Madelung’s constant

Summation of the series gives a value 1 -747 and is called Madelung s constant A. The expression Ae2jr is the energy which is obtained when one ion is surrounded by all the others, or, equally well, the energy necessary to remove one ion from the crystal. In order to remove 2N ions in a gram mol of the crystal, in other words to... 

The result for four lattices of the NaCl type are given in the following table. Madelung s constant is 1 742. The experimental values of the refractive index are taken from the Landolt-Bornstein tables 167 and 170. The value extrapolated to zero frequency from the visible range is given in the table. 

TahU CXLV. Values of Madelung s Constant for Various Ctystals attraction and repulsion... 

Crystal structure Coordination number Madelung s constant... 

Here is the bound energy linked oxygen and hydrogen in the water molecule and g is the distance between O and H in the molecule A,K)n rep is the constant that describes short-range repulsion and zo is a typical radius of this repulsion force a is Madelung s constant, which, as a mle, falls within the range from unit to two (see, e.g., Kittel [177]) is the dielectric constant (which, however, is absent in the case of ionic crystals). Having chosen the pair potential (577), we can now represent the repulsive potential U and the attractive potential V, which enter the Hamiltonian (552), as follows ... 

Let us now assign numerical values to the parameters g (lattice constant), a (Madelung s constant), s (permittivity), and 2,(fugacity) (other parameters are elementary charge e, dielectric constant o, and the temperature T, which can be set at 300 K). For liquid water we can set g = 0.281 nm for the estimation let a be 1.3 and let s be 4 (though s = 81 for bulk water). Stobbe and Peschel [405], based on literature data [417—419] and their own studies, have concluded that the dielectric constant s varies from 5 to 10 for the absorbed water layer with thickness from 1 nm to 12 nm. A cluster of water molecules with the radius on the order of lOg just falls under the aforementioned water layer thickness. 

If we now consider the effect of the crystal lattice, by introducing Madelung constant M and multiplying by Avogadro s number Nq (i.e., the number of atoms per weight formula unit), we get... 

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