The Madelung Constant and Crystal Lattice

The Elements, 4-1 to 42 The Genetic Code, 7-6 The Madelung Constant and Crystal Lattice Energy, 12-32... 

The Madelung constant and Born exponent appearing in Eq. 1 are related to the specific arrangement of ions in the crystal lattice. The Madelung constant may be considered as a decreasing series, which takes into account the repulsions among... 

Ewald 37) developed a method for evaluating the Madelung constants and other sums for crystal lattices which depends on the use of a 5-function transformation. This is probably the method of most general application though it suffers from the disadvantage that the convergence of the series obtained is less rapid than that obtained by the Emersleben method. 

One of the disadvantages of the fully theoretical approach is that it is necessary to know the crystal structure and the interionic distances to estimate the lattice energy. The Kapustinskii equations overcome this limitation by making some assumptions. The Madelung constant and the repulsive parameter n are put equal to average values, and it is also assumed that the interionic distance can be estimated as the sum of anion and cation radii r+ and r (see Topic D4Y The simpler... 

Madelung constant A constant arising in calcuiations of the cohesion of Ionic crystals. The electrostatic interaction per ion pair, U, is given by U[r) = -a lr, where a is the Madelung constant and e /ris the Coulomb interaction between the ions, with rbeing the lattice constant. The value of a depends on the type of lattice. For the sodium chloride lattice, a has a value of about 1.75. A more realistic calculation of cohesion is obtained if short-range repulsions with an inverse power law are included, i.e. 

We have already mentioned that for sodium chloride approximately 1.78 times as much energy is released when the crystal lattice forms as when ion pairs form. This value, the Madelung constant (A) for the sodium chloride lattice, could be incorporated to predict the total energy released when 1 mole of NaCl crystal is formed from the gaseous Na+ and Cl- ions. The result would be... 

When we have an ordered assembly of atoms called a lattice, there is more than one bond per atom, and we must take into account interactions with adjacent atoms that result in an increased interionic spacing compared to an isolated atom. We do this with the Madelmg constant, ckm. This parameter depends on the structure of the ionic crystal, the charge on the ions, and the relative size of the ions. The Madelung constant fits directly into the energy expression (Eq. 1.25) ... 

The electrostatic (Madelung) part of the lattice energy (MAPLE) has been employed to define Madelung potentials of ions in crystals (Hoppe, 1975). MAPLE of an ionic solid is regarded as a sum of contributions of cations and anions the Madelung constant. A, of a crystal would then be the sum of partial Madelung constants of cation and anion subarrays. Thus,... 

The series 6 - 12/21/2 + 8/31/2 - 6/41/2 + 24/51/2 -. .. eventually becomes convergent and gives the value for the Madelung constant for the sodium chloride lattice (the standard description of lattices which have the same form as that adopted by sodium fluoride). The values of Madelung constants for some common crystal lattices are given in Table 7.5. 

The crystal lattice energy can be estimated from a simple electrostatic model When this model is applied to an ionic crystal only the electrostatic charges and the shortest anion-cation intermiclear distance need be considered. The summation of all the geometrical interactions be/Kveeti the ions is called the Madelung constant. From this model an equatitWjor the crystal lattice energy is derived ... 

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 three-dimensional crystals the lattice summation converges only slowly, and in any brute-force computational scheme one must make sure that, as one sums outward from the "zeroth unit cell" at the center of the crystal, the ions included at any stage should have as close to zero net charge as possible. For some crystals the Madelung constants a have been evaluated (Table 8.4), using component potentials obtained by summing certain infinite series. 

These equilibria are theoretically particularly simple, since the energy of the conversion depends only on the differences in the lattice energies all other terms in the heat of formation have no influence since these are equal for the substances on the left hand side and on the right hand side of the equation. If in addition the crystal lattice is of the same type for all of them, in this case of the rock-salt type, then the Madelung constant is also the same and only the difference of the reciprocal ionic separations has any influence. 

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