Sommerfeld theory of metals

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... 

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. 

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. 

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

Examples of other work on research schools M. Eckert, "Sommerfeld s School and the Electron Theory of Metals," HSPS 17 (1987) 191234 Gerald Geison, Michael Foster and the Cambridge School of Physiology The Scientific Enterprise in Late Victorian Society (Princeton Princeton University Press, 1978) L. J. Klosterman, "A Research School of Chemistry in the Nineteenth Century Jean Baptiste Dumas and His Research Students," Annals of Science 43 (1985) 180 H. A. M. Snelders, "J. H. van t Hoffs Research School in Amsterdam (18771895)," Janus 71 (1984) 130 F. L. Holmes, "The Formation of the Munich School of Metabolism," in William Coleman and F. L. Holmes, eds., The Investigative Enterprise Studies on Nineteenth-Century Physiology and Medicine (Berkeley, Los Angeles, and London University of California Press, 1988). 

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. 

The quantum-mechanical theory of metals has been extensively developed hy Sommerfeld and many other investigators.3 Discussion of it is beyond the scope of this book, however, and instead we shall consider the problem of the structure of metals from a more chemical point >f view. The treatment given in the following sections is not to be interpreted as being a rival of the quantum-mechanical theory, but rather as offering an alternative avenue of approach toward the same goal as that of the theoretical physicists. 

From the theorist s point of view, the work of Sommerfeld on the Electron Theory of Metals was most seminal. It was eventually reviewed on a quantum mechanical basis in a famous article in the Handbuch der Physik , Vol. XXIV/2 [A. Sommerfeld, H. Bethe (1933)]. Two years before, Heisenberg had introduced the electron hole . A. H. Wilson worked on the Lheory of semiconductors, and it was understood that at T - OK their valence band was completely filled with electrons, whereas the conduction band was empty. At T> 0 K, electrons are thermally excited from the valence band into the conduction band. 

Further development of Sommerfeld s theory of metals would extend well outside the intended scope of this textbook. The interested reader may refer to any of several books for this (e.g. Seitz, 1940). Rather, this book will discuss the band approximation based upon the Bloch scheme. In the Bloch scheme, Sommerfeld s model corresponds to an empty lattice, in which the electronic Hamiltonian contains only the electron kinetic-energy term. The lattice potential is assumed constant, and taken to be zero, without any loss of generality. The solutions of the time-independent Schrodinger equation in this case can be written as simple plane waves, = exp[/A r]. As the wave function does not change if one adds an arbitrary reciprocal-lattice vector, G, to the wave vector, k, BZ symmetry may be superimposed on the plane waves to reduce the number of wave vectors that must be considered ... 

According to Sommerfeld s theory of metallic conduction, Wi, the kinetic energy of the conducting electrons, is connected with 0, the internal potential, and A, the work of escape or work ijpnction measured thermionically or photo-electrically, by the simple equation... 

Arnold Sommerfeld (1868-1951), German physicist and professor at the Mining Academy in Clausthal, then at the Technical University of Aachen, in the key period 1906-1938, was professor at Munich University. Sommerfeld considered not only circular (Bohr-like) orbits, but also elliptical ones, and introduced the angular quantum number. He also investigated X-rays and the theory of metals. The scientific father of many Nobel Prize winners, he did not earn this distinction himself. 

Fowler proposed a theory in 1931 which showed that the photoelectric current variation with light frequency could be accounted for by the effect of temperature on the number of electrons available for emission, in accordance with the distribution law of Sommerfeld s theory of metals. Sommerfeld s theory (1928) had resolved some of the problems surrounding the original models for electrons in metals. In classical Drude theory, a metal had been envisaged as a three-dimensional potential well (or box) containing a gas of freely mobile electrons. This adequately explained their high electrical and thermal conductivities. However, because experimentally it is found that metallic electrons do not show a gaslike heat capacity, the Boltzman distribution law is inappropriate. A Fermi-Dirac distribution function is required, consistent with the need that the electrons obey the Pauli exclusion principle, and this distribution function has the form... 

A. Sommerfeld and H. Bethe, "Handbuch der Physik," Voi. 24, second edition N. F. Mott and H. Jones, "The Theory of the Properties of Metals and Alloys," Oxford University Press, Oxford, 1936 F. Seitz, "The Modern Theory of Solids," McGraw-Hill Book Co., New York, N. Y., 1940. 

The theory fails to explain the molar specific heat of metals since the free electrons do not absorb heat as a gas obeying the classical kinetic gas laws. This problem was solved when Sommerfeld (1) applied quantum mechanics to the electron system. 

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... 

In the range 0 - Ta one generally resorts to use of a theory first proposed by Debye (1907) for nonmetallic solids and extended by Sommerfeld (1926) to metals. At low temperature and in zero order approximation,... 

Fano U (1941) The theory of anomalous diffraction gratings and of quasi-stationary waves on metallic surfaces (Sommerfeld s waves). J Opt Soc Am A 31 213-222... 

Ideally, the specific heat of conduction electrons (or holes) in a metal is a linear function of temperature C = yT, where y, known as the Sommerfeld constant, is in the range 0.001 to 0.01 J/(molK ) for normal materials. In HF compounds, y reaches values up to 10 times larger (see tables 9, 10 and 11). In the basic theory of the specific heat of itinerant electrons (free Fermi gas), y is proportional to the effective mass m of the charge carriers, and so the name heavy fermions has come to be attached to these high-y materials (see Stewart 1984). The linear relation between C and T is strictly fulfilled only in the limit of a free degenerate electron gas. In real materials, weak non-linearities show up that can be encompassed by, for example, allowing y to be temperature dependent, y T). The Sommerfeld constant of interest is then the extrapolation of y for... 

Sommerfeld modified the Drude theory by introducing the laws of quantum mechanics. According to quantum mechanics, electrons are associated with a wave character, the wavelength A being given by A = /i/p where p is the momentum, mv. It is convenient to introduce a parameter, k, called the wave vector, to specify free electrons in metals the magnitude of the wave vector is given by... 

Between 0 and Ta one frequently resorts to the Debye theory for the heat capacity of a nonconducting solids, and extended to metals by Sommerfeld. As a first approximation one uses the relation... 

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