The Madelung energy

We provide here a detailed calculation of the ion-ion interaction contributions to the total energy of a solid, the so called Madelung energy  

The method we will discuss here for obtaining accurate values of the Madelung energy 

We use spherical coordinates to perform the integration dt = t dt sin 6d6d(p, with the convenient choice of r defining the z axis of the integration variable t which gives 

The last expression is related to the error function, the definition of which is 

Notice that for large values of r the error function becomes 

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Table I. The electrostatic energies of the central silicon site and one nearest-neighbor oxygen site for a 977 point-ion cluster, and the Madelung energies for silicon and oxygen in alpha-quartz...

Fig. 3.3 shows a similar ionic-model construction of the electronic energies for the isostructural transition-metal oxide MnO. In this case, the Madelung energy lifts the Mn 3d level above the top of the... 

A metallic band structure is realized when the CT solids have a partial CT state and molecules form uniform segregated columns or layers. Figure 1 shows electrical conductivity data for 1 1 low-dimensional TTF TCNQ system, as a function of redox potentials [82]. The two lines a and b are related to the equation expressing the relationship between 7d, Ea, and the Madelung energy M 5) (5 = degree of CT) between partially charged component molecules (eq. 2) [83], where and Ea are... 

The Madelung energy, eqn (6.61), can also be expressed as a pairwise sum over coulomb interactions between the point ions, plus a q - 0 contribution arising from the electron-ion and electron-electron interactions. Grouping this together with the band-structure contribution we have... 

Fig. 8.15 The Madelung energy in ionic compounds as a function of the radius ratio for CsCl, NaCl and cubic ZnS lattices (assuming the anion radius, / , is held constant).

The Madelung energy, m. arising from the coulombic attraction of the partial... 

In order for an ionic compound to dissolve, the Madelung energy or electrostatic attraction between the ions in the lattice must be overcome. In a solution in which the ions are separated by molecules of a solvent with a high dielectric constant ( H 0 81.7 ) the attractive force will be considerably less. The process of solution of an ionic compound in water may be considered by a Bom-Haber type of cycle. The overall enthalpy of the process is the sum of two terms, the enthalpy of dissociating the ions from the lattice (the lattice energy) and the enthalpy of introducing the dissociated ions into the solvent (the solvation energy) ... 

The infinite series converges to A, the so-called Madelung constant. The interaction for an Avogadro number of formula units consisting of n ions (in this case n = 2) each, amounts to the Madelung energy ... 

The formula is converted into dimensionless units by defining a dimensionless distance based on a characteristic spacing for each compound. It is noted that the closest interionic distance can be specified as the sum of two ionic radii, d = r + r+. Using the well established anionic radii for halide and chalconide ions [75], a conversion factor R = /r 1 r is calculated in each case, and used to define the dimensionless distance d = d/R, such that the Madelung energy,... 

The neutral-ionic transition (NIT) at t = 0 occurs abruptly[94] when the Madelung energy M of the ionic lattice exceeds the energy I — A to transfer an electron form D to A. Long-range Coulomb interactions are treated self-consistently as part of A in the modified Hubbard model[95],... 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

Although TABLE 1 shows a small increase in the Madelung energy of Mg-doped GaN compared with that of Be-doped GaN, we find, from the calculated results, that the impurity states due to doped Mg are more localised on the neighbouring N atoms than those due to doped Be. This suggests that Mg is a... 

The Madelung constant a is defined as the ratio (a = EM/H) of the Madelung energy to the Coulomb attractive energy H between the nearest-neighbor anion and cation. Thus, for the monoatomic one-dimensional lattice of equidistant cations and anions of Eq. (8.10.3), a = 1.385294361. 

Experimental Born-Haber cycle for sodium chloride. The experimental binding energy AH = —8.0 eVis reasonably close to the Madelung energy Em= -8.923446 eV (after one adds to Em a relatively small ad hoc positive rep). 

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