Sommerfeld model

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. 

A more general relation between potential and electronic pressure for a density-functional treatment of a metal-metal interface has been given.74) For two metals, 1 and 2, in contact, equilibrium with respect to electron transfer requires that the electrochemical potential of the electron be the same in each. Ignoring the contribution of chemical or short-range forces, this means that —e + (h2/ m)x (3n/7r)2/3 should be the same for both metals. In the Sommerfeld model for a metal38 (uniformly distributed electrons confined to the interior of the metal by a step-function potential), there is no surface potential, so the difference of outer potentials, which is the contact potential, is given by... 

According to the Sommerfeld model electrons in a metal electrode are free to move through the bulk of the metal at a constant potential, but not to escape at the edge. Within the metal electrons have to penetrate the potential barriers that exist between atoms, as shown schematically below. 

The assumption of ZDO introduces periodicity into an otherwise constant potential free electron scheme, in the same way that the Kronig-Penney potential modifies the simple Sommerfeld model. 

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. 

Figure 8. A Bohr-Sommerfeld model of the xenon atom. (From H. A. Kramers and H. Horst, The Atom and the Bohr Theory of its Structure, 1924).

The Sommerfeld model for Ne is shown in figure 2.8. The He atom presented a special problem as the quantum numbers restrict the two electrons to the same circular orbit, on a collision course. One way to overcome this dilemma was by assuming an azimuthal quantum number k = for each electron, confining them to coplanar elliptic orbits with a common focal point. To avoid interference they need to stay precisely out of phase. This postulate, which antedates the discovery of electron spin was never seen as an acceptable solution to the problem which eventually led to the demise of the Sommerfeld model. 

For a brief period the Sommerfeld model enjoyed general acceptance in chemistry. The most powerful argument in its favour was the obvious agreement with the periodic law, e.g. by providing for 2 and Figure 2.8 Sommer-8 electrons in the first two shells, respectively. The feld model of the Ne predicted number of orbits for higher values of n are atom. also in agreement with the periodic law. 

The decay of the nanoparticle plasmons can be either radiative, ie by emission of a photon, or non-radiative (Figure 7.5). Within the Drude-Sommerfeld model the plasmon is a superposition of many independent electron oscillations. The non-radiative decay is thus due to a dephasing of the oscillation of individual electrons. In terms of the Drude-Sommerfeld model this is described by scattering events with phonons, lattice ions, other conduction or core electrons, the metal surface, impurities, etc. As a result of the Pauli exclusion principle, the electrons can be excited into empty states only in the CB, which in turn results in electron-hole pair generation. These excitations can be divided into inter- and intraband excitations by the origin of the electron either in the d-band or the CB (Figure 7.5) [15]. 

FERMI—DIRAC STATISTICS FOR ELECTRON GAS SOMMERFELD MODEL... 

About one month before Heisenberg laid the formal foundation of quanmm mechanics, he wrote a letter to Bohr in which he revealed, Recently I have been occupied with the intensities [of spectral lines], notably in the case of hydrogen. The present conditions are still not entirely sufficient to obtain the intensities. The intellectual context underlying Heisenberg s remark was the Bohr-Sommerfeld model of the hydrogen atom specifically, whether the light emitted in a spectral transition is bright or dim. 

The Sommerfeld model is a simple quantum-mechanical model which takes the Pauli principle into account. It is sufficient for developing a model for the probability of electron tunneling events. It will therefore be discussed in some detail in the next section. A more detailed discussion can be found in Ref. [11]. 

Figure 3. Energy diagram for free electrons in a metal. The positive background charge of the core ions leads to a potential energy well with respect to the energy of the electron in vacuum vac- The averaged kinetic energy of the free electrons is indicated with dashed lines 3/2 k%T according to the Drude model, and 3/5 according to the Sommerfeld model. The electrochemical potential of the electrons in the metal [Fermi level] is also indicated.

The Sommerfeld Model for Free Electrons in a Metallic Phase... 

We consider a metallic phase of volume V, with N free electrons, and hence a free electron density n given by n = NjV. The charge of the core ions is smeared out and leads to a potential energy well keeping the free electrons in the metallic phase (Figure 3). Since in the Sommerfeld model the electrons do not interact with each other, we can describe the electron energy levels by one-electron wave functions. An independent electron can be described by a single-electron wave function ij/ x,y,z) which satisfies... 

The following table contains information pertinent to the Sommerfeld model for some metals. The magnitudes of Tg are calculated using the expression... 

Although the four quantum numbers n, 1, m, and s, the Pauli Exclusion Principle, and Hund s rules were developed in the context of the Bohr-Sommerfeld model, they all found immediate application to Schrodinger s new quantum mechanics. The first three numbers specified atomic orbitals (replacing Bohr s orbits). Physicist Max Bom (1882-1970) equated the square of the wave functions, to regions of probability for finding electrons in each orbital. Werner Heisenberg (1901-76), whose mathematics provide the foundation of quantum mechanics, developed the uncertainty principle the product of the uncertainty in position (Ax) of a tiny particle such as an atom (or an electron) and the uncertainty in its momentum (Ap) is larger than the quantum (h/47t) ... 

Although the simplification is rather drastic, it may still yield a qualitatively instructive picture for the solid state, the famous Sommerfeld model of the free electrons [53] with its many useful conclusions belongs to this class of theories. Quantitatively, however, a Hamilton operator as given in Equation (2.68) does not lead anywhere because it is far too primitive for chemical questions, which can be easily illustrated for, say, the caesium atom. The above hi obviously means that the 6s valence electron moves in a nuclear potential generated by 55 protons but this very electron does not sense the remaining 54 core-like electrons which (should) lie deeper in energy, which must be quantitatively incorrect. 

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