Sommerfeld free-electron model

The first application of quantum mechanics to electrons in solids is contained in a paper by Sommerfeld published in 1928. In this the free-electron model of a metal was introduced, and for so simple a model, it was outstandingly successful. The assumptions made were the following. All the valence electrons were supposed to be free, so that the model neglected both the interaction of the electrons with the atoms of the lattice and with one another, which is the main subject matter of this book. Therefore each electron could be described by a wave function j/ identical with that of an electron in free space, namely... 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

Figure 4. (a) The Fermi surface in -space according to the free electron model of Sommerfeld. (b) The allowed free electron states are represented by discrete points [k, ky, k,). The electrons with the highest kinetic energy are on the Fermi-surface. 

Figure 5. The density of electron states, g E), (dashed line), and the density of occupied states, g(E)f(E), (full line, r > 0 K.) as a function of the energy, according to the free electron model of Sommerfeld.

Since we need the energy to decrease, showing a cohesive energy, we must allow the valence electrons to move throughout the box. They now move in a constant potential due to the nuclei, the inner-shell electrons and the other valence-shell electrons. This is the free-electron model for metals, due to Sommerfeld. ... 

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 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. 

The simplest model of metals is the Sommerfeld theory of free-electron metals (Ashcroft and Mermin 1985, Chapter 2), where a metal is described by a single parameter, the conduction electron density n. A widely used measure of... 

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. 

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 important fact that the quantum theory explains the extraordinarily small specific heat of electrons at normal temperatures, as is well known, was pointed out by Sommerfeld. Our model likewise explains this fact, though in a somewhat different way. If, as is permissible here, we work with the spins in the same way as we previously did with the free electrons, it follows by what we have said above that we have to deal with a degenerate Bose-Einstein gas, the specific heat of which at low temperatures is, as we know, given by... 

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. 

In 1933, Arnold Sommerfeld and Hans Bethe revised the Drude model. Their more complete, quantum mechanical theory goes under the name of the free electron... 

This expression will be used later on when we are calculating the contribution of the free electrons to the molar specific heat capacity of the metal at constant volume (see section 1.8.2.1). However, in order to do that, we need to know the term , which is the number of free electrons per atom of the metal. Sommerfeld s model does not provide us with this number, but Brillouin s band theory, or zone theory, can be used to evaluate it. 

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. 

In analogy to the model used by Sommerfeld in the theory of electron states of crystals Schmidt, Kuhn, and others have developed the so-called free-elect ron theories. 

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