Ionic lattices coulombic interactions

Temkin was the first to derive the ideal solution model for an ionic solution consisting of more than one sub-lattice [13]. An ionic solution, molten or solid, is considered as completely ionized and to consist of charged atoms anions and cations. These anions and cations are distributed on separate sub-lattices. There are strong Coulombic interactions between the ions, and in the solid state the positively charged cations are surrounded by negatively charged anions and vice versa. In the Temkin model, the local chemical order present in the solid state is assumed to be present also in the molten state, and an ionic liquid is considered using a quasi-lattice approach. If the different anions and the different cations have similar physical properties, it is assumed that the cations mix randomly at the cation sub-lattice and the anions randomly at the anion sub-lattice. 

Lattice enthalpies of ionic solids can be predicted from several equations, which account for the coulombic interactions [45-47,49]. The estimates can then be used to derive the standard enthalpies of formation, by equation 2.47. However,... 

A similar calculation can be done for ionic crystals. In this case the Coulomb interaction is taken into account, in addition to the van der Waals attraction and the Pauli repulsion. Although the van der Waals attraction contributes little to the three-dimensional lattice energy, its contribution to the surface energy is significant and typically 20-30%. The calculated surface energy depends sensitively on the particular choice of the inter-atomic potential. 

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],... 

Their weak interatomic interaction is responsible for the condensation of the normal inert gases into solids. Atoms of normal inert gas are brought together until the repulsive terms in the overlap interaction prevent further contraction. The attraction favors a close-packed structure, and all of the normal inert gases form face-centered cubic lattices. These two contributions to the total interaction will remain almost the same in the ionic crystals, but with added Coulomb interactions, so it is desirable to understand all of these contributions with some care. 

Madelung constant A term that accounts for the particular structure of an ionic crystal when the lattice energy is evaluated from the coulombic interactions. The value is different for each crystalline structure. 

Equation (6.13) was derived on a model of covalent bonds between nearest neighbors. It is not strictly applicable to ionic solids. The repulsion part of the potential energy must be similar for ionic and covalent cases, but the attraction part for ionic solids must also include the sum of the coulombic interactions with the remainder of the lattice. In effect, the number of bonds is increased. To see the magnitude of this effort, compare Equations (6.7) and (6.8) with their counterparts for a diatomic molecule, or ion-pair. 

Coulombic interactions are not the only forces operating in a real ionic lattice. The ions have finite size, and electron-electron and nucleus-nucleus repulsions also arise these are Born forces. Equation 5.12 gives the simplest expression for the increase in repulsive energy upon assembling the lattice from gaseous ions. 

In order to write an expression for the lattice energy that takes into account both the Coulombic and Bom interactions in an ionic lattice, we combine equations 5.11 and... 

The situation for alloys is more complex. In theoretical calculations of lattice energies one has to take into account the different contributions from Coulomb interactions (ionic bonds), localized orbitals (covalent bonds), and delocalized orbitals (metal bonds). Therefore an ab initio calculation is difficult. Solutions of the problem are under development and one can expect that in the near future reliable data will be presented. 

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