Madelung constants calculation

Values of the Madelung constant calculated from the geometry of the symmetrical structures are given in Table 28.1. 

Carter (1972) established a tetragonal model of ordered cationic vacancies in the R3- cX4 compounds from considerations involving the stereohedral metal sites or Wigner-Seitz cells, and tested it by Madelung constant calculations. 

Derive the first four terms of the series to calculate the MADELUNG constant for CsCl (Fig. 7.1). 

Details of the calculation of Madelung constants for all of the common types of crystals are beyond the scope of this book. When the arrangement of ions differs from that present in NaCl, the number of ions surrounding the ion chosen as a starting point and the distances between them may be difficult to determine. They will most certainly be much more difficult to represent as a simple factor of the basic distance between a cation and an anion. Therefore, each arrangement of ions (crystal type) will have a different value for the Madelung constant. The values for several common types of crystals are shown in Table 7.3. 

For ionic crystals > = 1, and the A are known (Madelung constants). For van der Waals crystals m — 6 (though small terms in and exist) but in view of the difficulties of calculation we obtain B from the observed heat of vapourisation (from A. 3). The repulsion exponent n varies from about 6 for LiF to 12 for Csl, for gases (Lennard-Jones) a value of about 12 seems the best. We assume a constant value of 11 throughout. 

Calculate the value of the Madelung constant for the structure in Figure 1.61. All bond lengths are equal and all bond angles are 90°. Assume that no ions exist other than those shown in the figure, and that the charges on the cations and anion are +1 and -1, respectively. 

The Madelung constant,. 4=-3,99. There are seven ions in the structure in Figure 1.61 six cations and one anion. First, calculate the contribution to the potential energy of interactions of the six cations with the central anion. Each cation is... 

The relative stability of the different structure types may now be calculated as a function of the radius ratio, R+/R, where R+ and R. are the hard-core radii of the positive and negative ions respectively. Figure 8.15 shows that when the ions are of equal size, the CsCl lattice with eight nearest neighbours and the largest Madelung constant is the most stable. However, as the radius ratio decreases, the structural trend from CsCl - NaCl -> ZnS is found. The structural transition from CsCl to NaCl is a direct consequence of the fact that the volume of the CsCl lattice is determined solely by the second nearest-neighbour anion-anion interactions for... 

As long as we do not neglect to understand each or the factors in the Born-Lande equation (4.13), we can simplify the calculations. It should be realized that the only variables in the Bom-Lande equation are the charges on the ions, the inlernuclear distance, the Madelung constant, and the value of n. Equation 4.13 may thus be simplified with no loss of accuracy by grouping the constants to give ... 

The enthalpy of formation of an ionic compound can be calculated with an accuracy of a few percent by means of the Born-Land6 equation (Eq. 4.13) and the Bcrn-Haber cycle. Consider NaCI. for example. We have seen that by using the predicted internuclear distance of 283 pm (or the experimental value of 281.4 pm), the Madelung constant of 1.748, the Rorn exponent, n, and various constants, a value of — 755 kJ mol-1 could be calculated for the lattice energy. The heat capacity correction is 2.1 kJ mol-1, which yields U 9i = —757 kJ mol-1. The Bom-Haber summation is then... 

Alternatively, we might examine the radius ratio of Oj BF and get a crude estimate of = 0.8. The accuracy of our values does not permit us to choose between coordination number 6 and 8, but since the value of the Madelung constant does not differ appreciably between the sodium chloride and cesium chloride structures, a value of 1.75 may be taken which will suffice for our present rough calculations. We may then use the Bom-Lande equation (Eq. 4.13), which provides an estimate of —616 kJ mor1 for the attractive energy, which will be decreased by about 10% (if... 

Calculate the Madelung constant for a two-dimensional crystal with a square unit cell having a positive ion at the center and negative ions on the comers. 

Kapustinskii noted that if the Madelung constant A is divided by the number of ions per formula unit for a number of crystal structures, nearly the same value is obtained. Furthermore, as both A/n and re increase with the coordination number, their ratio A/nre is expected to be approximately the same from one structure to another. Therefore, Kapustinskii proposed that the structure of any ionic solid is energetically equivalent to a hypothetical rock-salt structure and its lattice energy can be calculated using the Madelung constant of NaCl and the appropriate ionic radii for (6,6) coordination. 

For crystals containing unequal numbers of multiple cations and anions, the definition of a Madelung constant becomes more complicated, but the Madelung (or lattice) energy can always be calculated by simple computerized methods [76]. In principle, a dimensionless interaction curve can be derived for any structure type. 

Madelung constant — is the factor by which the ionic charges must be scaled to calculate the electrostatic interaction energy of an ion in a crystal lattice with given... 

A serious limitation of (but not an objection to) the hydridic model is its inability to rationalize the nonstoichiometry of metallic hydrides. The difficulty arises in part because of the use of the Madelung constant (or an approximate equivalent) and a Born-Lande or exponential repulsion term. It is not yet clear how these may be calculated for a defect structure where ions are randomly missing from their sites. It is reasonable that mutual repulsion of outer electrons will be less in such a structure, but no quantitative interpretation has been made. 

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