Free-electron gas

For a free electron gas, it is possible to evaluate the Flartree-Fock exchange energy directly [3, 16]. The Slater detemiinant is constructed using ftee electron orbitals. Each orbital is labelled by a k and a spin index. The Coulomb... 

Using the above expression and equation Al.3.19. the total electron energy, for a free electron gas... 

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

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... 

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... 

Hubbard, J., Proc. Roy. Soc. London) A243, 336, The correlation energy of a free-electron gas. ... 

As a result they form an almost free electron gas that spreads out over the entire metal. Hence, the atomic sp electron wave functions overlap to a great extent, and consequently the band they form is much broader (Fig. 6.10). 

Figure 6.12. Energy as a function ofthe reciprocal wave vector and the density of states for a free electron gas.

The hypothetical metal jellium consists of an ordered array of positively charged metal ions surrounded by a structureless sea of electrons that behaves as a free electron gas (Fig. 6.13). 

Our representation of a metal is shown in Fig. 6.18. It possesses a block-shaped, partly filled sp band behaving as a free electron gas and a d band that is filled to a certain degree. The sp band is broad as it consists of highly delocalized electrons smeared out over the entire lattice. In contrast, the d band is much narrower because the overlap between d states, which are more localized on the atoms, is much smaller. 

The free-electron gas was first applied to a metal by A. Sommerfeld (1928) and this application is also known as the Sommerfeld model. Although the model does not give results that are in quantitative agreement with experiments, it does predict the qualitative behavior of the electronic contribution to the heat capacity, electrical and thermal conductivity, and thermionic emission. The reason for the success of this model is that the quantum effects due to the antisymmetric character of the electronic wave function are very large and dominate the effects of the Coulombic interactions. 

The free-electron gas exerts a pressure on the walls of the infinite potential well in which it is contained. If the volume v of the gas is increased slightly by an amount dv, then the energy levels in equation (8.56) decrease... 

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. 

If the average exchange potential is assumed to depend only on the local electronic charge density, its value at a point r is equal to the VEx(p) for a free-electron gas, and... 

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 8.19 Energy distribution for a free electron gas at 0 K (shaded) and an elevated temperature (dashed line), T.

Using this simple argument, the electronic heat capacity, CE, of a free electron gas is... 

Figure 8.20 Heat capacity of a free electron gas. The population of the electronic states at different temperatures is shown in the insert. Tp is typically of the order of 103 K.

The electronic heat capacity for the free electron model is a linear function of temperature only for T Tp = p / kp. Nevertheless, the Fermi temperature Tp is of the order of 105 K and eq. (8.46) holds for most practical purposes. The population of the electronic states at different temperatures as well as the variation of the electronic heat capacity with temperature for a free electron gas is shown in Figure 8.20. Complete excitation is only expected at very high temperatures, T>Tp. Here the limiting value for a gas of structureless mass points 3/2/ is approached. 

The distinctive feature of the catalytically active metals is that they possess between 6 and 10 d-electrons, which are much more localized on the atoms than the s-electrons are. The d-electrons certainly do not behave as a free electron gas. Instead they spread over the crystal in well-defined bands which have retained characteristics of the atomic d-orbitals. 

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