Modified Born-Lande Equation to calculate Lattice Energy in a theoretical approach

Defects in ionic solid are very much common, which is increased with the rise in temperature. It causes the change in the value of many physical properties and varieties of physical parameters and the Lattice Energy is one such parameter to control the physical properties of the crystals. Considering the loss of ions from lattice points as random, the examination of each of the defects individually is going to be unpredictable, thus leading to almost nonattainment of the correct crystal structure with the theoretical calculations applying for available models. Here, in this present work, we have used some statistical methods and probabilistic approximation to introduce a novel idea of calculating the Madelung constant, and then Lattice Energy analytically. To make the understanding more lucid, we have taken one of the very common crystals, very popular in the crystallographic community, NaCl crystal having 6:6 co-ordination number, for which a significant number of Schottky defects are observed. During this study, we are bound to assume the random distribution of defects as Poisson distribution due to the fact that the number of defects is very less with respect to the total numbers of lattice points present in the crystal to calculate the Madelung Constant.

Cesaro summation by spheres of lattice sums and Madelung constants

<p style='text-indent:20px;'>We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.</p>

Correlation of the Madelung constant and I—M—I bonding angle with cohesive energy contributions in layered metal diiodides (MI2) with CdI2 (2H polytype) structure

This study uses theoretically methods to investigate, for metal diiodides MI2 (M = Mg, Ca, Mn, Fe, Cd, Pb) with CdI2 (2H polytype) structure, the mutual correlation between the structure-characterizing parameters (the flatness parameter of monolayers f, the Madelung constant A, and bonding angle I—M—I) and correlation of these parameters with contributions of the Coulomb and covalent energies to cohesive energy. The energy contributions to cohesive energy are determined with the use of empirical atomic potentials. It is demonstrated that the parameters f and A, and the bonding angle I—M—I are strictly correlated and increase in the same order: FeI2 < PbI2 < MnI2 < CdI2 < MgI2 < CaI2. It is found that with an increase of parameter A and bonding angle I—M—I the relative contribution of the Coulomb energy to cohesive energy increases, whereas the relative contribution of the covalent energy decreases. For a hypothetical MX 2 layered compound with the CdI2 (2H polytype) structure, composed of regular MX 6 octahedra (angle X—M—X = 90°), the flatness parameter and the Madelung constant are found to be f reg = 2.449 and A reg = 2.183, respectively. Correlation of the covalent energy with the type of distortion of MI6 octahedra (elongation or compression) with respect to regular configuration (angle I—M—I = 90°) is also analyzed.