Metals uniform background model

Figure 38 Electron density at a metal surface versus distance x normal to the surface, as computed by N. D. Lang [76) at two r, values, using the planar uniform-background model.

In most treatments of the surface electron distribution, the so-called "jellium model has been used, for which the atomic cores are smeared into a planar uniform background. The jellium model has been found, e.g., Lang and Kohn (1973), to work well for simple metals and, with regard to the surface electron response, has been employed to treat the behavior of some of the noble metals (Equiluz, 1984). 

The reason for this terminology is that in the simplest possible treatment of metallic electrons, they can be considered as free particles in the positive uniform background of the ions (the jellium model), in which case is indeed the kinetic energy in the following we adopt this terminology to conform to literature conventions. 

It turns out that for the s and p block metals, a simple model, namely, thejdlium model provides useful insight. In this model, the discrete nature of the ionic lattice is replaced with a smeared out uniform positive background exactly equal to that of the valence electron gas. In jellium, each element is completely specified by just the electron density n = N/V, where N is the number of electrons in the crystal and V is its volume. Often, the electron density is given in terms of the so-called Wigner-Seitz radius, tj, where tg = which corresponds to the spherical... 

An entirely different approach to the correlation problem is taken in the plasma model (Bohm and Pines 1953, Pines 1954, 1955), in which the electrons in a metal are approximated by a free-electron gas moving in a uniform positive background. According to classical discharge theory, such a plasma is characterized by an oscillatory behavior having a frequency... 

The dependence of dx on qM is central in a model, proposed by Price and Halley,93 for the metal surface in the double layer which is related to that discussed above. The positively charged ion background profile p+(z) is assumed uniform, with a value equal to the bulk density pb, from z = -oo to z = 0, with the electronic density profile n(z) more diffuse. In contrast to the previous model30 which emphasizes penetration by the conduction electrons of the region of solvent, this model93 supposes that the density profile n(z) is zero for z > dx, where z > dx defines the region of the electrolyte. Then the potential at dx is given by... 

The jellium model for the surface electronic structure of free-electron metals was introduced by Bardeen (1936) for a treatment of the surface potential. In the jellium model, the lattice of positively charged cores is replaced by a uniform positive charge background, which drops abruptly to zero at the... 

In the bulk, the charge density of electrons n equals in magnitude the charge density of the uniform positive charge background +, thus to preserve charge neutrality. The only parameter in the jellium model, r,, is the same as in the Sommerfeld theory of free-electron metals. 

Fig. 4.3. Position of the image plane in the jellium model. The surface potential of an electron in the jellium model is calculated using the local-density approximation. By fitting the numerically calculated surface potential with the classical image potential, Eq. (4.7), the position of the image plane is obtained as a function of r, and z. The results show that the classical image potential is accurate down to about 3 bohrs from the boundary of the uniform positive charge background. For metals used in STM, r, 2 — 3 bohr, zo 0.9 bohr. (Reproduced from Appelbaum and Hamann, 1972, with permission.

Experiments show that the values for the double layer capacity at single crystal metal surfaces depend on the nature of the metal. This indicates that the metal surface cannot be considered as a perfect conductor, as was done in classical theories. It is well known that an overspill of metal electrons can occur at the boundary of a metal with vacuum. A similar overspill expected at a metal-solution interface would alter the double layer capacity by an amount depending on the type of metal. Models have been constructed in which the metal is represented by an electronic plasma in a uniformly, positively charged background, which is known as the jellium model [81,82]. The inclusion of the electron overspill into the integral equation theories of the electric double layer has been performed basically with the HAB model. 

The role of the metal in double layer properties can be understood in greater detail when the system is examined on the basis of the jellium model. This model was developed to describe the electron gas within sp metals. It can be used to estimate several properties of interest, including the chemical potential of an electron in the metal, the extent of electron overspill, and the work function of the metal. More recently, it has been extended to describe metal surfaces in contact with polar solvents [26]. In its simplest form, the metal atoms in the metal are modeled as a uniform positive background for the electron gas, no consideration being given to their discrete nature and position in the metal lattice. The most important property of the system is the average electron density, N ), which depends on the number of metal atoms per unit volume and the number of valence electrons per atom, n. Thus, if pjj, is the mass density of the metal, and M, its atomic mass... 

Let us consider a metallic carbon nanotube of the radius r as a coaxial conductor, filled with the gas of noninteracting electrons moving on a uniform positive neutralizing background (Jelly model). The mean field for moving electrons can be represented as a coaxial well with infinite walls. Thus, electrons cannot leave the well. T he width o f t his well (the r adius d ifference o f t wo c ylindrical su rfaces) i s chosen to be approximately equal to the layer-layer separation 2A of graphite. 

The jeUium model of a metal surface takes into account, in a simplified though significant way, the problem of electron-electron interaction. The atomic potential is smeared out into a uniform positive background extending over the region z < 0. The electron density is taken into account by a dimensionless parameter rg, defined by the equation... 

A simple model for bulk metals and metal surfaces. The atomic nuclei are replaced by a uniform positive background with a sharp drop to zero at the surface. The distribution of electrons in the resulting potential is calculated using density functional theory yielding properties such as the work function. Muffin-tin-potential... 

The simplest electronic theory of metals regards a metallic object as a box filled with noninteracting electrons. (A slightly more elaborate picture is the jellium model in which the free electrons are moving on the background of a continuous positive uniform charge distribution that represents the nuclei.) The Drude model, built on this picture, is characterized by two parameters The density of electrons n (number per unit volume) and the relaxation time t. The density n is sometimes expressed in terms of the radius of a sphere whose volume is the volume per electron in the metal... 

In order to obtain the potential energy surface, one needs of course to solve an electronic structure problem. Several methods have been used, ranging from methods treating the electrons of the metal as a free electron gas to traditional ab initio calculations. One of the simplest models treats the solid as a jellium, i.e., a uniform elecron gas on a positive background. The hamiltonian for the electrons is then just the kinetic energy — ( /2m, )V. For z > 0 the one-electron wavefunctions are approximated by... 

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