Metals cohesive energies

The trend in the f-pressure is almost parabolic with band filling and this is typical for a transition metal (with d replaced by f). The physical basis was given by the Friedel who assumed that a rectangular density of states was being filled monotonically and thus was able to reproduce the parabolic trend in transition metal cohesive energies analytically. Pettifor has shown that the pressure formula can similarly be integrated analytically. 

Fig. 2. Calculated properties of the 3d and 4d transition metals-cohesive energy, lattice constant, and bulk modulus- compared with experiment (crosses). This represents a milestone in the development of the methods to calculate with sufficient accuracy to find these quantities. (From refs. 25 and 123, figure courtesy of V. MoruzzO... 

Note that the theoretical spall strength now depends upon the cohesive energy as well as the bulk modulus. Representative values for selected metals are shown in Table 8.1. These can be compared with experimental spall strengths in later sections. 

We will limit ourselves here to transition metals. It is well known that in these metals, the cohesive properties are largely dominated by the valence d electrons, and consequently, sp electrons can be neglected save for the elements with an almost empty or filled d valence shelP. Since the valence d atomic orbitals are rather localized, the d electronic states in the solid are well described in the tight-binding approximation. In this approximation, the cohesive energy of a bulk crystal is usually written as ... 

The materials for solid solutions of transition elements in j3-rh boron are prepared by arc melting the component elements or by solid-state diffusion of the metal into /3-rhombohedral (/3-rh) boron. Compositions as determined by erystal structure and electron microprobe analyses together with the unit cell dimensions are given in Table 1. The volume of the unit cell (V ) increases when the solid solution is formed. As illustrated in Fig. 1, V increases nearly linearly with metal content for the solid solution of Cu in /3-rh boron. In addition to the elements listed in Table 1, the expansion of the unit cell exceeds 7.0 X 10 pm for saturated solid solutions " of Ti, V, (2o, Ni, As, Se and Hf in /3-rh boron, whereas the increase is smaller for the remaining elements. The solubility of these elements does not exceed a few tenths at %. The microhardness of the solid solution increases with V . Boron is a brittle material, indicating the accommodation of transition-element atoms in the -rh boron structure is associated with an increase in the cohesion energy of the solid. 

Figure 6.19. Cohesive energy for the three groups oftransition metals. Note that the maximum cohesive energy ofthe 4d and 5d metals occurs when the d band is approximately halffull, as predicted by Eq. (49).

Another interesting feature explained by Eqs. (49) or (50) is the increase in cohesive energy in going from 3d to 4d to 5d metals (Fig. 6.19). As we go down the periodic system the orbitals become larger and the overlap increases. This implies that the band becomes broader, leading to larger value of tv in Eqs. (49) and (50). 

In the Introduction the problem of construction of a theoretical model of the metal surface was briefly discussed. If a model that would permit the theoretical description of the chemisorption complex is to be constructed, one must decide which type of the theoretical description of the metal should be used. Two basic approaches exist in the theory of transition metals (48). The first one is based on the assumption that the d-elec-trons are localized either on atoms or in bonds (which is particularly attractive for the discussion of the surface problems). The other is the itinerant approach, based on the collective model of metals (which was particularly successful in explaining the bulk properties of metals). The choice between these two is not easy. Even in contemporary solid state literature the possibility of d-electron localization is still being discussed (49-51). Examples can be found in the literature that discuss the following problems high cohesion energy of transition metals (52), their crystallographic structure (53), magnetic moments of the constituent atoms in alloys (54), optical and photoemission properties (48, 49), and plasma oscillation losses (55). 

A special class ofblock copolymers with blocks of very different polarity is known as amphiphilic (Figure 10.1). In general, the word amphiphile is used to describe molecules that stabilize the oil-water interface (e.g., surfactants). To a certain extent, amphiphilic block copolymers allow the generalization of amphi-philicity. This means that molecules can be designed that stabilize not only the oil-water interface but any interface between different materials with different cohesion energies or surface tensions (e.g., water-gas, oil-gas, polymer-metal, or polymer-polymerinterfaces). This approach is straightforward, since the wide variability of the chemical structure of polymers allows fine and specific adjustment of both polymer parts to any particular stabilization problem. 

Theory for the Size and Structural Dependendence of the Ionization and Cohesive Energy of Transtion Metal Clusters. 

On the practical side, we note that nature provides a number of extended systems like solid metals [29, 30], metal clusters [31], and semiconductors [30, 32]. These systems have much in common with the uniform electron gas, and their ground-state properties (lattice constants [29, 30, 32], bulk moduli [29, 30, 32], cohesive energies [29], surface energies [30, 31], etc.) are typically described much better by functionals (including even LSD) which have the right uniform density limit than by those that do not. There is no sharp boundary between quantum chemistry and condensed matter physics. A good density functional should describe all the continuous gradations between localized and delocalized electron densities, and all the combinations of both (such as a molecule bound to a metal surface a situation important for catalysis). 

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