Valence fractional

Another application of the concept of fractional valence bonds has been made in the field of metals and alloys. In the usual quantum mechanical discussion of metals, initiated by W. Pauli (Z. Physik, 41, 81 (1927)) and Sommerfeld (Naturwiss., 15, 825 (1927)), the assumption was made that only a small number of electrons contribute significantly to the binding together of the metal... 

Lanthanides with fractional valences have II, III and IV valences, as well as mixed II/III and III/IV valences. Depending on temperature and pressure, the degree of oxidation can change. This effect may result in a change in the different properties of nanoparticles, such as the stability, heat capacity, conductivity and magnetic susceptibility [218]. Valence fluctuation phenomena have been reported to occur... 

The compounds discussed above are not nonstoichiometric in the strict sense of the word, since they form well defined, ordered phases which can be assigned precise stoichiometric formulas. Accordingly, they are not berthollides. They may, however, be regarded as nonstoichiometric in the wider sense of the term which has become current, since fractional valence numbers must be assigned to the metal atoms and the ordered structures which they exhibit constitute a preferred alternative to the formation of disordered phases of variable composition. 

Formerly he had assumed instead fractional valencies. He assumed 5.78 bonding electrons, the other being present in atomic non-bonding 3d levels in which they are unpaired as far as possible (Hund s rule, p. 148). The numbers of these latter electrons were thus given by him as Cr 0.22, Mn 1.22,... 

Nonstoichiometry can be caused by oxygen deficiency (or excess) or by fractional valences of the metal components. For example, the existence of Cu " in nonstoichiometric cuprates has been widely discussed [9,10]. It is essential that in nonstoichiometric oxides the microscopic fluctuations of the composition should proceed (the so-called phase separation). The characteristic size of heterogeneities induced can exceed atomic dimensions by an order of magnitude. This phenomenon is attributable to the fact that the electron-nonuniform state of such chemically singlephase materials appears to be energetically more advantageous. 

In this section are displayed graphically the numerically exact results that have been obtained for unbiased, nearest-neighbor random walks on finite d = 2,3 dimensional regular. Euclidean lattices, each of uniform valency v, subject to periodic boundary conditions, and with a single deep trap. These data allow a quantitative assessment of the relative importance of changes in system size N, lattice dimensionality d, and/or valency v on the efficiency of diffusion-reaction processes on lattices of integral dimension, and provide a basis for understanding processes on lattices of fractal dimension or fractional valency. 

The term mixed valence is widely used in the literature to describe a phenomenon rather different from that considered here. Typically it refers to a metal (or a rare-earth alloy, or a compound such as SmS) which features a broad conduction band (formed by the overlap of s, p, or d orbitals), and a very narrow band such as an /-band, slightly above the Fermi level. This band becomes partially populated, hence the mixed, or fractional, valence. Fluctuations with time, and various degrees of localization in space, result from electron-phonon interaction. A useful review appeared recently.Many of the ideas used in this field parallel those used in other more chemical types of electron transfer. Recent articles on mixed-valence as a polaronic effect and on local polaronic effects exemplify this, and the dynamical properties have been discussed. Other recent reviews of this area deal with spectroscopic techniques and with mixed valency in rare-earth compounds. ... 

As pointed out by Helgeson (1991), it is quite possible to assign nominal charge contribution of -1 for each C-H bond, zero for each C-C bond, and -1-1 for each C-0, C-S, or C-N bond, and arrive at fractional valences that give a consistent accounting of electron transfers in reactions, regardless of the actual... 

The most important and characteristic feature of 3d-4f transitions in Sm-group elements and compounds is the absence of any observable change in the widths of the absorption lines when the valence change occurs. Quantitative estimates of the fractional valence in mixed-valence compoimds (Kaindl et al. 1985) for this group of elements can be performed by using the weighted contributions firom corresponding pure valent, divalent or trivalent compounds. 

M,v 363,2 core level fractional valence obtaine6 from the... 

Accordingly we define fractional valence by the time-averaged ratio of the occupation probabilities of the two integral valence states. If the population of the higher valence state is v, the fractional valence is defined by... 

The information about local binding energies, line intensities, 5d crystal field splittings and 5d bandwidths will become essential in the future, once one wishes to obtain quantitative control over why certain materials are mixed valent and why the value of the fractional valence is what one measures. At this time, however, the methods of quantitative extraction of chemical binding oiergies and the theory of the mixed valent state are insufficiently developed to make use of this information contained in the Lm spectra. Accordingly, in the following we shall address this aspect only occasionally. 

It is however obvious from the above, that extraction of numbers for the fractional valence has to be done with careful consideration of the various chemical effects contained in the spectra. For instance, a double-peaked spectral shape alone cannot be taken as an unambiguous signature of the mixed valent state, since splittings can also be produced by crystal field effects or by the formation of metal-ion ligands. Therefore we state as a rule that the proper spectroscopic analysis of Lm absorption in potentially mixed valent materials should be based on both, the measurement of the transition energy with respect to those in well-characterized isomorphous integral valent materials and the detailed analysis of the double-peaked spectra, i.e. their intensities, splittings and line widths. 

CeCujSij as obtained from the spectra, exhibited in fig. 33 (cf. also fig. 16). The linewidths are pressure (volume) dependent as expected from the relationship of atomic volume and spectral density (section 7.3.). W4. exhibits a larger pressure derivative than IF3. Plotting W3, W4 vs the occupation number v of the tetravalent state, we find a correlation of the linewidth differences W4 — W3 and the fractional valence. The derivative 0(1 4— lF3)/0v = 8.6 1 eV. This number seems to be a universal number in Ce systems. It has been found in elemental cerium under high pressure and Ce (Pdi, tT,t)3 (T = Ag,Rh,Y) also. From the Lm spectra in 2 jZ mixed valent materials (Pr,Sm,Eu) one extracts 2.6eV (cf. e.g. Rohler et al. (1983a), Liibcke (1985)). At the time there is no sound interpretation of these empirical relationships between valence and Lm linewidths. 

The lattice parameter is a direct indicator of what is happening at the microscopic level in these systems. It reflects the change in the size of the Sm ion, which in turn is directly related to its valence state. The lattice parameter that characterizes the divalent SmS (Sm S) and fully trivalent SmS (Sm S) are respectively 5.97 A and 5.62 A. Any lattice parameter intermediate in value suggests an intermediate or fractional valence state for the Sm ion. The lattice parameter data especially as a function of temperature has been very revealing. The abrupt jumps are due to a first-order valence transition. 

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