Metals free electron theory

Based on the metal free electron theory, Fermi velocity is defined as the free electron movement velocity at the highest energy Ep. The relation between Vp, Ep and lattice distortion rj may be given by... 

This rule conforms with the principle of equipartition of energy, first enunciated by Maxwell, that the heat capacity of an elementary solid, which reflected the vibrational energy of a three-dimensional solid, should be equal to 3RJK-1 mol-1. The anomaly that the free electron theory of metals described a metal as having a three-dimensional structure of ion-cores with a three-dimensional gas of free electrons required that the electron gas should add another (3/2)R to the heat capacity if the electrons behaved like a normal gas as described in Maxwell s kinetic theory, whereas the quantum theory of free electrons shows that these quantum particles do not contribute to the heat capacity to the classical extent, and only add a very small component to the heat capacity. 

The free-electron theory of metals was developed in three main stages (1) classical free-electron theory, (2) quantum free-electron theory, and (3) band theory. 

Classical Free-Electron Theory, Classical free-electron theory assumes the valence electrons to be virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are that (1) an electron can pass from one atom to another, and (2) in the absence of an electric field, electrons move randomly in all directions and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field, electrons drift toward the positive direction of the field, producing an electric current in the metal. The two main successes of classical free-electron theory are that (1) it provides an explanation of the high electronic and thermal conductivities of metals in terms of the ease with which the free electrons could move, and (2) it provides an explanation of the Wiedemann-Franz law, which states that at a given temperature T, the ratio of the electrical (cr) to the thermal (k) conductivities should be the same for all metals, in near agreement with experiment ... 

A theory for the metallic state proposed by Drude at the turn of this century explained many characteristic features of metals. In this model, called the free-electron theory, all the atoms in a metallic crystal are assumed to take part collectively in bonding, each atom providing a certain number of (valence) electrons to the bond. These free electrons belong to the crystal as a whole. The crystal is considered to be... 

An expression for the electrical conductivity of a metal can be derived in terms of the free-electron theory. When an electric field E is applied, the free carriers in a solid are accelerated but the acceleration is interrupted because of scattering by lattice vibrations (phonons) and other imperfections. The net result is that the charge carriers acquire a drift velocity given by... 

The molecular orbital treatment of a crystalline solid considers the outer electrons as belonging to the crystal as a whole (10,11). Sommer-feld s early free electron theory of metals neglected the field resulting... 

Lorentz1 advanced a theory of metals that accounts in a qualitative way for some of their characteristic properties and that has been extensively developed in recent years by the application of quantum mechanics. He thought of a metal as a crystalline arrangement of hard spheres (the metal cations), with free electrons moving in the interstices.. This free-electron theory provides a simple explanation of metallic luster and other optical properties, of high thermal and electric conductivity, of high values of heat capacity and entropy, and of certain other properties. 

ELECTRON GAS. The term electron gas is used to denote a system of mobile electrons, as. for example, the electrons in a metal that are free to move. In the free electron theory of metals, these electrons move through the metal in the region of nearly uniform positive potential created by the ions of the crystal lattice. This theory when modified by the Pauli exclusion principle, serves to explain many properties of metals, especially the alkali metals. For metals with more complex electronic structure, and semiconductors, the band theory of solids gives a better picture. 

The success of the simple free electron theory of metals was so staking that it was natural to ask how the same ideas could be apphed to other types of solids, such as semiconductors and insulators. The basic assumption of the free electron theory is that the atoms may be stopped of their outer electrons, the resulting ions arranged in the crystalline lattice, and the electrons then poured into the space between. 

The Classical Free-Electron Theory. The classical free-electron theory considers that the valence electrons are virtually free everywhere in the metal. The periodic lattice field of the positively charged ions is evened out into a uniform potential inside the metal. The major assumptions of this model are (1) an electron can pass from one atom to another and (2) in the absence of an electric field elec-rons move randomly in all directions, and their movements obey the laws of classical mechanics and the kinetic theory of gases. In an electric field electrons... 

THE MODERN THEORY OFSOLIDS, Frederick Seitz. First inexpensive edition of classic work on theory of ionic crystals, free-electron theory of metals and semiconductors, molecular binding, much more. 736pp. 55 x 814. 

The electrical resistivity of Na W03, Li WOs, and K WOg has been measured at 300° K. The range of x values was 0.25 < x < 0.9. All resistivities were characteristic of a metal and lie on a single curve. Extrapolation of the conductivity curve to zero conductivity indicated that the tungsten bronzes should be semiconductors for x < 0.25. The resistivities measured for tungsten bronzes with x < 0.25 showed semiconducting behavior. The resistivity of Li WOg exhibited an anomalous peak in the p vs. T curve. The Hall coefficient of Li0 37WO3 indicated one free electron per alkali atom, as previously found for Na WOg. The Seebeck coefficient of Na WOg depended linearly on x 2/3, as expected from free electron theory. The implications of these and other data are discussed. 

The Dmde free-electron theory can be used to model the optical properties of noble metals. This theory provides the following expression for relative permittivity as a function of frequency in the visible regime (with electron scattering neglected because the electromagnetic frequency is high) ... 

The central assumption of free-electron theory is that each atom gives up its valence electrons to the metal, and the states of these electrons are unaffected by the metallic ions formed from the atoms thus stripped of their electrons. The number Z of electrons each atom gives to the metal is unambiguous it is the number in excess of the last inert-gas shell or in excess of the last completed d shell, whichever is less. These electrons form a uniform electron gas in the metal. We may thus proceed to a discussion of such a gas and obtain the consequences for the properties of the metal. In Section 16-F we shall introduce the modification of the electron states caused by the metallic ions, describing the influence of those ions by a pscudopotential. 

The theory of the electronic properties of the simple metals that has been built from simple free-electron theory is extraordinary. It extends to thermal properties such as the specific heat, magnetic properties such as the magnetic susceptibility, and transport properties such as thermal, electrical, thermoelectric, and galvano-magnetic effects. This theory is discussed in standard solid state physios texts (see, for example, Harrison, 1970) and will not be discussed here. As a universal theory for all metals, it is not sensitive to the electronic structure it depends only upon the composition of the metals through simple parameters such as those of Table... 

The electronic structure of metals and metallic surfaces are interpreted by the free-electron theory of metals [1, 2]. 

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