Sp-valent metals

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

The above model of an sp-valent metal as a gas of free electrons would exhibit no bonding because the only contribution to the energy is the repulsive kinetic energy. It takes an average value per electron, which is given by... 

We have so far made two implicit assumptions. The first of these is that the gas of electrons is not scattered by the underlying ionic lattice. This can be understood by imagining that the ions are smeared out into a uniform positive background The second assumption is that the electrons move independently of each other, so that each electron feels the average repulsive electrostatic field from all the other electrons. This field would be completely cancelled by the attractive electrostatic potential from the smeared-out ionic background. Thus, we are treating our sp-valent metal as a metallic jelly or jellium within the independent particle approximation. 

The free-electron gas model is a good starting point for the sp-valent metals where the loosely bound valence electrons are stripped off from their ion cores as the atoms are brought together to form the solid. However, bonding in the majority of elements and compounds takes place through saturated... 

The variation in equilibrium bulk properties between one sp-valent metal and the next cannot be understood within the jellium model, since it has obscured the chemical behaviour of the elements by smearing out the ion... 

Fig. 5.13 The densities of states of sp-valent metals. (After Moruzzi et at.

It follows from eqn (5.66) and Table 5.2 that the presence of the ion core is crucial for obtaining realistic values of the bulk modulus of sp-valent metals. However, we see that the sd-valent noble metals Cu, Ag, and Au are not describable by the NFE approximation, the theoretical bulk modulus being a factor of five too small. We will return to the noble metals at the end of the next chapter. 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity... 

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