Free electron theory

The three most prominent theories to date to explain metal bonding include free-electron theory, the valence bond theory, and band theory. They represent a progression in scientists understanding of metals over time. 

This model may account for various properties observed in metals such as  

1 Malleability and ductility The sea of electrons act like a sponge, so when the metal is hammered the composition of the metal structure is not harmed or changed. The positively charged metal nuclei may be rearranged, but the sea of electrons simply adjust to the new formation and keep the metal intact. 

Luster The free electrons in the sea can absorb photons making them opaque-looking. However, the surface electrons also reflect light back at same frequency of incidence making the surface shiny. 

1/ Conductivity Because the electrons are free, the electrons move 

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One further effect of the formation of bands of electron energy in solids is that the effective mass of electrons is dependent on the shape of the E-k curve. If this is the parabolic shape of the classical free electron theory, the effective mass is the same as the mass of the free electron in space, but as this departs from the parabolic shape the effective mass varies, depending on the curvature of the E-k curve. From the definition of E in terms of k, it follows that the mass is related to the second derivative of E with respect to k thus... 

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 ... 

Quantum Free-Electron Theory Constant-Potential Model, The simple quantum free-electron theory (1) is based on the electron-in-a-box model, where the box is the size of the crystal. This model assumes that (1) the positively charged ions and all other electrons (nonvalence electrons) are smeared out to give a constant background potential (a potential box having a constant interior potential), and (2) the electron cannot escape from the box boundary conditions are such that the wavefunction if/ is... 

Because n can take all integral values, this means an infinite number of energy levels exist with larger and larger gaps between each level. Most solids of course are three-dimensional (although we shall meet some later where conductivity is confined to one or two dimensions) and so we need to extend the free electron theory to three dimensions. 

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

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... 

Let us turn now to the other conclusions which can be based on free electron theory. The Hall effect measurements of Kyser and Thompson permitted the computation of the free electron concentration. The Hall effect is produced by a balance between the magnetic force (Lorentz force) on a current carrier and the electric force produced by a displaced charge density within a conductor. For a charge, q, moving... 

The dotted line shows the free electron theory. Note the absence of data between... 

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. 

Table XXIX-2.—Electronic Specific Heat, Free Electron Theory...

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