Constante de Madelung

The Madelung constant is used in determining the electrostatic potential of a single ion in a crystal by approximating the ions by point charges. It is named after Erwin Madelung, a German physicist. Because the anions and cations in an ionic solid attract each other by virtue of their opposing charges, separating the ions requires a certain amount of energy. This energy must be given to the system in order to break the anion–cation bonds. The energy required to break these bonds for one mole of an ionic solid under standard conditions is the lattice energy. The Madelung constant allows for the calculation of the electric potential V_i of all ions of the lattice felt by the ion at position r_i where is the distance between the ith and the jth ion. In addition, z_j = number of charges of the jth ion e = the elementary charge, 1.6022 C 4πε_0 = 1.112e-10C2/(J⋅m); ε_0 is the permittivity of free space. If the distances r_ij are normalized to the nearest neighbor distance r_0, the potential may be written with M_i being the (dimensionless) Madelung constant of the ith ion Another convention is to base the reference length on the cubic root w of the unit cell volume, which for cubic systems is equal to the lattice constant. Thus, the Madelung constant then reads The electrostatic energy of the ion at site r_i then is the product of its charge with the potential acting at its site There occur as many Madelung constants M_i in a crystal structure as ions occupy different lattice sites. For example, for the ionic crystal NaCl, there arise two Madelung constants – one for Na and another for Cl. Since both ions, however, occupy lattice sites of the same symmetry they both are of the same magnitude and differ only by sign. The electrical charge of the and ion are assumed to be onefold positive and negative, respectively, z_Na = 1 and z_Cl = –1. The nearest neighbour distance amounts to half the lattice constant of the cubic unit cell and the Madelung constants become The prime indicates that the term is to be left out.