Madelung field

Influence of the electron correlation and the Madelung field on charge and spin distributions... 

Before examining the properties of the Madelung field, it is necessary to show that it is possible to partition the electron density of the crystal into atoms that... 

Fig. 2.1. A representation of the Madelung field of rutile (Ti02, 202240) in the (110) plane (x,y,z) x + y — 1. The light lines represent the field lines, the heavy lines show the zero-flux boundary that partitions space into bonds (from Preiser et al. 1999).

Fig. 2.2. The Madelung field of the same projections as Fig. 2.1 with one ion removed from the top right showing the rearrangement of the flux lines (from Preiser et al. 1999).

Fig. 3.1. The line shows correlation between bond valence and bond length for Ca-O bonds given by eqn (3.1). The circles represent bond fluxes calculated from the Madelung field (Preiser et al. 1999).

The similarity between eqns (2.7) and (3.3) (given the equality of g, and F,) is a necessary but not sufficient condition that bond fluxes, "hy, and bond valences, Sij, are the same. A theoretical proof of their equality is not possible, but it can be demonstrated by comparing the bond valences calculated using eqn (3.1) (the line in Fig. 3.1) with bond fluxes calculated from the Madelung fields in particular compounds (the points in Fig. 3.1). This figure shows that the flux and bond valence are essentially the same for Ca-0 bonds and similar agreement is found for other types of bond provided that electronic anisotropies of the kind discussed in Chapter 8 are not present (Preiser et al. 1999). 

In many compounds, the experimental bond valences, S, and the theoretical bond valences, s, are both found to be equal to the bond fluxes, <1>, within the limits of experimental uncertainty. This is an empirical observation that is not required by any theory. For this reason, and because there are occasions when the differences between them are significant and contain important information about the crystal chemistry, it is convenient to retain a different name for each of these three quantities to indicate the ways in which they have been determined. The bond flux is determined from the calculation of the Madelung field, the theoretical bond valence is calculated from the network equations (3.3) and (3.4), and the experimental bond valence is determined from the observed bond lengths using eqn (3.1) or (3.2). 

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. 

The coordination number of a cation is defined as the number of bonds that it forms. Although it was shown in Section 2.6 that the number of bonds is uniquely defined by the partitioning of the Madelung field, extensive calculation is needed to extract this information. It is, therefore, convenient to use a simpler, if more arbitrary, definition. 

MnO is approximately 0.1 eV. The results of Nesbet s calculations are given in Table XIII. Although the agreement between observed and calculated N6el temperatures is quite good, it is important to realize that Uaa has been estimated only for a three-atom cluster, and considerable modification of this term can be anticipated for a solid where the clusters are located in the electrostatic Madelung field of the crystal. 

The Ewald series for the three-dimensional crystal can also be differentiated. The first derivative yields expressions for the Madelung electric field Fm (due to local charges). The second derivative yields the Madelung field gradient, or, equivalently, the internal or dipolar or Lorentz field FD (due to local dipoles) [68-71],This second derivative can also generates the dimensionless 3x3 Lorentz factor tensor L with its nine components Lv/t ... 

It seems of interest to compare the covalent and ionic cluster models as applied to the same system. Such a comparison was made within the scope of CNDO/2 calculations of some characteristics of surface hydroxyls of Si02 (3magnetic resonance parameters of surface sites (39), and chemisorption (40). Though somewhat different, the results obtained within these models were quite close. Nevertheless, to make a more categorical conclusion, more extensive studies are required. It is likely that in several cases the close-packed oxide structures would be better represented by stoichiometric clusters (e.g., Mg2 02n), although even in these cases it seems appropriate to take into account the Madelung field of the nearest part of the resting lattice. 

The very ionic nature of MgO implies that the Madelung potential is explicitly included. Indeed, several properties of MgO are incorrectly described if the long-range Coulomb interactions are not taken into account [51]. A simple approach is to surround the cluster of Mg and O ions by a large array of point charges (PC) of value + 2 to reproduce the Madelung field of the host at the central region of the cluster... 

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