Ionic crystals, formation energy

Energy Changes in the Formation of Ionic Crystals— Lattice energies of ionic crystals can be related to certain atomic and thermodynamic properties by means of the Born-Fajans-Haber cycle (Fig. 12-51). 

Results have shown that the properties of solids can usually be modeled effectively if the interactions are expressed in terms of those between just pairs of atoms. The resulting potential expressions are termed pair potentials. The number and form of the pair potentials varies with the system chosen, and metals require a different set of potentials than semiconductors or molecules bound by van der Waals forces. To illustrate this consider the method employed with nominally ionic compounds, typically used to calculate the properties of perfect crystals and defect formation energies in these materials. 

Fig. 1. Born-Haber cycle for the formation of solvated ions from an ionic crystal [M+X ]w. U lattice energy, Affsoiv. enthalpy of ion solvation...

Born-Haber s cycle — Hess s law establishes that the enthalpy of a reaction is the same independently whether the reaction proceeds in one or several steps. It is a consequence of the first law of thermodynamics, which states the conservation of energy. Born and -> Haber applied Hess s law to determine the - enthalpy of formation of an ionic solid. The formation of an ionic crystal from its elements according to Born-Haber s cycle can be represented by the following diagram. 

Point (microscopic) defects in contrast from the macroscopic are compatible with the atomic distances between the neighboring atoms. The initial cause of appearance of the point defects in the first place is the local energy fluctuations, owing to the temperature fluctuations. Point defects can be divided into Frenkel defects and Schottky defects, and these often occur in ionic crystals. The former are due to misplacement of ions and vacancies. Charges are balanced in the whole crystal despite the presence of interstitial or extra ions and vacancies. If an atom leaves its site in the lattice (thereby creating a vacancy) and then moves to the surface of the crystal, it becomes a Schottky defect. On the other hand, an atom that vacates its position in the lattice and transfers to an interstitial position in the crystal is known as a Frenkel defect. The formation of a Frenkel defect therefore produces two defects within the lattice—a vacancy and the interstitial defect—while the formation of a Schottky defect leaves only one defect within the lattice, that is, a vacancy. Aside from the formation of Schottky and Frenkel defects, there is a third mechanism by which an intrinsic point defect may be formed, that is, the movement of a surface atom into an interstitial site. Considering the electroneutrality condition for the stoichiometric solid solution, the ratio of mole parts of the anion and cation vacancies is simply defined by the valence of atoms (ions). Therefore, for solid solution M X, the ratio of the anion vacancies is equal to mJn. 

---