F-electron dependence

The expected f-electron dependence for these two conventional mechanisms versus the experimental values obtained for 0 of RE>44Si2 are plotted in Figure 17. As can be seen, the f-electron dependence observed for RB44Si2 does not match either mechanism and it suggests another model is responsible for the observed results (Mori, 2006a). 

Regarding the mechanism of the magnetic interaction of the Bi2 icosahedral compoimds, Figure 3 shows a plot of some of the characteristic temperatures of the magnetism plotted versns f-electron number, together with expected dependencies in the case of conventional f-electron magnetism mechanisms such as the RKKY interaction and the dipole-dipole interaction (normalized with the value for Er arbitrarily set as 4.5 K). It can be seen that the f-electron dependencies for the higher borides do not simply follow... 

Figure 3 Rare earth (f-electron) dependence of the characteristic magnetic temperatures...

For many purposes, electron configurations are sufficient to describe the arrangements of electrons in atoms. Sometimes, however, it is useful to go a step further and show how electrons are distributed among orbitals. In such cases, orbital diagrams are used. Each orbital is represented by parentheses (), and electrons are shown by arrows written f or, depending on spin. 

The second term of the product is a reduced matrix element which contains the /-state dependence of the f electrons since only 4f" configuration, for which 1 = 3,... 

Ishikawa etal. proposed an approach for the determination of the ligand-field (LF) parameters of a set of isostructural lanthanide complexes. This method consists of a simultaneous fit of the temperature dependence of magnetic susceptibilities and NMR spectra for the whole isostructural series [18]. In order to avoid over-parametrization a key restriction is imposed each parameter is expressed as a linear function of the number of f electrons, n ... 

The number and type of basis functions strongly influence the quality of the results. The use of a single basis function for each atomic orbital leads to file minimal basis set. In order to improve the results, extended basis sets should be used. These basis sets are named double-f, triple-f, etc. depending on whether each atomic orbital is described by two, three, etc. basis functions. Higher angular momentum functions, called polarization functions, are also necessary to describe the distortion of the electronic distribution due... 

The colors in rare earth glasses are caused by the ion being dissolved and they behave uniquely because the 4 f electrons are deeply buried. Their colors depend on transitions taking place in an inner electronic shell while in other elements such as the transition metals, the chemical forces are restricted to deformation and exchanges of electrons within the outer shell. Since the rare earth s sharp absorption spectra are insensitive to glass composition and oxidation-reduction conditions, it is easy to produce and maintain definite colors in the glass making process. ( )... 

The exact position of the crossover depends however, on the type of compound that is formed from an actinide. It appears usually for the radioactive heavier actinides. In order to study it, therefore, compoimds of Pu, Am, Cm have to be investigated. Compounds of elements preceding them in the actinide series present properties due to the itineracy of the 5 f electrons, which are somewhat similar to the d-transition elements compounds (especially 3d). Heavier actinides are lanthanide-like although their properties may depart from a true lanthanide behaviour unfortunately, their rarity and the difficulty of their handling is such that very few photoemission results are available for them. 

It is possible to characterize f-electron states in the actinides in quite a simple manner and to compare them with the states of other transition metal series. To this, we employ some simple concepts from energy band theory. Firstly, it is possible to express the real bandwidth in a simple elose-packed metal as the product of two parts . One factor depends only upon the angular momentum character of the band and the structure of the solid but not upon its scale. Therefore, since we shall use the fee structure throughout, the scaling factor X is known once and for all. 

Although some of the lanthanides form compounds with oxidation states other than + 3, the vast majority of stable species involve the trivalent state. Due to the nature of the weak bonding f electrons, complexes in solution are normally quite labile, and as described below, the preparation and the isolation of pure enantiomers containing a central lanthanide ion and achiral ligands are extremely rare. The coordination number of lanthanide(III) ions is somewhat variable and depends on the size and charge of the coordinated ligands, and the size of the lanthanide ion that varies slightly... 

High pressure techniques, in contrast, offer a more powerful method for such studies. By applying pressure one can vary the interatomic distances continuously and thereby study the crystal structure dependence of the f-electron states directly. A second, equally interesting possibility is based on the different sensitivities of the energies of electronic states on interatomic distances. These specific sensitivities allows the experimentalist to alter the energy difference between two states and thus to get unique information about their mutual influence. 

An interesting improvement of the SRC model has been discussed by Wang and Bulou (1995). They considered simplified expansion factors used in the Hartree-Fock radial wave-functions for the f-electrons. With these factors it was possible to introduce a k dependence for the pressure-induced change of different Slater parameters. This procedure would remove the weak point of the model which could not account for the observed -dependence of the parameters under pressure. 

In summary, the analysis of the free-ion parameters of different materials under pressure have led to a coherent picture. The nephelauxetic effect can be described by a combination of the CFC and SRC models and depends on the expansion of the f-electron wavefunctions. Only at very high pressures or in the case of strongly expanded wavefunctions the SRC and CFC models could not provide a completely adequate description of the observed variations. 

The eigenfunctions of Hare written as W, and the eigenvalues as E. Although H contains no electron spin interactions, the F do depend upon both the spatial and spin co-ordinates of the electrons ... 

The potential distribution across the interface under depletion conditions is illustrated in Fig. 8.2. It can be seen that the majority of the potential drop is located in the solid under depletion conditions, and consequently changes in electrode potential have little effect on the potential drop across the Helmholtz layer, AH. The existence of a parabolic potential drop across the space charge region means that the potential energy of electrons depends on position. This band bending is shown in Fig. 8.2. 

The value of k is limited to k 21, where / is the orbital angular momentum of the electrons. For f electrons k 6 k must also be even based on parity of the matrix elements involved in the crystal field potential. Thus k = 2,4 or 6 for f electrons. Allowed values of q have to follow the rule q k. Any further restrictions on q are dependent on the symmetry... 

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