Nearly free electron gas

Wang Y A, Govind N and Carter E A 1998 Orbital-free kinetic energy functionals for the nearly-free electron gas Phys. Rev. B 58 13 465... 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

The outer electrons in metals such as Li and Na have a very low ionization energy, and are largely delocalized. Such electrons are described as constituting a nearly free electron gas. It may be noted, though, that this description is somewhat misleading as the behavior of the electrons is dominated by the exclusion principle, while the molecules in normal gases can be described by classical statistical mechanics. 

The required 2D nearly free electron gas is realized in Shockley type surface states of close-packed surfaces of noble metals. These states are located in narrow band gaps in the center of the first Brillouin zone of the (lll)-projected bulk band structure. The fact that their occupied bands are entirely in bulk band gaps separates the electrons in the 2D surface state from those in the underlying bulk. Only at structural defects, such as steps or adsorbates, is there an overlap of the wave functions, opening a finite transmission between the 2D and the 3D system. The fact that the surface state band is narrow implies extremely small Fermi wave vectors and consequently the Friedel oscillations of the surface state have a significantly larger wave length than those of bulk states. 

The electronic excitations caused by electron energy loss processes can be divided into two main classes collective electronic excitations and single electron excitation. The collective excitations may be regarded as plasma oscillations of a free or nearly free electron gas embedded in a homogeneously distributed positive charge. An extensive treatment of collective excitation losses is given by Lucas and Sunjic (1972). In a classical treatment the electron plasma frequency a>p is given by... 

In the microscopic theory of Bardeen et al. (1957) the electrons are treated as a nearly free electron gas, individual electrons interacting pairwise via an attractive phonon-mediated potential. The superconducting transition temperature is given by the well-known expression... 

Wang, Y. A. Govind, N. Carter, E. A. orbital-free kinetic-energy frmctionals for the nearly free electron gas. Phys. Rev. 1998, 58, 13465-13471. 

The glass-coloring experiments have been performed with gold, silver, nickel and other metals, which are much more difficult to handle theoretically than the alkalis. Among the latter, sodium is the best representative of the nearly free electron gas or jellium model which forms the basic assumption of some of the articles found here. Therefore this review is restricted to sodium clusters, and more specifically to their optical and thermal properties. 

The NFE theory describes a simple metal as a collection of ions that are weakly coupled through the electron gas. The potential energy is volume-dependent but is independent of the position of the electrons. This is valid for both solids and dense liquids. At densities well above that of the MNM transition, we can use effective pair potentials and find the thermophysical properties of metallic liquids with the thermodynamic variational methods usually employed in theoretical treatments of normal insulating liquids. One approach is a variational method based on hard sphere reference systems (Shimoji, 1977 Ashcroft and Stroud, 1978). The electron system is assumed to be a nearly-free-electron gas in which electrons interact weakly with the ions via a suitable pseudopotential. It is also assumed that the Helmholtz free energy per atom can be expressed in terms of the following contributions ... 

Kinetic-Energy Functionals for the Nearly Free Electron Gas. 

The decay is much faster into the Au metal owing to the screening effect of the nearly free electron gas. Relevant for our spectroscopic purposes, however, is only the evanescent character of the surface mode, which leads to an extension of the optical field into the aqueous phase of about 150 nm (defined by the 1/e decay of the peak intensity). This means that only chromophores that are within this exponential decay of the excitation light will be reached for fluorescence excitation and emission. However, this field then will be much stronger than that of the in-coupling laser beam. 

The most important ingredient in ELF seems to be the curvature of the Fermi hole (respectively the value of Pauli kinetic energy density). However, the respective functions itself do not show the rich structure so typical for ELF. It is exclusively the calibration with respect to the unifonn electron gas that generates all the desired features. Thus, the ELF values depend on the function used for the calibration, which was arbitrarily chosen to be the kinetic energy density of the uniform electron gas. This arbitrariness of the choice was often criticized, e.g., by Bader [73]. Another calibration function was examined by Ayers [54]. He used nearly free electron gas, but found the results much less satisfactory. 

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. 

Fig. 6.5 Scattering between filled and empty states near the Fermi surface. For the Fermi sphere of a free-electron gas the maximum number of such events occurs for q = 2kf. For a Fermi surface with flat regions the number of such events is dramatically enhanced for q = Q, the spanning wave vector.

Nearly-free electron bands. See Free-electron bands Neon, properties of. See Inert gas solids Nesting of Fermi surfaces, 490 Nickel... 

Nearly free-electron model of metals. A gas of free electrons into which a lattice of positive ions is immersed. 

Figure 5.6 Scheme of the near-free electron model of simple metals. The white circles represent the Wigner-Seitz cells, in which the point-positive charges are located. The lattice of the cells Is immersed in a free-electron gas. 

We have seen in previous chapters that the good examples of metallic crystals are the alkali metals, which can be correctly described by the near-free electron model. The valence electrons in these metals are completely separated from their ion cores and form a nearly uniform gas. 

The work function plays an important role in catalysis. The actual value of the work function depends not only on the nature of the solid (bulk contribution), but also on the surface structure, i.e. on the density of atoms at the surface. To understand this surface contribution to the work function we need to take into account the electron distribution near the metal surface. One of simplest and most widely used models for this electron distribution is the jellium model. This model describes the metal as a jellium , consisting of an ordered array of positive metal ions surrounded by a sea of electrons whose properties are those of a free-electron gas. 

The same spectrum of electronic states can be generated conceptually by increasing the density of a fixed number N of atoms. In either case, the overlap of the wave functions is very large in the condensed state and the individual bands superimpose to give a single broad continuum of states as shown in Fig. 2.8(d). Near the band edges, the density of states per unit energy, N(E), resembles that of a free electron gas (see, e.g., Ashcroft and Mermin, 1976) but with the electron mass ntg replaced by an effective mass m ff, namely. 

A consequence of the cancellation between the two terms of (6.47) is the surprisingly good description of the electronic structure of solids given by the nearly-free electron approximation. The fact that many metal and semiconductor band structures are a small distortion of the free electron gas band structure suggests that the valence electrons do indeed feel a weak potential. The Phillips and Kleinman potential explains the reason for this cancellation. 

The simple metals, whose conduction bands correspond to s and p shells in isolated atoms, include the alkali metals, the divalent metals Be, Mg, Zn, Cd, and Hg, the trivalent metals Al, Ga, In, and Tl, and the tetravalent metals (white) Sn and Pb. Almost all of their properties which are related to electronic band structure are explicable by nearly-free-electron theory using pseudopotentials (Sections 3.5 and 3.6). The extent to which they conform in detail to this generalization varies from one case to another. For all the metals cited simple pseudopotential theory is fairly successful in predicting or fitting Fermi surface properties. This will be evident from a consideration of the comparisons of theoretical and fitted pseudopotential parameters already shown in Figure 12. However, the use of perturbation theory is not very critical in this context [i.e., the contribution of screening to the values of v q) which are of interest is not large]. In other contexts the validity of perturbation theory is more critical, and indeed the use of pseudopotential-perturbation theory is then not always so successful. An example is the calculation of phonon dispersion relations by such methods, which has enjoyed remarkable success for Na, Mg, and j(i2i,i22) jjjjQ difficulties for the heavier metals and those... 

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