Localized behaviour

Many different types of networks have been developed. They all consist of small units, neurons, that are interconnected. The local behaviour of these units determines the overall behaviour of the network. The most common is the multi-layer-feed-forward network (MLF). Recently, other networks such as the Kohonen, radial basis function and ART networks have raised interest in the chemical application area. In this chapter we focus on the MLF networks. The principle of some of the other networks are explained and we also discuss how these networks relate with other algorithms, described elsewhere in this book. 

The reason why it is a useful concept in actinide solid state physics (as in d-transition metals) is that it describes the effect of overlapping in forming a band therefore, in d or f unfilled bands, a critical value of the inter-atomic distance should determine whether the electron states will be acting as localized (insulator s) states, or as an unfilled (metallic) band. If we refer to the concept of f-f-overlapping, it is thinkable that across a series such as the actinides the f-f-overlapping might come to a critical value distinguishing between band and localized behaviour. 

Armed with the general concepts developed in the previous section, we proceed now to the experimental evidence of 5f itinerant vs. localized behaviour in actinide metals. 

The Mott-like transition, a central concept for the description of the actinide metal series, causes the sudden increase of the atomic volumes, encountered when between Pu and Am (Fig. 3). All other properties indicate the onset of a 5f localized behaviour at Am (see Part V) the 5 f pressure, which had contained to smaller values the equilibrium interactinide distance, suddenly gives in, with the withdrawal of the 5f s within the atomic core. The occurrence of such a transition within a series characterized by an unsaturated shell, is a unique phenomenon of the actinide series. In lanthanides, it does not occur except perhaps under pressure in cerium metal the approaching of cerium atoms induces suddenly the itineracy of 4f orbitals and a sudden volume collapse - see Chap. C. Neither it occurs in d-transition metal series, where the atomic volumes have an almost parabolic behaviour when plotted vs. Z (see Fig. 3 and Chap. C). The current... 

In Fig. 6, it is seen that the metallic radius of americium metal is not obtained when applying the simplified Friedel-type model which, on the contrary, explains well the metallic radii of Pa, U, and, to some extent, of Np and Pu. Between Pu and Am we have indicated the Mott-like transition proposed by Johansson which was discussed in Chapt. A. This transition from itinerant to localized behaviour occurs because of the... 

After a survey of the basic theory and some experimental aspects of photoelectron spectroscopy which are relevant to actinide solids, two systems are illustrated elemental actinide metals, in which the Mott transition between plutonium and americium is evidenced in a photographic way by photoemission, and strongly ionic oxides, in which the 5f localized behaviour is clearly seen, and indications of f-p or d-p covalent mixing are investigated. 

Photoelectron spectroscopy is however able to give valuable information on the localized behaviour of open shell electrons. The second parameter, the Coulomb interaction energy Uh, to be balanced (Chap. A) with W to give a judgement on the localization vs. itineracy problem, has been shown to be experimentally accessible when XPS and BIS are applied on the same compound. 

A sign of the onset of a localized behaviour of 5 f states is considered to be the appearance of a compact structure, such as hep, fee or dhep. These structures are encountered in lanthanides and in heavy actinides (starting from Am). a-Pu is monoclinic, but 6-Pu has already the fee structure. 5 f electron localization is then expected to occur for this phase. In this case, the UPS valence band (e.g., for 40.8 eV excitation) of the 6-phase should change considerably from that of the a-phase, and a final state multiplet structure (similar to the one for bulk Sm) might be expected ... 

On the basis of the known electronic properties of actinides (which have been discussed elsewhere in this book), theoreticians had distinguished the 5f itinerant behaviour of light actinide metals from the 5 f localized behaviour of heavy actinide metals from Am on. The crossover, presented often as a Mott transition, had been predicted to occur between Pu and Am metal, due to the localized character of the 5f state in the latter. Photoemission spectroscopy demonstrates this phenomenon directly with the observation of a 5 f multiplet away from the Fermi level. The detailed description of this peak is certainly complicated, as often happens for response of localized states in photoemission on the other hand (Fig. 17) the contrast to the emission of Pu metal is convincing. 

