The Charge Density in Extended Solids

The bonding features in the charge density are pronounced in crystals with extended covalent networks. The availability of perfect silicon crystals has allowed the measurement of uncommonly accurate structure factors, of millielectron accuracy. The data have served as a test of experimental formalisms for charge density analysis, and at the same time have provided a stringent criterion for quantum-mechanical methods. 

We will start the discussion in this chapter with silicon and its analogues, diamond and germanium, and proceed with the treatment of silicates, and metallic and ionic crystals. 

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The classical idea of molecular structure gained its entry into quantum theory on the basis of the Born Oppenheimer approximation, albeit not as a non-classical concept. The B-0 assumption makes a clear distinction between the mechanical behaviour of atomic nuclei and electrons, which obeys quantum laws only for the latter. Any attempt to retrieve chemical structure quantum-mechanically must therefore be based on the analysis of electron charge density. This procedure is supported by crystallographic theory and the assumption that X-rays are scattered on electrons. Extended to the scattering of neutrons it can finally be shown that the atomic distribution in crystalline solids is identical with molecular structures defined by X-ray diffraction. 

The Chapman theory is briefly as follows. Suppose that the solid surface has a charge density of a. A net charge equal in magnitude but opposite in sign will be carried by the ionic atmosphere, which extends into the solution. The value of the charge density p per unit volume at any point in the solution portion of the electric double layer can be derived by the method employed in Chapter 6. Provided that e /kT is small in comparison with unity the result is (see equation (6.22))... 

It is important to note here that the situations depicted by cases 1 and 2 are rather generic in nature and can be judiciously extended to analyze the EDL formation in between two parallel plates separated by a distance of 2 h, for example. When the characteristic EDL thickness (A ) is much less than h, the location of the midplane (centerline) can mathematically be treated as a far-field one, since the charge density gradients are only confined within the EDLs adjacent to the solid surfaces, beyond which the effects of nearwall potential distribution cannot effectively penetrate. In such situations, the mathematical description introduced by case 2 readily applies. On the other hand, case 1 is the representative of a more general situation, in which the EDLs formed in the vicinity of the two plates can penetrate into the midplane and in fact may interfere with each other. We shall discuss about this situation more carefully, later in this article. 

Metal-metal bonding in transition metal complexes of low nuclearity (i.e., with only a few metal atoms) tends to be more directed and therefore stronger than the bonding in metals discussed in chapter 11. Accordingly, the metal-metal bonds in transition metal complexes are often localized and considerably shorter than those in most extended solids. Charge accumulations are frequently observed in metal-metal bonding regions of deformation density maps. 

At equilibrium, when Fermi levels are equal the contact on both sides of the interfacial distribution of charges is that shown in Fig. 3 a Qsc is the space charge in the semiconductor and extends over a distance w of several hundred nanometers, since the density of electronic charges available in the solid is several orders of magnitude... 

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