Madelung Constant - Generalization

Generalization

It is assumed for the calculation of Madelung constants that an ion’s charge density may be approximated by a point charge. This is allowed, if the electron distribution of the ion is spherically symmetric. In particular cases, however, when the ions reside on lattice site of certain crystallographic point groups, the inclusion of higher order moments, i.e. multipole moments of the charge density might be required. It is shown by electrostatics that the interaction between two point charges only accounts for the first term of a general Taylor series describing the interaction between two charge distributions of arbitrary shape. Accordingly, the Madelung constant only represents the monopole-monopole term.

The electrostatic interaction model of ions in solids has thus been extended to a point multipole concept that also includes higher multipole moments like dipoles, quadrupoles etc. These concepts require the determination of higher order Madelung constants or so-called electrostatic lattice constants. In their case, instead of the nearest neighbor distance another standard length like the cube root of the unit cell volume is appropriately used for purposes of normalization. For instance, the Madelung constant then reads

The proper calculation of electrostatic lattice constants has to consider the crystallographic point groups of ionic lattice sites; for instance, dipole moments may only arise on polar lattice sites, i. e. exhibiting a C₁, C_1h, C_n or C_nv site symmetry (n = 2, 3, 4 or 6). These second order Madelung constants turned out of having significant effects on the lattice energy and other physical properties of heteropolar crystals.