Electronic structure free-electron theory

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. 

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 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... 

Energy bands and a number of properties have been described here in terms of L( A() theory and matrix elements given by formulae such as Eq. (20-6). We turn now to the origin of those formulae and to a description of the electronic structure that proves useful for other properties. The formulae for the matrix elements will in fact be obtained from transition-metal pseudopotential theory, but the principal results can be obtained from the theory of Miiflin-Tin Orbitals, which we discu.ss first. Moreover, one of the central concepts of Muffin-Tin Orbital theory is necessary for using transition metal pseudopotential theory to obtain the formulae for the interatomic matrix elements. The analysis in this section and the next is somewhat analogous to the use of free-clectron theory to obtain the form and estimates of the magnitudes of the matrix elements used in the LCAO theory, and here the consequences arc just as rich. 

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

Many of the new tasks would be at the boundary with materials science. There are some that are obviously applications-oriented, like the electronic theory of high temperature superconduction in the layered copper-oxide perovskites, and other aspects of nanotechnology. There are also fundamental valence problems, such as accounting for the structures and properties of quasiciystals. Why is the association of transition metals and aluminium apparently of central importance How do we deal with the valence properties of systems where the free energy of formation or phase transition is dominated by the entropy term ... 

The bond electrons in covalent bond are very locked in the hybrid orbitals which gives very poor electrical conductance. This is in contrast to the bonds in metals. These bonds can be described by an electron sea model that tells us that the valence electrons freely can move around in the metal structure. The band theory tells us that the valence electrons move around in empty anti-bond orbitals that all lie very close in energy to the bond orbitals. The free movement of electrons in metals explain the very high electrical and thermal conductivity of metals. Metal atoms are arranged in different lattice structures. We saw how knowledge about the lattice structure and atomic radius can lead to calculation of the density of a metal. 

The main conclusion is that the rotational and electronic fine structure distributions can be understood quantitatively in terms of a parameter free FC-theory. This is the first case in which the selective population of A-doublet states could be understood quantitatively. As a matter of fact, it is the only case in which the FC-theory with electronic fine structure has been applied to a triatomic system. 

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. 

In fig. 3.22 the data on holmium in two different directions and at two different temperatures are shown. The near parabolic shape for the data along the b axis indicates that there is more symmetry in the Fermi surface around the hexagonal axis. However, the surface is not spherical because the data deviate considerably from the free electron theory. The slight temperature dependence of the c axis data for Ho reflects the change in band structure and Fermi surface due to magnetic ordering. 

In Eq. 3.4, a relates to the electron structure and scattering in the metal. From the free electron theory of metal, Fermi velocity is defined as the fi ee electron movement velocity at the highest energy (E ), and the relaxation time r is the time between the first and second collisions of the electron. So the electron mean free path Ip nearby the Fermi surface can be expressed as Ip = v t. In addition, o can be also expressed as below " ... 

---