Jellium approximation

Figure 4. Trends with impact parameter for the asymptotic values of basic observables for a collision of an Ar8+ projectile on a Nazto cluster at two projectile velocities, as indicated. The upper panel shows the excitation energy, the middle panel the number of escaping electrons and the lower panel dispalys ionization probabilities in the case of v = vp velocity. All quantities are plotted as a. function of impact parameters. Ionic background is treated in the sort jellium approximation.

Figure 7. Dipole signal along the direction of laser polarization (Dz, upper part) and number of emitted electrons (Nesc, lower part) as a function of time during the interaction of a femtosecond laser pulse withNaJ. The laser frequency and fluency are = 2,7oeV and J = 6 10uW/cm2, and the pulse envelope is indicated in dashed line. Both Vlasov and VUU results arc plotted for comparison. Ionic background is treated in the soft jellium approximation. From [44],...

FIG. 1. A schematic overview of the jellium approximation for an eight atom cluster. 

An alternative to the spherical jellium approximation just described is to use the tried and tested methods of theoretical chemistry, namely the energy variational principle, to determine the most probable geometrical structure for atomic clusters. This is the basis of the Hiickel method, a rough outline of which is as follows. 

Figure 1.7 Experimental mass spectra by Bj0mholm et al. [23] for several hundreds of Na atoms. Note the absence of magic numbers between 58 and 92 and between 92 and 138. This can be considered as a direct confirmation of the Jellium approximation. Reproduced with permission from Reference [23]. Copyright 1990 by the American Physical Society...

Figure 1.20 Optical absorption of Nag in its ground-state structure D2d. This result from pseudopotential perturbation theory is explained in the main text. The spherical plasmon line at about 2.5 eV (see Figure 1.21) is split into two components which can be understood as follows. The moments of inertia of the structure D2d point to a prolate spheroid within the jellium approximation to the distribution of ions. In such a system there are two collective excitations one at higher frequencies (perpendicular to the axis of symmetry) and one for the motion along the axis of symmetry. Because the motion perpendicular is twofold degenerate its intensity is twice that of the low-frequency motion (with the cluster being statistically oriented in the beam (see [30])...

At a metal-vacuum interface the most important contribution comes from the electrons located outside the metal, which are less screened than in the bulk. Their interaction with the correlation hole left behind them yields a positive term to the surface energy - typically 1.46 J/m for a magnesium-vacuum interface, in the jellium approximation. 

CM = -Wzim- It should be noted that the Budd-Vannimenus theorems are obeyed by the electron density n(z) which is the exact solution to the jellium problem, and not necessarily by any approximate (e.g., variational) solution. There are extensions of the Budd-Vannimenus theorems to a bimetallic surface.74... 

This was averaged over the total distribution of ionic and dipolar spheres in the solution phase. Parameters in the calculations were chosen to simulate the Hg/DMSO and Ga/DMSO interfaces, since the mean-spherical approximation, used for the charge and dipole distributions in the solution, is not suited to describe hydrogen-bonded solvents. Some parameters still had to be chosen arbitrarily. It was found that the calculated capacitance depended crucially on d, the metal-solution distance. However, the capacitance was always greater for Ga than for Hg, partly because of the different electron densities on the two metals and partly because d depends on the crystallographic radius. The importance of d is specific to these models, because the solution is supposed (perhaps incorrectly see above) to begin at some distance away from the jellium edge. 

The interfacial solution layer contains h3 ated ions and dipoles of water molecules. According to the hard sphere model or the mean sphere approximation of aqueous solution, the plane of the center of mass of the excess ionic charge, o,(x), is given at the distance x. from the jellium metal edge in Eqn. 5-31 ... 

FIG. 15. A comparison between the distance dependence of the tunneling barrier between a jellium tip and substrate immersed in solution versus vacuum under zero bias conditions. The apparent barrier height is derived from the WKB approximation. (From Ref. 110.)... 

Abstract This chapter reviews the methods that are useful for understanding the structure and bonding in Zintl ions and related bare post-transition element clusters in approximate historical order. After briefly discussing the Zintl-Klemm model the Wade-Mingos rules and related ideas are discussed. The chapter concludes with a discussion of the jellium model and special methods pertaining to bare metal clusters with interstitial atoms. 

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.

We have so far made two implicit assumptions. The first of these is that the gas of electrons is not scattered by the underlying ionic lattice. This can be understood by imagining that the ions are smeared out into a uniform positive background The second assumption is that the electrons move independently of each other, so that each electron feels the average repulsive electrostatic field from all the other electrons. This field would be completely cancelled by the attractive electrostatic potential from the smeared-out ionic background. Thus, we are treating our sp-valent metal as a metallic jelly or jellium within the independent particle approximation. 

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