Crystal-field Hamiltonian

We note that three spin-allowed electronic transitions should be observed in the d-d spectrum in each case. We have, thus, arrived at the same point established in Section 3.5. This time, however, we have used the so-called weak-field approach. Recall that the adjectives strong-field and weak-field refer to the magnitude of the crystal-field effect compared with the interelectron repulsion energies represented by the Coulomb term in the crystal-field Hamiltonian,... 

So, ligand-field theory is the name given to crystal-field theory that is freely parameterized. The centrally important point is that ligand-field calculations, whether numerical or merely qualitative, explicitly or implicitly employ a ligand-field Hamiltonian, very much like the crystal-field Hamiltonian, operating upon a basis set of pure d orbitals. Instead of the crystal-field Hamiltonian (Eq. 6.15),... 

The first published work on the pressure dependence of optical spectra of solids seems to be of Paetzold (1940), who has studied the effect of pressure on absorption spectra of praseodymium nitrate, ruby, and other minerals between 1938 and 1939. To generate a maximum pressure of 0.1 GPa the samples were subjected to pressurised nitrogen. Using the same high pressure apparatus, Hellwege and Schrock-Vietor (1955) studied the pressure dependence of the absorption spectra of EuZn-nitrate. These authors, for the first time, applied the crystal-field Hamiltonian formalism for the analysis of the high pressure spectroscopic results. 

The terms in the Hamiltonian that represent the non-spherical part of the interaction with the crystal are modeled using the so-called crystal-field Hamiltonian. It is important to recognize that this Hamiltonian is not restricted to electrostatic effects, which form a minor part of the total crystal-field effect (Ng and Newman, 1987). When the parameters are fitted to experimental energies, their values reflect all one-electron non-spherical interactions. The crystal-field Hamiltonian is expressed in Wyboume (1965) notation as,... 

The one-electron crystal field Hamiltonian does not take into account electron correlation effects. For some systems, it has been useful to augment the crystal field Hamiltonian with additional terms representing the two-electron, correlated crystal field. The additional terms most commonly used (see, for example, Peijzel et al., 2005b Wegh et al., 2003) are from the simplified delta-function correlation crystal field model first proposed by Judd (1978) that assumes electron interaction takes place only when two electrons are located at the same position (hence the name delta-function ). This simplified model, developed by Lo and Reid (1993), adds additional terms, given as,... 

At about the same time, Dieke developed and published optical spectra of lanthanides in 1956. After 1960 the raw data were converted into crystal field parameters by the methods developed by Stevens, Elliot and Judd. This was followed by a large body of data on crystal field parameters derived from optical spectra of lanthanides from all parts of the world. The crystal field Hamiltonian as given in equation (8.9) for an f electron is... 

The energy levels for / = 3 with complete crystal field Hamiltonian given in equation (8.15) and literature data [5] with the notation A rk) = B )k come out to be (equa-... 

Another orbital triplet " Ti, which appears in the energy level schemes of the chosen ions, represents somewhat more sophisticated case. The matter is that there are two " Ti states arising from two different LS terms (" F and " P) of the d (d ) electron configurations. These states are mixed up by both crystal field Hamiltonian... 

For the reference axis lying along the < 110> and <111) directions, die crystal field Hamiltonian may be expressed as (79)... 

In calculations involving higher J multiplets, matrix elements of the crystal field Hamiltonian between states belonging to different J multiplets are needed. Although these can be calculated by the method of operator equivalents extended to elements non-diagonal in J, it is convenient to use a more general approach, utilizing Racah s tensor operator technique (26). In this method the crystal field interaction may be written as... 

It is now well established that almost all transitions within the f shell are electric dipole in nature. The breakdown of the Laporte parity rule is brought about by non-centro-symmetric terms in the crystal-field Hamiltonian V, which have the effect of mixing d and g states into the f shell. Transitions which are nominally f to f take place because of the permitted transitions f- d and f->g. An early attempt (6) to explain the hypersensitivity used the fact that for rare-earth or actinide ions at site symmetries of the types... 

In an octahedral coordination we must consider the interaction represented by the crystal field Hamiltonian,... 

Using modern computers it is possible to diagonalize a complete free-ion and crystal-field Hamiltonian. However, for the lanthanides, if not for the actinides, the effect of the crystal field can usually be treated as a perturbation on the unsplit levels. The levels are split... 

Crystal field interactions in lanthanide systems are treated in the weak field limit and are normally expressed by the one-electron crystal field Hamiltonian Hcf given by... 

Another alternative is to use the crystal quantum number scheme of Hellwege (1949), which has a more direct connection with the crystal-field Hamiltonian. This Hamiltonian contains parameters Bim, where the allowed values of k and m are determined by the point-group symmetry. Each point group is characterized by an integer parameter q such that allowed values of m are given by... 

Because of the technological importance of ions as active elements in solid-state lasers, optical spectra of lanthanides in sites of low symmetry have been studied extensively. For these sites, the constraints imposed by group theory are weak, and selection rules are often nonexistent (such is the case with Kramers ions in C2 symmetry, for example). Thus, it becomes difficult to unravel the optical spectrum unambiguously. Even if this has been done, it is not straightforward to fit the data with a crystal-field Hamiltonian. The parameter space is large (14 crystal-field parameters in C2 symmetry) and several minima may exist that are indistinguishable from one another insofar as the quality of the fit is concerned. 

The Aicm are called crystal-field components. The parametrized crystal-field Hamiltonian is valid for a wide range of single electron-lattice interactions the decoupling represented by eq. (34), however, breaks down when nonelectrostatic effects such as overlap of neighboring ions (ligands), exchange, and covalency are included. In interpreting experimental data, the Bt are usually treated as variable parameters in a least-squares fit. 

Historically, the first extensive developments in crystal-field theory made use of the fact that the crystal-field coupling in lanthanide ions is small. In the operator-equivalent method (Stevens, 1952 Elliot and Stevens, 1953), the coupling of different free-ion levels by the crystal-field interaction is ignored and the crystal-field splitting of each Lj level is treated separately. Traditionally, in this method, the crystal-field Hamiltonian is written as... 

The crystal-field Hamiltonian that describes the interaction of a lanthanide ion with the host crystal lattice has been chosen in a number of different forms. Two of these have already been described one, eq. (33), in terms of the irreducible spherical tensors Ctm and the second, eq. (36), in terms of the V polynomials. A third notation, less common than the first two, is (Morrison et al., 1970)... 

Symmetry considerations have a profound effect on the interpretation of lanthanide spectra. We have already mentioned the effect of symmetry on selection rules for electric and magnetic dipole transitions and on the classification of crystal-field split energy levels. In this section, we consider the effect of symmetry on the crystal-field Hamiltonian itself. 

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