqf < In this case, the crystalline field is stronger than the spin-orbit interaction, but it is still less important than the interaction between the valence electrons. Here, the crystalline field is considered aperturbation on the terms. This approach is applied for transition metal ion centers in some crystals (see Section 6.4). 

Strong crystalline field Hso < H e < Hqf- In this approach, the crystalline field term dominates over both the spin-orbit and the electron-electron interactions. This applies to transition metal ions in some crystalline environments (see Section 6.4). 

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. 

One of the simplest descriptions of the crystalline field occurs for the d outer electronic configuration (i.e., for a single d valence electron). This means that //ee = 0 and, consequently, there is no distinction between intermediate and strong crystalline fields. 

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. 

On the other hand, the crystalline field due to main symmetries other than Oh symmetry can be also related to this same case. For this purpose, it is useful to represent the octahedral structure of our reference ABe center as in Figure 5.5(a). In this representation, the B ions lie in the center of the six faces of a regular cube of side 2a and the ion A (not displayed in the figure) is in the cube center the distance A-B is equal to a. 

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) ... 

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. 

Thus, MO theory is generally applied to interpretation of the so-called charge transfer spectra. However, for a great variety of centers in solids, crystalline field theory suffiees to provide at least a qnalitative interpretation of spectra. 

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. 

Figure A2.2 The effect of an octahedral crystalline field on a d energy level.

In Chapter 5, we discuss in a simple way static (crystalline field) and dynamic (coordinate configuration model) effects on the optically active centers and how they affect their spectra (the peak position, and the shape and intensity of optical bands). We also introduce nonradiative depopulation mechanisms (multiphonon emission and energy transfer) in order to understand the ability of a particular center to emit light in other words, the competition between the mechanisms of radiative de-excitation and nonradiative de-excitation. 

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... 

Investigation of structure and properties of crystal is one of the most important problems in solid state physics and chemistry. Thus study of the features of electron diffraction (ED) and their relation to the inner crystalline field and establishment of their link to physical properties is one of the major requests of modem stmcture analysis (SA). 

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.

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