Free-electron bands simple metals

In the conventional band picture of solid noble metals, the valence band contains the localized d-electrons as well as the extended s-electrons, and s-d mixing is substantial [9]. The picture of valence electrons is far from that of the free electrons in simple metals. It is, therefore, intriguing how well the shell model also works in noble metals. 

As a last example we consider two metals, A1 and Ag. The first is a simple metal, in the sense that only s and p orbitals are involved the corresponding atomic states are 3, 3p. Its band structure, shown in Fig. 4.11, has the characteristics of the free-electron band structure in the corresponding 3D FCC lattice. Indeed, A1 is the prototypical solid with behavior close to that of free electrons. The dispersion near the bottom of the lowest band is nearly a perfect parabola, as would be expected for... 

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. 

The high density of states found in the 3c/bands of Fe, Co, and Ni leads to a reduction of the mean free path of the electrons in this band. This causes a decrease in their mobility and hence in the electrical conductivity of these elements compared to simple metals and copper where the conduction electrons are in s/p bands. 

The simple picture for the chemisorption of an atom on a metal with d-electrons in Figure A. 14 arises as follows. First, we construct molecular orbitals from the atomic orbital of the adsorbate atom and the entire d-band. This produces a pair of bonding and antibonding chemisorption orbitals. Second, these new orbitals are broadened and perhaps shifted by the interaction with the free electron s-band of the metal. 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

Band Structure for the Free-Electron Case. If the electron is free, then the Bloch functions are simple plane waves, because the wavef unctions nk(r) used for the expansion Eq. (8.4.2) are themselves plane waves. For an electron gas with no lattice and no imposed symmetry, Fermi-Dirac statistics apply At 0 K all electrons pair up (spin-up and spin-down), with an occupancy of 2 for every k value from k 0 to the Fermi wavevector kF =1.92/rs = 3.63 a0/rs, and from zero energy up to the Fermi energy eF = h2kF2 /rn 50.1 eV rs/a0) 2, where rs is the radius per conduction electron and a0 is the Bohr radius, and the energy levels are spherically symmetric in k-space. The Fermi surface is a sphere of radius kF. Note that the ratio (rs/a0) varies from 0.2 to 1.0 nm for metals (Table 8.3). 

Let us note some qualitative aspects of electron dynamics. If the bands are narrow in energy, electron velocities will be small and electrons will behave like heavy particles. These qualities arc observed in insulator valence bands and in transition-metal d bands. In simple metals and semiconductors the bands tend to be broader and the electrons arc more mobile in metals the electrons typically behave as free particles with masses near the true electron mass. 

This does not mean that the LCAO approach of the type we have used is incorrect or not useful. Recent applications of LCAO theory, based only upon electron orbitals that are occupied in the free atom, have been made to the study of simple metals (Smith and Gay, 1975), noble-metal surfaces (Gay, Smith, and Arlinghaus, 1977), and transition metals (Rath and Callaway, 1973). In fact, the LCAO approach seems a particularly effective way to obtain self-consistent calculations. The difficulty from the point of view taken in this book is that, as with many other band-calculational techniques, LCAO theory has not provided a means for the elementary calculations of properties emphasized here, but pseudo-potentials have. 

The electronic structure is reformulated in terms of free electrons and a d resonance in order to relate the band width W, to the resonance width T, and is then reformulated again in terms of iransilion-metal pseudopotential theory, in which the hybridization between the frce-electron states and the d state is treated in perturbation theory, The pseudopotential theory provides both a definition of the d-state radius and a derivation of all interatomic matrix elements and the frce-electron effective mass in terms of it. Thus it provides all of the parameters for the L.CAO theory, as well as a means of direct calcidation of many properties, as was possible in the simple metals. ... 

It is clearly seen that 4d-4d, 4d-5d, 5d-4d and 5d-5d combinations exhibit similar patterns of "+" and signs. This is because the bonding in transition metals as well as in transition metal alloys is mainly determined by the valence d-electrons [29,31], which form quite localized bonds in contrast to the free-electron like bonding found in the simple metals. As a result the d-band occupation is the main parameter for the characterization of the bonding in this case. 

In the other extreme case the Fermi energy is found in the interior of a band and we are dealing with a metal. As long as it is far (relative to ksT) from the band edges, the situation is not much different from that described by the free electron model discussed in Sections 4.3.1 and 4.3.2, and this model provides a reasonable simple approximation forthe transport properties. 

All carbon-carbon bonds in the skeleton have 50% double bond character. This fact was later confirmed by X-ray diffraction studies. A simple free-electron model calculation shows that there is no energy gap between the valence and conduction bands and that the limit of the first UV-visible transition for an infinite chain is zero. Thus a simple free-electron model correctly reproduces the first UV transition with a metallic extrapolation for the infinite system. Conversely, in the polyene series, CH2=CH-(CH=CH) -CH=CH2, he had to disturb the constant potential using a sinusoidal potential in order to cover the experimental trends. The role of the sinusoidal potential is to take into account the structural bond alternation between bond lengths of single- and double-bond character. When applied to the infinite system, in this type of disturbed free-electron model or Hiickel-type theory, a non-zero energy gap is obtained (about 1.90 eV in Kuhn s calculation), as illustrated in Fig. 36.9. 

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