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.
Formal expression The Madelung constant allows for the calculation of the electric potential Vi of the ion at position ri due to all other ions of the lattice
V i = e 4 π ε 0 ∑ j ≠ i z j r i j {\displaystyle V_{i}={\frac {e}{4\pi \varepsilon _{0}}}\sum _{j\neq i}{\frac {z_{j}}{r_{ij}}}\,\!}
where r i j = | r i − r j | {\displaystyle r_{ij}=|r_{i}-r_{j}|} is the distance between the ith and the jth ion. In addition,
zj = number of charges of the jth ion e = the elementary charge, 1.6022×10−19 C 4πε0 = 1.112×10−10 C2/(J⋅m); ε0 is the permittivity of free space. If the distances rij are normalized to the nearest neighbor distance r0, the potential may be written
V i = e 4 π ε 0 r 0 ∑ j z j r 0 r i j = e 4 π ε 0 r 0 M i {\displaystyle V_{i}={\frac {e}{4\pi \varepsilon _{0}r_{0}}}\sum _{j}{\frac {z_{j}r_{0}}{r_{ij}}}={\frac {e}{4\pi \varepsilon _{0}r_{0}}}M_{i}}
with Mi being the (dimensionless) Madelung constant of the ith ion
M i = ∑ j z j r i j / r 0 . {\displaystyle M_{i}=\sum _{j}{\frac {z_{j}}{r_{ij}/r_{0}}}.}
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
M ¯ i = ∑ j z j r i j / w = M i w r 0 . {\displaystyle {\overline {M}}_{i}=\sum _{j}{\frac {z_{j}}{r_{ij}/w}}=M_{i}{\frac {w}{r_{0}}}.}
The electrostatic energy of the ion at site ri then is the product of its charge with the potential acting at its site
E e l , i = z i e V i = e 2 4 π ε 0 r 0 z i M i . {\displaystyle E_{el,i}=z_{i}eV_{i}={\frac {e^{2}}{4\pi \varepsilon _{0}r_{0}}}z_{i}M_{i}.}
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