Madelung constant, potential

Table 4.17 establishes the fact that increasing of coordination corresponds to the increase of the Madelung constant (potential). 

The Madelung constant,. 4=-3,99. There are seven ions in the structure in Figure 1.61 six cations and one anion. First, calculate the contribution to the potential energy of interactions of the six cations with the central anion. Each cation is... 

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,... 

The covalent curve of Figure 3-5 has been drawn in the same way as for the HX curves. The ionic curve represents a Coulomb attractive potential and a repulsive potential b/R9f in which the constant b is given values which lead (with use of the Madelung constant and corresponding constant in the repulsive potential) to the correct interatomic distance in the corresponding crystal (Chap. 13). Polarization is neglected. 

For three-dimensional crystals the lattice summation converges only slowly, and in any brute-force computational scheme one must make sure that, as one sums outward from the "zeroth unit cell" at the center of the crystal, the ions included at any stage should have as close to zero net charge as possible. For some crystals the Madelung constants a have been evaluated (Table 8.4), using component potentials obtained by summing certain infinite series. 

Yosio Sakamoto, Madelung constants of simple crystals expressed in terms of Bom s basic potentials of 15 figures, /. Chem. Phys. 28 164-165 (1958). 

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) ... 

Madelung constants can be directly related to the potential at one ionic site due to all other ions. In fact, the potential at a site of type one (taken for convenience to be a positive ion) can be written, for two ions per cell, as... 

The potential taking into account the Madelung constant the Madelimg constant is a representation of the distance of the ions from one another and thus depends on their geometric arrangement in a crystal structure or on a surface. 

If such a calculation is carried out for a real three-dimensional crystal, the result is a series (such as that just given in brackets) whose value sums to a dimensionless number that depends upon the crystal structure. That number is called the Madelung constant, M, and its value is independent of the unitcell dimensions. Table 21.5 lists the values of the Madelung constant for several crystal structures. The lattice energy is again the opposite of the total potential energy. Expressed in terms of the Madelung constant, it is... 

T3.2 The Madelung constant is in a sense a geometrical factor that depends on the position of ions within a unit cell. The electrostatic potential in which Na ions resides is given by ... 

Tables (4.2-4.4) compare ion potentials and Madelung constants in different layers computed for cation-anion pairs of three different surfaces of Ti02 the rutile (001) and (110) surfaces and the anatase (001) surface as a function of layer distance with respect to the surface. The anatase (001) surface is the one with the lowest energy of the anatase polymorph of Ti02 The (110) surface of rutile has a lower surface energy than the corresponding (001) surface. The anatase (001) surface is the one with the lowest energy of the anatase polymorph of Ti02< The (110) surface of rutile has a lower surface energy than the corresponding (001) surface.

Madelung constant in the surface layer has decreased relative to that of the bulk, but less than the decrease in first-shell coordination number. Interestingly, the ion potential in the layer of ions next to the surface the Madelung constant increases. The reason is that in this layer there is no change in coordination number in the first coordination shell, but there is a change in the second coordination shell. Due to the presence of the surface the number of the next-nearest-neighbor cations or anions... 

Table 4.3. Electrostatic potentials Ti ) and (0 ) and Madelung constants (to) (to — 0.1946i nm) as a function of the layer number for the rutile (UO) face(<...

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