Metals free electron model

Arguments based on a free electron model can be made to explain the conductivity of a metal. It can be shown that the k will evolve following a Newtonian law [1] ... 

Another important accomplislnnent of the free electron model concerns tire heat capacity of a metal. At low temperatures, the heat capacity of a metal goes linearly with the temperature and vanishes at absolute zero. This behaviour is in contrast with classical statistical mechanics. According to classical theories, the equipartition theory predicts that a free particle should have a heat capacity of where is the Boltzmann constant. An ideal gas has a heat capacity consistent with tliis value. The electrical conductivity of a metal suggests that the conduction electrons behave like free particles and might also have a heat capacity of 3/fg,... 

Simple metals like alkalis, or ones with only s and p valence electrons, can often be described by a free electron gas model, whereas transition metals and rare earth metals which have d and f valence electrons camiot. Transition metal and rare earth metals do not have energy band structures which resemble free electron models. The fonned bonds from d and f states often have some strong covalent character. This character strongly modulates the free-electron-like bands. 

Slater s Xa method is now regarded as so much history, but it gave an important stepping stone towards modem density functional theory. In Chapter 12, I discussed the free-electron model of the conduction electrons in a solid. The electrons were assumed to occupy a volume of space that we identified with the dimensions of the metal under smdy, and the electrons were taken to be non-interacting. 

It is well known that the energy profiles of Compton scattered X-rays in solids provide a lot of important information about the electronic structures [1], The application of the Compton scattering method to high pressure has attracted a lot of attention since the extremely intense X-rays was obtained from a synchrotron radiation (SR) source. Lithium with three electrons per atom (one conduction electron and two core electrons) is the most elementary metal available for both theoretical and experimental studies. Until now there have been a lot of works not only at ambient pressure but also at high pressure because its electronic state is approximated by free electron model (FEM) [2, 3]. In the present work we report the result of the measurement of the Compton profile of Li at high pressure and pressure dependence of the Fermi momentum by using SR. 

In the free electron model, the electrons are presumed to be loosely bound to the atoms, making them free to move throughout the metal. The development of this model requires the use of quantum statistics that apply to particles (such as electrons) that have half integral spin. These particles, known as fermions, obey the Pauli exclusion principle. In a metal, the electrons are treated as if they were particles in a three-dimensional box represented by the surfaces of the metal. For such a system when considering a cubic box, the energy of a particle is given by... 

These three structures are the predominant structures of metals, the exceptions being found mainly in such heavy metals as plutonium. Table 6.1 shows the structure in a sequence of the Periodic Groups, and gives a value of the distance of closest approach of two atoms in the metal. This latter may be viewed as representing the atomic size if the atoms are treated as hard spheres. Alternatively it may be treated as an inter-nuclear distance which is determined by the electronic structure of the metal atoms. In the free-electron model of metals, the structure is described as an ordered array of metallic ions immersed in a continuum of free or unbound electrons. A comparison of the ionic radius with the inter-nuclear distance shows that some metals, such as the alkali metals are empty i.e. the ions are small compared with the hard sphere model, while some such as copper are full with the ionic radius being close to the inter-nuclear distance in the metal. A consideration of ionic radii will be made later in the ionic structures of oxides. 

A representation of the free-electron model of metallic bonding. This model applies to metal alloys as well as to metallic elements. 

The free-electron model explains many properties of metals. For example ... 

The free-electron model is a simplified representation of metallic bonding. While it is helpful for visualizing metals at the atomic level, this model cannot sufficiently explain the properties of all metals. Quantum mechanics offers a more comprehensive model for metallic bonding. Go to the web site above, and click on Web Links. This will launch you into the world of molecular orbitals and band theory. Use a graphic organizer or write a brief report that compares the free-electron and band-theory models of metallic bonding. 

The simple free electron model (the Drude model) developed in Section 4.4 for metals successfully explains some general properties, such as the filter action for UV radiation and their high reflectivity in the visible. However, in spite of the fact that metals are generally good mirrors, we perceive visually that gold has a yellowish color and copper has a reddish aspect, while silver does not present any particular color that is it has a similarly high reflectivity across the whole visible spectrum. In order to account for some of these spectral differences, we have to discuss the nature of interband transitions in metals. 

Similar to the failures of the free-electron model of metals (Ashcroft and Mermin, 1985, Chapter 3), the fundamental deficiency of the jellium model consists in its total neglect of the atomic structure of the solids. Furthermore, because the jellium model does not have band structure, it does not support the concept of surface states. Regarding STM, the jellium model predicts the correct surface potential (the image force), and is useful for interpreting the distance dependence of tunneling current. However, it is inapplicable for describing STM images with atomic resolution. 

Fig. 4.8. Field-emission spectrum of Mo(lOO). The quantity displayed, Jf, is the ratio between the observed field-emission current and the prediction based on a free-electron model, Eq. (4.20). As shown, the field-emission spectrum of Mo(lOO) near the Fermi level is substantially different from a free-electron-metal behavior. (After Weng, 1977.)...

