The Free Electron Model of a Metal

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

A number of properties of the electrons in a metal can be found from our model. In particular, we can find the specific heat of the electrons and can see in a qualitative way what their contribution to the equation of state should be. For the heat capacity, Eq. (5.9), Chap. V, gives 

This is a heat capacity proportional to the temperature, and in Sec. 5, Chap. V, we computed it for a particular case, showing that it amounted to only about 1 per cent of the corresponding specific heat of free electrons on the Boltzmann statistics, at room temperature. In Table XXIX-2 we show the value of the electronic specific heat at 300° abs., computed from the values of Wi which we have already found, in calories per mole. We verify the fact that this specific heat is small, and for ordinary purposes it can be neglected, so that the specific heat of a metal can be found from the Debye theory, considering only the atomic vibrations. At low temperatures, however, Eq. (2.4) gives a specific heat varying as the first power of the temperature, while Debye s theory, as given in Eq. (3.8), 

Observations are taken from Jones and Mott, Proc. Roy. Soc., 162, 49 (1937). Calculations in that paper assume different numbers of electrons per atom, instead of one per atom as we have done, and secure somewhat better agreement with experiment. Specific heats are tabulated in calories per mole. 

shows that the specific heat of atomic vibrations varies as the third power. At temperatures of a few a fthird pmuor 

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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... 

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. 

Let us summarize the important differences between the free electron model of a metal and models based on the electronic band structure as discussed above. The... 

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. 

Flo. XXIX-7.—Diagram of occupied levels in momentum space, in free electron model of a metal. The points of the displaced sphere (shaded) are occupied when the electrons have been accelerated by an external field. 

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. 

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 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. 

Free Electron Model for Electronic Entropy Using the free electron model of chap. 3, estimate the entropy of metals. Begin by obtaining an expression for the electronic density of states for the three-dimensional electron gas. Then, make a low-temperature expansion for the free energy of the electron gas. Use your results to derive eqn (6.12). 

Solid state physicists are familiar with the free- and nearly free-electron models of simple metals [9]. The essence of those models is the fact that the effective potential seen by the conduction electrons in metals like Na, K, etc., is nearly constant through the volume of the metal. This is so because (a) the ion cores occupy only a small fraction of the atomic volume, and (b) the effective ionic potential is weak. Under these circumstances, a constant potential in the interior of the metal is a good approximation—even better if the metal is liquid. However, electrons cannot escape from the metal spontaneously in fact, the energy needed to extract one electron through the surface is called the work function. This means that the potential rises abruptly at the surface of the metal. If the piece of metal has microscopic dimensions and we assume for simplicity its form to be spherical - like a classical liquid drop, then the effective potential confining the valence electrons will be spherically symmetric, with a form intermediate between an isotropic harmonic oscillator and a square well [10]. These simple model potentials can already give an idea of the reason for the magic numbers the formation of electronic shells. 

A theory developed in the 1900s that depicts metallic cations, surrounded by a sea of electrons. The oldest theory used, the free- electron theory explains a great deal about the properties of the materials, but is still only a qualitative assessment and doesn t offer any method for testing or quantifying the theory. Figure 8-9 illustrates the free-electron model of metals. 

Figure 5.6 Scheme of the near-free electron model of simple metals. The white circles represent the Wigner-Seitz cells, in which the point-positive charges are located. The lattice of the cells Is immersed in a free-electron gas. 

We showed the existence conditions for volume plasmons in the framework of the free-electron model but a generalization for surface plasmons can also be made in the framework of other models. Since the plasmon theory can be derived for the RPA, the existence of a plasmon can easily be checked by the RPA calculations. In agreement with the general theory presented, the ab initio RPA calculations carried out on alkali metal clusters of the size presented in this work and of considerably larger sizes do not exhibit plasmon-like excitations. 

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,... 

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... 

H2, N2, or CO dissociates on a surface, we need to take two orbitals of the molecule into account, the highest occupied and the lowest unoccupied molecular orbital (the HOMO and LUMO of the so-called frontier orbital concept). Let us take a simple case to start with the molecule A2 with occupied bonding level a and unoccupied anti-bonding level a. We use jellium as the substrate metal and discuss the chemisorption of A2 in the resonant level model. What happens is that the two levels broaden because of the rather weak interaction with the free electron cloud of the metal. 

We will now analyze the general optical behavior of a metal using the simple Lorentz model developed in the previous section. Assuming that the restoring force on the valence electrons is equal to zero, these electrons become free and we can consider that Drude model, which was proposed by R Drude in 1900. We will see how this model successfully explains a number of important optical properties, such as the fact that metals are excellent reflectors in the visible while they become transparent in the ultraviolet. 

The simplest model of a metal is therefore the one in which the metal is depicted as an array of ions glued together by conduction (quasi-free) electrons. If this is the case, one may define a metallic valence" as being, essentially, the charge left in the ion cores when outer electrons have been stripped off. Conversely, the metallic valence can be defined as the contribution of outer electrons each atom gives to the sea of bonding conduction electrons. 

Finally, a special type of primary bond known as a metallic bond is found in an assembly of homonuclear atoms, such as copper or sodium. Here the bonding electrons become decentralized and are shared by the core of positive nuclei. Metallic bonds occur when elements of low electronegativity (usually found in the lower left region of the periodic table) bond with each other to form a class of materials we call metals. Metals tend to have common characteristics such as ductility, luster, and high thermal and electrical conductivity. All of these characteristics can to some degree be accounted for by the nature of the metallic bond. The model of a metallic bond, first proposed by Lorentz, consists of an assembly of positively charged ion cores surrounded by free electrons or an electron gas. We will see later on, when we... 

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... 

If one considers an elementary model of a metal consisting of a latlice of fixed positive ions immersed in a sea of conduction electrons that are free to move through the lattice, every direction of electron motion will be equally probable. Since the electrons fill the available quantized energy states staring with the lowest, a three-dimensional picture in momentum coordinates will show a spherical distribution of electron momenta and, hence, will yield a spherical Fermi surface. In this model, no account has been taken of the interaction between Ihe fixed posilive ions and the electrons. The only restriction on the movement or "freedom" of the electrons is the physical confines of the metal itself. 

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