The Free Electron Model

4 Modification of WitFs Terms and Classification of Colored Organic Compounds 

Although very popular even today, Witt s color terms, chromophore, chromogen and auxochrome, have no theoretical definition as such. Some broader definitions of these terms that will help bridge the gap between empirical observation and semi-quantitative understanding are the following  

Using these suggested terms, any atom that possesses lone pair electrons in conjugation with a TT-electron system can be regarded as an auxochrome. 

Using these revised definitions, one can conveniently classify the large diversity of colored organic molecules into four broad classes  

The vast majority of colored organic compounds are based on donor-acceptor chromogens, and with the exception of the polycyclic quinones and the phthalo-cyanines, all the commercially important synthetic dyes are of this type [34, 35]. 

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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,... 

The resolution of this issue is based on the application of the Pauli exclusion principle and Femii-Dirac statistics. From the free electron model, the total electronic energy, U, can be written as... 

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. 

Electrons do of course interact with each other through their mutual Coulomb electrostatic potential, so an alternative step to greater sophistication might be to allow electron repulsion into the free-electron model. We therefore start again from the free-electron model but allow for the Coulomb repulsion between the electrons. We don t worry about the fermion nature of electrons at this point. 

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. 

Although the free electron model leads to a simple understanding of electrochemical phenomena, even in solution, it offers no explanation of the different conduction properties of different types of solid. In order to understand the conduction of solids it is necessary to extend the free electron model to take account of the periodic lattice of a solid. 

The electronic heat capacity for the free electron model is a linear function of temperature only for T Tp = p / kp. Nevertheless, the Fermi temperature Tp is of the order of 105 K and eq. (8.46) holds for most practical purposes. The population of the electronic states at different temperatures as well as the variation of the electronic heat capacity with temperature for a free electron gas is shown in Figure 8.20. Complete excitation is only expected at very high temperatures, T>Tp. Here the limiting value for a gas of structureless mass points 3/2/ is approached. 

This consideration excludes the free-electron model for example, and the Floating Spherical Gaussian Orbital approach in its simplest form. 

The use of effective mass to understand the state of the microstructure is chosen to conserve the application of the free electron model by letting the mass of the electron incorporate the electron interactions with the lattice, which are experiencing potential energy interactions. Considering the total energy, E, of an electron in a solid, based on wave mechanics, then ... 

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. 

Thus, the free-electron model is not valid when Eq. (3.7) applies since the wave is reflected. The E,k) curve constructed on this basis is like that obtained from the Kronig-Penney model bands of allowed and forbidden energy regions. 

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. 

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. 

FIGURE 4.1 A density of states curve based on the free electron model. The levels occupied at 0 K are shaded. 

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. 

As an example of extinction by spherical particles in the surface plasmon region, Fig. 12.3 shows calculated results for aluminum spheres using optical constants from the Drude model taking into account the variation of the mean free path with radius by means of (12.23). Figure 9.11 and the attendant discussion have shown that the free-electron model accurately represents the bulk dielectric function of aluminum in the ultraviolet. In contrast with the Qext plot for SiC (Fig. 12.1), we now plot volume-normalized extinction. Because this measure of extinction is independent of radius in the small size... 

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 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... 

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.)...

In collisions with Rb nd states the fact that N2 is a molecule becomes more apparent. The Rb nd state cross sections, measured in the same way, are smaller, with cross sections of 150 A2, independent of n for 24 < n < 46. These cross sections can also be compared to those calculated using the free electron model. Consider the 35d state as a typical example. The computed cross sections for transfer to the nearby 35 s 3 and 34( > 3 states by the short range e -N2... 

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. 

Equation (2) is clearly true for the free-electron model and is true in general if G(k) contains the inversion operator (7 0) (Exercise 16.4-1), but eq. (2) shows that the energy curves Ek are always symmetrical about k=0 and so need only be displayed for k> 0. 

The two methods have a definite relation which comes from the fact that the structure of the molecule, and particularly its dimensions, is regulated by the value of the exchange energy. We thought it possible to specify this relation by using the essential peculiarity of the free-electron model, that is,. the elimination of all dynamic constants by the LCAO MO method. 

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