Electrons jellium model

Figure 5.7. Schematic representation of the definitions of work function O, chemical potential of electrons i, electrochemical potential of electrons or Fermi level p = EF, surface potential %, Galvani (or inner) potential

In this contribution we will deal with electron-electron correlation in solids and how to learn about these by means of inelastic X-ray scattering both in the regime of small and large momentum transfer. We will compare the predictions of simple models (free electron gas, jellium model) and more sophisticated ones (calculations using the self-energy influenced spectral weight function) to experimental results. In a last step, lattice effects will be included in the theoretical treatment. 

Figure 11 shows the influence of correlation and lattice effects on the shape of n (k) for the case of lithium. The short dashed line shows (k) according to the jellium model with no electron-electron interaction included. Inclusion of correlation effects can be described using a model-w(k) ... 

Data taken in another mass range also showed the inertness of AlJ7. While it was not produced by reaction of larger clusters, of which there are few, it was, however, found to be unreactive over a wide oxygen concentration range. In general, our results support the electron droplet Jellium model, although some anomalies (for... 

With respect to the thermodynamic stability of metal clusters, there is a plethora of results which support the spherical Jellium model for the alkalis as well as for other metals, like copper. This appears to be the case for cluster reactivity, at least for etching reactions, where electronic structure dominates reactivity and minor anomalies are attributable to geometric influence. These cases, however, illustrate a situation where significant addition or diminution of valence electron density occurs via loss or gain of metal atoms. A small molecule, like carbon monoxide,... 

Since each atom in a f.c.c. array of purely metallic atoms is the same as every other atom (except at the surface), only a representative positive ion needs to be considered. Let it interact with a spherical portion (radius = R) of the electron gas which has a density of one electron per ion. This is called the jellium model. 

Figure 3.4 Distribution of the electronic density in the jellium model the metal occupies the region x < 0. The unmarked curve is for an uncharged surface, the other two curves are for the indicated surface-charge densities. The distance along the x axis is measured in atomic units (a.u.), where 1 a.u. of length = 0.529 A.

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. 

Figure 17.3 Electronic density and charge distribution in the jellium model (schematic).

To explain where the surface contribution to the work function comes from, we need a model for the electron distribution in the surface region of a metal. One of the simplest is the jellium model [18]. 

The role of the metal electrons in determining properties of the metal/electrolyte solution interface was reviewed by Komyshev, who presented and discussed an alternative approach to the jellium model. As stated by Komyshev, the jellium model, which is correct for the interpretation of capacity measurements, appears to be too rough to explain the data obtained with modem techniques, such as many spectroscopic and miaoscopic techniques (see later discussion). 

Fig. 2-10. Profile of electron density and electronic potential energy across a metal/vacuum interface calculated by using the jellium model of metals MS = jellium surface of metals Xf = Fermi wave length p. - average positive charge density P- s negative charge density V = electron exchange and correlation energy V, - kinetic energy of electrons. [From Lange-Kohn, 1970.]...

Kg. 2-11. Work function, 4>, observed and calculated by using the jellium model as a function of Wigner-Seitz radius, rs, for various metals rs = 3 / (4 n n, = electron... 

The order of filling of the free electronic states in a jellium sphere is quite different from that of a free atom (Fig. 6). Although in a free atom the 2s level immediately follows the Is level, in a jellium sphere the p and lfree atom only the 2p level follows the 2s level before the 3s level is reached. However, in the jellium model the 2s level is followed by the 1/, 2p, Ig, and 2d levels before the 3s level is reached. 

In some cases, macroscopic models are used for simplified discussions of certain phenomena without atomic resolution. A macroscopic tip-sample distance should be defined. To avoid confusion, we use the term barrier thickness instead. Throughout the book, the barrier thickness is always denoted by a upper-case letter, such as W or L. In the Sommerfeld model of the free-electron metals, the barrier thickness is the distance between the surface of the metal pieces. In the jellium model (see Chapter 4), the barrier thickness is defined as the distance between the image-force planes. 

The simplest model of a metal surface is the jellium model, which is a Sommerfeld metal with an abrupt boundary. In provides a useful semiquanti-tative description of the work function and the surface potential (Bardeen, 1936). It validates the independent-electron picture of surface electronic structure Essentially all the quantum mechanical many-body effects can be represented by the classical image force, which has been discussed briefly in Section... 

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. 4.2. Charge distribution and surface potential in a jellium model, (a) Distribution of the positive charge (a uniform background abruptly drops to zero at the boundary) and the negative charge density, determined by a self-consistent field calculation. (b) Potential energy as seen by an electron. By including all the many-body effects, including the exchange potential and the correlation potential, the classical image potential provides an adequate approximation. (After Bardeen, 1936 see Herring, 1992.)...

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.

Similar to the failures of the free-electron model of metals (Ashcroft and Mermin, 1985, Chapter 3), the fundamental deficiency of the jellium model consists in its total neglect of the atomic structure of the solids. Furthermore, because the jellium model does not have band structure, it does not support the concept of surface states. Regarding STM, the jellium model predicts the correct surface potential (the image force), and is useful for interpreting the distance dependence of tunneling current. However, it is inapplicable for describing STM images with atomic resolution. 

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