The free electron model regards a metal as a box in which electrons are free to roam, unaffected by the atomic nuclei or by each other. The nearest approximation to this model is provided by metals on the far left of the Periodic Table—Group 1 (Na, K, etc.), Group 2 (Mg, Ca, etc.)—and aluminium. These metals are often referred to as simple metals. 

In the free electron model, the electron energy is kinetic. Using the formula E=Vz m, calculate the velocity of electrons at the Fermi level in sodium metal. The mass of an electron is 9.11 xlO" kg. Assume the band shown in Figure 4.2a starts at 0 energy. 

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... 

A major achievement of the free-electron model was to show why the contributions of the free electrons to the heat capacity and magnetic susceptibility of a metal are so small. According to Boltzmann statistics, the contribution to the former should be nkB per unit volume. According to Fermi-Dirac statistics, on the other hand, only a fraction of order kBT/ F of the electrons acquire any extra energy at temperature T, and these have extra energy of order kBT. Thus the specific heat is of order nfcBT/ F, and an evaluation of the constant gives... 

Fig. 9.7. The density of electronic states as a function of energy on the basis of the free electron model and the density of occupied states dictated by the Fermi-Dirac occupancy law. At a finite temperature, the Fermi energy moves very slightly below its position for T = 0 K. The effect shown here is an exaggerated one the curve in the figure for 7">0 would with most metals require a temperature of thousands of degrees Kelvin. (Reprinted from J. O M. Bockris and S. U. M. Khan, Quantum Electrochemistry, Plenum, 1979, P- 89.)...

Models for the electronic structure of polynuclear systems were also developed. Except for metals, where a free electron model of the valence electrons was used, all methods were based on a description of the electronic structure in terms of atomic orbitals. Direct numerical solutions of the Hartree-Fock equations were not feasible and the Thomas-Fermi density model gave ridiculous results. Instead, two different models were introduced. The valence bond formulation (5) followed closely the concepts of chemical bonds between atoms which predated quantum theory (and even the discovery of the electron). In this formulation certain reasonable "configurations" were constructed by drawing bonds between unpaired electrons on different atoms. A mathematical function formed from a sum of products of atomic orbitals was used to represent each configuration. The energy and electronic structure was then... 

Landau derivation of the diamagnetism no longer applies, and contributions from the orbital motion of the electrons may be expected. Here, as before, the free electron model only reproduces the broad trends of the data. Both model and the sparse data are essentially flat at concentrations above two mole % the free electron model should not be used at lower concentrations. The magnitudes agree as well as for the pure solid metal. 

One of the most important theoretical contributions of the 1970s was the work of Rudnick and Stern [26] which considered the microscopic sources of second harmonic production at metal surfaces and predicted sensitivity to surface effects. This work was a significant departure from previous theories which only considered quadrupole-type contributions from the rapid variation of the normal component of the electric field at the surface. Rudnick and Stern found that currents produced from the breaking of the inversion symmetry at the cubic metal surface were of equal magnitude and must be considered. Using a free electron model, they calculated the surface and bulk currents for second harmonic generation and introduced two phenomenological parameters, a and b , to describe the effects of the surface details on the perpendicular and parallel surface nonlinear currents. In related theoretical work, Bower [27] extended the early quantum mechanical calculation of Jha [23] to include interband transitions near their resonances as well as the effects of surface states. 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

Experimentally it is observed that specific adsorption occurs more with anions than with cations. This is in agreement with chemical models of the interfacial region. Since, according to the free electron model, a metallic lattice can be considered as a cation lattice in a sea of electrons in free movement, it is logical to expect a greater attraction for anions in solution. 

Several models based on the electronic properties of mixtures of metals and molten salts have been proposed, i.e., the localized electron model, the free electron model and the band model. A model which gives a good description of the properties of alkali metal-alkali halide mixtures at low metal concentrations is the model of trapped electrons or the so-called model of F-centers [76,77], An F-center may be regarded as a localized state, and the electron is then trapped in a cavity with octahedral coordination with the neighboring cations. On average, the F-center may be considered as an M65+ species. 

Fio. XXIX-4.—Simplified potential energy function for the free electron model of a metal. 

To understand the relation of the electrons to the equation of state of the metal, we may consider the internal energy at the absolute zero as a function of volume. This quantity, nf nnnrae, should havf> p. minimum for the actual volume of the metal, rising as it is compressed or expanded. In the free electron model, the energy will depend on volume in two ways. In the first place, the Iringtip mil Append nn vnlimiPJ account of... 

Fiq. XXIX-6.—Occupied energy levels for the free electron model of a metal, at the absolute zero, illustrating the relation between Wa, W%, and the thermionic work function or latent heat of vaporization of electrons. 

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