Free electrons in metals

In order to understand the properties of metab, it is necessary to understand the role of electrons in metab. We already know that the metallic bond has to do with electrons that become delocalized from their parent ion cores and are more or less free to move around in the metal. To the first approximation, treating the electrons as an ideal Fermi gas does a fairly good job of explaining most of the properties of metab, but there b more to it as we shall see. 

The first application of quantum mechanics to electrons in solids is contained in a paper by Sommerfeld published in 1928. In this the free-electron model of a metal was introduced, and for so simple a model, it was outstandingly successful. The assumptions made were the following. All the valence electrons were supposed to be free, so that the model neglected both the interaction of the electrons with the atoms of the lattice and with one another, which is the main subject matter of this book. Therefore each electron could be described by a wave function j/ identical with that of an electron in free space, namely 

The vector k is here the wave vector describing the momentum of the electron. But, unlike those for electrons in free space, the values that k can have are quantized in a cube of side L (Q=L3), if we write 

The density of states per unit energy range and per unit volume, for given spin direction, is written N(E), where E denotes the energy. Thus from (2), setting 2=1 cm3, we can write 

The successes of the free-electron model came from combining it with Fermi-Dirac statistics, according to which the number of electrons in each orbital state cannot be greater than two, one for each spin direction. Thus at the absolute zero of temperature all states are occupied up to a maximum energy F given by 

At a finite temperature T the number of electrons with energies between and + d is 

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For insulators, Z is very small because p is very high, ie, there is Htde electrical conduction for metals, Z is very small because S is very low. Z peaks for semiconductors at - 10 cm charge carrier concentration, which is about three orders of magnitude less than for free electrons in metals. Thus for electrical power production or heat pump operation the optimum materials are heavily doped semiconductors. 

As described above, quantum restrictions limit tire contribution of tire free electrons in metals to the heat capacity to a vety small effect. These same electrons dominate the thermal conduction of metals acting as efficient energy transfer media in metallic materials. The contribution of free electrons to thermal transport is very closely related to their role in the transport of electric current tlrrough a metal, and this major effect is described through the Wiedemann-Franz ratio which, in the Lorenz modification, states that... 

The electrochemical potential of single ionic species cannot be determined. In systems with charged components, all energy effects and all thermodynamic properties are associated not with ions of a single type but with combinations of different ions. Hence, the electrochemical potential of an individual ionic species is an experimentally undefined parameter, in contrast to the chemical potential of uncharged species. From the experimental data, only the combined values for electroneutral ensembles of ions can be found. Equally inaccessible to measurements is the electrochemical potential, of free electrons in metals, whereas the chemical potential, p, of the electrons coincides with the Fermi energy and can be calculated very approximately. 

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... 

A further proof of the correctness of the idea of free electrons in metals and of the applicability of the Fermi-Dirac statistics is furnished... 

The phonon contribution to the heat capacity is the most important one, but others also occur. As noted in Section 2.3.7, the heat capacity due to free electrons in metals is small but significant heat capacity changes accompany phase changes, such as order-disorder changes of the type noted with respect to ferroelectrics (Sections 11.3.5 and 11.3.6), or when a ferromagnetic solid becomes paramagnetic (Sections 12.1.2 and 12.3.1). 

Hybridization between )r-conjugated polymers and metals at the nanoscale level has been used for luminescence enhancement and to realize biosensing through SPR coupling [6, 64,65,101,102]. The surface plasmon defines a coherent excitation of free electrons in metal nanostructures interacting with an incident electromagnetic... 

To understand why a stable equilibrium state of two metals in contact includes a contact potential, we can consider the chemical potential of the free electrons. The concept of chemical potential (i.e., partial molar Gibbs energy) applies to the free electrons in a metal just as it does to other species. The dependence of the chemical potential of free electrons in metal phase a on the electric potential 0 of the phase is given by the relation of Eq. 10.1.6 on page 287, with the charge number zi set equal to -1 ... 

Surface plasma wave represents the oscillations of surface charge (free electrons in metal) which are stimulated by an external electric field. The amplitude of the wave shows a maximum intensity on the metallic surface and an exponential decay when it is propagated inside the sample. Surface plasmons are quantized oscillations of the wave. The dispersion pattern of a plasmon contains characteristic properties of a thin film and its interfacial properties. The dispersion relation for a nonradiative plasmon is given in the following expression for a plasmon wavevector. 

The interpretation of ELS spectra from oxides is considerably more complex. In oxides the electrons are tightly bound, and there are no free electrons in metal-... 

Plasmonic excitations involve the coherent collective response of free electrons in metals or doped semiconductors to external particles or fields. The collective response of a solid-state plasma to external fields or particles occurs... 

Are these electrons indeed the gas of free electrons The answer to this question has already been obtained no, there are no free electrons in metals electrons form energy bands that can either be overlapping, or be separated by the forbidden energy gap. Besides, moving in a periodic crystal field, electrons are symmetry-dependent. It is necessary to answer one more question how are electrons distributed among these bands and how are they distributed inside the band ... 

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