The Madelung field of a crystal

These link regions, which we can identify with chemical bonds, are characterized by the electrostatic flux, defined by eqn (2.6), that links the two ions i and j  

Section 14.1 gives an alternative view of why the choice of the atomic charge is not critical. 

The law of conservation of energy, which states that the sum of the potential differences around any closed loop is zero, can also be applied to this system if the potential differences between the ions can be calculated. To determine these, it is convenient to recognize that each bond acts as a capacitor, C,y, with the atoms acting as the plates that carry the charges and the bond providing the field linking them. This capacitor then supports the potential difference, P,y, according to the capacitor eqn (2.8)  

Since the law of conservation of energy requires that the sum of the potentials, Py, around any closed loop be zero, 

The values of Cy, of course, depend on which equipotential surface is used to represent the ion. Since these surfaces can be arbitrarily chosen, it might be supposed that all the values of Cy can also be arbitrarily chosen. However, the number of ions is always less than the number of bonds. If there are ions in the array, it is only possible to assign arbitrary values of Cy to - 1 bonds, those in the spanning tree described in Section 2.5 below. For the remaining bonds, those that close the loops in the network, a knowledge of the bond topology alone is insufficient to determine Cy. To find these values of Cy, the geometry of the array, i.e. the positions of the ions, must also be known. 

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In Chapter 2 it was shown that the Madelung field of a crystal is equivalent to a capacitive electric circuit which can be solved using a set of Kirchhoff equations. In Sections 3.1 and 3.2 it was shown that for unstrained structures the capacitances are all equal and that there is a simple relationship between the bond flux (or experimental bond valence) and the bond length. These ideas are brought together here in a summary of the three basic rules of the bond valence model, Rules 3.3, 3.4, and 3.5. 

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