Nearly-Free-Electron Theory

The nearly-free-electron theory, once thought of by many as largely irrelevant to real solids, has achieved a degree of respectability with the development of pseudopotential theory. For the moment the necessity of the use of a pseudopotential will be ignored and it will be assumed that the crystal potential F(r) is sufficiently weak to justify this point of view. Plane waves 

What do we mean by sufficiently weak First of all, if we are going to use perturbation theory, we would like 

Also shown are the results of perturbation theory and the free-electron approximation. Perturbation theory gives a divergent result when the 

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In Chapter 2, it was mentioned that there is a strong resemblance between bands obtained from nearest-neighbor interactions in the LCAO approximation and the bands obtained from nearly-free-electron theory. In fact, formulae for the interatomic matrix elements based upon that similarity were used to estimate properties of covalent and ionic solids in the chapters that followed, Now that a... 

The level of agreement between the near-free electron theory and experiments for metals with one electron in s shell may generally be considered as satisfactory. 

The nearly free electron theory developed by Faber and Ziman (1964) is an obvious starting point for discussing liquid alloys of type I. For those cases in which information is available about the three partial interference functions which characterize the structure of binary alloys, close quantitative agreement between theory and experiment has been obtained. We emphasize that a positive da/dT is entirely consistent with metaUic behaviour in Hquid alloys on account of the temperature dependence of the partial interference functions. For this reason many liquid alloys which have in the past been thought of in terms of a semiconducting framework should more properly be regarded as metallic. (It may, in certain cases, be necessary to introduce the Mott g factor but there is little evidence either way on this important point at the present time). Alloys of the second type will form the subject for section 7.7. 

The simple metals, whose conduction bands correspond to s and p shells in isolated atoms, include the alkali metals, the divalent metals Be, Mg, Zn, Cd, and Hg, the trivalent metals Al, Ga, In, and Tl, and the tetravalent metals (white) Sn and Pb. Almost all of their properties which are related to electronic band structure are explicable by nearly-free-electron theory using pseudopotentials (Sections 3.5 and 3.6). The extent to which they conform in detail to this generalization varies from one case to another. For all the metals cited simple pseudopotential theory is fairly successful in predicting or fitting Fermi surface properties. This will be evident from a consideration of the comparisons of theoretical and fitted pseudopotential parameters already shown in Figure 12. However, the use of perturbation theory is not very critical in this context [i.e., the contribution of screening to the values of v q) which are of interest is not large]. In other contexts the validity of perturbation theory is more critical, and indeed the use of pseudopotential-perturbation theory is then not always so successful. An example is the calculation of phonon dispersion relations by such methods, which has enjoyed remarkable success for Na, Mg, and j(i2i,i22) jjjjQ difficulties for the heavier metals and those... 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

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 ... 

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. 

How to proceed with these matrix elements will depend upon which property one wishes to estimate. Let us begin by discussing the effect of the pseudopotential as a cause of diffraction by the electrons this leads to the nearly-free-electron approximation. The relation of this description to the description of the electronic structure used for other systems will be seen. We shall then compute the screening of the pseudopotential, which is necessary to obtain correct magnitudes for the form factors, and then use quantum-mechanical perturbation theory to calculate electron scattering by defects and the changes in energy that accompany distortion of the lattice. 

Let us now complete the derivation of formulae for the interatomic matrix elemenfis, which was described in Section 2-D, by equating band energies obtained from LCAO theory and those obtained from nearly-free-electron bands. This analysis follows a study by Froyen and Harrison (1979). The band energies obtained from nearest-neighbor LCAO theory at symmetry points were given in... 

Band theory provides a picture of electron distribution in crystalline solids. The theory is based on nearly-free-electron models, which distinguish between conductors, insulators and semi-conductors. These models have much in common with the description of electrons confined in compressed atoms. The distinction between different types of condensed matter could, in principle, therefore also be related to quantum potential. This conjecture has never been followed up by theoretical analysis, and further discussion, which follows, is purely speculative. 

In the microscopic theory of Bardeen et al. (1957) the electrons are treated as a nearly free electron gas, individual electrons interacting pairwise via an attractive phonon-mediated potential. The superconducting transition temperature is given by the well-known expression... 

The Nearly Free Electron (NFE) Theory for Liquid Alloys... 

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