The localized-electron model or the ligand-field approach is essentially the same as the Heitler-London theory for the hydrogen molecule. The model assumes that a crystal is composed of an assembly of independent ions fixed at their lattice sites and that overlap of atomic orbitals is small. When interatomic interactions are weak, intraatomic exchange (Hund s rule splitting) and electron-phonon interactions favour the localized behaviour of electrons. This increases the relaxation time of a charge carrier from about 10 s in an ordinary metal to 10 s, which is the order of time required for a lattice vibration in a polar crystal. 

NMR is rather limited in its frequency range. It has however the important advantage that it is site-specific and allows the detailed determination of the local behaviour of a given elementary dipole. Blinc et al.4 has carried out NMR experiment to verify the deuteron intra-bond motion in deuteron glasses is a pure Arrhenius-like thermally activated process or whether deuteron tunnelling effects become important at low temperatures. 

However, the mobile or localized behaviour of the electrons changes so dramatically with only minor changes in the stoichiometry (Figure 7.11) that the Zener mechanism is thought to be too simple. It also fails to explain the magnetoresistance in compounds in which there are no Jahn-Teller distorted ions. 

In CeCu2Si2, UBei3 and UPt3, 1/7) shows a localized behaviour at high temperatures and starts to show a linear dependence below TK. Below 7) all the 1/7) values decrease sharply, followed by a T3 dependence without the coherent peak just below Tc indicating the gap zero on the line. The T3 behaviour is also seen in other HF superconductors, URu2Si2,13 UPd2Al31415... 

Similar to other trivalent and tetravalent metals, the actinides can absorb a large amount of hydrogen forming various phases. As the presence of hydrogen generally reduces the 5f-5f overlap, a more localized behaviour of the 5f states is found in the... 

On the other hand, the discussion of covalent bonds in the last three chapters has emphasized the bonding of atoms in pairs. The covalent bonding of more than two atoms into a polyatomic molecule has been pictured as occurring by the formation of links between adjacent atoms, each welded by the localized behaviour of electrons that remain associated with no more than two atoms. 

The linear differential system which describes the local behaviour in the vicinity of a point (cj, c, ..Cj), is associated with the system (1) ... 

Lack of space prevents us from describing in any detail other properties of pods. It should be mentioned though that pods may be characterized as repulsive or attractive according to the local behaviour of trajectories in their vicinity. Loosely speaking, repulsive pods repel trajectories in their vicinity, attractive pods attract them. These properties have enabled the development of a lower bound to the classical reaction probability, the construction of a new theory of direct reactions as well as new numerical methods for computation of classical product state distributions.The repulsive attractive properties provide a global picture of the classical flow of trajectories at a given energy. 

Figure 2 shows the PN-model for the loo2 system. The system is down, when Cl as well as C2 are not in their OK -state (see Figure 3). One can see at first sight that, in contrast to the Markov model, in PN the global behaviour of the system is modeled through the local behaviour of its elements - here, of its components On the left and right hand side of the PN, the behaviour of the two components is modeled, the structure on the top and at the bottom model the hehaviour of the system in case of a CCF.

A test plan must be drawn up and agreed by all parties before work begins. Of particular Importance is the instrumentation that should be used. There are various methods for measuring the surface strains on the components and the deformations. The choice will depend on the type of structure, access to critical locations during the test, the provision of a fixed frame of reference and, above all, on the purpose of the test. The major limitations are likely to be time and cost. It may be appropriate to concentrate on the overall behaviour of the structure, for example by mainly measuring deflections under load, and to determine the local behaviour of components or connections as a separate exercise, using the methods in the previous sections. 

A second comparison of the three types of 3D warp interlock fabric helps to reveal a more local behaviour at each point of the final preform due to their different weave diagrams. 

Fleischmann, M. and Kelsall, G.H. (1975) An investigation of the local behaviour of a copper fluidised bed electrode. Chemistry Industry, 8,320. 

---