Free-electron bands transition metals

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

Let us return to the d stales which, for each transition metal, at a given / o, are characterized by an average energy Ej with respect to the minimum of the free-electron band and a width Wj. The corresponding model density of stales is illustrated in Fig. 20-8,b. Given Wj and E, we can easily compute the value ofE,. that is consistent with the number of electrons present. 

Finally, lei us use the transition-metal pseudopotential theory to estimate matrix elements between d states and s and p states. These are not so useful in the transition metals themselves since the description of the electronic structure is better made in terms of d bands coupled to free-electron bands, ti k /(2m) d-<01 IT 10>, rather than in terms of d bands coupled to s and p bands. However, the matrix elements and so forth, directly enter the electronic structure of the transition-metal compounds, and it is desirable to obtain these matrix elements in terms of the d-state radius r. We do this by writing expressions for the bands in terms of pseudopotentials and equating them to the LCAO expressions obtained in Section 20-A. 

For analysis of the transition metals themselves, the use of free-electron bands and LCAO d states is preferable. The analysis based upon transition-metal pseudopotential theory has shown that the interatomic matrix elements between d states, the hybridization between the free-electron and d bands, and the resulting effective mass for the free-electron bands can all be written in terms of the d-state radius r, and values for have been listed in the Solid State Table. 

The outer-shell electrons of transition metals are separated into two bands. The filling of the narrow d-band corresponds to the occupation of the t/-shell, whereas a broad 5j 7-band is produced by the valence electrons of the free atom, rf-states are comparatively non-dispersive, the electrons being rather localized, whereas 5/ -states are more free-electron like. In real space this means that whereas rf-electrons to a large proportion reside around a certain atomic core or within a certain Wigner-Seitz cell sp-electrons penetrate the entire bulk [5]. Hence a comprehensive theoretical model of a metal surface has to account for delocalized electron states as well as for the discrete atomic character of the adsorbent. 

A general picture of the valence electronic structure of compounds of rare earths (which are early transition metals) and late 3d transition metals is shown in fig. 40, where the localized 4f states are not included. The lanthanide (R) contributes 5d, 6s and 6p valence states and the 3d transition metal (M) contributes 3d, 4s and 4p states. Since in the solid the s and p projected orbitals will form broad and rather feature-less free-electron bands, these may be neglected as a first approximation, leaving the 5d and 3d states. In the free atom the energy of the M 3d states lies far lower than that of the R 5d states, as shown on the left in fig. 40. The first... 

In noble or coinage metals (Cu, Ag, Au), the optical response does not reduce to the response of the free electron gas. Noble metals consist of atoms with completely filled 3d, 4d, and 5d shells and just a single electron in the 4s, 5s, and 6s bands, respectively this last electron in not completely free to move and the dielectric response is essentially influenced by optical transitions of electrons in deeper (e.g. core) levels. These inter-band excitations alter the dielectric function considerably. This contribution can be described using a full quantum mechanical treatment, which is introduced in the next section. 

The transition metals are also good conductors as they have a similar sp band as the free-electron metals, plus a partially filled d band. The Group IB metals, copper, silver and gold, represent borderline cases, as the d band is filled and located a few eV below the Fermi level. Their sp band, however, ensures that these metals are good conductors. 

Figure 6.28. Schematic illustration of the change in local electronic structure of an oxygen atom adsorbing on the late transition metal rhodium, the DOS of which is shown on the right-hand side. The interaction of the oxygen 2p orbital with the sp band of the transition metal is illustrated through interaction with the idealized free-electron...

Jellium is a good model for sp metals. This group of metals comprises, amongst others, the elements Hg, Cd, Zn, Tl, In, Ga and Pb, all of which are important as electrode materials in aqueous solutions. They possess wide conduction bands with delocalized electrons, which form a quasi-free-electron gas. The jellium model cannot be applied to transition metals, which have narrow d bands with a localized character. The sd metals Cu, Ag and Au are borderline cases. Cu and Ag have been successfully treated by a modified version of jellium [3], because their d orbitals are sufficiently low in energy. This is not possible for gold, whose characteristic color is caused by a d band near the Fermi level. 

Transition metals are characterized by a fairly tightly bound d band of width W that overlaps and hybridizes with a broader nearly-free-electron sp band as illustrated in Fig. 7.4. This difference in behaviour between the valence sp and d electrons arises from the d shell lying inside the outer valence s shell, thereby leading to small overlap between the d orbitals in the bulk. 

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