Madelung energy crystal electrostatic

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. 

Further, it is observed experimentally that electron-pair bonds are frequently associated with anisotropic, i.e. directed, atomic orbitals. This gives rise to open structures. However, the electrostatic (Madelung) energy associated with ionic crystals favors close packing Therefore largely ionic crystals favor more close-packed, two-sublattice structures such as rock salt versus zinc blende. In the case of two-sublattice structures induced by d electrons, electron-pair bonds are generally prohibited by the metallic or ionic outer s and p electrons that favor close packing. Nevertheless, it will be found in Chapter III, Section II that, if transition element cations are small relative to the anion interstice and simultaneously have Rti RCf electron-pair bonds may be formed below a critical temperature. 

The materials may be in a quasi-ionic phase when 1 > p > 0.5, or in a quasi-neutral phase when 0.5 > p > 0. In the simplest theoretical approach, the value of p depends on only three parameters D) the ionization potential of D Aa, the electron affinity of A and M, the electrostatic Madelung energy of the crystal lattice. A fully ionic lattice (p = 1) is then realized when /D - Aa > M, and a fully neutral lattice (p = 0) when /D — Aa < M. This result is, however, greatly obscured by the neglect of transfer integral t and of other relevant parameters [44]. 

The energy terms relevant for this crystal structure are the electrostatic Madelung energy obtained for the ionic crystal and the energy of charge... 

Madelung potential — refers to the electrostatic interaction per ion pair in a crystal. The electrostatic energy per ion pair is the Madelung energy,... 

The crystal cohesive energy, E, in Fig. 21 could also be plotted along an ionicity axis, q, normal to the paper. The crystal electrostatic (Madelung) energy of mixed stacks clearly increase with q, so that the second valley may deepen with increasing q. The first valley for segregated stacks may initially deepen but becomes far less favorable... 

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,... 

Ionic radii are discussed thoroughly in Chapters 4 and 7. For the present discussion it is only necessary to point out that the principal difference between ionic and van der Waals radii lies in the difference in the attractive force, not the difference in repulsion. The interionic distance in UF, for example, represents the distance at which the repulsion of a He core (Li+) and a Ne core (F ) counterbalances the strong electrostatic or Madelung force. The attractive energy for Lt F"is considerably over 500 kJ mol"1 anti the London energy of He-Ne is of the order of 4 kJ mol-1. The forces in the LiF crystal are therefore considerably greater and the interioric distance (201 pm) is less than expected for the addition of He and Ne van der Waals radii (340 pm). 

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 ... 

The theorem has the important implication that intramolecular interactions can be calculated by the methods of classical electrostatics if the electronic wave function (or charge distribution) is correctly known. The one instance where it can be applied immediately is in the calculation of cohesive energies in ionic crystals. Taking NaCl as an example, the assumed complete ionization that defines a (Na+Cl-) crystal, also defines the charge distribution and the correct cohesive energy is calculated directly by the Madelung procedure. 

In the bulk, cations and anions are sixfold coordinated. The total energy of the crystal is nearly completely given by the Madelung (electrostatic) energy. On the surfaces of the microcrystals, different local coordinations are encountered, with coordination numbers varying from five on the (100), (010), and (001) faces to four on edges and steps and three on corners. Threefold coordinated ions are also present on reconstructed (111) faces. 

Madelung constant — is the factor by which the ionic charges must be scaled to calculate the electrostatic interaction energy of an ion in a crystal lattice with given... 

The sums may be carried out with respect to the atomic positions in direct (real) space or to lattice planes in reciprocal space, an approach introduced in 1913 by Paul Peter Ewald (1888-1985), a doctoral student under Arnold Sommerfeld (Ewald, 1913). In reciprocal space, the structures of crystals are described using vectors that are defined as the reciprocals of the interplanar perpendicular distances between sets of lattice planes with Miller indices (hkl). In 1918, Erwin Rudolf Madelung (1881-1972) invoked both types of summations for calculating the electrostatic energy of NaCl (Madelung, 1918). 

The determination of the Madelung constant for a crystal structure requires the evaluation of electrostatic self-potentials of the structure. Any lattice energy can also be expressed using lattice site self-potentials (which is what Ewald actually calculated) ... 

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