Jellium solid

Figure 6.27. Charge density contours for the adsorption of Cl, Si, and Li on jellium. (a) Total charge and (b) induced charge solid lines indicate an increase in electron density, dashed...

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. 

The electrodynamic forces proposed for stabilizing jellium provide the principal type of bonding in molecular crystals such as solid methane, rare gas crystals, solid anthracene, and the like. These forces also form the inter-chain bonding of long-chain molecules in polymeric materials (the intra-molecular bonding within the chains is usually covalent). 

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. 

Fig. 6.4. Image profile with a Na-atom tip. (a) Geometry of the simulation. Two flat and. structureless jellium surfaces, each with an extra Na atom adsorbed on it, represent the tip and the sample, respectively. The net current from these Na atoms is kept constant while moving the Na atoms across each other. The path is generated numerically, (b) The simulated image (.solid curve) is in good agreement with the contour of the bermi-level LDOS and the total-charge-dcnsity contour. The apparent radius, as determined from the curve, is about 12 A. (After Lang, 1986.)...

Fig. 3. Approximate electron density variation on jellium surfaces with periodic positive charge boundaries. The solid line gives the edge of the uniform ionic charge density. The dashed line indicates the contour where the electron density is equal to one-half its interior value.

Figure 13 Plot of electronic charge density as a function of distance across a solid-vacuum interface as calculated from a jellium model...

Fig. 6. N(E, x) is the local density of states of energy E, at distance x from the surface of a semi-infinite jellium n(x) is the electron charge density at x (cf. Fig. 5, right). The quantity plotted along the vertical axis is the ratio N(E, x)ln(x) for several values of x. Negative x are inside the solid and positive are outside. The dashed vertical lines show the LDOS of states near the Fermi level, normalized by the electron charge density n(x) falls exponentially outside the surface. A large fraction of the charge outside the surface is due to Fermi-level electrons. [Reproduced with permission from Werner et al. (27). Copyright 1975 Institute of Physics Publishing.]...

Fig. 3. Stopping power as a function of the distance to the top-most layer for V = 1 a.u. protons traveling parallel to the Cu (111) surface. The solid line is the result of the model in which the surface band structure of the target is considered, the short-dashed line is obtained neglecting the surface state in the calculation, the long-dashed line is the result of using the jellium model and constructing XoCG, z, z , within the RPA using the self-consistent solutions for a finite step potential (see text), and the dot-dashed line is obtained in the jellium model within the SRM.

In either the Born-Oppenheimer (fixed-nuclei) or jellium (smeared-nuclei) approximations, the electronic part of a finite-molecular or infinite-solid... 

Fig. 6. Self-consistent surface charge density in the jellium model for K (solid line) and A1 (dashed line) (Lang and Kohn, 1 y70). Distance is measured in Fermi wavelengths from the positive background edge charge density is measured relative to the bulk density pa-...

Fig. 7. Surface energy for jellium with structureless pseudopotential (solid line), compared with experiment (dots) (Pcrdcw ct al., 1990). The dashed line gives results for jellium without the pseudopotential.

Fig. 22. Contours of electron density difference, for Na adsorbed on jellium (Al), compared with clean jellium plus isolated Na (Ishida, 1990). The different plots are for different Na spacings (ap). Solid and dashed contours correspond to positive and negative changes in density, respectively.

These are determined from the self-consistent-field local density (SCF-LD) functional calculations (Puska et al. 1981) on the embedding energy of an atom in jellium. The values as a function of jellium densities are represented by a Morse-like form. (It is not strictly a Morse potential since density is the variable instead of distance.) The variable, r depends upon the density of the solid at the position of A ... 

Assume that the binding curves for the asymptotic fragments are available and have been used to determine the relevant parameters (7>ab> ab> ab)> (Das, Kas. l As) (Dj,s, bs. bs)- The parameters for the interaction with jellium, (Dah, ah. ah) and (Dbh, bh. bh). are not adjustable, being determined by the SCF-LD values. Thus, the remaining variables in the PES, the so-called Sato parameters, Aab>A s, and A s, are undetermined and available for flexible representation of the full molecule-solid reactive PES. We consider the effect of these on the PES later. At this point, we do want to emphasize that the basic physics and chemistry— (1) interactions with localized and delocalized metal electrons and (2) nonadditive chemical bonding—are correct. We should also note that the representation of an interaction in metals in terms of an embedding function (in jellium) plus two-body terms is identical in spirit to the embedded atom method (EAM) (Daw and Baskes 1984, 1988). The distinction here is that we do not use the EAM for the A-B interaction and explicitly incorporated nonadditive energies via the LEPS prescription, both of which are important for the accurate representation of the reactive PES. 

The agreement was established for liquid metals, considering the jellium model in the case of solid or high-density materials, the criteria do not give good results and more corrections for the interaction energies are needed [21]. 

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