Sommerfeld theory

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... 

According to the Pauli exclusion principle, the conduction electrons occupy the states from the bottom of the conduction band up to an energy level where the metal becomes neutral. This highest energy level occupied by an electron is the Fermi level, /.. In the Sommerfeld theory of metals, a natural reference point of the energy level is the bottom of the conduction band. The Fermi level with respect to that reference point is... 

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. 

After the first theoretical work of Tamm (1932), a series of theoretical papers on surface states were published (for example, Shockley, 1939 Goodwin, 1939 Heine, 1963). However, there has been no experimental evidence of the surface states for more than three decades. In 1966, Swanson and Grouser (1966, 1967) found a substantial deviation of the observed fie Id-emission spectroscopy on W(IOO) and Mo(lOO) from the theoretical prediction based on the Sommerfeld theory of metals. This experimental discovery has motivated a large amount of theoretical and subsequent experimental work in an attempt to explain its nature. After a few years, it became clear that the observed deviation from free-electron behavior of the W and Mo surfaces is an unambiguous exhibition of the surface states, which were predicted some three decades earlier. 

With these new, accurate data, Smekal,4 Coster,8 Dauvillier8 and Wentzel6 have constructed atomic systems to explain the source of each known emission line. The systems are based on the Bohr-Sommerfeld theory of the origin of characteristic X-radiation. In developing and discussing their systems, these investigators have used the values of the L absorption limits measured by us. Certain criticisms have been made of a few of our values. It has therefore seemed desirable to measure some of the limits again. 

We shall frequently encounter cases with angular momentum values of / = and / = 1. For J =, such as for electron spin, there are only two possible orientations of the angular momentum they are depicted in Figure 9. Commemorative of the old Bohr-Sommerfeld theory, we say that the angular momentum is oriented parallel to the z axis in the case of M = + and antiparallel for M = - j. 

The Pauli-Sommerfeld theory of metals is the extension of this simple quantum mechanical picture to three dimensions, and it already enables us to calculate some properties reasonably well. 

The theory of Bohr and Sommerfeld has been applied extensively to atomic spectra with considerable success but it has proved of little use when applied to molecules and the problem of the chemical bond attempts by Pauli and Niessen to apply the theory to the simplest molecular system i.e. the hydrogen molecule ion were unsuccessful. This stable molecule, which has a bond enei of 61 kcal/gmmol, was shown on the basis of the Bohr-Sommerfeld theory to be unstable. More recently, defects have become apparent in the application of this theory to atomic spectra. 

In the Bohr-Sommerfeld theories of the atom, the electrons are moving in orbits which are precisely specified, and the velocities are given exactly. Those theories are therefore concerned with properties which cannot be measured precisely. This difficulty is avoided, however, if one develops theories based on the wave properties of electrons we have already seen, with reference to Figure 1.1, that such theories remove some of the arbitrariness inherent in the Bohr-Sommerfeld approach. Modern theories of atoms and molecules are, therefore, wave theories, which have led to a very considerable increase in our understanding. In the remainder of this chapter we will describe aspects of wave mechanics, or quantum mechanics, that will be of help to biologists in appreciating the nature of the molecular structures with which they are concerned. 

The Bohr-Sommerfeld theory provided a reasonably satisfactory explanation of the spectra of atoms having only one valence electron. With two or more electrons discrepancies occur and certain arbitrary selection rules for atomic transitions were required. In an attempt to solve this problem, several persons, including de Broglie, Schrodinger, Heisenberg, and Born, combined quantum mechanical and wave mechanical concepts. A detailed account of these efforts is beyond the scope of this book. However, some qualitative understanding of these concepts will be helpful and thus a brief account is included. 

The possible combinations of n, /, and m are shown in Table 2-1. What were assumptions of the Bohr-Sommerfeld theory for n, /c, and m become, through the solution of the Schrodinger wave equation, n, /, and m. Thus all three quantum numbers arise from the solution of the Schrodinger equation. 

More serious was the inability of even the Bohr-Sommerfeld theory to account for the spectral details of the atoms that have several electrons. But these were the 1920s and theoretical physics was enjoying its greatest period. Soon the ideas of de Broglie, Schro-dinger, and Heisenberg would put atomic theory on a sound foundation. 

The next step is the computation of the specific heat and the y value. The essential point being again that with such narrow f-like subbands near and around Ep, the density of states Pd(E) cannot be pulled in front of the integral of the total energy, as is done in classical Sommerfeld theory. Thus... 

A simple calculation in the spirit of Sommerfeld theory of metals for the two-dimensional case leads to the equation ... 

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