Band Structure of Transition Metals

The elevated catalytic activities of transition metals have been attributed to their partially unoccupied d-electron states. Then, information regarding the band structure of these metals is obviously of interest for understanding the catalytic action of these metals [30-34], 

In general terms, transition metals are those which have incompletely filled d-bands. The progression in the filling of the d-band in the first long-transition metal series is as follows Ti(HCP), V(BCC), Cr(BCC), Fe(BCC), Co(FCC), Ni(FCC), Cu(FCC), Zn(HCP), and is not highly influenced by the structural difference between the body-centered cubic (BCC) and face-centered cubic (FCC) lattices. However, this is not the case for the hexagonal close-packed (HCP) lattice [10], An analogous pattern is expected for the second and third series. 

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For example, from eqn (2.55), the average radial distance of the hydrogenic 3d and 4s wave functions are in the ratio 0.44 1. Thus we expect the band structure of transition metals to be represented accurately by a hybrid NFE-TB secular equation of the form... 

Calais, J.-L. (1977). Band structure of transition metal compounds. Adv. in Physics 26, 847-85. 

Sugiura, C., I. Suzuki, J. Kashiwakura, and Y. Gohshi (1976). Sulfur K x-ray emission bands and valence band structures of transition metal disulfides. J. Phys. Japan 40, 1720-24. 

O.K. Andersen "Band Structure of Transition Metals", in Mont Tremblant International Summer School 1973, unpublished... 

Various methods for the calculation of band structures have been devised. The augmented plane wave (APW) method (23,24) and the Green s function (GF) method of Korringa, Kohn, and Rostocker (KKR) (25-28) were used for most of the early calculations of the band structures of transition metal compounds. A common approximation in both methods is the use of the so-called muffin tin potential. In this approximation it is assumed that the crystal potential is spherically symmetric within nonoverlapping spheres around the atomic sites and constant in the region between the atomic spheres. 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

Analysis of the valence-band spectrum of NiO helped to understand the electronic structure of transition-metal compounds. It is to be noted that th.e crystal-field theory cannot explain the features over the entire valence-band region of NiO. It therefore becomes necessary to explicitly take into account the ligand(02p)-metal (Ni3d) hybridization and the intra-atomic Coulomb interaction, 11, in order to satisfactorily explain the spectral features. This has been done by approximating bulk NiO by a cluster (NiOg) ". The ground-state wave function Tg of this cluster is given by,... 

Figure 1.3 Electronic structure of transition metal carbides and nitrides. Band calculation reproduced from Hasegawa (Ref. 18).

The crystal structures of transition metal compounds and minerals have either cubic or lower symmetries. The cations may occur in regular octahedral (or tetrahedral) sites or be present in distorted coordination polyhedra in the crystal structures. When cations are located in low-symmetry coordination environments in non-cubic minerals, different absorption spectrum profiles may result when linearly polarized light is transmitted through single crystals of the anisotropic phases. Such polarization dependence of absorption bands is illustrated by the spectra ofFe2+ in gillespite (fig. 3.3) and of Fe3+in yellow sapphire (fig. 3.16). 

In Pd and other transition metals, hydrogen has a high solubility and diffuses very fast, possibly because of the high d-electron density in the band structure of these metals. During absorption, the hydrogen molecule is first dissociated in the Pd surface subsequently, the adsorbed hydrogen atoms are ionized, and are incorporated directly into the material as protons and electrons, e, as follows [31,32]... 

Some transition metal nitrides,10 MN, of Ti, Zr, and Hf have cubic (Nad type) structures. Others which are often not exactly stoichiometric (being N deficient), are chemically very inert and extremely hard with high melting points. The electronic band structure of the metal persists, the appearance is metallic and the compounds are electrically conducting. As an example, VN has mp 2570°C and hardness 9-10. 

J. Zaanen, G. A. Sawatzky, and J. W. Allen, Band gaps and electronic structure of transition-metal compounds, Phys. Rev. Letters 55, 418 (1985). 

The electronic structure of transition metal clusters is difficult to probe by electronic absorption spectroscopy, which has proved so fhiitful in revealing the d orbital orderings of classical complexes, because of the multiplicity of overlapping absorption bands. The one-to-one correlation between occupied orbitals and PE bands means that PES can give valuable information on the smaller clusters. The information obtained has given support to the semiempirical theoretical treatments of this class of molecules. 

The electronic state calculations of transition metal clusters have been carried out to study the basic electronic properties of these elements by the use of DV-Xa molecular orbital method. It is found that the covalent bonding between neighboring atoms, namely the short range chemical interaction is very important to determine the valence band structure of transition element. The spin polarization in the transition metal cluster has been investigated and the mechanism of the magnetic interaction between the atomic spins has been interpreted by means of the spin polarized molecular orbital description. For heavy elements like 5d transition metals, the relativistic effects are found to be very important even in the valence electronic state. 

We now wish to extend the dimer model to encompass an infinitely large three-dimensional crystal lattice this is very similar to the transition from the covalent bonding of two atoms to the band structure of a metal or a semiconductor with delocalised states. Starting from the more or less sharp energy levels in the two-body system, we arrive at a band of energy states whose width depends on the interactions of the individual molecules or the overlap of the molecular orbitals in the lattice. We must then take the interaction of an excited molecule with aU the other molecules in the crystal and with the periodic lattice potential into account The levels and E in the dimer model of Fig. 6.7 are transformed into a more or less broad band of energy levels. These are the excitonic bands of the crystal, which we shall treat in this section. 

It is unnecessary to provide details of the results of such calculations, or of their comparison with experimental determinations by for example soft X-ray spectroscopy band structures for Transition Metals can adopt quite complex forms, so we must content ourselves with a few qualitative observations. For the metals of catalytic interest, the nrf-electron band is narrow but has a high density of states (Figure 1.8), because these electrons are to some degree localised about each ion core, whereas the (n + l)s band is broad with a much lower density of states because s-electrons extend further and interact more. On progressing from iron through to copper, the d-band occupancy increases quickly, and the level density at the Fermi surface falls. The extent of vacancy of the d-band is provided by the saturation moment of magnetisation thus for example the electronic structure of metallic nickel is (Ar core) and is said to have 0.6 holes in the d-band . 

The area under the EPR curve (the intensities of the bands) is proportional to the number of unpaired electrons. Owing to the high sensitivity of the method, it may be used to determine the number of unpaired spins even in very dilute solutions and hence, for example, the oxidation numbers and low- or high-spin electronic structures of transitional metal ions. 

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